## 4 Application Cases
### Results
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Case 5**} & \multicolumn{2}{c}{**Case 6**} \\ \hline
**Eigenvalues** & **Damping Ratio (\%)** & **Eigenvalues** & **Damping Ratio (\%)** \\ \hline \(-100.0005\pm 0.2310944i\) & 99.9997 & \(-100.0007\pm 0.2700379i\) & 99.9996 \\ \(-42.3731\pm 0.114598i\) & 42.3728 & \(-42.3731\pm 0.129702i\) & 42.3727 \\ \(-38.9105\pm 0.0353774i\) & 38.9102 & \(-38.9583\) & 38.9579 \\ \(-0.838697\pm 12.7151i\) & 0.8387 & \(-38.8626\) & 38.8622 \\ \(-0.20069\pm 8.3749i\) & 0.2007 & \(-1.6603\pm 15.7714i\) & 1.6603 \\ \(-4.9831\) & 4.9830 & \(-1.59423\pm 12.0215i\) & 1.5942 \\ \(-4.3417\pm 1.8695i\) & 4.3417 & \(-4.7451\) & 4.7450 \\ \(-4.2221\pm 1.7196i\) & 4.2221 & \(-4.4188\pm 1.9184i\) & 4.4187 \\ \(-4.2777\pm 1.8013i\) & 4.2777 & \(-4.2204\pm 1.7038i\) & 4.2203 \\ \(-3.3397\) & 3.3397 & \(-4.294\pm 1.8199i\) & 4.2940 \\ \(-1.3942\) & 1.3941 & \(-2.0724\) & 2.0724 \\ \(-1.2578\) & 1.2578 & \(-1.4928\) & 1.4928 \\ \(-0.7910\) & 0.7910 & \(-0.78907\pm 0.3786i\) & 0.7891 \\ \(-0.5069\) & 0.5069 & \(-1.2080\) & 1.2080 \\ \(-0.3246\) & 0.3246 & 0.1771 & \(-0.1771\) \\ \(-0.2232\) & 0.2232 & 0.0002 & \(-0.0002\) \\ \(-0.0045\) & 0.0045 & \(-0.3588\) & 0.3588 \\ \(-0.0282\) & 0.0282 & \(-0.2231\) & 0.2231 \\ \(-0.1003\) & 0.1003 & \(-0.1001\) & 0.1001 \\ \(-0.1001\) & 0.1001 & \(-0.1001\) & 0.1001 \\ \(-3.2258\) & 3.2258 & \(-3.2258\) & 3.2258 \\ \hline \multicolumn{2}{c}{**Case 7**} & \multicolumn{2}{c}{**Case 8**} \\ \hline
**Eigenvalues** & **Damping Ratio (\%)** & **Eigenvalues** & **Damping Ratio (\%)** \\ \hline \(-100.0007\pm 0.2706928i\) & 99.9996 & \(-100.0006\pm 0.248591i\) & 99.9997 \\ \(-42.3732\pm 0.132069i\) & 42.3727 & \(-42.3731\pm 0.106511i\) & 42.3727 \\ \(-38.9450\) & 38.9446 & \(-39.0035\) & 39.0031 \\ \(-38.8759\) & 38.8755 & \(-38.8172\) & 38.8169 \\ \(-1.66024\pm 15.7902i\) & 1.6602 & \(-1.76227\pm 15.8307i\) & 1.7623 \\ \(-1.54966\pm 12.1486i\) & 1.5496 & \(-2.05173\pm 11.6027i\) & 2.0517 \\ \(-4.7810\) & 4.7809 & \(-4.5189\) & 4.5189 \\ \(-4.4198\pm 1.9231i\) & 4.4198 & \(-4.4769\pm 1.9239i\) & 4.4768 \\ \(-4.2206\pm 1.7084i\) & 4.2206 & \(-4.2252\pm 1.686i\) & 4.2251 \\ \(-4.2954\pm 1.8199i\) & 4.2954 & \(-4.3212\pm 1.8441i\) & 4.3212 \\ \(-2.1211\) & 2.1211 & 0.6634 & \(-0.6634\) \\ \(-1.5075\) & 1.5075 & \(-1.8076\) & 1.8076 \\ \(-1.2391\) & 1.2391 & \(-1.4787\) & 1.4787 \\ \(-0.76142\pm 0.27743i\) & 0.7614 & \(-0.80976\pm 0.61076i\) & 0.8098 \\ \(0.1164\) & \(-0.1164\) & \(-0.8088\) & 0.8088 \\ \(0.0003\) & \(-0.0003\) & \(-0.3860\) & 0.3860 \\ \(-0.3678\) & 0.3678 & 0.1447 & \(-0.1447\) \\ \(-0.2336\) & 0.2336 & 0.0001 & \(-0.0001\) \\ \(-0.1001\) & 0.1001 & \(-0.1001\pm 1.5159\times 10^{-5}i\) & 0.1001 \\ \(-0.1001\) & 0.1001 & \(-3.2258\) & 3.2258 \\ \(-3.2258\) & 3.2258 & & 3.2258 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Damping ratio percentages of Scenario 2. Although Case 5 is stable despite the operation condition, it has very low damping percentages and is close to instability with the analytical model. The real time domain simulation of Scenario 2 showed that all of the systems were unstable in all conditions. This reinforces the idea that the classical model does not properly represents unbalanced conditions and that the proposed approach does a better performance. To verify the instability in the time domain simulations, Figure 10 shows the frequency response for Cases 2 and 6. It is evident that Case 6 has an unstable condition as the frequency drops even without the short-circuit at Second 10. This behavior is identical in Cases 7 and 8. Figure 10: Angles and frequency of Machine 2 in Case 2 versus Case 6. Figure 9: Eigenvalues resulting from the modeling of the case with a 45% increased load.