Electric Power Systems Research 57 (2001) 133–139www.elsevier.com/locate/epsrPricing and operation in deregulated electricity market bynoncooperative gameL. Geerli a, *, L. Chen b , R. Yokoyama aa Tokyo Metropolitan Uniersity, Minamiosawa 1-1, Hachioji, Tokyo 192-03, Japanb Osaka Sangyo Uniersity, Nakagaito 3-1-1, Daito, Osaka 574-8530, JapanReceived 7 June 2000; received in revised form 3 October 2000; accepted 9 October 2000AbstractPricing structure is becoming considerably important for both electric utility industries and their customers. This paper derivesan operation rule for a market model with an electric utility and independent power producers (IPPs) as players of thenoncooperative game. The derived operation rules reflecting the competition can be viewed as an extension of the conventionalequalizing incremental cost method for the deregulated power systems. As indicated in this paper, the prices of electricity forpurchases and sales are equal to the incremental costs of the generators of IPPs but are generally cheaper than the incrementalcost of the generators belonging to the utility. To examine the proposed approach, several systems are used as the demonstratedexamples in this paper. © 2001 Elsevier Science B.V. All rights reserved.Keywords: Game theory; Equalizing increment cost; Pricing; Power systems; Deregulation; Market1. IntroductionElectricity markets are experiencing widespreadchanges that are significantly altering the industry.When new economic mechanisms are being consideredfor use in the electricity market, there is much talk of‘gaming’ and whether the mechanism will ‘be gamed’. Due to the emergence of independent power producers(IPP), as well as the changing structure of theelectricity supply industry, the electric power industryhas entered an increasingly competitive environmentunder which it becomes more realistic to improve economicsand reliability of power systems by enlistingmarket forces [4,8,9]. For example, to maximize hispayoffs, a player (e.g. a utility) seeks to displace expensivegeneration by importing power from neighboringplayers with lower cost energy. Likewise, a player (anIPP or a utility) with excess generation capacity canchoose to export power and receive an immediate returnon its investment . Up to now, a considerablenumber of literatures addressing the competition andderegulation issues have been published [4,8,7,9–12].* Corresponding author.This paper aims to propose a game theoretical negotiationmethod to provide rational decisions of the powerexchange for purchases and sales between the utilitiesand IPPs, as well as prices of the electrical energy undercompetitive environment, in contrast to the conventionalequalizing incremental cost rule. Generally, thegame theory is divided into cooperative game andnoncooperative game, depending on whether or not thedecisions are based on the unanimous agreementamong all players [5,10]. For the noncooperativegames, the Nash equilibrium solution, saddle solution,minimax principle, Stackelberg strategy and others aremostly used to make the decisions, in contrast to coresolution, nucleolus, Shapley value and Nash bargainingsolution, which are the solutions of cooperative games[5,6]. Although the solutions of cooperative games belongto the Pareto set from the viewpoint of socialwelfare, they generally are not easily realized due torequirement of unanimous agreement among all playersunder competitive environment, comparing with decisionsof noncooperative games, which are actually compromisesolutions for players.It is well known that the equalizing increment costrule for a simplified network model is not only simplebut also optimal due to the optimality of social welfare,0378-7796/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0378-7796(01)00094-3
134L. Geerli et al. / Electric Power Systems Research 57 (2001) 133–139as far as each participant in operations of a powersystem is completely cooperative . However, for thederegulated power systems, this rule may not holdbecause participants intend to pursue their ownbenefits, thereby may not always take cooperative attitude.In this paper in order to compare with theequalizing incremental cost rule, we mainly use theStackelberg strategy and the Nash equilibrium solutionto analyze the negotiation process between utilities andIPPs since they are widely recognized as rational decisionsfor competitive markets in terms of axiom [5,10].In Section 2, we first derive a common price ofelectricity for purchases and sales between a utility andIPPs when assuming that all of IPPs construct a coalitionwho negotiates with the utility for bargaining. Bythe Stackelberg strategy , the Section 2 also providesvery clear and simple operation rules, which are similarto the well-known equalizing incremental cost  butdifferent due to the fact of competitive environment. InSection 3, we give the prices of electricity, as well as theoperation decisions for the situation when each IPPnegotiates with the utility independently, which meansthat IPPs sell their electricity at different prices dependingon their negotiations with the utility. In Section 4,a Nash equilibrium  is used to analyze the negotiationbetween IPPs and the utility, which shows that theNash equilibrium and the Stackelberg strategy have thedifferent solutions when assuming that the purchasedelectric energy depends on its price (or is a function ofthe price). Section 5 extends the results of the earliersections by taking account of transmission network. Aswill be shown in this paper, all obtained results can beviewed as an extension of the equalizing incrementalcost from regulated power systems to competitivepower markets. In Section 6, we derive an optimalsolution by maximizing total welfare of electric powerindustry society, for which the operation rule obviouslycoincides with that of the conventional equalizing incrementalcost. To demonstrate the proposed model, severalexamples are simulated and used to compare withthe conventional approaches  in Section 7. Finally,we give several general remarks to conclude this paperin Section 8.2. Negotiation between a utility and a coalition ofIPPsThis section derives the price of electricity and operationdecisions on the basis of the noncooperative game by supposing that all of IPPs form a coalition whojointly negotiates with a utility. That means all of theelectricity sells at the same price from IPPs to theutility. Since the price of electricity for sales directlyinfluences the payoffs of IPPs, it is reasonable for theutility to assume that the quantity of the purchasedelectricity energy from IPPs depends on its price. Thebasic structure of power market in this paper is thatthere are a utility with transmission network and severalIPPs without the transmission network. Note thatall results of this paper can straightforwardly be extendedto the multi-utility negotiation game, althoughonly one utility is analyzed in this paper for the sake ofsimplification .Let Z=(z 1 ,…,z m ) be supply power vector of theutility, where z j ,forj=1,…,m stands for real poweroutput (MWH) of generator-j, andm is total number ofgenerators for the utility. Define g j (z j ) for j=1,…,m asthe incremental cost ($ per MWH) of generator-j of theutility. Then z j0g j (z)dz is the cost function ($) for generator-jof the utility. Assume that there are n IPPs anddefine X i =(x i1 ,…,x ini ) to be supply power of IPP-i=1,…,n, where z ij for j=1,…,n i represents real poweroutput (MWH) of generator-j and n i is the total numberof generators for the IPP-i. Letf ij (x ij )fori=1,…,n;j=1,…,n i be the incremental cost ($ per MWH) ofgenerator-j for the IPP-i, and f ij /x ij 0. Then x ij0 f ij (x)dx is the cost function ($) for generator-j of theIPP-i. Since this section analyses, the negotiation betweena coalition of IPPs and a utility, all of electricityis assumed to be sold at the same price from IPPs to theutility. Then let y be the price ($ per MWH) (or theincremental cost) at which the utility purchases theelectricity from IPPs. Let d be total demand (MWH) ofconsumers and h(d) is earning ($) of the utility fromselling electricity to customers and is not related withZ,X i . Then IPPs will sell electric power to utility bymaximizing their profits, which can be formulated asthe following problems [3,15]:IPP−i:(i=1,…,n)nMaximize y i nj=1 x ij − ij=1x ij ;j=1,…,n ix ij0f ij (x)dx (1)where x ij0 f ij (x)dx is the cost function ($) for generator-jof the IPP-i, and IPP-i is assumed to decide x ij , j=1,…,n.On the other hand, the utility intends to maximize itspayoffs by utilizing existing generators and lowering thepurchasing price of electricity from IPPs, which can bedescribed as the following problem:Utility:Maximizey,z j ; j=1,…mn nh(d)−y i=1 i mj=1 x ij (y)− j=1z jj=1g j (z)dz(2)n ns.t. i=1 ij=1 x ij (y)+ mj=1 z j =d (3)where Eq. (3) means that the utility has the obligationto serve all customers’ load, and the utility is assumedto decide y, z i , j=1,...,m. Note that we use a presumedcondition in Eqs. (2) and (3), i.e. the purchased electricpower x ij (y) is supposed to be a function of y. Since the
L. Geerli et al. / Electric Power Systems Research 57 (2001) 133–139 135price of electricity for sales directly influences the payoffsof IPPs, it is reasonable for the utility to assumethat the quantity of the purchased electricity energyfrom IPPs depends on its price. Notice that x ij (y) isnota predetermined function but an implicit function of y,which is decided by the game at the equilibrium.The optimality conditions for Eq. (1) is straightforwardby differentiating the objective function with respectto x ij , i.e.y=f ij (x ij ) (i=1,…,n;j=1,…,n i ) (4)According to the Implicit Function Theorem and Eq.(4), there exists a differentiable function x ij (y) attheneighborhood of x 0 when df/dx0. DifferentiatingEq. (4) again, we have the following relationsdx ijdy = 1(i=1,…,n;j=1,…,ndf ij (x ij )/dxi ) (5)ijOn the other hand, for Eqs. (2) and (3) of the utility,we establish a Lagrangian functionnL(Z,y,)=h(d)−y n+ n ii=1j=1mx ij (y)+ n ii=1j=1j=1mx ij (y)− j=1z j0g j (z)dzz j −d n (6)Note that h(d) is a constant, then the optimalityconditions for Eqs. (2) and (3) can be obtained byL/z j for j=1,…,m, L/y and L/=g j (z j ) (j=1,…,m) (7)ni=1ni=1n ij=1n ij=1nx ij (y)+(y−) mx ij (y)+ j=1n ii=1j=1dx ij=0 (8)dyz j =d (9)Substituting Eq. (5) into Eq. (8) and combining Eq. (4)with Eq. (9), we have the solution y=f ij (x ij ) (i=1,…,n i ) (10)=g j (z j ) (j=1,…,m) (11)−y=ni=1n ij=1n n i=1 ij=1 x ijn n i=1 i j=1 1/df ij /dx ij(12)mx ij + j=1z j =d (13)which are actually conditions of the equilibrium of thegame.Observing Eqs. (10) and (11), the generators for IPPsand the utility obviously have the same incrementalcost y and , respectively. As shown in Eq. (12),however, their incremental costs are different due tocompetition between the utility and the coalition ofIPPs. Since df ij /dx ij is assumed to be positive, theincremental cost for utility is generally larger than thatfor IPPs according to Eq. (12). That implies that thegenerators of IPPs have to supply power economicallyat the operation point, comparing to those of the utilitybecause the utility owns the transmission network andis generally burdened with the obligation to serve thecustomers’ loads. The price y of electricity for purchasesand sales is taken as the same value of theequalizing incremental cost for IPPs as indicated in Eq.(10). It is evident that the discrepancy between the twoincremental costs narrows if df ij /dx ij have small value,which means that the IPPs can benefit more if theyhave generators with smaller f ij /x ij (i.e. flat cost function).Therefore, Eqs. (10)–(13) can be viewed as anextended equalizing incremental cost rule for the deregulatedpower systems.Note that we can have similar equations as Eqs.(10)–(13) when taking the upper and lower limits ofeach variable into consideration, even though moreLagrangan multipliers have to be introduced. In thispaper, we assume that the utility decides the purchasedprices based on the outputs of IPPs while IPPs determinetheir outputs depending on the purchased prices.Therefore, this model mainly targets the system wherethe utility is large enough comparing to each IPP andalmost dominates the market.3. Negotiation between a utility and individual IPPsThis section gives a noncooperative solution by supposingthat each IPP individually negotiates with theutility . That means each IPP sells its electricity to theutility at the price different from those of other IPPs.Let Y=(y 1 ,…,y m ) where y i for i=1,...,m represents theprice at which the utility purchases the electricity fromIPP-i. All other definitions are the same as those inSection 2.Assuming that IPP-i and the utility decide x ij ( j=1,…,n j )andY,Z, respectively, then we have the followingproblems for IPPs:IPP−i:(i=1,…,n)and for the utility:Utility: MaximizeY,Zm− j=1z j0Maximizex ij ;j=1,…n iy i j=1g j (z)dzn i nx ij − ij=1x ijn nh(d)− i=1 y i ij=1 x ij (y i )n ns.t. i=1 ij=1 x ij (y i )+ mj=1 z j =d0f ij (x)dxHence in the same way as the derivations of theSection 2, we have the following conditions for anoncooperative solution
136y i f ij (x ij ) (i=1,…,n;j=1,…,n i ) (14)=g j (z j ) (j=1,…,m) (15)−y i =ni=1n ij=1n ij=1 x ijL. Geerli et al. / Electric Power Systems Research 57 (2001) 133–139(i=1, …,n) (16)n i j=1 1/df ij /dx ijmx ij + j=1z j =d. (17)Observing Eq. (15), it is evident that the generatorsfor the utility have the same incremental cost , whichcoincides with the conventional results. However, accordingto Eq. (14), the generators for IPPs have differentincremental costs because IPPs individuallynegotiate with the utility by using their own strategies,while the equalizing incremental cost is still hold for thegenerators within an IPP. The price y i of electricity forpurchases and sales are also different for each IPP fromEq. (14). Eq. (16) indicates that the flatter the fuel costfunctions of the generators for IPP-i is (i.e. the smallerdf ij /dx ij is), the less the discrepancy of the incrementalcosts between the utility and IPP-i is liable to be. Onthe other hand, Eqs. (14) and (16) also show that thecheaper the incremental cost of generators for the IPPis, the more an IPP can sell power. Note that we canderive a similar non cooperative solution when assumingevery generator in IPPs sells power at its price y ij ,analogue to the derivation of Eqs. (14)–(17). Therefore,Eqs. (14)–(17) can also be viewed as an extendedequalizing incremental cost rule for the deregulatedpower systems.4. Nash equilibriumIn this section, we only analyze the negotiation betweena utility and individual IPPs by using Nashequilibrium [5,3]. Therefore, we use the same definitiondescribed in Section 3.IPP-i assumes that the price y i is a function of theelectric power X i selling to the utility. Assuming thatIPP-i decides x ij for j=1,…,n j , then the strategy ofIPP-i can be formulated as:IPP−i:(i=1,…,n)ni− j=1x ij0nMaximize y i (X i ) ij=1x ij ;j=1,…,n if ij (x)dxOn the other hand, the utility assumes that thepurchased electric power x ij , from IPP-i is a function ofy i .Then the strategy of the utility can be described as:Utility:MaximizeY,Zx ijn nh(d)− i=1 y i ij=1 x ij (y i )zm− j=1 j0g j (z)dzn ns.t. i=1 ij=1 x ij (y i )+ mj=1 z j =dwhere the utility decides Y and Z.Supports y i /x ij 1/dx ij /dy i , we have the followingNash equilibrium solution .y i = 1n i j=1 x ijx ij0f ij (x)dx+n iji x ijy i (0)n i j=1 x ij(i=1,…,n;j=1,…n i ) (18)=g j (z j )(j=1,…,m) (19)−y i =ni=1n ij=1n ij=1 x ij(i=1,…,n) (20)n i nj=1 i j=1 x ij /f ij −y imx ij + j=1z j =d (21)where y i (0) is the electricity price when X i =0.The equations above are obviously different from theStackelberg strategy shown in Eqs. (14)–(17). For theStackelberg strategy, as indicated in Eq. (14), eachgenerator in IPPs operates at equalizing incrementalcost, which is also taken as the purchased price, whilethe purchased prices in the Nash game are determinedby not only the incremental costs but also outputs ofgenerators as shown in Eq. (18). Besides, comparingEq. (16) with Eq. (20), the discrepancies of the incrementalcosts between the utility and IPPs are alsodifferent for the Stackelberg strategy and the Nashgame. Although this section only considers the negotiationbetween the utility and individual IPPs, the Nashequilibrium solution can be derived as the same way forthe negotiation between the utility and a coalition ofIPPs by assuming y/x ij 1/dz ij /dy.5. Consideration of transmission networkThis section only deals with the negotiation betweenthe utility and individual IPPs. Assume that only theutility owns transmission network. Let H(Z,X 1 ,...,X n )be a transmission loss function (MWH), then we havethe following noncooperative game [3,15]:IPP−i: Maximize y i j=1(i=1,…,n) x ij ;=1,…,n iUtility:m− j=1z j0MaximizeY,Zg j (z)dzn i nx ij − ij=1x ijn nh(d)− i=1 y i ij=1 x ij (y i )0f ij (x)dx
L. Geerli et al. / Electric Power Systems Research 57 (2001) 133–139 137Table 1Coefficients of generators for 8-generator power systemGenerator numberab c Capacity (MW)Utility Unit 1 0.002 11 105 660Unit 2 0.002 11 105 660Unit 3 0.0019 10 100 960Unit 4 0.0019 10 100 960IPP IPP 1 0.0026 9 100 435IPP 2 0.0027 9 100 420IPP 3 0.0028 10 100 310IPP 4 0.0029 10 100 3006. Socially optimal behaviorNext, we will show that the conventional equalizingincremental cost rule optimizes the social welfare and isalso a Pareto solution. Consider a power industry societyincluding a utility and several IPPs. Then the problemto maximize the welfare of the society can beformulated as follows :Society:x ij0MaximizeX i ,Z;i=1,…,nxm jf ij (x)dx− j=10n nh(d)− i=1 ij=1g j (x)dzn ns.t. t=1 ij=1 x ij (y i )+ mj=1 x j −H(Z,X 1 ,…,X n )=d.n ns.t. i=1 ij=1 x ij (y i )+ mj=1 z j −H=dwhere H=H(z,X 1 (Y),...,X n (Y)). Then in the same wayas the derivations of the Section 3, we have a noncooperativesolution, which can be considered as an extendedequalizing incremental cost ruley i f ij (x ij )(i=1,…,n;j=1,…,n i ) (22)= g j(z j )( j=1,…,m) (23)1−H/z j1−n ij=1H/x ij /df ij /dx ij −y i =n i 1/df /dx ij ij j=1n ij=1 x ijn i j=1 1/df ij /dx ij(i=1,…,n) (24)ni=1n ij=1mx ij + j=1z j −H(Z,X 1 ,…,X n )=d. (25)Obviously, for the generators of the utility, Eq. (23)coincides with the conventional economic dispatch withthe consideration of the network loss. Generally, thetransmission loss H, which is expressed in quadraticforms of the generation powers can be estimated by theB-coefficient method .Thus the optimality conditions satisfy= g k(z k )= f ij(x ij )1−H/z k 1−H/x ij(i=1,…,n, j=1,…,n i ,k=1,…,m) (26)ni=1n ij=1mx ij + j=1z j −H(Z,X 1 ,…,X n )=d. (27)Eq. (26) is the equalizing incremental cost with theconsideration of the transmission loss for all of thegenerators in the society, which means that the bestsolution to the economic dispatch problem is the equalizingincremental cost if there is no competition butcooperation.7. Numerical simulation7.1. An 8-generator power systemA power system with 8 thermal generators shown inTable 1 is used to illustrate the application, where theutility owns four thermal generators and four IPPs haveone generator, respectively. This paper assumes that allof cost function for generators, i.e. z 0g j (z)dz=a j z 2 +b j z+c j or x 0f ij (x)dx=a ij x 2 +b ij x+c ij , has thequadratic forms, as shown in Tables 1 and 2. TheTable 2Coefficients of generators for 40-generator power systemGenerator numberabcCapacity (MW)Utility Unit 15 0.00211 105 325Unit 6150.0019 10100 610Unit 1620 0.0018 10 100 360Unit 2125 0.0029 9 100 635Unit 2630 0.0039100885IPP IPP 120.00269100235IPP 35 0.0027 9 100 370IPP610 0.0028 9 100 500
138L. Geerli et al. / Electric Power Systems Research 57 (2001) 133–139Table 3Simulation results for 8-generator power system aDemandCoalitionIndividual4500 (MW) Output (MW) Cost ($ per h) Benefit I ($ per Output (MW) Cost ($ per h) Benefit IC ($ per($ per h) MWH)($ per h) MWH)Utility 3206.5 38 918 4722 13.6 3051 36 813 4253 13.46IPP-1 434.1 4496.9 391.1 11.26 428.84437.7 378.1 11.23IPP-2 4184334 372.9 11.26 412.9 4277.1 360.5 11.23IPP-3 224.5 2486.5 41.7 11.26 308.8 3456.5 167.2 11.73IPP-4 216.8 2404.2 36.9 11.26 298.33340.7 157.9 11.73Total 4500 52 639.6 5564.6 450052 325 5316.7a IC: incremental cost.Table 4Simulation results for 40-generator power system aDemand Coalition Individual21 000 (MW)Output(MW)Cost($ per h)Benefit IC ($ per Output Cost Benefit($ per h) MWH) (MW) ($ per h) ($ per h)IC ($ perMWH)Utility Unit 15 1601.8 19 120.5 19 669.5 12.28 1648.3 19 743 20 307 12.32Unit 615 6003.3 67 885.4 73 725.4 12.28 6101.7 69 091 75 172.9 12.32Unit 1620 1779.8 21 272.5 21 855.5 12.28 1831.5 21 908.5 22 563.5 12.32Unit 2125 3155.2 34 073.5 38 746 12.28 3191 34 514 39 313 12.32Unit 2630 4401.2 42 780.5 54 046.5 12.28 4432.2 43 162 54 604.5 12.32IPP IPP 12 460.8 4623.4 4700.4 10.2638.2 6473.2 6803.2 10.66IPP 35 1099.1 10210 11 210.3 10.2 1079.6 10 018 10 969 10.16IPP 610 2498.5 21 834.5 25 484.7 10.2 2077.6 17 655.7 20 069.5 9.66Total21 000 221 850.3 249 438.3 21000 222 560.4 249 802.6a IC: incremental cost.simulation results are summarized in Table 3. Accordingto Table 3, the electricity prices of IPPs are lowerthan the operation cost of the utility because the utilityhas the obligation to serve all customers load. Whilethe total outputs of generators for IPPs and the utilityare the same for both coalition and individual. Actually,it can be proven that total outputs of IPPs are thesame for the negotiations of Sections 2 and 3 as far asthe cost functions x j0 f ij (z)dx and z j0g j (z)dz arequadratic. For the sake of the simplification, let h(d)=d where =80 ($ per MWH) is the electricity rate toconsumers.Since the df ij /dx ij IPP-1 are smaller than those ofIPP-2, IPP-3, IPP-4, the IPP-1 supplies more power tothe utility, especially for the case of the ‘individual’according to Table 3. The results also show that thetotal benefit for the case of the coalition is larger thanthe case of the individual due to their jointly bargainingwith the utility.7.2. A 40-generator power systemThe proposed method also applied to a 40-generatorpower system with one utility and 10 IPPs. The utilityhas 30 generators and 10 IPPs own one generator,respectively. Table 4 shows the simulation results whereh(d)=80d is used. In addition to the price y, obviouslythe earning of the utility also depends on the h(d) toconsumers. According to Table 4, the total outputs ofgenerators for IPPs and the utility are the same forboth coalition and individual. The incremental costs ofthe utility (12.8 and 12.32$ per h) are higher than theincremental costs of IPPs (10.2 and 10.66$ per h). Asshown in Table 4, total benefit for the case of theindividual is actually more than that for the case of thecoalition, but the benefit of each is not all increased dueto their different incremental costs.8. ConclusionsGame theory is a useful tool for the analysis ofelectric power market. The derived operation rulesreflecting the competition can be viewed as an extensionof the conventional equalizing incremental cost method.Besides, we also provided the prices of electricity forpurchases and sales, and as indicated in this paper,these prices are generally cheaper than the incremental
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