Chaos Induced Coyote Algorithm (CICA) for Extracting the Parameters in a Single, Double, and Three Diode Model of a Mono-Crystalline, Polycrystalline, and a Thin-Film Solar PV Cell

Abstract

The design of a solar PV system and its performance evaluation is an important aspect before going for a mass-scale installation and integration with the grid. The parameter evaluation of a solar PV model helps in accurate modeling and consequently efficient designing of the system. The parameters appear in the mathematical equations of the solar PV cell. A Chaos Induced Coyote Algorithm (CICA) to obtain the parameters in a single, double, and three diode model of a mono-crystalline, polycrystalline, and a thin-film solar PV cell has been proposed in this work. The Chaos Induced Coyote Algorithm for extracting the parameters incorporates the advantages of the conventional Coyote Algorithm by employing only two control parameters, making it easier to include the unique strategy that balances the exploration and exploitation in the search space. A comparison of the Chaos Induced Coyote Algorithm with some recently proposed solar photovoltaic cell parameter extraction algorithms has been presented. Analysis shows superior curve fitting and lesser Root Mean Square Error with the Chaos Induced Coyote Algorithm compared to other algorithms in a practical solar photovoltaic cell.

1. Introduction

Renewable energy, particularly from the sun, has increased in usage due to the increasing population, ever-growing industrial needs, depletion of fossil fuel reserves, and many environmental problems [1,2]. Solar energy does not cause any kind of pollution, i.e., it is a clean form of energy, it can also be readily available at a properly chosen site [3].

The complexity and non-linearity of the solar cell equations, in addition to various instability problems, act as a hurdle to the PV system [4,5,6,7]. Several other reasons, including temperature, solar radiation, cable losses, dust accumulation, soiling, and shading, also affect I–V characteristics [8,9]. Hence, to evaluate the solar PV system’s performance in actual working conditions, a thorough analysis of the solar PV system is essential. While it may not be possible to practically study these effects, as physical factors change rapidly, an accurate model of the solar PV system will give a close idea of the requirements of the practical implementation [10]. Thus, it becomes necessary to develop accurate models for it. The literature proposes different models to represent a solar cell differently from each other by the number of diodes they incorporate. The single diode models (SDM) and the double diode models (DDM) are the more famous models [11,12]. The models represent the current-voltage characteristics (I–V), but only for domestic purposes. Hence a three diode model (TDM) [12] was developed, which can also be used for industrial applications [13]. With the change in the number of diodes in the solar cell’s equivalent circuits, there is also a change in the number of extracted parameters. Although the single-diode model is accurate, the saturation current of the solar PV cell varies linearly with charge diffusion and recombination in the space charge layer [14], and thus more accurate representation is possible by adding two diodes in parallel to the ideal current source and associated series and shunt resistors [15]. DDM is more relevant for the operation of PV cells at low irradiance [14]. In order to include the effect of current due to the leakage through the peripheries, a third diode can be included in parallel [16].

To achieve the characteristics of the actual PV system, the simulation model must provide accurate parameter estimation, i.e., an optimized parameter estimation technique is required. There are many techniques available that can be used to optimize the values of these parameters. The various techniques include the classical techniques (analytical and numerical methods), as shown in

Table 1. Various classical optimization methods.

S. No.	Optimization Methods
1.	Least square-based Newton Raphson [17]
2.	Fitting the curve based on iteration [18]
3.	Lambert W based optimization [19,20,21,22,23,24]
4.	Identification by linear least square [25,26]
5.	Linear extrapolation and linear interpolation [27]
6.	Taylor series expansion [28]
7.	Chebyshev polynomials [29]
8.	Padé approximations [30]
9.	Symbolic function [31]
10.	Analytical mathematical method [32]

, and stochastic techniques (heuristic and meta-heuristic methods) shown in

Table 2. Various metaheuristic methods for parameter extraction reported in the literature.

S. No.	Metaheuristic Methods
1.	Flower Pollination Algorithm (FPA) [33]
2.	Firefly Algorithm (FA) [34]
3.	Simulated Annealing (SA) [35]
4.	Particle Swarm Optimization (PSO) [36]
5.	Hybrid Particle Swarm Optimization and Simulated Annealing (HPSOSA) [37]
6.	Differential Algorithm (DE) [38]
7.	Particle Swarm Optimization and Global Search Algorithm (PSOGSA) [39]
8.	Modified and Multiobjective Firefly Algorithm (FA) [40,41,42,43,44,45]
9.	Coyote Optimization Algorithm (COA) [46,47,48,49,50]

. In this paper, a novel chaotic coyote optimization-based parameter extraction has been shown to perform better than the recently developed coyote optimization discussed in section IV of the paper. The significant contribution of the paper can be summarized as:

• An exhaustive review of the techniques for solar PV cell parameter extraction is available in the literature.
• Detailed modeling for a solar PV cell for one, two, and three diodes.
• A novel chaotic coyote optimization method for the parameter extraction of the solar PV cell.
• Analysis and the performance evaluation of the novel chaotic coyote algorithms and comparison with other popular metaheuristic methods for parameter extraction of solar PV cells.

Due to the nonlinear relationship of a solar cell’s parameters, the classical methods cannot be efficient enough to extract these parameters, but the meta-heuristics are more efficient in optimizing the nonlinear and complex systems. Hence the meta-heuristic methods provide better alternatives to extract the solar PV parameters than the classical methods.

1.1. Classical Optimization Methods

The least-squares technique helps in better initialization of Newton’s method [17], providing the benefit that only two of the nonlinear parameters of the single diode model of a solar PV cell need to be initialized. However, this method of using Newton’s method based on the least-squares technique has slow convergence, with a convergence time of the order of a few minutes using any compiled language. The curve fitting techniques benefit from using all the points in the curve, which provides more confidence in the extracted parameters [18]. Although the curve fitting technique provides a very low error value, the solar PV cell’s resistances are not accurately estimated. The limitation that analytical optimization techniques cannot obtain the explicit solution of the characteristic equation of solar cell is overcome by Lambert W function [20], which helps the analytical optimization techniques transform the solar PV equation into an explicit one with higher accuracy, but it takes more time. The linear least square identification method proposed in [25] can extract solar cell parameters from a single I–V curve simultaneously without requiring any iterative searching or approximation. To represent the solar cell equation, [28] uses Taylor’s series expansion of second-order with the help of shunt resistances. This method, however, can only be used for the solar cell model involving one diode. An explicit method of modeling the solar PV cell characteristics involving Chebyshev polynomial is proposed in [29] that involves approximating the solar cell’s nonlinear exponential function to obtain an explicit solar cell equation. Two methods of modeling the solar PV cell based on Padé approximants are proposed in [30], namely the basic Padé approximation model and the modified Padé approximation model. The latter is a more generalized model that can be used to extract the parameters from both the implicit and explicit form of solar cell equation. However, it is difficult to provide the correct solution using the Padé approximation whenever the denominator of the Padé approximant becomes zero. In [31], Taylor’s series model based on symbolic function is used to achieve an explicit function of the solar cell’s characteristics. This method requires less operating time than any other numerically analytical method.

1.2. Metaheuristic Optimization Methods

Metaheuristic methods are nature-inspired stochastic methods that have proved to be a great alternative to the classical optimization techniques for parameter extraction of solar PV systems [40]. A metaheuristic is a guided random search technique that explores and exploits the entire search space. However, the solution may get trapped at a local optimum point, and also, the solution is not exact. In the metaheuristic techniques, generally, a trade-off has to be made between the solution’s quality and the time taken to obtain the solution. Sometimes to obtain the solution quickly, the quality of the solution has to be compromised.

Flower Pollination (FP) is an algorithm inspired by the process in the flowering plants known as pollination that aims to select the fittest. While extracting a solar cell’s parameters using DDM by applying FPA, the iteration converges at 421 iterations with RMSE = 9.154 × 10⁻⁴, and for 800 iterations, it reaches a value of RMSE = 7.8425 × 10⁻⁴. Moreover, the SDM converges at 419 iterations with RMSE = 7.7301 × 10⁻⁴. [33]. The Firefly Algorithm (FA) [41,42,43,44] uses the flashing patterns of fireflies in the dark [41]. FA is flexible, easy to use, and can converge at a global solution to any optimizing problem. While extracting a solar cell’s parameters using DDM by applying FA, the iteration converges with RMSE = 4.5484 × 10⁻⁶, and for the SDM, converges with RMSE = 5.138 × 10⁻⁴ [34]. The Simulated Annealing algorithm (SA) [45] is based on annealing in metal to obtain lower energy states [35,36]. In SA point to point optimization occurs, and the updated value of the solution is always in the proximity of the existing solution. Particle Swarm Optimization (PSO) is used to solve computationally difficult optimization problems. This technique is robust and based on the way swarms move. During iteration, each particle tries to update its previous experience and also the experience of its neighbors. Due to the problem of convergence at optimum local value, PSO, along with SA, is used in order to achieve the best quality solution. The global best solution that was calculated using PSO is processed again and calculated using SA at every iteration. Hence the solution obtained now will be significantly improved [37]. While extracting a solar cell’s parameters using DDM by applying HPSOSA, the iteration converges with RMSE = 7.453 × 10⁻⁴ and for the SDM converges with RMSE = 7.730 × 10⁻³ [37]. Differential Evolution (DE) uses the search and selection mechanism as a mutation operation to provide the right search direction in the entire search space region. With the help of the data in the manufacturer’s datasheet, the DE technique can extract the solar PV parameters at any value of the solar radiation and temperature. Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA) is a hybridization of the two techniques, PSO and GSA. It is based on taking the better of the two techniques, i.e., the ability to exploit PSO and the ability to explore from GSA. Using PSOGSA provides an improved chance of escaping the local optimum point and faster convergence.

In this paper, the Coyote Optimization Algorithm (COA) with the aid of chaotic maps is proposed for the extraction of solar PV parameters. COA is a metaheuristic technique inspired by nature by the social structure of the species known as coyotes/canis latrans. COA is the latest optimization technique developed in 2018 by Pierezan and Coelho [46]. It is a very efficient algorithm used to extract the parameters of single diode models, double diode models, and three diode models for different types of PV modules. This strategy provides a balance between both the exploitation process and the exploration process. The algorithm is straightforward in structure and implemented with only two control parameters (Np and Nc). COA can perform better than many nature-inspired stochastic techniques with a limited number of iterations and in lesser time. The paper compares COA’s performance with other stochastic techniques, such as DE, PSO, PSOGSA, and the COA technique shows better results compared to other techniques. Many stochastic algorithms face the problem of getting trapped in the local optima and slow convergence. Chaotic maps aid the optimization algorithms to overcome these problems. This paper uses ten chaotic maps aiding COA and shows that the algorithm can converge and achieve the optimum value in significantly less time than when performed without using the chaotic maps. Chaotic Optimization Algorithms also provide the ability to search locally as well as globally, which can enable the optimization method quite effective [50,51].

2. Materials and Methods

2.1. PV Cell Modelling

A precise model of the solar cell is necessary to represent all the parameters. An ideal model of a solar cell includes a current source and an antiparallel diode across it. However, practically, a resistance R_sh is connected in parallel that accounts for leakage currents and a resistance R_s that is connected in series, representing the material’s resistivity and the copper losses.

2.1.1. Single Diode Model (SDM) of a Solar Cell

The equivalent circuit of SDM is shown in Figure 1. The output current I_L can be found out by using Equations (1)−(3).

$${\mathrm{I}}_{\mathrm{L}}={\mathrm{I}}_{\mathrm{ph}} - {\mathrm{I}}_{\mathrm{D}} - {\mathrm{I}}_{\mathrm{sh}}$$

(1)

$${\mathrm{I}}_{\mathrm{D}}={\mathrm{I}}_0\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{\mathrm{nkT}}\right)- 1\right]$$

(2)

$${\mathrm{I}}_{\mathrm{sh}}=\frac{{\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{sh}}}{{\mathrm{R}}_{\mathrm{sh}}}$$

(3)

where I_L, I_ph, I_sh, I₀, and I_D are the output current from the solar cell equivalent circuit, the photo-current from the solar cell, current through the shunt resistance, reverse saturation current of the diode, and current through the diode, respectively. R_s and R_sh are the resistances connected in series and shunt branches, respectively. V_t is the voltage across the output terminal, n is the ideality factor, while k is the Boltzmann’s constant (1.380 × 10⁻²³ (J/K)). q (1.602 × 10⁻¹⁹ Coulumbs) is the magnitude of electronic charge. T is the absolute temperature of the solar cell in Kelvin.

The current I_L can be found by substituting the values of the current through the diode (I_D) from Equation (2) and the current through the shunt branch I_sh from Equation (3) in Equation (1) as:

$${\mathrm{I}}_{\mathrm{L}}={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_0\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{\mathrm{nkT}}\right)-1\right]-\frac{({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{sh}})}{{\mathrm{R}}_{\mathrm{sh}}}$$

(4)

Equation (4) helps in finding out the unknown parameters, i.e., R_s, R_sh, I₀, n₁, and I_ph of the solar cell with a single diode and obtain the characteristics of the cell closer to the actual one.

2.1.2. Double Diode Model (DDM) of a Solar Cell

The equivalent circuit of the double diode model can be represented as in Figure 2. It is different from SDM, with two diodes connected across the ideal current source. The output current of the solar cell, I_L, can be found out using Kirchhoff’s Current Law and is expressed as:

$${\mathrm{I}}_{\mathrm{L}}={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{\mathrm{D}1}-{\mathrm{I}}_{\mathrm{D}2}-{\mathrm{I}}_{\mathrm{sh}}$$

(5)

$${\mathrm{I}}_{\mathrm{L}}={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{01}\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{{\mathrm{n}}_1\mathrm{kT}}\right)-1\right]-{\mathrm{I}}_{02}\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{{\mathrm{n}}_2\mathrm{kT}}\right)-1\right]-\frac{{\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{sh}}}{{\mathrm{R}}_{\mathrm{sh}}}$$

(6)

where I₀₁ and n₁ are the reverse saturation current and ideality factor of diode D₁ and I₀₂, and n₂ is the reverse saturation current and ideality factor of the diode D₂.

2.1.3. Three Diode Model (TDM) of a Solar Cell

The TDM can be represented as in Figure 3. This model has three diodes connected in parallel to the ideal current source along with resistances R_s and R_sh connected in series and parallel, respectively. The output current of the solar cell, I_L, can be found out using KCL as:

$${\mathrm{I}}_{\mathrm{L}}={\mathrm{I}}_{\mathrm{ph}} - {\mathrm{I}}_{\mathrm{D}1} - {\mathrm{I}}_{\mathrm{D}2} - {\mathrm{I}}_{\mathrm{D}3} - {\mathrm{I}}_{\mathrm{sh}}$$

(7)

$${\mathrm{I}}_{\mathrm{L}}={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{01}\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{{\mathrm{n}}_1\mathrm{kT}}\right)-1\right]-{\mathrm{I}}_{02}\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{{\mathrm{n}}_2\mathrm{kT}}\right)-1\right]-{\mathrm{I}}_{03}\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{{\mathrm{n}}_3\mathrm{kT}}\right)-1\right]-\frac{{\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{sh}}}{{\mathrm{R}}_{\mathrm{sh}}}$$

(8)

where I₀₁, I₀₂, and I₀₃ are the reverse saturation currents, while n₁, n₂, and n₃ are the ideality factors of diodes D₁, D_2, and D₃, respectively.

The nine parameters of DDM, i.e., R_s, R_sh, I₀₁, I₀₂, I₀₃, n₁, n₂, n₃, and I_ph, are aimed to be found out using Equation (8), with an aim to achieve the solar cell characteristics closer to the actual one.

2.2. Problem Formulation

The modeling of the solar PV systems is mainly aimed to obtain the unknown solar cell parameters of their respective equivalent circuits, i.e., SDM, DDM, or TDM. There must be only a minimum difference between the experimental data and data extracted from the algorithm.

The vital step in any optimization algorithm is determining the vector of solutions (X_i), the search range, and the objective function. The various vectors of the solution for SDM and DDM are shown in

Table 3. Vectors of solutions (X_i) for SDM and DDM.

Model	Parameters (Xi)
Single Diode Model (SDM)	Rs, Rsh, I0, n, Iph
Double Diode Model (DDM)	Rs, Rsh, I01, I02, n1, n2, Iph
Three Diode Model (TDM)	Rs, Rsh, I01, I02, I03, n1, n2, n3, Iph

.

This paper’s objective function is the Root Mean Square Error (RMSE) function and can be defined as the difference between the value of the estimated current from the algorithm and the value of the actual current extracted from the experiments.

For SDM the error function can be written as:

$$\mathrm{J}\left({\mathrm{V}}_{\mathrm{t}},{\mathrm{I}}_{\mathrm{L}},\mathrm{X}\right)={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_0\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{\mathrm{nkT}}\right)-1\right]-\frac{\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{sh}}\right)}{{\mathrm{R}}_{\mathrm{sh}}}-{\mathrm{I}}_{\mathrm{L}}$$

(9)

For DDM the error function can be written as:

$$\mathrm{J}\left({\mathrm{V}}_{\mathrm{t}},{\mathrm{I}}_{\mathrm{L}},\mathrm{X}\right)={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{01}\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{{\mathrm{n}}_1\mathrm{kT}}\right)-1\right]-{\mathrm{I}}_{02}\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{{\mathrm{n}}_2\mathrm{kT}}\right)-1\right]-\frac{{\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{sh}}}{{\mathrm{R}}_{\mathrm{sh}}}-{\mathrm{I}}_{\mathrm{L}}$$

(10)

For TDM the error function can be written as:

$$\mathrm{J}\left({\mathrm{V}}_{\mathrm{t}},{\mathrm{I}}_{\mathrm{L}},\mathrm{X}\right)={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{01}\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{{\mathrm{n}}_1\mathrm{kT}}\right)-1\right]-{\mathrm{I}}_{02}\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{{\mathrm{n}}_2\mathrm{kT}}\right)-1\right]-{\mathrm{I}}_{03}\left[\exp \left(\frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{S}}\right)}{{\mathrm{n}}_3\mathrm{kT}}\right)-1\right]-\frac{{\mathrm{V}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{L}}{\mathrm{R}}_{\mathrm{sh}}}{{\mathrm{R}}_{\mathrm{sh}}}-{\mathrm{I}}_{\mathrm{L}}$$

(11)

The Root Mean Square Error (RMSE) is the objective function and is mathematically defined as:

$$\mathrm{RMSE}=\sqrt{\frac{1}{\mathrm{N}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{K}}\mathrm{J}{({\mathrm{V}}_{\mathrm{t}},{\mathrm{I}}_{\mathrm{L}},\mathrm{X})}^2}$$

(12)

where N is the no. of readings of the measurement. The iteration stops once a decided number of steps is reached, or the error tolerance reaches a predefined value.

2.3. Chaotic Coyote Optimization Algorithm

In this work, a novel chaotic coyote optimization algorithm (COA) has been used for solving the objective function obtained in Equation (12). It is a metaheuristic algorithm and is inspired by incorporating chaos in the population generation in the coyote optimization presented in [46]. The coyote optimization is itself a social behavior and adaption of a species known as coyote/canis latrans presented in [46]. In the COA algorithm, the coyote population’s arrangement is so that there are Np groups of coyotes, and each group has N_c coyotes. The product of N_p and N_c gives the total population of the species. Every individual coyote is a possible optimum solution for the optimization problem and its social conditions set (soc), which has all the decision variables included [46,47,48,49]. The Coyote Optimization Algorithm provides better parameter estimation of the solar PV module with lesser values of Root Mean Square Error than other mentioned techniques. However, when the algorithm gets aided with chaotic maps, there can be an appreciable improvement in the convergence speed of the results observed. The problem of getting trapped at the local optimum values, which is faced by other metaheuristic techniques, is overcome with the help of chaotic COA. The improvement of COA with the aid of ten different chaotic maps is shown in Section 3.2 of this paper for SDM, DDM, and TDM of various types of solar cell modules. The description of the coyote optimization and its subsequent modification to chaotic coyote has been discussed below:

The social condition of the cth coyote from the pth group at any instant is given in Equation (13) [47] as:

$$\mathrm{x}=\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,\dots \dots {\mathrm{x}}_{\mathrm{J}}\right)={\mathrm{soc}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}}$$

(13)

where J represents the dimension of the search space.

The adaptation of the coyote to its environment is the fitness function. The COA is started with the first step being the initialization of the global coyote population and are given by [47]:

$${\mathrm{SOC}}_{\mathrm{c},\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}={\mathrm{Lb}}_{\mathrm{j}}+{\mathrm{r}}_{\mathrm{j}}({\mathrm{Ub}}_{\mathrm{j}} - {\mathrm{Lb}}_{\mathrm{j}})$$

(14)

where Lb_j is the lower boundary, and Ub_j is the upper boundary of the design variable j, and r_j is any real random number between 0 and 1. The random number can better be replaced by chaotic numbers for better exploration of the search space. Thus, the following modification is done to Equation (14) [47].

$${\mathrm{SOC}}_{\mathrm{c},\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}={\mathrm{Lb}}_{\mathrm{j}}+{\mathrm{g}}_{\mathrm{j}}({\mathrm{Ub}}_{\mathrm{j}} - {\mathrm{Lb}}_{\mathrm{j}})$$

(15)

where ${\mathrm{g}}_{\mathrm{j}}$ is the chaotic map function. For each coyote, the fitness function is written as [47]:

$${\mathrm{fit}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}}=\mathrm{f}({\mathrm{soc}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}})$$

(16)

Sometimes at the beginning of COA, the coyotes might become solitary or join other groups, the probability of which is as follows [47]:

$${\mathrm{P}}_{\mathrm{l}}=0.005{\mathrm{N}}_{\mathrm{C}}{}^2$$

(17)

To avoid P_l exceeding the value of 1, N_c should be less than 14. In each pack, there is an alpha coyote that proves to have the maximum adaptation capability to the environment, is designated as ‘α’ and can be mathematically represented as [47]:

$${\mathsf{\alpha }}^{\mathrm{p},\mathrm{t}}={\mathrm{soc}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}} \mathrm{for} \min ({\mathrm{fit}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}})$$

(18)

It is assumed that for the survival of the group, the coyote has to share its culture with other coyotes of the group, and this cultural tendency can be mathematically written as [47]:

$${\mathrm{cult}}_{\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}=\{\begin{array}{ll}{\mathrm{O}}_{\frac{{\mathrm{N}}_{\mathrm{C}}+1}{2},\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}& ; \mathrm{if} {\mathrm{N}}_{\mathrm{C}} \mathrm{is} \mathrm{odd}\hfill \\ \frac{{\mathrm{O}}_{\frac{{\mathrm{N}}_{\mathrm{C}}}{2},\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}+{\mathrm{O}}_{\frac{{\mathrm{N}}_{\mathrm{C}}+1}{2},\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}}{2}& ; \mathrm{else}\hfill \end{array}$$

(19)

where O_p^t is the ranked social status of the coyotes in group ‘p’ at time instant ‘t’ for j = 1, 2, 3, …D. The life events, such as the birth and the coyotes’ death, are taken into consideration by the COA. The birth of the coyotes is influenced by the social behavior of the two randomly selected parent coyotes of the same group and the environmental factors and can be mathematically represented as [47]:

$${pup}_{\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}=\{\begin{array}{ll}{soc}_{\mathrm{r}1,\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}& ; {rand}_{\mathrm{j}} < {\mathrm{P}}_{\mathrm{s}} or \mathrm{j} = {\mathrm{j}}_1\\ {soc}_{\mathrm{r}2,\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}& ; {rand}_{\mathrm{j}} \ge {\mathrm{P}}_{\mathrm{s}} + {\mathrm{P}}_{\mathrm{a}} or \mathrm{j} = {\mathrm{j}}_2\\ {\mathrm{R}}_{\mathrm{j}}& ; else\end{array}$$

(20)

where ${\mathrm{soc}}_{\mathrm{r}1,\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}$ and ${\mathrm{soc}}_{\mathrm{r}2,\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}$ are the coyotes selected randomly from the p-th group. j₁ and j₂ are the two random dimensions of the search space. P_s represents the probability of scattering, P_a represents the probability of association, R_j is a vector that is generated randomly, and rand_j is any number in the interval [0, 1] chosen randomly. The scattering probability and the association probability are represented as [47]:

$${\mathrm{P}}_{\mathrm{s}}=\frac{1}{\mathrm{D}}$$

(21)

and

$${\mathrm{P}}_{\mathrm{a}}=\frac{1 - {\mathrm{P}}_{\mathrm{s}}}{2}$$

(22)

It has been found that 10% of the young coyotes die at the time of birth. While the mortality rate of coyotes also increases with age. Algorithm 1 [48] shows the synchronism of the birth and the death of the coyotes (where ρ is the group that adjusts least to the environment than the young ones and ζ is the group of coyotes that adjusts less to the group size).

Algorithm 1. Synchronism of the birth and the death of the coyotes.

Calculate ρ and ζ if ζ = 1 then                   retain the young coyote and eliminate the only coyote in ρ else if ζ > 1 then                   retain the young coyote and eliminate the oldest coyote in ρ else eliminate the young coyote end if

The cultural interaction of coyotes within groups is shown with the help of d₁ (it shows how the alpha influences a random coyote cr₁) and d₂ (showing how any random coyote cr₂ is influenced by the cultural tendency of the group). cr₁ and cr₂ are selected using random probability distribution function, while d₁ and d₂ can be mathematically represented as:

$${\mathrm{d}}_1={\mathrm{alpha}}^{\mathrm{p},\mathrm{t}} - {\mathrm{soc}}_{\mathrm{cr}1}{}^{\mathrm{p},\mathrm{t}}$$

(23)

and

$${\mathrm{d}}_2={\mathrm{cult}}^{\mathrm{p},\mathrm{t}} - {\mathrm{soc}}_{\mathrm{cr}2}{}^{\mathrm{p},\mathrm{t}}$$

(24)

The alpha coyote, as well as the other members of the group, affect the coyote’s social behavior, and the below equation gives the updated social condition:

$${\mathrm{socnew}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}}={\mathrm{soc}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}}+{\mathrm{r}}_1{\mathrm{d}}_1+{\mathrm{r}}_2{\mathrm{d}}_2$$

(25)

where r₁ and r₂ are any real random numbers such that 0 ≤ r₁, r₂ ≤ 1. The new social behavior of the coyotes is estimated by:

$${\mathrm{fitnew}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}}=\mathrm{f}({\mathrm{socnew}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}})$$

(26)

$${\mathrm{soc}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}+1}=\{\begin{array}{ll}{\mathrm{socnew}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}}& ; \mathrm{if} {\mathrm{fitnew}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}}<{\mathrm{fit}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}}\\ {\mathrm{soc}}_{\mathrm{c}}{}^{\mathrm{p},\mathrm{t}}& ; \mathrm{else}\end{array}$$

(27)

Parameter Constraints

To extract the solar cell parameters, Equation (25) possesses an issue of causing the coyotes (parameters) to remain out of the already defined boundaries of the search and give impractical solutions. Therefore, to avoid this issue, an operator is introduced as in Equation (28). It helps in reinitializing the violating elements randomly within the search space.

$${\mathrm{socnew}}_{\mathrm{c},\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}=\{\begin{array}{ll}{\mathrm{Lb}}_{\mathrm{j}}+{\mathrm{rand}}_{\mathrm{j}}({\mathrm{Ub}}_{\mathrm{j}}-{\mathrm{Lb}}_{\mathrm{j}})& ; \mathrm{if} {\mathrm{socnew}}_{\mathrm{c},\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}<{\mathrm{Lb}}_{\mathrm{j}}\\ & \mathrm{or} {\mathrm{socnew}}_{\mathrm{c},\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}>{\mathrm{Ub}}_{\mathrm{j}}\\ {\mathrm{socnew}}_{\mathrm{c},\mathrm{j}}{}^{\mathrm{p},\mathrm{t}}& ; \mathrm{else}& \end{array}$$

(28)

Algorithm 2 [48] shows the overall pseudo code to summarize the Chaotic COA. Chaotic COA is reiterated till a predefined criterion for stopping is met and the selection of that coyote is made, which adapts to the environment in a manner better than other coyotes. This coyote is selected as the Global Optimum Solution. The algorithm can also be observed from the flow chart shown in Figure 4.

Algorithm 2. Chaotic COA.

Create Np packs with Nc coyotes in each pack using Equation (15) Assess the adaption of each coyote Equation (16) while stopping criteria satisfied do          for each p pack do                         Find the pack’s alpha coyote Equation (18)                                     for each cth coyote of pth pack do                                                Generate new social conditions Equation (25)                                                Check the boundary conditions Equation (28)                                                Estimate the new social conditions Equation(26)                                                Decide whether to move or not Equation (27)                                     end for                                            Simulate the birth and the death of the coyotes using Equation (20) and Algorithm 1 end for          Impose the probabilities of coyotes leaving their packs using Equation (20)          Update the coyotes’ ages endwhile Select the coyote with most potential to adapt to the environment.

3. Results and Discussions

In Section 3.1, the COA algorithm is applied to the parameters of different solar PV systems. The results are compared with other recent optimization techniques to prove the COA technique’s accuracy in the process of extraction of the parameters. Finally, in Section 3.2, chaotic coyote optimization has been taken for parameter extraction, and the results are obtained by comparing the various chaotic maps.

3.1. Results of COA with the Data from Experiments

The COA technique’s performance to extract solar PV parameters is done with the data observed from the experiments of [48] for a single diode model of PV cells. The COA algorithm is tested with three types of solar cells: mono-crystalline Leibold Solar Module (LSM 20) with 20 cells in series and at a temperature of 24 °C and irradiance of 360 W/m², a Photowatt-PWP201 solar cell module with 36 polycrystalline silicon solar cells in series and a temperature of 45 °C and irradiance of 1000 W/m², and a PVM752 GaAs thin-film solar cell at a temperature of 25 °C and irradiation of 1000 W/m².

The extracted parameters are shown in

Table 4. Parameters extracted for a mono-crystalline LSM20 solar PV module by COA and its comparison with other techniques for SDM.

Parameters	COA	DE	PSO	PSOGSA
Rs	0.00	0.00	0.00	0.00
Rsh	100	100	96.76659	82.205
IL	0.2023	0.2023	0.20259	0.2026
I0	0.00	0.00	0.0001300	0.0001
n	1	1	1.08115	1.0527
RMSE	0.17668	0.1767446	0.1862443	0.1802
Time (sec.)	1.330	1.156	1.138	10.605

,

Table 5. Parameters extracted for a Photowatt-PWP201 solar PV module by COA and its comparison with other techniques for SDM.

Parameters	COA	DE	PSO	PSOGSA
Rs	0	0	0	0.0222
Rsh	0.6379	0.635	0.6356339	3.1774
IL	1.1201	1.1207	1.1207088	0.7173
I0	1	1	1	0.6874
n	2	1.9100	1.9109	1.6331
RMSE	0.54732	0.55444	0.561334	0.7224
Time (sec.)	1.300	1.674	1.087	10.504

and

Table 6. Parameters extracted for PVM752 GaAs thin-film solar PV cell by COA compared with other techniques for SDM.

Parameters	COA	DE	PSO	PSOGSA
Rs	0	0	0	0.05
Rsh	10.4707	10.4707	10.4707052	61.0883
IL	0.1816	0.1816	0.18156	0.1677
I0	0	0	0	0.0006
n	1.5848	1	1.26827	2
RMSE	0.221817	0.22179	1.89115	0.50266
Time (sec.)	1.2920	1.178	1.088	10.24

for the single diode model (SDM) of mono-crystalline LSM20 module, Photowatt-PWP201 module, and PVM752, respectively. The parameter extraction performance by COA is compared with various recently proposed techniques, such as Differential Evolution (DE), Particle Swarm Optimization (PSO), and Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA).

As evident from Table 4, Table 5 and Table 6, COA provides lower RMSE values than other optimization techniques.

The comparison of various parameters along with their corresponding RMSE values and time of convergence for the double diode model and three diode model is shown in the Appendix A and Appendix B.

The I–V characteristics for the experimentally measured and calculated data are shown in Figure 5, Figure 6 and Figure 7 for the single diode model of mono-crystalline LSM20 solar cell module, Photowatt-PWP201 solar cell module, and GaAs thin-film solar cell.

When the convergence curves of COA, DE, PSO, and PGSOSA are compared for the mono-crystalline LSM20 solar cell module (SDM), as shown in Figure 8, it can be observed that COA has the fastest convergence. The best values of optimal solution achieved are 0.1752, 0.175378, 0.17539, and 0.17667 in 215, 345, 465, and 500 iterations for COA, DE, PSO, and PSOGSA, respectively.

When the convergence curves of COA, DE, PSO, and PGSOSA are compared for the Photowatt-PWP201 solar cell module (SDM), as shown in Figure 9, it can be observed that COA has the fastest convergence. The best values of optimal solution achieved are 0.5439, 0.5439, 0.5439, and 0.7424 in 70, 100, 280, and 8 iterations for COA, DE, PSO, and PSOGSA, respectively.

When the convergence curves of COA, DE, PSO, and PGSOSA are compared for GaAs thin-film solar cell-PVM752 (SDM), as shown in Figure 10, it can be observed that COA has the fastest convergence. The best values of optimal solution achieved are 0.185986, 0.85986, 0.22175, and 0.185986 in 45, 65, 130, and 385 iterations for COA, DE, PSO, and PSOGSA, respectively.

When the convergence curves of COA, DE, PSO, and PGSOSA are compared for a mono-crystalline LSM20 solar cell module (DDM), as shown in Figure 11, it can be observed that COA has the best convergence. The best values of optimal solution achieved are 0.02098, 0.030434, 0.05705, and 0.08679 in 400, 600, 130, and 10 iterations for COA, DE, PSO, and PSOGSA, respectively.

When the convergence curves of COA, DE, PSO, and PGSOSA are compared for Photowatt-PWP201 solar cell module (DDM), as shown in Figure 12, it can be observed that COA has the fastest convergence and lowest RMS error. The best values of optimal solution achieved are 0.5675, 0.56901, 0.57014, and 0.71649 in 150, 180, 300, and 150 iterations for COA, DE, PSO, and PSOGSA, respectively.

When the convergence curves of COA, DE, PSO, and PGSOSA are compared for GaAs thin-film solar cell-PVM752 (DDM), as shown in Figure 13, it can be observed that COA has the best convergence. The best values of optimal solution achieved are 0.034061, 0.090718, 0.034931, and 0.041737 in 120, 50, 300, and 350 iterations for COA, DE, PSO, and PSOGSA, respectively.

When the convergence curves of COA, DE, PSO, and PGSOSA are compared for a mono-crystalline LSM20 solar cell module (TDM), as shown in Figure 14, it can be observed that COA has the best convergence. The best values of optimal solution achieved are 0.026525, 0.034356, 0.04397, and 0.14497 in 400, 700, 730, and 10 iterations for COA, DE, PSO, and PSOGSA, respectively.

When the convergence curves of COA, DE, PSO, and PGSOSA are compared for Photowatt-PWP201 solar cell module (TDM), as shown in Figure 15, it can be observed that COA has the fastest convergence and lowest RMS error. The best values of optimal solution achieved are 0.569033, 0.567506, 0.569564, and 0.93278 in 125, 200, 250, and 10 iterations for COA, DE, PSO, and PSOGSA, respectively.

When the convergence curves of COA, DE, PSO, and PGSOSA are compared for GaAs thin-film solar cell-PVM752 (DDM), as shown in Figure 16, it can be observed that COA has the best convergence. The best values of optimal solution achieved are 0.03483, 0.034061, 0.3503, and 0.088164 in 150, 450, 250, and 10 iterations for COA, DE, PSO, and PSOGSA, respectively.

3.2. Parameter Extraction Using Coyote Optimization Algorithm with Chaotic Maps

With the help of chaotic methods, the performance of the COA technique can be improved considerably as the convergence speed can be increased, and the problem of local optimum can be avoided. In this paper, ten chaotic maps are used in order to improve the performance of the COA [52].

Table 7. Chaos Table.

S. No.	Name	Chaotic Maps	Range
C 1	Chebyshev [53]	${\mathrm{x}}_{\mathrm{i}+1}=\cos ({\mathrm{icos}}^{-1}({\mathrm{x}}_{\mathrm{i}}))$	(−1, 1)
C 2	Circle [54]	${\mathrm{x}}_{\mathrm{i}+1}=\mathrm{mod}\left({\mathrm{x}}_{\mathrm{i}}+\mathrm{b} -\left(\frac{\mathrm{a}}{2\mathsf{\pi }}\right)\sin \left({2\mathsf{\pi }\mathrm{x}}_{\mathrm{k}}\right),1\right) , \mathrm{a}=0.5 \mathrm{and} \mathrm{b}=0.2$	(0, 1)
C 3	Gauss/mouse [55]	${\mathrm{x}}_{\mathrm{i}+1}=\{\begin{array}{cc}1& {\mathrm{x}}_{\mathrm{i}}=0\hfill \\ \frac{1}{\mathrm{mod}({\mathrm{x}}_{\mathrm{i}},1)}& \mathrm{otherwise}\hfill \end{array}$	(0, 1)
C 4	Iterative [56]	${\mathrm{x}}_{\mathrm{i}+1}=\sin \left(\frac{\mathrm{a}\mathsf{\pi }}{{\mathrm{x}}_{\mathrm{i}}}\right) \mathrm{a}=0.7$	(−1, 1)
C 5	Logistic [56]	${\mathrm{x}}_{\mathrm{i}+1}={\mathrm{ax}}_{\mathrm{i}}\left(1-{\mathrm{x}}_{\mathrm{i}}\right) ,\mathrm{a}=4$	(0, 1)
C 6	Piecewise [57]	${\mathrm{x}}_{\mathrm{i}+1}=\{\begin{array}{ll}\frac{{\mathrm{x}}_{\mathrm{i}}}{\mathrm{P}}& 0 \le {\mathrm{x}}_{\mathrm{i}} < \mathrm{P}\\ \frac{{\mathrm{x}}_{\mathrm{i}}-\mathrm{P}}{0.5-\mathrm{P}}& \mathrm{P} \le {\mathrm{x}}_{\mathrm{i}} < 0.5\\ \frac{1-\mathrm{P}-{\mathrm{x}}_{\mathrm{i}}}{0.5-\mathrm{P}}& 0.5 \le {\mathrm{x}}_{\mathrm{i}} < (1-\mathrm{P})\\ \frac{1-{\mathrm{x}}_{\mathrm{i}}}{\mathrm{P}}& (1-\mathrm{P}) \le {\mathrm{x}}_{\mathrm{i}} < 1\end{array}$	(0, 1)
C 7	Sine [58]	${\mathrm{x}}_{\mathrm{i}+1}=\frac{\mathrm{a}}{4}\sin \left({\mathsf{\pi }\mathrm{x}}_{\mathrm{i}}\right) , \mathrm{a}=4$	(0, 1)
C 8	Singer [59]	${\mathrm{x}}_{\mathrm{i}+1}=\mathsf{\mu }\left(7.86{\mathrm{x}}_{\mathrm{i}} - 23.31{\mathrm{x}}_{\mathrm{i}}{}^2+28.75{\mathrm{x}}_{\mathrm{i}}{}^3 - 13.302875{\mathrm{x}}_{\mathrm{i}}{}^4\right) , \mathsf{\mu }=1.07$	(0, 1)
C 9	Sinusoidal [60]	${\mathrm{x}}_{\mathrm{i}+1}={\mathrm{ax}}_{\mathrm{i}}{}^2\sin \left(\mathsf{\pi }{\mathrm{x}}_{\mathrm{i}}\right) , \mathrm{a}=2.3$	(0, 1)
C 10	Tent [61]	${\mathrm{x}}_{\mathrm{i}+1}=\{\begin{array}{ll}\frac{{\mathrm{x}}_{\mathrm{i}}}{0.7}& {\mathrm{x}}_{\mathrm{i}} < 0.7\\ \frac{10}{3}\left(1-{\mathrm{x}}_{\mathrm{i}}\right)& {\mathrm{x}}_{\mathrm{i}} \ge 0.7\end{array}$	(0, 1)

shows the enumeration of the chaotic functions used in this paper [52]. It is also observed that COA with chaos is faster with a very high convergence speed, which is evident from the convergence curves for the ten chaotic functions in Figure 17, Figure 18 and Figure 19, for a mono-crystalline LSM20 solar cell module, Photowatt-PWP201 solar cell module, and GaAs thin-film solar cell PVM752 respectively, where the chaos maps are named as chaos1, chaos2, …, chaos10 in accordance with Table 7.

The applications of various metaheuristic methods in Solar PV parameter extraction and other industrial application has been discussed in [62,63,64,65,66,67,68].

3.2.1. Chaotic COA for Single Diode Model

Figure 17 shows that the COA with chaos achieves the optimal value of 0.175235 for the LSM20 solar module in 310 iterations, which is much less than the COA without the chaos. Furthermore, the time taken by each chaotic map to converge and reach the optimal value of 0.175235 is shown in

Table 8. Time taken by COA with the ten chaotic functions for LSM20 solar module.

S. No.	Chaotic Maps	Time (Seconds)
C 1	Chebyshev	0.8600
C 2	Circle	0.7540
C 3	Gauss/mouse	0.8690
C 4	Iterative	0.8240
C 5	Logistic	0.8560
C 6	Piecewise	0.8650
C 7	Sine	0.8260
C 8	Singer	0.8530
C 9	Sinusoidal	0.8340
C 10	Tent	0.6500

.

Figure 18 shows that the COA with chaos achieves the optimal value of 0.5447 for the Photowatt-PWP 201 solar cell module in 60 iterations, which is much less than the COA without the chaos. Furthermore, each chaotic map’s time is shown in

Table 9. Time taken by COA with the ten chaotic functions for Photowatt-PWP201 solar cell module (SDM).

S. No.	Chaotic Maps	Time (Seconds)
C 1	Chebyshev	0.5510
C 2	Circle	0.4300
C 3	Gauss/mouse	0.4100
C 4	Iterative	0.4020
C 5	Logistic	0.4200
C 6	Piecewise	0.4340
C 7	Sine	0.4250
C 8	Singer	0.4280
C 9	Sinusoidal	0.4460
C 10	Tent	0.4150

.

Figure 19 shows that the COA with chaos achieves the optimal value of 0.1866 for GaAs thin-film solar cell PVM752 in 70 iterations, which is much less than the COA without the chaos. Furthermore, the time taken by each chaotic map is shown in

Table 10. Time taken by COA with the ten chaotic functions for GaAs thin-film solar cell PVM752 (SDM).

S. No.	Chaotic Maps	Time (seconds)
C 1	Chebyshev	0.5420
C 2	Circle	0.5250
C 3	Gauss/mouse	0.4860
C 4	Iterative	0.6490
C 5	Logistic	0.6670
C 6	Piecewise	0.6700
C 7	Sine	0.6550
C 8	Singer	0.7920
C 9	Sinusoidal	0.6430
C 10	Tent	0.6370

.

Chaotic functions are used with the Coyote Optimization Algorithm in order to attain improved results with a higher convergence speed. The effect of the ten chaotic functions on the RMSE values using COA are shown with the help of boxplots for single diode models of mono-crystalline LSM20 solar module in Figure 20, Photowatt-PWP201 solar cell module in Figure 21, and for GaAs thin-film solar cell module in Figure 22, which shows that the results are better than using COA alone. The various chaotic functions mentioned in Table 7 are represented as C-N’ in the boxplot, such that 1 ≤ N’ ≤ 10, (where N’ is the serial number of the chaotic maps mentioned in Table 7).

For a mono-crystalline LSM20 solar cell module (SDM), the median value of RMSE were found as 0.17635, 0.17652, 0.17645, 0.17608, 0.17671, 0.17663, 0.17678, 0.17659, 0.17672, and 0.17563, respectively, for various chaotic functions as in the order given in Table 7.

For a polycrystalline Photowatt-PWP201 solar cell module (SDM), the median RMSE values were 0.54566, 0.5452, 0.54483, 0.54447, 0.54483, 0.54513, 0.54473, 0.54566, 0.54577, and 0.54469, respectively, for various chaotic functions as in the order given in Table 7.

In the case of thin-film solar cell PVM752 (SDM), the median RMSE values were obtained as 0.22546, 0.18823, 0.18739, 0.18665, 0.32785, 0.23471, 0.18718, 0.33825, 0.24396, and 0.18645, respectively, for various chaotic functions as in the order of Table 7.

3.2.2. Chaotic COA for Double Diode Model

Figure 23 shows that the COA with chaos achieves the optimal value of 0.020908 for the LSM20 solar module (DDM) in 800 iterations, which is much less than the COA without the chaos. Furthermore, each chaotic map’s time to converge and reach the optimal value of 0.020908 is shown in

Table 11. Time taken by COA with the ten chaotic functions for LSM20 solar module (DDM).

S. No.	Chaotic Maps	Time (Seconds)
C 1	Chebyshev	1.32
C 2	Circle	1.26
C 3	Gauss/mouse	1.269
C 4	Iterative	1.272
C 5	Logistic	1.258
C 6	Piecewise	1.246
C 7	Sine	1.266
C 8	Singer	1.251
C 9	Sinusoidal	1.308
C 10	Tent	1.200

.

Figure 24 shows that the COA with chaos achieves the optimal value of 0.5697 for the Photowatt-PWP 201 solar cell module (DDM) in 127 iterations, which is much less than the COA without the chaos. Furthermore, each chaotic map’s time is shown in

Table 12. Time taken by COA with the ten chaotic functions for Photowatt-PWP201 solar cell module (DDM).

S. No.	Chaotic Maps	Time (Seconds)
C 1	Chebyshev	0.6390
C 2	Circle	0.6180
C 3	Gauss/mouse	0.5940
C 4	Iterative	0.6020
C 5	Logistic	0.6160
C 6	Piecewise	0.580
C 7	Sine	0.598
C 8	Singer	0.592
C 9	Sinusoidal	0.601
C 10	Tent	0.596

.

Figure 25 shows that the COA with chaos achieves the optimal value of 0.03468 for GaAs thin-film solar cell PVM752 in 140 iterations, which is much less than the COA without the chaos. Furthermore, each chaotic map’s time is shown in

Table 13. Time taken by COA with the ten chaotic functions for GaAs thin-film solar cell PVM752 (DDM).

S. No.	Chaotic Maps	Time (Seconds)
C 1	Chebyshev	1.292
C 2	Circle	0.661
C 3	Gauss/mouse	1.268
C 4	Iterative	0.912
C 5	Logistic	1.314
C 6	Piecewise	1.020
C 7	Sine	1.052
C 8	Singer	1.314
C 9	Sinusoidal	1.092
C 10	Tent	1.295

.

The effect of the ten chaotic functions on the RMSE values using COA is shown with the help of boxplots for double diode models of mono-crystalline LSM20 solar module in Figure 26, Photowatt-PWP201 solar cell module in Figure 27, and for GaAs thin-film solar cell module in Figure 28, which shows that the results are better in most of the chaotic maps when compared to using COA alone.

For a mono-crystalline LSM20 solar cell module (DDM), the median value of RMSE were found as 0.049869, 0.071838, 0.039428, 0.038175, 0.073488, 0.056864, 0.041471, 0.071894, 0.069284, and 0.04194, respectively, for various chaotic functions as in the order given in Table 7.

For a polycrystalline Photowatt-PWP201 solar cell module (DDM), the median RMSE values were 0.58195, 0.58233, 0.58031, 0.57613, 0.57982, 0.58613, 0.58026, 0.58301, 0.5898, and 0.58297, respectively, for various chaotic functions as in the order given in Table 7.

In the case of thin-film solar cell PVM752 (DDM), the median RMSE values were obtained as 0.11121, 0.051876, 0.06504, 0.035617, 0.064541, 0.1275, 0.1251, 0.14145, 0.15479, and 0.035838, respectively, for various chaotic functions as in the order of Table 7.

3.2.3. Chaotic COA for the Three Diode Model

Figure 29 shows that the COA with chaos achieves the optimal value of 0.026525 for the LSM20 solar module (TDM) in 750 iterations, which is much less than the COA without the chaos. Furthermore, each chaotic map’s time to converge and reach the optimal value is shown in

Table 14. Time taken by COA with the ten chaotic functions for LSM20 solar module (TDM).

S. No.	Chaotic Maps	Time (Seconds)
C 1	Chebyshev	1.237
C 2	Circle	1.26
C 3	Gauss/mouse	1.244
C 4	Iterative	1.195
C 5	Logistic	1.232
C 6	Piecewise	1.273
C 7	Sine	1.232
C 8	Singer	1.247
C 9	Sinusoidal	1.262
C 10	Tent	1.268

.

Figure 30 shows that the COA with chaos achieves the optimal value of 0.5692 for the Photowatt-PWP 201 solar cell module (DDM) in 100 iterations, which is much less than the COA without the chaos. Furthermore, each chaotic map’s time is shown in

Table 15. Time taken by COA with the ten chaotic functions for Photowatt-PWP201 solar cell module (TDM).

S. No.	Chaotic Maps	Time (Seconds)
C 1	Chebyshev	0.6030
C 2	Circle	0.5640
C 3	Gauss/mouse	0.5790
C 4	Iterative	0.5470
C 5	Logistic	0.5480
C 6	Piecewise	0.5340
C 7	Sine	0.5000
C 8	Singer	0.5250
C 9	Sinusoidal	0.5240
C 10	Tent	0.5310

.

Figure 31 shows that the COA with chaos achieves the optimal value of 0.034863 for GaAs thin-film solar cell PVM752 in 150 iterations, which is much less than the COA without the chaos. Furthermore, each chaotic map’s time is shown in

Table 16. Time taken by COA with the ten chaotic functions for GaAs thin-film solar cell PVM752 (TDM).

S. No.	Chaotic Maps	Time (Seconds)
C 1	Chebyshev	0.7410
C 2	Circle	0.7430
C 3	Gauss/mouse	0.7680
C 4	Iterative	0.7110
C 5	Logistic	0.7470
C 6	Piecewise	0.7890
C 7	Sine	0.7470
C 8	Singer	0.7550
C 9	Sinusoidal	0.7490
C 10	Tent	0.7360

.

The effect of the ten chaotic functions on the RMSE values using COA is shown with the help of boxplots for double diode models of mono-crystalline LSM20 solar module in Figure 32, Photowatt-PWP201 solar cell module in Figure 33, and for GaAs thin-film solar cell module in Figure 34, which shows that the results are better than using COA alone.

For a mono-crystalline LSM20 solar cell module (TDM), the median values of RMSE were found as 0.25648, 0.35002, 0.39102, 0.042622, 0.30653, 0.22045, 0.26169, 0.13361, 0.27937, and 0.0895, respectively, for various chaotic functions as in the order given in Table 7.

For a polycrystalline Photowatt-PWP201 solar cell module (TDM), the median RMSE values were 0.77655, 0.86682, 0.61008, 0.80594, 1.0487, 0.84171, 1.0181, 0.9137, and 0.63599, respectively, for various chaotic functions as in the order given in Table 7.

In the case of thin-film solar cell PVM752 (DDM), the median RMSE values were obtained as 86.284, 57.1823, 143.9872, 0.2632, 120.2435, 92.8693, 113.6986, 100.0146, and 146.8776, respectively, for various chaotic functions as in the order of Table 7.

3.3. Effects on the Results of COA with Variation in Temperature and Irradiance

The COA technique is used to extract a single diode model’s optimal parameters for different types of solar cells and modules (mono-crystalline solar cell module-LSM20, polycrystalline Photowatt-PWP201 solar cell module, and GaAs thin-film solar cell-PVM752). In the case of some type of solar PV system, the I–V characteristics can be directly achieved using the data that is provided by the manufacturer at different solar radiations and temperature conditions. It is also possible for some cases of solar PV that the I–V characteristics can be obtained at different environmental conditions by using the parameters at Standard Test Conditions (STC) provided by the manufacturer, such as V_0C-STC (open circuit voltage), I_SC-STC (short circuit current), V_MPP-STC (maximum power point voltage), I_MPP-STC (maximum power point current), and temperature coefficients α and β, from the following equations:

$${\mathrm{I}}_{\mathrm{SC}}\left(\mathrm{G},\mathrm{T}\right)={\mathrm{I}}_{\text{SC-STC}}\ast \frac{\mathrm{G}}{{\mathrm{G}}_{\mathrm{STC}}}+\mathsf{\alpha }\left(\mathrm{T}-{\mathrm{T}}_{\mathrm{STC}}\right)$$

(29)

$${\mathrm{I}}_{\mathrm{MPP}}\left(\mathrm{G},\mathrm{T}\right)={\mathrm{I}}_{\text{MPP-STC}}\ast \frac{\mathrm{G}}{{\mathrm{G}}_{\mathrm{STC}}}$$

(30)

$${\mathrm{V}}_{\mathrm{OC}}\left(\mathrm{G},\mathrm{T}\right)={\mathrm{V}}_{\text{OC-STC}}-\mathsf{\beta }(\mathrm{T}-{\mathrm{T}}_{\mathrm{STC}})+\frac{\mathrm{nkT}}{\mathrm{q}}\ast \ln (\frac{\mathrm{G}}{{\mathrm{G}}_{\mathrm{STC}}})$$

(31)

$${\mathrm{V}}_{\mathrm{MPP}}\left(\mathrm{G},\mathrm{T}\right)={\mathrm{V}}_{\text{MPP-STC}}-\mathsf{\beta }\left(\mathrm{T}-{\mathrm{T}}_{\mathrm{STC}}\right)$$

(32)

Table 17. Extracted parameters for a mono-crystalline solar cell module (LSM20) at different temperatures and irradiance of 1000 W/m².

Parameters	25 °C	60 °C	80 °C
Rs	0.2861	0.2973	0.2880
Rsh	6485.3459	3711.1515	2790.0845
IL	5	1.3536	0.5467
I0	2.0231	0	3.873
n	1.6185	1.4837	1.4178
RMSE	1.164391	1.1592703	1.156579

,

Table 18. Extracted parameters for Photowatt-PWP201 solar cell module at different temperatures and irradiance of 1000 W/m².

Parameters	25 °C	60 °C	80 °C
Rs	0.05	0.05	0.05
Rsh	8308.43	5804.069	1107.783
IL	1.9204	5	0.7208
I0	1.2713	0	1.1682
n	1.5441	1.4156	1.3528
RMSE	0.88267	0.877021	0.872806

and

Table 19. Extracted parameters for GaAs thin-film solar cell at different temperatures and irradiance of 1000 W/m².

Parameters	25 °C	60 °C	80 °C
Rs	1	1	1
Rsh	9765.4351	5061.5258	2720.594
IL	3.7317	5	0.1392
I0	0.3085	0.820	3.4280
n	2.7484	2.5197	2.4074
RMSE	1.45346	1.4455	1.44154

show the extracted parameters at an irradiance of 1000 W/m² at different temperatures for mono-crystalline solar cell module (LSM20), Photowatt-PWP201 solar cell module and GaAs thin-film solar cell.

Table 20. Extracted parameters for a mono-crystalline solar cell module (LSM20) at different irradiances and a temperature of 25 °C.

Parameters	200 W/m2	500 W/m2	800 W/m2	1000 W/m2
Rs	0.2450	0.2979	0.3075	0.3102
Rsh	1878.296	3095.537	2990.48	7249.32
IL	0.0464	4.9285	1.2624	5
I0	1.8635	2.5201	0.7712	2.8617
n	1.68986	1.5814	1.5311	1.5084
RMSE	0.55417	1.701389	2.8567	3.6297

,

Table 21. Extracted parameters for a polycrystalline solar cell module (PWP201) at different irradiances and a temperature of 25 °C.

Parameters	200 W/m2	500 W/m2	800 W/m2	1000 W/m2
Rs	0.05	0.05	0.05	0.05
Rsh	454.525	1713.430	3776.4653	9997.4275
IL	4.0432	5	3.9312	1.4427
I0	3.1249	0.9070	5	5
n	1.9233	1.6266	1.5544	1.5250
RMSE	1.15172	0.83263	0.75804	0.88636

and

Table 22. Extracted parameters for GaAs thin-film solar cell at different irradiances and a temperature of 25 °C.

Parameters	200 W/m2	500 W/m2	800 W/m2	1000 W/m2
Rs	1	1	1	1
Rsh	2253.8709	3304.608	7208.2173	8214.6883
IL	2.4687	0.4442	2.4873	5
I0	0.2695	0.7437	2.8613	0.5273
n	3.151	2.898	2.7851	2.3494
RMSE	0.1164	0.61287	1.11641	1.45343

show the extracted parameters at an irradiance of at 25 °C and different irradiance for mono-crystalline solar cell module (LSM20), Photowatt-PWP201 solar cell module and GaAs thin-film solar cell.

3.4. Consistency of the Algorithm

The algorithm was run, and the decision variables, along with the objective values, were computed over 25 times independently. The Root Mean Square Error (RMSE) indicates that the algorithm can produce the same result when the algorithm is run multiple times. The boxplots in Figure 20, Figure 21, Figure 22, Figure 26, Figure 27, Figure 28, Figure 32, Figure 33 and Figure 34, comparing the RMSE values for SDM, DDM, and TDM for various types of cells, show that the algorithm converges consistently over the entire search space.

4. Conclusions

In this paper, the Coyote Optimization Algorithm is used with chaos to enhance the optimization technique, and it is verified in the results for three different types of solar PV cells/modules, namely, mono-crystalline LSM20 solar PV module, polycrystalline Photowatt-PWP201 solar PV module, and GaAs thin-film solar cell PVM752. The results show that the COA technique with chaos takes less time to converge with a lesser number of iterations. On comparing the results with various other parameter extraction techniques, it was observed that COA attains the best value of the objective function (RMSE) when used with chaotic functions. This is evident from the mono-crystalline solar cell LSM20 module (SDM) results, for which the COA technique converged to the best objective value of 0.1752 in 1.330 s, which is better than the DE, PSO, and PGSOSA techniques. Furthermore, when the COA was aided with chaotic maps, the time taken to attain the objective function value was further reduced to the minimum of 0.7540 s, which is an appreciable improvement in the algorithm’s performance. This improvement in the convergence speed can also be observed for all types of photovoltaic cells discussed in the paper. Maximum Power Point Tracking of solar photovoltaics may be achieved in the future with the help of this algorithm. Other applications of COA include extracting the parameters of fuel cells and other renewable energy systems.