Eliminating Binary Variables in Battery Scheduling: LP Reformulations for Energy Arbitrage

Battery energy storage scheduling for price arbitrage is typically formulated as a Mixed-Integer Linear Program (MILP) to preclude simultaneous charging and discharging. This paper demonstrates that the MILP structure is unnecessarily complex: the original problem can be transformed into an equivalent Linear Program (LP) that solves in polynomial time. Two LP reformulations are presented. The first optimizes net energy variation under ideal efficiency, reducing the number of variables and eliminating binary constraints. The second, more realistic formulation explicitly incorporates distinct charging and discharging efficiencies while preserving a purely linear constraint set. Matrix representations and dimension analyses are provided to facilitate direct implementation with standard LP solvers. Numerical simulations confirm that both formulations produce identical net power flows under ideal conditions, and that the efficiency-aware LP correctly captures losses. Consequently, the computational burden of battery scheduling for real-time energy management is reduced from exponential to polynomial complexity without sacrificing physical accuracy. The reformulations offer a lightweight, solver-agnostic alternative to MILP for day-ahead and intraday bidding, and they lay a foundation for handling degradation, multi-battery coordination, and stochastic extensions.