An enhanced grey wolf optimizer with fusion strategies for identifying the parameters of photovoltaic models

Abstract

Identifying photovoltaic (PV) parameters accurately and reliably can be conducive to the effective use of solar energy. The grey wolf optimizer (GWO) that was proposed recently is an effective nature-inspired method and has become an effective way to solve PV parameter identification. However, determining PV parameters is typically regarded as a multimodal optimization, which is a challenging optimization problem; thus, the original GWO still has the problem of insufficient accuracy and reliability when identifying PV parameters. In this study, an enhanced grey wolf optimizer with fusion strategies (EGWOFS) is proposed to overcome these shortcomings. First, a modified multiple learning backtracking search algorithm (MMLBSA) is designed to ameliorate the global exploration potential of the original GWO. Second, a dynamic spiral updating position strategy (DSUPS) is constructed to promote the performance of local exploitation. Finally, the proposed EGWOFS is verified by two groups of test data, which include three types of PV test models and experimental data extracted from the manufacturer’s data sheet. Experiments show that the overall performance of the proposed EGWOFS achieves competitive or better results in terms of accuracy and reliability for most test models.

Keywords

Grey wolf optimizer solar cell parameter identification multimodal optimization photovoltaic models meta-heuristics

1. Introduction

To mitigate problems associated with the lack of natural resources and to protect the environment, the development of renewable and clean energy has become the focus of current fields of research. Solar energy is generally recognized as a promising resource that can be applicable in the real world because it can be switched to daily electricity through photovoltaic (PV) systems [1, 2].

However, PV systems are easily affected by harsh outdoor environments. Constructing accurate PV models is conducive to promoting the efficiency of converting solar energy into electrical energy [3, 4, 5]. Currently, although some effective PV mathematic models, including double and single diodes, have been proposed [3, 5, 6, 7], the accuracy of these models is restricted by PV parameters, which need to be obtained through identification technology. Therefore, developing an effective parameter identification technology is important.

In addition, the identification of PV parameters is typically regarded as a multimodal problem [3, 7, 8], which is a challenging and difficult optimization problem. According to [7, 8, 9, 10], the traditional deterministic technique and meta-heuristic algorithm are two primary methods for solving PV parameter identification. Although deterministic technology has excellent local search ability on the entire, its disadvantage still may affect its application in PV parameter identification, which requires better proficiency in global exploration [11]. However, with the development of nature-inspired computing [12], many meta-heuristic optimization algorithms that are inspired by nature have appeared and include neural dynamic models [13], genetic algorithms [14], spiral dynamics algorithms [15], water drop algorithms [16], harmony search algorithms [17], and gravitational search algorithms [18]. Because this type of algorithm is not constrained by the form of the problem, it is suitable for complex application problems, such as reducing memory demand [19], renewable energy systems [20], servo systems [21, 22, 23], 3D mesh simplification [24], online scheduling [25], and smart grids [26, 27]. Considering these advantages, many scholars are committed to the development and improvement of meta-heuristic algorithms [28, 29, 30, 31, 32, 33], particularly the application of meta-heuristic algorithms in the parameter identification problem of the PV model. These algorithms include particle swarm optimization (PSO) [34], differential evolution (DE) [35], backtracking search algorithm (BSA) [36], whale optimization algorithm (WOA) [37], Jaya [38], grasshopper optimization algorithm (GOA) [39], equilibrium optimizer (EO) [40], sine cosine algorithm (SCA) [41], and multi-verse optimizer (MVO) [42]. Many other meta-heuristic algorithms have been recommended for PV parameter identification [43], and these meta-heuristics have achieved good results in terms of accuracy and reliability for PV parameter identification. However, most of the existing meta-heuristics still have difficulty determining the global optimal parameters [38]. Thus, there is still a lot of room to develop a potential meta-heuristic algorithm to identify PV parameters effectively.

Recently, a new meta-heuristic method called the grey wolf optimizer (GWO) was proposed in [44]. Due to its efficiency and few adjustment parameters, GWO has been widely used in many fields, including path planning [45], multidimensional knapsack problems [46], feature selection [47], image processing [48], economic load dispatch [49], and other optimization problems [50, 51, 52]. In addition, GWO has been selected to manage PV parameter identification [7], and promising results have been obtained.

However, as a recent meta-heuristic algorithm, GWO still has the potential to enhance accuracy and reliability in the problem of PV parameter identification. Therefore, it is worth considering how to improve GWO to solve the problem of PV parameter identification more effectively. Specifically, the search process of GWO is guided by three levels of wolves and performs exploitation well, but the resulting global exploration ability still has room for improvement during position updating [53, 54]. In addition, although GWO achieves good exploitation ability, it still faces the problem of insufficient reliability when solving PV parameter identification. Therefore, we focus on the exploration and exploitation mechanism of GWO to improve its PV parameter identification ability. Overall, no algorithm can be universally considered the best optimization algorithm according to [55], but developing optimization algorithms to solve the problem of PV parameter identification more effectively should be encouraged.

Therefore, an enhanced grey wolf optimizer with fusion strategies (EGWOFS) is constructed in this study to obtain the PV parameters more effectively. In EGWOFS, first, to improve the insufficient exploration ability of the GWO algorithm when solving PV parameter identification, a modified multiple learning backtracking search algorithm (MMLBSA) is constructed to enhance the exploration performance of GWO, and then, the accuracy of the identified PV parameters can also be improved. Second, although the global exploration of the GWO algorithm has been improved, the reliability of GWO improved by MMLBSA is still insufficient at obtaining the PV parameters; thus, we design a dynamic spiral updating position strategy (DSUPS) to improve the reliability of GWO by promoting the exploitation performance. Finally, to measure the effectiveness of EGWOFS, experiments are performed out with two groups of data, including three types of benchmark PV models and measured data extracted from the manufacturer’s data sheet. Results comprehensively demonstrate that the overall performance of the proposed EGWOFS achieved better or competitive results in terms of accuracy and reliability for most PV models.

The four contributions of this study are as follows:

(1)
EGWOFS is designed to better identify PV parameters in terms of accuracy and reliability;
(2)
MMLBSA is constructed to promote the exploration performance of GWO;
(3)
DSUPS is proposed to ameliorate the exploitation performance;
(4)
The performance of EGWOFS is tested with comprehensive experiments.

The remainder of this paper is organized as follows. Section 2 introduces the solar cell model, PV module model and their parameter optimization. Section 3 describes the fundamentals of GWO. Section 4 describes the proposed EGWOFS. Section 5 describes the results and analysis. Section 6 provides conclusions of this study.
2. Problem formulation

According to the literature [8, 36, 56, 57, 58, 59], there are three primary PV models: the double diode model (DDM), PV module model, and single diode model (SDM). Their parameter identification problems have been transformed into an objective function that can be optimized by algorithms. The three models are illustrated in the following subsections.

2.1 Solar cell model

According to [8, 36, 56, 57, 58, 59], two models including SDM and DDM, are described in the following two subsections.

2.1.1 SDM

The SDM has five unknown parameters, and its current output can be handled by the following formulae:

$\displaystyle I_{L}=I_{ph}-I_{d}-I_{sh}$ (1) $\displaystyle I_{d}=I_{sd}\cdot\left[\exp\left(\frac{q\cdot(V_{L}+R_{s}\cdot I% _{L})}{n\cdot k\cdot T}\right)-1\right]$ (2) $\displaystyle I_{sh}=\frac{V_{L}+R_{s}\cdot I_{L}}{R_{sh}}$ (3) $\displaystyle I_{L}=I_{ph}-I_{sd}\cdot\!\left[\exp\left(\frac{q\cdot(V_{L}+R_{% s}\cdot I_{L})}{n\cdot k\cdot T}\right)\!-\!1\right]$ $\displaystyle\qquad\,-\frac{V_{L}+R_{s}\cdot I_{L}}{R_{sh}}$ (4)

where output current $I_{L}$ is calculated by three variables (shunt resistor current $I_{sh}$ , photo-generated current $I_{ph}$ , and diode current $I_{d}$ ); $V_{L}$ denotes the output voltage; $I_{sd}$ represents the reverse saturation current; the parameters $n$ and $T$ are the diode ideality factor and temperature (Kelvin), respectively; $R_{s}$ is the series resistance; the shunt resistance is denoted by $R_{sh}$ ; Boltzmann constant $k=$ 1.3806503 $\times$ 10 ${}^{-23}$ J/K; and $q=$ 1.60217646 $\times$ 10 ${}^{-19}$ C.

2.1.2 DDM

The DDM has seven unknown parameters, and its output current is described by the following formulae:

$\displaystyle I_{L}=I_{ph}-I_{d1}-I_{d2}-I_{sh}$ (5) $\displaystyle I_{d1}=I_{sd1}\!\cdot\!\left[\exp\left(\frac{q\cdot(V_{L}+R_{s}% \cdot I_{L})}{n_{1}\cdot k\cdot T}\right)-1\right]$ (6) $\displaystyle I_{d2}=I_{sd2}\!\cdot\!\left[\exp\left(\frac{q\cdot(V_{L}+R_{s}% \cdot I_{L})}{n_{2}\cdot k\cdot T}\right)-1\right]$ (7) $\displaystyle I_{L}=I_{ph}\!-\!I_{sd1}\!\cdot\!\left[\exp\left(\frac{q\cdot(V_% {L}+R_{s}\cdot I_{L})}{n_{1}\cdot k\cdot T}\right)\!-\!1\right]$ $\displaystyle\qquad\,-\,I_{sd2}\!\cdot\!\left[\exp\left(\frac{q\cdot(V_{L}+R_{% s}\cdot I_{L})}{n_{2}\cdot k\cdot T}\right)\!-\!1\right]$ $\displaystyle\qquad\,-\,\frac{V_{L}+R_{s}\cdot I_{L}}{R_{sh}}$ (8)

where the diffusion current $I_{d1}$ and saturation current $I_{d2}$ are used to calculate the diode current, and two ideality factors $n_{1}$ and $n_{2}$ are also essential parameters for the diode current.

2.2 PV module model

The current output of the single diode PV module is calculated using the following formula [8, 36, 56, 57, 58, 59]:

$\displaystyle I_{L}/N_{p}=I_{ph}-I_{sd}\cdot$ $\displaystyle\qquad\quad\left[\exp\left(\frac{q\cdot(V_{L}/N_{s}+R_{s}\cdot I_% {L}/N_{p})}{n\cdot k\cdot T}\right)\!-\!1\right]$ $\displaystyle\qquad\quad-\,\frac{V_{L}/N_{s}+R_{s}\cdot I_{L}/N_{p}}{R_{sh}}$ (9)

where the number of both parallel and series solar cells are represented by two variables $N_{p}$ and $N_{s}$ . When $N_{p}$ is equal to 1, Eq. (9) simplifies into Eq. (10):

$\displaystyle I_{L}=I_{ph}-I_{sd}\cdot\left[\exp\left(\frac{q\cdot(V_{L}+R_{s}% \cdot I_{L}\cdot N_{s})}{n\cdot k\cdot T\cdot N_{s}}\right)-1\right]-\,\frac{V% _{L}+R_{s}\cdot I_{L}\cdot N_{s}}{R_{sh}\cdot N_{s}}$ (10)

2.3 Parameter optimization of the PV model

2.3.1 Objective function

Identifying PV parameters needs to be converted into a problem that can be optimized by an algorithm. Therefore, Eq. (14), which has been widely used in the literature [8, 36, 56, 57, 58, 59], is considered to be the objective function in this study. The calculation result of the objective function primarily depends on the error functions shown in Eq. (11), Eq. (12), and Eq. (13):

$\displaystyle f_{k}(V_{L},I_{L},\mathbf{x})=I_{ph}-I_{sd1}\cdot$ $\displaystyle\qquad\left[\exp\left(\frac{q\cdot(V_{L}+R_{s}\cdot I_{L})}{n% \cdot k\cdot T}\right)-1\right]$ $\displaystyle\qquad-\frac{V_{L}+R_{s}\cdot I_{L}}{R_{sh}}-I_{L}$ (11) $\displaystyle f_{k}(V_{L},I_{L},\mathbf{x})=I_{ph}-I_{sd1}\cdot$ $\displaystyle\qquad\left[\exp\left(\frac{q\cdot(V_{L}+R_{s}\cdot I_{L})}{n_{1}% \cdot k\cdot T}\right)-1\right]$ $\displaystyle\qquad-I_{sd2}\cdot\left[\exp\left(\frac{q\cdot(V_{L}+R_{s}\cdot I% _{L})}{n_{1}\cdot k\cdot T}\right)-1\right]$ $\displaystyle\qquad-\frac{V_{L}+R_{s}\cdot I_{L}}{R_{sh}}-I_{L}$ (12) $\displaystyle f_{k}(V_{L},I_{L},\mathbf{x})=I_{ph}-I_{sd}\cdot$ $\displaystyle\qquad\left[\exp\left(\frac{q\cdot(V_{L}+R_{s}\cdot I_{L}\cdot N_% {s})}{n\cdot k\cdot T\cdot N_{s}}\right)-1\right]$ $\displaystyle\qquad-\frac{V_{L}+R_{s}\cdot I_{L}\cdot N_{s}}{R_{sh}\cdot N_{s}% }-I_{L}$ (13)

where Eq. (11), Eq. (12), and Eq. (13) are the error functions of the SDM, DDM, and single diode PV module model ( $N_{p}=1$ ), respectively. The function $f_{k}(V_{L},I_{L},\mathbf{x})$ is employed to calculate the error between the simulated current and the measured current in the corresponding PV model, and $I_{L}$ is the measured current of the corresponding PV model:

$\displaystyle\text{RMSE}(\mathbf{x})=\sqrt{\frac{1}{N}\sum_{k=1}^{N}f_{k}(V_{L% },I_{L},\mathbf{x})^{2}}$ (14)

where $\text{RMSE}(\mathbf{x})$ is the objective function and $N$ is the amount of experimental data.

2.3.2 Variables

In addition to the above objective function, parameter variables are an important part of the optimization problem.

For SDM, five parameter variables need to be obtained by the meta-heuristic algorithm in Eq. (11): $I_{ph}$ , $I_{sd}$ , $R_{s}$ , $R_{sh}$ , and $n$ .

For DDM, seven parameter variables need to be obtained by the meta-heuristic algorithm in Eq. (12): $I_{ph}$ , $I_{sd1}$ , $I_{sd2}$ , $R_{s}$ , $R_{sh}$ , $n_{1}$ , and $n_{2}$ .

For the PV module, five parameter variables need to be obtained by the meta-heuristic algorithm in Eq. (13): $I_{ph}$ , $I_{sd}$ , $R_{s}$ , $R_{sh}$ , and $n$ .

2.3.3 Constraints

Constraints are another core component of the optimization problem. The constraint intervals of the specific parameter variables for the three models are listed in Table 1 and were used in [7, 60, 61]. More details are described below.

Table 1
Search interval of identified parameters on three types of PV model

Parameter	SDM/DDM		STP6-120/36		STM6-40/36
	LB	UB	LB	UB	LB	UB
$I_{ph}(A)$	0	1	0	8	0	2
$I_{sd},I_{sd1},I_{sd2}(\mu A)$	0	1	0	50	0	50
$n,n_{1},n_{2}$	1	2	1	50	1	60
$R_{s}(\Omega)$	0	0.5	0	0.36	0	0.36
$R_{sh}(\Omega)$	0	100	0	1500	0	1000

3. Fundamentals of the grey wolf optimizer

3.1 Social hierarchy

According to the literature [44], the ideology of the grey wolf optimizer (GWO) is primarily dependent on the daily capturing behaviour of grey wolves. The grey wolf population is generally delimited into four types: the first three levels grey wolves $\alpha$ , $\beta$ , and $\delta$ , and other grey wolves. The basic GWO with the cooperation of the four levels of grey wolf individuals is described below [44].

3.2 Encircling prey

The behavior of encircling prey is primarily represented by the following model [44]:

$\displaystyle\vec{D}=|\vec{C}\cdot\vec{X}_{p}(t)-\vec{X}(t)|$ (15) $\displaystyle\vec{X}(t+1)=\vec{X}_{p}(t)-\vec{A}\cdot\vec{D}$ (16) $\displaystyle\vec{A}=2\vec{a}\cdot\vec{r}_{1}-\vec{a}$ (17) $\displaystyle\vec{C}=2\cdot\vec{r}_{2}$ (18)

where $\vec{X}_{p}$ is the position of the $p$ -th prey; $\vec{A}$ and $\vec{C}$ are determined by random $\vec{r}_{1}$ and $\vec{r}_{2}$ , respectively; $\vec{a}$ decreases from 2 to 0; and $\vec{r}_{1}$ and $\vec{r}_{2}$ are in the interval [0, 1].

Figure 1.

The flowchart of PV parameter identification based on EGWOFS.

3.3 Hunting

At this stage, the population depends on three levels of grey wolves to obtain the prey, and the positions of the remaining grey wolves are renewed via the prey position obtained by $\alpha$ , $\beta$ , and $\delta$ . Therefore, hunting behavior is primarily achieved through cooperation among the highest level and the second- and third-level grey wolves as follows [44]:

$\displaystyle\vec{D}_{\alpha}=|\vec{C}_{1}\cdot\vec{X}_{\alpha}-\vec{X}|$ (19) $\displaystyle\vec{D}_{\beta}=|\vec{C}_{2}\cdot\vec{X}_{\beta}-\vec{X}|$ (20) $\displaystyle\vec{D}_{\delta}=|\vec{C}_{3}\cdot\vec{X}_{\delta}-\vec{X}|$ (21) $\displaystyle\vec{X}_{1}=\vec{X}_{\alpha}-\vec{A}_{1}\cdot(\vec{D}_{\alpha})$ (22) $\displaystyle\vec{X}_{1}=\vec{X}_{\beta}-\vec{A}_{2}\cdot(\vec{D}_{\beta})$ (23) $\displaystyle\vec{X}_{1}=\vec{X}_{\delta}-\vec{A}_{2}\cdot(\vec{D}_{\delta})$ (24) $\displaystyle\vec{X}(t+1)=\frac{\vec{X}_{1}+\vec{X}_{2}+\vec{X}_{3}}{3}$ (25)

where the vector $\vec{X}(t+1)$ saves the average of the first three best grey wolf positions obtained thus far.

3.4 Attacking prey (exploitation)

In GWO, according to [44], the process of exploitation is achieved by the change of some variables, and the value of vector $\vec{a}$ is used to affect the behaviour of attacking prey. When vector $\vec{a}$ is decreased from 2 to 0, the variable $\vec{A}$ also changes and can guide the wolves to capture prey when $|\vec{A}|$ is less than 1.

3.5 Prey search (exploitation)

Based on [44], although GWO primarily uses three levels of grey wolves to guide other grey wolves to seek prey, it is still insufficient for the exploration effect of GWO. Therefore, the deficiency of GWO is enhanced by adjusting the condition $\vec{A}>1$ to oblige the grey wolves to find better prey. In addition, the value $|\vec{C}|$ is introduced to provide a random value at any time to emphasize the global exploration capability during the iteration.

4. Enhanced grey wolf optimizer with fusion strategies (EGWOFS)

Because the original GWO algorithm has deficiencies when solving PV parameters, an enhanced grey wolf optimizer with fusion strategies (EGWOFS) is proposed to more effectively identify PV parameters. A flowchart of PV parameter identification based on EGWOFS is shown in Fig. 1. In EGWOFS, we improved the process in two stages: global exploration (GE) and local exploitation (LE), where GE is achieved by reconstructing a modified multiple learning backtracking search algorithm (MMLBSA), and LE is realized by designing a dynamic spiral updating position strategy (DSUPS). Next, we examine how to better and effectively integrate the MLBSA and GWO, and fully utilize their respective advantages to compensate for each other’s shortcomings.

4.1 MMLBSA

In GWO, the search process primarily depends on three levels of grey wolves $\alpha$ , $\beta$ , and $\delta$ . Although this process may lead to better local exploitation capacity, the other individuals of the population may easily move to three levels of grey wolves during evolution, which may affect the extensive exploration and the accuracy of obtaining the optimal solution. Therefore, GWO still has room for improvement in global exploration. According to [36], the multiple learning backtracking search algorithm (MLBSA) has a strong global exploration ability when obtaining PV parameters due to its diversified search mechanisms, including the multiple learning strategy (MLS) shown in Eq. (28) and the elite method based on the chaotic local search (EMBCLS) shown in Eq. (32). Inspired by these facts, we consider employing two primary mechanisms of MLBSA to promote the exploration performance of GWO.

However, solving this problem is more difficult than it appears. Directly combining MLBSA with GWO cannot fully utilize the exploration proficiency of MLBSA and may reduce the accuracy of PV parameter identification. Therefore, the proposed motivation is to flexibly integrate both MLBSA and GWO to better promote the exploration performance of GWO. To solve this problem, we primarily make the following modifications to the original MLBSA.

First, according to MLBSA [36], the historical population, current population, and best individual of the current population are all considered in Eq. (28), which is important for the exploration and convergence performance of MLBSA. However, the information of the current worst individual is still ignored. Although the current best individual can promote convergence speed, it may not help with extensive exploration under the leadership of the best individual. Therefore, we reconstruct Eq. (28) into Eq. (29) and call it the modified multiple learning strategy (MMLS). Here, the historical population, current population, and best and worst grey wolves of the current population are all considered during the exploration. Next, the greedy selection strategy used in MLBSA is selected to renew the population, and the updated population is sorted in ascending order so that we can obtain the first three best individuals to update the original first three levels of grey wolves.

Second, according to MLBSA [36], the current population has better individuals than the historical population, which means that the information diversity of the current population can affect the evolutionary trend of the population. Although MMLS has been considered to ameliorate the convergence speed and exploration performance of GWO, the quality of the current population also must be considered. Therefore, the EMBCLS of MLBSA is also employed to update the current best grey wolf. In addition, to minimize the impact on the accuracy and reliability of parameter identification and reduce unnecessary computational cost, we design diversified constraints shown in Eq. (31) to control the execution times of EMBCLS. Next, we perform the elite mechanism used in MLBSA to update the worst individual of the current population once there is a better solution generated by Eq. (32). Then, the updated population is sorted in ascending order so that we can obtain the first three best individuals to update the original first three levels of grey wolves.

According to the above description, the related models of the MMLBSA are as follows.

$\displaystyle F=3\cdot\textit{randn}$ (26) $\displaystyle hX=\begin{cases}X,\quad\text{if}\ (q_{1}<q_{2})\\ hX(\textit{randperm}(N),:)\end{cases}$ (27) $\displaystyle X^{\prime}_{i,j}=\begin{cases}X_{i,j}+F\cdot((hX_{i,j}-X_{i,j})+% \\ \qquad(X_{g,j}-X_{i,j})),\text{if}\ (b_{1}<b_{2})\\ X_{i,j}+\textit{rand}\cdot(X_{B,j}-X_{i,j}),\\ \qquad\text{if}\ (b_{1}\geqslant b_{2})\end{cases}$ (28) $\displaystyle X^{\prime}_{i,j}=\begin{cases}X_{i,j}+F\cdot((hX_{i,j}-X_{i,j})+% \\ \qquad(X_{g},j-X_{i,j})),\text{if}\ (b_{1}<b_{2})\\ X_{i,j}+\textit{rand}\cdot((X_{B,j})-X_{W,j}),\\ \qquad\text{if}\ (b_{1}\geqslant b_{2})\end{cases}$ (29) $\displaystyle z_{m+1}=4\cdot z_{m}\cdot(1-z_{m})$ (30) $\displaystyle\text{constraints}:\begin{cases}r_{3}>(1-\textit{FE}/\textit{% MaxFE})\\ r_{3}>k\\ r_{e}>\exp(-\textit{FE}/\textit{MaxFE})\end{cases}$ (31) $\displaystyle X^{\prime\prime}_{B,k}=\begin{cases}X_{B,k},\\ \qquad\text{if}\ (r_{4}\geqslant 1-(\textit{FE}/\textit{MaxFE}))\\ X_{B,k}+r_{5}\cdot(2\cdot z_{m+1}-1),\\ \qquad\text{otherwise}\end{cases}$ (32)

where $F$ is the scale factor; randn $\sim N(0,1)$ , which can generate a normally distributed pseudorandom number; $hX_{i}$ is the $i$ -th individual of the historical population; the randperm function is the random permutation function; the $g$ -th individual $X_{g}$ is randomly selected from the current population; $g$ is not equal to $i$ ; $q_{1}$ , $q_{2}$ , $b_{1}$ , $b_{2}$ , $r a n d$ , $r_{3}$ , $k$ , $r_{4}$ , and $r_{5}$ are uniformly distributed random numbers between 0 and 1; and the temporary variables $X^{\prime}_{i}$ and $X^{\prime\prime}_{B}$ can save the updated $i$ -th individual $X_{i}$ and updated best individual $X_{i}$ in the current population, respectively. $X_{W}$ is the worst individual of the current population. Equation (30) is a chaotic map used in MLBSA, where the initial $z_{m}$ is also a random number between 0 and 1. The function $\exp()$ of Eq. (31) is the exponential function. The selection of constraint expressions can be determined according to specific problems. We choose expression $r_{3}>(1-\textit{FE}/\textit{MaxFE})$ as the constraint in this study. When the condition is met, Eq. (32) can be performed.

4.2 DSUPS

After these improvements, the MMLBSA can better assist GWO at improving its global exploration capability and obtain an accurate solution when solving PV parameter identification. However, this alone is insufficient: if the search mechanism of GWO is directly used to update the population obtained by MMLBSA, it may affect the reliability of the solution. Therefore, continuing to effectively combine MMLBSA and the search mechanism of GWO to improve the reliability of the solution is more difficult than it seems. To meet this challenge, we propose a dynamic spiral position update strategy (DSPUS) to strengthen the local search around the first three grey wolves. In DSPUS, there are three primary steps as follows.

First, inspired by the spiral position update strategy Eq. (36) of the whale optimization algorithm (WOA) [62], which has good flexibility and local search ability, the spiral position update strategy may be considered a promising method to further enhance the local search of GWO. However, if we directly employ Eq. (36) instead of updating mechanisms Eq. (22), Eq. (23), and Eq. (24), the three updating mechanisms of the first three wolves will become the same. This result means that the search of GWO will be guided only by the first-best wolf instead of the first three best wolves, which may easily affect its exploration performance and may also change the basic characteristics of GWO. Therefore, we modify Eq. (36) into the new Eq. (40), Eq. (42), and Eq. (44) to update the first three grey wolves.

Second, if we only employ Eq. (50) as the local search mechanism, this process does not fully consider the exploitation capabilities of Eq. (40), Eq. (42), and Eq. (44). Inspired by the hierarchy of GWO and the group search of improved JAYA (IJAYA) [8]. We design different updating mechanisms for different types of wolf individuals. Eq. (51), which is based on the idea of the DE mutation operator [63], is designed to update individual $X_{i}$ where $i$ is not equal to any one of $\textit{ind}(\textit{end})$ , $ind(1)$ , $ind(2)$ , or $ind(3)$ . Thus, Eq. (51) is not allowed to update the first three grey wolves and the worst wolf of the current population. Eq. (53), which is based on the Cauchy mutation operation [64, 65, 66, 67], is constructed to renew the worst individual. Eq. (55), which is also based on the Cauchy mutation operation, is designed to update the first three best wolves.

Third, we change the range of $a_{2}$ into $a^{\prime\prime}_{2}$ to balance local and global search more effectively.

Finally, adjust the boundary of the new individual generated by any one of Eq. (51), Eq. (53), and Eq. (55), and calculate the fitness of new individual. When the new individual is better than the current individual, the new individual is accepted. Then, the new population is sorted to obtain the first three best individuals, and the original first three grey wolves are updated with them.

Under the close cooperation of the above steps, the reliability of the solution can be improved.

$\displaystyle a_{2}=-1+t\cdot((-1)/\textit{Max\_iter})$ (33) $\displaystyle ll=(a_{2}-1)\cdot\textit{rand}+1$ (34) $\displaystyle\textit{disL}=\text{abs}(\textit{LeaderX}_{j}-X_{i,j})$ (35) $\displaystyle X=\textit{disL}\cdot e^{(b\cdot ll)}\cdot\cos(2\pi ll)$ $\displaystyle\qquad+\,\textit{LeaderX}_{j}$ (36) $\displaystyle a^{\prime\prime}_{2}=-100+t\cdot((-100)/\textit{Max\_iter})$ (37) $\displaystyle ll^{\prime\prime}=(a^{\prime\prime}_{2}-1)\cdot\textit{rand}+1$ (38) $\displaystyle\textit{disL}_{1}=\text{abs}(X_{\alpha,j}-X_{i,j})$ (39) $\displaystyle X_{1}=\textit{disL}_{1}\cdot e^{(b\cdot ll^{\prime\prime})}\cdot% \cos(2\pi ll^{\prime\prime})+X_{\alpha,j}$ (40) $\displaystyle\textit{disL}_{2}=\text{abs}(X_{\beta,j}-X_{i,j})$ (41) $\displaystyle X_{2}=\textit{disL}_{2}\cdot e^{(b\cdot ll^{\prime\prime})}\cdot% \cos(2\pi ll^{\prime\prime})+X_{\beta,j}$ (42) $\displaystyle\textit{disL}_{3}=\text{abs}(X_{\delta,j}-X_{i,j})$ (43) $\displaystyle X_{3}=\textit{disL}_{3}\cdot e^{(b\cdot ll^{\prime\prime})}\cdot% \cos(2\pi ll^{\prime\prime})+X_{\delta,j}$ (44) $\displaystyle h_{4}=\textit{randi}(N)$ (45) $\displaystyle h_{5}=\textit{randi}(N)$ (46) $\displaystyle tv=\bigl{(}r_{6}\cdot r_{7}\cdot r_{8}\!\cdot(\exp(-t/$ $\displaystyle\qquad\,\textit{Max\_iter}))\bigr{)}^{(3\cdot\cos(r_{9}))}$ (47) $\displaystyle\textit{sub}_{1}=X_{h_{4},j}-X_{i,j}$ (48) $\displaystyle\textit{sub}_{2}=X_{h_{5},j}-X_{i,j}$ (49) $\displaystyle\textit{mid}_{1}=(X_{1}+X_{2}+X_{3})/3$ (50) $\displaystyle\textit{newX}_{i,j}=(\textit{mid}_{1})+tv\cdot(\textit{sub}_{2})$ $\displaystyle\hskip 50.0pt+\,(1-tv)\cdot(\textit{sub}_{1})$ (51) $\displaystyle k_{2}=\textit{rand}\cdot(\text{exp}(-t/\textit{Max\_iter}))$ (52) $\displaystyle\textit{newX}_{i,j}=X_{\textit{ind}(\textit{end}-1),j}+k_{2}\cdot$ $\displaystyle\hskip 50.0pt\tan((\textit{rand}-0.5)\cdot\pi)$ (53) $\displaystyle\textit{mid}_{2}=(X_{2}+X_{3})/2$ (54) $\displaystyle\textit{newX}_{i,j}=(\textit{mid}_{2})+k_{2}\cdot$ $\displaystyle\hskip 50.0pt\tan((\textit{rand}-0.5)\cdot\pi)$ (55)

where $a_{2}$ linearly decreases between $-$ 1 and $-$ 2; LeaderX ${}_{j}$ is the $j$ -th element of the best whale individual in the WOA; and $X_{\alpha,j}$ , $X_{\beta,j}$ , and $X_{\delta,j}$ are the $j$ -th elements of the first three best grey wolves. The function $\textit{randi}(N)$ can randomly generate integers between 1 and $N$ , and random variables $r_{6}$ , $r_{7}$ , $r_{8}$ , and $r_{9}$ are in [0, 1]. The $\cos()$ function is the cosine function used in Eq. (47). Also, $X_{h_{4},j}$ and $X_{h_{5},j}$ are the $j$ -th elements of the $h_{4}$ -th individual $X_{h_{4}}$ and $h_{5}$ -th individual $X_{h_{5}}$ , respectively; the function $\exp()$ is the exponential function; $h_{4}\neq h_{5}\neq i$ ; and $X_{\textit{ind}(\textit{end}-1)}$ is the second worst individual of the current population. Equation (51), Eq. (53), and Eq. (55) are used to update different types of individuals.

5. Results and analysis of the experiment

Related tests and results are performed and analyzed to demonstrate the effectiveness of the proposed EGWOFS in this section. First, the parameter settings of related algorithms are introduced, and two popular PV models are employed to test the generalization ability of PV parameter identification for the proposed EGWOFS. Then, various statistical indicators and visualizations are used to show the effect in Subsection 5.1. Second, Subsection 5.2 compares the proposed EGWOFS with more PV models and established algorithms reported in the literature to verify its advanced performance in PV parameter identification. Third, Subsection 5.3 verifies the effectiveness of the two introduced mechanisms of the proposed EGWOFS. Fourth, EGWOFS is applied to measured data extracted from the manufacturer’s data sheet at different irradiance conditions to test its practical application ability and reliability in Subsection 5.4. The test data of the SDM, DDM and PV module model are obtained from [60, 61, 68, 69] and are widely employed to prove the proficiency of meta-heuristic algorithms in solving the PV parameter identification problem [7, 36, 38, 70], and the measured data of the Multi-crystalline KC200GT are obtained from [71].

5.1 Test on two popular PV models

5.1.1 Experimental settings

Table 2
Set parameters for seven algorithms

Algorithm	Parameters
EGWOFS	$N=20$ ; $a^{\prime\prime}_{2}$ decreases from $-$ 100 to $-$ 200; $b=1$
GWO	$N=20$ ; $a$ decreases linearly from 2 to 0
MVO	$N=20$ ; $\textit{WEP}\in[0.2,1]$ ; $g=6$
PSO	$N=20$ ; $c_{1}=c_{2}=2$ ; $\textit{vMax}=(ub-lb)\cdot 0.2$ ; $\textit{wMax}=0.9$ ; $\textit{wMin}=0.4$
WOA	$N=20$ ; $a$ decreases from 2 to 0; $a_{2}$ decreases from $-$ 1 to $-$ 2; $b=1$
SSA	$N=20$ ; $c_{2}\in[0,1]$ ; $c_{3}\in[0,1]$
EO	$N=20$ ; $a_{1}=2$ ; $a_{2}=1$ ; $\textit{GP}=0.5$ ; $V=1$

Two types of PV models are first used to verify the accuracy and reliability of the proposed EGWOFS in solving PV parameter identification within an acceptable convergence speed in Section 5.1.

For the evaluation methods, we first use the average (Mean), minimum (Min), and maximum (Max) of the RMSE values and curve characteristics to test the accuracy. Then, we use both boxplot and standard deviation (SD) to verify the method’s reliability. The statistics of four indicators (‘Mean’, ‘Min’, ‘SD’ and ‘Max’) are dependent on the best RMSE gained through each algorithm with 30 independent runs. The experimental results in boldface indicate the best effects gained by related algorithms for a certain indicator. In addition, the convergence speed is confirmed by a curve comparison of each algorithm in the convergence graphs. Finally, Friedman’s rank test is performed to demonstrate the overall ranking performance of EGWOFS.

Moreover, to provide a fair comparison of related algorithms, the number of evaluations of both the proposed EGWOFS and comparison algorithms, including GWO, MVO [72], PSO [73], WOA, Salp Swarm Algorithm (SSA) [74], and EO [75], are all set to 48,000. The population size of each algorithm is 20. The other parameter settings primarily depend on the original literature or the suggestions of the relevant literature. In Table 2, for EGWOFS, $a^{\prime\prime}_{2}$ decreases adaptively from $-$ 100 to $-$ 200 as iterations continue, and $b$ is a constant that is set to be consistent with the WOA. For GWO, $a$ decreases adaptively from 2 to 0. For MVO, the wormhole existence probability WEP and coefficient $g$ should be set by the user. For PSO, $c_{1}$ and $c_{2}$ are cognitive and social constants, respectively. wMax and wMin are the maximum and minimum inertia weights, respectively, and vMax is the velocity limit, which should be set by the user. For WOA, $a$ decreases adaptively from 2 to 0, and $a_{2}$ decreases adaptively from $-$ 1 to $-$ 2. For SSA, $c_{2}$ and $c_{3}$ are random numbers. For EO, constants $a_{1}$ , $a_{2}$ , generation probability GP and unit $V$ should be selected by the user.

5.1.2 Test on case 1: SDM

Table 3
Statistical results of seven algorithms on SDM

Algorithm	Max	Min	Mean	Std
EGWOFS	9.8602E-04	9.8602E-04	9.8602E-04	4.3149E-17
GWO	4.5470E-02	1.0811E-03	8.9116E-03	1.2664E-02
MVO	1.4931E-02	1.4195E-03	4.6543E-03	3.3336E-03
PSO	2.4480E-03	9.8617E-04	1.5264E-03	4.7205E-04
WOA	2.2362E-01	1.1961E-03	1.6817E-02	4.1338E-02
SSA	7.2035E-03	1.0579E-03	2.3040E-03	1.3368E-03
EO	3.8151E-02	9.8800E-04	6.2842E-03	1.2716E-02

Figure 2.

The $I$ - $V$ characteristic comparisons between simulated data achieved by EGWOFS and measured data on SDM.

In Table 3, the proposed EGWOFS achieves the best results in all four indicators for SDM. In particular, six comparison algorithms are much worse than the proposed EGWOFS in two indicators ‘Max’ and ‘Std’. The primary reason for these results is that the six comparison algorithms can easily fall into the local optimum prematurely and lack of the adequate exploitation capacity during the search for five optimal PV parameters, which can easily lead to poor accuracy of the maximum value and unstable solutions. While the proposed EGWOFS has a better global exploration ability for optimal SDM parameters, primarily due to the effectiveness of the MMLBSA mechanism in assisting EGWOFS in globally searching SDM parameters.

The curve feature of Fig. 2 indicates that the data gained by EGWOFS are generally coincident with the real data on SDM, which also verifies that the PV parameters identified by the proposed EGWOFS have high accuracy on SDM, primarily due to the effectiveness of MMLBSA mechanism in improving the global exploration capability of EGWOFS.

The boxplot in Fig. 3 shows that the smallest fluctuation range of the best RMSE is achieved by EGWOFS, which means that EGWOFS has the best reliability among the 7 algorithms on SDM, due to its better exploitation ability to search SDM parameters.

Figure 3.

The fluctuation comparison of best RMSE in 30 runs of seven algorithms on SDM.

Figure 4.

The convergence effect of seven algorithms on SDM.

Figure 4 shows that EGWOFS achieves the best convergence speed among the seven algorithms for SDM, primarily due to the EMBCLS employed by EGWOFS, in which the search process is performed under the guidance of elite individuals to improve the convergence speed. In addition, we find that EGWOFS, PSO and SSA have a better ability to jump out of the local optimum than the other four algorithms on SDM, which is primarily due to their better global search ability.

Therefore, compared with the original GWO algorithm, the global search ability and convergence speed of the proposed EGWOFS have been markedly improved, which also demonstrates the effectiveness of the improved mechanism.

5.1.3 Test on case 2: DDM

In Table 4, the proposed EGWOFS achieves the best results in all four indicators for the DDM. The values obtained by the six comparison algorithms are much worse than those obtained by the proposed EGWOFS on the three indicators ‘Max’, ‘Mean’ and ‘Std’. Compared with the SDM, there are more parameters to be identified by the algorithms on the DDM, and the six comparison algorithms have insufficient global exploration and local exploitation in solving the parameter identification of the DDM, which easily leads to their insufficient accuracy and stability. While the introduced MMLBSA and DSUPS can enhance the ability of the EGWOFS to globally explore and locally exploit DDM parameters, respectively. So the proposed EGWOFS has a better ability to jump out of the local optimum on the DDM.

Table 4
Statistical results of seven algorithms on DDM

Algorithm	Max	Min	Mean	Std
EGWOFS	9.8604E-04	9.8248E-04	9.8375E-04	1.4075E-06
GWO	3.9927E-02	1.3096E-03	8.9372E-03	1.1776E-02
MVO	5.9674E-03	1.3504E-03	3.3071E-03	1.3382E-03
PSO	4.3495E-02	9.9169E-04	7.2350E-03	1.2934E-02
WOA	5.0571E-02	1.1975E-03	1.3628E-02	1.7554E-02
SSA	8.5337E-03	1.0852E-03	2.9387E-03	1.5943E-03
EO	3.3392E-02	9.8447E-04	6.8350E-03	1.2085E-02

Figure 5.

The $I$ - $V$ characteristic comparisons between simulated data achieved by EGWOFS and measured data on DDM.

Figure 5 shows that EGWOFS has produced satisfactory data that are generally coincident with real data on the DDM, which means that the proposed EGWOFS achieves high accuracy in PV parameter identification on the DDM. Because the MMLBSA is conducive to improving the global search ability of the proposed EGWOFS, the accuracy of the solution can be improved.

The boxplot of Fig. 6 shows that the smallest fluctuation range of the best RMSE values is achieved by EGWOFS among the 7 algorithms, which means that the proposed EGWOFS has the best reliability among the 7 algorithms on the DDM. Because the DSUPS mechanism improves the ability of the EGWOFS to locally search the optimal solution of the DDM, the reliability of the solution is better enhanced.

Figure 6.

The fluctuation comparison of best RMSE in 30 runs of seven algorithms on DDM.

Figure 7.

The convergence performance of each algorithm on DDM.

Figure 7 shows that EGWOFS gains better convergence precision than the other six algorithms on the DDM, while the WOA, EO and PSO achieve faster convergence speeds than EGWOFS, SSA, GWO and MVO on the DDM, and the convergence speeds among WOA, EO and PSO are relatively close. The primary reason is that the DDM needs to identify more parameters than SDM, which makes it more difficult for the algorithms to search the optimal solution. In addition, Fig. 7 shows that the proposed EGWOFS has the best ability to jump out of the local optimum. Therefore, the proposed EGWOFS can obtain the best convergence precision but requires a longer convergence time to globally search the optimal solution than the WOA, EO, and PSO on the DDM.

According to the above analysis, the proposed EGWOFS achieves the best convergence accuracy within the accepted convergence speed on the DDM.

5.1.4 Statistical hypothesis test

To further verify the advantage of the EGWOFS in terms of the overall ranking, we perform a well-known statistical hypothesis test called Friedman’s rank test. In addition, based on the previous two test cases, the specific Friedman ranking test results of EGWOFS and six comparison algorithms on two PV models are shown in Table 5.

Table 5
Ranking result obtained by the Friedman test of seven algorithms on two PV models

Algorithm	Friedman’s rank	Final rank
EGWOFS	1.0000	1
GWO	6.0000	5
MVO	3.5000	3
PSO	3.5000	3
WOA	7.0000	6
SSA	2.5000	2
EO	4.5000	4

Table 6

Comparison with established algorithms on SDM

Algorithm

Max

Min

Mean

Std

EGWOFS

9.8602E-04

4.3149E-17

EABOA

9.8784E-04

9.8602E-04

9.8678E-04

9.3036E-07

GWOCS

9.9095E-04

9.8607E-04

9.8874E-04

2.4696E-06

AgGWO

7.1973E-03

6.7662E-03

6.9266E-03

2.3492E-04

IGWO

4.3283E-03

2.3038E-03

3.2492E-03

1.0188E-03

MLBSA

9.8602E-04

9.1461E-12

IJAYA

1.0622E-03

9.8603E-04

9.9204E-04

1.4033E-05

PGJAYA

9.8603E-04

9.8602E-04

1.4485E-09

JAYA

1.4783E-03

9.8946E-04

1.1617E-03

1.8796E-04

GOTLBO

1.2067E-03

9.8856E-04

1.0450E-03

5.0218E-05

LETLBO

1.1593E-03

9.8738E-04

1.0333E-03

4.6946E-05

TLABC

1.0308E-03

9.8602E-04

9.9417E-04

1.1896E-05

CLPSO

1.3196E-03

9.9633E-04

1.0581E-03

7.4854E-05

DE/BBO

2.2258E-03

9.9922E-04

1.2948E-03

2.5074E-04

CMM-DE

/BBO

1.3475E-03

9.8605E-04

1.0486E-03

8.1679E-05

Table 5 shows that the proposed EGWOFS ranks first, followed by SSA, while MVO and PSO rank third concurrently, EO ranks fourth, GWO ranks fifth, and WOA ranks last. Therefore, according to the ranking, the overall performance of the proposed EGWOFS is better than that of the comparison algorithms in SDM and DDM.

5.2 Comparison with more PV models and established algorithms

To strengthen the test on the advantage of EGWOFS, we compare the proposed EGWOFS with 16 well-established meta-heuristic algorithms: EABOA [70], GWOCS [7], alpha-guided (AgGWO) [76], improved GWO (IGWO) [77], PGJAYA [38], MLBSA [36], IJAYA [8], basic JAYA [78], GOTLBO [57], LETLBO [79], TLABC [61], CLPSO [80], DE/BBO [81], CMM-DE/BBO [82], TAPSO [83] and CMAES [84]. These algorithms were utilized to check the recognition effect of the PV parameter for the meta-heuristic algorithms in literature [7, 8, 36, 38, 70, 85]. The comparative data of Tables 6, 7, 8 and 9 are obtained from their corresponding literature [7, 8, 36, 38, 70, 85].

Table 7
Comparison with established algorithms on DDM

Algorithm

Max

Min

Mean

Std

EGWOFS

9.8604E-04

9.8248E-04

9.8375E-04

1.4075E-06

EABOA

1.0012E-03

9.8607E-04

9.9190E-04

6.6247E-06

GWOCS

1.0017E-03

9.8334E-04

9.9411E-04

9.5937E-06

AgGWO

3.8146E-02

1.1646E-03

1.4705E-02

2.0382E-02

IGWO

3.8871E-03

1.7576E-03

2.7192E-03

1.0796E-03

MLBSA

9.8798E-04

9.8249E-04

9.8518E-04

1.3482E-06

IJAYA

1.4055E-03

9.8293E-04

1.0269E-03

9.8325E-05

PGJAYA

9.9499E-04

9.8263E-04

9.8582E-04

2.5375E-06

JAYA

1.4793E-03

9.8934E-04

1.1767E-03

1.9356E-04

GOTLBO

1.3947E-03

9.8742E-04

1.1475E-03

1.1330E-04

LETLBO

1.4870E-03

9.8565E-04

1.0869E-03

1.5360E-04

TLABC

1.9826E-03

1.0012E-03

1.2116E-03

2.1100E-04

CLPSO

1.5494E-03

9.9894E-04

1.1458E-03

1.4367E-04

DE/BBO

2.4042E-03

1.0255E-03

1.5571E-03

3.6297E-04

CMM-DE

/BBO

2.0589E-03

1.0088E-03

1.5487E-03

2.9413E-04

Table 8

Comparison with established algorithms on STM6-40/36

Algorithm	Max	Min	Mean	Std
EGWOFS	1.7298E-03	1.7298E-03	1.7298E-03	1.3950E-17
GWOCS	1.7528E-03	1.7337E-03	1.7457E-03	1.0447E-05
AgGWO	1.1577E-02	5.8391E-03	8.7168E-03	2.8691E-03
IGWO	9.8971E-03	6.3448E-03	7.8755E-03	1.8263E-03
MLBSA	1.7885E-03	1.7298E-03	1.7338E-03	1.08E-05
IJAYA	2.0908E-03	1.7353E-03	1.8306E-03	9.33E-05
PGJAYA	1.7349E-03	1.7298E-03	1.7301E-03	9.37E-07
JAYA	1.4475E-02	1.7298E-03	5.1602E-03	4.76E-03
GOTLBO	5.7358E-03	1.7298E-03	2.6835E-03	1.04E-03
TAPSO	3.3278E-01	1.3423E-02	3.9292E-02	5.69E-02
CMAES	1.5029E-01	2.0958E-03	7.7961E-02	5.64E-02

Table 9

Comparison with established algorithms on STM6-120/36

Algorithm	Max	Min	Mean	Std
EGWOFS	1.6601E-02	1.6601E-02	1.6601E-02	2.4722E-16
MLBSA	1.8437E-02	1.6601E-02	1.6769E-02	4.62E-04
IJAYA	1.6904E-02	1.6744E-02	1.6812E-02	4.59E-05
PGJAYA	1.6634E-02	1.6601E-02	1.6603E-02	6.77E-06
JAYA	6.0207E-02	1.6603E-02	2.9335E-02	1.42E-02
GOTLBO	5.4115E-02	1.6601E-02	2.6330E-02	1.18E-02
TAPSO	1.1985E+00	5.4467E-02	3.3776E-01	2.99E-01
CMAES	1.4131E+00	1.6615E-02	5.7660E-01	5.75E-01

In Table 6, EGWOFS achieves the best results on four indicators; MLBSA achieves the best results on three indicators; PGJAYA achieves the best results on two indicators; and EABOA and TLABC achieve the best result on one indicator. In Table 7, EGWOFS achieves the best results on three indicators, while MLBSA achieves the best results on one indicator. In Table 8, EGWOFS achieves the best results on four indicators, while MLBSA, PGJAYA, JAYA, and GOTLBO achieve the best results on one indicator. In Table 9, EGWOFS achieves the best result on four indicators, while MLBSA, PGJAYA, and GOTLBO achieve the best results on one indicator.

In short, for accuracy indicators including ‘Max’, ‘Min’, and ‘Mean’, the proposed EGWOFS achieves the best results of three accuracy indicators on four models concurrently, while MLBSA achieves the best results of three accuracy indicators only on SDM concurrently. For the reliability indicator ‘Std’, the proposed EGWOFS achieves the best results on three models, including SDM and two PV module models, while MLBSA achieves the best reliability only on DDM.

Table 10

The verification of the introduced mechanisms on SDM

Algorithm	Max	Min	Mean	Std
EGWOFS1	1.4640E-03	9.8625E-04	1.0528E-03	1.0477E-04
EGWOFS2	9.8602E-04	9.8602E-04	9.8602E-04	1.7772E-16
EGWOFS	9.8602E-04	9.8602E-04	9.8602E-04	4.3149E-17

Table 11

The verification of the introduced mechanisms on DDM

Algorithm	Max	Min	Mean	Std
EGWOFS1	4.2190E-03	1.0815E-03	2.6412E-03	9.2629E-04
EGWOFS2	1.0445E-03	9.8250E-04	9.8646E-04	1.1051E-05
EGWOFS	9.8604E-04	9.8248E-04	9.8375E-04	1.4075E-06

Therefore, Tables 6, 7, 8 and 9 further demonstrate that the overall performance of the proposed EGWOFS achieves better or competitive results in terms of accuracy for the corresponding PV model, primarily due to the effectiveness of MMLBSA mechanism. In terms of reliability, the proposed EGWOFS achieves the best results in three of the four models, primarily due to the effectiveness of DSPUS mechanism.

5.3 Verification of two introduced mechanisms

Some tests will be further performed to verify the two introduced mechanisms in EGWOFS. The selection of algorithm parameters is consistent with the previous section. For this purpose, EGWOFS without MMLBSA is denoted as EGWOFS1, while EGWOFS without DSPUS is denoted as EGWOFS2. Then, EGWOFS1 and EGWOFS2 are compared with EGWOFS to describe the effectiveness of both MMLBSA and DSPUS. Results are shown in Tables 10, 11, 12, and 13.

Tables 10, 11, 12, and 13 show that EGWOFS simultaneously achieves better stability and precision than the compared algorithms on the four PV models. Although the recognition precision of EGWOFS2 is ameliorated, the reliability of EGWOFS2 is insufficient. When EGWOFS1 is combined with MMLBSA, accuracy and reliability improve, and when EGWOFS2 is combined with DSPUS, reliability further enhances. Therefore, these two mechanisms complement each other. We can also see that the primary reason for the accuracy improvement of EGWOFS lies in the MMLBSA mechanism, and the primary reason for the reliability improvement of EGWOFS lies in the DSPUS mechanism.

Table 12
The verification of the introduced mechanisms on STM6-40/36

Algorithm	Max	Min	Mean	Std
EGWOFS1	5.8720E-03	1.9337E-03	3.2626E-03	8.0715E-04
EGWOFS2	1.7298E-03	1.7298E-03	1.7298E-03	3.6677E-10
EGWOFS	1.7298E-03	1.7298E-03	1.7298E-03	1.3950E-17

Table 13

The verification of the introduced mechanisms on STP6-120/36

Algorithm	Max	Min	Mean	Std
EGWOFS1	3.7863E-02	1.6756E-02	2.0168E-02	4.6624E-03
EGWOFS2	1.6601E-02	1.6601E-02	1.6601E-02	9.9496E-09
EGWOFS	1.6601E-02	1.6601E-02	1.6601E-02	2.4722E-16

Table 14

The results obtained by each algorithm at certain conditions on KC200GT

Algorithm	200 W/m ${}^{2}$		400 W/m ${}^{2}$		600 W/m ${}^{2}$		800 W/m ${}^{2}$		1000 W/m ${}^{2}$
	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std
EGWOFS	1.4185E-03	1.0366E-10	1.4262E-03	5.7613E-10	1.2977E-03	7.4781E-09	1.6334E-03	7.9686E-06	1.5391E-03	5.7892E-07
GWO	4.9636E-02	4.9618E-02	1.1033E-01	1.4994E-01	1.0964E-01	1.9425E-01	1.4722E-01	2.3067E-01	1.7070E-01	2.7071E-01
MVO	5.9777E-02	1.3402E-02	9.4660E-02	2.7156E-02	1.0566E-01	1.6299E-02	1.0900E-01	1.1735E-02	1.0575E-01	6.2493E-03
PSO	1.5514E-02	3.7551E-02	2.7305E-02	2.9201E-02	8.4602E-02	1.4061E-01	1.1505E-01	1.6636E-01	1.7456E-01	2.6946E-01
WOA	6.1073E-02	4.7474E-02	1.5944E-01	1.6917E-01	2.3904E-01	3.1907E-01	3.6482E-01	4.5101E-01	6.0664E-01	1.4193E+00
SSA	5.6228E-02	1.6377E-02	9.2146E-02	1.9580E-02	9.5792E-02	2.1466E-02	1.0328E-01	1.4471E-02	1.0388E-01	6.5659E-03
EO	1.8383E-02	4.8001E-02	5.8894E-02	1.2876E-01	1.4958E-01	2.6900E-01	1.3138E-01	2.3506E-01	2.5605E-01	4.1469E-01

5.4 Test on data from the manufacturer’s data sheet

Different from the previous four test cases, the experimental data of the Multi-crystalline KC200GT [71] are used in this subsection to further test the application potentiality of EGWOFS in different irradiance cases, including 200 W/M ${}^{2}$ , 400 W/M ${}^{2}$ , 600 W/M ${}^{2}$ , 800 W/M ${}^{2}$ , and 1000 W/M ${}^{2}$ at 25 ${}^{\circ}$ C. Because the irradiance conditions of the experimental data of this subsection are variable, the reliability of the proposed EGWOFS can also be further tested. In addition, Multi-crystalline KC200GT is a PV module, so Eq. (13) and Eq. (14) of the previous section are used as the error function and objective function, respectively. The statistical results of the RMSE values obtained by each algorithm on the single diode PV module are listed in Table 14. The interval setting of identified variables are as follows: $I_{ph}$ is between $[0,2I_{sc}]$ (A); $I_{sd}$ is between $[0,100]$ ( $\mu$ A); $R_{s}$ is between $[0,2]$ ( $\Omega$ ); $R_{sh}$ is between $[0,5000]$ ( $\Omega$ ); $n$ is between $[1,4]$ ; and $I_{SC}$ is processed by Eq. (56):

$\displaystyle I_{\textit{SC}}(G,T)=I_{\textit{SC\_STC}}\frac{G}{G_{\textit{STC% }}}+\alpha(T-T_{\textit{STC}})$ (56)

where $T$ and $G$ are the temperature levels and irradiance, respectively; $\alpha$ denotes the temperature coefficient; and the short circuit current is represented by $I_{\textit{SC\_STC}}$ under standard conditions.

Table 14 shows both the average RMSE and its standard deviation produced by each algorithm with 30 independent runs under different irradiance cases. With an increase in irradiance, the overall performance of all algorithms decreases, but compared with the comparison algorithms, the performance of the proposed EGWOFS decreases less, and the proposed EGWOFS still maintains the best results in the values on ‘Mean’ and ‘Std’. These results also demonstrate the application potential and reliability of EGWOFS on the Multi-crystalline KC200GT. Because the DSUPS mechanism is conducive to improving the reliability of EGWOFS. While six comparison algorithms are prone to fall into local optimum to solve this problem and have a weak ability to locally search PV parameters, so the average accuracy and reliability of the solution are weak on the Multi-crystalline KC200GT under different irradiance cases.

6. Conclusion and future work

Due to the insufficient accuracy and reliability of the GWO algorithm when identifying photovoltaic parameters, this study proposed an enhanced grey wolf optimizer with fusion strategies to effectively solve these problems. In EGWOFS, the MMLBSA was first designed to promote the exploration performance of the original GWO. Then, the proposed DSUPS was designed to continue to ameliorate the local exploitation ability. Finally, numerous experiments were performed to show that the overall performance of EGWOFS achieves better or competitive results in terms of accuracy and reliability for most PV models. Therefore, the proposed EGWOFS is suitable for solving the parameter identification of popular single diode, double diode and PV module models, and has the potential to solve parameter identification of single diode PV modules in different irradiance cases. Because EGWOFS has a better ability to jump out of local optimum when identifying PV parameter.

In future work, we plan to continue to improve the proposed EGWOFS or develop other meta-heuristics to obtain PV parameters with lower evaluation cost. In addition, we plan to design an appropriate variant of EGWOFS to manage multi-objective or many-objective problems [86, 87, 88, 89, 90, 91, 92]. We also plan to study the application of other meta-heuristic algorithms to classification problems [93, 94], feature selection [95, 96], minimization of data size [97], deep learning [98], optimization of model training [99], multi-criteria optimization[100], compaction quality assessment[101], and other complex problems [102, 103, 104, 105, 106].

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An enhanced grey wolf optimizer with fusion strategies for identifying the parameters of photovoltaic models

Abstract

Keywords

1. Introduction

2.1 Solar cell model

2.1.1 SDM

2.3.1 Objective function

2.3.3 Constraints

Table 1 Search interval of identified parameters on three types of PV model

3.1 Social hierarchy

3.2 Encircling prey

3.5 Prey search (exploitation)

4. Enhanced grey wolf optimizer with fusion strategies (EGWOFS)

4.1 MMLBSA

5.1 Test on two popular PV models

5.1.1 Experimental settings

Table 2 Set parameters for seven algorithms

Table 3 Statistical results of seven algorithms on SDM

Table 4 Statistical results of seven algorithms on DDM

Table 5 Ranking result obtained by the Friedman test of seven algorithms on two PV models

Table 7 Comparison with established algorithms on DDM

Table 12 The verification of the introduced mechanisms on STM6-40/36

References

Table 1
Search interval of identified parameters on three types of PV model

Table 2
Set parameters for seven algorithms

Table 3
Statistical results of seven algorithms on SDM

Table 4
Statistical results of seven algorithms on DDM

Table 5
Ranking result obtained by the Friedman test of seven algorithms on two PV models

Table 7
Comparison with established algorithms on DDM

Table 12
The verification of the introduced mechanisms on STM6-40/36