## 5 Simulation and Results
### Variable Order Fractional PMSM Model  
In this section, EPSO and novel C.EPSO variants are utilized to estimate parameters of variable order fractional model using data in [2]. Where the PMSM system 2 behaves \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithms** & \(\boldsymbol{\sigma}_{mean\pm STD}\) & \(\boldsymbol{\gamma}_{mean\pm STD}\) & \(\boldsymbol{q}_{mean\pm STD}\) & \(\boldsymbol{MSE}_{best\pm,mean\pm STD}\) \\ \hline \multirow{7}{*}{
\begin{tabular}{c} CEPSO \\ \end{tabular} } & EPSO & \(10.00\pm 1.317\times 10^{-9}\) & \(100.00\pm 9.621\times 10^{-9}\) & \(0.950\pm 5.968\times 10^{-12}\) & \(3.859\times 10^{-20},1963\times 10^{-16}\pm 5.242\times 10^{-16}\) \\ \cline{2-6}  & C.EPSO & \(10.00\pm 2.777\times 10^{-15}\) & \(100.00\pm 1.498\times 10^{-14}\) & \(0.950\pm 0.00\) & \(4.416\times 10^{-26},4511\times 10^{-26}\pm 8.140\times 10^{-28}\) \\ \cline{2-6}  & C.EPSO2 & \(10.00\pm 1.872\times 10^{-15}\) & \(100.00\pm 9.474\times 10^{-15}\) & \(0.950\pm 0.00\) & \(4.416\times 10^{-26},4436\times 10^{-26}\pm 5.016\times 10^{-28}\) \\ \cline{2-6}  & C.EPSO3 & \(\boldsymbol{10.00\pm 1.776\times 10^{-15}}\) & \(\boldsymbol{100.00\pm 9.474\times 10^{-15}}\) & \(\boldsymbol{0.950\pm 0.00}\) & \(4.416\times 10^{-26},4432\times 10^{-26}\pm 4.985\times 10^{-28}\) \\ \cline{2-6}  & C.EPSO4 & \(10.00\pm 2.512\times 10^{-15}\) & \(100.00\pm 1.340\times 10^{-14}\) & \(0.950\pm 0.00\) & \(4.416\times 10^{-26},4448\times 10^{-26}\pm 6.646\times 10^{-28}\) \\ \cline{2-6}  & C.EPSO5 & \(10.00\pm 3.077\times 10^{-15}\) & \(100.00\pm 1.641\times 10^{-14}\) & \(0.950\pm 0.00\) & \(4.416\times 10^{-26},4448\times 10^{-26}\pm 6.646\times 10^{-28}\) \\ \cline{2-6}  & C.EPSO7 & \(10.00\pm 3.077\times 10^{-15}\) & \(100.00\pm 1.641\times 10^{-14}\) & \(0.950\pm 0.00\) & \(4.416\times 10^{-26},4448\times 10^{-26}\pm 6.646\times 10^{-28}\) \\ \cline{2-6}  & C.EPSO9 & \(10.00\pm 2.512\times 10^{-15}\) & \(100.00\pm 1.340\times 10^{-14}\) & \(0.950\pm 0.00\) & \(4.416\times 10^{-26},4448\times 10^{-26}\pm 6.646\times 10^{-28}\) \\ \cline{2-6}  & C.EPSO10 & \(10.00\pm 2.512\times 10^{-15}\) & \(100.00\pm 1.340\times 10^{-14}\) & \(0.950\pm 0.00\) & \(4.416\times 10^{-26},4448\times 10^{-26}\pm 6.646\times 10^{-28}\) \\ \cline{2-6}  & C.EPSO10 & \(10.00\pm 2.5277\times 10^{-15}\) & \(100.00\pm 1.641\times 10^{-14}\) & \(\boldsymbol{0.950\pm 0.00}\) & \(4.416\times 10^{-26},4448\times 10^{-26}\pm 4.738\times 10^{-28}\) \\ \cline{2-6}  & \(\boldsymbol{10.00\pm 1.776\times 10^{-15}}\) & \(\boldsymbol{100.00\pm 9.474\times 10^{-15}}\) & \(\boldsymbol{10.00\pm 9.474\times 10^{-15}}\) & \(\boldsymbol{4.416\times 10^{-26},4432\times 10^{-26}\pm 4.985\times 10^{-28}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(mean\pm STD\) of the extracted parameters for commensurate fractional order PMSM Model amd corresponding fitness function by proposed algorithms. Figure 2: Average convergence curves by C.EPSO variants versus the standard EPSO for the commensurate fractional order PMSM model of ten chaos maps. chaotically at values of parameters \(\gamma\), \(\sigma\), \(q_{1}\), \(q_{2}\) and \(q_{3}\) equal to 50, 4, 0.99, 1 and 0.98 respectively as well as initial values of [\(\dot{t}_{d}\), \(\dot{t}_{q}\), \(\omega\)] equal to [2.5, 3, 1]. The number of samples is 100 in the sampling vector with a step equal to 0.001 s.  
The number of iterations is 500, the number of search agents are 50 while the minimum and maximum limits of variables are in range of [2, 8] for \(\sigma\), [40, 60] for \(\gamma\) and [0.9, 1] for all \(qs\). A comparison is accomplished among the results of all variants over 20 independent runs to determine the most efficient one. The identified parameters' Mean values and STD as well as the cost function's value are tabulated in Table 3. Table 3 indicates that consolidating chaotic maps with the basic algorithms has a sensible impact on the accuracy as well as the homogeneity of the results.