## 3 Simplification and Performance Analysis  
The emulation of IR and PFR is derived from the mechanism imitation of the swing equation and the speed governor. The effect of the closed-loop transfer function of the VSI based on the cascade control structure is assumed to be fast and accurate. The high-order transfer function of IR and PFR can be simplified by the dominant poles and zeros according to control theory. By the inverse Laplace transformation, the performance indices of the dynamic responses in the time domain can be deduced. As the frequency is a common factor, when there is a load change, it is reflected instantaneously throughout the whole system by a change in frequency. For a step change of demand \(\Delta P_{L}\), the transfer function from \(\Delta P_{L}\) to the virtual angular speed of rotor \(\Delta\omega_{m}\) is  
\[G(s)=\frac{\Delta\omega_{m}}{\Delta P_{L}}=\frac{-R(1+sT_{G})(1+sT_{CH})(1+sT_ {RH})}{(2 Hs+D)(1+sT_{G})(1+sT_{CH})(1+sT_{RH})R+sF_{HP}T_{RH}+1} \tag{5}\]  
where \(s\) is the Laplace operator. For traditional SGs, the typical parameters are shown in Table 2.  
Figure 1: Control structure of active power-frequency control. Figure 2: Multiple temporal stages of frequency response. The pole-zero map of the transfer function is shown in Figure 3.  
From the transfer function \(G(s)\), the zeros are fixed on the real axis. With the increasing of H, the poles P3 and P4 tend to approach the zeros Z2 and Z3. As P3 and P4 are fast poles and very close to Z2 and Z3, the dynamic responses are determined by the dominant zeros Z1 and the slow poles P1 and P2 [34]. Simplifying the system by Z1, P1, and P2, the system transfer function is  
\[G(s)=\frac{-R(1+sT_{RH})}{(2 Hs+D)(1+sT_{RH})R+sF_{HP}T_{RH}+1}=K\frac{s+z_{1}}{s ^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}} \tag{6}\]  
where \(\omega_{n}\) and \(\zeta\) are the undamped natural frequency and damping ratio,  
\[ \begin{array}{l}K=-\frac{1}{2H},z_{1}=\frac{1}{T_{RH}},\,\omega_{n}=\sqrt{ \frac{1+RD}{2 HRT_{RH}}}\\ \zeta=\frac{2 HR+T_{RH}DR+F_{HP}T_{RH}}{4 HRT_{RH}}\times\sqrt{\frac{2 HRT_{RH} }{1+RD}}\end{array}  \tag{7}\]  
When the system is subjected to a step increase in demand,  
\[\Delta P_{L}=\frac{\Delta p}{s}. \tag{8}\]  
By taking the inverse Laplace transformation, the dynamic response of the frequency deviation in the time domain is  
\[\Delta f(t)=K\Delta p\times[\frac{z_{1}}{\omega_{n}^{2}}+Ae^{-\zeta\omega_{n}t }\sin(\omega_{a}t+\beta)] \tag{9}\]  
where  
\[ \begin{array}{l}\omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}}\\ A=\sqrt{\left(\frac{z_{1}\zeta-\omega_{n}}{\omega_{a}\omega_{n}}\right)^{2}+ \left(\frac{z_{1}}{\omega_{n}}\right)^{2}}\\ \beta=\arctan[\frac{z_{1}\omega_{d}}{\omega_{n}(z_{1}\zeta-\omega_{n})}]+\pi \end{array}  \tag{10}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Unit** & **Parameter** & **Value** & **Unit** \\ \hline \(R\) & 0.05 & pu & \(F_{HP}\) & 0.3 & pu \\ \(H\) & 5 & pu & \(T_{RH}\) & 7 & s \\ \(D\) & 1 & pu & \(T_{CH}\) & 0.3 & s \\ \(T_{G}\) & 0.2 & s & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Typical parameters of an SG with a reheat steam turbine. Figure 3: Pole-zero map of the transfer function. Assuming the original state of the system frequency is 1.0 pu, the dynamic response of the system frequency is  
\[f(t)=1+K\Delta p\times[\frac{z_{1}}{\omega_{n}^{2}}+Ae^{-\zeta\omega_{n}t}\sin( \omega_{d}t+\beta)]. \tag{11}\]  
From Equation (11), system performance indices during IR and PFR can be derived.