## 3 Analysis of SSO Inhibition Mechanism in the DFIG System
### Back-Stepping Control Design Based on DFIG Decoupling Model  
Traditional RSC controls use PI controllers, but the control performance of PI controllers is poor when a grid oscillates, so they cannot track the command value. Moreover, when an RSC current loop uses a PI controller, it is necessary to calculate the amount of decoupling compensation in order to eliminate the coupling relationship between the d- and q-axis components of the rotor current and realize independent control over active and reactive power. When the grid sub-synchronous current is input into the DFIG system, the amount of decoupling compensation in the inner loop of the RSC current is disturbed, resulting in the failure of decoupling. Therefore, it is necessary to decouple the RSC control model, which can not only eliminate the influence of the disturbance of the decoupling compensation in the case of oscillation, reducing the number of SSO propagation paths, but also facilitate the design of a back-stepping controller that can improve the system stability. The DFIG control model decoupling process is shown in Appendix A. The back-stepping method is adopted to control the decoupled DFIG system. Assume that \(-\frac{\eta_{\mathrm{p}}L_{\mathrm{m}}U_{\mathrm{s}}}{\
u_{1}L_{\mathrm{s} }}=\gamma\), \(\frac{\sigma L_{\mathrm{r}}L_{\mathrm{m}}U_{\mathrm{s}}}{\omega_{1}L_{\mathrm{ s}}}=\mu\), \(\frac{D}{\gamma}=F\), \(\frac{T_{\mathrm{l}}}{\gamma}=T\) and \(\sigma L_{\mathrm{r}}=g\). Equation (A8) in Appendix A can then be rewritten as:  
\[\left\{ \begin{array}{c}\dot{\dot{f}}_{\mathrm{rd}}=-\mu\omega_{\mathrm{s}}+ gu_{\mathrm{rd}}^{*}\\ \dot{\dot{\mathrm{r}}}_{\mathrm{rq}}=gu_{\mathrm{rq}}^{*}\\ \dot{\omega}_{\mathrm{r}}=\gamma\dot{\mathrm{r}}_{\mathrm{rd}}-F\omega_{ \mathrm{r}}+T\end{array} \right. \tag{6}\]Let \(i_{\rm rd}^{*}\) and \(i_{\rm rg}^{*}\) be the instruction values of the d and q components of the rotor current, respectively. According to the form of Equation (4), the error relationship between the actual value and the expected value can be obtained as shown in Equation (7):  
\[ \begin{cases}e_{1}=\widetilde{i}_{\rm rd}-i_{\rm rd}^{*}\\ e_{2}=\widetilde{i}_{\rm rg}-i_{\rm rg}^{*}\end{cases}  \tag{7}\]  
Taking the derivation of Equation (7), we get:  
\[\left\{ \begin{array}{c}\dot{e}_{1}=\dot{\widetilde{i}}_{\rm rd}-i_{\rm rd}^{ *}=-\mu\omega_{\rm s}+gu_{\rm rd}^{*}\\ \dot{e}_{2}=\dot{\widetilde{i}}_{\rm rg}-i_{\rm rg}^{*}=gu_{\rm rg}^{*}\end{array} \right. \tag{8}\]  
For the \(e_{1}\) construction states of Equations (7) and (18):  
\[ \begin{array}{c}\dot{e}_{1}\qquad=-k_{1}e_{1}+k_{1}e_{1}+gu_{\rm rd}^{*}- \mu\omega_{\rm s}\\ =-k_{1}e_{1}+gu_{\rm rd}^{*}-\mu\omega_{\rm s}+k_{1}\widetilde{i}_{\rm rd}-k_ {1}i_{\rm rd}^{*}\end{array}  \tag{9}\]  
Constructing the \(u_{\rm rd}^{*}\) control rate:  
\[gu_{\rm rg}^{*}=k_{1}i_{\rm rd}^{*}-k_{1}\widetilde{i}_{\rm rd}+\mu\omega_{ \rm s} \tag{10}\]  
and therefore: \(u_{\rm rg}^{*}=\frac{k_{1}}{g}(i_{\rm rd}^{*}-\widetilde{i}_{\rm rd}^{*})+ \frac{\mu\omega_{\rm s}}{g},k_{1}>0\). In the above equation, \(k_{1}\) is the control gain, and the Lyapunov function is constructed according to Equation (4). \[\lambda_{1}=\frac{1}{2}e_{1}^{2} \tag{11}\]  
In accordance with Equation (5), we can take the derivative of Equation (11) and insert Equations (9) and (10) into Equation (11) to obtain the following formula:  
\[\dot{\lambda}_{1}=-k_{1}e_{1}^{2}\leq 0 \tag{12}\]  
It can be seen that \(\lambda_{1}\geq 0\) and \(\dot{\lambda}_{1}\leq 0\); the subsystem is stable and can control the rotor d-axis current \(\widetilde{i}_{\rm rd}\). The same process is used for the \(e_{2}\) construction states in Equations (7) and (8):  
\[\dot{e}_{2}=-k_{2}e_{2}+k_{2}e_{2}+gu_{\rm rg}^{*}=-k_{2}e_{2}+gu_{\rm rg}^{*} +k_{2}\widetilde{i}_{\rm gr}-k_{2}i_{\rm qr}^{*} \tag{13}\]  
Constructing \(u_{\rm rg}^{*}\) control rate:  
\[gu_{\rm rg}^{*}=k_{1}i_{\rm qr}^{*}-k_{1}\widetilde{i}_{\rm qr} \tag{14}\]  
and therefore: \(u_{\rm rg}^{*}=\frac{k_{2}}{g}(i_{\rm qr}^{*}-\widetilde{i}_{\rm gr}),k_{2}>0\).