⚡ electrical engineering

A novel hybrid model to describe hysteresis in Li-Ion batteries of electric vehicles

This paper proposes a novel, computationally simple hybrid model combining an asymmetric hysteresis approach with the Bouc-Wen model to accurately predict and describe the significant hysteresis losses in Li-ion batteries for electric vehicles, thereby enabling more precise State of Charge (SoC) estimation.

Original authors: Khogesh K Rathore, Saurabh Biswas

Published 2026-09-02

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CC BY 4.0

Original authors: Khogesh K Rathore, Saurabh Biswas

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Electric vehicles have become a familiar sight on roads worldwide, driven by a global push to reduce the air pollution caused by traditional gasoline engines. At the heart of every electric car lies a battery, a complex chemical system that stores energy and releases it to power the vehicle. Among the various types available, lithium-ion batteries are the standard choice because they are efficient, durable, and easy to recharge. However, these batteries are not perfect containers of energy; they possess a subtle but significant flaw known as hysteresis. This phenomenon occurs when the battery's voltage—the electrical pressure that pushes current through a circuit—does not follow the same path when the battery is being charged as it does when it is being discharged. For a specific amount of stored energy, the voltage is slightly higher during charging than during discharging. This discrepancy creates a loop in the data, making it difficult to know exactly how much energy remains in the battery at any given moment.

Accurately tracking this remaining energy, known as the state of charge, is critical for the safety and longevity of an electric vehicle. If a car's computer cannot precisely determine how full the battery is, it might cut power unexpectedly or, worse, allow the battery to be overcharged, leading to damage or fire. To solve this, engineers rely on battery management systems that use mathematical models to predict the battery's behavior. Yet, existing models often struggle with the specific shape of the hysteresis loop in lithium-ion batteries. These loops are not symmetrical; they are lopsided and complex, defying the simple, balanced shapes that many traditional equations are designed to describe. This complexity has made it difficult to create a model that is both accurate enough to be useful and simple enough to run quickly on a car's computer.

In a recent study, researchers Khogesh K. Rathore and Saurabh Biswas proposed a new way to tackle this problem by combining two different mathematical approaches into a single, hybrid model. Rather than trying to force the battery's behavior into one rigid shape, they merged a well-known model used for mechanical systems with a newer model designed to handle asymmetry. The first part of their combination, the Bouc-Wen model, is a tool originally developed to describe how materials deform and recover under stress, such as metal bending. The second part is a specialized model that accounts for the uneven nature of the hysteresis loop, specifically the way the curve pinches and shifts differently depending on whether the battery is gaining or losing power. By adding the outputs of these two models together, the researchers created a new framework that could mimic the unique, lopsided loops seen in real batteries.

To test whether this new approach worked, the team applied it to four different sets of experimental data collected from previous studies on lithium-ion batteries. These data sets represented real-world charging and discharging cycles. Using a standard computer method to adjust the model's settings, the researchers fine-tuned the hybrid system until its predictions matched the experimental data as closely as possible. The results were striking. The new model was able to trace the complex, asymmetrical loops of all four test cases with high precision. The maximum percentage errors in fitting were reported as 2.7%, 1.2%, 3.9%, and 0.9% across the different cases. Crucially, the researchers achieved this accuracy using the exact same set of nine internal parameters for every single test, proving that the model was robust and not just a lucky fit for one specific battery.

The study demonstrates that this hybrid approach offers a practical solution to a persistent engineering challenge. Unlike many existing models that are either too complicated to run in real-time or too simple to capture the battery's true behavior, this new method strikes a balance. It is computationally simple, meaning it requires very little processing power, yet it captures the key features of the hysteresis loop, including its asymmetry. The researchers note that while their model successfully describes the main charging and discharging cycles, future work will be needed to refine it for more complex scenarios, such as partial charges where the battery is topped off before it is fully empty. For now, however, this work provides a clear, reliable tool that could help engineers build better battery management systems, ensuring that electric vehicles remain safe, efficient, and ready to go.

Technical Summary: A Novel Hybrid Model to Describe Hysteresis in Li-Ion Batteries

Problem Statement
Electric vehicles (EVs) rely heavily on Lithium-ion (Li-ion) batteries, yet accurate State-of-Charge (SoC) estimation remains a bottleneck due to hysteresis losses. Hysteresis in Li-ion batteries manifests as a discrepancy where the discharge equilibrium potential is consistently lower than the charge equilibrium potential for the same SoC. This phenomenon creates asymmetric hysteresis loops, complicating the development of precise Battery Management Systems (BMS) required for cell monitoring, balancing, and safety. While existing mathematical models (electrochemical, data-driven, and equivalent circuit) exist, many are either overly complex or fail to accurately capture the specific qualitative features of Li-ion hysteresis, particularly the significant asymmetry observed in the loops.

Methodology
To address these limitations, the authors propose a computationally simple hybrid model that combines two distinct hysteresis frameworks:

The Bouc-Wen Model: A well-established model typically used for mechanical systems, defined by a differential equation involving input $x(t)$ and output $f_1(t)$ with parameters $A, \beta, \gamma,$ and $n$ .
A Scalar Asymmetric Hysteresis Model: A newer model originally developed for elastic bodies with random microcracks, defined by an internal variable $\theta(t)$ and input $x(t)$ , utilizing parameters $\kappa, A_1, \theta_a, \beta_o,$ and $\alpha$ .

The hybrid approach generates the total hysteretic output by summing the outputs of the Bouc-Wen model and the asymmetric model. The model was validated against four sets of published experimental data representing Li-ion battery charging and discharging cycles (cited from references [25], [20], [15], and [16]). The authors employed a nonlinear least squares method using MATLAB's fminsearch function to fit the model to the experimental data. Notably, the authors utilized a single, fixed set of nine fitted parameters across all four experimental cases specifically to establish the robustness of the model.

Key Contributions and Results

Novel Application: The paper demonstrates a novel application of combining the Bouc-Wen model with a specific asymmetric hysteresis model to capture Li-ion battery behavior, a combination not previously reported for this specific application.
Computational Simplicity: The proposed framework is described as computationally simple compared to existing complex models.
Accurate Fitting: The hybrid model successfully captured the key hysteretic characteristics, including the significant asymmetry of the loops, across all four experimental datasets.
Quantitative Performance: The model achieved low maximum percentage errors in fitting the experimental data, ranging from 0.9% to 3.9% across the four cases (Table 1).
Parameter Robustness: By using the same parameter set for all cases, the study demonstrates the model's robustness, showing that a consistent parameter configuration can effectively fit diverse experimental conditions.

Significance and Claims
The authors claim that their primary contribution is a "simple but novel approach" that effectively balances computational efficiency with the ability to capture the complex, nonlinear, and asymmetric nature of Li-ion battery hysteresis. They assert that while existing literature offers models that are either too complex or insufficient in capturing qualitative behaviors, this hybrid model offers a viable alternative for accurate SoC prediction.

The paper modestly frames its scope, noting that the current work focuses on major hysteresis loops. The authors state that future work will aim to extend this approach to capture minor loops resulting from partial charging and discharging cycles, acknowledging that the current model does not yet address these specific partial-cycle dynamics.

Technical Summary: A Novel Hybrid Model to Describe Hysteresis in Li-Ion Batteries

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