An Efficient Electric Charged Particles Optimization Algorithm for Numerical Optimization and Optimal Estimation of Photovoltaic Models

Abstract

The electric charged particles optimization (ECPO) technique is inspired by the interaction (exerted forces) between electrically charged particles. A developed version of ECPO called MECPO is suggested in this article to enhance the capability of searching and balancing the exploitation and exploration phases of the conventional ECPO. To let the search agent jumps out from the local optimum and avoid stagnation in the local optimum in the proposed MECPO, three different strategies in the interaction between ECPs are modified in conjunction with the conventional ECPO. Therefore, the convergence rate is enhanced and reaches rapidly to the optimal solution. To evaluate the effectiveness of the MECPO, it is executed on the test functions of the CEC’17. Furthermore, the MECPO technique is suggested to estimate the parameters of different photovoltaic models, such as the single-diode model (SDM), the double-diode model (DDM), and the triple-diode model (TDM). The simulation results illustrate the validation and effectiveness of MECPO in extracting parameters from photovoltaic models.

1. Introduction

To deal with the increase in energy shortage as well as the several disadvantages of fossil fuels, increased research studies on renewable energy sources (RES) are urgently required [1,2]. Solar energy is one of the most important RES that researchers focus on due to its many advantages such as cleanness, safeness, and abundance [3]. The principal kind of solar energy is photovoltaic power generation [4]. PV solar installations around the world increased from about 217 GW in 2015 to just over 707 GW in 2020 [5]. However, using photovoltaic systems to produce electric energy is a big challenge because of their reliance on the climate and other factors [1]. Consequently, a precise model that almost represents the non-linear current–voltage (I–V) and the power–voltage (P–V) output characteristics of the PV model under normal operation is required for simulation and evaluation of photovoltaic systems [6]. These non-linearity challenges are difficult for any possibility and approximation to boost efficiency [7].

A solar PV cell is the main unit of the photovoltaic system. Therefore, it is important to estimate its unknown parameters to acquire a relative analysis of the PV panel performance. The equivalent circuits for the single-, double- and triple-diode model for parameter identification are well-known and most extensively used ways [8,9]. There are five parameters required in the single-diode model (SDM), while there are seven unknown parameters in the double-diode model (DDM) and nine unknown parameters in the triple-diode model (TDM). In many works of literature, the SDM and DDM are generally employed, in which the researchers have used several techniques to estimate the parameters of the PV model based on analytical and numerical methods and optimization techniques [10,11]. Despite the analytical approaches are the simplest and they reach the result rapidly, they do not have the precision under normal conditions [7].

Several metaheuristic techniques have been improved and employed to reach the optimum solution for the PV parameters extraction problem. An artificial electric field algorithm (AEFA) was proposed to extract nine parameters of the triple-diode photovoltaic model [12]. Meiying Ye employed the particle swarm algorithm (PSO) to solve the solar cell parameters [13]. In [14], the direction of movement (DM) in the gradient-based optimizer (GBO) algorithm is used to converge around the area of the solution, and it was suggested to change the DM value gradually in the modified gradient-based optimizer (MGBO) to improve the balancing between exploration and exploitation phases in the search process. The flower pollination algorithms (FPA) were used to extract the global parameters of both the single-diode and the double-diode models based on the experimental data [15]. Additionally, other recent optimization algorithms were used to find the best values of PV parameters, including transient search optimization (TSO) [16], cuckoo search (CS) [17], whale optimization algorithm (WOA) [18], supply demand-based optimization (SDO) [19], salp swarm algorithm (SSA) [20], improved bonobo optimizer (IBO) [21], multiverse optimizer (MVO) [22], tree growth algorithm (TGA) [23], grey wolf optimization (GWO) [24], triple-phase teaching-learning-based optimization (TPTLBO) [25], ant lion optimization (ALO) [26], chaos game optimization (CGO) [27], Harris hawk optimization (HHO) [28], Rao algorithm [29], slime mould algorithm (SMA) [30], and hybrid techniques, such as hybrid adaptive TLBO with DE algorithm (ATLDE) [31], GWOCS [1], PSOGWO [32], Marine predator algorithm (MPA) [33], Coyote optimization [34], and Jaya algorithm and its variants [35]. Each technique has various strategies to achieve a certain objective, and the power of each technique depends on the precision of the estimated parameters, computation time, and computational complexity.

In this article, a development version of ECPO called MECPO is suggested to enhance the search capability of searching and to increase the balance between the exploitation and exploration for the conventional ECPO. In addition, this new algorithm is utilized for extracting the PV module parameters. The performance of the MECPO technique is compared with the recent techniques, such as COOT [36], equilibrium optimizer (EO) [37], Giza pyramids construction (GPC) [38], and MPA [39] algorithms, as well as the original ECPO algorithm [40]. However, the major contributions of this paper are summarized as follows:

• Proposing an effective optimizer called MECPO with the aim of improving the performance of the original algorithm.
• The CEC’17 test suite is used to assess the MECPO efficiency.
• The proposed MECPO algorithm is employed for the PV models: SDM, DDM, and TDM of PV modules.
• The obtained statistical results using the proposed algorithm are compared with state-of-the-art techniques.
• The superiority and reliability of the MECPO-based methodology in solving the parameter estimating for the PV model’s problem are verified.

The rest of the article is organized as follows: Section 2 of the paper analyzes the problem formulation, including SDM, DDM, and TDM, and presents the objective function for identifying the parameters of the solar PV module. Section 3 presents an overview of the ECPO technique and the proposed MECPO technique. Section 4 describes the results, discussions, and comparisons. Section 5 gives the conclusion of this article with the final remarks.

2. PV Models

In this section, the PV models are presented. The PV models include the single-diode model (SDM), the double-diode model (DDM), and the triple-diode model (TDM).

2.1. Single-Diode Model (SDM)

SDM is simple, and Figure 1 illustrates its equivalent circuit diagram. In this figure, there are many significant parameters such as photocurrent $I_{Ph}$, the current of diode $I_{D1}$, shunt resistance $R_{sh}$, series resistance $R_s$, and output current $I$. It is observed from Figure 1 that it is easy to compute the output current for this model as follows:

$$I=I_{Ph}-I_{D1}-I_{sh}$$

(1)

where $I_{sh}$ is the shunt resistor current.

The modeling of SDM is represented in the following equation:

$$I=I_{Ph}-I_{sd}\left[\exp \left(\frac{q\left(V+I{R}_S\right)}{NKT}\right)-1\right]-\frac{V+I{R}_s}{R_{sh}}$$

(2)

where

▪

$I_{sd}$ is the reverse saturation current of D₁.

▪

The variable $q=1.60217646\times {10}^{-19} C$ is the charge of the electron.

▪

$V$ is the total voltage.

▪

$K=1.3806503\times {10}^{-23}$ (J/k) is the Boltzmann’s constant.

▪

$T$ is the temperature in Kelvin.

▪

N is the ideality factor of D₁.

2.2. The Double-Diode Model (DDM)

Compared to the first case (SDM), the impact of compound current loss is taken into consideration in the DDM. Figure 2 displays the structure of DDM. It can be seen from this figure that there are two diodes, in this case, in parallel with the source of current. The output current of the DDM is calculated from the following equation:

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

(3)

$$I=I_{Ph}-I_{sd1}\left[\exp \left(\frac{q\left(V+I{R}_S\right)}{N_{1}KT}\right)-1\right]-I_{sd2}\left[\exp \left(\frac{q\left(V+I{R}_S\right)}{N_{2}KT}\right)-1\right]-\frac{V+I{R}_s}{R_{sh}}$$

(4)

where

▪

$I_{sd1}$ and $I_{sd2}$ are the reverse saturation current of D₁ and D₂, respectively.

▪

N₁, and N₂ are the ideality factor of D₁ and D₂, respectively.

2.3. The Triple-Diode Model (TDM)

TDM has three diodes in parallel with the current source, as shown in Figure 3. Therefore, the output current can be given by the following equations:

$$I=I_{Ph}-I_{D1}-I_{D2}-I_{D3}-I_{sh}$$

(5)

$$I=I_{Ph}-I_{sd1}\left[\exp \left(\frac{q\left(V+I{R}_S\right)}{N_{1}KT}\right)-1\right]-I_{sd2}\left[\exp \left(\frac{q\left(V+I{R}_S\right)}{N_{2}KT}\right)-1\right]-I_{sd3}\left[\exp \left(\frac{q\left(V+I{R}_S\right)}{N_{3}KT}\right)-1\right]-\frac{V+I{R}_s}{R_{sh}}$$

(6)

where

▪

$I_{sd1}$, $I_{sd2}$, and $I_{sd3}$ are the reverse saturation current of D₁, D_2, and D₃, respectively.

▪

N₁, N₂, and N₃ are the ideality factor of D₁, D₂, and D₃, respectively.

2.4. The Objective Function

The objective function is to extract the best parameter values of the PV models by reducing the variance between the estimated data and the measured data. The objective function of SDM, DDM, and TDM is given as follows [33]:

• The objective function of SDM is given as:

  $$f_{SD}\left(V,I,x\right)=I-x_3+x_4\left[\exp \left(\frac{q\left(V+I{x}_1\right)}{x_{5}KT}\right)-1\right]-\frac{V+I{x}_1}{x_2}$$

  (7)
• The objective function of DDM is given as:

  $$f_{DD}\left(V,I,x\right)=I-x_3+x_4\left[\exp \left(\frac{q\left(V+I{x}_1\right)}{x_{6}KT}\right)-1\right]+x_5\left[\exp \left(\frac{q\left(V+I{x}_1\right)}{x_{7}KT}\right)-1\right]-\frac{V+I{x}_1}{x_2}$$

  (8)
• The objective function of TDM is given as:

  $$f_{TD}\left(V,I,x\right)=I-x_3+x_4\left[\exp \left(\frac{q\left(V+I{x}_1\right)}{x_{6}KT}\right)-1\right]+x_5\left[\exp \left(\frac{q\left(V+I{x}_1\right)}{x_{7}KT}\right)-1\right]+x_8\left[\exp \left(\frac{q\left(V+I{x}_1\right)}{x_{9}KT}\right)-1\right]-\frac{V+I{x}_1}{x_2}$$

  (9)

  where

  ▪

  $V$ and $I$ values can be given from a solar cell.

  ▪

  $x$ = [$x_1$, $x_2$, …$x_n$] is the estimating parameters vector for n-parameters for each solar cell model and can be defined as:

  (1)

  $x$ = [$R_s$, $R_{sh}$, $I_{ph}$, $I_{sd},$ $n$] for SDM.

  (2)

  $x$ = [$R_s$, $R_{sh}$, $I_{ph}$, $I_{sd1}$, $I_{sd2}$, $n_1$, $n_2$] for DDM.

  (3)

  $x$ = [$R_s$, $R_{sh}$, $I_{ph}$, $I_{sd1}$, $I_{sd2}$, $I_{sd3}$, $n_1$, $n_2$, $n_3$] for TDM.

The lower and upper boundaries of the unknown parameters are presented in

Table 1. The limits of extracted photovoltaic parameters.

Parameter	Lower Limit	Upper Limit
$I_{ph}$ (A)	0	1
$I_{sd1}$$, I_{sd2}$$, I_{sd3}$ (μA)	0	1
$R_s$ (Ω)	0	0.5
$R_{sh}$ (Ω)	0	100
$n_1$$, n_2$$, n_3$	1	2

.

In this paper, the root mean square error (RMSE) is the error between the estimated and the measured data of the diode model. Therefore, The RMSE can be calculated from the following equation [41]:

$$RMSE=\sqrt{\frac{1}{N}{\displaystyle \sum }_{i=1}^N{\left(f_h\left(V,I,x\right)\right)}^2}$$

(10)

where

▪

h is the solar cell model to be used.

▪

N identifies a set of samples is employed to verify the best optimum.

3. The Proposed MECPO

In this section, an overview of the MECPO algorithm is presented in detail, which improves the original ECPO as (1) a new parameter called transfer factor (TF) is added to transfer from exploration to exploitation and vice versa; (2) the three different strategies in the interaction between ECPs are modified to avoid gradually move closer to better individuals who easily make the algorithm stagnation in local optimum and convergence prematurely, and (3) the diversification phase is modified to let the search agent jumps out from local optimum and avoid stagnation in local optimum efficiently.

3.1. Advanced Transfer Factor (TF)

The balance between the exploration and exploitation phases is regarded as the major process to the success of any optimization algorithm and is controlled by a specific parameter. The performance of ECPO is affected by the transferring parameter from exploration to exploitation, which is missing. To resolve this issue, the non-linear parameter called transfer factor (TF) is injection into ECPO as follows:

$$TF=C_1\times cos{\left(1-\frac{t}{T}\right)}^{C_2}$$

(11)

where $C_1$ and $C_2$ are constant 3 and 0.25, respectively.

This enhances the switching phase between exploration and exploitation in a proper manner and maintains the diversity of the solutions.

3.2. Modified Interaction Phase

In this subsection, some setting will be proposed to avoid gradually moving closer to better individuals, which easily make the algorithm stagnation in local optimum and convergence prematurely.

3.2.1. Strategy 1

In this strategy, two new ECPs are created, called ${\mathrm{ECP}}_{\mathrm{inew}1}$ and ${\mathrm{ECP}}_{\mathrm{inew}2},$ while i denotes the index of the chosen ECP). This results from three interacting ECPs.

For $EC{P}_1 ,EC{P}_2, \mathrm{and} EC{P}_3$:

Where ECP₁ is the best agent, ECP₂ is the second-best agent, and ECP₃ is the third-best agent.

$EC{P}_1 \mathrm{is} \mathrm{affected} \mathrm{by} EC{P}_2$ and $EC{P}_{best}$ to move from $ECP$ to ${\mathrm{ECP}}_{1\mathrm{new}1}$. The total force exerts on $EC{P}_1$ become:

$$F=F_{21}+D\times F_{best1}$$

(12)

$D=\pm 1$—this parameter is proposed to give high opportunities with different values on resulting $F$ to give high opportunities to change the direction of agents that results in a good scan of the given search space in all possible directions.

The total force that pushes ECP₁ to the move to ECP_1new1 and it illustrates as below:

$$\begin{gathered} {\mathrm{ECP}}_{1\mathrm{new}1}=EC{P}_1+F \\ =EC{P}_1+F_{best1}+F_{21} \\ =EC{P}_1+D\times TF\times \beta \times \left(EC{P}_{best}-EC{P}_1\right)+\beta \times \left(EC{P}_1-EC{P}_2\right) \end{gathered}$$

(13)

where $\beta$ is a random number. F₂₁ refers to the force on ECP₁ of ECP₂ and F_best1 is the force on ECP₁ of ECP_best.

Then, ECP₁ is affected simultaneously by ECP₃ and ECP_best to move to ECP_1new2.

$${\mathrm{ECP}}_{1\mathrm{new}2}=EC{P}_1+D\times TF\times \beta \times \left(EC{P}_{best}-EC{P}_1\right)+\beta \times \left(EC{P}_1-EC{P}_3\right)$$

(14)

3.2.2. Strategy 2

To resolve the issue of the lack of interaction between ECP_best and the rest of ECPs, in the proposed method of three interacting ECPs, each ECP creates one new ECP called ECP_inew.

$$\begin{gathered} {\mathrm{ECP}}_{1\mathrm{new}1}=EC{P}_1+F \\ =EC{P}_1+F_{21}+F_{31} \\ =EC{P}_1+D\times TF\times \beta \times \left(EC{P}_1-EC{P}_2\right)+\beta \times \left(EC{P}_1-EC{P}_3\right) \end{gathered}$$

(15)

For $EC{P}_2$: The second particle ECP₂ is influenced at the same time by the first and third particles (i.e., ECP₁ and ECP₃), which moves to ECP_2new. The resulting force to move ECP₁ to ECP_1new can be defined as:

$$\begin{gathered} F=F_{12}+F_{32} \\ {\mathrm{ECP}}_{2\mathrm{new}1}=EC{P}_2+F \\ =EC{P}_2+F_{12}+F_{32} \\ =EC{P}_2+D\times TF\times \beta \times \left(EC{P}_2-EC{P}_1\right)+\beta \times \left(EC{P}_2-EC{P}_3\right) \end{gathered}$$

(16)

where F₁₂ is the force of ECP₁ on ECP₂, and F₃₂ is the force of ECP₃ on ECP₂.

For $EC{P}_3$:

The ECP₃ is affected by ECP₁ and ECP₂ at the same time with the force given by:

$$\begin{gathered} F=F_{13}+F_{23} \\ {\mathrm{ECP}}_{3\mathrm{new}1}=EC{P}_3+F \\ =EC{P}_3+F_{13}+F_{23} \\ =EC{P}_3+D\times TF\times \beta \times \left(EC{P}_3-EC{P}_1\right)+\beta \times \left(EC{P}_2-EC{P}_3\right) \end{gathered}$$

(17)

where F₁₃ is the force of ECP₁ on ECP₃ and F₂₃ is the force of ECP₂ on ECP₃.

3.2.3. Strategy 3

In the previous strategies, new ECPs are generated. Illustratively, the three interacting ECPs (nECPI = 3) will generate nine new ECPs (six ECPs from strategy 1 and three from strategy 2). In strategy 3, the equations defined in the previous two strategies are used.

3.3. Modified Diversification

The diversification phase updates the portion of ECPs population based on a variable called probability of diversification (Pd = 0.2). The new ECPs update their positions by selecting a random archive pool previously created. In the proposed enhancement, the new ECPs update their positions with five cases, including selecting a random archive pool, the remaining proposed four cases are (1) new ECPs update their positions based on boundary constraints to create new ECPS in new regions that may be not visited before; (2) new ECPs update their position with the best ECP in the population; (3) new ECPs update based on information from the archive and random ECPs with controlling parameter $a_j$ that initializes with a random value and decreases with problem size by multiplying it with another random number [0, 1]; (4) the new ECPS will equal the old ECPs (no change).

The proposed diversification phase jumps out from the local optimum and avoids stagnation in the local optimum.

For i = 1: newECP

$$a_j=rand$$

(18)

For j = 1: ProblemSize

$${\mathrm{newEPC}}_{i,j}=\{\begin{array}{cc}{\mathrm{newEPC}}_{i,j}=L_j+r_1\times (U_j-L_j)& r_2<0.3\hfill \\ {\mathrm{newEPC}}_{i,j}={ECP}_{best,j}& r_2<0.4\hfill \\ (1-a_j)\times {\mathrm{archiveECP}}_{K,j}+a_j\times {\mathrm{randECP}}_j& r_2<0.5 where a_j=a_j\ast rand\hfill \\ {\mathrm{newEPC}}_{i,j}={\mathrm{archiveECP}}_{K,j}& r_2<0.8\hfill \\ {\mathrm{newEPC}}_{i,j}={\mathrm{EPC}}_{i,j}& othewise\hfill \end{array}$$

(19)

End If;

End For;

Eventually, the pseudo-code of the MECPO method is reported in Algorithm 1, while Figure 4 illustrates the flowchart.

Algorithm 1: Pseudo-code of the MECPO technique

Randomly initialize an nECP charged particles (solutions) using normal distribution Calculate the fitness and sort them from the best to the worst Create Archive pool with the best ECPs Update transfer factor (TF) using Equation (11) Selection: nECPI particles are randomly selected Interaction: based on the specified strategy Strategy 1 using Equations (12)–(14) Strategy 2 using Equations (15)–(17) Strategy 3 using both of Strategy 1 and Strategy 2 For each individual Xi if (Xi < Xmin)   Xi = Xmin else if (Xmax < Xi)   Xi = Xmax End If End For Diversification For individual newECP (using Equation (19)) ▪ If r < 0.3 then update its solution ▪ Else If r < 0.4 then update its solution ▪ Else If r < 0.5 then update its solution ▪ Else If r < 0.8 then update its solution ▪ Else update its solution End If End for 9. Termination criterion

4. Performance Evaluation of MECPO

To assess the performance of the MECPO technique, the 2017 IEEE Congress on Evolutionary Computation (CEC’17) test suite [42] is solved. In addition, several statistical metrics are extracted, including mean value and standard deviation (STD) for optimal values achieved. The achieved results are compared with seven recent metaheuristic techniques, for instance, gravitational search algorithm (GSA) [43], grey wolf optimizer (GWO) [44], WOA [45], sine cosine algorithm (SCA) [46], EO [37], HHO [47], and the original electric charged particles optimization (ECPO) [40].

4.1. Parameter Settings Values

Table 2. The settings of the parameter for the MECPO technique and the counterparts.

Techniques	The Setting of Parameters
Public settings	Number of independent runs: 30 Maximum iterations: 1000 Population size: N = 30 Dim = 30,
GSA	α = 20, G0 = 100, Rnorm = 2, Rpower = 1
GWO	a linearly decrease from 2 to 0
WOA	a2 linearly decreases from −1 to −2 α variable decreases linearly from 2 to 0
SCA	A = 2
EO	a_1 = 2, a_2 = 1, GP = 0.5
HHO	E0 ∈ [−1, 1], β = 1.5
ECPO	Strategy = 1, archSize = 30 and NPI = 2
MECPO	$\mathrm{Strategy}=1, \mathrm{archSize}=30, \mathrm{NPI}=2, {\mathrm{C}}_1=3, {\mathrm{C}}_2=0.25, \mathrm{and} \beta =random number$

reports the parameters setting values; all the counterparts are evaluated on 30 independent runs and 1000 iterations (max_Itr) for every function for fair benchmarking comparison. We choose the default parameters values for the counterparts to decrease the risk of bias in the comparison, as illustrated in [48].

4.2. Definition of CEC’17 Benchmark Functions

CEC’17 test suite [42] including 29 functions is divided into three categories: (1) unimodal functions (F1–F3) has a single optimum solution and is used to evaluate the exploitation capability; (2) multimodal functions, which are from function F4 to function F10, have multiple local minima, and are employed to test the exploration capability; (3) hybrid functions from F11 to F20, and finally, the composition functions, which are from function F21 to function F30, have a huge number of local minima and are employed to evaluate the capability of the technique to avoid the local minima and try to maintain the stability between exploitation and exploration stages.

4.3. Statistical Results

The mean and STD of the optimum value acquired by the MECPO method and the counterparts are reported in

Table 3. The mean and STD of optimal values of the OF for thirty trials achieved with the various techniques on the CEC’17 benchmark with dimension = 30.

#	Measure	GSA	GWO	WOA	SCA	EO	HHO	EPCO	MEPCO
1	Mean	7.46 × 107	1.99 × 109	1.08 × 109	1.84 × 1010	6.40 × 103	3.17 × 107	7.14 × 108	3.78 × 103
	STD	1.57 × 108	1.30 × 109	3.15 × 108	3.07 × 109	6.54 × 103	7.47 × 106	1.31 × 109	4.94 × 103
3	Mean	9.62 × 104	5.33 × 104	2.95 × 105	6.66 × 104	2.69 × 104	3.99 × 104	2.23 × 104	1.06 × 105
	STD	1.10 × 104	1.32 × 104	7.46 × 104	1.33 × 104	8.38 × 103	6.79 × 103	2.29 × 104	2.52 × 104
4	Mean	6.54 × 102	5.96 × 102	8.74 × 102	2.35 × 103	4.99 × 102	5.55 × 102	5.18 × 102	4.90 × 102
	STD	1.25 × 102	5.81 × 101	4.64 × 101	6.92 × 102	2.48 × 101	1.46 × 101	4.36 × 101	3.48 × 101
5	Mean	7.39 × 102	6.16 × 102	8.58 × 102	8.14 × 102	6.77 × 102	7.40 × 102	6.16 × 102	5.55 × 102
	STD	2.43 × 101	2.30 × 101	5.22 × 101	2.64 × 101	2.44 × 101	4.90 × 101	2.55 × 101	5.50 × 101
6	Mean	6.58 × 102	6.11 × 102	6.91 × 102	6.60 × 102	6.01 × 102	6.73 × 102	6.19 × 102	6.00 × 102
	STD	3.99	4.34	1.52 × 101	6.28	8.31 × 10−1	9.36	6.71	1.88 × 10−1
7	Mean	9.62 × 102	8.86 × 102	1.29 × 103	1.21 × 103	8.40 × 102	1.30 × 103	9.68 × 102	8.40 × 102
	STD	5.50 × 101	5.02 × 101	5.18 × 101	4.61 × 101	3.33 × 101	5.43 × 101	7.65 × 101	6.25 × 101
8	Mean	9.63 × 102	8.96 × 102	1.07 × 103	1.09 × 103	8.86 × 102	9.68 × 102	9.01 × 102	8.35 × 102
	STD	1.64 × 101	1.56 × 101	4.81 × 101	2.13 × 101	1.81 × 101	3.46 × 101	2.20 × 101	2.73 × 101
9	Mean	4.17 × 103	2.26 × 103	1.16 × 104	7.92 × 103	1.26 × 103	7.55 × 103	2.28 × 103	9.48 × 102
	STD	4.17 × 102	6.75 × 102	5.87 × 103	1.54 × 103	4.48 × 102	1.90 × 103	1.01 × 103	4.73 × 101
10	Mean	5.39 × 103	4.80 × 103	6.73 × 103	8.66 × 103	5.21 × 103	6.14 × 103	4.68 × 103	8.04 × 103
	STD	4.92 × 102	1.15 × 103	9.07 × 102	3.33 × 102	7.53 × 102	3.57 × 102	7.77 × 102	5.16 × 102
11	Mean	3.97 × 103	2.04 × 103	6.90 × 103	3.45 × 103	1.29 × 103	1.29 × 103	4.04 × 101	1.17 × 103
	STD	1.12 × 103	8.34 × 102	3.18 × 103	1.08 × 103	4.29 × 101	3.41 × 101	7.49 × 102	1.68 × 103
12	Mean	1.25 × 108	1.34 × 108	3.17 × 108	2.15 × 109	7.58 × 105	3.69 × 107	2.99 × 106	1.23 × 106
	STD	1.29 × 108	2.48 × 108	1.29 × 108	5.75 × 108	7.77 × 105	2.63 × 107	3.71 × 106	6.53 × 105
13	Mean	3.40 × 104	3.51 × 105	1.21 × 106	9.15 × 108	1.79 × 104	6.68 × 105	2.16 × 105	1.39 × 104
	STD	9.35 × 103	6.03 × 105	3.84 × 105	4.21 × 108	1.68 × 104	4.26 × 105	8.31 × 105	1.37 × 104
14	Mean	1.03 × 106	5.06 × 105	3.17 × 106	4.70 × 105	5.14 × 104	4.12 × 105	1.19 × 105	9.76 × 104
	STD	3.52 × 105	4.37 × 105	2.59 × 106	3.85 × 105	3.70 × 104	2.54 × 105	4.22 × 105	8.76 × 104
15	Mean	1.78 × 104	5.95 × 105	2.27 × 106	4.26 × 107	6.10 × 103	1.11 × 105	1.08 × 104	6.10 × 103
	STD	5.93 × 103	1.23 × 106	3.67 × 106	3.22 × 107	5.89 × 103	1.13 × 105	9.09 × 103	5.50 × 103
16	Mean	3.47 × 103	2.53 × 103	4.06 × 103	3.95 × 103	2.41 × 103	3.23 × 103	2.65 × 103	2.57 × 103
	STD	3.34 × 102	2.52 × 102	1.09 × 103	2.38 × 102	3.42 × 102	2.66 × 102	2.89 × 102	5.61 × 102
17	Mean	2.88 × 103	2.07 × 103	2.91 × 103	2.64 × 103	2.12 × 103	2.50 × 103	2.26 × 103	1.93 × 103
	STD	2.44 × 102	2.00 × 102	3.23 × 102	2.88 × 102	2.19 × 102	3.09 × 102	2.51 × 102	2.20 × 102
18	Mean	5.16 × 105	2.43 × 106	1.24 × 107	9.49 × 106	7.54 × 105	2.54 × 106	1.95 × 105	1.88 × 106
	STD	4.03 × 105	5.15 × 106	6.47 × 106	4.99 × 106	5.47 × 105	1.02 × 106	2.42 × 105	2.75 × 106
19	Mean	1.35 × 105	3.06 × 106	6.09 × 106	7.10 × 107	5.99 × 103	8.94 × 105	1.19 × 104	9.67 × 103
	STD	1.08 × 105	8.77 × 106	5.06 × 106	4.12 × 107	2.03 × 103	5.59 × 105	1.16 × 104	1.15 × 104
20	Mean	3.07 × 103	2.45 × 103	2.82 × 103	2.90 × 103	2.29 × 103	2.85 × 103	2.49 × 103	2.21 × 103
	STD	2.66 × 102	1.73 × 102	1.98 × 102	1.43 × 102	1.26 × 102	1.86 × 102	2.40 × 102	1.85 × 102
21	Mean	2.62 × 103	2.40 × 103	2.60 × 103	2.59 × 103	2.437 × 103	2.61 × 103	2.40 × 103	2.36 × 103
	STD	3.59 × 101	2.46 × 101	2.00 × 101	2.62 × 101	2.02 × 101	4.52 × 101	2.44 × 101	5.82 × 101
22	Mean	7.25 × 103	4.99 × 103	6.55 × 103	9.01 × 103	3.51 × 103	5.77 × 103	4.53 × 103	7.79 × 103
	STD	5.08 × 102	1.84 × 103	2.45 × 103	2.49 × 103	1.97 × 103	2.99 × 103	1.88 × 103	3.09 × 103
23	Mean	3.80 × 103	2.80 × 103	3.07 × 103	3.05 × 103	2.73 × 103	3.25 × 103	2.81 × 103	2.68 × 103
	STD	1.85 × 102	6.09 × 101	1.06 × 102	2.96 × 101	1.92 × 101	6.49 × 101	4.82 × 101	8.28
24	Mean	3.44 × 103	2.94 × 103	3.23 × 103	3.24 × 103	2.99 × 103	3.43 × 103	2.96 × 103	3.00 × 103
	STD	1.32 × 102	5.36 × 101	6.98 × 101	3.01 × 101	2.34 × 101	1.17 × 102	3.88 × 101	1.47 × 101
25	Mean	2.99 × 103	3.01 × 103	3.10 × 103	3.48 × 103	2.95 × 103	2.94 × 103	2.93 × 103	2.89 × 103
	STD	2.57 × 101	4.51 × 101	3.28 × 101	1.43 × 102	2.73	1.62 × 101	3.42 × 101	1.27 × 101
26	Mean	7.85 × 103	4.90 × 103	9.31 × 103	7.60 × 103	5.87 × 103	8.21 × 103	5.21 × 103	4.08 × 103
	STD	6.30 × 102	3.12 × 102	1.01 × 103	3.93 × 102	7.24 × 102	1.18 × 103	8.11 × 102	1.79 × 102
27	Mean	5.10 × 103	3.25 × 103	3.48 × 103	3.51 × 103	3.22 × 103	3.65 × 103	3.27 × 103	3.22 × 103
	STD	4.19 × 102	1.78 × 101	8.82 × 101	4.41 × 101	9.77	3.28 × 102	3.14 × 101	1.33 × 101
28	Mean	3.56 × 103	3.51 × 103	3.56 × 103	4.23 × 103	3.22 × 103	3.35 × 103	3.30 × 103	3.23 × 103
	STD	2.08 × 102	1.98 × 102	1.50 × 102	2.36 × 102	1.93 × 101	3.48 × 101	8.87 × 101	2.34 × 101
29	Mean	5.28 × 103	3.95 × 103	5.42 × 103	5.05 × 103	3.68 × 103	4.90 × 103	3.94 × 103	3.46 × 103
	STD	3.33 × 102	1.80 × 102	3.70 × 102	1.93 × 102	2.27 × 102	6.31 × 102	2.23 × 102	1.43 × 102
30	Mean	3.18 × 106	9.29 × 106	3.98 × 107	1.41 × 108	9.95 × 103	7.82 × 106	2.52 × 104	9.01 × 103
	STD	4.09 × 106	8.10 × 106	3.06 × 107	4.86 × 107	3.45 × 103	3.85 × 106	5.26 × 104	2.89 × 103

The optimal values obtained are in bold.

for each CEC’17 function; the optimal results are highlighted in boldface. The obtained results proved that the MECPO technique is achieved the best values in solving 21 of CEC’17 functions. Moreover, MECPO gained the first rank.

4.4. Convergence Analysis

The convergence characteristics curves analysis of the MECPO technique and the other recent techniques over the CEC’17 functions are illustrated in Figure 5. It is clear from the charts that the proposed MECPO reached a stable point; therefore, it reaches the lowest average of the best so-far solutions on all functions. The achieved fast convergence qualifies the proposed MECPO to be in the first rank and solves problems that require fast computation.

4.5. Boxplot’s Analysis

Hence, the CEC’17 test suite has many local optima, so to know the distribution of best fitness, the boxplot analysis is drawn to describe data distributions into quartiles for each technique and each function in Figure 6. The box plots of the MECPO technique are very narrow for most functions compared to various recent methods distributions and, therefore, with the best values. In fact, the MECPO technique performs better than the other methods in most test functions.

4.6. The Wilcoxon Signed-Rank Test

Although metaheuristics are stochastic methods, the predicted performance should be more accurate. To verify the obtained statistical results and evaluate the significance, Wilcoxon’s rank-sum test was performed. More details about Wilcoxon’s test are given in [49]. The average pair-wise comparison of the optimal solutions with a significance level equal to 5% is reported in

Table 4. Results of Wilcoxon’s test (p ≥ 0.05) between the MECPO technique and the various recent techniques for the CEC’17 functions with Dim = 30.

#	GSA vs. MECPO	GWO vs. MECPO	WOA vs. MECPO	SCA vs. MECPO	EO vs. MECPO	HHO vs. MECPO	EPCO vs. MECPO
1	0.0030	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	0.2226	3.0199 × 10−11	8.9934 × 10−11
3	0.0748	7.3891 × 10−11	5.0723 × 10−10	8.4848 × 10−9	3.0199 × 10−11	3.0199 × 10−11	1.4643 × 10−10
4	6.6955 × 10−11	4.1997 × 10−10	3.0199 × 10−11	3.0199 × 10−11	0.5011	3.1967 × 10−9	0.0251
5	3.0199 × 10−11	4.2175 × 10−4	3.0199 × 10−11	3.0199 × 10−11	0.0025	3.0199 × 10−11	4.2175 × 10−4
6	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	0.0013	3.0199 × 10−11	3.0199 × 10−11
7	3.4971 × 10−9	0.0017	3.0199 × 10−11	3.0199 × 10−11	0.8883	3.0199 × 10−11	6.0459 × 10−7
8	7.7725 × 10−9	9.0632 × 10−8	4.5043 × 10−11	3.0199 × 10−11	9.0632 × 10−8	8.8910 × 10−10	7.6950 × 10−8
9	3.0199 × 10−11	3.6897 × 10−11	3.0199 × 10−11	3.0199 × 10−11	0.2519	3.0199 × 10−11	3.3384 × 10−11
10	1.6132 × 10−10	6.5183 × 10−9	8.1200 × 10−4	9.8329 × 10−8	1.2057 × 10−10	8.1014 × 10−10	8.1527 × 10−11
11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	6.9125 × 10−4	1.6132 × 10−10	6.6955 × 10−11
12	2.1544 × 10−10	6.0658 × 10−11	3.0199 × 10−11	3.0199 × 10−11	0.1120	3.0199 × 10−11	0.0339
13	4.3106 × 10−8	3.6897 × 10−11	3.0199 × 10−11	3.0199 × 10−11	0.0292	3.0199 × 10−11	0.0850
14	6.6955 × 10−11	0.0191	7.7725 × 10−9	2.3897 × 10−8	0.0030	2.6784 × 10−6	8.8411 × 10−7
15	1.3111 × 10−8	8.9934 × 10−11	3.0199 × 10−11	3.0199 × 10−11	0.8073	3.6897 × 10−11	0.0117
16	3.4971 × 10−9	0.1907	6.0658 × 10−11	3.0199 × 10−11	0.5895	2.0283 × 10−7	0.5493
17	3.0199 × 10−11	6.0459 × 10−7	4.0772 × 10−11	3.0199 × 10−11	5.8587 × 10−6	4.0772 × 10−11	2.1947 × 10−8
18	2.8389 × 10−4	0.3183	3.5708 × 10−6	2.9215 × 10−9	6.3560 × 10−5	0.0501	2.1947 × 10−8
19	6.6955 × 10−11	8.1014 × 10−10	3.0199 × 10−11	3.0199 × 10−11	0.1809	3.0199 × 10−11	0.7731
20	4.0772 × 10−11	5.9706 × 10−5	6.0658 × 10−11	9.9186 × 10−11	0.0080	9.7555 × 10−10	1.6062 × 10−6
21	3.0199 × 10−11	6.9125 × 10−4	3.0199 × 10−11	3.0199 × 10−11	0.0176	4.9752 × 10−11	0.0014
22	0.0080	0.0080	0.1580	5.5329 × 10−8	5.8737 × 10−4	0.0170	0.0080
23	3.0199 × 10−11	2.6695 × 10−9	3.0199 × 10−11	3.0199 × 10−11	2.1540 × 10−6	3.0199 × 10−11	5.5727 × 10−10
24	3.0199 × 10−11	1.7836 × 10−4	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.3384 × 10−11	6.2027 × 10−4
25	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	0.0519	9.7555 × 10−10	2.1540 × 10−6
26	3.0199 × 10−11	6.1210 × 10−10	3.0199 × 10−11	3.0199 × 10−11	4.4592 × 10−4	1.8731 × 10−7	1.0666 × 10−7
27	3.0199 × 10−11	1.1058 × 10−4	3.0199 × 10−11	3.0199 × 10−11	0.1453	3.0199 × 10−11	1.2860 × 10−6
28	3.0199 × 10−11	3.3384 × 10−11	3.0199 × 10−11	3.0199 × 10−11	0.0053	5.4941 × 10−11	8.3520 × 10−8
29	3.0199 × 10−11	1.8500 × 10−8	3.0199 × 10−11	3.0199 × 10−11	0.0018	3.0199 × 10−11	8.1014 × 10−10
30	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	0.0315	3.0199 × 10−11	6.2828 × 10−6

. In the same context, the results are statistically significant for all CEC’17 functions for most functions. The obtained results in Table 4 reveal that the performance of MECPO is superior over the other counterparts.

5. The Results and Discussion of Real-World Application

The numerical simulation of the proposed MECPO algorithm for identifying parameters of SDM, DDM, and TDM is illustrated in this section. Figure 7 displays the estimation process for the PV models and their validation procedure. The seven recent techniques COOT, EO, GPC, and MPA, as well as the conventional ECPO, are used in the comparison.

Table 5. Parameter settings of the selected techniques.

Algorithms	Parameters Setting
Common settings	Population size: nPop = 100 for SDM and nPop = 200 for DDM and TDM Maximum iterations: Max_iter = 1000 Number of independent runs: 20
GPC	G = 9.8, Tetha = 14, MuMin = 1, MuMax = 10, pSS = 0.5
EO	$a_1=2, a_2$ = 1
MPA	FADs = 0.2, P = 0.5, C = 0.05, e = 0.25

presents the parameter settings of the selected algorithms. All mentioned techniques have been executed for 20 independent runs using MATLAB 2016a platform with an Intel^® core TM i5-4210U CPU, 1.70 GHz, 8 GB RAM Laptop.

5.1. Case 1: Single-Diode Model (SDM)

In the first case,

Table 6. Calculated parameter in case of the SDM obtained by the proposed algorithm and other recent techniques.

Algorithm	MECPO	ECPO	COOT	GPC	EO	MPA
$R_s$ (Ω)	0.036377	0.151767	0.033574	0.000748	0.027151	0.038619
$R_{sh}$(Ω)	53.71852	35.53006	100	17.97128	99.52477	40.87662
$I_{ph}$ (A)	0.760776	0.677648	0.760476	0.776792	0.761999	0.761135
$I_{sd}$ (A)	3.23 × 10−7	0.000155	6.57 × 10−7	1.87 × 10−5	2.07 × 10−6	1.79 × 10−7
$n$	1.215672	1.984596	1.277162	1.684806	1.39214	1.168594
$RMSE$	9.86 × 10−4	0.717356	0.001731	0.018806	0.004147	0.00149

tabulated the optimal parameter values and the fitness values obtained by the MECPO algorithm, original ECPO, and four well-known algorithms for the SDM. The convergence characteristics for all techniques are displayed in Figure 8. The value of the parameters in the case of the SDM achieved using the five algorithms are presented in Table 6. Moreover, Figure 9 presents the boxplot of COOT, ECPO, EO, GPC, MPA, and the proposed MECPO techniques for the single diode, which illustrates the distribution of results achieved by various techniques in 20 trails. Table 6 tabulates the measured data’s voltage and current values and the simulated current value, simulated power value, relative error, and absolute error using the proposed MECPO for SDM.

In

Table 7. Statistical results of the proposed MECPO algorithm and other recent algorithms in the case of the SDM.

Algorithm	Best	Mean	Median	Worst	STD
MECPO	0.000986	0.000986	0.000986	0.000986	2.31 × 10−16
ECPO	0.717356	1.395899	1.390179	1.985481	0.316597
COOT	0.001731	0.004135	0.004434	0.005851	0.001238
GPC	0.018806	0.081121	0.076739	0.171743	0.035797
EO	0.004147	0.211926	0.222861	0.222861	0.048906
MPA	0.00149	0.009937	0.011736	0.012607	0.003505

The optimal values obtained are in bold.

, the fitness value (9.86 × 10⁻⁴) obtained by the proposed MECPO technique is less than any other algorithm. As shown in Figure 8, The MECPO algorithm has a high convergence speed compared with the other recent algorithms from the initial search stage to the end of iterations, and this confirms the high precision of the proposed algorithm. The five metrics results after 20 independent runs of all algorithms in Table 7 indicate that the effectiveness and robustness of the proposed algorithm are better than other techniques.

According to the comparisons on the solution distribution in Figure 9, it can be seen that the MECPO technique gives the best performance compared with the other techniques in terms of precision and strength. As can be seen from

Table 8. Measured and simulated data of voltages, currents, and power and the absolute errors values using MECPO for SDM.

Rank	Experimental Data			Simulated Current Data		Simulated Power Data
	V (V)	I (A)	P (W)	Isim (A)	IAEI (A)	Psim (W)	IAEP (W)
1	−0.2057	0.764	−0.157155	0.76409	8.80 × 10−5	−0.15717	2.00 × 10−5
2	−0.1291	0.762	−0.098374	0.76266	6.63 × 10−4	−0.09846	9.00 × 10−5
3	−0.0588	0.7605	−0.044717	0.76136	8.55 × 10−4	−0.04477	5.00 × 10−5
4	0.0057	0.7605	0.004335	0.76015	3.46 × 10−4	0.00433	0.00 × 10+00
5	0.0646	0.76	0.049096	0.75906	9.45 × 10−4	0.04903	6.00 × 10−5
6	0.1185	0.759	0.089942	0.75804	9.58 × 10−4	0.08983	1.10 × 10−4
7	0.1678	0.757	0.127025	0.75709	9.20 × 10−5	0.12704	2.00 × 10−5
8	0.2132	0.757	0.161392	0.75614	8.59 × 10−4	0.16121	1.80 × 10−4
9	0.2545	0.7555	0.192275	0.75509	4.13 × 10−4	0.19217	1.10 × 10−4
10	0.2924	0.754	0.22047	0.75366	3.36 × 10−4	0.22037	1.00 × 10−4
11	0.3269	0.7505	0.245338	0.75139	8.91 × 10−4	0.24563	2.90 × 10−4
12	0.3585	0.7465	0.26762	0.74735	8.54 × 10−4	0.26793	3.10 × 10−4
13	0.3873	0.7385	0.286021	0.74012	1.62 × 10−3	0.28665	6.30 × 10−4
14	0.4137	0.728	0.301174	0.72738	6.18 × 10−4	0.30092	2.60 × 10−4
15	0.4373	0.7065	0.308952	0.70697	4.73 × 10−4	0.30916	2.10 × 10−4
16	0.459	0.6755	0.310055	0.67528	2.20 × 10−4	0.30995	1.00 × 10−4
17	0.4784	0.632	0.302349	0.63076	1.24 × 10−3	0.30175	5.90 × 10−4
18	0.496	0.573	0.284208	0.57193	1.07 × 10−3	0.28368	5.30 × 10−4
19	0.5119	0.499	0.255438	0.49961	6.07 × 10−4	0.25575	3.10 × 10−4
20	0.5265	0.413	0.217445	0.41365	6.49 × 10−4	0.21779	3.40 × 10−4
21	0.5398	0.3165	0.170847	0.31751	1.01 × 10−3	0.17139	5.50 × 10−4
22	0.5521	0.212	0.117045	0.21215	1.55 × 10−4	0.11713	9.00 × 10−5
23	0.5633	0.1035	0.058302	0.10225	1.25 × 10−3	0.0576	7.00 × 10−4
24	0.5736	−0.0100	−0.005736	−0.00872	1.28 × 10−3	−0.005	7.40 × 10−4
25	0.5833	−0.1230	−0.071746	−0.12551	2.51 × 10−3	−0.07321	1.46 × 10−3
26	0.59	−0.2100	−0.1239	−0.20847	1.53 × 10−3	−0.123	9.00 × 10−4

, the calculated data of the SDM obtained by the proposed MECPO technique are compatible with the measured data.

Figure 10 displays the individual absolute error for the current and power of the SDM using the proposed technique. Furthermore, the characteristics curve of I–V and P–V for SDM is redrawn according to the best-optimized parameters obtained by executing the proposed MECPO algorithm and presented in Figure 11. This figure confirms that our algorithm could significantly estimate the parameter values that could concretely predict the curve of experimental data.

5.2. Case 2: Double-Diode Model (DDM)

In the second case,

Table 9. Calculated parameter in case of the DDM obtained by the proposed algorithm and other recent techniques.

Algorithm	MECPO	ECPO	COOT	GPC	EO	MPA
$R_s$ (Ω)	0.03687	0.229884	0.0389	0.000206	0.037022	0.042933
$R_{sh}$(Ω)	57.12581	73.68355	72.85675	100	52.11317	37.18875
$I_{ph}$ (A)	0.760789	0.233682	0.760767	0.712533	0.76082	0.76005
$I_{sd1}$ (A)	3.07 × 10−6	0.433524	5.79 × 10−6	1.28 × 10−10	3.41 × 10−7	2.84 × 10−19
$I_{sd2}$ (A)	2.31 × 10−7	0.91553	3.76 × 10−8	1.47 × 10−5	1.36 × 10−9	5.55 × 10−8
$n_1$	2	1.932754	1.754744	1	1.228865	1.00292
$n_2$	1.190954	1.912091	1.073828	1.659771	1.000002	1.085403
$RMSE$	0.000977	1.515062	0.001115	0.041377	0.001001	0.003574

presents the best values of the control variables regarding the optimal performance for the compared techniques. As shown, MECPO reaches the lowest RMSE of 0.000977 compared to the other algorithms. According to the results of the proposed MECPO algorithm, the series resistance is 0.03687 Ω; shunt resistance is 57.12581 Ω; the photo-generated current is 0.760789 A; the reverse saturation currents of D1 and D2 are 3.07 and 0.231 µA, respectively; the ideality factor of the diode of D1 and D2 is 2.190954. Figure 12 shows the convergence characteristics curves of the algorithms and demonstrates that the capability of the MECPO technique to obtain the smallest RMSE is the fastest.

Apparently, it is seen in

Table 10. Statistical results of the MECPO technique and other recent techniques in the case of the DDM.

Algorithm	Best	Mean	Median	Worst	STD
MECPO	0.000977	0.00098	0.000977	0.000989	4.26 × 10−6
ECPO	1.515062	2.285728	2.343209	3.069671	0.404779
COOT	0.001115	0.004534	0.004441	0.009303	0.002359
GPC	0.041377	0.080475	0.072482	0.131687	0.023745
EO	0.001001	0.00141	0.001138	0.003264	0.000674
MPA	0.003574	0.011005	0.012211	0.012964	0.002805

The optimal values obtained are in bold.

that MECPO reaches the minimum STD of 4.26 × 10⁻⁶, which is less than ECPO, COOT, GPC, EO, and MPA that acquired STD of 0.404779, 0.002359, 0.023745, 0.000674, and 0.002805, respectively. Figure 13 shows the box plot of the RMSE values for DDM in 20 individual runs using the MECPO and other recent algorithms.

Table 11. Experimental and simulated data of voltages, currents, and power and the absolute errors values using MECPO for DDM.

Rank	Experimental Data			Simulated Current Data		Simulated Power Data
	V (V)	I (A)	P (W)	Isim (A)	IAEI (A)	Psim (W)	IAEP (W)
1	−0.2057	0.764	−0.157155	0.7639	1.00 × 10−4	−0.15713	2.10 × 10−5
2	−0.1291	0.762	−0.098374	0.76256	5.60 × 10−4	−0.09845	7.20 × 10−5
3	−0.0588	0.7605	−0.044717	0.76133	8.30 × 10−4	−0.04477	4.90 × 10−5
4	0.0057	0.7605	0.004335	0.7602	3.00 × 10−4	0.004333	2.00 × 10−6
5	0.0646	0.76	0.049096	0.75916	8.40 × 10−4	0.049041	5.50 × 10−5
6	0.1185	0.759	0.089942	0.75819	8.10 × 10−4	0.089845	9.60 × 10−5
7	0.1678	0.757	0.127025	0.75726	2.60 × 10−4	0.127069	4.40 × 10−5
8	0.2132	0.757	0.161392	0.75632	6.80 × 10−4	0.161246	1.46 × 10−4
9	0.2545	0.7555	0.192275	0.75523	2.70 × 10−4	0.192207	6.80 × 10−5
10	0.2924	0.754	0.22047	0.75375	2.50 × 10−4	0.220396	7.40 × 10−5
11	0.3269	0.7505	0.245338	0.75139	8.90 × 10−4	0.245628	2.90 × 10−4
12	0.3585	0.7465	0.26762	0.74725	7.50 × 10−4	0.267889	2.69 × 10−4
13	0.3873	0.7385	0.286021	0.73993	1.43 × 10−3	0.286576	5.55 × 10−4
14	0.4137	0.728	0.301174	0.72716	8.40 × 10−4	0.300827	−3.47 × 10−4
15	0.4373	0.7065	0.308952	0.70678	2.80 × 10−4	0.309076	1.24 × 10−4
16	0.459	0.6755	0.310055	0.67518	3.20 × 10−4	0.309908	1.47 × 10−4
17	0.4784	0.632	0.302349	0.63077	1.23 × 10−3	0.301761	5.87 × 10−4
18	0.496	0.573	0.284208	0.57204	9.60 × 10−4	0.283731	4.77 × 10−4
19	0.5119	0.499	0.255438	0.49976	7.60 × 10−4	0.255828	3.90 × 10−4
20	0.5265	0.413	0.217445	0.41378	7.80 × 10−4	0.217853	4.08 × 10−4
21	0.5398	0.3165	0.170847	0.31756	1.06 × 10−3	0.171419	5.72 × 10−4
22	0.5521	0.212	0.117045	0.2121	1.00 × 10−4	0.117101	5.60 × 10−5
23	0.5633	0.1035	0.058302	0.10212	1.38 × 10−3	0.057522	7.80 × 10−4
24	0.5736	−0.0100	−0.005736	−0.00883	1.17 × 10−3	−0.00507	6.70 × 10−4
25	0.5833	−0.1230	−0.071746	−0.12556	2.56 × 10−3	−0.07324	1.49 × 10−3
26	0.59	−0.2100	−0.1239	−0.20832	1.68 × 10−3	−0.12291	9.90 × 10−4

shows the values of measured, simulated current and the absolute errors between them (IAE_I). Furthermore, it displays the values of measured, simulated power, and the absolute errors between them (IAE_P) when applying the proposed MECPO on DDM. Moreover, Figure 14 illustrates the absolute errors of measured and simulated current and the absolute errors of experimental and simulated power using the MECPO for the DDM. Figure 15 presents the estimated parameters obtained by the proposed algorithm for the DDM that lead to the high closeness between the estimated I–V and P–V characteristics.

5.3. Case 3: Triple-Diode Model (TDM)

In this case, in the case of the TDM, RMSE and the extracted nine parameters’ values are displayed in

Table 12. Calculated parameter in case of the TDM obtained by the proposed algorithm and other recent techniques.

Algorithm	MECPO	ECPO	COOT	GPC	EO	MPA
$R_s$ (Ω)	0.03687	0.04303	0.040051	0.001854	0.037781	0.03846
$R_{sh}$(Ω)	57.12579	100	99.89583	10.52881	54.91819	85.03041
$I_{ph}$ (A)	0.760789	0.721768	0.760434	0.776297	0.760872	0.76019
$I_{sd1}$ (A)	6.97 × 10−20	0.462181	9.59 × 10−9	1.23 × 10−5	6.19 × 10−7	7.53 × 10−8
$I_{sd2}$ (A)	3.07 × 10−6	0.288811	5.14 × 10−6	1 × 10−9	8.63 × 10−9	1.4 × 10−8
$I_{sd3}$ (A)	2.31 × 10−7	0.992376	2.08 × 10−7	1 × 10−9	9.67 × 10−9	1.85 × 10−6
$n_1$	1.220835	2	1.000105	1.650069	1.322086	1.982829
$n_2$	2	1.886728	1.664525	1	1.836014	1.030268
$n_3$	1.190954	2	1.999564	1	1.031254	1.486103
$RMSE$	0.000977	1.360341	0.001311	0.027075	0.001013	0.001181

. It can be clearly seen from this table that the MECPO algorithm achieves the optimal RMSE value (0.00097) among the five algorithms; further, the EO algorithm acquires the second-best RMSE value (0.001031), followed by MPA, COOT, GPC, and ECPO. Figure 16 presents the convergence curve for this case study using MECPO and other algorithms for the TDM. It is obvious that MECPO has a faster convergence rate than other techniques for single-, double-, and triple-diode models.

Table 13. Statistical results the MECPO technique and other recent algorithms in the case of the TDM.

Algorithm	Best	Mean	Median	Worst	STD
MECPO	0.000977	0.000979	0.000977	0.000997	5.1 × 10−6
ECPO	1.360341	1.978315	2.019703	2.895647	0.551247
COOT	0.001311	0.004247	0.004408	0.00884	0.001786
GPC	0.027075	0.101053	0.078046	0.410851	0.085461
EO	0.001013	0.001631	0.001267	0.003046	0.000735
MPA	0.001181	0.009682	0.011312	0.012131	0.003687

The optimal values obtained are in bold.

gives the statistical results for the TDM. We can conclude that our algorithm can achieve the best RMSE statistical values (Min, Mean, Median, Max, and STD) in the case of the TDM compared with the other recent techniques. Moreover, to display the distribution results obtained from various algorithms, the boxplot of TDM is presented in Figure 17. It can be seen that MECPO achieves superior performance in terms of precision and robustness. The statistical results of the IAE values based on the current and power for the measured and simulated data on TDM are tabulated in

Table 14. Measured and simulated data of voltages, currents, and power and the absolute errors values using MECPO for TDM.

Rank	Experimental Data			Simulated Current Data		Simulated Power Data
	V (V)	I (A)	P (W)	Isim (A)	IAEI (A)	Psim (W)	IAEP (W)
1	−0.2057	0.764	−0.157155	0.7639	8.30 × 10−4	−0.15713	2.00 × 10−5
2	−0.1291	0.762	−0.098374	0.76256	3.00 × 10−4	−0.09845	7.00 × 10−5
3	−0.0588	0.7605	−0.044717	0.76133	8.40 × 10−4	−0.04477	5.00 × 10−5
4	0.0057	0.7605	0.004335	0.7602	8.10 × 10−4	0.00433	0.00 × 10+00
5	0.0646	0.76	0.049096	0.75916	2.60 × 10−4	0.04904	5.00 × 10−5
6	0.1185	0.759	0.089942	0.75819	6.80 × 10−4	0.08985	1.00 × 10−4
7	0.1678	0.757	0.127025	0.75726	2.70 × 10−4	0.12707	4.00 × 10−5
8	0.2132	0.757	0.161392	0.75632	2.50 × 10−4	0.16125	1.50 × 10−4
9	0.2545	0.7555	0.192275	0.75523	8.90 × 10−4	0.19221	7.00× 10−5
10	0.2924	0.754	0.22047	0.75375	7.50 × 10−4	0.2204	7.00× 10−5
11	0.3269	0.7505	0.245338	0.75139	1.43 × 10−3	0.24563	2.90× 10−4
12	0.3585	0.7465	0.26762	0.74725	8.40 × 10−4	0.26789	2.70× 10−4
13	0.3873	0.7385	0.286021	0.73993	2.80 × 10−4	0.28658	5.50× 10−4
14	0.4137	0.728	0.301174	0.72716	3.20 × 10−4	0.30083	3.50× 10−4
15	0.4373	0.7065	0.308952	0.70678	1.23 × 10−3	0.30908	1.20× 10−4
16	0.459	0.6755	0.310055	0.67518	9.60 × 10−4	0.30991	1.50× 10−4
17	0.4784	0.632	0.302349	0.63077	7.60 × 10−4	0.30176	5.90× 10−4
18	0.496	0.573	0.284208	0.57204	7.80 × 10−4	0.28373	4.80× 10−4
19	0.5119	0.499	0.255438	0.49976	1.06 × 10−4	0.25583	3.90× 10−4
20	0.5265	0.413	0.217445	0.41378	1.00 × 10−4	0.21785	4.10× 10−4
21	0.5398	0.3165	0.170847	0.31756	1.38 × 10−4	0.17142	5.70× 10−4
22	0.5521	0.212	0.117045	0.2121	1.17 × 10−4	0.1171	6.00× 10−5
23	0.5633	0.1035	0.058302	0.10212	2.56 × 10−4	0.05752	7.80× 10−4
24	0.5736	−0.0100	−0.005736	−0.00883	1.68 × 10−3	−0.00507	6.70× 10−4
25	0.5833	−0.1230	−0.071746	−0.12556	8.30 × 10−4	−0.07324	1.49× 10−3
26	0.59	−0.2100	−0.1239	−0.20832	3.00 × 10−4	−0.12291	9.90× 10−4

. In addition, the IAEs for current and power using MECPO for TDM are displayed in Figure 18.

The I–V and P–V characteristics achieved by the proposed technique between the measured data and simulated data are shown in Figure 19. The results obviously show that the simulated data achieved by the MECPO algorithm extremely coincide with the measured data in the different PV models, which means that the identified parameters of the proposed MECPO are highly precise. The above-mentioned comparisons illustrate that the MECPO algorithm has better searching precision, reliability, and more speed convergence rate for determining the parameters extraction problems of various PV models (SDM, DDM, TDM), and its performance is outstanding or competitive in contrast with all other recent techniques.

6. Conclusions

This paper has proposed an improved version of the ECPO called MECPO to solve global optimization problems well. This modification has been implemented to escape dropping on the local minima in the conventional ECPO. Therefore, the convergence speed is improved. First, the proposed MECPO’s effectiveness has been tested on the different functions of the CEC’17 test suite. The MECPO achieved better performance compared with seven recent optimization algorithms, including GSA, GWO, WOA, SCA, EO, HHO, and the original ECPO. Following that, the suggested strategy based on MECPO has been employed to extract the many parameters of three models of PV cells. An evaluation study is used to test the ability of the MECPO compared with the other optimizers. The comparative study was carried out for the same data set. The statistical results were employed to analyze the efficacy of the MECPO technique. The greatest proximity between the simulated P–V and I–V curves was obtained by the MECPO algorithm in comparison with the measured data. Furthermore, the proposed technique has strong performance and better convergence rates for all tested cases. In future work, the uncertainty of the climate condition as well as the shading effect will be incorporated in the PV models. Additionally, the proposed algorithm is planned to be applied to other fields of energy such as maximum power point tracking (MPPT) and the energy scheduling problem of PV systems.