# Optimal Power Flow Solutions for Power System Considering Electric Market and Renewable Energy
## 4 Numerical Results
### Obtained Results for the IEEE 30-Node System  
#### 4.2.1 Results Obtained for Case 1  
In Case 1, all applied algorithms are successful to reach valid solutions. One valid solution shows one profit value and fifty profit values for each algorithm are summarized in Figure 4. In the figure, six profit boxplots with different colors are plotted to compare the performance of algorithms. The box and whisker of JSA are shorter than those of five other ones. The upper extreme, the lower extreme, and the median of JSA are, respectively, higher than those of others to confirm that JSA can reach better maximum profit, better mean profit, and better minimum profit than others. Median of JSA is much higher than the upper quartile of others, and even the lower quartile of JSA is also higher than the upper quartile of MSA and SSA. Notice here the deviation between the median of JSA and the first quartile of others. The height of yellow box is smaller than other boxes, and the yellow whiskers are also shorter than other whiskers. Concerning a clear number for comparisons, JSA reaches more profit than MSA, SSA, WCA, PSO, and GA by 0.4%, 1.2%, 0.2%, 0.7%, and 0.75%, respectively. The special result here is that the minimum profit of JSA is $6610.7, but the maximum profit of SSA, PSO, and GA is $6572.1, $6599.9, and $6558.5. Additionally, the mean profit of JSA is $6629.1, but the maximum profit of MSA is $6623.2. The comparisons indicate the following remarks:  
1. All solutions of JSA are much better than those of SSA, GA and PSO
2. Almost all solutions of JSA are better than those of MSA,
3. JSA has many better solutions than WCA.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**System**} & \multirow{2}{*}{**Study Cases**} & **Applied** & \multirow{2}{*}{\(S_{size^{j}}\) \(G_{max}\)} & **Number of** & **Optimized** \\  & & & **Algorithms** & & **WTs/WFs** & **Parameters of WTs** \\ \hline \multirow{8}{*}{IEEE 30-node system} & Case 1 & 6 & 40; 200 & 0 & No \\  & Case 2 & 6 & 40; 200 & 0 & No \\  & Case 3 & 6 & 40; 400 & 1 & Yes \\  & Case 4 & 1 (JSA) & 40; 200 & 1 & No \\  & & 5.1 & & 100; 1000 & 2 & \\  & Case 5 & 5.2 & 1 (JSA) & 100; 1500 & 3 & Yes \\  & & 5.3 & & 100; 2000 & 4 & \\  & Case 1 & & & 100; 2500 & 0 & No \\  & Case 2 & & & 100; 3000 & 1 & Yes \\ IEEE 118-node system & Case 3 & & 100; 3500 & 2 & Yes \\  & Case 4 & & 100; 4000 & 3 & Yes \\  & Case 5 & & 100; 4500 & 4 & Yes \\  & Case 6 & & 100; 5000 & 5 & Yes \\ \hline \hline \end{tabular}
\end{table}
Table 2: Information of two employed systems and applied algorithms.  
Figure 5 presents the best run among the fifty trials reaching the maximum profit obtained by six applied algorithms. In the figure, JSA can reach greater profit than SSA (from the 18 th iteration to the last iteration), MSA (from the 60 th iteration to the last iteration), WCA (from the 140 th iteration to the last iteration), PSO (from the 40 th iteration to the last iteration), and GA (from the 40 th iteration to the last iteration). On the other hand, the profit of JSA found at the 40 th iteration, the 80 th iteration, the 50 th iteration, the 130 th iteration, and the 160 th iteration is, respectively, greater than the profit of GA, PSO, SSA, MSA, and WCA found at the last iteration (iteration 200). Clearly, JSA is much faster than others in searching for high-quality solutions.  
Optimal power output of each thermal unit and total generation of all thermal units are given in Figures 6 and 7, respectively. In general, JSA has different generation values with others for each thermal units. Additionally, the total generation of JSA is also different from that of others. The power loss of JSA is the same as WCA with 10 MW, whereas that of others is different. SSA has the smallest power loss, but it has not reached the best profit. The phenomenon is because of the ineffectiveness of other control parameters from SSA.  
Figure 4: Summary of results obtained for Case 1.