## 4 Description of Search Algorithms
### Overview of Moth-Flame Optimization Algorithm
From these analyses, CSA has the best average \begin{table}
\begin{tabular}{l c c} \hline \hline
**Description** & **Symbol** & **Value** \\ \hline Inertia moment & \(J\) & \(3.5\times 10^{-5}\,\mathrm{kg/m^{2}}\) \\ Nominal voltage & \(v\) & \(120\) V \\ Rated current & & \(4\) A \\ Stator resistance & \(r_{s}\) & \(2.6\)\(\Omega\) \\ Stator inductance \(d\) & \(L_{d}\) & \(6.73\) mH \\ Stator inductance \(q\) & \(L_{q}\) & \(6.73\) mH \\ Stator dispersion inductance & \(L_{ls}\) & \(0.1L_{q}\) \\ Magnetic flux & \(\lambda_{m}\) & \(0.319\) Wb \\ Viscous friction coefficient & \(B\) & \(1.0\times 10^{-4}\) Nms \\ Direct current voltage bus & \(V_{CD}\) & \(250\) V \\ Pole pairs & \(n_{p}\) & \(2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: PMSM nominal parameters. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Algorithm** & \multicolumn{2}{c}{**WOA**} & \multicolumn{2}{c}{**CSA**} & \multicolumn{2}{c}{**MOA**} & \multicolumn{2}{c}{**DA**} & \multicolumn{2}{c}{**FPA**} & \multicolumn{2}{c}{**MOD-FPA**} \\ \hline  & **Fmin** & **Time (s)** & **Fmin** & **Time (s)** & **Fmin** & **Time (s)** & **Fmin** & **Time (s)** & **Fmin** & **Time (s)** \\ \hline



computation (8.92225 s); on the other hand, WOA has the worst fitness value than MOD-FPA, MOA, DA, and FPA. ### Robustness Study

#### 5.1.1 Simulation Result for Speed Tracking Task without Load and Nominal Parameters Condition

The velocity reference trajectory planned for the PMSM operation was specified to efficiently track the smooth motion profile defined as [45]. \[\omega^{*}(t)=\left\{ \begin{array}{cl}0&\text{for }0\leq t\leq T_{i}\\ \tilde{\omega}\left(t,T_{1},T_{2}\right)\tilde{\omega_{1}}&\text{for }T_{i}\leq t \leq T_{f}\\ \tilde{\omega_{3}}&\text{for }t>T_{f}\end{array} \right. \tag{31}\]

where \(\tilde{\omega_{1}}=500\) rpm, \(T_{i}=0\) s, \(T_{f}=0.02\) s, \(\tilde{\omega}\left(t,T_{i},T_{f}\right)\) is a Bezier interpolation polynomial, with \(\tilde{\omega}\left(T_{i},T_{i},T_{i+1}\right)=0\) and \(\tilde{\omega}\left(T_{i+1},T_{i},T_{i+1}\right)=1\), given by

\[\tilde{\omega}(t,T_{i},T_{i+1}) = \left(\frac{t-T_{i}}{T_{i+1}-T_{i}}\right)^{5}\left[d_{1}-d_{2} \left(\frac{t-T_{i}}{T_{i+1}-T_{i}}\right)\right.\] \[\left.+d_{3}\left(\frac{t-T_{i}}{T_{i+1}-T_{i}}\right)^{2}-...-d_ {6}\left(\frac{t-T_{i}}{T_{i+1}-T_{i}}\right)^{5}\right]\]

with \(d_{1}=252\), \(d_{2}=1050\), \(d_{3}=1800\), \(d_{4}=1575\), \(d_{5}=700\), \(d_{6}=126\). Figure 3 shows the closed-loop tracking response of the velocity reference trajectory without load condition with set speed of 500 rpm of six algorithms. The performance parameter of the speed tracking task response is shown in Table 5. The performance of the optimized PI controller is evaluated for trajectory tracking control in terms of time domain specifications in speed response and performance indices ( MSE, JISE, JIAE, JITSE, JITAE, \(t_{s}\), \(t_{r}\), \(M_{p}\) and \(e_{ss}\)). The main parameters, such as the overall error rate (0.46720) and overshoot (0.20369) are in favor of the MOD-FPA controller. The MOD-FPA controller has a better MSE, JISE, JIAE, JITSE, JITAE, and steady state error than other considered controllers, as shown in Table 5. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Algorithm** & \(k_{p,u,}\) & \(k_{i\omega_{r}}\) & \(k_{p,i_{q}}\) & \(k_{i\omega_{q}}\) & \(k_{p,i_{q}}\) & \(k_{i,i_{q}}\) & \(F_{min}\) \\ \hline
**CSA** & 1.0262 & 8.8369 & 6.9248 & 23.4701 & 7.1267 & 2.2591 & 12.2005 \\
**WOA** & 0.1104 & 46.4259 & 62.3245 & 0.1756 & 47.8563 & 0.1756 & 19.9996 \\
**MOA** & 0.2727 & 97.3000 & 11.1623 & 25.1522 & 1.3187 & 18.6495 & 15.0166 \\
**DA** & 0.0411 & 17.1440 & 28.1149 & 7.1869 & 12.9935 & 5.0739 & 12.1259 \\
**FPA** & 1.5970 & 88.4017 & 23.5742 & 185.9878 & 74.7438 & 194.9891 & 14.3185 \\
**MOD-FPA** & 0.4004 & 70.8571 & 39.7645 & 207.3428 & 51.864 & 180.338 & **11.6044** \\ \hline \hline \end{tabular}

The best results of each study and comparative analysis are highlighted in bold in the corresponding table.

\end{table}