FITTING ELECTRICITY MARKET MODELS. A CONJECTURAL VARIATIONS APPROACH

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1 FITTING LCTRICITY MARKT MODLS. A CONJCTURAL VARIATIONS APPROACH Antonio García-Alcald Mariano Vntosa Michl Rivir Andrés Ramos Grgorio Rlaño INSTITUTO D INVSTIGACIÓN TCNOLÓGICA Univrsidad Pontificia Comillas Albrto Aguilra Madrid, SPAIN antonio.garcia@iit.upco.s Dircción d Mrcados d lctricidad Ibrdrola Hrmosilla 3, 2800 Madrid, SPAIN grlano@ibrdrola.s Abstract Th worldwid lctricity industry has bn mbddd in a significant rstructuring procss toward drgulation and comptition during th last dcad. Simultanously, an important rsarch ffort has bn mad to proprly incorporat mart faturs within gnration opration modls. Cournot quilibrium has bn on of th thortical framwors most widly usd to modl mart bhavior. Howvr, it prsnts rlvant wanss rlatd to its high snsitivity to th dmand lasticity. This papr proposs th us of firms conctural variations to ovrcom this difficulty and shows a procdur to dtrmin thm. Th mthod, dsignd for long-trm opration modls, infrs th implicit valus associatd to th firms rsidual dmand mploying only public mart information. Kywords: Comptitiv lctricity mart, gnration schduling, quilibrium modls, Complmntarity Problm. INTRODUCTION In th nw drgulatd powr marts, lctric firms assum much mor ris and bcom highly rsponsibl for thir own conomic dcisions. Thrfor, firms nd dcision-support modls that fulfill ths nw rquirmnts. An important dvlopmntal ffort has bn mad to dsign original modls and tools that xplicitly rprsnt th lctricity mart bhavior. In this contxt, th mart mchanisms such as individual profit maximization and ris managmnt ar th driving forcs that xplain th gnration units opration and th nrgy xpctd pric. A grat numbr of mart modls that try to rprsnt GNCOs long-trm bhavior ar basd on Cournot mart quilibrium. Although advancs in mart quilibrium modls hav bn notabl, limitations prsist in stimating svral rlvant mart issus. This papr proposs a fitting procdur dsignd for long-trm oprational modls basd on conctural variations. This approach allows a mor flxibl rprsntation of firms bhavior and a mor accurat pric gnration procss than Cournot-basd modls. Ths improvmnts ar achivd by mans of modling firms rsidual dmand functions, which provid infor- Mart quilibrium provids a st of dcisions such that no firm, taing its comptitors dcisions as givn, wishs to chang its own unilatrally. mation about th nrgy that firms ar abl to sll at ach pric. Ths rsidual dmand functions ar difficult to stimat bcaus thy ar rlatd to th othr firms supply functions. Th proposd mthodology charactrizs ach rsidual dmand function by its lasticity and stimats th implicit valus of this paramtr. Thus, this procdur computs th long-trm rsidual dmand lasticity that ach firm infrs whn bidding. Th infrnc about firms bidding prcptions is basd on claring mart information providd by th Mart Oprator, instad of th ral supply functions submittd by th firms. An stimation procss basd on short-trm supply functions has bn dismissd bcaus thy do not rflct proprly firms long-trm stratgis but ust firms short-trm tactics. This papr is organizd as follows. Sction 2 provids an ovrviw of svral lctricity mart modls basd on Cournot quilibrium that ar availabl in th litratur. Sction 3 dtails th conctural variations approach whil sction 4 shows how this approach can b considrd in a MCP-basd schm. Sction 5 stats th stimation procdur of th implicit lasticitis that allows to fit mart modls basd on conctural variations. Sction 6 dscribs an application of th proposd approach to th Spanish lctricity Mart, and finally, sction 7 provids th conclusions drawn from th study. 2 COURNOT-BASD MARKT MODLS Th Cournot quilibrium suitability to modl comptition among fw firms has bn widly discussd in conomics litratur []. Th vast maority of th modls usd in an imprfct comptitiv contxt somhow considr firms' Cournot bhavior. In particular, th Cournot quilibrium approach applid to lctricity marts allows a mor ralistic modling of lctrical gnration faturs than any othr complx mthodologis, such as Supply Function quilibria (SF) 2. An intrsting wor dvotd to a concturd supply function approach can b found in [2]. Th aim of this 2 Supply Function quilibria (SF) considrs firms compt through out thir bidding supply curvs. Although this approach has bn provd to b solid, its application to ral marts is avoidd du to th multiplicity of solutions and th poor rprsntation of gnration opration.

2 papr is to prsnt a mor ralistic mthod to modl imprfct comptition in powr ntwors than th traditional ons. This approach combins th computational tractability of Cournot modls with a solid way of comptition such as SF. Intrst in th rsarch community rgarding th dvlopmnt of Cournot-basd quilibrium modls as bhavior-pattrn of lctricity marts has grown and has bn dmonstratd as such in numrous publications. First dvlopmnts [3] hav bn xtndd latr by dvloping an itrativ procdur to obtain th mart quilibrium in th Californian Mart [4]. An altrnativ procdur to comput th Cournot mart quilibrium by an quivalnt optimization problm can b sn in [5]. Hydrothrmal coordination in an imprfct comptition framwor was first considrd in [6]. Th proposd modl uss dual dynamic programming in which a duopoly Cournot modl is solvd in ach stag. An analytic statmnt of this problm's quilibrium conditions is introducd in [7]. This approach provids intrsting conclusions about th rol of hydro gnration in a mart framwor. In [8] it is shown that th Mixd Complmntarity Problm (MCP) is a powrful and flxibl mthodology for addrssing th long-trm opration planning of a GNCO within a Cournot contxt. Th MCP structur allows to considr th simultanous firms' profit optimizations whil rprsnting th rlvant opration constraints of gnration units. Th ntwor congstion influnc in Cournot GNCO's bhavior has also brought about intrsting dvlopmnts. Modling lctricity marts as spatial pric discrimination procss is proposd in [9], [0] and []. Rcntly, Complmntarity Problm and Variational Inquality approach hav bn usd to modl imprfct comptition among producrs, including a congstion-pricing schm for transmission [2] and [3]. 3 CONJCTURAL VARIATIONS APPROACH As can b sn in th prvious sction, thr xists a numbr of lctricity mart modls basd on Cournot comptition. Howvr, it is wll nown that ths modls provid barly crdibl prics. Ths high prics ar du to nglcting comptitors supply functions. Th conctural variations approach considrs th raction of comptitors whn a firm is dciding its optimal production. This raction coms out from firms supply functions and dmand curv. This raction can b modld by th so-calld firm s rsidual dmand function. This function is diffrnt for ach firm and rlats ach firm s production with th mart pric. It can b obtaind by mans of subtracting th rmaining comptitors supply functions from th mart dmand curv. In th nxt subsction, th optimization problm of ach firm is statd as its profit maximization facing its own rsidual dmand function. Hnc, conctural variations ar considrd in firms stratgic bhavior. 3. Firms optimization problm In this sction th optimization problm of ach firm is dscribd, whos obctiv function can b dfind as its profit maximization Π = P -C () whr Π is th firm s mart profit, is th systm marginal pric, P is th firm s total production and C rprsnts firm s oprational costs. Sinc th systm marginal pric is st by th dcisions mad by th supply and dmand sid, it is possibl to rlat this pric with th aggrgatd production by mans of th dmand function. = f P (2) In ordr to calculat th th firm optimal production, th drivativ of th profit function with rspct to th dcision variabl (P ) is qualizd to zro. Π = + P MC ( P) Th first two trms of th quation ma up th firms marginal rvnu MR. Th marginal rvnu masurs how firm s rvnu incrass (or dcrass) whn th firm incrass (or dcrass) its production in on unit. Liwis, firm s marginal cost can b dfind as how firm s cost incrass (or dcrass) whn th firm incrass (or dcrass) its production in on unit. Thrfor, firm s optimal production is achivd whn its marginal rvnu is qual to its marginal cost. Firm s marginal rvnu can b xprssd in trms of its rsidual dmand lasticity ε instad of th rsidual dmand slop MR( P) = + P = + ε whr rsidual dmand lasticity is dfind as quotint btwn th unitary chang on firm s production causd by an unitary chang on mart pric. P = (5) ε Not that rsidual dmand lasticity is diffrnt for ach firm. This paramtr tas into account how th mart pric changs whn ach firm changs its production unilatrally. To conclud, this paramtr xprsss th mart conctur of ach firm. Th variation of this paramtr ncompasss diffrnt typ of comptition, providing grat flxibility of modling. 3.2 Flxibility of modling Dpnding on th valu of firms rsidual dmand lasticity, a diffrnt assumption is mad about how (3) (4)

3 firms marginal rvnu is concivd, and consquntly, about how firms bhav. In this way, som widly usd mart modls can b statd in trms of th rsidual dmand lasticity as follows. Prfct comptition. Prfct comptition can b dfind as a mart situation whr firms ar not abl to chang th mart pric by modifying thir production unitrally. In trms of mart modling, ach firm s rsidual dmand function bcoms horizontal. Th lasticity valu of a fixd-pric rsidual dmand function (s quation(5)) is infinity. From quation (4), firm s marginal rvnu is qual to th mart pric. As xpctd, firm s optimal production tas plac whn its marginal cost rachs th mart pric. Cournot modl. As prviously mntiond, Cournot modl is on of th most common approachs to rprsnt comptition among fw firms. In contrast to prfct comptition, firms ar abl to modify th mart pric by mans of changing its own production. Th mart modl conctur that stats a Cournot quilibrium can b writtn in trms of a dmand lasticity qual to vry firm compting in th mart scald by its mart shar (ε = ε d / α ), whr ε d is th dmand function lasticity and α is th th firm mart shar. Cournot prics ar usually far highr than ral mart prics. Th diffrnt valus of th rsidual dmand lasticity dpnding on th thortical mart modl ar summarizd in Tabl. In this sction it has bn prsntd a flxibl approach that allows to rprsnt diffrnt optimal mart bhaviors using firms rsidual dmand lasticity. Modl Mart Conctur Prfct Comptition ε = Cournot ε = ε d /α Conctural Variations ε ε Tabl : Mart Conctur 4 MARKT QUILIBRIUM MODL Sinc firms compt within a mart contxt, firm s optimal nrgy schdul must considr not only its own optimization program but also th intraction among th optimization programs of th rmaining firms. Th mathmatical structur of a mart quilibrium problm concptually corrsponds to various simultanous optimizations (on for ach firm) lind togthr through th mart pric rsulting from th intraction of all of thm. This schm is shown in Figur, whr Π rprsnts th mart profit of ach company [,...,], x th dcision variabls and th st of constraints h and g ar particularizd for ach company. Th lctricity mart is modld by th dmand function that rlats th supplid dmand to th lctricity pric. Optimization Program of Firm maximiz : Π ( x ) subct to : h Optimization Program of Firm maximiz : Π Figur : Mart quilibrium m(x)=0 ( x ) subct to : h lctricity Mart Optimization Program of Firm maximiz : Π ( x ) subct to : h Thr ar svral mthodologis that ar abl to dal with this mart quilibrium structur. On of ths mthodologis is basd on th Mixd Complmntarity Problm (MCP). xampls of ths MCP-basd modls can b found in [8] and [2]. Th mart quilibrium statd in trms of an MCP schm rquirs stting th first ordr optimality conditions of Karush-Kuhn-Tucr [4]. Ths optimality conditions ar associatd to th st of maximization programs, as shown in Figur 2. Optimality Conditions of Firm Optimality Conditions of Firm Optimality Conditions of Firm λ L ( x, λ, µ ) = L ( x,, ) 0 x λ µ = = ( ) L x, λ, µ ( x, λ, µ ) µ g x L x = = h λ µ 0 L λ ( x, λ, µ ) µ g x = = h λ µ 0 λ µ g x L ( x, λ, µ ) = x = = h λ g 0 µ 0 m(x)=0 lctricity Mart Figur 2: Mart quilibrium as a Mixd Complmntarity Problm

4 In Figur 2, L rprsnts th Lagrangian function of th corrsponding optimization problm and λ and µ rprsnt th dual variabls associatd to th st of h and g constraints rspctivly. Th optimality conditions can b statd as thr sts of quations. Th first on cancls th gradint of th Lagrangian function with rspct to th dcision variabls x. Th scond st (th gradint of th Lagrangian function with rspct to th dual variabls λ) coincids with th quality constraints h. Th third on is mad up by th complmntary slacnss conditions associatd to th inquality constraints g. Grouping togthr all companis systm of quations lads to a mixd complmntarity problm formulation. Th mart quilibrium formulation as an MCP allows to modl firms comptition as th conctural variations approach prviously dtaild in sction 2. For th sa of clarity, it is considrd as ach firm s dcision variabl its portfolio production (P ). Thrfor, quation (3) xprssd in trms of rsidual dmand lasticity (ε ) corrsponds to th first st of th KKT optimality conditions within th MCP schm. xl ( ) = = + CM( P) ε (6) Not that in (6) appars th rsidual dmand lasticity. Hnc, an MCP-basd modl allows to considr xplicitly th conctural variations approach. In mor complx modls, singl units productions ar statd as dcision variabls instad of firms portfolio production. Still, firms marginal rvnu is always an lmnt of th optimality condition associatd to ach dcision variabl. 5 IMPLICIT LASTICITY STIMATION In this sction, a mthodology suitabl to stimat rsidual dmand lasticity is prsntd. Th rsulting stimations will allow th fitting of lctricity mart modls. 5. Ovrviw As has bn prviously dtaild in sction 2, th mart concturs slctd to modl th firms bhavior hav a rlvant impact on firms productions and mart prics. In sction 2 it has also bn prsntd a conctural variations approach to modl firms bhavior mor flxibly than th traditional Cournot modling. This approach is basd on using firms rsidual dmand functions instad of mart s aggrgatd dmand curv. Thrfor, it bcoms ncssary a mthodology abl to dal with th fitting and stimation procss of th firms rsidual dmand. Th proposd mthodology is basd on fitting th rsidual dmand lasticity by mans of an valuation of th conctural variations modl on past data. It is first supposd that firms bhavd via a conctural variations pattrn during th fitting priod. Thrfor, th dcisions (productions and mart prics) thy mad can b assumd to b optimal. This optimal schdul can b xprssd qualizing firm s marginal rvnu with its marginal cost (s quation (3)). Whn fitting with past data, firms dcisions hav alrady bn mad. Consquntly, w can infr th past rsidual dmand lasticity by mans of using ths dcision variabls as acnowldgd valus. This procdur is nown as implicit valuation. Thus, th rsidual dmand lasticity obtaind is calld implicit lasticity. It is important to rmar that th implicit valus of th lasticity masur firms prcption about thir mart positions in a conctural variations contxt. In fact, this prcption dos not nd to coincid with th ral rsidual dmand computd by th supply and dmand curvs submittd to th mart by all th firms. 5.2 Implicit lasticity Th procdur to obtain th implicit valu of firms rsidual dmand in a conctural variations contxt is statd along ths lins. Lt b th actual systm mart pric and MR b th marginal rvnu of th firm. Th implicit valu of th rsidual dmand lasticity ( ε ) is drivd from xprssion (4) and can b writtn as follows. ε = (7) MR Th valu of firm s marginal rvnu is impossibl to stimat rgarding mart data. Howvr, it can b approximatd to firm s marginal cost by mans of considring that ach firms is gnrating optimally (3). Firm s marginal cost can asily b stimatd as th gnration cost of th most xpnsiv committd unit 3. Thrfor, th implicit xprssion of th rsidual dmand lasticity can b statd in trms of th mart pric and firm s marginal cost as follows. ε (8) MC 5.3 stimation By mans of valuing th implicit rsidual dmand lasticity, it is possibl to infr firms long-trm bhavior. Th stimation procdur uss only public mart information (prics, production, stimation of firms marginal costs) hourly providd. Applying quation (8) to this information, th hourly tim sris of th implicit rsidual lasticity of ach firm is dirctly acnowldgd. Ths implicit valus not only considr longtrm firms bhavior, but also ar vry influncd by short-trm uncrtainty sourcs and oprational constraints. Som rlvant short-trm uncrtainty sourcs that hav som baring on th implicit dmand lasticity ar plants outags, watr spills, dviations of th forcastd dmand and supply curvs submittd by th rst of th firms. Short-trm oprational constraints as plant ramps, minimum stabl output, limitd rsrvoir capacity and 3 A mor prcis marginal cost stimation rquirs considring units gnration constraints (i.. ramps and minimum stabl output) and othr rlvant mart issus (i.. capacity paymnts).

5 ntwor constraints also hav a significant ffct on implicit valus. Thrfor, th prvious short-trm issus must b sivd in ordr to proprly stimat implicit valus that will modl firms long-trm bhavior. It is ncssary to procss all th information about th implicit rsidual dmand volution looing for a trnd valu. Statistics ma availabl svral data-analysis mthodologis that ar abl to dal with long-trm bhavior stimation from short-trm data. Dpnding on th natur of th variabls usd to do so, statistical mthods can b classifid as follows. Rlational modls. By mans of ths modls, th implicit lasticity of th rsidual dmand is rlatd to on or svral rlvant variabls (xplanatory variabls) whos long-trm valus ar asy to infr. This rlationship is xprssd by a function among th variabls. Having a long-trm stimation of th st of xplanatory variabls (.g. dmand, hydraulic inflows) a trnd valu of th implicit lasticity can asily b obtaind by valuating th formr function. Som of th statistical mthods that can b dfind as rlational modls ar rgrssion, nural ntwors and dcision trs. Classification modls. In contrast to rlational modls, th st of valus of implicit lasticitis ar catgorizd by th diffrnt lvls of svral discrt factors. Th long-trm valu of th implicit lasticity for ach factor combination (i.. class) can b xprssd as a statistical masur (avrag, mdian or mod) of th past data distribution. Varianc analysis and clustring ar som statistical mthodologis basd on classification procsss. Tim sris modls. Ths modls considr implicitly th volution of firm s lasticity and infrs a trnd valu of th tim sris. Tim sris mthodology allows not to ta into account othr variabls or factors whn stimating th long-trm valus of th implicit rsidual dmand lasticity. Som of th tim sris mthodologis dvotd to dal with long-trm trnds ar tim sris dcomposition and xponntial smoothing mthods. 6 CAS STUDY An MCP-basd quilibrium modl [8] in which firms compt considring conctural variations has bn applid to th Spanish lctricity Mart to validat th stimation and fitting procdurs dtaild in sctions 3, 4 and 5. A comparison with firms Cournot bhavior is also shown in this sction. 6. Systm dscription Th Spanish lctricity Mart was stablishd in 998 and sinc thn four firms compt: ndsa, Ibrdrola, Unión Fnosa and Hidrocantábrico. Th systm mts a maximum pa load clos to MW and a yarly nrgy dmand of GWh, and th avrag hydro nrgy availabl is 3076 GWh. Th annual scop of th modl has bn dividd into twlv priods (months) with 5 load lvls for ach on. Typ # Units Powr (MW) Ibdr. ndsa Unión Hidroc. Thrmal % 54 % 3 % 5 % Hydro % 36 % 3 % 0 % Tabl 2: Spanish firms production structur Thr ar 82 thrmal gnrators groupd into 42 thrmal plants. Th hydro units hav bn groupd into 28 quivalnt units plus 0 pumpd-storag units. All th information that has bn usd in this cas study is public. Units productions and mart prics hav bn obtaind through th Spanish Mart Oprator. Ful costs and units thrmal rats ar rspctivly basd on intrnational ful prics and standard tchnology rats. 6.2 Numrical rsults Two diffrnt Cournot scnarios hav bn considrd by changing th aggrgatd dmand s lasticity valu. For th first on, ε d is qual to 0.3, whil for th scond on ε d is qual to 0.5. A conctural variations approach has also bn implmntd. Th valu of firms rsidual dmand lasticity has bn stimatd using a classification modl whr ach valu was labld dpnding on th firm, dmand lvl and th xpctd hydraulic inflows. Data usd to fit this stimation modl is basd on th mart rsults of th yar 999. Th modl-obtaind prics ( /MWh) and th ral mart rsults can b sn in Tabl 3. Not that conctural variations provid a ralistic stimation of th ral mart prics for vry dmand lvl. In contrast, Cournot quilibrium givs dficint rsults whn dmand lasticity dcrass. Modl Assumption Cournot ε d 0.3 Cournot ε d 0.5 Conctural Variations Mart Prics (2000) On-On- Pa On- Pa Platau Off- Pa Off-Off- Pa Tabl 3: Modl s avrag prics ( /MWh) for ach dmand lvl For th first Cournot scnario, stimatd prics ar vn ight tims highr than th ral ons. It is important to rmar that a dmand lasticity valu of 0.3 is vry high in lctricity marts. For instanc, th avrag dmand lasticity in th Spanish lctricity Mart in 999 was Thrfor, th us of a Cournot modl in which its dmand lasticity stimation is basd

6 on th submittd dmand curvs is compltly inappropriat. Not that although only four firms compt in th Spanish lctricity Mart, actual claring prics ar much lowr than thos providd by a thortical Cournot modl. In fact, pric-shocs as occurrd in California hav not bn obsrvd in th Spanish lctricity Mart. Rgarding th rsults obtaind of th cas study, th conctural variations approach and th proposd implicit stimation mthodology of th rsidual dmand lasticitis provids a flxibl and accurat tool to infr ralistic firms productions and mart prics. Th advantags with rspct to Cournot-basd modls ar notabl. 7 CONCLUSIONS In this papr it has bn prsntd a conctural variation approach to fit lctricity mart modls. This approach allows a mor flxibl and accurat mart dscription than Cournot-basd modls whn modling firms bhavior in a libralizd contxt. Th diffrnt firms bhaviors, ranging from prfct comptition to Cournot, ar considrd by mans of its rsidual dmand lasticity. Du to th importanc of an accurat stimation of firms rsidual dmand function, an infrnc procdur of thir lasticity has bn dvlopd. It is basd on an implicit stimation procss that can asily b computd rgarding only public mart information. Ths implicit valus of rsidual dmand lasticity somhow modl firms prcption about thir mart position. A cas study applid to th Spanish lctricity Mart has bn prsntd to illustrat how th proposd mthodology is abl to provid crdibl rsults. Ths rsults also show that bhavior of firms compting in th Spanish Mart clarly diffrs from th thortical Cournot approach. 8 ACKNOWLDGMNTS This wor has bn dvlopd undr a rsarch proct fundd by Ibrdrola, S.A. and with th support of a grant from th Ministrio d ducación, Cultura y Dport (Rf. AP ). RFRNCS [] Vivs, X., Oligopoly Pricing. Th MIT Prss 999. [4] Bornstin, S. and Bushnll, J. An mpirical Analysis of th Potntial for Mart Powr in California s lctricity Industry. Journal of Industrial conomics, Vol. 47, No. 3, Sptmbr, 999. [5] Ramos, A., Vntosa, M., Rivir, M. Modling Comptition in lctric nrgy Marts by quilibrium Constraints. Utilitis Policy, Vol. 7 Issu 4 pp , Dc. 98. [6] Scott, T.J. and Rad,.G. Modlling Hydro Rsrvoir Opration in a Drgulatd lctricity Mart. Intrnational Transactions in Oprational Rsarch. Vol. 3 pp [7] Bushnll, J. Watr and Powr: Hydrolctric Rsourcs in th ra of Comptition in th Wstrn US. POWR Confrnc on lctricity Rstructuring. Univrsity of California nrgy Institut. Brly [8] M. Vntosa, M. Rivir, A. Ramos and A. García- Alcald, "An MCP Approach for Hydrothrmal Coordination in Drgulatd Powr Marts", I-PS Summr Mting Procdings, Vol 4, pp , Sattl WA, [9] Hogan, W.W. A Mart Powr Modl with Stratgic Intraction in lctricity Ntwors. Th nrgy Journal 8 (4), pp. 07-4, 997. [0] Orn, S. conomic Infficincy of Passiv Transmission Rights in Congstd lctrical Systm with Comptitiv Gnration, Th nrgy Journal 8 (), pp , 997. [] Hobbs B. F., Mtzlr C., Pang J. S. Stratgic Gaming Analysis for lctric Powr Ntwors: An MPC Approach. I Trans. Powr Systms [2] Hobbs B. F. LCP Modls of Nash-Cournot Comptition in Bilatral and POOLCO-Basd Powr Marts. Procdings, I Wintr Powr Mting, NY City, Fb [3] Wi. Jing-Yuan, Y. Smrs. Spatial Oligopolistic lctricity Modls with Cournot Gnrators and Rgulatd Transmission Prics. Oprations Rsarch, Vol. 47, No., January - Fbruary 999, pp [4] W.W. Cottl, J.S. Pang, and R.. Ston. Th Linar Complmntarity Problm. Acadmic Prss, Boston, 992. [2] Day, C. J., Hobbs B.F. Oligopolistic Comptition in Powr Ntwors: A Concturd Supply Function Approach. Tntativly accptd, I Trans. P.S. [3] Bornstin, S., Bushnll, J., Kahn,. and Stoft, S. Mart Powr in California lctricity Marts. Utilitis Policy. Vol. 5 No. 3/4 pp