## 2 Research on the Static Security Region (SSR) of Power Systems Based on New Energy
\tag{5}\]  
Simplifying Equation (3), we obtain  
\[\left(P_{\rm w}+\frac{V_{\rm pcc}^{2}R_{s}}{R_{s}^{2}+X_{s}^{2}}\right)^{2}+ \left(Q+\frac{V_{\rm pcc}^{2}X_{s}}{R_{s}^{2}+X_{s}^{2}}\right)^{2}=\frac{V_{ \rm pcc}^{2}E^{2}}{R_{s}^{2}+X_{s}^{2}} \tag{6}\]  
Organizing Equation (6), we obtain  
\[\left(V_{\rm pcc}^{2}\right)^{2}+\left[2P_{\rm w}R_{s}+2 QX_{s}-E^{2}\right]V_ {\rm pcc}^{2}+\left(R_{s}^{2}+X_{s}^{2}\right)\left(P_{\rm w}^{2}+Q^{2}\right)=0 \tag{7}\]  
Figure 1: The equivalent schematic diagram of wind farm integration into the grid. Solving for \(P\), \(Q\), and \(V\) in Equation (7), we obtain the expression for the static security domain as follows:  
\[\left\{ \begin{array}{l}P=-\frac{R_{\mathrm{s}}V_{\mathrm{pc}}^{2}\mp\sqrt{-Q^{2 }\left(R_{\mathrm{s}}^{2}+X_{\mathrm{s}}^{2}\right)^{2}+\left(R_{\mathrm{s}}^{ 2}+X_{\mathrm{s}}^{2}\right)\left(\mu_{\mathrm{pc}}^{2}-2Q\mathrm{x}_{\mathrm{s }}\right)V_{\mathrm{pc}}^{2}-X_{\mathrm{s}}^{2}V_{\mathrm{pc}}^{4}}\\ Q=-\frac{\mathrm{x}_{\mathrm{s}}V_{\mathrm{pc}}^{2}\pm\sqrt{-P^{2}\left(R_{ \mathrm{s}}^{2}+X_{\mathrm{s}}^{2}\right)^{2}+\left(R_{\mathrm{s}}^{2}+X_{ \mathrm{s}}^{2}\right)\left(V_{\mathrm{pc}}^{2}-2 PR_{\mathrm{s}}\right)V_{ \mathrm{pc}}^{2}-R_{\mathrm{s}}^{2}V_{\mathrm{pc}}^{4}}}{R_{\mathrm{s}}^{2}+X_ {\mathrm{s}}^{2}}\\ V=\sqrt{-(PR_{\mathrm{s}}+Q\mathrm{x}_{\mathrm{s}})+\frac{V_{\mathrm{pc}}^{2}}{2 }\pm\sqrt{\frac{V_{\mathrm{pc}}^{4}}{4}-\left(PR_{\mathrm{s}}-Q\mathrm{x}_{ \mathrm{s}}\right)^{2}-V_{\mathrm{pc}}^{2}(PR_{\mathrm{s}}+Q\mathrm{x}_{ \mathrm{s}})}}\end{array} \right. \tag{8}\]  
The assignment of values in Equation (8) should satisfy the following two constraint conditions. (1) Equality constraint condition  
The expression \(f(x,y)=0\) represents the equality constraint, ensuring power balance between supply and demand, where \(N\) denotes the total number of system nodes and \(G\) denotes the number of generator nodes, and it can be represented as follows:  
\[\left\{ \begin{array}{l}P_{i}-V_{i}\sum\limits_{j=1}^{N}V_{j}\big{(}G_{ij} \cos\theta_{ij}+B_{ij}\sin\theta_{ij}\big{)}=0\\ Q_{i}-V_{i}\sum\limits_{j=1}^{N}V_{j}\big{(}G_{ij}\sin\theta_{ij}-B_{ij}\cos \theta_{ij}\big{)}=0\end{array} \right. \tag{9}\]  
In Expression (9), \(x\) represents the system's state variables, \(y\) denotes the injected power at nodes, and \(P_{i}\) and \(Q_{i}\) represent the active and reactive power injection at node \(i\), respectively. \(G_{ij}\) and \(B_{ij}\) are the real and imaginary parts of the admittance of branch \(ij\), and \(\theta_{ij}\) is the phase angle difference between the two terminal nodes of branch \(ij\). (2) Inequality constraints  
When the system experiences voltage fluctuations caused by disturbances, the voltage at the grid connection point is maintained by dynamically adjusting the reactive power sources within the wind farm. The introduced constraint for the voltage deviation at the wind farm connection point is as follows:  
\[V_{\mathrm{PCC}}^{\mathrm{ref}}-V_{\mathrm{PCC}}^{\mathrm{err}}\leq V_{ \mathrm{PCC}}\leq V_{\mathrm{PCC}}^{\mathrm{ref}}+V_{\mathrm{PCC}}^{\mathrm{ err}} \tag{10}\]  
where \(V_{\mathrm{PCC}}\) represents the actual voltage at the wind farm connection point; \(V_{\mathrm{PCC}}^{\mathrm{ref}}\) denotes the commanded voltage at the connection point; and \(V_{\mathrm{PCC}}^{\mathrm{err}}\) signifies the permissible control error. The reactive regulation capacity constraint of wind turbines is as follows:  
\[[Q_{g_{\mathrm{imm}}},Q_{g_{\mathrm{imm}}}]= \begin{cases}[0,Q_{g_{\mathrm{ immax}}}]\ V_{\mathrm{pcc}}^{\mathrm{ref}}>V_{\mathrm{pcc}}+V_{\mathrm{pcc}}^{ \mathrm{dead}}\\ \ [Q_{g_{\mathrm{imm}}},0]\ V_{\mathrm{pcc}}^{\mathrm{ref}}<V_{\mathrm{pcc}}-V_{ \mathrm{pcc}}^{\mathrm{dead}}\end{cases}  \tag{11}\]  
The symbol \(U_{\mathrm{pcc}}^{\mathrm{dead}}\) represents the dead band for PCC bus voltage control. The inequality constraints for the wind farm also include voltage magnitude constraints for each node.