## 4 Transient Stability Analysis
### Analysis of Condition for Considering Only PLL Dynamics  
Based on the full-order model in this paper and the fourth-order models in [25], the system's transient response can be obtained. We introduce the positive- and negative-sequence angular frequency peak error coefficients, denoted as \(e^{+}\) and \(e^{-}\), respectively, to quantitatively describe the deviation between the two models. The expressions for \(e^{+}\) and \(e^{-}\) are given in Equations (25) and (26). The schematic diagram of \(e^{+}\) is illustrated in Figure 12.  
\[e^{+} =\left|\left(\Delta\omega_{cmax}{}^{+}-\Delta\omega_{ncmax}{}^{+} \right)/\Delta\omega_{cmax}{}^{+}\right| \tag{25}\] \[e^{-} =\left|\left(\Delta\omega_{cmax}{}^{-}-\Delta\omega_{ncmax}{}^{- }\right)/\Delta\omega_{cmax}{}^{-}\right| \tag{26}\]  
When both \(e^{+}\) and \(e^{-}\) are not greater than \(\varepsilon\), it can be considered that the dynamic responses of the fourth-order model and the full-order model are consistent. Therefore, the fourth-order model, which only considers PLL dynamics, can be used to analyze the transient synchronization stability of the system under asymmetric faults. Here, \(\varepsilon\) represents an acceptable model error, typically less than 5%. Thus, the conditions under which the impacts of DSOGI//PNSC and PR-based current control dynamics can be ignored are \(e^{+}\leq\varepsilon\) and \(e^{-}\leq\varepsilon\).  
Through the above analysis, this paper delineates the PLL bandwidth threshold \(f_{c}\), as shown in Figure 13. Mathematically, \(f_{c}\) can be described by (27). When the PLL bandwidth is less than \(f_{c}\), the impacts of DSOGI//PNSC and PR-based current control dynamics can be ignored when analyzing the transient synchronization stability of the grid-connected VSC system.  
\[max\big{\{}e^{+}(f_{c}),e^{-}(f_{c})\big{\}}=\varepsilon \tag{27}\]  
Considering a DLG fault applied on phases b and c, the current references are selected as \(I_{\text{tref}}{}^{+}=0.4\) pu, \(I_{\text{tref}}{}^{+}=0.5\) pu, \(I_{\text{tref}}{}^{-}=0\) pu, and \(I_{\text{tref}}{}^{-}=0.4\) pu. Then, by enumerating, the detailed values of \(f_{c}\) in Figure 13 can be calculated to be 15 rad/s. To validate the effectiveness of the PLL bandwidth threshold delineated in this paper, the PLL bandwidth was reduced to 14.5 rad/s (\(k_{p}=10\), \(k_{i}=50\)). This case is referred to as Case 2. Figure 14 compares the phase trajectories of the full-order and fourth-order models. As shown in Figure 14, the trajectories of the full-order and fourth-order nonlinear models in the \(\delta^{+}\)-\(x_{\text{pll}}{}^{+}\)-\(\delta^{-}\) plane are highly consistent. Additionally, Figure 15 compares the transient responses of the fourth-order, the full-order, and the detailed electromagnetic transient switching models. The figure shows that the responses of these three models are almost identical.  
Figure 12: The schematic diagram of the positive-sequence angular frequency peak error coefficients.  
The results of Figures 14 and 15 indicate that, when the PLL bandwidth is within the threshold defined by Equation (27), the impacts of DSOGI//PNSC and current control dynamics can be ignored. Consequently, the simplified fourth-order model can be used to analyze the transient synchronization stability of the system under asymmetric faults.