## 4 Simulations and Results
### Time Domain Verification of the Optimal UFLS Schemes  
In this part, the designed optimal UFLS schemes in the afore-mentioned section were applied to the full scale small receiving-end system. The system dynamic frequency responses of the full scale system for disturbance events are simulated using the DSA Tool software [30].  
For this simulation, the performances of Scheme A designed using the SE and LS based SFR model, Scheme B designed using the GA based identified SFR model, and Scheme B designed using the PSO based identified SFR model are compared. The system frequency responses of the full scale system with the designed optimal UFLS schemes for the contingencies computed with the DSA Tool software are shown in Figure 12. The numerical comparisons of different UFLS schemes are also given in Table 6.  
By means of the designed optimal UFLS schemes utilizing the proposed PSO-based model, the system frequency response trajectory for two contingencies is shown in Figure 12. It is shown in Figure 12 that the designed optimal UFLS schemes bring the system back to the safe region of [49.8 Hz,  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Condition** & \multicolumn{3}{c}{**Peak Demand**} & \multicolumn{3}{c}{**Valley Demand**} \\ \hline
**Scheme** & **A** & **B** & **C** & **A** & **B** & **C** \\ \hline Amount of Load Shed (MW) & 781.2 & 809.7 & 829.8 & 642.3 & 628.6 & 607.9 \\ Steady Frequency (Hz) & 49.88 & 79.89 & 49.90 & 49.96 & 49.91 & 49.88 \\ Minimum Frequency (Hz) & 48.92 & 48.85 & 48.81 & 49.09 & 49.10 & 49.02 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Performance of the different UFLS schemes.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number of Load** & **Best** & **Worst** & **Mean** & **Standard** & **Average Run** \\
**Shedding Stages** & **Best** & **Worst** & **Mean** & **Deviation** & **Time (s)** \\ \hline
2 & 1.864 & 2.061 & 1.962 & 0.047 & 2.27 \\
3 & 1.285 & 1.468 & 1.307 & 0.032 & 6.63 \\
4 & 1.517 & 1.736 & 1.594 & 0.054 & 10.78 \\
5 & 1.717 & 2.014 & 1.885 & 0.068 & 15.32 \\
6 & 2.134 & 2.387 & 2.205 & 0.073 & 20.25 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Statistics of the optimization results for the UFLS design.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Stage** & \multicolumn{3}{c}{**Number of Load Shedding Stages**} \\ \cline{2-7}  & \(f_{th}\) & \(\Delta d\) & \(t_{d}\) & \(f_{th}\) & \(\Delta d\) & \(t_{d}\) \\ \cline{2-7}  & **(Hz)** & & **(s)** & **(Hz)** & & **(s)** \\ \hline
1 & 49.50 & 7.52\% & 0.1 & 49.5 & 6.62\% & 0.13 \\
2 & 49.31 & 9.70\% & 0.1 & 49.3 & 7.11\% & 0.15 \\
3 & 49.11 & 6.25\% & 0.1 & 49.1 & 3.88\% & 0.10 \\
4 & \(-\) & \(-\) & \(-\) & 48.9 & 5.89\% & 0.14 \\
5 & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) \\
6 & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) & \(-\) \\ \hline Obj & & 1.285 & & & 1.517 & \\ \hline \hline \end{tabular}
\end{table}
Table 5: Optimized UFLS schemes.  
50.2 Hz] (the shadowed part in Figure 12) after islanding. The results also show that the system frequency does not settle above 50 Hz and does not exceed the safe range.  
As shown in Figure 12, the optimized UFLS schemes derived using the PSO based, identified simplified SFR model for two disturbances in the small receiving-end system performs better than the UFLS schemes designed using the SE and LS based SFR model. On the other hand, as depicted in Table 6, the optimized UFLS schemes obtained using the PSO based, identified simplified SFR model have advantages of the set rules to guard the security system and thus minimize the total load that is needed to be shed for the islanded system. The results also show that the optimal UFLS schemes derived from the PSO based, identified simplified SFR model have better robustness.  
In general, the optimized UFLS schemes derived utilizing the PSO based, identified simplified SFR model and the developed PSO-based UFLS schemes optimizing model gave better performances for the same contingency event, as shown in Figure 12a,b.