# Parameter Extraction of Photovoltaic Cells and Panels Using a PID-Based Metaheuristic Algorithm
## 3 Results and Discussion
### STM6-40 Panel  
The third solar panel under consideration is an STM6-40 monocrystalline panel manufactured by Schutten Solar. These panels are made up of 36 cells connected in series [50]. The I-V characteristic of the STM6-40 is measured at 1000 W/m\({}^{2}\) irradiance and 51 \({}^{\circ}\)C. The dataset of 20 points [35], which is shown in Table S5, used to extract the parameters, includes the current values derived using the SDMs and DDMs, as well as their corresponding errors. The STM6-40 photovoltaic panel's PSA-obtained parameters are displayed in Table 19 for both the SDM and DDM types. The SDM took 2.56053 s to execute, while the DDM version took 4.4231 s. RMSE, MBE, and STD were used to evaluate the effectiveness of the algorithm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of Runs** & **Population** & **Number of** **Iterations** & **RMSE** & **Run Time (s)** \\ \hline \multirow{8}{*}{5} & 20 & 100 & 5.5364 \(\times\)\(10^{-3}\) & 0.1895 \\ \cline{2-5}  & 30 & 100 & 7.6733 \(\times\)\(10^{-3}\) & 0.2784 \\ \cline{2-5}  & 50 & 100 & 8.9201 \(\times\)\(10^{-4}\) & 0.4655 \\ \cline{2-5}  & 60 & 100 & 3.3552 \(\times\)\(10^{-3}\) & 0.5530 \\ \cline{2-5}  & 80 & 100 & 2.9206 \(\times\)\(10^{-3}\) & 0.7356 \\ \cline{2-5}  & 100 & 100 & 5.9449 \(\times\)\(10^{-4}\) & 0.9250 \\ \cline{2-5}  & 50 & 300 & 5.1178 \(\times\)\(10^{-3}\) & 1.3636 \\ \cline{2-5}  & 50 & 500 & 4.8415 \(\times\)\(10^{-4}\) & 2.2742 \\ \cline{2-5}  & 50 & 700 & 2.3892 \(\times\)\(10^{-4}\) & 3.1928 \\ \cline{2-5}  & 50 & 900 & 3.2303 \(\times\)\(10^{-4}\) & 4.1243 \\ \cline{2-5}  & 50 & 1100 & 6.6249 \(\times\)\(10^{-4}\) & 5.0614 \\ \cline{2-5}  & 50 & 1300 & 1.6835 \(\times\)\(10^{-4}\) & 5.8600 \\ \cline{2-5}  & 50 & 1500 & 1.6579 \(\times\)\(10^{-4}\) & 6.8158 \\ \hline \hline \end{tabular}
\end{table} 
Table 17: PVM 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 & 2.2780 \(\times\)\(10^{-4}\) & 5.2231 & 9.8966 \\ \cline{2-5}  & DDM & 100/1100 & 1.6579 \(\times\)\(10^{-4}\) & 8.5112 & 12.1566 \\ \hline \hline \end{tabular}
\end{table} 
Table 18: PVM CPU execution time. The PSA is compared with ELADE, HSDA, GSK, IMPA, DE/BSA, reinforcement learning-based differential evolution (RLDE) [51], rnhanced leader particle swarm optimization (ELPSO) [52], conventional PSO (CPSO) [52], backtracking search algorithm (BSA) [52], and artificial bee colony (ABC) [52]. The PSA and other algorithms produce comparable RMSE values for the SDM, showing no variation in performance. However, when comparing PSA with ELPSO, it is roughly 20.6% better. The DDM's RMSE value is 1.83% better than HSDA's result and 7.72% better than it was when ELPSO was used. For both models, the RMSE performance of the PSA beats the other algorithms. RMSE for SDM shows that the PSA yielded the same results (1.72981 \(\times\) 10\({}^{-3}\)) as the ELADE, HSDA, RLDE, IMPA, and GSK algorithms, showing that the differences between them were quite small. In the case of the DDM, when compared to other algorithms like HSDA and ELPSO, however, certain key differences become apparent. Table 19 shows that, in contrast, the PSA consistently achieves the highest statistical success rates in the DDM. When compared to other methods, the PSA's study of all obtained parameters reveals significant discrepancies. The PSA's parameters displayed slightly bigger variations in values as compared to HSDA in the SDM, leading to a slightly better RMSE value than that produced by other algorithms. On the other hand, the PSA was better in the case of the DDM with an RMSE of 1.6892 \(\times\) 10\({}^{-3}\) compared to either HSDA with 1.72072 \(\times\) 10\({}^{-3}\) or ELPSO with 1.8307 \(\times\) 10\({}^{-3}\).