1. Introduction
In recent decades, due to their inexhaustibility, nonpolluting nature, and highly adaptable properties to decentralized generation, renewable energies have been the ecological alternative to fossil fuels and nuclear energy [
1,
2]. For these reasons, advanced technologies are currently being developed to benefit from these types of energy sources. Photovoltaic (PV) panels, which generate electricity using the sun’s energy as a renewable energy source, are one of the most prevalent forms of renewable energy [
3]. Solar energy is growing exponentially. Its main characteristic is to be a form of decentralized production, making it possible to meet strong demand from citizens and local authorities and to produce energy where it is consumed. Consequently, significant losses can be avoided during energy transportation. The PV industry has been overgrowing in recent years [
4] because it is not only inexhaustible but also silent and nondisturbing for residents, unlike wind turbines which cause visual and acoustic disturbances. In addition, the market to produce electricity from solar energy is proliferating [
5,
6]. In this context, the importance of photovoltaic generators connected to the electricity distribution network is growing rapidly [
7]. Hence, assessing and studying the performance of the photovoltaic module, which is the fundamental component of these generators, appears to be highly significant [
8]. The manufacturers typically tend to provide only limited operational data for PV panels. These data are only available under standard conditions of 1000 W/m
^{2} irradiation, 25 °C cell temperature, and air mass of 1.5 [
9]. Therefore, it is essential to understand each cell element’s physical properties and electrical characteristics before developing an equivalent circuit for a photovoltaic cell. Performance evaluation of PV modules and the design of energy systems are derived from the electrical characteristic current–voltage (I–V) of the modules under different radiation levels and different temperatures of the PV cell [
10,
11]. There are three forms of solar cell technologies available on the market: amorphous, monocrystalline, and polycrystalline [
12]. Monocrystalline and polycrystalline cells are found in rigid panels. The difference between the two types is mainly based on their efficiency. To achieve maximum performance, crystalline panels should be installed perpendicular to the sun’s rays. Generally abbreviated aSi, amorphous silicon is the noncrystallized allotropic variety of silicon; crystalline structures of the aSi are formed from disordered atoms that are not arranged regularly. Thin layers of amorphous silicon can be deposited at low temperatures on a wide variety of substrates. Hence, a wide range of microelectronic applications can be envisaged. The advantage of amorphous silicon cells is that they are environmentally friendly because they do not use toxic heavy metals, such as cadmium or lead. Compared to amorphous cells, crystalline panels do not perform as well in partial shadowing, and they lose a tiny percentage of their output as the temperature rises over 25 °C. Various equations can be used to model PV cells and modules approximated to differing degrees of accuracy from the actual device. This modeling offers essential advantages, such as ease of use, thanks to the equivalent electrical circuit and the popularization of the system properties. Therefore, the understanding of complex phenomena will be facilitated. Therefore, solar cells are considered power generators and will be modeled by equivalent circuits and electric models. The most commonly used are the singlediode model [
13], the model with two diodes [
14], and the one with three diodes [
15]. Each of these models has some unknown parameters that characterize and describe the behavior of a PV generator. In addition, the behavior of PV generators is influenced by various parameters related to electrical modeling [
16]. The power output of a photovoltaic (PV) cell is influenced by several factors such as the orientation of the panels, quality factor, kind of material, absorbent layer, and optical window. The optimal orientation of panels should be perpendicular to the sun’s direction to maximize the power output. The quality factor of the cell is a measure of its efficiency to convert sunlight into electricity, and it involves a tradeoff between efficiency and cost. The choices of material and the thickness and composition of the absorbent layer also play a significant role in determining the power output. Additionally, optimizing the optical window requires a balance between light transmission and absorption by the window. The PV cell’s performance is interdependent on various parameters, such as efficiency, opencircuit voltage, fill factor, shortcircuit current, and maximum power point. These parameters are interdependent, and there are constraints between them that must be considered to optimize the cell’s performance. Hence, understanding the constraints between the PV cell parameters is vital for designing efficient PV systems.
In order to optimize the various characteristics and simulate the behavior of a PV generator, it is crucial to identify the physical mechanisms at play within it. The complexity of the model is determined by the number of parameters that need to be identified. The ideal model includes a current source for solar power input and a diode for the PN junction, but additional components can be added to better represent the PV cell’s behavior in specific operational situations. Various methods of parameter identification have been studied in the literature, including numerical, analytical, deterministic, and metaheuristic methods. Numerical methods utilize mathematical algorithms to iteratively optimize parameter values using measured or simulated data. These methods employ numerical optimization techniques, including iterative algorithms and metaheuristic approaches, to minimize the discrepancy between model predictions and observed data. Numerical methods offer flexibility in handling complex and nonlinear problems [
17,
18]. Analytical methods involve analyzing mathematical formulas to identify the parameters of PV models. These methods are characterized by their short execution time and simplicity. However, their solutions are not precise [
19,
20]. The deterministic methods have major drawbacks, such as the high sensitivity to the initial hypotheses and the tendency of these algorithms to converge to the local optimum [
21]. Moreover, they depend on the convexity of the model [
22]. However, the models of photovoltaic cells are multimodal and characterized by nonlinearities. Recently, metaheuristic methods seem to be good potential alternatives for extracting parameters from PV models [
23]. Indeed, they overcome the shortcomings of the analytical and deterministic methods already cited. In the following, we mention some of the most popular metaheuristic methods: Genetic Algorithm (GA) [
24], artificial bee colony algorithm (ABC) [
25], differential evolution algorithm (DE) [
26], bird mating optimization (BMO) [
27], Ant Lion Optimizer (ALO) [
28], bacterial foraging optimization (BFO) [
29], gray wolf optimization (GWO) [
30], whale optimization algorithm (WOA) [
31], Slime Mould Algorithm (SMA) [
32], Sal Swarm Algorithm (SSA) [
33], and Coyote Optimization Algorithm (COA) [
15].
The primary objective of this study is to investigate and analyze the efficiency of a novel algorithm called Modified Social Group Optimization (MSGO) [
34] for the extraction and identification of the parameters of photovoltaic cells. To provide a comprehensive assessment, a comparative study was conducted, incorporating various technologies and models of photovoltaic panel cells.
In the initial phase of this investigation, the proposed algorithm was applied to the monocrystalline photovoltaic panel of RTC France Company, considering both single and doublediode cell models. The outcomes obtained through the MSGO algorithm were compared with results from previous studies utilizing alternative metaheuristic algorithms, such as the Nelder–Mead method and modified particle swarm optimization (NM–MPSO) [
35], Levenberg–Marquardt algorithm combined with Simulated Annealing (LMSA) [
36], ABC [
21], biogeographybased optimization algorithm with mutation strategies (BBOM) [
37], improved adaptive differential evolution (RcrIJADE) [
38], artificial bee swarm optimization algorithm (ABSO) [
39], and chaotic asexual reproduction optimization (CARO) [
40]. All these algorithms were tested on the same photovoltaic panel, under identical lighting and temperature conditions (temperature of 33 °C and irradiation of 1000 W/m
^{2}).
Subsequently, the proposed algorithm was also evaluated on a flexible photovoltaic panel composed of hydrogenated amorphous silicon (aSi: H). The obtained results were compared with the findings presented by authors from [
41], who based their research on the optimization algorithms QuasiNewton Method (QNM) and the SelfOrganizing Migrating Algorithm (SOMA).
To validate the results obtained by the proposed MSGO algorithm, an experimental study was performed on the TITAN1250 panel, utilizing polycrystalline cells [
42]. Finally, the paper concludes with a comparative analysis between different optimization algorithms employed for photovoltaic parameter extraction. The results obtained through the proposed MSGO algorithm are compared with those derived from other algorithms such as the WOA, SSA, Sine Cosine Algorithm (SCA), Virus Colony Search Algorithm (VCS), Gravitational Search Algorithm (GSA), and Ant Lion Optimizer (ALO). Throughout the remainder of this paper, three sections are described:
Section 2 introduces PV models and problem formulation.
Section 3 details the proposed MSGO algorithm.
Section 4 treats the study of the MSGO algorithm efficiency by testing various pieces of technology and PV cell models. In the last section, the obtained results are compared with those given in previous studies.
2. Mathematical PV Model Analysis
The evaluation of the PV module performance and the power system design is based on the current–voltage electrical characteristic of the modules under different radiation levels and various temperatures of the PV cells. It is possible to model PV cells and modules by means of equations that approximate the physical cell to varying degrees. Several electrical models are proposed in the literature for simulating PV cells under different conditions. The model’s complexity varies depending on the number of parameters (R_{s}, R_{sh}, etc.) to be considered. Every model is basically refinements of the ideal model, which consists of a diode that represents the PN junction and a current source that represents incident solar power.
It is possible to add several additional elements to provide a better representation of the behavior of PV cells in some operating areas [
43]. Singlediode models (SDMs), doublediode models (DDMs), and threediode models (TDMs) are the most used models.
Figure 1a represents the singlediode model, which is regarded as the most popular model. It is widely used because of its simplicity. It also provides high precision and simplicity in the power generation quadrant.
The singlediode model has undergone various advancements that have led to the development of more accurate models, such as the Bishop model, which explains the behavior of the PV cell under reverse polarization. The doublediode model, shown in
Figure 1b, considers losses due to various resistances and devices in the different electric components that constitute the circuit [
44]. An enhanced model considers the effects of grain boundaries and leakage currents. This model involves three diodes as it is shown in
Figure 1c. Although this model meets most of the physical requirements of solar cells, it involves computing nine parameters that require exceptionally high numerical execution. In addition, dynamic models are proposed by introducing the capacity to model the dynamic behavior of the PV cell. This model type is shown in
Figure 1d. All these models differ in the number of parameters required for computing the I–V characteristic [
45].
${I}_{ph}$ is the photogenerated current source; ${I}_{D1},{I}_{D2},{I}_{D3}$ are the currents of diodes D_{1}, D_{2}, and D_{3}; R_{p} is the shunt resistance; R_{s} is the series resistance; $I$ is the output current; and $V$ is the output voltage.
2.1. Mathematical Development
2.1.1. Crystalline Cells
From the equivalent circuit (
Figure 1a), it is evident that the current produced by the solar cell is equal to that produced by the current source (
I_{ph}), minus that which flows through the diode (
I_{d}), minus that which flows through the shunt resistor (
I_{p}).
where
${K}_{i}=\{\begin{array}{c}{K}_{1}=\left[\begin{array}{ccc}1& 0& 0\end{array}\right],i=1\hspace{0.17em}\mathrm{in}\mathrm{case}\mathrm{of}\hspace{0.17em}\mathrm{SDM}\\ {K}_{2}=\left[\begin{array}{ccc}1& 1& 0\end{array}\right],i=2\hspace{0.17em}\mathrm{in}\mathrm{case}\mathrm{of}\hspace{0.17em}\mathrm{DDM}\\ {K}_{3}=\left[\begin{array}{ccc}1& 1& 1\end{array}\right],i=3\hspace{0.17em}\mathrm{in}\mathrm{case}\mathrm{of}\hspace{0.17em}\mathrm{TDM}\end{array}$, where
${I}_{d}=\left[\begin{array}{c}{I}_{d1}\\ {I}_{d2}\\ {I}_{d3}\end{array}\right]$, and
${I}_{p}=\frac{V+{R}_{s}I}{{R}_{p}}$.
The current in the
j^{th} diode is given by
Then, the current given in Equation (1) is given by Equation (3), where
I_{SDM},
I_{DDM}, and
I_{TDM} are the total output current when considering the SDM, DDM, and TDM, respectively.
where
n_{1},
n_{2}, and
n_{3} are the ideality factors of the diodes
D_{1},
D_{2}, and
D_{3};
K is the Boltzmann constant (1.380649 × 10
^{−23} Joule/Kelvin);
T is the temperature of the PV panel (Kelvin); and
q is the charge of the electron (1.602176634 × 10
^{−19} Coulomb).
The TDM does not seem suitable for fast computations and has complex nonlinear analytic expressions; therefore, this model will be excluded from the parametric identification tests.
2.1.2. Amorphous Silicon Cell
Equation (4) defines the current–voltage characteristic for an amorphous silicon cell:
d_{i} denotes the width of the i^{th} layer in the (aSi) p_i_n diode, μ_{eff} represents the mobilitylifetime product of the electron and hole, and V_{bi} is the builtin field voltage.
The diode reverse saturation current and the photogenerated current of an (aSi) cell under constant light and temperature are given, respectively, by
where
A is the
p_{n} junction area,
L_{p} is the carrier diffusion length of the
ptype area,
L_{n} is the carrier diffusion length of the
ntype area,
W is the depletion layer,
D_{p} and
D_{n} are the holes and electrons diffusion coefficient,
p_{n}_{0} and
n_{p}_{0} are the minority carrier concentration in the
P region and
N region, and
g(
x) is the electron hole formation ratio.
2.2. The Objective Functions
The term objective function is used in mathematical optimization and operations research to refer to a function that acts as a criterion for identifying the best solution to an optimization problem.
The objective function of the SDM may be written as
For the DDM, the error function is expressed by
where
$X=\left[\begin{array}{ccccccc}{I}_{ph}& {I}_{s{d}_{1}}& {R}_{s}& {n}_{1}& {I}_{s{d}_{2}}& {n}_{2}& {R}_{p}\end{array}\right]$.
Whereas, the error function for the TDM is defined by
where
$X=\left[\begin{array}{ccccccccc}{I}_{ph}& {I}_{s{d}_{1}}& {R}_{s}& {n}_{1}& {I}_{s{d}_{2}}& {n}_{2}& {I}_{s{d}_{3}}& {n}_{3}& {R}_{p}\end{array}\right]$.
It is necessary to use
Ne samples (data points number) to widen the scope of the search and reach the global optimum. Equation (10) gives us a description of the cost function:
3. Procedure of Social Group Optimization for PV Parameters Estimation
The past twenty years have seen a remarkable rise in interest in metaheuristic optimization algorithms. The research work developed has enabled the appearance of new algorithms which are generally based on the following:
 
A new idea inspired by a natural, physical, chemical phenomena;
 
A modification of an existing algorithm to improve its performances;
 
The hybridization of two methods allows the strengths to merge and the weaknesses to be eliminated of the two algorithms.
However, no algorithm can be adapted to all types of problems. In 2016, a new metaheuristic optimization algorithm appeared, known as Social Group Optimization (SGO) [
46]. To solve complex problems, the new algorithm was inspired by the social behavior of individuals in groups. Each individual’s knowledge is mapped by its fitness. The algorithm contains two phases. The first phase is called the improving phase in which each individual interacts with the best person (best solution) to improve his knowledge by interacting. The second phase is named the acquiring phase, during which the individuals acquire knowledge when they interact with the best person and randomly selected individuals simultaneously. A comparative study is carried out to show the performance of the new method. Detailed information on the SGO algorithm can be found in the following articles [
47,
48]. The SGO algorithm is described with the following:
P_{i}, (i = 1, 2, 3, ..., N): Pi is the social group persons, and N is the total number of people in the social group.
P_{ij}, (j = 1, 2, ..., D): D is the traits number related to a person which allows us to determine the dimensions of a person.
f_{i}, (i = 1, 2, ..., N) is their corresponding fitness value.
In each social group, the role of the best person (
P_{best}) is to propagate knowledge between all persons. As a result, others in the group enhance their knowledge.
The following algorithm (Algorithm 1) can be used to calculate how often each person’s knowledge is updated:
Algorithm 1 Improving phase 
for i = 1: N 
for j = 1: D 
Pnew_{ij} = c * Pij + r * ( Pbest(j) − Pij) 
end for 
end for 
r: random number, and
r ∈ [0, 1]. If
Pnew provides higher fitness than
P_{old}, it is accepted [
34].
c is the parameter of selfintrospection
c ∈ [0, 1].
In the acquiring phase (Algorithm 2), a person acquires new knowledge by interacting with other persons of the group. The interaction can be with the best person (
Pbest) or randomly with other persons who have more knowledge. To acquire knowledge, a person always interacts with the
Pbest and with any other person of the group who has more knowledge than him. The ability to obtain a quantity of knowledge from another person is defined by the selfawareness probability (SAP). The modified acquiring phase is computed as
where
Pi is the updated value at the completion of the improving phase.
Algorithm 2 Acquiring phase 
for i = 1: N 
Randomly select one person P_{r} where i ≠ r 
If f(P_{i}) < f(P_{r})

If rand > SAP 
for j = 1: D 
Pnew_{i,j} = P_{i,j} + rand_{1} * (P_{i,j} − P_{r,j}) + rand_{2} * (best_{p} (j) − P_{i,j}) 
end for 
else 
for j = 1: D 
Pnew_{i,j} = lb + rand_{2} * (ub − lb) 
end for 
end if 
else 
for j = 1: D 
Pnew_{i,j} = P_{i,j} + rand_{1} * (P_{r,j} – P_{r,j}) + rand_{2} * (best_{p} (j) − P_{i,j}) 
end for 
end if 
end for 
Pnew is accepted if it provides a higher level of fitness than P.
The general steps to use the MSGO algorithm to extract parameters of a PV cell include:
Step 1: Define the objective function which describes the behavior of the PV cell under different conditions. This function takes input parameters, such as the cell’s temperature, irradiance, and voltage, and outputs a value that represents the cell’s performance. The goal is to find the values of these input parameters that maximize the output value of the objective function.
Step 2: Define the parameter space. The parameter space defines the objective function constraints for each parameter, such as the temperature may range from −10 °C to 100 °C, the irradiance may range from 0 W/m^{2} to 1000 W/m^{2}, and the voltage may range from 0 V to 1 V. All other range parameters are declared in Equations (15)–(17).
Step 3: Initialize the population. The population (P_{i}) is a set of solutions that are randomly generated within the parameter space. Each solution corresponds to a set of input parameters that are used to evaluate the objective function.
Step 4: Evaluate the fitness. The fitness is a measure of how well each solution performs with respect to the objective function. The fitness function takes as input the output value of the objective function and returns a scalar value that represents the quality of the solution. The higher the fitness, the better the solution.
Step 5: Update the population. The MSGO algorithm updates the population in two phases: the improving phase and the acquiring phase. In the improving phase, each individual interacts with the best person in the social group to improve its knowledge. In the acquiring phase, each individual acquires knowledge by interacting with the best person and randomly selected individuals. The updating of each person’s knowledge can be calculated using the formula described in the improving phase.
Step 6: Repeat steps 4 and 5 until convergence. The optimization process continues until the fitness values converge to a satisfactory level or the maximum number of iterations is reached. The best solution found during the optimization process corresponds to the set of input parameters that maximizes the output value of the objective function. These parameters can be used to characterize the behavior of the PV cell under the given conditions.