Participation Factors for Singular Systems of Differential Equations

In this article, we provide a method to measure the participation of system eigenvalues in system states, and vice versa, for a class of singular linear systems of differential equations. This method deals with eigenvalue multiplicities and covers all cases by taking into account both the algebraic and geometric multiplicity of the eigenvalues of the system matrix pencil. A Möbius transform is applied to determine the relative contributions associated with the infinite eigenvalue that appears because of the singularity of the system. The paper is a generalization of the conventional participation analysis, which provides a measure for the coupling between the states and the eigenvalues of systems of ordinary differential equations with distinct eigenvalues. Numerical examples are given including a classical DC circuit and a 2-bus power system dynamic model.