Capturing State‐of‐Charge‐Dependent Efficiency in Battery Energy Storage Systems: A Bernoulli Model Approach

Accurate modelling of battery energy storage systems (BESSs) is critical for optimising their integration into power grids with high penetration of renewable energy. Conventional stochastic models often assume constant charging and discharging efficiencies, an oversimplification that neglects the significant dependence of battery performance on its state-of-charge (SOC). This paper introduces a novel analytical model for BESS based on a discrete-time discrete-state Markov chain that explicitly incorporates SOC-dependent efficiencies and charge acceptance limitations. The BESS is modelled as a finite-state buffer subject to stochastic energy inflows and outflows, where the transition probabilities are functions of the current SOC. We derive a closed-form analytical solution for the steady-state probability distribution of the battery's charge levels. Through an illustrative numerical study, the comparative analysis demonstrates that, when charge acceptance is limited at high SOC (e.g., for State Of Health preservation), our model predicts a significant shift in operational behaviour compared to ideal or constant-efficiency models. The system's probability mass tends to concentrate in an intermediate range, a behaviour that simpler models often overlook. This has significant implications for BESS sizing and control strategies, indicating that the selected operating strategy greatly affects the appropriate usable capacity. The proposed model provides a computationally efficient and more realistic framework for analysing these dependencies.