Accelerated Particle Swarm Optimization Algorithms Coupled with Analysis of Variance for Intelligent Charging of Plug-in Hybrid Electric Vehicles

Abstract

Plug-in hybrid electric vehicles (PHEVs) and plug-in electric vehicles (PEVs) have gained enormous attention for their ability to reduce fuel consumption in transportation and are, thus, helpful in the reduction of the greenhouse effect and pollution. However, they bring up some technical problems that should be resolved. Due to the ever-increasing demand for these PHEVs, the simultaneous connection of large PEVs and PHEVs to the electric grid can cause overloading, which results in disturbance to overall power system stability and quality and can cause a blackout. Such situations can be avoided by adequately manipulating power available from the grid and vehicle power demand. State of charge (SoC) is the leading performance parameter that should be optimized using computational techniques to charge vehicles efficiently. In this research, an efficient metaheuristic algorithm, accelerated particle swarm optimization (APSO), and its five variants were applied to allocate power to vehicles connected to the grid intelligently. For this, the maximization of average SoC is considered a fitness function, and each PHEV can be connected to the grid once a day so that the maximum number of cars can be charged daily. To statistically compare the performance of these six algorithms, one-way ANOVA was used. Simulation and statistical results obtained by maximizing this highly non-linear objective function show that accelerated particle swarm optimization with Variant 5 achieved some improvements in terms of computational time and best fitness value. The APSO-5 solution has a considerable percentage increase compared with the solution of other variants of APSO for the four PHEV datasets considered. Moreover, after 30 trials, APSO 5 gives the highest possible fitness value among all the algorithms.

1. Introduction

Many problems are plaguing society today, including a lack of energy and pollution brought on by fossil fuels. Combustion of coal, oil, and natural gas to provide energy for various human activities, mainly for transportation, emits greenhouse gases that pollute the environment and could be health hazards. Since the first energy crisis, which prompted the transition of the power grid from a traditional centralized grid to a distributed smart grid (SG), renewable energy sources (RESs) have been used in an increasing proportion globally [1,2]. Yet, because of their unstable and intermittent character as well as their potential to be disrupted or destroyed by faults and natural catastrophes, the deployment of RESs in power systems may result in the creation of new peaks in the energy system demand profile, which has an impact on the stability of the power grid [3,4]. Transportation electrification has gained much attention in the last decade [5]. Vehicles can be categorized into three main types [6]: (1) internal combustion engine vehicles (ICEVs), (2) hybrid electric vehicles (HEVs), and (3) all-electric vehicles (AEVs).

Recently, plug-in hybrid electric vehicles (PHEVs) and plug-in electric vehicles (PEVs) have gained much attraction due to their batteries that can be charged from the traditional power grid, and thus assist automobiles to work independently in the “all-electric range” (AER). These vehicles use electric power as the only source to charge their batteries [7]. Thus, PHEVs shift energy demand from fossil fuel to electric power for the transportation sector [8]. Moreover, there can be a two-way power flow between the grid and vehicles; i.e., to meet peak power demand and provide auxiliary service to the grid, PHEVs can transfer power to the grid [9]. Therefore, advancement in technology is revamping the conventional view of power systems. Due to the enormous increase in electric vehicles, the Electric Power Research Institute (EPRI) proposed that, by 2050, 62% of US vehicles will be PHEVs or PEVs [10].

Compared with conventional vehicles, AEVs have many advantages, as discussed in [11], i.e., zero emissions of pollutants, simplicity, reliability, cost, comfort, efficiency, and accessibility. Along with all these advantages, some significant battery-related challenges for plug-in electric vehicles should be resolved. (1) Driving range: Electric vehicles face limitations in their driving range; usually, it is limited to 200 to 250 km with a fully charged battery. Research is being carried out to improve this range; e.g., the Nissan leaf has an extended range of up to 364 km [12], and the Tesla Model S can reach up to 500 km [13]. (2) Charging time: As electric vehicles have a bank of batteries, they usually take an average of 4 to 8 h to fully charge; e.g., by using the Tesla supercharger, one can charge a Tesla Model S up to 50% in 20 minutes and up to 80% in half an hour [13]. (3) Cost and weight of batteries: Electric vehicles require large battery packs, which are expensive and heavy and need considerable space in the car. Depending on the battery capacity, such vehicles have an approximate weight of 200 kg, but this can vary for different vehicle types [14]. (4) Energy management: When many PHEVs or PEVs connect to the national grid for charging, this could threaten the grid’s power stability. This could result in a blackout situation. Thus, efficient algorithms and mechanisms should be used to properly distribute the power from the grid to vehicles by considering different objective functions and system constraints [15,16]. To maximize customer satisfaction and minimize the load on the grid, a control appliance would be required to properly govern different battery loads from several electric vehicles [17].

The charging infrastructure for PHEVs or PEVs consists of three major substations, which are shown in Figure 1 [18]:

• Conventional power grids and renewable resources
• Intelligent charging controllers
• PHEV or PEV battery chargers and consumers

By applying an optimization technique to the model, power can be allocated efficiently to the connected vehicles. For smart charging of PHEVs, energy demand, energy price, charging time, and battery and grid capacity should be considered. APSO and its five variants were applied to maximize the state of charge and, thus, allocate power intelligently and efficiently to vehicles. There are some constraints to be met; i.e., the state of charge and power of the vehicle, ramp rate of SoC, and sum of power supplied to vehicles should be within limits.

Due to the variation in the number of vehicles plugged in for charging at different times, there is a need for an overall demand arrangement, which will help minimize the grid strain and enhance power production and supply [19].

Charging of PHEVs can be classified into two types, i.e., household charging and public charging. In this research, public charging stations are considered, as vehicles are usually expected to be charged in public stations [20]. A systematized charging infrastructure is required for charging such a large number of cars, because these new loads are an extra burden on the existing grid [21]. Thus, to facilitate this charging infrastructure, much research is being carried out [22,23]. Moreover, research to enable adequate energy storage, reduction in cost, service quality, and optimization of power allocation from charging infrastructure to vehicles is also in progress [24].

The type of optimization problem determines the kind of optimization algorithm to be applied to obtain the best fitness value. The manufacturing and design of a product are engineering optimization problems with specific constraints to be met. Optimization problems can be categorized in terms of several constraints: the type of function, determinacy, variable types, the landscape of the objective function, and the number of objectives to be met [25,26,27,28].

There are many optimization algorithms, but the nature of the algorithm and the optimization problem will decide which algorithm will solve a problem. Optimization algorithms are of three types, i.e., deterministic optimization algorithms, stochastic optimization algorithms, and metaheuristic optimization algorithms [28]. This research incorporated metaheuristic optimization algorithms to allocate a charge to the electric vehicles’ batteries from the grid efficiently. These algorithms use stochastic and deterministic approaches to obtain optimal global results. This algorithm has two main steps: the first step is selecting the best solution, which will lead to convergence at the end of the iterations, and the second step is to include randomness to ensure that the solution will not stick to the local solution and thus increase diversity. These algorithms use an approach followed by nature: i.e., the processes used by birds and fish to find food, genetic mutation, repulsion, attraction within the species, etc. PSO, APSO, and their variants are metaheuristic algorithms.

As mentioned previously, metaheuristic algorithms always follow different trajectories when applied to the same optimization problem with the same initial conditions. Thus, if we consider two metaheuristic algorithms, say A and B, and use them on the same problem, we cannot say that algorithm A outperformed algorithm B because it gives a better global result. This is due to the stochastic (random) nature of these algorithms. Thus, to properly compare the two algorithms, we must perform statistical tests on the results obtained by applying both algorithms to the same problem many times. One-way ANOVA is a statistical test that can compare more than two metaheuristic algorithms [27,28]. The advantages of using ANOVA over other statistical tests are that ANOVA uses simple algebra for its calculations, making it easier to implement, and that more than two groups with different observations can be compared at a time [29].

One-way ANOVA (“analysis of variance”) is a parametric test that compares the means of two or more independent groups to observe whether there is some statistical evidence to prove that the associated group means are significantly different. ANOVA’s method of implementation and hypotheses are given in Appendix A.

In reference [30], plug-in hybrid electric vehicles were charged using PSO and APSO algorithms. It was concluded that the PSO algorithm has certain drawbacks, like early convergence, requiring memory to update particles, and a low-quality solution, while, on the other hand, APSO outperformed PSO. APSO is a more efficient algorithm with a single update equation, a high-quality solution, and local exploitation capability. Reference [31] also highlights the significance of utilizing the APSO technique, since it infinitesimally reduces the Big O time complexity for the common benchmark test case of the CSTHTS optimization issue. Thus, the accelerated particle swarm optimization (APSO) approach and its five variants are employed in this research project to provide real-time and large-scale optimization for intelligently distributing available grid electricity to cars while maintaining the stability of the grid system. State of charge (SoC) is the main parameter that should be maximized to charge the vehicles effectively. Six metaheuristic algorithms are employed using four automobile datasets, including 50 PHEVs, 100 PHEVs, 500 PHEVs, and 1000 PHEVs. Statistical comparisons of six algorithms are made using a one-way ANOVA test.

The paper is organized in the following way. In Section 2, the problem statement is described. Accelerated PSO and its variants are exploited in Section 3. Section 4 pinpoints the methodology adopted in the research comprehensively. Section 5 is dedicated to results and discussion, considering four datasets of PHEVs. The conclusions of the presented study are drawn in Section 6.

2. Problem Statement

The purpose of this research was to intelligently allocate available power from the grid to plugged-in vehicles. Consider a charging infrastructure of total power capacity P and N PHEVs required to be served overall, 24 h a day. The vehicle should leave the charging station before the estimated leaving time, and each PHEV can be plugged in only once a day. State of charge is the main performance parameter that needs to be maximized to intelligently provide power to each connected vehicle from the grid without risking a blackout.

The fitness function, as given in reference [20], is given in (1).

$$MaxJ(k)=\sum _i\left(w{t}_i\times So{C}_i(k+1)\right)$$

(1)

$$w{t}_i(k)=f\left(C{a}_{r,i}(k),T_{r,i}(k),D_i(k)\right)$$

(2)

$$C{a}_{r,i}(k)=\left(1-So{C}_i(k)\right)\times C{a}_i$$

(3)

where $C{a}_{r,i}(k)$ is the remaining battery capacity required to be filled for the ith PHEV at a time step k. This is illustrated in Figure 2.

$T_{r,i}(k)$ is the remaining time left for fully charging the ith vehicle at time step k, $D_i(k)$ is the difference between real-time electricity price and the price which the ith vehicle’s owner agrees to pay, $C{a}_i$ is the rated battery capacity of the ith vehicle, $So{C}_i(k)$ and $So{C}_i(k+1)$ are the states of charge of the ith car at time steps k and k + 1, respectively, and $w{t}_i(k)$ is the weighting term of the ith vehicle, which is a function of time of charging, current SoC, and energy price. If a particular car has significantly less current SoC and less time to charge, but its owner has agreed to pay an extra cost, then the charging system will provide more power to that charger. In this research, the weighting term is randomly selected for the vehicles, and all concentration is kept to distribute the available power from the grid to the maximum number of vehicles. The optimization of weighting terms is another area for research. This example indicates the relation between $w{t}_i(k)$, $D_i(k)$, $T_{r,i}(k)$, and $C{a}_{r,i}(k)$. That is to say:

$$w{t}_i(k)\propto \left(C{a}_{r,i}(k)+D_i(k)+\frac{1}{T_{r,i}(k)}\right)$$

(4)

The terms in (4) need to be normalized because their scales are not similar.

$$c{a}_{r,i}(k)=\frac{C{a}_{r,i}(k)-Min[C{a}_{r,i}(k)]}{Max[C{a}_{r,i}(k)]-Min[C{a}_{r,i}(k)]}$$

(5)

$$d_i(k)=\frac{D_i(k)-Min[D_i(k)]}{Max[D_i(k)]-Min[D_i(k)]}$$

(6)

$$t_{r,i}(k)=\frac{T_{r,i}(k)-Min[T_{r,i}(k)]}{Max[T_{r,i}(k)]-Min[T_{r,i}(k)]}$$

(7)

Depending on preferences for these parameters, each has different important factors, as shown in (8). In this work, charging current is assumed to be a constant quantity over a time Δt.

$$w(k)={\alpha }_{1}c{a}_{r,i}(k)+{\alpha }_2{t}_{r,i}(k)+{\alpha }_3{d}_i(k)$$

(8)

$$C{a}_i\times \left\{So{C}_i(k+1)-So{C}_i(k)\right\}=Q_i=I_i(k)\times \Delta t$$

(9)

$$So{C}_i(k+1)=So{C}_i(k)+\left\{\frac{I_i(k)}{C{a}_i}\times \Delta t\right\}$$

(10)

where $I_i(k)$ is charging current over a period Δt and the sample time assigned by the operator of the parking station is Δt. $Q_i$ is the charge of the battery of the ith PHEV.

Considering the battery model, a capacitor circuit with $C{a}_i$ denoting the battery capacitance in Farads, the equation for charging the battery is given as

$$C{a}_i\times \frac{d{V}_i}{dt}=I_i$$

(11)

For a short time period, a voltage change can be taken to be linear.

$$\left[\frac{V_i(k+1)-V_i(k)}{\Delta t}\right]\times C{a}_i=I_i(k)$$

(12)

$$V_i(k+1)-V_i(k)=\frac{I_i(k)\times \Delta t}{C{a}_i}$$

(13)

Now, replacing $I_i(k)$ with $P_i(k)$ because power has to be allocated intelligently, this will be the main decision parameter:

$$\frac{2\times P_i(k)}{V_i(k+1)+V_i(k)}\times C{a}_i=I_i(k)$$

(14)

Put $I_i(k)$ from (14) into (12):

$$V_i(k)=\frac{2\times P_i(k)}{I_i}-\frac{I_i\times \Delta t}{C{a}_i}$$

(15)

Putting (13) into (15) yields a formula for $V_i(k)$, which will be used in algorithm coding to find the battery’s voltage at time step k.

$$V_i(k+1)=\sqrt{\frac{P_i(k)\times \Delta t}{0.5\times C{a}_i}+V_i{(k)}^2}$$

(16)

Substituting (12) into (10) gives $So{C}_i(k+1)$, which, along with (15), is put into (1), which will provide the final form of the objective function, which is to be maximized by applying APSO and its variants.

$$So{C}_i(k+1)=So{C}_i(k)+\left\{\frac{P_i(k)\times \Delta t}{0.5\times C{a}_i\times \left(V_i(k+1)+V_i(k)\right)}\right\}$$

(17)

$$MaxJ(k)=So{C}_i(k)+{\displaystyle \sum _i\left(w{t}_i\times \left\{\left(\frac{P_i(k)\times \Delta t}{0.5\times C{a}_i\times \left(\sqrt{\frac{2\times P_i(k)\times \Delta t}{C{a}_i}+V_i{(k)}^2}+V_i(k)\right)}\right)\right\}\right)}$$

(18)

The above equation is the final form of the objective function, which needs to be maximized.

2.1. Constraints

Rate of charging (i.e., slow, medium, or fast required SoC), the time during which the PHEV is connected to the charger, the price of electricity that the owner of the vehicle has agreed to pay, and battery conditions and requirements are some real-world constraints that should be taken in account and solved. Furthermore, sampling time is limited by communications bandwidth, affecting the processing ability of PHEV.

Primary energy constraints include utility power, i.e., the power available from the charging grid, and maximum power charged by the vehicle. The overall efficiency of the charging station is assumed to be constant at any time step k and denoted as η. $So{C}_{i,\max }(k)$ is the maximum limit of battery SoC for the ith PHEV, and is user-defined. When the $So{C}_i$ of the ith vehicle reaches its $So{C}_{i,\max }$ limit, the battery is fully charged, and the charger switches to standby mode. $\Delta So{C}_{\max }$ limits the SoC ramp rate. The control system information updates when:

• Utility information changes
• A new PHEV/PEV is plugged-in
• Δt has passed (Δt is sample time)

System constraints are defined as [32]:

$${\displaystyle \sum _i{P}_i}(k)\le P_{utility}(k)\times \eta$$

(19)

$$0\le P_i(k)\le P_{i,\max }(k)$$

(20)

$$0\le So{C}_i(k)\le So{C}_{i,\max }(k)$$

(21)

$$0\le So{C}_i(k+1)-So{C}_i(k)\le \Delta So{C}_{\max }$$

(22)

The constraint in (19) ensures that a blackout situation cannot happen; i.e., the sum of power delivered to all the PHEVs should be less than the product of utility power $P_{utility}$ and charging station efficiency η so that there is always some power in the charging station and the risk of blackout is reduced. Equations (20) and (21) are the constraints that each PHEV power and state of charge should always be within some specified limit. The constraint presented in (22) limits the ramp rate of the states of charge of car batteries and avoids sudden or abrupt changes in charge that could damage the battery.

2.2. Variables

The following variables are taken from reference [32] in this research:

$$0.2\le SoC\le 0.8$$

(23)

$$\mathrm{Waiting}\mathrm{time}\le 30\min \left(1800\mathrm{s}\right)$$

(24)

$$16\mathrm{kWh}\le \mathrm{Capacity}\mathrm{of}\mathrm{Battery}\mathrm{of}\mathrm{PHEVs}\le 40\mathrm{kWh}$$

(25)

2.3. Fixed Parameters

The fixed parameters remain constant throughout the use of the algorithm, as taken from reference [31].

• The charger’s maximum limit is $P_{i,\max }=6.7\mathrm{kW}$, meaning a particular vehicle’s power cannot exceed the 6.7 kW limit.
• Overall charging grid efficiency is taken as η = 0.9 all the time.
• $P_{utility}(k)=\eta \times n(k)\times P_{i,\max }(k)$, where η = 0.9, n is the no. of PHEVs plugged in at time step k and $P_{i,\max }=6.7\mathrm{kW}$ at any time step k.
• Sample time equals Δt = 20 min or 1200 s.
• Charging one vehicle multiple times in a day is not considered. Each vehicle is plugged in once a day.

Usually, a level-two charging method is considered for electric chargers of vehicles in both public and private stations. In this research, a level -two charging method is used: a single-phase outlet with 240 VAC, 32 A, and 7.7 kVA [20]. Figure 3 summarizes the problem statement.

3. Accelerated Particle Swarm Optimization Algorithm (APSO)

In 2007, at Cambridge University, to obtain convergence of the algorithm quickly by using the global best only, Yang developed the accelerated particle swarm optimization (APSO) algorithm [28]. This algorithm falls into the metaheuristic algorithms category and has been applied to many problems, as listed in [30,31,32,33,34,35,36,37,38,39,40,41]. One of the advantages of using APSO compared with other metaheuristic algorithms is its simplicity, as it uses only one update per iteration to reach convergence in a relatively short time. Thus, this algorithm requires less computational time and also uses less memory for storing data [28]; in references [26,27] a canonical version of APSO is proposed, which is simple and gives a robust solution in less iterations.

When applying the APSO algorithm to any problem, the first step is generating a set of solutions randomly initialized in each search space. Then, these particles move taking the influence of the global best solution, which is obtained by applying an updated equation to each particle. Equation (26) gives a revised equation for the APSO algorithm; i.e., the equation which will change the position of particles in each iteration using the global best so that particles converge to the final solution.

$$x_i^{n+1}=\left\{\left(1-\beta (n)\times x_i^n\right)+\left(\beta (n)\times g^{*n}\right)+\left(\alpha (n)\times (\epsilon -0.5)\right)\right\}$$

(26)

where $x_i^{n+1}$ is the ith particle value at iteration n + 1, $x_i^n$ is the ith particle value at iteration n, $g^{*n}$ is the global best among all the particles, $\alpha (n)$ and $\beta (n)$ are the tuning parameters of APSO at the nth iteration, and $\epsilon$ is a random number between 0 and 1.

The tuning parameters play a crucial role in the performance of the APSO algorithm, as the stochastic or random behavior of the APSO algorithm increases by increasing the α value and the deterministic behavior increases by increasing the β value. Thus, when changing the tuning parameters and values the algorithm behaves differently. In this research, six variant APSO algorithms and a simple APSO algorithm were applied to the same problem. Usually, the values of α and β are between 0 and 1, but reference [28] proposed the most efficient performance range for these tuning parameters, which is 0.1 to 0.4 for α and 0.1 to 0.7 for β. It is recommended that α and β values do not exceed one because the solution diverges. According to reference [38], the values of alpha (α) and beta (β) are constant or fixed in the canonical form. Due to randomness in the algorithmic procedure, there is a low chance that the algorithm will follow the same trajectory when applied again to the same problem [42].

The optimized result found iteratively is given as an output by the optimizer. In the first iteration, particles are initialized, then these particles are influenced by the global best particle, which is shown as a blue dot with a red asterisk in each iteration. This procedure is followed in all iterations, and particles move under the influence of the global best particle. When the stopping criteria are met, particles ultimately converge to the global best result. However, the global best particle, a blue dot with a red asterisk, will show the global best solution [27].

Variants of APSO

By controlling the values of the tuning parameters, i.e., alpha and beta, in the update equation, it is possible to achieve a better result by changing the search space and the exploitation strategies for the simple APSO algorithm. The two metaheuristic algorithms can only be compared statistically. Results obtained by a single algorithm run cannot be compared statistically. Instead, each algorithm should be simulated several times, and their results can be compared. The following are the variants used in this research to achieve better results for allocating power to PHEVs. Increasing the alpha value will increase the randomization of the APSO algorithm, while changing the beta will affect the weight of both the local best of every particle and the global best particle of each iteration. There are many other variants of APSO, i.e., Levi flights, chaotic map-based methods, etc. In this research, linear changes in alpha and beta are considered. The following variants have been taken from [43] and are presented in

Table 1. Variants of the APSO algorithm [43].

Algorithm	Alpha	Beta
APSO Variant 1	$\alpha (n)={\alpha }_{\max }-\left(\frac{{\alpha }_{\max }-{\alpha }_{\min }}{N}\right)\times n$	$\beta (n)=\beta {}_{\min }+\left(\frac{\beta {}_{\max }-\beta {}_{\min }}{N}\right)\times n\times \sin \left(\frac{\pi }{2}\right)$
APSO Variant 2	$\alpha (n)=\alpha {}_{\min }+\left(\frac{\alpha {}_{\max }-\alpha {}_{\min }}{N}\right)\times n\times \cos \left(\frac{\pi }{2}\right)$	$\beta (n)={\beta }_{\max }-\left(\frac{{\beta }_{\max }-{\beta }_{\min }}{N}\right)\times n$
APSO Variant 3	$\alpha (n)=\alpha {}_{\min }+\left(\frac{\alpha {}_{\max }-\alpha {}_{\min }}{N}\right)\times n\times \cos \left(\frac{\pi }{2}\right)$	$\beta (n)={\beta }_{\min }+\left(\frac{{\beta }_{\max }-{\beta }_{\min }}{N}\right)\times n\times \cos \left(\frac{\pi }{2}\right)$
APSO Variant 4	$\alpha (n)=\alpha {}_{\max }-\left(\frac{\alpha {}_{\max }-\alpha {}_{\min }}{N}\right)\times n$	$\beta (n)={\beta }_{\max }-\left(\frac{{\beta }_{\max }-{\beta }_{\min }}{N}\right)\times n$
APSO Variant 5	$\alpha (n)=\alpha {}_{\min }+\left(\frac{\alpha {}_{\max }-\alpha {}_{\min }}{N}\right)\times n\times \cos \left(\frac{\pi }{2}\right)$	$\beta (n)={\beta }_{\min }+\left(\frac{{\beta }_{\max }-{\beta }_{\min }}{N}\right)\times n$

N = the total number of iterations, n = the current iteration number, [α_max α_min] is the range of α, and [β_max β_min] is the range of β.

.

In this research, the α range is taken as [0.1 to 0.4] and the β range is taken as [0.2 to 0.5] in all variants. The APSO algorithm itself gives better results, but these results can be improved by adding variations in the alpha and beta values that are actually tuning the algorithmic search procedure. Randomization can be increased by increasing the alpha value, and the weight of the local best of every particle and the global best particle of each iteration can be varied by varying the beta value. Thus, by changing the tuning parameters, the search trajectory varies and better results can be achieved.

4. Methodology

The mathematical model of the problem solved in this research is explained. The objective function in (1) is optimized using the APSO algorithm and its five variants. Following are the main steps of these algorithms used to achieve results.

• First, generate a group of particles, where each particle is a set of random solutions; i.e., in this case, each particle is a set of powers allocated to PHEVs, but these powers should be within the limits given in the constraint equation. Moreover, it should also fulfill the constraint equation, which is that the sum of all the powers should be equal to or less than a product of P_utility and efficiency of the grid. Therefore, P = {P₁, P₂, P₃, …, P_n}, where, P₁ = {p₁, p₂, …, p_i}, P₂ = {p₁, p₂, …, p_i}, P₃ = {p₁, p₂, …, p_i}, and P_n = {p₁, p₂, …, p_i} are generated. Here, n is the number of particles set and i is the number of vehicles connected to the grid for charging.
• Find the voltages of batteries corresponding to the power values of each PHEV using (16).
• Find the best fitness value by putting power and voltage values into (18) and considering the weighing terms and battery capacities.
• Apply the APSO algorithm technique to the particles using (1).
• All the constraints are checked at each iteration.
• The local best fitness value is the maximum value obtained by each particle.
• The global best solution is the maximum value obtained at the end of the iteration.
• Update the power values using the local best and global best.
• Apply Step 2 onward until the stop criteria are met.
• Generate the results.
• Apply a one-way ANOVA test to the dataset obtained after running the algorithm 30 times for each PHEV set to compare the algorithms statistically.
• Generate the ANOVA result.

5. Results and Discussion

In this section, APSO and its five techniques are simulated to obtain the best fitness value for the fitness function stated in (18). Results for 50, 100, 500, and 1000 PHEVs are given. It can be seen that convergence occurs very rapidly. All the simulations were run on a computer with the following configuration:

• System type: 64-bit operating system, ×64-based processor
• Processor: Intel(R) Core (TM) i3-4010U CPU @ 1.70 GHz
• RAM: 8.00 GB
• Software: MATLAB R2015a

An APSO algorithm is applied to each 50-, 100-, 500-, and 1000-PHEV set; the results of 100 iterations using 100 particles are given below. Furthermore, the same data is simulated using five more algorithms, an extension of the simple APSO algorithm. In the APSO variants algorithm, i.e., from APSO Variant 1 to APSO Variant 5, the tuning parameters alpha (α) and beta (β) are recalculated in each iteration. The parameter values used for APSOs are summarized in

Table 2. Parameters for APSO and its variants.

Parameters	Values
Size of swarms	100
Maximum iterations	100
Alpha, α for APSO	0.2
Alpha, α for APSO variants	0.1 to 0.4
Beta, β for APSO	0.5
Beta, β for APSO variants	0.2 to 0.5

.

5.1. Dataset for 50 PHEVs

The dataset of 50 PHEVs was simulated by APSO and its five variants, and the results obtained are given in

Table 3. Summarized results of APSO and its variants for 50 PHEVs.

Algorithm Type	Best Fitness Value	Computational Time (s)
APSO	10.8615	10.336
APSO Variant 1	12.6598	10.538
APSO Variant 2	12.8692	10.326
APSO Variant 3	13.2891	10.667
APSO Variant 4	14.2314	10.819
APSO Variant 5	15.7580	11.055

. All these results fulfilled the constraints, and it was observed that APSO 5 provides the highest possible fitness value.

To make comparison easier, all the algorithm results are combined in Figure 4. It is shown that the convergence behaviors of these six algorithms are the same. It can be noted that convergence occurred within ten iterations. Regarding the fitness value, all APSO variants give better-optimized values than a simple APSO algorithm. Due to the tuning of the parameters in the APSO variants, provided in Table 2, alpha and beta values are taken between 0.1 and 0.4 and 0.2 and 0.5, respectively, and their value is tuned in each iteration until we obtain the best-optimized result. APSO Variant 5 gives the highest optimized value for state of charge compared with all the other simulated algorithms, as shown in Figure 4. Table 3 summarizes the results for all algorithms, and the computational time for APSO 5 is 0.2 s more than for APSO 4, but the fitness value is increased from 14.2314 to 15.758 (1.5266 difference), and this difference has a strong impact on the charging scenario.

Table 4. Comparison of the APSO 5 solution with other algorithm’s solutions in terms of percentage increase for the 50-PHEV dataset.

Algorithms Compared with APSO 5	Percentage Increase in the APSO Variant 5 Solution w.r.t Other Algorithms
APSO	45.49%
APSO 1	24.47%
APSO 2	22.44%
APSO 3	18.57%
APSO 4	10.72%

represents the comparison between APSO 5 and other algorithmic solutions in terms of percentage increase in fitness values.

APSO and its variants are meta-heuristic algorithms that are stochastic in nature. Thus, to compare these algorithms’ performance, a statistical test should be performed, and, for this, each algorithm should be applied to the dataset approximately 30 times. We cannot say anything about the superiority of any algorithm by seeing only one result. Thus, the results obtained after applying six algorithms to the 50-PHEV dataset 30 times are given in

Table 5. APSO and its variants’ results for 50 PHEVs (30 Trials).

Runs	APSO	APSO 1	APSO 2	APSO 3	APSO 4	APSO 5
1	13.502	13.5175	13.34854	14.19331	12.53391	12.72591
2	12.19284	11.75398	14.39	14.59802	11.55284	13.59895
3	12.16518	10.85215	11.00314	9.831956	13.77892	13.49392
4	12.65903	12.59184	14.46577	14.23699	11.82024	12.28426
5	12.54002	12.10285	13.22647	13.17717	13.26892	11.29777
6	14.14823	11.71262	14.03275	12.7529	13.71459	11.99627
7	12.75165	11.88622	14.28892	10.13385	13.3613	13.49167
8	13.61946	13.78939	13.917	11.49369	14.89828	13.40129
9	12.93798	12.34256	11.66381	12.73868	13.5866	16.65707
10	13.37426	10.85414	14.51048	10.03001	12.93882	13.83871
11	12.315	14.80563	12.88733	12.66867	14.51979	14.1858
12	10.48265	14.09458	11.78633	12.94528	13.11282	12.43806
13	14.1308	11.66965	9.381821	10.36878	12.41153	12.97832
14	12.33227	15.11519	13.37119	12.32318	10.64276	13.06213
15	12.56853	14.44043	11.32293	13.17804	14.57611	13.56715
16	13.65387	11.55932	13.33428	13.96601	13.11313	12.36803
17	11.2915	13.1983	14.46242	12.19765	11.53929	11.81985
18	11.11521	13.5135	13.57153	9.139856	13.07747	12.31658
19	12.28165	13.45304	12.42848	12.6213	12.32869	12.08295
20	12.45477	13.59552	13.3488	14.21477	12.03432	11.86018
21	12.49387	11.53808	14.02768	13.03311	13.90078	13.84535
Average	12.64302	12.82533	13.11153	12.74944	12.97046	13.04176
Max.	14.14823	15.11519	14.51048	14.59802	14.57611	16.65707

, and the maximum value is colored blue.

After obtaining 30 trial results, a one-way ANOVA statistical test was applied to observe whether all six algorithms were statistically similar in performance or not. As discussed earlier, to compare the means of more than two independent groups, one-way analysis of variance (ANOVA) is used instead of a t-test. If the sigma value is greater than 0.05, we can say that all the groups have equal means and are statistically similar in performance, and vice versa. Statistical testing was performed on SPSS software, and the results are shown in

Table 6. One-way ANOVA test on the 50-PHEV dataset.

	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	4.905	5	0.981	0.593	0.705
Within Groups	287.709	174	1.653
Total	292.614	179

; the sig value was 0.705, much greater than 0.05, and, thus, it was concluded that all the algorithms are statistically performing the same. However, with respect to the average value of the fitness function among the 30 trials and the maximum value among the 30 trials, it is not wrong to say that APSO Variant 5 performed better than the others because it gave the largest fitness value. Results of the one-way ANOVA test on the 50-PHEV dataset are presented in Table 6.

5.2. Dataset of 100 PHEVs

After applying the six algorithms to the 100-car set, the results obtained by all six algorithms, along with computational time, are given in

Table 7. Summarized results of APSO and its variants for 100 PHEVS.

Algorithm Type	Best Fitness Value	Computational Time (s)
APSO	24.227	19.226
APSO Variant 1	25.134	18.523
APSO Variant 2	27.518	17.711
APSO Variant 3	25.109	17.702
APSO Variant 4	26.288	17.720
APSO Variant 5	29.395	18.738

.

Per Table 7, simple APSO gives 24.227, while APSO 5 gives 29.395, which will be highly beneficial when charging the vehicles because we need to charge vehicles to the maximum given utility power and must follow the constraints. The results obtained from all six algorithms are shown in Figure 5 so that they can be compared easily.

In Figure 5, it can be noted that there is a similarity in the convergence behavior for the 100- and 50-car sets. All variants outperformed the simple APSO algorithm, and APSO 5 gives the highest possible fitness value with a computational time trade-off.

Table 8. Comparison of the APSO 5 solution with other algorithms’ solutions regarding percentage increase for the 100-PHEV dataset.

Algorithms Compared with APSO 5	Percentage Increase in the APSO Variant 5 Solution w.r.t Other Algorithms
APSO	21.33%
APSO 1	16.94%
APSO 2	6.82%
APSO 3	17.06%
APSO 4	11.81%

compares APSO 5 with the other algorithms and shows how much the percentage of the solution increases with APSO 5. Compared statistically, the 30 trial results for the 100-car set as simulated by six algorithms are given in

Table 9. APSO and its variants’ results for 100 PHEVs (30 Trials).

Runs	APSO	APSO 1	APSO 2	APSO 3	APSO 4	APSO 5
1	25.72988	24.27926	25.76255	25.76255	25.69987	29.71674
2	24.66268	27.02372	23.55225	23.55225	23.32775	23.61872
3	25.98954	27.48761	24.49497	24.49497	27.6624	25.01369
4	25.74936	25.68086	26.57952	26.57952	26.42648	25.8907
5	23.57134	23.36277	25.68215	25.68215	23.89143	28.48542
6	24.51917	27.70655	25.41083	25.41083	24.88746	26.01486
7	26.57295	26.49121	26.97476	26.97476	21.93518	26.0455
8	25.72195	23.92126	25.6395	25.6395	24.33336	23.07433
9	25.39879	24.73768	22.27257	22.27257	26.39556	23.99511
10	26.96351	22.05293	24.50455	24.50455	26.4875	24.36713
11	25.54414	25.11393	24.7606	24.7606	25.08445	24.49567
12	22.16602	26.41224	22.14677	22.14677	25.48106	25.20646
13	24.49963	26.46526	24.36882	24.36882	27.40725	26.10468
14	24.79176	26.46526	25.21517	25.21517	24.27485	26.72789
15	22.25561	25.48106	26.00881	26.00881	28.32577	27.81361
16	24.52954	27.3858	27.63619	27.63619	25.42645	23.63633
17	25.21217	24.25455	22.87731	22.87731	27.21134	27.81298
18	26.00402	28.30697	25.85988	25.85988	27.81529	24.34076
19	27.48829	25.41936	26.44606	26.44606	25.99717	25.45669
20	22.91014	27.16994	24.7946	24.7946	23.58674	25.97135
21	25.95945	27.80481	27.59583	27.59583	21.13038	24.53901
22	26.42808	25.9822	25.0354	25.0354	21.13038	24.39439
23	24.84809	23.71353	26.40167	26.40167	24.79599	25.46869
24	27.7008	21.11408	25.56867	25.56867	25.41023	26.96501
25	25.05716	24.64604	26.52091	26.52091	22.19488	24.1428
26	26.39884	24.81851	25.14769	25.14769	26.81314	24.06652
27	25.56719	25.40324	23.28052	23.28052	26.77089	27.75539
28	26.69622	21.59593	24.29273	24.29273	26.88282	27.83387
29	25.08511	26.78964	27.0289	27.0289	26.33804	23.38254
30	23.28059	26.55734	27.51446	27.51446	23.32128	25.19494
Average	25.2434	25.45478	25.31249	25.31249	25.21485	25.58439
Max.	27.7008	28.30697	27.63619	27.63619	28.32577	29.71674

.

The 30 trial results show that APSO Variant 5 gives the maximum average value. We are applying one-way ANOVA to see whether these algorithms perform differently or not. It can be seen in

Table 10. One-way ANOVA test on the 100-PHEV dataset.

	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	2.948	5	0.590	0.214	0.956
Within Groups	478.649	174	2.751
Total	481.597	179

that the sig. value is 0.956, which is much higher than 0.05, and, thus, it can be said that all perform statistically the same.

5.3. Dataset of 500 PHEVs

Results obtained after simulating the 500-car set data are given in

Table 11. Summarized results of APSO and its variants for 500 PHEVs.

Algorithm Type	Best Fitness Value	Computational Time (s)
APSO	126.292	84.989
APSO Variant 1	128.471	80.755
APSO Variant 2	129.330	97.386
APSO Variant 3	127.682	80.271
APSO Variant 4	127.540	87.832
APSO Variant 5	132.003	81.951

.

For this dataset, APSO 5 gives a maximum fitness value with comparatively less computational time. APSO 2 also shows a better maximum fitness value, but its computational time is higher than for all other algorithms. Therefore, the stochastic nature of the algorithms produces a beneficial result, as seen in this set. The convergence behaviors for all these algorithms are similar, as shown in Figure 6.

The maximum fitness value and lowest computational time are achieved in the 500-PHEV set by APSO 5.

Table 12. Comparison of the APSO 5 solution with other algorithms solutions regarding percentage increase for the 500-PHEV dataset.

Algorithms Compared with APSO 5	Percentage Increase in the APSO Variant 5 Solution w.r.t Other Algorithms
APSO	4.52%
APSO 1	2.74%
APSO 2	2.06%
APSO 3	3.38%
APSO 4	3.49%

represents the percentage increase with APSO 5 w.r.t each algorithm and proves that APSO 5 outperforms other algorithms. But to compare statistically, all six algorithms simulated this data 30 times, and their results are shown in

Table 13. APSO and its variants’ results for 500 PHEVs (30 Trials).

Runs	APSO	APSO 1	APSO 2	APSO 3	APSO 4	APSO 5
1	126.5735	121.9186	126.49897	131.9917	123.1114	121.9186
2	118.5763	125.0634	122.96496	132.6757	126.8516	125.0634
3	125.7073	129.5977	124.11509	125.0797	124.4189	129.5977
4	129.8414	127.6419	131.99168	125.9139	121.9661	127.6419
5	132.9401	127.4631	132.6757	122.8347	124.9691	127.4631
6	120.7045	129.9951	125.07965	121.9275	130.079	129.9951
7	115.1843	126.3374	125.91388	124.988	127.5952	126.3374
8	132.0033	121.4333	122.83467	129.5959	127.5402	130.3004
9	122.437	121.9186	127.10125	127.9723	129.9992	128.4759
10	121.4333	125.0634	123.29089	127.4973	127.2541	129.6517
11	126.337	129.5977	122.04113	129.9619	121.4614	127.1907
12	129.9952	127.6419	124.50352	126.4876	122.4556	124.8892
13	127.463	127.4631	128.31973	121.4392	132.0014	131.7331
14	127.6419	129.9951	125.22181	122.4328	114.9656	124.1206
15	129.5977	126.3374	134.83523	131.998	120.7139	126.4998
16	125.0634	121.4333	125.66249	115.0261	133.0082	122.8755
17	121.9186	122.437	123.21541	120.6972	130.9115	124.1174
18	126.337	132.0033	126.32302	132.953	125.7167	132.0569
19	129.9952	115.1843	124.42171	129.9211	118.5398	132.6094
20	127.463	120.7045	122.99828	125.6997	126.5188	125.0891
21	127.6419	132.9401	124.47612	118.5409	130.4012	125.9237
22	129.5977	129.8413	119.60667	126.5255	128.4612	122.8309
23	125.0634	125.7073	127.97136	128.0473	129.6462	127.0979
24	121.9186	118.5763	131.32843	130.3305	127.127	123.3213
25	121.9508	126.5735	126.36098	128.4446	124.8817	122.0304
26	124.6762	127.6401	126.22204	129.6345	132.0327	124.5046
27	132.306	130.3004	125.2958	127.683	124.2326	128.334
28	126.0091	128.4759	130.28276	124.872	126.5118	125.2105
29	122.2824	129.6517	123.83581	131.811	123.2037	134.859
30	126.3052	127.1907	129.99217	124.1028	124.1148	125.6579
Average	125.8321	126.2042	126.17937	126.5695	126.023	126.9132
Max.	132.9401	132.9401	134.83523	132.953	133.0082	134.859

. On applying one-way ANOVA on 30 trials, the obtained results are given in

Table 14. One-way ANOVA test on the 500-PHEV dataset.

	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	23.010	5	4.602	0.293	0.917
Within Groups	2736.162	174	15.725
Total	2759.172	179

below.

As can be seen from Table 14, the significant value is 0.917, which is much greater than 0.05, and, thus, this dataset also follows the trend that all these algorithms are statistically performing the same, but APSO with Variant 5 gives the highest possible fitness value.

However, for this dataset, APSO Variant 5 has the lowest computational time along with the highest fitness value, while, for the other datasets, APSO 5 gives the highest fitness value with little compromise on computational time.

5.4. Dataset of 1000 PHEVs

APSO and its variant algorithms were applied to the 1000-PHEV set. Keeping the constraints satisfied, the following results were gathered.

All six algorithms have exemplary convergence behavior and have very little chance of getting stuck in local maxima due to the algorithm’s stochastic nature. Additionally, all variants work better than the simple APSO algorithm due to the tuning of the parameters. It can be seen from

Table 15. Summarized results for APSO and its variants for 1000 PHEVS.

Algorithm Type	Best Fitness Value	Computational Time (s)
APSO	246.048	170.259
APSO Variant 1	252.678	163.379
APSO Variant 2	240.319	170.626
APSO Variant 3	253.034	167.090
APSO Variant 4	255.659	174.643
APSO Variant 5	258.104	182.413

that APSO Variant 5 gives the maximum value among all the variants with a compromise on computational time. However, here, we are trying to provide full power from available utility power to the vehicles and maximize the state of charge. Therefore, a small difference in computational time can be ignored, and only the maximum fitness value should be considered.

Table 16. Comparison of the APSO 5 solution with other algorithms regarding percentage increase for the 1000-PHEV dataset.

Algorithms Compared with APSO 5	Percentage Increase in the APSO Variant 5 Solution w.r.t Other Algorithms
APSO	4.81%
APSO 1	2.14%
APSO 2	7.40%
APSO 3	2.00%
APSO 4	1.01%

shows the importance of using APSO 5, which improves the solution compared to other algorithms. The convergence behavior of the algorithms is shown in Figure 7.

Due to the stochastic nature of these algorithms, it is required to run each algorithm a certain number of times (say for 30 trials) on each PHEV set to compare their performance statistically. Therefore, again, six algorithms were applied to the 1000-car set 30 times, and the results are given in

Table 17. APSO and its variants’ results for 1000 PHEVS (30 trials).

Runs	APSO	APSO 1	APSO 2	APSO 3	APSO 4	APSO 5
1	256.13764	249.7973	240.3178	250.4989	261.9885	249.0351
2	246.0487	245.60438	254.088	252.7105	246.6598	242.1332
3	247.61726	248.15366	250.3368	253.0247	250.8615	243.6935
4	255.83371	252.3311	250.3702	244.8549	255.0794	249.8352
5	258.10479	252.6764	258.0784	249.7282	251.2514	249.8802
6	248.84371	247.45823	248.6636	259.9551	250.8295	256.8304
7	247.95574	248.84369	247.9065	254.3202	249.0221	259.1609
8	243.80545	247.95577	247.5528	250.6186	242.0723	245.4554
9	258.42472	243.80521	255.7673	252.0285	243.5746	255.4172
10	248.68239	247.61727	258.0784	250.6758	247.571	263.5139
11	250.32604	255.83371	248.6636	253.4435	255.91	245.5679
12	256.58346	258.10481	247.9065	247.5528	258.0109	259.9388
13	256.01452	248.84369	243.7365	255.7673	248.6194	240.236
14	250.74767	247.95577	258.4053	258.0784	247.8989	247.6173
15	253.11394	247.61727	248.6774	248.6636	244.4708	255.8337
16	247.61726	255.83371	250.2822	247.9065	259.6944	258.1048
17	255.83371	258.10481	256.4984	243.7365	248.7005	248.8437
18	258.10479	248.84369	255.9761	258.4053	256.0186	247.9558
19	248.84371	247.95577	250.7484	248.6774	256.4801	243.8052
20	247.95574	243.80521	253.1931	250.2822	255.9998	248.6824
21	243.80545	247.61727	244.7054	256.4984	250.8999	250.326
22	258.42472	247.61727	261.1004	255.9761	253.3815	256.5835
23	248.68239	255.83371	252.965	250.7484	245.3576	256.0145
24	250.32604	258.10481	243.8777	253.1931	261.2543	250.7629
25	256.58346	248.84369	248.9531	244.7054	252.9813	253.1139
26	256.01452	247.95577	253.6329	261.1004	243.8997	244.4732
27	250.74767	243.80521	262.0778	252.965	248.6261	261.1325
28	247.61726	247.61727	245.6205	243.8777	249.2437	253.8278
29	255.83371	247.61727	246.1537	248.6053	251.2514	244.3531
30	258.10479	255.83371	255.0697	248.7633	250.8295	248.653
Average	252.09117	249.93292	251.3135	251.5787	251.2813	251.0260
Max.	258.42472	258.10481	262.0778	261.1004	261.9885	263.5139

. Comparing the performance of APSO and its variants on the 1000-PHEV dataset using one-way ANOVA gives the following results shown in

Table 18. One-way ANOVA test on the 1000-PHEV dataset.

	Sum of Squares	Df	Mean Square	F	Sig.
Between Groups	77.783	5	15.557	0.594	0.704
Within Groups	4556.047	174	26.184
Total	4633.830	179

.

The sig. value is 0.704, which is greater than 0.05; thus, it is concluded that these six algorithms are statistically performing the same for datasets of 1000 cars.

The aim of this research was to find the algorithm that will most intelligently and efficiently allocate available grid power to the vehicles that are plugged into the charging station. Table 3, Table 7, Table 11, and Table 15 give the best fitness values, which represent the maximization of the states of charge of the batteries connected to the station so that power available from the grid will charge the vehicle batteries to the maximum. The computational times for all six algorithms applied to the 50-, 100-, 500-, and 1000-PHEV sets are not much different, while small differences in fitness value will have a large impact on the charging scenario. APSO with Variant 5 gives the highest possible fitness value with a small increase in computational time. To compare these independent algorithms statistically with each other, one-way ANOVA was used. Table 4, Table 8, Table 12, and Table 16 show the percentage increase in fitness value using APSO 5 compared with other algorithms.

The results for 30 trials are given in Table 5, Table 9, Table 13, and Table 17. After applying the ANOVA test, it can be seen in Table 6, Table 10, Table 14 and Table 18 that the value of sig. is 0.705 for 50 PHEVs, 0.956 for 100 PHEVs, 0.917 for 500 PHEVs, and 0.704 for 1000 PHEVs; these are greater than 0.05 so the hypothesis that all the population or group means are similar is correct, and we can say that all six algorithms are statistically similar in performance. However, by seeing the average and maximum values for 30 trials given in Table 5, Table 9, Table 13, and Table 17, it can be noted that APSO 5 provides the highest value; i.e., it maximizes the objective function more and is closer to the global maximum.

6. Conclusions

In this research, four sets of plug-in hybrid electric vehicles, i.e., a 50-PHEV set, 100-PHEV set, 500-PHEV set, and 1000-PHEV set, are simulated by six algorithms to allocate power efficiently to the vehicles from the grid. These six algorithms are accelerated particle swarm optimization (APSO) and its five variants. State of charge is the main performance parameter that needs to be maximized so that batteries can be charged to the maximum with the available power from the grid. The constraints that should be met during the allocation of power to the vehicles are that the power of each PHEV must be less than or equal to 6.7 kW, the state of charge of each car must be within the 0.2 to 0.8 range, and the sum of allocated power to the vehicles should be less than or equal to the product of utility power and charging station efficiency.

These constraints will avoid the risk of overcharging cars’ batteries and also avoid a blackout situation. After meeting the constraints, APSO and its variants converge within ten iterations using 100 particles in each algorithm. APSO with Variant 5 gives the highest possible fitness value, which means that, with respect to other algorithms, APSO 5 will maximize the state of charge, and batteries could be charged more efficiently with the available power. There is a small increase in computational time, but this can be ignored because the increase in fitness value is much greater and has a larger impact on the charging scenario than an increase in computational time. The APSO 5 solution achieves percentage increases of 45.49%, 24.47%, 22.44%, 18.57%, and 10.72% over the solutions of APSO, APSO 1, APSO 2, APSO 3, and APSO 4, respectively, for the 50-PHEV dataset. For 100 PHEVs, the APSO 5 solution achieves increases of 21.33%, 16.94%, 6.82%, 17.06%, and 11.81% over the solutions of APSO, APSO 1, APSO 2, APSO3, and APSO 4, respectively. Similarly, for 500 PHEVs, the percentage increases in APSO 5’s fitness value over the fitness values of APSO, APSO 1, APSO 2, APSO 3, and APSO 4 are 4.52%, 2.74%, 2.06%, 3.38%, and 3.49%, respectively. For 1000 PHEVs, these increases are 4.8%, 2.14%, 7.40%, 2.0%, and 1.01% for the APSO 5 value over APSO, APSO 1, APSO 2, APSO 3, and APSO 4, respectively.

To compare these algorithms statistically, a one-way ANOVA test was used, and 30 trials of each algorithm were required for each car set. It was proven that all these algorithms performed statistically the same. Taking an average of 30 trial results and a maximum among them, it was noted that APSO Variant 5 gives the highest possible fitness value from among all the algorithms.

This problem belongs to the domain of operational research, and finding better and better optimization algorithms for attaining better solutions is a requirement in this research area. Previously, this research problem was solved using the canonical versions of PSO and APSO. This research paper discusses some better and improved variants of the APSO algorithm that can help in providing better solutions to SOC problems. This will help researchers gain insight into the various possible solution strategies for the SOC problem.