Utilities Policy 7 (1998) 233–242

Modeling competition in electric energy markets by equilibriumconstraints

Andres Ramos*, Mariano Ventosa, Michel RivierInstituto de Investigacio´n Tecnolo´gica, Universidad Pontificia Comillas, Alberto Aguilera 23, 28015 Madrid, Spain

Abstract

Throughout the world, the electricity industry is currently undergoing significant restructuring towards deregulation and compe-tition. Under this new framework, electric firms assume more risk, and are more responsible for their own decisions. Utilities needoriginal models that fulfil these new requirements. This paper presents a novel conceptual approach to modeling the newly deregu-lated power markets. It combines powerful traditional tools related to the detailed system operation with techniques for modelingeconomic market equilibria. The proposed approach models the competitive behavior of the electric firms by incorporating a setof constraints, namely the equilibrium constraints, into a traditional production cost model. These constraints reproduce the firstorder optimality conditions of the strategic companies. Thus the approach achieves a profit maximization objective while keepingthe system operation details. This model has been implemented in GAMS. An application to a sample case study is also presented. 1999 Elsevier Science Ltd. All rights reserved.

Keywords:Electricity market equilibrium; Mathematical programming with equilibrium constraints; Unit commitment

1. Introduction

Throughout the world, the electricity industry is cur-rently undergoing significant restructuring towardsderegulation and competition. In the new context, thegeneration of electricity becomes an unbundled activitysubject to a strong liberalization in which both expansionand operation decisions no longer depend upon adminis-trative and centralized procedures, but rather on themanagerial decisions of the generation companies. Theproduction path, and moreover, the economic remuner-ation of each individual generation unit increasinglydepend on the ability of its managers to ‘sell’ its product(electricity) on a wholesale electricity market. Eitherunder a more organized market scheme in which a mar-ket operator centralizes generation and demand bids,clears the market and leads the settlement process, orunder a more decentralized one where physical bilateralcontracts prevail, electric firms must assume much morerisk and responsibility for their own decisions. For gen-eration firms, it is now very important to be able to ana-lyze and to model the behavior of the market in orderto make decisions with the highest level of information.

* Corresponding author. E-mail: andres.ramos@iit.upco.es

0957-1787/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S0957-1787 (98)00016-2

This is true for existing firms, potential new investors orany entity interested in the electricity market.

Outcomes in the electrical system will no longerdepend upon a traditional total cost minimizationscheme, but rather on the interaction of individual, profitmaximizing firms. Each firm must look to maximize itsproduction surplus (market revenues minus operationcosts) in an uncertain context where its perception ofrisk, the behavior of the competitors, the ownershipstructures, the technology mix, as well as a multitudeof other external, technical, economical and managerialfactors heavily condition the market. The systembehavior will therefore be characterized by the economicmarket equilibrium as a result of the interaction of allthis factors. Market equilibrium defines a point of con-vergence of the market, provided that each participantbehaves looking forward to maximizing its own profits.

An important developmental effort is needed to adaptthe traditional models to the new context or to designnew models and tools devoted to the modeling of econ-omic market equilibria in the electricity generation sec-tor. Several areas of knowledge converge in modelingmarket equilibria: microeconomic theory (Varian, 1992)(Cournot and Bertrand models among others (Sherali etal., 1983; Kolstad & Mathiesen, 1991)), game theory(non-cooperative games (Lucas & Taylor, 1993)), mixed

234 A. Ramos et al. /Utilities Policy 7 (1998) 233–242

complementary problems (MCP) (Cottle et al., 1992;Ferris & Pang, 1995) or mathematical programming withequilibrium constraints (MPEC) (Luo et al., 1996).

Recently, several papers have addressed the compu-tation of the market equilibrium in the electric sector,see (Kahn, 1998b). These approaches are described inmore detail in Section 3. The theoretical and practicaladvances shown so far have been considerable. Howeverthese models contain some limitations in the represen-tation of the technical behavior and the production costsof each market player. First, they simplify the technicalcharacteristics of the different technologies: nuclear,thermal, hydro and pumping. Second, many of themassume a continuous production cost function for eachutility while variable costs are more closely representedby stepwise functions. Third, because of the huge com-putation burden even a quite simple representation of themodel has been required up to now, being for examplethe number of hydro reservoirs limited to one or two atmost even in a deterministic hydro inflows context.

This paper presents a novel conceptual approach tothis problem. It combines the powerful and well testedexisting tools for modeling the detailed operation of ther-mal and hydroelectric units, with techniques devoted tothe modeling of economic market equilibria. The goal isto incorporate additional constraints that reproduce themarket equilibrium into a traditional production costmodel. Individual units are modeled with their ownparticular technical constraints and their own productioncost functions. For the sake of simplicity, in this paperwe do not consider any stochastic behavior of the waterinflows, although it can be taken into account by usingthis method.

The paper is organized as follows. First, a brief reviewof the Spanish situation is used as an example to describehow a wholesale electricity market is organized and tojustify the need for new models which represent the mar-ket behavior. Section 3 reviews the different modelingapproaches proposed in the technical literature. Section4 presents a general overview of a production cost modeland shows how the way the equilibrium constraints areincorporated into it. Section 5 describes an applicationof the model to a sample case study and discusses thenumerical results. Finally, Section 6 presents the con-clusions extracted from the modeling approach.

2. The Spanish electricity market

The Spanish electricity industry is currently immersedin deep changes that have led to a completely new regu-latory framework as of January 1998. The Law Act(BOE, 1997) establishes, among other things, a day-ahead wholesale trading pool for selling and buyingenergy. A market operator (MO) determines the actualoperation of the generating units, based on a simple

hour-by-hour merit order of their bids. These are simplebids in which all fixed operating costs must beadequately internalized.1 The market clearing price is sethour-by-hour at the highest accepted bid. Several so-called intra-daily markets (five at the moment) allowboth the generation and the demand agents to react tounforeseen changes (unit total or partial outage, errorsin forecasted demand…) that arise after the daily markethas closed. A system operator (SO) is in charge of thetechnical security operation of the system. It controlsseveral ancillary markets (second and tertiary reserve)and several procedures devoted to meeting capacity andvoltage network constraints.

The former regulatory regime in Spain was anincurred-costs revenue scheme based on ‘standard’ costs.This regime provided some incentive to operateefficiently, since minimizing the actual costs meantgreater profits.2 A system operator centrally dispatchedthe units in order to achieve a global variable cost mini-mization. Several production cost models with differenttime horizons were hierarchically used for this purpose.Hydro scheduling for the short, medium and long term(with the economic impact and technical restrictionscoupling different time periods) as well as all the techni-cal constraints of thermal units were considered in thischain of models.

Under the new framework, electric firms assume muchmore risk and responsibility for their own decisions. Inparticular, they now have to estimate their own unit com-mitment in order to decide, based on costs, prices, andquantities, the bids that they will finally submit to themarket operator. These bids decide the actual operationof their units and their incomes. Utilities need modelsthat fulfil these new requirements. Such models mustconsider in detail the technical operation constraints stillprevailing in the system, and also model the new com-petitive framework. Most significantly, the new explicitobjective of the companies is to maximize their profits(revenues minus operation costs), rather than minimizecosts.

The Spanish electricity market presents some featuresthat make it especially difficult to model. A strong hori-zontal and vertical concentration remains in place. A 10year transition period has been adopted in order to lib-eralize the market fully. During this period, a compe-tition transition charge (CTC) will be collected in orderto pay the stranded costs of the utilities (all privateexcept one that has also been recently privatized). TheCTC remuneration mechanism interferes with the market

1 An ‘income constraint’, which sets a minimum daily income fora bid to be accepted, can voluntarily accompany each generationunit bid.

2 An excellent review of the Spanish electricity industry can befound in (Kahn, 1996).

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so that market prices and productions cannot beexplained without them.3

Although our work has been motivated by the needto adapt the proposed model to these specific Spanishconditions, this paper will focus on a general approach.This approach is applicable to a broad variety of whole-sale electricity markets.

3. Modeling approaches

Recently, several papers have addressed the compu-tation of the market equilibrium in the electric sector.Two main approaches have been explored so far. Oneis called ‘supply function equilibria’. The other isimplicitly or explicitly based on the Cournot equilibriumframework, where quantities are the decision variablesand the resulting price is set using a price-responsivedemand function. Borenstein et al. (1995) defined a gen-eral classification of the different markets and competi-tive equilibria in the electric industry.

Green & Newbery (1992) were the first to tackle theseissues. They use a simplified supply function equilibriumapproach. Rudkevich et al. (1998) recently extend thistechnique to the use of a stepwise supply function.

Borenstein & Bushnell (1997) later used a simulationmodel, which heuristically evaluates the California marketunder competition. The electric energy market is modeledusing the Cournot equilibrium framework, where the com-panies are considered strategic or competitive fringedepending on their institutional characteristics. Each mar-ket equilibrium is calculated using an iterative algorithmthat sets each strategic firm’s production at its optimallevel while holding constant the output of the other stra-tegic firms. This process is repeated until no strategic com-pany has the incentive to modify its production level giventhe production levels of the other strategic firms. Bushnell(1998) extends this simulation model to include interper-iod elements. He represents the equilibrium conditionsanalytically and his model achieves market equilibriumtaking into account hydro scheduling decisions, whichregard planning resources for multiple periods.

Hogan (1997) models the profit maximization prob-lem of each strategic firm as a nonlinear optimizationproblem that takes into account network constraints. Theprofit maximization of fringe companies is representedby incorporating their first order optimality conditionsinto the optimization problem of the strategic companies.

Scott & Read (1996) have developed a medium-termmodel with emphasis on hydro operation, focused on theNew Zealand conditions. The model utilizes stochasticdual dynamic programming, where at each stage the

3 A detailed review of the factors that condition the Spanish elec-tricity market can be found in Kahn (1998a).

hydro optimization problem is superimposed upon aCournot market equilibrium. The functions reflecting theproduction costs over the rest of the planning horizon,and the derivatives of these functions with respect to thereservoir levels, i.e. the marginal water value, are calcu-lated through backward induction. The main contributionof the paper is the introduction of equilibrium conceptsinto the hydrothermal coordination methodology.

The approach proposed here avoids any iterative pro-cedure to reach the market equilibrium. It models the com-petitive behavior of the electric generation energy marketby incorporating a set of constraints, namely the equilib-rium constraints, into a traditional production cost model.These constraints reproduce the first-order optimality con-ditions of the strategic companies. Thus, our approach rep-resents the objective of profit maximization, subject to oli-gopoly competition, while also keeping a high level ofoperational detail and without resorting to any kind of iter-ative procedure. This paper describes the evolution of adetailed traditional production cost model into a model thatdetermines both the short- and medium-term system oper-ation and the market equilibrium. All the system agents arerepresented in this model: the market operator, the electricgeneration firms, and the demand bidders.

4. Model description

4.1. Preamble

Traditional production cost models have two charac-teristics that remain relevant in the new regulatoryframework and lend themselves to a Cournot analysis:

I They provide a detailed representation of the electricsystem operation.

I The decision variables of production cost models areunit output levels (i.e., quantities).

The first characteristic allows us to model the techni-cal and economic constraints that affect the system oper-ation. Such constraints include ramp rates, minimum upand down times in the short-term scope or stochasticinflows, hydro reservoir limits, hydro, maintenance andfuel scheduling in the medium term, and new entrantsin the long term. By incorporating oligopoly equilibriumconditions into an existing production cost model, weare able to represent the system with the same level ofdetail and to preserve the natural time division in per-iods, subperiods and load levels.4 We can then observetheir influence on the market equilibrium.

4 Although this time division is flexible, for the short term a periodis typically a day and a load level is an hour. For the medium term aperiod can be a week and a load level can last several similar hours.For the long term a period corresponds to a month, a subperiod to allthe working days or weekends and a load level to peak, shoulder andoff-peak hours, for example. These time periods correspond to real-world time intervals where operating decisions can be taken or statevariables are registered and consequently are represented in the model.

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The second characteristic fits nicely with one of thestandard oligopoly equilibrium concepts, namely theCournot framework. Indeed, since the Cournot frame-work is based upon quantity rather than price compe-tition, we can therefore retain generation unit outputs asthe main decision variables of the model. Representingthe market equilibrium via a Cournot model has specificparticularities that must be taken into account whileusing the model. Probably one of the most relevant isits high sensitivity to the elasticity of the demand, whichis always a difficult parameter to estimate. This willimply a careful analysis of sensitivity with respect to thisparameter to correctly interpret the results. It should benoted that any oligopoly equilibrium framework is atbest a stylized representation of the actual competitiveinteraction between firms and should therefore be treatedwith caution.

4.2. Production cost models

The production cost model upon which we base ouranalysis (Ramos et al., 1995) performs hydro, mainte-nance and fuel scheduling, seasonal operation ofpumped-hydro units, weekly/daily operation of pumped-storage units, and thermal unit commitment for a gener-ation system. Theobjective functionto be minimized is

Table 1Scheme of the production cost model with equilibrium constraints

the total variable costs for the scope of the model andis subject to operating constraints. These can be classi-fied into inter- and intraperiod constraints, according tothe periods that are involved.Interperiod constraintsareassociated with the coordination of limited productionresources (minimum quotas of fuel consumption, hydroinflows, and seasonal pumping, storage and generation).Intraperiod constraintsdeal with the system operationin each period (balance between generation and demand,thermal unit commitment, weekly/daily pumping andstorage and all the generation limits).

One of the main characteristics of this model is itsflexibility, allowing different types of use. For example,it can be utilized as a short-term unit commitment modelor as a medium-term production cost model for annualeconomic planning, where the representation of the elec-tric system changes dramatically. Several options havebeen provided to customize the use of the model to dif-ferent needs.

I The hydro units of the system can be aggregated intoa single reservoir, or all the plants can be modeledindividually. Water resources can be represented interms of units of electrical energy or in terms ofhydrological units that include detailed modeling ofhydro cascades.

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Fig. 1. Direct utility function.

I Thermal generation costs can be modeled using differ-ent heat consumption curves (linear, piecewise linear)and multiple fuel types.

I Seasonal pumped hydro and/or weekly/daily pumpedstorage can be specified for any period, subperiod orload level.

I Production and consumption can be assumed to occurat a single transmission network node. Alternatively,a complete transmission network can be modeledusing the DC load flow approximation, see Rivier &Perez Arriaga (1993), that can include transmissionlosses.

Schematically, this classical production cost model isoutlined in Table 1, considering only the white areas.The introduction of market equilibrium constraints,which are to be discussed in the following section,implies only some minor modifications to the previousoptimization problem. The shaded areas correspond tothe new equilibrium constraints. The resulting optimiz-ation problem has been implemented in GAMS (Brookeet al., 1992) as a mixed integer programming problem,hard to solve for a large-scale electric energy system.

An important change in a traditional production costmodel is the introduction of elasticity of the demand, i.e.the response of the demand to the energy price. In classicproduction cost models the demand was inelastic andhad to be met (subject to a penalty for unserved load).Now, the equilibrium quantity is obtained by maximiz-ing the total surplus, defined as the sum of consumer’sand producer’s surplus. In fact, maximizing the total sur-plus is exactly equal to minimizing the area below thesupply curve on the left of the equilibrium quantity(accepted generation) and below the demand curve onthe right of this quantity (i.e. non-served demand), seeFig. 1.

4.3. Equilibrium constraints

The equilibrium constraints model the behavior of themarket agents. Their objective is to maximize their pro-fits. The producers’ surplus for a given load level is cal-

culated as the difference between revenues and costs.Revenues are calculated as the short run marginal price(SMP) times the energy produced by the firm.

profits 5 revenues2 costs5 SMP·Pi 2 Ci(Pi) (1)

where Pi is the generation of the firmi, C(Pi) is thefirm’s total variable cost as a function ofPi.

The equilibrium constraints represent the first-orderoptimality conditions of Eq. (1) with respect the firm’soutput, that corresponds directly to the Cournot marketequilibrium. For each firm in each load level the deriva-tive of the profit with respect to the power generated bythe firm is equal to zero.

SMP1 Pi

∂SMP∂P

2 MC(Pi) 5 0 (2)

whereMC(Pi) is the firm’s marginal cost as a functionof Pi, ∂SMP/∂P is the change in theSMPdue to a changein the output of the firm, corresponding to the slope ofthe demand curve, which is negative.

The first two terms of Eq. (2) form the marginal rev-enue of the firm and the last term corresponds to themarginal cost. So Eq. (2) is equivalent to

marginal revenue5 marginal cost (3)

Eq. (2) can be alternatively expressed as the gener-ation level that, for each firm, maximizes its profit as afunction of theSMP, its marginal cost and the slope ofthe demand curve

Pi 5SMP2 MC(Pi)

2 ∂SMP/∂P(4)

Fig. 2 shows the theoretical market equilibria reachedunder both the cost minimization (CM) and the profitmaximization (EC) frameworks. The upper side presentsthe profit as a function of the firm’s output, explicitlyshowing that the output in the oligopoly case is reducedto increase the profits. The lower side presents the differ-ent market equilibria obtained at the intersection pointof the marginal cost staircase curve and the demandfunction.

The previous market equilibrium model implicitlyassumes that there are no operating constraints. Thefirm’s profit maximization problem, subject to theoperating constraints, can be solved by constructing theLagrangian and then formulating the Karush–Kuhn–Tucker (KKT) first order optimality conditions. In theKKT equation corresponding to the first order derivativeof the Lagrangian with respect to the firm’s output wecan neglect the terms—all positive—associated with thederivative of the operating constraints and the con-straint becomes:

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Fig. 2. Market equilibrium.

Pi #SMP2 MC(Pi)

2 ∂SMP/∂P(5)

The inequality sign from Eq. (5) can be understoodintuitively. The objective function of cost minimization(or, equivalently, perfect competition) leads each firm’soutput to levels greater than those of the profit maximiz-ation problem which appears in an oligopoly market. Theinequality in Eq. (5) acts therefore as a constraint on theoutput levels of the firms, keeping them below perfectcompetitive levels.

The other first-order optimality conditions of the Lag-rangian correspond to satisfying the same operating con-straints already imposed on the production cost model.

An increasing stepwise function represents each firm’smarginal cost as a function of its own generation. Thesteps represent the variable (fuel, consumables, andoperation and maintenance) costs of the different com-mitted generating units. The marginal cost of each firmis greater than or equal to the variable cost of any firm’scommitted unit and can be expressed as a function ofthe commitment binary variables:

MCi $ vgag ∀g P i (6)

whereMCi is the marginal cost of firmi, vg is the vari-

able cost of unitg and ag is the commitment state ofunit g (ag 5 {0,1}).

The market-clearing price (SMP) is represented in twodifferent ways. As a constraint, it is modeled by a linearfunction of the electricity demand. In the objective func-tion (i.e., the term of non-served demand costs), SMP istransformed into a decreasing stepwise function (withthe same slope of the linear function) where each stepis a fictitious demand bid. This second modelingapproach is used to avoid nonlinearities in the objec-tive function.

SMP5 SMP0 1∂SMP

∂P Oi

Pi (7)

The previous equilibrium constraints Eqs. (5)–(7)incorporate the market behavior into the classical vari-able costs minimization problem while keeping all thesystem operation details.

4.4. Overview

It is interesting to analyze the complementary featuresof the cost minimization side, still explicitly representedin the model, and the firm’s profit maximization objec-

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tive, which are incorporated implicitly through the Cour-not equilibrium constraints. While the latter determines,for each strategic firm considered, an output level thatmaximizes its profits, it is the cost minimization whichdecides the specific unit commitment that achieves thatoutput level. It will do so by looking for the cheapestcommitment of their thermal units and the cheapesthydro scheduling, exactly as each firm would have doneif its output requirements had been set exogenously tothe model. With this approach, where market behaviorand operating constraints are simultaneously considered,the equilibrium solution accounts for all the technicaloperating constraints modeled, thereby achieving arealistic system dispatch.

There are still some issues that merit further investi-gation. The sensitivity of the results to parameters suchas the slope and elasticity of the demand function shouldbe examined. Moreover, further refinement of the sol-ution will be needed also to consider stochasticity in thecompetitor behavior.

5. Case study

In this section, we present a small case study, whichwas developed to test the results of the model. The timescope represented is a day. The main goal of thisexample is to show how the model solves for the marketequilibrium while taking into account some unit commit-ment constraints. These constraints include:

I Maximum committed capacity at peak hoursI Minimum load capacity at off-peak hoursI Thermal ramp rates at ramping hours

5.1. Suppliers

The generation system is composed of 18 generatingunits belonging to three companies. This case study con-siders firms of different size, structure and technologymix. Each firm has several thermal units, but only onefirm has a limited energy hydro plant. Firm A has seventhermal units with 7000 MW of installed generationcapacity. The production costs go from 15 to 32.5US$/MWh. Firm B owns six thermal units with acapacity of 5500 MW and another hydro unit of 3000MW. The thermal production cost ranges between 17.5

Table 2Summary of generation capacities by firm

Firms No. of units Thermal capacity (MW) Hydro capacity (MW)

A 7 7000 –B 6 5500 3000C 5 3500 –

Fig. 3. Variable production costs of each firm.

and 35 US$/MWh. Finally, firm C manages five thermalunits of 3500 MW and a cost range between 25 and35 US$/MWh.

Fig. 3 shows the aggregated variable cost curves ofeach firm. Table 2 shows the summary of thermal andhydro capacity of each firm while Table 3 shows thecharacteristics of the generating units: variable cost,maximum and minimum capacity, ramp rate and energylimit for the only hydro plant available.

5.2. Demand

The hour-by-hour demand profile corresponds to aweekday with a maximum of 19 000 MW and a mini-mum of 8000 MW. Each demand level is represented bya linear function of the price. The slope of the demandat every hour is 5 US$/MWh/GW which provides priceelasticities of around2 0.6. Fig. 4 shows the dailydemand and price profiles obtained by the model in aperfect competition framework.

Fig. 4. Hourly demand profile.

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Table 3Summary of the characteristics of the generation units

Unit Variable cost Maximum capacity Minimum capacity Ramp rate Energy limit(US$/MWh) (MW) (MW) (MW/h) (GWh)

Firm A 1 15 1000 833 50 –2 15 1000 833 50 –3 15 1000 833 50 –4 20 750 250 250 –5 21.25 500 250 250 –6 22.5 750 250 375 –7 32.5 2000 500 1250 –

Firm B 8 0 3000 0 1500 299 17.5 1500 1000 150 –

10 25 500 350 250 –11 28.5 500 350 250 –12 32.5 500 150 150 –13 35 2500 500 1250 –

Firm C 14 25 1500 1000 150 –15 27.5 500 250 250 –16 30 500 150 150 –17 32.5 500 150 150 –18 35 500 100 250 –

5.3. Analysis of market results

The model has been run with and without equilibriumconstraints. When run without equilibrium constraints,the model represents both the former regulatory frame-

Fig. 5. SMP and total output with and without equilibrium constraints (EC and CM, respectively).

Fig. 6. Output of each firm without and with equilibrium constraints.

work based on cost minimization and a perfectly com-petitive electricity market. When run with equilibriumconstraints, it incorporates the firms’ strategic profitmaximization objective. As it can be observed from Fig.5, in the first case the price is 35 US$/MWh for peak

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hours and around 20 US$/MWh for off-peak hours.However, in the second case, the generation is restrictedto increase the price. Producers’ surplus therefore alsoincreases. In the second case, the generation system onlyprovides about 75% of the energy of the first case. Theresulting price reaches a level of 80 US$/MWh for peakhours and 50 US$/MWh for off-peak hours.

The results show how firms A and B increase theirprofits in the second case by restricting their generation(Fig. 6). Their large size allows them to compensate theoutput decrement with an increment in revenuesobtained by their remaining generation. However, com-pany C (of smaller size) increases its output in thesecond case, because the optimum for this firm is to pro-

Fig. 7. Output of firm A without and with equilibrium constraints.

Fig. 8. Output of firm B without and with equilibrium constraints.

Fig. 9. Output of firm C without and with equilibrium constraints.

duce as much as possible. Because of its smaller size, apotential increase in price does not compensate for therevenues lost from the remaining restricted generation.Its profits are maximized from behaving like a price-taking company.

5.4. Analysis of production results.

The major contribution of this modeling approach isthe introduction of the operating constraints in the com-putation of the market equilibrium. Figs. 7–9 show theproduction results for each firm with equilibrium con-straints.

In the first case (just with cost minimization), com-

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pany B uses its hydro unit to avoid the shutdown andstartup of all the thermal units belonging to any firmequalizing the system marginal cost to 35 US$/MWh inall peak hours, see Fig. 5. This marginal cost is definedby the last thermal unit of firm B. Companies A and Chave marginal costs of 32.5 US$/MWh.

In the second case (cost minimization with equilib-rium constraints), companies A and C, without hydrounits, must turn off some thermal units during off-peakhours to satisfy the minimum load constraint. Theseunits are started up again to meet the demand duringpeak hours. Company B uses hydro generation to followthe demand profile without resorting to the shutdown ofsome units. This result reflects the intuition described inBushnell (1998) that optimal hydro energy use equalizesmarginal costs over the model time scope for the ownfirm. This company is not forced to incur in shutdownor startup of its units.

Finally, in the figures it can be observed how thermalunits with lower ramp rates limit their output during off-peak hours due to these ramp rates.

6. Conclusions

We have presented a novel and practical approachbased on mathematical programming with equilibriumconstraints for obtaining the electric power market equi-librium. The major contribution of this model is itsability to represent both the technical constraints and theproduction cost function of the electric power systemwith a high level of detail in a computationally tractablemarket equilibrium model.

This paper addresses the evolution of a detailed tra-ditional production cost model to determine simul-taneously both the system operation and the market equi-librium. The constraints determining the marketequilibrium among firms have been introduced into adetailed production cost model. As a consequence, theproducer’s surplus of each firm is maximized and all theoperation constraints are taken into account.

This model is an extremely useful tool for economicoperations planning in deregulated power markets. It canbe used to study system marginal prices and the differentmarket equilibria achieved under various demandcharacterizations and assumptions about the firm’sbehavior. The results can also be helpful in defining bid-ding strategies or tactics of individual firms, dependingon the behavior of their competitors.

Acknowledgements

This paper has benefited greatly from discussions anduseful comments by James Bushnell and Edward Kahn.All errors are the sole responsibility of the authors.

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