A Stackelberg Game Model for Plug-in Electric Vehicles in a Smart Grid

Transcription

1 Fifty-first Annual Allerton Conferene Allerton House, UIUC, Illinois, USA Otober 2-3, 2013 A Stakelberg Game Model for Plug-in Eletri Vehiles in a Smart Grid Shuvomoy Das Gupta Lara Pavel Abstrat In this paper, we onsider and analyze a Stakelberg game model for Plug-in Eletri Vehiles followers) harging from a Smart Grid leader). Our model attempts to aount for the time-of-use priing of the Smart Energy Meter using an indiret penalty approah. We show that a unique Stakelberg Equilibrium exists for the game under realisti onditions. To better understand the evolution of the game, we solve a monopolisti version of the game and then we solve the game for the general ase. The solution we obtain is in losed form and tratable, yet reveals several important aspets of the game arising from the interplay between the vehiles and the grid. The results an be applied to both individual vehiles and vehile groups. We show that, if the battery apaity of a partiular vehile model falls below the threshold battery apaity of the game, that model will be out of the market in the long run. We disuss the relation between the monopolisti game and the general game. Bakground I. ITRODUCTIO A Plug-in Eletri Vehile PEV), whih uses eletriity as its soure of energy, is the most promising alternative to petroleum vehiles [1]. Many developed ountries are moving towards large sale penetration of PEVs in the market. For example, by 2020, one in every 20 ars in Ontario, Canada will be PEV and 20% of total vehile purhases will be PEVs [2]. By then, the power grid is projeted to aquire the apability to operate and interat autonomously to effiiently ontrol the generation and distribution of eletri power. Suh a yber-physial grid is alled Smart Grid SG) [3]. For example, by 2030, all of the power generation and distribution in Ontario, Canada will be ontrolled by SG [4]. In suh a senario, PEVs onneted to the SG ompete among themselves to onsume as muh eletrial energy as possible subjet to their battery apaities, whereas the SG sells eletriity at a partiular prie to PEVs with the objetive to maximize its revenue without overloading the grid. So, a relevant question in this regard is how to determine a suitable priing strategy for PEVs onneted to the SG. Related Works Reently, there has been muh interest in the design of optimal harging strategies for PEVs onneted to the SG. In [5], two algorithms are introdued to determine the eonomially optimal solution for the PEVs in a deregulated S. D. Gupta and L. Pavel are with the Edward S. Rogers Department of Eletrial and Computer Engineering, University of Toronto shuvomoy.dasgupta@mail.utoronto.a, pavel@ontrol.utoronto.a eletriity market. The work in [6] augments the optimal harging problem of PEVs into the non onvex optimal power flow problem and solves the orresponding onvex dual problem. In [7], the non-ooperative interation between PEVs in a Cournot market is analyzed using a mean field game. A priing strategy to obtain optimal frequeny regulation servie from PEVs to the power grid is proposed in [8]. The work in [9] is one of the first few papers that apture the interation between the PEVs and the SG and the orresponding deision making proess in a grid-to-vehile senario. The authors have developed a Stakelberg game model for a number of PEV groups onneted to the SG using a Lagrangian priing approah. They have proposed an algorithm based on S-S hyperplane projetion method [10] to solve a soially stable refinement of the proposed game. However, the mehanism of a realisti Smart Energy Meter SEM), that implements time-of-use priing [11], is not onsidered in [9]. Time-of-use priing is the rate struture used by SEM that enables the SG to inrease or derease prie of eletriity depending on its demand and availability. Also, in their model the harging is assumed to be provided by harging stations and the ost funtion is proposed for PEV groups rather than individual PEVs, whih is not always realisti. For example, it is projeted that, in Ontario, Canada the PEV owners will harge their PEVs mainly at home [12], [2]. Our Contributions Our ontributions in this paper an be summarized as follows: Inspired by [13], we propose a game theoreti model in the form of a Stakelberg game for PEVs followers) harging under SG leader) onsidering realisti onditions. Our model attempts to aount for the time-ofuse priing of the SEM by using an indiret penalty approah and is appliable to both individual PEVs and PEV groups. Based oealisti assumptions, we show that, the PEVs game admits a unique and inner E and the game in general admits a unique Stakelberg game. We solve the game for a monopolisti market ondition, beause it is related to the general game. We show that, the prie of eletriity and revenue of the SG in the general ase ompared to the monopolisti ase depend on the ratio of the weighted average of battery apaity in the general ase to the single battery apaity in the monopolisti ase /13/$ IEEE 398

2 We solve the general game in expliit and tratable losed form using an impliit programming approah [14]. Suh a losed form reveals the interrelation between different parameters of the game in a preise manner. An important result in this setion is the ondition for survival of a PEV model in a ompetitive market based on its battery apaity. Organization The rest of the paper is organized as followed. In Setion II, a Stakelberg game model for PEVs operating under an SG is formulated. In Setion III, we show that, under realisti onditions the PEVs game admits unique and inner ash Equilibrium. Then, in Setion IV, a monopolisti version of the game is solved. In setion V, we determine the analytial solution of the game for the general ase in losed form. In the same setion, we make important remarks regarding the survival of a partiular PEV model, and the relation between the game for the general ase and the game for the monopolisti ase. In setion VI, we present and explain our results graphially and ompare them to that of [9]. Setion VII presents the onlusion. II. GAME THEORETIC FORMULATIO Consider a senario where the SG is harging PEVs at a partiular time. The SG sets a prie p > 0 for onsumption of 1 unit of eletrial energy. The PEVs are lassified into different PEV models in the desending order of their battery apaities, where {1,2,...,} is the set of the PEV models. PEV model number i has n i PEVs belonging to it, and i {1,2,...,n i } is the set of PEVs who are of the same PEV model number i. So, n i. Any PEV belonging to the same model number i has the same battery apaity b i, whih is in a ompat set B i [b min,b max ] R ++, where R ++ is the set of positive real numbers. R ++ is the total amount of eletrial energy provided by the SG for the harging of the PEVs. The onsumed energy by the jth PEV of model number i is u ij Ω ij, where Ω ij [0,u max ] R + is the set of onsumed energy of the mentioned PEV, and R + is the set ni j1 Ω ij be the of nonnegative real numbers. Let Ω ation spae in the absene of any onstraint. Total eletrial energy onsumed by all the PEVs with same model i is ū i n i j1 u ij. Total eletrial energy onsumed by all the PEVs under the SG is then, ū i ū i ni j1 u ij, and it annot exeed. We term the inequality ū as the energy apaity onstraint of the game. Let arithmeti mean of onsumed energy of all the PEVS be ũ ūi ū n i. Let the total eletrial energy onsumed by all PEVs under the SG exept those belonging to the model number i be u i ū ū i, and the total eletrial energy onsumed by all PEVs with same model number i exept the jth one be u ij ū i u ij. Thus we an write the energy apaity onstraint as ū u i +u ij +u ij. The overall ation spae of the PEVs is, Ω {{{u ij } ni j1 } Ω : ni j1 u ij 0} and it is ompat and onvex. The feasible ation set of the jth PEV of model number i is, ˆΩij u i,u ij ) {ζ Ω ij : ζ + u i + u ij }. Ω is oupled, beause harging of jth PEV of model number i from the SG will effet the feasible ation sets {{ˆΩ rk } nr } r1 \ˆΩ ij of other PEVs. All the PEVs try to minimize their ost funtions due to the onsumption of eletrial energy in a non ooperative manner at a fixed prie set by the SG, and the SG attempts to set a realisti prie to maximize its revenue. Suh a senario an be modeled as a Stakelberg game with a single leader and multiple ash followers. The solution to the PEVs minimization problem is attained by finding the orresponding ash Equilibrium E) at a fixed prie set by the game and the SG s maximization problem is solved by finding the prie that maximizes its revenue. The solution pair that simultaneously fulfills both the objetives of the PEVs and the SG is alled the Stakelberg Equilibrium SE). Inspired by [15], we make the following assumption: Assumption 1. Compared to the aggregate onsumed energy of all the PEVs harging from the SG ū), the onsumed energy of any single PEV u ij j i ) i ) is so negligible that it will have no effet on the SE of the game and u i +u ij an be onsidered equal to ū, i.e., u i +u ij u i +u ij +u ij ū j i ) i ) 1) We justify the assumption as followed. Aording to Table 2.1 of [16], in 2007 number of light vehiles in Ontario was 6,957, % of them were ars page 21, [16]), so in 2007 number of ars in Ontario was 3,687,255. Even if the number of ars stays the same, aording to [4], in 2020 the number of PEVs in Ontario operating under an SG will be 3,687, ,362. Even if only 10% of them are onneted to the SG as in off peak hour), the number of PEVs partiipating in the Stakelberg game will be 18,436. If the jth PEV of ith model onsumes maximum possible 65 kwh i.e. u ij MWh, and the rest of the PEVs onsume 10 kwh/pev for running 45 miles less than half of needed average energy 22 kwh for running 100 miles [9]), thenu i +u ij MWh, andū MWh with a hange of %. So, even in off peak hour, with the rest of the PEVs onsuming less than half of the needed energy, the assumption is quite aurate. ext, we formulate the ost funtion for the jth PEV of model number i, J ij : ˆΩij R. It an be written as the differene between a priing funtion P : ˆΩ ij R and a utility funtion U ij : ˆΩ ij R: J ij P ij U ij 2) The SG implements an approah alled time-of-use priing via the SEMs, where the prie is inreased when the eletri energy demand is high and is dereased in the opposite situation[11]. To reflet suh a poliy, using an 399

3 indiret penalty approah [17], we ompatly write the prie per unit energy p as a ratio of a positive priing parameter α set by the SG to a penalty term: p α ū The denominator in the prie penalizes the violation of the energy apaity onstraint by approahing zero when ū approahes, onsequently the prie inreases without any bound. So, the struture of p fores the PEVs to derease their energy onsumption when the load ondition demands suh, thus indiretly satisfying the energy apaity onstraint and aounting for the time-of-use priing. Hene, we write the priing funtioepresenting the payment due to the onsumption of eletrial energy as : 3) P ij u ij,u ij,u i ;α) pu ij α ū u ij 4) Aording to [9] the utility funtionu ij u ij,u ij,u i ;α) U should satisfy the following properties i) ij u ij 0, ii) 2 U ij U 0, iii) ij U u 2 b ij i > 0 and iv) ij s ij < 0. So, onsidering all these properties, in ontrast to the quadrati utility funtion of [9], we model the utility funtion for the jth PEV of model number i as a logarithmi funtion beause of its relation with the onept of proportional fairness [17] and nie onformity to the law of diminishing marginal utility[18]. U ij u ij,u ij,u i ;α) b i logu ij +1) s ij 5) The dissatisfation parameter s ij R ++, is individual user dependent, and is subtrated from the utility beause of the probable dissatisfation of onsumer psyhe that arises from onsumer onfusion [19]. s ij has no effet on the minimization of the ost funtions of the PEVs. So, the ost funtion for the jth PEV of model number i is: J ij u ij,u ij,u i ;α) α u ij ū b ilogu ij +1)+s ij 6) Revenue of the SG is the prie p > 0) per unit eletrial energy times the total energy ū) onsumed by all of the PEVs harging from the grid: Lp,ū) p ū α ū ū 7) We all the game in its totality a Stakelberg game and express it as G{ SG},{{J ij } ni j1 },L). The ash followers PEVs ) game is expressed as G, ˆΩ ij,j ij ;α), where we impose two assumptions on the E solution {{u ij }ni j1 }. Assumption 2. {{u ij }ni j1 } must obey the energy apaity onstraint stritly beause if the SG supplies all its eletri energy ) to the PEVs, it might lead to overloading and onsequently power outage. So, ū u i +u ij +u ij < 8) Assumption 3. At E of the PEVs game, every partiipating PEV is able to onsume a positive amount of eletrial energy from the SG: i ) j i ) u ij > 0 9) 8) and 9) implies an inner E solution. ow we define the E of the game G, ˆΩ ij,j ij ;α), revenue maximizing ondition for the SG and SE of the Stakelberg gameg{ SG},{{J ij } ni j1 },L) respetively: Definition 1 Condition for ahieving E for the PEVs). Consider the ash followers PEVs ) game G, ˆΩ ij,j ij ;α), where J ij is given by 6). For α > 0, {{u ij }ni } is alled the E of the mentioned game if besides obeying 8) and 9), u ij satisfies the following ondition: i ) j i ) u ij ˆΩ ij ) J ij u ij,u ij,u i ;α) J ij uij,u ij,u i ;α) 10) Definition 2 Revenue maximizing ondition for the SG). If the ash followers PEVs ) gameg, ˆΩ ij,j ij ;α) ahieves a unique E as haraterized by Definition 1, the leader s SG s) objetive is to find a priing parameter α > 0 suh that it maximizes its revenue funtion L given by 7), i.e.: α > 0) Lα,ū ) Lα,ū ) 11) Definition 3 SE of the game). The pair {{u ij }ni j1 }, p ) is alled the SE of the game G{ SG},{{J ij } ni j1 },L), if it satisfies 10) and 11) simultaneously. III. EXISTECE AD UIQUEESS OF E FOR THE GEERAL CASE To show the existene and uniqueness of E, we follow the approah in [20]. We use the following theorem Theorem 4.4 of [21]) to show that the PEVs game admits an E solution. Theorem 1. Let Ω be ompat and onvex and let the ost funtion J : Ω R be ontinuous on Ω and onvex in u ij ˆΩ ij, for j i, i. Then the assoiated ash game G, Ω,J;α ) admits an E solution at a fixed positive priing parameter α. The following lemma is used to show the existene and uniqueness of E and also to alulate E: Lemma 1. Under Assumption 1,J ij u ij,u ij,u i ;α) in 6) an be onsidered equal to approximated by) the following equivalent augmented ost funtion that is idential for all PEVs: J ū 1,ū 2,...,ū ;α) αū ū + r1 s rk r1 logu rk +1) 12) 400

4 and the game G, ˆΩ ij,j ij ;α) is equivalent to G, Ω,J;α). Proof: The term below : α n i Ψ u ik + u ij u i b i + n i ;k j r1;r i ;k j logu ik +1) s rk + n i ;k j u rk r1;r i r1;r i logu rk +1) s ik 13) is not a funtion of u ij, so an be added to both sides of 6), yet the optimization of resultant augmented ost funtion with respet to u ij will yield the same E of the followers game. Let nr r1 s rk s. We add 13) to both sides of 6) and then use 1) under Assumption 1 to obtain: J ij +Ψ α ū n i ;k j u ik + n i + s b i logu ik +1) αū ū + s r1 u rk r1;r i r1;r i logu rk +1) + αu ij ū logu rk +1) We express the resultant augmented ost funtion J ū 1,ū 2,...,ū ;α) as: J ū 1,...,ū ;α) αū ū + s r1 logu rk +1) Also J ū 1,...,ū ;α) takes values from Ω to R, so the PEVs game G, ˆΩ ij,j ij ;α) is equivalent to G, Ω,J;α ). We now give the following theorem regarding the uniqueness and existene of E of the PEVs game: Theorem 2. The PEVs gameg, Ω,J;α) admits a unique inner E satisfying Assumptions 1, 2 and 3, if 0 < α <, where weighted mean of all battery apaities bini ni. Proof: Outline : The proof has three parts. At first, we prove the existene of the E by showing that all onditions mentioned in Theorem 1 are met. Then, we find out the ondition for the E to be inner as imposed by Assumption 2 and Assumption 3. Finally, we show that the E is unique. For this, we exploit the struture of the Hessian of J evaluated at a onvex ombination of supposedly two different E solutions, integrate it with respet to the orresponding onvex parameter and show that the resultant matrix is nonsingular. This nonsingularity implies that the E solutions are in fat equal and hene the E is unique. Part 1: We alulate first and seond order derivative of the augmented ost funtion J with respet to u ij respetively: J u ij ū) 2 b i u ij +1 14) 2 J u 2 2 ij ū) 3 + b i u ij +1) 2 > 0 15) For any α > 0, 15) is greater than zero aording to 8), thus proving the onvexity of J in u ij. So, G, Ω,J;α ) admits an E solution. Part 2: Aording to Assumption 2 and Assumption 3 the E is inner, but only α > 0 does not imply that, so we find the ondition for the existene of an inner E by using Lemma 1. Due to the onvexity of J, the orresponding E is found by optimizing J with respet to all u ij -s, i.e., by equating 14) to zero and then solving the resultant equation gū)for eah u ij at a fixed α: gū) ū) 2 b i u ij ) u ij b i ū) 2 1 j i ) i ) 17) So, 17) implies that all PEVs belonging to the same model will have the same energy requirements at the E of the followers game: ū+ n i ū i u ij n i u ij ūi b i ū) 2 1 n i ū i ū)2 b in i ū) 2 n i b i u ij +1 j i) i ) So, the funtion gū) in 16) an be written as: gū) 18) ū) 2 ū+ 0 19) ow, aording to Assumption 2 and Assumption 3, ū 0,). Again, limū gū) and gū 0) α, so for the E to satisfy both Assumption 2 and Assumption 3, there must be a zero ofgū) in0,), yieldinggũ 0) < 0, i.e. α <. So, the ondition for the existene of an inner E is: 0 < α < 20) 401

5 Part 3: ow we prove the uniqueness of E. Let, u i1 u i2... u ij... u ini v i.then 14) and 15) an be written as: J v i ū) 2 b i 21) v i +1 2 J vi 2 2 ū) 3 + b i v i +1) 2 B i > 0 22) 2 J and 2 v i v χ ū) 3 A 23) The gradient vetor of J : Ω R is: b1 ū) 2 v Jv) ū) 2 b v +1 and the Hessian matrix of J is: B 1 A... A 2 Jv) A B A A... B 24) 25) Let v 1 v v 1 )T and v 0 v v 0 )T be two inner E solutions of the followers game, then Jv 1 ) 0, Jv 0 ) 0. Let, 0 < θ < 1 and vθ) be a onvex ombination of v 1 and v 0, i.e. vθ) θv 1 +1 θ)v 0. Jvθ))) 2 Jvθ)) vθ) 2 Jv)v 1 v 0 ) θ θ 1 1 d Jvθ))) 2 Jvθ))v 1 v 0 )dθ θ0 θ0 J v 1) J v 0) 1 ) 2 Jvθ))dθ v 1 v 0 ) θ0 0 26) Let Ā 1 θ0 Avθ))dθ, Bi 1 θ0 B ivθ))dθ. i )B i > A > 0 i ) B i > Ā. Then, B 1 Ā... Ā Ā B γ 1 θ0 2 Jvθ))dθ Ā Ā... B Using the definition of γ 26) beomes : γ. v 1 v 0) 0 27) Let, y v 1 v 0. Then, i-th row of 27) is: y i + Ā B i Ā y j 0 28) j1 As Ā B i Ā 0, 28) is true if and only if γ is a full rank matrix. So, the only solution to γ.y 0 is the trivial one, i.e., y 0 v 1 v 0. So, the E is unique. IV. SOLUTIO FOR A MOOPOLISTIC VERSIO OF THE GAME In this setion we solve the game for a monopolisti ase, where eah PEV has the same battery apaity parameter b i b, eah model type has same number of PEVs n i n, thus n. We express the orresponding Stakelberg game by G m { SG},{{J ij } ni j1 },L). The following theorem aptures the full solution to the monopolisti version of the game: Theorem 3. The monopolisti version of the Stakelberg game G m { SG},{{J ij } ni j1 },L) admits a unique SE given by {{u ij }ni j1 },p ) +2 {{ } ni j1 }, b + 2 ) ) Proof: For the monopolisti ase, we put b i b in 17), that gives: u ij b ū)2 1 j i ) i ). The right hand side of this equation is independent of i and j, so at the E all PEVs will be onsuming equal amount of eletrial energy i.e.,u ij u j i) i ). The simplified ost funtion for the monopolisti ase is obtained by modifying 6): J ij αuij u ij blogu ij +1) + s ij j i ) i ). There is one-to-one orrespondene betweenj ij andu ij, and the terms u ij and u i are absent in J ij. Then for the monopolisti ase, the ondition for ahieving E for the PEVs 10) and the revenue maximizing ondition for the SG 11) are simplified as: i ) j i ) u ij ˆΩ ij ) and J ij u ij ;α ) J ij u ij ;α) 29) α > 0) L α,u ij) Lα,uij ) 30) respetively. Also, the ondition for the existene of an inner E for the monopolisti ase an be found by substituting b in 20): 0 < α < b 31) So, determination of the SE for the monopolisti ase is equivalent to the optimization of J ij with respet to u ij at a fixed α, and L with respet to α. We take the derivative of J ij with respet to u ij, set it equal to zero and then find the priing parameter α at E: α b2 2bu ij +b 2 u 2 ij u ij +1) 32) We substitute this value of α from 32) in 7) to get: L u ij b 2 2bu ij +b 2 uij) 2 33) u ij +1) u ij ) Strit onavity of L ensures the existene of a global maximum and unique SE. To find the SE, we alulate the 402

6 first order derivative of L with respet to u ij, equate it to zero and then solve for u ij : u ij +2 34) From 32) the priing parameter set by the SG at SE is: α b + + 2) ) For an inner E, α must obey 31). Clearly α > 0. ow, let assume α b i.e., α b+ + 2 ) 2 b, + 2 resulting in 2 +) ) 0, whih is a ontradition, as left hand side of the inequality is positive. So, 0 < α < b is satisfied, establishing an inner E. The optimal prie p of the SG is: p b + 2 ) 36) The total onsumed energy at SE is: ū u ij ) We substitute the values of α and ū in 33) to determine the revenue of the SG at the SE of the monopolisti ase: Lα,ũ ) pū b + 2 ) 2 V. COMPLETE SOLUTIO OF THE GAME FOR THE GEERAL CASE 38) To find the SE of the game for the general ase, we use the fat that, any Stakelberg game an be reformulated as a Mathematial Programming with Equilibrium Constraints MPEC) problem [14].Thus, the objetive of finding the SE in this ase an be reformulated as followed: maximize Lα,ū ) subjet to 0 < α <, n i ū j1 u ij, {{u ij }ni j1 } interior Ω), and i ) j i ) u ij ˆΩ ij ) J ij u ij,u ij,u i;α ) J ij uij,u ij,u i;α ) 39) If there is a one-to-one relation between α and u ij j i ) i ), and α is Lipshitz ontinuous and diretionally differentiable in u ij j i ) i ), the Stakelberg game an be solved using impliit programming IMP) approah [14]. In this method, in the SG s revenue L, we substitute the value of α in terms of ũ, and then maximize the resultant funtion with respet to ũ. Finally, we substitute the value of the resultant maximizer ũ in 41) and 42) and thus obtain the SE. At first, we find the relation between α and u ij j i ) i ) to find out if the IMP approah an be applied. We drop the supersript. Then substituting ū ũ into 19): gũ) ũ) 2 ũ ) α ũ) 2 ũ+1) 41) The relation between u ij and ũ an be found out from 18): u ij ũ+1)b i 1 j i ) i ) 42) So, the relation between α and u ij satisfies all the neessary ondition to apply IMP approah. Using 41), the leader s revenue Lα,ū) an be written as: then Lα,ũ) ũ ũ) ũ+1) 43) L ũ ũ 2 2ũ+ ) ũ+1) 2 44) 2 L ũ 2 2+) ũ+1) 3 < 0 45) Strit onavity of L in ũ implies the existene of a global maxima in 0, ). We equate 44) to zero and solve the resultant quadrati equation for the positive ũ : ũ +2 46) To obtain the priing parameter set by the SG at the SE, we substitute the value of ũ in 41): α ũ ) 2 ũ +1) + + 2) ) For an inner E, α must obey 20). Clearly α > 0. ow, let assume α i.e., α ) implying 2 +) ) 0, whih leads to a ontradition as left hand side is positive. So, 0 < α < b is satisfied, resulting in an inner E. The optimal prie p that maximizes the revenue of the SG is: p α ũ +2 ) 48) We substitute the values of α and ũ in 43) to get the revenue of the SG at the SE: ) 2 + Lα,ũ ) pū 2 From 42) the E solution at the SE is: 49) 403

7 p USD KWh u ij ũ +1)b i 1 b i j i ) i ) 50) The orresponding ost funtion at SE is: p at 225 MWh p at 350 MWh p at 500MWh J ij u ij,u ij,u i ;α ) αu ij ū b ilog u ij +1) +s ij b i + 2 )+b i ) b i log b i + 2 )+s ij j i ) i ) 51) The following theorem summarizes the omplete solution to the general ase of the game: Theorem 4. Under Assumption 1, Assumption 2 and Assumption 3, the general ase of the Stakelberg gameg{ SG},{{J ij } ni j1 },L) admits a unique SE given by: u MWh umber of PEVs Fig. 1. Change of p with respet to for different -s {{u ij }ni j1 },p ) {{ b i + 2 1} ni j1 }, + 2 ) ) Proof: Proof follows readily from the disussion above. Remark 1: From the positivity ondition 9): u at 225MWh u at 350MWh u at 500MWh umber of PEVs u ij b i + 2 > 0 j i ) i ) b i > +1 i ) We interpret as the average eletrial energy per PEV supplied by the SG, and b +1 th as the threshold battery apaity. Then the inequality above beomes: b i > b th +1 52) 52) signifies that, if the battery apaity of a partiular model falls below the threshold battery apaity, that model an never ahieve SE and in the long run will be out of the market. Remark 2: Comparison between 34) and 46) reveals that, the arithmeti average of all the PEVs onsumed energy at SE is equal to the energy onsumed by a PEV at the SE in a monopolisti market, i.e.: u ij) monopolisti ũ j i ) i ) 53) From 48), 36), 38) and 49) we see that: p p ) monopolisti Lα,ū ) Lα,ū )) monopolisti b 54) Fig. 2. Change of ũ with respet to for different -s So, ompared to a monopolisti market with same number of PEVs, in a ompetitive market both the prie set and the revenue earned by the SG at SE inrease as the ratio of the weighted average of the battery apaity of the PEVs in the ompetitive market to the battery apaity of the only PEV model for the monopolisti ase inreases. VI. GRAPHICAL REPRESETATIO OF RESULTS In this setion we represent our results graphially, explain their impliations and auses and ompare them to [9] when possible. The weighted average of battery apaities is taken as, 50 kwh. We sale the parameters aordingly. Figure 1 shows the hange in the optimal prie per unit eletrial energy USD/KWh) set by the SG at SE p ) as number of PEVs inreases for different total eletrial energy supplied by the SG 225, 350 and 500 MWh. The figure shows that, p inreases with inreasing at a fixed and dereases with inreasing at a fixed. Intuitively, inreased number of PEVs result in inreased energy demand and it fores the SG to inrease the prie to disourage onnetions of more PEVs to the SG. On the other hand, as the total eletrial energy supplied by the SG inreases, the SG has surplus energy to sell and thus redues the prie per unit eletrial energy to enourage the onneted PEVs 404

8 b th MWh ACKOWLEDGMET The authors would like to aknowledge helpful disussions with Mr. Ron Groves and Dr. Wayes Tushar Fig. 3. b th at 225MWh b th at 350MWh b th at 500MWh umber of PEVs Change of b th with respet to for different -s to onsume more energy. Both of these observations agree with [9]. Figure 2 shows the hange in the average onsumed energy at SE MWh) ũ ) as inreases for different apaities, 225, 350 and 500 MWh. The figure shows that, ũ inreases with inreasing at a fixed and dereases with inreasing at a fixed. The first behavior of the PEVs is due to the fat that, asinreases with fixed, the prie per unit energy drops and on the average the PEVs take the opportunity to onsume more energy by paying less. [9] does not disuss suh a senario. The latter behavior an be explained by the fat that, as inreases with fixed, the prie per unit energy inreases, and on aggregate the PEVs redue their energy onsumptions to avoid extra payment. This behavior agrees with [9]. Figure 3 shows, the hange in threshold battery apaity as number of PEVs inreases for different apaities, 225, 350 and 500 MWh. Threshold battery apaity inreases with inreasing at fixed and dereases with inreasing at fixed. As inreases at a fixed, all PEV models are fored to derease their energy onsumptions and those with small battery apaities will obtain even smaller utilities eventually leading to their disonnetions from the SG. Similarly, the inrease in onsumed energy with inreasing at fixed results in more PEV models ahieving E, in effet dereasing b th. VII. COCLUSIO In this paper, a simple but realisti game theoreti model of PEVs harging from an SG is proposed and analyzed. At first, the game is solved for a monopolisti ase, where all the PEVs are essentially of the same model. Then, the game is analytially solved for any number of PEVs. We have derived an important inequality regarding the battery state of a PEV to reah SE state hene to survive in the market and an equation orresponding to the hange in the SG s revenue as it evolves from a monopolisti to ompetitive market. REFERECES [1] D. Sandalow, Plug-in Eletri Vehiles: What Role for Washington? Brookings Institution Press, [2] Towards an ontario ation plan for plug-in-eletri vehiles pevs), University of Waterloo, Teh. Rep., [Online]. Available: grid/materials/ / WATERLOO%20PHEV Report%20Final%20May 17.pdf [3] W. Saad, Z. Han, H. Poor, and T. Başar, Game-theoreti methods for the smart grid: an overview of mirogrid systems, demand-side management, and smart grid ommuniations, IEEE Signal Proessing Magazine, vol. 29, no. 5, pp , [4] Modernizing ontario s eletriity system: ext steps, Teh. Rep., 2011, seond Report. [Online]. Available: pubs/smart grid/smart Grid Forum-Report-May 2011.pdf [5]. Rotering and M. Ili, Optimal harge ontrol of plug-in hybrid eletri vehiles in deregulated eletriity markets, IEEE Transations on Power Systems, vol. 26, no. 3, pp , [6] S. Sojoudi and S. H. Low, Optimal harging of plug-in hybrid eletri vehiles in smart grids, IEEE Power and Energy Soiety General Meeting, pp. 1 6, [7] R. Couillet, S. M. Perlaza, H. Tembine, and M. Debbah, Eletrial vehiles in the smart grid: A mean field game analysis, IEEE Journal on Seleted Areas in Communiations, vol. 30, no. 6, pp , [8] C. Wu, H. Mohsenian-Rad, and J. Huang, Vehile-to-aggregator interation game, IEEE Transations on Smart Grid, vol. 3, no. 1, pp , [9] W. Tushar, W. Saad, H. V. Poor, and D. B. Smith, Eonomis of eletri vehile harging: A game theoreti approah, IEEE Transations on Smart Grid, vol. 3, no. 4, pp , [10] M. Solodov and B. Svaiter, A new projetion method for variational inequality problems, SIAM Journal on Control and Optimization, vol. 37, no. 3, pp , [11] Smart meters and time-of-use pries, Website of Ontario Ministry of Energy. [Online]. Available: on.a/en/smart-meters-and-tou-pries/ [12] Website of plug ndrive. [Online]. Available: a [13] T. Başar and R. Srikant, Revenue-maximizing priing and apaity expansion in a many-users regime, Twenty-First Annual Joint Conferene of the IEEE Computer and Communiations Soieties IFOCOM), vol. 1, pp , [14] J. Outrata, M. Kovara, and J. Zowe, onsmooth approah to optimization problems with equilibrium onstraints: theory, appliations and numerial results. Springer, [15] J.. Tsitsiklis and Y. Xu, Priing of flutuations in eletriity markets, IEEE 51st Annual Conferene on Deision and Control CDC), pp , [16] Summary report, anadian vehile survey, Teh. Rep., [Online]. Available: vs07/pdf/vs07.pdf [17] L. Pavel, Game theory for ontrol of optial networks. Springer, [18] A. Asano, An Introdution to Mathematis for Eonomis. Cambridge University Press, [19] G. Walsh and M. Yamin, Towards a oneptual model of onsumer onfusion, Advanes in onsumer Researh, vol. 32, pp , [20] T. Alpan and T. Başar, Distributed algorithms for nash equilibria of flow ontrol games, Advanes in Dynami Games, pp , [21] T. Başar and G. J. Olsder, Dynami nonooperative game theory. SIAM,