## 5 Numerical Implementation
### Analysis of Scenario 4 (S4)  
This scenario analyzes the impact of installing different FACTS technologies on the test systems, namely the SVC, TCSC, and UPFC devices. This analysis and comparison only consider a variable operation. Table 8 presents the results obtained for the objective function as well as the location and size of the FACTS for all test systems. These results only pertain to the proposed convex model, as it guarantees the global optimum of the problem.  
From Table 8, the following remarks can be made:  
1. The nodes found for locating the different types of FACTS are the same in all test systems. This means that these nodes are more sensitive to reducing the costs of  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Location** & **Size [Mvar]** & \(z\) **(USD/Year)** & **Reduction [\%]** \\ \hline \multicolumn{5}{c}{Fixed Operation} \\ \hline Convex model & [12; 34; 67] & [0.2490, 0.3930, 0.3289] & 113,619.97 & 26.53 \\ \hline \multicolumn{5}{c}{Variable operation} \\ \hline Convex model & [12; 34; 67] & [0.3094, 0.5085, 0.4221] & 107,777.80 & 30.31 \\ \hline \hline \end{tabular}
\end{table}
Table 7: _Cont._  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**FACTS Devices** & **Location** & **Size [Mvar]** & \(z\) **(USD/Year)** & **Reduction [\%]** \\ \hline \multicolumn{5}{c}{IEEE 33-Bus Test System} \\ \hline Benchmark case & – & – & 112,740.90 & – \\ UPFC & [14; 30; 32] & [0.1569, 0.3486, 0.1364] & 101,078.70 & 10.34 \\ TCSC & [14; 30; 32] & [0.1786, 0.4022, 0.1365] & 98,729.21 & 12.42 \\ SVC & [14; 30; 32] & [0.1962, 0.4150, 0.3990] & 96,676.76 & 14.24 \\ \hline \multicolumn{5}{c}{IEEE 69-bus test system} \\ \hline Benchmark case & – & – & 119,715.63 & – \\ UPFC & [21; 61; 64] & [0.0403, 0.4811, 0.1456] & 105,316.40 & 12.03 \\ TCSC & [21; 61; 64] & [0.0683, 0.5380, 0.1450] & 102,861.80 & 14.07 \\ SVC & [21; 61; 64] & [0.0929, 0.5774, 0.1481] & 100,806.50 & 15.79 \\ \hline \multicolumn{5}{c}{IEEE 85-bus test system} \\ \hline Benchmark case & – & – & 154,651.95 & – \\ UPFC & [12; 34; 67] & [0.2483, 0.4770, 0.3836] & 114,920.20 & 25.69 \\ TCSC & [12; 34; 67] & [0.2820, 0.4951, 0.4055] & 110,951.00 & 28.25 \\ SVC & [12; 34; 67] & [0.3094, 0.5085, 0.4221] & 107,777.80 & 30.31 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Numerical results for scenario 4.  
energy losses by injecting or absorbing reactive power. SVC devices maintain the same behavior in all test systems, reducing the costs of energy losses to a greater extent.
2. For the IEEE 33-bus test system, the annual operating costs were minimized to values ranging from USD 101,078.70 (for installed UPFC devices) to USD 96,676.76 (for installed SVCs). The reductions in the objective function were 10.34%, 12.42%, and 14.24% for the UPFC, TCSC, and SVC devices, respectively.
3. For the IEEE 69-bus test system, the annual equivalent operating costs were reduced to values ranging from USD 105,316.40 (for installed UPFC devices) to USD 100,806.50 (for installed SVCs). The objective function was reduced by 12.03%, 14.07%, and 15.79% with respect to the benchmark case (i.e., the case without FACTS integration) for the UPFC, TCSC, and SVC devices, respectively.
4. For the IEEE 85-bus test system, the annual energy loss costs decreased to values ranging from USD 114,920.20 (for installed UPFC devices) to USD 107,777.80 (for installed SVCs). The annual equivalent operating costs were reduced by 25.69%, 28.25%, and 30.31% with respect to the benchmark case for the UPFC, TCSC, and SVC devices, respectively.