A Study of Supply Function Equilibrium with
Applications to Electricity Markets
Mustafa Momen
Department of Electrical and Computer Engineering,
McGill University, Montreal
December, 2014
A thesis submitted to McGill University in partial fulfillment of the requirements
of the degree of Doctor of Philosophy
© Mustafa Momen, 2014
Abstract
The quest for economic efficiency has driven the restructuring of power systems from
vertically integrated monopolies to unbundled systems where the generating resources are owned
by multiple independent companies (gencos). This transition has resulted in oligopolistic
electricity markets where a few dominant gencos can exercise market power, that is, manipulate
the market price to a level higher than the perfectly competitive price. Hence, accurate
assessment of such market influence through modeling of strategic genco market behavior is of
paramount interest to various stakeholders such as genco owners, consumer welfare groups and
power system regulators.
As a contribution towards this goal, this thesis develops an equilibrium model where the
market solution is a one-shot pure strategy Nash Equilibrium where the gencos’ market offers are
supply functions which are continuous and differentiable in price but whose shape is not
otherwise pre-specified. While within the supply function equilibrium (SFE) paradigm, multiple
equilibrium candidates exist, this thesis shows that one is a focal equilibrium under which all
gencos earn the highest possible profits. If gencos, being rational, choose to game under this
focal SFE equilibrium, case studies suggest that the resulting genco profits will generally be
significantly higher than under Cournot and other forms of SFE.
i
Résumé
La poursuite de l'efficacité économique a entraîné la restructuration des réseaux électriques
de monopoles verticalement intégrés à des réseaux dégroupés où les ressources de production
appartiennent à plusieurs sociétés indépendantes (gencos). Cette transition a donné lieu à des
marchés de l'électricité qui sont oligopolistiques avec certaines entreprises de production
dominantes pouvant exercer un pouvoir de marché, à savoir, manipuler le prix du marché à un
niveau plus élevé que le prix parfaitement concurrentiel. Par conséquent, une évaluation précise
d'un tel pouvoir à travers la modélisation du comportement stratégique des gencos est d'un intérêt
primordial pour les différentes parties prenantes tels que les propriétaires des gencos, les groupes
de protection des consommateurs et les régulateurs des réseaux électriques.
Comme contribution à ce but, cette thèse présente un modèle d'équilibre où la solution de
marché est un équilibre de Nash avec une stratégie pure non-répété où les offres de marché des
gencos sont des fonctions d'approvisionnement continues et différentiables dans le prix, mais
dont la forme n’est pas autrement pré-spécifiée. Même si ce paradigme d’équilibre avec
fonctions d'alimentation (SFE) en général mène à nombreux candidats d'équilibre, cette thèse
démontre qu’il y a un seul équilibre focal sous lequel tous les gencos maximisent leurs bénéfices.
Si les gencos, étant rationnels, choisissent cette forme d’équilibre focal, des simulations
suggèrent que les bénéfices des gencos seront généralement bien plus élevés que sous l’équilibre
de Cournot ou autres formes de SFE.
ii
Acknowledgements
First, I want to express my gratitude to my supervisor Prof. Francisco Galiana for his
excellent advice and guidance during the tenure of my graduate studies. I will also remember his
continuous encouragement and motivation, which has been a never-ending source of inspiration
for me since the beginning of the research leading to this thesis.
I also thank my family members for their support during my years of study. Finally, I am
grateful to my colleagues with whom I spent many cherished hours on discussions about
academics and other topics of interest.
iii
Table of Contents
Abstract ............................................................................................................................................ i
Résumé............................................................................................................................................ ii
Acknowledgements ........................................................................................................................ iii
Table of Contents ........................................................................................................................... iv
List of Figures ............................................................................................................................... vii
List of Tables ................................................................................................................................. ix
List of Symbols ................................................................................................................................x
List of Acronyms ......................................................................................................................... xvi
Chapter 1 : Introduction ...................................................................................................................1
1.1 Deregulation in the Power Industry: Transition from Monopoly to Oligopoly .................... 2
1.2 Thesis Motivation.................................................................................................................. 3
1.3 Claims to Originality ............................................................................................................. 5
1.4 Thesis Organization............................................................................................................... 7
Chapter 2 : Electricity Market and Genco Gaming .........................................................................8
2.1 Operation of Electricity Markets ........................................................................................... 9
2.1.1 Market Price ................................................................................................................. 10
2.2 Strategic Genco Offers ........................................................................................................ 11
2.3 Cooperative and Non-Cooperative Gaming ........................................................................ 12
2.4 Nash Equilibrium under Non-Cooperative Gaming ........................................................... 13
2.5 Oligopolistic Gaming Models in Electricity Markets ......................................................... 14
2.5.1 Bertrand ........................................................................................................................ 14
2.5.2 Cournot ......................................................................................................................... 15
2.5.3 Supply Function Equilibrium ....................................................................................... 15
2.5.3.1 SFE with Symmetric Gencos ................................................................................. 17
2.5.3.2 SFE with Asymmetric Gencos and Affine Supply Functions ............................... 19
2.5.3.3 SFE with Asymmetric Gencos and Non-Affine Supply Functions ....................... 21
2.6 Comparing the Gaming Models .......................................................................................... 22
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2.7 Contrasting Existing Literature with the Thesis's Claims to Originality ............................ 24
Chapter 3 : Equilibrium Model without Pre-Specified Supply Function Shapes ..........................31
3.1 Assumptions ........................................................................................................................ 31
3.1.1 Predicted Day-Ahead Hourly Demand ......................................................................... 32
3.1.2 Unit Commitment and Temporal Constraints .............................................................. 34
3.1.3 Leading Genco .............................................................................................................. 34
3.1.4 Gaming Strategies......................................................................................................... 35
3.1.5 Market Operation and Shape of Supply Functions ....................................................... 36
3.1.6 Transmission Network .................................................................................................. 38
3.2 At Low Demand, the Leading Genco Monopolizes the Market ......................................... 38
3.2.1 Examples of Low Demand Monopolistic Market ........................................................ 41
3.3 Oligopolistic Markets .......................................................................................................... 44
3.3.1 Demand Sharing by the Leading Genco ....................................................................... 44
3.3.2 Demand Sharing under SFE according to the Klemperer-Meyer Differential Equations
............................................................................................................................................... 47
3.3.3 Supply Function Continuity and Numerical Singularity Resolution when Solving KM
............................................................................................................................................... 50
3.3.4 Initial Condition of Numerical Integration under the Control of the Leading Genco .. 53
3.3.5 Upper and Lower Bounds on Initial Condition S 0 ...................................................... 54
3.3.6 Market Equilibrium at the Tangent Condition ............................................................. 59
3.3.7 Tangent Condition, S0  S0t , Yields Maximum Profits for all Gencos Compared to any
other Feasible S0 .................................................................................................................. 61
3.3.8 Right-Bending Supply Function Equilibrium............................................................... 67
3.3.9 Supply Functions for Prices  above t ...................................................................... 71
3.3.10 Supply Function Equilibrium Path ............................................................................. 74
3.3.11 Effect of Demand Prediction Uncertainty on SFE ..................................................... 77
3.3.12 Procedure to Compute Supply Functions Numerically .............................................. 81
Chapter 4 : Case Studies of SFE in Electricity Markets ...............................................................84
4.1 Comparing Market Behaviour under SFE, Cournot and Perfect Competition ................... 84
4.2 Comparative Market Behaviour under SFE with and without Carbon Tax ...................... 103
4.3 Supply Functions for Large Demands ............................................................................... 109
v
4.4 Comparing Markets under SFE, Cournot and Perfect Competition for Uncertain Demand
................................................................................................................................................. 115
Chapter 5 : SFE with Genco Capacity Constraints .....................................................................117
5.1 Modified KM Equations in CSFE ..................................................................................... 117
5.2 First Order Nash Equilibrium Optimality near Price where one Genco Begins to Offer its
Maximum Capacity ................................................................................................................. 119
5.3 Release Condition for Constrained Genco ........................................................................ 121
5.4 Change of Leading Genco under CSFE ............................................................................ 123
5.5 Examples ........................................................................................................................... 124
Chapter 6 : Conclusions ..............................................................................................................130
6.1 Extensions ......................................................................................................................... 133
References ....................................................................................................................................135
Appendix A : Cournot Gaming ....................................................................................................143
Appendix B : Characteristics of the Genco Supply Functions ....................................................144
Appendix C : Non-Fulfillment of Nash Equilibrium Optimality Condition if Genco Supply
Functions Continue to Follow KM above the Market Clearing Price .........................................155
Appendix D : Nash Equilibrium Condition for Market Clearing at Tangent Condition with
Modified Supply Functions..........................................................................................................157
Appendix E : KM Equations with Slope of the Genco Supply Functions Decoupled for a System
with more than 2 Gencos .............................................................................................................165
Appendix F : Calculating the Slope of Genco Supply Functions at the Entry Price of a NonLeading Genco in an Oligopoly with more than 2 Gencos ..........................................................168
Appendix G : Analysis of CSFE for Triopolies and above .........................................................173
Appendix H : Condition at which a Supply Function Releases from its Capacity Limit ............178
vi
List of Figures
Figure 3.1: Demand curve as a function of price .......................................................................... 33
Figure 3.2: Market solutions for d0  65 MW and d0  95 MW .................................................. 41
Figure 3.3: Cournot equilibrium path 65  d0  95 MW and supply functions for d0  65 MW
and d0  95 MW ............................................................................................................................ 43
Figure 3.4: A monopolizing leading genco supply function for d0  150 MW............................ 45
Figure 3.5: Genco 1 monopolizing generation up to S0  30 MW and thereafter sharing with
genco 2 .......................................................................................................................................... 46
Figure 3.6: Example of a duopoly switching to a triopoly at   35 $/MWh................................ 52
Figure 3.7: Genco supply functions for S0  35 MW under SFE ................................................ 57
Figure 3.8: Demand curve and aggregate supply functions for varying S 0 ................................. 58
Figure 3.9: Variation in price due to increase in S 0 near tangent condition ................................. 63
Figure 3.10: Variation of genco profits with S 0 .......................................................................... 67
Figure 3.11: Profits of gaming gencos around tangent condition equilibrium ............................. 72
Figure 3.12: Modified genco supply functions ............................................................................. 73
Figure 3.13: Gaming genco profits with original (dotted) and modified (solid) supply functions74
Figure 3.14: Supply function and Cournot equilibrium paths of genco 1 .................................... 75
Figure 3.15: Supply function and Cournot equilibrium paths of genco 2 .................................... 76
Figure 3.16: Profit paths of gencos 1 and 2 under SF and Cournot equilibria ............................. 77
Figure 3.17: Plot of d 0 versus S0t ............................................................................................... 83
Figure 4.1: SFE market solution for d0  100 MW ...................................................................... 85
Figure 4.2: RBSFE market solution for d0  100 MW ................................................................. 87
Figure 4.3: Market solution under Cournot for d0  100 MW ..................................................... 88
Figure 4.4: Market solution for d0  100 MW assuming perfect competition ............................. 88
Figure 4.5: SFE market solution for d0  250 MW ...................................................................... 91
Figure 4.6: RBSFE market solution for d0  250 MW ................................................................. 91
Figure 4.7: Market solution with Cournot offers for d0  250 MW ............................................. 92
Figure 4.8: Market solution under perfect competition for d0  250 MW ................................... 92
Figure 4.9: Equilibrium paths of genco 1 in the various markets ................................................. 95
Figure 4.10: Equilibrium paths of genco 2 in the various markets ............................................... 95
vii
Figure 4.11: Equilibrium paths of genco 3 in the various markets ............................................... 96
Figure 4.12: Equilibrium paths of genco 4 in the various markets ............................................... 96
Figure 4.13: Equilibrium paths of genco 5 in the various markets ............................................... 97
Figure 4.14: Variation of genco 1 profit with demand ................................................................. 98
Figure 4.15: Variation of genco 2 profit with demand ................................................................. 99
Figure 4.16: Variation of genco 3 profit with demand ................................................................. 99
Figure 4.17: Variation of genco 4 profit with demand ............................................................... 100
Figure 4.18: Variation of genco 5 profit with demand ............................................................... 100
Figure 4.19: Variation with demand of the aggregate difference between genco profit loss and
gain when switching from SFE to Cournot................................................................................. 102
Figure 4.20: Variation of demand profit under SFE and Cournot .............................................. 103
Figure 4.21: Market solution with no carbon tax ........................................................................ 105
Figure 4.22: Market solution with carbon tax of 10 $/t .............................................................. 106
Figure 4.23: Market solution with carbon tax of 20 $/t .............................................................. 107
Figure 4.24: Market solution for d0  2000 MW ....................................................................... 111
Figure 4.25: Genco SFs till   100 for market solution in Figure 4.24 ................................... 112
Figure 4.26: Genco supply functions following Green's equation.............................................. 112
Figure 4.27: Market solution of the 5-genco system for d0  2500 MW ................................... 113
Figure 4.28: Magnification of Figure 4.27 for low price levels .................................................. 114
Figure 5.1: SFE with a single active genco capacity constraint ................................................. 125
Figure 5.2: SFE with multiple active genco capacity constraints for d0  300 MW .................. 127
Figure 5.3: SFE with multiple genco capacity constraints and leadership change for d0  350
MW ............................................................................................................................................. 129
viii
List of Tables
Table 3.1: Genco participation factors for varying demand uncertainty ...................................... 80
Table 4.1: Cost parameters ........................................................................................................... 84
Table 4.2: Market outputs of gencos for d0  100 MW................................................................ 89
Table 4.3: Genco profits for d0  100 MW .................................................................................. 89
Table 4.4: Genco outputs for d0  250 MW ................................................................................. 93
Table 4.5: Genco profits for d0  250 MW .................................................................................. 93
Table 4.6: Demand profit for d0  250 MW ................................................................................. 94
Table 4.7: Cost and emission parameters for 5-genco system .................................................... 104
Table 4.8: Cost parameters with carbon tax................................................................................ 105
Table 4.9: Genco outputs and profits with various carbon tax levels ......................................... 107
Table 4.10: Pollution and tax revenue with various carbon tax levels ....................................... 108
Table 4.11: Genco leadership hierarchy with and without carbon tax ....................................... 109
Table 4.12: Demand profit for various tax levels ....................................................................... 109
Table 4.13: Market equilibrium over a wide range of demands ................................................. 110
Table 4.14: Expected value and standard deviation of genco profits due to random changes in d 0
..................................................................................................................................................... 116
Table 5.1: Cost and capacity parameters for 5-genco system ..................................................... 124
Table 5.2: Genco outputs and profits for SFE with a single genco capacity constraint for
d0  260 MW .............................................................................................................................. 126
Table 5.3: Genco outputs and profits for SFE with multiple genco capacity constraints for
d0  300 MW .............................................................................................................................. 128
Table 5.4: Genco outputs and profits for the SFE with multiple genco capacity constraints ..... 129
ix
List of Symbols

Price of electricity in $/MWh
d
Small change in electricity price in $/MWh
d ( )
Electricity demand in MW when the price is 
d0
Component of demand, in MW, independent of the electricity price

Parameter measuring demand elasticity in $/MW2h
i, j, k
Genco indices
gi
Output of genco i in MW
dgi
Small change in output of genco i in MW
Ci ( gi )
True cost of electricity generation by genco i in $/h
dCi ( gi )
Small change in $/h of the true cost of electricity generation by genco i
ICi ( gi )
True incremental (or marginal) cost of electricity generation by genco i in
$/MWh
ai
True incremental cost of genco i at zero output in $/MWh
bi
Slope of the true incremental cost of genco i in $/MW2h
x
gimax
Maximum output capacity of genco i in MW
Si (  )
Supply function of genco i defining the quantity, in MW, that genco i is
willing to produce at price 
dSi   
Small change in MW in the supply function of genco i at the price 
Si (  )
First derivative of the supply function of genco i with respect to price 
c
Market clearing price under Cournot equilibrium in $/MWh
d0cmax
Maximum value of d 0 , in MW, for which the electricity market remains a
monopoly under general supply function equilibrium
S0
Value in MW of S1 (a2 ) , that is, the supply function of genco 1 at the
market entry price of genco 2, that is, at   a2
dS0
Small change, in MW, in S 0
S0t
Value, in MW, of S 0 when the market clears by satisfying the tangent
condition
S0e
Arbitrary value of S 0 in MW
e
Market clearing price in $/MWh corresponding to S0  S0e
t
Market clearing price in $/MWh at the tangent condition
xi
hi , low
The high and low prices, in $/MWh, when, under supply function
equilibrium, there are two market clearing prices
S0r
Value in MW of S 0 which corresponds to market clearing under rightbending supply function equilibrium
r
Market clearing price in $/MWh under right-bending supply function
equilibrium
pri ( )
Profit in $/h of genco i at price 
dpri ( )
Small change in the profit of genco i at price  in $/h
Si   , S0 
Supply function of genco i at price  corresponding to S 0
Si  (  , S 0 )
First derivative of Si   , S0  with respect to price 
Si   , S0 
Second derivative of Si   , S0  with respect to price 
Si   , S0 
S0
Partial derivative of the supply function Si   , S0  with respect to S 0
S1* ( )
Supply function of genco 1 under perfect competition
d 0rmin
Minimum value of the demand parameter d 0 in MW for which the market
under right-bending supply function equilibrium clears through the
participation of more than one genco
d
Deviation of the demand from its predicted value in MW
xii
d 0
Deviation of d 0 from its predicted value in MW
gi
Change in output of genco i in MW from its set-point
f
Deviation, in Hz, of the system frequency from its nominal value
B
Sensitivity of the system frequency to an imbalance between generation
and demand
i
Participation factor of genco i
d̂ 0
Predicted d 0 in MW
d 0min
Minimum value of d̂ 0 in MW
d0max
Maximum value of d̂ 0 in MW
S0tmin
Value of S0 in MW which clears the market at the tangent condition for a
demand with d0  d0min
S0tmax
Value of S0 in MW which clears the market at the tangent condition for a
demand with d0  d0max
tmin
Market clearing price at the tangent condition in $/MWh for a demand
with d0  d0min MW
tmax
Market clearing price at the tangent condition in $/MWh for a demand
with d0  d0max MW
xiii
t
Difference in $/MWh between the clearing prices at tangent condition for
a demand with d0  d0min MW and a demand with d0  d0max MW
Si
Difference in MW between the market output of genco i for a demand
with d0  d0min MW and the market output of the same genco for a demand
with d0  d0max MW
ng
Number of gencos in the market
ri   , S0 
The quantity
Si (  , S 0 )
, that is, at price  , the ratio of the
  ai  bi Si ( , S0 )
supply function of genco i to the difference between the price and the true
incremental cost corresponding to the supply function
i
CO2 emission rate of genco i in tons/MWh
tc
Carbon tax rate in $/ton
big
Slope of Green’s supply function of genco i in $/MW2h
S0tm
Maximum value of S0t in MW to which S0t converges for large demands
g ix
Output of genco i in MW for the xth sample of a randomized d 0
g i0
Output set-point in MW of genco i corresponding to the predicted value of
d0
xiv
d x
Deviation in MW of the demand from its predicted value due to a random
change in d 0 represented by the xth sample of a randomized d 0
k
Price in $/MWh at which the supply function of genco k reaches the
genco’s capacity limit
kr
Price in $/MWh at which the supply function of genco k is released from
the genco’s capacity limit
xv
List of Acronyms
Genco
Generating Company
NE
Nash Equilibrium
SFE
Supply Function Equilibrium
ISO
Independent System Operator
HHI
Herfindahl-Hirschman Index
FERC
Federal Energy Regulatory Commission
Disco
Distribution Company
IC
Incremental Cost
RBSFE
Right-Bending Supply Function Equilibrium
AGC
Automatic Generation Control
SF
Supply Function
KM
Klemperer-Meyer
CSFE
Constrained Supply Function Equilibrium
RC
Release Condition
LHS
Left Hand-Side
xvi
RHS
Right Hand-Side
UK
United Kingdom
xvii
Chapter 1: Introduction
The electricity industry has been operating since the early 1880s [1]. During its early stage
private companies managed all facets of the industry without any regulation governing their
activities. At that time, in areas with large demand, the electricity industry was characterized by
intense competition among a large number of companies, as well as by inefficiency due to
companies building their own transmission lines [1]. Due to a continuously increasing service
area and customer base, from the beginning of the 20th century the industry shifted towards a
regulated and more centralized mode of operation. By the end of the Second World War, the
electricity industry was widely regarded as a natural monopoly [2]. Hence, a power system
typically operated either as a nationwide public entity or as a private regional monopoly under
the supervision of a regulatory body.
However, this traditional monopolistic operation began to come under challenge in the late
1970s [3]. Technological advances in turbine manufacturing led to efficient small-to-medium
sized gas propelled turbines becoming a viable source of electricity generation [4]. Additionally,
innovations in transmission enabling efficient power delivery over long distances meant that
generators no longer had to confine themselves to locations relatively close to their customer
base [1]. These factors, along with the quest to improve the electricity sector efficiency, a goal
thought to be largely incompatible with the prevalent monopolistic setting of the industry [5], led
to the gradual abandonment of the notion that the electricity industry is a natural monopoly.
1
Hence, in 1978 the USA passed a law requiring utilities to buy power from small generators
[6]. Then in 1982, Chile introduced limited competition in its power industry through a
wholesale market where large electricity consumers were allowed to participate (although
competition in one of the two regional power systems was largely stifled by the dominance of a
single supplier) [7]. Thereafter, in 1990, the United Kingdom (UK), starting with the England
and Wales power system, deregulated its electricity sector by privatizing electricity generation
and introducing a pool market for electricity trading [7]. Following the pioneering example of
the UK, many countries around the world began to deregulate their electricity sector.
1.1 Deregulation in the Power Industry: Transition from Monopoly to Oligopoly
Deregulation in the electricity sector was mainly motivated by the desire to improve the
financial performance of electric utilities by introducing competition. In contrast to an
unregulated monopoly where a utility can charge any price of its choice (hence the need for
regulatory price ceilings), the price in a competitive electricity market is determined by the
degree of competition.
In a market with a large number of generating companies (gencos), no genco can influence
the market price, that is, all gencos are price takers. In such a so-called perfect market, gencos
offer to sell power at their true costs of production [8] and the market clears (generation and
demand balance) at the marginal cost of generation, that is, at a price equal to the cost of
producing the last and most expensive unit of electric energy required to clear the market. The
alternative market type is that which exhibits imperfect competition where a few gencos
2
dominate the market. In such a market, known as an oligopoly, the dominant gencos possess
market power, that is, they exhibit the ability to influence the market outcome so as to increase
their profits [9].
Since the objective of an electricity market is to achieve efficiency through competition, the
aim of market design is to come as close as possible to a perfect market [1]. However, electricity
is characterized by a number of distinguishing features that make this goal difficult to achieve.
From the consumer perspective, electricity does not have a proper short-term substitute that can
compete for a consumer's share of the electricity market. Also, variation in the demand for
electricity is mostly due to external factors (for example, weather and holiday patterns) rather
than strategic decisions of the consumers who generally do not influence the real-time electricity
market price but purchase the required power from a load serving entity at a contracted price.
From the supplier perspective, an electricity market also stands out from a typical commodity
market in that it presents a significant entry barrier to any new genco. This is because generating
units are capital intensive and any new genco entering the market risks a large investment whose
full return can only be expected after a significant period of time. Hence, with prospective
gencos discouraged by the entry barrier, electricity markets tend to remain oligopolistic [10].
1.2 Thesis Motivation
In oligopolistic electricity markets, gencos exercise market power by gaming, that is, by
strategically setting their market offers to be different from their true production costs. However
in setting its strategy, a gaming genco must also take into account the fact that other gencos can
3
and probably also game. Otherwise, it might seem that the gaming strategy of a genco should be
aimed at driving the market price to as high a value as possible. However, such a simplistic
approach would backfire if the competing gencos chose to adopt a strategy with comparatively
less aggressive offers. In such a case, owing to excessive greediness, a gaming genco would
receive a small share of market generation, or worse, kept completely out of the market. This
would severely decrease the gaming genco profit from what it would have earned had it
tempered its greediness and gamed with a lesser degree of aggressiveness. Therefore, it is
imperative for a genco to adopt a more rational gaming approach in which the opportunity to
increase its profit by gaming is considered alongside the associated risk of loss of profit due to
the gaming behaviour of the competing gencos.
A widely used prism in market analysis that captures the abovementioned balance between
greed and risk factors is the concept of Nash equilibrium (NE) [11]. Two commonly used
methods employing NE to model gaming in the electricity market are Cournot equilibrium, under
which gencos game with the quantity they offer to produce, and the more general supply function
equilibrium (SFE) under which gencos game with their supply function offers relating generation
output to price. Modeling the Cournot equilibrium is mathematically more tractable [12],
however, as argued in this thesis, SFE is more reflective of the realities of an electricity market.
One major drawback of the SFE model is that it generally results in multiple equilibria,
making the model of little predictive value in anticipating the market behaviour of gencos. This
thesis therefore develops a SFE model that, in contrast to most existing models, does not impose
any restriction on the shape of genco supply functions (other than continuity and differentiability
assumptions), and systematically identifies a unique market equilibrium under which the profits
of all gencos are higher compared to other supply function equilibria. Hence, it is argued that,
4
being rational entities, gencos would prefer to game according to this SFE. The SFE model
developed in this thesis contains a number of significant innovations summarized next.
1.3 Claims to Originality
The original contributions to knowledge of this thesis are the following:
Claim 1: Out of the multiple SFE candidates for a predicted demand valid for a single pricing
period, a focal SFE candidate is identified under the following assumptions (i) gencos are
asymmetric (non-identical) with respect to their true costs of production, generation limits and
offered supply functions (ii) genco supply functions are not necessarily affine (iii) genco
capacity constraints are non-binding and (iv) the market price cap is non-binding. This SFE
candidate is focal in the sense that it is unique and leads to higher profits for all gencos compared
to other SFE candidates. The focal SFE candidate is based on the numerical integration of the
differential equations governing the first order genco profit optimality conditions under the Nash
equilibrium.
Claim 2: It is shown that while the supply functions obtained by numerically integrating the
first order profit optimality conditions do not satisfy the second order NE conditions, the
resulting SFE candidate becomes a true equilibrium (that is, satisfies the global NE condition)
when the genco supply functions are set to their market clearing outputs for all prices above the
market clearing price. The generality of this equilibrium is established through analytic proof and
numerical simulations.
5
Claim 3: The traditional power system load-following method in power systems operation is
applied to adjust the SFE found in Claim 2 to account for errors in demand prediction. This
approach has the advantage of handling demand prediction errors in an ex-post manner (that is,
at the time when the prediction error materializes) avoiding the ex-ante method of dealing with
demand uncertainty requiring that genco supply functions be valid over the entire range of
demand uncertainty.
Claim 4: A variation of the focal SFE in Claim 2 is found by constraining gencos to offer
supply functions that are non-decreasing in price, also known as right-bending supply functions.
However, it is shown that under this right-bending supply function equilibrium (RBSFE) all
gencos lose profit when compared to the focal SFE that allows supply functions that may be
increasing or decreasing in price, that is, supply functions that may be right or left-bending.
Furthermore, simulations consistently show that under RBSFE all non-leading gencos lose profit
compared to the Cournot equilibrium.
Claim 5: While preserving the features of Claims 1, 2 and 3, the focal SFE model is extended
to cases in which the capacity limits of one or more gencos actively constrain the supply
functions.
Claim 6: The simulations on sample power systems comparing SFE with RBSFE and
Cournot also contribute to knowledge in this field, suggesting that SFE would have a significant
impact on genco profits and price, usually to the detriment of the consumers.
Claim 7: Applications of the SFE model to power systems subject to a carbon tax show that
gencos with lower CO2 emissions increase their market dominance relative to those with higher
6
emissions and that cleaner gencos can exploit this increased dominance by gaming to increase
their relative profits.
These claims to originality are contrasted with the existing literature in section 2.7.
1.4 Thesis Organization
This thesis contains five more chapters. Chapter 2 describes the operation of electricity
markets and gaming models used to study genco behaviour. Chapter 3 describes the SFE model
proposed in this thesis. Chapter 4 shows some applications of the SFE model to power systems.
Chapter 5 extends the SFE model to include cases where the capacity limits of gencos actively
constrain their supply functions, and Chapter 6 completes the thesis with concluding remarks and
some suggestions for future research.
7
Chapter 2: Electricity Market and Genco Gaming
In the context of a monopoly, the system operator has control over the operation of all system
resources. In such a system, power output set-points are allocated on an hourly basis to the
generating units so as to balance the hourly demand while ensuring that social welfare is
maximized subject to satisfying all technical constraints of the power system. In contrast, with
the advent of competition, electricity is supplied by multiple competing gencos each of which
aims to maximize its individual profit. In this restructured system, an independent system
operator (ISO) is responsible for satisfying the demand by scheduling generation levels based on
the offers to produce submitted by each genco where essentially, the cheaper the offer the higher
the level of generation scheduled.
The operation and analysis of an electricity market differs from other commodity markets
due to some technical issues intrinsic to electricity. First, electricity cannot be efficiently stored
on a large scale. As such, the ISO must ensure that system generation and demand balances
instantly. Second, electricity cannot be freely exchanged between gencos and consumers. Rather,
power must be traded through the medium of a transmission grid governed by line flow limits
and by Kirchhoff’s current and voltage laws. Hence, at certain times of the day, the ISO may
have to deny some gencos full access to the grid (and thereby curtail their trading ability) to
ensure safe operation of the system. The abovementioned issues also affect the strategic
behaviour of dominant market players as shown in [13] which notes that the standard HerfindahlHirschman index (HHI) to determine the degree of market competition does not work well for
8
electricity markets. This has led the Federal Energy Regulatory Commission (FERC) to express
interest in models which explicitly measure market power by direct representation of the pricing
behaviour arising out of the interaction of market participants [12].
Motivated by the above peculiarities, the rest of the chapter describes the operation of an
electricity market and the gaming models used to study genco market behaviour.
2.1 Operation of Electricity Markets
The electricity market acts as a medium at the wholesale level through which gencos sell
electricity to distribution companies (discos) and large consumers. Discos are entities that
purchase power from the gencos and then use their distribution network to deliver the power to
retail level users. The electricity market operates under the authority of the ISO to whom gencos
and discos periodically submit their respective offers to sell and bids to purchase. The ISO
ensures that these offers and bids conform to market regulations, and subsequently clears the
market while ensuring that no system technical constraint is violated.
In clearing the market, the ISO resorts to the steps of unit commitment and economic
dispatch [14]. Through the unit commitment step, the ISO decides (typically on a day-ahead
basis) the hourly on/off status of the generating units [15]. The goal of unit commitment is to
schedule enough generation to meet the system demand while minimizing the system operational
cost for the upcoming day including the cost of turning generating units on or off. This process is
subject to constraints on the generating units' capacities and ramp limits and on their minimum
up and down times. Unit commitment must also meet the minimum spinning reserve criteria and
9
satisfy transmission line constraints. On the other hand, economic dispatch sets the generating
units to the optimal output level that satisfy the hourly system demand and minimize the overall
cost of generation [14]. Economic dispatch is typically embedded within the unit commitment
process, however, it may also be carried out independently on an hour-ahead basis by
considering only the generating units turned on for that hour. It is noted that not all system
generation resources can be dispatched at a set output level. For example, since wind is variable
and somewhat unpredictable, wind power cannot be dispatched with precision. As such,
adaptations have been incorporated within the unit commitment and economic dispatch to
accommodate power output from random sources such as wind [16].
2.1.1 Market Price
Electricity markets are commonly based on a uniform pricing scheme under which all gencos
are paid the same price for every unit of electric energy sold and all consumers pay the same
price for every unit bought. Under marginal pricing, when transmission network constraints are
negligible, or equivalently, when all generating units and the demand are assumed to be located
at one node, the single system-wide electricity price is equal to the marginal cost of the last unit
of power needed to satisfy the demand. In contrast, when transmission line losses are considered
or power flow constraints are active, each node has its own electricity price determined by the
marginal cost offered for the last unit of electricity needed to balance the demand at that node.
On the other hand, some markets [17] use a discriminatory pricing system where a genco is
paid at the rate of its average cost offer (offered cost divided by the quantity of electricity
10
produced). Since, under this so-called pay-as-bid pricing each genco defines its own price,
gencos must set their market offers higher than their true costs in order to earn a profit.
The choice between marginal and pay-as-bid pricing schemes is a subject of debate. An
argument in favour of pay-as-bid pricing is that, since the scheme is based on average price, it
may result in lower market price volatility than under marginal pricing [18]. Additionally,
studies such as [19] suggest that, in comparison to marginal pricing, the risk of tacit collusion is
lower under pay-as-bid pricing. However, others [20] argue that the advantages of pay-as-bid
over marginal pricing may be temporary and the benefits of switching from marginal to pay-asbid pricing would disappear as gencos adapt and change their gaming strategy. Moreover, it has
been pointed out that only through marginal pricing can the market send a reliable economic
signal to investors about the adequacy of the system generation resources, that is, whether the
system generation capacity is sufficient or needs to be expanded [18].
2.2 Strategic Genco Offers
As surveyed in [9], in an oligopolistic market a genco can choose its strategic market offer in
several ways. One involves using historic market price data to predict the electricity price in the
next trading period and then offering to produce at a price just below the predicted price.
However, this method assumes that the strategic market behaviour of competing gencos does not
influence the market price, an assumption not valid in oligopolistic electricity markets. Another
way is by first modelling the market behaviour of the competing gencos via fuzzy logic and
probability analysis methods, and then developing a market offer to maximize profit given the
11
competitors' estimated offer [21]. Game theory techniques have also been used to develop
optimal offer strategies. This approach can be further categorized into two groups. In the first
one [22], each genco discretizes the space of its bidding strategy into different levels. Then, by
enumerating all possible combinations of genco offer strategies, payoff matrices are built based
on which the optimal offer strategy of the gencos are obtained. However, this method becomes
intractable when many gencos are considered. Additionally, the method requires gencos to
choose among discrete bidding strategies from a continuous bidding space, a restriction unlikely
to be realized in practice. In the second gaming approach [23], an oligopolistic gaming model is
used to simulate genco strategic behaviour. Different type of gaming models are used for this
purpose as discussed in section 2.5.
2.3 Cooperative and Non-Cooperative Gaming
In a competitive electricity market, gencos game by strategically choosing their market
offers in two main ways [24]. A genco may choose to game according to a coordinated
agreement reached with one or more of its market competitors. This type of gaming based on
mutual cooperation among gencos is illegal. Alternatively, the genco may game without any
explicit coordination with its competitors, relying solely on publicly available information to
formulate its gaming strategy. Non-cooperative games, which are permitted in electricity
markets, are categorized as either one-shot or repeated. In the former, players take part in the
game once, while in repeated games, players participate in the same game more than once using
the knowledge gained from previous iterations to predict the behavior of their competitors [25].
12
2.4 Nash Equilibrium under Non-Cooperative Gaming
Non-cooperative games have been widely studied through the concept of Nash equilibrium
(NE) [11]. A NE occurs when the strategic offers of all players are optimal in the sense that no
player can increase its payoff by changing its strategy unilaterally. This can occur through either
a pure or a mixed strategy game. The former means that each player games by taking a single
strategic action with certainty while in the latter case the players game by considering a set of
pure strategies where each strategy has a specified probability of being pursued [25].
In models representing either pure or mixed strategy games, multiple NE may exist. In such a
case, a unique NE does not generally stand out as being preferred by all gencos over all other
equilibria. Hence, unless all players choose the same equilibrium the resulting market solution
will not be a NE [25]. However, if all players have an incentive to prefer one equilibrium over
all others, the market converges to a single NE known as the focal equilibrium. An example of a
focal equilibrium occurs in electricity markets requiring that gencos offer blocks of finite
generating capacity at constant incremental cost (IC) where such IC blocks vary continuously
[26]. From among the various gaming models using NE to study the behaviour of electricity
markets, the existence of multiple equilibria is a major concern when considering supply
function equilibrium (SFE) games. This concern is resolved in the SFE model proposed in
Chapter 3 of this thesis.
13
2.5 Oligopolistic Gaming Models in Electricity Markets
The most prominent gaming models used to study electricity markets, principally, Bertrand,
Cournot and supply function equilibrium (SFE) are reviewed in this section1.
2.5.1 Bertrand
In the Bertrand model, players compete on the basis of price. Each player games with the
price at which it offers to produce its output while anticipating that, in reaction to any change in
the player's price offer, its competitors will not change their price offers [28]. Several early
studies [29, 30] used Bertrand models to represent deregulated operation of power systems. In
[29] gencos game according to the Bertrand model in the deregulated operation of a
representation of the upstate New York power system. The study compares the regulated and
deregulated operation of the system and concludes that if the regulated price is equal to the
average production cost, then under deregulation, consumers would in general be worse off as
they would pay more for using less power whereas the generation side would be better off as it
would earn positive profit (as opposed to zero profit in the regulated case). Another study [30]
compares mill pricing and spatial discriminate (nodal) pricing in a deregulated system. Under the
1
Stackelberg competition [1] is another form of gaming that this thesis does not consider since it is not generally used in the
modeling of electricity markets. There are two reasons for this. One is that electricity markets are oligopolies with more than one
dominant genco and Stackelberg assumes the existence of a single dominant firm called the leader. The second reason is that
Stackelberg competition is less aggressive than Cournot, favouring the profit of the leader at the expense of the followers. It is
not likely therefore that a second dominant genco would acquiesce to take on the role of a follower.
14
former pricing scheme, a power plant stipulates a price at the location of generation and
consumers pay for the cost of delivery, whereas, under the latter pricing scheme, the price of
electricity is dependent on the node of the transmission grid at which electricity is consumed.
The study shows that when firms compete according to Bertrand the system operates at a higher
social welfare than under spatial discriminate pricing.
2.5.2 Cournot
The Cournot model [31] requires that each player game with its output quantity while
conjecturing that its rivals will not change their quantity offers. Studies such as [32] noted that in
markets with long-term genco capacity commitment, strategic market behaviour over an
extended time horizon tends to resemble the prediction from the Cournot model even if gencos
compete on the basis of price in the short-term. The Cournot model has been widely employed to
analyze the impact of genco gaming [12]. Such studies include investigations into: genco market
power potential such as the empirical study reported in [33], impact of transmission constraint on
market behaviour [34], influence of the choice of transmission network model (that is, DC
versus AC) [35], as well as the effect of price cap, forward contracts and non-smooth demand
function on market equilibrium [36, 37]. References to further studies using the Cournot model
can be found in the review papers [38, 39].
15
2.5.3 Supply Function Equilibrium
Supply function equilibrium (SFE) was proposed in 1989 as a new model to study the
gaming behaviour of participants in an oligopolistic market with uncertain demand [40]. In this
model, each player games with a supply function relating the quantity it is willing to produce to
the market price, rather than gaming with only quantity or price alone as in the respective
Cournot or Bertrand games. If the demand is not uncertain, there exists a continuum of market
equilibria under SFE, making it difficult to investigate the market behavior of the participants. If
a demand uncertainty range is considered, then for symmetric firms (that is, firms with identical
cost functions) the set of supply functions forms a connected continuum where, the market price
ranges from the Cournot price to the price under perfect competition for the highest demand
level considered. However, to reach a unique equilibrium, the uncertainty range needs to be
extended so that the demand is unbounded which is unrealistic for an electricity market.
Numerous variations of the SFE model in [40] have since been proposed for electricity markets
many of which reduce the range of feasible equilibria under different assumptions. The first
applications of the SFE model to electricity markets were carried out by Green and Newberry
[41] and Bolle [42]. A common approach in SFE restricts gaming to affine (linear with non-zero
intercept) supply functions where the gencos game by varying the slope and/or intercept of their
supply functions [43]. In general, the SFE models may be categorized into three types, that is, (i)
those which assume that the electricity market contains only symmetric gencos (ii) those which
allow only affine supply functions and (iii) those which model markets with asymmetric gencos
and also allow non-affine supply functions. SFE models from these three categories are
described next.
16
2.5.3.1 SFE with Symmetric Gencos
Symmetric gencos are those that have identical cost functions and capacity limits. In many
SFE models, the generation side of electricity markets has been represented by symmetric
gencos. Green and Newberry [41] used such a representation to analyze the operation of the
restructured England and Wales electricity market. In [41], a symmetric duopoly is considered to
represent the two dominant gencos in the England and Wales market, with each genco offering a
single supply function on a day-ahead basis for the 48 time intervals in the next 24 hours. The
range of daily demand variation is considered mathematically equivalent to the uncertainty in
demand discussed in [40]. Their analysis is based on the least competitive equilibrium (that is,
the one giving the Cournot price for the peak demand) motivated by the fact that this choice of
equilibrium yields the highest profits for the firms. The study shows that the generation sector of
England and Wales would have wielded considerably less market power had it been unbundled
into five gencos instead of two. This study also notes that the effect of considering generation
capacity limits is to reduce the range of feasible equilibria.
Bolle [42] considers an electricity market where the shape of the genco supply functions is
not restricted. The demand and genco marginal costs are linear and the genco capacity limits are
not active. However, this work considers only symmetric supply functions and chooses the
equilibrium that corresponds to the highest genco profits while allowing for tacit collusion. Two
forms of games are considered, namely, one where the demand has to pay a price fixed by the
auctioneer and the other where the demand pays the spot price. In the former game, it is shown
17
that the monopoly situation (that is, the joint maximization of genco profits) is the chosen
equilibrium. Finally, [42] also shows that when consumers pay the spot price, the market price
decreases with increasing number of gencos and eventually the price converges to the marginal
cost value.
Using the model in [41] and assuming linear demand and constant marginal genco costs,
Newberry [44] derives analytical expressions for the equilibria when symmetric supply functions
are considered. The paper shows that the range of the continuum of feasible equilibria narrows
with increasing number of gencos. In addition, by considering coordination among gencos in the
contract market and the threat of potential entry of new gencos, [44] shows that the continuum of
equilibria reduces to a unique equilibrium if the incumbent gencos offer so that the average
market price is equal to the minimum price which would deter the entry of new gencos. A similar
study [45] investigates SFE for a symmetric duopoly in the presence of contract market, price
cap and genco capacity limits while the gencos maximize the expected profit over the uncertain
demand range.
Closed form equilibrium solutions are obtained in [46] for a market with symmetric firms
when the demand is perfectly inelastic. This study then chooses the most competitive equilibrium
solution to model the Pennsylvania electric market of 1995 and analyzes the effect on market
power of varying the number of firms, accuracy of demand forecast and the amount of system
capacity that is not available. In contrast, [47] establishes a unique supply function equilibrium in
a market with symmetric gencos for a perfectly inelastic stochastic demand whose maximum
value is such that it exceeds the system capacity, hence requiring the capacity limit of all gencos
to bind with positive probability. Also shown in [47] is that under this unique equilibrium, all
genco capacity limits bind at the market price cap.
18
2.5.3.2 SFE with Asymmetric Gencos and Affine Supply Functions
Reference [43] is the first study to analyze a power system using affine genco supply
functions. In [43], gencos are assumed to have symmetric marginal cost at zero supply and to
game using the slope of their supply functions as the strategic gaming parameter. When the
genco supply functions need to be valid over multiple pricing periods, [48] shows that gaming
according to [43] results in unique affine supply functions. Reference [49] develops the affine
SFE model further to include cases of asymmetric marginal cost intercepts and genco capacity
limits. Thereafter, many affine SFE models have been developed which can be classified into
four categories based on the strategic gaming parameter available to the gencos [50, 51]. These
four categories are:
1) Gaming with the supply function slope while keeping the intercept fixed (generally to the
marginal cost intercept): For markets where genco supply functions need to be valid for at least
two pricing periods, this category of affine supply function offers results in market equilibria that
fulfill the conditions of Nash equilibrium [52].
2) Gaming with the supply function intercept while keeping the slope fixed (generally to the
marginal cost slope): Studies show [53] that such affine supply functions benefit from the fact
that gencos owning multiple units can build an aggregate supply function whose slope can be set
strategically by a linear combination of the units' supply function intercepts. It has also been
noted that gaming with the supply function intercept is particular useful for assessing the extent
of genco market power [54].
19
3) Gaming with a parameter that relates the genco supply function slope and intercept to their
marginal cost counterparts: In this form of game, each genco offers an affine supply function
whose slope and intercept is related to the respective slope and intercept of its marginal cost
function by a single scaling factor. This factor is non-negative and is used by the genco to game
strategically. Studies [55] which employ this form of gaming demonstrate to a limited degree the
effect of strategically changing both the slope and the intercept of a genco supply function while
avoiding the complexity associated with representing these two parameters separately.
4) Gaming with both slope and intercept of the supply function: Studies [56] using this form of
game represent the fullest extent of flexibility available to gencos in exercising market power
within the affine SFE paradigm. However, allowing gencos to game with the slope and the
intercept of the supply function independently generally does not lead to unique market
equilibrium.
Further studies for each of the above categories are discussed in [51] which surveys papers
on affine SFE incorporating transmission flow constraints through the DC or AC network model.
However, these studies generally make simplifying assumptions about the network operation
since, as concluded in [57], the existence of a pure strategy Nash equilibrium in the presence of
transmission flow constraint is not guaranteed and is dependent on the position of the constrained
line(s) in the network topology with respect to the generation and load busses.
20
2.5.3.3 SFE with Asymmetric Gencos and Non-Affine Supply Functions
For asymmetric firms, [58] proposes an iterative method over the function space of
continuous, non-decreasing supply functions. The supply functions are piecewise linear over a
price range discretized by many breakpoints. From a set of starting functions, the proposed
solution uses a local search direction at each iteration to choose the best piecewise affine supply
function (that is, the one which yields maximum profit) of each genco in response to the supply
functions of the other gencos at the previous iteration. It is noted that because of the possibility
of the existence of several local optima, the convergence of the algorithm does not guarantee that
the resulting supply functions globally optimize the genco profits, and therefore the supply
functions may not correspond to the Nash equilibrium. In [58] and [59], it is shown that if gencos
have affine marginal costs and no price cap and genco capacity constraints are considered then
there is a single stable equilibrium when repeated games are considered. Here, an equilibrium is
considered to be stable if gencos deviate slightly from their equilibrium supply functions and the
best response of the firms to the perturbed supply functions is closer to the equilibrium. Hence, it
is concluded that only the stable equilibrium is likely to be observed in the learning environment
when the firms iterate over the space of supply functions to converge to an equilibrium.
Reference [60] uses a similar iterative approach as [58] but a larger step size at each iteration
results in non-convergence when the model is applied to a representation of the England and
Wales electricity market.
For a stochastic demand, [61] proposes a method which discretizes the genco supply
functions over the demand range with the supply functions being linear in each interval of
demand uncertainty within the range. Then, for each possible demand realization, the first order
21
profit optimality conditions as well as other inequalities (such as the power balance requirement)
are expressed as a set of complementarity and other equations. These equations then represent
the inner optimization problem of each firm maximizing its profit for all discrete demand levels.
The outer optimization is represented by various objective functions, each one resulting in a
different set of genco supply functions leading to a different market outcome. Reference [62]
extends the work of [33] to a numerical integration algorithm for asymmetric gencos that is valid
for an inelastic stochastic demand such that for the maximum demand considered all gencos
except for possibly one reach their capacity limits before the price cap. The initial condition for
the integration is the set of prices at which the gencos bind at their capacity limits as well as the
amount offered by the largest genco at the price cap. From these initial conditions, the numerical
process involves integrating downwards in price to form the supply functions, a process that
results in multiple sets of genco supply functions. An optimization algorithm is then used to
choose the set whose termination price is closest to the marginal cost at zero supply. It is
graphically shown that this set of supply functions obtained forms an actual equilibrium for the
example considered.
2.6 Comparing the Gaming Models
According to Bertrand, in the absence of other constraints (such as transmission line flow
constraints) a genco cannot exercise market power for any demand level corresponding to which
the market has at least two gencos with sufficiently large capacities so as to appear unlimited.
22
Oligopolistic electricity markets do not however experience such perfectly competitive
outcomes.
Comparing the Cournot and SFE models, the former is mathematically more tractable and
flexible, facilitating easier modeling of complex aspects of power systems such as transmission
network constraints. On the other hand, market rules and empirical observations of market
outcomes appear to lend preference to SFE models. Bidding rules in electricity markets require
that a genco offer a schedule of quantity against price resembling a supply function rather than
only a quantity offer as required by the Cournot strategy. Also, empirical studies [63, 64]
analyzing the balancing electricity market of ERCOT in Texas for 2002- 2003 provide at least
partial credence to the SFE approach being more realistic. These studies find that for demands
higher than the predicted demand in the day-ahead market, the market offers of two of the three
largest gencos are moderately consistent with ex-post optimal prediction of the SFE model
corresponding to the actual market offers of the other gencos. It is also noted in [63] that the
behaviour of the third largest genco can be explained under the SFE model when one considers
repeated games rather than a one-shot game. For the same market and demands, in [63] a SFE
model is developed with only the three largest gencos (with the rest of the gencos assumed to be
non-strategic) which shows that the actual market offers of these gencos are within the ballpark
of the Nash offers predicted by the SFE model. This suggests that the SFE model can represent
well the offers of strategic gencos in this balancing market. In [65], it is shown that for demands
exceeding the predicted day-ahead demand, prices from a linear SFE model are a reasonable
approximation of the actual market prices of ERCOT for 2002-2003 (average deviation being
less than 4%). These studies mention that the high operating and adjustment costs not considered
in the models (such as those arising due to ramping generation) or inefficiencies related to
23
adjusting forward contracts may have been responsible for the otherwise unaccountable
difference between the SFE model predictions and actual observations of market price and genco
offer behaviour when the demand falls below the day-ahead prediction. On the other hand, as
reported in [66], the Cournot model is not known to have yielded accurate prediction of prices in
any market.
2.7 Contrasting Existing Literature with the Thesis's Claims to Originality
To clarify the contributions of this thesis, this section compares the claims to originality in
section 1.3 to the existing literature. From section 2.5.3, the majority of the contributions on SFE
relate to either symmetric gencos or to markets that allow only affine supply functions. While
symmetric gencos can be modelled by dividing all generating resources equally among the
considered gencos, real markets tend to be characterized by various degrees of asymmetry.
Results from SFE models based on symmetric gencos may therefore lead to inaccurate
predictions about real markets. Similarly, the applicability of SFE models requiring gencos to
offer affine supply functions is also questionable since such a constraint arbitrarily restricts the
free competition expected in a real market.
The SFE model of this thesis is more general than the models under the two categories
discussed above in the sense that the SFE model described here can accommodate both
asymmetric gencos and non-affine supply functions. As such, the claims to originality of this
thesis are contrasted only with SFE models from the existing literature that also allow both
asymmetric gencos and non-affine supply functions.
24
Claim 1: Out of the multiple SFE candidates for a predicted demand valid for a single pricing
period, a focal SFE candidate is identified under the following assumptions (i) gencos are
asymmetric (non-identical) with respect to their true costs of production, generation limits and
offered supply functions (ii) genco supply functions are not necessarily affine (iii) genco
capacity constraints are non-binding and (iv) the market price cap is non-binding. This SFE
candidate is focal in the sense that it is unique and leads to higher profits for all gencos compared
to other SFE candidates. The focal SFE candidate is based on the numerical integration of the
differential equations governing the first order genco profit optimality conditions under the Nash
equilibrium.
Contrast with literature: In [58], multiple sets of starting functions are used where, after the
iterative process, each set leads to a SFE candidate. Hence, multiple SFE candidates are obtained
and when only one-shot games are considered the study does not offer any criteria to prefer a
single SFE candidate to the others. In contrast, in this thesis a focal SFE candidate is obtained
from a one-shot game. In [61], a unique SFE candidate is identified for a stochastic demand only
for the severely restrictive case where the maximum demand level considered requires that all
but one genco reach the capacity limit. In comparison, in arriving at the focal SFE candidate
proposed in this thesis, no restriction is invoked on the demand level and the capacity limits of
the gencos are not required to be binding on their supply functions. In [62], the termination of the
integration process (in the direction of decreasing price) at as close a price as possible to the
marginal cost at zero system output is used as the criteria to select a unique SFE candidate. The
integration process in [62] also begins with all gencos but one at the maximum outputs, stopping
due to numerical instability before the price reaches the desired marginal cost at zero output. In
25
this thesis, neither one of these restrictive assumptions are invoked in establishing the unique
nature of the candidate SFE.
Finally, it is observed in [40] that supply functions are redundant for a single demand level
and gencos need only specify their output at the clearing price since the market operating point is
directly defined by the solution to the equations governing first order profit optimality. While
this argument is valid with respect to satisfying the first order condition, we note that gencos still
need to offer supply functions to ensure that the market equilibrium remains a Nash Equilibrium
for large gaming deviations in supply.
The focal SFE candidate is characterized by what is termed in this thesis as the tangent
condition, which occurs when the market is cleared at an operating point with the slope of the
aggregate genco supply function equal to the slope of the system demand curve. The claim that
this SFE candidate is focal is proved in section 3.3.7. To obtain this focal SFE candidate, the
SFE model of this thesis builds the genco supply functions by numerically integrating the
differential equations governing the first order genco profit optimality conditions from a unique
initial condition.
The impediments of using the numerical integration approach as mentioned in the existing
literature are now discussed. In [58], it is pointed out that the integration process can be started
from a set of initial conditions each of which leads to a candidate SFE. This multiplicity of SFE
limits the ability to analyze or predict genco strategic behaviour. However, in this thesis, as
discussed in section 3.3.6, the genco with the cheapest marginal cost at zero output, the sotermed leading genco, is shown to uniquely define the initial condition of the numerical
integration leading to the focal SFE candidate. Hence, under this thesis' SFE model, the dilemma
26
of choosing from multiple initial conditions does not exist. References [58, 61, 62] mention that
the supply functions obtained from the numerical integration process may be decreasing in price
and hence should not be considered as valid market offers. In response to this concern, section
3.1.5 of this thesis argues in detail that genco supply functions which decrease with price may be
permitted in the prevailing oligopolistic electricity markets where the market equilibrium
objective is not to maximize true social welfare but individual profits. In [61] and [62], it is
pointed out that the integration process encounters singularities at the lowest portion of a genco
supply function when the offered quantity approaches zero with the price approaching the
corresponding marginal cost. These singularities cause numerical instability disrupting the
integration process. This issue of numerical instability is addressed and resolved in section 3.3.3
of this thesis where it is shown that the singularities occur when a particular term in the
differential equations governing SFE approaches the 0/0 form. As shown, this term can be
explicitly evaluated by a straightforward application of l'Hopital's rule. Finally, as pointed out in
[61], the supply functions resulting from the numerical integration approach satisfy only the local
first order profit optimality conditions and may not fulfill the conditions of the global Nash
equilibrium. This criticism does not however apply to the focal SFE candidate proposed in this
thesis, which is always a true equilibrium. This assertion forms the basis of the next claim of
originality.
Claim 2: It is shown that while the supply functions obtained by numerically integrating the
first order profit optimality conditions do not satisfy the second order NE conditions, the
resulting SFE candidate becomes a true equilibrium (that is, satisfies the global NE condition)
when the genco supply functions are set to their market clearing outputs for all prices above the
27
market clearing price. The generality of this equilibrium is established through analytic proof and
numerical simulations.
Contrast with literature: As previously mentioned, the iterative method of [58] does not
guarantee that the resulting supply functions form an equilibrium. In [61], the genco supply
functions obtained from the bi-level optimization method are checked for global optimality for
all discrete demand levels considered. If any genco violates the global optimality condition, the
supply functions are discarded and the outer level objective function is changed to generate a
new set of genco supply functions corresponding to a new SFE candidate. Hence, in general the
method requires a trial-and-error approach to determine a SFE. In our SFE model, the genco
supply functions are guaranteed to form a focal SFE that is not dependent on the heuristic choice
of any system parameters. In [62], for the example considered, contours of constant genco
profits are plotted and, using the shape of these contours, it is shown that no firm can earn a
profit higher than the equilibrium for both small and large gaming deviations. However, no
argument is provided to show that this conclusion is not specific to the example considered and
is generally valid.
Claim 3: The traditional power system load-following method in power systems operation is
applied to adjust the SFE found in Claim 2 to account for errors in demand prediction. This
approach has the advantage of handling demand prediction errors in an ex-post manner (that is,
at the time when the prediction error materializes) avoiding the ex-ante method of dealing with
demand uncertainty requiring that genco supply functions be valid over the entire range of
demand uncertainty.
28
Contrast with literature: When considering asymmetric SFEs, the literature generally uses
one of two assumptions to construct genco supply functions, namely:
1) Supply functions are valid for predicted demand levels spanning multiple pricing periods,
for example, a day.
2) Supply functions are applicable to a single pricing period but for a stochastic demand with
positive support (probability distribution) over a significant range of demand levels.
If the predicted demand range considered for the multiple pricing periods under the first
assumption is such that the range encompasses uncertainty arising from demand prediction errors
then under both assumptions the treatment of such errors follows an equivalent approach
requiring genco supply functions to be valid over the demand uncertainty range. However, as
explained in discussing the originality of Claim 2, such an approach may not result in a SFE.
In contrast, in this thesis, a SFE path is constructed for each genco representing the locus of
its output versus price for a varying demand. Then, the traditional power system load–following
approach is used to adjust the genco outputs when the realized demand deviates from the
predicted one. The details of these adjustments can be found in sections 3.3.10 and 3.3.11.
Claim 4: A variation of the focal SFE in Claim 2 is found by constraining gencos to offer
supply functions that are non-decreasing in price, also known as right-bending supply functions.
However, it is shown that under this right-bending supply function equilibrium (RBSFE) all
gencos lose profit when compared to the focal SFE that allows supply functions that may be
increasing or decreasing in price, that is, supply functions that may be right or left-bending.
Furthermore, simulations consistently show that under RBSFE all non-leading gencos lose profit
compared to the Cournot equilibrium.
29
Contrast with literature: The originality of the RBSFE can be established from the
discussion under Claims 1 and 2 pertaining to the originality of the focal SFE. This is because,
with the exception of the additional non-decreasing constraint imposed on the genco supply
functions under RBSFE, these two SFEs share the same basic features. The claim that the focal
SFE yields higher genco profits than RBSFE is discussed in theoretical terms in section 3.3.8 and
demonstrated through numerical examples in section 4.1.
Claim 5: While preserving the features of Claims 1, 2 and 3, the focal SFE model is extended
to cases in which the capacity limits of one or more gencos actively constrain the supply
functions.
Contrast with literature: The preceding discussion regarding Claims 1, 2 and 3 clarifies the
originality of these claims with respect to the existing literature without distinguishing whether
or not the genco capacity limits constrain the supply functions under the SFE model of thesis.
Hence, extending the SFE model to cases where one or more genco capacity limits are binding
on the supply functions while preserving the features of Claims 1, 2 and 3 is also an original
contribution of the thesis.
Finally, simulations on typical power systems comparing SFE to RBSFE and Cournot
present unique and original numerical results. Particularly innovative is the numerical study
analyzing the impact of a carbon tax on the comparative gaming behaviour of power plants with
different emission rates.
30
Chapter 3: Equilibrium Model without Pre-Specified Supply Function Shapes
This thesis proposes a supply function equilibrium (SFE) model to clear an oligopolistic
electricity market where asymmetric generating companies (gencos) sell power by submitting to
the market operator offers of hourly supply functions of power output versus price in order to
meet a predicted hourly demand. In contrast to existing SFE alternatives, the model presented
here, while ensuring market stability in the Nash Equilibrium (NE) sense, does not pre-specify
any shape on the supply functions other than continuity and differentiability assumptions. One
principal characteristic of the proposed SFE model is that it generally yields higher profits for all
gencos and generates a lower demand profit when compared to the more commonly assumed
Cournot equilibrium or perfect market case.
3.1 Assumptions
This section describes the main assumptions of the SFE model pertaining to the demand
function, the gencos’ true cost functions, the market operational principles, the gencos’ strategic
market behaviour, and the transmission network.
31
3.1.1 Predicted Day-Ahead Hourly Demand
In the context of the day-ahead market clearing exercise, the predicted system hourly demand
in MW, d ( ) , is related to the price of electricity in $/MWh,  , via a known function typically
taken to be of the form,
d ( )  d 0 


(3.1)
where d 0 is the component of demand in MW independent of the electricity price and  is a
parameter measuring demand elasticity in $/MW2h. For the sake of simplicity, we represent
hourly variations in the predicted day-ahead system demand through changes in the parameter
d 0 only, keeping the elasticity parameter  fixed. Figure 3.1 shows an example of demand as a
function of price for a particular hour of the day.
32
40
Demand function
35
30
Price ($/MWh)
25
20
15
10
5
0
0
10
20
30
40
50
60
70
80
90
100
Demand (MW)
Figure 3.1: Demand curve as a function of price
During power system operation, real-time demand prediction errors are taken into account by
slightly correcting the minute-to-minute genco outputs relative to the hourly set-points calculated
by the SFE scheme in the day-ahead mode. These real-time generation adjustments are
implemented via the classical mechanism of participation factors and the measured real-time area
control error [67]. Under SFE, the genco participation factors are calculated from the supply
function equilibrium paths, the details of which are presented in section 3.3.10.
Finally, we note that the proposed SFE model considers only genco gaming and assumes that
there is no strategic demand-side gaming.
33
3.1.2 Unit Commitment and Temporal Constraints
The SFE model studied in this thesis is used in the hourly economic dispatch of the
generating units. The SFE model assumes that the day-ahead unit commitment (UC) procedure
has been carried out and that the on/off status of the gencos is known for every hour of a typical
24-hour day-ahead horizon. The UC on/off schedule ensures that a number of constraints such as
generation ramp limits and minimum up and down times are satisfied [68]. Multi-period gaming
that would explicitly consider such temporal constraints in the gaming strategy is to our
knowledge an intractable problem.
3.1.3 Leading Genco
For an arbitrary genco i, its true cost of generation, Ci ( gi ) , in $/h, is represented, as is
1
often the case in practice, by a quadratic function Ci ( gi )  ai gi  bi gi2 where g i is the output in
2
MW of genco i or, equivalently, by its true incremental cost in $/MWh, ICi ( gi ) 
dCi ( gi )
=
dgi
ai  bi gi . In addition, the generation output is restricted to lie within the capacity limits,
0  gi  gimax .
In this thesis it is assumed that no two true incremental costs at zero output ( ICi (0)  ai )
are identical and, therefore, that they can be ordered so that ai 1  ai . By numerically indexing
the gencos in order of increasing ai , genco 1 (that is, the genco with the lowest zero output true
34
incremental cost ai ) is denoted as the leading genco or leader, while all others are denoted as
non-leading gencos. Later on, we will show that the leading genco has a competitive advantage
in defining all supply functions.
More generally, a generating company could own and control the offers of a set of
generators with different cost functions and capacity limits, however, for simplicity, in this thesis
we assume that each genco owns and controls only one generating unit.
Lastly, the market offer submitted by genco i is a supply function (SF), Si ( ) , which
defines the quantity of electricity that genco i is willing to produce at the price  . A genco i
enters the market when the market price reaches ai , its supply function being equal to zero for
all prices lower than or equal to ai , and greater than zero for all higher prices.
As a result, when the price is between a1 and a2 , the leading genco 1 is the only genco
able to make a profit and, therefore, the only one willing to participate in the market. This
monopoly situation persists until the demand increases to the point where the correspondingly
increasing price reaches a2 , which is the market entry price of the first non-leading genco. As
will be seen in section 3.3.4, the value of the leader’s supply function at the entry price of its
nearest competitor, S1 (a2 ) , determines the supply functions of all gencos subsequently entering
the market at higher prices.
3.1.4 Gaming Strategies
35
It is assumed that gencos as rational entities try to maximize their own profits considering
that all competitors do likewise. Hence, each gaming genco will willingly prefer a market
solution based on the Nash Equilibrium (NE) because of the resulting high profits and market
stability in the sense that no single genco has an incentive to game unilaterally.
It is also assumed that gencos do not cooperate, and that each genco can obtain accurate
cost parameter and capacity limit estimates of all competitors from public knowledge about the
type of power plant and fuel used.
Additionally, the gencos’ strategic offers are based on a one-shot game rather than on
repeated games.
3.1.5 Market Operation and Shape of Supply Functions
For every hour of the day-ahead horizon, gencos submit SF offers to the market operator
in order to participate in supplying the predicted demand. The market operator then “clears the
market” by setting the hourly price  so that the total generation and demand balance,
 S    d  
j
(3.2)
j
In this thesis, other than continuity and differentiability assumptions, no other restriction
is imposed on the shape of the genco supply functions. Gencos may therefore offer SFs that may
be right or left-bending, where the former means that gencos increase their MW output with
increasing electricity price while left-bending SFs are such that gencos decrease their MW output
36
with increasing electricity price. In this thesis, over its range of prices, a SF may have portions
that are right-bending and other portions that are left-bending.
This is an uncommon generalization but, in support of it, we argue that, in an
oligopolistic market where profit is the main goal (and not social welfare), given that leftbending SFs lead to higher individual genco profits, such functions should be examined as a
valid option.
We emphasize that the use of left-bending supply functions is a gaming strategy for a
particular predicted demand that, if followed by every genco, generally leads to all-around higher
profits; however it is important to note that the locus of the market clearing price versus demand
(or the so-called equilibrium path) resulting from the use of left-bending supply functions
follows the normal tendency under which price increases with demand. This point is discussed in
detail in section 3.3.10.
A further justification of left-bending supply functions is as follows. In traditional
monopolistic power systems, gencos offer true cost functions rather than supply functions, and
so, to clear the market, the operator maximizes social welfare (demand benefit minus aggregate
generation cost). For this maximization to yield a unique market solution, it is computationally
necessary that the gencos’ incremental cost offers be monotonically increasing with output (or
equivalently that the costs be convex), a requirement equivalent to the supply functions being
right-bending. However, since in an oligopolistic market the gaming gencos’ cost offers differ
from their true costs, the demand benefit minus the offered aggregate generation cost is a
measure that no longer corresponds to the social welfare. The market equilibrium resulting from
maximizing such a measure is therefore of questionable value to society. Moreover, in a
37
competitive world, by enforcing the monotonicity requirement on the gencos’ offers (or
equivalently by excluding left-bending supply functions), gencos are arbitrarily and, arguably,
unjustifiably restricted in the profits they can earn. As such, this thesis examines and solves the
supply function equilibrium problem allowing for both right and left-bending supply functions.
The thesis also compares this general SFE to the SFE obtained when the SFs are restricted to be
right-bending but otherwise free, denoted as RBSFE.
Finally, it is assumed that the market follows the commonly used marginal pricing
method to define the electricity price (in contrast to pay-as-bid pricing [18]).
3.1.6 Transmission Network
The SFE model assumes that the transmission network remains uncongested and that
transmission losses are sufficiently low so as to be negligible. In addition, it is assumed that the
network has enough reactive power sources to maintain all bus voltage magnitudes near their
nominal levels.
3.2 At Low Demand, the Leading Genco Monopolizes the Market
38
When the predicted demand is cleared at a price lower than the price at which the first
non-leading genco enters the market (that is, when the market equilibrium price is below a2 ), the
leading genco 1 monopolizes the market. Under this low demand condition, since the
competitors of genco 1 are inactive, the profit maximizing supply function of genco 1 is defined
by the Cournot strategy [39] detailed in Appendix A, that is, by offering a constant supply
function for all  given by,
 g1max
; c  a1   b1    g1max

  a
S1 ( )   c 1 ; a1  c  a1   b1    g1max
 b1  
0
; a1  c

(3.3)
where c is obtained from the power balance relation,
S1 (c )  d0 
c

(3.4)
However, since at such low demand levels there is no competition from other gencos,
genco 1 may obtain the same profit by offering any other supply function as long as the market
clears at the Cournot price, c found from (3.3) and (3.4). An example of such an alternative
supply function is the Cournot equilibrium path shown next,
 g1max
;   a1   b1    g1max

   a1
S1 ( )  
; a1    a1   b1    g1max
 b1  
0
; a1  

(3.5)
39
In this thesis, we use the constant supply function of (3.3) whenever the demand is such that
genco 1 monopolizes the market.
For the market to remain a monopoly, the clearing price c must not exceed a2 , that is,
c  a2
However, since d0 
(3.6)
c
 S1 (c ) , an equivalent condition to (3.6) is that the demand parameter

d 0 satisfy,
d0  d0max
c 
a2

 S1 (a2 )
(3.7)
Moreover, at the lower demand levels, when the parameter d 0 falls sufficiently below
d0cmax , at some point, the clearing price c reaches a1 and the leading genco 1 produces zero
output, thus losing its incentive to participate in the market. This minimum level of d 0 is found
by requiring that the clearing price be greater than a1 , giving,
a1
b1  
c 
 a1
1
1

b1   
d0 
(3.8)
Thus, combining (3.7) and (3.8), the market is monopolized by the leading genco 1 if the
demand parameter d 0 lies in the range,
a1

 d 0  d 0cmax
(3.9)
40
3.2.1 Examples of Low Demand Monopolistic Market
Consider a 2-genco system where the true cost parameters of gencos 1 and 2 are a1  20
$/MWh, a2  30 $/MWh, b1  0.1 $/MW2h and b2  0.1 $/MW2h, the demand elasticity is
characterized by   0.4 $/MW2h, and the genco capacities are very large.
Then, from (3.9), genco 1 monopolizes the market for 50  d0  95 MW. Considering two
arbitrary demands in this range (that is, d0  65 MW and d0  d0max
c  95 MW), Figure 3.2 shows
the market equilibrium solutions, where the vertical lines are the corresponding Cournot supply
functions.
38
Supply function of Genco 1 for d0=65 MW
Demand curve for d0 = 65 MW
36
Market solution for d = 65 MW (23.33 $/MWh, 6.66 MW)
0
Supply function of Genco 1for d = 95 MW
34
0
Demand curve for d = 95 MW
0
Market solution for d0 = 95 MW (30.00 $/MWh, 20.00 MW)
Price ($/MWh)
32
30
28
26
24
22
20
0
5
10
15
20
25
30
35
40
45
Generation/Demand (MW)
Figure 3.2: Market solutions for d0  65 MW and d0  95 MW
41
From Figure 3.2, it is noted that for d0  65 MW the market clears at 23.33 $/MWh, a
price significantly below a2  30 $/MWh, with genco 1 producing 6.67 MW to satisfy the
demand. In contrast, for d0  d0max
c  95 MW (that is, the maximum value of d 0 for which the
market remains a monopoly), the market clears at 30 $/MWh, the value of a2 , with genco 1
producing 20 MW and genco 2 producing zero.
As the demand parameter d 0 varies, the locus of the corresponding market solutions
forms the Cournot equilibrium path. Figure 3.3 shows such a path for the market equilibrium
price over the range, 65  d0  95 MW, together with two supply functions corresponding to
d0  65 MW and d0  95 MW. Note that while the Cournot supply functions may be vertical,
the equilibrium path is right-bending, that is, the equilibrium price increases with demand.
42
32
Cournot equilibrium path of Genco 1
Supply function of Genco 1 for d = 65 MW
0
Supply function of Genco 1 for d = 95 MW
0
30
Price ($/MWh)
28
26
24
22
20
6
8
10
12
14
16
18
20
Generation (MW)
Figure 3.3: Cournot equilibrium path 65  d0  95 MW and supply functions for d0  65
MW and d0  95 MW
For values of d 0 higher than 95 MW, as shown in the next section, both gencos 1 and 2
have an incentive to participate in the market, which then becomes oligopolistic.
43
3.3 Oligopolistic Markets
3.3.1 Demand Sharing by the Leading Genco
Even for demands with d 0 higher than d 0cmax (the demand level defined in (3.7) below
which the leading genco is the only supplier), the leading genco could continue to monopolize
generation. However, were the leading genco to make such a choice, it would have to offer the
entire demand at the entry price of the next competing genco 2,   a2 , thus removing any
incentive for genco 2 to generate.
Figure 3.4 shows an example of the leading genco 1 continuing to monopolize the
demand with d0  150 MW, which is higher than d0max
c  95 MW. One way of achieving this, as
seen in the figure, is by offering a SF that follows the Cournot path till   a2 and thereafter
offering a horizontal line till the intersection point with the demand curve.
44
60
Supply function of Genco 1
Maximum demand monopolized by Cournot offer of Genco 1
Higher demand curve
55
Price ($/MWh)
50
d0 = 150 > 95 MW
45
40
max
d0c = 95 MW
35
30
25
20
0
10
20
30
40
50
60
70
80
90
100
Generation/Demand (MW)
Figure 3.4: A monopolizing leading genco supply function for d0  150 MW
However, as we will show, for demands with d 0 higher than d 0cmax , it is always preferable
for the leading genco 1 to relinquish some of its monopolistic advantage and allow the
competition (in this case, genco 2) to share in supplying the demand. Sharing not only increases
the price, but also the profit of the leading genco relative to not sharing, as shown in section
3.3.7. Clearly, the profit of the non-leading genco 2 is higher when genco 1 allows demand
sharing since the profit of genco 2 then goes from zero to a positive amount.
The degree to which the leading genco 1 relinquishes its monopolistic advantage and
shares the demand with its competitor is determined by the output of genco 1 at   a2 . This
 
quantity, S1 a2 , is in this thesis denoted by S 0 .
45
Figure 3.5 shows an illustration where genco 1 monopolizes the generation up to S0  30
MW and thereafter, for higher prices, shares the demand with genco 2. In this illustrative
example, when sharing the demand, both gencos offer supply functions of an arbitrary rightbending nature.
60
Supply function of genco 1
Supply function of genco 2
Aggregate supply function
Demand curve for d = 150 MW
55
0
Price ($/MWh)
50
45
40
35
30
S1(a2) = S0 = 30 MW
25
20
0
10
20
30
40
50
60
70
80
90
100
Generation/Demand (MW)
Figure 3.5: Genco 1 monopolizing generation up to S0  30 MW and thereafter sharing
with genco 2
Demand sharing could also be carried out according to Cournot, in which case gencos
would offer supply functions that are constant in price. Recall that, under Cournot, deviations in
the output of a single gaming genco are compensated solely by demand elasticity.
46
3.3.2 Demand Sharing under SFE according to the Klemperer-Meyer Differential
Equations
Under the proposed SFE, the market equilibrium is stable in the Nash Equilibrium (NE)
sense, that is, no genco can increase its profit by being the only one to alter its supply function.
Supply functions that lead to a market equilibrium in the Nash sense must satisfy the first order
profit optimality conditions defined by the Klemperer-Meyer (KM) differential equations [40] as
described next.
To derive the KM necessary conditions, recall that, given a differentiable supply function
of price  , Si    , the profit of genco i, pri ( ) , is,
pri ( )   Si ( )  Ci (Si ( ))
(3.10)
Now, if a single gaming genco i varies its supply function incrementally by dSi    while
all other gencos keep their supply functions unchanged at S j ( ); j  i , there results a price
variation, d  , defined by the power balance,
dS ( ) 

  d 
Si ( )  dSi ( )    S j ( )  j
 d   d0 
d 

j i 
(3.11)
dS ( ) 
 1
dSi ( )      j
 d

d

j

i


(3.12)
from which,
47
To a first order, the profit of the gaming genco i then varies by,
dpri ( )  d  Si ( )   dSi ( )  ICi  Si ( )  dSi ( )
    ICi  Si ( )   dSi ( )  Si ( )d 
where ICi  Si  
(3.13)
dCi  Si 
is the true incremental cost of genco i. Then, substituting (3.12) into
dSi
(3.13), we obtain the first derivative of the profit of the single gaming genco, (where
S j  ( ) 
dS j ( )
d
),
dpri ( )
    ICi  Si ( )   Si ( )  Si ( )
d
 1

    ICi  Si ( )       S j ( )   Si ( )
  j i

(3.14)
To ensure that the market operates at a NE, the profit of each single gaming genco must
be at a maximum, at which point it is necessary that its first derivative be zero. From (3.14), this
condition leads to the Klemperer-Meyer (KM) differential equations,
1
Si (  )
 S  ( )      IC  S ( ) 
j i
j
i
i
(3.15)
i
the solution of which defines candidate sets of supply functions satisfying the first order NE
optimality condition.
In section 3.3.5, we show that, for a given demand, the KM equations in (3.15) have
infinitely many solutions corresponding to local maxima, minima or saddle points of the genco
profits. However, we also show that: Among the infinitely many SF solutions of (3.15), there
exists a unique set that, after a simple modification, leads to a global maximum profit in the NE
sense for all gencos.
48
In this thesis, without significant loss of generality, the true incremental genco costs are
assumed to be given by, ICi  Si ( )   ai  bi Si ( ) . Then, KM takes the form,
1
Si (  )
 S  (  )      a  b S ( )
j i
j
i
i
i
(3.16)
i
In an oligopolistic market where two or more gencos share in supplying the demand,
candidate SFs can be obtained by numerically integrating the set of KM differential equations
 
(3.16) over the price  . This integration starts at the initial condition S1 a2 , denoted by the
symbol S 0 , in other words, the output of the leading genco at the market entry price of the first
non-leading genco,   a2 . It is important to recall that the initial condition S 0 is under the sole
control of the leading genco 1. The numerical integration of KM runs until a clearing price is
reached at which point the total generation from all SFs and the given demand curve balance.
Note that before reaching the market clearing price, the numerical integration of KM may
reach the entry price of another more expensive genco, which, as indicated next, then joins the
market and introduces an extra degree of competition. Additionally, when considering genco
capacity limits, as the price increases during the integration process, some gencos may reach and
remain at their generation capacities while others may be released from their capacities. The
numerical integration of KM with and without genco capacity limits is analyzed in detail in later
sections of the thesis.
49
3.3.3 Supply Function Continuity and Numerical Singularity Resolution when
Solving KM
When integrating the KM equations (3.16), singularities arise whenever  reaches the
market entry price of a non-leading genco i, that is, when   ai i  1 . This occurs since at
  ai , Si  ai   0 , and therefore the right-hand side of (3.16),
Si ( )
, becomes a
  ai  bi Si ( )
fraction with both numerator and denominator equal to 0.
To resolve this singularity, l’Hopital’s rule is invoked by approaching the entry price ai
from above, so that in (3.16) the contentious term takes the form,
lim
 ai
Si  ai 
Si   

  ai  bi Si    1  bi Si  ai

(3.17)
Now, in an oligopoly, when the price reaches   ai , the gencos already in the market
could, in theory, change their outputs discontinuously, just as genco 1 does at   a2 when
shifting from a monopoly to a duopoly. However, this possibility is discounted on the grounds
that, for triopolies and beyond, the two or more gencos already in the market would have to
break the non-collaboration assumption and agree (collude) on specific discontinuous changes in
their outputs at   ai ; i  3 . This situation does not arise in going from a monopoly to a duopoly
since the decision to change the output discontinuously rests solely with genco 1 through the
parameter S 0 . Consequently, the condition is imposed that, with the sole exception of the SF of
50
the leading genco when going from a monopoly to duopoly at   a2 , all SFs are continuous in
 , including at the entry prices of new gencos.
Thus, when genco i enters the market and the price is marginally above   ai ,
considering (3.17) and the SF continuity requirement, the KM equations require that,
1
 S  (a )   
j i
j
i
Si  ai
1  bi Si  ai
1
 S  (a )    a
j k
j

i
i

Sk (ai )
 ak  bk Sk (ai )
(3.18)
k  i
Note that SF continuity implies that, for triopolies and beyond, the values of the SFs at a
price marginally below   ai are known from the integration of KM up to that price. Thus,
equation (3.18) represents a set of linear equations plus a quadratic equation in the slopes
S j  ai  ; j (including Si  ai  ) at a price marginally above   ai . The numerical solution of
(3.18) for the new SF slopes is detailed in Appendix F. Basically, as each new genco enters the
market, the SFs remain continuous but their slopes undergo a discontinuity. Knowing the new
SF slopes marginally above   ai , the set of KM equations can continue to be integrated for
  ai in its original form as per (3.16), but now including genco i.
Figure 3.6 shows an example of the entry of a third genco at   a3 into what was a
duopoly for   a3 .
51
37
Supply function of Genco 1
Supply function of Genco 2
Supply function of Genco 3
36.5
36
Price ($/MWh)
35.5
35
34.5
34
33.5
33
32.5
32
0
5
10
15
20
25
30
35
40
45
Generation (MW)
Figure 3.6: Example of a duopoly switching to a triopoly at   35 $/MWh
As Figure 3.6 shows, upon the market entry of genco 3, the SFs of gencos 1 and 2
become steeper (bending more to the left), indicating a greater SF aggressiveness in the sense
that the price increases at a faster rate per unit change in output. At first glance, this may seem
counter-intuitive on the argument that with more competition the SFs should become less rather
than more aggressive. However, we note that, in order to earn a profit, as the price increases, the
new entry has to increase its output from its initial zero level. Anticipating this necessary
behaviour by genco 3, the competing gencos 1 and 2 already in the market take advantage to
reduce their output and thus increase price and profits.
52
3.3.4 Initial Condition of Numerical Integration under the Control of the Leading
Genco
As discussed in section 3.3.1, since the leading genco 1 is a monopoly for prices between
a1 and a2 , at the latter price genco 1 is free to choose the value of its supply function
S0  S1  a2  within certain limits. In this section, we show that the choice of S 0 by the leading
genco defines the behaviour of all SFs for higher prices.
From equation (3.18), in going from a monopoly to a duopoly, at a price just above
  a2 the SF slopes are given by,
S1 (a2 ) 
1


1
S2 (a2 )  

S2  a2 
1  b2 S 2  a2 
(3.19)
S1 (a2 )
a2  a1  b1 S1 (a2 )
 
Since S0  S1 a2 , we can express the above slope conditions as,
S1 (a2 ) 
1


S2  a2 
1  b2 S 2  a2 
(3.20)
S0
S2 (a2 )  
 a2  a1  b1 S0
1
Thus,
53
S0
1

a2  a1  b1 S0 
1
S1 (a2 ) 


S0
1 
1  b2 
 
 a2  a1  b1 S0  
S2 (a2 ) 
(3.21)
S0
1

a2  a1  b1 S0 
Equation (3.21) defines the SF slopes marginally above   a2 as functions of the
 
parameter S0  S1 a2 under the control of the leading genco 1. Since, according to KM, these
initial slopes uniquely define the SFs for higher prices, it follows that the SFs under a duopoly
are implicit functions of S 0 . Note that the SF dependence on the single parameter S 0 continues
for triopolies and beyond since, as new gencos enter the market, they do so at zero output with
the SFs of all gencos already in the market being continuous in price and implicitly dependent on
S0 .
In this thesis, from now on, we indicate this implicit SF dependence on S 0 by denoting
the SFs by the symbol Si   , S0  .
3.3.5 Upper and Lower Bounds on Initial Condition S 0
At   a2 , the leading genco is restricted to choose its output S0 from within a range of
values as shown next.
54
First, since at its market entry price,   a2 , the supply function of the non-leading genco
2, S2 (a2 , S0 ) , is zero, and since genco outputs cannot be negative, the slope of the supply
function of genco 2 must be positive at   a2 , that is, S2 (a2 , S0 )  0 .
Now, in the range of prices a2    a3 , when only the two cheapest gencos 1 and 2
participate, the KM equations are of the form,
S1 ( , S0 ) 
S 2 ( , S 0 )
1

  a2  b2 S2 ( , S0 ) 
S2 ( , S0 ) 
S1 ( , S0 )
1

  a1  b1 S1 ( , S0 ) 
(3.22)
Imposing the condition S2 (a2 , S0 )  0 , it follows from (3.22) that,
S1 (a2 , S0 )
1
 0
a2  a1  b1 S1 (a2 , S0 ) 
(3.23)
or,
S0  S1 (a2 , S0 ) 
a2  a1
b1  
(3.24)
Additionally, for all  , the offered supply function of any genco cannot exceed its offer
under perfect competition, the least aggressive gaming strategy, under which gencos offer at
their true incremental cost. For genco 1, the perfect competition SF takes the form,
S1* ( ) 
  a1
. Therefore, at   a2 ,
b1
55
S1  a2 , S0   S0  S1*  a2  
a2  a1
b1
(3.25)
From (3.24) and (3.25), the set of lower and upper bounds within which S 0 must lie is
then,
a2  a1
a a
 S0  2 1
b1  
b1
(3.26)
Note that the lower bound in (3.26) is the value of the Cournot SF when the demand is at
its maximum level under the monopoly of the leading genco, that is, when d0  d0cmax . This lower
bound is necessary to ensure that genco 2 produces a positive output as the price increases. If this
lower bound were not respected, genco 2 would not participate in the market, as it would have to
operate at a negative output. As the numerical results presented later in this thesis show, the
values of S 0 that lead to market equilibrium are typically much lower than the upper bound in
(3.26).
Considering the same duopoly example as in section 3.2.1, the range of valid S 0 in (3.26)
becomes 20  S0  100 MW. Choosing an arbitrary value S0  35 MW, Figure 3.7 shows the
genco supply functions after numerically integrating KM using Matlab. Note how the SFs are
initially right-bending but subsequently become left-bending as the price increases and the
gencos become more aggressive.
56
50
Supply function of Genco 1
Supply function of Genco 2
Aggregated supply function
S = 35 MW
48
0
46
Price ($/MWh)
44
42
40
38
36
34
32
30
0
5
10
15
20
25
30
35
40
45
50
Generation (MW)
Figure 3.7: Genco supply functions for S0  35 MW under SFE
For the same duopoly example, Figure 3.8 shows the aggregate (the sum of the genco
SFs) supply functions for different choices of S 0 , namely, 31, 35.5, 35.7 and 40.2 MW, all of
which are candidate SFs in the Nash sense to balance a specified demand versus price curve.
Considering a demand curve with d0  150 MW and   0.4 $/MW2h., note that for small values
of S 0 , such as the left-most curve with S0  31 MW, the supply function and the demand curve
do not intersect and there is no market equilibrium. However, as S 0 increases within its allowed
range, the aggregate supply function eventually does intersect the demand curve. The smallest
value of S 0 for which the market has an equilibrium solution is S0  35.5 MW, corresponding to
the second aggregate supply function from the left in Figure 3.8. In this case, the market clears
57
with the aggregate supply function meeting the demand tangentially at a unique market
equilibrium.
50
Demand
Aggregated supply functions
48
46
Price ($/MWh)
44
42
40
38
36
34
32
30
20
30
40
50
60
70
80
Generation/Demand (MW)
Figure 3.8: Demand curve and aggregate supply functions for varying S 0
For still higher levels of S 0 , for example, the third curve from the left at S0  35.7 MW,
we observe that the aggregate supply function intersects the demand at two prices, corresponding
to two market solutions, the lower priced one yielding a higher profit. However, as S 0 continues
to increase (e.g. for S0  40.2 MW), the aggregate supply function is totally right-bending and
the market returns to a single equilibrium solution as shown by the rightmost curve in Figure 3.8.
58
From these examples, we see that although there exists infinitely many SF candidate sets
satisfying KM, this continuum of SFs is parameterized by a single quantity S 0 under the control
of the leading genco 1. The logical next step is therefore to find a value of S 0 that yields a set of
SFs whose aggregate supply balances the demand curve and maximizes the profit of the leading
genco.
3.3.6 Market Equilibrium at the Tangent Condition
In this section, we argue that within the range of values of the parameter S 0 defined by
inequality (3.26), the leading genco prefers to choose a value S0  S0t whose resultant SFs,
S  , S  , when compared to all other feasible choices of S
j
0t
0
, maximize the leading genco’s
profit. Furthermore, it will be shown that the condition S0  S0t corresponding to the maximum
leading genco profit also maximizes the profits of all non-leading gencos with respect to S 0 .
The maximum profit value of S 0 , denoted by S0t , corresponds to the tangent condition,
meaning that, in addition to balancing supply and demand,
 S  , S   d
i
t
i
0t
0

t

(3.27)
the aggregate supply function is such that its slope is equal to the slope of the price-sensitive
system demand, that is,
59
1
 S   , S    
i
t
0t
(3.28)
i
As illustrated in Figure 3.7, if the free parameter is chosen so that S0  S0t , the resultant
aggregate supply curve does not intersect the demand curve and there is no market equilibrium.
However if S0  S0t , the power balance relation is satisfied at two prices,  hi , low  , such that
low  t  hi . Since the resultant SFs under both prices satisfy the power balance, it follows
that,
hi

i

i Si  low , S0   d0  low
 S 
i
hi
, S0   d 0 
(3.29)
From the illustration in Figure 3.7, we also note that the slopes of the aggregate supply
function under the high and low solutions satisfy,
 S  
i
hi
, S0   
i
 S  
i
i
low
1

, S0   
1
(3.30)

In the next section, we examine two fundamental properties of the tangent condition:
Tangent condition property 1: The SFs found when S0  S0t , yield the highest genco
profits compared to all other choices of S 0 within its feasible range.
Tangent condition property 2: Although the SFs found by solving KM under S0  S0t
satisfy the first order optimality for a NE, the second order NE conditions show that these SFs
60
correspond to a saddle point and not to a global maximum. Nonetheless, via a simple
modification of the tangent condition SFs, in section 3.3.9 we show that the modified SFs retain
the maximum profit property while corresponding to a NE in the global sense.
3.3.7 Tangent Condition, S0  S0t , Yields Maximum Profits for all Gencos Compared
to any other Feasible S0


In this section we prove that the SFs corresponding to the tangent condition, S j   , S0t
yield maximum genco profits compared to all other choices of S 0 in its feasible range.
To prove this, we first let S 0 increase2 from S0t to S0t  dS0 and show that under the


new set of SFs, S j   , S0t  dS0  , the profit of an arbitrary genco i decreases. Next, we show
that if we change S 0 from an arbitrary value S0e  S0t to S0e  dS0  S0t then, under the new set


of SFs, S j   , S0e  dS0  , the profit of an arbitrary genco i decreases if dS0  0 and increases if
dS0  0 . Proving this therefore demonstrates that it is always more profitable to decrease an
initial feasible S 0 towards its tangent condition level S0t .
To prove the first proposition, we let S 0 increase from S0t to S0t  dS0 , at which level
the corresponding change in the market equilibrium price, d  , with respect to t is found from
the supply and demand balance,
2
Since S0 = S0t corresponds to the tangent condition, if S0 decreases with respect to S0t , supply
and demand cannot balance.
61
 S   d , S
j
t
0t
 dS0   d0 
j
t  d 

(3.31)
Expanding the SFs S j  t  d  , S0t  dS0  to a second order in d  and to a first order in dS0 ,
d
S j  t  d  , S0t  dS0   S j  t , S0t   S j  t , S0t  d   S j  t , S0t 
2
S   , S 
 j t 0t dS0
S0
2
(3.32)
Substituting (3.32) into (3.31) and using (3.27) gives,
1
 S    , S  d   2  S    , S  d   
2
j
j
t
0t
j
t
0t
j
j
S j  t , S0t 
S0
Furthermore, from the tangent condition,  Si  t , S0t   
i
1

dS0  
d

(3.33)
, equation (3.33) requires that,
S j  t , S0t 
1
S j  t , S0t  d  2  
dS0  0

2 j
S0
j
(3.34)
Now, from Property 2.1 of Appendix B, we know that the second derivative term
 S    , S  is strictly negative while the term 
j
j
t
0t
j
S j  t , S0t 
S0
is strictly positive. Thus, from
(3.34), we conclude that a small positive change dS0 with respect to S0t leads to two prices, one
below and the other above t , as illustrated in Figure 3.9.
62
Aggregated supply functions
Demand
Price ($/MWh)


hi
>
t
t

S
0t
S + dS
0t
0
low
<
t
Generation/Demand (MW)
Figure 3.9: Variation in price due to increase in S 0 near tangent condition
Next, we prove that for both of these prices the profits of all gencos decrease relative to
the tangent condition price. To see this, consider the change in profit of an arbitrary genco i for
small price variations in the neighbourhood of the tangent condition. Using equations (3.10) and
(3.32),


 Si (t , S0t )d 



d2 
 
dpri (t , Si (t , S0t ))   t  ai  bi Si (t , S0t )    Si (t , S0t )
2 


  Si (t , S0t ) dS 
0

S0


 Si (t , S0t )d  
(3.35)
bi
2
 Si(t , S0t )  d  2
2
63
However, since the SFs follow the KM differential equations for all i, it follows that,
1
Si (  , S 0 )
i  bi Si ( , S0 )
 S  ( , S )    S  (  , S )    a
j
0
i
0
j
i
(3.36)
or, in particular, at the tangent condition, from (3.28),
Si (t , S0t ) 
Si (t , S0t )
 0 i
t  ai  bi Si (t , S0t )
(3.37)
Combining (3.34) and (3.37), after some algebraic manipulations, the change in profit in (3.35)
becomes,
dpri (t , Si (t , S0t ))  


2
bi
Si (t , S0t ) d  2
2
(3.38)
which is always negative. Thus, the profit of any genco decreases if S 0 increases from S0t to
S0t  dS0 .
A similar result holds if we start with an arbitrary value of S0  S0e  S0t and increase it
by dS0 , in other words, the profit also decreases. Conversely, if we decrease S0  Se by dS0 , the
profit increases. To prove these two results, consider a value of S0  S0e  S0t but within its
allowable range as per section 3.3.5. Let e denote the corresponding equilibrium price so that,
 S  , S   d
j
e
j
0e
0

e

(3.39)
Now, if S0  S0e  dS0  S0t , from the power balance,
 S 
j
j
e
 d  , S0e  dS0   d 0 
e  d 

(3.40)
64
which, after expanding the SFs to a first order, gives,
 S   , S  d  
j
e
S j  e , S0e
0e
j
S0
j

dS0  
d

(3.41)
or,
d  

j
S j  e , S0 e

S0
1
 S j  e , S0e  
j
dS0
(3.42)

Next, using (3.36) and (3.42), the change in the profit of genco i can be expressed by,
 S  ( , S ) d 

 i e 0e

dpri (e , Si (e , S0 e ))   e  ai  bi Si (e , S 0 e )   Si (e , S0 e )

dS0 

S0


bi
2
 Si(t , S0t )  d  2
2

1 
   S j (e , S0 e )   d  
 
  e  ai  bi Si (e , S0 e )   
 Si (e , S0 e )

dS0
 

S0


 Si (e , S0 e )d  
(3.43)
S ( , S )


  e  ai  bi Si (e , S0 e )    j e 0 e dS0 
S0
 j i

which is negative if dS0  0 and positive otherwise. This is true irrespective of whether e in
(3.43) represents the lower or the higher equilibrium price corresponding to S 0e . We therefore
conclude that the SFs yielding the highest profit for all gencos are those corresponding to
S0  S0t .
65
Since this conclusion says that any feasible S0  S0t yields a lower profit for the leading
genco 1, we also conclude that when d0  d0cmax , it is more profitable for the leading genco to
share the demand according to the KM equations rather than monopolize it. This follows from
the fact that to monopolize the demand the leading genco would have to choose S0  d  a2   S0t .
For the duopoly example of section 3.2.1, Figure 3.10 shows the behaviour of genco
profits as S0 varies with respect to S0t . From the figure it is clear that, as S0 increases with
respect to S0t , the two possible equilibria yield lower profits than those at the tangent condition.
66
700
600
Profit ($/h)
500
Profit of genco 1 at lower clearing price
Profit of genco 1 at higher clearing price
Profit of genco 1 at tangent condition
Profit of genco 2 at lower clearing price
Profit of genco 2 at higher clearing price
Profit of genco 2 at tangent condition
400
300
200
100
0
35
35.5
36
36.5
37
37.5
38
38.5
39
39.5
40
S0 (MW)
Figure 3.10: Variation of genco profits with S 0
3.3.8 Right-Bending Supply Function Equilibrium
If the market rules require that SFs not be left-bending, the corresponding equilibrium,
here called right-bending SFE or RBSFE, can be found by a modification of the numerical
integration procedure presented above for the more general SFE that allows both right and leftbending SFs. In this section we derive the numerical integration approach to find the RBSFE and
67
show that such equilibrium results in genco profits that are lower than those found under the
general SFE.
Basically, to find RBSFE, as the parameter S 0 is varied, if any of the SFs reaches a point
where its slope is about to become negative, then the integration is discontinued and restarted
with a higher value of S 0 . This procedure is repeated until the aggregate SF balances the demand
without any left-bending SF. The lowest value of S 0 that meets this condition, termed S0r , must
clearly be greater than the tangent condition S 0 , that is, S0t . Thus, in contrast to SFE, where
market equilibrium is characterized by the tangent condition where
1

m
  S j (t , S0t )  0 , under
j 1
RBSFE, market equilibrium is characterized by one SF being vertical, that is, S j (r , S0 r )  0 .
From section 3.3.7, when the market is cleared through SFs corresponding to S0r , genco
profits are higher than the profits corresponding to S0  S0r . Hence, RBSFE uses S0  S0r to
obtain the genco SFs by numerically integrating the KM equations. In the same section 3.3.7, we
proved that the tangent condition with S0  S0t yields maximum profits for all gencos compared to
any other S 0 . However, Property 1 of Appendix B shows that at the tangent condition the slope
of all genco SFs are negative, that is, the SFs are left-bending. However, since from (3.43) we
concluded that the SFs yielding the highest profit for all gencos are the left-bending SFs
corresponding to S0  S0t , it follows that with S0  S0 r  S0t , that is, under RBSFE, gencos earn
lower profits than under SFE.
68
We also know that under RBSFE, at   a2 the slope of the leading genco 1 must be
positive, that is, S1 (a2 , S0 r )  0 . However, since from (3.21) the SF slopes of the first two
gencos are,
S0 r
1

a2  a1  b1 S0 r 
1
S1 (a2 , S0 r ) 


S0 r
1 
1  b2 
 
 a2  a1  b1 S0 r  
S 2 (a2 , S0 r ) 
(3.44)
S0 r
1

a2  a1  b1 S0 r 
it follows that requiring that S1 (a2 , S0 r ) be positive means that,
S0 r
1

a2  a1  b1 S0 r 
1
 0

S0 r
1 
1  b2 
 
 a2  a1  b1 S0 r  
or that,
S0 r 
a2  a1
1
b1 
1
1

 b2  
(3.45)
Now, genco 2 enters the market only if the demand at   a2 is greater than the supply
function of the leading genco 1 at that price, that is, only if d  a2   S0r . Hence, under RBSFE,
d  a2   d 0 
a2

 S0 r 
a2  a1
1
b1 
1
1

 b2  
(3.46)
69
From (3.46), it then follows that d 0rmin , the minimum value of d 0 above which the market
becomes oligopolistic under RBSFE, is defined by,
d 0min
r 
a2


a2  a1
1
b1 
1
1

 b2  
(3.47)
However, recalling from (3.7) that if g1  g1max is not a binding constraint on the supply
function of genco 1, then,
d0max
c 
a2  a1 a2

b1   
(3.48)
which means that,
max
d0min
r  d0 c
(3.49)
min
Consequently, under RBSFE, for demands in the range d0max
c  d0  d0 r the leading
genco is forced to act as a monopoly and clear the market by offering to produce d 0 
a2

at
  a2 . This leads to a loss of profit compared to SFE since, as shown in section 3.3.7, for
min
demands in the range d0max
c  d0  d0 r , the leading genco can earn more profit by sharing than by
monopolizing the demand. A numerical example comparing RBSFE with the general SFE model
proposed in this thesis is presented in section 4.1.
70
3.3.9 Supply Functions for Prices  above t
Under SFE, at the tangent condition, three conditions are met: (i) the given demand is
satisfied; (ii) the genco profits are the highest over the initial condition parameter S 0 ; (iii) the
KM equations (the first order optimality conditions for the NE) are satisfied.
However, without further modification of the supply functions for   t , the market
equilibrium does not satisfy the NE condition. In other words, if the supply functions continue to
be defined by KM for prices above t , then for such prices each genco would be able to earn a
higher profit by gaming on its own. Figure 3.11 illustrates this property using the duopoly
example of section 3.2.1. The figure shows that, at the tangent condition, the genco profits by
gaming correspond to NE saddle points, with such profits decreasing with decreasing price but
increasing with increasing price up to a maximum defined by the non-gaming genco reaching its
zero output. Recall that “profit by gaming” means that only one genco games while all others
continue to offer according to their supply functions, which, being left-bending for prices above
t , eventually reach zero output. This saddle-point characteristic is examined in further detail in
Appendix C.
71
55
Profit of Genco 1 when it games unilaterally
Profit of Genco 2 when it games unilaterally
Maximum profit of Genco 1 when it
games unilaterally occurs at 50.16 $/MWh
Maximum profit of Genco 2 when it
games unilaterally occurs at 50.16 $/MWh
Profit of Genco 1 at the tangent condition price
Profit of Genco 2 at the tangent condition price
Price ($/MWh)
50
45
40
35
30
0
100
200
300
400
500
600
700
Profit ($/h)
Figure 3.11: Profits of gaming gencos around tangent condition equilibrium
Hence, to ensure that the supply functions yield a market equilibrium that fulfills the
Nash Equilibrium (NE) requirements but retains the high genco profits of the tangent condition,
the numerical integration of the KM equations is discontinued when the price reaches t . For
higher prices, the SFs are instead set to constants equal to their market clearing output at t ,
leading to a set of modified SFs defined by,

 Si   , S 0 t  ;
Simod   , S0t   

 Si  t , S0t  ;
  t
  t
(3.50)
mod
The modified SFs, Si   , S0t  , illustrated in Figure 3.12, have three properties: (i) They
balance the demand at the same genco outputs and price as the original SFs; (ii) The genco
72
profits at equilibrium are still equal to the maxima found under the tangent condition; (iii) As
illustrated in Figure 3.13, the market equilibrium satisfies the NE in a global sense, that is, no
genco can increase its profit by gaming unilaterally. This global optimality is proven in
Appendix D.
50
Supply function of Genco 1
Supply function of Genco 2
Aggregate supply function
Demand
Clearing point (42.95 $/MWh; 42.62 MW)
48
46
Price ($/MWh)
44
42
40
38
36
34
32
30
0
10
20
30
35.5
40
50
60
70
80
Generation/Demand (MW)
Figure 3.12: Modified genco supply functions
73
55
Price ($/MWh)
50
Profit of Genco 1 when it games unilaterally with
unmodified supply functions
Profit of Genco 2 when it games unilaterally with
unmodified supply functions
Profit of Genco 1 when it games unilaterally with
modified supply functions
Profit of Genco 2 when it games unilaterally with
modified supply functions
Profit of Genco 1 at the tangent condition price
Profit of Genco 2 at the tangent condition price
45
40
35
30
0
100
200
300
400
500
600
700
Profit ($/h)
Figure 3.13: Gaming genco profits with original (dotted) and modified (solid) supply
functions
From this point on, in this thesis the term supply function will mean modified supply
function. In addition, the same symbol, that is, Si   , S0t  , will be used to represent a modified
supply function.
3.3.10 Supply Function Equilibrium Path
The supply function equilibrium path of an arbitrary genco i is the locus of its output at
the tangent condition equilibrium, Si  t , S0t  , versus t as the demand parameter, d 0 , varies.
74
The equilibrium path of a genco i is not to be confused with the genco’s supply function,
Si   , S0t  , which is a specific function of  corresponding to a specific value of the demand
parameter d 0 .
Figure 3.14 and Figure 3.15 show an equilibrium path example from the duopoly case of
section 3.2.1. Figure 3.14 shows in blue the path of the output of the leading genco 1, S1  t , S0t 
, over a set of values of d 0 together with the corresponding supply function, S1   , S0t  . Figure
3.15 shows a similar graph for the non-leading genco 2. The lower dotted blue line represents the
Cournot equilibrium path (see section 3.2.1). Additional examples of equilibrium paths can be
found in section 4.1.
55
Price ($/MWh)
50
45
Supply functions of genco 1
for d0 = 100,120 and 140 MW
40
SFE path of genco 1
for 95 < d0 < 200 MW
Cournot equlibrium path of genco 1
for 95 < d0 < 200 MW
35
30
20
25
30
35
40
45
50
55
60
65
70
Generation (MW)
Figure 3.14: Supply function and Cournot equilibrium paths of genco 1
75
55
Price ($/MWh)
50
45
40
Supply functions of genco 2 for d =100, 120 and 140 MW
0
35
SFE path of genco 2 for 95 < d < 200 MW
0
Cournot equilibrium path of genco 2 for 95 < d < 200 MW
0
30
0
5
10
15
20
25
30
35
40
45
50
Generation (MW)
Figure 3.15: Supply function and Cournot equilibrium paths of genco 2
Of note in the above equilibrium paths is that they are right-bending even though the
defining SFs are, in part, left-bending. Thus, under SFE, both the price and the equilibrium
outputs of gencos increase with increasing demand, which is an intuitively reasonable and
expected behaviour.
Figure 3.16 compares the behavior of the equilibrium price versus profit between SFE
and Cournot in the range 95  d0  200 MW, showing that SFE yields higher profits for both
gencos except when the price is near the entry price of the non-leading genco 2, in which case
genco 2 earns slightly lower profits under SFE. The leading genco 1 however always earns
significantly higher profits under SFE and has no incentive to offer according to Cournot.
76
1400
Profit of genco 1 under SFE
Profit of genco 1 under Cournot equilibrium
Profit of genco 2 under SFE
Profit of genco 2 under Cournot equilibrium
1200
Profit ($/h)
1000
800
600
400
200
0
100
110
120
130
140
150
160
170
180
190
200
Demand intercept, d (MW)
0
Figure 3.16: Profit paths of gencos 1 and 2 under SF and Cournot equilibria
3.3.11 Effect of Demand Prediction Uncertainty on SFE
The SFE paths offer a systematic manner of dealing with uncertainty in the predicted
demand.
In this thesis, to track random real-time demand variations, a procedure is followed akin
to the classical power system automatic generation control or AGC [67]. Under AGC, the genco
outputs are automatically varied by gi in response to d , the real-time power imbalance
between demand and generation relative to the predicted levels. Such imbalance is generally
77
measured through f , the system frequency deviation from its nominal value (or more generally
through deviations in the Area Control Error [67]), where d   Bf and where B is the
sensitivity of the system frequency to power imbalances in the control area. In order to execute
the required real-time genco output corrections while also accounting for genco economics, the
values of gi are determined by the gencos’ participation factors,  i , typically through a
proportional control scheme, so that,
gi  i d
(3.51)
where,  i is a fraction between 0 and 1.
Generally, in a monopoly, the participation factor of a genco depends on its economic
efficiency with the more efficient gencos having higher participation factors, while in a market
environment, the participation factors depend on the competitive advantage of the genco. The
use of AGC and participation factors ensure that, despite the presence of random real-time
demand variations from the predicted value, the power system continues to operate close to
optimum economic efficiency.
In the SFE market being studied in this thesis, the participating factors are calculated
from the SFE paths as sensitivity coefficients of the SFs with respect to the demand d , when d 0
varies from its predicted value, d̂ 0 . More precisely the participation factors are given by
i 
Si
d
. To compute these sensitivities, recall that the SF paths are of the form
d0  dˆ0
Si  t  d0  , S0t  d0   , from which it follows that if d0  dˆ0  d0 , for small d 0 , the SF paths
78
vary by Si 
Si
d0 while the demand changes by d  d 0 
d 0
t
d 0
d 0

. The participation
factors are then,
Si
d 0
i 
t
d
1 0
(3.52)

The following numerical procedure is followed here to compute the participation factors
i :
(1) Estimate the range  d0min , d0max  within which the uncertain demand parameter d 0 is
predicted to lie in the upcoming dispatching interval of time (typically of the order of one hour).
Let d0  d0max  d0min ;
(2) For both extreme values of d 0 , find the corresponding SFE initial condition
parameters, S0tmin and S0tmax , tangent condition prices, tmin and tmax , as well as the corresponding




min
SFs, Si tmin , S0min
and Si tmax , S0max
. Let Si  Si  tmax , S0max
, S0min
t
t
t   Si  t
t  and t 
tmax  tmin ;
(3) Then from (3.52),  i , the participation factor of genco i when d 0 is within the
predicted range  d0min , d0max  , is estimated by,
79
Si
d 0
Si
i 

t

d 0  t
d

1 0
(3.53)

As an example, consider a duopoly with genco cost parameters a1  20 $/MWh, a2  30
$/MWh, b1  0.1 $/MW2h, b2  0.15 $/MW2h and demand elasticity   0.4 $/MW2h. Table
3.1 compares participation factors and other variables for varying prediction error in d 0 (denoted
by d 0 ) around a mean value of 120 MW.
Table 3.1: Genco participation factors for varying demand uncertainty
d 0min
118
116
114
112
110
max
0
122
124
126
128
130
4
8
12
16
20
d
d 0

35.82 35.35 34.88 34.41
33.93

36.74 37.20 37.65 38.11
38.56
0.92
1.85
2.77
3.70
4.63
1.67
3.37
5.05
6.75
8.42
30.90 30.46 29.96 29.41
28.78
31.65 31.97 32.26 32.52
32.76
25.26 24.81 24.36 23.90
23.45
26.16 26.62 27.09 27.54
28.00
0.90
1.81
2.73
3.64
4.55
3.20
2.82
2.45
2.08
1.73
3.98
4.37
4.78
5.18
5.59
0.78
1.55
2.33
3.10
3.86
0.54
0.54
0.54
0.54
0.54
0.46
0.46
0.46
0.46
0.46
min
t
max
t
t

d 0  t

S
S
S1  
min
0t
max
0t
min
t
S1  
max
t
S2  
,S
max
0t
S1
min
t
S2  
,S
min
0t
max
t
,S
min
0t
,S
S2
1
2
max
0t




80
Of note is that, under SFE, the genco participation factors are relatively unaffected by the
range of demand prediction error. In fact, simulations suggest that the participation factors
derived from the SFE paths are essentially constant over a broad range of demand parameters.
See, for example, Figure 3.14 and Figure 3.15 in section 3.3.10, which show the SF paths to be
close to straight lines. Thus, by using the participation factors defined in this section, the power
system will continue to operate near the NE equilibrium even in the presence of demand
prediction uncertainty.
3.3.12 Procedure to Compute Supply Functions Numerically
The numerical procedure to obtain the genco supply functions under SFE is as follows:
1) For triopolies and above, the genco supply function slopes, appearing in a coupled
form in the KM equations (3.16), are first converted into a standard decoupled form. This
algebraic decoupling process, shown in Appendix E, is based on [49] and takes the form,
ng
S i   , S 0  
 r  , S 
j 1
j
ng  1
0
 ri   , S0  
1
; i
  ng  1
(3.54)
where,
ri   , S0  is the ratio
Si (  , S 0 )
on the right hand-side (RHS) of the KM equations
  ai  bi Si ( , S0 )
(3.16) and ng is the number of gencos in the market
81
2) A graph (or table) relating the demand parameter, d 0 , to the corresponding S0t , is
obtained from the following steps:
i) Using Matlab’s ode45, an ordinary differential equation solver (numerical
experimentation shows that an integration step of 0.01 $/MWh is accurate), the standard form in
(3.54) is numerically integrated for a chosen value of S0t (from within the range of allowed S0t )
to obtain the corresponding genco supply functions and their slopes.
ii) The aggregate supply function and its slope are computed from (i).
iii) From (ii), find the price, t and the corresponding aggregate genco output,
 S 
i
, S0t  , under the tangent condition, that is, when the aggregate supply function slope
t
i
1
 S    , S  is equal to   , the slope of the demand function.
i
t
0t
i
iv) From the power balance relation at the clearing point, the parameter d 0 is calculated
as,
d0   Si  t , S0t  
i
t

(3.55)
v) Steps i) through iv) are repeated over a range of values of S0t sufficient to account for
the expected range of the demand parameter d 0 and such data is stored in graph form or as a look
up table. For the duopoly case of section 3.2.1 Figure 3.17 shows such a graph for d 0 between
95 and 200 MW.
82
220
Variation of S0t with demand intercept d0
Demand intercept, d0 (MW)
200
180
160
140
120
100
80
20
22
24
26
28
30
32
34
36
38
Initial condition, S0t (MW)
Figure 3.17: Plot of d 0 versus S0t
3) To find the SFs for a specific value of d 0 , the value of S0t corresponding to that d 0 is
obtained from the graph or table found in step 2 (after interpolation between data points if
required). Using this S0t as the initial condition, the KM equations are integrated numerically to
obtain the genco supply functions.
83
Chapter 4: Case Studies of SFE in Electricity Markets
A number of case studies are presented in this chapter to illustrate quantitatively the
potential impact of SFE in electricity markets. These studies are: (1) Comparison of SF, Cournot
and perfect market equilibria; (2) Behaviour of markets under SFE with and without carbon tax;
(3) Behaviour of SFE when the demand becomes very large; (4) Effect of random demand
variations.
4.1 Comparing Market Behaviour under SFE, Cournot and Perfect Competition
Consider a 5-genco system with the cost parameters shown in Table 4.1.
Table 4.1: Cost parameters
genco
ai ($/MWh)
bi ($/MW2h)
1
20
0.1
2
30
0.1
3
35
0.1
4
38
0.1
5
40
0.1
Let the system demand elasticity be   0.4 $/MW2h.
84
From (3.7), the value of d 0 at and below which the leading genco 1 monopolizes the
market, is,
d0max
c 
a2  a1 a2
  95 MW
b1   
(4.1)
For a d0  100 MW, slightly above d 0cmax , Figure 4.1 shows the SFE solution.
33
32.5
Supply function of genco 1
Supply function of genco 2
Aggregate supply function
Demand
Clearing point (31.40 $/MWh; 21.51 MW)
Price ($/MWh)
32
31.5
31
30.5
30
0
5
10
15
20
25
Generation/Demand (MW)
Figure 4.1: SFE market solution for d0  100 MW
For this low demand level, the condition that the SFs should not be left-bending can only
be respected if the leading genco does not share the demand and continues to act as a monopoly.
85
That this is correct, can also be verified by recalling from (3.47) that d 0rmin , the value of d 0 below
which the market becomes oligopolistic under RBSFE is,
d 0min
r 
a2  a1
a
 2  106.03 MW
1

b1 
1
1

 b2  
(4.2)
which is a number higher than d0  100 MW.
Hence, for d0  100 MW, the market solution under RBSFE is a monopoly where the
leading genco supplies the entire demand at the entry price of genco 2,   30 $/MWh. Both
supply and demand are then given by d0 

 25 MW . To produce this quantity, the leading

genco can offer its Cournot path till   30 $/MWh and continue to offer at   30 $/MWh the
difference between the demand curve and the Cournot path as shown in Figure 4.2.
86
40
Supply function of genco 1
Demand
Clearing point ( 30.00 $/MWh; 25.00 MW)
38
36
Price ($/MWh)
34
32
30
28
26
24
22
20
0
5
10
15
20
25
30
35
40
45
50
Generation/Demand (MW)
Figure 4.2: RBSFE market solution for d0  100 MW
For the same demand, Figure 4.3 and Figure 4.4 respectively show the genco offers and
the clearing point under Cournot and perfect competition (where gencos offer their true
incremental costs). Note that under Cournot, gencos 1 and 2 share the demand, while under
perfect competition, only genco 1 supplies power.
87
35
Cournot offer of genco 1
Cournot offer of genco 2
Aggregate Cournot offer
Demand
Clearing point
(30.77 $/MWh; 23.08 MW)
Price ($/MWh)
34
33
32
31
30
0
5
10
15
20
25
30
Generation/Demand (MW)
Figure 4.3: Market solution under Cournot for d0  100 MW
30
True incremental cost of genco 1
Demand
Clearing point (24.00 $/MWh; 40.00 MW)
29
28
Price ($/MWh)
27
26
25
24
23
22
21
20
0
10
20
30
40
50
60
Generation/Demand (MW)
Figure 4.4: Market solution for d0  100 MW assuming perfect competition
88
The genco outputs and profits under the four markets considered are compared in Table
4.2 and Table 4.3 respectively.
Table 4.2: Market outputs of gencos for d0  100 MW
Output (MW)
genco
SFE
RBSFE
Cournot
True IC
1
21.22
25
21.54
40
2
0.29
0
1.54
0
3
0
0
0
0
4
0
0
0
0
5
0
0
0
0
Table 4.3: Genco profits for d0  100 MW
Profit ($/h)
genco
SFE
RBSFE
Cournot
True IC
1
219.29
218.75
208.79
80
2
0.40
0
1.07
0
3
0
0
0
0
4
0
0
0
0
5
0
0
0
0
89
From Table 4.3, we see that for the low demand condition being examined, under SFE
both gencos earn higher profits than under RBSFE or perfect competition. However, the profit of
genco 2 under Cournot is 0.67 $/h higher than under SFE. In contrast, the profit of genco 1 is
10.50 $/h higher under SFE compared to Cournot. By comparing SFE to Cournot, since under
SFE the 10.50 $/h increase in profit of the leading genco 1 is much higher than the 0.67 $/h loss
of profit of the non-leading genco, it is expected that genco 2 will offer according to SFE
recognizing that genco 1, attracted by the high profit gain from a SFE offer, will not offer
according to Cournot. This is a result that has shown to be consistently true for low demands.
Now consider a higher demand with d0  250 MW. The following four figures show the
genco offers and market solution under SFE, RBSFE, Cournot and perfect competition
respectively. Note that for this relatively high demand, with the exception of the perfect market,
under all other equilibria, all five gencos participate in supplying the demand. Note that under
SFE the SFs have portions that are both right and left-bending and that the equilibrium is defined
by the tangent condition. In contrast, under RBSFE, the SFs are only right-bending and the
equilibrium is defined by the SF on one genco (in this example, genco 1) being vertical. Finally,
of note is that under Cournot all SFs are vertical.
90
Supply function of genco 1
Supply function of genco 2
Supply function of genco 3
Supply function of genco 4
Supply function of genco 5
Aggregate supply function
Demand
Clearing point (65.66 $/MWh; 85.86 MW)
80
75
70
Price ($/MWh)
65
60
55
50
45
40
35
30
0
20
40
60
80
100
120
140
160
180
Generation/Demand (MW)
Figure 4.5: SFE market solution for d0  250 MW
Supply function of genco 1
Supply function of genco 2
Supply function of genco 3
Supply function of genco 4
Aggregate supply function
Demand
Clearing point
(39.12 $/MWh; 152.20 MW)
46
44
Price ($/MWh)
42
40
38
36
34
32
30
0
50
100
150
200
Generation/Demand (MW)
Figure 4.6: RBSFE market solution for d0  250 MW
91
Cournot offer of genco 1
Cournot offer of genco 2
Cournot offer of genco 3
Cournot offer of genco 4
Cournot offer of genco 5
Aggregate Cournot offer
Demand
Clearing point
(46.08 $/MWh; 134.80 MW)
50
48
46
Price ($/MWh)
44
42
40
38
36
34
32
30
0
50
100
150
200
Generation/Demand (MW)
Figure 4.7: Market solution with Cournot offers for d0  250 MW
38
True incremental cost of genco 1
True incremental cost of genco 2
Aggregate true incremental cost
Demand
Clearing point
(33.33 $/MWh; 166.60 MW)
37
Price ($/MWh)
36
35
34
33
32
31
30
0
20
40
60
80
100
120
140
160
180
Generation/Demand (MW)
Figure 4.8: Market solution under perfect competition for d0  250 MW
92
The genco outputs and profits for d0  250 MW under the four markets are shown in
Table 4.4 and Table 4.5 respectively.
Table 4.4: Genco outputs for d0  250 MW
Output (MW)
genco
SFE
RBSFE
Cournot
True IC
1
40.52
90.81
52.16
133.30
2
18.82
40.64
32.16
33.30
3
11.42
16.66
22.16
0
4
8.34
4.09
16.16
0
5
6.76
0
12.16
0
Table 4.5: Genco profits for d0  250 MW
Profit ($/h)
genco
SFE
RBSFE
Cournot
True IC
1
1767.67
1323.85
1224.30
888.44
2
653.39
288.01
465.42
55.44
3
343.65
54.73
220.98
0
4
227.16
3.74
117.52
0
5
171.18
0
66.54
0
93
For this higher demand level, from Table 4.5 we observe that all gencos earn higher
profits under SFE compared to Cournot, RBSFE and perfect market, a result that generally holds
for all demand levels except, as seen in the previous example, for very low demand. In addition,
all gencos except the leader earn lower profits under RBSFE than under Cournot. Finally, perfect
competition generally yields the lowest profits for all gencos.
On the other hand, from Table 4.6, we observe that the demand earns the highest profit
(defined by the difference between the demand benefit and payment) under perfect competition
followed respectively by RBSFE, Cournot and SFE.
Table 4.6: Demand profit for d0  250 MW
Demand profit ($/h)
SFE
RBSFE
Cournot
True IC
1474.44
4633.23
3634.21
5556.11
The following five figures show the genco SFE paths over the demand range
100  d0  250 MW and compare said paths with those found under Cournot and true IC offers.
94
70
SFE path of genco 1
RBSFE path of genco 1
Cournot equilibrium path of genco 1
True incremental cost curve of genco 1
65
60
Price ($/MWh)
55
50
45
40
35
30
25
20
0
50
100
150
200
250
300
350
400
450
500
Generation (MW)
Figure 4.9: Equilibrium paths of genco 1 in the various markets
70
SFE path of genco 2
RBSFE path of genco 2
Cournot equilibrium path of genco 2
True incremental cost curve of genco 2
65
Price ($/MWh)
60
55
50
45
40
35
30
0
50
100
150
200
250
300
350
400
Generation (MW)
Figure 4.10: Equilibrium paths of genco 2 in the various markets
95
70
SFE path of genco 3
RBSFE path of genco 3
Cournot equilibrium path of genco 3
True incremental cost curve of genco 3
65
Price ($/MWh)
60
55
50
45
40
35
0
50
100
150
200
250
300
350
Generation (MW)
Figure 4.11: Equilibrium paths of genco 3 in the various markets
70
SFE path of genco 4
RBSFE path of genco 4
Cournot equilibrium path of genco 4
True incremental cost curve of genco 4
65
Price ($/MWh)
60
55
50
45
40
35
0
50
100
150
200
250
300
Generation (MW)
Figure 4.12: Equilibrium paths of genco 4 in the various markets
96
70
SFE path of genco 5
Cournot equilibrium path of genco 5
True incremental cost curve of genco 5
65
Price ($/MWh)
60
55
50
45
40
0
50
100
150
200
250
300
Generation (MW)
Figure 4.13: Equilibrium paths of genco 5 in the various markets
From the preceding figures, we note that the SFE path is the most aggressive in the sense
that all gencos produce less and at higher prices than under RBSFE, Cournot or perfect
competition, a tendency that persists over the entire demand range.
We also observe that the RBSFE paths display a marked “wiggliness”. To explain this
behaviour, recall that when a new genco enters the market the SFs of those gencos already
participating have a tendency to bend to the left. In such a circumstance, the SF of a genco may
end up becoming left-bending, which would violate the right-bending-only rule. Consequently,
the leading genco has to set its initial condition to a level where either the incoming genco does
not come into the market (that is, the equilibrium price is at or below the new genco’s entry
97
price), or where the new genco enters the market without any existing SFs becoming leftbending.
The wiggliness of the SF equilibrium paths under RBSFE and their limited range are
significant drawbacks. The first characteristic requires repeated changes in the participation
factors as the demand changes. In comparison, the participation factors under SFE are practically
unchanged over a wide range of demand. Secondly, the right-bending requirement prevents more
expensive gencos from entering the market until the demand is very high.
The following five figures show the genco profit variations along the equilibrium paths
under the different markets as the demand varies within 100  d0  250 MW.
1800
Profit of genco 1 under SFE
Profit of genco 1 under RBSFE
Profit of genco 1 under Cournot
Profit of genco 1 under perfect competition
1600
1400
Profit ($/h)
1200
1000
800
600
400
200
0
80
100
120
140
160
180
200
220
240
260
Demand intercept, d0 (MW)
Figure 4.14: Variation of genco 1 profit with demand
98
700
Profit of genco 2 under SFE
Profit of genco 2 under RBSFE
Profit of genco 2 under Cournot
Profit of genco 2 under perfect competition
600
Profit ($/h)
500
400
300
200
100
0
100
120
140
160
180
200
220
240
260
Demand intercept, d0 (MW)
Figure 4.15: Variation of genco 2 profit with demand
350
Profit of genco 3 under SFE
Profit of genco 3 under RBSFE
Profit of genco 3 under Cournot
300
Profit ($/h)
250
200
150
100
50
0
120
140
160
180
200
220
240
260
Demand intercept, d0 (MW)
Figure 4.16: Variation of genco 3 profit with demand
99
250
Profit of genco 4 under SFE
Profit of genco 4 under RBSFE
Profit of genco 4 under Cournot
200
Profit ($/h)
150
100
50
0
120
140
160
180
200
220
240
260
Demand intercept, d0 (MW)
Figure 4.17: Variation of genco 4 profit with demand
180
Profit of genco 5 under SFE
Profit of genco 5 under Cournot
160
140
Profit ($/h)
120
100
80
60
40
20
0
140
160
180
200
220
240
260
Demand intercept, d0 (MW)
Figure 4.18: Variation of genco 5 profit with demand
100
From the preceding figures, we note that gencos generally earn higher profits under SFE
than under other markets. The only exception occurs when the demand is low, in which case
genco 2 earns a slightly higher profit under Cournot than under SFE as seen in Figure 4.15.
Additionally, the more expensive the genco, the more it gains under SFE compared to Cournot
and RBSFE. Finally, all non-leading gencos earn less profit under RBSFE than under Cournot.
Only the leading genco 1 earns higher profits under RBSFE than under Cournot. In fact, the
constraints imposed by the RBSFE are such that the more expensive gencos do not even enter the
market or only enter for very high demand.
However, for all demands, the following condition is true: If all gencos switch from SFE
to Cournot, those gencos making a lower profit under Cournot lose significantly more than the
gains made by those gencos that make a higher profit under Cournot. This can be seen from
Figure 4.19. Rational gencos are therefore expected to prefer offering according to SFE.
101
Aggregate difference between genco profit loss and gain
when switching from SFE to Cournot ($/h)
1200
Aggregate difference between genco profit loss and gain
when switching from SFE to Cournot
1000
800
600
400
200
0
100
120
140
160
180
200
220
240
260
Demand intercept, d (MW)
0
Figure 4.19: Variation with demand of the aggregate difference between genco profit loss
and gain when switching from SFE to Cournot
On the other hand, from the perspective of the demand, Cournot is preferred to SFE
because the demand profit is higher under Cournot compared to SFE as seen from Figure 4.20.
However, since the demand cannot game, it does not influence the gencos’ rational choice of
offering according to SFE.
102
4000
Demand profit under SFE
Demand profit under Cournot
3500
Demand profit ($/h)
3000
2500
2000
1500
1000
500
0
100
150
200
250
Demand intercept, d (MW)
0
Figure 4.20: Variation of demand profit under SFE and Cournot
4.2 Comparative Market Behaviour under SFE with and without Carbon Tax
The effect of a government carbon tax on power plants to penalize GHG emissions is
examined here in the context of electricity markets [69, 70].
Basically, by imposing a carbon tax, the more polluting plants become incrementally
more expensive, while the reverse holds for less polluting plants. A carbon tax therefore affects
the competitiveness of plants and alters the market equilibrium in ways that the SFE theory
developed in this thesis can quantify.
103
Here, we make the usual assumption that the GHG emissions of genco i are modeled by a
known constant carbon emission rate in tons per MWh,  i . The imposed carbon tax rate, tc , in
$/ton then has the effect of modifying the cost parameter ai to a new higher value
ai  ai  ai  tc i .
Consider the 5-genco system of Table 4.1 together with the carbon emission rates shown
in Table 4.7.
Table 4.7: Cost and emission parameters for 5-genco system
genco
ai ($/MWh) bi ($/MW2h)  i (t/MWh)
1
20
0.1
1.1
2
30
0.1
0.9
3
35
0.1
0.8
4
38
0.1
0.4
5
40
0.1
0
We also consider a system demand with d0  250 MW and   0.4 $/MW2h and examine the
effect of imposing a carbon tax by comparing the SFE solution with and without such a tax.
The SFE market solution without carbon tax is shown in Figure 4.21.
104
Supply function of genco 1
Supply function of genco 2
Supply function of genco 3
Supply function of genco 4
Supply function of genco 5
Aggregate supply function
Demand
Clearing point
(65.66 $/MWh; 85.86 MW)
80
75
70
Price ($/MWh)
65
60
55
50
45
40
35
30
0
20
40
60
80
100
120
140
160
180
Generation/Demand (MW)
Figure 4.21: Market solution with no carbon tax
Next, we consider two typical carbon tax rates of 10 and 20 $/t. Table 4.8 shows the cost
parameters of the system after imposing such taxes.
Table 4.8: Cost parameters with carbon tax
genco
ai
ai ($/MWh)
ai  ai ($/MWh)
Tax: 10
Tax: 20
Tax: 10
Tax: 20
($/t)
($/t)
($/t)
($/t)
($/MWh)
bi
i
($/MW2h) (t/MWh)
1
20
11
22
31
42
0.1
1.1
2
30
9
18
39
48
0.1
0.9
3
35
8
16
43
51
0.1
0.8
4
38
4
8
42
46
0.1
0.4
5
40
0
0
40
40
0.1
0
105
Figure 4.22 and Figure 4.23 show the market solution with carbon tax.
90
Supply function of genco 1
Supply function of genco 2
Supply function of genco 3
Supply function of genco 4
Supply function of genco 5
Aggregate supply function
Demand
Clearing point
(69.63 $/MWh; 75.92 MW)
85
80
Price ($/MWh)
75
70
65
60
55
50
45
40
0
20
40
60
80
100
120
140
160
Generation/Demand (MW)
Figure 4.22: Market solution with carbon tax of 10 $/t
106
Supply function of genco 1
Supply function of genco 2
Supply function of genco 3
Supply function of genco 4
Supply function of genco 5
Aggregate supply function
Demand
Clearing point
(72.83 $/MWh; 67.91 MW)
85
80
Price ($/MWh)
75
70
65
60
55
50
45
0
50
100
150
Generation/Demand (MW)
Figure 4.23: Market solution with carbon tax of 20 $/t
The market solutions for different tax rates are compared in Table 4.9 and Table 4.10.
Table 4.9: Genco outputs and profits with various carbon tax levels
Output (MW)
Profit ($/h)
genco No tax Tax: 10$/t Tax: 20$/t
No tax
Tax: 10$/t Tax: 20$/t
1
40.51
30.29
18.39
1767.67
1124.21
550.10
2
18.82
14.02
9.10
653.39
419.50
221.90
3
11.42
9.00
6.25
343.66
235.69
134.52
4
8.34
10.06
11.63
227.16
272.86
305.34
5
6.76
12.56
22.54
171.18
364.19
714.75
107
Table 4.10: Pollution and tax revenue with various carbon tax levels
Pollution (t/h)
genco
Tax revenue ($/h)
No tax Tax: 10$/t Tax: 20$/t No tax Tax: 10$/t Tax: 20$/t
1
44.56
33.32
20.23
0
333.19
404.58
2
16.94
12.62
8.19
0
126.18
163.80
3
9.14
7.20
5.00
0
72.00
100.00
4
3.34
4.02
4.65
0
40.24
93.04
5
0
0
0
0
0
0
Total
73.98
57.16
38.07
0
571.61
761.42
Table 4.11 shows how the carbon tax impacts the genco leadership hierarchy. Under the
10 $/t tax, genco 1 retains the leadership while gencos 3, 4 and 5 exchange positions. When the
taxation level is increased to 20 $/t, genco 5, which occupies the lowest position in the leadership
hierarchy in the absence of carbon tax, becomes the leading genco. This shows that for a
sufficiently high tax level, a “clean” genco can exercise leadership in the market and hence
acquire significant influence over the market clearing process.
108
Table 4.11: Genco leadership hierarchy with and without carbon tax
No tax Tax: 10$/t Tax: 20$/t
1
1
5
2
2
1
3
5
4
4
4
2
5
3
3
Finally, Table 4.12 shows that the demand profit decreases with increasing tax level.
Table 4.12: Demand profit for various tax levels
Demand profit ($/h)
No tax
Tax: 10$/t
Tax: 20$/t
1474.44
1152.71
922.44
4.3 Supply Functions for Large Demands
By experimentation and analysis we observed that as demand increases and all available
gencos enter the market, the initial output corresponding to the tangent condition specified by the
leader, that is, S0t , converges to a constant level that we denote as S0tm . Interestingly, as the
demand increases, S0t approaches but never reaches S0tm .
109
We first examine this behaviour through the example of the duopoly in section 3.2.1 over
a wide range of d 0 from 400 MW to 2000 MW in steps of 400 MW. Table 4.13 shows the main
characteristics of the resulting market equilibrium.
Table 4.13: Market equilibrium over a wide range of demands
d 0 (MW)
S0t (MW)
t ($/MWh)
g1 (MW)
g 2 (MW)
400
38.69
96.29
90.79
68.48
800
38.95
181.01
184.89
162.58
1200
38.99
265.84
278.85
256.54
1600
39.01
351.05
372.33
350.04
2000
39.02
435.27
467.06
444.76
In Table 4.13, as the demand increases, the clearing price and the genco outputs show a
not unexpected increasing trend. However, we also note that the initial condition of the leading
genco, S0t , converges to a value around 39 MW, specifically to 39.03 MW.
Also, by examining Figure 4.24 showing the market clearing solution for d0  2000 MW
and Figure 4.25 showing a close up of the same figure for lower prices, we note that the supply
functions closely approximate straight lines for all prices up until 100 $/MWh, beyond which
point they follow curved trajectories.
This behaviour is consistent with that of the so-called Green’s supply functions [43],
which are straight lines whose slopes big are given by Green’s equation,
110
1
1
1
   g ; i
b  bi  j i b j
(4.3)
g
i
The SFs found by our SFE integration scheme approach these straight lines during the early
portions of the trajectories since for large demands the initial condition of the leading genco, S0t
, approaches S0g , which is the initial condition under which the KM equations define SFs that are
straight lines.
Supply function of Genco 1
Supply function of Genco 2
Aggregate supply function
Demand
Clearing point
(435.27 $/MWh; 911.82 MW)
450
400
Price ($/MWh)
350
300
250
200
150
100
50
0
200
400
600
800
1000
1200
1400
1600
1800
Generation/Demand (MW)
Figure 4.24: Market solution for d0  2000 MW
The portion of Figure 4.24 for prices till   100 $/MWh is shown in Figure 4.25. Note that in
the figure the SFs are almost straight lines like the Green’s supply functions of Figure 4.26.
111
100
Supply function of Genco 1
Supply function of Genco 2
Aggregate supply function
90
Price ($/MWh)
80
70
60
50
40
30
0
50
100
150
200
250
300
Generation (MW)
Figure 4.25: Genco SFs till   100 for market solution in Figure 4.24
500
450
400
Price ($/MWh)
350
Supply function of Genco 1
according to Green's equation
Supply function of Genco 2
according to Green's equation
Aggregate supply function
according to Green's equation
300
250
200
150
100
50
0
500
1000
1500
2000
2500
3000
3500
4000
Generation (MW)
Figure 4.26: Genco supply functions following Green's equation
112
A similar behaviour under large demands was also observed in oligopolistic markets with
more than 2 gencos. To illustrate this, consider the example of the 5-genco system in Table 4.1.
Let a large system demand be represented by d0  2500 MW. Then, the market solution is shown
in Figure 4.27.
700
Supply function of genco 1
Supply function of genco 2
Supply function of genco 3
Supply function of genco 4
Supply function of genco 5
Aggregate supply function
Demand
Clearing point
(526.85 $/MWh; 1182.87 MW)
600
Price ($/MWh)
500
400
300
200
100
0
500
1000
1500
2000
Generation/Demand (MW)
Figure 4.27: Market solution of the 5-genco system for d0  2500 MW
Upon magnifying Figure 4.27, we observe in Figure 4.28 that the market clears with
S0t  41.15 MW, which is greater than 39.03 MW, the value of S 0 above which the SFs in the
duopoly section of the trajectories (that is, when the prices are between a2 and a3 ) are rightbending. This temporary right-bending characteristic of the SFs is seen in Figure 4.28.
113
As in the earlier duopoly example, in this 5-genco oligopoly, for demands with
d0  2500 MW, the initial S0t never exceeds the value of 41.16 MW. In general, every
oligopoly has a maximum value of S0t that is approached but never reached as the demand
increases.
Supply function of genco 1
Supply function of genco 2
Supply function of genco 3
Supply function of genco 4
Supply function of genco 5
Aggregate supply function
55
50
Sg(a ) = 39.03 MW
Price ($/MWh)
1
2
S = 41.15 MW
0t
45
40
35
30
10
20
30
40
50
60
70
Generation/Demand (MW)
Figure 4.28: Magnification of Figure 4.27 for low price levels
114
4.4 Comparing Markets under SFE, Cournot and Perfect Competition for
Uncertain Demand
Consider a predicted demand with d0  250 MW . Then, consider 100 random variations
of d 0 within the encompassing uncertainty range [240, 260] MW. For this range, we compute
the SFE genco participation factors as described in section 3.3.11. Using a similar process, we
obtain the participation factors when the markets are based on Cournot and perfect competition
respectively. Then, we compute the genco outputs as follows,
gix  gi0  i d x
(4.4)
where,
g ix is the output of genco i for the xth random variation of d 0 ;
g i0 is the output set-point of genco i for the predicted d 0 ;
 i is the participation factor of genco i;
d x is the deviation from the predicted demand due to the xth random change in d 0 .
Table 4.14 shows the expected value and the standard deviation of the genco profits
under SFE, Cournot and perfect competition over the set of random variations.
115
Table 4.14: Expected value and standard deviation of genco profits due to random changes
in d 0
Expected profit ($/h)
genco
SFE
Cournot
True IC
Standard deviation ($/h)
SFE
Cournot True IC
1
1767.70 1223.80
888.13 22.21
18.85
0.30
2
652.90
465.10
55.13
19.80
11.62
0.30
3
343.30
220.80
0
16.40
8.01
0
4
226.80
117.30
0
13.58
5.84
0
5
170.90
66.40
0
11.64
4.39
0
From Table 4.14, we observe that, even when we take into account the demand prediction
errors, SFE continues to yield higher expected genco profits in comparison to equilibria based on
Cournot and perfect competition. Furthermore, these expected profits are sufficiently higher
under SFE to counteract the higher standard deviations observed under SFE.
116
Chapter 5: SFE with Genco Capacity Constraints
In this chapter, we consider the constrained supply function equilibrium (CSFE) with finite
genco capacity constraints, gi  gimax . When these genco capacity constraints are active, the
unconstrained SFE model presented earlier in this thesis must be modified to ensure a feasible
Nash equilibrium. However, as will be seen next, many of the features developed for the
unconstrained SFE remain valid in the CSFE.
5.1 Modified KM Equations in CSFE
We begin by considering a general case of CSFE where the numerical integration process of
the KM equations starts at the initial condition S 0 , as in the unconstrained SFE, without any
genco reaching its capacity until the price   k , where the following holds:
(1) At least ng=3 gencos have entered the market. The special case with fewer gencos having
entered the market is relatively straightforward.
max
(2) Genco k reaches its capacity, that is, Sk  k , S0   gk ;
(3) The tangent condition corresponding to S 0 and the system demand has not yet been
satisfied, else the unconstrained SFE would have been found. From Property 3 of Appendix B,
this means that for a system with ng unconstrained gencos at a price slightly below k , we have,
117
1

ng
  S j  k , S0   0
(5.1)
j 1
(3) The SF slope of genco k is positive, that is,
Sk  k , S0   0
(5.2)
Thus, if the integration of the KM equations were to continue for   k as in the
unconstrained SFE, the SF of genco k would exceed its capacity limit.
To avoid such infeasibility, under CSFE, starting at k , genco k offers its capacity while the
remaining ng-1 unconstrained gencos offer according to KM, that is,
Sk   , S0   g kmax


1
 S    , S     S    , S     a  b S   , S  ; i  1,..., ng; i  k
Si  , S 0
ng
j 1
j k
j
0
i
(5.3)
0
i
i
i
0
The SFs under CSFE are defined by (5.3) until the price kr  k where the first order profit
optimality condition requires the SF of genco k to be released from its capacity g kmax . This
release condition (RC) is discussed in detail in section 5.3. For   kr , the SF of genco k is once
again defined by the KM equations together with all other unconstrained gencos, that is,


1
 S    , S     S    , S     a  b S   , S  ; i  1,..., ng
Si  , S 0
ng
j 1
j
0
i
0
i
i
i
(5.4)
0
The following sections describe in greater detail the above characteristics of the genco
supply functions under CSFE. For simplicity, we assume that only one genco reaches its limit.
118
However, the derivations remain valid when multiple gencos reach their capacity limits, one at a
time, and are then subsequently released.
5.2 First Order Nash Equilibrium Optimality near Price where one Genco Begins
to Offer its Maximum Capacity
In this section, we verify that (5.3) satisfies the NE first order optimality near and above
k . In this range, the supply functions of the unconstrained gencos clearly satisfy the NE first
order optimality since they continue to be defined by the KM equations as in (5.3). To show that
the constrained genco k also satisfies the first order NE optimality condition, we allow genco k to
game incrementally so that the price changes slightly with respect to k by d  . There are two
possibilities then, depending on whether d  is positive or negative. Consider first the case where
d   0 . Then, for the incremental change when gaming, dSk , to be feasible, it must satisfy,


1 ng  

dSk  
 S j  k , S0   d   0
  

j 1
j

k


(5.5)
However condition (5.1) and (5.5) are then incompatible with the assumption that d   0 . The
remaining alternative that d   0 results in a feasible dS k if and only if,
1

ng
  S j  k , S0   0
(5.6)
j 1
j k
119
To verify that inequality (5.6) holds, we first note that marginally below k , the KM equations
are,
ng
 S  
j
j 1

k
, S0  
1

 Si  k , S0  
Si  k , S0 
; i  1,..., ng (5.7)
k  ai  bi Si  k , S0 
In particular, when i  k , and recalling that Sk  k , S0   0 ,
ng
 S j  k , S0  
j 1
1

 Sk   k , S0  
g kmax
g kmax

k  ak  bk g kmax k  ak  bk g kmax
(5.8)
Now, from (5.7), summing over i and after some algebraic manipulation, we have,
ng
 S j  k , S0  
j 1
1


Si  k , S0 
1  ng
1



ng  1  i 1 k  ai  bi Si  k , S0   


(5.9)
Similarly, from (5.3), when  is slightly above k , we can write,
ng
1
 S j  k , S0   

j 1
j k


Si  k , S0 
1  ng
1


ng  2  i 1 k  ai  bi Si  k , S0   
 ik

(5.10)
Next, from (5.8) and (5.9),
Si  k , S0 
g kmax
1  ng
1
 

ng  1  i 1 k  ai  bi Si  k , S0    k  ak  bk g kmax
(5.11)
or, equivalently,
ng

i 1
ik
k
Si  k , S0 
 ai  bi Si  k , S0 

1

  ng  2 
g kmax
k  ak  bk g kmax
(5.12)
120
Combining (5.10) and (5.12), it follows that,
 ng

Si  k , S0 
1
1

S j   , S0   
 



 ng  2  i 1 k  ai  bi Si  k , S0   
j 1
j k
 ik

max
gk

0
k  ak  bk g kmax
ng

k
1
(5.13)
which proves that genco k can game by releasing its output from its capacity only by increasing
the price. Inequality (5.13) also proves that if the tangent condition is not met at a price slightly
below k , it is still not met at a price slightly above.
Finally, to prove that (5.3) characterizes a NE for prices above k , we combine (5.13) with
the incremental profit of genco k when gaming and show that such profit is negative,
dprk   k  ak  bk g kmax  dSk  g kmax d 
   k  ak  bk g
max
k
 m

 S   , S  1  d
  j  k 0   
 jj 1k



(5.14)
 g kmax d 
0
5.3 Release Condition for Constrained Genco
As the numerical KM integration progresses for   k , the release price, kr  k , is
eventually reached. At kr and above, genco k no longer has an incentive to offer its maximum,
121
once again preferring to offer a SF defined by KM. To prove this, we examine the profit of genco
k when gaming incrementally around an arbitrary   k ,
dprk     ak  bk g kmax  dSk  g kmax d 
     ak  bk g
max
k
 m

 S    , S   1  d   g max d 
  j 0  
k
 jj 1k



(5.15)
For genco k to continue to offer its maximum as  increases relative to k , ensuring that its
incremental profit when gaming defined by (5.15) remains negative,
ng
 S j   , S0  
j 1
j k
1


g kmax
   ak  bk gkmax 
(5.16)
ng
Condition (5.16) also ensures that the term
1
 S  , S   
j 1
j k
j
0
remains positive, ensuring that the
 ng

1


incremental gaming action dSk    S j   , S0  
d   0 is feasible;
 j 1
 
 j k

The release condition (RC) defining the price kr at which genco k is released from its
maximum output is then,
ng
1
 S j  kr , S0   
j 1
j k

g kmax
 kr  ak  bk gkmax 
(5.17)
Finally, we show that immediately above the RC price, the released genco k continues to
move away from its capacity. First, from KM and equation (5.17), we see that the first derivative
of the SF is zero,
122
1 m  r
r

Sk  k , S0     S j  k , S0  


1

j 1
m
  S j  kr  , S0  
j 1
Sk  kr , S0 
kr  ak  bk Sk  kr , S0 
g kmax
kr  ak  bk g kmax
(5.18)
0
Then, since experimental evidence shows that the second derivative of the SF of any genco is
negative, for all prices above kr , the SF of genco k moves progressively away from its maximum
capacity.
Finally, we note that any genco that becomes constrained during the numerical
integration process is eventually released, and so, under CSFE, at the tangent condition, all
gencos operate below their capacity limits. The details of this result are given in Appendix H.
5.4 Change of Leading Genco under CSFE
If during the numerical integration process the leading genco becomes constrained, it
may still retain its leadership. This occurs if the demand level is such that the leader can choose
an initial condition S0t that clears the market at the tangent condition after all SFs are released.
However, for a sufficiently large demand, it may happen that, in searching for an initial condition
that leads to the equilibrium, the SF trajectory of the leading genco reaches its capacity at a price
below the entry point of the next non-leading genco that is not constrained, in which case such a
genco takes over the leadership role.
123
5.5 Examples
Consider a 5-genco system with the cost and capacity parameters shown in Table 5.1.
Table 5.1: Cost and capacity parameters for 5-genco system
genco
ai ($/MWh)
bi ($/MW2h)
gimax (MW)
1
20
0.1
80
2
30
0.1
65
3
35
0.1
50
4
38
0.1
11
5
40
0.1
35
The system demand elasticity is   0.4 $/MW2h.
Example 1: Single genco capacity constraint
Figure 5.1 shows the market solution for a demand with d0  260 MW.
124
Supply function of genco 1
Supply function of genco 2
Supply function of genco 3
Supply function of genco 4
Supply function of genco 5
Aggregate supply function
Demand
Clearing point
(67.77 $/MWh; 90.59 MW)
80
70
Price ($/MWh)
gmax
= 11MW
4
60
50
40
30
0
20
40
60
80
100
120
140
160
180
Generation/Demand (MW)
Figure 5.1: SFE with a single active genco capacity constraint
In this market solution, the SF of genco 4 becomes constrained by its capacity limit when
the price reaches 48.27$/MWh. Above this price, its SF becomes its capacity limit of 11 MW
until the release condition is met at 55.38$/MWh at which price the SF of genco 4 is once again
freed. Hence, in the price range between 48.27 and 55.38$/MWh, the remaining four active
gencos (that is, gencos 1, 2, 3 and 5) offer their SFs according to KM for a 4 genco system. For
prices above 55.38$/MWh, genco 4 is released and the SFs are once again governed by the KM
equations of a 5 genco system.
Table 5.2 shows the corresponding genco outputs and profits at market equilibrium.
125
Table 5.2: Genco outputs and profits for SFE with a single genco capacity constraint for
d0  260 MW
genco Output (MW) Profit ($/h)
1
41.58
1899.61
2
19.98
734.63
3
12.53
402.54
4
8.80
258.27
5
7.69
210.67
Example 2: Multiple genco capacity constraints
Figure 5.2 shows the market solution for a higher demand level with d0  300 MW.
126
75
Supply function of genco 1
Supply function of genco 2
Supply function of genco 3
Supply function of genco 4
Supply function of genco 5
Aggregate supply function
Demand
Clearing point
(70.93 $/MWh; 122.67 MW)
70
65
Price ($/MWh)
60
gmax
= 80 MW
1
55
50
45
40
35
30
0
50
100
150
200
Generation/Demand (MW)
Figure 5.2: SFE with multiple active genco capacity constraints for d0  300 MW
In this case, with gencos 1, 2 and 3 in the market, the SF of genco 1 becomes constrained
by its capacity limit when the price reaches 37.06$/MWh. Above this price, its SF becomes its
capacity limit of 80 MW until genco 1 satisfies the RC at 40 $/MWh. Hence, between 37.06
$/MWh and 38 $/MWh the system operates as a duopoly with gencos 2 and 3, whereas between
38 and 40 $/MWh the system becomes a triopoly with the market entry of genco 4 at 38 $/MWh.
On the other hand, the SF of genco 4 becomes constrained at 43.24 $/MWh and satisfies the RC
at 65.52 $/MWh. Note that in this example the leading genco 1 retains its leadership despite the
fact that its SF is constrained by its capacity limit for part of the SF trajectory.
127
Table 5.3 shows the corresponding genco outputs and profits at market equilibrium.
Table 5.3: Genco outputs and profits for SFE with multiple genco capacity constraints for
d0  300 MW
genco Output (MW) Profit ($/h)
1
50.33
2436.84
2
28.43
1123.14
3
19.78
691.29
4
10.56
342.14
5
13.57
410.56
Example 3: Change of leadership
When the demand is increased further to a level with d0  350 MW, genco 1 cannot meet
the tangent condition as the leader and relinquishes its leadership to genco 2. The original
leader, genco 1, offers its capacity of 80 MW till the price at which its release condition is met
near 46 $/MWh. The SFs are shown in Figure 5.3 while Table 5.4 shows the corresponding
genco outputs and profits at market equilibrium.
128
Supply function of genco 1
Supply function of genco 2
Supply function of genco 3
Supply function of genco 4
Supply function of genco 5
Aggregate supply function
Demand
Clearing point
(86.57 $/MWh; 133.57 MW)
110
100
90
Price ($/MWh)
gmax
= 80 MW
1
80
70
60
50
40
0
50
100
150
200
250
Generation/Demand (MW)
Figure 5.3: SFE with multiple genco capacity constraints and leadership change for
d0  350 MW
Table 5.4: Genco outputs and profits for the SFE with multiple genco capacity constraints
genco Output (MW) Profit ($/h)
1
50.31
3222.85
2
32.07
1763.04
3
23.84
1201.27
4
9.87
474.65
5
17.47
798.21
129
Chapter 6: Conclusions
This thesis develops a supply function equilibrium model in which the genco supply
functions need only be differentiable and continuous in price but not otherwise restricted to a
pre-defined shape. It is assumed that gencos behave as rational entities and want to maximize
their profits while taking into account the risk of being undercut by the market offers of the
competing gencos. A judicious balance between profit maximization and associated risk is the
Nash Equilibrium (NE) paradigm and all gencos are assumed to prefer the NE market outcome.
It is further assumed that in setting their market offers gencos do not collude, relying only on
estimates of the true costs and generating capacities of their competitors.
Under the above assumptions, genco supply functions are built by numerically integrating the
KM equations from an initial condition shown to be under the sole control of the leading genco.
The resulting genco supply functions are unique, defined as those that clear the market under the
tangent condition where the slope of the aggregate genco supply function equals the slope of the
demand. The thesis shows that within the SFE paradigm such a clearing point not only
maximizes the profit of the leading genco but it is focal since it corresponds to higher profits for
all gencos compared to other SF equilibria. Furthermore, the tangent clearing point is a global
NE when, for prices above the market price, gencos offer to keep their outputs constant and
equal to their market clearing outputs.
The SFE model is tested on a number of power system examples, including a comparison of
the SFE model with three other gaming models, namely, RBSFE, which requires that genco
supply functions be non-decreasing in price, Cournot and perfect competition. Simulations show
130
that all genco profits under SFE are considerably higher than under Cournot, with a minor
exception at very low demand levels. Additionally, the right-bending SFE model generally yields
profits lower than Cournot for all gencos but the leader. It is therefore concluded that SFE would
be preferred by all gencos over Cournot and RBSFE, and that all gencos but the leader would
prefer Cournot over RBSFE.
The impact of demand prediction uncertainty on SFE has been examined via the notions of
SFE paths and participation factors. By pre-computing the genco SF paths over a predicted range
of demand levels, we are able to compute the genco participation factors, which simulations
show is approximately constant over a wide range of equilibrium prices. These participation
factors are then used to track the SFE under randomly varying demand levels in a manner akin to
automatic generation control. Simulations with large numbers of randomly generated demands
show that, even in the presence of prediction uncertainty, the dispatch of generation using
participation factors found from the SFE paths ensure that the system operates close to SFE.
The levying of a carbon tax on the pollutants emitted by gencos has been studied in the
context of SFE. Such a tax favours gencos using clean technology by reducing their incremental
costs (ICs) relative to the ICs of the more polluting gencos. Under SFE, this is shown to translate
into gaming opportunities for the cleaner gencos that lead to increased profit margins.
Additionally, a sufficiently high carbon taxation level can change the genco leadership hierarchy,
bestowing a clean genco with the market leverage associated with being the leading genco.
The thesis extends the SFE model to gencos that can become constrained by their capacity
limits (CSFE). The SFs then become a piecewise mix of trajectories either governed by the KM
equations or defined by the genco capacity limit. The latter persists over a range of prices until
131
the Release Condition (RC) is satisfied at which point the constrained genco is released and its
offer is once again defined by KM. Under the CSFE model, for an oligopoly with two or more
unconstrained gencos, all constrained gencos eventually release from their capacity limits. The
constrained SFE therefore also clears by satisfying the tangent condition as per the unconstrained
SFE model. If during its SF trajectory the leading genco becomes constrained, it can still retain
its leadership provided that it can set an initial condition such that the resulting market
equilibrium satisfies the tangent condition. For high demands this may not be feasible and the
incumbent leading genco then passes the leadership onto the free genco with the next cheapest
marginal cost at zero output. The new leader then determines the initial condition from which the
numerical integration of KM starts while the previous leading genco offers at its capacity limit
till the price at which it releases to become an unconstrained genco offering according to KM
equations.
In addition to the theoretical results presented in this thesis advancing the problem of SFE in
oligopolies with general supply function shapes, we anticipate a number of potential practical
impacts on electricity markets:
(a) If left-bending SFs were permitted, all gencos would increase their profits. This would
provide an economic stimulus to invest in new generating plants, which in due course
would lead to a corresponding decrease in market power and prices;
(b) Based on this SFE model, regulators could analyze whether genco profits would be
excessive. If that is the case, the regulator could examine the imposition of disincentives
such as a price cap or the splitting of gencos with excessive market power into smaller
companies;
132
(c) Where current policy forbids left-bending SFs, the thesis shows that all gencos except the
leader would prefer to game according to Cournot. In such cases, SFE need not even be
considered as a gaming option;
(d) To narrow down the range of multiple SFE equilibria, the standard approach today
considers a random spectrum of demands for a single time period or several predicted
demands for multiple time periods. This thesis takes a different tack by finding for an
arbitrary predicted demand the unique corresponding SFs that maximize individual genco
profits. As the predicted demand varies with time, the market operator receives the
corresponding genco offers, stores them and constructs the SF equilibrium path for each
genco. Demand prediction uncertainty is then accounted for by a mechanism similar to
the traditional power systems automatic generation control (AGC) via participation
factors, which as this thesis shows are computed by the market operator from the genco
SF equilibrium paths. The advantage of this approach is that if the actual demand differs
from the predicted demand, the genco outputs will be automatically varied by the AGC
and will continue to operate very close to the SFE for the actual demand. This approach
can also be used when the SFs are based on Cournot.
6.1 Extensions
The following are interesting future research topics arising from the SFE model described in
the thesis:
133
1) Gencos with multiple generating units. Here, each genco first finds its aggregate piecewise IC
function from the individual generating unit ICs. If the aggregate function is continuous, the
approach presented in the thesis can be used without major modification. On the other hand, if
the aggregate IC function is discontinuous due to plants with radically different technologies, a
number of issues have to analyzed and resolved, in particular when such discontinuities arise in
the IC function of the leading genco.
2) Demand-side gaming. If the market contains several demand aggregators and the served
demand has some flexibility, such aggregators may be able to bid strategically so as to maximize
their profit within the NE paradigm. Points that require further research are to determine whether
a set of conditions similar to KM apply to the gaming demand aggregators and whether the
tangent condition remains applicable.
3) Transmission flow limits. When transmission lines become active, the single market price
must be replaced by nodal prices, one per bus, that can be shown to be functions of the system IC
and the Lagrange multipliers of the active transmission constraints. Extensive research is
required here to formulate the first order NE conditions and to determine whether equilibria exist
and under what circumstances.
4) Uncertainty in genco cost and capacity parameters. The assumption made in the thesis that
such parameters are common knowledge and perfectly known is optimistic. One possible way of
examining the effect of this uncertainty may be by assuming that the zero output IC of each
genco lies within a known range. The maximum profit tangent condition would have to be
modified such that the most extreme combination of these uncertain parameters still leads to a
market equilibrium.
134
References
[1]
S. Stoft, Power system economics: designing markets for electricity. New York: IEEE
Press & Wiley-Interscience, 2002.
[2]
E. O. Taylor and G. A. Boa, Power system economics. London: Edward Arnold Ltd.,
1969.
[3]
International Energy Agency, Electricity market reform: an IEA handbook Paris:
OECD/IEA, 1999.
[4]
S. Hunt and G. Shuttleworth, " Unlocking the Grid," IEEE Spectrum, vol. 33, no. 7, pp.
20-25, 1996.
[5]
Energy Information Administration, Electricity Reform Abroad and U.S. Investment.
Washington: U.S. Department of Energy, 1997.
[6]
International Energy Agency, Competition in electricity markets. Paris: OECD/IEA,
2001.
[7]
S. Jia, "A study of the England and Wales power pool," M.S. thesis, Dept. Elec. and
Comp. Eng, McGill University, Montreal, 1998.
[8]
D. Finlay and G. Gross, "Generation Supply Bidding in Perfectly Competitive Electricity
Markets," Computational and Mathematical Organization Theory, special issue on user
competition in Operations Management, vol. 6, pp. 83-98, 2000.
[9]
A. K. David and F. Wen, "Strategic Bidding in Competitive Electricity Markets: A
Literature Survey," in IEEE Power Engineering Society Summer Meeting Seattle, 2000.
135
[10]
A. K. David and F. Wen, "Market Power in Electricity Supply," IEEE Power
Engineering Review, vol. 21, no. 12, pp. 67-68, 2001.
[11]
J. Nash, "Non-cooperative games," Annals of Mathematics, vol. 54, no. 2, pp. 286-295,
1951.
[12]
E. P. Kahn, "Numerical Techniques for Analyzing Market Power in Electricity," The
Electricity Journal, vol. 11, no. 6, pp. 34-43, 1998.
[13]
S. Borenstein, J. Bushnell, and C. R. Knittel, "Market power in electricity markets:
beyond concentration measures," The Energy Journal, vol. 20, no. 4, pp. 65-88, 1999.
[14]
A. .
ood, . F.
ollenberg, and G. . Sheblé, Power generation, operation, and
control, 3rd ed. Hoboken, New Jersey: Wiley-Interscience, 2014.
[15]
M. Carrion and J. M. Arroyo, "A computationally efficient mixed-integer linear
formulation for the thermal unit commitment problem," IEEE Transactions on Power
Systems, vol. 21, no. 3, pp. 1371-1378, 2006.
[16]
A. Kalantari, J. F. Restrepo, and F. D. Galiana, "Security-Constrained Unit Commitment
With Uncertain Wind Generation: The Loadability Set Approach," IEEE Transactions on
Power Systems, vol. 28, no. 2, pp. 1787-1796, 2013.
[17]
M. Motamedi, S. Moeeni, S. Gharakhani, and I. Keyfarokhi, "The Behavior of Iranian
Restructured Electricity Market in Supply Function Equilibrium Framework,"
International Journal of Academic Research in Business and Social Sciences, vol. 4, no.
1, pp. 178-191, 2014.
[18]
Y. Ren and F. D. Galiana, "Pay-as-bid versus marginal pricing-part I: Strategic generator
offers," IEEE Transactions on Power Systems, vol. 19, no. 4, pp. 1771-1776, 2004.
136
[19]
N. Fabra, "Tacit collusion in repeated auctions: uniform versus discriminatory," The
Journal of Industrial Economics, vol. 51, no. 3, pp. 271-293, 2003.
[20]
S. F. Tierney, T. Schatzki, and R. Mukerji, "Uniform-Pricing versus Pay-as-Bid in
Wholesale Electricity Markets: Does it Make a Difference?," New York ISO, March 2008.
[21]
G. B. Sheblé, Computational Auction Mechanisms for Restructured Power Industry
Operation, 1st ed. New York: Kluwer Academic Publisbers, 1999.
[22]
M. J. Exelby and N. J. D. Lucas, "Competition in the UK market for electricity
generating capacity: A game theory analysis," Energy Policy, vol. 21, no. 4, pp. 348-354,
1993.
[23]
M. entosa, . a llo, A. Ramos, and M. Rivier, "Electricity market modeling trends,"
Energy Policy, vol. 33, no. 7, pp. 897-913, 2005.
[24]
R. W. Ferrero, S. M. Shahidehpour, and V. C. Ramesh, "Transaction analysis in
deregulated power systems using game theory," IEEE Transactions on Power Systems,
vol. 12, no. 3, pp. 1340-1347, 1997.
[25]
D. Fudenberg and J. Tirole, Game Theory. Cambridge: MIT Press, 1991.
[26]
E. Hasan and F. D. Galiana, "Electricity Markets Cleared by Merit Order - Part II:
Strategic Offers and Market Power " IEEE Transactions on Power Systems, vol. 23, no.
2, pp. 372 - 379, 2008.
[27]
M. de Lujan Latorre and S. Granville, "The Stackelberg equilibrium applied to AC power
systems-a noninterior point algorithm," IEEE Transactions on Power Systems, vol. 18,
no. 2, pp. 611-618, 2003.
[28]
J. Bertrand, "Review of Walras's Théorie mathématique de la richesse sociale and
Cournot's Recherches sur les principes mathématiques de la théorie des richesses," in
137
Cournot oligopoly Characterization and Applications, A. F. Daughety, Ed., 1st ed
Cambridge: Cambridge University Press, 1989, pp. 73-81.
[29]
B. F. Hobbs, "Network Models of Spatial Oligopoly with an Application to Deregulation
of Electricity Generation," Operations Research, vol. 34, no. 3, pp. 395-409, 1986.
[30]
B. F. Hobbs, "Mill pricing versus spatial price discrimination under Bertrand and Cournot
spatial competition," The Journal of Industrial Economics, vol. 35, no. 2, pp. 173-191,
1986.
[31]
N. T. Bacon, Researches into the mathematical principles of the theory of wealth by
Augustin Cournot. 1838. New York: The MacMillan Company, 1897.
[32]
Z. Younes and M. Ilic, "Generation strategies for gaming transmission constraints: will
the deregulated electric power market be an oligopoly?," Decision Support Systems, vol.
24, no. 3–4, pp. 207-222, 1999.
[33]
S. Borenstein and J. Bushnell, "An empirical analysis of the potential for market power in
California’s electricity industry," The Journal of Industrial Economics, vol. 47, no. 3, pp.
285-323, 1999.
[34]
L. B. Cunningham, R. Baldick, and M. L. Baughman, "An empirical study of applied
game theory: transmission constrained Cournot behavior," IEEE Transactions on Power
Systems, vol. 17, no. 1, pp. 166-172, 2002.
[35]
G. Bautista, M. F. Anjos, and A. Vannelli, "Beyond the Use of Linear Approximations
for Modelling Nash-Cournot Equilibria," in Power Tech, 2007 IEEE Lausanne, 2007, pp.
831-836.
138
[36]
B. F. Hobbs and J. S. Pang, "Nash-Cournot Equilibria in Electric Power Markets with
Piecewise Linear Demand Functions and Joint Constraints," Operations Research, vol.
55, no. 1, pp. 113-127, 2007.
[37]
J. Yao, I. Adler, and S. S. Oren, "Modeling and Computing Two-Settlement Oligopolistic
Equilibrium in a Congested Electricity Network," Operations Research, vol. 56, no. 1,
pp. 34-47, 2008.
[38]
C. J. Day, B. F. Hobbs, and J. S. Pang, "Oligopolistic competition in power networks: a
conjectured supply function approach," IEEE Transactions on Power Systems, vol. 17,
no. 3, pp. 597-607, 2002.
[39]
B. F. Hobbs, "Linear complementarity models of Nash-Cournot competition in bilateral
and POOLCO power markets," IEEE Transactions on Power Systems, vol. 16, no. 2, pp.
194-202, 2001.
[40]
P. Klemperer and M. Meyer, "Supply function Equilibria in Oligopoly Under
Uncertainty," Econometrica, vol. 57, no. 6, pp. 1243-1277, 1989.
[41]
R. J. Green and D. M. Newbery, "Competition in the British electricity spot market,"
Journal of Political Economy, vol. 100, no. 5, pp. 929-953, 1992.
[42]
F. Bolle, "Supply function equilibria and the danger of tacit collusion: The case of spot
markets for electricity," Energy Economics, vol. 14, no. 2, pp. 94-102, 1992.
[43]
R. Green, "Increasing Competition in the British Electricity Spot Market," Journal of
Industrial Economics, vol. 44, no. 2, pp. 205-216, 1996.
[44]
D. M. Newbery, "Competition, contracts, and entry in the electricity spot market," The
RAND Journal of Economics, vol. 29, no. 4, pp. 726-749, 1998.
139
[45]
E. J. Anderson and H. Xu, "Supply Function Equilibrium in Electricity Spot Markets with
Contracts and Price Caps," Journal of Optimization Theory and Applications, vol. 124,
no. 2, pp. 257-283, 2005.
[46]
A. Rudkevich, M. Duckworth, and R. Rosen, "Modeling electricity pricing in a
deregulated generation industry: the potential for oligopoly pricing in a poolco," The
Energy Journal, vol. 19, no. 3, pp. 19-48, 1998.
[47]
P. Holmberg, "Unique supply function equilibrium with capacity constraints," Energy
Economics, vol. 30, no. 1, pp. 148-172, 2008.
[48]
A. Rudkevich, Supply Function Equilibrium in Power Markets: Learning All the Way
Tabors Caramanis & Associates, 1999.
[49]
R. Baldick, R. Grant, and E. P. Kahn. (2000). Linear Supply Function Equilibrium:
Generalizations, Application, and Limitations. Available:
www.ucei.berkeley.edu/ucei/PDF/pwp078.pdf
[50]
R. Baldick, "Electricity market equilibrium models: the effect of parametrization," IEEE
Transactions on Power Systems, vol. 17, no. 4, pp. 1170-1176, 2002.
[51]
A. G. Petoussis, Z. Xiao-Ping, S. G. Petoussis, and K. R. Godfrey, "Parameterization of
Linear Supply Functions in Nonlinear AC Electricity Market Equilibrium
Models&#x2014;Part I: Literature Review and Equilibrium Algorithm," IEEE
Transactions on Power Systems, vol. 28, no. 2, pp. 650-658, 2013.
[52]
R. Baldick, R. Grant, and E. Kahn, "Theory and Application of Linear Supply Function
Equilibrium in Electricity Markets," Journal of Regulatory Economics, vol. 25, no. 2, pp.
143-167, 2004/03/01 2004.
140
[53]
B. F. Hobbs, C. B. Metzler, and J. S. Pang, "Strategic gaming analysis for electric power
systems: an MPEC approach," IEEE Transactions on Power Systems, vol. 15, no. 2, pp.
638-645, 2000.
[54]
X. Hu and D. Ralph, "Using EPECs to Model Bilevel Games in Restructured Electricity
Markets with Locational Prices," Operations Research, vol. 55, no. 5, pp. 809-827, 2007.
[55]
J. D. Weber and T. J. Overbye, "An individual welfare maximization algorithm for
electricity markets," IEEE Transactions on Power Systems, vol. 17, no. 3, pp. 590-596,
2002.
[56]
H. Chen, K. P. Wong, C. Y. Chung, and D. H. M. Nguyen, "A coevolutionary approach
to analyzing supply function equilibrium model," IEEE Transactions onPower Systems,
vol. 21, no. 3, pp. 1019-1028, 2006.
[57]
L. Youfei and F. F. Wu, "Impacts of Network Constraints on Electricity Market
Equilibrium," IEEE Transactions on Power Systems, vol. 22, no. 1, pp. 126-135, 2007.
[58]
R. Baldick and W. Hogan. (2002). Capacity Constrained Supply Function Equilibrium
Models of Electricity Markets: Stability, Non-decreasing constraints, and Function Space
Iterations Available: http://www.ucei.berkeley.edu/PDF/pwp089.pdf
[59]
R. Baldick and W. Hogan, "Stability of supply function equilibria implications for daily
versus hourly bids in a poolco market," Journal of Regulatory Economics, vol. 30, no. 2,
pp. 119-139, 2006.
[60]
C. J. Day and D. W. Bunn, "Divestiture of Generation Assets in the Electricity Pool of
England and Wales: A Computational Approach to Analyzing Market Power," Journal of
Regulatory Economics, vol. 19, no. 2, pp. 123-141, 2001.
141
[61]
E. J. Anderson and X. Hu, "Finding Supply Function Equilibria with Asymmetric Firms,"
Operations Research, vol. 56, no. 3, pp. 697-711, 2008.
[62]
P. Holmberg, "Numerical calculation of an asymmetric supply function equilibrium with
capacity constraints," European Journal of Operational Research, vol. 199, no. 1, pp.
285-295, 2009.
[63]
R. Sioshansi and S. Oren, "How good are supply function equilibrium models: an
empirical analysis of the ERCOT balancing market," Journal of Regulatory Economics,
vol. 31, no. 1, pp. 1-35, 2007.
[64]
A. Hortaçsu and S. L. Puller, "Understanding strategic bidding in multi-unit auctions: a
case study of the Texas electricity spot market," The RAND Journal of Economics, vol.
39, no. 1, pp. 86-114, 2008.
[65]
N. Hui, R. Baldick, and Z. Guidong, "Supply function equilibrium bidding strategies with
fixed forward contracts," IEEE Transactions on Power Systems, vol. 20, no. 4, pp. 18591867, 2005.
[66]
J. Bushnell, "Looking for Trouble: Competition Policy in the U.S. Electricity Industry,"
in Electricity Restructuring: Choices and Challanges, James M. Griffin and S. L. Puller,
Eds., 1st ed: University of Chicago Press, 2005, pp. 256-296.
[67]
F. D. Galiana, F. Bouffard, J. M. Arroyo, and J. F. Restrepo, "Scheduling and pricing of
coupled energy and primary, secondary and tertiary reserves," Proceedings of the IEEE,
vol. 93, no. 11, pp. 1970-1983, 2005.
[68]
I.-K. Cho and S. P. Meyn, "Efficiency and marginal cost pricing in dynamic competitive
markets with friction," Theoretical Economics, vol. 5, no. 2, pp. 215-239, 2010.
142
[69]
S. El-Khatib, "Oligopolistic Electricity Markets under Cap-and-trade and Carbon Tax,"
Ph.D. dissertation, Dept. Elec. and Comp. Eng, McGill University, Montreal, 2011.
[70]
S. El-Khatib and F. D. Galiana, "An emission allowance auction for an oligopolistic
electricity market operating under cap-and-trade," IET Generation, Transmission &
Distribution, vol. 4, no. 2, pp. 191-200, 2010.
143
Appendix A : Cournot Gaming
Under Cournot, gencos game with a quantity offer. This means that no single genco can
improve its profit by varying its output while balancing the demand and while all remaining
gencos maintain their outputs constant.
To find the genco outputs under Cournot, we consider the profit of genco i, pri (c ) , at the
clearing price c , which is the difference between its revenue and cost of production,
pri (c )  c Si (c )  ai Si (c ) 
bi
Si (c ) 2
2
(A.1)
where Si (c ) is the supply function of genco i at the market clearing price   c according to
Cournot strategy. Thus, if genco i games by varying its output by dSi , the price will change by
d c and, since the gaming action must satisfy the power balance and all other gencos offer
constant outputs, we must have,
dSi  
1

d c
(A.2)
Thus, the incremental profit of genco i when gaming becomes,
dpri (c )   c  ai  bi Si (c )  dSi  Si (c )d c
 1

  c  ai  bi Si (c )    d c   Si (c )d c
 

(A.3)
144
Since the first order optimality condition for maximizing the profit of genco i requires that
dpri (c )
 0 , from (A.3),
d c
Si (c ) 
c  ai
bi  
(A.4)
Where, according to the power balance,
 S ( )  d
j
c
0

j
c

(A.5)
Appendix B : Characteristics of the Genco Supply Functions
Property 1: At the tangent condition, Si  t , S0t  , the slope of each genco supply function,
satisfies,
Si  t , S0t   
Si  t , S0t 
i
t  ai  bi Si  t , S0t 
(B.1)
Proof: The KM equations are,
1
Si   , S 0 
i
i  bi Si ( , S0 )
 S  , S       a
j i
j
0
(B.2)
We know that at the tangent condition, the slope of the aggregate supply function is equal to the
slope of the demand, that is,
145
1
 S   , S    
j
t
(B.3)
0t
j
from which we get,
1
 S    , S     S    , S 
j
j i
t
0t
i
t
(B.4)
0t
Substituting (B.4) into (B.2), we obtain,
Si  t , S0t   
Si  t , S0t 
i
t  ai  bi Si  t , S0t 
(B.5)
Property 2: When the slope of the aggregate supply function is less than or equal to the slope of
1
 S    , S     ), the second derivative of the aggregate genco supply
the demand (that is,
j
0
j
function is negative, that is,
 S    , S   0
j
0
j
Proof: The KM equations are,
Si   , S 0 
i
i  bi Si ( , S0 )
1
 S  , S       a
j
j i
0
(B.6)
Aggregating (B.6) over all gencos, we obtain,

S , S 
1
 S    , S       a  b S   , S    


i
i
j i
j
0
0
i
i
i
i
0

Si   , S 0 
1
or ,  ng  1  S j   , S0    
 

j
i    ai  bi Si   , S 0 

Si   , S 0 
1
1
or ,  S j   , S0  
 


ng  1 i    ai  bi Si   , S0   
j
(B.7)
146
where, ng is the total number of gencos in the market.
Differentiating (B.7) with respect to  , we obtain the second derivative of the aggregate genco
supply function,
     a  b S ( , S )  S    , S   
i
i i
0
i
0


   S ( , S ) 1  b S (  , S )

i
0 
i i
0 
1



j S j   , S0   ng  1 i 
2

   ai  bi Si ( , S0 ) 






    a  S    , S   S ( , S ) 
1
i
i
0
i
0




2
ng  1 i 
   ai  bi Si ( , S0 )  

Now, if
(B.8)
1
 S    , S     , using the same argument which proves Property 1, we obtain,
j
0
j
S i   , S 0   
Si   , S 0 
i
  ai  bi Si   , S0 
(B.9)
Substituting (B.9) into (B.8), the second derivative of the aggregate supply function becomes,
147
   a  S  , S   S , S  
1
i
i
0
i
0



2

ng  1 i


a

b
S

,
S
 i i i  0   





Si   , S 0 
    ai   
  Si   , S 0  
1


   ai  bi Si   , S0  


2


ng  1 i
   ai  bi Si   , S0  






    ai  Si   , S0 



3



a

b
S

,
S


 i i i 0  
1



ng  1 i 

Si   , S 0 

2 
    ai  bi Si   , S0   


 S j   , S0  
j
(B.10)
Since the term under summation on the left hand-side (LHS) of (B.10) is positive, the LHS of
(B.10) is negative. Therefore,
 S    , S   0 .
j
0
j
Property 2.1: At tangent condition, the second derivative of the aggregate supply function is
negative, that is,
 S    , S   0 .
j
t
0t
j
Proof: At the tangent condition,
1
 S    , S     . Therefore, from Property 2, it follows that
j
0
j
 S    , S   0 .
j
t
0t
j
Property 3: For prices below the tangent condition, the slope of the aggregate supply function is
greater than the slope of the demand, that is,
148
1
 S   , S     ;
j
0t
  t
(B.11)
j
Proof: Let us assume that (B.11) is false at a price a  t . Then, there are two cases:
(1) No genco enters the market at prices a    t
(2) At least one genco enters the market for a price within a    t
It will be shown that in both cases the assumption that (B.11) is false is not tenable.
Proof for case 1 (that is, when no genco enters the market at a price a    t ):
If (B.11) is false then for some a  t ,
1
 S   , S    
j
a
(B.12)
0t
j
Then, from Property 2,
 S    , S   0
(B.13)
1
 S  , S    
(B.14)
j
a
0t
j
Then for   a ,
j
0t
j
However, at   t  a , we require
1
 S   , S    
j
t
0t
which contradicts (B.14). Hence, we
j
conclude that (B.12) is false thereby establishing (B.11).
149
Proof for case 2 (that is, at least one genco enters the market for a price within a    t ):
We consider that the ngth genco enter the market at   ang where a  ang  t . Let us
  ang , S0t  and S new
  ang , S0t  the respective slopes of the gencos already active at
denote by S old
j
j
  ang , before and after the market entry of the ngth genco.
Before the ngth genco enters the market, we have ng  1 genco in the market and therefore, from
(B.7), the slope of the aggregate supply function is,
 S oldj  ang , S0t  
j
Si  ang , S0t 
1 ng 1 
1




ng  2 i 1  ang  ai  bi Si  ang , S0t   


(B.15)
After the market entry of the ngth genco, the KM equations become,
ng 1
S
j 1
ng
S
j 1
j k
S  a , S 
 (ang , S0t )  1  ng ng 0t
*
 1  bng
S ng  ang 
new
j
 (ang , S0t )  1 
new
j

(B.16)
Sk (ang , S0t )
ang  ak  bk Sk (ang , S0t )
k  ng
We can re-write the second equation in (B.16) as,
ng 1
S
j 1;
j k
 (ang , S0t ) 
new
j
Sk (ang , S0t )
ang  ak  bk Sk (ang , S0t )

1

 Sng (ang , S0t ); k  ng
(B.17)
Aggregating (B.17) over k and re-arranging,
150

Sk (ang , S0t )
1 ng 1 
1
  Sng  (ang ) 



ng  2 k 1  ang  ak  bk S k (ang , S0t ) 

ng 1
 (ang , S0t ) 
 S new
j
j 1

Sk (ang , S0t )
1 ng 1 
1
 


ng  2 k 1  ang  ak  bk S k (ang , S0t )  

ng  1 
Sng (ang , S0t )
ng  2
(B.18)
Adding the slope of the ngth genco to both sides of (B.18),
ng
 (ang , S0t ) 
 S new
j
j 1

Sk (ang , S0t )
1 ng 1 
1
 


ng  2 k 1  ang  ak  bk Sk (ang , S0t )  
1
Sng (ang , S0t )
ng  2
(B.19)
However, we know that the genco supply functions are continuous at   ang . Hence, we can
substitute the first term on the RHS of (B.19) with (B.15). Then, we obtain,
ng
ng 1
j 1
j 1
 (ang , S0t )   S old
 (ang , S0t ) 
 S new
j
j
1
Sng (ang , S0t )
ng  2
(B.20)
Since, Sng (ang , S0t ) must be positive, (B.20) shows that the slope of the aggregate supply
function becomes more negative with the market entry of the ngth genco.
Therefore, if for a price a  t ,
1
 S   , S    
j
a
0t
(B.21)
j
151
and thereafter a genco, for instance the ngth genco, enters the market before the tangent
condition, that is, a  ang  t , then (B.21) continues to remain valid. Hence, using the same
argument as case 1, we conclude (B.21) must be false thereby establishing (B.11).
Property 4: The aggregate supply function is less than the demand for prices below the price at
which the tangent condition occurs, that is,
 S , S   d
j
0t
0

j

;   t

(B.22)
Proof: Property 2.1 establishes that the second derivative of the aggregate supply function is
negative at the tangent condition. Therefore, to satisfy the tangent condition, the aggregate
supply function must approach the demand curve from below and hence, at least at a price just
below t , that is t , the aggregate supply function is less than the demand.
Now, suppose that for a price b  t , the aggregate supply function is higher than the demand.
Then, in order for the aggregate supply function to be less than the demand at t , the aggregate
supply function must intersect the demand with a slope lower than the slope of the demand at a
price  such that b    t , that is,
1
 S  , S     ;
j
0t
b    t
(B.23)
j
However, (B.23) contradicts (B.11), that is, Property 3. Hence, (B.23) must be false and,
therefore, (B.22) must be true.
152
Property 5: When genco supply functions follow KM, if a single genco i games at a price 
where the aggregate supply function is less than the demand, then the gaming profit of genco i,
pri gm ( , S0t ) , has a positive slope at  , that is,
pri gm ( , S0t )  0
if ,  S j   , S0t   d 0 
j
(B.24)


Proof: If the aggregate supply function is less than the demand, that is,  S j   , S0t   d 0 
j

,

gm
then, in order to meet the demand at  , the output of the gaming genco i, Si   , S0t  , must be
more than its supply function when it does not game, that is,
Sigm   , S0t   Si   , S0t 
(B.25)
However, we know that the market clears at t with the power balance relation,
Si (t , S0t )   S j (t , S0t )  d0 
j i
t

(B.26)
gm
Hence, if genco i unilaterally changes its offer to satisfy the demand at  , then Si   , S0t  must
satisfy,
Sigm ( , S0t )   S j ( , S0t )  d0 
j i


(B.27)
Subtracting (B.26) from (B.27) and re-arranging leads to,
153
Sigm ( , S0t )  Si (t , S0t )    S j (t , S0t )  S j ( , S0t )  
j i
t  

(B.28)
Differentiating (B.28) with respect to  , we get,
Sigm ( , S0t )    S j ( , S0t ) 
j i
1
(B.29)

We recall that the KM equations are,
1
Si   , S 0 
i
i  bi Si ( , S0 )
 S  , S       a
j i
j
0
(B.30)
Combining (B.29) with the KM equations in (B.30), we get,
Sigm ( , S0t )  
Si   , S 0 t 
  ai  bi Si ( , S0t )
(B.31)
gm
Now, by producing Si   , S0t  , the gaming genco i earns a profit, pri gm ( , S0t ) , given by,
pri gm ( , S0t )   Sigm ( , S0t )  ai Sigm ( , S0t ) 
bi gm
Si ( , S0t ) 2
2
(B.32)
By differentiating (B.32), we obtain,
pri gm ( , S0t )   Sigm ( , S0t )  Sigm ( , S0t )  ai Sigm ( , S0t )
bi Sigm ( , S0t ) Sigm ( , S0t )
(B.33)
    ai*  bi* Sires ( , S0t )  Sigm ( , S0t )  Sigm ( , S0t )
Putting (B.31) in(B.33), we obtain,
154
pri gm ( , S0t )      ai  bi Sigm ( , S0t ) 
Si (  , S 0 t )
 Sigm ( , S0t )
  ai  bi Si ( , S0t )
(B.34)
However, from (B.25), Sigm ( , S0t )  Si ( , S0t ) . So,
  ai  bi Sigm ( , S0t )
1
  ai  bi Si ( , S0t )
(B.35)
and, therefore, by putting (B.35) in (B.34), we obtain,
pri gm ( , S0t )   Si ( , S0t )  Sigm ( , S0t )
(B.36)
Replacing (B.25) in (B.36), we get,
pri gm ( , S0t )  0
(B.37)
which proves (B.24).
Property 6:
ng
 r   , S   1/   0;   
j 1
where, rj   , S0t  
j
0t
S j ( , S 0 t )
  a j  b j S j ( , S 0 t )
t
(B.38)
and ng is the number of genco in the market at price 
.Proof: We know that the KM equations are,
155
S , S 
1
 S    , S       a  b S ( , S
i
j
j i
0t
0t
i
i
i
0t
)
 ri   , S0t 
(B.39)
i
Aggregating (B.39) for these ng gencos, we obtain,

1
 
ng
 ng  1  S j   , S0t     ri   , S0t  
j

i
ng
or ,  ng  1  S j   , S0t    ri   , S0t  
i 1
j

(B.40)

1  ng
1
or ,  ng  1   S j   , S0t      ri   , S0t  
  i 1

 j
From Property 3,
1
 S    , S     ;    . Hence, the left hand-side of (B.40) is positive for
j
0t
t
j
  t . Therefore,
ng
 r   , S   1/   0;   
j 1
j
0t
t
which proves (B.38).
Appendix C : Non-Fulfillment of Nash Equilibrium Optimality Condition if Genco
Supply Functions Continue to Follow KM above the Market Clearing
Price
At the tangent condition,
1
 S   , S    
j
t
0t
(C.1)
j
156
Also, from Property 2.1,
 S    , S   0
j
t
(C.2)
0t
j
Hence, if the genco supply functions continue to follow KM equations for   t , then at   t
, the slope of the aggregate supply function must satisfy,
 S  

t
j
j
1
, S0t   
(C.3)

Then, from Property 2,
 S   

t
j
j
, S0t   0
(C.4)
From (C.3) and (C.4), we conclude that for prices above the tangent condition, the aggregate
supply function decreases faster than the demand,
1
 S   , S     ;
j
0t
  t
(C.5)
j
Now, at   t , we know that the power balance relation is satisfied, that is,
 S  , S   d
j
t
0t
0

j
t

(C.6)

;   t

(C.7)
By combining (C.5) and (C.6), we obtain,
 S , S   d
j
j
0t
0

157
Hence, from Property 5, we know that if genco i were to game unilaterally, its gaming profit,
pri gm ( , S0t ) , will have positive slope,
pri gm ( , S0t )  0;   t
(C.8)
From (C.8), we conclude that pri gm ( , S0t ) is an increasing function for   t if only genco i
games and all other gencos follow KM equations. Hence, genco i can earn a profit higher than
its market clearing profit by gaming unilaterally in the price range   t .
Appendix D : Nash Equilibrium Condition for Market Clearing at Tangent
Condition with Modified Supply Functions
For demands cleared above a2 , the market clears by satisfying the tangent condition (that
is, aggregate supply function is tangential to demand) at a clearing price of t . For   t , the
genco supply functions follow the KM equations whereas for   t the supply functions remain
equal to their market clearing output. For both of these ranges of price, it is shown that the
condition for Nash Equilibrium (NE) is satisfied.
NE for prices below the clearing price
We begin by dividing the prices below t into two ranges namely, one above a2 , that is,
a2    t and the other one below a2 , that is, a1    a2 . Then, we demonstrate that for both
these price ranges the NE condition is satisfied by all gencos.
158
1) NE for a2    t
For a2    t , all genco supply functions follow KM equations. Hence, Property 4
establishes that,
 S , S   d
j
0t
0

j

; a    t
 2
(D.1)
Hence, from Property 5, we know that if genco i were to game unilaterally, its gaming profit,
pri gm ( , S0t ) , will have positive slope,
pri gm ( , S0t )  0; a2    t
(D.2)
From (D.2), we conclude that pri gm ( , S0t ) is an increasing function for a2    t . Hence, for
this price range, pri gm ( , S0t )  pri (t , S0t ) meaning that the Nash Equilibrium condition is
satisfied for a2    t .
2) NE for a1    a2
Non-leading gencos do not participate in the market for a1    a2 , as in this price range
the non-leading gencos cannot earn profit. Hence, if the market is cleared within a1    a2 ,
the leading genco has to game and the entire demand will be satisfied by S1gm ( , S0t ) , the gaming
output of the leading genco, that is,
159
S1gm ( , S0t )  d0 


(D.3)
By gaming, the leading genco earns the profit,


b
pr1gm ( , S0t )   S1gm ( , S0t )   a1 S1gm ( , S0t )  1 S1gm ( , S0t ) 2 
2


(D.4)
Substituting (D.3) in (D.4), we obtain,



 b
pr ( , S0t )    d 0    a1  d 0    1

 2


gm
1
a b
2
  d 0   a1 d 0  1  1


2


 d0  




 d0  


2
2
(D.5)
Differentiating (D.5) with respect to  ,

  1 
2  d0     
 
   

2 a1 b1 

 d0 
   d0  
  

  a b 

 d0    1  1  d0  
   

 b   a 
 d 0  1  1     1 
      
  b      a1
 d0   1

  

 b   
     a1  
 1

   d 0    
   b1    
  
pr1gm ( , S0t )  d 0 
2

a1

b1
2
(D.6)
160
In (D.6), the first term
b1  

is always positive since both b1 and  are positive quantities.
Additionally, demands considered here are sufficiently high for the market to clear by satisfying
the tangent condition at a price   a2 . This means that for a1    a2 , the demand function,
d0 
  a1

, is greater than the Cournot strategic offer,
, of the leading genco 1 since,
b1  

otherwise, the market would have cleared within the price range a1    a2 . Therefore, for

     a1 
a1    a2 , the term  d0    
 in (D.6), is positive. Hence, from (D.6),
   b1   

pr1gm ( , S0t )  0
(D.7)
From (D.7), we conclude that pr1gm ( , S0t ) is an increasing function of  for a1    a2 .
Hence, pr1gm ( , S0t )  pr1gm (a2 , S0t ) for a1    a2 .
Moreover, we know that the supply functions of all non-leading gencos as well as the
demand are continuous at   a2 . Hence, the residual demand of the leading genco, and
consequently the corresponding profit function, pr1gm ( , S0t ) , are also continuous at   a2 .
Hence, combining (D.2) and (D.7), we get pr1gm ( , S0t )  0   t meaning that the gaming
profit of the leading genco at any price   t is less than the genco’s market clearing profit
pr1 (t , S0t ) .
For a2    t (D.2) shows that the gaming profit of a non-leading genco i,
pri gm (t , S0t ) , is also less than its market clearing profit, pri (t , S0t ) , whereas, for a1    a2 ,
161
no non-leading genco is active in the market. Hence, the gaming profit of a non-leading genco i
at any   t is also less than its market clearing profit pri (t , S0t ) .
Therefore, for   t , conditions for a NE is satisfied for both the leading and the nonleading gencos since no genco can unilaterally change its supply function and earn a profit
higher than its market clearing profit.
NE for prices above the clearing price
For prices higher than the clearing price , that is,   t , all gencos keep their supply
functions equal to their market clearing output Si (t , S0t ) .
At the market clearing point,
Si (t , S0t )   S j (t , S0t )  d0 
j i
t

(D.8)
The profit of genco i at the market clearing solution is,
pri (t , S0t )  t Si (t , S0t )  ai Si (t , S0t ) 
bi
Si (t , S0t ) 2
2
(D.9)
For   t , if genco i unilaterally changes its offer and satisfies the demand then its gaming
output, Sigm ( , S0t ) , must satisfy the power balance at  , that is,
Sigm ( , S0t )   S j (t , S0t )  d0 
j i


(D.10)
162
Subtracting (D.10) from (D.8) and re-arranging gives,
Sigm ( , S0t )  Si (t , S0t ) 
  t

(D.11)
The gaming profit of the gaming genco i , pri gm ( , S0t ) ,at  then becomes,
pri gm ( , S0t )   Sigm ( , S0t )  ai Sigm ( , S0t ) 
bi gm
Si ( , S0t ) 2
2
(D.12)
Putting (D.11) in (D.12) we obtain,

  t
pri gm ( , S0t )    Si (t , S0t ) 


b
 i
2

  ai


  t 
 Si (t , S0t ) 
 


  t 
 Si (t , S 0t ) 
 

2
 Si (t , S0t ) 


    t 





2
bi 
  t    t  
2
  Si (t , S0t )  2 Si (t , S0t )

 
2 

   
b
 t Si (t , S0t )  ai Si (t , S0t )  i Si (t , S 0t ) 2
2
  ai  bi Si (t , S 0t ) 
  t 

    t  Si (t , S0t ) 
bi    t 



  2   



   Si (t , S0t )    ai  bi Si (t , S 0t ) 
  t 

 pri (t , S0t ) 
bi    t 




 2   



*
  ai   bi    Si (t , S0t ) 

  t 
 pri (t , S0t ) 
 bi    t 

  


 2  


  t
  t     t    Si (t , S0t ) 



  ai

(D.13)
163


Representing the term   ai  bi   Si (t , S0t ) 
pri gm ( , S0t )  pri (t , S0t ) 
bi    t

2 

 in (D.13) by T, we obtain,

  t
T

(D.14)
From (D.14), since   t , iff T >0, pri gm ( , S0t )  pri (t , S0t ) .
We will show that T >0 by showing that it is necessary (and sufficient) to assume T >0 to
derive an expression which is necessarily true.
Assume T >0, then
T    ai   bi    Si (t , S0t ) 
bi    t

2 

0

(D.15)
Re-arranging (D.15), we obtain,
Si (t , S0t ) 
  ai
bi
   t 

bi    2 bi      
(D.16)
Since   t and both bi and  are positive, the second term on the right hand-side of (D.16) is
positive. Therefore, to show that both (D.15) and (D.16) are true, it suffices to show,
Si (t , S0t ) 
  ai
bi  
(D.17)
At   t , we know,
Si (t , S0t )   S j (t , S0t )  
j i
1

(D.18)
164
Additionally, from Property 1, we also know at t , supply function of genco i satisfies,
Si (t , S0t )  
Si (t , S0t )
t  ai  bi Si (t , S0t )
(D.19)
Since genco i is participating in the market at   t , its supply function, Si (t , S0 ) , is positive
and the market price, t , must be greater than the genco’s true incremental cost,
ICi* (Si  t )  ai  bi Si (t , S0 ) .
Since the right hand-side of (D.19) is positive, we know,
Si (t , S0t )  0
(D.20)
Combining (D.18) and (D.20),

Si (t , S0t )
1

t  ai  bi Si (t , S0t )

(  ai )
or , Si (t , S0t )  t
(bi   )
(D.21)
From (D.21), it follows that for   t ,
Si (t , S0t ) 
t  ai   ai

bi   bi  
(D.22)
which proves (D.17).
165
This, in turn, proves that (D.15) is true. Consequently, from (D.14), for   t we get
pri gm ( , S0t )  pri (t , S0t ) meaning that the profit of the gaming genco i will be less than the
profit it earns at the clearing price t . Hence, the NE condition is fulfilled for   t .
Appendix E : KM Equations with Slope of the Genco Supply Functions Decoupled
for a System with more than 2 Gencos
The KM equations are,
Si   , S 0 
i

b
S
(

,
S
)
i
i i
0
1
 S  , S       a
j i
j
0
(E.1)
We represent the RHS of (E.1) by,
ri   , S0  
Si   , S 0 
  ai  bi Si ( , S0 )
(E.2)
Hence the KM equations become,
1
 S   , S   r  , S   
j i
j
0
i
0
i
(E.3)
For a system with ng gencos, we can represent (E.3) in the matrix form as,
0
1



1
1
1


1   S1   , S0    r1 ( , S0 ) 
 
  r ( , S 0 )  1

  S2   , S0     2

 
1 




0  S   , S   rng ( , S0 ) 

0 
 ng 
1
1



1
(E.4)
166
or in vector form,
M S   , S0   r   , S0  
1

1
(E.5)
where,
r   , S0  is the vector of the ratios ri   , S0  in (E.2),
S   , S0  is the vector of the derivatives of the genco supply functions,
1 is a vector of dimension ng whose entries are all 1,
M is a ng by ng matrix with its diagonal entries equal to 0 and non-diagonal entries equal to 1.
From (E.4) M is,
M  11T  I
(E.6)
where,
1 is the vector with its entries equal to 1 and,
I is the ng by ng identity matrix
We know from (E.4),

1 
S     M 1  r     1 
 

(E.7)
167
Now, using the matrix inversion lemma,
1
1
M 1  11T  I     I  1 * 1T     I  1 * 1T 
11T

I
ng  1
 2  ng
 1
1 
 .

ng  1 
 .
 1
1
(E.8)
1 
1 
.
. 

.
1 
. 1 2  ng 
1
2  ng
.
1
.
Substituting (E.8) into (E.7), we obtain,
 2  ng
 1
1 
 .
S   , S 0  
ng  1 
 .
 1
1
2  ng
1
1 
1 

1 
.
.   r   , S0   1 
 

.
1 
. 1 2  ng 
.
.
(E.9)
From the matrix in (E.9), the slope of the supply function of genco i, Si   , S0  , is found as,
ng
 r , S 
S i   , S 0  
j 1
j i
j
0
ng  1
 ng  2 
1

 ri   , S0  
  ng  1
 ng  1 
ng

 r , S 
j 1
j
ng  1
0
 ri   , S0  
(E.10)
1
;
  ng  1
168
Appendix F : Calculating the Slope of Genco Supply Functions at the Entry Price of
a Non-Leading Genco in an Oligopoly with more than 2 Gencos
For a system with ng gencos where ng is greater than 2, the decoupled form of KM equations
derived in Appendix E, is,
ng
 r , S 
S i   , S 0  
j 1
j i
j
0
ng  1
 ng  2 
1

 ri   , S0  
  ng  1
 ng  1 
(F.1)
At   ang , the ngth genco enters the market. Therefore, the KM equation for the the ngth genco at
  ang* becomes,
ng 1
ng  2
Sng  ang , S0  
rng  ang , S0  
ng  1


 r a
j 1
j
ng
, S0   1/ 
ng  1
(F.2)
At   ang ,
rng  ang , S0  
Sng  ang , S0 
  ai  bi Sng  ang , S0 
(F.3)
However, since (F.3) is of the form 0/0, l’ Hopital’s rule is invoked so that,
rng  ang , S0  
Sng  ang , S0 
1  bng Sng  ang , S0 
(F.4)
Substituting (F.4) in (F.2),
169
ng 1
ng  2  Sng  ang , S0  
Sng  ang , S0  

ng  1  1  bng Sng   ang , S0  


 r a
j
j 1
ng
, S0   1/ 
(F.5)
ng  1
Multiplying both sides of (F.5) by ng  1 , we obtain,

Sng  ang , S0 
 ng 1
   r  a , S   1/ 
 1  bng Sng  ang , S0   j 1 j ng 0


 ng  1 Sng  ang , S0    ng  2  
ng 1
We denote
 r a
j 1
j
ng
(F.6)
, S0   1/  by T. Hence, (F.6) becomes,

Sng  ang , S0 

 T
 1  bng Sng  ang , S0  


 ng  1 Sng  ang , S0    ng  2  
(F.7)
Re-arranging (F.7), we obtain,
 ng  1 Sng  ang , S0  1  bng Sng  ang , S0     ng  2  Sng  ang , S0 
1  bng S ng  ang , S0 
T

or ,  2ng  3 S ng  ang , S0    ng  1 bng S ng2  ang , S 0   T 1  bng S ng  ang , S 0 

(F.8)
or ,   2ng  3  bngT  S ng  ang , S0    ng  1 bng S ng2  ang , S 0   T  0
or ,  ng  1 bng S ng2  ang , S0     2ng  3  bngT  S ng   ang , S0   T  0
The solution to the quadratic equation in (F.8) is,
 2ng  3  bngT    2ng  3  bngT 
Sng  ang , S0  
2  ng  1 bng
2
 4  ng  1 bngT
(F.9)
Multiplying throughout by bng , we obtain,
170
 2ng  3  bngT    2ng  3  bngT 
bng Sng  ang , S0  
2  ng  1
2
 4  ng  1 bngT
(F.10)
Simplifying (F.10), we obtain,
 2ng  3  2  2ng  3 bngT   bngT 
4  ng  1 bngT
2  ng  1
2
 2ng  3  bngT 
bng S ng  ang , S0  
 2ng  3  bngT   2ng  3

2  ng  1

 2ng  3  bngT 
b
2
 2bngT   bngT 
2
(F.11)
T  1   2ng  3  1
2
ng
2
2
2  ng  1
Now, from Property 6, T  0 and hence,
bngT  0
(F.12)
We now take the solution with the positive square-root term in (F.11).
bng Sng  ang , S0  
 2ng  3  bngT 
b
T  1   2ng  3  1
2
ng
2  ng  1
2
(F.13)
Substituting (F.12) and noting that  bngT  1  0 , we obtain,
2
 2ng  3   2ng  3
bng Sng  ang , S0  
2  ng  1
where,
 2ng  3
2
2
1
(F.14)
 1  1 for ng  2 .
171
Hence,
bng Sng   ang , S0   1
or , Sng  ang , S0  
(F.15)
1
bng
Next, we take the solution with the negative square-root term in (F.11).
bng Sng  ang , S0  
 2ng  3  bngT 
b
ng
T  1   2ng  3  1
2
2
2  ng  1
(F.16)
We note that for ng  2 ,
b
ng
T  1 
2
b
ng
T  1   2ng  3  1
2
2
(F.17)
Substituting (F.17) into (F.16), we obtain,
 2ng  3  bngT   bngT  1
*
bng Sng  ang
, S0  
2  ng  1
 2ng  3  bngT  bngT  1

2  ng  1
2
(F.18)
1
Hence,
bng Sng   ang , S0   1
or , Sng  ang , S0  
1
bng
(F.19)
172
Moreover, we show that the solution in (F.16) results in a positive value for bng Sng  ang , S0  . The
denominator of the right hand-side (RHS) of (F.16) is always as here we are considering ng  1.
Hence, RHS of (F.16) is positive if its numerator is positive, that is,
b
 2ng  3  bngT 
ng
T  1   2ng  3  1  0
2
2
(F.20)
Re-arranging (F.20), we obtain,
 2ng  3  b T   b
2
ng
ng
T  1   2ng  3  1
2
2
(F.21)
Simplifying (F.21) gives,
 2ng  3
2
 2bngT  2ng  3   bngT    bngT   2bngT  1   2ng  3  1
2
2
2
(F.22)
or , 4(ng  1)bngT  0
Here, since ng  1 and bngT  0 , (F.22) is always true. Therefore, from (F.16),
Sng  ang , S0   0
(F.23)
Combining (F.19) and (F.23), the solution of (F.16) lies in the range,
0  Sng  ang , S0  
1
bng
(F.24)
which shows that the solution of (F.16) results in the slope of the newly entering ngth genco at
  ang to be positive and steeper than the slope of the genco’s true incremental cost
1
.
bng
Additionally, as shown by (F.15), the alternative solution (F.13) results in a slope for the newly
173
entering genco greater than the slope of its true incremental cost
1
. Therefore, the slope of
bng
the ngth genco at the solution in (F.16) is chosen.
Using this slope of the ngth genco, we find the slope of the other gencos from the usual
form of KM,
ng
Si  ang , S0  
 r a
j 1
j
ng
, S0 
ng  1
 ri  ang , S0  
1
; i  ng thGenco
  ng  1
(F.25)
Appendix G : Analysis of CSFE for Triopolies and above
We begin by recalling the basic unconstrained case where the SFs of the marketparticipating gencos, S j   , S0  , do not reach their capacity limits. For a price   an , the SFs of
the n market-participating gencos satisfy the KM differential equations,
S  , S 
1
 S    , S     S    , S     a  b S   , S  ; k  1,..., n
n
j 1
k
j
0
k
0
0
i
i
k
(G.1)
0
Suppose however that at   i  an , the solution of (G.1) is such that the supply function (SF)
max
of the ith genco hits its capacity limit, that is, Si  i , S0   gi . Furthermore, suppose that at
  i the first derivative of the ith SF is positive, that is, Si  i , S0   0 . Then, for any small
price increase, d   0 , relative to i , the unconstrained SF of genco i would exceed gimax by
dgi  Si  i , S0  d   0 , which would be infeasible.
174
We therefore prove below that when an arbitrary genco i reaches its capacity limit at
some price   i , its SF for   i changes to,
Si   , S0   gimax
(G.2)
Meanwhile, the SFs of all other gencos for   i obey the set of KM differential equations
without genco i,
S , S 
1
 S    , S     S    , S     a  b S   , S  ; k  1,..., n; k  i
n
j 1
j i
k
j
0
k
0
0
i
i
k
(G.3)
0
where all SFs are assumed to be continuous at   i .
Proof: By fixing the SF of genco i to gimax for   i , the remaining free SFs undergo a
discontinuity in their slopes at   i from Sk   i , S0  at   i to Sk   i , S0  at   i ,
where,
n
 S  
j 1
j

i
, S0  
1

 Sk   i , S0  
Sk  i , S0 
; k  1, 2,..., n
i  ak  bk Sk  i , S0 
(G.4)
 Sk   i , S0  
Sk  i , S0 
; k  1,..., n; k  i
i  ak  bk Sk  i , S0 
(G.5)
and where,
n
 S  
j 1
j i
j

i
, S0  
1

175
Now, since Si  i , S0   gimax and since (G.4) is valid for all k, for the particular case where k  i ,
it follows that3,
n
 S j  i , S0  
j 1
j i
1


gimax
0
i  ai  bi gimax
(G.6)
Combining (G.6) with the assumption that Si  i , S0   0 , it follows that when the capacity limit
gimax is reached at   i , the necessary tangent condition to reach market equilibrium
 n

1
  S j  t , S0    0  , cannot have been met since,

 j 1

n
1
 S j  i , S0   
j 1

gimax
 Si  i , S0   0
max
i  ai  bi gi
(G.7)
max
The question now is whether or not the new SF trajectory of genco i, Si     gi ,
continues to satisfy the NE conditions for   i , that is, whether or not genco i can improve its
profit by offering something less than its capacity while all other gencos offer according to (G.3).
To test this, we examine the increment in profit of genco i for prices near i .
First, dgi , the incremental output of the sole gaming genco i must respect the power
balance relation,
3
In equation (G.6), li - ai - bi gimax is positive since ai + bi gimax is the price at which genco i offers gimax under the
least aggressive “perfect market” equilibrium. In contrast, since l i is the price at which genco i offers gimax under
the more aggressive SFE gaming strategy, it follows that li > ai + bi gimax .
176


n
  1  S     d  ; if d   0
j  i 
  

j 1


j i

dgi  

 1 n
    S j  i   d  ; if d   0

  j 1
j i


(G.8)
From (G.6) and (G.8), it follows that genco i cannot game so as to decrease the price as this
would make dgi  0 and violate the genco’s capacity limit. Gaming so that the price increases is
however feasible provided that,
1

n
  S j  i   0
(G.9)
j 1
j i
as the gaming action would then satisfy dgi  0 . To prove that (G.9) is valid, we begin by
summing both sides of (G.5) over k to arrive at,
n
 S j  i , S0  
j 1
j i
1


S j  i , S0 
1
1 n


 n  2   n  2 j 1 i  a j  b j S j  i , S0 
(G.10)
j i
Next, we sum up both sides of (G.4) to obtain,
n
 S  
j 1
j

i
, S0  
1


n
S j  i , S0 
1
1


 n  1   n  1 j 1 i  a j  b j S j  i , S0 
(G.11)
Now, from (G.7) and (G.11), we get,
S j  i , S0 
n
 a
j 1
i
j
 b j S j  i , S0 

1


 n  1 gimax
i  ai  bi gimax
(G.12)
177
or, equivalently, since Si  i , S0   gimax ,
S j  i , S0 
n
 a
j 1
j i
i
j
 b j S j  i , S0 

1


 n  2  gimax
i  ai  bi gimax
(G.13)
which combined with (G.10) gives,
n
1
 S j  i , S0    

j 1
j i

n  2  gimax 

1
1 1

 

 n  2   n  2   i  ai  bi gimax 
(G.14)
max
i
g
0
i  ai  bi gimax
Thus, it is always feasible for genco i to game near the price   i by decreasing its
output and increasing the price. However, the question is whether or not by gaming in this
manner genco i can increase its profit. To test this, we find the increment in profit of genco i for
a slight increase in price d  ,




1 n  

max 
max 

dpri    i  ai  bi gi     S j  i , S0   gi  d 
  j 1



j i




(G.15)
For this profit increment to be non-positive and satisfy the Nash requirement, it is necessary that,
n
 S  
j 1
j i
j

i
, S0  
1


gimax
i  ai  bi gimax
(G.16)
which from (G.14) is true.
178
Therefore, for prices near and above   i , genco i cannot increase its profit by
releasing its output from gimax . However, as shown in Appendix H, for sufficiently higher prices,
it again becomes more profitable for genco i to lower its output.
Appendix H : Condition at which a Supply Function Releases from its Capacity
Limit
Let genco i represent a genco whose SF is constrained by its capacity limit. We know that
pri ( , S0t ) , profit function of genco i is,
pri ( , S0t )   Si ( , S0t )  ai Si ( , S0t ) 
bi
Si (  , S 0 t ) 2
2
(H.1)
Also, from the power balance relation at  ,
dSi ( , S0t )  
d

  Si (  , S 0 t ) d 
(H.2)
j i
and from (H.1) we obtain,
dpri ( , S0t )     ai  bi Si ( , S0t )  dSi ( , S0t )  Si (, S0t )d 
(H.3)
Substituting (H.2) in (H.3) and simplifying, we obtain,


 1

dpri ( , S0t )      ai  bi Si ( , S0t )      Si ( , S0t )   Si ( , S0t )  d 


  j i



(H.4)
179
If genco i offers its capacity limit gimax at price  , then the genco will lower its output for a
marginally price if dpri ( , S0t ) >0, which from (H.4) means,


 1

    ai  bi Si ( , S0t )      Si ( , S0t )   Si ( , S0t )  d   0
  j i



(H.5)
Now, with genco i offering gimax , dSi ( , S0t ) in (H.2) must be non-positive, that is,
1

    Si ( , S 0 t )  d   0
  j i

(H.6)
However, from Property 3 of Appendix B, we conclude that for   t ,
1

  Si (  , S 0 t )  0
(H.7)
j i
So, from (H.6) and (H.7), we know that,
d  0
(H.8)
Substituting (H.8) in (H.5), genco i releases from its capacity limit gimax if,

   a  b g    1   S  (, S
i
i
max
i

j i
i
0t

)   gimax  0

(H.9)
Re-arranging (H.9), we obtain the release condition (RC) for genco i,
1

  S j   , S0t  
j i
gimax
  ai  bi gimax
(H.10)
180
We know that at the tangent condition the LHS of (H.10) is 0 whereas the RHS remains a
positive number. Hence, the RC is satisfied at the tangent condition. Moreover, at some price
  t sufficiently close to t , the LHS must be smaller than RHS (because, as the price
increases towards t , the LHS moves progressively closer to 0 whereas the RHS remains
positive). Hence, we conclude that the RC is satisfied at a price below t by any active genco i
whose SF is constrained by its capacity limit.
181