## 3 Influence of Inertia on Frequency Regulation
### Inertia Coefficient Analysis
#### 3.2.3 \(\zeta>1\) Case  
Considering a step-up load change for the case of \(\zeta>1\), the system frequency response can be reorganized as  
\[f_{r}(s)=\frac{G_{P_{1}\rightarrow\omega}(s)}{s}=G_{0}\cdot\bigg{[}\frac{1}{ s}\cdot\frac{z_{1}}{p_{1}p_{2}}+\frac{1}{s+p_{1}}\cdot\frac{z_{1}-p_{1}}{p_{2}-p_ {1}}\cdot\frac{1}{(-p_{1})}+\frac{1}{s+p_{2}}\cdot\frac{z_{1}-p_{2}}{p_{1}-p_ {2}}\cdot\frac{1}{(-p_{2})}\bigg{]} \tag{22}\]  
where \(p_{1}\) and \(p_{2}\) are the real poles of \(f_{r}\)(s), which can be represented as  
\[p_{1}=\zeta\omega_{n}-\omega_{n}\sqrt{\zeta^{2}-1},p_{2}=\zeta\omega_{n}+ \omega_{n}\sqrt{\zeta^{2}-1} \tag{23}\]  
The expression of frequency response in the time domain can be derived from (22) and listed as below:  
\[f_{r}(t)=L^{-1}[f_{r}(s)]=G_{0}\cdot\bigg{[}\frac{z_{1}}{p_{1}p_{2}}+\frac{z_ {1}-p_{1}}{p_{2}-p_{1}}\cdot\frac{1}{(-p_{1})}e^{-p_{1}t}+\frac{z_{1}-p_{2}}{ p_{1}-p_{2}}\cdot\frac{1}{(-p_{2})}e^{-p_{2}t}\bigg{]} \tag{24}\]  
The steady-state frequency refers to the first term of (24), whose form is the same as (9) and (18) and can be expressed as  
\[f_{r,\infty}=G_{0}\frac{z_{1}}{p_{1}p_{2}}=\frac{R}{DR+1} \tag{25}\]  
By differentiating (24) on both sides, the expression of ROCOF can be derived as follows:  
\[\frac{df_{r}(t)}{dt}=G_{0}\cdot\bigg{[}\frac{z_{1}-p_{1}}{p_{2}-p_{1}}e^{-p_{ 1}t}+\frac{z_{1}-p_{2}}{p_{1}-p_{2}}e^{-p_{2}t}\bigg{]} \tag{26}\]  
Once again, it can be concluded that the sign of (26) remains to be negative, and the relevant proof is provided in Appendix A. This implies that the frequency nadir, peak time, and overshoot are meaningless for the case of \(\zeta>1\). Moreover, the expression of ROCOF at \(t=0\) s remains unchanged, and it can be derived as  
\[\frac{df_{r}(t)}{dt}\Big{|}_{t=0s}=G_{0}=-\frac{1}{2H}. \tag{27}\]Finally, the expression of settling time is listed below:  
\[\left|\frac{f_{r}(t_{s})-f_{r,\infty}}{f_{r,\infty}}\right|=2\%\Rightarrow\left| \frac{z_{1}-p_{1}}{p_{2}-p_{1}}\cdot\frac{p_{2}}{(-z_{1})}e^{-p_{1}t_{s}}+\frac {z_{1}-p_{2}}{p_{1}-p_{2}}\cdot\frac{p_{1}}{(-z_{1})}e^{-p_{2}t_{s}}\right|=0.02 \tag{28}\]  
When \(\left|\,p_{2}\,\right|/\left|\,p_{1}\,\right|>1\), e.g., \(\left|\,p_{2}\,\right|/\left|\,p_{1}\,\right|>5\), the term regarding to \(e^{-p_{2}t_{s}}\) is relatively small, and hence, can be ignored. Under this condition, it is possible to derive the analytical solution of \(t_{s}\) as  
\[\left|\frac{z_{1}-p_{1}}{p_{2}-p_{1}}\cdot\frac{p_{2}}{(-z_{1})}e^{-p_{1}t_{s} }\right|=0.02\Rightarrow t_{s}=\frac{ln(0.02\cdot\frac{p_{2}-p_{1}}{z_{1}-p_{1 }}\cdot\frac{z_{1}}{p_{2}})}{-p_{1}} \tag{29}\]  
For the case of \(\zeta>1\), variations of performance indices versus the inertia coefficient \(H\) are demonstrated in Figure 8. As compared with Figure 7, performance indices of frequency nadir and overshoot are excluded from Figure 8 because they are not meaningful due to the elimination of oscillations. Additionally, only the case of \(T_{G}=0.1\) s is analyzed here as \(T_{G}=7\) s will always lead to \(\zeta<1\). As can be seen from Figure 7(a), the inverse proportional relationship between the ROCOF and inertia coefficient \(H\) remains unchanged. However, it is clear that the ROCOF is greatly reduced as compared with that shown in Figure 6(a). In other words, the ROCOF may not be a serious consideration for overdamped systems. Moreover, it is noted from Figure 7(b) that the settling time increases linearly with the increase of inertia coefficient \(H\) when \(\zeta>1\). In conclusion, a larger inertia coefficient \(H\) will increase the settling time and slightly reduce the ROCOF for the case of \(\zeta>1\).