## 4 Description of Search Algorithms
### Overview of Moth-Flame Optimization Algorithm
2.2 Case 2, Load Torque Is Varied from 0 to 2 Nm, \(L_{d}=2.5L_{d}\), \(R_{s}=2R_{s}\) and Nominal J, B, \(\lambda\)

The simulation results of the optimized PI controllers for closed-loop tracking response of the velocity under variations parameters (\(L_{d}=2.5L_{d}\), \(R_{s}=2R_{s}\)) and load torque of 0 to 2.0 Nm are shown in Figure 10. The performance parameter of the speed response is shown in Table 11. MOD-FPA controller has lower performance indices than the other considered controllers, as shown in Table 11. Nevertheless, the percentage increase in MSE for MOD-FPA is 45.487%, and for DF algorithm the increase is 33.27%. The percentage increase in JIAE for MOD-FPA is 28.78%, and for MOA the increase is 20.12%. 2.3 Case 3, Load Torque Is Varied from 0 to 2 Nm, \(L_{d}=0.8L_{d}\), \(R_{s}=3R_{s}\), \(J=3j\), \(B=3B\) and \(\lambda=0.9\lambda\)

The simulation results of the optimized PI controllers for closed-loop tracking response of the velocity under the said parameter variations are shown in Figure 11 and Table 12. The percentage increase in the MSE for MOD-FPA is 159.592%, and for DF algorithm, the growth was 180.279%. The percentage increase in the JIAE for MOD-FPA is 167.304% and for MFA, the increase is 170.688%; \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**Algorithm** & **MSE** & **JISE** & **JIAE** & **JITSE** & **JITAE** & **Rise Time** & **Overshoot** & **Settling Time** & \(\epsilon_{ss}\) & **Overall Error Rate** \\ \hline MFA & 12.50636 & 5.62799 & 0.38243 & 1.25215 & 0.08207 & 0.00013 & 1.72334 & 0.0202 & 0.00054 & 23.25371 \\ FPA & 9.93659 & 4.7156 & 0.30742 & 1.00099 & 0.06584 & 0.00009 & 3.01638 & 0.02026 & 0.00251 & 21.73004 \\ WOA & 16.66305 & 7.49854 & 0.52442 & 1.65803 & 0.11185 & 0.00017 & 2.19356 & 0.02042 & 0.00511 & 30.78568 \\ DF & 10.90970 & 4.99019 & 0.3587 & 1.09617 & 0.07615 & 0.00012 & **0.96234** & 0.0202 & 0.00192 & 19.25247 \\ CSA & 10.72061 & 4.82438 & 0.33895 & 10.77787 & 0.07271 & 0.00012 & 1.1434 & **0.02019** & 0.00091 & 28.996952 \\ MOD-FPA & **7.32725** & **3.29734** & **0.21939** & **0.74688** & **0.0731** & **0.00007** & 3.08115 & **0.02019** & **0.00053** & **17.71227** \\ \hline \hline \end{tabular} The best results of each study and comparative analysis are highlighted in bold in the corresponding table.

\end{table} 
Table 10: Performance analysis, Case 1. Figure 10: Simulation result for speed tracking task, Case 2. As a result, the PI controller tuned by MOD-FPA using the proposed objective function outperforms PI controllers tuned with WOA, CSA, MOA, DA, and FPA optimization techniques in angular velocity tracking of the PMSM under parameter uncertainty and external load disturbances. ## 6 Statistical Analysis

To statistically evaluate the results obtained by the algorithms used (MOD-FPA, WOA, CSA, MOA, DA, and FPA) for the estimation of the parameters of the PI controllers. When comparing pairs, through statistical tests, we seek to establish the performance of two algorithms when applied to a common set of problems [47]. Table 14 presents the critical number of wins with a \(\propto\) = 0.05 level of significance. We established that MOD-FPA as the comparison algorithm and the table shows that this algorithm is significantly better compared to the others. In this work, we used a non-parametric analysis called Wilcoxon test to compare two sets of ordinal data subject to different conditions [47]. For this statistical analysis, MOD-FPA algorithm is compared separately with other strategies proposed in order to test whether there is a significant difference between the proposed algorithm and the others with which it is compared. For this test, we have two hypotheses. :

1. Null hypothesis (\(H_{0}\)), there is no difference between the results of the compared strategies,
2. Alternative hypothesis (\(H_{1}\)), there is a difference between the results of the differentiated strategies, in other word when the null hypothesis is false.