## 4 Composite Performance Analysis and Adaptive Performance Tuning
### Supported Energy and Power during IR and PFR  
In the process of IR, the kinetic energy stored in the rotating equipment of SGs is provided to suppress the ROCOF. Correspondingly, when IBG provides IR, kinetic energy (such as rotating blades in wind turbines) or virtual kinetic energy (such as the spare generation availability, the electrostatic energy in capacitors, or the storage energy) is provided. The provided energy by the VSG-PEI can be expressed as  
\[\Delta E(t)=\frac{1}{2}J_{vir}(\omega_{t}^{2}-\omega_{0}^{2}) \tag{19}\]  
where \(\omega_{t}\) and \(\omega_{0}\) are virtual angular speed and its initial value, \(J_{vir}\) is the virtual moment of inertia. From Equation (19), the provided energy is related to the inertia constant \(H_{vir}\) and the frequency.  
For an IBG unit with a VA base of 15,000 watts, under different inertia constants, the provided energy is shown in Figure 11 (R = 0.05, D = 1, \(\Delta p\) = 0.03). From Figure 11, the IBG with larger virtual inertia constant provides more energy during the frequency response. This is because the IR activates instantaneously and faster than the activation of the PFR. Therefore, there is a tradeoff in the design of \(H\). Although the larger \(H\) indicates better nadir and settling frequency, it requires more energy from the PEIs.  
From the previous sections, the virtual moment of inertia \(J_{vir}\) is determined by the inertia constant \(H\). In steady state, the settling frequency is determined by the speed droop \(R\) (when \(D\) and \(\Delta p\) are considered as constants). Therefore, the provided energy at the settling time varies according to the  
Figure 11: Energy provided by inertia property.  
Figure 10: Composite influence on the peak frequency.  
\(H\) and \(R\) settings, as shown in Figure 12. From Figure 12, the supported energy increases with the increase of \(H\) and \(R\).  
In steady state, the provided power in the PFR loop can be expressed as  
\[p_{PFR}=\frac{1/R}{D+1/R}\times\Delta p \tag{20}\]  
where \(\Delta p\) is the PFR power and \(\Delta p\) is the step deviation of the load. From the expression, there is also a tradeoff between the speed droop \(R\) and the frequency deviation. In parallel operation with other PFR players, the speed droop \(R\) is inversely proportional to the supportive power. In a stand-alone scenario, the smaller speed droop \(R\) can improve the settling frequency, but it requires a little more power.  
It is noted that the supportive kinetic energy from traditional SGs is from and recharged by the primary energy input. The primary energy sources for the SGs (such as coal, nuclear) are abundant and persistent. However, the primary input for renewable generation is stochastic and fluctuant. The stored energy in capacitors or batteries can only provide frequency responses in a limited time-span, and after the responses, their operation status needs to be restored for self-stability. In the view of time, the IR and its support energy are short-term, while the PFR and its support power are long-term. Under a specific grid code, the provided energy and power from PEIs should be designed to be as low as possible to avoid the oversized headroom and financial costs. Consequently, the desired \(H\) and \(R\) should be set as the lowest values in their satisfying ranges.