## 5 Results and Discussion
When comparing HPSO-QN SLSM with HPSO-QN SM, percentage differences were around 20.01% for \(J_{1n}\), 10.90% for \(J_{2n}\), and 14.78% for \(K_{n}\). The friction coefficients (\(C_{1n}\), \(C_{2n}\), \(B_{1n}\), and \(B_{2n}\)) estimated by HPSO-QN SLSM showed percentage differences ranging from 20% to 45% compared to HPSO-QN SM. The percentage difference in \(Fcost\) between HPSO-QN SM and HPSO-QN SLSM was around 20.31%, with HPSO-QN SLSM providing a lower value. Comparing HPSO-QN MLSM with HPSO-QN SLSM, the percentage differences \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**PSO Methods**} & \multicolumn{8}{c}{**Parameters**} & \multirow{2}{*}{\(Fcost\) (\(\times\)10\({}^{-6}\))} \\ \cline{2-2} \cline{5-10}  & \(J_{1n}\) (s) & \(J_{2n}\) (s) & \(K_{n}\) (s\({}^{-1}\)) & \(C_{1n}\) (p.u) & \(C_{2n}\) (p.u) & \(B_{1n}\) (p.u) & \(B_{2n}\) (p.u) \\ \hline SPSO & 19.26 & 6.25 & 9.45 & 2.73 & 5.43 & 1.09 & 1.13 & 48.06 \\ HPSO-QN SM & 7.86 & 5.50 & 4.63 & 2.29 & 5.21 & 0.80 & 0.70 & 26.06 \\ HPSO-QN SLSM & 8.84 & 7.10 & 7.70 & 3.29 & 2.92 & 1.06 & 0.83 & 15.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Improvement ratio of the standard deviation of the HPSO-QN MLSM method compared to other methods. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**PSO Methods**} & \multicolumn{8}{c}{**Parameters**} & \multirow{2}{*}{\(Fcost\) (\(\times\)10\({}^{-6}\))} \\ \cline{2-2} \cline{5-10}  & \(J_{1n}\) (s) & \(J_{2n}\) (s) & \(K_{n}\) (s\({}^{-1}\)) & \(C_{1n}\) (p.u) & \(C_{2n}\) (p.u) & \(B_{1n}\) (p.u) & \(B_{2n}\) (p.u) \\ \hline SPSO & 1.2457 & 1.3193 & 16,554.7629 & 0.0158 & 0.0256 & 2.5953 & 3.0403 & 7.1430 \\ HPSO-QN SM & 1.0492 & 1.1201 & 14,035.9131 & 0.0211 & 0.0271 & 2.1197 & 2.7372 & 4.8995 \\ HPSO-QN SLSM & 0.8393 & 0.9980 & 11,961.5283 & 0.0289 & 0.0316 & 2.5762 & 1.4984 & 3.9044 \\ HPSO-QN MLSM & 0.8151 & 0.8974 & 11,265.7222 & 0.0287 & 0.0332 & 1.6844 & 1.7908 & 2.4341 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Performance comparison of the proposed method: average of identified parameters across five independent runs. obtained for \(J_{1n}\), \(J_{2n}\), and \(K_{n}\) were 2.88%, 10.08%, and 5.82%, respectively. For the friction coefficients (C\({}_{1n}\) and C\({}_{2n}\)), the average parameter differences between HPSO-QN MLSM and HPSO-QN SLSM were 0.69% and 5.06%, respectively. The average differences for \(B_{1n}\) and \(B_{2n}\) were 34.62% and 19.51%, respectively. The average \(Fcost\) estimated by HPSO-QN MLSM was 37.66% lower than that estimated by HPSO-QN SLSM. Finally, comparing the average parameters obtained from HPSO-QN MLSM to the SPSO method, HPSO-QN MLSM achieved a significant improvement in \(Fcost\), with a decrease of 63.92%. The percentage differences obtained for \(J_{1n}\) and \(J_{2n}\) were around 34.57% and 31.98%, respectively. For \(K_{n}\), the percentage difference obtained was approximately 31.95%. The percentage differences obtained for C\({}_{1n}\) and C\({}_{2n}\) were 81.65% and 29.69%, respectively. For \(B_{1n}\) and \(B_{2n}\), the percentage differences obtained were 35.10% and 41.10%, respectively. These results suggest that HPSO-QN MLSM outperforms SPSO in identifying the 2 MM parameters and minimizing \(Fcost\). Table 6 presents the accuracy of the estimated parameters compared to the benchmark parameters, measured by the absolute percentage error (APE). The APE is used as a measure of the deviation between the estimated and benchmark parameters, and it is defined as  
\[APE=\frac{|Estimated-Benchmark|}{Benchmark}\times 100 \tag{24}\]  
The HPSO-QN SM method shows the lowest error for C\({}_{2n}\) (APE of 0.18%) and performs well for \(B_{1n}\) (APE of 2.11%). The HPSO-QN SLSM method exhibits the lowest error for \(J_{1n}\) (APE of 3.68%) and performs well for C\({}_{1n}\) (APE of 4.93%).