## 7 Simulation Results
### Investigating the Performance of the Suggested Objective Functions  
Firstly, we investigate the proposed objective function given in (12). Two different cases (\(C_{1}\) & \(C_{2}\)) are considered in which these cases are different in terms of the pre-set minimum frequency at which a specific load to be shed is shown in Table 3. The optimization results in these cases are loaded to be shed at each minimum frequency and stage as shown in Table 3. In case 1, only 3-stages were activated to restore the frequency as shown in Figure 5. As can be seen from Figure 6, the frequency without load-shedding was below 46.5 Hz and at this frequency the system would have lost synchronism with the rest of the system. With load-shedding the frequency was restored to 50 Hz in 15 s in case 1 and within 10 s in case 2. The minimum system frequency was 49.1 Hz and 49.3 Hz for case 1 and 2, respectively. The differences in the minimum frequency reached was mainly due to two reasons, (i) the most minimum pre-set of the frequency was 48.8 Hz for case 1 while for case two was 49.0 Hz (ii) since the minimum set frequency was less for case 2, in optimization the algorithm shedding time was delayed compared to case 1. More load was shed in case 2 compared to case 1. Four load-shedding stages were activated in case 1 compared to the three activated stages in case 2 as shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{Stages (Pre-set)} & \((k)\) & 1 & 2 & 3 & 4 & 5 \\ Case1 & Frequency threshold settings (Pre-set) [Hz] & \(f_{th}\) & 49.3 & 49.2 & 49.1 & 48.95 & 48.8 \\  & Optimal load-shedding (Result) [p.u] & \(\left(\Delta P_{j}\right)\) & 0.1896 & 0.1451 & 0.1652 & 0.2000 & 0.1179 \\ \hline  & Stages & \((k)\) & 1 & 2 & 3 & 4 & 5 \\ Case 2 & Frequency threshold settings (Pre-set) [Hz] & \(f_{th}\) & 49.4 & 49.3 & 49.2 & 49.1 & 49.0 \\  & Optimal load-shedding (Result) [p.u] & \(\left(\Delta P_{j}\right)\) & 0.1891 & 0.0802 & 0.1064 & 0.1043 & 0.0586 \\ \hline  & Stages & \((k)\) & 1 & 2 & 3 & 4 & 5 \\  & Min frequency (Pre-set) [Hz] & \(f_{\text{min}}\) & 47.0 & 47.0 & 47.0 & 47.0 & 47.0 \\ Case 3 & Max frequency (Pre-set) [Hz] & \(f_{\text{max}}\) & 49.5 & 49.3 & 49.1 & 49.0 & 48.5 \\  & Optimal frequency threshold (Result) [Hz] & \(f_{th}\) & 49.49 & 49.30 & 49.10 & 48.49 & 47.03 \\  & Optimal load-shedding (Result) [p.u] & \(\left(\Delta P_{j}\right)\) & 0.1797 & 0.2000 & 0.1187 & 0.0004 & 0 \\ \hline  & Stages & \((k)\) & 1 & 2 & 3 & 3 & 5 \\ Case 4 & Max frequency (Pre-set) [Hz] & \(f_{\text{min}}\) & 48.8 & 48.8 & 48.8 & 48.8 & 48.8 \\  & Optimal frequency threshold (Result) [Hz] & \(f_{\text{max}}\) & 49.5 & 49.3 & 49.1 & 49.0 & 48.9 \\  & Optimal frequency threshold (Result) [Hz] & \(f_{th}\) & 49.19 & 49.15 & 48.98 & 48.96 & 48.50 \\  & Optimal load-shedding (Result) [p.u] & \(\left(\Delta P_{j}\right)\) & 0.1469 & 0.1501 & 0.0438 & 0.0057 & 0.2000 \\ \hline  & Stages & \((k)\) & 1 & 2 & 3 & 4 & 5 \\ Case 5 & Min frequency (Pre-set) [Hz] & \(f_{\text{min}}\) & 48.8 & 48.8 & 48.8 & 48.8 & 48.8 \\ Case 5 & Max frequency (Pre-set) [Hz] & \(f_{\text{max}}\) & 49.5 & 49.3 & 49.1 & 49.0 & 48.9 \\  & Optimal frequency threshold (Result) [Hz] & \(f_{th}\) & 49.49 & 48.97 & 48.93 & 48.91 & 48.90 \\  & Optimal load-shedding (Result) [p.u] & \(\left(\Delta P_{j}\right)\) & 0.2000 & 0.2000 & 0.1979 & 0.2000 & 0.2000 \\ \hline  & Stages & \((k)\) & 1 & 2 & 3 & 4 & 5 \\ Case 6 & Min frequency (Pre-set) [Hz] & \(f_{\text{min}}\) & 47.0 & 47.0 & 47.0 & 47.0 & 47.0 \\ Case 6 & Max frequency (Pre-set) [Hz] & \(f_{\text{max}}\) & 49.5 & 49.3 & 49.1 & 49.0 & 48.5 \\  & Optimal frequency threshold (Result) [Hz] & \(f_{th}\) & 49.06 & 47.90 & 47.66 & 47.20 & 47.00 \\  & Optimal load-shedding (Result) [p.u] & \(\left(\Delta P_{j}\right)\) & 0.2000 & 0.0985 & 0.2000 & 0.1708 & 0.2000 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Different simulation cases considered in this paper with upper/lower values of the constraints.