Electricity market modeling trends

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Energy Policy 33 (2005) 897–913

Electricity market modeling trends ! Mariano Ventosa*, Alvaro Ba!ıllo, Andre! s Ramos, Michel Rivier ! Tecnologica, ! Instituto de Investigacion 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 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. 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 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 *Corresponding author. Tel.: +34-91-542-28-00; fax: +34-91-54231-76. E-mail address: [email protected] (M. Ventosa). 0301-4215/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.enpol.2003.10.013

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 models.

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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 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 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 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 2 summarizes the classification scheme used in the survey. Section 3 describes the publications related to optimization 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 simulation 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 single optimization program in which one firm pursues its maximum profit. There is a single objective function to

Optimization Problem for One Firm

Electricity Market Modeling

Market Equilibrium Considering All Firms

Exogenous Price Demand-price Function

Cournot Equilibrium Supply Function Equilibrium

Equilibrium Models

Simulation Models

Agent-based Models

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

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Single-firm Optimization Model Optimization Program of firm f

maximize : Π

f

( x)

subject to : h f ( x ) = 0 f g (x ) ≤ 0

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Equilibrium Model Optimization Program of Firm 1

Optimization Program of Firm f

( xf )

maximize : Π 1 ( x1 )

maximize : Π

subject to : h1 ( x1) = 0

subject to : h f ( x f ) = 0

g (x ) ≤ 0 1

1

g

f

f

(x ) ≤ 0 f

Supply = Demand

Supply = Demand

Electricity Market

Electricity Market

Optimization Program of Firm F

maximize : Π

F

( xF )

subject to : h F ( x F ) = 0

g F (x F ) ≤ 0

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

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 Pf represents the profit of each firm f Af1; y; F g; xf are firm f’s decision variables; and hf ðxÞ and gf ðxÞ represent firm f’s constraints.

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 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.

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 as exogenous 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 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

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 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 firm’s decisions on the market clearing price.

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These models can again be classified into two subgroups, 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 is considered 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 production 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 price uncertainty is explicitly considered. For instance, Rajamaran et al. (2001) describe and solve the selfcommitment 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 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 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.

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’s 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 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-groups depending 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 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-side’s 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.

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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 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 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 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 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 collection 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 optimal output, is easier to compute than SFE because the mathematical structure of Cournot models turns out to

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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 equilibriumbased 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. 4.1.1. Market power analysis Market power measurement was the earliest application 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 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 models—most of them based on Cournot competition—for 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. 4.1.2. 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), in the context of New Zealand’s electricity market. Their model utilizes Dual Dynamic Programming (DDP), whereby at each stage the hydrooptimization 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 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 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 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 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. Barqu!ın et al. (2003) introduce an original approach to compute market equilibrium, by solving an equivalent minimization problem. This approach is oriented to the mediumterm 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’s 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 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 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 VI problem.

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 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 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-atrisk (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 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 9 A LCP is obtained when the KKT conditions are derived from an optimization problem with quadratic objective function and linear constraints. 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.

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reinforce the idea that the SFE approach is a better alternative to represent competition in electricity markets (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-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 (CV). Two recent publications (Garc!ıaAlcalde 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.

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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 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. 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 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

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 inelastic demand, 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. 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. 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 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.

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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 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 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. 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 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 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 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 12 In general, SFE-based approaches model the demand function as inelastic, which is the most suitable hypothesis in the case of electricity.

ARTICLE IN PRESS No E&W Pool E&W Pool IEEE 30-bus system Pennsylvania E&W Pool E&W Pool Four-node case 30-node case No No No Yes No No No Yes Yes Necessary conditions Numerical integration Closed-form expression Exhaustive enumeration Closed-form expression Heuristics Heuristics Heuristics MPEC No No Yes Yes No Yes Yes Yes Yes Klemperer and Meyer (1989) Green and Newbery (1992) Green et al. (1996) Ferrero et al. (1997) Rudkevich et al. (1998) Baldick et al. (2000) Baldick and Hogan (2001) Berry et al. (1999) Hobbs et al. (2000)

Convex Quadratic Linear Affine Stepwise Affine Affine Affine Affine

Concave Linear Linear Inelastic Inelastic Linear Linear Linear Linear

Twice continuously differentiable Twice continuously differentiable Affine Affine Differentiable Piecewise linear Piecewise linear non-decreasing Affine Affine

No Yes No Yes Yes Yes Yes Yes Yes

Transmission network Asymmetric firms Author

Table 1 A characterization of SFE models

Marginal costs

Demand curve

Supply functions

Capacity constraints

Solution method

Numerical application

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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 agent’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 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 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

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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.

Competition

Maximization of the Monopolist Firm

Oligopoly

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 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 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-asbid 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

Market Model Based on the Profit

Monopoly

Perfect Competition

Leader in Price

Nash Equilibrium (Cournot and SFE)

Nash Equilibrium (Cournot and Stackelberg)

Market Model Based on the Cost Minimization of the Whole System Short Term (Days)

Medium Term (Months)

Time Scope

Long Term (Years)

Fig 3. Theoretical electricity market models depending on competition and time scope.

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, optimizationbased 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 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

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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 (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; 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 MPEC due to the fact that there is only one leader firm. In contrast, the Stackelbergbased 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.

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 probabilistic— when the uncertain nature of random variables is incorporated using probabilistic distributions—and deterministic—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 uncertainty 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 framework.

6.3. Uncertainty modeling

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.

One of the most common applications of electricity market models is in the field of forecasting the market

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 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 intertemporal 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. 6.5. Transmission constraints

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Uncertainty Intraperiod constraints

Interperiod constraints

Single Node Probabilistic Transshipment Model DC Model

Deterministic

Interperiod Links

AC Model Single Node Transshipment Model DC Model Probabilistic AC Model Deterministic Intraperiod Constraints

Interperiod Constraints

Transmission Network

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

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 Kirchhoff’s voltage law (Wei and Smeers, 1999) although their model allows for intertemporal 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 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 et al. (2001) consider not only the hydroenergy constraints implicit in the management of water reserves but also the hydraulic inflow uncertainty. 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 generation system due to their numerical tractability limitations. 6.7. Market modeling The last attribute considered in this taxonomy is related to the market model under consideration. Pure

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Uncertainty

Exogenous Price

Single-firm Residual Demand

909

Imperfect Market Equilibrium

Low Probabilistic Medium High

Deterministic

Market Modeling Low

Probabilistic Medium Deterministic Exogenous Price

High Single-firm Residual Demand

Imperfect Market Equilibrium

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

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, 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 2, 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 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 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. 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 SFEbased 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 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 13 CSF: Conjectured Supply Function approach and CV: Conjectural Variations approach.

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910 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 Baldick and 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)

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, equilibrium models and simulation models. These models 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.

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

Table 3 Major uses and main features of the reviewed market models Year Major use

Main feature

Perfect Competition and Exogenous price

Gross and Finlay Fleten et al. Pereira et al. Rajamaran et al. Unger

1996 1997 1999 2001 2002

Generation scheduling Hydro and risk management Hydro and risk management Unit commitment Hydro and risk management

Deterministic prices LP Stochastic prices and inflows LP Solution method Benders Price uncertainty DP Risk modeling LP

Leader-in-price and residual demand function Garc!ıa et al. Ba!ıllo et al. Anderson and Philpott Ba!ıllo et al.

1999 2001 2002 2002

Unit commitment Thermal modeling Short-term hydrothermal coordination Non-convex profit Offer curve construction Stochastic demand function Offer curve construction Practical approach

Supply function equilibrium

Green and Newberry Bolle Rudkevich et al.

1992 Market power analysis 1992 Market power analysis 1998 Market power analysis

Linear supply function equilibrium

Green Ferrero et al. Berry et al. Hobbs et al. Baldick et al. Baldick et al. Day and Bunn

1996 1997 1999 2000 2000 2001 2001

Cournot equilibrium

Scott and Read Bushnell Borenstein and Bushnell Batlle et al. Otero-Novas et al. Kelman et al. Rivier et al. Barqu!ın et al.

1996 1998 1999 2000 2000 2000 2001 2003

Stackelberg

Ventosa et al. Murphy and Smeers

Spatial Cournot

CV CSF Agent-based

Solution method Size

Intended market

Large Large Large Large Large

E&W Nord Pool

MIP MIP NLP MIP

Large Large Small Large

Spain Spain New Zealand Spain

Symmetric firms Symmetric firms Closed-form solution

NI NI Analytic

Small Small Small

E&W E&W Pennsylvania

Market design Congestion management Congestion management Congestion management Market power analysis Market design Market power analysis

Closed-form solution AC Network Model DC Network Model DC Network Model Piecewise linear SFE Non-decreasing SFE Asymmetric firms

Analytic Enumeration Heuristic MPEC Heuristic Heuristic Enumeration

Small Small Small Medium Small Medium Medium

E&W

Hydrothermal coordination Hydrothermal coordination Market power analysis Risk analysis Yearly economic planning Long-term hydrothermal coordination Hydrothermal coordination Hydrothermal coordination

Hydro-interperiod links Analytic modeling Radial congestion Stochastic prices and inflows Agents’ behavior Stochastic inflows Hydrothermal modeling Stochastic inflows

DP DP Heuristic Simulation Heuristic DP MCP NLP

Medium Medium Medium Large Large Large Large Large

2002 Capacity expansion planning 2002 Capacity expansion planning

Investment decisions Investment decisions

MPEC EPEC

Medium Medium

Hogan Oren Wei and Smeers Hobbs

1997 1997 1999 2001

Network constraints Network constraints Transshipment model DC power flow

NLP Analytic VI LCP

Small Small Large Large

Garc!ıa-Alcalde et al. Day et al. Bower and Bunn

2002 Price forecasting 2002 Congestion management 2000 Market design

Fitting procedure DC power flow Learning procedure

LCP LCP Heuristic

Large Spain Large E&W Medium E&W

Congestion Congestion Congestion Congestion

management management management management

Nord Pool

E&W E&W E&W New Zealand California California Spain Spain Brazil Spain

Europe

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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 CV approach to overcome the high sensitivity of the priceclearing 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 decisionmaking 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.

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