## 4 Optimization Method
### Case Study Descriptions and Simulation Results
#### 5.1.2 Case B: Deterministic Multi-Objective RPP Considering the Loadability Factor  
In this part, the \(\varepsilon\)-constraint method is applied to the multi-objective RPP problem in a complex form. However, the uncertainties of the load demand and wind power generation are not considered in this case. In comparison with Case A, another objective, which is called the loadability factor, is added to the problem. Therefore, the main objectives in this part include minimizing the total VAR cost, reducing the active power losses, improving the voltage stability index, and maximizing the loadability factor. In order to solve a deterministic multi-objective RPP considering the loadability factor, it is assumed that the system is under full-load condition. The duration of the load is assumed to be 8760 h. \begin{table}
\begin{tabular}{c c} \hline
**Control Variable** & **Optimal Value** \\ \hline \(V_{g1}\) (p.u.) & 1.06940 \\ \(V_{g2}\) (p.u.) & 1.06150 \\ \(V_{g5}\) (p.u.) & 1.04110 \\ \(V_{g8}\) (p.u.) & 1.04260 \\ \(V_{g11}\) (p.u.) & 1.10000 \\ \(V_{g13}\) (p.u.) & 1.05550 \\ \(t_{6-9}\) (p.u.) & 1.03640 \\ \(t_{6-10}\) (p.u.) & 0.92960 \\ \(t_{4-12}\) (p.u.) & 0.97700 \\ \(t_{28-27}\) (p.u.) & 0.99910 \\ \(Q_{24}\) (MVAR) & 20.9529 \\ \(Q_{25}\) (MVAR) & 1.84190 \\ \(Q_{26}\) (MVAR) & 2.34290 \\ \(Q_{\ell 29}\) (MVAR) & 3.28580 \\ \(Q_{\ell 30}\) (MVAR) & 0.0000 \\ \hline \end{tabular}
\end{table} 
Table 6: The optimal values of the control variables for the deterministic multi-objective RPP without considering the loadability factor. \begin{table}
\begin{tabular}{c c c c} \hline
**Method** & \multicolumn{2}{c}{\(P_{loss}\) (MW)} & \(f_{1}\)**(S)** & \(f_{2}\) \\ \hline \(\varepsilon\)-constraint & Base Case & 5.4970 & \(2.8892\times 10^{6}\) & 0.16350 \\ Method & BCS & 4.9813 & \(2.7084\times 10^{6}\) & 0.12520 \\ Reduction (\%) & 9.3815 & 6.2578 & 23.4251 \\  & Base Case & 4.9630 & \(2.6085\times 10^{6}\) & 0.19780 \\ MODE Algorithm & BCS & 4.8300 & \(2.5387\times 10^{6}\) & 0.12040 \\  & Reduction (\%) & 2.6798 & 2.6759 & 39.1304 \\ \hline \end{tabular}
\end{table} 
Table 5: Comparison of the obtained results between the \(\varepsilon\)-constraint method and MODE algorithm for the BCS under the same operating conditions. Table 7 provides the simulation results of the deterministic multi-objective RPP problem considering the loadability factor. In addition, this table illustrates that among the 15 generated Pareto optimal solutions, the eighth solution (highlighted row) is the BCS through the min-max approach. The active power losses for the BCS is 9.3494 MW. Also, it can be observed that the total VAR cost and active power losses are dramatically increased for BCS in comparison with Case A. Moreover, the voltage stability index is not improved compared with Case A. Nevertheless, the loadability factor is improved in Case B. The main reason behind the deterioration of active power losses and voltage stability index is due to the enhancement of the loadability factor. It should be noted that the loadability factor can hugely affect the active power losses and voltage stability of power systems. Table 8 depicts the optimal values of the control variables for Case B. As it can be observed, three VAR sources have a value of zero. It should be noted that the fixed installation VAR cost is also considered for all VAR sources with the value of zero during the planning studies.