## 2 Modeling and Analysis of Wind Storage Systems
### Linearized Equivalence Model of Stand-Alone Infinity System with Wind Storage  
To analyze the low-frequency oscillation stability of the wind storage grid connected system under disturbance, the eigenvalue analysis method can be used to linearize the system [10] near the operating point. According to the matrix similarity transformation theory, the state matrix of the original system has the same eigenvalues as the similarity matrix, and the stability and oscillation mode information of the original nonlinear system at the operating point can be obtained by solving the equation of the Linear system. Establish the equations of state [11] and algebraic equation of the wind storage grid connected system:  
\[\left\{\begin{array}{l}p{x_{\text{SG}}}=f_{\text{SG}}({x_{\text{SG}}},{y_{ \text{SG}}},{y_{\text{SYS}}})\\ p{x_{\text{DFHG}}}=f_{\text{DFHG}}({x_{\text{DFHG}}},{y_{\text{DFHG}}},{y_{ \text{SYS}}})\\ p{x_{\text{BESS}}}=f_{\text{BESS}}({x_{\text{BESS}}},{y_{\text{BESS}}},{y_{ \text{SYS}}})\end{array}\right. \tag{19}\]  
\[\left\{\begin{array}{l}0={g_{\text{SG}}}({x_{\text{SG}}},{y_{\text{SG}}},{y _{\text{SYS}}})\\ 0={g_{\text{DFHG}}}({x_{\text{DFHG}}},{y_{\text{DFHG}}},{y_{\text{SYS}}})\\ 0={g_{\text{BESS}}}({x_{\text{BESS}}},{y_{\text{BESS}}},{y_{\text{SYS}}})\\ 0={g_{\text{SYS}}}({y_{\text{SYS}}})\end{array}\right. \tag{20}\]  
where \(p\) is a Differential operator;\(f\), \(g\) is an Equation of state and Algebraic equation, respectively; \(x\), \(y\) is a state variable and an algebraic variable, respectively.  
Figure 5: Wind Storage System Model.  
By linearizing the above two equations at steady-state operating points, a linearized model of the wind storage grid-connected system can be obtained [12]:  
\[\left\{\begin{bmatrix}p\left[\begin{array}{c}\Delta x_{\text{SG}}\\ \Delta x_{\text{DFIG}}\\ \Delta x_{\text{BESS}}\\ 0\end{array}\right]=\left[\begin{array}{c}AB\\ CD\end{array}\right]\left[\begin{array}{c}\Delta x_{\text{SG}}\\ \Delta x_{\text{DFIG}}\\ \Delta x_{\text{BESS}}\end{array}\right]\\ \left[\begin{array}{c}\Delta y_{\text{SG}}\\ \Delta y_{\text{DFIG}}\\ \Delta y_{\text{BESS}}\\ \Delta y_{\text{SYS}}\end{array}\right]\\ p\Delta x=(A-BD^{-1}C)=A_{\text{SYS}}\Delta x\end{bmatrix}\right. \tag{21}\]  
In the equation, \(A\), \(B\), \(C\), \(D\) are coefficient matrices; \(A_{\text{SYS}}\) is the system state matrix; \(\Delta\) represents increment. From \(A_{\text{SYS}}\), information such as oscillation mode, damping ratio, frequency, and participation factor can be calculated.  
Let the conjugate eigenvalue of the oscillation mode be \(\lambda=\sigma+j\omega\), and the damping ratio and oscillation frequency of the system be:  
\[\xi=-\frac{\sigma}{\sqrt{\sigma^{2}+\omega^{2}}} \tag{22}\]  
\[f=\frac{\omega}{2\pi} \tag{23}\]  
The left and right eigenvectors \(u\) and \(v\) related to the eigenvalues are defined as \(p_{\text{k}}=u_{\text{k}}v_{\text{k}}\) as participation factors, representing the degree of influence of the \(k\) state variable on the \(i\) oscillation mode. For the low-frequency oscillation problem studied in this article, it is necessary to study the roots of the strong correlation between \(\Delta\omega\) and \(\Delta\delta\), and define the electromechanical circuit-related participation factor of the \(i\) mode as:  
\[\rho_{\text{i}}=\left|\frac{\sum\limits_{\begin{subarray}{c}x_{\text{k}}\in( \Delta\omega,\Delta\delta)\end{subarray}}p_{\text{Ki}}}{\sum\limits_{ \begin{subarray}{c}x_{\text{k}}\in(\Delta\omega,\Delta\delta)\end{subarray}} p_{\text{Ki}}}\right| \tag{24}\]  
Based on the results of small signal analysis, inter-region oscillation modes can be identified, and damping control can be carried out using a POD controller. At the same time, the controller gain should be adjusted reasonably so that the controller can not only increase inter-region damping but also have a positive effect on oscillation in other modes.