## 3 Influence of Inertia on Frequency Regulation  
### Damping Factor Analysis  
According to the classic control theory, the damping factor \(\zeta\) has a great influence on the second-order system described by (1). In order to illustrate this effect, the pole-zero map and step response of \(G_{PL\rightarrow\omega r}(s)\) are, respectively, demonstrated in Figure 4a,b. In Figure 4a, only zero \(z_{1}\) is denoted as a circle while the poles \(p_{1}\) and \(p_{2}\) are represented by crosses. As seen, for the case of \(\zeta<1\), \(p_{1}\) and \(p_{2}\) are conjugate complex poles. One special case is \(\zeta=1\), where \(p_{1}=p_{2}\) is satisfied. Under the condition that \(\zeta>1\), \(p_{1}\) and \(p_{2}\) become real poles. It is noted from Figure 4b that as \(\zeta\) increases from less than one to be greater than one, oscillations of system frequency gradually disappear.  
Figure 3: Structure of frequency regulation based on VSMs.  
Figure 2: Structure of frequency regulation based on the conventional frequency regulation model.  
The relationships of \(\zeta\) with \(H\) and \(T_{G}\) are presented in Figure 5. One observation from Figure 5 is that \(\zeta\geq 1\) may only occur when \(T_{G}\) is small enough. For a relatively large \(T_{G}\), e.g., 7 s, \(\zeta\) will never exceed one even when \(H\) is as large as 50 s. It should be mentioned that the conventional frequency regulation model shown in Figure 2 can be replaced by the simplified model shown in Figure 3 as long as the effect of the turbine is considered. This effect can be emulated by \(T_{G}\), e.g., \(T_{G}=T_{R}=7\) s. Under this approximation, the damping factor \(\zeta\) will always be smaller than one [see Figure 5]. Consequently, frequency oscillations will exist and can only be attenuated with the increase of \(H\) (increase of \(\zeta\)). This was the basic motivation for previous researchers to increase the system inertia coefficient in power systems.  
Additionally, it is clear from Figure 5 that there exists one particular \(H\) for \(\zeta\) to be minimal. Its expression can be derived by first differentiating the fourth equation of (2) with respect to \(H\) and then letting it be zero:  
\[\frac{d\zeta}{dH}=\sqrt{\frac{2 RT_{G}}{DR+1}}\cdot\left(\frac{1}{4T_{G}\sqrt{ H}}-\frac{D}{8H\sqrt{H}}\right)=0\Rightarrow H=\frac{DT_{G}}{2} \tag{3}\]  
It should be highlighted that the design of \(H\leq DT_{G}/2\) is avoided in practice. Although this design may introduce the same or even larger values of \(\zeta\) as compared with the  
Figure 4: Influence of damping factor \(\zeta\) on \(G_{PL\rightarrow\omega r}(s)\).  
Figure 5: Variation of \(\zeta\) versus \(H\) and \(T_{G}\).  
design of \(H>DT_{G}/2\), it does not necessarily translate into fewer oscillations and smaller overshoots, and this statement will be proven in Appendix A. Henceforth, \(H>DT_{G}/2\) is always satisfied unless otherwise stated. Substituting (3) into the fourth equation of (2), the expression of minimal \(\zeta\) can be obtained as  
\[\zeta\sqrt{\frac{DR}{DR+1}_{min}} \tag{4}\]  
where \(\zeta_{min}\) represents the minimal value of \(\zeta\), which is independent of \(H\) and \(T_{G}\).  
The frequency responses of systems with \(T_{G}=7\) s and \(T_{G}=0.1\) s under a step-up load change are, respectively, demonstrated in Figure 5(a),b. As indicated by Figure 5(a), frequency oscillations are observable and can be attenuated with a larger inertia coefficient \(H\) because the damping factor \(\zeta\) increases as \(H\) increases, which is consistent with the conclusion drawn from Figure 5. In contrast, it can be noted from Figures 4(a) and 5(b) that the damping factor \(\zeta\) becomes larger than one for the case of \(T_{G}=0.1\) s when \(H=5\) or \(10\) s. In this case, no oscillation is observable. The only effect of increasing \(H\) is to prolong the dynamics of frequency regulation. Therefore, it is possible to employ a small \(H\) under this condition.