## 5 Numerical Results
### Comparison Results on Test System 1  
In this section, the MASCSA is tested on a large hydrothermal system with four hydropower plants and four thermal power plants, considering valve effects scheduled in twenty-four one-hour subintervals. In order to investigate the effectiveness of the MASCSA, CSA and SDCSA are implemented to compare the results. In the first simulation, \(P_{s}\) and \(Iter^{max}\) are set to 200 and 5000 for all methods, respectively, but CSA cannot reach successful runs for each of the 50 trial runs. Meanwhile, SDCSA reaches a very low success rate. Then, \(Iter^{max}\) is increased to 10,000 with a change of 1000 iterations. SDCSA and MASCSA can reach 100% successful runs at \(Iter^{max}\) = 10,000, but CSA only reaches 50 successful runs over 70 trial runs. Results obtained by the implemented methods are summarized in Table 1.  
It is noted that the results from CSA, SDCSA, and MASCSA are obtained at \(P_{s}\) = 200 and \(Iter^{max}\) = 10,000, with the aim of reaching a higher number of successful runs for CSA and SDCSA. In order to check the powerful searchability of MASCSA over CSA and SDCSA, Figures 4 and 5 are plotted to present less cost and the corresponding level of improvement. Figure 4 indicates that the reduced cost that ASCSA can reach is significant and much increased for average cost and maximum cost. Accordingly, the level of improvement of the minimum cost, average cost, and maximum cost are respectively 0.54%, 1.3% and 2.81% as compared to CSA and 0.29%, 0.92% and 2.75% as compared to SDCSA. Similarly, the improvement of standard deviation is also high, corresponding to 23% and 27.12%, as compared to CSA and SDCSA. The indicated numbers lead to the conclusion that MASCSA is superior over CSA and SDCSA, in terms of finding the best solution and reaching a more stable search process.  
In addition, the best run and the average run of 50 successful runs are also plotted in Figures 6 and 7 for search speed comparison. The two figures confirm that MASCSA is much faster than CSA and SDCSA for the best run and the average of all runs. In fact, in Figure 6, the best solution of MASCSA at the 5000 th iteration is much better than CSA and SDCSA, and the best solution of MASCSA at the 7000 th iteration is also better than that of CSA and SDCSA at the last iteration. This indicates that the speed of MASCSA can be nearly two times faster than CSA and SDCSA. In Figure 7, the average solution of 50 solutions found by MASCSA is also much more effective than that of CSA and SDCSA. The average solution of MASCSA at the 7000 th iteration is also better than that of CSA and SDCSA at the last iteration. Clearly, the stability of MASCSA is also nearly twice as good as that of CSA and SDCSA. The whole view of the 50 solutions comparison can be seen by checking Figure 8. Many solutions of MASCSA have lower cost than that of CSA and SDCSA.  
In summary, the proposed MASCSA is superior over CSA and SDCSA in finding optimal solutions and reaching a faster search speed for Test System 1. Hence, the proposed modifications of MASCSA are effective for large-scale power systems.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **CSA** & **SDCSA** & **MASCSA** \\ \hline Minimum Cost (\$) & 35640.09 & 35550.06 & 35447.25 \\ Average Cost (\$) & 36835.21 & 36694.27 & 36355.55 \\ Maximum Cost (\$) & 38616.82 & 38595.07 & 37533.4 \\ Std. Dev. (\$) & 595.36 & 628.65 & 458.1301 \\ Computation Time (s) & 437.30 & 498.71 & 457.92 \\ Success Rate & 50/70 & 50/50 & 50/50 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of results obtained by CSA, SDCSA, and MASCSA for Test System 1.  
Figure 4: Better cost in $ obtained by MASCSA, compared to CSA and SDCSA, for Test System 1.  
Figure 5: The level of improvement of MASCSA compared with CSA and SDCSA for Test System 1.  
Figure 6: The best convergence characteristics obtained by implemented CSA methods for Test System 1.