## 4 Results and Discussion
### Optimization Results for Four Degrees of Freedom  
In this section, the idea is to reproduce the effect of the ratio \(H_{0}/L_{0}\) over the three times minimized maximum excess of temperature, \((\theta_{max})_{3m}\), and the mean square errors (MSE) for \((\theta_{max})_{3m}\) and the corresponding optimal configurations, \((S_{1}/H_{0})_{0}\), \((\alpha)_{20}\), and \((\beta)_{30}\), for different versions of DE algorithm. The remaining degrees of freedom and constraints are kept constant, i.e., \(H/L\) = 1.0, \(\varphi_{\text{c}}\) = 0.01, and \(\varphi_{1}\) = \(\varphi_{2}\) =0.015. For geometric optimization it is assumed a discrete search space with \(\alpha=\beta\in[-80^{\circ};80^{\circ}]\) and interval of \(\Delta\alpha=\Delta\beta=1^{\circ}\) and \(S_{1}/H_{0}\) \(\in\) [0.1; 0.9] with an interval of \(\Delta S_{1}/H_{0}\) = 0.1. Based on the previous results, only DE1, DE2, and DE3 with \(F\) = 1.0 and \(CR\) = 0.0 are used here. An investigation performed here consists of the analysis of the influence of different combinations of the parameters \(NP\) and \(NG\) over the effect of the ratio \(H_{0}/L_{0}\) over the thermal performance and optimal configurations. It executed 30 runs of each algorithm version for the following discrete search space of \(H_{0}/L_{0}\):  
\[H_{0}/L_{0}=[\ 2.25;\ 4.72;\ 7.19;\ 9.66;\ 12.14;\ 14.61;\ 17.08;\ 19.55;\ 22.03;\ 24.5] \tag{30}\]  
Figure 11: Search space exploration with differential evolution algorithms for the three times optimized configuration: (**a**) \((S_{1}/H_{0})_{0}\), (**b**) \((\beta)_{30}\), (**c**) \((\alpha)_{20}\) (**d**) \((\theta_{max})_{3m}\). Table 12 shows the statistical metrics calculated from the results of DE algorithm versions. The first column shows some \(H_{0}/L_{0}\) search space values, and the second column shows the combinations of \(NP\times NG\) investigated. The other columns show the statistical metrics of the average and standard deviation of \((\theta_{max})_{3m}\) obtained with DE1, DE2, and DE3 versions. The first combination of \(NP=10\) and \(NG=50\) led to the worst results with high magnitudes of standard deviation and overall \((\theta_{max})_{3m}\) for all cases of \(H_{0}/L_{0}\). These results indicated the achievement of local points of optimum and dispersed magnitudes of thermal performance. In the second experiment, the magnitudes of \(NP\) and \(NG\) were increased two times (\(NP=20\) and \(NG=100\)). Table 12 data indicates a significant improvement in the results, especially in the magnitude of the standard deviation and the decrease in overall \((\theta_{max})_{3m}\) close to the global point of \((\theta_{max})_{3m}\). The following experiment tests the population size of 30 individuals and 2 total numbers of generations, 66 and 100. In this case, the results of DE2 and DE3 show some progress. On the other hand, the performance of DE1 for \(H_{0}/L_{0}=[2.25,24.5]\) has a slight decrease. The increase in \(NG\) from 66 to 100 does not significantly influence the results, except for the DE3 version in \(H_{0}/L_{0}=[12.14;24.5]\). For the final experiments with \(NP=40\) and \(NG=[75,112]\), the DE1 algorithm version showed better results with null values for standard deviation, which means that the algorithm reached the optimal solution in all execution and for all values of rate \(H_{0}/L_{0}\). The impact of the increase in \(NP\) was expected, but compared to other methods, the DE1 version shows better results even for \(NP=20\) and \(NG=100\). The DE2 version also achieved the optimal solution in all executions for \(H_{0}/L_{0}=2.25\). However, for other values of \(H_{0}/L_{0}\), the DE2 version does not reach the optimal solution in all runs. For the DE3 version, the parameters \(NP=40\) and \(NG=[75,112]\) improved the algorithm results, mainly for \(NG=112\), with null standard deviation values. Nevertheless, for \(H_{0}/L_{0}=2.25\), the results slightly decrease compared to \(NP=30\).