ventosa, baillo, ramos, rivier - electricity market modeling trends

Accepted for publication in Energy Policy

Electricity Market Modeling Trends

Mariano Ventosa*, lvaro Ballo, Andrs Ramos and Michel Rivier INSTITUTO DE INVESTIGACIN TECNOLGICA

Universidad Pontificia Comillas Alberto Aguilera 23

28015 Madrid, SPAIN [email protected]

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 classification according 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.

Index Terms Deregulated electric power systems, power generation scheduling, market

behavior.

* Corresponding author

1


1 Introduction

In the last decade, the electricity industry has experienced significant changes towards

deregulation and competition with the aim of improving economic efficiency. In many places,

these changes have culminated 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 characteristics 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

2


models.

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

modeling, 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 brevitys 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

liberalized energy markets. It also introduces the application of multistage-equilibrium

models in the context of investment 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, which assume oligopolistic

competition. Two kinds of equilibria 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 complementary 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 II summarizes the classification scheme used in

3


the survey. Section III describes the publications related to optimization models, whereas

Section IV focuses on those related to equilibrium models. Section V presents the

publications devoted to simulation models. Section VI details the proposed taxonomy for

electricity market models. Section VII points out the major uses of each modeling approach

and, finally, Section VIII 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.

Optimization Problem for One Firm

Exogenous Price

Demand-price Function

Electricity Market

Modeling

Market Equilibrium Considering

All Firms

Cournot Equilibrium

Supply Function Equilibrium

Simulation Models

Equilibrium Models

Agent-based Models

Fig. 1. Schematic representation of the electricity market modeling trends

Research developments follow three main trends: optimization models, equilibrium models

and simulation models. Optimization models focus on the profit maximization problem for a

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 VI), 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.

Mathematical structure: Optimization-based models are formulated as a single

4


optimization program in which one firm pursues its maximum profit. There is a single

objective function to be optimized subject to a set of technical and economic constraints. In

contrast, both equilibrium and simulation-based models consider the simultaneous profit

maximization program of each firm competing in the market. Both types of models are

schematically represented in Fig. 2, where f represents the profit of each firm { } Lf 1, ,F ; fx are firm fs decision variables; and ( )fh x and represent firm f's

constraints.

( )fg x

Optimization Programof Firm 1 f

Optimization Programof Firm

Optimization Programof Firm F

( )( )( )

1 1

1 1

1 1

00

=

maximize : x

subject to : h xg x

Electricity Market

Supply = Demand

( )( )( )

f f

f f

f f

maximize : x

subject to : h x 0g x 0

=

( )( )( )

F F

F F

F F

maximize : x


=

Optimization Program of firm f

( )( )( )

=

f

f

f

maximize : x


Single-firm Optimization Model Equilibrium Model

Electricity Market

Supply = Demand

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

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 as exogenous to the

optimization program or as dependent of the quantity supplied by the firm of interest.

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.

Major uses: The previously mentioned differences in mathematical structure, market

modeling and computational tractability 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

5


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 some controversy 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 firms optimization program, i.e., the system marginal price is an input parameter for the

optimization program. Consequently, as the price is fixed, the market revenueprice times the firms productionbecomes a linear function of the firms 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 employed to obtain the

solution of the model. Unfortunately, this type of optimization model can only properly

represent markets under quasi-perfect competition conditions because it neglects the influence

of the firms decisions on the market clearing price.

These models can again be classified into two sub-groups, depending on whether they use

a deterministic or probabilistic price representation.

Deterministic models: A good example is the model proposed in Gross and Finlay et al.

(1996).1 In this model, since the price is considered to be exogenous, it is shown that the

firms optimization problem can be decomposed into a set of sub-problemsone per generatorresembling the Lagrangian Relaxation approach.2 As expected in a case of perfect competition, deterministic price and convex costs, the simple comparison between each

1 Many later models are based on the same assumptions, thus leading to similar conclusions.

2 A large-scale problem with complicating constraints is amenable for a dual decomposition solution strategy,

commonly known as Lagrangian Relaxation approach.

6


generators marginal cost and the market price decides the production of each generator;

therefore, the best offer of each generation unit consists of bidding its marginal cost.

Stochastic models: The previous approach can be improved if price uncertainty 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 firms profit, based on the prices of energy and

reserve at the nodes where the firms units are located, which are assumed to be both

exogenously determined and uncertain. Similarly 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 a

solution, 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 penalizing 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 firms

maximization problem is decomposed into a financial master-problem and an operation sub-

problem. While the financial master-problem produces financial decisions 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 linear programming.

7


3.2 Price as a Function of the Firms Decisions

In contrast to the former approaches in which the price clearing process is assumed to be

independent of the firms decisions, there exists another family of models that explicitly

considers the influence of a firms 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-groups depending on whether a

probabilistic representation of the residual-demand function is used.

Deterministic models: The first publication on electricity markets based on the leader-in-

price model is Garca 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 firms total output, in order to allow for the use of powerful Mixed Integer

Linear Programming (MILP) solvers, a piecewise linearization procedure of the market

revenue is proposed. Likewise, Ballo et al. (2001) develop a MILP-based model focusing on

the problem of one firm with significant hydro resources. The Ballo model is more advanced

in that it allows non-concave market revenue functions by means of additional binary

variables. This approach is included in a recent monograph on new developments in unit

commitment models (Hobbs et al. 2001).

Stochastic models: Unlike previous approaches, Anderson and Philpott (2002) do not

formulate the problem of optimal production 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

3 From 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-sides buy bids. The term residual-demand function is also

known as effective demand function. 4 The 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.

8


short-term uncertainties in the marketplace. The thesis of Ballo (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 equilibria 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 firms compete in quantity strategies, whereas the

most complex type is based on Supply Function Equilibrium (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 equilibriumthe market reaches equilibrium when each firms 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 collection of essays regarding Cournot

competition, which links this approach with other later modelsincluding the SFE mentioned abovecan be found in (Daughety, 1988).

Cournot equilibrium, where firms choose their optimal output, is easier to compute than

supply function equilibrium 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

network and risk assessment.

Market power analysis: Market power measurement was the earliest application to

5 The 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 hydro reserves.

9


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 empirical 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 modelsmost of them based on Cournot competitionfor measuring market power in electricity can be found in (Bushnell et al. 1999). This paper

summarizes in tabular 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.

Hydrothermal coordination: Apart from market power analysis, Cournot competition has

also been considered in hydrothermal models. The first publication on this subject is by Scott

and Read (1996), which in the context of New Zealands electricity market. Their model

utilizes Dual Dynamic Programming (DDP), whereby at each stage the hydro optimization

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 most significant contribution is its

discussion about the meaning of the firms marginal water value in a deregulated framework.

Bushnell points out that the firms water value is related to the firms marginal revenue

instead of the traditional systems marginal cost. Although Bushnells analytical formulation

of the market equilibrium conditions is more elegant, the Scott and 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

10


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 algorithm is non-

convex in equilibrium problems. Barqun et al. (2003) introduce an original approach to

compute market equilibrium, by solving an equivalent minimization 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.

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

Kirchhoffs voltage 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 large problems. In all

these models it is assumed that the generation units of each firm are located at only one node

of the networkwhich is, obviously, a particular case. Unfortunately, since in the general

6 The 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 Mixed Complementarity

Problem (MCP) is a mixture of equations with a Complementarity Problem. 7 In the Hydrothermal Coordination Problem, the recourse function is known as the future water value. 8 KKT conditions can also be formulated as a Variational Inequality (VI) problem. 9 A Linear Complementarity Problem (LCP) is obtained when the KKT conditions are derived from an

optimization problem with quadratic objective function and linear constraints.

11


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).

Risk analysis: Finally, because of the difficulty in applying traditional risk management

techniques to electricity markets, only one publication has been identified that explicitly

addresses the risk management problem for generation firms under imperfect competition

conditions. 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 outcome 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 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 reinforce the idea that the Supply Function Equilibrium approach is a

better alternative to represent competition in electricity markets (Rudkevich, 1999).

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-based models. This approach changes the

conjectures that generators are expected to assume about their competitors strategic

decisions, in terms of the possibility of future reactions (conjectural variations). Two recent

publications (Garca-Alcalde et al. 2002; Day et al. 2002) suggest considering this approach

in order to improve Cournot pricing in electricity markets. Garca-Alcalde et al. (2002)

10 In 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.

12


assume that firms make conjectures about their residual demand elasticities, as in the general

conjectural variations 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 it offers different quantities to the market. This is the supply

function equilibrium (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 supply function equilibrium model and evaluation of the impact

of the electric power network.

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 is extremely inelasticdemand-side bidding

was almost non-existentand 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

13


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.

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 inelastic demand, further advances on the SFE theory

have been reported which increase its applicability. Rudkevich (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.

Convergence 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.

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

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

method to 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

11 According to Baldick (2000), the precise description would be affine demand, whereas the term linear

should be restricted to affine functions with zero intercept.

14


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.

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 Schweppes spot pricing theory, including the influence of

transmission constraints. A finite number of offering strategies are defined for each

generation 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 conditions 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.

15


TABLE 1. A CHARACTERIZATION OF SFE MODELS

Author Asymmetric Firms Marginal

Costs Demand Curve Supply Functions

Capacity Constraints

Solution Method

Transmission Network

Numerical Application

Klemperer, 1989 No Convex Concave Twice Continuously Differentiable No Necessary Conditions No No

Green, 1992 No Quadratic Linear Twice Continuously Differentiable Yes Numerical Integration No E&W Pool

Green, 1996 Yes Linear Linear Affine No Closed-form Expression No E&W Pool

Ferrero, 1997 Yes Affine Inelastic Affine Yes Exhaustive Enumeration Yes IEEE 30-bus

System

Rudkevich, 1998 No Stepwise Inelastic Differentiable Yes Closed-form Expression No Pennsylvania

Baldick, 2000 Yes Affine Linear Piecewise Linear Yes Heuristics No E&W Pool

Baldick, 2001 Yes Affine Linear Piecewise Linear Non-decreasing Yes Heuristics No E&W Pool

Berry, 1999 Yes Affine Linear Affine Yes Heuristics Yes Four-node Case

Hobbs, 2000 Yes Affine Linear Affine Yes MPEC Yes 30-node Case

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

function12, 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 assumptions 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, thus increasing

the computational requirements of this approach. Moreover, some of this systems 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

12 In general, SFE-based approaches model the demand function as inelastic, which is the most suitable

hypothesis in the case of electricity.

16


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 mathematically expressed in the form of a system of algebraic and/or differential

equations. This imposes limitations on the representation of competition between participants.

In addition, the resulting set of equations, if it has a solution, is frequently too hard to solve.

The fact that 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 each agents 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 theoretically 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 hydro generating 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 companys 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

17


demand, which is assumed to be inelastic.

Bunn and Day (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 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 hydro inflows). 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 participate 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 generation portfolio 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

the 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.

18


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 II to V, which is based on the

mathematical structure of each model, electricity market models can be categorized

considering more specific attributes. These characteristics are useful in understanding the

advantages and limits of each model surveyed in previous sections. The taxonomy presented

here considers the following issues: degree of competition, time scope of the model,

uncertainty modeling, interperiod links, transmission constraints, generating system

representation and market 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 perfectly competitive market can be modeled as

a cost minimization 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 competition

conditionsthe most common situationthe 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.

19


Competition

Time Scope

Oligopoly

Monopoly

Perfect Competition

Long Term (Years)

Medium Term(Months)

Short Term(Days)

Nash Equilibrium

(Cournot and Stackelberg)

Nash Equilibrium

(Cournot and SFE)

Leader in Price

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

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 maximum capacity of each generator is considered to be fixed.

As previously mentioned, under imperfect competition 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 this

paper suggests that the best way to represent the market is the leader-in-price model from

microeconomics theory (Garca et al. 1999; Ballo et al. 2001; Anderson and Philpott, 2002;

Ballo, 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 (one month to one year), the vast majority of the models are

20


based on both Cournot equilibrium (Scott and Read, 1996; Bushnell, 1998; Rivier et al. 2001;

Kelman et al. 2001; Barqun et al. 2003; Otero-Novas et al. 2000) and supply function

equilibrium (Green and Newberry, 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; the follower 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 equilibrium and Stackelberg

equilibrium for modeling investment 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 Mathematical Problem with Equilibrium

Constraints (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.

Moreover, in a competitive context, new sources of uncertainty must be considered due to

both strategic behavior of competitors and fuel price volatility.

According to the manner in which uncertainty is represented, models can be classified into

probabilisticwhen the uncertain nature of random variables is incorporated using

probabilistic distributionsand deterministicwhen 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 Supply Function

Equilibrium (SFE in Fig. 4) (Green and Newberry, 1992; Bolle, 1992; Rudkevich et al. 1998)

21


since they all consider uncertainty in demand. Based on a probabilistic version of the price-

leadership model, the Ballo model (2002) not only considers the uncertainty in demand but

also in competitors behavior. Finally, Fleten (2002), and Unger (2002) models focus on

uncertainty in prices and hydraulic inflows under pure competition assumptions, while

Kelman (2001) considers a Cournot framework.

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 management policies for hydro reserves 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 commitment 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 coordination and capacity expansion problems, focus on the

tradeoff of scheduling resources across time.

22


DC Model

Interperiod Links

Transmission Network

Uncertainty

Single Node

Probabilistic

Deterministic

Interperiod Constraints

Probabilistic

Deterministic Intraperiod Constraints

Interperiod constraints

Intraperiod constraints

DC Model

Single Node

Transshipment Model

Transshipment Model

AC Model

AC Model

Fig. 4. Characterization of some electricity market models according to the modeling of uncertainty,

transmission network and interperiod links

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.

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 transshipment network that omits Kirchhoffs voltage lawWei and

Smeers (1999)although their model allows for inter-temporal constraints regarding

investment decisions (Fig. 4). Other authors consider both of Kirchhoffs lawsBerry el al.

(1999), Hobbs et al. (2000) and 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 approachesHobbs (2001) and Wei and Smeers (1999)permit

solving realistically sized problems.

23


6.6 Generating System Modeling

A high degree of realism regarding the physical modeling of generating systems involves

the representation 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 (2001) consider not only the hydro energy constraints

implicit in the management of water reserves but also the hydraulic inflow uncertainty. On the

other hand, short-term models such as Garca et al. (1999) and Ballo (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 approachesthe Otero-Novas model

(2000), which combines a simulation algorithm with optimization techniques, and the Rivier

model (2001), which is solved by complementarity methodsreach 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 generation system due to their numerical tractability

limitations.

24


Probabilistic

Medium

Market Modeling

Generating System Modeling

Uncertainty

Low

Probabilistic

DeterministicHigh

Deterministic

Medium

Low

High

Single-firm Residual Demand

Exogenous Price

Imperfect Market

Equilibrium

Exogenous Price

Single-firm Residual Demand

Imperfect Market

Equilibrium

Fig. 5. Characterization of some electricity market models according to the treatment of uncertainty,

generation system modeling and market modeling

6.7 Market Modeling

The last attribute considered in this taxonomy is related to the market model under

consideration. Pure competition-based modelsFleten (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

conceptGarca (1999) and Ballo (2002)are considered to have an intermediate level of

complexity since they take into account the influence of the firms production on prices by

means of its residual demand function. Finally, the most 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 II, differences in mathematical 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 III, IV and V regarding the major uses of single-firm

25


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 variables

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. TABLE 2. MAJOR USES OF ELECTRICITY MARKET MODELS

Major Use One-firm Optimization Models Simulation

Models Imperfect Market Equilibrium Models

Risk Management (Fleten, 2002; Unger, 2002; Pereira, 1999)

Unit Commitment (Garca, 1999; Rajamaran, 2001)

Short-Term Hydrothermal Coordination (Ballo, 2001)

Strategic Bidding (Anderson & Philpott, 2002; Ballo, 2002)

Market Power Analysis (Day & Bunn, 2001)

(Green, 1992; Bolle, 1992; Rudkevich, 1998; Borenstein, 1995&1999; Baldick,

2000&2001)

Market Design (Bower & Bunn, 2000) (Green, 1996; Baldick, 2000 & 2001)

Yearly Economic Planning

(Otero-Novas, 2000) (Ramos, 1998)

Long-Term Hydrothermal Coordination

(Scott, 1996; Bushnell, 1998; Rivier, 2001; Kelman, 2001; Barqun, 2003)

Capacity Expansion Planning (Murphy & Smeers, 2002; Ventosa, 2002)

Congestion Management (Hogan, 1997; Oren, 1997; Hobbs, 2000 & 2001; Wei & Smeers, 1999; Berry, 1999)

In contrast, when long-term planning studies are conducted, equilibrium models are more

suitable because 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 yearly economic 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.

26


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 management 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 characteristics of the most significant models

referred to in previous sections. The models are classified into eight categories depending on

their market model.13 Within 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.

13 CSF: Conjectured Supply Function approach, and CV: Conjectural Variations approach. 14 Benders: 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 Programming, Simulation: Simulation Scenario Analysis, and VI: Variational

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

variables.

27


TABLE 3. MAJOR USES AND MAIN FEATURES OF THE REVIEWED MARKET MODELS

Market Model Authors Year Major Use Main Feature Solution Method

Size Intended Market

Gross & Finlay. 1996 Generation Scheduling Deterministic Prices LP Large E&W

Fleten et al. 1997 Hydro And Risk Management Stochastic Prices & Inflows LP Large Nord Pool

Pereira et al. 1999 Hydro And Risk Management Solution Method Benders Large

Rajamaran et al. 2001 Unit Commitment Price Uncertainty DP Large

Perfect Competition and Exogenous price

Unger 2002 Hydro And Risk Management Risk Modeling LP Large Nord Pool

Garca et al. 1999 Unit Commitment Thermal Modeling MIP Large Spain

Ballo et al. 2001 Short-Term Hydrothermal Coordination Non-Convex Profit MIP Large Spain

Anderson & Philpott 2002 Offer Curve Construction Stochastic Demand Function NLP Small New Zealand

Leader-in-price and Residual

Demand Function Ballo et al. 2002 Offer Curve Construction Practical Approach MIP Large Spain

Green & Newberry 1992 Market Power Analysis Symmetric Firms NI Small E&W

Bolle 1992 Market Power Analysis Symmetric Firms NI Small E&W Supply

Function Equilibrium Rudkevich et al. 1998 Market Power Analysis Closed-Form Solution Analytic Small Pennsylvania

Green 1996 Market Design Closed-Form Solution Analytic Small E&W

Ferrero et al. 1997 Congestion Management AC Network Model Enumeration Small

Berry et al. 1999 Congestion Management DC Network Model Heuristic Small

Hobbs et al. 2000 Congestion Management DC Network Model MPEC Medium

Baldick et al 2000 Market Power Analysis Piecewise Linear SFE Heuristic Small E&W

Baldick et al 2001 Market Design Non-Decreasing SFE Heuristic Medium E&W

Linear Supply

Function Equilibrium

Day & Bunn 2001 Market Power Analysis Asymmetric Firms Enumeration Medium E&W

Scott & Read 1996 Hydrothermal Coordination Hydro-Interperiod Links DP Medium New Zealand

Bushnell 1998 Hydrothermal Coordination Analytic Modeling DP Medium California

Borenstein & Bushnell 1999 Market Power Analysis Radial Congestion Heuristic Medium California

Batlle et al. 2000 Risk Analysis Stochastic Prices & Inflows Simulation Large Spain

Otero-Novas et al. 2000 Yearly Economic Planning Agents Behavior Heuristic Large Spain

Kelman et al. 2000 Long-Term Hydrothermal Coordination Stochastic Inflows DP Large Brazil

Rivier et al. 2001 Hydrothermal Coordination Hydrothermal Modeling MCP Large Spain

Cournot

Equilibrium

Barqun et al. 2003 Hydrothermal Coordination Stochastic Inflows NLP Large

Ventosa et al. 2002 Capacity Expansion Planning Investment Decisions MPEC Medium Stackelberg

Murphy & Smeers 2002 Capacity Expansion Planning Investment Decisions EPEC Medium

Hogan 1997 Congestion Management Network Constraints NLP Small

Oren 1997 Congestion Management Network Constraints Analytic Small

Wei & Smeers 1999 Congestion Management Transshipment Model VI Large Europe Spatial

Cournot

Hobbs 2001 Congestion Management DC Power Flow LCP Large CV Garca-Alcalde et al. 2002 Price Forecasting Fitting Procedure LCP Large Spain

CSF Day et al. 2002 Congestion Management DC Power Flow LCP Large E&W

Agent-based Bower & Bunn 2000 Market Design Learning Procedure Heuristic Medium E&W

28


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, equilibrium 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 present, 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 analyze realistic cases that include a

detailed representation 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 hydrothermal systems in the new regulatory framework. As in

the case of optimization models, the research community 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 conjectural variations

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

29


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 techniques

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), 82100.

Ballo, A., Ventosa, M., Ramos, A., Rivier, M., Canseco, A., 2001. Strategic Unit Commitment for

Generation Companies in Deregulated Electricity Markets. The Next Generation of Unit Commitment

Models. Hobbs, B., Rothkopf, M., ONeil, R., Chao, H. (Ed), Kluwer Academic Publishers. Boston.

pp. 227248.

Ballo, 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.

Baldick, R., Grant, R. and 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, California.

Baldick, R. and 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, California.

Barqun, J.,Centeno, E., Reneses, J. 2003. Medium-Term Generation Programming in Competitive

Environments: a New Optimization Approach for Market Equilibrium Computing. Accepted in IEE

Proceedings, Generation, Transmission and Distribution.

Batlle, C., Otero-Novas, I., Alba, J. J., Meseguer, C., Barqun, J. 2000. A Model Based in Numerical

Simulation Techniques as a Tool for Decision-Making and Risk Management in a Wholesale

30


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. and Stewart Jr. W. R. 1999. Understanding

How Market Power Can Arise in Network Competition: a Game Theoretic Approach. Utilities Policy

8, 139158.

Bolle, F. 1992. Supply Function Equilibria and the Danger of Tacit Collusion. Energy Economics.

94102.

Borenstein, S., Bushnell, J., Kahn, E. and Stoft, S. 1995. Market Power in California Electricity

Markets. Utilities Policy 5(3/4), 219236.

Borenstein, S. and Bushnell, J. 1999. An Empirical Analysis of the Potential for Market Power in

Californias Electricity Industry. Journal of Industrial Economics 47(3), 285323.

Bower, J. Bunn, D. 2000. Model-based Comparisons of Pool and Bilateral Markets for Electricity.

Energy Journal 21(3), 129.

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, California.

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), 123141.

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), 597607.

Ferrero, R., Shahidehpour, S. and Ramesh, V. 1997 Transaction Analysis in Deregulated Power

Systems Using Game Theory, IEEE Transactions on Power Systems 12, 13401347.

31


Fleten, S. E., Wallace, S. W., Ziemba, W. T. 1997. Portfolio Management 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 Decision Making Under Uncertainty: Energy and Power 128 of IMA Volumes on

Mathematics and Its Applications, Greengard, C. and Ruszczynski, A. (Ed), Springer-Verlag, New

York, pp. 7193.

Garca, J., Romn, J., Barqun, J., Gonzlez, A. 1999. Strategic Bidding in Deregulated Power

Systems. Proceedings 13th PSCC Conference, Norway, July.

Garca-Alcalde, A., Ventosa, M., Rivier, M., Ramos, A., Relao, G. 2002. Fitting Electricity Market

Models. A Conjectural Variations Approach. Proceedings 14th PSCC Conference, Seville. July.

Green, R. J. and Newbery, D. M. 1992. Competition in the British Electricity Spot Market. Journal of

Political Economy 100(5), 929953.

Green, R. J. 1996. Increasing Competition in the British Electricity Spot Market, Journal of Industrial

Economics (44), 205216.

Gross, G., Finlay, D. J. 1996. Optimal Bidding Strategies in Competitive Electricity Markets.

Proceedings 12th PSCC Conference, Germany. July.

Hobbs B. F., Metzler C., Pang J.S. 2000. Strategic Gaming Analysis for Electric Power Networks: An

MPEC Approach. IEEE Transactions on Power Systems 15(2), 638645.

Hobbs, B., Rothkopf, M., ONeil, R., Chao, H. 2001. The Next Generation of Unit Commitment

Models. Kluwer Academic Publishers. Boston.

Hobbs, B. F. 2001. Linear Complementarity Models of Nash-Cournot Competition in Bilateral and

POOLCO Power Markets. IEEE Transactions on Power Systems 16(2), 194202.

Hogan, W.W. 1997. A Market Power Model with Strategic Interaction in Electricity Networks. The

Energy Journal18(4), 107141.

Kahn, E. 1998. Numerical Techniques for Analyzing Market Power in Electricity. The Electricity

Journal. July 1998. 3443.

Kelman, R., Barroso, L., Pereira, M., 2001. Market Power Assessment and Mitigation in

Hydrothermal Systems. IEEE Transactions on Power Systems 16(3), 354359.

32


Klemperer, P. D., Meyer, M. A. 1989. Supply Function Equilibria in Oligopoly under Uncertainty.

Econometrica 57(6), 12431277.

Murphy, F., Smeers, Y., 2002. Capacity Expansion in Imperfectly Competitive Restructured

Electricity Market. Submitted to Operations Research.

Neuhoff, K., 2003 Integrating Transmission and Energy Markets Mitigates Market Power. CMI

Working paper 17. Massachusetts Institute of Technology, Center for Energy and Environmental

Policy Research.

Oren, S. 1997. Economic Inefficiency of Passive Transmission Rights in Congested Electrical System

with Competitive Generation. The Energy Journal 18(1), 6383.

Otero-Novas, I., Meseguer, C., Batlle, C., Alba, J. 2000. A Simulation Model for a Competitive

Generation Market. IEEE Transactions on Power Systems 15(1), 250257.

Pereira, M. V. 1999. Methods and Tools for Contracts in a Competitive Framework. Working paper

CICR Task Force 30-05-09.

Rajamaran, R., Kirsch, L., Alvarado, F. Clark, C. 2001. Optimal Self-commitment under Uncertain

Energy and Reserve Prices, in The Next Generation of Electric Power Unit Commitment Models,

International Series in Operations Research & Management Science, B. F. Hobbs, M. H. Rothkopf, R.

P. O'Neill and H.-P. Chao, (Ed.), Kluwer Academic Publishers. Boston, pp. 93116.

Ramos, A., Ventosa, M., Rivier, M. 1998. Modeling Competition in Electric Energy Markets by

Equilibrium Constraints. Utilities Policy 7(4), 233242.

Rivier, M., Ventosa, M., Ramos, A. 2001. A Generation Operation Planning Model in Deregulated

Electricity Markets based on the Complementarity Problem. In Applications and Algorithms of

complementarity. M. C. Ferris, O. L. Mangasarian and J.-S. Pang, (Ed), Kluwer Academic Publishers.

Boston. pp. 273298.

Rudkevich, A., Duckworth, M. and Rosen, R. 1998. Modeling Electricity Pricing in a Deregulated

Generation Industry: The Potential for Oligopoly Pricing in a Poolco. The Energy Journal 19(3),

1948.

Scott, T.J. and Read, E.G. 1996. Modelling Hydro Reservoir Operation in a Deregulated Electricity

Market. International Transactions in Operational Research 3, 243253.

33


Smeers, Y. 1997, Computable Equilibrium Models and the Restructuring of the European Electricity

and Gas Markets. The Energy Journal 18(4), 131.

Unger, G. 2002. Hedging Strategy and Electricity Contract Engineering. Ph.D. Thesis, Swiss Federal

Institute Of Technology, Zurich.

Varian, H.R. 1992. Microeconomic Analysis. W.W. Norton & Company. New York.

Ventosa, M., Denis, R., Redondo, C. 2002. Expansion Planning in Electricity Markets. Two Different

Approaches. Proceedings 14th PSCC Conference Seville, July.

Vives, X. 1999. Oligopoly Pricing. The MIT Press. Cambridge,

Wei, J.-Y., Smeers, Y. 1999, Spatial Oligopolistic Electricity Models with Cournot Generators and

Regulated Transmission Prices. Operations Research 47(1), 102112.

34

IntroductionElectricity Market Modeling TrendsSingle-Firm Optimization ModelsExogenous pricePrice as a Function of the Firms Decisions

Equilibrium ModelsCournot EquilibriumExtensions of Cournot EquilibriumSupply Function Equilibrium

Simulation ModelsSimulation Models related to Equilibrium ModelsAgent-Based Models

Taxonomy Of Electricity Market ModelsDegree of CompetitionTime ScopeUncertainty ModelingInterperiod LinksTransmission ConstraintsGenerating System ModelingMarket Modeling

Major UsesConclusion and Future Developmental TrendsReferences

ventosa, baillo, ramos, rivier - electricity market modeling trends

Documents