Explicit Sparse Polynomial Approximation of Multi-Parameter Eigenvalue Loci and Locus Pairs for Small Signal Stability Analysis

Many researches on analysis, uncertainty quantification, and control of small signal stability are essentially to study how stability indices (e.g., eigenvalue and damping ratio) vary with system stochastic or controllable parameters. The functional relationships between eigenvalues and parameters (termed as eigenvalue loci) are implicit, nonlinear, and elusive, hindering insights into complete portrait of system stability. This paper proposes a comprehensive method to accurately approximate multi-parameter simple eigenvalue loci and intersecting eigenvalue locus pairs with explicit polynomial functions, which will greatly facilitate further analysis. The proposed method follows the idea of sparse polynomial chaos expansion and best square approximation, and thus can ensure high accuracy over the whole considered parameter space, compared with existing Taylor expansion-based local eigenvalue sensitivity methods. On this basis, the single-parameter homotopy approach is subtly combined in order to avoid aliasing of relevant eigenvalue loci; a variable transformation approach is proposed to recover differentiability damaged by multiplicity 2 eigenvalue submanifolds so as to retain accuracy of approximating intersecting locus pairs. The accuracy and effectiveness of the proposed method are validated by computational results of low-frequency, ultra-low-frequency, and subsynchronous oscillation modes on the 15-parameter two-area system, and the low-frequency mode on the 30-parameter IEEE 145-bus system.