In this paper, we use the queuing theory idea to calculate the time loss of EV users waiting in line for charging at charging stations.

Assuming that the charging process of EVs at charging stations belongs to a multi-service station model, the corresponding average queuing time in a day can be calculated as:

(8)ti=∑i=1iHi(∑j=1jNijq)where q is the daily fast charging probability of a bicycle; For each charging station i, Nij is the charging demand from i to demand point j. The charging time expectation Hi is:

(9)Hi=s⋅ps+1⋅p0δs!(s−p)where p0 is a fixed value indicating the probability that all charging posts are idle, δ is the number of days admitted to the charging station, p is the charging station service intensity, s is the number of total charging posts.

(10)p0=[∑k=0s−1(sp)kk!+(sp)ss!(s−p)]−1

From the queuing time occupied per unit day, the total queuing loss cost can be calculated:

(11)h22=365⋅qa⋅t1

In determining the siting and capacity model for charging stations, the common interests of multiple parties need to be considered. Fig. 2 shows the total cost analysis [19].

Based on the above analysis, the model is proposed or established as follows. Refining the total cost, the objective function is:

(12)minS=∑i=1N(s1i+s2i+s3i+s4i+s5i+s6i+s7i)

Among them, the initial investment construction cost of charging equipment of the charging station (s1i) and the later maintenance cost to ensure the charging station can operate normally (s2i) belong to the construction and maintenance cost of the charging station; the total charging cost of user (s3i) and the additional loss cost of users annual charging (s4i) belong to the EV user side economic loss; the remaining part is the distribution side loss, including the distribution side network loss (s5i), the distribution side economic loss due to harmonics (s6i), the economic loss due to load fluctuation and the economic loss due to voltage shift on the distribution side (s7i).

The investment and construction costs of charging stations (s1i) include the purchase and installation costs of equipment for charging stations, the land price of the site, and infrastructure costs.

(13)s1i=aei+bmi+cmi2+diwhere for each charging station i, ei is the number of transformers to be purchased, mi is the number of chargers to be purchased; a and b are the price of each transformer and charger, respectively; c is the equivalent investment factor for the cost of associated equipment such as transmission lines, di is the cost of land and infrastructure.

The operation and maintenance costs of the charging station (s2i) include the repair and maintenance costs of the equipment, the refurbishment costs, and the salary costs of the personnel. This part of the cost is characterized by high randomness and uncertainty, which is difficult to quantify directly, so it is usually expressed as the ratio η of s1i (usually taken as 10%–20%), i.e.,

(14)s2i=s1iη

In the model of this paper, in addition to the construction and maintenance costs for the investors of charging stations, the usage costs on the user side of EVs also need to be considered.

For charging stations i, the annual charging cost of their users is

(15)s3i=p⋅Qi⋅365where Qi is the average of the charging demand in a day for the users covered by this charging station, and p is the charging tariff per unit load. Usually, this part of the demand and the cost it generates is a fixed value determined by the demand side.

In contrast, the additional loss cost of charging per year for the user (s4i) is related to the siting and capacity setting scheme and consists of two main components.

(16)s4i=h1+h2

One of them is the cost of idle power loss incurred by the user during the charging journey:

(17)h1=365⋅p⋅∑Ligwhere ∑Li is the sum of the distances from all charging demand points to the charging station within the service area of the charging station i, g is the distance traveled per unit of electricity of the EV.

The cost of time lost by the user h2 includes the time lost on the road and the time lost waiting in line at the charging station.

(18)h2=h21+h22

(19)h21=365⋅qa⋅∑Livwhere qa is the user’s travel time value; v is the average speed of the EV h22 can be calculated by solving the above queueing theory method.

The main adverse impact factors of EV charging load on the distribution network are shown in Fig. 3.

When EVs are charged and EV charging posts are interactively coupled with the distribution network, especially when charging stations are connected to the distribution network on a large scale, they will have a non-negligible impact on the local distribution network, thus causing certain cost losses, such as network loss, harmonic pollution, current and voltage fluctuations, etc.

Network loss related power:

(20)Ppw=∑iN∑jNGijUiUjcosδijwhere Ppw denotes the power loss of the distribution network; N is the total number of nodes; for any two nodes i and j, Ui and Uj are the node i and node j respectively voltage, Gij is the conductance between nodes i, j; δij is the power angle difference between nodes i, j.

From the power loss formula of the distribution network, the network loss at the distribution side can be quantified as

(21)s5i=365⋅tCD⋅p⋅Ppwwhere tCD is the total charging time of the charging station during the day; p is the tariff per unit load volume.

The final distribution-side economic loss under the influence of harmonics is

(22)s6i=Me+Mx+Mc+Ml

The economic loss caused by load volatility and voltage excursion is

(23)s7i=fbi⋅Rbi+fpi⋅Rpiwhere fbi is the evaluation index of load fluctuation severity, fpi is the evaluation index of voltage out-of-limit severity of node i.

The above model considers the operation efficiency of the charging station, the loss of the distribution network side, and the convenience of charging from the perspective of the whole society. In this model, variables with high uncertainty such as user queuing time and harmonic loss are modeled and quantified as specific values, which can be optimized by changing the location and capacity of the charging station, and added to a specific model to find a comprehensive optimal solution.

(24)PFDZ=PCDZ+Ui∑j=1NUj(Gijcos⁡θij+Bijsin⁡θij)

(25)QFDZ=QCDZ+Ui∑j=1NUj(Gijcos⁡θij+Bijsin⁡θij)

The above equation constraints can be interpreted as the tidal constraints of the charging station at the distribution network side Eq. (24) is the active power tidal constraint, PFDZ and PCDZ are the active power of the generator and charging station at grid node i. Eq. (25) is the reactive power tidal constraint, QFDZ and QCDZ are the reactive power of the generator and charging station at grid node i, Gij and Bij are the reactance and susceptance between nodes i and node j, respectively; θij is the voltage difference angle between nodes i and node j [20].

(1) Substation capacity constraint:

(26)Si≤Simaxwhere Si is the EV charging capacity load case; Simax is the load capacity constraint that each load in the charging station, such as transformer, can withstand.

(2) The maximum charging power constraint of EVs allowed to be connected to the distribution network:

(27)∑i=1NPCi≤PCmaxwhere for each charging station i, PCi is the charging power of charging station i; PCmax is the total maximum charging power reserved for EVs in this area.

(3) Feeder maximum current constraint:

(28)Iij≤Iijmax

Iij is the current of the feeder in the distribution network during charging, Iijmax is the maximum current flowing through the distribution network to keep it stable.

(4) Capacity constraint of charging station access point:

(29)Pcij≤Pjmaxwhere Pcij is the charging power of the charging station i connected to the grid node j; Pjmax is the maximum access power allowed by the grid node j, which is mainly determined by the load at node j and the transmission capacity of the line where it is located [21].

In the plane, there is a set of points where each vertex Pi (i = 1,2, …) is expanded in all directions according to a certain speed until they intersect at the plane, thus generating multiple regions, and finally generating a region division graph as shown in Fig. 4. Its mathematical definition can be described as

(30)Vorl=1,2,⋅⋅⋅,L(Hb)={u∈Vor(Hb)d(u,Hb)⩽d(u,Hl)}

After the Voronoi diagram related program runs, V(pi) is a certain region finally generated by pi, u is any point within the plane V(pi), d(u, pi) and d(u, pl) represent the geometric distances between u and pi, pl, respectively. Eventually, within the region V(pi) generated by each vertex pi, the distance between any point u and pi is less than the distance between u and other regions, and the dashed line in the figure is the dividing line between different regions V(pi).

Because different charging stations have different capacities, it is necessary to introduce a weight value to determine their specific responsible range, i.e., a weighted diagram is needed. Assuming that the corresponding weight value of each point pi is λi, Eq. (30) can be changed to:

(31)Vorl=1,2,⋅⋅⋅,L(pi)={u∈V(pi)|d(u,pi)λi⩽d(u,pl)λj}

The weights λi of charging stations i can be calculated by Eq. (32), where Sc is the reference capacity and Qi is the capacity of the re-charging station.

(32)λi=ScQi

From the above principle, each charging station is regarded as a certain point on the two-dimensional plane, and weighted by capacity, the charging stations with more charging posts and larger capacity are given higher weight values, and the corresponding weighted map is generated to finally arrive at the area responsible for each charging station, within which, according to the proximity principle, the total cost of the corresponding charging demand point going to the relevant charging station is smaller than the total cost of going to other charging stations.

Given that the siting and capacity determination of charging stations has the characteristics of a complex model and many variables, the traditional PSO algorithm has the problem of low efficiency, and may not be able to guarantee the search for the global optimal solution, thus causing certain economic and time losses.

Therefore, the immune algorithm is a stochastic search algorithm with jumping characteristics. In this algorithm, the optimal search problem of charging station siting is considered as the antigen and the feasible solution as the antibody, and the two interact with each other to obtain the final solution, and the antibody diversity is used to regulate itself, so that the particle population always maintains diversity and keeps the excellent particles with high adaptability, and iterates with the excellent particles saved by screening, so that the overall quality of the population particles is optimized, and the algorithm computing efficiency is enhanced, and the speed of the optimal search is improved, and finally the local convergence is effectively avoided.

In addition, it can be fused to join the simulated annealing algorithm and perform a collaborative search. SA algorithm has a strong ability of sudden jump, when the particle generates a new solution, the algorithm will judge the new solution according to the relevant acceptance criterion concerning the solid annealing process, so as to decide whether to update and optimize, in this way, the algorithm will not mention staying at the original local optimal solution, but will continue to search until finding the global optimal solution. At the same time, this cooling behavior likewise enhances the ability to determine the optimal solution, thus preventing the problem of missing the optimal solution due to particle swooping behavior.

In summary, the immune algorithm and the simulated annealing algorithm can be used simultaneously with the traditional particle swarm algorithm to improve the particle swarm algorithm for data processing work.

The specific steps of the EV charging pile siting and capacity planning using the Voronoi diagram and simulated annealing immune particle swarm algorithm together can be described in Fig. 5.

(1) The initialization of the model data is done by inputting the known parameter variables from the second part of the model.

(2) The upper limit of the total number of charging stations Nmax and Nmin is predicted according to the constraints Eqs. (33) and (34) and the number of charging stations in this range is used as a cyclic variable, N = Nmin as the initial value and thus as the initial value for iteration, with N = N + 1 for each iteration.

(33)Nmin=[QSmax]+1

(34)Nmax=[QSmin]

(3) The weights of each area are calculated using (32), and then the graph method considering the weights is used to divide the range situation responsible for each charging station and generate a specific area division graph from it.

(4) Initialization of algorithm parameters. This part can be divided into initialization of the basic particle swarm algorithm parameters, including the setting of the particle swarm dimension (each solution objective such as the horizontal and vertical coordinate positions of charging stations and the number of charging posts as one dimension), setting of the initial position x and initial velocity v under this dimension, particle swarm size μ, the maximum number of iterations tmax, learning factor c1, c2, inertia factor ω, etc.; initialization of parameters of the fused immune algorithm, including population size, suppression radius δ, weight coefficient, etc.; initialization of the fused simulated annealing algorithm, including cooling start temperature T0, cooling start temperature Tl, annealing rate α and setting the calculation accuracy ε.

(5) Calculate each cost and the total cost of each scheme for different number of charging stations, and use the total cost as the adaptation degree of the particle.

(6) The particle’s own and global extremes are updated according to the criterion. The fitness of each particle is compared with the individual extreme values, where the smaller value is the better fitness value.

(7) Compare whether the global extremes are less than the computational accuracy ε for x consecutive generations. If the condition is met, the process of simulated annealing is performed with gbest as the initial solution; otherwise, immune memory is performed directly and a memory bank is generated.

(9) Particle update. According to the introduction of linearly decreasing inertia weight update, M+N particles are randomly selected from the memory library to form a particle population for immune conditioning, and N particles are selected to form a new generation particle population.

(10) The convergence condition is judged. If the conditions are met, the data processing work is completed and the global optimal solution and location are output, at this time the obtained scheme is the three cost integrated optimal site selection and capacity planning scheme. Otherwise, return to Step (6) to iteratively update the particles again.

A street in Tianjin is selected as the planning object: The region covers an area of 63 km² and is estimated to have a total volume of about 9,500 EVs in 2022. It is assumed that the average driving speed of EVs is 35 km/h, and the one-day fast charging probability is 0.05. Each fast charging takes 15 min to be fully charged, the charging power is 96 kW, the consumption of 15 kW·h of electricity per 100 km, and the maximum range of a single driving is 160 km.

Given the large area of the area under study, a large number of car ownership, and the complex distribution of car charging demand load, 34 road network nodes are delineated in this area, and the specific location and load of each point are shown in Fig. 6 (the numbers marked at the nodes in the figure indicate the normalized load size).

For the distribution system, the IEEE33 node system is selected, and its distribution system wiring is shown in Fig. 7. The reference voltage and reference power are 12.66 kV and 10 MVA, respectively.

According to the above data, the simulated annealing-based immune particle swarm algorithm is used to solve the problem, and the number of algorithm populations is set to 50, the number of iterations is 300, the learning factor is set to 2, the range of inertia weights is set to [0.2 1.2], and the immune algorithm in α = β = 0.5; δ = 0.5; γ = 0.8; the simulated annealing process parameters are set to: T = 100; T0 = 0.01; K = 0.9.

The relationship between the number of charging stations and the total cost is shown in Fig. 8. It can be seen that: when the number of charging stations is 7, the total annual social cost is the smallest, which is 209 million. At this point, the total investment and maintenance cost is ¥100.6 million, of which the initial construction cost is ¥91.5 million and the later maintenance cost is ¥9.1 million. The cost on the user side and the cost on the distribution network side are ¥75.2 million and ¥33.2 million, accounting for 36% and 16% of the overall proportion, respectively.

The grid straight line is generated from the Fig. 9, the service area is divided into boundaries with the lowest cost for the demand point to go to the corresponding charging station in the respective area.

The corresponding station numbers and the respective number of charging posts configured for each charging station are shown in Table 1.

Based on the above siting and capacity setting of charging stations, indicators such as the average queue length and average stay time of each charging station can be calculated as shown in Table 2.

From Table 2, it can be seen that the average waiting time of customers at 7 charging stations does not exceed 1.8 min, the charging stations provide better service to customers, and the highest charger idle ratio is only 28.07%, i.e., the chargers of charging stations utilization rate reached 71.93% or more. The resources of the charging station are better utilized on the basis of satisfying the customers with high-quality services.

In order to compare the advantages of using the SA-IPSO algorithm, this paper compares the SA-IPSO algorithm with the PSO algorithm and the IPSO algorithm and obtains their respective convergence curves and operating results.

From Fig. 10, it can be seen that, in terms of convergence speed, the curves derived from the SA-IPSO and IPSO algorithms obviously converge faster than those corresponding to the PSO algorithm, indicating that the addition of the immune algorithm can improve the convergence speed and operational efficiency; it falls into the local optimum earlier The curves obtained by the SA-IPSO algorithm basically do not fall into the local optimum.

The final optimal solution results, number of iterations, solution time, and global optimal finding capability using different algorithms for data processing are shown in Table 3. The results show that, in terms of the integrated cost, the resulting optimal scheme has the lowest integrated total cost, followed by IPSO; among them, the scheme corresponding to SA-IPSO has the lowest total cost even though the number of charging posts is significantly more than that of IPSO, indicating that the scheme corresponding to this algorithm significantly saves the user of usage costs and economic losses on the grid side. This shows that the inclusion of two hybrid algorithms allows the site-setting solution with a low total cost and high user convenience not to be missed because it stays in the local optimal solution or the example swoops too fast, etc.

Furthermore, the SA-IPSO algorithm and IPSO algorithm with the addition of the immunization algorithm significantly reduce the solution time, with the SA-IPSO algorithm even reducing 18.1% of the time compared to the conventional PSO algorithm. This further illustrates the improvement in computing efficiency by introducing a hybrid algorithm.