Parameter Estimation of Static/Dynamic Photovoltaic Models Using a Developed Version of Eagle Strategy Gradient-Based Optimizer

Abstract

The global trend towards renewable energy sources, especially solar energy, has had a significant impact on the development of scientific research to manufacture high-performance solar cells. The issue of creating a model that simulates a solar module and extracting its parameter is essential in designing an improved and high performance photovoltaic system. However, the nonlinear nature of the photovoltaic cell increases the challenge in creating this model. The application of optimization algorithms to solve this issue is increased and developed rapidly. In this paper, a developed version of eagle strategy GBO with chaotic (ESCGBO) is proposed to enhance the original GBO performance and its search efficiency in solving difficult optimization problems such as this. In the literature, different PV models are presented, including static and dynamic PV models. Firstly, in order to evaluate the effectiveness of the proposed ESCGBO algorithm, it is executed on the 23 benchmark functions and the obtained results using the proposed algorithm are compared with that obtained using three well-known algorithms, including the original GBO algorithm, the equilibrium optimizer (EO) algorithm, and wild horse optimizer (WHO) algorithm. Furthermore, both of original GBO and developed ESCGBO are applied to estimate the parameters of single and double diode as static models, and integral and fractional models as examples for dynamic models. The results in all applications are evaluated and compared with different recent algorithms. The results analysis confirmed the efficiency, accuracy, and robustness of the proposed algorithm compared with the original one or the recent optimization algorithms.

1. Introduction

Recently, since the surge in fossil fuel prices, the world’s attention has turned to renewable energy sources, such as wind and solar energy. Solar energy has several advantages beyond resources. These advantages are simple in installation, low maintenance activities, and suitable for different sizes. Although the mentioned advantages of solar energy are still weak regarding high efficiency [1], increased interest in renewable energy sources has led to an increase in researchers’ interest in developing these systems and increasing their efficiency. Developing an optimal mathematical model that simulates the natural photovoltaic system is one of the biggest challenges for researchers. The difficulty of developing these models is due to the nonlinear properties of solar cells [2].

Different types of PV models are proposed in the literature. The static model based on the main characteristics of the PV cell as it consists of two semiconductor materials (p-type and n- type) to achieve simple P-N junction characteristics, is considered the base element of the PV system. The PV module is a connection of series and shunt PV cells [3]. The simplest static model is a single diode model (SDM), which has one diode connected with series and shunt resistance [4,5,6,7]. The development in this type of models is based on increasing the accuracy of the models by representing other effects. The double diode model (DDM) is developed to represent the recombination effect in the PV cell. DDM consists of two diodes with one series and one shunt resistance. The total estimated parameters in the DDM are seven parameters, making it more complex than SDM [8,9]. By increasing the number of diodes, other effects can be represented in the model, reflected in increasing the model accuracy; on the other hand, the estimated parameters for the model is also increased, reflected in increasing the model complexity. This idea is illustrated in the three diode model (TDM) [10,11,12]. TDM has three diodes to represent the effect of leakage current and grain boundaries. TDM has nine estimated parameters, so although the model is more accurate, it is considered more complex. The balance between accuracy and complexity is determined according to the application [13]. Although the static model has a wide range of developments in the literature and is more representative to the PV system, the representation of the load connection, variation, and switching are missed. The dynamic model has been proposed in the literature to overcome this problem by representing the load connection in the model [14,15,16,17]. The dynamic models proposed in the literature are the integer and fractional dynamic models. The integer dynamic model is considered the more popular dynamic model and the fractional model has been developed to increase the accuracy of the integer model [18]. All these models have different parameters. These parameters control the model output (dependent variable) based on the model input (independent variable). The well estimation of these parameters has a direct effect in the model accuracy, for which many researchers have proposed to discuss. Parameters estimation using optimization algorithms has been discussed by several researchers. A review about the applied optimization techniques to estimate the parameters of the PV models is proposed in [19]. The numerical/analytical methods are applied for the optimization problems but these methods achieve low accuracy solutions. Population based algorithms are applied widely for these problems as it is simple in application with more accurate results. Population based algorithms are too many to discuss in this paper, but one example includes the recently proposed Whippy Harris Hawks Optimization Algorithm (WHHO) [20]. WHHO is an enhancement for HHO proposed by [21]. HHO is inspired by a group of hawks in catching their prey. WHHO improves the weak in HHO by reducing the local optima problem occurrence besides increasing the algorithm search speed. This meta-heuristic method is a derivative-free optimization method. It has no restrictions on the objective function and has a great advantage in solving multimodal problems. Therefore, it is employed to solve many optimization problems, such as the high performance SPR sensor design [22], solar-based DG allocation [23], optimal reactive power dispatch [24], and parameter estimation of photovoltaic [25,26,27]. One recent application of meta-heuristic algorithm for PV parameters estimation is the application of wild horse optimizer (WHO) for parameter estimation of modified double-diode and triple-diode photovoltaic models proposed in [28]. Another interesting work is the application of an improved bald eagle search algorithm for parameter estimation of different photovoltaic models proposed in [29]. The improved algorithm has been tested through three static PV models SDM, DDM, and TDM. Moreover, three modified static PV models MSDM, MDDM, and MTDM. The study in [28,29] is interesting due to it using recent algorithms with modified static models, but the dynamic PV models are missed.

The GBO algorithm is considered a metaheuristic optimization algorithm inspired by gradient descent and the Newton method [30]. GBO has advantage of rapid convergence due to gradient search rules and also reducing probability of escaping from local optima. ESCGBO is a modified version of GBO proposed in this paper. The ESCGBO technique is an enhancement applied to the GBO using eagle strategy (ES) with chaotic method to enhance the balance between global search or exploration and local search or exploitation and then increasing the original algorithm performance. The ES has been used in several studies to improve the performance of the conventional algorithms, such as an improved particle swarm optimization (PSO) algorithm [31,32], adaptive Nelder–Mead simplex algorithm [33], and JAYA algorithm and Nelder–Mead simplex method [34], eagle strategy with flower algorithm [35], while chaotic maps were also used to enhance several optimization algorithms, such as developing chaotic HHO algorithm [36], chaotic artificial ecosystem-based optimization (CAEO) algorithm [37], chaotic JAYA algorithm [38], and chaotic salp swarm algorithm (CSSA) [39]. In this article, the ESCGBO algorithm is proposed and applied to estimate different PV static and dynamic PV models. The results from all applications are analyzed and evaluated.

The principal contributions of this article are summarized as follows:

• Proposing a developed version of GBO algorithm based on the eagle strategy with the chaotic method to enhance the performance and avoiding the local optima.
• Applying the conventional GBO and ESCGBO as well as other well-known optimization algorithms for parameter estimation of different PV models, such as static and dynamic PV models.
• The results confirm that the proposed ESCGBO has the capability to enhance the performance and increase the effectiveness of the conventional GBO and improve the convergence rate

The remainder of this paper is arranged as follow: Section 2 presents the static and dynamic PV models. ESCGBO is proposed in Section 3. The results and analysis is discussed in Section 4. Section 5 presents the conclusion.

2. PV Static and Dynamic Models

PV models should accurately describe the characteristics of the PV systems for different types of PV models proposed in literature in order to achieve this target. In this section, the most popular static and dynamic models are discussed.

2.1. Static SDM and DDM

SDM has one diode connected with series resistance and parallel resistance to the photo generated current, represented with current source connected parallel with the diode. SDM has five parameters; therefore, it is considered the simplest model. Consider x is a vector of model parameters x = (x₁, x₂, x₃, x₄, x₅) equivalent to (R_s, R_sh, I_ph, I_s, ɳ). The SDM equivalent circuit is shown in Figure 1 and represented by Equations (1) and (2). The DDM is developed to represent the effect of recombination in the PV cell, and this is achieved by adding a second diode to the SDM circuit as shown in Figure 2. The DDM has seven parameters x = (x₁, x₂, x₃, x₄, x₅, x₆, x₇) equivalent to (R_s, R_sh, I_ph, I_s1, I_s2, ɳ₁, ɳ₂). The DDM is described by Equations (3) and (4). The objective function for SDM and DDM is described by Equations (5) and (6) respectively.

$$I=I_{ph}-I_D-I_{sh}$$

(1)

$$I=I_{ph}-I_s\left[\exp (\frac{q(V+R_s\ast I)}{{\eta }_{}\ast K\ast T})-1\right]-\frac{(V+R_s\ast I)}{R_{sh}}$$

(2)

$$I=I_{ph}-I_{D1}-I_{D2}-I_{sh}$$

(3)

$$I=I_{ph}-I_{s1}\left[\exp (\frac{q(V+R_s\ast I)}{{\eta }_1\ast K\ast T})-1\right]-I_{s2}\left[\exp (\frac{q(V+R_s\ast I)}{{\eta }_2\ast K\ast T})-1\right]-\frac{(V+R_s\ast I)}{R_{sh}}$$

(4)

$$f_{DD}(V,I,X)=I-X_3+X_4\left[\exp (\frac{q(V+X_1\ast I)}{X_5\ast K\ast T})-1\right]+\frac{(V+X_1\ast I)}{X_2}$$

(5)

$$f_{DD}(V,I,X)=I-X_3+X_4\left[\exp (\frac{q(V+X_1\ast I)}{X_6\ast K\ast T})-1\right]+X_5\left[\exp (\frac{q(V+X_1\ast I)}{X_7\ast K\ast T})-1\right]+\frac{(V+X_1\ast I)}{X_2}$$

(6)

2.2. Dynamic PV Model

The integral and fractional dynamic PV models are two popular dynamic models, and are selected to be discussed here.

The integral dynamic model (IOM) is a second order model consists of constant voltage source $V_{oc}$ and series resistance $R_S$ to represent the static model as shown in Figure 3. The dynamic part is represented by capacitor $C$ for junction capacitance and resistance $R_c$ for conductance. The connected cables inductance is represented by the coil inductance. The load is represented by$R_L$.The total number of unknown parameters are three parameters ($R_c,C \mathrm{and} L$), the IOM is represented by Equations (7) and (8).

$$i_L\left(s\right)=\frac{V_{oc}}{s}\frac{a_{11}\left(s+b_1\right)+b_2\left(s-a_{11}\right)}{\left(s-a_{22}\right)\left(s-a_{11}\right)-a_{12}{a}_{21}}$$

(7)

$$\begin{array}{c}\left(\begin{array}{c}{a}_{11} a_{12}\\ a_{21} a_{22}\end{array}\right)=\left(\begin{array}{c}\frac{-1}{C\left(R_c+R_s\right)} \frac{-R_S}{C\left(R_c+R_s\right)}\\ \frac{R_S}{L\left(R_c+R_s\right)} \frac{-\left[R_L{R}_c+R_s{R}_c+R_L{R}_s\right]}{L\left(R_c+R_s\right)} \end{array}\right) ,\\ \left(\begin{array}{c}{b}_1\\ b_2\end{array}\right)=\left(\begin{array}{c}\frac{1}{C\left(R_c+R_s\right)} \\ \frac{R_c}{L\left(R_c+R_s\right)} \end{array}\right)\end{array}$$

(8)

The fractional order model (FOM) is developed to represent fractional capacitor in case of low values of $R_c$ due real frequency dependence on fractional capacitance impedance, as shown in Figure 4. The fractional order of capacitance and inductance are represented by α and β, respectively. The total number of FOM is five parameters ($R_c,C, L, \alpha \mathrm{and} \beta$), the IOM is represented by Equations (9) and (10).

$$i_L\left(s\right)=\frac{V_{oc}}{s}\frac{a_{11}\left(s^{\alpha }+b_1\right)+b_2\left(s^{\alpha }-a_{11}\right)}{\left(s^{\beta }-a_{22}\right)\left(s^{\alpha }-a_{11}\right)-a_{12}{a}_{21}}$$

(9)

$$\begin{array}{c}\left(\begin{array}{c}{a}_{11} a_{12}\\ a_{21} a_{22}\end{array}\right)=\left(\begin{array}{c}\frac{-1}{C_{\alpha }\left(R_c+R_s\right)} \frac{-R_S}{C_{\alpha }\left(R_c+R_s\right)}\\ \frac{R_S}{L_{\beta }\left(R_c+R_s\right)} \frac{-\left[R_L{R}_c+R_s{R}_c+R_L{R}_s\right]}{L_{\beta }\left(R_c+R_s\right)} \end{array}\right) ,\\ \left(\begin{array}{c}{b}_1\\ b_2\end{array}\right)=\left(\begin{array}{c}\frac{1}{C_{\alpha }\left(R_c+R_s\right)} \\ \frac{R_c}{L_{\beta }\left(R_c+R_s\right)} \end{array}\right) \end{array}$$

(10)

3. Optimization Methodology

This section describes the basic variant of the gradient-based optimizer (GBO). After, the process of the proposed EGCGBO algorithm is presented.

3.1. Original Gradient-Based Optimizer (GBO) Algorithm

The GBO algorithm is a proposed metaheuristic optimization algorithm, which is presented in reference [30]. This algorithm uses the Newton’s gradient-based method to explore the whole search space. This algorithm uses two main machinists, namely gradient search rule (GSR) and local escaping operator (LEO) to achieve the best solution.

3.1.1. GBO Initialization

Like many algorithms, the GBO comprises $N$ vector (members of populations) in the D-dimensional search space. The initial population is randomly generated by the following equation [40]:

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

(11)

where $X_n$ refers to the nth vector, $X_{min}, X_{max}$ are the limits of the solution space in each problem and $rand\left(0, 1\right)$ denotes a random number defined in the range of [0, 1].

3.1.2. Gradient Search Rule (GSR)

In the GBO algorithm, GSR is based on the gradient-based method where the aim of using the GSR is exploration tendency improvement and increasing the convergence rate. Therefore, the new position $X_{n+1}$ is defined as:

$$X_{n+1}=X_n-\frac{2∆x\times f\left(X_n\right)}{f\left(X_n+∆x\right)-f\left(X_n-∆x\right)}$$

(12)

The Equation (12) will be adjusted to include the population-based search theory, which is presented by Equation (13).

$$GSR=randn\times \frac{2∆x X_n}{\left(x_{worst}-x_{best}+\epsilon \right)}$$

(13)

where$randn$ is a random number with a normal distribution,$x_{worst},x_{best}$ denote the worst and best solutions obtained during the process of optimization, $\epsilon$ denotes a small number within the interval [0, 0.1], and $∆x$ refers to the change in position at each iteration. From the previous equations, the GSR can be defined as:

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

(14)

where ${\rho }_1$ denotes the randomly generated parameter and it calculated from the following equation:

$${\rho }_1=\left(2\times rand\times \alpha \right)-\alpha$$

(15)

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

(16)

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

(17)

where $\alpha$ represents a sine function for the transference from exploration to exploitation, ${\beta }_{min}$ and ${\beta }_{max}$ are constant values 0.2 and 1.2, respectively, $m$ is the current number of iterations, and $M$ represents the total number of iterations. The change $∆x$ between the best solution $x_{best}$ and a randomly chosen position $x_{r1}^m$ is given by:

$$∆x=rand\left(1:N\right)\times \left|step\right|$$

(18)

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

(19)

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

(20)

where $rand\left(1:N\right)$ denotes a random vector with N dimensions, $r1, r2, r3, \mathrm{and} r4 (r1\ne r2\ne r3\ne r4\ne n$) are different integers randomly chosen from [1, N], $step$ denotes a step size. The new position $X_{n+1}$ represents an updated version based on the GSR as follows:

$$X_{n+1}=X_n-GSR$$

(21)

The direction of movement (DM) is added for better exploitation of the nearby area of $X_n$ which can be calculated as:

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

(22)

$${\rho }_2=\left(2\times rand\times \alpha \right)-\alpha$$

(23)

Therefore, the new position $X{1}_n^m$ is calculated from the following equation after taking the GSR and DM into consideration:

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

(24)

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

(25)

The GBO used another position to increase the local search by putting the best-so-far solution ($x_{\mathit{best}}$) rather than the position $x_n^m$. The new position ($X{2}_n^m$) is calculated as follows:

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

(26)

where

$$y{p}_n=rand\times (\frac{\left[z_{n+1}+x_n\right]}{2}+rand\times ∆x)$$

(27)

$$y{q}_n=rand\times (\frac{\left[z_{n+1}+x_n\right]}{2}-rand\times ∆x)$$

(28)

According to the positions $X{1}_n^m$,$X{2}_n^m$, and the current position ($X_n^m)$, the new position at the next iteration ($x_n^{m+1}$) is defined as:

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

(29)

$$X{3}_n^m=X_n^m-{\rho }_1\times \left(X{2}_n^m-X{1}_n^m\right)$$

(30)

3.1.3. Local Escaping Operator (LEO)

The LEO is applied to enhance the performance of the GBO algorithm and to escape the local solutions for solving the complicated problems. The LEO generates an appropriate solution ($X_{LEO}^m$) by using several solutions, which include $x_{\mathit{best}}$, the solutions $X{1}_n^m$, and $X{2}_n^m$, two random solutions $x_{r1}^m$ and $x_{r2}^m$, and a new randomly generated solution ($x_k^m$). The solution $X_{LEO}^m$ is formulated as:

$$\begin{array}{l}if rand<pr\\ if rand<0.5\\ \hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{ll}{X}_{LEO}^m=& X_n^{m+1}+f_1\times \left(u_1\times x_{\mathrm{best}}-u_2\times x_k^m\right)+f_2\times {\rho }_1\times \left(u_3\times (X{2}_n^m-X{1}_n^m\right)\\ & +u_2\times \left(x_{r1}^m-x_{r2}^m\right))/2\end{array}\\ X_n^{m+1}=X_{LEO}^m\\ \mathrm{Else}\\ \begin{array}{c}{X}_{LEO}^m=x_{\mathrm{best}}+f_1\times \left(u_1\times x_{\mathrm{best}}-u_2\times x_k^m\right)+f_2\times {\rho }_1\times \left(u_3\times (X{2}_n^m-X{1}_n^m\right)\\ +u_2\times \left(x_{r1}^m-x_{r2}^m\right))/2\end{array}\\ X_n^{m+1}=X_{LEO}^m\\ \mathrm{End}\\ \mathrm{End}\end{array}$$

(31)

where $f_1$ denotes a uniform distributed random number in the range of [−1, 1], $f_2$ is a random number from a normal distribution with a mean of 0 and a standard deviation of 1, $pr$ is the probability, and $u_1$, $u_2$, and $u_3$ are random values generated as follows:

$$u_1=\{\begin{array}{c}2\times rand if {\mu }_1<0.5 \\ 1 otherwise \end{array}$$

(32)

$$u_2=\{\begin{array}{c}rand if {\mu }_1<0.5 \\ 1 otherwise \end{array}$$

(33)

$$u_3=\{\begin{array}{c}rand if {\mu }_1<0.5 \\ 1 otherwise \end{array}$$

(34)

where $rand$represents a random number in the range of [0, 1], and ${\mu }_1$ refers to a number in the range of [0, 1]. The above equations are simply explained as follows:

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

(35)

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

(36)

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

(37)

where $L_1$ denotes a binary parameter with a value of 0 or 1. If parameter ${\mu }_1$ < 0.5, the value of $L_1$ = 1, otherwise, $L_1$= 0. The solution $x_k^m$ is generated as follows:

$$x_k^m=\{\begin{array}{c}{x}_{rand} if {\mu }_2<0.5 \\ x_p^m otherwise \end{array}$$

(38)

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

(39)

where $x_{rand}$ is a random generated solution, $x_p^m$ is a randomly chosen solution of the population ($p\in [1, 2,\dots , N$]), and ${\mu }_2$ represents a random number in the range of [0, 1]. Equation (38) is simplified as:

$$x_k^m=L_2\times x_p^m+\left(1-L_2\right)\times x_{rand}$$

(40)

where $L_2$ denotes a binary parameter with a value of 0 or 1. If ${\mu }_2$ < 0.5, the value of $L_2$ = 1, otherwise, $L_2$ = 0.

3.2. Eagle Strategy and Chaotic with Gradient-Based Optimizer (ESCGBO) Algorithm

3.2.1. Eagle Strategy

The eagle strategy is proposed for solving real-world optimization problems that is developed by Yang et al. [35]. It is inspired by the foraging behavior of eagles that fly randomly in analogy to the Levy flights. It is the two-stage method: global search randomization stage and an intensive local search [31]. The first stage aims mainly to investigate the search space globally and rapidly obtain a promising solution, while the target of the second stage is to obtain the optimal solution through making an intensive local search based on the achieved solution in the first stage. The benefit of this strategy is that there is no limit to the kinds of techniques or algorithms used in each stage. Any technique that is able to achieve better results in a flexible way could be used in any stage [31].

During the iteration of the proposed technique, the new candidate solution is generated by Levy flight as follows:

$$X_{n+1}=X_n-\gamma \left(X_n-X_{best}\right)⨁Levy\left(\lambda \right)=X_n+\frac{0.01u}{{\left|v\right|}^{1/\lambda }}\left(X_n-X_{best}\right)$$

(41)

where $\gamma$ is the step scaling size, the $⨁$ refers to the process of element-wise multiplications, $\lambda$ is the Levy flight exponent, while u and v can be expressed as:

$$u~N\left(0,{\sigma }_u^2\right) , v~N\left(0,{\sigma }_v^2\right)$$

(42)

The standard deviations ${\sigma }_u$ and ${\sigma }_v$ are explained as:

$${\sigma }_u={\left[\frac{\sin \left(\frac{\lambda \pi }{2}\right)·\mathrm{\Gamma }\left(1+\lambda \right)}{2^{\left(\lambda -1\right)}\lambda · \mathrm{\Gamma }\left(\frac{1+\lambda }{2}\right)}\right]}^{1/\lambda }, {\sigma }_v=1$$

(43)

where $\mathrm{\Gamma }$ is the Gamma function.

3.2.2. Chaotic Maps

Chaotic systems are deterministic systems that present unpredictable conduct, whose action is complex and similar to that of randomness [41]. In [41], chaos-based exploration rate to enhance the performance of three well-known optimization algorithms was proposed. Based on this proposal, the real random number ($v$) in Equation (41) was replaced by a chaotic number in the eagle strategy. This modification makes the value of $v$ linearly decreased from 2 to 0 throughout the course of iterations. The steps of the proposed ESCGBO are presented in the flowchart illustrated in Figure 5.

4. Results

4.1. Performance of the Proposed ESCGBO Algorithm

In this subsection, the effective and performance of the proposed ESCGBO technique were evaluated on several benchmark functions, including statistical measurements, such as best values, mean values, worst values, and standard deviation (STD) for the solutions obtained by the ESCGBO technique, the original GBO algorithm as well as the two recent optimization algorithms, including the equilibrium optimizer (EO) algorithm [42] and wild horse optimizer (WHO) algorithm [43]. The results obtained with the proposed ESCGBO algorithm were compared with these well-known optimization algorithms. All mentioned techniques were executed for the maximum number of the iterations’ function of 500 and population size of 50 for 20 independent runs using Matlab R2016a’working on Windows 8.1, 64bit. All computations were performed on a Core i5-4210U CPU@ 2.40 GHz of speed, and 8 GB of RAM. Figure 6 shows the qualitative metrics on F1, F3, F4, F5, F8, F12, F15, and F18, including 2D views of the functions, search history, average fitness history, and convergence characteristics curve using the proposed ESCGBO algorithm.

Table 1. The statistical results of unimodal benchmark functions using the proposed technique and other well-known algorithms.

Function		ESCGBO	GBO	WHO	EO
F1	Best	2.20 × 10−136	1.44 × 10−136	2.83 × 10−57	3.89 × 10−51
	Worst	1.14 × 10−129	6.32 × 10−128	3.53 × 10−47	6.09 × 10−47
	Mean	1.33 × 10−130	7.41 × 10−129	1.78 × 10−48	4.11 × 10−48
	std	3.38 × 10−130	1.86 × 10−128	7.90 × 10−48	1.36 × 10−47
F2	Best	2.31 × 10−71	2.89 × 10−70	8.31 × 10−33	5.47 × 10−29
	Worst	2.33 × 10−65	1.84 × 10−66	9.82 × 10−29	1.3 × 10−27
	Mean	1.39 × 10−66	2.39 × 10−67	1.39 × 10−29	3.65 × 10−28
	std	5.2 × 10−66	4.92 × 10−67	2.48 × 10−29	3.47 × 10−28
F3	Best	8.7 × 10−116	1.6 × 10−115	6.42 × 10−36	5.41 × 10−17
	Worst	6.7 × 10−104	3.9 × 10−101	6.83 × 10−28	1.39 × 10−11
	Mean	3.4 × 10−105	1.9 × 10−102	5.22 × 10−29	2.34 × 10−12
	std	1.5 × 10−104	8.7 × 10−102	1.66 × 10−28	4.08 × 10−12
F4	Best	1.58 × 10−64	8.91 × 10−64	5.07 × 10−22	4 × 10−14
	Worst	3.03 × 10−58	1.63 × 10−59	3.71 × 10−19	6.87 × 10−12
	Mean	1.72 × 10−59	1.6 × 10−60	4.56 × 10−20	1.2 × 10−12
	std	6.75 × 10−59	3.65 × 10−60	9.74 × 10−20	1.77 × 10−12
F5	Best	19.99698	18.52388	23.60955	24.44135
	Worst	25.02491	24.02389	86.19444	25.0421
	Mean	21.68888	21.22658	34.30228	24.81639
	std	1.324568	1.169705	20.97713	0.201379
F6	Best	1.14 × 10−9	5.23 × 10−9	3.67 × 10−8	1.39 × 10−8
	Worst	1.61 × 10−7	1.07 × 10−6	2.02 × 10−5	3.8 × 10−7
	Mean	3.35 × 10−8	7.89 × 10−8	1.42 × 10−6	9.07 × 10−8
	std	3.91 × 10−8	2.35 × 10−7	4.46 × 10−6	9.27 × 10−8
F7	Best	0.000176	0.000111	4.5 × 10−5	9.73 × 10−5
	Worst	0.001732	0.001663	0.00184	0.001432
	Mean	0.000578	0.000528	0.000857	0.00068
	std	0.000427	0.000392	0.000514	0.000358

The best values obtained are in bold.

,

Table 2. The statistical results of multimodal benchmark functions using the proposed technique and other well-known algorithms.

Function		ESCGBO	GBO	WHO	EO
F8	Best	−1881.33	−1909.05	−1789.02	−1798.26
	Worst	−1668.99	−1659.76	−1600.49	−1715.16
	Mean	−1733.03	−1771.42	−1705.23	−1751.04
	Std	54.18392	83.01581	49.62292	21.99425
F9	Best	0.00	0.00	0.00	0.00
	Worst	0.00	0.00	0.00	0.00
	Mean	0.00	0.00	0.00	0.00
	Std	0.00	0.00	0.00	0.00
F10	Best	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16	20
	Worst	8.88 × 10−16	8.88 × 10−16	20.00111	20
	Mean	8.88 × 10−16	8.88 × 10−16	1.005259	20
	Std	0.00	0.00	4.471221	1.23 × 10−10
F11	Best	0.00	0.00	0.00	0.00
	Worst	0.00	0.00	0.00	0.00
	Mean	0.00	0.00	0.00	0.00
	Std	0.00	0.00	0.00	0.00
F12	Best	9.51 × 10−12	1.45 × 10−10	1.18 × 10−11	1.42 × 10−10
	Worst	1.62 × 10−7	0.103669	0.103669	1.98 × 10−7
	Mean	9.05 × 10−9	0.005183	0.01555	1.25 × 10−8
	Std	3.61 × 10−8	0.023181	0.037979	4.37 × 10−8
F13	Best	9.97 × 10−9	2.29 × 10−8	7.36 × 10−8	1.63 × 10−8
	Worst	0.054779	0.054779	0.397801	0.108359
	Mean	0.01043	0.016351	0.035814	0.018527
	Std	0.014444	0.019872	0.093881	0.037211

The best values obtained are in bold.

and

Table 3. The statistical Results of composite benchmark functions using the proposed technique and other well-known algorithms.

Function		ESCGBO	GBO	WHO	EO
F14	Best	0.998004	0.998004	0.998004	0.998004
	Worst	0.998004	0.998004	3.96825	0.998004
	Mean	0.998004	0.998004	1.543534	0.998004
	Std	0.00	0.00	0.936299	1.76 × 10−16
F15	Best	0.000307	0.000307	0.000307	0.000307
	Worst	0.001594	0.001223	0.020363	0.020363
	Mean	0.000528	0.000445	0.001676	0.002359
	Std	0.00046	0.000335	0.004424	0.006161
F16	Best	−1.03163	−1.03163	−1.03163	−1.03163
	Worst	−1.03163	−1.03163	−1.03163	−1.03163
	Mean	−1.03163	−1.03163	−1.03163	−1.03163
	Std	2.28 × 10−16	2.28 × 10−16	1.35 × 10−16	2.1 × 10−16
F17	Best	0.397887	0.397887	0.397887	0.397887
	Worst	0.397887	0.397887	0.397887	0.397887
	Mean	0.397887	0.397887	0.397887	0.397887
	Std	0.00	0.00	0.00	0.00
F18	Best	3.00	3.00	3.00	3.00
	Worst	3.00	3.00	3.00	3.00
	Mean	3.00	3.00	3.00	3.00
	Std	8.7 × 10−16	4.2 × 10−16	5.85 × 10−16	6.76 × 10−16
F19	Best	−0.30048	−0.30048	−0.30048	−0.30048
	Worst	−0.30048	−0.30048	−0.30048	−0.30048
	Mean	−0.30048	−0.30048	−0.30048	−0.30048
	Std	1.14 × 10−16	1.14 × 10−16	1.14 × 10−16	1.14 × 10−16
F20	Best	−3.322	−3.322	−3.322	−3.322
	Worst	−3.2031	−3.2031	−3.08668	−1.84092
	Mean	−3.28633	−3.28633	−3.31023	−3.12852
	Std	0.055899	0.055899	0.052619	0.370083
F21	Best	−10.1532	−10.1532	−10.1532	−10.1532
	Worst	−5.0552	−5.0552	−2.68286	−5.0552
	Mean	−8.8787	−8.3689	−9.52706	−8.88098
	Std	2.264846	2.494761	1.966723	2.260817
F22	Best	−10.4029	−10.4029	−10.4029	−10.4029
	Worst	−2.7659	−5.08767	−2.75193	−2.7659
	Mean	−7.82681	−8.5426	−8.96998	−9.75532
	Std	2.973254	2.601082	2.948831	2.02859
F23	Best	−10.5364	−10.5364	−10.5364	−10.5364
	Worst	−5.12848	−2.80663	−2.87114	−2.42173
	Mean	−9.18443	−8.25715	−9.148	−10.1307
	Std	2.402536	2.907055	2.855393	1.814497

The best values obtained are in bold.

present the statistical results of the proposed ESCGBO algorithm and other recent techniques when applied for unimodal benchmark functions, multimodal benchmark functions, and composite benchmark functions, respectively. The best values were obtained with the proposed ESCGBO, GBO, WHO, and EO algorithms shown in bold. It is clearly seen that the proposed ESCGBO technique achieves the optimal solution for most of the benchmark functions. The convergence curves of these algorithms for the unimodal benchmark functions are shown in Figure 7 and the boxplots for each algorithm for this type of function are presented in Figure 8. Additionally, Figure 9 displays convergence curves of all algorithms for the multimodal benchmark functions, named F8 to F13 and Figure 10 shows the boxplots for each algorithm for these functions. Furthermore, the convergence curves of these algorithms for the composite benchmark functions are presented in Figure 11 while Figure 12 illustrates the boxplots for each algorithm for this type of benchmark function. From those figures, it is obvious that the proposed algorithm reached a stable point for all functions and the boxplots of the proposed algorithm are narrow for most functions compared to the other algorithms.

Table 4. CPU time (s) of four algorithms on 23 benchmark functions.

	ESCGBO	GBO	WHO	EO
F1	2.24173	2.045234	1.666734	0.340013
F2	1.831693	1.549915	1.43607	0.3331
F3	2.615374	1.913063	1.671268	0.365052
F4	1.945816	1.773811	2.09404	0.35823
F5	2.032641	2.028365	1.704013	0.414698
F6	1.88023	1.661728	1.679613	0.326943
F7	2.077834	1.637886	1.454167	0.371617
F8	1.795602	1.540629	1.430021	0.341869
F9	1.842514	1.602524	1.657841	0.3381
F10	1.761122	1.503307	1.394121	0.35016
F11	1.780045	1.608481	1.64313	0.416369
F12	1.746382	1.505388	1.366705	0.33033
F13	1.99615	1.835235	1.43058	0.352283
F14	2.417839	1.952414	1.608765	0.561999
F15	1.985862	1.682823	2.037507	0.96656
F16	1.754978	2.052051	1.407174	0.342419
F17	1.72506	1.415434	1.564904	0.375495
F18	2.746123	1.617294	2.4928	0.506535
F19	1.891536	2.200207	1.470604	0.363025
F20	1.792942	1.509464	1.532523	0.360518
F21	2.009832	2.13849	1.56471	0.332705
F22	2.330221	2.10297	2.356706	0.375617
F23	2.127842	2.027598	1.530525	0.360371

shows the values of the average CPU time of different algorithms on the 23 benchmark functions.

4.2. Real-World Application

This results section is concerned with testing the proposed algorithm behavior from different sides and through different scenarios. Scenario 1 presents the results of the parameters estimation process for static SDM and DDM. Scenario 2 presents the results of the parameters estimation process for dynamic IOM and FOM.

4.2.1. Scenario 1

This scenario proposes the results and the analysis of the SDM and DDM parameters estimation process for has 57 mm diameter commercial silicon R.T.C France solar cell. The data captured from the cell at irradiance of 1000 W/m² and at temperature 33 °C [29]. The estimation process was carried out by ESCGBO and compared with the original algorithm GBO and some recent algorithms. The compared algorithms were artificial ecosystem-based optimization (AEO) [37] and jellyfish search optimizer (JS) [44]. The upper and lower constrains for all estimated parameters are presented in

Table 5. Upper and lower constrains for all estimated parameters.

Parameter	Solar Cell
	Lower Limit	Upper Limit
Rs	0	5
Rsh	0	100
Iph	0	2
Is1	0	1
Is2	0	1
ɳ 1	1	2
ɳ 2	1	2

. The control parameters for all the compared algorithms have been presented in

Table 6. Parameters setting for all compared algorithms.

Algorithm	Control Prameters
ESCGBO	nP = 50	pr = 0.5	CM = 4
GBO	nP = 50	pr = 0.5	CM = 4
AEO	PopSize = 50	r1 = rand
JS	Npop = 50
WHO	N = 50	PS = 0.2	PC = 0.13
EO	Particles_no = 50	a1 = 2, a2 = 1	GP = 0.5

. The estimated parameters by ESCGBO and other algorithms of SDM and DDM are presented in

Table 7. Estimated parameters and RMSE of ESCGBO and other algorithms for SDM model.

	ESCGBO	GBO	AEO	JS
Rs (Ω)	0.036377	0.036377	0.036377091	0.035989122
Rsh (Ω)	53.71853	53.71852	53.71853164	57.40304457
Iph (A)	0.760776	0.760776	0.76077553	0.760889273
Is (A)	3.23 × 10−7	3.23 × 10−7	3.23 × 10−7	3.61 × 10−7
ɳ	1.476894	1.476894	1.476894314	1.48800365
RMSE	0.000986022	0.000986022	0.000986022	0.001025784

and

Table 8. Estimated parameters and RMSE of ESCGBO and other algorithms for DDM model.

	ESCGBO	GBO	AEO	JS
Rs (Ω)	0.036742647	0.036375073	0.036706957	0.035338785
Rsh (Ω)	55.50056767	53.68896189	55.32835695	54.95529664
Iph (A)	0.760781168	0.76077892	0.760783208	0.760597151
Is1 (A)	7.64 × 10−7	3.78 × 10−9	2.34 × 10−7	3.51 × 10−7
Is2 (A)	2.26 × 10−7	3.20 × 10−7	7.00 × 10−7	2.02 × 10−7
ɳ 1	1.99999756	1.434376674	1.449582257	1.487619855
ɳ 2	1.446849934	1.477724492	1.999999984	1.916132114
RMSE	0.000982418	0.000986029	0.000982451	0.00119615

, respectively. The upper and lower limit used in the optimization are listed in Table 1. The best results of the compared algorithms were determined by the best RMSE values of the compared algorithms Equation (44). The obtained RMSE for parameters estimation of the SDM and DDM by all algorithms are listed in Table 7 and Table 8 respectively. For SDM the compared algorithms have the same RMSE except JS algorithm. For DDM, the ESCGBO had the best RMSE followed by the AEO algorithm. The convergence curve of all algorithms for SDM are displayed in Figure 13. The convergence curve of all algorithms for DDM are displayed in Figure 14. The best convergence behavior was achieved by ESCGBO for DDM, as can be seen in Figure 14. The robustness of the proposed and the compared algorithms are analyzed using statistical analysis. The statistical analysis of 30 independent runs of all algorithms for SDM and DDM are presented in

Table 9. The statistical results of SDM for all other algorithms.

	Minimum	Average	Maximum	STD
ESCGBO	0.000986022	0.000986026	0.000986032	5.507 × 10−9
GBO	0.000986022	0.000989455	0.000996022	5.688 × 10−6
AEO	0.000986022	0.000989555	0.000996022	5.608 × 10−6
JS	0.001025784	0.001525784	0.002025784	0.0005

and

Table 10. The statistical results of DDM for all algorithms.

	Minimum	Average	Maximum	STD
ESCGBO	0.000982418	0.000982454	0.000982518	5.507 × 10−8
GBO	0.000986029	0.000990032	0.000997039	6.088 × 10−6
AEO	0.000982451	0.000988451	0.000992451	5.291 × 10−6
JS	0.00110615	0.001469483	0.00210615	0.00055

, respectively, and graphically analyzed using boxplot figures in Figure 15. The best standard deviation (STDEV) is achieved by ESCGBO that refer to its stability and robustness. To check the behavior of the estimated models PV current—voltage characteristics and power—voltage characteristics are presented for SDM and DDM in Figure 16 and Figure 17, respectively. Further details on current absolute error (Equation (45)) and power absolute error (Equation (46)) for SDM and DDM are presented in Figure 18 and Figure 19 respectively.

$$RMSE=\sqrt{\frac{1}{N}{\displaystyle \sum _{K=1}^N{f}^2(V_{tm},I_{tm},X)}}$$

(44)

$$Current Absolut error=\sqrt[2]{{(I-I_{estimated})}^2}$$

(45)

$$Power Absolut error=\sqrt[2]{{(P-P_{estimated})}^2}$$

(46)

4.2.2. Scenario 2

This scenario is concerned with the application of ESCGBO for parameters estimation of dynamic PV models (IOM and FOM). The applied dataset were captured from PV module at a temperature of 25 °C and irradiance level of 655 W/m² through connected load of R_l = 23.1. The ranges for all the estimated parameters are presented in

Table 11. Upper and lower constrains for all estimated parameters.

Parameter	Solar Cell
	Lower Limit	Upper Limit
Rc	0	20
C	2 × 10−8	6 × 10−5
L	5 × 10−6	100 × 10−6
α	0.8	1.1
β	0.8	1.1

. The obtained results are also compared with other recent algorithms. The three estimated parameters for IOM ($R_c,C \mathrm{and} L)$and RMSE obtained by all algorithms are listed in

Table 12. Estimated parameters of IOM model for all algorithms.

	ESCGBO	GBO	AEO	JS
Rc	5.583588	5.624748753	5.624748647	5.6247490
C	8.30 × 10−6	8.16 × 10−6	8.16 × 10−6	8.15726 × 10−6
L	7.43 × 10−6	7.47 × 10−6	7.47 × 10−6	7.47323 × 10−6
RMSE	0.008259444	0.008493067	0.008493067	0.008493067262

. The five estimated parameters for FOM ($R_c,C, L, \alpha \mathrm{and} \beta$) and RMSE obtained by all algorithms are listed in

Table 13. Estimated parameters of FOM model for all algorithms.

	ESCGBO	GBO	AEO	JS
Rc	4.617127916	5.005984283	4.550201539	4.698924713
C	5.81 × 10−5	5.04 × 10−6	1.46 × 10−5	4.08 × 10−5
L	1.50 × 10−5	1.35 × 10−5	1.73 × 10−5	1.44 × 10−5
A	0.8	1.026120535	0.917230623	0.833404373
β	0.950840763	0.957165925	0.940654537	0.953192785
RMSE	0.007951289	0.008236017	0.00819598	0.007995872

. The best RMSE was achieved by ESCGBO in case of IOM and FOM. The convergence curves of all the compared algorithms are presented for IOM and FOM in Figure 20 and Figure 21, respectively. The load current curve for the real experimental data and all algorithms for IOM and FOM are presented in Figure 22 and Figure 23, respectively. The current absolute error curve of all algorithms for IOM and FOM are presented in Figure 24 and Figure 25 respectively. In the previous figures comparison, the results obtained by the ESCGBO are better than other algorithms and the results obtained for FOM are more accurate than IOM.

5. Conclusions

This work proposed a new modified metaheuristic optimization algorithm named ESCGBO. It is considered an enhancement for the original GBO to enhance the balance between exploration and exploitation and completely enhance the algorithm performance. First, the proposed ESCGBO’s performance was tested on the 23 benchmark functions. The proposed technique achieved better than three well-known optimization techniques, such as EO and WHO as well as the original GBO. Then, the new algorithm was applied to estimate the parameters of static SDM and DDM models through application 1, which used the real data of 57 mm diameter commercial silicon R.T.C France solar cell, and dynamic IOM and FOM models through application two,, using the dataset, were captured from the PV module at temperature 25 °C at an irradiance level of 655 W/m² through connected load of Rl = 23.1. The obtained results were analyzed in different ways to evaluate the performance of the proposed algorithm. The accuracy of the algorithm was tested through calculation of the RMSE and IAE, then by comparing it for all algorithms. The robustness was also checked by running the algorithms with 30 independent runs and analyzing the results through statistical analysis. From all the analysis, the proposed ESCGBO is more accurate and robust when compared with other recent algorithms. The static DDM is more accurate than SDM and dynamic IOM is more accurate than FOM. For future work, this study contributes to research that focuses on studying the applicability of ESCGBO PV parameters’ estimation for the large and complex PV system.