## 4 Optimization Method
### Case Study Descriptions and Simulation Results
#### 5.1.4 Case D: Probabilistic Multi-Objective RPP Considering the Wind Power Generation Uncertainty
After determining the load-wind scenarios, a probabilistic multi-objective RPP is performed to evaluate the existing objectives, while the number of random input variables increases. Taking the IEEE 30 bus-test system with initial settings as the benchmark, 18 combined load-wind scenarios are generated. In general, three load scenarios and six wind scenarios are used to generate 18 combined load-wind scenarios. The descriptions of generated load-wind scenarios are given in Table 15.  
Table 16 shows the obtained results for the probabilistic multi-objective RPP using the generated load-wind scenarios. As seen from Table 16, by applying the min-max method among the 15 generated Pareto optimal solutions using the \(\varepsilon\)-constraint approach, the eighth solution (highlighted row) is selected as the BCS. The associated value to expected active power losses is 8.5575 MW. It can be observed that the expected Total VAR cost is considerably reduced, while the expected voltage stability index and the expected loadability factor are not significantly improved towards Case B. In contrast with Case A and the Base Case, both the expected voltage stability index and the expected loadability factor are improved compared with case B. Considering the expected active power losses, no substantial decrease is observed in Case C when it is compared with Case B. Compared with Cases A and C, a reduction of about 1 MW in expected active power losses can be estimated. Due to enhancing the expected loadability factor, the expected active power losses escalate relative to the Base Case. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multicolumn{2}{c}{**Wind Power**} & \multirow{2}{*}{**Level of the Load**} & \multicolumn{2}{c}{**Duration of the**} & \multirow{2}{*}{**Probability**} \\  & **Generation (MW)** & & & **Load (h)** & \\ \hline \(S_{1}\) & 0.00000 & 0.95 & 400 & 0.0086 \\ \(S_{2}\) & 5.27050 & 0.95 & 400 & 0.0121 \\ \(S_{3}\) & 15.0917 & 0.95 & 400 & 0.0149 \\ \(S_{4}\) & 24.9726 & 0.95 & 400 & 0.0155 \\ \(S_{5}\) & 34.8784 & 0.95 & 400 & 0.0141 \\ \(S_{6}\) & 40.0000 & 0.95 & 400 & 0.0348 \\ \(S_{7}\) & 0.00000 & 1.00 & 730 & 0.0689 \\ \(S_{8}\) & 5.27050 & 1.00 & 730 & 0.0970 \\ \(S_{9}\) & 15.0917 & 1.00 & 730 & 0.1194 \\ \(S_{10}\) & 24.9726 & 1.00 & 730 & 0.1237 \\ \(S_{11}\) & 34.8784 & 1.00 & 730 & 0.1130 \\ \(S_{12}\) & 40.0000 & 1.00 & 730 & 0.2781 \\ \(S_{13}\) & 0.00000 & 1.05 & 330 & 0.0086 \\ \(S_{14}\) & 5.27050 & 1.05 & 330 & 0.0121 \\ \(S_{15}\) & 15.0917 & 1.05 & 330 & 0.0149 \\ \(S_{16}\) & 24.9726 & 1.05 & 330 & 0.0155 \\ \(S_{17}\) & 34.8784 & 1.05 & 330 & 0.0141 \\ \(S_{18}\) & 40.0000 & 1.05 & 330 & 0.0348 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Generated load-wind scenarios for probabilistic multi-objective RPP considering load demand and wind power generation uncertainties. The optimal values of the control variables among 18 generated load-wind scenarios for the BCS are represented in Table 17. As it can be observed from Table 17, the VAR sources gain the value of zero in most of the scenarios for the BCS. Hence, the fixed installation cost is calculated for the VAR sources that gain the value of zero. In order to investigate the impact of VAR planning in bus voltage magnitude over the different load-wind scenarios, the voltage profile of load buses is plotted for each load-wind scenario in Figure 4. This Figure shows that the voltage magnitude of the load buses is limited to the range of 0.95 p.u. and 1.05 p.u. for all scenarios. As a result, the voltage magnitude of the load buses is regulated within the predefined limits.