Abstract
This document presents a rigorous and self-contained mathematical framework for the application of adaptive resonant energy recovery principles to conventional alternating-current electrical power distribution systems. We identify three distinct hierarchical layers at which frequency mismatch between the system voltage and current waveforms produces measurable energy waste. At the \textbf{load compensation layer}, we prove the Frequency Mismatch Compensation Theorem, establishing that the steady-state power factor error of a switched shunt compensator is bounded above by a Lorentzian function $\Delta\text{PF} \le [1+Q_{\text{comp}}^2(\Gamma-\Gamma^{-1})^2]^{-1}$, where $Q_{\text{comp}}$ is the compensator quality factor and $\Gamma$ is the dimensionless ratio of the grid's natural resonant frequency at the point of common coupling to the effective compensation switching frequency. At the harmonic mitigation layer, we prove the Harmonic Energy Recovery Theorem, demonstrating that a nonlinear ferroresonant tank circuit incorporating a saturable-core inductor can recover a fraction $\eta_{\text{rec}} \ge 1 - [1+Q_{\text{tank}}^2(\omega_h/\omega_0-\omega_0/\omega_h)^2]^{-1}$ of the incident harmonic energy, rather than dissipating it resistively as conventional passive and active filters do. At the \textbf{distribution feeder layer}, we formulate the Resonant Distribution Feeder Optimization (RDFO) problem, prove its NP-hardness via a polynomial-time reduction from 3-Partition (Theorem~\ref{thm:rdf_np}), and provide a polynomial-time approximation algorithm with a proven asymptotic competitive ratio of $\frac32+\epsilon$ (Theorem~\ref{thm:rdf_approx}). Analytical projections derived from the proved efficiency bounds indicate that the integrated framework can reduce reactive power losses by 60--85\% and recover 40--70\% of harmonic energy currently dissipated as heat in industrial and commercial distribution systems. All theorems are proved from first principles under explicitly stated assumptions.