U.S. patent application number 13/895096 was filed with the patent office on 2013-11-21 for battery system and method with parameter estimator.
This patent application is currently assigned to Robert Bosch GmbH. The applicant listed for this patent is Robert Bosch GmbH. Invention is credited to Jasim Ahmed, Nalin Chaturvedi, Aleksandar Kojic, Michael Schoenleber.
Application Number | 20130311115 13/895096 |
Document ID | / |
Family ID | 48483241 |
Filed Date | 2013-11-21 |
United States Patent
Application |
20130311115 |
Kind Code |
A1 |
Chaturvedi; Nalin ; et
al. |
November 21, 2013 |
Battery System and Method with Parameter Estimator
Abstract
An electrochemical battery system in one embodiment includes at
least one electrochemical cell, a first sensor configured to
generate a current signal indicative of an amplitude of a current
passing into or out of the at least one electrochemical cell, a
second sensor configured to generate a voltage signal indicative of
a voltage across the at least one electrochemical cell, a memory in
which command instructions are stored, and a processor configured
to execute the command instructions to obtain the current signal
and the voltage signal, and to generate kinetic parameters for an
equivalent circuit model of the at least one electrochemical cell
by obtaining a derivative of an open cell voltage (U.sub.ocv),
obtaining an estimated nominal capacity (C.sub.nom) of the at least
one electrochemical cell, and estimating the kinetic parameters
using a modified least-square algorithm with forgetting factor.
Inventors: |
Chaturvedi; Nalin;
(Sunnyvale, CA) ; Schoenleber; Michael; (Mountain
View, CA) ; Ahmed; Jasim; (Mountain View, CA)
; Kojic; Aleksandar; (Sunnyvale, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Robert Bosch GmbH |
Stuttgart |
|
DE |
|
|
Assignee: |
Robert Bosch GmbH
Stuttgart
DE
|
Family ID: |
48483241 |
Appl. No.: |
13/895096 |
Filed: |
May 15, 2013 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
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61647904 |
May 16, 2012 |
|
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61647926 |
May 16, 2012 |
|
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61647948 |
May 16, 2012 |
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Current U.S.
Class: |
702/63 |
Current CPC
Class: |
G01R 31/382 20190101;
G01R 31/388 20190101; G01R 31/389 20190101; G01R 31/367 20190101;
G01R 31/3842 20190101; G06F 17/11 20130101; H01M 10/446 20130101;
Y02E 60/10 20130101 |
Class at
Publication: |
702/63 |
International
Class: |
G01R 31/36 20060101
G01R031/36; H01M 10/44 20060101 H01M010/44 |
Claims
1. An electrochemical battery system, comprising: at least one
electrochemical cell; a first sensor configured to generate a
current signal indicative of an amplitude of a current passing into
or out of the at least one electrochemical cell; a second sensor
configured to generate a voltage signal indicative of a voltage
across the at least one electrochemical cell; a memory in which
command instructions are stored; and a processor configured to
execute the command instructions to obtain the current signal and
the voltage signal, and to generate kinetic parameters for an
equivalent circuit model of the at least one electrochemical cell
by obtaining a derivative of an open cell voltage (U.sub.ocv) with
respect to a state of charge (SOC), obtaining an estimated nominal
capacity (C.sub.nom) of the at least one electrochemical cell, and
estimating the kinetic parameters using a modified least-square
algorithm with forgetting factor.
2. The system of claim 1, wherein the equivalent circuit model
comprises: an equivalent resistance (R.sub.e); and a parallel
circuit in series with the R.sub.e, the parallel circuit including
a parallel circuit resistance (R.sub.1) and a parallel circuit
capacitance (C.sub.1), with the kinetic parameters including
R.sub.e, R.sub.1, and C.sub.1.
3. The system of claim 2, wherein the equivalent circuit model in
continuous time is written as: ( x . 1 x . 2 ) = ( 0 0 0 - 1 / ( R
1 C 1 ) ) ( x 1 x 2 ) + ( - 1 / C nom 1 / R 1 C 1 ) u , and
##EQU00008## y = U O C V ( x 1 ) + ( 0 - R 1 ) ( x 1 x 2 ) + ( - R
e ) u , ##EQU00008.2## wherein "u" is a current applied to the at
least one electrochemical cell, "y" is a measured voltage across
the at least one electrochemical cell, "x.sub.1" is a SOC of the at
least one electrochemical cell, and "x.sub.2" is the current
(i.sub.1) through the R.sub.1.
4. The system of claim 2, wherein generating the kinetic parameters
is based upon defining a parametric form "" as: = s 2 y + s L { U O
C V ' ( x ^ 1 ) u C nom } ##EQU00009## wherein "(.cndot.)"
represents a Laplace transform, "u" is a current applied to the at
least one electrochemical cell, "y" is a measured voltage across
the at least one electrochemical cell, "s" represents a complex
number with real numbers .sigma. and .omega., "" represents a
higher order filter with a cut-off frequency that depends upon an
expected drive cycle, and "{circumflex over (x)}.sub.1" is an
estimate of a SOC of the at least one electrochemical cell.
5. The system of claim 4, wherein the higher order filter is a
4.sup.th order Butterworth filter.
6. The system of claim 4, wherein generating the kinetic parameters
is further based upon defining a vector (.PHI.) as: .PHI. = [ .PHI.
1 .PHI. 2 .PHI. 3 ] = [ su s 2 u sy + s L { U O C V ' ( x ^ 1 ) u C
nom } ] ##EQU00010##
7. The system of claim 6, wherein generating the kinetic parameters
comprises converting the vector (.PHI.) into the following
parametric form: z=.PHI..sup.T.THETA.+(U.sub.ocv(x.sub.1(0)),t)
wherein ".PHI..sup.T" is a transpose of the matrix.PHI., :
.times..sup.+.fwdarw. is a class function, .THETA. = [ .THETA. 1
.THETA. 2 .THETA. 3 ] .di-elect cons. 3 ##EQU00011## is a
non-linear transformation of the physical parameters (R.sub.e,
R.sub.1, C.sub.1).di-elect cons..sup.3, and the inverse transform
is defined as: [ R e R 1 C 1 ] = [ - .THETA. 2 .THETA. 1 + .THETA.
2 .THETA. 3 .THETA. 3 1 .THETA. 1 + .THETA. 2 .THETA. 3 ]
##EQU00012##
8. The system of claim 7, wherein generating the kinetic parameters
comprises executing the following parameter law: {circumflex over
({dot over (.THETA.)}(t)=.epsilon.(t)P(t).PHI.(t)
.epsilon.(t)=z(t)-.PHI..sup.T(t){circumflex over (.THETA.)}(t) {dot
over (P)}(t)=.beta.P(t)-P(t).PHI.(t).PHI.(t).sup.TP(t) wherein
".epsilon." is the output error, "P" is a covariance matrix, the
matrix P.di-elect cons..sup.3.times.3 is initialized as a positive
definitive matrix P.sub.o, and the initial kinetic parameters
estimate {circumflex over (.THETA.)}(0)=.THETA..sub.0 is used as an
initial value for the kinetic parameters (.THETA.).
9. A method of determining kinetic parameters for an equivalent
circuit model of at least one electrochemical cell, comprising:
obtaining a derivative of an open cell voltage (U.sub.ocv) with
respect to a state of charge (SOC); obtaining an estimated nominal
capacity (C.sub.nom) of the at least one electrochemical cell; and
estimating the kinetic parameters using a modified least-square
algorithm with forgetting factor.
10. The method of claim 9, wherein the equivalent circuit model
comprises: an equivalent resistance (R.sub.e); and a parallel
circuit in series with the R.sub.e, the parallel circuit including
a parallel circuit resistance (R.sub.1) and a parallel circuit
capacitance (C.sub.1), with the kinetic parameters including
R.sub.e, R.sub.1, and C.sub.1.
11. The method of claim 10, wherein the equivalent circuit model in
continuous time is written as: ( x . 1 x . 2 ) = ( 0 0 0 - 1 / ( R
1 C 1 ) ) ( x 1 x 2 ) + ( - 1 / C nom 1 / R 1 C 1 ) u , and
##EQU00013## y = U O C V ( x 1 ) + ( 0 - R 1 ) ( x 1 x 2 ) + ( - R
e ) u , ##EQU00013.2## wherein "u" is a current applied to the at
least one electrochemical cell, "y" is a measured voltage across
the at least one electrochemical cell, "x.sub.1" is a SOC of the at
least one electrochemical cell, and "x.sub.2" is the current
(i.sub.1) through the R.sub.1.
12. The method of claim 10, wherein estimating the kinetic
parameters comprises: defining a parametric form "" as: = s 2 y + s
L { U O C V ' ( x ^ 1 ) u C nom } ##EQU00014## wherein "(.cndot.)"
represents a Laplace transform, "u" is a current applied to the at
least one electrochemical cell, "y" is a measured voltage across
the at least one electrochemical cell, "s" represents a complex
number with real numbers .sigma. and .omega., "" represents a
higher order filter with a cut-off frequency that depends upon an
expected drive cycle, and "{circumflex over (x)}.sub.1" is an
estimate of a SOC of the at least one electrochemical cell.
13. The method of claim 12, wherein the higher order filter is a
4.sup.th order Butterworth filter.
14. The method of claim 12, wherein estimating the kinetic
parameters comprises: defining a vector (.PHI.) as: .PHI. = [ .PHI.
1 .PHI. 2 .PHI. 3 ] = [ su s 2 u sy + s L { U O C V ' ( x ^ 1 ) u C
nom } ] ##EQU00015##
15. The method of claim 14, wherein estimating the kinetic
parameters comprises: converting the vector (.PHI.) into the
following parametric form:
z=.PHI..sup.T.THETA.+(U.sub.OCV(x.sub.1(0)),t) wherein
".PHI..sup.T" is a transpose of the matrix.PHI., :
.times..sup.+.fwdarw. is a class function, .THETA. = [ .THETA. 1
.THETA. 2 .THETA. 3 ] .di-elect cons. 3 ##EQU00016## is a
non-linear transformation of the physical parameters (R.sub.e,
R.sub.1, C.sub.1).di-elect cons..sup.3, and the inverse transform
is defined as: [ R e R 1 C 1 ] = [ - .THETA. 2 .THETA. 1 + .THETA.
2 .THETA. 3 .THETA. 3 1 .THETA. 1 + .THETA. 2 .THETA. 3 ]
##EQU00017##
16. The method of claim 15, wherein estimating the kinetic
parameters comprises: executing the following parameter law:
{circumflex over ({dot over (.THETA.)}(t)=.epsilon.(t)P(t).PHI.(t)
.epsilon.(t)=z(t)-.PHI..sup.T(t){circumflex over (.THETA.)}(t) {dot
over (P)}(t)=.beta.P(t)-P(t).PHI.(t).PHI.(t).sup.TP(t) wherein
".epsilon." is the output error, "P" is a covariance matrix, the
matrix P.di-elect cons..sup.3.times.3 is initialized as a positive
definitive matrix P.sub.o, and the initial kinetic parameters
estimate {circumflex over (.THETA.)}(0)=.THETA..sub.0 is used as an
initial value for the kinetic parameters (.THETA.).
Description
[0001] This application claims the benefit of U.S. Provisional
Application No. 61/647,904 filed May 16, 2012, U.S. Provisional
Application No. 61/647,926 filed May 16, 2012, and U.S. Provisional
Application No. 61/647,948 filed May 16, 2012, the entirety of each
of which is incorporated herein by reference. The principles of the
present invention may be combined with features disclosed in those
patent applications.
FIELD OF THE INVENTION
[0002] This disclosure relates to batteries and more particularly
to electrochemical batteries.
BACKGROUND
[0003] Batteries are a useful source of stored energy that can be
incorporated into a number of systems. Rechargeable lithium-ion
(Li-ion) batteries are attractive energy storage systems for
portable electronics and electric and hybrid-electric vehicles
because of their high specific energy compared to other
electrochemical energy storage devices. In particular, batteries
with a form of lithium metal incorporated into the negative
electrode afford exceptionally high specific energy (in Wh/kg) and
energy density (in Wh/L) compared to batteries with conventional
carbonaceous negative electrodes. Li-ion batteries also exhibit
lack of hysteresis and low self-discharge currents. Accordingly,
lithium-ion batteries are a promising option for incorporation into
electric vehicles (EV), hybrid electric vehicles (HEV) and plug-in
hybrid electric vehicles (PHEV).
[0004] One requirement for incorporation of batteries including
Li-ion batteries into EV/HEV/PHEV systems is the ability to
accurately compute the state of charge (SOC) and state of health
(SOH) of the batteries in real time. SOC is a percentage which
reflects the available energy in a cell compared to the available
energy of the cell when fully charged. SOC is thus akin to the fuel
gauge provided on fossil fuel based vehicles.
[0005] SOH is a general term which encompasses a variety of
quantities and is in the form of a percentage which reflects the
presently available energy and power in a cell assuming the cell to
be fully charged compared to the available energy and power of the
cell when fully charged at beginning of cell life. SOH is thus akin
to the size of the fuel tank provided on fossil fuel based vehicles
and the health of the engine to provide the power. Unlike the
volume of a fuel tank and the power output of an engine, the SOH of
a cell decreases over cell life as discussed more fully below.
[0006] Both SOC and SOH are needed to understand, for example, the
available range of a vehicle using the cell and the available
power. In order to provide SOH/SOC data, a battery management
system (BMS) is incorporated into a vehicle to monitor battery
parameters and predict SOH/SOC.
[0007] Various algorithms have been proposed for use in a BMS to
maintain the battery system within safe operating parameters as
well as to predict the actual available power in the battery
system. One such approach based on an electrochemical paradigm is
described by N. Chaturvedi, R. Klein, J. Christensen, J. Ahmed, and
A. Kojic, "Algorithms for advanced battery-management systems,"
IEEE Control Systems Magazine, 30(3), pp. 49-68, 2010. Generally,
in order to accurately estimate the SOH of a system, the SOC of the
system must be accurately known. Conversely, in order to accurately
estimate the SOC of a system, the SOH of the system must be
accurately known.
[0008] SOC estimation, even when an accurate SOH is available, is
challenging since simple methods of predicting SOC, such as Coulomb
Integration, suffer from increased errors over increased
integration time. The increased errors result from biased current
measurements or discretization errors as reported by S. Piller, M
Perrin, and A. Jossen, "Methods for state-of-charge determination
and their applications," Journal of Power Sources, 96, pp. 113-120,
2001. Nonetheless, some approaches such as the approach described
by U.S. Pat. No. 7,684,942 of Yun et al. use pure current
integration to determine SOC and then derive SOH from the
determined SOC.
[0009] Other approaches avoid exclusive reliance upon current
integration by combining current integration with a form of SOC
estimation to obtain an SOC as a weighted sum of both methods as
disclosed in U.S. Pat. No. 7,352,156 of Ashizawa et al. In another
approach reported by K. Ng, C. Moo, Y. Chen, and Y. Hsieh,
"Enhanced coulomb counting method for estimating state-of-charge
and state-of-health of lithium-ion batteries," Journal of Applied
Energy, 86, pp. 1506-1511, 2009, the result obtained from current
integration is reset in accordance with an OCV/SOC look-up
table.
[0010] All of the foregoing approaches, however, rely upon
obtaining a dependable initial value for the cell SOC. If a
dependable initial value for cell SOC is not available, the
described methods fail. Unreliable SOC values are commonly
encountered during drive cycles or when switching off current. For
example, during driving cycles or when switching off current, the
dynamics of the battery may not decay to zero or settle at a
steady-state level at the precise moment that a measurement is
obtained. Thus a calculation depending upon an observed voltage may
be biased if the voltage is obtained during a transient.
[0011] Other approaches such as those described in U.S. Patent
Publication No. 2010/0076705 of Liu et al., U.S. Pat. No. 7,615,967
of Cho et al., and U.S. Patent Publication No. 2005/0231166 of
Melichar work only in discrete special cases and are not guaranteed
to work robustly during normal operation of a battery. These
approaches may further incur increased errors as a battery ages
with use.
[0012] Many advanced BMSs incorporate various forms of a Kalman
filter such as those reported by H. Dai, Z. Sun, and X. Wei,
"Online SOC Estimation of High-power Lithium-ion Batteries used on
HEV's," Vehicular Electronics and Safety, ICVES, 2006, and J. Lee,
O. Nam, and B. Cho, "Li-ion battery SOC estimation method based on
the reduced order extended Kalman Filtering," Journal of Power
Sources, 174, pp. 9-15, 2007. BMSs incorporating Kalman filters,
however, are based upon an assumption of known and time-invariant
parameters incorporated into a battery model. In a real battery
system the various parameters vary on both a long-term and
short-term basis. For example, battery aging alters the capacity
and internal resistance of the battery over the long term. Thus,
the SOH of the battery changes over cell lifetime introducing
errors into SOC calculations. Moreover, temperature and rate of
current draw vary over the short term and both temperature and rate
of current draw affect the SOC determination. Accordingly, while
accurate knowledge of the present SOH of the battery is a
prerequisite for accurate SOC determination in approaches
incorporating Kalman filters, such information may not be readily
available.
[0013] Accurate estimation of SOH is likewise challenging. A good
estimator has to be able to track battery model parameters on a
short time scale to account for the parameters' dependence or rate
of current draw, SOC, and temperature, and also on a long time
scale to account for changing health of the battery. Estimators
which operate when the battery is placed off-line have been
proposed. Placing a battery offline in order to determine remaining
driving range, however, is typically not possible. Moreover, this
approach is not recursive resulting in increased computational
expense. Thus, such off-line approaches are of limited value in
providing near real-time estimation which is needed during
operation of a vehicle.
[0014] Additionally, approaches which require stable input
parameters, which may be available when a system is offline, cannot
provide accurate estimates when presented with disturbances in the
measured battery parameter signals like voltage and current noise,
gain errors and/or measurement bias. Moreover, since the open
circuit voltage (OCV) of most batteries is nonlinear, a direct
application of standard parameter estimation theory which is
directed to estimating a constant value is not possible.
Accordingly, accurate knowledge of the present SOC of the battery
is a prerequisite for accurate SOH determination. U.S. Pat. No.
7,352,156 of Ashizawa et al. addresses this issue by assuming a
linearized model with an initially known OCV. As the actual SOC
diverges from the assumed linear model, however, estimation errors
are incurred and can eventually result in divergence of the
estimator. Thus, known systems rely on the actual SOC or
incorporate excess robustness into the SOH estimation to allow for
SOC errors.
[0015] Accordingly, accurately estimating SOH and SOC presents a
circular problem in known systems with accurate estimation of one
parameter depending upon accurate foreknowledge of the other of the
two parameters. Some attempts have been made to solve the circular
problem by performing a combined estimation of both parameters.
Such approaches have been reported by G. Plett, "Extended Kalman
Filtering for battery management systems of LiPB-based HEV battery
packs Part3. State and parameter estimation," Journal of Power
Sources, 134, pp. 277-292, 2004, and M. Roscher and D. Sauer,
"Dynamic electric behavior and open-circuit-voltage modeling of
LiFePO4-based lithium ion secondary batteries," Journal of Power
Sources, 196, pp. 331-336, 2011. These approaches, however, are
computationally expensive.
[0016] An alternative approach to solving the circular SOH/SOC
problem is to incorporate extended or unscented Kalman filters as
reported by G. Plett, "Sigma-point Kalman Filtering for battery
management systems of LiPB-based HEV battery packs. Part 2:
Simultaneous state and parameter estimation," Journal of Power
sources, 161, pp. 1369-1384, 2006. This approach, however, is also
computationally expensive.
[0017] What is needed therefore is a battery system incorporating a
BMS with SOH estimation which converges even for initially
inaccurate SOC parameters. A system which is much more robust than
known approaches given initial inaccuracies would be beneficial. A
system which accurately estimates SOH without relying upon initial
system assumptions regarding model noise would be further
advantageous.
SUMMARY
[0018] An electrochemical battery system in one embodiment includes
at least one electrochemical cell, a first sensor configured to
generate a current signal indicative of an amplitude of a current
passing into or out of the at least one electrochemical cell, a
second sensor configured to generate a voltage signal indicative of
a voltage across the at least one electrochemical cell, a memory in
which command instructions are stored, and a processor configured
to execute the command instructions to obtain the current signal
and the voltage signal, and to generate kinetic parameters for an
equivalent circuit model of the at least one electrochemical cell
by obtaining a derivative of an open cell voltage (U.sub.ocv) with
respect to SOC, obtaining an estimated nominal capacity (C.sub.nom)
of the at least one electrochemical cell, and estimating the
kinetic parameters using a modified least-square algorithm with
forgetting factor.
[0019] In accordance with another embodiment, a method of
determining kinetic parameters for an equivalent circuit model of
at least one electrochemical cell includes obtaining a derivative
of an open cell voltage (U.sub.ocv) with respect to SOC, obtaining
an estimated nominal capacity (C.sub.nom) of the at least one
electrochemical cell, and estimating the kinetic parameters using a
modified least-square algorithm with forgetting factor.
BRIEF DESCRIPTION OF THE DRAWINGS
[0020] FIG. 1 depicts a schematic of a battery system including a
lithium-ion cell, a processor, and a memory with command
instructions which, when executed by the processor, run a parameter
estimator which generates kinetic parameters of a model of the
battery system;
[0021] FIG. 2 depicts a schematic of an equivalent circuit of the
battery system of FIG. 1 including static parameters (open cell
voltage) and kinetic parameters, wherein the kinetic parameters
include an effective resistance, and a parallel circuit in series
with the effective resistance, the parallel circuit including a
resistance in parallel with a capacitance;
[0022] FIG. 3 depicts a schematic of a model executed by the
processor of FIG. 1 including an estimator which generates the
kinetic parameters of FIG. 2 based upon a sensed voltage and sensed
current of the battery system of FIG. 1, and based upon a received
SOC estimate from an observer; and
[0023] FIG. 4 depicts the results of a validation process in which
a lithium-ion cell is discharged and the model of FIG. 3 is used to
generate kinetic parameters.
DESCRIPTION
[0024] For the purposes of promoting an understanding of the
principles of the disclosure, reference will now be made to the
embodiments illustrated in the drawings and described in the
following written specification. It is understood that no
limitation to the scope of the disclosure is thereby intended. It
is further understood that the present disclosure includes any
alterations and modifications to the illustrated embodiments and
includes further applications of the principles of the disclosure
as would normally occur to one skilled in the art to which this
disclosure pertains.
[0025] FIG. 1 depicts an electrochemical battery system 100
including an electrochemical cell in the form of Li-ion cell 102, a
memory 104, and a processor 106. Various command instructions,
discussed in further detail below, are programmed into the memory
104. The processor 106 is operable to execute the command
instructions programmed into the memory 104.
[0026] The Li-ion cell 102 includes a negative electrode 108, a
positive electrode 110, and a separator region 112 between the
negative electrode 108 and the positive electrode 110. The negative
electrode 108 includes active materials 116 into which lithium can
be inserted, inert materials 118, electrolyte 120 and a current
collector 122.
[0027] The negative electrode 108 may be provided in various
alternative forms. The negative electrode 108 may incorporate dense
Li metal or a conventional porous composite electrode (e.g.,
graphite particles mixed with binder). Incorporation of Li metal is
desired since the Li metal affords a higher specific energy than
graphite.
[0028] The separator region 112 includes an electrolyte with a
lithium cation and serves as a physical and electrical barrier
between the negative electrode 108 and the positive electrode 110
so that the electrodes are not electronically connected within the
cell 102 while allowing transfer of lithium ions between the
negative electrode 108 and the positive electrode 110.
[0029] The positive electrode 110 includes active material 126 into
which lithium can be inserted, a conducting material 128, fluid
130, and a current collector 132. The active material 126 includes
a form of sulfur and may be entirely sulfur. The conducting
material 128 conducts both electrons and lithium ions and is well
connected to the separator 112, the active material 126, and the
collector 132. In alternative embodiments, separate material may be
provided to provide the electrical and lithium ion conduction. The
fluid 130, which may be a liquid or a gas, is relatively inert with
respect to the other components of the positive electrode 110. Gas
which may be used includes argon or nitrogen. The fluid 130 fills
the interstitial spaces between the active material 126 and the
conducting material 128.
[0030] The lithium-ion cell 102 operates in a manner similar to the
lithium-ion battery cell disclosed in U.S. Pat. No. 7,726,975,
which issued Jun. 1, 2010, the contents of which are herein
incorporated in their entirety by reference. In other embodiments,
other battery chemistries are used in the cell 102. In general,
electrons are generated at the negative electrode 108 during
discharging and an equal amount of electrons are consumed at the
positive electrode 110 as lithium and electrons move in the
direction of the arrow 134 of FIG. 1.
[0031] In the ideal discharging of the cell 102, the electrons are
generated at the negative electrode 108 because there is extraction
via oxidation of lithium ions from the active material 116 of the
negative electrode 108, and the electrons are consumed at the
positive electrode 110 because there is reduction of lithium ions
into the active material 126 of the positive electrode 110. During
discharging, the reactions are reversed, with lithium and electrons
moving in the direction of the arrow 136. While only one cell 102
is shown in the system 100, the system 100 may include more than
one cell 102.
[0032] During operation of the cell 102, cell voltage is monitored
using a voltage meter 138 and an amp meter 140 monitors current
flow into and out of the cell 102. Signals from the voltage meter
138 and the amp meter 140 are provided to the processor 106 which
uses the signals to estimate the SOH and, in this embodiment, SOC
of the cell 102. In general, the processor 106 uses a state space
equation which models the cell 102 to estimate SOH and SOC. By way
of background, a simple equivalent circuit for a known cell is
depicted in FIG. 2. In FIG. 2, open cell voltage (OCV), nominal
capacity (C.sub.nom), rest voltage, etc., are modeled as static
parameters 150. The internal resistance (R.sub.e) 152 and a
parallel circuit 154 including a resistor (R.sub.1) 156 and a
capacitor (C.sub.1) 158 represent kinetic parameters.
[0033] State space equations for the equivalent circuit of FIG. 2
can be written, in continuous time, as the following:
( x . 1 x . 2 ) = ( 0 0 0 - 1 / ( R 1 C 1 ) ) ( x 1 x 2 ) + ( - 1 /
C nom 1 / R 1 C 1 ) u , and ##EQU00001## y = U O C V ( x 1 ) + ( 0
- R 1 ) ( x 1 x 2 ) + ( - R e ) u , ##EQU00001.2##
wherein
[0034] "u" is the current applied to the battery,
[0035] "y" is the measured cell voltage,
[0036] "x.sub.1" is the cell SOC,
[0037] "x.sub.2" is the current (i.sub.1) through the impedance
(R.sub.1) 156,
[0038] "U.sub.OCV" is the open circuit voltage of the cell, and
[0039] "C.sub.nom" is the nominal capacity of the cell associated
with the U.sub.OCV.
[0040] In the foregoing state equations, the SOH battery parameters
R.sub.e, R.sub.1, and C.sub.1, in general terms, are functions of
the cell SOC, cell current, and cell temperature. Thus, the values
for those parameters can vary over time (kinetic parameters).
Consequently, the foregoing state equations are nonlinear.
Moreover, since the second state space equation above incorporates
the term "U.sub.OCV(.cndot.)" as a function of x.sub.1, it is
inherently nonlinear, even in situations with otherwise constant
parameters. Additionally, the first state space equation above
reveals that the system dynamics are Lyapunov stable, not
asymptotically stable. Accordingly, approaches which attempt to
predict SOH or SOC using linear systems are inherently
inaccurate.
[0041] In contrast with prior systems, the system 100 has a model
160 stored within the memory 104 which is executed by the processor
106 (see FIG. 1). The model 160 is schematically depicted in FIG.
3. The model 160 running within the processor 106 receives input
from the voltage meter 138 and the amp meter 140. Signals
indicative of the voltage of the cell 102 are provided to a
parameter estimator 162 and, in this embodiment, a reduced modified
observer 164. The parameters (.theta.) estimated by the parameter
estimator 162 are also provided as an input to the reduced modified
observer 164 while the output (SOC) of the reduced modified
observer 164 is provided as an input to the parameter estimator
162. The output parameters in this embodiment represent the values
for the kinetic parameters R.sub.e 152, R.sub.1 156, and C.sub.1
158 of FIG. 2.
[0042] Simply incorporating an adaptive observer does not
necessarily result in an algorithm which converges, however,
because small initial errors in the SOC estimate provided to the
parameter estimator can result in increasingly large SOH
estimations. This problem may be exacerbated by unknown offsets in
current and noise in current and voltage measurements.
[0043] In order to ensure convergence, the parameter estimator 162
estimates the kinetic parameters based upon voltage and current
measurements of the cell 102 while applying a modified least
squares algorithm with forgetting factor to data used in forming
the estimation. In other words, while historical data are used in
estimating present parameters, the older data are given
exponentially less weight in the estimation.
[0044] Additionally, rather than directly using an OCV reading as
an indication of SOC, the parameter estimator 162 uses a form of a
derivative with respect to SOC of the OCV signal. Using a
derivative of the OCV reduces the impact of an inaccurate SOC input
since the OCV for the cell 102 exhibits a nearly constant slope
over a wide range of SOC. Therefore, the impact of initial SOC
errors on the accuracy of the estimation is reduced.
[0045] The algorithm for the parameter estimator 162 in one
embodiment is derived from the above described state equations by
defining a parametric form "" in the following manner:
z = s 2 y + s L { U O C V ' ( x ^ 1 ) u C nom } ##EQU00002##
wherein
[0046] "(.cndot.)" represents a Laplace transform
[0047] "s" represents a complex number with real numbers .sigma.
and .omega.,
[0048] "" represents a higher order filter with a cut-off frequency
that depends upon the expected drive cycle (about 0.1 Hz in one
embodiment), such as a 4.sup.th order Butterworth filter with a
cut-off frequency of 0.1 rad/s, and
[0049] "{circumflex over (x)}.sub.1" is an estimate of the SOC from
the observer 164.
[0050] Next, a vector (.PHI.) is defined in the following
manner:
.PHI. = [ .PHI. 1 .PHI. 2 .PHI. 3 ] = [ su s 2 u sy + s L { U O C V
' ( x ^ 1 ) u C nom } ] ##EQU00003##
[0051] Converting the foregoing into parametric form results in the
following:
z=.PHI..sup.T.theta.+(U.sub.OCV(x.sub.1(0)),t)
wherein
[0052] ".PHI..sup.T" is a transpose of the matrix.PHI.,
[0053] : .times..sup.+.fwdarw. is a class function,
.THETA. = [ .THETA. 1 .THETA. 2 .THETA. 3 ] .di-elect cons. 3
##EQU00004##
is a non-linear transformation of the physical parameters (R.sub.e,
R.sub.1, C.sub.1).di-elect cons..sup.3, and the inverse transform
is defined as:
[ R e R 1 C 1 ] = [ - .THETA. 2 .THETA. 1 + .THETA. 2 .THETA. 3
.THETA. 3 1 .THETA. 1 + .THETA. 2 .THETA. 3 ] ##EQU00005##
[0054] In the equation above for the parametric form of "", the
last term accounts for effects resulting from an unknown state of
charge. For an asymptotically stable filter design, however, the
last two terms in the equation for the parametric form of "" vanish
asymptotically. Accordingly, by defining {circumflex over
(.THETA.)}(t) to be an estimate of the parameters at time "t", the
parameter estimator law is given by:
{circumflex over ({dot over
(.THETA.)}(t)=.epsilon.(t)P(t).PHI.(t)
.epsilon.(t)=z(t)-.PHI..sup.T(t){circumflex over (.THETA.)}(t)
{dot over (P)}(t)=.beta.P(t)-P(t).PHI.(t).PHI.(t).sup.TP(t)
wherein
[0055] ".epsilon." is the output error,
[0056] "P" is a covariance matrix,
[0057] the matrix P.di-elect cons..sup.3.times.3 is initialized as
a positive definitive matrix P.sub.o, and
[0058] the initial parameters estimate {circumflex over
(.THETA.)}(0)=.THETA..sub.0 is used as an initial value for the
parameters (.THETA.).
[0059] In the foregoing parameter algorithm, values for C.sub.nom
and an estimate for the SOC ({circumflex over (x)}{circumflex over
(x.sub.1)}) are needed. The SOC estimate can be provided in various
embodiments by any desired SOC estimator. The value of C.sub.nom
may be provided in any desired manner as well, although the model
160 in this embodiment includes an algorithm that provides a
C.sub.nom without the need for SOC or SOH inputs as described more
fully below.
[0060] In this embodiment, the system 100 includes a reduced
observer 164 which uses input from the parameter estimator 162 to
generate an estimated SOC. Given the foregoing parameter estimator
equations, the SOC for the cell 102 is defined by the following
equation in the reduced observer 164:
x 1 ^ . = - u C nom + L ( y - U O C V ( x ^ 1 ) + u R e + sy R 1 C
1 + u R 1 + su R e R 1 C 1 + U O C V ' ( x 1 ^ ) u R 1 C 1 C nom )
##EQU00006##
wherein "L" is the gain of the reduced observer 164.
[0061] The reduced observer 164 thus converges to a residual set,
i.e., a compact neighborhood of the desired values, for a bounded
error estimate of SOH. The SOC estimate is fed into the SOH
estimator 162 and modified parameters are generated by the
estimator 162 and fed back to the reduced observer 164.
Accordingly, the loop of FIG. 3 is closed.
[0062] In other embodiments, other observers are incorporated. By
way of example, in one embodiment the SOC for the cell 102 is
defined by the following equation in the reduced observer 164:
x 1 ^ . = - u C nom + L ( y - y ^ ) ##EQU00007##
wherein
[0063] "u" is the current applied to the battery,
[0064] "C.sub.nom" is the nominal capacity of the cell,
[0065] "L" is the gain of the reduced observer 164
[0066] "y" is the measured cell voltage,
[0067] "" represents a higher order filter with a cut-off frequency
that depends upon the expected drive cycle (about 0.1 Hz in one
embodiment), such as a 4.sup.th order Butterworth filter with a
cut-off frequency of 0.1 rad/s, and
[0068] "y" is the estimate of the output voltage.
[0069] The model 160 was validated using a commercial 18650 Li-ion
cell while estimating all parameters in real time. Actual values
for U.sub.ocv and nominal capacity C.sub.nom were obtained using
open cell voltage experiments prior to validation testing. During
validation testing, three consecutive drive cycles were applied to
the cell with intermediate rests. The results are shown in FIG. 4
which includes a chart 200 of the actual SOC of the cell versus
time. The three drive cycles resulted in voltage drop regions 202,
204, and 206 resulting in an ending SOC of 20%. The cell voltages
corresponding to 100% and 0% SOC were 4.1V and 2.8V,
respectively.
[0070] In running the model 160, a noise of 20 mV was introduced
into the voltage signal. A noise of C/50 A and an additional error
in the form of an offset of C/10 A was introduced on the current
signal. Additionally, the initial value for each of the kinetic
parameters was established at between 2 and 10 times the actual
value with an initial error of 20% for the SOC. The values for the
kinetic parameters and the SOC generated by the model 160 during
the validation testing are shown in FIG. 4 by charts 210, 212, 214,
and 216.
[0071] Chart 210 depicts the estimated value generated by the
parameter estimator 162 for the R.sub.e. The estimated R.sub.e
initially exhibits a large drop at 218 during the initial voltage
drop region 202 primarily because of the introduced 20% error in
the initial SOC estimate. The estimated R.sub.e quickly stabilizes
thereafter for the remainder of the voltage drop region 202. At the
initialization of the voltage drop regions 204 and 206, smaller
perturbations at 220 and 222 are exhibited because of changing
current, temperature, and SOC values. The estimated value of
R.sub.e is otherwise stable in the voltage drop regions 204 and
206.
[0072] Chart 212 depicts the estimated value generated by the
parameter estimator 162 for the resistor (R.sub.1) 156. The
estimated R.sub.1 is initially zero at 224 as the estimated R.sub.e
drops at 218 because of the large initial SOC error. As the
estimated R.sub.e begins to increase during the initial voltage
drop region 202, the estimated R.sub.1 increases at 226 and then
settles to a stable value for the remainder of the voltage drop
region 202. At the initialization of the voltage drop regions 204
and 206, smaller perturbations at 228 and 230 are exhibited because
of changing current, temperature, and SOC values. The estimated
value of R.sub.1 is otherwise stable in the voltage drop regions
204 and 206.
[0073] Chart 214 depicts the estimated value generated by the
parameter estimator 162 for the capacitor (C.sub.1) 158. The
estimated C.sub.1 initially exhibits a large perturbation at 232.
As the other estimated parameters and SOC stabilize during the
initial voltage drop region 202, the estimated C.sub.1 stabilizes
at 234 for the remainder of the voltage drop region 202. At the
initialization of the voltage drop regions 204 and 206, smaller
perturbations at 236 and 238 are exhibited because of changing
current, temperature, and SOC values. The estimated value of
C.sub.1 is otherwise stable in the voltage drop regions 204 and
206.
[0074] Chart 216 depicts the estimated SOC value 240 generated by
the reduced modified observer 164 along with the estimated SOC 242
based upon coulomb counting. The estimated SOH, initialized with a
20% error, rapidly converges to the SOC 242. The actual SOC error
of the estimated SOC value 240 is depicted in chart 250. Chart 250
reveals the actual SOC error decreases to less than 2% (line 252).
The variation in the SOC error during the rest periods of chart 200
result from changing temperature of the cell.
[0075] While the disclosure has been illustrated and described in
detail in the drawings and foregoing description, the same should
be considered as illustrative and not restrictive in character. It
is understood that only the preferred embodiments have been
presented and that all changes, modifications and further
applications that come within the spirit of the disclosure are
desired to be protected.
* * * * *