Analysis of the Leakage Queue: A Queueing Model for Energy Storage Systems with Self-discharge

Energy storage is a crucial component of the smart grid, since it provides the ability to buffer transient fluctuations of the energy supply from renewable sources. Even without a load, energy storage systems experience a reduction of the stored energy through self-discharge. In some storage technologies, the rate of self-discharge can exceed 50% of the stored energy per day. In this paper, we investigate the self-discharge phenomenon in energy storage using a queueing system model, which we refer to as leakage queue. When the average net charge is positive, we discover that the leakage queue operates in one of two regimes: a leakage-dominated regime and a capacity-dominated regime. We find that in the leakage-dominated regime, the stored energy stabilizes at a point that is below the storage capacity. Under suitable independence assumptions for energy supply and demand, the stored energy in this regime closely follows a normal distribution. We present two methods for computing probabilities of underflow and overflow at a leakage queue. The methods are validated in a numerical example where the energy supply resembles a wind energy source.