## 2 Analysis of Influence of SSO on DFIG System Operation  
### The Establishment of the Mathematical Model for an RSC with SSO  
The DFIG control model adopted in this study is shown in Figure 1. This double-closed-loop vector control (VC) model is one of the most widely used DFIG control models. The VC control model has the advantages of utilizing mature regulation technology and showing good steady-state performance and strong robustness [16]. The grid-side converter in the figure is represented by the term \"GSC\", while the SSO component of the power grid is marked with red arrows. As can be seen from Figure 1, the SSO is input into the RSC control system through three paths successively. These three paths are the power outer loop, the current inner loop and the feedforward decoupling compensation. All oscillations are indicated by the subscript \"sub\", and \"**\" represents the reference value for the input signal. Utilizing the voltage equation, the flux linkage equation and the power calculation for the DFIG system, the mathematical model for the RSC current control under SSO can be obtained through coordinate transformation [17; 18]:  
\[\left\{ \begin{array}{l}u_{\text{rd}}=R_{\text{r}}\widetilde{i_{\text{rd}}}+ \sigma L_{\text{r}}\frac{\widetilde{d}\widetilde{i_{\text{rd}}}}{\text{d}t}- \omega_{\text{s}}[-\frac{L_{\text{r}}}{\omega_{\text{s}}L_{\text{s}}}U_{\text {s}}+\sigma L_{\text{r}}\widetilde{i_{\text{rd}}}]\\ u_{\text{rq}}=R_{\text{r}}\widetilde{i_{\text{rq}}}+\sigma L_{\text{r}}\frac{ \widetilde{d}\widetilde{i_{\text{r}}}}{\text{d}t}+\omega_{\text{s}}\sigma L_{ \text{r}}\widetilde{i_{\text{rq}}}\end{array} \right. \tag{1}\]  
In Equation (1), \(u_{\text{rd}}\) and \(u_{\text{rq}}\) are the d and q components of the rotor voltage, respectively; \(R_{\text{r}}\) is the rotor winding resistance; \(\widetilde{i_{\text{rd}}}\) and \(\widetilde{i_{\text{rq}}}\) are the d and q components of the rotor current; \(L_{\text{s}}\) is the self-inductance of the stator winding; \(L_{\text{r}}\) is the rotor winding self-inductance; \(L_{\text{m}}\) is the mutual inductance between the equivalent windings of the stator and rotor; \(\sigma=1-L_{\text{m}}^{2}/L_{\text{s}}L_{\text{r}}\) is the leakage flux coefficient of the motor; \(\omega_{\text{s}}\) is the slip angle frequency, and \(\omega_{\text{s}}=\omega_{1}-\omega_{\text{r}}\); \(\omega_{1}\) is the synchronization angular frequency; and \(\omega_{\text{r}}\) is the rotor angular frequency. According to Figure 1, when the grid produces SSO, the sub-synchronous current in the grid is fed into the stator side of the DFIG. Through the electromagnetic induction effect, the stator sub-synchronous current induces the corresponding sub-synchronous current on the generator rotor side. At the same time, the oscillation of the stator current also causes the DFIG power oscillation, resulting in an RSC power outer-ring output instruction oscillation. The waveform distortion and phase shift of the stator and rotor side currents make the oscillating components affect the RSC control circuit, causing the control command signal generated by the RSC to oscillate, which further affects the actual current in the rotor and causes the DFIG output power to oscillate. In conclusion, when the grid SSO is generated, since the RSC control is disturbed by the oscillation in the stator current, the DFIG stator current produces the corresponding sub-synchronous component. The stator three-phase sub-synchronous current generated by the DFIG is calculated as follows:  
\[\left\{ \begin{array}{l}\Delta i_{\text{sa\_sub}}=\sqrt{2}|h|I_{\text{n}}\cos (\omega_{\text{n}}t+\phi_{\text{m}}+\pi-\phi)\\ \Delta i_{\text{sb\_sub}}=\sqrt{2}|h|I_{\text{n}}\cos(\omega_{\text{n}}t+\phi_ {\text{in}}+5\pi/3-\phi)\\ \Delta i_{\text{sc\_sub}}=\sqrt{2}|h|I_{\text{n}}\cos(\omega_{\text{n}}t+\phi_ {\text{in}}+\pi/3-\phi)\end{array} \right.