A Two Stage Stochastc Equlbrum Model for Electrcty Markets wth Two Way Contracts Dal Zhang and Hufu Xu School of Mathematcs Unversty of Southampton Southampton SO17 1BJ, UK Yue Wu School of Management Unversty of Southampton Southampton SO17 1BJ, UK December 12, 2008 Abstract Ths paper nvestgates generators strategc behavors n contract sgnng n the forward market and power transacton n the electrcty spot market. A stochastc equlbrum program wth equlbrum constrants (SEPEC) model s proposed to characterze the nteracton of generators competton n the two markets. The model s an extenson of a smlar model proposed by Gans, Prce and Woods [10] for a duopoly market to an olgopoly market. The man results of the paper concern the structure of a Nash-Cournot equlbrum n the forward-spot market: Frst, we develop a result on the exstence and unqueness of the equlbrum n the spot market for every demand scenaro. Then, we show the monotoncty and convexty of each generator s dspatch quantty n the spot equlbrum by takng t as a functon of the forward contracts. Fnally, we establsh some suffcent condtons for the exstence of an local and global Nash equlbrum n the forward-spot markets. Numercal experments are carred out to llustrate how the proposed SEPEC model can be used to analyze nteractons of the markets. 1 Introducton Over the past two decades, the electrcty ndustry n many countres has been deregulated. One of the man consequences of deregulaton s that the governments undertake ther efforts to develop fully compettve electrcty spot markets. In most of the wholesale spot markets (pool-type systems), generators make daly (or hourly) bds of generaton at dfferent prces, and then an ndependent system operator (ISO) decdes how actual demand s to be met by dspatchng cheaper power frst. In these pool-type electrcty markets (found n Australa, New Zealand, Norway, at one tme n UK, and some parts of US), a sngle market clearng prce s determned by a sealed-bd aucton and pad to each generator for all the power they dspatch. 1
Along wth the spot market emerges the forward market where generators and retalers may enter nto hedge contracts before bddng n the spot market. For example, n the early 90 s, durng the restructurng of the electrcty market n UK, some long term, take-or-pay contracts (or agreements) are stpulated by three man Scottsh electrcty generators, see [18]. Moreover, varous contract markets have also been establshed n Europe, Australa, New Zealand and North Amerca. By partcpatng n the forward markets, generators and retalers may share ther rsks assocated wth a fluctuatng pool prce for the real power dspatchng. The most common type of contract s known as a (two-way) contract-for-dfference (or hedge contract), whch operates between a retaler and a generator for a gven amount of power at a gven strke prce. The sgnng of ths type of contracts s separate from the market dspatchng mechansm and can be taken as fnancal nstruments wthout an actual transfer of power. In ths paper, we formulate generators competton n the forward-spot market mathematcally as a two stage stochastc equlbrum problem where each generator frst ams at maxmzng ts expected proft by sgnng a certan mount of long term contracts and then bds for dspatches n the spot market on a daly or hourly bass. Dfferng from the two stage competton model, a volume of prevous research has been performed to study the effect on the competton n the spot market from the contract quanttes, n whch the competton of sgnng contracts n the forward market s not consdered. Von der Fehr and Harhord [30] nvestgate the spot market by modelng t as a mult-unt aucton and demonstrate that contracts gve generators a strategc advantage n the spot market by allowng them to commt to dspatch greater quanttes durng peak demand perods. Powell [20] explores the nteracton between the forward market and the spot market by characterzng the competton n the spot market wthn a framework of Nash-Cournot equlbrum, and shows that rsk-neutral generators can rase ther profts by sellng contracts for more than the expected spot prce. Moreover, Green and Newbery [12] approprately look at the endogenous formaton of both pool and contract prces n a supply functon model, and apply ther analyss to the Brtsh electrcty market. By modelng the mechansm of the competton n the forward market as a Nash-Cournot game, prevous contrbutons, such as [1, 33, 10], focus on the mpact of the forward market on the spot prce and show that generators have ncentves to trade n the forward market whereas forward contractng reduces spot prces and ncreases consumpton levels. The exploraton of the blevel determnstc Nash-Cournot model for a duopoly forward-spot market s frst carred out by Allaz and Vla [1], whch dentfes two crtcal assumptons: One s the socalled Cournot behavor where producers (generators) act as though the quantty offered by the other compettors s fxed; the other s the connecton to the prsoner s dlemma where each producer (generator) wll sell forward so as to make them worse off and make consumers better off than would be the case f the forward market dd not exst. Applyng ths type of Nash- Cournot models of electrcty pools, Gans, Prce and Woods [10] demonstrate that the contract market can make the duopolstc spot market more compettve, and hence the exstence of the contract market lowers prces n pool markets. By replacng two way contracts wth call optons, Wllems [33] extends the results n Allaz and Vla [1] to the Cournot type market wth 2
optons, and compares t wth the market effcency effects of the Cournot game wth two way contracts. Instead of duopoly markets n [1], Bushnell [6] presents some estmaton of the mpact of forward contracts and load oblgatons on spot market prces for a Cournot type envronment wth multple generators. Dfferng from much of prevous work concernng on the nfluence on spot market effcency from contracts, our work provdes a new model for the entre forward-spot market by formulatng t as a two stage stochastc equlbrum problem wth equlbrum constrants (EPEC), whch refers to generators competton n the forward market as an equlbrum problem subject to the equlbrum n the spot market descrbed by a complementarty problem. Over the past few years, EPEC models have been appled to some herarchcal decson-makng problems n a wde doman n engneerng desgn, management, and economcs. Recently, a number of EPEC models have been performed for electrcty markets. In modelng the forward-spot market, Su [29] and Shanbhag [25, Chapter 5] study the Nash-Cournot equlbrum by modelng the blevel markets as an EPEC. Su [29] nvestgates the exstence results for the determnstc forward-spot market equlbrum ntroduced by Allaz and Vla [1]. Shanbhag [25, Chapter 5] ntroduces a 2-node forward-spot model and consders t as an expected proft maxmzaton problem subject to the complementarty constrants for every scenaro n the spot market. He also nvestgates exstence of the smultaneous stochastc Nash equlbrum (SSNE) n the context of the forwardspot electrcty market. Moreover, besdes the applcaton n the forward-spot market, the EPEC models are also used by Yao, Oren and Adler [36] to nvestgate the equlbrum n the spatal electrcty market, where they capture the congeston effects and blevel compettons by formulatng each generator s objectve as a maxmzaton problem n the forward market subject to the Karush-Kuhn-Tucker (KKT) optmal condtons n the spot market and the network constrants. More recently, Hu and Ralph [15] use EPEC to model a blevel electrcty market, where generators and customers bd cost and utlty functons n a nodal market and the ISO determnes the dspatch quanttes by mnmzng the overall socal cost n an upper optmzaton level. Apart from Cournot-type models, another well establshed approach s the supply functon equlbrum (SFE) model, whch clearly encapsulates the underlyng structure of bdders strategy on the quantty-prce relatonshp. SFE s orgnally proposed by Klemperer and Meyer [16] to model competton n a general olgopolstc market where the market demand s uncertan and each frm ams to develop a supply functon to maxmze ts proft n any demand scenaro. By applyng the SFE to predct the performance of the poneer England and Wales market, Green and Newbery [12] analyze the behavor of the duopoly and characterze the England and Wales electrcty market durng ts frst years of operaton under the SFE approach. Anderson and Phlpott [3] frst propose an optmal supply functon model wth dscontnuous supply functons to address the fact that supply functons n practce are not contnuous as assumed n SFE model and they use ths model to nvestgate generators optmal strateges of bddng a stack of prce-quantty offers nto electrcty markets n crcumstances where demand s unknown n advance. Anderson and Xu [4] extend the optmal supply functon approach to consder both 3
second order necessary condtons and suffcent condtons of the optmalty for each generator s prce-quantty offers gven ts rvals offers are fxed. Besdes the analyss on the optmalty condtons for the spot market, the SFE framework has also been appled to nvestgate the nteractons between the forward market and the spot market. Green [11] and Newbery [17] are among the frst researchers who study the mpact of two-way contracts n conjuncton wth the SFE model and observe that contracts provde ncentves for generators to supply more n a spot market. Anderson and Xu [5] make further nvestgatons n ths drecton by consderng the optmal supply functons n electrcty markets wth opton contracts and nonsmooth costs. However, calculatng an SFE requres solvng a set of dfferental equatons nstead of the typcal set of algebrac equatons as n Cournot models, whch presents consderable lmtatons on the equlbrum condtons and the numercal tractablty. Indeed, the exstence of the SFE has been proved only for lnear supply functon models (Rudkevch [23]) and for symmetrc model wthout capacty lmt (Klemperer and Meyer [16]), wth capacty constrans (Anderson and Xu [4], Holmberg [14]), and there s no dscusson about an SFE model for a two stage forward-spot market. Along the drecton of the research on EPEC and Cournot models, ths paper makes a number of contrbutons. Frst, we present mathematcal models for generator s optmal decsons and Nash-Cournot equlbrum problems n the forward-spot market. Second, we dscuss the exstence and unqueness of Nash-Cournot equlbrum n the spot market and nvestgate propertes of such equlbrum. Thrd, we show the exstence of Nash-Cournot equlbrum n the forward market. The rest of the paper s lad out as follows. In the next secton, we gve a detaled descrpton of an SEPEC model for the forward-spot market competton, and show that the equlbrum n the spot market depends on the contract quanttes rather than the strke prce. In Secton 3, we use a complementary program model to solve the equlbrum problem n the spot market, and obtan the exstence and unqueness results and the monotoncty of the supply functons wth respect to the contract quanttes. In Secton 4, we show the exstence of Nash-Cournot equlbrum of the forward-spot market nteracton, and the contnuty of each generator s proft n the forward market. In Secton 5, we present some numercal tests to llustrate the theoretcal results n ths paper.fnally, n Secton 6, we pont out the restrctons of the paper and further work. 2 Mathematcal descrpton of the problem In ths secton, we present mathematcal detals on modelng competton n the forward market and the spot market, and show that the optmzaton problem n the forward-spot market can be structured as a two stage stochastc equlbrum model. Ths model can be vewed as an extenson of a smlar model by Gans, Prce and Woods [10] n a duopoly to an olgopoly. 4
We suppose that there are M generators competng n a non-collaboratve manner for dspatch n the spot market on daly bass and these generators are economcally ratonal and rsk neutral. In the spot market, market demand s characterzed by an nverse demand functon p(q, ξ(ω)), where p(q, ξ) s the spot prce, Q s the aggregate dspatch quantty and ξ(ω) s a random shock, ξ : Ω IR s a contnuous random varable defned on probablty space (Ω, F, P ) wth known dstrbuton. To ease the notaton, we wll wrte ξ(ω) as ξ and the context wll make t clear when ξ should be nterpreted as a determnstc vector. We denote by ρ(ξ) the densty functon of the random shock and assume that ρ s well defned and has a support set Ξ. Snce the outcome of the clearng prce p(q, ξ) s fluctuatng n the spot market, both generators and retalers wshng to ensure a fxed or a stable electrcty prce to hedge the rsks rsng from the varaton of the spot prce can do so by sgnng forward electrcty contracts. Ths knd of contracts can be taken as a fnancal nstrument and does not nvolve actual transacton of power. There are essentally two types of contracts: a one-way contract such as a put opton or a call opton where only one sde of the contract commts to pay the dfference between the strke prce and the spot prce for the contracted quantty, and a two-way contract where both sdes of the contracts commt to pay the prces dfference as opposed to the one way contract. In ths paper, we smplfy the dscusson by focusng on two-way contract, that s, each generator sgns a two-way contract wth retalers. 2.1 Generator s optmal decson problem n the spot market We begn the model of the spot market by formulatng a generator s proft functon whch nvolves three terms: a revenue from sellng electrcty n the spot market, the cost of generatng the electrcty and the dfference due to the commtment to a contract. Frst, we look nto the term of each generator s commtment to ts contract by gvng detals on the contract sgnng and the mechansm of generators fulfllment n the spot market. We assume that, n the forward market, generator, = 1,, M, enters nto a two-way contract at a fxed prce z (x, x ) for an amount x, where x := (x 1,..., x 1, x +1,..., x M ) T denotes the vector of contract quanttes sgned by ts rvals and the superscrpt T denotes transpose. Here z s a functon of x and x. For the smplcty of notaton, we wrte z (x, x ) as z (x), where x := (x 1,..., x M ) T. We wll come back to nvestgate the property of functon z later on. Takng all forward contracts as fnancal nstruments, we may regard the fulfllment of these contracts equally as generators commtment to daly power supply over a certan tme perod. Under such contracts, generator gets pad x (p(q, ξ) z (x)) from the other party of the contract when the market clearng prce p(q, ξ) s greater than z (x) and pays the other party by x (z (x) p(q, ξ)) otherwse. Consder a spot market n whch generators set ther dspatch quanttes before the realzaton of the market demand uncertantes. If generator s dspatch quantty s q and the aggregate dspatch from ts rvals s Q, then at a demand scenaro p(, ξ), the market s cleared at the 5
prce p(q + Q, ξ) and each generator s pad at the prce for ther dspatch. Hence, we can formulate generator s revenue from sellng electrcty q by q p(q + Q, ξ). Note that n ths model, a generator can nfluence the market clearng prce and hence ts revenue by choosng a proper q. In realty, some markets allow generators to bd n a stack of quanttes at an ncreasng order of prces for dspatch and the ISO forms a schedule of aggregate quanttes at each prce by puttng them together. After the realzaton of the demand shock, the market clearng prce s determned and all bds below the prce get dspatched whch are pad at the same prce, see for nstance [3] and references theren. Our work smplfes the bddng and clearng mechansm n the real market by lookng at a generator s total dspatch/supply and amng to capture some nsghts on how a generator plays ts strategy to nfluence the spot market by adjustng ts total supply of power, whch s a Cournot model. Fnally, we assume that generaton of an amount q by generator ncurs a total cost of c (q ), whch s twce contnuously dfferentable for any q 0, = 1, 2,..., M. Accordngly, generator s proft n the spot market s R (q, x, Q, ξ) := q p(q + Q, ξ) c (q ) x (p(q + Q, ξ) z (x)). Therefore, generator s decson problem s to choose q to maxmze R (q, x, Q, ξ), where x, ξ, Q and z (x) treated as fxed parameters, that s, max R (q, x, Q, ξ) := q p(q + Q, ξ) c (q ) x p(q + Q, ξ) + x z (x). (2.1) q 0 In the followng, we state two assumptons on each generator s mplct capacty lmt, the dfferentablty of p(, ξ) for ξ Ξ and c ( ) for = 1, 2, M. We frst make the followng assumpton on generators capacty lmts. Assumpton 2.1 For each generator, = 1, 2,..., M, there s a capacty lmt q u, such that c (q ) p(q, ξ), for q q u, ξ Ξ. Observe that, Assumpton 2.1 s an mplct way of ensurng that each generator s dspatch quantty s upper bounded. Ths type of assumptons has been used by Sheral, Soyster and Murphy [28], DeWolf and Smeers [9] n a determnstc verson, and by DeMguel and Xu [8] n a stochastc verson, for the same purpose. The assumpton mples that even generator was a monopoly, ts margnal cost at output level q u or above would exceed any possble market prce. Therefore, none of the frms would wsh to supply more than q u. Moreover, we proceed to make some farly standard assumptons on the nverse demand functon and generators cost functons. Assumpton 2.2 For Q 0 and q 0, = 1, 2,..., M, the nverse demand functon p(q, ξ) and the cost functon c (q ) satsfy the followng: 6
(a) p(q, ξ) s twce contnuously dfferentable w.r.t. Q, and p(q, ξ) s a strctly decreasng and convex functon of Q for every fxed ξ Ξ. (b) p Q (Q, ξ) + Qp Q (Q, ξ) 0, for every Q 0 and ξ Ξ. (c) The cost functon c (q ), = 1, 2,..., M, s twce contnuously dfferentable and c (q ) 0 and c (q ) 0 for any q 0. The assumpton s farly standard and used n [28, 9, 34] except the convexty of the nverse demand functon. The convexty s requred to establsh some techncal results n Lemma 3.1 and t covers a varety of demand functons such as lnear multplcatve functon, soelastc functon and logarthmc functon. From the above assumptons and generators proft functons, we gve the followng proposton to show that each generator s optmal dspatch quantty n the spot market does not depend on the strke prce. Proposton 2.3 Generator s optmal soluton to (2.1) depends on the vector of contract quanttes x, the spot market scenaro ξ and the spot dspatches {q 1,..., q M } but not the strke prces {z 1 (x),..., z (x),..., z M (x)}. Moreover, f generator s contract quantty x s less than q u, then ts margnal proft s negatve for q > q u under Assumpton 2.1 and 2.2. Proof. Consder the dervatve of generator s proft maxmzaton problem (2.1). Snce ξ, x, Q and z (x) are fxed, dfferentatng R w.r.t. q, we have, R (q, x, Q, ξ) q = p(q + Q, ξ) + (q x )p q (q + Q, ξ) c (q ). (2.2) Snce the optmal soluton s determned by the above dervatve whch s ndependent of z (x), the frst part of the concluson follows. To show the second part of the proposton, note that p(q + Q, ξ) c (q ) < 0 for q q u under Assumpton 2.1 and (q x )p q (q + Q, ξ) < 0 when q q u as q u x and p q (q + Q, ξ) < 0. The concluson follows. By Proposton 2.3, we can add the capacty constrant explctly to the proft maxmzaton problem (2.1) : max R (q, x, Q, ξ) = q p(q + Q, ξ) c (q ) x p(q + Q, ξ) + x z (x). (2.3) q [0,q u] A referee rased a queston of whether we can replace the explct capacty lmt by assumng that c (q) ncreases steeply as q approaches q u but not mentonng q u explctly. The potental beneft of dong ths s that we don t need to consder the upper bound n the frst order optmalty condtons to be dscussed n Secton 3. The answer s yes. However, followng Proposton 2.3, we can gnore the upper bound n the dervaton of frst order optmalty condtons anyway because generator s optmum wll not be acheved beyond q u. The addtonal beneft of gvng an explct q u makes our proft maxmzaton problem (2.3) well defned wthout specfyng the propertes of the underlyng objectve functon. 7
2.2 Nash-Cournot equlbrum n the spot market In the spot market, when market demand s realzed, that s, every generator knows the nverse demand functon p(, ξ) gvng the relatonshp between the clearng prce and the aggregate dspatch quantty, and each generator sets ts optmal dspatch quantty to the pool market by solvng proft maxmzaton problem (2.1), whch means that generators play a Nash-Cournot game n the spot market, a stuaton that no generator can mprove ts proft n the spot market by changng ts dspatch unlaterally whle the other players keep ther bds fxed. Followng Proposton 2.3, f there exsts a Nash-Cournot equlbrum n the spot market, t must be ndependent of strke prce z (x), for = 1, 2,..., M. A formal defnton of such an equlbrum can be gven as follows. Defnton 2.4 A Nash-Cournot equlbrum n the spot market at demand scenaro p(, ξ) s an M-tuple (q 1 (x, ξ),, q M (x, ξ)) where q (x, ξ) solves (2.3) for = 1,, M. Remark 2.5 The dependence of q (x, ξ) on x s ntutve and follows from Proposton 2.3. However, the dependence of q (x, ξ) on x j needs some clarfcaton. Let us look at (2.2). If we change x j but q j s not changed accordngly (e.g., q j 0) for j = 1, 2,..., M and j, then Q does not change. In ths case, q (x, ξ) s not affected by the change of x j. Ths mples that only when the change of x j has an mpact on Q, t has an mpact on R (q,x,q,ξ) q, hence the optmal soluton q (x, ξ). Practcally, t means that a generator can nfluence a market equlbrum n the spot market only by changng ts dspatch quantty to the spot market. We wll use ths observaton n Proposton 3.7. From theoretcal pont of vew, there may exst multple equlbra although n practce only one of them s reached. We denote the set of these equlbra by q(x, ξ). We also use q(x, ξ) = (q 1 (x, ξ),, q M (x, ξ)) T to denote an equlbrum n the set q(x, ξ). Note also that the market clearng prce p(q(x, ξ), ξ) s determned by the market equlbrum at the end of competton because the aggregate dspatch s Q(x, ξ) = M =1 q (x, ξ). 2.3 Generator s optmal decson problem n the forward market In the forward market, when generators compete to sgn contracts, they do not know what market clearng prce wll be n the spot market. We assume here that each generator knows: (a) generators play a Nash-Cournot game n the spot market; (b) there s an equlbrum n every scenaro; (c) the nverse demand functon p(, ξ) and the dstrbuton of ξ. Under these assumptons, generator s expected proft can be wrtten as π (x, x ) := E [R (q (x, ξ), x, Q (x, ξ), ξ)], (2.4) where q (x, ξ) and Q (x, ξ) correspond to some equlbrum q(x, ξ) n the spot market, and generator ams to maxmze ts expected proft by choosng an optmal contract quantty x. 8
It s mportant to note that ths s a statstcal average that generator may expect before the competton n the spot market s realzed. Observe that f the spot market has multple equlbra, then each generator may have ts own predcton on an equlbrum q(x, ξ) q(x, ξ), and consequently q(x, ξ) n the term R (q (x, ξ), x, Q (x, ξ), ξ) n (2.4) may depend on, that s, t takes a value dependng on generator s vew about the market equlbra. For nstance, f generator s optmstc, then t may expect the best equlbrum stuaton, that s, to choose q(x, ξ) q(x, ξ) such that R (q (x, ξ), x, Q (x, ξ), ξ) s maxmzed. See a smlar dscusson by Pang and Fukushma [19] n a determnstc Nash equlbrum model and Shapro and Xu [26] n a stochastc mathematcal program wth equlbrum constrants (SMPEC) model. Therefore, the expected proft of generator at the forward market can be formulated as: [ ] ˆπ (x, x ) := E max q (x, ξ)p (Q(x, ξ), ξ) c (q (x, ξ)) x p(q(x, ξ), ξ) + x z (x). q(x,ξ) q(x,ξ) On the other hand, for a pessmstc generator, t may expect the worst equlbrum stuaton, that s, to choose q(x, ξ) q(x, ξ) such that R (q (x, ξ), x, Q (x, ξ), ξ) s mnmzed, and the expected proft of generator at the forward market can be formulated as: [ ] ˇπ (x, x ) := E mn q (x, ξ)p (Q(x, ξ), ξ) c (q (x, ξ)) x p(q(x, ξ), ξ) + x z (x). q(x,ξ) q(x,ξ) Let us now focus on the strke prce n the forward market. In practce, most generators are rsk neutral. That means, wth the perfect knowledge of the dstrbuton of the demand scenaro ξ, no generator wll sgn a contract at a strke prce lower than the expected spot prce, and smlarly retalers wll fnd no advantage to sgn a contract at a strke prce hgher than the expected spot prce. For the smplcty of dscusson, we assume that every generator and retaler s rsk neutral and they have the same vew on a market equlbrum. Ths leads to the followng assumpton. Assumpton 2.6 The strke prce n the forward market equals the expected spot market prce, that s, z (x) { E[p(Q(x, ξ), ξ)] : Q(x, ξ) = q T (x, ξ)e, q(x, ξ) q(x, ξ) }, (2.5) where e s an M-dmensonal vector wth unt components. Ths knd of assumpton s not new and has been made by Gans, Prce and Woods [10], Su [29] and Shanbhag [25, Chapter 5]. Under the rsk neutralty assumpton, f the spot market has a unque equlbrum n every demand scenaro, then we have an dentcal strke prce, that s, z 1 (x) = = z M (x). Of course, f the spot market has multple equlbra, and each generator has dfferent vew on a market equlbrum, then z (x), = 1,, M may take dfferent values and a contract can be agreed only when both partes of the contract have the same vew on spot market equlbrum. 9
2.4 Nash-Cournot equlbrum n the forward market For the smplfcaton of dscusson, we assume that z 1 (x) = = z M (x) ether because generators have the same vews on spot market equlbrum or there s a unque equlbrum n every scenaro. From a practcal perspectve, t means that, to each generator, every unt of contract defnes the same oblgaton of energy dspatchng n the spot market. Therefore, the expected profts of generators at the forward market can be rewrtten as π (x, x ) = E [q (x, ξ)p (Q(x, ξ), ξ) c (q (x, ξ))], for = 1,, M and ts decson problem n the forward market s max π (x, x ) = E [q (x, ξ)p (Q(x, ξ), ξ) c (q (x, ξ))], = 1,, M, (2.6) x 0 that s, generators play a Nash-Cournot game when they compete to sgn contracts n the forward market. We are nterested n the outcome of competton by lookng nto an equlbrum of the Nash-Cournot game. Defnton 2.7 A stochastc equlbrum n the forward-spot market s a 2M tuple (x 1,..., x M, q 1 (x, ξ),..., q M (x, ξ)) such that π (x, x ) = max π (x, x ), = 1,..., M, (2.7) x 0 q (x, ξ) arg max R (q (x, ξ), x, Q (x, ξ), ξ), = 1,..., M, ξ Ξ, (2.8) q 0 and (q 1 (x, ξ),..., q M (x, ξ)) s a Nash-Cournot equlbrum n demand scenaro p(, ξ). The problem s essentally an SEPEC. Recently DeMguel and Xu [8] propose a stochastc multple leader Stackelberg (SMS) model for a general olgopoly market where a group of frms compete to supply homogeneous goods to a future market and they model the problem as an SEPEC. The model extends Sheral s determnstc multple-leader model [27] and De Wolf and Smeers stochastc sngle-leader model [9]. However, there are some fundamental dfferences between ths model and the SMS model: (a) In the SMS model, only a few strategc frms (leaders) play a Nash-Cournot game at the frst stage and the non-strategc frms (followers) do not partcpate n the competton. In our model, all generators compete n the forward market. (b) In the SMS model, leaders do not compete at the second stage after market demand s realzed, and ther commtments (supply) at the frst stage are treated as gven and consequently followers only compete for a resdual demand. In our model, every generator must compete for dspatch n the spot market and ther optmal strategy s affected by ther commtments to forward contracts. 3 Equlbrum n the spot market In ths secton, we nvestgate n detal Nash-Cournot equlbrum n the spot market at demand scenaro p(, ξ). We are partcularly concerned wth exstence, unqueness of equlbrum and 10
propertes of equlbrum as a functon of forward contracts. 3.1 Exstence and unqueness of the equlbrum Frst, before presentng further analyss on the exstence and unqueness of the equlbrum, we gve some results on the strct concavty of each generator s proft functon. Lemma 3.1 Under Assumpton 2.2, for every Q 0 and ξ Ξ () Qp(Q + K, ξ) s a concave functon for any fxed K 0. () For any fxed K 0 and X 0, (Q X)p(Q + K, ξ) s a strctly concave functon of Q for Q 0. The proof to Lemma 3.1 s gven n the appendx. From the strct concavty of the functon (Q X)p(Q, ξ), we can verfy that each generator s objectve functon, R (q, x, Q, ξ), = 1, 2,..., M, s strctly concave w.r.t. q for fxed Q 0, x 0 and ξ Ξ. Proposton 3.2 Let R (q, x, Q, ξ) be defned as n (2.1). Under Assumptons 2.6 and 2.2, R (q, x, Q, ξ) s strctly concave w.r.t. q. The concluson follows straghtforwardly from the convexty of c (q ) and the concavty of (q x )p(q + Q, ξ) that s proved n Lemma 3.1 (). Proposton 3.3 Under Assumptons 2.1, 2.6 and 2.2, for every fxed x [0, + ), = 1, 2,..., M and ξ Ξ, there exsts a unque Nash-Cournot equlbrum n the spot market, q(x, ξ) = (q 1 (x, ξ),..., q M (x, ξ)) T, whch solves the followng problem q (x, ξ) arg max q 0 {R (q, x, Q, ξ) = (q x )p(q + Q, ξ) c (q ) + x z (x)}. Moreover, q (x, ξ) [0, max{q u, x }], for any fxed x and ξ wth = 1,..., M. Proof. Snce generator s objectve functon R (q, x, Q, ξ), s strctly concave n q (here x, ξ are parameters), the exstence of equlbrum follows from [22, Theorem 1] whle the unqueness follows from [22, Theorem 2] because the strct concavty mples the dagonally strct concavty of a weghted non-negatve sum of the objectve functons. Let us now look nto the boundedness of the equlbrum. Because, for any fxed ξ Ξ and x 0, R (q, x, Q, ξ) s strctly concave, we have dr (q, x, Q, ξ) dq = p(q + Q, ξ) + q p Q(q + Q, ξ) c (q ) x p Q(q + Q, ξ) p(q, ξ) + q p Q(q + Q, ξ) c (q ) x p Q(q + Q, ξ) (q x )p Q(q + q, ξ) 0, for any q max{q u, x }. Hence, R acheves maxmum n [0, max{q u, x }]. 11
3.2 Propertes of the equlbrum n the spot market We now nvestgate propertes of Nash-Cournot equlbrum q(x, ξ) n the spot market by takng t as a functon of x and ξ. We wll also nvestgate the monotoncty of aggregate dspatch functon Q(x, ξ) w.r.t.x for = 1, 2,..., M. We do so by reformulatng the Nash-Cournot equlbrum problem n the spot market as a nonlnear complementarty problem. The Karush- Kuhn-Tucker(KKT) condtons of the Nash-Cournot equlbrum problem can be wrtten as p(q, ξ) + (q x )p Q (Q, ξ) c (q ) + µ = 0, 0 µ q 0, (3.9) for = 1, 2,..., M, where 0 µ q 0 denotes that q 0, µ 0 and at least one of them s equal to zero. Denote generators cost functons n a vector-valued form as c(q) = (c 1 (q 1 ),..., c M (q M )) T and e = (1,..., 1) T wth an approprate dmenson. Defne a vector-valued functon G(q, x, ξ) := p(q T e, ξ)e (q x)p Q(q T e, ξ) + c(q), where c(q) := (c 1 (q 1),..., c M (q M)) T. The complementarty problem (3.9) can be rewrtten as 0 q G(q, x, ξ) 0. (3.10) Consequently, each generator s decson problem can be reformulated as a stochastc mathematcal program wth complementary constrants (SMPCC), where, for every = 1,..., M, generator s decson problem s max x 0 E [q (x, ξ)p(q(x, ξ), ξ) c (q (x, ξ))] s.t. q(x, ξ) solves 0 q G(q, x, ξ) 0, ξ Ξ. It s well known that (3.10) can be reformulated as a system of nonsmooth equatons as where mn s taken componentwse. F (q, x, ξ) := mn(g(q, x, ξ), q) = 0, (3.11) In what follows, we use equaton (3.11) to nvestgate the dependence of q on x and ξ. Observe that F s only pecewse smooth, therefore we need to use the Clarke generalzed mplct functon theorem rather than the classcal mplct functon theorem to derve the mplct functon q(x, ξ) defned by (3.11). Defnton 3.4 (Clarke generalzed Jacoban/subdfferental) Let H : R n R m be a Lpschtz contnuous functon. The Clarke generalzed Jacoban [7] of H at w R n s defned as { } H(w) conv lm H(w), y D H, y w where conv denotes the convex hull of a set and D H denotes the set of ponts n a neghborhood of x at whch H s Frechét dfferentable. 12
When m = 1 or n = 1, H s also called Clarke subdfferental. When n = m, the Clarke generalzed Jacoban H(x) s sad to be non-sngular f every matrx n H(x) s non-sngular. From Defnton 3.4, we can observe that the Clarke subdfferental concdes wth the usual gradent H(x) at the pont x where H( ) s strctly dfferentable. Note that a number of functons n ths paper are pecewse contnuously dfferentable, whch means that at most ponts, the Clarke subgradent concdes wth the classcal gradent. The addtonal beneft of the Clarke noton provdes us a dervatve tool to deal wth a few ponts where the classcal dervatves do not exst and tradtonal rght/left dervatve approach make dscussons complcated and ndeed not workng when dealng wth vector valued functons. By usng the Clarke noton, we have a unfed dervatve tool for both the dfferentable ponts and nondfferentable ponts. Theorem 3.5 Let F (q, x, ξ) be defned as n (3.11). Under Assumptons 2.1, 2.2 and 2.6, the followng results hold. () q F (q, x, ξ) s non-sngular for q 0 and x 0. () For every x 0 and ξ Ξ, there exsts a unque q such that F (q, x, ξ) = 0. () There exsts a unque Lpschtz contnuous and pecewse smooth functon q(x, ξ) defned on [0, + ) Ξ such that F (q(x, ξ), x, ξ) = 0. The theorem above shows that under Assumptons 2.1, 2.2 and 2.6, there exsts a unque Nash-Cournot equlbrum n the spot market for every x and ξ, and the equlbrum s a vector valued functon of x and ξ whch s Lpschtz contnuous and pecewse smooth. In what follows, we nvestgate the subdfferentals of the dspatch functon q(x, ξ) n the spot equlbrum and the aggregate dspatch Q(x, ξ) w.r.t. x and ξ. Ths s to examne the mpact of the changes of ndvdual generator s contract level and random shock ξ on the market equlbrum and the aggregate dspatch n the spot market. We need the followng assumpton to guarantee that, for every demand scenaro, there s at least one generator whose dspatch quantty to the spot market s strctly postve. Obvously, ths s always satsfed n the real electrcty market. Assumpton 3.6 Suppose that, for every ξ Ξ and x sgned n the forward market, the nverse demand functon p(, ξ) and the cost functons c ( ) satsfy mn =1,...,M c (0) < p(q(x, ξ), ξ). (3.12) The assumpton mples that at any demand scenaro, and for any contract quanttes x sgned n the forward market, there s at least one generator whose margnal cost of producng a very small amount of electrcty s strctly lower than the market clearng prce, whch means that there exsts at least one generator whch s proftable by supplyng a small amount of electrcty n the spot market. Ths assumpton excludes the case that no generator s wllng to sell electrcty n a partcular scenaro. 13
Proposton 3.7 Let F (q, x, ξ) be defned as n (3.11). Under Assumptons 2.1, 2.2, 2.6 and 3.6, we have the followng. () The Clarke generalzed Jacoban of q(x, ξ) w.r.t. x can be estmated as follows: x q(x, ξ) conv { W 1 U : (W, U, V ) F (q(x, ξ), x, ξ), W R M M, U R M, V R }. (3.13) () The Clarke subdfferental of the aggregate dspatch functon, Q(x, ξ), w.r.t. x, for = 1,..., M, can be estmated as x Q(x, ξ) [0, 1). The lower bound s reached only when q (x, ξ) = 0. () The Clarke subdfferental of generator s dspatch functon, q (x, ξ), w.r.t. estmated as x, can be x q (x, ξ) [0, 1). The lower bound s reached only when q (x, ξ) = 0. (v) The Clarke subdfferental of q (x, ξ), w.r.t. x j can be estmated as xj q (x, ξ) ( 1, 0]. The upper bound s reached only when q j (x, ξ) = 0. (v) If p Q,ξ (Q, ξ) = 0, then q (x, ξ) s an ncreasng functon of ξ; moreover, f there exsts a constant C 0 such that p Q(Q, ξ) + p Q(Q, ξ)(q x) T e < C, for Q 0, x 0 and ξ Ξ, then the Clarke subdfferental of Q(x, ξ) w.r.t. ξ can be estmated as follows: ξ Q(x, ξ) (0, 1C ] p ξ (Q(x, ξ), ξ). We provde a proof on these techncal results n the appendx. Moreover, some economc nterpretatons for these results can be gven as followng: Part () ndcates that every unt ncrease of contract quantty by a generator n the forward market wll result n an ncrease of the aggregate dspatch of all generators n the spot market by less than one unt. Part () has a smlar nterpretaton for an ndvdual generator. Part (v) means that generator s dspatch wll be reduced by less than one unt f one of ts rvals ncreases one unt n ts contract quantty. To gve an ntutve nterpretaton of the results n ths secton, we present a smple example of a duopoly market. 14
Example 3.8 Consder an electrcty market wth two generators, A and B. The generators cost functons are c A (q A ) = 0.8q A, c B (q B ) = q B, where q A and q B denote A and B s quanttes for dspatches n the spot market, respectvely. We assume that the nverse demand functon s p(q A + q B, ξ) = α(ξ) β(q A + q B ), where α(ξ) = 7 + ξ, β = 2, and the random shock ξ follows a unform dstrbuton on the set [0, 1]. Denote the contract postons of A and B n the forward market by x A and x B. The nverse demand functon after the realzaton of the random shock ξ s p(q A + q B, ξ) = 7 + ξ 2(q A + q B ). Let qa u = 3.6 and qu B = 3.5 be the capacty lmts of A and B. In the spot market, generator A and B s proft maxmzaton problems can be respectvely wrtten as max R A(q A, q B, x, ξ) = 2q 2 q A [0,qA u ] A + q A(6.2 2q B + ξ + 2x A ) x A (7 2q B + ξ), max R B(q B, q A, x, ξ) = 2q 2 q B [0,qB u ] B + q B(6 2q A + ξ + 2x B ) x B (7 2q A + ξ), (3.14) It s easly verfy that, for any ξ [0, 1], q A qa u and q B qb u, we have the followng nequaltes, p(q A, ξ) 7 + ξ 2q u A 0.8 = c A (q A), p(q B, ξ) 7 + ξ 2q u B 1 = c B (q B), (3.15) whch mples that Assumpton 2.1 holds n ths example. Accordng to our dscusson followng Assumpton 2.1, (3.15) mplctly ensures that the optmal soluton q (x, ξ) satsfy q (x, ξ) qu for = A, B and wll never go beyond qa u and qu B n every scenaro ξ Ξ. Therefore, the constrants q q u for = A, B n (3.14) are not actve, and generator A and B s proft maxmzaton problems can be respectvely reformulated as max R A(q A, q B, x, ξ) = 2qA 2 + q A (6.2 2q B + ξ + 2x A ) x A (7 2q B + ξ), q A 0 max R B(q B, q A, x, ξ) = 2qB 2 + q B (6 2q A + ξ + 2x B ) x B (7 2q A + ξ), q B 0 where R A and R B are quadratc functons. Therefore, the optmal dspatches n the spot market satsfy the followng frst-order condtons: 0 q A (x, ξ) 4q A (x, ξ) (6.2 2q B (x, ξ) + ξ + 2x A ) 0, 0 q B (x, ξ) 4q B (x, ξ) (6 2q A (x, ξ) + ξ + 2x B ) 0. (3.16) Note that, the case q A = q B = 0 s excluded by Assumpton 3.6 for (3.17). From (3.16), we have ( 0, 1 4 (2x B + 6 + ξ) ), f q A = 0; ( (q A (x, ξ), q B (x, ξ)) = 1 4 (2x A + 6.2 + ξ), 0 ), f q B = 0; ( 1 6 (4x A 2x B + ξ + 6.4), 1 6 (4x B 2x A + ξ + 5.8) ) (3.17), otherwse. 15
(3.17) mples that x q s a subset of [0, 1/2] or [1/2, 2/3] for = A, B,and xj q [ 1/3, 0] for, j = A, B and j, whch verfes the results () and (v) Proposton 3.7. Moreover, the aggregated dspatch quantty can be wrtten as 1 4 (6 + ξ + 2x B), f q A = 0; Q(x, ξ) = 1 4 (6.2 + ξ + 2x A), f q B = 0; 1 3 (6.1 + ξ + x A + x B ), otherwse, (3.18) whch mples x Q s a subset of [0, 1/2] or [1/3, 1/2], and hence the result () n Proposton 3.7. Observe that, (3.17) and (3.18) provde us wth a further propertes, that s, at the demand scenaro ξ, f q (x, ξ) 0 for every x, then xj q (x, ξ) {0} for, j = A, B and x Q(x, ξ) {0}. Ths fact verfes the lower bounds n the results () and (), and the upper bound n the result (v) n Proposton 3.7. 4 Equlbrum n the forward market In ths secton, we nvestgate the competton n the forward market. We do so by lookng nto the exstence of a Nash-Cournot equlbrum n the forward market as defned n Defnton 2.7. For the smplfcaton of dscusson, we assume that the spot market has a unque Nash-Cournot equlbrum, q(x, ξ) = (q 1 (x, ξ), q 2 (x, ξ),..., q M (x, ξ)) T for every x and ξ. Frst, from Proposton 3.7, we can establsh a relatonshp between the strke prce and the contract quanttes n the followng proposton. Proposton 4.1 Under Assumptons 2.1, 2.2 and 2.6, the strke prce z s a functon of the contract quanttes x, that s, z(x) = E[p(Q(x, ξ), ξ)]. Moreover, the elements n the set x z(x) are all non-postve. Proof. Under Assumpton 2.6 and the unqueness of the supply functons q (x, ξ) n the spot equlbrum, we have, z(x) = E[p(Q(x, ξ), ξ)]. The Clarke subdfferental of z(x) s x z(x) = x E[p(Q(x, ξ), ξ)]. Snce the nverse demand functon p(q, ξ) s a contnuously dfferentable functon of Q(see Assumpton 2.2), and Q(x, ξ) s a Lpschtz contnuous functon of each x proved n Proposton 3.5(), we have, p(q(x, ξ), ξ) s also a Lpschtz contnuous functon of x. Therefore, from [7, Theorem 2.7.5], x E[p(Q(x, ξ), ξ)] E[ x p(q(x, ξ), ξ)] E[p Q(x, ξ) x Q(x, ξ)]. Moreover, by Part () of Theorem 3.5, E[p Q(Q(x, ξ), ξ) x Q(x, ξ)] (p Q(Q(x, ξ), ξ), 0] (, 0]. 16
Ths completes the proof. Proposton 4.1 establshes a relatonshp between the strke prce and a generator s contract quantty n the forward market, n whch the negatvty of the elements n x z(x) mples that any unlateral ncrease of the contract quantty by a generator never results n an ncrease of the strke prce. 4.1 Dfferentablty of the expected proft We now dscuss the contnuty and dfferentablty of a generator s objectve functon n the forward market and nvestgate the change of the expected proft of an ndvdual generator aganst the change of ts contract quantty. To avod too much mathematcal detals and make our analyss more readable, we move all the detaled proofs of the lemmas and theorem n ths subsecton to the appendx. We start by consderng the frst order dervatve. Recall that π (x, x ) := [q (x, ξ)p(q(x, ξ), ξ) c (q (x, ξ))] ρ(ξ)dξ, for = 1, 2,..., M. ξ Ξ Obvously, the only component n the ntegrand whch may cause nondfferentablty of the ntegrand and hence π (x, x ) s q j (x, ξ), j = 1,..., M and j. In what follows, we demonstrate that under some moderate condton, the pecewse smoothness of q(x, ξ) may not cause nondfferentablty of π (x, x ). Assumpton 4.2 The nverse demand functon and the cost functons satsfy the followng. () For any fxed ξ Ξ, there exsts an L 1 (ξ) 0 such that max ( p Q(Q(x, ξ), ξ), p Q(Q(x, ξ), ξ), c (q (x, ξ)) ) L 1 (ξ), x 0, = 1, 2,..., M, and sup ξ Ξ L 1 (ξ) <. () There exsts a constant σ 0 such that p Q (Q(x, ξ), ξ) + c (q (x, ξ)) > σ, for all ξ Ξ and x 0 for = 1, 2,..., M. Under Assumpton 4.2, we need a couple of ntermedate results, Lemma 4.3 and Lemma 4.4, to obtan the man result on the twce contnuous dfferentablty of q (x, ξ) w.r.t. x n Theorem 4.5. For the clarty of notaton, we wrte q (x, ξ) as q (x, x, ξ) to dstngush x and x because x wll be treated as parameters when we analyze the senstvty of the quanttes w.r.t. x. Lemma 4.3 Under Assumptons 2.1, 2.6, 2.2, 3.6 and 4.2, the followng results hold. () For each = 1,..., M, q (x, x, ξ) s a pecewse contnuously dfferentable and ncreasng functon of x. 17
() For x j 0, j = 1, 2,..., M, j and ξ Ξ, q (x, x, ξ) s globally Lpschtz contnuous w.r.t. x ; that s, there exsts a functon L 2 (ξ), = 1,..., M, such that q (x (1), x, ξ) q (x (2), x, ξ) L 2(ξ) x (1) x (2), x (1), x (2) 0, where ξ Ξ L 2 (ξ)ρ(ξ)dξ <. From the part () of Lemma 4.3, we know that q (x, x, ξ) s a nondecreasng functon n x, and thus there exsts a unque pont at whch q (x, x, ξ) turns from zero to postve as x ncreases, and we denote ths pont by x (ξ). In economc terms, gven the contract poston x sgned by generator s rvals, for a realzed demand shock ξ Ξ, x (ξ) s the contract poston at whch generator s margnal proft n the spot market becomes from zero to postve, and ts dspatch quantty also becomes from zero to postve. Mathematcally, x (ξ) can be regarded as a degenerate pont of the complementarty problem (3.10) because at ths pont, both G (q(x, ξ), x, ξ) and q (x, x, ξ) are equal to zero. Note that q (x, x, ξ) s not dfferentable w.r.t. x at the pont x (ξ).from a practcal perspectve, part () of Lemma 4.3 mples that, the more contracts (n the sense of quanttes) a generator sgns n the forward market, the more dspatch the generator wll commt n the spot market. In what follows, we nvestgate the set of degenerate ponts x (ξ) for a gven x. Ths s because these degenerate ponts may result n non-dfferentablty of the ntegrand of π (x, x ) and potentally further result n the non-dfferentablty of π (x, x ) f there are too many such ponts (n the sense that the Lebesgue measure of the set of such ponts s non-zero). The followng lemma states that the number of degenerate ponts are actually fnte whch mples that they wll not cause non-dfferentablty of π (x, x ). Lemma 4.4 Let Ξ (x) := {ξ Ξ x = x (ξ)} and Ξ(x) := M =1 Ξ (x). Under Assumptons 2.1, 2.6, 2.2, 3.6 and 4.2, Ξ(x) s a fnte set for any x. Note that, from the defnton of Ξ(x), gven x, Ξ (x) φ means that there s a ξ such that generator s dspatch quantty q (x, x, ξ) turns from zero to strctly postve, that s, the th element of x s x (ξ). Therefore, Ξ (x) s the set of ponts ξ Ξ at whch generator s dspatch quantty turns from zero to postve, and Ξ(x) s the set of ponts ξ Ξ at whch the dspatch quantty of at least one of generators turns from zero to postve. As observed n the proof of Lemma 4.4 n the appendx, x (ξ) s a decreasng functon of ξ to mantan the property that q (x (ξ), x, ξ) 0 for x x (ξ) and q (x (ξ), x, ξ) > 0 for x > x (ξ). For gven x and x, let us defne v (x, ξ) := (q (x, x, ξ) x )p(q(x, ξ), ξ) c (q (x, x, ξ)). (4.19) The only values of ξ at whch v (, ξ) mght not be dfferentable w.r.t. x are ponts ξ at whch the dspatch of one of generator turns from postve to zero. These are only ponts at whch 18
Q(x, ξ) mght not be dfferentable w.r.t. x and thus v (, ξ) mght not be dfferentable w.r.t. x. By Lemma 3.4 n [8], Ξ (x ) s a fnte set, whch mples that Q(x, ξ) s dfferentable w.r.t. x for almost every ξ Ξ and thus v (x, ξ) s dfferentable w.r.t. x for almost every ξ Ξ. We are now able to address the man results of ths secton. Theorem 4.5 Suppose that there exsts L 3 (ξ) 0 such that Ξ L 3(ξ)ρ(ξ)dξ < and max ( p(q, ξ), p Q(Q, ξ), Q x (x, ξ) ) L 3 (ξ), for all Q 0, ξ Ξ and x wth = 1,..., M, at whch Q(x, ξ) s twce contnuously dfferentable w.r.t. x. Then, under Assumptons 2.1, 2.6, 2.2, 3.6 and 4.2, π (x, x ) s twce contnuously dfferentable. In what follows, we explan Theorem 4.5 through a smple example based on Example 3.8. Example 4.6 (Contnued from Example 3.8) Consder a duopoly market as descrbed n Example 3.8. From the defnton, for any fxed x B, the degenerate pont x A (ξ) (at whch q A (x A, x B, ξ) turns from zero to postve as x A ncreases) can be dentfed by solvng the followng equatons q A (x A (ξ), x B, ξ) = 0; and 4q A (x, ξ) (6.2 2q B (x A (ξ), x B, ξ) + ξ + 2x A (ξ)) = 0, where x A (ξ) 0. By solvng (3.16), we obtan x A (ξ) = 1 2 (x B 0.5ξ 3.2), for fxed x B. (4.20) Combnng the condton that x A (ξ) 0, we can see that for fxed x A there exsts at most one ξ such that q A (x A, x B, ξ) s possbly not dfferentable. Ths mples that the cardnalty of the set Ξ A (x) s at most 1, and hence verfes Lemma 4.4. In what follows, we look nto Theorem 4.5. For the sake of smplcty, we only verfy the dfferentablty of π A (x A, x B ) n x 1 A and x B 3.2 2. We can obtan q A and q B by solvng the followng complementarty problem: 4q A (x A, x B, ξ) (6.2 2q B (x A, x B, ξ) + ξ + 2x A ) = 0, 0 q B (x A, x B, ξ) 4q B (x A, x B, ξ) (6 2q A (x A, x B, ξ) + ξ + 2x B ) 0. (4.21) From (3.15) n Example 3.8, q A and q B can be expressed as: { (qa I, qi B ) = ( 1 4 (6.2 + ξ + 2x A), 0 ), f ξ [0, 2x A 4x B 5.8], (qa II, qii B ) = ( 1 6 (6.4 + ξ + 4x A 2x B ), 1 6 (5.8 + ξ + 4x B 2x A ) ), f ξ [2x A 4x B 5.8, 1], where the two smooth peces {(qa I, qi B )} ( and {(qii A, qii B ) )} ntersect at the pont x A = 2x B + 2.9 + 0.5ξ, where (qa I, qi B ) = (qii A, qii B ) = 6.2+ξ+2xA 4, 0. In other words, at any fxed pont x A 1 As we can do for x B n the same way. 2 There wll be two nondfferentable ponts when x B > 3.2. 19
and x B, there s at most one ξ such that q A and q B are not dfferentable w.r.t. varable x A. Consequently, generator A s expected proft n the forward market can be calculated as follows: 1 [ 0 q I A p(q I, ξ) c A (qa I )] ρ(ξ)dξ, f ξ 1; π A (x A, x B ) = ξ [ 0 q I A p(q I, ξ) c A (qa I )] ρ(ξ)dξ + [ 1 ξ q II A p(qii, ξ) c A (qa II)] ρ(ξ)dξ, f 0 < ξ < 1; [ q II A p(qii, ξ) c A (qa II)] ρ(ξ)dξ, f ξ 0, 1 0 where Q I = q I A + qi B, QII = q II A + qii B and ξ denotes ξ(x A ) := 2(x A 2x B 2.9). Calculatng the dervatve π A(x A,x B ) x A π A (x A, x B ) x A = ξ(xa ) 0 for the case 0 < ξ < 1, we have [ qa I p(qi, ξ) c A (qa I )] ρ(ξ)dξ x A + ξ(x A ) [ q I x A (x A, x B, ξ(x A ))p(q I (x, ξ(x A )), ξ(x A )) c A (qa(x I A, x B, ξ(x A ))) ] ρ( ξ(x A )) A + 1 ξ(x A ) ξ(x A ) x A [ qa IIp(QII, ξ) c A (qa II)] ρ(ξ)dξ x A Snce at the pont x A = 2x B + 2.9 + 0.5 ξ(x A ), [ q II A (x A, x B, ξ(x A ))p(q II (x, ξ(x A )), ξ(x A )) c A (q II A (x A, x B, ξ(x A ))) ] ρ( ξ(x A )). ( q I A (x A, x B, ξ(x A )), q I B(x A, x B, ξ(x A )) ) = ( q II A (x A, x B, ξ(x A )), q II B (x A, x B, ξ(x A )) ). and then π A (x A, x B )/ x A above can be smplfed as π A (x A,x B ) x A = ξ(xa ) [qa I p(qi,ξ) c A (q I )] A 0 x A ρ(ξ)dξ + [q 1 ξ(xa II ) A p(qii,ξ) c A (q II )] A x A ρ(ξ)dξ. (4.22) Moreover, for ξ 1 and ξ 0 we have π A (x A, x B ) = x A 1 [qa I p(qi,ξ) c A (q I )] A 0 1 [qa IIp(QII,ξ) c A (q II 0 x A ρ(ξ)dξ, f ξ 1; )] A x A ρ(ξ)dξ, f ξ 0, (4.23) Combnng both (4.22) and (4.23), we can see that π A(x A,x B ) x A s a contnuous functon of x A and hence π A (x A, x B ) s contnuously dfferentable w.r.t. x A. Repeatng the process above on dervatve π A(x A,x B ) x A, we can show that π A (x A, x B ) s twce contnuously dfferentable. Ths verfes the result n Theorem 4.5. 4.2 Exstence of the forward-spot equlbrum We now move on to dscuss the exstence of Nash-Cournot equlbrum n the forward-spot market. A well known suffcent condton for the exstence s the concavty or quas-concavty of each generator s objectve functon on ts strategy space. See for nstance [22, Theorem 1] and [37, Theorem 1]. It turns out, however, very dffcult to show ths knd of global concavty here. For ths reason, we look nto the local concavty and consequently nvestgate the exstence 20
of local Nash equlbrum. The noton s used by Hu and Ralph for modelng a blevel games n an electrcty markets wth locatonal prces. See [15] for detals. As noted n [15], the concept of local Nash equlbrum s proposed as a weaker alternatves to Nash equlbrum for the electrcty market. From a vewpont of n the real market, gven that the global optma of nonconcave maxmzaton problems are dffcult to dentfy, the lmtaton of knowledge of generators may lead to meanngful local Nash equlbra, n whch the local optmalty s suffcent for the satsfacton of generators. Moreover, gven the condton that the spot market s always proftable for every generator at every scenaro ξ, we establsh our man results on the exstence of the global Nash equlbrum n the forward-spot market. We start by gvng a defnton on local Nash equlbrum. Defnton 4.7 (Local Nash equlbrum) x s a local Nash-Cournot equlbrum of the forward market f for each, x s a local optmal soluton to the problem max π (x, x ) = E[q (x, x, ξ)p(q(x, x, ξ), ξ) c (q (x, x, ξ))], = 1, 2,..., M. x 0 Comparng to ther global counterparts, local Nash equlbra seem defcent. However, for some decson-makng problems, gven that global optma are dffcult to dentfy because of the nonconcave objectve functons, local optmalty may be suffcent for the satsfacton of players. For nstance, generators may only optmze ther contract postons locally due to lmted nformaton on the forward market or general conservatveness. To llustrate the exstence of the local Nash-Cournot equlbrum n the forward-spot market, we present the followng example based on the duopoly model n Example 3.8. Example 4.8 (Contnued from Example 4.6) Consder a duopoly market descrbed n Example 4.6, n whch the capacty lmts of generator A and B are qa u = 3.6 and qu B = 3.5, respectvely. Defne X = {(x A, x B ) x A > 0.4, x B > 0.6}, and x := (x A, x B ) X. Let X + = {x = (x A, x B ) q A (x, ξ) > 0, q B (x, ξ) > 0, ξ [0, 1]}. That s, f contract poston x = (x A, x B ) s n X +, then for all ξ Ξ, the dspatch of each generator n the spot market s always strctly postve. It s easy to verfy that X + s an open convex set. Let x X X + (the set X X + s nonempty, open and convex). It s easy to derve that the optmal dspatches n spot market satsfy the followng: q A (x, ξ) = 1 4 (6.2 2q B(x, ξ) + ξ + 2x A ), q B (x, ξ) = 1 4 (6 2q A(x, ξ) + ξ + 2x B ), q A (x, ξ) > 0, for all ξ Ξ, q B (x, ξ) > 0, for all ξ Ξ, (4.24) 21