## Appendix D Inferential Statistics of the Overall Performance
### Ranks Achieved by the Friedman Test  
The ranks obtained by the Friedman test for the three study cases are shown in Table 110.  
Ranks and \(p\)-value obtained by the Friedman test. A \(p\)-value less than 0.05 indicates that the group of values of the objective function for each algorithm are different.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{3}{c}{**Ranks**} \\  & **CHT** & **Study Case 1** & **Study Case 2** & **Study Case 3** \\ \hline DER1B & FR & 1.1 & 1.7 & 1.6 \\ DER1B & SR & 3.8 & 2.8 & 4 \\ DER1B & cC & 2.8 & 3.8 & 2.7 \\ DER1B & PF & 2.3 & 1.6 & 1.7 \\ \hline Statistic & & 67 & 59 & 66 \\ \(p\)-value & & \(2.24\times 10^{-14}\) & \(9.43\times 10^{-13}\) & \(2.83\times 10^{-14}\) \\ \hline DER1E & FR & 1.9 & 1.5 & 1.9 \\ DER1E & cC & 2.3 & 3.9 & 2.6 \\ DER1E & PF & 1.9 & 2.9 & 1.5 \\ \hline Statistic & & 53 & 71 & 64 \\ \(p\)-value & & \(1.51\times 10^{-11}\) & \(3.18\times 10^{-15}\) & \(6.87\times 10^{-14}\) \\ \hline DEB1B & FR & 2.2 & 2 & 1.3 \\ DEB1B & SR & 3.2 & 2.5 & 3.5 \\ DEB1B & cC & 3 & 2.9 & 3.5 \\ DEB1B & PF & 1.6 & 2.6 & 1.7 \\ Statistic & & 29 & 6.5 & 73 \\ \hline p-value & & \(2.67\times 10^{-8}\) & \(8.89\times 10^{-2}\) & \(7.82\times 10^{-16}\) \\ DEB1E & FR & 1.2 & 1.8 & 1.3 \\ DEB1E & SR & 3.8 & 3.5 & 4 \\ DEB1E & cC & 2.1 & 3 & 2.4 \\ DEB1E & PF & 2.9 & 1.7 & 2.3 \\ \hline Statistic & & 71 & 39 & 69 \\ \(p\)-value & & \(2.93\times 10^{-15}\) & \(1.48\times 10^{-8}\) & \(8.18\times 10^{-15}\) \\ \hline DCR & FR & 1.6 & 2 & 2 \\ DCR & SR & 3.9 & 1.6 & 1.9 \\ DCR & cC & 2.5 & 4 & 2.4 \\ DCR & PF & 2 & 2.4 & 3.7 \\ \hline Statistic & & 56 & 57 & 35 \\ \(p\)-value & & \(5.01\times 10^{-12}\) & \(2.67\times 10^{-12}\) & \(1.24\times 10^{-7}\) \\ \hline DCRB & FR & 1.9 & 1.8 & 2.3 \\ DECB & SR & 4 & 2.6 & 3.1 \\ DECB & eC & 2.5 & 3.4 & 1 \\ DECB & PF & 1.6 & 2.3 & 3.6 \\ \hline Statistic & & 61 & 24 & 69 \\ \(p\)-value & & \(3.81\times 10^{-13}\) & \(2.10\times 10^{-5}\) & \(5.52\times 10^{-15}\) \\ \hline DER1B & FR & 1.2 & 2.5 & 1.5 \\ DER1B & SR & 4 & 1.3 & 2.5 \\ DER1B & cC & 2 & 4 & 2.1 \\ DER1B & PF & 2.9 & 2.3 & 3.9 \\ \hline Statistic & & 80 & 69 & 58 \\ \(p\)-value & & \(3.01\times 10^{-17}\) & \(7.41\times 10^{-15}\) & \(1.67\times 10^{-12}\) \\ \hline DER1E & FR & 2.1 & 1.3 & 2 \\ DER1E & SR & 2.9 & 3 & 4 \\ DER1E & cC & 3.5 & 4 & 1 \\ DER1E & PF & 1.5 & 1.7 & 3 \\ \hline Statistic & & 42 & 80 & 88 \\ \(p\)-value & & \(4.34\times 10^{-9}\) & \(2.47\times 10^{-17}\) & \(7.04\times 10^{-19}\) \\ \hline GA & FR & 1.6 & 3.3 & 1.8 \\ GA & SR & 4 & 2.2 & 1.8 \\ GA & cC & 1.4 & 3.1 & 24 \\ GA & PF & 3 & 1.4 & 4 \\ \hline Statistic & & 81 & 42 & 58 \\ \(p\)-value & & \(1.73\times 10^{-17}\) & \(4.69\times 10^{-9}\) & \(1.88\times 10^{-12}\) \\ \hline PSO & FR & 2.1 & 2.9 & 3.8 \\ PSO & SR & 4 & 3.2 & 3 \\ PSO & eC & 1.9 & 2.5 & 1.3 \\ PSO & PF & 2 & 1.4 & 1.8 \\ \hline Statistic & & 52 & 33 & 68 \\ \(p\)-value & & \(2.82\times 10^{-11}\) & \(2.98\times 10^{-7}\) & \(1.12\times 10^{-14}\) \\ \hline \hline \end{tabular}
\end{table}
Table 10: Ranks and \(p\)-value obtained by the Friedman test. A \(p\)-value less than 0.05 indicates that the group of values of the objective function for each algorithm are different.