## 4 Numerical Examples
### Case 1  
In this section, a simple case was applied to test the functionalities of the algorithm. Three FLCs were selected and added as constraints to the FFLC-OPF. These FLCs are included to limit their corresponding bus fault currents levels. In this simulation, the FLC is the assigned maximum CB breaking duty (\(l_{CB,i}\)) for each bus. The set of FLCs for Case 1 is listed in Table 5. Also, the acceptable limit of bus fault current violation (\(\varepsilon_{i}\)) for the fuzzy membership function of each FLC is uniformly set as 20 [pu] for all three buses. After executing the initial steps of the procedure in Section 3.3, the three FLCs were identified to be critical. Although the FFLC-OPF subproblem was able to find a feasible solution the calculated bus fault current levels in all three buses still exceeded their corresponding FLCs as listed in Table 6. The most excessive of the three fault current levels is at bus 1 which was 12.21% above its \(\mathrm{I}_{\mathrm{CB,max}}\) which is 50 [pu]. Consequently, bus 1 has the lowest value among the three membership functions with 0.695. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Bus** & **Quadratic Term** & **Linear Term** & **Constant Term** \\ \hline
1 & 0.000 & 10.000 & 0.000 \\
2 & 0.000 & 10.000 & 0.000 \\
12 & 0.010 & 12.000 & 0.000 \\
13 & 0.010 & 10.833 & 0.000 \\
22 & 0.008 & 12.000 & 0.000 \\
23 & 0.007 & 11.669 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Candidate locations for countermeasures. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \# & from bus & to bus & \# & from bus & to bus \\ \hline \(X_{\mathrm{BS1}}\) & 1 & 2 & \(X_{\mathrm{BS4}}\) & 7 & 24 \\ \(X_{\mathrm{BS2}}\) & 3 & 25 & \(X_{\mathrm{BS5}}\) & 18 & 27 \\ \(X_{\mathrm{BS3}}\) & 4 & 26 & \(X_{\mathrm{BS6}}\) & 21 & 28 \\ \hline \hline \multicolumn{5}{c}{**Current Limiting Reactor**} \\ \hline \# & from bus & to bus & \# & from bus & to bus \\ \hline \(X_{\mathrm{CLR1}}\) & 3 & 4 & \(X_{\mathrm{CLR7}}\) & 14 & 16 \\ \(X_{\mathrm{CLR2}}\) & 5 & 19 & \(X_{\mathrm{CLR8}}\) & 16 & 18 \\ \(X_{\mathrm{CLR3}}\) & 6 & 5 & \(X_{\mathrm{CLR9}}\) & 25 & 26 \\ \(X_{\mathrm{CLR4}}\) & 6 & 9 & \(X_{\mathrm{CLR10}}\) & 26 & 8 \\ \(X_{\mathrm{CLR5}}\) & 9 & 11 & \(X_{\mathrm{CLR11}}\) & 27 & 19 \\ \(X_{\mathrm{CLR6}}\) & 11 & 9 & \(X_{\mathrm{CLR12}}\) & 27 & 28 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Generators’ quadratic cost function coefficients. \begin{table}
\begin{tabular}{c c} \hline \hline
**Bus \#** & **IC\({}_{\mathbf{B,max}}\) [pu]** \\ \hline
1 & 50 \\
26 & 45 \\
28 & 33 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Circuit breaker limits in Case 1. Buses 26 and 28 have better membership function values of 0.738 and 0.835, respectively. As stated in Equation (3), a value of 1.0 indicates a fault current level that is less than or equal to the IC\({}_{\text{B,max}}\). Furthermore, since \(z\) is the value of the lowest membership function among the given set of FLCs, the algorithm's stopping criterion is set such that all FLCs are satisfied Equation (11). To further reduce the fault current levels at the three critical buses, adequate countermeasures should be applied to the existing network. The algorithm proceeds to Step 6, wherein the sensitivities of each critical fault current level with respect to the countermeasures in Table 4 are evaluated. The sensitivities in Equation (9b) was calculated using an increment of 0.01 [pu] for the \(X\) variables (\(\Delta X_{i}\)). Table 7 shows the FLC sensitivities with respect to the incremental increase in the \(X\) variables particularly \(X_{BS}\) variables. For this case, the localized nature of the FLCs at buses 1, 26, and 28 resulted to negligible sensitivity values with respect to \(X_{CLR}\) variables and were excluded in the Table 7. The sensitivity matrix also reflects the impact of each \(X_{BS}\) variables to the fault current levels at the FLC buses. Note that the negative value indicates a reduction in fault current level as modeled in Equation (9b).