A Control Oriented Fractional-Order Model of Lithium-ion Batteries Based on Caputo Definition

TL;DR

This work addresses real-time Li-ion battery modeling by introducing a Caputo-based fractional-order model to overcome memory and initial-condition ambiguities of Grünwald--Letnikov discretizations. It derives a closed-form time-domain solution and a two-step recursive discretization, with the voltage evolution under constant current given by $U_1(t) = U_1(0) E_\alpha(-t^\alpha/(R_1 C_1)) + I_0 R_1 [1 - E_\alpha(-t^\alpha/(R_1 C_1))]$ and a practical discrete update $U_{1,k+1} = U_{1,k} E_\alpha(-T^\alpha/(R_1 C_1)) + I_k R_1 [1 - E_\alpha(-T^\alpha/(R_1 C_1))]$. Parameter identification uses HPPC data and a Trust-Region Reflective nonlinear least-squares fit of the zero-state response for $R_i$ and $\alpha_i$, with $R_0$ estimated from transients. Experimental validation on a 40.2 Ah NCM622 cell shows that a 1RC Caputo-based fractional-order model achieves voltage accuracy comparable to a 2RC integer-order model and to a G-L-based fractional model, while significantly reducing memory requirements due to the two-step update. The results highlight the model’s practical suitability for real-time onboard BMS, offering a favorable trade-off between accuracy and computational load.

Abstract

This letter proposes a fractional-order battery model based on the Caputo definition. A closed-form time-domain solution is derived, enabling a simple recursive expression for discrete-time implementation. The model requires only the current and previous time-step states in each iteration, significantly reducing memory usage compared to the conventional Grünwald--Letnikov (G-L) method. This recursive structure is highly compatible with filter design and online parameter identification. Experimental validation on a 40.2~Ah NCM622 cell shows that the proposed first-order model achieves voltage prediction accuracy comparable to a second-order integer-order model. The results demonstrate that the Caputo-based model offers a practical balance between accuracy and computational efficiency, making it well suited for real-time battery management systems (BMS).

Figures (4)

• Figure 1: Fractional-order nRC equivalent circuit model.
• Figure 2: The experimental platform
• Figure 3: (a) Single HPPC test segment illustrating zero-state responses. (b) Variation of Mittag--Leffler function $E_{\alpha}\left( -\frac{t^{\alpha}}{\tau} \right)$ with different $\alpha$.
• Figure 4: Voltage estimation under the FUDS profile. (a) Comparison between the 1RC fractional-order model and integer-order models. (b) Comparison between Caputo-based and G-L-based fractional-order models.