# Performance Tuning for Power Electronic Interfaces Under VSG Control
## 3 Simplification of the Dynamic System in IR and PFR Stages  
The emulation of IR and PFR is derived from the mechanism imitation of the swing equation and the turbine-speed governor system. The effect of the closed-loop transfer function of the VSI based on the cascade control structure is assumed to be fast and accurate. Then the high-order system involving virtual IR and PFR can be simplified into a second-order dynamic system by the dominant poles and zeros without too much error. The performance indices can be deduced from the simplified transfer function. Take a stand-alone VSG-PEI supplying isolated load for example, the block diagram involving IR and PFR is shown in Figure 3.  
For a step change of demand \(\Delta P_{L}\), the transfer function from \(\Delta P_{L}\) to the virtual angular speed of rotor \(\Delta\omega_{r}\) is  
\[G(s)=\frac{\Delta\omega_{r}}{\Delta P_{L}}=\frac{-R(1+sT_{G})(1+sT_{CH})(1+sT_ {RH})}{(2 Hs+D)(1+sT_{G})(1+sT_{CH})(1+sT_{RH})R+sF_{HP}T_{RH}+1} \tag{7}\]  
For SGs with reheat steam turbines, typical parameters are shown in Table 1.  
The pole-zero map of the transfer function is shown in Figure 4. 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 mainly determined by the dominant zeros Z1 and the slow poles P1 and P2 [37]. It is noted that the inertia constants of different SGs, which are shown in Table 2, are in a small range rather than accurately the same [16]. The concept of VSG is to enable the PEIs to provide approximately synchronous responses like traditional SGs. Moreover, fast poles influence the early part of the time history [37]. The frequency responses based on the transfer function of the original and simplified \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 1: Typical parameters of an SG with a reheat steam turbine [16]. Figure 3: Block diagram of the emulation of IR and PFR. systems are shown in Figure 5, which shows the maximum error between the two dynamic systems is less than 10% frequency deviation. Therefore, the estimation by the dominant zeros and poles are practically acceptable. According to the dominant zeros and poles (Z1, P1, and P2), the simplified transfer function of the dynamic system is derived from Equation (7), which can be expressed as  
\[ \begin{array}{ll}G(s)&=\frac{-R(1+sT_{RH})}{(2 Hs+D)+1+sT_{RH})R+sF_{HP}T_{RH }+1}\\ &=K\frac{s+z_{1}^{2}}{g^{2}+2\zeta_{\text{dB}}s+a_{0}^{2}}\end{array}  \tag{8}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type of Generation Unit** & **Inertia Constant** \\ \hline Thermal unit & (1) 2 poles & 2.5–6.0 \\ (2) 4 poles & 4.0–10.0 \\ Hydraulic unit & 2.0–4.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Typical inertia constant of traditional synchronous machines [16]. Figure 4: Pole-zero map of the transfer function. Figure 5: Frequency responses of dynamic systems based on original and simplified transfer functions (H = 5, 7, 10, 15 under a 3% step increase of the load). where \(\omega_{n}\) and \(\zeta\) are the undamped natural frequency and damping ratio [37],  
\[ \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}