## 4 Numerical Examples
### Case 2
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Bus \#** & \(\text{I}_{\text{CB}}\) **[pu]** & **Bus \#** & \(\text{I}_{\text{CB}}\) **[pu]** & **Bus \#** & \(\text{I}_{\text{CB}}\) **[pu]** \\ \hline
1 & 55 & 11 & 55 & 21 & 35 \\
2 & 55 & 13 & 40 & 22 & 30 \\
3 & 55 & 14 & 30 & 23 & 10 \\
4 & 50 & 15 & 25 & 24 & 50 \\
5 & 45 & 16 & 30 & 25 & 55 \\
6 & 55 & 17 & 35 & 26 & 50 \\
7 & 50 & 18 & 35 & 27 & 35 \\
8 & 45 & 19 & 35 & 28 & 35 \\
9 & 40 & 20 & 25 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Circuit breaker limits in Case 2.  
algorithm can search for other feasible solutions for the decision variables. This instance is illustrated in Figure 8 wherein \(X_{CLR11}\) increased in value and resulted to divergence during the fourth iteration, and with \(X_{CLR7}\) during the sixth iteration. Both \(X_{CLR}\) variables were limited to their previous \(X\) variable values during Step 5 of their respective iterations. Similarly, Figure 8 shows the intermediate values of the twelve \(X_{CLR}\) variables per iteration. The graph shows that the FLCs were sensitive to the incremental increase of only six \(X_{CLR}\) variables most especially with \(X_{CLR10}\) that reached a value of 0.0707 [pu] on the 13 th iteration. All \(X_{CLR}\) variables with intermediate values are considered as an additional 0.1 [pu] CLR countermeasures except for \(X_{CLR11}\) and \(X_{CLR7}\) variables since further increase in their values causes divergence on the subproblem. The discretization of \(X_{CLR}\) variables is illustrated in Figure 8.  
In summary, the list of \(X\) variables that were considered as bus splitting and CLR countermeasures for Case 2 is shown in Table 10. The \(X\) variables are also ranked based on their intermediate values and were applied on the network in the order as they were ranked. As seen in Figure 9, there is a general improvement in the \(z\) value as the number of applied countermeasures was increased. Also, the trend shows that applying one countermeasure and two countermeasures resulted to the same membership function. As discussed earlier, the \(z\) value is the least membership function in the given set of FLCs and this reflect the FLC bus with the highest excessive fault current level violation. Applying one and two bus splitting countermeasure have different effects on the fault current levels of different buses. Figure 8: Set of CLR \(X\) variables at each iteration in Case 2. Figure 7: Set of Bus splitting \(X\) variables at each iteration in Case 2. However, in this specific case, a bus split between buses 21-28 and an additional bus split in buses 1-2 did not affect the fault current level at bus 27 which remained at 38.48 [pu] yielding a membership function value of 0.8258 in both applications. A deeper look at the data revealed that the fault current level at bus 27 was significantly reduced after the application of CLR2 and CLR9, where additional 0.1 [pu] reactances were inserted between bus pairs 5-19 and 25-26, respectively. Almost a similar incident happened between applying five and six countermeasures as seen in Figure 9.  
Applying six countermeasures resulted to more reduction on the fault current levels at the FLC buses compared to only applying five countermeasures, except at bus 7 where six countermeasures only reduced the fault current level by 0.003 [pu]. This explains the very minimal improvement on the \(z\) value between the application of five and six countermeasures. In addition, it is important to note that all of these countermeasures and their combinations resulted to feasible solutions to the FFLC-OPF Ultimately, all twenty-six FLCs were satisfied after the application of the two bus splits (BS6 and BS1) and four current limiting reactors with 0.1 [pu] (CLR10, CLR1, CLR2, and CLR9). This is reflected with a \(z\) value equivalent to 1 as shown in Figure 9. Also, worth nothing that applying five countermeasures (BS6, BS1, CLR10, CLR1, and CLR2) is a good solution as well since it adequately satisfied the FLCs and has a \(z\) value of 0.9999.