## 2 Modeling and Analysis of Wind Storage Systems  
### Modeling of Wind Farm and Analysis of Its Working Characteristics  
Double-fed induction generator (DFIG) is a new type of generator based on traditional synchronous and asynchronous generators. However, unlike asynchronous generators, it has a three-phase excitation winding structure on the rotor side. Unlike synchronous generators, the rotor side is fed with variable frequency AC. In actual operation, the rotor speed of a doubly fed generator, coupled with the rotational magnetic field generated by the rotor excitation current, is equal to the synchronous speed of the rotor, thereby inducing an induced electromotive force at the synchronous frequency on the stator side. Figure 1 shows the basic structure diagram of DFIG, with the stator side connected to the power frequency grid and the rotor side connected to the grid through a partial power converter. By controlling the rotor side converter, the excitation current frequency, amplitude, and phase of the rotor winding are controlled to ensure stable power output on the stator side, while decoupling the rotor speed from the system frequency. The relationship between DFIG rotor speed and system frequency is as follows:  
\[\omega=2\pi f \tag{1}\]  
\(\omega\) is the rotor speed, and \(f\) is the system frequency. Figure 1: Basic structure diagram of doubly fed wind turbines. The mechanical power captured by DFIG is:  
\[\left\{ \begin{array}{l}P_{m}=\frac{1}{\rho}\rho SC_{\rm p}(\beta,\lambda)v^{3}\\ \lambda=\frac{v\hbar S}{v}\end{array} \right. \tag{2}\]  
where \(\rho\) is the Density of air; \(C_{\rm p}\) is the wind energy utilization coefficient, which is related to the tip speed ratio \(\lambda\) and the pitch angle \(\beta\); \(S\) is the area swept by the wind turbine rotor; \(v\) is the wind speed; \(\omega\) is the mechanical speed; \(R\) is the radius of the wind turbine. \(C_{\rm p}\) is represented as:  
\[\left\{ \begin{array}{l}\frac{1}{\lambda}=\frac{1}{\lambda+0.08}-\frac{0.035 }{1+\beta}\\ C_{\rm p}=C_{1}(\frac{C_{2}}{\lambda_{1}}-C_{3}\beta-C_{4})e^{-C_{5}\lambda_{1 }}+C_{6}\lambda\end{array} \right. \tag{3}\]  
In the equation \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\), \(C_{5}\) and \(C_{6}\) are constants. Figure 2 is the verification of the maximum power tracking characteristics of DFIG, and the output power of DFIG is affected by the change of pitch angle and wind speed. With the direction of the arrow on the right, it represents a gradual increase in wind speed, corresponding to the power tracking curve at the wind speed. The maximum power point tracking (MPPT) of the wind power generation system means that the wind turbine can obtain the maximum power output under any circumstances, that is, the wind energy can be converted into electrical energy to the maximum extent. DFIG consists of pitch angle control (2 nd order), wind turbine and transmission system (3 rd order), induction motor (4 th order), DFIG back-to-back converter, and DC capacitor model (9 th order) [8]. If the wind speed goes beyond the rated limit, it's important to ensure that the power absorbed by the wind turbine remains the same. This is because the wind turbine's mechanical strength and converter capacity have their limitations. To limit the aerodynamic efficiency of the wind turbine blades, the pitch angle is controlled. Thus, the wind turbine model uses a pitch angle control system, which can be represented by an equation as follows:  
\[\left\{ \begin{array}{l}\frac{d\beta}{dt}=\frac{1}{T_{\rm p}}(\beta_{\rm ref }-\beta)\\ \beta_{\rm ref}=K_{\rm p}(\omega_{\rm r}-\omega_{\rm ref})+K_{\rm i}\int( \omega_{\rm r}-\omega_{\rm ref})dt\end{array} \right.