## 4 Application of the CrSA for Parameter Extraction of the Solar Module
### Results for the SDiM of Mono PERC Solar PV Modules  
The parameter estimation process utilizes Mono PERC WSMD-545 and CS7L-590 MS solar PV modules. Table 5 presents the optimal parameters of the SDiM of the Mono PERC WSMD-545 solar PV module obtained from 20 runs, while Figure 3 illustrates the outcomes of the simulation. Among the five parameters in the SDiM, \(\rho\), \(\mathcal{R}_{sh}\), and \(\mathcal{R}_{s}\) are optimized using the CrSA method, while \(I_{ph}\) and \(I_{sa}\) are obtained from the derived analytical equations. Table 6 presents the optimal parameters of the SDiM of the Mono PERC CS7L-590 MS solar PV module obtained from 20 runs, while Figure 4 illustrates the outcomes of the simulation. Among the five parameters in the SDiM, \(\rho\), \(\mathcal{R}_{sh}\), and \(\mathcal{R}_{s}\) are optimized using the CrSA method, while \(I_{ph}\) and \(I_{sa}\) are obtained from the derived analytical equations. \begin{table}
\begin{tabular}{c c} \hline \hline
**SDiM** & **DDiM** \\ \hline \(\rho=[0.5,\,2.0]\), & \(\rho_{1},\rho_{2}=[0.5,\,2.0]\), \\ \(\mathcal{R}_{s}(\Omega)=[0.001,\,1]\), & \(\mathcal{R}_{s}(\Omega)=[0.001,\,1]\), \\ \(\mathcal{R}_{sh}(\Omega)=[50,\,200]\) & \(\mathcal{R}_{sh}(\Omega)=[50,\,200]\), \\  & \(I_{sal}(A)=\left[10^{-12},\,10^{-6}\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Limits of SDiM and DSiM parameters. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Run**} & \multicolumn{2}{c}{**CSrA**} & \multicolumn{2}{c}{**Analytical Approach**} & \multirow{2}{*}{**Error (\(\rho\))**} \\ \cline{2-2} \cline{5-6}  & \(\rho\) & \(\mathcal{R}_{s}[\Omega]\) & \(\mathcal{R}_{sh}[\Omega]\) & \(I_{sa}[\Lambda]\) & \(I_{ph}[\Lambda]\) \\ \hline
1 & 0.501029 & 0.022692 & 63.61758 & \(2.88\times 10^{-11}\) & 13.90496 & 3.11 \(\times\)\(10^{-11}\) \\
2 & 0 \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Run**} & \multicolumn{3}{c}{**CSrA**} & \multicolumn{3}{c}{**Analytical Approach**} & \multirow{2}{*}{**Error (\(\varphi\))**} \\ \cline{2-2} \cline{5-6}  & \(\rho\) & \(\mathcal{R}_{\text{s}}\)**[\(\Omega\)]** & \(\mathcal{R}_{\text{sh}}\)**[\(\Omega\)]** & \(\mathcal{I}_{\text{sa}}\)**[\(A\)]** & \(\mathcal{I}_{\text{ph}}\)**[\(A\)]** \\ \hline  
& 5.28 \(\times\)\(10^{-11}\) & 18.39403 & 1.02 \(\times\)\(10^{-10}\) \\ \hline \end{tabular}
\end{table} 
Table 6: The optimal estimated parameters for the SDiM of the Mono PERC CS7L-590 MS solar PV module using the CrSA and the analytical technique. Figure 3: (a) \(V\)-\(I\) characteristics of the SDiM of the Mono PERC WSMD-545 solar PV panel; (b) optimal solution set for parameter extraction of the SDiM of the Mono PERC WSMD-545 solar PV panel. Figure 4: (**a**) _V-I_ characteristics of the SDiM 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 SDiM of the Mono PERC CS7L-590 MS solar PV panel; (**c**) optimal solution set for parameter extraction of the SDiM of the Mono PERC CS7L-590 MS solar PV panel. Figures 3 and 4 demonstrate that the 20 \(V\)-\(I\) characteristics intersect at three specific points, which is the main consideration for developing the objective function. The error at three places is nearly \(10^{-14}\); however, in most investigations, the error is only considered at the _MPP_. Consequently, all 20 trials yield practically indistinguishable \(V\)-\(I\) characteristics when considering different possible parameter extractions. Among the five parameters in the SDiM, \(\rho\), \(\mathcal{R}_{sh}\), and \(\mathcal{R}_{s}\) are optimized using the CrSA method, while the \(I_{ph}\) and \(I_{sa}\) 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. The values of \(\mathcal{R}_{s}\) and \(\mathcal{R}_{sh}\) determine the slope of the curve at the _OC_ and _SC_ locations.