  This paper formulates and solves a multi-objective fast charging-minimum
degradation optimal control problem (OCP) for a lithium-ion battery module made
of series-connected cells equipped with an active balancing circuitry. The
cells in the module are subject to heterogeneity induced by manufacturing
defects and non-uniform operating conditions. Each cell is expressed via a
coupled nonlinear electrochemical, thermal, and aging model and the direct
collocation approach is employed to transcribe the OCP into a nonlinear
programming problem (NLP). The proposed OCP is formulated under two different
schemes of charging operation: (i) same-charging-time (OCP-SCT) and (ii)
different-charging-time (OCP-DCT). The former assumes simultaneous charging of
all cells irrespective of their initial conditions, whereas the latter allows
for different charging times of the cells to account for heterogeneous initial
conditions. The problem is solved for a module with two series-connected cells
with intrinsic heterogeneity among them in terms of state of charge and state
of health. Results show that the OCP-DCT scheme provides more flexibility to
deal with heterogeneity, boasting of lower temperature increase, charging
current amplitudes, and degradation. Finally, comparison with the common
practice of constant current (CC) charging over a long-term cycling operation
shows that promising savings, in terms of retained capacity, are attainable
under both the control (OCP-SCT and OCP-DCT) schemes.
