Dynamic Mode Decomposition (DMD)

DMD operates purely on measurement data—snapshots of the system state over time—without requiring a pre-existing physics-based model or differential equations. It identifies the best-fit linear operator A that maps the system from one timestep to the next. This makes it ideal for complex systems like power grids where deriving first-principles models is intractable. The algorithm computes the eigendecomposition of this approximated linear operator to extract dynamic modes.

DMD decomposes complex, high-dimensional data into a set of spatial modes, each associated with a specific temporal frequency and growth/decay rate. Each mode represents a coherent structure that oscillates, grows, or decays at a single fixed frequency. This is critical for isolating inter-area oscillations in synchrophasor data, where a mode's spatial shape reveals which generators are swinging against each other.

For each extracted mode, DMD provides direct estimates of key stability metrics without manual curve-fitting:

• Eigenvalue: Encodes the oscillation frequency (imaginary part) and damping/growth rate (real part).
• Mode Shape: The relative amplitude and phase of participation across measurement locations.
• Damping Ratio: Calculated directly from the eigenvalue, quantifying how quickly an oscillation decays. This is essential for small-signal stability assessment.

Although DMD fits a linear model, it is deeply connected to Koopman operator theory. The Koopman operator is an infinite-dimensional linear operator that exactly describes the evolution of nonlinear systems. DMD approximates the Koopman operator's eigenfunctions and eigenvalues from finite data. This provides a rigorous theoretical foundation for applying a linear analysis tool to inherently nonlinear power system dynamics.

DMD leverages Singular Value Decomposition (SVD) to project the high-dimensional snapshot data onto a low-dimensional subspace of dominant coherent features. By truncating to the first r singular values, DMD filters out measurement noise and redundant information, producing a reduced-order model. This rank truncation is the key to computational tractability when analyzing massive PMU data streams with thousands of channels.

Once the DMD modes and eigenvalues are computed, the original data can be reconstructed, and future states can be predicted by evolving the modes forward in time. The system state at any future timestep is a linear combination of the modes, each weighted by its eigenvalue raised to the appropriate power. This enables short-term oscillation forecasting and what-if scenario analysis for grid operators.

A classical signal processing method that fits a sum of exponentially damped sinusoids directly to a time-series signal. Unlike DMD, Prony operates on a single measurement channel and estimates frequency, damping, amplitude, and phase for each mode.

• Highly sensitive to measurement noise
• Requires careful model order selection
• Commonly applied to ringdown data from generator trips

A time-domain system identification technique that constructs a minimal-order state-space model from impulse response data using Hankel matrix decomposition. ERA generalizes DMD concepts to controlled systems.

• Uses singular value decomposition for noise rejection
• Produces a discrete-time linear model
• Widely used in aerospace and civil structural health monitoring

The mathematical foundation underlying DMD. The Koopman operator is an infinite-dimensional linear operator that advances observable functions of the system state forward in time, enabling globally linear representations of nonlinear dynamics.

• DMD approximates the Koopman operator's eigenfunctions
• Extended DMD enriches the observable space with nonlinear functions
• Connects fluid dynamics, power systems, and robotics under a unified framework

Also known as Principal Component Analysis, POD extracts spatial modes ranked by energy content rather than temporal dynamics. DMD extends POD by associating each mode with a specific growth rate and frequency.

• POD modes are purely spatial and energy-optimal
• DMD modes capture both spatial coherence and temporal evolution
• Often used together: POD for dimensionality reduction, DMD for dynamics

An adaptive time-frequency analysis method designed for non-stationary and nonlinear signals. HHT decomposes a signal into Intrinsic Mode Functions via empirical mode decomposition, then applies the Hilbert transform to extract instantaneous frequency.

• No assumption of linearity or stationarity
• Complements DMD for analyzing transient grid events
• Effective for forced oscillation detection where modal parameters shift over time

A disturbance-based technique that analyzes the transient oscillatory response of the grid immediately following a sudden event such as a line trip or generator outage. DMD excels at extracting damping ratios and mode shapes from ringdown data.

• Triggered by identifiable grid disturbances
• Provides high signal-to-noise ratio for modal identification
• Critical for validating small-signal stability models against real-world measurements