Power Distribution Scheduling for Electric Vehicles in Wireless Power Transfer Systems

Transcription

1 Power Distribution Scheduling for Electric Vehicles in Wireless Power Transfer Systes Chenxi Qiu College of Inforation Science and Technology Pennsylvania State University Ankur Sarker and Haiying Shen Departent of Coputer Science University of Virginia {as4z, Abstract Electric vehicles (EVs) will becoe a coponent of the future generation intelligent transportation syste. Because of EVs liited battery power, the wireless power transfer (WPT) syste has drawn uch attention in recent years. The WPT syste charges EVs in otion when they pass the charging lanes installed in roads without requiring physical contact between utility power supply and vehicle battery. A charging lane has liited power that can be transferred to EVs on the charging lane. A challenge here is how to allocate the liited power to the EVs so that they have sufficient power to arrive at the next charging lane or their destinations (when there are no charging lanes ahead). In this paper, we study this power distribution scheduling proble. We provide solutions to handle this challenge and also achieve each of the following goals as uch as possible: i) balancing the state of charge () of the EVs, ii) balancing the aount of stored power of the EVs, and iii) iniizing the total power charged. This paper is the first work that handles such a power distribution scheduling proble in WPT systes. Our extensive experients on MatLab and Siulation for Urban MObility (SUMO) show the effectiveness of our scheduling solutions in achieving the different goals copared with other scheduling ethods including first-coe-first-serve and equal share. I. INTRODUCTION An intelligent transportation syste, as a for of the future generation transportation syste, realizes real-tie sensing, coputing, and intelligence capabilities to increase traffic efficiency and provide better reliability and safety [8], [12], [15]. As a coponent of the intelligent transportation syste, electric vehicles (EVs) have drawn uch attention in recent years and will be widely used. For exaple, Tesla Motors will launch its new Model 3 EVs on March 31, 216 [3], which is a new and less-expensive of its kinds to ake EVs affordable for people. Several works already deonstrate the possible ipact of EVs on the road transportation syste [11], [16] such as petroleu consuption reduction and less environental pollution. As the total nuber of vehicles keeps rising worldwide and nationwide, the world and the nation are urged to switch fro gas-driven vehicles to EVs in order to decrease the consuption of oil and petroleu. The onboard energy storage of an EV supplies the power (power and energy are interchangeable ters in this paper) deands of the vehicle. EVs used for personal transportation can be charged by plugging the to standard power outlet sockets. However, EVs inherit soe battery-related drawbacks such as heavy weight, long charging tie, large size, and short driving range. To alleviate the battery-related probles, roadway-powered EVs that rely on the inductive wireless power transfer (WPT) systes for power charging have been developed. WPT allows charging procedure to take place on vehicles in otion autoatically without having any physical contact between utility power supply and electric battery. That is, power is transferred fro the power grid to a vehicle s battery while the vehicle passes charging lanes installed in roads. As an exaple, researchers fro KAIST [5] successfully deployed a 24k long WPT public transportation service in the city of Gui, Korea and have been developing the fifth generation roadway-powered EVs to increase the efficiency and functionality of the WPT syste. Fig. 1 shows an exaple of a WPT syste architecture. Each EV sends a charging request to the global charging controller (GCC) using vehicle-to-infrastructure (V2I) counication. GCC conducts the power distribution scheduling that allocates power to each EV when it passes each charging section of a charging lane. It then sends the schedule to grid side controllers (GSCs), each of which anages a charging lane. When an EV approaches a charging lane, it contacts GSC and then receives power as scheduled when it passes each charging section of the charging lane. In this paper, we consider a WPT syste in a highway scenario where vehicles follow a siilar velocity. An EV spends very little tie on top of a charging lane. The shorter duration of tie requires a higher power level (i.e., power transission rate) in the charging infrastructure so that a certain aount of power can be transferred to an EV in otion. Since usually there are a nuber of vehicles on a charging lane at a tie, the charging infrastructure needs to have a higher power level to eet the needs of all the vehicles at the sae tie. However, a higher power level requires higher investent on the charging infrastructure. For exaple, according to a recent study [14], the infrastructure setup cost is $7, $5 and $5, for the 1.44kW, 6kW and 9KW power transission rate to one vehicle, respectively. Therefore, in order to constrain the investent cost, the WPT designers tend to put liits on the power level of the charging infrastructure. A challenge here is when the infrastructure is not able to fulfill the deands fro all EVs on a charging lane, how to al-

2 request section energized Fig. 1. V2I lane Global charging controller Grid side controller... The wireless power transfer syste. locate the liited power to the EVs so that they have sufficient power to arrive at the next charging lane or their destinations (when there are no charging lanes ahead). In spite of several existing works that try to realize WPT systes considering different perspectives [18], [23], no previous works have been dedicated to handling this challenge. In this paper, we study this power distribution scheduling proble. We provide solutions to handle this challenge and also achieve each of the following goals as uch as possible: i) balancing the state of charge () of the EVs, ii) balancing the aount of stored power of the EVs, iii) iniizing the total power charged, and iv) iniizing the nuber of charges to increase the lifetie of batteries. We first forulate each power distribution scheduling proble with each of the above-stated objectives. We prove that the probles with goals i)-iii) are convex, and use the subgradient approach or greedy approach to solve the probles. We also prove that the proble with goal iv) is an NP-hard proble, and provide a greedy algorith to solve the proble. Our extensive experients on MatLab and Siulation for Urban MObility (SUMO) show the effectiveness of our scheduling solutions in achieving the different goals copared with other scheduling ethods including first-coe-first-serve and equal share. This paper is the first that handles such a power distribution scheduling proble in a WPT syste. The rest of this paper is organized as follows. Section II presents the background of the WPT syste and our otivation. Section III presents the proble forulation and solution for different power distribution scheduling probles for the WPT syste. Section IV presents the perforance evaluation of the proposed power distribution scheduling solutions for the WPT syste. Section V presents an overview of related work. Finally, Section VI concludes this paper with rearks on our future work. II. BACKGROUND AND MOTIVATION We first introduce the architecture of the WPT syste. As shown in Fig. 2, a charging lane is a portion of a road that consists of a nuber of charging sections. A charging section has charging coils ebedded which are used for charging EVs. The axiu aount of energy that a charging section can provide to an EV is deterined by the power transission rate of the charging coil and the tie that the EV spends on top of the charging section. The charging sections in each charging lane are controlled by a GSC. The GSC receives the schedule fro the global charging controller (GCC) in the cloud. Based on the schedule, the GSC delivers power to each charging section at every point of tie so that the EVs on the charging lane can be charged accordingly. The GSC can counicate with the vehicles and it is aware of the vehicles that are on top of its respective charging section. For a given vehicle speed, the charging lane can supply a axiu aount of energy which is the su of the axiu aount of energy that each charging section provides for that particular velocity. At every given tie t, there are a nuber of vehicles within the charging lane, which for a charging vehicle set. This set changes every tie that a vehicle checks in and/or checks out fro the charging lane. Each vehicle sends a charging request along with its own inforation (e.g.,, location, velocity) to the GCC. Accordingly, the GCC identifies the vehicles that are on the top of each charging section at each particular tie and groups the into a charging vehicle set. After deciding the charging vehicle set at each tie point, the GCC uses the inforation fro the set to decide the aount of power allocated to each vehicle at each particular tie. The GCC then sends the schedule to the GSC. When a vehicle enters the charging lane, it counicates with the GSC. Thus the GSC is aware of the vehicles entering its doain and transfers power to each charging section to charge each vehicle based on the schedule. The power capacity of a charging lane (or the GSC) is liited because a charging infrastructure with a higher power transission rate needs a higher investent cost. The aggregated value of the axiu power transission rates of all charging sections in a charging lance is equal to the capacity of the charging lane. We consider the WPT syste state as overload when the deands of all EVs on a charging lane is higher than the power capacity of the charging lane at a particular tie. That eans the deands of all EVs on the charging lane cannot be satisfied at the sae tie. When an overload occurs, it is iportant for the GCC to use a power distribution scheduling ethod to deterine the power allocated to each charging section to charge each EV at a tie in order to achieve certain goals. The scheduling ethod can be first-coe-first-serve, in which the vehicles that arrive at the charging lane earlier have higher priorities to receive their deanded power. It can also be equal share ethod, in which the vehicles in a charging vehicle set receive an equal portion of the total power that can be supplied by the GSC. When there is no overload, the GCC assigns the aount of power deanded by each vehicle in the charging vehicle set, regardless of the power distribution scheduling strategy selected. In the following sections, we present different power distribution scheduling probles and their solutions with different goals. III. POWER DISTRIBUTION SCHEDULING WPT enables EVs to increase their driving range. However, a charging lane has liited power that can be transferred to EVs on the charging lane. The power distribution scheduling schedules how to distribute liited power to EVs in the sae charging lane. In this section, we forulate different power

3 Sybol t s i t e i n c j a j A y i (t) z i (t) Paraeter Q batt,i V oc,i R int,i P batt,i Pbatt,i ax Pbatt,i in TABLE I SYMBOLS AND DEFINITIONS. Definition The nuber of EVs The start tie of EV i on the charging lane The end tie of EV i on the charging lane The nuber of charging sections The jth charging section The axiu power that can be provided by charging section j The axiu power that can be provided by the GSC The of EV i at tie t The aount of stored power of EV i at tie t TABLE II EVS PARAMETERS AND DEFINITIONS. Definition The battery capacity of EV i The voltage source of EV i The internal battery resistance of EV i EV i s battery power The axiu power of battery of EV i The iniu power of battery of EV i distribution scheduling probles for different goals. First, each EV ust receive sufficient power to enable it to drive to the next charging lane or its destination (when there are no charging lanes ahead). Besides this goal, we also consider each of the following goals based on different practical needs in the syste design: i) balancing the state of charge () of the EVs, ii) balancing the aount of stored power of the EVs, iii) iniizing the total power charged, and iv) iniizing the nuber of charges to increase the lifetie of batteries. The is easured by the percentage of energy stored in the battery. The for a vehicle will change when it passes along a charging lane. It will increase when the power received fro a charging section is higher than the power used for vehicle propulsion. A vehicle s indicates the reaining battery charge and it ust be aintained within a certain liit to ensure that the vehicle will not be out of power when it runs on the road. The above-stated goal i) ais to balance the of the EVs in a charging vehicle set when they leave the charging lane to achieve a certain fairness. Since different EVs have different battery capacities (i.e., sizes), rather than considering the relative percentage, we consider the absolute aount of stored power for another type of fairness. Thus, goal ii) ais to balance the absolute aount of stored power of the EVs. Because an EV does not have to receive ore power than what is needed to drive to the next charging lane or its destination (when there are no charging lanes ahead), goal iii) ais to iniize the total power charged to the EVs in a charging vehicle set. More battery charges lead to shorter lifetie of the battery. Therefore, goal iv) ais to iniize the nuber of charges in order to increase the lifetie of batteries. However, these probles are not trivial to solve because of the heterogeneous statuses of the EVs. That is, different EVs have different destinations, different s, different batteries, lane 1 section 1 Fig. 2. Cloud lane 2 section 2... Global charging controller Grid side controller... lane K section n The WPT syste architecture. voltage sources, and internal battery resistances when they enter a charging lane, which affect their states in ters of the above goals when they leave the charging lane. We forally forulate each proble and then find its solution. In the following, Section III-A introduces our scenario and soe definitions. Then, we present each of the above probles and its solution in the subsequent sections. A. EV Traffic Model In this paper, we consider a WPT syste in a highway scenario where vehicles follow a siilar velocity. We focus on scheduling power distribution to the charging sections within one charging lane, so that the EVs in a charging vehicle set can receive power when they pass each charging section for different goals when they leave the charging lane. Suppose there are n charging sections c 1, c 2,..., c n in a charging lane that charge heterogenous EVs {1, 2,..., } based on the EVs current stored energy in the batteries. Here, by heterogenous, we ean that different EVs have different EV paraeters as listed in Table II. Also, we assue that the EVs have siilar velocity when traveling through this charging lane. We assue that each charging section can have at ost one EV to be located at. We consider a discrete tie syste where tie t = 1, 2,... The start tie and the end tie for each EV i (i = 1, 2,..., ) on this charging lane are denoted by t s i and te i, respectively. The axiu capacity of the GSC is A and the axiu power that each charging section j can provide is a j, n j=1 a j = A. We represent the tie period that EV i is on charging section c j by t i,j = [t s i,j, te i,j ]. We use y i(t) to represent the of EV i at tie t, and let y(t) = [y 1 (t) y 2 (t)... y (t)]. Then, y i (t s i ) and y i(t e i ) represent the of EV i when it enters and leaves the charging lane, respectively. To guarantee that each EV i can drive to the next charging lane, or its destination when there are no charging lanes ahead, EV i needs to guarantee the its is higher than a value p req,i when it leaves the current charging lane, where p req,i is calculated based on inter-distance between the adjacent charging sections [27]. In the following sections, we forulate the proble for the power distribution scheduling with each of the aforeentioned goals and then provide proble solution.

4 B. Balancing the s of the EVs Given the EV traffic odel, in the following, we forulate a new proble, called the -Balanced Power Distribution Scheduling proble (-B). It ais to distribute the power to each charging section j in each tie slot t, x j (t), to guarantee all the EVs can finish their trips and the s of all the EVs are balanced when they leave the charging lane. To balance the s of the EVs when they leave the charging lane, we try to iniize the variance of the s [25] denoted by y 1 (t e 1), y 2 (t e 2),..., y( (t e ). That is, in y i (t e j=1 i) y j(t e j ) ) 2 (1) where each y i (t e i ) j=1 yj(te j ) (i = 1,..., ) easures how far y i (t e i ) is different fro j=1 yj(te j ), i.e., the ean value of y 1 (t e 1), y 2 (t e 2),..., y (t e ). Equ. (1) can be also written as in y (t e i)nn y(t e i) (2) where N R is defined by 1 1 n... 1 n 1 n n N = (3) 1 n 1 n... 1 In the following, we first introduce the notations we use and then list the constraints for this scheduling proble. P trac,i eans the energy consuption of EV i at each tie slot. T i eans the set of tie slots that EV i is located at a charging section. t i,j represents the tie slot that EV i is located at charging section j. x j (t) represents the aount of power allocated to charging section j at tie t. p th,i is the aount of stored power of EV i to guarantee that EV i can ove to the next charging section, and p req,i is the aount of stored power of EV i to guarantee that EV i can finish its trip; that is, it can ove 1) to the next charging lane if the current charging lane is not the last charging lane or 2) to the destination if the current charging lane is the last charging lane. Here, the forulas to calculate p th,i and p req,i are given in [24]. Then, the following constraints ust be satisfied for the -B power distribution scheduling proble aong the charging sections: x j (t) A, j, t (4) j=1 x j (t) a j, j, t (5) y i (t) p th,i, t, i (6) y i (t e i) p req,i, i (7) y i (t) 1, t, i (8) { zi (t) P z i (t + 1) = trac,i if t / T i (9) z i (t) + x j (t) P trac,i if t = t i,j Equ. (4) eans that the su of the allocated power of all the charging sections cannot exceed the axiu power provided by the GSC; Equ. (5) eans that the power allocated to each charging section j cannot exceed the axiu power provided by charging section j; Equ. (6) eans that the of each EV i should be at least p th,i in each tie point in order to guarantee that EV i can ove to the next charging section; Equ. (7) eans that when EV i leaves the charging lane at tie point t e i, its should be at least p req,i to guarantee that EV i can finish its trip; that is, it can ove 1) to the next charging lane if the current charging lane is not the last charging lane, or 2) to the destination if the current charging lane is the last charging lane; Equ. (8) eans that the of each EV i cannot exceed 1 because the axiu value is 1; Equ. (9) eans that at tie t, if EV i is not located at any charging section, then its power is reduced by P trac,i, which is the energy consuption at each tie slot; otherwise its power is added by x j (t) P trac,i, which eans that the power gained by EV i at tie slot t equals the charged power fro charging section j at tie slot t inus the power consued by EV i at tie slot t. The input of the -B proble includes all the paraeters of the EVs (listed in Table II) in a charging vehicle set, and the output records how uch power is allocated to each charging section to charge the EV on it at each tie slot fro the tie when the first EV enters the charging lane to the tie when the last EV leaves the charging lane. In the following, we will prove that the -B proble defined by Equ. (2)- (9) is a convex proble, i.e., the Hessian atrices of Equ. (2)-(9) are all nonnegative definite [21]. First, the Hessian atrix of the objective function (Equ. (2)) is given by: 2 f 2 f y 1 (t e i )2 y 1 (t e i ) y n(t e i ). H f(y (t e i )) = f 2 f y 1 (t e i ) y n(t e i ) y n (t e i )2 = NN, (1) which is nonnegative definite since for each x R, x NN x [21]. Then, we consider the constraint functions Equ. (4)-(8). They are all linear functions, which indicate that their Hessian atrices are all zero atrices and hence nonnegative definite. Now, we need to prove the convexity of Equ. (9). For this purpose, we first need to derive the quadratic for of Equ. (9), where the detailed process of the derivation is shown as follows. According to [24], the relationship between EV i s at tie t + 1 and tie t is given by y i (t + 1)2R int,i Q batt,i (11) = y i (t)2r int,i Q batt,i + Voc,i 2 int,iz i (t) V oc,i, fro which we can derive that z i (t) = w i (t)jw i (t) + b w i (t), (12)

5 where [ Rint,i Q 2 batt,i R int,i Q 2 batt,i J = R int,i Q 2 batt,i R int,i Q 2 batt,i [ ] [ Qbatt,i V b = oc,i and w Q batt,i V i (t) = oc,i ], y i (t) y i (t + 1) Then, based on Equ. (12), we can write Equ. (9) in its quadratic for, as shown by Equ. (13) and Equ. (15): If t / T i, If t T i and t = t i,j ]. w i (t + 1)Jw i (t + 1) + b w i (t + 1) (13) = w i (t)jw i (t) + b w i (t) P c (14) w i (t + 1)Jw i (t + 1) + b w i (t + 1) (15) = w i (t)jw i (t) + b w i (t) + x j (t) P c. (16) The Hessian atrices of both Equ. (13) and Equ. (15) are J, which is nonnegative definite since for any vector [x 1, x 2 ] (x 1, x 2 R), we have [x 1, x 2 ]J[x 1, x 2 ] = R int,i Q 2 batt,ix 2 1 2R int,i Q 2 batt,ix 1 x 2 + R int,i Q 2 batt,ix 2 2 = R int,i Q 2 batt,i (x 1 x 2 ) 2. (17) Algorith 1: The subgradient ethod for the -B and the Power-B probles. input : Paraeters of all the EVs; output: v ; // Power allocation of all charging sections 1 Select a starting solution: v (1) = ; 2 Put k = 1; 3 // Main step 4 repeat 5 ξ (k) (v (k) ) = f(v (k) ); 6 if ξ (k) (v (k) ) = then 7 Break; 8 else 9 Select a step size α (k) > and copute v (k+1) v (k) + α (k) ξ (k) (v (k) ); 1 k = k + 1; 11 until ξ (k) (v (k) ) < ɛ; 12 v v (k) ; 13 return v ; Accordingly, the -B proble defined by Equ. (2)-(9) is a convex proble, which can be directly solved by the subgradient ethod [21]. Algorith 1 shows the pseudocode of the subgradient ethod. Here, v is equivalent to w in - B. v (k) is the kth iterate, ξ (k) (v (k) ) is the subgradient of f at v (k), and α (k) > is the kth step size. At the beginning, we initiate v (1) by and set k by 1 (lines 1 2). Then, in each iteration, we first derive the subgradient of f(v (k) ) (line 4). If ξ (k) (v (k) ) =, which indicates that v (k) is the global optial solution, the algorith is finished; otherwise, we take a step in the direction of a negative subgradient α (k) ξ (k) (v (k) ). The above process is repeated until ξ (k) (v (k) ) < ɛ, i.e., the subgradient is close to. C. Balancing the Aount of the Stored Power of the EVs In Section III-B, we ainly consider the of EVs for fair power distribution. Since reflects the percentage of power stored in the battery while the EVs batteries have different capacities (i.e., sizes), balancing the absolute aount of stored power becoes an alternative for fairer power distribution. As a solution, we forulate the power distribution scheduling proble by taking into account the absolute stored power of each EV. In particular, we forulate another new proble, called the Power-Balanced Power Distribution Scheduling proble (Power-B). In this proble, we still need to guarantee that each EV can finish its trip. That is, each EV 1) has enough power to ove to the next charging section within the charging lane, and 2) has enough power to reach the next charging lane or its destination if there are no charging lanes ahead when it leaves the charging lane. The difference between -B and Power-B is that, in Power-B, we ai to balance the absolute aount of stored power of all the EVs when the EVs leave the charging lane. Before introducing the proble, we first odify the three constraints for each EV to finish its trip in -B, i.e., Equ. (6)-(8), to: s.t. z i (t) p th,i, t, i (18) z i (t e i) p req,i, i (19) z i (t e i) P batt,i, i. (2) Here, p th,i is the aount of EV i s stored power in each tie slot to guarantee that EV can ove to the next charging section, and it is calculated by: p th,i =.5ρ air,i C d,i v 2 i ax (21) where v i, ρ air,i, C d,i, and ax represent the velocity, the air density, the drag coefficient of EV i, and the axiu inter-distance between adjacent charging sections, respectively [24]. p req,i is the aount of EV i s stored power in each tie slot to guarantee that EV i can ove 1) to the next charging lane if the current charging lane is not the last charging lane, or 2) to the destination if the current charging lane is the last charging lane. p req,i is calculated by: p th,i =.5ρ air,i C d,i v 2 i next (22) where next represents the distance to the next charging section or the destination [24]. In Power-B, the objective function is to iniize the variance of the absolute aount of stored power of all the EVs when the EVs leave the charging lane. That is, ( in z i (t e j=1 i) z j(t e j ) ) 2 (23)

6 where each z i (t e i ) j=1 zj(te j ) (i = 1,..., ) easures how far z i (t e i ) is different fro j=1 zj(te j ), i.e., the ean value of z 1 (t e 1), z 2 (t e 2),..., z (t e ). Equ. (23) can be also written in a quadratic for: in z (t e i)nn z(t e i) (24) where N is given by Equ. (3). In addition, the constraints Equ. (4), (5), (9), (18)-(2) need to be satisfied. Since the objective function (Equ. (24)) is a quadratic function and all the constraints are linear functions, Power-B is a quadratic prograing proble, which is a convex proble and hence can be directly solved using the subgradient ethod [21]. D. Miniizing the Total Power Charged In the above, we focus on balancing the s or the aount of stored power of EVs for fair power distribution. In this section, we discuss another alternative of power distribution scheduling. Since the power deands of charging EVs ipose a high load on the power grid during peak hours, it is desirable to iniize the aount of energy received by each EV so that the power grid can satisfy the energy deands of as ore EVs as possible, especially during the peak hours. As an alternative solution, we forulate the power distribution scheduling proble by taking into account the iniization of the aount of charged power of each EV while enabling the EVs to arrive at the next charging lane or their destinations. We consider the proble as the Power Miniization Power Distribution Scheduling proble (). We again use the three constraints for each EV to finish its trip in -B, i.e., Equ. (18)-(2). The difference is that besides satisfying these constraints in, we ai to iniize the total power charged by all the charging sections in the charging lane, which is represented by: in t=1 T x i (t) (25) where T t=1 x i(t) eans the aount of charged power of EV i in the whole process and hence T t=1 x i(t) represents the aount of the total charged power of all the EVs. Siilar to Power-B, the constraints Equ. (4), (5), (9), (18)- (2) need to be satisfied. Since the objective function and all the constraints are linear, can be directly solved by the siplex ethod [21], which is a standard ethod of iniizing proble with a linear objective function and constraints. The basic idea of the siplex ethod is to explore the extree points of the feasible region constructed by the linear constraints to find the optial extree point that iniizes the objective function. To further increase the tie efficiency, in the following part, we will devise a greedy algorith that has lower tie coplexity than the siplex ethod. The basic idea of the greedy algorith is to iniize the power charged for each EV in each tie slot. Algorith 2 shows the pseudocode of this greedy algorith. More specifically, given a charging section j and an EV on it, say EV i, if charging section j is the last charging section in the Algorith 2: Greedy algorith for the proble. 1 for each charging section j at tie slot t do 2 if charging section j is the last charging section then 3 Charge each EV i with power 4 else x j (t) = ax{p req,i + P j trac,i z i(t), } 5 Charge each EV i with power x j (t) = ax{p th,i + P j trac,i z i(t), } current charging lane, then the power that charging section j provides to EV i should enable it to ove to the next charging lane or its destination. That is, we should guarantee that the power of EV i is at least p req,i when it leaves the charging lane (line 3). Otherwise, charging section j only needs to provide EV i with the power to enable EV i to ove to the next charging section. That is, we need to guarantee that EV i s power is at least p th,i after the EV is charged (line 5). Here, we assue that GSC has enough power to enable each EV i to leave the charging lane with power p req,i. In the following, we will prove the optiality of the solution obtained fro the greedy algorith in Theore 3.1. We first give Lea 3.1 for the aid of the proof of Theore 3.1. Lea 3.1: The total energy charged will not be lower than ( i T Ptrac,i + p req,i z i(t s i )). Proof: When each EV i enters the charging lane, the total energy required to reach the destination can be easily calculated: P total,i = T P trac,i +p req,i. Then, to guarantee that each EV i can reach its destination, the following condition needs to be satisfied: T x i (t) + z i (t s i) P total,i (26) t=1 where z i (t s i ) eans the energy stored in EV i enters the charging lane and T t=1 x i(t) eans the total energy charged on the charging lane. Accordingly, we can derive that the total power charged by all the charging sections in the charging lane is lower bounded: T t=1 x i (t) = (P total,i z i (t s i)) (27) ( T Ptrac,i + p req,i z i (t s i) ). (28) The proof is copleted. Theore 3.1: The greedy algorith can achieve the optial solution. Proof: We su the aount of power charged by all the

7 STD of the aount of stored power STD of the aount of stored power STD of STD of Avg aount of energy received Total aount of energy received Power-B -B (a) Standard deviation with error bars Fig Power-B -B (b) The average standard deviation Coparison perforance of balancing the s of the EVs. Power-B -B (a) Standard deviation of the aount(b) The average standard deviation of of stored power with error bars the aount of stored power Power-B -B Fig. 4. Coparison perforance of balancing the aount of the stored power of the EVs. Paraeters TABLE III EV PARAMETERS - NISSAN LEAF Value Air density ρ air kg/ 3 Drag Coefficient C d 1 EV frontal area A f EV Mass M 1521 kg Rolling Ressistace Coefficient C r.2 Transission Efficiency η tx.98 Gearbox Efficiency η GB.98 Electric Motor Efficiency η EM.99 Open circuit voltage V oc V Battery Capacity Q batt 66.2 Ah Battery internal resistance R int.1 Ω Battery Noinal Power Pbatt ax 24 kw charging sections in the charging lane: n T ( x j (t) = T Ptrac,i + p req,i z i (t s i) ). (29) j=1 t=1 According to Lea 3.1, there is no solution with lower total aount of charged power than the greedy algorith. Consequently, we can derive that the solution derived by the greedy algorith is the optial solution. IV. PERFORMANCE EVALUATION Experiental Settings. In this section, we evaluate the perforance of our proposed power distribution scheduling solutions. Here, we used MatLab to get the solution of our power scheduling optiization proble, and then used SUMO to apply the solution in the realistic traffic scenario. For the siulation, we varied the nuber of EVs fro 1 to 5 and set (a) The average aount of energy re-(bceived per EV with error bars all Total aount of energy received by EVs Fig Coparison perforance of iniizing the total power charged. the nuber of charging sections to 1. We randoly selected a value in [.4,.8] as the for each EV when it enters the charging lane. We randoly generated the value required for each EV to arrive at its destination. Three types of EVs were considered (Nissan Leaf, Toyota Prius, and Chevy Volt) in our experient [1], [2], [4]. Table III shows the paraeters of EVs and We set the charging section length L, the axiu capacity A, and the coil axiu power C by 2, 6kW, and 1kW, respectively. We repeat each experient for 2 ties. In each experient, the power capacity of the GSC was randoly chosen fro [4-1]Kw. Unless otherwise specified, the experiental result is the average of the 2 experients. We copared our solutions with the following power distribution ethods: sharing ethod. In the equal sharing ethod (denoted by ), suppose there are vehicles on top of a charging lane at a particular tie, the aount of power that each EV is scheduled to receive is A/ where A is the power capacity of that charging lane. First coe first serve ethod. In the First Coe First Serve power distribution ethod (denoted by ), the EVs are assigned to receive power in the order that they arrive at each charging section. Each EV s battery is fully charged until all power in the charging lane is transferred. State of charge ethod. The state of charge ethod (denoted by ) only tries to balance the s of the EVs when they leave the charging lane without considering any other factors (e.g., enough power to arrive at destinations, priorities, etc). A. Experiental Results Balancing the s of the EVs. In this experient, we calculated the standard deviation of s of the EVs for the six power distribution ethods:,,, - B, Power-B, and. Fig. 3(a) shows the edian, the 5th and the 95th percentiles of the standard deviation of s when different nuber of EVs are considered in the power distribution ethods. We see that the edian and the variance of the standard deviation of follow - B<Power-B<<<. The result eans that -B and can balance EVs better than other ethods. -B ais to balance the levels of EVs and also guarantee that the EVs can arrive at their destinations when they leave the charging lane. As a result, the deviation of -B is always sall. The power distribution ethod only considers balancing without considering the EVs

8 destinations. Thus, though its resultant variance is also sall, this ethod cannot guarantee that each EV can arrive at its destination with sufficient power. Our Power-B ethod considers balancing EVs absolute stored power and also their destinations. As a result, its deviation of is larger than and -B but saller than other ethods. The Share power distribution ethod only considers the equal distribution of energy and it does not pay attention to. Thus, its deviation is oderate copared to other ethod. Our proposed power distribution ethod ais to iniize the power distribution which causes larger deviation than our proposed -B and Power-B ethods. Finally, the power distribution ethod distributes power based on EV s arrival tie and its deviation is large. Fig. 3(b) shows the standard deviations of the six ethods. We can also ake the sae observations as Fig. 3(a) due to the sae reasons. The results confir that our -B solution can achieve its goal of balancing the s and also guarantee that the EVs can arrive at their destinations with enough power. Balancing the Aount of the Stored Power of the EVs. In this experient, we calculated the standard deviation of the aount of the stored power for the six power distribution ethods:,,, -B, Power-B, and. Fig. 4(a) shows the edian, the 5th, and the 95th percentiles of standard deviation of the aount of the stored power when different nuber of EVs are considered in the power distribution schee. We see that the edian and the variance of the standard deviation of the stored power follow Power- B< -B<<<. It eans that our proposed Power-B power distribution ethod can balance EVs stored power better than other ethods. The Power- B power distribution ethod considers balancing the stored power levels of EVs and their destinations when they leave the charging lane. Thus, the deviation of the stored power in Power-B is always sall. Since the share power distribution ethod and power distribution ethod do not consider to balance the stored power, their variance are large coparing with other ethods. The ethod and -B ethod consider balancing the s of EVs. Thus, their deviations of stored power are oderate copared to other ethods. Fig. 4(b) shows the standard deviations of the six ethods. We can find that the standard deviation of our Power-B is lower than the standard deviations of other ethods because of the reasons described above. We can also ake the sae observations as Fig. 4(a) due to the sae reasons. Miniizing the Total Power Charged. In order to show the effectiveness of our proposed solution in iniizing the total power charged of the EVs, we copared the perforance of with share,,. Fig. 5(a) shows the edian, the 5th, and the 95th percentiles of the average energy received per EV when different nuber of EVs are considered in the power distribution ethods. We can see that the edian and the variance of energy received per EV follow <<. The result eans that our proposed solution can iniize the energy received per EV than other three ethods. considers iniizing total power charged and the constraints to guarantee that the EVs can finish their trips at the sae tie. Thus, all EVs receive less aount of energy. The power distribution ethod only considers balancing without taking into account of the aount of received energy for EVs. In ethod, the aount of energy received per EV is higher than the energy received in. The share and power distribution ethods distribute all available energy aong EVs. Thus, the levels of energy received by EVs in share and are higher than other ethods. Fig. 5(b) shows the total energy received by all the EVs of the four ethods. We also find that the total energy received follows << due to the sae reasons entioned above. The results confir the effectiveness of our in iniizing the total power charged. V. RELATED WORK The ipleentation of efficient WPT systes and EVs techniques are critical to iprove energy efficiency and safety of in-otion EVs. Several works have studied on the WPT systes and EVs techniques. For exaple, Li et al. [2] presented an analytic study of the existing technologies in the WPT syste applicable to EV wireless charging. Siilarly, Lukic et al. [22] presented the background study, otivations, an overview of different charging coponents of EVs. Hori et al. [17] discussed different types of EVs with their future ipacts, and briefly discussed different EVs coponents with several technical and research challenges based on the existing technologies. Onar et al. [23] exained the technical aspects and charging topology of in-otion wireless power charging of EVs. The authors discussed several factors of power transfer procedures by considering highway surfacing aterials and presented an overview of WPT agnetic field easureents. There are also any efforts devoted to the design and ipleentation of the WPT systes for EVs in recent years. For exaple, Shin et al. [26] presented the design of an optiized core structure and electric coponents to ipleent the WPT syste for oving EVs [6], [9], [1]. Yilaz et al. [29] presented the general design requireents and analysis of WPT systes for online EVs. They presented and analyzed three different generic roadbed structures: 1) based on a long wire loop, 2) based on sectioned wire loops, and 3) based on spaced loops. Ahn et al. [7] presented the design ethodologies and the reduction of electroagnetic fields for high-efficiency WPT systes. The authors also suggested power pickup coils with optiized design paraeters. They also proposed passive and active plate shields to iniize the leakage electroagnetic field fro the WPT syste in online EVs. Lee et al. [19] presented a dynaic odel for identifying the axiu pickup in the WPT syste of the online EVs. In their odel, a siple second-order inductive circuit is obtained by applying the Laplace phasor transforation to the first order WPT syste. The discovered pickup current during the transient state is found to be relatively unchanged

9 for various load resistances. Power pulsations is another ajor technical challenge for the WPT systes of online EVs. The battery size and the positions of power transitters on the road are another ajor issues in the WPT syste design for online EVs. The study by Ko and Jang [18] presents an online charging EV syste and discuss these ajor issues. It tries to iniize the infrastructure setup cost by using an optiization odel, where the battery size and the nuber of power coils with their allocations are used as decision variables. Then, the solution is achieved using particle swap optiization technique. However, this work only considers static nuber of vehicles. Instead, we consider how a power grid controller distributes power so that the heterogeneous EVs have enough power to reach their destinations with different goals. VI. CONCLUSION In a WPT syste, a nuber of in-otion EVs on a charging lane are siultaneously charged by a GSC. Because the power capacity of the GSC is liited, the power deands fro all these EVs ay not be fully satisfied. In this paper, we studied the power distribution scheduling proble about how a GSC distributes power to enable the EVs to receive enough power to reach their destinations and eanwhile achieve a goal. In particular, considering the fairness aong EVs, we forally forulated two probles, called -B and Power-B, with the goals to balance the EVs and stored power, respectively. We showed that -B and Power-B are convex probles, which can be directly solved using an existing ethod, e.g., the subgradient ethod. Also, we forulated a proble, called, to iniize the total power charged to all the EVs, and also designed a greedy algorith that achieves the optial solution for this proble. We have conducted extensive experiental study on our proble solutions. 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