Multiobjective Electric Vehicle Charging Network Planning Considering Chance-Constraint on the Travel Distance for Charging

. Electric vehicle charging stations (EVCSs) are important infrastructures to support sustainable development of electric vehicles (EVs), by providing convenient, rapid charging services. Therefore, the planning of electric vehicle charging network (EVCN) has attracted wide interest from both industry and academia. In this paper, a multiobjective planning model for EVCN is developed, where a ﬁ xed number of EVCSs are planned in the traf ﬁ c network (TN) to achieve two objectives, i.e., minimizing both average travel distance for charging (TDfC) of EVs and investment costs of EVCN. According to the random characteristics of EVs ’ TDfC, its constraint is presented as a chance constraint in the developed EVCN planning model. The nondominated sorting genetic Algorithm II with the constraint domination principle (NSGA-II-CDP) is customized to solve the developed multiobjective EVCN planning model, by designing a special coding scheme, a crossover operator, and a mutation operator. Then, a maximum gradient principle of investment revenue is designed to select the optimal planning strategy from the Pareto-optimal solution set, when taking the investment return ratio as primary consideration. A 25-node TN is used to justify the effectiveness of the developed methodology.


Introduction
1.1.Background and Motivation.With the exacerbation of environmental pollution and the overuse of fossil fuels, electric vehicles (EVs) are receiving increasing attention in the auto market [1,2].Until now, rapid charging at EV charging stations (EVCSs) is one of the most commonly used charging models.The location of EVCSs can have dramatic impacts on the charging convenience for EV owners.Thus, it is critical to investigate the optimal planning strategy for EV charging networks (EVCN) in both urban areas [3] and highway networks [4,5], to boost the shares of EVs in the auto market [6,7].
In EVCSs, EV owners normally adopt rapid charging, whose rated power can be as high as tens or even hundreds of kW for each charger.By charging several EVs with rapid charging mode in an EVCS simultaneously, the total power could reach the level of MW, thus significantly affecting the secure and efficient operation of the power distribution system (PDS) [8][9][10][11].Therefore, it is important to consider PDS operation in EVCN planning, so as to ensure a reliable power supply to EVCN.In detail, it requires optimizing the location and rated power for the EVCSs to mitigate adverse impacts on the PDS resulting from charging large-scale EVs [12][13][14].
As important service facilities in the traffic network (TN), EVCSs play a role similar to "gas stations".The primary function of EVCSs is to provide high-quality and convenient charging services to EV owners [15,16].Therefore, planners should pay close attention to the needs by optimizing EVCS deployments when planning the EVCN [17,18].
Currently, EVCN is expanding at a very early stage, especially in some developing countries.It rarely exists the scenario that multiple EVCSs are connected to the same PDS and have significant impacts on the PDS.Therefore, at this stage, the industry should take the primary function of EVCSs, i.e., providing convenient charging service to customers, as the main task in planning the EVCN.The locations of EVCSs are directly related to the convenience of charging services.
1.2.Literature Review.When planning EVCN, two types of models are mainly used to optimize the locations of EVCSs, i.e., flow-based and spatial-based models.
As to the flow-based model, Hodgson [19] developed the flow-capturing location method (FCLM) to optimize the location of the service facilities within a TN.The optimal planning strategy could maximize captured traffic flows, i.e., providing a maximal quota of service to vehicles.Kuby and Lim [20] developed the flow-refueling location model (FRLM) method by improving the FCLM, which considered available driving distances of the vehicles, thus achieving better planning strategies.Furthermore, Capar et al. [21], Kadri et al. [22], and Lee and Han [23] improved the fundamental FRLM method by considering uncertain travel distance for charging (TDfC) of EVs in calculating the traffic flow captured by the EVCN, which is therefore more practical.A generalized FRLM was introduced in [21].This method measured the initial driving ranges of EVs as the distance between the origin and first EVCS on the EV driving paths.A multistage stochastic planning model was developed in [22] based on the generalized FRLM to derive the optimal sites of EVCSs, which was solved by the Benders decomposition approach and the genetic algorithm (GA).Lee and Han [23] developed a Benders-and-Price method to select the optimal locations for EVCSs.This method performed three advantages: (1) assuming that the distribution of maximum TDfC for EVs should be normally distributed; (2) considering uncertain characteristics of EV flows in TN, and; (3) EV owners might choose the nonshortest paths to suggested EVCSs.Convenient charging service means that EV owners could find an EVCS within their expected range when their EVs need to be charged, i.e., the shorter the average TDfC the better.This principle indicates that flow-based models [19][20][21][22][23] have limitations because they just devoted to maximizing the traffic flow captured by the EVCN, but have not taken the minimization of EVs' TDfC as a significant target of EVCN planning.
Different from the flow-based model, the spatial-based model was studied in [24][25][26][27][28][29][30] for optimizing the locations of EVCSs.Hakimi [24] developed the p-median model (PMM) to minimize the overall distance between demand and service facilities.And he improved the PMM by developing a pmedian competitive location model to solve the competitive locations between different charging service providers [25].Further, there are two other spatial-based models: the set covering model (SCM) and the maximal covering location model (MCLM).SCM is to derive the minimum number of service facilities by assuming all demands are met [26,27].By contrast, MCLM is to cover maximum demand under a given number of service facilities [28,29].PMM is compared with SCM and MCLM, based on case studies of EVCN planning in Beijing, China [30].The results show that both SCM and MCLM ignore the distance between the demands and EVCSs, although the TDfC of EVs is believed to be significant.This, in turn, justified the fact that PMM is more effective than SCM and MCLM in solving EVCN planning problems.Nevertheless, PMM also has limitations in implementation.For example, PMM defines EV charging demand and EVCSs as nodes in TN.This would turn the dynamic process into a "node-to-node" scenario, thus neglecting charging requirement of those EVs driving in paths.Furthermore, the TDfC of EVs is highly correlated to locations of EVs in TN, thus having random characteristics.However, there is no investigation into using spatial-based model to address the random characteristics of EVs' TDfC.

Contribution and Paper Organization.
In the current research, neither the flow-based model nor the spatial-based model takes EVs' TDfC, which seriously affects the charging convenience of EVs, as an important consideration when planning the EVCN.To address the limitations described above, this paper develops a novel multiobjective EVCN planning model with a chance constraint on the random TDfC of EVs.The proposed method has two objectives: (a) to minimize the average TDfC of EVs, (b) to minimize the investment in EVCN construction.The candidate sites are given beforehand as a planning boundary according to the topologies of TN and conditions of PDS.The optimization is solved by the nondominated sorting genetic Algorithm II with constraint domination principle (NSGA-II-CDP) to obtain the Pareto-optimal set.Then, a maximum gradient principle of investment revenue is proposed to select the optimal planning strategy from the derived Pareto-optimal solution set, when taking the investment return ratio as primary consideration.The effectiveness of the developed methodology is justified by applying it to a case TN with 25 nodes.
This paper brings the following key contributions:

TDfC Calculation for EVs
Compared to PMM, the FCLM could consider the charging demands of EVs in both scenarios of driving and stopping in the TN, as illustrated in Section 1.2.Therefore, a method to calculate random TDfC of EVs based on counting the EV flows in the TN by the FCLM is developed in this paper.In addition, the Floyd algorithm, also known as the interpolation method, is an algorithm that uses the idea of dynamic programing to find the shortest path between multiple source points in a given weighted graph with positive or negative edge weights [31].The set of the shortest path in TN is derived by the Floyd algorithm in this paper.Currently, with the rapid development of smart terminals and wireless communication technology, navigation software, such as Google Maps, has been widely used.Therefore, it can be assumed that EV owners follow the shortest path planned by the navigation software while driving [32].According to the realistic origin and destination flow structure, which is provided by a simple gravity spatial interaction model [19], the EV flow f q for the shortest path q, can be defined as follows: where W S,q , and W E,q are the weights of origin and destination of the shortest path q, respectively; Ω q is a set of the shortest path in TN derived by the Floyd algorithm; d q is the length of the shortest path q, which can be calculated by the set and length that the path q passes.When an EV needs to be charged at any point in TN, the owners will drive to the closest EVCS according to the navigation software.The scenario is shown below.Figure 1 is an example to present how to calculate TDfC of EVs.In Figure 1, both A and B are traffic nodes in TN and do not have EVCSs; AB is a road in TN with a length of l AB ; the closest EVCS to Node A is located at Node C in TN, AC is the shortest path from A to C with a length of l AC ; similarly, the closest EVCS to Node B is located at Node D in TN, BD is the shortest path from B to D with a length of l BD .Both AC and BD are elements in Ω q .An EV is assumed to be located between A and B with a distance of x to A. There will be two optional charging strategies for the EV: (1) driving through Node A to the EVCS at Node C with a TDfC of l AC + x; (2) driving through Node B to the EVCS at Node D with a TDfC of l BD + l AB −x.Assuming that the EV owner will drive to the closest EVCS for charging, the TDfC of the EV T(x) illustrated in Figure 1 can be calculated as follows: In addition, if there is an EVCS in Node A or B, l AC or l BD equals to 0 in Equation ( 2), respectively.Equation ( 2) demonstrates that EVs' TDfC is related to the multiple variables, e.g., the topology of TN, the deployment of EVCSs in TN, and the locations of EVs on the road.

EVCN Planning Model
Based on the EV flow on each shortest path in the TN, a multiobjective EVCN planning model is developed here to optimize locations of a fixed number of EVCSs for achieving two objectives: (1) to minimize average TDfC of EVs, and (2) to minimize investment of the EVCN construction.In the developed EVCN planning model, constraint on the TDfC of EVs is presented as a chance constraint according to the random characteristics of EVs' TDfC.

Objectives.
The developed EVCN planning model contains two objectives, which is to minimize both the average TDfC of all EVs in TN and the overall investments of the EVCN.
For an EVCN, its main function is to provide convenient charging services for EVs.In TN, EVs could drive on any road and stay at any location on different roads.It implies that the location of any EV is a random variable.When charging is required, EV owners would rush into the EVCS with the shortest distance or the least driving time.Therefore, Objective 1 of the developed EVCN planning model is to minimize the average TDfC of all EVs in TN, as shown by: where T ave is a weighted average from the average TDfC of EVs on each road in TN and represents the average TDfC of all EVs in TN, the EV flows on the roads are the weighting factors; the road set in TN is represented by Ω R ; f i is the EV flows on the i th road and summed from EV flows of the passing shortest path in the shortest paths set Ω q ; T av,i is the average TDfC of EVs on the i th road, and can be given by:

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where l d,i is the length of the i th road; x is the distance of the EV to be charged from the vertex of the i th road, it is assumed to follow a uniform distribution at the range of [0, l d,i ] as a random variable; T i (x) is defined in Equation ( 2), representing the TDfC of the EV on the i th road.Objective 2 of the developed EVCN planning model is to minimize the overall investments of the EVCNs, as given by: where C total is the overall investments of EVCNs; C con,j is the investment of deploying an EVCS at candidate site j, including land occupation cost, equipment cost, operationmaintenance cost, and extra energy losses cost in PDS resulting from supplying power for the EVCS; y j is a 0-1 binary optimization variable in the developed EVCN planning model: y j = 1, if deploying the EVCS at candidate site j; y j = 0, otherwise.

Constraints.
The developed EVCN planning model considers two constraints, i.e., chance constraint on TDfc and constraint on the number of planned EVCS.Currently, the range of most EVs is still less than that of traditional fossil fuel vehicles.Therefore, EVs' TDfC is expected to be less than a given threshold in the EVCN planning model.However, EVs' TDfC is random.If we manually set distances less than the given threshold, the optimization would generate unreasonable results.Therefore, the constraint on EVs' TDfC is defined using a chance constraint, as given by: where P r {•} is the probability of the occurrence of the event in the brace; φ d-char represents the TDfC of EVs in TN as a random variable; T cha-lim is the given range threshold.The lower the T cha-lim , the better the charging convenience provided by the EVCN; β is a confidence level of satisfying the constraint on the TDfC of EVs, the higher the β, the better the charging convenience provided by the EVCN.p i is the probability that the TDfC of EVs on the i th road is less than the given range threshold, which can be given by: where g i (x) is an auxiliary function to judge whether the TDfC of EV (which is x apart from the vertex of the i th road) is less than the given range threshold, which has two values: 0 or 1, as shown: In the developed EVCN planning model, planners predetermine the number of EVCSs and their candidate sites based on the boundary conditions, e.g., estimated investments, municipal planning, EV penetration rate, and so on.Hence, there is an additional constraint on the number of EVCS constructed in the model, as given by: where M is the number of EVCSs to be constructed.

Solution Method
The developed EVCN planning model is a chance-constraint binary programing problem with two optimization objectives.There is a lack of clear analytical expressions among optimization objectives, constraints, and control variables.Therefore, it is difficult to adopt traditional mathematical solvers, e.g., a general algebraic modeling system to solve the developed EVCN planning model.To achieve multiobjective optimization, it requires coordinating relationships among multiple objectives to obtain a solution set [33].The constrained multiobjective evolutionary algorithms (CMOEAs) can coordinate multiple objectives in the optimization, thus giving the solution set of the multiobjective optimization [34][35][36][37].NSGA-II-CDP is a famous CMOEA, which is customized and then utilized to solve the developed EVCN planning model in this paper.In Figure 2, N pop is the population size, i.e., the number of chromosomes in the population; g is an index of evolutional generation; T ave,k represents an average TDfC of all EVs to the closest EVCS when the EVCN is constructed according to the deployment strategy given by the k th chromosome; C total,k is an overall investment of the EVCN, and P ev,k represents a probability that the TDfC of EVs is less than the given threshold; G max is a predetermined maximum evolutionary generation.The customized NSGA-II-CDP stops when the evolution reaches the maximum generation.

Detailed Solution
Procedure.The detailed solution using the custom NSGA-II-CDP for the developed EVCN planning model is as follows: 4.2.1.Calculations of Parameters in TN.First, the shortest path set Ω q of the TN is derived by the Floyd algorithm according to the TN topology.Then, EV flows f q (q2Ω q ) on each shortest path are calculated according to Equation (1).At last, EV flows f i (i2Ω R ) on each road are calculated by summing the EV flows.

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Set NSGA-II-CDP parameters: population size N pop , crossover rate P c , mutation rate P m , and evolutionary generation G max etc.
Randomly initializing the parent population Q par ,0 that has N pop chromosomes Decode the k th chromosome in the pre-evolved population and calculate T ave, k , C total, k , and P ev, k The index of evolution generation g is initialized to be 0, i.e., g = 0 Let g = g + 1, start the g th evolutionary generation Output the first level of the feasible solution the Pareto-optimal solution set of the developed multiobjective EVCN planning model The developed EVCN planning model is a binary programing problem, we use a binary-coded approach for encoding and getting chromosomes, i.e., the solution is represented by a binary code string with the length of N, as illustrated in Figure 3.
The binary code string illustrated in Figure 3 represents a solution of the developed EVCN planning model.In Figure 3, the value at the j th coding point represents whether deploying an EVCS at the j th candidate site: if the value is equals to "1", an EVCS is deployed at the j th candidate site; otherwise, there is no deployment plan.To match the constraint on the number of EVCSs given by Equation ( 9), there should be only M coding points having the value of "1" in Figure 3.
In this paper, the parent population in the g th evolutionary generation is entitled as Q par,g .When initializing the parent population, the index of evolutional generation g is set to be 0. Therefore, the initial parent population can be entitled Q par, 0 and generated as the following steps: (a) Values of all coding points of each chromosome are set to be "0"; (b) For each chromosome, M coding points are randomly selected and their values are changed from "0" to "1".

Create a
Pre-Evolved Population.Two chromosomes from the parent population Q par,g are randomly selected, the better one is copied to the offspring population Q off,g .This process is repeated until the number of chromosomes in Q off,g reaches N pop .The basis for selecting better chromosomes can be found in 4.2.4.Q off, g is updated by applying crossover and mutation operations to chromosomes in Q off,g , respectively, at pretermitted probabilities.The operators of crossover and mutation are detailed as follows.Then, Q par,g and Q off,g are merged to create a pre-evolved population R g (R g = Q par,g ∪Q off,g ) which has 2N pop number of chromosomes.
(1) Crossover operator.To ensure that the postcrossover chromosome satisfies the constraint on the number of EVCSs as given by Equation ( 9), the conventional crossover operator is improved as detailed shown in Figure 4.
The improved crossover operator has three steps, detailed as follows: Step 1: two chromosomes are selected randomly from the population as the precrossed chromosomes.
Step 2: a coding point N cro (1 < N cro < N) is randomly selected as the precrossed point; if these two chromosomes have the same coding point number equal to "1" behind N cro , the coding point N cro is considered a feasible crossed point N ave ; otherwise, this process is repeated until the feasible crossed point N ave that satisfies the above demand is found.
Step 3: the coding string is exchanged behind N ave at a crossover probability of P c .
(2) Mutation operator.To ensure the postmutation chromosome satisfies the constraint on the number of EVCSs as given by Equation ( 9), a two-point mutation operator is developed by improving the conventional single-point mutation operator, as illustrated in Figure 5.
The developed two-point mutation operator has three steps, detailed as follows: Step 1: a chromosome is randomly selected from the population as the premutated chromosome.
Step 2: two different coding points N mut1 and N mut2 (1 ≤ N mut1 ≤ N, 1 ≤ N mut2 ≤ N) as the premutated coding points are randomly selected, the values of the coding points N mut1 and N mut2 should be different.
Step 3: the values of the coding points N mut1 and N mut2 are changed simultaneously at a mutation probability of P m .

Elitism Selection Strategy for the Pre-Evolved Population.
All chromosomes are decoded in the pre-evolved population R g to derive the EVCS deployment strategy represented by each chromosome in the pre-evolved population.2N pop chromosomes in the pre-evolved population correspond to their own solutions of the developed EVCN planning model.Both The improved cross-operator.
Mutation FIGURE 5: The developed two-point mutation operator.

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the parent and offspring populations satisfy the constraint on the number of EVCS, since the number of EVCS is set in advance and the improved crossover and mutation operators are designed for evolution.Thus, the pre-evolved population can be divided into two categories on whether the solution can meet the chance constraint on the TDfC of EVs.The solutions that satisfy the constraint are feasible, and the unsatisfied solutions are unfeasible.All chromosomes in the pre-evolved population are reordered according to the following rules [35,36].
(a) Any feasible solution should be better than an unfeasible solution.(b) According to the Pareto-dominance principle, nondominated sorting is executed for all feasible solutions, thus all feasible solutions are divided into several levels, as illustrated in Figure 6.If feasible solutions are on the different levels, the solutions at a more forward level are better than the others.(c) If solutions are on the same level, they can be evaluated by an assigned fitness, i.e., the solution with a greater assigned fitness is better.In order to calculate the assigned fitness, all feasible solutions on each level are first arranged according to the value of C total , as illustrated in Figure 6.Then, as introduced in [36], the assigned fitness of all feasible solutions on each level is determined according to Equation (10).
where n is the index of the solutions on the same level; r is the index of the level; F r,n is the assigned fitness of the n th solution on the r th level; n r is the number of solutions on the r th level; C r; n total and T r; n ave are, respectively, the C total and T ave of the n th solution on the r th level; K is a predetermined maximal number.
(d) For an unfeasible solution, the degree of violating against the chance constraint is entitled as C V , which is given by Equation (11).
The value of C V determines the order of unfeasible solutions, i.e., the less the violating degree, the higher the ranking in the order.
After reordering 2N pop chromosomes according to the principles described above, the top 50% of the chromosomes from the pre-evolved population are selected as the elitisms to constitute the parent population in the next evolutionary generation.
(e) When the evolution reaches the maximum evolutionary generation, the chromosomes in the first level of the feasible solution are outputted as the Paretooptimal solution set of the developed multiobjective EVCN planning model, thus ending the customized NSGA-II-CDP process.

4.3.
The Optimal Deployment Strategy.The set of Paretooptimal solutions derived by the custom NSGA-II-CDP contains multiple nondominated solutions.This leads to the problem that these nondominated solutions cannot be compared due to the conflict between two objectives of the developed EVCN planning model, that is, these two objectives cannot reach the optimum at the same time.
The choice of the optimal deployment strategy depends on the decision maker's subjective desires, which can be considered in three cases: (a) if the investment capital is very limited, the Pareto-optimal solution with the minimum C total is chosen; (b) if the goal is to minimize the average TDfC of EVs regardless of the cost, the optimal solution with the minimum T ave is chosen; (c) if these two objectives need to be taken into account at the same time, that is, taking the investment return ratio as primary consideration, a maximum gradient principle of investment revenue is proposed to select the optimal solution from the Pareto-optimal solution set.
The detailed steps of the maximum gradient principle of investment revenue are described below.
(a) All solutions in the Pareto-optimal solution set are sorted according to C total of the solution, from smallest to largest, thus the first solution is the one with the least investment.(b) Excluding the first solution in the Pareto-optimal solution set, investment revenue gradients of the other solutions are calculated by Equation (12).IET Electrical Systems in Transportation where h is the index of the solution in the Pareto-optimal solution set; H is the number of solutions in the Paretooptimal solution set; T ave,h and C total,h are, respectively, the average TDfC of all EVs in TN and the overall investments of the EVCN corresponding to the h th solution; E h is the gradient of investment revenue of the h th solution.
(c) The solution with the maximum gradient of investment revenue in the Pareto-optimal solution set is taken as the optimal deployment strategy of the EVCN.

Case Study
The developed EVCN planning model and NSGA-II-CDPbased solving method are validated by a case study based on a 25-node TN.
5.1.Case Detail.The topology of the 25-node TN is shown in Figure 7, which contains 25 nodes and 43 roads.In Figure 7, the red numbers indicate the node numbers, and the black numbers on the lines whose units are kilometers represent the link lengths between the nodes.Each node in the TN is a candidate site for EVCS.The weight of each node is introduced in Table 1 [19], and the construction costs of EVCS at each candidate site are presented in Table 2.
In the case study, each road is supposed to be bidirectional, i.e., the shortest path from the start node to the end node is the same as that from the end node to the start node.Therefore, the shortest path set Ω q contains 300 shortest paths (i.e., (25×(25 − 1))/2 = 300).All shortest paths in the TN together with the traffic nodes and roads passing by each shortest path can be determined by the Floyd algorithm.The overall EV flows on all the shortest paths and roads are 0.307 and 0.84, respectively.
The number of planned EVCSs is set to be four in this paper.The driving range of the mainstream EVs at the maximum state-of-charge (SOC) is between 300 and 500 km.Most EV owners would charge their EVs at about 20% SOC.Therefore, the threshold of EVs' TDfC T cha-lim and the confidence level β are, respectively, set as 80 km and 95%.

Solution of the Developed EVCN Planning Model by the
Customized NSGA-II-CDP.Before solving the developed EVCN planning model by the customized NSGA-II-CDP, the relevant parameters are set as follows: the population size N pop is 100; the maximum evolutionary generation G max is 150; the crossover probability P c ; and the mutation probability P m are, respectively, 0.05 and 0.1.
The planning strategies derived in the set of Paretooptimal solutions are shown in Table 3.All planning strategies have been reordered according to C total of each strategy.

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There are seven planning strategies in the set of Paretooptimal solutions.T ave is longest at 43.42 km and C total is 25 × 10 6 ¥ in the first planning strategy.Whereas, in the 7 th planning strategy, T ave and C total are, respectively, 34.08 km and 49 × 10 6 ¥.Compared to the first planning strategy, T ave in the 7 th planning strategy is 21.5% shorter, equivalent to 9.34 km.However, the investments C total increases by 96%, equivalent to 24 × 10 6 ¥.A short average TDfC of EVs brings a significant increase in the investment.Therefore, it is necessary to select the optimal strategy from the seven planning strategies based on the maximum marginal revenue principle.
According to the steps described in Section 4.3, the gradients of investment revenue of all planning strategies are calculated except the first one, which are shown in Figure 8.
Figure 8 presents that the 2 nd planning strategy has the highest marginal investment revenue, which is 11.17 km/ 10 6 ¥.The 2 nd planning strategy is therefore selected as the optimal planning strategy of the case study.
For the 2 nd planning strategy, the locations of the planned EVCSs are nodes 3, 11, 21, and 23, as illustrated in Figure 9. T ave is 36.72 km, C total is 25.6 × 10 6 ¥, and the p ev is 96.04%.Figures      the 2 nd planning strategy.This follows the chance constraint on EVs' TDfC given by Equation (6).

The Impact of Confidence Level on the Planning Results.
The confidence level β is an important parameter in the developed EVCN planning model, which is predetermined by the planners.When the confidence level β decreases from 95% to 90%, the derived planning strategies in the Paretooptimal solution set change accordingly, as illustrated in Table 4.The gradients of investment revenue of the planning strategies 2, 3, 4, 5, and 6 given in the Table 4 are calculated and presented in Figure 12.
Tables 3 and 4 show that a different Pareto-optimal solution set appears when reducing the confidence level β from 95% to 90%.The number of Pareto-optimal solutions reduces from 7 to 6 resulting from reducing the 5% confidence level.For this case, the minimum and maximum values of T ave are 33.99 and 45.14 km, respectively, and the corresponding C total are 44 × 10 6 ¥ and 24.2 × 10 6 ¥.
Figure 12 presents that the 3 rd planning strategy in the Pareto-optimal solution set is an optimal one, because it has the maximum value of E h , which is 6.01 km/10 6 ¥.Further observation shows that the 3 rd planning strategy in Table 4 is the same as the 2 nd one in Table 3.When the confidence level decreases from 95% to 90%, the optimal planning strategy does not despite a different Pareto-optimal solution.However, the comparison between Figures 8 and 12 shows that the advantage of the optimal solution over other solutions decreases with the decrease in confidence level.

The Planning Results in Different Numbers of EVCSs.
In previous cases, the number of planned EVCSs is set to 4. As an important boundary condition, the number of planned EVCSs may delivers a significant impact on the planning results.Table 5 shows the optimal planning strategies corresponding to the different numbers of planned EVCSs.

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Table 5 shows that T ave would decrease and C total would increase consistently with the increasing number of planned EVCSs.When the number of planned EVCSs increases from 9 to 10, C total remains unchanged, equals 59.2 × 10 6 ¥, however, T ave increases from 22.22 to 24.83 km.It shows that a greater number of the planned EVCSs is not always better.Therefore, planners should take a comprehensive consideration of the number of the planned EVCSs in the developed EVCN.
Figures 13 and 14 present the PDF and the CDF of the EVs' TDfC in the different numbers of planned EVCSs, respectively.These two figures demonstrate that the different number of planned EVCSs brings different probabilistic characteristics of the EVs' TDfC.The greater the number of planned EVCSs, the higher the probability that the TDfC of the EVs is less than the set threshold.When the number of planned EVCSs is more than 5, the p ev approaches 100%.When the number of planned EVCSs increases to 9, most of the EVs' TDfC is between 0 and 50 km with a probability of 94.44%.

Conclusion
To improve the charging convenience of EVs, this paper developed a chance-constraint multiobjective EVCN planning model.With a fixed number of planned EVCSs, the developed EVCN planning model desires to minimize the average TDfC of all EVs in the TN and the investment costs of EVCN by selecting locations of the EVCSs from candidate sites.NSGA-II-CDP is customized and utilized to solve the developed EVCN planning model, thus deriving the Paretooptimal solution set which contains several planning strategies.In addition, when taking the investment return ratio as primary consideration, a maximum gradient principle of investment revenue is proposed to select the optimal planning strategy from the derived Pareto-optimal solution set, i.e., to determine the optimal deployment strategy of the planned EVCSs.
In this paper, EV flows on each road in the developed EVCN planning model are calculated by the FCLM.In practice, these can also be determined according to the measured traffic data to ensure that the planning strategy obtained through the developed methodology is more suitable for the actual demand.IET Electrical Systems in Transportation

4. 1 .
Solving Framework.Figure 2 presents a flowchart for solving the EVCN planning model developed by the custom NSGA-II-CDP.

FIGURE 6 :
FIGURE 6: Nondominated sorting results for all feasible solutions.

FIGURE 7 :
FIGURE 7: The topology of the 25-node TN.

FIGURE 8 :
FIGURE 8: Gradient of investment revenue of all planning strategies except the first one (β = 95% and M = 4).

FIGURE 13 :
FIGURE 13: The PDF of the EVs' TDfC corresponding to different numbers of planned EVCSs.

FIGURE 14 :
FIGURE 14: The CDF of the EVs' TDfC corresponding to different numbers of planned EVCSs.

TABLE 1 :
Weight coefficients of nodes in the 25-node TN.

TABLE 2 :
Construction costs for EVCS at each candidate site.
10 and 11 are, respectively, the probability density function (PDF) and the cumulative distribution function (CDF) of random EVs' TDfC corresponding to the 2 nd planning strategy.Figures 10 and 11 present that the maximum TDfC of few EVs can exceed 110 km, but 96.04% of EVs have a TDfC less than the 80 km threshold, according to

TABLE 3 :
The planning results in the set of Pareto-optimal solutions (β = 95% and M = 4).

TABLE 4 :
The planning results in the Pareto-optimal solution set (β = 90% and M = 4).FIGURE 12: Gradient of investment revenue of all planning strategies except the first one (β = 90% and M = 4).