# Optimal Power Flow Solutions for Power System Considering Electric Market and Renewable Energy
## 4 Numerical Results
### Obtained Results for the IEEE 30-Node System
#### 4.2.2 Results Obtained for Case 2 and Case 3  
Results obtained by the six applied algorithms for Case 2 and Case 3 are reported in Figures 8 and 9, respectively. The two figures have the same manner that the box and the whisker of JSA are the shorter than those from other algorithms and are located above those from others. For Case 2, the maximum profit of JSA is $4998.8, while that from WCA, MSA, SSA, PSO, and GA is $4997.0, $4986.6, $4915.6, $4975.8, and $4948.0. JSA reaches greater maximum profit than WCA, MSA, SSA, PSO, and GA by $1.8, $12.2, $83.2, $23, and $50.8, respectively. The higher profit is equivalent to 0.04%, 2.44%, 1.7%, 0.46%, and 1.02% of the profit from these algorithms. Especially, the upper quartile of JSA is $4994.4. Meanwhile, that of these algorithms is, respectively, $4983.8, $4965.1, $4915.6, $4945.2, and $4948.0. The deviations indicate that the top solutions of JSA are more outstanding than others. Furthermore, the lower quartile of JSA is slightly smaller than the upper quartile of WCA and much higher than the upper quartile of others. This information indicates that the low-quality solution group of JSA is approximately equal to the top solutions of WCA and much better than the top solutions of MSA, SSA, PSO, and GA.  
The summary of results in Figure 9 for Case 3 also indicates the best performance of JSA as compared to PSO, GA, MSA, SSA, and WCA. The best profit of JSA is greater than that of PSO, GA, MSA, SSA, and WCA by 1.2%, 1.52%, 0.4%, 1.3%, and 0.1%, respectively. In addition, other comparisons show the outstanding performance of JSA. In fact, the worst profit of JSA is $4990.3, which is much greater than the best profit of SSA, $4972.6. The upper extremes of PSO, GA, MSA, SSA, and WCA with $4975.8, $4986.6, $05014.2, $4986.6, $15031.0 are smaller than the upper quartile of JSA with $5031.2. These comparisons indicate that JSA can reach more optimal solutions than others, and the number of high-quality solutions of JSA is also much greater than that of other. Thus, JSA is very effective for the study case of placing wind turbines in transmission power networks. Figures 10 and 11 present the best trial of algorithms over 50 independent trials for Case 2 and Case 3. The two figures have the same manner that WCA can reach greater profit than JSA from the first iteration to the 170 th iteration for Case 2 and from the first iteration to about the 280 th iteration for Case 3. For the last 30 iterations in Case 2 and the last 120 iterations, JSA can improve the profit more effectively than WCA. As compared to four remaining algorithms, JSA is much faster than them. Especially, JSA is about three times faster than PSO, SSA, and GA. For Case 2, the profit of JSA at the 100 th iteration is much greater than that of the three algorithms at the last iteration (iteration 200). For Case 3, the profit of JSA at the 150 th iteration is much greater than that of the three algorithms at the last iteration (iteration 400). In summary, JSA is very effective for the OPF problem with the placement of wind turbine and nodal prices. So, we only implement JSA for other cases in the following section. Figures 12-17 report optimal location and optimal size of wind turbine together with profit of each run over 50 executed runs obtained by the six applied algorithms. To plot the figures, the fifty solutions were sorted from the greatest to the smallest, and then location and size corresponding to each solution were added together with its profit. As shown in  
$4986.6, $2 Summary of results obtained for Case 3. Figure 12, the best location and the best size of wind turbine are, respectively, bus 5 and 10 MW. Similarly, the best factors of WCA, MSA, SSA, GA, and PSO are, respectively, bus 2 and 10 MW, bus 17 and 10 MW, bus 20 and 5 MW, bus 10 and 6 MW, and bus 7 and 10 MW, respectively. Clearly, 10 MW is the best size for the turbine and the three algorithms. JSA, WCA, MSA, and PSO have chosen it. However, the best location is bus 5, and only JSA could find it.