衡量潮流雅可比矩阵及其降阶阵不对称性和奇异性的指标

Indices for Evaluating the Unsymmetry and Singularity of Load Flow Jacobian Matrix and Its Reduced Ones

依据:①矩阵与其(反)对称部分范数间的关系;②矩阵的1-范数与∞-范数之间的关系;③矩阵特征值的绝对值的最大值与其最大奇异值之间的关系;④矩阵特征值的绝对值最大值和矩阵特征值的绝对值的最小值之比与矩阵的谱条件数之间的关系;⑤对最大奇异值,对应的奇异参与因子之和与1之间的关系,构造了衡量矩阵不对称性的指标,并依据:①矩阵特征值的绝对值最小值与最小奇异值之间的关系:②对最小奇异值而言,对应的奇异参与因子之和与1之间的关系,构造了衡量矩阵奇异性的指标。应用IEEE30系统算例和潮流雅可比矩阵及其相应的降阶雅可比矩阵对上述指标进行了分析,得出了潮流雅可比矩阵及其相应的降阶矩阵的谱条件数排序由相应的矩阵最小奇异值排序决定的结论。

The indices for evaluating the unsymmetry of a matrix are constructed according to the norm relation between matrix and its symmetrical part or its unsymmetrical part, the relation between 1- norm and ∞-norm, the relation between maximum of absolute eigenvalue and maximum singular value, the relation between spectrum condition number and the ratio of maximum of absolute eigenvalue to minimum of absolute eigenvalue, the relation between the sum of participation factors based on left and right singular vectors corresponding to maximum singular value and 1. At the same time, the new indices for evaluating the singularity of unsymmetrical matrix are constructed according to the relation between minimum of absolute eigenvalue and minimum singular value, the relation between the sum of participation factors based on left and right singular vectors corresponding to minimum singular value and one. These indices are verified using load flow Jacobian matrix and its reduced ones on IEEE 30 system, and the result shows that the ranking of spectrum condition number of load flow Jacobian matrix and its reduced ones are determined by their minimum singular values. At last, the relation between unsymmetry and singularity of load flow Jacobian matrix is discussed.