Performance of Gradient-Based Optimizer on Charging Station Placement Problem

Abstract

The electrification of transportation is necessary due to the expanded fuel cost and change in climate. The management of charging stations and their easy accessibility are the main concerns for receipting and accepting Electric Vehicles (EVs). The distribution network reliability, voltage stability and power loss are the main factors in designing the optimum placement and management strategy of a charging station. The planning of a charging stations is a complicated problem involving roads and power grids. The Gradient-based optimizer (GBO) used for solving the charger placement problem is tested in this work. A good balance between exploitation and exploration is achieved by the GBO. Furthermore, the likelihood of becoming stuck in premature convergence and local optima is rare in a GBO. Simulation results establish the efficacy and robustness of the GBO in solving the charger placement problem as compared to other metaheuristics such as a genetic algorithm, differential evaluation and practical swarm optimizer.

1. Introduction

The progress of energy is a main objective in human life [1,2,3,4,5]. Electrical vehicles (EV) are spread out in most countries due to the emissions of gases from internal combustion vehicles [6]. The management strategy and design of the placement of a charging station (CS) play an imperative part in distribution network reliability, power loss and voltage stability [7], the coordination between the distribution network in the road network and the layout of EV charging stations. Most researchers are interested in optimizing the layout of EV CSs. This problem can be solved with several optimization algorithms [8,9,10,11].

In [12], the geographic information system and greedy algorithm are used for locating fast-charging stations. In [13], an overview for the management of a charging station EV placement and its control aspects is discussed. In [14], a study integrating a power grid with EV charging stations is provided. In addition, an efficient solution is proposed for Public Fast-Charging Stations [15]. In [16], a genetic algorithm is applied in solving the placement of EV charging stations with the objective function of minimizing the cost. In [17], a comparison between a practical swarm and adaptive practical swarm optimization is discussed with the objective function of running and construction costs. In [18], an extraction of the location and number of CSs is performed using a hybridization of a genetic algorithm with k-means for clustering. In [19], a genetic algorithm is applied for solving the placement of EV charging stations with the objective function of minimizing the missed trip.

Various approaches have been applied to design the optimum placement of EV charging stations [20] and Electric vehicle charging station [21,22,23]. Charging stations can be designed optimally depending on the EV usage in an enhanced system performance and supplying a peak load. For that reason, the reduction in losses, better economics and the minimizing of voltage deviation are achieved using EVs [24]. The problem of charging stations placement was also solved using practical swarm optimization algorithms [25]. In this sense, the incorporation of CO${}_2$ in the optimum design of an EV charging station was discussed in [26]. The optimization of the layout for charging stations based on maximizing the reliability of a network, minimizing voltage deviation and power losses is applied using a K-means clustering algorithm [27]. The determination of optimum sizing and siting of charging stations and photovoltaic can avoid their negative effects [28]. Solar energy has also been used in charging stations to reduce the negative effect of EVs [29]. The distribution network voltage deviation, power losses and charging service have been used as a multi-objective function in the optimization placement of charging stations. This problem has been tested on a road network of 25 nodes and an IEEE 33 bus network by using a cross-entropy method and data envelopment analysis [30]. genetic algorithm was applied in solving the CS placement issue with the objective function dependent on the operation grid cost, the traffic circulation and the cost of station development [31].

The optimal design of a charging station has also been performed by using the TLBO and CSO algorithms with the objective function of minimizing the cost and the behaviour improvement of the distribution network was taken into consideration. The proposed method was tested on a case study of Guwahati City, India, with information from [32]. A modified primal dual interior point algorithm was applied in the estimation of an optimal charging station design based on the cost being a single-objective function. The proposed technique was tested on the IEEE 123 bus system [33]. A hierarchical genetic algorithm was applied in the estimation of an optimal charging station design based on cost as an objective function and the constraints of the maximum capacity ofa charging station and limits of power loss. Here, the proposed technique was tested on the IEEE 123 bus system [34]. An ant colony optimization was applied in the determination of the optimal charging station placement based on an objective function that maximizes the ability of the charging service [35]. A differential evolution was applied in designing the charging station placement based on the objective function of minimizing the cost and constraints of limiting voltage, current and power consumption [36]. Moreover, the information matrix from the household trip origin and dynamic vehicle model is the objective function used to determine the proper allocation of charging stations [37]. The recharging decisions, interactions of travels and the adjustments spontaneous of drivers are taken into consideration in the design of the placement of charging stations on a road network [38]. The determination of the charging infrastructure location was performed based on a cluster analysis for the urban area of Rome [39]. Based on the data of an EV operation and real trajectory of the trajectory-interception method, the facility designing of an EV taxi charging station was discussed [40].

A hybrid ant, lion and chicken swarm optimization algorithm (ALO CSO) was used for solving the single-objective charger placement problem with cost as the objective function [41]. It was observed that the proposed ALO-CSO performed better than the other metaheuristics. A multi-objective formulation of the charger placement problem is presented in [42] with cost, reliability, power loss as the objective function and the Teaching–Learning-Based Optimization (TLBO) was used for searching the optimal solution. A scheme for the placement of Level 1, Level 2 and Level 3 chargers in the active distribution network is presented in [43], with the installation cost and power losses as objective functions. Here, the PSO is implemented as a search strategy. A scheme for the charger allocation considering ride sharing was proposed in [44] considering the minimization of the vehicle idle time as the objective function. The optimization issue was handled by a surrogate-assisted optimization approach. In [45], a novel CSO-driven metaheuristic was proposed for the solving, planning and operation of charging stations. Meanwhile, in [46], a novel teaching–learning-based CSO is used for the charger placement problem. In [47], authors have proposed the implementation of the JAYA algorithm for solving the charger placement problem with cost as the objective function. In [48], a comprehensive framework for optimizing solar-powered charging stations is presented.

Recently, a Gradient-Based Optimizer (GBO) [49] was performed by Ahmadianfar et al. in a promising algorithm for solving the problem of CS placement in this work. The GBO is inspired by the gradient-based Newton method. In this paper, the performance of the GBO is evaluated to solve the problem of CS placement for a superimposed network of the 33 distribution bus and 25 road node. In addition, the effectiveness of the GBO is compared with other meta-heuristic algorithms such as the Genetic algorithm (GA) [50], Differential evolution (DE) [51] and Particle swarm optimization (PSO) [52].

The contributions of this work can be assembled in the following items:

• The use of the novel GBO to solve the charger placement problem.
• The charger placement problem is a combinatorial optimization issue that decides on three variables: the number of slow-charging stations, number of fast-charging stations and their places among a set of predefined nodes.
• A comparison of the GBO with other Metaheuristics such as the genetic algorithm, differential evaluation, practical swarm optimizer is discussed regarding the charger placement problem.
• GBO performance is studied based on a statistical analysis for 50 independent runs.

The organization of the paper of energy is as follows: Section 2 presents the charger placement problem. Section 3 presents the GBO algorithm. Section 4 presents the numerical analysis. Finally, the work concludes in Section 5.

2. Charger Placement Problem

The placement of a CS is a multi-dimensional issue where the output is the number and location of charging stations. Symbolically, the decision parameters were b, $N_{Fb}$ and $N_{Sb}$, b ∈ P, where b is the charging station placement bus, $N_{Sb}$ is the slow CS number at bus b, $N_{Fb}$ is the fast CS number at bus b and P is the set of CS nodes. Thus, the decision variables were position and the number of slow as well as fast-charging stations to be placed was determined.

The three variables were integers, and the initial solution generated was also an integer. Each solution was tested for constraint satisfaction. If constraints were not satisfied, the solutions were discarded, and a new solution was generated. The process continued until a feasible solution was generated.

The fitness function was the minimization of the overall cost of charging stations. Furthermore, the summation of the direct and indirect cost was the overall cost. The fitness function was defined mathematically as:

$$Min(C_{direct}+C_{indirect})$$

(1)

The operating and installation costs were the main direct cost ($C_{direct}$) of the charging station, which was elaborated as:

$$C_{direct}=C_{installation}+C_{operation}$$

(2)

$$C_{installation}=(N_{fastCS}\times C_{installationfast}+N_{slowCS}+C_{installationslow})$$

(3)

$$C_{operation}=(N_{fastCS}\times C{P}_{fast}+N_{slowCS}+C{P}_{slow})\times P_{electricity}\times T_i$$

(4)

$$N_{fastCS}=\sum _{i=1}^{N_D}{n}_i\times N_F^i$$

(5)

$$N_{slowCs}=\sum _{i=1}^{N_D}{n}_i\times N_S^i$$

(6)

$$n_i=1\forall i\in P$$

(7)

The indirect cost ($C_{indirect}$) was the sum of the travel time and cost of penalty paid. The mathematical form for the indirect cost discussed was as follows:

$$C_{indirect}=C_{penalty}+C_{travel}$$

(8)

$$C_{penalty}=V{D}_{penalty}+AEN{S}_{penalty}$$

(9)

$$V{D}_{penalty}=P_{VD}\ast \sum _{i=2}^{N_D}V{D}_i$$

(10)

$${VD}_i=V_i^{base}-V_i$$

(11)

In this work, the forward–backward sweep [53] was the method used in the computation of all bus voltages in the network.

$$AEN{S}_{penalty}=P_{AENS}\ast AENS$$

(12)

$$AENS=\frac{\sum L_i{U}_i}{\sum N_i}$$

(13)

$$C_{travel}=d_{CS}\times T_{costEV}$$

(14)

A penalty was assigned for violating the safe limits of AENS as shown in Equation (12). AENS is a reliability index of a power distribution network as shown in Equation (13). Further, the cost of travelling the distance from the point of charging demand to the charger location was also included as an objective function as shown in Equation (14).

The equality constraints could be expressed by the balance of the power flow equation, and the inequality constraints of the fitness function were formulated as follows:

$$0<N_{Fb}\le N_{fastCS}$$

(15)

$$0<N_{Sb}\le N_{slowCS}$$

(16)

$$S_{min}\le S_i\le S_{max}$$

(17)

$$L\le L_{max}$$

(18)

3. Gradient-Based Optimizer

The GBO algorithm was created by Ahmadianfar et al., and it is an algorithm that mimics population-based and gradient-based methods [49]. Newton’s method was used in the GBO to investigate the search space for a collection of search metrics. The GBO’s main steps are clarified in the following subsections.

3.1. The Initialization Process

The GBO balanced and switched between exploration and exploitation using the control parameters $\left(\alpha \right)$ and probability rate. The population and iteration counts were proportional to the complexity of the problem. In the GBO, X represents each member of the population and was used to describe the vector of N sub-vectors in D-dimensional space using Equation (19) as follows:

$$X_{n,d}=[X_{n,1},X_{n,2},\dots ,X_{n,D}],n=1,2,\dots ,N,d=1,2,\dots ,D$$

(19)

In the D-dimensional search space, the initial vectors for the GBO were typically generated by using a random distribution. The initialization in the GBO was then characterized as follows:

$$X_n=X_{min}+rand(0,1)\times (X_{max}-X_{min})$$

(20)

where $X_{min}$, and $X_{max}$ denote the bounds of the decision parameters X, and $rand(0,1)$ denotes a random number between the range 0 and 1.

3.2. Gradient Search Rule (GSR)

The GSR is a process used to ensure a balance between exploration and exploitation. To achieve near-global optimum points, the GBO employed the following significant factor ${\rho }_1$:

$${\rho }_1=2\times \mathit{rand}\times \alpha -\alpha$$

(21)

$$\alpha =\left|\beta \times sin\left(\frac{3\pi }{2}+sin\left(\beta \times \frac{3\pi }{2}\right)\right)\right|$$

(22)

$$\beta ={\beta }_{min}+\left({\beta }_{max}-{\beta }_{min}\right)\times {\left(1-{\left(\frac{m}{M}\right)}^3\right)}^2$$

(23)

where ${\beta }_{min}$ and ${\beta }_{max}$ are constant values of 0.2 and 1.2, respectively, and m denotes the current iteration and M denotes the total iterations number. Parameter ${\rho }_1$ in particular was responsible for balancing the exploration and exploitation using the sine function $\alpha$. The parameter’s value changed over time; it began with a large value during the initial optimization iterations to increase population diversity, and then decreased in value during the subsequent iterations to accelerate population convergence. The parameter value was increased over a defined number of iterations within the range [550, 750], in order to increase the diversity of solutions and to converge around the best obtained solution while also exploring additional solutions. This allowed the algorithm to avoid local sub-regions. As a result, the GSR could be calculated as follows:

$$GSR=randn\times {\rho }_1\times \frac{2\Delta x\times x_n}{\left(x_{\mathit{worst}}-x_{best}+\epsilon \right)}$$

(24)

where, randn is the normally distributed random number and $\epsilon$ is a small number.

The $GSR$ concept provided the GBO algorithm with random behaviour during iterations, enhancing exploration behaviour and allowing for escape from local optima. In Equation (24), the term $\Delta x$ refers to the difference between the optimal solution $\left(x_{best}\right)$ and a randomly chosen solution $x_{r1}^m$. Due to the following Equation (27), the parameter $delta$ was changed during iterations. Additionally, a random number ($randn$) was included to aid in exploration:

$$\Delta x=\mathit{rand}(1:N)\times |step|$$

(25)

$$\mathrm{step}=\frac{\left(x_{\mathit{best}}-x_{r1}^m\right)+\delta }{2}$$

(26)

$$\delta =2\times \mathit{rand}\times \left(\left|\frac{x_{r1}^m+x_{r2}^m+x_{r3}^m+x_{r4}^m}{4}-x_n^m\right|\right)$$

(27)

where $rand(1:N)$ is a vector containing N random values between 0 and 1. Additionally, four randomly selected integers from [1, N] were $r1$, $r2$, $r3$ and $r4$, such that $(r1\ne r2\ne r3\ne r4\ne n)$. $step$ represents a scaling factor defined by $x_{best}$ and $x_{r1}^m$. To accomplish convergence, directional movement was used to traverse the solution field $x_n$.

To avoid the local search convergence of the GBO, the  DM term selected the best vector from a set of suggested vectors and moved the current vector $\left(x_n\right)$ in the direction of the best vector $(x_{best}-x_n)$ as follows:

$$DM=rand\times {\rho }_2\times (x_{best}-x_n)$$

(28)

where rand is a uniformly distributed number between 0 and 1, a function of two parameters, and ${\rho }_2$ is a random parameter used to fine tune the phase size of each vector agent. Additionally, the ${\rho }_2$ parameter took into account important parameters in the GBO exploration process. This was the formula for calculating the ${\rho }_2$ parameter:

$${\rho }_2=2\times rand\times \alpha -\alpha .$$

(29)

Eventually, using the terms GSR and DM, we modified Equations (30) and (31) to account for the current vector location $\left(x_n^m\right)$.

$$X{1}_n^m=x_n^m-GSR+DM$$

(30)

where $X{1}_n^m$ is the modified vector as a result of the modification of $X{1}_n^m$. According to Equations (23) and (28), the transformation of $X{1}_n^m$ was as follows:

$$X{1}_n^m=x_n^m-randn\times {\rho }_1\times \frac{2\Delta x\times x_n^m}{\left(y{p}_n^m-y{q}_n^m+\epsilon \right)}+rand\times {\rho }_2\times \left(x_{\mathit{best}}-x_n^m\right)$$

(31)

where $y{p}_n^m, y{q}_n^m$ are equal to $y_n+\Delta x$ and $y_n-\Delta x$, $y_n$ vector is equal to the average of two vectors: the current solution $x_n$ and the $z_{n+1}$ vector, which were calculated by the following formula:

$$z_{n+1}=x_n-randn\times \frac{2\Delta x\times x_n}{\left(x_{\mathit{worst}}-x_{\mathit{best}}+\epsilon \right)}$$

(32)

Although $x_n$ represents the vector of current solution, $randn$ represents a vector of random solution of dimension n, $x_{worst}$ and $x_{best}$ denote the worst and best solutions, respectively, and $\Delta x$ is defined by Equation (25).

Using Equation (31), we obtained $X{2}_n^m$ by substituting the best solution vector $x_{best}$ for the current solution vector $x_n^m$:

$$X{2}_n^m=x_{\mathit{best}}-randn\times {\rho }_1\times \frac{2\Delta x\times x_n^m}{\left(y{p}_n^m-y{q}_n^m+\epsilon \right)}+randn\times {\rho }_2\times \left(x_{r1}^m-x_{r2}^m\right)$$

(33)

To be more precise, the GBO aimed to improve both the exploitation and exploration phases by utilising Equation (31) to enhance global search capabilities during the exploration phase and Equation (33) to enhance local search capabilities during the exploitation phase. Finally, the following procedure was used to generate the new solution for the next iteration:

$$x_n^{m+1}=r_a\times \left(r_b\times X{1}_n^m+\left(1-r_b\right)\times X{2}_n^m\right)+\left(1-r_a\right)\times X{3}_n^m$$

(34)

where $r_a$, and $r_b$ are randomly generated numbers in the range $[0,1]$, and $X{3}_n^m$ is defined as:

$$X{3}_n^m=X_n^{m+1}-\rho 1\times (X{2}_n^m-X{1}_n^m)$$

(35)

3.3. The Local Escaping Operator (LEO)

The LEO was used to boost the performance of an optimization technique by assisting in the solution of difficult engineering problems. The LEO operator assisted the algorithm in rapidly switching out of local optima points, which sped up the algorithm’s convergence. To develop a new solution that was more efficient, the LEO operator focused on $\left(X_{LEO}^m\right)$ by several solutions ($X_{best}$ was the best solution, $X{1}_n^m,X{2}_n^m$ were randomly selected from the population, and $X_{r1}^m,X{1}_{r2}^m$ were randomly generated solutions). The current solution was updated effectively by using the following process:

$${\displaystyle \begin{array}{c}\hfill X_{LEO}^m=\left\{\begin{array}{c}{X}_n^{m+1}+f_1\times \left(u_1\times x_{\mathit{best}}-u_2\times x_k^m\right)+f_2\times {\rho }_1\times \left(u_3\times \left(X{2}_n^m-X{1}_n^m\right)+u_2\times \left(x_{r1}^m-x_{r2}^m\right)\right)/2,\mathrm{if}\mathit{rand}<0.5\hfill \\ x_{\mathit{best}}+f_1\times \left(u_1\times x_{\mathit{best}}-u_2\times x_k^m\right)+f_2\times {\rho }_1\times \left(u_3\times \left(X{2}_n^m-X{1}_n^m\right)+u_2\times \left(x_{r1}^m-x_{r2}^m\right)\right)/2\mathrm{otherwise}\hfill \end{array}\right.\end{array}}$$

(36)

The procedure from Equation (36) was applied only if the probability condition $rand<pr$ was true. $pr$ was a probability value equal to 0.5, $f_1$, and $f_2$ were uniform distribution random numbers $\in [-1,1]$, and $u_1, u_2, u_3$ were generated randomly as follows:

$$u_1=\left\{\begin{array}{c}2\times \mathit{rand}\mathrm{if}{\mu }_1<0.5\hfill \\ 1\mathrm{otherwise}\hfill \end{array}\right.$$

(37)

$$u_2=\left\{\begin{array}{c}\mathit{rand}\mathrm{if}{\mu }_1<0.5\hfill \\ 1\mathrm{otherwise}\hfill \end{array}\right.$$

(38)

$$u_3=\left\{\begin{array}{c}\mathit{rand}\mathrm{if}{\mu }_1<0.5\hfill \\ 1\mathrm{otherwise}\hfill \end{array}\right.$$

(39)

where $rand$ is a random number between 0 and 1 and $m{u}_1$ is any value within the range $[0,1]$.

The preceding equations for $u_1, u_2$ and $u_3$ had the following explanation:

$$u_1=L_1\times 2\times rand+\left(1-L_1\right)$$

(40)

$$u_2=L_1\times rand+\left(1-L_1\right)$$

(41)

$$u_3=L_1\times rand+\left(1-L_1\right)$$

(42)

where $L_1$ is a binary parameter with a value of 0 or 1; for example, if ${\mu }_1<0.5$ was 0.5, the value of $L_1=1$, otherwise $L_1=0$.

$x_k^m$ was generated in the following manner:

$$\begin{array}{c}{x}_k^m=\left\{\begin{array}{cc}{x}_{\mathit{rand}}\hfill & \mathrm{if}{\mu }_2<0.5\hfill \\ x_p^m\hfill & \mathrm{otherwise}\hfill \end{array}\right.\hfill \end{array}$$

(43)

$x_{rand}$ generated a random solution using the following formula:

$$x_{\mathit{rand}}=X_{\mathit{min}}+\mathit{rand}(0,1)\times \left(X_{\mathit{max}}-X_{\mathit{min}}\right)$$

(44)

where $x_p^m$ is a randomly chosen solution from the population, while ${\mu }_2$ is a random number between 0 and 1. For additional information about the GBO, see [49]. The pseudo code of the GBO is described in [49]. The three variables were integers and the initial solution generated was also an integer. Each solution was tested for constraint satisfaction. If constraints were not satisfied, the solutions were discarded, and a new solution was generated. The process continued until a feasible solution was generated. Continuous values were mapped into discrete values based on probability. For each decision variable, the probability was calculated, and a random number was generated. If the random number was less than the probability value, the variable was rounded to the nearest integer value greater than the current value. If the random number was greater than or equal to the probability value, then the variable was rounded to the nearest integer value less than the current value. This preserved the structure of the GBO and moved the position of population to a discrete space.

The pseudo code of the GBO was described in the Methods [49] Algorithm 1.

Algorithm 1 The Pseudo code of the Gradient-based Optimizer

Step 1. Initialization Assign values for parameters $pr$, $\epsilon$, M Generate an initial population $X_0=[x_{0,1},x_{0,2},...,x_{o,D}]$ Evaluate the objective function value $f\left(X_0\right)=,n=1,...,N$ Specify the best and worst solutions $x_{best}^m and x_{worst}^m$ Step 2. Main loop while$m<M$do     for $n=1$ to $\mathit{N}$ do         for $n=1$ to $\mathit{D}$ do            Select randomly $r1\ne r2\ne r3\ne r4\ne n$ in the range of $[1,N]$            Calculate the position $x_{n,i}^{m+1}$ using Equation (34)         end for         if rand $<pr$ then          ▹ Local escaping operator            Calculate the position $x_{LEO}^m$ using Equation (36)            $x_n^{m+1}=x_{LEO}^m$         end if         Update the positions $x_{best}^m$ and $x_{worst}^m$     end for     $m=m+1$ end while Step 2. Return $x_{best}^m$

4. Experimental Results and Numerical Analysis

The charger placement problem was solved by the GBO for the superimposed network of 33 distribution bus and 25 road nodes as shown in Figure 1; the two routes were assumed for following the pass of EVs:

• Route 1-(1-2-3-4-5-6-7-8-9-10-13-11-12-15-16-17-18-20-21-14-22-23-24-25);
• Route 2-(1-2-3-4-5-6-7-8-9-10-13-11-12-15-16-17-19-20-21-14-22-23-24-25).

Figure 1. Test network [7].

Figure 1. Test network [7].

The performance of the GBO rivalled with several of the benchmark algorithms such as PSO, DE and GA, each with 50 independent runs. The settings of the parameters of the algorithms were as shown in

Table 1. Parameter setting.

Algorithm	Parameters
Common settings	Size of population : N = 30 Maximum iterations_Itr = 50
PSO	w = 0.1, c1 = c2 = 2
DE	F = 1.5, CR = 0.6
GBO	pr = 0.5

. The general specific parameters of the algorithm were selected by fine tuning that was achieved by trial and error. It was observed that the algorithm performed best for the settings considered in this work. For other problems or other test networks, the settings may change. Metaheuristics have a set of general and specific parameters [54]. The best settings of the algorithm-specific parameters were obtained by trial and error for the considered test network. For networks of other configurations, algorithm specific parameters could also be set by the trial-and-error method. Moreover, our future work will consider developing an adaptive version of the GBO.

Table 2. The charging station input parameters [7].

Parameter	Value
$C_{installationfast}$	3000 USD
$C_{installationslow}$	2500 USD
$C{P}_{fast}$	50 kW
$C{P}_{slow}$	19.2 kW
$P_{electricity}$	65 USD/MWhr
$P_{VD}$	${\left(VD\right)}^2\ast$ 1,000,000) USD
$P_{AENS}$	0.18 USD/MWhr

reports the values of input parameters of the CS problem.

Table 3. Optimal charging stations allocation.

Optimization Algorithm	Best Fitness Value	b	${\mathit{N}}_{\mathit{Fb}}$	${\mathit{N}}_{\mathit{Sb}}$
GBO	1.4898	23	1	3
		6	1	2
		3	1	3
PSO	1.4898	23	1	2
		6	1	3
		3	1	3
DE	1.4898	23	1	2
		6	1	3
		3	1	3
GA	1.5075	23	1	2
		3	1	3
		28	1	3

reports the optimal locations and numbers of chargers to be placed computed by the algorithms mentioned in Table 1. It was observed that the GBO, PSO and DE yielded the best fitness value of 1.4898 and performed better than the GA.

Table 4. Value of distribution network variables of after and before placement of CS.

Variables	Before	After
Deviation of voltage (pu)	0	0.0114
AENS in kWhr/yr	1.9369	2.5233
Loss of power (pu)	0.0021	0.0062

reports the values of operating distribution network parameters such as the voltage deviation, AENS and power losses after and before the placement of charging stations. It was noted that a voltage deviation of 0.0114 pu occurred post the placement of chargers at the locations mentioned in Table 3. The AENS value also changed from 1.9369 kWhr/yr to 2.5233 kWhr/yr post the placement of chargers at the locations mentioned in Table 3. Further, the power losses of the network also increased from 0.0021 pu to 0.0062 pu. Thus, the distribution network operating parameters degraded, but were still within safe limits.

Further, the statistical comparison ofthe GBO with the other benchmark algorithms listed in Table 1 was performed. The results of the statistical comparison of the GBO with other benchmark algorithms is presented in Table 4. It was noted that the performance of the GBO was competitive as compared to the other metaheuristics.

The robustness curve and the convergence curve of the algorithms in the case of solving the charger placement problem are shown in Figure 2 and Figure 3, respectively. It was observed that a good balance between exploitation and exploration was achieved by the GBO. Furthermore, the likelihood of becoming stuck in premature convergence and local optima was rare in the GBO.

Further, the impact of the charger placement on typical distribution network reliability indices such as SAIDI, SAIFI and CAIDI was analysed. SAIFI gave an idea about the frequency of interruption, SAIDI gave an idea about the duration of interruption and CAIDI was an index representing customer dissatisfaction because of interruption. Figure 4, Figure 5 and Figure 6 depict the values of SAIFI, SAIDI and CAIDI after and before the charger allocation, respectively. It could be inferred that the degraded values of the reliability indices were within the safe limits.

Table 5. Statistical analysis for solving problem of charging station placement.

Algorithm	Min Fitness	Max Fitness	Mean Fitness
GBO	1.4898	1.7116	1.5286
GA	1.5075	1.6199	1.5584
DE	1.4898	1.6199	1.5497
PSO	1.4898	1.5636	1.5341

discusses the evaluation of the GBO performance based on a statistical analysis of independent runs for this algorithm and other compared algorithms for the same case study. Based on this results, the GBO method achieved a better accuracy than all competitor algorithms.

5. Conclusions

The growing concerns regarding climate change, global warming and increased fuel price have initiated transportation electrification. For the adoption and acceptance of Electric Vehicles (EVs) amongst the masses, it is necessary to set up a sustainable and easily accessible charging infrastructure. The management strategy and the placement of the charging station play a significant role in maintaining the distribution network reliability, power loss and voltage stability. The charger placement formulation presented in this work was a single-objective formulation with cost as the objective function. The problem was solved considering the worst-case scenario. The uncertainty in road traffic, different scenarios of charging load, coordinated charging and the impact of vehicle grid integration were not considered in this work. A novel metaheuristic named GBO was used in this work for solving the charger placement problem. The Newton method was the main inspiration of the Gradient-based optimizer (GBO), that involves the LEO and GSR concepts. The performance of the GBO was compared with other metaheuristics such as DE, GA and PSO. The best fitness value of 1.4898 was achieved by the proposed GBO, PSO and DE algorithms. Furthermore, the values of SAIFI, SAIDI and CAIDI after and before the charger allocation were within the safe limits. A voltage deviation of 0.0114 occurred based on the best location extracted from the GBO algorithm. It was noted that the GBO presented comparatively well as competed to the aforesaid algorithms. Our future formulations will consider these aforementioned factors. Our future work will concentrate on testing the GBO behaviour on multi-objective charger placement problems, and several applications such as optimal load flow, the identification of a super-capacitor and fuel cell parameters, extraction of transformer parameters and optimization of a wind farm layout.