## 4 Numerical Examples
### Case 2  
For this case, the proposed algorithm is tested using the same number and types of countermeasures from Table 4 but with more constraints than Case 1. Twenty-six FLCs are implemented in this simulation and the list of maximum CB breaking duties (I\({}_{\text{CB,max}}\)) is shown in Table 9. The goal of the proposed algorithm is to find a feasible solution that has the optimal type and allocation of countermeasure/s that would reduce the bus fault current levels within the FLCs which is the maximum CB breaking duties for each bus. Figure 5: Fault currents at each of the three FLC buses after applying countermeasures in Case 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Reactance** & **Intermediate Value** & **Discretized Value** & **Countermeasure** & **Result with MFCC-OPF [16]** \\ \hline \(X_{\text{BS1}}\) & 0.03050 & 10.0000 & bus split & bus split \\ \(X_{\text{BS2}}\) & 0.00001 & 0.00001 & still integrated & still integrated \\ \(X_{\text{BS3}}\) & 0.02400 & 10.0000 & bus split & bus split \\ \(X_{\text{BS4}}\) & 0.00001 & 0.00001 & still integrated & still integrated \\ \(X_{\text{BS5}}\) & 0.00001 & 0.00001 & still integrated & still integrated \\ \(X_{\text{BS6}}\) & 0.11600 & 10.0000 & bus split & bus split \\ \hline \hline \end{tabular}
\end{table} 
Table 8: \(X\) variables of each bus splitting candidate (Case 1). MFCC-OPF: modified fault current constrained optimal power flow. Similar with the previous case, all the initial fault current levels at the FLC buses exceed their corresponding \(\text{I}_{\text{CB,max}}\). The worst of which is at bus 27 that initially exceeds its \(\text{I}_{\text{CB,max}}\) by 13.02% giving it the least membership function value of 0.772 as shown in Figure 6. In this case, the FLC buses are scattered throughout the network, therefore each incremental increase in the \(X\) variables have varied effects on the fault current level calculations of each bus. To be able to reduce the fault current level in a wider scale, the sensitivity matrix must reflect which \(\Delta X_{i}\) reduces the most bus fault current levels. The size of the sensitivity matrix for this case is twenty-six FLCs by eighteen \(X\) variables. As discussed in Section 3.2, there were \(X\) variables that were identified to be causing a rather large change in the generation cost and were causing infeasible solutions on the subproblem. In Case 2, it was identified that any increase in the settings of three \(X\) variables namely, \(X_{BS2}\), \(X_{BS4}\), and \(X_{BS5}\) caused the divergence on the subproblem with twenty-six FLCs. The modified Equations (9a) and (10) were implemented so the sensitivity values of the FLCs from these \(X\) variables were set to zero. Figure 6 shows the trajectory of the \(z\) values on each run. After 13 iterations within 50.34 seconds, the stopping criterion was satisfied. Also, the trajectories of the \(X\) variables for all iterations are shown in Figures 7 and 8. In Figure 7, only \(X_{BS6}\) and \(X_{BS1}\) have intermediate values which means that the incremental increase in the reactances of these bus splitting pairs contributed on the reduction of the fault current levels on the system. Both \(X_{BS}\) variables are considered for bus splitting while the other four bus pairs are kept integrated as indicated by the discrete points in Figure 7.  
During the simulation, there are also some instances wherein the subproblem diverges because of an \(X\) variable that reaches a reactance value after some time which results to an infeasible solution. Whenever this happens, Step 5 of the algorithm sets the previous \(X\) variables that yielded a feasible solution as the maximum limit before the algorithm proceeds to Step 6. In this way, the main problem  
Figure 6: Least membership functions value at each iteration Case 2.