Optimal Decentralized Protocol for Electric Vehicle Charging

Transcription

1 Optimal Decetralized Protocol for Electric Vehicle Chargig Ligwe Ga Ufuk Topcu Steve Low Abstract Motivated by the power-grid-side challeges i the itegratio of electric vehicles, we propose a decetralized protocol for egotiatig day-ahead chargig schedules for electric vehicles. The overall goal is to shift the load due to electric vehicles to fill the overight electricity demad valley. I each iteratio of the proposed protocol, electric vehicles choose their ow chargig profiles for the followig day accordig to the price profile broadcast by the utility, ad the utility updates the price profile to guide their behavior. This protocol is guarateed to coverge, irrespective of the specificatios e.g., maximum chargig rate ad deadlie) of electric vehicles. At covergece, the l orm of the aggregated demad is miimized, ad the aggregated demad profile is as flat as it ca possibly be. The proposed protocol eeds o coordiatio amog the electric vehicles, hece requires low commuicatio ad computatio capability. Simulatio results demostrate covergece to optimal collectios of chargig profiles withi few iteratios. I. INTRODUCTION Electric vehicles EVs) offer great potetial for icreasig eergy efficiecy, reducig greehouse gas emissios, ad relievig reliace o foreig oil for trasportatio []. Curretly, several types of EVs are either already i the U.S. market, or about to eter [], ad electrificatio of trasportatio is at the forefrot of may research ad developmet agedas [3]. O the other had, the potetial comes with a multitude of challeges icludig those i the itegratio with the electric power grid. EV chargig icreases the electric loads, ad potetially amplifies the peak demad or creates ew peaks [4]. It also icreases the demad side ucertaities, ad potetially reduces the distributio circuit ad trasformer lifespa [5]. Moreover, power losses ad voltage variatios become more likely [6]. The simulatio-based study i [7] suggests that, if o regulatio o EV chargig is implemeted, eve a 0% peetratio of EVs may cause uacceptable variatios i the voltage profiles. O the other had, may studies demostrate that adoptig smart chargig strategies ca mitigate some of the itegratio challeges, defer ifrastructure ivestmet eeded otherwise, ad eve stabilize the grid. For example, schedulig EV chargig so that aggregated EV load fills the overight demad valley may reduce daily cyclig of power plats ad operatioal cost of the electricity utility [8]. Furthermore, the eergy stored i EVs may be utilized as a alterative acillary service resource [9] for regulatig L. Ga is with the Departmet of Electrical Egieerig, Califoria Istitute of Techology, lga@caltech.edu. U. Topcu is with Cotrol ad Dyamical Systems, Califoria Istitute of Techology, utopcu@cds.caltech.edu. S. Low is with the Departmet of Computer Sciece ad the Departmet of Electrical Egieerig, Califoria Istitute of Techology, slow@caltech.edu. voltage profiles, ride-through support for fault protectio, ad eve compesatig fluctuatig reewable eergy geeratio [0]. Studies o EV chargig cotrol fall ito three categories: effect of time-of-use rates [], coordiated chargig schedulig [6], [0], [], ad decetralized schedulig [3] [4]. Referece [] explores the effect of higher price durig peak hours o shiftig EV load, but does ot provide strategies for settig the price. Referece [6], [0], [] study cetralized cotrol strategies that miimize power losses, load variace, or maximize load factor, allowable EV peetratio level. These strategies require a cetralized structure to collect all iformatio of all EVs ad optimize over the chargig profiles of all EVs, hece icur prohibitive commuicatio ad computatio requiremets. Referece [3] demostrates, through a simulatio-based study, that EV load ca be smoothed by itroducig load-side participatio ito the power market. They propose a decetralized strategy, but do ot provide ay aalytical optimality guaratees. Referece [4] proposes aother decetralized chargig strategy, ad proves its optimality i the case where all EVs plug i at the same time with the same state-of-charge SOC), ad have the same deadlie ad allowable chargig rates. For future referece, we call this settig homogeeous. I this paper, we partially alleviate some of the restrictios imposed i [4]. First, we defie optimal chargig profiles of EVs explicitly this defiitio geeralizes the implicit defiitio i [4]). Secod, we propose a decetralized chargig strategy that guaratees optimality i both homogeeous ad o-homogeeous cases, where EVs ca plug-i at differet times with differet SOC, have differet maximum chargig speed ad deadlies. Third, we remove the artificial trackig error pealty i EV owers objective i [4] by itroducig aother pealty term that vaishes at covergece. The rest of the paper is orgaized as follows. Sectio II formulates EV chargig cotrol as a global optimizatio problem. Sectio III explores properties of optimal chargig profiles, proposes a decetralized algorithm ad proves its covergece to optimal chargig profiles. Numerical simulatios are used to illustrate these results i sectio IV, ad coclusios ad future works are preseted i sectio V. II. PROBLEM FORMULATION We cosider a sceario where a electric utility egotiates with N electric vehicles idexed,..., N) for a day-ahead chargig schedule over T time slots idexed,..., T ), each of legth T. Let Dt) deote the aggregated o-ev demad) at time t, r t) deote the chargig rate of EV at time t, for all t {,..., T }.

2 TABLE I LIST OF SYMBOLS T time horizo t time, t {,..., T } N umber of EVs EV, {,..., N} Dt) at time t. D ) profile. r t) chargig rate of EV at time t r ) chargig profile of EV r ) chargig profile of all EVs R r ) aggregated chargig profile correspodig to r r t) chargig rate lower boud for EV at time t r t) chargig rate upper boud for EV at time t R chargig rate sum of EV D set of feasible chargig profiles for EV D set of feasible chargig profiles for all EVs S set of optimal chargig profiles if a, b R, a b a i b i for all i {,..., }. Let r := r ),..., r T )) deote the chargig profile of vehicle, for all {,..., N}. The, roughly speakig, a chargig profile r := r ),..., r N )) is valley-fillig, if it miimizes Dt) + r t)). A more rigorous discussio o valley-fillig will be give i sectio III-A. The overall goal is to achieve a valley-fillig chargig profile, subject to the followig costraits. For EV, due to battery specificatio, its chargig rate should be withi some iterval [r, r ]. Takig deadlie ito accout, we let r ad r be time-depedet ad set r t) = r t) = 0 for t outside the allowable chargig period of EV. Hece r t) r t) r t) for all {,..., N} ad all t {,..., T }. Let B, s 0), ad η deote the battery capacity, iitial SOC ad chargig efficiecy of EV. By its deadlie, EV should be fully charged, this is captured by the total eergy stored over the whole time horizo η T r t) T = B s 0)). Defie R := B s 0)) /η / T, it is equivalet to r t) = R. for all {,..., N}. Referece [5] summarizes some of the recetly aouced EV models, ad typical values of r ad R ca be derived from these models. r is usually set to be 0. Defiitio : Let D R T be the. A chargig profile r = r,..., r N ), where r R T for all {,..., N}, is optimal, or valley-fillig, if it solves miimize Dt) + r t)) ) r,...,r N subject to r r r, =,..., N, r t) = R, =,..., N, where r, r R T ad R R for all {,..., N}. Defiitio : A chargig profile is feasible if it satisfies the costraits i ). III. MAIN RESULTS I this sectio, we first explore properties of optimal chargig profiles, ad the propose a decetralized algorithm for solvig ). I the ed, we prove that the algorithm we propose coverges to the set of optimal chargig profiles. Give a chargig profile r := r,..., r N ), let R r := deote the aggregated chargig profile correspodig to r. A. Optimal Chargig Profiles Property : Let r be a feasible chargig profile. { If there exists A R, such that R r t) = N } max r t), A Dt) for all t {,..., T }, the r is optimal. Proof: Note that R r is the uique solutio to the problem miimize R subject to r Dt) + Rt)) ) Rt) = R r. For ay r feasible for ), R r is feasible for ). Furthermore, the objective fuctio i ) evaluated at R r is equal to the objective fuctio i ) evaluated at r. Hece, the optimal value d of ) is a lower boud for the optimal value p of ). The aggregated profile R r attais d, so r attais d p{. Sice r is feasible, it is optimal. Let D := r r r r, } T r t) = R deote the set of feasible chargig profiles of EV. The D := D D N is the set of feasible chargig profiles of all EVs. Property : If the set D of feasible chargig profiles is o-empty, the optimal chargig profiles exist. Proof: D is the feasible set of ) by Defiitio. Sice D is compact for each, D as a product of N compact sets is also compact. Furthermore, D is o-empty, ad the objective fuctio of ) is cotiuous. Hece, its optimal value is attaied at some r D. R

3 ormalized aggregated demad kw) ormalized aggregated demad kw) flat valley fillig o valley fillig 0:00 0:00 4:00 8:00 :00 6:00 o flat valley fillig o valley fillig 0:00 0:00 4:00 8:00 :00 6:00 Fig.. Base demad curve is the average residetial load i the service area of SCE from 0:00 o Feb. 3th to 9:00 o Feb 4th, 0 [6]. Valleyfillig curve correspods to the outcome of Algorithm ODC, ad ca be flat or o-flat. A hypothetical o-valley-fillig curve is show with the marker o. Property is directly related to our ituitive otio of valley-fillig. As show i Figure top), a valley-fillig chargig profile has the same aggregated demad durig periods i which some EVs charge 0:00-7:00). Ad the aggregated demad i this period is o higher tha that outside this period. We call such aggregated demad profile completely flat. However, uder certai coditios, complete flatess may ot be possible. For example, the valley may be so deep at time t that eve if every vehicle charges at its maximum rate, the valley may still ot be completely filled, e.g., at 4:00 i Figure bottom). Moreover, vehicles may have such striget deadlie costraits that the flexibility of schedulig is limited, ad complete flatess caot be achieved. Defiitio relaxes these restrictios as a result of Property. I cases where complete flatess is possible, Property guaratees that our costructive defiitio of valley-fillig agrees with complete flatess. I cases where complete flatess is ot possible, Property guaratees existece of a optimal chargig profile. Furthermore, as show i Figure bottom), a valley-fillig chargig profile ideed gives a smoother aggregated demad, compared to o-valley-fillig chargig profiles. We ow establish a equivalece relatio betwee chargig profiles, ad use it to describe a importat property of the set of optimal chargig profiles. Defiitio 3: Feasible chargig profiles r ad r are equivalet, deoted as r r, if R r = R r. That is, r ad r are equivalet if their aggregated chargig profiles are the same. It is easy to check that this is ideed a equivalece relatio. Now, we ca defie equivalece classes based o this equivalece relatio. Give D, defie S := {r D r optimal w.r.t D} as the set of optimal chargig profiles. Theorem : If feasible chargig profiles exist, the the set S of optimal chargig profiles is o-empty, compact, covex, ad a equivalece class. Proof: Property implies that S is oempty whe feasible chargig profiles exist. Let r be a optimal chargig profile, the r S. Defie S := {r D r r}. It is easy to show that S is closed ad covex. Sice S D, which is compact, S is also compact. Hece S is o-empty, compact, covex, ad a equivalece class by costructio). It is easy to see that S S. I order to prove S = S, we ow show that S S. For all r S, from the first order optimality coditio for ), D + R r, R r R r = 0, D + R r, R r R r = 0. Reverse R r R r i the first equatio ad subtract the secod oe to obtai R r R r, R r R r = 0. Hece R r = R r, r r, r S, S S. Cosequetly, S = S is o-empty, compact, covex, ad a equivalece class. Corollary : Optimal chargig profile is geerally ot uique. Corollary is a direct cosequece of the fact that S is a equivalece class. I geeral, two EVs ca exchage their chargig profiles i the opposite directio without affectig the objective value. For istace, EV raises its chargig rate at time t by r ad decreases its chargig rate at t by r, while ot violatig the costraits i ). Coversely, EV decreases its chargig rate at time t by r ad icreases its chargig rate at t by r, while ot violatig the costraits i ). If the origial chargig profile is optimal, the ew profile, which is equivalet to the origial oe, is also optimal, due to Theorem. B. Algorithm ODC We develop a decetralized algorithm to solve ). System diagram of the algorithm is show i Figure. Give broadcast electricity price profile, each EV makes a idepedet decisio ad chooses its chargig profile. The utility guides their behavior by settig the prices. Algorithm ODC optimal decetralized chargig): Give T, N, K, Dt), R, r t), r t), for all {,..., N} ad all t {,..., T }, defie α := N/.

4 p k ) r ) k K TABLE II LIST OF SYMBOLS IN ALGORITHM price profile i iteratio k chargig profile of vehicle i iteratio k maximum umber of iteratios Fig.. Schematic view of the iformatio flows i Algorithm ODC. Give predicted price profile, vehicles make idepedet decisios ad choose their ow optimal chargig profiles. The utility guides their behavior by settig the price based o the aggregated demad. ) Iitializatio: p 0 ) := D ), r 0 ) := 0, k 0. ) Utility broadcasts p k ). 3) EV calculates a ew chargig profile r k+ ) as argmi r ) p k t)r t) + α r t) rt) k ) subject to r ) r ) r ), r t) = R. 4) Utility collects ew chargig profiles of all EVs, ad updates the price accordig to p k+ ) := D ) + r k+ ). 3) 5) k k +, go to ) util k = K or covergece. I iteratio k +, the algorithm ca be split ito two parts. I the first part, EV chooses a chargig profile that miimizes its objective fuctio C r ) = p k t)r t) + α r t) rt) k ). 4) The first term is the electricity cost ad the secod term pealizes deviatios from the chargig profile computed i the previous iteratio. This extra pealty term esures covergece of Algorithm ODC. I sectio III-C, we prove Scalig factors used to cacel the uits are omitted for brevity. that r k+ t) r k t) 0 for all ad all t as k. Hece, the objective fuctio of each EV boils dow to its electricity cost at covergece. I the secod part of iteratio k +, the utility updates the price accordig to 3). It sets high prices for the periods with high aggregated demad. Ituitively, i the ext iteratio, vehicles are give the icetive to shift their chargig profiles. As a result, aggregated demad may be smootheed. As proved i sectio III-C, Algorithm ODC coverges to optimal chargig profiles. Furthermore, simulatio results demostrate fast covergece usually withi a few iteratios) of Algorithm ODC. C. Aalysis of Algorithm ODC We ow aalyze the covergece properties of Algorithm ODC. Sice optimal chargig profile is geerally ot uique, we eed to establish a appropriate otio of covergece to optimal chargig profiles. Let superscript k deote the respective value of i iteratio k of Algorithm ODC. For example, r k deotes the chargig profile of vehicle i iteratio k. Similarly, R k := N rk deotes the aggregated chargig profile i iteratio k, d k := D + R k )/N deotes the ormalized aggregated demad profile i iteratio k. Lemma : If feasible chargig profiles exist, the the iequality d k + r k+ r k d k r k+ r k 5) holds for all {,..., N} ad all k. Proof: Sice feasible chargig profiles exist, feasible chargig profiles for EV exist, ad r k D for all {,..., N} ad all k. Further, r k+ = argmi r D = argmi r D = argmi r D = argmi r D p k t)r t) + α r t) rt) k ) d k t)r t) + r t) rt) k ) d k t) + r t) rt) k ) d k + r r k. 6) From the first order optimality coditio for 6), d k + r k+ r k, r r k+ 0 7) for all r D. Choose r = r k to obtai d k + r k+ r k, r k r k+ 0. Split the first term to obtai Hece, d k, r k+ r k r k+ d k + r k+ = d k + r k+ d k r k+ r k r k. r k + dk, r k+ r k r k

5 for all {,..., N} ad all k. Lemma : If feasible chargig profiles exist, the for all {,..., N} ad all k, r k+ = r k if ad oly if d k, r r k 0 8) for all r D. Proof: Follows from 7) ad the strict covexity of 6). Theorem : Let r k := r, k..., rn k ) be the chargig profile i iteratio k of Algorithm ODC. If feasible chargig profiles exist, the r k S as k. Proof: The proof is based o a Lyapuov-type argumet. Defie Lr) := T Dt) + R rt)), the Lr k+ ) = Dt) + R k+ t) ) T = N d k t) + N T N N = N N d k + r k+ r k+ t) r k t) )) d k t) + r k+ t) r k t) ) r k d k r k+ ) r k ) 9) N d k = Lrk ). 0) The first iequality is due to Jesse s iequality, ad the secod iequality is due to Lemma. It is easy to show that Lr k+ ) = Lr k ) if ad oly if r k+ = r k for all, i.e., r k+ = r k. If r k+ = r k, it follows from Lemma that for all, d k, r r k 0 for all r D. Hece, D + R k, R r R k = N d k, r r k 0 ) for all r = r,..., r N ) D. This is the first order optimality coditio for ), hece r k S. O the other had, if r k S, Lr k ) Lr k+ ), hece Lr k+ ) = Lr k ). It follows that Lr k+ ) = Lr k ) r k+ = r k r k S. Now we have D is compact; S miimizes LD); Lr k+ ) < Lr k ) if r k / S. It follows that r k S as k. Corollary : A chargig profile r is a equilibrium poit of Algorithm ODC if ad oly if r S, the set of optimal chargig profiles. Proof: Sice r k+ = r k r k S showed i the proof of Theorem ), the set of equilibrium poits E := {r k r k+ = r k } = S. Theorem 3: Let r be a optimal chargig profile, p k be the price profile i iteratio k of Algorithm ODC. If feasible chargig profiles exist, the aggregated chargig profile coverges to that of r, i.e., lim k Rk = R r ; price profile coverges to that of r, i.e., lim k pk = D + R r ; for each EV, the chage i chargig profile betwee cosecutive iteratios of Algorithm ODC coverges to 0, i.e., lim k r k+ r k = 0 for all {,..., N}. Proof: Sice the set S of optimal chargig profiles is a equivalece class, for all r S, R r = R r. Let r k be the chargig profile i iteratio k. Theorem implies that we ca fid a sequece i S, such that the distace betwee r k ad the k th elemet of the sequece coverges to 0 as k. It follows that R k R r, as k. The price profile coverges as a direct cosequece of 3). Iequality 9) implies Lr k ) Lr k+ ) N r k+ r k. Sice Lr k ) coverges, Lr k ) Lr k+ ) 0. Hece r k+ r k 0, for all, as k.. Theorem 3 implies that i Algorithm ODC, the aggregated chargig profile ad price profile coverge to that of a optimal chargig profile. Furthermore, chage i the chargig profile for each EV betwee cosecutive iteratios vaishes as k. Hece, the objective fuctio 4) reduces to the electricity cost after certai umber of iteratios. Theorem 4: Algorithm ODC is a gradiet projectio algorithm for miimizig Lr). That is, r k+ = argmi r r k γ L ) r D r for all {,..., N} ad all k 0, where γ = /N). Proof: Sice γ L r r k ) = γ D + R k) = d k, Theorem 4 directly follows from 6). IV. SIMULATIONS We compare the covergece properties ad optimality of Algorithm ODC with the decetralized schedulig algorithm proposed i [4], i homogeeous ad o-homogeeous cases. For otatioal brevity, we call the algorithm i [4] DAP, stadig for Deviatio from Average Pealty. Recall that the optimality of DAP is oly guarateed i the homogeeous case. We choose the average residetial load profile i the service area of South Califoria Ediso from 0:00 o Feb. 3th 0 to 9:00 o Feb. 4th 0 as the ormalized profile. Accordig to the typical chargig characteristics of EVs i [5], we set r t) = 3.3 kw if EV ca be charged at time t, ad 0 kw otherwise. We assume chargig rate ca chage cotiuously i [0, r t)], hece choose r t) = 0 for all t. We set R = 0, 5, 40 for three differet types seda, compact,

6 ormalized aggregated demad kw) ODC DAP ormalized aggregated demad kw) ODC DAP 0:00 0:00 4:00 8:00 :00 6:00 0:00 0:00 4:00 8:00 :00 6:00 Fig. 3. Normalized aggregated demad profiles i each iteratio of Algorithm ODC ad DAP i a homogeeous case. All EVs are of type seda, plug i at 0:00 ad have deadlie 9:00 the ext day. Fig. 5. All EVs are of type seda ad plug i at 0:00. Half of the EVs has a deadlie at 6:00 o the ext day, ad the other half has a deadlie at 9:00 o the ext day. ormalized aggregated demad kw) ODC DAP 0:00 0:00 4:00 8:00 :00 6:00 Fig. 4. Oe third of the EVs belog to each of the three types: seda, compact, ad roadster. All EVs plug i at 0:00 ad have deadlie 9:00 the ext day. ad roadster) of EVs. We cosider the peetratio level of oe EV i every 5 households. Time is divided ito 4 slots, each of hour, durig which the chargig rate of each EV is ot chaged. As proposed i [4], we choose the price fuctio ad parameters for Algorithm DAP as followig: pd) = d, c =, δ =. Figure 3 shows the ormalized aggregated demad profile i each iteratio of Algorithm ODC ad DAP. I this case, all EVs are of type seda, plug i at 0:00, ad have deadlie 9:00 the ext day. Algorithm ODC ad DAP both coverge to a valley-fillig chargig profile. Moreover, ODC coverges after iteratio, while DAP takes several iteratios to coverge. It ca be easily show that algorithm ODC will always coverge after iteratio i homogeeous cases. Figure 4 shows the ormalized aggregated demad profiles at covergece i a o-homogeeous case, where oe third of the EVs belog to each of the three types: seda, compact, ad roadster. All EVs plug i at time 0:00 ad have deadlie 9:00 the ext day. ODC still coverges to valleyfillig chargig profiles withi few iteratios ormalized aggregated demad profile for each iteratio is ot show i Figure 4). DAP o loger coverges to valley-fillig results sice aggregated chargig rate at peak hour aroud 0:00) is o-zero. The optimality proof provided i [4] does ot exted straightforwardly to this o-homogeeous case. Figure 5 shows the ormalized aggregated demad profiles i aother o-homogeeous case. All EVs are of type seda, ad plug i at 0:00. Half of the EVs has a deadlie at 6:00 o the ext day, ad the other half has a deadlie at 9:00 o the ext day. Still, ODC coverges to valley-fillig chargig profiles withi few iteratios. For DAP, aggregated chargig profile is flat from 0:00 to 7:00, but lower later. This is because DAP uses a pealty term to prohibit all EV chargig profiles deviatio from the average chargig profile. Half of the EVs have early deadlies, but have to track the same chargig profile as the EVs with late deadlies. Ituitively, those EVs with early deadlies have to take larger chargig rates at the begiig hours, leadig to lower aggregated demad later. At higher EV peetratio levels, the differece becomes more sigificat. ODC chages deviatio from the average pealty to deviatio from the last iteratio pealty. While preservig covergece property, Algorithm ODC o loger requires differet EVs to track a commo chargig profile; hece, successfully deals with the deadlie issue. Meawhile, by choosig the objective fuctio for EVs, optimality is always guarateed, i both homogeeous ad o-homogeeous cases. A. Coclusios V. CONCLUSIONS AND FUTURE WORKS I this paper, we studied utilizig decetralized EV chargig cotrol to fill the overight electricity demad valley.

7 We defied optimality of a chargig profile as miimizig the l orm of the aggregated demad, ad proposed a decetralized EV chargig cotrol algorithm ODC). I each iteratio of ODC, each EV calculates its ow chargig profile accordig to the price profile broadcast by the utility, ad the utility guides their behavior by updatig the price profile. ODC requires low commuicatio ad computatio capability, ad is guarateed to coverge to the set of optimal chargig profiles, i both homogeeous ad ohomogeeous cases. Simulatios support these results, ad demostrate fast covergece of ODC. B. Future Works Algorithm ODC deals with day-ahead egotiatio of chargig profiles. That is, all EVs participate the egotiatio at the same time, ad implemet the schedules they commit to. I a more realistic sceario, EVs may participate the egotiatio at differet times, ot ecessarily kow to the utility beforehad. We are curretly extedig the methodology proposed here to study this real-time settig by icorporatig predictios o EV arrivals ad their eergy demad, ad accoutig for the ucertaities i these predictios. Distributed eergy storage, though curretly costly, offers alterative sources for acillary services to support for the itegratio of itermittet ad ucertai reewable eergy geeratio, as well as for regulatig voltage profiles ad fault protectio. Therefore, the eergy storage capacity embedded i EVs may provide a similar flexibility ad a log term research goal is to develop probably optimal schedulig protocols to exploit this potetial. [9] C. Qui, D. Zimmerle ad T. H. Bradley, The effect of commuicatio architecture o the availability, reliability, ad ecoomics of plug-i hybrid electric vehicle-to-grid acillary services, Joual of Power Sources, vol. 95, issue 5, 00, pp [0] J. A. P. Lopes, P. M. R. Almeida, ad A. M. M. da Silva, Smart chargig strategies for electric vehicles: Ehacig grid performace ad maximizig the use of variable reewable eergy sources, i Proceedigs of the 4th Iteratioal Battery Hybrid Fuel Cell Electrical Vehicle Symposium ad Exhibitio, Stavager, Norway, 009, pp. -. [] S. Shao, T. Zhag, M. Pipattaasompor ad S. Rahma, Impact of TOU rates o distributio load shapes i a smart grid with PHEV peetratio, Trasmissio ad Distributio Coferece ad Expositio, 00, pp. -6. [] E. Sortomme, M. M. Hidi, S. D. J. MacPherso ad S. S. Vekata, Coordiated chargig of plug-i hybrid electric vehicles to miimize distributio system losses, IEEE trasactios o smart grid, vol., o., 0, pp [3] M. C. Caramais ad J. M. Foster, Couplig of Day Ahead ad Real- Time Power Markets for Eergy ad Reserves Icorporatig Local Distributio Network Costs ad Cogestio, i 48th Aual Allerto Coferece o Commuicatio, Cotrol, ad Computig, 00, pp [4] Z. Ma, D. Callaway ad I. Hiskes, Decetralized Chargig Cotrol for Large Populatios of Plug-i Vehicles, i 49th Coferece o Decisio ad Cotrol, 00. [5] A. Ipakchi ad F. Albuyeh, Grid of the future, IEEE Power ad Eergy Magazie, vol. 7, o., 009, pp [6] SCE website [Olie]. Available: regul ifo/eca/domsm.dlp. VI. ACKNOWLEDGMENT The authors express their gratitude to Prof. K. Mai Chady from Califoria Istitute of Techology ad Dr. Sachi Adlakha from Califoria istitute of Techology for their valuable help ad advice. REFERENCES [] EPRI, Evirometal assessmet of plug-i hybrid electric vehicles - volume : atiowide greehouse gas emissios, Electric Power Research Istitute: Palo Alto, 007. [] Committee o Assessmet of Resource Needs for Full Cell ad Hydroge Techologies, Natioal Research Coucil, Trasitios to alterative trasportatio techologies plug-i hybrid electric vehicles, the atioal press, 00. [3] U.S. Departmet of Eergy website [Olie]. Available: [4] L. Kelly, A. Rowe ad P.Wild, Aalyzig the impacts of plugi electric vehicles o distributio etworks i British Columbia, Electrical Power & Eergy Coferece, -3 Oct. 009, pp. -6. [5] C. Roe, F. Evagelos, J. Meisel, S. Meliopoulos ad T. Overbye, Power system level impacts of PHEVs, i Proceedigs of the 4d Hawaii Iteratioal Coferece o System Sciece, 009, pp. -. [6] K. Clemet, E. Haese ad J. Driese, Coordiated chargig of multiple plug-i hybrid electric vehicles i residetial distributio grids, Power Systems Coferece ad Expositio, 5-8 March 009, pp. -7. [7] J. A. P. Lopes, F. J. Soares ad P. M. R. Almeida, Itegratio of electric vehicles i the electric power system, i Proceedigs of the IEEE, vol. 99, No., 0, pp [8] P. Deholm ad W. Short, A evaluatio of utility system impacts ad beefits of optimality dispatched plug-i hybrid vehicles, Natioal Reewable Eergy Laboratory, Techical Report NREL-TP , 006.