## 5 Numerical Results
### Investigation of the Proposed Method's Performance on the IEEE 118-Bus System
All results obtained for three single objectives are shown in Table 14. It can be seen that the best TPL, TVD, and \(L\) index from the method for the same comparison condition with methods in [19] (i.e., continuous variables, the same upper bound of load bus voltage of 1.1 pu) are 111.2344 MW, 0.1561 pu, and 0.5678 pu; meanwhile, these values of the best PSO-IPG method in [19] are 115.06 MW, 0.162 pu, and 0.0568 pu. Clearly, the proposed ICOA method can improve performance of up to 3.32% for TPL objective and 3.64% for TVD objective over PSO-IPG, while the proposed method has the same optimal solution quality with PSO-IPG for \(L\) index case. For comparison with other methods shown in Tables 11-13, the improvement level of the proposed method can be from 0.37% to 12.93% for TPL objective, from 0.93% to 92.82% for TVD, and from 0.15% to 56.33% for \(L\) index. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Method** & **Min.** & **Avg.** &
\begin{tabular}{c} **Average** \\ **Time (s)** \\ \end{tabular} & \(\mathbf{N_{\text{s}ape}}\) & **_Ti (pu)_** & **CV** & **DV** & **SR (\%)** \\ \hline ICA & 0.1605 & 0.2741 & 92.66 & 8000 & 1.00 & X & 92 \\ ICAO & 0.1608 & 0.2764 & 93.42 & 8000 & - & X & 93 \\ WCA & 0.2315 & 0.352345 & 126.37 & 8000 & 1.36 & X & 92 \\ SSA & 0.4883 & 0.582 & 93.66 & 8000 & 1.01 & X & 62 \\ SFOA & 0.6061 & 0.7848 & 102.58 & 8000 & 1.11 & X & 57 \\ COA & 0.2034 & 0.34003 & 121.91 & 8000 & 1.32 & X & 53 \\ COA1 & 0.1928 & 0.275772 & 107.04 & 8000 & 1.16 & X & 91 \\ COA2 & 0.1936 & 0.296056 & 115.96 & 8000 & 1.25 & X & 70 \\ CLPSO [16] & 1.6177 & - & - & 24,000 & - & X & - \\ PSO [16] & 2.2359 & - & - & 24,000 & - & X & - \\ PSO-AIW [19] & 0.1935 & 0.2291 & 78.49 & 8000 & 0.93 & X & - \\ PSO-AAC [19] & 0.3921 & 0.4724 & 78.7 & 8000 & 0.93 & X & - \\ SPSO-AAC [19] & 0.2074 & 0.2498 & 74.9 & 8000 & 0.89 & X & - \\ PSO-CF [19] & 0.1801 & 0.2143 & 78.13 & 8000 & 0.92 & X & - \\ PSO-PG [19] & 0.1658 & 0.2084 & 51.24 & 8000 & 0.61 & X & - \\ PSO-SWT [19] & 0.1658 & 0.2084 & 51.24 & 8000 & 0.61 & X & - \\ PSO-PGSWT [19] & 0.2355 & 0.2755 & 114.5 & 8000 & 1.35 & X & - \\ PSO-IPG [19] & 0.162 & 0.1923 & 47.86 & 8000 & 0.57 & X & - \\ QOTLBO [33] & 0.191 & 0.2043 & - & 10,000 & - & X & - \\ TLBO [33] & 0.2237 & 0.2306 & - & 10,000 & - & X & - \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Comparison of results obtained from the IEEE 118-bus system with voltage deviation optimization. All methods in [19] were run by setting \(N_{pop}\) = 40 and \(N_{Iter}\) = 200 corresponding to \(N_{sqe}\) = 8000, which is similar to that of the proposed method but their execution time was reported completely differently. For instance, the simulation time for TPL objective was approximately about 95 s. However, for TVD objectives, some methods reported about 75 s, and some other methods reported about 50 s, whilst one method, PSO-PGSWT, reported over 110 s. For \(L\) index, simulation time was over 110 s for five methods, but it was under 60 s for three other ones. The unreasonable issue is that methods using pseudogradient search reported the shortest time, even though pseudogradient search is complicated and it takes more time to determine the better direction for velocity. The comparisons of search speed easily led to the conclusion that the search speed of the proposed method is ranked in the leading method group, with once only methods in [19] having been executed for solving the IEEE 118-bus system by setting the same number of solution quality evaluations, which is less than that of all other methods. By considering \(Ti\), it can be indicated that \(Ti\) of other methods is higher than 1, i.e., they were slower to converge to their optimal solutions. Consequently, it can conclude that the proposed method is effective for solving the IEEE 118-bus system. Optimal solutions obtained by the proposed method for IEEE 118-bus system are given in Tables A3-A8 in Appendix A.