# Performance Tuning for Power Electronic Interfaces Under VSG Control
## 4 Simplification of the Dynamic System in SFR Stages  
The secondary control regulates the reference generation output and restores the system frequency to the nominal. The SFR is activated after the PFR has stabilized the system frequency. In VSG-PEI, the virtual SFR is executed by adding an integrator, shown in Figure 1b. After the frequency is stabilized, when the SFR kicks in, the transfer function of the dynamic system in the new stage is  
\[ \begin{array}{ll}G(s)=&(-R(1+sT_{G})(1+sT_{CH})(1+sT_{RH})s)\,/\\ &(s(1+sF_{HP}T_{RH})+K_{l}R(1+sT_{G})(1+sT_{CH})(1+sT_{RH})+\\ &(2 Hs+D)(1+sT_{G})(1+sT_{CH})(1+sT_{RH})Rs)\end{array}  \tag{22}\]  
The pole-zero map of the transfer function is shown in Figure 8.  
Figure 8: Pole-zero map of the transfer function in the SFR stage. Figure 7: Speed droop influence on performance indices: (**a**) maximum ROCOF, (**b**) peak frequency, (**c**) \(t_{peak}\), (**d**) \(t_{s}\), (**e**) overshot, and (**f**) setting frequency. From the pole-zero map, the zeros are fixed on the real axis. With the increasing of \(K_{l}\), P1 tends to approach to Z2, which results in the mutual cancelation of effects. The effect of poles P4 and P5, and the zeros Z3 and Z4, which are far away from the imaginary axis, would be neglected for simplifications. Therefore, the poles P2 and P3, and the zero Z1 are dominant in the dynamic processes. From the slow zeros and poles (Z1, P1, and P2), the system transfer function (Equation (22)) is simplified into the second-order, which can be expressed as  
\[G(s)=\frac{-Rs}{s(1+sF_{HP}T_{RH})+K_{l}R+R(Ms+D)s}=K\frac{s+z_{1}}{s^{2}+2 \zeta\omega_{n}s+\omega_{n}^{2}} \tag{23}\]  
where \(\omega_{n}\) and \(\zeta\) are the undamped natural frequency and damping ratio,  
\[ \begin{array}{l}K=-\frac{R}{F_{HP}T_{RH}+MR},\,z_{1}=0,\,\omega_{n}=\sqrt{ \frac{K_{l}R}{F_{HP}T_{RH}+MR}}\\ \zeta=\frac{DR+1}{2(F_{HP}T_{RH}+MR)}\times\sqrt{\frac{F_{HP}T_{RH}+MR}{K_{l}R }}\end{array}  \tag{24}\]  
It is noted that the initial status in the dynamic system is inherited from the IR and PFR stages. By taking the inverse Laplace transformation, the dynamic response of system frequency in time-domain is derived, which can be expressed as  
\[f(t)=1-\frac{K}{\omega_{d}}e^{-\zeta\omega_{n}t}\sin\omega_{d}t \tag{25}\]  
where  
\[\omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}} \tag{26}\]  
From Equation (25), system performance indices during SFR can be derived. First, the rate of change of frequency (ROCOF) can be expressed as  
\[ROCOF=\frac{df}{dt}=-\frac{K}{\omega_{d}}\times(-\zeta\omega_{n}e^{-\zeta \omega_{n}t}\sin\omega_{d}t+\omega_{d}e^{-\zeta\omega_{n}t}\cos\omega_{d}t) \tag{27}\]  
In Equation (27), when \(t=0\), ROCOF reaches the maximum value, which can be expressed as  
\[ROCOF_{\max}=-K \tag{28}\]  
The peak time \(t_{peak}\) is achieved when \(ROCOF=0\), then from Equation (27), the peak time can be expressed as  
\[t_{peak}=\frac{1}{\omega_{d}}\text{arctan}\frac{\sqrt{1-\zeta^{2}}}{\zeta} \tag{29}\]  
By substituting (29) into (25), the peak frequency \(f_{peak}\) can be expressed as  
\[f_{peak}=1-\frac{K}{\omega_{d}}e^{-\zeta\omega_{n}t_{peak}}\sqrt{1-\zeta^{2}} \tag{30}\]  
From Equation (25), the quasi-steady-state frequency \(f_{\rm ss}\) is  
\[f_{\rm ss}=1 \tag{31}\]  
For floating control, the frequency deviation can be evaluated by the frequency overshoot \(\sigma\), which can be derived as  
\[\sigma=\left|\frac{f_{\rm ss}-f_{\rm peak}}{f_{\rm ss}}\right|\times 100\%= \left|\frac{K}{\omega_{d}}\sqrt{1-\zeta^{2}}e^{-\zeta\omega_{n}t_{peak}} \right|\times 100\% \tag{32}\]The settling time \(t_{\rm s}\) can be defined when the decaying exponential \(e^{-\zeta\omega_{n}t}\) reaches 1% [37]. \[e^{-\zeta\omega_{n}t_{\rm s}}=0.01 \tag{33}\]  
then from Equation (33), the settling time can be derived as  
\[t_{\rm s}=\frac{4.6}{\zeta\omega_{n}} \tag{34}\]  
From the performance indices, the effect of separate parameters on system performance in the SFR stage can be derived.