# Multi-Timescale Control of Variable-Speed Wind Turbine for Inertia Provision
## 2 Principle of Inertia Emulation
### DFIG System Control
#### 3.2.2 RSC Control Scheme  
Similarly, the dynamic equations of the induction machine can be expressed as:  
\[\left\{\begin{array}{l}v_{\text{ds}}=R_{s}i_{\text{ds}}-\omega\lambda_{\text {qs}}+\frac{d}{dt}\lambda_{\text{ds}}\\ v_{\text{qs}}=R_{s}i_{\text{qs}}+\omega\lambda_{\text{ds}}+\frac{d}{dt}\lambda _{\text{qs}}\\ v_{\text{dr}}=R_{r}i_{\text{dr}}-(\omega-\omega_{r})\lambda_{\text{qr}}+\frac {d}{dt}\lambda_{\text{dr}}\\ v_{\text{qr}}=R_{r}i_{\text{qr}}+(\omega-\omega_{r})\lambda_{\text{dr}}+\frac {d}{dt}\lambda_{\text{qr}}\end{array}\right. \tag{15}\]  
The stator and rotor flux linkages are given by:  
\[\left\{\begin{array}{l}\lambda_{\text{ds}}=L_{\text{ls}}i_{\text{ds}}+L_{m }(i_{\text{ds}}+i_{\text{dr}})\\ \\ \lambda_{\text{qs}}=L_{\text{ls}}i_{\text{qs}}+L_{m}(i_{\text{qs}}+i_{\text{qr}} )\\ \\ \lambda_{\text{dr}}=L_{\text{lr}}i_{\text{dr}}+L_{m}(i_{\text{ds}}+i_{\text{dr}} )\\ \\ \lambda_{\text{qr}}=L_{\text{lr}}i_{\text{qr}}+L_{m}(i_{\text{qs}}+i_{\text{qr}} )\end{array}\right. \tag{16}\]  
Figure 9: Control block diagram of the GSC.  
where \(v_{\rm dr}\) and \(v_{\rm qr}\) are the rotor voltage, \(v_{\rm ds}\) and \(v_{\rm qs}\) are the stator voltage, \(\lambda_{\rm dr}\) and \(\lambda_{\rm qr}\) are the rotor flux, \(\lambda_{\rm ds}\) and \(\lambda_{\rm qs}\) are the stator flux, \(\omega_{e}\) is the angular frequency of the reference frame, \(\omega_{r}\) is the angular frequency of the rotor, \(L_{\rm fr}\) is the self-inductance, and \(L_{m}\) is the mutual inductance. Electromagnetic torque can be expressed as:  
\[T_{e}=\frac{3 PL_{m}}{2L_{r}}(i_{\rm qs}\lambda_{\rm dr}-i_{\rm ds}\lambda_{\rm qr })=K_{T}(i_{\rm qs}\lambda_{\rm dr}-i_{\rm ds}\lambda_{\rm qr}) \tag{17}\]  
By aligning the rotor flux vector with the \(d\)-axis of the synchronous reference frame, \(\lambda_{\rm qr}\) equals zero. In this case, the electromagnetic torque is proportional to the \(q\)-axis stator current, \(i_{\rm qs}\), whereas the \(d\)-axis stator current is responsible for producing the flux. Figure 10 shows the control block diagram of the RSC, where the indirect field-oriented control is implemented for the rotor flux orientation [37] and \(T_{\rm ref}\) is the reference electromagnetic torque.  
The determination of the reference electromagnetic torque \(T_{\rm ref}\) is displayed in Figure 11. It can be observed that the virtual inertia control, as well as the speed recovery control, are simultaneously incorporated into the control loop. \(\omega_{\rm r\_ref}\) represents the desired rotor speed that achieves the maximum power harvest, \(P_{\rm set}\) is the active power setpoint as determined by \(\omega_{\rm r\_ref}\), and \(H_{\rm v}\) is the virtual inertia coefficient. It should be mentioned that the PI control gain, \(k_{\rm p5}\) and \(k_{\rm Si}\), should be properly designed so that the virtual inertia control response time is much less than 10 s (the typical power system inertia response time). In addition, the PI control gain, \(k_{\rm p6}\) and \(k_{\rm i6}\), need to be designed so that the speed recovery control response time is much more than 10 s. In this regard, the conflict between two control loops can be avoided. The objectives of inertia provision and speed recovery can be realized under a multi-timescale approach.  
Figure 11: Control block diagrams for virtual inertia and speed recovery.  
Figure 10: Control block diagram of the RSC.