Parameter Estimation of Different Photovoltaic Models Using Hybrid Particle Swarm Optimization and Gravitational Search Algorithm

Abstract

The performance of a typical solar energy-based system can be improved by accurately modeling the current versus voltage characteristics of the involved solar cells. However, estimating the exact value of parameters related to solar cells is quite challenging. The optimization function, considering the current–voltage characteristics of solar cells, requires the solution of sophisticated non-linear and multi-modal optimization methods. So far, various optimization approaches have been reported. This paper proposes the application of a new hybrid algorithm, i.e., Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA), which is a combination of two algorithms, i.e., Gravitational Search Algorithm (GSA) and Particle Swarm Optimization (PSO) method. The hybrid PSOGSA algorithm is superior to other algorithms in terms of higher accuracy in searching for optimal solutions and better explorative capability. Moreover, the developed hybrid algorithm is benchmarked using ten standard test functions to verify its efficiency. In this manuscript, monocrystalline and polycrystalline solar cells are considered. The parameter optimization results are obtained using PSOGSA and further compared with those obtained using other algorithms presented in the literature, such as PSO, GSA, MVO, HBO, PO and SCA. The complete error analysis is carried out for the modified single-diode model (MSDM), the modified double-diode model (MDDM), and the modified three-diode model (MTDM) of photovoltaic (PV) cells to prove the superiority of the PSOGSA. Moreover, statistical results are carried out based on Friedman’s ranking and Wilcoxon’s rank sum test. The comparison results show that the proposed PSOGSA is better than other algorithms in estimating the unknown PV model parameters.

1. Introduction

1.1. Literature Review

Solar photovoltaic (PV) systems are eco-friendly and help to minimize the greenhouse gas (GHG) emission gases that would otherwise have emerged from the use of fossil fuels for generating electricity. Since solar energy is available in abundance, feeding electricity into the grid through solar PV plants can help in transmitting the much larger amounts of power that might be made from the GHG-intensive network. The increased cost of the PV modules restricts the optimum and maximum utilization of the available renewable energy. Thus, the system’s reliability and accuracy in simulated environments must be predicted before the system is actually installed.

Modeling solar cells is essential to predict the behavior of the actual solar cell under different environmental conditions and further acquire its characteristic curves; the current–voltage and power–voltage curves. The viable method is to use the analogous electrical circuit, which is based entirely on a source of light-generated current connected to a p–n junction diode. Many frameworks have been developed at different solar intensities and temperature conditions for the simulation of a single solar cell or a complete PV system [1,2,3,4]. Thus, effective and precise solar cell modeling is very crucial to predict its behavior under different atmospheric conditions. Effective modeling of the non-linear characteristics of the PV systems is very beneficial from the point of accurate simulation [5]. In literature, there are various types of stochastic approaches described in the past few decades, such as single-diode, two-diode, and three-diode models, designs with partial shading issues, and many more.

The mathematical model of I-V characteristics of PV cells contains a large number of unknown factors. The single and double-diode models are the leading theoretical frameworks for the precise modeling of the system [6]. The single-diode structure is represented by its simplicity and highly acceptable accuracy [7,8]. The I-V performance feature of the single-diode model has five unknown parameters, i.e., PV current, diode saturation current, ideality factor, parallel and series resistances [9]. This model, however, lacks precision at low irradiance open-circuit voltage as the carrier recombination losses in the depletion region seem to be ignored [10,11]. To overcome this problem, the double-diode model was introduced. In this model, to represent the recombination losses, an extra diode is used. This model shows better accuracy than the single-diode model. However, it is more complicated than a single-diode model because seven parameters need to be evaluated in the two-diode model in contrast to five parameters in a single-diode model [11,12,13]. The three-diode model exhibits better accuracy than the two-diode model, which affects the leakage current and grain boundaries [14]. The three-diode model involves the estimation of nine parameters which is somewhat more complicated than the two-diode model [15].

Hybridizing stochastic optimization algorithms is one of the best ways to develop superior algorithms and exploit numerous algorithm advantages for solving optimization problems [16,17,18,19,20]. In 2010, the PSOGSA algorithm was proposed, which is a novel hybrid of the PSO technique and GSA algorithm [21]. This algorithm shows a better outcome in solving optimization problems. In this algorithm, agents perform the search process, mimicking GSA behavior in the phase of exploration, and PSO in the exploitation phase. This algorithm was initially introduced to mitigate the GSA’s slow exploitation and also mitigate the PSO algorithm’s nature to get struck in local minima, which are their key drawbacks. Many studies have been undertaken to improve the performance of the GSA algorithm as well as the PSO algorithm. For solving the clustering problem, the hybrid PSOGSA proved to be the better method for solving the heuristic problems [22].

Many researchers are experimenting with different applications by developing new algorithms such as the Skill optimization algorithm [23], Equilibrium optimization [24] and so on. In this manuscript, an engineering application is presented using a hybrid algorithm, that is, the Particle Swarm Optimization and Gravity Search Algorithm (PSOGSA), to estimate the parameters of monocrystalline solar cells and polycrystalline cells. The results obtained using PSOGSA are compared with those obtained using Particle Swarm Optimization (PSO) [25], Gravitational Search Algorithm (GSA) [26], Multi-Verse Optimizer (MVO) [27], Sine Cosine Algorithm (SCA) [28], Heap Based Optimizer (HBO) [29], and Political Optimizer (PO) [30]-based techniques to check the effectiveness of the PSOGSA.

1.2. Paper Contribution

The main contributions of this work are summarized as follows:

• A new hybrid (PSOGSA) algorithm is proposed to estimate the unknown parameters of a single, double and triple-diode model for two different solar PV cells.
• Ten benchmark test functions are selected to justify the performance of the proposed hybrid algorithm.
• Different objective functions are implemented for extracting the solar PV cell parameters such as SSE, AE, MAE, MSE and RMSE functions.
• Comprehensive statistical analysis and comparisons of the hybrid algorithm with other metaheuristic optimization algorithms.
• Focusing on solution consistency results, I-V and P-V characteristic curves, non-parametric tests are also implemented.

1.3. Paper Organization

This work is arranged as follows: Section 2 presents the mathematical modeling of different PV cells. In Section 3, the problem formulation is proposed. Section 4 introduces the suggested hybrid PSOGSA algorithm and its steps. Section 5 depicts the simulation results, comparisons and statistical metrics. Finally, Section 6 draws conclusions and recommendations for further work.

2. Mathematical Modeling of PV Cell

The modeling of solar PV cells can be accomplished by a comprehensive study of mathematical equations based on the respective analogous circuits of solar PV models described in the following subsections [31,32,33,34].

2.1. Ideal Solar PV Cell Modeling

The ideal solar PV cells with photocurrent (I_ph) vary from the optimal outcome because of optical and electrical losses. Figure 1 represents a typical solar PV cell, which is the simplest PV model as the effect of parallel and series resistances is not reckoned.

The cell output is expressed by I-V characteristics and expressed in mathematical terms as follows:

$$I=I_{ph}-I_d$$

(1)

where I_d signifies the diode current, which represents the recombination and diffusion current in the quasi steady-state emitter region and PN junction region with excess concentration. This current in the diode is represented by the Shockley Equation as follows:

$$I_d=I_o\left(e^{\frac{V_d}{{NV}_t}}-1\right)$$

(2)

where I_o represents the saturation diode current, V_d is known as the diode voltage, V_t is the equivalent thermal voltage and N is the number of cells in series [35]. An ideal solar photovoltaic cell does not take into account the effects of internal resistance, and thus does not establish a stable relationship between the cell current and voltage.

2.2. Modified Equivalent Circuit of the Single-Diode PV Model

The ideal PV cell model can also be introduced by a series resistance to get accurate results. This model is simple because it reveals deficiencies when subjected to variations in temperature [36]. The refined version of the single-diode model (SDM) is the modified single-diode model (MSDM). In the MSDM, additional resistance is incorporated in series with the basic SDM which expresses the losses in the quasi-neutral region as shown in Figure 2. Equation (3) represents the modeling of a modified single-diode cell as follows:

$$I=I_{ph}-I_d\left(e^{\frac{V+I{R}_{se-I_d{R}_s}}{{NV}_T}}-1\right)-\frac{V+I{R}_{se}}{R_{sh}}$$

(3)

2.3. Modified Equivalent Circuit of the Double-Diode PV Model

The refined version of the double-diode model (DDM) is the modified double-diode model (MDDM). In the MDDM, additional resistance is incorporated in series with the basic DDM which represents that the resistance at grain boundaries is greater than resistance within crystallites. The accuracy of the modified single-diode model is uncertain; therefore, further investigation was carried out to search for more sophisticated models. A modified two-diode model is adopted to solve this problem, using two diodes attached parallel to the current source, as shown in Figure 3. The modified double-diode model is the modified version of the modified single-diode model, which takes the effect of recombination into account by introducing another diode in parallel.

In this model, two unknown diode-quality factors are introduced. As a result, there is an increase in the number of equations which makes calculations more complicated [37]. Hence, at a low insolation level, this model exhibits better accuracy. Equation (4) represents the performance equation of the modified double-diode cell model.

$$I=I_{ph}-I_d\left(e^{\frac{V+I{R}_{se}}{{VN}_T}}-1\right)-I_{d1}\left(e^{\frac{V+I{R}_{se}-I_{d1}{R}_s}{{NV}_T}}-1\right)-\frac{V+I{R}_{se}}{R_{sh}}$$

(4)

Therefore, considering all aspects of mathematical calculation, and the number of iterations, the modified single-diode model tends to have quick results due to fewer complex calculations.

2.4. Modified Equivalent Circuit of the Triple-Diode PV Model

The refined version of the triple-diode model (TDM) is the modified triple-diode model (MTDM). In the MTDM, additional resistance is incorporated in series with the basic TDM. Moreover, the losses in the defect area are represented by adding a modified series resistance i.e., R_s. Figure 4 represents the modified three-diode equivalent circuit model for a PV cell in which there are three diodes. The effect of the leakage current and grain boundaries represented by I_d3 has been taken into account in this model. In this model, the current through the shunt resistance branch is the leakage current. The semiconductor material resistance signifies a series resistance in the natural region of the solar cell [15]. Equation (5) represents the performance equation of the modified three-diode equivalent model of the solar PV cell.

$$I=I_{ph}-I_d\left(e^{\frac{V+I{R}_{se}}{{NV}_T}}-1\right)-I_{d1}\left(e^{\frac{V+I{R}_{se}}{{NV}_T}}-1\right)-I_{d2}\left(e^{\frac{V+I{R}_{se}-I_{d2}{R}_s}{N{V}_T}}-1\right)-\frac{V+I{R}_{se}}{R_{sh}}$$

(5)

3. Problem Formulation

This section presents the problem formulation that has been implemented to extract the parameters of different PV solar cells. To evaluate the characteristics of the solar cells, different objective functions have been carried out in this paper such as: the sum of square error (SSE), absolute error (AE), mean absolute error (MAE), mean square error (MSE) and the root mean square error (RMSE). However, RMSE is the most common formula used to estimate the unknown parameters of the PV cells [9,15]. The RMSE between the measured and calculated output current of the solar cell can be calculated as follows:

$$RMSE=\sqrt{\frac{1}{P}{{\displaystyle \sum }}_{i=1}^P{(I_{meas,i}-I_{calc,i})}^2}$$

(6)

where $P$ represents the number of measured points, $I_{meas,i}$ and $I_{calc,i}$ represent the measured and calculated solar cell current at point $i$, respectively. Moreover, the other error functions can be calculated using the following equations:

$$MSE=\frac{1}{P}{{\displaystyle \sum }}_{i=1}^P{(I_{meas,i}-I_{calc,i})}^2$$

(7)

$$MAE =\frac{1}{P}{{\displaystyle \sum }}_{i=1}^P\left|I_{meas,i}-I_{calc,i}\right|$$

(8)

$$AE=\left|\frac{I_{meas,i}-I_{calc,i}}{I_{calc,i}}\right|$$

(9)

$$SSE={{\displaystyle \sum }}_{i=1}^P{\left(I_{meas,i}-I_{av}\right)}^2$$

(10)

where $I_{av}$ is the mean value of all the measured cell currents. The prime objective of the presented technique is to extract the parameters of the modified-diode model by minimizing the root mean square error. Hence, the proposed approach will be used on the basis of minimizing this objective function with respect to the parameter limits.

4. The Proposed Hybrid PSOGSA Algorithm

This manuscript has identified the supremacy of the hybrid PSOGSA algorithm compared to other algorithms such as PSO [30], GSA [31], MVO [32], SCA [33], HBO [34], and PO [35] techniques. Using the modified single-diode, the modified double-diode, and the modified three-diode identical models, these algorithms have been used to optimize the parameters of monocrystalline and polycrystalline PV cells.

In general, the hybrid PSOGSA combines the exploitation ability of the PSO algorithm and the exploration ability of the GSA algorithm to obtain optimal results more effectively. This provides a better balance between the two algorithms for an optimal solution. Mirjalili and Mohd Hashim first introduced the hybrid PSOGSA algorithm [21]. The best solution achieved using the PSO algorithm is saved for further exploration in the GSA algorithm. The purpose of using the best solution is that GSA suffers from regression and exploitation delays in the initial iterations [38]. The PSOGSA has been shown to be effective enough to find the optimal solution compared to the PSO and GSA techniques. However, the primitive hybrid algorithm PSOGSA is a continuous algorithm that is not explicitly capable of solving binary problems. The flowchart of the hybrid PSOGSA algorithm is represented in Figure 5 and the steps are described below.

Step 1: In this step, the acceleration and position of all the particles are first initialized.

Step 2: According to Equation (11), the fitness values of all particles are found and updated for their constant gravity. In Equation (11), G_t and G_o stand for the gravitational constant of the universe at the moment t and t_o, respectively_. T stands for the maximum number of iterations.

$$G_t=G_o·e^{\frac{-\alpha t}{T}}$$

(11)

Step 3: In this step, the quality of the particle is obtained according to Equations (12)–(14). Using Equations (15)–(17), the acceleration of the particle is found. Lit stands for the fitness value in Equations (14) and (15).

$$n_j\left(t\right)=\frac{Li{t}_j\left(t\right)-worst\left(t\right)}{best\left(t\right)-worst\left(t\right)}$$

(12)

$$N_j\left(t\right)=\frac{n_j\left(t\right)}{{{\displaystyle \sum }}_N{n}_i\left(t\right)}$$

(13)

$$best\left(t\right)=ma{x}_{j=\left(1,2,\dots n\right)}Li{t}_j\left(t\right)$$

(14)

$$worst\left(t\right)=mi{n}_{j=\left(1,2,\dots n\right)}Li{t}_j\left(t\right)$$

(15)

F = G (M₁M₂/R)

(16)

a = F/M

(17)

Equation (16) obeys the law of gravity, and Equation (17) follows the law of motion. The maximum number of iterations is set to 1000 whereas the search agents are kept at 30 for all the algorithms.

Step 4: In this step, the position and velocity of the particle are calculated, and updated according to the position and velocity in Equation (19).

$$V_{di}\left(t+1\right)=\omega ·V_{di}\left(t\right)+M_1·rand()·\left(P_{di}-X_{di}\right)+M_2·rand()·\left(P_{dg}-X_{di}\right)$$

(18)

X_di (t + 1) = V_di (t + 1) + X_di (t)

(19)

where M₁ and M₂ are two positive constants of acceleration; rand () is a uniform random number (0, 1); P_di and P_dg are the best positions so far found by the ith particle and all the particles, respectively; t is the iteration count; and $\omega$ is an inertial weight that usually decreases linearly during the iterations. The weight of inertia is used to balance local and global searches [39].

Step 5: This is the last step of the algorithm. If the system has met the criteria, then the output is optimum; otherwise go back to step 2.

5. Results and Discussion

5.1. Benchmark Test Functions

In this section, the developed hybrid algorithm (PSOGSA) is benchmarked using ten standard test functions to verify its efficiency. The characteristics of these benchmark test functions are summarized in

Table 1. Definitions of benchmark functions.

Test Function	n	Range
$F_1\left(x\right)={\displaystyle \sum _{i=1}^n}{x}_i^2$	30	[−100, 100]
$F_2\left(x\right)={\displaystyle \sum _{i=1}^n}\left|x_i\right|+{\displaystyle \prod _{i=1}^n}\left|x_i\right|$	30	[−10, 10]
$F_3\left(x\right)={\displaystyle \sum _{i=1}^n}{\left({\displaystyle \sum _{j=1}^i}{x}_i\right)}^2$	30	[−100, 100]
$F_4\left(x\right)=max\left|x_i\right|, 1 \le x \le n$	30	[−100, 100]
$F_5\left(x\right)={\displaystyle \sum _{i=1}^{n-1}}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right]$	30	[−30, 30]
$F_6\left(x\right)={\displaystyle \sum _{i=1}^n}{\left(x_i+0.5\right)}^2$	30	[−100, 100]
$F_7\left(x\right)={\displaystyle \sum _{i=1}^n}i{x}_i^4+random\left[0,1\right]$	30	[−1.28, 1.28]
$F_8\left(x\right)={\displaystyle \sum _{i=1}^n}\left[-x_i\sin \left(\sqrt{\left|x_i\right|}\right)\right]$	30	[−500, 500]
$F_9\left(x\right)=10n+{\displaystyle \sum _{i=1}^n}{x}_i^2-10\cos \left(2\pi x_i\right)$	30	[−5.12, 5.12]
$F_{10}\left(x\right)=-20exp\left(-0.2\sqrt{\frac{1}{n} {\displaystyle \sum _{i=1}^n}{x}_i^2}\right)-exp\left(\frac{1}{n} {\displaystyle \sum _{i=1}^n}\cos \left(2\pi x_i\right)\right)+20+e$	30	[−32, 32]

[40,41]. The functions $F_1\left(x\right)$ to $F_7\left(x\right)$ are unimodal functions that are mainly used to investigate the convergence characteristics and optimization accuracy of a given algorithm. Moreover, the functions $F_8\left(x\right)$ to $F_{10}\left(x\right)$ are multimodal functions that are mainly used to check whether an algorithm can avoid precocity and find global optimal solutions [40,41]. Each function dimension (n) is set to 30 to avoid random interference.

For performance verification, the hybrid PSOGSA algorithm is compared with other optimization algorithms such as MVO, PSO, SCA, HBO, PO and GSA. The results of the benchmark functions are summarized in

Table 2. Results of benchmark test functions for different optimization algorithms.

	MVO		PSO		SCA		HBO		PO		GSA		PSOGSA
	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std
F1	0.203	0.135	0.0007	0.002	0.137	0.005	0.039	0.025	0.0022	0.014	0.2619	0.4904	4.74 × 10−12	2.65 × 10−11
F2	2.36	0.875	0.0620	0.0959	1.44	0.082	0.987	0.078	0.0044	0.013	0.7906	0.6496	3.89 × 10−6	1.46 × 10−5
F3	425.1	2597.2	0.0090	0.0547	265.8	0.105	90.86	115.4	392.35	2382.2	36.73	131.73	1.21 × 10−8	1.08 × 10−7
F4	1.68	1.87	0.0312	0.0732	0.753	0.874	0.561	0.521	0.3785	1.22	0.747	2.7602	1.07 × 10−5	4.42 × 10−5
F5	30.5	287.1	26.489	264.96	25.71	103.4	22.41	70.56	20.47	85.8	18.87	34.23	4.56	8.01
F6	0.001	0.0124	0.0006	0.0023	0.0025	0.0078	0.0036	0.0016	0.0019	0.0164	0.2884	0.054	6.79 × 10−12	5.54 × 10−11
F7	0.126	0.102	0.086	0.0239	0.0987	0.098	0.084	0.088	0.074	0.035	0.0605	0.042	0.0078	0.0269
F8	−2653.1	1648.2	−1815.6	1520.19	−1527.4	1571.3	−1312.9	1420.7	−1297.1	1350.2	−1634.1	976.14	−86.3	66.05
F9	35.6	38.3	8.026	19.61	20.4	30.78	16.45	25.12	8.35	30.21	20.47	22.41	7.93	22.31
F10	1.23	7.45	1.441	5.83	0.987	1.25	0.786	0.987	0.762	0.253	0.9115	0.7854	2.2 × 10−6	9.27 × 10−6

based on the mean value and standard deviation (std) values for each function. It can be observed from Table 2 that the hybrid PSOGSA algorithm is superior to other algorithms in terms of higher accuracy in searching for optimal solutions and better explorative capability.

5.2. Parameter Estimation Results

Two solar cells are used to estimate the parameters i.e., Monocrystalline and Polycrystalline, as shown in

Table 3. Datasheet for two different types of PV cells.

Company	H & T Gmbh	Solar World
Model	TS265D60	Pro.SW255
Cell Type	Monocrystalline	Polycrystalline
Vm [V]	30.90	30.90
Im [A]	8.58	8.32
Voc [V]	38.10	38.00
Isc [A]	9.19	8.88
Ns [Cells]	60	60
T [°C]	25	25

. In this section, the two cases are discussed in which the modified single-diode, the modified double-diode and the modified three-diode model parameter estimations have been carried out.

Each algorithm is set to a maximum number of 1000 iterations and with 40 search agents using MATLAB version 19b. All algorithms are repeated 30 times and the result of each is recorded. After estimating the parameters based on different errors, the statistical results such as the mean and standard deviation are calculated and compared with those obtained using different algorithms for 30 error values. The standard deviation can be calculated using Equation (20) as follows:

$$std. dev=\sqrt{\frac{{\displaystyle {\sum }_{i=1}^m}{\left(e_i-e_{av}\right)}^2}{m-1}}$$

(20)

where $m$ is the number of error values ($m=30$), and $e_i$ and $e_{av}$ represent the error calculated using each algorithm and the mean error value, respectively.

It has been observed that the developed hybrid PSOGSA algorithm gives better results compared to those obtained with the other algorithms considered in this work. Figure 6 and Figure 7 show the I-V and P-V curves at standard temperature conditions. After calculating the error, the non-parametric test is also conducted. The first non-parametric test performed is the Wilcoxon rank sum test which is explained in detail in [42,43,44]. After this test, the Friedman rank test is performed to see how accurate the newly developed hybrid algorithm is better than the other comparative algorithms. For parameter estimation of different types of the PV diode modeling, two cases are considered below:

Case 1: The solar cell is considered monocrystalline in this case. This case is further divided into three parts as follows:

• Modified Single-Diode Model (MSDM):

  Table 4. The monocrystalline MSDM parameter estimation results.

  Parameter/Algorithm	MVO	PSO	SCA	HBO	PO	GSA	PSOGSA
  Ipv	9.1903	9.3574	9.1752	9.1906	9.1550	5.6203	9.1908
  Alpha1	1.5430	1.3092	1.4853	15531	1.3825	1.3277	1.5429
  Rse	0.0301	0.0153	0.0081	0.0208	0.0325	0.2558	0.0331
  Rsh	291.508	71.542	219.539	250.550	290.002	274.016	320.478
  Io1	1.00 × 10−6	2.51 × 10−7	6.00 × 10−7	1.00 × 10−6	1.00 × 10−6	4.93 × 10−7	1.00 × 10−6
  Rs	0.0201	0.0253	0.0075	0.0107	0.0235	0.2485	0.0215
  Computation time per run (s)	2.157	3.548	3.479	2.987	2.147	1.589	1.658

  shows the results of parameter estimation using different optimization algorithms. Figure 8 represents the MSDM convergence curve. From this curve, we conclude that the efficiency of the proposed algorithm is better than the rest of the comparative algorithms. The errors, mean, and standard deviation of the MSDM circuit equivalent of a PV cell are shown in

  Table 5. The MSDM calculation error, mean and standard deviation for the monocrystalline PV cell.

  Algorithm		SSE	AE	MAE	MSE	RMSE
  GSA	Minimum	1.31 × 10−7	3.60 × 10−4	1.81 × 10−5	6.55 × 10−9	8.09 × 10−5
  	Maximum	7.87 × 10−7	8.84 × 10−4	4.43 × 10−5	3.93 × 10−8	1.96 × 10−4
  	Mean	4.32 × 10−7	6.57 × 10−4	3.28 × 10−5	2.16 × 10−8	1.40 × 10−4
  	Standard deviation	2.28 × 10−7	4.72 × 10−4	2.38 × 10−5	1.14 × 10−8	1.02 × 10−4
  MVO	Minimum	1.09 × 10−6	1.01 × 10−3	5.22 × 10−5	5.46 × 10−8	2.37 × 10−4
  	Maximum	8.41 × 10−6	2.95 × 10−3	1.43 × 10−4	4.21 × 10−7	6.46 × 10−4
  	Mean	3.74 × 10−6	1.96 × 10−3	9.66 × 10−5	1.87 × 10−7	4.38 × 10−4
  	Standard deviation	2.26 × 10−6	1.52 × 10−3	7.52 × 10−5	1.13 × 10−7	3.33 × 10−4
  PSO	Minimum	1.18 × 10−1	3.44 × 10−1	1.71 × 10−2	5.90 × 10−3	7.68 × 10−2
  	Maximum	7.85 × 10−1	8.86 × 10−1	4.43 × 10−2	3.92 × 10−2	1.98 × 10−2
  	Mean	3.12 × 10−1	5.59 × 10−1	2.79 × 10−2	1.56 × 10−2	1.24 × 10−1
  	Standard deviation	1.93 × 10−1	4.39 × 10−1	2.19 × 10−2	9.61 × 10−3	9.83 × 10−2
  SCA	Minimum	1.07 × 10−3	3.27 × 10−2	1.64 × 10−3	5.35 × 10−5	7.37 × 10−3
  	Maximum	9.05 × 10−3	9.53 × 10−2	4.76 × 10−3	4.55 × 10−4	2.13 × 10−2
  	Mean	5.23 × 10−3	7.21 × 10−2	3.63 × 10−3	2.68 × 10−4	1.61 × 10−2
  	Standard deviation	2.31 × 10−3	4.89 × 10−2	2.47 × 10−3	1.16 × 10−4	1.09 × 10−2
  HBO	Minimum	1.02 × 10−3	3.26 × 10−2	1.61 × 10−3	5.35 × 10−5	7.36 × 10−3
  	Maximum	8.41 × 10−3	9.18 × 10−2	4.51 × 10−3	4.29 × 10−4	2.05 × 10−2
  	Mean	4.07 × 10−3	6.37 × 10−2	3.15 × 10−3	2.09 × 10−4	1.42 × 10−2
  	Standard deviation	2.29 × 10−3	4.73 × 10−2	2.32 × 10−3	1.14 × 10−4	1.05 × 10−2
  PO	Minimum	1.01 × 10−3	3.32 × 10−2	1.66 × 10−3	5.25 × 10−5	7.41 × 10−3
  	Maximum	7.72 × 10−3	8.77 × 10−2	4.38 × 10−3	3.86 × 10−4	1.96 × 10−2
  	Mean	3.53 × 10−3	5.97 × 10−2	2.99 × 10−3	1.77 × 10−4	1.33 × 10−2
  	Standard deviation	2.37 × 10−3	4.79 × 10−2	2.41 × 10−3	1.14 × 10−4	1.07 × 10−2
  PSOGSA	Minimum	1.49 × 10−21	3.85 × 10−11	1.92 × 10−12	7.43 × 10−23	8.61 × 10−12
  	Maximum	5.41 × 10−21	7.35 × 10−11	3.67 × 10−12	2.70 × 10−22	1.64 × 10−11
  	Mean	3.22 × 10−21	5.67 × 10−11	2.83 × 10−12	1.61 × 10−22	1.26 × 10−11
  	Standard deviation	1.24 × 10−21	3.52 × 10−11	1.76 × 10−12	6.22 × 10−23	7.88 × 10−12

  . Based on the results, it can be inferred that parameter estimation using the developed hybrid PSOGSA is superior to other comparative algorithms.
• Modified Double-Diode Model (MDDM): 

  Table 6. The monocrystalline MDDM parameter estimation results.

  Parameter/Algorithm	MVO	PSO	SCA	HBO	PO	GSA	PSOGSA
  Ipv	9.1834	9.4699	9.1982	9.2130	9.2808	4.5140	9.1905
  Alpha1	1.6664	1.0193	1.2959	1.4758	1.5994	1.0457	1.5969
  Alpha2	1.4637	0.8275	1.4142	1.4570	1.6487	1.2393	1.5713
  Rse	0.0275	0.0023	0.0349	0.0901	0.0010	0.3198	0.0233
  Rsh	330.334	99.411	251.018	294.329	275.913	272.241	306.862
  Io1	8.00 × 10−7	3.49 × 10−7	1.73 × 10−7	3.01 × 10−7	2.66 × 10−7	5.61 × 10−7	0
  Io2	0	1.81 × 10−7	4.73 × 10−7	4.18 × 10−7	7.47 × 10−7	5.80 × 10−7	7.00 × 10−7
  Rs	0.0155	0.0048	0.0585	0.0658	0.0032	0.2589	0.0325
  Computation time per run (s)	2.015	3.248	3.142	2.723	2.001	1.358	1.421

  shows the results of parameter estimation using different optimization algorithms. Figure 9 represents the convergence curve for MDDM. It was concluded that the efficiency of the proposed algorithm is better than the rest of the comparative algorithms. The errors, mean, and standard deviation of the MDDM circuit equivalent to a PV cell are shown in

  Table 7. The MDDM calculation error, mean and standard deviation for the monocrystalline PV cell.

  Algorithm		SSE	AE	MAE	MSE	RMSE
  GSA	Minimum	1.30 × 10−5	3.61 × 10−3	1.84 × 10−4	6.50 × 10−7	8.05 × 10−4
  	Average	3.52 × 10−5	5.95 × 10−3	2.94 × 10−4	1.76 × 10−6	1.32 × 10−3
  	Maximum	9.21 × 10−5	9.54 × 10−3	4.88 × 10−4	4.61 × 10−6	2.14 × 10−3
  	Mean	3.52 × 10−5	5.97 × 10−3	2.93 × 10−4	1.76 × 10−6	1.32 × 10−3
  	Standard deviation	2.09 × 10−5	4.45 × 10−3	2.27 × 10−4	1.05 × 10−6	1.02 × 10−3
  MVO	Minimum	1.34 × 10−4	1.15 × 10−2	5.71 × 10−4	6.59 × 10−6	2.51 × 10−3
  	Average	3.28 × 10−4	1.80 × 10−2	9.04 × 10−4	1.62 × 10−5	4.02 × 10−3
  	Maximum	5.71 × 10−4	2.39 × 10−2	1.19 × 10−3	2.86 × 10−5	5.34 × 10−3
  	Mean	3.26 × 10−4	1.80 × 10−2	9.02 × 10−4	1.62 × 10−5	4.01 × 10−3
  	Standard deviation	1.23 × 10−4	1.10 × 10−2	5.55 × 10−4	6.11 × 10−6	2.45 × 10−3
  PSO	Minimum	2.93 × 10−3	5.40 × 10−2	2.74 × 10−3	1.44 × 10−4	1.20 × 10−2
  	Average	7.56 × 10−2	2.75 × 10−1	1.37 × 10−2	3.72 × 10−3	6.14 × 10−2
  	Maximum	2.30 × 10−1	4.79 × 10−1	2.39 × 10−2	1.15 × 10−2	1.07 × 10−2
  	Mean	7.56 × 10−2	2.75 × 10−1	1.37 × 10−2	3.74 × 10−3	6.14 × 10−2
  	Standard deviation	6.41 × 10−2	2.53 × 10−1	1.26 × 10−2	3.26 × 10−3	5.66 × 10−2
  SCA	Minimum	2.88 × 10−3	5.35 × 10−2	2.64 × 10−3	1.47 × 10−4	1.19 × 10−2
  	Average	4.57 × 10−2	2.14 × 10−1	1.06 × 10−2	2.22 × 10−3	4.78 × 10−2
  	Maximum	9.79 × 10−2	3.13 × 10−1	1.56 × 10−2	4.91 × 10−3	6.99 × 10−2
  	Mean	4.57 × 10−2	2.14 × 10−1	1.06 × 10−2	2.28 × 10−3	4.78 × 10−2
  	Standard deviation	3.11 × 10−2	1.76 × 10−1	8.89 × 10−3	1.54 × 10−3	3.94 × 10−2
  HBO	Minimum	1.05 × 10−3	3.26 × 10−2	1.66 × 10−3	5.35 × 10−5	7.36 × 10−3
  	Average	4.02 × 10−3	6.37 × 10−2	3.17 × 10−3	2.04 × 10−4	1.42 × 10−2
  	Maximum	8.44 × 10−3	9.18 × 10−2	4.51 × 10−3	4.25 × 10−4	2.05 × 10−2
  	Mean	4.10 × 10−3	6.37 × 10−2	3.17 × 10−3	2.03 × 10−4	1.42 × 10−2
  	Standard deviation	2.26 × 10−3	4.73 × 10−2	2.34 × 10−3	1.11 × 10−4	1.05 × 10−2
  PO	Minimum	1.09 × 10−3	3.37 × 10−2	1.65 × 10−3	5.69 × 10−5	7.51 × 10−3
  	Average	3.28 × 10−3	5.67 × 10−2	2.81 × 10−3	1.66 × 10−4	1.26 × 10−2
  	Maximum	1.01 × 10−2	1.01 × 10−1	5.00 × 10−3	5.07 × 10−4	2.25 × 10−2
  	Mean	3.28 × 10−3	5.67 × 10−2	2.87 × 10−3	1.64 × 10−4	1.26 × 10−2
  	Standard deviation	2.39 × 10−3	4.83 × 10−2	2.45 × 10−3	1.15 × 10−4	1.08 × 10−2
  PSOGSA	Minimum	1.03 × 10−21	3.21 × 10−11	1.60 × 10−12	5.17 × 10−23	7.19 × 10−12
  	Average	4.01 × 10−21	6.33 × 10−11	3.16 × 10−12	2.01 × 10−22	1.41 × 10−11
  	Maximum	9.36 × 10−21	9.67 × 10−11	4.83 × 10−12	4.68 × 10−22	2.16 × 10−11
  	Mean	4.01 × 10−21	6.33 × 10−11	3.16 × 10−12	2.01 × 10−22	1.41 × 10−11
  	Standard deviation	2.27 × 10−21	4.76 × 10−11	2.38 × 10−12	1.13 × 10−22	1.06 × 10−11

  . Based on the results, it can be concluded that parameter estimation using the developed hybrid PSOGSA is superior to other algorithms.
• The Modified Triple-Diode Model (MTDM): 

  Table 8. The monocrystalline MTDM parameter estimation results.

  Parameter/Algorithm	MVO	PSO	SCA	HBO	PO	GSA	PSOGSA
  Ipv	9.2263	9.4328	9.1648	9.2497	9.2717	4.7787	9.1906
  Alpha1	1.4762	0.7379	1.3331	1.7173	1.6719	1.2696	1.6424
  Alpha2	1.5427	0.8855	1.0370	1.5788	1.4993	1.2829	1.4497
  Alpha3	1.6127	0.8509	1.0940	1.6819	1.4359	1.3352	1.5069
  Rse	0.0116	0.0187	0.0363	0.0881	0.0071	0.2747	0.0252
  Rsh	236.681	73.639	268.011	226.724	192.211	236.425	331.022
  Io1	5.00 × 10−7	1.38 × 10−7	1.62 × 10−7	3.89 × 10−7	4.07 × 10−7	3.95 × 10−7	6.50 × 10−7
  Io2	5.00 × 10−7	2.18 × 10−7	1.53 × 10−7	3.61 × 10−7	4 × 10−7	4.13 × 10−7	0
  Io3	0	1.47 × 10−7	1.19 × 10−7	2.82 × 10−7	4.04 × 10−7	4.46 × 10−7	0
  Rs	0.0147	0.0158	0.0259	0.0754	0.0015	0.5847	0.0152
  Computation time per run (s)	2.002	3.124	3.058	2.524	1.951	1.215	1.327

  shows the results of parameter estimation using different optimization algorithms. Figure 10 represents the convergence curve of the MTDM that proves the efficiency of the proposed algorithm over other algorithms. The errors, mean, and standard deviation of the MTDM circuit equivalent to a PV cell are shown in

  Table 9. The MTDM calculation error, mean and standard deviation for the monocrystalline PV Cell.

  Algorithm		SSE	AE	MAE	MSE	RMSE
  GSA	Minimum	1.00 × 10−5	3.11 × 10−3	1.51 × 10−4	5.00 × 10−7	7.00 × 10−4
  	Maximum	9.34 × 10−5	9.68 × 10−3	4.80 × 10−4	4.67 × 10−6	2.17 × 10−3
  	Mean	4.05 × 10−5	6.34 × 10−3	3.17 × 10−4	2.03 × 10−6	1.45 × 10−3
  	Standard deviation	2.27 × 10−5	4.70 × 10−3	2.33 × 10−4	1.13 × 10−6	1.02 × 10−3
  MVO	Minimum	1.03 × 10−5	3.29 × 10−3	1.68 × 10−4	5.14 × 10−7	7.15 × 10−3
  	Maximum	9.87 × 10−5	9.35 × 10−3	4.91 × 10−3	4.97 × 10−6	2.20 × 10−3
  	Mean	4.48 × 10−5	6.66 × 10−3	3.84 × 10−4	2.24 × 10−6	1.43 × 10−3
  	Standard deviation	2.84 × 10−5	5.30 × 10−3	2.05 × 10−4	1.41 × 10−6	1.11 × 10−3
  PSO	Minimum	1.06 × 10−2	1.03 × 10−1	5.17 × 10−3	5.01 × 10−4	2.31 × 10−2
  	Maximum	7.19 × 10−1	8.48 × 10−1	4.23 × 10−2	3.59 × 10−2	1.89 × 10−1
  	Mean	2.39 × 10−1	4.89 × 10−1	2.44 × 10−2	1.19 × 10−2	1.09 × 10−1
  	Standard deviation	2.35 × 10−1	4.84 × 10−1	2.42 × 10−2	1.17 × 10−2	1.08 × 10−1
  SCA	Minimum	8.12 × 10−3	9.04 × 10−2	4.57 × 10−3	4.89 × 10−4	2.02 × 10−2
  	Maximum	9.71 × 10−2	3.12 × 10−1	1.55 × 10−2	4.82 × 10−3	6.96 × 10−2
  	Mean	4.08 × 10−2	2.02 × 10−1	1.01 × 10−2	2.07 × 10−3	4.51 × 10−2
  	Standard deviation	2.30 × 10−2	1.523 × 10−1	7.54 × 10−3	1.19 × 10−3	3.39 × 10−2
  HBO	Minimum	1.44 × 10−3	3.76 × 10−2	1.86 × 10−3	7.1 × 10−5	8.41 × 10−3
  	Maximum	6.85 × 10−3	8.27 × 10−2	4.18 × 10−3	3.46 × 10−4	1.85 × 10−2
  	Mean	4.31 × 10−3	6.59 × 10−2	3.33 × 10−3	2.21 × 10−4	1.47 × 10−2
  	Standard deviation	1.83 × 10−3	4.26 × 10−2	2.12 × 10−3	9.1 × 10−5	9.55 × 10−3
  PO	Minimum	1.07 × 10−3	3.17 × 10−2	1.59 × 10−3	5.05 × 10−5	7.15 × 10−3
  	Maximum	7.69 × 10−3	8.31 × 10−2	4.13 × 10−3	3.41 × 10−4	1.85 × 10−2
  	Mean	2.85 × 10−3	5.33 × 10−2	2.62 × 10−3	1.46 × 10−4	1.19 × 10−2
  	Standard deviation	2.01 × 10−3	4.55 × 10−2	2.29 × 10−3	1.09 × 10−4	1.01 × 10−2
  PSOGSA	Minimum	3.76 × 10−22	1.93 × 10−11	9.69 × 10−13	1.88 × 10−23	4.33 × 10−12
  	Maximum	7.76 × 10−21	8.81 × 10−11	4.40 × 10−12	3.88 × 10−22	1.97 × 10−11
  	Mean	3.76 × 10−21	6.13 × 10−11	3.06 × 10−12	1.88 × 10−22	1.37 × 10−11
  	Standard deviation	2.25 × 10−21	4.74 × 10−11	2.37 × 10−12	1.13 × 10−22	1.06 × 10−11

  . Based on the results, it can be inferred that parameter estimation using the developed hybrid PSOGSA is superior to other algorithms.
• Non-Parametric Test: Non-parametric tests are performed to justify the results obtained by all algorithms.

  Table 10. Statistical results based on Wilcoxon’s rank sum test for the monocrystalline PV Cell.

  PSOGSA vs.	GSA	PSO	SCA	HBO	PO	MVA
  MSDM	6.7957 × 10−8	6.7953 × 10−8	6.7574 × 10−8	6.7951 × 10−8	6.7940 × 10−8	6.7956 × 10−8
  MDDM	6.7860 × 10−8	6.7954 × 10−8	6.7956 × 10−8	6.7950 × 10−8	6.7960 × 10−8	6.0830 × 10−8
  MTDM	6.7956 × 10−8	6.7954 × 10−8	6.7860 × 10−8	6.7949 × 10−8	6.7944 × 10−8	6.7956 × 10−8

  represents the statistical result based on the Wilcoxon rank sum test. Friedman’s rank test is represented in

  Table 11. Friedman ranking test’s statistical result for the monocrystalline PV Cell.

  Algorithms	Friedman Ranking
  MVO	3
  PSO	7
  SCA	6
  HBO	5
  PO	4
  GSA	2
  PSOGSA	1

  . Based on the results of this test, it is clear that the newly generated hybrid algorithm is more reliable and superior to other similar algorithms. Based on the results of both tests, it is clear that PSOGSA is superior to other compared algorithms.

Case 2: This case is also divided into three parts as follows for the polycrystalline PV solar cell:

• Modified Single-diode Model (MSDM):

  Table 12. Parameter Estimation of the MSDM for the polycrystalline PV Cell.

  Parameter/Algorithm	MVO	PSO	SCA	HBO	PO	GSA	PSOGSA
  Ipv	8.9183	9.1173	8.9245	4.8097	4.8340	4.5256	8.8803
  Alpha1	1.5428	1.3272	1.4979	0.8534	0.8741	1.4562	1.5419
  Rse	0.0094	0.0010	0.0151	0.0490	0.0268	0.2300	0.0174
  Rsh	277.670	63.070	320.452	279.525	294.365	225.714	365.338
  Io1	1.00 × 10−6	1.87 × 10−7	6.99 × 10−7	4.19 × 10−7	5.91 × 10−7	4.54 × 10−7	1.00 × 10−6
  Rs	0.0045	0.0015	0.0125	0.0380	0.0214	0.1800	0.0258
  Computation time per run (s)	2.014	3.514	3.321	2.748	2.104	1.421	1.520

  shows the results of parameter estimation using different optimization algorithms. Figure 11 represents the convergence curve of the MSDM. From this convergence curve, it is concluded that the efficiency of the proposed algorithm is better than the rest of the compared algorithms. The errors, mean, and standard deviation of the MSDM circuit equivalent to a PV cell are shown in

  Table 13. The MSDM calculation error, mean, standard deviation for the polycrystalline PV Cell.

  Algorithms		SSE	AE	MAE	MSE	RMSE
  GSA	Minimum	1.04 × 10−6	1.06 × 10−3	5.09 × 10−5	5.20 × 10−8	2.87 × 10−4
  	Maximum	8.15 × 10−6	2.84 × 10−3	1.44 × 10−4	4.08 × 10−7	6.71 × 10−4
  	Mean	4.57 × 10−6	2.18 × 10−3	1.07 × 10−4	2.28 × 10−7	4.01 × 10−4
  	Standard deviation	2.32 × 10−6	1.57 × 10−3	7.60 × 10−5	1.16 × 10−7	3.06 × 10−4
  MVO	Minimum	1.09 × 10−5	3.25 × 10−3	1.61 × 10−4	5.44 × 10−7	7.65 × 10−4
  	Maximum	9.94 × 10−5	9.91 × 10−3	4.90 × 10−4	4.96 × 10−6	2.24 × 10−3
  	Mean	4.06 × 10−5	6.35 × 10−3	3.24 × 10−4	2.03 × 10−6	1.47 × 10−3
  	Standard deviation	2.98 × 10−5	5.04 × 10−3	2.60 × 10−4	1.48 × 10−6	1.25 × 10−3
  PSO	Minimum	4.67 × 10−2	2.16 × 10−1	1.08 × 10−2	2.33 × 10−3	4.83 × 10−3
  	Maximum	7.28 × 10−1	8.53 × 10−1	4.26 × 10−2	3.64 × 10−2	1.90 × 10−1
  	Mean	3.40 × 10−1	5.83 × 10−1	2.91 × 10−2	1.70 × 10−2	1.30 × 10−1
  	Standard deviation	2.02 × 10−1	4.49 × 10−1	2.24 × 10−2	1.01 × 10−2	1.01 × 10−1
  SCA	Minimum	3.14 × 10−3	5.62 × 10−2	2.89 × 10−3	1.74 × 10−4	1.25 × 10−2
  	Maximum	9.68 × 10−2	3.11 × 10−1	1.55 × 10−2	4.08 × 10−3	6.95 × 10−2
  	Mean	4.07 × 10−2	2.01 × 10−1	1.00 × 10−2	2.09 × 10−3	4.51 × 10−2
  	Standard deviation	2.67 × 10−2	1.64 × 10−1	8.21 × 10−2	1.37 × 10−3	3.66 × 10−2
  HBO	Minimum	1.32 × 10−3	3.64 × 10−2	1.88 × 10−3	6.63 × 10−5	8.10 × 10−3
  	Maximum	9.99 × 10−3	9.99 × 10−2	5.07 × 10−3	5.08 × 10−4	2.23 × 10−2
  	Mean	3.45 × 10−3	5.81 × 10−2	2.94 × 10−3	1.64 × 10−4	1.30 × 10−2
  	Standard deviation	2.30 × 10−3	4.52 × 10−2	2.32 × 10−3	1.01 × 10−4	1.01 × 10−2
  PO	Minimum	1.30 × 10−2	1.14 × 10−1	5.70 × 10−3	6.50 × 10−4	2.55 × 10−2
  	Maximum	5.63 × 10−2	2.37 × 10−1	1.18 × 10−3	2.81 × 10−3	5.30 × 10−2
  	Mean	2.49 × 10−2	1.57 × 10−1	7.68 × 10−3	1.24 × 10−3	3.52 × 10−2
  	Standard deviation	1.15 × 10−2	1.07 × 10−1	5.03 × 10−3	5.70 × 10−3	2.40 × 10−2
  PSOGSA	Minimum	1.07 × 10−21	3.26 × 10−11	1.64 × 10−12	5.34 × 10−23	7.30 × 10−12
  	Maximum	9.27 × 10−21	9.62 × 10−11	4.81 × 10−12	4.64 × 10−22	2.15 × 10−11
  	Mean	5.07 × 10−21	7.12 × 10−11	3.56 × 10−12	2.54 × 10−22	1.59 × 10−11
  	Standard deviation	2.81 × 10−21	5.30 × 10−11	2.65 × 10−12	1.41 × 10−22	1.18 × 10−11

  . Based on the results, it can be inferred that parameter estimation using the developed hybrid PSOGSA is superior to other algorithms.
• Modified Double-Diode Model (MDDM):

  Table 14. Results of parameter estimation of the MDDM for the polycrystalline PV Cell.

  Parameter/Algorithm	MVO	PSO	SCA	HBO	PO	GSA	PSOGSA
  Ipv	8.9060	9.0360	8.8406	4.8172	4.7933	4.8750	8.8803
  Alpha1	1.5764	0.8512	1.2455	1.1165	0.9294	1.1836	1.5774
  Alpha2	1.5476	0.9196	1.3272	1.1676	1.1838	1.2761	1.5603
  Rse	0.0053	0.0013	0.0422	0.0574	0.0261	0.2731	0.0155
  Rsh	256.488	69.109	259.831	267.516	299.723	247.912	372.649
  Io1	0	9.30 × 10−8	1.49 × 10−7	3.45 × 10−7	3.14 × 10−7	5.13 × 10−7	7.00 × 10−7
  Io2	6.52 × 10−7	1.26 × 10−7	1.59 × 10−7	3.41 × 10−7	2.47 × 10−7	4.94 × 10−7	0
  Rs	0.0047	0.0025	0.0458	0.0584	0.0162	0.3127	0.0514
  Computation time per run (s)	2.011	3.245	3.125	2.658	2.010	1.321	1.451

  shows the results of parameter estimation using different optimization algorithms. Figure 12 represents the convergence curve of the MDDM for this case. The errors, mean, and standard deviation of the MDDM circuit equivalent to a PV cell are shown in

  Table 15. The MDDM calculation error, mean and standard deviation for Case 2.

  Algorithm		SSE	AE	MAE	MSE	RMSE
  GSA	Minimum	1.02 × 10−5	3.17 × 10−3	1.63 × 10−4	5.10 × 10−7	7.81 × 10−4
  	Maximum	9.28 × 10−5	9.06 × 10−3	4.86 × 10−4	4.64 × 10−6	2.15 × 10−3
  	Mean	4.16 × 10−5	6.41 × 10−3	3.20 × 10−4	2.08 × 10−6	1.48 × 10−3
  	Standard deviation	2.67 × 10−5	5.17 × 10−3	2.55 × 10−4	1.34 × 10−6	1.15 × 10−3
  MVO	Minimum	1.52 × 10−4	1.07 × 10−2	5.05 × 10−4	5.72 × 10−6	2.37 × 10−3
  	Maximum	3.97 × 10−3	6.27 × 10−2	3.15 × 10−3	1.90 × 10−4	1.40 × 10−2
  	Mean	5.02 × 10−4	2.32 × 10−2	1.16 × 10−3	2.78 × 10−5	5.14 × 10−3
  	Standard deviation	4.21 × 10−4	2.00 × 10−2	1.14 × 10−3	2.00 × 10−5	4.44 × 10−3
  PSO	Minimum	1.59 × 10−2	1.26 × 10−1	6.31 × 10−3	7.04 × 10−4	2.82 × 10−2
  	Maximum	8.85 × 10−1	9.41 × 10−1	4.71 × 10−2	4.42 × 10−2	2.11 × 10−1
  	Mean	4.18 × 10−1	6.47 × 10−1	3.53 × 10−2	2.09 × 10−2	1.45 × 10−1
  	Standard deviation	2.57 × 10−1	5.06 × 10−1	2.53 × 10−2	1.28 × 10−2	1.13 × 10−1
  SCA	Minimum	3.00 × 10−3	5.50 × 10−2	2.70 × 10−3	1.58 × 10−4	1.23 × 10−2
  	Maximum	1.59 × 10−1	3.99 × 10−1	2.00 × 10−2	8.00 × 10−3	8.94 × 10−2
  	Mean	5.38 × 10−2	2.32 × 10−1	1.15 × 10−2	2.69 × 10−3	5.18 × 10−2
  	Standard deviation	4.38 × 10−2	2.09 × 10−1	1.04 × 10−2	2.19 × 10−3	4.68 × 10−2
  HBO	Minimum	1.00 × 10−3	3.20 × 10−2	1.65 × 10−3	5.13 × 10−5	7.11 × 10−3
  	Maximum	7.77 × 10−3	8.77 × 10−2	4.31 × 10−3	3.82 × 10−4	1.96 × 10−2
  	Mean	3.54 × 10−7	5.95 × 10−2	2.95 × 10−3	1.74 × 10−4	1.33 × 10−2
  	Standard deviation	2.29 × 10−3	4.69 × 10−2	2.34 × 10−3	1.81 × 10−5	1.04 × 10−2
  PO	Minimum	4.10 × 10−3	6.40 × 10−2	3.21 × 10−3	2.01 × 10−4	1.43 × 10−2
  	Maximum	4.39 × 10−2	2.09 × 10−1	1.04 × 10−2	2.19 × 10−3	4.68 × 10−2
  	Mean	1.67 × 10−2	1.29 × 10−1	6.49 × 10−3	8.37 × 10−4	2.89 × 10−2
  	Standard deviation	1.00 × 10−2	1.00 × 10−1	5.00 × 10−3	5.05 × 10−4	2.23 × 10−2
  PSOGSA	Minimum	7.28 × 10−22	2.69 × 10−11	1.34 × 10−12	3.64 × 10−23	6.03 × 10−12
  	Maximum	9.82 × 10−21	9.90 × 10−11	4.95 × 10−12	4.91 × 10−22	2.21 × 10−11
  	Mean	3.71 × 10−21	6.09 × 10−11	3.04 × 10−12	1.86 × 10−22	1.36 × 10−11
  	Standard deviation	2.90 × 10−21	5.38 × 10−11	2.69 × 10−12	1.45 × 10−22	1.20 × 10−11

  . Based on the results, it can be concluded that parameter estimation using the developed hybrid PSOGSA is superior to other algorithms.
• The Modified Three-Diode Model (MTDM): 

  Table 16. Results of parameter estimation of the MTDM for Case 2.

  Parameter/Algorithm	MVO	PSO	SCA	GSA	HBO	PO	PSOGSA
  Ipv	8.9392	9.1101	8.8629	5.8503	4.8215	4.8587	8.8802
  Alpha1	1.4659	0.7090	1.2986	1.2324	1.2278	1.0888	1.4959
  Alpha2	1.5157	0.7902	1.2520	1.2751	1.0856	1.0417	1.6972
  Alpha3	1.4794	0.8058	1.0996	1.2271	1.3308	1.2000	1.5183
  Rse	0.0086	0.0061	0.0170	0.2617	0.0554	0.0030	0.0140
  Rsh	248.788	74.256	296.161	248.43	280.642	184.906	380.081
  Io1	5.33 × 10−7	1.17 × 10−7	1.83 × 10−7	3.10 × 10−7	2.92 × 10−7	6 × 10−7	0
  Io2	0	1.66 × 10−7	1.87 × 10−7	5.13 × 10−7	3.64 × 10−7	3 × 10−7	0
  Io3	4.17 × 10−7	1.83 × 10−7	1.57 × 10−7	4.94 × 10−7	2.96 × 10−7	3.5 × 10−7	5.00 × 10−7
  Rs	0.0047	0.0087	0.0120	0.1762	0.4550	0.0052	0.0410
  Computation time per run (s)	2.015	3.112	3.035	2.415	1.851	1.112	1.200

  shows the results of parameter estimation using different optimization algorithms. Figure 13 represents the convergence curve for Case 2 that proves the efficiency of the proposed algorithm. The errors, mean, and standard deviation of the MTDM circuit equivalent to a PV cell are shown in

  Table 17. The MTDM calculation error, mean and standard deviation for Case 2.

  Algorithms		SSE	AE	MAE	MSE	RMSE
  GSA	Minimum	1.03 × 10−5	3.27 × 10−3	1.66 × 10−4	5.15 × 10−7	7.41 × 10−4
  	Maximum	7.71 × 10−5	8.70 × 10−3	4.33 × 10−4	3.86 × 10−6	1.98 × 10−3
  	Mean	3.18 × 10−5	5.64 × 10−3	2.81 × 10−4	1.59 × 10−6	1.22 × 10−3
  	Standard deviation	1.93 × 10−5	4.31 × 10−3	2.20 × 10−4	9.65 × 10−7	9.92 × 10−4
  MVO	Minimum	1.09 × 10−5	3.33 × 10−3	1.05 × 10−4	5.45 × 10−7	7.76 × 10−3
  	Maximum	8.25 × 10−5	9.07 × 10−3	4.14 × 10−4	4.13 × 10−6	2.02 × 10−3
  	Mean	3.66 × 10−5	6.01 × 10−3	3.62 × 10−4	1.83 × 10−6	1.36 × 10−3
  	Standard deviation	2.28 × 10−5	4.74 × 10−3	2.15 × 10−4	1.14 × 10−6	1.08 × 10−3
  PSO	Minimum	3.81 × 10−2	1.95 × 10−1	9.7 × 10−3	1.9 × 10−3	4.36 × 10−2
  	Maximum	8.74 × 10−1	9.35 × 10−1	4.67 × 10−2	4.37 × 10−2	2.09 × 10−1
  	Mean	4.14 × 10−1	6.43 × 10−1	3.21 × 10−2	2.07 × 10−2	1.43 × 10−1
  	Standard deviation	2.63 × 10−1	5.13 × 10−1	2.56 × 10−2	1.31 × 10−2	1.11 × 10−1
  SCA	Minimum	1.42 × 10−3	3.76 × 10−2	1.8 × 10−3	7.10 × 10−5	8.4 × 10−3
  	Maximum	1.40 × 10−1	3.74 × 10−1	1.87 × 10−2	7.0 × 10−3	8.37 × 10−2
  	Mean	4.0 × 10−2	2.0 × 10−1	1.0 × 10−2	2.0 × 10−3	4.47 × 10−2
  	Standard deviation	3.20 × 10−2	1.80 × 10−1	9.0 × 10−3	1.6 × 10−3	4.04 × 10−2
  HBO	Minimum	1.1 × 10−3	3.34 × 10−2	1.6 × 10−3	5.6 × 10−5	7.4 × 10−3
  	Maximum	7.5 × 10−3	8.65 × 10−2	4.3 × 10−3	3.7 × 10−4	1.93 × 10−2
  	Mean	3.4 × 10−3	5.86 × 10−2	2.9 × 10−3	1.7 × 10−4	1.31 × 10−2
  	Standard deviation	1.9 × 10−3	4.41 × 10−2	2.2 × 10−3	9.70 × 10−5	9.8 × 10−3
  PO	Minimum	1.0 × 10−3	3.27 × 10−2	1.6 × 10−3	5.35 × 10−5	7.3 × 10−3
  	Maximum	1.24 × 10−2	1.11 × 10−1	5.5 × 10−3	6.2 × 10−4	2.49 × 10−2
  	Mean	3.8 × 10−3	6.21 × 10−2	3.1 × 10−3	1.9 × 10−4	1.38 × 10−2
  	Standard deviation	3.1 × 10−3	5.60 × 10−2	2.8 × 10−3	1.5 × 10−4	1.25 × 10−2
  PSOGSA	Minimum	1.09 × 10−22	1.04 × 10−11	5.22 × 10−13	5.46 × 10−24	2.33 × 10−12
  	Maximum	9.77 × 10−21	9.88 × 10−11	4.94 × 10−12	4.89 × 10−22	2.21 × 10−11
  	Mean	3.97 × 10−21	6.30 × 10−11	3.15 × 10−12	1.99 × 10−22	1.40 × 10−11
  	Standard deviation	3.32 × 10−21	5.76 × 10−11	2.88 × 10−12	1.66 × 10−22	1.28 × 10−11

  . Based on the results, it can be inferred that parameter estimation using the developed hybrid PSOGSA is superior to other algorithms.
• Non-Parametric Test: Non-parametric tests are performed to justify the results obtained by all algorithms.

  Table 18. Wilcoxon’s rank sum test for Case 2.

  PSOGSA vs.	GSA	PSO	SCA	HBO	PO	MVA
  MSDM	6.7956 × 10−8	6.7954 × 10−8	6.7955 × 10−8	6.7952 × 10−8	6.7959 × 10−8	6.7956 × 10−8
  MDDM	6.7956 × 10−8	6.7955 × 10−8	6.7956 × 10−8	6.7956 × 10−8	6.7958 × 10−8	6.7957 × 10−8
  MTDM	6.7957 × 10−8	6.7954 × 10−8	6.7955 × 10−8	6.7952 × 10−8	6.7959 × 10−8	6.7956 × 10−8

  represents the statistical result based on the Wilcoxon rank sum test. The Friedman rank test is represented in

  Table 19. Friedman’s ranking test for Case 2.

  Algorithms	Friedman Ranking
  MVO	3
  PSO	7
  SCA	6
  HBO	4
  PO	5
  GSA	2
  PSOGSA	1

  . Based on the results of this test, it is clear that the newly generated hybrid algorithm is more reliable and superior to other similar algorithms. Based on the results of both tests, PSOGSA clearly outperforms the other algorithms.

6. Conclusions

The abundant availability of solar energy has attracted many efforts to exploit it as much as possible to meet the energy needs of humans. In this work, two types of solar photovoltaic cells were considered: the monocrystalline cell and the polycrystalline cell. Various mathematical equivalent models of the photovoltaic cell such as MSDM, MDDM and MTDM were explored in this work. For efficient parameter estimation in the considered models for both types of photovoltaics, a hybrid algorithm such as (PSOGSA) was tested, which is a combination of two algorithms, the particle swarm optimization and gravity search algorithm. In terms of exploration and exploitation, the hybrid algorithm is superior to both algorithms. Results obtained using PSOGSA were compared with those obtained using PSO, GSA, MVO HBO, PO and SCA techniques. The PSOGSA method clearly showed more promising results compared to any of the other algorithms considered in this work. Various errors such as SSE, AE, MAE, MSE, RMSE, were also calculated and the average, standard deviation is calculated after estimating the MSDM, MDDM and MTDM parameters of monocrystalline and polycrystalline solar PV cells. Full error analysis justified the superiority of PSOGSA in parameter estimation of solar PV cells over other algorithms considered in this work. To get a better accuracy of the hybrid algorithm, statistical results were also carried out, and through testing theses, it is clear that the proposed hybrid algorithm is better than the rest of the comparative algorithms. These studies have demonstrated that the hybrid algorithm (PSOGSA) is a promising and efficient method for modified solar PV parameter extraction as it increases the efficiency of the solar panel, proving that the same amount of power can be generated from the same solar panel by reducing the internal loses. For future studies, the hybrid algorithm can be used for establishing a dynamic model of the PV panel.