## 3 Influence of Inertia on Frequency Regulation
### Inertia Coefficient Analysis
#### 3.2.1 \(\zeta<1\) Case  
Considering the case of \(\zeta<1\), when a step-up load change occurs, the system frequency response can be expressed in the s domain as follows:  
\[f_{r}(s)=\frac{G_{P_{L}\rightarrow\omega}(s)}{s}=G_{0}\Bigg{\{}\frac{1}{ \omega_{d}}\cdot\frac{\omega_{d}}{\left(s+\zeta\omega_{n}\right)^{2}+\omega_{ d}{}^{2}}+\frac{z_{1}}{\omega_{n}{}^{2}}\cdot\left[\frac{1}{s}-\frac{s+\zeta \omega_{n}}{\left(s+\zeta\omega_{n}\right)^{2}+\omega_{d}{}^{2}}-\frac{\zeta \omega_{n}}{\omega_{d}}\cdot\frac{\omega_{d}}{\left(s+\zeta\omega_{n}\right)^ {2}+\omega_{d}{}^{2}}\right]\Bigg{\}} \tag{5}\]  
where  
\[\omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}} \tag{6}\]  
Figure 6: System frequency responses under a step-up load change. The time-domain expression of frequency response \(f_{r}(t)\) can be obtained by taking the inverse Laplace transform of (5):  
\[f_{r}(t)=L^{-1}[f_{r}(s)]=G_{0}\bigg{\{}\frac{z_{1}}{\omega_{n}{}^{2}}-e^{-\zeta \omega_{n}t}\cdot A\cdot\sin(\omega_{d}t+\beta)\bigg{\}} \tag{7}\]  
where  
\[A=\sqrt{(\frac{z_{1}}{\omega_{n}{}^{2}})^{2}+(\frac{-z_{1}\zeta+\omega_{n}}{ \omega_{n}\omega_{d}}^{2})},\beta=\arctan\bigg{[}\frac{\omega_{d}z_{1}}{(z_{1} \zeta-\omega_{n})\omega_{n}}\bigg{]} \tag{8}\]  
The first term of (7) represents the steady-state frequency, and it can be expressed as  
\[f_{r_{-}\infty}=G_{0}\frac{z_{1}}{\omega_{n}{}^{2}}=\frac{R}{DR+1} \tag{9}\]  
The derivative of \(f_{r}(t)\) stands for the changing rate of frequency, namely the ROCOF, which can be represented by the following equation:  
\[\frac{d}{dt}f_{r}(t)=G_{0}\Big{[}\zeta\omega_{n}e^{-\zeta\omega_{n}t}\cdot A \cdot\sin(\omega_{d}t+\beta)-e^{-\zeta\omega_{n}t}\cdot A\omega_{d}\cdot\cos (\omega_{d}t+\beta)\Big{]} \tag{10}\]  
Normally, the ROCOF at the instant of load change is more of concern, which takes the following form:  
\[\frac{d}{dt}f_{r}(t)\Big{|}_{t=0s}=G_{0}[\zeta\omega_{n}A\cdot\sin(\beta)-A \omega_{d}\cdot\cos(\beta)] \tag{11}\]  
The time of frequency nadir, i.e., the peak time, can be easily derived by equaling (10) to zero:  
\[t_{peak}=\frac{1}{\omega_{d}}\bigg{\{}\arctan(\frac{\sqrt{1-\zeta^{2}}}{\zeta} )-\beta\bigg{\}} \tag{12}\]  
Substituting (12) into (7), the expression of frequency nadir can be derived and represented as follows:  
\[f_{r_{-}peak}=G_{0}\bigg{\{}\frac{z_{1}}{\omega_{n}{}^{2}}-e^{-\zeta\omega_{n }t_{peak}}\cdot A\cdot\sqrt{1-\zeta^{2}}\bigg{\}} \tag{13}\]  
An alternative of (13) is the expression of frequency overshoot, which can be derived based on (9) and (13) as  
\[\omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}} \tag{6}\]0  
In order to measure the dynamic performances of frequency regulation, the expression of settling time (time for entering 2% steady-state error band) is derived and listed as follows:  
\[\omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}} \tag{6}\]1  
Based on (11)-(15), variations of important performance indices, including the ROCOF, frequency nadir, overshoot, and settling time, versus inertia coefficient \(H\), are presented in \[\omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}} \tag{6}\]2 In Figure 7, both the diagrams of \(T_{G}\) = 7 s and \(T_{G}\) = 0.1 s are provided. As mentioned before, for the case of \(T_{G}\) = 7 s, the damping factor \(\zeta\) < 1 will always be satisfied. In contrast, \(\zeta\) < 1 remains valid when \(H\) < 4.1 s for the case of \(T_{G}\) = 0.1 s, and this condition can be derived from Figure 5a. As can be observed from Figure 7a, the ROCOF \(|\,df_{r}/dt\,|\,t\) = 0 s decreases dramatically with the increase of \(H\). A similar conclusion is also applicable to the absolute value of frequency nadir \(|f_{r_{-}\text{peak}}|\) and overshoot \(\sigma\), which are inversely proportional to \(H\), as indicated by Figure 7b,c. Additionally, it is noted from Figure 7d that the settling time \(t_{s}\) increases with the increase of \(H\), which is reasonable since a large inertia coefficient would slow down the frequency dynamics.