## Appendix A DFIG Control Model Decoupling  
To date, various algorithms have been proposed to solve the problem of coupling different channels in a converter control. One study [26] proposed a d-q current decoupling control method for a single-phase grid-connected converter without a delay. This method took a single-phase pulse width modulation (PWM) converter as its research object. Byimproving the orthogonal signal generator, the virtual orthogonal component of the \(\beta\) axis corresponding to the grid current could be generated without a delay. In addition, the reference current signal mutation could be tracked without a delay, improving the dynamic response speed of the system. Another study [27] designed a two degrees of freedom PI controller by analyzing the influence of the converter decoupling control link on the stability of the control system. This method improved the equivalent motor stator resistance, the robustness of the link between the current ring and the decoupling parameters and the response characteristics of the current ring. The above methods can be used to improve the response speed of a control system, but the feedforward decoupling compensation still needs to be introduced into the current internal loop. When grid SSO occurs, the sub-synchronous component is input into the RSC control system through the feedforward decoupling. Therefore, in order to reduce the number of SSO propagation paths, a back-stepping controller was designed, combining state feedback decoupling and back-stepping control, to decouple the RSC current inner loop. The state feedback decoupling can reduce the complexity of the control system, thus simplifying the system equation of state, eliminating the coupling components and facilitating the design of subsequent controllers. The design of the back-stepping controller had to satisfy the stability equation of the system in order to improve the stability of the system. Considering the decoupling control ability of the back-stepping controller for nonlinear systems and that fact that it has no bandwidth limitations, the back-stepping controller can be applied to situations of unknown disturbance in power grids. Therefore, a back-stepping controller based on state feedback decoupling could not only improve the response speed and stability of control systems but also simplify the control strategy of the RSC and reduce the number of propagation paths of the SSO. Equation (A1) can be obtained from Equation (1) and the equation of motion of the DFIG system. In the equation, \(J\) is the moment of inertia of the generator; \(n_{\text{P}}\) is the number of pole pairs; \(D\) is the damping coefficient; and \(T_{\text{L}}\) is the input torque of the wind turbine. \(\widetilde{i}_{\text{rd}}=i_{\text{rd0}}+i_{\text{rd\_sub}}\), and \(\widetilde{i}_{\text{rq}}=i_{\text{rq0}}+i_{\text{rq\_sub}}\). \[\left\{ \begin{array}{c}\frac{\widetilde{d}_{\text{rd}}}{dt}=-\frac{R_{\text {r}}}{C}\widetilde{i}_{\text{rd}}+\omega_{\text{s}}\widetilde{i}_{\text{rq}}+ \frac{1}{C}{u_{\text{rd}}}-\frac{\omega_{\text{s}}L_{\text{m}}U_{\text{s}}}{ \sigma L_{\text{r}}L_{\text{s}}\omega_{\text{l}}}\\ \frac{\widetilde{d}_{\text{rn}}}{dt}=-\omega_{\text{s}}\widetilde{i}_{\text{rd }}-\frac{R_{\text{r}}}{\sigma L_{\text{r}}}\widetilde{i}_{\text{rq}}+\frac{ 1}{C}{u_{\text{rd}}}{u_{\text{rq}}}\\ \dot{\omega}_{\text{r}}=-\frac{n_{\text{P}}L_{\text{m}}U_{\text{s}}}{J\omega_{ \text{l}}\omega_{\text{s}}}\widetilde{i}_{\text{rd}}-\frac{D}{J}\omega_{\text {r}}+\frac{T_{\text{l}}}{J}\end{array} \right.\] (A1)  
Analysis of Equation (A1) shows that the state equation of the DFIG system is a third-order equation, and there is a coupling relationship between the first-order differential equations of \(\widetilde{i}_{\text{rd}}\) and \(\widetilde{i}_{\text{rq}}\). We can rewrite the two equations related to \(\widetilde{i}_{\text{rd}}\) and \(\widetilde{i}_{\text{rq}}\) in Equation (A1) into matrix form and use \(\widetilde{i}_{\text{rd}}\) and \(\widetilde{i}_{\text{rq}}\) as the output, so that we have:  
\[\left\{