## 5 Improved VSG Control Strategy Based on Active Transient Feedback Compensation
### APFBC-VSG Small-Signal Model Downscaling and Parameter Design  
A comparison of Equations (6) and (17) reveals that incorporating active power transient feedback compensation in the APFBC-VSG control enhances system stability. However, this modification also transforms the VSG control system from a second-order oscillatory system to a third-order system, making the determination of system control parameters more complex. Figure 9 illustrates that, with the compensation coefficient \(K_{\text{FB}}\) held constant, variations in the decay time constant \(T_{\text{FB}}\) have a negligible impact on the positioning of the dominant conjugate poles. \(K_{\text{FB}}\) predominantly influences the system's stability. Based on this analysis, this paper proposes a simplification approach for reducing the order of the small-signal model by excluding terms in the closed-loop transfer function \(G_{\text{PAFPBC}}(\text{s})\) that involve \(T_{\text{FB}}\) but not \(K_{\text{FB}}\). The small-signal model step-down simplification method for the APFBC-VSG, denoted as \(G_{\text{PFBC}}(\text{s})\), is provided in Equation (20). \[G_{\text{PFBC}}(s)=\frac{K_{\text{T}}}{J_{\omega}\omega_{0}s^{2}+(D_{\omega} \omega_{0}+K_{\omega}+K_{\text{T}}T_{\text{FB}}K_{\text{FB}})s+K_{\text{T}}} \tag{20}\]  
To validate the effectiveness of the small-signal model step-down simplification method proposed in this paper, Figure 11 presents the step response characteristic curves of the closed-loop small-signal model of the APFBC-VSG control before and after simplification, derived from Equations (17) and (20) respectively. Figure 11 shows the unit step response characteristics of \(G_{\text{PAPFBC}}(\text{s})\) and \(G_{\text{PFBC}}(\text{s})\) with \(T_{\text{FB}}\) = 0.006 as the compensation coefficient \(K_{\text{FB}}\) gradually increases. Figure 11b shows the unit step response characteristics of \(G_{\text{PAPFBC}}(\text{s})\) and \(G_{\text{PFBC}}(\text{s})\) with \(K_{\text{FB}}\) = 2, as the decay time constant \(T_{\text{FB}}\) gradually increases. Figure 11a illustrates that as \(K_{\text{FB}}\) increases, the step response characteristics of the reduced-order small-signal closed-loop transfer functions \(G_{\text{PAPFBC}}(\text{s})\) and \(G_{\text{PFBC}}(\text{s})\) remain nearly identical before and after the reduction. Figure 11b demonstrates that with a small \(T_{\text{FB}}\), the step response characteristics of the small-signal closed-loop transfer functions \(G_{\text{PAPFBC}}(\text{s})\) and \(G_{\text{PFBC}}(\text{s})\) before and after the step-down also almost coincide, confirming the efficacy of the proposed small-signal model reduction simplification method. However, with a large \(T_{\text{FB}}\), the zero points and poles in the APFBC-VSG control approach the imaginary axis, significantly impacting system stability and inducing oscillations in \(G_{\text{PAPFBC}}(\text{s})\) under active power disturbances. This observation is consistent with the earlier theoretical analysis, suggesting that \(T_{\text{FB}}\) should be kept small. Equation (20) shows that the simplified transfer function \(G_{\text{PFBC}}(\text{s})\) based on the small-signal model step-down method becomes a typical second-order oscillation system. The control parameters of the APFBC-VSG algorithm can be tuned and optimized according to the cut-off frequency requirements and phase angle constraints dictated by classical control theory for optimal system performance. From Equation (20), it can be deduced that the  
Figure 11: Step response characteristics of the system before and after order reduction: (**a**) the response curve with a change in \(T_{\text{FB}}\); (**b**) the response curve with a change in \(K_{\text{FB}}\). damping ratio and natural oscillation frequency corresponding to the APFBC-VSG system can be expressed by Equation (21). \[\left\{