## 2 Research on the Static Security Region (SSR) of Power Systems Based on New Energy  
The static security region of a power system is defined as the collection of all stable operating points. It approaches the problem from a domain perspective, describing the region where the system can operate safely and stably as a whole. The relative relationship between system operating points and the boundaries of the security region provides information on safety margins and optimal control, enabling more efficient analysis of the static stability of the power system. It is typically represented by a set of operating points that satisfy the power flow equations and various security constraints of the power system. Mathematically, the SSR can be represented as follows [20]:  
\[\Omega_{\text{SSR}}= \begin{cases}f(\mathbf{x},\mathbf{y})=0\\ V_{i}^{\min}\leq V_{i}\leq V_{i}^{\max},&\forall_{i}\in N\\ p_{\text{g},i}^{\min}\leq P_{\text{g},i}\leq p_{\text{g},i}^{\max},&\forall_{i }\in N_{\text{g}}\\ Q_{\text{g},i}^{\min}\leq Q_{\text{g},i}\leq Q_{\text{g},i}^{\max},&\forall_{ i}\in N_{\text{g}}\\ -P_{\text{l},i-j}^{\max}\leq P_{\text{l},i-j}\leq P_{\text{l},i-j}^{\max},& \forall_{i,j}\in N\end{cases}  \tag{1}\]  
In the provided description, \(\mathbf{x}\) represents the vector of system state variables. \(\mathbf{y}\) represents the vector of injected power at each node. \(f(\mathbf{x},\mathbf{y})=0\) denotes the system power flow equations. \(N\) and \(N_{\text{g}}\) are the sets of system nodes and generator nodes, respectively. \(V_{i}^{\min}\) and \(V_{i}^{\max}\) are the minimum and maximum allowable voltage magnitudes at each node. \(P_{\text{g},i}^{\min}\) and \(P_{\text{g},i}^{\max}\) are the minimum and maximum active power output limits of the generators. \(Q_{\text{g},i}^{\min}\) and \(Q_{\text{g},i}^{\max}\) are the minimum and maximum reactive power output limits of the generators. \(-P_{\text{l},i-j}^{\max}\) and \(P_{\text{l},i-j}^{\max}\) represent the active power limits in the forward and reverse directions, respectively, of transmission lines \(i-j\). Considering the integration of new energy sources and deriving an analytical expression for the system's static security region based on static power flow calculations, the equivalent schematic diagram of wind farm integration into the grid is illustrated in Figure 1.  
In the diagram, \(\dot{E}\) represents the voltage at the wind power access point in the system, \(\dot{V}_{pcc}\) represents the voltage at the wind farm grid connection point, \(Z_{s}=R_{s}+jX_{s}\) represents the equivalent impedance of the system, \(Z_{l}=R_{l}+jX_{l}\) represents the equivalent impedance of the transmission line between the wind farm grid connection point and the system equilibrium point, \(P_{w}+jQ_{w}\) represents the apparent power of the wind farm, and \(\dot{l}\) represents the injected current from the wind farm. The relationship between wind farm active power output and voltage curve is derived as follows:  
\[\dot{V}_{\rm pcc}=\dot{E}+(R_{s}+jX_{s})\dot{I} \tag{2}\]  
For the given load on the busbar, \(P_{w}+jQ_{w}\), \(\dot{I}=\frac{(P_{w}-jQ_{w})}{\
u_{\rm pcc}}\). The apparent power of the wind farm is given by  
\[ \begin{array}{ll}P_{\rm w}+jQ_{\rm w}&=\dot{V}_{\rm pcc}\frac{V_{\rm pcc}^{ 2}-\dot{E}}{R_{s}-jX_{s}}\\ &=\left(V_{\rm pcc}\cos\theta+jV_{\rm pcc}\sin\theta\right)\left(\frac{V_{\rm pcc }\cos\theta-jV_{\rm pcc}\sin\theta-E}{R_{s}-jX_{s}}\right)\end{array}  \tag{3}\]  
In Equation (3)  
\[\left\{ \begin{array}{ll}\dot{E}=E\angle 0^{\circ}\\ \dot{V}_{\rm pcc}=V_{\rm pcc}\angle\theta\end{array} \right. \tag{4}\]  
\[\left\{ \begin{array}{ll}P_{\rm w}=\frac{V_{\rm pcc}E(X_{s}\sin\theta-R_{s} \cos\theta)+V_{\rm pcc}^{2}R_{s}}{R_{s}^{2}+X_{s}^{2}}\\ Q_{w}=\frac{-V_{\rm pcc}E(X_{s}\cos\theta+R_{s}\sin\theta)+V_{\rm pcc}^{2}X_{ s}}{R_{s}^{2}+X_{s}^{2}}\end{array} \right.