## 5 Results and Discussion
### Parameter Estimation Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**PSOGSA vs.** & **GSA** & **PSO** & **SCA** & **HBO** & **PO** & **MVA** \\ \hline
**MSDM** & \(6.7957\times 10^{-8}\) & \(6.7953\times 10^{-8}\) & \(6.7574\times 10^{-8}\) & \(6.7951\times 10^{-8}\) & \(6.7940\times 10^{-8}\) & \(6.7956\times 10^{-8}\) \\
**MDDM** & \(6.7860\times 10^{-8}\) & \(6.7954\times 10^{-8}\) & \(6.7956\times 10^{-8}\) & \(6.7950\times 10^{-8}\) & \(6.7960\times 10^{-8}\) & \(6.0830\times 10^{-8}\) \\
**MTDM** & \(6.7956\times 10^{-8}\) & \(6.7954\times 10^{-8}\) & \(6.7860\times 10^{-8}\) & \(6.7949\times 10^{-8}\) & \(6.7944\times 10^{-8}\) & \(6.7956\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Statistical results based on Wilcoxon’s rank sum test for the monocrystalline PV Cell. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Algorithm** & & **SSE** & **AE** & **MAE** & **MSE** & **RMSE** \\ \hline
**GSA** & **Minimum** & \(1.00\times 10^{-5}\) & \(3.11\times 10^{-3}\) & \(1.51\times 10^{-4}\) & \(5.00\times 10^{-7}\) & \(7.00\times 10^{-4}\) \\
**GSA** & **Maximum** & \(9.34\times 10^{-5}  
Figure 8: The convergence curve of the monocrystalline MSDM. Figure 9: The convergence curve of the monocrystalline MDDM. **Case 2:** This case is also divided into three parts as follows for the polycrystalline PV solar cell:  
* _Modified Single-diode Model (MSDM):_ Table 12 shows the results of parameter estimation using different optimization algorithms. Figure 11 represents the convergence curve of the MSDM. From this convergence curve, it is concluded that the efficiency of the proposed algorithm is better than the rest of the compared algorithms. The errors, mean, and standard deviation of the MSDM circuit equivalent to a PV cell are shown in Table 13. Based on the results, it can be inferred that parameter estimation using the developed hybrid PSOGSA is superior to other algorithms. * _Modified Double-Diode Model (MDDM):_ Table 14 shows the results of parameter estimation using different optimization algorithms. Figure 12 represents the convergence curve of the MDDM for this case. The errors, mean, and standard deviation of the MDDM circuit equivalent to a PV cell are shown in Table 15. Based on the results, it can be concluded that parameter estimation using the developed hybrid PSOGSA is superior to other algorithms. * _The Modified Three-Diode Model (MTDM):_ Table 16 shows the results of parameter estimation using different optimization algorithms. Figure 13 represents the convergence curve for Case 2 that proves the efficiency of the proposed algorithm. The \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter/Algorithm** & **MVO** & **PSO** & **SCA** & **HBO** & **PO** & **GSA** & **PSOGSA** \\ \hline \(\mathbf{I_{pv}}\) & 8.9183 & 9.1173 & 8.9245 & 4.8097 & 4.8340 & 4.5256 & 8.8803 \\ \(\mathbf{Alpha_{1}}\) & 1.5428 & 1.3272 & 1.4979 & 0.8534 & 0.8741 & 1.4562 & 1.5419 \\ \(\mathbf{R_{ue}}\) & 0.0094 & 0.0010 & 0.0151 & 0.0490 & 0.0268 & 0.2300 & 0.0174 \\ \(\mathbf{R_{ch}}\) & 277.670 & 63.070 & 320.452 & 279.525 & 294.365 & 225.714 & 365.338 \\ \(\mathbf{I_{c1}}\) & \(1.00\times 10^{-6}\) & \(1.87\times 10^{-7}\) & \(6.99\times 10^{-7}\) & \(4.19\times 10^{-7}\) & \(5.91\times 10^{-7}\) & \(4.54\times 10^{-7}\) & \(1.00\times 10^{-6}\) \\ \(\mathbf{R_{u}}\) & 0.0045 & 0.0015 & 0.0125 & 0.0380 & 0.0214 & 0.1800 & 0.0258 \\
**Computation time** & 2.014 & 3.514 & 3.321 & 2.748 & 2.104 & 1.421 & 1.520 \\
**per run (s)** & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Parameter Estimation of the MSDM for the polycrystalline PV Cell. Figure 10: The convergence curve of the monocrystalline MTDM. errors, mean, and standard deviation of the MTDM circuit equivalent to a PV cell are shown in Table 17. Based on the results, it can be inferred that parameter estimation using the developed hybrid PSOGSA is superior to other algorithms. * _Non-Parametric Test:_ Non-parametric tests are performed to justify the results obtained by all algorithms.