Optimal control of batteries with fully and partially available rechargeability

Motivated by the increasing dependence of many systems on battery energy, we study the problem of optimally controlling how to discharge and recharge a non-ideal battery so as to maximize the work it can perform over a given time period and still maintain a desired final energy level. Modeling a battery as a dynamic system, we adopt a Kinetic Battery Model (KBM) and formulate a finite-horizon optimal control problem when recharging is always feasible under the constraint that discharging and recharging cannot occur at the same time. The solution is shown to be of bang-bang type with the property that the battery is always in recharging mode during the last part of the interval. When the length of the time horizon exceeds a critical value, we also show that the optimal policy includes chattering. Numerical results are included to illustrate our analysis. We then extend the problem to settings where recharging is only occasionally feasible and show that it can be reduced to a nonlinear optimization problem which can be solved at least numerically.