## 3 Frequency Response Analysis Considering the PFR Dead Zone
### Establishment and Quantification of a Frequency Response Model  
In order to derive the analytic expression of the frequency response during the whole process, the frequency of the power system is assumed to be equal, which means that the dynamics of frequency for any node as well as the generator are the same. The topology diagram of the power system is shown in Figure 5, which includes a synchronous generator (SG), a WTG, and a load. In the modeling, the classical frequency response model is modified by adding the frequency regulation control of WTG. In this case, the frequency regulation control block diagram of the system can be seen in Figure 6.  
Assuming that the system is in normal operation before time \(t_{0}\), which means that the power of the SG, WTG, and load are in a balanced state. At \(t=t_{0}\), the load increases \(\Delta P_{\text{L}}\), resulting in a decrease in the frequency of the power system. When the deviation in frequency (\(\Delta f\)) is detected, the SG and WTG will participate in the PFR. In this case, the equivalent rotor motion equation of the system is shown in (14):  
\[\Delta f(s)=\frac{1}{2 Hs+D_{\text{L}}}[\Delta P_{\text{SG}}(s)+\Delta P_{\text {W}}(s)-\Delta P_{\text{L}}(s)] \tag{15}\]  
Figure 5: The topology diagram of the power system. Figure 6: The frequency response control block diagram of the system. For the SG, \(\Delta P_{\text{SG}}\) is proportional to \(\Delta f\); consequently, the PFR of the SG can be reduced to a first-order inertia link. Therefore, the power equation for SG is as follows:  
\[\Delta P_{\text{SG}}(s)=-\frac{K_{\text{G}}}{1+T_{\text{GS}}}\Delta f(s) \tag{16}\]  
where \(T_{\text{G}}\) is the turbine reheat time constant and \(K_{\text{G}}\) represents the PFR coefficient of SG. Hence, the frequency response model of WTG based on VSG control when participating in frequency regulation of the system is shown in Figure 7. As can be seen from Figure 7, the control block diagram of the frequency response with the PFR dead zone can be considered as adding the constant number of dead zone to the input load disturbance. Based on the analysis of the transfer function block diagram in Figure 7, when the frequency response is disturbed by a step power (\(\Delta P_{\text{L}}\)), it is set that \(D_{p}=D_{\text{vgg}}+K_{\text{vgg}}\), then the expression of \(\Delta f\)(s) is shown in Equation (16):  
\[ \begin{array}{ll}\Delta f(s)&=\frac{\Delta P_{\text{L}}-K_{\text{vgg}}d}{s} \frac{(1+T_{\text{G}}s)(2H_{\text{WTG}}\omega_{\text{r0}}s+k)}{(1+T_{\text{G} }s)(2H_{\text{WTG}}\omega_{\text{r0}}s+k)(2H_{\text{S}}+D_{\text{L}})+2H_{ \text{WTG}}\omega_{\text{r0}}K_{\text{G}}s+K_{\text{G}}k+2H_{\text{WTG}} \omega_{\text{r0}}(J_{\text{vgg}}s+D_{\text{P}})(1+T_{\text{G}}s)}\\ &=\left(\Delta P_{\text{L}}-K_{\text{vgg}}d\right)\frac{p_{\text{vgg}}s+p_{1}s +p_{2}}{q_{0}s^{4}+q_{1}s^{3}+q_{2}s^{2}+q_{3}s}\end{array}  \tag{17}\]  
Variables in (16) are expressed as follows:  
\[ \begin{array}{ll}p_{0}=2H_{\text{WTG}}\omega_{\text{r0}}T_{\text{G}}\\ p_{1}=T_{\text{G}}k+2H_{\text{WTG}}\omega_{\text{r0}}\\ p_{2}=k\\ q_{0}=4H_{\text{WTG}}H\omega_{\text{r0}}T_{\text{G}}+2J_{\text{vgg}}T_{\text{G}} H_{\text{WTG}}\omega_{\text{r0}}\\ q_{1}=2J_{\text{vgg}}H_{\text{WTG}}\omega_{\text{r0}}+2T_{\text{G}}H_{\text{WTG }}\omega_{\text{r0}}D_{\text{P}}+2 HT_{\text{G}}k+4H_{\text{WTG}}H\omega_{\text {r0}}+2T_{\text{G}}H_{\text{WTG}}\omega_{\text{r0}}D_{\text{L}}\\ q_{2}=2K_{\text{G}}H_{\text{WTG}}\omega_{\text{r0}}+2H_{\text{WTG}}\omega_{ \text{r0}}D_{\text{P}}+2 Hk+T_{\text{G}}kD_{\text{L}}+2H_{\text{WTG}}\omega_{ \text{r0}}D_{\text{L}}\\ q_{3}=K_{\text{G}}k+D_{\text{L}}k\end{array}