Double-Sided Energy Auction: Equilibrium Under Price Anticipation

Transcription

1 IEEE TRANSACTIONS ON??? 1 Double-Sded Energy Aucton: Equlbrum Under Prce Antcpaton M.Nazf Faqry and Sanoy Das Electrcal & Computer Engneerng Department Kansas State Unversty Abstract Ths paper nvestgates the problem of proportonally far double-sded energy aucton nvolvng buyng and sellng agents. The grd s assumed to be operatng under slanded mode. A dstrbuted aucton algorthm that can be mplemented by an aggregator, as well as a possble approach by whch the agents may approxmate prce antcpaton s consdered. Equlbrum condtons arsng due to prce antcpaton s analyzed. A modfed aucton to mtgate the resultng loss n effcency due to such behavor s suggested. Ths modfed aucton allows the aggregate socal welfare of the agents to be arbtrarly close to that attanable wth prce takng agents. Next, equlbrum condtons when the aggregator collects a surcharge prce per unt of energy traded s examned. A bobectve optmzaton problem s dentfed that takes nto account both the agents socal welfare as well as the aggregator s revenue from the surcharge. Results of extensve smulatons, whch corroborate the theoretcal analyss, are reported. Index Terms Energy grd; aggregator; agents; tradng; aucton; bd; socal welfare. I. INTRODUCTION ETWORK economcs has made automated tradng feasble, Nwhere resources are exchanged for money through transactons that are done entrely through software agents and wthout the need for any human nterventon. It s often the case that the agents partcpatng n the trade are nvolved n drect competton wth one another, where the obectve of each agent s to maxmze ts own payoff. In these stuatons the trade proceeds through an aucton mechansm. In snglesded auctons all agents are ether buyers that compete to acqure a fnte resource, or sellers that compete to sell ther goods. Double auctons are mechansms nvolvng both buyers and sellers, whch smultaneously partcpate n the bddng process and are allocated ndvdual shares of the resource. Recent technologcal advancements n communcatons and renewable energy generaton have created much research nterest n energy aucton algorthms [SA+15, PD+15]. In these mechansms, the agents may represent ndvdual domestc unts wthn a mcrogrd, wth energy representng the resource, and wth PV-equpped homes experencng a surplus of energy actng as sellers, and the remanng domestc unts as buyers [MF+14]. An agent may also represent an ndvdual mcrogrd nvolvng a communty of homes that Ths proect was funded by NSF-CPS Award No. CNS to Pahwa (PI), Das, Deloach, Nataraan, Ou. All correspondence should be addressed to sdas@ksu.edu. collectvely behave as a sngle unt n the ensung aucton [WH15]. A buyer agent s payoff s typcally the dfference between ts utlty ganed from consumng a certan amount of energy and the prce that t has to pay n order to procure that energy. Lkewse a seller agent s payoff may be formulated as the sum of the monetary gan from supplyng an amount of energy and utlty t gans from retanng any surplus energy that s not traded. The Kelly mechansm refers to a class of aucton algorthms where agents are allowed to place ndvdual bds on the resource, whle a separate auctoneer that receves these bds, allocates the resource share of each bddng agent n proporton to the bd values [KMT98]. Wth a large number of agents, such a proportonal allocaton mechansm has been shown to maxmze the aggregate utltes of all agents, the latter commonly referred to as the socal welfare [MB03, JT04, CST15]. Such socal welfare maxmzng mechansms are effcent auctons. Unfortunately, the underlyng assumpton for proportonal allocaton to be effcent s that the agents be prce takers,.e. ones that assume that the bds that they place do not nfluence the market prce of the resource. Whle ths s approxmately true for auctons nvolvng a large number of agents, n smaller auctons, agents are aware of ther own market power, and accordngly, place strategc bds on the resource. Such a prce antcpatory bddng results n a loss n effcency where the resource allocaton of the aucton no longer maxmzes the socal welfare. It must be noted that there are other effcent auctons that explctly focus on elctng truthful bddng from the bdders; the most sgnfcant ones beng based on the well-known Vckrey-Clarke-Groves mechansm [NR07]. Furthermore, auctons desgned to maxmze the auctoneer s own earned revenue have been proposed n the vast body of lterature on ths subect. However, these mechansms are not of drect relevance n ths research, whch focuses on effcent double auctons along wth proportonal allocaton of electrcty among ts buyers. A. Current Research Proportonal allocaton n sngle-sded buyers auctons has been rgorously analyzed [JT04, CSS13]. It s shown that the aucton s effcent under the assumpton of prce takng buyers. Furthermore, when the agents utltes are strctly concave functons, the semnal study n [JT04] establshes a strong theoretcal upper lmt on the aucton s loss of effcency at ¼ of the maxmum attanable socal welfare.

2 IEEE TRANSACTIONS ON??? 2 More recently, t has been shown that even when buyers are prce antcpatng, proportonal allocaton allows the mechansm to attan the best possble outcome [JT09]. Smlar theoretcal lmts have been nvestgated for a more general class of auctons called smooth auctons wth proportonal allocaton [ST13]. Theoretcal propertes of a stuaton wth multple sellers partcpatng n a proportonal allocatng aucton, and wth nelastc (fxed) demand for the resource has been examned [JT11]. When the proportonal allocaton aucton takes nto account the costs of the network s lnks through whch the resource (data) flows, the effcency s shown to be at least of the socal welfare maxmum wth prce antcpatng buyers and wth convex costs [JMT05]. In a separate study t has been shown that when the costs are lnear, the mechansm s effcency loss s lower bounded at ⅓ of the maxmum value [JT06]. Unfortunately, the prevous studes are based on the assumpton that the utlty functons are convex. Under a more general settng where ths convexty assumpton s not true, the mechansm s effcency loss no longer enoys a theoretcal lmt, and could n fact be arbtrarly large [CST15]. A few studes have proposed schemes to address an aucton s effcency loss arsng from prce antcpaton. For nstance an aucton mechansm wth prce dfferentaton where each buyer has a dfferent prce, has been proposed [YML13]. Ths study also suggests a feedback control mechansm on the prce vector that drves the aucton to converge to the globally optmum socal welfare. Double aucton mechansms wth proportonal allocaton have been studed [KV10]. However, ths study consders only the case of prce takers. The research reported n ths paper ncludes a theoretcal study on both prce antcpatng buyers and prce antcpatng sellers partcpatng smultaneously n a double aucton. Research on energy auctons has closely followed the theoretcal advancement n mechansm desgn. A large body of recent lterature on energy aucton algorthms model large utlty company as the sellers [FAA14, JK15, Moh15, WLM15, ZX+15]. Many of these studes consder obectves and/or constrants that are applcable only to energy trade, such as generaton schedulng [FAA14, Moh15], economc dspatch [ZX+15], and transmsson losses [GM15]. Some energy aucton studes are desgned for revenue maxmzaton and are not socal welfare maxmzng (effcent) mechansms [JK15, ZX+15]. Effcent auctons have begun to gan research attenton n the present context energy trade. Several such nvestgatons do not consder proportonally far allocaton of energy [TS+12, TZ+14, WS+14]. One recent study proposes a VCGstyle aucton wth multple sellers and a sngle demand response aggregator as the buyer [NAC15]. Another study that uses the VCG mechansm reports a double aucton [FK+14]. The cake-cuttng algorthm has been appled to procure energy from multple sellers and for a communty of consumers actng n tandem as a sngle buyer [TY+15]. A truthful buyers aucton that makes use of the Arrow-d'Aspremont-Gerard- Varet mechansm has been suggested [MD+14]. Whle some exstng approaches as well as ths research explore Nash equlbrum, where all agents are assumed to act smultaneously [LCD15, NAC15, WH15], others use leaderfollower games under Stackelberg equlbrum [TS+12, TZ+14, LG+15, WLM15]. Ths equlbrum concept s applcable when the energy market s modeled as an olgopoly wth only a lmted number of supplers modeled as leader, or wth the ncluson of an upper level agent such as an aggregator [TS+12, TZ+14, LG+15, NAC15]. Ths s n contrast to the present work that treats both buyers and sellers wth equvalent market party. Many research papers n the exstng research on energy auctons report only sngle-sded ones [MD+14, LCD15, Moh15, NAC15, TY+15, WH15, WLM15]. However, there are some papers that do address some form of double aucton [DF+14, FK+14, GM15, JK15, LG+15, ZX+15]. One of them analyzes the loss of effcency arsng from Stackelberg equlbrum where the leadng agent enoys a frst-mover s advantage n the underlyng game [LG+15]. Another paper on double energy auctons, that proposes a prmal-dual algorthm, s based on the unrealstc assumpton that all agents buyers and sellers, act n concert to lower the overall grd cost [15]. Yet another such study consders a double aucton where the energy sellers are utlty companes whose goal s revenue maxmzaton [GM15]. A prelmnary study on double energy aucton based on the Kelly mechansm has been proposed [MF+14]. Another double aucton study that does not use proportonal allocaton addresses software ssues rather than the aucton [DF+14]. Moreover, none of these sngle or double energy auctons address the ssue of prce antcpatory behavor of the partcpatng agents. To the best of the authors knowledge, only one research n the lterature on effcent energy auctons does examne the adverse effect of prce antcpaton [LCD15]. Ths study s a sngle-sded aucton wth consumers of electrcty actng as prce antcpatng buyers. The theoretcal dervaton of the aucton s loss of effcency closely follows that n earler theoretcal studes on snglesded auctons [JT04, JT11, CSS13]. A few of the extant research on energy auctons apples some form of prce dfferentaton [FK+14, MF+14, TZ+14, GM15, TY+15]. Although ths paper does not explctly do so, t s worthwhle to menton that the research reported here can be readly extended to address ths scenaro as theoretcal research on such mechansms has been publshed [YML13]. B. Contrbuton Ths research proposes a double aucton mechansm that ncludes one set of agents as buyers, and another set as sellers. It also assumes the presence of a separate medatng agent called the aggregator whose role, unless ndcated otherwse, s to () receve monetary bds from the buyers and avalable energy for trade from the sellers; () proportonally allocate energy to the buyers; () teratvely converge to the market clearng prce. The man contrbutons of ths research are as follows. () It performs a theoretcal analyss of the equlbrum condtons arsng from prce antcpatng buyers and sellers, thereby extendng prevous studes on sngle-sded auctons to double auctons. It shows the exstence of a unque equlbrum for double auctons under prce antcpaton. () It proposes a dstrbuted teratve aucton algorthm, and suggests a possble realstc scheme through whch the selfsh

3 IEEE TRANSACTIONS ON??? 3 agents may use nformaton from pror teratons to approxmate prce antcpatng behavor wthout any knowledge of the other agents. () It shows that, unlke n sngle sded auctons, double auctons can readly mnmze the loss of effcency arsng from prce antcpaton, suggestng a smple extenson of proportonal allocaton n order to approach the mnmum. (v) It shows how the aggregator s own revenue can be ncorporated wthn the aucton framework, and proposes how ths can lead to a b-obectve optmzaton problem where the aggregator s no longer strctly selfless by establshng the presence of a Pareto front wthn specfc bounds. It must be emphaszed that although ths research consders energy as the traded resource, the underlyng theoretcal analyss s drectly applcable to other dvsble resource auctons. The rest of ths paper s organzed n the followng manner. Secton II presents the framework of the aucton. The double aucton under condtons of prce antcpaton and prce takng are outlned n sectons III and IV. The proofs of the propostons made n both sectons have been postponed untl the appendx. The results obtaned from smulatons are dscussed n secton V. Fnally, the concluson of ths research s derved n secton VI. II. FRAMEWORK A. Network Model Wth energy as the resource nvolved n the trade, the network of agents n our model conssts of a set D of buyers and another set S of sellers. Although grd energy auctons typcally nvolve the presence of prosumers that buy and sell energy, we assume for smplcty that D and S are dsont. The model also ncludes a separate entty, A, the aggregator (or auctoneer) that s responsble for communcatng wth the other agents and mplementng the aucton. Unless otherwse ndcated the aggregator acts as a selfless agent, requrng no separate parametrzaton of ts own, n whch case A =. Each agent, whether a buyer or a seller, has ts own utlty representng the gan (n monetary unts) t derves from consumng an amount of energy. The utlty of a buyer D s denoted as u, and that of a seller S, as v. As the sellers are capable of supplyng energy, the model assumes that each has a fxed amount of energy g, called ts generaton that s avalable both for ts own use and to sell. The underlyng physcal network that mplements the aucton mechansm can be completely defned as the followng 6-tuple Θ, Θ D, S, g, u, v, A. (1) The mathematcal treatment made throughout the rest of ths paper s based on the followng underlyng assumptons. () The utltes u and v are contnuous, dfferentable, monotoncally ncreasng and strctly concave functons wth non-negatve arguments. In other words, u, v > 0 and u, v < 0 when the argument les wthn the nterval (0, ). () There s at least one buyer and one seller,.e. D, S, and furthermore that at least one buyer D that can obtan energy from some seller S so that there some trade takes place. Ths assumpton can be summarzed as follows. D, S, u (0) > v g. (2) B. Aucton Process The buyers and sellers bddng processes are mplemented as separate steps n the aucton. Each buyer receves from the aggregator ts demand d, whch s the amount of energy that t s allocated for use. The buyer responds by communcatng to the aggregator ts bd b, whch s the amount of money that t s wllng to pay for t. Separately, each seller receves a per unt prce p of energy, and communcates back to the aggregator, ts avalablty a that t s wllng to supply. The schematc below (Fg. 1.) shows the layout of the entre aucton process. The aucton proceeds teratvely untl termnaton when p has converged to the market clearng prce. vrtual agent aggregator Fg.1: Schematc of the network model showng flow of nformaton durng aucton. III. DOUBLE AUCTION UNDER PRICE ANTICIPATION A. Aggregator It s assumed that there s no energy loss takng place durng transmsson. Thus, wth the network operatng under solaton as s also assumed n ths secton, the total amount of energy that s declared avalable by the sellers must be equal to the total amount demanded by the buyers, so that the followng energy balance equaton holds. d = a. (3) In ths secton the aggregator s also assumed to be selfless and plays no addtonal role other than that specfed earler (A = ), so that the money receved as the total buyers bds s exchanged for the total avalable energy sold by the sellers. Under these crcumstances, the per unt prce s gven by, p = a b. (4) As the aucton s based on proportonal allocaton of resources, the energy demand d that each buyer receves from the aggregator must be proportonal to ts bd b so that, d = b, D. (5) p B. Buyer Each buyer ams to maxmze ts payoff from the aucton mechansm. Notng that t has to pay an amount b n order to s

4 IEEE TRANSACTIONS ON??? 4 receve energy d, t places a bd b n accordance wth the followng optmzaton problem. Maxmze w.r.t. b : u (d ) b. (6) Proposton-1: The optmal bddng strategy of a buyer D s, b = d u (d )(1 β ). (7) Here the quantty β s the market power of buyer descrbed later n ths secton. C. Seller Each seller declares ts avalablty a at prce p to the aggregator to attan the maxmum of ts payoff, whch s the sum of the money that t receves from sellng energy as well as ts own utlty from consumng the remanng amount g a of energy. Notng that ts avalablty cannot exceed ts generaton g, ts partcpaton n the aucton s characterzed by means of the followng optmzaton problem. Maxmze w.r.t. a : v g a + pa. (8) Subect to: a g. (9) Proposton-2: The optmal bddng strategy of a seller S s gven by the expresson below. a = mn a, g, (10) where a s the soluton to the equaton, v g a = p 1 α. (11) The seller s market power α s descrbed below. D. Market Power The market power of an agent reflects ts ablty to nfluence the overall outcome of the aucton. When the aucton nvolves a large number of agents, an ndvdual agent s acton cannot exert a great deal of nfluence on the outcome; consequently the agent s market power s low. In the lmtng case when there are an nfnte number of agents, the market power approaches zero. It s ths lmtng case that prce takng condtons serves to approxmate. In the present case, the market power β of every buyer and α that of sellers can be defned through separate expressons, gven below, β = b b, D, (12) α = a a, S. (13) The level of awareness of each buyer or seller about the remanng agents can vary from complete unawareness (prce takng) to full awareness of the others bddng strateges (as requred n Eqns. (12) and (13) above). A realstc scenaro les somewhere n between. In such a case, the teratve aucton would allow the buyer or seller to approxmate ts market power from the nformaton gleaned from prevous teratons. The expressons below, whch are derved from Eqns. (7) (11) can be used as the means by whch each buyer or seller can obtan such estmates. Superscrpts (k 1) and (k) have been ntroduced for clarty to ndcate each teraton k and ts mmedately precedng teraton k 1. b (k) β (k) = 1 d (k) u d (k), (14) α (k) = 1 1 p (k) v g a (k) ρ (k). (15) Note that nether expresson above ncorporates quanttes pertanng to the other agents present n the network. The quantty ρ n Eqn. (15) s a dual varable obtaned from the constraned optmzaton problem n Eqns. (8) and (9). Further detals pertanng to ρ can be found n the appendx. It suffces to menton that ρ = 0 except n the case when the seller declares ts entre generaton as the avalablty,.e. a = g. At the onset of the aucton process (k = 1), when the agents lack pror nformaton, the market powers may be ntalzed to zero so that the agents act as smple prce takers. E. Dstrbuted Double Aucton Algorthm Before the teratve bddng process takes place, there are several ways by whch the aggregator can ntalze the aucton varables p and the d s that t communcates to the sellers and buyers so that they can place ther bds. An effectve way to mnmze the number of steps would be to use stored hstorcal nformaton from prevous rounds. Otherwse the aggregator may use heurstc means to do so. In the most smplstc case, these varables may be assgned entrely randomly. Ths ntalzaton and the subsequent aucton steps are outlned below. Dstrbuted Double Aucton Algorthm Intalze p (0), d (0) D //Buyer D: β 0 //Seller S: α 0 Set k 1 Whle (termnaton crteron = F ) Send p (k) to sellers S //Seller S bd Receve a (k) from sellers S (k) Send d to buyers D //Buyers D bd Receve b (k) from buyers D Increment k k + 1 (k) Obtan d Update p (k) (k) //Buyers estmate β (k) //Sellers estmate α Evaluate termnaton crteron End F. Equlbrum The aucton steps descrbed earler termnates when further updates of nether the prce p nor any of the bds submtted by the agents to the aggregator are changed. Ths s when generalzed Nash equlbrum [FK10] s establshed.

5 IEEE TRANSACTIONS ON??? 5 In order to characterze the equlbrum condtons under prce antcpaton, the functons π ( ) and π ( ) are ntroduced below, π (d ) = 1 a + a d u (d ) π g a = v g a a d u (z)dd, (16) 0 a a a v g z dd. (17) The effect of prce antcpaton can be examned n terms of the followng constraned optmzaton problem. Maxmze w.r.t. d, a : Π d, a Θ = π (d ) 0 + π g a. (18) Subect to constrants n Eqns. (3) and (9) whch are restated below, d = a, a g. Denotng the solutons of the above maxmzaton problem as d and a, the socal welfare s U U d, a Θ. Addtonally, the maxmum socal welfare correspondng to the effcent soluton s denoted as U. Proposton-3: () There exsts a unque equlbrum of the double aucton where the demand d of each buyer D and avalablty a of each seller S s the soluton to the optmzaton problem defned n Eqn. (18) wth Eqns. (3) and (9) as constrants. () The socal welfare attaned under prce antcpaton s no greater than that attanable under prce takng,.e., U U. (19) The above statement mples that there s a loss of effcency due to prce antcpaton. IV. DOUBLE AUCTION UNDER PRICE TAKING A. Effcent Soluton The effcent soluton can be obtaned from the followng constraned optmzaton problem. Maxmze w.r.t. d, a : U d, a Θ = u (d ) + v g a. (20) Subect to constrants n Eqns. (3) and (9) whch are restated below. d = a. a g, S. For the sake of convenence the buyer s bddng strategy, whch s that n Eqn. (7) wth β = 0, s provded below. b = d u (d ). (21) The seller s strategy s determned accordng to Eqns. (10) and (11) where α = 0, and gven below, a = mn a, g, (22) where a s the soluton to the equaton, v g a = p. (23) Proposton-4: Under the assumpton that the buyers and sellers are prce takers, the followng statements are true for the double aucton. () The buyer and seller strateges are defned accordng to Eqns. (21), (22) and (23). () The equlbrum demand d of each buyer and avalablty a of each seller after the termnaton of the aucton are unque solutons of Eqns. (3), (9) and (20). () There s no loss n effcency,.e. U U d, a Θ = max U d, a Θ. (24) d,a Thus the unque equlbrum of the double aucton s also the effcent soluton. The market prce at equlbrum from the aucton s denoted as p. At equlbrum, the dervatve of the utlty functon (also called margnal utlty) of each buyer and that of each seller that s not tradng ts entre generaton g s equal to the market prce; and for traders that trade all of t, more than the prce. Mathematcally, u (d ) = p ; v g a = p when a < g ; (25) v g a > p when a = g ; The equlbrum can be understood readly graphcally as shown n Fg. 2(a). Here, we defne the aggregate demand functon D(p) as the total amount of energy delvered to the buyers as a functon of the market prce p. Lkewse, we defne the avalablty functon A(p) as the total avalablty declared by the supplers as a functon of p. Thus, D(p) = d, (26) A(p) = a. (27) Proposton-5: Under the assumpton of prce takng, the followng statements are true for the double aucton. () The avalablty functon A(p) s zero when p mn v g, monotoncally ncreasng wth prce p n the nterval p mn v g, max v (0) and constant when p max v (0). In other words, A(p) = 0, A(p) mon. nc., A(p) constant, p mn v g ; mn v g < p < max v (0). p max v (0) ; (28) () The demand functon D(p) s monotoncally decreasng wth prce p n the nterval p (0, max u (0)) and zero when p max u (0). D(p) mon. dec., p < max u (0), D(p) = 0, p max u (29) (0). () At the unque equlbrum prce p, A(p ) = D(p ). B. Vrtual Bddng In general, the loss of effcency, when d s the demand of each buyer, and a, the allocaton of each seller, can be expressed as follows.

6 IEEE TRANSACTIONS ON??? 6 L Θ (d, a ) = U d, a Θ U d, a Θ U d, a. (30) Θ The loss that takes place when the agents partcpate n the aucton as prce antcpators s L Θ (d, a ). Ths secton shows how the basc proportonal allocaton double aucton mechansm can be extended to mtgate the loss of effcency. In order to mnmze the loss L Θ, a vrtual agent can be ntroduced to the network defned earler n Eqn. (1). The vrtual agent, whch s ndexed wth the subscrpt 0, partcpates n the aucton smultaneously as a buyer and a seller wth arbtrarly large avalablty a 0. As the vrtual agent s ncorporated wthn the aggregator, we let A = {a 0 }. Snce the vrtual agent does not have ts own generaton, t buys back the amount of energy a 0 declared as ts avalablty at the market prce defned n Eqn. (4), so that, b 0 = pa 0. (31) Proposton-6: As the vrtual agent s avalablty a 0 ncreases, the loss n effcency L Θ from prce antcpaton decreases. In the lmtng case, lm L Θ(d a 0, a ) = 0. (32) Snce the ncluson of vrtual bddng allows the aucton to behave lke a prce takng mechansm, for the remander of ths secton we assume that the buyers and sellers behave as prce takers. C. Surcharge We now consder the stuaton where the aggregator, A, s no longer a strctly selfless enabler n the aucton process but also has ts own ncentve to mplement the mechansm by levyng a surcharge prce p s per unt of energy traded. Thus n the model n Eqn. (1) the aggregator now ncludes the surcharge, whch we ndcate by lettng t be gven by A = {a 0, p s }. The total revenue earned by the aggregator from the aucton wth the ntroducton of surcharge s gven by, R = p s a. (33) The expresson for the prce n Eqn. (4) s modfed to account for the surcharge as follows, b = (p + p s ) a. (34) Wth proportonal allocaton, the demand d that each buyer receves s gven by the followng expresson that replaces the earler Eqn. (5), d = b p s + p. (35) Fg. 2(b) llustrates the effect of the surcharge. Eqn. (35) shows that the buyers purchase energy at an effectve per unt prce of p s + p whch s hgher than p that the sellers receve per unt of energy traded. The volume of energy traded s equal to D(p s + p) = A(p), whch s lower than A(p ) = D(p ). We show that the prce takng aucton s the soluton to the followng constraned optmzaton problem. Maxmze w.r.t. d, a : Ω d, a Θ = u (d ) + v g a p s a. (36) Subect to constrants n Eqns. (3) and (9), d = a, a g. Proposton-7: Under the assumpton of prce takng, the followng statements are true wth surcharge prce p s > 0. () The buyer and seller strateges are defned accordng to Eqns. (21), (22) and (23). () The equlbrum demand d of each buyer and avalablty a of each seller of the aucton are unque solutons of Eqns. (3), (9) and (36). () There exsts a Pareto front where any ncrease n revenue R s assocated wth a smultaneous decrease n the socal welfare n Eqn. (20). (v) There exsts an optmal surcharge p s OPT that maxmzes the aggregator A s revenue R. (a) (b) Fg.2: Plots of the A(p) and D(p) as functons of p. In the x- axs, l = mmm v g, m = mmm v (0), and n = mmm u (0). (a) Equlbrum condtons under prce takng. (b) The addton of surcharge prce wth revenue R beng the area shaded n grey. V. RESULTS A. Setup In order to complment the theoretcal consderatons n the earler sectons, several sets of smulatons were carred out. A total of fve scenaros were consdered, where the number of buyers and sellers were D = 2, S = 3, D = 2, S = 6, D = 2, S = 10, D = 3, S = 2 and D = 4, S = 4. In order to analyze the effect of prce antcpaton, the total number of agents were made relatvely small n comparson to other smulaton studes. Moreover, the frst three scenaros

7 IEEE TRANSACTIONS ON??? 7 contan only 2 sellers. Ths reflects the stuaton s a realstc mcrogrd, where the number of PV-equpped unts s usually lower than the number of those wthout t. The fourth and ffth scenaros were added to explore the performance of the double aucton under other potental stuatons. The utltes of the buyers and sellers assumed to follow logarthmc saturaton curves accordng to, u (d ) = x log(y d + 1), (42) and, v g a = x log(y (g a ) + 1). (43) The quanttes x, y, x and y were dfferent for each agent, and were generated randomly from a unform dstrbuton centered at unty. The generatons, g, for the sellers were also drawn n at random, unformly n the nterval [g mn, g max ]. B. Prce Antcpaton The frst set of smulatons was performed to examne the effect of prce antcpaton of the buyers and sellers upon the double aucton. The results of ths study are shown n Fg. 3. It can be seen that n each case there s a reducton n the socal welfare due to prce antcpaton. More detaled analyss shows that when consdered separately, whle the socal welfare of the buyers reduces due to prce antcpaton, the socal welfare of the sellers s ncreased. Ths s because Propostons 1 and 2 ndcate that prce antcpaton (β, α > 0) causes the values of d and a to be lower than wth prce takng (β, α = 0). Consequently, the volume of energy beng traded s also less so that the surplus amount of energy g a remanng wth each seller s hgher, whch also ncreases ts utlty, v g a. Although, due to prce antcpaton, the socal welfare n a double aucton s lower than ts optmal value, the utltes of the sellers change n the opposte drecton. Thus an observaton made from ths study s that the effect of prce antcpatory agents n double-auctons s less severe n comparson to sngle-sded auctons. C. Vrtual Bddng The effect of vrtual bddng was nvestgated through a second set of smulatons. The results of these smulatons are provded n Fg. 4, separately for each of the fve scenaros. It can be seen that the loss n effcency L Θ approaches zero as a 0 ncreases towards a 0. Ths observaton holds true for each of the fve scenaros that were smulated, and s consstent wth Proposton-6. D. Surcharge In order to examne the role of the surcharge prce p s on the double aucton, a set of smulatons were carred out for each of the fve scenaros descrbed earler. As prce takng condtons are assumed, the agents market powers were always set at β = 0, α = 0, throughout the teratve mechansm. The aucton was smulated untl equlbrum for dfferent values of the surcharge prce p s. Fg. 5 shows the Pareto front dscussed n the clam () n Proposton-7. In each scenaro, the extreme left end ponts of the fronts correspond to p s = 0 so that the socal welfare s maxmum U = U, whle the aggregator s revenue R = 0. socal welfare socal welfare under prce takng and prce antcpaton buyers(bs) welfare sellers(ss) welfare scenaro 1 (2Ss, 3Bs) scenaro 2 (2Ss, 6Bs) scenaro 3 (2Ss,10Bs) scenaro 4 (3Ss, 2Bs) scenaro 5 (4Ss, 4Bs) PT PA PT PA PT PA PT PA PT PA scenaro ndex Fg.3: Socal welfare U under prce takng (PT) and prce antcpaton (PA) for each scenaro. % loss n effcency loss n effcency vs vrtual bddng 2 sellers 3 buyers 2 sellers 6 buyers 2 sellers 10 buyers 3 sellers 2 buyers 4 sellers 4 buyers a 0 (vrtual avalablty bddng) Fg.4: Loss n effcency L Θ as a functon of a 0 for each scenaro. socal welfare Pareto front 2 sellers 3 buyers 2 sellers 6 buyers 2 sellers 10 buyers 3 sellers 2 buyers 4 sellers 4 buyers aggregator revenue Fg.5: The Pareto front of the revenue R and socal welfare U wth varyng surcharge prce p s, for each scenaro.

8 IEEE TRANSACTIONS ON??? 8 As p s progressvely ncreases untl p s OPT, so does R, whle U decreases. The rght ends of the Pareto fronts correspond to p s = p s OPT. When p s exceeds p s OPT, both U and R decrease, whch s not shown n Fg. 5. Fg. 6 shows how the aggregator s revenue R vares wth surcharge p s. In each scenaro, R ncreases wth p s untl t reaches ts maxmum when the surcharge s p s OPT. In all but one scenaro, the revenue R can be seen to decrease beyond ts correspondng maxmum. These results are consstent wth clam (v) of Proposton-7. R (aggregator revenue) aggregator revenue vs surcharge 2 sellers 3 buyers 2 sellers 6 buyers 2 sellers 10 buyers 3 sellers 2 buyers 4 sellers 4 buyers p (aggregator surcharge) s Fg.6: Aggregator s revenue R as a functon of surcharge prce p s, for each scenaro. VI. CONCLUSION The dstrbuted double aucton algorthm n Secton-III can be mplemented readly by the aggregator, even n the presence of a vrtual agent or wth surcharge prcng. The algorthm can optonally consder prce-antcpatory agents. A possble method by whch real world agents may use nformaton gathered from earler teraton to mtate prce antcpaton has been suggested. It s shown that wth prce antcpatng agents, the double aucton s equlbrum concdes wth that of a constraned optmzaton problem whose obectve functon Π s dfferent from the socal welfare functon U, resultng n a loss of effcency. It s shown that when the aggregator ncorporates a vrtual agent that s smultaneously both a buyer and a seller, can mnmze the loss of effcency, so that the double aucton can reach the effcent equlbrum. A generalzed aucton scenaro where the aggregator receves a surcharge prce s nvestgated where n the lmtng case, the aggregator may act as a selfsh agent tryng to maxmze ts revenue R from the aucton. Wth the socal welfare U and revenue R as ndependent obectves, a bobectve framework for the double aucton mechansm s suggested. The theoretcal analyss has been supplemented by several smulatons. The results of the smulatons are n complete agreement wth the theory. Let, A = a. N B B = b. =1 VII. APPENDIX Proof of Proposton-1: The statonary condton of Eqn. (6) s obtaned by dfferentaton wth respect to the bd b as shown below. u (d ) d = 1. b As buyer s prce antcpatng, d s dependent on b through the prce p. Hence replacng d approprately usng Eqn. (5) b and applyng the chan rule we get, u (d ) 1 p 1 b p = 1. (A1) p b Usng Eqn. (4) the above equalty yelds, u (d ) 1 b = p. B (A2) Whence from Eqn. (12), u (d )(1 β ) = p. (A3) Proposton-1 follows drectly from the above and Eqn. (5). Proof of Proposton-2: Introducng the dual varable ρ, the Lagrangan of the problem defned n Eqns. (8) and (9) s, L S a, ρ = v g a + pa + ρ a g. Ths yelds the followng KKT condtons. ρ a g = 0, (A4) v p g a = p + a + ρ a. (A5) Replacng p above approprately usng Eqn. (4), a v g a = p 1 α + ρ. (A6) The quantty α s defned n Eqn. (13). When a < g, Eqn. (A4) shows that ρ = 0. The correspondng avalablty n Eqn. (A6) s equal to a that solves Eqn. (11). The other stuaton n (A4) arses when ρ < 0, n whch case the entre generated energy s declared avalable,.e. a = g. We can rewrte the above observatons more concsely as, ρ = 0, a < g ; (A7) ρ < 0, a = g. Proof of Proposton-3: Observe that, usng Eqns. (3), (12) and (13), the dervatves of the functons defned earler n Eqns. (16) and (17) are, π d = (1 β )u (d ), (A8) π a = 1 v 1 α g a. (A9) From Eqn. (12) snce β > 0, whenever d > 0, π d > 0 n Eqn. (A8). The factor (1 β ) n Eqn. (A8) s also strctly

9 IEEE TRANSACTIONS ON??? 9 decreasng n b and hence d. Thus π d s also monotoncally decreasng. Therefore π s a strctly concave functon. In a smlar manner, from Eqn. (13) t s clear that α < 1 as long as 0 a g, so that π a < 0 n Eqn. (A9). Besdes as 1 s strctly ncreasng, the product s 1 α monotoncally decreasng. Therefore π s strctly concave. Thus there s a unque maxmum of Π d, a Θ as defned n Eqn. (18). The Lagrangan of the problem defned n Eqn. (18), wth Eqns. (3) and (9) actng as constrants, s gven by, L d, a, λ, μ = π (d ) + π g a + λ a g + μ a d. (A10) The quanttes μ and λ above are the dual varables ntroduced by the constrants n Eqns. (3) and (9). The prmal condtons from Eqns. (3) and (9) must be satsfed. Furthermore, complementary slackness condtons yeld, λ a g = 0. (A11) From Eqns. (A8) and (A9), the statonary condtons of Eqn. (A10) must satsfy, (1 β )u (d ) = μ, (A12) v g a = 1 α λ + μ. (A13) From Eqns. (A3) and (A12) t s observed that μ = p. Usng Eqn. (5) t s seen that the buyer s bddng strategy defned n Eqn. (7) s satsfed. When a < g, Eqn. (A11) shows that λ = 0. Replacng λ and μ wth 0 and p, Eqn. (11) s satsfed. On the other hand, when a = g, λ s set to an approprate value. From the concavty assumpton, v g a < p 1 α so that λ < 0. We summarze these observatons as follows. p = μ; λ < 0 when a = g ; (A14) λ = 0 when a < g. From the above consderatons, t s seen that n both cases Eqn. (A13) satsfes the seller s bddng strategy n Eqn. (10). Eqn. (19) s trvally true snce U whch s defned n Eqn. (24) s the maxmum socal welfare. Proof of Proposton-4: Frst, note that snce the utltes are strctly concave, there s a unque optmum of the optmzaton problem defned n Eqn. (20) wth constrants defned n Eqns. (3) and (9). The Lagrangan s, L d, a, λ, μ = u (d ) + v g a + λ a g + μ a The statonary condtons satsfy, λ a g = 0, d. (A15) (A16) u (d ) = μ, (A17) v g a = λ + μ. (A18) Comparng Eqn. (A17) wth Eqn. (21), under proportonal allocaton n Eqn. (5), we see that μ = p. Replacng p n Eqn. (A5) wth zero, the seller s statonary condtons satsfy (A4) and the followng, v g a = p + ρ. (A19) From Eqns. (A17), (A18) and (A19), p = μ; λ < 0 when a = g ; (A20) λ = 0 when a < g. Statements () and () follow from the above. Snce the socal welfare U d, a Θ s maxmzed, statement () holds. Proof of Proposton-5: For each seller, from Eqn. (A7) t s seen that ρ = 0 when a < g. Also α = 0 under prce takng. Hence from Eqn. (A6), a = g v (p). (A21) Snce v > 0, a s strctly ncreasng n the nterval p v g, v (0). Moreover as a [0, g ] and as Eqn. (26) shows, A(p) s the sum of all a s, statement () follows. For each buyer, from Eqn. (A3), wth β = 0, d = u (p). (A22) Snce u > 0, d s strctly decreasng wth p n the nterval p 0, u (0). Moreover d = 0 when p u (0). Hence statement () follows drectly from Eqn. (23) where D(p) s expressed as the sum of all d s. From Eqn. (2), there s a non-empty nterval p mn v g, max max u (0), max v (0) wthn whch D(p) s monotoncally decreasng or zero and A(p) s monotoncally ncreasng or fxed at a postve value. Thus there s a unque p such that A(p ) = D(p ). Proof of Proposton-6: Snce from Eqn. (31) b 0 = pa 0, the expresson for the prce n Eqn. (4) s replaced wth, p = a 0 + a b 0 + b. (A23) Wth the addton of vrtual bddng, Eqns. (12) and (13) pertanng to market powers are rewrtten as, β = b 0 + b b, D, a (A24) α = a 0 + a a, S. (A25) Eqns. (A24) and (A25) show that both β and α monotoncally decrease wth ncreasng a 0. From Eqns. (A8) and (A9) t follows that π d and π a monotoncally approach u and v. As we have shown that π and π are strctly concave, t follows that they ncrease monotoncally wth ncreasng values of a 0 ; whence from Eqn. (18), U also ncreases monotoncally. In the lmtng case, lm b0 β = 0, and lm a0 α = 0; whereupon t follows that

10 IEEE TRANSACTIONS ON??? 10 lm a0 U = U. Smultaneously, from Eqn. (30), the loss of effcency decreases monotoncally towards zero. Ths shows that nsertng the vrtual bdder nto the aucton allows the aucton to smulate prce-takng. Proof of Proposton-7: The equlbrum of the optmzaton problem defned n Eqn. (36) s unque because the addton of the lnear term nvolvng p s does not alter the concavty property. The Lagrangan of the problem defned n Eqn. (36), wth Eqns. (3) and (9) as constrants, s gven by, L d, a, λ, μ = u (d ) + λ a g + v g a p s a + μ a d. (A26) The statonary condtons satsfy, λ a g = 0, (A27) u (d ) = μ + p s, (A28) v g a = λ + μ. (A29) Analogous to the reasonng provded n the proof of Proposton-4, t can be establshed that at equlbrum we must have, p = μ; λ < 0 when a = g ; (A30) λ = 0 when a < g. Ths establshes the statements () and (). From Eqn. (33), when p s = 0, the revenue R = 0. However, from the assumpton n Eqn. (2), the aggregate avalablty A s nonzero. 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