## 3 Analysis of SSO Inhibition Mechanism in the DFIG System
### Introduction to the Principle of Back-Stepping Control  
At present, the control methods commonly used for nonlinear systems are back-stepping control and sliding mode control. Sliding mode control provides excellent immunity to external random excitation, system parameter variations and unconsidered external disturbances. However, sliding mode control needs to consider the chattering effect of the sliding mode structure, and the greater the amplitude of the control switching, the more obvious the chattering problem is. Most research on wind turbine sliding mode control focuses on improvements to power generation efficiency, often ignoring the system stability problem caused by the change in control strategy during the power tracking process, which easily increases the fatigue load on the system during the control process and reduces the operating life of the wind turbine [19]. In the process of reducing the order of the nonlinear system, the back-stepping method can also reduce errors. Controlling the system through the back-stepping method entails decomposing the more complex high-order system to obtain subsystems with less order compared to the system order. If the back-stepping method is combined with other control algorithms, dynamic control over the controlled system can be achieved despite the interference of unknown disturbance factors [20; 21; 22]. Compared with sliding mode control, the design process for a back-stepping controller strictly follows the system stability function and does not consider stability problems such as chattering. The structure of back-stepping controllers is clear, and the dynamic response speed is fast. Figure 3: Influence of SSO frequency variation on DFIG system: (**a**) rid oscillation frequency of 10 Hz; (**b**) grid oscillation frequency of 15 Hz. Set up the following system:  
\[\left\{ \begin{array}{c}\dot{\mathrm{x}}_{\mathrm{j}}=g_{\mathrm{j}}(\overline{ \mathrm{x}}_{\mathrm{j}})x_{\mathrm{j+1}}+f_{\mathrm{j}}(\overline{\mathrm{x}} _{\mathrm{j}})\\ \dot{\mathrm{x}}_{\mathrm{n}}=g_{\mathrm{n}}(x)u+f_{\mathrm{n}}(x),1\leq j\leq n -1\\ y=x_{1}\end{array} \right. \tag{3}\]  
In Equation (3), \(\overline{\mathrm{x}}_{\mathrm{j}}\) represents the state variable, and \(\overline{\mathrm{x}}_{\mathrm{j}}=(x_{1},x_{2},\cdots,x_{\mathrm{j}})^{T}\); \(g_{\mathrm{j}}\) and \(f_{\mathrm{j}}\) represent continuous functions; \(u\) represents the input control variable of the system; and \(y\) represents the system output variable. When using the back-stepping method, it is necessary to design a virtual control variable (e.g., \(x_{\mathrm{2d}}\)) for the control of the subsystem. Then, the error is set to \(e_{1}\) and a Lyapunov function is designed:  
\[\left\{ \begin{array}{c}e_{1}=x_{1}-x_{\mathrm{d}}\\ V_{1}=\frac{1}{2}e_{1}^{2}\end{array} \right. \tag{4}\]  
where \(x_{1}\) is the system output, \(x_{\mathrm{d}}\) is the output target value, and \(V_{1}\) is the Lyapunov function used to determine the stability of the system. It is necessary to determine the dummy control variables that make the derivative of the Lyapunov function less than zero, as follows:  
\[\left\{ \begin{array}{c}\dot{V}_{1}=e_{1}\cdot\dot{e}_{1}=e_{1}\cdot[g_{1}( \overline{\mathrm{x}}_{1})x_{\mathrm{2d}}+f_{1}(\overline{\mathrm{x}}_{1})- \dot{x}_{\mathrm{d}}]\\ x_{\mathrm{2d}}=[\dot{x}_{\mathrm{d}}-f_{1}(\overline{\mathrm{x}}_{1})-k_{1}e_{ 1}]g_{1}(\overline{\mathrm{x}}_{1})^{-1}\end{array} \right. \tag{5}\]  
\(x_{\mathrm{2d}}\) can make \(x_{1}\) approach and stabilize at \(x_{\mathrm{d}}\), and the virtual control variables \(x_{\mathrm{3d}}\), \(x_{\mathrm{4d}}\), \(\cdots\), \(u\) of other subsystems can be calculated with the same method, so that \(x_{2}\), \(x_{3}\), \(\cdots\), \(x_{\mathrm{n}}\) are gradually stabilized at \(x_{\mathrm{2d}}\), \(x_{\mathrm{3d}}\), \(\ldots\), \(x_{\mathrm{n}}\). Finally, all state variables in the system are eventually tracked smoothly.