Description
Planning involves the creation and maintenance of a plan. As such, planning is a fundamental property of intelligent behavior. This thought process is essential to the creation and refinement of a plan, or integration of planning it with other plans; that is, it combines forecasting of developments with the preparation of scenarios of how to react to them.
THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
OPTIMISATION OF
LONG-TERM INDUSTRIAL PLANNING
PETER FORSBERG
Department of Applied Mechanics
CHALMERS UNIVERSITY OF TECHNOLOGY
G¨ oteborg, Sweden 2006
Optimisation of Long-Term Industrial Planning
PETER FORSBERG
ISBN 91-7291-863-2
c PETER FORSBERG, 2006
Doktorsavhandlingar vid Chalmers tekniska h¨ ogskola
Ny serie 2544
ISSN 0346-718X
Department of Applied Mechanics
CHALMERS UNIVERSITY OF TECHNOLOGY
SE-412 96 G¨ oteborg
Sweden
Telephone +46 (0)31-772 1000
Printed in Sweden by
Chalmers Reproservice
G¨ oteborg, 2006
Till Flisan och Smulan
Abstract
In this thesis, long-term optimisation methods for industrial transition processes
have been developed, taking monetary and environmental considerations into ac-
count. Two different methods for investment optimisation have been developed.
First, an optimisation method comprising simultaneous calculation of the long-
term investment strategy and the short-term utilisation scheme for a deterministic
demand was developed. The method has been applied to the case of ?nding an
investment strategy for minimising the production cost for a single hydrogen refu-
elling station. The problem was shown to be convex; thus the resulting solution is
the global optimum. Second, an investment optimisation method using stochastic
demand scenarios and multi-objective optimal control to produce the Pareto front
of the two con?icting objectives expected production cost and expected unsatis-
?ed demand was developed. This method was applied to the case of ?nding the
optimal investment strategy for a combined hydrogen and hythane refuelling sta-
tion. Depending on the preferences of the decision-maker, many different feasible
solutions can be found. However, it was also found that, due to the uncertainty
of the stochastic demand function, satisfying all the estimated demands would re-
quire a production capacity well above the mean demand, which would be very
costly to maintain.
In addition to the two methods for investment optimisation, a modelling ap-
proach for systems combining economic and environmental aspects has been de-
veloped as well. This approach has been used for modelling cement production
facilities, taking both economic and environmental issues into consideration.
In order to deal with prediction uncertainties, time series prediction using ge-
netic algorithms was investigated as well. Discrete-time prediction networks, a
novel type of recurrent neural networks, were introduced, and were shown to pro-
vide one-step macro-economic time series prediction with greater accuracy than
several other methods.
Keywords: Transition strategy optimisation, Investment strategies, Multi-objective
decision making, Optimisation under uncertainty.
i
Acknowledgements
First I would like to thank my supervisor Mattias Wahde. At my moment of
despair, he took on the responsibility to be my supervisor, after which things got
considerably better.
I also would like to thank Competence Centre for Environmental Assessment
of Product and Materials Systems (CPM) at Chalmers University of Technology
for their ?nancial support of the project.
Magnus Karlstr¨ om and Karin G¨ abel have provided me with intriguing (and
real!) problems that needed solutions as well as a thorough understanding of
these problems as well as being co-authors for a number of papers, for which I
am grateful. If it was not for Raul Carlson and Anne-Marie Tillman, I would
de?nitely not have started my Ph.D. studies in the ?rst place. Even though fate
wanted me to continue elsewhere, I am thankful to them. I also want to thank
all my colleagues at the Department of Applied Mechanics for all the enjoyable
discussions and support.
Thank you Eva, So?a and Linn´ ea for support and love, even when I’ve been
absorbed by work.
Peter Forsberg
G¨ oteborg, 2006
iii
Contents
1 Introduction 1
1.1 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Transition processes 5
2.1 Economic and environmental aspects . . . . . . . . . . . . . . . . 7
3 Optimisation techniques 13
3.1 Deterministic optimisation techniques . . . . . . . . . . . . . . . 14
3.2 Stochastic optimisation techniques . . . . . . . . . . . . . . . . . 18
3.3 Dynamical optimisation . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Stochastic dynamical optimisation . . . . . . . . . . . . . . . . . 24
4 Assessing the future 27
4.1 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Decision-making under uncertainty . . . . . . . . . . . . . . . . . 32
5 Case studies 35
5.1 The cement production case . . . . . . . . . . . . . . . . . . . . 35
5.2 The hydrogen infrastructure case . . . . . . . . . . . . . . . . . . 40
5.3 The hythane infrastructure case . . . . . . . . . . . . . . . . . . . 44
6 Concluding remarks 51
6.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7 Summary of appended papers 55
7.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
v
vi CONTENTS
7.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
APPENDED PAPERS
List of publications
The work presented is based on the following publications, which are included in
the thesis.
I. Karin G¨ abel, Peter Forsberg and Ann-Marie Tillman, The design and build-
ing of a lifecycle-based process model for simulating environmental perfor-
mance, product performance and cost in cement manufacturing, Journal of
Cleaner Production, Volume 12, Issue 1, February 2004, pp. 77-93.
II. Peter Forsberg and Magnus Karlstr¨ om, On optimal investment strategies for
a hydrogen refueling station, International Journal of Hydrogen Energy, In
press, corrected proof available online 25 July 2006.
III. Peter Forsberg and Mattias Wahde, Macroeconomic and ?nancial time se-
ries prediction using networks and evolutionary algorithms, Proceedings of
Computational Finance 2006, London, 27-29 June 2006, pp. 403-411.
IV. Peter Forsberg and Magnus Karlstr¨ om, Optimization of the investment strat-
egy for a combined hydrogen and hythane refueling station, submitted to
International Journal of Hydrogen Energy.
The author has also contributed to research in the following related subjects.
V. Peter Forsberg, Modelling and Simulation in LCA, CPM Technical Report
2000:1, 2000.
VI. Wim Dewulf, Raul Carlson,
?
Asa Ander, Peter Forsberg and Joust Du?ou,
Integrating Pro-Active Support in Ecodesign of Railway Vehicles, Proceed-
ings of 7th CIRP Seminar on Life Cycle Engineering, Tokyo, 27-29 Nov.
2000, pp 111-118.
VII. Raul Carlson, Peter Forsberg, Wim Dewulf and Lennart Karlsson, A full
design for environment (DfE) data model, Proceedings of Product Data
Technology, Brussels, 25-26 April 2001, pp 129-135.
vii
viii List of publications
VIII. Raul Carlson, Maria Erixon, Peter Forsberg and Ann-Christin P? alsson, Sys-
tem for Integrated Business Environmental Information Management, Ad-
vances in Environmental Research, 5 2001, pp 369-375.
IX. Wim Dewulf, Joost Du?ou, Raul Carlson, Peter Forsberg, Lennart Karls-
son., Dag Ravemark,
?
Asa Ander and Gerold Spykman, Information Man-
agement of Rail Vehicle Design for Environment for the entire Product Life
Cycle, Proceedings of 1st International Conference on Life Cycle Manage-
ment, LCM 2001, Copenhagen, 27-29 August 2001, pp 69-72.
Nomenclature
Allele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, the possible settings for a gene.
Auto-regression, AR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
In time series prediction, using a linear combination
of past values to calculate the predicted future value.
Chromosome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
In a GA, a string of genes representing a potential so-
lution to a problem.
Convex curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
A curve that is bulging outward over its total exten-
sion.
Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, exchange of genetic material between two
individuals.
Crowding distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
The mean distance to the neighbouring solution for a
multi-objective optimisation problem.
Dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
A dynamic optimisation method that makes use of
Bellman’s principle of optimality to solve the prob-
lem by backward induction.
Dynamical optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
An optimisation problem de?ned over a time period.
Dynamical problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
A problem de?ned over time and containing time-
continuous or discrete dynamic parts.
ix
x Nomenclature
Feasible point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
A point satisfying all constraints for an optimisation
problem.
Flow semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
The connection between, and use of, the general vari-
ables intensity and ?ow..
Functional unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
In LCA, a reference to which the inputs and outputs
of a product system are related as parts of the normal-
isation.
Gene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, the smallest part of a chromosome.
Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, the procedure of evaluating the individuals in
a population and replacing the population by its off-
spring.
Genetic algorithm (GA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Genetic algorithm. A stochastic optimisation method
inspired by natural evolution.
Hydrogen reformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A device to produce hydrogen from hydrocarbons,
e.g. methane gas.
Hythane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Natural gas with a small ratio of hydrogen.
Individual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
In a GA, a member of a population, carrying one chro-
mosome.
LCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Life cycle assessment. A systematic method to as-
sess the environmental impact of a product or function
produced. Also called Life cycle analysis.
LCI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Life cycle inventory analysis. Quantifying the rele-
vant products, resources used and emissions released
for the entire life cycle of a product.
Nomenclature xi
Markov decision process (MDP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
A process where the decisions taken at a certain point
only depend on the state at the previous point in time,
and not states further back in time.
Moving average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
In time series prediction, to calculate the predicted fu-
ture value by averaging a number of past values.
Multi-objective optimisation problem (MOOP) . . . . . . . . . . . . . . . . . . . . . . . . 18
An optimisation problem having more than one ob-
jective function, see also Pareto front.
Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, the operation of probabilistically changing
one gene at random.
Normalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
In LCA, relating the ?ows in a product system to the
functional unit.
Objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
The function to be optimised in an optimisation prob-
lem.
Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
See Dynamical optimisation.
Pareto front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
A curve of points such that, for points on the curve, no
criterion can become better without making another
criterion worse.
Reference ?ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
A measure of the needed outputs from processes in a
given product system required to ful?ll the function
expressed by the functional unit.
Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
In stochastic optimisation, one outcome of the distur-
bance generated from a scenario.
Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, selection of individuals for crossover.
xii Nomenclature
Stochastic dynamic optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A dynamic optimisation method for systems under the
in?uence of a stochastic disturbance.
Stochastic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
An optimisation method for solving stochastic dynam-
ical optimisation problems.
Time series prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
The process of predicting future values in a time se-
ries using past data.
Unit process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
In LCA, the smallest part of a product system for
which data is collected when undertaking an LCA.
Chapter 1
Introduction
This thesis focuses on optimisation of industrial transition processes. A transi-
tion process can involve change of equipment, re-location of premises or some
other structural change in, for example, societal infrastructure. These processes
often involve large investments that are implemented over a long time, and in a
situation of uncertain future development. Under such circumstances, ?nding the
optimal investment strategy is not an easy task. A common approach in invest-
ment planning [1, 2] is to list a number of alternatives, and then to pick the best
one by hand or to use a rule-of-thumb technique, a procedure that, in the case of
highly complex systems, most often results in a sub-optimal solution being found.
A better approach for ?nding the optimal solution is to make use of mathematical
optimisation techniques such as dynamical optimisation to search for the optimal
investment strategy; Such approaches are considered in this thesis.
The research objective in this thesis is to develop methods for optimisation of
industrial transition processes with monetary and environmental considerations.
In doing so, three areas are investigated: (1) modelling of production systems,
(2) prediction of future behaviour using TSP and (3) optimisation of investment
strategies using optimal control.
In the ?eld of economics, investment planning is an important topic [3, 4].
Of highest interest is then, of course, to ?nd the optimal investment strategy.
This problem, which can be solved using optimal control theory, is in general
de?ned over some period of time. In other, related applications, optimal con-
trol theory is used for, e.g. maximising growth in national economics [5], ?nding
optimal investments in funds [6], production planning [7], optimisation of se-
quential investments [8], and maximising return on capital funds [9]. Under the
in?uence of a stochastic disturbance, here in the form of an uncertain future devel-
opment, the optimisation problem becomes a stochastic dynamic optimisation
problem. In economics, this type of problem is referred to as investment under
uncertainty [10, 11, 12] and in process engineering as process design under un-
1
2 CHAPTER 1. INTRODUCTION
certainty [13, 14]. In Paper IV, stochastic optimal control theory is used to ?nd
the lowest expected production cost for a combined hydrogen and hythane refu-
elling station under the in?uence of three stochastic demand scenarios. In both the
above cases, discussed in Papers II and IV, the developed methods are intended to
be used for decision support.
Recent studies have been made regarding the economical feasibility of hydro-
gen in regard to the infrastructure that must be built [15, 16, 17, 18, 19, 20, 21].
However, none of these studies investigates the implications of investments over
time. By contrast, in Paper II optimal control theory is used to ?nd a short-term
equipment variable utilization for one-week periods and, at the same time, a long-
term investment strategy for the whole investment period covering 20 years with
the aim of minimising the production cost for a plant. The method is exempli?ed
by a hydrogen dispensing infrastructure case.
Predicting future values of key variables is, of course, highly relevant in the op-
timisation of transition processes. Even short-term prediction is important. Such
prediction is considered in the ?eld of time series prediction (TSP). Due to the
often high level of noise present, standard procedures, such as the auto-regressive
and moving-average methods, are sometimes not fully successful [22, 23, 24]. In-
stead other, more adaptive methods based, for example, on neural networks can
be used [25, 26]. In Paper III of this thesis, a novel type of neural network is
developed for prediction of noisy time series.
When evaluating transition processes the economic consequences are impor-
tant. The environmental awareness in today’s society is constantly raising the
requirements of a cleaner production process. Thus, in this thesis, the environ-
mental effects are considered as well. At the same time the increasing complexity
of the production systems makes the environmental analysis more dif?cult to carry
out. In the 1990s more advanced methods were developed to assist environmen-
tal analysis of technical systems. One of these methods is life cycle assessment
(LCA). Much has been written about LCA. The ISO standards 14040-42 [27] give
very general guidelines on how an LCA should be performed. In fact, many stan-
dard papers on LCA, e.g. [28, 29, 30, 31, 32, 33], approach the topic in a rather
non-mathematical way. Until 1998 only one paper [34] was published regarding
guidelines on how to carry out the actual calculation, the so called normalisa-
tion. After 1998, the subject of normalisation in LCA has been considered in a
mathematical point-of-view by Heijungs [35, 36, 37]. All of the mathematical
methods presented by Heijungs only consider the standard LCA which includes
a linear and static model representation. For other approaches there is only a
limited number of texts available. Examples include linear optimisation of LCA
systems [38, 39], multi-objective optimisation [40] and dynamic life cycle inven-
tory models [41, 42]. Some cases with integration of economic cost objectives
have also emerged [43, 44]. There is still, however, a large potential for improve-
1.1. MAIN CONTRIBUTIONS 3
ments concerning, for example, the range of applicability of the models. This
thesis investigates a number of approaches, how they can be used, and possible
improvements. The ?ndings are exempli?ed by a cement production case [45]
in Paper I and are intended to be used in investment optimisations, such as those
presented in Papers II and IV.
1.1 Main contributions
Transition processes taking place in the societal infrastructure and in large in-
dustries are in general very complex systems. This is due to the fact that these
systems do not only have technical and economic aspects, but also social, envi-
ronmental, political and geographic aspects. To take all these considerations into
account when constructing a model is, of course, impossible. This thesis focuses
on economic and environmental aspects and aims at providing some examples of
general tools for carrying out structural transition optimisation. It is the aim of the
author that, when the tools presented here are used together, the effort of carrying
out the above-mentioned type of optimisations should be reduced considerably.
Mathematical models are generally speci?c to the application at hand. Using
the type of models discussed in this thesis, the ?exibility with regard to the types
of calculations that are possible to carry out can be increased considerably [46,
47, 48]. These models come from the study of physical systems [49], but can be
successfully applied to other types of systems, e.g. environmental systems [50].
The aim of the models is to provide the optimisation algorithm with the effects
of changes in the future strategy. In doing so, the model has to provide the future
behaviour of key parameters. Some of these can be modelled in detail but others,
where exact knowledge is lacking, must be predicted. One way of achieving this is
by time series prediction [23, 51, 52]. In this way the short-term future behaviour
can be estimated, something that is important for fast dynamics. Using the above-
mentioned tools, the transition strategy is then optimised using stochastic multi-
objective optimisation. To summarise, the main contributions of the presented
work are:
• A modelling approach for production systems with environmental measures
comprising separation of model and problem formulation, and leading to
more ?exible models (Paper I).
• Progress in time series prediction (TSP) using genetic algorithms (GAs)
resulting in increased accuracy for predictions of noisy time series (Paper
III).
• Methods for concurrent optimisation of investment strategies and run pat-
terns for long-term planning of industrial production sites, taking economic
4 CHAPTER 1. INTRODUCTION
and environmental considerations into account. In Paper II the model is
of the single-objective deterministic kind and in Paper IV of the multi-
objective stochastic kind.
• Speci?c results regarding optimal investments for hydrogen and hythane
refuelling stations (Papers II and IV).
The author was the main contributor to Papers II and IV. In Paper I, the author’s
contributions were to develop the modelling approach and the model framework,
to carry out all calculations needed for solving the problem, and to write a signif-
icant part of the paper. In Paper III both authors contributed equally.
Chapter 2
Transition processes
Structural transition processes occur in all industries. Examples include change
of machinery, moving of production units and change of production at current
sites. For decision support, a number of calculations are carried out regarding
the economic consequences and, at times, optimisations are done [53]. However,
only rarely are both economic and environmental aspects taken into account. This
thesis presents some methods for integrating economic and environmental aspects
when assessing industrial transition processes. The results are intended to be used
for decision support.
In particular, the problem of optimising investment strategies, i.e. selecting
when and to what extent investments are to be made for maximum performance,
is explored. In connection with this problem, the topic of predictability of vari-
ables has also been considered within the framework of time series prediction. In
most cases in this thesis the connections to economy are explicit, i.e. economic
measures appear in the model, and the environmental connections are implicit,
i.e. they are present through the use of an environmentally favourable technique. It
should be kept in mind that this implies a constraint in the sense that only environ-
mentally favourable techniques are considered to be part of the set of acceptable
solutions. However, the developed methods, such as the multi-objective optimi-
sation procedure described in Paper IV, have originally been designed for use in
cases with explicit connections between environmental and economic aspects.
When considering large industrial structures, the economic consequences are
distributed over a number of years. For a complete production line in a factory or
for a major societal infrastructure change, the consequences might be distributed
over a period exceeding 20 years [54, 55]. To be able to carry out an optimisation
of the economic results, the variation in a number of variables, e.g. the rate of
interest, the technical development within the ?eld and the yearly production and
demand, must be estimated. In order to improve the accuracy of such variables, a
study of time series prediction (TSP) has been undertaken in Paper III, aimed at
5
6 CHAPTER 2. TRANSITION PROCESSES
improving short-term (one-step) prediction of macroeconomic time series, where
the time step length often equals one year or a quarter of a year. However, long-
term predictions are much harder, and it is generally not possible to make optimi-
sations of investment strategies that will be valid for the whole investment period
(20 years, say). Instead, a dynamic approach must be taken to the optimisation
of transition processes, such that, when the assumed values of a parameter have
deviated substantially from the expected path, a new calculation is carried out,
based on the new, corrected behaviour of the parameter in question. This makes it
possible always to have the best and most up-to-date structural transition strategy
at hand.
Life cycle assessment (LCA) [31, 32, 33, 34] is a way to quantify environ-
mental in?uences of a product (or service) over its entire life. In the study pre-
sented in Paper I, similarities and differences between LCA and technical system
theory [56]
1
were investigated in order to make improvements in combining eco-
nomic and environmental modelling. Incorporating the new ?ndings, a model of a
cement production process was generated and calculations for improving the eco-
nomic and environmental performance were performed. The variables involved
express aspects of quality and economy, as well as resource use and emissions.
The model developed was intended for many types of calculations regarding,
e.g. economy, product quality and emissions.
Changes toward more environmentally favourable solutions frequently incor-
porate large investments in infrastructure. The cost and uncertainty of changing
these facilities are usually considered to be obstacles for the introduction of new
techniques. In Papers II and IV, methods for ?nding optimal investment strategies
for this type of environmentally favourable production facilities are investigated.
Based on an assumed future development scenario, optimal investment strategies
are calculated. In Paper IV, special emphasis is put on reducing the uncertainties
by using several different scenarios and stochastic optimisation. The applicability
of the developed method is exempli?ed in two studies on pro?table investment
strategies for a hydrogen station and a combined hydrogen and hythane refuelling
station.
Finding the optimal solution to the set of problems discussed above is far from
trivial. Nevertheless, in reality, many such problems are solved in an intuitive
manner based on experience [57, 58], in a large part due to the fact that the mathe-
matical formulation of the problems usually is very hard to ?nd for these complex
systems [59, 60]. A complex system here signi?es a system composed of a num-
ber of simpler subsystems with a large, often huge, number of interconnections.
Due to the large number of connections, these systems are usually very hard to
1
The term technical system theory is here used for the science of constructing models of
processes etc. as is done in control theory.
2.1. ECONOMIC AND ENVIRONMENTAL ASPECTS 7
understand and model. One example of this is emergence [61, 62, 63]. Emergent
properties are properties possessed by a system, which cannot be traced back to
any of its parts. A good explanation of emergence is given in [64] as ”A complex
system usually involves a large number of components. These components may be
simple, both in terms of their internal characteristics and in the way they interact.
Still, when the system is observed over longer time and length scales, there may
be phenomena that are not easily understood in terms of the simple components
and their interactions”.
Optimisation of complex systems can therefore play a very important role, by
revealing unforeseen solutions better than those currently available. One should
keep in mind though, that accurate optimisation over periods extending 20 years
into the future is not easy, even using techniques that reduce the effects of uncer-
tainty such as stochastic optimisation with multiple scenarios. When predictions
are generated using models built from time series data, like the DTPNs in Paper
III, an implicit assumption is that the future development follows a pattern similar
to that of the past data. In this case an unforeseen major event can disrupt all pre-
dictions instantaneously. Another option is to make prediction based on a detailed
model of the system combined with probable future developments, as is done in
Paper IV. However, even in this case, an unforeseen event not covered by the
model can disrupt all predictions. Since nothing in nature is discontinuous there
are, however, always precursor events to major events. To identify these events is,
of course, very important and therefore successful optimisation requires extensive
knowledge of the system under study. Ideally all possible future outcomes should
be included in the model. In reality this is impossible though, and one therefore
has to settle for less-than-perfect models.
2.1 Economic and environmental aspects
The objectives explored in this thesis are mostly related to economy. Therefore
this section starts with a discussion of the economic measures used in Papers II
and IV, after which some important aspects of one possible quanti?cation of en-
vironmental impact is presented, namely LCA. Both economic and environmen-
tal measures are considered, since it is desirable to ?nd environmentally viable
solutions that are still economically favourable. No company can support non-
pro?table environmental sustainability.
In the literature, future costs and incomes are usually discounted with regard
to the discount rate D > 0 using the net present value correction [65]
C
p
(t) =
1
(1 + D)
t
, (2.1)
8 CHAPTER 2. TRANSITION PROCESSES
where t is the time. The above equation re?ects investors’ preference of immedi-
ate return of cash in contrast to future returns. The actual discounting depends on
the length of time and the discount rate. Usually, the discount rate is the risk-free
interest rate added to an interest rate re?ecting the risk involved in the speci?c ven-
ture. As the name implies, the risk-free interest rate is the safe rate earned from a
completely risk-free investment. When optimising policies that span a long time,
as the calculations in Papers II and IV do, the value of the discount rate can have
a signi?cant effect. As is pointed out in Paper IV, for a discount rate of 0.1 and in
the case of an evenly distributed cost, the effective discounting is C(t) = 0.4466
for a time period of 20 years. C(t) is the mean value of C
p
(t) over the time period
considered. The high discount rate in Paper IV is motivated by the high risk; the
investments are made in a new technique with uncertain potential and acceptance.
In both Paper II and Paper IV, the loan for purchase of equipment is assumed
to be of the annuity type. In this case, it is possible to calculate a capital cost per
time unit for the refuelling station. The additional costs, e.g. for purchase of raw
materials and maintenance, are then added and a total instantaneous production
cost at time t per produced unit can be calculated, which is measured in USD per
kg H
2
.
In order to ?nd the mean production cost for the entire investment period,
one must integrate the instantaneous production cost. In Paper II, three ways of
integrating this cost are shown: (1) adding the costs at all times without discount-
ing, (2) discounting future costs using Eq. (2.1) above and (3) discounting future
costs using Eq. (2.1) and distributing the total cost evenly over the whole invest-
ment period. One should keep in mind that the third option does not re?ect the
sum of the real cost to the production facility. The discussed mean production
costs represent different ways of calculating the costs for production, depending
of preference. They are all candidates for an objective function that can be used
for optimisation. It should be noted that if the total capital cost is discounted to
present day value using the same interest rate as for an assumed annuity loan, the
result is the absolute investment cost. Since, in Paper II, only one week per in-
vestment is explicitly evaluated, the above ways of calculating the costs are used,
for simplicity. Details on how production costs have been calculated are given in
Paper II.
In Paper IV, the optimisation is carried out over the whole investment period
covering 20 years. Since loans are considered to be of the annuity type, the costs
for investments are not discounted. Investments in equipment are only subject to
a decreased cost due to increased production and technology development. This
is due to the fact that the effect of the annuity loan and the discounting will can-
cel each other out, provided that the discount rate is the same. In Paper IV, the
production cost was calculated over the entire investment period, taking the total
non-discounted purchase cost into account. This is in contrast to Paper II where
2.1. ECONOMIC AND ENVIRONMENTAL ASPECTS 9
the production cost was calculated for one week per investment, taking the non-
discounted cost for the annuity loan into account, a procedure that results in a
time-varying production cost, as can be seen in Paper II. Apart from investments,
all other production costs were discounted, however. The mean production costs
were calculated as the sum of costs divided by the sum of sold hydrogen and
hythane, respectively.
Environmental effects are often less tangible than are economic ones and
therefore harder to measure and quantify. One reason is that in the short perspec-
tive many environmental aspects tend to have little or no environmental effect.
Instead, at a certain level, there is an abrupt and sometimes unforeseen effect.
Another reason is that the causality of environmental effects is not always totally
understood. One example is the Greenhouse effect. Obviously the global mean
temperature is increasing in the short-term perspective, but is this caused by hu-
man activities? Economic aspects, on the other hand, tend to have a more direct
and immediate effect.
However, one way environmental in?uences can be quanti?ed is by LCA. The
life cycle usually starts with extraction of raw materials and continues with trans-
portation, manufacturing, use and possibly re-use. It then ends with waste man-
agement, recycling and disposal. There exists a vast literature on the concept of
LCA, see e.g. [30, 34, 66, 67, 68].
In 1997 the ISO standard 14040 on LCA was approved [27]. In this standard
LCA is de?ned as ‘...the environmental aspects and potential impact through-
out a product’s life (i.e. cradle-to-grave) from raw material acquisition through
production, use and disposal.’ Each of these stages consumes resources and pro-
duces emissions and waste. In LCA all these aspects are taken into account and
are related to the product or function produced. LCA further aims at assessing the
impact of the production on nature.
In the life cycle inventory analysis (LCI), which is one part of LCA, the re-
sources, emissions and products related to the production system are measured.
Then a model over the production system is created. In order to know the ef-
fects of each produced unit (in LCA called functional unit), the measured re-
sources etc. are scaled to the functional unit. This is done by aggregating all unit
processes in the product system and scaling the ?ows of these processes to match
the reference ?ow of the system, a process referred to as normalisation. The
data used in the inventory is based on time-averaged statistics and is hence in-
dependent of time. In addition a linear relation between resource use, emissions
and production is assumed. The resulting normalisation step is mathematically
equivalent to solving a linear equation system. The equation system is usually
well-posed by construction, i.e. having equal number of variables and constraints,
and hence possible to solve exactly. Publications on LCA cover handbooks, case
studies and theoretical studies on the concept that do not, in general give any
10 CHAPTER 2. TRANSITION PROCESSES
Figure 2.1: Example of a model for a small glass bottle production process. Note that,
since the data for the different processes (shown as boxes in the ?gure) are taken from
different sources, e.g. product data sheets, the amounts of ?ow do not match at this stage.
The objective of the normalization is to scale the processes so that the ?ows do match.
The results of the normalization procedure are shown in Figure 2.2 below.
directions on how to represent the ?ow model and form the resulting equation
system [33, 69]. Recently a number of publications on the computational part
have appeared [36, 37, 70, 71]. The result of the normalisation to the functional
unit presented there is expressed as
g = BA
?1
f, (2.2)
where A is a matrix describing the internal ?ows of the technical system (the
technology matrix), f the demand vector, i.e. a vector expressing what is being
produced, and B the intervention matrix, i.e. the external ?ows to and from the
technical system.
An example of a model for a small glass bottle production process is shown
in Figure 2.1. In this process sand is melted to glass, producing carbon dioxide
emissions. The glass is then cast to bottles which are scanned for defects before
delivery. Some bottles are discarded in the scanning process due to defects and
these bottles are returned to the casting process after crushing. Each of the above
processes are unit processes. Using the nomenclature introduced above, the tech-
2.1. ECONOMIC AND ENVIRONMENTAL ASPECTS 11
Figure 2.2: The resulting ?ow model for the small glass bottle production process intro-
duced in Figure 2.1.
nology matrix becomes
A =
_
¸
¸
_
92 ?0.2 0 1.9
0 1 ?10 0
0 0 1 ?10
0 0 9 0
_
¸
¸
_
, (2.3)
where the rows in A represent the ?ows of glass (row 1, measured in [kg]), bottles
([pcs], row 2), recycled bottles ([pcs], row 3), and delivered bottles ([pcs], row 4),
respectively. The columns represent the processes indicated by Roman numerals
in Figure 2.1. Note that negative numbers indicate a ?ow into a process, and
positive numbers a ?ow out from a process. The intervention matrix becomes
B =
_
?100 0 0 0
1 0 0 0
_
, (2.4)
where the ?rst row represents sand ([kg]) and the second row represents carbon
dioxide ([kg]). To calculate the intervention for a reference ?ow of one produced
bottle (after the calculation this ?ow will become the functional unit), f is set to
f = [0 0 0 1]
T
giving g = BA
?1
f = [?0.2186 0.0022]
T
. The resulting resource
use and emissions released are then 0.2186 kg sand and 0.0022 kg carbon dioxide,
respectively. The normalised ?ow model is shown in Figure 2.2.
Chapter 3
Optimisation techniques
The transition processes discussed in Chapter 2 are naturally de?ned over some
time period; they are dynamical problems. Industries are, of course, trying to
maximize performance, implying the need for optimisation.
In order to optimise a transition process a dynamical optimisation technique
must be utilized. In most cases, dynamical optimisation problems are solved by
transforming them into static optimisation problems. Therefore, this chapter starts
with a short overviewof unconstrained and constrained non-linear static optimisa-
tion which will lay the foundation for later discussions on dynamical optimisation
techniques in Section 3.3 below. In particular, the SQP algorithm used in Paper II
and the multi-objective genetic algorithm used in Paper IV will be examined. In
Section 3.4 optimisation of systems in?uenced by stochastic perturbations will be
discussed.
The goal of optimisation is to ?nd the optimal point x
?
for a given objective
function f(x), where x = (x
1
, x
2
, . . . , x
N
) ? R
N
. For the single-objective case,
f = f is a scalar, taking values in R
1
, whereas, for the multi-objective case,
f ? R
N
. There may also be equality and inequality constraints of the form
c(x) = 0, (3.1)
d(x) ? 0, (3.2)
as well as limits on the allowed intervals for x, i.e. x
l
? x ? x
u
1
. Despite the
modest appearance of this optimisation problem, ?nding the optimal point is, in
general, a dif?cult task.
A special type of optimisation problem is the convex problem. A function
f(x) : R
N
? R is convex if the domain of the function, denoted domf, is a
1
Throughout this thesis relations between vectors such as x
l
? x require equal dimensions and
are to be interpreted component-wise.
13
14 CHAPTER 3. OPTIMISATION TECHNIQUES
?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3.1: Example of a one-dimensional convex function.
convex set and
f(?x + (1 ??)y) ? ?f(x) + (1 ??)f
(3.3)
? x, y ? domf, 0 ? ? ? 1. A non-scienti?c geometrical interpretation is that if
f draws a curve that is bulging outward over its total extension, then it is convex.
An arbitrary example of a one-dimensional convex function is given in Figure 3.1.
A problem is called convex if both the objective function and the constraints are
convex. Such functions are important in optimisation since they make it possi-
ble to guarantee convergence [72]. Furthermore, many non-convex optimisation
problems can be transformed into convex ones [73].
3.1 Deterministic optimisation techniques
The term deterministic is here used to indicate that the optimisation algorithms
do not contain any stochastic parts. Thus, if such an algorithm is run twice, us-
ing the same set of inputs, the results will be identical to each other, i.e. they
are perfectly predictable. In deterministic optimisation there exists robust algo-
rithms with guaranteed convergence for the linear case, i.e. the case in which the
3.1. DETERMINISTIC OPTIMISATION TECHNIQUES 15
objective function and all constraints are linear. One such algorithm is the Sim-
plex method [74]. However, in most practical problems the objective functions
are non-linear. Such functions will usually have both a global optimal point and
many local optimal points. Since it is very dif?cult to distinguish between a lo-
cal and a global optimal point, simple gradient-descent algorithms are usually not
successful. Such algorithms tend to get stuck on a local optimal point instead of
?nding the true global optimal point.
Finding the global optimal point is the major task of non-linear programming
(NLP) [75, 76]. Consider the general NLP problem
min
x?R
N
f(x)
s.t. c(x) = 0 (3.4)
d(x) ? 0
where f(x) is the (scalar, i.e. single-objective) criterion function, c(x) the non-
linear equality constraints and d(x) the non-linear inequality constraints. The
functions f(x), c(x) and d(x) are assumed to be smooth, i.e. at least twice-
continuously differentiable. Let g(x) = ?
x
f(x) denote the gradient vector of
the objective function, C(x) =
?c
?x
the Jacobian matrix of the constraint vector
c(x), and D(x) =
?d
?x
the Jacobian matrix of the constraint vector d(x). Now
de?ne the (scalar-valued) Lagrangian function in the classical way [77]
L(x, ?) = f(x) ??
T
c(x) ?µ
T
d(x), (3.5)
where ? and µ are Lagrange multiplier vectors. In an optimal point the ?rst
derivative of the Lagrangian with respect to x is zero, i.e
?
x
L(x
?
, ?
?
, µ
?
) = g(x
?
) ??
?T
C(x
?
) ?µ
?T
D(x
?
) = 0 (3.6)
where (x
?
, ?
?
, µ
?
) is the optimal point. In addition, requirements have to be put
on the inequality part variables µ and d. At the optimal point, it is clear that an
inequality constraint d
i
(x
?
) can either be satis?ed as an equality, d
i
(x
?
) = 0 or
strictly satis?ed, d
i
(x
?
) > 0. In the former case the constraint is said to be active
and hence a part of the active set A, i.e. i ? A. In the latter case the constraint
is inactive and part of the inactive set A
?
, i.e. i ? A
?
. For the active set the
requirements equal those for equality constraints, i.e. µ ? 0. For the inactive set
the multiplier must be zero. This can also be formulated µ
?T
d(x
?
) = 0, which
is sometimes referred to as the complementary slackness condition. With these
requirements, the Karush-Kuhn-Tucker (KKT) [78] condition for optimality is
de?ned as
g(x
?
) ??
?T
C(x
?
) ?µ
?T
D(x
?
) = 0
µ
?T
d(x
?
) = 0 (3.7)
µ
?
? 0
16 CHAPTER 3. OPTIMISATION TECHNIQUES
and µ is sometimes referred to as the KKT multiplier. In addition the original
constraints from Eq. (3.4), c(x
?
) = 0 and d(x
?
) ? 0 must be satis?ed at the
optimal point. In order to solve the KKT for x
?
, the active inequality constraints
are treated as equality constraints and the inactive ones are ignored, giving
g(x
?
) ??
?T
J(x
?
) = 0,
r(x
?
) = 0, (3.8)
? ? 0,
where r ?
_
x ? R
N
|c(x) = 0, d
i
(x) = 0 ? i ? A
_
and J(x) = ?r/?x. Now
these re-de?ned requirements can be solved with Newton’s method by carrying
out a Taylor series expansion of Eq. (3.8). Letting H
L
= ?
2
xx
L, the expansion
becomes
g(x) ?J
T
(x)? +H
L
(x)(¯ x ?x) ?J
T
(x)( ¯ ? ??) = 0
r(x) +J(x)(¯ x ?x) = 0 (3.9)
which can be written as
_
H
L
J
T
J 0
_ _
?p
¯ ?
_
=
_
g
r
_
. (3.10)
Solving the above equation will yield the step p and the Lagrange multiplier at
the new point, ¯ ?. The new point is then obtained as ¯ x = x + p. Note that in
Eq. (3.10) the new Lagrange multiplier is calculated in an absolute way while, for
the new point ¯ x, the increment p is calculated. The Newton step is then iterated
until convergence.
The Newton method de?ned above is a local optimisation algorithm. In order
to improve the chances of ?nding the global optimum, one may use a globalization
strategy. One example is the line-search method which will adjust the step length
¯ x by a factor ? to ¯ x = x+?p. The value of ? is usually determined by the rate of
progress measured by a merit function [79, 80]. Another option is to adjust both
the magnitude and direction of the search step, so that the search direction p will
lie within a given radius which de?nes a trusted region [81].
A widely used algorithm to solve the above NLP problem in Eq. (3.10), i.e. to
?nd the global optimum, is sequential quadratic programming (SQP), see e.g. [82,
83]. When using SQP, one may observe that Eq. (3.10) represents the ?rst order
optimality conditions for the the optimisation problem
min
p
g
T
p +
1
2
p
T
H
L
p (3.11)
s.t. Jp = ?r,
3.1. DETERMINISTIC OPTIMISATION TECHNIQUES 17
0 2 4 6 8 10
1.8
2
2.2
2.4
2.6
2.8
3
3.2
x 10
5
Scenario 1
Hydrogen production cost [USD/kg]
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
k
g
] 1
5
9
13
17
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
%
o
f
t
o
t
a
l
]
2
4
55
60
65
70
75
80
85
90
Figure 3.2: Illustration of Pareto optimality. The objective is to minimise both the
unsatis?ed hydrogen demand and the hydrogen production cost. Solutions marked with
small dots and crosses are part of the Pareto front whereas the ones marked with circles
are not.
which is a quadratic programming (QP) problem. The SQP is a sequential al-
gorithm that makes use of inner and outer iterations. The objective of the inner
iteration is to ?nd a search direction p which is used in the outer one to ful?ll
the ?rst order conditions for optimality. The search direction p is found by solv-
ing the optimisation problem in Eq. (3.11). The outer iteration makes use of the
new search direction by taking the step ¯ x = x + ?p, where the magnitude of
the step (?) is determined by a line search method. This makes the SQP a global
optimisation algorithm.
In Paper II the resulting NLP optimisation problem was solved using the
NPSOL program [84], which is of the above SQP class. First the NPSOL al-
gorithm aims at calculating a point that is feasible, starting from a user-initiated
point. Then the SQP algorithm described above is used to ?nd the optimal point.
Calculating gradients for the investment problem in Paper II is not easy. One rea-
son is the fact that the objective function f(x) is not differentiable in the whole
of R
N
. Another reason is the complex structure of summations in f(x). Using
NPSOL, no algebraic expressions of gradients and Hessians needs to be given.
Instead, NPSOL can make use of ?nite-difference derivatives. The NPSOL algo-
18 CHAPTER 3. OPTIMISATION TECHNIQUES
rithmcan also deal with minor discontinuities if they are isolated and located away
from the solution. The SQL algorithm described above will converge to the global
optimum for convex problems [72]. Since the investment problem considered in
Paper II is convex, the algorithm will ?nd the global optimal point. The objective
function is discussed in Chapter 2 above and the considered optimisation problem
will be treated in more detail in Chapter 5.
In the above SQP algorithm, the objective function f : R
N
? R. If f :
R
N
? R
M
, M > 1, the problem is a multi-objective optimisation problem
(MOOP). The solution to such a problem does not consist of a single optimal
point, but instead a number of points lying on a curve or surface called the Pareto
front. These points all have in common that they are non-dominated. In short, it
means that no other solution exists where any of the objectives are strictly better
than those on the Pareto front without some other objective being equal or worse.
Figure 3.2, taken from Paper IV, illustrates the principles of Pareto optimality.
The objectives along the x and y axes are both to be minimised. The solutions 1,
13, 9, 5 and 17 all belong to the ?rst Pareto front. Solution 4, however, is said
to be dominated by solution 5 and therefore does not belong to the Pareto front.
Not taking the ?rst Pareto front into account, another, inferior non-dominated
set of solutions can be found. This set is called the second Pareto front. In the
same way a number of Pareto fronts can be found. From Figure 3.2 it is evident
that the two objectives are in con?ict with each other. This is typical for multi-
objective optimisation problems. Should the objectives not be in con?ict, at least
one dimension could be omitted. In the two-dimensional case illustrated above,
this would lead to a single-objective optimisation problem.
3.2 Stochastic optimisation techniques
The search methods employed by stochastic optimisation algorithms depend, in
part, upon computational procedures generating a random outcome. Therefore
these algorithms produce different paths towards the optimal solution each time
they are run, and in the general non-linear case, convergence cannot be guaran-
teed. However, in most practical applications, proof of optimality is not the most
important aspect; Instead, ?nding a solution better than any presently available
one is usually suf?cient.
One of the most widely used stochastic optimisation techniques is the genetic
algorithm (GA) [85, 86, 87]. This algorithm is inspired by biological evolution.
In a GA, a population of candidate solutions to the problem at hand, referred to
as individuals, is maintained. Each individual contains a chromosome that is a
representation of a potential solution to the problem. The chromosome can, for
example, consist of a string of discrete (e.g. binary) or decimal numbers. Mixed
3.2. STOCHASTIC OPTIMISATION TECHNIQUES 19
Figure 3.3: An encoding scheme for a genetic algorithm (GA) in which discrete numbers
are used. An example of a chromosome is shown at the bottom of the ?gure.
representations exist as well, in which the chromosome contains both discrete and
decimal numbers. An example of an encoding scheme used in Paper II can be
seen in Figure 3.3. Each part of the chromosome is called a gene, and each gene
may take different values, referred to as alleles. In this example, only discrete
numbers are used. Since the problem is combinatorial, i.e. consists of a number
of discrete alternative equipment sizes for a refueling station, it is natural to use
a discrete encoding scheme. The problem also contains the time for investment,
which is deliberately encoded as a discrete number in order to decrease the size of
the search space, for faster convergence.
After the initial population is created (usually randomly), all individuals are
evaluated. This involves a calculation of the ?tness measure, which is used to rank
individuals with respect to their performance. Usually, the calculation of ?tness
values is the most time-consuming part of the algorithm. The way the ?tness
measure is calculated depends entirely on the problem at hand. Two examples of
calculating ?tness measures are given in Papers III and IV.
When all individuals have been assigned ?tness values, generational replace-
ment is performed. First, the best individual is transferred unchanged to the next
iteration (referred to as a generation), and then the remaining individuals are cre-
ated from selection, crossover and mutation. The selection, usually of two indi-
viduals, is carried out in proportion to ?tness. The selected two individuals then
generate two new individuals by blending genes from both of them, a procedure
known as crossover. In the simple case the genes are exchanged between chromo-
somes by cutting the latter at a random crossover point. The mutation is a random,
lowprobability change to individual genes in the chromosome. Crossover and mu-
tation must normally be tailored to the speci?c problem. A ?ow chart for a simple
genetic algorithm can be seen in Figure 3.4.
In the problem considered in Paper IV, gradient information was very hard
20 CHAPTER 3. OPTIMISATION TECHNIQUES
Figure 3.4: Flow chart of a genetic algorithm.
to ?nd algebraically. Alternatively, numerical differentiation can be used; such a
technique was employed in connection with the deterministic optimisation carried
out in Paper II. However, in that paper it was shown that for longer optimisation
time periods, the deterministic optimisation algorithm, in which numerical differ-
entiation is an important part, led to unacceptably long computational times and
was therefore intractable. Therefore, in Paper IV, a GA was used instead.
One signi?cant advantage with using a GA is that it does not need any gradient
information. In fact, GAs only need the ?tness measure and can thus be applied
to any optimisation problem where ?tness can be quanti?ed. In addition, GAs do
not introduce any requirements on the function that is to be optimised (for exam-
ple, it need not be convex), but it should be kept in mind that extra precautions
may have to be taken for strongly non-linear and discontinuous functions. One
such technique is to increase the mutation rate at the beginning of the evaluation
in order for the solution not to get stuck on a local optimum but instead ?nd the
global one [85]. In all, the GA is a very ?exible and easy-to-use optimisation al-
gorithm. However, there are some drawbacks. One is the above-mentioned need
for tailoring the encoding scheme, and the operators for crossover and mutation.
Another, perhaps more serious, drawback is that since the search for better solu-
tions is stochastic, so is the evaluation time. Occasionally, it may be better to stop
the current calculation and start from the beginning by creating a new random
population again. Also, there is no guarantee that the found solution indeed is
the global optimum. However, in practice, these drawbacks are rarely signi?cant.
3.3. DYNAMICAL OPTIMISATION 21
The need for implementing extra operators is well compensated by the simplicity
of the algorithm. Since true global optimisation is such a hard task, not even the
most advanced deterministic algorithms can guarantee optimality in the general
case. Considering this, the GA is a good candidate for real-world optimisations.
GAs have been extended to the multi-objective case [88]. The resulting algo-
rithms are collectively known as multi-objective evolutionary algorithms (MOEA).
These algorithms aim at ?nding the ?rst Pareto-optimal front discussed above. In
Paper II, a particular type of the above MOEA, called NSGA-II [89], is used. This
is an elitist non-dominated sorting GA that uses an explicit diversity-preserving
mechanismto keep solutions separated fromeach other. Since solutions are spread
along the Pareto front, some measures need to be taken for the solutions not to clog
together in the same spot. This is done by calculating the distance to the nearest
neighbour, called the crowding distance. The new population is ?lled with so-
lutions from one front at a time and in ascending order of fronts. First solutions
from front number one is used, then solutions from front number two etc. If not all
solutions from a front can be used (i.e. if the population is about to be ?lled), the
remaining available positions in the new population are ?lled with the solutions
having the highest crowding distance, i.e. lying furthest apart from each other.
The NSGA-II algorithm can be summarized as follows
1. Randomly generate parent P
t
, (|P
t
| = N) and offspring Q
t
, (|Q
t
| = N)
populations.
2. Combine parent and offspring populations to form R
t
= P
t
? Q
t
.
3. Evaluate the combined population R
t
and sort it into a number of non-
dominated fronts F
i
, i = 1, 2, . . . , r.
4. Iteratively create a new population P
t+1
? P
t+1
? F
i
, |P
t+1
| + |F
i
| ?
N, i = 1, 2, . . .
5. Carry out a crowding distance sorting on the remaining fronts not included
in P
t+1
and include the most widely spread solutions until |P
t+1
| = N.
6. Create offspring population Q
t+1
from P
t+1
by crowded tournament selec-
tion, crossover and mutation.
7. Iterate Steps 2-6 until convergence.
3.3 Dynamical optimisation
The dynamical optimisation problem is de?ned as the problem of minimising a
cost function J over a given time period by ?nding the optimal control trajectory
22 CHAPTER 3. OPTIMISATION TECHNIQUES
u. Thus, this type of problem is also referred to as an optimal control problem
(OC) [90, 91, 92]. Common cost functions include energy, fuel, and time. The
dynamical system can be mechanical, electrical or any other type that can be de-
scribed mathematically. In this thesis, investment problems are considered. The
continuous-time deterministic optimal control problem can be de?ned generally
as
min
u(t)
J, where J = ?(x(t
f
)) +
_
t
f
t
0
L(x(t), u(t), t)dt, (3.12)
s.t. ? x = f(x, u, t),
c(x, u, t) ? 0,
where J is the objective function, f are the state equation constraints, c are the
path constraints and u is the control vector. The objective function consists of two
parts: ?, a cost based on the ?nal time and state, and an integral depending on the
time and state histories. In addition there may be simple bounds on the state and
control variables, i.e.
x
l
? x ? x
u
(3.13)
u
l
? u ? u
u
,
and also boundary conditions on x and u.
The above problemalso has a discrete version, in which the integral is replaced
by summation. The problems in Papers II and IV are both expressed in the discrete
form, mainly due to the type of data available and for simplicity of calculation.
The optimal control problem may be solved by any of the following four meth-
ods: dynamic programming, the indirect method, the direct method, or simulation-
based optimisation.
The dynamic programming approach makes use of Bellman’s principle of op-
timality to solve the problem by backward induction [93]. The resulting partial
differential equation is very hard to solve, except in very fortunate cases.
An indirect method aims at ful?lling the necessary conditions for an optimum,
the Euler-Lagrange equations and the adjoint equations, using variational calcu-
lus. Finding these expressions requires calculation of gradients and Hessians,
which usually is cumbersome. In addition, the indirect method is sensitive to the
choice of starting point, i.e. the ?rst estimate. A poor starting point may result in
divergence or wildly oscillating trajectories.
A direct method [90] uses a sequence of points to approximate the state and
control variables. The sequence may be a piecewise polynomial expansion. When
these approximations are inserted into the objective function and constraints, the
result is a static optimisation problem that can be solved using the methods dis-
cussed in Sections 3.1 and 3.2. Using the direct method [90], the integral in
3.3. DYNAMICAL OPTIMISATION 23
the objective function in Eq. (3.12) can be treated as an additional state ? x
n+1
=
L(x, u, t) with the initial condition x
n+1
(t
0
) = 0. It is thus possible to replace the
original objective function with one of the type J = ?(x(t
f
)). Now the interval
t
0
to t
f
is divided into n
s
segments where h
k
is the time span of one segment.
Furthermore, letting M = n
s
+1 be the number of points in the interval, the state
equations can be approximated with any numerical integration method, i.e. Euler,
Trapezoid and Runge-Kutta. For the simplest case using the Euler method, one
may de?ne ?
k
= x
k+1
? x
k
? h
k
f
k
, and the original optimal-control problem in
Eq. (3.12) can then be expressed as an NLP problem in each point 1, 2, 3 . . . M of
the time segments in the following way
min
(u
1
,y
1
,...u
M
,y
M
)
J, where J = ?(x
M
)
(?
1
, ?
2
, ..., ?
M?1
) = 0 (3.14)
(c
1
(x
1
, u
1
, t
1
), c
2
(x
2
, u
2
, t
2
), ..., c
M
(x
M
, u
M
, t
M
)) ? 0.
In this equation, the ?
1
, ?
2
, ..., ?
M?1
are the deviations, also referred to as the
defects, for the dynamics (augmented with the integral in the objective function
in Eq. (3.12)) approximated by the numerical integration method at each point.
(c
1
(x
1
, u
1
, t
1
), c
2
(x
2
, u
2
, t
2
), ..., c
M
(x
M
, u
M
, t
M
)) ? 0 are the original inequal-
ity constraints expressed at every point. The result of the optimisation is M control
and state vectors u and y.
This optimisation problem is of the static NLP type and can be solved with
the techniques discussed in the previous sections. The problem does, on the other
hand, have M?1 times more variables than the original dynamic optimal control
problem. For the case of equality constraints and boundary conditions, the number
of variables equals M ? 1 + M + 2 = 2 × M + 1. The method presented in
Eq. (3.14) is also referred to as the multiple shooting method. In the case where
n
s
= 1, it is called a single shooting method. In the case of a long interval t
f
?t
0
and small time constants, the resulting static problem will become hard to solve.
In Paper II the above direct transcription method was used; see Chapter 5 for a
further discussion.
The simulation-based approach uses a totally different technique to arrive at
an optimisation problem that can be treated as a static one. Here, the system under
study is simply simulated for the entire simulation period, during which the cost J
is also calculated. Based on the value of J a static optimisation algorithm adjusts
the control vector uin the direction of lower cost. The simulation-based technique
can be described as follows
1. Find a feasible control vector u.
2. Simulate the system using the control vector u and calculate the cost J.
24 CHAPTER 3. OPTIMISATION TECHNIQUES
3. Change the control vector for lower cost J using a static optimisation algo-
rithm.
4. Repeat Steps 2-3 until convergence.
The static optimisation technique in Step 3 can be of any kind. However, a GA,
as described in section 3.2, is particularly suitable.
3.4 Stochastic dynamical optimisation
In any real system, noise is present. In the systems discussed in this thesis, noise
is represented as an uncertainty about future development. This fact motivates the
usage of stochastic dynamical optimisation techniques [94, 95, 96], which is a
dynamical optimisation technique, as discussed above, applied to a problem under
the in?uence of a disturbance. The disturbance can be realized as a stochastic
variable with a prede?ned probability distribution. The discrete-time stochastic
optimal control problem can be de?ned generally as
min
U
J(U), where J(U) =
N?1
k=0
?(k, X
k
, U
k
, W
k
) + ?(X
N
, W
N
)
s.t. X
k+1
= f(k, X
k
, U
k
, W
k
) (3.15)
c
k
(X, U) ? 0 ? k = 1, . . . N,
where W is an independent random disturbance, ?(k, X
k
, U
k
, W
k
) is the cost
associated with each time step k, ?(X
N
, W
N
) is the terminal cost and c
k
(X, U)
represents simple limits of the state and control variables.
The above perturbed optimisation problem can be solved by methods analo-
gous to those used for the unperturbed problem in Section 3.3, that is by dynamic
programming, stochastic programming or the simulation-based approach. The
solution methods generally work in the same way as in the unperturbed case.
The differences compared with the unperturbed case are that in the perturbed
case the objectives are optimised with respect to expected values. The result is
one control strategy U that minimises the expected value (the mean value) of the
objectives. The de?nition of the disturbance can either be analytical, i.e. using
data from the distribution used, or scenario-based, i.e. using a number of sam-
ples from, e.g. a probability distribution. In the former case the optimisation is
done analytically and in the latter case it is done numerically. In Paper IV, several
scenarios were used for generating samples after which the scenario-based opti-
misation method was applied. Using this technique, each sample corresponds to
one possible future development of the disturbance, i.e., in this case, the number
3.4. STOCHASTIC DYNAMICAL OPTIMISATION 25
5.71 5.72 5.73 5.74 5.75 5.76 5.77 5.78 5.79 5.8
0
0.05
0.1
0.15
0.2
0.25
Histogram for investment strategy 89 in scenario 3, calculated for scenario 1.
Hydrogen production cost [USD/kg]
P
r
o
b
a
b
i
l
i
t
y
1300 1400 1500 1600 1700 1800 1900 2000 2100
0
0.05
0.1
0.15
0.2
0.25
Histogram for investment strategy 89 in scenario 3, calculated for scenario 1.
Hydrogen total unsatisfied demand [kg]
P
r
o
b
a
b
i
l
i
t
y
Figure 3.5: Examples of histograms representing the hydrogen production cost (left
panel) and unsatis?ed demand for hydrogen (right panel) for a given solution. The his-
tograms were generated using 100 samples.
of vehicles arriving at the refueling station. The solution is the expectation value
of the optimum for all samples.
In Paper IV, the problem is two-dimensional and, therefore, so is J. Applying
all samples in a given scenario
2
will result in a distribution of the two-dimensional
objective function J. In essence, for each solution U and set of samples there is
one distribution for each dimension in the objective function J. In Figure 3.5
an example from the hydrogen and hythane refueling case is shown. The two
objectives are hydrogen production and total hydrogen unsatis?ed demand
3
. Each
quantity has been calculated for one solution U, and for all available samples.
2
In Paper IV, 100 samples were generated for each scenario.
3
As shown in [97], the occurrence of an unsatis?ed demand is not uncommon.
Chapter 4
Assessing the future
This thesis focuses on long-term planning and optimisation, processes that require
an assessment of the future. Such assessments can be obtained in different ways,
and in this thesis (see Papers II and IV) the preferred method has been to generate
a number of possible outcomes, referred to as scenarios. A number of probability
distributions are associated with each scenario. Thus, once a scenario has been
de?ned, a large number of samples can be generated, each representing one pos-
sible future outcome. Needless to say, such scenarios will always have a certain
degree of arbitrariness.
Since the long-term effects of a scenario often may depend strongly on what
happens during the ?rst few time steps, short term prediction of time series, based
on past data, is certainly relevant. Thus, time series prediction (TSP), considered
in Paper III, can be used for reducing the prediction uncertainty
4.1 Forecasting
Forecasting time series is common in economics [23, 98]. In the technical domain
it is part of system identi?cation [99]. The underlying assumption is that the
data series have an internal structure that can be identi?ed. After identifying this
structure, a procedure known as model ?tting, a prediction of the future can be
made. In practice, the model is selected ?rst, after which the data sets are used
to estimate its parameters. Clearly, this requires that the model can represent
the data. If this is not the case, another model has to be found through structure
selection, i.e. by ?nding a model with a more suitable internal structure, which can
then be tuned by adjusting the values of the internal parameters so as to reduce the
error over the data set.
Another issue is the amount of noise present. White noise can, by de?nition,
not be predicted. Other types of noise can, to a certain and often limited degree,
27
28 CHAPTER 4. ASSESSING THE FUTURE
be predicted. In this case one can use models that have a noise part allowing
for multi-step predictions
1
[99]. For one-step prediction the expected increment
is zero in case of white noise, and the estimate is equal to that obtained from a
model without noise. However, the genuine information part of the time series is
deterministic and can, provided a good model is found, be subject to successful
prediction.
Traditionally, methods like the naive method, exponential smoothing and auto-
regressive integrated moving averages (ARIMA) have been used for time series
prediction [22, 52]. The naive method estimates the next value in the time series
by the present one. For very noisy time series, it is hard to beat the trivial pre-
diction obtained from the naive method [100]. The ARIMA techniques consist of
three parts: an auto-regressive part, an integrating part and a moving average part.
Exponential smoothing is a special type of ARIMA model. More recently, arti?-
cial neural networks (ANNs), of which feedforward neural networks (FFNNs) and
recurrent neural networks (RNNs) are examples, have been used in TSP [25, 101].
In Paper III, a novel kind of recurrent ANNs called discrete-time prediction
networks (DTPNs) was developed for time series prediction. This study was
a continuation and an improvement of earlier work on TSP using neural net-
works [26]. A DTPN contains inter-neuron connections as well as connections
to the inputs. In a DTPN, any neuron may be connected to any other neuron, and
to itself. Furthermore, each neuron has an individual squashing function ? which
is (in principle) arbitrary. In Paper III the logistic function
?
1
(z) =
1
1 + e
?cz
, (4.1)
where c is a positive constant, and the hyperbolic tangent
?
2
(z) = tanh cz, (4.2)
have been utilized. Since no gradient information is needed in the training proce-
dure, no restrictions exist on the functions that can be used. Therefore the squash-
ing functions
?
3
(z) = sgn(z), (4.3)
and
?
4
(z) =
_
_
_
tanh(z + c) if z < ?c
0 if ?c ? z ? c
tanh(z ?c) if z > c
(4.4)
were also used. In addition, the function
?
5
(z) =
cz
1 + (cz)
2
, (4.5)
1
A multi-step prediction is de?ned as x(t ?n), x(t ?n + 1), . . . , x(t) ? ˆ x(t + k), k > 1.
4.1. FORECASTING 29
Neuron1 Neuron2 Neuronn ...
Neuroni
w(interneuronweights) w (inputweights)
in
b c k(sigmoidtype)
...
EOT
Figure 4.1: A chromosome encoding a DTPN as used in Paper III.
was used. Even though DTPNs have arbitrary connections between neurons, the
order in which the neurons are updated is ?xed, and given by so called evaluation
order tags (EOTs), one for each neuron. In each time step, the neurons with lowest
EOT are updated ?rst according to
x
i
(t + 1) = ?
_
b
i
+
n
in
j=1
w
in
ij
I
j
(t) +
n
j=1
w
ij
x
j
(t)
_
, (4.6)
where w
in
ij
are the input weights, w
ij
the interneuron weights, and b
i
is the bias
term. I
j
are the inputs to the network which, in the case of time series prediction,
consist of earlier values of the time series Z(t), i.e. I
j
(t) = Z(t ? j + 1) For
neurons with the second lowest EOT, the equations look the same, except that
x(t) is changed to x(t + 1) for neurons with lowest EOT etc. Finally, the output
neuron (arbitrarily chosen as neuron 1) gives the following output
x
1
(t + 1) = ?
_
b
1
+
n
in
j=1
w
in
1j
I
j
(t) + w
11
x
1
(t) +
n
j=2
w
1j
x
j
(t + 1)
_
, (4.7)
since, at this stage, all neurons except neuron 1 have been updated. Like other
RNNs, DTPNs are capable of short-term memory, a feature which is important in
time series prediction (see also Paper III).
The optimisation of DTPNs is carried out by means of a genetic algorithm
(GA), which evolves not only the parameters of the network, but also its structure,
i.e. the number of neurons and their EOTs. The encoding scheme used for evolv-
ing DTPNs is shown in Figure 4.1. Note that the sigmoid type ?
k
is encoded by
the integer k in the chromosome.
In Paper III, DTPNs were evolved for one-step prediction of the Fed Funds
interest rate and US GDP, after ?rst rescaling the data to the range [?1, 1]. A
summary of the results is given in Table 4.1. As can be seen in the ?gure, the
prediction results (average prediction errors) for the DTPNs were better than those
of the other methods tested.
The DTPNs have been used for one-step predictions only. Of course, multi-
step predictions are possible, in principle, but will inevitably result in inaccurate
predictions due to the effects of cumulative errors [51, 102].
30 CHAPTER 4. ASSESSING THE FUTURE
Data set e
N
e
ES
e
ARMA
e
DTPN
Fed funds interest rate 0.2018 0.1901 0.1887 0.1837
GDP 0.1771 0.1490 0.1473 0.1305
Table 4.1: Average errors for one-step predictions carried out for two macroeconomic
time series: the Fed funds interest rate and the US GDP. The table shows the minimum
errors over the validation part of the data set, obtained using naive prediction (e
N
), expo-
nential smoothing (e
ES
), ARMA (e
ARMA
), and DTPNs (e
DTPN
). Only the results for the
very best DTPN are shown.
Also investigated were predictability measures, i.e. measures of the accuracy
of an individual prediction. Several empirical measures were investigated, as well
as one analytical measure. The empirical measures involved different ways of
augmenting the DTPNs to incorporate the predictability. The amount of genuine
information in a single time series can be analytically estimated using random
matrix theory [103, 104]. If the original observations are contained within the
Tx1 vector x(t), a T ?m×N delay matrix Z can be formed where the columns
are delayed observations, i.e. x(t), x(t ?1), x(t ?2) . . . x(t ?m). The parameter
m represents the maximum delay and should be chosen to cover the time period
of any cyclic behaviour. In the process of forming the delay matrix, m rows at
the end of Z will lack values and hence only T ? m rows can be further used in
Z. The number of columns, N, equals the maximum delay number, m, plus one
initial column. In order to use Z, each column has to be normalised to zero mean
and variance 1, which is done using
Z
m,n
?a Z
m,n
+ b, (4.8)
where
a =
¸
T ?m
T?m
i=1
Z
2
i,n
?1/(T ?m)(
T?m
i=1
Z
i,n
)
2
(4.9)
and
b = ?a
T?m
i=1
Z
i,n
T ?m
. (4.10)
The correlation matrix, C, is then de?ned as
C =
1
T
Z
T
Z. (4.11)
Furthermore, the eigenvalues of C can be used to estimate the information content
by comparing with the eigenvalues of a random matrix with the same dimensions.
4.1. FORECASTING 31
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
15
20
25
30
35
40
Information content Usa GDP (Total 20.0779%)
%
Figure 4.2: Information content in the US GDP time series using random matrix the-
ory [104]. The remaining part is noise. For the above case the delay m = 12 and the
window size 32. The total information content is calculated based on the eigenvalues for
the full, non-windowed, correlation matrix C.
Such a matrix, X [m×n], will, if it is scaled according to Eq. (4.8), have a density
of eigenvalues according to
?(?) =
Q
2?
_
(?
max
??)(? ??
min
)
?
(4.12)
for ? ? [?
min
, ?
max
], and zero otherwise, in the case where m, n ? ?, and
where Q = n/m ? 1 [105]. In Eq. (4.12), the minimum eigenvalue is ?
min
=
?
2
(1 ? 1/
?
Q)
2
and the maximum eigenvalue ?
max
= ?
2
(1 + 1/
?
Q)
2
. If the
eigenvalues of C lie outside the range [?
min
, ?
max
], the time series x(t) contains
real information. A numerical estimate of the minimum and maximum eigenval-
ues for matrices of ?nite size, can be obtained by taking the minimum and the
maximum eigenvalues obtained from a large number of generated random matri-
ces of the same dimension as C.
A way of estimating the percentage of information content is by computing
100?
max
/N, where ?
max
is the largest eigenvalue of the correlation matrix C. By
making use of a moving data window, the local information content in time series
may be estimated. Figure 4.2 shows such a calculation for the US GDP.
All results on the predictability measures were, however, negative. Compar-
ison between the analytical and empirical candidate measures showed no corre-
32 CHAPTER 4. ASSESSING THE FUTURE
lations. The reason may be that the DTPNs do extract most of the information
available in the time series.
4.2 Decision-making under uncertainty
Decision-making in industry as a scienti?c discipline can be regarded to be a
part of operations research (OR) [106, 107, 108]. OR originates from the mili-
tary sector, where it was used to make better decisions in, e.g. logistics and war
tactics [109, 110]. Nowadays OR is widely used in the industry to raise ef?-
ciencies and optimise performance. Here it is also known as management sci-
ence [111, 112]. The problems that occur in OR are real-world problems and
are often hard to solve using traditional deterministic methods. Therefore it is
not surprising that the ef?ciency of stochastic optimisation techniques was early
recognised [113].
An important ?eld where OR has contributed is decision-making under un-
certainty. In the industry, strategic decisions have to be made and in all practical
cases they are taken under uncertainty since they involve predicting, or guessing,
what will happen in the future. Since it is generally very dif?cult, not to say im-
possible, to make correct long-term decisions intuitively, this is an area where
much can be gained by applying advanced analytical techniques [114, 115, 116].
Application areas include logistics [117], ?nancial instruments, investment plan-
ning [11], risk management [118], water pollution problems [119], water resource
problems [120] etc. In all of the above areas, stochastic techniques, like GAs,
nowadays are common ways of solving the resulting optimisation problem.
For an investment decision problem, each decision to invest will, in the op-
timisation framework, result in one decision period. In practical cases there is
often a pre-de?ned number of occasions when investments are possible, e.g. once
a month or once a year, something that leads to a sequential decision problem. The
problems in Papers II and IV are both sequential decision problems. In addition,
the problem considered in Paper IV is also stochastic. Furthermore, the decision-
making can involve an inner loop where a number of decisions are made within
each outer decision period, leading to two stages of decisions. For a production
company, the ?rst stage typically involve decisions on investments in production
capacity and the second stage regards decisions concerning production planning
given the resulting constraints from the ?rst stage. The decision problem in Paper
IV is also of the above two-stage type and is solved by the use of a pre-de?ned
strategy for the ?rst investment stage and a combination of closed-loop regulation
2
and a pre-de?ned strategy for the second production stage.
2
The term closed-loop regulation is used in the standard way, as de?ned in control theory.
4.2. DECISION-MAKING UNDER UNCERTAINTY 33
Decision-making under uncertainty can be modelled as a Markov decision
process (MDP) [121]. In such a process, the decisions taken at a certain point
depend only on the state at the previous time point and not on states further back
in time. The MDP is a discrete-time stochastic control process that propagates
through a series of states. For each state the decision-maker takes actions based
on the information given by the previous state only. Next, a stochastic transition
function determines probabilities for transition to the next state. For each state
there is a reward, which depends on the new state. In Paper IV, the investment
decisions are parametrised and calculated in an open-loop way, i.e. they are pre-
de?ned. Uncertainties in the form of hydrogen and hythane demand are dealt with
in the second stage, where the amount of stored hydrogen is kept at a constant level
by adjusting the production. This second stage is a process of the Markov type.
The use of optimisation for ?nding optimal future investment strategies is a
decision that has to be carefully considered for each individual case. As with all
mathematical tools, optimisation methods also require a quanti?cation of the input
data, which is by no means trivial, since some data concern predictions of future
sales, prices etc. Other factors such as availability of skilled personnel and data
regarding the system under study also have to be taken into account. The methods
used in Papers II and IV are, from a mathematical point of view, very robust. The
sensitivities to disturbances can be determined by perturbation analysis, a tech-
nique used in Paper II. The quality of the optimisation results depends, of course,
on the quality of the input data. On the other hand, the investment methods pre-
sented in Paper II and IV are meant to be used to update the investment strategy
as soon as new and better predictions are found. In this case the optimal next in-
vestment decision can always be found, given the best available future estimates
at the time of calculation. It should also be stressed that the use of GAs for in-
vestment planning, as in Paper IV, reduces the calculation effort since no gradient
information is needed. It is the conviction of the author that, in the case of a com-
plex system and as soon as the objectives are quanti?able, the presented methods
are very powerful tools and will prove useful in many applications in addition to
those discussed in this thesis.
Chapter 5
Case studies
This chapter provides a background to the case studies presented in Papers I, II
and IV. Along with the general background, some details omitted from the papers
are given as well.
The three cases are (1) the cement production model, (2) the hydrogen refu-
elling station infrastructure investment optimisation and (3) the combined hydro-
gen and hythane infrastructure investment optimisation.
5.1 The cement production case
In the cement production case considered in Paper I, a highly ?exible model of a
cement production factory has been built. The model has been used in several dif-
ferent calculations, including process optimisations and environmental assessment
of new energy sources. The life cycle inventory analysis (LCI) model consists of
a foreground system which de?nes the on-site production over which the com-
pany has full control, and a background system comprising purchased services
and goods, see Figure 5.1. A more in-depth discussion of the production facility
is given in Paper I and in [45].
The raw materials, i.e. different sorts of sand, are transported to the production
site and ground depending on type. They are then mixed in relevant proportions
and burnt to clinker in the clinker production system. For the burning process,
fuel is, of course, required, and it may consist of coal, pet coke or an alternative
fuel. All fuels are transported to the site, ground and mixed in correct proportions,
before entering the burner. The produced clinker is then mixed with gypsum (and
possibly other materials), further ground, and stored as cement.
The problem is to ?nd the ratio of raw materials, fuels and the additional gyp-
sum to produce cement of a certain quality. The quality is measured using the fac-
tors indicated in Table 5.1. In addition the approximate monetary costs throughout
35
36 CHAPTER 5. CASE STUDIES
Figure 5.1: LCI model of the cement production line.
the production line must be calculated. Since the purchase costs of the raw mate-
rials and fuels are known, the production-related cost for each piece of equipment
in the line can be estimated and added to the product ?ow. The model can be
used as an aid in calculations for new types of raw materials, fuels and internal
settings, and for changes in the layout of the production line. In addition to static
solving, dynamic simulation and optimisation can be considered. It is therefore a
requirement that the model should be modular and highly ?exible.
One option would be to make a standard LCI model. When a standard LCI
is carried out, the linear technology matrix model (A) described in Section 2.1 is
suf?cient for describing the technical production system, since the underlying pro-
duction system is described as static and linear. Using this linear description, only
one type of calculation is made in an LCI, which is the normalisation to the func-
tional unit, obtained by solving a linear equation system. The developed mathe-
matical LCI methods are designed to achieve only this normalisation [31, 32, 34],
something that limits their usefulness. At times it is desirable to make extensions
to this type of LCI model. One such occasion is when the inherent physical be-
haviour of the production system is strongly non-linear when seen as a mapping
from resources and emissions to the product. A linear LCI model represents in
this case a linearisation around a speci?ed point and might lead to unacceptably
large deviations in the calculated resource use and emissions released. Another
5.1. THE CEMENT PRODUCTION CASE 37
Table 5.1: Cement product quality indicators. The notation indicates weight percentage
of the speci?ed material.
Name Symbol Description
Lime saturation factor LSF LSF =
100CaO
2.8SiO
3
+1.1Al
2
O
3
+0.7Fe
2
O
3
Silica ratio SR SR =
SiO
2
Al
2
O
3
+Fe
2
O
3
Alumina ratio AR AR =
Al
2
O
3
Fe
2
O
3
occasion occurs when dynamic aspects are relevant, for example when closing
down and starting up a production line in connection with maintenance. In the
situations just described, another modelling approach, making use of non-linear
and dynamic models, is needed. In addition, other types of calculations may be
desirable as well. Examples include optimisation, simulation over time etc. In or-
der to ful?ll the requirements, one needs a higher degree of ?exibility in the model
than is given by the LCI model. In short, the modelling approach has provide the
model with enough data to represent the underlying system in a correct way and
this data has to be arranged in such a way that it is possible to make the desired
calculations using the model.
In [122] the nature and effect of some different types of causality are dis-
cussed. The concept of causality is further applied to LCA in [50]. To recapitulate,
there are two types of causality of interest: (1) physical causality and (2) compu-
tational causality. The physical causality is the cause-effect connection inherent
in nature. The computational causality is the order in which an equation system
is solved. While the former is governed by the laws of nature, the latter is the
choice of the modeller. In [50], it was found that, by removing the computational
causality from the model, advantages in ?exibility can be achieved. The result is a
so called acausal [122] or non-causal model. In effect, the entity that is normally
regarded as the model can be split into three parts, namely:
• A computationally neutral (acausal) model, i.e. a model that maps the in-
terpretation of the production system onto a mathematical formulation, but
does not include any speci?c problem to be solved.
• A problem formulation, i.e. a description of which parameters should be
calculated and an explicit list of which parameters should be held constant
during a particular calculation as well as numerical values for each such
constant parameter.
38 CHAPTER 5. CASE STUDIES
• A method of calculation. This part can be considered as a part of the prob-
lem formulation.
In addition it was found that the modularity of the model, i.e. the ?exibility with
regard to both change and exchange of parts within the model, can be enhanced by
using an object-oriented modelling language in conjunction with physical entity
modelling. The intention with the latter is to keep real physical entities together
for ease of comprehension and transparency. This way of modelling also consti-
tutes a natural way to keep parts that are separate in reality as separate objects
in the model, so that the model resembles reality or a suitable representation of
reality. To summarise, the following requirements are considered:
• A computational acausal model that contains the structure and constants of
the system, but does not contain any information regarding computation.
• An object-oriented modelling language that makes use of encapsulation and
inheritance
1
.
• A physical property modelling approach that makes it possible to map the
real physical structure onto a similar model structure.
However, there are drawbacks with using an acausal model. Any mathematical
model consists of a number of equations. In the computationally causal case,
e.g. block diagrams and state-space models, these equations are ordered in a spe-
ci?c way to achieve the desired result. In the computationally acausal case the
equations are not ordered in any speci?c computational way. Instead they can
be regarded mathematically as a number of equilibrium equations connected to
each other, making them harder to understand. For models of physical systems
which are based on ?ow semantics, i.e. correlation between the general variables
intensity and ?ow, the model representation can be based on energy ?ow and is
usually relatively easy to construct. For the type of ?ow models used in LCI there
are also physical laws to consider, but not in the form of intensity-?ow related
connections. Under these circumstances acausal models can be structured in vari-
ous ways depending on the application, and therefore such models are dif?cult to
make both consistent and suf?ciently general to reach a high degree of ?exibility.
Another disadvantage is that in order to use an acausal model, a dedicated soft-
ware for sorting out the equations and ordering them computationally is needed.
In practice this is rarely a limitation, since such software is available.
As an illustration of the use of acausal models, consider the simple resistor
described by
u
R
= R i
R
. (5.1)
1
Encapsulation and inheritance are central concepts in object-oriented programming, see
e.g. [123, 124, 125] for details.
5.1. THE CEMENT PRODUCTION CASE 39
Figure 5.2: The separation of model and problem formulation that can be achieved by
the use of acausal models.
Is it the current ?owing through the resistor that causes the voltage drop or the
voltage drop that causes the current? What is the physical causality of the resis-
tor? Of course the order of calculation depends on the question at hand. If one
is interested in the voltage drop, one would use the computational order given in
Eq. (5.1). On the other hand, if one is interested in the current, one would re-order
the equation accordingly. If Eq. (5.1) is interpreted as a statement of equilibrium,
it can be regarded as an acausal model. A problem formulation might be to cal-
culate the voltage drop while keeping the current at a constant value. A suitable
method of calculation is then any static linear equation solver. Moreover, the
above example is a physical system and is based on ?ow semantics. In this case
the voltage is the intensity and the current is the ?ow [56]. For the simple example
discussed, the computational aspects are obvious and need no further formalisa-
tion. However, in cases with more than, say, 100,000 equations, much time can
be saved through the use of acausal modelling techniques.
When the project reported in Paper I was carried out (fall 1999), only a lim-
ited number of modelling languages and software programs for calculation were
available and able to ful?ll the requirements. Among them there were OmSim
(Omola) [126], Dymola (MODELICA) [48] and Ascend [46]. OmSim and Dy-
mola are specially made for modelling of physical systems and have a built-in sup-
port for ?ow semantic. Since the system considered does not have any intensity-
40 CHAPTER 5. CASE STUDIES
?ow dependency, it was decided not to use these programs (languages). Ascend
is both a calculation software and a modelling language and was originally de-
veloped for applications within chemistry. However, it can best be described as a
mathematical system modelling tool and is very ?exible in de?ning connections
and hence the structure of the system modelled, which is the main reason for using
it.
The model was built in a bottom-up manner according to Paper I. It should be
noted that the model is deliberately made redundant. In most cases redundancy
has a negative effect, but here it is used to enhance the ?exibility. The numerical
parameters in a calculation can be divided into the following categories:
• Constants. These are set, once and for all, when the model is built.
• Locked variables. Parameters set to a constant numerical value for a certain
calculation.
• Free variables. Parameters that will be calculated by the numerical algo-
rithm.
The number of parameters in each category depends on the speci?c calculation
considered. Providing information regarding these settings is part of the problem
formulation. In the model, the information needed for specifying one parame-
ter can be supplied in a number of ways. An example is the ingredients in raw
meal composition. These can be set by speci?cation of absolute masses or rela-
tive masses (percentage). The model contains the necessary mathematics to relate
these parameters at the time of calculation. At any given time, only one of the
two ways of specifying the parameter is used. The result, presented in Paper I,
is a highly ?exible calculation tool for the cement production process that has
been used by Cementa AB, for several different purposes. It should be noted that
the entire model of the cement production line was later transferred to MODEL-
ICA [127].
5.2 The hydrogen infrastructure case
The main task for the hydrogen refuelling station is to dispense hydrogen to ve-
hicles. Since the incentives for using hydrogen are environmental, an important
question to consider is where the hydrogen is to be produced. Producing the hy-
drogen is probably best done at large, centralised production facilities. It is then
easier to take care of the created emissions, e.g. CO
2
. The problem is to dis-
tribute the hydrogen to the local refuelling station. In order to do so ef?ciently,
the hydrogen gas has to be highly pressurised, which is expensive and can also
5.2. THE HYDROGEN INFRASTRUCTURE CASE 41
be dangerous. Another consideration is the vulnerability both to sabotage and
to accidents. In this thesis an alternative solution comprising local production
of hydrogen using a hydrogen reformer, i.e. a device that produces hydrogen
from hydrocarbons, is investigated. The input to the reformer can be any type of
methane gas and may originate from fossil or renewable resources. One disadvan-
tage is that the reformer will produce considerable amounts of CO
2
which will not
be easy to take care of. Probably it has to be released into the atmosphere. When
the natural gas comes from a renewable source of energy the net contribution of
CO
2
is nil. One obvious advantage with local production is that natural gas is
considerably easier to transport than hydrogen gas. In fact there is already a rather
small but growing number of natural gas refuelling stations in Sweden [128]. A
hydrogen production and refuelling part can then be added to the already existing
natural gas refuelling station. With such a refuelling station, it is also possible to
dispense natural gas as an intermediate alternative.
If the refuelling station is equipped with fuel cells, it can also be used as a lo-
cal electrical power station. This alternative might be useful in remote locations.
When hydrogen is produced from renewable energy sources, it might also be an
environmentally friendly alternative. If the refuelling station is located in a place
where electricity from the grid is cheap, it can be equipped with an electrolysis
part that can produce hydrogen gas directly from electricity. In this case it is im-
portant to keep track of how the electricity is produced. To begin by producing
electricity from coal and then using electrolysis to produce hydrogen is not, how-
ever, a good environmental solution. In addition to the hydrogen reformer, the
refuelling station layout that is considered in Paper II also has a local fuel cell
and an electrolysis plant. Figure 5.3 illustrates options investigated in this thesis.
The result is a refuelling station that is very ?exible in terms of resource use and
energy production.
The equipment for a hydrogen refueling station with the above layout is more
expensive than present-day petrol station parts. In addition, not all of the con-
?gurations are suitable for speci?c conditions. Under these circumstances it is
important to ?nd the most pro?table con?guration for the speci?c location, the
estimated number of customers, and general technical and economical develop-
ment. The problem is to ?nd the most pro?table con?guration. As described in
Paper II, this problem is equivalent to ?nding the least expensive mean production
cost for hydrogen. In Paper II, only the core parts (the parts within the shaded area
in Figure 5.3) are part of the optimisation. The remaining parts can be dealt with
separately, as discussed in Paper II.
In reality the choice of equipment is limited by supply. Let the set of available
equipment, which consists of a ?nite number of sizes for each equipment type, be
denoted by C and the control sequence u(t) be a vector of equipment sizes such
that ?u(t) ? C and u(t + ?t) ? u(t), ?t where ?t is a small time step. The
42 CHAPTER 5. CASE STUDIES
Figure 5.3: Refueling station layout. Natural gas is reformed to hydrogen at the site and
stored for delivery to vehicles. It is also possible to produce hydrogen from electricity by
electrolysis or electricity from hydrogen using a fuel cell.
implication is that u is only allowed to increase and to do so only with speci?c
increments, namely those in C. Further let f(u, x, w) be the description of the
core of the refuelling station in state-space form where x is accumulated volume
in each piece of equipment and w is the hydrogen refuelling demand. The most
general form of the problem of ?nding the most pro?table con?guration is, in the
continuous case,
min
u(t)
J = ¯ c(x(t), u(t), w(t))
s.t. ? x = f(x(t), u(t), w(t)) (5.2)
0 ? x(t) ? u(t),
where ¯ c is a cost function described in Paper II and further discussed in Section 2.1
in this thesis. There are two major dif?culties in solving this problem:
1. The problem is de?ned over the entire investment period of 20 years. At
the same time the assumed ?lling curve for hydrogen has a time step of one
hour. Dividing the interval of 20 years into one hour segments would lead to
2 ×M + 1 = 350, 401 variables, as is discussed in Section 3.3. This would
make numerical solving of the resulting non-linear optimisation problem
hard, not to say impossible.
5.2. THE HYDROGEN INFRASTRUCTURE CASE 43
2. The control sequence u can only increase in steps that are part of C. This
would make the problem discrete in u. Discrete problems are combinatoric
and in general harder to solve than continuous ones [129].
In order to solve the problem stated in Eq.(5.2) it is observed that, to be able to
satisfy the hydrogen demand, the ?rst investment has to take place initially, at
t = 0. Consecutive investments can be divided into separate cases. Since the
desired output, the ?lling curve f
w
, is given for one week (168h) and then scaled
using the S-curve R, it is suf?cient to consider only one week for each investment.
The S-curve is a purely exogenous estimate of the number of hydrogen vehicles
using the refuelling station and is de?ned as
R(t) =
1
1 + e
?B(t?Tx)
, (5.3)
where t is the time fromyear 2010, T
x
the S-curve in?ection point and B the slope.
For values of the constants T
x
and B, see Paper II. The week to consider for each
investment is when utilisation is at its maximum, namely the week right before
the next investment. At these points in time, the equipment is used at its maxi-
mum capacity and, in order to satisfy the increasing demand, a new investment
in capacity has to be made. By parametrising the set C using a scaling function
p
eq
(s
eq
) (explained in Paper II), where s
eq
is the size of equipment, the problem
will become continuous in u(t).
The driving signal for the fast dynamics is the ?lling curve ?owf
w
. This curve
is given for one week with the time step of one hour. The integral in the objective
functions in Section 2.1 can therefore be replaced by a sum. Also, the discrete
version of Eq. (5.2) can be used.
Using the direct transcription method in Section 3.3, the resulting NLP prob-
lem (see Section 3.1) becomes, in the single investment case
min
u
k
J = ¯ c(x
k
, u
k
, w
k
)
s.t. ?
k
= x
k+1
?x
k
?f(x
k
, u
k
, w
k
) = 0 (5.4)
0 ? x
k
? u
k
, k = 1, . . . , M ?1,
where ?
k
are the defect constraints and M = 168 the number of steps. Note that
in Eq. (5.4), the step length is one hour. Considering that the dynamics involved
is of the ?rst order and is stable, the multiple shooting method can be replaced by
a single shooting one, which would make the problem easier to solve numerically.
44 CHAPTER 5. CASE STUDIES
The above defect constraints can be replaced by a cumulative summation, giving
min
u
k
J = ¯ c(x
k
, u
k
, w
k
)
s.t. x
k
=
k
p=1
x
p
(5.5)
0 ? x
k
? u
k
, k = 1, . . . , M ?1,
This NLP problem can be solved using the methods in Section 3.1.
In Paper II two cases are considered: (1) variable utilisation as in Eq. (5.5) with
extra requirements on initial amount of hydrogen stored and periodic maintenance
and (2) constant utilisation, which is a special and simpler case of variable utili-
sation. In the variable utilisation case, the chosen special conditions are 100 kg
hydrogen storage initially and at the end, and a weekly stop for maintenance from
hour 75 to 87 during the week. Optimisations are done for the cases of one and
two investments during the investment period. The resulting problem formulation
is
min
u
k
J = ¯ c(x
k
, u
k
, w
k
)
s.t. x
hs,k
=
k
p=1
(x
i
hs,p
?x
o
hs,p
)
0 =
87
p=75
x
o
hr,p
. (5.6)
x
hs,1
= 100
x
hs,1
= x
hs,M
0 ? x
k
? u
k
, k = 1, . . . , M ?1,
The data resulting from the optimisation are (1) the size of the equipment, (2)
the running pattern of the facility, and (3) the price and utilization curve for the
hydrogen produced. Figs. 5.4, 5.5 and 5.6 show some of the results in the case of
two investments. The complete results are given in Paper II.
The problem in Paper II is on the edge of what is possible to solve. If the
number of investments becomes too large (> 2 in the variable utilisation case),
the computational time becomes prohibitively long.
5.3 The hythane infrastructure case
The hythane and hydrogen refueling station is a development of the previously
discussed hydrogen refueling station. The differences in the layout is the absence
5.3. THE HYTHANE INFRASTRUCTURE CASE 45
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
5
10
Hydrogen reformer output 1 [kg/h]
time [h]
k
g
/
h
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
50
100
150
200
Hydrogen stored 1 [kg]
time [h]
k
g
Figure 5.4: Variable utilisation case, two investments; throughput and stored hydrogen:
Investment 1 at t=0.
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
?10
0
10
20
30
40
50
60
Hydrogen reformer output 2 [kg/h]
time [h]
k
g
/
h
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
200
400
600
800
1000
Hydrogen stored 2 [kg]
time [h]
k
g
Figure 5.5: Variable utilisation case, two investments; throughput and stored hydrogen:
Investment 2 at t=5.4.
46 CHAPTER 5. CASE STUDIES
0 2 4 6 8 10 12 14 16 18 20
25
30
35
40
45
50
55
Absolute cost produced hydrogen [SEK/kg]
time [year]
S
E
K
/
k
g
0 2 4 6 8 10 12 14 16 18 20
0
2000
4000
6000
8000
Capacity ? Demand for hydrogen [kg/week]
time [year]
k
g
/
w
e
e
k
Figure 5.6: Variable utilisation case, two investments; hydrogen production cost and
capacity-demand.
of the fuel cell for electricity production and, of course, the presence of a hythane
dispenser, as can be seen in Figure 5.7. The major advantage with the above layout
is that hythane can be used as an intermediate alternative fuel and, possibly, help
introducing hydrogen by taking on some of the costs for the reformer, electrolysis
and hydrogen storage.
In this study, the investment strategy is optimised with the objectives of min-
imising production cost for hydrogen as well as the requested amount of hydrogen
that could not be satis?ed, also called hydrogen unsatis?ed demand. The optimi-
sation method is described in Section 3.4. It should be noted that, in Paper IV, the
results are expressed as Pareto fronts for the two objectives involved.
In Paper IV, the optimisation of the hydrogen and hythane station was carried
out using a GA (see Section 3.2). In the evaluation of each solution candidate
(individual), between one and 10 investments are allowed. This is accomplished
with a variable chromosome length, see Figure 5.8. Since the ?rst investment
is mandatory at year 1, no timing is needed. Instead the demand priority policy
?, is de?ned. Demand can exceed supply and the demand priority governs how
the available hydrogen is used to satisfy the demand at the hythane and hydrogen
dispensers. Following ? in the chromosome, is the size of each piece of equipment
for the refuelling station in investment 1. For consecutive investments, the demand
priority policy is replaced by the time for the investment.
The selection, crossover and mutation operators are specially designed to cope
5.3. THE HYTHANE INFRASTRUCTURE CASE 47
Figure 5.7: Hythane and hydrogen refueling station layout. Natural gas is reformed
to hydrogen on-site and stored for delivery to vehicles. It is also possible to produce
hydrogen from electricity by electrolysis. In Paper II, only the parts within the refueling
station are considered.
Figure 5.8: The encoding scheme used for the genetic algorithm in Paper IV. The priority
policy ?, reformer size S
hr
etc. are genes. Allowed values for genes (alleles) are given in
Paper IV. The grouping of genes in the ?gure is only done for clari?cation.
48 CHAPTER 5. CASE STUDIES
with the problemat hand, the variables of which are implemented in chromosomes
of the kind shown in Figure 5.8. The selection operator is of the crowded tour-
nament selection type [88] where ?tness is replaced with the inverse crowding
distance within each Pareto front. The crowding distance is the mean distance to
the nearest neighbour solutions. In this way the solutions lying furthest from each
other are retained and a better spread of solutions within the front is achieved. The
crossover operator is of the one-point type, in which a crossover point is chosen
randomly within the shortest mutual length of the chromosomes. This will allow
crossover between chromosomes of different length, i.e. different number of in-
vestments. An upper limit of 10 investments was set. In the mutation operator, the
number of investments is changed with a low probability.
Traditionally, the control u is expressed as one vector for each decision pe-
riod [93]. The above approach is a parametrization where investments not used
are not part of the control. This is done in order to reduce the variable space and
achieve a faster convergence.
The optimisations in Paper IV are all two-dimensional. Some experiments
were also made with three-dimensional optimisations, i.e. optimising three objec-
tives at the same time. Even though it is possible to extend the above algorithm
to any dimensionality, there are practical limitations governed by the calculation
time. Evaluation of one individual takes 75 s, using all scenario samples. Sce-
narios and sample generation are explained in Paper IV. In the two-dimensional
case, the population R
t
to be evaluated consists of 80 individuals. This number of
individuals is needed to populate the Pareto front curve. The evaluation thus takes
1.7 hours. In the three-dimensional case at least 80 × 80 = 6, 400 individuals
would be needed to populate the generated Pareto surface, implying an evaluation
time of 5.7 days per generation. Since at least 200-300 generations are needed,
the whole optimisation would take approximately four years!
Each optimisation will give 40 solutions along the Pareto front for the ob-
jectives hydrogen production cost and total hydrogen unsatis?ed demand. Each
solution is optimised for the lowest expected values of each objective, given 100
samples from one of three scenarios, and can therefore be expressed as a distri-
bution, as discussed in Section 3.4. In addition there are four other performance
measures de?ned in Paper IV. These are unsatis?ed demand for hydrogen x
h,u
,
production cost per kg for hythane p
yf
, unsatis?ed demand for hythane x
y,u
and
?exibility p
h?
. All of the above measures can be calculated for the two scenar-
ios not part of the optimisation (the passive scenarios). Also investigated was the
variance of all the above measures. In all, this represents a large amount of data to
take into consideration before making a decision. One way to handle the decision
process could be to follow the procedure:
1. Find a reasonable solution regarding hydrogen production cost and unsatis-
5.3. THE HYTHANE INFRASTRUCTURE CASE 49
?ed demand while checking that the distribution of each variable is not too
wide.
2. Check the location of the selected solution in the hythane production cost
curve and the variance of the selected solution in the histogram.
3. Check ?exibility, i.e. the result that would be obtained if another scenario
should become reality. This can either be done with the de?ned ?exibility
measure, or in more detail, by calculating, e.g. production costs and distri-
butions for the found solutions by applying the passive scenarios.
4. If production costs, unsatis?ed demands and ?exibility are all within rea-
sonable limits, choose the solution. If not, go back to step 1.
Chapter 6
Concluding remarks
A number of techniques involving optimisation of industrial transition processes
have been explored. In particular the problem of ?nding optimal long-term invest-
ment strategies taking economic an environmental considerations into account has
been considered.
The investment strategy optimisation methods described in Papers II and IV,
have been successfully applied to two cases concerning hydrogen dispensing in-
frastructure change. The ?rst optimisation method, presented in Paper II, com-
prises a simultaneous calculation of the long-term investment strategy and the
short-term utilisation scheme for a deterministic demand. The method has been
applied to the case of ?nding an investment strategy for minimising the produc-
tion cost for a single hydrogen refuelling station. The problem was shown to be
convex; thus the resulting solution is the global optimum. The second investment
optimisation method, presented in Paper IV, uses stochastic demand scenarios and
multi-objective optimal control to produce the Pareto front of the two con?ict-
ing objectives expected production cost and expected unsatis?ed demand. This
method was applied to the case of ?nding the optimal investment strategy for a
combined hydrogen and hythane refuelling station. Due to the uncertainty of the
stochastic demand function, satisfying all demand would require a production ca-
pacity well above the mean demand, which would be very costly to maintain.
New ways for modelling joint economic-environmental systems and predict-
ing future key parameters have been developed, in order to enhance the applica-
bility and accuracy of structural optimisation methods. The ?ndings are presented
in a production system modelling case in Paper I and a time series prediction case
in Paper III. The results obtained in Paper I have been applied in industry, by Ce-
menta AB, in the evaluation of the consequences of using new fuels in cement
production.
51
52 CHAPTER 6. CONCLUDING REMARKS
6.1 Future work
The optimisation method in Paper II was tested on the hydrogen refuelling station
case, which was shown to be convex. Other algorithms for solving the resulting
NLP problem could be investigated, e.g. interior point or cutting plane [72] meth-
ods, which are ef?cient for convex problems. It would also be interesting to test
the developed method on a non-convex case and still try to obtain a global solu-
tion. The performance of the method in the described hydrogen refuelling case can
most probably be improved. In case 2 with variable utilisation (see paper II) the
computational time is unrealistically high for more than two investments. Since
the most favourable solution probably lies between three and ?ve investments,
this limitation must be overcome. Even though computer hardware is constantly
gaining speed, this does not mean that efforts to improve optimisation techniques
should be neglected. On the contrary, in the author’s opinion, the improvement of
such techniques is more rewarding and useful than merely waiting for computers
to become faster.
Due to the sampled nature of the refuelling curve, the investigated test case
contained only time discrete dynamics. It would be interesting to try the method
in a continuous dynamic case, i.e. where all the driving signals are continuous.
Also, in order for the objective function to become more realistic, it should also
incorporate, for example, the cost of labour for the hydrogen part of the refuelling
station.
The investment problem in Paper IV was modelled as an open-loop system in
the sense that the entire investment strategy was decided upon in advance. Such a
system can, optimally, perform equally well as a closed-loop system [93]. How-
ever, in order to increase robustness, a closed-loop strategy could be used. This
would allow for the strategy to change in accordance with revealed uncertainty
in the demand. Furthermore, the large amount of data from stochastic multi-
objective optimisations can be a problem. Ef?cient use of the method for decision
support requires a higher degree of aggregation of the results than that done in
Paper IV.
In both Paper II and Paper IV, the environmental measures are implicit, i.e. present
through the use of an environmentally friendly technique. Another option would
be to have explicit environmental ?gures in the objective functions. In Paper IV,
which is a multi-objective problem solved with a GA, such objective functions
could easily have been used. If explicit environmental measures can be derived
and parametrised for different investment times, the techniques from Paper IV can
be applied.
Different options in the scenario generation procedures can be explored. At
present, a Poisson distribution is used to generate samples from the scenarios.
Other methods, e.g. ARIMA models or neural networks, could also be evaluated.
6.1. FUTURE WORK 53
Another possibility is to merge the three existing scenarios into one where the
long-term behaviour in terms of the number of vehicles is governed by a set of
stochastic variables. These variables can be tuned in accordance to expectations
regarding the future development.
Chapter 7
Summary of appended papers
7.1 Paper I
In cement manufacturing, according to the law, the effect of any change to the pro-
duction process must be investigated before the modi?ed process is implemented.
Such changes might involve type of sand, fuel or additives. Recently, Cementa
AB, a major cement producer in Sweden, started to investigate alternative, more
environmentally friendly types of fuel. In addition the company also started to
improve the understanding of the involved physics and chemistry, which turned
out to be complex. Today the veri?cation comprises a calculation of produced
emissions, but in the future other types of calculations would be needed.
In this paper a ?exible model is developed which ful?lls the requirements
above. A computationally acausal model made it possible to separate the model
describing the cement manufacturing process from the problem formulation. The
model was built in ASCEND [46], which is an object-oriented, mathematically
based modelling language as well as a multi-purpose simulation and calculation
environment. In order to further enhance ?exibility, the model was designed with
a high degree of redundancy, so that the quantity of one physical property is ex-
pressed through a number of linked equations. This gives the user freedom to
choose how to assign the physical property. In addition the model also fully traces
the total cost throughout the production line.
7.2 Paper II
Running vehicles on hydrogen rather than petrol could lead to less environmen-
tally hazardous emissions in a global perspective, especially if the hydrogen is
made from renewable energy. Techniques for producing and storing the hydro-
gen, as well as fuel cells to convert the hydrogen into electricity, are constantly
55
56 CHAPTER 7. SUMMARY OF APPENDED PAPERS
being improved. One of the most signi?cant dif?culties in the introduction of hy-
drogen vehicles today concerns the infrastructure that must be built. Considering
the fact that all present refuelling stations for petrol need to be replaced, the total
investment is huge. In this situation it is crucial to employ the most pro?table
investment strategy, given the probable future development.
In this paper the lowest production cost for a set of investments over a period of
20 years for an individual hydrogen refuelling station is found. For ?exibility and
convenience of transportation, the refuelling station utilises an on-site reformer for
natural gas. The ?rst case investigated assumes a constant production of hydrogen
and will yield the minimal cost, whereas the second one can be used when special
considerations like periodic stop for maintenance of the hydrogen reformer need
to be taken into account. Both optimisation problems are shown to be convex and
hence produce the global optimal point. The result is a hydrogen production cost
of 4-6 USD/kg, comparable to the results of other studies. The major difference
is that this study uses an increasing function to estimate the number of hydrogen
vehicles refuelling at the station, and the estimated production cost is obtained as
a time average. In other studies, the cost has been based on maximum utilization.
7.3 Paper III
In this paper, discrete-time prediction networks (DTPNs), a novel type of recurrent
neural networks, are introduced and applied to the problem of macro-economic
time series prediction. The DTPNs are optimized using a genetic algorithm (GA)
that allows both parametric and structural mutations. Due to the feedback cou-
plings present in the DTPNs, such networks are capable of a rudimentary short-
term memory.
The results from applications involving two time series, namely the Fed Funds
interest rate and US GDP, indicate that DTPNs are capable of one-step prediction
with higher accuracy than several other benchmark methods. Thus, even though
the data sets contain a large amount of noise, the study in Paper III indicates that
there is more information available in the time series than can be extracted using,
e.g. feedforward neural networks or ARIMA models.
In addition, an investigation of predictability was carried out. Here, the DTPNs
were required not only to make a one-step prediction, but also to provide an esti-
mate of the accuracy of the prediction. However, it was found that the discrepancy
between the predictions obtained from the DTPNs and the actual data points in the
time series consisted of noise, indicating that the DTPNs indeed extract almost all
the available information from the time series.
7.4. PAPER IV 57
7.4 Paper IV
Paper IV concerns the problem of ?nding the optimal investment strategy for a
single hydrogen and hythane refuelling station giving the minimum production
cost while trying to match the hythane and hydrogen capacity to a demand gen-
erated from three future stochastic scenarios over a 20-year period. Hydrogen is
a promising fuel for vehicles. However, one of the major barriers is the lack of a
hydrogen infrastructure. An important component of the hydrogen infrastructure
is the individual hydrogen refuelling station. The long-term pro?tability of the hy-
drogen ?lling station is a key issue for the success of the transition to a hydrogen
infrastructure. The resulting minimal expected production cost lies between 2-6
USD/kg for hydrogen and 1-1.5 USD/kg for hythane, depending on preferences
for unsatis?ed demand, ?exibility etc. The results are meant to be used as decision
support when planning new refuelling stations.
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Paper I
The design and building of a life cycle-based process
model for simulating environmental performance,
product performance and cost in cement
manufacturing
in
Journal of Cleaner Production, Volume 12, Issue 1, February 2004, pp. 77-93.
Journal of Cleaner Production 12 (2004) 77–93
www.cleanerproduction.net
The design and building of a life cycle-based process model for
simulating environmental performance, product performance and
cost in cement manufacturing
Karin Ga¨bel
a,b,?
, Peter Forsberg
c
, Anne-Marie Tillman
a
a
Environmental Systems Analysis, Chalmers University of Technology, S-412 96 Gothenburg, Sweden
b
Cementa AB, Research & Development, P.O. Box 144, S-182 12 Danderyd, Sweden
c
CPM—Centre for Environmental Assessment of Product and Material System, Chalmers University of Technology, S-412 96 Gothenburg,
Sweden
Received 3 April 2001; received in revised form 26 June 2002; accepted 11 December 2002
Abstract
State of the art Life Cycle Inventory (LCI) models are typically used to relate resource use and emissions to manufacturing and
use of a certain product. Corresponding software tools are generally specialised to perform normalisation of the ?ows to the
functional unit. In some cases it is, however, desirable to make use of the LCI model for other types of environmental assessments.
In this paper, an alternative modelling technique resulting in a more ?exible model is investigated. We exemplify the above by
designing and building a model of a cement plant. The commissioner’s, in this case Cementa AB, requirements on a ?exible model
that generates information on environmental performance, product performance and the economic cost were seen as important. The
work reported here thus has two purposes; on the one hand, to explore the possibility for building more ?exible LCI models, and
on the other hand, to provide the commissioner with a model that ful?ls their needs and requirements. Making use of a calculational
a-causal and object-oriented modelling approach satis?ed the commissioner’s special requirements on ?exibility in terms of modu-
larity and the types of calculations possible to perform. In addition, this model supports non-linear and dynamic elements for future
use. The result is a model that can be used for a number of purposes, such as assessment of cement quality and environmental
performance of the process using alternative fuels. It is also shown that by using the above modelling approach, ?exibility and
modularity can be greatly enhanced.
? 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Life-cycle-simulation; Predict; Consequences; Process model
1. Introduction
The interest in environmental issues, as well as the
pressure on industries to develop more environmentally
preferable products and processes, is constantly increas-
ing. This drives product and process development
towards more sustainable practices. However, products,
processes and production systems are always developed
taking cost and product performance into consideration.
Thus, there is a growing need for tools to predict and
?
Corresponding author. Tel.: +46-8-625-68-22; fax: +46-8-625-
68-98.
E-mail address: [email protected] (K. Ga¨bel).
0959-6526/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0959-6526(02)00196-8
assess both the environmental performance and the econ-
omic cost and the product performance of alternative
production operations.
The purpose of this paper is to describe how we
designed and built a ?exible model for process and pro-
duct development in the cement industry. The model
predicts the environmental performance, the economic
cost and the product performance by simulating different
operational alternatives for producing cement. The needs
and requirements were speci?ed by the cement industry.
These are outlined in Section 3. We give our interpret-
ations as a conceptual model in Section 4. We chose the
modelling approach and simulation tool and describe
how we designed and built the model in Section 5. We
end Section 5 by testing the tool in two real cases. The
78 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
results of these tests show that the modelling approach
used can generate a potentially powerful tool.
A life cycle perspective (“cradle to gate”) was used
to assess the environmental consequences of process and
product changes, in order to avoid sub-optimisation. The
conceptual model represents the cement manufacturing
process from cradle to gate. However, the model in this
paper, the construction of and test of we describe in
detail, represents the gate to gate part of the manufactur-
ing process. Environmental performance is described in
terms of environmental load (resource use and
emissions). Economic cost is described in terms of the
company’s own material cost and production cost. Pro-
duct performance is expressed as cement composition.
The product performance is used to determine whether
or not the operational alternative is feasible. Environ-
mental load and economic cost have to be related to a
feasible operational alternative and product.
Cementa AB, the cement manufacturer in Sweden and
the commissioner of the study, has previous experience
of Life Cycle Assessment (LCA) through a Nordic pro-
ject on Sustainable Concrete Technology [1]. In that pro-
ject, several LCA studies were carried out on cement,
concrete and concrete products [2,3,4,5,6]. One con-
clusion drawn from the project was that life cycle assess-
ment is a tool, with the potential for improvement, to be
used to avoid sub-optimisation in the development of
more environmentally adapted cement and concrete pro-
ducts and manufacturing processes [1]. Several other
LCA’s of cement, concrete and concrete products have
also been carried out [7,8,9,10].
However, there are limitations with today’s LCA. One
important limitation, from an industrial perspective, is
that social and economic bene?ts of industrial operations
are not taken into account. Another limitation of present
LCI modelling is its limited capability to perform differ-
ent types of simulations. There are limits on the possi-
bility of changing process variables without changing the
underlying model. Usually a new model is built for each
operational alternative simulated. In addition, LCI mod-
els are usually de?ned as linear and time independent.
2. Background
2.1. Cement manufacturing and related environmental
issues
The cement manufacturing process, shown in Fig. 1,
consists of the following main steps: limestone mining,
raw material preparation, raw meal grinding, fuel prep-
aration, clinker production, cement additives preparation
and cement grinding. Clinker is the intermediate product
in the manufacturing process. The following description
is based on the manufacturing process at Cementa’s Slite
plant. The cement manufacturing process at the Slite
Fig. 1. Cement manufacturing process.
plant is described in detail in the report “Cement Manu-
facturing — Process and Material Technology and
Related Environmental Aspects” [11].
Limestone, the main raw material is mined and
crushed. Other raw materials used may be sand, iron
oxide, bauxite, slag and ?y ash. The raw materials are
prepared and then proportioned to give the required
chemical composition, and ground into a ?ne and homo-
geneous powder called raw meal.
Various fuels can be used to provide the thermal
energy required for the clinker production process. Coal
and petroleum coke are the most commonly used fuels
in the European cement industry [12]. A wide range of
other fuels may be used, e.g. natural gas, oil and differ-
ent types of waste, e.g. used tyres, spent solvents, plas-
tics, waste oils. The fuels are processed, e.g. ground,
shredded, dried, before being introduced into the pro-
cess.
Clinker production is the “heart” of the cement manu-
facturing process. The raw meal is transformed into
glass-hard spherically shaped minerals clinker, through
heating, calcining and sintering. The raw meal enters the
clinker production system at the top of the cyclone tower
and is heated. Approximately half of the fuel is intro-
duced into the cyclone system, and at about 950° C the
carbon dioxide bound in the limestone is released, i.e.
the calcination takes place. The calcined raw meal enters
the rotary kiln and moves slowly towards the main
burner where the other half of the fuel is introduced.
Raw materials and fuels contain organic and inorganic
matters in various concentrations. Normal operation of
the kiln provides high temperature, a long retention time
and oxidising conditions adequate to destroy almost all
organic substances. Essentially all mineral input, includ-
ing the combustion ashes, is converted into clinker. How
metals entering the kiln behave depends largely on their
volatility. Most metals are fully incorporated into the
product, some precipitate with the kiln dust and are cap-
tured by the ?lter system, and some are present in the
exhaust gas.
Inter-grinding clinker with a small amount of gypsum
produces Portland cement. Blended cement contains, in
addition, cement additives such as granulated blast fur-
79 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
nace slag, pozzolanas, limestone or inert ?ller.
Depending on their origin, the additives require differ-
ent preparations.
The exhaust gases leaving the clinker production sys-
tem are passed through a dust reduction device before
being let out through the stack. The dust is normally
returned to the process. The clinker production system
is the most important part of the manufacturing process
in terms of environmental issues. The main use of energy
is the fuel for clinker production. Electricity is mainly
used by the mills and the exhaust fans. The emission to
air derives from the combustion of fuel and the trans-
formation of raw meal into clinker. Apart from nitrogen
and excess oxygen, the main components of kiln exhaust
gas are carbon dioxide from the combustion of fuel and
the calcination of limestone and water vapour from the
combustion process and raw materials. The exhaust gas
also contains dust, sulphur dioxide, depending on sul-
phur content of the raw materials, small quantities of
metals from raw material and fuel, and remnants of
organic compounds from the raw material.
The emissions to air from the clinker production sys-
tem largely depend on the design of the system and the
nature and composition of the raw material and fuel [11].
The raw material and fuel naturally vary in composition
and the content of different compounds have a different
standard deviation. The emissions of metals depend on
the content and volatility of the metal compound in the
raw material and fuel. The metal content varies over time
and consequently so does the metal emission.
The Nordic study “LCA of Cement and Concrete —
Main Report” points out emissions of carbon dioxide,
nitrogen oxides, sulphur dioxide and mercury, and the
consumption of fossil fuel as the main environmental
loads from cement production [6]. According to the Eur-
opean Commission, the main environmental issues asso-
ciated with cement production are emissions to air and
energy use [13]. The key emissions are reported to be
nitrogen oxides, sulphur dioxide, carbon dioxide and
dust.
2.2. Means and work done to minimise negative
environmental impact
The negative environmental impact from cement
manufacturing and cement can be minimised in numer-
ous ways. These can be grouped into four categories:
? Substituting input, raw materials, fuels and cement
additives, to the process.
? Process development; optimise and develop the exist-
ing process.
? End-of-pipe solutions; adding emission reduction sys-
tems.
? Product development; develop new products or
change cement composition and performance.
Many of these solutions have consequences outside
the actual cement manufacturing plant, both upstream as
well as downstream. Therefore, the life cycle perspective
is necessary to assess the environmental consequences
of process and production changes in order to avoid
sub-optimisation.
Examples of environmental improvement measures
taken at the Slite plant in recent years are given in the
following, in order to give examples of technical devices
and measures the model should be able to deal with.
Different types of waste are used, e.g. used tyres, plas-
tics, spent solvents, waste oils, as substitutes for tra-
ditional fuels to reduce the consumption of virgin fossil
fuels and the emission of carbon dioxide. The goal is to
replace 40% of the fossil fuel with alternative fuel [14]
by 2003. Cementa is also looking into the possibility of
using alternative raw materials, i.e. recovered materials,
to substitute for traditional, natural raw materials. The
alternative raw materials can either be used as raw
material in the clinker production process or as cement
additives, i.e. to substitute for clinker in cement grinding.
In 1999 a new type of cement, “building cement”, was
introduced on the Swedish market. Building cement is
a blended cement with about 10% of the clinker replaced
with limestone ?ller. The environmental bene?ts of sub-
stituting limestone ?ller for clinker are a reduction in the
amount of raw meal that has to be transformed into
clinker, and consequently less environmental impact
from the clinker production process, raw material and
fuel preparation. The environmental impact per ton
cement has been reduced by 10% [15].
The use of alternative material and fuel at the cement
plant requires pre-treatment, transport and handling, and
affects the alternative treatment of waste and by-pro-
ducts. New materials and fuels lead to new combinations
and concentrations of organic and inorganic compounds
in the clinker production system, which in turn lead to
new clinker- and exhaust gas compositions.
As an end of pipe-solution, a Selective Non Catalytic
Reduction system (SNCR) to reduce nitrogen oxide
emissions was installed at the Slite plant in 1996. In
1999, a scrubber was taken into operation to reduce sul-
phur dioxide emissions. In the scrubber, SO
2
is absorbed
in a slurry consisting of limestone and water. The separ-
ated product is used as gypsum in the cement grinding.
3. The commissioner’s needs and requirements on
the model
The commissioner’s, Cementa AB, needs and require-
ments, as interpreted from discussions with representa-
tives from different departments, are outlined in this sec-
tion.
Cementa AB needs a tool to predict and assess pro-
duct performance, environmental performance and econ-
80 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
omic cost of different operational alternatives for pro-
ducing cement. The tool is to be used to support
company internal decisions on product and process
development and strategic planning through generating
and assessing operational alternatives. Another speci?c
use is as a basis for government permits. To get permits
for test runs of new raw materials and fuels, information
on the expected outcome is needed.
Cementa intends to learn about the system and the
system’s properties regarding product performance,
environmental performance and economic cost and the
relations between these parameters. The life cycle per-
spective is seen as important. Cementa wants to be able
to simulate combinations of raw materials, fuels and
cement additives in combination with process changes
and end-of-pipe solutions. For all tested combinations,
information about the system’s predicted properties
should be generated and assessed in relation to feasibility
criteria, such as product performance, emission limits
and economic cost. Product performance is regarded as
the most important criterion.
The commissioner gave the following two examples
of how to use the tool. They asked for speci?c and
detailed information about the predicted consequences
for each alternative.
A Produce a given amount of cement, given the raw
material mix, the fuel mix and fuel demand, and the
cement additive mix. What is the product perform-
ance of the cement, the environmental performance
and the economic cost?
B Produce a given amount and type of cement, given
the fuel mix and fuel demand, the cement additive
mix and the available raw materials. What raw
material mix is required? What are the environmental
performance and the economic cost?
Concrete with different strength developments needs
different amounts of cement. Therefore, it should be
possible to state the amount of cement produced in the
operational alternative simulated. The environmental
performance should be described as environmental load,
i.e. as resource use, emissions to air and water, and
waste. The composition of the kiln exhaust gas from
clinker production should be described. The composition
of all raw material, fuel, intermediate products and pro-
ducts should be described and possible to evaluate. The
product performance should be described with three
ratios; the lime saturation factor (LSF), the silica ratio
(SR), and the alumina ratio (AR), used in the cement
industry as measures of cement composition. The ratios
describe the relation between the four main components
and are shown in Table 1. The total material and pro-
duction cost in “SEK” per amount cement produced
should be calculated. The accumulated material and pro-
duction cost should be available to study after each step
in the cement manufacturing process; both as cost per
amount cement produced and as cost per kilo of the
intermediate product.
Cementa produces cement at three plants in Sweden.
The different plants use the same main production pro-
cess as described in Section 2.1. However, there are vari-
ations between the plants, especially in the design of the
clinker production system. Variations are mainly due to
the nature of the available raw material, when the plant
was built, modi?cations done and the installation of dif-
ferent emission reduction systems. It should be easy to
adapt the tool to represent any of the commissioner’s
cement manufacturing processes, although the ?rst
model was intended to represent the Slite plant.
The content of metal compounds in the raw material,
and the standard deviation of the metal content, vary
depending on the location of the plant. Thus, the emis-
sions of metals to air vary from one plant to another.
Emission of metals from clinker production should be
included in the ?rst model, but they are not in focus.
However, in the next stage, when site-speci?c models of
each plant are developed, the level of detail with which
metal emissions are described, should be further
increased.
The cement manufacturing process is by nature non-
linear and dynamic. The tool should describe stable state
conditions and describe the static and linear transform-
ation of raw material and fuel into clinker. The tool has
to have development potential to include the non-linear
transformations in the process. In addition, there should
also be the potential to simulate dynamic behaviour, e.g.
during start-up and shut down of the kiln.
4. Conceptual model and system boundary
Based on the commissioner’s requirements, a concep-
tual model was constructed, as presented in the follow-
ing:
To avoid sub-optimisation, the model was to be from a
life cycle perspective. The raw material, fuel and cement
additives used are to be traced upstream to the point
where they are removed as a natural resource. Alterna-
tive raw materials, fuels and cement additives are by-
products or waste from other technical systems. The pro-
duction of these alternative products is not to be
included. However, the additional preparation, handling
and transport to make them ?t the cement industry is to
be included. The cement is to be followed to the gate
of the cement plant.
The cement manufacturing system has been divided
into a background system and a foreground system [16].
The foreground system represents Cementa’s “gate to
gate” part of the system. Cementa can, in detail, control
and decide on processes in the foreground system, but
can only make speci?cations and requirements on pro-
81 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
Table 1
Product performance (cement-, clinker-, raw meal ratios)
Ratio Denomination Formula
Lime saturation factor LSF LSF=(100CaO)/(2,8SiO
2
+1,1Al
2
O
3
+0,7Fe
2
O
3
)
Silica ratio SR SR=(SiO
2
)/(Al
2
O
3
+Fe
2
O
3
)
Alumina ratio AR AR=(Al
2
O
3
)/(Fe
2
O
3
)
Note: CaO, SiO
2
, Al
2
O
3
and Fe
2
O
3
are all expressed in weight percentage.
ducts from the background system. Depending on
whether the additional preparation, handling and trans-
port is done by Cementa or not, the processes are either
in the foreground system or the background system. The
conceptual model, in Fig. 2, shows the foreground and
background systems, and in addition a wider system. The
wider system shows consequences of actions taken at the
cement plant, which exist, but are not modelled.
The foreground system was divided into the follow-
ing processes:
? Lime- and marlstone extraction, mining and crushing;
? Sand grinding;
? Raw meal grinding;
? Coal and petroleum coke grinding;
? Clinker production;
? Cement grinding and storage.
Between each one of these processes, intermediate
Fig. 2. Conceptual model.
homogenisation, transportation and storage might take
place and, where applicable, are accounted for.
The background system consists of the following pro-
cesses:
? Production and transport of sand and other raw
material;
? Additional preparation of alternative raw materials
and transport to the cement plant;
? Production and transport of traditional fuels;
? Additional preparation of waste to convert them into
fuels for cement manufacture and transport to the
cement plant;
? Production and transport of cement additives;
? Additional preparation of alternative cement and
transport to the cement plant;
? Production of electricity.
The plant in Slite produces waste heat used for district
82 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
heating in Slite. The waste heat is accounted for as an
output, a product, but no credit is given to the cement
production through allocation or system enlargement. In
the same way, when alternative raw materials and fuels
are used in cement manufacturing, the amount of waste
thus disposed of is accounted for, but no allocation is
made. These consequences of the cement manufacturing
process are placed in the wider system in the concep-
tual model.
Not considered are:
? Production and maintenance of capital equipment for
manufacturing and transport;
? Extraction and production of alternative raw
materials, fuels and cement additives;
? Working material, such as explosives, grinding media
and refractory bricks;
? Iron-sulphate used in the cement milling to reduce
chromium;
? Of?ces.
The two systems were modelled with different tech-
niques and level of detail. The foreground system model
was built according to the techniques described in the
next section. For the background system, traditional life
cycle inventory (LCI) techniques [17] were used. Pro-
duct performance and economic cost were taken into
account by assigning the products entering the fore-
ground system a chemical composition and a cost. Sub-
sequently, ?ows entering the foreground system are
described as a ?ow of mass (kg/s), cost (SEK/s) and
thermal energy content (MJ/s) with a composition
according to Table 2, and in accordance with the pur-
chase deal. Flows of material in the background system
are de?ned and described as a ?ow of mass (kg/s).
The environmental load (resource use and emissions)
Table 2
Material and fuel composition
Compound Unit Compound Unit
CaO weight-share As, arsenic weight-share
SiO
2
weight-share Cd, cadmium weight-share
Al
2
O
3
weight-share Co, cobolt weight-share
Fe
2
O
3
weight-share Cr, chromium weight-share
MgO weight-share Cu, copper weight-share
K
2
O weight-share Hg, mercury weight-share
Na
2
O weight-share Mn, manganese weight-share
SO
3
(sulphides and organic in raw material) weight-share Ni, nickel weight-share
SO
3
(sulphates in raw material) weight-share Pb, lead weight-share
SO
3
(in fuel) weight-share Sb, antimony weight-share
Cl weight-share Se, selenium weight-share
C (in traditional fuel) weight-share Sn, tin weight-share
C (in alternative fuel) weight-share Te tellurium weight-share
C (in raw material) weight-share Tl, thallium weight-share
Organic (in raw material) weight-share V, vanadium weight-share
Moist (105° C) weight-share Zn, zinc weight-share
was described according to the parameters in Table 3.
The kiln exhaust gas from the clinker production system
was described using the parameters in emission to air in
Table 3. The transport was expressed both in ton kilo-
metres and as the related environmental load, according
to the parameters in Table 3.
Table 3
Environmental load, resource use and emissions to air and water
Resource use
Raw material, kg
Alternative raw material, kg
Fuel, kg and MJ
Alternative fuel, kg and MJ
Water, kg
Emission to air Hg, mercury
CO
2
, carbon dioxide Mn, manganese
NO
x
, nitrogen oxides (NO and NO
2
as NO
2
) Ni, nickel
SO
2
, sulphur dioxide Pb, lead
CO, carbon monoxide Sb, antimony
VOC, volatile organic compounds Se, selenium
Dust Sn, tin
As, arsenic Te, tellurium
Cd, cadmium Tl, thallium
Co, cobolt V, vanadium
Cr, chromium Zn, zinc
Cu, copper
Emission to water
BOD, biological oxygen demand
COD, chemical oxygen demand
Total N, total nitrogen content
Non elementary in-?ow, “?ows not followed to the cradle”
Alternative raw material and fuel
Non elementary out-?ows, “?ows not followed to the grave”
Industrial surplus heat, MJ
83 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
5. Modelling and simulation
This section starts by interpreting the commissioner’s
requirements in a system technical context. Only the
foreground system is considered in the following. The
result is a set of decisions on the modelling and the
simulation techniques. This is followed by a description
of how the model was built in accordance with these
techniques and, ?nally, how the constructed model
was validated.
5.1. System technical interpretation
To predict the performance of the desired type of
operational alternatives it was concluded that we had to
simulate them, i.e. perform calculations on a model rep-
resenting the cement manufacturing plant. A model is,
here, a mathematical description of any real subject. A
simulation is then any kind of mathematical experiment
carried out on the model.
The requirements on the model indicate the necessity
of keeping these simulations ?exible in the sense that it
should be possible to predict a number of aspects of the
plant, depending on the situation. Examples of static
equilibrium calculations that are given in Section 3
include:
A Setting the percentage of each raw material in the raw
meal and each fuel in the fuel mix used. Then calcu-
lating the percentage of raw meal mix and fuel mix,
the produced cement quality, emissions and economic
cost under the constraint that the fuel provides all the
thermal process energy. This means we give all the
materials necessary to produce cement and then
watch what comes out of the process.
B Setting properties of the produced cement and each
fuel in the fuel mix used. Then calculating the per-
centage of each raw material in the raw meal mix,
the percentage of raw meal mix and fuel mix, emis-
sions and economic cost under the constraint that the
fuel provides the process thermal energy. This means
we want to control properties of the cement produced
and calculate the proportions of the raw materials,
under the same constraint for the fuel to provide
enough thermal heat.
In a mathematical model, numerical parameters can
be divided into the following categories:
? Constants. Are set when the model is built and then
remain.
? Locked variables. Parameters set to a numerical value
throughout a certain simulation, in accordance to
input data.
? Free variables. Parameters that will be calculated in
the simulation. Some of these are internal variables
in the model and others are the ones we want to calcu-
late; the output.
The difference between the above cases is which para-
meters are locked and which are free. This controls how
the simulation is carried out, i.e. how the equations for
simulation are formulated. The two static equilibrium
cases above will result in different sets of equations. A
simultaneous solving of a respective set of equations will
render the result. It is indeed possible to make these cal-
culations with any general mathematical package avail-
able. If so, each of the cases has to be treated separately.
The result is a well functioning simulation for the spe-
ci?c case that cannot, however, be used for other differ-
ent simulations. If so, the equations need to be re-formu-
lated. Since a speci?c requirement was ?exibility in the
calculations that are possible to perform, we will re?ne
our modelling method by a separation of the model, or
what is normally thought of as the model, into three
parts, namely:
? A neutral model. Only the model, i.e. a description
of our system, in which the connecting equations are
expressed in a neutral form. The model maps our
interpretation of the plant onto a mathematical formu-
lation, but it does not include any speci?c problem to
be solved, hence it is called neutral.
? A problem formulation. An explicit list of which
parameters to lock and a value with which to desig-
nate each of them.
? A simulation method, which is the calculation method
chosen, can also be considered to be a part of the
problem formulation.
The most powerful way to achieve this separation is
to remove the calculational causality (CC) from the
model [18]. The CC determines the order in which the
equations included in a simulation are calculated. This
is merely a technical consideration and affects only the
order in which the calculations are done and does not
imply any restrictions or special considerations regard-
ing the nature or contents of the system behind the
model. The resulting model is said to be a-causal, or
non-causal, in that nothing is said about the order of
calculation in future simulations with the model. The
model can be regarded mathematically as a number of
equilibrium equations connected to each other.
Another important aspect of ?exibility for the model
is modularity. In order to be truly ?exible, according to
the requirement regarding adjustments to represent dif-
ferent cement plants, the model has to be easy to re-
build. In most practical cases, changes would probably
be limited to assigning different inputs and performing
different kinds of simulations, which would already be
part of the problem formulation. In some cases, this is
not enough and the underlying model structure needs to
84 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
be altered. Changing the number of raw materials or
fuels is one such case, and adopting it to ?t a cement
manufacturing plant with different designs is another. A
step to create modularity has already been taken by mak-
ing the model a-causal. This is merely a theoretical pre-
requisite and will not, in itself, produce a ?exible model.
On the other hand, if this is combined with an object
oriented modelling language, we will end up with a prac-
tical, easily re-combinable model. The paradigm of
object orientation is something that affects the language
the model is expressed in. This includes a natural way
to keep parts that are separate in reality as separate
objects in the model, so that the model resembles reality,
or a suitable picture of reality. Usually this feature is
used to group sub-parts of the model into objects, but it
is also useful to group ?ow entities together. Flows that
are made up of a number of substances can thus be
treated as an entity to enhance the transparency and ease
of comprehension.
The cement manufacturing process contains both parts
that vary over time and parts which cannot always be
suf?ciently described with a linear relation. One of the
requirements was to make it possible to account for these
properties in the future, so it must be possible to include
both dynamic and non-linear elements. The ?rst model
which is covered in this paper does not, however, contain
any dynamic or non-linear elements.
In addition to being able to include the above dynamic
elements of the model, we also need to perform dynamic
solving, i.e. calculate and trace (all) the variables in the
model over a certain time span. This simulation type can
be used for environmental predictions when, e.g., start-
ing up, shutting down or changing parameters in the
cement production process. The starting point for such
a simulation can be given values for a set of variables,
such as the start conditions for the plant when per-
forming a start up simulation. It can also be from a state
of equilibrium, which is the case when simulating a shut
down situation. In the latter case, we need a method to
determine this state of equilibrium, e.g. perform a steady
state solving. The steady state solving can, of course,
also be used on its own to ?nd stable points of operation
for the production plant. It is then equivalent to what in
LCI is generally called “normalisation of the life cycle”
or, speci?cally in ISO 14041 [17], “relating data to func-
tional unit”. In addition, another simulation type which
is mentioned for future use, is optimisation.
In summary, we have found that in order to ful?l the
requirements of the commissioner the model needs to be
?exible in terms of:
? Simulation — type of predictions that can be made:
static equilibrium, dynamic solving, etc.;
? Modularity — ease of combination into models of
other cement plants by re-arrangement of the parts;
? Transparency — all governing equations and resulting
?gures readily available to the user, even the
internal ones;
? Comprehension — easy to grasp and understand.
We have, thus, found that the following modelling
approach is needed:
? Calculational non-causal used to separate a neutral
model and the problem formulation;
? Physical modelling to keep physical entities together
in the model;
? Object oriented modelling language to enhance the
reusability of the model.
In addition, the model needs to support:
? Dynamic elements;
? Non-linear elements.
Simulation types the software tool needs to support:
? Steady state solving;
? Dynamic solving;
? Optimisation.
Not all of the requirements above are ful?lled with
state-of-the-art LCI techniques [19]. In LCI, it is gener-
ally enough to describe the life cycle with such a resol-
ution that it is suf?cient with a static and linear model.
Moreover, current LCA tools normally provide normal-
isation of the life cycle to the reference ?ow as the only
simulation alternative. Consequently, there are no LCA
related software tools available that can perform the
desired types of simulations. In the ?eld of general
simulation there are, however, a large number of tools
that can be used. Some equivalent examples include
OmSim [20], DYMOLA [21] and ASCEND [22]. These
software are of the kind that use computational non-cau-
sal models and allow a number of types of simulations
to be performed. For this application, ASCEND was
chosen based on the following criteria:
? It was possible to run on a PC, hence convenient
(DYMOLA, ASCEND);
? It had plug-in modules allowing user made simulation
types, hence ?exible (OmSim, DYMOLA,
ASCEND);
? It was freeware, hence economical (OmSim,
ASCEND).
5.2. Model construction
Building a model with the speci?cations and tech-
niques discussed above is more a matter of generalis-
85 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
ation than speci?cation. Most of the core components in
the model will hence re?ect the general behaviour of an
“object” or “function”. Later, these will be specialised
to the speci?c case, here the cement manufacturing plant.
This technique of extracting layers of behaviour is well
suited for object oriented implementation where the
mechanism of inheritance can be used for that purpose.
The general behaviours are implemented in base classes
and the more speci?c in inherited ones.
The ?rst step when building the model was to ?nd
the objects contained in our perception of the cement
manufacturing plant. This was already done in the con-
ceptual model. These objects then needed to be
abstracted into their general behaviour. Usually, this
reveals that a number of objects follow the same basic
rules, which then means that they can inherit from the
same base object.
First, the general functionality of parts in the concep-
tual model was extracted. Then, a number of general
objects were built to host the functionality. Focus was
put on the mechanisms behind the general functionality
and the correspondence with reality for the more speci?c
one. From the conceptual model, we found the objects
given in Table 4.
In the following, a detailed explanation of some of
these objects is given. The syntax used is based on the
ASCEND IV model language [22] but has been simpli-
?ed to only include the contents (semantic). All code is
given in another font (model). The word composition
thus means the model (object) composition as declared
in Table 5.
Table 4
Total listing of objects in the model
Name Inherits from Role
composition – Any kind of composition of a mixture
mass stream – Flow of material
materialfuel stream mass stream Flow of raw materials and fuels
kilnexhaustgas stream mass stream Flow of exhaust gas
chemical analyser – Test probe for speci?c cement ratios
materialfuel mixer – Mixer for n number of material fuel streams
rawmeal mixing materialfuel mixer Speci?c raw meal mixer at Slite
fuel mixing materialfuel mixer Speci?c fuel mixer at Slite
rawmealfuel mixing materialfuel mixer Speci?c raw meal fuel mixer at Slite
cement mixing materialfuel mixer Speci?c cement mixer at Slite
materialfuel grinder – General grinder for a material fuel stream
rawmeal grinder slite materialfuel grinder Speci?c grinder for raw meal at Slite
sand grinder slite materialfuel grinder Speci?c grinder for sand at Slite
lime grinder slite materialfuel grinder Speci?c grinder for lime at Slite
marl grinder slite materialfuel grinder Speci?c grinder for marl at Slite
coalpetcoke grinder slite materialfuel grinder Speci?c grinder for coal and pet coke mixture at Slite
cement grinder slite materialfuel grinder Speci?c grinder for cement at Slite
clinker production – General clinker production
clinker production slite clinker production Speci?c clinker production at Slite
cement model slite – Top level model over the Slite plant
Table 5
Syntax used in declaration of objects
Syntax Explanation
MODEL xyz Start declaration of the object xyz
Declarations: Part of object where declarations are given
abc IS A xyz; Declares abc as of type xyz
abc[n] IS A xyz; Declares abc as an array with n number of
elements of type xyz
Assignments: Part of object where constants are initiated
Rules: Part of object where the equations are given
FOR i IN abc END Loop where i get the contents of each
FOR; member in abc
SUM[abc] Compute the sum of all elements in abc
= Neutral equality. Used to express
equilibrium, i.e. that two expressions are
numerically equal. It is not an assignment
and does not imply any order of calculation,
e.g. left to right.
5.2.1. Composition
This object is used to represent any kind of compo-
sition of a mixture. A list is used to contain the name
of each component in the mixture (compounds). The
weight share of each component is given as a fraction
with the range of 0 to 1 (y[compounds]). To be able to
handle redundant descriptions (where the weight of the
parts differs from that of the whole), no limitation is put
on the fractions to sum up to 1.0. The object also con-
tains the cost (cost) and heat content (heat) per mass
unit of the total mixture. The typical usage of this object
is to declare the contents of a material, such as a raw
material, fuel or a product.
86 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
MODEL composition
Declarations:
compounds IS A set OF
symbol constant;
y[compounds] IS A fraction;
cost IS A cost per mass;
heat IS A
energy per mass;
Note: The contents of the compounds list is not yet
speci?ed.
5.2.2. Mass stream
The mass stream is a ?ow of material where the con-
tent is declared by a composition (state). The ?ow rate
is expressed both as total ?ow (quantity) and ?ow of
each of the contained components (f). For convenience
(easier access at higher levels), the list of components
in the ?ow is repeated (compounds). It is, then, declared
equivalent to the one already present within state to pre-
vent deviating values.
The two ways of describing the ?ow can be expressed
in terms of each other and, thus, are not independent of
each other. In fact, for all components the ?ow of each
component equals the total ?ow times the fraction for
the component in question (f = quantity?state.y).
MODEL mass stream
Declarations:
compounds IS A set OF
symbol constant;
state IS A composition;
quantity,f[compounds] IS A mass rate;
Rules:
compounds, ARE THE SAME;
state.compounds
FOR i IN compounds f def: f =
CREATE quantity?state.y;
END FOR;
5.2.3. Material–fuel stream
The material–fuel stream is a specialisation of the
mass-stream declared above. It represents the ?ow of
raw materials and fuels in the cement manufacturing pro-
cess. It takes all relevant materials into account, as
de?ned in Table 2, and permits these to be described
either as a share or mass per time. Here, the share option
is used to declare the weight share of each component.
The material–fuel stream also carries the associated cost
and heat.
MODEL materialfuel stream REFINES
mass stream
Declarations:
cost IS A cost per time;
heat IS A energy rate;
Assignments:
Compounds:= [‘CaO’,‘SiO2’,‘Al2O3’
,‘Fe2O3’,‘MgO’,‘K2O’
,‘Na2O’,‘SO3sulphides’
,‘SO3sulphates’,‘SO3fu
el’,‘Cl’,‘Ctrad’,‘Calt’
,‘Craw’,‘Moist’,‘Organi
c’,‘As’,‘Cd’,‘Co’,‘Cr’
,‘Cu’,‘Hg’,‘Mn’,‘Ni’
,‘Pb’,‘Sb’,‘Se’,‘Sn’,‘Te’
,‘Tl’,‘V’,‘Zn’];
Rules:
cost = quantity?state.cost;
heat = quantity?state.heat;
5.2.4. Kiln exhaust gas stream
The exhaust gas from the clinker production system
is modelled as a ?ow representation of its own. The
components are speci?ed with the mass ?ow, e.g. kg/s.
The components are de?ned in Table 3. The kiln exhaust
gas stream is a specialisation of the mass-stream, to
which the appropriate compounds have been added as
described below.
MODEL kilnexhaustgas stream REFINES
mass stream
Assignments:
Compounds:= [‘CO2raw’,‘CO2trad’
,‘CO2alt’,‘CO’,‘VOC’
,‘NOx’,‘SO2’,‘vapour’
,‘As’,‘Cd’,‘Co’,‘Cr’,‘Cu
’,‘Hg’,‘Mn’,‘Ni’,‘Pb’
,‘Sb’,‘Se’,‘Sn’,‘Te’,‘Tl’
,‘V’,‘Zn’];
5.2.5. Chemical analyser
A chemical analyser is a sort of test probe for product
performance. It describes the product performance in the
ratios used in the cement industry, i.e. Lime Saturation
Factor (LSF), Silica Ratio (SR) and Alumina Ratio (AR).
De?nitions of these are given in Table 1.
The analyser is modelled as a stand-alone object and
can be connected to any material fuel stream compo-
sition object in order to measure the performance.
MODEL chemical analyser
Declarations:
state IS A composition;
LSF IS A factor;
87 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
SR IS A factor;
AR IS A factor;
Rules:
LSF = 100?state.y[‘CaO’]/(2.8?state
.y[‘SiO2’]+1.1?state.y[‘Al2O
3’]+0.7?state.y[‘Fe2O3’]);
SR = state.y[‘SiO2’]/(state.y[‘Al2
O3’]+state.y[‘Fe2O3’]);
AR = state.y[‘Al2O3’]/state.y[‘Fe
2O3’];
The analyser can also be used to control the ratios of
a certain material–fuel stream. In such a case, the ratios’
parameters (LSF, SR and AR) can be set and there-
after locked.
5.2.6. Material–fuel mixer
A mixer object transforms two or more in?ows of
material into one out?ow and thus is an n-to-1 junction
for material–fuel streams. It can be used to mix a number
of material–fuel streams in ?xed percentages or to have
these percentages calculated, depending on settings. The
number of inputs (n inputs) must be set before the
object is used. The number of fractions
(mix part[1..n inputs]) equals the number of inputs.
Independent of the number of inputs, there is only one
output (out). The list of components (compounds) in
the inputs and the output are equivalent. For each
component, the output ?ow is the sum of the inputs
(out.f = SUM[in[1..n inputs].f]), or
f
out
?
?
n
inputs
i ? 1
f
in(i)
The mass balance for each individual component
must be maintained. (in[j].quantity =
mix part[j]?out.quantity). An additional constraint is
that the input fractions must sum up to 1.0
(SUM[mix part[1..n inputs]] = 1.0). The heat contents
and economic cost thus must be calculated separately.
Here, they are both expressed so that the respective cost
and heat for the output equals the sum of the input cost
and heat.
MODEL materialfuel mixer
Declarations:
n inputs IS A
integer constant;
in[1..n inputs], out IS A materialfuel
stream;
mix part[1..n inputs] IS A fraction;
Rules:
in[1..n inputs].compo ARE THE SAME
unds, out.compounds ;
FOR i IN cmb: out.f =
out.compounds SUM[in[1..n input
CREATE s].f];
END FOR;
FOR j IN mix[j]:
[1..n inputs] in[j].quantity =
CREATE mix part[j]?out
.quantity;
END FOR;
SUM[mix part[1..n inputs]]=1.0;
out.cost=SUM[in[k].cost | k IN
[1..n inputs]];
out.heat=SUM[in[k].heat | k IN
[1..n inputs]];
5.2.7. Material–fuel grinder
The material–fuel grinder represents grinding raw
meal, clinker, etc., and transforms one in?ow of coarse
material into one out?ow of ground material. Grinding
consumes electrical energy according to the mass
ground. The energy constant (ED) is used to calculate
total electrical power consumption
(electricity consumption). The quantity decreases due
to dust generation that is given by a dust-generating con-
stant (DG) de?ned as a fraction of the out quantity. A
total cost adding is modelled as a ?xed cost per mass
unit (COST) to cover maintenance and operation plus
the cost of electricity. This total cost is then added to
the cost for the material entering the grinder so that the
speci?ed material cost always corresponds to the cumu-
lated production cost at the speci?ed location.
The compositions of the input and output material–
fuel stream (in and out) are the same. The heat content
is not changed during grinding.
MODEL materialfuel grinder
Declarations:
in, out IS A
materialfuel stream;
electricity consu IS A energy rate;
mption
dust generation IS A mass rate;
cost adding IS A cost per mass;
ED IS A energy per mas
s constant;
DG IS A mass per mass c
onstant;
COST IS A cost per mass c
onstant;
ELECTRICITY IS A cost per energy
COST constant;
Rules:
in.compounds, ARE THE SAME;
88 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
out.compounds
in.state.y, ARE THE SAME;
out.state.y
dust generation = out.quantity ? DG;
out.quantity = in.quantity -
dust generation;
electricity consumption = out.quantity ?
ED; (? cost/s ?)
cost adding = COST +
ELECTRICITY COST ? ED; (?
cost/kg ?)
out.state.cost = in.state.cost +
cost adding; (? cost/kg ?)
out. state.heat = in.state.heat;
5.2.8. Clinker production
The clinker production transforms one in?ow of
material and fuel into one out?ow of material and one
out?ow of kiln exhaust gas. The module contains
relations and constants for cost adding, electricity-con-
sumption and dust-generation.
Clinker production requires a speci?ed amount of heat
per mass unit that must be supplied by the fuel. In this
model, a constant value per mass unit clinker entering
the clinker production is used. This amount was there-
fore calculated and set as a requirement on the heat con-
tents in the fuel entering the clinker production.
Fig. 3. Foreground system model.
The clinker production object contains equations that
relate input mixture, output clinker and emissions to
each other. From a modelling technique point of view,
clinker production does not contain any additional con-
cepts beyond what has already been discussed.
5.2.9. Cement plant
When all the objects were de?ned, they were connec-
ted to form a model of the foreground system: the
cement manufacturing plant at Slite. To start with, all
the necessary objects were instantiated and some of the
constants within them were set, such as the number of
inputs for all mixers and site speci?c values. Then they
were connected in accordance to the structure of the con-
ceptual model, which resulted in the model in Fig. 3.
5.3. Problem formulations
The model built is neutral in the sense that it does not
include any speci?c problem to be solved. Such a prob-
lem formulation, consequently, needs to be done separ-
ately. The formulation contains the following:
? A distinction between what to treat as locked vari-
ables and what to treat as free variables, depending
on the desired solution and the calculation method
chosen.
? A connection between input data and the model. Usu-
89 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
ally locked variables are initiated with suitable
input data.
? The calculation method to use, which sorts equations
and calculates the result by invoking a mathemat-
ical algorithm.
Problem formulations will, in the following, be exem-
pli?ed for the two speci?c operational alternatives dis-
cussed in Section 3. To be able to ?nd a solution, the
number of constraints (equations) needs to equal the
number of free variables. The number of equations is a
consequence of the model, and thus, the parts of the
model and how these are connected. Initially, all vari-
ables in the model are free. In the problem formulation,
some of them are locked so the desired simulations will
be possible to perform.
5.3.1. Case A
The requirements in Section 3, further interpreted in
Section 5.1, result in the locked variables, according to
Table 6. These variables are set to the values indicated,
which represent the input. With this problem formu-
lation, the number of variables will equal the number of
equations and the system, thus, becomes possible to
solve. The used solver in ASCEND is QRSlv, which is
a non-linear algebraic equation solver [23].
5.3.2. Case B
Here, variables are locked according to Table 7 and
constants are set to the values indicated. Even here the
Table 6
Constants and input data for Case A
Variable to lock Initiated data Comment
Quantity of cement 1000 kg/s Product quantity
Fraction gypsum for cement grinding 0.052
Fraction limestone for cement grinding 0.044 Implies 90.4% clinker for cement grinding
Fraction pet-coke in fuel mix 0.20 Implies 80% coal in fuel mix
Fraction sand in raw meal 0.02
Fraction marlstone in raw meal 0.71 Implies 27% limestone in raw meal
Heat required by clinker production 3.050 MJ/kg Related to the in?ow of raw meal fuel
Table 7
Constants and input data for Case B
Variable to lock Initiated data Comment
Quantity of cement 1000 kg/s Product quantity
Fraction gypsum for cement grinding 0.045
Fraction limestone for cement grinding 0.04 Implies 91.5% clinker for cement grinding
Fraction pet-coke in fuel mix 0.23
Fraction tyres in fuel mix 0.22 Implies 55% coal in fuel mix
Clinker LSF quality factor 97
Clinker SR quality factor 2.9 Only two out of three quality factors can be set
Heat required by clinker production 3.050 MJ/kg Related to the in?ow of raw meal fuel
number of variables will equal the number of equations
and the system will thus be possible to solve.
5.4. Model validation and simulation
To use the model, i.e. to predict the environmental
load, the product performance and the economic cost, a
prerequisite is that the model acts as the system it rep-
resents. Before using the model and accepting the infor-
mation generated, the model has to be validated. It has
to be determined whether or not the model gives a good
enough description of the system’s properties to be used
in its intended application. When satisfactory correspon-
dence between the situation, the model and the model-
ling purpose has been attained, then the use and
implementation are appropriate. However, validation of
the model will continue throughout the user phase. Once
a future operational alternative has been tested and
implemented, the simulated information will be com-
pared with the observations of the real system. It is then
possible to improve the model. Consequently, the val-
idity and relevance of the model may be continuously
improved.
Validation is an intrinsic part of model building and
the validity of the model has to be assessed according to
different criteria. Technical validation of the foreground
system model, i.e. to ensure that the model contains or
entails no logical contradictions and that the algorithms
are correct, was done as the model was built.
To validate the foreground-system-model, and in
90 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
addition show examples of model usage and results, we
performed simulations on two real operational alterna-
tives. These have actually been used at the plant, and
hence there were measurements to validate against. The
simulations are those given in Sections 3 and 5.1 and
are illustrated in Figs 4 and 5, respectively.
For each of the two operational alternatives, data gen-
erated with the model was compared with observations
and measurements of the real system. The simulated
values were related to the real values. A selection of
simulated values as a percentage of measured values is
shown in Figs 6 and 7 for the two real operational alter-
natives, respectively.
The two simulations show that the model can simulate
the desired operational alternative and generate the
desired information. The simulated and calculated infor-
mation shows, in comparison with the real system’s
properties, satisfactory correspondence. We have a valid
general model of the Slite plant that can be used to pre-
dict product performance, the economic cost and
environmental load.
For metals, the model has been technically validated.
But due to large variations in metal content in raw
material and fuel and insuf?cient empirical data to
describe the emissions of metals we did not achieve total
correspondence between simulated and real metal emis-
sions.
Fig. 4. Real operational alternative A to be simulated.
6. Discussion and future research
It has been shown that the modelling approach used,
i.e. a calculational non-causal model, physical modelling
and an object oriented modelling language can greatly
enhance modularity, ?exibility and comprehensiveness.
Together with an appropriate simulation tool, e.g.
ASCEND IV, this technique provided a ?exible and gen-
eral-purpose model of a cement manufacturing process
for process and product development purposes.
The tool generates the desired information, i.e. pre-
dicts the environmental load, product performance and
economic cost, by simulating the desired operation alter-
native. For the two operational alternatives tested, the
model generated information which shows satisfactory
agreement with the real system’s properties. We are of
the opinion that since all entities are described inde-
pendent of each other, they can easily be combined and
connected to represent another plant or manufacturing
process.
To avoid sub-optimisation, the model was to use a life
cycle perspective. The cement manufacturing process
from cradle to gate was divided into a foreground sys-
tem, the “gate to gate” part, and a background system.
To complete the model in the life-cycle aspect, the back-
ground system model, which is modelled using normal
LCI technique [17] and stored in the SPINE [24] format,
91 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
Fig. 5. Real operational alternative B to be simulated.
Fig. 6. Simulated values as a percentage of measured values. A selection for operational alternative A.
needs to be connected to the foreground model. Since
the background model is both linear and time inde-
pendent (static) it can be expressed with the techniques
and tools discussed in this paper.
As a result of the chosen modelling approach and
simulation tool the model, as such, has potential for
development. One especially interesting area for future
research is to develop the model and the problem formu-
lations so that it will be possible to perform optimisation
with the model. The library of re-usable problem formu-
lations and model parts can be developed and extended.
Other modelling developments would be adding non-lin-
ear and dynamic relations which transform input into
output, and increase the level of detail in the model,
where applicable.
Naturally, the validation process of the cement model
92 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
Fig. 7. Simulated values as a percentage of measured values. A selection for operational alternative B.
will continue to increase the validity and extend the
interval for which the model is valid. The next step thus
will be to use and implement site speci?c models,
including the emission of metals, in the cement industry.
Acknowledgements
We thank Cementa AB for ?nancing Karin Ga¨bel’s
industrial doctoral project. The project has been included
in the research program at the Centre for Environmental
Assessment of Product and Material Systems, CPM. We
also thank Bo-Erik Eriksson at Cementa AB for co-ordi-
nating and prioritising the commissioner’s needs and
requirements and for valuable recommendations on the
level of detail in the model.
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104
Paper II
On optimal investment strategies for a hydrogen
refueling station
in
International Journal of Hydrogen Energy,
In press, corrected proof available online 25 July 2006.
On optimal investment strategies for a hydrogen refueling station
Peter Forsberg
1
, Magnus Karlstr¨ om
2
1. Department of Applied Mechanics
Chalmers University of Technology, 412 96 G¨ oteborg, Sweden
e-mail: [email protected]
2. Environmental Systems Analysis
Chalmers University of Technology, 412 96 G¨ oteborg, Sweden
e-mail: [email protected]
Abstract
The uncertainty and cost of changing from a fossil-fuel based society
to a hydrogen based society are considered to be extensive obstacles to
the introduction of Fuel Cell Vehicles (FCVs). The absence of existing
pro?table refueling stations has been shown to be one of the major barri-
ers. This paper investigates methods for calculating an optimal transition
from a gasoline refueling station to future methane and hydrogen com-
bined use with an on site small-scale reformer for methane. In particular,
we look into the problem of matching the hydrogen capacity of a single
refueling station to an increasing demand. Based on an assumed fu-
ture development scenario, optimal investment strategies are calculated.
First a constant utilization of the hydrogen reformer is assumed in order
to ?nd the minimum hydrogen production cost. Second, when consid-
erations such as periodic maintenance are taken into account, optimal
control is used to concurrently ?nd both a short term equipment variable
utilization for one week and a long term strategy. The result is a mini-
mum hydrogen production cost of $4-6/kg, depending on the number of
re-investments during a 20 year period. The solution is shown to yield
minimum hydrogen production cost for the individual refueling station,
but the solution is sensitive to variations in the scenario parameters.
Keywords: Hydrogen; Infrastructure; Investment; Optimiza-
tion; Refueling station
1 Notations
Table 1 shows the symbolic conventions used in this paper. The
letters c, f etc. are variables and constants, which may be further
speci?ed using sub and superscripts. The symbol c
hr,w
indicates,
for example, the weekly capacity of the hydrogen reformer. Ad-
ditional scenario parameters starting with a capital letter are dis-
cussed in section 3.2 below. The currency used is USD ($).
2 Introduction
Hydrogen is considered a promising future fuel for vehicles [1, 2,
3]. Three main arguments are used to support this assertion: the
potential of reducing greenhouse gases from the transport sector;
greater energy supply security, i.e. hydrogen can be produced
from many energy sources and hence the risk of shortage of sup-
Table 1: Symbolic conventions.
Type Name Description Unit
Variables, c Capacity kg, kg/time unit
constants f Factor -
l Lifetime yr
p Cost, price $
r Pro?t $/time unit
s Size kg, kg/time unit
u Consumption kg/time unit
t Time yr
x Flow kg/time unit
? Ef?ciency -
Subscripts a Annuity
d Daily, per day
e Electricity
eq Equipment, any/all part(s)
f Filling, refueling
fc Hydrogen fuel cell
fp Hydrogen refueling pump
g Methane
h Hydrogen
hc Hydrogen compressor
he Hydrogen electrolysis
hf Hydrogen refueling
hr Hydrogen reformer
hs Hydrogen storage
i Investment
k Peak demand to average
m Maintenance
n Nominal
p Progress ratio
s Scale, scaling
t Technology development
w Weekly, per week
x In?exion point
Superscripts i Input ?ow
o Output ?ow
ply may be reduced; the potential of zero local emissions with the
use of fuel cells.
The absence of a hydrogen infrastructure is seen as a major ob-
stacle to the introduction of hydrogen fuel cell vehicles. A full-
scale hydrogen infrastructure with production facilities, a distri-
bution network and refueling stations is costly to build. The ven-
ture of constructing a hydrogen refueling infrastructure consti-
1
tutes a long-term, capital-intensive investment with great market
uncertainties for fuel cell vehicles. Therefore, reducing the ?-
nancial risk is a major objective of any long-term goal to build a
hydrogen infrastructure [4].
Ogden [5] has described several hydrogen supply options. In-
vestigations have also been made for large scale production of
hydrogen [6]. A number of studies of cost and technology for a
hydrogen infrastructure have also been carried out [7, 8, 9]. How-
ever, to the knowledge of the authors, no studies aimed at ?nding
the most pro?table investment strategy for the individual refuel-
ing station have been done.
There are several reasons to focus on the individual hydrogen
refueling station. Car owners are used to accessing a network
of stations. For the ordinary car owner to accept a hydrogen-
refueling infrastructure, accessibility of service stations will be
crucial [10]. Therefore, a network of hydrogen stations will need
to be built in order to reach the target of about 15-20% of the to-
tal number of refueling stations having a hydrogen fuelling option
for consumers. In the EU, the estimated need is 15-20,000 refu-
eling stations by 2020 (a maximum of 100,000 stations are pre-
dicted by 2020 for EU15
1
) [11]. At present there are only about
110-120 hydrogen stations around the world, some of which are
quite small [12]. Several researchers have proposed that small-
scale reforming of methane could be a feasible transition strategy
for the introduction of hydrogen fuel [13, 14].
This study aims at ?nding the most economic investment strat-
egy, i.e. the lowest cost for the hydrogen produced, for an in-
dividual hydrogen refueling station featuring on site small-scale
reforming of methane. The question is to what extent and when
to build the parts of the station, satisfying an increasing demand
of hydrogen. The method developed may be used to ?nd optimal
investment strategies in other cases in which the number of dy-
namic states is reasonably low. Our calculations begin in 2010
and cover 20 years, until 2030.
3 The refueling station
Methane is chosen as the main energy carrier for the refueling
station since:
1. Methane can be produced from fossil fuel, which is, and
will probably continue to be, one of the cheapest production
sources for hydrogen in the short term.
2. Methane can be produced from renewable resources, e.g.
from different types of wood and plants.
3. Methane is relatively easy to transport and can be re-formed
into hydrogen gas.
4. It is possible that running vehicles directly on methane
might be a favourable alternative as an intermediate step to-
ward hydrogen usage.
After reforming, the produced hydrogen gas can be com-
pressed and stored, or used directly in fuel cells on site for ei-
ther local consumption or distribution on the electricity grid when
electricity prices are high. On the other hand, when electricity
prices are low it might be more pro?table to produce hydrogen
1
The present 15 EU member states
by electrolysis using grid electricity. Taking this into consider-
ation, a large number of station con?gurations are possible, in-
cluding the one indicated in ?gure 1. The model developed does,
however, only consider the core components, i.e. reformer, com-
pressor and hydrogen storage. The model is ?exible with respect
to refueling station types, e.g. car, truck or bus, and refueling
station locations, e.g. central, suburb or countryside.
Figure 1: Possible refueling station con?gurations. Methane is re-
formed to hydrogen at the site and stored for delivery to vehicles. It
is also possible to produce hydrogen from electricity by electrolysis or
electricity from hydrogen using a fuel cell.
3.1 The parts of the refueling station
The parts of the refueling station, see ?gure 1, are considered to
have the characteristics given in table 2. This table gives data on
actual produced equipment in the year 2000. In this paper we
have used ?gures from the Simbeck [15] study. Another compa-
rable study is the GM Well-to-Wheel Analysis of Energy Use and
Greenhouse Gas Emissions of Advanced Fuel/Vehicle Systems -
A European Study (GM WtW) [16].
The purchase price is calculated using the scale function
p = p
n
c
1?fs
n
s
fs
, (1)
where f
s
is a scale factor, further discussed in section 3.2. Using
this function an existing piece of equipment with capacity c
n
and
purchase price p
n
is scaled to any size (s) to obtain an estimated
purchase price. The function (1) applies to all parts of the refu-
eling station except the ?lling pump, which is not scalable but
purchased on a piece-wise basis. Regardless of size, all equip-
ment is considered to have a certain expected lifetime, l. Using
the expected lifetime, the weekly annuity is calculated as
f
a,w
=
D
52(1 ?
1
(1+D/52)
52 l
)
, (2)
where D is the real rate of interest. The total weekly equipment
cost, including maintenance (f
m
), is then
p
w
= f
a,w
p(1 + f
m
). (3)
In reality each part of the refueling station is chosen from a
?nite number of available brands and sizes. By using scaling
2
Table 2: Refueling station parts data. Figures are from Simbeck [15] except lifetimes and progress ratio, which are assumed.
Part Reformer Compressor H
2
store Fill pump Electrolysis Fuel cell
Lifetime(l) 15 yr 15 yr 15 yr 20 yr 30 yr 11.4 yr
Nom. capacity(cn) 42 kg/h 42 kg/h 263 kg 48 kg/h 42 kg/h -
Nom. purchase cost(pn) 38,774 $ h/kg 7,792 $ h/kg 592 $/kg 83,117 $/pc 25,665 $ h/kg 12,987 $/kW
Scale factor(fs) 0.75 0.80 0.80 - 0.72 -
Maintenance cost(fm) 0.05 0.06 0.05 0.05 0.02 0.1
Ef?ciency(?) 0.286 kg H
2
/kg NG 0.99 0.99 0.99 0.02 kg H
2
/kWh 18.33 kWh/kg H
2
Electricity use(fe) 0.02 2.492 0 0 - -
Progress ratio(fp) 0.8 0.9 0.9 0.9 0.9 0.9
functions (1) for the purchase price, the set of available parts can
be replaced with one continuous variable. Compared to evaluat-
ing a number of discrete alternatives, this represents a signi?cant
saving in computational complexity.
The ef?ciency in table 2 indicates the relation between the
mass entering and leaving the equipment. In the case of the re-
former, the substance entering is methane and that leaving is hy-
drogen.
3.2 Scenario parameters
The scenario parameters re?ect assumed developments in the fu-
ture and are given in table 3 together with their respective values.
Table 3: Scenario parameters. The electricity price is assumed to be
higher during daytime (6 am-10 pm) than at night (10 pm-6 am).
Name Description Value Unit
B S-curve slope 0.3 -
D Real rate of interest 0.05 1/yr
Fcont Contingency cost factor 0.1 -
Feng Engineering permitting cost factor 0.1 -
F
f
Refueling characteristics factor -
F
f,k
Refueling ratio peak-demand to average 1.12 -
Fgen Include land cost factor 0.2 -
P
1
Cost of manufacturing 1
th
unit - $
Pe Electricity price vector (6am-10pm) 7.8 c/kWh
Electricity price vector (10pm-6am) 3.9 c/kWh
Pg Methane gas price[17] 47 c/kg
Pn Cost of manufacturing n
th
unit - $
R(t) Relative number of hydrogen vehicles at time t - -
Tx In?ection point of the S-curve 10 yr
U
h,d
Mean hydrogen consumption 1000 kg/day
Vn Cumulative production at n
th
unit - -
V (t) Number of vehicles at time t - -
Vtot Total nr of vehicles at t
end
- -
The number of hydrogen vehicles refueling at the station is
a crucial variable for optimization. It is probably also the most
dif?cult parameter to predict. In this study the S-curve
V (t) =
V
tot
1 + e
?B(t?Tx)
(4)
is used. V
tot
is the total number of vehicles using the refueling
station, T
x
the S-curve in?ection point, and B the slope. The
relative number of hydrogen vehicles at the station is thus
R(t) =
1
1 + e
?B(t?Tx)
. (5)
The function R(t) is purely exogenous and therefore uncertain.
This uncertainty will in?uence the results, as is discussed in sec-
tion 6.
Figure 2 shows the refueling characteristics during 24 hours of
operation for a typical gasoline station (F
f,d
). Together with the
daily mean hydrogen consumption (U
h,d
) and ratio peak-demand
to average (F
f,k
), an absolute demanded refueling capacity is cal-
culated. This sequence is the hydrogen demand when 100% of
the vehicles use hydrogen. To adjust the demand to intermedi-
ate situations the sequence is scaled using the S-curve (5), which
results in the daily maximum hydrogen demand curve
x
h,f,d
(t) = R(t) F
f,d
U
h,d
F
f,k
. (6)
0 5 10 15 20
0
0.02
0.04
0.06
0.08
0.1
0.12
Time [h]
F
r
a
c
t
i
o
n
o
f
2
4
h
c
o
n
s
u
m
p
t
i
o
n
Figure 2: Refueling curve for 24 hours (F
f,d
). This curve gives the
distribution of hydrogen demand in fractions of the total consumption
for one day. The ?gures are based on statistics for a typical gasoline
refueling station. [15]
The demand also differs between weekdays according to ta-
ble 4, which creates a periodic sequence of one week for the total
hydrogen demand (x
h,f,w
). Both the 24 hour refueling curve and
the variations between weekdays are based on statistics for a typ-
ical gasoline refueling station. We assume that this behavior is
independent of fuel type and therefore will persist when hydro-
gen is used in place of gasoline.
It is assumed that equipment becomes cheaper with increasing
production and technology development, which is adjusted in the
3
Table 4: Distribution of hydrogen demand in fractions of the total con-
sumption for one week (F
f,w
). The ?gures are based on statistics for a
typical gasoline refueling station. [15]
Day Mon Tue Wed Thu Fri Sat Sun
Fraction 0.14 0.14 0.14 0.15 0.16 0.14 0.13
cost to manufacture the n
th
unit P
n
according to
P
n
= P
1
V
(log fp/ log 2)
n
, (7)
where P
1
is the cost of manufacturing the ?rst unit, V
n
the cumu-
lative production at n
th
unit and f
p
the progress ratio factor. It is
assumed that the number of hydrogen refueling stations (cumu-
lative production) will be 5,000 in 2010 and 50,000 in 2030, and
will follow the S-curve (4). We have assumed an increase in the
number of refueling stations using steam reforming from5,000 to
50,000 in the world from the year 2010 to 2030. These numbers
are based on the predictions about when the fuel cell vehicle mar-
ket will open up and on the number of station needed. A report
presented by E4tech [18] and funded by the UK Department of
Trade and Industry and the Carbon Trust predicts that ”if the hur-
dles are overcome, the mainstream propulsion market is expected
to open up after 2010”. Melaina [19] made a preliminary analy-
sis of the suf?cient number of initial hydrogen stations in the US,
and concluded that between 4,500 and 17,700 hydrogen stations
would be required in the US to initiate a hydrogen infrastructure
for fuel cell vehicles. We justify our estimate of 50,000 hydro-
gen stations by the fact that we consider the whole world and in
a later stage than do Melaina.
The total decrease in relation to the present-day purchase price
owing to increased production and technology development is
thus
f
eq,t
(t) =
(50000R(t))
(log fp/ log 2)
5000
(log fp/ log 2)
(8)
= (10R(t))
(log(fp)/ log(2))
,
where t is the time from year 2010. Within any real mass
production-based learning process, there will be a trade-off be-
tween system standardization and modularity of system capacity.
However, in this model we have used a simpli?cation, as indi-
cated in equation 8.
With respect to interest rates, future costs can be calculated
from present day values using the Present Day Value Correction
pdc(t) =
1
(1 + D)
t
. (9)
3.3 Initial considerations
We assume that, from the outset, no economic costs, i.e. wages
and rent for land, from the gasoline part of the refueling station
are shared with the hydrogen part and vice versa. Hence, it does
not matter if the gasoline refueling station is present or not when
the hydrogen station is being built. In reality some resources can
probably be shared between the gasoline and hydrogen parts of
the refueling station.
The hydrogen part of the refueling station, see ?gure 1, can be
divided into a number of units that can be optimized separately.
The question of whether or not the local fuel cell and electrol-
ysis parts are pro?table depends on the price of electricity and
methane. If the total cost including maintenance (see (3)) is lower
than the difference between produced and bought electricity and
hydrogen respectively, it is pro?table to invest in the respective
equipment. For the fuel cell the pro?t is then
r
fc
= p
e
s
fc
?
p
g
s
fc
?
hr
?
fc
?
p
fc,w
+ ?p
hr,w
168
, (10)
i.e. almost linearly dependent on the investment, with no up-
per boundary. The same reasoning applies to the electrolysis.
An intermediate situation may appear when the purchase price of
electricity is high, whereas the selling price is low. It might then
only be pro?table to produce electricity for the consumption of
the refueling station.
The methane storage facility is sized in accordance with how
frequently the methane gas tank is ?lled at the refueling station.
Since the estimated methane consumption is known, the periodic
delivery can be calculated. A large volume of methane delivered
at the same time would cost less per kg, but requires a larger
storage tank. This is a separate problem that can be solved us-
ing optimization techniques. For the remainder of this study we
therefore assume a constant delivery of methane from a pipeline
or similar construction.
The refueling pump can be dimensioned according to the max-
imum amount refueled. The daily mean hydrogen consumption
U
h,d
is distributed throughout the day corresponding to the re-
fueling curve (?gure 2). A maximum rate of 0.11 is reached
between 3 pm and 5 pm. The busiest day of the week is Fri-
day, reaching 0.16 of the weekly consumption. The ratio peak-
demand to average (F
f,k
), which is estimated to 1.12, should also
be considered. All in all the number of refueling pumps required
is
c
fp
(t) = ceil(2.87R(t)), (11)
where the function ceil rounds to the nearest integer greater than
or equal to the operand.
The remaining parts of the refueling station are the ones within
the shaded area in ?gure 1 and are collectively called ”the core”.
This core consists of reformer, compressor and storage tank.
Considering only the core, hydrogen is delivered to vehicles only.
This means that all the hydrogen produced by the reformer will
go through the compressor to the storage tank. The size of the
reformer and compressor will thus have to be the same.
Owing to a non-linear price decrease in the equipment over
time (7), the core problem cannot be further split into size dis-
tribution between reformer and tank, plus time and extent of in-
vestments. The relative cost between reformer/compressor and
storage tank will change over time.
3.4 Core model
The model describing the core, see ?gure 1, is quite simple and
includes only one state, the hydrogen storage. It can be described
4
by
? x
hs
= x
i
hs
?x
o
hs
x
i
hs
= x
i
hr
?
hr
?
hc
x
o
hs
?
fp
= x
hf
x
hf
= Hf
w
x
i
hr
= x
ng
(12)
subject to the constraints
0 ? x
o
hr
? c
hr
0 ? x
i
hc
? c
hc
0 ? x
hs
? c
hs
0 ? x
i
fp
? c
fp
. (13)
The rest of this paper addresses the problem of choosing the
size of reformer/compressor versus storage volume over one or
many investments over time for the core of the refueling station.
The model developed will be used in the optimization in the sub-
sequent section.
4 The optimization problem
Optimal control can be used to optimize a system over a certain
time interval. Given a dynamic model of the system, an objective
function, and constraints, a path from one state to another can be
calculated where the objective function is at a minimum. Solving
non-dynamic design problems using optimization techniques is
common practice, see e.g. [20] or [21]. In the case where the
model is static, i.e. does not change over time, such techniques
are suf?cient.
In this study, however, we are interested in investment planning
and internal properties that change over time, such as utilization
curves and transients for hydrogen generation. Therefore a dy-
namic optimization technique is used, see e.g. [22].
4.1 The objective function
The criterion function to be minimized is based on the total pro-
duction cost for hydrogen, which consists of costs for equipment,
methane and electricity. In addition the number of investments
has to be taken into account.
The total weekly equipment cost is the sum of the cost for each
part of the refueling station (p
w
, see (3)), i.e
p
eq,w
(s
eq
, t
i
) =
? eq
p
w
(s
eq
) f
eq,t
(t
i
). (14)
The loans are of the annuity type, which make the equipment cost
independent of time.
Since consumption is given by the refueling demand (x
h,f,w
),
production during the given time frame can be calculated. The
total weekly methane gas cost is then
p
g,w
(t
w
) =
168
t=1
(x
g,w
) P
g
R(t
w
) =
168
t=1
(x
h,f,w
)P
g
R(t
w
)
?
fp
?
hs
?
hc
?
hr
, (15)
where t
w
indicates the time (in years) when the weekly cost is
calculated.
For electricity the price varies throughout the day and needs to
be evaluated on an hourly basis. The weekly cost is then scaled
using the S-curve (5). The total weekly electricity cost is thus
p
e,w
(t
w
) = x
e
P
T
e
R(t
w
) =
168
t=1
(x
h,f,w
) (16)
(f
fp,e
+
f
hs,e
?
fp
+
f
hc,e
?
fp
?
hs
+
f
hr,e
?
fp
?
hs
?
hc
p
T
el
R(t
w
).
Note that both the methane and electricity costs are independent
of the size of the equipment.
In this study the production cost for hydrogen is averaged over
the whole investment period as
p
h
=
1
N
N
tw=1
? seq,ti
p
eq,w
+ p
g,w
(t
w
) + p
e,w
(t
w
)
168
t=1
(x
h,f,w
) R(t
w
)
, (17)
where N is the number of weeks for the investment period. This
production cost takes into account the timing of the investments,
making the purchase cost of all parts of the refueling station de-
crease over time (8). It does not, however, correct future costs
to present-day values (9). It is possible to calculate the hydrogen
cost in other ways. One way is to use the formula above and add
the Present Day Value Correction (9), which gives
p
h
=
1
N
N
tw=1
pdc(t
w
)
? seq,ti
(p
eq,w
) + p
ng,w
+ p
el,w
168
t=1
(x
h,f,w
)R(t
w
)
. (18)
Another totally different approach is to distribute the total cost
evenly over the whole investment period, i.e
p
h
=
N
tw=1
? seq,ti
(
? eq
(p
eq
(1 + f
m
)pdc(t
w
)))f
a,w
168
t=1
(x
h,f,w
)R(t
w
)
. (19)
In this study the average production cost of hydrogen (17) is
used to ?nd the objective function. Expanding the function, it is
clear that not all the terms are necessary to generate the shape
of the production cost. The terms p
g,w
and p
e,w
can be summed
up separately and are independent of size of equipment (s
eq
) and
time for investment (t
i
), giving a constant contribution. In ad-
dition the sum over x
h,f,w
and N are constants. Omitting these
constant terms yields the objective function
p
obj
(t
i
) =
N
tw=1
? seq,ti
(p
eq,w
)
R(t
w
)
. (20)
Expanding the objective function, it can be written as
p
obj
?
N
tw=1
(1 + e
?B(tw?tx)
) (21)
? seq,ti
((1 + e
?B(ti?tx)
)
C1
? eq
s
fs
eq
),
where C
1
is a constant. Since this is a sum over exponentials of
convex functions, the objective function is also convex [23].
In the case when the objective function (20) does not provide
enough information to ?nd an unambiguous optimal point, it can
be augmented , e.g. by variations in utilization of equipment.
5
This would result in a smoother utilization curve. Adding e.g.
the quadratic variations in the hydrogen reformer output would
then give
p
obj2
= p
obj
+ ?
M?1
k=1
(x
o
hr
(k) ?x
o
hr
(k + 1))
2
, (22)
where ? is a weight factor and M the number of hours to con-
sider.
4.2 The constraints
The constraints for the optimization problem are developed from
(12) and (13). By integrating the ?rst equation of (12), which
controls the storage of hydrogen at the refueling station, the ?rst
constraint is found. Since consumption statistics for refueling are
given on an hourly basis for one week, it is possible to use a time-
discrete formulation where integration is replaced by summation.
The model (12) becomes
x
hs
=
t
f
t0
(x
i
hs
?x
o
hs
)
x
i
hs
= x
ng
?
hr
x
o
hs
?
fp
= x
h,f,w
. (23)
By assuming a periodicity of one week for all variables included,
it is only necessary to take 7 ? 24 = 168 points (hours) into ac-
count.
The method of transcription used in the time continuous case
to eliminate time [22] can be replaced in this discrete problem by
cumulative summation. This results in the following constraints:
t
t0
(x
i
hs
?x
o
hs
) ? 0, t
0
? t ? t
f
c
eq
? x
eq
. (24)
The ?rst equation ensures that the stored amount of hydrogen
does not become negative, while the second ensures that the ?ow
through each piece of equipment does not exceed the capacity.
A way of extinguishing transients of the state variables is to
require that the initial value equals the end value, i.e
x
hs
(t
0
) = x
hs
(t
f
). (25)
One inconvenience is that in order to precisely follow the cal-
culated path, the storage containers have to be initially ?lled to a
certain extent. In reality this is not very important since the initial
transients decay rapidly.
Other requirements, e.g. periodical maintenance stops of re-
former or required initial amount of hydrogen stored may also be
taken into consideration by adding one or more constraints.
The optimizations in cases 1 and 2 are carried out for a one
week operation for each investment. For intermediate invest-
ments, the demand is estimated using the S-curve (R(t)), and
equipment is assumed to have decreased in price according to
(7) with scenario parameters as in table 3. Note that the price de-
crease is faster for the reformer than for the rest of the equipment.
5 Results
This section presents the results from the optimization in the two
cases discussed. Both cases are solved using Tomlab [24]. No
gradients or Hessians are provided, instead estimates are made
using numerical differentiation within the optimization method.
5.1 Case 1: constant utilization
In case 1, constant utilization of reformer and compressor is con-
sidered. The capacity of the reformer/compressor (c
hr
) is deter-
mined fromthe weekly average demand and the hydrogen storage
capacity by ?nding the minimum of the sum of net input to the
hydrogen storage tank, see (26).
c
hr
=
168
t=1
(x
h,f,w
)
168?
hc
?
hs
?
fp
(26)
x
hs
(t
0
) = ?min
t
t
t0
(c
hr
?
hc
?x
h,f,w
/(?
fp
?
hs
))
c
hs
= max
t
t
t0
(c
hr
?
hc
?x
h,f,w
/(?
fp
?
hs
)) + x
hs
(t
0
)
The result is an unconstrained optimization problem that can be
described by
min
ti
p
obj
(t
i
). (27)
0 2 4 6 8 10 12 14 16 18 20
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
6.2
Second investment time [year]
H
y
d
r
o
g
e
n
p
r
o
d
u
c
t
i
o
n
c
o
s
t
[
U
S
D
/
k
g
]
Figure 3: Case 1, 2 investments, hydrogen production cost as a func-
tion of second investment time. Note the discontinuities at times 8.5 and
13.2. These are caused by an increase in the number of refueling pumps
(11).
If only one investment is made, the characteristics are calcu-
lated directly using (26) and no optimization is carried out. In
the case of 2 or more investments, the curves in ?gures 3 and 4
illustrate the effect of investment time.
The problem in case 1 is solved using a quasi-Newton method
implemented in the Tomlab function ucSolve. The results for 1-3
investments are as shown in table 6, ?gures 5 and 6.
6
Figure 4: Case 1, 3 investments, hydrogen production cost as a func-
tion of second and third investment time.
Table 5: Result of optimization, case 1. The total cost is the cost for
equipment, methane and electricity for the entire investment period. The
mean distance cost is calculated from the use of 0.1kg H2/10km for fuel
cell vehicles.
No of investments 1 2 3
Investment time [yr] 0 0, 5.7 0, 3.9, 8.4
Cost equipment [$] 3,868,763 2,961,677 2,793,208
Total cost [$] 16,296,295 15,026,375 14,791,149
Mean cost hydrogen[$/kg] 6.03 4.37 4.14
Mean distance cost[$/10km] 0.60 0.44 0.41
Reformer size [kg/h] 45.47 9.2+36.3 5.8+10.9+28.7
Storage size [kg] 606 123+484 77+146+383
Initial storage [kg] 271 55,271 35,100,271
Refueling pump no [pcs] 3 1+2 1+0+2
The mean distance cost is based on a consumption of 0.1kg
H
2
/10km for a fuel cell vehicles. This is an estimate of the hy-
drogen consumption for a small fuel cell vehicle and is only used
for comparison with petrol fueled cars.
Further investments have very little effect on the mean hydro-
gen production cost, as can be seen in ?gure 7.
5.2 Case 2: variable utilization
In Case 2 utilization of equipment is parametrized and deter-
mined by the optimization algorithm. The chosen special condi-
tions in this study are 100 kg hydrogen storage initially and at the
end of each week (periodic boundary conditions), and a weekly
stop for maintenance from hours 75 to 87 during the week. In-
vestments are done on 1 and 2 occasions during the investment
period. Further investments have not been investigated for case
2, owing to the computational complexity and time involved. The
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
100
200
300
400
500
600
700
Hydrogen stored 1 [kg]
time [h]
k
g
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
20
40
60
80
100
120
140
Hydrogen storage output 1 [kg/h]
time [h]
k
g
/
h
Figure 5: Case 1, 1 investment, stored hydrogen and storage output.
0 2 4 6 8 10 12 14 16 18 20
5
10
15
20
Absolute cost produced hydrogen [USD/kg]
time [year]
U
S
D
/
k
g
0 2 4 6 8 10 12 14 16 18 20
0
2000
4000
6000
8000
Capacity ? Demand for hydrogen [kg/week]
time [year]
k
g
/
w
e
e
k
Figure 6: Case 1, 1 investment, hydrogen production cost and capacity-
demand.
resulting constrained non-linear optimization problem
min
ti,s
hr
p
obj2
s.t.
t
t0
(x
i
hs
?x
o
hs
) ? 0, t
0
? t ? t
f
c
eq
? x
eq
x
hs
(t
0
) = x
hs
(t
f
)
x
hs
(t
0
) = 100
87
t=75
x
o
hr
= 0, (28)
was solved using a Sequential Quadric Programming (SPQ)
method [25], as part of the NPSOL [26] package running in Tom-
lab [24].
The results from the optimization give the size of equipment
7
1 2 3 4 5 6 7 8 9 10
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
6.3
6.4
6.5
Number of investments
H
y
d
r
o
g
e
n
p
r
o
d
u
c
t
i
o
n
c
o
s
t
[
U
S
D
/
k
g
]
Figure 7: Case 1, hydrogen production cost for 1-10 investments.
(table 6), running pattern of the facility (?gure 8) and produced
hydrogen price and utilization curves (?gure 9). The solution
shows good utilization of storage; the stored hydrogen amount
frequently drops to near zero.
Table 6: Result of optimization, case 2. The total cost is the cost for
equipment, methane and electricity for the entire investment period. The
mean distance cost is calculated from the use of 0.1kg H2/10km for fuel
cell vehicles.
No of investments 1 2
Investment time [yr] 0 0, 5.6
Cost equipment [$] 4,707,805 3,724,066
Total cost [$] 17,522,971 16,151,907
Mean cost hydrogen [$/kg] 6.74 4.72
Mean distance cost [$/10km] 0.67 0.47
Reformer size [kg/h] 57 10.0, 50.0
Storage size [kg] 939 199, 873
Initial storage [kg] 63 90, 64
Refuelling pump no [pcs] 3 1 + 2
5.3 Sensitivity of the solution
To evaluate the sensitivity of the solution in case 1 (with 2 invest-
ments) to changes in the scenario parameters, calculations were
made with slightly changed values from the settings in table 3.
Table 7 shows the results in relative sensitivities, i.e.
sens
y
=
min
i
(z + ?z)
min
i
(z)
, (29)
where sens
y
is the relative sensitivity value with respect to prop-
erty z. The optimization procedure is abbreviated min
i
.
Owing to the similarities in the objective function, the sensi-
tivities also are valid for case 2.
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
10
20
30
40
50
60
Hydrogen reformer output 1 [kg/h]
time [h]
k
g
/
h
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
200
400
600
800
1000
Hydrogen stored 1 [kg]
time [h]
k
g
Figure 8: Case 2, 1 investment, throughput and stored hydrogen.
0 2 4 6 8 10 12 14 16 18 20
0
5
10
15
20
25
Absolute cost produced hydrogen [USD/kg]
time [year]
U
S
D
/
k
g
0 2 4 6 8 10 12 14 16 18 20
0
2000
4000
6000
8000
Capacity ? Demand for hydrogen [kg/week]
time [year]
k
g
/
w
e
e
k
Figure 9: Case 2, 1 investment, hydrogen production cost and capacity-
demand.
Table 7: Relative sensitivity at the optimal point, case 1 with 2 invest-
ments. The numbers given are relative sensitivity at the optimal point,
see (29).
Property name Investment time H
2
production cost
sensitivity sensitivity
Real rate of interest (D) -0.04 0.10
Progress ratio (fp) 0.26 0.85
S-curve slope (B) -0.42 0.47
S-curve in?ection point(Tx) 0.34 0.58
Total units at t
end
(Vtot) 0.02 -0.23
Mean hydrogen cons.(U
h,d
) -0.01 0.002
6 Discussion
It is possible to use optimization and optimal control to determine
optimal investment strategies for a hydrogen refueling station for
8
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
2
4
6
8
10
12
Hydrogen reformer output 1 [kg/h]
time [h]
k
g
/
h
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
50
100
150
200
Hydrogen stored 1 [kg]
time [h]
k
g
Figure 10: Case 2, 2 investments, throughput and stored hydrogen:
Investment 1 at t=0.
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
10
20
30
40
50
60
70
Hydrogen reformer output 2 [kg/h]
time [h]
k
g
/
h
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
200
400
600
800
1000
1200
Hydrogen stored 2 [kg]
time [h]
k
g
Figure 11: Case 2, 2 investments, throughput and stored hydrogen:
Investment 2 at t=5.6.
vehicles. The results indicate a hydrogen production cost at a re-
fueling station with on site reforming of methane ranging from
$4.1 to 6.0/kg, depending on the number of investments and spe-
cial requirements of periodic maintenance, etc. This is in the
same range as previous ?ndings, see table 8. The main differ-
ence is that this study uses a function that increases over time (4)
to estimate the number of hydrogen vehicles refueling at the sta-
tion, which makes the estimated production cost an average over
time. In other studies, the cost is based on maximum utilization.
The idea underpinning the method developed is to be able to eas-
ily change assumptions and scenario parameters according to a
given case. The method can then be used for investment planning
in individual refueling station cases.
When one large investment is made, hydrogen produced will
initially become very expensive. Although the production cost
will have dropped to a more reasonable level after 10 years of
0 2 4 6 8 10 12 14 16 18 20
3
4
5
6
7
8
9
Absolute cost produced hydrogen [USD/kg]
time [year]
U
S
D
/
k
g
0 2 4 6 8 10 12 14 16 18 20
0
2000
4000
6000
8000
Capacity ? Demand for hydrogen [kg/week]
time [year]
k
g
/
w
e
e
k
Figure 12: Case 2, 2 investments, hydrogen production cost and
capacity-demand.
Table 8: Other studies of on site reforming of methane
Study $/kg Size
Schoenung [27] 5.7 400 kg/d
Knight [4] 1.79 250 cars/d
Thomas [7] 11-2.2 180 - 2720 kg/d
Simbeck [15] 4.4 470 kg/d
Ogden [5] 1.7-5.6 400 cars/d
production, the refueling station may not survive that long. A bet-
ter approach would be to start with a smaller capacity, and then
increase it over time. The results show that the most realistic eco-
nomic production cost situation can be achieved at approximately
4 to 5 investments (?gure 7) and that little is to be gained by fur-
ther increasing number of investments. Increasing the number of
investments can also be more favourable froma risk management
point of view. It is then possible to adjust the investment plan be-
fore the next investment is made, using the same method but with
more recent assumptions.
The sensitivity analysis shows that the H
2
production cost is
quite sensitive to changes in some of the scenario parameters.
The most sensitive one is progress ratio, i.e the price decrease
for equipment. Since the progress ratio is not known in advance,
large changes in the predicted production cost may result. One
way of handling this situation is to add uncertainty estimates to
all scenario parameters and make an optimization that takes these
uncertainties into account.
Some factors in the cost function can be improved to make the
results more realistic, e.g. maintenance of the equipment and re-
sources split between the gasoline and hydrogen parts of the refu-
eling station. Another area that can be improved is the ef?ciency
factors for equipment, ?
eq
. In reality, ef?ciency is dependent on
other factors, e.g. ?owthrough the equipment. Also, in reality fu-
ture development is not known. By using stochastic variables and
make a stochastic optimization, uncertainties can be expressed in
the result in terms of probability functions.
When the number of investments increases, so does the com-
9
putational time. For case 2 with variable utilization, calculations
with more than 2 investments already result in unrealistically long
computational time. Since the problem is convex in the objec-
tive function (but not in the constraints), other more specialised
optimization algorithms may be used. In addition, gradient and
Hessian information can be provided to further reduce computa-
tional time.
The above model can probably also be used as a starting
point when doing investment optimization for multiple refuel-
ing stations in a community. This optimization problem is not
as straightforward as the one discussed in this paper: here fac-
tors such as local competition between refueling stations and how
this affect sales (i.e. supply-demand curve) have to be taken into
account. Another option would be to investigate under what cir-
cumstances the complete station layout (?gure 1) would be prof-
itable.
7 Summary and conclusions
1. With the assumptions made in this study, it is possible to
produce hydrogen with on site reforming at a price ranging
from$6.0/kg for one investment to $4.1/kg for 3 investments
over 20 years when continuous production is considered.
2. Special requirements, e.g. speci?ed storage in the beginning
of the week and periodic maintenance stops of the reformer,
can be accounted for but will make the produced hydrogen
more expensive.
3. Investment timing is most sensitive (in order of magnitude)
to changes in the scenario parameter S-curve steepness (B),
S-curve in?ection point (t
x
) and progress ratio (f
p
). It is
less sensitive to changes in methane and electricity prices,
interest rates (D) and S-curve total number of units at t
end
(V
tot
).
4. The hydrogen production cost is most sensitive (in order of
magnitude) to changes in progress ratio (f
p
), scenario pa-
rameter S-curve in?ection point (t
x
) and S-curve steepness
(B). It is also quite sensitive to changes in S-curve total
number of units at t
end
(V
tot
) and interest rates (D).
5. The method developed in this paper can be used for optimal
investment planning in other areas with ?ow processes that
can be described with state equations.
8 Acknowledgments
Financial support from the Competence Center for Environmen-
tal Assessment of Product and Materials Systems (CPM) at
Chalmers University of Technology and the Swedish Foundation
for Strategic Environmental Research, MISTRA, is gratefully ac-
knowledged. Part of this work was done within the framework of
the Jungner Center.
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11
118
Paper III
Macroeconomic and ?nancial time series prediction
using networks and evolutionary algorithms
in
Proceedings of Computational Finance 2006,
London, 27-29 June 2006, pp. 403-411.
Macroeconomic time series prediction
using prediction networks and
evolutionary algorithms
Peter Forsberg
1
, Mattias Wahde
Department of Applied Mechanics,
Chalmers University of Technology, Sweden
1. Corresponding author
Abstract
The prediction of macroeconomic time series by means of a form of fully
recurrent neural networks, called discrete-time prediction networks (DTPNs),
is considered. The DTPNs are generated using an evolutionary algorithm,
allowing both structural and parametric modi?cations of the networks, as
well as modi?cations in the squashing function of individual neurons.
The results show that the evolved DTPNs achieve better performance on
both training and validation data compared to benchmark prediction meth-
ods. The importance of allowing structural modi?cations in the evolving
networks is discussed. Finally, a brief investigation of predictability mea-
sures is presented.
Key words: time series prediction, recurrent neural networks, evolutionary
algorithms
1 Introduction
Prediction of time series is an important problem in many ?elds, including
economics. Due to the high level of noise in macroeconomic time series,
models involving two parts, one deterministic and one stochastic, are often
used. One such method is ARIMA [1]. For one-step prediction, the results
obtained by these simple predictive methods (such as exponential smooth-
ing, which is a special case of ARIMA models), are di?cult to improve much
due to the high levels of noise present. However, even a small improvement
can translate into considerable amounts of money for data sets that concern
e.g. an entire national economy. The aims of this paper is (1) to introduce a
class of generalized, recurrent neural networks and an associated evolution-
ary optimization method and (2) to apply such networks to the problem
of deterministic prediction of macroeconomic time series, with the aim of
extracting as much information as possible, while keeping in mind that the
noise in the data introduces limits on the achievable performance.
2 Macroeconomic data
Two di?erent data sets were considered, namely US GDP (quarterly varia-
tion, from 1947, ?rst quarter to 2005, second quarter), and the Fed Funds
interest rate (monthly values, from July 1954 to July 2005). The raw GDP
and interest data were ?rst transformed to a relative di?erence series, using
the transformation
Z
RD
(t) =
Z
raw
(t) ?Z
raw
(t ?1)
Z
raw
(t ?1)
. (1)
Next, this series was further transformed using a hyperbolic tangent trans-
formation
Z(t) = tanh(C
TH
Z
RD
(t)). (2)
For the GDP and interest rate series transformations, the values C
TH
= 25
and C
TH
= 5 were used, respectively. The aim of the hyperbolic tangent
transformation was to make the data points as evenly distributed as possible
in the range [?1, 1].
Both data sets were divided into a training part with M
tr
data points, and
a validation part with M
val
data points. During training, only the results
(i.e. the error) over the training data set were used as feedback to the
optimization procedure (see below). The rescaled GDP data set contained
233 data points. For training, steps 16-115 were used (M
tr
= 100) and for
validation, steps 126-225 were used (M
val
= 100). During training, the ?rst
15 steps were used to initialize the short-term memory of the DTPN. A
similar initialization procedure was applied during validation. For the Fed
Funds data set, with 612 data points, steps 26-475 were used for training
(M
tr
= 450) and steps 486-605 (M
val
= 120) were used for validation.
3 Methods for prediction
3.1 Discrete-time prediction networks
Neural networks constitute a commonly used blackbox prediction model.
In most cases, feedforward neural networks (FFNNs) are used. In such net-
works, the computational elements (neurons) are placed in layers. The input
signals (i.e. earlier, consecutive values of the time series) are distributed to
the neurons in the ?rst layer, and the output signals of those neurons are
then computed and used as input in the second layer etc. The output of a
given neuron i is computed as
x
i
(t + 1) = ?
?
?
b
i
+
N
j=1
w
ij
y
j
?
?
, (3)
where b
i
is the bias term, w
ij
are the weights connecting neuron j in the
preceding layer to neuron i, N is the number of neurons in the preceding
layer, and ? is the squashing function, usually taken as the logistic function
?
1
(z) =
1
1 + e
?cz
, (4)
where c is a positive constant, or the hyperbolic tangent
?
2
(z) = tanh cz. (5)
Given a set of training data, i.e. a list of input vectors and their corre-
sponding desired output, such networks can be trained using gradient-based
methods, such as e.g. backpropagation.
However, there are fundamental limitations in the prediction that can be
achieved using FFNNs, due to their lack of dynamic (short-term) memory.
Stated di?erently, an FFNN will, for a given input, always give the same
output, regardless of any earlier input signals [2], [3]. Thus, such networks
are unable to deal with situations in which identical inputs to the network
(at di?erent times along the time series) require di?erent outputs. Earlier
work [2] has shown that dynamic short-term memory does make a di?erence
in neural network-based time series prediction.
Furthermore, the requirement that it should be possible to obtain a gra-
dient of the prediction error, in order to form the derivatives needed for
updating the weights (during training), restricts the shape of the squashing
functions. Without such restrictions, squashing functions such as e.g.
?
3
(z) = sgn(z), (6)
and
?
4
(z) =
?
?
?
?
?
tanh(z + c) if z < ?c
0 if ?c ? z ? c
tanh(z ?c) if z > c
(7)
could be used.
To overcome the limitations of FFNNs, it is possible to introduce feed-
back couplings in the networks, transforming them into recurrent neural
networks (RNNs). Such networks have been used in many ?nancial and
macroeconomic applications, see e.g. [3], [4]. A problem with many stan-
dard training techniques for neural networks is that they require that the
user should set the structure of the network (i.e. the number of neurons and
their position in the network), a procedure for which one often has to rely
on guesswork and rules-of-thumb [5]. An alternative training procedure is to
use an evolutionary algorithm (EA) which, if properly designed, can handle
both structural and parametric optimization [6].
In this paper, a new kind of network (and an associated evolutionary
optimization method), well suited for the problem of time series predic-
tion, will be used, with dynamical memory, arbitrary structure, and (in
principle) arbitrary squashing functions. Each of the n neurons in these
networks which, henceforth, will be called discrete-time prediction networks
or DTPNs for short) contains arbitrary connections from the n
in
input ele-
ments and from other neurons (including itself). In addition, each neuron
has an evaluation order tag (EOT) such that, in each time step, the output
of the neurons with the lowest EOT values is computed ?rst, followed by
the output of the neurons with the second lowest EOT values etc. The out-
put neuron, i.e. the neuron with highest EOT (arbitrarily chosen as neuron
1) is evaluated last. Thus, the equations for neurons with the lowest EOT
become
x
i
(t + 1) = ?
?
?
b
i
+
nin
j=1
w
in
ij
I
j
(t) +
n
j=1
w
ij
x
j
(t)
?
?
, (8)
where w
in
ij
are the input weights, w
ij
the interneuron weights, and b
i
is
the bias term. I
j
are the inputs to the network which, in the case of time
series prediction, consist of earlier values of the time series Z(t), i.e. I
j
(t) =
Z(t ?j + 1). The number of inputs can thus be referred to as the lookback
(L) of the DTPN. For neurons with the second lowest EOT, the equations
look the same, except that x(t) is changed to x(t+1) for neurons with lowest
EOT etc. Finally, the output neuron gives the following output
x
1
(t + 1) = ?
?
?
b
1
+
nin
j=1
w
in
1j
I
j
(t) + w
11
x
1
(t) +
n
j=2
w
1j
x
j
(t + 1)
?
?
, (9)
since, at this stage, all neurons except neuron 1 have been updated. It is
evident that the EOTs introduce the equivalent of layers. Thus, while most
DTPNs will contain many recurrent connections, an FFNN is a special case
of a DTPN. More precisely, a DTPN is equivalent to an ordinary FFNN if
and only if (1) all squashing functions are of the same type (either ?
1
or
?
2
), (2) only neurons with the lowest EOT values receive external input,
and (3) w
ij
(i.e. the weight connecting neuron j to neuron i) is equal to
zero if EOT(j) ? EOT(i).
3.2 Benchmark predictions
In order to evaluate the results obtained using DTPNs, a comparison will
be made with two standard prediction techniques, namely autoregressive
moving average (ARMA) and exponential smoothing. The general simple
ARMA(p, q) model
?(?)Z(t) = ?(?)(t), (10)
where ? is the lag operator, is the disturbance Z ?
ˆ
Z, and
?(?) = 1 ??
1
? ?. . . ??
p
?
p
, (11)
and
?(?) = 1 + ?
1
? + . . . + ?
q
?
q
, (12)
gives the one-step prediction
ˆ
Z(t + 1|t)
ˆ
Z(t + 1|t) =
p
i=0
?
i
Z(t ?i) +
q
i=0
?
i
(t ?i). (13)
?
i
and ?
i
are parameters to be estimated in order to ?nd the lowest error.
The exponential smoothing technique (without trend) is described by the
Neuron1 Neuron2 Neuronn ...
Neuroni
w(interneuronweights) w (inputweights)
in
b c k(sigmoidtype)
...
EOT
Figure 1: A chromosome encoding a DTPN.
ARIMA(0,1,1) equation
(1 ??)Z(t) = (1 ??
1
?)(t). (14)
This model gives the prediction
ˆ
Z(t + 1|t) =
1 ??
1
1 ??
1
?
Z(t) = ?
1
ˆ
Z(t|t ?1) + (1 ??
1
)Z(t). (15)
As a special case, if ?
1
= 0, the naive prediction
ˆ
Z(t+1|t) = Z(t) is obtained.
4 Evolutionary algorithm
The DTPNs were generated using an evolutionary algorithm (EA) [7]. The
EA used here employed a non-standard chromosomal representation, shown
in Fig. 1, in which each gene represented a neuron in the network, encoding
its interneuron weights (w
ij
), input weights (w
in
ij
), bias term (b
i
), sigmoid
parameter (c), sigmoid type, and EOT. During the formation of new individ-
uals, crossover was only allowed between individuals containing the same
number of neurons. Several di?erent forms of mutations were used, both
parametric mutations modifying the values of the parameters (including
the EOT) listed above, and structural mutations which could either add or
subtract a neuron from the DTPN. No upper limit was set on the number
of neurons. A lower limit of 2 neurons was introduced, however. In addition
to the mutations just listed, a sigmoid type mutation was introduced as
well, allowing a neuron to change its sigmoid type by randomly changing
the index k of the sigmoid ?
k
(see Sect. 3.1 and Eq. (17) below). Finally, in
order to allow (not force) the EA to produce sparsely connected networks,
some runs were carried out in which parametric mutations of interneuron
weights, input weights, and biases not only could modify the value of the
parameter in question, but also (with low probability) could set it exactly
to zero. Thus, these mutations essentially functioned as on-o? toggles, and
were therefore called zero-toggle mutations. The number of input elements
(and therefore the lookback L) was ?xed in each run. The ?tness measure
F used by the EA was taken as the inverse of the RMS prediction error over
the training set, i.e. F = 1/e
RMS
where
e
RMS
=
1
M
tr
Mtr
i=1
Z(i) ?
ˆ
Z(i)
2
(16)
Note that the use of an EA implies that any form of sigmoid function can
be used in the networks. In addition to the four functions ?
1
? ?
4
, a ?fth
sigmoid, namely
?
5
(z) =
cz
1 + (cz)
2
, (17)
was also allowed in the simulations reported below.
5 Prediction results
A large number of runs were carried out, using di?erent number of inputs
and di?erent EA parameters in order to test the ability of the evolutionary
algorithm to generate DTPNs with low prediction error for the two data
sets under consideration.
The results are summarized in Table 1. The table shows the prediction
error for the DTPN with lowest validation error. In addition, the prediction
errors obtained using naive prediction, exponential smoothing, and ARMA
(all with optimized parameter values), are shown.
As is evident from the table, the best DTPNs outperform the two other
prediction methods. Table 2 gives a more detailed description of the best
DTPNs, obtained with di?erent values of n
in
. For comparison, note that the
best training errors obtained with exponential smoothing were e
tr
ES
= 0.2512
for the GDP data and e
tr
ES
= 0.3477 for the Fed funds data. Using the ARMA
model, the best training errors were e
tr
ARMA
= 0.2108 and e
tr
ARMA
= 0.3248,
respectively.
6 Predictability measures
The fact that the DTPNs outperform the benchmark prediction methods
does not imply that these networks extract all the available information
in the time series under study. One way of determining whether addi-
tional information can be extracted would be to devise a measure P(t)
of predictability such that, in addition to the prediction
ˆ
Z(t + 1) of the
Data set e
N
e
ES
e
ARMA
e
DTPN
Fed funds interest rate 0.2018 0.1901 0.1887 0.1837
GDP 0.1771 0.1490 0.1473 0.1305
Table 1: Minimum errors over the validation part of the data set, obtained
using naive prediction (e
N
), exponential smoothing (e
ES
), ARMA
(e
ARMA
), and DTPNs (e
DTPN
). Only the results for the very best
DTPN are shown.
Data set n
IN
P
zero
n n
L
e
tr
DTPN
e
val
DTPN
Fed funds, run 1 2 0.00 7 5 0.3072 0.1837
Fed funds, run 2 2 0.25 5 5 0.2968 0.1881
GDP, run 1 5 0.00 4 4 0.2095 0.1423
GDP, run 2 4 0.00 6 4 0.2173 0.1399
GDP, run 3 3 0.00 5 4 0.2131 0.1360
GDP, run 4 3 0.20 11 5 0.2094 0.1305
Table 2: Examples of the performance of evolved DTPNs. The second col-
umn shows the number of inputs to the network, and the third col-
umn shows the probability of a mutation being of the zero-toggle
type, i.e. a mutation that sets the parameter in question to zero.
The fourth column shows the (evolved) number of neurons, and
the ?fth column shows the (evolved) number of layers (n
L
), i.e.
the number of distinct EOT values in the evolved network. The
two ?nal columns show the errors over the training and validation
parts of the data set.
next value in the time series, one would obtain an estimate of the error
e(t + 1) = Z(t + 1) ?
ˆ
Z(t + 1). Ideally, the measure should be such that
P(t) = f(e(t + 1)) where f is a known, monotonous function.
Several di?erent predictability measure can be formed. The amount of
(local) information in a time series can, for instance, be estimated analyti-
cally using random matrix theory, based on the correlation matrix formed
from the delay matrix D [8]. In addition, various empirical measures can
Figure 2: The best evolved network (run 4) for the prediction of the GDP
series. Input elements are shown as squares and neurons as ?lled
circles. The neurons are arranged in layers based on their EOT
values. For clarity, only the inputs to one neuron are shown. Solid
lines indicate positive weights and dotted lines negative ones.
also be generated, based on the prediction errors obtained in previous time
steps. An investigation was made involving both the analytical measure and
a few di?erent empirical measures, applied to the rescaled di?erence series
Z(t). However, in all cases, the results were negative, i.e. the proposed pre-
dictability measure showed near-zero correlation with the actual prediction
error, and therefore these measures will not be described further here.
7 Discussion and conclusion
This investigation has shown that it is possible to improve, albeit only
slightly, the predictions obtained from standard prediction methods using a
generalized version of neural networks (called discrete-time prediction net-
works, DTPNs) with the possibility of adding a short-term memory through
feedback couplings.
In earlier work [2], continuous-time recurrent neural networks were consid-
ered for time series prediction. The DTPNs introduced here do not require
continuous-time integration, i.e. the network output is obtained by discrete-
time equations rather than di?erential equations, making the evaluation of
the networks much faster, while still allowing a rich dynamical structure,
including dynamic short-term memory.
The use of an EA for the optimization of the networks removes all restric-
tions regarding both the behavior of individual neurons as well as the struc-
ture of the network as a whole, while still allowing standard feedforward
neural networks as a special case.
The importance of structural modi?cations in the network is illustrated
by the fact that, in any given run, the structure of the current best net-
work varied signi?cantly during the run. The ?nal networks often contained
rather few neurons and used only a few input elements, illustrating another
advantage of using recurrent networks: because of their ability to form a
short-term dynamic memory, such networks need not use as many inputs as
a feedforward network, thus also reducing the number of networks weights
and hence the risk of over?tting.
The best network for prediction of the GDP series, shown in Fig. 2, had
a slightly more complex structure. However, in the run generating that
network, zero-toggle mutations were used, and indeed the resulting network
was far from fully connected, and therefore had, in fact, a somewhat simpler
structure than would have been suspected on the basis of the number of
neurons involved.
The fact that the predictability measures all gave negative results was
expected, and it indicates that the DTPNs really do extract all, or almost
all, information available in the time series.
References
[1] Harvey, A., The econometric analysis of time series. London School of
Economics handbooks in economic analysis, New York; London: Philip
Allan, 2nd edition, 1990.
[2] Hulth´en, E. & Wahde, M., Improving time series prediction using evolu-
tionary algorithms for the generation of feedback connections in neural
networks. Proc. of Comp. Finance 2004, 2004.
[3] Giles, C.L., Lawrence, S. & Tsoi, A.C., Noisy time series prediction
using a recurrent neural network and grammatical inference. Machine
Learning, 44(1/2), pp. 161–183, 2001.
[4] Tino, P., Schittenkopf, C. & Dor?ner, G., Financial volatility trading
using recurrent neural networks. IEEE-NN, 12, pp. 865–874, 2001.
[5] Herbrich, R., Keilbach, M., Graepel, T., Bollmann-Sdorra, P. & Ober-
mayer, K., Neural networks in economics: Background, applications and
new developments. Advances in Computational Economics, 11, pp. 169–
196, 1999.
[6] Yao, X., Evolving arti?cial neural networks. Proc of the IEEE, 87, pp.
1423–1447, 1999.
[7] B¨ack, T., Fogel, D. & Michalewicz, Z., Handbook of Evolutionary Com-
putation. Institute of Physics Publishing and Oxford University, 1997.
[8] Ormerud, P., Extracting information from noisy time series data. Tech-
nical report, Volterra Consulting Ltd, 2004.
Paper IV
Optimization of the investment strategy for a
combined hydrogen and hythane refueling station
Submitted to
International Journal of Hydrogen Energy.
Optimization of the investment strategy for a combined hydrogen and
hythane refueling station
Peter Forsberg
1
, Magnus Karlstr¨ om
2
1. Department of Applied Mechanics
Chalmers University of Technology, 412 96 G¨ oteborg, Sweden
e-mail: [email protected]
2. ETC Battery and FuelCells Sweden AB
Box 2055, 449 11 Nol , Sweden
e-mail: [email protected]
Abstract
One of the major barriers to the widespread use of hydrogen
is the lack of a hydrogen infrastructure, an important com-
ponent of which is the individual hydrogen refueling station.
The long-term pro?tability of the hydrogen ?lling station is
a key issue for the success of the transition to a hydrogen in-
frastructure.
The topic of this paper is the problemof ?nding the optimal
investment strategy for a single hydrogen and hythane refu-
eling station giving minimum production cost, while match-
ing the hythane and hydrogen capacity to a demand generated
from three stochastic scenarios over a 20-year period.
A minimal resulting production cost between USD 2-6/kg
for hydrogen and USD 1-1.5/kg for hythane (depending on
preferences concerning unsatis?ed demand, ?exibility etc.)
was found. It was also found that the production cost and
the amount of unsatis?ed demand constitute con?icting ob-
jectives so that, for example, if the total hydrogen and hythane
demand is to be satis?ed, the production cost of hydrogen
will be unrealistically high. The effect of uncertainties for the
constructed scenarios is minimized by the use of stochastic
optimization techniques.
Keywords: Hydrogen; Hythane; Infrastructure; Invest-
ment; Optimization; Refueling station
1 Introduction
Hydrogen is a promising fuel for vehicles. Four main argu-
ments support this assertion: (1) The potential of reducing
greenhouse gases from the transport sector; (2) An increase
in energy supply security, since hydrogen can be produced
from many energy sources so that the risk of a shortage of
supply may be reduced; (3) Hydrogen has higher energy ef-
?ciency than do other fuels; (4) The use of hydrogen leads
to the possibility of zero local emissions with the use of fuel
cells [1, 2, 3]. The magnitude of the bene?ts of hydrogen fuel
cell vehicles has been assessed by Karlstr¨ om [4] and Sanden
and Karlstr¨ om [5].
However, these bene?ts can only be exploited if several
barriers to a large-scale introduction of hydrogen fuel cell ve-
hicles are reduced. One of the major barriers is the lack of a
hydrogen infrastructure [6]. The construction of a full-scale
hydrogen infrastructure with production facilities, a distribu-
tion network, and refueling stations is likely to be very costly.
The venture of constructing a hydrogen refueling infrastruc-
ture constitutes a long-term, capital intensive investment with
great market uncertainties for fuel cell vehicles. Therefore,
reducing the ?nancial risk is a major objective of any long-
term goal to build a hydrogen infrastructure [7].
Ogden [8] has described several hydrogen supply options.
Investigations have also been made for large scale production
of hydrogen [9, 10, 11]. Many studies of cost and technology
for a hydrogen infrastructure and for individual stations have
also been carried out [12, 13, 14, 15, 16, 17, 18, 19]. The
H2A analysis group at the US department of energy (DOE)
has recently developed two H2A delivery models: the H2A
Delivery Components Model and the H2A Delivery Scenario
Model [20].
The above studies mainly consider hydrogen as a fuel from
an environmental and economic standpoint in a large scale
perspective. By contrast, Forsberg and Karlstr¨ om [21] in-
vestigated the most pro?table investment strategy for the in-
dividual hydrogen refueling station featuring on-site small-
scale reforming of methane [21]. The question was when,
and to what extent, to build the parts of the station, satisfying
an increasing demand of hydrogen. The result was a mini-
mum hydrogen production cost of 4-6 USD/kg, depending on
the number of re-investments during the 20-year-period con-
sidered. This paper is a continuation and an improvement
of [21]. The improvements explored are mainly:
1. The earlier study relied on single-objective optimization
of the production cost and hence produced one optimal
investment strategy. In this study, multi-objective op-
timization is used for ?nding Pareto optimal fronts for
contradictory objectives. For decision-making it is more
favourable to have a set of solutions, each represent-
ing different possible investment alternatives. Preferably
1
these should lie on the pareto-optimal front of important
objectives.
2. The sensitivity analysis in the previous study showed
that the results were quite sensitive to variations in the
strategic parameters and in particular to the estimated
number of refueling vehicles (represnted by the S-curve,
i.e. a curve indicating the estimated technology adapta-
tion, see below). In this study we present three scenar-
ios with considerably different future developments. For
each scenario we generate a large number of samples and
make use of stochastic optimization to ?nd the best so-
lutions.
3. Hythane, i.e. natural gas mixed with a small fraction of
hydrogen, is a viable intermediate alternative fuel for ve-
hicles. Whereas the earlier study only considered hydro-
gen, in this study both hydrogen and hythane are taken
into account.
4. In reality equipment is available in a (?nite) number of
sizes. For simplicity, the earlier study made use of con-
tinuously sized equipment. By contrast, this study ap-
plies presently available sizes of equipment, thus achiev-
ing a higher degree of realism. In addition, the costs of
equipment, electricity, and natural gas have been updated
to current (2006) values.
The most important issues in this study are to reduce the ef-
fect of uncertainties for scenario parameters and to identify
connections between production cost and other results. The
calculations cover 20 years, from 2010 until 2030. If an in-
vestment is made, it takes place at the very beginning of the
year, i.e. an investment is in year 1 occurs on the 1
st
of Janu-
ary 2010.
2 Strategic parameters
The strategic parameters in?uence how the generated future
scenarios are calculated and are therefore crucial to the re-
sults. These parameters and their respective values are given
in Table 1.
The number of produced units of reformers, electrolysis
etc. is considered to equal the number of hydrogen refueling
stations, which is estimated to reach 5,000 in the year 2010
and 50,000 in the year 2030, and to follow the S-curve
R(t) =
1
1 + e
?B(t?Tx)
, (1)
in between. t is the time from year 2010, T
x
the S-curve in-
?ection point and B the slope. The estimation concerning the
growth of hydrogen fuel cell vehicles is assumed. The growth
of number of stations is based upon the estimated growth of
hydrogen fuel cell vehicles and their hydrogen demand. A
report presented by E4tech [14] and funded by the UK De-
partment of Trade and Industry and the Carbon Trust predicts
that ”if the hurdles are overcome, the mainstream propulsion
market is expected to open up after 2010”. Melaina [24] made
a preliminary investigation concerning the suf?cient number
Table 1: Strategic parameters. The electricity price is as-
sumed to be higher during daytime (6 am-10 pm) than at night
(10 pm-6 am). All parameter values are estimates except nat-
ural gas price, which is from [22] and electricity price, which
is from [23]
Name Description Value Unit
B S-curve slope 0.3 -
D Real rate of interest 0.1 1/year
Fcont Contingency cost factor 0.1 -
Feng Engineering permitting cost factor 0.1 -
Fgen Include land cost factor 0.2 -
F
h2y
Mass ratio hydrogen in hythane 0.03
N Number of time steps 175 200 -
Pe Electricity price vector (6am-10pm) 0.10 USD/kWh
Electricity price vector (10pm-6am) 0.08 USD/kWh
Png Natural gas price 0.97 USD/kg
t0 Start time of calculations (2010) 0
t
f
End time of calculations (2030) 175 200
Tx In?ection point of the S-curve 10 year
W Scenario sample - -
X
hf
Hydrogen demand (from scenario) - kg/h
X
yf
hythane demand (from scenario) - kg/h
of initial hydrogen stations in the US, and concluded that be-
tween 4,500 and 17,700 hydrogen stations would be required
to initiate a hydrogen infrastructure for fuel cell vehicles. The
estimate of 50,000 hydrogen stations in 2030 used here is mo-
tivated by the fact that this investigation takes the whole world
into account. Also, in this study, the market reaches a high
level of maturity, further motivating the estimate for the num-
ber of stations in 2030.
The decrease in purchase price, in relation to the present-
day purchase price, owing to increased production and tech-
nology development (f
t
) for a given type of equipment eq is
approximated as
f
t,eq
(t) =
(50000R(t))
(log fp,eq/ log 2)
5000
(log fp,eq/ log 2)
(2)
= (10R(t))
(log(fp,eq)/ log(2))
.
Here, f
p,eq
is a progress factor for the equipment in ques-
tion, i.e. a factor that determines the purchase cost decay
rate for the speci?ed equipment. Equation (2) is used for all
equipment parts of the refueling station, regardless of size. It
should be kept in mind that the function f
t,eq
is purely exoge-
nous and therefore uncertain. This uncertainty will in?uence
the results, as is discussed in Section 6.
Using the present day value correction factor, (C
p
), future
costs can be discounted to present day value as
C
p
(t) =
1
(1 + D)
t/8760
, (3)
where t is the number of hours from the start of calculation,
t
0
, and D is the real interest rate. Furthermore, the consecu-
tive present day value correction vector is de?ned as
C = [C
p
(1) C
p
(2) . . . C
p
(N)] . (4)
The average of the components of this column vector, i.e.
C(t) =
1
N
N
t=1
C
p
(t) (5)
can be used to calculate the present value of evenly distributed
costs. For D = 0.1, C(t) = 0.4466.
2
2.1 Scenario generation
The number of vehicles visiting the single refueling station is
a stochastic variable which is estimated in three scenarios. In
these scenarios, the following vehicles are considered:
1. Ordinary combustion engine powered buses running on
hythane.
2. Ordinary combustion engine powered cars running on
hythane.
3. Ordinary combustion engine powered buses running on
hydrogen.
4. Ordinary combustion engine powered cars running on
hydrogen.
5. Fuel cell driven buses running on hydrogen.
6. Fuel cell driven cars running on hydrogen.
7. Fuel cell driven scooters running on hydrogen.
The ?rst four vehicles represent intermediate solutions, used
until the fuel cell driven alternatives have become dominant.
The above vehicles are considered to have ?lling data and sta-
tistics in accordance with Table 2.
Using these data, three possible future scenarios are given
in Table 3. The ?rst scenario emphasizes hythane and hydro-
gen powered buses as an intermediate alternative. The second
scenario focuses on hydrogen cars, primarily with combustion
engines early on, and fuel cells toward the end of the period
considered. In the third scenario hydrogen fuel cell powered
scooters are in focus. In all three scenarios hydrogen fuel cell
cars are used in the longer perspective.
For interpolation between the three time periods speci?ed
in Table 3, the S-curve has been used, giving the smooth curve
shown in Figure 1. The smoothness obtained through the in-
terpolation is likely to be valid for the car purchases of groups
of individuals, but may, of course, be violated e.g. in the case
of large corporations that may acquire several vehicles (such
as buses) at the same time.
For each scenario a set of samples (W) is generated using
Poisson distributions with parameters from Tables 2 and 3.
With a time step length of one hour, the total number of steps
is N = 24×365×20 = 175, 200 for each sample. For the sce-
nario generation, hydrogen ?lling is separated from hythane
?lling. The resulting hydrogen and hythane demand is de-
noted X
hf
and X
yf
, respectively.
3 The refueling station
The task of the refueling station is to provide fuel for hydro-
gen and hythane vehicles. As the main energy carrier, natural
gas is chosen. One reason is the already present natural gas
refueling network. Moreover, natural gas is one of the cheap-
est production sources for hydrogen in the short term. The
hythane dispenser part of the refueling station is an interme-
diate alternative on the path to hydrogen vehicles.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0
5
10
15
20
25
30
35
40
45
Scenario1
Year
N
u
m
b
e
r
o
f
v
e
h
i
c
l
e
s
Hytan ic cars
Hytan ic buses
Hydro fc cars
Figure 1: Number of vehicles in scenario 1 as a function of
time. Data based on S-function-smoothened values from Ta-
ble 3.
After reforming, the produced hydrogen gas is compressed
and stored. The high storage pressure then makes it possi-
ble to refuel hydrogen without further compression. Some
of the hydrogen can be used for mixing with natural gas to
form hythane, which is refueled using a special hythane dis-
penser. The mixture of hydrogen in hythane is here set to 3%
by weight. When electricity prices are low (i.e. at night), it
might be more pro?table to produce hydrogen by electrolysis
than through the reformer. For this option an electrolyser can
be added. All in all, this is a ?exible layout capable of sim-
ulating many types of possible future hythane and hydrogen
refueling stations, see Figure 2. The model is also ?exible
with respect to refueling station types, e.g. car, truck, or bus,
and refueling station locations e.g. central, suburban, or coun-
tryside, by changing the strategic parameters.
The model developed and the optimisations made involve
the components within the refueling station. Components out-
side the station are considered to be already present.
3.1 The parts of the refueling station
Table 4 gives data on the parts of the refueling station from
Figure 2. Data are taken from actual produced equipment in
the year 2000 [25].
The reformer, electrolysis, and compressor are chosen from
a ?nite set of available sizes, while the H
2
store as well as
the hythane and H
2
dispensers are purchased on a piece-wise
basis depending on the required capacity. Some equipment
cannot be used below a certain minimum level, which is indi-
cated by the minimum usage rate (f
u
) and given as a fraction
of maximum capacity. For the reformer, the minimum usage
rate equals the minimum capacity. For the H
2
store, the min-
imum usage rate indicates the minimum storage level. Below
this minimum storage level the pressure drops too low to be
dispensed to vehicles. For sizes different from the nominal
capacity (c
n
), the purchase price is calculated using the scale
3
Table 2: Filling statistics for vehicles visiting the single refueling station. ic denotes internal combustion engine and fc denotes
fuel cell powered engine. ?T denotes the time between ?llings, and T
d
denotes the time of day at which ?lling takes place. The
numbers are estimates.
Vehicle type hythane [kg/?lling] H2 [kg/?lling] ?T [days] T
d
[h]
hythane ic bus 61 0 1 5-8
hythane ic car 6 0 3 1-24
Hydrogen ic bus 0 60 1 5-8
Hydrogen ic car 0 6 3 1-24
Hydrogen fc scooter 0 2 5 1-24
Hydrogen fc bus 0 40 1 5-8
Hydrogen fc car 0 5 5 1-24
Table 3: Number of vehicles visiting the single refueling station for each scenario. ic denotes internal combustion engine and fc
denotes fuel cell powered engine. Figures are based on assumptions of different future scenarios for the introduction of hydrogen
vehicles.
Time span Scenario 1 Scenario 2 Scenario 3
Year 1-5 10 hythane ic buses, 40 hythane ic cars 10 H2 ic cars, 2 H2 ic buses 30 hydro fc scooters
Year 5-10 20 hythane ic buses, 10 H2 fc cars 10 H2 ic cars, 2 H2 ic buses, 20 H2 fc cars 30 H2 fc scooters, 20 H2 fc cars
Year 10-20 40 H2 fc cars 40 H2 fc cars 40 H2 fc cars
Table 4: Data on the refueling station parts. Electricity use is the amount of electrical energy consumed for each kg of output for
the piece of equipment in question. ng denotes natural gas and pc number of pieces. Data on the hythane dispenser are estimated
from data on the hydrogen dispenser. Figures are from [25].
Part Reformer Electrolysis Compressor H
2
store Hythane dispenser H
2
dispenser
Life time (l) 10 year 20 year 10 year 20 year 10 year 10 year
Nom. capacity (cn) 4.2 kg/h 62.5 kg/h 4.2 kg/h 21 kg 96 kg/h 48 kg/h
Nom. purchase cost (pn) 100,000 USD h/kg 34,632 USD h/kg 12,143 USD h/kg 22,500 USD/pc 120,000 USD/pc 60,000 USD/pc
Scale factor (fs) 0.75 0.72 0.8 - - -
Available sizes 4.2, 12.5, 62.5 kg/h 4.20, 12.5, 62.5 kg/h 5, 15, 75 kg/h 21 kg 96 kg/h 48 kg/h
Maintenance cost (fm) 0.07 0.07 0.05 0.01 0.035 0.035
Ef?ciency (?) 0.26 kg H
2
/kg NG 0.02 kg H
2
/kWh 1.0 1.0 1.0 1.0
Electricity use (fe) 1.5 kWh/kg - 2.2 kWh/kg 0.0 kWh/kg 0.0 kWh/kg 0.0 kWh/kg
Progress ratio (fp) 0.9 0.9 0.9 0.9 0.9 0.90
Min. usage rate (fu) 0.25 0.0 0.0 0.56 0.0 0.0
4
Figure 2: Hythane and hydrogen refueling station layout.
Natural gas is reformed to hydrogen on-site and stored for
delivery to vehicles. It is also possible to produce hydrogen
from electricity by electrolysis. In this study, only the parts
within the refueling station are considered.
function
p
eq
(s) = p
n
s
c
n
s
1?fs
= p
n
c
1?fs
n
s
fs
. (6)
Here f
s
is a scale factor (see Section 2). Using this function
an estimated purchase price is obtained for a piece of equip-
ment of arbitrary size (s [kg/h]), using the purchase price p
n
for an existing piece of equipment, with capacity c
n
. The
function (6) applies to the reformer, electrolysis, and com-
pressor. The expected life time l is used to reduce the in-
vestment cost, should the investment period (2010-2030) end
before the end of life of the piece of equipment in question.
The reduction in purchase cost is approximated by a linear
function (in time). The ef?ciencies in Table 4 indicate the re-
lation between the mass entering and leaving the equipment.
In the case of the reformer, the substance entering is methane
and that leaving is hydrogen.
3.2 Initial considerations
The hydrogen/hythane refueling station is assumed to be built
in conjunction with an existing natural gas refueling station.
The supply of natural gas can be delivered by truck or, more
commonly, by pipeline. In any case, the supply is consid-
ered to be already established and only the cost for the pur-
chase of natural gas is taken into account. All costs for the
hythane/hydrogen part of the station, i.e. land use and wages,
are accounted for. In reality some resources can probably be
shared between the natural gas and hythane/hydrogen parts of
the refueling station. Initially, the hydrogen storage is consid-
ered to be empty.
3.3 The model
The model for the refueling station in Figure 2 only has one
state variable, i.e. a variable to be integrated, which is the
amount of stored hydrogen, x
hs
. Once the stored amount of
hydrogen is known, all other relevant quantities can be cal-
culated directly. Letting k denote the time step, one can ex-
press x
hs
in the form of the difference equation x
hs
(k +1) =
x
hs
(k) + x
o
hc
? x
o
hs
, where x
o
hc
is the amount leaving the
compressor (see Figure 2), which equals the amount enter-
ing the storage, and x
o
hs
the amount leaving the storage. The
compressor output comes from the reformer x
o
hr
and elec-
trolysis x
o
he
and taking the compressor ef?ciency ?
hc
into
account one can write x
o
hc
= (x
o
hr
+ x
o
he
)?
hc
. In order to
determine the natural gas consumption of the reformer x
i
hr
,
the equation x
o
hr
= x
i
hr
?
hr
is added, where ?
hr
is the ef-
?ciency of the reformer. The amount leaving the hydrogen
storage is the sum of dispensed hydrogen x
o
hd
and the hy-
drogen part F
h2y
of dispensed hythane x
yd
. Taking the dis-
penser’s ef?ciencies ?
hd
and ?
yd
into account, this amount
is obtained as x
o
hs
= x
o
hd
/?
hd
+ x
o
yd
F
h2ng
/?
yd
. Now the
total natural gas consumption x
ng
from both the hydrogen
x
i
hr
and hythane x
o
yd
(1 ? F
h2y
) part can be calculated as
x
ng
= x
i
hr
+ x
o
yd
(1 ? F
h2y
)/?
yd
. Thus, in summary, the
following equation system is obtained
x
hs
(k + 1) = x
hs
(k) + x
o
hc
?x
o
hs
,
x
o
hc
= (x
o
hr
+ x
o
he
)?
hc
,
x
o
hr
= x
i
hr
?
hr
,
x
o
hs
=
x
o
hd
?
hd
+
x
o
yd
F
h2ng
?
yd
,
x
ng
= x
i
hr
+
x
o
yd
(1 ?F
h2y
)
?
yd
. (7)
However, these equations are subject to some constraints.
First of all, there are minimum and maximum levels both for
the ?ow and for the amount stored. Second, the amount of
dispensed hydrogen and hythane, x
o
hd
and x
o
yd
, must be non-
negative and are limited from above by the scenario sample
demand (X
hf
and X
yf
, respectively). Note that the demand
is not necessarily totally satis?ed. All in all, the following
constraint equations are obtained
f
u,hr
s
hr
? x
o
hr
? c
hr
,
0 ? x
o
he
? c
he
,
0 ? x
o
hc
? c
hc
,
f
u,hs
s
hs
? x
hs
? c
hs
,
0 ? x
o
hd
? c
hd
,
0 ? x
o
yd
? c
yd
,
0 ? x
o
hd
? X
hf
,
0 ? x
o
yd
? X
yf
, (8)
where c
h
e denotes the hydrogen electrolysis capacity, c
hc
the compressor capacity, c
hs
the storage total capacity, c
hd
and c
yd
the hydrogen and hythane dispenser capacity, respec-
tively. Note that, for the reformer and storage, the minimum
utilization level is higher than zero. This is due to the fact that
the hydrogen dispenser cannot be run below a certain mini-
mum ?ow rate, given as a ratio f
u,hr
of the maximum capac-
ity s
hr
, giving f
u,hr
s
hr
as the minimum allowed ?ow rate.
For the storage, the hydrogen gas pressure falls below accept-
able limits for the dispenser if the stored amount is less than
5
the ratio f
u,hs
of the total capacity s
hs
, thus making f
u,hs
s
hs
the minimum amount to be stored.
For the optimization, the state variable is the hydrogen stor-
age (x
hs
), the control variables are the outputs of reformer
and electrolysis (x
o
hr
and x
o
he
, respectively), and the distur-
bances are the stochastic variables hydrogen and hythane de-
mand (X
hf
and X
yf
, respectively).
4 The optimization problem
The optimization problem under consideration can be for-
mulated as a discrete-time stochastic optimal-control prob-
lem [26, 27, 28]. In this type of problem the aim is to ?nd
the control U that minimises an objective function J(U) for a
dynamical system f(X, U, W) during a speci?ed time, in the
discrete case indexed by the time step variable k. The system
is also in?uenced by an independent random disturbance W.
The general formulation is
min
U
J(U) =
N?1
k=0
?(k, X
k
, U
k
, W
k
) + ?(X
N
, W
N
)
s.t. X
k+1
= f(k, X
k
, U
k
, W
k
) (9)
c
k
(X, U) ? 0 ?k = 1, . . . N,
where ?(k, X
k
, U
k
, W
k
) is the cost associated with each time
step k, ?(X
N
, W
N
) is the terminal cost and c
k
(X, U) rep-
resents simple limits of the state and control variables. The
controller makes use of the information set ?
k
, the contents
of which depend on the type of control system. For an open-
loop system, ?
k
= {X
0
} ?k, whereas for a feedback system,
?
k
= {X
0
, X
k
}, k = 0, 1, . . . , N ? 1. For a closed-loop
system, ?
k
= {X
0
, X
1
, . . . , X
k
, U
0
, U
1
, . . . , U
k?1
}. In this
study, the investment strategy is set prior to t
0
and then fol-
lowed until t
f
, thus de?ning an open-loop control system as
described above.
The problem is to ?nd the optimal investment strategy ?
?
that will subsequently minimise two objective functions, fur-
ther discussed in Section 4.1. An inner control loop is used
to keep the hydrogen storage level at a given amount. In this
loop, a control algorithm is implemented to keep the hydro-
gen storage at a speci?ed level. Since the time constants of
both reformer and electrolysis are very short (of the order of
minutes) compared to the time step (one hour), the desired
control action will be considered to take effect immediately.
Due to the rapid dynamics of both reformer and electrolysis,
these devices can be shut down fast and, therefore, there is
no need to keep the storage below 100% as a precaution to
avoid over?ow due to slow production adaptability for un-
expectedly low levels of demand. The control algorithm ?rst
calculates the deviation in hydrogen storage fromthe set point
(the error), then ?lls up the storage with available hydrogen,
which is the sum of reformer maximumcapacity and electrol-
ysis during the period when electricity is cheaper, i.e. 10pm -
6am. No electrolysis is used during the remaining expensive
hours. The calculated amount is then added to the storage.
Depending on the demand priority policy ?, either an attempt
is made to satisfy ?rst the hydrogen refueling demand and
then the hydrogen part of hythane, or vice versa. Tuning this
policy will have a signi?cant in?uence on the amount of un-
satis?ed demand. The total control vector is then
U = [? ?]. (10)
The inner control loop does not have any tunable parameters
and is thus not part of the optimal control problem. It is im-
plemented as an obvious optimal solution to keep the size of
the variable space at a minimum.
Direct control parameter mapping methods, i.e. methods
that will need dedicated control parameters for each step, are
not used. Such techniques are intractable due to the large
number (N = 175, 200) of steps involved.
No terminal cost ?(X
N
, W
N
) is used. Instead the total in-
vestment cost (for the entire life time) is scaled linearly, in
accordance to the usage time of the piece of equipment, see
Eq. (13) below. The system equation and the simple con-
straints have been given in Eqs. (7) and (8). The random
disturbance W is the hythane and hydrogen demand, further
described in Section 2.1 above.
4.1 Objective functions
In this study, the following performance measures are used
1. Production cost per kg for hydrogen p
hf
. This is the
production cost for hydrogen at the hydrogen dispenser
and is calculated as the sum of all hydrogen related costs
divided by the total amount of sold hydrogen x
o
hd
.
2. Unsatis?ed demand for hydrogen x
h,u
. This is the de-
mand that cannot be satis?ed at the hydrogen dispenser
and is a negative measure, i.e. a low amount of unsatis-
?ed demand is desirable.
3. Production cost per kg for hythane p
yf
. This is the pro-
duction cost for hythane at the hydrogen dispenser and is
calculated as the sum of all hythane-related costs divided
by the total amount of sold hythane x
o
yd
.
4. Unsatis?ed demand for hythane x
y,u
. This is the demand
that cannot be satis?ed at the hythane dispenser and is a
negative measure, i.e. a low amount of unsatis?ed de-
mand is desirable.
5. Unsatis?ed demand for all hydrogen x
ht,u
. The total
amount of hydrogen demand that cannot be satis?ed,
i.e. the sum of unsatis?ed demand at the hydrogen dis-
penser x
h,u
and as part of hythane F
h2y
at the hythane
dispenser x
y,u
.
6. Flexibility p
h?
. A measure used for quantifying the dif-
ference between the cost for the active scenario, i.e. the
scenario for which the present solutions have been op-
timized, and the passive ones, i.e. the scenarios that are
not part of the optimization.
For the optimization, the production cost per kg for hydro-
gen x
h,u
and the unsatis?ed demand for all hydrogen x
ht,u
are used in the objective function J which then takes the form
J(U) = [p
hf
x
ht,u
]. (11)
6
In order to compute the above performance measures, a num-
ber of costs and ?ows need to be calculated. Before carrying
out the calculation, however, an assumption is made (and ap-
plied to all calculations below) that each part of the refueling
stations takes on its own expenses. This implies that expenses
from the hythane part will be added to the hythane produc-
tion cost and the same for the hydrogen part. Parts used in
both hythane and hydrogen production, such as storage, are
charged according to the usage ratio
f
hpc
=
x
hf
(x
hf
+ F
h2y
x
yf
)
. (12)
For each part (eq) of the refueling station where the pur-
chase cost is scaled to the used size, i.e. reformer, electrolysis
and compressor, the cost is computed as
p
eq
(t
i
, s
eq
) = f
t,eq
(t
i
) p
eq
(s
eq
) max(
20 ?t
i
l
eq
, 1), (13)
where l
eq
is the estimated lifetime of the part in question, see
Table 4. For those items purchased on a piece-wise basis,
i.e. storage tanks and dispensers, the cost is computed as
p
eq
(t
i
, n
eq
) = f
t,eq
(t
i
) p
n
n
eq
max(
20 ?t
i
l
eq
, 1). (14)
This implies a linear scaling of the total investment cost for
the entire estimated life time to the time it is actually used.
The total cost consists of costs for purchase of equipment,
resources, maintenance and a factor to cover for construction,
land use and general expenses. It is estimated that the cost for
loans for equipment will cancel the effect of the present value
correction for the sum of instalments. This is exactly the case
of annuity loans. All other costs are discounted to present day
using equation (3).
The cost for equipment shared by the hythane and hydro-
gen parts is then
p
c,eq
=
?i
(p
hr
(t
i
, s
hr,i
) + p
he
(t
i
, s
he,i
) +
p
hc
(t
i
, s
hc,i
) + p
hs
(t
i
, n
hs,i
)). (15)
The cost for maintenance is estimated to a speci?ed fraction
(f
m
) of the equipment cost, that is
p
c,m
=
?
?
?
?
?
1
1
.
.
.
1
?
?
?
?
?
?i
f
m,hr
p
hr
(t
i
, s
hr
)
8760 l
hr
+
f
m,he
p
he
(t
i
, s
he
)
8760 l
he
+
f
m,hc
p
hc
(t
i
, s
hc
)
8760 l
hc
+
f
m,hs
p
hs
(t
i
, n
hs
)
8760 l
hs
, (16)
which is an N × 1 column vector. The factor 8760 is used
to convert the equipment life time from years to hours. The
electricity cost is calculated from equipment ?ows as
p
c,e
= P
e
?(f
e,hr
x
o
hr
+ f
e,he
x
o
he
+ f
e,hc
x
o
hc
+ f
e,hs
x
hs
) (17)
where ?is the element-wise multiplication operator. The nat-
ural gas cost is
p
c,mg
= P
ng
x
i
hr
. (18)
The total cost for the shared parts is then
p
c
= (1 + F
cont
+ F
eng
+ F
gen
) p
c,eq
+
pds ×(p
c,m
+ p
c,e
+ p
c,mg
). (19)
For the hydrogen and hythane speci?c parts, calculation of
costs follows the same pattern as in Eqs. (15)-(19) above, and
is therefore not listed here.
The resulting production cost per kg of hydrogen p
hf
and
hythane p
yf
is now
p
hf
=
p
c
f
hpc
+ p
h
x
hf
(20)
and
p
yf
=
p
c
(1 ?f
hpc
) + p
y
x
yf
, (21)
respectively.
It is clear from Eq. (8) that not all the demand from the
scenario samples need be satis?ed. The difference between
the demand and the actual sold amount is called unsatis?ed
demand. Three measures of unsatis?ed demand are used,
namely (1) hydrogen dispenser unsatis?ed demand
x
h,u
= X
hf
?x
o
hd
, (22)
(2) hythane dispenser unsatis?ed demand
x
y,u
= X
yf
?x
o
yd
, (23)
and (3) total unsatis?ed hydrogen demand from both the hy-
drogen dispenser and the hydrogen part of the hythane at the
hythane dispenser
x
ht,u
= x
h,u
+ F
h2y
x
y,u
. (24)
The variance of the above objectives p
hf
, p
yf
, x
h,u
and x
y,u
between samples is used as a measure of sensitivity, which
can also be interpreted as risk. A high variance would imply
a higher risk. Flexibility is a complex measure that can be
de?ned in a number of ways [29]. Here, it has been de?ned
as
p
h?
= p
a
h
?
p
p1
h
+ p
p2
h
2
, (25)
i.e. as the mean difference between the hydrogen production
cost for the active scenario and the passive ones. A positive
value indicates a lower cost for the passive scenarios and vice
versa.
4.2 Optimization strategy
The stochastic control problem (9) is solved with a
simulation-based optimization technique [30]. For each can-
didate solution U to the problem, the refueling station is eval-
uated in a number of discrete simulations under the stochas-
tic in?uences from samples generated from scenarios. When
all samples have been evaluated, the performance measures
are estimated and the solutions are tuned accordingly. In this
study a genetic algorithm (GA) has been used to optimise the
7
Figure 3: A schematic illustration of the optimization frame-
work.
solutions. GAs are optimization algorithms inspired by bio-
logical evolution. Such algorithms can easily be adapted to
a wide range of optimization problems [31], including multi-
dimensional problems [32]. The optimization algorithm used
in this study is an elitist non-dominated sorting GA, called
NSGA-II [33], which uses an explicit diversity-preserving
mechanism. For each Pareto-optimal front, this algorithmwill
remove solutions lying close to each other, while preserving
those far from each other. The result is a good spread of solu-
tions along the front. However, the longer the front, the more
solutions are needed to get a good picture of the details of
the curve. Since each solution corresponds to one individual
in the GA, more solutions means a larger population which
takes longer time to evolve.
The major parts in the optimization framework is the sce-
nario generator, simulator and optimiser, see Figure 3. The
steps of the evaluation are
1. A sequence of samples are generated for each scenario
in the scenario generator. The probability distributions
used are discussed in Section 2.1.
2. A number of initial individuals (candidate solutions, U)
are randomly generated.
3. For each individual (candidate solution, U), the simula-
tor simulates the refueling station (Section 3.3) over the
entire investment period. The simulator also contains a
control algorithm for proportional control of the amount
of hydrogen stored.
4. When all samples for all scenarios have been simulated,
the simulator estimates the performance measures and
objectives for the individual.
5. When all individuals in the population have been evalu-
ated, the front and objective distance (crowding) sorting
is carried out, followed by a generational replacement
with crossover and mutation.
6. Steps 3-5 are repeated until convergence.
0 1 2 3 4 5 6 7 8 9 10 11
0
0.5
1
1.5
2
2.5
3
3.5
x 10
5
Scenario 1
Hydrogen production cost [USD/kg]
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
k
g
]
1
2 3
4
6
10
20
35
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
%
o
f
t
o
t
a
l
]
0
10
20
30
40
50
60
70
80
90
100
Figure 4: The resulting Pareto-front when optimising U for
scenario 1.
5 Results
This section presents the results from the optimization for the
three scenarios discussed in Section 2.1. For each scenario the
optimization has been carried out using the objective function
de?ned in Eq. (11).
5.1 Scenario 1: hythane combustion engine
buses
Scenario 1 emphasizes hythane and hydrogen powered buses
as an intermediate alternative, after which fuel cell powered
cars take over. The hydrogen cost versus total hydrogen un-
satis?ed demand from both hydrogen and hythane dispenser
can be seen in Figure 4, in which several interesting solutions
(discussed below) are marked with their respective numbers.
It is evident that as the cost decreases, the unsatis?ed demand
increases and therefore that these objectives are in con?ict
with each other. The discontinuities represent stepwise in-
creases of capacity to the next available size as de?ned in Ta-
ble 4.
As can be seen in Figure 4, Solutions 1 and 2 represent
extreme points regarding the two optimization objectives.
For Solution 1, the production cost is at a minimum, 1.96
USD/kg. In this strategy, an investment in a small electrolysis
equipment is made in year 18. The strategy results in a large
amount of unsatis?ed demand, 1.4×10
5
kg (87%of the total)
and 6.0 × 10
6
(100% of the total) for hydrogen and hythane,
respectively. By contrast, in Solution 2 where the production
cost is at its maximum (10.5 USD/kg), investments are made
in years 0, 5, and 10. With this strategy, the unsatis?ed de-
mand is negligible for both hydrogen and hythane. Further-
more, the strategy exhibits a preference for reformer use in
the beginning of the period, and electrolysis towards the end.
Intermediate solutions include number 3, 10 and 35 in Fig-
ure 4. These solutions represent extreme points before the
next possible equipment size is used. If unsatis?ed demand is
to be kept at minimum, Solution 3 may be a good alternative.
If hydrogen production cost is to be kept low, while still not
8
Table 5: Investment strategy for solution 3 for scenario 1.
Investment no 1 at year 1 2 at year 11
Reformer 4.2 kg/h 0.0 kg/h
Electrolysis 0.0 kg/h 12.5 kg/h
Compressor 5.0 kg/h 15.0 kg/h
H2 store 84 kg 147 kg
H2 dispenser 48 kg/h 48 kg/h
Hythane dispenser 864 kg/h 864 kg/h
Investment cost 1.5 × 10
6
USD 9.5 × 10
5
USD
Prio. strategy Hydrogen
Total inv. cost 2.4 × 10
6
USD
Maint. cost 5.1 × 10
4
USD
6.02 6.03 6.04 6.05 6.06 6.07 6.08 6.09 6.1 6.11
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Histogram for investment strategy 3 in scenario 1, calculated for scenario 1.
Hydrogen production cost [USD/kg]
P
r
o
b
a
b
i
l
i
t
y
Figure 5: Histogram for hydrogen production cost for solu-
tion 3.
allowing a large amount of unsatis?ed amount of hydrogen,
Solution 10 may be considered. The details of the investment
strategy, including the investment cost, can be seen in Table 5.
If the hydrogen production cost is calculated for all sam-
ples, a cost distribution is generated. Figure 5 shows a his-
togram of the cost distribution for solution 3. The distrib-
ution shows only a relatively small variance. A calculation
of the cost for all solutions indicates a decrease in relative
variance (details omitted), i.e. var(x)/x, for lower production
cost. Since a high variance indicates a high uncertainty, the
variance does not con?ict with the hydrogen production cost.
This dependence is typical for all scenarios.
In Figure 6, which shows hydrogen versus hythane produc-
tion cost, a strong non-linear correlation with a minimum for
Solution 35 can be noticed. The non-linearity is most evident
for the region left of this minimum. These points correspond
to solutions to the left of Solution 35 in Figure 4. The lowest
hydrogen production cost, represented by Solution 1, utilizes
no hythane and is therefore not shown in Figure 6.
Each solution and sample corresponds to one trajectory of
the state variable x
hs
. Given the state variable and the refu-
eling ?ows X
hf
and X
hf
, all other ?ows can be calculated
from Eqs. (7). When these ?ows are calculated for Solution
3, a good utilisation of equipment is found for most of the 100
samples, and this is veri?ed by the small variance in produc-
tion cost in Figure 5.
If the optimal solutions for Scenario 1 are calculated using
0 1 2 3 4 5 6 7 8 9 10 11
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Scenario 1
Hydrogen production cost [USD/kg]
H
y
t
a
n
p
r
o
d
u
c
t
i
o
n
c
o
s
t
[
U
S
D
/
k
g
]
2
3
4
6 10
20
35
Figure 6: Hydrogen cost versus hythane cost, Scenario 1.
0 1 2 3 4 5 6 7 8 9 10 11
?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
Scenario 1
Hydrogen production cost [USD/kg]
H
y
d
r
o
g
e
n
f
l
e
x
i
b
i
l
i
t
y
[
U
S
D
/
k
g
]
1
2 3
4
6
10
20
35
Figure 7: Hydrogen cost versus hydrogen ?exibility for Sce-
nario 1.
Scenarios 2 and 3, it is found that hydrogen production cost
will most likely increase. The ?exibility index in Figure 7
shows that some solutions result in considerably higher hy-
drogen production cost, e.g. Solution 6, while others do not,
e.g. Solution 1. Note that a positive value indicates a lower
cost for the passive scenarios and vice versa. The solutions
are no longer part of the Pareto-optimal front since they orig-
inate from the solution for Scenario 1. However, the original
extreme Solutions 1 and 2 will still be the extreme ones.
5.2 Scenario 2: Hydrogen combustion engine
cars
The second scenario focuses on hydrogen cars, with primarily
combustion engines in the beginning and fuel cells towards
the end of the 20-year time period. No hythane is used in this
scenario.
In essence, the production cost versus hydrogen unsatis?ed
demand curve resembles that of Scenario 1. Figure 8 shows
that almost zero unsatis?ed demand can be maintained down
to a production cost of around $6/kg, belowwhich the amount
9
0 1 2 3 4 5 6 7 8 9 10 11 12 12
0
1
2
3
4
5
6
7
8
9
x 10
5
Scenario 2
Hydrogen production cost [USD/kg]
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
k
g
]
1
2
41
49
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
%
o
f
t
o
t
a
l
]
0
10
20
30
40
50
60
70
80
90
100
Figure 8: The resulting Pareto-front when optimising U for
Scenario 2.
0 1 2 3 4 5 6 7 8 9
0
2
4
6
8
10
12
14
16
18
x 10
4
Scenario 3
Hydrogen production cost [USD/kg]
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
k
g
]
1
2 21
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
%
o
f
t
o
t
a
l
]
0
10
20
30
40
50
60
70
80
90
Figure 9: The resulting Pareto-front when optimising U for
Scenario 3.
of unsatis?ed demand starts to rise.
The ?exibility of the solutions for scenario 2 is in general
lower than for other solutions, which is evident when the pas-
sive Scenarios 1 and 3 are used. The production cost for these
passive scenarios is at least double that obtained when the ac-
tive Scenario 2 is used.
5.3 Scenario 3: Hydrogen fuel cell cars
In the third scenario, hydrogen fuel cell powered scooters are
in focus in the beginning, and later fuel cell driven cars. No
hythane is used in this scenario.
The production cost versus unsatis?ed demand curve in
Figure 9 reveals a slightly more expensive production than
in the previous cases. This is even more obvious when the
solutions for Scenario 3 are applied to Scenario 1 and 2.
6 Discussion and conclusion
In this paper, it has been demonstrated that it is possible to use
stochastic optimization in order to ?nd investment strategies
for a combined hydrogen and hythane on-site reformer refu-
eling station. The resulting cost of hydrogen and hythane are
2-6 USD/kg and 1-1.5 USD/kg respectively, depending on the
preferences concerning unsatis?ed demand, ?exibility etc.
The results from this study can be used as decision support
when planning combined hydrogen and hythane refueling sta-
tions. Not only the production cost and unsatis?ed demand
for the present scenario are important, but also the ?exibil-
ity of the solution to unforeseen events and developments. In
addition, there are other performance measures such as vari-
ance and the comparison between hydrogen and hythane that
should be taken into account. The selected solution is a matter
of preference.
The problem of ?nding investment strategies involves a
considerable amount of information, and therefore aggregate
measures have been de?ned for ?exibility.
As observed from the connection between production cost
and unsatis?ed demand for all scenarios, these two measures
are in con?ict. One reason for this is the stochastic demand
curve which makes it unrealistic to achieve zero unsatis?ed
demand. This is so since, occasionally, a larger amount of
vehicles will come to the station than it can serve, which is
probably close to what would be observed in reality. Other
reasons for this con?ict in measures are the technology de-
velopment reduction in purchase price (see Eq. (2)) and the
discounted costs (see Eq. (3)). Since it is likely that it will be
cheaper to build and run the refueling station in the future, the
optimization tends to prefer future solutions to present ones.
An evenly distributed cost will be discounted to 0.4466 of the
original value, so the discount effect is not negligible.
Improvements can be made to the scenario data in Sec-
tion 2.1. It may be unrealistic having all buses refuelling in
the morning. Instead, a slow ?lling during night time might
be considered. Also the 24-hour car refueling curve, which
assumes a constant ?lling frequency throughout the day, may
be adjusted to a more realistic setting.
A key to successful investment planning is the minimiza-
tion of the uncertainty of future developments. For this rea-
son, three different future scenarios, with 100 samples each,
have been used. Within each scenario the uncertainties are
kept at minimum given the strategic parameters, by taking
all samples into account. The strategic parameters can eas-
ily be changed for other cases. For each solution, the effects
are easy to quantify should another scenario become reality.
However, the S-curve (Eq. (1)) has still been used for estimat-
ing the (uncertain) number of produced units. These numbers
are taken in a global perspective, which may make them less
sensitive.
In this study, linear scaling of equipment cost to usage time
is used. Another option, often used in the literature [26],
would be to use salvage value or terminal cost. Given that the
salvage value can be de?ned arbitrarily, these two approaches
can be considered identical.
In the optimisations presented here, the investment strat-
egy is set prior to the calculations. Another option would be
10
to de?ne control policies that use system information for de-
cisions, i.e. a fully closed-loop. Such a strategy would be
able to incorporate not only investments but also a quantita-
tive demand satisfaction calculation. At present, the only real
feedback loop has been implemented for the hydrogen storage
level.
An increase in the number of solutions will give better res-
olution for the Pareto curve. On the other hand, a larger pop-
ulation will be needed in the GA, which in turn will increase
the calculation time. At present, simulation of one scenario
sample takes about 0.25 s, giving a total of 75 s per individual
and hence 1 h 40 min. for a population of 80 individuals. By
aggregating calculated measures and coding more of the algo-
rithm in a low-level language, the simulation time can proba-
bly be shortened considerably. If this is done, the resolution
can be enhanced.
It should also be noted that, as for all heuristic methods,
convergence cannot be guaranteed. Instead, one has to settle
for a solution which is good enough and preferably better than
any other known solution.
To conclude, it has been found that it is possible to op-
timise the hydrogen production cost for a combined hydro-
gen and hythane refueling station, and that the resulting costs
lies between 2-6 USD/kg for hydrogen and 1-1.5 USD/kg for
hythane. The production cost and the amount of unsatis?ed
demand constitute con?icting objectives so that, for exam-
ple, if the total hydrogen and hythane demand is to be sat-
is?ed, the production cost of hydrogen will be unrealistically
high. However, an intermediate realistic solution can be found
along the curve of cost versus unsatis?ed demand. In all
cases, the lowest production cost for hydrogen and hythane
is achieved by satisfying the hydrogen demand ?rst and then
the hythane demand.
Acknowledgments
Financial support from the Competence Center for Environ-
mental Assessment of Product and Materials Systems (CPM)
at Chalmers University of Technology is gratefully acknowl-
edged.
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12
doc_692605009.pdf
Planning involves the creation and maintenance of a plan. As such, planning is a fundamental property of intelligent behavior. This thought process is essential to the creation and refinement of a plan, or integration of planning it with other plans; that is, it combines forecasting of developments with the preparation of scenarios of how to react to them.
THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
OPTIMISATION OF
LONG-TERM INDUSTRIAL PLANNING
PETER FORSBERG
Department of Applied Mechanics
CHALMERS UNIVERSITY OF TECHNOLOGY
G¨ oteborg, Sweden 2006
Optimisation of Long-Term Industrial Planning
PETER FORSBERG
ISBN 91-7291-863-2
c PETER FORSBERG, 2006
Doktorsavhandlingar vid Chalmers tekniska h¨ ogskola
Ny serie 2544
ISSN 0346-718X
Department of Applied Mechanics
CHALMERS UNIVERSITY OF TECHNOLOGY
SE-412 96 G¨ oteborg
Sweden
Telephone +46 (0)31-772 1000
Printed in Sweden by
Chalmers Reproservice
G¨ oteborg, 2006
Till Flisan och Smulan
Abstract
In this thesis, long-term optimisation methods for industrial transition processes
have been developed, taking monetary and environmental considerations into ac-
count. Two different methods for investment optimisation have been developed.
First, an optimisation method comprising simultaneous calculation of the long-
term investment strategy and the short-term utilisation scheme for a deterministic
demand was developed. The method has been applied to the case of ?nding an
investment strategy for minimising the production cost for a single hydrogen refu-
elling station. The problem was shown to be convex; thus the resulting solution is
the global optimum. Second, an investment optimisation method using stochastic
demand scenarios and multi-objective optimal control to produce the Pareto front
of the two con?icting objectives expected production cost and expected unsatis-
?ed demand was developed. This method was applied to the case of ?nding the
optimal investment strategy for a combined hydrogen and hythane refuelling sta-
tion. Depending on the preferences of the decision-maker, many different feasible
solutions can be found. However, it was also found that, due to the uncertainty
of the stochastic demand function, satisfying all the estimated demands would re-
quire a production capacity well above the mean demand, which would be very
costly to maintain.
In addition to the two methods for investment optimisation, a modelling ap-
proach for systems combining economic and environmental aspects has been de-
veloped as well. This approach has been used for modelling cement production
facilities, taking both economic and environmental issues into consideration.
In order to deal with prediction uncertainties, time series prediction using ge-
netic algorithms was investigated as well. Discrete-time prediction networks, a
novel type of recurrent neural networks, were introduced, and were shown to pro-
vide one-step macro-economic time series prediction with greater accuracy than
several other methods.
Keywords: Transition strategy optimisation, Investment strategies, Multi-objective
decision making, Optimisation under uncertainty.
i
Acknowledgements
First I would like to thank my supervisor Mattias Wahde. At my moment of
despair, he took on the responsibility to be my supervisor, after which things got
considerably better.
I also would like to thank Competence Centre for Environmental Assessment
of Product and Materials Systems (CPM) at Chalmers University of Technology
for their ?nancial support of the project.
Magnus Karlstr¨ om and Karin G¨ abel have provided me with intriguing (and
real!) problems that needed solutions as well as a thorough understanding of
these problems as well as being co-authors for a number of papers, for which I
am grateful. If it was not for Raul Carlson and Anne-Marie Tillman, I would
de?nitely not have started my Ph.D. studies in the ?rst place. Even though fate
wanted me to continue elsewhere, I am thankful to them. I also want to thank
all my colleagues at the Department of Applied Mechanics for all the enjoyable
discussions and support.
Thank you Eva, So?a and Linn´ ea for support and love, even when I’ve been
absorbed by work.
Peter Forsberg
G¨ oteborg, 2006
iii
Contents
1 Introduction 1
1.1 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Transition processes 5
2.1 Economic and environmental aspects . . . . . . . . . . . . . . . . 7
3 Optimisation techniques 13
3.1 Deterministic optimisation techniques . . . . . . . . . . . . . . . 14
3.2 Stochastic optimisation techniques . . . . . . . . . . . . . . . . . 18
3.3 Dynamical optimisation . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Stochastic dynamical optimisation . . . . . . . . . . . . . . . . . 24
4 Assessing the future 27
4.1 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Decision-making under uncertainty . . . . . . . . . . . . . . . . . 32
5 Case studies 35
5.1 The cement production case . . . . . . . . . . . . . . . . . . . . 35
5.2 The hydrogen infrastructure case . . . . . . . . . . . . . . . . . . 40
5.3 The hythane infrastructure case . . . . . . . . . . . . . . . . . . . 44
6 Concluding remarks 51
6.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7 Summary of appended papers 55
7.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
v
vi CONTENTS
7.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
APPENDED PAPERS
List of publications
The work presented is based on the following publications, which are included in
the thesis.
I. Karin G¨ abel, Peter Forsberg and Ann-Marie Tillman, The design and build-
ing of a lifecycle-based process model for simulating environmental perfor-
mance, product performance and cost in cement manufacturing, Journal of
Cleaner Production, Volume 12, Issue 1, February 2004, pp. 77-93.
II. Peter Forsberg and Magnus Karlstr¨ om, On optimal investment strategies for
a hydrogen refueling station, International Journal of Hydrogen Energy, In
press, corrected proof available online 25 July 2006.
III. Peter Forsberg and Mattias Wahde, Macroeconomic and ?nancial time se-
ries prediction using networks and evolutionary algorithms, Proceedings of
Computational Finance 2006, London, 27-29 June 2006, pp. 403-411.
IV. Peter Forsberg and Magnus Karlstr¨ om, Optimization of the investment strat-
egy for a combined hydrogen and hythane refueling station, submitted to
International Journal of Hydrogen Energy.
The author has also contributed to research in the following related subjects.
V. Peter Forsberg, Modelling and Simulation in LCA, CPM Technical Report
2000:1, 2000.
VI. Wim Dewulf, Raul Carlson,
?
Asa Ander, Peter Forsberg and Joust Du?ou,
Integrating Pro-Active Support in Ecodesign of Railway Vehicles, Proceed-
ings of 7th CIRP Seminar on Life Cycle Engineering, Tokyo, 27-29 Nov.
2000, pp 111-118.
VII. Raul Carlson, Peter Forsberg, Wim Dewulf and Lennart Karlsson, A full
design for environment (DfE) data model, Proceedings of Product Data
Technology, Brussels, 25-26 April 2001, pp 129-135.
vii
viii List of publications
VIII. Raul Carlson, Maria Erixon, Peter Forsberg and Ann-Christin P? alsson, Sys-
tem for Integrated Business Environmental Information Management, Ad-
vances in Environmental Research, 5 2001, pp 369-375.
IX. Wim Dewulf, Joost Du?ou, Raul Carlson, Peter Forsberg, Lennart Karls-
son., Dag Ravemark,
?
Asa Ander and Gerold Spykman, Information Man-
agement of Rail Vehicle Design for Environment for the entire Product Life
Cycle, Proceedings of 1st International Conference on Life Cycle Manage-
ment, LCM 2001, Copenhagen, 27-29 August 2001, pp 69-72.
Nomenclature
Allele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, the possible settings for a gene.
Auto-regression, AR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
In time series prediction, using a linear combination
of past values to calculate the predicted future value.
Chromosome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
In a GA, a string of genes representing a potential so-
lution to a problem.
Convex curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
A curve that is bulging outward over its total exten-
sion.
Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, exchange of genetic material between two
individuals.
Crowding distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
The mean distance to the neighbouring solution for a
multi-objective optimisation problem.
Dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
A dynamic optimisation method that makes use of
Bellman’s principle of optimality to solve the prob-
lem by backward induction.
Dynamical optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
An optimisation problem de?ned over a time period.
Dynamical problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
A problem de?ned over time and containing time-
continuous or discrete dynamic parts.
ix
x Nomenclature
Feasible point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
A point satisfying all constraints for an optimisation
problem.
Flow semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
The connection between, and use of, the general vari-
ables intensity and ?ow..
Functional unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
In LCA, a reference to which the inputs and outputs
of a product system are related as parts of the normal-
isation.
Gene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, the smallest part of a chromosome.
Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, the procedure of evaluating the individuals in
a population and replacing the population by its off-
spring.
Genetic algorithm (GA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Genetic algorithm. A stochastic optimisation method
inspired by natural evolution.
Hydrogen reformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A device to produce hydrogen from hydrocarbons,
e.g. methane gas.
Hythane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Natural gas with a small ratio of hydrogen.
Individual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
In a GA, a member of a population, carrying one chro-
mosome.
LCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Life cycle assessment. A systematic method to as-
sess the environmental impact of a product or function
produced. Also called Life cycle analysis.
LCI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Life cycle inventory analysis. Quantifying the rele-
vant products, resources used and emissions released
for the entire life cycle of a product.
Nomenclature xi
Markov decision process (MDP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
A process where the decisions taken at a certain point
only depend on the state at the previous point in time,
and not states further back in time.
Moving average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
In time series prediction, to calculate the predicted fu-
ture value by averaging a number of past values.
Multi-objective optimisation problem (MOOP) . . . . . . . . . . . . . . . . . . . . . . . . 18
An optimisation problem having more than one ob-
jective function, see also Pareto front.
Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, the operation of probabilistically changing
one gene at random.
Normalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
In LCA, relating the ?ows in a product system to the
functional unit.
Objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
The function to be optimised in an optimisation prob-
lem.
Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
See Dynamical optimisation.
Pareto front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
A curve of points such that, for points on the curve, no
criterion can become better without making another
criterion worse.
Reference ?ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
A measure of the needed outputs from processes in a
given product system required to ful?ll the function
expressed by the functional unit.
Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
In stochastic optimisation, one outcome of the distur-
bance generated from a scenario.
Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
In a GA, selection of individuals for crossover.
xii Nomenclature
Stochastic dynamic optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A dynamic optimisation method for systems under the
in?uence of a stochastic disturbance.
Stochastic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
An optimisation method for solving stochastic dynam-
ical optimisation problems.
Time series prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
The process of predicting future values in a time se-
ries using past data.
Unit process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
In LCA, the smallest part of a product system for
which data is collected when undertaking an LCA.
Chapter 1
Introduction
This thesis focuses on optimisation of industrial transition processes. A transi-
tion process can involve change of equipment, re-location of premises or some
other structural change in, for example, societal infrastructure. These processes
often involve large investments that are implemented over a long time, and in a
situation of uncertain future development. Under such circumstances, ?nding the
optimal investment strategy is not an easy task. A common approach in invest-
ment planning [1, 2] is to list a number of alternatives, and then to pick the best
one by hand or to use a rule-of-thumb technique, a procedure that, in the case of
highly complex systems, most often results in a sub-optimal solution being found.
A better approach for ?nding the optimal solution is to make use of mathematical
optimisation techniques such as dynamical optimisation to search for the optimal
investment strategy; Such approaches are considered in this thesis.
The research objective in this thesis is to develop methods for optimisation of
industrial transition processes with monetary and environmental considerations.
In doing so, three areas are investigated: (1) modelling of production systems,
(2) prediction of future behaviour using TSP and (3) optimisation of investment
strategies using optimal control.
In the ?eld of economics, investment planning is an important topic [3, 4].
Of highest interest is then, of course, to ?nd the optimal investment strategy.
This problem, which can be solved using optimal control theory, is in general
de?ned over some period of time. In other, related applications, optimal con-
trol theory is used for, e.g. maximising growth in national economics [5], ?nding
optimal investments in funds [6], production planning [7], optimisation of se-
quential investments [8], and maximising return on capital funds [9]. Under the
in?uence of a stochastic disturbance, here in the form of an uncertain future devel-
opment, the optimisation problem becomes a stochastic dynamic optimisation
problem. In economics, this type of problem is referred to as investment under
uncertainty [10, 11, 12] and in process engineering as process design under un-
1
2 CHAPTER 1. INTRODUCTION
certainty [13, 14]. In Paper IV, stochastic optimal control theory is used to ?nd
the lowest expected production cost for a combined hydrogen and hythane refu-
elling station under the in?uence of three stochastic demand scenarios. In both the
above cases, discussed in Papers II and IV, the developed methods are intended to
be used for decision support.
Recent studies have been made regarding the economical feasibility of hydro-
gen in regard to the infrastructure that must be built [15, 16, 17, 18, 19, 20, 21].
However, none of these studies investigates the implications of investments over
time. By contrast, in Paper II optimal control theory is used to ?nd a short-term
equipment variable utilization for one-week periods and, at the same time, a long-
term investment strategy for the whole investment period covering 20 years with
the aim of minimising the production cost for a plant. The method is exempli?ed
by a hydrogen dispensing infrastructure case.
Predicting future values of key variables is, of course, highly relevant in the op-
timisation of transition processes. Even short-term prediction is important. Such
prediction is considered in the ?eld of time series prediction (TSP). Due to the
often high level of noise present, standard procedures, such as the auto-regressive
and moving-average methods, are sometimes not fully successful [22, 23, 24]. In-
stead other, more adaptive methods based, for example, on neural networks can
be used [25, 26]. In Paper III of this thesis, a novel type of neural network is
developed for prediction of noisy time series.
When evaluating transition processes the economic consequences are impor-
tant. The environmental awareness in today’s society is constantly raising the
requirements of a cleaner production process. Thus, in this thesis, the environ-
mental effects are considered as well. At the same time the increasing complexity
of the production systems makes the environmental analysis more dif?cult to carry
out. In the 1990s more advanced methods were developed to assist environmen-
tal analysis of technical systems. One of these methods is life cycle assessment
(LCA). Much has been written about LCA. The ISO standards 14040-42 [27] give
very general guidelines on how an LCA should be performed. In fact, many stan-
dard papers on LCA, e.g. [28, 29, 30, 31, 32, 33], approach the topic in a rather
non-mathematical way. Until 1998 only one paper [34] was published regarding
guidelines on how to carry out the actual calculation, the so called normalisa-
tion. After 1998, the subject of normalisation in LCA has been considered in a
mathematical point-of-view by Heijungs [35, 36, 37]. All of the mathematical
methods presented by Heijungs only consider the standard LCA which includes
a linear and static model representation. For other approaches there is only a
limited number of texts available. Examples include linear optimisation of LCA
systems [38, 39], multi-objective optimisation [40] and dynamic life cycle inven-
tory models [41, 42]. Some cases with integration of economic cost objectives
have also emerged [43, 44]. There is still, however, a large potential for improve-
1.1. MAIN CONTRIBUTIONS 3
ments concerning, for example, the range of applicability of the models. This
thesis investigates a number of approaches, how they can be used, and possible
improvements. The ?ndings are exempli?ed by a cement production case [45]
in Paper I and are intended to be used in investment optimisations, such as those
presented in Papers II and IV.
1.1 Main contributions
Transition processes taking place in the societal infrastructure and in large in-
dustries are in general very complex systems. This is due to the fact that these
systems do not only have technical and economic aspects, but also social, envi-
ronmental, political and geographic aspects. To take all these considerations into
account when constructing a model is, of course, impossible. This thesis focuses
on economic and environmental aspects and aims at providing some examples of
general tools for carrying out structural transition optimisation. It is the aim of the
author that, when the tools presented here are used together, the effort of carrying
out the above-mentioned type of optimisations should be reduced considerably.
Mathematical models are generally speci?c to the application at hand. Using
the type of models discussed in this thesis, the ?exibility with regard to the types
of calculations that are possible to carry out can be increased considerably [46,
47, 48]. These models come from the study of physical systems [49], but can be
successfully applied to other types of systems, e.g. environmental systems [50].
The aim of the models is to provide the optimisation algorithm with the effects
of changes in the future strategy. In doing so, the model has to provide the future
behaviour of key parameters. Some of these can be modelled in detail but others,
where exact knowledge is lacking, must be predicted. One way of achieving this is
by time series prediction [23, 51, 52]. In this way the short-term future behaviour
can be estimated, something that is important for fast dynamics. Using the above-
mentioned tools, the transition strategy is then optimised using stochastic multi-
objective optimisation. To summarise, the main contributions of the presented
work are:
• A modelling approach for production systems with environmental measures
comprising separation of model and problem formulation, and leading to
more ?exible models (Paper I).
• Progress in time series prediction (TSP) using genetic algorithms (GAs)
resulting in increased accuracy for predictions of noisy time series (Paper
III).
• Methods for concurrent optimisation of investment strategies and run pat-
terns for long-term planning of industrial production sites, taking economic
4 CHAPTER 1. INTRODUCTION
and environmental considerations into account. In Paper II the model is
of the single-objective deterministic kind and in Paper IV of the multi-
objective stochastic kind.
• Speci?c results regarding optimal investments for hydrogen and hythane
refuelling stations (Papers II and IV).
The author was the main contributor to Papers II and IV. In Paper I, the author’s
contributions were to develop the modelling approach and the model framework,
to carry out all calculations needed for solving the problem, and to write a signif-
icant part of the paper. In Paper III both authors contributed equally.
Chapter 2
Transition processes
Structural transition processes occur in all industries. Examples include change
of machinery, moving of production units and change of production at current
sites. For decision support, a number of calculations are carried out regarding
the economic consequences and, at times, optimisations are done [53]. However,
only rarely are both economic and environmental aspects taken into account. This
thesis presents some methods for integrating economic and environmental aspects
when assessing industrial transition processes. The results are intended to be used
for decision support.
In particular, the problem of optimising investment strategies, i.e. selecting
when and to what extent investments are to be made for maximum performance,
is explored. In connection with this problem, the topic of predictability of vari-
ables has also been considered within the framework of time series prediction. In
most cases in this thesis the connections to economy are explicit, i.e. economic
measures appear in the model, and the environmental connections are implicit,
i.e. they are present through the use of an environmentally favourable technique. It
should be kept in mind that this implies a constraint in the sense that only environ-
mentally favourable techniques are considered to be part of the set of acceptable
solutions. However, the developed methods, such as the multi-objective optimi-
sation procedure described in Paper IV, have originally been designed for use in
cases with explicit connections between environmental and economic aspects.
When considering large industrial structures, the economic consequences are
distributed over a number of years. For a complete production line in a factory or
for a major societal infrastructure change, the consequences might be distributed
over a period exceeding 20 years [54, 55]. To be able to carry out an optimisation
of the economic results, the variation in a number of variables, e.g. the rate of
interest, the technical development within the ?eld and the yearly production and
demand, must be estimated. In order to improve the accuracy of such variables, a
study of time series prediction (TSP) has been undertaken in Paper III, aimed at
5
6 CHAPTER 2. TRANSITION PROCESSES
improving short-term (one-step) prediction of macroeconomic time series, where
the time step length often equals one year or a quarter of a year. However, long-
term predictions are much harder, and it is generally not possible to make optimi-
sations of investment strategies that will be valid for the whole investment period
(20 years, say). Instead, a dynamic approach must be taken to the optimisation
of transition processes, such that, when the assumed values of a parameter have
deviated substantially from the expected path, a new calculation is carried out,
based on the new, corrected behaviour of the parameter in question. This makes it
possible always to have the best and most up-to-date structural transition strategy
at hand.
Life cycle assessment (LCA) [31, 32, 33, 34] is a way to quantify environ-
mental in?uences of a product (or service) over its entire life. In the study pre-
sented in Paper I, similarities and differences between LCA and technical system
theory [56]
1
were investigated in order to make improvements in combining eco-
nomic and environmental modelling. Incorporating the new ?ndings, a model of a
cement production process was generated and calculations for improving the eco-
nomic and environmental performance were performed. The variables involved
express aspects of quality and economy, as well as resource use and emissions.
The model developed was intended for many types of calculations regarding,
e.g. economy, product quality and emissions.
Changes toward more environmentally favourable solutions frequently incor-
porate large investments in infrastructure. The cost and uncertainty of changing
these facilities are usually considered to be obstacles for the introduction of new
techniques. In Papers II and IV, methods for ?nding optimal investment strategies
for this type of environmentally favourable production facilities are investigated.
Based on an assumed future development scenario, optimal investment strategies
are calculated. In Paper IV, special emphasis is put on reducing the uncertainties
by using several different scenarios and stochastic optimisation. The applicability
of the developed method is exempli?ed in two studies on pro?table investment
strategies for a hydrogen station and a combined hydrogen and hythane refuelling
station.
Finding the optimal solution to the set of problems discussed above is far from
trivial. Nevertheless, in reality, many such problems are solved in an intuitive
manner based on experience [57, 58], in a large part due to the fact that the mathe-
matical formulation of the problems usually is very hard to ?nd for these complex
systems [59, 60]. A complex system here signi?es a system composed of a num-
ber of simpler subsystems with a large, often huge, number of interconnections.
Due to the large number of connections, these systems are usually very hard to
1
The term technical system theory is here used for the science of constructing models of
processes etc. as is done in control theory.
2.1. ECONOMIC AND ENVIRONMENTAL ASPECTS 7
understand and model. One example of this is emergence [61, 62, 63]. Emergent
properties are properties possessed by a system, which cannot be traced back to
any of its parts. A good explanation of emergence is given in [64] as ”A complex
system usually involves a large number of components. These components may be
simple, both in terms of their internal characteristics and in the way they interact.
Still, when the system is observed over longer time and length scales, there may
be phenomena that are not easily understood in terms of the simple components
and their interactions”.
Optimisation of complex systems can therefore play a very important role, by
revealing unforeseen solutions better than those currently available. One should
keep in mind though, that accurate optimisation over periods extending 20 years
into the future is not easy, even using techniques that reduce the effects of uncer-
tainty such as stochastic optimisation with multiple scenarios. When predictions
are generated using models built from time series data, like the DTPNs in Paper
III, an implicit assumption is that the future development follows a pattern similar
to that of the past data. In this case an unforeseen major event can disrupt all pre-
dictions instantaneously. Another option is to make prediction based on a detailed
model of the system combined with probable future developments, as is done in
Paper IV. However, even in this case, an unforeseen event not covered by the
model can disrupt all predictions. Since nothing in nature is discontinuous there
are, however, always precursor events to major events. To identify these events is,
of course, very important and therefore successful optimisation requires extensive
knowledge of the system under study. Ideally all possible future outcomes should
be included in the model. In reality this is impossible though, and one therefore
has to settle for less-than-perfect models.
2.1 Economic and environmental aspects
The objectives explored in this thesis are mostly related to economy. Therefore
this section starts with a discussion of the economic measures used in Papers II
and IV, after which some important aspects of one possible quanti?cation of en-
vironmental impact is presented, namely LCA. Both economic and environmen-
tal measures are considered, since it is desirable to ?nd environmentally viable
solutions that are still economically favourable. No company can support non-
pro?table environmental sustainability.
In the literature, future costs and incomes are usually discounted with regard
to the discount rate D > 0 using the net present value correction [65]
C
p
(t) =
1
(1 + D)
t
, (2.1)
8 CHAPTER 2. TRANSITION PROCESSES
where t is the time. The above equation re?ects investors’ preference of immedi-
ate return of cash in contrast to future returns. The actual discounting depends on
the length of time and the discount rate. Usually, the discount rate is the risk-free
interest rate added to an interest rate re?ecting the risk involved in the speci?c ven-
ture. As the name implies, the risk-free interest rate is the safe rate earned from a
completely risk-free investment. When optimising policies that span a long time,
as the calculations in Papers II and IV do, the value of the discount rate can have
a signi?cant effect. As is pointed out in Paper IV, for a discount rate of 0.1 and in
the case of an evenly distributed cost, the effective discounting is C(t) = 0.4466
for a time period of 20 years. C(t) is the mean value of C
p
(t) over the time period
considered. The high discount rate in Paper IV is motivated by the high risk; the
investments are made in a new technique with uncertain potential and acceptance.
In both Paper II and Paper IV, the loan for purchase of equipment is assumed
to be of the annuity type. In this case, it is possible to calculate a capital cost per
time unit for the refuelling station. The additional costs, e.g. for purchase of raw
materials and maintenance, are then added and a total instantaneous production
cost at time t per produced unit can be calculated, which is measured in USD per
kg H
2
.
In order to ?nd the mean production cost for the entire investment period,
one must integrate the instantaneous production cost. In Paper II, three ways of
integrating this cost are shown: (1) adding the costs at all times without discount-
ing, (2) discounting future costs using Eq. (2.1) above and (3) discounting future
costs using Eq. (2.1) and distributing the total cost evenly over the whole invest-
ment period. One should keep in mind that the third option does not re?ect the
sum of the real cost to the production facility. The discussed mean production
costs represent different ways of calculating the costs for production, depending
of preference. They are all candidates for an objective function that can be used
for optimisation. It should be noted that if the total capital cost is discounted to
present day value using the same interest rate as for an assumed annuity loan, the
result is the absolute investment cost. Since, in Paper II, only one week per in-
vestment is explicitly evaluated, the above ways of calculating the costs are used,
for simplicity. Details on how production costs have been calculated are given in
Paper II.
In Paper IV, the optimisation is carried out over the whole investment period
covering 20 years. Since loans are considered to be of the annuity type, the costs
for investments are not discounted. Investments in equipment are only subject to
a decreased cost due to increased production and technology development. This
is due to the fact that the effect of the annuity loan and the discounting will can-
cel each other out, provided that the discount rate is the same. In Paper IV, the
production cost was calculated over the entire investment period, taking the total
non-discounted purchase cost into account. This is in contrast to Paper II where
2.1. ECONOMIC AND ENVIRONMENTAL ASPECTS 9
the production cost was calculated for one week per investment, taking the non-
discounted cost for the annuity loan into account, a procedure that results in a
time-varying production cost, as can be seen in Paper II. Apart from investments,
all other production costs were discounted, however. The mean production costs
were calculated as the sum of costs divided by the sum of sold hydrogen and
hythane, respectively.
Environmental effects are often less tangible than are economic ones and
therefore harder to measure and quantify. One reason is that in the short perspec-
tive many environmental aspects tend to have little or no environmental effect.
Instead, at a certain level, there is an abrupt and sometimes unforeseen effect.
Another reason is that the causality of environmental effects is not always totally
understood. One example is the Greenhouse effect. Obviously the global mean
temperature is increasing in the short-term perspective, but is this caused by hu-
man activities? Economic aspects, on the other hand, tend to have a more direct
and immediate effect.
However, one way environmental in?uences can be quanti?ed is by LCA. The
life cycle usually starts with extraction of raw materials and continues with trans-
portation, manufacturing, use and possibly re-use. It then ends with waste man-
agement, recycling and disposal. There exists a vast literature on the concept of
LCA, see e.g. [30, 34, 66, 67, 68].
In 1997 the ISO standard 14040 on LCA was approved [27]. In this standard
LCA is de?ned as ‘...the environmental aspects and potential impact through-
out a product’s life (i.e. cradle-to-grave) from raw material acquisition through
production, use and disposal.’ Each of these stages consumes resources and pro-
duces emissions and waste. In LCA all these aspects are taken into account and
are related to the product or function produced. LCA further aims at assessing the
impact of the production on nature.
In the life cycle inventory analysis (LCI), which is one part of LCA, the re-
sources, emissions and products related to the production system are measured.
Then a model over the production system is created. In order to know the ef-
fects of each produced unit (in LCA called functional unit), the measured re-
sources etc. are scaled to the functional unit. This is done by aggregating all unit
processes in the product system and scaling the ?ows of these processes to match
the reference ?ow of the system, a process referred to as normalisation. The
data used in the inventory is based on time-averaged statistics and is hence in-
dependent of time. In addition a linear relation between resource use, emissions
and production is assumed. The resulting normalisation step is mathematically
equivalent to solving a linear equation system. The equation system is usually
well-posed by construction, i.e. having equal number of variables and constraints,
and hence possible to solve exactly. Publications on LCA cover handbooks, case
studies and theoretical studies on the concept that do not, in general give any
10 CHAPTER 2. TRANSITION PROCESSES
Figure 2.1: Example of a model for a small glass bottle production process. Note that,
since the data for the different processes (shown as boxes in the ?gure) are taken from
different sources, e.g. product data sheets, the amounts of ?ow do not match at this stage.
The objective of the normalization is to scale the processes so that the ?ows do match.
The results of the normalization procedure are shown in Figure 2.2 below.
directions on how to represent the ?ow model and form the resulting equation
system [33, 69]. Recently a number of publications on the computational part
have appeared [36, 37, 70, 71]. The result of the normalisation to the functional
unit presented there is expressed as
g = BA
?1
f, (2.2)
where A is a matrix describing the internal ?ows of the technical system (the
technology matrix), f the demand vector, i.e. a vector expressing what is being
produced, and B the intervention matrix, i.e. the external ?ows to and from the
technical system.
An example of a model for a small glass bottle production process is shown
in Figure 2.1. In this process sand is melted to glass, producing carbon dioxide
emissions. The glass is then cast to bottles which are scanned for defects before
delivery. Some bottles are discarded in the scanning process due to defects and
these bottles are returned to the casting process after crushing. Each of the above
processes are unit processes. Using the nomenclature introduced above, the tech-
2.1. ECONOMIC AND ENVIRONMENTAL ASPECTS 11
Figure 2.2: The resulting ?ow model for the small glass bottle production process intro-
duced in Figure 2.1.
nology matrix becomes
A =
_
¸
¸
_
92 ?0.2 0 1.9
0 1 ?10 0
0 0 1 ?10
0 0 9 0
_
¸
¸
_
, (2.3)
where the rows in A represent the ?ows of glass (row 1, measured in [kg]), bottles
([pcs], row 2), recycled bottles ([pcs], row 3), and delivered bottles ([pcs], row 4),
respectively. The columns represent the processes indicated by Roman numerals
in Figure 2.1. Note that negative numbers indicate a ?ow into a process, and
positive numbers a ?ow out from a process. The intervention matrix becomes
B =
_
?100 0 0 0
1 0 0 0
_
, (2.4)
where the ?rst row represents sand ([kg]) and the second row represents carbon
dioxide ([kg]). To calculate the intervention for a reference ?ow of one produced
bottle (after the calculation this ?ow will become the functional unit), f is set to
f = [0 0 0 1]
T
giving g = BA
?1
f = [?0.2186 0.0022]
T
. The resulting resource
use and emissions released are then 0.2186 kg sand and 0.0022 kg carbon dioxide,
respectively. The normalised ?ow model is shown in Figure 2.2.
Chapter 3
Optimisation techniques
The transition processes discussed in Chapter 2 are naturally de?ned over some
time period; they are dynamical problems. Industries are, of course, trying to
maximize performance, implying the need for optimisation.
In order to optimise a transition process a dynamical optimisation technique
must be utilized. In most cases, dynamical optimisation problems are solved by
transforming them into static optimisation problems. Therefore, this chapter starts
with a short overviewof unconstrained and constrained non-linear static optimisa-
tion which will lay the foundation for later discussions on dynamical optimisation
techniques in Section 3.3 below. In particular, the SQP algorithm used in Paper II
and the multi-objective genetic algorithm used in Paper IV will be examined. In
Section 3.4 optimisation of systems in?uenced by stochastic perturbations will be
discussed.
The goal of optimisation is to ?nd the optimal point x
?
for a given objective
function f(x), where x = (x
1
, x
2
, . . . , x
N
) ? R
N
. For the single-objective case,
f = f is a scalar, taking values in R
1
, whereas, for the multi-objective case,
f ? R
N
. There may also be equality and inequality constraints of the form
c(x) = 0, (3.1)
d(x) ? 0, (3.2)
as well as limits on the allowed intervals for x, i.e. x
l
? x ? x
u
1
. Despite the
modest appearance of this optimisation problem, ?nding the optimal point is, in
general, a dif?cult task.
A special type of optimisation problem is the convex problem. A function
f(x) : R
N
? R is convex if the domain of the function, denoted domf, is a
1
Throughout this thesis relations between vectors such as x
l
? x require equal dimensions and
are to be interpreted component-wise.
13
14 CHAPTER 3. OPTIMISATION TECHNIQUES
?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3.1: Example of a one-dimensional convex function.
convex set and
f(?x + (1 ??)y) ? ?f(x) + (1 ??)f
(3.3)? x, y ? domf, 0 ? ? ? 1. A non-scienti?c geometrical interpretation is that if
f draws a curve that is bulging outward over its total extension, then it is convex.
An arbitrary example of a one-dimensional convex function is given in Figure 3.1.
A problem is called convex if both the objective function and the constraints are
convex. Such functions are important in optimisation since they make it possi-
ble to guarantee convergence [72]. Furthermore, many non-convex optimisation
problems can be transformed into convex ones [73].
3.1 Deterministic optimisation techniques
The term deterministic is here used to indicate that the optimisation algorithms
do not contain any stochastic parts. Thus, if such an algorithm is run twice, us-
ing the same set of inputs, the results will be identical to each other, i.e. they
are perfectly predictable. In deterministic optimisation there exists robust algo-
rithms with guaranteed convergence for the linear case, i.e. the case in which the
3.1. DETERMINISTIC OPTIMISATION TECHNIQUES 15
objective function and all constraints are linear. One such algorithm is the Sim-
plex method [74]. However, in most practical problems the objective functions
are non-linear. Such functions will usually have both a global optimal point and
many local optimal points. Since it is very dif?cult to distinguish between a lo-
cal and a global optimal point, simple gradient-descent algorithms are usually not
successful. Such algorithms tend to get stuck on a local optimal point instead of
?nding the true global optimal point.
Finding the global optimal point is the major task of non-linear programming
(NLP) [75, 76]. Consider the general NLP problem
min
x?R
N
f(x)
s.t. c(x) = 0 (3.4)
d(x) ? 0
where f(x) is the (scalar, i.e. single-objective) criterion function, c(x) the non-
linear equality constraints and d(x) the non-linear inequality constraints. The
functions f(x), c(x) and d(x) are assumed to be smooth, i.e. at least twice-
continuously differentiable. Let g(x) = ?
x
f(x) denote the gradient vector of
the objective function, C(x) =
?c
?x
the Jacobian matrix of the constraint vector
c(x), and D(x) =
?d
?x
the Jacobian matrix of the constraint vector d(x). Now
de?ne the (scalar-valued) Lagrangian function in the classical way [77]
L(x, ?) = f(x) ??
T
c(x) ?µ
T
d(x), (3.5)
where ? and µ are Lagrange multiplier vectors. In an optimal point the ?rst
derivative of the Lagrangian with respect to x is zero, i.e
?
x
L(x
?
, ?
?
, µ
?
) = g(x
?
) ??
?T
C(x
?
) ?µ
?T
D(x
?
) = 0 (3.6)
where (x
?
, ?
?
, µ
?
) is the optimal point. In addition, requirements have to be put
on the inequality part variables µ and d. At the optimal point, it is clear that an
inequality constraint d
i
(x
?
) can either be satis?ed as an equality, d
i
(x
?
) = 0 or
strictly satis?ed, d
i
(x
?
) > 0. In the former case the constraint is said to be active
and hence a part of the active set A, i.e. i ? A. In the latter case the constraint
is inactive and part of the inactive set A
?
, i.e. i ? A
?
. For the active set the
requirements equal those for equality constraints, i.e. µ ? 0. For the inactive set
the multiplier must be zero. This can also be formulated µ
?T
d(x
?
) = 0, which
is sometimes referred to as the complementary slackness condition. With these
requirements, the Karush-Kuhn-Tucker (KKT) [78] condition for optimality is
de?ned as
g(x
?
) ??
?T
C(x
?
) ?µ
?T
D(x
?
) = 0
µ
?T
d(x
?
) = 0 (3.7)
µ
?
? 0
16 CHAPTER 3. OPTIMISATION TECHNIQUES
and µ is sometimes referred to as the KKT multiplier. In addition the original
constraints from Eq. (3.4), c(x
?
) = 0 and d(x
?
) ? 0 must be satis?ed at the
optimal point. In order to solve the KKT for x
?
, the active inequality constraints
are treated as equality constraints and the inactive ones are ignored, giving
g(x
?
) ??
?T
J(x
?
) = 0,
r(x
?
) = 0, (3.8)
? ? 0,
where r ?
_
x ? R
N
|c(x) = 0, d
i
(x) = 0 ? i ? A
_
and J(x) = ?r/?x. Now
these re-de?ned requirements can be solved with Newton’s method by carrying
out a Taylor series expansion of Eq. (3.8). Letting H
L
= ?
2
xx
L, the expansion
becomes
g(x) ?J
T
(x)? +H
L
(x)(¯ x ?x) ?J
T
(x)( ¯ ? ??) = 0
r(x) +J(x)(¯ x ?x) = 0 (3.9)
which can be written as
_
H
L
J
T
J 0
_ _
?p
¯ ?
_
=
_
g
r
_
. (3.10)
Solving the above equation will yield the step p and the Lagrange multiplier at
the new point, ¯ ?. The new point is then obtained as ¯ x = x + p. Note that in
Eq. (3.10) the new Lagrange multiplier is calculated in an absolute way while, for
the new point ¯ x, the increment p is calculated. The Newton step is then iterated
until convergence.
The Newton method de?ned above is a local optimisation algorithm. In order
to improve the chances of ?nding the global optimum, one may use a globalization
strategy. One example is the line-search method which will adjust the step length
¯ x by a factor ? to ¯ x = x+?p. The value of ? is usually determined by the rate of
progress measured by a merit function [79, 80]. Another option is to adjust both
the magnitude and direction of the search step, so that the search direction p will
lie within a given radius which de?nes a trusted region [81].
A widely used algorithm to solve the above NLP problem in Eq. (3.10), i.e. to
?nd the global optimum, is sequential quadratic programming (SQP), see e.g. [82,
83]. When using SQP, one may observe that Eq. (3.10) represents the ?rst order
optimality conditions for the the optimisation problem
min
p
g
T
p +
1
2
p
T
H
L
p (3.11)
s.t. Jp = ?r,
3.1. DETERMINISTIC OPTIMISATION TECHNIQUES 17
0 2 4 6 8 10
1.8
2
2.2
2.4
2.6
2.8
3
3.2
x 10
5
Scenario 1
Hydrogen production cost [USD/kg]
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
k
g
] 1
5
9
13
17
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
%
o
f
t
o
t
a
l
]
2
4
55
60
65
70
75
80
85
90
Figure 3.2: Illustration of Pareto optimality. The objective is to minimise both the
unsatis?ed hydrogen demand and the hydrogen production cost. Solutions marked with
small dots and crosses are part of the Pareto front whereas the ones marked with circles
are not.
which is a quadratic programming (QP) problem. The SQP is a sequential al-
gorithm that makes use of inner and outer iterations. The objective of the inner
iteration is to ?nd a search direction p which is used in the outer one to ful?ll
the ?rst order conditions for optimality. The search direction p is found by solv-
ing the optimisation problem in Eq. (3.11). The outer iteration makes use of the
new search direction by taking the step ¯ x = x + ?p, where the magnitude of
the step (?) is determined by a line search method. This makes the SQP a global
optimisation algorithm.
In Paper II the resulting NLP optimisation problem was solved using the
NPSOL program [84], which is of the above SQP class. First the NPSOL al-
gorithm aims at calculating a point that is feasible, starting from a user-initiated
point. Then the SQP algorithm described above is used to ?nd the optimal point.
Calculating gradients for the investment problem in Paper II is not easy. One rea-
son is the fact that the objective function f(x) is not differentiable in the whole
of R
N
. Another reason is the complex structure of summations in f(x). Using
NPSOL, no algebraic expressions of gradients and Hessians needs to be given.
Instead, NPSOL can make use of ?nite-difference derivatives. The NPSOL algo-
18 CHAPTER 3. OPTIMISATION TECHNIQUES
rithmcan also deal with minor discontinuities if they are isolated and located away
from the solution. The SQL algorithm described above will converge to the global
optimum for convex problems [72]. Since the investment problem considered in
Paper II is convex, the algorithm will ?nd the global optimal point. The objective
function is discussed in Chapter 2 above and the considered optimisation problem
will be treated in more detail in Chapter 5.
In the above SQP algorithm, the objective function f : R
N
? R. If f :
R
N
? R
M
, M > 1, the problem is a multi-objective optimisation problem
(MOOP). The solution to such a problem does not consist of a single optimal
point, but instead a number of points lying on a curve or surface called the Pareto
front. These points all have in common that they are non-dominated. In short, it
means that no other solution exists where any of the objectives are strictly better
than those on the Pareto front without some other objective being equal or worse.
Figure 3.2, taken from Paper IV, illustrates the principles of Pareto optimality.
The objectives along the x and y axes are both to be minimised. The solutions 1,
13, 9, 5 and 17 all belong to the ?rst Pareto front. Solution 4, however, is said
to be dominated by solution 5 and therefore does not belong to the Pareto front.
Not taking the ?rst Pareto front into account, another, inferior non-dominated
set of solutions can be found. This set is called the second Pareto front. In the
same way a number of Pareto fronts can be found. From Figure 3.2 it is evident
that the two objectives are in con?ict with each other. This is typical for multi-
objective optimisation problems. Should the objectives not be in con?ict, at least
one dimension could be omitted. In the two-dimensional case illustrated above,
this would lead to a single-objective optimisation problem.
3.2 Stochastic optimisation techniques
The search methods employed by stochastic optimisation algorithms depend, in
part, upon computational procedures generating a random outcome. Therefore
these algorithms produce different paths towards the optimal solution each time
they are run, and in the general non-linear case, convergence cannot be guaran-
teed. However, in most practical applications, proof of optimality is not the most
important aspect; Instead, ?nding a solution better than any presently available
one is usually suf?cient.
One of the most widely used stochastic optimisation techniques is the genetic
algorithm (GA) [85, 86, 87]. This algorithm is inspired by biological evolution.
In a GA, a population of candidate solutions to the problem at hand, referred to
as individuals, is maintained. Each individual contains a chromosome that is a
representation of a potential solution to the problem. The chromosome can, for
example, consist of a string of discrete (e.g. binary) or decimal numbers. Mixed
3.2. STOCHASTIC OPTIMISATION TECHNIQUES 19
Figure 3.3: An encoding scheme for a genetic algorithm (GA) in which discrete numbers
are used. An example of a chromosome is shown at the bottom of the ?gure.
representations exist as well, in which the chromosome contains both discrete and
decimal numbers. An example of an encoding scheme used in Paper II can be
seen in Figure 3.3. Each part of the chromosome is called a gene, and each gene
may take different values, referred to as alleles. In this example, only discrete
numbers are used. Since the problem is combinatorial, i.e. consists of a number
of discrete alternative equipment sizes for a refueling station, it is natural to use
a discrete encoding scheme. The problem also contains the time for investment,
which is deliberately encoded as a discrete number in order to decrease the size of
the search space, for faster convergence.
After the initial population is created (usually randomly), all individuals are
evaluated. This involves a calculation of the ?tness measure, which is used to rank
individuals with respect to their performance. Usually, the calculation of ?tness
values is the most time-consuming part of the algorithm. The way the ?tness
measure is calculated depends entirely on the problem at hand. Two examples of
calculating ?tness measures are given in Papers III and IV.
When all individuals have been assigned ?tness values, generational replace-
ment is performed. First, the best individual is transferred unchanged to the next
iteration (referred to as a generation), and then the remaining individuals are cre-
ated from selection, crossover and mutation. The selection, usually of two indi-
viduals, is carried out in proportion to ?tness. The selected two individuals then
generate two new individuals by blending genes from both of them, a procedure
known as crossover. In the simple case the genes are exchanged between chromo-
somes by cutting the latter at a random crossover point. The mutation is a random,
lowprobability change to individual genes in the chromosome. Crossover and mu-
tation must normally be tailored to the speci?c problem. A ?ow chart for a simple
genetic algorithm can be seen in Figure 3.4.
In the problem considered in Paper IV, gradient information was very hard
20 CHAPTER 3. OPTIMISATION TECHNIQUES
Figure 3.4: Flow chart of a genetic algorithm.
to ?nd algebraically. Alternatively, numerical differentiation can be used; such a
technique was employed in connection with the deterministic optimisation carried
out in Paper II. However, in that paper it was shown that for longer optimisation
time periods, the deterministic optimisation algorithm, in which numerical differ-
entiation is an important part, led to unacceptably long computational times and
was therefore intractable. Therefore, in Paper IV, a GA was used instead.
One signi?cant advantage with using a GA is that it does not need any gradient
information. In fact, GAs only need the ?tness measure and can thus be applied
to any optimisation problem where ?tness can be quanti?ed. In addition, GAs do
not introduce any requirements on the function that is to be optimised (for exam-
ple, it need not be convex), but it should be kept in mind that extra precautions
may have to be taken for strongly non-linear and discontinuous functions. One
such technique is to increase the mutation rate at the beginning of the evaluation
in order for the solution not to get stuck on a local optimum but instead ?nd the
global one [85]. In all, the GA is a very ?exible and easy-to-use optimisation al-
gorithm. However, there are some drawbacks. One is the above-mentioned need
for tailoring the encoding scheme, and the operators for crossover and mutation.
Another, perhaps more serious, drawback is that since the search for better solu-
tions is stochastic, so is the evaluation time. Occasionally, it may be better to stop
the current calculation and start from the beginning by creating a new random
population again. Also, there is no guarantee that the found solution indeed is
the global optimum. However, in practice, these drawbacks are rarely signi?cant.
3.3. DYNAMICAL OPTIMISATION 21
The need for implementing extra operators is well compensated by the simplicity
of the algorithm. Since true global optimisation is such a hard task, not even the
most advanced deterministic algorithms can guarantee optimality in the general
case. Considering this, the GA is a good candidate for real-world optimisations.
GAs have been extended to the multi-objective case [88]. The resulting algo-
rithms are collectively known as multi-objective evolutionary algorithms (MOEA).
These algorithms aim at ?nding the ?rst Pareto-optimal front discussed above. In
Paper II, a particular type of the above MOEA, called NSGA-II [89], is used. This
is an elitist non-dominated sorting GA that uses an explicit diversity-preserving
mechanismto keep solutions separated fromeach other. Since solutions are spread
along the Pareto front, some measures need to be taken for the solutions not to clog
together in the same spot. This is done by calculating the distance to the nearest
neighbour, called the crowding distance. The new population is ?lled with so-
lutions from one front at a time and in ascending order of fronts. First solutions
from front number one is used, then solutions from front number two etc. If not all
solutions from a front can be used (i.e. if the population is about to be ?lled), the
remaining available positions in the new population are ?lled with the solutions
having the highest crowding distance, i.e. lying furthest apart from each other.
The NSGA-II algorithm can be summarized as follows
1. Randomly generate parent P
t
, (|P
t
| = N) and offspring Q
t
, (|Q
t
| = N)
populations.
2. Combine parent and offspring populations to form R
t
= P
t
? Q
t
.
3. Evaluate the combined population R
t
and sort it into a number of non-
dominated fronts F
i
, i = 1, 2, . . . , r.
4. Iteratively create a new population P
t+1
? P
t+1
? F
i
, |P
t+1
| + |F
i
| ?
N, i = 1, 2, . . .
5. Carry out a crowding distance sorting on the remaining fronts not included
in P
t+1
and include the most widely spread solutions until |P
t+1
| = N.
6. Create offspring population Q
t+1
from P
t+1
by crowded tournament selec-
tion, crossover and mutation.
7. Iterate Steps 2-6 until convergence.
3.3 Dynamical optimisation
The dynamical optimisation problem is de?ned as the problem of minimising a
cost function J over a given time period by ?nding the optimal control trajectory
22 CHAPTER 3. OPTIMISATION TECHNIQUES
u. Thus, this type of problem is also referred to as an optimal control problem
(OC) [90, 91, 92]. Common cost functions include energy, fuel, and time. The
dynamical system can be mechanical, electrical or any other type that can be de-
scribed mathematically. In this thesis, investment problems are considered. The
continuous-time deterministic optimal control problem can be de?ned generally
as
min
u(t)
J, where J = ?(x(t
f
)) +
_
t
f
t
0
L(x(t), u(t), t)dt, (3.12)
s.t. ? x = f(x, u, t),
c(x, u, t) ? 0,
where J is the objective function, f are the state equation constraints, c are the
path constraints and u is the control vector. The objective function consists of two
parts: ?, a cost based on the ?nal time and state, and an integral depending on the
time and state histories. In addition there may be simple bounds on the state and
control variables, i.e.
x
l
? x ? x
u
(3.13)
u
l
? u ? u
u
,
and also boundary conditions on x and u.
The above problemalso has a discrete version, in which the integral is replaced
by summation. The problems in Papers II and IV are both expressed in the discrete
form, mainly due to the type of data available and for simplicity of calculation.
The optimal control problem may be solved by any of the following four meth-
ods: dynamic programming, the indirect method, the direct method, or simulation-
based optimisation.
The dynamic programming approach makes use of Bellman’s principle of op-
timality to solve the problem by backward induction [93]. The resulting partial
differential equation is very hard to solve, except in very fortunate cases.
An indirect method aims at ful?lling the necessary conditions for an optimum,
the Euler-Lagrange equations and the adjoint equations, using variational calcu-
lus. Finding these expressions requires calculation of gradients and Hessians,
which usually is cumbersome. In addition, the indirect method is sensitive to the
choice of starting point, i.e. the ?rst estimate. A poor starting point may result in
divergence or wildly oscillating trajectories.
A direct method [90] uses a sequence of points to approximate the state and
control variables. The sequence may be a piecewise polynomial expansion. When
these approximations are inserted into the objective function and constraints, the
result is a static optimisation problem that can be solved using the methods dis-
cussed in Sections 3.1 and 3.2. Using the direct method [90], the integral in
3.3. DYNAMICAL OPTIMISATION 23
the objective function in Eq. (3.12) can be treated as an additional state ? x
n+1
=
L(x, u, t) with the initial condition x
n+1
(t
0
) = 0. It is thus possible to replace the
original objective function with one of the type J = ?(x(t
f
)). Now the interval
t
0
to t
f
is divided into n
s
segments where h
k
is the time span of one segment.
Furthermore, letting M = n
s
+1 be the number of points in the interval, the state
equations can be approximated with any numerical integration method, i.e. Euler,
Trapezoid and Runge-Kutta. For the simplest case using the Euler method, one
may de?ne ?
k
= x
k+1
? x
k
? h
k
f
k
, and the original optimal-control problem in
Eq. (3.12) can then be expressed as an NLP problem in each point 1, 2, 3 . . . M of
the time segments in the following way
min
(u
1
,y
1
,...u
M
,y
M
)
J, where J = ?(x
M
)
(?
1
, ?
2
, ..., ?
M?1
) = 0 (3.14)
(c
1
(x
1
, u
1
, t
1
), c
2
(x
2
, u
2
, t
2
), ..., c
M
(x
M
, u
M
, t
M
)) ? 0.
In this equation, the ?
1
, ?
2
, ..., ?
M?1
are the deviations, also referred to as the
defects, for the dynamics (augmented with the integral in the objective function
in Eq. (3.12)) approximated by the numerical integration method at each point.
(c
1
(x
1
, u
1
, t
1
), c
2
(x
2
, u
2
, t
2
), ..., c
M
(x
M
, u
M
, t
M
)) ? 0 are the original inequal-
ity constraints expressed at every point. The result of the optimisation is M control
and state vectors u and y.
This optimisation problem is of the static NLP type and can be solved with
the techniques discussed in the previous sections. The problem does, on the other
hand, have M?1 times more variables than the original dynamic optimal control
problem. For the case of equality constraints and boundary conditions, the number
of variables equals M ? 1 + M + 2 = 2 × M + 1. The method presented in
Eq. (3.14) is also referred to as the multiple shooting method. In the case where
n
s
= 1, it is called a single shooting method. In the case of a long interval t
f
?t
0
and small time constants, the resulting static problem will become hard to solve.
In Paper II the above direct transcription method was used; see Chapter 5 for a
further discussion.
The simulation-based approach uses a totally different technique to arrive at
an optimisation problem that can be treated as a static one. Here, the system under
study is simply simulated for the entire simulation period, during which the cost J
is also calculated. Based on the value of J a static optimisation algorithm adjusts
the control vector uin the direction of lower cost. The simulation-based technique
can be described as follows
1. Find a feasible control vector u.
2. Simulate the system using the control vector u and calculate the cost J.
24 CHAPTER 3. OPTIMISATION TECHNIQUES
3. Change the control vector for lower cost J using a static optimisation algo-
rithm.
4. Repeat Steps 2-3 until convergence.
The static optimisation technique in Step 3 can be of any kind. However, a GA,
as described in section 3.2, is particularly suitable.
3.4 Stochastic dynamical optimisation
In any real system, noise is present. In the systems discussed in this thesis, noise
is represented as an uncertainty about future development. This fact motivates the
usage of stochastic dynamical optimisation techniques [94, 95, 96], which is a
dynamical optimisation technique, as discussed above, applied to a problem under
the in?uence of a disturbance. The disturbance can be realized as a stochastic
variable with a prede?ned probability distribution. The discrete-time stochastic
optimal control problem can be de?ned generally as
min
U
J(U), where J(U) =
N?1
k=0
?(k, X
k
, U
k
, W
k
) + ?(X
N
, W
N
)
s.t. X
k+1
= f(k, X
k
, U
k
, W
k
) (3.15)
c
k
(X, U) ? 0 ? k = 1, . . . N,
where W is an independent random disturbance, ?(k, X
k
, U
k
, W
k
) is the cost
associated with each time step k, ?(X
N
, W
N
) is the terminal cost and c
k
(X, U)
represents simple limits of the state and control variables.
The above perturbed optimisation problem can be solved by methods analo-
gous to those used for the unperturbed problem in Section 3.3, that is by dynamic
programming, stochastic programming or the simulation-based approach. The
solution methods generally work in the same way as in the unperturbed case.
The differences compared with the unperturbed case are that in the perturbed
case the objectives are optimised with respect to expected values. The result is
one control strategy U that minimises the expected value (the mean value) of the
objectives. The de?nition of the disturbance can either be analytical, i.e. using
data from the distribution used, or scenario-based, i.e. using a number of sam-
ples from, e.g. a probability distribution. In the former case the optimisation is
done analytically and in the latter case it is done numerically. In Paper IV, several
scenarios were used for generating samples after which the scenario-based opti-
misation method was applied. Using this technique, each sample corresponds to
one possible future development of the disturbance, i.e., in this case, the number
3.4. STOCHASTIC DYNAMICAL OPTIMISATION 25
5.71 5.72 5.73 5.74 5.75 5.76 5.77 5.78 5.79 5.8
0
0.05
0.1
0.15
0.2
0.25
Histogram for investment strategy 89 in scenario 3, calculated for scenario 1.
Hydrogen production cost [USD/kg]
P
r
o
b
a
b
i
l
i
t
y
1300 1400 1500 1600 1700 1800 1900 2000 2100
0
0.05
0.1
0.15
0.2
0.25
Histogram for investment strategy 89 in scenario 3, calculated for scenario 1.
Hydrogen total unsatisfied demand [kg]
P
r
o
b
a
b
i
l
i
t
y
Figure 3.5: Examples of histograms representing the hydrogen production cost (left
panel) and unsatis?ed demand for hydrogen (right panel) for a given solution. The his-
tograms were generated using 100 samples.
of vehicles arriving at the refueling station. The solution is the expectation value
of the optimum for all samples.
In Paper IV, the problem is two-dimensional and, therefore, so is J. Applying
all samples in a given scenario
2
will result in a distribution of the two-dimensional
objective function J. In essence, for each solution U and set of samples there is
one distribution for each dimension in the objective function J. In Figure 3.5
an example from the hydrogen and hythane refueling case is shown. The two
objectives are hydrogen production and total hydrogen unsatis?ed demand
3
. Each
quantity has been calculated for one solution U, and for all available samples.
2
In Paper IV, 100 samples were generated for each scenario.
3
As shown in [97], the occurrence of an unsatis?ed demand is not uncommon.
Chapter 4
Assessing the future
This thesis focuses on long-term planning and optimisation, processes that require
an assessment of the future. Such assessments can be obtained in different ways,
and in this thesis (see Papers II and IV) the preferred method has been to generate
a number of possible outcomes, referred to as scenarios. A number of probability
distributions are associated with each scenario. Thus, once a scenario has been
de?ned, a large number of samples can be generated, each representing one pos-
sible future outcome. Needless to say, such scenarios will always have a certain
degree of arbitrariness.
Since the long-term effects of a scenario often may depend strongly on what
happens during the ?rst few time steps, short term prediction of time series, based
on past data, is certainly relevant. Thus, time series prediction (TSP), considered
in Paper III, can be used for reducing the prediction uncertainty
4.1 Forecasting
Forecasting time series is common in economics [23, 98]. In the technical domain
it is part of system identi?cation [99]. The underlying assumption is that the
data series have an internal structure that can be identi?ed. After identifying this
structure, a procedure known as model ?tting, a prediction of the future can be
made. In practice, the model is selected ?rst, after which the data sets are used
to estimate its parameters. Clearly, this requires that the model can represent
the data. If this is not the case, another model has to be found through structure
selection, i.e. by ?nding a model with a more suitable internal structure, which can
then be tuned by adjusting the values of the internal parameters so as to reduce the
error over the data set.
Another issue is the amount of noise present. White noise can, by de?nition,
not be predicted. Other types of noise can, to a certain and often limited degree,
27
28 CHAPTER 4. ASSESSING THE FUTURE
be predicted. In this case one can use models that have a noise part allowing
for multi-step predictions
1
[99]. For one-step prediction the expected increment
is zero in case of white noise, and the estimate is equal to that obtained from a
model without noise. However, the genuine information part of the time series is
deterministic and can, provided a good model is found, be subject to successful
prediction.
Traditionally, methods like the naive method, exponential smoothing and auto-
regressive integrated moving averages (ARIMA) have been used for time series
prediction [22, 52]. The naive method estimates the next value in the time series
by the present one. For very noisy time series, it is hard to beat the trivial pre-
diction obtained from the naive method [100]. The ARIMA techniques consist of
three parts: an auto-regressive part, an integrating part and a moving average part.
Exponential smoothing is a special type of ARIMA model. More recently, arti?-
cial neural networks (ANNs), of which feedforward neural networks (FFNNs) and
recurrent neural networks (RNNs) are examples, have been used in TSP [25, 101].
In Paper III, a novel kind of recurrent ANNs called discrete-time prediction
networks (DTPNs) was developed for time series prediction. This study was
a continuation and an improvement of earlier work on TSP using neural net-
works [26]. A DTPN contains inter-neuron connections as well as connections
to the inputs. In a DTPN, any neuron may be connected to any other neuron, and
to itself. Furthermore, each neuron has an individual squashing function ? which
is (in principle) arbitrary. In Paper III the logistic function
?
1
(z) =
1
1 + e
?cz
, (4.1)
where c is a positive constant, and the hyperbolic tangent
?
2
(z) = tanh cz, (4.2)
have been utilized. Since no gradient information is needed in the training proce-
dure, no restrictions exist on the functions that can be used. Therefore the squash-
ing functions
?
3
(z) = sgn(z), (4.3)
and
?
4
(z) =
_
_
_
tanh(z + c) if z < ?c
0 if ?c ? z ? c
tanh(z ?c) if z > c
(4.4)
were also used. In addition, the function
?
5
(z) =
cz
1 + (cz)
2
, (4.5)
1
A multi-step prediction is de?ned as x(t ?n), x(t ?n + 1), . . . , x(t) ? ˆ x(t + k), k > 1.
4.1. FORECASTING 29
Neuron1 Neuron2 Neuronn ...
Neuroni
w(interneuronweights) w (inputweights)
in
b c k(sigmoidtype)
...
EOT
Figure 4.1: A chromosome encoding a DTPN as used in Paper III.
was used. Even though DTPNs have arbitrary connections between neurons, the
order in which the neurons are updated is ?xed, and given by so called evaluation
order tags (EOTs), one for each neuron. In each time step, the neurons with lowest
EOT are updated ?rst according to
x
i
(t + 1) = ?
_
b
i
+
n
in
j=1
w
in
ij
I
j
(t) +
n
j=1
w
ij
x
j
(t)
_
, (4.6)
where w
in
ij
are the input weights, w
ij
the interneuron weights, and b
i
is the bias
term. I
j
are the inputs to the network which, in the case of time series prediction,
consist of earlier values of the time series Z(t), i.e. I
j
(t) = Z(t ? j + 1) For
neurons with the second lowest EOT, the equations look the same, except that
x(t) is changed to x(t + 1) for neurons with lowest EOT etc. Finally, the output
neuron (arbitrarily chosen as neuron 1) gives the following output
x
1
(t + 1) = ?
_
b
1
+
n
in
j=1
w
in
1j
I
j
(t) + w
11
x
1
(t) +
n
j=2
w
1j
x
j
(t + 1)
_
, (4.7)
since, at this stage, all neurons except neuron 1 have been updated. Like other
RNNs, DTPNs are capable of short-term memory, a feature which is important in
time series prediction (see also Paper III).
The optimisation of DTPNs is carried out by means of a genetic algorithm
(GA), which evolves not only the parameters of the network, but also its structure,
i.e. the number of neurons and their EOTs. The encoding scheme used for evolv-
ing DTPNs is shown in Figure 4.1. Note that the sigmoid type ?
k
is encoded by
the integer k in the chromosome.
In Paper III, DTPNs were evolved for one-step prediction of the Fed Funds
interest rate and US GDP, after ?rst rescaling the data to the range [?1, 1]. A
summary of the results is given in Table 4.1. As can be seen in the ?gure, the
prediction results (average prediction errors) for the DTPNs were better than those
of the other methods tested.
The DTPNs have been used for one-step predictions only. Of course, multi-
step predictions are possible, in principle, but will inevitably result in inaccurate
predictions due to the effects of cumulative errors [51, 102].
30 CHAPTER 4. ASSESSING THE FUTURE
Data set e
N
e
ES
e
ARMA
e
DTPN
Fed funds interest rate 0.2018 0.1901 0.1887 0.1837
GDP 0.1771 0.1490 0.1473 0.1305
Table 4.1: Average errors for one-step predictions carried out for two macroeconomic
time series: the Fed funds interest rate and the US GDP. The table shows the minimum
errors over the validation part of the data set, obtained using naive prediction (e
N
), expo-
nential smoothing (e
ES
), ARMA (e
ARMA
), and DTPNs (e
DTPN
). Only the results for the
very best DTPN are shown.
Also investigated were predictability measures, i.e. measures of the accuracy
of an individual prediction. Several empirical measures were investigated, as well
as one analytical measure. The empirical measures involved different ways of
augmenting the DTPNs to incorporate the predictability. The amount of genuine
information in a single time series can be analytically estimated using random
matrix theory [103, 104]. If the original observations are contained within the
Tx1 vector x(t), a T ?m×N delay matrix Z can be formed where the columns
are delayed observations, i.e. x(t), x(t ?1), x(t ?2) . . . x(t ?m). The parameter
m represents the maximum delay and should be chosen to cover the time period
of any cyclic behaviour. In the process of forming the delay matrix, m rows at
the end of Z will lack values and hence only T ? m rows can be further used in
Z. The number of columns, N, equals the maximum delay number, m, plus one
initial column. In order to use Z, each column has to be normalised to zero mean
and variance 1, which is done using
Z
m,n
?a Z
m,n
+ b, (4.8)
where
a =
¸
T ?m
T?m
i=1
Z
2
i,n
?1/(T ?m)(
T?m
i=1
Z
i,n
)
2
(4.9)
and
b = ?a
T?m
i=1
Z
i,n
T ?m
. (4.10)
The correlation matrix, C, is then de?ned as
C =
1
T
Z
T
Z. (4.11)
Furthermore, the eigenvalues of C can be used to estimate the information content
by comparing with the eigenvalues of a random matrix with the same dimensions.
4.1. FORECASTING 31
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
15
20
25
30
35
40
Information content Usa GDP (Total 20.0779%)
%
Figure 4.2: Information content in the US GDP time series using random matrix the-
ory [104]. The remaining part is noise. For the above case the delay m = 12 and the
window size 32. The total information content is calculated based on the eigenvalues for
the full, non-windowed, correlation matrix C.
Such a matrix, X [m×n], will, if it is scaled according to Eq. (4.8), have a density
of eigenvalues according to
?(?) =
Q
2?
_
(?
max
??)(? ??
min
)
?
(4.12)
for ? ? [?
min
, ?
max
], and zero otherwise, in the case where m, n ? ?, and
where Q = n/m ? 1 [105]. In Eq. (4.12), the minimum eigenvalue is ?
min
=
?
2
(1 ? 1/
?
Q)
2
and the maximum eigenvalue ?
max
= ?
2
(1 + 1/
?
Q)
2
. If the
eigenvalues of C lie outside the range [?
min
, ?
max
], the time series x(t) contains
real information. A numerical estimate of the minimum and maximum eigenval-
ues for matrices of ?nite size, can be obtained by taking the minimum and the
maximum eigenvalues obtained from a large number of generated random matri-
ces of the same dimension as C.
A way of estimating the percentage of information content is by computing
100?
max
/N, where ?
max
is the largest eigenvalue of the correlation matrix C. By
making use of a moving data window, the local information content in time series
may be estimated. Figure 4.2 shows such a calculation for the US GDP.
All results on the predictability measures were, however, negative. Compar-
ison between the analytical and empirical candidate measures showed no corre-
32 CHAPTER 4. ASSESSING THE FUTURE
lations. The reason may be that the DTPNs do extract most of the information
available in the time series.
4.2 Decision-making under uncertainty
Decision-making in industry as a scienti?c discipline can be regarded to be a
part of operations research (OR) [106, 107, 108]. OR originates from the mili-
tary sector, where it was used to make better decisions in, e.g. logistics and war
tactics [109, 110]. Nowadays OR is widely used in the industry to raise ef?-
ciencies and optimise performance. Here it is also known as management sci-
ence [111, 112]. The problems that occur in OR are real-world problems and
are often hard to solve using traditional deterministic methods. Therefore it is
not surprising that the ef?ciency of stochastic optimisation techniques was early
recognised [113].
An important ?eld where OR has contributed is decision-making under un-
certainty. In the industry, strategic decisions have to be made and in all practical
cases they are taken under uncertainty since they involve predicting, or guessing,
what will happen in the future. Since it is generally very dif?cult, not to say im-
possible, to make correct long-term decisions intuitively, this is an area where
much can be gained by applying advanced analytical techniques [114, 115, 116].
Application areas include logistics [117], ?nancial instruments, investment plan-
ning [11], risk management [118], water pollution problems [119], water resource
problems [120] etc. In all of the above areas, stochastic techniques, like GAs,
nowadays are common ways of solving the resulting optimisation problem.
For an investment decision problem, each decision to invest will, in the op-
timisation framework, result in one decision period. In practical cases there is
often a pre-de?ned number of occasions when investments are possible, e.g. once
a month or once a year, something that leads to a sequential decision problem. The
problems in Papers II and IV are both sequential decision problems. In addition,
the problem considered in Paper IV is also stochastic. Furthermore, the decision-
making can involve an inner loop where a number of decisions are made within
each outer decision period, leading to two stages of decisions. For a production
company, the ?rst stage typically involve decisions on investments in production
capacity and the second stage regards decisions concerning production planning
given the resulting constraints from the ?rst stage. The decision problem in Paper
IV is also of the above two-stage type and is solved by the use of a pre-de?ned
strategy for the ?rst investment stage and a combination of closed-loop regulation
2
and a pre-de?ned strategy for the second production stage.
2
The term closed-loop regulation is used in the standard way, as de?ned in control theory.
4.2. DECISION-MAKING UNDER UNCERTAINTY 33
Decision-making under uncertainty can be modelled as a Markov decision
process (MDP) [121]. In such a process, the decisions taken at a certain point
depend only on the state at the previous time point and not on states further back
in time. The MDP is a discrete-time stochastic control process that propagates
through a series of states. For each state the decision-maker takes actions based
on the information given by the previous state only. Next, a stochastic transition
function determines probabilities for transition to the next state. For each state
there is a reward, which depends on the new state. In Paper IV, the investment
decisions are parametrised and calculated in an open-loop way, i.e. they are pre-
de?ned. Uncertainties in the form of hydrogen and hythane demand are dealt with
in the second stage, where the amount of stored hydrogen is kept at a constant level
by adjusting the production. This second stage is a process of the Markov type.
The use of optimisation for ?nding optimal future investment strategies is a
decision that has to be carefully considered for each individual case. As with all
mathematical tools, optimisation methods also require a quanti?cation of the input
data, which is by no means trivial, since some data concern predictions of future
sales, prices etc. Other factors such as availability of skilled personnel and data
regarding the system under study also have to be taken into account. The methods
used in Papers II and IV are, from a mathematical point of view, very robust. The
sensitivities to disturbances can be determined by perturbation analysis, a tech-
nique used in Paper II. The quality of the optimisation results depends, of course,
on the quality of the input data. On the other hand, the investment methods pre-
sented in Paper II and IV are meant to be used to update the investment strategy
as soon as new and better predictions are found. In this case the optimal next in-
vestment decision can always be found, given the best available future estimates
at the time of calculation. It should also be stressed that the use of GAs for in-
vestment planning, as in Paper IV, reduces the calculation effort since no gradient
information is needed. It is the conviction of the author that, in the case of a com-
plex system and as soon as the objectives are quanti?able, the presented methods
are very powerful tools and will prove useful in many applications in addition to
those discussed in this thesis.
Chapter 5
Case studies
This chapter provides a background to the case studies presented in Papers I, II
and IV. Along with the general background, some details omitted from the papers
are given as well.
The three cases are (1) the cement production model, (2) the hydrogen refu-
elling station infrastructure investment optimisation and (3) the combined hydro-
gen and hythane infrastructure investment optimisation.
5.1 The cement production case
In the cement production case considered in Paper I, a highly ?exible model of a
cement production factory has been built. The model has been used in several dif-
ferent calculations, including process optimisations and environmental assessment
of new energy sources. The life cycle inventory analysis (LCI) model consists of
a foreground system which de?nes the on-site production over which the com-
pany has full control, and a background system comprising purchased services
and goods, see Figure 5.1. A more in-depth discussion of the production facility
is given in Paper I and in [45].
The raw materials, i.e. different sorts of sand, are transported to the production
site and ground depending on type. They are then mixed in relevant proportions
and burnt to clinker in the clinker production system. For the burning process,
fuel is, of course, required, and it may consist of coal, pet coke or an alternative
fuel. All fuels are transported to the site, ground and mixed in correct proportions,
before entering the burner. The produced clinker is then mixed with gypsum (and
possibly other materials), further ground, and stored as cement.
The problem is to ?nd the ratio of raw materials, fuels and the additional gyp-
sum to produce cement of a certain quality. The quality is measured using the fac-
tors indicated in Table 5.1. In addition the approximate monetary costs throughout
35
36 CHAPTER 5. CASE STUDIES
Figure 5.1: LCI model of the cement production line.
the production line must be calculated. Since the purchase costs of the raw mate-
rials and fuels are known, the production-related cost for each piece of equipment
in the line can be estimated and added to the product ?ow. The model can be
used as an aid in calculations for new types of raw materials, fuels and internal
settings, and for changes in the layout of the production line. In addition to static
solving, dynamic simulation and optimisation can be considered. It is therefore a
requirement that the model should be modular and highly ?exible.
One option would be to make a standard LCI model. When a standard LCI
is carried out, the linear technology matrix model (A) described in Section 2.1 is
suf?cient for describing the technical production system, since the underlying pro-
duction system is described as static and linear. Using this linear description, only
one type of calculation is made in an LCI, which is the normalisation to the func-
tional unit, obtained by solving a linear equation system. The developed mathe-
matical LCI methods are designed to achieve only this normalisation [31, 32, 34],
something that limits their usefulness. At times it is desirable to make extensions
to this type of LCI model. One such occasion is when the inherent physical be-
haviour of the production system is strongly non-linear when seen as a mapping
from resources and emissions to the product. A linear LCI model represents in
this case a linearisation around a speci?ed point and might lead to unacceptably
large deviations in the calculated resource use and emissions released. Another
5.1. THE CEMENT PRODUCTION CASE 37
Table 5.1: Cement product quality indicators. The notation indicates weight percentage
of the speci?ed material.
Name Symbol Description
Lime saturation factor LSF LSF =
100CaO
2.8SiO
3
+1.1Al
2
O
3
+0.7Fe
2
O
3
Silica ratio SR SR =
SiO
2
Al
2
O
3
+Fe
2
O
3
Alumina ratio AR AR =
Al
2
O
3
Fe
2
O
3
occasion occurs when dynamic aspects are relevant, for example when closing
down and starting up a production line in connection with maintenance. In the
situations just described, another modelling approach, making use of non-linear
and dynamic models, is needed. In addition, other types of calculations may be
desirable as well. Examples include optimisation, simulation over time etc. In or-
der to ful?ll the requirements, one needs a higher degree of ?exibility in the model
than is given by the LCI model. In short, the modelling approach has provide the
model with enough data to represent the underlying system in a correct way and
this data has to be arranged in such a way that it is possible to make the desired
calculations using the model.
In [122] the nature and effect of some different types of causality are dis-
cussed. The concept of causality is further applied to LCA in [50]. To recapitulate,
there are two types of causality of interest: (1) physical causality and (2) compu-
tational causality. The physical causality is the cause-effect connection inherent
in nature. The computational causality is the order in which an equation system
is solved. While the former is governed by the laws of nature, the latter is the
choice of the modeller. In [50], it was found that, by removing the computational
causality from the model, advantages in ?exibility can be achieved. The result is a
so called acausal [122] or non-causal model. In effect, the entity that is normally
regarded as the model can be split into three parts, namely:
• A computationally neutral (acausal) model, i.e. a model that maps the in-
terpretation of the production system onto a mathematical formulation, but
does not include any speci?c problem to be solved.
• A problem formulation, i.e. a description of which parameters should be
calculated and an explicit list of which parameters should be held constant
during a particular calculation as well as numerical values for each such
constant parameter.
38 CHAPTER 5. CASE STUDIES
• A method of calculation. This part can be considered as a part of the prob-
lem formulation.
In addition it was found that the modularity of the model, i.e. the ?exibility with
regard to both change and exchange of parts within the model, can be enhanced by
using an object-oriented modelling language in conjunction with physical entity
modelling. The intention with the latter is to keep real physical entities together
for ease of comprehension and transparency. This way of modelling also consti-
tutes a natural way to keep parts that are separate in reality as separate objects
in the model, so that the model resembles reality or a suitable representation of
reality. To summarise, the following requirements are considered:
• A computational acausal model that contains the structure and constants of
the system, but does not contain any information regarding computation.
• An object-oriented modelling language that makes use of encapsulation and
inheritance
1
.
• A physical property modelling approach that makes it possible to map the
real physical structure onto a similar model structure.
However, there are drawbacks with using an acausal model. Any mathematical
model consists of a number of equations. In the computationally causal case,
e.g. block diagrams and state-space models, these equations are ordered in a spe-
ci?c way to achieve the desired result. In the computationally acausal case the
equations are not ordered in any speci?c computational way. Instead they can
be regarded mathematically as a number of equilibrium equations connected to
each other, making them harder to understand. For models of physical systems
which are based on ?ow semantics, i.e. correlation between the general variables
intensity and ?ow, the model representation can be based on energy ?ow and is
usually relatively easy to construct. For the type of ?ow models used in LCI there
are also physical laws to consider, but not in the form of intensity-?ow related
connections. Under these circumstances acausal models can be structured in vari-
ous ways depending on the application, and therefore such models are dif?cult to
make both consistent and suf?ciently general to reach a high degree of ?exibility.
Another disadvantage is that in order to use an acausal model, a dedicated soft-
ware for sorting out the equations and ordering them computationally is needed.
In practice this is rarely a limitation, since such software is available.
As an illustration of the use of acausal models, consider the simple resistor
described by
u
R
= R i
R
. (5.1)
1
Encapsulation and inheritance are central concepts in object-oriented programming, see
e.g. [123, 124, 125] for details.
5.1. THE CEMENT PRODUCTION CASE 39
Figure 5.2: The separation of model and problem formulation that can be achieved by
the use of acausal models.
Is it the current ?owing through the resistor that causes the voltage drop or the
voltage drop that causes the current? What is the physical causality of the resis-
tor? Of course the order of calculation depends on the question at hand. If one
is interested in the voltage drop, one would use the computational order given in
Eq. (5.1). On the other hand, if one is interested in the current, one would re-order
the equation accordingly. If Eq. (5.1) is interpreted as a statement of equilibrium,
it can be regarded as an acausal model. A problem formulation might be to cal-
culate the voltage drop while keeping the current at a constant value. A suitable
method of calculation is then any static linear equation solver. Moreover, the
above example is a physical system and is based on ?ow semantics. In this case
the voltage is the intensity and the current is the ?ow [56]. For the simple example
discussed, the computational aspects are obvious and need no further formalisa-
tion. However, in cases with more than, say, 100,000 equations, much time can
be saved through the use of acausal modelling techniques.
When the project reported in Paper I was carried out (fall 1999), only a lim-
ited number of modelling languages and software programs for calculation were
available and able to ful?ll the requirements. Among them there were OmSim
(Omola) [126], Dymola (MODELICA) [48] and Ascend [46]. OmSim and Dy-
mola are specially made for modelling of physical systems and have a built-in sup-
port for ?ow semantic. Since the system considered does not have any intensity-
40 CHAPTER 5. CASE STUDIES
?ow dependency, it was decided not to use these programs (languages). Ascend
is both a calculation software and a modelling language and was originally de-
veloped for applications within chemistry. However, it can best be described as a
mathematical system modelling tool and is very ?exible in de?ning connections
and hence the structure of the system modelled, which is the main reason for using
it.
The model was built in a bottom-up manner according to Paper I. It should be
noted that the model is deliberately made redundant. In most cases redundancy
has a negative effect, but here it is used to enhance the ?exibility. The numerical
parameters in a calculation can be divided into the following categories:
• Constants. These are set, once and for all, when the model is built.
• Locked variables. Parameters set to a constant numerical value for a certain
calculation.
• Free variables. Parameters that will be calculated by the numerical algo-
rithm.
The number of parameters in each category depends on the speci?c calculation
considered. Providing information regarding these settings is part of the problem
formulation. In the model, the information needed for specifying one parame-
ter can be supplied in a number of ways. An example is the ingredients in raw
meal composition. These can be set by speci?cation of absolute masses or rela-
tive masses (percentage). The model contains the necessary mathematics to relate
these parameters at the time of calculation. At any given time, only one of the
two ways of specifying the parameter is used. The result, presented in Paper I,
is a highly ?exible calculation tool for the cement production process that has
been used by Cementa AB, for several different purposes. It should be noted that
the entire model of the cement production line was later transferred to MODEL-
ICA [127].
5.2 The hydrogen infrastructure case
The main task for the hydrogen refuelling station is to dispense hydrogen to ve-
hicles. Since the incentives for using hydrogen are environmental, an important
question to consider is where the hydrogen is to be produced. Producing the hy-
drogen is probably best done at large, centralised production facilities. It is then
easier to take care of the created emissions, e.g. CO
2
. The problem is to dis-
tribute the hydrogen to the local refuelling station. In order to do so ef?ciently,
the hydrogen gas has to be highly pressurised, which is expensive and can also
5.2. THE HYDROGEN INFRASTRUCTURE CASE 41
be dangerous. Another consideration is the vulnerability both to sabotage and
to accidents. In this thesis an alternative solution comprising local production
of hydrogen using a hydrogen reformer, i.e. a device that produces hydrogen
from hydrocarbons, is investigated. The input to the reformer can be any type of
methane gas and may originate from fossil or renewable resources. One disadvan-
tage is that the reformer will produce considerable amounts of CO
2
which will not
be easy to take care of. Probably it has to be released into the atmosphere. When
the natural gas comes from a renewable source of energy the net contribution of
CO
2
is nil. One obvious advantage with local production is that natural gas is
considerably easier to transport than hydrogen gas. In fact there is already a rather
small but growing number of natural gas refuelling stations in Sweden [128]. A
hydrogen production and refuelling part can then be added to the already existing
natural gas refuelling station. With such a refuelling station, it is also possible to
dispense natural gas as an intermediate alternative.
If the refuelling station is equipped with fuel cells, it can also be used as a lo-
cal electrical power station. This alternative might be useful in remote locations.
When hydrogen is produced from renewable energy sources, it might also be an
environmentally friendly alternative. If the refuelling station is located in a place
where electricity from the grid is cheap, it can be equipped with an electrolysis
part that can produce hydrogen gas directly from electricity. In this case it is im-
portant to keep track of how the electricity is produced. To begin by producing
electricity from coal and then using electrolysis to produce hydrogen is not, how-
ever, a good environmental solution. In addition to the hydrogen reformer, the
refuelling station layout that is considered in Paper II also has a local fuel cell
and an electrolysis plant. Figure 5.3 illustrates options investigated in this thesis.
The result is a refuelling station that is very ?exible in terms of resource use and
energy production.
The equipment for a hydrogen refueling station with the above layout is more
expensive than present-day petrol station parts. In addition, not all of the con-
?gurations are suitable for speci?c conditions. Under these circumstances it is
important to ?nd the most pro?table con?guration for the speci?c location, the
estimated number of customers, and general technical and economical develop-
ment. The problem is to ?nd the most pro?table con?guration. As described in
Paper II, this problem is equivalent to ?nding the least expensive mean production
cost for hydrogen. In Paper II, only the core parts (the parts within the shaded area
in Figure 5.3) are part of the optimisation. The remaining parts can be dealt with
separately, as discussed in Paper II.
In reality the choice of equipment is limited by supply. Let the set of available
equipment, which consists of a ?nite number of sizes for each equipment type, be
denoted by C and the control sequence u(t) be a vector of equipment sizes such
that ?u(t) ? C and u(t + ?t) ? u(t), ?t where ?t is a small time step. The
42 CHAPTER 5. CASE STUDIES
Figure 5.3: Refueling station layout. Natural gas is reformed to hydrogen at the site and
stored for delivery to vehicles. It is also possible to produce hydrogen from electricity by
electrolysis or electricity from hydrogen using a fuel cell.
implication is that u is only allowed to increase and to do so only with speci?c
increments, namely those in C. Further let f(u, x, w) be the description of the
core of the refuelling station in state-space form where x is accumulated volume
in each piece of equipment and w is the hydrogen refuelling demand. The most
general form of the problem of ?nding the most pro?table con?guration is, in the
continuous case,
min
u(t)
J = ¯ c(x(t), u(t), w(t))
s.t. ? x = f(x(t), u(t), w(t)) (5.2)
0 ? x(t) ? u(t),
where ¯ c is a cost function described in Paper II and further discussed in Section 2.1
in this thesis. There are two major dif?culties in solving this problem:
1. The problem is de?ned over the entire investment period of 20 years. At
the same time the assumed ?lling curve for hydrogen has a time step of one
hour. Dividing the interval of 20 years into one hour segments would lead to
2 ×M + 1 = 350, 401 variables, as is discussed in Section 3.3. This would
make numerical solving of the resulting non-linear optimisation problem
hard, not to say impossible.
5.2. THE HYDROGEN INFRASTRUCTURE CASE 43
2. The control sequence u can only increase in steps that are part of C. This
would make the problem discrete in u. Discrete problems are combinatoric
and in general harder to solve than continuous ones [129].
In order to solve the problem stated in Eq.(5.2) it is observed that, to be able to
satisfy the hydrogen demand, the ?rst investment has to take place initially, at
t = 0. Consecutive investments can be divided into separate cases. Since the
desired output, the ?lling curve f
w
, is given for one week (168h) and then scaled
using the S-curve R, it is suf?cient to consider only one week for each investment.
The S-curve is a purely exogenous estimate of the number of hydrogen vehicles
using the refuelling station and is de?ned as
R(t) =
1
1 + e
?B(t?Tx)
, (5.3)
where t is the time fromyear 2010, T
x
the S-curve in?ection point and B the slope.
For values of the constants T
x
and B, see Paper II. The week to consider for each
investment is when utilisation is at its maximum, namely the week right before
the next investment. At these points in time, the equipment is used at its maxi-
mum capacity and, in order to satisfy the increasing demand, a new investment
in capacity has to be made. By parametrising the set C using a scaling function
p
eq
(s
eq
) (explained in Paper II), where s
eq
is the size of equipment, the problem
will become continuous in u(t).
The driving signal for the fast dynamics is the ?lling curve ?owf
w
. This curve
is given for one week with the time step of one hour. The integral in the objective
functions in Section 2.1 can therefore be replaced by a sum. Also, the discrete
version of Eq. (5.2) can be used.
Using the direct transcription method in Section 3.3, the resulting NLP prob-
lem (see Section 3.1) becomes, in the single investment case
min
u
k
J = ¯ c(x
k
, u
k
, w
k
)
s.t. ?
k
= x
k+1
?x
k
?f(x
k
, u
k
, w
k
) = 0 (5.4)
0 ? x
k
? u
k
, k = 1, . . . , M ?1,
where ?
k
are the defect constraints and M = 168 the number of steps. Note that
in Eq. (5.4), the step length is one hour. Considering that the dynamics involved
is of the ?rst order and is stable, the multiple shooting method can be replaced by
a single shooting one, which would make the problem easier to solve numerically.
44 CHAPTER 5. CASE STUDIES
The above defect constraints can be replaced by a cumulative summation, giving
min
u
k
J = ¯ c(x
k
, u
k
, w
k
)
s.t. x
k
=
k
p=1
x
p
(5.5)
0 ? x
k
? u
k
, k = 1, . . . , M ?1,
This NLP problem can be solved using the methods in Section 3.1.
In Paper II two cases are considered: (1) variable utilisation as in Eq. (5.5) with
extra requirements on initial amount of hydrogen stored and periodic maintenance
and (2) constant utilisation, which is a special and simpler case of variable utili-
sation. In the variable utilisation case, the chosen special conditions are 100 kg
hydrogen storage initially and at the end, and a weekly stop for maintenance from
hour 75 to 87 during the week. Optimisations are done for the cases of one and
two investments during the investment period. The resulting problem formulation
is
min
u
k
J = ¯ c(x
k
, u
k
, w
k
)
s.t. x
hs,k
=
k
p=1
(x
i
hs,p
?x
o
hs,p
)
0 =
87
p=75
x
o
hr,p
. (5.6)
x
hs,1
= 100
x
hs,1
= x
hs,M
0 ? x
k
? u
k
, k = 1, . . . , M ?1,
The data resulting from the optimisation are (1) the size of the equipment, (2)
the running pattern of the facility, and (3) the price and utilization curve for the
hydrogen produced. Figs. 5.4, 5.5 and 5.6 show some of the results in the case of
two investments. The complete results are given in Paper II.
The problem in Paper II is on the edge of what is possible to solve. If the
number of investments becomes too large (> 2 in the variable utilisation case),
the computational time becomes prohibitively long.
5.3 The hythane infrastructure case
The hythane and hydrogen refueling station is a development of the previously
discussed hydrogen refueling station. The differences in the layout is the absence
5.3. THE HYTHANE INFRASTRUCTURE CASE 45
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
5
10
Hydrogen reformer output 1 [kg/h]
time [h]
k
g
/
h
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
50
100
150
200
Hydrogen stored 1 [kg]
time [h]
k
g
Figure 5.4: Variable utilisation case, two investments; throughput and stored hydrogen:
Investment 1 at t=0.
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
?10
0
10
20
30
40
50
60
Hydrogen reformer output 2 [kg/h]
time [h]
k
g
/
h
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
200
400
600
800
1000
Hydrogen stored 2 [kg]
time [h]
k
g
Figure 5.5: Variable utilisation case, two investments; throughput and stored hydrogen:
Investment 2 at t=5.4.
46 CHAPTER 5. CASE STUDIES
0 2 4 6 8 10 12 14 16 18 20
25
30
35
40
45
50
55
Absolute cost produced hydrogen [SEK/kg]
time [year]
S
E
K
/
k
g
0 2 4 6 8 10 12 14 16 18 20
0
2000
4000
6000
8000
Capacity ? Demand for hydrogen [kg/week]
time [year]
k
g
/
w
e
e
k
Figure 5.6: Variable utilisation case, two investments; hydrogen production cost and
capacity-demand.
of the fuel cell for electricity production and, of course, the presence of a hythane
dispenser, as can be seen in Figure 5.7. The major advantage with the above layout
is that hythane can be used as an intermediate alternative fuel and, possibly, help
introducing hydrogen by taking on some of the costs for the reformer, electrolysis
and hydrogen storage.
In this study, the investment strategy is optimised with the objectives of min-
imising production cost for hydrogen as well as the requested amount of hydrogen
that could not be satis?ed, also called hydrogen unsatis?ed demand. The optimi-
sation method is described in Section 3.4. It should be noted that, in Paper IV, the
results are expressed as Pareto fronts for the two objectives involved.
In Paper IV, the optimisation of the hydrogen and hythane station was carried
out using a GA (see Section 3.2). In the evaluation of each solution candidate
(individual), between one and 10 investments are allowed. This is accomplished
with a variable chromosome length, see Figure 5.8. Since the ?rst investment
is mandatory at year 1, no timing is needed. Instead the demand priority policy
?, is de?ned. Demand can exceed supply and the demand priority governs how
the available hydrogen is used to satisfy the demand at the hythane and hydrogen
dispensers. Following ? in the chromosome, is the size of each piece of equipment
for the refuelling station in investment 1. For consecutive investments, the demand
priority policy is replaced by the time for the investment.
The selection, crossover and mutation operators are specially designed to cope
5.3. THE HYTHANE INFRASTRUCTURE CASE 47
Figure 5.7: Hythane and hydrogen refueling station layout. Natural gas is reformed
to hydrogen on-site and stored for delivery to vehicles. It is also possible to produce
hydrogen from electricity by electrolysis. In Paper II, only the parts within the refueling
station are considered.
Figure 5.8: The encoding scheme used for the genetic algorithm in Paper IV. The priority
policy ?, reformer size S
hr
etc. are genes. Allowed values for genes (alleles) are given in
Paper IV. The grouping of genes in the ?gure is only done for clari?cation.
48 CHAPTER 5. CASE STUDIES
with the problemat hand, the variables of which are implemented in chromosomes
of the kind shown in Figure 5.8. The selection operator is of the crowded tour-
nament selection type [88] where ?tness is replaced with the inverse crowding
distance within each Pareto front. The crowding distance is the mean distance to
the nearest neighbour solutions. In this way the solutions lying furthest from each
other are retained and a better spread of solutions within the front is achieved. The
crossover operator is of the one-point type, in which a crossover point is chosen
randomly within the shortest mutual length of the chromosomes. This will allow
crossover between chromosomes of different length, i.e. different number of in-
vestments. An upper limit of 10 investments was set. In the mutation operator, the
number of investments is changed with a low probability.
Traditionally, the control u is expressed as one vector for each decision pe-
riod [93]. The above approach is a parametrization where investments not used
are not part of the control. This is done in order to reduce the variable space and
achieve a faster convergence.
The optimisations in Paper IV are all two-dimensional. Some experiments
were also made with three-dimensional optimisations, i.e. optimising three objec-
tives at the same time. Even though it is possible to extend the above algorithm
to any dimensionality, there are practical limitations governed by the calculation
time. Evaluation of one individual takes 75 s, using all scenario samples. Sce-
narios and sample generation are explained in Paper IV. In the two-dimensional
case, the population R
t
to be evaluated consists of 80 individuals. This number of
individuals is needed to populate the Pareto front curve. The evaluation thus takes
1.7 hours. In the three-dimensional case at least 80 × 80 = 6, 400 individuals
would be needed to populate the generated Pareto surface, implying an evaluation
time of 5.7 days per generation. Since at least 200-300 generations are needed,
the whole optimisation would take approximately four years!
Each optimisation will give 40 solutions along the Pareto front for the ob-
jectives hydrogen production cost and total hydrogen unsatis?ed demand. Each
solution is optimised for the lowest expected values of each objective, given 100
samples from one of three scenarios, and can therefore be expressed as a distri-
bution, as discussed in Section 3.4. In addition there are four other performance
measures de?ned in Paper IV. These are unsatis?ed demand for hydrogen x
h,u
,
production cost per kg for hythane p
yf
, unsatis?ed demand for hythane x
y,u
and
?exibility p
h?
. All of the above measures can be calculated for the two scenar-
ios not part of the optimisation (the passive scenarios). Also investigated was the
variance of all the above measures. In all, this represents a large amount of data to
take into consideration before making a decision. One way to handle the decision
process could be to follow the procedure:
1. Find a reasonable solution regarding hydrogen production cost and unsatis-
5.3. THE HYTHANE INFRASTRUCTURE CASE 49
?ed demand while checking that the distribution of each variable is not too
wide.
2. Check the location of the selected solution in the hythane production cost
curve and the variance of the selected solution in the histogram.
3. Check ?exibility, i.e. the result that would be obtained if another scenario
should become reality. This can either be done with the de?ned ?exibility
measure, or in more detail, by calculating, e.g. production costs and distri-
butions for the found solutions by applying the passive scenarios.
4. If production costs, unsatis?ed demands and ?exibility are all within rea-
sonable limits, choose the solution. If not, go back to step 1.
Chapter 6
Concluding remarks
A number of techniques involving optimisation of industrial transition processes
have been explored. In particular the problem of ?nding optimal long-term invest-
ment strategies taking economic an environmental considerations into account has
been considered.
The investment strategy optimisation methods described in Papers II and IV,
have been successfully applied to two cases concerning hydrogen dispensing in-
frastructure change. The ?rst optimisation method, presented in Paper II, com-
prises a simultaneous calculation of the long-term investment strategy and the
short-term utilisation scheme for a deterministic demand. The method has been
applied to the case of ?nding an investment strategy for minimising the produc-
tion cost for a single hydrogen refuelling station. The problem was shown to be
convex; thus the resulting solution is the global optimum. The second investment
optimisation method, presented in Paper IV, uses stochastic demand scenarios and
multi-objective optimal control to produce the Pareto front of the two con?ict-
ing objectives expected production cost and expected unsatis?ed demand. This
method was applied to the case of ?nding the optimal investment strategy for a
combined hydrogen and hythane refuelling station. Due to the uncertainty of the
stochastic demand function, satisfying all demand would require a production ca-
pacity well above the mean demand, which would be very costly to maintain.
New ways for modelling joint economic-environmental systems and predict-
ing future key parameters have been developed, in order to enhance the applica-
bility and accuracy of structural optimisation methods. The ?ndings are presented
in a production system modelling case in Paper I and a time series prediction case
in Paper III. The results obtained in Paper I have been applied in industry, by Ce-
menta AB, in the evaluation of the consequences of using new fuels in cement
production.
51
52 CHAPTER 6. CONCLUDING REMARKS
6.1 Future work
The optimisation method in Paper II was tested on the hydrogen refuelling station
case, which was shown to be convex. Other algorithms for solving the resulting
NLP problem could be investigated, e.g. interior point or cutting plane [72] meth-
ods, which are ef?cient for convex problems. It would also be interesting to test
the developed method on a non-convex case and still try to obtain a global solu-
tion. The performance of the method in the described hydrogen refuelling case can
most probably be improved. In case 2 with variable utilisation (see paper II) the
computational time is unrealistically high for more than two investments. Since
the most favourable solution probably lies between three and ?ve investments,
this limitation must be overcome. Even though computer hardware is constantly
gaining speed, this does not mean that efforts to improve optimisation techniques
should be neglected. On the contrary, in the author’s opinion, the improvement of
such techniques is more rewarding and useful than merely waiting for computers
to become faster.
Due to the sampled nature of the refuelling curve, the investigated test case
contained only time discrete dynamics. It would be interesting to try the method
in a continuous dynamic case, i.e. where all the driving signals are continuous.
Also, in order for the objective function to become more realistic, it should also
incorporate, for example, the cost of labour for the hydrogen part of the refuelling
station.
The investment problem in Paper IV was modelled as an open-loop system in
the sense that the entire investment strategy was decided upon in advance. Such a
system can, optimally, perform equally well as a closed-loop system [93]. How-
ever, in order to increase robustness, a closed-loop strategy could be used. This
would allow for the strategy to change in accordance with revealed uncertainty
in the demand. Furthermore, the large amount of data from stochastic multi-
objective optimisations can be a problem. Ef?cient use of the method for decision
support requires a higher degree of aggregation of the results than that done in
Paper IV.
In both Paper II and Paper IV, the environmental measures are implicit, i.e. present
through the use of an environmentally friendly technique. Another option would
be to have explicit environmental ?gures in the objective functions. In Paper IV,
which is a multi-objective problem solved with a GA, such objective functions
could easily have been used. If explicit environmental measures can be derived
and parametrised for different investment times, the techniques from Paper IV can
be applied.
Different options in the scenario generation procedures can be explored. At
present, a Poisson distribution is used to generate samples from the scenarios.
Other methods, e.g. ARIMA models or neural networks, could also be evaluated.
6.1. FUTURE WORK 53
Another possibility is to merge the three existing scenarios into one where the
long-term behaviour in terms of the number of vehicles is governed by a set of
stochastic variables. These variables can be tuned in accordance to expectations
regarding the future development.
Chapter 7
Summary of appended papers
7.1 Paper I
In cement manufacturing, according to the law, the effect of any change to the pro-
duction process must be investigated before the modi?ed process is implemented.
Such changes might involve type of sand, fuel or additives. Recently, Cementa
AB, a major cement producer in Sweden, started to investigate alternative, more
environmentally friendly types of fuel. In addition the company also started to
improve the understanding of the involved physics and chemistry, which turned
out to be complex. Today the veri?cation comprises a calculation of produced
emissions, but in the future other types of calculations would be needed.
In this paper a ?exible model is developed which ful?lls the requirements
above. A computationally acausal model made it possible to separate the model
describing the cement manufacturing process from the problem formulation. The
model was built in ASCEND [46], which is an object-oriented, mathematically
based modelling language as well as a multi-purpose simulation and calculation
environment. In order to further enhance ?exibility, the model was designed with
a high degree of redundancy, so that the quantity of one physical property is ex-
pressed through a number of linked equations. This gives the user freedom to
choose how to assign the physical property. In addition the model also fully traces
the total cost throughout the production line.
7.2 Paper II
Running vehicles on hydrogen rather than petrol could lead to less environmen-
tally hazardous emissions in a global perspective, especially if the hydrogen is
made from renewable energy. Techniques for producing and storing the hydro-
gen, as well as fuel cells to convert the hydrogen into electricity, are constantly
55
56 CHAPTER 7. SUMMARY OF APPENDED PAPERS
being improved. One of the most signi?cant dif?culties in the introduction of hy-
drogen vehicles today concerns the infrastructure that must be built. Considering
the fact that all present refuelling stations for petrol need to be replaced, the total
investment is huge. In this situation it is crucial to employ the most pro?table
investment strategy, given the probable future development.
In this paper the lowest production cost for a set of investments over a period of
20 years for an individual hydrogen refuelling station is found. For ?exibility and
convenience of transportation, the refuelling station utilises an on-site reformer for
natural gas. The ?rst case investigated assumes a constant production of hydrogen
and will yield the minimal cost, whereas the second one can be used when special
considerations like periodic stop for maintenance of the hydrogen reformer need
to be taken into account. Both optimisation problems are shown to be convex and
hence produce the global optimal point. The result is a hydrogen production cost
of 4-6 USD/kg, comparable to the results of other studies. The major difference
is that this study uses an increasing function to estimate the number of hydrogen
vehicles refuelling at the station, and the estimated production cost is obtained as
a time average. In other studies, the cost has been based on maximum utilization.
7.3 Paper III
In this paper, discrete-time prediction networks (DTPNs), a novel type of recurrent
neural networks, are introduced and applied to the problem of macro-economic
time series prediction. The DTPNs are optimized using a genetic algorithm (GA)
that allows both parametric and structural mutations. Due to the feedback cou-
plings present in the DTPNs, such networks are capable of a rudimentary short-
term memory.
The results from applications involving two time series, namely the Fed Funds
interest rate and US GDP, indicate that DTPNs are capable of one-step prediction
with higher accuracy than several other benchmark methods. Thus, even though
the data sets contain a large amount of noise, the study in Paper III indicates that
there is more information available in the time series than can be extracted using,
e.g. feedforward neural networks or ARIMA models.
In addition, an investigation of predictability was carried out. Here, the DTPNs
were required not only to make a one-step prediction, but also to provide an esti-
mate of the accuracy of the prediction. However, it was found that the discrepancy
between the predictions obtained from the DTPNs and the actual data points in the
time series consisted of noise, indicating that the DTPNs indeed extract almost all
the available information from the time series.
7.4. PAPER IV 57
7.4 Paper IV
Paper IV concerns the problem of ?nding the optimal investment strategy for a
single hydrogen and hythane refuelling station giving the minimum production
cost while trying to match the hythane and hydrogen capacity to a demand gen-
erated from three future stochastic scenarios over a 20-year period. Hydrogen is
a promising fuel for vehicles. However, one of the major barriers is the lack of a
hydrogen infrastructure. An important component of the hydrogen infrastructure
is the individual hydrogen refuelling station. The long-term pro?tability of the hy-
drogen ?lling station is a key issue for the success of the transition to a hydrogen
infrastructure. The resulting minimal expected production cost lies between 2-6
USD/kg for hydrogen and 1-1.5 USD/kg for hythane, depending on preferences
for unsatis?ed demand, ?exibility etc. The results are meant to be used as decision
support when planning new refuelling stations.
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Paper I
The design and building of a life cycle-based process
model for simulating environmental performance,
product performance and cost in cement
manufacturing
in
Journal of Cleaner Production, Volume 12, Issue 1, February 2004, pp. 77-93.
Journal of Cleaner Production 12 (2004) 77–93
www.cleanerproduction.net
The design and building of a life cycle-based process model for
simulating environmental performance, product performance and
cost in cement manufacturing
Karin Ga¨bel
a,b,?
, Peter Forsberg
c
, Anne-Marie Tillman
a
a
Environmental Systems Analysis, Chalmers University of Technology, S-412 96 Gothenburg, Sweden
b
Cementa AB, Research & Development, P.O. Box 144, S-182 12 Danderyd, Sweden
c
CPM—Centre for Environmental Assessment of Product and Material System, Chalmers University of Technology, S-412 96 Gothenburg,
Sweden
Received 3 April 2001; received in revised form 26 June 2002; accepted 11 December 2002
Abstract
State of the art Life Cycle Inventory (LCI) models are typically used to relate resource use and emissions to manufacturing and
use of a certain product. Corresponding software tools are generally specialised to perform normalisation of the ?ows to the
functional unit. In some cases it is, however, desirable to make use of the LCI model for other types of environmental assessments.
In this paper, an alternative modelling technique resulting in a more ?exible model is investigated. We exemplify the above by
designing and building a model of a cement plant. The commissioner’s, in this case Cementa AB, requirements on a ?exible model
that generates information on environmental performance, product performance and the economic cost were seen as important. The
work reported here thus has two purposes; on the one hand, to explore the possibility for building more ?exible LCI models, and
on the other hand, to provide the commissioner with a model that ful?ls their needs and requirements. Making use of a calculational
a-causal and object-oriented modelling approach satis?ed the commissioner’s special requirements on ?exibility in terms of modu-
larity and the types of calculations possible to perform. In addition, this model supports non-linear and dynamic elements for future
use. The result is a model that can be used for a number of purposes, such as assessment of cement quality and environmental
performance of the process using alternative fuels. It is also shown that by using the above modelling approach, ?exibility and
modularity can be greatly enhanced.
? 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Life-cycle-simulation; Predict; Consequences; Process model
1. Introduction
The interest in environmental issues, as well as the
pressure on industries to develop more environmentally
preferable products and processes, is constantly increas-
ing. This drives product and process development
towards more sustainable practices. However, products,
processes and production systems are always developed
taking cost and product performance into consideration.
Thus, there is a growing need for tools to predict and
?
Corresponding author. Tel.: +46-8-625-68-22; fax: +46-8-625-
68-98.
E-mail address: [email protected] (K. Ga¨bel).
0959-6526/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0959-6526(02)00196-8
assess both the environmental performance and the econ-
omic cost and the product performance of alternative
production operations.
The purpose of this paper is to describe how we
designed and built a ?exible model for process and pro-
duct development in the cement industry. The model
predicts the environmental performance, the economic
cost and the product performance by simulating different
operational alternatives for producing cement. The needs
and requirements were speci?ed by the cement industry.
These are outlined in Section 3. We give our interpret-
ations as a conceptual model in Section 4. We chose the
modelling approach and simulation tool and describe
how we designed and built the model in Section 5. We
end Section 5 by testing the tool in two real cases. The
78 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
results of these tests show that the modelling approach
used can generate a potentially powerful tool.
A life cycle perspective (“cradle to gate”) was used
to assess the environmental consequences of process and
product changes, in order to avoid sub-optimisation. The
conceptual model represents the cement manufacturing
process from cradle to gate. However, the model in this
paper, the construction of and test of we describe in
detail, represents the gate to gate part of the manufactur-
ing process. Environmental performance is described in
terms of environmental load (resource use and
emissions). Economic cost is described in terms of the
company’s own material cost and production cost. Pro-
duct performance is expressed as cement composition.
The product performance is used to determine whether
or not the operational alternative is feasible. Environ-
mental load and economic cost have to be related to a
feasible operational alternative and product.
Cementa AB, the cement manufacturer in Sweden and
the commissioner of the study, has previous experience
of Life Cycle Assessment (LCA) through a Nordic pro-
ject on Sustainable Concrete Technology [1]. In that pro-
ject, several LCA studies were carried out on cement,
concrete and concrete products [2,3,4,5,6]. One con-
clusion drawn from the project was that life cycle assess-
ment is a tool, with the potential for improvement, to be
used to avoid sub-optimisation in the development of
more environmentally adapted cement and concrete pro-
ducts and manufacturing processes [1]. Several other
LCA’s of cement, concrete and concrete products have
also been carried out [7,8,9,10].
However, there are limitations with today’s LCA. One
important limitation, from an industrial perspective, is
that social and economic bene?ts of industrial operations
are not taken into account. Another limitation of present
LCI modelling is its limited capability to perform differ-
ent types of simulations. There are limits on the possi-
bility of changing process variables without changing the
underlying model. Usually a new model is built for each
operational alternative simulated. In addition, LCI mod-
els are usually de?ned as linear and time independent.
2. Background
2.1. Cement manufacturing and related environmental
issues
The cement manufacturing process, shown in Fig. 1,
consists of the following main steps: limestone mining,
raw material preparation, raw meal grinding, fuel prep-
aration, clinker production, cement additives preparation
and cement grinding. Clinker is the intermediate product
in the manufacturing process. The following description
is based on the manufacturing process at Cementa’s Slite
plant. The cement manufacturing process at the Slite
Fig. 1. Cement manufacturing process.
plant is described in detail in the report “Cement Manu-
facturing — Process and Material Technology and
Related Environmental Aspects” [11].
Limestone, the main raw material is mined and
crushed. Other raw materials used may be sand, iron
oxide, bauxite, slag and ?y ash. The raw materials are
prepared and then proportioned to give the required
chemical composition, and ground into a ?ne and homo-
geneous powder called raw meal.
Various fuels can be used to provide the thermal
energy required for the clinker production process. Coal
and petroleum coke are the most commonly used fuels
in the European cement industry [12]. A wide range of
other fuels may be used, e.g. natural gas, oil and differ-
ent types of waste, e.g. used tyres, spent solvents, plas-
tics, waste oils. The fuels are processed, e.g. ground,
shredded, dried, before being introduced into the pro-
cess.
Clinker production is the “heart” of the cement manu-
facturing process. The raw meal is transformed into
glass-hard spherically shaped minerals clinker, through
heating, calcining and sintering. The raw meal enters the
clinker production system at the top of the cyclone tower
and is heated. Approximately half of the fuel is intro-
duced into the cyclone system, and at about 950° C the
carbon dioxide bound in the limestone is released, i.e.
the calcination takes place. The calcined raw meal enters
the rotary kiln and moves slowly towards the main
burner where the other half of the fuel is introduced.
Raw materials and fuels contain organic and inorganic
matters in various concentrations. Normal operation of
the kiln provides high temperature, a long retention time
and oxidising conditions adequate to destroy almost all
organic substances. Essentially all mineral input, includ-
ing the combustion ashes, is converted into clinker. How
metals entering the kiln behave depends largely on their
volatility. Most metals are fully incorporated into the
product, some precipitate with the kiln dust and are cap-
tured by the ?lter system, and some are present in the
exhaust gas.
Inter-grinding clinker with a small amount of gypsum
produces Portland cement. Blended cement contains, in
addition, cement additives such as granulated blast fur-
79 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
nace slag, pozzolanas, limestone or inert ?ller.
Depending on their origin, the additives require differ-
ent preparations.
The exhaust gases leaving the clinker production sys-
tem are passed through a dust reduction device before
being let out through the stack. The dust is normally
returned to the process. The clinker production system
is the most important part of the manufacturing process
in terms of environmental issues. The main use of energy
is the fuel for clinker production. Electricity is mainly
used by the mills and the exhaust fans. The emission to
air derives from the combustion of fuel and the trans-
formation of raw meal into clinker. Apart from nitrogen
and excess oxygen, the main components of kiln exhaust
gas are carbon dioxide from the combustion of fuel and
the calcination of limestone and water vapour from the
combustion process and raw materials. The exhaust gas
also contains dust, sulphur dioxide, depending on sul-
phur content of the raw materials, small quantities of
metals from raw material and fuel, and remnants of
organic compounds from the raw material.
The emissions to air from the clinker production sys-
tem largely depend on the design of the system and the
nature and composition of the raw material and fuel [11].
The raw material and fuel naturally vary in composition
and the content of different compounds have a different
standard deviation. The emissions of metals depend on
the content and volatility of the metal compound in the
raw material and fuel. The metal content varies over time
and consequently so does the metal emission.
The Nordic study “LCA of Cement and Concrete —
Main Report” points out emissions of carbon dioxide,
nitrogen oxides, sulphur dioxide and mercury, and the
consumption of fossil fuel as the main environmental
loads from cement production [6]. According to the Eur-
opean Commission, the main environmental issues asso-
ciated with cement production are emissions to air and
energy use [13]. The key emissions are reported to be
nitrogen oxides, sulphur dioxide, carbon dioxide and
dust.
2.2. Means and work done to minimise negative
environmental impact
The negative environmental impact from cement
manufacturing and cement can be minimised in numer-
ous ways. These can be grouped into four categories:
? Substituting input, raw materials, fuels and cement
additives, to the process.
? Process development; optimise and develop the exist-
ing process.
? End-of-pipe solutions; adding emission reduction sys-
tems.
? Product development; develop new products or
change cement composition and performance.
Many of these solutions have consequences outside
the actual cement manufacturing plant, both upstream as
well as downstream. Therefore, the life cycle perspective
is necessary to assess the environmental consequences
of process and production changes in order to avoid
sub-optimisation.
Examples of environmental improvement measures
taken at the Slite plant in recent years are given in the
following, in order to give examples of technical devices
and measures the model should be able to deal with.
Different types of waste are used, e.g. used tyres, plas-
tics, spent solvents, waste oils, as substitutes for tra-
ditional fuels to reduce the consumption of virgin fossil
fuels and the emission of carbon dioxide. The goal is to
replace 40% of the fossil fuel with alternative fuel [14]
by 2003. Cementa is also looking into the possibility of
using alternative raw materials, i.e. recovered materials,
to substitute for traditional, natural raw materials. The
alternative raw materials can either be used as raw
material in the clinker production process or as cement
additives, i.e. to substitute for clinker in cement grinding.
In 1999 a new type of cement, “building cement”, was
introduced on the Swedish market. Building cement is
a blended cement with about 10% of the clinker replaced
with limestone ?ller. The environmental bene?ts of sub-
stituting limestone ?ller for clinker are a reduction in the
amount of raw meal that has to be transformed into
clinker, and consequently less environmental impact
from the clinker production process, raw material and
fuel preparation. The environmental impact per ton
cement has been reduced by 10% [15].
The use of alternative material and fuel at the cement
plant requires pre-treatment, transport and handling, and
affects the alternative treatment of waste and by-pro-
ducts. New materials and fuels lead to new combinations
and concentrations of organic and inorganic compounds
in the clinker production system, which in turn lead to
new clinker- and exhaust gas compositions.
As an end of pipe-solution, a Selective Non Catalytic
Reduction system (SNCR) to reduce nitrogen oxide
emissions was installed at the Slite plant in 1996. In
1999, a scrubber was taken into operation to reduce sul-
phur dioxide emissions. In the scrubber, SO
2
is absorbed
in a slurry consisting of limestone and water. The separ-
ated product is used as gypsum in the cement grinding.
3. The commissioner’s needs and requirements on
the model
The commissioner’s, Cementa AB, needs and require-
ments, as interpreted from discussions with representa-
tives from different departments, are outlined in this sec-
tion.
Cementa AB needs a tool to predict and assess pro-
duct performance, environmental performance and econ-
80 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
omic cost of different operational alternatives for pro-
ducing cement. The tool is to be used to support
company internal decisions on product and process
development and strategic planning through generating
and assessing operational alternatives. Another speci?c
use is as a basis for government permits. To get permits
for test runs of new raw materials and fuels, information
on the expected outcome is needed.
Cementa intends to learn about the system and the
system’s properties regarding product performance,
environmental performance and economic cost and the
relations between these parameters. The life cycle per-
spective is seen as important. Cementa wants to be able
to simulate combinations of raw materials, fuels and
cement additives in combination with process changes
and end-of-pipe solutions. For all tested combinations,
information about the system’s predicted properties
should be generated and assessed in relation to feasibility
criteria, such as product performance, emission limits
and economic cost. Product performance is regarded as
the most important criterion.
The commissioner gave the following two examples
of how to use the tool. They asked for speci?c and
detailed information about the predicted consequences
for each alternative.
A Produce a given amount of cement, given the raw
material mix, the fuel mix and fuel demand, and the
cement additive mix. What is the product perform-
ance of the cement, the environmental performance
and the economic cost?
B Produce a given amount and type of cement, given
the fuel mix and fuel demand, the cement additive
mix and the available raw materials. What raw
material mix is required? What are the environmental
performance and the economic cost?
Concrete with different strength developments needs
different amounts of cement. Therefore, it should be
possible to state the amount of cement produced in the
operational alternative simulated. The environmental
performance should be described as environmental load,
i.e. as resource use, emissions to air and water, and
waste. The composition of the kiln exhaust gas from
clinker production should be described. The composition
of all raw material, fuel, intermediate products and pro-
ducts should be described and possible to evaluate. The
product performance should be described with three
ratios; the lime saturation factor (LSF), the silica ratio
(SR), and the alumina ratio (AR), used in the cement
industry as measures of cement composition. The ratios
describe the relation between the four main components
and are shown in Table 1. The total material and pro-
duction cost in “SEK” per amount cement produced
should be calculated. The accumulated material and pro-
duction cost should be available to study after each step
in the cement manufacturing process; both as cost per
amount cement produced and as cost per kilo of the
intermediate product.
Cementa produces cement at three plants in Sweden.
The different plants use the same main production pro-
cess as described in Section 2.1. However, there are vari-
ations between the plants, especially in the design of the
clinker production system. Variations are mainly due to
the nature of the available raw material, when the plant
was built, modi?cations done and the installation of dif-
ferent emission reduction systems. It should be easy to
adapt the tool to represent any of the commissioner’s
cement manufacturing processes, although the ?rst
model was intended to represent the Slite plant.
The content of metal compounds in the raw material,
and the standard deviation of the metal content, vary
depending on the location of the plant. Thus, the emis-
sions of metals to air vary from one plant to another.
Emission of metals from clinker production should be
included in the ?rst model, but they are not in focus.
However, in the next stage, when site-speci?c models of
each plant are developed, the level of detail with which
metal emissions are described, should be further
increased.
The cement manufacturing process is by nature non-
linear and dynamic. The tool should describe stable state
conditions and describe the static and linear transform-
ation of raw material and fuel into clinker. The tool has
to have development potential to include the non-linear
transformations in the process. In addition, there should
also be the potential to simulate dynamic behaviour, e.g.
during start-up and shut down of the kiln.
4. Conceptual model and system boundary
Based on the commissioner’s requirements, a concep-
tual model was constructed, as presented in the follow-
ing:
To avoid sub-optimisation, the model was to be from a
life cycle perspective. The raw material, fuel and cement
additives used are to be traced upstream to the point
where they are removed as a natural resource. Alterna-
tive raw materials, fuels and cement additives are by-
products or waste from other technical systems. The pro-
duction of these alternative products is not to be
included. However, the additional preparation, handling
and transport to make them ?t the cement industry is to
be included. The cement is to be followed to the gate
of the cement plant.
The cement manufacturing system has been divided
into a background system and a foreground system [16].
The foreground system represents Cementa’s “gate to
gate” part of the system. Cementa can, in detail, control
and decide on processes in the foreground system, but
can only make speci?cations and requirements on pro-
81 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
Table 1
Product performance (cement-, clinker-, raw meal ratios)
Ratio Denomination Formula
Lime saturation factor LSF LSF=(100CaO)/(2,8SiO
2
+1,1Al
2
O
3
+0,7Fe
2
O
3
)
Silica ratio SR SR=(SiO
2
)/(Al
2
O
3
+Fe
2
O
3
)
Alumina ratio AR AR=(Al
2
O
3
)/(Fe
2
O
3
)
Note: CaO, SiO
2
, Al
2
O
3
and Fe
2
O
3
are all expressed in weight percentage.
ducts from the background system. Depending on
whether the additional preparation, handling and trans-
port is done by Cementa or not, the processes are either
in the foreground system or the background system. The
conceptual model, in Fig. 2, shows the foreground and
background systems, and in addition a wider system. The
wider system shows consequences of actions taken at the
cement plant, which exist, but are not modelled.
The foreground system was divided into the follow-
ing processes:
? Lime- and marlstone extraction, mining and crushing;
? Sand grinding;
? Raw meal grinding;
? Coal and petroleum coke grinding;
? Clinker production;
? Cement grinding and storage.
Between each one of these processes, intermediate
Fig. 2. Conceptual model.
homogenisation, transportation and storage might take
place and, where applicable, are accounted for.
The background system consists of the following pro-
cesses:
? Production and transport of sand and other raw
material;
? Additional preparation of alternative raw materials
and transport to the cement plant;
? Production and transport of traditional fuels;
? Additional preparation of waste to convert them into
fuels for cement manufacture and transport to the
cement plant;
? Production and transport of cement additives;
? Additional preparation of alternative cement and
transport to the cement plant;
? Production of electricity.
The plant in Slite produces waste heat used for district
82 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
heating in Slite. The waste heat is accounted for as an
output, a product, but no credit is given to the cement
production through allocation or system enlargement. In
the same way, when alternative raw materials and fuels
are used in cement manufacturing, the amount of waste
thus disposed of is accounted for, but no allocation is
made. These consequences of the cement manufacturing
process are placed in the wider system in the concep-
tual model.
Not considered are:
? Production and maintenance of capital equipment for
manufacturing and transport;
? Extraction and production of alternative raw
materials, fuels and cement additives;
? Working material, such as explosives, grinding media
and refractory bricks;
? Iron-sulphate used in the cement milling to reduce
chromium;
? Of?ces.
The two systems were modelled with different tech-
niques and level of detail. The foreground system model
was built according to the techniques described in the
next section. For the background system, traditional life
cycle inventory (LCI) techniques [17] were used. Pro-
duct performance and economic cost were taken into
account by assigning the products entering the fore-
ground system a chemical composition and a cost. Sub-
sequently, ?ows entering the foreground system are
described as a ?ow of mass (kg/s), cost (SEK/s) and
thermal energy content (MJ/s) with a composition
according to Table 2, and in accordance with the pur-
chase deal. Flows of material in the background system
are de?ned and described as a ?ow of mass (kg/s).
The environmental load (resource use and emissions)
Table 2
Material and fuel composition
Compound Unit Compound Unit
CaO weight-share As, arsenic weight-share
SiO
2
weight-share Cd, cadmium weight-share
Al
2
O
3
weight-share Co, cobolt weight-share
Fe
2
O
3
weight-share Cr, chromium weight-share
MgO weight-share Cu, copper weight-share
K
2
O weight-share Hg, mercury weight-share
Na
2
O weight-share Mn, manganese weight-share
SO
3
(sulphides and organic in raw material) weight-share Ni, nickel weight-share
SO
3
(sulphates in raw material) weight-share Pb, lead weight-share
SO
3
(in fuel) weight-share Sb, antimony weight-share
Cl weight-share Se, selenium weight-share
C (in traditional fuel) weight-share Sn, tin weight-share
C (in alternative fuel) weight-share Te tellurium weight-share
C (in raw material) weight-share Tl, thallium weight-share
Organic (in raw material) weight-share V, vanadium weight-share
Moist (105° C) weight-share Zn, zinc weight-share
was described according to the parameters in Table 3.
The kiln exhaust gas from the clinker production system
was described using the parameters in emission to air in
Table 3. The transport was expressed both in ton kilo-
metres and as the related environmental load, according
to the parameters in Table 3.
Table 3
Environmental load, resource use and emissions to air and water
Resource use
Raw material, kg
Alternative raw material, kg
Fuel, kg and MJ
Alternative fuel, kg and MJ
Water, kg
Emission to air Hg, mercury
CO
2
, carbon dioxide Mn, manganese
NO
x
, nitrogen oxides (NO and NO
2
as NO
2
) Ni, nickel
SO
2
, sulphur dioxide Pb, lead
CO, carbon monoxide Sb, antimony
VOC, volatile organic compounds Se, selenium
Dust Sn, tin
As, arsenic Te, tellurium
Cd, cadmium Tl, thallium
Co, cobolt V, vanadium
Cr, chromium Zn, zinc
Cu, copper
Emission to water
BOD, biological oxygen demand
COD, chemical oxygen demand
Total N, total nitrogen content
Non elementary in-?ow, “?ows not followed to the cradle”
Alternative raw material and fuel
Non elementary out-?ows, “?ows not followed to the grave”
Industrial surplus heat, MJ
83 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
5. Modelling and simulation
This section starts by interpreting the commissioner’s
requirements in a system technical context. Only the
foreground system is considered in the following. The
result is a set of decisions on the modelling and the
simulation techniques. This is followed by a description
of how the model was built in accordance with these
techniques and, ?nally, how the constructed model
was validated.
5.1. System technical interpretation
To predict the performance of the desired type of
operational alternatives it was concluded that we had to
simulate them, i.e. perform calculations on a model rep-
resenting the cement manufacturing plant. A model is,
here, a mathematical description of any real subject. A
simulation is then any kind of mathematical experiment
carried out on the model.
The requirements on the model indicate the necessity
of keeping these simulations ?exible in the sense that it
should be possible to predict a number of aspects of the
plant, depending on the situation. Examples of static
equilibrium calculations that are given in Section 3
include:
A Setting the percentage of each raw material in the raw
meal and each fuel in the fuel mix used. Then calcu-
lating the percentage of raw meal mix and fuel mix,
the produced cement quality, emissions and economic
cost under the constraint that the fuel provides all the
thermal process energy. This means we give all the
materials necessary to produce cement and then
watch what comes out of the process.
B Setting properties of the produced cement and each
fuel in the fuel mix used. Then calculating the per-
centage of each raw material in the raw meal mix,
the percentage of raw meal mix and fuel mix, emis-
sions and economic cost under the constraint that the
fuel provides the process thermal energy. This means
we want to control properties of the cement produced
and calculate the proportions of the raw materials,
under the same constraint for the fuel to provide
enough thermal heat.
In a mathematical model, numerical parameters can
be divided into the following categories:
? Constants. Are set when the model is built and then
remain.
? Locked variables. Parameters set to a numerical value
throughout a certain simulation, in accordance to
input data.
? Free variables. Parameters that will be calculated in
the simulation. Some of these are internal variables
in the model and others are the ones we want to calcu-
late; the output.
The difference between the above cases is which para-
meters are locked and which are free. This controls how
the simulation is carried out, i.e. how the equations for
simulation are formulated. The two static equilibrium
cases above will result in different sets of equations. A
simultaneous solving of a respective set of equations will
render the result. It is indeed possible to make these cal-
culations with any general mathematical package avail-
able. If so, each of the cases has to be treated separately.
The result is a well functioning simulation for the spe-
ci?c case that cannot, however, be used for other differ-
ent simulations. If so, the equations need to be re-formu-
lated. Since a speci?c requirement was ?exibility in the
calculations that are possible to perform, we will re?ne
our modelling method by a separation of the model, or
what is normally thought of as the model, into three
parts, namely:
? A neutral model. Only the model, i.e. a description
of our system, in which the connecting equations are
expressed in a neutral form. The model maps our
interpretation of the plant onto a mathematical formu-
lation, but it does not include any speci?c problem to
be solved, hence it is called neutral.
? A problem formulation. An explicit list of which
parameters to lock and a value with which to desig-
nate each of them.
? A simulation method, which is the calculation method
chosen, can also be considered to be a part of the
problem formulation.
The most powerful way to achieve this separation is
to remove the calculational causality (CC) from the
model [18]. The CC determines the order in which the
equations included in a simulation are calculated. This
is merely a technical consideration and affects only the
order in which the calculations are done and does not
imply any restrictions or special considerations regard-
ing the nature or contents of the system behind the
model. The resulting model is said to be a-causal, or
non-causal, in that nothing is said about the order of
calculation in future simulations with the model. The
model can be regarded mathematically as a number of
equilibrium equations connected to each other.
Another important aspect of ?exibility for the model
is modularity. In order to be truly ?exible, according to
the requirement regarding adjustments to represent dif-
ferent cement plants, the model has to be easy to re-
build. In most practical cases, changes would probably
be limited to assigning different inputs and performing
different kinds of simulations, which would already be
part of the problem formulation. In some cases, this is
not enough and the underlying model structure needs to
84 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
be altered. Changing the number of raw materials or
fuels is one such case, and adopting it to ?t a cement
manufacturing plant with different designs is another. A
step to create modularity has already been taken by mak-
ing the model a-causal. This is merely a theoretical pre-
requisite and will not, in itself, produce a ?exible model.
On the other hand, if this is combined with an object
oriented modelling language, we will end up with a prac-
tical, easily re-combinable model. The paradigm of
object orientation is something that affects the language
the model is expressed in. This includes a natural way
to keep parts that are separate in reality as separate
objects in the model, so that the model resembles reality,
or a suitable picture of reality. Usually this feature is
used to group sub-parts of the model into objects, but it
is also useful to group ?ow entities together. Flows that
are made up of a number of substances can thus be
treated as an entity to enhance the transparency and ease
of comprehension.
The cement manufacturing process contains both parts
that vary over time and parts which cannot always be
suf?ciently described with a linear relation. One of the
requirements was to make it possible to account for these
properties in the future, so it must be possible to include
both dynamic and non-linear elements. The ?rst model
which is covered in this paper does not, however, contain
any dynamic or non-linear elements.
In addition to being able to include the above dynamic
elements of the model, we also need to perform dynamic
solving, i.e. calculate and trace (all) the variables in the
model over a certain time span. This simulation type can
be used for environmental predictions when, e.g., start-
ing up, shutting down or changing parameters in the
cement production process. The starting point for such
a simulation can be given values for a set of variables,
such as the start conditions for the plant when per-
forming a start up simulation. It can also be from a state
of equilibrium, which is the case when simulating a shut
down situation. In the latter case, we need a method to
determine this state of equilibrium, e.g. perform a steady
state solving. The steady state solving can, of course,
also be used on its own to ?nd stable points of operation
for the production plant. It is then equivalent to what in
LCI is generally called “normalisation of the life cycle”
or, speci?cally in ISO 14041 [17], “relating data to func-
tional unit”. In addition, another simulation type which
is mentioned for future use, is optimisation.
In summary, we have found that in order to ful?l the
requirements of the commissioner the model needs to be
?exible in terms of:
? Simulation — type of predictions that can be made:
static equilibrium, dynamic solving, etc.;
? Modularity — ease of combination into models of
other cement plants by re-arrangement of the parts;
? Transparency — all governing equations and resulting
?gures readily available to the user, even the
internal ones;
? Comprehension — easy to grasp and understand.
We have, thus, found that the following modelling
approach is needed:
? Calculational non-causal used to separate a neutral
model and the problem formulation;
? Physical modelling to keep physical entities together
in the model;
? Object oriented modelling language to enhance the
reusability of the model.
In addition, the model needs to support:
? Dynamic elements;
? Non-linear elements.
Simulation types the software tool needs to support:
? Steady state solving;
? Dynamic solving;
? Optimisation.
Not all of the requirements above are ful?lled with
state-of-the-art LCI techniques [19]. In LCI, it is gener-
ally enough to describe the life cycle with such a resol-
ution that it is suf?cient with a static and linear model.
Moreover, current LCA tools normally provide normal-
isation of the life cycle to the reference ?ow as the only
simulation alternative. Consequently, there are no LCA
related software tools available that can perform the
desired types of simulations. In the ?eld of general
simulation there are, however, a large number of tools
that can be used. Some equivalent examples include
OmSim [20], DYMOLA [21] and ASCEND [22]. These
software are of the kind that use computational non-cau-
sal models and allow a number of types of simulations
to be performed. For this application, ASCEND was
chosen based on the following criteria:
? It was possible to run on a PC, hence convenient
(DYMOLA, ASCEND);
? It had plug-in modules allowing user made simulation
types, hence ?exible (OmSim, DYMOLA,
ASCEND);
? It was freeware, hence economical (OmSim,
ASCEND).
5.2. Model construction
Building a model with the speci?cations and tech-
niques discussed above is more a matter of generalis-
85 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
ation than speci?cation. Most of the core components in
the model will hence re?ect the general behaviour of an
“object” or “function”. Later, these will be specialised
to the speci?c case, here the cement manufacturing plant.
This technique of extracting layers of behaviour is well
suited for object oriented implementation where the
mechanism of inheritance can be used for that purpose.
The general behaviours are implemented in base classes
and the more speci?c in inherited ones.
The ?rst step when building the model was to ?nd
the objects contained in our perception of the cement
manufacturing plant. This was already done in the con-
ceptual model. These objects then needed to be
abstracted into their general behaviour. Usually, this
reveals that a number of objects follow the same basic
rules, which then means that they can inherit from the
same base object.
First, the general functionality of parts in the concep-
tual model was extracted. Then, a number of general
objects were built to host the functionality. Focus was
put on the mechanisms behind the general functionality
and the correspondence with reality for the more speci?c
one. From the conceptual model, we found the objects
given in Table 4.
In the following, a detailed explanation of some of
these objects is given. The syntax used is based on the
ASCEND IV model language [22] but has been simpli-
?ed to only include the contents (semantic). All code is
given in another font (model). The word composition
thus means the model (object) composition as declared
in Table 5.
Table 4
Total listing of objects in the model
Name Inherits from Role
composition – Any kind of composition of a mixture
mass stream – Flow of material
materialfuel stream mass stream Flow of raw materials and fuels
kilnexhaustgas stream mass stream Flow of exhaust gas
chemical analyser – Test probe for speci?c cement ratios
materialfuel mixer – Mixer for n number of material fuel streams
rawmeal mixing materialfuel mixer Speci?c raw meal mixer at Slite
fuel mixing materialfuel mixer Speci?c fuel mixer at Slite
rawmealfuel mixing materialfuel mixer Speci?c raw meal fuel mixer at Slite
cement mixing materialfuel mixer Speci?c cement mixer at Slite
materialfuel grinder – General grinder for a material fuel stream
rawmeal grinder slite materialfuel grinder Speci?c grinder for raw meal at Slite
sand grinder slite materialfuel grinder Speci?c grinder for sand at Slite
lime grinder slite materialfuel grinder Speci?c grinder for lime at Slite
marl grinder slite materialfuel grinder Speci?c grinder for marl at Slite
coalpetcoke grinder slite materialfuel grinder Speci?c grinder for coal and pet coke mixture at Slite
cement grinder slite materialfuel grinder Speci?c grinder for cement at Slite
clinker production – General clinker production
clinker production slite clinker production Speci?c clinker production at Slite
cement model slite – Top level model over the Slite plant
Table 5
Syntax used in declaration of objects
Syntax Explanation
MODEL xyz Start declaration of the object xyz
Declarations: Part of object where declarations are given
abc IS A xyz; Declares abc as of type xyz
abc[n] IS A xyz; Declares abc as an array with n number of
elements of type xyz
Assignments: Part of object where constants are initiated
Rules: Part of object where the equations are given
FOR i IN abc END Loop where i get the contents of each
FOR; member in abc
SUM[abc] Compute the sum of all elements in abc
= Neutral equality. Used to express
equilibrium, i.e. that two expressions are
numerically equal. It is not an assignment
and does not imply any order of calculation,
e.g. left to right.
5.2.1. Composition
This object is used to represent any kind of compo-
sition of a mixture. A list is used to contain the name
of each component in the mixture (compounds). The
weight share of each component is given as a fraction
with the range of 0 to 1 (y[compounds]). To be able to
handle redundant descriptions (where the weight of the
parts differs from that of the whole), no limitation is put
on the fractions to sum up to 1.0. The object also con-
tains the cost (cost) and heat content (heat) per mass
unit of the total mixture. The typical usage of this object
is to declare the contents of a material, such as a raw
material, fuel or a product.
86 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
MODEL composition
Declarations:
compounds IS A set OF
symbol constant;
y[compounds] IS A fraction;
cost IS A cost per mass;
heat IS A
energy per mass;
Note: The contents of the compounds list is not yet
speci?ed.
5.2.2. Mass stream
The mass stream is a ?ow of material where the con-
tent is declared by a composition (state). The ?ow rate
is expressed both as total ?ow (quantity) and ?ow of
each of the contained components (f). For convenience
(easier access at higher levels), the list of components
in the ?ow is repeated (compounds). It is, then, declared
equivalent to the one already present within state to pre-
vent deviating values.
The two ways of describing the ?ow can be expressed
in terms of each other and, thus, are not independent of
each other. In fact, for all components the ?ow of each
component equals the total ?ow times the fraction for
the component in question (f = quantity?state.y).
MODEL mass stream
Declarations:
compounds IS A set OF
symbol constant;
state IS A composition;
quantity,f[compounds] IS A mass rate;
Rules:
compounds, ARE THE SAME;
state.compounds
FOR i IN compounds f def: f =
CREATE quantity?state.y;
END FOR;
5.2.3. Material–fuel stream
The material–fuel stream is a specialisation of the
mass-stream declared above. It represents the ?ow of
raw materials and fuels in the cement manufacturing pro-
cess. It takes all relevant materials into account, as
de?ned in Table 2, and permits these to be described
either as a share or mass per time. Here, the share option
is used to declare the weight share of each component.
The material–fuel stream also carries the associated cost
and heat.
MODEL materialfuel stream REFINES
mass stream
Declarations:
cost IS A cost per time;
heat IS A energy rate;
Assignments:
Compounds:= [‘CaO’,‘SiO2’,‘Al2O3’
,‘Fe2O3’,‘MgO’,‘K2O’
,‘Na2O’,‘SO3sulphides’
,‘SO3sulphates’,‘SO3fu
el’,‘Cl’,‘Ctrad’,‘Calt’
,‘Craw’,‘Moist’,‘Organi
c’,‘As’,‘Cd’,‘Co’,‘Cr’
,‘Cu’,‘Hg’,‘Mn’,‘Ni’
,‘Pb’,‘Sb’,‘Se’,‘Sn’,‘Te’
,‘Tl’,‘V’,‘Zn’];
Rules:
cost = quantity?state.cost;
heat = quantity?state.heat;
5.2.4. Kiln exhaust gas stream
The exhaust gas from the clinker production system
is modelled as a ?ow representation of its own. The
components are speci?ed with the mass ?ow, e.g. kg/s.
The components are de?ned in Table 3. The kiln exhaust
gas stream is a specialisation of the mass-stream, to
which the appropriate compounds have been added as
described below.
MODEL kilnexhaustgas stream REFINES
mass stream
Assignments:
Compounds:= [‘CO2raw’,‘CO2trad’
,‘CO2alt’,‘CO’,‘VOC’
,‘NOx’,‘SO2’,‘vapour’
,‘As’,‘Cd’,‘Co’,‘Cr’,‘Cu
’,‘Hg’,‘Mn’,‘Ni’,‘Pb’
,‘Sb’,‘Se’,‘Sn’,‘Te’,‘Tl’
,‘V’,‘Zn’];
5.2.5. Chemical analyser
A chemical analyser is a sort of test probe for product
performance. It describes the product performance in the
ratios used in the cement industry, i.e. Lime Saturation
Factor (LSF), Silica Ratio (SR) and Alumina Ratio (AR).
De?nitions of these are given in Table 1.
The analyser is modelled as a stand-alone object and
can be connected to any material fuel stream compo-
sition object in order to measure the performance.
MODEL chemical analyser
Declarations:
state IS A composition;
LSF IS A factor;
87 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
SR IS A factor;
AR IS A factor;
Rules:
LSF = 100?state.y[‘CaO’]/(2.8?state
.y[‘SiO2’]+1.1?state.y[‘Al2O
3’]+0.7?state.y[‘Fe2O3’]);
SR = state.y[‘SiO2’]/(state.y[‘Al2
O3’]+state.y[‘Fe2O3’]);
AR = state.y[‘Al2O3’]/state.y[‘Fe
2O3’];
The analyser can also be used to control the ratios of
a certain material–fuel stream. In such a case, the ratios’
parameters (LSF, SR and AR) can be set and there-
after locked.
5.2.6. Material–fuel mixer
A mixer object transforms two or more in?ows of
material into one out?ow and thus is an n-to-1 junction
for material–fuel streams. It can be used to mix a number
of material–fuel streams in ?xed percentages or to have
these percentages calculated, depending on settings. The
number of inputs (n inputs) must be set before the
object is used. The number of fractions
(mix part[1..n inputs]) equals the number of inputs.
Independent of the number of inputs, there is only one
output (out). The list of components (compounds) in
the inputs and the output are equivalent. For each
component, the output ?ow is the sum of the inputs
(out.f = SUM[in[1..n inputs].f]), or
f
out
?
?
n
inputs
i ? 1
f
in(i)
The mass balance for each individual component
must be maintained. (in[j].quantity =
mix part[j]?out.quantity). An additional constraint is
that the input fractions must sum up to 1.0
(SUM[mix part[1..n inputs]] = 1.0). The heat contents
and economic cost thus must be calculated separately.
Here, they are both expressed so that the respective cost
and heat for the output equals the sum of the input cost
and heat.
MODEL materialfuel mixer
Declarations:
n inputs IS A
integer constant;
in[1..n inputs], out IS A materialfuel
stream;
mix part[1..n inputs] IS A fraction;
Rules:
in[1..n inputs].compo ARE THE SAME
unds, out.compounds ;
FOR i IN cmb: out.f =
out.compounds SUM[in[1..n input
CREATE s].f];
END FOR;
FOR j IN mix[j]:
[1..n inputs] in[j].quantity =
CREATE mix part[j]?out
.quantity;
END FOR;
SUM[mix part[1..n inputs]]=1.0;
out.cost=SUM[in[k].cost | k IN
[1..n inputs]];
out.heat=SUM[in[k].heat | k IN
[1..n inputs]];
5.2.7. Material–fuel grinder
The material–fuel grinder represents grinding raw
meal, clinker, etc., and transforms one in?ow of coarse
material into one out?ow of ground material. Grinding
consumes electrical energy according to the mass
ground. The energy constant (ED) is used to calculate
total electrical power consumption
(electricity consumption). The quantity decreases due
to dust generation that is given by a dust-generating con-
stant (DG) de?ned as a fraction of the out quantity. A
total cost adding is modelled as a ?xed cost per mass
unit (COST) to cover maintenance and operation plus
the cost of electricity. This total cost is then added to
the cost for the material entering the grinder so that the
speci?ed material cost always corresponds to the cumu-
lated production cost at the speci?ed location.
The compositions of the input and output material–
fuel stream (in and out) are the same. The heat content
is not changed during grinding.
MODEL materialfuel grinder
Declarations:
in, out IS A
materialfuel stream;
electricity consu IS A energy rate;
mption
dust generation IS A mass rate;
cost adding IS A cost per mass;
ED IS A energy per mas
s constant;
DG IS A mass per mass c
onstant;
COST IS A cost per mass c
onstant;
ELECTRICITY IS A cost per energy
COST constant;
Rules:
in.compounds, ARE THE SAME;
88 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
out.compounds
in.state.y, ARE THE SAME;
out.state.y
dust generation = out.quantity ? DG;
out.quantity = in.quantity -
dust generation;
electricity consumption = out.quantity ?
ED; (? cost/s ?)
cost adding = COST +
ELECTRICITY COST ? ED; (?
cost/kg ?)
out.state.cost = in.state.cost +
cost adding; (? cost/kg ?)
out. state.heat = in.state.heat;
5.2.8. Clinker production
The clinker production transforms one in?ow of
material and fuel into one out?ow of material and one
out?ow of kiln exhaust gas. The module contains
relations and constants for cost adding, electricity-con-
sumption and dust-generation.
Clinker production requires a speci?ed amount of heat
per mass unit that must be supplied by the fuel. In this
model, a constant value per mass unit clinker entering
the clinker production is used. This amount was there-
fore calculated and set as a requirement on the heat con-
tents in the fuel entering the clinker production.
Fig. 3. Foreground system model.
The clinker production object contains equations that
relate input mixture, output clinker and emissions to
each other. From a modelling technique point of view,
clinker production does not contain any additional con-
cepts beyond what has already been discussed.
5.2.9. Cement plant
When all the objects were de?ned, they were connec-
ted to form a model of the foreground system: the
cement manufacturing plant at Slite. To start with, all
the necessary objects were instantiated and some of the
constants within them were set, such as the number of
inputs for all mixers and site speci?c values. Then they
were connected in accordance to the structure of the con-
ceptual model, which resulted in the model in Fig. 3.
5.3. Problem formulations
The model built is neutral in the sense that it does not
include any speci?c problem to be solved. Such a prob-
lem formulation, consequently, needs to be done separ-
ately. The formulation contains the following:
? A distinction between what to treat as locked vari-
ables and what to treat as free variables, depending
on the desired solution and the calculation method
chosen.
? A connection between input data and the model. Usu-
89 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
ally locked variables are initiated with suitable
input data.
? The calculation method to use, which sorts equations
and calculates the result by invoking a mathemat-
ical algorithm.
Problem formulations will, in the following, be exem-
pli?ed for the two speci?c operational alternatives dis-
cussed in Section 3. To be able to ?nd a solution, the
number of constraints (equations) needs to equal the
number of free variables. The number of equations is a
consequence of the model, and thus, the parts of the
model and how these are connected. Initially, all vari-
ables in the model are free. In the problem formulation,
some of them are locked so the desired simulations will
be possible to perform.
5.3.1. Case A
The requirements in Section 3, further interpreted in
Section 5.1, result in the locked variables, according to
Table 6. These variables are set to the values indicated,
which represent the input. With this problem formu-
lation, the number of variables will equal the number of
equations and the system, thus, becomes possible to
solve. The used solver in ASCEND is QRSlv, which is
a non-linear algebraic equation solver [23].
5.3.2. Case B
Here, variables are locked according to Table 7 and
constants are set to the values indicated. Even here the
Table 6
Constants and input data for Case A
Variable to lock Initiated data Comment
Quantity of cement 1000 kg/s Product quantity
Fraction gypsum for cement grinding 0.052
Fraction limestone for cement grinding 0.044 Implies 90.4% clinker for cement grinding
Fraction pet-coke in fuel mix 0.20 Implies 80% coal in fuel mix
Fraction sand in raw meal 0.02
Fraction marlstone in raw meal 0.71 Implies 27% limestone in raw meal
Heat required by clinker production 3.050 MJ/kg Related to the in?ow of raw meal fuel
Table 7
Constants and input data for Case B
Variable to lock Initiated data Comment
Quantity of cement 1000 kg/s Product quantity
Fraction gypsum for cement grinding 0.045
Fraction limestone for cement grinding 0.04 Implies 91.5% clinker for cement grinding
Fraction pet-coke in fuel mix 0.23
Fraction tyres in fuel mix 0.22 Implies 55% coal in fuel mix
Clinker LSF quality factor 97
Clinker SR quality factor 2.9 Only two out of three quality factors can be set
Heat required by clinker production 3.050 MJ/kg Related to the in?ow of raw meal fuel
number of variables will equal the number of equations
and the system will thus be possible to solve.
5.4. Model validation and simulation
To use the model, i.e. to predict the environmental
load, the product performance and the economic cost, a
prerequisite is that the model acts as the system it rep-
resents. Before using the model and accepting the infor-
mation generated, the model has to be validated. It has
to be determined whether or not the model gives a good
enough description of the system’s properties to be used
in its intended application. When satisfactory correspon-
dence between the situation, the model and the model-
ling purpose has been attained, then the use and
implementation are appropriate. However, validation of
the model will continue throughout the user phase. Once
a future operational alternative has been tested and
implemented, the simulated information will be com-
pared with the observations of the real system. It is then
possible to improve the model. Consequently, the val-
idity and relevance of the model may be continuously
improved.
Validation is an intrinsic part of model building and
the validity of the model has to be assessed according to
different criteria. Technical validation of the foreground
system model, i.e. to ensure that the model contains or
entails no logical contradictions and that the algorithms
are correct, was done as the model was built.
To validate the foreground-system-model, and in
90 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
addition show examples of model usage and results, we
performed simulations on two real operational alterna-
tives. These have actually been used at the plant, and
hence there were measurements to validate against. The
simulations are those given in Sections 3 and 5.1 and
are illustrated in Figs 4 and 5, respectively.
For each of the two operational alternatives, data gen-
erated with the model was compared with observations
and measurements of the real system. The simulated
values were related to the real values. A selection of
simulated values as a percentage of measured values is
shown in Figs 6 and 7 for the two real operational alter-
natives, respectively.
The two simulations show that the model can simulate
the desired operational alternative and generate the
desired information. The simulated and calculated infor-
mation shows, in comparison with the real system’s
properties, satisfactory correspondence. We have a valid
general model of the Slite plant that can be used to pre-
dict product performance, the economic cost and
environmental load.
For metals, the model has been technically validated.
But due to large variations in metal content in raw
material and fuel and insuf?cient empirical data to
describe the emissions of metals we did not achieve total
correspondence between simulated and real metal emis-
sions.
Fig. 4. Real operational alternative A to be simulated.
6. Discussion and future research
It has been shown that the modelling approach used,
i.e. a calculational non-causal model, physical modelling
and an object oriented modelling language can greatly
enhance modularity, ?exibility and comprehensiveness.
Together with an appropriate simulation tool, e.g.
ASCEND IV, this technique provided a ?exible and gen-
eral-purpose model of a cement manufacturing process
for process and product development purposes.
The tool generates the desired information, i.e. pre-
dicts the environmental load, product performance and
economic cost, by simulating the desired operation alter-
native. For the two operational alternatives tested, the
model generated information which shows satisfactory
agreement with the real system’s properties. We are of
the opinion that since all entities are described inde-
pendent of each other, they can easily be combined and
connected to represent another plant or manufacturing
process.
To avoid sub-optimisation, the model was to use a life
cycle perspective. The cement manufacturing process
from cradle to gate was divided into a foreground sys-
tem, the “gate to gate” part, and a background system.
To complete the model in the life-cycle aspect, the back-
ground system model, which is modelled using normal
LCI technique [17] and stored in the SPINE [24] format,
91 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
Fig. 5. Real operational alternative B to be simulated.
Fig. 6. Simulated values as a percentage of measured values. A selection for operational alternative A.
needs to be connected to the foreground model. Since
the background model is both linear and time inde-
pendent (static) it can be expressed with the techniques
and tools discussed in this paper.
As a result of the chosen modelling approach and
simulation tool the model, as such, has potential for
development. One especially interesting area for future
research is to develop the model and the problem formu-
lations so that it will be possible to perform optimisation
with the model. The library of re-usable problem formu-
lations and model parts can be developed and extended.
Other modelling developments would be adding non-lin-
ear and dynamic relations which transform input into
output, and increase the level of detail in the model,
where applicable.
Naturally, the validation process of the cement model
92 K. Ga¨bel et al. / Journal of Cleaner Production 12 (2004) 77–93
Fig. 7. Simulated values as a percentage of measured values. A selection for operational alternative B.
will continue to increase the validity and extend the
interval for which the model is valid. The next step thus
will be to use and implement site speci?c models,
including the emission of metals, in the cement industry.
Acknowledgements
We thank Cementa AB for ?nancing Karin Ga¨bel’s
industrial doctoral project. The project has been included
in the research program at the Centre for Environmental
Assessment of Product and Material Systems, CPM. We
also thank Bo-Erik Eriksson at Cementa AB for co-ordi-
nating and prioritising the commissioner’s needs and
requirements and for valuable recommendations on the
level of detail in the model.
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104
Paper II
On optimal investment strategies for a hydrogen
refueling station
in
International Journal of Hydrogen Energy,
In press, corrected proof available online 25 July 2006.
On optimal investment strategies for a hydrogen refueling station
Peter Forsberg
1
, Magnus Karlstr¨ om
2
1. Department of Applied Mechanics
Chalmers University of Technology, 412 96 G¨ oteborg, Sweden
e-mail: [email protected]
2. Environmental Systems Analysis
Chalmers University of Technology, 412 96 G¨ oteborg, Sweden
e-mail: [email protected]
Abstract
The uncertainty and cost of changing from a fossil-fuel based society
to a hydrogen based society are considered to be extensive obstacles to
the introduction of Fuel Cell Vehicles (FCVs). The absence of existing
pro?table refueling stations has been shown to be one of the major barri-
ers. This paper investigates methods for calculating an optimal transition
from a gasoline refueling station to future methane and hydrogen com-
bined use with an on site small-scale reformer for methane. In particular,
we look into the problem of matching the hydrogen capacity of a single
refueling station to an increasing demand. Based on an assumed fu-
ture development scenario, optimal investment strategies are calculated.
First a constant utilization of the hydrogen reformer is assumed in order
to ?nd the minimum hydrogen production cost. Second, when consid-
erations such as periodic maintenance are taken into account, optimal
control is used to concurrently ?nd both a short term equipment variable
utilization for one week and a long term strategy. The result is a mini-
mum hydrogen production cost of $4-6/kg, depending on the number of
re-investments during a 20 year period. The solution is shown to yield
minimum hydrogen production cost for the individual refueling station,
but the solution is sensitive to variations in the scenario parameters.
Keywords: Hydrogen; Infrastructure; Investment; Optimiza-
tion; Refueling station
1 Notations
Table 1 shows the symbolic conventions used in this paper. The
letters c, f etc. are variables and constants, which may be further
speci?ed using sub and superscripts. The symbol c
hr,w
indicates,
for example, the weekly capacity of the hydrogen reformer. Ad-
ditional scenario parameters starting with a capital letter are dis-
cussed in section 3.2 below. The currency used is USD ($).
2 Introduction
Hydrogen is considered a promising future fuel for vehicles [1, 2,
3]. Three main arguments are used to support this assertion: the
potential of reducing greenhouse gases from the transport sector;
greater energy supply security, i.e. hydrogen can be produced
from many energy sources and hence the risk of shortage of sup-
Table 1: Symbolic conventions.
Type Name Description Unit
Variables, c Capacity kg, kg/time unit
constants f Factor -
l Lifetime yr
p Cost, price $
r Pro?t $/time unit
s Size kg, kg/time unit
u Consumption kg/time unit
t Time yr
x Flow kg/time unit
? Ef?ciency -
Subscripts a Annuity
d Daily, per day
e Electricity
eq Equipment, any/all part(s)
f Filling, refueling
fc Hydrogen fuel cell
fp Hydrogen refueling pump
g Methane
h Hydrogen
hc Hydrogen compressor
he Hydrogen electrolysis
hf Hydrogen refueling
hr Hydrogen reformer
hs Hydrogen storage
i Investment
k Peak demand to average
m Maintenance
n Nominal
p Progress ratio
s Scale, scaling
t Technology development
w Weekly, per week
x In?exion point
Superscripts i Input ?ow
o Output ?ow
ply may be reduced; the potential of zero local emissions with the
use of fuel cells.
The absence of a hydrogen infrastructure is seen as a major ob-
stacle to the introduction of hydrogen fuel cell vehicles. A full-
scale hydrogen infrastructure with production facilities, a distri-
bution network and refueling stations is costly to build. The ven-
ture of constructing a hydrogen refueling infrastructure consti-
1
tutes a long-term, capital-intensive investment with great market
uncertainties for fuel cell vehicles. Therefore, reducing the ?-
nancial risk is a major objective of any long-term goal to build a
hydrogen infrastructure [4].
Ogden [5] has described several hydrogen supply options. In-
vestigations have also been made for large scale production of
hydrogen [6]. A number of studies of cost and technology for a
hydrogen infrastructure have also been carried out [7, 8, 9]. How-
ever, to the knowledge of the authors, no studies aimed at ?nding
the most pro?table investment strategy for the individual refuel-
ing station have been done.
There are several reasons to focus on the individual hydrogen
refueling station. Car owners are used to accessing a network
of stations. For the ordinary car owner to accept a hydrogen-
refueling infrastructure, accessibility of service stations will be
crucial [10]. Therefore, a network of hydrogen stations will need
to be built in order to reach the target of about 15-20% of the to-
tal number of refueling stations having a hydrogen fuelling option
for consumers. In the EU, the estimated need is 15-20,000 refu-
eling stations by 2020 (a maximum of 100,000 stations are pre-
dicted by 2020 for EU15
1
) [11]. At present there are only about
110-120 hydrogen stations around the world, some of which are
quite small [12]. Several researchers have proposed that small-
scale reforming of methane could be a feasible transition strategy
for the introduction of hydrogen fuel [13, 14].
This study aims at ?nding the most economic investment strat-
egy, i.e. the lowest cost for the hydrogen produced, for an in-
dividual hydrogen refueling station featuring on site small-scale
reforming of methane. The question is to what extent and when
to build the parts of the station, satisfying an increasing demand
of hydrogen. The method developed may be used to ?nd optimal
investment strategies in other cases in which the number of dy-
namic states is reasonably low. Our calculations begin in 2010
and cover 20 years, until 2030.
3 The refueling station
Methane is chosen as the main energy carrier for the refueling
station since:
1. Methane can be produced from fossil fuel, which is, and
will probably continue to be, one of the cheapest production
sources for hydrogen in the short term.
2. Methane can be produced from renewable resources, e.g.
from different types of wood and plants.
3. Methane is relatively easy to transport and can be re-formed
into hydrogen gas.
4. It is possible that running vehicles directly on methane
might be a favourable alternative as an intermediate step to-
ward hydrogen usage.
After reforming, the produced hydrogen gas can be com-
pressed and stored, or used directly in fuel cells on site for ei-
ther local consumption or distribution on the electricity grid when
electricity prices are high. On the other hand, when electricity
prices are low it might be more pro?table to produce hydrogen
1
The present 15 EU member states
by electrolysis using grid electricity. Taking this into consider-
ation, a large number of station con?gurations are possible, in-
cluding the one indicated in ?gure 1. The model developed does,
however, only consider the core components, i.e. reformer, com-
pressor and hydrogen storage. The model is ?exible with respect
to refueling station types, e.g. car, truck or bus, and refueling
station locations, e.g. central, suburb or countryside.
Figure 1: Possible refueling station con?gurations. Methane is re-
formed to hydrogen at the site and stored for delivery to vehicles. It
is also possible to produce hydrogen from electricity by electrolysis or
electricity from hydrogen using a fuel cell.
3.1 The parts of the refueling station
The parts of the refueling station, see ?gure 1, are considered to
have the characteristics given in table 2. This table gives data on
actual produced equipment in the year 2000. In this paper we
have used ?gures from the Simbeck [15] study. Another compa-
rable study is the GM Well-to-Wheel Analysis of Energy Use and
Greenhouse Gas Emissions of Advanced Fuel/Vehicle Systems -
A European Study (GM WtW) [16].
The purchase price is calculated using the scale function
p = p
n
c
1?fs
n
s
fs
, (1)
where f
s
is a scale factor, further discussed in section 3.2. Using
this function an existing piece of equipment with capacity c
n
and
purchase price p
n
is scaled to any size (s) to obtain an estimated
purchase price. The function (1) applies to all parts of the refu-
eling station except the ?lling pump, which is not scalable but
purchased on a piece-wise basis. Regardless of size, all equip-
ment is considered to have a certain expected lifetime, l. Using
the expected lifetime, the weekly annuity is calculated as
f
a,w
=
D
52(1 ?
1
(1+D/52)
52 l
)
, (2)
where D is the real rate of interest. The total weekly equipment
cost, including maintenance (f
m
), is then
p
w
= f
a,w
p(1 + f
m
). (3)
In reality each part of the refueling station is chosen from a
?nite number of available brands and sizes. By using scaling
2
Table 2: Refueling station parts data. Figures are from Simbeck [15] except lifetimes and progress ratio, which are assumed.
Part Reformer Compressor H
2
store Fill pump Electrolysis Fuel cell
Lifetime(l) 15 yr 15 yr 15 yr 20 yr 30 yr 11.4 yr
Nom. capacity(cn) 42 kg/h 42 kg/h 263 kg 48 kg/h 42 kg/h -
Nom. purchase cost(pn) 38,774 $ h/kg 7,792 $ h/kg 592 $/kg 83,117 $/pc 25,665 $ h/kg 12,987 $/kW
Scale factor(fs) 0.75 0.80 0.80 - 0.72 -
Maintenance cost(fm) 0.05 0.06 0.05 0.05 0.02 0.1
Ef?ciency(?) 0.286 kg H
2
/kg NG 0.99 0.99 0.99 0.02 kg H
2
/kWh 18.33 kWh/kg H
2
Electricity use(fe) 0.02 2.492 0 0 - -
Progress ratio(fp) 0.8 0.9 0.9 0.9 0.9 0.9
functions (1) for the purchase price, the set of available parts can
be replaced with one continuous variable. Compared to evaluat-
ing a number of discrete alternatives, this represents a signi?cant
saving in computational complexity.
The ef?ciency in table 2 indicates the relation between the
mass entering and leaving the equipment. In the case of the re-
former, the substance entering is methane and that leaving is hy-
drogen.
3.2 Scenario parameters
The scenario parameters re?ect assumed developments in the fu-
ture and are given in table 3 together with their respective values.
Table 3: Scenario parameters. The electricity price is assumed to be
higher during daytime (6 am-10 pm) than at night (10 pm-6 am).
Name Description Value Unit
B S-curve slope 0.3 -
D Real rate of interest 0.05 1/yr
Fcont Contingency cost factor 0.1 -
Feng Engineering permitting cost factor 0.1 -
F
f
Refueling characteristics factor -
F
f,k
Refueling ratio peak-demand to average 1.12 -
Fgen Include land cost factor 0.2 -
P
1
Cost of manufacturing 1
th
unit - $
Pe Electricity price vector (6am-10pm) 7.8 c/kWh
Electricity price vector (10pm-6am) 3.9 c/kWh
Pg Methane gas price[17] 47 c/kg
Pn Cost of manufacturing n
th
unit - $
R(t) Relative number of hydrogen vehicles at time t - -
Tx In?ection point of the S-curve 10 yr
U
h,d
Mean hydrogen consumption 1000 kg/day
Vn Cumulative production at n
th
unit - -
V (t) Number of vehicles at time t - -
Vtot Total nr of vehicles at t
end
- -
The number of hydrogen vehicles refueling at the station is
a crucial variable for optimization. It is probably also the most
dif?cult parameter to predict. In this study the S-curve
V (t) =
V
tot
1 + e
?B(t?Tx)
(4)
is used. V
tot
is the total number of vehicles using the refueling
station, T
x
the S-curve in?ection point, and B the slope. The
relative number of hydrogen vehicles at the station is thus
R(t) =
1
1 + e
?B(t?Tx)
. (5)
The function R(t) is purely exogenous and therefore uncertain.
This uncertainty will in?uence the results, as is discussed in sec-
tion 6.
Figure 2 shows the refueling characteristics during 24 hours of
operation for a typical gasoline station (F
f,d
). Together with the
daily mean hydrogen consumption (U
h,d
) and ratio peak-demand
to average (F
f,k
), an absolute demanded refueling capacity is cal-
culated. This sequence is the hydrogen demand when 100% of
the vehicles use hydrogen. To adjust the demand to intermedi-
ate situations the sequence is scaled using the S-curve (5), which
results in the daily maximum hydrogen demand curve
x
h,f,d
(t) = R(t) F
f,d
U
h,d
F
f,k
. (6)
0 5 10 15 20
0
0.02
0.04
0.06
0.08
0.1
0.12
Time [h]
F
r
a
c
t
i
o
n
o
f
2
4
h
c
o
n
s
u
m
p
t
i
o
n
Figure 2: Refueling curve for 24 hours (F
f,d
). This curve gives the
distribution of hydrogen demand in fractions of the total consumption
for one day. The ?gures are based on statistics for a typical gasoline
refueling station. [15]
The demand also differs between weekdays according to ta-
ble 4, which creates a periodic sequence of one week for the total
hydrogen demand (x
h,f,w
). Both the 24 hour refueling curve and
the variations between weekdays are based on statistics for a typ-
ical gasoline refueling station. We assume that this behavior is
independent of fuel type and therefore will persist when hydro-
gen is used in place of gasoline.
It is assumed that equipment becomes cheaper with increasing
production and technology development, which is adjusted in the
3
Table 4: Distribution of hydrogen demand in fractions of the total con-
sumption for one week (F
f,w
). The ?gures are based on statistics for a
typical gasoline refueling station. [15]
Day Mon Tue Wed Thu Fri Sat Sun
Fraction 0.14 0.14 0.14 0.15 0.16 0.14 0.13
cost to manufacture the n
th
unit P
n
according to
P
n
= P
1
V
(log fp/ log 2)
n
, (7)
where P
1
is the cost of manufacturing the ?rst unit, V
n
the cumu-
lative production at n
th
unit and f
p
the progress ratio factor. It is
assumed that the number of hydrogen refueling stations (cumu-
lative production) will be 5,000 in 2010 and 50,000 in 2030, and
will follow the S-curve (4). We have assumed an increase in the
number of refueling stations using steam reforming from5,000 to
50,000 in the world from the year 2010 to 2030. These numbers
are based on the predictions about when the fuel cell vehicle mar-
ket will open up and on the number of station needed. A report
presented by E4tech [18] and funded by the UK Department of
Trade and Industry and the Carbon Trust predicts that ”if the hur-
dles are overcome, the mainstream propulsion market is expected
to open up after 2010”. Melaina [19] made a preliminary analy-
sis of the suf?cient number of initial hydrogen stations in the US,
and concluded that between 4,500 and 17,700 hydrogen stations
would be required in the US to initiate a hydrogen infrastructure
for fuel cell vehicles. We justify our estimate of 50,000 hydro-
gen stations by the fact that we consider the whole world and in
a later stage than do Melaina.
The total decrease in relation to the present-day purchase price
owing to increased production and technology development is
thus
f
eq,t
(t) =
(50000R(t))
(log fp/ log 2)
5000
(log fp/ log 2)
(8)
= (10R(t))
(log(fp)/ log(2))
,
where t is the time from year 2010. Within any real mass
production-based learning process, there will be a trade-off be-
tween system standardization and modularity of system capacity.
However, in this model we have used a simpli?cation, as indi-
cated in equation 8.
With respect to interest rates, future costs can be calculated
from present day values using the Present Day Value Correction
pdc(t) =
1
(1 + D)
t
. (9)
3.3 Initial considerations
We assume that, from the outset, no economic costs, i.e. wages
and rent for land, from the gasoline part of the refueling station
are shared with the hydrogen part and vice versa. Hence, it does
not matter if the gasoline refueling station is present or not when
the hydrogen station is being built. In reality some resources can
probably be shared between the gasoline and hydrogen parts of
the refueling station.
The hydrogen part of the refueling station, see ?gure 1, can be
divided into a number of units that can be optimized separately.
The question of whether or not the local fuel cell and electrol-
ysis parts are pro?table depends on the price of electricity and
methane. If the total cost including maintenance (see (3)) is lower
than the difference between produced and bought electricity and
hydrogen respectively, it is pro?table to invest in the respective
equipment. For the fuel cell the pro?t is then
r
fc
= p
e
s
fc
?
p
g
s
fc
?
hr
?
fc
?
p
fc,w
+ ?p
hr,w
168
, (10)
i.e. almost linearly dependent on the investment, with no up-
per boundary. The same reasoning applies to the electrolysis.
An intermediate situation may appear when the purchase price of
electricity is high, whereas the selling price is low. It might then
only be pro?table to produce electricity for the consumption of
the refueling station.
The methane storage facility is sized in accordance with how
frequently the methane gas tank is ?lled at the refueling station.
Since the estimated methane consumption is known, the periodic
delivery can be calculated. A large volume of methane delivered
at the same time would cost less per kg, but requires a larger
storage tank. This is a separate problem that can be solved us-
ing optimization techniques. For the remainder of this study we
therefore assume a constant delivery of methane from a pipeline
or similar construction.
The refueling pump can be dimensioned according to the max-
imum amount refueled. The daily mean hydrogen consumption
U
h,d
is distributed throughout the day corresponding to the re-
fueling curve (?gure 2). A maximum rate of 0.11 is reached
between 3 pm and 5 pm. The busiest day of the week is Fri-
day, reaching 0.16 of the weekly consumption. The ratio peak-
demand to average (F
f,k
), which is estimated to 1.12, should also
be considered. All in all the number of refueling pumps required
is
c
fp
(t) = ceil(2.87R(t)), (11)
where the function ceil rounds to the nearest integer greater than
or equal to the operand.
The remaining parts of the refueling station are the ones within
the shaded area in ?gure 1 and are collectively called ”the core”.
This core consists of reformer, compressor and storage tank.
Considering only the core, hydrogen is delivered to vehicles only.
This means that all the hydrogen produced by the reformer will
go through the compressor to the storage tank. The size of the
reformer and compressor will thus have to be the same.
Owing to a non-linear price decrease in the equipment over
time (7), the core problem cannot be further split into size dis-
tribution between reformer and tank, plus time and extent of in-
vestments. The relative cost between reformer/compressor and
storage tank will change over time.
3.4 Core model
The model describing the core, see ?gure 1, is quite simple and
includes only one state, the hydrogen storage. It can be described
4
by
? x
hs
= x
i
hs
?x
o
hs
x
i
hs
= x
i
hr
?
hr
?
hc
x
o
hs
?
fp
= x
hf
x
hf
= Hf
w
x
i
hr
= x
ng
(12)
subject to the constraints
0 ? x
o
hr
? c
hr
0 ? x
i
hc
? c
hc
0 ? x
hs
? c
hs
0 ? x
i
fp
? c
fp
. (13)
The rest of this paper addresses the problem of choosing the
size of reformer/compressor versus storage volume over one or
many investments over time for the core of the refueling station.
The model developed will be used in the optimization in the sub-
sequent section.
4 The optimization problem
Optimal control can be used to optimize a system over a certain
time interval. Given a dynamic model of the system, an objective
function, and constraints, a path from one state to another can be
calculated where the objective function is at a minimum. Solving
non-dynamic design problems using optimization techniques is
common practice, see e.g. [20] or [21]. In the case where the
model is static, i.e. does not change over time, such techniques
are suf?cient.
In this study, however, we are interested in investment planning
and internal properties that change over time, such as utilization
curves and transients for hydrogen generation. Therefore a dy-
namic optimization technique is used, see e.g. [22].
4.1 The objective function
The criterion function to be minimized is based on the total pro-
duction cost for hydrogen, which consists of costs for equipment,
methane and electricity. In addition the number of investments
has to be taken into account.
The total weekly equipment cost is the sum of the cost for each
part of the refueling station (p
w
, see (3)), i.e
p
eq,w
(s
eq
, t
i
) =
? eq
p
w
(s
eq
) f
eq,t
(t
i
). (14)
The loans are of the annuity type, which make the equipment cost
independent of time.
Since consumption is given by the refueling demand (x
h,f,w
),
production during the given time frame can be calculated. The
total weekly methane gas cost is then
p
g,w
(t
w
) =
168
t=1
(x
g,w
) P
g
R(t
w
) =
168
t=1
(x
h,f,w
)P
g
R(t
w
)
?
fp
?
hs
?
hc
?
hr
, (15)
where t
w
indicates the time (in years) when the weekly cost is
calculated.
For electricity the price varies throughout the day and needs to
be evaluated on an hourly basis. The weekly cost is then scaled
using the S-curve (5). The total weekly electricity cost is thus
p
e,w
(t
w
) = x
e
P
T
e
R(t
w
) =
168
t=1
(x
h,f,w
) (16)
(f
fp,e
+
f
hs,e
?
fp
+
f
hc,e
?
fp
?
hs
+
f
hr,e
?
fp
?
hs
?
hc
p
T
el
R(t
w
).
Note that both the methane and electricity costs are independent
of the size of the equipment.
In this study the production cost for hydrogen is averaged over
the whole investment period as
p
h
=
1
N
N
tw=1
? seq,ti
p
eq,w
+ p
g,w
(t
w
) + p
e,w
(t
w
)
168
t=1
(x
h,f,w
) R(t
w
)
, (17)
where N is the number of weeks for the investment period. This
production cost takes into account the timing of the investments,
making the purchase cost of all parts of the refueling station de-
crease over time (8). It does not, however, correct future costs
to present-day values (9). It is possible to calculate the hydrogen
cost in other ways. One way is to use the formula above and add
the Present Day Value Correction (9), which gives
p
h
=
1
N
N
tw=1
pdc(t
w
)
? seq,ti
(p
eq,w
) + p
ng,w
+ p
el,w
168
t=1
(x
h,f,w
)R(t
w
)
. (18)
Another totally different approach is to distribute the total cost
evenly over the whole investment period, i.e
p
h
=
N
tw=1
? seq,ti
(
? eq
(p
eq
(1 + f
m
)pdc(t
w
)))f
a,w
168
t=1
(x
h,f,w
)R(t
w
)
. (19)
In this study the average production cost of hydrogen (17) is
used to ?nd the objective function. Expanding the function, it is
clear that not all the terms are necessary to generate the shape
of the production cost. The terms p
g,w
and p
e,w
can be summed
up separately and are independent of size of equipment (s
eq
) and
time for investment (t
i
), giving a constant contribution. In ad-
dition the sum over x
h,f,w
and N are constants. Omitting these
constant terms yields the objective function
p
obj
(t
i
) =
N
tw=1
? seq,ti
(p
eq,w
)
R(t
w
)
. (20)
Expanding the objective function, it can be written as
p
obj
?
N
tw=1
(1 + e
?B(tw?tx)
) (21)
? seq,ti
((1 + e
?B(ti?tx)
)
C1
? eq
s
fs
eq
),
where C
1
is a constant. Since this is a sum over exponentials of
convex functions, the objective function is also convex [23].
In the case when the objective function (20) does not provide
enough information to ?nd an unambiguous optimal point, it can
be augmented , e.g. by variations in utilization of equipment.
5
This would result in a smoother utilization curve. Adding e.g.
the quadratic variations in the hydrogen reformer output would
then give
p
obj2
= p
obj
+ ?
M?1
k=1
(x
o
hr
(k) ?x
o
hr
(k + 1))
2
, (22)
where ? is a weight factor and M the number of hours to con-
sider.
4.2 The constraints
The constraints for the optimization problem are developed from
(12) and (13). By integrating the ?rst equation of (12), which
controls the storage of hydrogen at the refueling station, the ?rst
constraint is found. Since consumption statistics for refueling are
given on an hourly basis for one week, it is possible to use a time-
discrete formulation where integration is replaced by summation.
The model (12) becomes
x
hs
=
t
f
t0
(x
i
hs
?x
o
hs
)
x
i
hs
= x
ng
?
hr
x
o
hs
?
fp
= x
h,f,w
. (23)
By assuming a periodicity of one week for all variables included,
it is only necessary to take 7 ? 24 = 168 points (hours) into ac-
count.
The method of transcription used in the time continuous case
to eliminate time [22] can be replaced in this discrete problem by
cumulative summation. This results in the following constraints:
t
t0
(x
i
hs
?x
o
hs
) ? 0, t
0
? t ? t
f
c
eq
? x
eq
. (24)
The ?rst equation ensures that the stored amount of hydrogen
does not become negative, while the second ensures that the ?ow
through each piece of equipment does not exceed the capacity.
A way of extinguishing transients of the state variables is to
require that the initial value equals the end value, i.e
x
hs
(t
0
) = x
hs
(t
f
). (25)
One inconvenience is that in order to precisely follow the cal-
culated path, the storage containers have to be initially ?lled to a
certain extent. In reality this is not very important since the initial
transients decay rapidly.
Other requirements, e.g. periodical maintenance stops of re-
former or required initial amount of hydrogen stored may also be
taken into consideration by adding one or more constraints.
The optimizations in cases 1 and 2 are carried out for a one
week operation for each investment. For intermediate invest-
ments, the demand is estimated using the S-curve (R(t)), and
equipment is assumed to have decreased in price according to
(7) with scenario parameters as in table 3. Note that the price de-
crease is faster for the reformer than for the rest of the equipment.
5 Results
This section presents the results from the optimization in the two
cases discussed. Both cases are solved using Tomlab [24]. No
gradients or Hessians are provided, instead estimates are made
using numerical differentiation within the optimization method.
5.1 Case 1: constant utilization
In case 1, constant utilization of reformer and compressor is con-
sidered. The capacity of the reformer/compressor (c
hr
) is deter-
mined fromthe weekly average demand and the hydrogen storage
capacity by ?nding the minimum of the sum of net input to the
hydrogen storage tank, see (26).
c
hr
=
168
t=1
(x
h,f,w
)
168?
hc
?
hs
?
fp
(26)
x
hs
(t
0
) = ?min
t
t
t0
(c
hr
?
hc
?x
h,f,w
/(?
fp
?
hs
))
c
hs
= max
t
t
t0
(c
hr
?
hc
?x
h,f,w
/(?
fp
?
hs
)) + x
hs
(t
0
)
The result is an unconstrained optimization problem that can be
described by
min
ti
p
obj
(t
i
). (27)
0 2 4 6 8 10 12 14 16 18 20
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
6.2
Second investment time [year]
H
y
d
r
o
g
e
n
p
r
o
d
u
c
t
i
o
n
c
o
s
t
[
U
S
D
/
k
g
]
Figure 3: Case 1, 2 investments, hydrogen production cost as a func-
tion of second investment time. Note the discontinuities at times 8.5 and
13.2. These are caused by an increase in the number of refueling pumps
(11).
If only one investment is made, the characteristics are calcu-
lated directly using (26) and no optimization is carried out. In
the case of 2 or more investments, the curves in ?gures 3 and 4
illustrate the effect of investment time.
The problem in case 1 is solved using a quasi-Newton method
implemented in the Tomlab function ucSolve. The results for 1-3
investments are as shown in table 6, ?gures 5 and 6.
6
Figure 4: Case 1, 3 investments, hydrogen production cost as a func-
tion of second and third investment time.
Table 5: Result of optimization, case 1. The total cost is the cost for
equipment, methane and electricity for the entire investment period. The
mean distance cost is calculated from the use of 0.1kg H2/10km for fuel
cell vehicles.
No of investments 1 2 3
Investment time [yr] 0 0, 5.7 0, 3.9, 8.4
Cost equipment [$] 3,868,763 2,961,677 2,793,208
Total cost [$] 16,296,295 15,026,375 14,791,149
Mean cost hydrogen[$/kg] 6.03 4.37 4.14
Mean distance cost[$/10km] 0.60 0.44 0.41
Reformer size [kg/h] 45.47 9.2+36.3 5.8+10.9+28.7
Storage size [kg] 606 123+484 77+146+383
Initial storage [kg] 271 55,271 35,100,271
Refueling pump no [pcs] 3 1+2 1+0+2
The mean distance cost is based on a consumption of 0.1kg
H
2
/10km for a fuel cell vehicles. This is an estimate of the hy-
drogen consumption for a small fuel cell vehicle and is only used
for comparison with petrol fueled cars.
Further investments have very little effect on the mean hydro-
gen production cost, as can be seen in ?gure 7.
5.2 Case 2: variable utilization
In Case 2 utilization of equipment is parametrized and deter-
mined by the optimization algorithm. The chosen special condi-
tions in this study are 100 kg hydrogen storage initially and at the
end of each week (periodic boundary conditions), and a weekly
stop for maintenance from hours 75 to 87 during the week. In-
vestments are done on 1 and 2 occasions during the investment
period. Further investments have not been investigated for case
2, owing to the computational complexity and time involved. The
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
100
200
300
400
500
600
700
Hydrogen stored 1 [kg]
time [h]
k
g
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
20
40
60
80
100
120
140
Hydrogen storage output 1 [kg/h]
time [h]
k
g
/
h
Figure 5: Case 1, 1 investment, stored hydrogen and storage output.
0 2 4 6 8 10 12 14 16 18 20
5
10
15
20
Absolute cost produced hydrogen [USD/kg]
time [year]
U
S
D
/
k
g
0 2 4 6 8 10 12 14 16 18 20
0
2000
4000
6000
8000
Capacity ? Demand for hydrogen [kg/week]
time [year]
k
g
/
w
e
e
k
Figure 6: Case 1, 1 investment, hydrogen production cost and capacity-
demand.
resulting constrained non-linear optimization problem
min
ti,s
hr
p
obj2
s.t.
t
t0
(x
i
hs
?x
o
hs
) ? 0, t
0
? t ? t
f
c
eq
? x
eq
x
hs
(t
0
) = x
hs
(t
f
)
x
hs
(t
0
) = 100
87
t=75
x
o
hr
= 0, (28)
was solved using a Sequential Quadric Programming (SPQ)
method [25], as part of the NPSOL [26] package running in Tom-
lab [24].
The results from the optimization give the size of equipment
7
1 2 3 4 5 6 7 8 9 10
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
6.3
6.4
6.5
Number of investments
H
y
d
r
o
g
e
n
p
r
o
d
u
c
t
i
o
n
c
o
s
t
[
U
S
D
/
k
g
]
Figure 7: Case 1, hydrogen production cost for 1-10 investments.
(table 6), running pattern of the facility (?gure 8) and produced
hydrogen price and utilization curves (?gure 9). The solution
shows good utilization of storage; the stored hydrogen amount
frequently drops to near zero.
Table 6: Result of optimization, case 2. The total cost is the cost for
equipment, methane and electricity for the entire investment period. The
mean distance cost is calculated from the use of 0.1kg H2/10km for fuel
cell vehicles.
No of investments 1 2
Investment time [yr] 0 0, 5.6
Cost equipment [$] 4,707,805 3,724,066
Total cost [$] 17,522,971 16,151,907
Mean cost hydrogen [$/kg] 6.74 4.72
Mean distance cost [$/10km] 0.67 0.47
Reformer size [kg/h] 57 10.0, 50.0
Storage size [kg] 939 199, 873
Initial storage [kg] 63 90, 64
Refuelling pump no [pcs] 3 1 + 2
5.3 Sensitivity of the solution
To evaluate the sensitivity of the solution in case 1 (with 2 invest-
ments) to changes in the scenario parameters, calculations were
made with slightly changed values from the settings in table 3.
Table 7 shows the results in relative sensitivities, i.e.
sens
y
=
min
i
(z + ?z)
min
i
(z)
, (29)
where sens
y
is the relative sensitivity value with respect to prop-
erty z. The optimization procedure is abbreviated min
i
.
Owing to the similarities in the objective function, the sensi-
tivities also are valid for case 2.
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
10
20
30
40
50
60
Hydrogen reformer output 1 [kg/h]
time [h]
k
g
/
h
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
200
400
600
800
1000
Hydrogen stored 1 [kg]
time [h]
k
g
Figure 8: Case 2, 1 investment, throughput and stored hydrogen.
0 2 4 6 8 10 12 14 16 18 20
0
5
10
15
20
25
Absolute cost produced hydrogen [USD/kg]
time [year]
U
S
D
/
k
g
0 2 4 6 8 10 12 14 16 18 20
0
2000
4000
6000
8000
Capacity ? Demand for hydrogen [kg/week]
time [year]
k
g
/
w
e
e
k
Figure 9: Case 2, 1 investment, hydrogen production cost and capacity-
demand.
Table 7: Relative sensitivity at the optimal point, case 1 with 2 invest-
ments. The numbers given are relative sensitivity at the optimal point,
see (29).
Property name Investment time H
2
production cost
sensitivity sensitivity
Real rate of interest (D) -0.04 0.10
Progress ratio (fp) 0.26 0.85
S-curve slope (B) -0.42 0.47
S-curve in?ection point(Tx) 0.34 0.58
Total units at t
end
(Vtot) 0.02 -0.23
Mean hydrogen cons.(U
h,d
) -0.01 0.002
6 Discussion
It is possible to use optimization and optimal control to determine
optimal investment strategies for a hydrogen refueling station for
8
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
2
4
6
8
10
12
Hydrogen reformer output 1 [kg/h]
time [h]
k
g
/
h
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
50
100
150
200
Hydrogen stored 1 [kg]
time [h]
k
g
Figure 10: Case 2, 2 investments, throughput and stored hydrogen:
Investment 1 at t=0.
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
10
20
30
40
50
60
70
Hydrogen reformer output 2 [kg/h]
time [h]
k
g
/
h
0 12 24 36 48 60 72 84 96 108 120 132 144 156 168
0
200
400
600
800
1000
1200
Hydrogen stored 2 [kg]
time [h]
k
g
Figure 11: Case 2, 2 investments, throughput and stored hydrogen:
Investment 2 at t=5.6.
vehicles. The results indicate a hydrogen production cost at a re-
fueling station with on site reforming of methane ranging from
$4.1 to 6.0/kg, depending on the number of investments and spe-
cial requirements of periodic maintenance, etc. This is in the
same range as previous ?ndings, see table 8. The main differ-
ence is that this study uses a function that increases over time (4)
to estimate the number of hydrogen vehicles refueling at the sta-
tion, which makes the estimated production cost an average over
time. In other studies, the cost is based on maximum utilization.
The idea underpinning the method developed is to be able to eas-
ily change assumptions and scenario parameters according to a
given case. The method can then be used for investment planning
in individual refueling station cases.
When one large investment is made, hydrogen produced will
initially become very expensive. Although the production cost
will have dropped to a more reasonable level after 10 years of
0 2 4 6 8 10 12 14 16 18 20
3
4
5
6
7
8
9
Absolute cost produced hydrogen [USD/kg]
time [year]
U
S
D
/
k
g
0 2 4 6 8 10 12 14 16 18 20
0
2000
4000
6000
8000
Capacity ? Demand for hydrogen [kg/week]
time [year]
k
g
/
w
e
e
k
Figure 12: Case 2, 2 investments, hydrogen production cost and
capacity-demand.
Table 8: Other studies of on site reforming of methane
Study $/kg Size
Schoenung [27] 5.7 400 kg/d
Knight [4] 1.79 250 cars/d
Thomas [7] 11-2.2 180 - 2720 kg/d
Simbeck [15] 4.4 470 kg/d
Ogden [5] 1.7-5.6 400 cars/d
production, the refueling station may not survive that long. A bet-
ter approach would be to start with a smaller capacity, and then
increase it over time. The results show that the most realistic eco-
nomic production cost situation can be achieved at approximately
4 to 5 investments (?gure 7) and that little is to be gained by fur-
ther increasing number of investments. Increasing the number of
investments can also be more favourable froma risk management
point of view. It is then possible to adjust the investment plan be-
fore the next investment is made, using the same method but with
more recent assumptions.
The sensitivity analysis shows that the H
2
production cost is
quite sensitive to changes in some of the scenario parameters.
The most sensitive one is progress ratio, i.e the price decrease
for equipment. Since the progress ratio is not known in advance,
large changes in the predicted production cost may result. One
way of handling this situation is to add uncertainty estimates to
all scenario parameters and make an optimization that takes these
uncertainties into account.
Some factors in the cost function can be improved to make the
results more realistic, e.g. maintenance of the equipment and re-
sources split between the gasoline and hydrogen parts of the refu-
eling station. Another area that can be improved is the ef?ciency
factors for equipment, ?
eq
. In reality, ef?ciency is dependent on
other factors, e.g. ?owthrough the equipment. Also, in reality fu-
ture development is not known. By using stochastic variables and
make a stochastic optimization, uncertainties can be expressed in
the result in terms of probability functions.
When the number of investments increases, so does the com-
9
putational time. For case 2 with variable utilization, calculations
with more than 2 investments already result in unrealistically long
computational time. Since the problem is convex in the objec-
tive function (but not in the constraints), other more specialised
optimization algorithms may be used. In addition, gradient and
Hessian information can be provided to further reduce computa-
tional time.
The above model can probably also be used as a starting
point when doing investment optimization for multiple refuel-
ing stations in a community. This optimization problem is not
as straightforward as the one discussed in this paper: here fac-
tors such as local competition between refueling stations and how
this affect sales (i.e. supply-demand curve) have to be taken into
account. Another option would be to investigate under what cir-
cumstances the complete station layout (?gure 1) would be prof-
itable.
7 Summary and conclusions
1. With the assumptions made in this study, it is possible to
produce hydrogen with on site reforming at a price ranging
from$6.0/kg for one investment to $4.1/kg for 3 investments
over 20 years when continuous production is considered.
2. Special requirements, e.g. speci?ed storage in the beginning
of the week and periodic maintenance stops of the reformer,
can be accounted for but will make the produced hydrogen
more expensive.
3. Investment timing is most sensitive (in order of magnitude)
to changes in the scenario parameter S-curve steepness (B),
S-curve in?ection point (t
x
) and progress ratio (f
p
). It is
less sensitive to changes in methane and electricity prices,
interest rates (D) and S-curve total number of units at t
end
(V
tot
).
4. The hydrogen production cost is most sensitive (in order of
magnitude) to changes in progress ratio (f
p
), scenario pa-
rameter S-curve in?ection point (t
x
) and S-curve steepness
(B). It is also quite sensitive to changes in S-curve total
number of units at t
end
(V
tot
) and interest rates (D).
5. The method developed in this paper can be used for optimal
investment planning in other areas with ?ow processes that
can be described with state equations.
8 Acknowledgments
Financial support from the Competence Center for Environmen-
tal Assessment of Product and Materials Systems (CPM) at
Chalmers University of Technology and the Swedish Foundation
for Strategic Environmental Research, MISTRA, is gratefully ac-
knowledged. Part of this work was done within the framework of
the Jungner Center.
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ery, storage, conversion, application, public education and
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(2002).
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term co-operation (2003).
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Tech. rep., EU high level group for hydrogen and fuel cells
(2003).
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C. Thomas, Bringing Fuel Cell Vehicles to Market: Scenar-
ios and Challenges with Fuel Alternatives Consultant Study
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between today and 2020 - time scale & investment costs
- while considering regulatory aspects, ghg reduction po-
tentials, and renewable energy supply potentials for h2 fuel
production, Presentation at Fuel Cell Teach-in European
Commission DGTren, Brussels (July 11/12 2002).
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infrastructure, Tech. rep., Fuel Cell Today (2005).
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Systems Studies, Tech. rep., National Renewable Energy
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membrane (pem), fuel cell systems for transportation
applications: Hydrogen infrastructure report, Tech. rep.,
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for Hydrogen Pathways - Scoping Analysis, Tech. Rep.
NREL/SR-540-32525, National Renewable Energy Labo-
ratory, Golden, Colorado (2002).
10
[16] Well-to-wheel analysis of energy use and greenhouse gas
emissions of advanced fuel/vehicle systems - a european
study, Tech. rep., L-B-Systemtechnik Gmbh (2002).
[17] Nordleden: Gemensamma ber¨ akningsf¨ oruts¨ attningar (in
Swedish), Tech. rep., Nordleden, www.nordleden.nu
(2001).
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Carbon Trust (Final Report), Tech. rep., DTI and Carbon
trust report (2003).
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nary Analysis of a Suf?cient Number of Initial Hydrogen
Stations in the US, International Journal of Hydrogen En-
ergy 28 (7) (2003) 743–755.
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Academic Press, New York, 1981.
[21] F. Lewis, Optimal Control, Wiley, 1995.
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linear Programming, SIAM, 2001.
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University, 2002.
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timization laboratory, Stanford University, Stanford, CA
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Program Review, 2001.
11
118
Paper III
Macroeconomic and ?nancial time series prediction
using networks and evolutionary algorithms
in
Proceedings of Computational Finance 2006,
London, 27-29 June 2006, pp. 403-411.
Macroeconomic time series prediction
using prediction networks and
evolutionary algorithms
Peter Forsberg
1
, Mattias Wahde
Department of Applied Mechanics,
Chalmers University of Technology, Sweden
1. Corresponding author
Abstract
The prediction of macroeconomic time series by means of a form of fully
recurrent neural networks, called discrete-time prediction networks (DTPNs),
is considered. The DTPNs are generated using an evolutionary algorithm,
allowing both structural and parametric modi?cations of the networks, as
well as modi?cations in the squashing function of individual neurons.
The results show that the evolved DTPNs achieve better performance on
both training and validation data compared to benchmark prediction meth-
ods. The importance of allowing structural modi?cations in the evolving
networks is discussed. Finally, a brief investigation of predictability mea-
sures is presented.
Key words: time series prediction, recurrent neural networks, evolutionary
algorithms
1 Introduction
Prediction of time series is an important problem in many ?elds, including
economics. Due to the high level of noise in macroeconomic time series,
models involving two parts, one deterministic and one stochastic, are often
used. One such method is ARIMA [1]. For one-step prediction, the results
obtained by these simple predictive methods (such as exponential smooth-
ing, which is a special case of ARIMA models), are di?cult to improve much
due to the high levels of noise present. However, even a small improvement
can translate into considerable amounts of money for data sets that concern
e.g. an entire national economy. The aims of this paper is (1) to introduce a
class of generalized, recurrent neural networks and an associated evolution-
ary optimization method and (2) to apply such networks to the problem
of deterministic prediction of macroeconomic time series, with the aim of
extracting as much information as possible, while keeping in mind that the
noise in the data introduces limits on the achievable performance.
2 Macroeconomic data
Two di?erent data sets were considered, namely US GDP (quarterly varia-
tion, from 1947, ?rst quarter to 2005, second quarter), and the Fed Funds
interest rate (monthly values, from July 1954 to July 2005). The raw GDP
and interest data were ?rst transformed to a relative di?erence series, using
the transformation
Z
RD
(t) =
Z
raw
(t) ?Z
raw
(t ?1)
Z
raw
(t ?1)
. (1)
Next, this series was further transformed using a hyperbolic tangent trans-
formation
Z(t) = tanh(C
TH
Z
RD
(t)). (2)
For the GDP and interest rate series transformations, the values C
TH
= 25
and C
TH
= 5 were used, respectively. The aim of the hyperbolic tangent
transformation was to make the data points as evenly distributed as possible
in the range [?1, 1].
Both data sets were divided into a training part with M
tr
data points, and
a validation part with M
val
data points. During training, only the results
(i.e. the error) over the training data set were used as feedback to the
optimization procedure (see below). The rescaled GDP data set contained
233 data points. For training, steps 16-115 were used (M
tr
= 100) and for
validation, steps 126-225 were used (M
val
= 100). During training, the ?rst
15 steps were used to initialize the short-term memory of the DTPN. A
similar initialization procedure was applied during validation. For the Fed
Funds data set, with 612 data points, steps 26-475 were used for training
(M
tr
= 450) and steps 486-605 (M
val
= 120) were used for validation.
3 Methods for prediction
3.1 Discrete-time prediction networks
Neural networks constitute a commonly used blackbox prediction model.
In most cases, feedforward neural networks (FFNNs) are used. In such net-
works, the computational elements (neurons) are placed in layers. The input
signals (i.e. earlier, consecutive values of the time series) are distributed to
the neurons in the ?rst layer, and the output signals of those neurons are
then computed and used as input in the second layer etc. The output of a
given neuron i is computed as
x
i
(t + 1) = ?
?
?
b
i
+
N
j=1
w
ij
y
j
?
?
, (3)
where b
i
is the bias term, w
ij
are the weights connecting neuron j in the
preceding layer to neuron i, N is the number of neurons in the preceding
layer, and ? is the squashing function, usually taken as the logistic function
?
1
(z) =
1
1 + e
?cz
, (4)
where c is a positive constant, or the hyperbolic tangent
?
2
(z) = tanh cz. (5)
Given a set of training data, i.e. a list of input vectors and their corre-
sponding desired output, such networks can be trained using gradient-based
methods, such as e.g. backpropagation.
However, there are fundamental limitations in the prediction that can be
achieved using FFNNs, due to their lack of dynamic (short-term) memory.
Stated di?erently, an FFNN will, for a given input, always give the same
output, regardless of any earlier input signals [2], [3]. Thus, such networks
are unable to deal with situations in which identical inputs to the network
(at di?erent times along the time series) require di?erent outputs. Earlier
work [2] has shown that dynamic short-term memory does make a di?erence
in neural network-based time series prediction.
Furthermore, the requirement that it should be possible to obtain a gra-
dient of the prediction error, in order to form the derivatives needed for
updating the weights (during training), restricts the shape of the squashing
functions. Without such restrictions, squashing functions such as e.g.
?
3
(z) = sgn(z), (6)
and
?
4
(z) =
?
?
?
?
?
tanh(z + c) if z < ?c
0 if ?c ? z ? c
tanh(z ?c) if z > c
(7)
could be used.
To overcome the limitations of FFNNs, it is possible to introduce feed-
back couplings in the networks, transforming them into recurrent neural
networks (RNNs). Such networks have been used in many ?nancial and
macroeconomic applications, see e.g. [3], [4]. A problem with many stan-
dard training techniques for neural networks is that they require that the
user should set the structure of the network (i.e. the number of neurons and
their position in the network), a procedure for which one often has to rely
on guesswork and rules-of-thumb [5]. An alternative training procedure is to
use an evolutionary algorithm (EA) which, if properly designed, can handle
both structural and parametric optimization [6].
In this paper, a new kind of network (and an associated evolutionary
optimization method), well suited for the problem of time series predic-
tion, will be used, with dynamical memory, arbitrary structure, and (in
principle) arbitrary squashing functions. Each of the n neurons in these
networks which, henceforth, will be called discrete-time prediction networks
or DTPNs for short) contains arbitrary connections from the n
in
input ele-
ments and from other neurons (including itself). In addition, each neuron
has an evaluation order tag (EOT) such that, in each time step, the output
of the neurons with the lowest EOT values is computed ?rst, followed by
the output of the neurons with the second lowest EOT values etc. The out-
put neuron, i.e. the neuron with highest EOT (arbitrarily chosen as neuron
1) is evaluated last. Thus, the equations for neurons with the lowest EOT
become
x
i
(t + 1) = ?
?
?
b
i
+
nin
j=1
w
in
ij
I
j
(t) +
n
j=1
w
ij
x
j
(t)
?
?
, (8)
where w
in
ij
are the input weights, w
ij
the interneuron weights, and b
i
is
the bias term. I
j
are the inputs to the network which, in the case of time
series prediction, consist of earlier values of the time series Z(t), i.e. I
j
(t) =
Z(t ?j + 1). The number of inputs can thus be referred to as the lookback
(L) of the DTPN. For neurons with the second lowest EOT, the equations
look the same, except that x(t) is changed to x(t+1) for neurons with lowest
EOT etc. Finally, the output neuron gives the following output
x
1
(t + 1) = ?
?
?
b
1
+
nin
j=1
w
in
1j
I
j
(t) + w
11
x
1
(t) +
n
j=2
w
1j
x
j
(t + 1)
?
?
, (9)
since, at this stage, all neurons except neuron 1 have been updated. It is
evident that the EOTs introduce the equivalent of layers. Thus, while most
DTPNs will contain many recurrent connections, an FFNN is a special case
of a DTPN. More precisely, a DTPN is equivalent to an ordinary FFNN if
and only if (1) all squashing functions are of the same type (either ?
1
or
?
2
), (2) only neurons with the lowest EOT values receive external input,
and (3) w
ij
(i.e. the weight connecting neuron j to neuron i) is equal to
zero if EOT(j) ? EOT(i).
3.2 Benchmark predictions
In order to evaluate the results obtained using DTPNs, a comparison will
be made with two standard prediction techniques, namely autoregressive
moving average (ARMA) and exponential smoothing. The general simple
ARMA(p, q) model
?(?)Z(t) = ?(?)(t), (10)
where ? is the lag operator, is the disturbance Z ?
ˆ
Z, and
?(?) = 1 ??
1
? ?. . . ??
p
?
p
, (11)
and
?(?) = 1 + ?
1
? + . . . + ?
q
?
q
, (12)
gives the one-step prediction
ˆ
Z(t + 1|t)
ˆ
Z(t + 1|t) =
p
i=0
?
i
Z(t ?i) +
q
i=0
?
i
(t ?i). (13)
?
i
and ?
i
are parameters to be estimated in order to ?nd the lowest error.
The exponential smoothing technique (without trend) is described by the
Neuron1 Neuron2 Neuronn ...
Neuroni
w(interneuronweights) w (inputweights)
in
b c k(sigmoidtype)
...
EOT
Figure 1: A chromosome encoding a DTPN.
ARIMA(0,1,1) equation
(1 ??)Z(t) = (1 ??
1
?)(t). (14)
This model gives the prediction
ˆ
Z(t + 1|t) =
1 ??
1
1 ??
1
?
Z(t) = ?
1
ˆ
Z(t|t ?1) + (1 ??
1
)Z(t). (15)
As a special case, if ?
1
= 0, the naive prediction
ˆ
Z(t+1|t) = Z(t) is obtained.
4 Evolutionary algorithm
The DTPNs were generated using an evolutionary algorithm (EA) [7]. The
EA used here employed a non-standard chromosomal representation, shown
in Fig. 1, in which each gene represented a neuron in the network, encoding
its interneuron weights (w
ij
), input weights (w
in
ij
), bias term (b
i
), sigmoid
parameter (c), sigmoid type, and EOT. During the formation of new individ-
uals, crossover was only allowed between individuals containing the same
number of neurons. Several di?erent forms of mutations were used, both
parametric mutations modifying the values of the parameters (including
the EOT) listed above, and structural mutations which could either add or
subtract a neuron from the DTPN. No upper limit was set on the number
of neurons. A lower limit of 2 neurons was introduced, however. In addition
to the mutations just listed, a sigmoid type mutation was introduced as
well, allowing a neuron to change its sigmoid type by randomly changing
the index k of the sigmoid ?
k
(see Sect. 3.1 and Eq. (17) below). Finally, in
order to allow (not force) the EA to produce sparsely connected networks,
some runs were carried out in which parametric mutations of interneuron
weights, input weights, and biases not only could modify the value of the
parameter in question, but also (with low probability) could set it exactly
to zero. Thus, these mutations essentially functioned as on-o? toggles, and
were therefore called zero-toggle mutations. The number of input elements
(and therefore the lookback L) was ?xed in each run. The ?tness measure
F used by the EA was taken as the inverse of the RMS prediction error over
the training set, i.e. F = 1/e
RMS
where
e
RMS
=
1
M
tr
Mtr
i=1
Z(i) ?
ˆ
Z(i)
2
(16)
Note that the use of an EA implies that any form of sigmoid function can
be used in the networks. In addition to the four functions ?
1
? ?
4
, a ?fth
sigmoid, namely
?
5
(z) =
cz
1 + (cz)
2
, (17)
was also allowed in the simulations reported below.
5 Prediction results
A large number of runs were carried out, using di?erent number of inputs
and di?erent EA parameters in order to test the ability of the evolutionary
algorithm to generate DTPNs with low prediction error for the two data
sets under consideration.
The results are summarized in Table 1. The table shows the prediction
error for the DTPN with lowest validation error. In addition, the prediction
errors obtained using naive prediction, exponential smoothing, and ARMA
(all with optimized parameter values), are shown.
As is evident from the table, the best DTPNs outperform the two other
prediction methods. Table 2 gives a more detailed description of the best
DTPNs, obtained with di?erent values of n
in
. For comparison, note that the
best training errors obtained with exponential smoothing were e
tr
ES
= 0.2512
for the GDP data and e
tr
ES
= 0.3477 for the Fed funds data. Using the ARMA
model, the best training errors were e
tr
ARMA
= 0.2108 and e
tr
ARMA
= 0.3248,
respectively.
6 Predictability measures
The fact that the DTPNs outperform the benchmark prediction methods
does not imply that these networks extract all the available information
in the time series under study. One way of determining whether addi-
tional information can be extracted would be to devise a measure P(t)
of predictability such that, in addition to the prediction
ˆ
Z(t + 1) of the
Data set e
N
e
ES
e
ARMA
e
DTPN
Fed funds interest rate 0.2018 0.1901 0.1887 0.1837
GDP 0.1771 0.1490 0.1473 0.1305
Table 1: Minimum errors over the validation part of the data set, obtained
using naive prediction (e
N
), exponential smoothing (e
ES
), ARMA
(e
ARMA
), and DTPNs (e
DTPN
). Only the results for the very best
DTPN are shown.
Data set n
IN
P
zero
n n
L
e
tr
DTPN
e
val
DTPN
Fed funds, run 1 2 0.00 7 5 0.3072 0.1837
Fed funds, run 2 2 0.25 5 5 0.2968 0.1881
GDP, run 1 5 0.00 4 4 0.2095 0.1423
GDP, run 2 4 0.00 6 4 0.2173 0.1399
GDP, run 3 3 0.00 5 4 0.2131 0.1360
GDP, run 4 3 0.20 11 5 0.2094 0.1305
Table 2: Examples of the performance of evolved DTPNs. The second col-
umn shows the number of inputs to the network, and the third col-
umn shows the probability of a mutation being of the zero-toggle
type, i.e. a mutation that sets the parameter in question to zero.
The fourth column shows the (evolved) number of neurons, and
the ?fth column shows the (evolved) number of layers (n
L
), i.e.
the number of distinct EOT values in the evolved network. The
two ?nal columns show the errors over the training and validation
parts of the data set.
next value in the time series, one would obtain an estimate of the error
e(t + 1) = Z(t + 1) ?
ˆ
Z(t + 1). Ideally, the measure should be such that
P(t) = f(e(t + 1)) where f is a known, monotonous function.
Several di?erent predictability measure can be formed. The amount of
(local) information in a time series can, for instance, be estimated analyti-
cally using random matrix theory, based on the correlation matrix formed
from the delay matrix D [8]. In addition, various empirical measures can
Figure 2: The best evolved network (run 4) for the prediction of the GDP
series. Input elements are shown as squares and neurons as ?lled
circles. The neurons are arranged in layers based on their EOT
values. For clarity, only the inputs to one neuron are shown. Solid
lines indicate positive weights and dotted lines negative ones.
also be generated, based on the prediction errors obtained in previous time
steps. An investigation was made involving both the analytical measure and
a few di?erent empirical measures, applied to the rescaled di?erence series
Z(t). However, in all cases, the results were negative, i.e. the proposed pre-
dictability measure showed near-zero correlation with the actual prediction
error, and therefore these measures will not be described further here.
7 Discussion and conclusion
This investigation has shown that it is possible to improve, albeit only
slightly, the predictions obtained from standard prediction methods using a
generalized version of neural networks (called discrete-time prediction net-
works, DTPNs) with the possibility of adding a short-term memory through
feedback couplings.
In earlier work [2], continuous-time recurrent neural networks were consid-
ered for time series prediction. The DTPNs introduced here do not require
continuous-time integration, i.e. the network output is obtained by discrete-
time equations rather than di?erential equations, making the evaluation of
the networks much faster, while still allowing a rich dynamical structure,
including dynamic short-term memory.
The use of an EA for the optimization of the networks removes all restric-
tions regarding both the behavior of individual neurons as well as the struc-
ture of the network as a whole, while still allowing standard feedforward
neural networks as a special case.
The importance of structural modi?cations in the network is illustrated
by the fact that, in any given run, the structure of the current best net-
work varied signi?cantly during the run. The ?nal networks often contained
rather few neurons and used only a few input elements, illustrating another
advantage of using recurrent networks: because of their ability to form a
short-term dynamic memory, such networks need not use as many inputs as
a feedforward network, thus also reducing the number of networks weights
and hence the risk of over?tting.
The best network for prediction of the GDP series, shown in Fig. 2, had
a slightly more complex structure. However, in the run generating that
network, zero-toggle mutations were used, and indeed the resulting network
was far from fully connected, and therefore had, in fact, a somewhat simpler
structure than would have been suspected on the basis of the number of
neurons involved.
The fact that the predictability measures all gave negative results was
expected, and it indicates that the DTPNs really do extract all, or almost
all, information available in the time series.
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Paper IV
Optimization of the investment strategy for a
combined hydrogen and hythane refueling station
Submitted to
International Journal of Hydrogen Energy.
Optimization of the investment strategy for a combined hydrogen and
hythane refueling station
Peter Forsberg
1
, Magnus Karlstr¨ om
2
1. Department of Applied Mechanics
Chalmers University of Technology, 412 96 G¨ oteborg, Sweden
e-mail: [email protected]
2. ETC Battery and FuelCells Sweden AB
Box 2055, 449 11 Nol , Sweden
e-mail: [email protected]
Abstract
One of the major barriers to the widespread use of hydrogen
is the lack of a hydrogen infrastructure, an important com-
ponent of which is the individual hydrogen refueling station.
The long-term pro?tability of the hydrogen ?lling station is
a key issue for the success of the transition to a hydrogen in-
frastructure.
The topic of this paper is the problemof ?nding the optimal
investment strategy for a single hydrogen and hythane refu-
eling station giving minimum production cost, while match-
ing the hythane and hydrogen capacity to a demand generated
from three stochastic scenarios over a 20-year period.
A minimal resulting production cost between USD 2-6/kg
for hydrogen and USD 1-1.5/kg for hythane (depending on
preferences concerning unsatis?ed demand, ?exibility etc.)
was found. It was also found that the production cost and
the amount of unsatis?ed demand constitute con?icting ob-
jectives so that, for example, if the total hydrogen and hythane
demand is to be satis?ed, the production cost of hydrogen
will be unrealistically high. The effect of uncertainties for the
constructed scenarios is minimized by the use of stochastic
optimization techniques.
Keywords: Hydrogen; Hythane; Infrastructure; Invest-
ment; Optimization; Refueling station
1 Introduction
Hydrogen is a promising fuel for vehicles. Four main argu-
ments support this assertion: (1) The potential of reducing
greenhouse gases from the transport sector; (2) An increase
in energy supply security, since hydrogen can be produced
from many energy sources so that the risk of a shortage of
supply may be reduced; (3) Hydrogen has higher energy ef-
?ciency than do other fuels; (4) The use of hydrogen leads
to the possibility of zero local emissions with the use of fuel
cells [1, 2, 3]. The magnitude of the bene?ts of hydrogen fuel
cell vehicles has been assessed by Karlstr¨ om [4] and Sanden
and Karlstr¨ om [5].
However, these bene?ts can only be exploited if several
barriers to a large-scale introduction of hydrogen fuel cell ve-
hicles are reduced. One of the major barriers is the lack of a
hydrogen infrastructure [6]. The construction of a full-scale
hydrogen infrastructure with production facilities, a distribu-
tion network, and refueling stations is likely to be very costly.
The venture of constructing a hydrogen refueling infrastruc-
ture constitutes a long-term, capital intensive investment with
great market uncertainties for fuel cell vehicles. Therefore,
reducing the ?nancial risk is a major objective of any long-
term goal to build a hydrogen infrastructure [7].
Ogden [8] has described several hydrogen supply options.
Investigations have also been made for large scale production
of hydrogen [9, 10, 11]. Many studies of cost and technology
for a hydrogen infrastructure and for individual stations have
also been carried out [12, 13, 14, 15, 16, 17, 18, 19]. The
H2A analysis group at the US department of energy (DOE)
has recently developed two H2A delivery models: the H2A
Delivery Components Model and the H2A Delivery Scenario
Model [20].
The above studies mainly consider hydrogen as a fuel from
an environmental and economic standpoint in a large scale
perspective. By contrast, Forsberg and Karlstr¨ om [21] in-
vestigated the most pro?table investment strategy for the in-
dividual hydrogen refueling station featuring on-site small-
scale reforming of methane [21]. The question was when,
and to what extent, to build the parts of the station, satisfying
an increasing demand of hydrogen. The result was a mini-
mum hydrogen production cost of 4-6 USD/kg, depending on
the number of re-investments during the 20-year-period con-
sidered. This paper is a continuation and an improvement
of [21]. The improvements explored are mainly:
1. The earlier study relied on single-objective optimization
of the production cost and hence produced one optimal
investment strategy. In this study, multi-objective op-
timization is used for ?nding Pareto optimal fronts for
contradictory objectives. For decision-making it is more
favourable to have a set of solutions, each represent-
ing different possible investment alternatives. Preferably
1
these should lie on the pareto-optimal front of important
objectives.
2. The sensitivity analysis in the previous study showed
that the results were quite sensitive to variations in the
strategic parameters and in particular to the estimated
number of refueling vehicles (represnted by the S-curve,
i.e. a curve indicating the estimated technology adapta-
tion, see below). In this study we present three scenar-
ios with considerably different future developments. For
each scenario we generate a large number of samples and
make use of stochastic optimization to ?nd the best so-
lutions.
3. Hythane, i.e. natural gas mixed with a small fraction of
hydrogen, is a viable intermediate alternative fuel for ve-
hicles. Whereas the earlier study only considered hydro-
gen, in this study both hydrogen and hythane are taken
into account.
4. In reality equipment is available in a (?nite) number of
sizes. For simplicity, the earlier study made use of con-
tinuously sized equipment. By contrast, this study ap-
plies presently available sizes of equipment, thus achiev-
ing a higher degree of realism. In addition, the costs of
equipment, electricity, and natural gas have been updated
to current (2006) values.
The most important issues in this study are to reduce the ef-
fect of uncertainties for scenario parameters and to identify
connections between production cost and other results. The
calculations cover 20 years, from 2010 until 2030. If an in-
vestment is made, it takes place at the very beginning of the
year, i.e. an investment is in year 1 occurs on the 1
st
of Janu-
ary 2010.
2 Strategic parameters
The strategic parameters in?uence how the generated future
scenarios are calculated and are therefore crucial to the re-
sults. These parameters and their respective values are given
in Table 1.
The number of produced units of reformers, electrolysis
etc. is considered to equal the number of hydrogen refueling
stations, which is estimated to reach 5,000 in the year 2010
and 50,000 in the year 2030, and to follow the S-curve
R(t) =
1
1 + e
?B(t?Tx)
, (1)
in between. t is the time from year 2010, T
x
the S-curve in-
?ection point and B the slope. The estimation concerning the
growth of hydrogen fuel cell vehicles is assumed. The growth
of number of stations is based upon the estimated growth of
hydrogen fuel cell vehicles and their hydrogen demand. A
report presented by E4tech [14] and funded by the UK De-
partment of Trade and Industry and the Carbon Trust predicts
that ”if the hurdles are overcome, the mainstream propulsion
market is expected to open up after 2010”. Melaina [24] made
a preliminary investigation concerning the suf?cient number
Table 1: Strategic parameters. The electricity price is as-
sumed to be higher during daytime (6 am-10 pm) than at night
(10 pm-6 am). All parameter values are estimates except nat-
ural gas price, which is from [22] and electricity price, which
is from [23]
Name Description Value Unit
B S-curve slope 0.3 -
D Real rate of interest 0.1 1/year
Fcont Contingency cost factor 0.1 -
Feng Engineering permitting cost factor 0.1 -
Fgen Include land cost factor 0.2 -
F
h2y
Mass ratio hydrogen in hythane 0.03
N Number of time steps 175 200 -
Pe Electricity price vector (6am-10pm) 0.10 USD/kWh
Electricity price vector (10pm-6am) 0.08 USD/kWh
Png Natural gas price 0.97 USD/kg
t0 Start time of calculations (2010) 0
t
f
End time of calculations (2030) 175 200
Tx In?ection point of the S-curve 10 year
W Scenario sample - -
X
hf
Hydrogen demand (from scenario) - kg/h
X
yf
hythane demand (from scenario) - kg/h
of initial hydrogen stations in the US, and concluded that be-
tween 4,500 and 17,700 hydrogen stations would be required
to initiate a hydrogen infrastructure for fuel cell vehicles. The
estimate of 50,000 hydrogen stations in 2030 used here is mo-
tivated by the fact that this investigation takes the whole world
into account. Also, in this study, the market reaches a high
level of maturity, further motivating the estimate for the num-
ber of stations in 2030.
The decrease in purchase price, in relation to the present-
day purchase price, owing to increased production and tech-
nology development (f
t
) for a given type of equipment eq is
approximated as
f
t,eq
(t) =
(50000R(t))
(log fp,eq/ log 2)
5000
(log fp,eq/ log 2)
(2)
= (10R(t))
(log(fp,eq)/ log(2))
.
Here, f
p,eq
is a progress factor for the equipment in ques-
tion, i.e. a factor that determines the purchase cost decay
rate for the speci?ed equipment. Equation (2) is used for all
equipment parts of the refueling station, regardless of size. It
should be kept in mind that the function f
t,eq
is purely exoge-
nous and therefore uncertain. This uncertainty will in?uence
the results, as is discussed in Section 6.
Using the present day value correction factor, (C
p
), future
costs can be discounted to present day value as
C
p
(t) =
1
(1 + D)
t/8760
, (3)
where t is the number of hours from the start of calculation,
t
0
, and D is the real interest rate. Furthermore, the consecu-
tive present day value correction vector is de?ned as
C = [C
p
(1) C
p
(2) . . . C
p
(N)] . (4)
The average of the components of this column vector, i.e.
C(t) =
1
N
N
t=1
C
p
(t) (5)
can be used to calculate the present value of evenly distributed
costs. For D = 0.1, C(t) = 0.4466.
2
2.1 Scenario generation
The number of vehicles visiting the single refueling station is
a stochastic variable which is estimated in three scenarios. In
these scenarios, the following vehicles are considered:
1. Ordinary combustion engine powered buses running on
hythane.
2. Ordinary combustion engine powered cars running on
hythane.
3. Ordinary combustion engine powered buses running on
hydrogen.
4. Ordinary combustion engine powered cars running on
hydrogen.
5. Fuel cell driven buses running on hydrogen.
6. Fuel cell driven cars running on hydrogen.
7. Fuel cell driven scooters running on hydrogen.
The ?rst four vehicles represent intermediate solutions, used
until the fuel cell driven alternatives have become dominant.
The above vehicles are considered to have ?lling data and sta-
tistics in accordance with Table 2.
Using these data, three possible future scenarios are given
in Table 3. The ?rst scenario emphasizes hythane and hydro-
gen powered buses as an intermediate alternative. The second
scenario focuses on hydrogen cars, primarily with combustion
engines early on, and fuel cells toward the end of the period
considered. In the third scenario hydrogen fuel cell powered
scooters are in focus. In all three scenarios hydrogen fuel cell
cars are used in the longer perspective.
For interpolation between the three time periods speci?ed
in Table 3, the S-curve has been used, giving the smooth curve
shown in Figure 1. The smoothness obtained through the in-
terpolation is likely to be valid for the car purchases of groups
of individuals, but may, of course, be violated e.g. in the case
of large corporations that may acquire several vehicles (such
as buses) at the same time.
For each scenario a set of samples (W) is generated using
Poisson distributions with parameters from Tables 2 and 3.
With a time step length of one hour, the total number of steps
is N = 24×365×20 = 175, 200 for each sample. For the sce-
nario generation, hydrogen ?lling is separated from hythane
?lling. The resulting hydrogen and hythane demand is de-
noted X
hf
and X
yf
, respectively.
3 The refueling station
The task of the refueling station is to provide fuel for hydro-
gen and hythane vehicles. As the main energy carrier, natural
gas is chosen. One reason is the already present natural gas
refueling network. Moreover, natural gas is one of the cheap-
est production sources for hydrogen in the short term. The
hythane dispenser part of the refueling station is an interme-
diate alternative on the path to hydrogen vehicles.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0
5
10
15
20
25
30
35
40
45
Scenario1
Year
N
u
m
b
e
r
o
f
v
e
h
i
c
l
e
s
Hytan ic cars
Hytan ic buses
Hydro fc cars
Figure 1: Number of vehicles in scenario 1 as a function of
time. Data based on S-function-smoothened values from Ta-
ble 3.
After reforming, the produced hydrogen gas is compressed
and stored. The high storage pressure then makes it possi-
ble to refuel hydrogen without further compression. Some
of the hydrogen can be used for mixing with natural gas to
form hythane, which is refueled using a special hythane dis-
penser. The mixture of hydrogen in hythane is here set to 3%
by weight. When electricity prices are low (i.e. at night), it
might be more pro?table to produce hydrogen by electrolysis
than through the reformer. For this option an electrolyser can
be added. All in all, this is a ?exible layout capable of sim-
ulating many types of possible future hythane and hydrogen
refueling stations, see Figure 2. The model is also ?exible
with respect to refueling station types, e.g. car, truck, or bus,
and refueling station locations e.g. central, suburban, or coun-
tryside, by changing the strategic parameters.
The model developed and the optimisations made involve
the components within the refueling station. Components out-
side the station are considered to be already present.
3.1 The parts of the refueling station
Table 4 gives data on the parts of the refueling station from
Figure 2. Data are taken from actual produced equipment in
the year 2000 [25].
The reformer, electrolysis, and compressor are chosen from
a ?nite set of available sizes, while the H
2
store as well as
the hythane and H
2
dispensers are purchased on a piece-wise
basis depending on the required capacity. Some equipment
cannot be used below a certain minimum level, which is indi-
cated by the minimum usage rate (f
u
) and given as a fraction
of maximum capacity. For the reformer, the minimum usage
rate equals the minimum capacity. For the H
2
store, the min-
imum usage rate indicates the minimum storage level. Below
this minimum storage level the pressure drops too low to be
dispensed to vehicles. For sizes different from the nominal
capacity (c
n
), the purchase price is calculated using the scale
3
Table 2: Filling statistics for vehicles visiting the single refueling station. ic denotes internal combustion engine and fc denotes
fuel cell powered engine. ?T denotes the time between ?llings, and T
d
denotes the time of day at which ?lling takes place. The
numbers are estimates.
Vehicle type hythane [kg/?lling] H2 [kg/?lling] ?T [days] T
d
[h]
hythane ic bus 61 0 1 5-8
hythane ic car 6 0 3 1-24
Hydrogen ic bus 0 60 1 5-8
Hydrogen ic car 0 6 3 1-24
Hydrogen fc scooter 0 2 5 1-24
Hydrogen fc bus 0 40 1 5-8
Hydrogen fc car 0 5 5 1-24
Table 3: Number of vehicles visiting the single refueling station for each scenario. ic denotes internal combustion engine and fc
denotes fuel cell powered engine. Figures are based on assumptions of different future scenarios for the introduction of hydrogen
vehicles.
Time span Scenario 1 Scenario 2 Scenario 3
Year 1-5 10 hythane ic buses, 40 hythane ic cars 10 H2 ic cars, 2 H2 ic buses 30 hydro fc scooters
Year 5-10 20 hythane ic buses, 10 H2 fc cars 10 H2 ic cars, 2 H2 ic buses, 20 H2 fc cars 30 H2 fc scooters, 20 H2 fc cars
Year 10-20 40 H2 fc cars 40 H2 fc cars 40 H2 fc cars
Table 4: Data on the refueling station parts. Electricity use is the amount of electrical energy consumed for each kg of output for
the piece of equipment in question. ng denotes natural gas and pc number of pieces. Data on the hythane dispenser are estimated
from data on the hydrogen dispenser. Figures are from [25].
Part Reformer Electrolysis Compressor H
2
store Hythane dispenser H
2
dispenser
Life time (l) 10 year 20 year 10 year 20 year 10 year 10 year
Nom. capacity (cn) 4.2 kg/h 62.5 kg/h 4.2 kg/h 21 kg 96 kg/h 48 kg/h
Nom. purchase cost (pn) 100,000 USD h/kg 34,632 USD h/kg 12,143 USD h/kg 22,500 USD/pc 120,000 USD/pc 60,000 USD/pc
Scale factor (fs) 0.75 0.72 0.8 - - -
Available sizes 4.2, 12.5, 62.5 kg/h 4.20, 12.5, 62.5 kg/h 5, 15, 75 kg/h 21 kg 96 kg/h 48 kg/h
Maintenance cost (fm) 0.07 0.07 0.05 0.01 0.035 0.035
Ef?ciency (?) 0.26 kg H
2
/kg NG 0.02 kg H
2
/kWh 1.0 1.0 1.0 1.0
Electricity use (fe) 1.5 kWh/kg - 2.2 kWh/kg 0.0 kWh/kg 0.0 kWh/kg 0.0 kWh/kg
Progress ratio (fp) 0.9 0.9 0.9 0.9 0.9 0.90
Min. usage rate (fu) 0.25 0.0 0.0 0.56 0.0 0.0
4
Figure 2: Hythane and hydrogen refueling station layout.
Natural gas is reformed to hydrogen on-site and stored for
delivery to vehicles. It is also possible to produce hydrogen
from electricity by electrolysis. In this study, only the parts
within the refueling station are considered.
function
p
eq
(s) = p
n
s
c
n
s
1?fs
= p
n
c
1?fs
n
s
fs
. (6)
Here f
s
is a scale factor (see Section 2). Using this function
an estimated purchase price is obtained for a piece of equip-
ment of arbitrary size (s [kg/h]), using the purchase price p
n
for an existing piece of equipment, with capacity c
n
. The
function (6) applies to the reformer, electrolysis, and com-
pressor. The expected life time l is used to reduce the in-
vestment cost, should the investment period (2010-2030) end
before the end of life of the piece of equipment in question.
The reduction in purchase cost is approximated by a linear
function (in time). The ef?ciencies in Table 4 indicate the re-
lation between the mass entering and leaving the equipment.
In the case of the reformer, the substance entering is methane
and that leaving is hydrogen.
3.2 Initial considerations
The hydrogen/hythane refueling station is assumed to be built
in conjunction with an existing natural gas refueling station.
The supply of natural gas can be delivered by truck or, more
commonly, by pipeline. In any case, the supply is consid-
ered to be already established and only the cost for the pur-
chase of natural gas is taken into account. All costs for the
hythane/hydrogen part of the station, i.e. land use and wages,
are accounted for. In reality some resources can probably be
shared between the natural gas and hythane/hydrogen parts of
the refueling station. Initially, the hydrogen storage is consid-
ered to be empty.
3.3 The model
The model for the refueling station in Figure 2 only has one
state variable, i.e. a variable to be integrated, which is the
amount of stored hydrogen, x
hs
. Once the stored amount of
hydrogen is known, all other relevant quantities can be cal-
culated directly. Letting k denote the time step, one can ex-
press x
hs
in the form of the difference equation x
hs
(k +1) =
x
hs
(k) + x
o
hc
? x
o
hs
, where x
o
hc
is the amount leaving the
compressor (see Figure 2), which equals the amount enter-
ing the storage, and x
o
hs
the amount leaving the storage. The
compressor output comes from the reformer x
o
hr
and elec-
trolysis x
o
he
and taking the compressor ef?ciency ?
hc
into
account one can write x
o
hc
= (x
o
hr
+ x
o
he
)?
hc
. In order to
determine the natural gas consumption of the reformer x
i
hr
,
the equation x
o
hr
= x
i
hr
?
hr
is added, where ?
hr
is the ef-
?ciency of the reformer. The amount leaving the hydrogen
storage is the sum of dispensed hydrogen x
o
hd
and the hy-
drogen part F
h2y
of dispensed hythane x
yd
. Taking the dis-
penser’s ef?ciencies ?
hd
and ?
yd
into account, this amount
is obtained as x
o
hs
= x
o
hd
/?
hd
+ x
o
yd
F
h2ng
/?
yd
. Now the
total natural gas consumption x
ng
from both the hydrogen
x
i
hr
and hythane x
o
yd
(1 ? F
h2y
) part can be calculated as
x
ng
= x
i
hr
+ x
o
yd
(1 ? F
h2y
)/?
yd
. Thus, in summary, the
following equation system is obtained
x
hs
(k + 1) = x
hs
(k) + x
o
hc
?x
o
hs
,
x
o
hc
= (x
o
hr
+ x
o
he
)?
hc
,
x
o
hr
= x
i
hr
?
hr
,
x
o
hs
=
x
o
hd
?
hd
+
x
o
yd
F
h2ng
?
yd
,
x
ng
= x
i
hr
+
x
o
yd
(1 ?F
h2y
)
?
yd
. (7)
However, these equations are subject to some constraints.
First of all, there are minimum and maximum levels both for
the ?ow and for the amount stored. Second, the amount of
dispensed hydrogen and hythane, x
o
hd
and x
o
yd
, must be non-
negative and are limited from above by the scenario sample
demand (X
hf
and X
yf
, respectively). Note that the demand
is not necessarily totally satis?ed. All in all, the following
constraint equations are obtained
f
u,hr
s
hr
? x
o
hr
? c
hr
,
0 ? x
o
he
? c
he
,
0 ? x
o
hc
? c
hc
,
f
u,hs
s
hs
? x
hs
? c
hs
,
0 ? x
o
hd
? c
hd
,
0 ? x
o
yd
? c
yd
,
0 ? x
o
hd
? X
hf
,
0 ? x
o
yd
? X
yf
, (8)
where c
h
e denotes the hydrogen electrolysis capacity, c
hc
the compressor capacity, c
hs
the storage total capacity, c
hd
and c
yd
the hydrogen and hythane dispenser capacity, respec-
tively. Note that, for the reformer and storage, the minimum
utilization level is higher than zero. This is due to the fact that
the hydrogen dispenser cannot be run below a certain mini-
mum ?ow rate, given as a ratio f
u,hr
of the maximum capac-
ity s
hr
, giving f
u,hr
s
hr
as the minimum allowed ?ow rate.
For the storage, the hydrogen gas pressure falls below accept-
able limits for the dispenser if the stored amount is less than
5
the ratio f
u,hs
of the total capacity s
hs
, thus making f
u,hs
s
hs
the minimum amount to be stored.
For the optimization, the state variable is the hydrogen stor-
age (x
hs
), the control variables are the outputs of reformer
and electrolysis (x
o
hr
and x
o
he
, respectively), and the distur-
bances are the stochastic variables hydrogen and hythane de-
mand (X
hf
and X
yf
, respectively).
4 The optimization problem
The optimization problem under consideration can be for-
mulated as a discrete-time stochastic optimal-control prob-
lem [26, 27, 28]. In this type of problem the aim is to ?nd
the control U that minimises an objective function J(U) for a
dynamical system f(X, U, W) during a speci?ed time, in the
discrete case indexed by the time step variable k. The system
is also in?uenced by an independent random disturbance W.
The general formulation is
min
U
J(U) =
N?1
k=0
?(k, X
k
, U
k
, W
k
) + ?(X
N
, W
N
)
s.t. X
k+1
= f(k, X
k
, U
k
, W
k
) (9)
c
k
(X, U) ? 0 ?k = 1, . . . N,
where ?(k, X
k
, U
k
, W
k
) is the cost associated with each time
step k, ?(X
N
, W
N
) is the terminal cost and c
k
(X, U) rep-
resents simple limits of the state and control variables. The
controller makes use of the information set ?
k
, the contents
of which depend on the type of control system. For an open-
loop system, ?
k
= {X
0
} ?k, whereas for a feedback system,
?
k
= {X
0
, X
k
}, k = 0, 1, . . . , N ? 1. For a closed-loop
system, ?
k
= {X
0
, X
1
, . . . , X
k
, U
0
, U
1
, . . . , U
k?1
}. In this
study, the investment strategy is set prior to t
0
and then fol-
lowed until t
f
, thus de?ning an open-loop control system as
described above.
The problem is to ?nd the optimal investment strategy ?
?
that will subsequently minimise two objective functions, fur-
ther discussed in Section 4.1. An inner control loop is used
to keep the hydrogen storage level at a given amount. In this
loop, a control algorithm is implemented to keep the hydro-
gen storage at a speci?ed level. Since the time constants of
both reformer and electrolysis are very short (of the order of
minutes) compared to the time step (one hour), the desired
control action will be considered to take effect immediately.
Due to the rapid dynamics of both reformer and electrolysis,
these devices can be shut down fast and, therefore, there is
no need to keep the storage below 100% as a precaution to
avoid over?ow due to slow production adaptability for un-
expectedly low levels of demand. The control algorithm ?rst
calculates the deviation in hydrogen storage fromthe set point
(the error), then ?lls up the storage with available hydrogen,
which is the sum of reformer maximumcapacity and electrol-
ysis during the period when electricity is cheaper, i.e. 10pm -
6am. No electrolysis is used during the remaining expensive
hours. The calculated amount is then added to the storage.
Depending on the demand priority policy ?, either an attempt
is made to satisfy ?rst the hydrogen refueling demand and
then the hydrogen part of hythane, or vice versa. Tuning this
policy will have a signi?cant in?uence on the amount of un-
satis?ed demand. The total control vector is then
U = [? ?]. (10)
The inner control loop does not have any tunable parameters
and is thus not part of the optimal control problem. It is im-
plemented as an obvious optimal solution to keep the size of
the variable space at a minimum.
Direct control parameter mapping methods, i.e. methods
that will need dedicated control parameters for each step, are
not used. Such techniques are intractable due to the large
number (N = 175, 200) of steps involved.
No terminal cost ?(X
N
, W
N
) is used. Instead the total in-
vestment cost (for the entire life time) is scaled linearly, in
accordance to the usage time of the piece of equipment, see
Eq. (13) below. The system equation and the simple con-
straints have been given in Eqs. (7) and (8). The random
disturbance W is the hythane and hydrogen demand, further
described in Section 2.1 above.
4.1 Objective functions
In this study, the following performance measures are used
1. Production cost per kg for hydrogen p
hf
. This is the
production cost for hydrogen at the hydrogen dispenser
and is calculated as the sum of all hydrogen related costs
divided by the total amount of sold hydrogen x
o
hd
.
2. Unsatis?ed demand for hydrogen x
h,u
. This is the de-
mand that cannot be satis?ed at the hydrogen dispenser
and is a negative measure, i.e. a low amount of unsatis-
?ed demand is desirable.
3. Production cost per kg for hythane p
yf
. This is the pro-
duction cost for hythane at the hydrogen dispenser and is
calculated as the sum of all hythane-related costs divided
by the total amount of sold hythane x
o
yd
.
4. Unsatis?ed demand for hythane x
y,u
. This is the demand
that cannot be satis?ed at the hythane dispenser and is a
negative measure, i.e. a low amount of unsatis?ed de-
mand is desirable.
5. Unsatis?ed demand for all hydrogen x
ht,u
. The total
amount of hydrogen demand that cannot be satis?ed,
i.e. the sum of unsatis?ed demand at the hydrogen dis-
penser x
h,u
and as part of hythane F
h2y
at the hythane
dispenser x
y,u
.
6. Flexibility p
h?
. A measure used for quantifying the dif-
ference between the cost for the active scenario, i.e. the
scenario for which the present solutions have been op-
timized, and the passive ones, i.e. the scenarios that are
not part of the optimization.
For the optimization, the production cost per kg for hydro-
gen x
h,u
and the unsatis?ed demand for all hydrogen x
ht,u
are used in the objective function J which then takes the form
J(U) = [p
hf
x
ht,u
]. (11)
6
In order to compute the above performance measures, a num-
ber of costs and ?ows need to be calculated. Before carrying
out the calculation, however, an assumption is made (and ap-
plied to all calculations below) that each part of the refueling
stations takes on its own expenses. This implies that expenses
from the hythane part will be added to the hythane produc-
tion cost and the same for the hydrogen part. Parts used in
both hythane and hydrogen production, such as storage, are
charged according to the usage ratio
f
hpc
=
x
hf
(x
hf
+ F
h2y
x
yf
)
. (12)
For each part (eq) of the refueling station where the pur-
chase cost is scaled to the used size, i.e. reformer, electrolysis
and compressor, the cost is computed as
p
eq
(t
i
, s
eq
) = f
t,eq
(t
i
) p
eq
(s
eq
) max(
20 ?t
i
l
eq
, 1), (13)
where l
eq
is the estimated lifetime of the part in question, see
Table 4. For those items purchased on a piece-wise basis,
i.e. storage tanks and dispensers, the cost is computed as
p
eq
(t
i
, n
eq
) = f
t,eq
(t
i
) p
n
n
eq
max(
20 ?t
i
l
eq
, 1). (14)
This implies a linear scaling of the total investment cost for
the entire estimated life time to the time it is actually used.
The total cost consists of costs for purchase of equipment,
resources, maintenance and a factor to cover for construction,
land use and general expenses. It is estimated that the cost for
loans for equipment will cancel the effect of the present value
correction for the sum of instalments. This is exactly the case
of annuity loans. All other costs are discounted to present day
using equation (3).
The cost for equipment shared by the hythane and hydro-
gen parts is then
p
c,eq
=
?i
(p
hr
(t
i
, s
hr,i
) + p
he
(t
i
, s
he,i
) +
p
hc
(t
i
, s
hc,i
) + p
hs
(t
i
, n
hs,i
)). (15)
The cost for maintenance is estimated to a speci?ed fraction
(f
m
) of the equipment cost, that is
p
c,m
=
?
?
?
?
?
1
1
.
.
.
1
?
?
?
?
?
?i
f
m,hr
p
hr
(t
i
, s
hr
)
8760 l
hr
+
f
m,he
p
he
(t
i
, s
he
)
8760 l
he
+
f
m,hc
p
hc
(t
i
, s
hc
)
8760 l
hc
+
f
m,hs
p
hs
(t
i
, n
hs
)
8760 l
hs
, (16)
which is an N × 1 column vector. The factor 8760 is used
to convert the equipment life time from years to hours. The
electricity cost is calculated from equipment ?ows as
p
c,e
= P
e
?(f
e,hr
x
o
hr
+ f
e,he
x
o
he
+ f
e,hc
x
o
hc
+ f
e,hs
x
hs
) (17)
where ?is the element-wise multiplication operator. The nat-
ural gas cost is
p
c,mg
= P
ng
x
i
hr
. (18)
The total cost for the shared parts is then
p
c
= (1 + F
cont
+ F
eng
+ F
gen
) p
c,eq
+
pds ×(p
c,m
+ p
c,e
+ p
c,mg
). (19)
For the hydrogen and hythane speci?c parts, calculation of
costs follows the same pattern as in Eqs. (15)-(19) above, and
is therefore not listed here.
The resulting production cost per kg of hydrogen p
hf
and
hythane p
yf
is now
p
hf
=
p
c
f
hpc
+ p
h
x
hf
(20)
and
p
yf
=
p
c
(1 ?f
hpc
) + p
y
x
yf
, (21)
respectively.
It is clear from Eq. (8) that not all the demand from the
scenario samples need be satis?ed. The difference between
the demand and the actual sold amount is called unsatis?ed
demand. Three measures of unsatis?ed demand are used,
namely (1) hydrogen dispenser unsatis?ed demand
x
h,u
= X
hf
?x
o
hd
, (22)
(2) hythane dispenser unsatis?ed demand
x
y,u
= X
yf
?x
o
yd
, (23)
and (3) total unsatis?ed hydrogen demand from both the hy-
drogen dispenser and the hydrogen part of the hythane at the
hythane dispenser
x
ht,u
= x
h,u
+ F
h2y
x
y,u
. (24)
The variance of the above objectives p
hf
, p
yf
, x
h,u
and x
y,u
between samples is used as a measure of sensitivity, which
can also be interpreted as risk. A high variance would imply
a higher risk. Flexibility is a complex measure that can be
de?ned in a number of ways [29]. Here, it has been de?ned
as
p
h?
= p
a
h
?
p
p1
h
+ p
p2
h
2
, (25)
i.e. as the mean difference between the hydrogen production
cost for the active scenario and the passive ones. A positive
value indicates a lower cost for the passive scenarios and vice
versa.
4.2 Optimization strategy
The stochastic control problem (9) is solved with a
simulation-based optimization technique [30]. For each can-
didate solution U to the problem, the refueling station is eval-
uated in a number of discrete simulations under the stochas-
tic in?uences from samples generated from scenarios. When
all samples have been evaluated, the performance measures
are estimated and the solutions are tuned accordingly. In this
study a genetic algorithm (GA) has been used to optimise the
7
Figure 3: A schematic illustration of the optimization frame-
work.
solutions. GAs are optimization algorithms inspired by bio-
logical evolution. Such algorithms can easily be adapted to
a wide range of optimization problems [31], including multi-
dimensional problems [32]. The optimization algorithm used
in this study is an elitist non-dominated sorting GA, called
NSGA-II [33], which uses an explicit diversity-preserving
mechanism. For each Pareto-optimal front, this algorithmwill
remove solutions lying close to each other, while preserving
those far from each other. The result is a good spread of solu-
tions along the front. However, the longer the front, the more
solutions are needed to get a good picture of the details of
the curve. Since each solution corresponds to one individual
in the GA, more solutions means a larger population which
takes longer time to evolve.
The major parts in the optimization framework is the sce-
nario generator, simulator and optimiser, see Figure 3. The
steps of the evaluation are
1. A sequence of samples are generated for each scenario
in the scenario generator. The probability distributions
used are discussed in Section 2.1.
2. A number of initial individuals (candidate solutions, U)
are randomly generated.
3. For each individual (candidate solution, U), the simula-
tor simulates the refueling station (Section 3.3) over the
entire investment period. The simulator also contains a
control algorithm for proportional control of the amount
of hydrogen stored.
4. When all samples for all scenarios have been simulated,
the simulator estimates the performance measures and
objectives for the individual.
5. When all individuals in the population have been evalu-
ated, the front and objective distance (crowding) sorting
is carried out, followed by a generational replacement
with crossover and mutation.
6. Steps 3-5 are repeated until convergence.
0 1 2 3 4 5 6 7 8 9 10 11
0
0.5
1
1.5
2
2.5
3
3.5
x 10
5
Scenario 1
Hydrogen production cost [USD/kg]
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
k
g
]
1
2 3
4
6
10
20
35
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
%
o
f
t
o
t
a
l
]
0
10
20
30
40
50
60
70
80
90
100
Figure 4: The resulting Pareto-front when optimising U for
scenario 1.
5 Results
This section presents the results from the optimization for the
three scenarios discussed in Section 2.1. For each scenario the
optimization has been carried out using the objective function
de?ned in Eq. (11).
5.1 Scenario 1: hythane combustion engine
buses
Scenario 1 emphasizes hythane and hydrogen powered buses
as an intermediate alternative, after which fuel cell powered
cars take over. The hydrogen cost versus total hydrogen un-
satis?ed demand from both hydrogen and hythane dispenser
can be seen in Figure 4, in which several interesting solutions
(discussed below) are marked with their respective numbers.
It is evident that as the cost decreases, the unsatis?ed demand
increases and therefore that these objectives are in con?ict
with each other. The discontinuities represent stepwise in-
creases of capacity to the next available size as de?ned in Ta-
ble 4.
As can be seen in Figure 4, Solutions 1 and 2 represent
extreme points regarding the two optimization objectives.
For Solution 1, the production cost is at a minimum, 1.96
USD/kg. In this strategy, an investment in a small electrolysis
equipment is made in year 18. The strategy results in a large
amount of unsatis?ed demand, 1.4×10
5
kg (87%of the total)
and 6.0 × 10
6
(100% of the total) for hydrogen and hythane,
respectively. By contrast, in Solution 2 where the production
cost is at its maximum (10.5 USD/kg), investments are made
in years 0, 5, and 10. With this strategy, the unsatis?ed de-
mand is negligible for both hydrogen and hythane. Further-
more, the strategy exhibits a preference for reformer use in
the beginning of the period, and electrolysis towards the end.
Intermediate solutions include number 3, 10 and 35 in Fig-
ure 4. These solutions represent extreme points before the
next possible equipment size is used. If unsatis?ed demand is
to be kept at minimum, Solution 3 may be a good alternative.
If hydrogen production cost is to be kept low, while still not
8
Table 5: Investment strategy for solution 3 for scenario 1.
Investment no 1 at year 1 2 at year 11
Reformer 4.2 kg/h 0.0 kg/h
Electrolysis 0.0 kg/h 12.5 kg/h
Compressor 5.0 kg/h 15.0 kg/h
H2 store 84 kg 147 kg
H2 dispenser 48 kg/h 48 kg/h
Hythane dispenser 864 kg/h 864 kg/h
Investment cost 1.5 × 10
6
USD 9.5 × 10
5
USD
Prio. strategy Hydrogen
Total inv. cost 2.4 × 10
6
USD
Maint. cost 5.1 × 10
4
USD
6.02 6.03 6.04 6.05 6.06 6.07 6.08 6.09 6.1 6.11
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Histogram for investment strategy 3 in scenario 1, calculated for scenario 1.
Hydrogen production cost [USD/kg]
P
r
o
b
a
b
i
l
i
t
y
Figure 5: Histogram for hydrogen production cost for solu-
tion 3.
allowing a large amount of unsatis?ed amount of hydrogen,
Solution 10 may be considered. The details of the investment
strategy, including the investment cost, can be seen in Table 5.
If the hydrogen production cost is calculated for all sam-
ples, a cost distribution is generated. Figure 5 shows a his-
togram of the cost distribution for solution 3. The distrib-
ution shows only a relatively small variance. A calculation
of the cost for all solutions indicates a decrease in relative
variance (details omitted), i.e. var(x)/x, for lower production
cost. Since a high variance indicates a high uncertainty, the
variance does not con?ict with the hydrogen production cost.
This dependence is typical for all scenarios.
In Figure 6, which shows hydrogen versus hythane produc-
tion cost, a strong non-linear correlation with a minimum for
Solution 35 can be noticed. The non-linearity is most evident
for the region left of this minimum. These points correspond
to solutions to the left of Solution 35 in Figure 4. The lowest
hydrogen production cost, represented by Solution 1, utilizes
no hythane and is therefore not shown in Figure 6.
Each solution and sample corresponds to one trajectory of
the state variable x
hs
. Given the state variable and the refu-
eling ?ows X
hf
and X
hf
, all other ?ows can be calculated
from Eqs. (7). When these ?ows are calculated for Solution
3, a good utilisation of equipment is found for most of the 100
samples, and this is veri?ed by the small variance in produc-
tion cost in Figure 5.
If the optimal solutions for Scenario 1 are calculated using
0 1 2 3 4 5 6 7 8 9 10 11
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Scenario 1
Hydrogen production cost [USD/kg]
H
y
t
a
n
p
r
o
d
u
c
t
i
o
n
c
o
s
t
[
U
S
D
/
k
g
]
2
3
4
6 10
20
35
Figure 6: Hydrogen cost versus hythane cost, Scenario 1.
0 1 2 3 4 5 6 7 8 9 10 11
?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
Scenario 1
Hydrogen production cost [USD/kg]
H
y
d
r
o
g
e
n
f
l
e
x
i
b
i
l
i
t
y
[
U
S
D
/
k
g
]
1
2 3
4
6
10
20
35
Figure 7: Hydrogen cost versus hydrogen ?exibility for Sce-
nario 1.
Scenarios 2 and 3, it is found that hydrogen production cost
will most likely increase. The ?exibility index in Figure 7
shows that some solutions result in considerably higher hy-
drogen production cost, e.g. Solution 6, while others do not,
e.g. Solution 1. Note that a positive value indicates a lower
cost for the passive scenarios and vice versa. The solutions
are no longer part of the Pareto-optimal front since they orig-
inate from the solution for Scenario 1. However, the original
extreme Solutions 1 and 2 will still be the extreme ones.
5.2 Scenario 2: Hydrogen combustion engine
cars
The second scenario focuses on hydrogen cars, with primarily
combustion engines in the beginning and fuel cells towards
the end of the 20-year time period. No hythane is used in this
scenario.
In essence, the production cost versus hydrogen unsatis?ed
demand curve resembles that of Scenario 1. Figure 8 shows
that almost zero unsatis?ed demand can be maintained down
to a production cost of around $6/kg, belowwhich the amount
9
0 1 2 3 4 5 6 7 8 9 10 11 12 12
0
1
2
3
4
5
6
7
8
9
x 10
5
Scenario 2
Hydrogen production cost [USD/kg]
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
k
g
]
1
2
41
49
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
%
o
f
t
o
t
a
l
]
0
10
20
30
40
50
60
70
80
90
100
Figure 8: The resulting Pareto-front when optimising U for
Scenario 2.
0 1 2 3 4 5 6 7 8 9
0
2
4
6
8
10
12
14
16
18
x 10
4
Scenario 3
Hydrogen production cost [USD/kg]
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
k
g
]
1
2 21
T
o
t
a
l
u
n
s
a
t
i
s
f
i
e
d
h
y
d
r
o
g
e
n
d
e
m
a
n
d
(
H
2
a
n
d
h
y
t
a
n
)
[
%
o
f
t
o
t
a
l
]
0
10
20
30
40
50
60
70
80
90
Figure 9: The resulting Pareto-front when optimising U for
Scenario 3.
of unsatis?ed demand starts to rise.
The ?exibility of the solutions for scenario 2 is in general
lower than for other solutions, which is evident when the pas-
sive Scenarios 1 and 3 are used. The production cost for these
passive scenarios is at least double that obtained when the ac-
tive Scenario 2 is used.
5.3 Scenario 3: Hydrogen fuel cell cars
In the third scenario, hydrogen fuel cell powered scooters are
in focus in the beginning, and later fuel cell driven cars. No
hythane is used in this scenario.
The production cost versus unsatis?ed demand curve in
Figure 9 reveals a slightly more expensive production than
in the previous cases. This is even more obvious when the
solutions for Scenario 3 are applied to Scenario 1 and 2.
6 Discussion and conclusion
In this paper, it has been demonstrated that it is possible to use
stochastic optimization in order to ?nd investment strategies
for a combined hydrogen and hythane on-site reformer refu-
eling station. The resulting cost of hydrogen and hythane are
2-6 USD/kg and 1-1.5 USD/kg respectively, depending on the
preferences concerning unsatis?ed demand, ?exibility etc.
The results from this study can be used as decision support
when planning combined hydrogen and hythane refueling sta-
tions. Not only the production cost and unsatis?ed demand
for the present scenario are important, but also the ?exibil-
ity of the solution to unforeseen events and developments. In
addition, there are other performance measures such as vari-
ance and the comparison between hydrogen and hythane that
should be taken into account. The selected solution is a matter
of preference.
The problem of ?nding investment strategies involves a
considerable amount of information, and therefore aggregate
measures have been de?ned for ?exibility.
As observed from the connection between production cost
and unsatis?ed demand for all scenarios, these two measures
are in con?ict. One reason for this is the stochastic demand
curve which makes it unrealistic to achieve zero unsatis?ed
demand. This is so since, occasionally, a larger amount of
vehicles will come to the station than it can serve, which is
probably close to what would be observed in reality. Other
reasons for this con?ict in measures are the technology de-
velopment reduction in purchase price (see Eq. (2)) and the
discounted costs (see Eq. (3)). Since it is likely that it will be
cheaper to build and run the refueling station in the future, the
optimization tends to prefer future solutions to present ones.
An evenly distributed cost will be discounted to 0.4466 of the
original value, so the discount effect is not negligible.
Improvements can be made to the scenario data in Sec-
tion 2.1. It may be unrealistic having all buses refuelling in
the morning. Instead, a slow ?lling during night time might
be considered. Also the 24-hour car refueling curve, which
assumes a constant ?lling frequency throughout the day, may
be adjusted to a more realistic setting.
A key to successful investment planning is the minimiza-
tion of the uncertainty of future developments. For this rea-
son, three different future scenarios, with 100 samples each,
have been used. Within each scenario the uncertainties are
kept at minimum given the strategic parameters, by taking
all samples into account. The strategic parameters can eas-
ily be changed for other cases. For each solution, the effects
are easy to quantify should another scenario become reality.
However, the S-curve (Eq. (1)) has still been used for estimat-
ing the (uncertain) number of produced units. These numbers
are taken in a global perspective, which may make them less
sensitive.
In this study, linear scaling of equipment cost to usage time
is used. Another option, often used in the literature [26],
would be to use salvage value or terminal cost. Given that the
salvage value can be de?ned arbitrarily, these two approaches
can be considered identical.
In the optimisations presented here, the investment strat-
egy is set prior to the calculations. Another option would be
10
to de?ne control policies that use system information for de-
cisions, i.e. a fully closed-loop. Such a strategy would be
able to incorporate not only investments but also a quantita-
tive demand satisfaction calculation. At present, the only real
feedback loop has been implemented for the hydrogen storage
level.
An increase in the number of solutions will give better res-
olution for the Pareto curve. On the other hand, a larger pop-
ulation will be needed in the GA, which in turn will increase
the calculation time. At present, simulation of one scenario
sample takes about 0.25 s, giving a total of 75 s per individual
and hence 1 h 40 min. for a population of 80 individuals. By
aggregating calculated measures and coding more of the algo-
rithm in a low-level language, the simulation time can proba-
bly be shortened considerably. If this is done, the resolution
can be enhanced.
It should also be noted that, as for all heuristic methods,
convergence cannot be guaranteed. Instead, one has to settle
for a solution which is good enough and preferably better than
any other known solution.
To conclude, it has been found that it is possible to op-
timise the hydrogen production cost for a combined hydro-
gen and hythane refueling station, and that the resulting costs
lies between 2-6 USD/kg for hydrogen and 1-1.5 USD/kg for
hythane. The production cost and the amount of unsatis?ed
demand constitute con?icting objectives so that, for exam-
ple, if the total hydrogen and hythane demand is to be sat-
is?ed, the production cost of hydrogen will be unrealistically
high. However, an intermediate realistic solution can be found
along the curve of cost versus unsatis?ed demand. In all
cases, the lowest production cost for hydrogen and hythane
is achieved by satisfying the hydrogen demand ?rst and then
the hythane demand.
Acknowledgments
Financial support from the Competence Center for Environ-
mental Assessment of Product and Materials Systems (CPM)
at Chalmers University of Technology is gratefully acknowl-
edged.
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