## 1 Introduction
### Stability Margin Evaluation Indexes  
Once a PSR is obtained, the stability margin evaluation indexes of controller parameters for SSO prevention and control are proposed as follows  
\[F(\mathbf{X})=\left.\left(\sum_{s=1}^{K}d_{s}\right)\right/d_{b} \tag{5}\]  
\[d_{s}=\frac{|a_{s1}X_{1}+a_{s2}X_{2}+\cdots+a_{sn}X_{n}+b_{s}|}{\sqrt{a_{s1}^{2 }+a_{s2}^{2}+\cdots+a_{sn}^{2}}} \tag{6}\]  
where \(s=1\)-\(K\) represents the \(s\)-th fitting hyperplane that constitutes the PSR, respectively, and its hyperplane expression is \(a_{s1}X_{1}+a_{s2}X_{2}+\cdots+a_{sn}X_{n}+b_{s}=0\), \(\mathbf{X}=(X_{1},\,\ldots\,\,X_{n})\) are the dominant controller parameters of the PSR. \(d_{s}\) is the distance from the operating point of the dominant controller parameter set to the \(s\)-th fitting hyperplane. \(d_{b}\) is used as the evaluation benchmark of other parameter points. For the PSR, \(d_{b}\) is the value of \(\Sigma_{s=1}^{K}d_{s}\) calculated at the parameter equilibrium point (the controller parameters when starting PSR construction). If the parameter set is inside the PSR, then the closer the controller parameters are to the center of the PSR, the greater the index is, and the closer to the PSR boundary, the smaller the index is. If the index \(F\) is greater than 1, it shows that the parameter set is farther away from the boundary, and when the index \(F\) is less than 1, it shows that the parameter set is closer to the boundary, and system small signal stability decreases, so as to provide guidance for the reasonable selection and online refreshing of the controller parameters according to the current operation mode, fulfilling the prevention and control of SSO, thus securing the safe and stable operation of the high-proportion renewable power grid. Preconditions of Gerschgorin Disk Theorem's Application on Parameter Stability Region and the Construction Method of Transition Matrix  
Based on the analysis in Section 2, the GDT guarantees that the eigenvalues of a matrix are located in the disks corresponding to the matrix, and the PSR can be constructed based on the stability constraints on the Gerschgorin disks. However, according to the stability constraints during PSR construction, the centers of the Gerschgorin disks must be on the left side of the imaginary axis, that is, the diagonal elements of the system state matrix must be negative, which is a tough condition and is hardly satisfied in the normal operation of the actual power system. To meet the above requirements and increase the feasibility of online rapid construction of the PSR, a method to structure the transition matrix is proposed, and the approach to construct the PSR with system transition matrix containing parameter variables instead of the system state matrix is given. The detailed procedure is as follows. Assuming that under the initial value of the parameter set (\(a_{0}\), \(b_{0}\)), the state matrix is \(A0\). The eigenvalues and eigenvectors of the matrix are calculated once, and the eigenvalue diagonal matrix is recorded as \(D0\) and the right eigenvector matrix is recorded as \(\mathbf{P}\); \(\mathbf{D}0\) and \(\mathbf{P}\) are known. When the parameter set changes (denoted as \((a,\,b)\)), the eigenvalue diagonal matrix \(\mathbf{D}\) of the new state matrix \(\mathbf{A}\) is as follows  
\[\mathbf{D}=\left[ \begin{array}{ccccccccc}\lambda_{1}&&&&\\ &\lambda_{2}&&&&\\ &&\bullet&&\\ &&&&\bullet&&\\ &&&&\bullet&\\ &&&&\lambda_{n}\end{array} \right] \tag{7}\]  
where \(\lambda_{1}\cdots\lambda_{n}\) are eigenvalues (\(\lambda_{1}\cdots\lambda_{n}\) are unknown) of the matrix \(\mathbf{A}\), the right eigenvector matrix is recorded as \(\mathbf{X}\), \(\mathbf{X}=[x_{1},x_{2},\cdots,x_{n}]\) (which is unknown), and \(x_{i}\) is a column vector.