Data-Driven Participation Factors for Nonlinear Systems Based on Koopman Mode Decomposition

This paper develops a novel data-driven technique to compute the participation factors for nonlinear systems based on the Koopman mode decomposition. Provided that certain conditions are satisfied, it is shown that the proposed technique generalizes the original definition of the linear mode-in-state participation factors. Two numerical examples are provided to demonstrate the performance of our approach: one relying on a canonical nonlinear dynamical system, and the other based on the two-area four-machine power system. The Koopman mode decomposition is capable of coping with a large class of nonlinearity, thereby making our technique able to deal with oscillations arising in practice due to nonlinearities while being fast to compute and compatible with real-time applications.

💡 Research Summary

The paper introduces a data‑driven framework for computing participation factors in nonlinear dynamical systems, leveraging the Koopman operator theory and the Extended Dynamic Mode Decomposition (EDMD). Participation factors—both mode‑in‑state and state‑in‑mode—are essential metrics in power‑system analysis for assessing modal contributions to system states, guiding model reduction, and designing stabilizers. Traditional approaches rely on linearization around an operating point, which fails to capture the true modal behavior under stressed or highly nonlinear conditions.

The authors first review the classical linear definitions of participation factors (pᵢⱼ = vᵢⱼ uᵢⱼ) and the stochastic extensions proposed by Hashlamoun et al., where the initial state is treated as a random vector and expectations replace deterministic products. They then introduce the Koopman operator, a linear but infinite‑dimensional operator that acts on observable functions of the state. By focusing on the point spectrum (eigenvalues µⱼ) and associated eigenfunctions ϕⱼ, any observable g(xₖ) can be expressed as a linear combination of Koopman modes φⱼ weighted by the eigenfunctions evaluated at the initial condition.

To make the theory computationally tractable, the paper adopts EDMD. Given a set of snapshot pairs {xₖ, xₖ₊₁}, a library of observables γ(x) (which may include the original state and nonlinear functions such as squares or trigonometric terms) is constructed. The finite‑dimensional approximation of the Koopman operator is K = ΓX′ ΓX†, where ΓX and ΓX′ contain the lifted snapshots. The eigen‑decomposition of K yields approximations of the eigenvalues, left eigenvectors ξⱼ, and right eigenvectors. The Koopman modes for the original state are then obtained via Φ = B Ξ⁻¹, where B maps lifted observables back to the physical state.

The core contribution lies in the new definitions of participation factors for nonlinear systems:

• Data‑driven mode‑in‑state participation factor
  pᵢⱼ = ξᵢⱼ φᵢⱼ + Σ_{r≠i} ξᵣⱼ φᵢⱼ E