Geometric Time-Domain Identification of Three-Phase Load Equivalents from Terminal Measurements

This paper presents a geometric time-domain method for identifying three-phase load equivalents from instantaneous voltage and current measurements at the point of common coupling. Measured waveforms are interpreted as trajectories in Euclidean signal spaces, and load-equivalent parameters are recovered from the geometry of those trajectories. The method extends a previously published single-phase geometric identification formulation to three- and four-wire systems and places special emphasis on the three-wire case, where no neutral voltage is measured and the terminal data must satisfy coupled Kirchhoff constraints. The main advance over the earlier analytical formulation is a sampled-data implementation based on local time windows, normalized matrix equations, harmonic-projection derivative and primitive coordinates, explicit geometric identifiability tests, passivity constraints, and energy/Kirchhoff residuals. The method does not force a model when the measured trajectory lacks enough information; instead, it reports low-rank or ill-conditioned windows as low-confidence evidence. Numerical simulations with clean data, measurement noise, window-length sweeps, and sensor delay show that the method accurately identifies informative three-phase trajectories and exposes structurally degenerate cases such as pure single-frequency excitation for higher-order three-wire models. For a given admissible topology the identified circuit closes the instantaneous terminal energy balance of the measured load over the analysis window.