# Multi-Timescale Control of Variable-Speed Wind Turbine for Inertia Provision
## 4 Simulation Results  
To verify the theoretical analysis that the rotor speed recovery bandwidth will affect the inertia emulation effect and the effectiveness of the proposed multi-timescale controller, simulations were conducted in the PLECS (piece-wise linear simulation of electrical circuits) environment. The DFIG wind power generation system (schematic diagram shown in Figure 1) is connected to a practical power grid represented by an SM (primary frequency control framework illustrated in Figure 2). The power grid primary frequency regulation framework is illustrated in Figure 5. The main circuit and control parameters are listed in Table 1.  
Figure 12 shows the simulation results when the DFIG generation system does not provide inertia support to the grid. At \(t=5\) s, a 5% step load change happens, and the power system frequency drops. As shown in Figure 12, the frequency nadir is 49.73 Hz, and the maximum RoCoF is \(-0.2\) Hz/s. It can also be observed in Figure 12 that the wind turbine rotor speed, \(\omega_{r}\), is decoupled from the grid frequency, as well as the RoCoF value (i.e., there is almost no change in \(\omega_{r}\) during the frequency event). This indicates that the mechanical energy stored in the rotor was not released to support the primary frequency response of the power system. Figure 13 shows the simulation results when the DFIG system provides inertia support. The speed recovery control bandwidth is designed to be 0.03 Hz according to the multi-timescale control concept discussed in Section 3. Similarly, a 5% step load change happens when \(t=5\) s. It is observed that the wind turbine rotor speed is reduced during the inertia response period (\(5\) s \(<t<15\) s). As a result, the rotor mechanical energy is released to support the grid frequency and limit the RoCoF. The frequency nadir is improved to 49.76 Hz, and the maximum RoCoF is reduced to \(-0.16\) Hz/s. For a larger timescale (\(15\) s \(<t<40\) s), the speed recovery control gradually takes effect, and the rotor speed comes back to the desired value of 300 rad/s. Note that the DC-link voltage is also under good control and remains close to the nominal value. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(R\) & Frequency droop slope (p.u.) & 0.05 \\ \(T_{G}\) & Speed governor constant & 0.1 s \\ \(F_{\text{HP}}\) & Turbine HP constant & 0.3 s \\ \(T_{\text{RH}}\) & Reheater time constant & 7.0 s \\ \(T_{\text{CH}}\) & Inlet volume time constant & 0.2 s \\ \(D\) & Load-damping coefficient (p.u.) & 1.0 \\ \(H_{M}\) & Synchronous machine inertia (p.u.) & 5.0 s \\ \(V_{S}\) & Grid voltage magnitude & 1000 V \\ \(\omega_{0}\) & Nominal grid frequency & 100\(\cdot\pi\) rad/s \\ \(V\Delta_{\text{base}}\) & System base power (1 p.u.) & 10 MW \\ \(J_{r}\) & Moment of inertia (turbine rotor) & 75 kg\(\cdot\)m\({}^{2}\) \\ \(H_{\text{p}}\) & Virtual inertia (p.u.) & 2.0 s \\ \(\omega_{r\_{\text{ref}}}\) & Reference rotor speed & 300 rad/s \\ \(P_{\text{set}}\) & Active power set power & 2.5 MW \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Circuit and control parameters for the simulation. For the next scenario, the speed recovery control bandwidth was intentionally adjusted to 0.1 Hz, which is close to that of the inertia response. Figure 14 displays the corresponding simulation results. Although the wind turbine rotor speed is decreased at the start of the frequency event, the coupling between the inertia control and the speed recovery control makes \(\omega_{r}\) recover to the expected value within around 10 s. As a consequence, the inertia  
Figure 12: Simulation results when the DFIG system does not have an inertia-provision function. (**a**) Grid frequency; (**b**) grid RoCoF; (**c**) rotor angular speed; (**d**) DC-link voltage. Figure 13: Simulation results when DFIG provides inertia support and the speed recovery control bandwidth is 0.03 Hz. (**a**) Grid frequency; (**b**) RoCoF; (**c**) rotor speed; (**d**) DC-link voltage. provision effect is compromised.