# Electro-Thermal and Aging Lithium-Ion Cell Modelling with Application to Optimal Battery Charging
## 1 Introduction  
A long lifetime is a very important requirement for the design of electric and hybrid-electric vehicles. To meet this requirement, battery lifetime must be optimized. Discharge phases are not considered here, as they mainly depend on user demands. However, during charging it is simultaneously possible to optimize the charging time and the battery degradation. It is crucial to monitor battery degradation during charging, as hard charging profiles are usually used to quickly arrive at the desired State of Charge (SOC). With such charging profiles, battery internal temperatures, and thus the rate of unwanted side reactions, increase. Lithium ions are consumed by side reactions and can thus no longer participate in the intercalation process. This process causes the aging of the battery and hence poor performance. A fast charging controller can be implemented to deal with this problem. Using information on the battery state, through direct measurement and using a model for some internal variables that cannot be measured, the charging controller can be used to adjust operating limitation to prevent ageing. The battery model solves the compromise between accuracy (predominant electrochemical phenomena must be taken into account) and parsimony (efforts for model parameterization and simulation must be limited). Such a compromise can be reached using the single particle fractional model that was introduced in [1], or in [2; 3] for simplified versions. These models involve fractional transfer functions that are particularly efficient at modelling the diffusion phenomena that take place in batteries [2] and also in several other electrochemical devices. Several papers have demonstrated the efficiency of these models [3; 4; 5], particularly when they are used for charge estimation. Extensions of these models to take into account cell aging and cell thermal behaviour can be found in [6]. Compared to other models proposed in the literature, the model in [6] is a physics-based model that takes into account degradation mechanisms in terms of temperature, SOC and C-rate (discharge or charge current normalised by the cell capacity), thus capturing experimental measures well. Several control strategies have been proposed for fast charging control taking battery state into account. The strategies can be classified into two categories: open-loop and closed-loop strategies. Model based open-loop optimal control has been considered in several papers. In [7], a Taguchi type algorithm was used to compute an optimal charging profile that minimises a cost function involving energy efficiency, charging time and temperature variation. In this approach, the model used is an Equivalent Circuit Model (ECM), which does not take cell degradations into account. In [8], an optimal open loop current profile, analytically defined, was computed using an optimization problem that minimizes a multi-objective function involving energy loss, temperature rise and charging time to charge. Such an approach is also based on an ECM. In [9], a more complex battery ECM that takes into account aging and thermal behavior was implemented. It was used to solve an open-loop multi-objective optimization problem based on a weighted sum of charging time and aging. The open-loop approaches previously discussed suffer from a high sensitivity to parametric uncertainties coming from the approximations made to derive a battery model. To overcome these modelling errors, a closed loop approach is required. Some closed-loop optimal charging strategies have been proposed in the literature. In [10], charging time was minimised using a feedback controller. Excessive anode surface concentration and the increase of temperature were assumed to be the reason for aging, and were thus considered as optimisation constraints. A physics-based electro-thermal model was used for the battery state estimation, but suffered from a poor state model state observability (mainly for the estimation of the electrodes' solid-phase concentrations).