Optimal Planning of Electric Vehicle Fast-Charging Stations Considering Uncertain Charging Demands via Dantzig–Wolfe Decomposition

Abstract

This study investigates the planning problem of fast-charging stations for electric vehicles with the consideration of uncertain charging demands. This research aims to determine where to build fast-charging stations and how many charging piles to be installed in each fast-charging station. Based on the multicommodity flow model, a chance-constrained programming model is established to address this planning problem. A scenario-based approach as well as a big-M coefficients generation algorithm are applied to reformulate the programming model into tractable one, then the Dantzig–Wolfe decomposition method is leveraged to find its optimal solution. Finally, a numerical experiment is conducted in a 25-node network to assess the efficiency of the proposed model and solution approach.

1. Introduction

Electric vehicles (EVs) are widely acknowledged as a promising solution for reducing greenhouse gas emissions from road transportation and reducing reliance on finite fossil energy sources. Consequently, numerous governments have implemented incentive policies to encourage widespread adoption of EVs, resulting in significant progress towards this goal. However, there are several major obstacles that hinder the adoption of EVs on a larger scale. Among them, the “range anxiety” may be the most momentous challenge, which reflects the fear of running out of energy before reaching the destination or charging station. In practice, an efficient way to alleviate such negative psychology for popularizing EVs is to reasonably plan fast-charging stations in transportation networks. This situation motivates us to investigate the optimal planning problem of fast-charging stations, that is, answer the questions of where to build fast-charging stations and how many charging piles are required in one fast-charging station [1,2,3].

The planning of fast-charging stations has been studied in various contexts, with the charging demands of EVs being a critical component. In this paper, we address the challenge of uncertain charging demands, for which probability distributions are unknown. Instead, we rely on large amounts of collected data on charging demands and adopt a data-driven approach to formulate a siting and sizing model for fast-charging stations. Our approach involves formulating data-driven chance constraints for planning decisions, by incorporating sample data on uncertain charging demands. Specifically, we aim to determine the appropriate number of charging piles to be installed at fast-charging stations, such that they can accommodate the allocated charging demands while meeting a prescribed service level.

Chance constraints are known to be intractable due to their general nonconvexity [4]. To overcome this computational difficulty, we adopt a scenario-based reformulation approach based on a finite number of scenarios extracted from samples of the charging demands. By considering these scenarios, we reformulate the chance-constrained program into a mixed-integer linear program (MILP) that can be efficiently solved using Dantzig–Wolfe decomposition. However, in practice, the MILP is often weak due to the introduction of big-M coefficients through the scenario-based approach. To address this, we propose a big-M coefficient strengthening method to identify appropriate big-M coefficients that result in better computational performance for the subproblems.

The remainder of this paper is organized as follows. Section 2 reviews the literature on the chance-constrained siting and sizing problem of fast-charging stations and highlights the main contributions of this paper. Section 2.5 formulates the chance-constrained programming model for such problem. Section 3 presents a mixed-integer linear reformulation using scenario-based approach and Dantzig–Wolfe decomposition, and provides an efficient way to strengthen the big-M coefficients that can be exploited to accelerate the computation. Section 4 conducts numerical experiments and illustrates managerial insights in the context of both virtual and real world transportation networks. Section 5 concludes this paper with a summary of the most important findings and opportunities for future work.

2. Literature Review

This section reviews the literature on the fast-charging station planning problem and on Dantzig–Wolfe decomposition. We close this section with a statement of the main contributions of this paper.

2.1. Fast-Charging Station Planning Problem

The optimization models that tackle the refueling station planning problem are generally extended from the facility location problem. There are two dominant modeling approaches: the node-based approach and the flow-based approach [5]. The node-based approach considers the node-based demands and utilizes the generalization of p-median model, whose goal is to maximally cover the potential charging demands from a given area by planning a minimum number of fast-charging stations [6,7,8,9,10]. However, the limited driving ranges of EVs make the complete coverage impossible, since EVs may require to be charged multiple times to sustain their long trips, and there always exist unexpected charging demands from other areas. In this regard, the flow-based approach provides an alternative modeling way which considers the charging demands on the path between O-D pairs. The flow-based modeling idea originates from [11], who proposes a flow-capturing location model (FCLM) which aims to plan a given number of stations to capture as much flow as possible. To account for the limited driving ranges, [12] proposes the flow refueling location model (FRLM), where a sequence of refueling stations along a path between an O-D pair that enables a successful trip is defined with respect to a feasible combination of refueling facilities. The FRLM has been extended to be applicable for various theoretical and engineering aspects ([13] and the references therein).

The aforementioned works merely consider the deterministic charging demands; however, the charging demands usually encounter the uncertain fluctuation in reality, which brings about difficulties in obtaining accurate charging demands. The pioneering work that handles the stochastic charging demands is reported by [14], in which the charging demands are assumed to follow Poisson distribution. They develop a cost-concern model as well as a goal-driven model by the distributionally robust optimization approach, where the constraints that relate to the worst-case expected cost and the worst-case occurring probability are approximated by the standard normal distribution. Following this work, Refs. [15,16] consecutively consider the fast-charging station planning problem with the consideration of Poisson-distribution-based charging demands. Such two works simultaneously formulate a second-order cone programming planning model, where the differences appear in the method of solving: Ref. [15] directly uses the CPLEX solver, while Ref. [16] leverages Benders decomposition as well as the CPLEX solver.

In practice, the distribution of uncertain charging demands is often unknown, which has posed challenges to the stochastic modeling approach [17,18,19]. Instead, massive amounts of data are routinely available in transportation networks, for which the distributionally robust optimization and chance-constrained programming approach is more suitable. Ref. [20] formulates a two-stage distributionally robust optimization planning model that incorporates the uncertain charging demands, which is a generalization of FCLM. They employ a sample average approximation method and an averaged two-replication procedure to solve the planning model. Ref. [21] formulates a distributionally robust optimization planning model based on $\varphi$-divergence, where the potential distributions of uncertain charging demands are included in a $\varphi$-divergence-based ambiguous set that is constructed using the available data of charging demands. In order to be efficiently solved via off-the-shelf commercial solvers, they reformulate the planning model into its tractable counterpart using standard reformulation techniques of distributionally robust programs. In addition, Ref. [22] formulates a bi-level chance-constrained programming (CCP) model to tackle the uncertain charging demands, in which the chance constraints guarantee that the probability the EVs successfully get charged in a fast-charging station at least reaches a specified service level. The chance-constrained planning model is finally reformulated into the second-order cone counterparts that can be efficiently solved by off-the-shelf commercial solvers.

2.2. Dantzig–Wolfe Decomposition

Although the planning models have concise formulations, the massive number of decision variables related to large-scale transportation networks makes the solving process computationally demanding, and considering an arbitrary subset of them may bring about a suboptimal solution [23,24,25]. Thereby, the column generation method is introduced to screen out the variables that do not have the potential to consist the optimal solution. Ref. [26] considers the refueling station location problem with routing. The authors apply a branch-and-price algorithm, in which the column generation procedure is employed to manage the size of the location model so as to be solved efficiently. Ref. [27] investigates the optimal deployment problem of charging stations, in which a branch-and-price algorithm is devised for the nonlinear deployment model whose linear relaxations are solved by column generation. Ref. [28] develops a Benders-and-Price algorithm to solve the developed charging station location model, where the column generation procedure is proposed to solve the Benders subproblem. It should be pointed out that Benders decomposition is not applicable for the scenario-based reformulation of chance constraints, because this approach introduces the auxiliary binary variables and big-M coefficient, which are obstacles for obtaining Benders cuts through the duality method.

In this aspect, Dantzig–Wolfe decomposition is applicable, which also employs a column generation procedure [29,30]. Different from Benders decomposition that adds new constraints to the Benders master problem according to the solution of the Benders subproblem, Dantzig–Wolfe decomposition adds new variables to Dantzig–Wolfe master problem according to Dantzig–Wolfe subproblems, which are column generation (pricing) subproblems that aim to find the variables with positive/negative reduced costs [31]. Briefly speaking, Benders decomposition adds rows to the master problem, while Dantzig–Wolfe decomposition adds columns to the master problem. Ref. [32] adapts Dantzig–Wolfe decomposition to the multiple depot vehicle scheduling problem. They apply two types of Dantzig–Wolfe decomposition to the inventory formulation: one for each depot, the other for each trip. Ref. [33] considers the facility location and production planning problem, in which Dantzig–Wolfe decomposition is proposed to detect and compare the lower bounds between the two formulated mathematical models. Ref. [34] describes two versions of the chance-constrained bin packing problem, the first one is a scenario-based reformulation, and the second one is a distributionally robust reformulation. Both versions are solved by a Dantzig–Wolfe formulation suited to a branch-and-price algorithm. Ref. [35] utilizes Dantzig–Wolfe decomposition as well as a branch-and-price procedure to solve an MILP model for the vehicle allocation problem.

2.3. Summary of Main Contributions

When implementing the standard Dantzig–Wolfe decomposition, one master problem and multiple subproblems should be formulated. The master problem should gather the optimal solutions (extreme points) of all the subproblems to represent its feasible solution as the convex combination of such extreme points, which may give rise to unexpected demanding computation in a large-scale network (so-called “dimensional curse”), because massive decision variables have to be introduced corresponding to the extreme points of the subproblems. To overcome this “dimensional curse”, we intend to separate the original planning problem into disjoint sub-components corresponding to the O-D pairs, then leverage a Dantzig–Wolfe decomposition to find their optimal solutions, in which the master problem and subproblem are one-to-one with respect to each O-D pair. Thus, a moderate scale of decision variables are involved in each pair of master problem and subproblem, which is conducive to improving the computational performance of Dantzig–Wolfe decomposition. As long as all the sub-components are solved, we are able to obtain the optimal solution of the original problem by synthesizing the optimal solution of all the sub-components. In addition, the disjoint property of the sub-components makes their facilitation to be implemented in a decentralized manner using parallel machines, which helps improve the computational efficiency as well. To the best of our knowledge, this is the first attempt to leverage Dantzig–Wolfe decomposition for the optimal planning of electric vehicle fast-charging stations considering uncertain charging demands. We contribute to existing works as follows:

(i)

The data-driven chance constraints are introduced to address the uncertain charging demands, since it is difficult to extract the exact values and accurate probability distributions of the uncertain charging demands in practice. In this circumstance, date-driven chance constraints are viable alternatives to measure the service abilities of fast-charging stations.

(ii)

The original planning model is separated into multiple disjoint sub-components corresponding to the O-D pairs in the transportation network, which are facilitated to be implemented in a decentralized manner using parallel machines. Then, the Dantzig–Wolfe decomposition is utilized to solve the planning model, where the master problems and subproblems are one-to-one. Each possesses a moderate scale of decision variables, which makes it conducive to improve the computational performances.

(iii)

The computational performance is tested using the 25-node benchmark network, in which the proposed Dantzig–Wolfe decomposition is shown to be efficient for solving the planning problem over a large transportation network.

2.4. Expanded Network Generation

Prior to establishing the planning model, a finite set of candidate siting nodes should be identified according to the driving ranges of EVs, a part of which is then chosen to build fast-charging stations and install charging piles so as to satisfy the potential charging demands of EVs. Here, we use the expanded network approach. The expanded network approach is originally proposed in [36], where the charging constraints that express the charging demands on the paths are presented by the flow balance constraints, that is, an available flow on the expanded network ensures a feasible charging pattern. Although the expanded network expands the scale of the original network, its numerical efficiency is demonstrated to outperform some existing models, and simultaneously, it removes the big-M constants from the formulations that yield poor linear relaxation bounds [28]. Due to the above-mentioned advantages, the expanded network model is widely employed and extended in the literature, such as in [28,37,38,39]. On the other hand, another modeling idea called sub-path is also proposed in [14] and employed and extended in [15,16,26,27]. As the name suggests, the sub-path model segments a complete path into several sub-paths with the consideration of driving ranges, then the optimization model for siting and sizing the fast-charging stations is formulated based on such sub-paths. Owing to the fact that the paths in the original transportation network are segmented into small pieces, the flow balance constraints that are considered in the expanded network model cannot be incorporated. Therefore, the sub-path model often considers the charging demands on each sub-path or individual nodes instead. In practice, however, the transportation networks are usually composed by a large amount of nodes that may result in countless sub-paths, which leads to computational difficulties. To avoid this situation, we employ the expanded network model, which is generated based on the following assumptions:

(A1)

More than half of the battery is full when travel starts at the original node.

(A2)

More than half of the battery is full when travel ends at the destination node.

(A3)

The driving ranges of electric vehicles are deterministic and fixed.

When finishing their previous trips, the drivers of electric vehicles always have the willingness to reserve enough remaining mileage for their next trips. Owing to the driving range anxiety of electric vehicle drivers, there exists a threshold of remaining mileage for the drivers: If the remaining mileage stage is above the threshold, the drivers would continue their next trip; otherwise, they would choose to charge in fast-charging stations. Assumption (A1) describes the situation in which the threshold of remaining mileage is set to be half of the full driving range. Assumption (A2) describes another situation that, when a driver needs to complete a round trip, they should reserve adequate remaining mileage after finishing a single-loop trip. Assumption (A3) indicates that the driving ranges are always a constant R that describes the driving distance of a fully charged electric vehicle.

Let $\mathcal{G}\left(V,A,Q\right)$ be a given transportation network, where V and A are the set of nodes and arcs, respectively. We consider an O-D pair q on $\mathcal{G}$, which starts at the origin node O and terminates at the destination node D. We denote the set of nodes and arcs on O-D pair q as $V^q$ and $A^q$, respectively, and the driving distance on arc $(i,j)\in A^q$ as $d_{ij}^q$. Then, the generation of the expanded network ${\overline{\mathcal{G}}}^q\left({\overline{V}}^q,{\overline{A}}^q\right)$ based on O-D pair q involves the following steps:

Step 1 

Before the original node O, add a source node $s^q$. After the destination node D, add a sink node $t^q$. Note that ${\overline{V}}^q$ is ${\overline{V}}^q\leftarrow V^q\bigcup \{s^q,t^q\}$.

Step 2 

Connect $s^q$ and O, D and $t^q$. Note that ${\overline{A}}^q\leftarrow A^q\bigcup \{\left(s^q,O\right),\left(D,t^q\right)\}$.

Step 3 

Ordering the nodes in ${\overline{V}}^q$ sequentially. Let $ord(·)$ denote the ordering index of the nodes.

Step 4 

Add arc $(s^q,j)$ to ${\overline{A}}^q$ if $d_{Oj}^q$ is strictly less than $\frac{R}{2}$; add arc $(i,t^q)$ to ${\overline{A}}^q$ if $d_{iD}^q$ is strictly less than $\frac{R}{2}$. Add arc $(i,j)$ to ${\overline{A}}^q$ if $ord\left(i\right)<ord\left(j\right)$ and $d_{ij}^q$ is strictly less than R.

We take a toy network in Figure 1, Figure 2 and Figure 3 as an example to clarify how to generate the expanded network according to Step 1–Step 4, where the driving range R is set to be 120 km.

2.5. Mathematical Model

Based on the expanded network, an optimization model for the fast charging station planning problem is proposed, which is based upon a generalized multicommodity flow model. The conventional multicommodity flow model considered in [28] utilizes binary variables for the arcs to represent whether the arcs are used by the EVs, while we utilize continuous variables for the arcs to represent the proportion of charging demands that allocated on the arcs. In practice, due to the rapid growth of electric vehicle ownership, their charging demands have emerged a salient feature of uncertainty; even their probability distributions are difficult to be accurately obtained. Thus, we consider the uncertain charging demands ${\tilde{F}}^q$ on path $q\in Q$. For any path $q\in Q$, we generate the corresponding expanded network ${\overline{\mathcal{G}}}^q\left({\overline{V}}^q,{\overline{A}}^q\right)$. Let $c_{1,i}$ and $c_{2,i}$ be the building cost of a fast-charging station and installing cost of charging piles at node $i\in V$. Then, the mathematical model for the fast-charging station planning problem is formulated as follows:

$$\begin{array}{}\mathrm{(1a)}& \left(\mathbf{P}\right)\hfill & min\sum _{i\in V}\left(c_{1,i}{u}_i+c_{2,i}{y}_i\right),\hfill \mathrm{(1b)}& \hfill & s.t.\sum _{j\in N_i^{-}}{x}_{ij}^q-\sum _{j\in N_i^{+}}{x}_{ji}^q=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}i=s^q,\hfill \\ -1,\hfill & \mathrm{if}i=t^q,\hfill \\ 0,\hfill & \mathrm{if}i\ne s^q,i\ne t^q,\hfill \end{array}\right.\forall q\in Q,\hfill \mathrm{(1c)}& \hfill & \mathbb{P}\left(\sum _{q\in Q}{\tilde{F}}^q\sum _{j\in N_i^{+}}{x}_{ji}^q\le y_i\right)\ge \alpha ,\forall i\in V,\hfill \mathrm{(1d)}& \hfill & y_i\le {\rm Y}{u}_i,\forall i\in V,\hfill \mathrm{(1e)}& \hfill & u_i\in \{0,1\},\forall i\in V,\hfill \mathrm{(1f)}& \hfill & y_i\in {\mathbb{Z}}_{+},\forall i\in V,\hfill \mathrm{(1g)}& \hfill & x_{ij}^q\ge 0,\forall (i,j)\in {\overline{A}}^q,q\in Q.\hfill \end{array}$$

where $N_i^{+}$ and $N_i^{-}$ are the set of in-neighbors and out-neighbors of node i on the expanded network of O-D pair q, that is, $N_i^{+}=\left\{j|(j,i)\in {\overline{A}}^q,q\in Q\right\}$ and $N_i^{-}=\left\{j|(i,j)\in {\overline{A}}^q,q\in Q\right\}$. Objective function (1a) aims to minimize the total setup cost of fast-charging stations, which includes building the fast-charging stations and installing the charging piles. Constraints (1b) are the flow balance constraints that allocate the charging demands to every node that the path traverses, which describe the process that the charging demands flow losslessly from the source node to the sink node through the intermediate nodes. Constraints (1c) are the chance constraints that ensure that the allocated charging demands are accommodated by building fast-charging stations and installing charging piles that at least reach a specified service level $\alpha$. Constraints (1d) enforce the upper bound of installed charging piles at a fast-charging station. Constraints (1e)–(1g) restrict the domains of the siting, sizing, and allocation decision variables, respectively. Figure 4 illustrates a toy network to clarify the generalized multicommodity flow model. The flow conservation constraints (1b) ensures that the out-flow and in-flow at each node are balanced, that is, $x_{O1}=x_{13}+x_{41}=\frac{1}{2}$, $x_{O2}=x_{2D}=\frac{1}{2}$, $x_{13}=x_{3D}=\frac{1}{4}$, $x_{14}=x_{4D}=\frac{1}{4}$, which indicates that $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{4}$, and $\frac{1}{4}$ of total charging demands are allocated at Node 1–Node 4, respectively. Thus, fast-charging stations should be planned properly at these nodes to satisfy such allocated charging demands.

Even though the number of charging piles installed in a fast-charging station has to be an integer, it can be relaxed as a continuous variable without affecting the optimality of P. The following theorem supports this claim.

Theorem 1. 

Denote RP the relaxed model of P, where variables $y_i$ in model P are relaxed to be continuous variables $v_i$ in RP for all $i\in V$. Let $\left({\left(u_i^{*},v_i^{*}\right)}_{i\in V},{\left\{x_{ji}^{q*}\right\}}_{q\in Q,(j,i)\in {\overline{A}}^q}\right)$ be the optimal solution of  RP. Then, $\left({\left(u_i^{*},y_i^{*}\right)}_{i\in V},{\left\{x_{ji}^{q*}\right\}}_{q\in Q,(j,i)\in {\overline{A}}^q}\right)$ with $y_i^{*}=⌈v_i^{*}⌉$ is an optimal solution of  P, where $⌈·⌉$ stands for the ceiling function of real numbers.

Proof. 

We consider the following optimization model

$$\begin{array}{}\mathrm{(2a)}& \left(\tilde{\mathbf{P}}\right)\hfill & min\sum _{i\in V}{c}_{1,i}{u}_i,\hfill & \hfill & s.t.\left(1\mathrm{b}\right),\left(1\mathrm{e}\right),\left(1\mathrm{g}\right),\hfill \mathrm{(2b)}& \hfill & \mathbb{P}\left(\sum _{q\in Q}{\tilde{F}}^q\sum _{j\in N_i^{+}}{x}_{ji}^q\le {\rm Y}{u}_i\right)\ge \alpha ,\forall i\in V.\hfill \end{array}$$

Suppose the optimal solution of $\tilde{\mathbf{P}}$ to be $\left({\left(u_i^{*}\right)}_{i\in V},{\left\{x_{ji}^{q*}\right\}}_{q\in Q,(j,i)\in {\overline{A}}^q}\right)$. Notice that we are capable of equivalently transforming Constraints (2b) into Constraints (1c) and (1d) by introducing the sizing variables $y_i$ for all $i\in V$. Therefore, the optimal values of variables $u_i$ and $x_{ij}^q$ in RP and P must be identical to that in model ($\tilde{\mathbf{P}}$). Thus, we suppose the optimal solution of RP and P to be $\left({\left(u_i^{*},v_i^{*}\right)}_{i\in V},{\left\{x_{ji}^{q*}\right\}}_{q\in Q,(j,i)\in {\overline{A}}^q}\right)$ and $\left({\left(u_i^{*},y_i^{*}\right)}_{i\in V},{\left\{x_{ji}^{q*}\right\}}_{q\in Q,(j,i)\in {\overline{A}}^q}\right)$, respectively. Since the sizing variables are restricted to be integers according to Constraints (1f), in order to minimize the objective (1a), $y_i^{*}$ has to be the lowest integer not less than $v_i^{*}$, that is, $y_i^{*}=⌈v_i^{*}⌉$ for all $i\in V$.   □

Hereafter, we consider the relaxed model RP instead of P

$$\begin{array}{}\mathrm{(3a)}& \left(\mathbf{RP}\right)\hfill & min\sum _{i\in V}\left(c_{1,i}{u}_i+c_{2,i}{v}_i\right),\hfill & & s.t.\left(1\mathrm{b}\right),\left(1\mathrm{e}\right),\left(1\mathrm{g}\right),\mathrm{(3b)}& \hfill & \mathbb{P}\left(\sum _{q\in Q}{\tilde{F}}^q\sum _{j\in N_i^{+}}{x}_{ji}^q\le v_i\right)\ge \alpha ,\forall i\in V,\hfill \mathrm{(3c)}& \hfill & v_i\le {\rm Y}{u}_i,\forall i\in V,\hfill \mathrm{(3d)}& \hfill & v_i\ge 0,\forall i\in V.\hfill \end{array}$$

Although RP provides a concise formulation of planning the fast-charging stations, the nonconvexity of chance constraints as well as the tremendous candidate nodes give rise to its intractability. In the next section, we initially reformulate the chance constraints (3b) into their tractable counterparts, then propose a Dantzig–Wolfe-decomposition-based approach to solve the reformulations to optimality.

3. Scenario-Based Approach

Solving optimization model RP requires tackling chance constraints (3b) prior to this. Here, we leverage the scenario-based technique to do such work. The Abbreviations section lists the nomenclatures of the used symbols and notations of this section.

3.1. Scenario-Based Reformulation

Denote $\mathrm{\Omega }$ as the finite set of all possible scenarios based on the sample average approximation (SAA), where $\omega$ indexes the realizations of charging demands ${\tilde{F}}^q$ for all $q\in Q$ as $\left\{{\widehat{F}}_1^q,{\widehat{F}}_2^q,\dots ,{\widehat{F}}_{\left|\mathrm{\Omega }\right|}^q\right\}$. With the consideration of scenarios of charging demands, we have to further introduce auxiliary binary variables $z_{i\omega }$ to reformulate Constraints (3b) into their linear counterparts, where $z_{i\omega }=1$ if fast-charging station at node i provides adequate service abilities for the allocated charging demands in scenario $\omega$, that is, the ith chance constraint is satisfied, and $z_{i\omega }=0$ otherwise. Then, a tractable reformulation of RP can be expressed as follows

$$\begin{array}{}\mathrm{(4a)}& \left(\mathbf{SRP}\right)\hfill & min\sum _{i\in V}\left(c_{1,i}{u}_i+c_{2,i}{v}_i\right),\hfill & & s.t.\left(1\mathrm{b}\right),\left(1\mathrm{e}\right),\left(1\mathrm{g}\right),\left(3\mathrm{c}\right),\left(3\mathrm{d}\right),\mathrm{(4b)}& \hfill & \sum _{q\in Q}{F}_{\omega }^q\sum _{j\in N_i^{+}}{x}_{ji}^q\le v_i+M_{i\omega }\left(1-z_{i\omega }\right),\forall i\in V,\omega \in \mathrm{\Omega },\hfill \mathrm{(4c)}& \hfill & \sum _{\omega \in \mathrm{\Omega }}{z}_{i\omega }\ge ⌈\alpha \left|\mathrm{\Omega }\right|⌉,\forall i\in V,\hfill \mathrm{(4d)}& \hfill & z_{i\omega }\in \{0,1\},\forall i\in V,\omega \in \mathrm{\Omega },\hfill \end{array}$$

where $M_{i\omega }$ are big-M coefficients for all $i\in V$, $\omega \in \mathrm{\Omega }$. Constraints (4b) and (4c) linearize Constraints (3b), which jointly enforce the service abilities of fast-charging stations.

The big-M coefficients $M_{i\omega }$ in Constraints (4b) have to be not less than ${\sum }_{q\in Q}{F}_{\omega }^q{\sum }_{j\in N_i^{+}}$$x_{ji}^q-{\rm Y}$ for all $\omega \in \mathrm{\Omega }$ and $i\in V$, so as to ensure the feasibility of SRP-SP for all $i\in V$, since Constraints (4b) can be satisfied when $z_{i\omega }$ equals 0. However, the big-M coefficients must not be unnecessarily large, which degrades the performance of SRP-SP. Accordingly, if the columns are fixed to satisfy chance constraints (Constraints (3c)), $M_{i\omega }$ could be strengthened more tightly.

We recall $\tilde{\mathbf{P}}$ as a simplified version of RP, where the sizing decisions are not involved. The scenario-based reformulation $\widetilde{\mathbf{SP}}$ of $\tilde{\mathbf{P}}$ is stated as follows:

$$\begin{array}{cc}\hfill \left(\widetilde{\mathbf{SRP}}\right)& min\sum _{i\in V}{c}_{1,i}{u}_i,\hfill \\ \hfill & s.t.\left(1\mathrm{b}\right),\left(1\mathrm{e}\right),\left(1\mathrm{g}\right),\hfill \\ \hfill & \sum _{q\in Q}{F}_{\omega }^q\sum _{j\in N_i^{+}}{x}_{ji}^q\le {\rm Y}{u}_i+{\tilde{M}}_{i\omega }\left(1-z_{i\omega }\right),\forall i\in V,\omega \in \mathrm{\Omega },\hfill \\ \hfill & \sum _{\omega \in \mathrm{\Omega }}{z}_{i\omega }\ge ⌈\alpha \left|\mathrm{\Omega }\right|⌉,\forall i\in V,\hfill \\ \hfill & z_{i\omega }\in \{0,1\},\forall i\in V,\omega \in \mathrm{\Omega }.\hfill \end{array}$$

Indeed, although $\widetilde{\mathbf{SRP}}$ does not include the sizing variables $v_i$, the sizing decisions can also be made according to the binary variable $z_{i\omega }$. For $\omega \in \mathrm{\Omega }$ and $i\in V$, if $z_{i\omega }=1$, then the sizing decision at node i could be ${\sum }_{q\in Q}{F}_{\omega }^q{\sum }_{j\in N_i^{+}}{\widehat{x}}_{ji}^q$, where ${\widehat{x}}_{ji}^q$ is the optimal value of variable $x_{ji}^q$; otherwise, the sizing decision at node i could be ${\rm Y}$. Consequently, $\widetilde{\mathbf{SRP}}$ is actually equivalent to SRP.

In the following, we have to find a valid ${\tilde{M}}_{i\omega }$ in $\widetilde{\mathbf{SRP}}$ for all $\omega \in \mathrm{\Omega }$ and simultaneously make their values as small as possible. We are able to achieve such a goal by solving the following optimization problem for a particular $\omega \in \mathrm{\Omega }$,

$$\begin{array}{cc}\hfill {\overline{M}}_{i\omega }=& max\sum _{q\in Q}{F}_{\omega }^q\sum _{j\in N_i^{+}}{x}_{ji}^q-{\rm Y},\hfill \\ \hfill & s.t.\left(1\mathrm{b}\right),\left(1\mathrm{g}\right),\hfill \\ \hfill & \sum _{q\in Q}{F}_{{\omega }^{\prime }}^q\sum _{j\in N_i^{+}}{x}_{ji}^q\le {\rm Y}+{\tilde{M}}_{i{\omega }^{\prime }}\left(1-z_{i{\omega }^{\prime }}\right),\forall i\in V,{\omega }^{\prime }\in \mathrm{\Omega },\hfill \\ \hfill & \sum _{{\omega }^{\prime }\in \mathrm{\Omega }}{z}_{i{\omega }^{\prime }}\ge ⌈\alpha \left|\mathrm{\Omega }\right|⌉,\forall i\in V,\hfill \\ \hfill & z_{i{\omega }^{\prime }}\in \{0,1\},\forall i\in V,{\omega }^{\prime }\in \mathrm{\Omega }.\hfill \end{array}$$

Then, ${\tilde{M}}_{i\omega }$ can be valued greater or equal to ${\overline{M}}_{i\omega }$ for all $\omega \in \mathrm{\Omega }$. In practice, however, it is computationally intensive to determine the exact value of ${\overline{M}}_{i\omega }$. There exist several approaches reported that are efficiently compute moderate upper bounds of ${\overline{M}}_{i\omega }$. We adapt the approach established in [40], which proposes a simple yet efficient algorithm that performs how to generate ${\tilde{M}}_{i\omega }$. This approach is a generalization of the quantile approximation approach utilized in [34,41]. We consider the following continuous knapsack problem for particular $\omega \in \mathrm{\Omega }$

$$\begin{array}{}\mathrm{(5a)}& max\sum _{q\in Q}{F}_{\omega }^q\sum _{j\in N_i^{+}}{x}_{ji}^q,\hfill \mathrm{(5b)}& s.t.\sum _{q\in Q}{F}_{{\omega }^{\prime }}^q\sum _{j\in N_i^{+}}{x}_{ji}^q\le {\rm Y},\forall {\omega }^{\prime }\in \mathrm{\Omega },\hfill \mathrm{(5c)}& x_{ji}^q\ge 0,\forall j\in N_i^{+},q\in Q.\hfill \end{array}$$

Because $x_{ji}^q$ are continuous variables, we leverage the greedy-based algorithm to obtain the optimal solution of problem (5), in which we denote $L=\lfloor (1-\alpha )\left|\mathrm{\Omega }\right|\rfloor$ as the upper bound of the total number of violated scenarios for $i\in V$, where $\lfloor ·\rfloor$ is the flooring function of real numbers. Then, we present the big-M coefficients generation algorithm as follows:

In Algorithm 1, we follow a decreasing order of $F_{\omega }^q/F_{{\omega }^{\prime }}^q$ for all $q\in Q$ (O-D pair q should traverse node i) to fill up the value of allocation variables $x_{ji}^q$ to their upper bounds 1, which correspond to Steps 3–15. Since the chance constraints can be violated in at most L scenarios, we merely require the ($L+1$)-th smallest value among all ${\tilde{m}}_{i\omega }^{{\omega }^{\prime }}$’s, rather than ${max}_{{\omega }^{\prime }\in \mathrm{\Omega }}\left\{{\tilde{m}}_{i\omega }^{{\omega }^{\prime }}\right\}$. This corresponds to Step 18 in Algorithm 1.

Algorithm 1 Big-M Coefficients Generation algorithm.

for $i\in V$do  for $\omega \in \mathrm{\Omega }$ do   for ${\omega }^{\prime }\in \mathrm{\Omega }$ do     for $q\in Q$ do      if q traverses node i then       Compute ${\zeta }_{i_q}=F_{\omega }^q/F_{{\omega }^{\prime }}^q$.      end if     end for     Sort all ${\zeta }_{i_q}$’s as a descending order, ${\zeta }_{i_{q_1}}\ge {\zeta }_{i_{q_2}}\ge \dots \ge {\zeta }_{i_{q_{|Q_i|}}}$.     $\ell \leftarrow 1$, $a\leftarrow 0$, $b\leftarrow {\rm Y}$.     while $b>0$ do      $c\leftarrow min\left\{1,b/F_{{\omega }^{\prime }}^{q_{\ell }}\right\}$.      $a\leftarrow a+c{F}_{\omega }^{q_{\ell }}$, $b\leftarrow b-c{F}_{{\omega }^{\prime }}^{q_{\ell }}$.      $\ell \leftarrow \ell +1$.     end while     ${\tilde{m}}_{i\omega }^{{\omega }^{\prime }}\leftarrow \lfloor a\rfloor$.   end for   ${\tilde{M}}_{i\omega }\leftarrow \mathrm{the}(L+1)-\mathrm{th}\mathrm{smallest}\mathrm{number}\mathrm{of}{\tilde{m}}_{i\omega }^{{\omega }^{\prime }}$ for all ${\omega }^{\prime }\in \mathrm{\Omega }$.  end for end for

It is worth mentioning that [28] considers the fast-charging station location problem using Benders decomposition. However, Benders decomposition is inapplicable for SRP, because the big-M coefficients $M_{i\omega }$ and integer variables $z_{i\omega }$ introduced in Constraints (4b) and (4c) impede obtaining Benders cuts through the duality method. To tackle this obstacle, we exploit Dantzig–Wolfe decomposition [30], which decomposes SRP into a master problem and several subproblems, and simultaneously, the column generation procedures are employed to obtain the optimal solutions.

3.2. Dantzig–Wolfe Decomposition for SRP

In this section, we apply Dantzig–Wolfe decomposition for solving SRP. Different from the existing works that employ Dantzig–Wolfe decomposition, which usually generate one master problem and several subproblems [29,30,31,32,33,34,42], we separate SRP into Q sub-components corresponding to the Q O-D pairs, then generate Q pairs of master problem and subproblem using Dantzig–Wolfe decomposition to find their optimal solutions, where the sub-components are decomposed by the available arcs between O-D pairs. Lastly, we obtain the optimal solution of SRP by synthesizing the optimal solutions of the sub-components. The sub-components of SRP are as follows

$$\begin{array}{}\mathrm{(6a)}& \left(\mathbf{SRP}\right(q\left)\right)\hfill & min\sum _{i\in V^q}\left(c_{1,i}{u}_i^q+c_{2,i}{v}_i^q\right),\hfill \mathrm{(6b)}& & s.t.\sum _{j\in N_i^{-}}{x}_{ij}^q-\sum _{j\in N_i^{+}}{x}_{ji}^q=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}i=s^q,\hfill \\ -1,\hfill & \mathrm{if}i=t^q,\hfill \\ 0,\hfill & \mathrm{if}i\ne s^q,i\ne t^q,\hfill \end{array}\right.\hfill \mathrm{(6c)}& & F_{\omega }^q\sum _{j\in N_i^{+}}{x}_{ji}^q\le v_i+M_{i\omega }\left(1-z_{i\omega }^q\right),\forall i\in V,\omega \in \mathrm{\Omega },\hfill \mathrm{(6d)}& & v_i^q\le {\rm Y}{u}_i^q,\forall i\in V^q,\hfill \mathrm{(6e)}& & \sum _{\omega \in \mathrm{\Omega }}{z}_{i\omega }^q\ge ⌈\alpha \left|\mathrm{\Omega }\right|⌉,\forall i\in V^q,\hfill \mathrm{(6f)}& & z_{i\omega }^q\in \{0,1\},\forall i\in V^q,\omega \in \mathrm{\Omega },\hfill \mathrm{(6g)}& & u_i^q\in \{0,1\},\forall i\in V^q,\hfill \mathrm{(6h)}& & v_i^q\ge 0,\forall i\in V^q.\hfill \mathrm{(6i)}& & x_{ij}^q\ge 0,\forall (i,j)\in {\overline{A}}^q.\hfill \end{array}$$

In the master problem for O-D pair q, the allocation variable $x_{ij}^q$’s are represented by the convex combination of the extreme points of the feasible region in subproblems. Let ${\mathcal{H}}^q$ be the index set of extreme points of allocation variables on available arc $(i,j)$ between O-D pair q. We then express the allocation variables in the master problem for O-D pair q as

$$\begin{array}{}\mathrm{(7a)}& \sum _{h\in {\mathcal{H}}^q}\sum _{j\in N_i^{-}}{\lambda }_{ijh}^q{\widehat{x}}_{ijh}^q=\sum _{j\in N_i^{-}}{x}_{ij}^q,\forall (i,j)\in {\overline{A}}^q,\hfill \mathrm{(7b)}& \sum _{h\in {\mathcal{H}}^q}\sum _{j\in N_i^{+}}{\lambda }_{jih}^q{\widehat{x}}_{jih}^q=\sum _{j\in N_i^{+}}{x}_{ji}^q,\forall (j,i)\in {\overline{A}}^q,\hfill \mathrm{(7c)}& \sum _{h\in {\mathcal{H}}^q}{\lambda }_{ijh}^q=1,\forall (i,j)\in {\overline{A}}^q,\hfill \mathrm{(7d)}& {\lambda }_{ijh}^q\ge 0,\forall h\in {\mathcal{H}}^q,(i,j)\in {\overline{A}}^q.\hfill \end{array}$$

where ${\widehat{x}}_{jih}^q$ represents the extreme points of allocation variables of subproblems. We then apply Dantzig–Wolfe decomposition to SRP using (6), and denote the multiscenario, column-oriented master problem as SRP-MP(q), where ${\mathcal{H}}^q$ is the index set of all corner points of O-D pair q.

$$\begin{array}{}\mathrm{(8a)}& \left(\mathbf{SRP}-\mathbf{MP}\right(q\left)\right)\hfill & min\sum _{h\in {\mathcal{H}}^q}{\widehat{\Delta }}_h^q\sum _{(i,j)\in {\overline{A}}^q}{\lambda }_{ijh}^q,\hfill \mathrm{(8b)}& & s.t.\sum _{h\in {\mathcal{H}}^q}\sum _{j\in N_i^{-}}{\lambda }_{ijh}^q{\widehat{x}}_{ijh}^q-\sum _{h\in {\mathcal{H}}^q}\sum _{j\in N_i^{+}}{\lambda }_{jih}^q{\widehat{x}}_{jih}^q=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}i=s^q,\hfill \\ -1,\hfill & \mathrm{if}i=t^q,\hfill \\ 0,\hfill & \mathrm{if}i\ne s^q,i\ne t^q,\hfill \end{array}\right.\hfill \mathrm{(8c)}& & \sum _{h\in {\mathcal{H}}^q}{\lambda }_{ijh}^q=1,\forall (i,j)\in {\overline{A}}^q,\hfill \mathrm{(8d)}& & {\lambda }_{ijh}^q\ge 0,\forall h\in {\mathcal{H}}^q,\forall (i,j)\in {\overline{A}}^q,\hfill \end{array}$$

where ${\widehat{\Delta }}_h^q={\sum }_{i\in V}{c}_{1,i}{\widehat{u}}_{ih}^q+c_{2,i}{\widehat{v}}_{ih}^q$ denotes the planning cost at node i for corner point $h\in {\mathcal{H}}^q$. Objective function (7a) minimize the combination of the planning costs of all paths. Constraints (7b) ensure that the charging demands of each path allocated at each node are balanced. Constraints (7c) and (7d) are convex combination constraints.

The huge number of extreme points of $x_{ji}^q$ result in countless variables ${\lambda }_{jih}^q$ in SRP-MP(q), which makes it difficult to explicitly investigate all of them. In addition, considering any arbitrary subset of index set ${\mathcal{H}}^q$ may cause a suboptimal solution. Thus, it is necessary to employ the column generation method to screen out the indexes that are useful to obtain the optimal solution. Accordingly, we consider a restricted master problem SRP-RMP(q) that is identical to SRP-MP(q) except the index set ${\mathcal{H}}^q$ is replaced by a subset of it. Then, we solve the SRP-RMP(q) to extract the optimal solutions and dual variables. Lastly, we generate new columns corresponding to allocation variables with negative reduced cost according to the solutions of subproblem. We generate $\left|Q\right|$ subproblems with respect to the reduced cost of ${\lambda }_{jih}^q$ for all $i\in V$ and $j\in N_i^{+}$, which are referred to as SRP-SP(q). Moreover, the independency among the subproblems for different paths make them ideal to be parallelized. For simplicity, we suppose that SRP-SP(q) is always feasible, that is, ${\mathcal{H}}^q\ne \varnothing$ and $q\in Q$.

$$\begin{array}{}\hfill & \left(\mathbf{SRP}-\mathbf{SP}\left(\mathbf{q}\right)\right)\hspace{1em}\hspace{1em}{\zeta }_q^{SRP-SP}\hfill & =min{\Delta }^q-{\widehat{\pi }}_{i_{s_q}}^q\sum _{j\in N_{i_{s_q}}^{-}}{x}_{i_{s_q}j}^q-{\widehat{\pi }}_{i_{t_q}}^q\sum _{j\in N_{i_{t_q}}^{+}}{x}_{j{i}_{t_q}}^q\hfill \mathrm{(9a)}& & -\sum _{i\in V}{\widehat{\pi }}_i^q\left(\sum _{j\in N_i^{-}}{x}_{ij}^q-\sum _{j\in N_i^{+}}{x}_{ij}^q\right)-\sum _{(i,j)\in {\overline{A}}^q}{\widehat{\mu }}_{ij}^q,\hfill \mathrm{(9b)}& & s.t.F_{\omega }^q\sum _{j\in N_i^{+}}{x}_{ji}^q\le v_i^q+M_{i\omega }\left(1-z_{i\omega }^q\right),\forall i\in V,\omega \in \mathrm{\Omega },\hfill \mathrm{(9c)}& & \sum _{\omega \in \mathrm{\Omega }}{z}_{i\omega }^q\ge ⌈\alpha \left|\mathrm{\Omega }\right|⌉,\forall i\in V,\hfill \mathrm{(9d)}& & v_i^q\le {\rm Y}{u}_i^q,\forall i\in V,\hfill \mathrm{(9e)}& & \sum _{j\in N_i^{-}}{x}_{ij}^q\le 1,\forall i\in V,\hfill \mathrm{(9f)}& & \sum _{j\in N_i^{+}}{x}_{ji}^q\le 1,\forall i\in V,\hfill \mathrm{(9g)}& & \sum _{j\in N_{i_{s_q}}^{-}}{x}_{i_{s_q}j}^q=1,\hfill \mathrm{(9h)}& & \sum _{j\in N_{i_{t_q}}^{+}}{x}_{j{i}_{t_q}}^q=1,\hfill \mathrm{(9i)}& & z_{i\omega }^q\in \{0,1\},\forall i\in V,\omega \in \mathrm{\Omega },\hfill \mathrm{(9j)}& & u_i^q\in \{0,1\},v_i^q\ge 0,\forall i\in V,\hfill \mathrm{(9k)}& & x_{ij}^q\ge 0,\forall (i,j)\in {\overline{A}}^q,\hfill \end{array}$$

where ${\Delta }^q={\sum }_{i\in V}\left(c_{1,i}{u}_i^q+c_{2,i}{v}_i^q\right)$ is defined by the aggregated planning cost of fast-charging stations on O-D pair q, and ${\widehat{\pi }}_i^q$ and ${\widehat{\mu }}_{ij}^q$ are the value of dual variables of Constraints (7b) and (7c), respectively. Objective (8a) minimizes the total planning cost minus the summation of dual prices associated with the column of O-D pair q, which is the reduced cost of variable ${\lambda }_{ijh}^q$ in SRP-MP(q). Constraints (9e)–(9h) originate from the flow conservation Constraints (1b), which ensures that the unitary allocation on the arcs be respected.

After subproblem SRP-SP(q) is solved with optimal values $u_i^{q*}$, $v_i^{q*}$, and $x_{ij}^{q*}$, and it generates an improving column $h+1$ to SRP-MP(q) for negative ${\xi }_q^{SRP-SP}$, we then solve SRP-MP(q) with new column that $u_{i,h+1}^q=u_i^{q*}$, $v_{i,h+1}^q=v_i^{q*}$, and ${\widehat{x}}_{ij,h+1}^q=x_{ij}^{q*}$. The repeated procedures between solving SRP-MP(q) and SRP-SP(q) operate until no column is favorable, that is, ${\xi }_q^{SRP-SP}$ is positive for all $q\in Q$. If none of the columns is improving toward SRP-MP(q), it indicates SRP is already solved to optimality with $u_i^{*}={max}_{q\in Q_i}\left\{u_i^{q*}\right\}$ and $v_i^{*}={\sum }_{q\in Q_i}{v}_i^{q*}$. In fact, we have multiple improving columns for SRP-MP(q), such as adding the first improving column, the most improving column, or all of these columns [43]. In practice, however, $u_i^{*}$ and $v_i^{*}$ may not satisfy Constraints (3c), because the sizing bound ${\rm Y}$ can not be separated with respect to the paths. A feasible way to overcome this obstacle is to choose a proper ${\rm Y}$, such that the fast-charging station is able to accommodate the allocated charging demands from all possible paths and simultaneously retains an acceptable solution time of SRP-SP(q).

Remark 1. 

When conducting the numerical experiment, we are aware that  SRP  is computationally demanding in a large-scale network using off-the-shelf solves. Even when we use the conventional Dantzig–Wolfe decomposition, it still consumes massive amount of CPU time to find the optimal solution of the master problem because  SRP  incorporates a multitude of decision variables ${\lambda }_{ijh}^q$ all over the expanded network. Compared with the existing works, our proposed approach enables us to improve the computational efficiency twofold: one is that  SRP  is separated into Q disjoint sub-components, which are facilitated to be solved using parallel machines; the other is that the sub-components merely incorporate the decision variables corresponding to available arcs between specific O-D pairs, which contain far fewer decision variables in the Dantzig–Wolfe master problem, and thus simplifies the computation to a extremely large extent.

4. Numerical Experiments

In this section, a network topology has been used to evaluate the performance of the proposed model and Dantzig–Wolfe decomposition method. The solving algorithms are coded in Matlab R2017b called YALMIP (version: R20210331) [44] using CPLEX 12.10 on a Windows machine with Intel(R) Core(TM) i9-11950H and 64 GB RAM. The building cost of the fast-charging stations and the installing cost of the charging piles are set to be 100 and 10, respectively.

We test the proposed model in a hypothetical 25-node network shown in Figure 5, which is originally proposed by [45] and has been extensively used as a benchmark network for the charging station location problem [26,27,36]. The circled numbers denote the labels of the nodes, the line between two nodes denotes the link that connects them, and the number in each link denotes the length of the link. This network possesses 25 nodes and 86 arcs (43 undirected links). Every node is considered as an origin, destination, or candidate site for the fast-charging stations, yielding in total 300 O-D pairs and 25 candidate sites. We suppose that the EVs start their trips with fully charged batteries, since they can be charged through slow chargers during their leisure time at home or work place [46]. For the scenario-based reformulation, we investigate 10 scenarios of charging demands between O-D pairs. In each scenario $\omega$, the charging demands between each O-D pair are equal, $F_{\omega }^q=5+w$ for all $q\in Q$.

To evaluate the performance of Dantzig–Wolfe decomposition reformulations, we generate 120 instances by considering 4 types of driving ranges, 5 levels of service, abilities and 6 values of sizing limits:

• Driving ranges: $R=8,9,10,11$.
• Service levels: $\alpha =10\%,20\%,30\%,40\%,50\%$.
• Sizing limits: ${\rm Y}=800,900,1000,1100,1200,1300$.

Table A1. Results of scenario-based reformulation in the 25-node network with $\alpha =10\%$, where superscript * indicates that optimal solutions cannot be found within 14,400 s.

$\mathit{\alpha }=10\%$		Performance of SRP(q)					Performance of Dantzig–Wolfe Decomposition
$\mathit{R}$	$\mathit{{\rm Y}}$	Obj.	Runtime (in Sec.)				Obj.	Runtime of MP (in Sec.)				Runtime of SP (in Sec.)				Iter. (in Num.)
			Max.	Med.	Ave.	Min.		Max.	Med.	Ave.	Min.	Max.	Med.	Ave.	Min.	Max.	Med.	Ave.	Min.
8	800	54,180	7364.90	14.27	100.28	0.16	38,760	0.22	0.16	0.16	0.13	0.28	0.21	0.21	0.16	7	4	4	1
	900	54,180	7383.03	22.66	97.67	0.21	38,760	0.29	0.13	0.13	0.08	0.65	0.24	0.24	0.14	9	5	6	1
	1000	54,180	7901.77	14.91	106.17	0.15	38,760	0.30	0.22	0.22	0.17	0.52	0.34	0.34	0.22	9	4	3	1
	1100	54,180	7880.42	17.23	117.96	0.14	38,760	0.32	0.16	0.19	0.13	0.45	0.22	0.25	0.16	7	3	2	1
	1200	54,180	7533.70	16.50	98.95	0.17	38,760	0.31	0.14	0.16	0.12	0.40	0.25	0.25	0.15	5	3	4	1
	1300	54,180	7750.69	23.94	123.61	0.24	38,760	0.36	0.17	0.18	0.14	0.42	0.21	0.24	0.13	8	6	6	1
9	800	46,420	10,551.68	6.72	150.30	0.14	36,920	0.32	0.19	0.20	0.16	0.39	0.25	0.25	0.20	7	3	2	1
	900	46,420	10,266.49	29.46	174.91	0.11	36,920	0.34	0.18	0.14	0.11	0.34	0.24	0.24	0.20	6	3	4	1
	1000	46,420	10,365.19	10.93	150.00	0.15	36,920	0.32	0.19	0.19	0.13	0.31	0.25	0.27	0.21	5	3	4	1
	1100	46,420	10,206.27	34.75	186.17	0.17	36,920	0.33	0.16	0.19	0.11	0.36	0.26	0.27	0.22	9	4	4	1
	1200	46,420	10,451.60	41.91	164.01	0.23	36,920	0.31	0.18	0.16	0.12	0.36	0.26	0.26	0.21	7	5	6	1
	1300	46,420	10,553.18	16.10	198.16	0.18	36,920	0.35	0.12	0.12	0.10	0.39	0.22	0.23	0.19	9	6	5	1
10	800	37,380	13,421.00	4.45	214.82	0.16	35,220	0.36	0.24	0.25	0.21	0.46	0.30	0.30	0.24	7	2	3	1
	900	37,380	13,137.68	4.54	222.84	0.06	35,220	0.35	0.26	0.23	0.20	0.42	0.31	0.32	0.23	5	2	2	1
	1000	37,380	13,108.08	13.84	273.30	0.10	35,220	0.34	0.29	0.29	0.22	0.47	0.33	0.34	0.27	9	2	3	1
	1100	37,380	13,633.3	22.03	220.92	0.14	35,220	0.37	0.25	0.25	0.21	0.44	0.34	0.36	0.26	9	3	4	1
	1200	37,380	13,863.36	25.54	275.85	0.08	35,220	0.38	0.26	0.26	0.23	0.47	0.33	0.35	0.25	8	2	3	1
	1300	37,380	13,654.73	16.97	220.27	0.17	35,220	0.32	0.28	0.25	0.20	0.46	0.32	0.34	0.27	5	3	3	1
11	800	—	14,400 *	—	—	0.11	33,540	0.46	0.28	0.29	0.26	0.49	0.34	0.34	0.28	9	2	1	1
	900	—	14,400 *	—	—	0.23	33,540	0.42	0.26	0.25	0.22	0.43	0.37	0.37	0.27	7	5	5	1
	1000	—	14,400 *	—	—	0.12	33,540	0.45	0.29	0.28	0.24	0.47	0.38	0.37	0.26	5	3	4	1
	1100	—	14,400 *	—	—	0.24	33,540	0.43	0.27	0.27	0.23	0.47	0.36	0.35	0.29	9	6	7	1
	1200	—	14,400 *	—	—	0.20	33,540	0.42	0.26	0.26	0.22	0.49	0.34	0.34	0.26	7	5	4	1
	1300	—	14,400 *	—	—	0.13	33,540	0.47	0.28	0.26	0.21	0.48	0.33	0.32	0.24	5	2	1	1

,

Table A2. Results of scenario-based reformulation in the 25-node network with $\alpha =20\%$.

$\mathit{\alpha }=20\%$		Performance of SRP(q)					Performance of Dantzig–Wolfe Decomposition
$\mathit{R}$	$\mathit{{\rm Y}}$	Obj.	Runtime (in Sec.)				Obj.	Runtime of MP (in Sec.)				Runtime of SP (in Sec.)				Iter. (in Num.)
			Max.	Med.	Ave.	Min.		Max.	Med.	Ave.	Min.	Max.	Med.	Ave.	Min.	Max.	Med.	Ave.	Min.
8	800	62,860	5308.90	10.34	32.17	0.15	44,720	0.47	0.35	0.35	0.30	0.74	0.41	0.41	0.34	5	2	2	1
	900	62,160	5283.03	15.60	36.77	0.12	44,720	0.44	0.37	0.37	0.33	0.71	0.42	0.42	0.39	6	3	4	1
	1000	62,160	5101.77	10.62	34.79	0.17	44,720	0.41	0.35	0.34	0.30	0.75	0.43	0.43	0.34	8	2	1	1
	1100	62,160	5280.42	10.59	32.39	0.12	44,720	0.42	0.38	0.37	0.32	0.76	0.48	0.46	0.35	8	4	5	1
	1200	62,160	5233.70	11.59	32.39	0.12	44,720	0.49	0.32	0.34	0.31	0.78	0.45	0.45	0.32	5	2	3	1
	1300	62,160	5250.69	9.79	36.09	0.16	44,720	0.47	0.32	0.32	0.28	0.70	0.47	0.47	0.39	9	7	8	1
9	800	52,950	5520.18	9.79	41.94	0.13	42,690	0.54	0.39	0.39	0.35	0.79	0.45	0.45	0.37	5	3	3	1
	900	52,950	5533.75	13.38	42.05	0.15	42,690	0.57	0.33	0.33	0.30	0.74	0.44	0.44	0.37	9	6	6	1
	1000	52,950	5512.58	7.82	44.12	0.14	42,690	0.54	0.36	0.36	0.32	0.78	0.41	0.41	0.36	5	2	3	1
	1100	52,950	5490.68	12.29	38.28	0.17	42,690	0.52	0.39	0.38	0.31	0.72	0.40	0.40	0.38	6	3	3	1
	1200	52,950	5512.49	9.85	44.09	0.09	42,690	0.51	0.38	0.36	0.33	0.71	0.44	0.43	0.39	9	7	8	1
	1300	52,950	5510.95	8.77	46.17	0.09	42,690	0.56	0.35	0.35	0.30	0.70	0.45	0.44	0.34	6	2	3	1
10	800	41,860	5639.59	9.74	56.85	0.10	40,690	0.60	0.42	0.42	0.38	0.74	0.47	0.48	0.42	6	2	3	1
	900	41,860	5519.59	10.26	54.11	0.14	40,690	0.66	0.40	0.40	0.34	0.79	0.48	0.49	0.42	5	3	2	1
	1000	41,860	5561.04	7.44	57.17	0.15	40,690	0.68	0.42	0.42	0.32	0.75	0.46	0.47	0.42	7	3	4	1
	1100	41,860	5656.06	8.59	57.12	0.13	40,690	0.63	0.48	0.48	0.33	0.70	0.47	0.47	0.41	6	4	5	1
	1200	41,860	5513.64	10.41	53.25	0.11	40,690	0.64	0.47	0.47	0.32	0.72	0.41	0.41	0.36	8	2	3	1
	1300	41,860	5659.92	10.09	52.61	0.10	40,690	0.67	0.41	0.40	0.36	0.71	0.43	0.44	0.40	7	3	4	1
11	800	36,680	5759.50	10.38	66.79	0.14	38,730	0.68	0.47	0.48	0.43	0.81	0.52	0.53	0.46	7	3	3	1
	900	36,680	5644.64	6.46	65.52	0.12	38,730	0.69	0.48	0.49	0.43	0.89	0.57	0.57	0.44	7	2	3	1
	1000	36,680	5618.28	10.64	59.03	0.09	38,730	0.63	0.47	0.47	0.43	0.84	0.54	0.54	0.46	5	2	1	1
	1100	36,680	5638.48	16.09	66.79	0.09	38,730	0.66	0.49	0.49	0.42	0.88	0.50	0.51	0.44	5	3	3	1
	1200	36,680	5500.02	12.58	61.66	0.15	38,730	0.67	0.48	0.48	0.42	0.84	0.57	0.56	0.42	7	5	4	1
	1300	36,680	5632.61	13.33	67.85	0.11	38,730	0.65	0.48	0.48	0.43	0.85	0.56	0.56	0.45	8	5	6	1

,

Table A3. Results of scenario-based reformulation in the 25-node network with $\alpha =30\%$.

$\mathit{\alpha }=30\%$		Performance of SRP(q)					Performance of Dantzig–Wolfe Decomposition
$\mathit{R}$	$\mathit{{\rm Y}}$	Obj.	Runtime (in Sec.)				Obj.	Runtime of MP (in Sec.)				Runtime of SP (in Sec.)				Iter. (in Num.)
			Max.	Med.	Ave.	Min.		Max.	Med.	Ave.	Min.	Max.	Med.	Ave.	Min.	Max.	Med.	Ave.	Min.
8	800	71,640	137.77	8.85	15.12	0.08	50,880	0.76	0.51	0.52	0.46	0.75	0.56	0.57	0.50	9	3	4	1
	900	71,640	129.82	7.15	13.84	0.08	50,880	0.70	0.59	0.59	0.49	0.78	0.52	0.52	0.48	5	2	3	1
	1000	71,640	139.59	8.93	12.73	0.13	50,880	0.71	0.51	0.58	0.45	0.77	0.54	0.54	0.51	9	7	6	1
	1100	71,640	103.66	5.22	14.23	0.12	50,880	0.79	0.52	0.50	0.42	0.79	0.57	0.57	0.53	8	3	2	1
	1200	71,640	119.82	6.53	15.35	0.13	50,880	0.75	0.58	0.57	0.41	0.70	0.52	0.52	0.51	8	2	3	1
	1300	71,640	127.83	5.90	12.55	0.09	50,880	0.77	0.50	0.50	0.44	0.75	0.54	0.54	0.50	7	4	4	1
9	800	60,660	193.99	10.19	19.38	0.12	48,560	1.34	0.57	0.57	0.50	0.89	0.63	0.63	0.55	9	5	4	1
	900	60,660	195.61	13.55	16.26	0.09	48,560	1.38	0.57	0.57	0.51	0.88	0.65	0.65	0.56	5	2	3	1
	1000	60,660	200.70	12.29	17.89	0.13	48,560	1.36	0.58	0.58	0.52	0.87	0.64	0.64	0.54	9	5	4	1
	1100	60,660	196.69	14.88	16.27	0.17	48,560	1.30	0.54	0.54	0.50	0.88	0.62	0.62	0.56	8	5	4	1
	1200	60,660	203.79	12.21	18.31	0.16	48,560	1.31	0.55	0.55	0.48	0.82	0.64	0.65	0.59	6	2	1	1
	1300	60,660	190.07	10.25	16.69	0.12	48,560	1.31	0.57	0.57	0.52	0.89	0.65	0.65	0.52	7	2	3	1
10	800	48,950	279.19	14.98	20.68	0.13	46,160	0.55	0.06	0.06	0.04	0.44	0.12	0.13	0.07	5	2	2	1
	900	48,950	276.79	14.18	22.73	0.13	46,160	0.59	0.11	0.12	0.07	0.46	0.14	0.14	0.10	8	4	4	1
	1000	48,950	268.56	13.86	21.07	0.13	46,160	0.52	0.09	0.09	0.06	0.40	0.18	0.17	0.12	8	3	3	1
	1100	48,950	263.13	13.22	21.56	0.11	46,160	0.50	0.12	0.11	0.05	0.46	0.12	0.12	0.07	6	2	1	1
	1200	48,950	262.90	19.42	22.64	0.14	46,160	0.51	0.08	0.07	0.03	0.47	0.13	0.14	0.06	7	2	2	1
	1300	48,950	267.42	14.83	21.02	0.10	46,160	0.58	0.07	0.07	0.04	0.42	0.17	0.16	0.09	6	3	2	1
11	800	41,620	336.85	19.67	25.84	0.16	43,920	0.16	0.10	0.10	0.08	0.30	0.16	0.17	0.10	6	2	3	1
	900	41,620	323.50	20.07	26.28	0.10	43,920	0.17	0.12	0.12	0.06	0.31	0.15	0.14	0.05	6	4	3	1
	1000	41,620	313.97	24.15	25.11	0.10	43,920	0.19	0.13	0.12	0.05	0.30	0.16	0.16	0.09	9	5	5	1
	1100	41,620	314.81	18.31	26.15	0.10	43,920	0.23	0.10	0.12	0.04	0.32	0.17	0.17	0.07	8	4	4	1
	1200	41,620	331.38	18.63	26.55	0.13	43,920	0.19	0.11	0.12	0.04	0.39	0.13	0.14	0.04	7	2	1	1
	1300	41,620	324.50	24.69	24.18	0.13	43,920	0.17	0.10	0.12	0.08	0.35	0.12	0.11	0.07	8	2	1	1

,

Table A4. Results of scenario-based reformulation in the 25-node network with $\alpha =40\%$.

$\mathit{\alpha }=40\%$		Performance of SRP(q)					Performance of Dantzig–Wolfe Decomposition
$\mathit{R}$	$\mathit{{\rm Y}}$	Obj.	Runtime (in Sec.)				Obj.	Runtime of MP (in Sec.)				Runtime of SP (in Sec.)				Iter. (in Num.)
			Max.	Med.	Ave.	Min.		Max.	Med.	Ave.	Min.	Max.	Med.	Ave.	Min.	Max.	Med.	Ave.	Min.
8	800	80,320	42.14	12.90	15.40	0.09	56,940	0.21	0.14	0.15	0.12	0.32	0.20	0.21	0.14	6	3	4	1
	900	79,520	42.41	12.83	17.82	0.13	56,940	0.25	0.11	0.11	0.09	0.36	0.22	0.22	0.10	5	2	1	1
	1000	79,520	39.14	10.74	13.51	0.15	56,940	0.23	0.14	0.14	0.11	0.31	0.24	0.24	0.11	7	5	4	1
	1100	79,520	41.00	9.54	13.09	0.13	56,940	0.28	0.15	0.16	0.12	0.39	0.21	0.21	0.18	6	2	3	1
	1200	79,520	40.69	9.43	16.07	0.16	56,940	0.22	0.10	0.10	0.07	0.38	0.28	0.27	0.17	5	3	2	1
	1300	79,520	43.40	11.42	14.33	0.17	56,940	0.21	0.12	0.12	0.08	0.37	0.25	0.25	0.18	8	6	5	1
9	800	67,080	48.81	10.72	14.36	0.11	54,230	0.29	0.19	0.19	0.16	0.33	0.25	0.25	0.18	7	2	2	1
	900	66,980	47.68	10.74	17.49	0.18	54,230	0.24	0.14	0.14	0.11	0.39	0.25	0.24	0.17	6	4	3	1
	1000	66,980	48.39	12.38	13.19	0.09	54,230	0.27	0.18	0.18	0.14	0.30	0.22	0.21	0.13	6	2	3	1
	1100	66,980	49.41	9.88	14.66	0.15	54,230	0.28	0.17	0.17	0.12	0.35	0.21	0.22	0.14	8	6	5	1
	1200	66,980	47.39	9.57	17.78	0.13	54,230	0.23	0.19	0.18	0.13	0.43	0.26	0.25	0.15	8	3	3	1
	1300	66,980	48.06	9.69	17.96	0.15	54,230	0.28	0.15	0.15	0.10	0.31	0.25	0.25	0.11	6	4	5	1
10	800	53,720	52.84	10.45	13.77	0.10	51,630	0.73	0.23	0.23	0.20	0.62	0.29	0.29	0.22	6	2	2	1
	900	53,720	52.90	11.92	13.48	0.11	51,630	0.74	0.26	0.26	0.21	0.62	0.26	0.26	0.20	6	3	4	1
	1000	53,620	53.39	12.41	15.47	0.13	51,630	0.70	0.24	0.24	0.20	0.61	0.28	0.28	0.22	6	4	5	1
	1100	53,620	54.21	9.63	16.78	0.16	51,630	0.73	0.27	0.26	0.22	0.62	0.25	0.25	0.20	6	3	3	1
	1200	53,620	55.87	10.30	17.66	0.11	51,630	0.75	0.26	0.26	0.22	0.61	0.25	0.25	0.21	9	3	4	1
	1300	53,620	55.26	10.65	17.84	0.16	51,630	0.70	0.22	0.22	0.17	0.69	0.24	0.24	0.21	8	3	4	1
11	800	46,760	57.13	9.07	10.92	0.14	49,110	0.37	0.27	0.27	0.24	0.47	0.33	0.34	0.27	6	2	3	1
	900	46,760	56.04	10.48	9.44	0.09	49,110	0.34	0.24	0.24	0.20	0.40	0.30	0.31	0.22	8	3	2	1
	1000	46,760	58.50	10.24	15.01	0.08	49,110	0.35	0.21	0.21	0.17	0.45	0.38	0.38	0.31	9	7	7	1
	1100	46,760	55.52	10.69	10.10	0.16	49,110	0.35	0.22	0.22	0.16	0.42	0.32	0.32	0.25	9	6	6	1
	1200	46,760	58.39	10.13	17.79	0.15	49,110	0.39	0.23	0.23	0.14	0.48	0.35	0.35	0.27	9	3	2	1
	1300	46,760	59.84	9.72	13.27	0.10	49,110	0.32	0.22	0.22	0.18	0.41	0.32	0.32	0.21	5	2	1	1

and

Table A5. Results of scenario-based reformulation in the 25-node network with $\alpha =50\%$.

$\mathit{\alpha }=50\%$		Performance of SRP(q)					Performance of Dantzig–Wolfe Decomposition
$\mathit{R}$	$\mathit{{\rm Y}}$	Obj.	Runtime (in Sec.)				Obj.	Runtime of MP (in Sec.)				Runtime of SP (in Sec.)				Iter. (in Num.)
			Max.	Med.	Ave.	Min.		Max.	Med.	Ave.	Min.	Max.	Med.	Ave.	Min.	Max.	Med.	Ave.	Min.
8	800	89,000	34.48	12.13	14.28	0.15	62,800	0.68	0.47	0.47	0.42	0.76	0.52	0.51	0.44	7	4	4	1
	900	88,600	33.63	12.63	14.75	0.17	62,800	0.63	0.49	0.48	0.46	0.72	0.56	0.56	0.46	5	3	3	1
	1000	88,200	32.77	9.93	14.98	0.13	62,800	0.60	0.46	0.46	0.41	0.73	0.54	0.54	0.42	7	2	1	1
	1100	88,200	32.89	12.22	13.13	0.13	62,800	0.67	0.45	0.45	0.40	0.78	0.59	0.59	0.43	9	7	7	1
	1200	88,200	32.93	10.91	13.73	0.12	62,800	0.63	0.49	0.49	0.45	0.74	0.50	0.50	0.45	6	3	3	1
	1300	88,200	32.09	13.14	12.85	0.08	62,800	0.66	0.41	0.42	0.38	0.73	0.56	0.56	0.44	6	2	3	1
9	800	75,430	35.65	13.38	15.16	0.11	60,000	0.51	0.07	0.07	0.04	0.59	0.13	0.13	0.07	5	2	1	1
	900	75,430	34.83	12.05	14.40	0.1	60,000	0.55	0.08	0.08	0.04	0.59	0.14	0.14	0.07	5	2	2	1
	1000	75,430	32.98	11.76	13.46	0.12	60,000	0.53	0.09	0.09	0.03	0.51	0.10	0.11	0.05	8	4	4	1
	1100	75,430	32.32	10.49	9.00	0.17	60,000	0.55	0.08	0.08	0.05	0.55	0.15	0.14	0.07	5	3	3	1
	1200	75,430	32.20	7.52	11.27	0.16	60,000	0.53	0.07	0.07	0.03	0.52	0.13	0.13	0.05	5	2	1	1
	1300	75,430	32.99	7.73	12.17	0.17	60,000	0.50	0.09	0.09	0.06	0.50	0.19	0.19	0.08	6	2	1	1
10	800	60,900	33.77	12.00	14.24	0.15	57,100	0.59	0.11	0.11	0.08	0.88	0.17	0.17	0.11	7	3	4	1
	900	60,900	32.83	10.67	14.48	0.11	57,100	0.58	0.14	0.14	0.08	0.81	0.18	0.17	0.14	7	3	4	1
	1000	60,900	32.51	10.82	12.81	0.18	57,100	0.56	0.12	0.12	0.04	0.86	0.15	0.15	0.07	7	3	3	1
	1100	60,900	32.48	8.14	13.32	0.10	57,100	0.59	0.12	0.12	0.07	0.81	0.24	0.24	0.13	8	5	6	1
	1200	60,900	33.69	10.93	15.37	0.17	57,100	0.56	0.10	0.10	0.06	0.86	0.22	0.22	0.16	5	2	2	1
	1300	60,900	33.76	7.41	14.15	0.17	57,100	0.53	0.11	0.11	0.09	0.88	0.19	0.19	0.13	6	2	1	1
11	800	51,500	37.42	7.36	7.79	0.09	54,300	0.53	0.06	0.07	0.04	5.26	0.14	0.17	0.09	6	3	4	1
	900	51,500	37.87	7.19	13.46	0.14	54,300	0.53	0.15	0.15	0.11	4.39	0.24	0.24	0.14	5	3	4	1
	1000	51,500	36.11	9.27	14.33	0.10	54,300	0.59	0.19	0.19	0.15	4.59	0.21	0.21	0.14	6	2	3	1
	1100	51,500	38.28	11.28	14.57	0.08	54,300	0.54	0.13	0.13	0.10	4.45	0.24	0.24	0.11	7	5	5	1
	1200	51,500	35.54	13.16	15.16	0.12	54,300	0.53	0.12	0.13	0.09	4.37	0.23	0.24	0.19	6	3	2	1
	1300	51,500	38.68	11.91	12.36	0.18	54,300	0.26	0.18	0.18	0.15	4.41	0.25	0.24	0.19	5	2	2	1

in Appendix A illustrate the results of numerical experiments in the 25-node network for the five different service levels $\alpha =10\%$, $20\%$, $30\%$, $40\%$, $50\%$, respectively. In each table, the rows correspond to the results of a particular instance indicated by the service level ($\alpha$), the driving ranges (R), and the sizing limit (${\rm Y}$). These tables compare the numerical performances between the original model SRP(q) and Dantzig–Wolfe decomposition SRP-MP(q) and SRP-SP(q), in which the total planning costs of fast-charging stations (Obj.) and the runtime (in seconds) of such two models as well as the iteration rounds (in Num.) of Dantzig–Wolfe decomposition are provided. Especially, the maximal (Max.), median (Med.), average (Ave.), and minimal (Min.) runtime and iteration rounds are reported in detail. In general, the CPLEX solver terminates with a relatively optimal tolerance in ${10}^{-4}$. The upper bound of runtime is set to be 14,400 s (4 h); the results marked by superscript ${14,400}^{ *}$ indicate that optimal solutions cannot be found within 14,400 s, while the results marked by—means that we cannot obtain the exact output within 14,400 s. Finally, the initial values of the allocation variable are selected randomly with respect to Constraints (1b) and (1g).

According to the results in Table A1, Table A2, Table A3, Table A4 and Table A5 in Appendix A, we observe that Dantzig–Wolfe decomposition SRP-MP(q) and SRP-SP(q) can significantly reduce the runtime of solving the original model SRP(q) directly. For instance, in Table A3, the row under service level $\alpha =10\%$, driving range $R=8$, and sizing bound ${\rm Y}=800$ manifests that, after leveraging Dantzig–Wolfe decomposition, the maximal runtime is reduced from $7364.90$ s to $0.22$ s, the median runtime is reduced from $14.27$ s to $0.16$ s, and the average runtime is reduced from $100.28$ s to $0.16$ s. The similar runtime reduction effect can also be observed through the numerical results shown in other tables, which demonstrates that Dantzig–Wolfe decomposition gives rise to higher numerical efficiency than solving the original model directly. In addition, for some instances, such as service level $\alpha =10\%$ and driving range $R=11$, we cannot obtain the optimal planning strategies directly, whereas Dantzig–Wolfe decomposition can still effectively obtain the optimal planning results. Table A1, Table A2, Table A3, Table A4 and Table A5 also show the iteration rounds of SRP-MP(q) and SRP-SP(q), which depend on the selected initial values of the allocation variables. We observe that Dantzig–Wolfe decomposition can find the optimal planning strategy within 10 rounds of iteration and thus is able to guarantee its numerical efficiency.

In addition to the computational efficiency of Dantzig–Wolfe decomposition, Table A1, Table A2, Table A3, Table A4 and Table A5 also report the effects of service level $\alpha$ and the driving range R on the optimal planning costs. Although the optimal planning costs of the original model and Dantzig–Wolfe decomposition are nonidentical, they share a common changing tendency with respect to the change of $\alpha$ and R. On the whole, the optimal planning costs grow with the increase of service level $\alpha$. For instance, under driving range $R=8$, the optimal planning costs grow with respect to the increasing $\alpha$. The reason is that larger $\alpha$ usually implies more charging piles should be provided to the EVs, which stimulates the increase of planning costs. Moreover, the optimal planning costs decline with the increase of driving range R. For instance, under driving range $\alpha =20\%$, the optimal planning costs decline with respect to the increasing R. The reason is that larger R means EVs are able to drive a longer distance without refueling, which is conducive to reducing the charging demands and thus decreasing the planning costs. On the other hand, the sizing bound ${\rm Y}$ seems to not affect the planning costs of fast-charging stations according to Table A1, Table A2, Table A3, Table A4 and Table A5. In practice, however, ${\rm Y}$ determines the feasibility of the planning models, that is, some fast-charging stations may be unable to accommodate the allocated charging demands due to the limitation of installing charging piles. Thus, it is important to set a proper ${\rm Y}$ before establishing the planning model. Finally, the number of scenarios $\left|\mathrm{\Omega }\right|$ profoundly affects the runtime of our models. It is obvious that, if $\mathrm{\Omega }$ contains more scenarios, the models will encounter more constraints, and solving the model thus requires more runtime. Nevertheless, the models with more scenarios can reflect the real world more closely. Hence, a moderate size of scenarios that makes a good trade-off between the computational efficiency and the practical significance plays an indispensable role in establishing the planning models.

5. Conclusions

We addressed the modeling and solution approaches of the fast-charging station planning problem. We formulated a CCP model that exploits the multicommodity flow model to handle the uncertain charging demands with unknown probability distributions. We first applied a scenario-based reformulation approach that reformulates the CCP model into a mixed-integer linear program, and further enhanced the performance of the scenario-based formulation using a coefficients-strengthening algorithm, which aimed to find the appropriate big-M coefficients. We then proposed Dantzig–Wolfe decomposition to find the optimal solutions of the planning model. Finally, we showed that the proposed Dantzig–Wolfe decomposition method considerably improved the computational efficiency for solving a large-scale problem instance in reality. It should be mentioned that Dantzig–Wolfe decomposition can also be applied to the real-world transportation network, such as the Texas highway network and the California Road network. Such applications will be considered as a future work.