electricity market modeling trends

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Energy Policy 33 (2005) 897–913

Electricity market modeling trends

Mariano Ventosa*, !Alvaro Ba!ıllo, Andr!es Ramos, Michel Rivier

Instituto de Investigaci !on Tecnol !ogica, Universidad Pontificia Comillas, Alberto Aguilera 23, 28015 Madrid, Spain

Abstract

The trend towards competition in the electricity sector has led to efforts by the research community to develop decision and

analysis support models adapted to the new market context. This paper focuses on electricity generation market modeling. Its aim is

to help to identify, classify and characterize the somewhat confusing diversity of approaches that can be found in the technical

literature on the subject. The paper presents a survey of the most relevant publications regarding electricity market modeling,

identifying three major trends: optimization models, equilibrium models and simulation models. It introduces a classificationaccording to their most relevant attributes. Finally, it identifies the most suitable approaches for conducting various types of

planning studies or market analysis in this new context.

r 2003 Elsevier Ltd. All rights reserved.

Keywords: Deregulated electric power systems; Power generation scheduling; Market behavior

1. Introduction

In the last decade, the electricity industry has

experienced significant changes towards deregulationand competition with the aim of improving economic

efficiency. In many places, these changes have culmi-

nated in the appearance of a wholesale electricity

market. In this new context, the actual operation of

the generating units no longer depends on state- or

utility-based centralized procedures, but rather on

decentralized decisions of generation firms whose goals

are to maximize their own profits. All firms compete to

provide generation services at a price set by the market,

as a result of the interaction of all of them and the

demand.

Therefore, electricity firms are exposed to significantly

higher risks and their need for suitable decision-support

models has greatly increased. On the other hand,

regulatory agencies also require analysis-support models

in order to monitor and supervise market behavior.

Traditional electrical operation models are a poor fit

to the new circumstances since market behavior, the new

driving force for any operation decision, was not

modeled in. Hence, a new area of highly interesting

research for the electrical industry has opened up.

Numerous publications give evidence of extensive effort

by the research community to develop electricity market

models adapted to the new competitive context.This paper focuses on electricity generation market

modeling. Two main technical features determine the

complexity of such models: the product ‘‘electricity’’

cannot be stored and its transportation requires a

physical link (transmission lines).

On the one hand, these features explain why

electricity market modeling usually requires the

representation of the underlying technical character-

istics and limitations of the production assets. Pure

economic or financial models used in other kind

of activities do a poor job of explaining electrical

market behavior. This paper deals specifically with those

models that combine a detailed representation of the

physical system with rational modeling of the firms’

behavior.

On the other hand, unless a high interregional or

international capacity interconnection exists or a very

proactive divestiture program is prompted (and this is

true for very few countries), only a handful of firms are

expected to participate in each wholesale electricity

market. Proper market models, in most cases, must deal

with imperfectly competitive markets, which are much

more complex to represent. This paper focuses on these

kinds of models.

ARTICLE IN PRESS

*Corresponding author. Tel.: +34-91-542-28-00; fax: +34-91-542-

31-76.

E-mail address: [email protected] (M. Ventosa).

0301-4215/$- see front matterr 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enpol.2003.10.013


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The aim of this paper is to help to identify, classify

and characterize the somewhat confusing diversity of

approaches that can be found in the technical literature

on the subject. The paper presents a survey of the most

relevant publications regarding electricity market mod-

eling, identifying three major trends: optimization

models, equilibrium models and simulation models.Although there is a large number of papers devoted to

modeling the operation of deregulated power systems, in

this survey only a selection of the most relevant has been

considered for brevity’s sake. An original taxonomy of

these models is also introduced in order to classify them

according to specific attributes: degree of competition,

time scope, uncertainty modeling, interperiod links,

transmission constraints and market representation.

These specific characteristics are helpful to understand

the advantages and limits of each model surveyed in this

paper. Finally, the paper identifies which approaches are

most suitable for each purpose (i.e., planning studies or

market analysis), including risk management, which is

an increasingly important market issue.

Four articles, Smeers (1997), Kahn (1998), Hobbs

(2001) and Day et al. (2002), have already addressed the

classification of these approaches. The first points out

how game theory-based models can be used to explore

relevant aspects of the design and regulation of liberal-

ized energy markets. It also introduces the application of

multistage-equilibrium models in the context of invest-

ment in deregulated electricity markets. Kahn (1998)

surveys numerical techniques for analyzing market

power in electricity focusing on equilibrium models,

based on profit maximization of participants, whichassume oligopolistic competition. Two kinds of equili-

bria are mentioned in this survey. The commonest one is

based on Cournot competition, where firms compete in

quantity. In contrast, in the supply function equilibrium

approach (SFE), firms compete both in quantity and

price. The conclusion is that Cournot is more flexible

and tractable, and for this reason it has attracted more

interest. More recently, Hobbs (2001) presents a brief

overview of the related literature, concentrating on

Cournot-based models. Finally, Day et al. (2002)

perform a more detailed survey of the power market

modeling literature with emphasis on equilibrium

models. The new survey presented in this paper does

not offer a significantly different vision of the existing

electricity market modeling trends, but rather a com-

plementary and unifying one. It constitutes an effort to

organize and characterize the existing proposals so as to

clarify their main contributions and shortfalls and pave

the way toward future developments.

The paper is organized as follows. Section 2

summarizes the classification scheme used in the survey.

Section 3 describes the publications related to optimiza-

tion models, whereas Section 4 focuses on those related

to equilibrium models. Section 5 presents the publica-

tions devoted to simulation models. Section 6 details the

proposed taxonomy for electricity market models. Section

7 points out the major uses of each modeling approach

and, finally, Section 8 provides some conclusions.

2. Electricity market modeling trends

From a structural point of view, the different

approaches that have been proposed in the technical

literature can be classified according to the scheme

shown in Fig. 1.

Research developments follow three main trends:

optimization models, equilibrium models and simula-

tion models. Optimization models focus on the profit

maximization problem for one of the firms competing in

the market, while equilibrium models represent the

overall market behavior taking into consideration

competition among all participants. Simulation models

are an alternative to equilibrium models when the

problem under consideration is too complex to be

addressed within a formal equilibrium framework.

Although there are many other possible classifications

based on more specific attributes (see Section 6), the

different mathematical structures of these three modeling

trends establish a clearer division. Their various purposes

and scopes also imply distinctions related to market

modeling, computational tractability and main uses.

2.1. Mathematical structure

Optimization-based models are formulated as a singleoptimization program in which one firm pursues its

maximum profit. There is a single objective function to

ARTICLE IN PRESS

OptimizationProblem forOne Firm

ExogenousPrice

Demand-priceFunction

ElectricityMarket

Modeling

MarketEquilibriumConsidering

All Firms

CournotEquilibrium

Supply FunctionEquilibrium

SimulationModels

EquilibriumModels

Agent-basedModels

Fig. 1. Schematic representation of the electricity market modeling

trends.

M. Ventosa et al. / Energy Policy 33 (2005) 897–913898


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be optimized subject to a set of technical and economic

constraints. In contrast, both equilibrium and simula-

tion-based models consider the simultaneous profit

maximization program of each firm competing in the

market. Both types of models are schematically repre-sented in Fig. 2, where P f represents the profit of each

firm f Af1;y; F g; x f are firm f ’s decision variables; and

h f ðxÞ and g f ðxÞ represent firm f ’s constraints.

2.2. Market modeling

Equilibrium and simulation-based models represent

market behavior considering competition among all

participants. On the contrary, optimization models only

represent one firm. Consequently, in the latter models,

the market is synthesized in the representation of the

price clearing process, which can be modeled asexogenous to the optimization program or as dependent

of the quantity supplied by the firm of interest.

2.3. Computational tractability

While complex mathematical programming methods

are required to deal with equilibrium-based models,

powerful and well-known optimization algorithms

bestowing a more detailed modeling capability can be

applied to solve optimization-based models. Simulation

models provide a more flexible way to address the

market problem than equilibrium models although, in

general, they are based on assumptions that are

particular to each study.

2.4. Major uses

The previously mentioned differences in mathematical

structure, market modeling and computational tract-

ability provide useful information in order to identify

the major uses of each modeling trend. For example, the

better computational tractability of optimization models

enables them to deal with difficult and detailed

problems, such as building daily bid curves in the

short-term. On the contrary, equilibrium models are

more suitable to long-term planning and market power

analysis since they consider all participants. The

modeling flexibility of simulation models allows for a

wide range of purposes although there is still somecontroversy as to the appropriate uses of agent-based

models. The major uses of existing electricity models are

presented in more detail in Section 7.

3. Single-firm optimization models

In this paper, approaches based on the profit

maximization problem of one firm are grouped together

into the single-firm optimization category. These models

take into account relevant operational constraints of the

generation system owned by the firm of interest as well

as the price clearing process. According to the manner in

which this process is represented, these models can be

classified into two types: price modeled as an exogenous

variable and price modeled as a function of the demand

supplied by the firm of study.

3.1. Exogenous price

The lowest level of market modeling represents the

price clearing process as exogenous to the firm’s

optimization program, i.e., the system marginal price

is an input parameter for the optimization program.

Consequently, as the price is fixed, the market revenue—

price times the firm’s production—becomes a linear

function of the firm’s production, which is the main

decision variable in this approach. In view of that,

traditional Linear Programming (LP) and Mixed Integer

Linear Programming (MILP) techniques can be em-

ployed to obtain the solution of the model. Unfortu-

nately, this type of optimization model can only

properly represent markets under quasi-perfect competi-

tion conditions because it neglects the influence of the

firm’s decisions on the market clearing price.

ARTICLE IN PRESS

Optimization Programof Firm 1

Optimization Programof Firm f

Optimization Programof Firm F

( )

( )( )

1

1 1

1 1

0

0

Π

=

≤

maximize : x1

subject to : h x

g x

Electricity Market

Supply = Demand

( )

( )( )

f f

f f

f f

maximize : x

subject to : h x 0

g x 0

Π

=

≤

( )

( )( )

F F

F F

F F

maximize : x

subject to : h x 0

g x 0

Π

=

≤

Optimization Programof firm f

( )

( )( )

Π

=

≤

f

f

f

maximize : x

subject to : h x 0

g x 0

Single-firmOptimization Model Equilibrium Model

Electricity Market

Supply = Demand

Fig 2. Mathematical structure of single-firm optimization models and equilibrium-based models.

M. Ventosa et al. / Energy Policy 33 (2005) 897–913 899


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These models can again be classified into two sub-

groups, depending on whether they use a deterministic

or probabilistic price representation.

3.1.1. Deterministic models

A good example is the model proposed in Gross and

Finlay (1996).1 In this model, since the price isconsidered to be exogenous, it is shown that the firm’s

optimization problem can be decomposed into a set of

sub-problems—one per generator—resembling the

Lagrangian Relaxation approach.2 As expected in a case

of perfect competition, deterministic price and convex

costs, the simple comparison between each generator’s

marginal cost and the market price decides the produc-

tion of each generator; therefore, the best offer of each

generation unit consists of bidding its marginal cost.

3.1.2. Stochastic models

The previous approach can be improved if priceuncertainty is explicitly considered. For instance,

Rajamaran et al. (2001) describe and solve the self-

commitment problem of a generation firm in the

presence of exogenous price uncertainty. The objective

function to be maximized is the firm’s profit, based on

the prices of energy and reserve at the nodes where the

firm’s units are located, which are assumed to be both

exogenously determined and uncertain. Similar to the

Gross and Finlay approach, the authors correctly

interpret that, in this setting, the scheduling problem

for each generating unit can be treated independently,

which significantly simplifies the process of obtaining asolution, thus permitting a detailed representation of

each unit. The problem is solved using backward

Dynamic Programming and several numerical examples

illustrate the possibilities of this approach.

A number of recent models represent the price of

electricity as an uncertain exogenous variable in the

context of deciding the operation of the generating units

and at the same time adopting risk-hedging measures.

Fleten et al. (1997, 2002) address the medium-term risk

management problem of electricity producers that

participate in the Nord Pool. These firms face significant

uncertainty in hydraulic inflows and prices of spot

market and contract markets. Considering that prices

and inflows are highly correlated, they propose a

stochastic programming model coordinating physical

generation resources and hedging through the forward

market. They model risk aversion by means of penaliz-

ing risk through a piecewise linear target shortfall cost

function. More recently, Unger (2002) improves the

Fleten approach by explicitly measuring the risk as

conditional value at risk (CVaR). Similar to the models

proposed by Fleten and Unger, another stochastic

approach, which focuses on the solution method, is

presented in Pereira (1999). The resulting large-scale

optimization program is solved using the Benders

decomposition technique, in which the entire firm’smaximization problem is decomposed into a financial

master-problem and an operation sub-problem. While

the financial master-problem produces financial deci-

sions related to the purchase of financial assets

(forwards, options, futures and so forth), the operation

sub-problems decide both the dispatch of the physical

generation system and the exercise of financial assets

providing feedback to the financial problem. The

master-problem and sub-problems are solved using LP.

3.2. Price as a function of the firm’s decisions

In contrast to the former approaches in which the

price clearing process is assumed to be independent of

the firm’s decisions, there exists another family of

models that explicitly considers the influence of a firm’s

production on price. In the context of microeconomic

theory, the behavior of one firm that pursues its

maximum profit taking as given the demand curve and

the supply curve of the rest of competitors is described

by the so-called leader-in-price model (Varian, 1992). In

such a model the amount of electricity that the firm of

interest is able to sell at each price is given by its

residual-demand function.3 Electricity market models of

this type can also be classified in two sub-groupsdepending on whether a probabilistic representation of

the residual-demand function is used.

3.2.1. Deterministic models

The first publication on electricity markets based on

the leader-in-price model is Garc!ıa et al. (1999). They

address the unit commitment4 problem of a specific firm

facing a linear residual-demand function. Given that the

market revenue is a quadratic function of the firm’s total

output, in order to allow for the use of powerful MILP

solvers, a piecewise linearization procedure of the

market revenue is proposed. Likewise, Ba!ıllo et al.

(2001) develop a MILP-based model focusing on the

problem of one firm with significant hydroresources.

The Ba!ıllo model is more advanced in that it allows

non-concave market revenue functions by means of

ARTICLE IN PRESS

1Many later models are based on the same assumptions, thus

leading to similar conclusions.2A large-scale problem with complicating constraints is amenable

for a dual decomposition solution strategy, commonly known as

Lagrangian Relaxation approach.

3From the point of view of one firm, its residual-demand function is

obtained by subtracting the aggregation of all competitors’ selling

offers from the demand-side’s buy bids. The term residual-demand

function is also known as effective demand function.4The Unit Commitment Problem deals with the short-term schedule

of thermal units in order to supply the electricity demand in an efficient

manner. In this type of model, the main decision variables are

generators start-ups and shut-downs.



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additional binary variables. This approach is included in

a recent monograph on new developments in unit

commitment models (Hobbs et al., 2001).

3.2.2. Stochastic models

Unlike previous approaches, Anderson and Philpott

(2002) do not formulate the problem of optimalproduction but rather the problem of constructing the

optimal offer curve of a generation firm. In order to

obtain the optimal shape of that offer curve, the

uncertain behavior of both competitors and consumers

must be taken into account. For this reason, they

represent uncertainty in the residual-demand function

by a probability distribution. This approach constitutes

an interesting starting point for the development of new

models that convert the offer curve into a profitable risk

hedging mechanism against short-term uncertainties in

the marketplace. The thesis of Ba!ıllo (2002) advances the

Anderson and Philpott approach by incorporating a

detailed modeling of the generating system which

implies that offer curves of different hours are not

independent.

4. Equilibrium models

Approaches which explicitly consider market equili-

bria within a traditional mathematical programming

framework are grouped together into the equilibrium

models category. As mentioned earlier, there are two

main types of equilibrium models. The commonest type

is based on Cournot competition, in which firmscompete in quantity strategies, whereas the most

complex type is based on SFE, where firms compete in

offer curve strategies. Although both approaches differ

in regard to the strategic variable (quantities vs. offer

curves), both are based on the concept of Nash

equilibrium—the market reaches equilibrium when each

firm’s strategy is the best response to the strategies

actually employed by its opponents.

4.1. Cournot equilibrium

Although the theoretical support of applying Cournot

equilibrium model to electricity markets is controversial,

the economic research community tends to agree that, in

the case of imperfect competition, this is a suitable

market model. In addition, it has frequently been used

to support market power studies. A thoughtful collec-

tion of essays regarding Cournot competition, which

links this approach with other later models—including

the SFE mentioned above—can be found in (Daughety,

1988).

Cournot equilibrium, where firms choose their opti-

mal output, is easier to compute than SFE because the

mathematical structure of Cournot models turns out to

be a set of algebraic equations, while the mathematical

structure of SFE models turns out to be a set of

differential equations. As a result, most equilibrium-

based models stem from the Cournot solution concept.

The publications devoted to these models concentrate

on four areas: market power analysis, hydrothermal

coordination,5

influence of the transmission networkand risk assessment.

4.1.1. Market power analysis

Market power measurement was the earliest applica-

tion to electricity markets of a Cournot-based model.

Borenstein et al. (1995) employed this theoretical market

model to analyze Californian electricity market power

instead of using the more traditional Hirschman–

Herfindahl Index (HHI) and Lerner Index, which

measure market shares and price-cost margins,

respectively. Later, Borenstein and Bushnell (1999)

have extended this approach by developing an em-pirical simulation model that calculates the Cournot

equilibrium iteratively: the profit-maximizing output

of each firm is obtained assuming that the production

of the remaining firms is fixed. This is repeated for

each supplier until no firm can improve its profit.

Although this model has been successfully applied

to the Californian market, it shows some algorithmic

deficiencies regarding convergence properties as

well as a simplistic representation of the hydroelectric

plants operation. Finally, a collection of models—most

of them based on Cournot competition—for mea-

suring market power in electricity can be found in

Bushnell et al. (1999). This paper summarizes intabular format these models, which have been

applied to the analysis of some of the most relevant

deregulated power markets: California, New England,

England and Wales, Norway, Ontario, and New

Zealand.

4.1.2. Hydrothermal coordination

Apart from market power analysis, Cournot competi-

tion has also been considered in hydrothermal models.

The first publication on this subject is by Scott and Read

(1996), in the context of New Zealand’s electricity

market. Their model utilizes Dual Dynamic Program-ming (DDP), whereby at each stage the hydrooptimiza-

tion problem is superimposed on a Cournot market

equilibrium. In this dual version of the dynamic

programming algorithm, the state space is defined by

the marginal water value (value of water) instead of the

storage level of the reservoir. Bushnell (1998) proposes a

similar model for studying the California market. Its

ARTICLE IN PRESS

5The Hydrothermal Coordination Problem provides the optimal

allocation of hydraulic and thermal generation resources for a specific

planning horizon by explicitly considering the fuel cost savings that

can be obtained due to an intelligent use of hydroreserves.



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most significant contribution is its discussion about the

meaning of the firm’s marginal water value in a

deregulated framework. Bushnell points out that the

firm’s water value is related to the firm’s marginal

revenue instead of the traditional system’s marginal

cost. Although Bushnell’s analytical formulation of the

market equilibrium conditions is more elegant, the Scottand Read model contains a more detailed representation

of the physical system. Similar to the Bushnell approach,

Rivier et al. (2001) state the market equilibrium using

the equations that express the optimal behavior of

generation companies, i.e., by means of the firms’

optimality conditions. Unlike both the Scott and Read

model and the Bushnell model, the Rivier et al. (2001)

approach takes advantage of the fact that the optimality

conditions can be directly solved due to its Mixed

Complementarity Problem6 (MCP) structure, which

allows for the use of special complementarity methods

to solve realistically sized problems. Kelman et al. (2001)

combine the Cournot concept with the Stochastic

Dynamic Programming technique in order to cope with

hydraulic inflow uncertainty problems. However, they

do not mention how they deal with the fact that the

recourse function7 of the Dynamic Programming algo-

rithm is non-convex in equilibrium problems. Barqu!ın

et al. (2003) introduce an original approach to compute

market equilibrium, by solving an equivalent minimiza-

tion problem. This approach is oriented to the medium-

term planning of large-size hydrothermal systems,

including the determination of water value and other

sensitivity results obtained as dual variables of the

optimization problem.

4.1.3. Electric power network

Congestion pricing in transmission networks is

another area in which Cournot-based models have also

played a significant role. Both Hogan (1997) and Oren

(1997) formulate a spatial electricity model wherein

firms compete in a Cournot manner. Wei and Smeers

(1999) use a variational inequality8 (VI) approach for

computing the spatial market equilibrium including

generation capacity expansion decisions. They model

the electrical network considering only power-flow

conservation-equations since they omit Kirchhoff’svoltage law. This type of electric network model is

known as transshipment model.

More recently, Hobbs (2001) models imperfect

competition among electricity producers in bilateral

and POOLCO-based power markets as a Linear

Complementarity Problem (LCP).9 His model includes

a congestion-pricing scheme for transmission in which

load flows are modeled considering both the first and the

second Kirchhoff laws by means of a linearized

formulation. This type of electric network model is

known as DC model. In contrast to previous models, the

VI and LCP approaches are able to cope with largeproblems. In all these models, it is assumed that the

generation units of each firm are located at only one

node of the network—which is, obviously, a particular

case. Unfortunately, since in the general case in which

each firm is allowed to own generation units in more

than one node, a pure-strategy equilibrium does not

exist, as it is pointed out by Neuhoff (2003).

4.1.4. Risk analysis

Finally, because of the difficulty in applying tradi-

tional risk management techniques to electricity mar-

kets, only one publication has been identified that

explicitly addresses the risk management problem for

generation firms under imperfect competition condi-

tions. Batlle et al. (2000) present a procedure capable of

taking into account some risk factors, such as hydraulic

inflows, demand growth and fuel costs. Cournot market

behavior is considered using the simulation model

described in Otero-Novas et al. (2000), which computes

market prices under a wide range of scenarios. The

Batlle model provides risk measures such as value-at-

risk (VaR) or profit-at-risk (PaR).

4.2. Extensions of cournot equilibrium

The assumption of generation companies behaving as

Cournot players has been extensively used to conduct a

diversity of analysis concerning the medium-term out-

come of a variety of electricity market designs. The

possibility of formulating these models under the MCP/

VI framework and benefiting from specific commercial

solvers capable of tackling large-scale problems has

significantly contributed to the popularity of this

approach.

However, a number of drawbacks seem to question

the applicability of the Cournot model. The most

important one stems from the fact that under the

Cournot approach, generators’ strategies are expressed

in the terms of quantities and not in the terms of offer

curves. Hence, equilibrium prices are determined only

by the demand function being therefore highly sensitive

to demand representation and usually higher to those

observed in reality.10 This shortcoming seems to

ARTICLE IN PRESS

6The Karush–Kuhn–Tucker (KKT) optimality conditions of any

optimization problem can be formulated making use of a special

mathematical structure known as Complementarity Problem. A MCP

is a mixture of equations with a Complementarity Problem.7In the Hydrothermal Coordination Problem, the recourse function

is known as the future water value.8KKT conditions can also be formulated as a VI problem.

9A LCP is obtained when the KKT conditions are derived from an

optimization problem with quadratic objective function and linear

constraints.10In some respects, the models’ predicted prices are too high because

they do not take into account some of the external circumstances such

as stranded cost payments, new entry aversion or regulatory threats.



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reinforce the idea that the SFE approach is a better

alternative to represent competition in electricity mar-

kets (Rudkevich et al., 1998). Incorporating the

conjectural variations (CV) approach described in

traditional microeconomics theory (Vives, 1999) is

another way to overcome this limitation. The CV

approach is easy to introduce into Cournot-basedmodels. This approach changes the conjectures that

generators are expected to assume about their compe-

titors’ strategic decisions, in terms of the possibility of

future reactions (CV). Two recent publications (Garc!ıa-

Alcalde et al., 2002; Day et al. 2002) suggest considering

this approach in order to improve Cournot pricing in

electricity markets. Garc!ıa-Alcalde et al. (2002) assume

that firms make conjectures about their residual demand

elasticities, as in the general CV approach, whereas Day

et al. (2002) assume that firms make conjectures about

their rivals’ supply functions. In the context of electricity

markets, this approach is already labeled as the

Conjectured Supply Function (CSF) approach.

4.3. Supply function equilibrium

Klemperer and Meyer (1989) showed that, in the

absence of uncertainty and given the competitors’

strategic variables (quantities or prices), each firm has

no preference between expressing its decisions in terms

of a quantity or a price, because it faces a unique

residual demand. On the contrary, when a firm faces a

range of possible residual demand curves, it expects, in

general, a bigger profit expressing its decisions in terms

of a supply function that indicates the price at which itoffers different quantities to the market. This is the SFE

approach which, originally developed by Klemperer and

Meyer (1989), has proven to be an extremely attractive

line of research for the analysis of equilibrium in

wholesale electricity markets.

Calculating an SFE requires solving a set of

differential equations, instead of the typical set of

algebraic equations that arises in traditional equilibrium

models, where strategic variables take the form of

quantities or prices. SFE models have thus considerable

limitations concerning their numerical tractability. In

particular, they rarely include a detailed representation

of the generation system under consideration. The

publications devoted to these models concentrate on

four topics: market power analysis, representation of

electricity pricing, linearization of the SFE model and

evaluation of the impact of the electric power network.

4.3.1. Market power analysis

The SFE approach was extensively used to predict the

performance of the pioneering England & Wales (E&W)

Pool, whose revolutionary design did not seem to fit into

more conventional oligopoly models. The relatively

unimportant role played by the transmission network

in this particular power system increased the relevance

of these studies. Green and Newbery (1992) analyze the

behavior of the duopoly that characterized the E&W

electricity market during its first years of operation

under the SFE approach. It is assumed that each

company submits a daily smooth supply function. The

demand curve faced by generation companies isextremely inelastic—demand-side bidding was almost

non-existent—and varies over time since in the E&W

Pool offers were required to be kept unchanged

throughout the day. Interesting conclusions were

reached. For instance, in the case of an asymmetric

duopoly, it is shown that the large firm finds price

increases more profitable and therefore has a greater

incentive to submit a steeper supply function. The small

firm then faces a less elastic residual demand curve and

also tends to deviate from its marginal costs. This was

previously pointed out by Bolle (1992), where the large

generation company suffers the consequences of the

curse of market power and indirectly causes an increase

of its rivals’ profits.

4.3.2. Electricity pricing

The possibility of obtaining reasonable medium-term

price estimations with the SFE approach is considerably

attractive, particularly when conventional equilibrium

models based on the Cournot conjecture have proven to

be unreliable in this aspect mainly due to their strong

dependence on the elasticity assumed for the demand

curve. Indeed, the SFE framework does not require the

residual demand curve either to be elastic or to be

known in advance. Based on the assumption of inelasticdemand, further advances on the SFE theory have been

reported which increase its applicability. Rudkevich et al.

(1998) has obtained a closed-form expression that

provides the price for a SFE given a demand realization

under the assumption of an n-firm symmetric oligopoly

with inelastic demand and uniform pricing. Conver-

gence problems due to the numerical integration of the

SFE system of differential equations are thus overcome.

This approach also allows to consider stepwise marginal

cost functions, which is more realistic than the convex

and differentiable cost functions typical of previous SFE

models.

4.3.3. Linear supply function equilibrium models

For numerical tractability reasons, researchers have

recently focused on the linear SFE model, in which

demand is linear,11 marginal costs are linear or affine

and SFE can be obtained in terms of linear or affine

supply functions. Green (1996) considers the case of an

asymmetric n-firm oligopoly with linear marginal costs

ARTICLE IN PRESS

11According to Baldick (2000), the precise description would be

‘‘affine demand’’, whereas the term ‘‘linear’’ should be restricted to

affine functions with zero intercept.



8/17

facing a linear demand curve whose slope remains

invariable over time. An SFE expressed in terms of

affine supply functions is obtained. Baldick et al. (2000)

extend previous results to the case of affine marginal

cost functions and capacity constraints. Solutions for

the SFE are provided in the form of piecewise affine

non-decreasing supply functions. They use this methodto predict the extent to which structural changes in the

E&W electricity industry may affect wholesale electricity

spot prices. Baldick and Hogan (2001) perform a

comprehensive review of the SFE approach. The

authors first revisit the general SFE problem of an

asymmetric n-firm oligopoly facing a linear demand

curve (no explicit assumption is made concerning the

firms’ marginal costs) and show the extraordinary

complexity of obtaining solutions for the system of

differential equations that results. In particular, they

highlight the difficulty of discarding infeasible solutions

(e.g., equilibria with decreasing supply functions). An

iterative procedure to calculate feasible SFE solutions is

proposed and extensively used to analyze the influence

of a variety of factors such as capacity constraints, price

caps, bid caps or the time horizon over which offers are

required to remain unchanged.

4.3.4. Electric power network

In Ferrero et al. (1997), generation companies are

assumed to offer one affine supply curve at each of the

nodes in which their units are located. Transaction costs

are calculated based on Schweppe’s spot pricing theory,

including the influence of transmission constraints. A

finite number of offering strategies are defined for eachgeneration company and an exhaustive enumeration

solution process is proposed. Berry et al. (1999) use an

SFE model to predict the outcome of a given market

structure including an explicit representation of the

transmission network. Forcing supply functions to be

affine typically alleviates the complexity of searching for

a solution. Different conceptual approaches have been

adopted to obtain numerical solutions for this family of

models. In general, no existence or uniqueness condi-

tions are derived. Hobbs et al. (2000) propose a model in

which the strategy of each firm takes the form of a set of

nodal affine supply functions. The problem is structured

in two optimization levels and therefore the solution

procedure is based on Mathematical Programming with

Equilibrium Constraints (MPEC).

In spite of the variety of modeling proposals, it is

possible to identify a number of attributes that can be

used to establish a comparison between different SFE

approaches. Some of these attributes refer to the market

representation adopted by each author, such as the

possibility of considering asymmetric firms and the

assumptions made about the shape of the marginal cost

curves, the supply functions or the demand curve.

Others attributes refer to the model of the generation

system (e.g., capacity constraints) or the transmission

network (e.g., transmission constraints). Finally, the

solution method used by each author and the numerical

cases addressed are also two relevant features. In order

to illustrate the evolution of this line of research, Table 1

presents a summary of the works that have been

reviewed in this section.In conclusion, the SFE approach presents certain

advantages with respect to more traditional models of

imperfect competition. In particular, it appears to be an

appropriate model to predict medium-term prices of

electricity, given that it does not rely on the demand

function,12 as the Cournot model, but on the shape of

the equilibrium supply functions decided by the firms. In

addition to this, firms’ strategies do not need to be

modified as demand evolves over time. Quite the

opposite, supply functions are specifically conceived to

represent the firms’ behavior under a variety of demand

scenarios. This flexibility, however, is accompanied by

significant practical limitations concerning numerical

tractability. To date, only under very strong assump-

tions have SFE problems been solved when applied to

real cases. Given that SFE shortcomings are well

documented, only the main disadvantages will be cited

here. Firstly, in general, multiple SFE may exist and it is

not clear which of them is more qualified to represent

firms’ strategic behavior. Secondly, except for very

simple versions of the SFE model, existence and

uniqueness of a solution are very hard to prove. Thirdly,

closed-form expressions of a solution are very rarely

obtained. Consequently, numerical methods are needed

to solve the system of differential equations, thusincreasing the computational requirements of this

approach. Moreover, some of this system’s solutions

may violate the non-decreasing constraint that supply

functions must observe. This leads to ad hoc solution

procedures that usually present convergence problems.

Needless to say, transmission constraints are only

considered in extremely simplified versions of the SFE

model. Nevertheless, research efforts have recently

produced encouraging results that may ultimately

increase the applicability of this approach.

5. Simulation models

As indicated above, equilibrium models are based on

a formal definition of equilibrium, which is mathema-

tically expressed in the form of a system of algebraic

and/or differential equations. This imposes limitations

on the representation of competition between partici-

pants. In addition, the resulting set of equations, if it has

a solution, is frequently too hard to solve. The fact that

ARTICLE IN PRESS

12In general, SFE-based approaches model the demand function as

inelastic, which is the most suitable hypothesis in the case of electricity.



9/17

power systems are based on the operation of generation

units with complex constraints only contributes to

complicate the situation.

Simulation models are an alternative to equilibrium

models when the problem under consideration is too

complex to be addressed within a formal equilibrium

framework. Simulation models typically represent eachagent’s strategic decision dynamics by a set of sequential

rules that can range from scheduling generation units to

constructing offer curves that include a reaction to

previous offers submitted by competitors. The great

advantage of a simulation approach lies in the flexibility

it provides to implement almost any kind of strategic

behavior. However, this freedom also requires that the

assumptions embedded in the simulation be theoreti-

cally justified.

5.1. Simulation models related to equilibrium models

In many cases, simulation models are closely related

to one of the families of equilibrium models. For

example, when in a simulation model firms are assumed

to take their decisions in the form of quantities, the

authors will typically refer to the Cournot equilibrium

model in order to support the adequacy of their

approach.

Otero-Novas et al. (2000) present a simulation model

that considers the profit maximization objective of each

generation firm while accounting for the technical

constraints that affect thermal and hydrogenerating

units. The decisions taken by the generation firms are

derived with an iterative procedure. In each iteration,given the results obtained in the previous one, every firm

modifies its strategic position with a two-level decision

process. First, each firm updates its output for each

planning period by means of a profit maximization

problem in which market clearing prices are held fixed

and a Cournot constraint is included limiting the

company’s output. Subsequently, the price at which

the company offers the output of each generating unit in

each planning period is modified, according to a

descending rule. New market clearing prices are

calculated based on these offers and on the evolution

of demand, which is assumed to be inelastic.

Day and Bunn (2001) propose a simulation model,

which constructs optimal supply functions, to analyze

the potential for Market Power in the E&W Pool. This

approach is similar to the SFE scheme, but it provides a

more flexible framework that enables us to consider

actual marginal cost data and asymmetric firms. In this

model, each generation company assumes that its

competitors will keep the same supply functions that

they submitted in the previous day. Uncertainty about

the residual demand curve is due to demand variation

throughout the day. The optimization process to

construct nearly optimal supply functions is based on

ARTICLE IN PRESS

T a b l e 1

A c h a r a c t e r i z a t i o n o f S F E m o d e l s

A u t h o r

A s y m m e

t r i c

fi r m s

M a r g i n a l

c o s t s

D e m a n d

c u r v e

S u p p l y f u

n c t i o n s

C a p a c i t y

c o n s t r a i n t s

S o l u t i o n

m e t h o d

T r a n s m i s s i o n

n e t w o r k

N u m e r i c a l

a p p l i c a t i o n

K l e m p e r e r a n d M e y e r ( 1 9 8 9 )

N o

C o n v e x

C o n c a v e

T w i c e c o n t i n u o u s l y d i f f e r e n t i a b l e

N o

N e c e s s a r y c o n d i t i o n s

N o

N o

G r e e n a n d N e w b e r y ( 1 9 9 2 )

N o

Q u a d r a t i c

L i n e a r

T w i c e c o n t i n u o u s l y d i f f e r e n t i a b l e

Y e s

N u m e r i c a l i n t e g r a t i o n

N o

E & W

P o o l

G r e e n e t a l . ( 1 9 9 6 )

Y e s

L i n e a r

L i n e a r

A f fi n e

N o

C l o s e d - f o r m e x p r e s s i o n

N o

E & W

P o o l

F e r r e r o e t a l . ( 1 9 9 7 )

Y e s

A f fi n e

I n e l a s t i c

A f fi n e

Y e s

E x h a u s t i v e e n u m e r a t i o n

Y e s

I E E E 3 0 - b u s s y s t e m

R u d k e v i c h e t a l . ( 1 9 9 8 )

N o

S t e p w i s e

I n e l a s t i c

D i f f e r e n t i a b l e

Y e s

C l o s e d - f o r m e x p r e s s i o n

N o

P e n n s y l v a n i a

B a l d i c k e t a l . ( 2 0 0 0 )

Y e s

A f fi n e

L i n e a r

P i e c e w i s e

l i n e a r

Y e s

H e u r i s t i c s

N o

E & W

P o o l

B a l d i c k a n d H o g a n ( 2 0 0 1 )

Y e s

A f fi n e

L i n e a r

P i e c e w i s e

l i n e a r n o n - d e c r e a s i n g

Y e s

H e u r i s t i c s

N o

E & W

P o o l

B e r r y e t a l . ( 1 9 9 9 )

Y e s

A f fi n e

L i n e a r

A f fi n e

Y e s

H e u r i s t i c s

Y e s

F o u r - n o d e c a s e

H o b b s e t a l . ( 2 0 0 0 )

Y e s

A f fi n e

L i n e a r

A f fi n e

Y e s

M P E C

Y e s

3 0 - n o d e c a s e



10/17

an exhaustive search, rather than on the solution of a

formal mathematical programming problem. The

authors compare the results of their model for a

symmetric case with linear marginal costs to those

obtained under the SFE framework, which turns out to

be extraordinarily similar.

5.2. Agent-based models

Simulation provides a more flexible framework to

explore the influence that the repetitive interaction of

participants exerts on the evolution of wholesale

electricity markets. Static models seem to neglect the

fact that agents base their decisions on the historic

information accumulated due to the daily operation of

market mechanisms. In other words, agents learn from

past experience, improve their decision-making and

adapt to changes in the environment (e.g., competitors’

moves, demand variations or uncertain hydroinflows).

This suggests that adaptive agent-based simulation

techniques can shed light on features of electricity

markets that static models ignore.

Bower and Bunn (2000) present an agent-based

simulation model in which generation companies are

represented as autonomous adaptive agents that parti-

cipate in a repetitive daily market and search for

strategies that maximize their profit based on the results

obtained in the previous session. Each company

expresses its strategic decisions by means of the prices

at which it offers the output of its plants. Every day,

companies are assumed to pursue two main objectives: a

minimum rate of utilization for their generationportfolio and a higher profit than that of the previous

day. The only information available to each generation

company consists of its own profits and the hourly

output of its generating units. As usual in these models,

demand side is simply represented by a linear demand

curve. This setting allows the authors to test a number of

market designs relevant for the changes that have

recently taken place in E&W wholesale electricity

market. In particular, they compare the market outcome

that results under the pay-as-bid rule to that obtained

when uniform pricing is assumed. Additionally, they

evaluate the influence of allowing companies to submit

different offers for each hour, instead of keeping them

unchanged for the whole day. The conclusion is that

daily bidding together with uniform pricing yields the

lowest prices, whereas hourly bidding under the pay-as-

bid rule leads to the highest prices.

6. Taxonomy of electricity market models

In addition to the classification presented in Sections

2–5, which is based on the mathematical structure of

each model, electricity market models can be categorized

considering more specific attributes. These character-

istics are useful in understanding the advantages and

limits of each model surveyed in previous sections. Thetaxonomy presented here considers the following issues:

degree of competition, time scope of the model,

uncertainty modeling, interperiod links, transmission

constraints, generating system representation and mar-

ket modeling.

6.1. Degree of competition

Markets can be classified into three broad categories

according to their degree of competition: perfect

competition, oligopoly and monopoly.

Since microeconomic theory proves that a perfectlycompetitive market can be modeled as a cost minimiza-

tion or net benefit maximization problem, optimization-

based models are usually the best way to model this type

of market. Similarly, a monopoly can be modeled by the

profit maximization program of the monopolistic firm

(see Fig. 3). In these models the price is derived from the

demand function. In contrast, under imperfect competi-

tion conditions—the most common situation—the profit

maximization problem of each participant must be

solved simultaneously. In addition, as discussed in the

next subsection, the suitability of each oligopolistic

model depends on the time scope of the study.

6.2. Time scope

The time scope is a basic attribute for classifying

electricity models since each time scope involves both

different decision variables and different modeling

approaches. For example, when long-term planning

studies are conducted, capacity-investment decisions are

the main decision variables while unit-commitment

decisions are usually neglected. On the contrary, in

short-term scheduling studies, start-ups and shut-downs

become significant decision variables, while the

ARTICLE IN PRESS

Competition

TimeScope

Oligopoly

Monopoly

PerfectCompetition

Long Term(Years)

Medium Term(Months)

Short Term(Days)

NashEquilibrium

(Cournot andStackelberg)

NashEquilibrium

(Cournot andSFE)

Leader inPrice

Market Model Based on the Cost

Minimization of the Whole System

Market Model Based on the Profit

Maximization of the Monopolist Firm

Fig 3. Theoretical electricity market models depending on competition

and time scope.



11/17

maximum capacity of each generator is considered to be

fixed.

As previously mentioned, under imperfect competi-

tion conditions, the time scope of the model defines

different market modeling approaches. To be specific, in

the case of short-term operation (one day to one week),

the experience drawn from the literature surveyed in thispaper suggests that the best way to represent the market

is the leader-in-price model from microeconomics theory

(Garc!ıa et al., 1999; Ba!ıllo et al., 2001; Anderson and

Philpott, 2002; Ba!ıllo, 2002). In the leader-in-price

model, the incumbent firm pursues its maximum profit

taking into account its residual demand function that

relates the price to its energy output. The most

controversial assumption of this theoretical model lies

on the static perspective that the residual demand

function provides about other agents. An intuitive

explanation about the suitability of this conjecture in

short-term models is that the shorter the time scope

considered, the more consistent this conjecture becomes.

In the medium-term case (1 month to 1 year), the vast

majority of the models are based on both Cournot

equilibrium (Scott and Read, 1996; Bushnell, 1998;

Rivier et al., 2001; Kelman et al., 2001; Barqu!ın et al.,

2003; Otero-Novas et al., 2000) and SFE (Green and

Newbery, 1992; Bolle, 1992; Rudkevich et al., 1998;

Baldick and Hogan, 2001).

Finally, microeconomics suggests that the Stackelberg

equilibrium may fit better than other oligopolistic

models with the long-term investment-decision problem

due to its sequential decision-making process. There is a

leader firm that first decides its optimal capacity; thefollower firms then make their optimal decisions

knowing the capacity of the leader firm (Varian, 1992).

Up to now, there are only a few articles (Ventosa et al.,

2002; Murphy and Smeers, 2002) devoted to represent

investment in imperfect electricity markets. In both

publications, a comparison between the Cournot equili-

brium and Stackelberg equilibrium for modeling invest-

ment decisions is conducted. One conclusion is that

although from a theoretical point of view both models

are based on different assumptions, from a practical

point of view there are minor differences in most results.

The Stackelberg model of Ventosa et al. turns out to

have the structure of a MPEC due to the fact that there

is only one leader firm. In contrast, the Stackelberg-

based model of Murphy and Smeers has the structure of

an Equilibrium Problem with Equilibrium Constraints

(EPEC) because more that one leader firm may exist.

The EPEC model is more general although it is also

more difficult to manage.

6.3. Uncertainty modeling

One of the most common applications of electricity

market models is in the field of forecasting the market

outcome under a wide range of scenarios since prices

depend on random variables such as generators’ forced

outages, hydraulic inflows and levels of demand. More-

over, in a competitive context, new sources of un-

certainty must be considered due to both strategic

behavior of competitors and fuel price volatility.

According to the manner in which uncertainty isrepresented, models can be classified into probabilistic—

when the uncertain nature of random variables is

incorporated using probabilistic distributions—and de-

terministic—when only the expected value of such

variables is considered. Needless to say, probabilistic

models result in large-scale stochastic problems that

require complex solution techniques.

In regard to the representation of the stochasticity of

demand within the context of electricity markets, the

best examples are those models based on the SFE

(Fig. 4) (Green and Newbery, 1992; Bolle, 1992;

Rudkevich et al. 1998) since they all consider uncer-

tainty in demand. Based on a probabilistic version of the

price-leadership model, the Ba!ıllo model (2002) not only

considers the uncertainty in demand but also in

competitors’ behavior. Finally, Fleten et al. (2002) and

Unger (2002) models focus on uncertainty in prices and

hydraulic inflows under pure competition assumptions,

while Kelman et al. (2001) considers a Cournot frame-

work.

6.4. Interperiod links

The time scope considered in planning studies is

typically split into intervals commonly known as periods. In electricity generation, there are many costs

and decisions that, when addressed within a certain time

scope, involve the scheduling of resources in the multiple

intermediate periods. For example, long-term studies

are typically oriented to derive optimal annual manage-

ment policies for hydroreserves that must consider the

dynamic process of inflows and thus take the form of a

set of monthly or weekly operation decisions. Similarly,

short-term models must take into account the inter-

temporal constraints implicit in thermal unit commit-

ment decisions.

SFE-based models do not usually consider these

interperiod effects. In contrast, almost all the rest of

models reviewed in this paper, such as those devoted to

optimal offer curve construction, hydrothermal coordi-

nation and capacity expansion problems, focus on the

tradeoff of scheduling resources across time.

6.5. Transmission constraints

As in the case of previous attributes, the consideration

of transmission constraints divides electricity market

models into two main types: single-node models and

transmission network models.

ARTICLE IN PRESS



12/17

The majority of models surveyed in this paper do not

consider the transmission network; nevertheless, there

are good examples of transmission models. In terms of

network modeling, some authors consider a transship-

ment network that omits Kirchhoff’s voltage law (Wei

and Smeers, 1999) although their model allows for inter-temporal constraints regarding investment decisions

(Fig. 4). Other authors consider both of Kirchhoff’s

laws (Berry et al., 1999; Hobbs et al., 2000; Hobbs, 2001)

by means of a linearized DC network whereas Ferrero

et al. (1997) use a nonlinear AC network model. From a

computational point of view, only two of these

approaches (Hobbs, 2001; Wei and Smeers, 1999)

permit solving realistically sized problems.

6.6. Generating system modeling

A high degree of realism regarding the physical

modeling of generating systems involves the representa-

tion of technical limits affecting generators as well as the

consideration of accurate production cost functions of

thermal units.

As shown in Fig. 5, optimization-based models for

individual firms achieve a high level of accuracy in

system modeling due to the powerful LP and MILP

techniques available to solve them. These models

consider in detail the relevant technical constraints

affecting generation units. In addition, these models

consider every individual generation unit of interest in a

non-aggregated manner. For instance, medium-term

models such as those proposed in Fleten et al. (2002),

Unger (2002) and Kelman et al. (2001) consider not only

the hydroenergy constraints implicit in the management

of water reserves but also the hydraulic inflow un-

certainty. On the other hand, short-term models such as

Garc!ıa et al. (1999) and Ba

!ıllo (2002) consider in detail

inter-temporal constraints, such as ramp-rate limits, and

incorporate binary variables to deal with decisions such

as the start-up and shut-down of thermal units.

In the case of equilibrium models, two of the revised

approaches—the Otero-Novas model (2000), which

combines a simulation algorithm with optimization

techniques, and the Rivier model (2001), which is solved

by complementarity methods—reach a degree of realism

similar to that of optimization models. Both models are

able to manage realistically sized problems considering

every generation unit as independent with its particular

constraints. Scott and Read (1996) and Bushnell (1998)

are considered to have an intermediate level in terms of

generation system modeling since they take into account

independent units but they are not capable of solving

large problems. Finally, it is very rare that SFE-based

models include a detailed representation of the genera-

tion system due to their numerical tractability limita-

tions.

6.7. Market modeling

The last attribute considered in this taxonomy is

related to the market model under consideration. Pure

ARTICLE IN PRESS

DCModel

Interperiod

Links

TransmissionNetwork

Uncertainty

SingleNode

Probabilistic

Deterministic

InterperiodConstraints

Probabilistic

Deterministic

IntraperiodConstraints

Interperiodconstraints

Intraperiodconstraints

DCModel

SingleNode

TransshipmentModel

TransshipmentModel

ACModel

ACModel

Fig. 4. Characterization of some electricity market models according to the modeling of uncertainty, transmission network and interperiod links.



13/17

competition-based models— Fleten et al. (2002), Unger

(2002), Pereira (1999) —are the simplest in terms of

market modeling since they consider the price clearing

process as exogenous to the optimization problem.

Models based on the leader-in-price concept— Garc!ıa

et al. (1999) and Ba!ıllo (2002) —are considered to have

an intermediate level of complexity since they take into

account the influence of the firm’s production on prices

by means of its residual demand function. Finally, themost complex market models are those based on

imperfect market equilibrium as they take into account

the interaction of all participants.

7. Major uses

As mentioned in Section 2, differences in mathema-

tical structure, market modeling and computational

tractability provide useful information in order to

identify the major use of each modeling trend. This

section summarizes the experience and conclusions

drawn from the publications referred to in Sections 3–

5 regarding the major uses of single-firm optimization

models, imperfect market equilibrium models and

simulation models (see Table 2).

One-firm optimization models are able to deal with

difficult and detailed problems because of their better

computational tractability. Good examples of such

models are those related to short-term hydrothermal

coordination and unit commitment, which require

binary variables, and optimal offer curve construction

under uncertainty, which not only needs binary vari-

ables but also involves a probabilistic representation of

the competitors’ offers and demand-side bids. Usually,

risk management models are also based on optimization

due to their complexity and size.

In contrast, when long-term planning studies are

conducted, equilibrium models are more suitable be-

cause the longer the time scope of the study, the lower

the requirement for detailed modeling capability, and

the more significant the response of all competitors.

Therefore, the majority of models devoted to yearlyeconomic planning and hydrothermal coordination are

Cournot-based approaches, which provide more realism

in the representation of physical constraints than SFE-

based approaches, that have numerical tractability

limitations.

As in the case of long-term studies, in market power

analysis and market design, it is also necessary to

consider the market outcome resulting from competition

among all participants. Consequently, equilibrium

models and simulation models are the best alternative

to traditional anti-trust tools based on indices such as

Hirschman–Herfindahl Index (HHI) and Lerner Index.

Finally, regarding the analysis of congestion manage-

ment in transmission networks, Cournot and SFE

equilibrium models are able to simultaneously consider

power flow constraints and the competition of several

firms at each node.

In conclusion, Table 3 summarizes the main char-

acteristics of the most significant models referred to in

previous sections. The models are classified into eight

categories depending on their market model.13 Within

ARTICLE IN PRESS

Probabilistic

Medium

Market

Modeling

GeneratingSystem Modeling

Uncertainty

Low

Probabilistic

DeterministicHigh

Deterministic

Medium

Low

High

Single-firmResidualDemand

ExogenousPrice

ImperfectMarket

Equilibrium

ExogenousPrice

Single-firmResidualDemand

ImperfectMarket

Equilibrium

Fig. 5. Characterization of some electricity market models according to the treatment of uncertainty, generation system modeling and market

modeling.

13CSF: Conjectured Supply Function approach and CV: Conjectur-

al Variations approach.



14/17

each category, models are listed by year of publication.

Other columns are related to major use, main features of

the model, numerical solution method,14 problem size15

of the case study and the regional market considered.

8. Conclusion and future developmental trends

This paper presents a survey of the literature on

electricity market models showing that there are three

main lines of development: optimization models, equili-

brium models and simulation models. These models

differ as to their mathematical structure, market

representation, computational tractability and major

uses.

In the case of single-firm optimization models,

researchers have been developing models that address

problems such as the optimization of generation

scheduling or the construction of offer curves under

perfect and imperfect competition conditions. At pre-

sent, they are working on two different challenges. On

the one hand, they are tackling the cutting edge problem

of converting the offer curve of a generating firm into a

robust risk hedging mechanism against the short-term

uncertainties due to changes in demand and competitors

behavior. On the other hand, they are developing risk

management models that help firms to decide their

optimal position in spot, future and over-the-counter

markets with an acceptable level of risk.

Models that evaluate the interaction of agents in

wholesale electricity markets have persistently stemmed

from the game-theory concept of equilibrium. Some of

these models represent the equilibrium in terms of

variational inequalities or, alternatively, in the form of a

complementarity problem, providing a framework to

ARTICLE IN PRESS

Table 2

Major uses of electricity market models

Major use One-firm optimization

models

Simulation models Imperfect market equilibrium models

Risk management Fleten et al. (2002), Unger

(2002) and Pereira (1999)

Unit commitment Garc!ıa et al. (1999) and

Rajamaran et al. (2001)

Short-term hydrothermal

coordination

Ba!ıllo et al. (2001)

Strategic bidding Anderson and Philpott

(2002) and Ba!ıllo (2002)

Market power analysis Day and Bunn (2001) Green (1996), Bolle (1992) Rudkevich et al.

(1998), Borenstein et al. (1995), Borenstein and

Bushnell (1999), Baldick et al. (2000) and

Baldick and Hogan (2001)

Market design Bower and Bunn (2000) Green (1996), Baldick et al. (2000) and Baldickand Hogan (2001)

Yearly economic planning Otero-Novas et al. (2000) Ramos et al. (1998)

Long-term hydrothermal

coordination

Scott and Read (1996), Bushnell (1998), Rivier

et al. (2001), Kelman et al. (2001) and Barqu!ın

et al. (2003)

Capacity expansion planning Murphy and Smeers (2002) and Ventosa et al.

(2002)

Congestion management Hogan (1997), Oren (1997), Hobbs et al. (2000),

Hobbs (2001), Wei and Smeers (1999) and Berry

et al. (1999)

14Benders: Benders Decomposition, DP: Dynamic Programming,

Enumeration: Exhaustive Enumeration, EPEC: Equilibrium Program

with Equilibrium Constraints, Heuristic: Ad hoc Heuristic Algorithm,

LCP: Linear Complementarity Problem, LP: Linear Programming,

MCP: Mixed Complementarity Problem, MIP: Mixed Integer

Programming, MPEC: Mathematical Programming with Equilibrium

Constraints, NI: Numerical Integration, NLP: Non-Linear Program-

ming, Simulation: Simulation Scenario Analysis, and VI: Variational

Inequality.15Small: less than 100 variables, Medium: between 100 and 10,000

variables, and Large: more than 10,000 variables.



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analyze realistic cases that include a detailed representa-

tion of the generation system and the transmission

network. This line of research has also provided

theoretical results relative to the design of electricity

markets or to the medium-term operation of hydro-

thermal systems in the new regulatory framework. As in

the case of optimization models, the research commu-nity is now trying to develop a new generation of

equilibrium models capable of taking risk management

decisions under imperfect competition.

On the subject of market representation, there are

recent publications devoted to the improvement of

existing Cournot-based models. They propose the CV

approach to overcome the high sensitivity of the price-

clearing process with respect to demand representation

typical of such models. Obviously, there are still

questions to be resolved. For instance, even when the

simple Cournot conjecture is assumed, pure strategy

solutions may not exist if there are transmission

constraints. Another example is that non-decreasing

supply functions may be unstable when generating

capacity constraints are considered.

The contribution of simulation models has been

significant as well, on account of their flexibility to

incorporate more complex assumptions than those

allowed by formal equilibrium models. Simulation

models can explore the influence that the repetitive

interaction of participants exerts on the evolution of

wholesale electricity markets. In these models, agents

learn from past experience, improve their decision-

making and adapt to changes in the environment. This

suggests that adaptive agent-based simulation techni-ques can shed light on certain features of electricity

markets ignored by static models and therefore these

techniques will be helpful in the analysis of new

regulatory measures and market rules.

As a concluding remark, it should be pointed out that

the impressive advances registered in this research field

underscore how much interest this matter has drawn

during the last decade.

References

Anderson, E.J., Philpott, A.B., 2002. Optimal offer curve construction

in electricity markets. Mathematics of Operations Research 27 (1),

82–100.

Ba!ıllo, A., 2002. A methodology to develop optimal schedules and

offering strategies for a generation company operating in a short-

term electricity market. Ph.D. Dissertation, ICAI, Universidad

Pontificia Comillas, Madrid.

Ba!ıllo, A., Ventosa, M., Ramos, A., Rivier, M., Canseco, A., 2001.

Strategic unit commitment for generation companies in deregulated

electricity markets. In: Hobbs, B., Rothkopf, M., O’Neil, R., Chao,

H. (Eds.), The Next Generation of Unit Commitment Models.

Kluwer Academic Publishers, Boston, pp. 227–248.

Baldick, R., Hogan, W.W., 2001. Capacity Constrained Supply

Function Equilibrium Models of electricity markets: stability,

non-decreasing constraints, and function space iterations. Program

on Workable Energy Regulation (POWER) PWP-089. University

of California Energy Institute, Berkeley, CA.

Baldick, R., Grant, R., Kahn, E., 2000. Linear supply function

equilibrium: generalizations, application and limitations, Program

on Workable Energy Regulation (POWER) PWP-078. University

of California Energy Institute, Berkeley, CA.

Barqu!ın, J.,Centeno, E., Reneses, J., 2003. Medium-term generation

programming in competitive environments: a new optimization

approach for market equilibrium computing. IEE Proceedings,

Generation, Transmission and Distribution (in press).

Batlle, C., Otero-Novas, I., Alba, J.J., Meseguer, C., Barqu!ın, J., 2000.

A model based in numerical simulation techniques as a tool for

decision-making and risk management in a wholesale electricity

market. Part I: general structure and scenario generators.

Proceedings PMAPS00 Conference, Madeira.

Berry, C.A., Hobbs, B.F., Meroney, W.A., O’Neill, R.P., Stewart Jr.,

W.R., 1999. Understanding how market power can arise in network

competition: a game theoretic approach. Utilities Policy 8, 139–158.

Bolle, F., 1992. Supply function equilibria and the danger of tacit

collusion. Energy Economics 14 (2), 94–102.

Borenstein, S., Bushnell, J., 1999. An empirical analysis of the

potential for market power in California’s electricity industry.Journal of Industrial Economics 47 (3), 285–323.

Borenstein, S., Bushnell, J., Kahn, E., Stoft, S., 1995. Market power in

California electricity markets. Utilities Policy 5 (3/4), 219–236.

Bower, J., Bunn, D., 2000. Model-based comparisons of pool and

bilateral markets for electricity. Energy Journal 21 (3), 1–29.

Bushnell, J., 1998. Water and power: hydroelectric resources in the era

of competition in the Western US. Program on Workable Energy

Regulation (POWER) PWP-056. University of California Energy

Institute, Berkeley, CA.

Bushnell, J., Day, C., Duckworth, M., Green, R., Halseth, A., Grant

Read, E., Scott Rogers, J., Rudkevich, A., Scott, T., Smeers, Y.,

Huntington, H., 1999. An international comparison of models for

measuring market power in electricity. Energy Modeling Forum,

Stanford University. Available from World Wide Web: (http://

www.stanford.edu/group/EMF/research/doc/EMF17.1.PDF ).

Daughety, A.F., 1988. Cournot Oligopoly. Cambridge University

Press, Cambridge.

Day, C., Bunn, D., 2001. Divestiture of generation assets in the

electricity pool of England and wales: a computational approach to

analyzing market power. Journal of Regulatory Economics 19 (2),

123–141.

Day, C.J., Hobbs, B.F., Pang, J.-S., 2002. Oligopolistic competition in

power networks: a conjectured supply function approach. IEEE

Transactions on Power Systems 17 (3), 597–607.

Ferrero, R., Shahidehpour, S., Ramesh, V., 1997. Transaction analysis

in deregulated power systems using game theory. IEEE Transac-

tions on Power Systems 12, 1340–1347.

Fleten, S.E., Wallace, S.W., Ziemba, W.T., 1997. Portfolio manage-

ment in a deregulated hydropower based electricity market.Proceedings Hydropower97, Norway, July.

Fleten, S.E., Ziemba, W.T., Wallace, S.W., 2002. Hedging electricity

portfolios via stochastic programming. In: Greengard, C., Ruszc-

zynski, A. (Eds.), Decision Making Under Uncertainty: Energy and

Power 128 of IMA Volumes on Mathematics and Its Applications.

Springer, New York, pp. 71–93.

Garc!ıa, J., Rom!an, J., Barqu!ın, J., Gonz!alez, A., 1999. Strategic

bidding in deregulated power systems. Proceedings 13th PSCC

Conference, Norway, July.

Garc!ıa-Alcalde, A., Ventosa, M., Rivier, M., Ramos, A., Relan ˜ o, G.,

2002. Fitting electricity market models. A conjectural variations

approach. Proceedings of the 14th PSCC Conference, Seville. July.

Green, R.J., 1996. Increasing competition in the British electricity spot

market. Journal of Industrial Economics (44), 205–216.

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