Research on the Collaborative Optimization of the Power Distribution Network and Traffic Network Based on Dynamic Traffic Allocation

Abstract

With the increasing penetration rate of electric vehicles, the spatiotemporal coupling relationship between the power distribution network and traffic network is stronger than ever before. Under the dynamic wireless charging mode, traffic jam charging is introduced and the dynamic loading process of traffic flow is described using a cellular transmission model. The charging load is related to traffic flow and serves as a bond between the power distribution network and traffic network. The traffic flow achieves balanced allocation under dynamic user equilibrium conditions, and cooperatively optimizes the power flow of the power distribution network in conjunction with charging loads. Numerical analysis shows that this model can accurately depict the congestion situation during peak travel periods, and alleviate traffic congestion and distribution network voltage out of range.

1. Introduction

With the progress and maturity of the electric vehicle industry, issues such as short driving distance, limited battery capacity, and long charging time have been solved through high-density charge storage technology and fast charging technology. However, the issues of mileage anxiety and inconvenient charging still exist [1,2]. The dynamic wireless charging technology of electric vehicles achieves the transfer of electrical energy through non-electrical contact, which can effectively alleviate mileage anxiety and reduce battery volume, providing the possibility of unlimited driving range and zero charging parking. The promotion of this technology requires the support of dynamic traffic allocation theory.

For the collaborative optimization problem of transportation and distribution networks, efficient embedded charging facilities [3], setting road congestion charges [4], and setting fast charging prices [5] can be used on the traffic network side to guide users in their travel paths and charging choices. On the distribution network side, dispatching distributed generation output [6] and dispatching reactive compensation equipment [7] can be used to strengthen the control ability of power flow [8], alleviate voltage out of range, and reduce energy imbalance [9]. The real-time jam determines the spatiotemporal distribution of electric vehicles and affects the power flow of the distribution network through charging stations. Fluctuations in charging prices can also alter the charging decisions and driving paths of electric vehicles. Noon and evening are peak periods for people’s transportation and electricity consumption, during which traffic jams and voltage out of range can occur [10].

Traffic flow assignment can be divided into static traffic assignment (STA), semi-dynamic traffic assignment (SDTA), and dynamic traffic assignment (DTA) [11,12,13]. STA and SDTA are often used to allocate traffic loads in the traffic planning process, but traffic flow changes in real-time during traffic jams. Scholars prefer using DTA to accurately depict users’ dynamic changes in traffic flow [14,15,16]. The DTA problem is composed of two basic parts: dynamic network loading (DNL) and travel selection criteria [17,18,19]. The DNL process uses the Cell Transmission Model (CTM) to further refine the road impedance estimation process, which is more in line with the actual traffic flow changes. Currently, few studies have applied CTM to the collaborative optimization of traffic networks and distribution networks. In recent years, the paper [20] simultaneously considered the grid economic operation and traffic flow assignment. The paper [21] took hydrogen energy into account when considering the power distribution network and traffic network. The paper [22] proposed a differentiated pricing scheme to regulate traffic flow.

Based on the above considerations, a traffic network-distribution network collaborative optimization model under dynamic wireless charging is first established. The dynamic traffic flow assignment process is depicted with CTM. At the same time, independent operators are introduced to charge, which is based on nodal voltage and road jam. Finally, the collaborative optimization model is validated by simulation.

2. Collaborative Optimization Framework

The framework of traffic network-power distribution network collaborative optimization is shown in Figure 1. Within this framework, independent operators, which act as a coupling unit between the transportation network and power distribution network, are designed to alleviate traffic jams by levying the lowest possible traffic jam charges and charging electricity prices.

On the premise of mature dynamic wireless charging technology, wireless charging vehicles in the transportation network can charge when they pass along the road with charging coils. The adjacent dynamic wireless charging section can be considered as a dynamic wireless cluster charging station. The dynamic charging behavior of electric vehicles is merged into the corresponding grid nodes through the charging station. The charging price and jam charge can be used as a means of regulating the coordinated optimal operation of the two networks.

At the same time, attention should be paid to the time overlap between the traffic network and power distribution network during the noon and evening rush hours. In this case, the dynamic wireless charging car in the traffic network will affect the distribution network through the wireless charging section when congestion occurs. If no action is taken, the node voltage may be out of bounds.

3. Collaborative Optimization Model

This chapter establishes a traffic network-power distribution network collaborative optimization model based on dynamic traffic distribution.

3.1. Dynamic Traffic Assignment

3.1.1. DNL Model

The key problem in dynamic traffic assignment is calculating the route travel time through the road impedance function. Since Professor Daganzo of the University of California proposed the CTM model, many researchers have considered it as the basis for considering the dynamic traffic assignment problem. The CTM model is a discretized approximation of the Lighthill-Whitham-Richards (LWR) model for road traffic. Traffic flows satisfy the following continuous equations:

$$\frac{\partial q}{\partial x}+\frac{\partial p}{\partial t}=0$$

(1)

$$q=p{u}_e(p)$$

(2)

where $q$ is the traffic flow; $p$ is the traffic flow density; $x$ is the spatial position; and $t$ is time. $u_e(p)$ denotes the function relationship between traffic flow speed and density under balanced conditions.

Consider a road as one or more cells which can quickly load the dynamic traffic of network nodes in medium-sized traffic networks. Section boundaries are shown in Figure 2. In a transportation network, a one-way path starts at $x_{a0}$ and ends at $x_{aL}$, consisting of multiple one-way roads. The length of the road $L_a$ is not fixed, and traffic flows are dynamically loaded on the starting cells, then spread to other cells, and eventually leave the network from emission cells.

In order to calculate the traffic volume, the cumulative vehicle time function is introduced and there is the continuous numbering of vehicles as they pass $x_{a0}$. $N_{a0}(x,t)$ is the number of the last vehicle recorded at the time $t$. When the vehicle leaves this road, traffic flow capacity and travel time are derived from the cumulative number of vehicles.

As shown in Figure 3, the travel time ${\tau }_a$ in the road $a$ can be expressed by the horizontal distance of the $x$-axis between the two curves $N(x_{a0},t)$ and $N(x_{aL},t)$ with height $h$, and the vertical distance is the traffic flow at time $t_1$. In this way, the traffic flow meets the first-in and first-off criteria. $N_r(x_{a0},t)$ represents the cumulative number of vehicles on the path $r$ at the time $t$, and the travel time ${\tau }_r$ of the path $r$ can also be represented by the horizontal distance of the $x$-axis between the two curves $N_r(x_{a0},t)$ and $N_r(x_{aL},t)$. Considering that the nodes do not have the actual physical length, the time spent on the nodes is approximately zero. The travel time of the path depends only on the travel time of the roads.

When traffic demand is equally allocated to a path, the total traffic flow in the cell is the sum of the traffic flow through the path. The traffic flow at the location $x$ in the network can be expressed as:

$$N(x,t)={\displaystyle \sum _{r\in R}{N}_r(x,t)}$$

(3)

Here, $N_r(x,t)$ denotes the cumulative traffic flow through $x$ on path $r$.

The total flow conservation constraint can be expressed as:

$$x_{i,t}=d_{i,t-1}+x_{i,t-1}-y_{ij,t} \forall i\in C_R,j\in {\Gamma }_i^{},t=1,\cdots ,T$$

(4)

$$x_{i,t}=x_{i,t-1}+y_{ki,t-1} \forall i\in C_S,k\in {\Gamma }_i^{-1},t=1,\cdots ,T$$

(5)

where $x_{i,t}$ is the traffic flow of cell $i$ at time $t$; $d_{i,t-1}$ is the traffic flow of cell i at time $t-1$; ${\Gamma }_i^{}$ is the downstream cell collection of cell $i$; ${\Gamma }_i^{-1}$ is the upstream cell collection of cell $i$; $y_{ij,t}$ is the traffic flow of cell $i$ to cell $j$ at time $t$; and C_R and C_S are the source cell collection and elimination cell collection, respectively.

The conservation of traffic for ordinary cells can be expressed as:

$$x_{i,t}-x_{i,t-1}-{\displaystyle \sum _{k\in {\Gamma }_i^{-1}}{y}_{ki,t-1}}+{\displaystyle \sum _{j\in {\Gamma }_i^{}}{y}_{ij,t-1}=0} \forall i\in C\backslash \{C_R{C}_S\},t=1,\cdots ,T$$

(6)

where $C$ denotes the set of all cells.

The link flow conservation constraints for ordinary cells can be expressed as:

$$y_{ij,t}-x_{i,t}\le 0,y_{ij,t}\le Q_{i,t},y_{ij,t}\le Q_{j,t}$$

(7)

$$y_{ij,t}\le \delta (N_{j,t}-x_{j,t}) \forall (i,j)\in E_o,t=1,\cdots ,T$$

(8)

where is the maximum outflow rate of the cell $i$; $N_{j,t}$ is the blocking density of the cell $j$; $E_o$ denotes the set of ordinary cell links.

Link flow conservation constraints for shunt cells can be expressed as:

$${\displaystyle \sum _{j\in {\Gamma }_i}{y}_{ij,t}}-x_{i,t}\le 0,{\displaystyle \sum _{j\in {\Gamma }_i}{y}_{ij,t}}\le Q_{i,t},y_{ij,t}\le Q_{j,t}$$

(9)

$$y_{ij,t}\le \delta (N_{j,t}-x_{j,t}) \forall (i,j)\in E_D,t=1,\cdots ,T$$

(10)

where $E_D$ denotes the set of shunt cell links.

The link flow conservation constraints for confluent cells can be expressed as:

$$y_{ki,t}-x_{k,t}\le 0,y_{ki,t}\le Q_{k,t},{\displaystyle \sum _{k\in {\Gamma }_i^{-1}}{y}_{ki,t}}\le Q_{i,t}$$

(11)

$${\displaystyle \sum _{k\in {\Gamma }_i^{-1}}{y}_{ki,t}}\le \delta (N_{i,t}-x_{j,t}) \forall (i,j)\in E_M,t=1,\cdots ,T$$

(12)

where $E_M$ denotes the set of confluent cell links.

At the start, the traffic in the cell is set to zero. To initialize cells, the general cell link update equation can be expressed as:

$$y_{ij,t}^r={\delta }_i^r \cdot \left[\min ({\overline{x}}_{i,t},Q^i,Q^j,\gamma (N_j-{\overline{x}}_{j,t})\cdot \frac{x_{i,t}^r}{{\overline{x}}_{i,t}+\mu })\right] \forall (i,j)\in {\mathrm{E}}_0, j\in {\Gamma }_i^{},t=1,\cdots ,T$$

(13)

Here, ${\overline{x}}_{i,t}$ denotes the sum traffic flow of cell $i$ at time $t$, ${\overline{x}}_{i,t}={\displaystyle \sum _r{x}_{i,t}^r}$; $\gamma$ denotes the ratio of reverse velocity to free velocity, and $\mu$ is an infinitesimal positive real number that does not have a denominator of 0.

The updated equation for the merge link is:

$$y_{ki,t}^r={\delta }_{ki}^r \cdot \left[\min (Q^k,{\overline{x}}_{i,t})\cdot \min \left(1,\chi \right)\right]\cdot \frac{x_{k,t}^r}{{\overline{x}}_{k,t}+\mu }$$

(14)

$$\chi =\frac{\min (Q_i,\gamma (N_i-{\overline{x}}_{i,t}))}{{\displaystyle {\sum }_{k^{\prime }\in {\Gamma }_i^{-1}}(\min (Q_{k^{\prime }},{\overline{x}}_{k^{\prime },t}))}}\forall i\in C_M, t=1,\cdots ,T$$

(15)

where ${\overline{x}}_{k^{\prime },t}$ is the traffic flow of the cell $k$ in the next stage, $Q_{k^{\prime }}$ is the maximum outflow rate of the cell $k$ in the next stage.

The updated equation for the diversion link is:

$$\begin{array}{l}{y}_{ij,t}^r={\delta }_{ij}^r \cdot \left[\min (Q_j,{\tilde{x}}_{ij,t},\gamma (N_j-{\overline{x}}_{j,t}))\cdot \min (1,\frac{Q_i}{{\displaystyle {\sum }_{k^{\prime }\in {\Gamma }_t^{}}(\min ({\tilde{x}}_{i{j}^{\prime },t},Q_{j^{\prime }},\gamma (N_{j^{\prime }}-{\overline{x}}_{j^{\prime },t})))+\mu }})\right]\\ \cdot \frac{x_{i,t}^r}{{\tilde{x}}_{ij,t}+\mu }\forall i\in C_D, \mathrm{j}\in {\Gamma }_i^{},t=1,\cdots ,T\end{array}$$

(16)

where $Q_{j^{\prime },t}$ is the jam density of the cell $j$ in the next stage, ${\overline{x}}_{j^{\prime },t}$ is the traffic flow of the cell $j$ in the next stage.

The average travel time ${\eta }_{r,t}^w$ of all electric vehicles departing on the path $r$ can be expressed as:

$${\eta }_t^{r,w}=\frac{{\displaystyle {\int }_{N_{t-1}^{r,o}}^{N_t^{r,o}}\left[N_v^{r,d^{-1}}-N_v^{r,o^{-1}}\right]dv}}{N_t^{r,o}-N_{t-1}^{r,o}}$$

(17)

where $N_t^{r,o}$ and $N_t^{r,s}$ are the cumulative departure flow and cumulative arrival flow, respectively, $v$ is the free flow speed. ${\eta }_t^{r,w}$ can be calculated from the inverse function of the cumulative traffic flow curve.

3.1.2. User Travel Criterions

Under the condition of following dynamic user optimization, traffic congestion should be alleviated and travel comfort for traffic users should be improved. As a travel criterion, the dynamic user equilibrium can be expressed as:

$$f_t^{r,w}({\eta }_t^{r,w}-{\pi }_t^w)=0\hspace{1em}\forall \omega \in W,r\in R^w,t\in T$$

(18)

$${\eta }_t^{r,w}-{\pi }_t^w\ge 0\hspace{1em}\forall \omega \in W,r\in R^w,t\in T$$

(19)

where $f_t^{r,w}$ is the traffic flow on path $r$ and ${\pi }_t^w$ is the smallest traffic flow among all paths in the OD pairs. $W$ denotes the set of all OD pairs in the traffic network.

The conditions that path traffic needs to meet can be expressed as:

$${\displaystyle \sum _{r\in R^w}{f}_t^{r,w}}=q_t^w\hspace{1em}\forall \omega \in W,t\in T$$

(20)

$$\mathrm{f}\ge 0,\hspace{1em}\mathrm{u}\ge 0$$

(21)

In Formula (20), $q_t^w$ denotes the actual traffic demand of $w$; $\mathrm{f}$ denotes the column vector of the number of vehicles in the path, $\mathrm{f}=\left\{\mathrm{f}|f_t^{r,w}, \forall \omega \in W,r\in R^w,t\in T\right\}$; and $\mathrm{u}$ denotes the column vector of the travel time of the path under equilibrium conditions, $\mathrm{u}=\left\{\mathrm{u}|{\pi }_t^w, \forall \omega \in W,t\in T\right\}$.

While ensuring macro traffic allocation, it is also necessary to accurately depict the micro-dynamic characteristics of traffic flow to provide accurate information for users’ travel. The user optimal variational inequality problem for dynamic path selection can be expressed as:

$$({\mathrm{f}-\mathrm{f}}^{*}{)}^T\cdot {\mathrm{t}}_{\mathrm{n}}{}^{*}\ge 0\hspace{1em}\forall \mathrm{f}\in \Omega$$

(22)

where ${\mathrm{f}}^{*}$ is the optimal path traffic flow; ${\mathrm{t}}_{\mathrm{n}}{}^{*}$ is the optimal solution for the actual travel time; and $\Omega$ is the set of feasible solutions.

Set jam charges on bottleneck roads in the traffic network to maintain normal traffic flow on these roads. Jam charges can be set to:

$${\beta }_t^{r,w}={\tau }^{rate}\cdot \Delta t_p^w$$

(23)

where ${\beta }_{r,t}^w$ is the jam charge for Route $r$; ${\tau }^{rate}$ is the cost ratio related to jam delay time; and $\Delta t_p^w$ is the delay time compared to free passage time.

The joint charging based on the voltage level and road jam level of dynamic wireless charging station cluster nodes can be expressed as:

$${\tau }_t^{r,w}={\theta }_1{\beta }_t^{r,w}+{\psi }^r{\theta }_2{\tau }_{{}^t}^{DWC}$$

(24)

where ${\tau }_{{}^t}^{DWC}$ is the node voltage congestion level; ${\tau }_{r,t}^w$ is the joint fee price; and ${\psi }^r$ is the 0–1 sign position, and when a vehicle passes through a toll road section, it is 1. Otherwise, it is 0. ${\theta }_1/{\theta }_2$ denotes the weights of traffic jam charges and voltage congestion charges, respectively.

Under joint pricing, the travel cost is minimized, and the objective function can be expressed as:

$$Min{F}_{\mathrm{road}}={\displaystyle \sum _t{\displaystyle \sum _r{\displaystyle \sum _w{\tau }_{r,t}^w}}}{\eta }_{r,t}^w$$

(25)

3.2. Distribution Network Model

This article is based on the Distflow branch power flow form and calculates the power flow distribution based on the second-order cone relaxation model.

The constraint conditions of the second-order cone relaxation model in the power distribution network can be expressed as:

$$P_{ij,t}^{}+{\displaystyle \sum _{DG}{P}_{j,t}^G}-I_{ij}{r}_{ij}-{\displaystyle \sum _h{P}_{jh,t}^{}}-P_{j,t}^L=0$$

(26)

$$P_{j,t}^L=P_{s,t,s\in j}^{ev}+P_{j,t}^{Ld}$$

(27)

$$Q_{ij,t}^{}+{\displaystyle \sum _{DG}{Q}_{j,t}^G}-I_{ij}{x}_{ij}-{\displaystyle \sum _h{Q}_{jh,t}^{}}-Q_{j,t}^L=0$$

(28)

$$Q_{j,t}^L=Q_{s,t,s\in j}^{ev}+Q_{j,t}^{Ld}$$

(29)

where $P_{ij,t}^{}$ is the active power of the branch $ij$; $P_{j,t}^G$ is the active power of the distributed power source at node $j$; $P_{jh,t}^{}$ is the active power of branch $jh$ connected to node $j$; $I_{ij,t}$ is the square of the amplitude of the branch current; $r_{ij}$ is the resistance value; $P_{j,t}^L$ is the active power of node $j$; $P_{s,t,s\in j}^{ev}$ and $P_{j,t}^{Ld}$ are the active power of the dynamic wireless charging station of node $j$ and other conventional loads; $Q_{ij,t}^{}$ is the reactive power of the branch $ij$ in the distribution network; $Q_{j,t}^G$ is the reactive power of the distributed power source; $Q_{jh,t}^{}$ is the reactive power of the branch $jh$ connected to node $j$; $x_{ij}$ is the reactance value; $Q_{j,t}^L$ is the reactive power of node $j$; and $Q_{s,t,s\in j}^{ev}$ and $Q_{j,t}^{Ld}$ represent the reactive power of the dynamic wireless charging station set of node $j$ and the reactive power of other conventional loads, respectively.

The goal of the optimal power flow model for distribution networks is to minimize the operating cost of DG.

The cost of purchasing electricity for DG is expressed as:

$$F_G={\displaystyle \sum _t{\displaystyle \sum _g(a_g{(P_{g,t}^G)}^2+b_g(P_{g,t}^G))}}$$

(30)

In formula, $a_g/b_g$ denotes the generation cost coefficients of distributed power sources.

The DG power ramp constraint is expressed as:

$$\begin{array}{l} P^{G,\min }\le P_{g,t}^G\le P^{G,\max }\\ Q^{G,\min }\le Q_{g,t}^G\le Q^{G,\max }\\ -P_g^{G,ramp}\le P_{g,t}^G-Q_{g,t}^G\le P_g^{G,ramp}\end{array}$$

(31)

where $P^{G,\max }$, $P^{G,\min }$, $Q^{G,\max }$, and $Q^{G,\min }$ are the maximum and minimum values of the active and reactive power output of DG, respectively, and $P_g^{G,ramp}$ is the climbing limit value of DG.

Taking the congestion of the nodes’ voltage as one of the pricing considerations, the charging price based on voltage congestion can be defined as:

$${\tau }_{{}^t}^{DWC}={\tau }_{{}^t}^{base}+{\beta }_l{\tau }_{{}^t}^{con}$$

(32)

where ${\tau }_{{}^t}^{base}$ is the basic electricity price, usually 0.6; ${\tau }_{{}^t}^{con}$ is the voltage congestion value of the wireless charging station node; and ${\beta }_l$ is the conversion function between the voltage congestion and charging price, and as the degree of node voltage deviation increases, the congestion price increases exponentially and is superimposed on the base price. When the voltage is high, it is positive, and when the voltage is low, it is negative.

The objective function can be expressed as:

$$\min F_{grid}={\displaystyle \sum _t{\displaystyle \sum _g(a_g{(P_{g,t}^G)}^2+b_g(P_{g,t}^G))}}$$

(33)

3.3. Collaborative Optimization

The traffic network and power distribution network are coupled together by charging electricity prices and charging loads. The traffic flow in the transportation network is linearly correlated with the load power of wireless charging station nodes in the power distribution network achieving interaction, which can be expressed as:

$$P_t^{\mathrm{ev}}={\tau }_1{\displaystyle \sum _{a\in DWC}{x}_{a,t}^w},Q_t^{\mathrm{ev}}={\tau }_2{\displaystyle \sum _{a\in DWC}{x}_{a,t}^w}$$

(34)

where ${\tau }_1$ and ${\tau }_2$ denote the conversion coefficient between traffic flow and charging power, respectively.

Independent operators consider charging for node voltage congestion and road jam. Road jam charges are not applicable to all roads, and setting jam charges on bottleneck roads can effectively alleviate traffic problems. It should be pointed out that jam charging is based on delay time rather than actual travel time. If jam charging is based on actual travel time, it overlaps normal travel time, so the cost is calculated based on delay time.

To solve variational inequalities in dynamic traffic allocation, the projection algorithm requires $t_n^{*}$ to be strictly monotonic. But the impedance function characteristics are complex. Therefore, a modified projection algorithm is used to solve the problem. The algorithm is as follows:

1. Initialization. Set convergence criteria to $\epsilon >0$. Given any initial value ${\mathrm{f}}^0$, set $k=1$.

2. Solve the distribution of road traffic flow at any time period using Equations (4)–(24). Where $\rho$ is a fixed coefficient, the formula can be expressed as:

$$\min \frac{1}{2}{\mathrm{f}}^T\mathrm{f}+[\rho {\mathrm{t}}_n^{k-1}{(\mathrm{f}}^{k-1})-{\mathrm{f}}^{k-1}{]}^T\mathrm{f}$$

(35)

3. Correction, solve ${\mathrm{f}}^j$.

$$\min \frac{1}{2}{\mathrm{f}}^T\mathrm{f}+[\rho {\mathrm{t}}_n^{j-1}{(\mathrm{f}}^{j-1})-{\mathrm{f}}^{j-1}{]}^T\mathrm{f}$$

(36)

4. Check stop conditions. If the iteration stop condition $\Vert {\mathrm{f}}^j-{\mathrm{f}}^{j-1}\Vert \le \epsilon$ is met, the operation is terminated. Otherwise, $\epsilon >0$ and return to step 2 and iterate again.

The solution algorithm for the collaborative optimization model of the traffic network and power distribution network is as follows:

1. Initialization. Set convergence criteria $\epsilon >0$. Set $k=1$. The charging electricity price is given an initial value ${\tau }^{k,0}$, the jam charge is given an initial value, and the traffic flow is given an initial value of 0.

2. Balanced traffic network allocation. Update the charging electricity price to ${\lambda }^k$ and update the jam charge. Obtain $P_{s,t}^{ev,k}$ and $Q_{s,t}^{ev,k}$ by solving the traffic network objective function.

3. Optimal power flow of distribution network. Load the distribution network data and solve the distribution network objective function based on $P_s^{ev,k}$ and $Q_s^{ev,k}$ obtained in step 2 to obtain the distribution network voltage distribution and charging price ${\tau }^{k+1}$.

4. Check stop conditions. If the iteration stop condition is met, the operation is terminated. Otherwise, $k=k+1$ and return to step 2 and iterate again. The iteration stop condition can be expressed as:

$$\frac{{\Vert {\tau }^{k+1}-{\tau }^k\Vert }^2}{{\Vert {\tau }^k\Vert }^2}\le \epsilon$$

(37)

The solution process for the two objective functions of the coupled model is shown in Figure 4.

4. Example Analysis

The traffic network adopts an improved Nguyen Baran&Wu13 nodes network, and the power distribution network adopts an IEEE33 nodes system, as shown in Figure 5. After the transportation network is cellular, the coupling system is shown in Figure 6. Three DGs with the same parameters are installed in the distribution network at nodes 4, 11, and 29, with a capacity of 1.5 p.u and cost coefficient sets of $a_g=200$ and $b_g=1200$, respectively. The upper and lower voltage limits of nodes are 1.05 p.u and 0.95 p.u, respectively. The impedance parameters are referred to in reference [23]. The scheduling cycle is 2 h, with a time interval of 6 min. The system sets 700 vehicles, and the O-D pair allocation and requirements of the vehicles are shown in

Table 1. OD pairs demand for transportation.

OD Pairs	Traffic Demand at Each Time Period (p.u.)
	11:00–11:30	11:30–12:00	12:00–12:30	12:30–13:00
(1,2)	70	43	30	19
(1,3)	69	58	47	40
(4,2)	63	51	43	32
(4,3)	65	53	41	32
(1,2)	70	43	30	19

. The topology structure of the traffic network consists of 13 nodes, 19 roads, and 25 paths. Among them, roads 7, 8, 10, 13, 14, and 18 are set as toll roads, and the entire roads are set as wireless charging roads. Therefore, roads 7, 8, 10, 13, 14, and 18 are set as a combined charging price, while other road sections are only charged using charging prices. The solution will be achieved by using a 3.0 GHz PC with 32 GM RAM.

To verify the rationality of the proposed model, four cases are set up for comparison:

Case 1: The traffic network only introduces the charge price, and the two networks cooperate to optimize under dynamic traffic allocation.

Case 2: Based on the introduction of the charging electricity price, the traffic network uses road jam charging to optimize the two networks under dynamic traffic allocation.

Case 3: The traffic network uses combined toll collection, power distribution network control resources are insufficient, and dynamic traffic allocation under the collaborative optimization of the two networks.

Case 4: The traffic network only introduces the charging power price, and the two networks work together under static traffic allocation.

Figure 7 shows the distribution of total traffic demand based on static and dynamic traffic allocation in Cases 1 and 4. Within 2 h of the scheduling cycle, the traffic demand in Case 4 was roughly depicted, and the real-time changes in traffic demand within half an hour were not described, which cannot meet the demand for precision in traffic flow modeling during jam periods. Furthermore, through the distribution map of traffic flow hotspots at cellular nodes in Case 1 of Figure 8, we can see the changes in traffic flow and the transmission process of a jam at each cellular node. Dynamic traffic allocation can more finely depict the real-time changes of traffic flow during traffic jam periods than static traffic allocation, and node electricity prices are also more sensitive, which is conducive to the collaborative optimization of the two networks.

Figure 9 shows the distribution of traffic flow on the paths under Case 1 and 2. When the dynamic wireless charging of electric vehicles enters the noon peak, the spatiotemporal distribution of charging load changes more significantly. In Case 1, there is no joint charging, and the price of electric vehicle wireless charging only depends on the degree of voltage congestion on the node bus where the wireless charging station is located. When there is congestion on the road during peak travel hours, charging vehicles gather on wireless charging roads and change the voltage of corresponding nodes in the power distribution network through dynamic wireless charging. The congestion of node voltage requires a certain time process, and the charging price cannot provide timely feedback on the traffic flow in the traffic network. Therefore, the phenomenon of traffic jam cannot be alleviated in a timely manner. In Case 1, roads 2, 7, 11, 14, and 15 all experienced a certain degree of congestion at 12:30 pm during the noon rush hour, especially in road path 11 where there were multiple periods of high congestion. In Case 2, the congestion on road 11 was alleviated. Prior to the formation of congestion, some traffic flow was diverted in advance through road congestion charges on roads 18 and 7, and the traffic flow on roads 2, 7, 14, and 15 remained at normal levels. Figure 7 shows the distribution of traffic path flow in Case 1 and 2, further illustrating the changes in path flow after adopting joint pricing. Path 3 (roads traversal numbers 1, 5, 7, 10, 15) takes on the main task of diverting the OD pair (1,2). At 12:30, roads 7 and 10 of the path have set higher jam charging rates, which significantly optimizes traffic flow. Path 25 (roads traversal numbers 3, 6, 12, 14, 16) has set a congestion fee rate on road path 14, but its effect is not as effective as Path 3. It should also be pointed out that the traffic flow at 11:00 and 1:00 is 0 because the loading of the cellular transmission model starts from 0, and after loading, the traffic flow is also 0. However, after loading for a period of time, it can generally describe the basic propagation law of traffic flow. Based on Figure 9 and Figure 10, it can be seen that after adopting joint pricing, higher joint pricing reduces the attractiveness of high load nodes to electric vehicle users, and joint pricing can alleviate a jam.

Figure 11 shows the bus voltage levels of power distribution network nodes at different time periods in Cases 1 and 2. Select the time periods of 11:00, 11:30, 12:00, 12:30, and 13:00 to study the changes in node voltage levels. In Case 1, without the use of joint pricing, there is a double peak superposition phenomenon of traffic and electricity peaks at 12:30, resulting in the voltage of node 18 exceeding the boundary, and the voltage of other line end nodes is also at a lower level. Case 2 evacuates the traffic flow of 18 nodes adjacent to the charging station through congestion charging, reallocates the traffic flow at the time and space levels, and coordinates the dispatching of the control equipment in the distribution network under the condition of balanced traffic flow distribution, eliminating node voltage congestion and avoiding the chain congestion of the traffic network and power distribution network.

The distribution of traffic flow on certain roads under Cases 1, 2, and 3 is shown in Figure 12. Case 3 represents a situation where the capacity of regulatory equipment may be insufficient during peak transportation and electricity load periods. Under Case 3, roads 7, 11, and 12 continued to experience congestion during consecutive periods without any relief. It can be seen that even with the use of joint pricing in Case 3, the traffic flow in the traffic network has not been evenly distributed. When a jam is about to occur in the traffic network, the dynamic wireless charging vehicles on a certain section of the road will continue to stay, leading to a significant increase in the load of nearby dynamic wireless charging stations. At this time, it is also the peak electricity load in the distribution network, and the node voltage is easily reduced or even crossed. The voltage drop of nodes in the power distribution network cannot be alleviated, leading to an increase in charging prices. The corresponding charging station nodes’ charging prices will continue to rise. Even if congestion charges increase the joint charging prices and cause some traffic users to change their travel strategies, the problems of traffic congestion and low node voltage will not be alleviated quickly in the short term. Due to the insufficient capacity of flexible devices, the charging electricity price and charging load, as the coupling link between the two networks, have a hysteresis effect, and the bottleneck road inhibition effect of joint charging costs has been greatly weakened. There is no flexible feedback between the two networks, which may even exacerbate congestion.

Figure 13 shows the charging prices of charging stations in Case 2 and the combined charging prices. During the peak travel period from 12:00 to 12:30, jam is relatively high on roads 7 and 18, and the delay time for electric vehicle users passing through these two roads is higher. Jam fees are also higher compared to other periods. Although the charging prices from 11:00 to 13:00 do not fluctuate significantly, adding jam charges for joint charging can result in higher travel costs during peak hours on roads 7 and 18. Users who originally traveled on these two roads chose to travel on other roads without jam charges, which alleviated the traffic jam bottleneck on these two jams and helped improve user travel comfort.

5. Conclusions

In dynamic wireless charging scenarios, this article establishes a collaborative optimization model to simulate the spatiotemporal distribution of the charging load and power flow of power distribution networks during traffic jams. Numerical simulations demonstrated that: (1) DTA is more accurate in depicting traffic flow and jam changes compared to SDTA. The voltage level of key nodes (such as node 18 in the case study) has also been improved. (2) By introducing wireless charging prices and setting a joint charge for bottleneck road jams, electric vehicles can be more efficiently guided to choose their travel time and path than when using charging prices alone, therefore alleviating traffic congestion. The next research direction is to consider the topology flexibility of the power distribution network; that is, to use dynamic reconstruction or soft switching technology to achieve a spatial dynamic allocation of charging loads.