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Finite-volume method and observability analysis for core-shell enhanced single particle model for lithium iron phosphate batteries

Le Xu, Simone Fasolato, Simona Onori

Abstract

The increasing adoption of Lithium Iron Phosphate (LFP) batteries in Electric Vehicles is driven by their affordability, abundant material supply, and safety advantages. However, challenges arise in controlling/estimating unmeasurable LFP states such as state of charge (SOC), due to its flat open circuit voltage, hysteresis, and path dependence dynamics during intercalation and de-intercalation processes. The Core Shell Average Enhanced Single Particle Model (CSa-ESPM) effectively captures the electrochemical dynamics and phase transition behavior of LFP batteries by means of Partial Differential-Algebraic Equations (PDAEs). These governing PDAEs, including a moving boundary Ordinary Differential Equation (ODE), require a fine-grained spatial grid for accurate and stable solutions when employing the Finite Difference Method (FDM). This, in turn, leads to a computationally expensive system intractable for the design of real-time battery management system algorithms. In this study, we demonstrate that the Finite Volume Method (FVM) effectively discretizes the CSa-ESPM and provides accurate solutions with fewer than 4 control volumes while ensuring mass conservation across multi ple operational cycles. The resulting control-oriented reduced order FVM-based CSa-ESPM is experimentally validated using various C-rate load profiles and its observability is assessed through nonlinear observability analysis. Our results reveal that different current inputs and discrete equation numbers influence model observability, with non-observable regions identified where solid-phase concentration gradients are negligible.

Finite-volume method and observability analysis for core-shell enhanced single particle model for lithium iron phosphate batteries

Abstract

The increasing adoption of Lithium Iron Phosphate (LFP) batteries in Electric Vehicles is driven by their affordability, abundant material supply, and safety advantages. However, challenges arise in controlling/estimating unmeasurable LFP states such as state of charge (SOC), due to its flat open circuit voltage, hysteresis, and path dependence dynamics during intercalation and de-intercalation processes. The Core Shell Average Enhanced Single Particle Model (CSa-ESPM) effectively captures the electrochemical dynamics and phase transition behavior of LFP batteries by means of Partial Differential-Algebraic Equations (PDAEs). These governing PDAEs, including a moving boundary Ordinary Differential Equation (ODE), require a fine-grained spatial grid for accurate and stable solutions when employing the Finite Difference Method (FDM). This, in turn, leads to a computationally expensive system intractable for the design of real-time battery management system algorithms. In this study, we demonstrate that the Finite Volume Method (FVM) effectively discretizes the CSa-ESPM and provides accurate solutions with fewer than 4 control volumes while ensuring mass conservation across multi ple operational cycles. The resulting control-oriented reduced order FVM-based CSa-ESPM is experimentally validated using various C-rate load profiles and its observability is assessed through nonlinear observability analysis. Our results reveal that different current inputs and discrete equation numbers influence model observability, with non-observable regions identified where solid-phase concentration gradients are negligible.

Paper Structure

This paper contains 9 sections, 15 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Core-shell average ESPM
  3. CSa-ESPM: Governing equations
  4. CSa-ESPM: FVM scheme
  5. Parameters identification
  6. Observability analysis
  7. Non-linear observability analysis
  8. CSa-ESPM observability analysis
  9. Conclusion

Figures (4)

  • Figure 1: C/4 discharge. The moving boundary ($r_p$) is plotted to illustate the positive electrode one-phase ($r_p/R_{s,p}=0$) and two-phase ($r_p/R_{s,p}>0$) regions.
  • Figure 2: Comparison results at C/4, C/2, and 1C for (a) voltage, (b) moving boundary, (c) volume-average concentraion, (d) moving boundary during cycling.
  • Figure 3: Observability analysis results containing phase transition dynamics, rank number and the common logarithm (i.e., $Log_{10}$) of condition number for (a) C/4 and 1C charge with $N_r=2$, (b) 1C charge with $N_r$=2 and $N_r$=3. The zoom plot shows the rank number between 10.9% - 12.5% SOC, (c) FDM with $N_r=2$ and FVM with $N_r=2$ under 1C charge, and (d) UDDS current profile with $N_r=2$.
  • Figure 4: Positive electrode concentration distribution in two-phase region for FVM with $N_r=3$ during 1C charge