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Research Papers

# Adaptive Partial Differential Equation Observer for Battery State-of-Charge/State-of-Health Estimation Via an Electrochemical Model

[+] Author and Article Information
Scott J. Moura

UC President's Postdoctoral Fellow
Mechanical and Aerospace Engineering,
University of California, San Diego,
San Diego, CA 92093
e-mail: smoura@ucsd.edu

Nalin A. Chaturvedi

Senior Research Engineer
Research and Technology Center,
Robert Bosch LLC,
Palo Alto, CA 94304
e-mail: nalin.chaturvedi@us.bosch.com

Miroslav Krstić

Professor
Mechanical and Aerospace Engineering,
University of California, San Diego,
San Diego, CA 92093
e-mail: krstic@ucsd.edu

To be technically correct, the cathode concentration should depend on the anode concentration summed over the spherical volume: $css+(t)=(1/ɛs+L+A)[nLi-(3ɛs-L-A/4πRs-3)∫0Rs-4πr2cs-(r,t)dr]$. However, this results in a nonlinear output equation which depends on the in-domain states, as well as the boundary state. This would create additional complexity to the backstepping approach we employ in this paper.

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received July 18, 2012; final manuscript received June 8, 2013; published online October 15, 2013. Assoc. Editor: Yang Shi.

J. Dyn. Sys., Meas., Control 136(1), 011015 (Oct 15, 2013) (11 pages) Paper No: DS-12-1228; doi: 10.1115/1.4024801 History: Received July 18, 2012; Revised June 08, 2013

## Abstract

This paper develops an adaptive partial differential equation (PDE) observer for battery state-of-charge (SOC) and state-of-health (SOH) estimation. Real-time state and parameter information enables operation near physical limits without compromising durability, thereby unlocking the full potential of battery energy storage. SOC/SOH estimation is technically challenging because battery dynamics are governed by electrochemical principles, mathematically modeled by PDEs. We cast this problem as a simultaneous state (SOC) and parameter (SOH) estimation design for a linear PDE with a nonlinear output mapping. Several new theoretical ideas are developed, integrated together, and tested. These include a backstepping PDE state estimator, a Padé-based parameter identifier, nonlinear parameter sensitivity analysis, and adaptive inversion of nonlinear output functions. The key novelty of this design is a combined SOC/SOH battery estimation algorithm that identifies physical system variables, from measurements of voltage and current only.

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## Figures

Fig. 1

Each electrode is idealized as a single porous spherical particle. This model results from assuming the electrolyte concentration is constant in space and time.

Fig. 2

DFN predictions of the solid and electrolyte concentrations as functions of space. The DFN model retains electrolyte and spatial dynamics. State values are depicted after 50 s of 5 C discharge. Symbol cavg is the solid concentration averaged over a spherical particle and css is the surface concentration. Note the non-negligible concentration gradients in the electrolyte.

Fig. 3

Voltage response for several discharge rates, for the SPM and DFN model. The SPM exhibits increasing error as C-rate increases, but identical discharge capacities.

Fig. 4

Block diagram of the adaptive observer. It is composed of the backstepping state estimator (blue), PDE parameter identifier (green), output function parameter identifier (red), and adaptive output function inversion (orange). The observer furnishes estimates of SOC (i.e., c∧s-(r,t)) and SOH (i.e., ɛ∧,q∧,θ∧h) given measurements of I(t) and V(t), only.

Fig. 5

Bode plots of the transcendental transfer function (42) and Padé approximates in Table 1

Fig. 6

Evolution of state and parameter estimates for UDDSx2 charge/discharge cycle. Zero mean Gaussian noise with a 10 mV variance was added to the voltage measurement. The DFN model provides the “measured” plant data. State and parameter estimates were initialized with incorrect values.

Fig. 7

Evolution of state and parameter estimates for a 20 min 1 C discharge and 10 min relaxation

Fig. 8

Evolution of state estimates for UDDSx2 charge/discharge cycle with no parameter adaptation. Accurate parameter values and/or online adaptation are critical for unbiased estimates.

Fig. 9

Relationship between c∧(r,t) (i.e., SOC∧) and n∧Li, for varying Ph011 and λ = -1. An improper selection between these two gains results in biased estimates.

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