A Multi-Objective Hybrid BESSA Optimization Scheme for Parameter Extraction from PV Modules

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Abstract

In this work, a multi-objective Hybrid Bald Eagle Search Simulated Annealing (Hybrid BESSA) parameter extraction technique for photovoltaic (PV) modules is discussed. First, the efficacy of the Hybrid BESSA was proved via testing on unimodal functions, multimodal functions, and fixed dimensional multimodal functions and the results were compared with the Bald Eagle Search (BES) and other recently proposed optimization techniques. Then, a multi-objective Hybrid Bald Eagle Search Simulated Annealing (Hybrid BESSA) parameter extraction technique was devised for photovoltaic (PV) module parameter extraction. The Hybrid BESSA parameter extraction technique was simulated and analyzed in the MATLAB/SIMULINK environment and in a practical experimental setup for the PV Module AS-M3607-S (G1 CELLS). It was found that the Hybrid BESSA possessed better exploration and exploitation capabilities as compared to the BES and other state-of-the-art techniques. It was found that the fitness function value derived by the Hybrid BESSA technique was less than that of the BES technique when tested under various weather conditions. The percentage error for open circuit voltage, output power, and short circuit current was lower when derived by the Hybrid BESSA in comparison with the BES technique. From the results obtained from modeling the PV Module AS-M3607-S (G1 CELLS) based on Hybrid BESSA-based extracted parameters and BES-based extracted parameters, it was seen that percentage improvement in the combined objective function for the condition of keeping irradiance fixed at 1000 W/m² at a temperature varying from −30 °C, 0 °C, 25 °C, 30 °C, 50 °C, and 70 °C were 0.9%, 8.5%, 29.2%, 0.03%, 5.7%, and 0.5%, respectively. When the temperature was kept fixed at 250 °C and irradiance varied from 1000 W/m², 800 W/m², 600 W/m², and 400 W/m², the percentage improvement in combined objective function was found to be 0.5%, 8.1%, 0.5%, and 0.8%, respectively. By analyzing the simulation as well as the experimental results, it was established that the PV model parameter extraction method based on the Hybrid BESSA is more accurate than the BES technique. This analysis is based on a single-diode PV module. A double-diode PV module analysis still needs to be explored.

1. Introduction

Predictive performance tools determine the success of any technology. Consumers can now obtain and analyze optimized performance evaluations of their PV modules. The high growth rate in the utilization of PV energy has led researchers to develop novel photovoltaic cells and to develop theoretical analyses for forecasting the performance estimation, sizing, and optimization of PV energy systems [1]. In order to extract maximum power, the PV module must to be operated at its Maximum Power Point (MPP). This requires an MPP Tracker, which then demands a meticulous and accurate analytical model of PV cells for analysis and testing. A photovoltaic module operating at a temperature of 25 °C and irradiance of 1000 W/m² is said to be operating in Standard Operating Condition (SOC) [2]. Again, it is well known that the manufacturer’s data sheet provides limited information regarding the operational data of a PV module [2], as they only include PV module performance information operating in SOC. The data sheet provides information on short circuit current (ISC), open circuit voltage (VOC), maximum power (PMPP), voltage at PMPP (i.e., VMPP), and current at PMPP (i.e., IMPP) only when operating in SOC. In actuality, PV panels are subjected to operating conditions which differ from SOC. This requires a reliable PV model for apprehending electrical performance other than that achieved in SOC [2].

It has been reported that PV modules can be modelled theoretically via many methods [1]. It has also been reported that PV modelling based on two diode models is a mathematically rigorous and difficult process. As a result, the single-diode model has been adopted by most researchers worldwide for research and analysis [3].

Without illumination, PV cells operate as diodes. When the PV cell is operated in illumination, it yields photocurrent. This leads the PV cell to be modelled in the form of a current source parallelly connected with a diode in series with resistance ($ℛ$_S) and in shunt to resistance($ℛ$_ɡ) [4]. The above-mentioned resistances model the dissipative effect and constructional defects, if any, that can cause a parasitic current.

Figure 1 shows the electrical equivalent of single-diode PV model incorporating its five parameters, i.e., $ℛ$_ɡ, δ, $ℛ$_S, I_∆, and I_ƥ. The PV cell performance is evaluated by visualizing it in the form of an electrical equivalent circuit including five parameters; these are I_ƥ, I_∆, $ℛ$_S, $ℛ$_ɡ, and δ.

It is to be noted that the manufacturer’s datasheet does not provide information on these parameters. Hence, these parameters must be extracted using suitable techniques for the subsequent correct modelling and analysis of a PV module.

Researchers have detailed many techniques for the parameter extraction and modeling of PV modules. These include the Modified Gromov, Derivative α, and Simple Conductance [5] methods, the penalty-based differential evolution algorithm [6], the Newton–Raphson technique [7], and using the Ant Lion Optimizer [8].

Optimization algorithms [9,10] are an efficient method for finding solutions to many nonlinear real-world [11,12] problems. As the NFL (No Free Lunch) theory explains that one single optimization algorithm cannot provide satisfactory results for all types of problems, there is always the opportunity to discover and advance new optimization techniques that suit a particular problem [13]. This is the motivation for this present research work.

Optimization techniques [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] are nature-inspired optimisation technique are broadly categorized into three categories. These are evolutionary methods, trajectory methods, and swarm methods [14,15,16]. Alsattar et al. [17] introduced Bald Eagle Search optimization technique. In 1983, Kirkpatrick [18] developed Simulated Annealing technique to solve real world problem. Saha et al. implements optimization technique for parameter extraction of PV module. The performance efficiency of any optimization technique can be judged by applying on standard mathematical test functions [20]. Many efficient optimization algorithm are proposed and tested by researcher such as Hybrid Water Cycle—Moth Flame Optimization algorithm (WCMFO) [21], Artificial Bee Colony (ABC) algorithm [22], Cuckoo Search (CS) [23], Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA) [24], Moth Flame Optimization (MFO) [25], Gravitational Search Algorithm (GSA) [26], Water Cycle Algorithm (WCA) [27], Dragonfly Algorithm (DA) [28], and Whale Optimization Algorithm (WOA) [29] for finding solution to many real world problem.

Finding solution to problem based on multi-objective optimization process and single objective optimization process. It is found that multi-objective optimization process has better problem-solving ability as compared to single objective optimization process [30].

Researchers all over the world suggest that [16] there are three methods for improving optimization techniques. These are: (i) putting forward new optimization techniques, (ii) improving present techniques, and (iii) the hybridization of optimization techniques. The third one, hybridization of optimization techniques [16], is one of the most popular and efficient methods. In this paper, a multi-objective PV module parameter extraction scheme depending upon the Hybrid Bald Eagle Search Simulated Annealing technique (Hybrid BESSA) is discussed. This technique is a hybridization of the Bald Eagle Search and Simulated Annealing techniques.

Problem-solving with a multi-objective optimization process has many advantages in comparison to solving problems with a single-objective problem formulation [17]. It is reported in [10] that multi-objective optimization can deal with many objectives at one time. Multi-objective optimization-solving schemes output better results when the objectives are correlated to each other as compared to a single-objective optimization problem-solving method. To reiterate, the single-objective optimization problem-solving method takes more time and is also a cumbersome process for finding the correct parameters, whereas a multi-objective optimization problem-solving method takes less time but is difficult to frame and requires a greater depth of optimization knowledge.

Section 2 focuses on the analytical modeling of the PV module. Section 3 focuses on the BES technique. Section 4 explains the Simulated Annealing technique. Section 5 discusses the Hybrid BESSA technique and the improvement in the results obtained by implementing the Hybrid BESSA technique as compared to the BES for 23 benchmark functions. Section 6 discusses a multi-objective parameter extraction scheme and problem formulation based on the Hybrid BESSA technique. Results are discussed in Section 7. The conclusion and future scope for research are discussed in Section 8.

2. Analysis and Modeling of Analytical PV Cell Introduction

2.1. Mathematical Modelling of the PV Module

Accurate forecasting of PV module performance under different weather conditions only depends upon the degree of exactness in the modeling of the photovoltaic(PV) cell, which again depends upon correctness in the values of the extracted parameters.

The current(I)–voltage(V) characteristics of a PV cell [1] during dark operating conditions are governed by (1).

$$I_{∆}=I_{ɤ}\left(e^{\raisebox{1ex}{$cV$}\!\left/ \!\raisebox{-1ex}{$\delta k{T}^{\prime }$}\right.}-1\right)$$

(1)

In (1), the diode current and reverse saturation current are designated as I_∆ and I_ɤ, respectively. The symbol $c$ isthe charge of an electron in coulombs ($1.602\times {10}^{-19}C$) and V and δ signify output voltage and diode ideality factor, respectively. Diode ideality factor (δ) depends on the quality of the semiconductor material used and has a value between 1 and 2 [1]. In the above equation, k represents the Boltzmann constant. The value of k is $1.381\times {10}^{-23}$ J/K. T′ represents junction temperature.

Photon illumination of the PV module produces photon current I_ƥ. Thus, an ideal PV cell is considered as one which has a current source with an intensity of $I_{ƥ}$ and is in parallel connection with the diode. The PV characteristics [4] of a diode are explained via the Shockley equation. The Shockley equation is mathematically expressed by (2).

$$I_{∆}=I_{ƥ}-I_{ɤ}\left(e^{\raisebox{1ex}{$cV$}\!\left/ \!\raisebox{-1ex}{$\delta kT$}\right.}-1\right)$$

(2)

All of the analysis discussed above is for the ideal case. The effect of electrodes is not taken into account. However, practically, the current conducted from each elementary diode moves across the semiconductor slabs using different paths which have varying resistances and voltage drops. Thus, the electrical equivalent of the PV cell is envisaged as a current source connected in parallel with a diode and shunt resistance ($ℛ$_ɡ) and in series with resistance($ℛ$_S).

The constructional defects and dissipative effects [4] are taken into account by these resistances. The constructional defects and dissipative effects cause a flow of parasitic current in the PV panel.

Five parameters of the single-diode model are represented in Figure 1. Here, I_ƥ is photon current, I_∆ is diode current, I_l is diode leakage current, I₀ is output current, and I_ɤ is reverse saturation current. The shunt and series resistance are represented by $ℛ$_S and $ℛ$_ɡ, respectively, whereas the diode ideality factor is represented by δ.

$${I_0=I}_{ƥ}-I_{\Delta }-I_{sh}$$

(3)

Equation (3) formulates output current (I₀) [2]. Equation (5) explains the shunt current and Equation (4) explains the diode current. Output current, $I_0\left(G,T\right)$, is derived in (6) from (3)–(5).

$$I_{\Delta }=(e^{\frac{V+I{ℛ}_s}{{\mathit{\delta }V}_T}}-1)$$

(4)

$$I_{sh}=\frac{V+I{ℛ}_s}{{ℛ}_g}$$

(5)

$${I_0\left(G,T\right)=I}_{ƥ}\left(G\right)-I_{ɤ}\left(T\right)\left[e^{\frac{V+I\left(\mathrm{G},\mathrm{T}\right){ℛ}_S\left(\mathrm{G},\mathrm{T}\right)}{n{V}_T}}-1\right]-\frac{V+I{(G,T)ℛ}_s}{{ℛ}_g}$$

(6)

Equation (6) explains the relationship between the photo current and the reverse saturation current ($I_{ɤ}$), photocurrent ($I_{ƥ}$), solar temperature (T), and irradiance (G). Here, ${ℛ}_s$, n, and $ℛ$_ɡ are constants [4]. The output current at any value of T and G is expressed by (6). Open circuit voltage is formulated using (7), Equation (8) formulates short circuit current, and (9) formulates current at MPP, i.e., I_MPP at any value of $T$ and $G$. Power at MPP ($P_{MPP}$) for any value of $G$ and $T$ is expressed in (10)

$$V_0\left(G,T\right)={ℛ}_g\left(G,T\right)\left[I_{ƥ}\left(G,T\right)-I_{ɤ}\left(T\right)\left[e^{\frac{V_0\left(G,T\right)}{n{V}_T}}-1\right]\right]$$

(7)

$${I_{SC}\left(G,T\right)=I}_{ƥ}\left(G,T\right)-I_{ɤ}\left(T\right)\left[e^{\frac{I_{SC}\left(G,T\right){ℛ}_S\left(G,T\right)}{n{V}_T}}-1\right]-\frac{I_{SC}{(G,T)ℛ}_s}{{ℛ}_g}$$

(8)

$${I_{MPP}\left(G,T\right)=I}_{ƥ}\left(G\right)-I_{ɤ}\left(T\right)\left[e^{\frac{V_{MPP}(G,T)+I_{MPP}\left(G,T\right){ℛ}_S\left(G,T\right)}{n{V}_T}}-1\right]-\frac{V_{MPP}(G,T)+I_{MPP}{(G,T){ℛ}_S\left(G,T\right)}_{}}{{ℛ}_l(G,T)}$$

(9)

$$P_{MPP}\left(G,T\right)=V_{MPP}\left(G,T\right)\times I_{MPP}\left(G,T\right)$$

(10)

2.2. Solar Radiation and Temperature Effects on Reference Parameters

Environmental conditions like temperature, shading, and soiling affect the efficiency of the PV system [31]. In the equations below,$I_{SC\_ref}$ and $I_{SC\_SOC}$ represent the reference short circuit and short circuit current in SOC. $V_{OC\_ref}$ and $V_{OC\_SOC}$ represent the reference open circuit voltage and actual open circuit voltage in SOC. $P_{MPP\_ref}$ and $P_{MPP\_SOC}$ represent reference power and actual power at MPP in SOC [5,6,7] when G and T are at any value of irradiance and temperature.

$$I_{SC\_ref}\left(G,T\right)=\frac{G}{G_{SOC}}[I_{SC\_SOC}+{\sigma }_{SC}(T-T_{SOC}\left)\right]$$

(11)

$$V_{OC\_ref}\left(G,T\right)=V_{OC\_SOC}+{\sigma }_{OC}(T-T_{SOC})$$

(12)

$$P_{MPP\_ref}\left(G,T\right)=\frac{G}{G_{SOC}}[P_{MPP\_SOC}+{\sigma }_{MPP}(T-T_{SOC}\left)\right]$$

(13)

In the above equations, $I_{SC\_SOC}, V_{OC\_SOC}$, and $P_{MPP\_SOC}$ represent short circuit current, open circuit voltage, and power at MPP, respectively, in SOC. The terms ${\sigma }_{SC}$, ${\sigma }_{OC}$ and ${\sigma }_{MPP}$ represent the temperature coefficient at short circuit current, open circuit voltage, and maximum power point, respectively. The following terms’ information, $V_{OC\_ref}$, $I_{SC\_ref}$, $P_{MPP\_ref}\left(G,T\right)$, ${\sigma }_{OC}$, ${\sigma }_{SC}$, and ${\sigma }_{MPP}$, are provided by the manufacturer’s data sheet.

2.3. Mathematical Formulation for Photo Current and Dark Saturation

Photon illumination of the PV module produces photon current I_ƥ. Thus, an ideal PV cell is considered as one which has a current source with an intensity of I_ƥ. and is in parallel connection with the diode. The dark saturation current is a measure of recombination in a device. The values of the parameters I_ƥ, $I_0$, ${ℛ}_g$, ${ℛ}_s$, and δ are temperature- and irradiance-dependent.

Equation (14) formulates the dark saturation current $I_{ɤ}\left(G,T\right)$. The photo current $I_{ƥ}\left(G\right)$ is formulated using (15).

$$I_{\mathit{ɤ}}\left(G,T\right)=\frac{I_{SC}\left(\mathrm{G},\mathrm{T}\right)+{\sigma }_{SC}\left(T-T_{SOC}\right)}{e^{\left(\frac{V_{OC}\left(G,T\right)+{\sigma }_{OC}(T-T_{SOC})}{n{V}_T}\right)}-1}$$

(14)

$$I_{ƥ}\left(G\right)=\frac{G}{G_{SOC}}[I_{SC}\left[\frac{{ℛ}_g\left(G,T\right)+{ℛ}_s\left(G,T\right)}{{ℛ}_g\left(G,T\right)}\right]+{\sigma }_{SC}(T-T_{SOC}\left)\right]$$

(15)

For any value of irradiance and temperature, the corresponding open circuit voltage short circuit current, and maximum power point can be obtained from Equations (7)–(9), whereas the respective reference short circuit current, reference open circuit voltage, and reference maximum power can be obtained from Equations (11)–(13). The difference between the reference values of the voltage, current, and power (i.e., $V_{OC\_ref}$, $I_{SC\_ref}$, and $P_{MPP\_ref}$, respectively) and the actual values of the voltage, current, and power (i.e.,$V_0,I_{SC}$, and $P_{MPP}$, respectively) gives the voltage error (E_V), current error (E_I), and power error (E_p).

3. Bald Eagle Search (BES) Technique

The Bald Eagle Search Algorithm imitates the predatory adroitness of the bald eagle species [17]. It is abbreviated as BES and is a recently proposed metaheuristic algorithm. Bald Eagle accomplishes hunting in three steps. In the first step, the search space is selected. In the second step, it searches for suitable prey in the selected search space. The third step is accomplished with a swooping phase.

Selection Phase

In the selection phase, the bald eagle identifies the best region for predation. It is governed by (16):

S_new(i) = S_best + 𝛇 ∗ 𝓇 (S_mean − S_i)

(16)

In (16), $𝓇$ is a random number chosen between 0 and 2. Here, position change is regulated by parameter 𝛇, which can range from 1.5 to 2. Having accumulated information from the previous stage, the bald eagle identifies the area. Let S_best be present searching space. Selection of S_best depends on the selected best position from the previous search. S_mean signifies that all the previous information in the previous search space is utilized. The present advancement/motion of the eagle is formulated with the product of 𝛇 ∗ $𝓇$.

Search Phase

In the search phase, the bald eagle identifies prey in the region selected for searching and moves in diverse directions in a spiral path to speed up the search process. The best position secured by this movement can be understood via the equations below:

S_new(i) = S_i + y(i) ∗ (S_i − S_i+1) + x(i) ∗ (S_i − S_mean)

(17)

$$y\left(i\right)=\frac{y𝓇\left(i\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left|y𝓇\right|}$$

(18)

$$x\left(i\right)=\frac{x𝓇\left(i\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left|x𝓇\right|}$$

(19)

$$x𝓇\left(i\right)=𝓇\left(i\right)\ast sin\left(\mathsf{\theta }\right(i\left)\right)$$

(20)

$$y𝓇\left(i\right)=𝓇\left(i\right)\ast cos\left(\mathsf{\theta }\right(i\left)\right)$$

(21)

$$\mathsf{\theta }\left(i\right)=a\ast \pi \ast rand$$

(22)

$$𝓇\left(i\right)=\mathsf{\theta }\left(i\right)+R\ast rand$$

(23)

The variable $a$ takes any value within the range of 5 to 10. The corner within the point search is determined by $a$. Parameter $R$ controls the number of chases cycles and takes values within the range of 0.5 to 2. With changing values of parameters, a and R, the algorithm augments diversification in order to escape from the local optimum and to continuously acquire an efficient solution. The best point is considered an improved position in comparison with the mean point. As pictured in Figure 2, the bald eagle advances in a spiral path and identifies the best position to swoop down on its prey. All points in the polar plot assume a value between −1 and 1.

Swooping Phase

The eagle swoops down from the best point to focus in on its prey. This results in the moving of all the points towards the best position.

$$S_{new\left(i\right)} = rand\ast S_{best} + y1\left(i\right)\ast (S_i-m1\ast S_{best})+x1\left(i\right)\ast (S_i-m2\ast S_{best})$$

(24)

$$x1\left(i\right)=\frac{x𝓇\left(i\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left|x𝓇\right|}$$

(25)

$$y1\left(i\right)=\frac{y𝓇\left(i\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left|y𝓇\right|}$$

(26)

$$x𝓇\left(i\right)=𝓇\left(i\right)\ast sinh\left(\mathsf{\theta }\right(i\left)\right)$$

(27)

$$y𝓇\left(i\right)=𝓇\left(i\right)\ast cosh\left(\mathsf{\theta }\right(i\left)\right)$$

(28)

$$\mathsf{\theta }\left(i\right)=a\ast \pi \ast rand$$

(29)

$$𝓇\left(i\right)=\mathsf{\theta }\left(i\right)$$

(30)

In the above equations, m1,m2 ∈ [1, 2]. This intensifies the bald eagle’s swing intensity. During this condition, the mean population escalates and diversifies the algorithm, which again results in reaching the best solution.

The movement of the eagle is computed by using a polar equation. The best point is computed from the multiplication of the polar point on the x-axis with the difference between the present location and the center location and from the multiplication of the polar point on the y-axis with the difference between the present location and the center location. In (24), the best solution is multiplied with a random number as the parameters of m1, m2 intensify the movement of the bald eagle towards the center point and best point. The mean population assists the algorithm with attaining diversification and intensification. All solutions approach the best solution.

4. Simulated Annealing

The Simulated Annealing (SA) search technique is described by Kirkpatrick et al. [18]. The SA metaheuristic method is formulated based on the ascent of a hill. In the first attempt, a random solution is produced. In every iteration, an attempt is made to reach a solution close to the present solution. The solution which has a better fit when compared to original solution is accepted and the juxtaposition of the solutions is permitted using the Boltzmann probability (p). The Boltzmann probability, p, is given as p = e^−θ/T, where θ measures the fit between the best solution and the generated solution. The temperature parameter is represented by T and is decreased systematically in the search process according to a cooling scheme. The beginning temperature is defined as 2 ∗ |N|. Here, |N| embodies the total number of attributes in the data set. T is the cooling schedule and is taken to be 0.93 ∗ T.

Tournament Selection

Goldberg et al. [12] proposed tournament selection. This method starts with the selection of n random solutions from the population. After that, comparisons of the solutions are made, and then the winner is announced. For selection, the comparisons are always conducted based on probability. This results in a suitable method for accommodating a selection pressure set at 0.5. A solution with high fit value is selected, otherwise, a feeble solution is selected. The tournament selection process utilizes probability to select the most suitable solution.

5. Hybrid BESSA Technique

As mentioned in Section 1, the hybridization of optimization techniques [16] is one of the most popular and efficient methods for improving upon them. In [16], it is explained that two algorithms can be fused together and hybridized to evolve into a novel algorithm at a high level or at a low level with co-evolutionary methods or relays. The idea here is to combine the algorithmic strength of both the BES and SA, which includes the exploitation capability of SA and the exploration capability of the BES in a population-based Hybrid BESSA algorithm to arrive at different optimal parameter values required for the modeling of solar photovoltaic modules.

The proposed Hybrid BESSA technique hybridized the local search Simulated Annealing (SA) Technique and the global search BES technique. The metaheuristic algorithm reaches a final stage depending upon its initial population.

In the proposed BESSA technique, in the first step, the BES is implemented to obtain an initial solution, then SA is implemented to arrive at an ultimate solution. This process embodies the hybridization of a global search with local searches. In this process of hybridizing the BES and SA, the capability of social thinking, i.e., finding the best solution, is merged with the local search capability of SA. The task of the global search is conducted with the BES technique and the local search is conducted with the SA technique. Here, the BES technique uses a random selection process to choose the initial solutions. The improvement in the final solution results from the application of SA. Thus, this process leads to a better, more accurate, and finer search for final solutions.

6. Parameter-Setting of the Optimization Technique

The performance of any optimization algorithm builds upon the involved control parameters. Again, these parameters play a vital role in asserting an accurate solution. In the BESSA technique, the below parameters are considered. The position control parameter is 𝛇, and it varies from 1.5 to 2. The corner between the search is determined by parameter $a$, which takes any value within a range of 5 to 10. The search cycle number is controlled by parameter $R$, which takes any value between 0.5 and 2. In the swooping stage, the swing intensity of the bald eagle is increased by parameter m1, m2, which takes any value between 1 and 2. After generating an initial population with the BES technique, SA is implemented to reach the best solution. The solution which is a better fit is acceptable by a probability defined as a Boltzmann Probability(p). Here, θ determines degree of fit between the generated solution and the best solution. The temperature parameter is explained by T, which decreases in an organized way in accordance to the cooling scheme. The starting temperature is defined as 2 ∗ |N|. Here, N is the total number of attributes in the data set. In the tournament selection stage, the selection pressure is set at 0.5.

By nature, a metaheuristic algorithm is stochastic. The metaheuristic algorithm starts with a randomly generated initial population. The algorithm gradually proceeds and reaches the final stage depending on this initial population. Thus, multiple runs must be executed to derive the best, average, and standard values. To apprehend the performance appraisal of a metaheuristic algorithm, it is investigated with a number of benchmark test functions. In this work, the BESSA is tested on 23 benchmark functions [20] which are used by researchers worldwide. These benchmark functions include multimodal functions, fixed dimensional multimodal functions, and unimodal functions. The exploitation capability of a metaheuristic optimization algorithm is investigated by testing it on unimodal functions, as these functions are characterized by possessing unique global optima but no local optima.

The multimodal functions and fixed dimensional multimodal functions are used for investigating the exploration capability of a metaheuristic optimization algorithm. The above functions are characterized by possessing unique global optima and many local optima. To determine the performance efficiency of the BESSA technique, a comparison was made with the recently proposed BES [17] and CamWOA techniques [9]. The Hybrid BESSA technique was also compared with the Hybrid Water Cycle—Moth Flame Optimization algorithm (WCMFO) [21], Artificial Bee Colony (ABC) algorithm [22], Cuckoo Search (CS) [23], Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA) [24], Moth Flame Optimization (MFO) [25], Gravitational Search Algorithm (GSA) [26], Water Cycle Algorithm (WCA) [27], Dragonfly Algorithm (DA) [28], and Whale Optimization Algorithm (WOA) [29], as described in the literature [9]. The mentioned algorithms’ control parameters are detailed in the Appendix A.

For all the above analyses, the maximum number of search particles taken and the total iteration counts were limited to 100 and 1000, respectively, which are common parameters for all the algorithms mentioned above. To reach the right conclusion, the number of runs for each algorithm was fixed at 30. The residuum is analyzed in following section.

Fan et al. [32] developed a random reselection PSO for the optimal design of solar PV modules. There are different optimization models which have been developed to handle the model parameters of PV solar panels [33,34,35,36,37,38,39,40,41], which include the mutation-based PSO, slime mold algorithm, Gorilla Troops optimizer, Harris Hawk technique, gradient-based optimizer, and Coyote optimization technique. The application of such metaheuristic techniques has also been observed in crack detection for cantilever beams [42].

Analysis and Appraisal of Exploitation Capability ($f_1\left(y\right)$–$f_7\left(y\right)$)

The unimodal functions ascertain and evaluate the exploitation capability of an optimization technique. These unimodal functions are tabulated in

Table 1. Unimodal benchmark functions.

Functions	Range	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$
$f_1\left(p\right)={\displaystyle \sum _{i=1}^n}{p}_i^2$	${\left[-100,100\right]}^{10}$	0
$f_2\left(p\right)={\displaystyle \sum _{i=1}^n}\left|p_i\right|+{\displaystyle \prod _{i=1}^n}{p}_i$	${\left[-10,10\right]}^{10}$	0
$f_3\left(p\right)={\displaystyle \sum _{i=1}^n}{\left({\displaystyle \sum _{j-1}^i}{p}_j\right)}^2$	${\left[-100,100\right]}^{10}$	0
$f_4\left(p\right)={max}_i\{\left|p_i\right|,1\le i\le n\}$	${\left[-100,100\right]}^{10}$	0
$f_5\left(p\right)={\displaystyle \sum _{i=1}^{n-1}}\left[100{\left(p_{i+1}-p_i^2\right)}^2+{\left(p_i-1\right)}^2\right]$	${\left[-30,30\right]}^{10}$	0
$f_6\left(p\right)={\sum }_{i=1}^n{\left(\left[p_i+0.5\right]\right)}^2$	${\left[-100,100\right]}^{10}$	0
$f_7\left(p\right)={\displaystyle {\sum }_{i=1}^n}i{p}_i^4+random[0,1)$	${\left[-1.28,1.28\right]}^{10}$	0

.

To appraise the exploitation capability of the proposed BESSA Technique, it is tested on seven unimodal functions, i.e., $f_1\left(y\right)–f_7\left(y\right)$, as presented in [9]. The results are depicted in

Table 2. Statistical results for unimodal functions.

Function	BESSA		BES		CamWOA		WOA
	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev
$f_1\left(p\right)$	0	0	0	$0$	0	0	$1.4\times {10}^{-195}$	0
$f_2\left(p\right)$	0	0	0	0	$2.1\times {10}^{-3}$	0	$5.3\times {10}^{-117}$	$2.3\times {10}^{-116}$
$f_3\left(p\right)$	0	0	0	0	0	0	0.10019	0.23771
$f_4\left(p\right)$	0	0	0	0	$4.5\times {10}^{-306}$	0	0.00054	0.00224
$f_5\left(p\right)$	$5.91\times {10}^{-13}$	$0.027\times {10}^{-16}$	$23.52\times {10}^{-13}$	$0.12\times {10}^{-16}$	6.8177	0.602	4.7626	0.346
$f_6\left(p\right)$	0	0	0	0	0.001415	0.000982	$7.03\times {10}^{-7}$	$6.62\times {10}^{-7}$
$f_7\left(p\right)$	$1.672\times {10}^{-5}$	$1.21\times {10}^{-5}$	$2.5\times {10}^{-5}$	$2.701\times {10}^{-5}$	$1.77\times {10}^{-5}$	$1.38\times {10}^{-5}$	0.00041	0.00046
Function	WCMFO		ABC		PSOGSA		CS
	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev
$f_1\left(p\right)$	$1.10\times {10}^{-95}$	$6.05\times {10}^{-95}$	$1.93\times {10}^{-2}$	$1.46\times {10}^{-2}$	$1.24\times {10}^{-20}$	$3\times {10}^{-21}$	$9.48\times {10}^{-13}$	$1.46\times {10}^{-2}$
$f_2\left(p\right)$	$3.52\times {10}^{-33}$	$1.35\times {10}^{-32}$	$3.42\times {10}^{-2}$	$1.3\times {10}^{-2}$	$2.58\times {10}^{-10}$	$4.57\times {10}^{-11}$	$1.67\times {10}^{-5}$	$1.3\times {10}^{-2}$
$f_3\left(p\right)$	$1.62\times {10}^{-32}$	$8.88\times {10}^{-32}$	848.34	227.45	$2.48\times {10}^{-20}$	$9.34\times {10}^{-21}$	$2.17\times {10}^{-6}$	227.45
$f_4\left(p\right)$	$3.29\times {10}^{-24}$	$1.31\times {10}^{-23}$	7.939	1.938	$6.35\times {10}^{-11}$	$1.26\times {10}^{-11}$	$4.9\times {10}^{-3}$	1.938
$f_5\left(p\right)$	2.4259	3.467	44.513	15.196	1.1607	2.0941	0.68946	0.57142
$f_6\left(p\right)$	$6.97\times {10}^{-29}$	$3.29\times {10}^{-28}$	$1.13\times {10}^{-2}$	$7.3\times {10}^{-3}$	$1.38\times {10}^{-20}$	$3.37\times {10}^{-21}$	$1.48\times {10}^{-12}$	$8.3\times {10}^{-13}$
$f_7\left(p\right)$	0.4987	0.305	$3.93\times {10}^{-2}$	$1.09\times {10}^{-2}$	$2.3\times {10}^{-3}$	$1.2\times {10}^{-3}$	$4.735\times {10}^{-3}$	$1.754\times {10}^{-3}$
Function	MFO		GSA		WCA		DA
	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev
$f_1\left(p\right)$	$1.65\times {10}^{-31}$	$4.91\times {10}^{-31}$	$1.02\times {10}^{-18}$	$3.3\times {10}^{-19}$	$3.1\times {10}^{-14}$	$5.8\times {10}^{-14}$	$5.303\times {10}^{-1}$	1.3180
$f_2\left(p\right)$	$2.69\times {10}^{-19}$	$6.22\times {10}^{-19}$	$2.33\times {10}^{-9}$	$4.39\times {10}^{-10}$	$2.11\times {10}^{-7}$	$3.9\times {10}^{-7}$	2.392	3.912
$f_3\left(p\right)$	$2.05\times {10}^{-11}$	$4.21\times {10}^{-11}$	$1.0\times {10}^{-5}$	$5.5\times {10}^{-5}$	$3.56\times {10}^{-12}$	$9.56\times {10}^{-12}$	215.45	935.17
$f_4\left(p\right)$	$5.79\times {10}^{-6}$	$3.17\times {10}^{-5}$	$4.76\times {10}^{-10}$	$8.44\times {10}^{-11}$	$1.08\times {10}^{-11}$	$5.73\times {10}^{-11}$	1.153	2.702
$f_5\left(p\right)$	133.11	555.57	5.423	0.1238	1.252	1.831	6784.5	21,974.5
$f_6\left(p\right)$	$4.78\times {10}^{-32}$	$1.27\times {10}^{-31}$	$6.40\times {10}^{-19}$	$2.3\times {10}^{-19}$	$4.6\times {10}^{-18}$	$2.26\times {10}^{-17}$	2.2023	5.528
$f_7\left(p\right)$	$1.23\times {10}^{-3}$	$7.2\times {10}^{-4}$	$1.86\times {10}^{-3}$	$6.7\times {10}^{-4}$	0.5155	0.2552	$6.9\times {10}^{-3}$	$7.6\times {10}^{-3}$

. These unimodal functions are characterized by having single global optima. The statistical outcomes of the BESSA, original BES, CamWOA, and the other 10 algorithms presented in [9] are also tabulated. It was observed that the Hybrid BESSA produced better results and was able to find global optima for all of the unimodal functions as compared to the BES, CamWOA, and other techniques mentioned in [9].

Appraisal and Analysis of Exploration Capability (p8(y)–p13(y))

Multimodal functions are characterized by having a number of local optima, thus testing on these functions appraise the exploration capability of any metaheuristic algorithm.

Table 3. Multimodal benchmark functions.

Function	Range	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$
$f_8\left(p\right)={\sum }_{i=1}^n-p_{i}sin\left(\sqrt{\left|p_i\right|}\right)$	${\left[-500,500\right]}^{10}$	$-418.9829\times \mathrm{n}$
$f_9\left(p\right)={\sum }_{i=1}^n\left[p_i^2-10\mathit{cos}\left(2\pi p_i\right)+10\right]$	${\left[-5.12,5.12\right]}^{10}$	0
$f_{10}\left(p\right)=-20\mathit{exp}\left(-0.2\sqrt{\frac{1}{n}{\sum }_{i=1}^n{p}_i^2}\right)-exp\left(\frac{1}{n}{\sum }_{i=1}^{n}cos\left(2\pi p_i\right)\right)+20+e$	${\left[-32,32\right]}^{10}$	0
$$\begin{gathered} f_{11}\left(p\right)=\frac{1}{4000}{\sum }_{i=1}^n{p}_i^2-{\prod }_{j=1}^{n}cos\left(\frac{p_i}{\sqrt{i}}\right)+1 \\ {f}_{12}\left(p\right)=\frac{\pi }{n}\left\{10\mathit{sin}\left(\pi p_i\right)+{\sum }_{i=1}^{n-1}{\left(p_i^{-}1\right)}^2\left[1+10{sin}^2\left(\pi p_{i+1}\right)\right]\right\} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{+(p_n-1)}^2+{\sum }_{i=1}^{n}u(p_i,10,100,4) \\ {f}_i=1+\frac{p_i+1}{4} \\ u\left(y_i,a,k.m\right)=\left\{\begin{array}{c}k{\left(p_i-a\right)}^m{p}_i>a\\ 0-a{<p}_i<a\\ k{(-p_i-a)}^m{p}_i<a\end{array}\right. \\ {f}_{13}\left(p\right)=0.1\{{sin}^2\left(3\pi p_i\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^n}{(p_i-1)}^2\left[1+{sin}^2(3\pi p_i+1)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{(p_n-1)}^2\left[1+{sin}^2\left(2\pi p_n\right)\right]\}+{\displaystyle \sum _{i=1}^n}u(p_i,5100.4) \end{gathered}$$	${\left[-600,600\right]}^{10}$ ${\left[-50,50\right]}^{10}$ ${\left[-50,50\right]}^{10}$	0 0 0

lists the multimodal functions.

Table 4. Multimodal benchmark functions statistical outcomes.

Function	BESSA		BES		CamWOA		WOA
	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev
$f_8\left(p\right)$	−3449.9	330.9	−3222.5	391.2	−7339.503	947.3066	−12,269.5	543.21289
$f_9\left(p\right)$	0	0	0	0	0	0	0	0
$f_{10}\left(p\right)$	$8.882\times {10}^{-16}$	0	$8.88\times {10}^{-16}$	0	$8.88\times {10}^{-16}$	0	$3.84\times {10}^{-15}$	$2.1035\times {10}^{-15}$
$f_{11}\left(p\right)$	0	0	0	0	0	0	0.042819	0.12435
$f_{12}\left(p\right)$	$4.712\times {10}^{-3}$	0	$4.701\times {10}^{-32}$	0	0.0022626	0.003269	$1.18\times {10}^{-6}$	$7.6666\times {10}^{-7}$
$f_{13}\left(p\right)$	0.9934	0.4249	0.2628	0.4697	0.0086723	0.006867	$7.15\times {10}^{-6}$	$6.449\times {10}^{-6}$
Function	WCMFO		ABC		PSOGSA		CS
	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev
$f_8\left(p\right)$	−3729.7	96.325	−3922.73	88.61857	−3271.6	278.08	−3712.01	167.4447
$f_9\left(p\right)$	2.089	1.508	3.677	1.0365	23.281	12.968	6.574	1.367
$f_{10}\left(p\right)$	$8.88\times {10}^{-16}$	$1.0\times {10}^{-31}$	$1.21\times {10}^{-6}$	$9.37\times {10}^{-7}$	$4.94\times {10}^{-12}$	$2.26\times {10}^{-12}$	$1.24\times {10}^{-15}$	$1.08\times {10}^{-15}$
$f_{11}\left(p\right)$	$9.91\times {10}^{-2}$	$5.31\times {10}^{-2}$	0.281	0.1086	0.2004	0.1141	$3.96\times {10}^{-2}$	$8.88\times {10}^{-3}$
$f_{12}\left(p\right)$	$2.0\times {10}^{-29}$	$6.44\times {10}^{-29}$	$1.9\times {10}^{-3}$	$1.3\times {10}^{-3}$	0.2491	0.581	$9.77\times {10}^{-5}$	$1.3\times {10}^{-4}$
$f_{13}\left(p\right)$	$4.49\times {10}^{-22}$	$2.06\times {10}^{-21}$	$8.3\times {10}^{-3}$	$5.1\times {10}^{-3}$	$3.11\times {10}^{-21}$	$1.066\times {10}^{-21}$	$1.31\times {10}^{-9}$	$1.39\times {10}^{-9}$
Function	MFO		GSA		WCA		DA
	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev	Avg.	Std. Dev
$f_8\left(p\right)$	−3329.13	288.317	−1694.53	190.6721	−3422.55	304.572	−3213.66	431.748
$f_9\left(p\right)$	12.8372	7.352	1.392	1.214	20.993	10.524	11.561	10.177
$f_{10}\left(p\right)$	$8.8\times {10}^{-16}$	$1.0\times {10}^{-31}$	$1.28\times {10}^{-10}$	$6.71\times {10}^{-11}$	$2.42\times {10}^{-15}$	$1.79\times {10}^{-15}$	$3.14\times {10}^{-5}$	$1.7\times {10}^{-4}$
$f_{11}\left(p\right)$	$1.78\times {10}^{-1}$	$8.43\times {10}^{-2}$	$1.67\times {10}^{-2}$	$2.79\times {10}^{-2}$	0.1502	$9.44\times {10}^{-2}$	0.3846	0.3826
$f_{12}\left(p\right)$	$3.11\times {10}^{-2}$	$9.487\times {10}^{-2}$	$7.95\times {10}^{-21}$	$3.23\times {10}^{-2}$	$1.036\times {10}^{-2}$	$5.67\times {10}^{-2}$	0.5296	0.6912
$f_{13}\left(p\right)$	$1.10\times {10}^{-3}$	$3.33\times {10}^{-3}$	$5.67\times {10}^{-20}$	$1.88\times {10}^{-20}$	$7.3\times {10}^{-4}$	$2.7\times {10}^{-3}$	0.5292	0.7173

enumerates the statistical results of the different algorithms gathered for 30 runs on every benchmark function.

The statistical outcomes of the Hybrid BESSA, BES, CamWOA, and the other ten optimization techniques [9] tested on the multimodal functions ($f_8\left(p\right)-f_{13}\left(p\right)$), are reported in Table 4, and it was found that the Hybrid BESSA technique produced better results for all of the mentioned functions except function $f_{13}\left(p\right)$.

For function $f_{13}\left(p\right)$, the BES produced a better result than the Hybrid BESSA orsCamWOA. For function $f_{10}\left(p\right)$, the results are comparable.

Appraisal and Analysis of Exploration Capability (Testing on Fixed Dimension Multimodal Functions, f_14(p) − f_23(p))

The fixed dimensional multimodal functions ($f_8\left(p\right)-f_{13}\left(p\right))$ and their outcomes are displayed in

Table 5. Fixed dimension multimodal benchmark functions.

Function	Range	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$
$f_{14}\left(p\right)={\left(\frac{1}{500}+{\displaystyle \sum _{j=1}^{25}}\frac{1}{j+\sum _{i=1}^2{\left(p_i-a_{ij}\right)}^2}\right)}^{-1}$	${\left[-65,65\right]}^2$	1
$f_{15}\left(p\right)={\displaystyle \sum _{i=1}^{11}}{\left[a_i-\frac{p_1(b_i^2+{b_{i}p}_2)}{b_i^2+b_i{p}_3+p_4}\right]}^2$	${\left[-5,5\right]}^4$	0.00030
$f_{16}=4p_1^2-2.1p_1^4=\frac{1}{3}p+p_1{y}_2-4p_2^2+4p_2^4$	${\left[-5,5\right]}^2$	−1.0316
$f_{17}\left(p\right)={\left(p_2-\frac{5.1}{4p^2}{p}_1+\frac{5}{\pi }{p}_1-6\right)}^2+10\left(1-\frac{1}{8\pi }\right)cos\left(p_1\right)+10$	${\left[-5,5\right]}^2$	0.398
$$\begin{gathered} f_{18}\left(p\right)=\left[1+{\left(p_1+p_2+1\right)}^2\left(19-14p_1+3p_1^2-14p_2+6p_1{p}_2+3p_2^2\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\times \left[30+{\left(2p_1-3p_2\right)}^2\times (18-32p_1+12p_1^2+48p_2-36p_1{p}_2+27p_2^2)\right] \end{gathered}$$	${\left[-2,2\right]}^2$	3
$f_{19}\left(p\right)=-{\sum }_{i=1}^4{c}_{i}exp\left(-{\sum }_{j=1}^3{ɑ}_{ij}{\left(p_j-m_{ij}\right)}^2\right)$	${\left[1,3\right]}^3$	−3.86
$f_{20}\left(p\right)=-{\sum }_{i=1}^4{c}_{i}exp\left(-{\sum }_{j=1}^6{ɑ}_{ij}{\left(p_j-m_{ij}\right)}^2\right)$	${\left[0,1\right]}^6$	−3.32
$f_{21}\left(p\right)=-{\displaystyle \sum _{i=1}^5}{\left[\left(P-{ɑ}_i\right){\left(P-{ɑ}_i\right)}^T+c_i\right]}^{-1}$	${\left[0,10\right]}^4$	−10.1532
$f_{22}\left(p\right)=-{\displaystyle \sum _{i=1}^7}{\left[\left(P-{ɑ}_i\right){\left(P-{ɑ}_i\right)}^T+c_i\right]}^{-1}$	${\left[0,10\right]}^4$	−10.4028
$f_{23}\left(p\right)=-{\displaystyle \sum _{i=1}^{10}}{\left[\left(P-{ɑ}_i\right){\left(P-{ɑ}_i\right)}^T+c_i\right]}^{-1}$	${\left[0,10\right]}^4$	−10.5363

and

Table 6. Statistical outcomes of fixed dimensional multimodal benchmark test functions.

Function	BESSA		BES		CamWOA		WOA
	Av.	St. Dev	Av.	St. Dev	Av.	St. Dev	Av.	St. Dev
$f_{14}\left(p\right)$	3.70141	4.3351	3.27674	4.1862	0.9980	3.86634	0.9980	0.60541
$f_{15}\left(p\right)$	0.000402	0.00031	0.000423	0.000321	$4.765\times {10}^{-4}$	0.0003808	0.00055	0.0003073
$f_{16}\left(p\right)$	−1.03164	0	−1.03162	0	−1.0316	$7.228\times {10}^{-5}$	−1.03163	$4.67\times {10}^{-13}$
$f_{17}\left(p\right)$	0.3979	0.0	0.3979	0.0	0.3978	0.010865	0.3978	$1.05\times {10}^{-8}$
$f_{18}\left(p\right)$	3.0	0.0	3.0	0.0	3	$3.377\times {10}^{-5}$	3	$2.51\times {10}^{-7}$
$f_{19}\left(p\right)$	−3.8628	0.0	3.8628	0.0	−3.8587	0.003693	−3.8626	0.000201
$f_{20}\left(p\right)$	−3.2765	0.05993	3.26252	0.0605	−3.273	0.10283	−3.2	0.068717
$f_{21}\left(p\right)$	−7.43431	2.58685	−7.94413	2.5694	−10.153	0.0069855	−10.153	$7.76\times {10}^{-5}$
$f_{22}\left(p\right)$	−8.27682	2.64851	−8.10100	2.6773	−10.4025	0.0049065	−10.402	1.6218
$f_{23}\left(p\right)$	−8.19300	2.7256	−9.63512	2.0499	−10.5336	0.0068874	−5.1	0.987257
Function	WCMFO		ABC		PSOGSA		CS
	Av.	St. Dev	Av.	St. Dev	Av.	St. Dev	Av.	St. Dev
$f_{14}\left(p\right)$	0.998	$5.36\times {10}^{-16}$	0.998	$1.02\times {10}^{-13}$	1.06	0.252	0.998	$3.39\times {10}^{-16}$
$f_{15}\left(p\right)$	$3.0\times {10}^{-4}$	$1.07\times {10}^{-15}$	$7.1\times {10}^{-14}$	$1.3\times {10}^{-4}$	$3.79\times {10}^{-3}$	$7.5\times {10}^{-3}$	$3\times {10}^{-4}$	$4.23\times {10}^{-9}$
$f_{16}\left(p\right)$	−1.03	0	−1.03	$7.36\times {10}^{-11}$	−1.03	0	−1.03	0
$f_{17}\left(p\right)$	$3.98\times {10}^{-1}$	$1.13\times {10}^{-16}$	$3.98\times {10}^{-1}$	$5.68\times {10}^{-9}$	$3.98\times {10}^{-1}$	$1.13\times {10}^{-16}$	$3.98\times {10}^{-1}$	$1.13\times {10}^{-16}$
$f_{18}\left(p\right)$	3	$9.57\times {10}^{-15}$	3	$8.64\times {10}^{-5}$	3	$4.52\times {10}^{-16}$	3	$4.52\times {10}^{-16}$
$f_{19}\left(p\right)$	−3.86	$2.71\times {10}^{-15}$	−3.86	$7.89\times {10}^{-11}$	−3.86	$2.71\times {10}^{-15}$	−3.86	$2.71\times {10}^{-15}$
$f_{20}\left(p\right)$	−3.25	$6.027\times {10}^{-2}$	−3.32	$4.82\times {10}^{-6}$	−3.26	$6.032\times {10}^{-2}$	−3.32	$1.26\times {10}^{-13}$
$f_{21}\left(p\right)$	−8.89	2.361515	−10.1	$6.784\times {10}^{-3}$	−5.9	3.421068	−10.2	$1.81\times {10}^{-15}$
$f_{22}\left(p\right)$	−10.4	$1.59\times {10}^{-12}$	−10.4	$2.886\times {10}^{-3}$	−5.76	3.454976	−10.4	$6.18\times {10}^{-14}$
$f_{23}\left(p\right)$	−10.5	$4.09\times {10}^{-14}$	−10.5	$4.07\times {10}^{-3}$	−6.99	3.890193	−10.5	$2.15\times {10}^{-12}$
Function	MFO		GSA		WCA		DA
	Av.	St. Dev	Av.	St. Dev	Av.	St. Dev	Av.	St. Dev
$f_{14}\left(p\right)$	1.03	0.1814836	3.4	2.578637	0.998	$3.39\times {10}^{-16}$	1.1	0.303
$f_{15}\left(p\right)$	$8.374\times {10}^{-4}$	$2.54\times {10}^{-4}$	$1.8\times {10}^{-3}$	$4.9\times {10}^{-4}$	$3.69\times {10}^{-4}$	$2.32\times {10}^{-4}$	$1.34\times {10}^{-3}$	$5.11\times {10}^{-4}$
$f_{16}\left(p\right)$	−1.03	0	−1.03	0	−1.03	0	−1.03	$2.55\times {10}^{-11}$
$f_{17}\left(p\right)$	$3.98\times {10}^{-1}$	$1.13\times {10}^{-16}$	$3.98\times {10}^{-1}$	$1.13\times {10}^{-16}$	$3.98\times {10}^{-1}$	$3.79\times {10}^{-16}$	$3.98\times {10}^{-1}$	$7.6\times {10}^{-13}$
$f_{18}\left(p\right)$	3	$1.95\times {10}^{-15}$	3	$4.02\times {10}^{-15}$	3	$1.79\times {10}^{-14}$	3	$1.38\times {10}^{-6}$
$f_{19}\left(p\right)$	−3.86	$2.71\times {10}^{-15}$	−3.86	$2.71\times {10}^{-15}$	−3.86	$2.71\times {10}^{-15}$	−3.86	$1.587\times {10}^{-3}$
$f_{20}\left(p\right)$	−3.22	$4.5066\times {10}^{-2}$	−3.32	$1.36\times {10}^{-15}$	−3.26	$6.04\times {10}^{-2}$	−3.25	$6.72\times {10}^{-2}$
$f_{21}\left(p\right)$	−7.56	3.323037	−7.45	3.381188	−8.31	2.718	−9.81	1.28
$f_{22}\left(p\right)$	−9.35	2.423664	−10.4	0	−9.52	2.002	−10.4	0.192
$f_{23}\left(p\right)$	−10.3	1.39948	−10.5	$9.03\times {10}^{-15}$	−9.82	2.235	−10.3	1.06

. It was found that, when tested upon the fixed dimensional multimodal functions ($f_{14}\left(p\right)-f_{23}\left(p\right)$), the proposed Hybrid BESSA technique and the BES were both able to find global optima for functions $f_{16}\left(p\right)-f_{19}\left(p\right)$. For functions $f_{14}\left(p\right)$, $f_{15}\left(p\right)$, $f_{17}\left(p\right)$, $f_{19}\left(p\right)$, and $f_{20}\left(p\right)$, the Hybrid BESSA produced better results than the BES, CamWOA, and all of the other mentioned 10 optimization techniques [9]. For functions $f_{16}\left(p\right)$ and $f_{18}\left(p\right)$, the proposed BESSA was able to find global optima. For functions $f_{21}\left(p\right)$, $f_{22}\left(p\right)$, and $f_{23}\left(p\right)$, CamWOA provided better results. From the above analysis, it is evident that the Hybrid BESSA has a better exploration capability when compared to the BES, CamWOA, and other techniques.

Examining all of the above analysis for the unimodal functions, multimodal functions, and fixed dimensional multimodal functions, the proposed Hybrid BESSA produced better results for 19 out of the 23 functions. Hence, the proposed BESSA technique was proved to be a more efficient technique compared to the BES [17], CaMWOA [9], ABC [23], CS [24], GSA [27], PSOGSA [25], WCA [28], MFO [26], DA [29], and WCMFO [22] as detailed in [9].

Convergence Characteristics Analysis

Convergence characteristics play a vital role, as they offer information about how population-based optimization search techniques converge to a best solution. It is seen from the Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 that the proposed Hybrid BESSA has better convergence characteristics compared to the BES.

7. Multi-Objective Parameter Extraction Scheme and Problem Formulation Based on the Hybrid BESSA Technique

A multi-objective parameter extraction scheme based on the Hybrid BESSA approach was devised for the simultaneous extraction of the parameters of a photovoltaic module from a combined objective function. The optimal combinations of three operating parameters were then used for the mathematical modeling of the photovoltaic module. The accuracy of the PV modeling was judged based on the ISE of power (ε_P), ISE of voltage (ε_V), and ISE of current(ε_I) values obtained from simulation via the BESSA and BES techniques. For any value of irradiance and temperature, the corresponding open circuit voltage, short circuit current, and maximum power point can be obtained using Equations (7)–(9), whereas the respective reference short circuit current, reference open circuit voltage, and reference maximum power can be obtained using Equations (11)–(13). The difference between reference values of the voltage, current, and power (i.e., $V_{OC\_ref}$, $I_{SC\_ref}$, and $P_{MPP\_ref}$, respectively) and the actual values of the voltage, current, and power (i.e., $V_0, I_{SC}$, and $P_{MPP}$, respectively) gives the voltage error (E_V), current error (E_I), and power error (E_p), as demonstrated in Figure 9.

A multi-objective optimization problem statement is framed with the problem of computing the optimum combination of three PV parameter values that include the shunt($ℛ$_ɡ) and series resistance ($ℛ$_s) and diode ideality factor, which is represented by δ, with respect to obtaining the percentage error of the ISE of power (ε_P),the percentage error of the ISE of voltage (ε_V), and the percentage error of the ISE of current (ε_I).The combined objective was to minimize the ISE of power (ε_P), ISE of voltage (ε_V), and ISE of current (ε_I) from (31) to (36) and the combined fitness function (37).

Equation (37) formulates a multi-objective optimization problem based on the Hybrid BESSA with the objective of minimizing errors and extracting these unknown parameters necessary for the modeling of PV cells. By studying the obtained results, it was found that the Hybrid BESSA was capable of correctly formulating the PV panel model when compared to the BES technique.

$${\epsilon }_{\mathrm{I}}=\int {(I_{SC\_ref}-I_{SC})}^{2}dt$$

(31)

$${\epsilon }_{\mathrm{V}}=\int {(V_{OC\_ref}-V_{OC})}^{2}dt$$

(32)

$${\epsilon }_{\mathrm{P}}=\int {(P_{MPP\_ref}-I_{MPP})}^{2}dt$$

(33)

Equations (34)–(36) represent the minimization of the ISE of I_sc, the minimization of the ISE of V_oc, and the minimization of the ISE of P_MPP.

y₁ = min(ε_I)

(34)

y₂ = min(ε_V)

(35)

y₃ = min(ε_P)

(36)

$$\mathit{min}\left(Y\right)={\omega }_1{y}_1+{\omega }_2{y}_2+{\omega }_3{y}_3$$

(37)

${\omega }_1, {\omega }_2, \mathrm{a}\mathrm{n}\mathrm{d} {\omega }_3$ are declared as weighting factors, and their values are selected in such a way as to cause each term to compete with the others in the course of the optimization search process. This normalizes all three objectives (i.e., y₁, y₂, and y₃) in one uniform scale.

8. Results and Discussion

Simulation Results

The efficacy as well as the applicability of the Hybrid BESSA-based PV parameter extraction technique was tested on the PV module AS-M3607-S. The PV AS-M3607-S module was experimented on at different operating temperatures (i.e., −30 °C, 0 °C, 25 °C, 30 °C, 50 °C, and 70 °C) when keeping irradiance constant at 1000 W/m² and was compared at different irradiances (i.e., at 1000 W/m², 800 W/m², 600 W/m², and 400 W/m²) at a constant temperature of 25 °C. The bounds of the gains of the excavated parameters are listed in

Table 7. Lower and upper bounds of the parameters.

PV Module	$ℛ$S_min	$ℛ$S_max	$ℛ$ɡ_min	$ℛ$ɡ_max	δmin	δmax
AS-M3607-S (G1 CELLS)	0.00	1	30	4000	1	2

.

Table 8. Parameters of the Photovoltaic Module AS-M3607-S (G1 CELLS) obtained using the Hybrid BESSA- and BES-based parameter extraction technique with G fixed at 1000 W/m² and T varying from −30 °C to 70 °C.

T	Hybrid BESSA-Based Parameter Extraction Technique
	$ℛ$S	$ℛ$ɡ	δ	Y
−30 °C	0.29012	3999.6	1.99	3.6199 × 105
0 °C	0.29379	3998.4	1.8896	2.73 × 105
25 °C	0.2341	3999.94	2	2.12 × 105
30 °C	0.3426	2702.553	1.6299	2.01 × 105
50 °C	0.1967	4000	2	1.61 × 105
70 °C	0.4229	3709.568	1.3032	1.26 × 105
T	BES Technique
	$ℛ$S	$ℛ$ɡ	δ	Y
−30 °C	0.29026	4000	2	3.654 × 105
0 °C	0.2948	4000	1.8898	298,578.7
25 °C	0.4226	3730.083	1.4108	253,863.1
30 °C	0.3426	2519.003	1.6287	200,937.4
50 °C	0.1988	4000	1.998	170,735.5
70 °C	0.4238	3638.63	1.3002	126,727.9

provides the values of the parameters extracted by the Hybrid BESSA and BES scheme for the PV Module AS-M3607-S (G1 CELLS) when keeping irradiance fixed at 1000 W/m² and varying temperature from −30 °C to 70 °C.

Table 9. Reference values and percentage error obtained for Module AS-M3607-S with G fixed at 1000 W/m² and T varying from −30 °C to 70 °C.

T	Hybrid BESSA-Based Parameter Extraction Technique
	${\mathit{P}}_{\mathit{M}\mathit{P}\mathit{P}\_\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{V}}_{\mathit{O}\mathit{C}\_\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{I}}_{\mathit{S}\mathit{C}\_\mathit{r}\mathit{e}\mathit{f}}$	${\%\mathit{e}}_{\mathit{P}}$	${\%\mathit{e}}_{\mathit{V}}$	${\%\mathit{e}}_{\mathit{I}}$
−30 °C	474.8	56.85	9.897	0	0.703	0.01
0 °C	434	52.84	10.02	0	0.378	0
25 °C	400	49.5	10.12	0	0	0
30 °C	393.2	48.83	10.14	0	0.081	0
50 °C	366	46.16	10.22	0	0.476	0
70 °C	338.8	43.49	10.3	0	1.01	0
T	BES Technique
	${\mathit{P}}_{\mathit{M}\mathit{P}\mathit{P}\_\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{V}}_{\mathit{O}\mathit{C}\_\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{I}}_{\mathit{S}\mathit{C}\_\mathit{r}\mathit{e}\mathit{f}}$	${\%\mathit{e}}_{\mathit{P}}$	${\%\mathit{e}}_{\mathit{V}}$	${\%\mathit{e}}_{\mathit{I}}$
−30 °C	474.8	56.85	9.897	0.201	1.49	0
0 °C	434	52.84	10.02	0.161	0.738	0
25 °C	400	49.5	10.12	0.025	0	0
30 °C	393.2	48.83	10.14	0.101	0.204	0
50 °C	366	46.16	10.22	0.191	0.974	0
70 °C	338.8	43.49	10.3	0.088	1.21	0

shows the reference values and percentage error obtained for Module AS-M3607-S with G fixed at 1000 W/m² and T varying from −30 °C to 70 °C.

Table 10. Parameters of the PV Module AS-M3607-S (G1 CELLS) extracted using the Hybrid BESSA and BES techniques with G varying from 400 W/m² to 1000 W/m² while keeping T fixed at 25 °C.

G	Hybrid BESSA Technique
	$ℛ$S	$ℛ$ɡ	δ	Y
1000 W/m2	0.425	3800	1.4	2.108 × 105
800 W/m2	0.669	3800	1	1.3425 × 105
600 W/m2	0.799	3400	1.029	7.64534 × 104
400 W/m2	0.981	3450	1.17	3.4316 × 104
G	BES Technique
	$ℛ$S	$ℛ$ɡ	δ	Y
1000 W/m2	0.4336	3730.083	1.4108	211,863.1
800 W/m2	0.726	4000	1.2304	146,235.2
600 W/m2	0.3314	4000	1.8073	76,846.52
400 W/m2	0.878	4000	1.1779	34,623.14

provides parameters extracted by applying the Hybrid BESSA and BES technique to the PV Module AS-M3607-S (G1 CELLS) when keeping temperature fixed at 25 °C and varying irradiance changing from 1000 W/m² to 800 W/m², 600 W/m², and 400 W/m².

Table 9 and

Table 11. Reference values and percentage error for the PV Module AS-M3607-S(G1 CELLS) obtained using the Hybrid BESSA- and BES-based parameter extraction techniques with G varying from 1000 W/m² to 400 W/m² while keeping T fixed at 25 °C.

G	Hybrid BESSA Technique
	${\mathit{P}}_{\mathit{M}\mathit{P}\mathit{P}\_\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{V}}_{\mathit{O}\mathit{C}\_\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{I}}_{\mathit{S}\mathit{C}\_\mathit{r}\mathit{e}\mathit{f}}$	${\%\mathit{e}}_{\mathit{P}}$	${\%\mathit{e}}_{\mathit{V}}$	${\%\mathit{e}}_{\mathit{I}}$
1000 W/m2	400	49.5	10.12	0.01	0	0
800 W/m2	320	49.5	8.096	0	0.424	0.01
600 W/m2	240	49.5	6.071	0	0.969	0.01
400 W/m2	160	49.5	4.048	0	2	0.01
G	BES Technique
	${\mathit{P}}_{\mathit{M}\mathit{P}\mathit{P}\_\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{V}}_{\mathit{O}\mathit{C}\_\mathit{r}\mathit{e}\mathit{f}}$	${\mathit{I}}_{\mathit{S}\mathit{C}\_\mathit{r}\mathit{e}\mathit{f}}$	${\%\mathit{e}}_{\mathit{P}}$	${\%\mathit{e}}_{\mathit{V}}$	${\%\mathit{e}}_{\mathit{I}}$
1000 W/m2	400	49.5	10.12	0.026	0	0
800 W/m2	320	49.5	8.095	0.1869	0.612	0.01
600 W/m2	240	49.5	6.071	0.310	1.87	0
400 W/m2	160	49.5	4.048	0.81	2.018	0.01

provide information on the reference values for power, current, and voltage, as well as the percentage error for the open circuit voltage, power, and short circuit current of the PV Module AS-M3607-S simulated using the parameters extracted using the Hybrid BESSA technique and the BES technique, respectively, at different temperatures when keeping irradiance fixed at 1000 W/m².

In Table 11, the reference values and error values are obtained at different irradiances and while keeping temperature fixed at 25 °C.

Figure 10 shows the simulated IV curve of Module AS-M3607-S (G1 CELLS) at different irradiances at a fixed temperature of 25 °C using the Hybrid BESSA technique. Figure 11 shows the simulated PV curve using the BES technique for Module AS-M3607-S (G1 CELLS) at varying irradiances at a fixed temperature of 25 °C. Figure 12 shows the IV curve obtained via simulation for Module AS-M3607-S (G1 CELLS) at various temperatures at a fixed irradiance of 1000 W/m² by utilizing the Hybrid BESSA technique. Figure 13 shows the PV curve obtained via simulation for Module AS-M3607-S (G1 CELLS) at various temperature ranging from −30 °C to 70 °C at a fixed irradiance of 1000 W/m² using the Hybrid BESSA technique.

Experimental results

For the validation and practicability of proposed Hybrid BESSA technique, an experimental test was conducted on monocrystalline PV Module AS-M3607-S, as shown in Figure 14.The experimental results were obtained via the experimental set up, as shown in Figure 15. Current sensors were used to detect current and to convert it to an easily measurable output voltage. The voltage was monitored by voltage sensors. The Module AS-M3607-S was seated on roof top of laboratory. Reading of the inputs was performed using ARDUINO boards. ARDUINO is an open-source electronics platform based on easy-to-use hardware and software. Figure 14 displays the obtained experimental results measured at an irradiance of 480 W/m² and a temperature of 28 °C.

Figure 15 shows a comparison of the experimental curve with the Hybrid BESSA simulated power–voltage characteristics and the BES simulated power–voltage characteristics.

The simulation results obtained by applying the Hybrid BESSA and BES techniques were discussed in the earlier section. It is evident that the PV module simulated parameters extracted using the Hybrid BESSA technique yielded a better result and better fit with a lower degree of error when compared against the hardware results and those to obtained using the BES technique. The Hybrid BESSA simulated power–voltage characteristics more closely resembled the hardware results. This was also evident from the simulation result analyses. As depicted in Table 10 and Table 11, the fitness function value derived using the Hybrid BESSA technique was less than that of the BES technique. In the same way, the percentage of error obtained for the open circuit voltage, output power, and short circuit current was less when derived using the Hybrid BESSA in comparison with the BES technique. Thus, it has been proven that extracted parameter values derived using the Hybrid BESSA technique have better modeling capability when compared to those of the BES technique, proving its superiority both theoretically and experimentally. This is due to the hybridization of the BES and SA techniques. This process embodies the hybridization of a global search with local searches.

9. Discussion

A multi-objective parameter extraction scheme based on the Hybrid BESSA and BES for PV modules is discussed in this work. In the first stage, the efficacy of the Hybrid BESSA was analyzed with testing on unimodal benchmark functions, fixed dimensional benchmark functions, and fixed dimensional multimodal benchmark functions, as discussed in Section 6. The statistical results of the Hybrid BESSA technique was compared with those of the BES, Hybrid Water Cycle—Moth Flame Optimization Algorithm (WCMFO) [21], Artificial Bee Colony (ABC) algorithm [22], Cuckoo Search (CS) [23], Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA) [24], Moth Flame Optimization (MFO) [25], Gravitational Search Algorithm (GSA) [26], Water Cycle Algorithm (WCA) [27], Dragonfly Algorithm (DA) [28], and Whale Optimization Algorithm(WOA) [29] as detailed in [9]. For all of the above analyses, the maximum number of search particles taken and total iteration counts were limited to 100 and 1000, respectively, and these are common parameters for all of the algorithms mentioned above. To reach the right conclusion, the number of runs for each algorithm was fixed at 30.

It was observed that the Hybrid BESSA produced better results and was able to find global optima for all of the unimodal functions when compared to the BES, CamWOA, and all of the other above-mentioned techniques [9]. The statistical outcomes from the Hybrid BESSA technique obtained by testing multimodal functions revealed that the Hybrid BESSA technique produced better results for all of the mentioned functions except function $f_{13}\left(p\right)$. For function $f_{13}\left(p\right)$, the BES produced better results than the Hybrid BESSA and CamWOA. For function $f_{10}\left(p\right)$, the results were comparable.

When tested upon the fixed dimensional multimodal functions, it was found that the Hybrid BESSA and BES techniques were both able to find global optima for functions $f_{16}\left(p\right)-f_{19}\left(p\right)$. For functions $f_{14}\left(p\right)$, $f_{15}\left(p\right)$, $f_{17}\left(p\right)$, $f_{19}\left(p\right)$, and $f_{20}\left(p\right)$, the Hybrid BESSA produced better results than the BES, CamWOA, and all of the other mentioned 10 optimization techniques [9]. For functions $f_{16}\left(p\right)$ and $f_{18}\left(p\right)$, the proposed BESSA was able to find global optima. For functions $f_{21}\left(p\right)$, $f_{22}\left(p\right)$, and $f_{23}\left(p\right)$, CamWOA provided better results. From the above analysis, it is evident that the Hybrid BESSA has a better exploration capability when compared to the BES, CamWOA, and other techniques.

Thus, it was concluded that the Hybrid BESSA has better exploration and exploitation capabilities when compared to the BES, CamWOA, WCMFO, ABC, CS, PSOGSA, MFO, GSA, WCA, DA, and WOA.

Again, by studying the convergence characteristics in Section 6, it was found that the convergence capability of the Hybrid BESSA was better and faster than that of the BES. This is due to the hybridization of a global search (i.e., BES) with local search (i.e., SA) techniques.

The Hybrid BESSA was also used for parameter extraction for the PV Module AS-M3607-S in a MATLAB/SIMULINK environment.

It was found that single-diode PV modeling based on parameters extracted from the Hybrid BESSA was more robust and accurate than that based on the BES-based extracted parameters. The efficacy of the modeling was tested in two ways. The first was by varying temperature from−30 °C to 0 °C, 25 °C, 30 °C, 50 °C, and 70 °C while keeping irradiance fixed at 1000 W/m². In the second set of tests, the temperature was kept fixed at 25 °C and the irradiance varied by four values, i.e., 400 W/m², 600 W/m², 800 W/m², and 1000 W/m².

For exactness and efficacy in evaluation, the lower bound and upper bound of the extracted parameters are specified in Table 7. Table 8 provide the values of the parameters extracted using the Hybrid BESSA and BES techniques for the PV Module AS-M3607-S (G1 CELLS) when keeping irradiance fixed at 1000 W/m² and varying temperature from −30 °C to 70 °C. Table 10 provides the parameters extracted by applying the Hybrid BESSA and BES techniques to the PV Module AS-M3607-S (G1 CELLS) when keeping temperature fixed at 25 °C and varying irradiance from 1000 W/m² to 800 W/m², 600 W/m², and 400 W/m².

Table 9 and Table 11 provide information about the reference values for power, current, and voltage, as well as the percentage error for the open circuit voltage, percentage error of power, and percentage error for the short circuit current of the PV Module AS-M3607-S when simulated using parameters extracted with the Hybrid BESSA and BES techniques, respectively, at different temperatures while keeping irradiance fixed at 1000 W/m².

It can be seen from Table 9 that when the modeling of the PV cell was conducted with the Hybrid BESSA-based extracted parameters with irradiance at 1000 W/m² and temperature at −30 °C, the combined objective function (Y) obtained was 3.6199 × 10⁵ and the percentage error for the open circuit voltage (${\%e}_V$), percentage error of power(${\%e}_P$), and percentage error for the short circuit current (${\%e}_I$)were 0.703, 0, and 0.01, respectively, whereas when the parameters extracted from the Hybrid BESSA resulted in a combined objective function of 3.654 × 10⁵, the obtained values for ${\%e}_V$, ${\%e}_P$, and ${\%e}_I$ were 1.49, 0.201, and 0, respectively. The improvement in the combined objective function using the Hybrid BESSA-based extraction technique was 0.9$\%$ as compared to that of the BES.

When irradiance was fixed at 1000 W/m² with a temperature of 0 °C, the corresponding values of Y, ${\%e}_P$, ${\%e}_V$, and ${\%e}_I$ were 2.73 × 10⁵, 0, 0.378, and 0, respectively, for the modeling conducted with the Hybrid BESSA-based extracted parameters, whereas those same values were 298,578.7, 0.161, 0.738, and 0, respectively, for the modeling conducted with the BES-based extracted parameters. The improvement in the combined objective function using the Hybrid BESSA-based extraction technique was 8.5$\%$ as compared to that of the BES.

When irradiance was fixed at 1000 W/m² with a temperature of 25 °C, the corresponding values of Y, ${\%e}_V$, ${\%e}_P$, and ${\%e}_I$ were 2.12 × 10⁵, 0, 0, and 0 for the modeling conducted with the Hybrid BESSA-based extracted parameters, whereas those same values were 298,578.7, 0.025, 0, 0, and 0 for the modeling conducted with the BES-based extracted parameters. The improvement in the combined objective function using the Hybrid BESSA-based extraction technique was 29.2$\%$ as compared tothat of the BES.

When irradiance was fixed at 1000 W/m² with a temperature of 30 °C, the corresponding values of Y, ${\%e}_P$, ${\%e}_V$, and ${\%e}_I$ were 2.01 × 10⁵, 0, 0.081, and 0 for the modeling conducted with the Hybrid BESSA-based extracted parameters, whereas those same values were 200,937.4, 0.204, 0.101, and 0 for the modeling conducted with the BES-based extracted parameters. The improvement in the combined objective function using the Hybrid BESSA-based extraction technique was 0.03$\%$ as compared to that of the BES.

When irradiance was fixed at 1000 W/m² with a temperature of 50 °C, the corresponding values of Y, ${\%e}_P$, ${\%e}_V$, and ${\%e}_I$ were 1.61 × 10⁵, 0, 0.476, and 0 for the modeling conducted with the Hybrid BESSA-based extracted parameters, whereas those same values were170,735.5, 0.101, 0.204, and 0 for the modeling conducted with the BES-based extracted parameters. The improvement in the combined objective function using the Hybrid BESSA-based extraction technique was 5.7$\%$ as compared to that of the BES.

When irradiance was fixed at 1000 W/m² with a temperature of 70 °C, the corresponding values of Y, ${\%e}_P$, ${\%e}_V$, and ${\%e}_I$ were 1.26 × 10⁵, 0, 0.101, and 0 for the modeling conducted with the Hybrid BESSA-based extracted parameters, whereas those same values were 126,727.9, 0.088, 1.21, and 0 for the modeling conducted with the BES-based extracted parameters. The improvement in the combined objective function using the Hybrid BESSA-based extraction technique was 0.5$\%$ as compared to that of the BES.

It can be seen from Table 11 that when the modeling of the PV cell was conducted with the Hybrid BESSA-based extracted parameters with irradiance at 1000 W/m² and a temperature of 25 °C, the combined objective function(Y) obtained was 2.108 × 10⁵, and the percentage error for the open circuit voltage (${\%e}_V$), percentage error of power (${\%e}_P$), and percentage error for the short circuit current (${\%e}_I$)were 0, 0.01, and 0, respectively, whereas the parameters extracted using the Hybrid BESSA resulted in a combined objective function 211,863.1 and the ${\%e}_V$, ${\%e}_P$, and ${\%e}_I$ obtained were 0, 0.26, and 0, respectively. The improvement in the combined objective function using the Hybrid BESSA-based extraction technique was 0.5$\%$ as compared to that of the BES.

When temperature was fixed at 25 °C with irradiance at 800 W/m², the corresponding values of Y, ${\%e}_P$, ${\%e}_V$, and ${\%e}_I$ were 1.3425 × 10⁵, 0, 0.424, 0.1 for the modeling conducted with the Hybrid BESSA-based extracted parameters, whereas they were 146,235.2, 0.1869, 0.612, and 0.1 for the modeling conducted with the BES-based extracted parameters. The improvement in the combined objective function using the Hybrid BESSA-based extraction technique was 8.1$\%$ as compared to that of the BES.

When temperature was fixed at 25 °C with irradiance at 600 W/m², the corresponding values of Y, ${\%e}_P$, ${\%e}_V$, and ${\%e}_I$ were 7.64534 × 10⁴, 0, 0.969, and 0.01 for the modeling conducted with the Hybrid BESSA-based extracted parameters, whereas they were 76,846.52, 0.310, 1.87, and 0 for the modeling conducted with the BES-based extracted parameters. The improvement in the combined objective function using the Hybrid BESSA-based extraction technique was 0.5$\%$ as compared to that of the BES.

When temperature was fixed at 25 °C with irradiance at 400 W/m², the corresponding values of Y, ${\%e}_P$, ${\%e}_V$, and ${\%e}_I$ were 3.4316 × 10⁴, 0, 2, and 0.01 for the modeling conducted with the Hybrid BESSA-based extracted parameters, whereas they were 34,623.14, 0.81, 2.018, and 0.01 for the modeling conducted with the BES-based extracted parameters. The improvement in the combined objective function using the Hybrid BESSA-based extraction technique was 0.8$\%$ as compared to that of the BES.

From the above analysis, it is evident that the combined objective function of ${\%e}_P$, ${\%e}_V$, and ${\%e}_I$ demonstrated significant improvements in the PV cell modeling conducted with the Hybrid BESSA-based extracted parameters as compared to the BES-based extracted parameters. This proves the robustness and high accuracy of the Hybrid BESSA technique.

The modeling was simulated based on the practical hardware PV Module AS-M3607-S. The multi-objective parameter extraction scheme based on the Hybrid BESSA provided a more accurate modeling output as compared to that of the BES. This is evident from the lower percentage error of power, lower percentage error of the open circuit voltage, and lower percentage error of the short circuit current. The percentage error was calculated by comparing the results with the respective reference values. The combined objective function was also found to be lower for the Hybrid BESSA-based multi-objective parameter extraction technique as compared to that of the BES.

The PV modeling output obtained using the multi-objective parameter extraction scheme based on the Hybrid BESSA and the BES was also tested on a hardware platform using ARDUINO boards, as depicted in Figure 13. It is evident from Figure 15 that the PV curve simulated using the parameters extracted via the multi-objective parameter extraction scheme based on the Hybrid BESSA was a better fit to the hardware-obtained results as compared to those of the BES. The PV Module AS-M3607-S was the basis for both the simulation and hardware results.

10. Conclusions and Future Scope of Work

Our main conclusions are as follows:

• The Hybrid BESSA has better exploitation capabilities, as the Hybrid BESSA is able to find global optima for all the unimodal functions when compared to the BES, CamWOA, WCMFO, ABC, CS, PSOGSA, MFO, GSA, WCA, DA, and WOA.
• The Hybrid BESSA has better exploration capabilities, as it produced better results for all of the multimodal functions except function $f_{13}\left(p\right)$.
• When tested on fixed dimensional multimodal functions, it was found that the Hybrid BESSA provided better results for six functions out of the ten functions.
• The multi-objective parameter extraction scheme based on the Hybrid BESSA yielded a more accurate and robust modeling output as compared to that of the BES, as was evident from the lower percentage error of power, lower percentage error of the open circuit voltage, and lower percentage error of the short circuit current obtained as compared to the modeling conducted with the parameters using the BES-based extraction scheme. Again, the combined objective function of the Hybrid BESSA-based scheme was lower than that of the BES-based scheme.
• To validate the theoretical analysis, the Hybrid BESSA was also tested on the hardware PV Module AS-M3607-S.The Hybrid BESSA was a better fit and was more accurate when compared against the hardware-obtained results as compared to the BES results.

From all of these studies, it has been concluded that the Hybrid BESSA-based multiobjective parameter extraction scheme is more accurate than the BES-based multi-objective parameter extraction scheme. The Hybrid BESSA is more accurate and robust as compared to the BES.

These current research findings are limited to a comparison of the BESSA and BES algorithms with experimental results which achieved less than 1% error rates. However, the inclusion of Wilcoxon’s rank-sum test would be a better tool for proving the reliability of the results at a higher level of accuracy and is one of our suggestions for future work. This future scope of work consists of deducing and analyzing results based on different types of PV modules, i.e., thin film modules, polycrystalline modules, and monocrystalline modules, working in different weather conditions. It is also necessary to complete performance comparisons between a dual-diode PV model and a single-diode PV model both with regards to actual hardware results and their theoretical bases.