## 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} \hline \hline \multirow{2}{*}{**Scenario**} & **Wind Power** & \multirow{2}{*}{**Probability**} & \multirow{2}{*}{**Level of the Load**} & **Duration of the** \\  & **Generation (MW)** & & & **Load (h)** \\ \hline \(S_{1}\) & 0.00000 & 0.0861 & 1 & 1460 \\ \(S_{2}\) & 5.27050 & 0.1212 & 1 & 1460 \\ \(S_{3}\) & 15.0917 & 0.1492 & 1 & 1460 \\ \(S_{4}\) & 24.9726 & 0.1546 & 1 & 1460 \\ \(S_{5}\) & 34.8784 & 0.1413 & 1 & 1460 \\ \(S_{6}\) & 40.0000 & 0.3476 & 1 & 1460 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Obtained Pareto optimal solutions for the probabilistic multi-objective RPP considering the wind generation uncertainty. The optimal values of the control variables for the BCS over the wind scenarios are represented in Table 14. Taking a look at Table 14, it is shown that the VAR sources gain the value of zero in almost all scenarios. Therefore, the variable VAR investment cost for those VAR sources equals to zero. However, the VAR sources are allocated and their fixed VAR investment costs are taken into account during the planning horizon. The voltage profile of the load buses for the BCS is plotted in Figure 3. As it can be observed, the voltage magnitude of the load buses is kept in the interval of 0.95 p.u. and 1.05 p.u. during all wind scenarios. Hence, it can be concluded that based on a proper RPP and having adequate reactive power reserve, the voltage magnitude of the load buses are restricted with specific limits. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Control** & \multirow{2}{*}{\(S_{1}\)} & \multirow{2}{*}{\(S_{2}\)} & \multirow{2}{*}{\(S_{3}\)} & \multirow{2}{*}{\(S_{4}\)} & \multirow{2}{*}{\(S_{5}\)} & \multirow{2}{*}{\(S_{6}\)} & **Expected** \\
**Variable** & & & & & & & **Value** \\ \hline \(V_{g1}\) (p.u.) & 1.0630 & 1.0627 & 1.0619 & 1.0611 & 1.0604 & 1.0599 & 1.0611 \\ \(V_{g2}\) (p.u.) & 1.0531 & 1.0530 & 1.0527 & 1.0523 & 1.0520 & 1.0518 & 1.0523 \\ \(V_{g5}\) (p.u.) & 1.0767 & 1.0778 & 1.0773 & 1.0769 & 1.0766 & 1.0765 & 1.0769 \\ \(V_{g8}\) (p.u.) & 1.0536 & 1.0525 & 1.0528 & 1.0531 & 1.0533 & 1.0534 & 1.0531 \\ \(V_{g11}\) (p.u.) & 1.1000 & 1.1000 & 1.1000 & 1.1000 & 1.1000 & 1.1000 & 1.1000 \\ \(V_{g13}\) (p.u.) & 1.0867 & 1.0893 & 1.0887 & 1.0883 & 1.0882 & 1.0883 & 1.0883 \\ \(t_{6-9}\) (p.u.) & 1.0264 & 1.0266 & 1.0289 & 1.0311 & 1.0330 & 1.0338 & 1.0310 \\ \(t_{6-10}\) (p.u.) & 0.9000 & 0.9000 & 0.9000 & 0.9000 & 0.9000 & 0.9000 & 0.9000 \\ \(t_{4-12}\) (p.u.) & 1.0084 & 1.0121 & 1.0094 & 1.0071 & 1.0052 & 1.0043 & 1.0070 \\ \(t_{28-27}\) (p.u.) & 0.9657 & 0.9499 & 0.9490 & 0.9480 & 0.9468 & 0.9462 & 0.9491 \\ \(Q_{c24}\) (MVAR) & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ \(Q_{c25}\) (MVAR) & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ \(Q_{c26}\) (MVAR) & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ \(Q_{c29}\) (MVAR) & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ \(Q_{c30}\) (MVAR) & 3.8586 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.3321 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: The optimal values of the control variables among the wind scenarios for probabilistic multi-objective RPP considering the wind power generation uncertainty. Figure 3: Voltage profile of the load buses for probabilistic multi-objective RPP considering the wind power generation uncertainty. 1.5 Case E: Probabilistic Multi-Objective RPP Considering Load Demand and Wind Power Generation Uncertainties  
To have a more comprehensive overlook of the probabilistic RPP, in Case E, it is preferred to perform RPP in the presence of two stochastic input variables, including the load demand and wind power generation. Hence, considering the level of the load and wind power generation as the stochastic input variables, combined load-wind scenarios are generated via the proposed technique.