2.1

LTM road segment model

The road transmission model uses the Cumulative Vehicle Number (CVN) N(x,t) (cumulative number of vehicles passing through a certain observation point) describing the propagation of traffic flow on the road net, where x and t represent the spatial location and time on the road net, respectively. Taking part differentiation of CVN yields the flowing function q(x,t) and dens function p(x,t).

In addition, the second-order partial differential of N(x,t) is equal, i.e

Substituting equation (1) into equation (2) yields the LWR conservation equation

The road section transmission model LTM adopts a fundamental diagram (FD) of triangles, as shown in Figure 1. This basic diagram can be described in terms of free flow velocity (v_f), congestion density (p_jam), and maximum capacity (q_max). Through these three parameters, the critical density (p_crit) and shock wave backward propagation velocity (w) can be calculated. The LTM road segment model is used to calculating the receiving and sending ability of the road segment.

Figure 1.

Triangle basic diagram

The transmission capacity of section a is represented by the time period [t, t+ Δt]. The maximum quantity of vehicles that woll flow downstream of section a, denoted by S_a(t). It is constrained by the upstream entrance traffic flow and downstream exit flow capacity of the section, and the calculation formula is as follows:

In the equation, 、、 L_a、v_a、 represent the upstream position, downstream position, length, flow, and flow capacity of section a.

The reception capacity of section a is represented by the maximum number of vehicles that can flow into the road section at the time period [t, t+ Δt], denoted by R_a(t). It is constrained by the downstream exit traffic flow and the upstream inlet inflow capacity of the section, and the calculation formula is as follows:

In the equation, w_a and q⁰a(t) respectively represent the transitory wave back propagating velocity and upstream inflow capacity of section a.

2.2

LTM node model

In the transportation network G(N, A) represented by graph theory, N is the set of nodes; A is the set of edges; For node n (n ∈N), use A_n to represent the set of road segments entering node n; Use B_n to represent the set of road segments leaving node n. The LTM is used to calculate time period [t, t+ Δt]. Traffic flow from section a to b, G_ab(t), is constrained by the sending capacity S_a(t) of section a and the receiving capacity R_b(t) of section b. It is possible to calculate the diversion S_ab(t) of the sending capacity S_a(t) of section a:

In the formula: S_ab(t) is the flow sent from section a to section b; p is the path; δ_bp is the correlation matrix between road segments and paths δ If segment b is on path p, then δ_bp =1, otherwise δ_bp =0; represents the moment when the vehicle passes through the upstream entrance boundary of section a at point .

R_ab(t) represents the traffic flow allocated from section b to section a, while p_ab(t) represents the priority coefficient for the traffic flow from section a to enter section b, satisfying . Therefore, the diversion R_ab(t) of the receiving capacity R_b(t) of section b can be calculated

Assuming that the propagation of traffic flow on a road segment meets FIFO principle.

2.3

CVN update method

By using node models and link models, the cumulative inflow and outflow of the road segment can be updated

By using node models and link models, the cumulative inflow and outflow of the road segment can be updated

2.4

Calculation of traffic density

Consider any point in the road network space, assuming x is on section a, i.e.. The transmission capacity S(x,t) and reception capacity S(x,t) at this point as follows

According to equations (11), the traffic flow at any position and time on the road section can be calculated:

2.5

M/M/C queuing model

Vehicles arriving at charging piles to receive services are mutually independent, satisfying smoothness, independence, and commonality. Therefore, the Poisson process can be used to describe the law of vehicles arriving at charging piles. This article assumes that:

3.1

Objective Functions

3.2

Restrictive Conditions

3.3

Model Solutions

The two objective functions represent the goals of the two interested parties, the charging pile builder expects to minimise the construction investment cost under the constraints, while the EV users want to have enough charging facilities and the travel time cost to be as small as possible. Therefore, the two objective functions cannot be optimal at the same time. In this paper, the multiple objective particle swarming algorithm is used to solving the above multiple objective planning problem on the Pareto frontier:

Figure 2.

Multi-objective Particle Swarm Algorithm for Pareto Frontier Flowchart

4.1

Basic Data

The circular expressway network is shown in Figure 3, with a total of 5 toll gates (entrances and exits) and a total mileage of 465km.

Figure 3.

Example of Ring Expressway Network Calculation

The average distance between charging piles is 50km, and it is estimated that a total of 20 charging piles will need to be built on the highway network. The penetration rate of vehicles in the planned target year is 10%. Assuming that electric vehicle owners randomly choose highway charging piles for charging, in order to meet the travel needs of vehicles and reduce concerns about electric vehicle battery depletion, this article assumes that on average, highway electric vehicle owners need to charge once every three charging piles, which is the comprehensive driving rate of vehicles at highway charging piles. At this point, the comprehensive driving rate matches the range of vehicles, which can almost 100% meet the charging needs of vehicles.

The simulation node parameters of the high-speed road network in this article are shown in Table I.

Table Ⅰ.

Node parameters of highway network.

The parameters of the highway network section are shown in Table II.

Table Ⅱ.

Highway network section parameters.

4.2

Analysis of results

Using the spatiotemporal distribution results of traffic flow obtained from the above traffic simulation as input, the previous model establishes a multi-objective planning model for charging piles. Taking an external archive size of 100, a population of 50 particles, and a maximum number of iterations of 500, the Pareto frontier of the non inferior solution planning scheme is obtained by solving the model using multi-objective particle swarm optimization algorithm, as shown in Figure 4.

Figure 4.

Pareto Frontiers and Ideal Solutions for Charging pile Planning Schemes

From the above figure, it can be seen that the time of waiting cost and annual construction and operation cost of vehicles cannot reach the optimal level simultaneously. As the annual construction and operation costs of charging piles increase, the time of waiting cost of vehicles first significantly decreases (Zone I), but further increases the annual construction and operation costs of charging piles. The improvement in waiting time cost of vehicles is relatively small but has not converged to zero (Zone I).

Use the VIKOR method to rank the Pareto frontiers mentioned above and find the optimal solution. Use f1 to represent the average annual construction and operation cost of highways, and f2 represent the daily waiting charging time cost of vehicles. When planning highway charging piles, the decision-maker is the construction party, who tends to consider the convenience of user charging while minimizing the construction and operation costs. Therefore, the weight coefficients β=[0.75,0.25]^T, v=0.5 used in this article. The ideal solution F*=(2086.08, 271.26) was calculated using equation (2-49), as shown by the red star in Figure 4. The S, R, and Q values of each non inferior solution on the Pareto front were sequentially calculated. Sorting based on the Q values, including the comparison of the six optimal solutions, is shown in Table III.

Table Ⅲ.

VIKOR method sorting results.

From the analysis of the calculation results in Tables III, it can be seen that Scheme 1 ranks best according to the comprehensive value Q, and Scheme 1 still ranks first in the S and R values. According to VIKOR theory, Scheme 1 can be selected as the optimal solution among all non inferior schemes.

The position of the optimal solution (2156.56, 467.47) in the Pareto front is shown in the red circle in Figure 5. From Figure 4, it can be seen that Option 1 is not the point on the Pareto front with the shortest distance from the ideal solution, because the weights of each criterion are different; At a weight of β=[0.75,0.25]^T, the weight of construction and operation costs is greater than the weight of waiting time for vehicles. Therefore, according to the VIKOR theory, the optimal solution obtained is closer to the vertical axis, where the annual construction and operation costs are lower and the daily charging waiting time is larger. It is the feasible solution closest to the ideal solution on the Pareto front under the weight of this criterion.

Figure 5.

Location Map of the Optimal Planning Scheme

The station location results corresponding to the above optimal planning scheme 1 are shown in Figure 5. The number of charging piles configured under the optimal planning scheme is shown in Table IV, and the station location coordinates are shown in Table V.

Table Ⅳ.

Number of charging pile charging piles configured.

Table Ⅴ.

Location coordinates of charging piles under optimal planning scheme.

In actual transportation networks, due to differences in users’ travel OD needs, they are coupled and affect each other in the network, resulting in different distribution of traffic flow on different road sections at different times. Based on the results of traffic flow simulation in Figure 4, it can be seen that in places with high traffic flow, such as stations 1,2,7.8,15,16 on road sections near large cities, there are more chargers configured; In places with low traffic flow, such as stations 11,12,19,20 on the road leading to small cities, there are fewer charging pile configurations.

In summary, this article takes into account the coupling effects of dynamic traffic simulation and OD, and the number of charging piles obtained matches the traffic flow. The planning results are more economical and reasonable, and can well meet the charging needs of vehicles.