# Electro-Thermal and Aging Lithium-Ion Cell Modelling with Application to Optimal Battery Charging
## 3 Trajectory Planning  
The fast charging strategy developed in this paper, which can be integrated in a battery management system (BMS), is described globally by Figure 2. In this diagram, the model presented in Section 2 is used in the battery observer to simulate the battery behavior under various operating conditions. Using measures of the cell current \(I_{Charge}\), voltage \(U_{cell}\) and temperature \(T_{cell}\), this observer estimates (among other information) the cell capacity fade \(\Delta C_{fade}\). This information is used by a fast charging controller that adapts the charge current magnitude to follow a charge profile \(I_{ref}\) while reducing the cell capacity fade during recharge. The function of the charging controller is to determine the optimal charging current profile imposed on the battery. The optimal current is that which minimizes the recharging time (therefore the highest possible) while limiting a variable upper limit on the current. This makes it possible to meet a rapid recharging target without increasing battery aging too much. The strategy adopted in this work to determine the optimal charging profile thus consists in computing offline the side reaction current before battery charging. The side reaction current changes over time and is a function of the SOC and of the charging current. If the charging process is split into a finite number of time slots, a lookup table can provide the limit on the side reaction current on each interval as a function of SOC. The side reaction current limit provided on each slot defines the side reaction current trajectory. The controller thus computes the corresponding charging current and defines in real time the optimal profile of the charging current to meet the calculated side reaction current trajectory. In practice, to acheive the optimal charging trajectory, the capacity loss during a fixed charging time \(t_{f}\) of twenty minutes is minimised under the following constraints:  
* bounds on the charging current (Ich), in order to avoid exceeding the maximum charge current limits (3.5 C) for safety reasons,
* an increase in the SOC from 5% to 80% (but other ranges of SOC variation can be defined). Denoted in the sequel is a Multi-stage Constant Current (MCC) of duration \(t_{f}\), a charge current profile made of a series of \(N\) pulses (or slots) of the same duration (\(t_{f}/N\)), in which each pulse can have a different magnitude (thus \(N\) degree of freedom in the current profile that can be optimised). An example of an MCC profile is represented in Figure 3. It is used for its flexibility in the definition of a profile in order to maximize energy efficiency and to improve battery life during fast charging [25]. The optimal control problem can thus be analytically defined by  
\[\min_{l_{ch}(t)}\int_{0}^{t_{f}}\Delta C_{loss}(t)dt \tag{20}\]  
subject to  
\[\dot{x}(t)=f(x(t),l_{ch}(t)),x(0)=x_{0} \tag{21}\]  
\[\frac{1}{3600}\int_{0}^{t_{f}}l_{ch}(t)dt\geq Q_{trans} \tag{22}\]  
\[0\leq l_{ch}(t)\leq 3.5C \tag{23}\]  
in which \(Q_{trans}\) denotes the Li-ion charges stored in (Ah), \(\Delta C_{loss}\) denotes the capacity loss (Ah), and \(t_{f}\) (s) is the charging time. Sequential Quadratic Programming (SQP) is used to solve the optimal control problem defined by Relations (20) to (23) for various operating conditions. Figure 2: Description of the implementation of the fast charging functions in a battery management system (BMS). Variations of SOC, anode potential, side reaction and capacity loss variation at 35 \({}^{\circ}\)C obtained after optimisation with only 4 slots are shown in Figure 4. These curves are compared to those obtained with a conventional Constant Current Constant Voltage (CCCV) profile. This kind of charge profile has been chosen for comparison purposes because it is very commonly used for recharging batteries (lithium ion or not). This figure shows a decrease in capacity loss close to 15% with the optimal current profile in comparison with the CCCV profile.