A Mathematical Model for Managing Renewable Energy Fluctuations and Automated Battery Activation - Final version

This technical note introduces a novel mathematical stability index (𝚽) designed for real-time monitoring of phase shifts and wave stability in inverter-based renewable energy grids. By establishing a reference state of dynamic equilibrium from harmonic oscillations and normalising derivative perturbations, the proposed model isolates structural wave consistency independently of external amplitude fluctuations. The mathematical framework is validated across two bounds: an ideal complex exponential baseline where 𝚽 = 1, and real-time oscillatory dynamics where the index maps into a localized response, exhibiting a selective sensitivity range (𝟎. 𝟎𝟎𝟎𝟒𝟐° ≤ 𝚫𝜽 ≤ 𝟒. 𝟎𝟓°). Furthermore, the algorithm inherently achieves seamless boundary continuity (phase unwrapping) from 360° to 0° without requiring conditional runtime code. To ensure high computational efficiency and noise resilience for practical microgrid deployment, we outline an embedded hardware implementation pipeline utilizing a Look-Up Table (LUT) with a customized angular offset (epsilon), coupled with a Kalman Filter and time-debouncing thresholds for automated, transient-proof battery storage activation