# Parameter Extraction of Photovoltaic Cells and Panels Using a PID-Based Metaheuristic Algorithm
## 3 Results and Discussion
### RTC France Cell  
The first solar cell being examined is the 57 mm diameter RTC France silicon commercial cell. The majority of studies that extract photovoltaic cell parameters and apply metaheuristic algorithms for the SDMs and DDMs include this specific cell, which has been extensively examined. Although the changes could be slight, they are an essential way to show the new algorithm's effectiveness and accuracy. At 1000 W/m\({}^{2}\) and 33 \({}^{\circ}\)C, the RTC silicon photovoltaic cell's I-V characteristic is measured. The associated dataset [34] is shown in Table S1, and it contains values for the current calculated using the SDMs and DDMs, along with their respective errors. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Parameter** & \multicolumn{2}{c}{**RTC Cell**} & \multicolumn{2}{c}{**mSi Cell**} & \multicolumn{2}{c}{**aSi Cell**} & \multicolumn{2}{c}{**MJSC Cell**} \\ \hline  & **LB** & **UB** & **LB** & **UB** & **LB** & **UB** & **LB** & **UB** \\ \hline \(I_{ph}\) (A) & 0 & 1 & 0 & 1 & 0 & 1 & 0.001 & 0.2 \\ \hline \(I_{sd1},I_{sd2}\)(A) & \(10^{-12}\) & \(10^{-5}\) & \(10^{-12}\) & \(10^{-5}\) & \(10^{-12}\) & \(10^{-5}\) & \(10^{-40}\) & \(10^{-5}\) \\ \hline \(R_{s}(\Omega)\) & 0 & 1 & 0 & 1 & 0 & 0.5 & 0.001 & 3 \\ \hline \(R_{sh}(\Omega)\) & 0 & 100 & 0 & 200 & 0 & 1000 & 0 & 100,000 \\ \hline \(n_{1}\), \(n_{2}\) & 1 & 3 & 1 & 3 & 1 & 5 & 1 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: PV cells lower and upper bounds. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \multicolumn{2}{c}{**PVM**} & \multicolumn{2}{c}{**STM6-40**} \\ \hline  & **LB** & **UB** & **LB** & **UB** \\ \hline \(I_{ph}\) (A) & 0 & 1 & 0 & 2 \\ \hline \(I_{sd1},I_{sd2}\)(A) & \(10^{-12}\) & \(10^{-5}\) & \(10^{-12}\) & \(10^{-5}\) \\ \hline \(R_{s}(\Omega)\) & 0 & 1 & 0 & 1 \\ \hline \(R_{sh}(\Omega)\) & 0 & 1000 & 0 & 1000 \\ \hline \(n_{1}\), \(n_{2}\) & 1 & 10 & 1 & 2 \(\times\) 36 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: PV modules lower and upper bounds. Both the SDM's and DDM's execution times of 3.3512 and 6.5292 s, respectively, are shown in Table 3, along with the RTC photovoltaic cell parameters retrieved using the PSA. The performance of the proposed algorithm was compared using RMSE, MBE, and STD with BWOA, ISHTS, NSGNDO, ISOA, AOS, HSDA, GSK, DIWJAYA, memory-based improved gorilla troops optimizer (MIGTO) [32], growth optimizer (GO) [41], improved marine predators algorithm (IMPA) [22], improved moth flame algorithms with local escape operators (IMFOL) [42], modified stochastic fractal search algorithm (MSFS) [43], a logistic chaotic JAYA algorithm (LCJAYA) [44], hybrid backtracking search optimization algorithm with differential evolution (DE/BSA) [45], leveraging the opposition-based exponential distribution optimizer (OBEDO) [46], and improved differential evolution algorithm (DE) [47]. As all of them produce the same RMSE value for the SDM, there is almost no difference in RMSE between the PSA and other top-performing algorithms. The RMSE value for the DDM, however, has improved to 0.67% when compared to BWOA and 1.15% when compared to ISOA, which demonstrates that PSA obtained the best RMSE value. Despite this, the RMSE for the DDM produced with the ASO, MSFS, MIGTO, IMFOL, and GSK algorithms is higher than that obtained with the PSA. In the case of the SDM, because the RMSE values derived from the majority of algorithms are essentially identical, the difference in the retrieved parameters is negligible. This is due to the use of the best algorithms from the specialized literature. Algorithms such as DE and LCJAYA, on the other hand, give limited results because of their tendency to converge to local minima, preventing them from discovering optimal solutions and giving inferior results when compared to other top-performing algorithms for both models. This means that not all the algorithms can obtain the minimum value for RMSE.