# 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

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## Abstract

**:**

## 1. Introduction

- 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.

#### 1.1. Classical Optimization Methods

#### 1.2. Metaheuristic Optimization Methods

^{−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.

## 2. Materials and Methods

#### 2.1. PV Cell Modelling

_{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

_{L}can be found out by using Equations (1)−(3).

_{L}, I

_{ph}, I

_{sh}, I

_{0}, 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.

_{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:

_{s}, R

_{sh}, I

_{0}, n

_{1}, 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

_{L}, can be found out using Kirchhoff’s Current Law and is expressed as:

_{01}and n

_{1}are the reverse saturation current and ideality factor of diode D

_{1}and I

_{02}, and n

_{2}is the reverse saturation current and ideality factor of the diode D

_{2}.

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

_{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:

_{01}, I

_{02}, and I

_{03}are the reverse saturation currents, while n

_{1}, n

_{2}, and n

_{3}are the ideality factors of diodes D

_{1}, D

_{2,}and D

_{3}, respectively.

_{s}, R

_{sh}, I

_{01}, I

_{02}, I

_{03}, n

_{1}, n

_{2}, n

_{3}, 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

_{i}), the search range, and the objective function. The various vectors of the solution for SDM and DDM are shown in Table 3.

#### 2.3. Chaotic Coyote Optimization Algorithm

_{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:

_{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].

_{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]:

_{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]:

_{1}and j

_{2}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]:

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 |

_{1}(it shows how the alpha influences a random coyote cr

_{1}) and d

_{2}(showing how any random coyote cr

_{2}is influenced by the cultural tendency of the group). cr

_{1}and cr

_{2}are selected using random probability distribution function, while d

_{1}and d

_{2}can be mathematically represented as:

_{1}and r

_{2}are any real random numbers such that 0 ≤ r

_{1}, r

_{2}≤ 1. The new social behavior of the coyotes is estimated by:

#### Parameter Constraints

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 dofor each p pack doFind the pack’s alpha coyote Equation (18) for each cth coyote of pth pack doGenerate 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 forSimulate the birth and the death of the coyotes using Equation (20) and Algorithm 1 end forImpose the probabilities of coyotes leaving their packs using Equation (20) Update the coyotes’ ages endwhileSelect the coyote with most potential to adapt to the environment. |

## 3. Results and Discussions

#### 3.1. Results of COA with the Data from Experiments

^{2}, 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/m

^{2}, and a PVM752 GaAs thin-film solar cell at a temperature of 25 °C and irradiation of 1000 W/m

^{2}.

#### 3.2. Parameter Extraction Using Coyote Optimization Algorithm with Chaotic Maps

#### 3.2.1. Chaotic COA for Single Diode Model

#### 3.2.2. Chaotic COA for Double Diode Model

#### 3.2.3. Chaotic COA for the Three Diode Model

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

_{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:

#### 3.4. Consistency of the Algorithm

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Comparison between RMSE using COA and other techniques for a mono-crystalline LSM20 solar cell module (SDM).

**Figure A2.**Comparison between RMSE using COA and other techniques for Photowatt-PWP201 solar cell module (SDM).

**Figure A3.**Comparison between RMSE using COA and other techniques for GaAs thin-film solar cell (SDM).

**Figure A4.**Comparison between RMSE using COA and other techniques for a mono-crystalline LSM20 solar cell module (DDM).

**Figure A5.**Comparison between RMSE using COA and other techniques for Photowatt-PWP201 solar cell module (DDM).

**Figure A6.**I–V curves for the experimentally measured data and the estimated results for GaAs thin-film solar cell (DDM).

**Figure A7.**Comparison between RMSE using COA and other techniques for a mono-crystalline LSM20 solar cell module (TDM).

**Figure A8.**Comparison between RMSE using COA and other techniques for Photowatt-PWP201 solar cell module (TDM).

**Figure A9.**Comparison between RMSE using COA and other techniques for GaAs thin-film solar cell (TDM).

## Appendix B

**Table A1.**Parameters extracted for a mono-crystalline LSM20 solar PV module by COA and its comparison with other techniques for DDM.

Parameters | COA | DE | PSO | PSOGSA |
---|---|---|---|---|

R_{s} | 0.05 | 0.05 | 0.05 | 0.05 |

R_{sh} | 140.498 | 999.987 | 1000 | 689.6236 |

I_{L} | 0.15017 | 0.1563 | 0.159019 | 0.1608 |

I_{01} | 0 | 0 | 0 | 0 |

I_{02} | 0 | 0.0003 | 0.3879 | 1.3719 |

n_{1} | 0.60272 | 1.510 | 4.381 | 1 |

n_{2} | 0.49903 | 1.123 | 1.7654 | 2 |

RMSE | 0.04083 | 0.04655 | 0.6169 | 0.048612 |

Time (sec.) | 2.5570 | 1.012 | 0.9699 | 7.029 |

**Table A2.**Parameters extracted for a Photowatt-PWP201 solar PV module by COA compared with other techniques for DDM.

Parameters | COA | DE | PSO | PSOGSA |
---|---|---|---|---|

R_{s} | 0.05 | 0.05 | 0.05 | 0.05 |

R_{sh} | 0.6614 | 0.6549 | 0.66173 | 999.987 |

I_{L} | 1.16773 | 1.1716 | 1.16692 | 0.7555 |

I_{01} | 5 | 3 | 5 | 3.8744 |

I_{02} | 5 | 3 | 2.2291 | 3.2104 |

n_{1} | 1.96225 | 1.8770 | 2.1457 | 1.7894 |

n_{2} | 1.96225 | 1.8770 | 1.7413 | 1.7894 |

RMSE | 0.5675 | 0.5856 | 0.7268 | 0.7128 |

Time (sec.) | 2.6330 | 1.146 | 1.040 | 11.650 |

**Table A3.**Parameters extracted for PVM752 GaAs thin-film solar PV cell by COA compared with other techniques for DDM.

Parameters | COA | DE | PSO | PSOGSA |
---|---|---|---|---|

R_{s} | 1 | 0.05 | 0.0469 | 0.0554 |

R_{sh} | 1000 | 13.662 | 1000 | 998.6647 |

I_{L} | 0.12291 | 0.1316 | 0.1228 | 0.1118 |

I_{01} | 0 | 0 | 0.002 | 0.0001 |

I_{02} | 0.01216 | 0 | 0.012 | 0 |

n_{1} | 2.92677 | 2 | 0.4085 | 1.8807 |

n_{2} | 2.39519 | 1.781 | 0.34401 | 1.0829 |

RMSE | 0.04005 | 0.14569 | 0.21708 | 33.3819 |

Time (sec.) | 2.39 | 1.027 | 1.2265 | 11.977 |

**Table A4.**Parameters extracted for a mono-crystalline LSM20 solar PV module by COA and its comparison with other techniques for TDM.

Parameters | COA | DE | PSO | PSOGSA |
---|---|---|---|---|

R_{s} | 0.05 | 0.05 | 0.05 | 0.9745 |

R_{sh} | 902.3729 | 940.92 | 1000 | 863.6476 |

I_{L} | 0.151626 | 0.1536 | 0.16504 | 0.1406 |

I_{01} | 9 × 10^{−4} | 0 | 0.002803 | 4.4411 |

I_{02} | 0 | 0.0054 | 0.000050 | 1.9226 |

I_{03} | 0 | 0.0887 | 0.005 | 1.4669 |

n_{1} | 0.74364 | 0.9445 | 5 | 2.7619 |

n_{2} | 2 | 3.8352 | 1.115 | 4.9773 |

n_{3} | 1.67313 | 3.4296 | 5 | 2.7642 |

RMSE | 0.03192 | 0.0478 | 0.053484 | 0.13897 |

Time (sec.) | 2.420 | 1.0980 | 1.54 | 14.08 |

**Table A5.**Parameters extracted for a Photowatt-PWP201 solar PV module by COA compared with other techniques for TDM.

Parameters | COA | DE | PSO | PSOGSA |
---|---|---|---|---|

R_{s} | 0.05 | 0.05 | 0.05 | 0.3571 |

R_{sh} | 0.66614 | 0.6640 | 0.664016 | 629.6271 |

I_{L} | 1.17658 | 1.1694 | 1.16924 | 0.8324 |

I_{01} | 2.02460 | 5 | 5 | 3.5722 |

I_{02} | 4.31185 | 5 | 5 | 2.6484 |

I_{03} | 4.21045 | 4.0206 | 2.8013 | 0.4331 |

n_{1} | 2 | 2.0939 | 2.0939 | 2.8021 |

n_{2} | 2 | 2.0939 | 2.0939 | 2.6180 |

n_{3} | 2 | 4.6802 | 5 | 2.6855 |

RMSE | 0.58053 | 0.5675 | 0.60298 | 0.92678 |

Time (sec.) | 2.230 | 1.321 | 1.437 | 14.518 |

**Table A6.**Parameters extracted for PVM752 GaAs thin-film solar PV cell by COA compared to other techniques for TDM.

Parameters | COA | DE | PSO | PSOGSA |
---|---|---|---|---|

R_{s} | 1 | 1 | 0.05 | 0.2001 |

R_{sh} | 1000 | 1000 | 1000 | 743.355 |

I_{L} | 0.1197 | 0.1266 | 0.12654 | 0.1532 |

I_{01} | 0.00049 | 0.0122 | 0.005 | 4.9820 |

I_{02} | 0 | 0 | 0.0049 | 2.8344 |

I_{03} | 0 | 2.9404 | 0.00025 | 4.3603 |

n_{1} | 2 | 2.3952 | 4.1086 | 4.9048 |

n_{2} | 0.5142 | 4.0086 | 4.1086 | 3.9848 |

n_{3} | 2 | 4.9094 | 5 | 3.8023 |

RMSE | 0.02594 | 0.046074 | 0.0532 | 0.0821 |

Time (sec.) | 2.286 | 1.258 | 1.457 | 13.719 |

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**Figure 5.**I–V curves for the experimentally measured data and the estimated results for a mono-crystalline LSM20 solar cell module (SDM).

**Figure 6.**I–V curves for the experimentally measured data and the estimated results for Photowatt-PWP201 solar cell module (SDM).

**Figure 7.**I–V curves for the experimentally measured data and the estimated results for GaAs thin-film solar cell (SDM).

**Figure 8.**Convergence curves of mono-crystalline LSM20 solar cell module (SDM) comparing the convergence characteristics of COA, DE, PSO, and PSOGSA.

**Figure 9.**Convergence curves of Photowatt-PWP201 solar cell module (SDM) comparing the convergence characteristics of COA, DE, PSO, and PSOGSA.

**Figure 10.**Convergence curves of GaAs thin-film solar cell-PVM752 (SDM) comparing the convergence characteristics of COA, DE, PSO, and PSOGSA.

**Figure 11.**Convergence curves of mono-crystalline LSM20 solar cell module (DDM) comparing the convergence characteristics of COA, DE, PSO, and PSOGSA.

**Figure 12.**Convergence curves of Photowatt-PWP201 solar cell module (DDM) comparing the convergence characteristics of COA, DE, PSO, and PSOGSA.

**Figure 13.**Convergence curves of GaAs thin-film solar cell-PVM752 (DDM) comparing the convergence characteristics of COA, DE, PSO, and PSOGSA.

**Figure 14.**Convergence curves of mono-crystalline LSM20 solar cell module (TDM) comparing the convergence characteristics of COA, DE, PSO, and PSOGSA.

**Figure 15.**Convergence curves of Photowatt-PWP201 solar cell module (TDM) comparing the convergence characteristics of COA, DE, PSO and PSOGSA.

**Figure 16.**Convergence curves of GaAs thin-film solar cell-PVM752 (TDM) comparing the convergence characteristics of COA, DE, PSO, and PSOGSA.

**Figure 17.**Convergence curves of chaotic COA optimization technique during parameter extraction for a mono-crystalline solar cell LSM20 module (SDM).

**Figure 18.**Convergence curves of chaotic COA optimization technique during parameter extraction for Photowatt-PWP201 solar cell module (SDM).

**Figure 19.**Convergence curves of chaotic COA optimization technique during parameter extraction for GaAs thin-film solar cell PVM752 (SDM).

**Figure 20.**Boxplot comparing RMSE values using COA with the help of chaotic maps for a mono-crystalline LSM20 solar cell module (SDM).

**Figure 21.**Boxplot comparing RMSE values using COA with the help of chaotic maps for a polycrystalline PWP-201 solar cell module (SDM).

**Figure 22.**Boxplot comparing RMSE values using COA with the help of chaotic maps for a thin-film solar cell PVM752 (SDM).

**Figure 23.**Convergence curves of chaotic COA optimization technique during parameter extraction for a mono-crystalline solar cell LSM20 module (DDM).

**Figure 24.**Convergence curves of chaotic COA optimization technique during parameter extraction for Photowatt-PWP201 solar cell module (DDM).

**Figure 25.**Convergence curves of chaotic COA optimization technique during parameter extraction for GaAs thin-film solar cell PVM752 (DDM).

**Figure 26.**Boxplot comparing RMSE values using COA with the help of chaotic maps for a mono-crystalline LSM20 solar cell module (DDM).

**Figure 27.**Boxplot comparing RMSE values using COA with the help of chaotic maps for a polycrystalline PWP-201 solar cell module (DDM).

**Figure 28.**Boxplot comparing RMSE values using COA with the help of chaotic maps for a thin-film solar cell PVM752 (DDM).

**Figure 29.**Convergence curves of chaotic COA optimization technique during parameter extraction for a mono-crystalline solar cell LSM20 module (TDM).

**Figure 30.**Convergence curves of chaotic COA optimization technique during parameter extraction for Photowatt-PWP201 solar cell module (TDM).

**Figure 31.**Convergence curves of chaotic COA optimization technique during parameter extraction for GaAs thin-film solar cell PVM752 (TDM).

**Figure 32.**Boxplot comparing RMSE values using COA with the help of chaotic maps for a mono-crystalline LSM20 solar cell module (TDM).

**Figure 33.**Boxplot comparing RMSE values using COA with the help of chaotic maps for a polycrystalline PWP-201 solar cell module (TDM).

**Figure 34.**Boxplot comparing RMSE values using COA with the help of chaotic maps for a thin-film solar cell PVM752 (TDM).

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] |

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] |

Model | Parameters (X_{i}) |
---|---|

Single Diode Model (SDM) | R_{s}, R_{sh}, I_{0}, n, I_{ph} |

Double Diode Model (DDM) | R_{s}, R_{sh}, I_{01}, I_{02}, n_{1}, n_{2}, I_{ph} |

Three Diode Model (TDM) | R_{s}, R_{sh}, I_{01}, I_{02}, I_{03}, n_{1}, n_{2}, n_{3}, I_{ph} |

**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 |
---|---|---|---|---|

R_{s} | 0.00 | 0.00 | 0.00 | 0.00 |

R_{sh} | 100 | 100 | 96.76659 | 82.205 |

I_{L} | 0.2023 | 0.2023 | 0.20259 | 0.2026 |

I_{0} | 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 |
---|---|---|---|---|

R_{s} | 0 | 0 | 0 | 0.0222 |

R_{sh} | 0.6379 | 0.635 | 0.6356339 | 3.1774 |

I_{L} | 1.1201 | 1.1207 | 1.1207088 | 0.7173 |

I_{0} | 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 |

**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 |
---|---|---|---|---|

R_{s} | 0 | 0 | 0 | 0.05 |

R_{sh} | 10.4707 | 10.4707 | 10.4707052 | 61.0883 |

I_{L} | 0.1816 | 0.1816 | 0.18156 | 0.1677 |

I_{0} | 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 |

S. No. | Name | Chaotic Maps | Range |
---|---|---|---|

C 1 | Chebyshev [53] | ${\mathrm{x}}_{\mathrm{i}+1}=\mathrm{cos}({\mathrm{icos}}^{-1}({\mathrm{x}}_{\mathrm{i}}))$ | (−1, 1) |

C 2 | Circle [54] | ${\mathrm{x}}_{\mathrm{i}+1}=\mathrm{mod}\left({\mathrm{x}}_{\mathrm{i}}+\mathrm{b}-\left(\frac{\mathrm{a}}{2\mathsf{\pi}}\right)\mathrm{sin}\left({2\mathsf{\pi}\mathrm{x}}_{\mathrm{k}}\right),1\right),\mathrm{a}=0.5\mathrm{and}\mathrm{b}=0.2$ | (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] | ${\mathrm{x}}_{\mathrm{i}+1}=\mathrm{sin}\left(\frac{\mathrm{a}\mathsf{\pi}}{{\mathrm{x}}_{\mathrm{i}}}\right)\mathrm{a}=0.7$ | (−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] | ${\mathrm{x}}_{\mathrm{i}+1}=\frac{\mathrm{a}}{4}\mathrm{sin}\left({\mathsf{\pi}\mathrm{x}}_{\mathrm{i}}\right),\mathrm{a}=4$ | (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] | ${\mathrm{x}}_{\mathrm{i}+1}={\mathrm{ax}}_{\mathrm{i}}{}^{2}\mathrm{sin}\left(\mathsf{\pi}{\mathrm{x}}_{\mathrm{i}}\right),\mathrm{a}=2.3$ | (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) |

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 |

**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 |

**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 |

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 |

**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 |

**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 |

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 |

**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 |

**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 |

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

^{2}.

Parameters | 25 °C | 60 °C | 80 °C |
---|---|---|---|

R_{s} | 0.2861 | 0.2973 | 0.2880 |

R_{sh} | 6485.3459 | 3711.1515 | 2790.0845 |

I_{L} | 5 | 1.3536 | 0.5467 |

I_{0} | 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/m

^{2}.

Parameters | 25 °C | 60 °C | 80 °C |
---|---|---|---|

R_{s} | 0.05 | 0.05 | 0.05 |

R_{sh} | 8308.43 | 5804.069 | 1107.783 |

I_{L} | 1.9204 | 5 | 0.7208 |

I_{0} | 1.2713 | 0 | 1.1682 |

n | 1.5441 | 1.4156 | 1.3528 |

RMSE | 0.88267 | 0.877021 | 0.872806 |

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

^{2}.

Parameters | 25 °C | 60 °C | 80 °C |
---|---|---|---|

R_{s} | 1 | 1 | 1 |

R_{sh} | 9765.4351 | 5061.5258 | 2720.594 |

I_{L} | 3.7317 | 5 | 0.1392 |

I_{0} | 0.3085 | 0.820 | 3.4280 |

n | 2.7484 | 2.5197 | 2.4074 |

RMSE | 1.45346 | 1.4455 | 1.44154 |

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

Parameters | 200 W/m^{2} | 500 W/m^{2} | 800 W/m^{2} | 1000 W/m^{2} |
---|---|---|---|---|

R_{s} | 0.2450 | 0.2979 | 0.3075 | 0.3102 |

R_{sh} | 1878.296 | 3095.537 | 2990.48 | 7249.32 |

I_{L} | 0.0464 | 4.9285 | 1.2624 | 5 |

I_{0} | 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/m^{2} | 500 W/m^{2} | 800 W/m^{2} | 1000 W/m^{2} |
---|---|---|---|---|

R_{s} | 0.05 | 0.05 | 0.05 | 0.05 |

R_{sh} | 454.525 | 1713.430 | 3776.4653 | 9997.4275 |

I_{L} | 4.0432 | 5 | 3.9312 | 1.4427 |

I_{0} | 3.1249 | 0.9070 | 5 | 5 |

n | 1.9233 | 1.6266 | 1.5544 | 1.5250 |

RMSE | 1.15172 | 0.83263 | 0.75804 | 0.88636 |

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

Parameters | 200 W/m^{2} | 500 W/m^{2} | 800 W/m^{2} | 1000 W/m^{2} |
---|---|---|---|---|

R_{s} | 1 | 1 | 1 | 1 |

R_{sh} | 2253.8709 | 3304.608 | 7208.2173 | 8214.6883 |

I_{L} | 2.4687 | 0.4442 | 2.4873 | 5 |

I_{0} | 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 |

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**MDPI and ACS Style**

Khan, S.A.; Ahmad, S.; Sarwar, A.; Tariq, M.; Ahmad, J.; Asim, M.; Soliman, A.T.; Hossain, M.A.
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. *Electronics* **2021**, *10*, 2094.
https://doi.org/10.3390/electronics10172094

**AMA Style**

Khan SA, Ahmad S, Sarwar A, Tariq M, Ahmad J, Asim M, Soliman AT, Hossain MA.
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. *Electronics*. 2021; 10(17):2094.
https://doi.org/10.3390/electronics10172094

**Chicago/Turabian Style**

Khan, Shoeb Ahmad, Shafiq Ahmad, Adil Sarwar, Mohd Tariq, Javed Ahmad, Mohammed Asim, Ahmed T. Soliman, and Md. Alamgir Hossain.
2021. "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" *Electronics* 10, no. 17: 2094.
https://doi.org/10.3390/electronics10172094