An Online Mechanism for Multi-Speed Electric Vehicle Charging

Transcription

1 An Onlne Mechansm for Mult-Speed Electrc Vehcle Chargng The Harvard communty has made ths artcle openly avalable. Please share how ths access benefts you. Your story matters. Ctaton Publshed Verson Accessed Ctable Lnk Terms of Use Valentn, Robu, Sebastan Sten, Enrco H. Gerdng, Davd C. Parkes, Alex Rogers and Ncholas R. Jennngs. Forthcomng. An onlne mechansm for mult-speed electrc vehcle chargng. In Auctons, Market Mechansms and Ther Applcatons. Lecture Notes of the Insttute for Computer Scences, Socal-Informatcs and Telecommuncatons Engneerng: Second Internatonal ICST Conference, AMMA 2011, New York, NY, USA, August 22-23, 2011, Revsed Selected Papers. Berln; New York; Hedel: Sprnger Verlag. do: / January 7, :53:24 PM EST Ths artcle was downloaded from Harvard Unversty's DASH repostory, and s made avalable under the terms and condtons applcable to Open Access Polcy Artcles, as set forth at (Artcle begns on next page)

2 An Onlne Mechansm for Mult-Speed Electrc Vehcle Chargng Valentn Robu 1, Sebastan Sten 1, Enrco H. Gerdng 1, Davd C. Parkes 2, Alex Rogers 1, and Ncholas R. Jennngs 1 1 Unversty of Southampton, SO17 1BJ, Southampton, UK {vr2,ss2,eg,acr,nrj}@ecs.soton.ac.uk 2 Harvard Unversty, Cambrdge, MA 02138, USA parkes@eecs.harvard.edu Abstract. As plug-n electrc vehcles become more wdespread, ther chargng needs to be coordnated, n order to ensure that capacty constrants are not exceeded. Ths s becomng partcularly crtcal as new fast-chargng technologes are beng developed that place addtonal burden on local transformers. To address ths problem, we propose a novel onlne mechansm n whch agents representng vehcle owners are ncentvsed to be truthful not only about ther margnal valuatons for electrcty unts, but also about ther arrval, departure and maxmum chargng speeds. The work extends the state of the art n several ways. We develop an onlne, model-free mechansm that handles mult-unt demand per perod, thus accommodatng vehcles wth heterogeneous and flexble chargng speeds; we provde compettve worst-case bounds for our mechansm; fnally, we smulate the proposed onlne mechansm usng data from a real-world tral of electrc vehcles n the UK, showng that usng fast chargng leads to sgnfcant cost savngs. 1 Introducton Recent advances n battery technology, and pressures to reduce the carbon emssons of transport, have stmulated renewed nterest n electrc vehcles (EVs). New hybrd desgns, equpped wth both an electrc motor and an nternal combuston engne (that can be used to drve or charge the battery), address common worres about the lmted range of such vehcles, and EVs are expected to represent close to 10% of all vehcle sales by 2020, accordng to a recent Gartner report [7]. However, ths potental growth has generated concerns that f many of these vehcles are plugged n, and charged smultaneously, they rsk overloadng local electrcty dstrbuton networks (Shao et al[10]). To address ths problem, a number of researchers have begun to nvestgate mechansms to schedule the chargng of EVs, such that the local constrants of the dstrbuton network are not exceeded. For example, Clement, Haesen & Dresen [1] propose a centralsed scheduler whch makes optmal use of the network capacty when vehcle owners truthfully report ther expected future vehcle use to the system. However, such approaches fal to address the fact that owners wll lkely msreport ths nformaton f t s n ther nterest to do so (for example, reportng that they requre ther vehcle earler than s actually the case to receve preferental chargng). To ths end, n ths paper we use onlne mechansm desgn, n order to engneer payment mechansms that provde

3 2 Robu, V., Sten, S., Gerdng, E.H,, Parkes, D.C., Rogers, A., Jennngs, N.R. ncentves for these owners to report truthfully ther value for recevng electrcty, ther wllngness to wat and ther maxmum chargng rate. Onlne mechansm desgn s an mportant topc n dstrbuted AI and economcs. For example, Parkes and Sngh [8] propose an onlne varant of the Vckrey-Clarke- Groves (VCG) mechansm, whch uses an MDP-type framework to predct future arrvals. Unlke ther work, the mechansm proposed here s model-free (.e., assumes no knowledge about the future). Model-free onlne settngs have been consdered by Porter [9] and Hajaghay et al. [5], who study the problem of onlne schedulng of a sngle, reusable resource over a fnte tme perod. We extend ths work by consderng mult-unt domans, wth preferences descrbed by a non-ncreasng vector of values. A dfferent approach for dynamc problems s consdered by Juda and Parkes [6]. They consder a mechansm n whch agents are allocated optons (a rght to buy) for the goods, nstead of the goods themselves. The concept of optons s promsng, but would need modfcatons to apply to our settng wth pershable electrcty unts. In addton to theoretcal results, several applcatons have been suggested for onlne mechansms, ncludng: the allocaton of W-F bandwdth (Fredman & Parkes [3]), schedulng of jobs on a server (Porter [9]) and the reservaton of dsplay space n onlne advertsng (Constantn et al. [2]). In recent work (Gerdng et al. [4]), we propose the frst onlne mechansm to deal wth the problem of coordnatng the chargng of a set of plug-n hybrd electrc vehcles (PHEVs) under lmted supply. Ths earler model, however, has several lmtatons that we address n ths work. Frst, all vehcles partcpatng n the system are assumed to have the same chargng speed. In fact, gven the large numbers of competng manufacturers enterng ths space, t s lkely that domestc EV chargers wth a wde range of maxmum chargng speeds wll become avalable. 1 Hgh performance chargers may pose addtonal burdens on the local transmsson network and, for a real-world deployment, the allocaton model and market desgn needs to be able to deal wth such a challenge. Moreover, the presence of multple, asymmetrc chargng speeds may consderably affect the dynamcs of such a market, as they enable dfferent allocatons of the lmted network capacty to become feasble. Fnally, no theoretcal worst-case bound for the mechansm was presented, whereas such a guarantee s hghly desrable, especally as onlne allocaton leaves some tems unallocated to guarantee truthfulness. Aganst ths background, ths paper makes the followng contrbutons to the state of the art: We develop a novel onlne mechansm that deals wth mult-unt demands per tme step, by extendng the mechansm proposed by Gerdng et al. [4] to accommodate heterogeneous and flexble chargng speeds. We provde compettve bounds on allocatve effcency compared to the optmal offlne allocaton (whch assumes pror knowledge of future arrvals). We smulate the proposed mechansm usng data from a real-world tral of electrc vehcles n the UK. We show that the use of fast chargers can lead to sgnfcant savngs n fuel consumpton, and s benefcal both from the perspectve of ndvdual vehcle owners, and for the allocatve effcency of the whole market. 1 provdes an example of a domestc charger that already offers a 2.5 tmes speed ncrease over standard chargers.

4 An Onlne Mechansm for Mult-Speed Electrc Vehcle Chargng 3 2 Mult-Speed EV Chargng Model We consder a settng n whch multple unts of electrcty are perodcally sold at fxed tme steps, t (e.g., once every hour). A unt of electrcty s defned as the amount of kwh when chargng at the lowest rate durng that nterval. Importantly, we assume that the chargng rate can be flexble, and each vehcle has a maxmum chargng rate (whch depends on the battery and ts charger). We assume unts to be ndvsble, whch means that the chargng rate s a multple of the lowest rate (ths s not a lmtaton snce unts can be defned to be arbtrarly small). Moreover, we denote by S(t) the supply of electrcty avalable for EV chargng,.e. the number of unts to be sold at tme t. Note that, snce our allocaton s essentally greedy (.e. unts are allocated just before they are they are charged and there s no pre-allocaton), ths allows us to dstrbute electrcty comng from uncertan sources such as a shared renewable generator (e.g. a shared neghbourhood wnd turbne). For now, we consder a market for electrcty for EV chargng that s separate from that for household consumpton, and so S(t) can be consdered the resdual supply once household consumpton has been removed. Beyond provdng a manageable model, a practcal reason for ths separaton of concern s to protect one neghbour from hgher electrcty prces for runnng regular jobs (lghtnng, domestc applances) n the case that hs neghbours have purchased EVs. Let I = {1,2,...,N} denote the set of agents, each of whch operates on behalf of a sngle EV and ts owner. Vehcles come and go and are not always avalable for chargng. Furthermore, they can have dfferent maxmum chargng rates, and ther owners have dfferent valuatons for the electrcty. Gven ths, an agent s type s descrbed by the tuple θ = v,a,d,r, where v s the margnal valuaton vector, a and d are the arrval and departure tmes (the earlest and latest tmes that the EV s avalable for chargng) and r s the maxmum chargng rate (.e., the maxmum number of unts agent can charge at any tme t). Each element v,k of v represents the agent s wllngness to pay for the k th unt of electrcty. We assume non-ncreasng margnal valuatons,.e., v,k v,k+1. Ths s a realstc assumpton for PHEVs snce the frst few unts of electrcty are always more lkely to be used [4]. Furthermore, a and d defne when the agent s present n the market, where a s the agent s arrval or earlest tme the vehcle can be charged, and d s the pont of departure, after whch the vehcle s requred by the owner. Gven ths, a mechansm asks the agents to report ther types and decdes on an approprate allocaton and payment. We denote the reported type by ˆθ = ˆv,â, ˆd, ˆr. In practce, the arrval report â s the tme at whch the owner plugs a vehcle nto the electrcty network. At the tme of arrval, an agent s also requred to report ts margnal valuaton vector ˆv. The departure report ˆd s not requred n advance and t smply represents the tme when the vehcle s unplugged from ts unt. A vehcle owner may decde to change ts (reported) departure tme, by smply unpluggng her vehcle. Agents (or ther owners) can msreport ther avalablty, for example, by unpluggng the vehcle early or pluggng n the vehcle some tme after arrval to try and get a better prce. Also, they could report a lower maxmum chargng rate or hgher valuatons. Our am s to develop a mechansm whch s domnant strategy ncentve compatble (DSIC),.e., agents are best off reportng ˆθ = θ, no matter what other agents report. Formally, a mechansm s gven by the allocaton polcy π t (ˆθ I k t ), I, whch determnes the number of unts allocated to agent at tme t, and payment polcy x (ˆθ k ), I, whch calculates the total payment on (reported) departure of an agent. We denote by

5 4 Robu, V., Sten, S., Gerdng, E.H,, Parkes, D.C., Rogers, A., Jennngs, N.R. θ I = {θ I} the types of all agents, and θ = {θ j j I,j } all agents except. Payments are ndependent of an agent s own report.here, k s agent s endowment (number of tems allocated) on departure, and k t = k t 1,...,k t N denotes the endowment of all agents at tme t, where k t+1 3 The Onlne Mechansm = k t + π t (ˆθ I k t ). Essentally, our mechansm uses a greedy allocaton polcy, whch allocates avalable unts to the agents wth the current hghest margnal valuatons (gven ther current endowments). Whle ths s not always optmal (snce t does not consder the fact that some agents leave earler than others), t s generally applcable, as t does not requre nformaton about future arrvals or departures (.e., t s model free ). A key problem, dentfed n our prevous work [4], s that greedy allocaton n onlne domans sometmes overallocates (gven the prces t has to pay, the agent would prefer to get fewer unts than t would be allocated). To address ths problem, we correct the allocaton by leavng some unts unallocated under well specfed condtons, whch we refer to as burnng unts. We consder two types of burnng, dscussed below: mmedate burnng, where unts are smply left unallocated (.e., none of the agents receve the unt, even f there s a demand for them), and on-departure, where unts are ntally allocated usng the greedy approach (.e., the battery s beng charged), but then on departure of the agent, any overallocated unts are dscharged from the battery. Both of these have advantages/dsadvantages. The model wth on-departure burnng s more effcent (.e., generally burns fewer unts) and s also more computatonally effcent n computng the payments. However, t may not be realstc to expect that we can partally dscharge a car s battery on the departure, so the model wth mmedate burnng may be more realstc, gven the applcaton. Formally, let the vector b t = ˆv,k t +1,...,ˆv,k t +ˆr denote agent s reported at tme t. It s convenent to thnk of ths as the agent s bds for the unts avalable at tme t. Furthermore, let B t denote the multset of such bds from all agents that are present n the market at tme t,.e., from all I s.t. â t ˆd. The allocaton rule s: Defnton 1 (Greedy Allocaton). At each tme step t, allocate the S(t) unts of electrcty to the hghest bds n B t. To ensure that the payment s report-ndependent, the mechansm reruns the market wthout agent (from â onwards, snce agent dd not affect the market pror to ths margnal values for the next ˆr unts, gven ts endowment k t tme). Let B t denote the multset of the bds placed by all agents n the market at tme t f agent were removed and the market were rerun from â t onwards. In case B t < S(t), we add a number of zero-valued bds and refer to ths enlarged set as B t 0, to ensure that B t 0 S(t). Next, we defne set operators max k B and mn k B to return the hghest and, respectvely, lowest k elements of multset B (or, f B < k, to return B). Then, we defne the externalty that agent would mpose on other agents f t won mn(r,s(t)) out of S(t) unts at tme t as: = mn(max r S(t) B t 0 ) here contans the bds from other agents that would lose out E t Intutvely, the multset E t f agent were to wn r unts at tme t. Note that the ntuton here s the same as n the

6 An Onlne Mechansm for Mult-Speed Electrc Vehcle Chargng 5 agent 1 agent 2 agent 3 v 1= 10,8,3 v 2= 7 v 3= 1 S(t1)=2 S(t )=1 2 Fg. 1. Example showng arrvals, departures, and valuaton vectors of 2 tme unts and 3 agents. regular Vckrey-Clarke-Groves (VCG) mechansm, as the total payment corresponds to the sum of the externaltes. However, to compute the overall payments onlne, we need to combne these externaltes across all tme steps n the agent s actve perod up to current tme t. To do ths, we defne an ordered vector of prces, p t, as follows: p t = ncr ( t t =a E t where ncr s an operator that orders elements from a multset n ncreasng order, and we use the unon symbol to denote the unon of multsets (and so the same element can appear multple tmes). Now, p t,k s the prce that agent s charged for the k-th unt of electrcty. Intutvely, ths s the mnmum valuaton that the agent could report for wnnng ths unt by tme t. Note we use p,k to denote ths prce at tme ˆd. Gven ths, the mechansm s as follows. Decson Polcy The decson conssts of two stages. Stage 1 At each tme pont t, pre-allocate usng Greedy Allocaton (see Defnton 1). Stage 2 We consder two varatons of when to decde to burn pre-allocated unts: Immedate Burnng. Leave any unt unallocated whenever the prce for ths unt s greater than the margnal value,.e., whenever: ), ˆv,k < p t,k for k t < k k t + π t On-Departure Burnng. For each departng agent, dscharge any unt k k where ˆv,k < p,k. Payment Polcy Payment always occurs on reported departure (.e., when the owner/agent unplugs the vehcle). Gven that k unts are allocated to agent, the payment collected from s: x (ˆθ k ) = k k=1 p,k (1) Fgure 1 llustrates the mechansm through an example wth 2 tme steps and 3 agents, showng the agents arrval, departure and valuatons. Suppose furthermore, that supply s S(t 1 ) = 2 and S(t 2 ) = 1. Now, consder 2 dstnct cases: The maxmum chargng speed of agent 1 s r 1 = 1. 2 In ths case, at most one margnal value s taken from each agent. At tme t 1, margnal valuatons v 1,1 = 10 of agent 1, and v 2,1 = 7 of agent 2 are allocated, whle at tme t 2, margnal value v 1,2 = 8 of agent 1 s allocated. The prces charged to agent 1 are: p 1 = 0,1, because wthout agent 1 n the market, there would be a free, spare unt at tme t 1 and the avalable unt at t 2 would sell to agent 3 for 1. No unts get burnt n ths case, and the actual allocaton s actually equvalent to the optmal offlne allocaton. 2 Note that the other 2 agents only desre one unt, so ther maxmum chargng speed s rrelevant n ths example.

7 6 Robu, V., Sten, S., Gerdng, E.H,, Parkes, D.C., Rogers, A., Jennngs, N.R. The maxmum chargng speed of agent 1 s r 1 = 2. Then, at tme t 1, the greedy mechansm descrbed above allocates the 2 margnal values of agent 1: v 1,2 = 10 and v 1,2 = 8, as they are both hgher than v 2,1 = 7, and agent 2 drops out of the market. At tme t 2, the unt s agan allocated to agent 1 (due to the margnal value of 3 beng hgher than 1). However, now the margnal payments vector requred from agent 1 s p 1 = 0,1,7, whle the margnal valuatons are v 1 = 10,8,3. Gven the prces, agent 1 prefers 2 unts to 3 (because > ), so the thrd s burnt. The overall market effcency s lower, as the thrd avalable unt s now burnt, whereas wth r 1 = 1 t was allocated to agent 2. Note, however, that even though the effcency s much lower, agent 1 has an ncentve to declare ts true maxmum chargng speed r 1 = 2 as, n ths case, ts payment does not change. 4 Truthfulness Propertes In ths secton, we dscuss the ncentve compatblty propertes of our mechansm under the followng assumpton: Lmted Msreports: Agents cannot report an earler arrval, a later departure, or a hgher chargng rate,.e., â a, ˆd d, ˆr r must hold. In our doman, ths s a reasonable assumpton snce a vehcle owner cannot physcally plug n her vehcle before t s avalable or unplug t after the actual departure. Note that the assumpton ˆr r s natural for EV chargng. Whle most electrc batteres can be confgured to charge at a slower rate, chargng them at a faster rate than the one allowed by the manufacturer mght destroy them. As the battery s an ntegral and expensve part of an EV, ths by tself acts as a natural deterrent. An EV agent may strategse by reportng a ˆr lower than ts true maxmal speed, but we show truthful reportng of r s a domnant strategy. Gven ths, we can state the followng: Theorem 1. Assumng lmted msreports, and gven the decson and payment polcy as descrbed above, reportng ˆθ = θ s a domnant strategy for I. Proof Sketch:The proof for ths theorem has 3 parts: 1. Showng that, after fxng any tuple of â, ˆd, ˆr, reportng the valuaton vector ˆv = v truthfully s domnant strategy ncentve compatble (DSIC). 2. Showng that for any fxed ˆv, ˆr (and under the lmted msreports assumpton â a, ˆd d ), reportng truthfully â = a, ˆd = d s DSIC. 3. Showng that for any fxed ˆv,â, ˆd, usng the lmted msreports assumpton ˆr r, reportng truthfully the maxmum chargng rate ˆr = r s DSIC. The frst two parts can be shown by usng the same proof technques as n [4]. For the thrd part, we can show that the vector of margnal payments p ˆr (computed gven the report ˆr ) wll always contan a subset of the elements from the vector p r (computed gven the truthful report r ), because ˆr r. Thus, ether an agent gets the same allocaton, but pays less by reportng a hgher speed (because the p vectors are ncreasngly ordered, and the payment s the sum of the frst k unts allocated), or the agent s allocated more unts but then those addtonal unts are needed, gven the prces (otherwse, burnng would apply). 5 Theoretcal Bounds on Allocatve Effcency An mportant queston gven the onlne allocaton wth burnng proposed above s how the allocatve effcency compares to that of an optmal offlne allocaton (.e., assumng

8 An Onlne Mechansm for Mult-Speed Electrc Vehcle Chargng 7 full knowledge of the future). To answer ths, we consder the cases wth mmedate burnng and on-departure burnng separately. For the case of mmedate burnng, we can show that no lower bound can be establshed through an example: Example 1: Let an agent A syn wth margnal valuaton vector v syn = v 1,v 2,...,v n, where margnal valuatons are strctly decreasng,.e., v 1 > v 2 >... > v n. Assume supply s S(t) = 1 for all t, and at each tme step, one other local agent s present for that tme step only and desres a sngle unt. The valuaton of the frst local bdder s v 1 = v 1 ǫ, the valuatons of the next two are v 2,3 = v 2 ǫ, the next three v 3,4,5 = v 3 ǫ, and so on, where ǫ s an arbtrarly small quantty. In other words, each margnal value v k ǫ appears exactly k tmes. In ths example, agent A syn s pre-allocated every unt, but for each valuaton v k, the frst k 1 pre-allocated unts are burnt, wth only the last unt beng allocated (due to the expandng p t vector). Thus, n the lmt, the fracton of unts burnt goes to 1, whle the effcency goes to 0. For on-departure burnng, Theorem 2 provdes the followng worst-case guarantee: Theorem 2. The mechansm wth on-departure burnng s 2-compettve wth the optmal offlne allocaton, for a settng wth non-ncreasng margnal values. Proof Sketch: In order to establsh a compettve bound wth the optmal offlne allocaton, we use a chargng argument smlar to Hajaghay et al. [5]. 3 The basc dea s to charge (or match) all the margnal values of each agent that are allocated n the offlne case wth one or more values allocated onlne. Specfcally, let v,k off denote the k-th margnal value of agent that s allocated n the offlne case. In the followng we say that a unt s satsfed (or allocated) n the onlne case f t s actually allocated to agent by the greedy mechansm and not burnt on departure. Consder each margnal value v,k off from agents that are allocated n the offlne case, and to charge the values as follows: 1. Margnal values allocated both n the onlne and offlne case are charged to themselves. 2. Margnal values that are allocated n the offlne case, but not n the onlne case wll always be dsplaced by some hgher value. Note that, snce these values could be allocated onlne and are hgher, they must necessarly also be allocated n the offlne case (but ths may occur at a dfferent tme than n the onlne case, when they would dsplace a lower valued unt). To complete the chargng argument, we need to show what happens to these unts w.r.t. burnng. vj,p off Lemma 1. A margnal utlty value v,k off that s allocated to agent both offlne and onlne cannot be burnt on the departure of agent n the onlne market. Moreover, f a unt v,k off s dsplaced by another unt voff j,p, the dsplacng unt voff j,p cannot be burnt on departure of agent j. Proof Sketch: The proof for ths lemma reles on comparng the vectors of decreasng margnal values v and the payment vector p as well as the condton for burnng specfed by the on-departure burnng decson polcy. In summary, one can show by contradcton that, f a unt v,k off s allocated both offlne and onlne, there could not be enough hgher valued margnal unts n agent s actve nterval [a,d ] for ths unt (or a hgher valued one) to be burnt. Gven Lemma 1, for all agents, each margnal value unt that s allocated offlne can be charged at most twce: 3 Here the term chargng does not refer to electrcty chargng, but represents the name of a proof devce used n onlne MD.

9 8 Robu, V., Sten, S., Gerdng, E.H,, Parkes, D.C., Rogers, A., Jennngs, N.R. - At most once to tself (f t s also allocated onlne). - At most once by another unt allocated offlne, wth a value less than ts own, that t dsplaced at some tme t when that unt was allocated n the offlne case (recall that n the mechansm wth on-departure burnng, each unt v,j can be the wnnng bd at most once, so t can dsplace at most one other unt). Thus each unt allocated offlne s charged at most twce, gvng the onlne allocaton wth on-departure burnng and nonncreasng margnal values a compettve bound of 2 wth the optmal offlne allocaton. Theorem 2 shows the mult-unt demand case wth on-departure burnng s no worse n terms of worst-case compettve bound than the case wth sngle unt demand (c.f. Hajaghay et al. [5]) whch may seem surprsng, gven the burnng. However, the only unts allocated onlne whch could be burnt onlne on departure of an agent are unts of such low value that would not be allocated n the correspondng offlne case. 6 Expermental Evaluaton In ths secton, we apply our mechansm to a range of settngs that are based on real data collected durng the frst large-scale tral of EVs n the UK. Our experments examne two man questons. Frst, we study how our onlne allocaton mechansm performs compared to a range of benchmarks as demand for electrcty ncreases n a neghbourhood wth lmted supply. Second, we look at how the gradual ntroducton of fastchargng EVs would affect the neghbourhood, both n terms of socal welfare (.e., overall fuel savngs) and the fnancal savngs of ndvduals. 6.1 Expermental Setup We base our experments on data gathered by the CABLED (Coventry And Brmngham Low Emssons Demonstraton) project, 4 whch s the frst large-scale endeavour n the UK to record and study the drvng and chargng behavours of EV owners. The arrval and departure tmes of vehcle owners are generated based on ths data. We also construct the agents valuaton vectors usng the expected travel dstances reported by the project, as well as the fuel and electrc effcences and battery capactes of typcal plug-n hybrd EVs, such as the Chevrolet Volt or Toyota Plug-In Prus. Such hybrd electrc vehcles have a dual engne, and a per mle drvng effcency, whch can be expressed ether n unts of electrcty or ltres of fuel (assumng a standard drvng speed). As such, an agent s margnal valuaton of a unt of electrcty corresponds to the savngs n fuel costs that the agent expects to make due to consumng ths unt nstead of regular fuel (based on a fuel prce of 1.3/ltre). For each confguraton, 500 or 1000 runs were performed, wth the expected drvng dstance for each vehcle for the next day beng and ts entry pont nto the market beng randomzed n each run. For full detals, see Gerdng et al. [4], where we use the same setup and dataset. For each experment, we consder a sngle day and, to calculate the capacty constrants (the supply functon S(t)), we dvde the day nto 24 hourly tme steps. For each tme step, we frst obtan the overall neghbourhood consumpton based on real data. 5 We then consder two possble scenaros. Frst, when supply s low, electrcty s hghly constraned, and we set the capacty lmt to 90% of the peak overall consumpton,.e., about 0.9 kw per household. Ths scenaro represents neghbourhoods where the local 4 See 5 We use the average data for domestc households n June 2010, as reported by SCE (http: //

10 An Onlne Mechansm for Mult-Speed Electrc Vehcle Chargng Hgh Supply Low Supply Number of 3 kwh Unts Avalable for EV Chargng 0 15:00 18:00 21:00 00:00 03:00 06:00 09:00 12:00 Tme Fg. 2. Unts of electrcty avalable for EV chargng. transformer can only just support domestc demand wth no spare capacty for vehcle chargng durng peak hours. Second, for hgh supply wth sgnfcant spare capacty, we set the lmt to 150% of peak consumpton,.e., approxmately 1.5 kw per household. In addton to our mechansm wth On-Departure and Immedate burnng, we consder a set of benchmark mechansms. (): Random allocates unts randomly to agents that have some postve margnal valuaton for them. (): Fxed Prce allocates unts at random to agents that have valuatons greater than a fxed prce and collects a payment equal to ths fxed prce. The prce s chosen a pror to optmse the expected socal welfare (.e., total fuel savngs), gven full knowledge of the agent type dstrbutons. (v): Heurstc s a common earlest-deadlne-frst schedulng heurstc that allocates unts to agents to maxmse a weghted sum of the agent s valuaton for the unt and ts deadlne. The weghts are chosen as for Fxed Prce. (v): Optmal allocates unts to optmse the overall socal welfare, assumng full knowledge of all future arrvals. Note that apart from the Random baselne, these benchmarks assume addtonal knowledge ether about agent type dstrbutons or even future arrvals, and so they should be seen as upper bounds for our approach rather than actual realstc alternatves. Furthermore, only Random and Fxed Prce are truthful (DSIC) lke our mechansm. 6.2 Result 1: General Trends wth Increasng Demand Frst we consder the effect of rsng electrcty demand wthn a neghbourhood. To ths end, Fgure 3 shows the performance of our mechansm n a neghbourhood of 25 households wth ncreasng numbers of EVs. 6 These results are for the low supply settng only, as the varaton of EV numbers also changes the supply/demand balance. We assume that electrcty s allocated n unts of 3 kwh (the chargng rate of a standard UK power socket), and that maxmum rates are chosen at random from {1,2,3} unts, correspondng to the rates of currently avalable fast chargers for domestc use. Due to the computatonal complexty of the Optmal and Immedate mechansms, we only plot those up to 30 and 15 agents, respectvely. 7 6 Note that as ths exceeds 25, some households wll have multple cars. We show ths to ndcate the effect of very hgh demand and assume there s no colluson wthn a household. 7 Brefly, Immedate s computatonally hard, because the prce vector p t has to be computed at every tme step by smulatng the market wthout agent, and, recursvely, for every agent wthn that market, as t s needed to compute when burnng takes place.

11 10 Robu, V., Sten, S., Gerdng, E.H,, Parkes, D.C., Rogers, A., Jennngs, N.R Fuel Savngs per Agent (n ) Optmal Heurstc On Departure Immedate Fxed Prce Random 100% 90% 80% Allocatve Effcency (% of Optmal) 1 70% % Number of EVs Number of EVs Fg. 3. Effect of ncreasng demand for electrcty n a small 25-household neghbourhood wth low supply. Frst, the graph on the left shows the daly fuel savng (excludng payments) for EV owners under the dfferent mechansms (all results reported are averaged over at least 500 samples and plotted wth 95% confdence ntervals). Ths shows that there s lttle dfference between the varous mechansms when competton s low, as most agents are allocated; but as demand ncreases, our mechansm starts to outperform the other truthful benchmarks (by almost 50%) and acheves 95% or more of the optmal throughout. Interestngly, our mechansm sustans hgher savngs per agent for neghbourhoods wth far hgher EV penetraton than the truthful benchmarks. For example, to guarantee a savng of at least 0.7 (just over half a ltre of fuel), the Random benchmark can support up to 35 EVs, whle our mechansm can cope wth 50. The graph on the rght shows the overall allocatve effcency acheved n the same settng, normalsed to the optmal (beyond 30 EVs, ths s normalsed to the Heurstc as a close approxmaton). Our prncpal fndng s that onlne allocaton wth On-Departure burnng acheves almost the same performance as the Optmal, calculated usng full pror nformaton about departures (the dfference s not statstcally sgnfcant). Moreover, the dfference between our onlne allocaton wth Immedate burnng and wth On-Departure burnng s also not sgnfcant. Surprsng, gven the dfferent theoretcal performance bounds of these polces, and ndcates that burnng s not a large problem n realstc settngs. 6.3 Result 2: Prolferaton of Fast-Chargng EVs We now consder n more detal how the presence of fast-chargng vehcles affects market performance. To ths end, we examne a larger neghbourhood of 50 households wth one EV each. We choose ths sze, because t s realstc and leads to more nterestng results, due to ncreasng lkelhood of competton at each tme step. Due to ther computatonal cost, we no longer run the Optmal and Immedate mechansms (but ther performance s smlar to the Heurstc and On-Departure, respectvely, as shown prevously). To nvestgate the mpact of fast-chargng, we assume there are two agent types the frst, normal, can charge a sngle unt of 3 kwh per tme step, whle the second, fast, are equpped wth fast chargers that can charge up to three such unts per tme step. Throughout the experments, we vary the number of fast-chargng EVs (out of the total 50). Results for ths settng are shown n Fgure 4. Frst, we note that the trends for the two scenaros are dfferent when supply s low, the ntroducton of more fast chargng

12 An Onlne Mechansm for Mult-Speed Electrc Vehcle Chargng 11 Socal Welfare / Fuel Savngs (n ) Number of Fast Chargng EVs Socal Welfare / Fuel Savngs (n ) Heurstc On Departure Fxed Prce Random Number of Fast Chargng EVs 0.5 Utlty Per Agent (n ) Utlty Per Agent (n ) On Departure (Fast) On Departure (Normal) Number of Fast Chargng EVs Number of Fast Chargng EVs 3% 2% Unts Burnt (% of Total Allocated) 3% 2% Unts Burnt (% of Total Allocated) 1% On Departure 1% 0% 0% Number of Fast Chargng EVs Number of Fast Chargng EVs Fg. 4. Socal welfare, ndvdual utltes and unts burnt n scenaros wth low supply (left) and hgh supply (rght), as more fast-chargng EVs are ntroduced. vehcles results only n a small overall mprovement n socal welfare. However, when t s hgh, there s a very marked mprovement, wth an addtonal ltres of fuel per day beng saved by our mechansm as more fast-chargers are present (correspondng to a savng of around 5000 ltres of fuel per year over the whole neghbourhood). Ths s because, at low demand tmes, there s some spare capacty that remans unallocated, unless vehcles have a hgh chargng speed. Wth respect to the utlty of ndvdual agents (ncludng payments to the mechansm), we see that agents n both settngs always have an ncentve to swtch to fastchargng EVs (e.g., by purchasng a domestc fast charger). Wth low supply, the expected daly savng when swtchng to a fast-chargng EV s approxmately 0.05, whle wth hgh supply, ths s around In both cases, ths beneft s the result of ncreasng avalable supply per tme step, as well as ncreasng the sze of the prce vector. Fnally, another beneft of ntroducng fast-chargng vehcles s the reduced proporton of unts of electrcty that are burnt by our mechansm. In the low supply settng, the percentage of burnt unts s more than halved to less than 1% of all unts allocated

13 12 Robu, V., Sten, S., Gerdng, E.H,, Parkes, D.C., Rogers, A., Jennngs, N.R. (on average 1.5 kwh) when all cars are fast-chargng. Wth hgh supply, ths s even more sgnfcant as only about a thrd of unts are burnt. Ths reducton s a result of reduced competton and a larger prce vector. 7 Conclusons and Further Work Ths work makes several key contrbutons to the exstng lterature. On the theoretcal sde, we extend the mult-unt, onlne mechansm proposed n Gerdng et al. [4] to also allow for modelng mult-unt demand per tme step, n conjuncton wth multunt demand over tme. For our EV applcaton, ths allows us to model more realstc markets, whch nclude vehcles wth heterogeneous chargng capabltes. Moreover, we provde the frst theoretcal worst-case bounds for mult-unt onlne mechansms. On the practcal sde, we buld a detaled smulaton of an EV chargng market and show that our onlne mechansm performs close to the offlne optmal n a varety of realstc settngs. We also show that faster-chargng batteres lead to savngs n fuel consumpton and ncreased allocatve effcency. In future work, we plan to compare the model-free mechansm presented n ths paper wth onlne mechansms that use a model of expected future arrvals. Moreover, we are nterested n studyng whether there are truthful mechansms for ths problem that do not requre monetary payments. Acknowledgements Ths work was carred out as part of the IDEaS project ( funded by a company actve wthn the energy doman. References 1. K. Clement, E. Haesen, and J. Dresen. Coordnated chargng of multple plug-n hybrd electrc vehcles n resdental dstrbuton grds. In Proc. of the IEEE/PES Power Systems Conference and Exposton, pages 1 7, F. Constantn, J. Feldman, S. Muthukrshnan, and M. Pal. An onlne mechansm for ad slot reservatons wth cancellatons. In Proc. SODA 09, pages , E. Fredman and D. C. Parkes. Prcng WF at Starbucks Issues n onlne mechansm desgn. In 4th ACM Conference on Electronc Commerce, pages , E. H. Gerdng, V. Robu, S. Sten, D. C. Parkes, A. Rogers, and N. R. Jennngs. Onlne mechansm desgn for electrc vehcle chargng. In Proc. of the 10th Internetonal Conference Autonomous Agents and Mult-Agent Systems (AAMAS 11), M. Hajaghay, R. Klenberg, M. Mahdan, and D. C. Parkes. Onlne auctons wth re-usable goods. In 6th ACM Conference on Electronc Commerce (EC 05), pages , A.I. Juda and D. C. Parkes. An optons-based soluton to the sequental aucton problem. Artfcal Intellgence, 173: , T. Koslowsk. The electrc vehcle s value chan and technology evoluton. Techncal report, Gartner Inc., D. C. Parkes and S. Sngh. An MDP-Based approach to Onlne Mechansm Desgn. In Proc. of NIPS 03, pages , R. Porter. Mechansm desgn for onlne real-tme schedulng. In Proc. 5th ACM Conference on Electronc Commerce (EC 04), pages 61 70, S. Shao, M. Ppattanasomporn, and S. Rahman. Challenges of PHEV penetraton to the resdental dstrbuton network. In IEEE Energy Socety General Meetng, 2009.