Extreme Nash Equilibrium of Polymatrix Games in Electricity Market

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1 Extreme Nash Equlbrum of Polymatrx Games n Electrcty Market Kalash Chand Sharma, Roht Bhakar and Harpal Twar Department of Electrcal Engneerng, Malavya Natonal Insttute of Technology, Japur, Inda Faculty of Engneerng & Desgn, Unversty of Bath, Bath, UK kalashsharma3889@gmal.com, r.bhakar@bath.ac.uk, harpaltwar@yahoo.co.n Abstract Game theoretcal approaches are wdely used for the analyss of olgopolstc electrcty markets. Nash equlbrum s a soluton concept of game theoretcal approaches. Due to exstence of mxed strategy equlbrum and large number of multple players, fndng Nash equlbrum for problems n electrcty market s a dffcult task. To resolve these dffcultes, ths paper proposes a smplfed approach for fndng extreme Nash equlbrum, based on payoff matrx approach and mxed nteger lnear programmng (MILP). To llustrate the proposed approach, a practcal case study of Cournot poly-matrx game s consdered. Elmnatng constrants are appended on the proposed approach to fnd a global optmal soluton. Obtaned results show the strength of proposed approach, n terms of smplcty and computatonal tme. Index Terms Electrcty market, poly-matrx game, Nash equlbrum, payoff matrx approach, mxed-nteger lnear programmng. I. INTRODUCTION Electrc power sector s beng restructured for mprovng operatonal effcency and enhancng consumer benefts. Ths ntroduces competton between generaton companes and retalers. In a fully compettve market, market clearng prce s always close to generator s margnal cost and there s no scope for market power. However, electrcty markets are olgopolstc n nature, due to lmted number of market partcpants and other operatonal constrants. In olgopolstc electrcty markets, strategc behavor of domnatng generaton companes ncreases ther expected proft. Therefore, market prce s hgher than expected, and s undesrable. To prevent domnatng generaton companes to abuse market power, forecastng of future market outcomes s essental []. Game theory s an approprate tool to model strategc behavor of domnatng partcpants n the market. Each strategc frm n the game acts as a player, and has a set of strateges. Players select the strategy to be offered n the market, whch gves them maxmum payoff, dependng on other players strateges. Among the varety of game theoretcal models, Cournot and Supply functon equlbrum are the most popular for modelng olgopolstc electrcty markets []. Soluton of any game theoretcal model s Nash equlbrum s a stable pont of game where none of the player s nterested to change ts strategy unlaterally. Ths paper ams to fnd Nash equlbrum for a game that represents electrcty market n an effcent and smplfed manner. Due to transmsson constrants, and nonlnear and nonconvex payoff, fndng Nash equlbrum for games related to electrcty markets are dffcult. Varous approaches proposed to fnd Nash equlbrum are broadly classfed nto three categores: mathematcal programmng approaches [3-5], coevolutonary approaches [6-7] and payoff matrx approaches [8-0]. In mathematcal programmng approaches, equlbrum problem s formulated as an optmzaton problem, and then solved usng teratve and optmzaton technques. Cournot Nash equlbrum n both blateral and pool based electrcty market s obtaned by lnear complementarty approach [3]. Indvdual strategc bddng problem formulated as a mathematcal program wth equlbrum constrants can be solved by nteror pont algorthm [4]. Equlbrum problem wth equlbrum constrants transform nto mxed nteger lnear programmng problem usng bnary expanson approach [5]. Ths type of approach can be used for fndng multplayer game Nash equlbrum. However, teratve technques have convergence dffcultes and optmzaton technques may provde local optmal soluton that s not a true Nash equlbrum. In addton, these types of approaches are sutable only for fndng pure strategy equlbrum. The co-evolutonary programmng approaches employ artfcal ntellgence based optmzaton technques for fndng Nash equlbrum [6-7]. These approaches effectvely deal wth transmsson constraned non-convex payoff. However, co-evolutonary programmng approaches are desgned for fndng only pure strategy equlbrum and not for mxed strategy equlbrum. Payoff matrx approaches can fnd global soluton and are sutable for both pure and mxed strategy equlbrum. To reduce the sze of payoff matrx, contnuous strateges must be The frst author acknowledges fnancal support by UGC Grant No /00 (SR). [ /4/$ IEEE]

2 dscretzed. Several heurstc based on payoff matrx approach have been used to solve three-player Cournot and Bertrand game [9]. Nash equlbrum condton of payoff matrx s characterzed by the polynomal equatons that are solved by homotopy contnuaton algorthm [0]. Approaches proposed n [9-0] are complcated, ntutonal and do not provde optmal Nash equlbrum. Ths paper presents a soluton method to fnd extreme Nash equlbrum of poly-matrx games usng payoff matrx approach and mxed nteger lnear programmng (MILP). Nash equlbrum s extreme because t s expressed as a soluton of optmzaton problem. Lnearzaton technque s employed to transform the equlbrum problem nto MILP problem. Two numercal examples, one nvolvng b-matrx game and other poly-matrx game, are consdered to test the proposed approach. Results obtaned from the proposed method are verfed wth approaches proposed n [9] and GAMBIT [5]. Rest of the paper s organzed as follows: In Secton II, the basc poly-matrx game formulaton and Nash equlbrum are descrbed. Secton III provdes MILP formulaton of payoff matrx approach and the smulaton procedure. Secton IV ncludes numercal and graphcal results of testng the proposed approach through a comprehensve analyss on bmatrx and poly-matrx game. In Secton V, relevant conclusons are drawn. II. PROBLEM FORMULATION A. Polymatrx Cournot Game A poly-matrx game s a normal form mult-player noncooperatve game. If players compete n game based on ther producton quantty, games become poly-matrx Cournot game. In ths type of game, each player (Frm) selects ts producton level assumng known and constant behavor of rval players. Let N = {,,..., n} be a set of players that partcpate strategcally n electrcty market. The cost functon of each frm s expressed as: C( Pg) = apg + bpg + C( a > 0; N) () where, Pg s the output power of th power producer n MW, and a, b, c are the cost coeffcents. Players select ther offered power between mnmum P g and maxmum Pg generaton lmt n MW defned as: Pg Pg Pg N () The sum of offered power of each frm s assumed equal to the total market demand, neglectng transmsson losses such that P Pd = Pg (3) Market prce λ n $/MWh depends on total demand and s unform for each node of the system. The relatonshp between prce and demand s represented by nverse lnear demand curve, and s expressed as: λn = θ ρpd (4) where, θ ( $/MWh) and ρ ( $/MW h) are demand beneft coeffcents. Margnal cost of generators can be defned as: ( ) MC Pg apg b = + (5) Margnal cost of generators s equal to the market clearng prce, when no generaton capacty constrants and transmsson constrants are consdered. B. Equlbrum Problem Formulaton Let us consder that each player has a fnte set of pure k strateges S = { s, s,... s }, where, k s the total number of strateges. Payoff of player for each strategy s calculated usng the followng expresson (,,... k ) ( ) λ a s s s = npg C Pg (6) Payoff matrx A conssts of payoff for player for each strategc combnaton. If player chooses hs strategy s k and player j chooses hs strategy l k l s j, a partal payoff aj ( s, sj ) s assgned for player. Therefore, for any pure strategc choce ( k,..., l s s n ) of the n players, overall payoff of player at the end of game s A( s k,... sn l ) = aj ( s k, s l j) (7) For a mxed strategy game, let X be a probablty vector over the set of pure strategy S havng followng condton: { : t, 0} S = X e X = X (8) At the end of poly-matrx game, overall payoff of player s R ( ) ( ) T X = X AjX j (9) In a poly-matrx game, tuple X * = X *,..., Xn * of the mxed strategy s called Nash equlbrum f and only f, for any other n tuple X * = ( X *,..., X *, X, X * +..., Xn * ), satsfy followng condton: * T * T * ( X ) AjX j ( X) AjX j N (0) The proof of pure and mxed strategy Nash equlbrum exstence n fnte mult-player games can be found n [- ]. n ( )

3 III. PROPOSED SOLUTION APPROACH A. MILP Formulaton The problem of poly-matrx Nash equlbrum s expressed n (0). Ths s transformed to MILP problem usng lnearzaton technques [3-4]. The payoff maxmzaton objectve of player shown n ( 8) and ( 9) can be transformed nto lnear mnmzaton objectve usng dualty theory. mn { γ : e γ AjX j} γ () where, γ s a dual objectve varable of player, and equal to ts prmal payoff γ ( ) T = X AjX j. The fnal optmzaton problem to fnd Nash equlbrum can be formulated as follows: mn γ γ, X N Subject to t ex = (3) N γ e AjX j MU 0 (4) j N: X + U e (5) U = { 0,} (6) X 0 (7) X j 0 (8) γ (9) s equal to the sum of players Objectve functon ( ) expected payoff. Constrant ( 3) ensures that sum of probablty of mxed strategy s always equal to one. Equaton ( 4) consttutes the complementarty slackness condtons of objectve. In ths condton, M s constant, and should be large enough, as obtaned by the followng expresson: M = max aj mn aj (0), j I: j, j I: j Inequalty constrant ( 6) shows that the sum of pure mxed strategy varable and bnary varable s always less than unty. The soluton of formulated MILP problem n ()- (9) may be local optma. To fnd global optmal soluton, elmnate constrants added on formulated MILP problem. N + U{} U{ 0} L N () L = N M N + () Ths constrant elmnates the bnary varable combnatons found n ntal equlbrum soluton. B. Smulaton Procedure Ths secton descrbes the smulaton procedure used to obtan extreme Nash equlbrum by the proposed approach. Step : Model ntalzaton: Defne the number of generaton companes that partcpate as a strategc player n the electrcty market poly-matrx game. Collect ther cost parameters and respectve demand beneft coeffcents. Step : Defne strategy set: Contnuous pure strategy set of players s defned between ther maxmum and mnmum capacty. Usng dscretzaton contnuous strategy set transform nto dscrete strategy set. Step 3: Payoff matrx constructon: For each strategy combnaton, market-clearng prce and players proft s calculated usng ()-(6). Then construct a payoff matrx of each player to have ther proft for all strategy of opponents. Step 4: MILP formulaton: When all players payoff matrx s constructed, then transpose equlbrum problem n MILP usng ()-(). Step 5: Nash equlbrum: Soluton of MILP usng any commercal MILP solver provdes Nash equlbrum. Step 6: Add elmnaton constrants: To check whether obtaned Nash equlbrum s local or global optmal, add elmnaton constrants ()-() on formulated MILP n ()-(9). Agan, run MILP solver to obtan new Nash equlbrum. Step 7: Check global optmal: If new Nash equlbrum obtaned n Step 6 s equal to old Nash equlbrum obtaned n Step 5, ths means global optmal equlbrum s acheved, then go to next Step. If new Nash equlbrum dffers from old Nash equlbrum, t means soluton s local optmal, so go to Step 6. Step 8: End IV. CASE STUDY To llustrate the proposed soluton approach, two cases are consdered. Case I s a b-matrx Cournot game and Case II s a poly-matrx Cournot game. A. Case I: Bmatx Cournot Game In ths case, two generators G and G compete n olgopolstc market to meet demand. The strategc behavor of generators s modeled as a strategc player n a b-matrx Cournot game. Fg. shows system confguraton and provdes nformaton about generators locaton and network constrants. Fgure. Two bus system.

4 In ths case, generators mnmum and maxmum generaton capacty s consdered as 0 MW and 00 MW respectvely. The margnal cost of generators s assumed, as $0/MWh for generator G and $0/MWh for generator G. Players dscrete strategy set are formulated by an ncrement of 0 MW from generators mnmum to maxmum lmt. Therefore, each generator has 0 strateges to offer n the market. Demand s equal to the sum of generators offered power. Expected Payoff ($/h) G G Strategy Fgure. Generators expected payoff. Demand beneft coeffcents θ and ρ are equal to 00 and 0.5 respectvely. The sze of payoff matrx for ths game s0 0. The player s expected payoff corresponds to ther pure strateges, as shown n Fg.. Large coeffcents M and M values are equal to 4600 and 4800 for player G and G payoff, respectvely. Nash equlbrum obtaned by the proposed approach results n an offer of 60 MW for G and 50 MW for G, n the absence of network lmts. B. Case II: Polymatrx Cournot Game In ths case, three generators G, G and G 3 strategcally nteract n olgopolstc electrcty market. Ther strategc nteracton represents a poly-matrx Cournot game. For ths case, system confguraton s shown n Fg. 3. The transmsson lnes are assumed to be lossless and have equal reactance. Lne flow s obtaned usng DC power flow soluton. Detals about generators cost coeffcents are provded n Table I. TABLE I. GENERATORS COST COEFFICIENTS (SOURCE: TABLE, [8]) Generator G G G 3 a $/MW h Cost coeffcent ( ) Cost coeffcent b ( $/MWh) Cost coeffcent c ( $/h) Demand beneft coeffcents θ and ρ are and 0.006, respectvely. Mnmum and maxmum lmt of each generator s consdered as 600 MW and 500 MW, respectvely. The dscrete strategy set of each player s formulated by dscretzaton of 00 MW n the range 600~500 MW. The sze of payoff matrx for ths game s The expected player s payoff correspondng to ther pure strateges s shown n Fg. 4. Large coeffcents M, M and M 3 are equal to , and for player G, G and G 3 payoff, respectvely. Nash equlbrum obtaned by the proposed approach results n G offerng 00 MW, G offerng 000 MW and G 3 offerng 000 MW n the absence of network lmts. The obtaned results are smlar to unconstraned case of [9]. Expected Payoff ($/h).4 x G G G Strategy Fgure 4. Generators expected payoff. Fgure 3. Three bus system. To mnmze computatonal effort, payoff matrx constructon and power flow operaton s performed n MATLAB platform. MILP problem have been solved usng CPLEX.0 under GAMS. All smulatons are performed on a Wndows based Personal Computer,.73 GHz processor and.0 GB RAM.

5 TABLE II. COMPUTATIONAL TIME REQUIREMENTS Approach Dmenson Computaton Tme (Sec.) Proposed Approach GAMBIT [5] Comparson of computatonal tme taken by the proposed approach wth others s shown n Table II. Vsualzaton of ths table shows that proposed approach takes only 0.09 seconds wthout elmnaton of domnated strateges. Soluton of Case I b-matrx game s obtaned n 0.06 seconds. V. CONCLUSION The competton between strategc power producers n an olgopolstc electrcty markets s generally modeled through game theoretcal or equlbrum models. Due to presence of multple strategc power producers n electrcty markets fndng Nash equlbrum s dffcult task. Ths paper proposes a payoff matrx and MILP based soluton approach for fndng extreme Nash equlbrum of a poly-matrx game. Proposed approach has been tested on case studes of bmatrx and poly-matrx game. Smplcty, solvablty and less computatonal tme are the advantages of proposed approach. Implementaton on larger systems, wth sze reducton technques, s a major aspect to be nvestgated n future. REFERENCES [] X. Guan, Y. Ho and D. L. Pepyne, Gamng and prce spkes n electrc power markets, IEEE Trans. Power Syst., vol. 6, no. 3, pp , Aug. 00. [] R. Baldck and W. Hogan, Capacty Constraned Supply Functon Equlbrum Models of Electrcty Markets: Stablty, Non-Decreasng Constrants, and Functon Space Iteratons, Unv. Calf. Energy Inst., Dec. 00. [Onlne]. Avalable: pwp089.pdf. [3] B. F. Hobbs, Lnear complementarty models of Nash-Cournot competton n blateral and POOLCO power market, IEEE Trans. Power Syst., vol. 6, no., pp. 94 0, May 00. [4] B. F. Hobbs, C. B. Metzler, and J. S. Pang, Strategc gamng analyss for electrc power systems: An MPEC approach, IEEE Trans. Power Syst., vol. 57, no., pp , May 000. [5] M. V. Perera, S. Granvlle, M. H. C. Fampa, R. Dx and L. A. Barroso, Nash equlbrum n strategc bddng: a bnary expanson approach, IEEE Trans. Power Syst., vol., no., pp , May 006. [6] H. Chen, K. P. Wong, D. H. M. Nguyen, and C. Y. Chung, Analyzng olgopolstc electrcty market usng coevolutonary computaton, IEEE Trans. Power Syst., vol., no., pp. 43 5, Feb [7] T. C. Prce, Usng co-evolutonary programmng to smulate strategc behavor n markets, J. Evol. Econ., vol. 7, pp. 9 54, 997. [8] L. B. Cunnngham, R. Baldck, and M. L. Baughman, An emprcal study of appled game theory: Transmsson constraned Cournot behavor, IEEE Trans. Power Syst., vol. 7, no., pp. 66 7, Feb. 00. [9] K. H. Lee and R. Baldck, Solvng three-player games by the matrx approach wth applcaton to an electrc power market, IEEE Trans. Power Syst., vol. 8, no. 4, pp , Nov [0] Y. Yang, Y. Zhang, F. L and H. Chen, Computng all nash equlbra of multplayer games n electrcty markets by solvng polynomal equatons, IEEE Trans. Power Syst., vol. 7, no., pp. 8 9, Feb. 0. [] J. Nash, Non-cooperatve games, Ann. Math., vol. 54, no., pp , 95. [] D. Fudenberg and J. Trole, Game Theory. Cambrdge, MA: MIT Press, 99. [3] C. Audt, S. Belhaza and P. Hansen, Enumeraton of all extreme equlbrum n game theory: bmatrx and polymatrx games, J. Optmz. Theory App., vol. 9, no. 3, pp , Jun [4] M. R. Hesamzadeh and D. R. Bggar, Computaton of extremal-nash equlbra n a wholesale power market usng a sngle-stage mlp, IEEE Trans. Power Syst., vol. 7, no., pp , Aug. 0. [5] GAMBIT [Onlne]. Avalable: