## 3 Analysis of MRSMC Strategy
### Tracking Behavior with Distorted Grid  
The unbalanced, 5\(-\) and 7\(+\) harmonics distorted grid voltages will introduce disturbance \(dj\) (\(j\) = \(r\) or \(g\)) into the DFIG control system, which mostly contains the ac components at the frequencies of 2\(\omega\) and 6\(\omega\). If the voltages applied to the rotor and GSC are according to (8) and (15), the sliding surfaces functions with disturbance \(dj\) (\(j\) = \(r\) or \(g\)) are  
\[\frac{dS_{j}\big{(}\mathbf{x}_{j\mathrm{e}},t\big{)}}{dt}=-\mathbf{d}_{j}-\mathbf{k}_{j \mathrm{u}}sat\big{(}\mathbf{S}_{j}\big{)};\ j=r,g \tag{20}\]  
The system described as (20) contains the saturation functions, and is a nonlinear system. In order to investigate the system dynamical behavior, the phase plane trajectory is adopted. Figure 4 shows the phase plane trajectories with dc disturbance and ac disturbances at the frequencies of 2\(\omega\) and 6\(\omega\). It should be pointed out that, if the resonant coefficients \(k_{\mathrm{ri}}\) and \(k_{\mathrm{gi}}\) (\(i\) = 2, 6) are zero, the control strategy will be the ISMC strategy. Thus, Figure 4a, in which the resonant coefficients \(k_{\mathrm{ri}}\) (\(i\) = 2, 6) are equal to zero, shows the phase plane trajectory of ISMC. The resonant coefficient in Figure 4b is 20 and 50, and the Figure 4b shows the phase plane trajectory of MRSMC. In Figure 4a, outside the boundary layer \(\mathbf{\lambda}_{r}\), the \(\mathbf{S}_{\mathrm{rp}}\) moves toward to the boundary layer with the dc disturbance or ac disturbances \(\mathbf{d}_{r}\), whereas inside the boundary layer, due to the integral action, the \(\mathbf{S}_{\mathrm{rp}}\) achieves the origin only with the DC disturbance. With the ac disturbances, the \(\mathbf{S}_{\mathrm{rp}}\) deviates from the origin, which means that it exists steady state errors with the ac disturbances. Thus, the ISMC strategy will degrade the system operation performance under unbalanced, 5\(-\) and 7\(+\) harmonics distorted grid voltage conditions. In Figure 4b, outside the boundary layer, the \(\mathbf{S}_{\mathrm{rp}}\) moves toward to the boundary layer regardless of disturbances. Inside the boundary layer, with the integral action and resonant action, the \(\mathbf{S}_{\mathrm{rp}}\) achieves the origin regardless of the dc or ac disturbances, which means the steady state errors can be removed with the MRSMC. Thus, the MRSMC strategy features rejection on disturbances, and is insensitive to the distorted grid voltage. The analysis on phase plane trajectories of \(x_{\mathrm{geq}}\), \(x_{\mathrm{geq}}\) and \(x_{\mathrm{req}}\) are similar as \(x_{\mathrm{req}}\), which points out the similar conclusion. In order to analyze the tracking precision on dc and ac references with ISMC and MRSMC, the phase plane trajectory with different tracking references were investigated. According to control laws and the DFIG and GSC's equivalent circuits, the relationships of tracking references \(\mathbf{x}_{\mathrm{j}}^{*}\) (\(j\) = \(r\) or \(g\)) and tracking error \(\mathbf{x}_{\mathrm{je}}\) (\(j\) = \(r\) or \(g\)) are  
\[\left\{ \begin{array}{c}\mathbf{x}_{\mathrm{g}}=L_{\mathrm{g}}k_{\mathrm{gu}}sat \big{[}G_{\mathrm{gs}}(s)\mathbf{x}_{\mathrm{ge}}\big{]}\big{/}\big{(}sL_{ \mathrm{g}}+R_{\mathrm{g}}\big{)}\\ \mathbf{x}_{\mathrm{r}}=L_{\mathrm{m}}\mathbf{k}_{\mathrm{ru}}sat\big{[}G_{\mathrm{rs} }(s)\mathbf{x}_{\mathrm{re}}\big{]}/(s\sigma L_{\mathrm{r}}L_{\mathrm{s}}+R_{ \mathrm{r}}L_{\mathrm{s}})\end{array} \right. \tag{21}\]According to (21), the phase plane trajectory of \(\mathbf{x}_{\rm{gep}}\) with ISMC and MRSMC are shown as Figure 5a,b. Under unbalanced and \(5-\) harmonics and \(7+\) harmonics distorted grid voltage conditions, the tracking references will contain the ac components at the frequencies of \(2\omega\) and \(6\omega\). Thus, Figure 5 shows the phase plane trajectories with the DC reference and AC references at the frequencies of \(2\omega\) and \(6\omega\).