Pricing Mechanisms for Economic Dispatch: A Game-Theoretic Perspective

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1 Prcng Mechansms for Economc Dspatch: A Game-Theoretc Perspectve Wenyuan Tang a, Rahul Jan a a Unversty of Southern Calforna, Los Angeles, CA 90089, USA Abstract The economc dspatch problem s to determne a socally optmal allocaton of supply and demand subject to transmsson constrants. Wth strategc generators possessng market power, the locatonal margnal prcng mechansm may not nduce the economc dspatch. We nvestgate the market outcomes from a game-theoretc perspectve, employng pecewse constant offer curves as n practce, n contrast to affne offer curves as n the lterature. We present counterexamples to show that a Nash equlbrum may not exst, or that the prce of anarchy can be arbtrarly large. On the other hand, we provde suffcent condtons under whch there exst effcent Nash equlbra. We then study the LMP mechansm wth Cournot offers. As an alternatve, we propose the margnal contrbuton prcng mechansm that guarantees effcent outcomes. Keywords: Cournot competton, economc dspatch, electrcty market, game theory, locatonal margnal prcng, supply functon equlbrum 1. Introducton Electrc power s traded n a wholesale electrcty market that nvolves varous enttes: the generators who generate and sell power, the dstrbutors who buy power and sell to consumers, and the ndependent system operator (ISO) who coordnates the operaton of the market and the power system. The generators and the dstrbutors submt economc sgnals for supply and demand to the ISO, who then determnes an optmal dspatch that maxmzes the socal welfare whle satsfyng the physcal and operatonal constrants. Ths problem s referred to as the economc dspatch problem. Whle the economc dspatch s computed, the prce pad (or charged) to each generator (or dstrbutor) s determned accordngly. Locatonal margnal prcng (LMP) s a wdely used mechansm for prcng electrcty n the wholesale electrcty market [1, 2]. When the market s compettve, the LMP mechansm s effcent,.e., nducng the economc dspatch. Emal addresses: wenyuan@usc.edu (Wenyuan Tang), rahul.jan@usc.edu (Rahul Jan) Preprnt submtted to Elsever March 15, 2015

2 However, those market partcpants are strategc agents who potentally possess market power. They may not have ncentves to reveal ther true characterstcs, or may even provde msleadng nformaton for ther own nterests. For example, Enron s energy traders found ways to manpulate the congeston prces that led to the Calforna electrcty crss of [3]. Whle market power can be measured usng market power ndces [4], those ndces have ther own lmtatons and may not adequately characterze the stuaton of market manpulaton. In ths work, we adopt a game-theoretc approach to study the LMP mechansm through nvestgatng the equlbrum outcomes. We focus on the strategc bddng of the supply sde, assume general cost functons, and take the transmsson constrants nto account. Our model dffers from the supply functon equlbrum (SFE) models [5, 6], n that we employ pecewse constant offer curves as n practce, n contrast to affne offer curves as n the SFE lterature. Moreover, we explore the exstence and the effcency of Nash equlbra, whle most of the exstng work focuses on computng SFEs [7, 8, 9, 10]. Our man results are twofold. On the one hand, we show that a Nash equlbrum may not exst, or that the prce of anarchy can be arbtrarly large. On the other hand, we provde suffcent condtons under whch there exst effcent Nash equlbra. We then study the LMP mechansm wth Cournot offers, n whch generators submt quantty offers nstead of prce/quantty offer curves. Due to the tractablty, Cournot models have ganed popularty n electrcty market analyss [11, 12, 13]. We establsh the exstence of Nash equlbra under certan mld condtons. Furthermore, there are other mechansms as alternatves to the LMP mechansm. For example, the pay-as-bd prcng mechansm s compared wth the LMP mechansm n [14]. We propose the margnal contrbuton prcng (MCP) mechansm whch always nduces effcent Nash equlbra, usng pecewse constant offer curves, Ths paper s organzed as follows. In Secton 2, we ntroduce the economc dspatch problem and the LMP mechansm. We then formulate the economc dspatch game n Secton 3. We provde counterexamples n Secton 4 to llustrate the phenomena of market manpulaton, and suffcent condtons n Secton 5 whch guarantee market effcency. In Secton 6, we study the LMP mechansm wth Cournot offers. In Secton 7, we propose the MCP mechansm as an alternatve to the LMP mechansm. Fnally, we present a case study n Secton 8 to llustrate the equlbrum outcomes under varous prcng mechansms and offer formats. Secton 9 concludes the paper. 2. Model 2.1. Economc Dspatch We consder a connected power network whch conssts of I buses ndexed by = 1,..., I, and N generators ndexed by n = 1,..., N. There can be zero, one, or more generators located at each bus, the set of whch s denoted by 2

3 N. For each generator n, the cost of supplyng x n s c n (x n ). The cost functon c n : R + R + s assumed to be strctly ncreasng, convex and pecewse dfferentable. The compettve demand at each bus s modeled by an nverse demand functon p (y ), whch maps the quantty y to the maxmum prce that the consumers are wllng to pay. The nverse demand functon p : R + R + s assumed to be contnuous and decreasng. It s sometmes convenent to work wth the valuaton functon v (y ) = y 0 p (z)dz, whch s ncreasng, concave and contnuously dfferentable. In ths paper, we wll also consder nelastc demand, n whch case all the y s are fxed. For analytcal and computatonal smplcty, we adopt a DC power flow model as a common practce [15]. In the DC flow model, a branch -j s characterzed by B j, the negatve of ts susceptance, whch satsfes B j = B j 0. Let θ be the voltage phase angle at bus. Then the actve power flow over branch -j s gven by f j = B j (θ θ j ). The bus power balance equaton for bus s the followng: x n y = f j. n N j Let C j be the flow lmt of branch -j, whch satsfes C j = C j 0. The branch power flow constrant for branch -j s the followng: f j C j. Let x = (x 1,..., x N ), y = (y 1,..., y I ), and θ = (θ 1,..., θ I ). The economc dspatch problem s to determne an optmal allocaton of supply and demand that maxmzes the socal welfare whle satsfyng the transmsson constrants. Formally, t s modeled as a convex programmng problem: maxmze x,y,θ subject to v (y ) c n (x n ) n x n y = B j (θ θ j ),, n N j B j (θ θ j ) C j, (, j), x n 0, n, y 0,. (1a) (1b) (1c) (1d) The prmal optmal soluton to (1) s called the economc dspatch, denoted by (x, y, θ ) throughout the paper. (1e) 3

4 Snce ( n N x n y ) = j B j(θ θ j ) = 0, the system of lnear equatons (1b) over s underdetermned wth respect to θ. In fact, only the phase angle dfferences matter. Thus, for computatonal purposes, we choose bus 1 as the slack bus by settng θ 1 = 0. The economc dspatch problem s based on the gven on/off states of the generators, n comparson wth the unt commtment problem. In the proposed model, the N generators have been started up, so that the fxed cost of each generator does not affect the dspatch. Thus, we further assume c n (0) = 0 for all n wthout loss of generalty. For a more detaled DC power flow model (e.g., where lne losses are taken nto account), the reader can refer to [16] Locatonal Margnal Prcng Assocate the dual varables λ wth (1b) and µ j wth (1c). Let λ = (λ 1,..., λ I ) and µ = [µ j ] I I. It s easly seen that the economc dspatch problem (1) s always feasble, and that the (refned) Slater s condton s automatcally satsfed. Therefore, strong dualty holds, and the Karush-Kuhn-Tucker (KKT) condtons provde necessary and suffcent condtons for optmalty [17]: (c n(x n ) λ )x n = 0, n N, (2a) c n(x n ) λ 0, n N, (2b) (v (y ) λ )y = 0,, (2c) v (y ) λ 0,, B j (λ λ j + µ j µ j ) = 0,, j x n y B j (θ θ j ) = 0,, n N j (2d) (2e) (2f) µ j (B j (θ θ j ) C j ) = 0, (, j), (2g) B j (θ θ j ) C j 0, (, j), x n 0, n, y 0,, (2h) (2) (2j) µ j 0, (, j). (2k) Note that when c n s not dfferentable at x n, c n(x n ) should be nterpreted as a subdervatve, whch always exsts gven the assumptons of the cost functon. Throughout the paper, denote by (λ, µ ) the dual optmal soluton to (1). Then (x, y, θ, λ, µ ) satsfes the KKT condtons (2). The locatonal margnal prce (LMP) at each bus s gven by λ, whch s the cost of servng an addtonal ncrement of load at that bus. Thus, the payoff of generator n located at bus s gven by π n = λ x n c n (x n ). 4

5 The LMP mechansm s effcent n a compettve envronment,.e., when generators are prce takers. If the prce at each bus s fxed as λ, and each generator determnes ts supply to maxmzes ts own payoff, then the resultng allocaton s the economc dspatch. Ths can be seen from the KKT condtons, whch state that the margnal cost of a generator wth postve generaton s exactly the LMP at that bus. 3. Game-Theoretc Formulaton To solve (1) for the economc dspatch, the ISO needs to know the cost functons. However, strategc generators may not have ncentves to reveal ther true cost functons. They are often not even able to do so, snce the offer s typcally a low-dmensonal sgnal that may not fully parametrze the cost functon. Ths motvates the formulaton of the economc dspatch game. We frst specfy the strategy space, or the offer format Offer Format An offer curve p n (x n ) maps the quantty x n to the mnmum prce that generator n s wllng to accept. It s supposed to approxmate the margnal cost functon c n(x n ), but not necessarly. The offer curve p n (x n ) corresponds to a reported cost ĉ n (x n ) = xn 0 p n (z)dz. The offer format refers to the parametrzaton of the offer curve, or equvalently, the parametrzaton of the reported cost. In the SFE lterature, the offer curve often takes an affne form: p n (x n ) = a n + b n x n, a n > 0, b n > 0, whch corresponds to a quadratc reported cost ĉ n (x n ) = a n x n + (b n /2)x 2 n. In practce, however, most ISOs adopt pecewse constant (or step) offer curves. Such an offer curve corresponds to a pecewse lnear reported cost. For example, the Calforna ISO asks for 10 prce/quantty pars [18], whch correspond to a 10-segment pecewse lnear reported cost. In ths paper, we employ the realstc offer format. For smplcty, we consder a two-segment pecewse lnear reported cost parametrzed by a four-dmensonal sgnal (rn, s n, r n +, s + n ), where 0 rn r n +, 0 s n s + n : { rn x n, x n [0, s n ], ĉ n (x n ) = rn s n + r n + (x n s n ), x n (s n, s + n ]. 5

6 p n (x n ) ĉ n (x n ) O p n (x n ) x n (a) Affne offer curve and quadratc reported cost. ĉ n (x n ) O x n r + n r + n r n r n O x n s n s + n O s n s + n x n (b) Pecewse constant offer curve and pecewse lnear reported cost. Fgure 1: Comparson of offer formats used n the SFE lterature and n ths paper. Fg. 1 llustrates the offer format used n the SFE lterature and that used n ths paper. We note that quadratc costs provde smooth dspatch, revenue and proft curves that facltate calculus-based analyss, whle pecewse lnear costs do not produce contnuously dfferentable dspatch, revenue and proft curves, requrng dfferent analyss technques [19]. Ths dstngushes our work from the SFE lterature. 6

7 3.2. Economc Dspatch Game Let ĉ = (ĉ 1,..., ĉ N ) be the offer profle,.e., the collecton of reported cost. In realty, the ISO solves the followng problem: maxmze v (y ) ĉ n (x n ) (3a) x,y,θ n subject to x n y = B j (θ θ j ),, (3b) n N j B j (θ θ j ) C j, (, j), x n 0, n, y 0,. (3c) (3d) Compared wth (1), the true costs are replaced by the reported costs n (3). Let (x, y, θ ) be the prmal optmal soluton to (3). The LMP at each bus s gven by λ, the optmal dual varable assocated wth the bus power balance equaton (3b). Thus, the payoff of generator n located at bus s gven by π n = λ x n c n (x n). Ths completes the specfcaton of the economc dspatch game. We adopt the pure strategy Nash equlbrum as the soluton concept. An offer profle s a Nash equlbrum f no generator can get a hgher payoff by a unlateral devaton. A Nash equlbrum s effcent f t nduces the economc dspatch (x, y, θ ). (3e) 4. Market Manpulaton We have formulated an economc dspatch game to study the LMP mechansm wth the realstc offer format. In ths secton, we llustrate the phenomena of market manpulaton. Specfcally, we present counterexamples to show that a Nash equlbrum may not exst, and that even when a Nash equlbrum exsts, the prce of anarchy can be arbtrarly large Nonexstence of Nash Equlbra The followng example s adapted from [7], where SFEs are computed wth affne offer curves. Our focus, however, s on the exstence of Nash equlbra wth pecewse lnear offer curves. Example 1. Consder the network as shown n Fg. 2. There are two buses, wth generator 1 at bus 1 and generator 2 at bus 2. The cost functons are c 1 (x 1 ) = 0.01x x 1 and c 2 (x 2 ) = 0.01x x 2. There s no demand at bus 1. The nverse demand functon at bus 2 s p(y) = y. The branch flow lmt s C = 100. We prove by contradcton that a Nash equlbrum does not exst n ths example. 7

8 1 2 x 1 1 C 2 x 2 y Fgure 2: Example of a two-bus network n whch a Nash equlbrum does not exst. Suppose that a Nash equlbrum (ĉ 1, ĉ 2) exsts, whch nduces the outcome (x 1, x 2, λ 1, λ 2). Snce there s no demand at bus 1, λ 1 λ 2. If λ 1 < λ 2, t s easly seen that generator 1 has an ncentve to devate. Thus, λ 1 = λ 2. Snce c 1(C) = 12 < p(2c) = 14, the branch must be congested; otherwse, t can be shown that at least one of the generators has an ncentve to devate. Thus, x 1 = 100, λ 1 = λ Then (x 2, λ 2) must solve the followng problem: maxmze x 2,λ 2 λ 2 x 2 c 2 (x 2 ) subject to λ 2 = p(x 1 + x 2 ), x 2, λ 2 0, whch gves x 2 = 66.67, λ 2 = Snce a porton of x 1 s suppled at a reported margnal cost λ 1 = λ 2 = > c 2(x 2) = 11.33, t can be shown that generator 2 has an ncentve to devate: by decreasng λ 2 by an arbtrarly small number ɛ > 0 and ncreasng x 2, generator 2 can get a hgher payoff. Therefore, there does not exst a Nash equlbrum. Remark 1. In ths example, we llustrate a stuaton of market manpulaton. The result s closely related to the concept of market power. Indeed, the nonexstence of Nash equlbra s due the exercse of market power of generator 2. However, not all market power ndces are relevant. For example, the nodal must-run share (NMRS) [20] s undefned here, snce the demand s elastc Bad Prce of Anarchy There are also stuatons n whch Nash equlbra exst but some of them are undesrable n terms of effcency. The prce of anarchy s a metrc that measures how the effcency degrades due to the strategc behavor, compared wth the socally optmal outcome. Consder the case of nelastc demand, n whch y s fxed for all. Let (x, θ ) be any equlbrum dspatch. The prce of anarchy s defned as the rato between the cost of the worst equlbrum and the socally optmal cost: PoA = max n c n(x n) n c n(x, n ) where the maxmum s taken over the set of equlbrum dspatch. The followng example shows that the prce of anarchy can be arbtrarly large. 8

9 1 2 x 1,k 1 C 2 x 2,k x 4,k 4 3 x 3,k y = 2C Fgure 3: Example of a two-bus network n whch the prce of anarchy can be arbtrarly large. Example 2. Consder the network as shown n Fg. 3. The network has two buses, wth generator 1 and 4 at bus 1, and generator 2 and 3 at bus 2. The branch flow lmt s C. The nelastc demand at bus 1 s y = 2C. There s no demand at bus 2. Consder a sequence of games {Γ k, k 1}. In game Γ k, the cost functons are gven by The economc dspatch s c 1,k (x 1,k ) = x 1,k, c 2,k (x 2,k ) = kx 2,k, c 3,k (x 3,k ) = kx 3,k, c 4,k (x 4,k ) = 2kx 4,k. (x 1,k, x 2,k, x 3,k, x 4,k) = (2C, 0, 0, 0), wth a socal cost 2C. One Nash equlbrum s whch nduces the outcome ĉ 1,k(x 1,k ) = 2kx 1,k, x 1,k 2C, ĉ 2,k(x 2,k ) = kx 2,k, x 2,k 2C, ĉ 3,k(x 3,k ) = kx 3,k, x 3,k 2C, ĉ 4,k(x 4,k ) = 2kx 4,k, x 4,k 2C, (x 1,k, x 2,k, x 3,k, x 4,k) = (C, C, 0, 0), (λ 1,k, λ 2,k) = (2k, k), wth a socal cost C + kc. For example, generator 1 has no ncentve to report ĉ 1,k (x 1,k ) = kx 1,k, x 1,k 2C, snce ts new payoff, (k 1)2C, would be smaller than ts current payoff, (2k 1)C. Thus, the prce of anarchy s at least whch s unbounded as k. C + kc 2C = k + 1, 2 9

10 Remark 2. In ths example, we llustrate another stuaton of market manpulaton. Whle a Nash equlbrum always exsts, t can be arbtrarly neffcent. Ths s due to the fact that generator 1 possesses the market power. However, the market power ndex NMRS s not nformatve here, whch s zero for each generator. 5. Market Effcency On the other hand, the LMP mechansm may work well. We propose two suffcent condtons under ether of whch there exsts an effcent Nash equlbrum. In the game-theoretc language, the prce of stablty (defned as the rato between the cost of the best equlbrum and the socally optmal cost) s equal to Congeston-Free Condton Defnton 1 (Congeston-Free Condton). No branch power flow constrant (1c) s bndng at the economc dspatch (x, y, θ ). Lemma 1. Under the congeston-free condton, all the LMPs are equal at the economc dspatch. Proof. Under the congeston-free condton, µ j = 0 for all (, j). From the KKT condtons (2), we have j B j (λ λ j ) = 0,. Let I = arg max λ be the set of buses wth the largest LMPs. For each I, snce λ λ j and B j 0 for all j, there must be λ j = λ for j connected to such that B j > 0. It follows by the connectedness of the network that all the buses belong to I. That s, λ = λ 1 for all. It s mmedate to prove by constructon the exstence of effcent Nash equlbra under the congeston-free condton, whch also tells how to compute such equlbra drectly. In the followng, let Q be a large enough constant. Theorem 1. Under the congeston-free condton, there exsts an effcent Nash equlbrum n the economc dspatch game. Proof. By Lemma 1, λ = λ 1 for all. Let ĉ be an offer profle where ĉ n(x n ) = λ 1 x n, x n Q for all n. Clearly, (x, y, θ, λ, µ ) s also a soluton to (3). It remans to show that ĉ s a Nash equlbrum. Consder generator n located at bus. From the KKT condtons, x n arg max xn λ 1 x n c n (x n ). Its current payoff s πn = λ 1 x n c n (x n ). Suppose that t changes ts offer to ĉ n, resultng a new dspatch (ˆx, ŷ, ˆθ, ˆλ, ˆµ) gven 10

11 (ĉ n, ĉ n). In lght of ĉ n, we have ˆλ λ 1. Its new payoff would be ˆπ n = ˆλ ˆx n c n (ˆx n ) λ 1 ˆx n c n (ˆx n ) λ 1 x n c n (x n ) = πn. Thus, t has no ncentve to devate. profle s a Nash equlbrum. Ths proves that the constructed offer Remark 3. Snce each ĉ n s lnear (wth a large enough cap Q), t s essentally one-segment so that the value of s n does not matter. On the other hand, f we set s n = x n for all n, ths provdes a suggested dspatch pont for the ISO for te-breakng purposes. To put t another way, f we further set r n + = λ 1 +δ wth a small enough δ > 0 for all n, then we obtan an effcent ɛ-nash equlbrum Monopoly-Free Condton Defnton 2 (Monopoly-Free Condton). There are at least two, or no generators at each bus. Ths condton s easer to use than the congeston-free condton, snce we only need to know the placement of the generators. The constructve proof s smlar as before. Theorem 2. Under the monopoly-free condton, there exsts an effcent Nash equlbrum n the economc dspatch game. Proof. Let ĉ be an offer profle where ĉ n(x n ) = λ x n, x n Q for all n N. Then (x, y, θ, λ, µ ) s also a soluton to (3). It remans to show that ĉ s a Nash equlbrum. arg max xn λ n ). Suppose that t changes ts offer to ĉ n, resultng a new dspatch (ˆx, ŷ, ˆθ, ˆλ, ˆµ) gven (ĉ n, ĉ n). Under the monopoly-free condton, there s at least one another generator m located at the same bus, whose offer s ĉ m(x m ) = λ x m, x m Q. Consder generator n located at bus. From the KKT condtons, x n x n c n (x n ). Its current payoff s πn = λ x n c n (x So we have ˆλ λ. Generator n s new payoff would be ˆπ n = ˆλ ˆx n c n (ˆx n ) λ ˆx n c n (ˆx n ) λ = πn. x n c n (x n ) Thus, t has no ncentve to devate. profle s a Nash equlbrum. Ths proves that the constructed offer 11

12 Example 3. In Example 2, the monopoly-free condton holds, so that an effcent Nash equlbrum exsts: whch nduces the outcome ĉ 1,k(x 1,k ) = x 1,k, x 1,k 2C, ĉ 2,k(x 2,k ) = kx 2,k, x 2,k 2C, ĉ 3,k(x 3,k ) = kx 3,k, x 3,k 2C, ĉ 4,k(x 4,k ) = x 4,k, x 4,k 2C, (x 1,k, x 2,k, x 3,k, x 4,k) = (2C, 0, 0, 0), (λ 1,k, λ 2,k) = (1, 1). Remark 4. The ntuton of the two suffcent condtons s to lmt the exercse of market power. On the other hand, nether of them s necessary for the exstence of Nash equlbra, whether effcent or not. That sad, they may be used together for a larger class of scenaros (see Secton 8 for an example). As a sde motvaton of our work, the offer format may have a sgnfcant effect on the equlbrum outcomes. It has been shown that dfferent parametrzatons of affne offer curves lead to dfferent outcomes [21]. We further examne the case of pecewse constant offer curves. The followng example shows that whle pecewse constant offer curves yeld an effcent Nash equlbrum, affne offer curves may not do so. Example 4. Consder two dentcal generators located at a sngle bus, wth cost functons c n (x n ) = x 2 n for n = 1, 2. The nverse demand functon s p(y) = 2 y. Snce both the congeston-free and the monopoly-free condtons hold, an effcent Nash equlbrum exsts wth pecewse lnear reported costs: whch nduces the outcome ĉ 1(x 1 ) = x 1, x 1 1, ĉ 2(x 2 ) = x 2, x 2 1, (x 1, x 2, y ) = (0.5, 0.5, 1), λ = 1. Now consder quadratc reported costs whch take the followng form: ĉ n (x n ) = γ n x 2 n. Suppose that there exsts an effcent Nash equlbrum (γ 1, γ 2 ). There must be γ 1 = γ 2 = 1. Then ether of them has an ncentve to devate. For example, gven γ 2 = 1, the best response of generator 1 s γ 1 = 4/3. Thus, there s no effcent Nash equlbrum wth quadratc reported cost. 12

13 6. Economc Dspatch Game wth Cournot offers We now study the economc dspatch game wth Cournot offers. Note that the prcng mechansm s stll the LMP mechansm. In ths game, each generator n submts a scalar offer x n 0 whch specfes the quantty to be generated. Let x = (x 1,..., x N ) be the offer profle. Then the ISO solves the followng problem: maxmze y,θ subject to v (y ) x n y = B j (θ θ j ),, n N j B j (θ θ j ) C j, (, j), x n 0, n, y 0,. (4a) (4b) (4c) (4d) Compared wth (1), x n s are the nputs of (4) nstead of the decson varables. Let y be the prmal optmal soluton to (4). The LMP at each bus s gven by λ (x), the optmal dual varable assocated wth the bus power balance equaton (4b). Thus, the payoff of generator n located at bus s gven by π n (x) = λ (x)x n c n (x n ). The proposed game s dfferent from the one n [22], n whch the ISO also acts as a strategc agent who moves smultaneously wth the generators. Snce the acton set of the ISO s constraned by the generators actons, the generalzed Nash equlbrum s adopted as the soluton concept. Our model s more realstc, n whch the ISO determnes the dspatch after the generators submt the offers. In the next, we establsh the exstence of Nash equlbra under certan mld assumptons. Ths result s smlar to that n [22]. Assumpton 1. For each, the nverse demand functon p (y ) s strctly decreasng on [0, D ] wth p (D ) = 0 for some constant D > 0. Assumpton 2. For any x 0, the dual optmal soluton λ (x) s unque. For x = 0 (so that y = 0), we set λ = max j p j (0) for all. Assumpton 2 elmnates degenerate dual optmal solutons to ensure that λ s a functon of x, and s contnuous at x = 0. Lemma 2. There exsts some constant M > 0 such that n any Nash equlbrum x, x n M for all n. Proof. Let M = D. Suppose that x n > M for some n N. Then λ (x) = 0, so that π n < 0. Thus, x cannot be a Nash equlbrum. Lemma 2 mples that the strategy space of each generator can be restrcted to a compact set [0, M]. (4e) 13

14 Lemma 3. In any Nash equlbrum, y D for all. Proof. Suppose y > D for some. Then π = 0, whch mples that there must be some generator n who can reduce x n to get a hgher payoff. Thus, y > D cannot happen n a Nash equlbrum. Lemma 3 shows that the demand n equlbrum s bounded, so that the objectve functon v (y ) s strctly concave n y. To study the relatonshp between λ (x) and x, we consder the (partal) Lagrangan: L(y, θ, λ) = v (y ) + λ x y B j (θ θ j ), n N j wth dom L = Y Θ Λ, where Y = {y 0 y D, }, Θ = {θ B j (θ θ j ) C j, (, j)}, Λ = {λ 0 λ max j p j (0), }. The dual functon g : Λ R s defned as the maxmum value of the Lagrangan over (y, θ): g(λ) = max L(y, θ, λ) = h(λ) + λ x, y Y,θ Θ n N where h(λ) = max y Y (v (y ) λ y ) + max θ Θ λ B j (θ θ j ). Lemma 4. h(λ) s strctly convex and contnuously dfferentable on Λ. Proof. h(λ) (as well as g(λ)) s convex, snce t s the pontwse maxmum of a famly of affne functons of λ. By Danskn s theorem [23], t can be shown that h(λ) s contnuously dfferentable on Λ. To prove the strct convexty, t s enough to show that for λ ˆλ, the correspondng maxmzers (y, θ) and (ŷ, ˆθ) are not equal. Suppose (y, θ) = (ŷ, ˆθ). Then x = ˆx, whch means λ = ˆλ by Assumpton 2. Now we consder the dual problem: The frst order condton gves mn λ Λ h(λ) + dh = dλ λ n N x. n N x,. j 14

15 Lemma 5. λ s contnuous n x. Proof. Snce h s contnuously dfferentable, h s contnuous n λ. Snce h s strctly convex, h s nvertble. Therefore, λ s contnuous n x. Lemma 6. λ s decreasng n x n for all and for all n. Proof. dh/dλ s decreasng n x n for n N. Snce h s convex, dh/dλ j s ncreasng n λ j for all j. Thus, λ j s decreasng n x n for all j, whether j equals or not. Theorem 3. Under Assumpton 1 and 2, there exsts a Nash equlbrum n the economc dspatch game wth Cournot offers. Proof. The payoff of generator n located at bus s gven by π n (x) = λ (x)x n c n (x n ). By Lemma 5, π n s contnuous n x, whch mples nonempty, closed-graph reacton correspondences. By Lemma 6, t can be shown that π n s quasconcave n x n, whch mples the reacton correspondences are convex-valued. It follows from Kakutan s fxed pont theorem that a Nash equlbrum exsts [24]. 7. Margnal Contrbuton Prcng Mechansm We have studed the LMP-based economc dspatch game wth varous offer formats. As an alternatve, we propose the MCP mechansm, n whch each generator s pad a sngle amount of ts margnal contrbuton, nstead of a unt prce of the locatonal margnal cost as n the LMP mechansm. The proposed MCP mechansm s adapted from the Vckrey-Clarke-Groves (VCG) mechansm, a canoncal mechansm n the mechansm desgn theory that mplements effcent allocatons n domnant strateges [25]. However, the standard VCG mechansm does not apply drectly, snce we requre a fntedmensonal (and preferably low-dmensonal) offer format whle the true cost functon can be nfnte-dmensonal. In the MCP mechansm, we stll use the realstc offer format, n whch the reported cost s a two-segment pecewse lnear functon. The dspatch rule s also gven by (3). We only change the payment rule, so that the payoff of generator m s π m = w m c m (x m ), (5) where w m s the payment made to generator m. Let (x m, y m ) be the dspatch when generator m s excluded (so that x m m = 0). Then w m s gven by w m = v (y ) ĉ n (x n) n m v (y m ) ĉ n (x m n m n ), (6) 15

16 whch s the (postve) externalty that generator m mposes on the other generators by ts partcpaton. Theorem 4. There exsts an effcent Nash equlbrum n the MCP-based economc dspatch game. Proof. Let ĉ be a bd profle where ĉ n(x n ) = λ x n, x n Q for all n N. Then (x, y, θ, λ, µ ) s also a soluton to problem (3). It remans to show that ĉ s a Nash equlbrum. Consder generator m located at bus. Its current payoff s πm = v (y ) ĉ n(x n m n ) v (y m ) ĉ n(x m n m n ) c m (x m ). Suppose that t changes ts offer to ĉ m, resultng a new dspatch (ˆx, ŷ, ˆθ, ˆλ, ˆµ) gven (ĉ m, ĉ m). Its new payoff would be ˆπ m = v (ŷ ) ĉ n(ˆx n ) n m v (ŷ m ) ĉ n(ˆx m n m n ) c m (ˆx m ). So ts payoff changes by ˆπ m πm = v (ŷ ) ĉ n(ˆx n ) c m (ˆx m ) n m v (y ) ĉ n(x n m n ) + c m (x m ) ĉ m(ˆx m ) ĉ m(x m ) + c m (x m ) c m (ˆx m ) = λ 0. ˆx m λ x m + c m (x m ) c m (ˆx m ) The frst equalty follows from the fact that (x m, y m ) = (ˆx m, ŷ m ). The frst nequalty follows snce v (y ) ĉ n(x n ) v (ŷ ) ĉ n(ˆx n ). n n The last nequalty follows snce x m arg max x m λ x m c m (x m ). Thus, t has no ncentve to devate. profle s a Nash equlbrum. Ths proves that the constructed offer 16

17 x C 3 3 x 3 x x 4 y 1 y 4 Fgure 4: Example of a four-bus network whch llustrates dfferent equlbrum outcomes under dfferent prcng mechansms and offer formats. 8. Case Study We present an example to llustrate the dfferent equlbrum outcomes under dfferent prcng mechansms and offer formats. Ths example also llustrates how to apply the congeston-free and the monopoly-free condtons together n a hybrd scenaro. Consder the network as shown n Fg. 4. The network has four buses, wth generator n at bus n, for n = 1, 2, 3, 4. All the generators have the same cost functon c n (x n ) = x 2 n. The nverse demand functon at bus 1 s p 1 (y 1 ) = 1 y 1, and the one at bus 4 s p 4 (y 4 ) = 1 0.5y 4. There s no demand at bus 2 and 3. The flow lmt of branch 2-3 s C = The other two branches have suffcently large flow lmts Economc Dspatch The economc dspatch s x 1 = x 2 = , x 3 = x 4 = 0.325, y 0.475, y4 = 0.7, wth a socal welfare LMP Mechansm wth Pecewse Lnear Reported Cost 1 = Whle the suffcent condtons cannot be appled drectly, we observe that branch 1-2 and branch 3-4 are always congeston-free. Thus, bus 1 and bus 2 can be vewed as a sngle bus, and the same for bus 3 and bus 4, n whch case monopoly-free condton holds. Indeed, an effcent Nash equlbrum exsts, whch nduces the economc dspatch. Moreover, the LMPs are λ 1 = λ 2 = 0.525, λ 3 = λ 4 = 0.65, and the payoffs are π1 = π2 = , π3 = π4 = LMP Mechansm wth Cournot Offers In ths case, a Nash equlbrum s guaranteed to exst. In fact, the Nash equlbrum s unque n ths example: x 1 = x 2 = 0.21, x 3 = x 4 = 0.279, whch nduces the dspatch y 1 = 0.37, y 2 = , wth a socal welfare Clearly, the Nash equlbrum s not effcent. Moreover, the LMPs are λ 1 = λ 2 = 0.63, λ 3 = λ 4 = , and the payoffs are π 1 = π 2 = , π 3 = π 4 =

18 One can see that the LMPs wth Cournot offers are greater than those wth pecewse lnear reported cost. Ths llustrates the exercse of market power, whch leads to the neffcency of the outcome MCP Mechansm wth Pecewse Lnear Reported Cost In ths case, an effcent Nash equlbrum always exsts, whch nduces the economc dspatch. There s no noton of LMPs n the MCP mechansm. The payoffs are π 1 = π 2 = , π 3 = π 4 = The payoffs n the MCP mechansm are the hghest. The nterpretaton s that ths mechansm needs to provde enough economc ncentves to ensure that an effcent Nash equlbrum always exsts. 9. Concluson We developed a game-theoretc framework for analyss and desgn of market mechansms n the context of economc dspatch. Some mmedate extensons nclude consderng demand sde to be strategc as well. We focused on the LMP mechansm wth pecewse constant offer curves as n practce. On the one hand, we show that a Nash equlbrum may not exst, or that even when a Nash equlbrum exsts, the prce of anarchy can be arbtrarly large. On the other hand, we provde the congeston-free and the monopoly-free condtons under ether of whch there exsts an effcent Nash equlbrum. Our fndngs concde wth the polcy proposed n [26]: ensurng enough competton to lmt the exercse of market power. The presented counterexamples also suggest that the results n the SFE lterature may not apply drectly to practce. Moreover, for a general transmsson-constraned Cournot model n whch the generators submt quantty offers, we show that there exsts a Nash equlbrum under certan mld condtons. The proposed MCP mechansm always nduces an effcent Nash equlbrum, at the expense of prce dscrmnaton. Specfcally, the MCP mechansm assgns a total payment to each generator, whle the LMP mechansm assgns a unform prce at each bus. Lke the LMP mechansm, there may be undesrable Nash equlbra n the MCP mechansm. Future work s needed to tackle these ssues. References [1] S. Stoft, Power system economcs: Desgnng markets for electrcty, Wley- IEEE Press, [2] D. S. Krschen, G. Strbac, Fundamentals of power system economcs, John Wley & Sons, [3] McCullough Research, Congeston manpulaton n ISO Calforna (2002). URL 18

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