Minimization of circuit power losses using a variational approach with time dependent constraints

This work investigates the minimization of ohmic losses of generic electrical circuits during charge and discharge processes using variational techniques. The optimal charge/discharge laws are obtained by solving the Euler-Lagrange equations, based on the current or output voltage, subjected to time dependent constraints. Fora given electrical system, these restrictions correspond to the conservation of charge and energy, which must be imposed at each time step, resulting in a constrained Lagrangian. This methodology addresses a significant gap in previous works, that either: (i) focused on simple circuits where the inclusion of constraints was not needed, since they could be removed from the formulation; (ii) or analyzed more complex systems without enforcing the necessary conservation laws, resulting in non-physical solutions. To verify the proposed methodology, analytical and numerical solutions are obtained for several circuits, with an increasing level of complexity, including the dependence of the capacitance on the voltage. The analyzed circuits have been studied in recent works and the optimal laws derived in this paper always show better performance than the usual reported constant current strategies, leading to significant energy savings.