## 2 Research on the Static Security Region (SSR) of Power Systems Based on New Energy
\[V_{i}^{\mathrm{min}}\leq V_{i}\leq V_{i}^{\mathrm{max}},\forall_{i}\in N \tag{12}\]  
The symbols \(V_{i}^{\mathrm{min}}\) and \(V_{i}^{\mathrm{max}}\) represent the minimum and maximum values of the node voltage magnitude, respectively. The active power constraint for the branch is given by  
\[-P_{ij}^{\mathrm{max}}\leq P_{ij}\leq P_{ij}^{\mathrm{max}},\forall_{i,j}\in B \tag{13}\]  
\(-P_{\mathrm{ij}}^{\mathrm{min}}\) and \(P_{\mathrm{ij}}^{\mathrm{max}}\) are the positive and negative active output limits of branch \(ij\), respectively. The reactive power constraint for branches is as follows:Let \(\left|y_{ij}\right|=\left|G_{ij}+jB_{ij}\right|\), where \(\left(G_{ij}+jB_{ij}\right)\) is the (\(i\),\(j\))-th element of the nodal admittance matrix. Therefore  
\[\dot{I_{ij}} =\left(G_{ij}+jB_{ij}\right)\left(\dot{V_{i}}-\dot{V_{j}}\right) \tag{14}\] \[=\left(G_{ij}+jB_{ij}\right)\left(V_{i}e^{i\theta_{i}}-V_{j}e^{i \theta_{j}}\right)\] \[=-y_{ij}e^{-j\left(90^{\circ}-\beta\right)}V_{i}e^{i\theta_{j}} \left(e^{i\theta_{ij}}-V_{j}/V_{i}\right)\] \[=-y_{ij}V_{i}e^{-j\left(\theta_{j}+\beta-90^{\circ}\right)}V_{i}e^ {i\theta_{j}}\left(\cos\theta_{ij}+j\sin\theta_{ij}-V_{j}/V_{i}\right)\]  
Assuming \(V_{j}/V_{i}=1\); \(\theta_{ij}\) is small; therefore,\(\cos\theta_{ij}\approx 1\) and \(\sin\theta_{ij}\approx\theta_{ij}\). Equation (14) can be written as follows:  
\[\dot{I_{ij}}\approx-y_{ij}V_{i}e^{-j\left(\theta_{j}+\beta-90^{\circ}\right)} \left(j\theta_{ij}\right) \tag{15}\]  
Taking the modulus of both sides in Equation (15), we obtain  
\[\left|\dot{I_{ij}}\right|=y_{ij}V\left|\theta_{ij}\right| \tag{16}\]  
The constraint expressed by the voltage phase difference at both ends of branch \(ij\) can be obtained from the current constraint, namely  
\[-\theta_{ij}^{\max}\leq\theta_{ij}\leq\theta_{ij}^{\max},\forall_{i,j}\in B \tag{17}\]  
\[\theta_{ij}^{\max}=\frac{I_{ij}^{\max}}{y_{ij}V_{i}^{\max}} \tag{18}\]  
In conclusion, the reactive power flow constraint is as follows:  
\[-P_{ij}^{\mathrm{max}}\leq P_{ij}\leq P_{ij}^{\mathrm{max}},\forall_{i,j}\in B \tag{13}\]0  
The static security domain obtained in Equation (8) and its constrained conditions, (9) to (19), requires an optimization algorithm for global parameter identification. In this paper, we employ the gradient descent algorithm, which offers fast computational speed and excellent global optimization capability for addressing large-scale mathematical optimization search problems, as detailed in Sections 2 and 3. The equation for the safety domain hyperplane can be expressed in the following form:  
\[-P_{ij}^{\mathrm{max}}\leq P_{ij}\leq P_{ij}^{\mathrm{max}},\forall_{i,j}\in B \tag{13}\]1  
In Equation (20), \(a_{i},i=1,2,\ldots,n\) represents the coefficients of the hyperplane to be determined, y denotes the dimension of the system, and y is the observed variable (usually 1). For the m critical points found during the search \(\left(x_{j1},x_{j2},\ldots,x_{jn}\right),j=1,2,\ldots,m\), and their errors can be represented as follows:  
\[-P_{ij}^{\mathrm{max}}\leq P_{ij}\leq P_{ij}^{\mathrm{max}},\forall_{i,j}\in B \tag{13}\]2  
In Equation (21)  
\[\left\{ \begin{array}{l}Y=\left[y_{1},y_{2},\ldots,y_{m}\right]^{T}\\ X=\left[\begin{array}{cccc}x_{11}&x_{12}&\ldots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ \vdots&\vdots&&\vdots\\ x_{m1}&x_{m2}&\ldots&x_{mn}\\ \end{array}\right]\\ \alpha=\left[\alpha_{1},\alpha_{2},\cdots,\alpha_{v}\right]^{T}\\ \varepsilon=\left[\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{m} \right]^{T}\end{array} \right. \tag{22}\]The error vector \(\varepsilon\) should satisfy Equation (23)  
\[\left\|\varepsilon\right\|^{2}=\left(Y-Xa\right)^{T}\!\!\left(Y-Xa\right)\to \min \tag{23}\]  
Taking the partial derivative with respect to \(a\) to \(\left\|\varepsilon\right\|^{2}\), we have  
\[ \begin{array}{c}\frac{\partial\left\|\varepsilon\right\|^{2}}{\partial x}= \frac{\partial\left(Y^{T}Y-2Y^{T}Xa+\varepsilon^{T}X^{T}Xa\right)}{\partial x }\\ =-2Y^{T}X+2\alpha^{T}X^{T}X=0\end{array}