Owing to their high energy density, light weight, and strong durability, lithium-ion batteries are ideal for applications such as electric vehicles and renewable energy storage. As the energy grid becomes increasingly electrified, they are now playing an increasingly important role in modern energy infrastructures.

In order to maintain safety and efficiency, operation of such batteries is controlled by a battery management system, or BMS. Among the most important roles of a BMS is to assess various ‘battery states’, such as the energy remaining in the battery, its overall condition and capacity compared with its original state, and the maximum amount of power it can safely deliver or accept.

By monitoring these states in real time, the BMS can maintain the battery within safe temperature limits, regulate discharge and charge rates to prevent overloading or damaging the battery, and ensure that the charge levels remain equal across all cells in a battery pack.

Although these functions are essential for most battery designs, there are multiple ways to design a BMS. One of the most precise approaches relies on physics-based models, which use detailed mathematical frameworks to describe the processes unfolding inside the battery. Yet although this approach is highly accurate, the vast complexity involved in these processes require large amounts of computing power to simulate, making it impractical for real-time BMSs applications.

The main solution to this setback is to use simplified models with lower computational requirements, at the cost of some accuracy in simulating physical processes. However, these simplified models themselves can be implemented in various ways, each offering a different trade-off between computational speed and accuracy. This variability makes it difficult for battery operators to decide which models are best suited to their needs in practical BMSs.

In their study, Luc Raijmakers and his team carried out an in-depth comparison of simplified, physics-based lithium-ion battery models to determine which models offer the best possible combination of high computational speed and relatively good accuracy.

To make their comparisons, Raijmakers’ team focused on several possible simplifications to the Doyle-Fuller-Newman, or DFN model: a physics-based model that describes key processes within the battery, including the diffusion of lithium ions within the solid electrodes and liquid electrolyte, the charge transfer reactions, and the overall transport of charge and mass throughout the cell.

While the DFN model provides detailed insights into battery behaviour, it is computationally expensive. One way to overcome this challenge is to simplify the DFN using Reduced-Order Models, or ROMs: simplified mathematical representations of battery dynamics that can simulate the behaviour of lithium-ion batteries with significantly reduced complexity. In their study, Raijmakers and his team considered two different ROMs.

Firstly, the Single Particle Model, or SPM, simplifies the battery’s dynamics by representing each of its electrodes as a single spherical, particle, while ignoring the transport of ions within the electrolyte. With such a high level of simplification, computational speeds can be extremely fast. The SPM is fairly accurate when simulating lower rates of discharge and charge, where processes in the electrolyte are less important. Yet at higher rates, when electrolyte effects become more pronounced, the model becomes less accurate.

Alternatively, the extended Single Particle Model, or ESPM, is a modified form of the SPM that also accounts for electrolyte dynamics taking place between the battery. As a result, the ESPM significantly enhances simulation accuracy compared with the SPM. While this improvement comes at the cost of slightly higher computational demands with respect to the SPM, the ESPM still offers a reasonable balance between computational speed and accuracy.

In addition to ROMs, Raijmakers’ team investigated a method for mathematically approximating the relationships that describe the diffusion of lithium ions within the particles.

These approximations are based on polynomial functions: equations made up of constants, variables, and exponents, which can be added or subtracted together to fit a curve that closely defines the relationships between variables, making them both simple and computationally efficient. By incorporating more parameters into the polynomial functions, physical processes can be approximated more accurately, though this comes at the cost of increased complexity.

Alternatively, Padé approximations use ratios between two different polynomial functions. This allows them to approximate the physical processes more closely while using fewer parameters than pure polynomials. As a result, Padé approximations offer a more efficient way to model the relationships between variables.

To compare the advantages of these simplification methods, Raijmakers’ team assessed their performance when simulating the behaviour of high-energy and high-power batteries under different rates of discharge and charge, using simulations based on the full DFN model as a benchmark.

The results showed that, among the two ROMs, the ESPM consistently offered the best combination of high computational speed and relatively high accuracy under most rates of discharge and charge when compared with the DFN model.

For the approximation method, the team found that higher-order polynomial functions, which incorporated more parameters into their equations, provided the biggest advantage in terms of computational speed, while still offering relatively good accuracy. The highest accuracy was achieved when using Padé approximations.

Altogether, the team’s results provide valuable insights into which approaches are best suited for simplifying physics-based models of lithium-ion batteries under different discharge and charge rates. These findings could help improve battery management systems, allowing them to operate more safely and efficiently. By using the best model for specific conditions, operators can ensure optimal performance and longevity for lithium-ion batteries.

Ultimately, the insights gained from this study could play a crucial role in accelerating the practical rollout of lithium-ion batteries. This, in turn, would contribute to the development of a more efficient and sustainable energy grid, helping to meet the increasing demand for cleaner, more reliable energy storage solutions.