# Prediction of Traffic Flow considering Electric Vehicle Market Share and Random Charging

Authors: Yunjuan Yan; Weixiong Zha; Jungang Shi; Liping Yan
Journal: Journal of Advanced Transportation
 (2023)

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## Abstract

This paper mainly studies the multiclass stochastic user equilibrium problem considering the market share of battery electric vehicles (BEVs) and random charging behavior (RCB) in a mixed transport network containing electric vehicles and gasoline vehicles (GVs). In order to analyze the random charging and path choice behaviors of BEV users and extract the differences in travel behaviors between BEV and GV users, an improved logit-based model, multilabel algorithm, and queuing theory are applied. The influencing factors of charging possibility mainly include the initial state of charge (SOC), the SOC at the beginning of charging, and the psychologically acceptable safe SOC threshold arriving at the destination. Diversity choices of user paths and charging locations will result in changes in queuing traffic and differences in queuing time. Conversely, different stations have different queuing dwell times, which will also affect the routing and charging locations for BEVs with RCB. The path-based method of successive averages (MSA) is adopted to solve the model. Through the simulation of the test network Sioux Falls, the equilibrium traffic flow and possible charging flow under different market shares and initial SOC are predicted, and the properties of the model and the feasibility of the algorithm are verified.


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## Body

## 1. Introduction

Since the 21st century, the global energy and environment have become extremely severe, and the contradiction between rapid economic growth and resources has become increasingly prominent. BEVs deliver many benefits, such as low greenhouse gas emissions and high energy efficiency, compared with GVs. Therefore, BEVs are considered one of the main competitors in reducing emissions in the transportation sector, which has grown rapidly in recent years.However, there are several unsolved problems to be addressed, one of which is the “range anxiety” problem [1]. Electric vehicle (EV) users can often feel anxious about depleting their battery before they reach their destinations. As a matter of fact, the deployment of charging stations (CS) has been recognized as a crucial factor for the wide adoption of EVs [2]. In the recent decade, numerous efforts have been devoted to selecting the strategical locations and sizing of CSs for EVs.Erdoğan et al. [3] generated the best alternative solution for the location of charging stations with the goal of maximizing the total traffic flow. Morán et al. [4] used the “SUMO” simulator interface to simulate the vehicle traffic behavior in the different traffic scenarios and use it as the main variable to determine the location of the electric vehicle charging center or charging station. Campaña et al. [5] presented the multicommodity flow problem (MCFP) algorithm for sizing and allocating resources to CS infrastructures for EVs in a heterogeneous transport system, considering trajectories and restrictions of capacity in each section of the road. From the perspective of microscopic analysis of traffic flow, the above-given two references can identify travel patterns in vehicles and their effect on different levels of traffic, so as to make the best decision when carrying out large-scale projects including electric vehicles. Gao et al. [6] proposed a bilevel model to depict the interaction between traffic flow distribution and the location of CSs in the EVs and GVs hybrid network. With BEV’s increasing share in the automobile market, it will inevitably affect the traffic network. Kong et al. [7] used dynamic charging demand in the location planning of EVCS, taking into account the dynamic traffic flow and the relationship between the vehicle’s current location, destination, and candidate location of the charging station. The above-given studies have completed the deployment of CS either from the perspective of microtraffic flow simulation or network traffic flow.Considering the charging station capacity, charging time, and waiting time, some researchers combined queuing theory to build the CS planning models and determined the optimal locations of charging stations and the optimal scale (the quantity of chargers and waiting spaces). Xiao et al. [8] proposed an optimal location model to determine the optimal locations and capacities of EV charging infrastructure to minimize the comprehensive total cost, which considers the charging queuing behavior with finite queue length and various siting constraints. Zhang et al. [9] controlled CS service quality by a two-stage priority queue to optimize the station, battery inventory level, and the number of superchargers at selected stations. Choi and Lim [10] proposed a queuing model and two congestion control policies based on EV queue length thresholds. Wang et al. [11] used the M/G/k system to model the queuing of charging stations and incorporate it into an expanded network to capture the optimal recharging strategies for EV drivers.Similarly, the route choice of BEV users in the transportation network is also affected by charging demand, the deployment of CSs, and queuing time. Li et al. [12] proposed a public charging infrastructure localization and route planning strategy for logistics companies based on a bilevel program. Froger et al. [13] introduced a framework to solve EV routing problems with CS capacity constraints in general. Ashkrof et al. [14] aimed to analyze the influencing factors of route selection and charging behavior of BEV drivers. In addition to classic route attributes and variables related to rapid charging (such as charging time and waiting time), the SOC at the start and end of the journey is also an important indicator that affects the driver’s decision on route and charging behavior. EV routing problem with backhauls was solved considering the location of charging stations and the operation of the electric power distribution system [15]. Koç et al. [16] presented an electric vehicle routing problem with a shared charging station (E-VRP-SCS), optimal charging station location, and delivery route planning based on ALNS and mixed-integer linear programming. Therefore, quantifying the above variables and analyzing their impact on BEV routing and charging selection has become a new challenge for traffic flow allocation models in mixed transport networks.In the past, BEV users always worried about the return trip and generated range anxiety due to short driving range, inadequate charging infrastructure, and long charging time. BEV users determine their feasible path by implementing mileage constraints and then generate them for traffic assignment. Since a feasible path may thus consist of several relays, the problem is called the optimal path problem with relays (OPPR). BEVs that need to be charged immediately must be charged via CS. In recent years, the range of BEVs has increased considerably, which allows the BEV routes to be freely chosen; i.e., not all BEV routes have to go through CSs. In China, most users choose BEVs with a driving range of 120 km to 160 km because the price of vehicles within this range is relatively low and can meet the travel demand of medium-sized cities (diameter of 80 kilometers). Power consumption is very persistent when SOC is between 50% and 100%, but very fast when the SOC is below 50%. BEV users can selectively charge based on a psychologically acceptable safety SOC and have more flexible charging behavior. Charging behavior has changed from the inevitability of the past to the randomness of the present. RCB plays a role in influencing drivers’ personal travel choices, corresponding road congestion, and network performance. With the interaction of BEVs and GVs, the route choice behavior of users will become more complicated. The traffic assignment problem of the mixed traffic network with RCB deserves our study.In a mixed network with both BEVs and GVs, the layout of CSs and the distribution of traffic flow interact with each other. The detailed analysis of BEV users’ behavior and the accurate prediction of the charging demand are very important for designing a reasonable layout of CSs. With the location of alternative charging facilities known, the path selection of BEV users and the resulting changes in traffic patterns can be studied by combining the above variables to analyze the rationality of CS layout and the evolutionary trend of link traffic flow. This paper is devoted to analyze BEV users’ charging and route choice behaviours and random charging behavior based on an improved logit discrete choice model. Travel time cost, SOC, queuing dwell time cost, and electricity consumption cost are considered in calculating the path choice generalized cost. A mixed stochastic user equilibrium model is established to study the interaction of BEV users and GV users in an urban network with random charging demand, to predict the changing trend of network traffic flow and charging flow, and to verify the rationality of charging layout from the perspective of demand to provide a basis for decision makers.The rest of this paper is organized as follows. Section2 describes the related work. Section 3 describes the model’s assumptions, notations, the method of determining the set of alternative paths, and constructs an improved logit stochastic user equilibrium model. In Section 4, we apply the algorithm to the test network Sioux Falls, comparing and analyzing the equilibrium traffic flows and charging traffic flows at different market shares. Conclusions and future research are provided in Section 5.


## 2. Related Works

The traffic assignment problem (TAP) describes how the traffic demand is distributed on the available routes in the traffic network. The core of any traffic assignment method is the route choice model. The deterministic user equilibrium problem (DUE) was the most studied example in which the route choice assumption states that drivers behave as if they have perfect knowledge of route costs and select the best route to minimize their travel costs. However, the DUE principle is recognized to be unrealistic because it assumes that all travelers have accurate perceptions of the transportation network. Daganzo and Sheffi [17] proposed the principle of stochastic user equilibrium (SUE) to capture travelers’ perception errors of travel time, to compensate for the absence of the UE principle that all travelers have an accurate perception of the traffic network. Although the Probit model can obtain a path flow solution that is more in line with the actual situation, it needs to estimate the path selection probability through analytical, simulation, numerical integration, and other methods, and the calculation is relatively complex, so it is not commonly used in practice.The logit model is widely used in traffic network analysis due to its simple structure and strong interpretability. Iraganaboina et al. [18] considered various attributes that affect routing selection and established a path selection set based on the panel mixed multinomial logit model of regret minimization (RRM). As for the route choice behavior when BEV has charging demand, the charging station attributes such as charging time and charging station’s location have significant influences on BEV drivers’ decision-making process. Balakrishnan [19] applied a mixed logit (ML) model to analyze the safer route choices carried out by two-wheeler road users and account for the heterogeneity and explain its impact on the willingness to pay (WTP) for accident reduction. Wang et al. [20] adopted the dlogit model to account for captive mode travelers in the modalsplit problem, and the path-size logit (PSL) model was used to capture route overlapping effects in the traffic assignment problem. Yang et al. [21] proposed nested logit model to analysis for BEV drivers’ charging and route choice behaviour. In view of the random charging demand of BEVs and the influence of safety threshold on their path selection, this paper constructs an improved logit model by adding a correction term to the fixed utility to determine the path choice probability.In the existing literature, the mixed assignment model combined with user equilibrium routing criteria is widely used to reflect the equilibrium routing behavior of BEVs with limited mileage. Yuan et al. [22] proposed a coordinated optimization method of electric vehicle flow and charging demand based on traffic-user equilibrium. Chen et al. [23] extended the equilibrium-embedded charging station location problem to allow multiple en-route charges and include the queuing effect at en-route charging stations, but their model does not explicitly present the relationship between the path feasibility and driving range limit and fails to state how feasible paths could be determined. Ferro et al. [24] extended the classic UE traffic allocation method to the case where an EV generates a certain amount of traffic to consider the location and size of charging stations on the transportation network. Geng et al. [25] proposed an intelligent charging management system for electric vehicles. In this system, a multiclass user traffic equilibrium allocation model with elastic charging demand is established to capture the link flow distribution of vehicles in the urban traffic network and estimate the charging demand of each fast charging station. By transforming the user equilibrium into a variational inequality problem based on the extended network, Wang et al. [11] established an expanded network structure to model the set of valid charging strategies for EV drivers, and then a variational inequality (VI) is formulated to capture the equilibrated route-choice and charging behaviors of EVs by incorporating an approximated queuing time function for a capacitated charging facility. Duell et al. [26] introduced a constrained shortest-path algorithm that accounts for the distance limitations imposed on EV drivers. Cen et al. [27] proposed the concept of charging rate and developed a mathematical model to explore how a mixture of EVs and GVs affects an urban network under user equilibrium, and how the EV demand is under different initial states or subsidy strategies. Xu et al. [28] decomposed the UE-BEV&GV model into two submodels equivalently and applied the Frank–Wolfe algorithm and the multilabel method to solve the model. The model considers the fixed EV demand of OD pairs, where EV minimizes its individual path cost and charging cost, and GV minimizes its individual path cost, regardless of the difference between fuel and power costs. Wang et al. [29] presented a distance-constrained traffic assignment problem that incorporates trip chains, as a more realistic modeling tool than the ones proposed by Jiang et al. [30, 31]. Xie and Jiang [32] first defined subpath, pure subpath, and feasible subpath. They applied the Bender decomposition algorithm and gradient projection algorithm to solve the traffic equilibrium assignment problem with edge constraints. Jiang et al. [33] proposed a new modeling dimension to address the network equilibrium problems, aiming to solve the joint selection destination, route, and parking. The common feature of these three papers is that a distance-constrained model was proposed. He et al. [30] formulated three mathematical models to describe the network equilibrium flow distributions and first defined a charging-depleting path to ensure that the trip was completed without running out of battery. Jiang et al. [31] proposed a path-constrained model to limit the flow of the path to zero if the path distance is greater than the vehicle’s distance limit.The premise of the above-given research is that BEV will inevitably charge due to its limited mileage. If the minimum distance between the node pair exceeds the range limit, the vehicle must be refueled at designated stations. BEV charging demand modeling can be roughly divided into two categories. The first category evaluates the charging demand from the perspective of a single BEV driver. The self-interested behavior of BEV users may aggravate traffic congestion and charging congestion (long queuing time). The second category aims to determine the user equilibrium assignment problem of traffic flow. Table1 summarizes characteristics comparison of related work on the UE problems for BEVs. From the perspective of the traffic system, it considers the traffic link and node congestion caused by the aggregation movement of electric vehicles and the selfish behaviour of the single electric vehicle in route selection. As the driving range of BEVs has dramatically increased, the current mileage of BEVs is usually enough to cover the daily requirements. Therefore, the driver’s charging behavior during a commute is not inevitable. Whether a BEV user chooses to charge the battery depends on the safe range they can take, which makes charging behavior randomly. Users with different initial SOC will choose different ratios of charging; that is, the charging behaviour of the BEV is random. At the same time, the UE principle is recognized to be unrealistic, because it assumes that all travelers have accurate perceptions of the condition of the transportation network. A more realistic and general situation is that generalized travel times are random variables or generalized travel times are perceived by travelers in an imperfect stochastic manner.Table 1
Characteristics comparison of related work.
StudiesProblem settingModelSolution algorithmTest examplesDwell time cost at FCSYuan et al. (2022)The optimal EV charging demand and EV flows in the TUE solutionEV stochastic traffic assignment problem-traffic user equilibriumDL-based surrogate model124-node TN and 118-node PDN system of Nanjing cityYesChen et al. (2020)Location problems considering UEBilevel mathematical model considering the equilibrium of route choice and waiting time for chargingNLP algorithmsSioux-falls network Nguyen and Dupuis networkYesWang et al. (2019)Large-scale location and capacity design problemThe charging equilibrium model along with the optimal recharging strategiesA customized neighbourhood search strategyYangtze river Delta networkYesGeng et al. (2019)The coordinated operation of urban transportation and power distribution network with elastic charging demandA multiclass user traffic equilibrium assignment model with elastic charging demandAlternating direction multiplier methodThe topology of the UTN and PDNNoFerro et al. (2019)Determining jointly the traffic and the charging service assignmentThe electric vehicles user equilibrium assignment modelLagrange multiplierGenoa municipality (Liguria region)NoThis paperPrediction of traffic flow considering electric vehicle market share and random chargingMulticlass stochastic user equilibrium model based on improved logit modelThe method of successive averageSioux Falls networkYes


## 3. Problem Formulation and Methodology

### 3.1. Basic Assumptions

Assuming that the range of BEVs considered in this paper is 160 km and the battery capacity isω0=24kwh. When the battery capacity is greater than or less than this assumption, it will be converted into the corresponding percentage. Many BEVs’ users will choose to quickly recharge their batteries along the way according to the acceptable safe state of charge SOCsafe, and the charging behavior is random. The CS is located at the network node, and the charging piles are all fast-charging piles. The number of charging piles and service rate of each charging station are given in advance.According to the China EV big data [34] survey sample, the range of BEVs is between 150 km and 200 km, and the initial SOC0 follows the blue normal distribution curve in Figure 1, denoted as fsoc0x∼Nμ1,σ1. The initial SOC0 is discretized into m groups, denoted by the set M. Each BEV in group m∈M has the same range of state of charge, SOCm0∈φ1m,φ2m. With GV users, there are m + 1 groups of users in the traffic network. These m + 1-type users choose the path according to their perception of the generalized cost of the path.Figure 1
Recharging probability density.


### 3.2. Notations and Variables

The mathematical notations and variables used in the paper are listed in Table2.Table 2
Notations.
SetsISet of nodes in the transport network, wherei∈IASet of links, wherea∈AUSet of CSs, whereu∈URSet of all origin nodes, wherer∈RSSet of all destination nodes, wheres∈SKgrsSet of all paths from the originr to the destination s for GVs, where kgrs∈KgrsKe,mrsSet of all paths from the originr to the destination s for BEVs in group, where ke,mrs∈Ke,mrsMSet of BEV group, wherem∈MParametersqers and qgrsThe travel demands of BEVs in groupm∈M and GVs between O–D pair r,s respectivelyCg and CeGasoline or electricity cost per kilometre for GVs and BEVsβg and βeTime value conversion coefficient of GVs and BEVsθg and θeGVs and BEVs user perceived discrete parameterlaThe length ofaωaverageThe average electricity consumption per kilometreμ1, and σ1Mean and variance of initial SOC distributionμ2, and σ2Mean and square of SOC distribution at the beginning of RCBta0Free-flow travel time on linkataTravel time function of linkaCaCapacity of linkasui and τuiThe number of servers and average service rate at the CSuiSOCm0Initial state of charge in groupm∈MγCorrection coefficientφ1m,φ2mInitial SOC range of typem BEV, SOCm0∈φ1m,φ2mηBEV market shareDecision variablesfkgrs and fke,mrsThe GVs traffic flow on the pathkgrs∈Kgrs and the BEVs traffic flow on the path ke,mrs∈Ke,mrs, respectivelyyui,mThe possible charging flow in them-group BEVs at the CS uixaThe aggregated traffic flow on linka∈Axag and xaeThe GV flow on linka and the BEV flow on link aPkgrs and Pke,mrsThe routekgrs and ke,mrs choice probabilityνm,krsCorrection termpke,mrsuiThe possibility of charging at the charging stationui∈U on the path ke,mrspui0Idle probability of CSuiρuiSystem utilization of CSuilkemrsuiThe distance from origins to charging station ui on the path ke,mrs∈Ke,mrs for BEVs in the group m∈MLui and duiρuiThe average queue length and the average dwell time at the CSuihke,mrsGeneralized cost of BEVs on pathke,mrshkgrsGeneralized cost of GVs on pathkgrsδa,kgrs and δa,ke,mrsBinary variable, equals to 1 if linka is on the path kgrs∈Kgrs (ke,mrs∈Ke,mrs), 0 otherwiseδu,ke,mrsiBinary variable, equals to 1 if CSui is on the path ke,mrs∈Ke,mrs, 0 otherwise


### 3.3. Path Choice and Stochastic Equilibrium Flow of GVs

The GVs’ generalized route cost in the mixed network includes travel time and travel cost. The travel time of GVs on the link is limited by the link flow and link capacity, which is given by the BPR function. The travel cost of the GVs driving is mainly the fuel cost, which can be calculated by the length of the link and the fuel cost per unit length. It is assumed that the fuel cost of GVs is positively related to travel distance. Thek-shortest loopless path algorithm (Yen algorithm) [35] is used to find the path choice set of GVs. The actual generalized path cost of GVs on path kgrs is(1)hkgrs=∑a∈Ataxδa,kgrs+∑a∈ACglaβgxagδa,kgrs.Faced with a random actual road network, GV users can only estimate a minimum cost trip according to their preferences. In this paper, we assume that the GV users’ route selection criteria are the estimated minimum generalized travel cost and that there is a random errorεgkrs between the estimated generalized cost and the actual generalized cost. The random errors are independent of each other and obey the Gumbel distribution with mean zero and identical standard deviations. The estimated generalized travel cost of GV users is expressed as(2)Hkgrs=hkgrs+εkgrs,where θg is a user-perceived discrete parameter to measure the GV users’ understanding of travel time on the road choice. The greater the value of θg, the higher the sensitivity of route choice to generalized travel cost. The route choice probability model can be specified as the following logit-based formula:(3)Pkgrs=exp−θghkgrs∑lg∈Kgrsexp−θghlgrs.The route flow distributions of GVs users can be estimated as follows:(4)fkgrs=qgrs⋅Pkgrs.


### 3.4. Path Choice and Stochastic Equilibrium Flow of BEVs

#### 3.4.1. Radom Charging Probability (RCP)

The incidence of BEVs’ RCB is related to the initialSOCm0 and the SOCu at the beginning of charging. Department of Energy and led by ECOtality North America presented data on the SOC distribution of batteries at the beginning of recharging [36]. Meanwhile, Yagcitekin et al. [37] studied the statistical law of the SOC level at the beginning of the charging process and gave a similar distribution of SOC. Figure 1 shows the distribution of SOCu at the start of RCB, denoted as fsocux∼Nμ2,σ2. It is assumed that the SOCm0 range of Class m BEV users is φ1m,φ2m. The possibility of charging at the charging station ui∈U on path ke,mrs, can be calculated by the following equation:(5)pke,mrsui=∫φ1mφ2mfsoc0xdx∫x−lkemrsuiωaveragexfsocuxdx,where lkemrsui is the distance from origin s to charging station ui on the path ke,mrs∈Ke,mrs for BEVs in the group m∈M.


#### 3.4.2. Dwell Time at the Charging Station (DTCS)

Since the ratio of charging flow and the selection of charging stations are random, the dwell time of charging stations is also random. The dwell time function of BEV at CSs with RCB includes two parts: the charging time and the queuing waiting time. It is assumed that there aresui servers at the CS ui, and each charging server with the average service rate τui works independently. The dwell time at a charging station is assumed in direct proportion to the average response time in an M/M/s/FCFS queue model [38], in which the number of arrivals per unit time equals the charging flow yui and the service capacity rate of the charging station is suiτui (the condition that yui<suiτui should be held in the queue model to ensure stability). By using Little’s law, the probability that all servers are idle can be expressed, respectively, as(6)pui0=∑k=0sui−1suiρuikk!+suiρuisuisui!1−ρui−1,where ρui=yui/suiτui, which is system utilization.Then, the average queue length and the average dwell time at the CSu can be given by the following equation:(7)Lui=suisuiρuisui+1pui0sui!1−ρui2+suiρui,(8)duiρui=Luisuiρui.In each network traffic flow assignment, the possible charging flow at the CSui is variable, i.e., the system arrival rate yui is uncertain. If the system arrival rate is greater than the product of the average service rate and the number of servers, the system utilization rate is greater than 1, and the dwell time is infinite. Conversely, when system utilization is less than 1, the average dwell time is calculated according to equation (8).


#### 3.4.3. Reasonable Path Choice Set of BEV

The generalized route cost components of BEVs include travel time and is mainly composed of three parts: travel time, travel cost, and charging dwell time. The travel cost of BEVs driving is mainly the electricity consumption cost, which can be calculated from the electricity cost of the unit length and link length. The actual generalized path cost of BEVs on pathke,mrs(9)he,mrs=∑a∈A∑m∈Mtaxaδa,ke,mrs+∑a∈A∑m∈MCelaβexaeδa,ke,mrs+∑m∈M∑u∈Uduiyuiδui,ke,mrspke,mrsui.In this paper, we assume that the BEV users’ route selection criteria are the estimated minimum generalized cost, and there is a random errorεmkrs between the estimated generalized cost and the actual generalized cost, which are independent of each other and obey the Gumbel distribution with mean zero and identical standard deviations. The estimated generalized travel cost of GV users is expressed as(10)Hke,mrs=hke,mrs+εke,mrs.There are three factors that determine the alternative path and charging choice of BEV: the safety thresholdSOCsafe, SOCui arriving at the CS ui, and the initial SOC0. Let SOCd be the state of charge at the destination. When it is greater than SOCsafe, BEV users prefer to choose a shorter path, regardless of whether the path passes through the charging station. When it is less than SOCsafe, BEV users are more inclined to choose the path through the charging station and the charging probability is related to SOCui. Combined with SOC0 and SOCsafe, we can determine the path length threshold Dsafem. Considering the dynamic changes of each path flow and dwell time in CSs, the alternate path set of BEV users is given based on the multilabel algorithm. We present a network by a directed weighted graph G=I,A,T,L,D, where I is a set of n vertices, which contains CS nodes, namely, SCS nodes; A is a set of edges, one for each link; T is a n × n link travel time function matrix from A; L is a n × n length of link matrix from A; D is a 1 × n DTCS vector from I. Each directed edge in set of edges A is denoted by an ordered pair of nodes from I. If directed edge auv=1, node u is said to be reachable from node v in I. The travel time of edge uv is denoted by tuv, which is in row u and column v of the matrix T. The length of edge uv is denoted by luv, which is in row u and column v of the matrix L.Let the path between two nodes v0 and vk be represented by a finite sequence: p=v0v1…vi…vk.The generalized cost of the path p is denoted as dis=∑0≤i<ktvi−1vi+∑0≤i<kCeβelvi−1vi+∑0≤i≤kdvi. The weight of edge vivi+1 is denoted by ωvivi+1=tvivi+1+Ceβelvivi+1xvivi+1. If node vi is CS node, dvi which is the component of vector D is greater than 0; otherwise, it is equal to 0. In order to find the reasonable path choice set of BEV paths of BEVs-based RCB, it is not only necessary to determine the path generalized cost but also to record the length of the path and whether the path passes over the CS node.Let us findk reasonable path choice set of BEV based on a multilabel algorithm (RPS-ML). Each label (or path) is identified by a number and stored in a working variable X until it is scanned. Let x,y be the element of X, then the correspondent label has the form lxu=disuf,Luf,parentuf,εuf, where disuf is the shortest generalized cost of path f from origin v0 to node u; Luf is the existing path length from origin v0 to node u; εuf is denoted as ∑v∈v0…uδvf , where δvf is 1 if v is CS node; otherwise, it is 0. It is supposed that the node v is the immediate neighbour of the node u, then a label denoted by lyv at node v. The correspondent label will be generated from label x, namely, lyv=disuf+ωuv+dvδvf,Luf+luv,u,εuf+δvf. The multilabel algorithm and yen algorithm are used to find the k generalized shortest paths of BEV in group m∈M, and mark them as pathkm=diskm,Lkm,εkm,Dsafem−Lkm.Dsafem represents the psychological safety driving distance corresponding to BEV users in the group m∈M under the psychological safety SOCsafe, denote as Dsafem=SOCm0−SOCsafe/ωverage⋅ω0. If Lkm>Dsafem, BEVs with possible charging demand (CD) need to choose the route through the charging station on the way to the destination, that is, εkm≥1. In this way, reasonable k path choice set of BEV is determined. BEV users prefer to use alternative paths that do not exceed the psychological safety distance. To express this intention, we add a correction term νm,krs=expγ⋅Dsafe−Lm,krs to the deterministic part of the utility function to adjust the choice probability. θe is a user-perceived discrete parameter to measure the difference of BEV users’ understanding of travel time on the road choice. The path choice probability model can be specified as the following improved logit-based formula:(11)Pke,mrs=exp−θehke,mrs+νm,krs∑le,m∈Ke,mrsexp−θehle,mrs+νm,krs.The route flow distributions of BEVs users can be estimated as follows:(12)fke,mrs=qe,mrs⋅Pke.mrs.


### 3.5. Multiclass Stochastic User Equilibrium Model Based on Improved Logit Model (LSUE-RCB)

The equivalent entropy-type mathematical programming (MP) for multiclass stochastic user equilibrium models based on an improved logit model can be formulated as follows:(13)minZf=∑a∈A∫0xatxdx+∑a∈A∫0xagCgβglaxdx+∑a∈A∫0xaeCe,mβe,mlaxdx+1θg∑rs∑kgrsfkgrslnfkgrs+∑m∈M∑rs∑ke,mrs1θm,efke,mrslnfke,mrs−νm,krs.Subject to:(14)∑rs∑kgrs∈Kgrsfkgrs=1−ηqrs,(15)∑rs∑m∈M∑ke,mrs∈Ke,mrsfke,mrs=ηqrs,(16)fkgrs=qgrs⋅Pkgrskgrs∈Kgrs,(17)fke,mrs=qe,mrs⋅Pke.mrske,mrs∈Ke,mrs,(18)qe,mrs=1−ηqrs⋅∫φ1mφ2mfSOC0dx,(19)xag=∑rs∑kgrs∈Kgrsfkgrsδa,kgrs,(20)xae=∑m∈M∑rs∑ke,mrs∈Ke,mrsfke,mrsδa,ke,mrs,(21)xa=xag+xae,(22)yui,m=∑r∈R∑s∈S∑ke,mrs∈Ke,mrsfke,mrsδui,ke,mrspke,mrsui,(23)yui=∑m∈Myui,m=∑m∈M∑r∈R∑s∈S∑ke,mrs∈Ke,mrsfke,mrsδui,ke,mrspke,mrsui,(24)fkgrs≥0,fke,mrs≥0.Equation (13) is the minimization objective function, which has no intuitive economic significance. Equations (14) and (15) are the relationship between path flow and OD demand; equations (16) and (17) indicate the path traffic logit probability loading. Equation (18) represents the proportion of category m BEV in the total flow of BEV. Equations (19) and (20) are the relationship between the link flow and path flow of GV and BEV users, respectively; equation (21) is the relationship between the total link traffic and the link traffic of users in group m + 1; equations (22) and (23) represent the total possible charging flow of BEV at CS ui. Equation (24) is the non-negative constraint of path flow.Theorem 1.
The mathematical programming model proposed in this paper is equivalent to stochastic user equilibrium allocation based on the improved logit.Proof.
The Lagrange for the minimization Model LSUE-RCB can be formulated as(25)Lf,π=Zf+∑rs∑kgrs∈Kgrsπgrsqgrs−fkgrs+∑rs∑m∈M∑ke,mrs∈Ke,mrsπe,mrsqers−fke,mrs,where πgrs and πe,mrs are Lagrange multipliers associated with constraints (14) and (15), respectively. Take the derivative of equation (25) and obtain its corresponding Kuhn–Tucker condition as follows:(26)∂L∂fkgrs⋅fkgrs=0,fkgrs≥0,∂L∂fkgrs≥0,∂L∂fke,mrs⋅fke,mrs=0,fke,mrs≥0,∂L∂fke,mrs≥0,∂L∂πe,mrs=0,∂L∂πgrs=0,
For each effective pathke,mrs of BEV users between O-D pair r,s, there is fke,mrs>0, so we can obtain the following equation:(27)∂L∂fke,mrs=hke,mrs+1θe,mlnfke,mrs+1+νm,krs−πgrs=0,(28)fke,mrs=exp−θe,mhke,mrs+νm,krs⋅expθe,mπe,mrs−1.
All paths between the O-D pairr,s are summed in equation (28), and the path choice probability of ke,mrs can be obtained:(29)Pke,mrs=fkemrs∑lflemrsexp−θehke,mrs+νm,krs∑le,m∈Ke,mrsexp−θehle,mrs+νm,krs.
For each effective pathkgrs of GV users between the O-D pair r,s, there is fkgrs>0, so we can obtain the following equation:(30)∂L∂fkgrs=hkgrs+1θglnfkgrs+1−πgrs=0,(31)fkgrs=exp−θghkgrs⋅expθgπgrs−1.
All paths between the O-D pairr,s are summed up in equation (31), and the path choice probability of kgrs of GV users can be obtained:(32)Pkgrs=fkgrs∑lgflgrsexp−θghkgrs∑lg∈Kgrsexp−θghlgrs.
The above-given proof shows that the mathematical programming model proposed in this paper is equivalent to the multiclass stochastic user equilibrium assignment based on improved logit.Theorem 2.
The solution of the mathematical programming model proposed in this paper is unique.Proof.
The second derivative of the objective function is as follows:(33)∂2L∂fke,mrs∂fle,mrs=1θe,mfke,mrs,ifl=k,0,ifl≠k,∂2L∂fkgrs∂flgrs=1θgfkgrs,ifl=k,0,l≠k.
It can be seen that the Hessian matrix corresponding to the objective function is positive definite, so the model constructed has a unique solution.


### 3.6. Solution Algorithm Design

This paper uses the method of successive average (MSA) [39] based on the path to solve the multiuser stochastic user equilibrium model based on the improved logit (MSA-LSUE-RCB). The pseudocode of the MSA-LSUE-RCB algorithm is shown in Algorithm 1. The algorithm flow chart is shown in Figure 2. The main steps of this algorithm are as follows.Figure 2
Flowchart for MSA-LSUE-RCB algorithm.Algorithm 1: Pseudocode of MSA-LSUE-RCB algorithm.
(1)
Input road network information (OD demand; capacity, free flow, and length of each link; nodes of alternative charging station; SOC of BEV); and other parameters(2)
Initializexag0=0,xae0=0,yu0=0, for all a∈A,u∈U(3)
Calculate the generalized cost ofk paths of BEV and GV path choice set respectively hkgrs0hke,mrs0, k∈l1,l2⋯,lk and correction term νm,krs0=expγ⋅Dsafe−Lm,krs. Use equations (3)–(6) to yields k path flows set {fkgrs0, fke,mrs0}, k∈l1,l2⋯,lk and yu0. Set counter n = 1(4)
Update. According to the path flowfkgrs0,fke,mrs0, k∈l1,l2⋯,lk and equation (21)–(23) the link flow and charging flow can xagn,xaen=0,yun be obtained. So as to obtain the generalized cost of k paths of BEV and GV path choice set respectively hkgrsnhke,mrsn, k∈l1,l2⋯,lk and correction term νm,krsn=expγ⋅Dsafe−Lm,krs. Use equations (3)–(6) to yields k path flows set fkgrsn,fke,mrsn, k∈l1,l2⋯,lk and yun(5)
Descent-direction finding. Use equations (3)–(6) to yields k auxiliary path flows set {gkgrsn, gke,mrsn}, k∈l1,l2⋯,lk and y˜un(6)
Move. Setfkgrsn+1=fkgrsn+1/ngkgrsn−fkgrsn, fke,mrsn+1=fke,mrsn+1/ngke,mrsn−fke,mrsn, and yun+1=yun+1/ny˜un−yun(7)
Convergence test. If a convergence criterion is not met, setn: = n + 1 and go to step 4; otherwise, stop and the set of stochastic user equilibrium flows is xagn+1,xaen+1,xagn+1+xaen+1,yun.Step 1.
Initialization. According to the road network, the related parameter values are calculated, and the generalized travel costhkgrs0, hke,mrs0, and correction term νm,krs0 for each user of the reasonable path with zero initial flow is obtained. Calculate the initial path flow fkgrsn,fke,mrsn according to equations (16) and (17). Set counter n := 1.Step 2.
Update. Set countern: = 1 + n. According to the path flow fkgrsn,fke,mrsn, the link flow xan and charging flow yun can be obtained according to equations (19)–(23). Find the k path choice set of GV by combining the k-shortest loopless path algorithm. Find the k path choice set of BEV reasonable routing set by the multilabel algorithm. At the same time, correction terms νm,krsn for each BEV user of the reasonable path can be obtained. Calculate the generalized path travel cost hkgrsn, hke,mrsn of each new reasonable path.Step 3.
Determine the descent direction. Calculate the additional path flowgkgrsn,gke,mrsn according to equations (16) and (17). Calculate the additional changing flow yun according to equations (22) and (23). The GV and BEV path flow descent direction are gkgrsn−fkgrsn, gke,mrsn−fke,mrsn, and y˜un−yun, respectively.Step 4.
Move. Improve the path flow with the iterative weighting method:(34)fkgrsn+1=fkgrsn+1ngkgrsn−fkgrsn,fke,mrsn+1=fke,mrsn+1ngke,mrsn−fke,mrsn,yun+1=yun+1ny˜un−yun,Step 5.
Convergence test. This paper uses root mean squared error (RMSE) to judge convergence:(35)RMSEn=1K∑rs∑k∈l1,l2⋯,lkgkrsn−fkrsn2+∑u∈Uyun−yun−12,where K is the number of paths between all OD pairs. When RMSEn≤ε, stop. Otherwise, let n = n + 1, and go to Step 2.


## 3.1. Basic Assumptions

Assuming that the range of BEVs considered in this paper is 160 km and the battery capacity isω0=24kwh. When the battery capacity is greater than or less than this assumption, it will be converted into the corresponding percentage. Many BEVs’ users will choose to quickly recharge their batteries along the way according to the acceptable safe state of charge SOCsafe, and the charging behavior is random. The CS is located at the network node, and the charging piles are all fast-charging piles. The number of charging piles and service rate of each charging station are given in advance.According to the China EV big data [34] survey sample, the range of BEVs is between 150 km and 200 km, and the initial SOC0 follows the blue normal distribution curve in Figure 1, denoted as fsoc0x∼Nμ1,σ1. The initial SOC0 is discretized into m groups, denoted by the set M. Each BEV in group m∈M has the same range of state of charge, SOCm0∈φ1m,φ2m. With GV users, there are m + 1 groups of users in the traffic network. These m + 1-type users choose the path according to their perception of the generalized cost of the path.Figure 1
Recharging probability density.


## 3.2. Notations and Variables

The mathematical notations and variables used in the paper are listed in Table2.Table 2
Notations.
SetsISet of nodes in the transport network, wherei∈IASet of links, wherea∈AUSet of CSs, whereu∈URSet of all origin nodes, wherer∈RSSet of all destination nodes, wheres∈SKgrsSet of all paths from the originr to the destination s for GVs, where kgrs∈KgrsKe,mrsSet of all paths from the originr to the destination s for BEVs in group, where ke,mrs∈Ke,mrsMSet of BEV group, wherem∈MParametersqers and qgrsThe travel demands of BEVs in groupm∈M and GVs between O–D pair r,s respectivelyCg and CeGasoline or electricity cost per kilometre for GVs and BEVsβg and βeTime value conversion coefficient of GVs and BEVsθg and θeGVs and BEVs user perceived discrete parameterlaThe length ofaωaverageThe average electricity consumption per kilometreμ1, and σ1Mean and variance of initial SOC distributionμ2, and σ2Mean and square of SOC distribution at the beginning of RCBta0Free-flow travel time on linkataTravel time function of linkaCaCapacity of linkasui and τuiThe number of servers and average service rate at the CSuiSOCm0Initial state of charge in groupm∈MγCorrection coefficientφ1m,φ2mInitial SOC range of typem BEV, SOCm0∈φ1m,φ2mηBEV market shareDecision variablesfkgrs and fke,mrsThe GVs traffic flow on the pathkgrs∈Kgrs and the BEVs traffic flow on the path ke,mrs∈Ke,mrs, respectivelyyui,mThe possible charging flow in them-group BEVs at the CS uixaThe aggregated traffic flow on linka∈Axag and xaeThe GV flow on linka and the BEV flow on link aPkgrs and Pke,mrsThe routekgrs and ke,mrs choice probabilityνm,krsCorrection termpke,mrsuiThe possibility of charging at the charging stationui∈U on the path ke,mrspui0Idle probability of CSuiρuiSystem utilization of CSuilkemrsuiThe distance from origins to charging station ui on the path ke,mrs∈Ke,mrs for BEVs in the group m∈MLui and duiρuiThe average queue length and the average dwell time at the CSuihke,mrsGeneralized cost of BEVs on pathke,mrshkgrsGeneralized cost of GVs on pathkgrsδa,kgrs and δa,ke,mrsBinary variable, equals to 1 if linka is on the path kgrs∈Kgrs (ke,mrs∈Ke,mrs), 0 otherwiseδu,ke,mrsiBinary variable, equals to 1 if CSui is on the path ke,mrs∈Ke,mrs, 0 otherwise


## 3.3. Path Choice and Stochastic Equilibrium Flow of GVs

The GVs’ generalized route cost in the mixed network includes travel time and travel cost. The travel time of GVs on the link is limited by the link flow and link capacity, which is given by the BPR function. The travel cost of the GVs driving is mainly the fuel cost, which can be calculated by the length of the link and the fuel cost per unit length. It is assumed that the fuel cost of GVs is positively related to travel distance. Thek-shortest loopless path algorithm (Yen algorithm) [35] is used to find the path choice set of GVs. The actual generalized path cost of GVs on path kgrs is(1)hkgrs=∑a∈Ataxδa,kgrs+∑a∈ACglaβgxagδa,kgrs.Faced with a random actual road network, GV users can only estimate a minimum cost trip according to their preferences. In this paper, we assume that the GV users’ route selection criteria are the estimated minimum generalized travel cost and that there is a random errorεgkrs between the estimated generalized cost and the actual generalized cost. The random errors are independent of each other and obey the Gumbel distribution with mean zero and identical standard deviations. The estimated generalized travel cost of GV users is expressed as(2)Hkgrs=hkgrs+εkgrs,where θg is a user-perceived discrete parameter to measure the GV users’ understanding of travel time on the road choice. The greater the value of θg, the higher the sensitivity of route choice to generalized travel cost. The route choice probability model can be specified as the following logit-based formula:(3)Pkgrs=exp−θghkgrs∑lg∈Kgrsexp−θghlgrs.The route flow distributions of GVs users can be estimated as follows:(4)fkgrs=qgrs⋅Pkgrs.


## 3.4. Path Choice and Stochastic Equilibrium Flow of BEVs

### 3.4.1. Radom Charging Probability (RCP)

The incidence of BEVs’ RCB is related to the initialSOCm0 and the SOCu at the beginning of charging. Department of Energy and led by ECOtality North America presented data on the SOC distribution of batteries at the beginning of recharging [36]. Meanwhile, Yagcitekin et al. [37] studied the statistical law of the SOC level at the beginning of the charging process and gave a similar distribution of SOC. Figure 1 shows the distribution of SOCu at the start of RCB, denoted as fsocux∼Nμ2,σ2. It is assumed that the SOCm0 range of Class m BEV users is φ1m,φ2m. The possibility of charging at the charging station ui∈U on path ke,mrs, can be calculated by the following equation:(5)pke,mrsui=∫φ1mφ2mfsoc0xdx∫x−lkemrsuiωaveragexfsocuxdx,where lkemrsui is the distance from origin s to charging station ui on the path ke,mrs∈Ke,mrs for BEVs in the group m∈M.


### 3.4.2. Dwell Time at the Charging Station (DTCS)

Since the ratio of charging flow and the selection of charging stations are random, the dwell time of charging stations is also random. The dwell time function of BEV at CSs with RCB includes two parts: the charging time and the queuing waiting time. It is assumed that there aresui servers at the CS ui, and each charging server with the average service rate τui works independently. The dwell time at a charging station is assumed in direct proportion to the average response time in an M/M/s/FCFS queue model [38], in which the number of arrivals per unit time equals the charging flow yui and the service capacity rate of the charging station is suiτui (the condition that yui<suiτui should be held in the queue model to ensure stability). By using Little’s law, the probability that all servers are idle can be expressed, respectively, as(6)pui0=∑k=0sui−1suiρuikk!+suiρuisuisui!1−ρui−1,where ρui=yui/suiτui, which is system utilization.Then, the average queue length and the average dwell time at the CSu can be given by the following equation:(7)Lui=suisuiρuisui+1pui0sui!1−ρui2+suiρui,(8)duiρui=Luisuiρui.In each network traffic flow assignment, the possible charging flow at the CSui is variable, i.e., the system arrival rate yui is uncertain. If the system arrival rate is greater than the product of the average service rate and the number of servers, the system utilization rate is greater than 1, and the dwell time is infinite. Conversely, when system utilization is less than 1, the average dwell time is calculated according to equation (8).


### 3.4.3. Reasonable Path Choice Set of BEV

The generalized route cost components of BEVs include travel time and is mainly composed of three parts: travel time, travel cost, and charging dwell time. The travel cost of BEVs driving is mainly the electricity consumption cost, which can be calculated from the electricity cost of the unit length and link length. The actual generalized path cost of BEVs on pathke,mrs(9)he,mrs=∑a∈A∑m∈Mtaxaδa,ke,mrs+∑a∈A∑m∈MCelaβexaeδa,ke,mrs+∑m∈M∑u∈Uduiyuiδui,ke,mrspke,mrsui.In this paper, we assume that the BEV users’ route selection criteria are the estimated minimum generalized cost, and there is a random errorεmkrs between the estimated generalized cost and the actual generalized cost, which are independent of each other and obey the Gumbel distribution with mean zero and identical standard deviations. The estimated generalized travel cost of GV users is expressed as(10)Hke,mrs=hke,mrs+εke,mrs.There are three factors that determine the alternative path and charging choice of BEV: the safety thresholdSOCsafe, SOCui arriving at the CS ui, and the initial SOC0. Let SOCd be the state of charge at the destination. When it is greater than SOCsafe, BEV users prefer to choose a shorter path, regardless of whether the path passes through the charging station. When it is less than SOCsafe, BEV users are more inclined to choose the path through the charging station and the charging probability is related to SOCui. Combined with SOC0 and SOCsafe, we can determine the path length threshold Dsafem. Considering the dynamic changes of each path flow and dwell time in CSs, the alternate path set of BEV users is given based on the multilabel algorithm. We present a network by a directed weighted graph G=I,A,T,L,D, where I is a set of n vertices, which contains CS nodes, namely, SCS nodes; A is a set of edges, one for each link; T is a n × n link travel time function matrix from A; L is a n × n length of link matrix from A; D is a 1 × n DTCS vector from I. Each directed edge in set of edges A is denoted by an ordered pair of nodes from I. If directed edge auv=1, node u is said to be reachable from node v in I. The travel time of edge uv is denoted by tuv, which is in row u and column v of the matrix T. The length of edge uv is denoted by luv, which is in row u and column v of the matrix L.Let the path between two nodes v0 and vk be represented by a finite sequence: p=v0v1…vi…vk.The generalized cost of the path p is denoted as dis=∑0≤i<ktvi−1vi+∑0≤i<kCeβelvi−1vi+∑0≤i≤kdvi. The weight of edge vivi+1 is denoted by ωvivi+1=tvivi+1+Ceβelvivi+1xvivi+1. If node vi is CS node, dvi which is the component of vector D is greater than 0; otherwise, it is equal to 0. In order to find the reasonable path choice set of BEV paths of BEVs-based RCB, it is not only necessary to determine the path generalized cost but also to record the length of the path and whether the path passes over the CS node.Let us findk reasonable path choice set of BEV based on a multilabel algorithm (RPS-ML). Each label (or path) is identified by a number and stored in a working variable X until it is scanned. Let x,y be the element of X, then the correspondent label has the form lxu=disuf,Luf,parentuf,εuf, where disuf is the shortest generalized cost of path f from origin v0 to node u; Luf is the existing path length from origin v0 to node u; εuf is denoted as ∑v∈v0…uδvf , where δvf is 1 if v is CS node; otherwise, it is 0. It is supposed that the node v is the immediate neighbour of the node u, then a label denoted by lyv at node v. The correspondent label will be generated from label x, namely, lyv=disuf+ωuv+dvδvf,Luf+luv,u,εuf+δvf. The multilabel algorithm and yen algorithm are used to find the k generalized shortest paths of BEV in group m∈M, and mark them as pathkm=diskm,Lkm,εkm,Dsafem−Lkm.Dsafem represents the psychological safety driving distance corresponding to BEV users in the group m∈M under the psychological safety SOCsafe, denote as Dsafem=SOCm0−SOCsafe/ωverage⋅ω0. If Lkm>Dsafem, BEVs with possible charging demand (CD) need to choose the route through the charging station on the way to the destination, that is, εkm≥1. In this way, reasonable k path choice set of BEV is determined. BEV users prefer to use alternative paths that do not exceed the psychological safety distance. To express this intention, we add a correction term νm,krs=expγ⋅Dsafe−Lm,krs to the deterministic part of the utility function to adjust the choice probability. θe is a user-perceived discrete parameter to measure the difference of BEV users’ understanding of travel time on the road choice. The path choice probability model can be specified as the following improved logit-based formula:(11)Pke,mrs=exp−θehke,mrs+νm,krs∑le,m∈Ke,mrsexp−θehle,mrs+νm,krs.The route flow distributions of BEVs users can be estimated as follows:(12)fke,mrs=qe,mrs⋅Pke.mrs.


## 3.4.1. Radom Charging Probability (RCP)

The incidence of BEVs’ RCB is related to the initialSOCm0 and the SOCu at the beginning of charging. Department of Energy and led by ECOtality North America presented data on the SOC distribution of batteries at the beginning of recharging [36]. Meanwhile, Yagcitekin et al. [37] studied the statistical law of the SOC level at the beginning of the charging process and gave a similar distribution of SOC. Figure 1 shows the distribution of SOCu at the start of RCB, denoted as fsocux∼Nμ2,σ2. It is assumed that the SOCm0 range of Class m BEV users is φ1m,φ2m. The possibility of charging at the charging station ui∈U on path ke,mrs, can be calculated by the following equation:(5)pke,mrsui=∫φ1mφ2mfsoc0xdx∫x−lkemrsuiωaveragexfsocuxdx,where lkemrsui is the distance from origin s to charging station ui on the path ke,mrs∈Ke,mrs for BEVs in the group m∈M.


## 3.4.2. Dwell Time at the Charging Station (DTCS)

Since the ratio of charging flow and the selection of charging stations are random, the dwell time of charging stations is also random. The dwell time function of BEV at CSs with RCB includes two parts: the charging time and the queuing waiting time. It is assumed that there aresui servers at the CS ui, and each charging server with the average service rate τui works independently. The dwell time at a charging station is assumed in direct proportion to the average response time in an M/M/s/FCFS queue model [38], in which the number of arrivals per unit time equals the charging flow yui and the service capacity rate of the charging station is suiτui (the condition that yui<suiτui should be held in the queue model to ensure stability). By using Little’s law, the probability that all servers are idle can be expressed, respectively, as(6)pui0=∑k=0sui−1suiρuikk!+suiρuisuisui!1−ρui−1,where ρui=yui/suiτui, which is system utilization.Then, the average queue length and the average dwell time at the CSu can be given by the following equation:(7)Lui=suisuiρuisui+1pui0sui!1−ρui2+suiρui,(8)duiρui=Luisuiρui.In each network traffic flow assignment, the possible charging flow at the CSui is variable, i.e., the system arrival rate yui is uncertain. If the system arrival rate is greater than the product of the average service rate and the number of servers, the system utilization rate is greater than 1, and the dwell time is infinite. Conversely, when system utilization is less than 1, the average dwell time is calculated according to equation (8).


## 3.4.3. Reasonable Path Choice Set of BEV

The generalized route cost components of BEVs include travel time and is mainly composed of three parts: travel time, travel cost, and charging dwell time. The travel cost of BEVs driving is mainly the electricity consumption cost, which can be calculated from the electricity cost of the unit length and link length. The actual generalized path cost of BEVs on pathke,mrs(9)he,mrs=∑a∈A∑m∈Mtaxaδa,ke,mrs+∑a∈A∑m∈MCelaβexaeδa,ke,mrs+∑m∈M∑u∈Uduiyuiδui,ke,mrspke,mrsui.In this paper, we assume that the BEV users’ route selection criteria are the estimated minimum generalized cost, and there is a random errorεmkrs between the estimated generalized cost and the actual generalized cost, which are independent of each other and obey the Gumbel distribution with mean zero and identical standard deviations. The estimated generalized travel cost of GV users is expressed as(10)Hke,mrs=hke,mrs+εke,mrs.There are three factors that determine the alternative path and charging choice of BEV: the safety thresholdSOCsafe, SOCui arriving at the CS ui, and the initial SOC0. Let SOCd be the state of charge at the destination. When it is greater than SOCsafe, BEV users prefer to choose a shorter path, regardless of whether the path passes through the charging station. When it is less than SOCsafe, BEV users are more inclined to choose the path through the charging station and the charging probability is related to SOCui. Combined with SOC0 and SOCsafe, we can determine the path length threshold Dsafem. Considering the dynamic changes of each path flow and dwell time in CSs, the alternate path set of BEV users is given based on the multilabel algorithm. We present a network by a directed weighted graph G=I,A,T,L,D, where I is a set of n vertices, which contains CS nodes, namely, SCS nodes; A is a set of edges, one for each link; T is a n × n link travel time function matrix from A; L is a n × n length of link matrix from A; D is a 1 × n DTCS vector from I. Each directed edge in set of edges A is denoted by an ordered pair of nodes from I. If directed edge auv=1, node u is said to be reachable from node v in I. The travel time of edge uv is denoted by tuv, which is in row u and column v of the matrix T. The length of edge uv is denoted by luv, which is in row u and column v of the matrix L.Let the path between two nodes v0 and vk be represented by a finite sequence: p=v0v1…vi…vk.The generalized cost of the path p is denoted as dis=∑0≤i<ktvi−1vi+∑0≤i<kCeβelvi−1vi+∑0≤i≤kdvi. The weight of edge vivi+1 is denoted by ωvivi+1=tvivi+1+Ceβelvivi+1xvivi+1. If node vi is CS node, dvi which is the component of vector D is greater than 0; otherwise, it is equal to 0. In order to find the reasonable path choice set of BEV paths of BEVs-based RCB, it is not only necessary to determine the path generalized cost but also to record the length of the path and whether the path passes over the CS node.Let us findk reasonable path choice set of BEV based on a multilabel algorithm (RPS-ML). Each label (or path) is identified by a number and stored in a working variable X until it is scanned. Let x,y be the element of X, then the correspondent label has the form lxu=disuf,Luf,parentuf,εuf, where disuf is the shortest generalized cost of path f from origin v0 to node u; Luf is the existing path length from origin v0 to node u; εuf is denoted as ∑v∈v0…uδvf , where δvf is 1 if v is CS node; otherwise, it is 0. It is supposed that the node v is the immediate neighbour of the node u, then a label denoted by lyv at node v. The correspondent label will be generated from label x, namely, lyv=disuf+ωuv+dvδvf,Luf+luv,u,εuf+δvf. The multilabel algorithm and yen algorithm are used to find the k generalized shortest paths of BEV in group m∈M, and mark them as pathkm=diskm,Lkm,εkm,Dsafem−Lkm.Dsafem represents the psychological safety driving distance corresponding to BEV users in the group m∈M under the psychological safety SOCsafe, denote as Dsafem=SOCm0−SOCsafe/ωverage⋅ω0. If Lkm>Dsafem, BEVs with possible charging demand (CD) need to choose the route through the charging station on the way to the destination, that is, εkm≥1. In this way, reasonable k path choice set of BEV is determined. BEV users prefer to use alternative paths that do not exceed the psychological safety distance. To express this intention, we add a correction term νm,krs=expγ⋅Dsafe−Lm,krs to the deterministic part of the utility function to adjust the choice probability. θe is a user-perceived discrete parameter to measure the difference of BEV users’ understanding of travel time on the road choice. The path choice probability model can be specified as the following improved logit-based formula:(11)Pke,mrs=exp−θehke,mrs+νm,krs∑le,m∈Ke,mrsexp−θehle,mrs+νm,krs.The route flow distributions of BEVs users can be estimated as follows:(12)fke,mrs=qe,mrs⋅Pke.mrs.


## 3.5. Multiclass Stochastic User Equilibrium Model Based on Improved Logit Model (LSUE-RCB)

The equivalent entropy-type mathematical programming (MP) for multiclass stochastic user equilibrium models based on an improved logit model can be formulated as follows:(13)minZf=∑a∈A∫0xatxdx+∑a∈A∫0xagCgβglaxdx+∑a∈A∫0xaeCe,mβe,mlaxdx+1θg∑rs∑kgrsfkgrslnfkgrs+∑m∈M∑rs∑ke,mrs1θm,efke,mrslnfke,mrs−νm,krs.Subject to:(14)∑rs∑kgrs∈Kgrsfkgrs=1−ηqrs,(15)∑rs∑m∈M∑ke,mrs∈Ke,mrsfke,mrs=ηqrs,(16)fkgrs=qgrs⋅Pkgrskgrs∈Kgrs,(17)fke,mrs=qe,mrs⋅Pke.mrske,mrs∈Ke,mrs,(18)qe,mrs=1−ηqrs⋅∫φ1mφ2mfSOC0dx,(19)xag=∑rs∑kgrs∈Kgrsfkgrsδa,kgrs,(20)xae=∑m∈M∑rs∑ke,mrs∈Ke,mrsfke,mrsδa,ke,mrs,(21)xa=xag+xae,(22)yui,m=∑r∈R∑s∈S∑ke,mrs∈Ke,mrsfke,mrsδui,ke,mrspke,mrsui,(23)yui=∑m∈Myui,m=∑m∈M∑r∈R∑s∈S∑ke,mrs∈Ke,mrsfke,mrsδui,ke,mrspke,mrsui,(24)fkgrs≥0,fke,mrs≥0.Equation (13) is the minimization objective function, which has no intuitive economic significance. Equations (14) and (15) are the relationship between path flow and OD demand; equations (16) and (17) indicate the path traffic logit probability loading. Equation (18) represents the proportion of category m BEV in the total flow of BEV. Equations (19) and (20) are the relationship between the link flow and path flow of GV and BEV users, respectively; equation (21) is the relationship between the total link traffic and the link traffic of users in group m + 1; equations (22) and (23) represent the total possible charging flow of BEV at CS ui. Equation (24) is the non-negative constraint of path flow.Theorem 1.
The mathematical programming model proposed in this paper is equivalent to stochastic user equilibrium allocation based on the improved logit.Proof.
The Lagrange for the minimization Model LSUE-RCB can be formulated as(25)Lf,π=Zf+∑rs∑kgrs∈Kgrsπgrsqgrs−fkgrs+∑rs∑m∈M∑ke,mrs∈Ke,mrsπe,mrsqers−fke,mrs,where πgrs and πe,mrs are Lagrange multipliers associated with constraints (14) and (15), respectively. Take the derivative of equation (25) and obtain its corresponding Kuhn–Tucker condition as follows:(26)∂L∂fkgrs⋅fkgrs=0,fkgrs≥0,∂L∂fkgrs≥0,∂L∂fke,mrs⋅fke,mrs=0,fke,mrs≥0,∂L∂fke,mrs≥0,∂L∂πe,mrs=0,∂L∂πgrs=0,
For each effective pathke,mrs of BEV users between O-D pair r,s, there is fke,mrs>0, so we can obtain the following equation:(27)∂L∂fke,mrs=hke,mrs+1θe,mlnfke,mrs+1+νm,krs−πgrs=0,(28)fke,mrs=exp−θe,mhke,mrs+νm,krs⋅expθe,mπe,mrs−1.
All paths between the O-D pairr,s are summed in equation (28), and the path choice probability of ke,mrs can be obtained:(29)Pke,mrs=fkemrs∑lflemrsexp−θehke,mrs+νm,krs∑le,m∈Ke,mrsexp−θehle,mrs+νm,krs.
For each effective pathkgrs of GV users between the O-D pair r,s, there is fkgrs>0, so we can obtain the following equation:(30)∂L∂fkgrs=hkgrs+1θglnfkgrs+1−πgrs=0,(31)fkgrs=exp−θghkgrs⋅expθgπgrs−1.
All paths between the O-D pairr,s are summed up in equation (31), and the path choice probability of kgrs of GV users can be obtained:(32)Pkgrs=fkgrs∑lgflgrsexp−θghkgrs∑lg∈Kgrsexp−θghlgrs.
The above-given proof shows that the mathematical programming model proposed in this paper is equivalent to the multiclass stochastic user equilibrium assignment based on improved logit.Theorem 2.
The solution of the mathematical programming model proposed in this paper is unique.Proof.
The second derivative of the objective function is as follows:(33)∂2L∂fke,mrs∂fle,mrs=1θe,mfke,mrs,ifl=k,0,ifl≠k,∂2L∂fkgrs∂flgrs=1θgfkgrs,ifl=k,0,l≠k.
It can be seen that the Hessian matrix corresponding to the objective function is positive definite, so the model constructed has a unique solution.


## 3.6. Solution Algorithm Design

This paper uses the method of successive average (MSA) [39] based on the path to solve the multiuser stochastic user equilibrium model based on the improved logit (MSA-LSUE-RCB). The pseudocode of the MSA-LSUE-RCB algorithm is shown in Algorithm 1. The algorithm flow chart is shown in Figure 2. The main steps of this algorithm are as follows.Figure 2
Flowchart for MSA-LSUE-RCB algorithm.Algorithm 1: Pseudocode of MSA-LSUE-RCB algorithm.
(1)
Input road network information (OD demand; capacity, free flow, and length of each link; nodes of alternative charging station; SOC of BEV); and other parameters(2)
Initializexag0=0,xae0=0,yu0=0, for all a∈A,u∈U(3)
Calculate the generalized cost ofk paths of BEV and GV path choice set respectively hkgrs0hke,mrs0, k∈l1,l2⋯,lk and correction term νm,krs0=expγ⋅Dsafe−Lm,krs. Use equations (3)–(6) to yields k path flows set {fkgrs0, fke,mrs0}, k∈l1,l2⋯,lk and yu0. Set counter n = 1(4)
Update. According to the path flowfkgrs0,fke,mrs0, k∈l1,l2⋯,lk and equation (21)–(23) the link flow and charging flow can xagn,xaen=0,yun be obtained. So as to obtain the generalized cost of k paths of BEV and GV path choice set respectively hkgrsnhke,mrsn, k∈l1,l2⋯,lk and correction term νm,krsn=expγ⋅Dsafe−Lm,krs. Use equations (3)–(6) to yields k path flows set fkgrsn,fke,mrsn, k∈l1,l2⋯,lk and yun(5)
Descent-direction finding. Use equations (3)–(6) to yields k auxiliary path flows set {gkgrsn, gke,mrsn}, k∈l1,l2⋯,lk and y˜un(6)
Move. Setfkgrsn+1=fkgrsn+1/ngkgrsn−fkgrsn, fke,mrsn+1=fke,mrsn+1/ngke,mrsn−fke,mrsn, and yun+1=yun+1/ny˜un−yun(7)
Convergence test. If a convergence criterion is not met, setn: = n + 1 and go to step 4; otherwise, stop and the set of stochastic user equilibrium flows is xagn+1,xaen+1,xagn+1+xaen+1,yun.Step 1.
Initialization. According to the road network, the related parameter values are calculated, and the generalized travel costhkgrs0, hke,mrs0, and correction term νm,krs0 for each user of the reasonable path with zero initial flow is obtained. Calculate the initial path flow fkgrsn,fke,mrsn according to equations (16) and (17). Set counter n := 1.Step 2.
Update. Set countern: = 1 + n. According to the path flow fkgrsn,fke,mrsn, the link flow xan and charging flow yun can be obtained according to equations (19)–(23). Find the k path choice set of GV by combining the k-shortest loopless path algorithm. Find the k path choice set of BEV reasonable routing set by the multilabel algorithm. At the same time, correction terms νm,krsn for each BEV user of the reasonable path can be obtained. Calculate the generalized path travel cost hkgrsn, hke,mrsn of each new reasonable path.Step 3.
Determine the descent direction. Calculate the additional path flowgkgrsn,gke,mrsn according to equations (16) and (17). Calculate the additional changing flow yun according to equations (22) and (23). The GV and BEV path flow descent direction are gkgrsn−fkgrsn, gke,mrsn−fke,mrsn, and y˜un−yun, respectively.Step 4.
Move. Improve the path flow with the iterative weighting method:(34)fkgrsn+1=fkgrsn+1ngkgrsn−fkgrsn,fke,mrsn+1=fke,mrsn+1ngke,mrsn−fke,mrsn,yun+1=yun+1ny˜un−yun,Step 5.
Convergence test. This paper uses root mean squared error (RMSE) to judge convergence:(35)RMSEn=1K∑rs∑k∈l1,l2⋯,lkgkrsn−fkrsn2+∑u∈Uyun−yun−12,where K is the number of paths between all OD pairs. When RMSEn≤ε, stop. Otherwise, let n = n + 1, and go to Step 2.


## 4. Analysis of Results

In order to validate the proposed model and algorithm, take Sioux Falls as an example, which is composed of 24 nodes, 76 road sections, and 576 OD pairs. The free flow and capacity of each link are from He et al. [30], as shown in Figure 3. O-D demand comes from Bar Gear [39]. The numbers in brackets at the arrow point take values of link length (km), free flow (min), and link capacity (veh/h), respectively. The numbers at the connection point represent the link number. There are five fast charging stations in the network, located at nodes 5, 11, 15, 16, and 24. The SOC of BEV users at the beginning of driving is 60% or above, accounting for 97% [34]. Discrete initial SOC0, BEV users are divided into four categories: SOC10=60%,70%, SOC20=70%,80%, SOC30=80%,90%, and SOC40=90%,100%. Considering the gradual increase of BEV market share, we divide the market share into three situations η=10%,20%,30% to analyse its impact on the transportation network and charging stations. SOCsafe is set to 30%. Other corresponding parameters are given in Table 3.Figure 3
Sioux Falls network (colorfilled node denotes CS).Table 3
Corresponding parameters.
ParametersValueμ10.64σ10.12μ20.35σ10.08ωaverage0.153 kwh/kmCe0.22 yuan/kmCg2.67 yuan/kmβeβg0.6 min/yuanvi,i=1,⋯58 veh/hs5s2460s11s1580s16100ε0.001θg2θe1γ0.01Figures4–6 compare the equilibrium BEV link flow, the equilibrium GV link flow, and the total equilibrium link flow for the three market shares. For links 25, 26, 27, and 32, BEV traffic demand increases rapidly with the increase of BEV market share; at the same time, the traffic demand of GV in these links correspondingly reduces the demand, so the total flow of the links will not fluctuate too much. The total traffic flow of each link is shown in Table 4 and Figure 6. With the increase of BEV market share, the total traffic demand for sections 15, 47, 49, 50, and 55 increases significantly by an average of 1,000 to more than 1,200 vehicles per hour, an increase of between 6% and 9%. These links are concentrated near the charging station 16, and the traffic demand of other sections near the charging station increases little. At the same time, the total traffic demand of links 17, 18, 20, and 54 also decreased significantly by an average of 380 to more than 440 vehicles per hour; the flow change of other links is not significant. Figure 7 shows the comparison of the possible charging flow of five charging stations under different BEV market shares. With the increase of BEV market share, the random charging flow of charging station 16 has significantly increased from 437 to more than 1093 vehicles per hour, indicating that the location of the charging station is more appropriate. However, the random charging flow of charging station 24 has little change with the increase of BEV market share, which indicates that its utilization is not sufficient and the site selection needs to be reconsidered. The random charging flow of charging stations 5, 11, and 15 increases with the increase of BEV market share, indicating that the site selection is appropriate.Figure 4
The equilibrium BEV flow under different BEV market shares.Figure 5
The equilibrium GV flow under different BEV market shares.Figure 6
The total equilibrium flow under different BEV market shares.Table 4
Generalized link flow (veh/h).
No.Market share10%20%30%13590338031702613062606390336003400320046470634062105612062406360610380105610740768106720663081159011780119709138801406014240106920704071601115170153401551012980010000102001364606620678014648063606240151073011060113901617210170201683017116601122010780181031099209530191805018000179502016970165401611021103701024010110221061010820110302368206840686024157701564015510252349023680238702623650237002375027234302346023490281418014260143402913620138401406030580058005800316740688070203222830228602289033105201064010760341903019060190903567906680657036114401158011720371133011360113903812230122601229039109301096010990401823018260182904162106220623042119201194011960431438014460145404468006800680045153101542015530461668016860170404711160115201188048133801346013540491269012980132705011880124601304051640064006400521229012580128705313350133001325054103009900950055123901298013570561551015420153305715100152001530058134501340013350596120604059606016110160201593061591058205730626140618062206367806560634064534053805420651050010520105406612660127201278067177801796018140687970774075106911020110401106070844083808320711053010560105907296309560949073583058605890741083010860108907512960130201308076573057605790Figure 7
Comparison of possible charging different BEV market shares.


## 5. Conclusions

In this paper, the stochastic user model based on the improved logit model is proposed considering the BEV market share and random charging behavior. The choice of the alternative path is affected by the travel time, the path energy consumption, and the queue dwell time of charging. Based on the multilabel algorithm and Yen algorithm, the alternative path set of BEV is determined and multiple attributes of the path are recorded. In the set of alternative paths, BEV users prefer to choose paths whose path length does not exceed the psychological safety distance, so we add a correction term to the logit model. In the interaction between BEVs’ users and GVs’ users, the traffic net finally reaches the equilibrium state. The MSA based on the path for solving the mixed model is verified by simulating different scenarios.Under the condition that the location of the alternative charging station is known, 364680 vehicles and 24 OD pairs are simulated and evaluated, and the indicators related to the increase of BEV market share are determined. In the simulation scenario, the market share of BEV starts to increase from 10%, with a total of 36468 vehicles, 72936 vehicles accounting for 20%, and gradually increases to 109404 vehicles accounting for 30%. With the gradual increase of BEV market share, the sharp increase of charging demand leads to the rapid growth of traffic flow on the road near the charging station and the key road to the charging station, so it is necessary for traffic planners to adopt corresponding traffic dispersion schemes on these roads. At the same time, the predicted growth rate of the charging demand of the five charging stations has increased from 50% to 66%, and the growth rate of the charging demand of the charging stations at the 16th node is more than twice the growth rate of the number of BEVs. According to this trend, the number of charging stations and the number of charging pile configurations is far from enough, and great efforts should be made to build and develop. By simulating the random charging behavior of BEV users and the interaction of BEV and GV users’ random path selection, we can determine the impact of charging demand on the driving behavior of electric vehicles and how the insertion of electric vehicles affects the traffic flow. Combined with the random distribution of initial SOC, the random distribution of SOC during charging, and the safe SOC threshold of BEV users, the average dwell time and average length at alternative charging stations can be determined, so that the traffic network users can randomly choose the feasible path according to their perception of the generalized path cost, and finally reach the network equilibrium.Forecasting and analyzing the changing trend of traffic flow and charging flow can provide a basis for decision makers and the CS operator from the perspective of demand. Through the simulation analysis of the traffic flow of each link of the traffic network under different market shares and initial SOC, it can provide a reference for traffic managers to formulate management plans, such as improving some links’ grades and adding new links. Similarly, the utilization degree of the charging station can be inversely deduced by predicting the possible charging demand, which also provides a reference scheme for the layout of civil infrastructure. The expansion research of this paper can also build a double-layer model to analyze the relationship between charging supply and demand in combination with the charging station location problem.



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*Source: 7649689-2023-03-23.xml*