Equilibrium Market Power Modeling

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Equilibrium Market Power Modeling forLarge Scale Power SystemsBenjamin F. Hobbs, M.IEEE, Udi Hehnan, and Jong-Shi Pang

,4bstract-A review of economic equilibrium models forsimulating imperfect competition among electricity producers ispresented in this paper. The presentation will emphasize theapplication of a model using a linearized DC grid is formulatedusing the concept of Cournot and supply-function games, andthe resulting models are solved as a linear complementarilyprogram (LCP). Both POOLCO and bilateral-type marketscan be simulated. Properties of the solutions are to these modelsare analyzed. The presentation will include applications to theEastern Interconnection that explore how transmissionlimitations and possible mergers might affect the equilibriumprices and market shares calculated by the model.Imfex TermsElectricity competition, Electricity generation,

Market models, Strategic pricing, Complementarity, Cournot,Eastern interconnection.

1, INTRODUCTIONTHEbility to unilaterally manipulate pricesmarketpoweris an important concern in restructured powermarkets. Transmission limitations are an important source ofthis market power [44]. This panel presentation describesapplications and properties of one approach to modeling suchmarkets: equilibrium modeling. To provide context for thispresentation, this presentation summary presents a detailedreview of approaches to equilibrium modeling and theirapplication. (For other reviews, see [34,47 ].)

H.EQUILIBRIUM MODEL FORMULATIONSA. De$nition of Equilibrium bfodeling Approach

The models applied in this presentation are based upon ageneral approach of defining a market equilibrium as a set ofprices, producer input and output decisions, transmissionflows, and consumption that simultaneously satisfj eachmarket participants first order conditions for maximization oftheir net benefits while clearing the market (suppIy =demand). A solution satis$ing those conditions will havethe property that no participant will want to alter theirdecision unilaterally (a Nash equilibrium). Smeers [47]concludes his survey of energy market models by arguing thatexplicit statement and solution of equilibrium conditions is apromising theoretical and computational approach to modelingstrategic behavior. Although it is well recognized that nomodeling approach can precisely predict prices in imperfectmarkets, there appears to k agreement that such models are

indispensable for gain 1ing insights on modes of behavior andrelative differences in efficiency, price levels, and other marketoutcomes of alternative market designs.

Equilibria for the models are obtained by deriving first-order conditions for each player, adding market clearingconditions, and solving them simultaneously. The firstorder (Kuhn-Karesh-Tucker/KKT) conditions for anoptimization problem MAX F(x,y) subject to G(x,y) = 0,H(x,y)~O, x> Oare:

O~. J_ dF/& - J8G/& @H/& ~ 0;dF/2y - A3G/dy - p3H/dy = O;G(,y) = O; O SF lH(x,y)z O

where 4 and ~ are the dual variables for constraints G and H,respectively, and J_is interpreted thus:

{0< x J-f(x) ~ 0) is shorthand for:{x> 0: f(x) 50: and xf(x) =0)The equations associated with the nonnegative variables x

and P are called complementarily conditions. Manyequilibrium models of power markets created by combiningthe ICKT conditions for all the market participants and thenadding equality conditions to represent clearing of themarket. A problem that includes both equalities andcomplementarities is termed a mixed complementarily prob-lem, or MCP, Iff(r) is afflne, then the MCP is a linear MCP(or mixed LCP),Note that the use of first-order conditions to defineequilibria implies that each players optimization problem isconvex. This is not true of many power operations andplanning problems, such as, unit commitment or planningdiscrete facilities. In general, when there are lumpydecisions, KKT conditions defining a solution do not existand neither will market equilibria. Nonetheless, we willusually assume that the problems can be approximated asbeing convex. If this approximation is at times unrealistic, itis at least partially compensated for by the ability to analyzelarge systems.

The direct solution of the market equilibrium conditionsby complementarily methods has important computationaladvantages. Large LCPS and even nonlinear MCPS withthousands of variables and complementarily conditions can besolved using available MCP software. Examples includeimplementations of Lemkes algorithm and the MILES andPATH solvers within GAMS [21], as well as manycontemporary algorithms based on advanced nonsmoothNewton methods. For a summary of various possiblealgorithms, see Ferris and Pang [19]. These algorithms

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permit application of strategic market models to large systemswith thousands of power plants and hundreds or eventhousands of constrained! transmission interfaces.B.Applications of Equilibrium Modeling Approach

Numerous studies have used equilibrium models toaddress market power in power markets, many consideringcompetition in both energy and transmission services. Thestudies can be classifieti by the market clearing mechanismused (centralized/POOLCO or decentralizedbilateral); therepresentation of the physical characteristics of the electricnetwork; and the assumed type of interaction between rivalpower producers.

Regarding market clearing mechanisms, most studieshave implicitly or explicitly assumed a centralized biddingprocess supervised by a PX or ISO [e.g., 12,37]. This processresults in a set of publicly disclosed market clearing prices.There have also been shldies that model bilateral trading [28]and the market power that large power traders might exercise[49]. It has been shown, however, that if there is perfectcompetition among power traders so that they arbitrage awayany non-cost based price differences between differentlocations, then POOLCO and bilateral trading systems yieldthe same equilibria under either perfect competition [7] orCoumot competition [36].

Turning to the physical representation of the electricnetwork, many studies disregard transmission constraintsaltogether [e.g., 1,22, 54]. Others use a transshipmentnetwork that ignores KirchhofTs voltage law [2,6,28]. Thesesimplifications may eliminate the opportunity to analyze theunique market manipulation opportunities in electricnetworks. To correct this shortcoming, some studies haveused AC [18,55] or linearized DC [e.g., 12,26,27,33 a,37]load flow models. DC models are more widely applied notonly because of their ,inearity, but also because numericaltests have found that their congestion costs are excellentapproximations if thermal constraints are the main concern[31].

The final classification-the type of interaction assumedamong rival generators and other playershas a crucialimpact on model results. Power producers can be intenselycompetitive or they may collude. Seemingly arcanedistinctions in assum~tions concerning player interactionscan result in large changes in economic equilibria and policyimplications. For example, there has been extensive debate[32,38,50] regarding the proper way to measure and analyzecompetition in networks and how strategic behavior byproducers will manifist itself. What conclusions resultdepend heavily on the assumptions made. Thus, there is anadvantage to modeling frameworks that can accommodate arange of degrees of competitiveness.C. Types of Strategic Interaction in Equilibrium Models

We next define several types of strategic interaction,most of them being familiar concepts from game theory andindustrial organization [20,43,53]. They differ in how eachplayer f anticipates that rivals will react to its decisions

concerning either prices p or quantities q. Our models will bedesigned to represent the fill range of these behaviors.

The definitions below refer to competition amongsuppliers, so q is referred to as sales or output. But moregenerally, analogous games can be defined between suppliersand consumers, and among consumers; q can therefore alsorefer to purchases. Also, we temporarily disregard the factthat demand is temporally and spatially distributed. Inaddition, these definitions omit the effect of contracts fordifferences upon marginal revenues [23]. Finally, thesedefinitions assume that all players get the market clearingprice; however, a pay your bid (first price) auction is soonto implemented in the UK. Strategic models for suchauctions can base revenue on the players bid [8,41].

G Pure Competition CNoMarket Power)/Bertrand: Just q~in firm fs revenue pq~ is a decision variable; p is takenas fixed. Thus marginal revenue MR (= 6jq~&) in firmfs first-order profit maximization conditions equalsp.Generalized Bertrand Stratew (or Game in Prices):Here, pqf = pf qfi~ , p.f*), where pf is f s decisionvariable, p< is the vector of prices offered by other firms,and q~is a function of all prices. The asterisk on p+ *indicates that facts as if its rivals prices wont change inreaction to changes in f s prices. For a homogeneousgood, f can sell as much qf as it wants to (up to themarket demand) if p~ s (lowest delivered price amongrival producers); otherwise, q~ = O. But forheterogeneous goods (such as green and non-greenpower), there may be nonzero cross price elasticities, andq,@fip.J will take on other forms.Cournot Strategy (or Game in Quantities): pqf = p(qf+q


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iimction of q~ The marginal revenue term in the firstorder condition forj- S profit maximization is:

MR = $q{&,= p + (+/@(l +@#@J= p i- (@/~)(1 + e)where I3is the cor,jectural variation. If 0 is constantand equals O, the Cournot game results. Meanwhile, 6 =-1 yields the pure competition/Bertrand game, while 6 =+1 is equivalent to collusion if~and g are identical firms(and the Coumot conjecture is applied to the output ofother firms). If 0 equals the actual local response ofrivals, then this is a consistent conjectures model [9].Recent theoretical work [11] shows that some conjecturalvariations models are the reduced form of equilibriumstrategies in games involving repeated playsuch asdailv Dower auctions.. .

G General Rival Supply Function (or Rival Price ReactionEw.@iL In this =, outputW rivals isanticbated(perhaps incorrect] y) to respond to price followingfunction q.~); thus, pq~ = p(qJ +q.j@))qj This can beviewed as a generalization of Stackelberg models in thatthe conjectured response may not equal the true response9Y-TW It can also be seen as generalizing the SFEmethod, next. The general rival supply ti.mctionapproach is rarely used in market power simulations;however, it has several advantages that make it worthconsidering. One is that q.fi) might be modeled as asmooth finction, ~implifing calculation of equilibria.Others are discusseli below.

. SUPPIYFunction Ecluilibria (SFE) [35~ In this specialcase of the general rival supply fimction model, thedecision variables for each firm~are the parameters qyofits bid function Pb(i~j/@. A market clearing mechanism(e.g., the California PX) then determines p, and sets qf =Pbb/91). AS a result, the revenue term in fs objectivefimction is (with some abuse of notation) p@~(jJ/qJ +Zz *J Pi] @J/9g*)) pi(j /@. The asterisk indicates thatbids from other firms qg are treated as if they are fixed.

Let us now define the equilibrium of a game involving theabove strategies [20,53]:

Nash Equilibrium for a Game in X. Let Xl = & bevariable(s) (strategies) under control of firm~ ~f be thespace of feasible strategies for j X+ = {Xg, Vgzj); andfljfljx-~ be the payoff to f given the decisions of allfirms. Then {Xj*,X+*} is a Nash Equilibrium (in purestrategies) ifiIqxf,xf2IZj(y,x.j)xf % ~fFor Coumot games, Xf = q~; for generalized Bertrand games,

Xf = pJ , and for supply function equilibria, XJ = qfi Ageneralized Nash equilibrium results if& = &(Xg) [50].Important questions include whether the equilibria exist andare unique, and how they can be calculated. Equilibriumconcepts can also be defined for games involving Stackeibergplayers, general conjectural variations, and general rivalsupply functions by dei-1ning {Xf*,Xf*}such that:

fl,.v. .Yjfll)) ~ flf(xj; ~.jfl~) ~xf E & ~f andX.J(X~*)=X4*, df

where X


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has been chosen as the basis of several power market models[e.g., 4,16,18,22,24,27,41,42,54,55,57]. The resultingequilibria generally lie between the Bertrand and Coumotresultsan intermediate level of competition. Butsometimes equilibria are not unique, a large range ofoutcomes is possible; in general, the Cournot equilibrium willbe their upper bound [3,24,35,52]. A drawback of thesemodels, however, is that it is difficult to calculate anequilibrium; indeed, it may not even exist [4]. Most supplyfimction studies have been designed for very simple systems(e.g., 1 to 4 nodes). ,kltematively, when a more realisticnetwork has been consiclered, the model searched over only ahandful of strategies to find the optimal strategy for each oftwo firms [18] or the bids have been restricted to a linearfimction with either jixed slope or intercept [27]. Afi.rrrdamentai problem is that the optimization problem eachfirm faces is nonconvex, and can possess multiple localoptima,

To our knowledge, there are no published power marketmodels based on the general conjectural variations or generalrival supply functions models. The major reasons appear tobe the conceptual simplicity of Coumot models and theperceived appropriateness of SFE modeis for POOLCOmarkets. However, the two latter models also have seriouslimitations that make it worthwhile to consider alternativeapproaches. First, Coumot models do not give meaningfidequilibria when price elasticities are low or zer~as theyoflen are for short run power demands, ancillary services, andshort run supplies of transmission capacity. It is notreasonable, for example, to assume that a supplier will be ablepush prices arbitrarily high without any response whatsoeverfrom rival suppliers. (Conjectural variation models share thisproblem with Coumot models when price elasticities are verylow or zero; unless @


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network [a la 37]. Transmission tariffs can be set by aregulated body to recover grid costs using either distance-dependent or postage-stamp fees. In general, the authorspoint out that such a system can have multiple equilibria withwidely diverging effects on the profits and outputs ofindividual firms. That is, their MCP generally has multiplesolutions. They then use a variational inequality (IV) solutionapproach to solve for one of the possible equilibria; the VIsolution is proven to exist and be unique (even if there areactually multiple Coumot equilibria). In contrast, the MCPmodels defined in [26,48] have unique solutions, implyingthat the market equilibria are unique. This is made possibleby making the simpli@ing assumption that each generatordoes not anticipate how its actions will affect transmissionpricesi.e., the market for transmission services is effectivelycompetitive.

Smeers & Jing-Yuan [48] and Hobbs [26]present models ofmarkets for energy and transmission services in which generatorsbehave strategically in the energy market (Cournot), transmissioncapacity is rationed competitive y (a la Hogan [30] /Schweppe[45] or Cha&Peek [14]), power flows over a linearized DCnetwork and no arbitragers exist to erase non-cost-baseddifferences in energy prices at different lccations. These models[like 33a] can also include the possibility of generation capacityexpansion. Existence and uniqueness properties of the marketsolution can be proven [36,48]. These formulations permitsolution of large problems; e.g., Day and Hobbs [15] apply [26] tothe UK power system, which involves 252 busses, 68 plants, and352 lines.

Two other market models explicitly include KKTconditions for Coumot producers for intertemporal powerproduction decisions while omitting transmission constraints.Bushnell [10] considers how such producers would allocatehydropower over time. Meanwhile, Ramos et al. [40] modelunit commitment over the course of a day.Other mcdels consider arbitragers/marketers. In [49],generators are competitive but a small set of Cournot arbitragerswield oligopsonist market power with respect to generators andoligopolistic market power with respect to power consumers.Versions of this model wtth thousands of variables have beensolved for large systems in the EU. In contrast, the arbitragedbilateral model of Hobbs [26] represents Cournot generators, withthe assumption that low barriers to entry mean that arbitragersbehave competitively. A large scale version of the latter modelhas been solved for the Eastern Interconnection [25], considering2728 plants, 829 producers, and 814 transmission flowgates.

III.The

LCP MODELSFORANALYSISOFMARKETPOWERONLARGESCALELINEARIZEDDC POWERSYSTEMSmodeling approach presented here is based on the

Cournot modej by Hobbs [26]. Versions includemodifications introduced by Day and Hobbs [15] to includegeneral rival supply responses that may deviate from the fixedsupply conjecture of the Coumot model.In the presentation, summaries will be presented oftheoretical results. The most important result is that for the

basic Hobbs and Day & Hobbs models, the POOLCO andbilateral transaction (with perfect arbitrage) models yield thesame equilibrium prices, profits, and total production by eachfirm The existence and uniqueness of these solutions hasbeen proven. Also investigated are properties of solutions ofmodels in which generating firms anticipate that prices fortransmission services, allowances, and other inputs willchange depending on much of those inputs are used by thefirm. This relaxes the Hobbs [26] assumption that thatgenerators naively believe that transmission fees will notchange.The presentation will also summarize recent applications of

the Helman et al. [25] version of the model to the EasternInterconnection in which several issues are addressed,including the following:1. In what control areas might transmission limitationsbetween control areas result in significant local madcetpower?2. How do changes in the amount of transmissioncapability among control areas affect the ability toexercise market power?3. How would mergers among generating companies aflkctequilibrium prices?

IV. ACKNOWLEDGMENTThe contributions by our collaborators Carolyn Metzler and

Christopher Day are gratefully acknowledged. The opinicmsexpressed are the responsibility of the authors, and do notnecessarily represent the positions of the finding agencies oremployers of the authors,

[1]

[2]

[3][4]

[5]

[6]

[7][8]

[9][10][11][12]

V, REFERENCESB. Arrdersson and L. Bergman, Market Structure and the Price ofElectnclty: .&r i+ Ante Analysis of Deregulated Swedish Markets,Energy J., 16(2), 1995,97-109X. Bai, S M Shahidehpour, V C, Ramesh, and E. Yu, Transmis:,ionAnalysis by Nash Game Method, IEEE Trarrs Power Jwfems 12(3),1997, 1046-1052R. Baldick, Electnclty Market Models Coumot versus Supply FunctionEquilibrium, Dratl, Umverslty of Texas, Oct. 1999.C.A, Berry, B.F. Hobbs, W,A. Meroney, R,P, ONedl, and W.R, Stewart,Jr., Analyzing Strategic Bldddlg Behavior m Tmnsmisslon Networks,Ufdifles Policy, 2000,S. Borenstein and J. Bushnell An Empirical ArralysNof the Potential forMarket Power in Californias Electricity Industry, J. IndustrialEconomics, 47(3), Sept. 1999,285-323.S. Borenstein, J. Bushnell, and S. Stoft, The Competuive Effects ofIransmission Capacity m a Deregulated E1ectnctty Industry, PWP-040R,Umv. Calif. Energy Inst., Berkeley, CA, 1998,J. Boucher and Y. Smeers, Alternative Models of Restructured ElectricitySystems, 1: No Market Power, Universlte Catholique de Louvam, Mwch1999J. Bower and D. Bunn, AOAgent-Based Model Comparison of Pool andBdateral Market Mechamsrns for Electricity Trading m England andWales, Energy Modeling Forum 17, Washington, DC, Nov. 4.5, 1999.T.F. Bresnahan, Duopoly Models With Consistent Conjectures, ,4mer.Economic Review, 71(5), 1981,934-945.J, Bushnell, Water and Power, Hydroelectric Resources m the Em ofCompetition in the Western US; U, Ca, Energy Inst., Berkeley, 1998.L.M.B. Cabral, Conjectural Vmations as a Reduced Form, EcorronrIcsLetters, 49(4), Oct. 1995,397-402,J. Cardell, C.C. Hltt, and W.W. Hogan Market Power and StrategicInteraction in Electricity Networks, Resource and Energy Econ., 19(I-2), 1997, 109-137.

0-7803-7031-7/01/$10.00 (C) 2001 IEEE0-7803-7173-9/01/$10.00 2001 IEEE 562


6/6

[13] E.G. Cazalet, Power Market Decision Analysm Model MethodologyReport, Report to BPA, Consulting Decls~on Analysts, LOSAltos Hills,[14][15]

[16]

[17][18]

[19][20][21][22][23]

[24][25]

[26][27]

[28]

[29]

[30][31]

[32]

[33]

CA, 1991.H.P. Chao and S. Peck, A Market Mechanism for Electric PowerTransmlsslort, J Regulatory Economics, 10(1), July 1996,25-59.C.J. Day and B F. Hobbs, LCP Models ofNash-Coumot Competition inthe England and Wales Electricity Market, INFORMS National Meeting,Phdadelphia, Nov. 7-10, 1999,J. Falk, Detenninmg Market Power m Deregulated Generation Marketsby Measuring Price-Cost Margins, The Electricity J., 11(6), July 1998,44-50.J. Farrell and C. Shapiro, Horizontal Mergers An Equilibrium Analysis;,4rnertcan Ecorromlc Review, 80(1), March 1990,107-126.R.W. Ferrero, S.M. Shahidehpour, and V.C. Ramesh. TransactionAnalysls in Deregulated Power Systems Using Game Theory, IEEETrans. Power Systems, 12(3), 1997, 1340-1347.M.C. Ferns and J S. Pang, Engineenng And Economic Applications OfComplementarrty Problems, SI.4.i4Review, 39(4), Dec. 1997,669-713.D. Fudenberg and J. Tirole, Game Theop, MIT Press, Cambridge, SbPrinting, 1996.General Algebraic Modeling System, www.gams.com.R. Green. Increasing Competition in the British Electricity Spot Market,J Ind. Econ, 44, 1996,205-216.R. Green, The Electrici ty Contract Market in England and Wales; J . ojIndustrial Economics, 47(l), Mar. 1999, 107-124.R. Green and D Newbery, Competitmn in the British Electric SpotMarket, J Pol. Econ., 100, 1992,929-953.U. Helman, B.F. Hobbs, J.B. Cardell, T. Luong, and M.T. Wander,Modehng Strategic Pncmg in Bdateral and Poolco Electricity Markets ANash-Coumot Approach with Applications to US Eastern Interconnection,ENFORMSNanonal Meeting, Philadelphia, Nov. 7-10, 1999BF Hobbs. LCP Models of Nash-Coumot Competit ion in Bilateral andPOOLCO-Based Power Markets, IEEE Trans. Power Systems, m press.B.F. Hobbs, C. Metzler, and J.-S. Pang, Calculating Equdibna in[imperfectly Competitive Power Markets: An MPEC Approach; IEEETrans. Power S,vstems, May 2000.B F. Hobbs and R,E. Schuler, Assessment of the Deregtdatlon of ElectricPower Generation Using Network Models of Imperfect SpatialCompetmon, Papers Reg. Sci. As$ocia[ton, 57, 1985,75-89.B.F. Hobbs and R.E. Schuler, Deregulating the Distribution of Electricity:Price and Welfare Consequences of Spatial Oligopoly with UmformDelwered Prices. J Regional Science, 26(2), 1986,235-265,WW. Hogan, Contract Networks for Electric Power Transmission, J.Regulatory Ecorr,, 4, 1992,211-242.W W Hogan, Markets m Real Elcctnc Networks Require ReactivePncesY m M. Einhom and R. Slddiql , L?lectr tclty Transmission Pricingand Technology, KJuwer Academic Publishers, Boston, 1995, 143-182.W.W Hogan, Transmmlon Rights, Through the Looking-Glass: A FalseDichotomy Between Actwe and Passive Trading, The Elecmciy Journal,10(8), October 1997.86-91.S, Hunt and G. Shuttleworrh, Competmon and Chotce m Electrlc/ty,Wdey, NY, 1996.

[33al W. Jing-Yuan and Y. Smeers, Spatial Ohgopolistic Electricity Models

[34][35][36]

[37]

[38][39][40]

with Coumot Generators and Regulated Transmission Prices, Operatiorr$Res., 47(l) , 1999, 102-I 12.E. Kahn, Numerical Techniques for Analyzing Market Power inElectnclty, Electricity Journal, July 1998. 34-+3P.D IUemperer and M A. Lieyer, Supply Function Equihbna,Econometrics, 57, 1989, 1243-1277,C Metzler, B F. Hobbs. and J.S. Pang, Nash-Coumot Equilibria in PowerMarkets on a Lmearrzed DC Network with Arbitrage Formulations andPropemes, Networks and Spema[ Theory. m process.S .S. Oren, Economic InetXclency of Passwe Transmission Rights mCongested Electricity Systems with Competitive Generation, Energy J.,fc?(l), 1997,63-83.S.S. Oren, Passive Transmission Rights Will Not Do the Job, TheElectr[ctty Journal, /O(5), June 1997,22-33,R. Qi and P Harker, Generalized Spatial Price Equilibria with Semi-Contmuous Market Structures; Umverslty of Pennsylvania, 1997.A. Ramos, M. Ventosa, and M. Rlvier, Modeling Competition in ElectricEnergy Markets by Eqcuhbnum Constraints, L7d. Policy, 7, 1998, 233-242.

[41]

[42]

[43]

A. Rudkevich, Using Supply Function Equilibrium Method to ArralWeElectricity Markets with What YOUBid ISWhat YOUGet Payment Rule,Energy Modeling Forum 17, Meeting 2. Washington. DC, Nov. 5, 1999.A, Rudkevich, M. Duckworth, and R. Rosen, Modehrrg Electricl~Pricing in a Deregulated Generation Industry: The Potential for OligopolyPricing in a Poolco: Energy J., 20(3), July 1998.F.M. Seherer and D. Ross. IndustrialMarketStructureand EconomicPerformance, 3d cd., Houghton-Mifflin, Boston, 1990.[43a] R. Schmalensee and B. Golub, Estimating Effective Concentrat ion in

[44]

[45][46]

[47][48]

[49][50][51][52][53][54][55]

[56][57]

Deregulated Wholesale Electric@ Markets, Rand J. Econ, 1985, 12-26.R.E. Schuler, Analytic and Experimentally Derived Estimates of MalketPower in Deregulated Electrici ty Systems: Pohcy Implications for theManagement and Institutional Evolution of the Industry, DecisionSupport Sysrems, 30(3), Jan. 2001,341-355.F.C. Schweppe, MC. Caramams, R.E. Tabors, and R.E. Bohrr, ~,potPricing of Elecrncity, KJuwer, Norwell, MA, 1988.H. Singh, cd., Game Theory Applications in Electric Power Markets,Tutorial Pubhcation TP-136-O, IEEE Power Engmeenng Society,Piscataway, NJ, Jan, 1999.Y, Smeers, Computable Equdibrium Models and the Restructuring oftheEuropean Electricity and Gas Markets, Energy J., 18(4), 1997, 1-31.Y. Smeers and W. Jing-Yuan, Spatially Oligopolistic Model withOpportoruty Cost Pricing for Transmission Capacity Reservatiorrs+AVariational Inequality Approach, CORE Disc. Paper 9717, Universi tyCatholique de Louvain, Feb. 1997.Y. Smeers and W. Jing-Yuan , Do We Need a Power Exchange if ThereAre Enough Power Marketers?, CORE Disc. Paper 9760, UmversiteCatholique de Louvain, Aug. 1997.S. Stoff, Transmission Rights and Wrongs, The Electrmy Journal,10(8), October 1997b, 91-95,S. StoR, Price CongestIon! Semmar, Office of Economic Policy, FederalEnergy Regulatory Corrrrmssion, March 20, 1998.C. Supatgiat, R.Q, Zhang, J.R. Buge, Equdlbnum Values in aCompetitive Power Exchange Market, Department of Industrial andOperations Engnreermg, The Umverslty ofMichigan, 1999.J.Twole, The Theory of Industrial Orgarri:ation, MIT Press, Cambridge,MA, 1988,N.M. von der Febr and D, Harbord, Spot Market Competition in the UKElecmclty Industry, Econ. J., 103, 1993,531-546,J.D. Weber and T.J. Overbye, A Two-Level Optlmlzatlon Problem forAnalysis of Market Blddmg Strategies, Paper WE099, IEEE PowerEngineering Meeting, Edmonton, July 1999.CD. Wolfram, Strategic Bidding in a Muhi-Umt Auction: An EmpiricalAnalysjs of Bids to Supply Electricity in England and Wales, Rand .I, ofEcOtlOl?llCS,9(4). 1998,703-725,Z. Younes and M. Ihc, Generation Strategies For Gammg TransmissionConstraints: WIII The Deregulated Electric Power Market Be AnOligopoly?, Dec~smrrSupport Sysrems, 24(3-4), Jan. 1999,207-222,

W. BIOGRAPHIESBenjamin F. Hobbs has been on the faculty of the Johns Hopkins Universitysince 1995. Prewously, he was on the faculty of the Department of Systems,Control, and Industrial Engmeerirrg at Case Western Reserve Umverslty. He Malso o consultant to the FERC OtXce of the Economic Adwsor,Udi Helman ISan economist with the Federal Energy Regulatory Commlsslon.He is also a Ph.D student in the Department of Geography & EnwrorrmentalEngmeermg at The Johns Hopkins Umverslty,Jong-Shi Pang is a professor in the Mathematical Sciences Department at TheJohns Hopkins Umverslry, Prewously, he was a professor in the Schoc~lofManagement at the University of Texas, Dallas,

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Equilibrium Market Power Modeling

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