Abstract-- The paper describes the application of a real-time, adaptive battery modelling methodology to Li-ion batteries. This methodology allows accurate estimation of the State-of-Function (SoF) of batteries for an Electric or Hybrid-Electric vehicle. Through use of a Kalman Estimator and online battery model parameter estimation, the voltages associated with monitoring the State of Charge (SoC) of the battery system are shown to be accurately estimated, even given erroneous initial conditions in both the model and parameters. In this way, problems such as self-discharge during non-use of the cells and SoC drift (as usually incurred by coulomb-counting methods due to over-charging or ambient temperature fluctuations) are overcome. A further benefit of the adaptive nature of the parameter estimation allows battery ageing (State of Health - SoH) to be monitored and, in the case of safety-critical systems, cell failure may be predicted in time to avoid inconvenience to passenger networks. Moreover, the ability to accurately predict the SoF and changes in battery parameters allows charging scenarios to be optimized to extend lifetime and facilitate future “Vehicle-to-Grid” (V2G) implementation.

Index Terms-- “Battery Modelling”, “Parameter

Estimation”, “Energy Storage”

I. INTRODUCTION The desire for miniaturisation, reduction in weight,

and increase in time between charges has led to the prevalence of Li-ion cells for use in portable electronic equipment. The attraction of this energy-dense technology has also been reorganised by the auto industry, and the take-up of Li-ion chemistries by automotive OEMs for Plug-in Hybrid Electric Vehicles (PHEVs) and 2nd generation fully Electric Vehicles (EVs) has increased rapidly over the last 3 years. As increasingly more variations of the Li-ion chemistry and anode/cathode technology are investigated and developed (see Fig. 1), the difficulty in maximising the battery pack lifetime, and preventing catastrophic failure due to under/over-charging places greater demands on a Battery Monitoring System (BMS) for such a vehicle. For increased consumer uptake of EVs, these BMS must be capable of accurately predicting the State-of-Charge (SoC - amount of charge available in the battery at any time to

sink/source power) so that intelligent charging scenarios can be employed to maximise battery lifetime (or State-of-Health - SoH – the ability of the battery to repeatedly provide its rated capacity over its lifetime), whilst enabling the user to reach their destination without any “Range-anxiety”.

Fig. 1. Example Li-ion cylindrical-type cells for automotive use

Unfortunately, there is no quantitative method of measuring SoC and SoH directly when the battery is in dynamic use, i.e. in-situ on the vehicle, necessitating the use of system modelling based on available, quantifiable data.

Literature reviews of existing battery modelling techniques, [1-3] for example categorise these techniques into the following main areas:

1. Electrochemical models 2. Empirical/Mathematical models 3. Equivalent circuit models

Considering electrochemical models [4-7], the battery

chemistry charge/discharge carrier mechanisms are described by a high number of Partial Differential Equations (PDEs) which must be solved simultaneously with a computational expense that generally precludes use in real-time online control. Alternatively, these high numbers of PDEs can be represented by a lower number of ‘reduced order’ PDEs and by substituting boundary conditions and discretisation, real-time application is possible at the expense of reduced SoC accuracy [7].

� EV/HEV Li-ion Battery Modelling and State-of-Function Determination

Chris Gould†, Jiabin Wang, Dave Stone, and Martin Foster UNIVERSITY OF SHEFFIELD

Department of Electronic and Electrical Engineering Mappin Street, Sheffield, S1 3JD, UK

Tel. +44 (0) 114 2225132, Fax. +44 (0) 114 2225196,

†E-mail: c.gould@sheffield.ac.uk

978-1-4673-1301-8/12/$31.00 ©2012 IEEE

2012International Symposium on Power Electronics,Electrical Drives, Automation and Motion

353

Empirical or mathematical techniques are here used to describe models that rely on some form of battery characterisation through experimental data fitting. Most often these techniques express the battery’s terminal voltage as a function of demand current, ambient temperature, SoC and a DC gain, such as the Shepherd Model, Unnwehr Universal Model or Nernst Model as described in [1]. Other SoC techniques can range from the simple pulsed-discharge and relaxation techniques that allow the Open-Circuit Voltage (OCV) to be assessed and linearized for given SoC points [8], to the more complicated Neural Network approach of [9], requiring input-output data for training sets. Each of these methods suffers from an inability to be generally applicable to batteries from the same chemistry family, let alone a range of cells from different chemistries, since production variations would require each battery to be pre-tested/evaluated for accurate SoC estimation.

Equivalent circuit models operate along similar principles to the previously mentioned empirical techniques, in that the terminal voltage is modelled as a function of demand current, ambient temperature and SoC. However, these functions are structured into discrete passive electronic component networks. This enables the battery to be modelled using circuit theory (and hence circuit simulation software).

One of the simplest forms of equivalent circuit modelling is to treat the battery as a large capacitive charge store that can accumulate charge by integrating current over time. This method is more commonly known as coulomb-counting or Ampere-hour (Ah [3]) technique, and is an effective first-order approximation to the charge stored in a battery, provided that accurate current sensing technology is available. The technique is however subject to drift, as any measurement errors are accumulated, and also presumes that the demand current has no physical effect on the battery i.e. no internal temperature change or unwanted chemical transition effects. These effects can be taken into consideration through the use of higher order networks such as the well-known Randles’ [10] model (Fig. 2) originally adapted for Lead-acid batteries.

Fig. 2. Randles 2nd order equivalent circuit model

Here, the main capacitive charge store Cb is augmented with a transient impedance network Cs and Rt (to account for small time-constant electrochemical transitions), a terminal and interconnection resistance Ri and a large self-discharge resistance Rd. Further transient

responses with different time-constants can be included by the further introduction of R-C networks such as [11-13], requiring filtering of the input-output response to obtain suitable static circuit parameters.

As with the empirical techniques, these parameters must be evaluated with test data, and are largely cell-specific, requiring much a priori information before incorporation into a BMS. In [14-16], the authors have demonstrated that the 2nd order Randles’ model can be adapted to provide a more suitable state-variable circuit model that allows accurate online circuit parameter estimation, and when coupled to Kalman Estimator techniques [17], forms a system of algorithms able to converge on circuit variables linked to SoF indicators.

Whilst these techniques were developed primarily on Lead-acid batteries, the underlying adaptive nature of the methodology was proposed to be suitable for a wide range of battery topologies. With the adoption of Li-ion as the most common battery chemistry by electric vehicle manufacturers, the authors apply similar techniques to commercially available Li-ion cells (Fig. 1) in order to investigate the range of parameters observed by the system, and its ability to correctly converge on the SoC voltages traditionally linked to the main charge store of a battery.

II. BATTERY MODELLING The proposed equivalent circuit model shown in Fig. 3

is derived from the Randles’ 2nd order model shown in Fig. 2 through use of a star-delta transform as per [15], resulting in the parameter transforms of (1) where it can be seen that if the main charge store Cb >> Cs then Cb becomes analogous to Cn, and similarly Cs analogous to Cp. The network state-variable equations are given in (2).

Fig. 3. Remapped circuit model proposed by Gould et al [15]

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In the remapped circuit, the voltage across the main charge store (VCb in the Randles’ model) can be obtained from (3), where it can be seen that if Cn >> Cp, the SoC voltage becomes analogous to VCn. However, the only simplification made to the system for parameter estimation is that the discharge resistance, Rp, is of such a size and near constant value that the self-discharge rate is negligible, resulting in the simplified state-variable description of (4).

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In order to obtain suitable input-output data to validate

both the parameter estimation techniques and the Kalman Estimator theory in the following sections, a commercially available Lithium-ion Iron Phosphate (LiFePO4) cylindrical cell such as may be found in EV battery packs is used as the candidate Li-ion chemistry. Fig. 4 lists the manufacturer data for the 3000mAh 3.2V nominal LIF26650E cell used in testing.

Fig. 4. Manufacturer data for a 3000mAh LiFePO4 cell [18]

A current-demand drive-cycle profile has been adapted from track data taken from a Honda Insight HEV equipped with a bespoke Lead-acid battery pack used in the Reliable Highly Optimised Lead Acid Battery (RHOLAB) project [19], in order to provide dynamic test-data. The cycle has a predominant discharge characteristic and is continuously cycled from 100% SoC until the cell is fully discharged to evaluate the response of the algorithms over the full SoC range. The resulting measured demand current and cell terminal voltage are shown in Figs. 5 and 6.

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Fig. 6. Resulting cell terminal voltage variation from drive cycle of Fig. 5 applied to candidate LiFePO4 cell

III. PARAMETER ESTIMATION Considering the input-output data of Figs. 5 and 6,

black-box state-space modelling [20] may be used to estimate suitable model orders and parameters, where the general structure of estimation can be described by (5).

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estimate an nth order model to the input-output data (in MATLAB’s System Identification toolbox for example). However, considering the equivalent circuit model derived from Randles’ model above (Fig. 3), a structured parameter estimation model may be formulated to better estimate the parameters associated with the electrochemical reactions.

Considering the state-space description given in (4), a second order model is desirable, and considering that the model parameter associated with RnCn will be large, and can be considered to be time-invariant over suitably sized buffers of data (due to the inherent nature of Li-ion discharge characteristics), the model can be structured

355

with both time-invariant and time-variant parameters. The Prediction-Error Minimisation (PEM) algorithm in MATLAB’s System Identification toolbox allows exactly this kind of structuring. The PEM algorithm operates in a similar manner to an ARMAX model, however, utilises the cost function VN(G,H) in (6) to minimise the error e(t) between the predicted model output G(q)u(t) for a set of estimated model parameters and the measured output data y(t).

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output data of Figs. 5 and 6 online by buffering 90seconds of measured input-output data in a FIFO buffer that is updated every 10seconds. Initial conditions for the state estimations (i.e. capacitor voltages VCn and VCp) at each update stage are taken from the outputs of the Kalman Estimator predictions of state voltages as will be described in the next section.

IV. KALMAN ESTIMATOR THEORY The Kalman Estimator [17] is particularly useful for

this methodology, since it optimally estimates states affected by broadband noise within the system bandwidth (i.e. tries to minimise the sum-of-squared errors between actual and estimated states), which may not be able to be filtered classically. The recursive use of predictor-corrector algorithms is also beneficial, since at each stage, the updated parameter estimates may be used to aid the KE in convergence.

The implementation of the KE follows the method introduced in [15] (repeated here for clarity), whereby a discrete-time equivalent model of a system’s state-variable description (7) is generated, using a first-order Taylor series expansion, resulting in (8) and (9) where Tc is the sampling rate.

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that at each sample step ‘k’ the algorithms operate on the input-output data collected up to and including the kth sample as a prediction/correction flow diagram depicted in Fig 7. Intuitively this will require initial estimates for the states and covariance matrices considered by the model, and a ‘best guess’ scenario of these values is prompted by values taken from previously characterised Lead-acid cells (12) for investigation into the robustness of the estimation algorithms.

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V. RESULTS AND DISCUSSIONS

The parameter estimation technique described in the previous sections has been applied to the measured battery input-output data of Figs. 5 and 6. The KE is initialised with Ri = 30mΩ, Rn = 10mΩ, Cp = 1000F, Cn = 100000F, as would be expected from Lead-acid cells, and the parameter estimation algorithms are used to update these model parameters every 10s. Since the parameter Ri is available from the estimation algorithms, the terminal voltage is pre-processed to remove IinRi, beneficially obtaining the state associated with Cp.

The resulting model parameter variations for the candidate LiFePO4 cell are shown in Figs. 8-10.

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of 30mΩ from Fig. 4, the estimated internal resistance, Ri, of ~40mΩ in Fig. 8, reflects the temperature increase (after model convergence) expected during continuous cycling, whilst the decay in capacitance Cp (Fig. 9) and

increase in resistance Rn (Fig. 10) models the expected electrochemical variation as the cell approaches deep discharge, and is unable to source the transient power requirements of the drive-cycle current-demand.

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Further application of the KE to the input-output data of Figs. 5 and 6 with the time-varying parameters from Figs. 8-10 is shown in Fig. 11. To provide a reference to the expected SoC voltage variation, pulsed-discharge and relaxation techniques are applied to the same cell using the methods of [8] to obtain the Open-Circuit Voltage (OCV) estimation of SoC (normalised to the time to total discharge). It can be seen that despite initialisation with an erroneous initial voltage condition, the KE rapidly converges onto the OCV estimation with excellent accuracy.

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Fig. 11. Comparison of SoC voltage estimation techniques demonstrating convergence ability of proposed method

Considering the OCV-SoC relationship derived from the pulsed-discharge and relaxation techniques, a conversion from SoC voltage to SoC percentage can be applied to Fig. 11 and compared to typical coulomb-counting techniques that utilise a ‘best-guess’ estimation to the actual cell capacity by assuming the nominal capacity is 100% SoC. The results shown in Fig. 12 demonstrate excellent correlation between the 3Ah

357

nominal coulomb-counting method (measured capacity of 3.025Ah) and the KE with adaptive parameters. Moreover, Fig. 12 demonstrates the convergence ability of the KE over the coulomb-counting technique which depends upon accurate a priori knowledge of the initial capacity. If, for example, an unknown cell had been tested that was at 50% SoC, the KE would have converged to this value, whilst the coulomb-counting technique would have to assume it had started at the erroneous nominal 3Ah, or 100%SoC.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-20

0

20

40

60

80

100

Time (s)

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%)

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Fig. 12. Comparison of SoC techniques demonstrating

convergence ability of proposed method

CONCLUSIONS

Novel modelling techniques have been applied to Li-ion battery chemistries in order to estimate their State-of-Function. It is shown that convergence on, and accurate estimation of the voltage across the main charge store of a battery (which is commonly associated with State-of-Charge), is achievable, and that State-of-Health variation can be inferred from excessive variation in the resulting parameter estimations. These techniques have been adapted from algorithms developed on Lead-acid cells, demonstrating their ability to model widely varying battery chemistries.

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