Optimal Depth of Discharge for Electric Batteries with Robust Capacity-Shrinkage Estimator

A user’s benefit from the energy stored in a battery over its lifetime depends on the time-varying characteristics of the battery, which are in turn affected by the chosen usage behavior. Both the capacity shrinkage and the number of lifetime cycles are strongly impacted by the depth of discharge as a key decision variable. Available models of battery lifetime are rather complex and depend on many factors, such as temperature and other physical particularities, which complicate the user’s decision problem. We propose a simple, relatively robust approach for determining an optimal robust depth of discharge, based on a cycle-discharge curve and an unknown exponential capacity shrinkage curve. We characterize the optimal robust depth of discharge and describe the implied performance guarantees. A relative cost of robustness is obtained as the boundary of the uncertainty set varies. The method provides an example of parameter estimation based on minimizing the relative regret caused by the implied decisions. In the special case where the cycle-discharge curve is exponential as well, we find a closed-form solution.