## 3 Frequency Response Analysis Considering the PFR Dead Zone
### Establishment and Quantification of a Frequency Response Model
\]  
Arrange Equation (16) as follows:  
\[\Delta f(s)=\frac{\left(\Delta P_{L}-K_{\text{vgg}}d\right)p_{0}}{q_{0}}\frac{ s^{2}+\frac{p_{1}}{p_{0}}s+\frac{p_{2}}{p_{0}}}{s^{4}+\frac{q_{1}}{q_{0}}s^{3} +\frac{q_{2}}{q_{0}}s^{2}+\frac{q_{3}}{q_{0}}s} \tag{18}\]  
Equation (17) can be decomposed into:  
\[\Delta f(s)=\frac{\left(\Delta P_{L}-K_{\text{vgg}}d\right)p_{0}}{q_{0}}\bigg{(} \frac{m_{0}}{s}+\frac{m_{1}}{s-\lambda}+\frac{n_{0}s+n_{1}}{s^{2}+2\xi\omega s +\omega^{2}}\bigg{)} \tag{19}\]  
Figure 7: The control block diagram of the frequency response of the system. By comparing Equation (17) with Equation (18) and making the numerator and denominator coefficients equal, the following relation can be obtained:  
\[ \begin{array}{l}m_{0}+m_{1}+n_{0}=0\\ 2(m_{0}+m_{1})\xi\omega-m_{0}\lambda+n_{1}-n_{0}\lambda=1\\ (m_{0}+m_{1})\omega^{2}-2m_{0}\lambda\xi\omega-n_{1}\lambda=\frac{p_{1}}{p_{0}} \\ -m_{0}\lambda\omega^{2}=\frac{p_{2}}{p_{0}}\\ 2\xi\omega-\lambda=\frac{q_{1}}{q_{0}}\\ \omega^{2}-2\xi\omega=\frac{q_{2}}{q_{0}}\\ \omega^{2}\lambda=\frac{q_{3}}{q_{0}}\end{array}  \tag{20}\]  
Variables in (19) are expressed as follows:  
\[ \begin{array}{l}m_{0}=-\frac{q_{0}p_{2}}{q_{0}}\\ m_{1}=\frac{\frac{q_{0}^{2}p_{2}}{q_{0}}\lambda^{3}-q_{0}\lambda^{2}+\left( \frac{q_{0}^{2}p_{2}}{q_{0}}\frac{-q_{0}p_{1}}{p_{0}}\right)\lambda+\frac{q_{0 }^{2}p_{2}}{q_{0}\lambda^{3}+q_{1}\lambda^{2}+q_{3}}}{+\frac{q_{0}p_{2}}{q_{ 0}}}\\ n_{0}=\frac{q_{1}}{q_{0}}\frac{q_{0}^{2}p_{2}}{q_{0}}\lambda^{3}-q_{0}\lambda^ {2}+\left(\frac{q_{0}^{2}p_{2}}{q_{0}}\frac{-q_{0}p_{1}}{p_{0}}\right)\lambda+ \frac{q_{0}^{4}p_{2}}{q_{0}\lambda^{3}}\\ 2q_{0}\lambda^{3}+q_{1}\lambda^{2}+q_{3}}\\ \xi=\sqrt{\frac{q_{3}}{q_{0}}}\\ \omega=\frac{1}{\left(\frac{q_{1}}{q_{0}}+\lambda\right)}\sqrt{\frac{q_{3}}{q_ {0}}}\\ \lambda=\sqrt{-\frac{Q}{2}+\sqrt{\left(\frac{Q}{2}\right)^{2}+\left(\frac{M}{ 3}\right)^{3}}}+\sqrt{-\frac{Q}{2}-\sqrt{\left(\frac{Q}{2}\right)^{2}+\left( \frac{M}{3}\right)^{3}}}-\frac{q_{1}}{3q_{0}}\\ M=\frac{q_{2}}{q_{0}}-\frac{q_{1}^{2}}{3q_{0}}\\ Q=-\frac{q_{3}}{q_{0}}+\frac{2I_{3}}{27q_{0}^{3}}-\frac{q_{1}q_{2}}{3q_{0}^{ 2}}\end{array} \]  
By taking the inverse Laplace transform of Equation (17), Equation (21) is obtained:  
\[\Delta f(t)=\frac{\left(\Delta P_{\rm L}-K_{\rm vsg}d\right)p_{0}}{q_{0}}\left( m_{0}+m_{1}e^{\lambda t}+C_{1}e^{\lambda_{1}t}+C_{2}e^{\lambda_{2}t}\right) \tag{21}\]  
where \(m_{0}\), \(m_{1}\), \(C_{1}\), \(C_{2}\), \(\lambda\), \(\lambda_{1}\), and \(\lambda_{2}\) are the time-domain expression coefficients of the frequency response for the system, respectively. Maximum frequency deviation \(\Delta f_{\rm max}\) and maximum frequency change rate can be obtained from the above equation, as (22) shows:  
\[\left\{ \begin{array}{l}\Delta f_{\rm max}=\frac{\left(\Delta P_{\rm L}-K_{ \rm vsg}d\right)p_{0}}{q_{0}}\left(m_{0}+m_{1}e^{\lambda t_{1}}+C_{1}e^{ \lambda_{1}t_{1}}+C_{2}e^{\lambda_{2}t_{1}}\right)\\ \frac{d\Delta f(t)}{dt}\Big{|}_{t=t_{0}}=\underset{s\rightarrow\infty}{\lim} s^{2}\Delta f(s)=\left(\Delta P_{\rm L}-K_{\rm vsg}d\right)\frac{p_{0}}{q_{0}} \end{array} \right. \tag{22}\]  
where \(t_{1}\) is the time corresponding to the lowest point of the frequency. Equations (21) and (22), respectively, represent the time-domain characteristics and key indicators of the frequency variation in the power system after considering the PFR dead zone. This analytical expression will lay a quantitative foundation for further optimization of the frequency regulation parameter in the following paper.