Online Mechanism Design for Electric Vehicle Charging

Transcription

1 Onlne Mechansm Desgn for Electrc Vehcle Chargng Enrco H. Gerdng Davd C. Parkes Valentn Robu Alex Rogers Unversty of Southampton, SO17 1BJ, Southampton, UK Harvard Unversty, Cambrdge, MA 02138, USA Sebastan Sten Ncholas R. Jennngs ABSTRACT Plug-n hybrd electrc vehcles are expected to place a consderable stran on local electrcty dstrbuton networks, requrng chargng to be coordnated n order to accommodate capacty constrants. We desgn a novel onlne aucton protocol for ths problem, wheren vehcle owners use agents to bd for power and also state tme wndows n whch a vehcle s avalable for chargng. Ths s a mult-dmensonal mechansm desgn doman, wth owners havng non-ncreasng margnal valuatons for each subsequent unt of electrcty. In our desgn, we couple a greedy allocaton algorthm wth the occasonal burnng of allocated power, leavng t unallocated, n order to adjust an allocaton and acheve monotoncty and thus truthfulness. We consder two varatons: burnng at each tme step or on-departure. Both mechansms are evaluated n depth, usng data from a real-world tral of electrc vehcles n the UK to smulate system dynamcs and valuatons. The mechansms provde hgher allocatve effcency than a fxed prce system, are almost compettve wth a standard schedulng heurstc whch assumes non-strategc agents, and can sustan a substantally larger number of vehcles at the same per-owner fuel cost savng than a smple random scheme. Categores and Subject Descrptors I.2.11 [AI]: Dstrbuted AI - multagent systems General Terms Algorthms, Desgn, Economcs Keywords electrc vehcle, mechansm desgn, prcng 1. INTRODUCTION Promotng the use of electrc vehcles (EVs) s a key element n many countres ntatves to transton to a low carbon economy [4]. Recent years have seen rapd nnovaton wthn the automotve ndustry [10], wth desgns such as plug-n hybrd vehcles (PHEVs, whch have both an electrc motor and an nternal combuston engne) and range-extended electrc vehcles (whch have an electrc motor and an on-board generator drven by an nternal combuston engne) promsng to overcome consumers range anxety 1 and thereby ncreasng manstream EV use 2. However, there 1 Fear that a car wll run out of electrcty n the mddle of nowhere. 2 The Toyota plug-n Prus and the Chevrolet Volt are commercal Cte as: Ttle, Author(s), Proc. of 10th Int. Conf. on Autonomous Agents and Multagent Systems (AAMAS 2011), Yolum, Tumer, Stone and Sonenberg (eds.), May, 2 6, 2011, Tape, Tawan, pp. XXX XXX. Copyrght c 2011, Internatonal Foundaton for Autonomous Agents and Multagent Systems ( All rghts reserved. are sgnfcant concerns wthn the electrcty dstrbuton ndustres regardng the wdespread use of such vehcles, snce the hgh chargng rates that these vehcles requre (up to three tmes the maxmum current demand of a typcal home) could overload local electrcty dstrbuton networks at peak tmes [5]. Indeed, street-level transformers servcng between homes may become sgnfcant bottlenecks n the wdespread adopton of EVs [11]. To address these concerns, electrcty dstrbuton companes that are already seeng sgnfcant EV use (such as the Pacfc Gas and Electrc Company n Calforna) have ntroduced tme-of-use prcng plans for electrc vehcle chargng that attempt to dssuade owners from chargng ther vehcles at peak tmes, when the local electrcty dstrbuton network s already close to capacty 3. Whle such approaches are easly understood by customers, they fal to fully account for the constrants on the local dstrbuton networks, and they are necessarly statc snce they requre that vehcle owners ndvdually respond to ths prce sgnal and adapt ther behavour (.e., manually changng the tme at whch they charge ther vehcle). Lookng further ahead, researchers have also begun to nvestgate the automatc schedulng of EV chargng. Typcally, ths work allows ndvdual vehcle owners to ndcate the tmes at whch the car wll be avalable for chargng, allowng automatc schedulng whle satsfyng the constrants of the dstrbuton network [15, 2]. However, snce these approaches separate the schedulng of the chargng from the prce pad for the electrcty (typcally assumng a fxed per unt prce plan), they are unable to preclude the ncentve to msreport (e.g., an owner may ndcate an earler departure tme or further travel dstances n order to receve preferental chargng). To address the above shortcomngs, we turn to the feld of onlne mechansm desgn [12]. Specfcally, we focus on mechansms that are model-free (whch make no assumptons about future demand and supply of electrcty), and that allocate resources as they become avalable (electrcty s pershable snce nstallng alternatve storage capacty can be very costly). Now, exstng mechansms of ths knd assume that the preferences of the agents (representng the vehcle owners) can be descrbed by a sngle parameter, so-called sngle-valued domans. However, ths assumpton s not approprate for our problem, where agents have mult-unt demand wth margnal non-ncreasng valuatons for ncremental klowatt hours (kwh) of electrcty. 4 To ths end, we extend the state of the art n dynamc mechansm desgn as follows: examples of both, whch wll be on the road n See for example pge/electrcvehcles/fuelrates/. 4 Margnal valuatons are non-ncreasng n our doman because dstance and energy usage are uncertan, and therefore the frst few unts of electrcty are more lkely to be used, and (n the case of plug-n hybrd electrc vehcles) any shortfall can be made up by usng the vehcle s nternal combuston engne.

2 We develop a formal framework and soluton for the EV chargng problem, and show that t can be naturally modeled as an onlne mechansm desgn problem where agents have mult-unt demand wth non-ncreasng margnal valuatons. We develop the frst model-free onlne mechansm for pershable goods, where agents have mult-unt demand wth decreasng margnal valuatons. To ensure truthfulness, we show that ths mechansm occasonally requres unts to reman unallocated (we say that these unts are burned ), even f there s demand for these unts. Ths burnng can be done n two ways: at the tme of allocaton, or on departure of the agent. The latter results n hgher allocatve effcency and allocatons are easer to compute, but occasonally requres the battery to be dscharged whch may not always be feasble n practce. Both varants are (weakly) domnant-strategy ncentve compatble (DSIC), whch means that no agent has an ncentve to msreport ther demand vector and the vehcle avalablty, regardless of the others reports. We evaluate our mechansm through numercal smulaton of electrc vehcle chargng usng vehcle use data taken from a recent tral of EVs n the UK. In dong so, we show how the agent valuatons can be derved from real monetary costs to the vehcle owners, by consderng factors such as fuel prces, the dstance that the owner expects to travel, and the energy effcency of the vehcle. Experments conducted n ths realstc settng show that the mechansm wth on-departure burnng s hghly scalable (t can handle hundreds of agents), and both varants outperform any fxed prce mechansm for ths problem n terms of allocatve effcency, whle performng only slghtly worse than a well known schedulng heurstc, whch assumes non-strategc agents. Throughout ths paper, we focus on measurng allocatve effcency rather than seller proft, snce our man desgn goal s to assure that the capacty of the dstrbuton network s not exceeded, and that agents that need electrcty most are allocated, rather than on maxmzng profts. 2. RELATED WORK Onlne mechansm desgn s an mportant topc n the mult-agent and economcs lterature and there are several lnes of research n ths feld. One of these ams to develop onlne varants of Vckrey- Clarke-Groves (VCG) mechansms [13, 7]. Whle these frameworks are qute general, ther focus s on (a slght strengthenng of) Bayesan-Nash ncentve compatblty, whereas n ths paper we focus on the stronger concept of DSIC. Moreover, these works rely on a model of future avalablty, as well as future supply (e.g., Parkes and Sngh [13] use an MDP-type framework for predctng future arrvals), whle the mechansm proposed here s model-free. Such models requre fewer assumptons, and make computng allocatons more tractable than VCG-lke approaches. Model-free settngs are consdered by both Hajaghay et al. [8] and Porter [14], who study the problem of onlne schedulng of a sngle, re-usable resource over a fnte tme perod. They characterse truthful allocaton rules for ths settng and derve lower bound compettve ratos. A lmtaton of ths work [12, 8, 14] s that they consder sngle-valued domans and, as we show, these exstng approaches are no longer ncentve compatble for our settng where agents preferences are descrbed by a vector of values. Another related drecton of work concerns desgnng truthful mult-unt demand mechansms for statc settngs. A semnal result n ths area s the suffcent charactersaton of DSIC n terms of weak monotoncty (WMON) [1]. Although ths work s relevant to our model (we brefly dscuss the relatonshp between our mechansm and WMON n Secton 4.3), t does not propose any specfc mechansm, and, more mportantly, exstng results do not mmedately apply to onlne domans where agents arrve over tme and report ther arrval and departure tmes, as well as ther demand. A dfferent approach for dynamc problems s proposed by Juda and Parkes [9]. They consder a mechansm n whch agents are allocated optons (a rght to buy) for the goods, nstead of the goods themselves, and agents can choose whether or not to exercse the optons when they ext the market. The concept of optons would need to be modfed to our settng wth pershable goods, wth power allocated and then burned so that the fnal allocaton reflects only those optons that would be allocated. It s not clear how our onlne burnng mechansm maps to ther method. In addton to theoretcal results, several applcatons have been suggested for onlne mechansms, ncludng: the allocaton of W- F bandwdth at Starbucks [6], schedulng of jobs on a server [14] and the reservaton of dsplay space n onlne advertsng [3]. However, ths s the frst work that proposes an onlne mechansm for electrc vehcle chargng, and we show how our theoretcal framework naturally maps nto ths doman. 3. EV CHARGING MODEL In ths secton we present a model for our problem, formally defnng t as an onlne allocaton problem. (Supply) We consder a model wth dscrete and possbly nfnte tme steps (e.g., hourly slots) t T. At each tme step, a number of unts of electrcty are avalable for vehcle chargng as descrbed by the supply functon S : T N + 0,whereS(t) descrbes the number of unts avalable at tme t. Supply can vary over tme due to changes n electrcty demand for purposes other than vehcle chargng, as well as changeable supply from renewable energy sources, such as wnd and solar. Importantly, we assume that all vehcle batteres are charged at the same rate. 5 Thus, a unt of electrcty corresponds to the total energy consumed for chargng a sngle vehcle n a sngle tme step. Note that, whle there are multple unts of supply at each tme step (and agents have demand for multple unts), each agent can be allocated at most a sngle unt per tme step. These unts are allocated usng a perodc mult-unt aucton, one per tme step. Unts of electrcty are pershable, meanng that any unallocated unts at each tme step wll be lost. (Agents and Preferences) Each vehcle owner s represented by an agent. Let I = {1,...,n} denote the set of all agents. An agent s (true) avalablty for chargng s gven by ts arrval tme a T (.e., the earlest possble tme the vehcle can be plugged n), and departure tme d a,d T (.e., after whch the vehcle s needed by the owner). We wll sometmes use T = {a,...,d } to ndcate agent s avalablty and we say that agent s actve n the market durng ths perod. An agent has a postve value for unts allocated when the agent s actve, and has zero value for any unts allocated outsde of ts actve perod. Furthermore, agents have preferences whch determne ther value or utlty for a certan number of unts of electrcty. These preferences can change from one agent to another, and depend on factors such as the effcency of the vehcle, travel dstance, uncertanty n usage, battery capacty and local fuel prces. Formally, preferences are descrbed by a valuaton vector v = v,1,v,2,...,v,m,wherev,k denotes the margnal value for the k th unt and m s the maxmum demand from agent. That s, v,k =0for k>m. We wll often use v,k+1, whch descrbes the value for the next unt when an agent already has k unts of electrcty. Note that the agent s ndfferent 5 We beleve that our approach can be extended to address settngs wth varable charge rates, but leave ths for future work.

3 w.r.t. the precse allocaton tmes, and merely cares about the total number of unts receved over the entre actve perod. These components together descrbe agent s type θ = a,d, v. We let θ = {θ 1,...,θ n}, andθ s the types of all agents except. We wll often use the notaton (θ,θ ) =θ. We assume that agents have non-ncreasng margnal valuatons,.e., v,k 0 and v,k+1 v,k. As we wll show n Secton 5, ths assumpton s realstc n a settng wth plug-n hybrd and rangeextended EVs, where the more a vehcle battery s charged, the less t needs to rely on the fuel-consumng nternal combuston engne. (Reports and Mechansm) Importantly, we allow agents the opportunty to msreport ther types. Let ˆθ = {â, ˆd, ˆv } denote an agent s report. 6 Gven ths, a mechansm takes the agents reported (or observed) types as nput as they enter the system, and based on these reports determnes the allocaton of resources, as well as the payments to the agents. Our goal s then to desgn a mechansm whch ncentvses truthful reportng. The decson polcy then specfes an allocaton π t (ˆθ; k t ) at each tme pont t T and for each agent I, wherek t =(k t 1,...,k t n ) denotes the total endowments of the agents at tme t before the start of the aucton at tme t. That s: k t = t 1 π t t =â (ˆθ, ˆθ k t ). The polcy π s subject to the constrant that unts can only be allocated to agents wthn ther reported actvaton perod. In what follows, we wll use the abbrevated notaton π t (ˆθ), leavng any dependence on the current endowments mplct. Furthermore, let π (ˆθ, ˆθ ) = ˆd t=â π t (ˆθ, ˆθ ) denote the total number of unts allocated to agent n ts (reported) actve tme perod. We wll sometmes omt the arguments when ths s clear from the context. Furthermore, the payment polcy specfes a payment functon x (ˆθ, ˆθ π ) for each agent. Importantly, whle allocatons occur at each tme pont t T (snce unts are pershable), payments are calculated at the reported departure tme ˆd (.e., when the owner physcally unplugs the vehcle). (Lmted Msreports) As n [12], we assume that the agents cannot report an earler arrval, noralater departure. Formally, â a and ˆd d, and we say such a par â, ˆd s admssble. Ths s a vald assumpton n our doman because the agent s vehcle has to be physcally plugged nto the system, and ths cannot be done f the vehcle s not avalable. However, t can stll report an earler departure snce the vehcle can be unplugged before the vehcle s truly needed. Smlarly, t can delay ts effectve arrval (.e., after havng arrved, the vehcle owner can delay actually pluggng n the vehcle). (Agent Utlty) Gven ts preferences, an agent s utlty by the departure tme s gven by the valuaton for ts obtaned unts of electrcty, mnus the payments to the mechansm. Formally: u (ˆθ ; θ )= π (ˆθ,ˆθ ) k=1 v,k x (ˆθ, ˆθ π (ˆθ, ˆθ )) 6 In practce, reported arrval and departure correspond to tmes when the vehcle s physcally plugged nto, and, respectvely, unplugged from the network (whch could dffer from when the vehcle s truly avalable), whch can typcally be observed by the system. Ths s because we use a greedy-lke schedulng approach (see Secton 4) whch does not requre agents to report ther types, nor have knowledge of ther true types, n advance. Consequently, t s straghtforward to apply our approach to settngs where agents do not know ther exact avalablty or ths changes due to unexpected events. (1) agent 1 agent 2 agent 3 v 1 = 10, 4 v 2 = 5 v 3 = 2 t=1 t=2 Fgure 1: Example showng arrvals, departures, and valuaton vectors of 3 agents. 4. THE ONLINE MECHANISM In ths secton we consder the problem of desgnng a model-free mechansm for the above settng. Now, n the case of sngle-unt demand, a smple greedy mechansm wth an approprate payment polcy s DSIC [12]. However, we wll show through an example, that ths s no longer the case n a mult-unt demand settng that we consder. A greedy allocaton s formally defned as follows: DEFINITION 1 (GREEDY ALLOCATION). At each step t allocate the S(t) unts to the actve agents wth the hghest margnal valuatons, v t,k, where tes are broken randomly. +1 Consder the example wth 2 tme steps and 3 agents n Fgure 1, showng the agents arrval, departure and valuatons. Suppose furthermore that supply s S(t) =1at each tme step. Greedy would then allocate both unts to agent 1, becauseagent1 has the hghest margnal valuaton n both auctons. Now, consder the queston of fndng a payment scheme that makes greedy allocaton truthful. How much should agent 1 pay? To answer ths, note that the payment for the unt allocated at tme t =1has to be at least 5. Otherwse, f agent 1 were present n the market only at tme t =1and had a valuaton v 1,1 (5 ɛ, 5), t would not be truthful, because t could report ˆv 1,1 > 5 and stll wn. Smlarly, the payment for the unt allocated at tme t =2has to be at least 2. Thus, the mnmum payment of agent 1 f allocated 2 unts s x 1(ˆθ π 1 =2)=7. On the other hand, how much should agent 1 pay f t were allocated only 1 unt nstead? We argue no more than 2. If x 1(ˆθ π 1 = 1) = 2 + ɛ (where ɛ>0), then f the agent s frst margnal value was nstead v 1,1 (2, 2+ɛ), wth remanng margnal values zero, then t would wn n perod 2, but t would pay 2+ɛ and hence have negatve utlty. However, f x 1(ˆθ π 1 =2) 7 and x 1(ˆθ π 1 =1) 2, then agent 1 wants only 1 unt, not 2, as allocated by the greedy mechansm (ts utlty for one unt s greater than for two, as 10 2 > ). Hence, onlne greedy allocaton cannot be made truthful. 7 In order to address ths, n our mechansm we extend the Greedy decson polcy by allowng the system to occasonally burn unts of electrcty when necessary, n order to mantan ncentve compatblty. By burnng we mean that ths unt s not allocated to any agent, even when there s local demand. We consder two approaches: mmedate burnng, where the decsons to leave a unt unallocated s made at each tme step before chargng, and ondeparture burnng, where allocated unts can be reclamed by the system when the agent leaves the market (.e., the correspondng amount of electrcty s dscharged from the battery on departure). Each of these approaches has ther own advantages and dsadvantages. Burnng on departure generally requres burnng fewer unts n some cases, and thus t leads to a hgher effcency. Moreover, the current method we use to determne payments for mmedate burnng can have a computatonal cost exponental n the 7 Formally, ths s because the decson polcy volates a property called weak monotoncty [1]. In ths paper, we omt a detaled dscusson of ths relatonshp, due to space restrctons.

4 number of the agents present, whereas for on-departure burnng, the cost of determnng payments s lnear. However, n terms of the applcaton doman, fast dschargng of a vehcle s battery may not be practcal. Note that, for both approaches, the energy that s burnt s not necessarly wasted, but t s smply returned to the grd, to be used for other purposes than electrc vehcle chargng. For mmedate burnng, the unallocated electrcty unts are returned to the grd before t s actually charged by the agent. For the mechansm wth on-departure burnng, unts may be charged frst and then rapdly dscharged when the agent leaves the market. Whle ths may result n some loss, ths s probably neglgble w.r.t. the overall amount of electrcty allocated. 4.1 The Mechansm Before we ntroduce the decson polcy, we show how we can compute a set of threshold values, whch are used both to calculate the payments and to decde when to burn a unt of electrcty. Let k t,j = t 1 t =a j π t j (θ ) denote the endowment of an actve agent j at start tme t, under the allocaton we would have n the absence of agent (note that calculatng ths value requres recomputng allocatons wthout agent n the market from a untl the current tme t). Then v t j,k s the margnal valuaton of agent,j +1 j at tme t n the absence of agent. Gven ths, we defne v (n),t to be the n th hghest of such valuatons from all actve agents j. Then v (S(t)),t, for supply S(t), s the lowest value that s stll allocated a unt at tme t, f agent were not present. Henceforth, we refer to v (S(t)),t as the margnal clearng value for agent n perod t, and we wll often use v,t = v (S(t)),t for brevty. Now, let p t = ncr(v,a,v,a +1,...v,t) denote agent s prce vector at tme t, wherea s the reported arrval tme of agent and ncr(.) s an operator whch takes a vector of real values as nput and returns t n ncreasng order. In addton, let p = denote the value of ths vector at tme d when agent leaves the market. Intutvely, n any round t, the prce p t,k that agent s charged for the k-th unt s the mnmum valuaton the agent could report for the k-th unt and wn t by tme t, gven the greedy allocaton polcy wth burnng descrbed below. Gven ths, the decson and payment polces of our mechansm are as follows. p d Decson Polcy The decson conssts of two stages. Stage 1 At each tme pont t, pre-allocate usng Greedy (see Defnton 1). Stage 2 We consder two varatons n terms of when to decde to burn pre-allocated unts: Immedate Burnng. Burn a unt whenever: v t,k +1 <p t,k t +1 On-Departure Burnng. Ths type of burnng occurs on reported departure. For each departng agent, burn any unt k π where v,k <p,k. Payment Polcy Payment occurs on reported departure. Gven that π unts are allocated to agent at tme t = ˆd, the payment collected from s: x (ˆθ, ˆθ π )= π k=1 p,k (2) Burnng occurs whenever the margnal value for an addtonal unt s smaller than the margnal payment for that unt. Thus these values are effectvely agent-specfc threshold values, below whch no agent 1: agent 2: agent 3: T 1 = {1, 2, 3} T 2 = {1} T 3 = {2, 3} v 1 = 10, 4 v 2 = 5 v 3 = 2 t =1 k 1 1 =0 v 1,1 =5 k 1 2 =0 v 2,1 =10 p 1 1 = 5 π 1 1 =1 π 1 2 =0 p 1 2 = 10 t =2 k 2 1 =1 v 1,2 =2 k 2 v 3,2 =4 p 2 1 = 2, 5 p 2 3 = 4 π 2 1 =0(IM) π 2 π 2 1 =1(OD) 1 =1 k 3 t =3 k 3 IM v 1, v 3,3 =4 p 3 1 = 0, 2, 5 p 3 3 = 4, 4 π 3 1 =1 π 3 t =3 k 3 1 =2 k 3 OD v 1, v 3, p 3 1 = 0, 2, 5 p 3 3 = 0, 4 π 3 1 =0 π 3 3 =1 Table 1: Example run of the mechansm wth 3 agents and 3 tme perods for mmedate (IM) and on-departure (OD) burnng. Grey cells ndcate dfferent values for IM and OD burnng. unt s allocated to that agent. Moreover, t s mportant to note that the mechansm used for computng the prces mrrors the actual allocaton mechansm. So, for example, f mmedate burnng s used n the decson polcy, then for each agent and for all tmes t, the values of the p t vector are computed by re-runnng the market, n the absence of agent usng mmedate burnng, based on the reports of the other agents. Conversely, f on-departure burnng s used for the decson polcy, the same mechansm should be used n computng the p prces. 4.2 Example To demonstrate how the mechansm works, we extend the prevous example shown n Fgure 1 to nclude a thrd tme step, t =3. Both agents 1 and 3 reman n the market at t =3(.e., d 1 = d 3 =3) and no new agents arrve. Furthermore, S(t) =1n t {1, 2, 3}, and so there are now 3 unts to be allocated n total. Table 1 shows the endowments k t, the margnal clearng values v,t, the p t vectors, and the allocaton decsons π t at dfferent tme perods. We start by consderng the allocatons and payment usng mmedate burnng. At tme t =1, Stage 1 of the mechansm allocates the unt to agent 1, and snce v 1,1 =10 p 1 1,1 =5, ths unt s not burnt n the second stage. At tme t =2, the unt agan gets pre-allocated to agent 1 snce v 1,2 =4>v 3,1 =2. However, the margnal clearng value v 1,2 s nserted at the begnnng of the vector, and as a result v1,2 =4<p 2 1,2 =5. Consequently, ths unt gets burnt and s allocated to nether of the agents. At tme t =3, therefore, the margnal value of agent 1 s stll 4 (snce ts endowment s unchanged), and ths value s added to agent 3 s margnal clearng values. To calculate the margnal clearng value of agent 1, recall that the decson polcy needs to be recomputed wth agent 1 entrely removed from the market. In that case agent 3 would have been allocated a unt at tme t =2, and thus at tme t =3the margnal value of ths agent s 0. Thus, the value of 0 p 2 1 s nserted n the p 3 vector. At t =3, snce agent 1 stll has the hghest margnal value, t s agan pre-allocated the unt. However, now v 1,2 =4 p 3 1,2 =2, and therefore the unt s not burnt. So, n case of mmedate burnng, 2 out of 3 unts are allocated to agent

5 1, and that agent pays p 3 1,1 + p 3 1,2 =2. Now consder the same settng but wth on-departure burnng. The frst two tme steps are as before, except that there s no burnng at t = 2 (snce ths wll be done on departure f needed). Ths changes the endowment state of agent 1 at t =3, and therefore the margnal value of agent 1 at t =3s equal to v 1,.Therefore, the unt s allocated to agent 3, and the payment for ths unt s p 3,1 =0. The vector p 3 1 remans unchanged compared to the mmedate burnng case. At ths pont, there s no longer a need to burn one of the unts of agent 1, snce t has receved k =2 unts, the same allocaton as wth mmedate burnng, and note that v 1,2 >p 1,2. Stll, t s possble to construct examples where, both wth ondeparture and mmedate burnng, half of the unts need to be burnt. Furthermore, note that ths unt cannot go to agent 3, becausepayment would have been p 3 3,1 =4, whch would result n a negatve utlty for agent Propertes In ths secton we prove that the above mechansm s DSIC. We wll frst establsh DSIC wth respect to valuatons only, and prove truthful reportng of arrval and departure tmes separately. In more detal, we proceed n the followng 3 stages: () We defne the concept of a threshold polcy, and show that, when coupled wth an approprate payment functon, and gven any admssble par â, ˆd, f a decson polcy s a threshold polcy, then the mechansm s DSIC wth respect to the valuatons (Lemma 1). () We show that our decson polcy s a threshold polcy (Lemma 2). () Fnally, we show that, f agents truthfully report ther valuatons, reportng â = a, ˆd = d s a weakly domnant strategy (Lemma 3). These results are combned n Theorem 1 to show that our polcy s DSIC. DEFINITION 2 (THRESHOLD POLICY). A decson polcy π s a threshold polcy f, for a gven agent wth fxed â, ˆd and ˆθ, there exsts a margnally non-decreasng threshold vector τ, ndependent from the report ˆv made by agent, such that followng holds: k, ˆv : π (ˆθ, ˆθ ) k f and only f ˆv,k τ k. In other words, a threshold polcy has a (potentally dfferent) threshold τ k for each k, such that agent wll receve at least k unts f and only f ts (reported) valuaton for the k th tem s at least τ k. 8 Importantly, the vector τ has to be non-decreasng,.e., τ k+1 τ k, and should be ndependent of the reported valuaton vector ˆv. Note that both of these propertes are satsfed by the p vector, and we wll use ths to show that our mechansm s a threshold polcy. Frst, however, we show that a threshold polcy wth approprate payments s DSIC wth respect to the valuatons. LEMMA 1. Fxng admssble â, ˆd and ˆθ, f π s a threshold polcy coupled wth a payment polcy: x (ˆθ, ˆθ ) = π (ˆθ,ˆθ ) k=1 τ k, then f v s margnally non-ncreasng, reportng v truthfully s a weakly domnant strategy. 8 A threshold polcy satsfes weak-monotoncty (WMON) [1], and s therefore suffcent for truthfulness n ths doman snce we have bounded agent valuatons and the doman s completely ordered, meanng that all payoff types agree on the same weak preference orderng on all allocatons (.e., more s always weakly better than less), and ndfference to the way goods are allocated to other agents. We show that our decson polcy has the threshold property, and thus the WMON, and that t also handles msreports of arrvals and departures. PROOF. Agent s utlty can be rewrtten as: u (ˆθ ; θ )= π (ˆθ,ˆθ ) k=1 (v,k τ k ) Snce τ s ndependent of ˆv,agent can only potentally beneft by changng the allocaton, π (ˆθ, ˆθ ). Snce the values of τ k+1 τ k (non-decreasng threshold vector) and v,k+1 v,k (non-ncreasng margnal values), by defnton 2 we have v,k τ k 0 for any k π (θ ) and v,k τ k 0 for any k>π (θ ). Suppose that, by msreportng agent s allocated π (ˆθ ) >π (θ ), then u (ˆθ ; θ ) <u (θ ; θ ) snce: π (ˆθ,ˆθ ) k=π (θ,ˆθ )+1 (v,k τ k ) < 0 Smlarly, msreportng such that π (ˆθ, ˆθ ) <π (θ, ˆθ ) results n u (ˆθ ; θ ) <u (θ ; θ ) snce: π (θ,ˆθ ) k=π (ˆθ,ˆθ )+1 (v,k τ k ) 0 If msreportng has no effect on the allocaton, the utlty remans the same. Therefore, there s no ncentve for agent to msreport ts valuatons. Note that Greedy (as per Defnton 1) s not a threshold polcy. To see ths, consder the example from Fgure 1. As we saw earler, Greedy allocates 2 unts to agent 1, and the requred threshold τ 2 for wnnng the second unt s 2 (below whch Greedy would allocate 1 unt). However, f agent 1 had valuaton v 1 = 4, 4,Greedy would allocate only 1 unt, even though v 2 >τ 2, whch conflcts wth the requrement of a threshold polcy. The next lemma shows that the threshold condton holds f we nclude burnng, and f we set the threshold values to τ k = p,k. LEMMA 2. Gven non-ncreasng margnal valuatons, the decson polcy π n Secton 4.1 s (for ether burnng polcy) a threshold polcy where τ k = p,k. PROOF. Frst, from the defnton of vector p t and p from Secton 4.1, the values of p t are ndependent of the reports ˆv made by agent. Ths s because each of ts component values v,a,...v,t are computed based only on the reports of the other agents, by frst removng agent from the market. Second, we need to show two nequaltes, thus the proof s done n two parts. Part 1: Whenever v,k p,k, π allocates at least k unts to agent. Part 2: Whenever v,k <p,k, π allocates strctly less than k unts to agent. Part 1: Let v,k p,k. Suppose that agent has the same margnal values, v,k, for the frst k unts (.e., v,1 = v,2 =...= v,k ), then t wll wn exactly those auctons where v,k v,t, t T n Stage 1 of the mechansm (gnorng te breakng). Note that even when, by wnnng an aucton, agent dsplaces the losng margnal value to a future aucton, snce ths value s less or equal to v,k, t wll not affect the future auctons for agent snce t wll stll outbd that agent n the next aucton. Now, because p,j p,k for j k (by defnton), there must be at least k auctons where p,k v,t n the perod t T, and snce v,k p,k,agent wns at least k auctons n Stage 1. Furthermore, each tme an aucton s won, the clearng values appear as one of the j frst elements of the p t vector, where j s the number of auctons so far (snce these are the auctons wth the lowest clearng values, and the clearng values are ordered ascendngly). Because agent wns an aucton n Stage 1 f and only f v,k v,t, t follows that v,k = v,j p,j whenever t wns an aucton n Stage 1. Therefore, there s no burnng n Stage 2.

6 The above holds f agent has unform margnal values of v,k for the frst k unts. In fact, however, because of non-ncreasng valuatons, we have v,j v,k, for all 1 j k, and thus the decson polcy wll allocate at least k unts to agent. Part 2: Let v,k <p,k. Frst consder the on-departure burnng case. As per the defnton of Stage 2 of the mechansm, unt k s burnt. However, we stll need to show that any unts j > k are burnt as well. Snce p,j p,k and v,j v,k for all j>k, t follows that v,j <p,j for all j>k.thereforeeven f Stage 1 allocates k or more unts, these wll be burnt n Stage 2, and thus strctly less than k unts reman. Now consder the mmedate burnng case. Note that p,k p t,k for (a + k 1) t d. That s, threshold values can only decrease over tme. Thus t follows that v,k <p t,k for any (a + k 1) t d. Consder a case where, at tme t k, the k th unt s allocated n Stage 1. Because v,k <p t k,k, ths unt wll always be burnt n Stage 2 of the decson polcy. Therefore, the fnal result s an allocaton of strctly less than k unts. By settng τ k = p,k, the payment functon n Equaton 2 corresponds to the payment functon n Lemma 1. Therefore the proposed mechansm s shown to be DSIC n valuatons. We now complete the proof by showng that truthful reportng of the arrval and departure tmes are also DSIC (gven lmted msreports), gven truthful reportng of v. LEMMA 3. Gven lmted msreports, and assumng that truthfully reportng ˆv = v s a domnant strategy for any gven par of arrval/departure reports â, ˆd, then t s a domnant strategy to report â = a and ˆd = d. PROOF. Let p â, ˆd denote the vector of ncreasngly ordered margnal clearng values (computed wthout ), gven the agent reports ˆθ = â, ˆd, v. By reportng type ˆθ, the agent s allocated π (ˆθ ) tems, and ts total payment s: π (ˆθ ).Foreach j=1 p â, ˆd,j agent, msreportng from θ to ˆθ results n one of two cases: π (ˆθ )=π (θ ): Msreportng by agent has no affect on the margnal clearng values v,t, but can only decrease the sze of the p vector. In partcular, due to lmted msreports we have â a and ˆd d, and thus p â, ˆd contans a subset of the elements from p a,d. As these vectors are by defnton ncreasngly ordered, t follows that p â, ˆd,j p a,d,j, j ( ˆd â +1). Snce the payment conssts of the frst k = ˆk elements, ths can only ncrease by msreportng. π (ˆθ ) π (θ ): Frst, we show that π (ˆθ ) >π (θ ) could never occur. Snce the margnal clearng values reman the same, but the number of auctons n whch the agent partcpates decreases by msreportng, Stage 1 of the mechansm can only allocate fewer or equal tems. Furthermore, snce p â, ˆd,j p a,d,j, the possblty of burnng can only ncrease n Stage 2. Thus, t always holds that π (ˆθ ) π (θ ). Now, we consder the case π (ˆθ ) <π (θ ). Frst, as shown for the case π (ˆθ )=π (θ ) above, we know that π (ˆθ ) π (ˆθ ) j=1 p â, ˆd,k j=1 p a,d,j (.e., the payment for those unts won can only ncrease by msreportng arrval and/or departure). Furthermore, we know that the allocaton π (θ ) s preferable to any other allocaton π (ˆθ ) <π (θ ), otherwse reportng the true valuaton vector v would not be a domnant strategy. Snce the payment for these tems s potentally even hgher when msreportng, the agent cannot beneft by wnnng fewer tems. We are now ready to present the man theoretcal result: THEOREM 1. Gven non-ncreasng margnal valuatons and lmted msreports, Greedy wth on-departure and mmedate burnng and wth payment functon accordng to Equaton 2 are DSIC. PROOF. The proof of ths theorem follows drectly from the above lemmas. Lemmas 1 and 2 show that, for any par of arrval/departure (ms)-reports â ˆd the decson polcy s truthful n terms of the valuaton vector v, gven an approprate payment polcy. Furthermore, the payments n Equaton 2 correspond to those n Lemma 2, and therefore they truthfully mplement the mechansm. Fnally, Lemma 3 completes ths reasonng, by showng that for a truthful report of valuaton vector v, agents cannot beneft by msreportng arrvals/departures. 5. EXPERIMENTAL EVALUATION In ths secton, we evaluate our proposed mechansm emprcally. In dong so, we seek to answer a number of pertnent questons. Frst, snce our greedy approach does not generally fnd the optmal allocaton, we are nterested n how close t comes to ths n realstc settngs. Second, we nvestgate the extent to whch unt burnng occurs n practce (.e., how often unts of electrcty need to be burned by our decson polces, n order to ensure truthfulness). Ths s crtcal, as t may negatvely affect effcency. Fnally, we compare our mechansm to a range of smpler truthful mechansms that employ fxed prcng, as well as to a well-known onlne schedulng approach. These serve as benchmarks for our mechansm fxed prcng s a common mechansm for sellng goods n a wde range of settngs, whle the schedulng approach hghlghts what a non-truthful mechansm could acheve. 5.1 Expermental Setup Our expermental setup s based on data collected durng the frst large-scale UK tral of EVs. In December 2009, 25 EVs were provded to members of the publc as part of the CABLED (Coventry And Brmngham Low Emssons Demonstraton) project. 9 The am of ths tral was to nvestgate real-world usage patterns of EVs. To ths end, they were equpped wth GPS and data loggers to record comprehensve usage nformaton, such as trp duratons and dstances, home chargng patterns and energy consumpton. We use the data publshed by ths project for the frst quarter of 2010 to realstcally smulate typcal behavour patterns. More specfcally, n each of our experments, we smulate a sngle 24 hour day, where chargng perods are dvded nto hourly tme ntervals. For the purpose of the experments, a smulated day starts at 15:00, as vehcle owners begn to arrve back from work. To determne the arrval tme of each agent, we randomly draw samples from the home chargng start tmes reported by the project. These are hghest after 18:00 and then quckly drop off durng the nght. Lkewse, to smulate departures, we sample from data recordng journey start tmes. In order to smulate realstc margnal valuaton vectors for the agents, we combne data from the project about journey dstances wth a prncpled approach for calculatng the expected economc beneft of vehcle chargng. In partcular, we can calculate the expected utlty of a gven amount of charge (n kwh), c e,gvena prce of fuel (n /ltre), p p, an nternal combuston engne effcency (n mles/ltre), e p, an electrc effcency (n mles/kwh), e e, and a probablty densty functon, p(m), that descrbes the dstance to be drven the next day: E(u(c e)) = 0 9 See p p p p m p(m)dm m p(m)dm, (3) e p c e e e e p

7 where the frst term s the expected fuel cost wthout any charge, and the second term s the expected cost wth a battery charge of c e. Gven ths, and a chargng rate (n kw), r e, t s straght-forward to calculate the margnal valuaton of the kth hour of chargng tme: v k = E(u(k r e)) E(u((k 1) r e)). To generate a varety of margnal valuatons, we note that e e and e p depend on the specfc make and type of the EV and thus vary between households, whle p(m) depends on the drvng behavour of the car owner. We draw e e unformly at random from 2 4 mles/kwh and e p s drawn from 9 18 mles/ltre. Furthermore, we create p(m) from daly drvng dstances presented n the CABLED report. These dstances are typcally short, wth a daly mean of 23 mles, but the dstrbuton has a long tal wth a maxmum of 101 mles. Next, we draw the capacty of a car battery from kwh and set the chargng rate to 3 kw. These and earler specfcatons are all based on the Chevrolet Volt, the frst massproduced range-extended EV to be on the road n However, we nclude some varance to account for other vehcle types. Fnally, to derve the supply functon S, we consder a realstc neghbourhood-based supply functon usng the average energy consumpton of a UK household over tme. 10 In ths settng, the total energy avalable for chargng depends on the number of households n the neghbourhood and the constrants of the local transformer. Hence, avalable supply durng the nght s sgnfcantly hgher than durng the day. Furthermore, we tested a range of other supply functons and valuaton dstrbutons, where we observed the same general trends as dscussed n the remander of ths secton. However, we omt the detals here for brevty. 5.2 Benchmark Mechansms In addton to the two decson polces developed wthn ths paper Greedy wth Immedate Burnng (Immedate) and Greedy wth On-Departure Burnng (On-Departure) we benchmark the followng strateges that have been wdely appled n smlar settngs: Fxed Prce allocates unts to those agents that value them hgher than a fxed prce p. The prce they pay for ths unt s p. Whendemand s greater than supply, unts are allocated randomly between all agents wth a suffcently hgh valuaton. Ths mechansm s DSIC and so t consttutes a drect comparson to our mechansms. However, to optmse the performance of the fxed prce mechansm, p must be carefully chosen. Thus, we test all possble values (n steps of 0.01) and select the p that acheves the hghest average effcency (over 1000 trals) for a gven settng. Thus, when showng the results of Fxed Prce, ths consttutes an upper bound of what could be acheved wth ths mechansm. We use the specal case p =0as a baselne benchmark and denote ths as Random. Heurstc allocates unts such that a weghted combnaton of an agent s valuaton and urgency (proxmty to ts departure tme) s maxmsed. Here, an α [0, 1] parameter denotes the mportance of the urgency, such that α = 1 corresponds to the well-known earlest-deadlne-frst heurstc n schedulng, whle α = 0 ndcates that unts are always allocated to the agent wth the hghest valuaton. Ths s not a truthful mechansm and we do not mpose payments here, as ts prmary purpose s as a benchmark for our approach. Agan, we always select the best α. Optmal allocates unts to agents to maxmse the overall allocaton effcency, assumng complete knowledge of future arrvals and supply. Clearly, ths mechansm s not practcal and t s also not truthful (agan we mpose no payments), but t serves as an upper bound for the effcency that could be acheved. Havng descrbed the valuaton calculaton, the expermental settng, and the benchmarks, we now descrbe our results. 10 We use the average evaluated durng a work day n wnter, avalable at 100% 90% 80% 70% 60% 50% % 5% 0% Allocatve Effcency (% of Optmal) Heurstc On Departure Immedate Fxed Prce Random Utlty Per Agent (n ) Unts Burnt (% of Total Allocated) Immedate On Departure 100% 90% 80% 70% 60% 50% % 5% 0% (a) Allocatve Effcency (% of Optmal) Utlty Per Agent (n ) Unts Burnt (% of Total Allocated) (b) Fgure 2: Results for a small neghbourhood wth 30 houses (a) and a large one wth 200 houses (b). 5.3 Results For our experments, we consder two possble neghbourhood szes one wth 30 households and one wth 200 households. In these settngs, the capacty of the local transformer s constraned, so that only a couple of cars can charge at the same tme n the 30 household case and up to 16 wth 200 households. We choose such hghly constraned settngs here, because they are ntrnscally more challengng and nterestng than settngs where all cars can be fully charged overnght. Across the experments, we vary the number of these households that own an EV. Note here that we only show results for Immedate burnng up to 15 agents, because our current mplementaton of ths s computatonally expensve. Ths s because the vector of margnal clearng values p t at tme t depends on whch unts are burned n s absence (and as ths vector s used to determne when burnng takes place, t recursvely depends on the correspondng vectors of all agents that are allocated n s absence). Thus, we may potentally need to evaluate all subsets of agents, whch grows exponentally wth n. Although t may be possble to prune the search space effcently n practce, we leave these computatonal aspects to future work. It s nterestng that ths does not apply to On-Departure burnng, because here burnng does not nfluence the agents margnal clearng values. The results for both settngs are gven n Fgure 2. Frst, the top row shows the average 11 effcency, normalsed to the performance of Optmal (when there are more than 30 EV owners, Optmal becomes ntractable and so we normalse results to the performance of Heurstc n those cases as a close approxmaton). Here, we note that our two burnng polces consstently outperform (or match) all other truthful benchmarks. The mprovement compared to Random s partcularly pronounced, but our approach stll acheves a sgnfcant mprovement over the Fxed Prce mechansm. For small neghbourhoods, ths s almost 10%, whle n larger neghbour- 11 All results are averaged over 1000 trals. We plot 95% confdence ntervals, and sgnfcant dfferences reported are at t<0.05 level.

8 hoods, t s up to 5%. Ths s a promsng result, because settng the optmal prce for the fxed prce strategy requres knowledge about the dstrbutons of agents types, but our approach makes no such assumptons. Ths mprovement s due the ablty of our mechansm to allocate the agents wth the hghest margnal valuatons, whle Fxed Prce randomses over those that meet ts prce. Our approach s also responsve to changes n demand over tme, consstently allocatng unts even when the hghest valuatons are low. In contrast, Fxed Prce must be tuned to operate at any partcular balance of supply and demand. Thus, t does not allocate when ts prce s unmet. It performs better n the larger settng because t s more lkely that at least some of the agents meet the fxed prce n ths case. Next, our mechansm also performs close to the Optmal and Heurstc, consstently achevng 95% or better, whch ndcates that our greedy approach performs well n realstc settngs even wthout havng access to complete nformaton (such as departure tmes or even future arrvals). The lowest relatve effcency to the optmal s acheved when there are few EVs (about 20% of the neghbourhood). Here, schedulng constrants are most crtcal, as t may sometmes be optmal to prortse an agent wth lower valuatons over one wth hgher valuatons, but a longer deadlne. Ths becomes less crtcal when there are more agents, as there are typcally suffcently many wth hgh valuatons. Fnally, we see that Immedate burnng acheves a slghtly lower average effcency than On-Departure. Ths s due to hgher levels of burnng, but the dfference s small (and, n fact, not statstcally sgnfcant). In the second row of Fgure 2, the average utlty of each EV owner s allocaton (not ncludng the payments to the mechansm) s shown. Ths corresponds drectly to the fuel costs that a sngle EV owner saves by usng electrcty nstead of fuel. Intally, ths s hgh (around 2), as there s lttle competton, but starts droppng as more EV owners compete for the same amount of electrcty. Of key nterest here s the horzontal separaton between the dfferent mechansms. For a gven fuel savng per agent, our mechansm can sustan a sgnfcantly larger number of agents than the other ncentve-compatble mechansms. For example, to save at least 1 per agent n the small neghbourhood, Random can support up to 10 EV owners, whle Immedate and On-Departure acheve the same threshold for up to 14 EV owners (a 40% mprovement). In the large neghbourhood, our mechansm can support around 60 addtonal vehcles n some cases (to acheve a 0.65 threshold). Fnally, the last row shows the average number of unts that are burned by our two decson polces, as a percentage of the overall (tentatvely) allocated unts. Agan, due to computatonal lmtatons, full results for the Immedate burnng polcy are only shown up to 15 agents. For up to 18 agents, results from only 100 trals are shown (resultng n larger confdence ntervals). On-Departure burnng clearly burns sgnfcantly fewer unts than Immedate, as the latter sometmes unnecessarly burns unts. There s also a clear maxmum n the number of burned unts when around 20% of households are EV owners. Ths s because there s a sgnfcant amount of competton, wth many agents that have smlar margnal valuatons, and ths nduces burnng. However, when the number of agents rses further, burnng drops agan. Ths s because agents are ncreasngly less lkely to be allocated more than a sngle unt n these very compettve settngs and so there s no need for burnng. It should be noted that burnng s generally low (for On-Departure burnng), wth typcally only 1-2% of allocated unts beng burned (and always less than 10%). 6. CONCLUSIONS Ths paper proposes a novel onlne allocaton mechansm for a problem that s of great practcal nterest for the smart grd communty, that of ntegratng EVs nto the electrcty grd. Our contrbuton to exstng lterature s two-fold. On the theoretcal sde, we extend model-free, onlne mechansm desgn wth pershable goods to handle mult-unt demand wth decreasng margnal valuatons. On the practcal sde, we emprcally evaluate our mechansm n a real-world settng, and showed that the proposed mechansm s hghly robust, and acheves better allocatve effcency than any fxed-prce benchmark, whle only beng slghtly suboptmal w.r.t. an establshed cooperatve schedulng heurstc. For future work we plan to look at several ssues. Frst, n ths paper we assumed all EVs have a unform chargng rate, but n the future we plan to extend the allocaton model to deal wth heterogeneous maxmal chargng rates (correspondng to dfferent types of EVs). Second, t would be nterestng to compare the performance of the model-free onlne mechansm proposed n ths paper to a model-based approach, such as the one n [13]. Fnally, ths paper looked at performance n terms of a realstc applcaton scenaro, but we also plan to study the worst-case bounds on allocatve effcency and number of tems our mechansm burns n future work. 7. REFERENCES [1] S. Bkhchandan, S. Chatterj, R. Lav, A. Mu alem, N. Nsan, and A. Sen. Weak monotoncty characterzes determnstc domnant-strategy mplementaton. Econometrca, 74(4): , [2] K. Clement-Nyns, E. Haesen, and J. Dresen. The mpact of chargng plug-n hybrd electrc vehcles on a resdental dstrbuton grd. IEEE Transactons on Power Systems, 25(1): ,2010. [3] F. Constantn, J. Feldman, S. Muthukrshnan, and M. Pal. Anonlne mechansm for ad slot reservatons wth cancellatons. In Proc. ACM-SIAM Symposum on Dscrete Algorthms (SODA 09), pages , [4] Department of Energy and Clmate Change. The UK low carbon transton plan: Natonal strategy for clmate and energy. HM Government, [5] P. Farley. Speed bumps ahead for electrc-vehcle chargng. IEEE Spectrum, Jan.2010.Avalableat [6] E. Fredman and D.C. Parkes. Prcng WF at Starbucks Issues n onlne mechansm desgn. In Proc. of the 4th ACM Conf. on Electronc Commerce, pages ,2003. [7] A. Gershkov and B. Moldovanu. Effcent sequental assgnment wth ncomplete nformaton. Games and Economc Behavor, 68(1): , [8] M. Hajaghay, R. Klenberg, M. Mahdan, and D.C. Parkes. Onlne auctons wth re-usable goods. In Proc. of the 6th ACM Conf. on Electronc Commerce (EC 05),pages ,2005. [9] A.I. Juda and D.C. Parkes. An optons-based soluton to the sequental aucton problem. Artfcal Intellgence, 173: , [10] W.J. Mtchell, C.E. Borron-Brd, and L.D. Burns. Renventng the automoble: Personal urban moblty for the 21st century. MIT Press, [11] Royal Academy of Engneerng. Electrc Vehcles: Charged wth potental. Royal Academy of Engneerng, [12] D.C. Parkes. Onlne mechansms. In N. Nsan, T. Roughgarden, E. Tardos, and V. Vazran, edtors, Algorthmc Game Theory, pages , [13] D.C. Parkes and S. Sngh. An MDP-Based approach to Onlne Mechansm Desgn. In Proc. of NIPS 03,2003. [14] R. Porter. Mechansm desgn for onlne real-tme schedulng. In Proc. the 5th ACM Conf. on Electronc Commerce (EC 04),pages 61 70, [15] S. Vandael, N. Boucke, T. Holvoet, and G. Deconnck. Decentralzed demand sde management of plug-n hybrd vehcles n a smart grd. In Proc. of 1st Int. Workshop on Agent Technologes for Energy Systems, pages67 74,2010.