## 3 Dynamic Response Characteristics of the TVSG  
### Transient Oscillation Mechanism of TVSG  
The low-frequency characteristics of APL significantly impact the transient response performance of TVSG. Frequency domain analysis indicates that the voltage control loop has good followability in the low-frequency band, and the response speed of the voltage-current inner loop is significantly faster than that of the power outer loop. In this paper, the impact of the inner loop on the output characteristics of the APL in the VSG is not considered, as the focus is directed towards the system's transient performance [23]. Based on the power transmission principle illustrated in Figure 1, the electromagnetic power output of the TVSG can be formulated as follows:  
\[\left\{ \begin{array}{l}P_{\rm{e}}=\frac{3 EU_{\rm{g}}\sin\delta}{X}\approx \frac{3 EU_{\rm{g}}\delta}{X}\\ Q_{\rm{e}}=\frac{3U_{\rm{g}}\left(E\cos\delta-U_{\rm{g}}\right)}{X}\approx \frac{3U_{\rm{g}}\left(E-U_{\rm{g}}\right)}{X}\end{array} \right. \tag{4}\]  
where \(U_{\rm{g}}\) is the phase voltage of the grid; \(K_{\rm{T}}=3E{U_{\rm{g}}}/X\) is the synchronization voltage coefficient; \(\delta\) is the phase angle difference between the VSG and the grid voltage. Typically, \(\delta\) is small, resulting in \(\sin\delta\approx\delta\) and \(\cos\delta\approx 1\); \(X\) is the equivalent impedance of the line. Combining Equations (1), (2) and (4), the control block diagram for TVSG grid-connected APL is illustrated in Figure 2.  
Based on Figure 2, the small-signal model of the closed-loop transfer function of the TVSG's APL can be derived as follows:  
\[ \begin{array}{rl}\Delta P_{\rm{e}}&=G_{\rm{PP}}(s)\Delta P_{\rm{ref}}+G_{ \rm{Pu}}(s)\Delta\omega_{\rm{g}}\\ &=\frac{K_{\rm{T}}}{\int_{\omega}\omega_{0}s^{2}+\left(D_{\omega}\omega_{0}+K_{ \omega}\right)s+K_{\rm{T}}}\Delta P_{\rm{ref}}+\frac{-K_{\rm{T}}\left(I_{\omega }\omega_{0}s+D_{\omega}\omega_{0}+K_{\omega}\right)}{I_{\omega}\omega_{0}s^{2} +\left(D_{\omega}\omega_{0}+K_{\omega}\right)s+K_{\rm{T}}}\Delta\omega_{\rm{g} }\end{array}  \tag{5}\]  
Figure 2: Small-signal control block diagram of TVSG. In Equation (6),  
\[\left\{ \begin{array}{l}G_{\text{PP}}(s)=\left.\frac{\Delta P_{\text{e}}}{\Delta P _{\text{rot}}}\right|_{\Delta\omega_{\text{g}}=0}=\frac{k_{\text{T}}}{J_{\omega }\omega_{\text{g}}s^{2}+(D_{\omega}\omega_{\text{0}}+K_{\omega})s+K_{\text{T}}} \\ G_{\text{P}\omega}(s)=\left.\frac{\Delta P_{\text{e}}}{\Delta\omega_{\text{g}}} \right|_{\Delta P_{\text{rot}}=0}=\frac{-K_{\text{T}}(J_{\omega}\omega_{\text {0}}s+D_{\omega}\omega_{\text{0}}+K_{\omega})}{J_{\omega}\omega_{\text{0}}s^{2} +(D_{\omega}\omega_{\text{0}}+K_{\omega})s+K_{\text{T}}}\end{array} \right. \tag{6}\]  
where \(G_{\text{PP}}\)(s) is the closed-loop transfer function of the output power for the given power disturbance; \(G_{\text{P}\omega}\)(s) is the closed-loop transfer function of the output power for the grid frequency disturbance; \(\Delta P_{\text{e}}=P_{\text{e}}-P_{\text{ref}}\); \(\Delta\omega_{\text{g}}=\omega_{\text{g}}-\omega_{\text{0}}\); \(\Delta P_{\text{ref}}\) is the given power disturbance step. Equation (6) demonstrates that incorporating the virtual inertia \(J_{\omega}\) transforms \(G_{\text{PP}}\)(s) into a typical second-order oscillatory system. In instances of active power disturbances or grid frequency dips, the grid-connected output active power of the TVSG undergoes transient oscillations and power overshoot issues. The damping ratio \(\xi\) and the natural oscillation angular frequency \(\omega_{\text{n}}\) of this second-order oscillatory system are defined by Equation (7). \[\left\{ \begin{array}{l}\xi=\frac{D_{\omega}\omega_{\text{0}}+K_{\omega}}{2 \sqrt{J_{\omega}K_{\text{T}}\omega_{\text{0}}}}\\ \omega_{\text{n}}=\sqrt{\frac{K_{\text{T}}}{I\omega\omega_{\text{0}}}}\end{array} \right. \tag{7}\]  
The PFR coefficient \(K_{\omega}\) is determined by relevant standards. Consequently, Equation (7) indicates that the system damping ratio in the APL is related to \(J_{\omega}\) and \(D_{\omega}\).