A tracking problem for the state of charge in an electrochemical Li-ion battery model

Esteban Hernández; Christophe Prieur; Eduardo Cerpa

doi:10.3934/mcrf.2021041

Article Contents

2022, Volume 12, Issue 3: 709-732. Doi: 10.3934/mcrf.2021041

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A tracking problem for the state of charge in an electrochemical Li-ion battery model

1.
Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso, Chile
2.
Gipsa-Lab, Université Grenoble Alpes, 11 rue des Mathématiques, Grenoble Campus BP46, Saint Martin D'Heres, France
3.
Instituto de Ingeniería Matemática y Computacional, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avda. Vicuña Mackenna 4860, Santiago, Chile

^* Corresponding author: Esteban Hernández
^* Corresponding author: Esteban Hernández

Received: June 2019

Revised: November 2020

Early access: September 2021

Published: July 2022

This work has been partially financed by Fondecyt 1180528 (E. Cerpa), ECOS-CONICYT C16E06, PIIC Universidad Técnica Federico Santa María, Basal Project FB0008 AC3E and ANID BECAS/DOCTORADO NACIONAL/2017-21171188.
This article was recruited by Sergey Dashkovskiy..

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Abstract

In this paper the Single Particle Model is used to describe the behavior of a Li-ion battery. The main goal is to design a feedback input current in order to regulate the State of Charge (SOC) to a prescribed reference trajectory. In order to do that, we use the boundary ion concentration as output. First, we measure it directly and then we assume the existence of an appropriate estimator, which has been established in the literature using voltage measurements. By applying backstepping and Lyapunov tools, we are able to build observers and to design output feedback controllers giving a positive answer to the SOC tracking problem. We provide convergence proofs and perform some numerical simulations to illustrate our theoretical results.

Keywords:

Mathematics Subject Classification: Primary: 35A22, 35B35; Secondary: 35A24, 35B30.

Citation:

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Figure 1. The continuous function $ \tau $ is positive in $ [2+\sqrt{6}, \lambda_{\sup} ) $

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Figure 2. $ (Left) $ The input $ I_{ref}(t) = 0.5C $. $ (Right) $ We compare $ SOC(t) $ for the controlled system with the reference $ SOC_{ref}(t) $, generated by $ I_{ref}(t) $

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Figure 3. $ (Left) $ The input $ I_{ref}(t) = 4.5\, square(\frac{64}{900\pi}t)C $. $ (Right) $ We compare $ SOC(t) $ for the controlled system with the reference $ SOC_{ref}(t) $, generated by $ I_{ref}(t) $

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Figure 4. (a) We compare $ SOC(t) $ for the controlled system with the reference $ SOC_{ref}(t) $ generated by $ I_{ref}(t) = 0.5C $ (Left) and $ I_{ref}(t) = 4.5square(\frac{64}{900\pi}t)C $ (Right)

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Table 1. Model variables and electrochemical parameters

Model states, inputs and outputs
$ c^{\pm} $	Lithium concentration in solid phase $ [mol/m^3] $
$ c_{s}(t) $	Lithium concentration at solid particle surface$ [mol/m^2] $
$ c_{e} $	Lithium concentration in electrolyte phase $ [mol/m^3] $
$ T $	Temperature $ [K] $
$ I $	Applied current, $ [A/m^2] $
$ V $	Output Voltage $ [V] $
Electrochemical model parameters
$ D^{\pm} $	Diffusivity $ [m^2/s] $
$ R_{\pm} $	Particle radius in solid phase $ [m] $
$ F $	Faraday Constant $ [C/mol] $
$ R $	Universal gas constant $ [J/mol\cdot K] $
$ \alpha $	Charge transfer coefficient $ [-] $
$ c^{\pm}_{\max} $	Maximum concentration of solid material $ [mol/m^3] $
$ U^{\pm} $	Open circuit potential of solid material $ [V] $
$ R_{f} $	Solid interphase films resistance $ [\Omega\cdot m^2] $
$ L^{\pm} $	Length of region $ [m] $
$ A $	Area $ [m^2] $
$ \phi_{1} $	Heat transfer coefficient $ [1/s] $
$ \phi_{2} $	Inverse of heat capacity $ [J/K]^{-1} $
$ \varepsilon^{\pm} $	Volume fraction of solid phase $ [-] $

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Table 2. Parameter simulations

Parameters	Values
$ c(r, 0) $	$ 1.5c_{0} $
$ \widehat{c}(r, 0) $	$ 1.5c_{0} $
$ c_{\max} $	$ 2.5\cdot 10^4 $
$ \lambda $	$ 5 $
$ \gamma $	$ 70 $

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