Lithium-Ion Battery Modeling: When Physics Joins Hands with Machine Learning

By Huazhen Fang and Hao Tu

Acclaimed as “the technology of our time” [1], lithium-ion batteries impact numerous aspects of contemporary life and industry. They power billions of consumer electronics devices; enable electrified transportation, smart grids, and the adoption of renewable energy; and drive the world towards a decarbonized energy future. Their outstanding features—including high energy/power density, high voltage, lack of memory effect, low self-discharge rates, and long service life—make them quite popular. Yet with the surging use of lithium-ion batteries comes an ever-growing demand for higher power performance and safety — a trend that is continuously reinforced by many well-publicized battery fire incidents with cell phones, commercial electric vehicle models, and Boeing 787 Dreamliners. The optimal, safe operation of lithium-ion batteries has thus become an essential challenge, involving a range of tasks like energy state monitoring, fast charging control, degradation analysis, and fault diagnostics. These tasks unanimously rely on accurate and computationally efficient dynamic models of lithium-ion batteries.

Figure 1. Comparison of different approaches to lithium-ion battery modeling. Figure courtesy of Huazhen Fang and Hao Tu, with “battery superman” image courtesy of PNG EGG.

There are two main types of physics-based lithium-ion battery models: (i) electrochemical models, which use electrochemical principles to describe electrochemical reactions, lithium-ion diffusion, and current and potential changes in a cell’s electrodes and electrolyte; and (ii) equivalent circuit models, which are circuit analogs that are composed of resistors, capacitors, and voltage sources to simulate a cell’s electrical dynamics in a physically interpretable way. However, both types of models woefully struggle between computational complexity and predictive accuracy more than usual. In general, electrochemical models can provide fine-grained descriptions of processes that underlie charging/discharging to achieve decent accuracy — at the cost of heavy computation. Equivalent circuit models are just the opposite. To further complicate the situation, the dynamics of lithium-ion batteries are subject to much complexity and uncertainty. For example, ambient temperatures that are too high or too low, as well as currents that are too large, will trigger a large number of complicated side reactions. These phenomena often make accurate physical modeling elusive, and turn a lithium-ion battery cell into a “grey box”.

To mitigate the aforementioned struggles, we let physics-based models join hands with machine learning (see Figure 1). Because machine learning can powerfully extract relations from data, we want it to work together with a relatively simple, approximate physical model to make sense of battery-generated data and make predictions. Such a hybrid way of modeling yields several benefits. First, a hybrid model—if trained on rich and informative data—can attain high accuracy under uncertainty and across a broad array of operating conditions, including vastly wide current, temperature, and power load ranges. It may also preserve physical interpretability to a certain extent, thanks to the use of the physical model. Second, a hybrid model can still be satisfactorily quick despite requiring somewhat more computation than the physical model. This is true because a concise, simple machine learning model may suffice if it can effectively utilize the physical model and runs once with only fixed computational costs after being trained. Finally, plentiful open-source machine learning tools like Keras, TensorFlow, and PyTorch allow practitioners to develop hybrid models efficiently and conveniently.

Figure 2. Schematic and evaluation of hybrid modeling. 2a. A proposed structure of hybrid modeling for lithium-ion batteries. The structure combines a physical model (either the single particle model or the nonlinear double-capacitor model) with a feedforward neural network. The main characteristic lies in feeding the physical model’s state information to the neural network. 2b. Predictive accuracy comparison between the single particle model (circle) and a hybrid model that integrates the single particle model with a feedforward neural network (plus sign), relative to the truth (solid curve). The single particle model loses performance at high C rates (a measure of the current rate at which a battery is charged/discharged relative to its maximum capacity). By contrast, the hybrid model retains high accuracy from low to high C rates. Figure courtesy of Huazhen Fang and Hao Tu.

Moving forward, our notion of hybrid modeling is characterized by informing a machine learning model of a physical model’s state information. As such, the machine learning model becomes aware of the ongoing state evolution of the physical model and more effectively learns what the physical model misses in comparison to measurement data. We develop a catalog of hybrid models based on this notion [4] and highlight a structure design here. In the structure, a physical model cascades with a feedforward neural network (see Figure 2a). For the physical model, one can choose either an electrochemical model or an equivalent circuit model. Our study considers the single particle model and nonlinear double capacitor model, respectively. The former is a simple electrochemical model that represents each electrode of a battery cell as a spherical particle and describes the lithium-ion diffusion inside the particles [2]; the latter is a novel equivalent circuit model [3]. The feedforward neural network can capture the physical model’s residual error with respect to the true voltage while taking certain state variables of the physical model as part of its input. Figure 2b demonstrates that such a hybrid model is considerably more accurate than only the physical model at different current rates.

If we liken lithium-ion batteries to a grey box, then cell aging and degradation make the box even darker. This occurs because aging causes and drives model mismatch — an initially well-fitted model gradually loses its accuracy throughout a cell’s cycle life, causing a vexing issue for battery modeling and management. We further expand the hybrid modeling notion to keep the neural network informed of the cell’s state of health based on capacity fade measurements. The upgraded model in Figure 3a accurately predicts voltage under different aging conditions, as shown by the comparison in Figures 3b-3c. Figure 3 also shows that the model can even transfer its predictive capabilities between cells of the same type. Specifically, we train the model on a Samsung INR18650-25R cell and find that it can successfully predict the voltage behavior of another cell. This type of “transferrable” model can allow users to skip tedious model calibration for individual cells and facilitate real-world battery model deployment.

Figure 3. Integration of aging awareness into hybrid modeling and experimental validation. 3a. Structure of an aging-aware hybrid model in which a feedforward neural network works with the nonlinear double-capacitor model and takes capacity fade (i.e., state of health) as an input variable. 3b. and 3c. Comparison of voltage prediction errors by the original nonlinear double-capacitor model (3b) and the hybrid model (3c) under different state-of-health conditions and current profiles. The hybrid model consistently presents high predictive accuracy throughout the cycle life. Figure courtesy of Huazhen Fang and Hao Tu.

In summation, the ever-increasing adoption of lithium-ion batteries across various sectors presents a pressing demand for accurate and computationally efficient models. This demand has motivated us to explore interfaces between physics-based modeling and data-driven machine learning. Our exploration has led to hybrid models that are based on the concept of feeding a physical model’s state information to a machine learning model. We also note that this idea can find potential use in more scientific and engineering domains to advance the use of physics and machine learning for the modeling of other complex systems.

References
[1] A plug for the battery. (2016, January 16). The Economist. Retrieved from https://www.economist.com/leaders/2016/01/16/a-plug-for-the-battery.
[2] Guo, M., Sikha, G., & White, R.E. (2011). Single-particle model for a lithium-ion cell: Thermal behavior. J. Electrochem. Soc., 158(2), A122.
[3] Tian, N., Fang, H., Chen, J., & Wang, Y. (2021). Nonlinear double-capacitor model for rechargeable batteries: Modeling, identification, and validation. IEEE Trans. Control Syst. Tech., 29(1), 370–384.
[4] Tu, H., Moura, S., Wang, Y., & Fang, H. (2021). Integrating physics-based modeling with machine learning for lithium-ion batteries. Preprint, arXiv:2112.12979.

	Huazhen Fang is an associate professor in the Department of Mechanical Engineering at the University of Kansas. His research interests lie in control and automation of complex systems, with application to energy management and robotics.
	Hao Tu is a Ph.D. student in mechanical engineering at the University of Kansas. His research focuses on advanced battery management at the intersections of modeling, control theory, and machine learning.