## 3 Transient Oscillation Mechanism Analysis of the TVSG  
### VSG Steady-State Power Characteristics and Existing Problems  
This article focuses on the analysis of the APL with low-frequency characteristics. It can be inferred from the frequency analysis that the voltage control loop presents followability at the low frequency [22], and the adjustment time of the voltage and current inner loop is significantly faster than that of the APL. Hence, the influence of the inner loops on APL outputs is not considered in this article.  
Predicated on the TVSG grid-connected operation circuit depicted in Figure 1, the output power expression [27] is derived as follows:  
\[\left\{\begin{array}{c}P_{\rm out}=3\frac{E\mu_{\rm G}\sin\delta}{X}\approx 3 \frac{E\mu_{\rm G}\delta}{X}\\ Q_{\rm out}=3U_{\rm g}\frac{E\cos\delta-U_{\rm g}}{X}\approx 3U_{\rm g}\frac{E-U_{ \rm g}}{X}\end{array}\right. \tag{6}\]  
where \(U_{\rm g}\) is the designator for the grid's phase voltage; \(\delta\) is the angular disparity in the phase between the TVSG and the grid voltage; and \(X\) is the notation for the line's equivalent impedance. Constructed by integrating Equations (1), (2), and (6), the small-signal control block diagram of the APL system in the TVSG is illustrated in Figure 3.  
As illustrated in Figure 3, \(K_{\rm P}=3E\mu_{\rm G}/X\), when the TVSG operates in grid-connected mode, the closed-loop transfer function of its active output power \(P_{\rm out}\)[28] is presented in the following equation:  
\[\begin{array}{rl}P_{\rm out}&=\frac{K_{\rm P}}{J_{\omega}\omega_{0}\delta^ {2}+(K_{\omega}+D_{\omega})\omega_{0}+K_{\rm P}}P_{\rm ref}+\frac{-K_{\rm P} \omega_{0}(J_{\omega}\delta+K_{\omega}+D_{\omega})}{J_{\omega}\omega_{0}\delta ^{2}+(K_{\omega}+D_{\omega})\omega_{0}\delta^{2}+K_{\rm P}}(\omega-\omega_{0}) \\ &=G_{\rm P}(s)P_{\rm ref}+G_{\omega}(s)(\omega-\omega_{0})\end{array} \tag{7}\]  
where \(G_{\rm P}\)(s) is the closed-loop transfer function of the APL under given power disturbances; \(G_{\omega}\)(s) is the closed-loop transfer function of the APL when the grid frequency is disturbed.  
Figure 3: Small-signal model of the APL in the TVSG.  
Based on Equation (7), when the TVSG operates in grid-connected mode, the steady-state output \(P_{\text{outs}}\) is expressed by the following equation.  
\[\begin{array}{ll}P_{\text{outs}}&=\lim_{s\to 0}\Bigl{(}\frac{K_{\text{P}}}{I_{ \omega\omega}\omega_{0}s^{2}+(K_{\omega}+D_{\omega})\omega_{0}+K_{\text{P}}}P_{ \text{ref}}+\frac{-K_{\text{P}}\omega_{0}(I_{\omega}s+K_{\omega}+D_{\omega})}{ I_{\omega\omega}\omega_{0}s^{2}+(K_{\omega}+D_{\omega})\omega_{0}s+K_{\text{P}}}( \omega-\omega_{0})\Bigr{)}\\ &=P_{\text{ref}}-(K_{\omega}\omega_{0}+D_{\omega}\omega_{0})(\omega-\omega_{0} )\end{array} \tag{8}\]  
As illustrated in Equation (8), under the influence of the damping coefficient, the \(P_{\text{outs}}\) of the TVSG include a steady-state active power deviation (\(D_{\omega}\omega_{0}+K_{\omega}\omega_{0})(\omega_{0}-\omega)\) resulting from the coupling of \(D_{\omega}\) and \(K_{\omega}\). The damping power feedback control of the TVSG essentially simplifies the PFR and damping feedback control into a droop control mechanism. Given the differing purposes of the PFR and damping feedback control, a reasonable \(K_{\omega}\) may lead to the TVSG lacking necessary damping characteristics, resulting in significant power oscillations within the system. An appropriate \(D_{\omega}\) may also result in an undesired PFR coefficient, failing to satisfy the PFR requirement of the TVSG.