## 5 Improved VSG Control Strategy Based on Active Transient Feedback Compensation
### Transient Oscillation Mechanism of APFBC-VSG Control  
Rearranging Equation (17) gives the following:  
\[\left\{\begin{array}{l}G_{\text{APFBC}}(s)=\frac{K_{\text{T}}(T_{\text{FB}}s +1)}{c(T_{\text{FB}}s+1)+T_{\text{FB}}K_{\text{T}}K_{\text{FB}}s}\\ G_{\omega\text{APFBC}}(s)=\frac{-K_{\text{T}}(T_{\text{FB}}s+1)(\int_{\omega} \omega_{\text{b}}\beta^{+}+D_{\text{w}}\omega_{\text{b}}+K_{\text{w}})}{c(T_{ \text{FB}}s+1)+T_{\text{FB}}K_{\text{FB}}K_{\text{FB}}s}\end{array}\right. \tag{19}\]  
Figure 8: Small-signal model control block diagram of APFBC-VSG.  
where \(c=J_{\omega}\omega_{0}s^{2}+(D_{\omega}\omega_{0}+K_{\omega})s+K_{\mathrm{T}}\).  
From Equation (19), it is evident that if \(T_{\mathrm{FB}}K_{\mathrm{T}}K_{\mathrm{FB}}\) is relatively small, Equation (19) approximates Equation (6), resulting in weaker feedback compensation and diminished effectiveness in suppressing transient oscillations in the VSG. Consequently, \(T_{\mathrm{FB}}K_{\mathrm{T}}K_{\mathrm{FB}}\) should not be excessively small. Furthermore, since \(K_{\mathrm{T}}\) is generally large, selecting a smaller \(T_{\mathrm{FB}}K_{\mathrm{FB}}\) can enhance the system's transient stability performance.  
Based on Equation (19), Figure 9 depicts the system pole distribution curves for APFBC-VSG control, varying with the decay time constant \(T_{\mathrm{FB}}\) and the compensation coefficient \(K_{\mathrm{FB}}\). Figure 9a illustrates the pole distribution for \(T_{\mathrm{FB}}\) = 0.004 as \(K_{\mathrm{FB}}\) increases from 0 to 20. Compared to the pole distribution of TVSG control, APFBC-VSG control introduces a negative real pole \(s_{\mathrm{FB3}}\) that is initially positioned far from the imaginary axis. As \(K_{\mathrm{FB}}\) increases, \(s_{\mathrm{FB3}}\) gradually approaches the imaginary axis but remains relatively distant, thus exerting only a minor impact on system stability. With increasing \(K_{\mathrm{FB}}\), the dominant poles \(s_{\mathrm{FB1}}\) and \(s_{\mathrm{FB2}}\) of APFBC-VSG control progressively move closer to the real axis, enhancing system stability. When \(K_{\mathrm{FB}}\) = 15.1, these poles transition into negative real poles, shifting the system from underdamped to overdamped. Consequently, APFBC-VSG enhances the system's equivalent damping characteristics and effectively mitigates transient oscillations.  
Figure 9b illustrates the pole distribution at \(K_{\mathrm{FB}}\) = 0.8 and the decay time constant \(T_{\mathrm{FB}}\) increases from 0 to 2. Initially, the newly introduced dominant poles \(s_{\mathrm{FB1}}\) and \(s_{\mathrm{FB2}}\) in the APFBC-VSG control move away from the imaginary axis. However, as \(T_{\mathrm{FB}}\) increases, these poles gradually approach the imaginary axis. The displacement range of the conjugate poles is relatively small, resulting in a minor impact on the system's damping ratio. This indicates that the choice of \(T_{\mathrm{FB}}\) has a limited effect on system stability. Additionally, although the conjugate poles \(s_{\mathrm{FB1}}\) and \(s_{\mathrm{FB2}}\) are close to the imaginary axis and primarily influence the system's stability, the newly introduced real pole \(s_{\mathrm{FB3}}\) in APFBC-VSG control approaches the imaginary axis as \(T_{\mathrm{FB}}\) increases. Despite this, \(s_{\mathrm{FB3}}\) does not become a dominant pole and thus has a minimal effect on system stability. As \(T_{\mathrm{FB}}\) increases, the newly introduced real pole \(s_{\mathrm{FB3}}\) in APFBC-VSG control approaches the imaginary axis.