## 2 Sliding Mode Control Law of the DFIG System
### Sliding Mode Control of DFIG (RSC)  
The DFIG equivalent circuit in the \(\alpha\beta\) frame is shown in Figure 2.  
In order to regulate the active power and reactive power directly, the instantaneous power errors are defined as the DFIG's state variables \(\mathbf{x}_{\text{re}}=[\mathbf{x}_{\text{rep}}\ \mathbf{x}_{\text{req}}]^{\text{T}}\)  
\[\mathbf{x}_{\text{re}}=\mathbf{x}_{\text{r}}^{*}-\mathbf{x}_{\text{r}} \tag{1}\]  
where, \(\mathbf{x}_{\text{r}}=[P_{S}\ Q_{\text{s}}]^{\text{T}};\mathbf{x}_{\text{r}}^{*}=[P_{ \text{s}}^{*}\ Q_{\text{s}}^{*}]^{\text{T}}\). Then, according to the mathematical model, the behavior of the DFIG can be described by the following state-space form  
\[\frac{d\mathbf{x}_{\text{re}}}{dt}=\frac{d\mathbf{x}_{\text{r}}^{*}}{dt}-\mathbf{f}_{ \text{r}}+\delta\mathbf{B}_{0}\mathbf{v}_{\text{r}}+\mathbf{d}_{\text{r}} \tag{2}\]  
\[\mathbf{f}_{\text{r}}=\left[ \begin{array}{c}\omega Q_{\text{s}}-1.5\delta\mu_{ \text{g}}^{2}\\ \omega P_{\text{s}}\end{array} \right]+\delta\mathbf{x}_{\text{r}}+\frac{j\delta \omega_{\text{r}}L_{\text{r}}}{L_{\text{m}}^{2}}\mathbf{B}_{0}\mathbf{\varphi}_{\text{ s}\alpha\beta} \tag{3}\]  
\[\mathbf{B}_{0}=1.5\left[ \begin{array}{cc}u_{\beta\alpha}&u_{\beta\beta}\\ -u_{\beta\beta}&u_{\beta\alpha}\end{array} \right]\mathbf{d}_{\text{r}}=-1.5\left[ \begin{array}{cc}\zeta_{\alpha}&\zeta_{\beta}\\ -\zeta_{\beta}&\zeta_{\alpha}\end{array} \right]\mathbf{i}_{\text{s}\alpha\beta} \tag{4}\]  
where, \(\delta=1/(\sigma L_{\text{m}})\); \(\zeta_{\alpha}=2\omega u_{\beta\beta}\pm 6\omega u_{\beta\beta 5}-6\omega u_{ \beta\beta 7}\), \(\zeta_{\beta}=-2\omega u_{\beta\alpha}-6\omega u_{\beta\alpha 5}-6\omega u_{ \beta\alpha 7}\), which are related to the unbalanced and harmonic voltage;\(f_{\text{r}}\) is the known disturbance of system; due to the \(\zeta_{\alpha}\) and \(\zeta_{\beta}\) are unknown items without the unbalanced and harmonic separation, \(\mathbf{d}_{\text{r}}\) is the unknown disturbance. And \(\mathbf{d}_{\text{r}}\) contains ac components at the frequencies of \(2\omega\), \(4\omega\), \(6\omega\), \(8\omega\) and \(12\omega\) produced by the corresponding currents and voltages. The AC components at the frequencies of \(2\omega\) and \(6\omega\) are produced by the fundamental, unbalanced, \(5-\), \(7+\) harmonics currents and voltages. While the ac components at the frequencies of \(4\omega\), \(8\omega\) and \(12\omega\) are produced by the unbalanced, \(5-\), \(7+\) harmonics currents and voltages. Thus, the ac components at the frequencies of \(4\omega\), \(8\omega\) and \(12\omega\) can be neglected compared to the ac components at the frequencies of \(2\omega\) and \(6\omega\). In the conventional SMC, the sliding surface contains a proportional and an integral item, which provides accurate tracking on dc reference and rejection to dc disturbance. However, the AC errors will  
Figure 1: Schematic diagram of a DFIG-based (doubly-fed induction generator) wind power system. RSC: rotor-side converter; GSC: grid-side converter. Figure 2: DFIG equivalent circuit in the stationary \(\alpha\beta\) reference frame. arise when the tracking reference or disturbance is the AC component. In order to provide sufficient tracking precision, multi-resonant items are added to the conventional integral sliding surface, which can be called \"MRSMC\". Then, the DFIG's sliding surfaces \(S_{\text{r}}=[S_{\text{rp}}\ S_{\text{rq}}]^{\text{T}}\) are defined as follows  
\[S_{\text{r}}(s)=G_{\text{rs}}(s)\mathbf{x}_{\text{re}}(s);\ G_{\text{rs}}(s)=1+ \frac{k_{\text{rs}}}{s}+\sum_{i=2,6}\frac{sk_{\text{ri}}}{s^{2}+2e_{\text{c}} (i\omega)s+\left(i\omega\right)^{2}} \tag{5}\]  
where \(k_{\text{rs}}\) is the integral coefficient related to dc error elimination; \(k_{\text{ri}}\) (\(i=2,6\)) is the resonant coefficient related to ac error elimination; \(e_{\text{c}}\) is related to cutoff frequency and inserted into the resonant part to increase its frequency bandwidth.