## 5 Numerical Results
### Comparison Results on Test System 2  
In this section, the implemented methods are tested on a wind-hydro-thermal system. The system is the combination of the hydrothermal system in Test System 1 and two wind farms. The system is optimally scheduled in twenty-four one-hour subintervals. Similar to Test System 1, three CSA methods, including CSA, SDCSA, and MASCSA, are successfully implemented considering all constraints of the system with the initial settings of \(P_{s}\) = 200 and \(Iter^{max}\) = 10,000. Accordingly, Table 2 shows the obtained results by CSA, SDCSA, and MASCSA. The key information in this table is the success rate comparison. Meanwhile, the comparison of cost is shown in Figures 9 and 10 for reporting less cost and the corresponding level of improvement of MASCSA over CSA and SDCSA, respectively. It should be emphasized that MASCSA can reach 50 successful runs over 50 trial runs, but the number of trial runs for CSA and SDCSA is much higher, which is 72 runs for CSA and 65 runs for SDCSA. Obviously, the constraint solving performance of MASCSA is much better than CSA and SDCSA. Figure 9 shows the significant cost reduction that MASCSA can reach as compared to CSA and SDCSA. The exact calculation, as compared to CSA and SDCSA, of MASCSA can reduce minimum cost by $685.51 and $422.90, mean cost by $572.95 and $466.75, maximum cost by $447.48 and $291.97, and standard deviation by 49.53 and 72.62. As can be observed from Figure 10, the level of improvement is also high and can be up to 2.46% for minimum cost and 14.69% for standard deviation.  
Figure 8: Fitness functions of 50 successful runs obtained by CSA methods for Test System 1.  
Figure 7: The mean convergence characteristics of 50 successful runs obtained by implemented CSA methods for Test System 1.  
Figures 11 and 12 illustrate the faster search performance of MASCSA than CSA and SDCSA for the best run and the whole search process of 50 successful runs. The pink curves of MASCSA in the two figures are always below the black and blue curves of CSA and SDCSA. The best solution and the mean solution of MASCSA are always more promising than those of CSA and SDCSA at each iteration. Namely, the best solution and the mean solution of MASCSA at the 7000 th iteration have lower fitness functions than those of CSA and SDCSA at the 10,000 th iteration. Fifty valid solutions shown in Figure 13 indicate that MASCSA can find a high number of better solutions than the best solution of CSA and SDCSA.  
In summary, the proposed MASCSA can reach a higher success rate, better solutions, and faster speed than CSA and SDCSA for Test System 2. Consequently, the proposed MASCSA is really effective for the system.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **CSA** & **SDCSA** & **MASCSA** \\ \hline Minimum Cost (\$) & 27890.67 & 27628.06 & 27205.16 \\ Average Cost (\$) & 28682.37 & 28576.17 & 28109.42 \\ Maximum Cost (\$) & 29793.52 & 29638.01 & 29346.04 \\ Std. Dev. & 471.41 & 494.50 & 421.88 \\ Computation Time (s) & 440.5 & 499. 1 & 462.4 \\ Success Rate (\%) & 50/72 & 50/65 & 50/50 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of results obtained by CSA, SDCSA, and MASCSA for Test System 2.  
Figure 10: The level of improvement of MASCSA, compared to CSA and SDCSA for Test System 2.  
Figure 9: Better cost in \(\$\) obtained by MASCSA, compared to CSA and SDCSA for Test System 2.