### Minimum-time optimal charging of lithiumion batteries

We have been working on a master thesis for Combine with the goal of finding a minimum-time optimal charging strategy for lithium-ion batteries. There is a number of things to take into consideration while charging to make sure that the battery isn't negatively effected. As most of the harmful effects can't be measured directly on a physical battery we used two different models, one for optimization and one for evaluation, which allowed us to see what happened inside the battery in detail.

The first model was a simple Equivalent Circuit Model (ECM) whose parameters first were identified to fit the model used for evaluation and then the ECM was used to perform the optimization on. The circuit can be seen in figure 1. The model used for evaluation was an advanced Electrochemical Model (EM) implemented in a framework called LIONSIMBA, which models the chemical reactions inside the battery with partial differential equations and therefore isn’t suitable for optimal control. The method used to fit the ECM to the EM could also be applied to fit the ECM to a physical battery, making it useful in real world applications as well.

Figure 1: ECM of a lithium-ion battery cell

The system of equations seen in equation 1 shows what the dynamics of the ECM looks like as well as the models used for temperature and State of Charge (SoC) estimation.

Since the goal was to charge as fast as possible, we wanted to minimize the charging time, which was done through minimum-time optimization. One way to solve minimum-time optimization problems, also the one used by us, can be seen in equation 2.

As there are a number of harmful phenomenon’s that can occur in a battery, additional constraints were needed as well. Two of the most important effects are lithium plating and overcharging, both of which we take into consideration. Both lead to decreased capacity, increased internal resistance and a higher rate of heat generation. It is known that there is some kind of connection between these effects and the voltage over the RC-pairs, vs, however not linear. This is why we applied a constraint to this voltage because without it, the solver would only take the temperature constraint into consideration which would lead to damaging the battery.

The EM allows us to see what happens inside of the battery in regard to the harmful effects when we input the current achieved through solving the optimization problem. One of the evaluated cases can be seen in figure, where both the result from the ECM and the EM are included. This case is for charging from 20-80% at an initial temperature of 15 C.

Figure 2: Results and model comparison for the EM and ECM.

The top left plot in the figure above shows the lithium plating voltage which has to be kept above 0 and is controlled by the linear constraint put on vs, which is also shown. The top right plot shows if the battery is being overcharged, which also controlled by the constraint on vs. The bottom left plot shows the temperature and the bottom right one shows the current which is the result from solving the optimization problem.

The next thing we did was to compare our fast charging to a conventional charging method, namely the constant current-constant voltage (CC-CV) method. The constant current part was maximized for all cases to reach the same maximum values to make the comparison fair. The following plots are the same ones as above but compares our fast charging with CC-CV charging instead, showing that the fast charging is 22% faster and does not come as close to zero in terms of lithium plating voltage as the CC-CV method, although it has a higher average temperature due to the higher average input current.

Figure 3: Comparison between the optimized fast charging and CC-CV charging.

A summary of the charging times and the improvement over CC-CV can be seen in table 1 & 2 for charging from 20-80% and 10-95% for different temperatures respectively.

**Conclusion
**By performing optimization on an equivalent circuit model of a lithium-ion cell simulated in LIONSIMBA it was possible to achieve charging times that for some cases were up to 40% faster than with traditional CC-CV charging while still keeping the battery within the same constraints. To control the charging and avoid both lithium plating and overcharging a linear constraint was applied to the voltage over the two RC-pairs in the equivalent circuit model. The result clearly shows that the method has potential and that it should be possible to apply it on a physical battery even though it will be more difficult to choose constraints for the optimization.