Sparse PMUs, full-machine trajectories: a physics-informed graph neural operator estimates every rotor in one forward pass and beats per-window fitting on 118 buses
Synopsis
With phasor measurement units installed at only a few generator buses and reporting intermittently, a physics-informed graph neural operator maps a measurement window directly to the rotor-angle and speed trajectories of all machines, including those without sensors, and trains without state labels. It imposes the swing equations as a hard constraint, or as a soft one with a learned correction that absorbs model error; on the 9-bus, Kundur, 39-bus and 118-bus systems it is the most accurate estimator that neither optimizes per scenario nor uses state labels, except on the 9-bus system at large model error, where a tuned Kalman smoother is better.
Interpretation
Dynamic state estimation can now produce the rotor-angle and speed trajectories of every machine, including those without a phasor measurement unit, in one forward pass from sparse, intermittent phasor measurements at a few generator buses. Model-based methods need an accurate model and enough measurements to determine the state, and moving-horizon estimation must solve a new optimization for every estimate; purely data-driven methods depend on state labels that field data do not provide. On the 9-bus, Kundur two-area, IEEE 39-bus and IEEE 118-bus systems, the operator is the most accurate estimator that neither optimizes per scenario nor uses state labels, exceeded only on the 9-bus system at large model error by a tuned Kalman smoother.
The measurement window is encoded into one latent vector per machine, and a physics decoder integrates the swing equations from a learned initial state to produce the whole trajectory. The hard variant integrates only the nominal model, so its output always obeys the model; the soft variant adds a small learned correction to the speed equation to absorb the mismatch between nominal and true parameters. On the 9-bus system, hard-constrained fitting is almost exact with a perfect model and both operators are slightly less accurate; as model error grows, the soft correction absorbs error the hard fit cannot, tying the smoother up to a stated error level.
When a window carries several phasor measurement units, training randomly hides one measured machine from the input while keeping its samples in the loss, so the operator learns to predict machines it does not see. Through the coupling of the trajectory integration alone, some unmeasured initial states barely affect the measurement residual, and masked training constrains exactly those combinations. On the 118-bus system, masking lowers the error of both operators significantly, by about a tenth, and makes the soft variant significantly more accurate than the hard one.
Perspective
Validation is entirely simulation-based, covering the 9-bus, Kundur two-area, 39-bus and 118-bus systems plus a detailed machine model with an exciter and automatic voltage regulator; in the detailed model the machine is observed only through its terminal voltage phasor.
Amortized inference is tuned to the training measurement density and loses ground at dropout rates not seen in training; masked training needs at least two phasor measurement units per window and varying placements, and a machine that is never measured is never a target; topology errors, false-data injection and model error in the detailed model are not studied; the estimates carry no calibrated uncertainty.