RENEWABLE energy increasingly constitutes a greater

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1 IEEE TRANSACTIONS ON SMART GRID, VOL. 0, NO. 0, JANUARY 0000 Aggregatng Correlated Wnd Power wth Full Surplus Extracton Wenyuan Tang, Member, IEEE, and Rahul Jan, Member, IEEE Abstract We study the problem of desgnng proftmaxmzng mechansms for an aggregator who aggregates wnd power from a group of wnd power producers WPPs). The WPPs have more refned forecasts of the wnd power generaton than the aggregator. Such forecasts are ther prvate nformaton, whch also gve the reservaton utltes of the WPPs. The goal of the aggregator s to elct the prvate nformaton truthfully, whle payng them as lttle as possble. Insprng by the fact that those forecasts are typcally correlated due to the geographcal proxmty of the WPPs, we formally defne the full correlaton condton, whch holds ubqutously n practce. Under that condton, we construct an optmal mechansm whch yelds the truthful elctaton, whle extractng the full surplus.e., wth mnmum payments equal to the reservaton utltes) n expectaton. Fnally, we conduct a case study based on the realworld data, whch emprcally valdates the results. Index Terms Electrcty market, mechansm desgn, wnd power aggregaton. NOMENCLATURE N Number of WPPs. WPP ndex takng values from to N. W Random varable representng the wnd power generaton of WPP. W Random vector of all W s. W Random vector of all W s except W. g w ) Probablty densty functon of W. K Number of components for the Gaussan mxture dstrbuton. k Index takng values from to K. L Constant equal to K N. l Index takng values from to L. α,k Pror probablty of pckng the kth component for WPP. µ,k Mean of the kth component for WPP. σ,k 2 Varance of the kth component for WPP. θ Type of WPP. Θ Space of θ. Manuscrpt receved Aprl 8, 206; revsed July 20, 206, November 2, 206, and February 0, 207. A prelmnary verson of ths paper appears n SmartGrdComm 204 [. W. Tang s wth the Department of Electrcal Engneerng and Computer Scences, Unversty of Calforna, Berkeley, CA 94720, USA, and the Department of Cvl and Envronmental Engneerng, Stanford Unversty, Stanford, CA 94305, USA e-mal: tangwenyuan@berkeley.edu). R. Jan s wth the Department of Electrcal Engneerng, Unversty of Southern Calforna, Los Angeles, CA 90089, USA e-mal: rahul.jan@usc.edu) θ Type of all WPPs. θ Type of all WPPs except WPP. Θ Space of θ. φθ) Pror dstrbuton of θ. φ θ θ ) Condtonal dstrbuton of θ gven θ. p Day-ahead contract prce. q Real-tme penalty prce. F X ) Cumulatve dstrbuton functon of a random varable X. f X ) Probablty densty functon of a random varable X. V θ ) Expected proft when WPP partcpates n the market ndvdually. x θ ) Optmal power contract when WPP partcpates n the market ndvdually. V θ) Expected proft when all WPPs partcpate n the market as an aggregate. x θ) Optmal power contract when all WPPs partcpate n the market as an aggregate. V θ ) Expected proft when all WPPs except WPP partcpate n the market as an aggregate. x θ ) Optmal power contract when all WPPs except WPP partcpate n the market as an aggregate. R θ ) Reservaton utlty of WPP. ˆθ Reported type of WPP. ˆθ Reported type of all WPPs. ˆθ Reported type of all WPPs except WPP. t ˆθ, W ) Payment to WPP. t ˆθ, W ) Payment to WPP n the stochastc VCG mechansm. A Condtonal dstrbuton matrx for WPP. h θ ), t ˆθ ) Auxlary varables for contructng the optmal mechansm. t ˆθ, W ) Payment to WPP n the optmal mechansm. I. INTRODUCTION RENEWABLE energy ncreasngly consttutes a greater fracton of the energy portfolo. Calforna has set an ambtous goal of requrng 33 percent of retal sales from

2 2 IEEE TRANSACTIONS ON SMART GRID, VOL. 0, NO. 0, JANUARY 0000 renewable energy by On the other hand, due to the varablty and unpredctablty of renewables such as wnd and solar, many system challenges arse n ntegratng renewable energy nto the current grd and the electrcty market. The wholesale electrcty markets n the Unted States are manly two-settlement markets, each of whch conssts of a day-ahead forward market and a real-tme spot market. Consder a wnd power producer WPP) who partcpates n the two-settlement market ndvdually. In the day-ahead market, the WPP chooses a power contract whch specfes the amount of supply n the real-tme market. In the real-tme market, the WPP pays a penalty for the devaton from the day-ahead commtment. The tradng problem for a WPP has been addressed n the lterature [2, [3. The system operaton problem nvolvng uncertan wnd power, has also been studed [4, [5, n the framework of stochastc programmng or robust optmzaton. Underlyng those tradng and operaton problems s the task of wnd power forecastng, whch s challengng and stll developng. In [6, a statstcal dstrbuton model for the forecast errors s developed. Advanced statstcal learnng technques have been employed n varous forecastng scenaros [7 [0. It s well known that aggregatng dverse wnd power sources can reduce the overall varablty, whch benefts both the system and the members of the aggregaton. Consequently, the proft under aggregaton can be greater than the total proft under ndvdual partcpaton. The queston s how to farly allocate the proft under aggregaton among the WPPs wth random generaton. In [, they address the queston n the framework of cooperatve game theory. They show that there always exsts a payoff allocaton that stablzes the grand coalton. As related work, [2 ntroduces rsky power contracts n addton to frm power contracts to enable flexble and effcent wnd power aggregaton. In ths work, we follow a dfferent approach. We assume that there s an aggregator who aggregates wnd power from the WPPs. Such an aggregator can be a thrd-party fnancal entty, who may not own any physcal resources n the grd. It s not new that purely fnancal enttes partcpate n the electrcty market. Some common fnancal nstruments nclude fnancal transmsson rghts and vrtual bds. Recently, demand response provders e.g., OhmConnect) have been enterng the market. Those provders assst retal customers to lower electrcty blls, and then get pad from the market through load curtalment. They play a smlar role to an aggregator. Clearly, the aggregator can potentally extract the surplus from aggregaton. But what are the ncentves for a WPP to contract wth the aggregator? Ths can be justfed as follows. Frst, t can be costly for a small WPP to partcpate n the market drectly, due to regulaton requrements. Second, a WPP s more rsk averse and senstve to the hghly volatle real-tme penaltes, whle the aggregator assumes the rsk to make a proft. Thrd, n the proposed mechansm, each WPP s guaranteed to make at least the proft under ndvdual partcpaton, whch we call the reservaton utlty. Therefore, as a proft-maxmzng aggregator, ts objectve s to extract as much surplus as possble, subject to makng some mnmum payments to the WPPs. On the other hand, such reservaton utltes are prvate nformaton, whch the WPPs, as strategc players, may not have ncentves to reveal. The goal of ths work s to desgn mechansms whch elct truthful nformaton of the WPPs and maxmze the proft, on behalf of the aggregator. Achevng ncentve compatblty by tself s not dffcult, and the well-known Vckrey-Clarke-Groves VCG) mechansm [3 provdes a wdely applcable soluton. As an extenson, we propose the stochastc VCG mechansm n our earler work [4 to address the uncertanty of wnd power generaton. In that settng, the aggregator s a welfare-maxmzng socal planner. However, for a proft-maxmzng aggregator, the stochastc VCG mechansm suffers from the ssue of budget defct. Snce the WPPs are n close proxmty to each other, ther forecasts of ther generaton are possbly correlated. Such correlaton can be exploted by the aggregator to elct the forecasts truthfully. Combnng the deas of the exstng work on mechansm desgn wth correlated types [5 [8 and our work on mechansm desgn wth stochastc resources [4, we show that under the full correlaton condton, the optmal mechansm extracts the full surplus n expectaton. The proposed mechansm naturally handles prvate reservaton utltes, whch s an open problem n a general settng [9. The paper s organzed as follows. In Secton II, we ntroduce a Gaussan mxture model of wnd power generaton, whch motvates the mechansm desgn problem for wnd power aggregaton. In Secton III, we formulate the problem. The man results are presented n Secton IV. In Secton V, we conduct a case study to valdate the theoretcal results. Secton VI concludes the paper. II. STATISTICAL MODEL OF WIND POWER GENERATION Throughout the paper, we use the wnd power data from GEFCom202 [20. The data contans hourly wnd power generaton for seven wnd farms WPPs hereafter) n the same regon of the world from July, 2009 to December 3, 200. Fg. shows the dstrbuton of wnd power generaton for WPP durng hour the frst hour of each day). Note that the generaton n the data has been normalzed by the respectve nomnal capactes of the WPPs. Let W be a random varable denotng the wnd power generaton. There are many ways to model the dstrbuton of W. We use the Gaussan mxture model, under whch the probablty densty functon of W s gven by gw) = K α k N w; µ k, σk), 2 ) k= where K s the number of mxture components, α k s the pror probablty of pckng the kth component, and N ; µ k, σk 2) s the Gaussan densty of the kth component. We use K-means clusterng to partton the support of W nto K clusters. Fx µ k as the center of each cluster k. For smplcty, µ k s are sorted such that µ < < µ K. Then we use the expectaton-maxmzaton algorthm to obtan α k and σk 2 for each k. When K = 5, the result s shown n Fg..

3 TANG et al.: AGGREGATING CORRELATED WIND POWER WITH FULL SURPLUS EXTRACTION 3 5 WPP 7 Correlaton Coeffcent 4 WPP WPP Densty 2 WPP WPP WPP Wnd Power Generaton WPP WPP WPP 2 WPP 3 WPP 4 WPP 5 WPP 6 WPP 7 Fg.. The dstrbuton of wnd power generaton for WPP durng hour, ftted by a Gaussan mxture model. Fg. 2. Correlaton matrx heat map of wnd power generaton for the seven WPPs durng hour. Such a mxture model has a meanngful nterpretaton n the context of wnd power aggregaton. At the day-ahead stage, the aggregator can only tell that the wnd power generaton W follows a Gaussan mxture dstrbuton, whle the WPP has a more refned forecast. In partcular, the WPP knows whch of the K Gaussans W s drawn from,.e., the value of k. Reportng k can be translated nto predctng whether the generaton of the followng day s, say K = 5, very low, low, medum, hgh, or very hgh. In other words, k gves the posteror dstrbuton of W. We emphasze that we use the mxture model to hghlght the layered nformaton structure between the aggregator and the WPPs. The choce of the base dstrbuton s not crucal t does not have to be Gaussan. The selecton of the parameters, ncludng determnng the value of K, s a modelng ssue, whch does not affect our man analyss. The objectve of the proft-maxmzng aggregator s to elct the value of k for each WPP whle makng mnmum payments to them. In general, ths task s challengng, as the WPPs may not have ncentves to reveal such prvate nformaton to the aggregator. Fortunately, besdes the mxture model, the aggregator can also explot the correlaton structure of the ndvdual forecasts. Ths s the central dea of ths paper. Fg. 2 shows the correlaton matrx heat map of wnd power generaton for the seven WPPs. It can be seen that the generaton s hghly correlated. Furthermore, assumng that each realzaton s drawn from the nearest component, the component ndces can also be shown to be hghly correlated. That s, when the generaton of one WPP s drawn from the kth component, t s lkely that the other s s also drawn from the kth component. In prncple, the correlaton can be weak or even negatve, but our man results do not rely on the sgn of correlaton. III. PROBLEM STATEMENT The conceptual framework of ths work s llustrated n Fg. 3. Consder an aggregator who aggregates wnd power WPP Two-Settlement Market p, q W V θ) Aggregator W V φθ) θ ) ˆθ W t ˆθ, W ) WPP θ, R θ ), φ θ θ ) WPP N Fg. 3. An aggregator aggregates wnd power from N WPPs. Each WPP s type θ, as ts prvate nformaton, gves more refned forecast of ts generaton W, whle the aggregator only knows the pror dstrbuton of θ. The type θ also gves the reservaton utlty R θ ) and the condtonal dstrbuton φ θ θ ). The goal of the aggregator s to nduce each WPP to report truthfully,.e., ˆθ = θ, whle mnmzng the payment t ˆθ, W ) to the WPPs. In that case, the expected proft under aggregaton s V θ), whch s greater than or equal to the sum of the expected profts V θ ) s under ndvdual partcpaton. from N WPPs, ndexed by =,..., N. The generaton of WPP s a random varable W, followng a Gaussan mxture dstrbuton wth K components: g w ) = K α,k N w ; µ,k, σ,k), 2 2) k= where α,k s the pror probablty of pckng the kth component for WPP, and N ; µ,k, σ,k 2 ) s the Gaussan densty of the kth component for WPP. Let W = W,..., W N ) and W = W,..., W, W,..., W N ). Whle the realzaton of W cannot be known a pror, WPP learns the value of k, denoted by θ {,..., K}. Then the

4 4 IEEE TRANSACTIONS ON SMART GRID, VOL. 0, NO. 0, JANUARY 0000 posteror dstrbuton of W gven θ s N ; µ,θ, σ,θ 2 ). We refer to θ as the type of WPP, and Θ = {,..., K} as the type space same for all ). Let θ = θ,..., θ N ), θ = θ,..., θ, θ,..., θ N ), and Θ = {,..., K} N. We assume that θ s drawn from a commonly known dstrbuton φ ), whch s understood as a probablty mass functon n ths paper. Note that θ,..., θ N are not necessarly ndependent. In fact, they are hghly correlated n our context. In other words, condtoned on θ, each WPP regards φ θ θ ) as the dstrbuton of θ, whereas the aggregator only knows the pror jont dstrbuton φ ). We consder a two-settlement system of the wholesale electrcty market. In the day-ahead market, a market partcpant offers a power contract.e., a constant amount of power) at a prce p > 0; n the real-tme market, when the random generaton s realzed, the market partcpant s charged at a prce q > p for any shortfall.e., negatve devaton from the contract). For smplcty, we assume that any surplus.e., postve devaton from the contract) s splled at no cost. Note that the man analyss can be easly generalzed when p and q are random varables. A. Rsk Poolng We frst llustrate the advantage of aggregaton under uncertanty, whch s termed rsk poolng n fnance. Defne x = max{x, 0}. When WPP partcpates n the market ndvdually, the expected proft s the optmal value of the followng optmzaton problem: V θ ) = max x px E W [qx W ) θ ), 3) where x s the power contract. It s easy to derve the optmal power contract. Proposton : The optmal power contract when WPP partcpates n the market ndvdually, as the soluton to 3), s gven by ) p x θ ) = F W θ, 4) q where F X ) the cumulatve dstrbuton functon of a random varable X. Proof: The result follows from the frst-order condton: d px E W [qx W ) θ ) dx = p q d x ) x w )f W θ dx w )dw = p q x 0 0 = p qf W θ x ) f W θ w )dw = 0, 5) where f X ) the probablty densty functon of a random varable X. The dea of rsk poolng s to aggregate the ndvdual WPPs to form a pool, so that the total varablty, or rsk, s mnmzed. When the N WPPs partcpate n the market as an aggregate, the expected proft s the optmal value of the followng optmzaton problem: [ V θ) = max px E W q x ) ) W x θ. 6) Smlarly as before, the optmal power contract under aggregaton, as the soluton to 6), s gven by ) x θ) = F p W θ. 7) q The advantage of aggregaton s formally proved n the followng result. Proposton 2: The expected proft under aggregaton s greater than or equal to the sum of the expected profts under ndvdual partcpaton: V θ) V θ ) 0. 8) [ V θ) = max px E W q x ) ) W x θ p x θ ) E W [q x θ ) ) W p [ x θ ) E W q x θ ) W ) θ = = px θ ) E W [qx θ ) W ) θ ) θ V θ ), 9) where the second nequalty follows from the fact that x y) x y for any x and y. When all the WPPs except WPP partcpate n the market as an aggregate, the expected proft s the optmal value of the followng optmzaton problem: V θ ) = max px E W q x ) x j θ. 0) Smlarly as before, the optmal power contract under aggregaton except WPP, as the the soluton to 0), s gven by ) x θ ) = F p. ) j W θ q The next result shows that the ndvdual contrbuton to the aggregate s at least as much as the ndvdual value. Proposton 3: The margnal contrbuton of WPP to the expected proft under aggregaton s greater than or equal to the expected proft under ndvdual partcpaton: V θ ) V θ ) V θ) V θ ) V θ ). 2)

5 TANG et al.: AGGREGATING CORRELATED WIND POWER WITH FULL SURPLUS EXTRACTION 5 = px θ ) E W [qx θ ) W ) θ px θ ) E W q x θ ) ) j θ px θ ) x θ )) E W [q x θ ) x θ ) ) W θ px θ) E W [q x θ) ) W θ = V θ). 3) Moreover, the total nvdual contrbuton to the aggregate s at least as much as the aggregate value. Proposton 4: The sum of margnal contrbutons s greater than or equal to the expected proft under aggregaton: V θ) V θ )) V θ). 4) V θ ) = px θ ) E W q x θ ) ) j px θ ) E W q x θ ) ) j θ θ = p x θ ) E W [q x θ ) N ) ) W θ = N )p x θ ) ) N N )E W [q x θ ) ) W N θ [ N ) px θ) E W q x θ) ) ) W θ = N )V θ). 5) B. Reservaton Utlty and Mechansm Desgn By aggregatng the WPPs, the aggregator can partcpate n the market and make an expected proft V θ). On the other hand, to provde ncentves for the WPPs to contract wth the aggregator, the aggregator needs to make a mnmum payment to each WPP. Such mnmum payment s called the reservaton utlty, denoted by R θ ) for WPP, whch depends on ts type. To motvate the problem formulaton, we frst consder R θ ) = V θ ) for all, where the reservaton utlty reflects the opportunty cost, whle the margnal cost s neglgble. If the aggregator knows θ, t can make a payment R θ ) to each WPP, and makes a non-negatve expected proft as mpled by 8): V θ) R θ ) 0. 6) Also, by 2), we have V θ) V θ ) R θ ). 7) However, the type θ of each WPP s not known by the aggregator. Our goal s to desgn mechansms whch elct the true types of the WPPs and make a maxmum proft, on behalf of the aggregator. Now we formalze the problem n the mechansm desgn framework. In a mechansm, each WPP reports a type ˆθ, whch may not be the true type θ. Smlarly as before, let ˆθ = ˆθ,..., ˆθ N ) and ˆθ = ˆθ,..., ˆθ, ˆθ,..., ˆθ N ). The key component s the payment t ˆθ, W ) made by the aggregator to each WPP, whch depends on the reported types of all the WPPs and the realzed generaton of WPP. Defnton Domnant Strategy Incentve Compatblty): A mechansm s domnant strategy ncentve compatble f reportng truthfully s a domnant strategy for each WPP : E W [t θ, ˆθ, W ) θ, ˆθ E W [t ˆθ, ˆθ, W ) θ, ˆθ, θ, ˆθ, ˆθ. 8) Defnton 2 Interm Indvdual Ratonalty): A mechansm s nterm ndvdual ratonal f each WPP has no ncentve to wthdraw from the mechansm after t learns ts own type but before the others types are revealed: E θ,w [t θ, θ, W ) θ R θ ), θ. 9) Defnton 3 Optmal Mechansm): A mechansm s optmal f t maxmzes the expected proft of the aggregator, or equvalently, mnmzes the expected total payment to the WPPs, subject to the constrants of domnant strategy ncentve compatblty and nterm ndvdual ratonalty: mnmze E θ,w t θ, W ) t ), subject to 8) and 9). C. Stochastc VCG Mechansm 20) The stochastc VCG mechansm [4 characterzes a feasble soluton to 20). In that mechansm, the payment functon s gven by t ˆθ, W ) = px ˆθ) V ˆθ ) E W q x ˆθ) W ) j ˆθ. 2)

6 6 IEEE TRANSACTIONS ON SMART GRID, VOL. 0, NO. 0, JANUARY 0000 Proposton 5: The stochastc VCG mechansm s domnant strategy ncentve compatble. E W [ t ˆθ, ˆθ, W ) θ, ˆθ = px ˆθ, ˆθ ) V ˆθ ) E W q x ˆθ, ˆθ ) W ) j px θ, ˆθ ) V ˆθ ) E q x θ, ˆθ ) W ) j θ, ˆθ θ, ˆθ = E W [ t θ, ˆθ, W ) θ, ˆθ. 22) Such domnant strategy mplementaton s very robust, n the sense that each WPP s best response only depends on ts own type, regardless of the others true or reported types. In partcular, the best response does not depend on φ ), the pror dstrbuton of the true types. Gven the constrant of domnant strategy ncentve compatblty, we can wrte t ˆθ, W ) = t θ, W ). Proposton 6: The stochastc VCG mechansm s nterm ndvdual ratonal. E W [ t θ, W ) θ = px θ) V θ ) E W [q x θ) = V θ) V θ ) W ) θ R θ ), 23) and therefore E θ,w [ t θ, θ, W ) θ = E θ [E W [ t θ, θ, W ) θ, θ θ E θ [R θ ) θ = R θ ). 24) Whle the VCG mechansm s a feasble soluton to 20), t s not optmal. In fact, the total payment made to the WPPs s too hgh to be recovered by the proft under aggregaton: E W [ t θ, W ) θ = V θ) V θ )) V θ), 25) followng from 4). Such budget defct s undesrable for the proft-maxmzng aggregator. Thus, the aggregator needs to seek for alternatve mechansms whch make smaller payments to the WPPs, yet stll elct truthful nformaton. Ths can be acheved by explotng the correlaton structure of the types, whch s not used n the stochastc VCG mechansm. IV. OPTIMAL MECHANISM DESIGN In the prevous secton, we consdered a partcular reservaton utlty, R θ ) = V θ ) for all. From now on, we consder general reservaton utltes, and take 6) and 7) as practcal constrants. Assumpton : The reservaton utlty R θ ) for each satsfes 6) and 7). A. Full Surplus Extracton As shown n 6), the maxmum expected proft of the aggregator s acheved when the aggregator has full nformaton about θ, and makes a payment equal to the reservaton utlty to each WPP. In that case, we have t θ, W ) = R θ ). 26) In general, 26) s too strong for a practcal mechansm to hold, snce the aggregator does not have full nformaton. Instead, we consder a relaxed property whch states that 26) holds n expectaton, wth respect to both W and θ. Defnton 4 Full Surplus Extracton): A mechansm s sad to extract the full surplus, f E θ,w t θ, W ) = E θ R θ ). 27) By Defnton 3, f a mechansm satsfyng 8) and 9) also satsfes 27), then t s an optmal mechansm. B. Full Correlaton Snce the WPPs are geographcally close together, ther types can be hghly correlated. We ntroduce a defnton that characterzes the correlaton structure of the types. Recall that for each, we have Θ = {,..., K}, Θ = K, Θ = {,..., K} N, and Θ = K N. Let θ,..., θk ) be a permutaton of Θ, and θ,..., θl ) be a permutaton of Θ, where we defne L = K N. Defnton 5 Condtonal Dstrbuton Matrx): The condtonal dstrbuton matrx A for each s defned as a K by L matrx: A = φ θ θ ) φ θ L θ )..... φ θ θk ) φ θ L θk ), 28) where the rows are ndexed by the elements n Θ, the columns are ndexed by the elements n Θ, and φ θ l θk ) s the probablty of θ = θ l condtoned on θ = θ k. The concept of full correlaton s based on the condtonal dstrbuton matrx. Defnton 6 Full Correlaton): When A has rank K for all, we say that the types are fully correlated, or that the jont dstrbuton φ ) satsfes the full correlaton condton. One specal case of full correlaton s perfect correlaton,.e., for any, the value of θ corresponds to a unque value of θ. In that case, each A has exactly one entry n each row and 0s elsewhere. Another specal case s establshed n the followng.

7 TANG et al.: AGGREGATING CORRELATED WIND POWER WITH FULL SURPLUS EXTRACTION 7 Proposton 7: When N = 2, defne the jont dstrbuton matrx A as φθ, θ2) φθ, θ2 K ) A = ) φθ K, θ2) φθ K, θ2 K ) If A s nvertble, then φ ) satsfes the full correlaton condton. Proof: Snce A s nvertble, det A 0. Note that A s obtaned by dvdng each row k of A by the margnal dstrbuton φθ k, ). Thus, we have det A = det A K 0. 30) k= φθk, ) So A s nvertble. Smlarly, A 2 s also nvertble. The result follows mmedately. We note that the full correlaton condton s farly mld, whch only requres that knowng the type of one WPP gves more refned nformaton about the types of other WPPs. C. Optmal Mechansm under Full Correlaton Under the full correlaton condton, we can show that the optmal mechansm extracts the full surplus. Ths s proved by constructon based on the stochastc VCG mechansm. Defne h θ ) = θ φ θ θ )V θ) V θ )) R θ ). 3) For each, snce A has rank K, there exsts a t = t θ ),..., t θl ))T such that t θ ) h θ ) A. =.. 32) t θl ) h θ K) Subtractng t ˆθ ) from the payment functon of the stochastc VCG mechansm, we obtan the payment functon of the proposed mechansm: t ˆθ, W ) = t ˆθ, W ) t ˆθ ). 33) Proposton 8: When the jont dstrbuton φ ) satsfes the full correlaton condton, the proposed mechansm specfed by 33) s an optmal mechansm. Proof: Frst, the mechansm s domnant strategy ncentve compatble, snce t ˆθ ) does not depend on ˆθ : E W [t ˆθ, ˆθ, W ) θ, ˆθ = E W [ t ˆθ, ˆθ, W ) θ, ˆθ t ˆθ ) E W [ t θ, ˆθ, W ) θ, ˆθ t ˆθ ) = E W [t θ, ˆθ, W ) θ, ˆθ. 34) Second, the mechansm s nterm ndvdual ratonal: E θ,w [t θ, θ, W ) θ = E θ [E W [t θ, θ, W ) θ, θ θ = E θ [E W [ t θ, θ, W ) θ, θ t θ ) θ = E θ [V θ) V θ ) t θ ) θ = θ TABLE I PARTITION OF WPP NOMINAL CAPACITY WPP µ µ 2 µ 3 µ 4 µ φ θ θ )V θ) V θ ) t θ )) = φ θ θ )V θ) V θ )) h θ ) θ = R θ ). 35) It follows that the mechansm extracts the full surplus: E θ,w t θ, W ) = = = = E θ E θ,w [t θ, W ) E θ [E θ,w [t θ, W ) θ E θ [R θ ) V. CASE STUDY R θ ). 36) In ths secton, we present a case study based on the GEFCom202 data to valdate the man results. Recall that the seven WPPs n the data are n close proxmty to each other, and the wnd power generaton has been normalzed. For smplcty, we focus on the data for hour. As ntroduced n Secton II, for each WPP, we use K-means clusterng to partton the nomnal capacty nto K clusters. We choose K = 5 n the study. The centers of the clusters are lsted n Table I. As an approxmaton, we assume that each realzaton s drawn from the nearest component n the mxture model. For example, when the realzed generaton for WPP s 0.5, t s consdered as drawn from the component centerng at 0.6, and therefore θ = 2, whch s the prvate nformaton of WPP at the day-ahead stage. Let p = $50/MWh and q = 0/MWh n the study. We calculate the proft under ndvdual partcpaton when varous nformaton s avalable: ) ex ante) only the mxture model s known; ) type) the component ndex based on the realzaton s known; and ) ex post) the realzaton s known, whch s clarvoyant. The results are shown n Fg. 4. The proft under ex post nformaton s the hghest, and that

8 8 IEEE TRANSACTIONS ON SMART GRID, VOL. 0, NO. 0, JANUARY 0000 $6,000 Ex Ante $8,000 Ex Ante Cumulatve Proft $4,000 $2,000 Type Ex Post Cumulatve Surplus $6,000 $4,000 $2,000 Type Ex Post Jul 2009 Jan 200 Jul 200 Jan 20 Jul 2009 Jan 200 Jul 200 Jan 20 Fg. 4. The cumulatve proft for WPP when varous nformaton s avalable. Fg. 6. The cumulatve surplus when varous nformaton s avalable. $50,000 Ex Ante $50,000 Proft Cumulatve Proft $40,000 $30,000 $20,000 Type Ex Post Cumulatve Sum $40,000 $30,000 $20,000 Payment,000,000 Jul 2009 Jan 200 Jul 200 Jan 20 Jul 2009 Jan 200 Jul 200 Jan 20 Fg. 5. The cumulatve proft under aggregaton when varous nformaton s avalable. Fg. 7. The cumulatve proft under aggregaton and the total payment n the stochastc VCG mechansm when varous nformaton s avalable. under ex ante nformaton s the lowest. When the type s known, the proft s close to the ex post scenaro. Smlarly, the proft under the aggregaton of the seven WPPs s shown n Fg. 5. It exhbts a smlar pattern as the proft under ndvdual partcpaton. Consder the surplus as defned n 8), whch s shown n Fg. 6. There s no surplus under ex post nformaton, snce there s no uncertanty. Dfferent from the prevous plots, the surplus based on ex ante nformaton s hgher than that based on types. That s, aggregaton adds more value when less nformaton s avalable. Next, we examne the emprcal condtonal dstrbuton matrces. The matrx A for each s a 5 by 5 6 matrx. It can be verfed that all the A s have rank 5,.e., full rank. In fact, the full correlaton condton s farly mld and holds n most crcumstances. To demonstrate the ssue of budget defct for the stochastc VCG mechansm, we plot the proft under aggregaton and the total payment made to the WPPs n Fg. 7. It can easly seen that the total payment exceeds the proft under aggregaton, whch s undesrable for the aggregator. Now we study the optmal mechansm. We assume that the reservaton utlty of each WPP s the proft under ndvdual partcpaton. Snce A s not square, we consder the Moore- Penrose pseudonverse of A when computng t. The results are presented n Fg. 8. Whle the payment n the stochastc Cumulatve Sum $6,000 $4,000 $2,000 Reservaton Utlty VCG Payment Optmal Payment Jul 2009 Jan 200 Jul 200 Jan 20 Fg. 8. The cumulatve reservaton utlty for WPP, and the payments made to WPP n the stochastc VCG mechansm and n the optmal mechansm. VCG mechansm exceeds the reservaton utlty, that n the optmal mechansm s approxmately equal to the reservaton utlty. VI. CONCLUSION Aggregatng dverse wnd power sources can reduce the overall varablty. The surplus provdes an ncentve for a thrd-party aggregator to enter the market, by assumng the rsk due to the uncertanty of wnd power generaton. To

9 TANG et al.: AGGREGATING CORRELATED WIND POWER WITH FULL SURPLUS EXTRACTION 9 extract the surplus, the aggregator needs to elct the prvate nformaton of the WPPs. Whle the objectve of truthful elctaton can be acheved by the stochastc VCG mechansm, the total payment made to the WPPs cannot be recovered by the proft under aggregaton. On the other hand, the aggregator can explot the correlaton structure of the types. Under the full correlaton condton, the proposed mechansm extracts the full surplus, subject to the constrants of domnant strategy ncentve compatblty and nterm ndvdual ratonalty. It can be shown that there does not exst a mechansm wth stronger propertes. The results reveal an nterestng trade-off n wnd power aggregaton: aggregaton acheves the maxmum proft when the types are ndependent, whle the surplus can be fully extracted when the types are fully correlated. [20 T. Hong, P. Pnson, and S. Fan, Global energy forecastng competton 202, Internatonal Journal of Forecastng, vol. 30, no. 2, pp , 204. REFERENCES [ W. Tang and R. Jan, Buyng random yet correlated wnd power, n Smart Grd Communcatons SmartGrdComm), 204 IEEE Internatonal Conference on, 204, pp [2 J. M. Morales, A. J. Conejo, and J. Pérez-Ruz, Short-term tradng for a wnd power producer, IEEE Transactons on Power Systems, vol. 25, no., pp , 200. [3 E. Y. Btar, R. Rajagopal, P. P. Khargonekar, K. Poolla, and P. Varaya, Brngng wnd energy to market, Power Systems, IEEE Transactons on, vol. 27, no. 3, pp , 202. [4 Q. Wang, Y. Guan, and J. Wang, A chance-constraned two-stage stochastc program for unt commtment wth uncertan wnd power output, IEEE Transactons on Power Systems, vol. 27, no., pp , 202. [5 C. Zhao, J. Wang, J.-P. Watson, and Y. Guan, Mult-stage robust unt commtment consderng wnd and demand response uncertantes, IEEE Transactons on Power Systems, vol. 28, no. 3, pp , 203. [6 S. Tewar, C. J. Geyer, and N. Mohan, A statstcal model for wnd power forecast error and ts applcaton to the estmaton of penaltes n lberalzed markets, IEEE Transactons on Power Systems, vol. 26, no. 4, pp , 20. [7 C. Wan, Z. Xu, and P. Pnson, Drect nterval forecastng of wnd power, IEEE Transactons on Power Systems, vol. 28, no. 4, pp , 203. [8 C. Wan, Z. Xu, P. Pnson, Z. Y. Dong, and K. P. Wong, Probablstc forecastng of wnd power generaton usng extreme learnng machne, IEEE Transactons on Power Systems, vol. 29, no. 3, pp , 204. [9, Optmal predcton ntervals of wnd power generaton, IEEE Transactons on Power Systems, vol. 29, no. 3, pp , 204. [0 C. Wan, J. Ln, J. Wang, Y. Song, and Z. Y. Dong, Drect quantle regresson for nonparametrc probablstc forecastng of wnd power generaton, IEEE Transactons on Power Systems, 206. [ E. Baeyens, E. Y. Btar, P. P. Khargonekar, and K. Poolla, Coaltonal aggregaton of wnd power, IEEE Transactons on Power Systems, vol. 28, no. 4, pp , 203. [2 Y. Zhao, J. Qn, R. Rajagopal, A. Goldsmth, and H. V. Poor, Wnd aggregaton va rsky power markets, Power Systems, IEEE Transactons on, vol. 30, no. 3, pp , 205. [3 A. Mas-Colell, M. Whnston, and J. Green, Mcroeconomc theory. Oxford, 995. [4 W. Tang and R. Jan, Market mechansms for buyng random wnd, Sustanable Energy, IEEE Transactons on, vol. 6, no. 4, pp , 205. [5 J. Crémer and R. P. McLean, Optmal sellng strateges under uncertanty for a dscrmnatng monopolst when demands are nterdependent, Econometrca, vol. 53, no. 2, pp , 985. [6, Full extracton of the surplus n bayesan and domnant strategy auctons, Econometrca, vol. 56, no. 6, pp , 988. [7 M. H. Rordan and D. E. Sappngton, Optmal contracts wth publc ex post nformaton, Journal of Economc Theory, vol. 45, no., pp , 988. [8 R. P. McAfee and P. J. Reny, Correlated nformaton and mechansm desgn, Econometrca, vol. 60, no. 2, pp , 992. [9 D. Fudenberg and J. Trole, Game theory. MIT Press, 99.