COURNOT COMPETITION UNDER UNCERTAINTY
IN POWER MARKETS
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee.
This dissertation does not include proprietary or classified information.
Pavee Siriruk
Certificate of Approval:
Pavee Siriruk
Jorge F. Valenzuela, Chair
Associate Professor
Industrial and Systems Engineering
Robert L. Bulfin
Professor
Industrial and Systems Engineering
Emmett J. Lodree
Assistant Professor
Industrial and Systems Engineering
George T. Flowers
Dean
Graduate School
COURNOT COMPETITION UNDER UNCERTAINTY
IN POWER MARKETS
Pavee Siriruk
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
December 18, 2009
iii
COURNOT COMPETITION UNDER UNCERTAINTY
IN POWER MARKETS
Pavee Siriruk
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon request of individuals or institutions and at their expense. The author
reserves all publication rights.
Signature of Author
Date of Graduation
iv
VITA
Pavee Siriruk, son of Dr. Pisit Siriruk and Kulthanee Siriruk, was born June 13,
1982, in Nakhonratchasima, Thailand. He graduated with a Bachelor of Science degree in
Industrial Engineering (2003) from Kasetsart University in Bangkok, Thailand. In 2006,
he received his Master of Science degree in Industrial and Systems Engineering from
Auburn University, Auburn, Alabama, USA. Following his graduation, he entered the
doctoral program in Industrial and Systems Engineering at Auburn University. During his
years at Auburn University, he served as a graduate teaching assistant at both the
undergraduate and graduate levels, as well as a graduate research assistant.
v
DISSERTATION ABSTRACT
COURNOT COMPETITION UNDER UNCERTAINTY
IN POWER MARKETS
Pavee Siriruk
Doctor of Philosophy, December 18, 2009
(M.S., Auburn University, 2006)
(B.S., Kasetsart University, 2003)
137 Typed Pages
Directed by Jorge F. Valenzuela
As deregulation of the electric industry has come into effect in many parts of the
world, the price of electricity is no longer determined by regulatory agencies. In contrast,
price is determined by market demand, supply conditions, load elasticity, and strategic
behavior. Firms, nowadays, face much greater risks and have become more responsible
for their own economic decisions in deregulated power markets. Therefore, decision
support models can help firms fulfill these new requirements.
vi
This research is aimed at developing analytical models for longterm markets to
assess the effect of uncertainties on electricity market prices. A multiperiod Cournot
model was developed for this purpose. Specifically, two significant uncertainty factors,
the availability of the generating units and fuel price uncertainty, are considered in this
model. An impact analysis of these two factors on firms? expected profits is also carried
out. Finally, a sensitivity analysis is performed to determine the parameters that have the
most significant impact on the Nashequilibrium solutions.
vii
ACKNOWLEDGEMENTS
The author is grateful to Dr. Jorge Valenzuela for his guidance and support
throughout this dissertation. Special thanks go to Dr. Robert L. Bulfin and Dr. Emmett J.
Lodree for serving on the dissertation committee. The author is also indebted to his
parents, Dr. Pisit Siriruk and Kulthanee Siriruk and to his brother, Akawut Siriruk for
their valuable advice and continued support. This dissertation could not have been
completed without them.
viii
Style manual or journal used: Institute of the Electrical and Electronics Engineers
Transaction (IEEE) on Power Systems
Computer software used: MATLAB, AMPL, CPLEX 8.0, PATH SOLVER, Microsoft
Excel, and Microsoft Word
ix
TABLE OF CONTENTS
LIST OF FIGURES??????????????????????????...xii
LIST OF TABLES?????????????????????????...?xiii
CHAPTER 1
INTRODUCTION???????????????????????????...1
1.1. Contributions??????????????????????????...3
1.2. Dissertation Organization??????????????????????5
CHAPTER 2
LITERATURE REVIEW?????????????????????????7
2.1. Deregulation in the United States???????????????????7
2.2. Capacity Market?????????????????????????.11
2.3. Cournot Model?????????????????????????...13
2.4. Mixed Complementarity Problems??????????????????18
2.4.1. Linear Complementarity Problem?????.??...?????????.20
CHAPTER 3
COURNOT MODEL IGNORING UNCERTAINTY?????????????...23
3.1. Nomenclature??????????????????????????.24
3.2. Introduction???????????????????????????25
3.3. Model Description????????????????????????.27
3.3.1. Model Assumptions??????????????????????..27
3.3.1.1. Power Producer?????????????????????...28
3.3.1.2. Demand (load)??????????????????????28
3.3.1.3. Market Operation?????????????????????29
3.3.2. Mathematical Model??????????????????????.30
3.4. Experimental Results???????????????????????.34
3.5. Conclusions???????????????????????????36
CHAPTER 4
STOCHASTIC COURNOT MODEL INCLUDING GENERATOR OUTAGES??...38
4.1. Nomenclature??????????????????????????.39
4.2. Introduction???????????????????????????40
4.3. Model Description????????????????????????.42
x
4.3.1. Model Assumptions??????????????????????..42
4.3.1.1. Market Operation???????..?????????????..42
4.3.1.2. Power Producer?????????????????????...43
4.3.2. Mathematical Model??????????????????????.44
4.3.2.1. Demand????????????????????????...44
4.3.2.2. Profit Function??????????????????????45
4.3.2.3. Reliability????????????.??????...?????46
4.3.2.4. Cost Function??????????????????????..47
4.3.2.5. Modeling Competition..??????????????????..51
4.4. Experimental Results???????????????????????.52
4.4.1. Effect of Generator Availability?????????????????...59
4.5. Conclusions???????????????????????????61
CHAPTER 5
STOCHASTIC COURNOT MODEL INCLUDING
GENERATOR OUTAGES AND FUEL PRICE UNCERTAINTY????????.62
5.1. Nomenclature??????????????????????????.63
5.2. Introduction???????????????????????????64
5.3. Model Description????????????????????????.67
5.3.1. Model Assumptions??????????????????????..67
5.3.2. Mathematical Model??????????????????????.68
5.3.2.1. Reliability????????????????????????68
5.3.2.2. System Load???????????????????????68
5.3.2.3. Profit Function??????????????????????69
5.3.2.4. Cost Function??????????????????????..70
5.3.2.5. Cournot Model??????????????????????74
5.4. Experimental Results???????????????????????.77
5.4.1. Effect of Generator Availability?????????????????...85
5.5. Conclusions???????????????????????????88
CHAPTER 6
TOLERANCE APPROACH TO SENSITIVITY ANALYSIS
IN THE STOCHASTIC COURNOT MODEL????????????????.89
6.1. Nomenclature??????????????????????????.90
6.2. Introduction???????????????????????????91
6.3. Model Description????????????????????????.92
6.3.1. Linear Complementarity Problem (LCP)??????????????..92
6.3.2. Cournot Model????????????????????????..93
6.3.3. Tolerance Approach to Sensitivity
Analysis in the Stochastic Cournot Model??????????????96
6.3.3.1. A single parameter ( )
k
q k? B is perturbed???????????98
6.3.3.2. A single parameter ( )
k
q k?N is perturbed??????????.101
6.3.3.3. Numerical Examples???????????????????.102
xi
6.3.4. Tolerance approach to sensitivity analysis
when all values of q vary simultaneously?????????????.107
6.3.4.1. Algorithm???????????????????????..107
6.3.4.2. Numerical Example???????????????????..110
6.4. Conclusions??????????????????????????..116
CHAPTER 7
CONCLUSIONS AND
RECOMMENDATIONS FOR FUTURE RESEARCH?????..???????117
7.1. Conclusions??????????????????????????..117
7.2. Directions for Future Research???????????????????119
BIBLIOGRAPHY???????????????????????????121
xii
LIST OF FIGURES
Figure 1. Consumer Demand Curve ............................................................................... 29
Figure 2. Market operation in the deterministic model ................................................... 30
Figure 3. Market operation in the stochastic model ........................................................ 43
Figure 4. The expected cost curve of 12 generating units ............................................... 49
Figure 5. Cost curves of supplier 1 ................................................................................. 56
Figure 6. Cost curves of supplier 2 ................................................................................. 56
Figure 7. Cost curves of supplier 3 ................................................................................. 57
Figure 8. U.S. retail gasoline prices, Regular grade (Source: EIA 2009)......................... 65
Figure 9. Flowchart of the algorithm used to calculate an approximation of the expected
cost function .................................................................................................................. 75
Figure 10. Expected cost curves of supplier 1 for given marginal costs .......................... 80
Figure 11. Expected cost curves of supplier 2 for given marginal costs .......................... 81
Figure 12. Expected cost curves of supplier 3 for given marginal costs .......................... 81
Figure 13. Estimate of expected cost curves before and after applying the slope reduction
algorithm for firm 1 ....................................................................................................... 82
Figure 14. Estimate of expected cost curves before and after applying the slope reduction
algorithm for firm 2 ....................................................................................................... 82
Figure 15. Estimate of expected cost curves before and after applying the slope reduction
algorithm for firm 3 ....................................................................................................... 83
xiii
LIST OF TABLES
Table 1. Market composition and generating unit Data .................................................. 35
Table 2. The results of Nashequilibrium ignoring uncertainty ....................................... 36
Table 3. The total profit when ignoring uncertainty........................................................ 36
Table 4. Market composition and generating unit data including unit availability ........... 55
Table 5. The data list of hypothetical generating units for firm 1 .................................... 57
Table 6. The data of hypothetical generating units for firm 2 ......................................... 58
Table 7. The data of hypothetical generating units for firm 3 ......................................... 58
Table 8. The results of Nashequilibrium including generator outages ........................... 59
Table 9. Expected profits of firms when ignoring uncertainty and including outages...... 60
Table 10. The distribution of oil and coal price ($/MBTU) ............................................ 78
Table 11. Market composition and generating unit data including unit availability ......... 79
Table 12. Data for the hypothetical generating units of firm 1 ........................................ 83
Table 13. Data for the hypothetical generating units of firm 2 ........................................ 84
Table 14. Data for the hypothetical generating unit of firm 3 ......................................... 84
Table 15. Results of Nashequilibrium including generator outages ............................... 85
Table 16. A comparison of firm expected profits when the uncertainty of fuel prices and
outages are ignored and when the uncertainty of generator and fuel costs are included .. 87
Table 17. Market composition and generating unit data ............................................... 103
Table 18. The solutions associated with set B
*
( )
B
z ...................................................... 104
Table 19. The maximum allowable range of each marginal cost (
,f j
c ) ........................ 105
Table 20. The maximum allowable range of each capacity (
max
,f j
P ) .............................. 106
Table 21. The list of hypothetical generating unit data associated with estimated expected
cost function for firm 1 ................................................................................................ 112
Table 22. The list of hypothetical generating unit data associated with estimated expected
cost function for firm 2 ................................................................................................ 113
Table 23. The list of hypothetical generating unit data associated with estimated expected
cost function for firm 3 ................................................................................................ 114
Table 24. The new optimal solutions using
new
q
associated with set B ......................... 115
1
CHAPTER I
INTRODUCTION
Power supply systems have been changing their economic modes of operation to
systems based on a market mechanism as deregulation of electric industries has become
the norm in many parts of the world since the 1970s (Fujii, Okamura, Inagaki, and
Yamaji, 2004). The deregulation of the electricity industry is essentially changing the
way in which suppliers do business. A firm?s decisions now depend, to a large extent, on
market electricity prices. The price of electricity under deregulation is determined by
market demand, supply conditions, load elasticity, and strategic behavior. It also depends
on physical factors such as production cost, load, unit commitment, and transmission
constraint (Valenzuela and Mazumda, 2005). Moreover, considerations of uncertainty
factors such as generator reliability, demand uncertainty, and fuel price volatility are
inevitable when making decisions. In general, most companies handle uncertainty in
power markets by making hasty decisions about sensitivity, comparing scenarios,
performing worstcase analyses, etc. This is, however, not an effective way to cope with
uncertainties (Krukanont and Tezuka, 2006). Electricity is different from other products
because it has yet to become efficiently storable. Therefore, its demand and supply must
be matched every second, and failure to do so may result in a costly system collapse.
2
Firms face much more risk and become responsible for their own economic decisions in
deregulated power markets. Those firms, hence, need decisionsupport models that fulfill
these new requirements. In other words, decisionsupport models need to incorporate the
uncertainties and other factors involved in deregulated power markets.
Recent attempts to model the structure of deregulated electricity markets via
utility system production simulation models have continued to rely on models used in the
past for planning and regulatory purposes (Kahn, Bailey, and Pando, 1996), but many
analysts believe that the Cournot model is better able to represent the electricity market as
it has evolved (Borenstein and Bushnell, 1999). However, in the existing literature the
Cournot model usually assumes perfect information about the salient factors such as
generator outages and fuel cost uncertainty. This is difficult and poses risk for decision
makers, especially in longterm analyses that involve large uncertainties in the decision
making process (Krukanont and Tezuka, 2006).
This dissertation develops models for the longterm markets to assess the effect of
uncertainties (generator failure and fuel price uncertainty) on electricity market prices
under Cournot competition. Specifically, the stochastic singleperiod model is extended
to a multiperiod model. Uncertainty factors, the availability of the generating units and
fuel price uncertainty, are then added to the model as well as their sensitivity analysis,
and the model shows their effects on market prices. Moreover, the effects of those factors
on a firm?s expected profits are studied in this research. Transmission congestion and
demand uncertainty are not considered in this dissertation.
3
1.1. Contributions
The main contribution of this dissertation is within the study of market prices in
the longterm power markets when uncertainties take place. In particular, this research
focuses on developing a stochastic Cournot model to evaluate the effects of vital
uncertainty factors on electricity market prices. In reality, most companies cope with
uncertainty by performing simple methods such as sensitivity and worstcase analysis.
Those techniques, however, could lead to inaccurate results. As the electricity prices in
deregulation have major effects on firms? profits, companies are responsible for their own
economic decisions. The development of an analytical model in this research which
incorporates uncertainty and other crucial factors in deregulated power markets will help
power companies make the precise decisions which they need to operate.
Another contribution of this dissertation is an approach used to cope with
generator failures. In the existing literature, the approach used to consider generator
outages into production cost models is to derate plant capacity. However, this method
could lead to inaccurate results (Valenzuela and Mazumda, 2007). In this research, the
expected production cost function, including generator outages, is modeled. This
approach yields more accurate results when power producers consider the uncertainty of
generator availability. When incorporating generator outages in the model, the expected
cost function becomes a piecewise linear function. The piecewise linear function in some
cases generates a large number of slopes which has a direct link to computational
complexity. The algorithm to reduce a number of slopes of a piecewise linear function is
implemented. The small number of slopes means less computational complexity, as the
4
number of slopes of an expected cost function grows exponentially with the number of
generators. In general, the algorithm developed in this research can reduce the number of
slopes efficiently and ease computational complexity which will help support other
research.
This dissertation contributes to the optimization theory and applications. The
tolerance approach to sensitivity analysis in a linear complementarity problem is here
implemented and applied to the stochastic Cournot model. It can be broadly applied to
numerous applications of the linear complementarity problem such as game theory and
equilibrium problems. In addition, power companies can employ this approach to study
the effect of input data on the output results. It might be useful for companies if they can
detect which input data are sensitive and have a significant impact on firms? optimal
strategic planning and operations.
Finally, this research also provides a valuable new tool for all participants in
power markets. The tolerance approach to sensitivity analysis is applied to determine
whether the new input data affects the optimal solutions. If the perturbed problems still
have the same index set of solutions, the new optimal solutions can be calculated without
directly solving a linear complementarity problem. Since solving a large scale linear
complementarity problem may take long computational time and the input data such as
the fuel price may change every minute, power producers can make a decision swiftly
with this approach, as new input data are obtained.
5
1.2. Dissertation Organization
The remainder of this dissertation is organized as follows: Chapter 2 provides a
brief history of the development of the power market. The structure, volume, and market
concentration of the capacity market are also discussed as is the basic Cournot model.
Finally, a background and review of the significant literature concerning the Cournot
model, mixed complementarity problems (MCP), and linear complementarity problems
(LCP), which are a subset of mixed complementarity problems, are provided.
Chapter 3 presents the multiperiod deterministic Cournot model in the long term
market which is extended from a single period model. The time value of money is also
considered in the model while demand and fuel cost are assumed to be constant.
Moreover, the availability of generating units is ignored in this chapter. The Nash
equilibrium quantities are calculated by combining the KKT first order optimality
conditions of the extended model. The KKT conditions of the deterministic Cournot
model are considered as an LCP. Finally, the market prices and each firm?s expected
profit are calculated. These results are used as standard results to show the effect of
uncertainties in power markets when we consider those uncertainties in the model.
Chapter 4 presents an approach to determining market prices when generator
outages are taken into consideration. Specifically, the expected cost function including
generator availability is developed. This expected cost function yields a large number of
slopes. Each slope represents one marginal cost and maximum capacity including
generator availability which is used to compute the Nashequilibrium quantities. To
consider all of them would take long computational time. Therefore, an algorithm to
6
reduce a number of slopes without losing precision is developed. Finally, the effect of
generator outages on the market prices is analyzed.
Chapter 5 provides an approach for determining the price of electricity when
generator outages and fuel price uncertainty are in effect. Generally, each power
company owns capacity resources in different fuel technologies. Four types of fuel
technology including oil, coal, hydro, and nuclear are assumed in this dissertation. The
most recent data on fuel prices obtained from reliable sources are used to generate the
distribution for each generator?s marginal cost. The effects of generator availability and
fuel price uncertainty are investigated.
Chapter 6 describes the tolerance approach to sensitivity analysis in the stochastic
Cournot model is proposed. Moreover, an algorithm to compute new optimal solutions
when all parameters on the righthand side vary simultaneously without directly solving
the LCP is also presented in this chapter. The maximum allowable range which does not
affect firms? optimal strategy for each marginal cost and maximum capacity is computed
in order to detect the sensitive parameters.
Chapter 7 summarizes the study, discusses the conclusion of this research, and
suggests directions for future research.
7
CHAPTER II
LITERATURE REVIEW
This chapter presents the review of the current literature in four sections. The first
section presents the history and current state of deregulation in the United States. The
second section provides information about deregulation and the basic structure and
operation of the capacity market as well as the current state of the capacity market. The
third section presents a Cournot model, originally developed by Augustin Cournot along
with a literature review of how the Cournot model is applied to the power markets. The
fourth section focuses on the literature review of mixed complementarity problems. The
general form of a model and the available software which can be used to solve mixed
complementarity problems are discussed. Finally, a linear complementarity problem as
well as its general form is highlighted. Some crucial algorithms, which are able to solve
the linear complemetarity problem, are also presented.
2.1. Deregulation in the United States
The report prepared by the Energy Information Administration (1997) gives an
interesting perspective on the history of the topic from the beginning of power markets in
the United States until deregulation. In 1882, Thomas Edison?s Pearl Street Station began
8
supply electricity to 85 customers for the first time in New York City. By 1916, 33 states
had established regulatory agencies to organize the utilities in their jurisdictions, with the
authority to franchise utilities, regulate their rates, financing, and service, and establish
utility accounting systems. State regulation provided protection to consumers from the
possibility of monopolistic practices by the utilities and ensured the reliability of
electricity supplies. Moreover, they allowed utilities to receive a fair rate of return but
there was debate at that period of time whether state regulation of electric power emerged
to protect the consumers or to protect the profits of the electric utilities.
By the early 1930s, the price of electricity had fallen and service had been
extended to twothirds of the U.S. population which meant the demand for electricity
increased. Consequently, ownership of operating companies was centralized under
holding companies which facilitated access to the capital required for expansion and for
exploiting economies of scale. As many states regulated local operating companies, there
was no effective regulation of the increasingly expansive holding companies. As a result,
when the worldwide economic downturn called Great Depression arrived in the early
1930s, many holding companies failed because of highleverage, unsecured financing,
and investments in business unrelated to energy services. In return, Congress passed the
Public Utility Holding Company Act of 1935 (PUCHA). The purpose of this legislation
was to give control of electricity service to local operating companies. In other words,
PUCHA restricted the electric power generating business to local utilities which built and
operated power plants to serve service territories. This meant that there was no
competition in supplying electricity under PUCHA. Title II of PUCHA granted the
Federal Government explicit authority over most interstate wholesale electric power
9
sales. Therefore, by the end of 1935, nearly all electric power transactions in the U.S.
were regulated under a scheme called ?rate of return.? This scheme facilitated the
production and sale of lowcost, reliable electricity in the U.S. for about 50 years.
Before deregulation, there was a belief that electricity is a national asset.
Therefore, electric sectors in most regions of the country were subject to full regulation.
The generation, transmission, distribution and retail segments were controlled by state
governments playing a dual role as electricity service providers and as regulators. As
economic growth became more and more dependent on sufficient electricity suppliers,
the importance of electricity increased tremendously. Consequently, many governments
have started to realize that this growth may be impeded by full regulation because of
suppliers? slow response to technological progress in electricity operations. Furthermore,
the successful deregulation in oil and gas supports the belief that electricity is a service
and it can be accomplished by deregulation (Yao, 2006).
Thus, the first step of restructuring the market began in 1978 when congress
passed the Public Utility Regulatory Policies Act of 1978 (PURPA). PURPA not only
opened the door to competition in the U.S. but also promoted greater use of renewable
energy. This law created a market for nonutility electric power producers by forcing
electric utilities to purchase power from these firms at the ?avoided cost? rate, which is
determined by bids from nonutility electric power producers. This is the first time in the
United States that organizations other than public utilities were allowed to sell electric
power.
10
The Federal Energy Regulatory Commission (FERC) was given the responsibility
by Congress of implementing open transmission access under the Energy Policy Act of
1992 (EPACT) in order to spur competition in the wholesale electricity market. On April
24, 1996, the Federal Energy Regulatory Commission issued order 888 establishing a
guideline to provide open access to transmission lines. This policy removed restrictions
on ownership of power generation facilities, which allowed nonutility electric power
producers to access transmission lines. This was a major step toward electricity
deregulation in the United States.
The main purpose of deregulation is to reduce operational cost, to increase
efficiency, and to encourage competition among electricity suppliers with the medium
and longterm goal of combating high prices. Deregulation gives consumers more
alternatives because consumers are not held to only one service provider. Availability of
power from various suppliers guarantees supply reliability in case of a peak demand or
unexpected outages. In other words, the more there is available, the greater the
competition will be to produce and sell power in an efficient way, leading to lower prices
and more energy efficiency. Moreover, deregulation is believed to provide better
economic incentives and opportunities to both consumers and suppliers because the
existence of a large number of consumers and suppliers reduces market power which
prevents a firm from dominating a market. Consequently, it enables a company to enter
or exit the markets which allows competitors to take advantage of any economic
opportunity.
11
2.2. Capacity Market
Capacity markets have proven to be one of the most contentious elements of
electricity restructuring. However, Cramton and Stoft (2005) argue that a capacity market
is needed in most restructured electricity markets.
In competitive markets, consumers can easily decrease their demand in response
to prices and other market signals for most products. In this case, extra capacity is not
necessary because prices give consumers the right signals when supplies are tight
(NEPA, 2008). In power markets, however, there is little demand response to price,
primarily because load neither sees nor pays the realtime price. Realtime meters and
demand management control systems are not yet in place for most electricity consumers.
This absence prevents consumers? willingness to limit demand during times of supply
scarcity. As a result, the price can reach extreme values. Price caps are set by market
administrators in order to limit the peak prices, which occur at peak demand periods or
with unexpected outages. As the price caps are in effect, the investors do not see the
opportunity for them to make an investment in new resources. The imperfectly
competitive market structure is the other issue in power markets. Consequently, there are
instances when one or more suppliers have substantial market power, especially at peak
times or during an outage of a large generator or transmission line. Addressing these two
issues typically results in price peaks that are too infrequent and insufficient to motivate
efficient investment in new capacity (Cramton and Stoft, 2005) which leads to the failure
of electricity markets.
12
Moreover, the economic consequences of running out of electric generating
capacity are severe. The best example is an event of the blackout that took place in the
US East coast in August 2003 which shows that the value of the losses caused by a
system imbalance may be substantially large (Creti and Fabra 2004). Fixed cost is also an
important factor in electricity markets. Since power cannot be stored, there has to be
significant extra capacity available to meet both peak demand periods, and because some
generators will not be available due to outages and maintenance. To stay reliable, a power
system has to have this builtin reserve margin. Some source of money is necessary to
cover the cost of the extra plants required for this purpose, which is not very often.
Normal revenues in the energy market will not cover the costs of this extra reserve
margin, especially with prices subject to various regulatory controls (NEPA, 2008).
For those reasons, the capacity markets must be introduced to the restructured
electricity markets. In capacity markets, each retail supplier is required to produce its
share of the responsibility for ensuring there is sufficient generating capacity in the
region. Retail suppliers can purchase capacity either from generators that meet certain
standards of availability or on the spot market to meet their requirements. The main
purpose of introducing capacity markets into the restructured electricity markets is to
ensure that sufficient generating capacity will be available to meet peak demands while
providing investment incentives for power suppliers. In addition, capacity markets will
cover the cost to keep adequate generation available. The capacity market payments
reflect the costs of keeping sufficient capacity (plus the reserve) ready and available to
the region. In other words, they represent the option to call on generators as and when
needed (NEPA, 2008).
13
Creti and Fabra (2004) discussed in their paper that capacity markets can be
classified as either price based or quantity based. In price based systems, the capacity
availability is paid either via lumpsum payments or increase to energy payments
depending on the probability of outages. The price based systems are not working as well
as expected because producers are able to increase capacity payments by making fewer
capacity resources available instead of increasing capacity resources. This increases the
probability of shortages. Quantity based systems have been introduced in several power
markets in the U.S. such as New England Power Generators Association Inc., New York
ISO, and PJM. Nowadays, installed capacity markets, which are one of the quantity based
systems, are the focus of the policy debate in the United States. The purpose of
introducing the installed capacity markets is to ensure that adequate capacity is
committed on a daily or seasonal basis to meet system loads and reserve requirements.
The Load Serving Entities (LSEs) that sell electricity to enduser consumers must satisfy
the expected peak loads plus a reserve margin. LSEs can buy through internal
transactions, bilateral transactions, or capacity markets in the event of shortages. The
equilibrium price in the capacity markets should be related to the overall capacity in the
system.
2.3. Cournot Model
The Cournot model was named after Antoine Augustin Cournot, the nineteenth
century French mathematician, who first examined its implications. Augustin Cournot
was born in 1801. His book, Researches into the Mathematical Principles of the Theory
of Wealth, was published in 1838. The basic Cournot Model is a oneperiod model in
14
which each firm has to forecast the other firm?s output decision. Given its forecast, each
firm then chooses a profitmaximizing output for itself (Varian, 2006). Cournot
competition, in other words, is a form of quantity competition which means firms must
choose profitmaximizing output levels instead of prices in the belief that each competing
firm maximizes its expected profits. It also assumes that the quantities supplied by other
producers are fixed and do not react to price changes. The competition then seeks
equilibrium. The Cournot equilibrium refers to a situation where each firm finds its
beliefs about the other firms to be confirmed. The Nash equilibrium solution for the
optimum quantities to be generated by each producer is provided by Cournot model. The
market price is determined by the Nash equilibrium solution given the price elasticity of
demand.
The following is the standard Cournot model that will be used later as a
fundamental model in this dissertation. Daughety (2005) discussed the basic onestage
Cournot model for an industry comprised of n firms. Each firm chooses its output level.
Firm i?s output level is denoted as
i
q where i?1, ?, n and let the vector of firm outputs
be denoted as
1 2
( , ,..., )
n
q q q?q . Let Q refers to the aggregate industry output level (i.e.,
1
n
i
i
Q q
?
?
?
). We will refer the ( 1)n? vector of output levels chosen by other firms as
i
q .
Thus, ( )
i, i
qq also denotes to the vector of firm outputs, q. The inverse market demand is
denoted as ( )pQ . Furthermore, firm i?s cost function can be denoted as ( )
i i
c q .
Therefore, firm i?s profit function can be written as ( ) ( ) ( )
i i i i
p Q q c q? ? ?q . The
Cournot equilibrium consists of a vector of output levels,
CE
q , such that no firm wishes
15
to change its output level when other firms produce the output levels assigned to them in
the equilibrium. A Cournot equilibrium can be alternatively called a NashCournot
equilibrium because it is a Nash equilibrium with quantities as strategies chosen from a
compact space. Hence,
CE
q is a Cournot equilibrium if ( ) ( , )
CE
i i i i
q q? ??
CE
q for all
values of
i
q , for i = 1, 2,?, n.
An immense effort has been made to design several models and tools that
specifically represent the electricity market behavior. The Cournot model has been one of
the theoretical frameworks most widely used to model strategic behavior in electricity
markets. In this research, our main goal is to model electricity markets under competition
conditions and several uncertain factors. Our approach considers a market in which firms
compete in quantity as in the Cournot model.
Browning and Browning (1989) discussed the definition of the Cournot model, an
excellent way to introduce the nature of oligopolistic interdependence. Each firm takes
into account how price changes as the firm or its competitors change quantity and choose
their quantity to maximize their profits given the quantity that their competitor is
producing. The Cournot model shows how uncoordinated output decisions between rival
firms could interact to produce an outcome that lies between the competitive and
monopolistic equilibria. However, the final equilibrium reflects their interdependence
although each firm explicitly ignores the other. In the last several years, the topic of
strategic behavior in electricity markets has received a great deal of attention. Several
oligopoly models have been proposed, notable among which is the Cournot model. Many
analysts believe that the Cournot model is better able to represent the electricity bilateral
16
market than most other models. Borenstein and Bushnell (1999) give the arguments about
various approaches and explain why the Cournot model is the best approach for market
power in electricity markets. In addition, competing firms have longterm commitments
to capacity although they may compete on price in the short term. The perfect
competition model is based on the assumption that any firm can capture the entire market
by pricing below other suppliers and supplying the entire demand but generating capacity
constraints and increasing marginal costs make this assumption invalid. Thus, the
centralized pricing mechanism and the capacity constraints support the case for adopting
the Cournot model. The Cournot model also enables the analysis of situations in which
producers unilaterally decide to withhold supply from the market by declaring some of
their generators to be unavailable. Furthermore, The Cournot model leads to a simple
analytical expression for the market price that renders itself easily to analytical
manipulations.
Restructured power markets take a wide variety of forms. A wide range of models
have been proposed for simulating the interaction of competing power generation
companies. Benjamin Hobbs (2001) presents two Cournot models of imperfect
competition among electricity producers. The first model presents the producers? and grid
owner?s optimization problems. It includes a congestion pricing scheme for transmission.
After combining their KKT conditions with the market clearing condition, they yield a
mixed linear complementarity problem. The second model differs from the first model in
that the first one has no arbitrage between nodes of the network, while in the other model,
arbitragers erase any noncostbased differences in price. In other words, power
generators recognize that marketers will buy and resell power where price differences
17
exceed the cost of transmission. Finally, a simple example is presented to illustrate their
application.
Most previous NashCournot models of competition among electricity suppliers
have assumed smooth demand functions. However, nonsmooth demand functions are an
important feature of real power markets due to many factors such as transmission
constraints. Pang and Hobb (2005) developed the complementaritybased model of Nash
Counot oligopolistic electric power markets to include concave demand functions that are
piecewise linear. These models also include linear joint constraints within generator
profit maximization problems. Furthermore, they begin with a multivalued
complementarity formulation of the equilibrium problem, from which an equivalent
singlevalued linear complementarity problem formulation is derived. They mentioned in
the paper that this new model is computationally challenging. For instance, they
immediately invalidate the solution methods employed for the previous models.
However, they successfully solved this problem by using a specific algorithm. As
mentioned earlier, transmission constraints are another important factor in real power
markets. Yuan, Liu, Jiang, and Hou (2005) proposed the Cournot model taking into
account transmission constraints based on DC power flow. They analyzed the effects of
simple twobus network and threebus network transmission constraints on the pure
strategies of suppliers. The results show that there may exist different pure strategy
equilibriums if transmission constraints are considered. Cunningham, Baldick, and
Baughman (2002) also investigated perfect competition equilibrium and Cournot
equilibrium in a simple example, triangular connection. Both cases are examined on
transmission unconstrained and transmission constrained in order to compare the results.
18
Results show that a pure strategy equilibrium can break down even when a transmission
constraint exceeds the value of the unconstrained Cournot equilibrium line flow.
One of the major problems in the power markets is assuring that generators,
which independently decide about their outputs, will not produce more than the available
transmission capacity. Willems (2000) developed a model for the power markets of
generators when transmission capacity is scarce. The purpose of this model is to apply
the different models to a simple electricity market with one transmission line. They apply
different Cournot concepts and explain the implicit assumptions about the behavior of the
system operator. They also show that these implicit assumptions are not realistic.
According to implicit assumptions, they formulate some alternative assumptions for the
behavior of the System Operator and examine the results.
2.4. Mixed Complementarity Problems
Complementarity problems are a natural format for expressing a variety of
economic models and arise frequently in the general equilibrium theory of economics
(Ferris and Kanzow, 1998 and Rutherford, 2002). Optimization may be viewed as a
special case of complementarity problems, since the standard optimality conditions for
linear and smooth nonlinear optimization are complementarity problems (AMPL, 2008).
Many computable general equilibrium models are used for various aspects of policy
design and analysis, including carbon abatement, trade reform, and game theory (Ferris
and Kanzow, 1998). One example of this area is the deregulation of electricity markets.
19
According to Rutherford (2002), the mixed complementarity problem is defined
as
Given: :
N N
F R R? , ,
N
l u R?
Find: , ,
N
z w v R?
s.t. ( ) 0F z w v? ? ?
, 0, 0l z u w v? ? ? ?
( ) 0, ( ) 0
T T
w z l v u z? ? ? ?
in which .l u??? ? ???
Complementarity problems consist of complementarity conditions and each of
them requires that the product of two or more decision variables be zero. Michael Ferris
and Munson (1998) present how these problems are modeled within the GAMS modeling
language and provide details about the PATH solver for finding a solution. Specifically,
they develop the complementarity framework by looking at the transportation model. The
transportation model is a simple linear program where demands for a single product must
be satisfied by suppliers at minimal transportation cost. In other words, they show how to
convert a linear program into a linear complementarity problem which can be recognized
as the complementary slackness conditions of the linear program. One popular solver for
this problem is called PATH which can be found in GAMS and AMPL. Like AMPL,
GAMS is a highlevel modeling system for mathematical programming and optimization.
20
Ferris and Munson show how to implement a linear complementarity problem in GAMS
and discuss the available options and output of the PATH solver. Finally, some
extensions of complementarity problems and additional uses of the solver are given.
Another is Billups and Murty (1999). They provide an introduction to
complementarity problems. Various forms of complementarity problems are described
along with a few sample applications such as: piecewise linear equations; an application
of a small size convex QP model; obstacle with free boundary problems; and traffic
equilibrium. The important algorithms are presented with a discussion of when they can
be used effectively. They also present other interesting algorithms for solving linear
complementarity problem. The first one is Pivotal Method which tries to obtain a basic
feasible complementary vector through a series of pivot steps. The next algorithm is the
interior point method. This algorithm follows a path in an effort to reduce the constraints
to zero. Furthermore, they provide a brief introduction to the study of matrix classes and
their relation to linear complementarity problems.
2.4.1. Linear Complementarity Problem
A linear complemetarity problem (LCP) is a subset of mixed complementarity
problems. Linear complementarity problems are problems where a given m m? matrix
M and a compatible vector q
are given. The task is to find the value of a vector z that
satisfies a set of constraints. The general form of LCP can be written as
21
Given: ,
m m m?
? ?M R q R
Find:
m
?z R
s.t. 0? ?Mz q
0?z
and ( ) 0.
T
? ?z Mz q
LCP is truly nonlinear and tools of nonlinear analysis can be successfully applied
to LCP (Stewart, 2008 and Thomas, 2002). They can be represented in various ways in
terms of nonlinear systems of equations such as quadratic programs.
Many algorithms which can be used to solve linear complementarity problem
have been proposed including one by He, Li, and Pan (2005). They developed a self
adjusting interior point algorithm for linear complementarity problems. This algorithm is
based on constructing a new proximity measure function instead of using the primaldual
interior point methods. As a result, they get a new centering equation with a set of
parameters which play the important role of being selfadjusting in this algorithm.
Numerical comparison is made between the proposed algorithm and the primaldual
interior point methods. Results show that the proposed algorithm has the efficiency as
well as some other advantages. For example, the number of iterations increases very
slowly as the number of variables increases.
Li and Dai (2007) introduced a generalized AOR method for solving a linear
complementarity problem whose general case is reduced to a generalized SOR method.
22
These two methods are considered to be an iterative algorithm. Some computational
results are presented in this paper.
Liao and Wang (2003) proposed a selfadaptive projection and contraction
method for linear complementarity problems. They claim their algorithm is better than
He?s algorithm (1992) in the sense that their algorithm improves the practical
performance of the modified projection method. The proof of global convergence of this
new method is also included in the end of their paper.
Sun and Huang (2006) developed a smoothing Newton algorithm for the LCP
with a sufficient matrix. First, they applied a smoothing function to LCP which leads to a
new formulation called a parameterized smooth equation. Then a Newton method with a
projection type testing procedure is used to solve that equation. They also show that this
algorithm will terminate in a finite number of steps as long as the LCP has a solution.
23
CHAPTER III
COURNOT MODEL IGNORING UNCERTAINTY
Abstract  Since deregulation of the electric industry has become the norm in many
parts of the world, the price of electricity under deregulation is no longer determined by
regulatory agencies but by market demand, supply conditions, load elasticity, and
strategic behavior. Firms now face much more risk and are responsible for their own
economic decisions. The firms, therefore, need decisionsupport models that support
these new requirements. In this research, we have developed the multiperiod
deterministic Cournot model for the longterm market which is extended from a single
period model. However, demand uncertainty, generator outages, and fuel price
uncertainty are ignored in this chapter. The Nashequilibrium quantities are calculated by
combining the KKT first order optimality conditions of the extended model. The KKT
conditions of the deterministic Cournot model are considered as linear complementarity
problems (LCP). The market prices and each firm?s profit are calculated. Results in this
chapter are used as the standard results to show the effect of uncertainties in power
markets when we consider them in the model.
24
3.1. NOMENCLATURE
The notation used in this chapter is given below for reference.
n Number of firms
N Total number of generators
f
N Number of generators of firm f
max
fj
P The capacity of the
th
j unit of firm f (MW)
fj
c The marginal cost of the
th
j unit of firm f ($/MWh)
*
f
s The Nashequilibrium quantities (MWh)
*
S The Nashequilibrium total bid (MWh)
*
p The Cournot price ($/MWh)
K Nominal demand (MWh)
h Number of contracted hours
0
t The beginning of a contract
t The present time
i Interest rate (%)
? Compoundamount factor
f
? Profit of firm f ($)
? Slope parameter for demand
25
3.2. INTRODUCTION
Before deregulation, there was a belief that electricity is a national asset.
Therefore, electric sectors in most countries were subject to full regulation. The
generation, transmission, distribution and retail segments were controlled by state
governments which played a dual role as electricity service providers and regulators. As
economic growth became more and more dependent on sufficient electricity suppliers,
the importance of electricity increased tremendously. Consequently, many governments
started to realize that this growth may be impeded by being fully regulated because of the
regulators? slow response to technological progress in electricity operations. In addition,
the successful deregulation in oil and gas established the belief that electricity is a service
which can also be improved by deregulation (Yao, 2006).
Deregulation in the United States took place at both the federal and state levels in
1996. The Federal Energy Regulatory Commission (FERC) encouraged deregulation of
the electricity market by issuing order 888 and establishing guidelines to provide open
access to transmission lines. This policy removed restrictions on ownership of power
generation facilities which allowed nonutility electric power producers to access
transmission lines.
The main purpose of deregulation is to reduce operational cost, to increase
efficiency, and to encourage competition among electricity suppliers with the medium
and longterm goal of combating high prices. Deregulation gives consumers more choices
because they are then not held to only one power provider. Availability of power from
diverse suppliers ensures supply reliability throughout the operation in case of a peak
demand or unexpected outages. It can be said that the greater the availability, the greater
26
the competition will be to produce and sell power in an efficient way, leading to lower
prices and more energy efficiency. Moreover, deregulation is believed to provide better
economic incentives and opportunities to both consumers and suppliers because the
existence of a large number of consumers and suppliers reduces market power in which a
firm is prevented from dominating a market. Consequently, it enables any company to
enter or exit the market which in turn allows competitors to take advantage of any
economic opportunity.
The deregulation of the electricity industry is significantly changing the way in
which suppliers do their business. Firms? optimal decisions will now be dependent on
market electricity prices. The price of electricity under deregulation is determined by
market demand, supply conditions, load elasticity, and strategic behavior. It also depends
on physical factors such as production cost, load, unit commitment, and transmission
constraint (Valenzuela and Mazumda, 2005). Electricity is different from other products
because it has yet to become efficiently storable. Therefore, its demand and supply must
be matched every second. Otherwise, a costly system collapse may result. Firms,
therefore, are faced with much more risk, and they become greatly responsible for their
own economic decisions in deregulated power markets. Hence, these firms need decision
support models that fulfill these new requirements. The decisionsupport models need to
incorporate the uncertainties and other important factors involved in deregulated power
markets.
Recent attempts have been made to model the structure of deregulated electricity
markets via utility system production simulation models that have been used in the past
for planning and regulatory purposes (Kahn, Bailey, and Pando, 1996) but many analysts
27
believe that the Cournot model is better able to represent the electricity market as it has
evolved (Borenstein and Bushnell, 1999).
In this chapter, the multiperiod deterministic model which is extended from a
single period model is presented. Longterm power agreements and the Cournot
competition is assumed for firms? bidding structure in the market. The time value of
money is also considered in the model. Demand and fuel cost are assumed to be constant.
Moreover, the availability of generating units is ignored in this chapter.
The remainder of this chapter is organized as follows: Section 3 provides a
methodology to develop the multiperiod deterministic Cournot model in the long term
markets. In section 4, a numerical example, Nashequilibrium quantities, market prices
and, each firm?s profits are presented. The conclusions are given in section 5.
3.3. MODEL DESCRIPTION
In this section, model assumptions and a methodology to develop the multiperiod
deterministic Cournot model in the long term markets are presented.
3.3.1. Model Assumptions
In the restructured wholesale market, power generators can trade power in both
shortterm and longterm markets. Shortterm refers to a day or hours, while longterm
refers to weeks or years. In the longterm market, generators and consumers agree in
private at time 0 through a central exchange on the delivery of specified power quantities
at some specific time in the future (at time
0
t ) through a long term power purchase
agreement. In this research, the main focus is to model the generators in longterm power
28
agreements and study their effects on market prices and firms? profits. The price is set at
the time of the agreement and remains unchanged for the period of the contract in the
longterm market. The following additional assumptions apply to the model:
3.3.1.1. Power Producer
A total of n competing asymmetric firms with firm f having a set of 1
f
N ? units
available for production at time 0 are assumed. The total number of units available in the
market at time 0 is denoted byN . In reality, there are other available power sources.
Instead of producing its own electricity, a firm has the ability to purchase energy at a
higher price from outside sources. Therefore, the last unit of production or a unit
th
f
N
represents the available power sources which are considered to have infinite capacity and
to be always available. The following are assumptions related to each unit:
? The capacity of the
th
j unit of firm f is represented by
max
fj
P (MW).
? Each firm?s units are dispatched according to an ascending order of their
marginal costs, which is denoted by
fj
c .
? The unit commitment and transmission constraints are ignored.
? Fuel cost is assumed to be a constant and generator outages are ignored.
3.3.1.2. Demand (load)
To represent the behavior of the loadserving entities, a linear inverse demand
function is assumed. In Figure 1, the curve shows a general demand function for the long
term market and indicates the price responsiveness of consumers. The quantity K is the
nominal demand and is assumed to be a constant in this model. The actual realized
29
demand function, which is denoted by L, is affected by the price elasticity of demand
0? ? , which is also known by the producers. p is the electricity price ($/MWh).
Thus, the actual demand of the system is represented by the following linear relationship:
L K p?? ? (3.1)
Figure 1. Consumer Demand Curve
3.3.1.3. Market Operation
As mentioned earlier, generators and consumers sign a longterm contract at time
0. Therefore, the production amounts are determined at time 0 and the actual generation
will occur at time
0
t . This scenario is depicted in Figure 2.
p
30
Figure 2. Market operation in the deterministic model
3.3.2. Mathematical Model
To model the Cournot competition, each supplier simultaneously determines a
quantity
f
s that it is willing to supply at each period t. Suppliers choose their quantities
in order to maximize their total profit at each period over the duration of the contract,
assuming that the total of other firms? bids
f
s
?
is known. Hence, the profit of firm f at
hour
0
t t? can be written as
0 0
, ,
(  ) ( , ) ( )s s
f t t f f f f f f t t f
s s s p K Cost s? ?
? ? ? ?
? ?
(3.2)
where
0
t is time at the beginning of the contract,
0
,
( )
f t t f
Cost s
?
is the cost at hour
0
t t?
for supplier f to produce the quantity
f
s and ( , )
f f
p K s s
?
? refers to the nonnegative
price at this period when the nominal demand is K, and the total market supply is the
Nashequilibrium total bid (
*
S ). Note that t can be any number between 0 and the
number of contracted hours ( 0,1,2,...,t h? ). The cost function,
0
,
( )
f t t f
Cost s
?
, is not a
random variable as fuel cost is assumed to be a constant and generator outages are
ignored in this chapter.
31
As we know, the value of money now is different from the value of money in the
future, the profit at every hour t must take the time value of money into consideration
before calculating the total profit. Therefore, the net present value must be applied to the
profit at every hour tin order to obtain the total profit at time
0
t . The total profit function
after applying the net present value is shown below:
0
,
(  )
f t t f f t
f
t D
f
Max s s
s
? ? ?
? ?
?
? ?
? ??
? ?
? ?
(3.3)
where
1
(1 )
t t
i
? ?
?
,
D is the set of hours at which the supplier will deliver the contracted quantity
f
s (MW)
in the future, ? is compound amount factor, and i is the discount rate (%).
Considering that after the production amounts are determined at time 0 and the
actual generation will occur at time
0
t . In a deterministic model, the hourly profit remains
the same at every hour. Hence, the total profit function can be written as follows:
(  )
(  )
f f f t
f
t D
f
f f f
Max s s
s
s s
? ? ?
? ?
?
?
?
? ?
? ?
? ?
? ?
? ?
?
? ?
where
t
t D
? ?
?
?
?
.
Therefore, the total profit function (3.3) is shown below:
( , ) ( )
f f f f f
f
f
Max s p K s s Cost s
s
? ?
?
? ?
? ? ?
? ?
. (3.4)
32
When the Cournot decision maker assumes that generators are always available and fuel
costs remain unchanged over a period, the function ( , )
f f
p K s s
?
? in (3.4) based on (3.1)
can be written as
1
( )
f f
K s s
?
?
? ? . Note that K is assumed to be a constant due to
constant demand. The cost function in (3.4) is represented by ( )
f f
C s when generator
outages are ignored and fuel prices are assumed to remain unchanged. Thus, the total
profit function for each firm becomes
( ; ) ( )
where
1
( ; ) ( )
f
f f f f f f
s
f f f f f f
Max R s s C s
R s s s K s s
? ? ?
?
?
? ?
? ?
? ? ?
(3.5)
where h is the contracted number of hours, the cardinality of D, and
1
h
t
t
? ?
?
?
?
. The
function ( )
f f
C s is the production cost curve of supplier f assuming all generators are
available. The production cost curve of supplier f can be calculated by
1
( )
f
N
f f fj fj
j
C s c g
?
?
?
where
fj
c is the variable cost of the
th
j unit of firm f in $/MWh and
fj
g is the power
generated by the
th
j generator of firm f in MWh.
Therefore, the optimization problem given by (3.5) can be written as the
following programming model:
33
1
1
max
( ; )
subject to
 0 ( )
( ) for 1,.....,
, 0
f
f
f
N
f f f f fj fj
s
j
N
fj f f
j
fj fj fj f
fj f
Max R s s c g
g s
g P j N
g s
? ? ?
?
?
?
?
?
? ?
?
? ?
? ?
?
?
(3.6)
Where the values of
and
f fj
? ? are the dual variables of the corresponding constraints.
The above model (3.6) is a quadratic programming model.
The Nashequilibrium quantities that solve the set of problems
f
s ( f ?1, ? , n)
can be computed by combining the KKT first order optimality conditions of system
equation (3.6) of all suppliers. The KKT conditions of optimization problem (3.6) can be
written as the following equations:
1
max
for 1,...,
( 2 ) 0 0
0 0
for 1,..., and 1,...,
0 0
0
f
f f f f
N
fj f f
j
f
fj fj f fj
fj fj f
f n
K s s s
g s
f n j N
c g
P g
?
?
?
?
? ? ?
?
?
?
?
? ? ? ? ? ?
? ? ? ?
? ?
? ? ? ? ?
? ? ?
?
0
j
?
(3.7)
The result of these KKT optimality conditions is a linear complementary problem
(LCP). By using available software to solve it as a complementarity problem, the Nash
equilibrium quantity for each firm in MWh,
*
f
s , is obtained. The total demand can be
34
calculated as
* *
1
n
f
f
S s
?
?
?
. The demand relationship (3.1) is used to determine the Cournot
price,
*
p , which becomes
*
*
K S
p
?
?
? . (3.8)
3.4. EXPERIMENTAL RESULTS
For a numerical illustration of results, we consider a market that consists of three
firms. The composition of each firm is given in Table 1 which includes the capacity,
marginal cost, and net plant heat rate. The characteristics of the unit types in Table 1 are
taken from the IEEE reliability test system (Grigg, 1996). Firms 1, 2, and 3 have 11, 7,
and 9 generators respectively.
The last row of each firm in the table corresponds to the assumptions that the
th
f
N
unit of generator for firm f has infinite capacity and is perfectly reliable due to other
available sources in the markets. Each firm is assumed to operate 12 hours per day (off
peak hour) and 30 days per month. The annual percentage rate (APR) is assumed to be
7% for all firms. It is also assumed each firm and the consumers agree through a central
exchange on the delivery of specified power quantities for 2 months and they receive the
payment from the customers at the end of each month. Furthermore, the values of
parameters and K? are assumed to be 15 and 3000 respectively. Based on the given
numerical data, the value of the discount coefficient (? ) in (3.5) is 713.75.
35
Table 1. Market composition and generating unit Data
Firm Unit Fuel Type
Capacity
(MW)
Net Plant Heat Rate
(Mbtu/MWh)
Marginal Cost
($/MWh)
Fi
rm
1
1 Coal 350 9.5 52.45
2 Coal 350 9.5 52.45
3 Coal 155 9.72 52.76
4 Coal 155 9.72 52.76
5 Coal 76 11.9 55.86
6 Coal 76 11.9 55.86
7 Gas 48 10.23 62.56
8 Gas 48 10.23 62.56
9 Gas 78 11.63 68.41
10 Gas 78 11.63 68.41
11 Gas 149 12.87 73.59
12  ?
 999
Fi
rm
2
1 Hydro 50  0.07
2 Hydro 50  0.07
3 Coal 155 9.72 52.76
4 Coal 76 11.9 55.86
5 Gas 48 10.23 62.56
6 Gas 48 10.23 62.56
7 Gas 78
11.63
68.41
8  ?

999
Fi
rm
3
1 Uranium 400

0.017
2 Hydro 50  0.07
3 Hydro 50

0.07
4 Coal 350 9.5 52.45
5 Coal 76
11.9
55.86
6 Gas 48 10.23 62.56
7 Gas 78
11.63
68.41
8 Gas 149 12.87 73.59
9 Gas 149 12.87 73.59
10  ?

999
The Nashequilibrium quantities are obtained by solving the linear
complementarity problem in (3.7) using PATH solver called by AMPL. The Cournot
price is calculated according to (3.8). The results are shown in Table 2.
36
Table 2. The results of Nashequilibrium ignoring uncertainty
Firm 1
(MWh)
Firm 2
(MWh)
Firm 3
(MWh)
Total
(MWh)
p*
($)
595.42 427 595.42 1,617.84 92.14
The total profit for each firm can be computed by substituting the Nash
equilibrium quantities and the power generated by each generator into (3.5). The result is
shown in Table 3.
Table 3. The total profit when ignoring uncertainty
Firm Total Profit ($)
1 16,869,300
2 14,924,400
3 35,577,600
These results will be used to show the effect of uncertainties in power markets
when we consider those uncertainties in chapters 4 and 5.
3.5. CONCLUSIONS
The multiperiod deterministic Cournot model for the longterm market was
developed in this research. The deterministic Cournot model in this chapter belongs to a
class of a quadratic programming. The Nashequilibrium quantities were calculated by
combining the KKT first order optimality conditions of the extended model. The KKT
conditions of the deterministic Cournot model are considered as linear complementarity
problems (LCP). The linear complementarity problem can be solved by using the PATH
solver in AMPL. The market prices and each firm?s profit were calculated by substituting
37
the Nashequilibrium quantities and the power generated by each generator into the profit
function. The deterministic model and results in this chapter will be used to show the
effect of uncertainties in power markets when we consider these uncertainties in the next
two chapters.
38
CHAPTER IV
STOCHASTIC COURNOT MODEL INCLUDING GENERATOR OUTAGES
Abstract  The uncertainty in generator availability is a crucial factor which
should be considered in a mediumterm or longterm planning process. An approach to
determining market prices considering generator outages is proposed in this chapter. The
multiperiod Cournot model in the previous chapter is modified by replacing its cost
function with the expected cost function. Specifically, the expected cost function in terms
of generator availability is developed. The expected cost curve is a piecewise linear
function with a large number of slopes. Each slope represents the marginal cost and
capacity of a hypothetical generator. Since considering all slopes could take long
computational time and make the problem difficult to solve, an algorithm for reducing the
number of slopes without losing accuracy is developed. In addition, the effect of
generator outages on the firms? expected profits is analyzed.
39
4.1. NOMENCLATURE
The notation used in this chapter is given below for reference.
n Number of firms
N Total number of generators
f
N Number of generators of firm f
max
fjP The capacity of the
th
j hypothetical unit of firm f considering outages (MW)
fjc
The marginal cost of the
th
j hypothetical unit of firm f including generator
outages ($/MWh)
*
f
s The Nashequilibrium quantities (MWh)
*
S The Nashequilibrium total bid (MWh)
*
p The Cournot price ($/MWh)
K Nominal demand
h Number of contracted hours
i Interest rate (%)
? Compoundamount factor
f
? Profit of firm f ($)
? Slope parameter for demand
t The present time
fj
? The failure rate of the
th
j unit of firm f per hour
fj
? The repair rate of the
th
j unit of firm f per hour
40
4.2. INTRODUCTION
Power outages play an important role in both developing and industrialized
countries. When the outages take place, they can be momentary or last for several days
affecting only a few small areas or entire cities. Although most of the power outages are
manmade, there are some outages caused by nature such as hurricanes, flooding, and
earthquakes.
Unplanned power outages are the situation that power plants in both developing
and industrialized countries try to avoid because the economic consequences of electric
power outages are severe. Blackouts affect not only economics but also our daily life.
The best example took place in the Northeastern United States in August 2003, which
was the worst blackout in U.S. history.
The blackout in August 2003 started shortly after 4 PM EDT and resulted in the
loss of 61,800 MW of electric load that served more than 50 million people in the U.S.
and Canada including large urban centers that are heavily industrialized and important
financial centers (e.g., New York City and Toronto, Ontario). Nearly half the Canadian
economy is located in Ontario and was affected by the blackout. Most areas were fully
restored within two days but parts of Ontario took more than a week before power was
restored (Electricity Consumers Resource Council (ELCON), 2004). In addition, the
blackout also affected industry as well as infrastructure such as water supply,
transportation, and communication. All manufacturers in the Northeastern United States
reporting indicated that the blackout caused a complete shutdown in operation. The Ohio
manufacturers? Association estimated the direct cost of the blackout on Ohio
manufacturers to be $1.08 billion (Electricity Consumers Resource Council (ELCON),
41
2004). The U.S. Department of Energy (2008) has also published a total economic cost
estimate of about $6 billion due to the blackout. Electricity is a very special product in
the sense that it is not storable over extended periods of time. It is generally consumed in
less than a second after being produced. The power generation and demand must be
matched every second in the power supply networks. The mismatch between supply and
demand, either overload of a power line or underload/overload of a generator, can cause
severe damage to a network component which may lead to a cascading failure of a large
section. The reality is that generators are not always available. They may fail any second
and the next cheapest available generators will replace them in order to meet the demand.
Generator failure is believed to be one of the common causes for long duration outages in
power markets. In addition, analysis has shown that the outage of a generator is the
initiating event of cascading faults which may rapidly lead to a catastrophic failure, i.e. a
major blackout (Genesi, Granelli, Innorta, Marannino, Montagna, and Zanellini, 2007).
Since only a few minutes of blackout can cost millions of dollars to the whole system,
uncertainty of generator availability is one of the significant factors in power markets
which must be considered in a decisionsupport model.
The main purpose of this chapter is to develop a model to assess the effects of
generator outages on electricity prices under Cournot competition. The extension of the
deterministic version of the Cournot model in chapter 3 is developed by including the
reliability of the generating units. Fuel price uncertainty is ignored in this chapter. The
method generally used in the literature to deal with the issue of generator failure is to
derate plant capacities. This approach, however, may lead to inaccurate results
(Valenzuela and Mazumda, 2007). Some literature simply ignores this factor. Unlike
42
other literature, the expected production cost function including generator failure is
implicitly modeled in this chapter rather than the cost of the expected production
quantities obtained after the generators are derated. When incorporating generator
outages, the cost function is modified by introducing the operating state of each generator
which will lead to a new expected cost function.
This chapter is organized as follows: A stochastic Cournot model in the long term
markets including generator outages is developed in section 3. In section 4, numerical
results as well as the comparison of both firm?s expected profit considering generator
failure and ignoring outages are presented. The conclusions are outlined in section 5.
4.3. MODEL DESCRIPTION
In this section, a procedure for computing the market prices taking generator
failure in a deregulated longterm power market into consideration is described. Instead
of derating plant capacities, the expected production cost function including generator
outages is developed.
4.3.1. Model Assumptions
The following assumptions apply to this model:
4.3.1.1. Market Operation
In a longterm market, the price is set at the time of the agreement (time 0) and
remains unchanged for the period of the contract. In other words, the production amounts
are determined at time 0 and the actual generation will occur long after this time (at time
43
0
t ). The Markov process is assumed to reach its steady state when the contracted amount
will be generated. This scenario is illustrated in Figure 3.
Figure 3. Market operation in the stochastic model
4.3.1.2. Power Producer
A total of n competing asymmetric firms are considered in this chapter, in which
all firms make their decisions according to the Cournot model. That each firm f
possesses a set of 1
f
N ? units available for production at time 0 is assumed. In addition,
the last unit of production or a unit
th
f
N represents the available power sources. In this
chapter, the power sources are considered to have infinite capacity and to be always
available. The following are assumptions related to each unit:
? The capacity of the
th
j unit of firm f is represented by
max
fj
P (MW).
? Each firm?s units are dispatched according to an ascending order of their
marginal costs, which is denoted by
fj
c .
? Generator outages are assumed in this chapter.
44
? The unit commitment and transmission constraints are ignored and fuel
price is assumed to be a constant.
4.3.2. Mathematical Model
4.3.2.1. Demand (load)
To represent the behavior of the loadserving entities, a linear inverse demand
function is assumed. The actual realized demand function of the system, denoted by
*
S ,
is affected by the price elasticity of demand 0? ? , which is known by the power
producers. The actual demand of the system is represented by the following linear
relationship:
L K p?? ? . (4.1)
The quantity K stands for the nominal demand and is assumed to be a constant in
this section. The generation quantity for firm f (MWh),
f
s , is the sum of generation
quantities of all generators of firm f ( 1, 2, ... , f n? ) and is calculated as follows:
1
=
f
N
f fj
j
s g
?
?
.
The actual demand of the system is equal to the total of all firms? generation
quantities and it can be written as follows:
* *
1
n
f
f
S s
?
?
?
.
45
4.3.2.2. Profit Function
Power producers that are bidding in the market own a set of generators. When
those generators are not always available, the cost function becomes a random variable.
Therefore, the total profit function from the previous chapter becomes the expected total
profit function in this chapter. After considering the time value of money and generator
outages, the expected total profit function, based on (3.3), becomes
0
,
(  )
f t t f f t
f
t D
f
Max E s s
s
? ? ?
? ?
?
? ?
? ??
? ?
? ?
where
1
(1 )
t t
i
? ?
?
.
Considering that after the production amounts are determined at time 0 and the
actual generation will occur long after time
0
t , the Markov process of the unit availability
is assumed to reach the steady state (see Figure 3). In the steady state (
0
t ??), hourly
expected profits remain the same at every hour. The total expected profit function in the
steady state can be simplified as follows:
0
0
0
0
,
(  )
(  )
(  )
f t t f f t
f
t
t D
f
f f f t
t
t D
f f f
t
Max s sE
s
s sE
s sE
? ? ?
? ?
? ?
? ?
??
?
?
??
?
?
??
? ?
? ??
? ?
? ?
? ?
? ?
?
?
? ?
? ?
? ?
? ?
?
? ?
where
t
t D
? ?
?
?
?
and
0
t
E
??
represents the expected value considering that the stochastic
process that reigns the unit availability is in the steady state.
46
Hence, the expected total profit function for supplier f , in the steady state, can
be rewritten as
0 0
(  ) ( )( ; )
f f f f f f f f
t t
s s R s s Cost s
E E
? ? ? ?
? ?
?? ??
? ? ? ?? ?
? ? ? ?
.
In Cournot competition, firm f assumes the total quantity, produced by other
firms
? ?
f
s
?
, and each firm f then determines the quantity
f
s
in order to maximize its
total expected profit at each period over the duration of the contract. Thus, the total
expected profit function for supplier f in the steady state can be calculated as follows:
? ?
( , ) ( )
f f f f f f
s p K s s E Cost s? ?
?
? ?? ? ?
? ?
. (4.2)
4.3.2.3. Reliability
To introduce the operating state of a generator j of firm f , we begin by defining
a twostate continuoustime Markov process which is 1 if unit j of firm f is available at
time t and 0 for otherwise. The Markov process has a failure rate per hour
,f j
? and a
repair rate per hour
,f j
? . Since in our model the production amounts are determined at
time 0 and the actual generation will occur long after this time (at time
0
t ), the Markov
process is assumed to reach its steady state when the contracted amount is generated. The
steady state probability that the generator j of firm f will be available is denoted by
,
,
, ,
f j
f j
f j f j
r
?
? ?
?
?
(4.3)
and, the steady state probability that generator j of firm f will not be available is
denoted by
47
, ,
1
f j f j
q r? ? . (4.4)
4.3.2.4. Cost function
To compute the expected cost function, we first define , ( )f jC s as the expected
cost function of supplier f producing s units of energy when generators 1, 2, ... , 1j?
are not available. The expected production cost function including generator outages is
developed below.
For unit 1, 2,..., 1
f
j N? ? , the following recursive relationship is used:
? ?
, 1
, , , ,
,
, 1 , 1
, , , , , ,
( ) for 0
( )
( ) ( ) for
Max
f j
f j f j f j f j
f j
Max Max Max
f j f j
f j f j f j f j f j f j
r c s q C s s P
C s
r c P C s P q C s s P
?
? ?
? ? ?
?
? ? ? ?
The last unit of generator (
f
N ), which is always available, can be represented by
the following relationship:
,
,
( )
f
f
f N
f N
C s s c? ? .
Owing to the recursive relationship, the calculation starts from the first unit of
generator to the last unit (
f
N ) in order to obtain the total expected cost. Therefore, the
expected production cost function considering generator outages can be written as
follows:
,1( ) ( )f
f f f
E Cost s C s? ??
? ?
where the expected cost of firm f , given by ,1( )f
f
C s , is a piecewise linear function with
respect to
f
s . The slopes of this piecewise linear function are always increasing, as each
48
firm?s units are dispatched according to an ascending order of their marginal costs. Each
slope represents a marginal cost and capacity of a generator including outages. The set of
all combinations of unit capacities determines where the function changes its slope. For
example, one firm has three units with a capacity of 12, 20, and 50 MW. There are
3
2 1?
combinations in total with these three generators which mean the expected cost function
changes its slope at 12, 20, 32, 50, 62, 70, and 82 MW. In other words, this firm actually
owns a set of seven hypothetical generators with the maximum capacity of 12, 20, 32, 50,
62, 70, and 82 MW respectively when considering generator outages. In the aspect of
complexity, this may not seem to be much different from assuming that generators are
always available. However, if a firm owns a total of n generators, it could have 2 1
n
?
different slopes in the worst case which means this firm has to consider a set of 2 1
n
?
different generators and maximum capacities. Since this is an exponential function, when
n is a big number, it will make a huge difference in computational complexity. An
example of the expected cost curve when a power company has 12 units of generators is
depicted in Figure 4.
In Figure 4, the graph seems to show only 3 or 4 different slopes but the expected
cost curve actually contains roughly
12
2 1? different slopes. The reason that the graph
displays only 3 to 4 slopes is some slopes may differ from others by a small amount.
Considering the fact that some slopes may differ from others by a small amount, it is
better to combine them and represent the cost function with a lesser number of slopes
without losing accuracy. Moreover, the complexity of calculations can be eased with a
smaller number of slopes.
49
Figure 4. The expected cost curve of 12 generating units
To reduce the number of slopes, we define and s ?? to be the increasing amount
of capacity and the difference between slopes, respectively. Both are constant values and
must be set at the beginning. Let
0
N be the maximum range in which we want to
combine slopes such that
0 max
,
1
f
N
f j
j
N P
?
?
?
for each firm f.
The following pseudocode describes the slope reduction algorithm for firm f :
1:
0
to 2 FOR l N?
2: Set and ,
l
X l s? ?? ,1( )f
l l
C C X?
3:END FOR
4:
1 1
Set and 1, 2, 0, 0r K C X? ? ? ?
50
5:WHILE
0
K N?
6:
0
1 1
Min such and ( 2 )
i i i
j i K i N C C C ?
? ?
? ? ? ? ? ?
7:
1
,
2
1
j
l l
l K
f r j
l
l K
XC
m
X
? ?
? ?
?
?
?
8:
,
and
Max
f r j j
NewP X Temp X? ?
9:
0
to FOR l j N?
10:
l l
X X Temp? ?
11:
,l l f r
C C m Temp? ? ?
12: END FOR
13: 1K j? ?
14: 1r r? ?
15:END
The slope reduction algorithm produces an estimation of the expected cost
function which is used to generate estimated expected cost curve. A slope in estimated
expected cost curve is associated with each hypothetical generator (r) which has a
maximum capacity (
,
Max
f r
NewP ) and marginal cost
,
( )
f r
m . Let
max
r
represent a value of r
when the slope reduction algorithm terminates. The values of
,f r
m and
,
Max
f r
NewP are
similar in nature to those
,f j
c and
max
,f j
P described in chapter 3 but they take generator
outages into account. In order to simplify the presentation, the values of marginal cost
,
( )
f r
m and maximum capacity (
,
Max
f r
NewP ) will be represented as
,f jc
and
max
,f jP where
(
max
1, 2, ... , j r? ).
An estimation of the total expected cost for each firm f , ,1( )f
f
C s , can be
calculated as follows:
51
,1 ,
,
1
( )
f
N
f f j
f f j
j
C s c g
?
?
?
where
,f jc
is the marginal cost obtained from the slope reduction algorithm.
4.3.2.5. Modeling Competition
When the Cournot decision maker assumes demand remains unchanged over a
period of time, the function ( , )
f f
p K s s
?
? in (4.4) based on (4.1) can be written as
1
( , ) ( )
f f f f
p K s s K s s
?
? ?
? ? ? ? .
Thus, each supplier f solves the following programming model:
,1( ; ) ( )
f
f
f f f f f
s
Max R s s C s? ? ?
?
? ?
1
max
max
subject to
 0 ( )
( ) for 1,.....,
, 0
f
N
fj f f
j
fj
fj fj
fj f
g s
g P j r
g s
?
?
?
?
? ?
? ?
?
(4.5)
1
where
1
( ; ) ( )
and .
f f f f f f
h
t
t
R s s s K s s
?
? ?
? ?
?
? ? ?
?
?
The Nashequilibrium quantities
f
s ( f ?1, 2, ?, n) that solve the set of
problems can be computed by combining the KKT first order optimality conditions of
52
system equation (4.5) of all firms. The KKT conditions of optimization problem (4.5) can
be written as the following equations:
1
max
max
for 1,...,
( 2 ) 0 0
0 0
for 1,..., and 1,...,
0 0
0
f
f f f f
N
fj f f
j
fj
fj f fj
fj
fj
f n
K s s s
g s
f n j r
c g
P g
?
?
?
?
? ? ?
?
?
?
? ? ? ? ? ?
? ? ? ?
? ?
? ? ? ? ?
? ? ?
?
0
fj
? ?
(4.6)
These KKT conditions are still considered as a linear complementary problem
(LCP). The Nashequilibrium quantity for each firm in MWh (
*
f
s ) is obtained by using
available software to solve those optimality conditions. As mentioned earlier, the actual
demand of the system or total demand can be calculated as
* *
1
n
f
f
S s
?
?
?
. The linear
relationship in demand (4.1) is used to determine the Cournot price,
*
p , which becomes
*
*
K S
p
?
?
? . (4.7)
4.4. EXPERIMENTAL RESULTS
In this section, the methodology explained in section 3 is implemented for a
market that consists of three firms. For a numerical illustration of results, the composition
of each firm is given in Table 4 including the capacity, marginal cost, net plant heat rate,
and availability of each generator. The characteristics of the unit types in Table 4 are
53
taken from the IEEE reliability test system (Grigg, 1996). Firms 1, 2, and 3 have 11, 7,
and 9 generators respectively. The last row of each firm in the table corresponds to the
assumption that the
th
f
N generating unit of firm f has infinite capacity and is assumed to
be perfectly reliable due to other available sources in the markets. The available generator
types are coal, oil, hydro, and nuclear. The marginal cost of one type of technology is
calculated based on its heat rate. Although generators consume the same type of fuel,
they do not always have the same marginal cost. Furthermore, the values of parameters
and K? are assumed to be 15 and 3000 respectively. Each firm is assumed to operate 12
hours per day (offpeak hours) and 30 days per month. It is also assumed that all firms
and consumers agree in private through a central exchange on the delivery of specified
power quantities for 2 months and the firms receive the payment from customers at the
end of each month. The annual percentage rate (APR) is assumed to be 7% for all firms.
Next, the discount coefficient in (4.5), ? , can be computed in order to consider time
value of money in the model. Based on the given APR, the value of the discount
coefficient is 713.75.
Based on the numerical data in Table 4, the expected cost function of each firm
contains a large number of slopes. The slope reduction algorithm plays a vital role in
reducing the number of slopes (those marginal costs and maximum capacities). The value
of s? is assumed to be 1 in this section. Moreover, a value of the difference between
slopes ? is chosen by determining the smallest number that yields the value of estimated
cost as close as the original value of expected cost. The greater the value of ? , the more
inaccurate the results will be. The value of ? can be simply judged by comparing the
54
expected cost curve and the estimated cost curve. If those two graphs are similar, the
value of ? can be used without making the final results imprecise. After applying the
slope reduction algorithm to the numerical data in Table 4, the expected cost curve and
the estimated cost curve for firms 1, 2, and 3 are shown in Figures 5, 6, and 7,
respectively.
The value of ? for firms 1, 2, and 3 is chosen to be 6, 3, and 5, respectively.
Comparing the expected cost curve and the estimated cost curve in each of Figures 5, 6,
and 7, the two graphs are almost identical. Therefore, it can be said that this new set of
hypothetical generators can be used to compute the equilibrium quantities without losing
accuracy. The curve of the normal cost function is also included in Figures 5, 6, and 7.
Note that for a given quantity, a value of the expected cost is higher than the normal cost
described in chapter 3 because of generator outages.
Marginal costs and maximum capacities of the hypothetical generators for firms
1, 2, and 3 are shown in Tables 5, 6, and 7, respectively.
55
Table 4. Market composition and generating unit data including unit availability
Firm Unit
Fuel
Type
Capacity
(MW)
Net Plant
Heat Rate
(Mbtu/MWh)
Marginal Cost
($/MWh)
Availability (hours)
1/? 1/?
Fi
rm
1
1 Coal 350 9.5 52.45 1150 100
2 Coal 350 9.5 52.45 1150 100
3 Coal 155 9.72 52.76 960 40
4 Coal 155 9.72 52.76 960 40
5 Coal 76 11.9 55.86 1960 40
6 Coal 76 11.9 55.86 1960 40
7 Gas 48 10.23 62.56 1340 26
8 Gas 48 10.23 62.56 1340 26
9 Gas 78 11.63 68.41 1720 30
10 Gas 78 11.63 68.41 1720 30
11 Gas 149 12.87 73.59 1505 42
12  ?
 999  
Fi
rm
2
1 Hydro 50  0.07 1980 20
2 Hydro 50  0.07 1980 20
3 Coal 155 9.72 52.76 960 40
4 Coal 76 11.9 55.86 1960 40
5 Gas 48 10.23 62.56 1340 26
6 Gas 48 10.23 62.56 1340 26
7 Gas 78
11.63
68.41 1720 30
8  ?

999  
Fi
rm
3
1 Uranium 400

0.017 1100 150
2 Hydro 50  0.07 1980 20
3 Hydro 50

0.07 1980 20
4 Coal 350 9.5 52.45 1150 100
5 Coal 76
11.9
55.86 1960 40
6 Gas 48 10.23 62.56 1340 26
7 Gas 78
11.63
68.41 1720 30
8 Gas 149 12.87 73.59 1505 42
9 Gas 149 12.87 73.59 1505 42
10  ?

999  
56
Figure 5. Cost curves of supplier 1
Figure 6. Cost curves of supplier 2
57
Figure 7. Cost curves of supplier 3
Table 5. The data list of hypothetical generating units for firm 1
Unit
Capacity
(MW)
Marginal Cost
($/MWh)
1 1058 53.19
2 155 99.85
3 195 211.47
4 71 294.29
5 28 346.29
6 48 372.44
7 2 403.11
8 6 415.63
9 ? 999
58
Table 6. The data of hypothetical generating units for firm 2
Unit
Capacity
(MW)
Marginal Cost
($/MWh)
1 100 0.41
2 155 53.18
3 76 57.91
4 19 69.47
5 77 101.18
6 26 139.00
7 2 152.01
8 2 158.68
9 48 187.40
10 ? 999
Table 7. The data of hypothetical generating units for firm 3
Unit
Capacity
(MW)
Marginal Cost
($/MWh)
1 100 0.06
2 400 6.50
3 100 55.95
4 201 64.93
5 125 75.68
6 24 107.94
7 50 181.97
8 201 249.86
9 2 264.23
10 71 294.10
11 26 317.07
12 2 325.95
13 48 342.38
14 ? 999
The Nashequilibrium quantities or firms? quantity bids are calculated according
to the KKT first order optimality conditions in (4.6) using the PATH solver. The Cournot
price is calculated according to (4.7) assuming a linear relationship. The results of Nash
equilibrium quantities and market prices are shown in Table 8.
59
Table 8. The results of Nashequilibrium including generator outages
Firm 1
(MWh)
Firm 2
(MWh)
Firm 3
(MWh)
Total
(MWh)
p*
($)
631.16 350 589.78 1570.94 95.27
4.4.1. Effect of generator availability
For ease of explaning, the deterministic model described in chapter 3 is called
Model A and the stochastic model concerning outages is called Model B. To evaluate
whether the availability of generators has an effect on market prices, first the model profit
and adjusted profit ignoring uncertainty (Model A) for each firm are compared. Next, the
adjusted profits of Model A and the expected profits of Model B are compared.
For model A, the model profit is the amount that firms believe they will make
when ignoring outages, while the adjusted expected profit is the amount that firms would
obtain in reality because generator outages do occur.
The model profits of Model A are computed by substituting the Nashequilibrium
quantities and the power generated by each generator into the objective function of
Model A (3.5). Let
*A
f
s represent the Nashequilibrium quantities of Model A. Model A?s
profit can be written as follows:
* * *
( ; ) ( )
A A A
f f f f f f
R s s C s? ?
?
? ?? ?
? ?
where
* * * * *
1
( ; ) ( )
A A A A A
f f f f f f
R s s s K s s
?
? ?
? ? ? .
Unlike the model profit, the adjusted expected profit of Model A is calculated
from the difference between the revenue function and the expected cost function
60
including outages,
*
,1( )f
f
C s . For Model A, the adjusted profit can be calculated as
follows:
? ?
* * *
,1
[ ] ( ; ) ( )
A A A
f
f f f f f
E R s s C s? ?
?
? ?
where
*A
f
s stands for the Nashequilibrium quantities obtained from Model A.
The model expected profit and expected profit of Model B are the same because
the model considers outages. The model expected profit and expected profit when
considering generator failures can be computed as follows:
? ?
* * *
,1
( ; ) ( )
B B B
f
f f f f f
R s s C s? ?
?
? ?
where
*B
f
s represents the Nashequilibrium quantities obtained from Model B.
Results of the model profits and expected profits for each firm in both cases are
shown in Table 9.
Table 9. Expected profits of firms when ignoring uncertainty and including outages
Model A
Ignoring Uncertainty ($/Hour)
Model B
Including Outages ($/Hour)
Firm Model Profit
Adjusted Expected
Profit
Model Expected
Profit
Expected Profit
1 23,634.67 23,194.05 26,557.68 26,557.68
2
20,909.78 17,549.88 19,340.47 19,340.47
3
49,845.87 46,919.70 48,560.83 48,560.83
According to Table 9, the adjusted profits and the model profits of firm 1 for
model A are comparable. For firm 2 and 3, however, the adjusted profit is less than the
model profit, which means, for example, firm 3 believes that it will make
$49,845.87/hour, but in reality it would make only $46,919.70/hour on average.
61
When comparing the expected profits using Model B firms 1, 2, and 3 make more
profit than Model A. These results indicate market participants, who make decisions
without considering uncertainty in generator availability, could be led to false decision
making and an inaccurate planning process.
4.5. CONCLUSIONS
An approach to determining market prices that considers generator outages was
developed in this chapter. The multiperiod Cournot model in the previous chapter was
modified by replacing its cost function with the expected cost function. Specifically, the
expected cost function in regard to generator availability was developed by defining a
twostate continuoustime Markov process. The expected cost function generated a large
number of slopes. Each slope represents one value of marginal cost and maximum
capacity which includes the uncertainty in generator availability. To ease computational
complexity, the slope reduction algorithm was developed.
Results showed that the slope reduction algorithm can efficiently reduce a number
of slopes and aid computational complexity. In addition, the model and expected profits
were computed to evaluate whether generator outages have an effect on firms? profit.
Results indicated that generator availability is a crucial factor, as it has effects on both
market prices and firms? profit.
62
CHAPTER V
STOCHASTIC COURNOT MODEL INCLUDING GENERATOR OUTAGES
AND FUEL PRICE UNCERTAINTY
Abstract  Nowadays, the volatility associated with generation and fuel prices
places a new emphasis on modeling uncertainties in power markets. It is essential for all
companies to account for uncertainty. Each firm operates a set of generators which use
different types of fuels to produce electricity. The fluctuation in fuel costs significantly
impacts the firm?s longterm operation. In this chapter, the stochastic Cournot model is
extended to consider not only the availability of generators but also fuel price uncertainty.
Since each generator consumes a fuel type whose price is not known in advance, the
marginal cost of each generator becomes a random variable. A convenient way to
calculate the Nashequilibrium quantities when considering the randomness in fuel price
is the use of a Monte Carlo simulation. A numerical example is given, where the market
prices and firms? expected profits are computed. In addition, the effects of generator
outages and fuel price uncertainty on power markets are analyzed.
63
5.1. NOMENCLATURE
The notation used in this chapter is given below for reference.
n Number of firms
N Total number of generators
f
N Number of generators of firm f
max
fjP The capacity of the
th
j hypothetical unit of firm f considering outages and fuel
price uncertainty (MW)
fj
c The marginal cost of the
th
j hypothetical unit of firm f including generator
outages and fuel price uncertainty ($/MWh)
*
f
s The Nashequilibrium quantities (MWh)
*
S The Nashequilibrium total bid (MWh)
*
p The Cournot price ($/MWh)
K Nominal demand
h Number of contracted hours
i Interest rate (%)
? Compoundamount factor
f
? Profit of firm f ($)
? Slope parameter for demand
fj
? The failure rate of the
th
j unit of firm f per hour
fj
? The repair rate of the
th
j unit of firm f per hour
64
5.2. INTRODUCTION
It is widely accepted that an increase in fuel price volatility has a great impact on
the whole of economic activity and creates an uncertain situation for power producers,
consumers, investors, and legislators. It may slow down economic growth. It may delay
producers? decisions on making new investments which may result in lost market
opportunities and inefficient longrun resource allocations. Daily or hourly fluctuations in
wholesale prices may be almost irrelevant to the consumers, but it is vital to power
trading companies since increase or decrease in prices can change the way companies do
their business. Moreover, fuel price uncertainty may create pressures for regulatory
intervention which can bias the power markets and penalize market participants by
generating wide and unpredictable revenue swings (Henning, Sloan, and Leon, 2003).
Hence, volatility in fuel costs has become a new issue that power companies must be able
to handle in order to guarantee appropriate power system planning and operation.
The price of gasoline is the best example to show why fuel price is so difficult to
forecast. In October 2006, the average retail price for a gallon of gasoline in the U.S. was
around $2.20 per gallon. Since then, the price of gasoline has risen dramatically. The
maximum average price per gallon was over $4. This trend continued until late 2008.
This situation is depicted in Figure 8 (EIA, 2008). In August 2008, the average price of
gasoline decreased sharply. The average price decreased from $4.10 to $1.60 in only 4
months. Demand is not the only factor that affects the price of gasoline; other factors
include worldwide economics and politics. Thus, the price of gasoline is considered
highly volatile similar to those of natural gas and electricity.
65
Figure 8. U.S. retail gasoline prices, Regular grade (Source: EIA 2009)
Most electricity in U.S. is generated by rotating turbines which are most
commonly driven by steam. Steam is typically produced from water that is boiled by
burning coal, natural gas, or petroleum. In 2008, coalfired plants contributed 48.3% of
the U.S electric power. Nuclear plants contributed 19.3%, while 21.5% was generated at
natural gasfired plants. Hydroelectric provided 6.7% of the total while petroleum and
renewable energy generated the remaining electric power (EIA, 2008). Owing to lower
fuel costs, nuclear plants are operating at a much higher utilization and they supply the
base load in general.
The Energy Information Administration (EIA) forecasts in the Energy Outlook
2009 early release that the price of petroleum, natural gas, and coal in U.S. will increase
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
3.75
4
4.25
4.5
Doll
ar
s
per
gallon
66
about 80.55%, 33.8%, and 14.4% respectively in the next 20 years. Moreover, EIA also
predicts in the annual energy outlook that the price of electricity will only increase about
13%. Power producers have to sell electricity cheaper due to severe competition, and they
have to generate electricity cheaper than other firms despite the increase in the price of
fuel. For a thermal generationbased electric utility, fuel costs comprise approximately
80% of the system operating cost. Consequently, a small percentage savings in fuel costs
represents significant monetary value. In the era of intense competition in power markets,
a small savings may be crucial (Lee, Liao, and Breipohl, 1992). As it appears fuel costs
will be volatile in the future and fuel savings will be vital to stay in the business; power
producers in the U.S. are facing a new set of challenges. In the last decades, uncertainty
in fuel costs and generator availability have become a structural element in this new
environment that all power companies must be able to cope with in order to guarantee
appropriate power system planning and operation as well as their economical liquidity
(Gomes, Saraiva, and Neves, 2008). In addition, company profits are also influenced by
the fluctuation of market prices of electricity that varies concomitantly depending on fuel
market prices (Bannai and Tomita, 2005).
In this research, the primary aim is to develop a model to evaluate the effects of
the uncertainty of fuel costs and generator availability on firms? profits. The
determination of market price is achieved by the Cournot competition of firms in the
market. Each firm operates a set of generators which use different types of fuels whose
marginal costs are subject to uncertainty. Thus, the marginal costs become random
variables which affect the expected cost function. When considering both generator
outages and fuel price uncertainty, a convenient way to calculate the Nashequilibrium
67
quantities is to use a Monte Carlo simulation (MCS) based technique. The major
advantage in an MCSbased approach is that it can capture the spatial distribution of the
uncertainties of generation, which is very important where marginal cost differentials
play a leading role (Wijayatunga and Cory, 2003).
The remainder of this chapter is organized as follows: Section 3 provides a
methodology to develop the stochastic Cournot model considering generator outages and
fuel cost uncertainty. In section 4, numerical example, Nashequilibrium quantities,
market prices and each firm?s expected profit are presented. The conclusions are given in
section 5.
5.3. MODEL DESCRIPTION
In this section, model assumptions and a methodology to develop the stochastic
Cournot model including generator availability and fuel price uncertainty are presented.
5.3.1. Model Assumptions
The main focus of this chapter is to model the generators in longterm power
agreements and to study the effects of uncertainty on market price and firms? profits.
Hence, the price of electricity is set at the time of the agreement and remains unchanged
for the period of the contract in the longterm market. A total of n competing firms with
firm f having a set of 1
f
N ? units available for production at time 0 is assumed, and the
last unit of production or a unit
th
f
N represents the available power sources and is
considered to have infinite capacity and to be always available. The total number of units
available in the market at time 0 is denoted by N . The capacity of the
th
j unit of firm f
68
is represented by
max
fj
P (MW). Each firm?s units are dispatched according to an ascending
order of their marginal costs, which is denoted by
fj
c . Unit commitment and transmission
constraints are ignored. Fluctuation of fuel costs and generator outages are assumed in
this chapter.
5.3.2. Mathematical Model
5.3.2.1. Reliability
To introduce the operating state of a generator j of firm f , we begin by defining
a twostate continuoustime Markov process, which is 1, if unit j of firm f is available
at time t and 0 for otherwise. The Markov process is assumed to reach its steady state
when the contracted amount is generated. The Markov process has a failure rate per hour
,f j
? and repair rate per hour
,f j
? . The steady state probability that generator j of firm f
will be available is denoted by
,
,
, ,
,
f j
f j
f j f j
r
?
? ?
?
?
and the steady state probability that the generator j of firm f will not be available
is denoted by
, ,
1 .
f j f j
q r? ?
5.3.2.2. System load
To represent the behavior of the loadserving entities, a linear inverse demand
function is assumed. The actual demand of the system is represented by the following
linear relationship:
69
L K p?? ? (5.1)
The actual realized demand function of the system, which is denoted by L, is
affected by the price elasticity of demand 0? ? , which is also known by the power
producers. The quantity K stands for the nominal demand (MWh), and it is assumed to
be a constant in this section. The generation quantity for firm f
is
f
s , which is the sum
of generation quantities of all generators of firm f ( 1,2, ... , f n? ) in MWh and it is
calculated as follows:
1
=
f
N
f fj
j
s g
?
?
.
The actual demand of the system is equal to the total of all firms? generation
quantities and it can be written as follows:
*
1
n
f
f
S s
?
?
?
.
5.3.2.3. Profit function
Power companies that are bidding in the market operate a set of generators. When
those generators are not always available and their marginal costs are subject to
uncertainty because of fuel price fluctuation, the cost function is a random variable. The
revenue function, however, is not affected by those uncertainties. Owing to generator
outages and fuel price uncertainty, the expected total profit function for supplier f in the
steady state can be written as follows:
? ?
( , ) ( )
f f f f f f
s p K s s E Cost s? ?
?
? ?? ? ?
? ?
(5.2)
70
where ( , )
f f f
s p K s s
?
? is the revenue function and is not affected by generator
availability and fuel cost uncertainty. When the Cournot decision maker assumes demand
remains unchanged over a period of time, the function ( , )
f f
p K s s
?
? based on
relationship in (5.1) can be written as
1
( , ) ( )
f f f f
p K s s K s s
?
? ?
? ? ? ? .
5.3.2.4. Cost function
The expected production cost function including only generator outages of
supplier f producing s units of energy (MWh), when generating units 1, 2, ... , 1j? are
not available, can be written as follows:
For unit 1, 2,..., 1
f
j N? ? , the following recursive relationship is used:
? ?
, 1
, , , ,
,
, 1 , 1
, , , , , ,
( ) for 0
( )
( ) ( ) for
Max
f j
f j f j f j f j
f j
Max Max Max
f j f j
f j f j f j f j f j f j
r c s q C s s P
C s
r c P C s P q C s s P
?
? ?
? ? ?
?
? ? ? ?
(5.3)
The last unit of generator (
f
N ), which is always available, can be represented by
the following relationship:
,
,
( )
f
f
f N
f N
C s s c? ? . (5.4)
When the price of fuel becomes volatile, the marginal costs of generators are
subject to uncertainty. In other words, marginal costs of all generators (
,f j
c ) are random
variables except for the last unit (
f
N ) whose marginal cost is assumed to be a constant. It
is assumed that each marginal cost is a continuous random variable which has an
associated probability density function. Since the marginal costs in (5.3) are random
variables, it becomes difficult to analytically compute the expected production cost.
71
A convenient way to calculate Nashequilibrium quantities, when considering fuel
price uncertainty in the expected cost function, is to use a Monte Carlo simulation based
technique.
The following procedure describes the algorithm to compute a MonteCarlo
estimate of the expected cost function for firm f (see Fig. 9):
Step 1) Sample the costs of fuel and calculate the marginal cost of each unit.
It is assumed that the fuel costs are continuous random variables with associated
probability density functions. The Monte Carlo simulation algorithm is employed to
sample the cost of each fuel type. When the values of all fuel types are generated, one
sample is obtained. Each generator is assumed to use only a specific fuel type. The
marginal cost for each generator is obtained by the multiplication of a sample value of the
fuel cost ($/MBTU) and its net plant heat rate (MBTU/MWh). Note that the net plant heat
rate is a given constant value. The maximum number of samples is denoted by R. The
parameter z is used to count the number of samples ( 1,2, ...,z R? ).
Step 2) Sort the values of marginal costs in an ascending order.
After randomly generating the values of marginal costs, the generators are
dispatched according to an ascending order of their marginal costs.
72
Step 3) Calculate the expected cost.
Since the values of marginal costs are known after step 1 and sorted in step 2, the
expected cost can be calculated by using (5.3) and (5.4) with s in increments of ? MWh.
A single expected cost curve is obtained after completing step 3.
Step 4) Set z = z + 1. Repeat step 1, 2, and 3 until z > R.
Step 5) Calculate the mean value of all expected costs.
R sampled expected cost curves were obtained in step 4. Next, the average cost of
production s (in increments of ? MWh) is computed. The result is an estimate of the
expected cost curve. Note that the values of costs after computing the average include
both the generator outages and the fuel price uncertainty. However, the problem of a
large number of slopes of the estimated expected cost curve as experienced in the
previous chapter arises.
Step 6) Apply the slope reduction algorithm.
The pseudocode below describes the slope reduction algorithm used to reduce
the number of slopes of the expected cost curve for firm f:
1:
0
to 2 FOR l N?
2: Set and ,
l
X l s? ?? ,1( )f
l l
C C X?
3:END FOR
4:
1 1
Set and 1, 2, 0, 0r K C X? ? ? ?
5:WHILE
0
K N?
6:
0
1 1
Min such and ( 2 )
i i i
j i K i N C C C ?
? ?
? ? ? ? ? ?
73
7:
1
,
2
1
j
l l
l K
f r j
l
l K
XC
m
X
? ?
? ?
?
?
?
8:
,
and
Max
f r j j
NewP X Temp X? ?
9:
0
to FOR l j N?
10:
l l
X X Temp? ?
11:
,l l f r
C C m Temp? ? ?
12: END FOR
13: 1K j? ?
14: 1r r? ?
15:END
where and s ?? are the increasing amount of capacity and the difference between slopes,
respectively. Both are constant values and must be set at the beginning. Let
0
N be the
maximum range in which we want to combine slopes such that
0 max
,
1
f
N
f j
j
N P
?
?
?
for each
firm f.
Step 7) Obtain an estimation of the expected cost function.
The slope reduction algorithm produces an estimation of the expected cost
function which is used to generate the estimated expected cost curve. A slope in the
estimated expected cost curve is associated with each hypothetical generator (r) which
has a maximum capacity (
,
Max
f r
NewP ) and marginal cost
,
( )
f r
m . Let
max
r
represent a value
of r when the slope reduction algorithm terminates. The values of
,f r
m and
,
Max
f r
NewP are
similar in nature to those
,f jc
and
max
,f jP described in chapter 4, but they take the generator
74
outages and the fuel price fluctuation into account. In order to clarify the presentation, the
values of marginal cost (
,f r
m ) and maximum capacity (
,
Max
f r
NewP ) will be represented as
,f jc and
max
,f jP where
max
( 1, 2, ... , )j r? .
Thus, the approximation of a total expected cost considering generator availability
and fuel price uncertainty for each firm f , which is denoted by ( )f
f
C s , can be
calculated as follows:
,
,
1
?
( ) ( )
f
N
f j
f f f f f j
j
E Cost s C s c g
?
? ?? ?
? ?
?
where ,f jc is the marginal cost obtained from the slope reduction algorithm in step 6.
5.3.2.5. Cournot Model
Each firm f solves the following programming model:
( ; ) ( )
f
f f f f f f
s
Max R s s C s? ? ?
?
? ?
1
max
max
subject to
 0 ( )
( ) for 1,.....,
, 0
f
N
fj f f
j
fj
fj fj
fj f
g s
g P j r
g s
?
?
?
?
? ?
? ?
?
(5.5)
1
where
1
( ; ) ( )
and .
f f f f f f
h
t
t
R s s s K s s
?
? ?
? ?
?
? ? ?
?
?
75
Figure 9. Flowchart of the algorithm used to calculate an approximation of the expected
cost function
Start
z = 1
Sample the costs of fuel and compute the
marginal cost of each unit
z > R?
Sort marginal costs in an ascending order
Calculate the expected cost function for the
given marginal costs
No
z = z+1
Yes
Calculate mean values of all expected costs
Reduce the number of slopes
Obtain an estimation of the expected cost
function
76
The Nashequilibrium quantities
f
s ( f ?1, 2, ? , n) that solve the set of
problems can be computed by combining the KKT first order optimality conditions of
system equation (5.5) of all firms. The KKT conditions of the optimization problem
above can be written as the following equations:
1
max
,
max
for 1,...,
( 2 ) 0 0
0 0
for 1,..., and 1,...,
0 0
0
f
f f f f
N
fj f f
j
f j
fj f fj
fj
fj
f n
K s s s
g s
f n j r
c g
P g
?
?
?
?
? ? ?
?
?
?
? ? ? ? ? ?
? ? ? ?
? ?
? ? ? ? ?
? ? ?
?
0
fj
? ?
(5.6)
These KKT conditions are considered as a linear complementary problem (LCP).
The Nashequilibrium quantity for each firm in MWh (
*
f
s ) is obtained by using available
software to solve those optimality conditions. As mentioned earlier, the actual demand of
the system or total demand can be calculated as
* *
1
n
f
f
S s
?
?
?
. The linear relationship in
demand (3.1) is used to determine the Cournot price,
*
p , which becomes
*
*
K S
p
?
?
? . (5.7)
77
5.4. EXPERIMENTAL RESULTS
A market that consists of three firms is considered in this section. For a numerical
explanation of results, the composition of each firm is given in Table 11 including the
capacity, net plant heat rate, and availability of each generator. The characteristics of the
unit types in table 11 are taken from the IEEE reliability test system (Grigg, 1996). Firms
1, 2 and 3 have 11, 7, and 9 generators, respectively. The last row of each firm in the
table corresponds to the assumption that the
th
f
N generator for firm f has infinite
capacity and is perfectly reliable due to other available sources in the markets.
Furthermore, the values of parameters and K? are assumed to be 15 and 3000
respectively. Each firm is assumed to work 12 hours per day (offpeak hours) and 30 days
per month. The annual percentage rate (APR) is assumed to be 7% for all firms. It is also
assumed that all firms and consumers agree in private through a central exchange on the
delivery of specified power quantities for 2 months and that the firms receive the
payment from customers at the end of each month. The discount coefficient, ? , in (5.5) is
computed in order to assess the time value of money in the model. Based on a given
APR, it can be shown that the value of the discount coefficient is 713.75.
The fuel sources of these generators are coal, oil, hydro, and nuclear. Since fuel
price uncertainty is assumed in this chapter, the fuel costs are continuous random
variables which have an associated probability density function. The probability density
function for each fuel type is determined by collecting the daily price ($/MBTU) of each
fuel for 3 months. All daily marginal costs in 3 months for each fuel type are then fit into
a distribution. The distribution is selected by choosing the minimum value of the Chi
78
square goodness of fit test. As a result, a lognormal distribution is selected for coaltype
generators, and a uniform distribution is selected for oiltype generators. The means and
standard deviations of each fuel type are displayed in Table 10. Generators which use the
same fuel type do not always have the same marginal cost, as each may have a different
heat rate. If generators, however, use the same fuel type and have the same net plant heat
rate, they are assumed to have the same marginal costs. The capture costs of CO
2
, which
will be added to the marginal costs, are assumed to be $19.77/MWh and $38.96/MWh for
gas and coal, respectively. Since the marginal cost of uranium and hydro are not notably
volatile within 3 months, they are assumed to be constant. The marginal costs of uranium
and hydro are assumed to be 0.0168 and 0.07 $/MWh respectively.
Table 10. The distribution of oil and coal price ($/MBTU)
Type Distribution
Gas Lognormal (? = 1.30, ? = 0.51)
Coal Lognormal (? = 0.34, ? = 0.15)
After the distribution of marginal costs for all generators are defined, the Monte
Carlosimulation algorithm flowcharted in Figure 9 is performed to sample the values of
fuel costs. A value of the expected cost function can then be computed in increments of
?. The value of R is set to 5,000 in this experiment, which means we perform the
simulation algorithm defined in the previous section with 5,000 sample sets. As a result,
the total of 5,000 cost curves is obtained after completing step 4 for each firm as shown
in Figures 10, 11, and 12.
79
Table 11. Market composition and generating unit data including unit availability
Firm Unit
Fuel Type
Capacity
(MW)
Marginal Cost
($/MWh)
Net Plant
Heat Rate
(Mbtu/MWh)
Availability (hours)
1/? 1/?
Fi
rm
1
1 Coal 350 9.5  1150 100
2 Coal 350 9.5  1150 100
3 Coal 155 9.72  960 40
4 Coal 155 9.72  960 40
5 Coal 76 11.9  1960 40
6 Coal 76 11.9  1960 40
7 Gas 48 10.23  1340 26
8 Gas 48 10.23  1340 26
9 Gas 78 11.63  1720 30
10 Gas 78 11.63  1720 30
11 Gas 149 12.87  1505 42
12  ?
 999  
Fi
rm
2
1 Hydro 50  0.07 1980 20
2 Hydro 50  0.07 1980 20
3 Coal 155 9.72  960 40
4 Coal 76 11.9  1960 40
5 Gas 48 10.23  1340 26
6 Gas 48 10.23  1340 26
7 Gas 78 11.63  1720 30
8  ?
 999  
Fi
rm
3
1 Uranium 400  0.017 1100 150
2 Hydro 50  0.07 1980 20
3 Hydro 50  0.07 1980 20
4 Coal 350 9.5  1150 100
5 Coal 76 11.9  1960 40
6 Gas 48 10.23  1340 26
7 Gas 78 11.63  1720 30
8 Gas 149 12.87  1505 42
9 Gas 149 12.87  1505 42
10  ?
 999  
Because of generator outages and the fluctuation of fuel prices, the estimated cost
curve has a large number of slopes upon completion of step 5. To reduce the number of
slopes, the slope reduction algorithm in step 6 is applied. The value of s? is assumed to
80
be 1 in this section. The value of ? for firms 1, 2, and 3 is selected to be 6, 3, and 5
respectively. The estimated cost curves before and after applying the slope reduction
algorithm for firms 1, 2, and 3 are illustrated in Figures 13, 14, and 15, respectively.
By analyzing Figures 13, 14, and 15, a new set of hypothetical generators can be
used to calculate the Nashequilibrium quantities without losing accuracy. Specifically,
the estimated cost curves before and after applying the slope reduction algorithm are
nearly indistinguishable. Based on the estimated production cost curves, the lists of the
new set of hypothetical generators which have a smaller number of marginal costs and
maximum capacities for firms 1, 2, and 3 are shown in Tables 12, 13, and 14,
respectively.
Figure 10. Expected cost curves of supplier 1 for given marginal costs
81
Figure 11. Expected cost curves of supplier 2 for given marginal costs
Figure 12. Expected cost curves of supplier 3 for given marginal costs
82
Figure 13. Estimate of expected cost curves before and after applying the slope reduction
algorithm for firm 1
Figure 14. Estimate of expected cost curves before and after applying the slope reduction
algorithm for firm 2
83
Figure 15. Estimate of expected cost curves before and after applying the slope reduction
algorithm for firm 3
Table 12. Data for the hypothetical generating units of firm 1
Unit
Capacity
(MW)
Marginal Cost
($/MWh)
1 1058 51.84
2 51 75.75
3 104 110.94
4 195 220.26
5 71 299.98
6 6 329.62
7 28 349.45
8 2 361.91
9 48 374.52
10 ? 999
84
Table 13. Data for the hypothetical generating units of firm 2
Unit
Capacity
(MW)
Marginal Cost
($/MWh)
1 100 0.38
2 174 49.89
3 57 59.91
4 19 68.98
5 77 105.96
6 26 143.03
7 2 146.76
8 2 161.70
9 48 191.08
10 ? 999
Table 14. Data for the hypothetical generating unit of firm 3
Unit
Capacity
(MW)
Marginal Cost
($/MWh)
1 100 0.06
2 400 6.23
3 100 53.32
4 201 63.93
5 125 74.03
6 24 111.71
7 50 185.42
8 201 252.04
9 2 269.02
10 71 295.83
11 26 319.52
12 2 328.68
13 48 344.79
14 ? 999
Note that each firm still owns the original set of generators listed in Table 11, but each
firm in reality pays the amount of money given in Tables 12, 13, and 14, when generator
outages and fuel price uncertainty are considered.
85
The Nashequilibrium quantities or firms? quantity bids are calculated according
to the KKT first order optimality conditions in (5.6) using the PATH solver. The market
price is calculated according to (5.7) assuming a linear relationship. The results of the
Nashequilibrium quantities and market prices are displayed in Table 15.
Table 15. Results of Nashequilibrium including generator outages
and fuel price uncertainty
Firm 1
(MWh)
Firm 2
(MWh)
Firm 3
(MWh)
Total
(MWh)
p*
($)
636.20 350 600 1586.2 94.25
5.4.1. Effect of uncertainty of generator availability and fuel price
In order to simplify the presentation, the model in each case is named as follows:
Model A: the deterministic model described in chapter 3
Model B: the stochastic model considering outages described in chapter 4
Model C: the stochastic model including outages and fuel price uncertainty described in
this chapter
For model A, the model profit is the amount firms believe they will make when
ignoring outages, while the adjusted expected profit is the amount that firms will make
because generator outages and fuel price uncertainty do occur.
The model profits of Model A are computed by substituting the Nashequilibrium
quantities and the power generated by each generator into the objective function of
Model A (3.5). Let
*A
f
s represent the Nashequilibrium quantities of Model A. The Model
A?s profit can be written as follows:
86
* * *
( ; ) ( )
A A A
f f f f f f
R s s C s? ?
?
? ?? ?
? ?
where
* * * * *
1
( ; ) ( )
A A A A A
f f f f f f
R s s s K s s
?
? ?
? ? ? .
Unlike the model profit, the adjusted expected profit of Model A is calculated
from the difference between the revenue function and the estimation of a total expected
cost function including outages and fuel price uncertainty,
*
( )f
f
C s . For Model A, the
expected profit can be calculated as follows:
? ?
* * *
[ ] ( ; ) ( )
A A A
f
f f f f f
E R s s C s? ?
?
? ?
where
*A
f
s stands for the Nashequilibrium quantities obtained from Model A.
For model B, the model expected profits are calculated by substituting the Nash
equilibrium quantities and the power generated by each generator into the objective
function of Model B (4.5). Let
*B
f
s represent the Nashequilibrium quantities of Model B.
Then Model B?s profit can be written as follows:
* * *
,1[ ] ( ; ) ( )
B B B
f
f f f f f
E R s s C s? ?
?
? ?
? ?
? ?
where
* * * * *
1
( ; ) ( )
B B B B B
f f f f f f
R s s s K s s
?
? ?
? ? ? .
The adjusted expected profit of Model B is computed from the difference between
the revenue function and an estimation of a total expected cost function including outages
and fuel price uncertainty,
*
( )f
f
C s . For Model B, the adjusted expected profit can be
calculated as follows:
87
? ?
* * *
[ ] ( ; ) ( )
B B B
f
f f f f f
E R s s C s? ?
?
? ?
where
*B
f
s stands for the Nashequilibrium quantities obtained from Model A.
The model expected profit and expected profit of Model C are the same because
the model considers uncertainty in generator availability and fuel prices. The model profit
and expected profit when considering generator failures can be computed as follows:
? ?
* * *
[ ] ( ; ) ( )
C C C
f
f f f f f
E R s s C s? ?
?
? ?
where
*C
f
s represents the Nashequilibrium quantities obtained from Model C.
Results of the model profit and expected profit in all three cases for each firm are
shown in Table 16.
Table 16. A comparison of firm expected profits when ignoring the uncertainty of fuel
prices and outages and when including the uncertainty of generator and fuel costs
Model A
Ignoring Uncertainty
($/Hour)
Model B
Including Outages
($/Hour)
Model C
Including Outages &
Fuel Cost Uncertainty
($/Hour)
Firm
Model
Profit
Adjusted
Expected
Profit
Model
Expected
Profit
Adjusted
Expected
Profit
Model
Expected
Profit
Expected
Profit
1 23,634.67 23,997.59 26,557.54 27,411.56 26,983.03 26,983.03
2 20,909.78 17,742.00 19,340.41 19,899.82 19,543.76 19,543.76
3 49,845.87 47,279.01 48,560.82 48,903.96 48,722.51 48,722.51
According to Table 16, the adjusted expected profits and the model profits of firm
1 for Model A are similar. For firms 2 and 3, however, the adjusted expected profit is
significantly less than the model profit, which means, for example, firm 3 anticipates
making $49,845.87/hour, but in reality it would make only $47,279.01/hour on average.
88
For Model B, the model expected profits of all firms are slightly less than the adjusted
expected profits.
When comparing the adjusted expected profit of Model A with Model C firms 1,
2, and 3 make more profits when using Model C than when using Model A. However, all
firms make fewer profits when the adjusted expected profit of Model B is compared with
Model C.
These results indicate that making decisions without considering the uncertainties
in generator availability and fuel prices could lead to bad decisionmaking and an
inaccurate planning process.
5.5. CONCLUSIONS
In this chapter, the stochastic Cournot model was extended to consider not only
the availability of generators but also fuel price uncertainty. Some generators used a type
of fuel whose price was unpredictable thereby affecting the cost function. The Monte
Carlo simulation based technique was used to calculate an estimate of the expected cost
function. Then the Nashequilibrium quantities, market prices, and firms? expected
profits were computed.
The model expected profits and expected profits were computed for models A, B,
and C to evaluate whether the availability of generators and the volatility of fuel prices
have an effect on the expected total profit for all firms. Results indicate that both
generator outages and fuel price uncertainty are vital factors in power markets and they
must be considered in both the power system planning and operation.
89
CHAPTER VI
TOLERANCE APPROACH TO SENSITIVITY ANALYSIS IN
THE STOCHASTIC COURNOT MODEL
Abstract  The Cournot model is used to describe the behavior of generating
companies in power markets. Two major uncertainty factors, generator outages and fuel
price uncertainty, are considered in the model. One way to compute the Nashequilibrium
quantities when considering both factors is the use of a Monte Carlo simulation based
technique. Due to its random processes, this simulation technique yields slightly different
results each time it is run. Accordingly, it is doubtful that companies should make a
decision based on those results. In addition, running the simulation several times in order
to certify the results may take considerable computational time. Therefore, a sensitivity
analysis is performed to determine which parameter is having a significant impact on the
Nashequilibrium quantities. Since the KKT conditions of the Cournot model represent a
linear complementarity problem (LCP), the theory of tolerance approach to sensitivity
analysis in LCP is applied. The maximum tolerance gives the maximum allowable
fluctuation of marginal costs and capacities without affecting the firms? strategic
planning and operation.
90
6.1. NOMENCLATURE
The notation used in this chapter is given below for reference.
n Number of firms
N Total number of generators
f
N Number of generators of firm f
max
fjP The capacity of the
th
j hypothetical unit of firm f considering outages and fuel
price uncertainty (MW)
fjc The marginal cost of the
th
j hypothetical unit of firm f including generator
outages and fuel price uncertainty ($/MWh)
*
f
s The Nashequilibrium quantities (MWh)
*
S The Nashequilibrium total bid (MWh)
*
p The Cournot price ($/MWh)
K Nominal demand
h Number of contracted hours
i Interest rate (%)
? Compoundamount factor
f
? Profit of firm f ($)
? Slope parameter for demand
91
6.2. INTRODUCTION
The stochastic Cournot model described in chapter 5 considers uncertainty in
generator availability and fuel price. The Monte Carlo simulation based technique was
used to estimate the expected cost function. A set of marginal costs ( ,f jc ) and the
maximum capacity (
max
,f jP ) for each generator j were obtained after performing the
simulation method. These results were used to compute the Nashequilibrium quantities.
However, the simulation technique yielded slightly different results each time it was run.
Due to the fluctuation in simulation output, it is doubtful that firms should make a
decision based on those results. Moreover, it is also difficult for a company to run the
simulation several times in order to certify those results due to the long computational
time.
Therefore, a sensitivity analysis is performed to determine which parameters have
the most significant impact on the optimal solutions. The company can then concentrate
on acquiring accurate data for those sensitive parameters. Sensitivity analysis will help a
company determine whether the optimal solution is sensitive to small changes in some of
the input data used in the simulation so that the company can use the results with
confidence.
The main goal of this chapter is to find the maximum tolerance on the crucial
parameters of the stochastic Cournot model. Since the stochastic Cournot model
considering generator outages and fuel price uncertainty in chapter 5 is a linear
complementarity problem, the theory of tolerance approach to sensitivity analysis in
linear complementarity problem is used. The tolerance approach is mainly applied to the
92
marginal costs and maximum capacities as they are believed to have significant impacts
on the Nashequilibrium quantities. The maximum tolerance gives the maximum
allowable fluctuation of marginal costs and capacities without affecting the firms?
strategic planning and operation. Moreover, it also specifies which generators have
significant impact on the Nashequilibrium quantities.
This chapter is organized as follows: The tolerance approach to sensitivity
analysis in LCP is developed in section 3. In section 4, numerical results and the analysis
of the maximum tolerance on each parameter are presented. The conclusions are outlined
in section 5.
6.3. MODEL DESCRIPTION
This section begins with an introduction to the concept of the linear
complementarity problem (LCP). The theory of tolerance approach to sensitivity analysis
in LCP is then developed to find the maximum tolerance such that the perturbed
problems have the same index set of nonzero elements as the original problems.
6.3.1. Linear Complementarity Problem (LCP)
The linear complementarity problem is to find the value of a vector z that satisfies
a set of constraints for a given m m? matrix M and compatible vector q. The general
form of an LCP can be written as (Stewart, 2008 and Thomas, 2002):
93
Given: ,
m m m?
? ?M R q R
Find:
m
?z R
Subject to 0? ?Mz q
0?z
and ( ) 0.
T
? ?z Mz q
Throughout this chapter, we shall refer to this problem as LCP(M, q).
6.3.2. Cournot Model
Since matrices M and q play an important role in developing the tolerance
approach to sensitivity analysis in LCP, the KKT conditions of the stochastic Cournot
model described in chapter 5 must be transformed into the matrix form. For the sake of
completeness, the stochastic Cournot model described in chapter 5 is shown below.
( ; ) ( )
f
f
f f f f f
s
Max R s s C s? ? ?
?
? ?
1
max
subject to
 0 ( )
( ) for 1,.....,
, 0
f
N
fj f f
j
fj
fj fj f
fj f
g s
g P j N
g s
?
?
?
?
? ?
? ?
?
1
where
1
( ; ) ( ) and .
h
f f f f f f t
t
R s s s K s s ? ?
?
? ?
?
? ? ? ?
?
94
The KKT first optimality conditions of this optimization problem are:
1
for 1,...,
( 2 ) 0 0 (6.1)
0 0
f
f f f f
N
fj f f
j
f n
K s s s
g s
?
?
?
?
?
?
?
? ? ? ? ? ?
? ? ? ?
?
,
max
(6.2)
for 1,..., and 1,...,
0 0 (6.3)
0 0.
f
f j
fj f fj
fj
fj fj
f n j N
c g
P g
? ? ?
?
? ?
? ? ? ? ?
? ? ? ? (6.4)
These KKT conditions (6.1.6.4) are represented in the matrix form as follows:
1 1 11 11
... ... ... ...
T
f f fj fj
s s g g? ? ? ?? ??
? ?
z
max max
1111... 0 ... 0 ... ...
T
fjfj
K K
c c P P
? ?
? ?
? ?
? ?
? ? ?
? ?
? ?
q
Due to the hefty size of matrixM , this matrix is separated into five parts for ease
of presentation. Each part is associated with a constraint in the KKT conditions. All of
the components of the matrix M are summarized below.
(Equation 5.1)
(5.1)
2
1 0 0 0 0
2
0 1 0
2
0 0 1 0 0
? ? ?
? ? ?
? ?
? ?
? ? ?
? ? ?
? ?
? ?
? ?
? ?
? ?
?
? ?
? ?
? ?
? ?
? ?
? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ?
M
n n
n N? n N?
95
(Equation 5.2)
(5.2)
1 0 0 0 0 1 1 0 0 0 0 0 0
0 1 0 0 0 1 1 0 0
0 0 1 0 0 0 0 0 0 1 1 0 0
?? ?
? ?
?
? ?
?
? ?
? ?
?
? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
M
(Equation 5.3)
(5.3)
0 0 1 0 0 0 0 1 0
1 0 0
0 1 0 0
0 1 0 0
0 0 1
0 0 0 0 1 0 0 0 1
?
? ?
? ?
? ?
? ??
? ?
?
? ?
? ?
?? ?
?
? ?
? ?
? ?
?
? ?
? ?
? ?
? ??
? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
? ? ? ?
M
(Equation 5.4)
(5.4)
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
?? ?
? ?
?
? ?
? ??
? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
M
Hence, the matrix M is equivalent to
(5.1)
(5.2)
(5.3)
(5.4)
? ?
? ?
? ?
?
? ?
? ?
? ?
? ?
M
M
M
M
M
.
96
6.3.3. Tolerance Approach to Sensitivity Analysis in the Stochastic Cournot model
The results obtained from the simulation will be used as an input to the LCP
model (6.16.4) in order to calculate the Nashequilibrium quantities. One of the
drawbacks of the simulation technique is that it yields slightly different results each time
it is run. Since those results will be used as an input to the LCP, it would be best to know
the effects of input data perturbation on the optimal solutions. Because M does not affect
firms? strategic planning and operation, the main focus of this section is to discover
whether marginal costs ( ,f jc ) and capacities (
max
,f jP ) in vector q have a major impact on
the Nashequilibrium quantities. The algorithm developed in (Ha and Narula, 1992) is
modified in order to calculate the range within which an entry value of vector q can vary
independently such that the perturbed problem has the same index set of nonzero
elements as the original problem.
Let
*
z be a solution of the linear complementarity problem, LCP(M, q). Assume
that
*
z is locally unique and a nondegenerate solution. Note that the size of matrix M is
m m? . According to the definition of a nodegenerate LCP,
either
? ?
* * *
0 or 0
i i
i
z x Mz q? ? ? ? .
Define the sets B and N as follows:
? ?
*
: 0
i
i z? ?B and
? ?
*
: 0
i
i x? ?N
where the cardinality of sets B and N are assumed to be u and v, respectively. Define
vector
B
z as the vector whose components are the components
i
z of vector z, for i?B. In
97
addition, define matrix
BN
M as the submatrix of M whose entries are
ij
m , for
and i j? ?B N. Other vectors and matrices are defined in the same way.
The problem LCP(M, q), using the index sets B and N, can be written as
,? ? ? ?0
B BB B BN N B
x M z M z q
,? ? ? ?0
N NB B NN N N
x M z M z q
, , .? ? ?0 0 0
T
B N
z z z x
By the definition of sets B and N, at the solution
*
z we know
* *
and .? ?0 0
B N
z z (6.5)
Thus, from (6.5)
* *
,? ? ?0
B BB B B
x M z q (6.6)
* *
.? ? ?0
N NB B N
x M z q (6.7)
The equation (6.6) and (6.7) can be rewritten as
* 1
,
?
??
B BB B
z M q (6.8)
* 1
.
?
?? ?
N NB BB B N
x M M q q (6.9)
Note that (6.8) and (6.9) are used later in this section when the tolerance approach is
implemented.
We introduce the perturbed linear complementarity problem in order to develop
the tolerance approach in LCP(M, q). As mentioned earlier, only vector q is disturbed.
Therefore, the perturbed linear complementarity problem can be written as follows:
? ? ?0Mz q ? ,
?0z , (6.10)
98
( )
T
? ?z Mz q ? ,
where ? is the parameter vector in
m
R .
6.3.3.1. A single parameter ( )
k
q k?B is perturbed.
This section begins by developing the tolerance approach to sensitivity analysis in
the LCP in which a single parameter in vector q is perturbed. As marginal costs and
maximum capacities have significant impacts on the firms? operations, it would be
beneficial for a company to know which parameters are more sensitive. To perform the
sensitivity analysis of a single parameter in vector q by the tolerance approach, we first
define
k
? (size 1u? ) to be the parameter vector which has a value
k
? in the
th
k position
( 0
k
? ? ) and the value of 0 in all other positions. For example, the parameter vector
k
? ,
when 3k ? , can be written as
3
3
1
0
0
0
0
u
?
?
? ?
? ?
? ?
? ?
?
? ?
? ?
? ?
? ?
? ?
?
? .
Using the index sets B and N, the perturbed problem (6.10) can be rewritten as
,
k
? ? ? ? ?0
B BB B BN N B
x M z M z q ?
,? ? ? ?0
N NB B NN N N
x M z M z q (6.11)
, , , .? ? ? ?0 0 0 0
T T
B N B B N N
z z z x z x
99
The parameter vector
k
? is allowable if problem (6.11) has a solution
?
z such
that
? ? ? ?
, , ,and ? ? ? ?0 0 0 0
B N B N
z z x x .
Thus, if
k
? is allowable, then (6.11) becomes
? ?
,
k
? ? ? ?0
B BB B B
x M z q ? (6.12)
? ?
,? ? ?0
N NB B N
x M z q (6.13)
?
.?0
B
z
We can solve for
?
B
z in (6.12) as we did in (6.8). It can be written as
?
1
( )
k
?
?? ? ?0
B BB B
z M q ? ,
or
1 1? ?
??
BB k BB B
M M q? . (6.14)
Substituting this value of
?
B
z in (6.13), equation (6.13) becomes
1
( ) ,
k
?
? ? ? ?0
NB BB B N
M M q q?
or
1 1
.
k
? ?
?? ?
NB BB NB BB B N
M M M M q q? (6.15)
Using the expression in (6.8) and (6.9), equation (6.14) and (6.15) can be
rewritten in terms of
* *
and
B N
z x as follows:
1 *
k
?
?
BB B
M z? , (6.16)
100
1 *
.
k
?
?
NB BB N
M M x? (6.17)
Equation (6.16) and (6.17) can be combined and written as linear inequality
system,
k
?A b? , (6.18)
where A and b are given by
1
1
?
?
? ?
?
? ?
? ?
BB
NB BB
M
A
M M
,
*
*
? ?
?
? ?
? ?
B
N
z
b
x
.
Note that b is the vector in
m
R whose components are
i
b, for ? ?1,2, ...,i m? . Matrix A
and vector b are outlined as follows:
11 12 1
21 22 2
1 2
u
u
v v vu
a a a
a a a
a a a
? ?
? ?
? ?
?
? ?
? ?
? ?
?
? ? ?
?
A ,
1
2
u v
b
b
b
?
? ?
? ?
? ?
?
? ?
? ?
? ?
?
b .
Define
k
UB as an upper bound of the maximum allowable range in
k
? and
k
LB
as a lower bound of the maximum allowable range in
k
? . In other words,
k k k
LB UB?? ?
where k?B.
The values of
k
UB and
k
LB are obtained by solving a linear inequality system
(6.18) for each value of k ( 1, 2, ..., )k u? and they can be determined by the following
equations:
1
: 0 ,
i
k ik
i m
ik
b
UB minimum a
a
? ?
? ?
? ?
? ?
? ?
(6.19)
101
1
: 0 .
i
k ik
i m
ik
b
LB maximum a
a
? ?
? ?
? ?
? ?
? ?
(6.20)
6.3.3.2. A single parameter ( )
k
q k?N is perturbed.
In this section, the main focus is to perform sensitivity analysis on a single
parameter
k
q in set N. Thus, we define
k
? to be the parameter vector which has a value
k
? in the
th
k position ( 0
k
? ? ) and the value of 0 in all other positions. The size of the
vector
k
? is 1v? . Note that the cardinality of set N is v.
Using the index sets B and N, the perturbed problem (6.10) is equivalent to the
following systems:
,? ? ? ?0
B BB B BN N B
x M z M z q
,? ? ? ? ?0
N NB B NN N N k
x M z M z q ? (6.21)
, , , .? ? ? ?0 0 0 0
T T
B N B B N N
z z z x z x
if
k
? is said to be allowable, then (6.21) becomes
? ?
,? ? ?0
B BB B B
x M z q (6.22)
? ?
,? ? ? ?0
N NB B N k
x M z q ? (6.23)
?
.?0
B
z
We can solve for
?
B
z in (6.12) as we did in (6.8). It can be written as
?
1?
??
B BB B
z M q . (6.24)
102
By substituting this value of
?
B
z in (6.23), equation (6.23) becomes
1
,
?
? ? ? ?0
NB BB B N k
M M q q ?
or
1
.
?
? ?
k NB BB B N
M M q q? (6.25)
Using the expression in (6.9), equation (6.25) can be rewritten as follows:
*
.??
k N
x? (6.26)
Since the values of all components in
*
N
x are always positive, a lower bound of
the maximum allowable range in
k
? (or
k
? ) can be instantly determined by the value of
*
k
x in equation (6.26), for .k?N According to (6.26), the upper bound of the maximum
allowable range in
k
? will be ?. In other words, the maximum allowable range for each
index k can be written as
*
k k
x ?? ?? (6.27)
where k?N .
6.3.3.3. Numerical Example
The main purpose of this section is to perturb one parameter at a time in either set
B or N. If the parameter is in set B, equations (6.19) and (6.20) are used to determine the
maximum allowable range of that parameter. Equation (6.27) is used, if the parameter is
in set N. As mentioned earlier, only marginal costs ( ,f jc ) and capacities (
max
,f jP ) in vector
q are perturbed.
103
The optimal solutions, obtained from solving the stochastic Cournot model using
data in Table 17, are used to perform the sensitivity analysis to the LCP and shown in
Table 18.
Table 17. Market composition and generating unit data
Firm Unit
Capacity
(MW)
Marginal Cost
($/MWh)
Fi
rm
1
1 812 17.06
2 195 52.99
3
79 60.97
4 76 77.61
5
44 215.80
6 24 227.08
7
20 232.94
8 133 242.57
9
173 264.04
10 24 328.35
11 20 345.67
12
? 400
Fi
rm
2
1 100 0.23
2
26 25.31
3 50 28.66
4
24 222.45
5 76 229.10
6
24 277.24
7 20 338.82
8
? 400
Fi
rm
3
1 176 0.80
2
324 25.87
3 76 48.52
4 194 214.90
5 400 242.69
6
? 400
104
The table includes values of the dual variables in the variable vector z. Due to a
large number of variables, only the optimal solutions in set B are shown. All other
variables, based on the definition of the LCP, have a value of 0. The values of the
parameters and K? are assumed to be 38.5 and 1972, respectively. The APR is assumed
to be 7%. The value of ? is calculated to be 713.75. It is assumed that the first six values
in the parameter vector q do not change.
The maximum allowable ranges of all parameters in vector q are shown in Tables
19 and 20.
Table 18. The solutions associated with set B
*
( )
B
z
Variable Value Variable Value
*
1
s
506.54 2,1
g
100
*
2
s
126 2,2
g
26
*
3
s
176 3,1
g
176
1
?
12178.81
2,1
? 19070.62
2
?
19233.57
2,2
? 1170.82
3
?
18306.62
3,1
? 17734.89
1,1
g
506.54
 
105
Table 19. The maximum allowable range of each marginal cost ( ,f jc )
Parameter Set B Current Value ? ?+ %? %?+ LB
k
UB
k
c
1,1
Yes 17.06 3.28 0.44 19.23 2.61 13.78 17.51
c
1,2
52.99 35.93 ? 67.80 ? 17.06 ?
c
1,3
60.97 43.91 ? 72.01 ? 17.06 ?
c
1,4
77.61 60.54 ? 78.01 ? 17.06 ?
c
1,5
215.80 198.74 ? 92.09 ? 17.06 ?
c
1,6
227.08 210.01 ? 92.49 ? 17.06 ?
c
1,7
232.94 215.88 ? 92.68 ? 17.06 ?
c
1,8
242.57 225.51 ? 92.97 ? 17.06 ?
c
1,9
264.04 246.97 ? 93.54 ? 17.06 ?
c
1,10
328.35 311.28 ? 94.80 ? 17.06 ?
c
1,11
345.67 328.61 ? 95.06 ? 17.06 ?
c
1,12
400.00 382.94 ? 95.73 ? 17.06 ?
c
2,1
Yes 0.23 "?" 26.72 "?" 11703.77 0.00 26.95
c
2,2
Yes 25.31 "?" 1.64 "?" 6.48 0.00 26.95
c
2,3
28.66 1.72 ? 5.99 ? 26.95 ?
c
2,4
222.45 195.50 ? 87.89 ? 26.95 ?
c
2,5
229.10 202.15 ? 88.24 ? 26.95 ?
c
2,6
277.24 250.29 ? 90.28 ? 26.95 ?
c
2,7
338.82 311.87 ? 92.05 ? 26.95 ?
c
2,8
400.00 373.05 ? 93.26 ? 26.95 ?
c
3,1
Yes 0.80 "?" 24.85 "?" 3102.00 0.00 25.65
c
3,2
25.87 0.22 ? 0.86 ? 25.65 ?
c
3,3
48.52 22.88 ? 47.14 ? 25.65 ?
c
3,4
214.90 189.25 ? 88.06 ? 25.65 ?
c
3,5
242.69 217.04 ? 89.43 ? 25.65 ?
c
3,6
400.00 374.35 ? 93.59 ? 25.65 ?
106
Table 20. The maximum allowable range of each capacity (
max
,f jP )
Parameter set B Current Value ? ?+ %? %?+ LB
k
UB
k
,
812.00 305.46 ? 37.62 ? 506.54 ?
,
195.00 195.00 ? 100.00 ? 0.00 ?
,
79.00 79.00 ? 100.00 ? 0.00 ?
,
76.00 76.00 ? 100.00 ? 0.00 ?
,
44.00 44.00 ? 100.00 ? 0.00 ?
,
24.00 24.00 ? 100.00 ? 0.00 ?
,
20.00 20.00 ? 100.00 ? 0.00 ?
,
133.00 133.00 ? 100.00 ? 0.00 ?
,
173.00 173.00 ? 100.00 ? 0.00 ?
,
24.00 24.00 ? 100.00 ? 0.00 ?
,
20.00 20.00 ? 100.00 ? 0.00 ?
,
30000.00 30000.00 ? 100.00 ? 0.00 ?
,
Yes 100.00 17.12 42.10 17.12 42.10 82.88 142.10
,
Yes 26.00 17.12 42.10 65.86 161.94 8.88 68.10
,
50.00 50.00 ? 100.00 ? 0.00 ?
,
24.00 24.00 ? 100.00 ? 0.00 ?
,
76.00 76.00 ? 100.00 ? 0.00 ?
,
24.00 24.00 ? 100.00 ? 0.00 ?
,
20.00 20.00 ? 100.00 ? 0.00 ?
,
30000.00 30000.00 ? 100.00 ? 0.00 ?
,
Yes 176.00 5.71 126.31 3.24 139.34 170.29 302.31
,
324.00 324.00 ? 100.00 ? 0.00 ?
,
76.00 76.00 ? 100.00 ? 0.00 ?
,
194.00 194.00 ? 100.00 ? 0.00 ?
,
400.00 400.00 ? 100.00 ? 0.00 ?
,
30000.00 30000.00 ? 100.00 ? 0.00 ?
107
6.3.4. Tolerance approach to sensitivity analysis when all values of q vary
simultaneously
The major drawback in running a simulation is the computational time. One of the
most important factors that affect the simulation time is the sample size. In this research,
we sample the prices of fuel and compute the expected cost curve based on those prices.
Since the results from the simulation will be used as input data to the LCP to compute the
Nashequilibrium quantities, the simulation results obtained from a large sample size are
preferred. However, a large sample size may a take long computational time because of
the process of calculating the expected cost curves. Furthermore, a linear complemetarity
problem (LCP) is equivalent to quadratic programming. The linear complementarity
problem belongs to a class of NPcomplete problems (Murty, 2008). There are several
methods for solving an LCP, such as iterative methods and pivoting methods, but solving
the LCP in polynomial time is not expected in these algorithms.
The tolerance approach to sensitivity analysis in the LCP is applied to determine
whether the perturbed problem has the same index set of nonzero elements as the original
problem. If both problems still have the same index set of nonzero elements, then the new
solutions can be calculated without directly solving the LCP.
6.3.4.1. Algorithm
Since the main interest of this section is to determine whether the perturbed vector
q has effects on an index set of the optimal solutions, we define
new
q to be the perturbed
vector and q to be the original vector.
108
Based on (6.10), W is defined as the perturbed amount vector where
?
new
W = q q. (6.28)
The elements of vector W
are
1 1 1
2 2 2
1
new
new
new
m m m
m
q q
q q
q q
?
?
?
?
? ?? ?
? ?
? ?
? ?
?
? ?
? ?
? ?
? ?
? ?
?
W .
Note that the size of vector W is 1m? . Thus, the perturbed problem (6.10) using the
index sets B and N can be written in the following equivalent forms:
,0? ? ? ? ?
B BB B BN N B B
x M z M z q W
,0? ? ? ? ?
N NB B NN N N N
x M z M z q W (6.29)
, , , .? ? ? ?0 0 0 0
T T
B N B B N N
z z z x z x
The vector W is said to be allowable if problem (6.11) has a solution z
?
such that
? ? ? ?
, , ,and ? ? ? ?0 0 0 0
B N B N
z z x x .
Thus, if ? is allowable, then (6.29) becomes
? ?
,0? ? ? ?
B BB B B B
x M z q W (6.30)
? ?
,0? ? ? ?
N NB B N N
x M z q W (6.31)
?
.?0
B
z
We can solve for
?
B
z in (6.30) which is
?
1
( ) 0
?
?? ? ?
B BB B B
z M q W ,
or
109
1 1? ?
??
BB B BB B
M M q? . (6.32)
Substituting this value of
?
B
z in (6.31), equation (6.31) can be written as
1
( ) ,0
?
? ? ? ? ?
NB BB B B N N
M M q W q W
or
1 1
.
? ?
? ?? ?
NB BB B N NB BB B N
M M W W M M q q (6.33)
Using the expressions in (6.8) and (6.9), equations (6.32) and (6.33) can be rewritten in
terms of
* *
and
B N
z x as follows:
1 *?
?
BB B B
M W z , (6.34)
1 *
.
?
? ?
NB BB B N N
M M W W x (6.35)
Equations (6.34) and (6.35) can be written as the following linear inequality system,
?CW b, (6.36)
where C and b are given by
1
1
0
?
?
? ?
?
? ?
?
? ?
BB
NB BB
M
C
IM M
,
*
*
? ?
?
? ?
? ?
B
N
z
b
x
,
where I is the identity matrix of size v v? .
To determine effects of
new
q on the optimal solutions, the following procedure is
applied:
110
START
Given M, q, and z
*
.
Sets B and N are defined as
? ?
*
: 0
i
i z? ?B and
? ?
*
: 0
i
i x? ?N where
? ?
* *
? ?x Mz q .
* 1
and
?
??
B BB B
z M q
* 1
.
?
?? ?
N NB BB B N
x M M q q
1
1
0
?
?
? ?
?
? ?
?
? ?
BB
NB BB
M
C
IM M
and
*
*
? ?
?
? ?
? ?
B
N
z
b
x
.
IF ?CW b THEN
The new optimal solutions can be calculated as
?
? ?
1 new?
??
new
B BB B
z M q , (6.37a)
?
?0
new
N
z . (6.37b)
ELSE
The index set of nonzero elements has changed and the LCP needs to be
resolved with the value
new
q .
END
END
6.3.4.2. Numerical Example
This numerical example shows how the algorithm described in this section is
applied. The main objective is to determine whether a small sample size can be used to
compute the Nashequilibrium quantities. For the numerical example, data in Table 17
111
are used. The values given in those tables correspond to an estimation of the expected
cost curve and are associated with the vector q. These values were obtained by using a
sample size equal to 10,000. It took roughly 2 weeks, 15 hours, and 3 days to obtain the
results in those three tables, respectively. Running a simulation this long is not practical
since in real operation the decisions must be made on a daily basis. However, sampling
with a small sample size yields volatile results each time the simulation is run. When
using a small sample size, the tolerance approach to sensitivity analysis developed in this
section is employed to determine if a smaller sample is practical.
The results obtained from simulation using a sample size equal to 10,000 are
compared with the results using a sample size equal to 100. Results obtained from the
simulation with sample size equal to 100 are displayed in Tables 21, 22, and 23. The
procedure to determine effects of
new
q on the optimal solutions described in section 3.4.1
is then employed. The Nashequilibrium quantities using
new
q associated with set B are
shown in Table 24. The process of running the simulation and determining the effect of
new
q is repeated 10 times.
The results in Table 24 indicate that the results obtained from the simulation with
sample size equal to 100 break the optimal condition 3 times out of 10.
112
Table 21. The list of hypothetical generating unit data associated with estimated expected
cost function for firm 1
Firm 1
Run Parameter Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10 Unit 11 Unit 12
1
c
1j
17.06 52.99 60.97 77.60 215.80 227.08 232.94 242.57 264.03 328.34 345.67 400.00
812 195 79 76 44 24 20 133 173 24 20 ?
2
c
1j
17.02 52.45 60.39 76.89 213.01 224.16 230.01 240.12 261.82 326.51 342.61 400.00
812 195 79 76 44 24 20 133 173 24 20 ?
3
c
1j
17.05 53.24 61.24 77.95 217.29 228.62 234.50 243.87 265.21 329.32 347.30 400.00
812 195 79 76 44 24 20 133 173 24 20 ?
4
c
1j
17.09 53.09 61.08 77.73 216.23 227.53 233.40 242.95 264.38 328.63 346.14 400.00
812 195 79 76 44 24 20 133 173 24 20 ?
5
c
1j
17.01 52.80 60.77 77.37 215.01 226.25 232.11 241.87 263.41 327.83 344.80 400.00
812 195 79 76 44 24 20 133 173 24 20 ?
6
c
1j
17.06 52.99 60.98 77.61 215.86 227.14 233.01 242.62 264.08 328.39 345.74 400.00
812 195 79 76 44 24 20 133 173 24 20 ?
7
c
1j
17.11 52.65 60.60 77.14 213.64 224.82 230.67 240.68 262.32 326.93 343.30 400.00
812 195 79 76 44 24 20 133 173 24 20 ?
8
c
1j
17.07 53.25 61.25 77.96 217.26 228.60 234.48 243.85 265.19 329.30 347.27 400.00
812 195 79 76 44 24 20 133 173 24 20 ?
9
c
1j
17.14 52.69 60.64 77.18 213.74 224.93 230.78 240.76 262.40 326.99 343.41 400.00
812 195 79 76 44 24 20 133 173 24 20 ?
10
c
1j
17.01 52.77 60.73 77.32 214.83 226.06 231.92 241.71 263.26 327.71 344.60 400.00
812 195 79 76 44 24 20 133 173 24 20 ?
113
Table 22. The list of hypothetical generating unit data associated with estimated expected
cost function for firm 2
Firm 2
Run Parameter Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8
1
c
2j
0.23 25.17 28.49 220.00 226.71 274.71 335.79 400.00
100 26 50 24 76 24 20 ?
2
c
2j
0.23 25.29 28.69 224.65 231.24 279.51 341.54 400.00
100 26 50 24 76 24 20 ?
3
c
2j
0.23 25.33 28.70 223.50 230.12 278.32 340.11 400.00
100 26 50 24 76 24 20 ?
4
c
2j
0.23 25.19 28.55 222.01 228.67 276.79 338.27 400.00
100 26 50 24 76 24 20 ?
5
c
2j
0.23 25.31 28.69 223.77 230.39 278.60 340.45 400.00
100 26 50 24 76 24 20 ?
6
c
2j
0.23 25.29 28.61 220.42 227.13 275.15 336.31 400.00
100 26 50 24 76 24 20 ?
7
c
2j
0.23 25.32 28.71 224.46 231.06 279.31 341.30 400.00
100 26 50 24 76 24 20 ?
8
c
2j
0.23 25.33 28.65 220.69 227.38 275.42 336.63 400.00
100 26 50 24 76 24 20 ?
9
c
2j
0.23 25.20 28.56 222.09 228.76 276.87 338.38 400.00
100 26 50 24 76 24 20 ?
10
c
2j
0.23 25.18 28.49 219.90 226.61 274.61 335.66 400.00
100 26 50 24 76 24 20 ?
114
Table 23. The list of hypothetical generating unit data associated with estimated expected
cost function for firm 3
Firm 3
Run Parameter Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6
1
c
3j
0.80 26.13 48.79 217.03 244.57 400.00
176 324 76 194 400 ?
2
c
3j
0.80 25.59 48.12 212.58 240.65 400.00
176 324 76 194 400 ?
3
c
3j
0.80 25.66 48.28 213.18 241.18 400.00
176 324 76 194 400 ?
4
c
3j
0.80 26.00 48.67 215.94 243.61 400.00
176 324 76 194 400 ?
5
c
3j
0.80 25.84 48.41 214.66 242.48 400.00
176 324 76 194 400 ?
6
c
3j
0.80 25.95 48.58 215.55 243.27 400.00
176 324 76 194 400 ?
7
c
3j
0.80 25.67 48.31 213.24 241.23 400.00
176 324 76 194 400 ?
8
c
3j
0.80 26.13 48.80 217.01 244.54 400.00
176 324 76 194 400 ?
9
c
3j
0.80 25.68 48.35 213.35 241.33 400.00
176 324 76 194 400 ?
10
c
3j
0.80 25.72 48.35 213.64 241.58 400.00
176 324 76 194 400 ?
115
Table 24. The new optimal solutions using
new
q
associated with set B
Parameter
Run
*
1
s
*
2
s
*
3
s
1
?
2
?
3
?
1,1
g
2,1
g
2,2
g
3,1
g
2,1
?
2,2
?
3,1
? CW ? b
1 505.49 126.00 176.00 12217.59 19252.96 18326.01 505.49 100.00 26.00 176.00 19090.62 1286.14 17754.04 Yes
2              No
3 506.71 126.00 176.00 12172.26 19230.29 18303.34 506.71 100.00 26.00 176.00 19067.25 1153.53 17732.54 Yes
4 506.07 126.00 176.00 12196.06 19242.20 18315.24 506.07 100.00 26.00 176.00 19079.76 1261.23 17743.02 Yes
5 507.51 126.00 176.00 12142.76 19215.54 18288.59 507.51 100.00 26.00 176.00 19052.59 1153.59 17719.00 Yes
6 506.67 126.00 176.00 12173.83 19231.08 18304.13 506.67 100.00 26.00 176.00 19068.22 1181.11 17732.93 Yes
7              No
8 506.46 126.00 176.00 12181.57 19234.95 18308.00 506.46 100.00 26.00 176.00 19071.91 1155.96 17735.71 Yes
9              No
10 507.55 126.00 176.00 12141.31 19214.82 18287.87 507.55 100.00 26.00 176.00 19052.44 1241.15 17716.65 Yes
116
6.4. CONCLUSIONS
In this chapter, the theory of tolerance approach to sensitivity analysis in LCP was
applied. The maximum tolerance within which the righthand side (vector q) of the
problem can vary independently and simultaneously such that the perturbed problems
have the same index set of nonzero elements as the original problems was established.
An algorithm was developed to find the maximum tolerance, in which the right
hand side of the problem is perturbed independently. As the major parameters within the
righthand side are marginal costs and maximum capacities, the maximum tolerance on
those two factors was evaluated. The maximum tolerance indicates the maximum range
of marginal costs and maximum capacities that can be perturbed without affecting the
firms? strategic planning and operation.
In addition, the algorithm to find the maximum allowable range when the right
hand sides of the problems vary simultaneously was developed. This algorithm also can
be used to determine whether the new input data within vector q affect the index set of
nonzero elements. If the perturbed problems still have the same index set of nonzero
elements, the new optimal solutions can be calculated without directly solving LCP.
117
CHAPTER VII
CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH
7.1. Conclusions
This research proposed a model and solution for evaluating the effects of uncertainty
in deregulated electricity markets. Two essential factors in power markets, generator
outages and fuel price uncertainty, were considered in this research. First, a multiperiod
deterministic Cournot model was developed to resemble the structure of longterm
deregulated electricity markets. The Cournot model then incorporated generator outages
by replacing the cost functions with the expected cost functions. The Cournot model then
becomes a stochastic model that considers the availability of generators. Next, both
generator outages and fuel price uncertainties were included in the Cournot model.
Finally, the tolerance approach to sensitivity analysis was implemented to determine the
sensitive parameters of the stochastic Cournot model when considering both uncertainty
factors.
In the deterministic model, uncertainty is disregarded, but the model takes the time
value of money into account. The model belongs to a class of quadratic programming
models. The KKT first order optimality conditions of the model were considered as a
linear complementarity problem (LCP). The Nashequilibrium quantities were computed
by combining the KKT first order optimality conditions.
118
To incorporate generator outages into the model, the expected production cost
function that includes the availability of generators was developed. The resulting
expected production cost function was a piecewise linear function. It was shown that the
set of all combinations of unit capacities determines where the expected production cost
function changes its slope. The number of slopes grows exponentially as the number of
generating units increases. This issue has a direct link to the computational complexity of
the problem. Hence, an algorithm to reduce the number of slopes without losing
computational accuracy was devised. The results showed that the proposed algorithm is
able to reduce the number of slopes effectively and thus simplifies the computations. The
algorithm produced a set of hypothetical generators with a smaller number of units when
taking generator outages into account. The results also showed that generator outages
have an important effect on firms? expected profits and that they should be considered in
any mediumterm or longterm planning process.
The consideration of stochastic fuel costs in the stochastic Cournot model
provided more accurate decisions to power producers as the fluctuation of fuel costs
significantly impacts a firm?s longterm operation. Each firm was assumed to operate a
set of generators which used different types of fuels whose marginal costs are subject to
uncertainty. Therefore, the marginal costs were considered random variables. The Monte
Carlo simulation based technique was employed to sample the cost of each fuel type. The
slope reduction algorithm was applied in order to aid the computational complexity as the
estimated production cost curve contains a large number of slopes. The Nashequilibrium
quantities were then calculated. The model expected profits and expected profits of all
three cases (models A, B, and C) were computed to show the effects of both factors. The
119
results showed that the availability of generators and the volatility of fuel prices have a
significant impact on firms? expected profits and that they should be considered in the
planning and operation of a power system.
The theory of tolerance approach to sensitivity analysis in LCP was applied to the
stochastic Cournot model. Specifically, a method was devised to find the maximum
tolerance within the righthand side of the problem that can vary independently or
simultaneously such that the perturbed problems have the same index set of nonzero
elements as the original problems. The maximum tolerance indicates the maximum range
of each parameter in the righthand side that can be perturbed without affecting firms?
strategic planning and operation.
The algorithm to find the maximum range when the righthand sides of the
problems vary simultaneously can be used to determine whether the new input data
affects the index set of nonzero elements. If the perturbed problems still have the same
index set of nonzero elements, the new optimal solutions can be calculated without
directly solving a linear complementarity problem. This approach is intended to lessen
computational complexity in a largescale linear complementarity problem. It is also
useful for market participants when they have a new set of input data or the input data in
the righthand side are perturbed.
7.2. Directions for future research
This research proposed a Cournot model to evaluate two major uncertainty factors
in power markets. However, there are many other factors which affect the price of
electricity and market participants, and among them are transmission constraints. The
120
capacity of transmission lines is restricted by technical constraints. As a result, power
market trading can be limited and controlled by transmission constraints. An extension
of this dissertation could be to investigate the effect of network configuration on market
prices and firms? profits as the constraints could limit the competition because of
congestion.
Another research opportunity is the study of the effects of demand uncertainty on
electricity prices. In general, the demand of electricity follows daily or seasonal cycles.
Demand also depends on the lifestyle of the consumers and weather conditions. The price
of electricity, however, does not follow the same pattern as demand. Thus, the effect of
demand uncertainty on market prices and the behavior of market participants are two
interesting topics worth investigating.
This research is one of the very first attempts to apply the tolerance approach to
sensitivity analysis to the stochastic Cournot model. Future research in developing the
theory of tolerance approach is still wide open. Since all significant parameters in the
proposed model are in the righthand side of the problem formation, the algorithm
developed in this research considered only the perturbation of the right hand side of the
equation. One major improvement opportunity to the algorithm is the perturbation of the
other parameters of the problem as models may have some vital parameters in the left
hand side.
121
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