On the Eigenvalue Tracking of Large-Scale Systems
Andreas Bouterakos, Joseph McKeon, Georgios Tzounas
TL;DR
The paper addresses tracking eigenvalue trajectories in large-scale power-system models as a parameter varies. It proposes a continuation-based framework that traces targeted eigenvalues using the matrix pencil $P(s,p)=s\,\mathbf{E}(p)-\mathbf{A}(p)$ and a differential system in $y=(\Re\{\boldsymbol{\phi}\},\Im\{\boldsymbol{\phi}\},s_r,s_i)$ with mass matrix $\mathbf{M}(\mathbf{y})$. Key contributions include a general formulation compatible with dense and sparse representations and both explicit and semi-implicit DAEs, robust handling of large parameter steps and defective eigenvalues, and practical guidance on step-size, matrix updates, and derivatives; the method is validated on the FR mode of the IEEE 39-bus system and benchmarked on a 1,502-bus Irish transmission model. Adaptive step-size strategies and optional Newton correctors maintain accuracy while achieving significant speedups over repeated QR factorizations for large-scale pencils. Overall, the approach enables scalable small-signal stability analysis and mode-tracking in modern power networks with DERs and dynamic controllers.
Abstract
The paper focuses on the problem of tracking eigenvalue trajectories in large-scale power system models as system parameters vary. A continuation-based formulation is presented for tracing any single eigenvalue of interest, which supports sparse matrix representations and accommodates both explicit and semi-implicit differential-algebraic models. Key implementation aspects, such as numerical integration, matrix updates, derivative approximations, and handling defective eigenvalues, are discussed in detail and practical recommendations are duly provided. The tracking approach is demonstrated through a comprehensive case study on the IEEE 39-bus system, as well as on a realistic dynamic model of the Irish transmission system.