Optimal Scheduling for Charging and Discharging of Electric Vehicles

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1 1 Optial Scheduling for Charging and Discharging of Electric Vehicles Yifeng He, Meber, IEEE, Bala Venatesh, Senior Meber, IEEE, and Ling Guan, Fellow, IEEE Abstract The vehicle electrification will have a significant ipact on the power grid due to the increase in electricity consuption. It is iportant to perfor intelligent scheduling for charging and discharging of Electric Vehicles (EVs). However, there are two ajor challenges in the scheduling proble. First, it is challenging to find the globally optial scheduling solution which can iniize the total cost. Second, it is difficult to find a distributed scheduling schee which can handle a large population and the rando arrivals of the EVs. In this paper, we propose a globally optial scheduling schee and a locally optial scheduling schee for EV charging and discharging. We first forulate a global scheduling optiization proble, in which the charging powers are optiized to iniize the total cost of all EVs which perfor charging and discharging during the day. The globally optial solution provides the globally inial total cost. However, the globally optial scheduling schee is ipractical since it requires the inforation on the future base loads and the arrival ties and the charging periods of the EVs that will arrive in the future tie of the day. To develop a practical scheduling schee, we then forulate a local scheduling optiization proble, which ais to iniize the total cost of the EVs in the current ongoing EV set in the local group. The locally optial scheduling schee is not only scalable to a large EV population but also resilient to the dynaic EV arrivals. Through siulations, we deonstrate that the locally optial scheduling schee can achieve a close perforance copared to the globally optial scheduling schee. Index Ters Optial scheduling, electric vehicle, charging and discharging, Vehicle-to-Grid (V2G), convex optiization, distributed solution, sart grid N M M CHG M V 2G x i T τ E ini E cap E fin P ax γ F z i L b i L bf i NOMENCLATURE Interval set Set of Electric Vehicles (EVs) Charging-only EV set Vehicle-to-Grid (V2G) EV set Charging power of EV in interval i Charging period of EV Length of an interval Initial energy of EV Battery capacity of EV Final energy of EV Maxiu charging power Final energy ratio of EV Charging-interval atrix Total load in interval i Real base load in interval i Forecasted base load in interval i Yifeng He, Bala Venatesh, and Ling Guan are with the Departent of Electrical and Coputer Engineering, Ryerson University, Toronto, Ontario, M5B2K3, Canada. E-ail: yhe@ee.ryerson.ca, bala@ryerson.ca, and lguan@ee.ryerson.ca y i 0 1 C i Q (i) B H (i) W (i) H (i)chg (i)v 2G H t arr t dep t C s t C e Charging load in interval i Intercept in the real-tie pricing odel Slope in the real-tie pricing odel Cost for EV charging in interval i Previous-interval set of interval i Group set Ongoing EV set at the beginning of interval i in group Sliding window at the beginning of interval i in group Charging-only EV set at the beginning of interval i in group V2G EV set at the beginning of interval i in group Arrival tie of EV Departure tie of EV Start tie of the charging period of EV End tie of the charging period of EV I. INTRODUCTION The autootive industry is heavily investing in Plug-in Hybrid Electric Vehicles (PHEVs) and fully Electric Vehicles (EVs) ainly in order to reduce the CO2 eissions and oil dependency of current autootive technology. The vehicle electrification will have significant ipacts on the power grid due to the increase in electricity consuption. The overall load profile of electric syste will be changed due to the introduction of EV charging and discharging. The charging of a large population of EVs has a significant ipact on the power grid. It have been estiated that the total charging load of the EVs in US can reach 18% of the US suer pea at the EV penetration level of 30% [1]. On the other hand, an EV can also provide energy to the power grid by discharging the battery, which is nown as Vehicle-to-Grid (V2G) [2]. An intelligent scheduling schee can optially schedule the EV charging patterns such that the load profile of the electric syste can be effectively flattened. This will reduce potential capital costs and iniize operational costs. Intelligent scheduling for EV charging and discharging has becoe a vital step towards sart grid ipleentation [3][4]. The essential principle in intelligent scheduling is to reshape the load profile by charging the EV battery fro the grid at the tie when the deand is low and discharging the EV battery to the grid when the deand is high. However, it is challenging to schedule the patterns of EV charging and discharging in an optial way. First, it is difficult to find the globally optial scheduling solution which can iniize the

2 2 overall charging cost, especially in the presence of a large EV population. Second, the scheduling schee is required to have the capacity to efficiently handle the rando arrivals of the EVs. In the recent literature, a nuber of scheduling schees for EV charging and discharging have been proposed [5][6][7][8]. However, the scheduling schees in [5][6] only dealt with battery charging without V2G function. Though the existing wor on V2G scheduling [7][8] tried to optiize the charging and discharging powers to iniize the cost, their ethods are essentially centralized algoriths, which ay not be suitable for the EV charging and discharging systes with a large population and dynaic arrivals. In this paper, we propose a globally optial scheduling schee and a locally optial scheduling schee for EV charging and discharging. Our contributions are suarized as follows. We forulate a global scheduling optiization proble, which ais to iniize the total cost for charging all EVs within the day. The optiization proble is a convex optiization proble, which can be solved efficiently. The globally optial scheduling schee deterines the optial charging powers for all EVs for all intervals by solving a single global scheduling optiization proble, thus obtaining the globally inial total cost. We forulate a local scheduling optiization proble for the EVs in the local group. Based on the local scheduling optiization proble, we develop a locally optial scheduling schee, which is perfored in an independent and distributed way. The locally optial scheduling schee is very appropriate for the EV charging and discharging systes with a large population and dynaic arrivals. The perforance of the locally optial scheduling schee is lower than but very close to that of the globally optial scheduling schee. The globally optial scheduling schee provides the globally inial total cost. However, the globally optial scheduling schee is ipractical since it requires the inforation on the future base loads and the arrival ties and the charging periods of the EVs that will arrive in the future tie of the day. Though the locally optial scheduling schee perfors a little worse than the globally optial scheduling schee, it it is a practical schee which can efficiently handle a large EV population and dynaic EV arrivals. Therefore, the locally optial scheduling schee is the final solution suggested in the paper. With the globally inial total cost provided by the globally optial scheduling schee, we can find out the optiality gap between the two schees. The reainder of the paper is organized as follows. Section II discusses the related wor. In Section III, we forulate and solve the global scheduling optiization proble. In Section IV, we forulate and solve the local scheduling optiization proble. The siulation results are presented in Section V, and the conclusions are drawn in Section VI. II. RELATED WORK Depending on the direction of energy flow, existing wor on EV charging scheduling can be classified into two classes: 1) scheduling for charging only, and 2) scheduling for both charging and discharging. In charging-only scheduling, the scheduler tries to optiize the energy flow fro the grid to the battery of the EV. In [5], Shrestha et al. optiized the EV battery charging during the low-cost off-pea period to iniize the charging cost in the context of Singapore. The paper in [9] exained the proble of optiizing the charge trajectory of a PHEV, defined as the tie and the rate with which the PHEV obtains electricity fro the power grid. In [1], a decentralized charging control algorith was proposed to schedule charging for large populations of EVs. The paper in [10] optiized EV battery charging behavior to iniize charging costs, achieving satisfactory state-of-energy levels, and optial power balancing. Mets et al. in [6] presented sart energy control strategies for charging residential PHEVs, aiing to iniize the pea load and flatten the overall load profile. The ipact of different battery charging rates of EVs on the power quality of sart grid distribution systes was studied in [11]. In [12], Cleent et al. proposed coordinated charging with stochastic prograing, which was introduced to represent the error in the load forecasting. In charging and discharging scheduling, the scheduler tries to optiize the bidirectional energy flows: fro the grid to the EV battery and fro the EV battery to the grid. Binary particle swar ethods were eployed to optiize the V2G scheduling in a paring lot to axiize the profit [7][8][13]. Sortoe et al. proposed an unidirectional regulation at the aggregator, in which several sart charging algoriths were exained to set the point about which the rate of charge varies while perforing regulation [14]. The paper in [16] developed an aggregator for V2G frequency regulation with the optial control strategy, which ais to axiize the revenue. Jang et al. proposed a ethod for an analytic estiation of the probability distribution of the Procured Power Capacity (PPC), based on which the optial contract size was decided [17]. The paper in [18] presented a real-tie odel of a fleet of plug-in vehicles perforing V2G power transactions. In [19], Singh et al. deonstrated that the coordinated charging and discharging of EVs can iprove the voltage profile and reduce the power transission loss. The paper in [15] discussed the vehicle to grid integration and described the vehicle-to-grid counication interface. III. GLOBAL SCHEDULING OPTIMIZATION In this section, we forulate a global scheduling optiization for EV charging and discharging based on a real-tie pricing odel. The solution to the optiization proble provides a globally optial scheduling schee which iniizes the total cost. A. Syste Models We study the battery charging and discharging of EVs during a day, which is evenly divided into a set of intervals. The interval set is denoted by N. The length of an interval is denoted by τ. We assue that the charging or discharging power in an interval is ept unchanged. In this paper, we divide

3 3 Arrival tie of EV Fig Departure tie of EV tie Charging period of EV Charging period of EV the day into 24 intervals such that the interval length is given by τ = 1 hour. The set of the EVs, which perfor charging and discharging during the day, is denoted by M. The EV set M consists of two sets: 1) the charging-only EV set M CHG, which includes the EVs that only charge their battery and do not provide the battery energy to the grid, and 2) the V2G EV set M V 2G, which includes the EVs that perfor both battery charging and battery discharging. We have M = M CHG +M V 2G. The charging or discharging power of EV in interval i is denoted by x i ( M, i N). In order to unify the notation, we just call x i the charging power of EV in interval i. If x i > 0, it eans that EV charges its battery in interval i. If x i < 0, it eans that EV discharges its battery in interval i. The EVs in the charging-only set M CHG always satisfy x i 0 since they do not discharge their battery at any tie. On the other hand, the EVs in the V2G set M V 2G ay have a positive, zero, or negative charging power x i in interval i ( i N) since they have bidirectional energy flows between the battery and the power grid. The arrival tie of EV, denoted by t arr, is the tie when EV is plugged into the charging station. The departure tie of EV, denoted by t dep, is the tie when EV is plugged out of the charging station. The charging period of EV, denoted by T, is the period in which EV charges or discharges its battery. Since we divide the tie into ultiple intervals, we define the charging period T of EV as the set of continuous intervals that fall between the arrival tie t arr and the departure tie tdep of EV, as illustrated in Fig. 1. The initial energy of EV, denoted by E ini, is defined as the battery energy at the arrival tie t arr. The battery capacity of EV is denoted by Ecap. The final energy of EV, denoted by E fin, is defined as the battery energy at the departure tie t arr. The final energy E fin is no larger than the battery capacity Ecap. We define a final energy ratio of EV as γ = E fin /E cap where 0 γ 1. The charging station can autoatically detect the arrival tie, the initial energy and the battery capacity of EV when the EV is connected to the charging station. The departure tie and the final energy ratio of EV are provided to the charging station by the user of EV before charging is started. The charging station can deterine the charging period T of EV fro the paraeters t arr and tdep. EV perfors charging and discharging activities during the charging period T. To represent the relationship between the charging/discharging activities and the intervals, we define a charging-interval atrix F {0, 1} M N where M and N denote the nuber of eleents in the set M and the set N, respectively. The eleents of F are defined as f i = 1, if interval i falls within the charging period T of EV, 0, otherwise. (1) In this paper, we consider the scheduling of EV charging and discharging in a sall geographic area. In our realtie pricing odel, we ae two assuptions: 1) the losses between nodes are sall and thus neglectable, and 2) there is no congestion in transission. The two assuptions allow us to neglect the spatial variation of the electricity prices. The electricity price at a tie instant is the sae regardless of the charging location. The optiizations of EV charging based on only teporal variation but not spatial variation of the price have be seen in [1][6]. The electricity price is odeled as a linear function of the instant load [1], which is given as follows. g(z t ) = z t, (2) where 0 is the intercept and 1 is the slope, which are both non-negative real nuber, and z i is the total load at tie t. The total load in interval i consists of two parts: 1) the base load L b i, which represents the load of all electricity consuptions in interval i except EV charging, and 2) the charging load y i, which represents the load of EV charging in interval i. We assue that the base load L b i reains constant in interval i. The charging load in interval i is given by y i = M x if i. If the load fro the grid to the batteries of the EVs is greater than that fro the batteries of the EVs to the grid in interval i, the charging load y i is positive. Otherwise, it is negative. The total load in interval i is given by z i = L b i + y i = L b i + M x if i. Since both the base load L b i and the charging power x i ( M, i N) reain constant in interval i, the total load z i is constant in interval i. In this paper, we define the charging cost in interval i, denoted as C i, as the total aount of the oney that the custoers pay for charging and discharging of their EVs in interval i. Based on the pricing odel, the charging cost in interval i ( i N) is given by C i = z i L b i( z t )dz t = ( 0 z i z2 i ) ( 0L b i (Lb i )2 ). As shown in Equation (3), the charging cost C i can be positive or negative. If the charging load y i, given by y i = z i L b i, in interval i is positive, the charging cost C i is positive. Otherwise, it is negative. B. Proble Forulation and Solution In order to find a globally optial scheduling schee for the EVs that perfor charging and discharging during the day, we ae the following assuptions: 1) the arrival tie and the departure tie of each EV in the EV set M are nown (this is realistic in the case where each EV user signs the charging contract and bring in the EV at a designated tie); 2) the initial energy and the final energy of the battery for each EV in the EV set M are nown; 3) the base load in each interval (3)

4 4 of the day is nown; and 4) a central controller collects all the inforation and then perfors the scheduling optiization. The total cost is defined as the su of the charging costs over the interval set N. The total cost is then given by Local Controller (LC) 1 LC1 Utility copany Central controller LCn C tot = i N C i = i N (( 0z i z2 i ) ( 0L b i + (4) 1 2 (Lb i )2 )). The global scheduling optiization proble can be stated as to iniize the total cost of the EVs which perfor charging and discharging during the day, by optiizing the total load z i in interval i ( i N) and the charging power x i ( M, i N), subject to the relationship between the total load in an interval and the charging power of an individual EV, the instant energy constraints, the final energy constraints, and the lower bound and the upper bound of the charging power. Matheatically, the optiization proble can be forulated as follows. Miniize x,z (( 0 z i z2 i ) ( 0L b i (Lb i )2 )) (5a) i N subject to z i = L b i + M x i f i, i N, (5b) 0 E ini + τx f E cap, M, i N, Q (i) (5c) E ini + τx i f i γ E cap, M, (5d) i N 0 x i P ax, M CHG, i N, (5e) P ax x i P ax, M V 2G, i N. (5f) In the optiization proble (5), the objective function (5a) to be iniized is the total cost of the EVs which perfor charging and discharging during the day. Constraints (5b) represent the relationship between the total load in an interval and the charging power of an individual EV. Constraints (5c) are the instant energy constraints, which require the energy of EV ( M) at the end of interval i ( i N), given by E i = E ini + Q τx f (i), to be no less than 0 and no larger than the battery capacity E cap of EV. Constraints (5d) are the final energy constraints, which require the final energy of EV ( M), given by E fin = Eini + i N τx if i, to be no less than the specified energy level, which is given by γ E cap. Constraints (5e) specify the lower bound 0 and the upper bound P ax of the charging power x i for the EVs in the charging-only set M CHG. Constraints (5f) specify the lower bound ( P ax ) and the upper bound P ax of the charging power x i for the EVs in the V2G set M V 2G. In the optiization proble (5), the objective function (5a) is convex, and all the constraint functions are linear. Therefore the optiization proble (5) is a convex optiization proble, which can be solved efficiently with the interior point ethods [20]. The solution to the optiization proble (5) provides the globally optial scheduling schee for EV charging and discharging during the day. Group 1 Charging station LC2 Group 2 Group n Fig. 2. Illustration of counications and controls in the locally optial scheduling schee IV. LOCAL SCHEDULING OPTIMIZATION The globally optial scheduling schee gives the globally inial total cost. However, the globally optial scheduling schee is ipractical due to the following reasons. First, the EVs that will arrive in the future tie of the day are unnown at the current oent. Second, the base load in the future tie of the day is unnown at the current oent. Third, it is not scalable for a centralized scheduling schee in which the central controller ay be overrun by a large nuber of EVs. In this section, we forulate a local scheduling optiization proble, which relaxes the assuptions used in the global scheduling optiization proble (5). The solution to the local scheduling optiization proble is a locally optial scheduling schee, which can achieve the perforance close to that in the globally optial scheduling schee. Copared to the globally optial scheduling schee, the locally optial scheduling schee is practical and scalable. A. Proble Forulation and Solution In the globally optial scheduling schee, since we assue that we have the global nowledge of the inforation about the EVs and the base load within the day, we can find the optial charging powers at each interval by solving the global scheduling optiization proble (5) only once. In the locally optial scheduling schee, we do not now the inforation of the future load and the future EVs. We propose a locally optial scheduling schee to find the optial charging powers in the next interval for the local EVs by using a sliding window echanis. In the locally optial scheduling schee, we perfor the scheduling optiization based on groups. A group of EVs includes the EVs in one location or ultiple nearby locations. For exaple, the EVs which perfor charging and discharging in a paring lot can be classified into a group, and the EVs in a residential garage can be classified into another group. There is a Local Controller (LC) for each group. The counications and controls in the locally optial scheduling schee are illustrated in Fig 2. The local controller establishes counication connections with the central controller located in the utility copany and the charging stations at the local site. The local controller receives the forecasted loads for the day fro the

5 5 Charging period of EV 1 Charging period of EV 4 Charging period of EV 3 Charging period of EV Sliding window Current tie instant 24 tie Fig. 3. Illustration of the ongoing EV set and the sliding window in the locally optial scheduling schee central controller. The local controller counicates with each charging station in real tie to collect the EV inforation, based on which it perfors scheduling optiization and then instructs each local EV to charge or discharge the battery with the optial charging powers. We denote the group set by B. Since each local controller perfors scheduling independently, we will just study the scheduling optiization in group ( B). The local controller does not now the future arrivals of the EVs in the group. Therefore, we propose to update the charging powers at the beginning of each interval by using a sliding window. At the beginning of interval i ( i N), we need to first deterine the current ongoing EV set H (i) and the current sliding window W (i). Let the current tie tcur be the beginning of interval i ( i N). Each EV has a charging period. The start tie and the end tie of the charging period of EV is denoted by t C s and t C e, respectively. If EV satisfies t C s t cur and t C e > t cur, we say that EV belongs to the current ongoing EV set H (i). The current sliding window W (i) at the beginning of interval i is defined as the set of the consecutive intervals between the start tie t W s i and the end tie t W e i of the sliding window. The start tie of the sliding window is always given by t W s i = t cur, and the end tie of the sliding window is defined by t W e i = ax{t C e H (i) }. Fig. 3 illustrates the ongoing EV set and the sliding window at the beginning of interval 2. As shown in Fig. 3, EV 1 has copleted charging since t1 C s t cur and t C e 1 t cur. EVs 2, 3, and 4 satisfy t C s tcur and t C e > tcur. Therefore the current ongoing EV set is given by H (2) = {EVs 2, 3, 4}, and the current sliding window is given by W (2) = {intervals 2, 3, 4, 5, 6}. EV ( H (i) ) perfors charging and discharging activities during its charging period. At the beginning of interval i ( i N), we define a charging-interval atrix F (i) {0, 1} H(i) W(i) whose eleents are given by 1, if interval j falls within W (i) and within f (i) j = the charging period of EV, 0, otherwise. (6) In order to deterine the charging powers in the current sliding window, we need to now the base loads in the sliding window W (i), which can be forecasted using siilar-day approach, regression ethods or tie-series ethods [21]. In this paper, we adopt the siilar-day approach [21], in which the base load in each interval of the sliding window is estiated by averaging the base loads of the sae interval of the recent days with siilar weather conditions. The forecasted base load is denoted by L bf j for j W (i). Based on the current ongoing EV set H (i) and the current sliding window W (i), we forulate the local scheduling optiization proble for the current oent in group. The optiization proble can be stated as to iniize the total cost of the EVs in the current ongoing EV set H (i) during the current sliding window W (i), by optiizing the total load z j in interval j ( i W (i) ) and the charging power x j ( H (i), j W(i) ), subject to the relationship between the total load in an interval and the charging power of an individual EV, the instant energy constraints, the final energy constraints, and the lower bound and the upper bound of the charging power. Matheatically, the optiization proble can be forulated as follows. Miniize x,z j W (i) subject to z j = L bf j 0 E (i)ini E (i)ini (( 0 z j z2 j) ( 0 L bf j (LbF j ) 2 )) + + H (i) + j W (i) s Q (j) x j f (i) j, j W(i), (7a) (7b) τx s f (i) s Ecap, H(i), j W(i), τx j f (i) j γ E cap, H(i) 0 x j P ax, H (i)chg P ax x j P ax, H, j W (i) (i)v 2G,,, j W (i). (7c) (7d) (7e) (7f) In the local scheduling optiization proble (7), the objective function (7a) to be iniized is the total cost of the EVs in the current ongoing EV set H (i) during the current sliding window W (i). Constraints (7b) represent the relationship between the total load and the charging power of an individual EV in an interval of the current sliding window W (i). Constraints (7c) are the instant energy constraints, which require the energy of EV ( H (i) ) at the end of interval j ( j W (i) ), given by E(i)j = E (i)ini + τx s Q (j) hs f (i) hs, to be no less than 0 and no larger than the battery capacity E cap of EV. In Constraints (7c), E (i)ini denotes the energy at the beginning of interval i, Q (j) denotes the current previousinterval set, defined as the set of intervals that belong to the current sliding window W (i) but are no later than interval j. Constraints (7d) are the final energy constraints, which require the final energy of EV ( H (i) ) to be no less than γ E cap. Constraints (7e) specify the lower bound 0 and the upper bound P ax of the charging power x j for the EVs in the current charging-only EV set H (i)chg. Constraints (7f) specify the lower bound ( P ax ) and the upper bound P ax of the charging power x j for the EVs in the current V2G (i)v 2G EV set H.

6 6 The local scheduling optiization proble (7) at the beginning of interval i is a convex optiization proble, which can be solved efficiently with the interior point ethods [20]. By solving the optiization proble (7), we obtain optial charging powers x j ( H(i), j W(i) ), aong which we only accept and execute the optial charging powers x i ( H (i) ) for interval i, and discard the other charging powers x j ( H(i), j W(i), j > i) which will be finally updated at the beginning of interval j (j > i). B. Distributed Scheduling Protocol Based on the local scheduling optiization proble (7), we develop a distributed scheduling protocol to ipleent the locally optial scheduling schee. The protocol for locally optial scheduling executed at the local controller is shown in Table I. The functionalities of the central controller are to perfor load forecasting and collect the actual charging load for each EV. At the beginning of the day, the central controller forecasts the base loads of the day using siilarday approach, and then broadcasts the forecasted base loads to all the local controllers. After receiving the forecasted base loads of the day, the local controller for group ( B) perfors the scheduling optiization at the beginning of each interval, starting fro the first interval until the last interval in the interval set N in sequence. In group at the beginning of interval i ( i N), the local controller deterines the current ongoing EV set H (i) and the current sliding window W (i). Since the local controller does not now the real base loads in the future intervals, the price in interval j ( j ) is deterined based on the forecasted base load and the charging load of the local EVs in the interval. The price in interval j varies fro ( L bf j ) at the beginning of interval W (i) j to ( (L bf j + x H (i) j f (i) j )) at the end of interval j. At the end of each interval, each local controller reports the actual charging load of each local EV in this interval to the central controller. There are two ajor advantages for the locally optial scheduling schee. First, it is scalable. Even when the nuber of the total EVs is large, each local controller only needs to tae care of the scheduling optiization for the local EVs. Second, it is resilient to the dynaics of EV arrivals. The locally optial scheduling schee collects the EV inforation and then updates the charging powers at the beginning of each interval, thus responding quicly to the dynaic arrivals of the EVs. C. Considering the Cost of Battery Lifetie Reduction The lifetie of the battery of an EV will be reduced due to frequent charging and discharging. In this section, we consider the the cost of battery lifetie reduction caused by EV charging and discharging. We odel the cost of battery lifetie reduction for EV, denoted by ψ, as the su of two cost coponents: the cost coponent ψ A caused by the aount of charging and discharging power in each interval, and the cost coponent ψ F caused by the fluctuation of charging and discharging power between any two consecutive intervals. The cost coponent ψ A of EV depends on the aount of charging and discharging power of EV in each interval of the day, and it is given by ψ A = i N βx2 i, (8) where β is a odel paraeter, and x i is the charging power of EV in interval i. As shown in Equation (8), the cost coponent ψ A of EV is proportional to the su of the squares of charging powers over the interval set N. Given the sae initial energy and the sae final energy, the EV which discharges ore energy during the day will have a higher cost copared to the one which discharges less energy. The cost coponent ψ F of EV depends on the fluctuations of charging and discharging powers of EV during the day, and it is given by ψ F = N i=2 η(x i x (i 1) ) 2, (9) where N represents the nuber of the intervals in the interval set N, and η is a odel paraeter. As shown in Equation (9), the cost coponent ψ F of EV is proportional to the su of the squared differences of the charging powers between two consecutive intervals over the interval set N. If the charging powers in the two consecutive intervals, intervals i and (i 1) for i = 2,..., N, have opposite signs, a higher value will be added to the cost coponent ψ F of EV, copared to the case in which the charging powers in the two consecutive intervals have the sae sign. In other words, a change of charging direction of EV in interval i (i = 2,..., N ) adds a higher value to the cost coponent ψ. F Given the sae initial energy and the sae final energy, the EV which frequently switches between charging and discharging during the day will have a higher cost copared to the one which does not frequently switch between charging and discharging. The cost of battery lifetie reduction for all EVs in the EV set M during the day is given by ψ = M ψ = M (ψa + ψf ) = M ( i N βx2 i + N i=2 η(x i x (i 1) ) 2 ). (10) By adding the cost of battery lifetie reduction to the total cost, the objective function of the global scheduling optiization proble is changed to f gso = i N (( 0z i z2 i ) ( 0L b i (Lb i )2 )) + M i N βx2 i + N M i=2 η(x i x (i 1) ) 2. (11) The global scheduling optiization proble considering the the cost of battery lifetie reduction is the sae as the optiization proble (5) except a different objective function given by f gso in Equation (11). The local scheduling optiization proble considering the cost of battery lifetie reduction, at the beginning of interval i ( i N) in group ( B), is the sae as the optiization proble (7) except a different

7 7 TABLE I PROTOCOL FOR LOCALLY OPTIMAL SCHEDULING EXECUTED AT THE LOCAL CONTROLLER Initialize: At the beginning of the day, the central controller forecasts the base loads of the day using siilar-day approach, and then broadcasts the forecasted base loads to all the local controllers. At the beginning of interval i, i = 1, 2,..., N where N represents the nuber of intervals in the set N, the local controller does the following: 1. Counicate with each charging station respectively to collect the EV inforation, 2. Deterine the current ongoing EV set H (i) and the current sliding window W(i), 3. Deterine the current charging-interval atrix F (i), 4. Find the optial charging powers x i ( H(i) ) by solving the local scheduling optiization proble (7), 5. Instruct EV ( H (i) ) to perfor charging with the optial charging power x i in interval i. objective function given by f (i) lso = (( j W (i) 0 z j zj 2) ( 0L bf j (LbF j ) 2 )) + βx 2 H (i) j W (i) j + (i) W H (i) j=2 η(x j x (j 1) ) 2, where H (i) is the ongoing EV set and W (i) is the sliding window at the beginning of interval i. The global scheduling optiization proble and the local scheduling optiization proble, which consider the cost of battery lifetie reduction, are both convex optiization probles. Therefore, they can be solved efficiently with the interior point ethods [20]. Fig. 4. Load [W] Real base load Forecasted base load 800 Tie (h) Coparison of the real base load and the forecasted base load V. SIMULATIONS We perfor extensive siulations to evaluate the proposed scheduling schees for EV charging and discharging. A. Siulation Setting We consider the electric load in a icrogrid. We exaine EV charging and discharging during a day (24 hours) starting fro 12:00 AM in idnight. The day is evenly divided into 24 intervals. Each interval has a length of 1 hour. The base load at each interval is siulated by scaling the real load in Toronto on August 21, 2009 (Friday) by a factor of 1/1500 [22]. The unit of the electricity price is Canadian dollar (C$)/Wh, and the unit of the cost is C$. In the pricing odel shown in Equation (2), we set 0 = 10 4 C$/Wh and 1 = C$/Wh/W. The battery paraeters of the EVs are based on the specifications of the Chrevolet Volt [23]. The battery capacity is 16 Wh with electric range up to 64.0 M [23]. We assue the sae specifications for every EV. The battery energy is required to reach at least 90% of the battery capacity at the end of the charging period. The axiu charging power for all EVs is set to P ax = 5.0 W. The arrival ties, the charging periods, and the initial energy of the EVs are odeled as follows. The total nuber of the EVs is set to 200 by default. The arrival ties of the EVs are uniforly distributed across the day, and the percentage of arriving vehicles in any hour of the day is less than 15%. The charging periods of the EVs are uniforly distributed between 4 and 12 hours. The initial energy of the EVs is uniforly distributed between 0 and 80% of the battery capacity. To solve the optiization probles (5) and (7), we use CVX, a pacage for specifying and solving convex progras [24][25]. B. Siulation Results The globally optial scheduling schee is a globally optial solution which requires the perfect inforation. Therefore, the real base loads are used in the global scheduling optiization proble. However, in practical systes, the real base loads in the future intervals are unavailable. The locally optial scheduling schee is a practical solution. Therefore, the forecasted base loads are used in the local scheduling optiization proble. The coparison of the real and forecasted base loads is shown in Fig. 4. The real base load is obtained by scaling the load in Toronto on August 21, 2009 (Friday) by a factor of 1/1500. The forecasted base load is obtained with a siilar-day approach, in which we average the loads of 8 weedays in Toronto fro August 11, 2009 to August 20, 2009 [22]. The ean relative error between the forecasted and real base loads, defined as ǫ = (1/ N ) i N LbF i L b i /Lb i, is 0.041, which is quite sall. We copare three scheduling schees: 1) the globally optial scheduling schee, which is the optial solution to the global scheduling optiization proble (5), 2) the locally optial scheduling schee, which is the optial solution to the local scheduling optiization proble (7), and 3) the equal allocation schee, in which the charging power of an EV in an interval is allocated based on the following criteria: a) charging or discharging of an EV in an interval is deterined based on the electricity price on the previous day, and b) the absolute value of the charging power of the EV is equal in each interval. The coparison is perfored under the following siulation setting. The nuber of the total EVs is 200, and all EVs can perfor both charging and discharging. The total EVs are divided into two groups, and each group consists of 100 EVs. In order for fair coparison, the total costs in the three schees are all calculated based on

8 8 Charging load [W] Total load [W] Globally optial schee Locally optial schee Equal allocation schee Tie [h] (a) Base load 1000 Total load in globally optial schee Total load in locally optial schee Total load in equal allocation schee 800 Tie [h] (b) Fig. 5. Variation of charging load and total load in each interval: (a) the charging load, and (b) the total load Energy [Wh] Charging power [W] Globally optial schee Locally optial schee Equal allocation schee Tie [h] (a) Globally optial schee Locally optial schee Equal allocation schee 2 Tie [h] (b) Fig. 6. Variation of energy and charging power of EV 5 in each interval: (a) the energy, and (b) the charging power the real base loads. The total costs in the globally optial scheduling schee, the locally optial scheduling schee and the equal allocation schee are C$, C$, and C$, respectively. The globally optial scheduling schee and the locally optial scheduling schee reduce the total cost by 9.40% and 8.16%, respectively, copared to the equal allocation schee. The variation of the charging load and the total load in each interval in the three schees is shown in Fig. 5. We can see fro Fig. 5(a) that the globally optial scheduling schee and the locally optial scheduling schee charge the battery fro the grid in the intervals with a lower deand and discharge the battery to the grid in the intervals with a higher deand to achieve a low total cost. The globally optial scheduling schee and the locally optial scheduling schee can reshape the total load profile, as shown in Fig. 5(b). The globally optial scheduling schee flattens the total load profile in intervals 1-7 and intervals to iniize the total cost. The globally optial scheduling schee deterines the optial charging powers for all EVs for all intervals by solving a single global scheduling optiization proble, thus obtaining the globally inial total cost. The locally optial scheduling schee deterines the optial charging powers for a group of EVs for interval i (i N) by solving the local scheduling optiization proble for interval i, respectively. The local scheduling optiization proble is forulated based on the local nowledge, while the global scheduling optiization proble is forulated based on the global nowledge. Therefore, the total cost obtained in the locally optial scheduling schee is close to but always larger than that in the globally optial scheduling schee. We next exaine the scheduling of charging power for a randoly chosen EV (e.g., EV 19) in Fig. 6. The charging period of EV 19 is fro interval 16 to interval 24. As shown in Fig. 6(b), the equal allocation schee discharges the Fig. 7. Total cost [C$] Charging only ratio Globally optial schee Locally optial schee Equal allocation schee Variation of total cost with different charging-only ratio battery in interval 16, and then charges the battery in intervals 17-24, with a constant charging or discharging power. The globally optial scheduling schee and the locally optial scheduling schee deterine the charging powers by solving the optiization probles (5) and (7), respectively. All the three schees enable EV 19 to reach the sae final energy, as shown in Fig. 6(a). Each EV decides whether it is willing to discharge the battery to the grid before starting charging. Therefore, each EV is classified into either the charging-only set M CHG or the V2G set M V 2G. We define a charging-only ratio as the ratio between the nuber of EVs in the charging-only set M CHG and the nuber of the total EVs. Fig. 7 shows the ipact of the charging-only ratio to the total cost. The increase of the charging-only ratio eans ore EVs in the charging-only set M CHG and less EVs in the V2G set M V 2G, thus causing a higher total cost in all three schees, as shown in Fig. 7. In the locally optial scheduling schee, the local controller schedules the EVs in the local group in an independent and distributed way. We define the group size as the nuber of the EVs in the group, and evaluate the perforance under different average group size in Fig. 8. The total nuber of EVs is fixed at 200. Therefore, a larger average group size indicates

9 9 Total cost [C$] Total load [W] Globally optial schee Locally optial schee Equal allocation schee Average group size [EVs] (a) Globally optial schee Locally optial schee (group size=1) Locally optial schee (group size=100) Locally optial schee (group size=200) 800 Tie [h] (b) Fig. 8. Perforance evaluation under different group size: (a) the total cost, and (b) the total load Total load [W] Base load Total load in globally optial schee Total load in locally optial schee 800 Tie [h] Fig. 9. Coparison of total load when considering the cost of battery lifetie reduction Fig. 10. Total cost [C$] Mean relative error Globally optial schee Locally optial schee Variation of total cost with different load forecasting error a saller nuber of groups. The locally optial scheduling schee deterines the optial charging powers for a group of EVs for interval i (i N) based on the local nowledge. A larger group size eans ore local nowledge available at the local controller, thus leading to a lower total cost, as shown in Fig. 8(a). The highest total cost is obtained in the case of group size of 1 EV, in which each local controller has the least local nowledge (e.g., only the inforation of one EV) and optiizes the charging power of one EV. The lowest total cost is obtained in the case of group size of 200 EVs, in which there is only one central controller, which has the inforation of all EVs. If the installation cost of the local controllers is considered, the case with a larger nuber of groups (or equivalently a saller average group size) will have a higher installation cost. However, in the case that there are a saller nuber of groups, each local controller needs to control ore EVs in a larger area, thus introducing a higher cost in data counications between the local controller and the EVs in the group. In Fig. 8(b), we can see that the total load profile in the locally optial scheduling schee is changed closer to that in the globally optial scheduling schee as the average group size is increased fro 1 to 200 EVs. Fig. 9 shows the total load profiles in both the globally optial scheduling schee and the locally optial scheduling schee considering the cost of battery lifetie reduction. The odel paraeters are set as: β = C$/Wh 2 and η = 10 3 C$/Wh 2. The charging powers in each interval are obtained by solving the global scheduling optiization proble or the local scheduling optiization proble with the revised objective function, as described in Section IV-C. The total costs in the globally optial scheduling schee and the locally optial scheduling schee considering the cost of battery lifetie reduction are C$ and C$, respectively. We show the ipact of load forecasting error in the locally optial scheduling schee in Fig. 10. In siilar-day approach for load forecasting, the forecasting error depends on the loads of the chosen siilar days. In the siulation, we choose three sets of siilar days. The future load in an interval is estiated by averaging the loads in this interval over the set of the siilar days. The three sets lead to three ean relative errors, which are 0.023, 0.041, and 0.089, respectively. We also find the total cost in the locally optial scheduling schee in the siulation when the forecasted loads are assued to be exactly equal to the real loads (e.g., the ean relative error is 0). As shown in Fig. 10, a lower forecasting error leads to a lower total cost in the locally optial scheduling schee. We can see fro Fig. 10 that the total cost in the locally optial scheduling schee approaches closer to that in the globally optial scheduling schee when the forecasting error approaches 0. We can also see fro Fig. 8(a) that the total cost in the locally optial scheduling schee approaches closer to that in the globally optial scheduling schee when the average group size approaches the axiu (e.g., 200 EVs). In an extree case where the forecasting error is 0 and the group size is 200, the total cost obtained in the locally optial scheduling schee is C$, which is higher than globally optial result ( C$) by 0.43%. The reason is that the globally optial scheduling schee deterines the optial charging powers for all EVs for all intervals by solving a single global scheduling optiization proble, while the extree case of the locally optial scheduling schee deterines the optial charging powers for all EVs for interval i (i N) by solving the local scheduling optiization proble for interval i, respectively. In the default siulation setting, the nuber of the total EVs is set to 200. In Fig. 11, we vary the nuber of the EVs

10 10 Total cost [C$] Total load [W] Globally optial schee Locally optial schee Equal allocation schee Nuber of EVs (a) 1200 Base load 100 EVs EVs 300 EVs 400 EVs 800 Tie [h] (b) Fig. 11. Perforance evaluation under different nuber of EVs: (a) the total cost, and (b) the total load fro 100 to 400, and then copare the total cost and the total load. All EVs are required to reach 90% of the battery capacity at the end of the charging period. A higher nuber of EVs eans that a higher aount of energy is required to fill the battery, thus leading to a higher total cost. The globally optial scheduling schee provides the lowest cost under different nuber of EVs. As shown in Fig. 11(a), the locally optial scheduling schee outperfors the equal allocation schee, and perfors very close to the globally optial scheduling schee, under different nuber of EVs. Fig. 11(b) shows the coparison between the base loads without EV charging and the total loads with different nuber of EVs using the globally optial scheduling schee. As shown in Fig. 11(b), a higher nuber of EVs can reshape the total load profile to be flatter. VI. CONCLUSIONS In this paper, we study the scheduling optiization proble for EV charging and discharging. We first forulate a global scheduling optiization proble, in which the charging powers are optiized to iniize the total cost of all EVs which perfor charging and discharging during the day. The globally optial solution provides the globally inial total cost. However, the globally optial scheduling schee is ipractical since it assues that the arrivals of all EV and the base loads during the day are nown in advance. To develop a practical scheduling schee, we forulate a local scheduling optiization proble, which ais to iniize the total cost of the EVs in the current ongoing EV set in the local group. 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