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Journal of Advanced Transportation
, Volume 2023 – Mar 23, 2023

/lp/hindawi-publishing-corporation/prediction-of-traffic-flow-considering-electric-vehicle-market-share-Xqntx0PjNM

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- Publisher
- Hindawi Publishing Corporation
- ISSN
- 0197-6729
- eISSN
- 2042-3195
- DOI
- 10.1155/2023/7649689
- Publisher site
- See Article on Publisher Site

Hindawi Journal of Advanced Transportation Volume 2023, Article ID 7649689, 16 pages https://doi.org/10.1155/2023/7649689 Research Article Prediction of Traffic Flow considering Electric Vehicle Market Share and Random Charging 1 2 2 3 Yunjuan Yan , Weixiong Zha , Jungang Shi, and Liping Yan School of Mechatronics and Vehicle Engineering, East China Jiaotong University, Nanchang 330013, China School of Transportation and Logistics, East China Jiaotong University, Nanchang 330013, China School of Software, East China Jiaotong University, Nanchang 330013, China Correspondence should be addressed to Weixiong Zha; jxzhawx@sina.com Received 5 September 2022; Revised 26 February 2023; Accepted 1 March 2023; Published 23 March 2023 Academic Editor: Jing Dong-O’Brien Copyright © 2023 Yunjuan Yan et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tis 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 diferences in travel behaviors between BEV and GV users, an improved logit-based model, multilabel algorithm, and queuing theory are applied. Te infuencing 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 trafc and diferences in queuing time. Conversely, diferent stations have diferent queuing dwell times, which will also afect the routing and charging locations for BEVs with RCB. Te path-based method of successive averages (MSA) is adopted to solve the model. Trough the simulation of the test network Sioux Falls, the equilibrium trafc fow and possible charging fow under diferent market shares and initial SOC are predicted, and the properties of the model and the feasibility of the algorithm are verifed. Erdogan ˘ et al. [3] generated the best alternative solution 1.Introduction for the location of charging stations with the goal of max- Since the 21st century, the global energy and environment imizing the total trafc fow. Mor an ´ et al. [4] used the have become extremely severe, and the contradiction be- “SUMO” simulator interface to simulate the vehicle trafc tween rapid economic growth and resources has become behavior in the diferent trafc scenarios and use it as the increasingly prominent. BEVs deliver many benefts, such as main variable to determine the location of the electric vehicle low greenhouse gas emissions and high energy efciency, charging center or charging station. Campaña et al. [5] compared with GVs. Terefore, BEVs are considered one of presented the multicommodity fow problem (MCFP) al- the main competitors in reducing emissions in the trans- gorithm for sizing and allocating resources to CS in- portation sector, which has grown rapidly in recent years. frastructures for EVs in a heterogeneous transport system, However, there are several unsolved problems to be considering trajectories and restrictions of capacity in each addressed, one of which is the “range anxiety” problem [1]. section of the road. From the perspective of microscopic Electric vehicle (EV) users can often feel anxious about de- analysis of trafc fow, the above-given two references can pleting their battery before they reach their destinations. As identify travel patterns in vehicles and their efect on dif- a matter of fact, the deployment of charging stations (CS) has ferent levels of trafc, so as to make the best decision when been recognized as a crucial factor for the wide adoption of EVs carrying out large-scale projects including electric vehicles. [2]. In the recent decade, numerous eforts have been devoted Gao et al. [6] proposed a bilevel model to depict the in- to selecting the strategical locations and sizing of CSs for EVs. teraction between trafc fow distribution and the location of 2 Journal of Advanced Transportation relays (OPPR). BEVs that need to be charged immediately CSs in the EVs and GVs hybrid network. With BEV’s in- creasing share in the automobile market, it will inevitably must be charged via CS. In recent years, the range of BEVs has increased considerably, which allows the BEV routes to afect the trafc network. Kong et al. [7] used dynamic charging demand in the location planning of EVCS, taking be freely chosen; i.e., not all BEV routes have to go through into account the dynamic trafc fow and the relationship CSs. In China, most users choose BEVs with a driving range between the vehicle’s current location, destination, and of 120 km to 160 km because the price of vehicles within this candidate location of the charging station. Te above-given range is relatively low and can meet the travel demand of studies have completed the deployment of CS either from medium-sized cities (diameter of 80 kilometers). Power the perspective of microtrafc fow simulation or network consumption is very persistent when SOC is between 50% and 100%, but very fast when the SOC is below 50%. BEV trafc fow. Considering the charging station capacity, charging users can selectively charge based on a psychologically ac- ceptable safety SOC and have more fexible charging be- time, and waiting time, some researchers combined queuing theory to build the CS planning models and determined the havior. Charging behavior has changed from the inevitability of the past to the randomness of the present. optimal locations of charging stations and the optimal scale (the quantity of chargers and waiting spaces). Xiao et al. [8] RCB plays a role in infuencing drivers’ personal travel proposed an optimal location model to determine the op- choices, corresponding road congestion, and network per- timal locations and capacities of EV charging infrastructure formance. With the interaction of BEVs and GVs, the route to minimize the comprehensive total cost, which considers choice behavior of users will become more complicated. Te the charging queuing behavior with fnite queue length and trafc assignment problem of the mixed trafc network with various siting constraints. Zhang et al. [9] controlled CS RCB deserves our study. In a mixed network with both BEVs and GVs, the layout service quality by a two-stage priority queue to optimize the station, battery inventory level, and the number of super- of CSs and the distribution of trafc fow interact with each other. Te detailed analysis of BEV users’ behavior and the chargers at selected stations. Choi and Lim [10] proposed a queuing model and two congestion control policies based accurate prediction of the charging demand are very im- portant for designing a reasonable layout of CSs. With the on EV queue length thresholds. Wang et al. [11] used the M/G/k system to model the queuing of charging stations and location of alternative charging facilities known, the path incorporate it into an expanded network to capture the selection of BEV users and the resulting changes in trafc optimal recharging strategies for EV drivers. patterns can be studied by combining the above variables to Similarly, the route choice of BEV users in the trans- analyze the rationality of CS layout and the evolutionary portation network is also afected by charging demand, the trend of link trafc fow. Tis paper is devoted to analyze deployment of CSs, and queuing time. Li et al. [12] proposed BEV users’ charging and route choice behaviours and random charging behavior based on an improved logit a public charging infrastructure localization and route planning strategy for logistics companies based on a bilevel discrete choice model. Travel time cost, SOC, queuing dwell time cost, and electricity consumption cost are considered in program. Froger et al. [13] introduced a framework to solve EV routing problems with CS capacity constraints in gen- calculating the path choice generalized cost. A mixed sto- eral. Ashkrof et al. [14] aimed to analyze the infuencing chastic user equilibrium model is established to study the factors of route selection and charging behavior of BEV interaction of BEV users and GV users in an urban network drivers. In addition to classic route attributes and variables with random charging demand, to predict the changing related to rapid charging (such as charging time and waiting trend of network trafc fow and charging fow, and to verify time), the SOC at the start and end of the journey is also an the rationality of charging layout from the perspective of important indicator that afects the driver’s decision on demand to provide a basis for decision makers. route and charging behavior. EV routing problem with Te rest of this paper is organized as follows. Section 2 describes the related work. Section 3 describes the model’s backhauls was solved considering the location of charging stations and the operation of the electric power distribution assumptions, notations, the method of determining the set of alternative paths, and constructs an improved logit sto- system [15]. Koç et al. [16] presented an electric vehicle routing problem with a shared charging station (E-VRP- chastic user equilibrium model. In Section 4, we apply the SCS), optimal charging station location, and delivery route algorithm to the test network Sioux Falls, comparing and planning based on ALNS and mixed-integer linear pro- analyzing the equilibrium trafc fows and charging trafc gramming. Terefore, quantifying the above variables and fows at diferent market shares. Conclusions and future analyzing their impact on BEV routing and charging se- research are provided in Section 5. lection has become a new challenge for trafc fow allocation models in mixed transport networks. 2.Related Works In the past, BEV users always worried about the return trip and generated range anxiety due to short driving range, Te trafc assignment problem (TAP) describes how the inadequate charging infrastructure, and long charging time. trafc demand is distributed on the available routes in the BEV users determine their feasible path by implementing trafc network. Te core of any trafc assignment method is mileage constraints and then generate them for trafc as- the route choice model. Te deterministic user equilibrium signment. Since a feasible path may thus consist of several problem (DUE) was the most studied example in which the relays, the problem is called the optimal path problem with route choice assumption states that drivers behave as if they Journal of Advanced Transportation 3 charging station. By transforming the user equilibrium into have perfect knowledge of route costs and select the best route to minimize their travel costs. However, the DUE a variational inequality problem based on the extended network, Wang et al. [11] established an expanded network principle is recognized to be unrealistic because it assumes that all travelers have accurate perceptions of the trans- structure to model the set of valid charging strategies for EV portation network. Daganzo and Shef [17] proposed the drivers, and then a variational inequality (VI) is formulated principle of stochastic user equilibrium (SUE) to capture to capture the equilibrated route-choice and charging be- travelers’ perception errors of travel time, to compensate for haviors of EVs by incorporating an approximated queuing the absence of the UE principle that all travelers have an time function for a capacitated charging facility. Duell et al. accurate perception of the trafc network. Although the [26] introduced a constrained shortest-path algorithm that accounts for the distance limitations imposed on EV drivers. Probit model can obtain a path fow solution that is more in line with the actual situation, it needs to estimate the path Cen et al. [27] proposed the concept of charging rate and developed a mathematical model to explore how a mixture selection probability through analytical, simulation, nu- merical integration, and other methods, and the calculation of EVs and GVs afects an urban network under user equilibrium, and how the EV demand is under diferent is relatively complex, so it is not commonly used in practice. Te logit model is widely used in trafc network analysis initial states or subsidy strategies. Xu et al. [28] decomposed due to its simple structure and strong interpretability. Ira- the UE-BEV&GV model into two submodels equivalently ganaboina et al. [18] considered various attributes that afect and applied the Frank–Wolfe algorithm and the multilabel routing selection and established a path selection set based method to solve the model. Te model considers the fxed on the panel mixed multinomial logit model of regret EV demand of OD pairs, where EV minimizes its individual minimization (RRM). As for the route choice behavior when path cost and charging cost, and GV minimizes its individual path cost, regardless of the diference between fuel and BEV has charging demand, the charging station attributes such as charging time and charging station’s location have power costs. Wang et al. [29] presented a distance- constrained trafc assignment problem that incorporates signifcant infuences on BEV drivers’ decision-making process. Balakrishnan [19] applied a mixed logit (ML) trip chains, as a more realistic modeling tool than the ones proposed by Jiang et al. [30, 31]. Xie and Jiang [32] frst model to analyze the safer route choices carried out by two- wheeler road users and account for the heterogeneity and defned subpath, pure subpath, and feasible subpath. Tey explain its impact on the willingness to pay (WTP) for applied the Bender decomposition algorithm and gradient accident reduction. Wang et al. [20] adopted the dlogit projection algorithm to solve the trafc equilibrium as- model to account for captive mode travelers in the signment problem with edge constraints. Jiang et al. [33] modalsplit problem, and the path-size logit (PSL) model was proposed a new modeling dimension to address the network used to capture route overlapping efects in the trafc as- equilibrium problems, aiming to solve the joint selection destination, route, and parking. Te common feature of signment problem. Yang et al. [21] proposed nested logit model to analysis for BEV drivers’ charging and route choice these three papers is that a distance-constrained model was proposed. He et al. [30] formulated three mathematical behaviour. In view of the random charging demand of BEVs and the infuence of safety threshold on their path selection, models to describe the network equilibrium fow distribu- this paper constructs an improved logit model by adding tions and frst defned a charging-depleting path to ensure a correction term to the fxed utility to determine the path that the trip was completed without running out of battery. choice probability. Jiang et al. [31] proposed a path-constrained model to limit In the existing literature, the mixed assignment model the fow of the path to zero if the path distance is greater than combined with user equilibrium routing criteria is widely the vehicle’s distance limit. used to refect the equilibrium routing behavior of BEVs Te premise of the above-given research is that BEV will with limited mileage. Yuan et al. [22] proposed a co- inevitably charge due to its limited mileage. If the minimum distance between the node pair exceeds the range limit, the ordinated optimization method of electric vehicle fow and charging demand based on trafc-user equilibrium. Chen vehicle must be refueled at designated stations. BEV charging demand modeling can be roughly divided into two et al. [23] extended the equilibrium-embedded charging station location problem to allow multiple en-route charges categories. Te frst category evaluates the charging demand and include the queuing efect at en-route charging stations, from the perspective of a single BEV driver. Te self- but their model does not explicitly present the relationship interested behavior of BEV users may aggravate trafc between the path feasibility and driving range limit and fails congestion and charging congestion (long queuing time). to state how feasible paths could be determined. Ferro et al. Te second category aims to determine the user equilibrium [24] extended the classic UE trafc allocation method to the assignment problem of trafc fow. Table 1 summarizes characteristics comparison of related work on the UE case where an EV generates a certain amount of trafc to consider the location and size of charging stations on the problems for BEVs. From the perspective of the trafc system, it considers the trafc link and node congestion transportation network. Geng et al. [25] proposed an in- telligent charging management system for electric vehicles. caused by the aggregation movement of electric vehicles and the selfsh behaviour of the single electric vehicle in route In this system, a multiclass user trafc equilibrium allocation model with elastic charging demand is established to capture selection. As the driving range of BEVs has dramatically the link fow distribution of vehicles in the urban trafc increased, the current mileage of BEVs is usually enough to network and estimate the charging demand of each fast cover the daily requirements. Terefore, the driver’s 4 Journal of Advanced Transportation Table 1: Characteristics comparison of related work. Dwell time Studies Problem setting Model Solution algorithm Test examples cost at FCS Yuan et al. Te optimal EV charging demand and EV EV stochastic trafc assignment 124-node TN and 118-node PDN DL-based surrogate model Yes (2022) fows in the TUE solution problem-trafc user equilibrium system of Nanjing city Bilevel mathematical model considering the Chen et al. Sioux-falls network Nguyen and Location problems considering UE equilibrium of route choice and waiting time NLP algorithms Yes (2020) Dupuis network for charging A customized Wang et al. Large-scale location and capacity design Te charging equilibrium model along with neighbourhood search Yangtze river Delta network Yes (2019) problem the optimal recharging strategies strategy Te coordinated operation of urban A multiclass user trafc equilibrium Geng et al. Alternating direction Te topology of the UTN and transportation and power distribution assignment model with elastic charging No (2019) multiplier method PDN network with elastic charging demand demand Ferro et al. Determining jointly the trafc and the Te electric vehicles user equilibrium Genoa municipality (Liguria Lagrange multiplier No (2019) charging service assignment assignment model region) Prediction of trafc fow considering electric Multiclass stochastic user equilibrium model Te method of successive Tis paper Sioux Falls network Yes vehicle market share and random charging based on improved logit model average Journal of Advanced Transportation 5 charging behavior during a commute is not inevitable. between the estimated generalized cost and the actual Whether a BEV user chooses to charge the battery depends generalized cost. Te random errors are independent of each on the safe range they can take, which makes charging other and obey the Gumbel distribution with mean zero and behavior randomly. Users with diferent initial SOC will identical standard deviations. Te estimated generalized choose diferent ratios of charging; that is, the charging travel cost of GV users is expressed as behaviour of the BEV is random. At the same time, the UE rs rs rs H � h + ε , (2) k k k g g g principle is recognized to be unrealistic, because it assumes that all travelers have accurate perceptions of the condition where θ is a user-perceived discrete parameter to measure of the transportation network. A more realistic and general the GV users’ understanding of travel time on the road situation is that generalized travel times are random vari- choice. Te greater the value of θ , the higher the sensitivity ables or generalized travel times are perceived by travelers in of route choice to generalized travel cost. Te route choice an imperfect stochastic manner. probability model can be specifed as the following logit- based formula: 3.Problem Formulation and Methodology rs exp−θ h 3.1. Basic Assumptions. Assuming that the range of BEVs rs P � . (3) considered in this paper is 160 km and the battery capacity is rs rsexp−θ h l ∈K g g g g ω � 24kwh. When the battery capacity is greater than or less than this assumption, it will be converted into the corresponding Te route fow distributions of GVs users can be esti- percentage. Many BEVs’ users will choose to quickly recharge mated as follows: their batteries along the way according to the acceptable safe rs rs rs state of charge SOC , and the charging behavior is random. f � q · P . (4) safe k g k g g Te CS is located at the network node, and the charging piles are all fast-charging piles. Te number of charging piles and service rate of each charging station are given in advance. 3.4. Path Choice and Stochastic Equilibrium Flow of BEVs According to the China EV big data [34] survey sample, the range of BEVs is between 150 km and 200 km, and the 3.4.1. Radom Charging Probability (RCP). Te incidence of initial SOC follows the blue normal distribution curve in BEVs’ RCB is related to the initial SOC and the SOC at m0 u Figure 1, denoted as f (x)∼ N(μ , σ ). Te initial SOC is the beginning of charging. Department of Energy and led by soc 1 1 0 discretized into m groups, denoted by the set M. Each BEV ECOtality North America presented data on the SOC dis- tribution of batteries at the beginning of recharging [36]. in group m ∈ M has the same range of state of charge, SOC ∈ (φ , φ ). With GV users, there are m + 1 groups Meanwhile, Yagcitekin et al. [37] studied the statistical law of m0 1m 2m the SOC level at the beginning of the charging process and of users in the trafc network. Tese m + 1-type users choose the path according to their perception of the generalized cost gave a similar distribution of SOC. Figure 1 shows the of the path. distribution of SOC at the start of RCB, denoted as f (x)∼ N(μ , σ ). It is assumed that the SOC range of soc 2 2 m0 Class m BEV users is [φ , φ ]. Te possibility of charging 1m 2m 3.2. Notations and Variables. Te mathematical notations rs at the charging station u ∈ U on path k , can be calculated i e,m and variables used in the paper are listed in Table 2. by the following equation: φ x 2m p � f (x)dx f (x)dx, rs (5) 3.3. Path Choice and Stochastic Equilibrium Flow of GVs. soc soc k 0 u rs e,m φ x−l u ω ( ) 1m kem i average Te GVs’ generalized route cost in the mixed network in- rs cludes travel time and travel cost. Te travel time of GVs on where l (u ) is the distance from origin s to charging kem rs the link is limited by the link fow and link capacity, which is rs station u on the path k ∈ K for BEVs in the group i e,m e,m given by the BPR function. Te travel cost of the GVs driving m ∈ M. 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. 3.4.2. Dwell Time at the Charging Station (DTCS). Since the Te k-shortest loopless path algorithm (Yen algorithm) [35] ratio of charging fow and the selection of charging sta- is used to fnd the path choice set of GVs. Te actual tions are random, the dwell time of charging stations is rs generalized path cost of GVs on path k is also random. Te dwell time function of BEV at CSs with RCB includes two parts: the charging time and the rs rs rs h � t (x)δ + C l β x δ . a a,k g a g ag a,k g g g (1) queuing waiting time. It is assumed that there are s a∈A a∈A servers at the CS u , and each charging server with the Faced with a random actual road network, GV users can average service rate τ works independently. Te dwell only estimate a minimum cost trip according to their time at a charging station is assumed in direct proportion preferences. In this paper, we assume that the GV users’ to the average response time in an M/M/s/FCFS queue route selection criteria are the estimated minimum gener- model [38], in which the number of arrivals per unit time rs alized travel cost and that there is a random error ε equals the charging fow y and the service capacity rate of gk u i 6 Journal of Advanced Transportation 0 0.2 0.4 0.6 0.8 1 SOC Initial SOC Charging SOC Figure 1: Recharging probability density. Table 2: Notations. Sets I Set of nodes in the transport network, where i ∈ I A Set of links, where a ∈ A U Set of CSs, where u ∈ U R Set of all origin nodes, where r ∈ R S Set of all destination nodes, where s ∈ S rs rs rs K Set of all paths from the origin r to the destination s for GVs, where k ∈ K g g g Set of all paths from the origin r to the destination s for BEVs in group, where rs rs e,m rs k ∈ K e,m e,m M Set of BEV group, where m ∈ M Parameters Te travel demands of BEVs in group m ∈ M and GVs between O–D pair (r, s) rs rs q and q e g respectively C and C Gasoline or electricity cost per kilometre for GVs and BEVs g e β and β Time value conversion coefcient of GVs and BEVs g e θ and θ GVs and BEVs user perceived discrete parameter g e l Te length of a ω Te average electricity consumption per kilometre average μ , and σ Mean and variance of initial SOC distribution 1 1 μ , and σ Mean and square of SOC distribution at the beginning of RCB 2 2 t Free-fow travel time on link a a0 t Travel time function of link a C Capacity of link a s and τ Te number of servers and average service rate at the CS u u u i i SOC Initial state of charge in group m ∈ M m0 c Correction coefcient (φ , φ ) Initial SOC range of type m BEV, SOC ∈ (φ , φ ) 1m 2m m0 1m 2m η BEV market share Decision variables rs rs Te GVs trafc fow on the path k ∈ K and the BEVs trafc fow on the path g g rs rs f and f rs k k rs g e,m k ∈ K , respectively e,m e,m y i Te possible charging fow in the m-group BEVs at the CS u u ,m x Te aggregated trafc fow on link a ∈ A x and x Te GV fow on link a and the BEV fow on link a ag ae rs rs rs rs P and P Te route k and k choice probability k k g e,m g e,m rs ] Correction term m,k probability density Journal of Advanced Transportation 7 Table 2: Continued. rs Te possibility of charging at the charging station u ∈ U on the path k p rs i e,m e,m p Idle probability of CS u u 0 ρ System utilization of CS u rs rs Te distance from origin s to charging station u on the path k ∈ K for BEVs in rs i e,m e,m l (u ) kem the group m ∈ M L and d (ρ ) Te average queue length and the average dwell time at the CS u u u u i i i rs rs h Generalized cost of BEVs on path k k e,m e,m rs rs h Generalized cost of GVs on path k k g rs rs rs rs rs rs δ and δ Binary variable, equals to 1 if link a is on the path k ∈ K (k ∈ K ), 0 otherwise a,k a,k g g e,m e,m g e,m i rs i rs δ rs Binary variable, equals to 1 if CS u is on the path k ∈ K , 0 otherwise u,k e,m e,m e,m the charging station is s τ (the condition that y < s τ generalized cost and the actual generalized cost, which are u u u u u i i i i i should be held in the queue model to ensure stability). By independent of each other and obey the Gumbel distribution using Little’s law, the probability that all servers are idle with mean zero and identical standard deviations. Te es- can be expressed, respectively, as timated generalized travel cost of GV users is expressed as − 1 k s s −1 u i rs rs rs s ρ s ρ u u u u H � h + ε . ⎢ ⎥ (10) ⎡ ⎢ i i i i ⎤ ⎥ ⎢ ⎥ k k k ⎢ ⎥ (6) ⎣ ⎦ e,m e,m e,m p � + , u 0 k! s !1 − ρ u u k�0 i i Tere are three factors that determine the alternative where ρ � y /s τ , which is system utilization. path and charging choice of BEV: the safety threshold u u u u i i i i Ten, the average queue length and the average dwell SOC , SOC arriving at the CS u , and the initial SOC . Let safe u i 0 time at the CS u can be given by the following equation: SOC be the state of charge at the destination. When it is s s +1 greater than SOC , BEV users prefer to choose a shorter u u i i safe s ρ p i u u 0 i i path, regardless of whether the path passes through the L � + s ρ , (7) u u u i i i charging station. When it is less than SOC , BEV users are s !1 − ρ safe u u i i more inclined to choose the path through the charging station and the charging probability is related to SOC i. L u d ρ � . (8) Combined with SOC and SOC , we can determine the u u i i 0 safe s ρ u u m i i path length threshold D . Considering the dynamic safe changes of each path fow and dwell time in CSs, the alternate In each network trafc fow assignment, the possible path set of BEV users is given based on the multilabel algo- charging fow at the CS u is variable, i.e., the system arrival rithm. We present a network by a directed weighted graph rate y is uncertain. If the system arrival rate is greater than G � (I, A, T, L, D), where I is a set of n vertices, which the product of the average service rate and the number of contains CS nodes, namely, SCS nodes; A is a set of edges, one servers, the system utilization rate is greater than 1, and the for each link; T is a n × n link travel time function matrix from dwell time is infnite. Conversely, when system utilization is A; L is a n × n length of link matrix from A; D is a 1 × n DTCS less than 1, the average dwell time is calculated according to vector from I. Each directed edge in set of edges A is denoted equation (8). by an ordered pair of nodes from I. If directed edge a � 1, uv node u is said to be reachable from node v in I. Te travel time 3.4.3. Reasonable Path Choice Set of BEV. Te generalized of edge uv is denoted by t(uv), which is in row u and column v route cost components of BEVs include travel time and is of the matrix T. Te length of edge uv is denoted by l(uv), mainly composed of three parts: travel time, travel cost, and which is in row u and column v of the matrix L.Let the path charging dwell time. Te travel cost of BEVs driving is mainly between two nodes v and v be represented by a fnite se- 0 k the electricity consumption cost, which can be calculated from quence: p � v v . . . v . . . v .Te generalized cost of the path p 0 1 i k the electricity cost of the unit length and link length. Te actual is denoted as di s � t(v v ) + C β l(v v )+ rs 0≤i<k i−1 i 0≤i<k e e i−1 i generalized path cost of BEVs on path k e,m d(v ). Te weight of edge v v is denoted by ω � 0≤i≤k i i i+1 v v i i+1 rs h � t x δ rs t(v v )+ C β l(v v )x(v v ). If node v is CS node, d(v ) e,m a a a,k i i+1 e e i i+1 i i+1 i i e,m a∈A m∈M which is the component of vector D is greater than 0; otherwise, it is equal to 0. In order to fnd the reasonable path choice set of rs + C l β x δ e a e ae a,k (9) e,m BEV paths of BEVs-based RCB, it is not only necessary to a∈A m∈M determine the path generalized cost but also to record the length rs + d y δ p rs . u u u ,k i i i e,m e,m of the path and whether the path passes over the CS node. m∈M u∈U Let us fnd k reasonable path choice set of BEV based on In this paper, we assume that the BEV users’ route se- a multilabel algorithm (RPS-ML). Each label (or path) is lection criteria are the estimated minimum generalized cost, identifed by a number and stored in a working variable X rs and there is a random error ε between the estimated until it is scanned. Let x, y be the element of X, then the mk 8 Journal of Advanced Transportation rs rs correspondent label has the form l (u) � term ] � exp(c · (D − L )) to the deterministic part x safe m,k m,k f f f f [dis , L , parent , ε ], where dis is the shortest generalized u u u u of the utility function to adjust the choice probability. θ is cost of path f from origin v to node u; L is the existing a user-perceived discrete parameter to measure the dif- 0 u path length from origin v to node u; ε is denoted as u ference of BEV users’ understanding of travel time on the f f δ , where δ is 1 if v is CS node; otherwise, it is 0. road choice. Te path choice probability model can be v∈(v ...u) v v specifed as the following improved logit-based formula: It is supposed that the node v is the immediate neighbour of the node u, then a label denoted by l (v) at node v. Te rs rs exp −θ h + ] e k m,k e,m correspondent label will be generated from label x, namely, rs P � . (11) f f f f f e,m rs rs l (v) � [dis + ω + d δ , L + l , u, ε + δ ]. Te multi- u u y uv v uv rs exp −θ h + ] v u v l ∈K e l m,k e,m e,m e,m label algorithm and yen algorithm are used to fnd the k generalized shortest paths of BEV in group m ∈ M, and Te route fow distributions of BEVs users can be es- m m m m m m m mark them as path � [dis , L , ε , D − L ].D rep- timated as follows: k k k k safe k safe resents the psychological safety driving distance corre- rs rs f rs � q · P . (12) k e,m k sponding to BEV users in the group m ∈ M under the e,m e.m psychological safety SOC , denote as safe m m m D � SOC − SOC /ω · ω . If L > D , BEVs safe m0 safe verage 0 k safe 3.5. Multiclass Stochastic User Equilibrium Model Based on with possible charging demand (CD) need to choose the ImprovedLogitModel(LSUE-RCB). Te equivalent entropy- route through the charging station on the way to the type mathematical programming (MP) for multiclass sto- destination, that is, ε ≥ 1. In this way, reasonable k path chastic user equilibrium models based on an improved logit choice set of BEV is determined. BEV users prefer to use model can be formulated as follows: alternative paths that do not exceed the psychological safety distance. To express this intention, we add a correction x x x a ag ae min Z(f) � t(x)dx + C β l xdx + C β l xdx g g a e,m e,m a 0 0 0 a∈A a∈A a∈A (13) 1 1 rs + f rs ln f rs + f rs ln f rs − ] . k k k k m,k g g e,m e,m θ θ rs rs g m,e rs rs k m∈M k g e,m y � f rs δ rs p , i rs u ,m k u ,k i k e,m e,m (22) e,m Subject to: rs rs r∈R s∈S k ∈K e,m e,m rs rs f � (1 − η)q , g y � y (14) u u ,m rs i i rs rs k ∈K g g m∈M rs rs � f δ p rs , k u ,k rs e,m i e,m e,m f rs � ηq , rs rs m∈M r∈R s∈S k k ∈K e,m e,m e,m (15) rs rs rs m∈M k ∈K e,m e,m (23) rs rs rs rs q · P k ∈ K f rs � , (16) g k g g k g f rs ≥ 0, (24) rs rs rs rs f rs ≥ 0. q · P k ∈ K e,m f rs � , (17) e,m e,m e,m k k e,m e.m Equation (13) is the minimization objective function, 2m rs rs which has no intuitive economic signifcance. Equations q � (1 − η)q · f dx, (18) e,m SOC φ (14) and (15) are the relationship between path fow and 1m OD demand; equations (16) and (17) indicate the path rs rs trafc logit probability loading. Equation (18) represents x � f δ , ag k a,k g g (19) rs rs rs the proportion of category m BEV in the total fow of BEV. k ∈K g g Equations (19) and (20) are the relationship between the link fow and path fow of GV and BEV users, respectively; x � f rs δ rs , ae k a,k e,m e,m (20) rs equation (21) is the relationship between the total link rs rs m∈M k ∈K e,m e,m trafc and the link trafc of users in group m + 1; equations (22) and (23) represent the total possible charging fow of x � x + x , (21) a ag ae BEV at CS u . Equation (24) is the non-negative constraint of path fow. Journal of Advanced Transportation 9 Theorem 1. Te mathematical programming model pro- zL 1 rs rs � h + ln f rs + 1 − π � 0, (30) k k g posed in this paper is equivalent to stochastic user equilibrium g g zf rs θ k g allocation based on the improved logit. rs rs rs f � exp−θ h · expθ π − 1. (31) k g g Proof. Te Lagrange for the minimization Model k g g g LSUE-RCB can be formulated as All paths between the O-D pair (r, s) are summed up in rs rs rs rs L(f , π) � Z(f) + π q − f equation (31), and the path choice probability of k of GV k g g g rs rs rs k ∈K users can be obtained: g g (25) rs rs rs rs exp−θ h + π q − f , rs e,m e k g k e,m g rs g rs rs rs m∈M P � . (32) k ∈K e,m e,m k rs rs f l l g g rsexp−θ h l ∈K g l g g rs rs where π and π are Lagrange multipliers associated with g e,m constraints (14) and (15), respectively. Take the derivative of Te above-given proof shows that the mathematical equation (25) and obtain its corresponding Kuhn–Tucker programming model proposed in this paper is equivalent to condition as follows: the multiclass stochastic user equilibrium assignment based on improved logit. □ zL · f rs � 0, zf rs Theorem 2. Te solution of the mathematical programming model proposed in this paper is unique. rs f ≥ 0, zL Proof. Te second derivative of the objective function is as ≥ 0, zf rs follows: ⎧ ⎪ zL , if l � k, 2 ⎪ · f rs � 0, z L rs θ f k e,m k e,m e,m zf rs � k ⎪ e,m zf rs zf rs ⎪ k l ⎪ (26) e,m e,m 0, if l ≠ k, rs f ≥ 0, e,m (33) ⎧ ⎪ , if l � k, 2 ⎪ zL z L θ f rs g k ≥ 0, rs ⎪ zf rs rs k zf zf ⎪ e,m k l ⎪ g g 0, l ≠ k. zL � 0, rs It can be seen that the Hessian matrix corresponding to zπ e,m the objective function is positive defnite, so the model zL constructed has a unique solution. □ � 0, rs zπ rs 3.6. Solution Algorithm Design. Tis paper uses the method For each efective path k of BEV users between O-D e,m of successive average (MSA) [39] based on the path rs pair (r, s), there is f > 0, so we can obtain the following e,m to solve the multiuser stochastic user equilibrium equation: model based on the improved logit (MSA-LSUE-RCB). zL 1 rs rs rs Te pseudocode of the MSA-LSUE-RCB algorithm is rs � h + ln f + 1 + ] − π � 0, (27) k m,k g e,m e,m rs zf θ k e,m shown in Algorithm 1. Te algorithm fow chart is e,m shown in Figure 2. Te main steps of this algorithm are as rs rs rs follows. rs f � exp−θ h + ] · expθ π − 1. (28) k e,m k m,k e,m e,m e,m e,m Step 1. Initialization. According to the road network, the All paths between the O-D pair (r, s) are summed in rs related parameter values are calculated, and the generalized equation (28), and the path choice probability of k can be e,m rs(0) rs(0) rs(0) travel cost h , h , and correction term ] for each k k m,k obtained: g e,m user of the reasonable path with zero initial fow is obtained. (n) (n) rs rs Calculate the initial path fow f rs , f rs according to exp−θ h + ] k k rs e f k m,k g e,m e,m rs em equations (16) and (17). Set counter n :�1. P � . k (29) e,m rs rs rs f em exp−θ h + ] l m,k e,m rs l ∈K e,m e,m Step 2. Update. Set counter n: �1 + n. According to the rs (n) (n) (n) For each efective path k of GV users between the O-D path fow f , f , the link fow x and charging fow rs rs g a k k g e,m (n) rs pair (r, s), there is f > 0, so we can obtain the following y can be obtained according to equations (19)–(23). k u equation: Find the k path choice set of GV by combining the k- 10 Journal of Advanced Transportation (1) Input road network information (OD demand; capacity, free fow, and length of each link; nodes of alternative charging station; SOC of BEV); and other parameters (0) (0) (0) (2) Initialize x � 0, x � 0, y � 0, for all a ∈ A, u ∈ U ag ae u rs(0) rs(0) (3) Calculate the generalized cost of k paths of BEV and GV path choice set respectively h h , k ∈ l l · · · , l and correction 1, 2 k k k g e,m rs(0) rs(0) rs(0) rs (0) term ] � exp(c · (D − L )). Use equations (3)–(6) to yields k path fows set {f , f }, k ∈ l l · · · , l and y . Set safe 1, 2 k m,k m,k k k u g e,m counter n � 1 rs(0) rs(0) (4) Update. According to the path fow f , f , k ∈ l l · · · , l and equation (21)–(23) the link fow and charging fow can k k 1, 2 k g e,m (n) (n) (n) rs(n) rs(n) x , x � 0, y be obtained. So as to obtain the generalized cost of k paths of BEV and GV path choice set respectively h h , ag ae u k k g e,m rs(n) rs k ∈ l l · · · , l and correction term ] � exp(c · (D − L )). Use equations (3)–(6) to yields k path fows set 1, 2 k safe m,k m,k rs(n) rs(n) (n) f , f , k ∈ l l · · · , l and y k k 1, 2 k u g e,m rs(n) rs(n) (n) (5) Descent-direction fnding. Use equations (3)–(6) to yields k auxiliary path fows set {g , g }, k ∈ l l · · · , l and y 1, 2 k u k k g e,m (n+1) (n) (n) (n) (n+1) (n) (n) (n) (n+1) (n) (n) (n) (6) Move. Set f � f + 1/n(g − f ), f � f + 1/n(g − f ), and y � y + 1/n(y − y ) rs rs rs rs rs rs rs rs k k k k k k k k u u u u g g g g e,m e,m e,m e,m (7) Convergence test. If a convergence criterion is not met, set n: � n + 1 and go to step 4; otherwise, stop and the set of stochastic user (n+1) (n+1) (n+1) (n+1) (n) equilibrium fows is x , x , x + x , y . ag ae ag ae u ALGORITHM 1: Pseudocode of MSA-LSUE-RCB algorithm. Set n=1 Generate initial k path flows sets and charging flow GV path choice BEVpath choice probability probability Generate k auxiliary path flow sets and auxiliary charging flow Find the descending direction and update the float flow n=1+n No Convergence test Yes End Figure 2: Flowchart for MSA-LSUE-RCB algorithm. shortest loopless path algorithm. Find the k path choice Step 3. Determine the descent direction. Calculate the ad- (n) (n) set of BEV reasonable routing set by the multilabel al- ditional path fow g rs , g rs according to equations (16) and k k g e,m rs(n) (n) gorithm. At the same time, correction terms ] for each (17). Calculate the additional changing fow y according m,k u BEV user of the reasonable path can be obtained. Cal- to equations (22) and (23). Te GV and BEV path fow rs(n) rs(n) (n) (n) (n) (n) culate the generalized path travel cost h , h of each descent direction are g rs − f rs , g rs − f rs , and k k k k k k g e,m g g e,m e,m (n) (n) new reasonable path. y − y , respectively. u u (24,3,10,15) (26,4.2,9.99) Journal of Advanced Transportation 11 1 2 (16,3.6,6.02) 4 14 (28,2.4,13.86) (16,2.4,25.82) (24,1.2,28.25) 3 4 5 6 6 9 8 11 13 23 16 19 (20,1.2,10.1) (16,1.8,15.68) 9 8 10 31 7 35 7 24 20 21 17 18 54 22 47 25 26 (14,3.6,9.82) (18,3,20) (22,3,10.27) (36,1.8,39.36) 12 36 11 32 10 29 16 50 33 27 48 55 (14,3.6,9.82) (18,3,20) 28 43 34 40 (16,3,10.26) (16,2.4,4.42) 14 19 41 15 45 44 57 37 38 42 71 46 67 59 61 23 22 73 76 69 64 62 (12,2.4,10.18) (12,1.8,9.77) (12,3.6,10.12) 13 24 21 20 39 75 65 74 66 63 Figure 3: Sioux Falls network (colorflled node denotes CS). Step 4. Move. Improve the path fow with the iterative Step 5. Convergence test. Tis paper uses root mean squared weighting method: error (RMSE) to judge convergence: (n+1) (n) (n) (n) f rs � f rs + g rs − f rs , k k k k g g g g (n+1) (n) (n) (n) (34) f rs � f rs + g rs − f rs , k k k k e,m e,m e,m e,m (n+1) (n) (n) (n) y � y + y − y , u u u u ����������������������������������������������� � 1 2 2 (n) rs(n) rs(n) (n) (n−1) ⎛ ⎜ ⎜ ⎠ ⎜ ⎞ RMSE � g − f + y − y , (35) k k u u |K| rs u∈U k∈ l l ···,l { 1, 2 k} (22,2.4,8.11) (12,2.4,46.8) (30,1.8,51.8) (20,2.4,46.8) (12,1.2,10.16) (18,2.4,51.8) (28,2.4,9.75) (32,3.6,9.04) (24,1.2,10.46) (14,2.4,20.63) (34,3.6,27.02) (32,3.6,9.04) (8,3,10.25) (24,1.8,39.36) (26,2.4,10.01) (20,1.2,21.62) (20,3,15.92) (26,2.4,6.05) (18,3,10.09) (24,1.8,39.36) (18,1.2,46.81) 12 Journal of Advanced Transportation Table 3: Corresponding parameters. ×10 2.5 Parameters Value μ 0.64 σ 0.12 μ 0.35 σ 0.08 1.5 ω 0.153 kwh/km average C 0.22 yuan/km C 2.67 yuan/km β (β ) 0.6 min/yuan e g v , i � 1, · · · 5 8 veh/h s (s ) 60 0.5 5 24 s (s ) 80 11 15 s 100 ε 0.001 10 20 30 40 50 60 70 80 θ 2 link θ 1 c 0.01 10% 20% 30% Figure 5: Te equilibrium GV fow under diferent BEV market shares. ×10 2.5 1.5 10 20 30 40 50 60 70 80 link 0.5 10% 20% 30% 10 20 30 40 50 60 70 80 Figure 4: Te equilibrium BEV fow under diferent BEV market Link shares. 10% where |K| is the number of paths between all OD pairs. When 20% (n) RMSE ≤ ε, stop. Otherwise, let n = n + 1, and go to Step 2. 30% Figure 6: Te total equilibrium fow under diferent BEV market 4.Analysis of Results shares. 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. Te free fow and capacity categories: SOC � [60%, 70%], SOC � [70%, 80%], 10 20 of each link are from He et al. [30], as shown in Figure 3. O- SOC � [80%, 90%], and SOC � [90%, 100%]. Consid- 30 40 D demand comes from Bar Gear [39]. Te numbers in ering the gradual increase of BEV market share, we divide brackets at the arrow point take values of link length (km), the market share into three situations (η � 10%, 20%, 30%) free fow (min), and link capacity (veh/h), respectively. Te to analyse its impact on the transportation network and numbers at the connection point represent the link number. charging stations. SOC is set to 30%. Other corre- safe Tere are fve fast charging stations in the network, located at sponding parameters are given in Table 3. nodes 5, 11, 15, 16, and 24. Te SOC of BEV users at the Figures 4–6 compare the equilibrium BEV link fow, the beginning of driving is 60% or above, accounting for 97% equilibrium GV link fow, and the total equilibrium link fow [34]. Discrete initial SOC , BEV users are divided into four for the three market shares. For links 25, 26, 27, and 32, BEV The equilibrium BEV flow Total equilibrium fow The equilibrium GV flow Journal of Advanced Transportation 13 Table 4: Generalized link fow (veh/h). Table 4: Continued. Market share Market share No. No. 10% 20% 30% 10% 20% 30% 1 3590 3380 3170 60 16110 16020 15930 2 6130 6260 6390 61 5910 5820 5730 3 3600 3400 3200 62 6140 6180 6220 4 6470 6340 6210 63 6780 6560 6340 5 6120 6240 6360 64 5340 5380 5420 6 10380 1056 10740 65 10500 10520 10540 7 6810 6720 6630 66 12660 12720 12780 8 11590 11780 11970 67 17780 17960 18140 9 13880 14060 14240 68 7970 7740 7510 10 6920 7040 7160 69 11020 11040 11060 11 15170 15340 15510 70 8440 8380 8320 12 9800 10000 10200 71 10530 10560 10590 13 6460 6620 6780 72 9630 9560 9490 14 6480 6360 6240 73 5830 5860 5890 15 10730 11060 11390 74 10830 10860 10890 16 17210 17020 16830 75 12960 13020 13080 17 11660 11220 10780 76 5730 5760 5790 18 10310 9920 9530 19 18050 18000 17950 20 16970 16540 16110 21 10370 10240 10110 22 10610 10820 11030 23 6820 6840 6860 24 15770 15640 15510 25 23490 23680 23870 26 23650 23700 23750 27 23430 23460 23490 28 14180 14260 14340 29 13620 13840 14060 30 5800 5800 5800 31 6740 6880 7020 32 22830 22860 22890 33 10520 10640 10760 34 19030 19060 19090 35 6790 6680 6570 5 11 15 16 24 36 11440 11580 11720 Charging station 37 11330 11360 11390 38 12230 12260 12290 10% 39 10930 10960 10990 20% 40 18230 18260 18290 30% 41 6210 6220 6230 Figure 7: Comparison of possible charging diferent BEV market 42 11920 11940 11960 shares. 43 14380 14460 14540 44 6800 6800 6800 45 15310 15420 15530 trafc demand increases rapidly with the increase of BEV 46 16680 16860 17040 47 11160 11520 11880 market share; at the same time, the trafc demand of GV in 48 13380 13460 13540 these links correspondingly reduces the demand, so the total 49 12690 12980 13270 fow of the links will not fuctuate too much. Te total trafc 50 11880 12460 13040 fow of each link is shown in Table 4 and Figure 6. With the 51 6400 6400 6400 increase of BEV market share, the total trafc demand for 52 12290 12580 12870 sections 15, 47, 49, 50, and 55 increases signifcantly by an 53 13350 13300 13250 average of 1,000 to more than 1,200 vehicles per hour, an 54 10300 9900 9500 increase of between 6% and 9%. Tese links are concentrated 55 12390 12980 13570 near the charging station 16, and the trafc demand of other 56 15510 15420 15330 sections near the charging station increases little. At the 57 15100 15200 15300 same time, the total trafc demand of links 17, 18, 20, and 54 58 13450 13400 13350 59 6120 6040 5960 also decreased signifcantly by an average of 380 to more than 440 vehicles per hour; the fow change of other links is Te possible charging fow 14 Journal of Advanced Transportation stations can be determined, so that the trafc network users not signifcant. Figure 7 shows the comparison of the possible charging fow of fve charging stations under dif- can randomly choose the feasible path according to their perception of the generalized path cost, and fnally reach the ferent BEV market shares. With the increase of BEV market share, the random charging fow of charging station 16 has network equilibrium. signifcantly increased from 437 to more than 1093 vehicles Forecasting and analyzing the changing trend of trafc per hour, indicating that the location of the charging station fow and charging fow can provide a basis for decision is more appropriate. However, the random charging fow of makers and the CS operator from the perspective of demand. charging station 24 has little change with the increase of BEV Trough the simulation analysis of the trafc fow of each market share, which indicates that its utilization is not link of the trafc network under diferent market shares and sufcient and the site selection needs to be reconsidered. Te initial SOC, it can provide a reference for trafc managers to random charging fow of charging stations 5, 11, and 15 formulate management plans, such as improving some links’ grades and adding new links. Similarly, the utilization degree increases with the increase of BEV market share, indicating that the site selection is appropriate. of the charging station can be inversely deduced by pre- dicting the possible charging demand, which also provides a reference scheme for the layout of civil infrastructure. Te 5.Conclusions expansion research of this paper can also build a double- In this paper, the stochastic user model based on the im- layer model to analyze the relationship between charging proved logit model is proposed considering the BEV market supply and demand in combination with the charging share and random charging behavior. Te choice of the station location problem. alternative path is afected by the travel time, the path energy consumption, and the queue dwell time of charging. Based Data Availability on the multilabel algorithm and Yen algorithm, the alter- native path set of BEV is determined and multiple attributes Te data used to support the fndings of this study are in- of the path are recorded. In the set of alternative paths, BEV cluded within the article and are from Nguyen, S., Dupuis, users prefer to choose paths whose path length does not C., 1984. An Efcient Method for computing trafc equi- exceed the psychological safety distance, so we add a cor- libria in networks with asymmetric transp. costs. Transp. Sci. rection term to the logit model. In the interaction between 18, 185–202. BEVs’ users and GVs’ users, the trafc net fnally reaches the equilibrium state. Te MSA based on the path for solving the Conflicts of Interest mixed model is verifed by simulating diferent scenarios. Under the condition that the location of the alternative Te authors declare that they have no conficts of interest. charging station is known, 364680 vehicles and 24 OD pairs are simulated and evaluated, and the indicators related to the Acknowledgments increase of BEV market share are determined. In the sim- ulation scenario, the market share of BEV starts to increase Tis research was supported by the National Natural Science from 10%, with a total of 36468 vehicles, 72936 vehicles Youth Fund Project (Grant nos. 71801093 and 62002117) accounting for 20%, and gradually increases to 109404 ve- and the Project of Jiangxi Provincial Department of Edu- hicles accounting for 30%. With the gradual increase of BEV cation (Grant no. 190306). market share, the sharp increase of charging demand leads to the rapid growth of trafc fow on the road near the charging References station and the key road to the charging station, so it is necessary for trafc planners to adopt corresponding trafc [1] C. 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Journal of Advanced Transportation – Hindawi Publishing Corporation

**Published: ** Mar 23, 2023

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