# Parameter Extraction of Photovoltaic Cells and Panels Using a PID-Based Metaheuristic Algorithm
## 4 Conclusions  
This research introduced a novel metaheuristic algorithm, the PID-based search algorithm (PSA), for extracting parameters of PV cells and panels. By optimizing the proportion, integral, and differential parameters and relocating initial error calculations outside the iterative loop, the PSA demonstrated enhanced exploration, exploitation, and computational \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{\begin{tabular}{c} **Number of** \\ **Runs** \\ \end{tabular} } & \multirow{2}{*}{**Model**} & \begin{tabular}{c} **Population Size/** \\ **Number of** \\ **Iterations** \\ \end{tabular} & \multirow{2}{*}{**RMSE**} & \multirow{2}{*}{**Run Time**} & \multirow{2}{*}{\
\begin{tabular}{c} **Intel Core i9** \\ **(s)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **AMD Ryzen 5** \\ **5600H CPU** \\ **Time (s)** \\ \end{tabular} } \\ \cline{5-6}  & & & & Case A: MJSC at 41.5 \({}^{\circ}\)C & & \\ \hline SDM & 100/1100 & 7.8273 \(\times\) 10\({}^{-5}\) & 3.1843 & 3.0124 & 4.9071 \\ \hline DDM & 100/1100 & 7.8031 \(\times\) 10\({}^{-5}\) & 4.099 & 3.8558 & 8.0901 \\ \hline \multirow{2}{*}{5} & \multicolumn{4}{c}{Case B: MJSC at 51.3 \({}^{\circ}\)C} \\ \cline{2-6}  & SDM & 100/1100 & 7.5387 \(\times\) 10\({}^{-5}\) & 3.3770 & 3.1109 & 4.5467 \\ \hline DDM & 100/1100 & 7.5161 \(\times\) 10\({}^{-5}\) & 4.6077 & 4.441 & 8.2144 \\ \hline \multicolumn{6}{c}{Case C: MJSC at 61.6 \({}^{\circ}\)C} \\ \cline{2-6}  & SDM & 100/1100 & 6.7754 \(\times\) 10\({}^{-5}\) & 3.0998 & 2.9817 & 4.0331 \\ \hline DDM & 100/1100 & 6.7754 \(\times\) 10\({}^{-5}\) & 4.1012 & 3.9912 & 8.5455 \\ \hline \end{tabular}
\end{table} 
Table 24: MJSC CPU execution time for all cases. Figure 24: RMSE for PSA with regard to temperature. efficiency. The algorithm's performance was validated using both SDMs and DDMs across datasets, including RTC France, aSi, mSi, and various photovoltaic panels. Compared to 26 state-of-the-art algorithms using four statistical tests, the PSA consistently achieved superior or comparable accuracy. In particular, the RTC France DDM yielded RMSE improvements of 0.67-2.10%, while the mSi cell demonstrated up to 40.9% RMSE improvement compared with the ISOA, which obtained 4.4383 \(\times\) 10\({}^{-4}\), while PSA obtained 2.6216 \(\times\) 10\({}^{-4}\) with execution times of 3.94 s for the SDM and 5.57 s for the DDM. Additionally, the PSA required smaller population sizes and fewer iterations, underscoring its practicality. Testing the PSA under diverse conditions, including the multijunction solar cell (MJSC) at three different temperatures (41.5 \({}^{\circ}\)C, 51.3 \({}^{\circ}\)C, and 61.6 \({}^{\circ}\)C), demonstrated its reliability and effectiveness. It achieved average RMSE improvements of 12.17-12.42% across BWOA and CSA and execution times as low as 2.98 s for SDMs and 4.10 s for DDMs. The PSA was also tested using two CPUs--Intel Core i9 and AMD Ryzen 5 5600H--to evaluate CPU efficiency, with Intel Core i9 consistently achieving faster results. The extracted parameters, such as \(I_{\text{ph}}\), \(I_{\text{sd1}}\), \(R_{\text{sh}}\), and \(n_{1}\), were highly accurate and stable, even under high thermal stress. To enhance the robustness and practical applicability of the PID-based Search Algorithm (PSA) in real-world environments, future work will integrate strategies to handle uncertainties such as sensor noise, environmental variability, and data imperfections commonly encountered in field conditions. One promising direction is the use of stochastic perturbation models that simulate noisy I-V measurements, enabling PSA to be tested and calibrated for resilience under such conditions. Moreover, the control-based nature of PSA allows for adaptive adjustment of the PID coefficients (\(K_{p}\), \(K_{i}\), \(K_{d}\)) in response to the algorithm's convergence behavior, which can improve stability and prevent premature convergence.