Online Routing and Charging Schedule of Electric Vehicles With Uninterrupted Charging Rates

With the development of electric vehicle technology and the Internet of Things (IoT) technology, IoT-Based electric vehicles (IoEVs) will become a main public transposition in smart city. The increasing penetration of IoEV will have a great impact on the transportation network and power grid. In this work, we aim to study the routing and charging scheduling problem of IoEVs with the constraint of power limit. This paper proposes a clustered rolling framework and a mathematical model for the IoEVs optimal routing and charging problem to minimize the waiting time of customers and get a good converge ratio of the energy block with power limit. This framework can effectively deal with the multi-step pickup and delivery of each vehicle over the time horizon in a dynamic environment. We give a criterion to judge whether pickup and delivery tasks can be assigned to the IoEV. Then, we design a new rectangle packing method for IoEV charging dispatch with the constraint of power limit. Moreover, we consider that the charging process of IoEVs cannot be interrupted before IoEVs meet the electricity demand. We propose the IoEV Routing and Charging Algorithm and analyze its computational complexity. Simulation results show that IoEVs can offer the shuttle service to customers with the minimum waiting time and IoEV charging energy block has a good coverage ratio of the total energy block. Simulation results can verify the effectiveness of our proposed framework, compared with two benchmark algorithms.

can offer shuttle service and other emerging transportation 28 The associate editor coordinating the review of this manuscript and approving it for publication was Francisco Perez-Pinal .
network services, such as Uber, Lyft, Zipcar, and Didi. The 29 development of IoEVs facilitates to reduce the energy con-30 sumption [5]. Accordingly, it is important to design high 31 efficient algorithms and models to boost the utilization rate 32 of IoEVs. 33 In this work, we aim study an IoEV pickup and delivery 34 problem in a clustered rolling framework with the constraint 35 of battery limit. We consider the one-step pickup and delivery 36 problem when there is a customer to pick up and deliver 37 in current time slot. Due to the different pickup and deliv-38 ery service, the electricity demands of IoEVs are differ-39 ent. We set the charging criterion and then collect the idle 40 IoEVs that lack electricity to charge in the charging point. 41 We design a new rectangle packing method for IoEV charging 42 dispatch with the constraint of power limit from power grid.
for IoEV charging dispatch are proposed in Section III. 71 Online IoEV routing and charging algorithm in a clustered 72 rolling framework is proposed in Section IV. The case study 73 and performance evaluation are in Section V. Section VI 74 concludes this paper. 76 For traditional EVs, some deterministic and probabilistic 77 prediction methods have been studied to predict the future 78 power demand [6]. The radial basis function neural network 79 in a deterministic model in [7], only considered EVs' his-80 torical velocity for the prediction. Some methods are not 81 available in the general transportation system and some are 82 mainly utilized in simple traffic scenarios. Compared with 83 traditional EV, IoEV is easier to perceive the surrounding 84 environment equipped with numerous sensors, which can be 85 fully or partially driverless [8]. Without direct human inter-86 vention, IoEVs are easy to dispatch and schedule, which is a 87 tendency of EVs [9], [10]. The traditional centralized man-88 agement method for Internet of Vehicles is difficult to solve 89 the real-time schedule problem [11], [12]. The core models 90 and algorithms for IoEVs sharing service in smart cities 91 are studied to improve the routing and charging strategies. 92 The vehicle routing problem over the time horizon can be 93 formulated as a large scale mixed integer linear programming 94 (MILP) problem, which is very challenged to be solved in 95 real time [13]. The problem of a capacitated vehicle with 96 shuttle service is studied in [14] to find a minimum route. 97 Wang et al. [13] investigated a vehicle routing problem (VRP) 98 with simultaneous delivery and pickup and time windows 99 for five objectives in the logistics industry. Zhou et al. [15] 100 proposed a decomposition-based local search algorithm for 101 large-scale VRP with simultaneous delivery and pickup and 102 time windows. A Hamiltonian graph-guided algorithm is 103 designed in [16] to solve the vehicle routing problem. Deep 104 reinforcement learning method is utilized in [17] to minimize 105 the total travel cost of a selected EV charging station and the 106 charging cost by the real time information. Dubois et al. [18] 107 proposed an algorithm with data from the past crisis to solve 108 this VRP for the rescue vehicles according to operation time 109 and rapid intervention. Inspired by these researches, we can 110 solve the pickup and delivery problem of IoEV routing with 111 time windows.

112
The IoEV charging problem will become increasingly 113 important to the stability and power quality of electricity 114 grid with the increasing penetration of IoEVs into power 115 systems. Some researches are studied about the stability of 116 power grid. Zhang et al. [19] studied delay-tolerant charg-117 ing problem of EVs at a charging station with multiple 118 charge points, which are equipped with renewable generation. 119 Yu et al. [20], [21] studied the time-based pricing mecha-120 nism for heterogeneous charging stations and the charging 121 station allocation mechanism for EVs, and proposed a match-122 ing game framework for the charging stations allocation. 123 ElGhanam et al. [22] proposed an online EV allocation algo-124 rithm with a dynamic wireless charging coordination strategy, 125 VOLUME 10, 2022 can be classified into multiple levels from the fast charging 139 rate to the slow charging rate. Ding et al. [26] formulated  There are some works that combine the vehicle routing 149 problem with the EV charging problem. Zuo et al. [27] 150 studied the VRP of EVs with time-window with logis- A. IoEV PICKUP AND DELIVERY MODEL 181 We consider a set of IoEVs traveling from the starting points 182 to different destinations in a road network, in which there 183 are many pickups and deliveries. The road network can be 184 described as a graph G(N , R), where N represents all the 185 nodes in the network, and R denotes the set of paths con-186 necting nodes in N . In graph G(N , R), there are five types 187 of nodes in N , i.e., starting points of IoEVs N 1 , end points 188 of the IoEVs N 2 , charging stations N 3 , pickup nodes N 4 and 189 delivery nodes N 5 , which are shown in Fig. 1. In actual case, 190 IoEVs start from the starting points and end at the ending 191 points. Starting points N 1 and ending points N 2 are certain 192 but the number of customer is not certain and sets of pickup 193 and delivery are scalable when we assign IoEVs to pick up 194 and deliver the customers. The starting point and end point 195 are special nodes because IoEVs only can flow out of the 196 starting point and flow into the end point. We define the travel 197 distance between nodes i and j as D ij , if nodes i and j are 198 directly connected. Notice that node i is starting point or node 199 j is end point, D ji = inf, which means that node j cannot flow 200 into node i and no node can flow into starting node or flow 201 out of the ending node. Otherwise, The routing indicator matrix D can be formed by D ij accord-204 ing to the network topology structure. A small example for six 205 nodes which are not starting point or ending point is depicted 206 in Fig. 2 with the corresponding matrix D as follows,  To formulate the IoEV routing and charging problem, 215 we define the binary variable x k,u ij to indicate if IoEV k is 216 scheduled to route customer u between node i and j as binary 1 IoEV k is scheduled to customer u between nodes i and j, 0 otherwise.
(3) 219 The starting node i ∈ N 1 must have only one path flowing 220 out of it, The end node j ∈ N 2 must have only one path flowing in it, For nodes which are neither the starting and end nodes, there 225 is less than one path flowing out of or into the node, which 226 satisfies the following two constraints, The left hand side of constraint (6) and constraint (7) can be 232 zero, which implies that there is none IoEV passing through is no constraint on this node i.
Constraint (9) indicates that if there is an IoEV leaving a node 242 j that is neither a starting node nor an end node, it should flow 243 into this node beforehand. Otherwise, there is no constraint on 244 this node j. (10) indicates that an IoEV has supported a pickup 250 and deliver service to one customer, The sum of service flag y k u of customer u should be one to 254 ensure that every customer can be served by only one IoEV, 255 where U is the set of all customers. In this work, we study 257 the case where each IoEV k can only offer service to one 258 customer when it finishes delivering, i.e., where IoEV k may not offer service when u∈U y k u = 0. 261 Given customer u, each starting point s ∈ N 1 (u) must have 262 one way to leave from starting node s, where N 1 (u) is the 263 starting node set, so we have, where s ∈ N 1 (u) represents the starting node that has access 266 to connect with customer u and N out (s) = N out NC (s) ∪ N out 3 (s) 267 denotes all outflow nodes connected to node s. N out NC (s) and 268 N out 3 (s) denote nodes that are not available charging nodes 269 and are available charging nodes in N out (s), respectively. 270 Similarly, for given customer u, each end point e ∈ N 2 (u) 271 must have only one way to reach ending node e, where N 2 (u) 272 is the end node set, so we have, where e ∈ N 2 (u) represents the ending node that has access 275 to connect with customer u and N in (e) = N in NC (e) ∪ N in 3 (e) 276 denote the all inflow nodes connected to node e. N in NC (e) and 277 N in 3 (e) denote the nodes that are not available charging nodes 278 and are available charging nodes in N in (e), respectively. 279 When node i is neither the starting point nor the end point, 280 inflow should equal outflow at this node, i.e., 281 m∈N in (i)   IoEV k between nodes i and j as v k ,

331
where V k is the ideal speed of IoEV k, Q i,j is traffic flow of 332 nodes i and j, C r is road capacity, β and γ are experimental 333 parameters from actual observations. Then the equivalent 334 traveling time of nodes i and j can be given as follow, We set IoEV k assigned to customer u at node i at time 337 slot T k,u and we need to minimize the total waiting time of 338 customers as follows, where D i,u p is the distance between node i of IoEV k and the 341 pickup node u p of customer u. For the reference time for the 342 start node s, we set the initial time stamp of IoEV k as T k,0 = 343 0. IoEV k is assigned to customer u at time slot T k,u . After the control center receives the signal from the cus-349 tomers, it will schedule IoEVs to pick up and deliver the 350 customers in a time frame. There is a charging optimization 351 problem of idle IoEVs, where there are two problems: when 352 and which charging station to charge. In time slot t, the 353 control center can make out which IoEV k is out of service, 354 i.e., u∈U y k u = 0. We denote the set of IoEVs out of service 355 in time slot t as H t . When an IoEV is out of service, then this 356 IoEV is qualified to be charged. By charging strategy, IoEV k 357 sends a charging signal to the control center, which includes 358 the requested charging amount B c k .

359
We study a two-dimensional-rectangle packing problem 360 for the uninterrupted charging problem of IoEVs with the 361 constraint of power limits from power grid. Due to the unin-362 terrupted charging characteristic, we can see the EV charging 363 as a energy block. The two-dimensional-rectangle packing 364 problem can be efficient to solve by the optimization toolbox. 365 IoEVs can support the pickup and delivery service when 366 they have enough electricity. For those idle IoEVs that lack 367 electricity, we need to do the charging schedule of IoEVs. 368 The maximum charging rate is limited by power limit of 369 the charging station (i.e., P max ). We formulate our charging 370 optimization model under a maximum allowed parking time 371 T max . When it reaches the time limit T max , the IoEV should 372 stop charging and leave the charging station. During the 373 time interval [0, T max ], the base blocks are represented as 374 histogram with B blocks. In Fig. 3, we show that there are 375 a certain number of IoEVs coming to charging station during 376 time period [0, T max ]. Due to the limitation of current charg-377 ing technology, the charging rate should be discrete in the 378 charging process and the charging process is not interrupted 379 until the electricity demand is met. Therefore, we skillfully 380 consider the charging amount in a time interval of IoEV k 381 as a rectangle energy block. The rectangle energy packing 382 problem is challenging to solve. The n−th (n ∈ {1, . . . , N }) 383 type rectangle energy block can be described by a triplet 384 (CT n , P n , X n ), where CT n is the charging duration time, P n is 385 the charging rate and X n = P n * CT n is the charging amount.

386
The IoEVs can be charged at N type of charging rates, e.g., 387 if IoEVs charge at fast, medium and slow charging rate, can be expressed as follows,

400
where (25) and (26) show the X-axis and Y-axis relationship 401 between τ K +b and T K +b about left-bottom corner of the base 402 load b. We denote a binary variable z i,n that indicates whether 403 n-th type rectangle energy block is charged at time slot τ i .

404
We need to guarantee that the four corners of each rectangle 405 energy block i is within the rectangular area with allowed 406 parking time T max and power limit P max . The constraints can 407 be represent in the following constraints.

419
As shown in Fig. 4, the location of two energy blocks i 420 and j has four possible non-overlapping relationship in the 421 following expression when the packing is feasible.
where we can set M 1 = P max and M 2 = T max . To satisfy the 465 energy limits from power grid, we need to schedule as many 466 IoEVs as possible to charge in a fixed time frame T max with 467 the power upper limit P max of the charging station, which is 468 formulated as follows,   Fig. 6. The white dots are IoEV's 504 starting points, the brown dots are the couple of pickup and 505 delivery points of customers, the dark dots are the ending 506 points, and the green dots are the charging nodes. In time slot 507 t 2 , we need to solve a larger optimization problem in Fig. 5, 508 compared with the optimization model in Fig. 6. Therefore, 509 we study the routing part of this optimization problem in each 510 current time slot by the rolling windows method.

511
If pickup and delivery services of customers come up one 512 by one, it is easy to figure out the routing problem of IoEVs. 513 We focus on the optimal routing problem that the pickup 514 and delivery services of customers come up simultaneously. 515 First, we generate a regional map partitioned from the whole 516 map with clustering method, which contains some pickup and 517 delivery nodes of customers and charging nodes. We schedule 518 IoEVs to study the optimization problem in the small region. 519 Second, we calculate the minimum distance from the delivery 520 nodes of customers to all the charging stations in the regional  drop-off node u d , we can assign IoEV k to customer u. If the 549 energy of IoEV k is not sufficient to pick up and deliver any 550 customer and reach the nearest charging node connected to 551 drop-off node, IoEV k cannot be arranged to support pickup 552 and delivery service. When an IoEV is out of service, this 553 IoEV is qualified to be charged. After checking with the 554 charging strategy, IoEV k sends the control center a charging 555 signal, which includes the requested charging amount B c k .

556
After the control center receives the signal from the cus-557 tomers, it will schedule IoEVs with plenty of electricity to 558 pick up and deliver the customers. For the idle IoEVs with 559 insufficient energy, there is a charging optimization prob-560 lem. If there are a large number of IoEVs to charge, it will 561 influence the stability of power grid. We consider that there 562 are a certain number K of idle IoEVs to charge in a fixed 563 time frame T max with power upper limit P max of the charging 564 station. It is a two-dimensional rectangle packing problem for 565 IoEV charging dispatch. We need to decide when to charge 566 those idle IoEVs. Here, the worst-case computational com-567 plexity of IoEV charging dispatch has the following theorem. 568 Theorem 2: The worst-case computational complexity of 569 two-dimensional rectangle packing problem for IoEV charg-570 ing dispatch is O Ψ x (T max )Ψ y (P max )(Ψ x (T max )+Ψ y (P max )) .  Ψ y (P max )(Ψ x (T max )+Ψ y (P max )) for each recursively sub-cut      Fig. 8. The maximum power is set as 614 133 KW and the time frame is set as 12 h, which is broken 615 down into 720 min. The charging efficiency is set to be 100%. 616 The routing simulation runs in a Matlab environment on a 617 general computer.

618
The optimal solution of U1 is that IoEV 1 at node 1 serves 619 the pickup and delivery {3, 17} of customer 1, IoEV 2 at 620 node 2 serves the pickup and delivery {4, 22} of customer 2, 621 IoEV 3 at node 12 serves the pickup and delivery {11, 15} 622 of customer 5, IoEV 4 at node 24 serves the pickup and 623 delivery {23, 16} of customer 4, IoEV 5 at node 21 serves 624 the pickup and delivery {5, 14} of customer 3, and IoEV 6 at 625 node 20 serves the pickup and delivery {19, 10} of customer 6. 626 The route solutions and waiting time of customers are shown 627 in Table 3 IoEVs that need to charge are out of routing service. 642 They wait in the charging station to charge when there are 643 37 IoEVs. We decide when these IoEVs are charged under the 644 constraint of power limit 133 kW and time limit 12 h. We con-645 sider three charging rate: fast charging 60 kW (C1-type), 646 medium charging 6 kW (C2-type) and slow charging 3 kW 647    Fig. 9. 652 We evaluate the performance of ERC algorithm by using   The rectangle packing problem for IoEV charging dispatch 663 is simulated in Gurobi 9.0 [38] and the result is shown in 664 Fig. 9 and the total load is shown in Fig. 10. The left-bottom 665 corner of each rectangle shows the optimal IoEV charging 666 time in Fig. 9. There are 25 C1-type, 11 C2-type and 1 667 C3-type energy blocks. The running time for rectangle pack-668 ing problem of IoEV charging dispatch is 43.67 s. We can 669 see that the coverage ratio of available IoEV charging energy 670 block is 85.65% from Fig. 10, where the coverage ratio is the 671 rate of the available IoEV charging energy block to the total 672 energy block. If IoEVs are charged by 60% of the battery 673 capacity, the running time for rectangle packing problem of 674 IoEV charging dispatch is 13.39 s. There are 34 C1-type and 675 15 C2-type energy blocks. The scheduling result for 49 IoEVs 676 by 60% battery capacity is shown in Fig. 11 and the coverage 677 ratio of available IoEV charging energy block is 85.21%. 678 If we charge less, we can arrange more IoEVs to satisfy the 679 energy limits from power grid. We need to change the planned 680 number of IoEVs when the charging amount of IoEVs out of 681 routing service is changed. 682 We set the electricity price for sale as a constant 0.2$/kWh, 683 then the result of four charging scenarios for two power 684 limits 133kW and 200kW are shown in Table 4. Scenario 'A' 685 (only C1) allows the fast charging rate, scenario 'B' (only 686 C2) allows the medium charging rate, scenario 'C' (only 687 C3) allows the slow charging rate and scenario 'D' allows 688 the mixed kinds of charging rates. The simulation results 689 show that coordinating three types of charging rates can 690 achieve the optimal charging dispatch. The optimal solution 691 of scenario 'D' increases by 32.43%, 13.51%, 10.81% for 692 the case where the power limit is 133 kWh and increases by 693 19.35%, 12.90%, 11.29% for the case where the power limit 694 is 200 kWh, compared with that in scenario 'A', 'B', and 'C'. 695 For the coverage ratio of the total energy block, we com-696 pare the performance of our ERC algorithm with that of 697 LLABF and BLF algorithms in two scenarios, which is shown 698 in Table 5. We can see that in the scenario where EVs need to 699 charge up to 80% of the battery capacity, the coverage ratio 700 of the total energy block of our ERC algorithm is 85.65%. 701 In this scenario, the coverage ratio of the total energy block of 702 LLABF and BLF algorithm are 66.35% and 76.88%, which 703 is 22.53% and 10.24% lower than ERC algorithm. In the 704 scenario where EVs need to charge up to 60% of the battery 705 capacity, the coverage ratio of the total energy block of our 706 ERC algorithm is 85.21%. In this scenario, the coverage ratio 707 of the total energy block of LLABF and BLF algorithm are 708 64.16% and 80.90%, which is 24.70% and 5.06% lower than 709 ERC algorithm. We can see that ERC algorithm has the better 710 performance than LLABF and BLF algorithms. Our ERC 711 algorithm has the largest coverage ratio of the total energy 712 block with power limits. 713 VOLUME 10, 2022

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In this work, we propose a clustered rolling framework for 715 optimal routing and charging of IoEVs. We take the waiting 716 time of customers and power limit of power grid into con-