## 5 Comparison of the LSPMSM Parameters Identification Results Obtained by DE and NM
The best result out of six performed repetitions for the DE and DE-3 is shown in Table 4.  
Figure 8 shows the objective function values for the considered optimization algorithms with the number of objective function calls, starting from the 120 th NM call (1 st generation for DE). Table 5 shows the mean values and RMS deviation of identified parameters obtained using the DE and DE-3 algorithms. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Real Value** & **DE** & **DE-3** & **NM** & **NM-3** \\ \hline \(L_{d}\), H & 0.26 & 0.244 & 0.253 & 0.256 & 0.257 \\ \(L_{q}\), H & 0.15 & 0.162 & 0.155 & 0.150 & 0.15 \\ \(L_{vd}\), H & 0.038 & 0.039 & 0.038 & 0.038 & 0.038 \\ \(L_{vdg}\), H & 0.041 & 0.042 & 0.044 & 0.041 & 0.041 \\ \(r^{\prime}_{d}\), Ohm & 9.24 & 10.23 & 8.90 & 9.24 & 9.26 \\ \(r^{\prime}_{d}\), Ohm & 10.1 & 9.70 & 9.69 & 10.09 & 10.1 \\ \(\chi_{0}\), Wb & 0.76 & 0.76 & 0.75 & 0.76 & 0.76 \\ \(J\), kg-m\({}^{2}\) & 0.003 & 0.003 & 0.003 & 0.003 & 0.003 \\ Objective function value & - & 3.288 & 0.177 & 0.0095 & 0.0016 \\ Number of iterations & - & 5135 & 3547 & 2661 & 1200 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Identification results obtained by the considered algorithms and real motor parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of a DE Run** & **DE** & **DE-3** & **NM** & **NM-3** \\ \hline
1 & 3.473/5383 & 0.423/3323 & & \\
2 & 11.973/5399 & 0.627/3314 & & \\
3 & 10.974/5398 & 0.388/3457 & 0.0095/2661 & 0.0016/1200 \\
4 & 7.525/5399 & 0.462/3475 & & \\
5 & 16.68/3996 & 0.177/3547 & & \\
6 & 3.288/5135 & 0.196/3942 & & \\ Mean deviation & 8.986 & 0.379 & - & - \\ RMS deviation & 5.236 & 0.17 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Final values of the objective function/number of iterations for different algorithms. With successful parametric identification, the objective function was reduced by about an order of magnitude when using DE and by two orders of magnitude when using DE-3. In addition, DE-3 required about two times fewer function calls than DE. Although NM is a local search algorithm, and DE is a global one, in both cases, being stuck in a local minimum was possible (for example, see NM-3, start 1 and DE-3 start 6 in Table 6). Further, DE and DE-3 required more objective function calls compared to NM and NM-3. From the above, we can conclude that in the considered application the Nelder-Mead method provides significant advantages over differential evolution. Therefore, it is preferable to run NM multiple times to find the best initial approximation rather than using DE. This does not preclude the use of DE to automatically find an initial approximation for NM. The faster convergence of DE-3 compared to DE and NM-3 compared to NM confirms the effectiveness of the \"stop and restart\" technique used in the case of a random initial approximation, as well. Furthermore, even though NM is an unrestricted method, the results of identification using NM and NM-3 satisfy the conditions that all identified \begin{table}
\begin{tabular}{c c c c c} \hline
**Execution No.** & **DE** & **DE-3** & **NM** & **NM-3** \\ \hline
1 & 3.3/5399 & 0.4/3891 & \(7.8\times 10^{-10}/3002\) & 6.5/1203 \\
2 & 6.4/5396 & 0.61/3511 & \(7.7\times 10^{-13}/2674\) & \(6.5\times 10^{-4}/1203\) \\
3 & 4.1/5396 & 0.4/3891 & \(8.3\times 10^{-13}/2195\) & \(2.1\times 10^{-4}/1203\) \\
4 & 10.6/5398 & 0.34/3951 & \(4.7\times 10^{-12}/2809\) & \(1.8\times 10^{-10}/1203\) \\
5 & 5.4/5395 & 0.61/3511 & \(8\times 10^{-13}/2770\) & \(5.5\times 10^{-3}/1203\) \\
6 & 4.8/5400 & 24/2878 & \(8\times 10^{-13}/2650\) & \(1.7\times 10^{-9}/1203\) \\ Mean deviation & 5.8 & 4.4 & \(1.3\times 10^{-10}\) & 1.1 \\ RMS deviation & 2.6 & 9.6 & \(3.2\times 10^{-10}\) & 2.7 \\ \hline \end{tabular}
\end{table} 
Table 6: The results of 6 executions for the considered identification algorithms with a random initial approximation: the objective function final values and the number of iterations.