## 4 Optimization Method
### Case Study Descriptions and Simulation Results
#### 5.1.4 Case D: Probabilistic Multi-Objective RPP Considering the Wind Power Generation Uncertainty
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\chi\) & \(f_{1}(\$)\) & \(f_{2}\) & \(f_{3}\) & \(\hat{F}_{1}\) & \(\hat{F}_{2}\) & \(\hat{F}_{3}\) & \(\min(\hat{F}_{1},\hat{F}_{2},\hat{F}_{3})\) \\ \hline
1 & \(2.0134\times 10^{5}\) & 0.1172 & 0.0011 & 1.0000 & 1.0000 & 0.0000 & 0.0000 \\
2 & \(2.1190\times 10^{5}\) & 0.1224 & 0.0307 & 0.9718 & 0.9277 & 0.0717 & 0.0717 \\
3 & \(2.2563\times 10^{5}\) & 0.1277 & 0.0607 & 0.9352 & 0.8555 & 0.1441 & 0.1441 \\
4 & \(2.4456\times 10^{5}\) & 0.1330 & 0.0907 & 0.8847 & 0.7832 & 0.2165 & 0.2165 \\
5 & \(2.6724\times 10^{5}\) & 0.1382 & 0.1207 & 0.8242 & 0.7110 & 0.2889 & 0.2889 \\
6 & \(2.9247\times 10^{5}\) & 0.1435 & 0.1507 & 0.7569 & 0.6387 & 0.3614 & 0.3614 \\
7 & \(3.1866\times 10^{5}\) & 0.1487 & 0.1807 & 0.6871 & 0.5667 & 0.4338 & 0.4338 \\
8 & \(3.4683\times 10^{5}\) & 0.1539 & 0.2107 & 0.6119 & 0.4946 & 0.5062 & 0.4946 \\
9 & \(3.7578\times 10^{5}\) & 0.1592 & 0.2407 & 0.5347 & 0.4224 & 0.5786 & 0.4224 \\
10 & \(4.0375\times 10^{5}\) & 0.1644 & 0.2707 & 0.4601 & 0.3503 & 0.6511 & 0.3503 \\
11 & \(4.3143\times 10^{5}\) & 0.1697 & 0.3007 & 0.3863 & 0.2782 & 0.7235 & 0.2782 \\
12 & \(4.6080\times 10^{5}\) & 0.1749 & 0.3307 & 0.3079 & 0.2061 & 0.7959 & 0.2061 \\
13 & \(4.9207\times 10^{5}\) & 0.1801 & 0.3607 & 0.2245 & 0.1351 & 0.8684 & 0.1351 \\
14 & \(5.2537\times 10^{5}\) & 0.1850 & 0.3910 & 0.1357 & 0.0672 & 0.9414 & 0.0672 \\
15 & \(5.7623\times 10^{5}\) & 0.1899 & 0.4153 & 0.0000 & 0.0000 & 1.0000 & 0.0000 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Obtained Pareto optimal solutions for the probabilistic multi-objective RPP considering load demand and wind power generation uncertainties. 1.6. Case F: Probabilistic Multi-Objective RPP Considering Load Demand and Wind Power Generation Uncertainties Incorporating Reactive Power from Wind Farms  
In order to evaluate the impact of generated reactive power by wind farms on RPP, the reactive power of wind farms is taken into account during the planning process. Assuming the constant PF operation for the proposed wind farms, the generated reactive power is calculated for each scenario based on Table 18. The value of PF for the proposed wind farms is taken to be 0.98. It should be noted that the generated reactive power by the proposed wind farms is calculated using Equation (47), as follows:  
\[Q_{W_{iS}}=P_{W_{iS}}\tan\left(\cos^{-1}(PF)\right) \tag{47}\]  
where \(Q_{W_{iS}}\) and \(P_{W_{iS}}\) indicate the generated reactive and active power by the proposed wind farms, respectively. \begin{table}
\begin{tabular}{c c} \hline
**Scenario** & **Wind Power Generation (MVAR)** \\ \hline \(S_{1}\) & 0.0000 \\ \(S_{2}\) & 1.0702 \\ \(S_{3}\) & 3.0645 \\ \(S_{4}\) & 5.0709 \\ \(S_{5}\) & 7.0824 \\ \(S_{6}\) & 8.1223 \\ \(S_{7}\) & 0.0000 \\ \(S_{8}\) & 1.0702 \\ \(S_{9}\) & 3.0645 \\ \(S_{10}\) & 5.0709 \\ \(S_{11}\) & 7.0824 \\ \(S_{12}\) & 8.1223 \\ \(S_{13}\) & 0.0000 \\ \(S_{14}\) & 1.0702 \\ \(S_{15}\) & 3.0645 \\ \(S_{16}\) & 5.0709 \\ \(S_{17}\) & 7.0824 \\ \(S_{18}\) & 8.1223 \\ \hline \end{tabular}
\end{table} 
Table 18. The characteristics of generated reactive power by the proposed wind farm for different scenarios. Figure 4. Voltage profile of the load buses for probabilistic multi-objective RPP considering the wind power generation uncertainty. It can be inferred from Table 3 that in almost all scenarios, the VAR sources gain the value of zero, except for the installed VAR compensator at bus 30. In addition, it is revealed that the expected value of the required VAR compensator device at bus 30 reduces while the generated reactive power of the hypothetical wind farms is taken into account. As a result, wind farms have the capability to participate in VAR planning. This leads to a reduction in the size and amount of VAR sources. Therefore, practical power systems show less desire to install new VAR support while numerous large-scale wind farms with sufficient generated reactive power are available.