Optimum Policies for an Energy Harvesting Transmitter Under Energy Storage Losses

We consider an energy harvesting network where the transmitter harvests energy from nature, and the harvested energy can be saved in an imperfect battery which suffers from charging/discharging inefficiency. In particular, when E units of energy is to be stored in the battery, only eta E units is saved and (1-eta) E is lost due to charging/discharging inefficiency, where 0 <= eta <= 1 represents the storing efficiency. We determine the optimum offline transmit power schedule for such a system for single-user and broadcast channel models, for static and fading channels, with and without a finite battery size. We show that the optimum policy is a double-threshold policy: specifically, we store energy in the battery only when the harvested energy is above an upper threshold, and retrieve energy from the battery only when the harvested energy is below a lower threshold; when the harvested energy is in between these two thresholds, we use it in its entirety in the current slot. We show that the two thresholds remain constant unless the battery is depleted or full. We provide an algorithm to determine the sequence of optimum thresholds. For the case with fading, we develop a directional water-filling algorithm which has a double-threshold structure. Finally, we formulate the online problem using dynamic programming, and numerically observe that the online policy exhibits a double-threshold structure as well.