Koopman Operator Theory

The Koopman operator K acts on observable functions g(x) rather than the state x itself. While the underlying system ẋ = f(x) is nonlinear, the operator satisfies Kg(x) = g ∘ F(x), where F is the flow map. This lifts the dynamics into an infinite-dimensional space where evolution is perfectly linear, enabling spectral decomposition of complex rotor angle trajectories.

The eigenvalues λ and eigenfunctions φ of the Koopman operator encode the fundamental modes of the system:

• Point spectrum: Captures isolated frequencies and growth/decay rates
• Continuous spectrum: Represents chaotic or mixing dynamics
• Koopman modes: Spatial structures associated with each eigenvalue

For transient stability, the dominant eigenvalues directly reveal oscillation frequencies and damping ratios of inter-area modes.

The choice of observable dictionary is critical for practical computation:

• Monomials and polynomials: Capture nonlinear interactions between state variables
• Radial basis functions: Provide universal approximation in high dimensions
• Delay coordinates: Embed temporal information for partially observed systems
• Physics-informed observables: Incorporate known energy functions or Lyapunov candidates

In power systems, observables often include kinetic energy terms and voltage magnitude functions.

Dynamic Mode Decomposition (DMD) and Extended DMD provide finite-dimensional approximations of the Koopman operator from trajectory data:

• EDMD constructs a matrix representation using a predefined dictionary of observables
• Kernel DMD implicitly uses infinite-dimensional feature spaces via the kernel trick
• Deep DMD learns optimal observables using neural network autoencoders

These methods extract Koopman eigenvalues and modes directly from PMU measurement streams without requiring explicit system models.

Unlike local linearization around an operating point, the Koopman operator captures global nonlinear behavior:

• Region of attraction estimation: Eigenfunctions define stability boundaries in state space
• Transient energy prediction: Koopman modes decompose post-fault energy into constituent components
• Invariant subspace identification: Isolates coherent generator groups that swing together

This enables assessment of large-disturbance stability without time-domain simulation of every contingency.

DMD is the primary computational algorithm for approximating the Koopman operator:

• Standard DMD assumes linear observables and extracts spatial-temporal modes
• Exact DMD provides a numerically robust formulation using the pseudoinverse
• Optimized DMD fits a low-order linear model to nonlinear data via variable projection
• Sparsity-promoting DMD selects a minimal set of modes for interpretable models

For transient stability, DMD applied to post-fault PMU data can identify poorly damped modes within seconds of a disturbance.

A data-driven, equation-free algorithm that approximates the Koopman operator by extracting spatio-temporal coherent structures from high-dimensional simulation or PMU data. DMD decomposes complex transient trajectories into a superposition of modes, each associated with a specific frequency and growth rate.

• Computes eigenvalues and eigenvectors directly from snapshot pairs
• Identifies dominant oscillation modes without linearizing the underlying system
• Enables real-time stability assessment from streaming synchrophasor data
• Variants like Extended DMD use nonlinear observable dictionaries to improve approximation accuracy

Scalar-valued functions of the system state that form the lifted space in which the Koopman operator acts linearly. Selecting an appropriate dictionary of observables is the central design challenge in applying the theory to power systems.

• Identity observable: Maps state to itself, preserving physical interpretation
• Polynomial observables: Capture nonlinear interactions between state variables
• Radial basis functions: Provide universal approximation in infinite-dimensional spaces
• Physics-informed observables: Embed known energy functions or Lyapunov candidates
• The choice directly determines the accuracy of spectral decomposition for rotor angle dynamics

The Koopman spectrum—eigenvalues, eigenfunctions, and Koopman modes—provides a complete decomposition of nonlinear transient behavior into linear components. This enables global stability characterization beyond the reach of local linearization.

• Eigenvalues on the unit circle indicate neutrally stable oscillatory modes
• Eigenvalues inside the unit circle correspond to damped, decaying transients
• Koopman modes reveal spatial coherence patterns of generator groups
• Enables identification of inter-area oscillation modes directly from fault response data
• Provides a rigorous mathematical bridge between time-domain simulation and modal analysis

The set of all post-fault initial states from which trajectories converge to a stable equilibrium. Koopman operator theory enables data-driven estimation of this stability boundary without exhaustive time-domain simulation.

• Stable manifold of the closest unstable equilibrium defines the boundary
• Koopman eigenfunctions with positive growth rates indicate proximity to instability
• Enables real-time monitoring of transient stability margins from PMU streams
• Complements direct methods like the Transient Energy Function approach
• Critical for online contingency screening and preventive control decisions

The process of grouping synchronous machines that exhibit identical rotor angle swings following a disturbance. Koopman mode analysis provides a rigorous, data-driven alternative to traditional slow coherency methods.

• Generators sharing the same Koopman mode are dynamically coherent
• Enables automated model order reduction through dynamic equivalencing
• Identifies coherent groups directly from PMU data without system model knowledge
• Facilitates controlled islanding scheme design for preventing cascading blackouts
• Reveals how coherency patterns evolve with changing operating conditions

An advanced variant of DMD that lifts the system state into a higher-dimensional space using a dictionary of nonlinear observable functions before applying linear regression. EDMD converges to the true Koopman operator as the dictionary size increases.

• Approximates the infinite-dimensional Koopman operator with a finite matrix
• Dictionary selection is critical: polynomials, Fourier features, or kernel functions
• Enables global linear representations of strongly nonlinear swing dynamics
• Computationally tractable for large-scale power systems using randomized algorithms
• Forms the algorithmic backbone for data-driven transient stability assessment