Second-order variational analysis of PV–battery energy management using jacobi equations

Abstract

The increasing penetration of photovoltaic (PV) generation requires energy management strategies for PV–battery systems that are not only optimal but also stable and robust under variable solar generation and load demand. Classical optimal control approaches based on first-order optimality conditions ensure stationarity of solutions but provide limited observation into local stability and sensitivity to operational perturbations. This paper introduces a second-order variational framework based on Jacobi equations to analyze and design optimal PV–battery energy management trajectories. The proposed methodology quantifies state-of-charge (SOC) trajectory stability, explicitly identifies conjugate points that signal loss of local optimality, and characterizes time-dependent sensitivity to PV and load fluctuations through Jacobi fields. Numerical experiments on a representative 24-hour PV–battery system demonstrate the practical effectiveness of the approach, revealing critical periods of vulnerability and providing quantitative guidance for battery sizing, control-weight selection, and predictive operational planning. Results show that incorporating second-order optimality conditions enables rigorous stability verification and enhanced robustness compared with classical first-order methods. By extending conventional optimal control frameworks with stability-aware analysis, this work provides a mathematically grounded and practically relevant foundation for resilient energy management in renewable microgrids and residential PV–battery systems