**5. Numerical Results**

In this section, the performance of the proposed MASCSA is investigated by comparing the results of the proposed method to those from other implemented methods, such as CSA and SDCSA. Two test systems are employed as follows:


one-hour subintervals. The data of the hydrothermal system are taken from Test System 1 while wind data are taken from [45] and also reported in Table A3 in the Appendix A.

The implemented methods are coded on MATLAB and a personal computer with the CPU of Intel Core i7-2.4GHz, RAM 4GB for obtaining 50 successful runs. The optimal generations of two systems are reported in Tables A4 and A5 in the Appendix A.

### *5.1. 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 e ffects scheduled in twenty-four one-hour subintervals. In order to investigate the e ffectiveness of the MASCSA, CSA and SDCSA are implemented to compare the results. In the first simulation, *Ps* and *Itermax* 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, *Itermax* is increased to 10,000 with a change of 1000 iterations. SDCSA and MASCSA can reach 100% successful runs at *Itermax* = 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 *Ps* = 200 and *Itermax* = 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 5000th iteration is much better than CSA and SDCSA, and the best solution of MASCSA at the 7000th 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 e ffective than that of CSA and SDCSA. The average solution of MASCSA at the 7000th 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 e ffective for large-scale power systems.


**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.

**Figure 7.** The mean convergence characteristics of 50 successful runs obtained by implemented CSA methods for Test System 1.

**Figure 8.** Fitness functions of 50 successful runs obtained by CSA methods for Test System 1.

### *5.2. 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 *Ps* = 200 and *Itermax* = 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.


**Table 2.** Summary of results obtained by CSA, SDCSA, and MASCSA for Test System 2.

**Figure 9.** Better cost in \$ obtained by MASCSA, compared to CSA and SDCSA for Test System 2.

**Figure 10.** The level of improvement of MASCSA, compared to CSA and SDCSA for Test System 2.

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 7000th iteration have lower fitness functions than those of CSA and SDCSA at the 10,000th 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.

**Figure 11.** The best convergence characteristics obtained by implemented CSA methods for Test System 2.

**Figure 12.** The mean convergence characteristics of 50 successful runs obtained by implemented CSA methods for Test System 2.

**Figure 13.** Fitness functions of 50 successful runs obtained by CSA methods for Test System 2.
