# Electro-Thermal and Aging Lithium-Ion Cell Modelling with Application to Optimal Battery Charging
## 1 Introduction
Extensive work has been performed to address optimal charging for aging minimization with Model Predictive Control (MPC) for various application fields. An aging-aware predictive control strategy of PV-battery was presented in [11]. In this work, a convex MPC problem was solved to seek the optimal balance between the building utility cost and the battery life-cycle cost. In [12], a MPC scheme based on a linearized version of a Pseudo-two Dimensional (P2D) model was proposed in order to track a reference value of the SOC, while taking into account the aging dynamics of the system, as well as temperature and voltage constraints. Ref. [13] presents an MPC application to minimize the charging time of a lithium-ion battery subject to electrochemical and thermal constraints. The satisfaction of these constraints ensures that the battery degradation is minimized, or at least mitigated. A new methodology for battery charging control enabling an optimal tradeoff between the charging time and battery state-of-health (SOH) was also proposed in [14]. Using a model reduction approach, a physics-based low-order battery model was first proposed and used to formulate a model-based charging strategy. The optimal fast charging problem was formulated in the framework of tracking MPC. In [15], a state-feedback nonlinear MPC approach was introduced, with the objective of minimising charging time. Constraints on the temperature and on the Solid Electrolyte Interphase (SEI) overpotential (assumed to be the causes of aging) were also taken into account. An MPC strategy was also used in [16] for charging control. A genetic algorithm was combined with MPC for charging time minimisation and to keep the battery temperature above a fixed limit to reduce aging. In [17], a nonlinear MPC was used for the energy management of a power-split hybrid electric vehicle to improve battery aging while maintaining the fuel economy at a reasonable level. In all these closed loop charging strategies, in spite of their valuable contribution, the MPC approach does not ensure robustness with respect to rapid variations in state variables (for instance, anode potential). As a battery is a highly nonlinear system, any linear approximation depends strongly on the signal levels (states and inputs). With MPC, the model parameters can be time-varying. However, the control design is based on the nominal plant and needs information on the system disturbances and on the exact dynamic behaviour in each control sequence. This leads to a lack of efficiency in relation to uncertainties and disturbances. Fast charging closed loop strategies also suffer from another defect. No study considers the battery aging produced by the magnitude of the charging current applied. This magnitude is, however, known to greatly aggravate battery aging. Referring to this state of the art in the field of lithium-ion fast charging strategies and using a physics-based model of the cell, this paper addresses the following main challenges:  
* the design of a charging strategy that permits fast charging and a minimisation of aging resulting from battery temperature increase and large current magnitude,
* the robustness of the strategy with respect to large dynamic behaviour variations. The paper is organized as follows. Section 2 describes the electro-thermal aging model that will be used in the sequel for the design of an optimal fast charging strategy. The optimal charging problem for a Li-ion battery of PHEVs (Plug-in Hybrid Electric Vehicles) is formulated in Section 3. An example of an optimal current profile and the corresponding battery model response are described in Section 4. The optimal charging problem formulated in Section 3 is solved in Section 5 with two imbricated control loops that involve robust controllers. The design of these controllers is based on a family of models that capture the nonlinear behaviour of the cell around several operating points. This family of models is computed in Section 4.2.