Indirect Mechanism Design for Efficient and Stable Renewable Energy Aggregation

Transcription

1 1 Indrect Mechansm Desgn for Effcent and Stable Rewable Ergy Aggregaton Hossen Khazae, Student Member, IEEE, and Yue Zhao, Member, IEEE arxv: v1 [cs.gt] 6 Oct 2018 Abstract Mechansm desgn s studed for aggregatng rewable power producers RPPs) n a two-settlement power market. Employng an ndrect mechansm desgn framework, a payoff allocaton mechansm PAM) s derved from the compettve equlbrum CE) of a specally formulated market wth transferrable payoff. Gven the desgd mechansm, the strategc behavors of the partcpatng RPPs ental a non-cooperatve game: It s proven that a unque pure ash equlbrum E) exsts among the RPPs, for whch a closed-form expresson s found. Moreover, t s proven that the desgd mechansm acheves a number of key desrable propertes at the E: these nclude effcency.e., an deal Prce of Anarchy of o), stablty.e., n the core from a coaltonal game theoretc perspectve), and no colluson. In addton, t s shown that a set of desrable ex-post propertes are also acheved by the desgd mechansm. Extensve smulatons are conducted and corroborate the theoretcal results. Index Terms Cost allocaton, ash equlbrum, mechansm desgn, coaltonal game, rewable ergy, electrcty market I. ITRODUCTIO Rewable erges play a central role n achevng a sustanable ergy future. However, rewable erges such as wnd and solar power are nherently non-dspatchable, and yet hghly uncertan and varable. As a result, ntegratng rewable erges nto power systems to serve loads rases sgnfcant relablty and effcency challenges [1], [2]. A varety of approaches have been proposed to compensate for the uncertanty of rewable erges, such as mprovng rewable power geraton forecast [2], employng better geraton dspatch methods [3], ergy storage deployment and control [4], [5], [6], and demand response programs [7], [8], [9]. Another soluton that has receved consderable attenton s to aggregate statstcally dverse rewable ergy sources [1], [10], [11]. In an aggregaton, rewable power producers RPPs) pool ther geraton together so as to reduce the aggregate uncertanty and rsk, and hence the correspondng cost of compensaton for ther uncertantes. Accordngly, by formng an aggregaton, RPPs can n total earn a hgher payoff. A key queston n aggregatng RPPs s thus how to allocate the total payoff of an aggregaton among ts member RPPs. otably, aggregatng rewable erges has been studed extensvely n the context of a two-settlement power market model, consstng of a forward power market and a real tme o. As such, RPPs partcpate n these markets n the same way as conventonal gerators do. Wth ths model, allocatng payoffs n an aggregaton of RPPs has been studed n a coaltonal game framework based on the jont probablty H. Khazae and Y. Zhao are wth the Dept. of Electrcal and Computer Engerng, Stony Brook Unversty, Stony Brook, Y, USA e-mals: {hossen.khazae, yue.zhao.2}@stonybrook.edu). dstrbuton of all the RPPs uncertan geraton [10], [11]. The prmary nterest n ths settng s to fnd a payoff allocaton soluton that s stable/n the core of the game. Ths s n geral computatonally hard n the sense that the number of constrants of the correspondng optmzaton problem grows expontally wth the number of RPPs. To ths end, the core s proven to be non-empty n [10], and a closed-form soluton of a payoff allocaton n the core s found n [11]. Whle ths l of works acheve effcency wth an optmal forward contract) and stablty wth a payoff allocaton n the core) n aggregatng RPPs, an underlyng assumpton s that the aggregator knows the jont probablty dstrbuton of the RPPs geraton. In practce, however, an aggregator typcally does not have the best or full knowledge of such nformaton about the RPPs: not only the amount of relevant nformaton can be overwhelmng to glean, but also the RPPs themselves often have better nformaton prvately about ther own geraton than the aggregator does. As a result, to aggregate rewables n practce, t s essental to consder an nformaton collecton step by the aggregator wth the RPPs. Consequently, wth ths step, aggregatng rewable erges n a two-settlement market consttutes a mechansm desgn problem as wll be shown n detals n Secton II-C). In short, the prmary goal of such mechansm desgn s the followng: Granted that all the RPPs behave strategcally for ther own nterests under ths mechansm, the aggregaton can stll acheve the same desrable outcome as f all the RPPs nformaton are ndeed known to the aggregator. In geral, there s complete freedom n desgnng the nformaton collecton step of the mechansm. In partcular, when the aggregator does not elct all avalable nformaton from the RPPs, the mechansm s called an ndrect o, n contrast to a drect o when all nformaton from the RPPs are requested by the aggregator upfront) [12]. Indeed, we would lke an aggregator to elct as lttle nformaton as possble from the RPPs, whle stll guaranteeng the performance of the overall mechansm. To ths end, a smple nterface between aggregator and RPPs has been proposed n [13]: each RPP submts just a sngle number to the aggregator, and the aggregator smply passes on the sum of these numbers as the forward power contract for the entre aggregaton. Based on ths smple nterface, the central desgn task s agan on the payoff allocaton among the RPPs, for whch a number of payoff allocaton mechansms PAMs) have been proposed [13], [14], [15], [16], [17], [18]. Under any gven PAM, the RPPs strategc decson makng entals a non-cooperatve game as wll be descrbed later n Secton II-E), and propertes of the ash Equlbra of ths game have been studed n [13], [14], [15], [16]. The exstng PAMs n the lterature, however, have only gad lmted success, as some essental

2 and hghly desred propertes stll cannot be acheved. In partcular, achevng effcency and stablty/n the core at the ash Equlbra remans to be an open queston. Lastly, we note that mechansm desgn methods have also been employed n power markets for problems other than rewable ergy aggregaton, e.g., for ncentvzng conventonal gerators to reveal truthful nformaton [19]. In ths paper, we nvestgate ndrect mechansm desgn under the framework of the above smple nterface. We propose a w payoff allocaton mechansm, and show that all the essental desrable propertes are acheved by ths PAM. We frst show that, gven the desgd mechansm, the outcome of the mechansm can be predcted by a unque ash equlbrum E) among the RPPs, for whch we provde a closed-form expresson. Moreover, ths unque E s effcent, meanng that t acheves the maxmum total payoff as f all nformaton are known a-pror to the aggregator. ext, we show that, the proposed payoff allocaton s stable/n the core at ths unque E, meanng that no subset of the RPPs have any ncentve to leave the aggregaton as they cannot possbly earn a hgher payoff on ther own. Furthermore, we show that the desgd mechansm guarantees no colluson among the RPPs at the unque E, as they cannot earn a hgher payoff by colludng. Lastly, we show that a set of ex-post propertes are acheved by the proposed mechansm wth results reported n part n [17]). We note that smlar ex-post results have also been ndependently developed n a recent work [18]. The remander of the paper s organzed as follows. The problem s formulated n Secton II, and the ndrect mechansm desgn framework for aggregatng RPPs s ntroduced. The desgn goals,.e., the desred propertes of the PAM are descrbed n Secton III. The man results are presented n Secton IV, n whch we show that the proposed PAM acheves all the desred propertes. Analyss and proofs of the man results are provded n Secton V. Another set of propertes acheved n an ex-post sense by the proposed PAM are presented n Secton VI. Smulatons are conducted n Secton VII. Conclusons are drawn n Secton VIII. A. System Model II. PROBLEM FORMULATIO We consder RPPs partcpatng n a two-settlement power market consstng of a day-ahead DA) and a real tme RT) market. As a basel case, we frst consder an RPP who partcpates n the market separately from the other RPPs. In the DA market, RPP s geraton at the tme of nterest n the xt day s modeled as a random varable, denoted by X. We assume that the jont probablty densty functon for the vector of random varables X 1,..., X exsts.) RPP then determs a forward power supply contract n the amount of c to sell n the DA-market. Interchangabely, c s also termed a day-ahead commtment. RPP gets a payoff of p f c where p f denotes the prce n the DA market. At the delvery tme n the xt day, RPP obtans ts realzed geraton x : a) If t faces a shortfall,.e., c x > 0, t eds to purchase the remanng power from the RT market at a real-tme buyng prce p r,b, b) f t has excess power,.e. x c > 0, t can sell t n the RT-market at a real-tme sellng prce p r,s. In case excess power eds to be penalzed as opposed to rewarded, we model such cases by havng p r,s < 0. We make the assumpton that p r,s p r,b, whch must hold for no arbtrage. Intutvely, the hgher uncertanty an RPP s geraton has at DA, the more cost t ncurs to the RPP. Specfcally, the realzed payoff of an RPP who separately partcpates n the market s gven by p f c p r,b c x ) + + p r,s x c ) + 1) where ) + max0, ) 1. We denote the expected payoff of RPP at the tme when c s determd o day ahead by π sep c ) E[ ], 2) where the expectaton s taken over the random geraton X. Remark 1 Model Assumptons on Prces): In ths paper, we consder a prce takng envronment for the RPPs n the DA market, and thus the prce p f s gven. We also consder that some fxed values for the RT buyng and sellng prces p r,b and p r,s are assumed at DA. These values can be nterpreted as the RPPs expectatons of the RT prces f they were to experence a shortfall or a surplus, respectvely. We note that the prce takng assumpton s a smplfyng o, whch assumes that no of the RPPs can affect the prce sgnfcantly at DA due to ts relatvely small sze. Ths provdes a frst order approxmaton of the problem that allows us to perform effectve analyss and gan nsght. The results wll lay the foundaton for further nvestgaton of more geral scenaros. In partcular, we would lke to note that our latest result followng ths paper has acheved some success n relaxng these assumptons, and has addressed the prcemakng scenaros n both DA and RT markets [20]. B. Aggregatng Rewable Erges We consder an aggregator that aggregates the power geraton from a set of RPPs, denoted by, and partcpates n the DA-RT market on behalf of the RPPs. Intutvely, aggregaton reduces the total uncertanty for the RPPs due to the statstcal compensaton among the random power geraton at DA, and hence brngs economc beft to them. In ths paper, transmsson twork constrants are not consdered, and are left for future work. In geral, the aggregator takes actons n the DA and RT markets as follows: a. In the DA market, the aggregator determs an amount of forward power contract to sell, denoted by c. b. At the delvery tme, the aggregator collects all the realzed geraton from the RPPs, denoted by x = x, to meet the commtment c. The devaton s settled n the RT market n the same way as n Secton II-A. The realzed payoff of the aggregator s thus P p f c p r,b c x ) + + p r,s x c ) + 3) ext, the aggregator returns a payoff P to each RPP. 1 We use the symbol to def notatons.

3 Remark 2 Budget Balance): In ths paper, we requre the aggregator s budget balance be satsfed n all crcumstances: P = P. 4) =1 We note that ths s a stronger condton than just requrng budget balance be satsfed n expectaton. Accordngly, there are two decsons an aggregator eds to make: a) the total commtment c at DA, and b) the set of payoff allocatons {P } at RT. In makng these decsons, two fundamental goals an aggregator would lke to acheve are: Effcency: The total expected payoff of the aggregaton π = E[P ] s maxmzed. Farss / Stablty: The payoff allocaton wthn the aggregaton {P } are far to each RPP. In ths paper, we nterpret farss usng the noton of stablty/n the core from a coaltonal game perspectve, as wll be descrbed n detal n Secton III. In partcular, a) achevng effcency depends on the aggregator s decson on the total DA commtment c, and b) wth the optmal c, achevng stablty depends on the decsons on the RT payoff allocaton {P }. The Ideal Case of Aggregator Havng Full Informaton: To acheve effcency and stablty, makng decsons on c and {P } requres the aggregator to know suffcent nformaton from the RPPs. The deal case would be an aggregator wth full nformaton from the RPPs, n partcular, the DA jont probablty dstrbuton of all the random geraton {X }. Based on the jont probablty dstrbuton of {X }, closedform solutons of c and {P } that acheve effcency and stablty have been found n [11], and wll be used for numercal comparsons later. C. The Mechansm Desgn Problem In practce, however, t s unlkely for an aggregator to precsely know the DA jont probablty dstrbuton of {X } for a number of reasons: a) the best nformaton on future power geraton may only be prvately known to the RPPs, and b) the amount of nformaton can be overwhelmngly large and dffcult to glean for a sngle aggregator, especally when the number of RPPs becomes large, consder, e.g., hundreds of thousands of dstrbuted ergy resources n a power dstrbuton system). In ths paper, we do not assume the aggregator knows any nformaton a-pror at DA on the RPPs random geraton {X }. Instead, we consder a geral framework n whch the aggregator elcts nformaton from the RPPs, based on whch decsons on c and {P } are then made. As such, the aggregator s actons nvolve the followng three geral steps: a. Informaton Collecton: At DA, the aggregator elcts certan nformaton from the RPPs. b. Commtment: At DA, the aggregator determs a total DA commtment c. c. Payoff Allocaton: At RT, the aggregator allocates a payoff P to RPP, = 1,...,. Specfyng how these three steps are performed consttutes a mechansm desgn problem. It s mportant to note the geralty of ths desgn problem, as there s complete freedom n choosng what nformaton to request from the RPPs, how they are used to determ c, and how payoffs are allocated. D. An Indrect Mechansm Desgn Framework In ths mechansm desgn problem, t s not mperatve for the aggregator to elct all nformaton from the RPPs. When the aggregator does elct all nformaton upfront, such a mechansm s called a drect mechansm ; Otherwse, t s called an ndrect mechansm [12]. Rather than restrctng ourselves to drect mechansms, more gerally, we wll nvestgate ndrect mechansm desgn: We would lke to elct as lttle nformaton from the RPPs as possble, whle stll guaranteeng effcency and stablty of the aggregaton. In partcular, we wll nvestgate the followng framework of ndrect mechansms employng a smple desgn of Steps a. Informaton Collecton) and b. Commtment) [13]. a. At DA, the aggregator elcts a sngle number c from each RPP. b. At DA, the aggregator commts c = =1 c. c. At RT, the aggregator allocates a payoff P to RPP, = 1,...,. We term the number c submtted by RPP ts DA commtment. Accordngly, the aggregator smply passes on the sum of the RPPs DA commtments as the aggregate commtment. As Steps a. and b. are now fully specfed, the central desgn task s Step c. Payoff Allocaton. As such, the desgn and analyss of Payoff Allocaton Mechansms PAMs) that determs P would be the focus of the remander of the paper. E. Mechansm s Outcome n a on-cooperatve Game of Strategc RPPs Gven any PAM that specfes the rule of determnng P, a key queston s how o predcts the outcome under ths mechansm. To answer ths queston, we must understand the behavor of the RPPs gven any mechansm. As an RPP s free to submt any DA commtment c to the aggregator, a ratonal and strategc RPP would submt a c at DA that maxmzes ts expected allocated payoff E[P ]. It s mportant to note that, gven a PAM, E[P ] can also depend on the other RPPs submssons of commtments. Accordngly, we denote the expected payoff of RPP by π c, {c }) E[P ], 5) where {c } denotes the set of the commtments of the RPPs other than. ote that π c, {c }) depends on the partcular desgn of the PAM. Therefore, the strategc decson makng of the RPPs on ther submssons {c } at DA can be studed under a noncooperatve game framework, termed a contract game n [13].) To predct the outcome of any desgd mechansm, a natural soluton concept s the ash equlbra of ths non-cooperatve game 2. Specfcally, gven a PAM, a set of 2 We note that, n practce, E may not be acheved n a dynamc market. Investgaton of other soluton concepts s left for future work.

4 commtments {c } s at a pure ash equlbrum E) f they satsfy c argmax π c, { c }),. 6) c As such, an E offers a stable 3 outcome of the RPPs decson makng on ther commtments, as no RPP has any ncentve to devate from ts already best respondng commtment. In the remander of the paper, we devote the notaton {c } to denotng a set of commtments at a pure E. As a mechansm desgr for aggregatng RPPs, we are nterested n desgnng a PAM so that a set of essental and desrable propertes can be acheved at equlbra of ths noncooperatve game, gven the desgd PAM. III. DESIG GOALS: DESIRED EFFICIECY AD STABILITY PROPERTIES In ths secton, we provde a detaled descrpton of the desred propertes n desgnng PAM for aggregatng RPPs. 1) Exstence and unquess of pure ash equlbrum: For a mechansm to have predctable outcomes, t s desred that the non-cooperatve game among the RPPs nduced by the mechansm has a unque pure E, whch would be the unque outcome that o shall predct from the RPPs strategc decson makng. 2) Effcent computaton of E: In partcular, we are nterested n whether the unque pure E, f exsts, can be computed n closed form. 3) Effcency: A PAM s effcent f, at the E, the aggregaton acheves the maxmum expected payoff for the entre group of RPPs. Specfcally, ths means that c =1 c s equal to the optmal commtment for the entre aggregaton c argmax c E[P ] 7) cf. 3)). Ths optmal commtment can n fact be computed as a soluton to a ws-vendor problem for whch we refer the readers to [21] for more detals): c = F 1 p f p r,s ), where F p r,b p r,s x ) s the cumulatve dstrbuton functon cdf) of the aggregate random geraton X = =1 X. In ths paper, we assume that the nverse functon F 1 ) exsts. 4) Indvdual ratonalty: At the E, the expected payoff of RPP should be at least as hgh as the maxmum payoff t could have gotten had t separately partcpated n the DA-RT market. Specfcally, where, π c c,sep, { c }) π sep c,sep), 8) argmax π sep c ) 9) c cf. 1) and 2)). It s mportant to note that, the separately) optmal commtment c,sep s n geral not equal to the equlbrum commtment c. Wth 3 Ths noton of stablty from E s not the coaltonal game theoretc stablty whch wll be descrbed as Property 5) n the xt secton. ndvdual ratonalty satsfed 8), not a sngle RPP has any ncentve to leave the aggregaton. 5) Stablty / In the core: A geralzaton of ndvdual ratonalty to a much stronger sense of stablty s beng n the core, a property celebrated n coaltonal game theory [22]. Specfcally, beng n the core means that the RPPs expected payoffs satsfy the followng condton: f any subset T of the RPPs leave the aggregaton, separately form ther own aggregaton, and then partcpate n the market based on ther aggregate geraton X T T X, ther hghest possble expected payoff would be no hgher than the sum of ther expected payoffs orgnally from the PAM at the E. Specfcally, T, T where T π c, {c } ) π sep T c,sep T ), 10) π sep T c T ) E [ p f c T p r,b c T X T ) + + p r,s X T c T ) + ], 11) and c,sep T argmax ct π sep T c T ). As such, beng n the core means that, not only every sngle RPP, but all subsets of RPPs do not have any ncentve to leave the aggregaton. Ths mples a very strong sense of stablty of an aggregaton. 6) o colluson: Suppose a subset of RPPs jon together as a sngle player before partcpatng n the aggregaton wth the remanng RPPs. For now, we assume the remanng RPPs know the jonng of these RPPs as a sngle player. Later, we wll show that the DA commtments of the remanng RPPs at the E actually do not depend on whether or not they know there s a colluson. Because of the change of the set of players, a w game, and hence w E would arse. The expected payoff of ths jont player at the w E should be no hgher than the sum of these RPPs expected payoffs at the E of the orgnal game. Otherwse, some RPPs could have ncentves to collude, jon together, and collectvely nterface wth the aggregator as a sngle and larger) RPP n order to earn a hgher total payoff. Rgorously, no colluson s defd as follows: T, π c, { c }) }) πt c T, { c T, 12) }} where { c T, { c T s the w E of the w noncooperatve game for the case when the RPPs n T jon as a sngle player. IV. MAI RESULTS We now present the man results of ths paper: a proposed payoff allocaton mechansm, and how t acheves all the above desred propertes 1) - 6). Frst, we propose the followng payoff allocaton mechansm: p f c + p r,b x c ) f x c < 0 P = p f c + p x c ) f x c = 0, 13) p f c + p r,s x c ) f x c > 0

5 where p r,s p p r,b, and p can be chosen arbtrarly wthn ths range. Remark 3 on-concavty of the on-cooperatve Game): Gven the proposed PAM 13), t s worth notng that π c, {c }) s not a concave functon n c. Thus, the non-cooperatve game of DA commtments among the RPPs s not a concave game. As such, the behavor of the game e.g., whether an E exsts) cannot be predcted from exstng theores of concave games [23]. otheless, we analyzed the game wth w technques, and the man results are descrbed xt. Due to the non-concavty of the non-cooperatve game among the RPPs, the exstence of a pure E s not always guaranteed. In the followng, we frst show the closed form of the unque pure E f any pure E exsts at all, and then show a cessary and suffcent condton for a pure E to always exst regardless of the DA and RT prces. Theorem 1: Employng the PAM 13), f the noncooperatve game among the RPPs cf. Secton II-E) possesses any pure E, t must be unque, and s gven by the followng DA commtments: = 1,...,, c = E [ X X = c ], 14) where X = =1 X, c argmax c E[P ] = F 1 p f p r,s ), and F p r,b p r,s x ) s the cdf of X. Remark 4: To compute ts equlbrum commtment, an RPP eds only to know the bvarate probablty dstrbuton functon pdf) of ts own geraton and the total geraton of the aggregaton,.e., f XX x, x ). As such, knowledge of the complete jont pdf of all the RPPs s not eded. Theorem 2: The non-cooperatve game among the RPPs always possesses a pure E f the followng condton holds, de X X, α, [X X = α] 1. 15) Conversely, f the condton 15) does not hold, then there exsts a set of DA and RT prces such that a pure E does not exst. Remark 5: We argue that the condton 15) s a reasonable o, as t holds when no sngle RPP domnates the entre aggregaton. Consder the case when k, α, s.t., de X k X [X k X = α] > 1. 16) Because E Xk X [X k X ] + k E X X [X X ] = X, we have that de Xk X [X k X = α] + de X k X [X k X = α] where X k k X. From 16) and 17), we have = 1, 17) de X k X [X k X = α] < 0. 18) Ths means that the aggregaton of all the RPPs other than k s gatvely correlated wth the entre aggregaton. Intutvely speakng, ths means that the sngle RPP k s not only gatvely correlated wth the aggregaton of all the other RPPs, but also domnates them, and hence domnates the entre aggregaton. However, ths s an unlkely stuaton especally when the number of RPPs s relatvely large, and no sngle RPP can domnate the entre aggregaton. For the rest of ths paper, we assume that the condton 15) holds, and thus the unque pure E of the game s gven by 14) as n Theorem 1. The closed form of ths pure E mmedately mples the followng: Corollary 1: Gven the PAM 13), effcency of the aggregaton s acheved at the unque pure E,.e., c = c, 19) where c =1 c. In other words, Corollary 1 mples that the desgd mechansm acheves an deal Prce of Anarchy of o [12]. Furthermore, we have that all the remanng desred propertes ntroduced n Secton III are also acheved: Theorem 3: Employng the PAM 13), ndvdual ratonalty, stablty / n the core, and no colluson are acheved at the unque pure E specfed n 14). As a result of Theorems 1, 2, 3 and Corollary 1, we conclude that the proposed PAM 13) nduces a unque E among the RPPs, expressed n closed form 14), whch s both effcent and stable.e., n the core from a coaltonal game perspectve) for the entre aggregaton, and guarantees no colluson. We further note that, nterestngly, results and technques smlar to our fndngs have ndependently been developed for an ergy storage sharng problem n the recent works [24] and [25]. Lastly, whle the proposed PAM 13) s shown to acheve all the desred propertes, an nterestng queston s how ts specfc form s dscovered. For ths, we refer the readers to Appendx D, n whch we show the PAM 13) can be derved from a compettve equlbrum of a specally formulated market wth transferrable payoff [22]. V. AALYSIS AD PROOFS OF THE MAI RESULTS A. Understandng the Proposed PAM 1) The Excess Payoff from Aggregaton: We frst exam the excess payoff from aggregatng the RPPs gven a set of DA commtments {c },.e., the dfference between a) the realzed payoff of the aggregaton, and b) the sum of the realzed payoffs of the RPPs had they separately partcpated n the DA-RT market usng the same DA commtments {c }. We def the followng notatons for the realzaton dependent) sets of RPPs wth geraton surpluses and shortfalls: S + { x c 0}, S { x c < 0} c S + c, x S + x, c S c, x S x. S + S + S S For convence, we def c = x = 0. We then have the followng lemma on expressng the excess payoff n terms of the above notatons, whose proof s relegated to Appendx A. Lemma 1: The excess payoff from aggregatng the RPPs s P = p r,b p r,s) mn x S + c S +), c S x S )) 20)

6 Lemma 1 mples that the excess payoff from aggregaton s always non-gatve. The excess payoff s zero f the RPPs ether all have excesses or all have shortfalls,.e., no compensaton happens among the RPPs. 2) Intuton of the Proposed PAM: To understand the proposed PAM 13), let us consder the followng two cases: Case 1: The aggregaton has a shortfall n total,.e. x c < 0. In ths case, P =, S,.e., those RPPs wth a shortfall earns exactly the same as f they each partcpates n the market separately. In comparson, S +, P = p r,b p r,s )x c ). As a result, only those RPPs n S + can gan extra earnngs compared to f they partcpate n the markets separately. In other words, when the aggregaton has a shortfall, all the excess payoff 20) are allocated to those RPPs wth a surplus. Case 2: The aggregaton has a surplus n total,.e. x c > 0. In ths case, P =, S +,.e., those RPPs wth a surplus earn exactly the same as f they partcpate n the market separately. In comparson, S, P = p r,b p r,s )c x ). As a result, when the aggregaton has a surplus, all the excess payoff 20) are allocated to those RPPs wth a shortfall. Whle ths may seem unntutve at frst glance, t can be understood as only rewardng those RPPs who reduce the total devaton, even when the total devaton s a surplus and reducng t means havng a shortfall. Remark 6 Margnal Proft): The proposed PAM 13) can also be understood as follows: At RT, gven the total possbly gatve) extra geraton x c from the aggregaton of RPPs, f an addtonal nfntesmal unt of ergy s gerated by RPP, the resultng addtonal proft the aggregaton earns dctates the prce of possbly gatve) extra geraton x c for RPP. B. Proofs of the Man Results We relegate the proof of Theorems 1 and 2 to Appendces B and C due to ther manly algebrac nature. In the followng, we present the proof of Theorem 3. Proof of Theorem 3: We wll frst show that the proposed mechansm 13) acheves ndvdual ratonalty and nocolluson, whch would then be used to prove that stablty / n the core s further acheved. a. Indvdual Ratonalty: Comparng 1) wth 13), we mmedately have the followng qualty:, {c } and {x }, P {c }, {x }) c, x ). 21) As a heads up, we term 21) ex-post restrcted ndvdual ratonalty, whch we wll descrbe n detal later n Secton VI-B cf. Property 1 theren). By takng expectaton of 21) over {X }, we have and {c }, π c, {c }) π sep c ). 22) We then apply ths qualty for the followng specfc choce of {c } : c = c,sep cf. 9)) and {c } = {c }:, π c,sep, { c }) π sep c,sep). 23) On the other hand, from the best respondng property n the defnton of E 6), we have π c, { c }) π c,sep, { c }) 24) Combnng 24) and 23), we have π c, { c }) π,sep c,sep), 25) whch completes the proof of ndvdual ratonalty. b. o Colluson: To prove no colluson 12), we begn wth showng that c T = c, T. 26) T Wthout loss of geralty WLOG), consder RPPs 1, 2,, T form as a jont player T. Based on Theorem 1, from the closed form expresson of the unque pure E 14), we have c T [ ] X = E X T = c = j T = E X X j = c j T [ ] X E X j = c = c. 27) T Smlarly, n the w game wth RPPs n T jonng as a sngle player, the remanng RPPs commtments at the w E stay the same as at the orgnal game s E: / T, c = E [ X X = c ] = c. 28) ow, from the specal pece-wse lar structure of the proposed PAM 13), t s straghtforward to verfy that } P T c T, { c T, {xj } ) = P {c }, {x j }). T In other words, the total payoff allocated to the RPPs n T remans the same before and after they form as a jont player. As ths holds n all crcumstances, t also holds n expectaton: π T c T, { c T }) = T π c, { c }). 29) As a result, the proposed PAM 13) acheves the no colluson property 12): In partcular, the qualty s always acheved by equalty 29). We term equaton 29) the o Colluson Equaton. c. Stablty / In the Core: We now show that ndvdual ratonalty and no colluson collectvely mples stablty of the proposed PAM cf. Property 5 n Secton III). For any subset of RPPs T, consder the hypothetcal case of them jonng as a sngle player to aggregate wth the remanng RPPs T \ under the proposed PAM 13). Applyng ndvdual ratonalty cf. 8)) specfcally to ths jont player, we have π T c T, c T ) π,sep T c,sep) T. 30) From 30) and the o Colluson Equaton 29), we have π c, { c }) π,sep T c,sep) T, 31) T completng the proof of Property 5) - Stablty / In the Core - of the proposed PAM 13).

7 VI. EX-POST PROPERTIES OF THE PROPOSED MECHAISM In ths secton, we present another set of desrable propertes acheved by the proposed PAM 13): These propertes are termed ex-post propertes because they are acheved for all possble realzatons of the random power geraton {X }. These propertes, however, are dstngushed from those dscussed n Secton III by a key caveat an assumpton on the RPPs DA commtments, as wll be descrbed xt. A. A Specalzed Coaltonal Game Gven any set of DA commtments {c } and realzatons of geraton {x }, smlar to the realzed payoffs 1) and 3), we def a functon v ) for the value of a coalton of any subset of RPPs as follows: T, v T ) = p f ĉ T p r,b ĉ T x T ) + + p r,s x T ĉ T ) + 32) where ĉ T T c and x T T x. Wth the above value functon v ), desgnng the PAM {P {c }, {x }), } can be studed n a well-defd coaltonal game [22]. In partcular, n a coaltonal game, a PAM {P } s sad to be stable/n the core f and only f t satsfes the followng set of qualtes: T, T P v T ). 33) In other words, the total payoffs allocated to any subset of the players should be no less than the value of ths subset. In partcular, here the value of a subset of RPPs T 32) has a specfc meanng the realzed payoff of the subset of RPPs T had they left the aggregaton and collectvely partcpate n the DA-RT markets wth a specfc total commtment ĉ T = T c. As such, f a PAM {P } s n the core cf. 33)), any subset of RPPs T n the aggregaton earn at least as much as they would otherwse earn outsde the aggregaton provded that they stck to the same total DA commtments as they do nsde the aggregaton. Remark 7 Restrctve Assumpton on DA Commtments): As wth the prevously dscussed stablty property n Secton III, o would deally lke stablty / n the core be satsfed wthout any restrcton on how a subset of RPPs determ ther DA commtments. The requrement of ĉ T = T c, T n ths secton s hence a restrctve o, as t does not allow a subset of RPPs leavng the aggregaton to re-adjust ther DA commtments. Ths assumpton notheless leads to a set of ex-post propertes of the proposed PAM 13) n the followng. B. Ex-post Restrcted Stablty and o Colluson We now descrbe the propertes acheved by the proposed PAM 13) n an ex-post sense, meanng that they hold for all possble {c } and rewable geraton realzatons {x }. 1) Ex-post restrcted ndvdual ratonalty: P {c }, {x }) c, x ),, cf. 1) and 13), and mentod earler as 21)). In other words, the payoff of RPP s at least as hgh as the payoff t could have gotten had t separately partcpated n the DA-RT market wth the same DA commtment c as orgnally submtted to the aggregator. 2) Ex-post restrctedly stable/n the core: Beng restrctedly stable/n the core n an ex-post sense s defd by 33) and 32). In other words, f any subset of the RPPs T leave the aggregaton, separately form ther own aggregaton, and then partcpate n the market wth the same sum of DA commtments ĉ T = T c as orgnally submtted to the aggregator, they would get a realzed payoff no hgher than the sum of ther realzed payoffs orgnally from the PAM. 3) Ex-post restrcted no colluson: Suppose any subset of RPPs T jon together as a sngle player before partcpatng n the aggregaton wth the remanng RPPs, and submt the same sum of DA commtments ĉ T = T c to the aggregator. Ther total realzed payoff would be no hgher than the sum of ther orgnal realzed payoffs from the PAM. Rgorously, T, P {c }, {x }) P T ĉ T, {c T }, {x }), 34) T where P T ĉ T, {c T }, {x }) s the w payoff of a subset of RPPs T f they a) jon as a sngle player, and then b) aggregate wth the remanng RPPs \T under the proposed PAM 13), employng the orgnal total commtment of ĉ T = T c. We note that the reason for the above propertes to be termed as restrcted os s the assumpton of ĉ T = T c, T cf. Remark 7). We now have the followng theorem. Theorem 4: Employng the PAM 13), ex-post restrcted ndvdual ratonalty, stablty / n the core, and no colluson are acheved for all possble {c } and {x }. It s straghtforward to verfy from 1) and 13) that expost restrcted ndvdual ratonalty and no colluson both hold. To prove ex-post restrcted stablty/n the core, we agan refer the readers to Appendx D, n whch stablty/n the core s mpled by the compettve equlbrum of a specally formulated market wth transferrable payoff. We further note that, nterestngly, results smlar to the proposed PAM 13) n achevng ex-post propertes have ndependently been developed n a recent work [18], usng a cost causaton based analyss. VII. SIMULATIO A. Data Descrpton and Smulaton Setup We perform smulatons usng the REL dataset [26] for ten wnd power producers WPPs) located n the PJM nterconcton. For each WPP, both the hourly DA forecasts and the actual realzed geraton are avalable from the data set. The geraton of the WPPs for each hour t are modeled as W t) = Ŵ t) + ɛ t),, where Ŵ s the pont) forecast geraton of WPP, and ɛ s the forecast error. For smplcty, we consder the WPPs modelng ther forecast errors usng a zero mean jontly Gaussan dstrbuton, 0, Σ). We ft the covarance matrx Σ usng the real data of these ten WPPs n Jan The smulatons are then performed based on the real data of these ten WPPs n Feb We note that the WPPs Gaussan) probablstc belefs are only ther crude statstcal models of

8 Table I: Total payoff of all the WPPs Cases 1 and 2 Case 3 Case 4 Total Payoff $) 10, 428, , 352, 581 9, 148, 024 ther geraton, and all the data used n the actual smulatons are real data as opposed to Gaussanly dstrbuted). Ths, however, s already suffcent to provde nstructve numercal results as wll be shown n the remander of the secton. To smulate the WPPs nteractons wth the DA-RT markets, we employ the hourly DA and RT locatonal margnal prces LMPs) n Feb from where the ten WPPs are located all n the PJM nterconcton). In partcular, p f n 1) s obtad from the hourly day ahead market prce p DA t). To obtan p r,b and p r,s, the same approach as n [11] s employed: we let p r,b = max 1.2p DA t), 2p RT t) ) and p r,s = mn p DA t)/1.2, p RT t)/2 ), where p RT t) s the hourly real tme market prce. We evaluate four dfferent cases of WPPs partcpatng n the DA-RT market, wth and wthout aggregaton: Case 1: An aggregator employs an effcent and stable/n the core PAM prevously derved n [11] to aggregate the WPPs. Ths PAM assumes the knowledge of the jont probablty dstrbuton of the WPPs random geraton. Case 2: An aggregator employs the proposed mechansm cf. Secton II-D and 13)) to aggregate the WPPs. At the unque E, each WPP submts c cf. 14)). Case 3: An aggregator employs the proposed mechansm to aggregate the WPPs. Each WPP submts the DA commtment that would be optmal had t separately partcpated n the markets,.e., c,sep = argmax c π sep c ). Case 4: Wthout an aggregator, each WPP separately partcpates n the DA-RT market, and makes the optmal DA commtment c,sep = argmax c π sep c ). In partcular, for Case 1, the smulated payoff allocaton mechansm based on the PAM n [9] s gven by P case 1 = p x + P j=1 p j x j, 35) where p s the compettve prce gven by eq. 15) n [11]. B. Smulaton Results The total payoffs of all the WPPs for the four cases are summarzed n Table. I. As expected from Corollary 1, the total payoffs for Cases 1 and 2 are the same snce both cases acheve effcency,.e., maxmum expected proft for the aggregaton. In comparson, snce Case 3 does not acheve effcency for the aggregaton, a lower total payoff s acheved than that n Cases 1 and 2. Lastly, Cases 1, 2, and 3 all acheve sgnfcantly hgher total payoffs than Case 4, demonstratng the beft of aggregatng the WPPs. Breakng down the total payoffs across WPPs and hours, a) the daly average payoffs of the WPPs are shown n Fgure 1, and b) the hourly average payoffs of the aggregaton are shown n Fgure 2. It s observed that the payoffs n Cases 1 and 2 are consstently hgher than that n Case 3, whch are further always hgher than that n Case 4. For all the WPPs, ndvdual ratonalty s confrmed. Daly Average Payoff of WPPs $) Hourly Average Payoff of Aggregator $) #10 4 Case 1 Case 2 Case 3 Case WPPs Fgure 1: Comparson of the daly average payoffs of the WPPs. # Case 1 & 2 Case 3 Case Hours Fgure 2: Comparson of the hourly average payoffs of the aggregaton. Furthermore, we plot the real-tme payoff traces of o of the ten WPPs #05711) n Fgure 3. In Fgure 3a), we compare Case 2 aganst Case 4: t s observed that, whle for most of the tmes the WPP earns a hgher payoff n Case 2, there are a small number of hours e.g., hour #144 and #189) n whch the WPP earns a hgher payoff n Case 4. Ths s not uxpected for two reasons: a) Case 2 acheves ndvdual ratonalty n expectaton cf. Secton III), and therefore does not preclude some realzed scenaros n whch Case 4 turns out better, and b) The WPPs and the aggregator only employ a crude Gaussan model for the forecast errors n our smulatons cf. Secton VII-A), and thus the WPPs payoffs determd accordngly can potentally devate from the deal os had the ground truth probablty dstrbutons are employed. In Fgure 3b), we compare Case 3 aganst Case 4: t s observed that for all tmes the WPP earns a hgher payoff n Case 3. Ths s consstent wth the ex-post restrcted ndvdual ratonalty cf. Secton VI-B), n partcular because each WPP s n Case 3 makes the DA commtments n the same way as n Case 4 c,sep ). Smlar observatons have been made for the realzed real-tme payoff traces for the entre aggregaton: a) In Case 2, for most but not all tmes, the aggregator s total payoff s hgher than that n Case 4, and b) In Case 3, for all tmes the aggregator s total payoff s hgher than that n Case 4. In lght of these, t s worth re-emphaszng that effcency,.e., maxmum expected proft for the aggregaton s n fact acheved n Case 2, but not n Case 3 cf. Table I), even though for some realzed scenaros the payoffs n Case 2 are worse than Case 3.

9 payoff of SITE $) payoff of SITE $) Case 2 Case Hour a) Case 3 Case Hour b) Fgure 3: The payoffs traces over tme of a WPP. VIII. COCLUSIO An ndrect mechansm desgn framework s employed for aggregatng rewable power producers n a two settlement power market. We have desgd a payoff allocaton mechansm by solvng the compettve equlbrum of a specally formulated market wth transferrable payoff. We have proved that the outcome of the desgd mechansm s predcted by a unque ash equlbrum among the RPPs partcpatng n the aggregaton, characterzed n closed from. Moreover, at ths E nduced by the mechansm, the entre aggregaton acheves effcency,.e., the maxmum expected proft as f all the RPPs fully cooperate. Ths mples an deal Prce of Anarchy of o. We have then proved that the E s n the core, and s hence stable from a coaltonal game theoretc perspectve. Furthermore, we have proved that the E guarantees no colluson among the RPPs. In addton, a set of ex-post propertes are also acheved by the desgd mechansm. We have smulated the desgd mechansm wth data from 10 wnd power producers n the PJM nterconcton. umercal results consstent wth theoretcal predctons are observed. ACKOWLEDGMET The authors would lke to thank the anonymous revewers for ther helpful comments and suggestons. REFERECES [1] orth Amercan Electrc Relablty Corporaton, Accommodatng hgh levels of varable geraton, Tech. Rep., Aprl [2] P. Pnson, Wnd ergy: Forecastng challenges for ts operatonal management, Statstcal Scence, vol. 28, no. 4, pp , [3] P. P. Varaya, F. F. Wu, and J. W. Balek, Smart operaton of smart grd: Rsk-lmtng dspatch, Proceedngs of the IEEE, vol. 99, no. 1, pp , [4] E. Btar, R. Rajagopal, P. Khargokar, and K. Poolla, The role of co-located storage for wnd power producers n conventonal electrcty markets, n Proc. Amercan Control Conference, 2011, pp [5] P. Harsha and M. Dahleh, Optmal management and szng of ergy storage under dynamc prcng for the effcent ntegraton, IEEE Transactons on Power Systems, vol. 30, no. 3, pp , May [6] M. Chowdhury, M. Rao, Y. Zhao, T. Javd, and A. Goldsmth, Befts of storage control for wnd power producers n power markets, IEEE Transactons on Sustanable Ergy, vol. 7, no. 4, pp , Oct [7] A. J. Cojo, J. M. Morales, and L. Barngo, Real-tme demand response model, IEEE Transactons on Smart Grd, vol. 1, no. 3, pp , [8]. L, L. Chen, and S. H. Low, Optmal demand response based on utlty maxmzaton n power tworks, n Proc. IEEE Power and Ergy Socety Geral Meetng, 2011, pp [9] J. Comden, Z. Lu, and Y. Zhao, Harssng flexble and relable demand response under customer uncertantes, n Proc. of the Eghth Internatonal Conference on Future Ergy Systems ACM e-ergy), 2017, pp [10] E. Baeyens, E. Btar, P. Khargokar, and K. Poolla, Coaltonal aggregaton of wnd power, IEEE Transactons on Power Systems, vol. 28, no. 4, pp , Feb [11] Y. Zhao, J. Qn, R. Rajagopal, A. Goldsmth, and H. V. Poor, Wnd aggregaton va rsky power markets, IEEE Transactons on Power Systems, vol. 30, no. 3, pp , May [12]. san, T. Roughgarden, T. Tardos, and V. V. Vazran, Algorthmc Game Theory. Cambrdge Unversty Press, [13] A. ayyar, K. Poolla, and P. Varaya, A statstcally robust payment sharng mechansm for an aggregate of rewable ergy producers, n Proc European Control Conference ECC), 2013, pp [14] W. Ln and E. Btar, Forward electrcty markets wth uncertan supply: Cost sharng and effcency loss, n Proc. IEEE 53rd Conference on Decson and Control CDC), 2014, pp [15] F. Harrch, T. Vncent, and D. Yang, Optmal payment sharng mechansm for rewable ergy aggregaton, n Proc IEEE Internatonal Conference on Smart Grd Communcatons SmartGrdComm), ov 2014, pp [16], Effect of bonus payments n cost sharng mechansm desgn for rewable ergy aggregaton, n Proc. IEEE Conference on Decson and Control CDC), Dec [17] H. Khazae and Y. Zhao, Ex-post stable and far payoff allocaton for rewable ergy aggregaton, n Proc. IEEE PES Innovatve Smart Grd Technologes Conference ISGT), Aprl 2017, pp [18] P. Chakraborty, E. Baeyens, and P. P. Khargokar, Cost causaton based allocatons of costs for market ntegraton of rewable ergy, IEEE Transactons on Power Systems, vol. 33, no. 1, pp , Jan [19] C. Slva, B. F. Wollenberg, and C. Z. Zheng, Applcaton of mechansm desgn to electrc power markets republshed), IEEE Transactons on Power Systems, vol. 16, no. 4, pp , [20] H. Khazae and Y. Zhao, Compettve market wth rewable power producers acheves asymptotc socal effcency, n Proc. IEEE Power and Ergy Socety Geral Meetng, [21] E. Btar, R. Rajagopal, P. Khargokar, K. Poolla, and P. Varaya, Brngng wnd ergy to market, IEEE Transactons on Power Systems, vol. 27, no. 3, pp , [22] M. J. Osbor and A. Rubnsten, A Course n Game Theory. MIT Press, [23] J. B. Rosen, Exstence and unquess of equlbrum ponts for concave n-person games, Econometrca: Journal of the Econometrc Socety, pp , [24] D. Kalathl, C. Wu, K. Poolla, and P. Varaya, The sharng economy for the electrcty storage, IEEE Transactons on Smart Grd, [25] C. Wu, D. Kalathl, K. Poolla, and P. Varaya, Sharng electrcty storage, n Proc. IEEE 55th Conference ondecson and Control CDC), 2016, pp [26] atonal Rewable Ergy Laboratory, Eastern wnd ntegraton and transmsson study, Tech. Rep., Jan Frst, we have that = APPEDIX A PROOF OF LEMMA 1 { p f c + p r,s x c ), f S + p f c p r,b c x ). f S

10 As a result, = S + + S = p f c S + + c S ) p r,b c S x S ) + p r,s x S + c S +) 36) We now consder the case of x S + c S + c S x S,.e., there s an excess power n total n the aggregaton. In ths case, P = p f c S + + c S ) + p r,s x S + + x S c S + c S ) 37) From 36) and 37), we have: P = p r,b p r,s ) c S x S ) = p r,b p r,s) mn x S + c S +), c S x S )) 38) The case when x S + c S + < c S x S can be proved smlarly. APPEDIX B PROOF OF THEOREM 1 From the best respondng condton of E 6), a cessary condton for a set of DA commtments {c } to be at a pure E s dπ = 0, = 1,...,. 39) dc {c}={c } Gven the proposed PAM, wth a lttle algebra, the above dervatve can be expressed as dπ = p r,b p r,s) c f X x = c j dc j + p r,b p r,s) j cj x f X,X x = c j, x dx 0 j + p f p r,s) p r,b p r,s) F x = j c j. 40) Wth 40), the sum of all the equatons 39) smplfes as 0 = dπ dc {c}={c } = p f p r,s) p r,b p r,s) F c From 41), we obtan that c = j c j = F 1 j j 41) p f p r,s ) p r,b p r,s = c, 42) whch n fact proves the effcency of the E cf. Corollary 1). ow, substtutng j c j = c nto 40) and 39), we get the unque soluton c c x 0 f X,X = x = c, x ) dx f X x = c ) = E [ X X = c ]. 43) APPEDIX C PROOF OF THEOREM 2 From Theorem 1, f a pure E exsts, t must take the form of 14). In ths proof, we show that 1) If condton 15) holds, then 14) s ndeed a pure E, and 2) If condton 15) does not hold, then there exsts a set of DA and RT prces such that a pure E does not exst. Part 1): Suppose condton 15) holds. From Theorem 1, [ the unque canddate ] for a pure E s gven by c X = E X = c, cf. 14)). The expected payoff of RPP at ths canddate pure E s π c, { c }) = p r,s µ + p r,b p r,s) c E X X [X X = x ] f X x ) dx x =0 44) For the strategy profle {c } to ndeed be a pure E, we must also have:, c R, π c, { c }) π c, { c }), 45) where π c, { c }) s the expected payoff of RPP f t chooses c as ts strategy.e., ts DA-commtment), and the other RPPs choose the strateges { } c. π c, { c }) can then be expressed n closed form as follows: π c, { c }) = p r,s µ + p f p r,s) c + p r,b p r,s) + c x =0 { c F c ) E X X [X X = x ] f X x ) dx } 46) where c = c + j c j. Substtutng 44) and 46), we have that 45) s equvalent to, c R, c } {E X X [X X = x ] c f X x ) dx x =c 0. 47) From Theorem 1, when c = c, we have that E X X [X X = x ] c = 0, and the left hand sde LHS) of 47) equals to zero. ow, from the condton 15), If c > c, then c > c, and E X X [X X = x ] c 0, x [c, c ]. Thus, 47) holds. If c < c, then c < c, and E X X [X X = x ] c 0, x [c, c ]. Thus, 47) holds. Part 2): Suppose condton 15) does not hold. In other words, there exsts an RPP k, for some α, de Xk X [X k X =α] > 1. Assumng contnuty of E Xk X [X k X = α] as a functon of α, there exsts an nterval [D 1, D 2 ], where de Xk X [x k X =α] > 1 for all α [D 1, D 2 ].

11 ow, under the mld techncal condton that F s nvertble, we can always fnd a set of prces { p f, p r,b, p r,s} that satsfy p c = F 1 f p r,s ) p r,b p r,s = D 1. 48) We now exam, for ths RPP k, any c k c k, c k + D 2 D 1 ). ote that, for c k = c k, the LHS of 47) s zero; and for any c k c k, c k + D 2 D 1 ), the LHS of 47) s postve. Therefore, under ths partcular set of prces, the unque canddate of a pure E s not a pure E, and thus a pure E does not exst. APPEDIX D DERIVIG THE PAYOFF ALLOCATIO MECHAISM FROM A COMPETITIVE EQUILIBRIUM In ths secton, we show that the proposed PAM 13) can n fact be derved from computng the compettve equlbrum of a specally formulated market wth transferrable payoff. Ths also offers a proof of Theorem 4. A. Market wth Transferrable Payoff We frst def the followng market wth transferrable payoff [22]: The RPPs, denoted by, are a fnte set of agents. There s o type of nput goods power geraton. Each agent has an endowment n the amount of x R + the realzed power of RPP. Each agent has a contnuous, nondecreasng, and concave producton functon f : R + R: f x ) = = p f c p r,b c x ) + + p r,s x c ) +. 49) Snce all the producton functons {f } produce the same type of transferrable output,.e., motary payoff, the above formulaton precsely defs a market wth transferrable payoff. ext, a coaltonal game can be defd based on a market wth transferrable payoff [22]. Specfcally, for any coalton of a subset of RPPs T, def v T ) = max f z ) 50) {z R +, T } s.t. T T z = T In other words, {z, T } denotes a redstrbuton of the total realzed power T x among the members of T. Ths vt ) represents the maxmum total payoff that the members of T can acheve among all possble redstrbutons, computed accordng to f defd n 49). The core of ths coaltonal game s also called the core of the market. We now prove that ths coaltonal game s exactly the same as the coaltonal game defd prevously n 32). Lemma 2: The values of coaltons 50) are the same as 32). Proof: Straghtforwardly, 32) 50) because 32) s the maxmum achevable payoff by the subset T after ther x. aggregaton. ext, we show that 32) can be acheved by 50),.e., 32) 50). We def T + { T x c 0} and T { T x c < 0}. The ntuton of a redstrbuton {z } to acheve 32) s the followng: We gve as much of the excess power of the RPPs n T + as possble to the RPPs n T to offset ther defct power. Specfcally, f x T c T < 0,.e., T x + c ), we let T c x ) > T +, z = c, 51) T, x z c, so that z x ) = x z ). 52) T T + As a result, f z ) = f z ) + f z ) T T + T = p f c + p f c p r,b c z ) ) T + T = p f c T p r,b c x ) z x )) T = p f c T p r,b T c x ) T + x c ) ) 53) = p f c T p r,b c T x T ) = 32), 54) where 53) s mpled by 51) and 52). The case of x T c T 0 can be proved smlarly. As a result, from the property of market wth transferrable payoff cf. Proposton n [22]), we mmedately have that ths coaltonal game has a non-empty core. Moreover, ths formulaton as a market enables us to compute a soluton n the core by dervng the compettve equlbrum CE) of the market, as follows. B. Compettve Equlbrum For the market wth transferrable payoff defd n the last subsecton, a compettve equlbrum s defd [22] as a prce-quantty par of p R + and z R +, such that, ) For each agent, z solves the followng problem: max f z ) p z x )). 55) z R + ) z s a redstrbuton,.e., z = x. The ntuton of a CE s the followng: At the prce p, ) to maxmze ts payoff, each agent can trade any amount of the nput realzed power) on the market wthout worryng whether there s enough supply or demand to fulfll ts trade request, and ) collectvely, the market of nput supply and demand stll clears,.e., the resultng z from the optmal trades s feasble. At a compettve equlbrum p, z ), p s called the compettve prce, and the value of the maxmum of 55) s called the compettve payoff of agent. We then have the followng theorem cf. Proposton n [22]) dctatng that all the CEs are n the core.