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