## 5 Results and Discussion
\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 & 0.9211 & 1.1142 & 13,323.9963 & 0.0093 & 0.0353 & 1.6971 & 4.5508 & 3.3859 \\ HPSO-QN SM & 0.8642 & 1.0453 & 12,497.3021 & 0.0299 & 0.0261 & 0.7666 & 3.9063 & 2.9167 \\ HPSO-QN SLSM & 0.7718 & 0.8310 & 10,532.9554 & 0.0197 & 0.0303 & 3.6192 & 1.4733 & 2.4787 \\ HPSO-QN MLSM & 0.7987 & 0.8983 & 11,169.0221 & 0.0272 & 0.0325 & 0.3083 & 3.7556 & 2.3200 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Selected identified parameters of the 2 MM system using the proposed methods with the lowest cost function. \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 & 0.4199 & 0.3198 & 4356.4730 & 0.0142 & 0.0152 & 1.3070 & 1.6753 & 3.7320 \\ HPSO-QN SM & 0.1714 & 0.2817 & 2137.6343 & 0.0119 & 0.0146 & 0.9651 & 1.0390 & 2.0512 \\ HPSO-QN SLSM & 0.1927 & 0.3636 & 3548.1141 & 0.0171 & 0.0082 & 1.2920 & 1.2398 & 1.2274 \\ HPSO-QN MLSM & 0.0218 & 0.0512 & 461.0188 & 0.0052 & 0.0028 & 1.2008 & 1.4837 & 0.0787 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: A comparison of the standard deviation of the identified parameters across five independent runs using the proposed methods. The improvement ratios of the HPSO-QN MLSM method over other methods for each identified parameter are summarized in Table 4. The HPSO-QN MLSM method consistently yields substantially lower standard deviation values across all parameters compared to the other methods. For example, the standard deviation of \(J_{1n}\) using the HPSO-QN MLSM is reduced by a factor of 19.26 compared to SPSO, 7.86 compared to HPSO-QN SM, and 8.84 compared to HPSO-QN SLSM. Other parameters also experience notable improvements. The standard deviation of \(J_{2n}\) is reduced by factors of 6.25, 5.50, and 7.10 compared to SPSO, HPSO-QN SM, and HPSO-QN SLSM, respectively, when using the HPSO-QN MLSM method. Similarly, the \(K_{n}\) parameter sees reduction factors of 9.45, 4.63, and 7.70. Similar improvements are observed for \(C_{1n}\), \(C_{2n}\), \(B_{1n}\), and \(B_{2n}\). It is important to note that for \(B_{1n}\) and \(B_{2n}\), HPSO-QN SM demonstrates slightly lower standard deviation values compared to HPSO-QN MLSM. However, HPSO-QN MLSM still outperforms the other PSO methods in terms of identifying parameters with lower standard deviation values for the majority of parameters. While the HPSO-QN SM method also shows lower standard deviation values for \(B_{1n}\) and \(B_{2n}\), HPSO-QN MLSM surpasses it for most parameters. Consequently, the HPSO-QN MLSM method proves to be a promising approach for parameter identification problems in similar systems. The results in Table 5 offer a comparison of the proposed methods based on the average identified parameters across five independent runs. The HPSO-QN MLSM method achieved the lowest \(Fcost\) value of 2.4341 \(\times\) 10\({}^{-6}\), indicating its effectiveness in accurately estimating the mechanical parameters of the 2 MM system. When comparing the average parameters obtained from HPSO-QN SM with those obtained from SPSO, the percentage difference for \(J_{1n}\) is around 15.77%, while for \(J_{2n}\), it is around 15.10%. The percentage difference obtained for \(K_{n}\) is around 15.22%, and for \(C_{1n}\) and \(C_{2n}\), they are 33.54% and 5.86%, respectively. As for \(B_{1n}\) and \(B_{2n}\), the percentage differences are 18.33% and 9.97%, respectively. Regarding \(Fcost\), the percentage difference obtained is 31.41% lower compared to SPSO.