## 4 SSO Suppression Strategy for DFIG System
### Improved Controller Performance Analysis  
The basic principle of the control strategy proposed in this paper is explained in the analysis in Section 4.1. This section analyzes the control performance of the strategy for sub-synchronous oscillation of a power grid. When the grid voltage is oriented along the d-axis, Equations (23) and (24) can be obtained from the model of the DFIG:  
\[\left\{ \begin{array}{c}u_{\text{sd}}=|u_{\text{s}}|=U_{\text{s}}\approx- \omega_{1}\psi_{\text{sq}}\\ u_{\text{sq}}=0\approx\omega_{1}\psi_{\text{sd}}\end{array} \right. \tag{23}\]  
\[\boldsymbol{I}_{\text{sdq}}=\frac{1}{L_{\text{s}}}\psi_{\text{sdq}}-\frac{L_{ \text{m}}}{L_{\text{s}}}\boldsymbol{I}_{\text{rdq}} \tag{24}\]  
Figure 5: Back-stepping SSO suppression strategy based on adaptive quasi-resonant controller. where \(u_{\rm sd}\) is the d-axis component of the grid voltage vector; \(u_{\rm sq}\) is the grid voltage of the q-axis component; \(u_{\rm s}\) is the grid voltage vector; \(u_{\rm s}\) is the grid voltage vector magnitude; \(\omega_{1}\) is the rotational angular velocity of the grid voltage vector; \(\psi_{\rm sq}\) is the q-axis component of the flux linkage vector; \(\psi_{\rm sd}\) is the d-axis component of the flux linkage vector; \(\mathbf{I}_{\rm sdq}\) is the dq-axis component of the stator current; \(L_{\rm s}\) is the stator inductance; \(L_{\rm m}\) is the mutual inductance between the stator and rotor; and \(\mathbf{I}_{\rm rdq}\) is the dq-axis component of the rotor current. The block diagram for the RSC current vector control embedded in the adaptive quasi-resonant (R) and back-stepping controller is shown in Figure 6.  
In the case of grid voltage orientation, utilizing Figure 6 and Equations (23) and (24), the equivalent transfer function of an RSC based on adaptive quasi-resonance and back-stepping control strategies can be written as:  
\[I_{\rm sdq}=F_{\rm I}(s)U_{\rm sdq}+H_{\rm I}(s)I_{\rm rdq}^{*} \tag{25}\]  
In the vector control of the d-axis grid voltage, we only consider the characteristics for the protection of the DFIG wind power system against sub-synchronous harmonic voltage disturbance along the d-axis. In the block diagram for the current vector control embedded in the adaptive quasi-resonant and back-stepping controller, \(F_{\rm I}(s)\) is used to measure the disturbance on the forward channel of the DFIG; that is, the anti-disturbance capability of the sub-synchronous harmonic voltage. \(H_{\rm I}(s)\) is used to represent the dynamic response capability of the current loop. From Figure 6, the following equation can be deduced:  
\[F_{\rm I}(s)=\frac{G_{\rm I}(s)/L_{\rm s}+G_{\rm I}(s)G_{\rm back}(s)G_{\rm P }(s)/L_{\rm s}+G_{\rm I}(s)G_{\rm Z}(s)G_{\rm P}(s)L_{\rm m}/L_{\rm s}}{1+G_{ \rm back}(s)G_{\rm P}(s)+G_{\rm R}(s)G_{\rm P}(s)L_{\rm m}/L_{\rm s}} \tag{26}\]  
\[H_{\rm I}(s)=\frac{G_{\rm back}(s)G_{\rm P}(s)L_{\rm m}/L_{\rm s}}{1+G_{\rm back }(s)G_{\rm P}(s)+G_{\rm R}(s)G_{\rm P}(s)L_{\rm m}/L_{\rm s}} \tag{27}\]  
We can substitute \(F_{\rm I}(s)\) and \(H_{\rm I}(s)\) into the parameters of Table 1 to draw the Bode diagrams of \(F_{\rm I}(s)\) and \(H_{\rm I}(s)\), as shown in Figures 7 and 8. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Rated power & 15 kW & Rotor leakage reactance \(L_{\rm nr}\) & 0.0022 H \\ Stator voltage & 200 V & Moment of inertia of the motor \(J\) & 0.39 Kg-m\({}^{2}\) \\ Mutual inductance resistance \(L_{\rm m}\) & 0.0427 H & Pole logarithm of motor \(n_{\rm P}\) & 3 \\ Stator resistance \(R_{\rm s}\) & 0.379 \(\Omega\) & DC side voltage \(V_{\rm dc}\) & 400 V \\ Stator leakage reactance \(L_{\rm s\sigma}\) & 0.0011 H & Inductance into the line \(L_{\rm g}\) & 0.005 H \\ Rotor resistance \(R_{\rm r}\) & 0.314 \(\Omega\) & Motor magnetic flux leakage coefficient \(\sigma\) & 0.07288 \\ \hline Rated frequency & 50 Hz & DC bus capacitance C & 2200 \(\mu\)F \\ \hline \end{tabular}
\end{table} 
Table 1: DFIG experimental platform-related parameters.