## 2 Basic Principles of the TVSG  
Combined with typical grid-connected converter topology, power swing equation, and electromagnetic equation of the SG, the TVSG main circuit topology and its simplified structure are constructed, as shown in Figure 1.  
Predicated on the second-order mathematical model of the traditional SG, the mechanical equation for the APL controller in the TVSG [19] is presented in Equation (1):  
Figure 1: TVSG main circuit topology and its simplified structure. \[P_{\rm m}-P_{\rm out}=J_{\omega}\omega_{0}\frac{\mathrm{d}\omega}{\mathrm{d}t}+D_{ \omega}\omega_{0}(\omega-\omega_{0}) \tag{1}\]  
where \(J_{\omega}\) is the virtual inertia; \(D_{\omega}\) is the damping coefficient; \(\omega\) is the virtual angular frequency; \(\omega_{0}\) is the rated angular frequency; and \(P_{\rm m}\) and \(P_{\rm out}\) are, respectively, designated as the VSG's virtual machine power and output active power. The virtual mechanical power \(P_{\rm m}\) is composed of the active power regulation instruction \(P_{\rm ref}\) and the frequency regulation power output by the PFR characteristic, as shown in the following equation:  
\[P_{\rm m}=P_{\rm ref}+K_{\omega}\omega_{0}(\omega_{0}-\omega) \tag{2}\]  
where \(K_{\omega}\) is designated as the FPR coefficient. Based on the operational principles of the traditional SG reactive power regulator, the primary voltage regulation (PVR) equation for the voltage amplitude command \(E\), emitted by the TVSG, is presented as follows:  
\[ \begin{array}{ll}E&=E_{0}+\Delta E_{\rm Q}+\Delta E_{\rm U}\\ &=E_{0}+K_{\rm q}(Q_{\rm ref}-Q)+K_{\rm u}(U_{\rm ref}-U)\end{array}  \tag{3}\]  
where \(E_{0}\) is the TVSG's no-load potential; \(\Delta E_{\rm Q}\) is the reactive power regulation deviation; \(\Delta E_{\rm U}\) is the voltage regulation deviation; \(K_{\rm q}\) is the power adjustment coefficient; \(Q_{\rm ref}\) is the reactive power adjustment instruction; \(Q\) is the instantaneous reactive power value of the TVSG output; \(K_{\rm u}\) is the voltage regulation coefficient; \(U_{\rm ref}\) is the voltage regulation instruction; and \(U\) is the instantaneous terminal voltage. This paper primarily optimizes the dynamics and transient stability of the APL for the TVSG, achieving decoupling control of active and reactive power in the VSG power control link under inductive line impedance conditions [10]. Consequently, the PVR characteristics of the VSG reactive power loop and voltage-current double closed-loop control strategy are not extensively discussed. Predicated on the above analysis, the TVSG control block diagram is illustrated in Figure 2.  
The virtual inertia and damping characteristics in the TVSG system are primarily reflected in the APL characteristics of its power swing equation, specifically the \(P\)-\(\omega\) characteristics. Based on Equations (1) and (2), the VSG's open-loop transfer function is derived as follows:  
\[G_{\rm P-\omega}=\frac{\Delta\omega}{\Delta P}=\frac{1}{J_{\omega}\omega_{0}s+ K_{\omega}\omega_{0}+D_{\omega}\omega_{0}}=\frac{K_{\rm A}}{\tau s+1} \tag{4}\]  
Figure 2: TVSG control block diagram. \[\left\{ \begin{array}{c}K_{\rm A}=\frac{1}{D_{\omega}\omega_{\rm B}+K_{\omega} \omega_{0}}\\ \tau=\frac{J_{\omega}}{D_{\omega}+K_{\omega}}\end{array} \right. \tag{5}\]  
where \(K_{\rm A}\) represents the power droop coefficient; \(\tau\) is the inertial time constant. Equation (4) illustrates that \(G_{\rm P}\)-\(\omega\) serves as a first-order inertial component. Considering the inherent inertia transition characteristic of \(\tau\), changes in \(\Delta\omega\) exhibit a lag relative to \(\Delta P\), with the extent of this lag contingent upon parameters such as \(J_{\omega}\), \(D_{\omega}\), and \(K_{\omega}\). The interaction between \(D_{\omega}\) and \(K_{\omega}\) determines the extent of steady-state deviation in the TVSG's grid-connected outputs.