## 5 Case Study
### Validation of the WSCC Practical Criterion  
As mentioned in Section 4, the system can be judged to be stable if the bus voltage is higher than 0.8 pu or the voltage sag duration is less than 0.33 s. On the contrary, if the voltage drops below 0.8 pu and remains low for more than 0.33 s, the system cannot be judged as unstable. These conclusions will be verified below.  
Firstly, the conclusion that the system will be stable if the bus voltage is higher than 0.8 pu was verified. A permanent short-circuit fault was applied to the Lucheng bus, and the Lucheng bus voltage dropped to 0.8 pu. The voltage no longer recovered. The Lucheng bus voltage curve is shown in Figure 8a, and the slip curves of the induction motors are shown in Figure 8b. As can be seen from Figure 8b, the induction motors operated at a new steady-state point after the Lucheng bus voltage dropped to 0.8 pu. This shows that the induction motors would not stall as long as the Lucheng bus voltage was higher than 0.8 pu.  
Then, the conclusion that the system will be stable if the voltage sag duration is less than 0.33 s was verified. In the Chuxiong region, if the bus voltage is less than 0.5 pu, the low voltage protection device will act to cut off the induction motor. Therefore, the lowest bus  
\begin{table}
\begin{tabular}{c c} \hline
**Bus** & **Value (MW)** \\ \hline Yuanmou & 15 \\ \hline Chuqian & 22 \\ \hline Xiejiahe & 11 \\ \hline Nanhua & 63 \\ \hline \end{tabular}
\end{table}
Table 2: Parameters of the loads.  
\begin{table}
\begin{tabular}{c c} \hline
**Item** & **Value (pu)** \\ \hline Stator resistance \(R_{\text{a}}\) & 0.007 \\ \hline Stator reactance \(X_{\text{a}}\) & 0.0818 \\ \hline Excitation reactance \(X_{\text{m}}\) & 3.62 \\ \hline Rotor resistance \(R_{\text{r}}\) & 0.0062 \\ \hline Rotor reactance \(X_{\text{r}}\) & 0.0534 \\ \hline Rotor inertia time constant \(H\) & 1.6 \\ \hline \end{tabular}
\end{table}
Table 3: Parameters of the induction motor.  
Figure 7: Single-line diagram of the Chuxiong region power grid.  
voltage that may cause motor stalling is 0.5 pu. A short-circuit fault was applied to the Lucheng bus, the fault lasted for 0.33 s, and the Lucheng bus voltage dropped to 0.5 pu during the fault. The bus voltage curve is shown in Figure 8c, and the slip curves are shown in Figure 8d. As can be seen from Figure 8d, the induction motors would not stall if the bus voltage dropped to 0.5 pu and remained low for 0.33 s. Therefore, it can be considered that the system will be stable if the voltage sag duration is less than 0.33 s.  
Finally, the conclusion that the system cannot be determined to be unstable if the bus voltage drops below 0.8 pu and remains low for more than 0.33 s was verified. Similarly, a short-circuit fault was applied to the Lucheng bus, the bus voltage dropped to 0.6 pu, and the sag lasted for 0.6 s. The voltage curve is shown in Figure 8e, and the slip curves are shown in Figure 8f. Obviously, the voltage dropped below 0.8 pu and remained low for more than 0.33 s. However, the induction motors could still return to the original steady-state point after fault clearance. Therefore, the system cannot be judged to be unstable if the voltage drops below 0.8 pu and remains low for more than 0.33 s.