## 4 SSO Suppression Strategy for DFIG System  
### Design of SSO Suppression Strategy for DFIG System  
From the mathematical model of the RSC control with SSO, it can be deduced that the voltage command value output by the RSC controller under the oscillation condition can be expressed as:  
\[\begin{cases}u_{\rm rd}^{*}=u_{\rm rd0}^{*}+u_{\rm rd\_sub}^{*}\\ u_{\rm rd0}^{*}=u_{\rm rd0}^{*}+u_{\rm rd\_sub}^{*}\end{cases} \tag{22}\]  
When the SSO is input into the DFIG system, the sub-synchronous current component in the stator current has the most serious impact on the RSC control system. To suppress the sub-synchronous component in the DFIG stator current, the output voltage command value of the RSC controller in the d-q coordinate system should be divided into two parts. One part comprises the fundamental wave components \(u_{\rm rd0}^{*}\) and \(u_{\rm rd0}^{*}\) and the other part comprises the secondary synchronous components \(u_{\rm rd0\_sub}^{*}\) and \(u_{\rm rd0\_sub}^{*}\). The fundamental wave component is the DC component, and the secondary synchronous component is the AC component. \(u_{\rm rd0}^{*}\) and \(u_{\rm rd0}^{*}\) can control the active and reactive power output by the DFIG, and \(u_{\rm rd0\_sub}^{*}\) and \(u_{\rm rd0\_sub}^{*}\) need to achieve the suppression of the sub-synchronous component in the DFIG stator current.  
In the previous analysis, the back-stepping controller could control the fundamental components \(u_{\rm rd0}^{*}\) and \(u_{\rm rd0}^{*}\). The AC components \(u_{\rm rd0\_sub}^{*}\) and \(u_{\rm rd0\_sub}^{*}\) were controlled by a quasi-resonant controller. In order to suppress the frequency-varying sub-synchronous current, an adaptive quasi-resonant controller can be used to track it. By optimizing the RSC control strategy, a back-stepping SSO suppression strategy can be designed for the adaptive quasi-resonant controller. The control block diagram is shown in Figure 5.  
The synergistic process of the RSC controller and quasi-resonant controller with SSO occurs as follows: when oscillation occurs, the rotor d-axis current \(\widetilde{i}_{\text{rd}}\) and q-axis current \(\widetilde{i}_{\text{rq}}\) containing oscillation components are input into the back-stepping controller and, at the same time, the collected synchronous angular frequency \(\omega_{1}\) and rotor angular frequency \(\omega_{\text{r}}\) are input into the back-stepping controller. The rotor voltage control commands \(u_{\text{rd}}^{*}\) and \(u_{\text{rq}}^{*}\) are obtained through the calculations of the back-stepping controller. \(u_{\text{rd}}^{*}\) and \(u_{\text{rq}}^{*}\) can realize decoupling control over the DFIG output active and reactive power.  
Due to the existence of the oscillating current, it is difficult for the back-stepping controller to make the DFIG output converge to the desired value. In order to suppress the stator sub-synchronous current, a direct resonant control was adopted in this study [25]. The stator current d and q components are input into the adaptive quasi-resonant controller, and the resonant controller controls the oscillating current. Since the resonance controller has no adjustment effect on the DC component, the final output resonance control voltages \(u_{\text{d}}^{*}\) and \(u_{\text{q}}^{*}\) can only control the corresponding stator oscillation current. Through the RSC current control, \(u_{\text{d}}^{*}\) and \(u_{\text{q}}^{*}\) interact with the sub-synchronous resonant electromotive force induced on the rotor side of the DFIG to generate a sub-synchronous harmonic current, thereby suppressing the sub-synchronous current in the DFIG stator and realizing the sinusoidal output of the stator current. The synergy of the RSC current controller and the adaptive quasi-resonant controller ensures that the RSC is not disturbed by the oscillation and can still maintain the normal operation of the DFIG when grid SSO occurs.