## 4 Optimization Method
### Case Study Descriptions and Simulation Results
#### 5.1.4 Case D: Probabilistic Multi-Objective RPP Considering the Wind Power Generation Uncertainty  
In Case D, it is assumed that a wind farm is located at a PQ node. After generating the wind speed scenarios using the Weibull distribution and power curve, a probabilistic multi-objective RPP is performed. In this case, the IEEE 30-bus test system is modified based on [45]. Hence, a wind farm with a rated power of 40 MW is added to bus 22. The wind farm data is derived from [45] and is presented in Appendix A. To evaluate the impact of the wind farm, six scenarios for the output power of the wind farm are generated. The duration of the load is assumed to be 1460 h and without changes in the load level. The generated wind scenarios and their details are given in Table 12.  
The obtained results from the probabilistic multi-objective RPP using the generated wind scenarios are represented in Table 13. As it is observed from this table, among the 15 generated Pareto optimal solutions using the \(\varepsilon\)-constraint technique, the eighth solution (highlighted row) is selected as the BSC after applying the min-max approach. The corresponding value of expected active power losses is 8.5777 MW. Compared with the Base Case and Case A, it is clear that the expected total VAR cost has had a remarkable reduction for the best compromise solution. In addition, the enhancement of expected voltage stability index and expected loadability factor is undeniable towards the Base Case and Case A. In addition, the expected active power losses are elevated in contrast with the Base Case. However, the expected active power losses show a reduction of roughly 1 MW, when it is compared with Case A. The main reason for this reduction is the existence of the wind farm in the case study. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{\chi}\) & \(\mathbf{f_{1}(\mathbf{s})}\) & \(\mathbf{f_{2}}\) & \(\mathbf{f_{3}}\) & \(\overset{\mathbf{\wedge}}{\mathbf{F_{1}}}\) & \(\overset{\mathbf{\wedge}}{\mathbf{F_{2}}}\) & \(\overset{\mathbf{\wedge}}{\mathbf{F_{3}}}\) & \(\overset{\mathbf{\wedge}}{\mathbf{min}}(\overset{\mathbf{\wedge}}{\mathbf{F_{1}}}\overset{\mathbf{ \wedge}}{\mathbf{F_{2}}}\overset{\mathbf{\wedge}}{\mathbf{F_{3}}})\) \\ \hline
1 & \(3.7924\times 10^{5}\) & 0.1222 & 0.0000 & 1.0000 & 1.0000 & 0.0000 & 0.0000 \\
2 & \(4.1835\times 10^{5}\) & 0.1282 & 0.0300 & 0.9538 & 0.9096 & 0.0724 & 0.0724 \\
3 & \(4.6485\times 10^{5}\) & 0.1335 & 0.0600 & 0.8988 & 0.8293 & 0.1448 & 0.1448 \\
4 & \(5.1663\times 10^{5}\) & 0.1386 & 0.0900 & 0.8376 & 0.7523 & 0.2172 & 0.2172 \\
5 & \(5.7321\times 10^{5}\) & 0.1442 & 0.1200 & 0.7707 & 0.6683 & 0.2896 & 0.2896 \\
6 & \(6.3071\times 10^{5}\) & 0.1480 & 0.1500 & 0.7028 & 0.6093 & 0.3620 & 0.3620 \\
7 & \(6.9212\times 10^{5}\) & 0.1525 & 0.1800 & 0.6302 & 0.5418 & 0.4344 & 0.4344 \\
8 & \(7.5741\times 10^{5}\) & 0.1570 & 0.2100 & 0.5530 & 0.4740 & 0.5068 & 0.4740 \\ \hline
9 & \(8.2287\times 10^{5}\) & 0.1615 & 0.2400 & 0.4757 & 0.4058 & 0.5792 & 0.4058 \\
10 & \(8.8964\times 10^{5}\) & 0.1661 & 0.2700 & 0.3968 & 0.3355 & 0.6515 & 0.3355 \\
11 & \(9.4995\times 10^{5}\) & 0.1706 & 0.3000 & 0.3255 & 0.2680 & 0.7239 & 0.2680 \\
12 & \(1.0169\times 10^{6}\) & 0.1752 & 0.3300 & 0.2463 & 0.1984 & 0.7963 & 0.1984 \\
13 & \(1.0876\times 10^{6}\) & 0.1798 & 0.3600 & 0.1628 & 0.1285 & 0.8687 & 0.1285 \\
14 & \(1.1621\times 10^{6}\) & 0.1845 & 0.3900 & 0.0747 & 0.0583 & 0.9411 & 0.0583 \\
15 & \(1.2253\times 10^{6}\) & 0.1883 & 0.4144 & 0.0000 & 0.0000 & 1.0000 & 0.0000 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Generated wind scenarios for the probabilistic multi-objective RPP considering the wind power generation uncertainty.