-
Journal of Power Sources 134 (2004) 262276
Extended Kalman filtering for battery management systemsof
LiPB-based HEV battery packsPart 2. Modeling and identification
Gregory L. Plett,1Department of Electrical and Computer
Engineering, University of Colorado at Colorado Springs,
1420 Austin Bluffs Parkway, P.O. Box 7150, Colorado Springs, CO
80933-7150, USAReceived 27 January 2004; accepted 26 February
2004
Available online 28 May 2004
Abstract
Battery management systems in hybrid electric vehicle battery
packs must estimate values descriptive of the packs present
operatingcondition. These include: battery state of charge, power
fade, capacity fade, and instantaneous available power. The
estimation mechanismmust adapt to changing cell characteristics as
cells age and therefore provide accurate estimates over the
lifetime of the pack.
In a series of three papers, we propose a method, based on
extended Kalman filtering (EKF), that is able to accomplish these
goals ona lithium ion polymer battery pack. We expect that it will
also work well on other battery chemistries. These papers cover the
requiredmathematical background, cell modeling and system
identification requirements, and the final solution, together with
results.
In order to use EKF to estimate the desired quantities, we first
require a mathematical model that can accurately capture the
dynamics ofa cell. In this paper we evolve a suitable model from
one that is very primitive to one that is more advanced and works
well in practice.The final model includes terms that describe the
dynamic contributions due to open-circuit voltage, ohmic loss,
polarization time constants,electro-chemical hysteresis, and the
effects of temperature. We also give a means, based on EKF, whereby
the constant model parametersmay be determined from cell test data.
Results are presented that demonstrate it is possible to achieve
root-mean-squared modeling errorsmaller than the level of
quantization error expected in an implementation. 2004 Elsevier
B.V. All rights reserved.
Keywords: Battery management system (BMS); Hybrid electric
vehicle (HEV); Extended Kalman filter (EKF); State of charge (SOC);
State of health (SOH);Lithium-ion polymer battery (LiPB)
1. Introduction
This paper is the second in a series of three that de-scribe
advanced algorithms for a battery management sys-tem (BMS) for
hybrid electric vehicle (HEV) application.This BMS is able to
estimate battery state of charge (SOC),power fade, capacity fade
and instantaneous available power,and is able to adapt to changing
cell characteristics over timeas the cells in the battery pack age.
The algorithms havebeen implemented on a lithium-ion polymer
battery (LiPB)
Tel.: +1-719-262-3468; fax: +1-719-262-3589.E-mail addresses:
[email protected], [email protected](G.L. Plett).URL:
http://mocha-java.uccs.edu.
1 The author is also consultant to Compact Power Inc., Monument,
CO80132, USA. Tel.: +1-719-488-1600; fax: +1-719-487-9485.
pack, but we expect them to work well for other
batterychemistries.
The method that we use to estimate these parameters isbased on
Kalman filter theory. (There have been other re-ported methods for
SOC estimation that use Kalman filter-ing [1,2], but the method in
this series of papers expands onthese results and also differs in
some important respects, aswill be outlined later.) Kalman filters
are an intelligentandsometimes optimalmeans for estimating the
state of a dy-namic system. By modeling our battery system to
includethe wanted unknown quantities in the state, we may usethe
Kalman filter to estimate their values. An additional ben-efit of
the Kalman filter is that it automatically providesdynamic
error-bounds on these estimates as well. We ex-ploit this fact to
give aggressive performance from our bat-tery pack, without fear of
causing damage by overcharge oroverdischarge.
0378-7753/$ see front matter 2004 Elsevier B.V. All rights
reserved.doi:10.1016/j.jpowsour.2004.02.032
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G.L. Plett / Journal of Power Sources 134 (2004) 262276 263
The first paper [3] is an introduction to the problem.
Itdescribes the HEV environment and the requirement spec-ifications
for a BMS. The remainder of the paper is a brieftutorial on the
Kalman filter theory necessary to grasp thecontent of the remaining
papers; additionally, a nonlinearextension called the extended
Kalman filter (EKF) isdiscussed.
This second paper describes some mathematical cell mod-els that
may be used with this method. The HEV applicationis a very harsh
environment, with rate requirements up to andexceeding 20C and very
dynamic rate profiles. This is incontrast to relatively benign
portable-electronic applicationswith constant power output and
fractional C rates. Methodsfor estimating SOC that work well in
portable-electronic de-vices may not work well in the HEV
application. If preciseSOC estimation is required by the HEV, then
a very accuratecell model is necessary.
Results of lab tests on physical cells are presented andcompared
with model prediction. The best modeling resultsobtained to date
are so precise that the root-mean-squared(RMS) estimation error is
less than the quantization noisefloor expected in our battery
management system design.More importantly, the model allows very
precise SOCestimation, therefore allowing the vehicle controller to
con-fidently use the battery packs full operating range withoutfear
of over- or under-charging cells. This paper also givesan overview
of other modeling methods in the literatureand shows how an EKF may
be used to adaptively identifyunknown parameters in a cell model,
in real time, given testdata.
The third paper [4] covers the real-time parameter estima-tion
problem; namely, how to dynamically estimate SOC,power fade,
capacity fade, available power and so forth. AnEKF is used in
conjunction with the cell model. The cellmodel may be fixed, or may
itself have adaptable parame-ters so that the model tracks cell
aging effects. Details for apractical implementation are
discussed.
We now proceed by briefly reviewing cell models in theliterature
that have been proposed for SOC estimation. Weexplain why these do
not meet the requirements presentedin [3]. Several models from
Refs. [5,6] do meet the require-ments, and they are described in
detail here, together withsome new models and results. A method for
identifyingmodel parameters using an extended Kalman filter is
pre-sented, followed by conclusions.
2. Standard cell-modeling methods for SOC estimation
The literature documents a number of cell-modeling meth-ods for
SOC estimation. An excellent summary, in greaterdetail than can be
presented here, may be found in refer-ence [7]. Here, we
investigate to see whether any of thesemethods meets our needs.
Recall that our application is tomodel cell dynamics for the
purpose of SOC estimation inan HEV battery pack.
For this application, the cell model must be accurate for
alloperating conditions. These include: very high rates (up
toabout20C, far exceeding the low rates considered by manypapers in
the literature for portable electronic applications),temperature
variation in the automotive range of 30 to50 C, very dynamic rates
(unlike the more benign portableelectronic and battery electric
vehicle application). Chargingmust be accounted for in the
model.
We also require non-invasive methods using only readilyavailable
signals. This requirement is imposed by the HEVenvironment where
the BMS has no direct control over cur-rent and voltage experienced
by the battery packthese arein the domain of the vehicle controller
and inverter. We mustrely on such measurements as instantaneous
cell terminalvoltage, cell current and cell external
temperature.
Our cell chemistry also limits the range of approaches wemight
consider. Techniques specific to lead-acid chemistries,for example,
are not appropriate for LiPB cells.
2.1. Laboratory and chemistry-dependent methods
Several methods for direct SOC estimation simply cannotbe used
in our application:
1. A laboratory method for determining SOC is tocompletely
discharge a cell, recording dischargedampere-hours, to determine
its present remaining capac-ity. This is the most accurate SOC
measurement tech-nique, but is impractical in HEV as the battery
energyis wasted by the test, and the test cannot
dynamicallyestimate SOC.
2. Chemistry-dependent methods for other chemistries,such as
Coup de Fouet measurement, or measurementof electrolyte physical
properties for lead-acid batteries,are all inappropriate (as our
application uses LiPB cells).
3. Open-circuit voltage (OCV) measurements: If the cell
isallowed to rest for a long period, its terminal voltage de-cays
to OCV, and OCV may be used to infer SOC (vialookup table, for
example). However, long periods (some-times hours) of battery
inactivity must occur before theterminal voltage approaches OCV.
This method may notbe used for dynamic SOC estimation. (Other
complica-tions with this method include the dependence of OCVon
temperature, and presence of terminal voltage hys-teresis,
especially at low temperatures.)
2.2. Electro-chemical modeling
One approach to modeling cell electrical dynamics isto carefully
consider, at the molecular level, the variousprocesses that occur
within the cell. Accurate terminalvoltage prediction may be
achieved by these models (seeRef. [8], for example). However, it
would be difficult (ifpossible) to measure the many required
physical param-eters on a cell-by-cell basis in a high-volume
consumerproduct. We have not pursued this approach, although
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264 G.L. Plett / Journal of Power Sources 134 (2004) 262276
we employ many of the macroscopic concepts in ourmodels.
2.3. Impedance spectroscopy
Another broad category of cell modeling involves mea-suring cell
impedances over a wide range of ac frequenciesat different states
of charge [913]. Values of the modelparameters are found by
least-squares fitting to measuredimpedance values. SOC may be
indirectly inferred by mea-suring present cell impedance and
correlating them withknown impedances at various SOC levels. We
must also dis-count this method for our application, as we have no
directmethod to inject signals into cells to measure impedances.We
rely on the vehicle to generate and dissipate all energyflowing
through the battery pack.
2.4. Circuit models
A number of papers present equivalent circuit models ofcells
[1417]. Typically, a high-valued capacitor or voltagesource is used
to represent the open-circuit voltage. The re-mainder of the
circuit models the cells internal resistanceand more dynamic
effects such as terminal voltage relax-ation. From the OCV
estimate, SOC may be inferred viatable lookup. Both linear- and
nonlinear-circuit models maybe used. Our model has many
similarities to a circuit model,except that our fundamental state
is SOC, not OCV.
2.5. Coulomb counting
The final method discussed in the literature involves
SOCestimation directly via Coulomb counting. This may be
doneopen-loop, which is very sensitive to current measurementerror,
or closed-loop which can be much more accurate.The feedback
mechanism may be empirically designed [18]or may use a
mathematically optimized approach such asthe Kalman filtering
method [1,2] to generate the feedback.All Kalman filtering-based
methods by other authors in theliterature (with which we are
familiar) use a circuit model ofthe cell with voltage sources and
capacitor voltages repre-senting OCV and relaxation effects. OCV
may be estimatedand SOC inferred from OCV.
Our approach is also based on the Kalman filteringmethod, but
the fundamental aspect of our model that sets itapart from those
reported in the literature is that SOC itselfis required to be a
state of the system. The direct benefit ofthis approach is that the
Kalman filter automatically givesa dynamic estimate of the SOC and
its uncertainty (thisis discussed in greater detail in Ref. [4]).
That is, insteadof reporting the SOC to the vehicle controller (at
somepoint in time) to be about 55%, the algorithm is able toreport
that the SOC is 55 3%, for example. This allowsthe vehicle
controller to confidently use the battery packsfull operating range
without fear of over- or under-chargingcells.
3. An evolution of cell model structures
In order to use Kalman-based methods for a batterymanagement
system, we must first have a cell model in adiscrete-time
state-space form. Specifically, we assume theform
xk+1 = f(xk, uk)+ wk, (1)yk = g(xk, uk)+ vk, (2)
where xk is the system state vector at discrete-time index
k,where the state of a system comprises in summary formthe total
effect of past inputs on the system operation sothat the present
output may be predicted solely as a functionof the state and
present input. Values of past inputs are notrequired. The vector uk
is the measured exogenous systeminput at time k and wk is
unmeasured process noise thataffects the system state. The system
output is yk and vk ismeasurement noise, which does not affect the
system state.Equation (1) is called the state equation, (2) is
called theoutput equation, and f(, ) and g(, ) are (possibly
non-linear) functions, specified by the particular cell model
used.All of the system dynamics are represented in (1). Equation(2)
is a static relationship. In the models to follow, wk isused to
account for current-sensor error and inaccuracy ofthe state
equation, and vk is used to account for voltage sen-sor error and
inaccuracy of the output equation.
In the case where we wish to model a cells dynamicsusing (1) and
(2), the vector uk contains the instantaneouscell current ik. It
may also contain the cell temperature Tk,an estimate of the cells
capacity C, and/or an estimate ofthe cells internal resistance Rk,
for example. The systemoutput is typically a scalar but may be
vector valued as well.Here we consider the output to be the cells
loaded terminalvoltage (not its at-rest OCV). Our method constrains
thestate vector xk to include SOC as one component.
There are many possible candidates for (1) and (2), and forthe
choice of xk and uk. Here, we describe the developmentof some
modeling equations in an evolutionary sense. Thatis, we start with
a very simple model, and gradually addcomplexity to better
represent the true cell dynamics. Inorder to justify the changes in
the model, we compare modeldynamics to cell dynamics based on data
collected from celltests. We first describe the cell tests, and
then develop themodel structures.
3.1. Cell tests for model fitting
In order to compare the abilities of the proposed models
tocapture a cells dynamics, we gathered data from a prototypeLiPB
cell. The cell comprises a LiMn2O4 cathode, an artifi-cial graphite
anode, is designed for high-power applications,has a nominal
capacity of 7.5 Ah and a nominal voltage of3.8 V. For the tests, we
used a Tenney thermal chamber set at25C and an Arbin BT2000 cell
cycler. Each channel of theArbin was capable of 20 A current, and
ten channels were
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G.L. Plett / Journal of Power Sources 134 (2004) 262276 265
Fig. 1. Plots showing SOC vs. time and rate vs. time for
pulsed-current cell tests. Discharge portion of test is shown in
(a); charge portion of test isshown in (b). Dark line is SOC, gray
line is current.
connected in parallel to achieve currents of up to 200 A.
Thecyclers voltage measurement accuracy was 5 mV and itscurrent
measurement accuracy was 200 mA.
Two types of cell tests were performed for the work re-ported in
this paper. The first type comprised a sequence ofconstant-current
discharge pulses and rests followed by a se-quence of
constant-current charge pulses and rests. The cellstarted fully
charged before the test began. Discharge currentpulses from 150
down to 1 A, and charge pulses from 150down to 1 A were used. The
current and SOC profiles forthis test are shown in Fig. 1(a) and
(b). Frame (a) shows thedischarge portion of the test and frame (b)
shows the chargeportion of the test. Data points (including
voltage, current,ampere-hours discharged and ampere-hours charged)
werecollected once per second.
The second test was a sequence of 16 urban dynamome-ter driving
schedule (UDDS) cycles, separated by 40 A dis-charge pulses and 5
min rests, and spread over the 9010%SOC range. The SOC as a
function of time is plotted inFig. 2(a), and rate as a function of
time for one of the UDDScycles is plotted in Fig. 2(b). We see that
SOC increases byabout 5% during each UDDS cycle, but is brought
downabout 10% during each discharge between cycles. The en-tire
operating range for these cells (1090% SOC) is excitedduring the
cell test.
0 50 100 150 200 250 300 350 400 4500
102030405060708090
100SOC as a function of time
Time (min.)
SOC
(perce
nt)
205 215 225 235 245-60
-40
-20
0
20
40
60
80Current for one UDDS cycle (zoom)
Time (min.)
Curre
nt (A
)
(b)(a)
Fig. 2. Plots showing SOC vs. time and rate vs. time for UDDS
cell tests. SOC is shown in (a); rate for one UDDS cycle is shown
in (b).
The data was used to identify parameters of the cell mod-els to
be described in the next sections. The goal is to havethe cell
model output resemble the cell terminal voltage un-der load as
closely as possible, at all times, when the cellmodel input is
equal to the cell current. Model fit was judgedby comparing
root-mean-squared estimation error (estima-tion error equals cell
voltage minus model voltage) over theportions of the cell tests
where SOC was between 5 and95%. Model error outside that SOC range
was not consid-ered as the HEV pack operation design limits are
1090%SOC.
3.2. SOC as a state-vector component
The one constant requirement for the cell model is thatwe
constrain SOC, denoted as zk, to be a member of thestate vector xk.
To be careful, we give a list of definitionsculminating in our
understood definition of SOC.
Definition: A cell is fully charged when its voltage reachesv =
vh after being charged at infinitesimal current levels.Here, we use
vh = 4.2 V at room temperature (25 C).
Definition: A cell is fully discharged when its voltagereaches v
= vl after being drained at infinitesimal currentlevels. Here, we
use vl = 3.0 V at room temperature.
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266 G.L. Plett / Journal of Power Sources 134 (2004) 262276
Definition: The capacity of a cell C is the maximum num-ber of
ampere-hours that can be drawn from the cell be-fore it is fully
discharged, at room temperature, startingwith the cell fully
charged.
Definition: The nominal capacity Cn of the cell is thenumber of
ampere-hours that can be drawn from the cellat room temperature at
the C/30 rate, starting with the cellfully charged.
Definition: The SOC of the cell is the ratio of the remain-ing
capacity to the nominal capacity of the cell, where theremaining
capacity is the number of ampere-hours thatcan be drawn from the
cell at room temperature at theC/30 rate before it is fully
discharged.
With these definitions in place, we can then investigatesome
mathematical relations involving SOC. Particularly,
z(t) = z(0) t
0
ii()
Cnd, (3)
where z(t) is the cell SOC, i(t) is instantaneous cell
current(assumed positive for discharge, negative for charge), andCn
is the cell nominal capacity. Cell Coulombic efficiencyi is i = 1
for discharge, and i = 1 for charge.
Using a rectangular approximation for integration and asuitably
small sampling periodt, a discrete-time approx-imate recurrence may
then be written as
zk+1 = zk (it
Cn
)ik. (4)
Eq. (4) is the basis for including SOC in the state vector ofthe
cell model as it is in state equation format already, withSOC as
the state and ik as the input.
The cell model may be completed by adding additionalstates, as
necessary, and an output equation. Here, we firstrevisit an output
equation from an earlier paper [5], and showhow it may be enhanced.
Next, we add a state to the modelto account for cell hysteresis.
Thirdly, we add dynamics tothe state to model cell terminal voltage
relaxation. Finally,we discuss adding temperature dependence to the
models.
3.3. Models with only SOC as a state
The first three model structures that we investigate havestate
vector xk = zk. That is, the only state in the stateequation (1) is
SOC. These models can estimate cell terminalvoltage in a limited
way, and are improved upon later usingmultiple states.
3.3.1. The combined modelWith SOC available as part of the model
state, terminal
voltage may be predicted in a number of different ways.Several
different forms are adapted from reference [19]. Shepherd model: yk
= E0 Rik Ki/zk. Unnewehr universal model: yk = E0 Rik Kizk. Nernst
model: yk = E0Rik+K2 ln(zk)+K3 ln(1zk).
In these models, yk is the cell terminal voltage, R is thecell
internal resistance (different values may be used
forcharge/discharge and at different SOC levels if desired),Ki
isthe polarization resistance and K1, K2 and K3 are constantschosen
to make the model fit the data well. All of the terms ofthese
models may be collected to make a combined modelthat performs
better than any of the individual models alone.This model is
zk+1 = zk (it
Cn
)ik,
yk = K0 Rik K1zk
K2zk +K3 ln(zk)+K4 ln(1 zk).
The unknown quantities in the combined model may be es-timated
using a system identification procedure. This modelhas the
advantage of being linear in the parameters; that is,the unknowns
occur linearly in the output equation. Givena set of N cell
inputoutput three-tuples {yk, ik, zk}, the pa-rameters may be
solved for in closed form using a resultfrom least-squares
estimation. This simple off-line (batch)method is as follows: We
first form the vector
Y = [y1, y2, . . . , yN]T,and the matrix
H = [h1, h2, . . . , hN]T.The rows of H are
hTj =[
1, i+j , ij ,
1zj, zj, ln(zj), ln(1 zj)
],
where i+j is equal to ij if ij > 0, ij is equal to ij if ij
,1, ik < ,sk1, |ik| .
M(zk) is half the difference between the two legs of
thecharge/discharge curve, minus the Rik loss, and is plotted
inFig. 5(b). Here, we use a constant value for M.
4 Successively smaller concentric minor loops may be obtained
byalternating shorter and shorter charge and discharge pulses,
eventuallyconverging on the mean of the two values of Fig. 5(a) at
each SOC.Therefore, we compute OCV as a function of SOC as the mean
of thetwo legs of the major hysteresis loop.
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G.L. Plett / Journal of Power Sources 134 (2004) 262276 269
This model type is also linear in the parameters. Off-linesystem
identification is done as follows: We first form thevector
Y= [y1 OCV(z1), y2 OCV(z2), . . . , yn OCV(zN)]T,
and the matrix
H = [h1, h2, . . . , hN]T.
The rows of H are
hTj = [i+j , ij , sj].
0 10 20 30 40 50 60 70
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling discharge: Combined model
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
0 10 20 30 40 50 60 70 80 90
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling charge: Combined model
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
0 10 20 30 40 50 60 70
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling discharge: Simple model
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
0 10 20 30 40 50 60 70 80 90
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling charge: Simple model
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
0 10 20 30 40 50 60 70
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling discharge: Zero state hysteresis model
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
0 10 20 30 40 50 60 70 80 90
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling charge: Zero state hysteresis model
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
(a) (d)
(b)
(c) (f)
(e)
Fig. 6. Results of cell modeling using models with only SOC as a
state for the pulsed-current cell tests. Discharge portion of test
is shown in (a)(c);charge portion of test is shown in (d)(f). The
gray line is the measured cell voltage, and the black line is the
model prediction.
Again, we see that Y = H, where T = [R+, R,M] is thevector of
unknown parameters. We solve for the parameters using the known
matrices Y and H as = (HTH)1HTY .
Results comparing the zero-state hysteresis model cellvoltage
estimation with the cells true voltage for thepulsed-current test
are shown in Figs. 6(c) and (f). Fig. 6(c),shows the comparison
over discharge pulses, and Fig. 6(f)shows the comparison over
charge pulses. The RMS cellmodel estimation error over the test
shown in Fig. 6 islisted in Table 1. Performance of the zero-state
hysteresismodel is consistently better than that of the simple
model.Results comparing the zero-state hysteresis model cell
volt-age estimation with the cells true voltage for one cycle
of
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270 G.L. Plett / Journal of Power Sources 134 (2004) 262276
the UDDS test are shown in Fig. 8(c). Similar commentsapply.
3.4. Models with SOC and additional states
In order to better estimate cell voltage effects that are
cou-pled to the history of the cells input current, we must
makemodifications to the model state equation (1). We examinetwo
additions in the following sections.
3.4.1. The one-state hysteresis modelThe zero-state hysteresis
model is an improvement over
the simple model, but only crudely approximates the under-lying
phenomenon. Whereas the level of hysteresis slowlychanges as the
cell is charged or discharged, the model es-timates hysteresis as
immediately flipping between its max-imum positive and negative
values when the sign of currentchanges.
The slow transition may be modeled by adding a hys-teresis state
to the model state equation (1). The hystere-sis state is not a
differential equation in time, but in SOC(or, ampere-hours). Let
h(z, t) be the hysteresis voltage as afunction of SOC and time, and
let z = dz/dt. Then,dh(z, t)
dz= sgn(z)(M(z, z) h(z, t)),
where M(z, z) is a function that gives the maximum po-larization
due to hysteresis as a function of SOC and therate-of-change of
SOC. Specifically, M(z, z) is positive forcharge (z > 0) and is
negative for discharge (z < 0). TheM(z, z) h(z, t) term in the
differential equation states thatthe rate-of-change of hysteresis
voltage is proportional to thedistance away from the major
hysteresis loop, leading to akind of exponential decay of voltage
to the major loop. Theterm in front of this has a positive constant
, which tunesthe rate of decay, and sgn(z), which forces the
equation tobe stable for both charge and discharge.
In order to fit the differential equation for h(z, t) into
ourmodel, we must manipulate it to be a differential equation
intime, not in SOC. We accomplish this by multiplying bothsides of
the equation by dz/dt
dh(z, t)dz
dzdt
= sgn(z)(M(z, z) h(z, t))dzdt.
Note that dz/dt = ii(t)/Cn, and that z sgn(z) = |z|. Thus,
h(t) = ii(t)Cn
h(t)+ii(t)Cn
M(z, z).This may be converted into a difference equation for
ourdiscrete-time application using standard techniques (assum-ing
that i(t) andM(z, z) are constant over the sample period):
hk+1 = exp(iiktCn
)hk
+(
1 exp(iiktCn
))
M(z, z).
Note that this is a linear-time-varying system as the
factorsmultiplying the state and input change with ik and hencewith
time. If we define F(ik) = exp(|iikt/Cn|), thenthe overall
state-space equations for the one-state hysteresismodel
are[hk+1zk+1
]=[F(ik) 0
0 1
][hk
zk
]
+ 0 (1 F(ik))it
Cn0
[ ik
M(z, z)
],
yk = OCV(zk) Rik + hk.
Results comparing the one-state hysteresis model cellvoltage
estimation with the cells true voltage for thepulsed-current test
are shown in Fig. 7(a) and (d). Fig. 7(a),shows the comparison over
discharge pulses, and Fig. 7(d)shows the comparison over charge
pulses. The RMS cellmodel estimation error over the test shown in
Fig. 7 islisted in Table 1. Performance of the one-state
hysteresismodel is consistently better than the simpler models.
Re-sults comparing the one-state hysteresis model cell
voltageestimation with the cells true voltage for one cycle of
theUDDS test are shown in Fig. 8(d). Similar comments apply.
3.4.2. The enhanced self-correcting (ESC) modelA significant
element missing from these models is a de-
scription of time constants during pulsed current events. Ifa
cell is allowed to rest, it takes some time for the voltageto
completely relax to its rest voltage. If a cell is pulsedwith
current, it takes time for the voltage to converge to
itssteady-state level. These time constants, which describe
thephenomenon we henceforth refer to as the relaxation effect,may
be implemented as a low-pass filter on ik. Since thecell model must
accurately predict its behavior in a dynamicHEV environment, we
find it is essential to include relax-ation effects.
Early attempts to model the relaxation effect includedfiltering
the state-of-charge as well as the input current (cf.the filter
state cell model [5]). While this model couldfit voltage data
reasonably well, it had the unfortunate sideeffect that SOC
estimation using an EKF was not reliable.The output equation had
the form
yk = OCV(zk)+ filt(zk)+ filt(ik) Rik,
where filt() is some dynamic operation filtering its operand.The
EKF had problems with this model because thecredit/blame portion of
the algorithm could not reliably de-termine whether model error was
due to bad SOC throughthe OCV() function or through the filt(zk)
function, ordue to bad filter states in the filt(ik) function. In
particular,some simple cell tests running the EKF showed a lack
ofrobustness:
-
G.L. Plett / Journal of Power Sources 134 (2004) 262276 271
0 10 20 30 40 50 60 70
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling discharge: One state hysteresis
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
0 10 20 30 40 50 60 70 80 90
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling charge: One state hysteresis
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
0 10 20 30 40 50 60 70
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling discharge: ESC, 2 filter states
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
0 10 20 30 40 50 60 70 80 90
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling charge: ESC, 2 filter states
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
0 10 20 30 40 50 60 70
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling discharge: ESC, 4 filter states
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
0 10 20 30 40 50 60 70 80 90
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Modeling charge: ESC, 4 filter states
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
(a) (d)
(b) (e)
(c) (f)
Fig. 7. Results of cell modeling using models with multiple
states for the pulsed-current cell tests. Discharge portion of test
is shown in (a)(c); chargeportion of test is shown in (d)(f). The
gray line is the measured cell voltage, and the black line is the
model prediction.
1. A constant-current discharge/charge should make theSOC ramp
down/up at the slope i/Cn (A/Ah). In practice,the slope using the
filter state model was often wrong.
2. During a rest period, cell terminal voltage converges toOCV
(neglecting hysteresis effects) and estimated SOCshould converge to
the SOC predicted by OCV. In theimplementation, we observed SOC to
drift considerably,not converging to the correct value.
The model that we will develop in this section, called
theenhanced self-correcting model, forces yk to converge toOCV
after a rest period and it forces yk to converge to
OCVRik for a constant-current discharge/charge. To meetthese
requirements, with hysteresis added, the output equa-tion needs to
have the form
yk = OCV(zk) fn(zk)
+ hkfn(zk,ik)
+ filt(ik) Rik fn(ik)
.
In this equation, SOC and hysteresis contribute the long-termdc
level (bias) to the output and ik and its history contributethe
short-term variation around this level. SOC itself is nolonger
filtered as in the filter state modelit makes nosense to have a
moving bias point.
-
272 G.L. Plett / Journal of Power Sources 134 (2004) 262276
205 215 225 235 2453.7
3.8
3.9
4.0
4.1
4.2Modeling (zoom): Combined model
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
205 215 225 235 2453.7
3.8
3.9
4.0
4.1
4.2Modeling (zoom): Simple model
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
205 215 225 235 2453.7
3.8
3.9
4.0
4.1
4.2Modeling (zoom): Zero state hysteresis model
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
205 215 225 235 2453.7
3.8
3.9
4.0
4.1
4.2Modeling (zoom): One state hysteresis
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
205 215 225 235 2453.7
3.8
3.9
4.0
4.1
4.2Modeling (zoom): ESC, 2 filter states
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
205 215 225 235 2453.7
3.8
3.9
4.0
4.1
4.2Modeling (zoom): ESC, 4 filter states
Time (min.)
Volta
ge (V
)
True voltageEstimated voltage
(a) (d)
(b) (e)
(c) (f)
Fig. 8. Results of cell modeling for the UDDS cell tests. The
gray line is the measured cell voltage, and the black line is the
model prediction.
The filter filt() must satisfy two criteria: (1) after along
rest period its output must be zero so that yk OCV + hk; (2) during
a constant-current discharge/charge,its output must converge to
zero so that yk OCV +hk Rik. The first criterion is satisfied by a
stable linearfilter, and the second is satisfied by a linear filter
with zerodc gain. Both of these may be enforced in the filter
de-sign.
A linear filter may be implemented in a state-space formas
fk+1 = Affk + Bf ik,yfk = Gfk,
where Af is the state-transition matrix of the filter, Bf is
theinput matrix of the filter, G is the output matrix of the
filter,and fk is the filter state. The eigenvalues of the Af
matrixare the poles of the filter and determine its stability.
Thefilter is stable if max|eig(Af)| < 1. The location of poles
de-termine the systems dynamic behavior: poles near +1 haveslowly
decaying dynamics, poles near zero decay quickly,negative poles
oscillate. Complex-conjugate poles also oscil-late, and do not
appear to improve performance by their in-clusion. Therefore, it is
sufficient to have an Af matrix of theform Af = diag(), where is a
vector comprising the polelocations. Stability is ensured if all 1
< j < 1. The Bfmatrix may be chosen arbitrarily so long as no
entry is zero.
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G.L. Plett / Journal of Power Sources 134 (2004) 262276 273
Now that we have guaranteed stability, yk OCV + hkduring rest.
By carefully selecting the G matrix, we canensure a zero dc-gain of
the filter so that yk OCV+hkRik during constant-current profiles.
The gain of the filter is(where nf is the number of filter states,
determined duringsystem identification to fit cell data):
G(I A)1B = 0,G
[diag
(1
1 k
)]B = 0,
nfk=1
gk
1 k = 0.
If we let g1 through gnf1 be found by a
system-identificationprocedure and assuming that Bf = [ 1 1 ]T,
then thezero dc-gain constraint fixes gnf as
gnf = nfk=1
gk(1 nf )(1 k) .
So, the full self-correcting model is fk+1hk+1zk+1
=
diag() 0 00 F(ik) 0
0 0 1
fkhkzk
+
1 00 (1 F(ik))
itCn
0
[
ik
M(z, z)
],
yk = OCV(zk) Rik + hk + Gfk.Results comparing the ESC model cell
voltage estimationwith the cells true voltage for the
pulsed-current test areshown in Figs. 7(b) and (e) for nf = 2 and
in Figs. 7(c)and (f) for nf = 4. Figs. 7(b) and (c), show the
comparisonover discharge pulses, and Figs. 7(e) and (f) show the
com-parison over charge pulses. The RMS cell model estimationerror
over the tests shown in Fig. 7 are listed in Table 1. Per-formance
is significantly improved by the addition of filter
30 25 20 15 10 5 0 5 10 15 20 25 30 35 40 450
102030405060708090
100
Temperature (C)
RM
S m
odel
ing
erro
r (mV
)
RMS modeling error versus temperatureCombined modelSimple
modelZero state hysteresisOne state hysteresisESC, nf=2ESC,
nf=4
0 10 20 30 40 50 60
11.8
19.1
22.9
35.0
53.8
32.0
Zero-state Hysteresis
Simple
Combined
ESC, nf=4
ESC, nf=2
One state Hysteresis
Average RMS modeling error (mV)
Mod
el s
truct
ure
Summary modeling performance over temperature
(a) (b)
Fig. 9. Results when modeling over a temperature range. Frame
(a) shows the individual modeling results, and frame (b) compares
the average modelingresults.
states. We do not see much improvement by increasing nf be-yond
4. Results comparing the ESC model cell voltage esti-mation with
the cells true voltage for one cycle of the UDDStest are shown in
Figs. 8(e) for nf = 2 and (f) for nf = 4.
3.5. Adding temperature dependence to the models
Thus far, we have discussed only how to model cell dy-namics at
one specific temperature. We now embark on abrief discussion on how
to incorporate temperature depen-dence into the models.
A very simple method, and the one we tried first, was touse a
table of different models, where each model had pa-rameters
optimized for a specific temperature. For example,we used sixteen
models over the temperature range 30 to45 C in increments of 5 C.
This worked well so long asthe cell under test had temperature
equivalent to one of thesixteen stored models. If the temperature
was between twostored model values, we linearly interpolated model
param-eters between the parameters of the models in the table.
Thisdid not work well.
We found that two adjacent models in the table did
notnecessarily have similar parameters. Individually optimiz-ing
model parameters at specific temperatures resulted invalues that
were over-fit to the data and did not generalizewell to cases not
previously seen. We remedied the problemby performing joint
optimization over the entire temperaturerange, where every
parameter was represented by a contin-uous polynomial of
temperature (fourth order). This forcednearby models to have
similar parameter values. Althoughjoint optimization did not result
in modeling errors as lowas when individually optimized, the
generalization perfor-mance was much better.
Fig. 9 shows the results for joint optimization overthe entire
temperature range, and Table 1 lists some nu-meric values
corresponding to the plots. The cell data wascollected from UDDS
tests similar to those described inSection 3.1, but performed at 16
controlled temperaturesfrom 30 to 45 C, in increments of 5C. At
lower temper-atures, the magnitude of the current had to be scaled
downso as not to exceed voltage limits (due to increased cell
-
274 G.L. Plett / Journal of Power Sources 134 (2004) 262276
resistance at lower temperatures), and hence more UDDScycles had
to be completed to cover the desired SOC range,but the tests were
otherwise the same. In frame (a), theRMS modeling errors for the
jointly optimized models areplotted versus temperature for the
different cell models.In frame (b), the average RMS modeling errors
over alltemperatures are presented as a bar graph.
We see that the combined model appears to perform well.However,
this is an artifact. The {K0, . . . , K4} points werefound to
over-fit the measured data, so that the resultingcurve plotted with
their values did not resemble OCV to anydegree of fidelity. While
the simple model has worse nu-meric indicators of performance, it
generalized better thanthe combined model. By adding the crude
model of hystere-sis, we see a significant performance jump,
especially at coldtemperatures where hysteresis is most evident in
our cellsdynamics. Adding the dynamics of a state to the
hysteresismodel also improves performance, again with the
greatestgains at low temperatures. Filter states also contribute to
per-formance gains. The final model, enhanced-self-correctingwith
nf = 4 gave the best performance in all cases. Byincreasing the
number of filter states we would expect con-tinued performance
gains, at the cost of greater complexity.Note that the
cold-temperature performance is not improvedas much, in relative
terms, by adding filter states, so it islikely that some
cold-temperature phenomena is not yetbeing modeled well.
Conceivably, a second hysteresis statecould be added to the model
to improve performance here.We have not investigated this
possibility as yet.
4. System identification
The first three system models introduced in this paper arelinear
in the parameters. This makes identifying the valuesof the model
parameters straightforward using least-squaresestimation, and has
been discussed earlier. When the modelis not linear in the
parameters, as in the remaining systemmodels, this method may not
be used. We must turn to moreadvanced methods.
Here we look at one method in particular. We know thata Kalman
filter or extended Kalman filter may be used toestimate the state
of a dynamic system given noisy mea-surements; e.g., to estimate
the cell SOC. We may also usean extended Kalman filter to perform
system identificationgiven clean measurements. To do so, we require
a state-spacemodel describing the dynamics of the parameters of
thesystem model. We will use the Kalman filter as an
optimumobserver of these parameter values, creating an estimate .In
electro-chemical cells, the true parameters will changeonly very
slowly, so we model them as constant with somesmall
perturbation:
k+1 = k + rk.The small white noise input rk is fictitious, but
modelsthe slow change in the parameters of the system plus the
Table 2Summary of the nonlinear extended Kalman filter for
system identification[21]Nonlinear state-space modela
k+1 = k + rkdk = g(xk, uk, k)+ ek
Definition
Ck =dg(xk, uk, )
d
=
k
InitializationFor k = 0, set+0 = E[0]+,0 = E[(0
+0 )(0 +0 )T]
ComputationFor k = 1, 2, . . . compute
State estimate time update: k = +k1Error covariance time
update:
,k= +
,k1 +rKalman gain matrix: Lk = ,k(C
k)
T[Ck,k(Ck)
T +e]1
State estimate measurement update:+k = k + Lk[yk g(xk, uk, k
)]
Error covariance measurement update: +,k
= (I LkCk),ka rk and ek are independent, zero-mean, Gaussian
noise processes of
covariance matrices r and e, respectively.
infidelity of the model structure to capture all of the
celldynamics.
The output equation required for Kalman-filter
systemidentification must be a measurable function of the
systemparameters. We usedk = g(xk, uk, k)+ ek,where g(, , ) is the
output equation of the system modelbeing identified, and ek models
the sensor noise and model-ing error. We compare dk computed using
k to the measuredcell output, and adapt k to minimize the
difference.
We can create an extended Kalman filter using thisstate-space
model and cell data to estimate the system pa-rameters as
summarized in Table 2. We initialize the stateestimate with our
best information re. the state value: +0 =E[0], and the state
estimation error covariance matrix:+,0
= E[( +0 )( +0 )T].The time update propagates the state estimate
as k =
+k1 since the parameters are assumed constant, and theerror
covariance as
,k= +
,k1 +r to account for theadded uncertainty due to the fictitious
noise input rk. Theeffect of adding r is to increase the estimates
uncertainty,and to allow adaptation to .
The extended Kalman filter gain matrix is computed bylinearizing
the state-space models output equation. We com-pute
Ck =dg(xk, uk, )
d
=
k
,
Lk = ,k(Ck)
T[Ck,k(Ck)
T +e]1.
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G.L. Plett / Journal of Power Sources 134 (2004) 262276 275
Next, we measure the true cell output yk and compare it to
themodel output g(xk, uk, k ). To do so, we need to simulatethe
model in parallel with the real cell to have an appropriatevalue of
xk available. The difference between cell outputand model output
can be attributed to noises and modelingerror. The extended Kalman
filter adapts to minimize thisdifference
+k = k + Lk[yk g(xk, uk, k )].Finally, the measurement update is
applied to the state errorcovariance matrix
+,k
= (I LkCk),k.This has the effect of decreasing the modeling
uncertaintydue to the measurement update. All terms are accounted
for,and the algorithm is complete.
4.1. Extended Kalman-filter system identification for
theone-state hysteresis model
The details for applying this method to any particular cellmodel
are differentiated by the calculation of Ck . For theone-state
hysteresis model, let the vector of parameters be
= [R+ R M ]T.R+ is the cell resistance when current is positive,
and Ris the cell resistance when current is negative. M is
themaximum hysteresis voltage, and is the hysteresis rateconstant,
which is part of F(ik). To calculate Ck we requiredg(xk, uk, )
d= g(xk, uk, )
+ g(xk, uk, )
xk
dxkd
, (6)
dxkd
= f(xk1, uk1, )
+ f(xk1, uk1, )xk1
dxk1d
. (7)
The derivative calculations are recursive in nature, andevolve
over time as the state evolves. The term dx0/d isinitialized to
zero unless side information gives a better es-timate of its value.
We see that in order to calculate Ck forany specific model
structure, we require methods to calcu-late the partial derivatives
in (6) and (7). For the one-statehysteresis model we have
g(xk, uk, )
= [ i+ i 0 0 ], g(xk, uk, )
xk=[
1OCV(zk)
zk
],
f(xk1, uk1, )
= 0 0 (1 Fk1) sgn(ik1) (M hk1)
iik1tCFk1
0 0 0 0
, f(xk1, uk1, )
xk1=[Fk1 0
0 1
].
Note that the OCV(zk)/zk term is never needed, as italways
multiplies zero. For this particular model, we cansimplify the
calculations by removing the multiplies by zero:
dg(xk, uk, )d
= [ i+ i 0 0 ] + dhkd
,dhkd
=[
0 0 (1 Fk1) sgn(ik1)(M hk1)iik1tC
Fk1]+ Fk1 dhk1d .
4.2. Extended Kalman-filter system identification for
theenhanced self-correcting model
For the enhanced self-correcting model, let the vector
ofparameters be
= [R+ R g1 gnf1 1 nf M ]T,where = tanh () and is the vector of
filter pole lo-cations. We use the tanh () function during system
identi-fication because it forces filter poles to remain within
1(i.e., stable) regardless of the value of . When calculatingthe
partial derivatives we must remember that since gnf =nf1i=1 gi(1 nf
)/(1 i), it is not independently iden-tified but is computed from
the g1, . . . , gnf1 terms. Thisalso forces the derivatives to be
more complicated than aquick glance would indicate. That is,
g(xk, uk, )
g1= fk,1 1 nf1 1 fk,nf ,
and so forth. Also, since i = tanh (i), it can never beunity, so
division by zero is impossible in the derivativecomputation. With
this in mind, the partial derivativesin (6) and (7) may be computed
in a straightforwardway.
5. Conclusions
This paper has proposed five mathematical state-spacestructures
for the purpose of modeling LiPB HEV cell dy-namics for their
eventual role in HEV BMS algorithms.Models with a single-state are
very simple, but perform thepoorest. Adding hysteresis and filter
states to the model aidsperformance, at some cost in
complexity.
We have also seen how to identify the parameters of thecell
models given cell-test data. Models that are linear in
theparameters may have their parameters fit in a very
straight-forward way using methods from
least-squares-estimationtheory. Models with more dynamics than
simply SOC re-quire more sophisticated techniques. One possibility
is to
use an extended Kalman filter to identify the cell parametersin
an on-line or off-line manner.
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276 G.L. Plett / Journal of Power Sources 134 (2004) 262276
In the third paper [4], we will employ extended Kalmanfiltering
from [3], using the cell models developed here, toimplement HEV BMS
algorithms. We will see how to useEKF to estimate SOC and all other
model states as the sys-tem operates. This model state will then
allow us to accu-rately compute a dynamic estimate of available
power. Wecan additionally employ a technique called dual
extendedKalman filtering to simultaneously estimate cell state
andparameters, allowing tracking of cell power fade and capac-ity
fade, for example. Finally, the parameter data and SOCestimate may
be combined to determine which cells in thepack must have their
charge levels modified in order to bringthe pack into
equalization.
Acknowledgements
This work was supported in part by Compact Power Inc.(CPI). The
use of company facilities, and many enlighteningdiscussions with
Drs. Mohamed Alamgir and Dan Riversand others are gratefully
acknowledged.
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Extended Kalman filtering for battery management systems of
LiPB-based HEV battery packsPart 2. Modeling and
identificationIntroductionStandard cell-modeling methods for SOC
estimationLaboratory and chemistry-dependent
methodsElectro-chemical modelingImpedance spectroscopyCircuit
modelsCoulomb counting
An evolution of cell model structuresCell tests for model
fittingSOC as a state-vector componentModels with only SOC as a
stateThe combined modelThe simple modelThe zero-state hysteresis
model
Models with SOC and additional statesThe one-state hysteresis
modelThe enhanced self-correcting (ESC) model
Adding temperature dependence to the models
System identificationExtended Kalman-filter system
identification for the one-state hysteresis modelExtended
Kalman-filter system identification for the enhanced
self-correcting model
ConclusionsAcknowledgementsReferences