## 4 TDC-VSG Control Strategy
### TDC-VSG Control Principle  
To fully reflect the damping characteristics of the VSG during transient oscillations and mitigate the steady-state deviations attributed to these characteristics, a transient damping link based on a first-order differential approach is integrated into the damping feedback loop to enhance the transient stability of the APL system in the TVSG. The specific expression is as follows:  
\[P_{\text{D}}=D_{\omega}\omega_{0}s(\omega-\omega_{0}) \tag{10}\]  
Introducing differential links renders the damping circuit more susceptible to high-frequency interference signals and facilitates an increase in the harmonic components of the control signal [20]. Therefore, a first-order lead-lag link is incorporated into the APL to enhance the dynamics and high-frequency disturbance rejection of the TVSG damping feedback loop. The specific expression is as follows:  
\[P_{\text{D}}=\frac{D_{\text{T}}\omega_{0}T_{\text{T}}s}{T_{\text{T}}s+1}( \omega-\omega_{0}) \tag{11}\]  
where \(T_{\text{T}}\) represents the attenuation time constant of the TDC-VSG; \(D_{\text{T}}\) is the transient damping coefficient. Predicated on Equation (11), the block diagram of the TDC-VSG is shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Parameter** & **Value** \\ \hline \(U_{\text{N}}\) & Peak voltage of the grid & 311 V \\ \(f\) & Rated frequency & 50 Hz \\ \(f_{\text{s}}\) & Switching frequency & 10 kHz \\ \(L_{\text{f}}\) & Filter inductance & 3.5 mH \\ \(R_{\text{f}}\) & Inductance spurious & 0.1 \(\Omega\) \\ \(L_{\text{g}}\) & Grid inductance & 1.2 mH \\ \(R_{\text{g}}\) & Spurious resistance & 0.02 \(\Omega\) \\ \(U_{\text{dc}}\) & DC bus voltage & 800 V \\ \(X\) & Equivalent impedance & 1.49 \(\Omega\) \\ \(K_{\omega}\) & PFR coefficient & 7.6 \\ \hline \end{tabular}
\end{table}
Table 1: TVSG main circuit parameters.  
The small-signal model of the APL in the TDC-VSG, considering given power and grid frequency variations, is derived from Figure 5 as follows:  
\[\left\{\begin{array}{l}G_{\text{PTIDc}}(s)=\frac{K_{\text{P}}(T_{\text{T}}s+1)}{ T_{\text{T}}l_{\omega}\omega\omega_{0}^{5}+a_{\text{P}}\omega_{1}^{2}+h_{\text{P}}s+K_{ \text{P}}}\\ G_{\omega\text{TDC}}(s)=\frac{-K_{\text{P}}(T_{\text{T}}s+1)(J_{\omega}\omega _{0}^{5}+D_{\text{T}}\omega_{0}+K_{\omega}\omega_{0})-D_{\text{T}}\omega_{0})}{ T_{\text{T}}l_{\omega}\omega\omega^{5}+a_{\text{P}}\omega_{1}^{2}+h_{\text{P}}s+K_{ \text{P}}}\end{array}\right. \tag{12}\]  
where \(G_{\text{PTIDc}}(s)\) is the closed-loop transfer function of the APL system in the TDC-VSG under given power variations, while \(G_{\omega\text{TDC}}(s)\) is the closed-loop transfer function of the APL under frequency variations. Coefficient \(a_{1}=J_{\omega}\omega_{0}+T_{\text{T}}\left(D_{\text{T}}\omega_{0}+K_{\omega }\omega_{0}\right)\), \(b_{1}=K_{\omega}+T_{\text{T}}K_{\text{P}}\).  
Based on Equation (12), the steady output active power \(\Delta P_{\text{outs}}\) of the APL system in the TDC-VSG during grid-connected operation can be expressed as follows:  
\[\begin{array}{ll}\Delta P_{\text{outs}}&=\lim_{s\to 0}(G_{\text{PTIDc}}(s) \Delta P_{\text{ref}}+G_{\omega\text{TDC}}(s)\left(\omega-\omega_{0}\right)) \\ &=\Delta P_{\text{ref}}-K_{\omega}\omega_{0}(\omega-\omega_{0})\end{array} \tag{13}\]  
By comparing Equation (8) with Equation (13), the damping characteristics of the TDC-VSG achieve a decoupling operation between the damping characteristics and the PFR characteristics during steady-state operation. As a result, the damping characteristics only affect the transient oscillation process, thus eliminating the steady-state deviation caused by the fixed damping control strategy.