# Parameter Extraction of Photovoltaic Cells and Panels Using a PID-Based Metaheuristic Algorithm
## 3 Results and Discussion
### Multijunction Solar Cells (MJSC)  
Three datasets were measured for an InGaP/InGaAs/Ge multijunction solar cell (MJSC) at three different temperatures. The MJSC, with a 1 cm\({}^{2}\) active area, was tested under one-sun irradiance at three temperature conditions: Case A (41.5 \({}^{\circ}\)C), Case B (51.3 \({}^{\circ}\)C), and Case C (61.6 \({}^{\circ}\)C). These measurements were conducted in natural sunlight using the RELab system. The datasets were utilized to extract either five or seven parameters of the MJSC based on the SDMs and DDMs [33]. The associated errors and current values derived from the three cases are presented in Tables S6-S8. The PSA was compared with other methods, including BWOA, the Chameleon Swarm Algorithm (CSA) [33], and the 5P method. The results obtained using the PSA for both the SDMs and DDMs are summarized in Table 23. To evaluate algorithm performance, the statistical tests RMSE and MBE were employed. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of Runs** & **Population** & **Number of** **Iterations** & **RMSE** & **Run Time (s)** \\ \hline \hline  & 20 & 100 & \(3.30\times 10^{-3}\) & 0.1077 \\ \cline{2-5}  & 30 & 100 & \(3.49\times 10^{-2}\) & 0.1556 \\ \cline{2-5}  & 50 & 100 & \(2.25\times 10^{-2}\) & 0.2524 \\ \cline{2-5}  & 60 & 100 & \(1.45\times 10^{-2}\) & 0.2968 \\ \cline{2-5}  & 80 & 100 & \(1.87\times 10^{-3}\) & 0.4020 \\ \cline{2-5}  & 100 & 100 & \(1.80\times 10^{-3}\) & 0.4867 \\ \cline{2-5}
5 & 50 & 300 & \(5.9485\times 10^{-3}\) & 0.7507 \\ \cline{2-5}  & 50 & 500 & \(2.7450\times 10^{-3}\) & 1.2540 \\ \cline{2-5}  & 50 & 700 & \(3.3232\times 10^{-3}\) & 1.7156 \\ \cline{2-5}  & 50 & 900 & \(2.2871\times 10^{-3}\) & 2.2240 \\ \cline{2-5}  & 50 & 1100 & \(2.7683\times 10^{-3}\) & 2.5609 \\ \cline{2-5}  & 50 & 1300 & \(1.7029\times 10^{-3}\) & 3.0451 \\ \cline{2-5}  & 50 & 1500 & \(1.6892\times 10^{-3}\) & 3.5337 \\ \hline \hline \end{tabular}
\end{table} 
Table 21: STM6-40 iteration vs. population size DDM case. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of Runs** & **Model** & \
\begin{tabular}{c} **Population Size/** **Number of Iterations** \\ \end{tabular} & **RMSE** &
\begin{tabular}{c} **Intel Core i9 CPU** \\ **Execution Time (s)** \\ \end{tabular} & **AMD Ryzen 5 5600H** \\ \hline \multirow{2}{*}{5} & SDM & 100/1100 & \(1.7298\times 10^{-3}\) & 2.6012 & 4.9071 \\ \cline{2-5}  & DDM & 100/1100 & \(1.6892\times 10^{-3}\) & 4.4091 & 8.0901 \\ \hline \hline \end{tabular}
\end{table} 
Table 22: STM6-40 CPU execution time. The PSA consistently outperformed BWOA and CSA across all temperatures and models. It achieved significant RMSE reductions, with improvements of up to 42.26% over BWOA and 88.11% over CSA. Additionally, the PSA demonstrated exceptional accuracy and stability in parameter extraction, particularly for \(I_{ph}\), \(I_{sd1}\), \(R_{sh}\), and \(n1\), even in complex scenarios and at higher temperatures. At Case A (41.5 \({}^{\circ}\)C), for the SDM, PSA achieved the lowest RMSE of 7.8273 \(\times\) 10\({}^{-5}\), improving over BWOA and CSA by 12.17% and 23.17%, respectively. The extracted parameters from PSA showed minimal variation, particularly for the photogenerated current (\(I_{ph}\)), which differed by only 0.12% compared to other algorithms. Moreover, PSA outperformed its counterparts in accurately estimating reverse saturation current (\(I_{sd1}\)) and shunt resistance (\(R_{sh}\)), with deviations significantly lower than those produced by CSA. In the DDM, PSA continued to lead with an RMSE of 7.803119 \(\times\) 10\({}^{-5}\), reflecting a 12.42% improvement over BWOA and a 22.67% improvement over CSA. The extracted parameters from PSA were more consistent, with \(R_{s}\) and \(n1\) values remaining stable, while \(R_{sh}\) and \(I_{sd1}\) exhibited less deviation than those obtained using the CSA. At a higher temperature of 51.3 \({}^{\circ}\)C (Case B), PSA maintained its leading performance.