⚡ electrical engineering

Lithium-ion cell modelling: an example of fractional order behavior modeling with distributed time delay models

This paper demonstrates that classical distributed time delay models can effectively replicate the power-law dynamics and memory effects typically associated with fractional-order models, offering a simpler and more physically interpretable alternative for applications such as lithium-ion battery cell modeling.

Original authors: Jocelyn SABATIER

Published 2026-09-30

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CC BY 4.0

Original authors: Jocelyn SABATIER

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

In the hidden world inside a battery, electricity does not always flow like water through a pipe. Sometimes, the movement of charged particles behaves more like a slow, spreading stain on a piece of paper, where the speed of the spread changes over time in a way that defies simple prediction. Scientists call this behavior "long memory," meaning the system remembers its past states and lets that history influence what happens next. For decades, researchers have used a specialized branch of mathematics, known as fractional calculus, to describe these complex, lingering effects. This mathematical tool has become the standard language for explaining why batteries, biological tissues, and certain materials react the way they do over long periods. However, this mathematical language is notoriously difficult to interpret physically, often leaving engineers with accurate predictions but no clear picture of the actual mechanisms at work.

A new study by Jocelyn Sabatier at Bordeaux University challenges the idea that this complex mathematics is the only way to describe these phenomena. The researcher set out to see if a much older, simpler class of models—ones that account for delays spread out over time rather than a single instant—could produce the exact same results. The focus was on a lithium-ion battery cell, the kind found in electric vehicles and smartphones. When a battery is charged or discharged, its voltage drops in a specific pattern that looks like a power law, a curve that scientists have traditionally attributed to the mysterious rules of fractional calculus. Sabatier proposed that this same curve might simply be the result of many different diffusion processes happening at once, each with its own speed, rather than a single, exotic mathematical rule.

To test this, the team analyzed the voltage response of a battery cell when a steady current was applied. They observed the voltage drop over time, which is caused by a combination of immediate resistance, surface reactions, and the slow diffusion of lithium ions deep inside the solid particles of the electrode. The researchers then built two different models to predict this behavior. The first was a standard fractional-order model, the current industry favorite, which uses non-integer powers to capture the long-memory effect. The second was a distributed time delay model, which treats the system as a collection of many simple delays, each weighted differently, to mimic the spread of time.

The results were striking. When the team fitted both models to the actual experimental data, they found that the simpler distributed delay model performed just as well as the complex fractional model. Both models predicted the battery's voltage with nearly identical precision, achieving a fit accuracy of over 94 percent. The distributed delay model required the same number of adjustable parameters to reach this level of accuracy, yet it did so without invoking fractional derivatives. In fact, the study showed that the mathematical coefficients of the delay model naturally evolved to look like the power-law curves that fractional calculus is famous for, proving that the complex math was not strictly necessary to generate the observed behavior.

This finding suggests that the power-law responses seen in batteries and other systems might not be evidence of a fundamental, exotic physical law, but rather the natural outcome of many classical processes overlapping. The distributed delay model offers a distinct advantage in clarity: while fractional calculus describes memory through abstract operators that are hard to visualize, the delay model describes it through a distribution of time lags that can be directly linked to physical transport times, diffusion rates, or residence times. The study does not claim that fractional models are wrong or useless, but it does argue that they may be an unnecessary complication when a classical approach can explain the same data with equal accuracy and greater physical intuition.

By demonstrating that a classical framework can replicate the behavior of a fractional one, the research invites a reevaluation of how scientists model memory-driven systems. It suggests that the increasing complexity of modern fractional frameworks, with their many variations and definitions, might be solving problems that do not exist. If a simple distribution of delays can capture the essence of long-memory phenomena, then the path to understanding these systems may lie not in more abstract mathematics, but in better descriptions of how physical processes unfold over time. The work does not declare a final victory for one method over the other, but it firmly establishes that the simpler, more interpretable approach is a viable and powerful alternative for understanding the slow, lingering dynamics of the real world.

Technical Summary: Lithium-ion Cell Modelling with Distributed Time Delay Models

Problem Statement
Fractional-order dynamics and power-law time responses are ubiquitous in complex systems, including electrochemical devices like lithium-ion batteries, viscoelastic materials, and biological tissues. Traditionally, these behaviors are modeled using fractional calculus, which naturally generates long-memory effects and power-law responses via fractional differential equations. However, the physical interpretation of fractional-order states, the definition of initial conditions, and the existence of multiple competing fractional operators (e.g., Riemann–Liouville, Caputo) present challenges.

Concurrently, distributed time delay models constitute a well-established class of infinite-dimensional systems capable of describing memory and hereditary effects through a continuum of delay values. While widely used in population dynamics and process control, their specific ability to reproduce fractional-order power-law behaviors has received limited attention. The central problem addressed by this paper is whether classical distributed time delay models can reproduce the power-law dynamics typically attributed to fractional operators, thereby offering a physically interpretable alternative to fractional modeling.

Methodology
The paper proposes a theoretical derivation linking power-law responses to discrete and continuous distributed time delay models, followed by an empirical validation using lithium-ion battery data.

Theoretical Derivation:
- Discrete Model: Starting from a power-law time response characteristic of a system exhibiting fractional behavior (specifically $Q(t) \propto t^\alpha$ ), the author derives a discrete distributed time delay model. By discretizing the integral form of the power law and applying Euler's explicit formula, the paper demonstrates that the resulting coefficients asymptotically reproduce the power-law decay ( $m^{-\nu}$ ). This leads to a recursive equation where the current state is a weighted sum of past states, effectively forming a distributed delay model.
- Continuous Model: The paper references a continuous approximation method where a fractional integrator is approximated by a transfer function involving a kernel $\eta_a(t)$ . This kernel is constructed using a product of poles and zeros to approximate $t^{-\nu}$ over a specific time interval. This allows the fractional behavior to be represented as a distributed time delay system within the classical theory of dynamical systems.
Application to Lithium-Ion Cells:
- The study utilizes experimental data from a lithium-ion cell (current input, voltage output) provided by PSA Peugeot-Citroën.
- The cell dynamics are modeled using a single-electrode framework where the voltage drop is decomposed into instantaneous resistance, double-layer capacitance effects, and diffusion-driven polarization.
- Three models are compared:
  - Fractional-Order Model: Uses fractional transfer functions ( $H_1(s)$ and $H_2(s)$ ) derived from diffusion equations in spherical geometry, approximated by interlaced poles and zeros.
  - Discrete Distributed Time Delay Model: Implements the derived discrete recursive equation.
  - Continuous Distributed Time Delay Model: Implements the continuous delay formulation using the kernel approximation.
- Parameter identification for all models is performed using a nonlinear Levenberg-Marquardt optimization algorithm to minimize the Root Mean Square Error (RMSE) between simulated and measured voltages.

Key Contributions

Theoretical Link: The paper establishes a direct mathematical connection showing that a classical integer-order distributed time delay model can generate responses exhibiting the same power-law properties as fractional calculus.
Alternative Framework: It demonstrates that distributed time delay models can serve as a physically interpretable alternative to fractional-order models for memory-driven phenomena.
Comparative Validation: The study provides a side-by-side comparison of fractional and distributed delay models applied to a real-world lithium-ion battery, showing that both approaches achieve comparable accuracy with a similar number of tuning parameters.

Results
The modeling results for the lithium-ion cell indicate that:

Accuracy: Both the fractional-order models and the distributed time delay models (both discrete and continuous) reproduce the measured voltage dynamics with high precision.
- The fractional model ( $H_2$ ) achieved a FIT of 95.15% and an RMSE of 0.00500.
- The discrete distributed delay model achieved a FIT of 95.08% and an RMSE of 0.00508.
- The continuous distributed delay model achieved a FIT of 94.93% and an RMSE of 0.00523.
Parameter Efficiency: All models required a limited and equivalent number of tuning parameters (4 to 6 parameters depending on the model complexity).
Error Analysis: The modeling errors ( $\epsilon$ ) for all approaches were visually and statistically comparable, confirming that the distributed delay models do not sacrifice accuracy for interpretability.

Significance and Claims
The paper claims that the observed power-law behaviors in electrochemical systems should not be automatically interpreted as evidence of an underlying fractional-order mechanism. Instead, these behaviors can emerge from the aggregation of classical distributed processes acting over multiple time scales.

The significance of this work lies in the following points:

Physical Interpretability: Unlike fractional models, which rely on abstract differentiation operators and face challenges regarding initial conditions, distributed time delay models explicitly represent the influence of past states through delay kernels. These kernels can be directly associated with physical quantities such as transport times, diffusion times, and residence-time distributions.
Mathematical Simplicity: Distributed delay models operate within the classical theory of dynamical systems, avoiding the complexities and competing definitions inherent in fractional calculus.
Paradigm Shift: The results suggest that distributed time delay models are not merely numerical approximations of fractional models but constitute a distinct modeling paradigm. They offer a way to capture long-memory and power-law behaviors while maintaining a clear connection to underlying physical mechanisms.
Re-evaluation of Complexity: The paper invites scrutiny of the increasing complexity in fractional modeling frameworks (e.g., variable-order, distributed-order). If classical distributed delay structures can capture essential dynamics, the necessity of these more sophisticated formulations for certain applications may need re-evaluation.

In conclusion, the paper posits that distributed time delay models constitute a relevant and attractive alternative for describing diffusion-driven and memory-dependent phenomena, offering a balance between modeling performance and physical interpretability.

Technical Summary: Lithium-ion Cell Modelling with Distributed Time Delay Models

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