## 5 Improved VSG Control Strategy Based on Active Transient Feedback Compensation
### APFBC-VSG Control and Its Small-Signal Analysis  
A compensation mechanism based on active power transient feedback is incorporated into the APL to mitigate transient oscillations in grid-connected power. Consequently, the enhanced APL of the VSG can be expressed by Equation (15).  
\[P_{\rm m}-P_{\rm e}-P_{\rm APFBC}=J_{\omega}\omega_{0}\frac{d\omega}{dt}+D_{ \omega}(\omega_{\rm g}-\omega_{0}) \tag{15}\]  
Figure 7: Transient response characteristics of TDP-VSG under different control parameters: (**a**) The response curve of \(G_{\rm PD}(s)\) (\(T_{\rm d}=0.5\)); (**b**) The response curve of \(G_{\omega\rm PD}(s)\) (\(T_{\rm d}=0.5\)); (**c**) The response curve of \(G_{\rm PD}(s)\) (\(D_{\rm s}=15\)); (**d**) The response curve of \(G_{\omega\rm PD}(s)\) (\(D_{\rm s}=15\)).  
where \(P_{\text{APFBC}}\) is the active power transient feedback compensation. Its precise formulation is provided in Equation (16).  
\[P_{\text{APFBC}}=K_{\text{FB}}(P_{\text{e}}-\frac{1}{T_{\text{FB}}+1}p_{\text{e}}) \tag{16}\]  
where \(T_{\text{FB}}\) is the decay time constant of the APFBC-VSG and \(K_{\text{FB}}\) is the transient power compensation coefficient. This transient compensation power is active only during transient oscillations, thus having no impact on the steady-state output of the VSG. Based on Equations (15) and (16), the active closed-loop control block diagram for the APFBC-VSG control is depicted in Figure 8.  
Referring to Figure 8, the closed-loop small-signal model of the APFBC-VSG control under active power disturbances or grid frequency disturbances is formulated as presented in Equation (17).  
\[\left\{\begin{array}{l}G_{\text{APFBC}}(s)=\frac{K_{\text{T}}(T_{\text{FB}}s +1)}{T_{\text{FB}}\omega_{\text{w}}\omega_{\text{b}}\beta^{+}+2\beta^{2}+b_{2 }s+K_{\text{T}}}\\ G_{\omega\text{APFBC}}(s)=\frac{-K_{\text{T}}(T_{\text{FB}}s+1)}{T_{\text{FB}} \omega_{\text{w}}\omega_{\text{b}}\beta^{+}+D_{\text{w}}\omega_{\text{b}}+K_{ \text{w}}})}{T_{\text{FB}}\omega_{\text{w}}\omega_{\text{b}}\beta^{+}+a_{2} \beta^{2}+b_{2}s+K_{\text{T}}}\end{array}\right. \tag{17}\]  
where \(G_{\text{APFBC}}\)(s) is the closed-loop small-signal model of the APFBC-VSG grid-connected output power under given power disturbances; \(G_{\omega\text{APFBC}}\)(s) is the closed-loop small-signal model of the VSG grid-connected output power under grid frequency disturbances, where \(a_{2}=\int_{\omega}\omega_{0}+T_{\text{FB}}(D_{\omega}\omega_{0}+K_{\omega})\), \(b_{2}=D_{\omega}\omega_{0}+K_{\omega}+T_{\text{FB}}K_{\text{T}}(1+K_{\text{FB}})\).  
The steady-state deviation of the APFBC-VSG control during grid-connected operation, as derived from Equation (17), is expressed by the following equation:  
\[\Delta P_{\text{es-FBC}}=\lim_{s\to 0}\bigl{(}G_{\text{APFBC}}(s)\Delta P_{ \text{ref}}+G_{\omega\text{APFBC}}(s)\Delta\omega_{\text{g}}\bigr{)}=\Delta P _{\text{ref}}-(D_{\omega}\omega_{0}+K_{\omega})\Delta\omega_{\text{g}} \tag{18}\]  
In summary, setting \(D_{\omega}=0\) can eliminate the steady-state deviation induced by the interaction between \(D_{\omega}\) and \(K_{\omega}\). Conversely, by comparing and analysing Equations (6) and (17), it is evident that similar to the TDP-VSG control, the APFBC-VSG control adds a zero point and a pole in its APL and changes the original distribution of zero points and poles, thereby modifying the original distribution of zero points and poles. With appropriate parameter configuration, this approach will effectively suppress transient oscillations in power and enhance system stability.