Article Contents
Article Contents

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

• * Corresponding author: Esteban Hernández

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.

• 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.

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

 Citation:

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

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 $I_{ref}(t)$

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 $I_{ref}(t)$

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)

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 $[-]$

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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