Sigma-point Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2: Simultaneous state and parameter estimation

doi:10.1016/j.jpowsour.2006.06.004

Journal of Power Sources

Volume 161, Issue 2, 27 October 2006, Pages 1369-1384

https://doi.org/10.1016/j.jpowsour.2006.06.004 Get rights and content

Abstract

We have previously described algorithms for a battery management system (BMS) that uses Kalman filtering (KF) techniques to estimate such quantities as: cell self-discharge rate, state-of-charge, nominal capacity, resistance, and others. Since the dynamics of electrochemical cells are not linear, we used a nonlinear extension to the original KF called the extended Kalman filter (EKF).

Now, we introduce an alternative nonlinear Kalman filtering technique known as “sigma-point Kalman filtering” (SPKF), which has some theoretical advantages that manifest themselves in more accurate predictions. The computational complexity of SPKF is of the same order as EKF, so the gains are made at little or no additional cost.

This paper is the second in a two-part series. The first paper explored the theoretical background to the Kalman filter, the extended Kalman filter, and the sigma-point Kalman filter. It explained why the SPKF is often superior to the EKF and applied SPKF to estimate the state of a third-generation prototype lithium-ion polymer battery (LiPB) cell in dynamic conditions, including the state-of-charge of the cell.

In this paper, we first investigate the use of the SPKF method to estimate battery parameters. A numerically efficient “square-root sigma-point Kalman filter” (SR-SPKF) is introduced for this purpose. Additionally, we discuss two SPKF-based methods for simultaneous estimation of both the quickly time-varying state and slowly time-varying parameters. Results are presented for a battery pack based on a fourth-generation prototype LiPB cell, and some limitations of the current approach, based on the probability density functions of estimation error, are also discussed.

Introduction

This paper applies results from the field of study known variously as sequential probabilistic inference or optimal estimation theory to advanced algorithms for a battery management system (BMS). This BMS is able to estimate battery state-of-charge (SOC), instantaneous available power, and parameters indicative of the battery state-of-health (SOH) such as power fade and capacity fade, and is able to adapt to changing cell characteristics over time as the cells in the battery pack age. The algorithms have been successfully implemented on a lithium-ion polymer battery (LiPB) pack for hybrid-electric-vehicle (HEV) application,² and we also expect them to work well for other battery chemistries and less demanding applications. We have previously reported work using extended Kalman filters (EKF) to solve the HEV BMS algorithm requirements [1], [2], [3], [4], [5], [6]. We have since explored a different form of Kalman filtering called sigma-point Kalman filters(SPKF), and have found them to have several important advantages. We introduce SPKF here in a two-part series, of which this is the second part. The companion to this paper [7] introduces the SPKF and applies it to estimating the state of an LiPB-based HEV battery cell, particularly its SOC. In this paper, we build on the introduction to first investigate the use of the SPKF method to estimate battery parameters. A numerically efficient “square-root sigma-point Kalman filter” (SR-SPKF) is described for this purpose. Additionally, we discuss two SPKF-based methods for simultaneous estimation of both the quickly time-varying state and slowly time-varying parameters. Applications to the HEV BMS algorithm requirements are outlined, example results given using fourth-generation prototype LiPB cells, and conclusions made.

We note before continuing that there are conflicting objectives in making this paper both self-contained and simultaneously reducing redundant material with respect to the first paper in this series. We have perhaps erred on the side of conciseness in several areas. For a more in-depth discussion of sequential probabilistic inference and the base-line SPKF algorithm, the reader is referred to Ref. [7].

Section snippets

Summary of sequential probabilistic inference

Very generally, any causal dynamic system (e.g., a battery cell) generates its outputs as some function of its past and present inputs. Often, we can define a state vector for the system whose values together summarize the effect of all past inputs. Present system output is a function of present input and present state only; past input values need not be stored. The system’s parameter vector comprises all quasi-static numeric quantities that describe how the system state evolves and how the

Summary of sigma-point Kalman filters

For nonlinear systems, a closed-form solution or even an algorithm to exactly implement the general probabilistic inference solution in Table 1 is not available. The EKF is one approach to approximating the solution using a first-order linearization of the system dynamics, which is based on some questionable assumptions. Sigma-point Kalman filters are an alternate approach to generalizing the Kalman filter to state estimation for nonlinear systems. SPKFs rely on numeric approximations rather

Computationally efficient square-root sigma-point Kalman filters

Sigma-point Kalman filters may be used directly for state estimation and we have shown that they produce better state estimates and much better covariance estimates than EKF [7]. The computational complexity is $O (L^{3})$ , which is of equivalent complexity to EKF state estimation, where L is the dimension of the augmented state. They may also be used for parameter estimation, as will be described in the sequel, but the computational complexity remains $O (L^{3})$ , whereas the corresponding EKF method has

Parameter estimation using SR-SPKF

The various methods for state estimation presented so far have assumed a known system model in the form of Eqs. (1) and (2). These equations will generally involve numeric values in their computations. Some of these values might be intrinsic constants, but others might be factors determined by the electrochemistry or construction of a particular cell. We refer to these latter factors as the “parameters” of the cell model.

To use the enhanced self-correcting (ESC) cell model as an example (cf.

Joint and dual sigma-point filtering

We have now shown how to estimate the state of a system given a known model and noisy measurements and how to estimate the parameters of the system given a known state and clean measurements. We proceed to give two methods whereby one can simultaneously estimate both the state and parameters of a system given noisy measurements.

The various methods of state estimation presented so far have assumed a constant system model. However, when applying these procedures to estimate battery SOC, for

The enhanced self-correcting cell model

In order to examine and compare performance of the proposed algorithms, we must first define a discrete-time state–space model of the form of (1) and (2) that applies to battery cells. Here, we briefly review the “enhanced self-correcting cell model” from Refs. [5], [3]. This model includes effects due to open-circuit-voltage, internal resistance, voltage time constants, and hysteresis. For the purpose of example, we will later fit parameter values to this model structure to model the dynamics

Cell and cell test description

The cells used in this paper differ electrochemically from those reported in previous work. We refer to the older cells as GEN3 cells, and to the newer cells as G4 cells. The GEN3 cells are high-power (¿20 C capable) 7.5Ah Mn spinel/graphite LiPB, and the G4 cells are very high-power ( $>$ 30 C capable) 5Ah Mn spinel/blended-carbon LiPB, both reported in Ref. [22].

In order to compare the various Kalman filtering methods’ abilities to estimate SOC and SOH, we gathered data from two prototype LiPB

Conclusions

This paper concludes a two-part series discussing the application of sigma-point Kalman filters to battery management algorithms. In the first paper, we introduced the general probabilistic inference solution to optimal estimation, and derived the KF, EKF, and SPKF from this solution using different sets of assumptions. The SPKF was shown to be theoretically more precise than EKF; testing with real cell data supported this analysis.

This paper showed how SPKF could be very closely approximated

Acknowledgements

This work was supported in part by Compact Power Inc. (CPI). The use of company facilities, and many enlightening discussions with Drs. Mohamed Alamgir, Bruce Johnson, Dan Rivers, and others are gratefully acknowledged.

I would also like to thank the reviewers for their helpful comments, which led to improvements in the clarity of this work with respect to what I originally wrote.

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¹: Also consultant to Compact Power Inc., 707 County Line Rd., P.O. Box 509, Palmer Lake, CO 80133. United States. Tel.: +1 719 488 1600x134; fax: +1 719 487 9485.

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Sigma-point Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2: Simultaneous state and parameter estimation

Abstract

Introduction

Section snippets

Summary of sequential probabilistic inference

Summary of sigma-point Kalman filters

Computationally efficient square-root sigma-point Kalman filters

Parameter estimation using SR-SPKF

Joint and dual sigma-point filtering

The enhanced self-correcting cell model

Cell and cell test description

Conclusions

Acknowledgements

J. Power Sources

J. Power Sources

J. Power Sources

Automatica

LiPB dynamic cell models for Kalman-filter SOC estimation

Kalman-filter SOC estimation for LiPB HEV cells

Advances in EKF SOC estimation for LiPB HEV battery packs

Numerical Recipes in C: The Art of Scientific Computing

A new approach for filtering nonlinear systems