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Hybrid low-dimensional limiting state of charge estimator for multi-cell lithium-ion batteries

Mira Khalil, Romain Postoyan, Stéphane Raël, Dragan Nešić

TL;DR

This work tackles the practical problem of estimating the minimum state of charge in a large series-connected lithium-ion battery pack without running an observer per cell. It introduces a low-dimensional hybrid estimator that actively switches among cells using online estimates of the open-circuit voltage function $V_{OCV}$, coupled with a single-cell observer and a switching logic based on z-score-like quantities to track the true minimum SOC. The stability analysis builds on non-smooth hybrid Lyapunov theory to prove practical exponential convergence of the estimated minimum SOC to the actual minimum, while guaranteeing absence of Zeno behavior and providing explicit bounds that quantify the effects of parameter mismatch and the switching threshold. Numerical simulations on a 200-cell pack demonstrate rapid and robust tracking of $SOC_{ ext{min}}$, illustrating the method’s potential for scalable battery management in resource-constrained BMS environments.

Abstract

The state of charge (SOC) of lithium-ion batteries needs to be accurately estimated for safety and reliability purposes. For battery packs made of a large number of cells, it is not always feasible to design one SOC estimator per cell due to limited computational resources. Instead, only the minimum and the maximum SOC need to be estimated. The challenge is that the cells having minimum and maximum SOC typically change over time. In this context, we present a low-dimensional hybrid estimator of the minimum (maximum) SOC, whose convergence is analytically guaranteed. We consider for this purpose a battery consisting of cells interconnected in series, which we model by electric equivalent circuit models. We then present the hybrid estimator, which runs an observer designed for a single cell at any time instant, selected by a switching-like logic mechanism. We establish a practical exponential stability property for the estimation error on the minimum (maximum) SOC thereby guaranteeing the ability of the hybrid scheme to generate accurate estimates of the minimum (maximum) SOC. The analysis relies on non-smooth hybrid Lyapunov techniques. A numerical illustration is provided to showcase the relevance of the proposed approach.

Hybrid low-dimensional limiting state of charge estimator for multi-cell lithium-ion batteries

TL;DR

This work tackles the practical problem of estimating the minimum state of charge in a large series-connected lithium-ion battery pack without running an observer per cell. It introduces a low-dimensional hybrid estimator that actively switches among cells using online estimates of the open-circuit voltage function , coupled with a single-cell observer and a switching logic based on z-score-like quantities to track the true minimum SOC. The stability analysis builds on non-smooth hybrid Lyapunov theory to prove practical exponential convergence of the estimated minimum SOC to the actual minimum, while guaranteeing absence of Zeno behavior and providing explicit bounds that quantify the effects of parameter mismatch and the switching threshold. Numerical simulations on a 200-cell pack demonstrate rapid and robust tracking of , illustrating the method’s potential for scalable battery management in resource-constrained BMS environments.

Abstract

The state of charge (SOC) of lithium-ion batteries needs to be accurately estimated for safety and reliability purposes. For battery packs made of a large number of cells, it is not always feasible to design one SOC estimator per cell due to limited computational resources. Instead, only the minimum and the maximum SOC need to be estimated. The challenge is that the cells having minimum and maximum SOC typically change over time. In this context, we present a low-dimensional hybrid estimator of the minimum (maximum) SOC, whose convergence is analytically guaranteed. We consider for this purpose a battery consisting of cells interconnected in series, which we model by electric equivalent circuit models. We then present the hybrid estimator, which runs an observer designed for a single cell at any time instant, selected by a switching-like logic mechanism. We establish a practical exponential stability property for the estimation error on the minimum (maximum) SOC thereby guaranteeing the ability of the hybrid scheme to generate accurate estimates of the minimum (maximum) SOC. The analysis relies on non-smooth hybrid Lyapunov techniques. A numerical illustration is provided to showcase the relevance of the proposed approach.
Paper Structure (19 sections, 4 theorems, 35 equations, 5 figures)

This paper contains 19 sections, 4 theorems, 35 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Battery pack modeling
  4. Individual cell
  5. Multi-cell
  6. Limiting cell
  7. Hybrid design
  8. Continuous-time dynamics
  9. Discrete-time dynamics
  10. Switching logic
  11. Hybrid model
  12. Stability guarantees
  13. Main Results
  14. Proofs
  15. Proof of Proposition \ref{['Th_StabilityProperty_delta_URC']}
  16. ...and 4 more sections

Key Result

Proposition 1

Given any $\ell >0$ and $\varepsilon>0$, consider system (compacthybridsystem), for any càdlàg input $u$, any corresponding solution $q$ to (compacthybridsystem) satisfies, for all $(t,j) \in$ dom $q$, where $\widehat{U}_{RC}:=(\widehat{U}_{RC,1},\hdots,\widehat{U}_{RC,N})$ and $d:=\underset{i \in \{1,\hdots,N\}}{\max}\{|\tfrac{1}{\tau_d}-\tfrac{1}{\tau_{d,i}} | \}$. $\Box$

Figures (5)

  • Figure 1: Schematic diagram for the lithium-ion battery pack consisting of $N$ cells in series modeled by a first order ECM.
  • Figure 2: $V_{OCV}$ function.
  • Figure 3: Input current profile.
  • Figure 4: The index of the cell having $SOC_{\min}$ and $\sigma$.
  • Figure 5: $SOC_{\min}$ et $\widehat{SOC}$ generated by the hybrid estimator (top) and the norm of the estimation error $|SOC_{\min}-\widehat{SOC}|$ (bottom).

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Proposition 3