## 4 Design of Inertia Coefficient  
This section focuses on the design of inertia coefficient \(H\) based on the performance indices derived from Section 3. It is desirable to have fewer frequency oscillations and deviations as well as faster frequency recovery during frequency events. Therefore, the objectives of inertia design can be summarized as minimized ROCOF, frequency nadir or overshoot, and settling time. It should be mentioned that the requirement of frequency nadir and overshoot are essentially the same. The only difference is that frequency nadir varies as the change of percentage of load change, while overshoot specifies only the one hundred percent load change. In practice, grid codes often refer to the requirement of frequency nadir instead of overshoot. In any case, the requirements of ROCOF and frequency nadir must be strictly complied with so as to avoid undesirable load shedding or blackout. Under this circumstance, the settling time should be as low as possible. When designing the inertia coefficient \(H\), it is assumed that the other parameters, \(T_{G}\), \(D\), and \(R\), in Figure 3 and the existing inertia coefficient \(H_{0}\) (\(H=H_{0}+\Delta H\)) are fixed and already known.  
Figure 8: Variations of performance indices versus inertia coefficient \(H\) for the case of \(\zeta>1\) (\(T_{G}=0.1\) s).  
Figure 9 show a design flowchart of inertia coefficient \(H\). Based on the system parameters, the first step of design is to plot the diagram of \(\zeta\) versus \(H\) according to the fourth equation of (2), as shown in Figure 5. From this figure, the boundary condition of \(H\) for \(\zeta\) = 1, denoted as \(H_{b}\), can be derived. The second step is to compare \(H_{0}\) with \(H_{b}\). If \(H_{0}\) < \(H_{b}\), the case of \(\zeta\) < 1 is satisfied. In contrast, if \(H_{0}\) > \(H_{b}\), the case of \(\zeta\) > 1 is satisfied. For the case of \(H_{0}\) = \(H_{b}\), \(\zeta\) = 1 can be obtained.  
The third design step will be different for different cases. In the \(\zeta\) < 1 case, the diagrams of ROCOF and frequency nadir or overshoot should be drawn according to (11) and (13) or (14), respectively, as shown in Figure 7, in the third step. From this figure, the minimum value of \(H\) required by ROCOF and frequency nadir or overshoot is finally selected in the fourth step; this \(H\) also guarantees a relatively small value of \(t_{s}\). Moreover, the maximum capacity of energy storage units dictates the maximum possible value of \(H\). If this maximum value is less than the selected \(H\), requirements should be modified, and \(H\) should be selected again. For the cases of \(\zeta\) = 1 and \(\zeta\) > 1, the frequency nadir or overshoot will not exist, as mentioned in Section 3. Moreover, the satisfaction of requirements dictated by ROCOF can be relatively easy. It is possible to validate this requirement through plotting the diagram of ROCOF versus \(H\) based on (27), as shown in Figure 8a. Considering that the settling time should be as low as possible, \(H\) = \(H_{0}\) is normally the result of design for these cases.  
Based on the system parameter values \(H_{0}\) = 5 s, \(D\) = 1.0, and \(R\) = 0.05 listed in Table 1, two cases of \(T_{G}\) = 7 s and \(T_{G}\) = 0.1 s are, respectively, designed to comply with the requirements of ROCOF \(\leq\) 0.5 Hz/s and frequency nadir within 50 \(\pm\) 0.2 Hz under a 3% step-up load change. These requirements are translated into \(\mid\)\(f_{r\_peak}\)\(\mid\) 13.3% and \(\mid\) d\(f_{r}\)(t)/d\(t\)\(\mid\) \(t\) = 0 s \(\leq\) 33.3% under a 100% load change. In the case of \(T_{G}\) = 7 s, according to (11) and (13), the minimum feasible value, i.e., the selected value, of \(H\) is 10 s. In contrast, for the case of \(T_{G}\) = 0.1 s, the requirements can be easily satisfied with \(H\) = \(H_{0}\) = 5 s.  
Figure 9: A design flowchart of inertia coefficient \(H\).