  When random disturbances are regularly introduced into a dynamical system
over time, its small-signal stability is determined by the energy of
perturbations accumulated in the system. To analyze this perturbation energy,
this paper proposes a novel physically motivated Lyapunov modal analysis (LMA)
framework, which combines selective modal analysis with the spectral
decompositions of specially chosen Lyapunov functions. This approach allows the
modal interactions in dynamical systems to be characterized and estimated in
connection with specific state variables. Conventional participation factors
characterize the relative contribution of the system modes and state variables
to the evolution of states and modes, respectively. In contrast, the proposed
Lyapunov participation factors characterize similar contributions to
corresponding Lyapunov functions, which determine the integral energy
associated with the states and modes on an infinite or finite time interval.
This allows the estimation of modal interactions in terms of total energy
produced by their mutual actions over time. Using a two-area four-machines
power system, we demonstrate that LMA reliably identifies resonant modal
interactions, merging of modes, and loss of stability, even for a linear model,
and associates them with certain state variables. The Lyapunov participation
factors corresponding to the selected part of the system spectrum can be
calculated independently and serve as a basis for rapid real-time calculations
of critical mode behaviors in large-scale dynamical systems.
