A fractional-order model for calendar aging with dynamic storage conditions


The invention of lithium-ion (Li-ion) batteries is a big milestone in electrochemical energy storage. Today, Li-ion batteries are increasingly being used in different renewable energy applications and electric cars. Given their importance in current and future applications, performance enhancement, especially their long-term health, has drawn significant research attention. This requires a thorough understanding of their aging characteristics.

Li-ion battery aging is broadly classified into two processes, namely, calendar aging and cycle aging. The former describes the degradation of battery performance with time in the absence of electric current, while the latter describes the degradation associated with the applied charge and discharge current. In some cases, however, both aging may occur simultaneously. Whereas there are no accurate approaches for quantifying the contribution of each to overall battery aging, the existing approaches require modeling calendar aging under dynamic conditions during cycle operation.

The state of charge (SOC), temperature, and structure and chemical composition of the electrodes are the main factors influencing calendar aging. Presently, the battery aging phenomena are modeled using two types of models: physical and empirical models. Laboratory experiments involving calendar aging are conducted under well-controlled static conditions, where SOC and ambient temperature are generally maintained except for short duration of time when resistance changes and capacity are being measured. However, in practical applications, batteries operate under non-static conditions.

A number of models have been developed to predict the calendar life of Li-ion batteries under dynamic storage conditions. Among them, semi-empirical modeling is commonly used for this purpose. To improve the effectiveness of existing models, developing effective and robust models is crucial. Consequently, fractional calculus order models can be used to represent the power law behavior of physical systems. This concept could be extended to modeling calendar life of batteries because calendar aging exhibits a power law dependence over time.

On this account, Professor Juan Antonio López-Villanueva, Dr. Pablo Rodríguez Iturriaga and Professor Salvador Rodríguez-Bolívar from the University of Granada in Spain proposed a fractional-order model for forecasting calendar aging in Li-ion batteries under dynamic storage conditions. They commenced their work by reviewing previous models to determine calendar aging-induced capacity loss under variable SOC and temperature conditions according to the power-law behavior. Their work is published in the Journal of Energy Storage.

The authors showed that the proposed model successfully overcame most limitations of the previous models, and they could predict features such as non-monotonic behaviors induced by significant changes in temperature and SOC. Compared to the existing models, it provided comparable or even better accuracy. The new model was validated by comparing its predictions to experimental results. This model had two main advantages. First, it did produce not only similar results under static conditions as other semiempirical models but also showed interesting qualitative and quantitative differences and varying storage conditions. Second, the results confirmed, using known facts like the fractional order differential equations, that the calendar aging depends on the whole battery history.

In summary, a fractional calculus-based model for predicting the capacity fading of Li-ion batteries due to calendar aging under dynamic conditions was presented. This model assumed that calendar aging depends on the temperature and SOC during battery storage and it accounted for experimental features not predicted using previous models. In a statement to Advances in Engineering, Professor Juan Antonio López-Villanueva said their findings provided more insights that would contribute to the expanded application of fraction calculus in modeling battery aging and overall performance.


López-Villanueva, J. A., Iturriaga, P. R., & Rodríguez-Bolívar, S. (2022). A fractional-order model for calendar aging with dynamic storage conditions. Journal of Energy Storage, 50, 104537.

Go To Journal of Energy Storage

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