## 4 Numerical Examples
### Case 1
Hence, the FLCs at bus 1, 26, and 28 are more sensitive to the incremental increase in \(X_{BS1}\), \(X_{BS3}\), and \(X_{BS6}\), respectively for this particular iteration. Afterwards, the algorithm proceeds to Step 7 where the main problem of minimizing the sum of all \(X\) variables is solved Equation (8a). The sensitivity matrix from the previous step is utilized as part of the formulation of constraints for this linear programming problem Equation (8b). The sensitivity matrix, which is essentially the rate of change in fault current levels, multiplied by the optimal minimum set of \(X\) variables must be less than or equal to the product of an acceleration coefficient (\(\alpha=2.50\)) and the difference of its circuit breaking duties and existing fault currents. Table 8 shows the intermediate values of the \(X_{BS}\) variables after running the main problem. The calculated fault current level after the first iteration is shown in the third column of Table 6. Subsequently, the algorithm returned to Step 4 to run the FFLC-OPF to get a feasible operating point after the addition of the new \(X_{BS}\) variables from the first iteration. The final fault current levels are shown in the last column of Table 6. Obviously, since all fault current levels are now below their \(I_{\text{CB}}\)s, the least membership function is now equivalent to 1.0 and has satisfied the algorithm's stopping criterion. The next step is to discretize the \(X\) variables that were identified to cause the reduction of the fault current level of the network. Since the algorithm only took one complete iteration of the subproblem and main problem, the intermediate values are quite low. However, these intermediate values still indicate its effect on the identified FLCs that we aim to satisfy. In Table 8, \(X_{BS}\) variables with intermediate values are discretized and assigned with 10.0 [pu] for bus splitting. For comparison, the result applying the modified fault current constrained optimal power flow (MFCC-OPF) to the case with bus splitting candidates was shown in the last column of \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**FLC**} & \multirow{2}{*}{**I\({}_{\text{CB,max}}\) [pu]**} & \multicolumn{2}{c}{**First Iteration**} & \multicolumn{2}{c}{**Second Iteration**} \\ \cline{3-4}  & & **Subproblem** & **Main Problem** & **Subproblem** \\ \hline
1 & 50 & 56.107 & 47.403 & 48.936 \\
26 & 45 & 50.020 & 41.125 & 41.569 \\
28 & 33 & 36.566 & 32.860 & 32.869 \\  & Time: & 0.194 (s) & 3.660 (s) & 0.176 (s) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Fault current level at each iteration. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**FLC** & \(X_{\text{BS1}}\) & \(X_{\text{BS2}}\) & \(X_{\text{BS3}}\) & \(X_{\text{BS4}}\) & \(X_{\text{BS5}}\) & \(X_{\text{BS6}}\) \\ \hline
1 & \(-484.43\) & \(-6.60\) & \(-16.12\) & \(-96.78\) & \(-5.82\) & \(-0.93\) \\
26 & 0.00 & \(-3.67\) & \(-519.60\) & \(-41.29\) & \(-17.27\) & \(-0.56\) \\
28 & 0.00 & \(-2.90\) & \(-16.67\) & \(-1.33\) & \(-7.92\) & \(-73.38\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Sensitivities with respect to the \(X\) variables. Figure 5 below shows the initial and I\({}_{\text{CB,max}}\) fault current levels of each of the three buses with FLCs. Also shown in the graph are the effects on the fault current levels when countermeasures are applied on the network. From Table 8, priority in bus splitting countermeasures can be ranked according to their corresponding intermediate values: \(X_{BS6}\), \(X_{BS1}\), and \(X_{BS3}\). The \(X_{BS6}\) countermeasure corresponds to splitting buses 21-28 and resulted to a fault level reduction in bus 28. Applying two bus splitting countermeasures involved \(X_{BS6}\) and \(X_{BS1}\) which resulted to a fault reduction in buses 28 and 1. Lastly, three bus splitting countermeasures resolved all excessive fault current levels in all three FLC buses.