UE-based Location Model of Rapid Charging Stations for EVs with Batteries that Have Different States-of-charge

1 The aim of this research is to develop a location model of rapid charging stations for electric vehicles (EV) 2 in urban areas considering the batteries’ state-of-charge (SOC) and the users’ charging and traveling 3 behaviors. EVs are developed to prepare for the energy crisis and reduce greenhouse gas emissions. In 4 order to help relieve range anxiety, an adequate number of EV charging stations must be constructed. In 5 urban areas, the construction of rapid charging stations is needed because there is inadequate space for 6 slow-charging equipment. The objective function of the model is to minimize EVs’ travel fail distance 7 and total travel time of the entire network when the link flow is determined by user equilibrium (UE) 8 assignment. The remaining fuel range (RFR) at the origin node is assumed to follow a probabilistic 9 distribution in order to reflect users’ charging behavior or technical development. The results indicate 10 that the location model described in this paper can identify locations for charging stations by using a 11 probabilistic distribution function for the RFR. And the location model, which is developed based on UE 12 assignment, is likely to consider the congested traffic conditions of urban areas in order to avoid locating 13 charging stations where they could cause further traffic congestion. The proposed model can assist 14 decision makers in developing policies that encourage the use of EVs, and it will be useful in developing 15 an appropriate budget for implementing the plan. 16 17 18 TRB 2014 Annual Meeting Paper revised from original submittal. Lee, Kim, Kho, and Lee 3 INTRODUCTION 1 The electric vehicle (EV) is one of the most popular alternative-fuel vehicles (1). However, range anxiety 2 has restricted the pace at which EVs have penetrated the market. The construction of an adequate number 3 of EV charging stations can help relieve this range anxiety (2). Considering the budget constraints, 4 choosing where such stations should be located is an important issue. Access to EV charging stations will 5 impact the use rates of EVs, decisions concerning their use, the percentage of miles attained with 6 electricity, the demand for petroleum, and power consumption at various times during the day (3, 4). So 7 the problem of properly locating EV charging stations is an essential topic, and some important studies 8 have been conducted in the past few years (5-10). 9 To formulate a practical model for determining the appropriate locations for EV charging 10 stations, serveral variables must be considered, including the vehicle range (VR), batteries’ state-of11 charge (SOC), users’ charging behavior, and travel preferences. In the early stages of EVs, the targeted 12 consumers were people who traveled almost exclusively within the urban area (11). In existing models for 13 locating EV charging stations in urban areas, slow-charging equipment was targeted, and the objectives of 14 the existing studies were to optimize the total usage of electrical power, maximize profit, and minimize 15 costs. The location of rapid charging stations in urban areas also is very important because adequate space 16 cannot be made available to accommodate the larger numbers of slow-charging equipment that would be 17 necessary. However, most studies are based on parking behavior, and there is a lack of research on 18 charging on route. Rapid charging stations in urban area can help increase accessibility to charging to a 19 greater extent than such stations could in rural areas. Therefore, current planning involves establishing 20 charging stations first in urban areas and then expanding their availability to intercity roads (12). However, 21 rapid charging stations, at which EV users can recharge during their trips, have not been considered in the 22 most of the studies. 23 Flow-refueling location models (FRLMs) have been developed to find adequate location of gas 24 station for vehicles that need refuel during their trip. FRLMs for alternative-fuel vehicles are extended 25 models of flow-capturing location models (FCLMs) that were developed for convenience stores by 26 Hodgson (13, 14). An FCLM is a maximum covering model, and it cannot handle the multiple refueling 27 stations needed for paths longer than the VR. Vehicle range (VR) is the distance that a vehicle can travel 28 when it is fully charged, and FRLMs can be extended by adding the vehicle-range constraint. 29 However, it is difficult to apply FRLMs to urban areas for two reasons. First, the travel paths 30 used in these models were determined exogenously. In the existing models, all of the vehicles from the 31 same origin-destination (OD) pair must be assigned to one path. This is reasonable for inter-city trips, but 32 it would rarely occur in urban areas, because numerous alternative paths are available. In practice, drivers 33 can detour to charge their vehicles they so desire. Some studies have considered such detours, but only 34 from the standpoint of the probability of trip availability (15, 16). Second, the SOC level was assumed to 35 be 0.5 because only the marginal case of alternative fuel vehicles was considered in the studies of FRLMs 36 TRB 2014 Annual Meeting Paper revised from original submittal. Lee, Kim, Kho, and Lee 4 (17). If each vehicle has a different remaining fuel range (RFR), the path used by each vehicle may be 1 different. RFR can be calculated by multiplying VR by the SOC level of the battery. When a constant 2 RFR is used, the number of stations that should be constructed can be underestimated in urban areas. 3 Summarizing the literature, two variations are required to develop the location model for EV 4 charging stations in an urban area. One is determining the travel path endogenously, and the other is 5 assuming probabilistic RFR. In this way, the model can simulate EV users’ behaviors reasonably and 6 extend the results by enhancing batteries’ volume or charging performances. The summary of the 7 literature review and the contribution of this study are presented in Table 1. 8 9 TABLE 1 Literature Review and the difference of this study (5-10, 14-24) 10 Previous studies Objective function Station Spatial scope RFR Travel path Kuby and Lim (2005) Minimum failure AFV INT D EXO Wang (2009) Minimum number of stations AFV D EXO Upchurch et al. (2009) Minimum failure cost AFV INT D EXO Wang and Lin (2009) Minimum number of stations AFV INT D EXO Kim (2010) Minimum failure AFV INT D EXO Ip et al. (2010) Minimum operational cost EV Wang and Wang (2010) Minimum failure + construction cost AFV INT D EXO Hanabusa and Horiguchi (2011) Entropy maximization EV Ge et al. (2011) Minimum users’ loss EV Wang et al. (2011) Maximum net income BS Frade et al. (2011) Maximum covering EV-S INN Kim and Kuby (2012) Minimum failure AFV INT D EXO Capar and Kuby (2012) Minimum failure AFV INT D EXO Chen et al. (2013) Minimum access cost EV-S INN Proposed Model Minimum failure cost + network cost EV-R INN P ENDO note: -= not considered, AFV = station for alternative fuel vehicle, EV = station for electric vehicle, 11 BS=battery switch station for EV, EV-S=slow-charging station for EV, EV-R=rapid charging station for 12 EV, INT = intercity, INN = inner-city, D = deterministic, P = probabilistic, EXO = exogenously 13 determined (all-or-nothing assignment), ENDO = endogenously determined (user equilibrium assignment) 14 15 TRB 2014 Annual Meeting Paper revised from original submittal. Lee, Kim, Kho, and Lee 5 The aim of the research reported in this paper is to develop a location model of rapid charging 1 stations considering vehicles’ ranges, batteries’ SOC, and users’ charging and travel behaviors. The 2 model was formed as a bi-level optimization model in which the main problem was formulated to 3 determine the locations of the stations and the patterns of use by EVs. The sub-problem was formulated to 4 determine link flow based on the user-equilibrium principle. To solve the problem in reasonable time, a 5 modified, simulated annealing algorithm was proposed. The applicability of the model was tested in a 6 network in the example networks 7 The structure of this paper is as follows. In section 2, the location model of rapid charging 8 stations based on the user-equilibrium principle is formulated, and a heuristic algorithm is proposed that 9 assists in efficiently determining the approximate solution of the problem. In section 3, potential 10 applications of the proposed model on a simulated network are performed, and the results are compared 11 with those provided by existing methods. Our conclusions and recommendations for future research are 12 presented in section 4. 13 14 MODEL FORMULATION 15 The proposed model is an uncapacitated facility location problem to minimize travelers’ costs. The model 16 is a modification of a P-median problem combined with a user-equilibrium problem. The proposed model 17 was based on the following considerations. First, the vehicle range is assumed to be longer than the 18 distance between the origin and destination for all OD pairs. This means that the EV’s battery would not 19 have to be recharged more than once during the trip. Thus, we can eliminate the vehicle-range constraint 20 from the FRLM. 21 Second, it was assumed that the remaining fuel range at the origin node followed a probabilistic 22 distribution. The state of technical development or the supply of slow-recharging equipment at the origin 23 node could affect the SOC. An EV that has a long range must be charged longer than an EV that has a 24 shorter range. If charging time becomes shorter or larger numbers of slow-recharging equipment are 25 offered, the SOC level can be higher. If the remaining fuel range is assumed to be constant, as Capar and 26 Kuby did (18), the RFR function is a unit impulse function. The cumulative distribution function of RFR, 27 which is used in the literature, can be written as shown in equation 1. In this study, the RFR function was 28 assumed to be a probabilistic distribution function, such as a uniform distribution, an increasing 29 distribution, and a triangular distribution. The distribution functions that were assumed in this study are 30 shown in equations 2-4. 31 Constant function : G(r) = �0 if 0 ≤ r < 0.5rV 1 if 0.5rV ≤ r ≤ rV 1) Uniform function : G(r) = 1 rV × r 2) TRB 2014 Annual Meeting Paper revised from original submittal. Lee, Kim, Kho, and Lee 6 Increasing function : G(r) = 1 rV2 × r2 3) Triangular function : G(r) = � 2 rV2 × r2 if 0 ≤ r < 0.5rV 4 rV × r − 2 rV2 × r2 − 1 if 0.5rV ≤ r ≤ rV 4) where: 1 r = remaining fuel range in distance 2 rV= vehicle range in distance 3 G(r) = cumulative distribution function of r 4 5 The probability of how many EVs can travel with or without charging can be calculated by the 6 RFR distribution function. Figure 1 shows the relationship between the type of RFR distribution function 7 and trip ratio. An increasing distribution may be found when people can easily charge their parked EVs. 8 However, a uniform or a triangular distribution may be found when sufficient slow-charging equipment is 9 not installed to accommodate the requirements of the EVs. 10 11 Flow from i to j : 100 Path used for trip without recharging : i → j Path used for trip with recharging : i → k(station) → j Distribution Constant distribution Uniform distribution Triangular distribution Increasing distribution rV = 20 Trip failure 0 35 25 13 Trip success 100 65 75 87 without recharging 0 25 12 43 with recharging 100 40 63 44 rV = 40 Trip failure 0 18 7 4 Trip success 100 82 93 96 without recharging 100 62 71 85 with recharging 0 20 22 11 FIGURE 1 RFR distribution functions and travel path by RFR. 12 13 Third, trips are classified by users’ recharging behaviors. An existing fossil-fuel vehicle can be refueled 14 easily, because there are many gas stations available. But there are far fewer charging stations for EVs 15 than there are gas stations for fossil-fuel vehicles, so it was assumed that, before departure, the drivers of 16 TRB 2014 Annual Meeting Paper revised from original submittal. Lee, Kim, Kho, and Lee 7 EVs chose where they would recharge their EVs. A user’s behavior is determined based on the remaining 1 fuel range displayed in the vehicle’s instrument panel. If the RFR is greater than the distance between the 2 origin of the trip and its destination, the travelers can go to their destination without recharging, while 3 they must locate a charging station or travel using gasoline when the RFR is less than the distance to the 4 destination. Here, we propose two new decision variables, y�ijk and y�ij, to divide the entire trip into 5 three groups. The decision variable y�ijk is trip ratio travels via station k among whole trip from origin 6 i to destination j. The decision variable y�ij is the ratio who cannot travel with EV. Travelers who can 7 no longer travel in their EVs may travel by public transportation, taxi, or incur the cost of an emergency 8 service to recharge their EV. In the flow refueling location model, travel paths are given as input data. As 9 shown in Figure 2, three paths are available with the same distance from node 1 to node 8, i.e., 1-2-4-8, 110 5-4-8, and 1-5-7-8. In the existing model, just one path is available among the three paths. Here, we 11 assume that the consumption of the battery’s charge is proportional to the distance traveled because the 12 RFR is displayed in kilometers, and users usually use this information to decide whether their EVs must 13 be charged on their routes or not. 14 15 Travel path FRLM (17-19) Proposed model Path used when the station is located at node 2 Path used when the station is located at node 5 note: The RFR distribution function is assumed to be a uniform function. The link travel time is assumed 16 to be constant. 17 FIGURE 2 Travel paths by location of charging station. 18 19 Finally, a traveler’s trip follows the user-equilibrium principle in terms of mean travel time, 20 while the purpose of locating rapid charging stations is to minimize social costs, including travel time cost 21 and the penalty associated with EV travel failure. The notation for formulating the model is as follows: 22 23 (a) Sets 24 N = node set, indexed by n (N ∋ n) 25 TRB 2014 Annual Meeting Paper revised from original submittal. Lee, Kim, Kho, and Lee 8 A = link set, indexed by a (A ∋ a) 1 I = set of origin nodes, indexed by i (I ⊆ N, I ∋ i) 2 J = set of destination nodes, indexed by j (J ⊆ N, J ∋ j) 3 H = set of paths connecting O-D pair −j , indexed by h (H ∋ h) 4 K = set of candidate nodes, indexed by k or k′ (K ⊆ N, K ∋ k, k′) 5 6 (b) Number of location 7 P = number of charging stations 8 9 (c) Weights 10 γ = additional penalty of failed travel 11 ω = weight of γ ; ω = γ/(1 + γ) 12 13 (d) Variables about remaining fuel range 14 r = remaining fuel range 15 gi(r) = probability distribution function about remaining fuel range on origin node i 16 Gi(r) = cumulative distribution function of gi(r) 17 G�ij = failure ratio travel between O-D pair −j ; G�ij = Gi(cij) 18 19 (e) Node-based variables 20 Qij = total demand between O-D pair i − j 21 y�ijk = charging ratio at the station k of travel between O-D pair i − j (0 ≤ y�ijk ≤ 1 ) 22 y�ij = failure ratio of travel between O-D pair i − j 23 zk = �1 if we locate at candidate node k 0 otherwise 24 vik = �1 if station k is the nearest from origin node i 0 otherwise 25 cij = minimum fuel consumption between O-D pair i − j 26 ξikk′ = � 1 if cik ≤ cik′ (∀i, k, k′) 0 otherwise 27 28 (f) Link-based variables 29 xa = flow on link a 30 ta = travel time on link a 31 ca = fuel consumption on link a 32 33 TRB 2014 Annual Meeting Paper revised from original submittal. Lee, Kim, Kho, and Lee 9 (g) Node/link/path-based variables 1 fh ij = flow on path h between O-D pair i − j 2 δah ij = indicator variable; δah ij = �1 if link a is on path h between O − D pair i − j 0 otherwise 3 4 The proposed model is a bi-level optimization model. The main problem is a location-allocation 5 problem to minimize the social cost, while the sub-problem is the trip assignment problem based on the 6 user-equilibrium principle. In the main problem, the EV trip failure ratio, the trip ratio via each station for 7 each origin-destination pair, and the location of charging station are determined. The equations were 8 formulated based on the general facility-location problem (25). In the sub-problem, link flow was 9 determined. The equations in the sub-problem are the modification of Beckmann’s mathematical 10 programming (26). The mathematical model is as follows. 11 12 Main Problem: 13 min �ω��Qijcijy�ij + (1 − ω)�xata(xa) a j i � 5)