Forty-Nnth Annual Allerton Conference Allerton House, UIUC, Illnos, USA September 28-30, 2011 Stochastc Resource Auctons for Renewable Energy Integraton Wenyuan Tang Department of Electrcal Engneerng Unversty of Southern Calforna wenyuan@usc.edu Rahul Jan EE & ISE Departments Unversty of Southern Calforna rahul.jan@usc.edu Abstract Among the many challenges of ntegratng renewable energy sources nto the the exstng grd, s the challenge of ntegratng renewable energy generators nto the power systems economy. Electrc markets currently are run n a way that partcpatng generators must supply contracted amounts. And yet, renewable energy generators such as wnd power generators cannot supply contracted amounts wth certanty. Thus, alternatve market archtectures must be consdered where there are aggregator enttes who partcpate n the electrcty market by buyng power from the renewable energy generators, and assumng rsk of any shortfall from contracted amounts. In ths paper, we propose aucton mechansms that can be used by the aggregators for procurng stochastc resources, such as wnd power. The nature of stochastc resources s dfferent from classcal resources n that such a resource s only avalable stochastcally. The dstrbuton of the generaton s prvate nformaton, and the system objectve s to truthfully elct such nformaton. We ntroduce a varant of the VCG mechansm for ths problem. We also propose a non-vcg mechansm wth a contracted-payment-pluspenalty payoff structure. We show generalzaton of the mechansms to general objectve functons, as well as multple wnners. We then consder the settng where the generators need to fulfll any shortfall from the contracted amount by buyng from the spot market. Index Terms Renewable energy ntegraton, smart grd, mechansm desgn, stochastc resource auctons. I. INTRODUCTION Renewable energy wll ncreasngly consttute a greater fracton of the energy portfolo. In fact, on Aprl 12, 2011, the Governor of Calforna sgned legslaton to requre one-thrd of the state s electrcty to come from renewable energy by 2020. Apart from the sgnfcant nvestment requred, there are many system challenges n ntegratng renewable energy nto the current power grd and electrcty markets. These manly arse due to the varablty and unpredctablty of such energy sources. For example, wnd power can vary from 0 to 100 MW (at a sngle plant) n a matter of a couple of hours. Solar The second author s research on ths project s supported by the NSF CAREER award CNS-0954116 and an IBM Faculty Award. power s equally unpredctable and hghly varable, whle tdal power has a cyclc nature, marked by extreme peaks (durng extreme events such as storms and hurrcanes). Ths ntroduces sgnfcant challenges n matchng supply wth demand, whch vares seasonally and by tme of day but s typcally qute nelastc. For ths purpose, demand response solutons are beng devsed where consumers are exposed to tme-varyng prces va smart-meters over a smart-grd nfrastructure. Massvely scalable energy storage solutons wll have to be a part of the soluton story f such a vson of a smart grd system s to succeed. And yet, ths s not all. Wth one-thrd of energy comng from renewable sources, challenges also arse n how the electrc power economy operates. Ths comprses the electrcty markets that operate at varous tmescales (spot, day-ahead, week-ahead, etc.), the transmsson capacty market, the generators, the utlty companes and the consumers, and prcng to them. An mportant ssue s how renewable energy generators such as of wnd power are to partcpate n and ntegrate wth the electrc power economy. How can they partcpate n, say day-ahead electrcty markets gven the uncertanty about ther generaton for the next day? What s the rght market archtecture that leads to effcency? Though the management of uncertanty n, say wnd power generaton seems dauntng, f the market s structured the rght way, enablng hundreds and thousands of geographcally dspersed renewable energy generators to aggregate, then the statstcal multplexng gans can make the managng of uncertanty much easer as the varance goes down. In ths paper, we focus on the problem of aggregatng power generated by renewable energy generators. Our mplct assumpton s that the current (day-ahead) electrcty market archtecture s not changed, but new enttes called aggregators wll be allowed to enter, who wll buy power from the renewable energy generators, and then sell t n the market, assumng any 978-1-4577-1818-2/11/$26.00 2011 IEEE 345
rsk arsng due to uncertanty n beng able to supply the contracted amount. We nvestgate desgn of aucton mechansms that aggregators can use n buyng power from the renewable energy generators. These auctons must be desgned n such a way that t nduces the generators to reveal the true dstrbutons of the amount of power they wll be able to generate the next day. Ths then provdes the rght nformaton to the aggregators to be able to plan optmally for the rsk they assume due to shortfall n meetng ther generaton commtment n the day-ahead electrcty market. Lterature Survey: Aucton desgn for electrcty markets s a well-studed problem. However, the problem we ntroduce n ths paper whch nvolves stochastc resources (e.g., the electrcty from renewable energy sources to be suppled the next day) s new. Almost all of economc and aucton theory deals wth classcal goods,.e., non-stochastc goods that can be exchanged wth certanty. In contrast, f a renewable energy generator contracts to supply Q MW of power the next day, t may be able to supply that only wth some probablty p. Wth probablty 1 p, t may fal to supply the contracted amount. The aucton desgn problem we ntroduce s for such stochastc goods, whch has receved only scant attenton, f at all. We now provde a bref overvew of some relevant work on aucton and market desgn for electrcty markets though all of t deals wth classcal goods. Green [4] studed a lnear supply functon market model to nvestgate methods of ncreasng competton (whch leads to reducton n dead-weght losses) n the power market n England and Wales. Baldck, et al [1] generalzed Green s model by usng affne functons and ntroducng capacty lmts. Such work prmarly focused on computatonal approaches to fndng the supply functon equlbrum. Johar, et al [5] proposed market mechansms based on revealng a class of supply functons parameterzed by a sngle scalar, whch s closely related to the proportonal allocaton mechansm frst studed by Kelly [6]. These mechansms are desgned for auctons among conventonal power generators, wth no uncertanty n generaton. Dynamc general equlbrum models wth supply frcton are studed n [3]. The most related work to ths paper s Btar, et al [2]. They nvestgated how an ndependent renewable energy generator mght bd optmally n a compettve electrcty pool. The energy market system consdered n that paper conssts of a sngle ex-ante day-ahead forward market wth an ex-post mbalance (shortfall) penalty for scheduled contract devatons. They derved analytcal expressons for the optmal contracted capacty and the correspondng expected proft. They also studed the role of a varety of factors ncludng mproved forecastng, local generaton, energy storage, etc. But what they consdered s a decson-makng problem: a sngle player chooses the contracted capacty to maxmze hs proft. In ths paper, we formulate an aucton desgn problem for renewable energy markets. The key ssue s that the generaton of each renewable energy generator s a random varable. Even the player hmself has no dea of the realzaton, but he knows the dstrbuton. In the desgned aucton, we requre each player to report (the parameters of) hs dstrbuton as hs bd. The auctoneer then pcks one or more players as the wnners who have the best dstrbutons. The man objectve of aucton desgn s to elct the players true types (.e., the true dstrbutons) so that the optmal socal welfare can be acheved. We call such an aucton a stochastc resource aucton. Ths s the frst work on stochastc resource auctons to the best of our knowledge. We propose a two-stage stochastc varant of VCG, as well as a stochastc non-vcg mechansm. We demonstrate the ncentve compatblty property of the two auctons desgned. We also do a revenue comparson. We show generalzaton of the mechansms to general objectve functons, as well as multple wnners. We then consder a settng wheren the burden of the any shortfall n generaton falls on the generators themselves. We propose a VCG-type mechansm for ths settng, and do ts equlbrum analyss. II. STOCHASTIC RESOURCE AUCTION DESIGN: PROBLEM STATEMENT Consder N renewable energy generators, denoted by player P s ( = 1,..., N). Player P s generaton for a gven future tme (e.g., the next day n the settng of day-ahead markets) s a random varable denoted by X, whch s normalzed so that X Ω := [0, 1]. All the X s are ndependent. Assume there s a type space Θ such that the dstrbuton of X (for all ) can be parameterzed by a K-dmensonal vector θ = (θ (1),..., θ (K) ) Θ. We refer to θ as player P s type. Let z = (z 1,..., z N ) be an outcome vector, where z = 1 f player P s a wnner and z = 0 otherwse. Denote the outcome space by Z. Defnton 1. A stochastc socal choce functon (SSCF) f : Θ N Z for each possble profle of agents types θ = (θ 1,..., θ N ) Θ N specfes the outcome f(θ) Z. Note that the SSCF s ndependent of the realzaton X = (X 1,..., X N ) Ω N, because our goal s smply to maxmze the expectaton of some functon of the re- 346
Tme Tme All undefned concepts are standard, and the reader can consult [8] for ther defntons. Players Fg. 1. Payment q (ˆθ, X) Realzaton of X Payment p (ˆθ) Outcome g(ˆθ) Bds ˆθ The stochastc resource aucton. Auctoneer alzatons. We next defne the aucton form for stochastc resources as follows. Defnton 2. A stochastc resource aucton M = (S, g, p, q) s specfed by 1) The strategy profle space S = N =1 S from whch players report ther bds s = (s 1,..., s N ) 2) The outcome functon g : S Z whch determnes the wnners 3) The payment p : S R made by the auctoneer to each player P before the realzaton of X 4) The payment q : S Ω N R made by each player P to the auctoneer after the realzaton of X. We wll consder drect stochastc resource auctons so that the strategy space S for each player s the same as the type space Θ. The players report ther bds ˆθ = (ˆθ 1,..., ˆθ N ) to the auctoneer. The auctoneer then determnes the outcome g(ˆθ) = z, ndcatng the wnners. A payment p (ˆθ) s now made to each player P whch only depends on the bds but not on the realzaton of the generaton. Ths can be nterpreted as the contractual payment. Upon realzatons, each player makes a payment q (ˆθ, X) to the auctoneer. Ths can be nterpreted as the penalty that each player pays for not fulfllng some contracted amount. Note that a negatve p or q ndcates a reversed payment. The process of a stochastc resource aucton s shown n Fg. 1. Snce the players are strategc, they may msreport ther prvate nformaton. Our goal s to desgn ncentve compatble mechansms that mplement the SSCF n domnant strateges,.e., that yeld truthful revelaton of the players types as the domnant strategy equlbrum. III. TWO BASIC MECHANISMS FOR STOCHASTIC RESOURCE ALLOCATIONS We frst consder the basc scenaro where there s a sngle wnner,.e., there exsts an such that z = 1 and z = 0 for all. Let F ( ) be the cumulatve dstrbuton functon (CDF) of X, whch s determned by the type θ. Let ˆF ( ) be the CDF correspondng to the reported type ˆθ. A basc objectve (represented by the SSCF) for a stochastc resource aucton could be to dentfy the player who yelds the hghest expected generaton (through learnng the true dstrbutons). We now propose two desgns wheren t s a domnant strategy for each player to reveal ther types truthfully. The frst s a varant of the well-known VCG mechansm. The second s not, but s n some sense more natural and lkely to be more acceptable to generators. A. The Stochastc VCG (SVCG) Mechansm We frst ntroduce a stochastc verson of the VCG mechansm. From the reported types, the assocated (reported) dstrbutons are determned, and the expected generatons can be computed. We choose the player wth the hghest expected generaton as the wnner P : arg max x d ˆF (x). (1) We also defne the margnal loser P as arg max x d ˆF (x). (2) Then, the wnner makes a payment p to the auctoneer before the realzaton of X : p = x d ˆF (x). (3) Ths can be nterpreted as the contractual or sgn-on amount the generator pays to the auctoneer. Upon the realzaton of X, the auctoneer makes a payment to the wnner: q = X. (4) Ths can be nterpreted as the payment for the supply that the generator actually makes to the aggregator (at prce 1). Then wnner s payoff s U = p q = X x d ˆF (x). The other players get zero payoffs. 347
Theorem 1. The SVCG mechansm specfed by (1)-(4) s ncentve compatble. Due to space constrants, we omt the proofs of Theorem 1 and 2. The reader can consult the proof of Theorem 3 for the general case. Ths mechansm s a stochastc varant of the Vckrey- Clarke-Groves (VCG) mechansm. The expectaton of X can be vewed as the counterpart of the valuaton n the standard VCG mechansm, and x d ˆF (x) as the counterpart of the payment (externalty). Unlke the classcal settng, the valuaton n our model s not ntrnsc. In fact, t s also n the form of a payment. Ths opens the possblty of other knds of ncentve compatble mechansms. B. The Stochastc Shortfall Penalty (SSP) Mechansm We now propose a non-vcg mechansm. The wnner P s chosen as n (1), and the margnal loser s defned as n (2). A payment p s made n advance to the wnner: p = 1. (5) The nterpretaton s that the auctoneer pays for the normalzed full capacty (at prce 1) before the realzaton of X. After realzaton, f there s a shortfall, the contracted generator has to pay a penalty q that depends on the shortfall: where q = λ(1 X ), (6) 1 λ = 1 x d ˆF (x) can be vewed as the penalty prce for the shortfall 1 X. As before, the wnner s payoff s U = p q = 1 λ(1 X ). The other players get zero payoffs. Theorem 2. The SSP mechansm specfed by (1)-(2) and (5)-(6) s ncentve compatble. The above mechansm s nspred by [2], whch adopts a contracted-payment-plus-penalty payoff structure for a renewable energy generator. However, they have consdered a decson-makng problem n whch the decson maker chooses the contracted capacty, whle we consder an aucton desgn problem n whch players bd ther dstrbutons. Note that λ 1, whch s necessary for the mechansm to be ncentve compatble. It s nterestng to observe a quas-dualty between the SVCG and the SSP mechansms: Before the realzaton, money flows from the wnner to the auctoneer n the SVCG mechansm (whch depends on the second hghest bd), whle t flows from the auctoneer to the wnner n the SSP mechansm (whch s a constant). After the realzaton, money flows from the auctoneer to the wnner n the SVCG mechansm (whch depends on the realzaton), whle t flows from the wnner to the auctoneer n the SSP mechansm (whch depends on both the second hghest bd and the realzaton). C. Revenue Comparson It s useful to compare the expected revenue obtaned wth the two mechansms. We assume that the auctoneer resells the acqured resource X at prce 1. Proposton 1. The auctoneer s expected revenue n the SVCG mechansm s greater than or equal to that n the SSP mechansm. Proof: Snce the auctoneer gets the same amount of power n both mechansms, hs revenue from resale s the same. We just need to compare hs payment to the wnner. Equvalently, we can compare the payment receved by the wnner, whch s just the wnner s payoff. Gven that truth-tellng s a domnant strategy for each player, the wnner s expected payoff n the SVCG mechansm s x df (x) x df (x), whle the wnner s expected payoff n the SSP mechansm s [ ] λ x df (x) x df (x). (Readers are referred to the proof of Theorem 3 for detals.) Snce λ 1, the wnner s expected payoff n the SVCG mechansm s smaller than or equal to that n the SSP mechansm. Therefore, the auctoneer s expected revenue n the SVCG mechansm s greater than or equal to that n the SSP mechansm. IV. GENERALIZATIONS OF THE BASIC MECHANISMS A. General Objectve Functons In the prevous analyss for the basc scenaro, we have assumed that the socal planner s objectve s to contract wth player P who yelds the hghest expected generaton (see (1)). Now we generalze the socal planner s objectve. Assume that the socal planner wants to contract wth the one who yelds the hghest E [h(x )], where h( ) s a functon of the random varable X. 348
We call h( ) the objectve functon, and we have E [h(x )] = h(x) df (x). For example, the socal planner s demand may be at a certan level D [0, 1]. That s, he only needs D amount of power and does not care about how much more would be generated. The objectve functon then s h(x) = mn {x, D}. (7) Now we propose the mechansms for the general objectve functon h( ). As before, let denote the wnner and the margnal loser. That s, arg max h(x) d ˆF (x), (8) arg max h(x) d ˆF (x). (9) Defne the functon space H := {h : [0, 1] R}. We also defne H p as a subset of H, whch only contans non-negatve, non-decreasng functons. We wll show that the generalzed SVCG mechansm s applcable to any h( ) H, whle the generalzed SSP mechansm s only vald for any h( ) H p. Although H p s a subset of H, t stll represents a large class of objectve functons. We now clam the followng and prove the ncentve compatblty of both mechansms n an ntegrated framework. Theorem 3. () For any h( ) H, the generalzed SVCG mechansm specfed by (8)-(9) wth p = h(x) d ˆF (x) and q = h(x ), s ncentve compatble. () For any h( ) H p, the generalzed SSP mechansm specfed by (8)-(9) wth and where λ = s ncentve compatble. p = h(1) q = λ[h(1) h(x )], h(1) h(1) h(x) d ˆF (x), Proof: We want to show that truth-tellng (ˆθ = θ) s a domnant strategy equlbrum. Fx a player P wth bd ˆθ = θ. We wll show that he cannot be better off by reportng ˆθ θ for any ˆθ. Suppose he s the wnner. Let the margnal loser be P. In the generalzed SVCG mechansm, player P s expected payoff s E [U ] = E [h(x )] h(x) d ˆF (x) = h(x) df (x) h(x) d ˆF (x) 0, whle n the generalzed SSP mechansm, t s E [U ] = h(1) E [λ(h(1) h(x ))] [ ] = h(1) λ h(1) h(x) df (x) [ ] = λ h(x) df (x) h(x) d ˆF (x) 0. In both mechansms, by changng hs bd, ether the outcome remans the same, or he loses the contract and then gets a zero payoff. Thus, he has no ncentve to devate. Suppose he s a loser. Let the wnner be P. If player P changes hs bd so that he outbds the wnner, n the generalzed SVCG mechansm, hs expected payoff would be E [U ] = h(x) df (x) h(x) d ˆF (x) 0, whle n the generalzed SSP mechansm, t would be [ ] E [U ] = λ h(x) df (x) h(x) d ˆF (x) 0, where λ = h(1) h(1) h(x) d ˆF (x). In both mechansms, f he does not outbd the wnner, he stll gets a zero payoff. Thus, he has no ncentve to devate. Ths proves the theorem. Note that the condton h( ) H p n the generalzed SSP mechansm ensures that the penalty prce λ 1. Example 1. It s easy to verfy that the objectve functon n (7) satsfes h( ) H p H. Thus, both the generalzed SVCG and SSP mechansms can be used. The wnner s payoff n the generalzed SVCG mechansm s U = mn {X, D} mn {x, D} d ˆF (x), 349
whle that n the generalzed SSP mechansm s U = mn {1, D} λ(mn {1, D} mn {X, D}) = D λ(d X ) +, where D λ = D mn {x, D} d ˆF (x), x+ := max {x, 0}. B. Elmnatng the Undesred Equlbra It s worth pontng out that gven an objectve functon h( ), any ˆθ (or equvalently, ˆF ( )) that satsfes h(x) d ˆF (x) = h(x) df (x) s a domnant strategy for player. That s, to maxmze hs own payoff, player P does not have to report θ but just ˆθ, as long as the above equlty holds. Thus, whle the proposed mechansms are domnant strategy ncentve compatble, ths laxty may be consderably undesrable n some scenaros. The auctoneer may not only want to elct the true measures for a gven objectve functon, but also want to elct the true types, so that ths nformaton can be used for other purposes. Hence, a stronger result would be more desrable: n addton to domnant strategy ncentve compatblty, each player P s strctly better off by reportng ˆθ = θ than any ˆθ θ for some ˆθ. Ths problem can be easly fxed f the auctoneer does not announce the objectve functon h( ) to be used n the mechansm but pcks t arbtrarly from H (or H p ). We thus clam the followng. Proposton 2. In the generalzed SVCG (or SSP) mechansm, f the objectve functon h( ) H (or H p ) s chosen arbtrarly and not revealed to the bdders untl the bds have been submtted, then truth-tellng s a unque domnant strategy for each player. C. Extenson to Multple Wnners Now we consder the case of multple wnners. Instances of such extenson of VCG mechansms n the classcal settng can be found n [7]. Suppose the socal planner wants to contract wth M (< N) players who yeld the hghest E [h(x )]. We rank the players n order of the bds,.e., h(x) d ˆF (1)(x) h(x) d ˆF (N)(x), (10) where ( (1),..., (N) ) s a permutaton of {1,..., N}. Then players P (1),..., P (M) are the wnners and player P (M+1) s the margnal loser. We now clam the followng, whose proofs are qute analogous to the prevous ones and are therefore omtted. Theorem 4. () For any h( ) H, the M-SVCG mechansm specfed by (10) wth p (m) = h(x) d ˆF (M+1)(x) and q (m) = h(x (m)), for m = 1,..., M, s ncentve compatble. () For any h( ) H p, the M-SSP mechansm specfed by (10) wth p (m) = h(1) and q (m) for m = 1,..., M, where λ = s ncentve compatble. = λ[h(1) h(x (m))], h(1) h(1) h(x) d ˆF (M+1)(x), V. RISK-AWARE GENERATION ASSIGNMENT AUCTIONS FOR STOCHASTIC RESOURCES In earler sectons, we have proposed ncentve compatble mechansms for stochastc resource auctons where the auctoneer does bear the rsk of not beng able to acqure some capacty D from the generators. The auctoneer may be an aggregator who s commtted to supplyng D amount of power n an electrcty market. In that case, he wll have to meet any shortfall by buyng power from the spot market (presumably at hgh prces). In ths secton, we consder a related aucton desgn problem, wheren the rsk of any shortfall s borne entrely by the generators themselves. That s, the generators must buy power from the spot market and delver t to the auctoneer f there s any shortfall. Ths may ndeed dstort the ncentves of the generators who may now become conservatve n ther bds, thus leadng to neffcency. We call such auctons rsk-aware generaton assgnment auctons snce the players must take the rsk of the shortfall nto account. A. Problem Statement Consder player P who s requred to meet a fxed demand y [0, 1]. As before, hs generaton at a future tme s a random varable X [0, 1] wth CDF F ( ) and probablty densty functon (PDF) f ( ). Let λ be a constant denotng the prce of the resource n the spot market, whch can also be vewed as the penalty prce. Defne the (expected) cost functon as c (y ) := E [λ(y X ) + ] = λ y 0 (y x)f (x) dx, 350
whch s the expected payment made by the generator to the spot market to make up for the shortfall f any, when the assgned generaton s y. We derve some propertes of the cost functon. It s easy to check the followng: dc (y ) dy = λ y 0 d 2 c (y ) dy 2 f (x) dx = λf (y ), = λf (y ) 0. Thus, c (y ) s convex on [0, 1]. Moreover, dc (y ) dy = λf (0) = 0, (11) y =0.e., the margnal cost at zero s zero. We also have dc (y ) dy = λf (1) = λ, (12) y =1.e., the margnal cost s at most λ, the spot market prce. The generaton assgnment aucton problem s to desgn a mechansm to satsfy a fxed demand D > 0 such that the followng socal welfare optmzaton problem s solved at equlbrum: mnmze y 1,...,y N,y subject to c (y ) + λ(d y) + y = y, 0 y 1,. (13) We consder two cases, dependng on whether F ( ) can be parameterzed or not. In both cases, player P s (expected) payoff s defned as U (y ) := w c (y ), where w s the payment receved by player P from the auctoneer. B. VCG Mechansm for Complete Parametrzaton In ths case, each player P can report the complete dstrbuton, or equvalently, the complete cost functon. Denote the reported cost functon by c ( ). In the mechansm, the assgnment (y 1,..., y N, y) s a soluton of the followng optmzaton problem: mnmze y 1,...,y N,y subject to c (y ) + λ(d y) + y = y, 0 y 1,. Let (y1,..., y 1, 0, y +1,..., y N, y ) denote the assgnment as a soluton of the above wth y = 0,.e., when player P s not present. Then player P s pad w = [ cj (y j ) c j (y j ) ] + λ(y y ), j whch s the reverse externalty that player P mposes on the other players by hs partcpaton. Ths s just a smple varant of the standard VCG mechansm, and the ncentve compatblty follows. C. -VCG Mechansm for Incomplete Parametrzaton Now we consder the more nterestng and dffcult case of non-parametrc F ( ). In ths case, t s mpossble for a player to communcate the cost functon exactly. Instead, we ask each player P to report a two-dmensonal bd b = (β, d ), where β s hs ask prce and d s the maxmum quantty that could be offered at prce β. In the mechansm, the assgnment (y 1,..., y N, y) s a soluton of the followng optmzaton problem: mnmze y 1,...,y N,y subject to Let (y 1,..., y 1 β y + λ(d y) + y = y, 0 y d,., 0, y +1,..., y N, y ) denote the assgnment as a soluton of the above wth d = 0,.e., when player P s not present. Then player P s pad w = j β j (y j y j ) + λ(y y ), whch s the reverse externalty that player P mposes on the other players by hs partcpaton. We call ths the -VCG mechansm (wth ncomplete parametrzaton). Although domnant strategy mplementaton s mpossble n ths case, we show that the - VCG mechansm s a (weak) Nash mplementaton,.e., there exsts a Nash equlbrum n whch the effcent assgnment s acheved. Theorem 5. There exsts an effcent Nash equlbrum n the -VCG mechansm. Proof: The socal welfare optmzaton problem (13) s a convex optmzaton problem that s easy to solve wth the propertes (11) and (12). Let the soluton be (y1,..., yn, y ), whch satsfes c 1(y 1 ) = = c N (y N ) := µ. Consder the strategy profle: β = µ and d = y for all. Clearly, t nduces the effcent assgnment (y1,..., yn, y ). It remans to show that t s a Nash 351
equlbrum. Consder player P wth bd b = (µ, y ). Hs current payoff s U (y ) = λy c (y ). If he changes hs bd to decrease hs contracted capacty y by a δ > 0, then the others assgnment does not change but hs payoff becomes U (y δ) = λ(y So hs payoff changes by δ) c (y U (y δ) U (y ) = λδ c (y δ) + c (y 0. c (y )δ c (y ) δ). δ) + c (y ) If he changes hs bd to ncrease hs contracted capacty y by a δ > 0, then y does not change but the assgnment of some player P j (j ) changes to yj. Player P s payoff becomes U (y + δ) = µ(yj yj ) + λ y j j c (y + δ) = µδ + λy c (y + δ) = c (y )δ + λy c (y + δ). So hs payoff changes by U (y = c (y )δ c (y 0. + δ) U (y ) y j + δ) + c (y ) Thus, player P has no ncentve to devate, whch proves that the constructed strategy profle s a Nash equlbrum. VI. CONCLUSION In ths paper, we have formulated aucton desgn problems for auctonng stochastc resources among renewable energy generators. The mechansms can be used by an aggregator who can then bd n a futures electrcty market. We have consdered two alternatve market archtectures. In the frst, the rsk due to uncertan generaton s assumed by the aggregator, and the generators compete for contract. The desgned mechansms are ncentve compatble, n whch the generators would truthfully reveal ther probablty dstrbutons. Ths s acheved va a two-part payment, an ex ante payment (before the realzaton) and an ex post payment (after the realzaton). Such mechansms are mportant and useful for the aggregator snce he can now hedge aganst hs rsk. In the second nstance, the rsk due to uncertan generaton s assumed by the generators, and the generators compete for assgnment. If there s any shortfall, the generators are responsble for buyng from the spot market and fulfllng ther contracts. Ths can possbly skew ncentves, and make the generators averse to truthfully reportng ther probablty dstrbutons, as well as make achevement of socal welfare optmzaton dffcult. It turns out that ths s not the case. In the parametrc case, the VCG mechansm stll yelds ncentve compatblty, whle n the non-parametrc case, the -VCG mechansm s a Nash mplementaton (domnant strategy mplementaton cannot be acheved n ths case). In the future, we wll consder a repeated verson of stochastc resource auctons wth spot market prces that vary accordng to a Markov process. We would also consder a double-sded market archtecture wheren there are buyers (the utlty companes) as well as both types of sellers (conventonal as well as renewable energy generators). It s an open queston whether t s even possble to desgn such a market wth desrable equlbrum propertes. If ths s mpossble, then ths would provde regulators wth a ratonale to consder alternatve market archtectures, as well as allow for partcpaton of newer enttes who wll act as aggregators. The presented work, along wth the proposed future work, wll potentally provde economc solutons for ntegratng renewable energy generators nto smart grd networks. REFERENCES [1] ROSS BALDICK, RYAN GRANT AND EDWARD KAHN, Theory and applcaton of lnear supply functon equlbrum n electrcty markets, Journal of Regulatory Economcs, 25:143-167, 2004. [2] E.Y. BITAR, R. RAJAGOPAL, P.P. KHARGONEKAR, K. POOLLA AND P. VARAIYA, Brngng wnd energy to market, Submtted to IEEE Transactons on Power Systems, 2010. [3] I. K-CHO AND S. MEYN, Effcency and Margnal Cost Prcng n Dynamc Compettve Markets wth Frcton, Pre-prnt, 2009. [4] RICHARD GREEN, Increasng competton n the Brtsh electrcty spot market, The Journal of Industral Economcs, 44(2):205-216, 1996. [5] R. JOHARI AND J.N. TSITSIKLIS, Parameterzed supply functon bddng: equlbrum and welfare, To Appear n Operatons Research, 2010. [6] FRANK P. KELLY, Chargng and rate control for elastc traffc, European Transactons on Telecommuncatons, 8:33-37, 1997. [7] VIJAY KRISHNA, Aucton Theory, Second Edton, Academc Press, 2009. [8] ANDREU MAS-COLELL, MICHAEL D. WHINSTON AND JERRY R. GREEN, Mcroeconomc Theory, Oxford Unversty Press, 1995. 352