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IEEE TRANSACTIONS ON POWER SYSTEMS 1

Nash-Cournot Equilibria in HydrothermalElectricity Markets

Juan Pablo Molina, Juan Manuel Zolezzi, Senior Member, IEEE, Javier Contreras, Senior Member, IEEE,Hugh Rudnick, Fellow, IEEE, and María José Reveco

Abstract—A method is presented that allows finding the marketclearance prices within a hydrothermal power exchange marketthat incorporates network constraints. An analysis is made ofthe degree of market power exerted by the various agents. TheNash-Cournot equilibrium solution of the market is achievedusing the Nikaido-Isoda function, which is derived from the profitmaximization functions calculated by the generating companies.Both thermal, hydro generation, and network constraints areconsidered in the model, with coupling constraints of the hydrounits also taken into account. The model is applied to the mainChilean interconnected power system, whose abundant hydroproduction and existing network bottlenecks make it a valuablesuitable case for study. The results for pool-based and bilateralcontract markets highlight the strength of the method and showthe contribution of the transmission system and hydro constraints.

Index Terms—Multimodal hydrothermal system, Nash-Cournotequilibrium, Nikaido-Isoda function, transmission system.

I. INTRODUCTION

I N recent years, use has been made of different types ofeconomic models, aimed at analyzing oligopolistic be-

havior subject to network constraints and, consequently, pricedynamics in pool-based electricity markets [1]–[4]. In 1982,Chile was a pioneer, introducing electricity reforms to increasecompetition and efficiency, especially in the hydro generationsector, which evolved from state-based to be vested in privatehands. The Chilean energy market operator is the power ex-change (PX), and the independent system operator (ISO) is incharge of the physical operations of the grid. The PX receiveshourly supply bids from the generators. With them, it buildsa merit order list, sorting the prices from the least to the mostexpensive ones until the generation matches up the demandcurve, obtaining the hourly market clearing prices as a result.

The Sistema Interconectado Central (SIC) is the most im-portant of the four Chilean electricity systems, reaching 75%

Manuscript received October 28, 2009; revised October 31, 2009, March 11,2010, and July 21, 2010; accepted September 08, 2010. This work was sup-ported by Fondecyt Project 1030067 and the Spanish Ministry of Education andScience of Spain grant ENE2006-02664. Paper no. TPWRS-00844-2009.

J. P. Molina is with the Electrical Department of Escondida copper mine,Antofagasta, Chile (e-mail: [email protected]).

J. M. Zolezzi is with the Department of Electrical Engineering, Universidadde Santiago de Chile, Santiago, Chile. (e-mail: [email protected]).

J. Contreras is with E.T.S. de Ingenieros Industriales, Universidad deCastilla-La Mancha, 13071 Ciudad Real, Spain (e-mail: [email protected]).

H. Rudnick is with Department of Electrical Engineering, Pontificia Univer-sidad Católica de Chile, Santiago, Chile (e-mail: [email protected]).

M. J. Reveco is with the Ministry of Energy of Chile, Santiago, Chile (e-mail:[email protected]).

Digital Object Identifier 10.1109/TPWRS.2010.2077313

of the demand in the country. Hydro units in the SIC accountfor 60% of its installed capacity, introducing a new factor, thestrategic use of water resources by hydro companies. Endesa,with an important market share in the SIC, has a monopolyover the hydro resources. This situation affects market compe-tition, since hydro units have much lower operating costs thattheir thermal counterparts, and their water constraints encom-pass much longer periods.

Two markets are comparable to the Chilean market, Colombiaand the South West of Brazil. Both of them show significantmarket power and the strategic use of water resources stored inthe companies’ reservoirs.

This work analyzes the tools provided by game theoryfor evaluating the behavior of the agents in deregulatedhydro-thermal markets in order to find the market balance,due to the interrelation that exists among the various agentswith each other and with the system constraints. In particular,the effect of the network constraints in a system with Cournotagents has been proposed by Jing-Yuan and Smeers [5] usinga variational inequalities model, and also by Hobbs [6], whodescribes in detail a mixed linear complementarity modelframework.

Although equilibrium models have been consistently usedin thermal-based market studies, there are relatively few ref-erences on hydrothermal systems. In this regard, Scott andRead [7] propose a dual dynamic programming model to findthe optimal hydro schedule of a unit that competes with otherCournot thermal producers. Kelman et al. [8] introduce amulti-year hydro scheduling model using stochastic dynamicprogramming. Ventosa et al. [9] define a hydrothermal coor-dination methodology by means of mixed complementaritymodel. Villar et al. [10] construct a hydrothermal simulator tofind possible equilibria considering the effects of market power.Market power issues in hydrothermal bid-based systems arealso dealt with in [11]. A Brazilian Nash-Cournot equilibriummodel where all types of units, including the hydro units, areconsidered is analyzed in [12], defining the optimal trans-mission capacity and its influence on market power. Finally,Bushnell [13] models a Cournot equilibrium framed as a mixedlinear complementarity model, showing that hydro units exertmarket power by reducing their production (increasing prices)in peak hours.

To the authors’ knowledge, a previous study [14] has solvedsimilar problems using dynamic programming. In addition, andwith a different scope, the problem of hydrothermal planningwith middle-term uncertainty has been addressed in [15].

To account for the presence of hydro and thermal units, theproposed approach extends the thermal-only model used in [1],

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2 IEEE TRANSACTIONS ON POWER SYSTEMS

which uses a Nikaido-Isoda function and a relaxation algorithm.This work adds another iterative algorithm within the Nikaido-Isoda function to allocate the water of the hydro units. The unitwith the lowest Lagrange multiplier corresponding to its hydroproduction limit increases its production until all the water isallocated for all units.

The mathematical software NIRA [16] is utilized, whose se-quential improvement of the Nikaido-Isoda function is obtainedthrough a relaxation algorithm that is shown to converge toa Nash-Cournot equilibrium for a wide class of problems, in-cluding nondifferential payoffs and coupled constraint games[17], [18]. A general reference for the Nikaido-Isoda method-ology is provided in [19].

The contributions of this paper are: 1) to apply theNikaido-Isoda methodology to find a hydrothermal marketequilibrium, 2) to introduce a numerical method based onLagrange multipliers to allocate water production among unitswithin the hydrothermal equilibrium, 3) to analyze the degreeof market power exerted by the units and its relationship withthe network constraints, and 4) to model another market model,the bilateral contract market, in addition to the pool-based one.To achieve the above, a Nash-Cournot model is used, where theequilibrium is reached iteratively.

The paper is structured as follows: Section II presents thegeneral Nikaido-Isoda methodology. Section III presents theNikaido-Isoda equilibrium model applied to a hydrothermalsystem subject to network constraints. Section IV depicts aflowchart of the equilibrium model. Section V shows severalillustrative examples of the proposed methodology to findequilibria in the main Chilean interconnected power system.Finally, Section VI presents the conclusions derived from thiswork.

II. NIKAIDO-ISODA METHODOLOGY

This work considers an electricity market with a hy-drothermal generating pool in which the generators use theamount to be generated as strategic variable. To carry out theoptimization, an iterative model is followed, where each playerknows its opponents’ offer in the previous iteration and, fromthat, it determines its optimum generation level. Since thetransmission system is considered, it is not possible to carry outa sequential game in which each player optimizes its objectivefunction separately, because when congestion problems appearin some of the lines, the player or players that face an initiallyuncongested system will have an advantage.

A. General Model of the Nikaido-Isoda Equilibrium

Formally, a game with a set of playersis defined as , where is the set ofstrategies of player so that the collective action set is

and is the payoff function of player . If itis assumed that each player makes individual decisions, , thevector belonging to the collective action setis defined as the joint action vector formed by the strategies ofeach player . Also, isdefined as the vector of strategies that player can take, with

the strategies of the other players remaining constant,.

One can then define point as Nash-Cournotequilibrium if for every , it is true that

(1)

The Nikaido-Isoda function is introduced to transform thesearch problem into one of simultaneous decision. The Nikaido-Isoda function is defined as

(2)

The term in brackets represents the change in the benefit ofplayer , since it decides to change its generation strategy to

, considering that the others will remain constant. When thisfunction becomes equal to zero, there will not be any strategythat will increase a player’s payoffs when it changes its strategyfrom to .

At this point, the optimum response function is introduced.It is the result of maximizing the Nikaido-Isoda function (2),where all players try to improve their payoffs. The optimumresponse function at point is

(3)

This function returns the set of players’ actions whereby theyall try to unilaterally maximize their respective payoffs, basedon the separability of the Nikaido-Isoda function with respectto actions . So, by “playing” actions rather than , theplayers approach the equilibrium. Note that, by doing that, aplayer maximizes its profit assuming that the competitors arefixed in their actions, which is the definition of Nash-Cournotequilibrium.

Finally, a relaxation algorithm that uses the Nikaido-Isodafunction to compute a Nash-Cournot equilibrium is presented.At each iteration of the algorithm, the players wish to move toa point that represents an improvement on the current player’spoint. The relaxation consists on adding the previous value ofto the optimal response function , both terms weighted bya coefficient, to find the iterative equilibrium.

B. Relaxation Algorithm

In order to find the Nash-Cournot equilibrium of a gamehaving an initial estimate , the relaxation algorithm of theoptimum response function, when is single-valued (everyinput is associated with one output only) and the concavity con-ditions (the Nikaido-Isoda function is weakly convex-concave;see [19, definition 2.9]) are satisfied, is

(4)

where . An iterative stepis constructed as a weighted average (also a convex combi-

nation) of the improvement point and the current point. The optimum response function is calculated after


MOLINA et al.: NASH-COURNOT EQUILIBRIA IN HYDROTHERMAL ELECTRICITY MARKETS 3

solving the optimization problem in (3). The averaging shownin (4) ensures convergence of the algorithm under certain condi-tions [18], [19]. At each stage, the optimum response of a playeris chosen, assuming that the rest will play as they did in the pre-vious period. Thus, by taking a sufficient number of iterations,the algorithm converges to the Nash-Cournot equilibrium .The convergence conditions are available in the relaxation algo-rithm convergence theorem in Appendix A [1]. The problem canbe considered either a centralized optimization model or a cal-culation of the succession of actions by the players at each stage,where players choose their optimum response given the actionsof the opponents in the previous period. Note that generationcost data must be known to perform the studies, as it is the caseof Chile and other centrally dispatched markets, but not in gen-eral. Since this is an equilibrium model, one must rely on goodapproximations from historical data, bids, etc. in market-basedsystems.

III. NIKAIDO-ISODA EQUILIBRIUM MODEL

OF A HYDROTHERMAL SYSTEM SUBJECT

TO NETWORK CONSTRAINTS

To determine each company’s benefit function, , as used in(1) and (2), considers company , which has power plantsand sells power to node NV, with the following price-demandfunction at node NV:

(5)

where:

price at node NV;

price intercept;

demand price inverse elasticity;

demand at node NV.

Equation (5) depicts a linear relationship between price anddemand at every node. This assumption can adequately repre-sent many actual situations in markets by changing the values ofthe inverse elasticity, as shown in [6] and [13]. If one considersthat the demand at node NV, , is equal to the generationprovided by company , plus what the other companiesgenerate, , i.e.,

(6)

then, the income function of company as a function of its owngeneration will be given by

(7)

Consider that in iteration of the relaxation algorithm (seeSection II-B), company assumes that its competitors willretain the generation of the previous iteration. Note that it isassumed that only the competitor’s overall supply functionis known by company , as it is the case in many markets toprotect the privacy of individual offers and bids. Then, the

following equation results:

(8)

The cost function of company will be given by the sum ofthe costs of the power plants that make up the company, i.e.,

(9)

where

power supplied by power plant of company ;

cost of power plant of company ;

quadratic coefficient of the cost of power plantbelonging to company ;linear coefficient of the cost of power plantbelonging to company ;fixed cost of power plant belonging to company ;

set of thermal plants connected to node NV.

All the individual cost functions must be known by the algo-rithm to find plausible equilibria. If cost information is missing,the algorithm proposed must rely on good approximations.

It must be noted that

(10)

From (8)–(10), one can solve the benefit function of companyselling at node NV for iteration as follows:

(11)

Equation (11) represents the benefit function of companyif one considers a system where there is a single selling node.Now, if one considers NV0 selling nodes, the benefit functionof company becomes

(12)



Therefore, if one considers companies, the sum of all ob-jective functions for all companies at iteration is

(13)

Note that the proposed model considers thermal and hydropower plants (as it is the case of the interconnected Chileanpower system), where the latter ones can move water blocksfrom one time period to another. Because of the above, andsince the players try to maximize their benefit, the objectivefunction is altered and now has a temporal characteristic.

With the above, and assuming that there are time periods ofstudy of duration each, company (which has thermal andhydro power plants) has the following benefit as a function oftime, starting from the generalization of (12):

(14)

subject to

(15)

(16)

(17)

(18)

where

set of hydro power plants belonging to company;

set of thermal power plants belonging to company;

power sold to node NV in period and iterationby the hydro power plant of company ;power sold to node NV in period and iterationby the thermal power plant of company ;maximum energy of the hydro power plant thatcan be generated in a lapse of time comprisingtime periods;minimum power of the hydro power plant ofcompany ;maximum power of the hydro power plant ofcompany ;minimum power of the thermal power plantof company .maximum power of the thermal power plantof company .

For the sake of completeness, one needs to include the net-work constraints, according to a linear power flow model. The

basic linearized equation that governs the behavior of powerwith respect to the angles of the nodes is

(19)

(20)

(21)

where matrix represents the injection of power at the nodesand vector symbolizes the angles of each node referred to thesystem’s slack node. It is also considered that the power flowalong any given line that joins nodes is given by

(22)

and in node NV

(23)

where is the demand at node NV and is the generationat node NV.

Since all the companies make their decisions as a function ofwhat company decides in the above iteration, they indirectlytake on a temporal characteristic, getting to the following objec-tive function that takes into account all companies:

(24)

Note that from (24), the Nikaido-Isoda function in (2), theoptimum response function in (3) and the relaxation algorithmin (4) can be obtained. Since (24) represents the sum of indi-vidual benefits of all the plants, the Nikaido-Isoda function (2)is formed by terms, one per plant, where, for each term, theproduction of the competitors is fixed, in (2), and each agenthas a production decision variable, in (2). Also note that ex-pression (3) is just the maximization of the Nikaido-Isoda func-tion (2). Finally, the relaxation algorithm in (4) means that, ateach iteration, the production of each plant is obtained as theweighted sum of two terms: the production value obtained fromthe previous iteration and the optimized production value ob-tained from the maximization in (3). See the Appendix for anexample to show how the Nikaido-Isoda function is obtained.

A. Hydro Units: Active Set Method1

Although the objective function (24) allows to determinethe Nash-Cournot equilibrium of an electric system with hy-drothermal characteristics, this function does not seem verypractical when one considers a real system that has a large

1A detailed presentation of the active set method is provided in the Appendix.



Fig. 1. Flowchart of the search algorithm for finding Nash-Cournot equilibriain a hydrothermal system.

TABLE IELECTRIC PARAMETERS OF THE SIC TRANSMISSION LINES

number of power plants, sales nodes, and/or requires the rep-resentation of a large number of periods, since in any of theabove cases, a large number of variables must be optimized.

In the present work, it was decided to carry out a discreteoptimization of the water use, which allows the uncoupling theoptimization problem2 solving (24) iteratively.

In (16), the term represents the power gen-erated and sold to all the nodes concerned by a generic hydropower plant of company in period and iteration . Now, ifone assumes that this value is set for every period to ,the following constraint is obtained:

(25)

Considering this constraint as active, one gets a Lagrangemultiplier, , associated with constraint (25), a value that rep-resents the penalty for increasing the value of by oneunit.

If the optimization problem (24) is solved for every period ,considering constraint (25), this work gets a multiplier vector

. Then, comparing these multipliers, one canidentify the period that gives the largest benefit by increasingthe water generation of the corresponding hydro power plant(for that same period). This Lagrange method converges if theproblem is convex, which is the case, since the objective func-tion is quadratic and the constraints are linear.

IV. FLOWCHART OF THE NIKAIDO-ISODA EQUILIBRIUM

ALGORITHM FOR HYDROTHERMAL SYSTEMS

WITH NETWORK CONSTRAINTS

The general scheme of the algorithm presented in Fig. 1 canbe subdivided into several steps. Previously, the data of thethermal and hydraulic power plants are established, with theirproduction limits as well as the characteristics of the lines andthe limits of their capacity.

The first step establishes that the generation of the hydraulicpower plants is set at the beginning at their minimum value forevery study period.

The second step begins the process of searching for thethermo-hydraulic equilibrium in period 1, setting the hydraulicproduction of all power plants, and using the relaxation of theNikaido-Isoda algorithm (4) for a single period. The Lagrangemultiplier associated with (25) is calculated for the hydro unitsin that period. The same is done successively for all the studyperiods.

In the third step, as long as there is residual capacity in thehydro units (no more MW can be added when the residualcapacity has been exhausted, i.e., when the maximum energyvalue of the plants has been reached), all the Lagrange multi-pliers (one per period) of the constraint imposed on the hydropower plants are compared. Thus, the period that presents thelowest multiplier value is the one that has the greatest earningsand its generation is increased by MW. This results from thedefinition of Lagrange multiplier that represents the increase in

2An alternative way of approaching the optimum available water use strategyis dynamic programming, but it is not very efficient and it would delay the cal-culations in realistic cases [14].



TABLE IISYSTEM DATA OF THE EIGHT-NODE SIC

profits if the right-hand side of (25) is increased. Therefore, thisperiod is allowed to generate an extra MW. Then the optimiza-tion is performed again using the Nikaido-Isoda relaxation onlyfor this period, and the multiplier for this period is recalculated.Again the Lagrange coefficients are compared and anotherMW is assigned to the hydro unit that has the lowest multipliervalue.

Finally, the procedure is ended as the residual capacity of allthe power plants is exhausted.

V. CASE STUDIES: INTERCONNECTED CHILEAN SYSTEM

The methodology presented above is used to determine theNash-Cournot equilibrium of the main Chilean system, theSistema Interconectado Central (SIC), assuming a pool-typemarket.

The following assumptions are made.1) The strategic companies include Colbún, Gener, and En-

desa, which have greater capacity to exert market powerdue to their large amount of installed power.

2) To construct the demand functions, price intercepts, ,and inverse demand price elasticities, , are given inTable III. We assume that the price elasticity of demand isequal to 0.8 and each bus in the system has allocated a cer-tain percent of the demand, as shown in Table III. Choosinga value of 0.8 as demand elasticity is consistent with otherstudies of the Chilean system, whose values range from 0.1to 1 focusing on the interval [0.4 0.7] [20], and around 0.4in [21].

3) With the purpose of reducing the number of variables inthe simulation, the hydro and run-of-river power plants ofthe strategic companies are grouped in equivalent powerplants depending on their geographic location and owner.

4) This paper considers only the Endesa generating com-plex located at the Charrúa node (according to a reducedmodel), with the capacity to displace power blocks, i.e.,acting as hydro power plant.

To model the transmission network of the SIC an eight-nodereduced model is used, with the data shown in Tables I and II.A graphical depiction of the SIC is shown in Fig. 2. A valuefor MW is assumed. The Nikaido-Isoda equilibriummodel is applied to a pool-based hydrothermal market and to abilateral contract framework.

A. Pool-Based Market Case Study Based on the SIC System

Two cases are simulated for the SIC. In case I, a single-nodesystem is considered, i.e., the problem is solved without networkconstraints, while case II uses the network. The generation re-sults obtained by type (hydraulic and thermal) in each case arepresented in Figs. 3 and 4, respectively.

Analyzing the generation strategies, the following conclu-sions can be established. In the case of Endesa (the only com-pany with the capacity to transfer power blocks), the strategyadopted by the company for its hydro power plant in case I ismore flexible, allowing better distribution of the water availableduring the day, with substantial differences between the hours ofmaximum and minimum demand, showing a 1000 MW differ-ence in generation, in contrast with case II, where a difference



Fig. 2. Single line diagram of the SIC.

Fig. 3. Generation of hydro and thermal power plants, case I, without network.

Fig. 4. Generation of hydro and thermal power plants, case II, with network.

Fig. 5. Generation of Endesa hydro storage power plants.

of only 400 MW is reached between the hours of maximum andminimum demand, as a direct consequence of line congestion.This is shown in Fig. 5. The effect on prices weighted by the

Fig. 6. Weighted prices for a power exchange market type, cases I and II.

Fig. 7. Nodal prices for case I, without network.

Fig. 8. Nodal prices for case II, with network.

demand per node due to the adopted strategies is illustrated inFig. 6.

The prices at all periods are higher, in both cases, than thoseof perfect competition (with an approximate value of US$14.68/MWh); in some cases, prices reached 200% of the competitivevalue.

To see the effect of line congestion on prices, their dynamicsfor each demand node are shown in Figs. 7 and 8.

It is seen, for example, that in case II, and as a consequenceof congestion, the prices at each node differ substantially eventhough the same equivalent elasticity of the system is used. Ac-cordingly, the effect can be reflected in a node price increaseor reduction. For example, when line congestion occurs at theCharrúa node, which has a large hydro hydraulic generation ca-pacity, the plants cannot transfer their power to other nodes thathave greater demand, such as Santiago, generating subsystemsor islands.

Analyzing the flow along the system’s lines in both cases, it ispossible to reach the following conclusions. When the line thatjoins the Ancoa and Alto Jahuel nodes is congested, there is anoversupply at the Charrúa node for two reasons: first, the powerplants located at Ancoa, which are mostly of the hydraulic type,



Fig. 9. Generation by holding, cases I and II, without and with network.

increase the flow toward the Charrúa node, and second, there is ahigh generation capacity by Endesa at the Charrúa node. Theseevents cause a decrease of the price of power at Charrúa. Thisfact is not reflected totally at the Temuco node, which showsquite high peak prices due to the congestion of line 7–8, espe-cially during peak hours.

Finally, when comparing the amount of power that the com-panies produce in both cases, the following conclusions can bereached. It is clearly seen how the companies change their pro-duction if the network is considered.

Endesa, in spite of being the company with the largest in-stalled power in the SIC, because of the location of its powerplants, is the only company that reduces its generation signif-icantly due to the large congestion that exists in the lines ofthe south. In contrast with this phenomenon, Gener increasesits production, combining its position in the SIC with a low costof the thermal power plants, with a great presence in the AltoJahuel node. It can be said that Gener has a strategic locationin the network. Fig. 9 shows a graph of the daily generation ofeach holding considered.

B. Bilateral Contracts Market Case Study Basedon the SIC System

A common practice in the electric sector is to make bilateralcontracts. The model presented in this paper allows the exis-tence of bilateral contracts to be considered.

If a company has a certain amount of energy com-mitted at a certain node and at an already fixed price , toachieve the maximum return, this company must take into ac-count the following equation to calculate the benefit from thesales at node NV:

(26)

where

price for energy at node NV as an inverse functionof demand;power that company produces at its own powerstations to sell to node NV;cost of generating for company .

In (26), the term represents a cost forthe company if , because it would have to buyenergy. On the contrary, when , it turns into an in-come for the company, given by the surplus between the energygenerated by the company and that sold through the bilateralcontract. An important observation is related to the term givenby , which, because it is constant, does not play an

Fig. 10. Total production considering bilateral contracts, case I.

Fig. 11. Weighted average of the prices considering bilateral contracts, case I.

important role. The last term corresponds to the operating costof the company’s generating plants. As can be seen in the pre-vious equation, incorporating a bilateral contract is very simple.

With this new benefit function of the company, similar inspirit to (11), the electric market is simulated again, assumingtwo bilateral contract situations:

• Case I: Endesa has 550 MW contracted at Charrúa node 4and Gener has 600 MW contracted at Santiago node 2.

• Case II: Endesa has 500 MW contracted at Charrúa node4, Gener has 500 MW contracted at Santiago node 2, andColbún has 500 MW contracted at Santiago node 2.

Results for Case I: Throughout the day, a mean energy priceof US$21.693/MWh is found.

Results for Case II: Throughout the day, a mean energy priceof US$20.068/MWh is found.

From the graphs of the different cases of bilateral contracts,one can conclude that case II, with a greater amount of energytransacted as compared to case I, shows lower prices, as shownin Figs. 11 and 14. See Table IV for a comparison of the energyproduced in a pool-based market (see Fig. 4) versus a bilateralcontract market (see Figs. 10 and 13). Prices at different nodesare depicted in Figs. 12 and 15 for cases I and II, respectively.

Running times for all cases solved range between 1 and 2 hof CPU. Every iteration of the Lagrange relaxation algorithmtakes between 2 to 4 min for a given period and hydro energyper plant. The are not constant, but optimally selected, asderived in [16]. There are 95 plant variables (19 plants times 5nodes) and 133 constraints.

VI. CONCLUSION

The model presented in this paper carries out an iterativeNash-Cournot equilibrium game that considers either a gener-



Fig. 12. Prices at each node considering bilateral contracts, case I.

Fig. 13. Total production considering bilateral contracts, case II.

Fig. 14. Weighted average of the prices considering bilateral contracts, case II.

Fig. 15. Prices at each node assuming bilateral contracts, case II.

ating pool market or a bilateral contract model. It allows forthe incorporation of the network constraints and hydro plants.It is possible to add independent demand curves for each de-mand node, thereby assigning different elasticity and consump-tion values at the node. Our model is a useful tool to analyzethe strategic behavior of the agents immersed in a competitive

hydrothermal market. From a practical standpoint, the use of La-grange multipliers has made possible to dramatically decreasethe simulation time.

For future development, the use of a stochastic model isproposed to incorporate the randomness of water resources formiddle- and long-term studies.

APPENDIX AILLUSTRATIVE EXAMPLE OF THE

NIKAIDO-ISODA METHODOLOGY

The example is a system of 30 buses with three generationcompanies. The first one owns one hydro plant and the other twoown two and three thermal plants, respectively. There are threeperiods of study of 8 h each and the network is not represented.The minimum power of each of the six units is set to 0 and themaximum power to 80, 80, 50, 55, 30, and 40 MW, respectively.The cost of each unit is of the quadratic type, as in (9), wherethe quadratic and linear cost coefficients, and , are (0,0.035, 0.125, 0.0166, 0.05, 0.05) and (0, 1.75, 1, 3.25, 3, 3), re-spectively. The hydro unit is supposed to have a limit on the totalenergy produced during the three periods equal to 800 MWh.The price as a function of the demand is set as in (5) is for eachof the three periods: , and

, respectively. For a company whose units aredenoted by , the income from sales in period and iterationis given by the matrix expression, which is equivalent to (8):

where and are the price intercept and inverse elasticity,is a vector whose entries are the powers produced by

each of the units of company is the totalamount of power produced by all the units of the other compa-nies different from is a unit square matrix of dimension ,and is a unit vector of dimension .

Similarly, the expression for the cost of company is givenby

where is a diagonal matrix of dimension whose elementsare , and is a vector of dimension whose elementsare .

Since profit is equal to income minus cost, one can start fromthe Nikaido-Isoda function as in (2):

where corresponds to the profit in the previous iter-ation, which is not necessary in the optimization process,since are not the decision variables, . Thus, the fol-lowing generic Nikaido-Isoda objective function results:

, which is the base for (24).



TABLE IIIPRICE INTERCEPT AND DEMAND PRICE INVERSE

ELASTICITY DATA OF THE EIGHT-NODE SIC

In this example, the Nikaido-Isoda function for three compa-nies and three periods becomes

TABLE IVENERGY PRODUCED: POOL-BASED VERSUS

BILATERAL CONTRACT CASES I AND II

The optimization of is achieved finding the best re-sponse function, as in (3). Once obtained, the best response asa function of the values of the previous iteration, a relaxationalgorithm is used, as in (4), until convergence. Note that the ob-jective function is subject to inequality constraints (min and maxpower limits) and one hydro energy equality constraint that tiesthe three periods. In general, even without the maximum dailyhydro constraint, the problem cannot be solved separately percompany because line constraints tie the companies’ produc-tions even if one decouples the problem by period.



APPENDIX BLAGRANGE METHOD: ACTIVE SET METHOD

To solve the quadratic optimization of the Nikaido-Isodafunction subject to linear equality and inequality constraintspresented in the previous section, a Lagrange method is used[22]. The general problem can be set as

where is the vector of variables, and are the equality andinequality constraints, and is a symmetric positive definitematrix. If one only considers the equality constraints, the first-order Lagrange necessary conditions are

where is a vector with the Lagrange multipliers of the equalityconstraints. The following solution results:

The general problem subject to inequality constraints issolved by an active set method [22]. It is necessary to knowthe active constraints and it is done iteratively. At iteration , apoint in the working set is feasible for all the equalityconstraints and for some of the inequality constraints. Thequadratic problem of the working set is of the form

where . This problem only has equality con-straints and can be solved for . If , the current point isoptimal with respect to the working set. If and isfeasible for all constraints, then becomes the new .If is not feasible, a search of the formis made, where is selected as large as possible to maintain fea-sibility. In particular

With this, a new inequality constraint is satisfied by equalityand it is adjoined to the working set . The process pro-gresses adjoining constraints to the working set until a minimumpoint is obtained. Then, the corresponding Lagrange multipliers

for constraints in the working set as defined above are exam-ined. If all are nonnegative, the current point is optimal; other-wise, usually the most negative is dropped from the working setand the process continues.

For the particular hydro-thermal setting of this work, thismethodology is applied starting from the Nikaido-Isoda func-

tion in (24). This problem is solved using the active set methoddefined before:

(24)

One can rewrite (24) as follows:

The objective function consists of a summation of terms forall periods, . Therefore, one can rewrite the problem and solveit for each period separately. To do that, a new constraint (25) isadded, that: 1) guarantees that constraint (15) is never active, and2) allows us to approximate the optimal use of water reservoirs.The problem becomes decoupled per period and per company:



Fig. 16. Hydro unit Lagrange multiplier versus available power for hour 11.

When this problem is solved by the active set method,one obtains the Lagrange multipliers for every period. Byincreasing the amount of water in the period with the mostnegative Lagrange multiplier by MW, the equality

is relaxed and is eventuallydropped from the active set until all the water is allocated.Since the problem is quadratic, matrix is positive semidefi-nite, and the constraints are linear, one can reach convergenceiteratively. In Fig. 16, one can observe that the behavior of theLagrange multiplier of constraint (25) for a gradual increase ofthe hydro generation in MW steps, showing that thevalue of the multiplier is inversely proportional to the amountof water used. It is also a piecewise linear function due to thechanges in the active set, since an increase in the amount ofwater makes some constraints to be active and they produce achange of slope.

REFERENCES

[1] J. Contreras, M. Klusch, and J. B. Krawczyk, “Numerical solutionsto Nash-Cournot equilibria in coupled constraint electricity markets,”IEEE Trans. Power Syst., vol. 19, no. 1, pp. 195–206, Feb. 2004.

[2] A. Maiorano, Y. H. Song, and M. Trovato, “Dynamics of noncollu-sive oligopolistic electricity markets,” in Proc. IEEE Power Eng. Soc.Winter Meeting, Singapore, 2000, vol. 2, pp. 838–844.

[3] S. Stoft, H. Singh, Ed., “Using game theory to study market power insimple networks,” in IEEE Tutorial on Game Theory Applications toPower Markets, 1999, 99TP-136.

[4] B. F. Hobbs, C. B. Metzler, and J.-S. Pang, “Strategic gaming analysisfor electric power systems: An MPEC approach,” IEEE Trans. PowerSyst., vol. 15, no. 2, pp. 638–645, May 2000.

[5] W. Jing-Yuan and Y. Smeers, “Spatial oligopolistic electricity modelswith Cournot generators and regulated transmission prices,” Oper.Res., vol. 47, no. 1, pp. 102–112, 1997.

[6] B. F. Hobbs, “Linear complementarity models of Nash-Cournot com-petition in bilateral and POOLCO power markets,” IEEE Trans. PowerSyst., vol. 16, no. 2, pp. 194–202, May 2001.

[7] T. J. Scott and E. G. Read, “Modelling hydro reservoir operation in aderegulated electricity market,” Int. Trans. Oper. Res., vol. 3, no. 3–4,pp. 243–253, 1996.

[8] R. Kelman, L. A. N. Barroso, and M. V. F. Pereira, “Market power as-sessment and mitigation in hydrothermal systems,” IEEE Trans. PowerSyst., vol. 16, no. 3, pp. 354–359, Aug. 2001.

[9] M. Ventosa, M. Rivier, A. Ramos, and A. García-Alcaide, “An MCPapproach for hydrothermal coordination in deregulated power mar-kets,” in Proc. IEEE Power Eng. Soc. Summer Meeting, Seattle, WA,Jul. 16–20, 2000, vol. 4, pp. 2272–2277.

[10] J. Villar and H. Rudnick, “Hydrothermal market simulator using gametheory: Assessment of market power,” IEEE Trans. Power Syst., vol.18, no. 1, pp. 91–98, Feb. 2003.

[11] L. A. N. Barroso, M. H. C. Fampa, R. Kelman, M. V. F. Pereira, andP. Lino, “Market power issues in bid-based hydrothermald dispatch,”Ann. Oper. Res., vol. 117, no. 1–4, pp. 247–270, 2002.

[12] L. G. T. Carpio and A. O. Pereira, Jr., “Economical efficiency of coordi-nating the generation by subsystems with the capacity of transmissionin the Brazilian market of electricity,” Energy Econ., vol. 29, no. 3, pp.454–466, May 2007.

[13] J. Bushnell, “A mixed complementarity model of hydrothermal elec-tricity competition in the Western United States,” Oper. Res., vol. 51,no. 1, pp. 80–93, Jan. 2003.

[14] A. Rubiales, P. Lotito, and F. Mayorano, “Numerical solutions tothe hydrothermal coordination problem considering an oligopolisticmarket structure,” in Proc. Int. Conf. Eng. Optim., EngOPT 2008, Riode Janeiro, Brazil, Jun. 1–5, 2008.

[15] M. Tesser, A. Pagès, and N. Nabona, “An oligopoly model for medium-term power planning in a liberalized electricity market,” IEEE Trans.Power Syst., vol. 24, no. 1, pp. 66–77, Feb. 2009.

[16] J. B. Krawczyk and J. Zuccollo, NIRA-3: An Improved MATLABPackage for Finding Nash Equilibria in Infinite Games, WorkingPaper, Munich Personal RePec Archive, 2006. [Online]. Available:http://mpra.ub.uni-muenchen.de/1119/1/MPRA_paper_1119.pdf.

[17] S. Uryasev and R. Y. Rubinstein, “On relaxation algorithms in compu-tation of noncooperative equilibria,” IEEE Trans. Autom. Control, vol.39, no. 6, pp. 1263–1267, Jun. 1994.

[18] J. B. Rosen, “Existence and uniqueness of equilibrium points forconcave n-person games,” Econometrica, vol. 33, no. 3, pp. 520–534,1965.

[19] J. B. Krawczyk and S. Uryasev, “Relaxation algorithms to find Nashequilibria with economic applications,” Environment. Model. Assess.vol. 5, no. 1, pp. 63–73, Jan. 2000. [Online]. Available: http://www.ise.ufl.edu/uryasev/relaxation_algorithms_for_Nash_equilibrium.pdf.

[20] R. Rojas, “Bolsa de energía en el SING—Simulación vía teoría dejuegos,” (in Spanish) M.S. thesis, Dept. Elect. Eng., Pontificia Univ.Católica de Chile, Santiago, Chile, 2001.

[21] A. Galetovic and C. M. Muñoz, “Estimating deficit probabilities withprice-responsive demand in contract-based electricity markets,” EnergyPol., vol. 37, pp. 560–569, 2009.

[22] D. G. Luenberger, Linear and Nonlinear Programming, 2nded. Reading, MA: Addison-Wesley, 1984.

Juan Pablo Molina received the Electrical Engineerand M.Sc. degrees from the Universidad de Santiagode Chile, Santiago, Chile. He is pursuing the M.B.A.degree at the Universidad de Chile, Santiago.

His research activities focus on the economic op-eration, planning, and regulation of power systems.He works at the Electrical Department of Escondidacopper mine, Antofagasta, Chile.

Juan Manuel Zolezzi (SM’03) received the Elec-trical Engineer degree from the Universidad deSantiago de Chile, Santiago Chile, the M.Sc. de-gree from the Universidad de Chile, Santiago, andthe Ph.D. degree from the Pontificia UniversidadCatólica de Chile, Santiago.

He is Rector and Professor of Engineering at theUniversidad de Santiago de Chile. His research activ-ities focus on the economic operation, planning, andregulation of electric power systems.

Javier Contreras (SM’05) received the ElectricalEngineer degree from the University of Zaragoza,Zaragoza, Spain, the M.Sc. degree from the Uni-versity of Southern California, Los Angeles, andthe Ph.D. degree from the University of California,Berkeley, in 1989, 1992, and 1997, respectively.

His research interests include power systems eco-nomics. He is a Full Professor at the University ofCastilla-La Mancha, Ciudad Real, Spain.



Hugh Rudnick (F’00) received the Electrical Engi-neer degree from the University of Chile, Santiago,Chile, and the M.Sc. and Ph.D. degrees from the Vic-toria University of Manchester, Manchester, U.K.

He is Professor of Engineering at Pontificia Uni-versidad Católica de Chile, Santiago, and the Directorof Systep Engineering. His research activities focuson the economic operation, planning, and regulationof electric power systems.

María José Reveco received the Electrical Engineerdegree from the Universidad de Santiago de Chile,Santiago, Chile, and the M.A. degree in economicsfrom Ilades/Georgetown University, Santiago.

She is an advisor at the Ministry of Energy ofChile, Santiago, in electric regulation and a Pro-fessor at the Universidad de Santiago de Chile. Herresearch activities focus on the economic operation,planning, and regulation of electric power systems.

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