## 4 Results and Discussion
### Optimization Results for Two Degrees of Freedom
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**DE Versions** & **Average (\(\theta_{max}\))\({}_{2m}\)** & **Variance (\(\times\) 10\({}^{-8}\))** & **Standard Deviation (\(\times\)10\({}^{-4}\))** & **RMSE (\(\times\)10\({}^{-4}\))** \\ \hline DE11; DE12; DE14; & & & \\ DE24; DE32; DE34; & 0.029969 & - & - & - \\ DE41; DE43; DE44 & & & \\ DE13; DE22; DE42 & 0.029972 & 0.0178 & 0.134 & 0.134 \\ DE23 & 0.030014 & 1.36 & 1.17 & 1.23 \\ DE33 & 0.030236 & 7.57 & 2.75 & 3.80 \\ DE21 & 0.030051 & 18.7 & 4.33 & 4.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Statistical metrics of the results achieved for achievement of \((\theta_{max})_{2m}\) obtained with \(NP\) = 30 and \(NG\) = 50. \begin{table}
\begin{tabular}{c c c} \hline \hline
**DE Versions** & \(p\)**-Value** & **Distribution** \\ \hline DE12; DE32; DE34 & - & Same results for all runs \\ DE13; DE14 & 4.402 \(\times\) 10\({}^{-11}\) & Non-normal \\ DE24; ED44 & 7.169 \(\times\) 10\({}^{-9}\) & Non-normal \\ ED23 & 1.479 \(\times\) 10\({}^{-8}\) & Non-normal \\ \hline \hline \multicolumn{3}{c}{**Kruskal–Wallis: Rank Sum Test**} \\ \hline
**DE Versions** & \(p\)**-Value** & \(\alpha_{\rm s}\) & **Differences** \\ \hline DE12-14; DE23, DE24; & & \\ DE34, DE44 & \(1.576\times\) 10\({}^{-8}\) & 0.05 & Significant \\ All versions & \(2.2\times\) 10\({}^{-16}\) & 0.05 & Significant \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results of statistical tests for achievement of \((\theta_{max})_{2m}\) for DE versions with \(NP\) = 20 and \(NG\) = 50. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Shapiro–Wilk: Normality Test**} \\ \hline
**DE Versions** & \(p\)**-Value** & **Distribution** \\ \hline DE11, DE12, DE14, & & & \\ DE24, DE32, DE34, & - & same results for all runs \\ DE41, DE43, DE44 & 7.76 \(\times\) 10\({}^{-12}\) & Non-normal \\ DE33 & 9.32 \(\times\) 10\({}^{-5}\) & Non-normal \\ ED23 & 8.97 \(\times\) 10\({}^{-12}\) & Non-normal \\ \hline \hline \multicolumn{3}{c}{**Kruskal–Wallis: Rank Sum Test**} \\ \hline
**DE Versions** & \(p\)**-Value** & \(\alpha_{\rm s}\) & **Differences** \\ \hline All versions & \(2.2\times\) 10\({}^{-16}\) & 0.05 & Significant \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Results of statistical tests for achievement of \((\theta_{max})_{2m}\) for DE versions with \(NP\) = 30 and \(NG\) = 50. The algorithms with intermediate performance are classified in group B. Most of the DE versions in this group have the mutation parameter _best/1/bin_ (ED2\(y\)) and show high variance. Group C is reserved for the algorithm DE33, which led to the worst performance in terms of achievement of the global point of optimum and variance of results for different runs. In general, the results of Figure 6 indicated that, among the DE versions with _rand/1/bin_ (DE1\(y\)), there is no significant difference. For _NP_ = 20 and _NG_ = 50, parameters \(F\) and _CR_ do not have significant influence over the optimization performance. However, for the DE2\(y\) versions (_best/1/bin_), the algorithm with parameters \(F\) = 1 and _CR_ = 0.7 show a better performance in comparison with others with the same mutation operator. In Figure 7, it is possible to notice the emergence of an intermediate group, AB, beyond the previous groups seen in Figure 6, which can be a result of the increase in population size from _NP_ = 20 to 30. In Figure 7 it is also notice the predominance of algorithms with the parameter \(F\) = 1.0 classified in group A. More precisely, six of nine algorithms with \(F\) = 1.0 are in this group. Results also indicated that changes in parameter CR do not lead to significant variations in the performance of different DE versions. The optimization process performed by the DEx4 versions, groups \(A\) and AB, is shown in Figure 8a,b. The optimization evolution of the degree of freedom \(\beta\) is illustrated in the Figure 8a, where it is possible to note the concentration of values near to the optimal angle of \((\beta)_{20}=36^{\circ}\).