# Performance Tuning for Power Electronic Interfaces Under VSG Control
## 3 Simplification of the Dynamic System in IR and PFR Stages
 \tag{9}\]  
When the system is subjected to a step increase in demand  
\[\Delta P_{L}=\frac{\Delta p}{s} \tag{10}\]  
By multiplying (8) and (10) and taking the inverse Laplace transformation, the dynamic response of frequency deviation in time-domain can be expressed as  
\[\Delta f(t)=K\Delta p\times\big{[}\frac{z_{1}}{\omega_{n}^{2}}+Ae^{-\zeta \omega_{n}t}\sin(\omega_{d}t+\beta)\big{]} \tag{11}\]  
where  
\[ \begin{array}{l}\omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}}\\ A=\sqrt{\big{(}\frac{z_{1}\zeta-\omega_{n}}{\omega_{d}a_{h}}\big{)}^{2}+\big{(} \frac{z_{1}}{\omega_{n}z}\big{)}^{2}}\\ \beta=\arctan\big{[}\frac{z_{1}\omega_{d}}{\omega_{n}(z_{1}\zeta-\omega_{n})} \big{]}+\pi\end{array}  \tag{12}\]  
Assuming the original state of the frequency is 1.0 pu, the frequency response is  
\[f(t)=1+K\Delta p\times\big{[}\frac{z_{1}}{\omega_{n}^{2}}+Ae^{-\zeta\omega_{n} t}\sin(\omega_{d}t+\beta)\big{]} \tag{13}\]  
From Equation (11), performance indices during IR and PFR can be derived. First, the rate of change of frequency (ROCOF) is  
\[\begin{array}{l}ROCOF= \begin{array}{l}\frac{d\Delta f}{dt}-K\Delta p\times \big{[}A\zeta\omega_{n}e^{-\zeta\omega_{n}t}\sin(\omega_{d}t+\beta)-\\ A\omega_{d}e^{-\zeta\omega_{n}t}\cos(\omega_{d}t+\beta)\big{]}\end{array}  \tag{14}\]  
It is noted that, when \(t=0\), ROCOF reaches maximum magnitude, which can be expressed as  
\[ROCOF_{\max}=-K\Delta p\times\big{(}A\zeta\omega_{n}\sin\beta-A\omega_{d}\cos\beta \big{)} \tag{15}\]  
From Equation (11), when \(ROCOF=0\), the peak time \(t_{\text{peak}}\) (the time when the frequency reaches a nadir or a peak) can be expressed as  
\[t_{peak}=\frac{1}{\omega_{d}}(\arctan\frac{\sqrt{1-\zeta^{2}}}{\zeta}-\beta) \tag{16}\]  
Substitute (16) to (13), then the peak frequency \(f_{peak}\) (or nadir) is  
\[f_{\text{peak}}=1+K\Delta p\times\frac{z_{1}}{\omega_{n}^{2}}+K\Delta pAe^{- \zeta\omega_{n}t_{\text{peak}}}\sqrt{1-\zeta^{2}} \tag{17}\]  
When the third item in the Equation (17) decays approximately to zero, the quasi-steady-state frequency \(f_{\text{ss}}\) is achieved, which can be expressed as  
\[f_{\text{ss}}=1+K\Delta p\times\frac{z_{1}}{\omega_{n}^{2}}=1-\frac{R}{DR+1}\Delta p \tag{18}\]The frequency deviation can also be evaluated by the frequency overshoot \(\sigma\), which can be expressed as  
\[\sigma=\left|\frac{f_{\text{ss}}-f_{\text{peak}}}{f_{\text{ss}}}\right|\times 100 \%=\left|\frac{A\omega_{n}^{2}\sqrt{1-\zeta^{2}}e^{-\zeta\omega_{n}t_{\text{ peak}}}}{z_{1}+\omega_{n}^{2}/K\Delta p}\right| \tag{19}\]  
The settling time \(t_{\text{s}}\) can be defined as the dynamic response enters the 2% quasi-steady-state error band [17],  
\[\left|\frac{f(t_{\text{s}})-f_{\text{ss}}}{1-f_{\text{ss}}}\right|=2\% \tag{20}\]  
and by submitting Equations (11) and (12) into Equation (20), the settling time can be deduced as  
\[t_{\text{s}}=-\frac{1}{\zeta\omega_{n}}\ln\frac{z_{1}}{50A\omega_{n}^{2}} \tag{21}\]  
From the performance indices, the effect of separate parameters on system performance can be derived. When subjected to a step increase of load by 0.03 pu, the trends of the above performance indices corresponding to different \(H\) and \(R\) are shown in Figures 6 and 7. In Figure 6, \(R\) is set as a constant, and in Figure 7, \(H\) is set as a constant. From Figure 6a,b,e, the increasing of inertia constant \(H\) leads to the decreasing of the magnitude of ROCOF, frequency nadir, and the overshoot. Therefore, it proves that inertia property could suppress the ROCOF and improve frequency nadir. From (c), the increasing of \(H\) also delays the peak time. However, the increasing \(H\) would slow down the dynamic responses, which is shown in (d). Therefore, there is a trade-off between the nadir improvement and the delivery time. From (f), the settling frequency is irrelative to \(H\). From Figure 7a, percentage \(R\) does not influence the ROCOF. From (b) and (e), the nadir gets worse, and the overshoot increases as the percentage \(R\) increases.