# Multi-Timescale Control of Variable-Speed Wind Turbine for Inertia Provision
## 2 Principle of Inertia Emulation  
### Power System Frequency Response  
Figure 2 shows the power system primary frequency regulation framework. For the convenience of a statement, all the variables are displayed in per-unit values. \(R\) is the frequency droop coefficient; \(T_{G}\) refers to the speed governor time constant; \(F_{\text{HP}}\), \(T_{\text{RH}}\), and \(T_{\text{CH}}\) are the reheat turbine time constants; \(\Delta P_{\text{ref\_pu}}\) is the input-power reference variation; \(\Delta P_{\text{m\_pu}}\) is the mechanical power variation; and \(\Delta P_{\text{l\_pu}}\) is the load power variation. According to the well-known swing equation [8], the power grid frequency, \(\Delta f_{\text{g\_pu}}\), is given by:  
\[\frac{d}{dt}\Delta f_{\text{g\_pu}}=\frac{1}{2H_{\text{M}}}\Big{(}\Delta P_{ \text{m\_pu}}-\Delta P_{\text{l\_pu}}\Big{)} \tag{1}\]  
where \(H_{\text{M}}\) is the inertia constant of the synchronous machine (SM). Equation (1) indicates that the power grid frequency deviates from the nominal value if there is a power imbalance between the power generation and the load demand. More specifically, Figure 3 illustrates  
Figure 1: Schematic diagram of the DFIG power conversion system.  
the typical frequency response when the power grid is subjected to a disturbance (taking a sudden load change as an example).  
Due to the inertia of the SM, the power grid frequency does not immediately drop or increase with a high ramp rate. The instantaneous power imbalance is compensated by the mechanical energy of the rotor, resulting in a comparatively small variation in the system frequency. This process is known as the inertia response and usually takes less than 10 s. For a longer timescale, the speed governor control takes effect, and the frequency finally reaches a new equilibrium point. There are two important metrics that require attention, from the power system frequency stability perspective. One is the frequency nadir (\(f_{\min}\) in Figure 3), and the other is the RoCoF value. When the frequency nadir goes beyond the normal operating range, proper measures, such as load shedding, should be taken, which causes a negative impact on electricity customers. In addition to this, the high RoCoF value may also cause a pole slip, trip the SM, and even lead to frequency instability. In this regard, many countries have set stringent requirements for the frequency nadir and the RoCoF.  
To illustrate the impact of inertia on the frequency nadir and the RoCoF, Figure 4 shows the grid frequency response with different inertia values. It can be observed that the increase in the power system inertia reduces the RoCoF value and also improves the frequency nadir. Consequently, the power system frequency stability is enhanced. However, when conventional SMs and SGs are gradually replaced by inertia-less power converters, the total inertia of the power grid will decrease, giving ride to frequency stability concerns. Therefore, it is of importance to enable power converters to make an active contribution to the power grid by properly providing grid support, e.g., emulating inertia.  
Figure 3: Typical power system frequency response after a disturbance (e.g., a load step change).  
Figure 2: Primary power system frequency regulation framework.