## 4 Application of the CrSA for Parameter Extraction of the Solar Module
### Results for the DDiM of Mono PERC Solar PV Modules  
Among the seven parameters in the DDiM, \(\rho_{1}\), \(\rho_{2}\), \(\mathcal{R}_{sh}\), \(\mathcal{R}_{s}\), and \(I_{sa1}\) are optimized using the CrSA method, while \(I_{ph}\) and \(I_{sa2}\) are obtained from the derived analytical equations. Table 7 presents the optimal parameters of the DDiM of the Mono PERC WSMD-545 obtained from 20 runs, while Figure 5 illustrates the outcomes of the simulation.  
Table 8 presents the optimal parameters of the DDiM of the Mono PERC CS/L-590 MS obtained from 20 runs, while Figure 6 illustrates the outcomes of the simulation.  
The 20 \(V\)-\(I\) characteristics flow through three locations in Figures 5 and 6, which is the primary factor considered in the objective function. The error at the three places is nearly \(10^{-14}\). Out of seven parameters in the DDiM, five parameters, \(\rho_{1}\), \(\rho_{2}\), \(\mathcal{R}_{sh}\), \(\mathcal{R}_{s}\), and \(I_{sa1}\), are optimized using the CrSA method, and the remaining two parameters, \(I_{ph}\) and \(I_{sa2}\), are obtained from the derived analytical equations. The \(V\)-\(I\) curves acquired in each run are optimum due to the optimized parameters within the defined search space bounds. This is because the provided approach is metaheuristic.  
From the simulation results of the SDiM and DDiM of Mono PERC WSMD-545 and CS/L-590 MS solar PV panels, it is evident that the parameters estimated using the suggested CrSA method guarantee the precision of the \(V\)-\(I\) characteristics and that the three points obtained align with the datasheet.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{Run} & \multicolumn{4}{c}{**CrSA**} & \multicolumn{4}{c}{**Analytical Approach**} \\ \cline{2-9}  & \(\rho_{1}\) & \(\rho_{2}\) & \(\mathcal{R}_{s}\)(\(\Omega\)) & \(\mathcal{R}_{sh}\)(\(\Omega\)) & \(I_{sa1}\)(\(A\)) & \(I_{sa2}\)(\(A\)) & \(I_{ph}\) & _Error(\(\rho\))_ \\ \hline
1 & 0.665423 & 0.5 & 0.044369 & 67.1908 & \(1.00\times 10^{-12}\) & \(2.74\times 10^{-11}\) & 13.90918 & \(2.62\times 10^{-16}\) \\
2 & 0.846636 & 0.511733 & 0.001693 & 62.4105 & \(1.00\times 10^{-12}\) & \(5.05\times 10^{-11}\) & 13.90038 & \(3.30\times 10^{-12}\) \\
3 & 1.263774 & 0.544015 & 0.046843 & 81.9756 & \(8.69\times 10^{-02}\) & \(2.43\times 10^{-10}\) & 13.90794 & \(8.48\times 10^{-13}\) \\
4 & 0.5 & 0.52135 & 0.001 & 63.7887 & \(1.00\times 10^{-12}\) & \(7.91\times 10^{-11}\) & 13.90022 & \(2.65\times 10^{-11}\) \\
5 & 1.264447 & 0.540702 & 0.037732 & 77.5655 & \(9.01\times 10^{-02}\) & \(2.08\times 10^{-10}\) & 13.90676 & \(4.21\times 10^{-11}\) \\
6 & 1.819716 & 0.5 & 0.013479 & 62.1266 & \(1.08\times 10^{-02}\) & \(2.72\times 10^{-11}\) & 13.90302 & \(3.26\times 10^{-13}\) \\
7 & 0 & 0.515526 & 0.001 & 62.8402 & \(10.00\times 10^{-12}\) & \(5.90\times 10^{-11}\) & 13.90022 & \(1.11\times 10^{-12}\) \\
8 & 0.918244 & 0.5 &  
Figure 5: (a) \(V\)-\(I\) characteristics of the DDiM of the Mono PERC WSMD-545 solar PV panel; (b) \(V\)-\(I\) characteristics with maximum \(\mathcal{R}_{sh}\) and \(\mathcal{R}_{s}\) of the DDiM of the Mono PERC WSMD-545 solar PV panel; (c) optimal solution set for parameter extraction of the DDiM of the Mono PERC WSMD-545 solar PV panel.  
Figure 6: (**a**) \(V\)-\(I\) characteristics of the DDiM of the Mono PERC CS7L-590 MS solar PV panel; (**b**) \(V\)-\(I\) characteristics with maximum \(\mathcal{R}_{sh}\) and \(\mathcal{R}_{s}\) of the DDiM of the Mono PERC CS7L-590 MS solar PV panel; (**c**) optimal solution set for parameter extraction of the DDiM of the Mono PERC CS7L-590 MS solar PV panel.