- Researcher: Valentin Sulzer
- Academic Supervisors: Jon Chapman, David Howey, Charles Monroe and Colin Please
- Industrial Supervisor: Chris Baker-Brian

## Background

As renewable energy sources become cheaper and more efficient, they are used increasingly to take over from fossil fuels in producing electricity. Since these sources are inherently fluctuating, the supply must be smoothed out by energy storage, in particular using batteries. Lead-acid batteries are a cheap, safe, reliable and recyclable solution for off-grid energy storage, and thus are used by BBOXX, who design, manufacture and distribute solar home systems to improve energy access in the developing world.

In our work, we are developing ways to track and manage the batteries, with the aim of improving their discharge and charge efficiency, and increasing their lifetime. To do this, we are developing a mathematical model of the physical and chemical processes occurring within the battery.

## Progress

We model the batteries as one-dimensional, with the dimension of interest being across the electrodes and separator. By considering the movement of electrically charged ions in the electrolyte, the reactions occurring in the porous electrodes, and the distribution of current across the system, we develop a homogenised system of Differential-Algebraic Equations (DAEs) known as the Newman model.

We solve the Newman model with a hierarchy of methods, from fastest but least accurate to slowest but most accurate: quasi-static, composite and numerical.

To solve the model numerically, we transform the DAE system into a system of Partial Differential Equations (PDEs), which we can solve numerically using the Method of Lines: we first discretise in space with the Finite Volume Method, and then solve the resulting system of Ordinary Differential Equations (ODEs). This takes roughly 10 seconds to solve for a single discharge.

To obtain a more computationally efficient solution and improve our understanding of the model, we note that the “Damköhler number”, Da (the ratio between the reaction rate and diffusion rate) is small at most practical current values. Searching for an asymptotic expansion of the solution in powers of Da gives a hierarchy of solutions with increasing accuracy, each of which is quasi-static and can be solved analytically. Thus the solution is very fast to compute (0.01 seconds for a single discharge). Further, this solution gives us a greater understanding of the model: at leading order, the voltage is the “open-circuit voltage” minus voltage drops due to the reactions in the electrodes. Voltage drops due to non-uniformities in the electrolyte concentration and current distribution only come in at first order.

The quasi-static solution is accurate for most of the discharge profile, but does not capture diffusion transients that occur when there is a jump in the current. To capture such transients, we develop a composite solution. At leading order, our composite solution is identical to the quasi-static solution, but at first order we solve a single linear diffusion equation for the concentration, and thus take into account diffusion transients while maintaining a small solution time (0.5 seconds for a single discharge).

We have evaluated our model by comparing it to current and voltage data from a series of constant-current discharges provided by BBOXX: the current data is used as an input, and we compare the voltage output by our model against the real voltage data. We perform a parameter fit using Matlab, and find very good agreement between our model and the data, as shown in Figure 1 (below).

*Figure 1: Model fit to BBOXX data. Circles are data and lines are model; different colours represent different discharges at different currents (0.5A - 3A).*

## Future Work

In the immediate future, we will extend our system to model charging behaviour, in particular by including side reactions such as hydrolysis. We will also solve our model in a two-dimensional setting, with the second dimension being vertically up the electrodes, to determine whether any additional effects occur that are not captured by the one-dimensional model.

Later on, we will include degradation effects in order to model the State of Health of the batteries. Finally, if time allows, we will develop control strategies to determine the State of Charge and State of Health of the batteries in real time.