## 5 Results and Discussion
### Parameter Estimation Results  
Two solar cells are used to estimate the parameters i.e., Monocrystalline and Polycrystalline, as shown in Table 3. In this section, the two cases are discussed in which the modified single-diode, the modified double-diode and the modified three-diode model parameter estimations have been carried out. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Test Function** & **n** & **Range** \\ \hline \(F_{1}(x)=\sum\limits_{i=1}^{n}x_{i}^{2}\) & 30 & \([-100,100]\) \\ \(F_{2}(x)=\sum\limits_{i=1}^{n}[x_{i}]+\prod\limits_{i=1}^{n}|x_{i}|\) & 30 & \([-10,10]\) \\ \(F_{3}(x)=\sum\limits_{i=1}^{n}\left(\sum\limits_{j=1}^{i}x_{i}\right)^{2}\) & 30 & \([-100,100]\) \\ \(F_{4}(x)=max|x_{i}|\), \(1\leq x\leq n\) & 30 & \([-100,100]\) \\ \(F_{5}(x)=\sum\limits_{i=1}^{n-1}\left[100\big{(}x_{i+1}-x_{i}^{2}\big{)}^{2}+ \left(x_{i}-1\right)^{2}\right]\) & 30 & \([-30,30]\) \\ \(F_{6}(x)=\sum\limits_{i=1}^{n}\left(x_{i}+0.5\right)^{2}\) & 30 & \([-100,100]\) \\ \(F_{7}(x)=\sum\limits_{i=1}^{n}ix_{i}^{4}+random[0,1]\) & 30 & \([-1.28,1.28]\) \\ \(F_{8}(x)=\sum\limits_{i=1}^{n}\left[-x_{i}\sin\left(\sqrt{|x_{i}|}\right)\right]\) & 30 & \([-500,500]\) \\ \(F_{9}(x)=10n+\sum\limits_{i=1}^{n}x_{i}^{2}-10\cos(2\pi x_{i})\) & 30 & \([-5.12,5.12]\) \\ \(F_{10}(x)=-20 exp\left(-0.2\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}x_{i}^{2}} \right)-exp\left(\frac{1}{n}\sum\limits_{i=1}^{n}\cos(2\pi x_{i})\right)+20+e\) & 30 & \([-32,32]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definitions of benchmark functions. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**MVO**} & \multicolumn{2}{c}{**PSO**} & \multicolumn{2}{c}{**SCA**} & \multicolumn{2}{c}{**HBO**} & \multicolumn{2}{c}{**PO**} & \multicolumn{2}{c}{**GSA**} & \multicolumn{2}{c}{**PSOGSA**} \\ \cline{2-13}  & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** \\ \hline
**F\({}_{1}\)** & 0.203 & 0.135 & 0.0007 & 0.002 & 0.137 & 0.005 & 0.039 & 0.025 & 0.0022 & 0.014 & 0.2619 & 0.4904 & \(\mathbf{\pi}4.7\times 10^{-  
**Case 1:** The solar cell is considered monocrystalline in this case. This case is further divided into three parts as follows:  
1. _Modified Single-Diode Model (MSDM):_ Table 4 shows the results of parameter estimation using different optimization algorithms. Figure 8 represents the MSDM convergence curve. From this curve, we conclude that the efficiency of the proposed algorithm is better than the rest of the comparative algorithms. The errors, mean, and standard deviation of the MSDM circuit equivalent of a PV cell are shown in Table 5. Based on the results, it can be inferred that parameter estimation using the developed hybrid PSOGSA is superior to other comparative algorithms. 2. _Modified Double-Diode Model (MDDM):_ Table 6 shows the results of parameter estimation using different optimization algorithms. Figure 9 represents the convergence curve for MDDM. It was concluded that the efficiency of the proposed algorithm is better than the rest of the comparative algorithms. The errors, mean, and standard deviation of the MDDM circuit equivalent to a PV cell are shown in Table 7. Based on the results, it can be concluded that parameter estimation using the developed hybrid PSOGSA is superior to other algorithms. 3. _The Modified Triple-Diode Model (MTDM):_ Table 8 shows the results of parameter estimation using different optimization algorithms. Figure 10 represents the convergence curve of the MTDM that proves the efficiency of the proposed algorithm over other algorithms. The errors, mean, and standard deviation of the MTDM circuit equivalent to a PV cell are shown in Table 9. Based on the results, it can be inferred that parameter estimation using the developed hybrid PSOGSA is superior to other algorithms. 4. _Non-Parametric Test:_ Non-parametric tests are performed to justify the results obtained by all algorithms.