# An Improved Dingo Optimization Algorithm Applied to SHE-PWM Modulation Strategy
## 4 Results  
In order to verify the mDOA capabilities to solve the non-linear set of equations of the selective harmonic elimination technique, three tests were conducted for each of the selected algorithms. As stated in Equation (4), the modulation index \(M\) can vary between 0 and 1. Thus, three modulation index analysis at 0.6, 0.8, and 1 are then proposed. It is worth emphasizing that say values are widely used in multilevel inverters. As previously mentioned, all tests were run for a population size of 100 and 250 iterations. Tables 1-3, summarize the experimental tests conducted, where the modified version of the dingo optimization algorithm outperforms the other algorithms by obtaining the best fitness for all of the selected modulation indexes. The near-optimal obtained angles are then used in a Fourier analysis to determine the total harmonic distortion and the corrected elimination of the fifth, seventh, eleventh, and thirteenth harmonics for the three selected modulation indexes. As the mDOA algorithm showed the best fitness, Figure 7 shows the Fourier spectrum calculated by using the mDOA angles summarized in Tables 1-3, where the correct elimination of the selected low order harmonics is appreciated. advantage for modulation indexes between 0.65 and 0.9. Moreover, the Wilcoxon analysis against the BWOA shows competitive results at modulation indexes of 0.5, 0.55, and 0.7. It is to be noted that at a modulation index of 1, the mDOA results are statistically significant compared with the BWOA, WOA, and GWOA. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{5}{c}{**Angles**} & \multirow{2}{*}{**THD**} & \multirow{2}{*}{**Fitness**} \\ \cline{2-2} \cline{5-8}  & \(\mathbf{\alpha_{1}}\) & \(\mathbf{\alpha_{2}}\) & \(\mathbf{\alpha_{3}}\) & \(\mathbf{\alpha_{4}}\) & \(\mathbf{\alpha_{5}}\) & \\ \hline mDOA & 7.84 & 19.37 & 29.63 & 47.66 & 62.90 & 5.01 & 1.22 \(\times\)\(10^{-28}\) \\ \hline BWOA & 7.86 & 19.37 & 29.65 & 47.68 & 63.21 & 5.01 & 1.29 \(\times\)\(10^{-28}\) \\ \hline GWOA & 0.49 & 14.74 & 25.61 & 40.57 & 89.16 & 5.71 & 16.04 \(\times\)\(10^{-02}\) \\ \hline WOA & 4.19 & 20.29 & 22.12 & 41.97 & 61.15 & 6.90 & 3.93 \(\times\)\(10^{-02}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Algorithms results for modulation index = 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{5}{c}{**Angles**} & \multirow{2}{*}{**THD**} & \multirow{2}{*}{**Fitness**} \\ \cline{2-2} \cline{5-8}  & \(\mathbf{\alpha_{1}}\) & \(\mathbf{\alpha_{2}}\) & \(\mathbf{\alpha_{3}}\) & \(\mathbf{\alpha_{4}}\) & \(\mathbf{\alpha_{5}}\) & \\ \hline mDOA & 35.42 & 46.94 & 58.57 & 72.67 & 87.85 & 6.79 & 4.15 \(\times\)\(10^{-27}\) \\ \hline BWOA & 35.44 & 46.95 & 58.58 & 72.61 & 87.86 & 6.82 & 4.19 \(\times\)\(10^{-27}\) \\ \hline GWOA & 35.29 & 46.80 & 58.45 & 72.46 & 87.74 & 6.87 & 2.78 \(\times\)\(10^{-04}\) \\ \hline WOA & 35.35 & 46.89 & 58.49 & 72.44 & 87.70 & 6.87 & 8.17 \(\times\)\(10^{-05}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Algorithms results for modulation index = 0.6. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{5}{c}{**Angles**} & \multirow{2}{*}{**THD**} & \multirow{2}{*}{**Fitness**} \\ \cline{2-2} \cline{5-8}  & \(\mathbf{\alpha_{1}}\) & \(\mathbf{\alpha_{2}}\) & \(\mathbf{\alpha_{3}}\) & \(\mathbf{\alpha_{4}}\) & \(\mathbf{\alpha_{5}}\) & \\ \hline mDOA & 35.42 & 46.94 & 58.57 & 72.67 & 87.85 & 6.79 & 4.15 \(\times\)\(10^{-27}\) \\ \hline BWOA & 35.44 & 46.95 & 58.58 & 72.61 & 87.86 & 6.82 & 4.19 \(\times\)\(10^{-27}\) \\ \hline GWOA & 35.29 & 46.80 & 58.45 & 72.46 & 87.74 & 6.87 & 2.78 \(\times\)\(10^{-04}\) \\ \hline WOA & 35.35 & 46.89 & 58.49 & 72.44 & 87.70 & 6.87 & 8.17 \(\times\)\(10^{-05}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The \(p\)-values of the Wilcoxon rank-sum test with 5% significance for mDOA vs. BWOA, WOA, and GWOA. \(p\)-values \(\leq\) 0.05 are shown in bold.