Dynamic Mode Decomposition (DMD)

DMD provides a finite-dimensional approximation of the Koopman operator, which governs the evolution of observables in a linear but infinite-dimensional space. This allows the analysis of nonlinear rotor angle dynamics using linear spectral techniques. The resulting eigenvalues characterize oscillation frequencies and growth/decay rates, while eigenvectors represent spatial mode shapes across the network. For power systems, this means identifying inter-area oscillation modes and their damping ratios directly from PMU data without linearizing the underlying differential-algebraic equations.

Unlike traditional small-signal stability analysis that requires explicit state-space models, DMD operates purely on snapshot data from simulations or measurements. This is critical for modern grids where accurate dynamic models of inverter-based resources may be unavailable. DMD discovers the governing dynamics from time-series data alone by:

• Extracting dominant temporal patterns from high-dimensional measurements
• Identifying coherent generator groups without prior system knowledge
• Reconstructing full-field dynamics from sparse sensor placements
• Bypassing the need for explicit differential-algebraic equation formulation

DMD enables online transient stability assessment by processing streaming PMU data to compute eigenvalues that indicate proximity to instability. As the system approaches the stability boundary, the dominant DMD eigenvalue migrates toward the unit circle. This provides a model-free stability margin that operators can monitor in real time. The method is computationally efficient, requiring only singular value decomposition of data matrices, making it suitable for wide-area monitoring systems with sub-second update requirements.

DMD decomposes complex grid responses into spatially coherent structures oscillating at distinct frequencies. Each DMD mode consists of:

• An eigenvalue determining the temporal behavior (frequency and damping)
• An eigenvector defining the spatial distribution across measurement locations This decomposition isolates phenomena like local plant modes (0.7-2.0 Hz) from inter-area modes (0.1-0.8 Hz), allowing targeted damping control strategies. The spatial component reveals which generators participate most strongly in each oscillation mode.

DMD informs sparse sensing strategies by identifying the most informative measurement locations for reconstructing full system dynamics. Using the DMD basis, operators can determine minimal sensor configurations that capture dominant coherent structures. This is achieved through:

• QR pivoting on the DMD mode matrix to rank sensor importance
• Reconstruction of unmeasured states from limited observations
• Guiding PMU deployment to maximize observability of critical inter-area modes
• Reducing infrastructure costs while maintaining full dynamic visibility

Once DMD modes are extracted, they form a reduced-order model capable of forecasting future grid states without running full time-domain simulations. The linear combination of DMD modes extrapolates system trajectories forward in time with minimal computational cost. This enables:

• Rapid prediction of post-fault rotor angle trajectories
• Early warning of growing oscillations before they become visible in raw measurements
• Evaluation of control action effectiveness within milliseconds
• Integration into remedial action schemes for predictive stability control

The mathematical foundation of DMD. The Koopman operator is an infinite-dimensional linear operator that governs the evolution of observables of a nonlinear dynamical system. DMD provides a finite-dimensional approximation of this operator from data.

• Lifts nonlinear dynamics into a globally linear framework
• Enables spectral analysis of nonlinear systems
• DMD eigenvalues approximate the Koopman spectrum

A classical signal processing technique that decomposes a transient waveform into a sum of damped complex exponentials. Prony analysis extracts modal frequencies, damping ratios, amplitudes, and phases from time-domain ringdown data.

• Directly estimates electromechanical mode parameters
• Applied to generator speed or power signals post-disturbance
• DMD generalizes Prony's method to high-dimensional spatio-temporal data

Also known as Principal Component Analysis (PCA) in statistics. POD extracts energetically dominant spatial modes from data but does not capture temporal dynamics. DMD extends POD by associating each mode with a specific frequency and growth/decay rate.

• POD modes are ranked by energy content
• DMD modes are ranked by dynamic significance
• Often used as a preprocessing step for reduced-order modeling

An extension of standard DMD that uses an L1-regularization penalty to select a parsimonious subset of modes that best reconstruct the full dataset. This identifies the most dynamically important coherent structures while discarding noise-induced modes.

• Formulated as a convex optimization problem
• Balances reconstruction accuracy against model complexity
• Critical for isolating dominant inter-area oscillation modes

Deep learning frameworks that embed the swing equation and other governing differential-algebraic equations directly into the neural network loss function. While DMD is purely data-driven, PINNs enforce physical consistency constraints during training.

• Combines measurement data with first-principles physics
• Useful when DMD modes violate known conservation laws
• Complementary approach for hybrid stability assessment

The real-time inference of a generator's internal dynamic states—rotor angle, transient voltage, and mechanical power—using streaming PMU data and Kalman filtering. DMD provides the linearized dynamic model used within the estimator's prediction step.

• Enables online transient stability monitoring
• DMD-derived models reduce computational burden
• Critical for wide-area control applications