Online Mechanisms for Charging Electric Vehicles in Settings with Varying Marginal Electricity Costs

Transcription

1 Proceedngs of the Twenty-Fourth Internatonal Jont Conference on Artfcal Intellgence (IJCAI 2015) Onlne Mechansms for Chargng Electrc Vehcles n Settngs wth Varyng Margnal Electrcty Costs Kechro Hayakawa Toyota Central R&D Labs., Inc. Ach, Japan ke-hayakawa@mosk.tytlabs.co.jp Sebastan Sten Unversty of Southampton Southampton, Unted Kngdom ss2@ecs.soton.ac.uk Abstract We propose new mechansms that can be used by a demand response aggregator to flexbly shft the chargng of electrc vehcles (EVs) to tmes where cheap but ntermttent renewable energy s n hgh supply. Here, t s mportant to consder the constrants and preferences of EV owners, whle elmnatng the scope for strategc behavour. To acheve ths, we propose, for the frst tme, a generc class of ncentve mechansms for settngs wth both varyng margnal electrcty costs and multdmensonal preferences. We show these are domnant strategy ncentve compatble,.e., EV owners are ncentvsed to report ther constrants and preferences truthfully. We also detal a specfc nstance of ths class, show that t acheves 98% of the optmal n realstc scenaros and demonstrate how t can be adapted to trade off effcency wth proft. 1 Introducton The wdespread adopton of electrc vehcles (EVs) s often seen as a vtal step for mtgatng clmate change. Coupled wth a shft towards clean renewable electrcty generaton, such as photovoltacs (PV) or wnd energy, EVs promse to dramatcally reduce emssons [Royal Academy of Engneerng, 2010]. However, these renewable energy sources are typcally ntermttent and depend heavly on weather condtons [Btar et al., 2011]. In partcular, peaks n renewable energy supply (e.g., at noon for PV) may not concde wth peaks n demand (e.g., n the evenng when EV owners plug n). Thus, more expensve and pollutng conventonal methods of electrcty generaton have to be used at those peak tmes. To address ths dsparty n supply and demand, t s possble to explot the flexblty of EV owners and delay chargng to perods of hgher renewable supply [Ipakch and Albuyeh, 2009; Clement-Nyns et al., 2011]. In partcular, recent work has suggested the ntroducton of demand response Enrco H. Gerdng Unversty of Southampton Southampton, Unted Kngdom eg@ecs.soton.ac.uk Takahro Shga Toyota Central R&D Labs., Inc. Ach, Japan t-shga@mosk.tytlabs.co.jp servces, whereby electrcty consumers are offered fnancal ncentves to shft ther consumpton [Albad and El- Saadany, 2007]. Often, ths s facltated by an aggregator, whch procures electrcty from the wholesale electrcty market and then coordnates the consumpton of ndvdual endusers [Gkatzks et al., 2013]. Ths allows the aggregator to acheve sgnfcant cost savngs. As the end-consumers (.e., EV owners) n these systems are self-nterested agents, whch balance the nconvenence of shftng ther consumpton wth the assocated fnancal ncentves, there s consderable work on modellng ths settng usng game theory [Mohsenan-Rad et al., 2010; Saad et al., 2012; Bhattacharya et al., 2014]. Typcally, coordnaton s acheved usng real-tme prcng mechansms, whch adjust electrcty prces dependng on demand and on the margnal costs of generaton [Conejo et al., 2010]. However, partcpatng n such a mechansm mposes a sgnfcant burden on EV owners, as they have to reason strategcally about future demand n order to decde when to charge. To address ths, recent work has looked at coordnatng the chargng of EVs usng onlne mechansm desgn [Parkes, 2007]. Specfcally, Gerdng et al. [2011] propose a schedulng and prcng mechansm that aggregates the multdmensonal preferences of dynamcally arrvng EV owners and that ensures that truthful reportng of these preferences maxmses each partcpant s utlty (.e., the mechansm s domnant strategy ncentve compatble, or DSIC). Ths removes the need for strategc behavour, allowng the mechansm to fnd a schedule wth hgh effcency. Other work uses the noton of pre-commtment to delay more flexble EV owners whle retanng the DSIC property, albet only for settngs wth sngle-dmensonal preferences [Sten et al., 2012], and ths s extended to deal wth a demand response settng wth varable renewable generaton n [Ströhle et al., 2014]. However, these mechansms do not consder the margnal cost of generaton, whch depends on the avalable supply of renewable energy and vares wth the amount of energy that s consumed. Addtonally, the mechansm n [Gerdng et al., 2610

2 2011] s based on a smple greedy allocaton and requres cancellng some allocatons. Ths leads to very neffcent allocaton decsons, especally when margnal costs are consdered. To address these shortcomngs, we make the followng novel contrbutons: We characterse, for the frst tme, a general class of DSIC onlne mechansms whch deal wth both margnal generaton costs and mult-dmensonal preferences. We descrbe one partcular nstance from ths class of mechansms whch, n addton to beng DSIC, s computatonally effcent, scalng to hundreds of agents. We emprcally benchmark ths mechansm and show that t acheves near-optmal performance and sgnfcantly outperforms the current state of the art, whch does not consder the margnal cost of generaton. 2 Smart Chargng System Model We consder a smart EV chargng system, whch s mplemented by a demand response aggregator. Ths aggregator acts as a broker between the EV owners and the electrcty market. Specfcally, t procures electrcty for EV chargng from a mxture of local renewable generators, possbly through long-term contracts, and the wholesale global electrcty market. The aggregator also collects the constrants and preferences of ndvdual EV owners and then schedules ther chargng to maxmse socal welfare and/or proft. Ths may nvolve shftng the chargng of flexble EV owners to tmes where electrcty s cheap or even curtalng consumpton when prces are too hgh. 2.1 EV Owner Model We consder a model wth dscrete and possbly nfnte tme steps t T. Each EV owner s represented by an agent, and we use I = 1,..., n} to denote the set of all agents. At every tme step, an agent can charge a sngle unt of electrcty (e.g., correspondng to 3 kwh), and we assume that all EVs charge at the same rate. Each agent I has a lmted avalablty for chargng, whch s gven by an arrval tme a T (.e., earlest possble tme for chargng) and a departure tme d T (.e., latest tme for chargng), wth d a. The agent s valuatons for chargng are gven as a vector v = v,1, v,2,...}, where v,k denotes the margnal value for the kth unt. We assume these are non-ncreasng,.e., k > j : v,j v,k, whch s a natural assumpton for plug-n hybrd EVs [Robu et al., 2013]. Gven ths, we use θ = a, d, v } to summarse agent s type. θ = θ 1,..., θ n } denotes the types of all agents and θ denotes the types of all agents except. Furthermore, we use I t and θ t to denote the agents and ther types n the market at or before tme t. From tme a, when agent becomes avalable for chargng (.e., when the EV arrves at home and s plugged n), he can report hs type to the aggregator, e.g., usng a communcaton devce that s ntegrated wth hs chargng equpment or va a smart phone app. Crucally, we assume that the agent could strategcally msreport hs type, f ths s n hs best nterest. Thus, we use ˆθ = â, ˆd, ˆv } to denote agent s report (here, â s gven mplctly by the tme the report s made). As s common n ths doman, we assume that agents cannot report an earler arrval or later departure tme,.e., â a and ˆd d. Ths s a natural assumpton n ths doman, as a vehcle cannot be plugged nto the chargng equpment when t s unavalable for chargng, but t s easy to delay pluggng n, or to unplug early [Robu et al., 2013]. 2.2 Aggregator Mechansm Gven the reported types of EV agents, the aggregator now uses a schedulng functon π,t (ˆθ t ), whch keeps track of how many unts of electrcty have been allocated to agent on or before tme t. Importantly, as ths s an onlne settng, ths functon can only depend on the types that have arrved on or before the current tme t. Here, t, â t ˆd : π,t (ˆθ t ) π,t 1 (ˆθ t 1 ) 0, 1}, and ths ndcates whether agent charges at tme t. When chargng EVs, the aggregator also has to procure the necessary amount of electrcty ether from the global wholesale market or from local renewable sources. The cost for ths depends both on the tme of chargng and on the amount of electrcty that s needed. To model ths, we use margnal costs c(t, m), whch s the margnal cost for chargng the mth vehcle at tme t. Here, we assume that supply s nfnte,.e., any number of vehcles can be charged concurrently, but the assocated margnal costs could be very hgh (reflectng, for example, the need to power up addtonal generators). Typcally, c(t, m + 1) c(t, m), but ths s not a requrement for our mechansm. We also assume that costs are determnstc and known n advance, but they could reflect expected costs wthout changng the mechansms presented here. Table 1 shows a part of the cost functon used n the experments. In addton to decdng on a schedule, the mechansm also determnes a payment x (ˆθ ˆd ) for each agent, whch has to be pad on hs departure. As before, ths can only depend on the reported types up to the current tme. Ths payment, along wth the schedulng decsons, can be desgned to ensure certan desrable propertes n strategc settngs, whch we brefly dscuss n the followng secton. Gven ths, the aggregator s goal could be to maxmse socal welfare, denoted by SW (θ), whch s the sum of margnal values mnus the sum of margnal costs. Formally, let M π t t T denote the number of unts allocated at tme t, then: π, ˆd (ˆθ) M π t SW (ˆθ) = I j=1 ˆv,j t T m=1 c(t, m). An alternatve goal s to maxmse proft, whch s the total payments receved mnus the total costs: I x (ˆθ ˆd ) M π t m=1 c(t, m). We consder both goals n Secton Strategc Behavour We assume that EV owners are self-nterested, and so we model them as ratonal utlty-maxmsers. Specfcally, the utlty of an agent wth type θ and who reports ˆθ (whle all other agents report ˆθ ) s gven by U (ˆθ, ˆθ }, θ ) = π, ˆd (ˆθ ˆd,ˆθ }) j=1 v,j x (ˆθ ˆd, ˆθ }). 2611

3 As a result, we must assume that agents wll msreport ther types f ths ncreases ther own utlty. To address ths, we am to desgn a schedulng functon π,t and payment x that ensure the mechansm s domnant strategy ncentve compatble (DSIC),.e., agents maxmse ther utlty when reportng ther own types truthfully. Formally, we want to ensure that θ, ˆθ, ˆθ : U (ˆθ, θ }, θ ) U (ˆθ, ˆθ }, θ ). 3 Proposed Mechansm In what follows we frst ntroduce a generc mechansm and then show that t satsfes the DSIC property n our settng. The mechansm s generc, as t can be used wth a varety of prcng rules and (possbly sub-optmal) schedulng algorthms. To acheve DSIC, t smply mposes certan constrants on these rules and algorthms. We then ntroduce a specfc prcng rule and schedulng heurstc that satsfy these constrants and whch are used n the experments. 3.1 Generc Mechansm Unlke standard mechansm desgn approaches, whch defne a (weakly) monotonc allocaton and then use crtcal value payments to ensure DSIC [Bkhchandan et al., 2006], we take a dfferent approach. Frst, for each agent, a mnmum prce for chargng at every tme step t s determned, usng a prcng functon f. These prces, combned wth an agent s valuaton functon and hs avalablty n the market, determne the maxmum number of unts, l, whch need to be allocated to the agent by hs departure tme. Importantly, prces can ncrease over tme f more agents (wth potentally hgher valuatons) enter the market later and so the allocaton to agent can decrease. However, ths has to be done carefully. Allocatons have to reman feasble by an agent s departure tme, agents wth a longer avalablty cannot be penalsed through these revsed allocatons, and, to avod cancellatons, the mechansm must never charge more unts than an agent would lke, gven the fnal prces. To ensure these propertes, we derve general condtons on prces and allocatons. The full process, whch s performed for each agent, s detaled below. Calculatng the mnmum margnal prce vector We start by computng the mnmum margnal payment vector at tme t, p t, where p t,j s the (mnmum) payment for the jth unt of electrcty. To ths end, we frst determne the prces of chargng at specfc tmes n the market. We dstngush between past unts, for whch the prces are fxed, and future unts, for whch the prces can stll ncrease. Let f(ˆθ t, t ) denote a functon whch determnes the prce of a unt of electrcty at any tme t t n the future. For brevty we use f t t,t = f(ˆθ, t ). We dscuss a specfc example functon n Secton 3.3, but the mechansm can support any functon as long as t does not depend on θ and t satsfes: t, t t + 1 : f t+1,t f t,t (1) That s, prces at specfc tmes n the future t > t can only ncrease or reman unchanged as the actual tme, t, approaches these tme ponts. 1 A trval example satsfyng ths constrant s where prces are always zero. However, n that case the aggregator wll make a loss f costs are strctly postve. Once we reach a certan tme step, the prces at that tme are fxed and no longer change. Formally, f t s the current tme step, and the current prce s gven by f t,t, then: t, t > t : f t,t = f t,t (2) Gven ths, we can compute a prce vector at tme t as follows: } η t = f â, f â+1,â,..., f t +1,t, f t,t+1,..., f t, ˆd (3) Note that ths vector conssts of two parts: the prces from the agent s reported arrval, â, up to the current tme, t, are fxed, whereas the remanng prces are future prces and so they can stll ncrease. Now, the prce vector η t determnes the prces at dfferent tme steps, but the agent s not nterested n when t s allocated the unts but would smply lke the cheapest ones wthn the perod that he s avalable n the market. To ensure that the agent always gets the best prce (whch s not necessarly the prce at the tme he s charged by the schedulng mechansm), we sort the prces n ascendng order to obtan the mnmum margnal payment vector: p t = ncr(η t ), (4) where ncr(.) s an operator whch arranges the nput vector n ascendng order. Determnng the schedulng constrants We now compute the number of unts to be assgned to agent by hs reported departure, ˆd, gven the current prces, p t, and the agent s valuaton v, such that hs utlty s maxmsed. Specfcally: k ( ) l t = argmax ˆv,j p t 0 k ˆd j=1,j (5) â +1 Note that ˆd â +1 s an upper bound on the number of unts whch can be allocated wthn the tme that the agent s avalable n the market. As a result, even f prces are zero, the mechansm never allocates more than s physcally possble (recall that we assume unlmted supply). Henceforth we refer to l t as the temporarly assgned number of unts, whch s mposed on the schedulng algorthm (as detaled below). The temporarly assgned number of unts ncludes possble future allocatons (up to the agent s reported deadlne). Therefore, f prces ncrease, the number of assgned unts could decrease over tme. In order to prevent the mechansm from over-allocatng unts, whch then later would need to be cancelled, we mpose an addtonal constrant on the number of unts whch the scheduler can allocate up to the current tme. Ths constrant uses only the fxed prces from the prce vector. Specfcally, let: } η t = f â, f â+1,â,..., f t +1,t 1 Ths s not a major restrcton as typcally new agents enterng the market n the future wll push up prces. 2612

4 denote the vector of fxed prces. ncr(η t ) and l t s gven by: k ( l t = argmax 0 k t â j=1 +1 Smlarly, p t ˆv,j p t,j Note that the upper bound of l s t â + 1. We refer to l t as the upper lmt allocaton for agent at tme t. Schedulng algorthm We now proceed wth the actual allocaton of unts to agents. As wth the prces, we consder both the past allocatons as well as future allocatons. Let π t,t = π (θ t, t ) denote a schedulng algorthm whch specfes the total number of unts allocated to agent by tme t when the current tme s t. Note that, snce we have that only a sngle unt can be allocated per tme step, necessarly t, t : π t,t +1 π t,t 1. Note that ths s an extenson of the notaton ntroduced n Secton 2.2, whch allows the schedulng algorthm to make (potentally temporary) allocaton decsons for future tme t > t, gven the nformaton avalable at tme t. Specfcally, π t,t = π,t. Naturally, the schedulng algorthm cannot change past allocatons and so: t, t > t : π t,t ) = (6) = π t,t = π,t (7) In terms of future allocatons, any allocaton functon can be used, as long as t meets the followng constrants at every t: π t, ˆd = l t That s, by the reported deadlne agent needs to be able to receve ts temporarly assgned number of unts. Note that ths future allocaton can stll decrease over tme (when prces ncrease), but t should be possble to allocate the necessary number of unts f prces and correspondng allocatons do not change. At the same tme, t needs to meet the constrant: (8) π t,t l t (9) That s, the scheduler cannot assgn more than the upper lmt allocaton by the current tme. In Secton 3.2 we show that, provded that the prcng condtons are met, these constrants can always be satsfed. Payment Fnally, the payment, whch s computed on the (reported) departure of agent, s gven by: 3.2 Theoretcal Propertes x (ˆθ ˆd ) = π ˆd, ˆd κ=1 p ˆd,κ (10) Before we show that the proposed class of mechansms s always DSIC, we frst show that Eqs. 8 and 9 can always be satsfed usng a feasble schedule. Defnton 1 (Feasble schedule). A schedule s feasble f t : π t,t π t 1,t 1 0, 1},.e., we assgn at most one unt per tme step, and we do not reduce the allocatons. Lemma 1. Gven unlmted supply and gven that ˆv, I are margnally non-ncreasng, f Eqs. 8 and 9 are satsfed at tme t 1, there always exsts a feasble schedule whch satsfes the equatons at tme t. Proof. Frst we show π t,t π t 1,t 1 0,.e., the constrants never lead to a decrease n actual allocatons. Because a new prce s added to the fxed prce vector at each tme step, we have that l t l t 1,.e., the upper lmt allocaton can only ncrease. At the same tme, because future prces t ncrease, we have that l l t 1,.e., the temporarly assgned number of unts can decrease. However, ths can never go below the upper lmt allocatons. In partcular: l t l t 1. To see ths, note that all fxed prces n p t 1 are also n p t. Therefore, wth more (possbly lower) prces to choose from, the allocaton whch maxmses the agent s utlty, Eq. 5, s at least as hgh Eq. 6. To ensure that π t,t π t 1,t 1 1, we consder two cases. Case 1: π t 1,t 1 < l t. In ths case, gven that the constrants are satsfed at t 1, because l t l t 1 we can always satsfy both constrants at tme t by chargng at tme t,.e., by settng π t,t = π t 1,t Case 2: π t 1,t 1 = l t. In ths case, n order to satsfy Eq. 9 we cannot charge at tme t. We have to show that we can stll satsfy Eq. 8. Note that such a schedule s feasble as long as l t l t ˆd t,.e., the total number of unts to be allocated by the deadlne s at most the number of remanng tme steps (snce we can only allocate at most one unt per tme step). Note that p t contans all the prces n p t plus exactly ˆd t addtonal values. In addton, the dfferences n the upper bounds s also exactly ˆd t (n partcular, note that p ˆd = p ˆd. Therefore, provded that the margnal valuatons are non-ncreasng, we have that the dfference between l t and l t s at most ˆd t. 2 Lemma 2. There always exsts a feasble schedule where Constrants 8 and 9 can be satsfed on arrval. Proof. Ths can be acheved by settng π â = l â. The above results not only mply that a feasble schedule always exsts but, more mportantly, any exstng schedulng algorthm can be adapted by smply ntroducng the necessary constrants at each tme step, whch automatcally ensures that the constrants can be satsfed n the next tme step. We now show that the constrants always lead to the mechansm beng DSIC. Theorem 1. Gven lmted msreports and non-ncreasng margnal valuatons ˆv of agents I, any mechansm wth 2 The assumpton that margnal valuatons are non-ncreasng s mportant here. Suppose otherwse, e.g., v = 0, 0, 10}, â = 1, ˆd = 3, p 2 = 1, 1} and p 2 = 1, 1, 1}, we have that l 2 = 0 and l 2 = 3, resultng n an nfeasble schedule snce π 2,3 π 2,2 = 3 0 =

5 prcng functon f(ˆθ t, t ) satsfyng Eq. 1 and schedulng algorthm satsfyng Eq. 8 s DSIC. Proof. (1) We show that, regardless of the reported arrval and departure, t s a domnant strategy to truthfully report v. (2) We then show that, gven v = ˆv, there s no ncentve to report a later arrval or an earler departure. (1) Gven that prcng functon f does not depend on θ, the margnal prces on departure of agent, p ˆd, only depend on â and ˆd, and not on ˆv. Therefore the agent cannot nfluence the prces by msreportng v but only the allocaton. For a gven allocaton π, and payments accordng to Eq. 10, the utlty of agent s gven by: π j=1 ( v,j p ˆd,j ). Due to Constrant 8, the allocaton on (reported) departure maxmses Eq. 5, whch corresponds to the agent s utlty when v = ˆv. Therefore, reportng v truthfully s optmal. (2) Suppose that the agent reports a later arrval, â > a. Accordng to Eq. 1 we have that f â,t f a,t, whch means that prces for the same correspondng tme perod n prce vector η ˆd can ncrease (and never decrease). In addton, η ˆd contans â a fewer prces. Because the margnal payment vector p ˆd s n ascendng order, the payment can only ncrease by havng fewer prces. Now suppose that ˆd < d. Let η ˆd = f â,..., f ˆd, ˆd } denote the fnal prce vector when msreportng, } and let η d = f â,..., f ˆd, ˆd, f ˆd +1, ˆd,..., f d +1,d denote the fnal prces usng the agent s true departure. Note that, n both cases, the frst ˆd â + 1 prces are dentcal. Therefore, by msreportng an earler departure, the prce vector only contans fewer prces. Ths means margnal payments can only ncrease. 3.3 Specfc Mechansm Although our approach guarantees DSIC for a range of prcng functons and schedulng algorthms, a poor choce can lead to poor effcency and a loss for the aggregator f t cannot recoup the ncurred margnal costs from the supplers (formally, our class of mechansms does not guarantee weak budget balance). In ths secton we demonstrate our approach for a specfc prcng functon and schedulng algorthm, whch we then evaluate emprcally n Secton 4. Prcng Functon In ths secton we nstantate the prcng functon f(ˆθ t, t ). Note that dfferent prces need to be calculated for each agent I, and we cannot use agent s type when dong so. In addton we need to calculate prces for each future tme pont t, where t t ˆd. To reduce the computatonal costs, we use a heurstc approach. In detal, we run a vrtual verson of the market wthout t agent and wth all known agents ˆθ, and we allocate the unts to those agents n a greedy manner as follows. 3 We take 3 The market needs to be run from before the frst arrval, e.g., when the market frst started or from the begnnng of each day (denoted by t 1). the agent wth the hghest margnal value, and allocate ths agent the unt wth the lowest avalable margnal cost gven the agent s arrval and departure tme, and provded that the margnal value s equal to or hgher than the margnal cost (otherwse no match s made). We remove the matched values and costs from the market, and we repeat ths process untl no more matches can be made. Let SW (ˆθ t ) denote the socal welfare of the resultng allocaton. Now, to compute the prces at tme t, we allocate a sngle unt at tme t to agent and we rerun the market as before. Let SW t (ˆθ t ) denote the socal welfare of the resultng allocaton, excludng agent s value for the unt. Gven ths, the prcng functon for agent s defned as follows. If t = t 1, f(ˆθ t, t ) = SW (ˆθ t ) SW t t (ˆθ ). Otherwse: f(ˆθ t, t ) = max SW (ˆθ t ) SW t t 1 t (ˆθ ), f,t } Intutvely, the payment s the externalty mposed on the system excludng agent f a unt at tme t s allocated to. 4 Note that the max operator ensures that Eq. 1 always holds. Schedulng Algorthm To determne the prces we used a fast heurstc schedulng algorthm snce ths needs to be repeated for each agent and future tme step. Also, for the prces we dd not need to consder the schedulng constrants. In contrast, the schedulng algorthm for the actual allocaton needs to be computed only once at every current tme step. Hence, for the allocaton we optmse the socal welfare consderng all agents, subject to Eqs. 8 and 9 for each agent. The optmzaton s executed by solvng a Mxed Integer Program (MIP) usng the Gurob 5 solver wth a tolerance of Note that, although ths algorthm s near-optmal, ths s not a requrement for the mechansm to satsfy DSIC. 4 Numercal Analyss To quantfy the performance of our proposed mechansm n realstc demand response settngs, we now evaluate t numercally, comparng t to exstng state-of-the-art mechansms. 4.1 Expermental Setup We determne the hourly margnal costs c(t, m) of a 24-hour perod of a typcal day as follows. For electrcty prces we use the data from the Japan Electrc Power Exchange 6 on the 5th of June, To determne the costs, n addton we use the typcal energy consumpton of Japanese households on a fne June day 7. Takng regular household consumpton nto account s mportant, snce EV chargng wll use energy on top of what s normally consumed. Furthermore, we assume that the chargng rate of all agents s 3kW, and so a sngle unt 4 The ntuton s smlar to the well-known Vckrey-Clarke- Groves mechansm for statc settngs, but our approach does not assume optmalty and only consders a sngle unt at a tme We use data from the Archtectural Insttute of Japan, see /database/ndex.htm. 2614

6 Table 1: A part of the margnal cost table [JPY] tme(t) number of agents (m) s 3kWh. Detals are omtted due to space constrants but a small sample s shown n Table 1. Here, c(13, 1) = 0.1 s the cost n JPY for chargng a sngle vehcle at tme step 13. The margnal cost for the second vehcle s c(13, 2) = 8.2, resultng n a total cost of 8.3 JPY for chargng both. To obtan the dstrbuton of the agents arrval and departure tmes, we use the results from a questonnare of 340 ctzens n Nagoya Cty n Japan, whch asked about daly movements. From ths questonnare we use the answers regardng when and for how long cars were parked at home durng weekdays. For the smulatons, each agent s arrval-departure par s randomly selected from the 340 samples. Fnally, the remanng capacty of each agent s randomly chosen between 1 and 6 unts of charge. For each unt, the agent s margnal valuaton s unformly drawn from 0 to 100 JPY and these are arranged n descendng order. 4.2 Benchmark Mechansms We compare the performance of the mechansm from Secton 3.3, called Proposed, wth those from the lterature. Greedy optmally allocates the avalable electrcty n each tme step wthout consderng future tme steps. As noted n [Gerdng et al., 2011], ths mechansm volates weak monotoncty and therefore s not truthful n settngs wth multdmensonal valuatons (even when costs are zero). Greedy wth cancellaton on departure (GCOD) works lke Greedy except that allocatons are sometmes cancelled on departure as specfed n [Gerdng et al., 2011] to ensure DSIC. Frst Come Frst Serve (FCFS) optmally schedules each agent n order of ther arrval. If multple agents arrve at the same tme, they are scheduled n order of ther ID, whch s assgned to all agents beforehand. Ths mechansm s DSIC when the payment s set equal to the ncurred margnal costs (but the aggregator makes no proft). Myopcally Optmal (MO) optmses socal welfare usng all nformaton currently avalable. The allocatons are recalculated whenever a new agent enters the market. Ths s the same schedulng mechansm used n Secton 3.3 wthout consderng the constrants, and s not DSIC. Optmal uses the same algorthm as MO but assumes perfect foresght of the agents arrvng n the future. 4.3 Results We frst compare the effcency, whch s the obtaned socal welfare as a proporton of the Optmal. To ths end, we vary the number of agents from 10 to 300, and for each settng we run 1000 trals. Each tral smulates a 48-hour perod, and the cost table s assumed to be the same for these two days. Effcency Proposed FCFS Greedy GCOD MO Number of Agents Effcency Number of Agents Proposed FCFS MO Fgure 1: Comparng effcency of all mechansms (left) and top 3 mechansms only (rght). 95% confdence ntervals are smaller than 0.05% of the effcency and therefore omtted. Effcency Proposed FCFS Proft of Aggregator [JPY] Fgure 2: Trade off between effcency and aggregator proft. The results n Fgure 1 show that Proposed s very close to Optmal. As expected, MO outperforms Proposed snce the former s not bound by the constrants. However, Proposed outperforms all other DSIC mechansms, especally GCOD and even Greedy. GCOD performs especally poorly due to the costs whch are stll ncurred even f the allocaton s cancelled (and so the electrcty s unused). Note that the effcency of the top three mechansms frst decreases when the number of agents s less than 30 agents, and then ncreases. Ths s because, when there are few agents, schedulng s relatvely easy snce there s lttle competton. When there are a lot of agents, many of them wll have smlar hgh margnal values and so t does not matter too much whch of them are scheduled. The fact that ths happens around 30 s manly due to the cost table used. Surprsngly, the smple mechansm FCFS also performs very well n terms of effcency, whlst also beng DSIC. However, as mentoned n Secton 4.2, no proft s made by the aggregator snce the payment s equal to the margnal costs. A smple way for the aggregator to ncrease profts of any mechansm (whlst mantanng DSIC) s to artfcally ncrease costs by a constant, and pocket the dfference. However, as wth reserve prces, ths lowers effcency. In ths part, we compare the mechansms Proposed and FCFS n terms of ther trade off between proft and effcency. Specfcally, we multply the costs used n the prcng rule by a constant, α. To ths end, Fgure 2 shows the relaton between effcency 2615

7 and aggregator proft for dfferent values of α, where α s vared from 1.0 to 2.5. The number of agents s set to 200, and each pont shows the average over 1000 trals. Here, effcency s w.r.t. the Optmal wth orgnal costs. The top left s where α = 1. As we ncrease α, as expected, ntally the proft ncreases for both mechansms, but eventually the proft decreases (snce the effcency becomes very low). Comparng the two mechansms, we can see that the Proposed mechansm s able to obtan 44% hgher profts than FCFS whlst havng the same effcency. Conversely, for the same level of proft, the effcency of Proposed s consderably hgher. Fnally, we brefly comment on the computatonal tractablty. Even though Proposed requres more computaton than the other approaches, t s scalable to hundreds of agents. For example, a 48-hour tral wth 300 agents takes 52 seconds on average on an Intel(R)Core(TM)7-4770K 3.50GHz usng a sngle core. 5 Conclusons We have proposed a novel approach for desgnng onlne mechansms for mult-dmensonal valuatons and margnal costs. The approach can be used n combnaton wth any prcng and allocaton functon, subject to a set of constrants beng met. We have shown that the constrants are always feasble n settngs wth margnally non-ncreasng valuatons and that the resultng mechansms are DSIC. We have emprcally compared an nstantaton and have shown that, n a realstc settng wth varyng margnal costs, our mechansm outperforms exstng DSIC mechansms n terms of effcency and proft. It s also computatonally effcent and scales to hundreds of agents. In future, we plan to extend the mechansms to allow for multple unts to be allocated to an agent n a sngle tme step (to capture settngs where EVs can be charged at dfferent rates). Moreover, we wll consder mechansms for settngs where the margnal valuatons of agents can be ncreasng. References [Albad and El-Saadany, 2007] M.H. Albad and E.F. El- Saadany. Demand response n electrcty markets: An overvew. In Power Engneerng Socety General Meetng, IEEE, pages 1 5, June [Bhattacharya et al., 2014] S. Bhattacharya, K. Kar, J.H. Chow, and A. Gupta. Extended second prce auctons for plug-n electrc vehcle (pev) chargng n smart dstrbuton grds. In Amercan Control Conference (ACC), 2014, pages IEEE, [Bkhchandan et al., 2006] S. Bkhchandan, S. Chatterj, R. Lav, A. Mu alem, N. Nsan, and A. Sen. Weak monotoncty characterzes determnstc domnant-strategy mplementaton. Econometrca, 74(4): , [Btar et al., 2011] E. Btar, P.P. Khargonekar, and K. Poolla. Systems and control opportuntes n the ntegraton of renewable energy nto the smart grd. In Proc. of IFAC World Congress, pages , [Clement-Nyns et al., 2011] K. Clement-Nyns, E. Haesen, and J. Dresen. The mpact of vehcle-to-grd on the dstrbuton grd. Electrc Power Systems Research, 81(1): , [Conejo et al., 2010] A.J. Conejo, J.M. Morales, and L. Barngo. Real-tme demand response model. Smart Grd, IEEE Transactons on, 1(3): , Dec [Gerdng et al., 2011] E.H. Gerdng, V. Robu, S. Sten, D.C. Parkes, A.C. Rogers, and N.R. Jennngs. Onlne mechansm desgn for electrc vehcle chargng. In The 10th Internatonal Conference on Autonomous Agents and Multagent Systems, pages , [Gkatzks et al., 2013] L. Gkatzks, I. Koutsopoulos, and T. Salonds. The role of aggregators n smart grd demand response markets. Selected Areas n Communcatons, IEEE Journal on, 31(7): , July [Ipakch and Albuyeh, 2009] A. Ipakch and F. Albuyeh. Grd of the future. Power and Energy Magazne, IEEE, 7(2):52 62, [Mohsenan-Rad et al., 2010] A.-H. Mohsenan-Rad, V.W.S. Wong, J. Jatskevch, and R. Schober. Optmal and autonomous ncentve-based energy consumpton schedulng algorthm for smart grd. In Innovatve Smart Grd Technologes (ISGT), 2010, pages 1 6. IEEE, [Parkes, 2007] D.C. Parkes. Onlne mechansms. In Nsan, N., Roughgarden, T., Tardos, E., and Vazran, V. (Eds.) Algorthmc Game Theory, pages , [Robu et al., 2013] V. Robu, E.H. Gerdng, S. Sten, D.C. Parkes, A.C. Rogers, and N.R. Jennngs. An onlne mechansm for mult-unt demand and ts applcaton to plug-n hybrd electrc vehcle chargng. Journal of Artfcal Intellgence Research, 48: , [Royal Academy of Engneerng, 2010] Royal Academy of Engneerng. Electrc Vehcles: Charged wth potental. Royal Academy of Engneerng, [Saad et al., 2012] W. Saad, Z. Han, H.V. Poor, and T. Basar. Game-theoretc methods for the smart grd: An overvew of mcrogrd systems, demand-sde management, and smart grd communcatons. Sgnal Processng Magazne, IEEE, 29(5):86 105, [Sten et al., 2012] S. Sten, E.H. Gerdng, V. Robu, and N.R. Jennngs. A model-based onlne mechansm wth precommtment and ts applcaton to electrc vehcle chargng. In Proceedngs of the 11th Internatonal Conference on Autonomous Agents and Multagent Systems (AA- MAS 12), pages , [Ströhle et al., 2014] P. Ströhle, E.H. Gerdng, M. Weerdt, S. Sten, and V. Robu. Onlne mechansm desgn for schedulng non-preemptve jobs under uncertan supply and demand. In 13th Internatonal Conference on Autonomous Agents and Mult-Agent Systems, pages , May