Definition of State-in-Mode Participation Factors for Modal Analysis of Linear Systems

Modal participation factors are widely used in the power industry and other applied areas. Hashlamoun, Hassouneh, and Abed (2009) observed that the definition of state-in-mode participation factors (SIMPFs) commonly used in the modal analysis of linear systems could lead to inadequate results. Accordingly, they proposed an alternative definition of SIMPFs. This article uncovered a weakness in their formulation in that it yields results highly sensitive to varying eigenvector normalizations for complex eigenvalues. Thus, we present a modified well-defined formulation and interpretation of SIMPFs in this article. The proposed approach combines the framework of averaging over an uncertain set of initial conditions, introduced by Hashlamoun, Hassouneh, and Abed, with the novel concept of modal energy. Following this approach, an alternative definition for mode-in-state participation factors was discovered, which differs from the conventional definition. Simple examples are used to show that new definitions provide more accurate measures of state-in-mode and mode-in-state participations compared to conventional participation factors in the case of an arbitrary distribution of initial conditions over state variables. In particular, the proposed energy-based interpretation of novel indicators precludes negative or complex measures.