Generation strategies for gaming transmission constraints. Will the deregulated electric power market be an oligopoly?

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1 See dscussons, stats, and author profles for ths publcaton at: Generaton strateges for gamng transmsson constrants. Wll the deregulated electrc power market be an olgopoly? Conference Paper n Decson Support Systems February 1998 DOI: /HICSS Source: IEEE Xplore CITATIONS 89 READS 48 authors, ncludng: Marja Ilc Carnege Mellon Unversty 389 PUBLICATIONS 5,864 CITATIONS SEE PROFILE Avalable from: Marja Ilc Retreved on: 18 September 016

2 /98 $10.00 (C) 1998 IEEE Generaton Strateges for Gamng Transmsson Constrants Wll the Deregulated Electrc Power Market Be an Olgopoly? Zad Younes Technology and Polcy Program, Massachusetts Insttute of Technology Econome Industrelle, Unversté de Pars IX e-mal: Marja Ilc Department of Electrcal Engneerng and Computer Scence Massachusetts Insttute of Technology e-mal: Abstract Constraned transmsson lnes are known to be able to economcally solate submarkets from the competton of players located elsewhere on the network. Ths paper examnes the type of olgopolstc competton that s lkely to take place n these submarkets. It shows, usng smple models, how statc or ntertemporal Nash equlbra can rse n a framework of prce or supply functon compettons, found to be more realstc than Cournot models n the partcular case of short term competton n the electrc power market. Ths paper shows also how transmsson constrants can play a drect role n the outcome of the olgopolstc competton and encourage strategc behavor by the generators. Transmsson lnes that would not be constraned f the players dd not know of ther thermal lmts may be strategcally drven to operate at these lmts n order to maxmze the profts of the players who have market power, leavng the others to cope wth the consequences of such behavor. 1. Introducton The nodal prcng proposal advocated by Hogan n [5] has been frequently crtczed because t gnores the potental market power that the market partcpants can have n such a framework. Sngh et al. [10] show how, wth locaton-dependent nodal spot prcng and transmsson constrants, a non-dscrmnatng aucton mechansm creates opportuntes for strategc behavor. Nevertheless, the problem of market power les beyond the aucton mechansm or the features of the nodal prcng model. More generally, n any market structure n whch the generators determne ther bds, those generators n areas constraned by weak transmsson lnes should see ther market power boosted because they are solated, by the constrants, from the competton of other generators. As shown n [13], locatonal market power wll exst because of loop flow and transmsson constrants that produce geographcally and temporally localzed relevant markets. One sngle constraned lne n a meshed 4 bus system was shown to be suffcent to create a relevant submarket of three buses whle ths submarket was stll lnked to the rest of the network by 4 unconstraned lnes. Moreover, and as shown by the experence of deregulaton n the Brtsh electrc power market [1], global market power can also exst, transmsson constrants put apart, because of the relatve sze of compettors. Therefore, one should not assume, a pror, that the market s perfect, but rather acknowledge the potental exstence of market power n any form of proposed deregulaton and try to lmt ths power. In order to acheve ths goal, we wll have to replace our tradtonal assumptons of a perfect market wth more realstc olgopolstc models. The objectve of ths paper s to present some of the potental conceptual consequences of abandonng these perfect market assumptons. The next secton revews the bascs of olgopolstc competton and statc or ntertemporal Nash equlbra; readers famlar wth these concepts mght want to skp t. In secton 3, we dscuss the choce of the olgopolstc model best suted for electrc power systems. In secton 4, we expose some of the consequences of olgopolstc competton pror to accountng for transmsson constrants; we dscuss the potental emergence of tact colluson and determne condtons under whch ths behavor s not sustanable. In secton 5, we ntroduce transmsson constrants n the smple three bus power system models and show how they can nduce new types of strategc behavor.

3 /98 $10.00 (C) 1998 IEEE. Olgopolstc competton: generaltes In a dynamc game wth dscrete stages, a number of players are makng a strategc decson at every stage. The combned decsons of all players determne, through a mddleman or n a decentralzed market settng, the quanttes of good exchanged between the players as well as the prces at whch they are exchanged, and, therefore, the profts of the players for the stage. Let n be the number of players k the stage ndex x the strategc varable whose value player chooses n the set E x (k) the strategc decson of player at stage k π the one-stage proft functon of player, π (x 1 (k),..,x n (k)) s ts proft at perod k r the dscount rate between two consecutve stages X the vector [x 1,..,x n ] X - the vector [x 1,.., x -1,x +1,...,x n ].1 One stage competton In a one stage competton framework, player assumes that all other players are takng fxed decsons Xo -. He wll gve to hs own decson varable x the value xo that maxmzes hs proft. (1) π ( xo, Xo ) = Max π ( α, Xo ) α E Or, when x takes real values and π s dfferentable, π () ( xo, Xo ) = 0 x When π has contnuous second dervatves, the mplct functon theorem [9] ensures locally the exstence of a reacton functon R, whose partal dervatves exst and are contnuous, so that, π ( R( X ), X ) (3) = 0 where, x (4) R( X ) = x gves the optmal choce of player for X - fxed. Defnton 1: We say that X* s a statc Nash equlbrum f: π ( X*) (5), = 0 x At a statc Nash equlbrum, every strategc varable x * s the optmal choce of player for X - * fxed. X* s also an ntersecton pont of all reacton functons: (6), x* = R( X *) In an olgopolstc stuaton wth one stage only, the statc Nash equlbrum s lkely to be obtaned f all players can observe the decsons of all other players and change ther own decsons n real tme. If the bds are revealed only after the market clears, attanng ths equlbrum s unlkely and depends on the nformaton that every player has snce he wll make hs decson as a functon of what he expects the other decsons to be.. Dynamc process and stablty of statc Nash equlbra In a dynamc framework, the players learn at every stage more about the decsons of the others and the strategc varables may converge to the statc Nash equlbrum even f the bds are not revealed before the market clears at any gven stage. Assumng that a player knows the bets of other players n the last stage, not knowng what they wll be at the current stage, the smplest strategy would be for hm to assume that the bets are stayng as they were n the last perod and to maxmze hs proft accordngly. In ths case, (7), k, x( k + 1) = R( X ( k)) Ths set of equatons descrbes dynamcally how the bets at the current stage depend on the bets at the prevous stage. It s straghtforward to see that f the process defned n (7) converges, ts lmt s a statc Nash equlbrum. The stablty of ths equlbrum s an mportant queston. Studyng the stablty of an equlbrum requres determnng whether the system s able to converge to ths equlbrum ndependently of the ntal condtons (general stablty), and /or whether t can return to t after a small perturbaton (local stablty). When X* s a statc-nash equlbrum, (8), x( k+ 1) x* = R( X ( k)) R( X *) (9) + = + R, k, x( k 1) x * ( X *).( xj( k) xj*) + j xj OX ( k) X * From (3) we get: (10), ( π + = xx dx R j ) π x. x dx j 0 j j j When π s dfferent from 0 for all, equaton (9) x can be rewrtten as: j

4 /98 $10.00 (C) 1998 IEEE (11), k, ( x ( k + 1) x *) = π xx j ( x *).( xj( k) xj*) + OX ( k) X * j π x Lnearzng equaton (11) gves: k (1) ( Xk ( ) X*) = A.( X X*) where the elements of matrx A are, π xx j (13) aj =, j and a j = 0, = j π x Theorem: The stablty of the lnearzed system, and, therefore, the local stablty of the statc Nash equlbrum, s ensured when the egenvalues of matrx A are less than 1 n module. Furthermore, when equaton (11) s lnear, the same condton mples general stablty. Ths theorem s an mmedate applcaton of [1]..3 Intertemporal Nash strateges Nash statc equlbrum, although useful for analyzng many stuatons, has a major weakness n multstage games snce t does not recognze that the decson of a player at one stage mght affect the decsons of other players at further stages and that ths player mght take hs decson accordngly. In order to recognze ths nterdependence n multstage games, we can defne Nash equlbra n terms of strateges rather then n terms of strategc varables, where a strategy chosen by a player determnes hs decson at a stage as a functon of what the other players dd at past stages of the game. We say that the functon S s the ntertemporal strategy of player when the decson of player at the stage k s gven by (14) x (k)=s (k, X(0),...X(k-1)) If S=[S 1,...S n ] s the set of strateges of all players, ther strategc decsons at every stage wll be defned recursvely by the strateges S and the vector X at stage k wll be X(k,S). Defnton : We say that S*=[S* 1,...,S* n ] s an ntertemporal Nash equlbrum f every strategy S* maxmzes the dscounted sum of the profts of player gven the other strateges S* - =[ S* 1,. S* -1,S* +1..,S* n ] as fxed. Or, π( XkS (, *)) π( XkS (,, S* )) (15), = Max k + r S k ( 1 ) ( 1+ r) k 0 k When X* s a statc Nash equlbrum, stayng at X* at every perod (strateges S* where X(k,S*)=X*) s also an ntertemporal Nash equlbrum. It mght however not be the unque or even a Pareto-optmal 1 ntertemporal Nash equlbrum..4 Cournot competton, Bertrand competton and perfect markets A Bertrand competton s an olgopolstc framework where the strategc varables (x ) n are the prces (p ) n of the goods produced by the players, a statc Nash equlbrum s then called a Bertrand-Nash equlbrum. In a Cournot competton, the strategc varables (x ) n are the quanttes (q ) n of goods produced by the players, and an equlbrum would be a Cournot-Nash equlbrum. A perfect market s an extreme case of Cournot competton where the number of players s nfnte and the prce of the market p s an exogenous constant rather than a functon of the quanttes. In a perfect market, equalty (5) shows how every player wll produce the quantty that equalzes ts margnal cost wth the market prce. 3. Choosng the strategc varables The type of olgopolstc model that s adapted to study an olgopolstc market for electrc power depends on the future structure of ths market. In a centralzed market, where a Power Exchange (PX) takes the bds of the generators and loads and determnes what the physcal dspatch wll be, the type of competton s exogenous and depends on the bddng procedure. When the generators are bddng prces and the PX n decdng on quanttes, the competton wll be a Bertrand competton. When the generators are to bd ther producton levels as a functon of the prce they receve, the equlbrum prces wll then be gven by the supply functon equlbrum developed n [6] and appled to the Brtsh power market n [4].The equlbrum prce wll then le above those yelded by a Bertrand competton and beneath those gven by a Cournot competton n the unlkely case where the generators are bddng quanttes only. In a decentralzed market where the transactons are settled n blateral and multlateral markets, the type of competton s endogenous and probably not unque. 1 An equlbrum s Pareto-optmal f there s no other equlbrum that makes all the players better off or ndfferent. However, t mght be affected by the rules adopted by the PX to have the players respect the transmsson

5 /98 $10.00 (C) 1998 IEEE One of the classcal olgopolstc models s the Cournot model where the frms compete by choosng the quantty they want to put on the market and an ndependent auctoneer sets the prce that clears the market. Well adapted to study long-term competton and barrers for entry, Cournot competton models are useful n scenaros where the frms frst commt themselves to a producton capacty and compete next by choosng prces n a second perod snce the competton by prces n the second perod yelds the same results gven by a onephase game where the strategc varable chosen by the frms s ther output [7]. Therefore, the Cournot competton model mght be adapted to examne generaton competton n a long term strategc nteracton framework where the generators have to choose ther generaton capacty à la Cournot before competng à la Bertrand every day. Nevertheless, besdes some specfc cases (the ol market n certan perods of ts hstory, for example), competton by quanttes s farly unrealstc to analyze short-term competton. Ths s especally true n decentralzed multlateral and blateral electrc power markets where frms wll bd rarely for quanttes only and where the central auctoneer that sets the prce does not exst. One could argue n favor of usng Cournot competton to determne what happens on a daly bass snce t s supposed to gve the expected output of a Bertrand competton n the second perod. Nevertheless, ths reasonng makes two very strong and unrealstc assumptons: (1) the demand characterstcs should be the same n the second perod as those expected when the capacty choces were made (frst perod); () the demand should also be farly stable n the very short term and very close to ts average to enable the longterm expected outcome (the outcome of the Cournot competton) to be used to nterpret what happens on an hourly bass. Furthermore, when transmsson lnes are constraned 3 and generators are competng for ther use, t s unrealstc to consder that when takng ts strategc decsons, a generator wll not expect hs compettors to react mmedately to any change n hs own output. For all these reasons, ths model seems unable to gve valuable nsghts on the short-term (hourly, daly) competton n the generaton market. An alternatve s the classcal Bertrand olgopolstc competton model where the strategc varables are the prces that each competng frm chooses to maxmze ts constrants: Curtalment of output, surcharges for usng constraned lnes, etc. 3 We use the term constraned for congested. The latter term has only been used recently and s not a standard engneerng term. proft, consderng as fxed the prces of ts compettors. Under ths model, the equlbrum prce wll be the margnal cost of producton when the products are undfferentated, the frms can serve all the demand they face at a constant margnal prce, and the players play once. Ths result, combned wth the observaton that the prces on the Brtsh electrc power market are above margnal prces has sometmes been used to reject ths competton model n favor of a Cournot competton [8] Nevertheless, Edgeworth [3] has shown that, f no sngle frm can serve all the demand, the output of a Bertrand competton wth producton capacty constrants s no longer compettve and the equlbrum prce can go above the margnal cost; as reported by [11], ths result s vald n the more general context of prce competton between frms wth ncreasng margnal costs. The prce competton model can gve nterestng results f used n the context of generaton capacty constrants and ncreasng margnal costs, but these results must be nterpreted carefully snce every player assumes n ths model that the prce decsons of hs compettors are weakly lnked to hs own decson. The supply functon model where generators bd functons lnkng ther prces to ther output can consttute a credble alternatve and a good compromse between Cournot and Bertrand competton n a hghly decentralzed market where the prces of the transacton are not publc and where large transactons are lkely to be made at dfferent prces than smaller ones. In a decentralzed market, we are lkely to observe a combnaton of Bertrand competton and supply functon competton n dfferent geographcally and temporally localzed relevant submarkets. However, none of the models we dscussed takes nto consderaton the repeated nature of the nteractons between the players. When ths nteracton s perodc, and especally n a centralzed scheme where the prces are publc, ntertemporal Nash strateges where tact colluson could be enforced by retalaton threats s a credble alternatve that must be nvestgated. An example s gven n secton Market power n small networks 4.1 An example of supply curve equlbrum: exstence and stablty of statc Nash equlbrum To llustrate statc Nash equlbrum and ts dynamc stablty, we consder the case of an nelastc load L wth a demand q L buyng power from two generators G1 and G wth quadratc cost functons:

6 /98 $10.00 (C) 1998 IEEE 1 (16) C = b. q + a. q, =, where q s the output of generator. The two generators are a duopoly and are assumed to compete by provdng lnear 4 supply functons: C (17) p = x., = 1, or, q (18) p = x.( b + a. q), = 1, where x s a strategc varable set by player. Wthout transmsson constrants, the market clears at a unque equlbrum prce p so that, (19) ql = q1 + q Equaltes (16) to (19) yeld: 1 x. b + xj. bj + ql. xj. aj (0) q =, j a. x + aj. xj Knowng the quantty functons, and, therefore, the prce as a functon of the strategc varables, t s easy to calculate π (x 1,x ), the profts of the generators as a functon of ther strategc varables. For the purposes of smulatons, we have used the software Maple V to derve the reacton functons R1 and R, the unque statc Nash equlbrum [x 1 *,x *], and the module (deta) 1/ of the egenvalues of matrx A as defned n secton. 5 ql. aj + aj. bj (1) R = xj a aj ql + bj a + ajb () x * = 1 aj. a. ql + aj. b + bj. a a. aj. ql + bj. a + ajb, j ql. a + aj. a. ql + a. b + bj. a + aj. b b. a + a b j j j 1/ (3) ( deta) = ( aa 1. ql + a. b1).( aa 1. ql + a1. b) b. a + ( aa. q + a. b). ( aa. q + a. b ) + b. a ( ) ( ) 1 1 L 1 1 L 1 1 Ths last term s smaller than 1 f the coeffcents a1, a, b1, b are strctly postve. We can conclude that a unque Nash equlbrum exsts for ths type of competton and that ths equlbrum s always stable f the coeffcents of the quadratc cost functons are postve. 1 / 4. Bertrand competton wth n symmetrc players: Statc and ntertemporal Nash equlbra We are consderng here a smple model smlar to that of [] where n generators, wth a constant margnal costs c and ndvdual producton capactes k, are facng a demand q = a - p, where p s the prce and a s a coeffcent larger than c 6. The generators are competng by prces n an nfntely repeated game. In a one-stage game, the equlbrum would be a Bertrand-Nash equlbrum (BNE) of pure or mxed strateges 7. In the repeated game, [S* 1,..S* n ] s a set of strateges that consst n bettng the monopoly prce as long as all other players do the same, and stckng to the Nash bet of the sngle stage game after observng any devaton. If S* s an ntertemporal Nash equlbrum as defned n secton, we say that the players can tactly collude at the monopoly prce. At every stage of the repeated game, f generator thnks that the other generators are playng the strateges S* -, he wll have to choose between colludng by bddng the monopoly prce and sharng the demand wth the others, defectng by bddng a prce that s slghtly under the monopoly prce and producng at ts full capacty, or playng the one-stage Bertrand Nash strategy f he s expectng other generators to do the same. When the profts from defectng n one perod are smaller than the dscounted future profts that a generator wll loose by defectng, he wll choose to collude f no one has defected untl now but he wll bet the BNE bets f anyone has defected because he knows that hs compettors wll follow ther strateges S* - and do the same. Therefore, n ths case, the strategy S* s hs optmal strategy and S* s a Nash ntertemporal equlbrum that satsfes (15). When r s the annual dscount rate, T the perod between two nteractons, C the proft from colludng, D the proft from defecton and B the profts at the BNE, tact colluson s an ntertemporal Nash equlbrum f: Cnk (, ) Bnk (, ) (4) Dnk (, ) Cnk (, ) e rt 1 It s clear from (4) that the perod T between two nteractons wll determne whether the strateges S* are Nash strateges and whether we are lkely to observe tact colluson at the monopoly prce. It s shown n [13], that tact colluson s not sustanable for any values of k and n f T satsfy: 4 Ths assumpton was make for the sake of smplfyng the problem and fndng algebrac solutons 5 Wth n=, A s a x matrx wth zeros on the dagonal. The egenvalues are +/- (deta) 1/ 6 For a more n depth analyss of ths model and ts mplcatons, see [13] 7 For (a-c)/(n+1)<k<(a-c)/(n-1) there exsts no pure strategy,.e., the equlbrum bddng prces are random varables

7 /98 $10.00 (C) 1998 IEEE (5) T ln( ) r In other words, for any number of generators of any sze facng any lnear demand and for a dscount rate of 10%, contracts for delvery of electrc power durng 7 years or more should always help hnder collusve behavor 8. On the other hand, spot market, where the perod between two nteractons s very small, are lkely to encourage colluson; ths ssue s dscussed n [13]. 5. Strategc behavor due to transmsson constrants We have seen n the prevous secton some examples where a reduced number of players nteractng n a small market can rase the prce above the compettve levels and therefore reduce the socal welfare. Transmsson constrants solate a reduced number of players n small markets and produce the type of mperfectons descrbed prevously. However, they can also have a more drect role n creatng mperfectons n the market for electrc power through gvng gamng opportuntes to some of the players. The goal of ths secton s to gve a sense of the type of strategc behavor that could take place through a seres of very smple smulatons of three bus power systems. Frst, to develop an ntutve sense of what could happen, we return to the example descrbed n secton 4.1. T=d/4 G1 G L Fgure 1 The three lnes of ths network have the same mpedance and the lne jonng the two generators has a thermal constrant of d/4, where d was the nelastc demand of load L. Because of the constrant, the generator G1 can decde to rase ts prce (va rasng x 1 ) and ts profts as much as t wants snce the constrant s protectng a mnmal output q 1 of at least d/8. Generator G could do the same. The mere exstence of a transmsson 8 Of course ths strong result gves only an order of magntude, and more realstc data should be collected to analyze ths phenomenon wth non-lnear demand, ncreasng and asymmetrc margnal cost functons and a potental supply functon competton scheme. constrant s transformng a duopoly that respected ths constrant at ts equlbrum n a quas monopoly. We wll consder now the case where the load s elastc and examne the results gven by dfferent topologes and transmsson network constrants. As n secton 4, the cost functons and the supply functons of the two generators are the followng: (6) C = b. q + a. q, C (7) p = x. = x.( b + a. q) q The load s elastc and has a utlty functon: (8) U = b.( q + q ) a.( q + q ) L 1 L 1 The load s assumed to be an aggregaton of smaller loads wthout market power. Its demand functon wll therefore be: (9) p = b. a.( q + q ) L L L 1 In a market wthout network externaltes, the unque market prce that makes supply match demand maxmzes the (apparent) socal welfare, total utlty of the consumers mnus the apparent total cost of producton: U x. C x. C or q1+ q q1 q 1 1 pl ( q) dq p1( q) dq p( q) dq. The presence of transmsson constrants may not allow such a unque prce to exst and the maxmum welfare s attaned for dfferent prces at dfferent nodes through a coordnatng PX that takes the supply functon bds of the generators and determnes the optmal outputs, or through an approprate set of tradng rules whch would make the system attan ths optmal scheme through decentralzed transactons. The way the outputs are determned s rrelevant to the smulaton as long as the outputs and prces are the same. Moreover, whether the lnes to be constraned and the outputs q 1 and q are set by a coordnatng power exchange (PX) organsm or through an approprate set of rules for decentralzed blateral or multlateral tradng, t s the choce of x 1 and x by the generators that wll effectvely determne the constraned lnes as well as the optmal outputs q1( x1, x) and q( x1, x)of the generators. Knowng these functons, the generators deduce ther profts as a functon of x 1 and x as well as ther reacton functons. More specfcally, dependng on what constrants are actve, the analytcal formulaton of the output functons q 1 and q could change. The players wll calculate prmary reacton functons assocated wth every type of constraned or unconstraned dspatch. Every player wll then buld hs global reacton functon by comparng the profts yelded by hs prmary reacton functons. These global reacton functons wll reflect when a player wll chose to

8 /98 $10.00 (C) 1998 IEEE constran the network and when he chooses to leave t unconstraned. 5.1 Topology 1: no loop flows G G1 T L Fgure The lne lnkng G1 and the load has a thermal constrant T and therefore, (30) q1 + q T In all the smulatons, the gray lne separates the area where the bds (x 1, x ) wll lead the lne to be operatng at ts thermal lmt (constraned area) from the area where t would be operatng below ths lmt (unconstraned area). The sold lnes are the reacton functons R1 and R and the dotted lnes are the reacton functon as they would have been f the lne had no thermal constrant. When the constrant s relatvely low (Fgure 3), the Nash equlbrum s the same as the one that would be obtaned wthout constrants. Fgure 4 The last case, shown n Fgure 5, s an ntermedate case where the unconstraned equlbrum would lye n the constraned area but the prces are not hgh enough to allow for a constraned equlbrum. We observe a pecular stuaton where a contnuum of Nash equlbra seem to be sustanable on the boundary separatng the constraned from the unconstraned area. Fgure 5 5. Topology : ntroducton of loop flow Fgure 3 We add n ths example an unconstraned lne between G and the load, gvng to G a compettve advantage on G1 because of loop flows. When the constrant s tghtened (Fgure 4), the Nash equlbrum s obtaned n the constraned area and for hgher prces because the generators, havng market power, can explot the constrant more effectvely than a prce taker load.

9 /98 $10.00 (C) 1998 IEEE G L Fgure 6 To respect the constrant we must have: (31) α. q1 + β. q T, wth (3) 0 < β < α < 1 where α and β reflect the physcal characterstcs of the network. In the frst smulaton, (Fgure 7) the frst generator had more power to affect the state of the system, constraned or unconstraned, than G because of loop flow. In ths example, because the transmsson constrant s not very tght, the unconstraned equlbrum s however sustanable. T G1 A hgher constrant on the network (Fgure 8) moves the Nash equlbrum on the boundary separatng the constraned from the unconstraned areas. When we rase the nelastcty of the load 9, (Fgure 9) we observe how R becomes dscontnuous and no longer ntersects R1, preventng a statc Nash equlbrum from exstng. Ths dscontnuty s due to the fact that generator G, because t has a better strategc locaton, wll prefer not to follow the frst generators when the bds of the latter are too low, but rather concentrate on rasng the prce and the profts from the market share that the constrant s protectng for hm. It s mostly notable that n ths example, an unconstraned equlbrum exsts n the unconstraned area and that t would be a feasble Nash equlbrum f the players dd not know of the exstence of the constrants. However, because G knows that the constrant s protectng a porton of hs market share, he wll devate from ths equlbrum and try to rase ts proft further. Fgure 7 Fgure Topology 3: two way constrants In ths example, t s the lne lnkng the two generators that has a thermal lmt T mposng two constrants on the system: (33) α. q1 β. q T, (34) β. q α. q T, wth 1 Fgure 8 9 To rase the nelastcty, we have changed the coeffcents of the utlty functon to make ths functon steeper, under the condton however that the outputs at the compettve equlbrum (when the generators are bddng the true cost functons) stay unchanged.

10 /98 $10.00 (C) 1998 IEEE (35) 0 < β < 1, 0 < α < 1, where α and β reflect the physcal characterstcs of the network. G T G1 L Fgure 10 When the lne lnkng the two generators s weak, t can be constraned n ts two drectons, whch creates two constraned areas: Constraned area 1 where (33) s an equalty and Constraned area where (34) s an equalty (Fgure 11). Fgure 1 Fgure 13 Fgure 11 The generators have now new opportuntes for strategc bddng: In the last example, only one generator had a market share protected by loop flow and constrants. Here, both do; and that s why we observe dscontnutes n both reacton functons n Fgure 11. Nevertheless, the constrant s not very restrctve n ths example and the reacton curves ntersect n the unconstraned area at the unconstraned Nash equlbrum. However, wth a tghter transmsson constrant, as n Fgure 1, or a more nelastc load, as n Fgure 13,, the reacton functons may no longer ntersect, even f, once more, the unconstraned equlbrum (where the market would settle f the generators dd not know the exstence of the constrant) les n the feasble unconstraned area. Interpretng what would happen n realty when the reacton functons do not ntersect and where there s no statc Nash equlbrum s not easy. A frst basc nterpretaton would be that every generator wll react myopcally to the bds of hs compettors, observng these bds and reactng accordngly to ts reacton functon, and leadng the market to an unstable stuaton. In ths case, the examples developed above seem to ndcate that the bds wll always reman above the bds of the unconstraned Nash equlbrum, ndependently on the physcal feasblty of ths equlbrum. Ths s due to the fact that n the cases where the reacton functons dd not ntersect, at least one of them les completely above the bet of the unconstraned equlbrum, the other havng ether the same property or beng monotonc. A more sophstcated nterpretaton s that the generators wll play mxed strateges,.e., that ther bets wll follow a random process maxmzng ther expected profts, and leadng also to nstablty. Nevertheless, myopc

11 /98 $10.00 (C) 1998 IEEE nstablty or mxed strategy equlbra may not be long term sustanable n a repeated game where contnuous nteractons (and maybe the fear that a hgh nstablty wll lead the regulator to nvestgate and change the rules) could lead to make a more sophstcated ntertemporal Nash equlbra sustanable. In all these scenaros, the load seems to be worse because the generators have more market power and more opportuntes for strategc bddng created by thermal lmts on some of the transmsson lnes. 6. Conclusons Transmsson constrants and loop flow consttute a double threat to the emergence of a compettve electrcty market. When these constrants are too tght, they separate sub-markets from the rest of the network and make them relevant markets from an economc pont of vew, even f these markets are stll lnked by other unconstraned lnes to the rest of the network [13]. The reducton n the number of players n nteracton leads to hgher prces as usually recognzed, and sometmes to tact colluson due to repeated perodc nteractons as seen n secton 4. But transmsson lnes can also lead to neffcent behavor even when they are not supposed to be constraned n a compettve framework. A transmsson lne that may have a reasonable capacty to allow normal operatons of the network, could become a source of neffcences and hgher market prces f the generators realze that they can make profts by strategcally constranng t. A farly nelastc load on a network wth comfortable transmsson capactes, or a more elastc load and low transmsson capactes, may gve the generators the opportunty to strategcally constranng the network n order to explot ther protected market share wth very hgh bds. The load elastcty, whch s known to affect the prces at the unconstraned equlbrum and to rase the market power of the players, s found to also be capable of affectng ther ablty to strategcally constran the network, create potental nstabltes, and rase the average prces. If the regulators fal to make the loads more elastc through the approprate prce sgnals, the network mght have to be largely overbuld, wth all the economc and envronmental problems that ths mght cause. Future research should focus on whether a polcy for expandng the grd, that gves approprate ncentves to overbuld a lne when market power s used before the expanson, consttute a credble threat aganst strategc bddng and excessve abuse of market power. Such a strategy could paradoxcally save the network from beng overbult [13]. Quanttatve analyss must also be conducted to determnng the sgnfcance of a problem that we have only shown to be conceptually possble. More complex networks must be used, wth realstc data and non lnear supply functons. 7. Acknowledgments The authors greatly apprecate the fnancal support by the M.I.T Consortum for Transmsson Provson and Prcng, that has made ths research possble. Dscussons wth Professors John Tstskls at M.I.T., Francsco Galana at Mc Gll Unversty and Dr. Rchard Green at Cambrdge Unversty were mportant as the authors plowed through ths new terran. 8. References [1] Aok, M (1976). Optmal Control and System Theory n Dynamc Economc Analyss. North-Holland: 13. [] Brock, W. and J. Schnekman (1985). "Prce Settng Supergames wth Capacty Constrants." Revew of Economc Studes 11: [3] Edgeworth, F. (1897). "La Teora Pura del Monopolo." Gornale degl Economst 40:13-3 or (195). "The Pure Theory of Monopoly." Papers Relatng to Poltcal Economy vol.1, ed. F. Edgeworth (London: Macmllan). [4] Green, R. and D. Newberry (199). Competton n the Brtsh Electrc Spot Market. Journal of Poltcal Economy 100: [5] Hogan, W. (199). "Contract Networks for Electrc Power Transmsson." Journal of Regulatory Economcs. 4: [6] Klemperer, P.D. and M.A. Meyer (1989). Supply Functon Equlbra n Olgopoly Under Uncertanty. Econometrca 57: [7] Kreps, D. and J. Schnekman (1983). "Quantty Precommtment and Bertrand Competton Yeld Cournot Outcomes." Bell Journal of Economcs 14: [8] Oren, S. (1997). "Economc Ineffcency of Passve Transmsson Rghts n Congested Electrcty Systems wth Compettve Generaton." The Energy Journal 18(1): [9] Rams, E., Cl. Deschamps, J. Odoux (1988). Cours de Mathematques Spécales Masson, Pars [10] Sngh, H., S. Hao and A. Papalexopoulos (1997). "Power Auctons and Network Constrants." IEEE [11] Trole, J. (1988). "The Theory of Industral Organzaton. The MIT Press. [1] Wolfram, C.D. (1995). Measurng Duopoly Power n the Brtsh Electrcty Market. MIT Department of Economcs, WP, November. [13] Younes, Z. and M. Ilc (1997). Transmsson System Constrants n Non-Perfect Electrcty Market. Proc. of 18th Annual North Amercan Conference USAEE/IAEE