## 3 Analysis of SSO Inhibition Mechanism in the DFIG System
### Quasi-Resonant Controller  
In this study, an RSC back-stepping controller was designed based on the DFIG decoupling model, simplifying the RSC control strategy. Although the back-stepping controller design could meet the system stability requirements, it did not completely suppress the sub-synchronous disturbance in the RSC system, so it was difficult make the output of the DFIG system converge to the expected value. In order to realize the suppression of the stator sub-synchronous current by the DFIG system, an adaptive quasi-resonant controller was used to track the stator sub-synchronous current, and the control voltage output by the quasi-resonant controller was input into the RSC control system.  
The transfer function of the quasi-resonant controller is shown in Equation (20) [23; 24]:  
\[G(s)=\frac{2K_{\text{R}}\omega_{\text{c}}s}{s^{2}+2\omega_{\text{c}}s+\omega_ {0}^{2}} \tag{20}\]  
In Equation (20), \(K_{\text{R}}\) is the gain coefficient, \(\omega_{\text{c}}\) is the cutoff frequency and \(\omega_{0}\) is the signal frequency that the quasi-resonant controller needs to control.  
A Bode diagram was used to analyze the performance of the resonant controller described by Equation (20), with \(\omega_{0}\) set to 50 Hz, and the result is shown in Figure 4.  
Figure 4: Bode diagrams of G(s) as \(K_{\text{R}}\) and \(\omega_{\text{c}}\) change: (**a**) the value of \(K_{\text{R}}\) changes; (**b**) the value of \(\omega_{\text{c}}\) changes.  
It can be seen from Figure 4 that, when \(\omega_{\rm c}\) is a fixed value, the amplitude gain in the quasi-resonant controller is positively correlated with the magnitude of \(K_{\rm R}\). When \(K_{\rm R}\) is a fixed value, the amplitude and phase of the quasi-resonant controller tend to be flat with the increase of \(\omega_{\rm c}\); that is, the magnitude of \(\omega_{\rm c}\) is negatively correlated with the influence of \(G(s)\), which is affected by the frequency change. However, no matter how \(\omega_{\rm c}\) and \(K_{\rm R}\) change, when \(\omega_{0}\) is determined, the quasi-resonant observation control has the best tracking effect on this frequency signal. Therefore, it is feasible to use the quasi-resonant controller to control the SSO component in the DFIG system.  
Traditional quasi-resonant controllers only control AC signals of specific frequencies. However, when the input signal frequency deviates greatly from the control frequency \(\omega_{0}\), the gain provided by the quasi-resonant controller is greatly reduced. An adaptive algorithm is required for sub-synchronous current frequency changes.  
A simplified transformation of Equation (20) can be obtained as follows:  
\[G(s)=\frac{Hs}{s^{2}+HKs+MH} \tag{21}\]  
In the equation, \(H=2K_{\rm R}\omega_{\rm c}\), \(K=\frac{1}{K_{\rm R}}\) and \(M=\frac{\omega_{0}^{2}}{2K_{\rm R}\omega_{\rm c}}\).  
It can be seen from Equation (21) that the value of the parameter \(M\) is directly related to the frequency of the control target signal; that is, \(M\) directly determines the tracking of the input signal frequency by the quasi-resonant controller. Therefore, the key link in designing a frequency-adaptive quasi-resonant controller is realizing the real-time adjustment of \(M\) when the frequency of the input signal changes through an adaptive design. At this time, the adaptive quasi-resonant controller outputs a control signal that changes with the frequency of the input signal in order to achieve control over the input signal.