Bi-Level Collaborative Optimization of Dynamic Wireless Charging Systems Considering Traffic Flow Distribution

Abstract

To address the challenges of facility–demand mismatch, aggravated congestion, and imbalanced benefit distribution caused by the interdependence between dynamic wireless charging systems (DWCS) and transportation networks, this study proposes an optimization scheme that coordinates DWCS planning, travel flow guidance for electric vehicle (EV) owners, and transportation network operations. We develop a bi-level dynamic collaborative optimization model. The upper-level model aims to maximize the annual net profit of DWCS operators and determines DWCS planning by optimizing the traffic flow distribution. The lower-level model, based on the user equilibrium principle, guides EV route choices via a traffic flow guidance mechanism to mitigate peak-hour congestion and minimize vehicle owners’ travel costs. We validate the model using a test network comprising 9 nodes and 13 links. Results indicate that, compared with a full-coverage planning scenario, the proposed bi-level optimization scheme significantly reduces operational losses by accounting for owners’ optimal travel flow distribution. Introducing a traffic flow guidance mechanism further improves traffic flow distribution, enhances operator revenue, and effectively reduces owners’ travel time costs. Sensitivity analysis reveals that increased battery capacity decreases construction and maintenance costs, thereby improving annual net profit, while lower energy consumption reduces charging demand and weakens dependence on charging infrastructure. These factors are interrelated; specifically, lower energy consumption implies reduced battery capacity requirements for the same driving range. Additionally, the effectiveness of the traffic flow guidance mechanism becomes more pronounced as traffic flow increases. Overall, the proposed framework integrates DWCS planning and traffic flow guidance to achieve a win–win outcome for both operators and owners. These findings demonstrate the practicality and economic feasibility of interactive optimization between DWCS and transportation networks.

1. Introduction

With the global popularization of electric vehicles (EVs), excessively long charging wait times at traditional stations and insufficient driving range have become common concerns for owners. These problems not only restrict the daily use of EVs but also trigger “range anxiety” [1]. To alleviate this concern, dynamic wireless charging systems (DWCS) have emerged as a promising solution with broad application prospects [2]. This technology embeds inductive coils beneath the road surface, utilizing electromagnetic induction to transfer energy to a receiving device installed on the vehicle’s underside. Consequently, vehicles can charge in real-time while in motion. This approach significantly reduces charging wait times, enhances driving range, effectively mitigates the limitations of traditional charging methods, and alleviates owner anxiety [3]. However, despite its considerable potential, practical application still faces challenges in optimizing charging infrastructure planning and resource allocation. Given the road-based nature of DWCS, planning must be grounded in charging demand. Yet owners’ travel route choices and traffic flow distribution dynamically influence the spatiotemporal characteristics of charging demand. Therefore, a critical problem requiring urgent attention is how to establish a dynamic synergistic relationship among optimal DWCS planning, owner travel choices, and the transportation network. Such a framework would balance operator planning profits against owner travel costs.

In recent years, the new charging mode enabled by wireless charging has prompted researchers to focus on the DWCS planning and the EVs charging scheduling. Currently, the research related to DWCS planning is mainly divided into single-level and bi-level planning. Single-level planning primarily addresses comprehensive cost control, benefit assessment, and system reliability. For instance, reference [4] reduces comprehensive costs for DWCS operators by minimizing the costs of dynamic wireless power transfer (DWPT) pads and inverters within the DWCS. Reference [5] proposes a distributed planning method that optimizes DWCS planning and scale through a mixed integer linear programming model to maximize economic benefits. Reference [6] conducts joint planning for DWCS of transport trucks and related power infrastructure, aiming to optimize DWCS planning, power supply strategies, and system upgrades. DWCS bi-level planning focuses on balancing the interests of multiple subjects and system collaborative optimization. This approach emphasizes coordinating demands from different parties, such as operators, users, and power grids. For instance, reference [7] explicitly trades off the relationship between the DWCS construction and EV battery manufacturing costs. The author proposes a bi-level programming model where the upper level aims to minimize the total social cost and the lower-level captures owners’ battery size selection behavior. The upper-level model of reference [8] aims to minimize the DWCS installation cost for operators, while the lower-level model focuses on energy conservation for EVs before they reach their destinations. Single-level planning typically targets a single core objective. Bi-level planning, however, places greater emphasis on balancing multiple stakeholder interests, making it more suitable for complex DWCS road planning scenarios. When coordinating these diverse interests, bi-level planning can effectively integrate vehicle operational characteristics, including key factors such as traffic flow distribution and charging demand. These characteristics directly affect the charging DWCS efficiency and planning rationality, thereby influencing the practical effectiveness of planning schemes. For complex DWCS road planning scenarios, therefore, adopting bi-level planning ensures both scientific rationality and practical adaptability of the scheme.

Indeed, relevant studies on charging stations have already explored the interactive mechanism between traffic flow distribution and charging facility planning. For instance, reference [9] proposes a multi-objective optimization framework for charging stations equipped with shunt capacitors. This framework captures traffic flow distribution based on EV driving range to guide station planning, while simultaneously stabilizing voltage and enhancing charging accessibility. Reference [10], on the other hand, developed a two-layer location planning method for fast-charging stations. By integrating dynamic real-time vehicle-traffic-power grid data and incorporating traffic flow constraints, this method achieves multi-objective integrated optimization. These studies emphasize the interaction mechanism for a critical reason: in the context of charging stations, traffic flow distribution directly affects facility utilization rates, while facility planning conversely guides traffic flow patterns. Exploring this bidirectional interaction is therefore essential for optimizing planning outcomes.

A fundamental distinction exists between DWCS and traditional charging stations. Unlike fixed stations where vehicles must stop, DWCS enables vehicles to charge in real-time while driving on equipped roads. This dynamic charging characteristic implies that vehicle owners’ route choices directly determine their charging duration, charging location, and charging power. Consequently, these choices profoundly affect charging demand distribution and DWCS operational efficiency. However, existing DWCS planning studies often fail to accurately capture the dynamic coupling among DWCS planning, owners’ travel choices, and traffic flow distribution. This coupling manifests as a triple interaction. First, under the constraint that vehicles meet the electricity threshold throughout their journey, owners’ route choices will dynamically restructure the traffic flow distribution in the road network. Second, such traffic flow distribution directly determines the spatial heterogeneity of charging demands. Third, the spatial characteristics of charging demands, in turn, affect the DWCS planning efficiency. Merely ensuring basic travel needs is insufficient; without proper coordination, operators face a mismatch between facility planning and dynamic charging demands. Such mismatches reduce operator benefits, while owners may incur additional travel costs due to limited reasonable route options. Thus, establishing a dynamic collaborative coupling mechanism is crucial. This approach achieves equilibrium between charging demand and facility planning, effectively balancing operator benefits with owner costs.

Furthermore, large-scale DWCS planning may exacerbate traffic congestion due to drivers’ route choices for charging. This affects traffic flow distribution and charging demand, ultimately reducing system operational efficiency. Tolling—long recognized as an effective economic instrument for congestion mitigation—has remained a central research focus in transportation studies. For instance, reference [11] investigates distance- and time-based tolling schemes in large-scale dynamic transportation networks. Reference [12] examines road pricing design for autonomous vehicles by constructing a bi-level framework: the upper level determines the tolling strategy, which then influences owners’ equilibrium route choices in the lower level. However, existing tolling research has predominantly focused on conventional internal combustion engine vehicles. The core logic of using price signals to actively manage traffic flow has not yet been systematically integrated into optimization frameworks for transportation networks equipped with DWCS. This gap leaves critical challenges unaddressed, including inefficient infrastructure planning and insufficient system-level coordination potentially induced by DWCS deployment.

This gap in active regulatory mechanisms renders current studies ill-equipped to handle the dynamic coupling among “DWCS planning, driver route choice, and traffic flow distribution” in DWCS-enabled networks. The absence of active guidance mechanisms limits the ability of current studies to handle the dynamic coupling among DWCS planning, EV owners’ route choice, and traffic flow distribution in DWCS-enabled transportation networks. Even the few bi-level DWCS planning studies attempting to address this coupling suffer from significant modeling limitations. For example, reference [8] treats traffic flow distribution as a static outcome determined exogenously by the classical User Equilibrium (UE) model, and facility planning is then based on this static outcome. Such approaches merely adapt passively to pre-existing traffic conditions. They cannot proactively regulate traffic flow redistribution caused by the spatial clustering of charging demand. Consequently, these methods fail to break potential negative feedback loops between infrastructure planning and congestion, hindering system-wide coordinated optimization.

To address the shortcomings of existing DWCS bi-level optimization research, which overlooks EV owners’ travel choices and lacks the ability to actively regulate traffic congestion, this paper proposes a bi-level optimization model incorporating a traffic flow guidance mechanism. This model characterizes users’ travel choice behavior and reveals the dynamic interaction mechanism among DWCS planning, owners’ travel choices, and traffic flow distribution, thereby generating a more coordinated planning scheme. Within this framework, the upper-level operator jointly optimizes DWCS planning and pricing strategies. This allows for active steering and reshaping of traffic flow in the lower level. Meanwhile, EVs in the lower level make route choices by considering travel time costs, charging costs, and traffic flow guidance fees. This design marks a significant shift in system optimization. It moves from the traditional passive approach of “planning infrastructure based on fixed traffic flows” to an active co-optimization mode.

Therefore, to address these limitations in dynamic collaboration mechanisms and active guidance strategies, this study proposes a bi-level optimization model for transportation networks integrated with DWCS. The key contributions are outlined as follows:

First, this study proposes a dynamic collaborative coupling mechanism. This approach overcomes the limitation of treating traffic flow as a static input. It clarifies the triple interactive relationship among DWCS optimal planning, owners’ travel choices, and traffic flow distribution. Consequently, it provides a new logical framework for balancing operators’ revenue and owners’ travel costs.

Second, we develop a bi-level optimization model embedded with a traffic flow guidance mechanism. The upper-level maximizes operators’ annual net profit by optimizing DWCS planning and guidance fee parameters. Meanwhile, the lower-level minimizes owners’ comprehensive travel time costs based on the user equilibrium. This structure effectively alleviates peak-hour congestion and facility-demand mismatch.

Third, a quantitative analysis framework is established to evaluate the synergistic effects of key technical parameters. It reveals the interactive impact of EV battery capacity and unit energy consumption on DWCS planning costs and changes in traffic flow. These insights provide decision support for operators to dynamically adjust planning strategies according to technological evolution.

The remainder of the paper is organized as follows: Section 2 elaborates on the bi-level optimization framework for the DWCS. Section 3 presents the network modeling that incorporates the DWCS. Section 4 details the lower-level travel model based on the user equilibrium principle. Section 5 formulates the upper-level planning model, which aims to maximize the operator’s annual net profit. Section 6 introduces the solution methodology for the bi-level model. Section 7 validates the proposed model through case studies and conducts a sensitivity analysis of key technical parameters. Section 8 summarizes the research conclusions and outlines directions for future work.

2. Two-Level Optimization Framework for DWCS

The bi-level optimization framework constructed in this study is grounded in a logic of dynamic synergy. The upper level focuses on long-term planning. Here, operators’ costs include the construction, maintenance, and electricity purchase expenses of inverters and DWPT in the DWCS. Revenue, conversely, derives from electricity sales and traffic flow guidance fees [13]; the lower level focuses on short-term travel, where the owners’ generalized travel cost consists of three parts: travel time cost calculated based on the monetary value of time, charging fees, and traffic flow guidance fees. The upper level transmits planning parameters and traffic flow guidance fees to the lower level, which in turn feeds back the actual traffic flow distribution to enable dynamic adaptation. Figure 1 illustrates the overall bi-level optimization framework.

3. Network Modeling Incorporating DWCS

3.1. Transportation Network Topology

Figure 2 shows a transportation network composed of ordinary links and DWCS links, where nodes 1–4 represent the network nodes, while O and D denote the origin and destination of all paths, respectively. The ordinary links and DWCS links in the upper part constitute the path set from O to D in the lower part, encompassing both non-wireless-charging ordinary paths and wireless-based wireless charging paths. The network is modeled as a connected graph $G_N = [T_N,L_N]$, where $T_N$ represents the set of nodes in the transportation network—including origin nodes, destination nodes, and intersection nodes—and $L_N$ represents the set of links connecting these nodes.

Each origin–destination (O-D) pair in the network is connected by a set of paths composed of links. For any given O-D pair, the travel demand $Q_S$ is known, and each O-D pair may correspond to one or multiple paths.

In a transportation network consisting of $T_N$ nodes, when performing quantitative assignment, the length of link $L_N$ in the road matrix $D$ is defined as follows:

$$D = \left\{\begin{array}{l}{k}_{ij}\hspace{1em}ij\in L_N\\ 0\hspace{1em} i=j\\ \infty \hspace{1em}ij\notin L_N\end{array}\right.$$

(1)

where $k_{ij}$ denotes the link length from node $i$ to node $j$; when $i=j$ the link length is 0; $\infty$ indicates that there is no direct connection between the two nodes.

3.2. EV Traffic Link Travel Time

The travel time of EVs is usually modeled using the statistical function of the U.S. Bureau of Public Roads (BPR) [14]:

$$t_l({\vartheta }_l)= t_l^0\left[1+0.15{\left({\displaystyle \frac{{\vartheta }_l}{C_l}}\right)}^4\right] \forall l\in T_L$$

(2)

where $t_l({\vartheta }_l)$ represents the travel time of vehicles on link $l$; $t_l^0$ represents the free-flow travel time of link $l$ under zero traffic flow, i.e., the ideal travel time when the traffic flow is zero; ${\vartheta }_l$ represents the traffic flow on link $l$, and $C_l$ represents the traffic capacity of link $l$. In the BPR function, DWCS links and ordinary links adopt the same expression, meaning that the charging process does not directly affect the vehicle travel speed.

3.3. DWPT Model

The construction cost of DWCS mainly includes DWPT and inverters [15]. Among them, DWPT is installed at the bottom of the link to provide charging services. In this study, the planning problem of DWPT is modeled as selecting a suitable installation location on a specific link, and a binary variable $X_{ij}$ is used to indicate whether the link is covered by DWPT.

$$X_{ij}=\left\{\begin{array}{l}1 X_{ij}\in \mathrm{DWPT} \\ 0 X_{ij}\notin \mathrm{DWPT}\end{array}\right.$$

(3)

When $X_{ij}=1$, it indicates that the link connecting node $i$ to node $j$ is a charging link covered by DWPT, which can provide wireless charging services for electric vehicles in motion.

3.4. DWCS Charging Power Model

The charging power of the DWCS is as follows:

$$\left\{\begin{array}{l}{q}_{lr}^e = \rho t_l({\vartheta }_l){\mu }_{lr}^e\\ {\mu }_{lr}^e=\min \{{\mathsf{\mu }}_{\max },{\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{cap}}{\mathcal{l}}_l}{{\vartheta }_l}}\}\end{array}\right. \forall l\in L_N,r\in R^e,e\in E$$

(4)

where we first define the physical meanings and hierarchical relationships of the core symbols to clarify the relationships among paths, links, and nodes. For any O-D pair $e\in E$, a path $r\in R^e$ is formed by a series of ordered links $l\in L_N$ connected in series. Each link $l\in L_N$ corresponds to a directed edge connecting the start node $i$ and the end node $j$, denoted as $(i,j)\in T_N$. $q_{lr}^e$, $\rho$ and ${\mu }_{lr}^e$ represent the actual charging power of each EV on charging link $l$, the DWCS charging efficiency, and the hourly charging power of the vehicle on charging link $l$, respectively; ${\mathsf{\mu }}_{\max }$ is the upper limit of the maximum charging power that the DWCS can output to a single EV [5]; ${\mathsf{\mu }}_{\mathrm{cap}}$ is the charging power density that the system can provide per unit length, representing the power output capacity per unit length of the DWPT embedded in the road surface; ${\mathcal{l}}_l$ is the length of link $l$ covered by DWPT. The value of ${\mu }_{lr}^e$ is constrained by ${\mathsf{\mu }}_{\max }$, and the power available to each vehicle after the total available power of the road segment is evenly distributed among all vehicles on that segment, and the charging power is taken as the smaller of the two values to ensure that neither the system capacity nor the road segment power distribution limit is exceeded. When ${\mu }_{lr}^e$ takes the upper limit of the maximum charging power ${\mathsf{\mu }}_{\max }$, the actual charging power of each EV $q_{lr}^e$ is proportional to the owner’s travel time $t_l({\vartheta }_l)$. When ${\mu }_{lr}^e$ is constrained by the road segment power allocation, it is no longer proportional to the $t_l({\vartheta }_l)$, but exhibits a nonlinear relationship determined by the traffic flow ${\vartheta }_l$.

4. Lower-Level Travel Model

Traffic flow distribution directly affects the charging choices of EVs on the transportation network with DWCS, which in turn influences the DWCS planning. Conversely, DWCS planning alters EV route decisions. Therefore, optimizing DWCS requires considering both travel distribution and charging demand to ensure vehicles always have sufficient power to reach their destinations during trips. Merely meeting basic traffic flow demand is insufficient for cost optimization. Comprehensive optimization of EV travel costs also requires strategic resource allocation and guidance measures.

4.1. EV Travel Model and Constraints in DWCS Transportation Network

4.1.1. Lower-Level Objective Function

(1)

Travel Time Cost

Based on the travel time characteristics of each link, the following link travel cost function is constructed:

$$C_l^t\left({\vartheta }_l\right) = \mathrm{\Phi }{t}_l({\vartheta }_l)\hspace{1em}\hspace{1em}\forall l\in L_N$$

(5)

where $C_l^t\left({\vartheta }_l\right)$ represents the travel time cost of vehicles traveling on link $l$ in the transportation network; $\mathrm{\Phi }$ is the monetary value of the owners’ travel time [16]. In this paper, the time value conversion method is adopted to convert travel time into economic cost.

(2)

Charging Fees

Charging costs are calculated based on the time EVs travel on DWCS, combined with charging power and electricity prices:

$$C_l^c\left({\vartheta }_l\right) ={\displaystyle \sum _{X_{ij}\in \mathrm{DWPT}}{X}_{ij}{q}_{lr}^e{\eta }_{ij}\hspace{1em}\hspace{1em}\forall l\in L_N,r\in R^e,e\in E}$$

(6)

where $C_l^c\left({\vartheta }_l\right)$ and represent the charging cost and the unit charging price of a single vehicle on the DWCS-covered link $l$, respectively.

(3)

Traffic Flow Guidance Fees

This study employs tolling as the regulatory instrument within the traffic flow guidance mechanism. The traffic flow guidance mechanism is directly tied to the actual travel time experienced by vehicles on each link. Specifically, fees are adjusted dynamically based on link travel time, following the fee structure determined by the upper level [17]. When congestion leads to longer travel times, the corresponding fee increases; conversely, it decreases when travel times are shorter. This design leverages pricing mechanisms to incentivize drivers to avoid highly congested road sections, guiding traffic flow toward smoother routes. Consequently, the overall travel efficiency of the road network is optimized. The specific calculation formula for these fees is as follows:

$$C_l^p\left({\vartheta }_l\right)={\tau }_l{t}_l({\vartheta }_l)\hspace{1em}\hspace{1em}\forall l\in L_N$$

(7)

where ${\tau }_l$ represents the unit time traffic flow guidance fee for the traffic link $l\in L_N$, which reflects the intensity of congestion pricing on that road segment. The larger its value, the higher the traffic flow guidance fee charged on that segment during congestion, and the stronger its guiding effect on traffic flow; $C_l^p\left({\vartheta }_l\right)$ is the traffic flow guidance fee for traffic link $l\in L_N$ in the transportation network, which dynamically changes with the degree of congestion on the road segment (i.e., travel time) and is directly included in the generalized travel cost of vehicle owners, thereby influencing their route choice behavior.

(4)

Travel Cost of EVs

Accordingly, the travel cost of EVs can be defined as follows:

$$C_l\left({\vartheta }_l\right)=C_l^t\left({\vartheta }_l\right) + C_l^c\left({\vartheta }_l\right) + C_l^p\left({\vartheta }_l\right)\hspace{1em}\hspace{1em}\forall l\in L_N$$

(8)

where $C_l\left({\vartheta }_l\right)$ is the generalized travel cost of link $l$ in the transportation network.

4.1.2. Constraint Conditions

(1)

Traffic Flow Constraints

The relationship between the total traffic flow of the transportation network and the individual link flows, along with the associated link flow constraints, is given as follows:

$${\beta }_{lr}^e\in \left\{0,1\right\}$$

(9)

$$Q_S={\displaystyle \sum _{e\in E}{\displaystyle \sum _{r\in R^e}{\displaystyle \sum _{l\in L_N}{\beta }_{lr}^e{\vartheta }_l}}}$$

(10)

$${\vartheta }_l\ge 0\hspace{1em} \forall l\in L_N$$

(11)

where ${\beta }_{lr}^e$ is a path-link binary variable. Constraint (9) denotes that ${\beta }_{lr}^e$ is 1 if link $l$ belongs to path $r\in R^e$ between O-D pair $e\in E$, and 0 otherwise. Constraint (10) is the flow conservation equation, indicating that the total road network flow $Q_S$ is equal to the sum of the flows on all links across all O-D pairs and all paths. Constraint (11) is the non-negativity constraint, specifying that the flow on each link must be greater than or equal to zero.

(2)

Path Feasibility Constraints

To define feasible paths in the network, the following constraint conditions are introduced:

$${\gamma }_r^e\in \left\{0,1\right\}\hspace{1em}\hspace{1em}\forall r\in R^e,e\in E$$

(12)

$${\displaystyle \sum _{l\in L_N}{\vartheta }_l}\le {\gamma }_r^{e}M\hspace{1em} \forall r\in R^e,e\in E$$

(13)

where ${\gamma }_r^e$ is a binary variable, which is 1 if path $r$ is feasible, and 0 otherwise. Constraint (13) is the feasibility-flow coupling constraint. If ${\gamma }_r^e=1$, the path flow ${\displaystyle \sum _{l\in L_N}{\vartheta }_l}$ can be greater than zero but shall not exceed $M$, where $M$ is a sufficiently large positive constant. If ${\gamma }_r^e=0$, it enforces ${\displaystyle \sum _{l\in L_N}{\vartheta }_l}=0$, ensuring that no flow is assigned to infeasible paths.

(3)

State of Charge Constraints

The DWCS power constraint targets the power requirements of EVs when traveling on charging roads:

$$SO{C}_{jr}^e-SO{C}_{ir}^e+\mathrm{J}{k}_{ij}-X_{ij}{q}_{lr}^e=0\hspace{1em} \forall l=(i,j)\in T_N,r\in R^e,e\in E$$

(14)

$$SO{C}_{or}^e=SO{C}^{initial}\hspace{1em} \forall r\in R^e,e\in E$$

(15)

$${\mathrm{SOC}}_{\min }\le SO{C}_{ir}^e\le {\mathrm{SOC}}_{\max }\hspace{1em} \forall i\in T_N,r\in R^e,e\in E$$

(16)

In Constraint (14), $SO{C}_{ir}^e$ and $SO{C}_{jr}^e$ represent the battery charge at the start node $i$ and end node $j$ of link $l$ on path $r$ for the EV of O-D pair $e$, respectively, and $\mathrm{J}$ is the electricity consumption per kilometer for the EV. This equation describes the dynamic update of node charge during the link traversal process. When an EV enters link $l$ from its start node $i$ on path $r$, traversing the link consumes electricity $\mathrm{J}{k}_{ij}$; meanwhile, if it traverses a charging link, it gains electricity $q_{lr}^e$. Upon reaching the end node $j$ of the link, the charge at node $j$, $SO{C}_{jr}^e$, is derived from the charge at the start node $i$, $SO{C}_{ir}^e$, after accounting for consumption and charging. Furthermore, the recursive rule of Constraint (14) also serves as the fundamental basis for the dynamic evolution of SOC at the path level. Specifically, for any path $r$ corresponding to O-D pair $e$, the path consists of a sequential series of ordered links $l\in L_N$: the electric vehicle departs from the origin node O (with the initial SOC given by Constraint (15)). Each time it traverses a link $l$ connecting node $i$ and $j$, the charge status is updated between nodes via Constraint (14). During this process, the SOC at all nodes $i\in T_N$ on the path must satisfy the upper and lower bound constraints specified in Constraint (16) until the vehicle reaches the path’s destination. This continuous recursive process characterizes the dynamic variation pattern of the EV’s SOC from the origin to the destination on path $r$. In Constraint (15), $SO{C}_{or}^e$ denotes the initial battery charge of the EV for O-D pair $e$ at the origin node O of the path $r$, and $SO{C}^{initial}$ is the preset initial battery charge value. This constraint serves as the initial charge input for the dynamic SOC evolution on the path $r$. In Constraint (16), ${\mathrm{SOC}}_{\min }$ is the minimum threshold of battery charge, and ${\mathrm{SOC}}_{\max }$ is the actual usable battery capacity. This constraint requires that the battery charge of the EV must remain within a reasonably specified range at all nodes $i\in T_N$ on the path $r$.

(4)

Charging Power Constraints

The charging power constraint defines the charging power range of EVs during the charging process:

$${\displaystyle \sum _{e\in E}{\displaystyle \sum _{r\in R^e}{\displaystyle \sum _{l\in L_N}{\beta }_{lr}^e{\vartheta }_l{q}_{lr}^e}\le {\displaystyle \sum _{l\in L_N}{\mathsf{\mu }}_{\mathrm{cap}}{\mathcal{l}}_l}}}$$

(17)

where the charging power constraint of DWCS ensures that the total power demand of all charging vehicles does not exceed the upper limit of the total power supply capacity that the DWCS system can provide based on its planning scale.

4.1.3. User Equilibrium Conditions

In a transportation network, when every traveler selects the route that minimizes their own travel cost, and no individual can further reduce their travel time or expense by unilaterally changing routes, the system is said to be in user equilibrium (UE) [18]. This equilibrium must satisfy the following nonlinear complementarity conditions:

$$0\le {\vartheta }_r\perp C_r\left({\vartheta }_r\right)-C^e\ge 0\hspace{1em} \forall r\in R^e,e\in E$$

(18)

where ${\vartheta }_r$, $C_r\left({\vartheta }_r\right)$ and $C^e$ represent the flow on path $r$, the generalized travel cost on path $r$, and the minimum generalized travel cost among all feasible paths between this O-D pair, respectively. If ${\vartheta }_r>0$, indicating that the path is used, the travel cost of EVs on path $r$, $C_r\left({\vartheta }_r\right)$, is equal to $C_r^e$, meaning the path cost is equal to the minimum cost. If ${\vartheta }_r=0$, indicating that the path is not used, the generalized travel cost of EVs on path $r$, $C_r\left({\vartheta }_r\right)$, is not lower than $C_r^e$, meaning the path cost is not lower than the minimum cost.

The complementarity constraints show that, in the equilibrium state, all paths actually used have the same and minimum generalized travel cost, while any path with a higher cost will not be chosen by vehicles. This mathematically rigorously characterizes the route choice behavior of drivers under selfish rationality and serves as the theoretical basis for the lower-level traffic flow assignment model.

4.1.4. Variational Inequality Equation of the User Equilibrium Model

To analyze the theoretical properties of the lower-level UE model, we formulate it as a Variational Inequality (VI) problem, i.e., find the equilibrium path flow vector ${\vartheta }^{\ast }= {\left({{\vartheta }_r}^{\ast }\right)}_{\mathrm{r}\in R^e}\in \mathrm{\Omega }$ such that

$${\displaystyle \sum _{r\in R^e}{C}_r({\vartheta }^{\ast })(C_r-C_r^{\ast })}\ge 0,\hspace{1em} \forall \vartheta ={\left({\vartheta }_r\right)}_{\mathrm{r}\in R^e}\in \mathrm{\Omega }$$

(19)

where $\mathrm{\Omega }\subset {ℝ}^{\left|R\right|}$ is the set of feasible path flows satisfying flow conservation, non-negativity, and SOC constraints, and $C_r(\vartheta )$ is the generalized travel cost of path $r$.

4.1.5. Analysis of the Existence, Uniqueness, Stability of Equilibrium Solutions and Algorithm Convergence

(1)

The assumptions supporting the existence of equilibrium are as follows:

(1.1.)

The feasible set $\mathrm{\Omega }$ is a non-empty, compact, and convex set.

(1.2.)

The path generalized cost mapping $C(\vartheta )$ is continuous on $\mathrm{\Omega }$.

Proposition 1 

(Existence of Equilibrium Solution). If the above assumptions hold, then the lower-level UE problem has at least one equilibrium solution.

Proof. 

First, we prove Assumption (1). The set $\mathrm{\Omega }$ is formed by the intersection of linear constraints (flow conservation, SOC constraints) and non-negativity constraints, which is convex. Since the O-D demand is positive and feasible paths exist, $\mathrm{\Omega }$ is non-empty. The flow constraints ensure that $\mathrm{\Omega }$ is a bounded and closed set (compact). Next, we prove Assumption (2). The path cost is $C_r(\vartheta )={\displaystyle \sum _{l\in L_N}{\vartheta }_l{C}_l\left({\vartheta }_l\right)}$. The link travel time $t_l({\vartheta }_l)$ is given by the continuously differentiable BPR function in Equation (2). The toll is a continuous function of time, and thus continuous. The charging cost is the product of constants (electricity price, charging efficiency) and continuous terms of travel time and charging power. The composition and product of continuous functions preserve continuity, so the charging cost is continuous with respect to flow. The path cost is a linear superposition of link costs, so the cost mapping is continuous.

In summary, the assumptions hold, and according to variational inequality theory, an equilibrium solution exists. Q.E.D. □

(2)

The assumptions supporting the uniqueness of the equilibrium link flow are as follows:

(2.1)

The feasible set $\mathrm{\Omega }$ is a non-empty, compact, and convex set; (proven by Proposition 1)

(2.2)

The generalized travel cost function $C_l\left({\vartheta }_l\right)$ of each link is strictly convex with respect to its own link flow ${\vartheta }_l$, i.e., ${\displaystyle \frac{d^2{C}_l\left({\vartheta }_l\right)}{d{\vartheta }_l^2}}>0$ holds for all $l\in L_N$.

Proposition 2 

(Uniqueness of Equilibrium Path Flows). If the above assumptions hold, then the equilibrium link flow vector ${\vartheta }^{\ast }$corresponding to the lower-level user equilibrium problem is unique.

Proof. 

By the Beckmann transformation, the user equilibrium problem is equivalent to minimizing the potential function $Z(\vartheta )={\displaystyle \sum _{l\in L_N}{\displaystyle \underset{0}{\overset{{\vartheta }_l}{\int }}{C}_l\left(w\right)}}dw$ over the feasible set $\mathrm{\Omega }$, where the integration variable $w$ denotes the flow value on the link. By Assumption (2), each $C_l({\vartheta }_l)$ is strictly convex, so its derivative ${\displaystyle \frac{d{C}_l}{d{\vartheta }_l}}>0$ is strictly increasing. The Hessian matrix of $Z(\vartheta )$ is diagonal, with diagonal elements ${\displaystyle \frac{{\mathsf{\partial }}^{2}Z}{\mathsf{\partial }{\vartheta }_L^2}}={\displaystyle \frac{d^2{C}_l}{d{\vartheta }_l^2}}>0$, implying that the Hessian is positive definite and $Z(\vartheta )$ is a strictly convex function. A strictly convex function has a unique minimizer over a non-empty, compact, and convex set, so the equilibrium link flow vector ${\vartheta }^{\ast }$ is unique. Q.E.D. □

(3)

The assumptions supporting the stability of the equilibrium solution and the convergence of the algorithm are as follows:

(3.1)

The feasible set $\mathrm{\Omega }$ is a non-empty, compact, and convex set (guaranteed by Proposition 1);

(3.2)

The generalized cost function $C_l\left({\vartheta }_l\right)$ of each link is strictly convex with respect to its own link flow ${\vartheta }_l$ (proven by Proposition 2);

(3.3)

The fitness function of the GA used to solve the lower-level problem is continuous and locally convex on $\mathrm{\Omega }$;

(3.4)

The parameters of the GA are reasonable.

Proposition 3 

(Stability of Equilibrium Solutions and Algorithm Convergence). By Assumptions 1 and 2, the lower-level equilibrium link flows change continuously with the continuous variation in the upper-level traffic flow guidance fees. Under perturbations, user paths gradually converge to the unique solution, so the equilibrium solution is locally stable. By Assumption 3, the optimization problem has a unique global optimal solution. By Assumption 4, the fitness value of the GA is monotonically decreasing and bounded below, so it converges to this unique equilibrium solution.

4.2. Lower-Level Model Under User Equilibrium

Therefore, the optimization problem characterized by the user equilibrium described above is referred to as the traffic assignment problem.

$$\left\{\begin{array}{l}\min C_{\mathrm{TAP}}={\displaystyle \sum _{l\in L_N}{\displaystyle {\int }_0^{{\vartheta }_l}{C}_l^t\left({\vartheta }_l\right) d\vartheta }}+{\displaystyle \sum _{l\in L_N}{\displaystyle {\int }_0^{{\vartheta }_l}{C}_l^p\left({\vartheta }_l\right) d\vartheta }}+{\displaystyle \sum _{e\in E}{\displaystyle \sum _{r\in R^e}{\displaystyle \sum _{l\in L_N}{\vartheta }_l{C}_l^c\left({\vartheta }_l\right)}}}\\ s.t.(9)~(18)\end{array}\right.$$

(20)

where $C_{\mathrm{TAP}}$ denotes the generalized travel cost of the transportation network. The first term on the right-hand side represents the travel time cost of the network, which is obtained by integrating the link-specific time cost functions and reflects the time loss due to congestion. The second term corresponds to the traffic guidance fees, and the third term represents the charging cost of the DWCS. Specifically, embedding the traffic guidance mechanism into the user equilibrium model influences the comprehensive travel cost of EVs. Constraints (9)–(18) include flow conservation, path feasibility, SOC constraints, charging power constraints, and user equilibrium complementarity conditions, ensuring that the solution is both physically and behaviorally feasible.

5. Upper-Level Planning Model

The objective of the upper-level planning model is to maximize the annual comprehensive net profit of the DWCS operator, taking into account both the operator’s annual revenue and annual expenditures. At the same time, the operator influences route choices at the lower level by optimizing traffic flow guidance fees, thereby indirectly regulating the state variable of travel time.

5.1. Traffic Flow Guidance Fees Model

To optimize the overall system performance and achieve active guidance of lower-level users’ route choices, the upper-level DWCS operator has established a traffic flow guidance fee regulation model. This model dynamically adjusts the unit time toll ${\tau }_l$ using the price lever to influence users’ travel costs, thereby guiding traffic flow towards a more efficient distribution. Its regulation mechanism follows a closed-loop iterative optimization process: First, based on the initial toll scheme and the existing facility planning, lower-level users make route choices according to their generalized travel costs and feedback travel times and flow distributions to the upper level. Subsequently, with the objective of minimizing the total travel time of the entire network, the upper level optimizes and adjusts the toll for each link within the feasible range $[{\tau }_{\min },{\tau }_{\max }]$. This process iterates repeatedly until the system reaches a stable state, ultimately obtaining the optimal travel time cost and equilibrium flow distribution under the indirect regulation of the traffic flow guidance fee.

$$\left\{\begin{array}{l}{S}_V={\displaystyle \sum _{l\in L_N}{t}_l({\vartheta }_l){\vartheta }_l}\\ s.t.\hspace{1em}{\tau }_{\min }\le {\tau }_l\le {\tau }_{\max }\end{array}\right.$$

(21)

where $S_V$ represents the total travel time of vehicle owners in the transportation network, which is the direct objective of upper-level control; $S_V$ is affected by the traffic flow guidance fees ${\tau }_l$.

5.2. Upper-Level Objective Function

The objective function of the upper-level model is as follows:

$$\left\{\begin{array}{l}\max I_{\mathrm{total}}=I_{\mathrm{income}}-\left(C_{\mathrm{Inverter}} + C_{\mathrm{DWPT}} + C_{\mathrm{maintain}}+ C_{\mathrm{purchase}}+C_{\mathrm{losses}}\right)\\ I_{\mathrm{income}} ={\displaystyle \sum _{t=1}^{365}{\displaystyle \sum _{e\in E}{\displaystyle \sum _{r\in R^e}{\displaystyle \sum _{l\in L_N}{\vartheta }_l{C}_l^m\left({\vartheta }_l\right)+{\displaystyle \sum _{t=1}^{365}{\displaystyle \sum _{l\in L_N}{\displaystyle {\int }_0^{{\vartheta }_l}{C}_l^p\left({\vartheta }_l\right) d\vartheta }}}}}}}\\ C_l^m\left({\vartheta }_l\right) ={\displaystyle \sum _{X_{ij}\in \mathrm{DWPT}}{X}_{ij}{q}_{lr}^e{m}_{ij}\hspace{1em}\hspace{1em}\forall l\in L_N,r\in R^e,e\in E}\end{array}\right.$$

(22)

where I_total represents the annual comprehensive net profit of the DWCS operator. This objective function consists of two major components: annual income and annual expenditure. The annual income I_income includes two parts: the first is the total annual charging profit, and the second is the total annual traffic guidance fee revenue, which reflects the additional benefits brought by the traffic flow guidance fee mechanism, where $C_l^m$(ϑ_l) and m_ij represent the charging profit and profit unit price of the operator on DWCS link l respectively. The annual expenditure includes the inverter acquisition cost C_inverter, DWPT acquisition cost C_DWPT, annual system maintenance cost C_maintain, annual electricity purchase cost C_purchase, and the power transmission loss cost C_losses caused by insufficient transmission efficiency.

5.3. Installation Location Constraints of DWCS

In DWCS engineering practice, to reduce equipment investment, adjacent DWPT links typically share the same inverter [19]. To clarify the inverter installation requirements, this paper introduces two binary variables to characterize the connection direction of nodes within the DWPT network, providing a topological basis for subsequently determining which nodes can serve as the starting points of independent charging paths.

$$y_i^{\mathrm{out}}\in \left\{0,1\right\}$$

(23)

$$Z_i^{\mathrm{in}} \in \left\{0,1\right\}$$

(24)

where $y_i^{\mathrm{out}}=1$ indicates that node $i$ is the starting node of at least one link covered by DWPT; $Z_i^{\mathrm{in}}=1$ indicates that node $i$ is the ending node of at least one link covered by DWPT.

Based on the definition of binary variables $y_i^{\mathrm{out}}$ and $Z_i^{i\mathrm{n}}$, to ensure the rationality of inverter arrangement, the following constraints are obtained in this study:

$$y_i^{\mathrm{out}}\le {\displaystyle \sum _{X_{ij}\in \mathrm{DWPT}}{X}_{ij}}$$

(25)

$$y_i^{\mathrm{out}}\le 1-X_{mi}\hspace{1em}\forall (m,i)\in T_N$$

(26)

$$Z_i^{\mathrm{in}}\le {\displaystyle \sum _{(m,i)\in T_N}{X}_{mi}}$$

(27)

$$Z_i^{\mathrm{in}}\ge X_{mi}\hspace{1em} \forall (m,i)\in T_N$$

(28)

where Constraint (25) denotes that if all outgoing links of node $i$ are not covered by DWCS, it cannot be used as the starting point; Constraint (26) denotes that if any incoming link of node $i$ is covered by DWCS, it cannot be used as the starting point of DWCS; Constraint (27) denotes that if node $i$ is not connected by any link covered by DWPT, it cannot be used as the ending node of DWCS; Constraint (28) denotes that if node $i$ is connected by any link covered by DWPT, it is the ending node of DWCS.

Based on the above variables and their constraints, the installation rules for inverters are as follows:

(a) The node has no incoming links, or all incoming links are not covered by DWPT.

(b) The node has at least one outgoing link covered by DWPT.

Nodes that meet the conditions are the starting points of independent charging paths and should be equipped with inverters.

The total number of inverters to be purchased is determined by the total number of independent path starting points, and the calculation formula is as follows:

$${\mathrm{N}}_{\mathrm{Inverter}}= {\displaystyle \sum _{i\in T_N}{y}_i^{\mathrm{out}}}-\left({\displaystyle \sum _{X_{mi}\in \mathrm{DWPT}}{X}_{mi}}-{\displaystyle \sum _{i\in T_N}{Z}_i^{\mathrm{in}}}\right)$$

(29)

where ${\mathrm{N}}_{\mathrm{Inverter}}$ represents the total number of inverters to be installed, calculated as the number of start nodes minus (the total number of DWPT links minus the number of end nodes). This formula computes the total number of independent charging start points in the network by excluding the intermediate nodes that are shared along the paths, which corresponds to the minimum number of inverters required.

5.4. Annual Expenditure Cost of DWCS Operators

(1)

Construction Cost

The purchase cost of inverters in DWCS is as follows:

$$C_{\mathrm{Inverter}}= {\mathsf{\partial }}_{\mathrm{Inverter}}{\mathrm{N}}_{\mathrm{Inverter}}$$

(30)

where $C_{\mathrm{Inverter}}$ represents the total purchase cost of inverters, ${\mathsf{\partial }}_{\mathrm{Inverter}}$ denotes the purchase cost of a single inverter, and ${\mathrm{N}}_{\mathrm{Inverter}}$ represents the total number of inverters purchased.

The purchase cost of DWPT is as follows:

$$C_{\mathrm{DWPT}} = {\mathsf{\partial }}_{\mathrm{DWPT}}{\displaystyle \sum _{X_{ij}\in \mathrm{DWPT}}{X}_{ij}{\mathcal{l}}_l}$$

(31)

where $C_{\mathrm{DWPT}}$ represents the total installation cost of DWPT, ${\mathsf{\partial }}_{\mathrm{DWPT}}$ denotes the installation cost per unit length of DWPT, and ${\displaystyle \sum _{X_{ij}\in \mathrm{DWPT}}{X}_{ij}{\mathcal{l}}_l}$ is the total laying length of all DWPT links.

(2)

Maintenance Cost

The maintenance cost of DWCS is as follows:

$$C_{\mathrm{maintain}}={\upsilon }_{\mathrm{DWCS}}{\displaystyle \sum _{X_{ij}\in \mathrm{DWPT}}{X}_{ij}{\mathcal{l}}_l}$$

(32)

where $C_{\mathrm{maintain}}$ represents the annual maintenance cost of the DWCS, and ${\upsilon }_{\mathrm{DWCS}}$ denotes the annual maintenance cost per unit length of the DWCS.

(3)

Power Purchase Cost

The annual electricity purchase cost of the DWCS operator is as follows:

$$C_{\mathrm{purchase}}={\displaystyle \sum _{t=1}^{365}{\displaystyle \sum _{e\in E}{\displaystyle \sum _{r\in R^e}{\displaystyle \sum _{l\in L_N}{\displaystyle \sum _{X_{ij}\in \mathrm{DWPT}}{\vartheta }_l{X}_{ij}{q}_{lr}^e{\mathrm{\Psi }}_{ij}}}}}}$$

(33)

where $C_{\mathrm{purchase}}$ represents the annual electricity purchase cost of the operator, and ${\mathrm{\Psi }}_{ij}$ represents the unit price of electricity purchased by the DWCS operator from the power grid.

(4)

Power Transmission Loss Cost

In the annual expenditure cost of DWCS, due to insufficient transmission efficiency, part of the electric energy is lost during transmission. The formula for power transmission loss cost is introduced as follows:

$$C_{\mathrm{losses}}={\displaystyle \sum _{t=1}^{365}{\displaystyle \sum _{e\in E}{\displaystyle \sum _{r\in R^e}{\displaystyle \sum _{l\in L_N}{\displaystyle \sum _{X_{ij}\in \mathrm{DWPT}}{\vartheta }_l{X}_{ij}\left(1-\rho \right)t_l({\vartheta }_l){\mu }_{lr}^e{\mathrm{\Psi }}_{ij}}}}}}$$

(34)

where $C_{\mathrm{losses}}$ represents the total annual cost of power transmission losses for the DWCS system, and $(1-\rho )$ represents the power transmission loss efficiency of the DWCS. This formula calculates the total economic cost incurred by insufficient transmission efficiency by summing the lost electrical energy across all deployed DWPT links, time periods, O-D pairs, and paths, and then multiplying by the unit electricity purchase price.

6. Solution Methodology for the Bi-Level Model

6.1. Reasons for Selection and Comparative Analysis of the Solution Algorithm

(1)

Applicability Limitations of Traditional Bi-level Optimization Methods

The lower-level user equilibrium problem in this model exhibits several non-standard features. These characteristics make it difficult to effectively implement classical bi-level optimization methods. First, the lower-level user equilibrium problem has a large dimension and is highly nonlinearly coupled with the upper-level DWCS planning. This coupling invalidates traditional analytical methods, such as KKT conditions and gradient-based algorithms. Second, although the path generalized cost function is strictly convex, the non-monotonic property of charging power leads to a non-convex and multi-modal structure in the lower-level objective function. Consequently, convexity-dependent methods, such as sequential quadratic programming, struggle to converge stably. In addition, an implicit black-box mapping exists between the upper and lower levels. The upper-level traffic flow guidance fee affects flow distribution indirectly through path costs. However, this mapping is implicitly defined by the user equilibrium iteration. It is non-differentiable and difficult to approximate using surrogate models. These limitations hinder the application of sensitivity analysis and gradient backpropagation.

(2)

Adaptive Advantages of the Heuristic-CPLEX Iterative Framework

To address these challenges, this study adopted an iterative solution strategy combining heuristic algorithms with CPLEX. The upper level involved a mixed-integer nonlinear programming problem with binary variables. CPLEX solved this exactly for DWCS facility location and capacity allocation, ensuring global optimality of the planning scheme. Conversely, the lower level handled the user equilibrium problem with nonlinear congestion via heuristic algorithms, ensuring quick convergence to the network flow equilibrium state. Through alternating iterations, the framework leveraged the robust search ability of heuristics and the computational efficiency of CPLEX. This effectively overcame difficulties such as the curse of dimensionality, non-convexity, multi-modality, and implicit mapping. Thus, it provided a stable and feasible approach for solving highly nonlinearly coupled bi-level programs.

6.2. Iterative Solution and Flow Chart

The model is solved using an iterative solution method [20]. In each iteration, the upper-level model first generates an initial DWCS planning scheme and a set of traffic flow guidance fees, which are then passed to the lower-level model. Upon receiving this information, the lower-level model solves for the resulting traffic flow distribution using a GA and obtains the corresponding total travel time as an initial solution. Subsequently, a joint iterative optimization based on SA-GA is implemented until the temperature requirement is met (i.e., the current temperature is lower than the preset termination temperature). At this point, the iterative optimization loop of the lower-level algorithm terminates, and the obtained annual travel flow is fed back to the upper-level model. The upper-level and lower-level models are solved alternately and iteratively until the upper-level model satisfies the convergence condition shown in Equation (35), at which point the process ends.

$$|C_s-C_{s-1}|\le \epsilon$$

(35)

where $S$ is the number of iterations for solving the bi-level model, and $\epsilon$ is the convergence threshold. The overall solution process of the DWCS bi-level optimization model proposed in this study is shown in Figure 3.

6.3. Feasibility-Preserving Mechanism for the Lower-Level User Equilibrium

To ensure that the lower-level user equilibrium solution always satisfies the physical and modeling constraints during the bi-level optimization process, a feasibility-preserving mechanism is incorporated into the algorithm design in this paper.

(1)

Feasibility of Chromosome Encoding and Initialization

The GA encoded the variables to be solved into chromosomes. Each chromosome corresponded to a complete link flow distribution scheme, with each gene representing the flow assigned to the corresponding path. The number of paths defines the length of a single chromosome. The population size was set to 60, and the maximum number of iterations was 80. The offspring size was 80% of the parent population, with a mutation probability of 0.01. The number of genes selected for crossover and mutation was 3 and 4, respectively. The initial assignment was randomly generated. The flow of the last path was then adjusted to satisfy the flow conservation equation, yielding an initial solution that met flow conservation requirements.

During population initialization, a heuristic random generation method was adopted. For each O-D pair, the algorithm randomly selected a path from its feasible path set and assigned it a random flow value. Subsequently, normalization was performed to ensure the total flow for each O-D pair matched the demand. This procedure guaranteed that the initial population satisfied all flow conservation constraints.

(2)

Feasibility of Crossover and Mutation

An improved multi-point crossover operator was adopted for the crossover operation. Chromosomes were encoded in segments based on the path flows of two O-D pairs (1–8 and 1–9), as shown in Figure 4 (The data in the first four light blue boxes represent the number of vehicles on each of the four paths in O-D pair 1–8, and the data in the next seven dark blue boxes represent the number of vehicles on each of the seven paths in O-D pair 1–9). During crossover, three crossover points were randomly generated, and the path flows of the parent chromosomes at these points were exchanged sequentially (In the figure, taking the crossover points located at the first three gene loci as an example, the red arrows point to the boxes indicating where the exchange occurs between the two parent chromosomes. The exchanged genes are shown in the middle and lower boxes, based on which the offspring are generated.). The crossover operation was conducted within the same O-D pair to ensure that the exchanged flows still belonged to the feasible path set of that O-D pair. After the crossover, the flows of each O-D pair were normalized so that their sum strictly equaled the traffic demand in the corresponding period, thereby satisfying the flow conservation constraints. If negative values or constraint violations occurred after adjustment, the crossover was abandoned, and the parent individual was retained.

The mutation operation maintained flow conservation through intra-O-D flow shifting, as shown in Figure 5. Two paths within the same O-D pair (1–8 or 1–9) were randomly selected. A portion of the flow (not exceeding a preset fluctuation value) was transferred from one path to the other, ensuring that both paths maintained non-negative flows and neither exceeded the total demand of the O-D pair (As shown in the figure, the flow from paths 2 and 3 in the parent O-D pair 1–8 is transferred to paths 1 and 4 to generate the offspring; the flow from paths 4 and 7 in the parent O-D pair 1–9 is transferred to paths 1 and 6 to generate the offspring). If the transfer violated flow constraints, the mutation was canceled.

(3)

Feasibility of State-of-Charge Constraints

A SOC feasibility check was performed for each candidate path: the recursive SOC process of the vehicle from origin to destination was simulated in accordance with Equation (14). Only paths where the SOC remained above the minimum threshold at all nodes were retained, forming the feasible path set for each O-D pair.

(4)

Feasibility of User Equilibrium Conditions

The lower-level model takes the minimization of the generalized travel cost of the transportation network as its objective function, which is directly used as the fitness function of the GA in this paper. Experiments show that the algorithm generally converges to a stable solution within approximately 11 iterations, and the solution always meets the feasibility requirements of UE under the protection of the aforementioned mechanisms.

6.4. Algorithm Flow

The collaborative simulation algorithm flow chart for the bi-level optimization problem is shown in Algorithm 1:

Algorithm 1: Collaborative Simulation Algorithm for Bi-Level Optimization Problems in DWCS

1. Algorithm Input Transportation network parameters: $G_N = [T_N,L_N]$, $k_{ij}$, $t_l^0$, $C_l$, $Q_S$, $\mathrm{\Phi }$ input via matrix, corresponding to Equations (1), (2) and (5) DWCS parameters: $\rho$, ${\mathsf{\mu }}_{\max }$, ${\mathsf{\mu }}_{\mathrm{cap}}$, ${\mathsf{\partial }}_{\mathrm{Inverter}}$, ${\mathsf{\partial }}_{\mathrm{DWPT}}$, ${\upsilon }_{\mathrm{DWCS}}$ parameter input, corresponding to Equations (4) and (30)–(32) Electricity price parameter: ${\eta }_{ij}$, ${\mathrm{\Psi }}_{ij}$, $m_{ij}$ input via matrix, corresponding to Equations (6), (22) and (33) Electricity quantity parameter $T_{soc}^{initial}$ input via matrix, corresponding to Equation (15) Algorithm parameter: convergence threshold $\epsilon$ parameter input, corresponding to Equation (35) 2. Algorithm Steps Step 1: Initialization       ● Initialize facility planning scheme by matrix input, corresponding to Equation (3)       ● Initialize the traffic flow guidance fee scheme ${\tau }_l$ input via matrix, corresponding to Equation (21) Step 2: Main Iteration Loop   while not converged do   (1) Lower-level optimization (vehicle owner operation layer)   Input: current planning scheme    (1.1) SA initialization: set current temperature and current traffic flow guidance scheme    (1.2) Search loop (temperature > termination temperature)         (a) Generate new solution: apply random perturbation to traffic flow guidance fees to generate a new traffic flow guidance fee scheme         (b) Solve equilibrium with fixed Traffic guidance fees (call GA)          i. GA framework initialization: chromosome length (number of paths), population size (path scale), maximum number of iterations, offspring ratio, offspring population size, number of genes selected for crossover operation, mutation probability, number of genes selected for mutation          ii. Population initialization: generate the initial upper-level population (initial traffic flow allocation scheme), ensuring that the total traffic flow of each O-D pair equals the demand to satisfy: the flow conservation Constraints (9) and (10) satisfied by normalization, non-negative flow Constraint (11), and path feasibility Constraints (12) and (13)          iii. Fitness evaluation:            - Validate each traffic flow individual: traverse each node to calculate the electricity quantity conservation Constraints (14) and (15), electricity quantity threshold Constraint (16), and charging power Constraint (17)            - Traverse each node to calculate the charging power (4)            - Calculate the generalized travel costs (5–8)            - Calculate the generalized travel cost of the system as the fitness value (20)            - Select individuals with better fitness as parents          iv. Genetic operation: select, crossover and mutate according to fitness to generate a new generation of the population          v. Iteration and convergence: repeat evaluation and evolution until the GA terminates and converges to obtain the user equilibrium condition (18). Output the optimal traffic flow distribution and the corresponding minimum system travel time         (c) SA acceptance decision:         Time cost difference: $\mathsf{\Delta }{S}_V = S_V(new)-S_V(current)$         If $\mathsf{\Delta }{S}_V<0$, accept the new solution;         else, Accept with probability         (d) Update and cooling:         If the new solution is accepted, update it         After completing L searches at the current temperature, decrease the temperature    (1.3) Output lower-level results   (2) Upper-level optimization (planning layer):   Input: Traffic flow distribution    (2.1) Construct MIP model:   - Decision variables: binary variables DWPT, Inverter   - Objective function: calculate the upper-level objective function (22)   - Constraints: determine the node inverter planning using logical constraint rules (23–29)    (2.2) Solve: Call CPLEX to solve the MIP, and obtain:              ● New DWPT planning       ● Inverter planning       ● Total profit of the operator (3) Check convergence condition:         If $|C_s-C_{s-1}|\le \epsilon$ then         The algorithm converges and exits the loop         Else pass the new facility planning scheme to the lower level and return to Step (1)     end if     end while 3. Output final results When the main iteration loop terminates due to convergence, the optimal facility planning scheme is obtained.

7. Case Study

In this section, to verify the performance of the proposed model, we conduct numerical simulation analysis based on the transportation network model adopted in references [13,17]. The network is shown in Figure 6, which contains 9 nodes and 13 links (The arrows in the figure indicate the direction of travel on each link, representing the permissible flow direction of traffic.). The data on link length, capacity, and free-flow travel time are shown in

Table 1. Link parameter.

Links	Length (km)	Link Capacity	Link Free-Flow Travel Time (min)
1	2	200	2
2	7	133	2
3	8	200	8
4	2	66	2
5	4	66	4
6	6	33	6
7	2	133	2
8	6	200	6
9	3	133	3
10	9	66	9
11	26	100	26
12	4	100	4
13	5	100	5

. The O-D pairs of vehicles are nodes 1–8 and 1–9 in the network. The shortest path method is used to generate valid path sets for each O-D pair, as shown in

Table 2. O-D path set.

O-D Pairs	Effective Node Set	Effective Link Set
1–8	1–8	11
	1–2–3–5–7–8	1–2–5–9–13
	1–2–5–7–8	1–3–9–13
	1–2–7–8	1–10–13
1–9	1–2–3–4–6–9	1–2–4–6–8
	1–2–3–5–6–9	1–2–5–7–8
	1–2–3–5–7–8–9	1–2–5–9–13–12
	1–2–5–6–9	1–3–7–8
	1–2–5–7–8–9	1–3–9–13–12
	1–2–7–8–9	1–10–13–12
	1–8–9	11–12

. The 24 h traffic demand of O-D pairs on a typical day is shown in

Table 3. O-D 24 h traffic flow demand.

Time	O-D 1–8	O-D 1–9	Time	O-D 1–8	O-D 1–9
1:00	75	90	13:00	210	330
2:00	60	90	14:00	240	270
3:00	90	150	15:00	180	270
4:00	120	180	16:00	210	330
5:00	150	210	17:00	330	390
6:00	210	330	18:00	390	510
7:00	330	390	19:00	330	390
8:00	390	510	20:00	270	390
9:00	270	330	21:00	210	270
10:00	210	270	22:00	150	210
11:00	210	240	23:00	90	150
12:00	270	330	24:00	90	90

.

The charging power capacity per kilometer of DWCS is 65.79 MW [21], the purchase cost of DWPT is 188.34 USD/m [4], the purchase cost of an inverter is 14,127.75 USD/unit, and the annual maintenance cost per unit length of DWCS is 65.93 USD/m [5].

According to statistics from the UK Department for Transport, the rated battery capacity of EVs is set to 50 kWh [22]. Based on a study on the lifecycle aging behavior of lithium-ion batteries, the actual usable battery capacity degrades to below 90% of the rated battery capacity at the 400–cycle aging inflection point, entering the accelerated aging stage [23]. Therefore, the actual usable battery capacity is set to 45 kWh. To control the battery depth of discharge, extend battery life, and ensure travel reachability, this study sets the minimum battery SOC threshold of the EV at 12 kWh. The battery energy consumption of the EV is 0.25 kWh/km, the maximum DWCS charging power is 60 kW [16], and the charging efficiency is 0.9. The initial SOC of the EV at departure is uniformly distributed between 15 kWh and 45 kWh. This study refers to the time cost model proposed in reference [16] and converts the monetary value of vehicles’ travel time via exchange rate and unit conversion, yielding 11.26 USD/h. The time-of-use electricity price is shown in

Table 4. Time-of-use tariff and charging tariff.

	Off-Peak Period	Flat Period	Peak Period
Time period	1:00–6:00 20:00–24:00	10:00–15:00	6:00–10:00 15:00–20:00
Electricity Purchasing Price (USD/kWh)	0.0540	0.1012	0.1578
Electricity Selling Price (USD/kWh)	0.1667	0.1957	0.2246

. The parameter values are shown in

Table 5. Parameter value table.

Parameter	Values	Parameter	Values
$\mathrm{\Phi }$	11.26 USD/h [16]	${\mathsf{\partial }}_{\mathrm{Inverter}}$	14,127.75 USD/unit [5]
$\rho$	0.9 [24]	${\mathsf{\partial }}_{\mathrm{DWPT}}$	188.34 USD/m [5]
${\mathsf{\mu }}_{\max }$	60 kW [16]	${\upsilon }_{\mathrm{DWCS}}$	65.93 USD/m [9]
${\mathsf{\mu }}_{\mathrm{cap}}$	65.79 MW [21]	$T_{or}^e$	Uniform (15, 45)
$T_{\min }$	12 kWh	$J$	0.25 kWh/km
$T_{\max }$	45 kWh [22,23]	$\epsilon$	1 × 10−3

. This study uses the MATLAB 2021b environment and combines the CPLEX solver for an iterative solution to obtain the optimization scheme.

7.1. Simulation Comparison

7.1.1. Two-Level Optimization Scenario Setting for DWCS

Based on the user equilibrium principle, in addition to considering the scenario of full coverage of DWCS, this study also constructs three simulation scenarios and conducts a comparative analysis:

Scenario 1: Considering the optimal travel plan for EV owners, a bi-level optimization scheme for DWCS is proposed.

Scenario 2: By introducing traffic flow guidance fees to regulate traffic flow and based on reducing travel time, considering the optimal travel plan for EV owners, a bi-level optimization scheme for DWCS is proposed.

Scenario 3: Adopting the same DWCS optimization scheme as Scenario 2, considering the optimal travel plan for EV owners with fixed planning, without introducing traffic flow guidance fees.

7.1.2. Planning Results of DWCS

The planning of DWCS in full coverage, Scenario 1, Scenario 2 and Scenario 3 are as follows:

Figure 7 presents the planning results of four DWCSs (The numbers 1–13 in the figure represent the link numbers. The purple boxes indicate links equipped with DWPT, the red nodes represent nodes equipped with inverters, and the green nodes represent ordinary nodes). The scheme shown in Figure 7a is equipped with a total of one inverter, located at node 1; a total of 13 DWPTs are deployed, on links 1 to 13, with the total deployed length accounting for 100% of the total length of the road network. The scheme shown in Figure 7b is equipped with a total of four inverters, located at nodes 1, 3, 5, and 7 respectively; a total of five DWPTs are deployed, on links 1, 4, 7, 11, and 13 respectively, with the total deployed length accounting for 44.05% of the total length of the road network. The scheme shown in Figure 7c is equipped with three inverters, located at nodes 1, 5, and 7; a total of six DWPTs are deployed, on links 1, 2, 4, 7, 11, and 13 respectively, accounting for 52.38% of the total length of the road network, which is the same as Scenario 3 (Figure 7d).

7.1.3. Comparative Analysis of Optimization Results Between Scenario 1 and the Full-Coverage Scenario

As shown in

Table 6. Bi-level optimization results under different scenarios.

Parameter	Full Coverage	Scenario 1	Scenario 2	Scenario 3
DWCS operators
DWPT Length/km	84	37	44	44
Construction Cost (‘000 USD)	15,834.69	7025.09	8329.34	8329.34
Maintenance Cost (‘000 USD)	5538.12	2439.41	2900.92	2900.92
Transmission Loss Cost (‘000 USD)	3125.55	2025.91	2035.97	2011.42
Annual Net Profit (‘000 USD)	−6163.06	−775.35	1966.78	−1750.90
Charging Net Profit (‘000 USD)	18,335.29	10,715.06	12,598.58	11,490.78
EV owners
Traffic Flow Guidance Fees (‘000 USD)	\	\	2634.43	\
Vehicle Owners’ Charging Cost (‘000 USD)	49,590.75	30,974.13	32,958.28	31,605.01
Travel Time Cost (‘000 USD)	19,699.07	20,279.31	13,538.73	19,959.91
Owners’ Travel Cost (‘000 USD)	69,289.82	51,253.44	49,131.44	51,564.92

, compared with the annual loss of −6163.06 thousand USD in the full-coverage scenario, the annual net loss of the operator in Scenario 1 is reduced to −775.35 thousand USD, with a loss reduction of 5387.71 thousand USD, achieving a narrowing of losses; the owners’ travel cost is reduced by 18,036.38 thousand USD.

This result is mainly attributed to the following optimization mechanisms: On the DWCS operator side, by optimizing facility planning while ensuring each vehicle can reach its destination smoothly with sufficient power, resource redundancy caused by the full-coverage mode was effectively avoided. This reduced the DWPT installation cost by 55.63% and the DWCS maintenance cost by 55.95%. On the owners’ side, Scenario 1 directly reduced owners’ charging expenses by 18,616.62 thousand USD by eliminating invalid charging in the full-coverage scenario, thereby significantly lowering their overall travel costs.

Compared with the full-coverage scenario, Scenario 1 optimized planning on the upper-level DWCS planning side, but this scheme did not significantly improve travel time costs for owners. Meanwhile, although losses were reduced, the operator remained in a state of loss. In fact, the travel time of vehicles in the transportation network with DWCS directly affects charging costs, overall travel costs, and road congestion levels, and indirectly changes traffic flow distribution. In turn, traffic flow distribution affects the planning profit of DWCS operators and the travel costs of owners. For this reason, we propose Scenario 2 to optimize travel time through a traffic flow guidance fee mechanism and analyze it by comparing Scenarios 1 and 3.

7.1.4. Comparative Analysis of Optimization Results Between Scenario 2 and Scenarios 1 and 3

As shown in Table 6, compared with Scenario 1, although the total construction and maintenance cost of Scenario 2 increases by 1765.76 thousand USD, the comprehensive net profit increases significantly by 2742.13 thousand USD. At the same time, owners’ charging expenses increase by 1984.15 thousand USD, while the travel time cost decreases substantially by 6740.58 thousand USD.

This is because Scenario 1 optimizes the DWCS planning to match traffic demand based on the traffic flow distribution formed by owners’ path choices, avoiding resource waste. However, this approach does not alter the underlying traffic congestion pattern, leading to relatively high time costs for owners. In contrast, Scenario 2 regulates traffic flow through traffic flow guidance fees, alleviates traffic congestion, and significantly reduces owners’ time costs. Although Scenario 2 involves higher construction and maintenance expenditures due to the installation of additional DWCS facilities, the regulatory effect of the guidance fees indirectly stimulates a significant increase in charging volume. As a result, it achieves a net gain in charging revenue of 1883.52 thousand USD. Combined with the traffic flow guidance fee revenue of 2634.43 thousand USD, Scenario 2 achieves a net profit growth of 2742.13 thousand USD. Although owners bear an additional charging expense of 2634.43 thousand USD, they gain a time cost saving of 6740.58 thousand USD, resulting in a net welfare increase of 2122 thousand USD. Furthermore, to isolate the direct contribution of the traffic flow guidance fee mechanism, this study introduces Scenario 3 as a control scheme. In this scenario, the guidance fee mechanism is removed, while the facility planning solution from Scenario 2 is held unchanged. The comparative analysis yields the following results:

Compared with Scenario 3, the annual comprehensive net profit of the operator under Scenario 2 increases by 3717.68 thousand USD, and the owners’ travel cost decreases by 2433.48 thousand USD. Among these, the owners’ travel time cost decreases by 6421.18 thousand USD, which becomes the main factor driving the reduction in owners’ travel cost.

Therefore, this study compares the estimated total travel time required to fulfill all owners’ travel demands under Scenarios 2 and 3 across different time intervals on a typical day, as shown in Figure 8. From the perspective of daily total travel time, Scenario 2 achieves a 48.89% reduction compared to Scenario 3, indicating a substantial optimization in travel efficiency. Examining specific time periods, during the low-traffic intervals from 1:00 to 6:00 and 20:00 to 24:00, the difference in travel time between the two scenarios is minimal, largely due to naturally smooth traffic conditions. In contrast, during the peak hours from 7:00 to 19:00, the traffic flow guidance fee mechanism significantly mitigates the additional travel time typically caused by congestion.

To further examine the regulatory effect of the traffic flow guidance mechanism during peak periods, this study selects the 18:00 peak hour on a typical day for analysis. It compares the travel time distribution characteristics of each link in the road network under two scenarios, as shown in Figure 9. The results indicate that compared with Scenario 3, the total travel time in Scenario 2 decreases from 381.49 h to 245.82 h, representing a reduction of 35.56%. Moreover, the standard deviation of travel time across links in Scenario 2 is 50.76% lower than that in Scenario 3, suggesting a more balanced spatiotemporal distribution of traffic flow. These findings demonstrate that the traffic flow guidance mechanism can effectively guide the rational allocation of traffic flow, reduce excessive congestion duration on certain links, and promote balanced vehicle distribution across the road network, thereby significantly alleviating peak-period congestion and reducing owners’ travel time costs. Based on the annualized calculation of average daily travel time, owners’ travel time costs are substantially reduced, while the operators’ annual comprehensive net profit is significantly increased.

7.2. Algorithm Performance Analysis

7.2.1. Guarantee of Repeatability and Robustness

To ensure the repeatability and robustness of the research results, the following measures are adopted in this paper. First, regarding repeatability, all numerical experiments are performed under a unified computing environment (MATLAB R2021b for programming, IBM ILOG CPLEX Optimization Studio 12.10 from IBM Corp., Armonk, NY, USA as the solver). All parameter values of the model, constraints, and algorithm initialization methods are disclosed in the paper. The core hyperparameters of the adopted heuristic algorithm and the CPLEX iterative solution are provided in Section 7.2.4.

Secondly, in terms of robustness, we conducted a parametric sensitivity analysis. As shown in the sensitivity verification table in Section 7.2.4, the algorithm’s performance is insensitive to variations in key parameters such as crossover rate and mutation rate, and the quality of the obtained solutions remains stable when these parameters are adjusted within reasonable ranges. Furthermore, the stability of the algorithm was verified through multiple independent runs, with average performance metrics reported in Section 7.2.2 (2). The convergence analysis (Figure 10) further demonstrates that the algorithm can efficiently converge to a stable solution within a limited number of iterations.

7.2.2. Convergence Analysis of the Lower-Level Algorithm

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Numerical Convergence Verification

Figure 10 shows the convergence curve of the generalized travel cost solved by the lower-level GA after the upper-level SA perturbs the traffic flow guidance fee at the 60th temperature decrease (the global convergence optimal stage). The maximum number of iterations is set to 80, and the GA converges to the lower-level optimal solution under the near-optimal traffic flow guidance fee after approximately 11 iterations.

(2)

Algorithm running time

The average time for the GA to converge to the lower-level optimal solution after about 11 iterations for the input traffic flow guidance fee is 2.3 s; the average total time for the SA to complete one perturbation of the traffic flow guidance fee and synchronously finish the whole lower-level GA solution is 2.4 s; when the SA achieves global convergence after 60 iterations with a chain length of 18 for each temperature decrease, the overall average running time of the lower-level algorithm is about 2592 s. With an upper-level convergence threshold of 1 × 10⁻³ and an average of 8 iterations, the total time of the bi-level programming is about 5.76 h.

7.2.3. Quantitative Analysis of Computational Complexity

Under the test scenario with 9 nodes, 13 links, and 2 O-D pairs, the time complexity of each module is analyzed as follows:

(1) SA component: With an initial temperature of 120, a cooling coefficient of 0.9, and termination temperature of 0.22, the number of temperature decay steps k ≈ 60, and 18 iterations within a single temperature step, the total operation complexity of SA is O (k × 18) = O (1080).

(2) GA component: With a population size of 60 and 80 iterations, a single GA run requires fitness evaluation for 60 individuals (including BPR function and SOC constraint verification), yielding a complexity of O (60 × 80) = O (4800). The total number of GA calls within a single bi-level iteration is 60 × 10 = 600, corresponding to a complexity of O (1080 × 4800) = O (5.18 × 10⁶).

(3) Bi-level iteration component: With an upper-level convergence threshold of 1 × 10⁻³, the average number of iterations is 8, and the total complexity of the overall algorithm is O (8 × 5.18 × 10⁶) = O (4.14 × 10⁷).

7.2.4. Parameter Sensitivity Verification

The parameter sensitivity verification table is shown below. As can be seen from

Table 7. Algorithm parameter sensitivity analysis.

Parameter	Test Range	Impact on Solution Quality	Value
Population size	40, 60, 80	<50: degraded solution quality	60
Number of iterations	20, 80, 140	=80: fully converged	80
Crossover rate	0.6, 0.7, 0.8	Little effect	0.8
Mutation rate	0.01, 0.05, 0.1	Little effect	0.01
Initial temperature	80, 100, 120	<100: easily trapped in local optimum	120
Cooling factor	0.88, 0.89, 0.9	Little effect	0.9
Termination temperature	0.2, 0.21, 0.22	Little effect	0.22
Chain length	10, 14, 18	Little effect	18
Upper-level convergence threshold	1 × 10−4, 1 × 10−3, 5 × 10−3	Little effect	1 × 10−3

, the number of iterations and initial temperature are the core sensitive parameters of the algorithm, while the others are secondary sensitive parameters:

7.3. Sensitivity Analysis

In the bi-level model, battery capacity and unit energy consumption determine the charging demand, which in turn affects the planning and capacity determination of upper-level facilities. Meanwhile, different traffic flows alter travel time and the distribution of charging demand and also exert a significant impact on the effectiveness of DWCS planning. For this reason, this section conducts a sensitivity analysis to quantify the impact of changes in the above factors on the optimization results, providing an adjustable planning basis for DWCS operators.

7.3.1. The Impact of Battery Capacity Changes on DWCS and Operational Benefits

As the battery capacity increases from 45 kWh to 50 kWh (with other parameters unchanged), Figure 11 shows the planning under the 50 kWh capacity, where the number and locations of inverters and DWPT have changed significantly. In Figure 12, compared with 45 kWh, the construction cost and maintenance cost of the DWCS operator under the 50 kWh capacity have decreased by 2462.55 thousand USD and 857.09 thousand USD respectively, and the annual net profit has increased by 624.45 thousand USD.

With each 1 kWh increase in battery capacity, the mean state of charge rises from 30 kWh to 30.5 kWh, thereby shifting the distribution of charging thresholds for marginal vehicles. Taking O-D pair 1–8 as an example, the charging threshold for its shortest path (16 km) is 16 kWh—calculated as the minimum energy threshold (12 kWh) plus the route length multiplied by the energy consumption rate (0.25 kWh/km). Vehicles with initial energy in the [15, 16) kWh interval, therefore, require mid-journey charging. Each 1 kWh capacity increment reduces the expected number of such vehicles by approximately 3.23%; when capacity increases to 50 kWh (raising the mean to 32.5 kWh), the cumulative reduction reaches about 14.29%. Similarly, for O-D pair 1–9, the expected number of vehicles in the [15, 18) kWh interval declines by the same proportion. The reduction in marginal vehicles directly lowers charging demand density on certain road segments. This adjustment is propagated from the lower-level traffic assignment to the upper-level planning layer through the bi-level framework, ultimately triggering a reconfiguration of the charging network planning.

Further analysis reveals that the impact of capacity expansion on operator profitability is nonlinear. As capacity increases incrementally from 45 kWh to 50 kWh, the annual profit growth rates decline sequentially: 14.5%, 6.2%, 3.79%, 2.43%, and 1.4%. This diminishing trend arises from two counteracting effects: on one hand, higher capacity reduces the need for charging infrastructure, lowering planning costs; on the other hand, reduced charging demand along routes decreases revenue from charging services. At lower capacity levels, cost savings dominate; however, as capacity continues to rise, marginal cost savings diminish while charging revenue keeps falling in line with demand. The interplay of these two forces leads to a gradual slowdown in profit growth.

It should be noted that while increasing battery capacity can reduce dependence on DWCS, the manufacturing cost of large-capacity batteries remains relatively high, potentially compromising vehicle economics. Therefore, future research must balance the construction and maintenance costs of charging facilities against vehicle costs to achieve overall benefit optimization.

7.3.2. The Impact of Energy Consumption and Battery Capacity on the Construction and Maintenance Cost

Figure 13 shows the significant impact of energy consumption and battery capacity on the total construction and maintenance costs of DWCS. Taking a battery capacity of 45 kWh as an example, as EV energy consumption decreases from 0.25 kWh/km to 0.20 kWh/km, charging demand declines accordingly. This reduced reliance on DWCS leads to a corresponding drop in total construction and maintenance costs.

When the battery capacity is 45 kWh, and the energy consumption rate is 0.20 kWh/km, the total construction and maintenance cost of the DWCS is substantially lower than in the scenario with a 44 kWh battery and a 0.25 kWh/km consumption rate. This finding suggests that, to maintain a given driving range, higher energy consumption necessitates a larger battery capacity. Energy consumption and battery capacity are closely interrelated, and this relationship directly influences the overall construction and maintenance costs.

7.3.3. Effectiveness of Traffic Flow Guidance Mechanism Under Varying Traffic Flows

To evaluate the planning effectiveness and control capability of introducing a traffic flow guidance mechanism under varying traffic demand conditions, this paper, based on the baseline flow distribution shown in Table 3, further establishes two scenarios: 80% of the baseline flow (low flow) and 120% of the baseline flow (high flow). While keeping the DWCS planning scheme consistent with the corresponding traffic flow scenario, a horizontal comparison is conducted between two modes: “with traffic guidance mechanism” and “without traffic guidance mechanism.”

Figure 14 presents the planning results for the six comparative scenarios. In the low-flow scenarios shown in Figure 14a,d, the road network is equipped with a total of two inverters, located at nodes 1 and 7 respectively; a total of four DWPTs are deployed, on links 1, 2, 11, and 13 respectively, with the total deployed length accounting for 47.61% of the total length of the road network. In the baseline flow scenarios shown in Figure 14b,e, the road network is equipped with a total of three inverters, located at nodes 1, 5, and 7 respectively; a total of six DWPTs are deployed, on links 1, 2, 4, 7, 11, and 13 respectively, with the total deployed length accounting for 52.38% of the total length of the road network. In the scenarios shown in Figure 14c,f, the road network is equipped with a total of three inverters, located at nodes 1, 3, and 7 respectively; a total of seven DWPTs are deployed, on links 1, 4, 5, 7, 8, 11, and 13 respectively, with the total deployed length accounting for 55.95% of the total length of the road network. This indicates that as traffic flow increases, the planning scale of DWCS (number of inverters, number of DWPTs, and total length) expands accordingly to meet higher charging demand.

As shown in

Table 8. DWCS planning cost results under different traffic flows.

Parameter	Low Flow with Traffic Guidance	Low Flow Without Traffic Guidance	Baseline Flow with Traffic Guidance	Baseline Flow Without Traffic Guidance	High Flow with Traffic Guidance	High Flow Without Traffic Guidance
DWCS operators
DWPT Length (km)	40	40	44	44	47	47
Construction Cost (‘000 USD)	7561.86	7561.86	8329.34	8329.34	8894.36	8894.36
Maintenance Cost (‘000 USD)	2637.2	2637.2	2900.92	2900.92	3098.71	3098.71
Transmission Loss Cost (‘000 USD)	1549.95	1446.54	2035.97	2011.42	2848.59	2598.75
Annual Net Profit (‘000 USD)	−1496.72	−3230.63	1966.78	−1750.90	5437.87	578.93
Charging Net Profit (‘000 USD)	9198.52	8414.97	12,598.58	11,490.78	16,445.14	15,170.75
EV owners
Traffic Flow Guidance Fees (‘000 USD)	1053.77	\	2634.43	\	3834.39	\
Vehicle Owners’ Charging Cost (‘000 USD)	24,698.03	22,880.32	32,958.28	31,605.01	44,931.03	41,158.25
Travel Time Cost (‘000 USD)	9885.13	14,053.85	13,538.73	19,959.91	16,796.25	29,690.75
Owners’ Travel Cost (‘000 USD)	35,636.93	36,934.17	49,131.44	51,564.92	65,561.67	70,849.00

, after the introduction of the traffic flow guidance mechanism, the user travel time costs under the low, baseline, and high flow scenarios decreased by 29.7%, 32.2%, and 43.4%, respectively. This trend indicates that the regulatory capability of the traffic guidance mechanism may strengthen with increasing traffic flow, with its effect on alleviating congestion and saving travel time being relatively more prominent under the high-flow scenario.

Based on this, combined with the comparison results of the 24 h cumulative travel time curves for a typical day under the six scenarios (Figure 15), it can be observed that in the 80% low-flow scenario, the curves corresponding to with and without traffic guidance fee are relatively close overall, with relatively small differences in travel time across different periods. Under low-flow conditions, the traffic guidance mechanism has a relatively limited effect on travel choices. As the flow increases to the baseline level, the two curves show a certain degree of separation during the morning and evening peak hours (6:00–10:00, 15:00–20:00), with the cumulative travel time in the scenario without a traffic guidance fee increasing. The impact of the presence or absence of the traffic guidance fee on travel time begins to gradually emerge. In the 120% high-flow scenario, the degree of separation between the curves further widens, and the differences in travel time during the morning and evening peak hours become more pronounced. This reflects that under high-flow conditions, the absence of a traffic guidance mechanism may exacerbate congestion, while the effectiveness of the traffic guidance mechanism in reducing travel time is better demonstrated in this scenario.

To further quantify the differences caused by the presence or absence of traffic guidance fees and variations in traffic flow from a statistical perspective, we analyze the travel time distributions under different flow scenarios using box plots (Figure 16). In the 80% low-flow scenario, the box plot is relatively compact, with a median of approximately 22.8 h, and the overall distribution is close to the zero line. This indicates that the traffic system still maintains a certain redundant capacity under low-flow conditions. The influence of traffic guidance fees on travel choices is relatively limited, resulting in a mild overall impact on the total travel time.

When the traffic flow increases to the baseline level, the height of the box increases noticeably, with the median rising to approximately 67.9 h and the upper whisker increasing sharply to about 140.8 h. This demonstrates that under the baseline flow, the traffic system gradually approaches saturation. The absence of price signals from traffic guidance fees may exacerbate congestion in certain periods, thereby significantly enhancing the impact of traffic guidance fees on travel time.

In the 120% high-flow scenario, both the box height and whisker range are relatively large, with the median rising to approximately 91.2 h and the maximum individual difference reaching about 177.4 h. The overall distribution of the boxplot is clearly far from the zero line. This indicates to some extent that the traffic system tends to become saturated under high flow. The absence of traffic guidance fees may lead to a significant increase in travel time, and the role of traffic guidance fees in alleviating congestion and saving travel time is more fully demonstrated in this scenario.

Table 8 also demonstrates the impact of the traffic flow guidance mechanism on system economic efficiency under different traffic flow scenarios. Under low traffic flow, the operator’s loss narrows from 3230.63 thousand USD to 1496.72 thousand USD after introducing tolls. Under the baseline traffic flow scenario, the system turns into a profit, with a net profit of 1966.78 thousand USD. Under high traffic flow, the operator’s net profit surges from 578.93 thousand USD to 5437.87 thousand USD. As traffic volume increases, the role of tolls in improving operator revenue becomes increasingly prominent. In addition, toll revenue also rises significantly with increasing traffic volume, from 1053.77 thousand USD to 3834.39 thousand USD, reflecting that the regulatory intensity of tolls strengthens with the growth of traffic flow.

7.4. Comprehensive Discussion

(1)

Discussion on More Complex Topologies

At the model structure level, the upper-level planning model aims to maximize the annual net profit of the DWCS operator. Its decision variables include the planning locations of DWPT and inverters, as well as the traffic flow guidance fee, all of which are defined on links and nodes. When the road network scale expands, only the number of decision variables corresponding to links and nodes needs to be increased, while the model structure itself remains unchanged.

At the computational level, to address the curse of dimensionality in large-scale road networks, the framework proposed in this paper can be adapted through parallel computing and distributed optimization architectures. The lower-level user equilibrium model involves flow assignment for multiple O-D pairs. Since the calculations for different O-D pairs are relatively independent, the problem can be decomposed into several subtasks for parallel solution and then converge iteratively via a boundary coordination mechanism. In the upper-level planning model, DWPT planning decisions and traffic flow guidance fees on different links can also be optimized in a partitioned manner. Under the premise of protecting local data privacy, regional operators only exchange boundary flows and coordination parameters to achieve distributed collaboration.

(2)

Discussion on More Diverse Travel Characteristics

At the travel chain level, the generalized travel cost function uniformly quantifies time cost, charging cost, and traffic flow guidance fee. Different travel types exist in real urban road networks, and the cost sensitivity of various users can be distinguished by introducing weight coefficients according to travel purposes. For instance, rigid commuting, rigid return trips, flexible leisure trips, and business trips have different weights for each cost component. At the temporal–spatial response level, time-dependent traffic flow guidance mechanisms can be adjusted to match travel characteristics during morning and evening peaks: for example, tolls can be appropriately raised to alleviate congestion when traffic is concentrated in the morning peak and lowered to reduce user expenses when traffic is dispersed in the evening peak. At the user heterogeneity level, the SOC constraints in the paper determine the battery level of vehicles at each node. In future work, more realistic distribution rules can be assigned to the initial SOC according to travel purposes (e.g., fully charged for commuting vehicles, large fluctuations for leisure vehicles). Furthermore, methods can be introduced to characterize users’ different sensitivities to traffic flow guidance fee and charging fee, simulating the differentiated choice behaviors where some users prefer detours for lower costs while others prioritize the shortest travel time.

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Discussion on the Fairness and Social Acceptance of Traffic Flow Guidance Mechanism

One of the core challenges in implementing a traffic flow guidance mechanism in practice is the issue of fairness. When users feel that they are “paying more than they gain,” questions of fairness may arise, making it difficult for the policy to gain public acceptance. This issue is particularly pronounced among different income groups, low-income groups are far more sensitive to charges than high-income groups and without complementary compensation mechanisms, it may exacerbate social inequality. Therefore, how to balance congestion management while considering the fairness of the burden on different groups is a critical issue that must be carefully addressed in policy implementation.

The implementation of the traffic flow guidance mechanism in practice also faces the challenge of user acceptance. Congestion charging policies are sometimes perceived by the public as an additional burden, which may trigger a certain degree of psychological resistance. Improving user acceptance depends on making people feel that they “have choices and receive returns” rather than simply being passive payers. Enhancing users’ understanding and recognition of the policy is an important aspect of improving its implementation effectiveness.

To address the above challenges, future research could draw on a charging and subsidy mechanism: by subsidizing alternative routes, users who choose non-congested sections can gain tangible benefits. This not only alleviates the financial pressure on low-income groups but also changes the public’s perception of the policy, thereby seeking a more balanced implementation path between efficiency and fairness.

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Discussion on Traffic Flow Guidance Mechanism and Existing Congestion Pricing

Currently, there are two main types of congestion charging schemes: one is per-entry charging, where vehicles pay a fixed fee each time they enter a charging zone, such as Singapore’s Electronic Road Pricing system; the other is daily charging, where vehicles pay only once a day for traveling within the charging zone, such as London’s congestion charge policy. In fact, both of these single charging models have the issue of undercharging long-distance vehicles and overcharging short-distance vehicles. The time-based traffic flow guidance mechanism proposed in this article can complement existing schemes: within per-entry charging zones, dynamic traffic flow guidance mechanism for specific sections can be introduced as a supplement to guide the reasonable distribution of vehicles within the zone; under the daily charging model, traffic guidance mechanism can be superimposed as a secondary adjustment mechanism for vehicles continuously traveling on congested sections.

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Discussion on the Regulatory Feasibility of the Traffic Flow Guidance Mechanism

The feasibility of traffic flow guidance mechanism policies lies in constructing a stable institutional framework that balances commercial interests and public interests. Private operators aim to maximize profits, and their willingness to participate directly depends on whether traffic flow guidance fee revenues can cover long-term operational costs such as road maintenance and repair, while also providing reasonable returns on investment. Therefore, the focus of policy design is to establish a clear and predictable regulatory system: in terms of charging, set reasonable upper limits and dynamic adjustment mechanisms to ensure operators’ profit margins while preventing excessive charges from harming public interests; in terms of repair standards, define the minimum requirements for road surface service quality to ensure that private operation does not compromise facility quality for profit; in terms of operation periods, stipulate stable concession periods to provide certainty for long-term private capital investment. Only with such institutional guarantees can the private operation model achieve both economic benefits and social benefits, thereby becoming truly feasible and sustainable.

8. Conclusions

8.1. Main Conclusions

This study proposes a bi-level collaborative optimization method for DWCS that considers traffic flow distribution. Through setting up multiple scenarios for simulation and comparative analysis, the following conclusions are drawn:

(1)

The construction of a dynamic collaborative coupling mechanism enables the deep integration of DWCS planning with the transportation system. This study employs a bi-level model to clarify the dynamic interactions among the three key elements: path guidance reshapes traffic flow distribution to alleviate congestion, which in turn determines the spatial distribution of charging demand, while charging demand feeds back into planning benefits. This framework offers a novel logical approach to balancing operator profitability with owner travel costs.

(2)

Compared with the full-coverage scenario of DWCS, the bi-level optimization scenario considering owners’ optimal travel can narrow the loss, with the loss reduced by 5387.71 thousand USD; on this basis, introducing traffic flow guidance fees to regulate traffic flow can turn losses into profits, with the comprehensive net profit significantly increased by 2742.13 thousand USD. Owners’ travel time cost and overall travel cost are reduced by 6740.58 thousand USD and 2122 thousand USD, respectively, realizing the collaborative optimization of operators’ benefits, owners’ travel time cost and overall travel cost.

(3)

The correlation between technical parameters and optimization schemes provides a quantitative basis for DWCS planning. Sensitivity analysis indicates that increasing battery capacity and reducing energy consumption can indirectly optimize facility planning costs by lowering charging demand; notably, these two factors exhibit a synergistic effect. Specifically, reduced energy consumption decreases the battery capacity required for a given driving range. This insight offers decision support for operators to dynamically adjust planning strategies in response to technological evolution. Moreover, the regulatory effect of the traffic flow guidance mechanism is shown to strengthen with increasing traffic flow.

8.2. Future Work

This study aims to propose and validate the effectiveness of the model. However, it is necessary to point out this work’s limitations in order to identify promising avenues for future work.

(1)

The validation cases adopted in this study are based on small-scale stylized transportation networks with only two sets of O-D pairs. Although this setting facilitates a clear interpretation of the model mechanism, it cannot fully reflect the dynamic interaction characteristics of traffic flow under multiple O-D pairs and complex topological structures in large-scale urban road networks. Therefore, in future research, the model can be extended to large-scale road networks with real topological structures, incorporating the complex attributes of actual roads. Meanwhile, efficient algorithms such as parallel computing and distributed optimization can be developed to address computational challenges, so as to systematically reveal the deep influence of network structure on the planning pattern of DWCS.

(2)

This paper assumes that traffic demand in each period is deterministic, without fully considering the randomness and uncertainty of travel demand. Meanwhile, key parameters such as the value of time are set as fixed values, which makes it difficult to reflect the heterogeneous characteristics of travelers. Therefore, future research can introduce stochastic programming or robust optimization methods to characterize the uncertainty of traffic demand and integrate multi-source data to further analyze user heterogeneity, so as to improve the practical guiding significance of the optimization results.