Branch-Level Energy Localization in Three-Phase Loads: Resolving Indeterminacy in Time-Domain
For power systems engineers and researchers, this framework provides a rigorous method to localize energy dissipation and storage in three-phase loads, complementing existing standards like IEEE Std. 1459 and resolving long-standing ambiguities.
The paper develops a branch-level energy localization framework for three-phase loads that uniquely decomposes instantaneous terminal power into Joule dissipation and stored-energy rates per branch, resolving indeterminacies in time-domain analysis. It demonstrates the framework on six test cases, including the de Leon–Cohen open-phase paradox and delta-versus-wye indeterminacy, showing that apparent paradoxes vanish when the question is posed precisely.
This paper develops a branch-level energy-localization framework for three-phase loads. The instantaneous terminal power of an admissible lumped equivalent is decomposed uniquely as Joule dissipation plus magnetic and electric stored-energy rates, branch by branch. Three formal results are established: a Branch-Level Localization Theorem (uniqueness given an admissible topology); a Topology-Indeterminacy Theorem (multiple admissible topologies reproduce identical terminal data with distinct localizations); and a Generalized Energetic Duality Theorem that organizes classical electrical dualities (Norton-Thevenin, series--parallel, L vs C, R vs G) as restrictions to Linear Time Invariant (LTI) sinusoidal regimes of a single time-domain principle in which constant-parameter equivalence is replaced by time-varying parameters. The framework is exercised on six test cases including the de Leon--Cohen open-phase paradox, switched-resistive loads, three-wire delta-versus-wye-virtual indeterminacy, fluctuating-phase loads, and a four-wire nonlinear load with hysteretic, linear, and switched branches. The framework is positioned as complementary to IEEE Std. 1459, CPC, instantaneous p-q, and Fryze-Buchholz-Depenbrock: each answers a different question, and the apparent paradoxes vanish once the question is posed precisely.