## 4 Dynamic Response Characteristics of the TDP-VSG
### Analysis of Active Power Overshoot Problem of TDP-VSG  
To further investigate the power overshoot phenomenon in the TDP-VSG system, the zero point distributions of both the TVSG system and the TDP-VSG system are derived from Equations (6) and (10), respectively, as presented in Table 2.  
As illustrated in Equation (6), the closed-loop transfer function of TVSG under active power disturbance contains no zero points. Under grid frequency disturbances, the closed-loop transfer function features a zero point \(-(D_{\rm s}\omega_{0}+K_{\omega})/(l_{\omega}\omega_{0})\) that is significantly distant from the imaginary axis, thereby exerting a minimal impact on system stability. Furthermore, as the damping coefficient \(D_{\rm s}\) increases, the system's zero points shift further from the imaginary axis. Consequently, for TVSG, increasing \(D_{\rm s}\) mitigates transient oscillations and resolves the power overshoot issue induced by grid frequency disturbances, consistent with the step response characteristics of TVSG presented in Figure 4.  
As illustrated in Table 2, the zero points \(z_{1}\) and \(z_{2}\) of \(\rm G_{\omega D}(s)\) are as follows:  
\[(T_{\rm d}s+1)(J_{\omega}\omega_{0}s+D_{\rm s}\omega_{0}+K_{\omega})-D_{\rm s }\omega_{0}=0 \tag{11}\]  
Rearranging the aforementioned equation yields the following:  
\[T_{\rm d}J_{\omega}\omega_{0}s^{2}+(J_{\omega}\omega_{0}+T_{\rm d}(D_{\rm s} \omega_{0}+K_{\omega}))s+K_{\omega}=0 \tag{12}\]  
The expression for the discriminant \(\Delta\) is derived from the above equation as follows:  
\[ \begin{array}{ll}\Delta&=b^{2}-4T_{\rm d}J_{\omega}\omega_{0}K_{\omega}\\ &=[T_{\rm d}(D_{\rm s}\omega_{0}+K_{\omega})+J_{\omega}\omega_{0}]^{2}-4T_{\rm d }J_{\omega}\omega_{0}K_{\omega}\\ &=[T_{\rm d}(D_{\rm s}\omega_{0}+K_{\omega})-J_{\omega}\omega_{0}]^{2}+4T_{ \rm d}J_{\omega}D_{\rm s}\omega_{0}^{2}\\ &\geq[T_{\rm d}(D_{\rm s}\omega_{0}+K_{\omega})-J_{\omega}\omega_{0}]^{2}\\ \end{array}  \tag{13}\]  
Based on the discriminant expression provided in Equation (13), where \(\Delta>0\), the two zero points \(z_{1}\) and \(z_{2}\) of \(\rm G_{\omega D}(s)\) are real zero points. Their explicit expressions are as follows:  
\[\left\{ \begin{array}{ll}z_{1}=\frac{-(I_{\omega}\omega_{0}+T_{\rm d}(D_{ \omega}\omega_{0}+K_{\omega}))+\sqrt{\Delta}}{2T_{\rm d}J_{\omega}=\omega_{0}} \geq-\frac{1}{T_{\rm d}}\\ z_{2}=\frac{-(I_{\omega}\omega_{0})+T_{\rm d}(D_{\omega}\omega_{0}+K_{\omega}) )-\sqrt{\Delta}}{2T_{\rm d}J_{\omega}=\omega_{0}}\leq-\frac{D_{\rm s}\omega_{0 }+K_{\omega}}{I_{\omega}\omega=0}\\ \end{array} \right. \tag{14}\]  
As illustrated in Table 2, the closed-loop transfer function of the TDP-VSG control under given power disturbances features a fixed zero point \(-1/T_{\rm d}\). An increased \(T_{\rm d}\) shifts this zero point closer to the imaginary axis, leading to the TDP-VSG control's ability to suppress transient oscillations. However, this also results in a significant overshoot in the response characteristics of \(G_{\rm PD}(s)\). The closed-loop transfer function \(\rm G_{\omega D}(s)\) of the TDP-VSG control under grid frequency disturbances contains zero points that vary with the transient damping coefficient \(D_{\rm s}\). Zero-point \(z_{2}\) is located further from the imaginary axis and continues to move further away as \(D_{\rm s}\) increases, indicating that \(z_{2}\) exerts a weaker influence on the overshoot of the response characteristics. In contrast, zero \(z_{1}\) is nearer to the imaginary axis and, as \(D_{\rm s}\) increases, \(z_{1}\) grows larger than \(-1/T_{\rm d}\) and progressively approaches the imaginary axis. This significantly affects the overshoot of the TDP-VSG output response characteristics. To validate the theoretical analysis, the unit step response curves of the TDP-VSG control under given power disturbances and grid frequency disturbances, derived from the closed-loop transfer function in Equation (10), are illustrated in Figure 7. As illustrated in