## 3 Simulation Results
### N-1 Contingency Analysis  
The effectiveness of the proposed approach to \(N\)-1 contingency situation, which is the normal system minus one component, was evaluated. In this work, the considered single contingency conditions are outages of lines and outages of generators.  
The IEEE 30-bus system was selected as an example to illustrate the performance of the proposed method to branch outages. Table 6 summarizes the contingency ranking list for the IEEE 30-bus test system. In this table, column 2 denotes different branch outage scenarios and these contingencies are ranked by severity order in terms of \(\lambda^{\max}\) given in column 3. A smaller value of \(\lambda^{\max}\) results in a more severe branch contingency with the higher rank. The obtained results indicate that outages of branches 1-2, 28-27, and 27-30 are identified as the most severe contingencies among the other branch outage cases.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{\(\boldsymbol{\lambda^{\max}}\)} & \multicolumn{2}{c}{**VSM (\%)**} & \multicolumn{2}{c}{**CPU Time (s)**} \\ \cline{2-7}  & **Proposed** & **CPFLOW** & **Proposed** & **CPFLOW** & **Proposed** & **CPFLOW** \\ \hline
1 & 1.4223 & 1.4223 & 23.16 & 23.16 & 125.36 & 506.22 \\
2 & 1.3839 & 1.3839 & 18.69 & 18.69 & 118.08 & 510.17 \\
3 & 1.2641 & 1.2641 & 9.56 & 9.56 & 127.28 & 511.98 \\
4 & 1.2022 & 1.2022 & 5.87 & 5.87 & 117.89 & 525.42 \\
5 & 1.3817 & 1.3817 & 17.14 & 17.14 & 128.76 & 523.11 \\
6 & 1.5732 & 1.5732 & 42.12 & 42.12 & 125.87 & 514.41 \\
7 & 1.2587 & 1.2587 & 15.68 & 15.68 & 120.22 & 520.35 \\
8 & 1.4059 & 1.4059 & 22.61 & 22.61 & 129.21 & 526.41 \\
9 & 1.2831 & 1.2831 & 11.89 & 11.89 & 129.97 & 529.56 \\
10 & 1.4164 & 1.4164 & 22.48 & 22.48 & 118.79 & 518.15 \\
11 & 1.5185 & 1.5185 & 38.53 & 38.53 & 130.51 & 530.76 \\
12 & 1.4611 & 1.4611 & 33.96 & 33.96 & 119.44 & 522.49 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Performance comparison between the proposed approach and CPFLOW approach for Taipower system.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Test System**} & \multirow{2}{*}{**No. of Generators**} & \multirow{2}{*}{**No. of Loads**} & \multirow{2}{*}{**No. of Lines**} & \multicolumn{2}{c}{**Average CPU Time (s)**} \\ \cline{4-5}  & & & & **Proposed** & **CPFLOW** \\ \hline IEEE 30-bus & 6 & 24 & 41 & 0.36 & 1.25 \\ IEEE 118-bus & 54 & 64 & 186 & 3.03 & 11.81 \\ Taipower & 271 & 1538 & 3319 & 121.27 & 502.08 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of average CPU time (s) under 500 simulation runs for the test system.  
The last two columns of Table 6 represent the execution times required for the proposed approach and conventional CPFLOW approach. It indicates that the proposed method can highly reduce the computation burden while maintaining \(\lambda^{\max}\) accuracy.  
Table 7 shows some selected simulation results of single contingency cases for the test systems. An inspection of this table clearly observes that the proposed method can work properly not only for line outage cases but also for generator outage cases.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Rank**} & \multirow{2}{*}{**Branch Outage**} & \multirow{2}{*}{\(\mathbf{\lambda^{\max}}\)} & \multicolumn{2}{c}{**CPU Time (s)**} \\ \cline{3-5}  & & & **Proposed** & **CPFLOW** \\ \hline
1 & 1-2 & 1.2575 & 0.57 & 2.26 \\
2 & 28-27 & 1.5173 & 0.58 & 2.27 \\
3 & 27-30 & 2.0271 & 0.55 & 2.24 \\
4 & 2-5 & 2.