U.S. patent application number 16/911179 was filed with the patent office on 2021-12-30 for interval estimation for state-of-charge and temperature in battery packs with heterogeneous cells.
This patent application is currently assigned to TOTAL S.A.. The applicant listed for this patent is TOTAL S.A., Universite Libre de Bruxelles, University of California - Berkeley. Invention is credited to Sebastien BENJAMIN, Luis COUTO, Preet GILL, Scott MOURA, Wente ZENG, Dong ZHANG.
Application Number | 20210405121 16/911179 |
Document ID | / |
Family ID | 1000004987966 |
Filed Date | 2021-12-30 |
United States Patent
Application |
20210405121 |
Kind Code |
A1 |
ZHANG; Dong ; et
al. |
December 30, 2021 |
INTERVAL ESTIMATION FOR STATE-OF-CHARGE AND TEMPERATURE IN BATTERY
PACKS WITH HETEROGENEOUS CELLS
Abstract
An interval observer based on an equivalent circuit-thermal
model for lithium-ion batteries is presented. State of
charge-temperature-dependent parameters are considered as unknown
but bounded uncertainties in a single cell model. A parallel and a
series arrangement of five cells are used for observer design,
where cell heterogeneity is accounted for through the uncertainty
bounding functions.
Inventors: |
ZHANG; Dong; (Albany,
CA) ; GILL; Preet; (El Sobrante, CA) ; MOURA;
Scott; (Berkeley, CA) ; COUTO; Luis; (Oxford,
GB) ; BENJAMIN; Sebastien; (Leognan, FR) ;
ZENG; Wente; (San Francisco, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
TOTAL S.A.
University of California - Berkeley
Universite Libre de Bruxelles |
Paris La Defense
Berkeley
Bruxelles |
CA |
FR
US
BE |
|
|
Assignee: |
TOTAL S.A.
Paris La Defense
CA
University of California - Berkeley
Berkeley
Universite Libre de Bruxelles
Bruxelles
|
Family ID: |
1000004987966 |
Appl. No.: |
16/911179 |
Filed: |
June 24, 2020 |
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G01R 31/396 20190101;
G01K 13/00 20130101; G01R 31/374 20190101; G01R 31/3842 20190101;
G05B 13/04 20130101; G01R 31/387 20190101; G01R 31/367
20190101 |
International
Class: |
G01R 31/374 20060101
G01R031/374; G01R 31/3842 20060101 G01R031/3842; G01K 13/00
20060101 G01K013/00; G01R 31/367 20060101 G01R031/367; G01R 31/396
20060101 G01R031/396 |
Claims
1. A method for operating batteries belonging in a battery pack
comprising a plurality of cells of batteries, each of the cells
being represented by an R-C pair comprising a resistor with
resistance R.sub.2 and a capacitor with capacitance C, the R-C pair
being in series with a resistor with resistance R.sub.1, the method
comprising: estimating a state of charge (SOC) of each of the cells
of the batteries belonging in the battery pack based on a coupled
electrical-thermal model, wherein the SOC of each battery in each
of the cells is derived from the non-linear differential equation
system: d.xi./dt=A.sub.0.xi.+.delta.A(.theta.).xi.+b(.xi.,.theta.)
dy/dt=H.xi.+v(.theta.,t) where .xi. = ( .xi. 1 .xi. 2 ) = ( OCV
.function. ( x 1 ) + x 2 - .tau. k , 0 .times. x 2 ) . .times. A 0
= [ 0 1 0 - .tau. k , 0 ] , .delta.A .function. ( .theta. ) = [ 0
.delta..tau. k / .tau. k , 0 0 - .delta..tau. k ] . .times. b
.function. ( .xi. , .theta. ) = ( ( 1 / Q k ) .times. .phi.
.function. ( .xi. 1 + .xi. 2 / .tau. k , 0 ) + 1 / C k - .tau. k ,
0 / C k ) .times. I , H = [ 1 , 0 ] . ##EQU00012## .phi.(
)=dOCV/dx(OCV.sup.-1( )), v(.theta., t)=.delta.h(.theta.)u, where
u=I.sub.k(t) R is a system input, where I.sub.k(t) is a current
flowing in the k-th cell, y R is a system output, x R.sup.n is a
state vector, having a state x.sub.1=z.sub.k and a state
x.sub.2=V.sub.c,k, where z.sub.k(t) represents the SOC for the k-th
cell, and V.sub.c,k(t) represents a voltage across the R-C pair in
the k-th cell, t represents a time variable, OCV is an open circuit
voltage across the k-th cell,
.tau..sub.k=1/R.sub.2,kC.sub.k=.tau..sub.k,0+.delta..tau..sub.k
where .tau..sub.k,0 is a deterministic scalar and
.delta..tau..sub.k represents an uncertain component, {dot over
(Q)}.sub.k(t) is an internal heat generation from resistive
dissipation, C.sub.k is the capacitance of the capacitor in the R-C
pair in the k-th cell, I is an identity matrix,
.delta.h(.theta.)=R.sub.1,k, R.sub.1,k is R.sub.1 in the k-th cell,
R.sub.2,k is R.sub.2 in the k-th cell, where unknown quantities
Q.sub.k, C.sub.k, .delta..sub..tau.k and R.sub.1,k have values
belonging within intervals [Q.sub.k, Q.sub.k.sup.-], [C.sub.k,
C.sub.k.sup.-], [.delta..sub..tau.k, .delta..sub..tau.k.sup.-], and
[R.sub.1,kR.sub.1,k.sup.-], respectively, with set upper and lower
limits; estimating a surface temperature T.sub.s and a core
temperature T.sub.c for each of the cells, wherein the T.sub.s and
the T.sub.c of each of the cells is derived from the non-linear
differential equation system: dx(t)/dt=Ax(t)+Bu(t)+D{dot over
(Q)}(t) y(t)=Cx(t) where x = ( T c T s ) , A = [ - 1 / R c .times.
C c 1 / R c .times. C c 1 / R c .times. C c - ( 1 R u .times. C s +
1 R c .times. C s ) ] , .times. B = ( 0 1 / R u .times. C s ) , D =
( 1 / C s 0 ) , C = [ 0 , 1 ] , u = T f ; ##EQU00013## where
R.sub.c, R.sub.u, C.sub.c, and C.sub.s represent heat conduction
resistance between core and surface, convection resistance between
ambient and surface, core heat capacity, and surface heat capacity,
respectively, and T.sub.f is a cool ant flow temperature at a cell;
and adjusting charging of the batteries based on the estimated SOC,
T.sub.s, and T.sub.c of the batteries.
2. The method according to claim 1, wherein the battery pack
comprises a string of a plurality of cells connected in series.
3. The method according to claim 2, wherein the battery pack
comprises a plurality of strings of cells connected in
parallel.
4. The method according to claim 1, wherein the battery pack
comprises 20 strings of cells connected in parallel, each string
comprising 400 blocks of cells connected in series, each of the
block comprising 5 cells connected in parallel.
Description
TECHNICAL FIELD
[0001] The disclosure relates to estimating the operation of
multiple lithium-ion batteries connected in series and in parallel,
including estimating the state of charge and temperature of the
batteries.
BACKGROUND
[0002] Lithium-ion (Li-ion) batteries play a key role in achieving
energy sustainability and reduction in emissions. Li-ion batteries
benefit from high energy density, which has motivated their wide
use in a variety of applications including electric vehicles and
grid energy storage. In recent years, a substantial body of
research on real-time control and estimation algorithms for
batteries has emerged. However, safe and efficient operation of
batteries remains a challenge, especially as the performance
requirements of these devices increase.
[0003] Energy storage systems for electric vehicles and large-scale
grid applications often require hundreds to thousands of cells
connected in series and in parallel to achieve the demanded power
and voltage. A battery pack's instantaneous power capability is
crucial for on-board management and safe operation. However,
real-time state of charge (SOC) estimation for a battery pack is a
very intricate task due to (i) limited measurements, (ii) complex
electrochemical-thermal-mechanical physics, and (iii) high
computational cost.
[0004] Different battery models for state observer design have been
proposed in the literature, which can be classified into
electrochemical white box models, equivalent-circuit gray box
models, and data driven black box models, sorted from low to high
physical interpretability. Although each modelling framework has
their own merits and drawbacks, equivalent circuit models (ECMs)
provide a reasonable trade-off between model complexity and
accuracy. ECMs can be made more accurate by increasing the system
order to account for additional electrochemical phenomena, like
charge transfer, but at the expense of computational cost that
grows polynomially with the number of states and linearly with the
number of cells.
[0005] An important fact often ignored during battery modeling is
the time-varying electrical parameters. In practice, internal
parameters, e.g. resistances and capacitance, are non-linearly
dependent on the cell's temperature and SOC. Therefore, it is
important to model cell temperature and its coupling with the
electrical dynamics. High-fidelity temperature models have more
accurate temperature predictions, but suffer from high
computational cost, rendering them of little use for on-board
thermal management systems. First principles-based two-state
thermal models for the cell's core and surface temperatures provide
a balanced trade-off between computational efficiency and
fidelity.
[0006] Lumped equivalent circuit-thermal models with temperature
dependent parameters have been studied and used for state
estimation via an adaptive observer. Existing techniques for
battery pack state estimation include Luenberger observers, Kalman
filters (KFs), unscented Kalman filters (UKFs), and sliding mode
observers, among others. However, all the previously mentioned
techniques require a state observer for each cell within the pack,
which makes them computationally intractable for large packs.
[0007] In the stochastic estimation/filtering framework, the
process and measurement noises are often assumed to be Gaussian.
The system characteristics, e.g., mean and variance, are required
by filtering algorithms. Nonetheless, the statistical and
calibration procedures to obtain these characteristics are often
tedious. In contrast, interval estimation assumes that the
measurement and process noises are unknown but bounded. The
interval observer literature derives a feasible set for state
estimation at every time instant. In a battery pack that consists
of thousands of cells, executing state estimation algorithms based
on highly nonlinear and coupled dynamics for every single cell in
real time becomes intractable. The interval observer benefits from
its scalability by deriving only upper and lower bounds that
enclose all unmeasured internal states for all cells in a pack.
Previously, only Perez et al. in "Sensitivity-based interval PDE
observer for battery SOC estimation" (American Control Conference,
pp. 323-328, 2015), had designed a sensitivity-based interval
observer for single cell SOC estimation from an electrochemical
perspective, but without provable observer stability and inclusion
properties.
SUMMARY
[0008] The internal states of Li-ion batteries, notably SOC, need
to be carefully monitored during battery operation in order to
prevent dangerous situations. SOC estimation in an electrically and
thermally coupled parallel connection of cells is a particularly
challenging problem because cells in parallel yield a system of
differential algebraic equations, which are more difficult to
handle than ordinary differential equations (e.g. a series string
of cells). For a large battery pack with thousands of cells,
applying an estimation algorithm on each and every cell would be
mathematically and computationally intractable. The present
application tackles these issues using an interval observer based
on a coupled equivalent circuit-thermal model, and considers cell
heterogeneity as well as state-dependent parameters as unknown, but
with bounded uncertainty. The resulting interval observer maps
bounded uncertainties to a feasible set of state estimation and is
independent of the number of cells in connection (in parallel or in
series). Stability and inclusion of the interval observer are
proven and validated through numerical studies.
[0009] A robust interval observer, using the measurements, to
determine the set of admissible values for cell SOC and temperature
at each time instant, when the plant model is subject to bounded
uncertainties in the parameters and states' initial conditions, is
used.
[0010] In the present application, heterogeneous battery cells
(where the cells are heterogeneous due to variations in health,
temperature, and/or charge levels) in parallel and in series
connection are both considered and analyzed, based on an interval
observer design, given uncertain model parameters, initial
conditions, and measurements. The approach is based upon the theory
of interval observers (a.k.a. set-membership estimation). Interval
observers provide an elegant approach to directly address the
heterogeneity and scalability problems. Namely, heterogeneity is
handled by conceptualizing parameters and inputs as existing within
some membership set. Scalability is handled by designing a single
observer, which estimates the range of SOC and temperature values
across the battery pack.
[0011] Li-ion batteries are vulnerable to overcharge and
overdischarge. Overcharge can lead to lithium deposition and
electrolyte solvent decomposition, resulting in fire or even
explosion. This issue is particularly critical for battery packs in
plug-in hybrid electric vehicles (PHEVs) and battery electric
vehicles (BEVs), since those batteries are usually charged to high
SOC. In the meantime, the charging (and discharging) capability of
a pack is determined by the cells with highest (lowest) SOC. Thus,
to prevent overcharging, the cells with lower SOC cannot be fully
charged after the cells with higher SOC have been fully charged. As
a result, cells are not fully utilized when they are not balanced.
In order to prolong the battery life cycle and increase the stack
utilization, it is recommended that all the cells in a pack have
the same SOC during battery operations. Cell equalization is
performed based on the knowledge of the best and worst SOCs in a
pack.
[0012] In real-world applications, e.g. BEVs, it is desirable to
charge the battery pack as fast as possible. However, charging the
battery with high current will significantly damage the Li-ion
cells which could lead to catastrophic failure. Ensuring safe
operating constraints is a basic requirement for batteries. For
instance, while charging, SOC, core and surface temperatures (along
with other important variables) of every single cell needs to be
maintained within an operational bound to ensure pack safety. The
interval observer provides such information that can be utilized to
infer whether these variables violate the safe constraints.
[0013] Embodiments disclosed herein allow the estimation of SOC and
temperature during operation of the batteries, followed by
appropriate charging of the batteries within safe limits.
BRIEF DESCRIPTION OF THE DRAWINGS
[0014] The application will be better understood in light of the
description which is given in a non-limiting manner, accompanied by
the attached drawings in which:
[0015] FIG. 1A shows a schematic for the ECM for a single cell.
[0016] FIG. 1B shows a schematic of five cells in parallel.
[0017] FIG. 1C shows a schematic of five cells in series.
[0018] FIG. 2A shows the input current profile for the simulation
study
[0019] FIGS. 2B-2C show the cell heterogeneity of two battery cells
in parallel under same initial conditions for the state of
charge.
[0020] FIGS. 2D-2E show the cell heterogeneity of two battery cells
in parallel under different initial conditions for the state of
charge.
[0021] FIGS. 3A-3C show the performance of interval observer when
estimating the state of charge of a single battery cell and five
cells in parallel for one set of initial conditions for the state
of charge.
[0022] FIG. 4A shows the performance of temperature-enhanced
interval observer when estimating the state of charge of five
battery cells in series for another set of initial conditions for
the state of charge.
[0023] FIG. 4B shows the performance of temperature-enhanced
interval observer when estimating the core temperature of five
battery cells in series.
[0024] FIG. 5 shows a schematic for the interrelation between the
interval observer for the electrical model and the interval
observer for the thermal model.
[0025] FIG. 6 shows a method according to embodiments disclosed
herein.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0026] Reference throughout this specification to "one embodiment"
or "an embodiment" means that a particular feature, structure,
material, or characteristic described in connection with the
embodiment is included in at least one embodiment of the
application, but do not denote that they are present in every
embodiment.
[0027] Thus, the appearances of the phrases "in one embodiment" or
"in an embodiment" in various places throughout this specification
are not necessarily referring to the same embodiment of the
application. Furthermore, the particular features, structures,
materials, or characteristics may be combined in any suitable
manner in one or more embodiments.
[0028] Throughout, the symbol I.sub.dn denotes the identity matrix
with dimension n.times.n. For a matrix A R.sup.n.times.n,
.parallel.A.parallel..sub.max=max.sub.i;j=1;2 . . . ;n|Ai,j| (the
element wise maximum norm). The relation Q>0 (Q<0) means that
the matrix Q R.sup.n.times.n is positive (negative) definite. The
inner product between x, y R.sup.n is given by <x,
y>=.SIGMA.x.sub.iy.sub.i, i=1 to n.
[0029] The embodiments described herein are widely applicable to
energy storage systems (e.g. Li-ion batteries) comprising hundreds
or thousands of dissimilar cells arranged in series and in
parallel. Knowledge of these largescale systems is limited by the
amount of sensing locations and cell heterogeneity. One of such
examples, among many others, is the high energy pack module
developed by Saft, which integrates 40,000 single cells of 18650
type. The module constitutes 20 strings of cells connected in
parallel, where each string is made of 400 blocks in a series
arrangement. Additionally, each block includes 5 cells in parallel.
An interval observer aims to estimate the range of SOC values
across the battery pack, with guaranteed provable inclusion and
stability properties, which will be discussed in the subsequent
sections.
[0030] This section reviews an ECM coupled with a two-state thermal
model for a single battery cell, which is then electrically and
thermally interconnected with other cell models to form a parallel
arrangement of cells.
[0031] Single Battery Cell
[0032] The ECM for a single cell, shown in FIG. 1A, is described by
the following continuous-time dynamical equations,
dz.sub.k(t)/dt=(1/Q.sub.k)I.sub.k(t); (1)
dV.sub.c,k(t)/dt=-(1/R.sub.2,k(z.sub.k,T.sub.k)C.sub.k(z.sub.k,T.sub.k))-
V.sub.c,k(t)+(1/C.sub.k(z.sub.k,T.sub.k))I.sub.k(t) (2)
V.sub.k(t)=OCV(z.sub.k(t))+V.sub.c,k(t)+R.sub.1,k(z.sub.k,T.sub.k)I.sub.-
k(t) (3)
[0033] where z.sub.k(t) represents the SOC for the k-th cell,
V.sub.c;k(t) denotes the voltage across the R-C pair for the kth
cell, V.sub.k(t) represents the voltage at the output of the k-th
cell, and I.sub.k(t) represents the current flowing across the
R.sub.1 and the R-C pair in the k-th cell. The symbol T.sub.k is
the cell temperature that will be defined later. The electrical
model parameters, namely, R.sub.1,k, F.sub.2,k, and C.sub.k, are
dependent on SOC and cell temperature, and such dependence can be
explicitly characterized via an offline experimental procedure for
a cell of interest (for example, for a LiFePO4/Graphite cell). The
output eq. (3) for the k-th cell provides the voltage response
characterized by a nonlinear open circuit voltage (OCV) as a
function of SOC, voltage from the R-C pair, and voltage associated
with an ohmic resistance R.sub.1;k. Herein, positive current is
specified for charging and negative current is specified for
discharging.
[0034] In a two-state thermal model of the k-th cell, the model
states are cell core temperature and surface temperature:
C.sub.cdT.sub.c,k(t)/dt={dot over
(Q)}.sub.k(t)+((T.sub.s,k(t)-T.sub.c,k(t))/R.sub.c (4)
C.sub.sdT.sub.s,k(t)/dt=((T.sub.f,k(t)-T.sub.s,k(t))/R.sub.u-((T.sub.s,k-
(t)-T.sub.c,k(t))/R.sub.c (5)
{dot over (Q)}.sub.k(t)=|I.sub.k(t)(V.sub.k(t)-OCV(z.sub.k(t)))|
(6)
T.sub.k(t)=1/2(T.sub.s,k(t)-T.sub.c,k(t)) (7)
[0035] where T.sub.c,k and T.sub.s,k are the core and surface
temperatures for the k-th cell. Symbols R.sub.c, R.sub.u, C.sub.c,
and C.sub.s represent heat conduction resistance between core and
surface, convection resistance between ambient and surface, core
heat capacity, and surface heat capacity, respectively. In this
framework, symbol {dot over (Q)}.sub.k(t).gtoreq.0 is the internal
heat generation from resistive dissipation. Note that the
electrical model, as represented by eqs. (1)-(3), and the thermal
model, as represented by eqs. (4)-(7), are coupled via internal
heat generation {dot over (Q)}.sub.k(t) in a nonlinear fashion.
[0036] The measured quantities for the coupled electrical-thermal
model (1)-(7) are the cell voltage and surface temperature:
y.sub.k(t)=[V.sub.k(t),T.sub.s,k(t)] (8)
[0037] Parallel Arrangement of Battery Cells
[0038] For a block of m cells in parallel (shown in FIG. 1B for
m=5), in order to reduce sensing effort, it is assumed that only
the total current and voltage for one of the cells are measured.
Electrically, Kirchhoff s voltage law indicates that a parallel
connection of cells constraints terminal voltage to the same value
for all cells. Kirchhoff s current law indicates that the overall
current is equal to the summation of cell local currents.
Mathematically, the following nonlinear algebraic constraints,
according to Kirchhoff s voltage law, need to be enforced:
OCV(z.sub.i)+V.sub.c,i+R.sub.1,iI.sub.i=OCV(zj)+V.sub.c,j+R.sub.1,jI.sub-
.j i,j {1,2, . . . m}, i.noteq.j (9)
Similarly, Kirchhoff s current law poses the following linear
algebraic constraint with respect to cell local currents,
.SIGMA.I.sub.k(t)=I(t). k=1 to m (10)
[0039] where I(t) is the measured total current, and I.sub.k(t)
represents the local current for cell k. It is worth highlighting
that eq. (9) imposes (m-1) nonlinear algebraic constraints with
respect to differential states and local currents, whereas eq. (10)
imposes one algebraic constraint with respect to local currents.
When only the total current is measured, the local cell currents
are unknown. Hence, the system of differential-algebraic eqs.
(1)-(10) must be solved such that the algebraic eqs. (9) and (10)
are fulfilled for all t. This is realized by augmenting the local
currents to the differential state vector to form a nonlinear
descriptor system.
[0040] The cells are thermally coupled through coolant flow and
heat exchange between adjacent cells.
[0041] For cell k, where k={1, 2, . . . , m},
C.sub.cdT.sub.c,k(t)/dt={dot over
(Q)}.sub.k(t)+((T.sub.s,k(t)-T.sub.c,k(t))/R.sub.c (11)
C.sub.sdT.sub.s,k(t)/dt=((T.sub.f,k(t)-T.sub.s,k(t))/R.sub.u-((T.sub.s,k-
(t)-T.sub.c,k(t))/R.sub.c+(T.sub.s,k-1(t)+T.sub.s,k-1(t)-2T.sub.s,k(t))/R.-
sub.cc (12)
T.sub.f,k(t)=T.sub.r,k-1(t)+(T.sub.s,k-1(t)-T.sub.r,k-1(t))/R.sub.uC.sub-
.r (13)
{dot over (Q)}.sub.k(t)=|.sub.k(t)(y.sub.k(t)-OCV(z.sub.k(t)))|
(14)
T.sub.k(t)=(T.sub.s,k(t)+T.sub.c,k(t))/2 (15)
where T.sub.r,k is the coolant flow temperature at the k-th cell,
and R.sub.cc denotes heat conduction resistance between adjacent
battery cell surfaces. Heat conduction between battery cells is
driven by the temperature difference between cell surfaces, and
this process is described by the third term on the right hand side
of eq. (12). Inside the block of m cells in parallel, it is assumed
that the coolant flows through individual cells, and the coolant
flow temperature at the k-th cell is determined by the flow heat
balance of the previous cell, as illustrated in eq. (13). Here it
is also assumed that all the battery cells have the same thermal
parameters.
[0042] Analysis for Cell Heterogeneity
[0043] In this section, the heterogeneity for cells in parallel via
an open-loop simulation study is presented. Without loss of
generality, two LiNiMnCoC.sub.2/Graphite (NMC) type cells with 2.8
Ah nominal capacity in parallel are considered. In this embodiment,
the cells have identical SOC-OCV relationship, and the
heterogeneity arises from: [0044] difference in SOC initialization
[0045] difference in electrical parameters due to SOC variation
[0046] unevenly distributed currents due to parameter variation
[0047] difference in temperature due to currents variation
[0048] A transient electric vehicle-like charge/discharge cycle
generated from the urban dynamometer driving schedule (UDDS) is
applied. Specifically, this total applied current (summation of
local currents) is plotted in FIG. 2A.
[0049] Two cases are examined here. In the first case, the cells
are initialized at the same SOC, z.sub.k(0)=0.25 for k {1,2}. Since
Cell 2 has larger resistance, its local current is smaller in
magnitude relative to the local current of Cell 1, while the
summation of the local currents equals to the total applied current
for all t, as shown in FIG. 2C and FIG. 2E. FIG. 2B shows the
estimation of the SOC for the two cells. FIGS. 2D and 2E
demonstrate the second case where the cells have different
z.sub.k(0): z.sub.1(0) (Cell 1) at 0.35 and z.sub.2(0) (Cell 2) at
0.25. It can be observed that even though the applied total current
is small (around zero) initially, Cell 1 takes large negative
current (around -10 A) and Cell 2 positions itself at a large
positive current (around 10 A). This occurs because z.sub.1(0) is
initialized higher, and even though the z values for two cells
follow a similar trend, they do not synchronize.
[0050] In a battery pack composed of hundreds or thousands of
cells, executing state estimation algorithms based on a highly
nonlinear and coupled model consist of differential-algebraic
equations for every single cell in real-time. This would be
mathematically intractable and involve tremendous computational
burden. This motivates the present study on interval observers that
generate an upper and a lower bound to enclose all z.sub.k(t)
trajectories, thus reducing computation and design complexity.
[0051] Consider the following nonlinear model dynamics:
dx/dt=f(x)+B(.theta.)u+.delta.f(x,.theta.) (16)
y=h(x)+.delta.h(.theta.)u (17)
where x R.sup.n is the state vector, and u R and y R are the system
input and output, respectively. The considered system is
single-input-single-output (SISO). The functions f(x) and h(x) are
deterministic and smooth, and .delta.f is uncertain and assumed to
be locally Lipschitz continuous with respect to x. It is noted that
the nominal terms f(x) and h(x) can be freely assigned by the
designer via the modification of .delta.f and .delta.h. The initial
conditions for the states belong to a compact set x.sub.0 [x.sub.0,
x.sub.0 ], where x.sub.0 and x.sub.0 are given. Suppose the
uncertain parameters .theta.(t) belong to a compact set .THETA.
R.sup.p, where p is the number of uncertain parameters. The values
of the parameter vector .theta.(t) are not available for
measurement, and only the set of admissible values .THETA. is
known.
[0052] From eqs. (16)-(17) by setting B=0, .delta.f=0, .delta.h=0,
one obtains
dx/dt=f(x) (18)
y=h(x) (19)
[0053] A coordinate transformation obtained from the locally
observable nominal system (18)-(19) is then utilized to transform
the original uncertain system (16)-(17) into a partial linear
system.
[0054] In particular, by denoting the gradient of a scalar field h
by dh, and the Lie derivative of h along a vector field f is
defined by the inner product L.sub.fh(x)=<dh(x), f (x)>.
High-order Lie derivatives are computed with the iteration
L.sub.f.sup.kh(x)=L.sub.f(L.sub.f.sup.k-1 h(x)) where L.sub.f.sup.0
h(x)=h(x). The local observability matrix around an equilibrium
point for the states x=x.sub.e is given by
O .function. ( x e ) = [ d .times. h .function. ( x e ) d .times. L
f .times. h .function. ( x e ) d .times. L f n - 1 .times. h
.function. ( x e ) ] ##EQU00001##
[0055] The nominal system (18)-(19) is locally observable around
x=x.sub.e if the matrix O(x.sub.e) is of rank n.
[0056] If the system (18)-(19) is locally observable, then the
vectors h(x), L.sub.fh(x), . . . , L.sub.f.sup.n-1h(x) form the new
coordinate for the sates in a neighborhood pf x defined by
.times. .PHI. .function. ( x ) = [ .phi. 1 .function. ( x ) .phi. 2
.function. ( x ) .phi. n .function. ( x ) ] = [ h .function. ( x )
L f .times. h .function. ( x ) L f n - 1 .times. h .function. ( x )
] .times. and .times. .times. the .times. .times. transformation
.times. .times. map .times. [ .xi. 1 .times. .times. .xi. 2 .times.
.times. .times. .times. .xi. n ] T = [ .phi. 1 .function. ( x )
.times. .times. .phi. 2 .function. ( x ) .times. .times. .times.
.times. .phi. n .function. ( x ) ] T ##EQU00002##
is a local diffeomorphism. The coordinate transformation obtained
from the locally observable nominal system (18)-(19) is then
utilized to transform the original uncertain system (16)-(17) into
a partial-linear expression
d.xi./dt=A.sub.0.xi.+.delta.A(.theta.).xi.+b(.xi.,.theta.) (20)
y=H.xi.+v(.theta.,t) (21)
[0057] where v(.theta., t)=.delta.h(.theta.)u. The matrix A.sub.0
R.sup.n is deterministic and the matrix .delta.A(.theta.) R.sup.n
represents the uncertain part inherited from the uncertain
nonlinear system (16)-(17). Symbol b(.xi., .theta.) indicates a
lumped uncertain nonlinear function.
[0058] Interval Observer for Batteries
[0059] In this section, the interval observer design introduced
previously is applied to the Li-ion battery state estimation
problem. Four scenarios are examined: (i) interval observer for SOC
of a single battery cell with temperature and SOC dependent
electrical parameters; (ii) interval observer for SOC of
electrically and thermally coupled cells in parallel, with SOC and
temperature-dependent electrical parameters; (iii) interval
observer for SOC of electrically and thermally coupled cells in
series, with SOC and temperature-dependent electrical parameters,
and (iv) temperature-enhanced interval observer for SOC and
temperature of electrically and thermally coupled cells in series,
with SOC and temperature-dependent electrical parameters.
[0060] SOC Interval Observer for a Single Battery Cell
[0061] It is hereby assumed that the input current, terminal
voltage and surface temperature of the k-th single cell are
experimentally measured. Ideally, a deterministic state observer is
supposed to be proposed and analyzed for the coupled nonlinear
electrical-thermal system (1)-(8) to reconstruct the internal
states, i.e. z.sub.k(t), V.sub.c;k(t), and T.sub.c;k. However, this
approach becomes intractable due to the nonlinear coupling between
electrical and thermal models, the nonlinear dependence of
electrical parameters on internal states, as well as nonlinear
voltage output function. To tackle this issue, the present
embodiment suppresses the electrical parameters' dependence on the
internal states, and treats these parameters as uncertain.
Specifically, .theta. .THETA..OR right.=R.sup.4, where
.theta.=[R.sub.1,k R.sub.2,k C.sub.k Q.sub.k].sup.T The objective
is to design a robust interval observer, using the measurements, to
determine the set of admissible values for cell SOC at each time
instant, when the plant model is subject to bounded uncertainties
in the parameters and states' initial conditions.
[0062] Let .tau..sub.k=1/(R.sub.2,kC.sub.k), and consider a known
nominal value .tau..sub.k,0 such that
.tau..sub.k=.tau..sub.k,0+.delta..sub..tau.k (22)
where .tau..sub.k,0 is a deterministic scalar and
.delta..sub..tau.k represents the uncertain component. The single
cell electrical system of eqs. (1)-(3) can thus be formulated in
terms of uncertain system of eqs. (16)-(17) with:
x = ( x 1 x 2 ) = ( z k V c , k ) , f .function. ( x ) = ( 0 -
.tau. k , 0 .times. x 2 ) .times. .times. .delta. .times. f
.function. ( x , .theta. ) = ( 0 - .delta..tau. k .times. x 2 ) , B
.function. ( .theta. ) = ( 1 / Q k 1 / C k ) , u = I k .function. (
t ) , .times. h .function. ( x ) = O .times. C .times. V .function.
( x 1 ) + x 2 , .delta. .times. h .function. ( .theta. ) = R 1 , k
( 23 ) ##EQU00003##
[0063] It is assumed that the following upper and lower bounds are
imposed on the uncertain parameters,
Q.sub.k [Q.sub.k,Q.sub.k ],C.sub.k [C.sub.k,C.sub.k
],.delta..sub..tau.k [.delta..sub..tau.k,.delta..sub..tau.k
],R.sub.1,k [R.sub.1,k,R.sub.1,k ], (24)
[0064] so that .THETA. is a four-dimensional polytope. The local
observability matrix for the nominal system is then given by
O .function. ( x ) = ( d .times. h .function. ( x ) d .times. L f
.times. h .function. ( x ) ) = [ d .times. O .times. C .times. V /
d .times. x 1 .function. ( x 1 ) 1 0 - .tau. k , 0 ] ( 25 )
##EQU00004##
[0065] whose rank is 2 if and only if the first derivative of the
OCV function with respect to SOC is non-zero around an equilibrium
point x.sub.1=x.sub.1,e and .tau..sub.k,0.noteq.0, i.e.,
dOCV/dx.sub.1(x.sub.1,e).noteq.0, .tau..sub.k,0.noteq.0 (26)
[0066] which aligns with existing results on local observability
for battery models. Hence, the coordinate transformation based on
Lie algebra
.PHI. .function. ( x ) = ( .xi. 1 .xi. 2 ) = ( OCV .function. ( x 1
) + x 2 .tau. k , 0 .times. x 2 ) ( 27 ) ##EQU00005##
transforms the system (16), (17) with (23) to the non-linear
parameter-varying system (20)-(21), with
A 0 = [ 0 1 0 - .tau. k , 0 ] , .delta. .times. A .function. (
.theta. ) = [ 0 .delta..tau. k / .tau. k , 0 0 - .delta..tau. k ] ,
( 28 ) b .function. ( .xi. , .theta. ) = ( ( 1 / Q k ) .times.
.phi. .function. ( .xi. 1 + .xi. 2 / .tau. k , 0 ) + 1 / C k -
.tau. k , 0 / C k ) .times. I , H = [ 1 , 0 ] , .times. where
.times. .times. .phi. .function. ( ) = dOCV / dx .function. ( OCV -
1 .function. ( ) ) ( 29 ) ##EQU00006##
[0067] An interval observer is designed according to Stanislav
Chebotarev et al. "Interval observers for continuous-time LPV
systems with L1/L2 performance", Automatica, 58:82-89, 2015. The
bounding functions .delta.A and .delta.A for .delta.A are obtained
by applying the parameter bounds in (24). The bounding functions
b(t) and b(t) for b(.xi., .theta.) are evaluated according to the
direction of current I(t) for all t.
[0068] SOC Interval Observer for Battery Cells in Parallel
[0069] As opposed to having one interval observer for each single
cell in the preceding discussion, the present design is generalized
for a cluster of battery cells connected in parallel. One practical
advantage for using an interval observer for a group of cells is
scalabilty. An interval observer, composed of only two dynamical
systems estimating upper and lower bounds that all trajectories of
unknown internal states live in, significantly reduces computation
and design effort. For instance, consider two cells in parallel as
studied previously and FIGS. 2A-2E. Due to cell heterogeneity, an
interval observer is capable of constructing two trajectories that
upper and lower bound all SOC trajectories, without dealing with
the differential-algebraic nature of the circuit dynamics.
[0070] The interval observer design for parallel cells inherits the
essence of the design for single cells. The only difference is to
compute a single set of bounding functions that bound uncertainties
from each cell in the parallel configuration.
[0071] A crucial step in designing interval observers for cells in
parallel is to find the bounding functions for the uncertainties.
Namely, the bounding functions are closely associated with the
instantaneous bounds on the local currents. However, unlike the
single cell scenario, the local currents of parallel cells are not
available for measurement. Here, it is assumed that appropriate
bounds on the local currents are given.
[0072] The width/tightness of the estimated intervals is dependent
on the magnitude of model uncertainties, and the knowledge of the
uncertainties when defining the bounding functions.
[0073] SOC Interval Observer for a Battery Cells in Series
[0074] According to the above, for a single cell modelled by
ECM
dz.sub.k/dt=(1/Q.sub.k)I(t); (30)
dV.sub.c,k/dt=-(1/R.sub.2,kC.sub.k)V.sub.c,k+(1/C.sub.k)I(t)
(31)
y.sub.k=OCV(z.sub.k)+V.sub.c,k+R.sub.1,kI(t) (32)
[0075] where k {1, 2, ,N} and N is the total number of cells
connected in series (see, for example, FIG. 1(C) for N=5). The
above model can be re-expressed into the following general
nonlinear system form:
dx.sub.k/dt=f(x.sub.k)+.delta.f(x.sub.k,.theta..sub.k)+B(.theta..sub.k)u
(33)
y.sub.k=h(x.sub.k)+.delta.h(.theta..sub.k)u, (34)
with
x k = ( x k , 1 x k , 2 ) = ( z k V c , k ) .times. f .function. (
x ) = ( 0 .times. .tau. 0 .times. x k , 2 ) , .times. .delta.
.times. .times. f .function. ( x k , .theta. k ) = ( 0 .delta..tau.
k .times. x k , 2 ) , B .function. ( .theta. k ) = ( 1 / Q k 1 / C
k ) , u = I k .function. ( t ) , ##EQU00007##
h(x.sub.k)=OCV(x.sub.k,1)+x.sub.k,2,.delta.h(.theta.k)=R.sub.1,k
and .tau..sub.k=1/R.sub.2,kC.sub.k=.tau..sub.0+.delta..tau..sub.k
(35)
where .tau..sub.0 is a user defined nominal value (assumed greater
than zero), .delta..tau..sub.k is an uncertain parameter, and
.theta..sub.k is the uncertain parameter vector
.theta..sub.k=[R.sub.1,k R.sub.2,k C.sub.k Q.sub.k], for all k {1,
2, ,N}.
[0076] By setting B=0, .delta.f=0 and .delta.h=0, in eqs.
(33)-(34), one obtains an uncertainty free nominal system
dx.sub.k/dt=f(x.sub.k) (36)
y.sub.k=h(x.sub.k) (37)
[0077] Using the transformation of coordinates .PHI.(x.sub.k),
where
.PHI. .function. ( x k ) = ( .xi. k , 1 .xi. k , 2 ) = ( .phi. 1
.function. ( x k ) .phi. 2 .function. ( x k ) ) = ( OCV .function.
( x k , 1 ) + x k , 2 - .tau. 0 .times. x k , 2 ) ( 38 )
##EQU00008##
[0078] obtained from the nominal locally observable system
(36)-(37), the original uncertain system (33)-(34) can be
transformed into the following dynamical equations:
d.xi..sub.k/dt=A.sub.0.xi..sub.k+.delta.A(.theta..sub.k).xi..sub.k+b(.xi-
..sub.k,.theta..sub.k,u) (39)
y.sub.k=H.xi.k+.delta.h(.theta..sub.k)u (40)
where
A 0 = [ 0 1 0 - .tau. 0 ] , .delta. .times. A .function. ( .theta.
k ) = [ 0 .delta..tau. k / .tau. 0 0 - .delta..tau. k ] , .times. b
.function. ( .xi. k , .theta. k , u ) = ( ( 1 / Q k ) .times. .phi.
.function. ( .xi. k , 1 + .xi. k , 2 / .tau. 0 ) + 1 / C k - .tau.
0 / C k ) .times. I , .times. H = [ 1 , 0 ] , and .times. .times.
.phi. .function. ( ) = dOCV / dx k , 1 .function. ( O .times. C
.times. V - 1 .function. ( ) ) ( 41 ) ##EQU00009##
[0079] Temperature Enhanced SOC Interval Observer--Coupled with an
Interval Observer for Battery Core and Surface Temperature
[0080] The temperature dynamics for a cell can be described by
(where the symbols are the same as in Eqs. (4)-(6))
C.sub.cdT.sub.c(t)/dt={dot over
(Q)}(t)+((T.sub.s(t)-T.sub.c(t))/R.sub.c (41)
C.sub.sdT.sub.s(t)/dt=(T.sub.f,-T.sub.s(t))/R.sub.u-((T.sub.s(t)-T.sub.c-
(t))/R.sub.c (42)
{dot over (Q)}(t)=I(t)[y(t)-OCV(z.sub.k(t))] (43)
y.sub.T(t)=T.sub.s(t) (44)
which is then re-arranged into
dx(t)/dt=Ax(t)+Bu(t)+D{dot over (Q)}(t) (45)
y(t)=Cx(t) (46)
[0081] where
x = ( T c T s ) , A = [ - 1 / R c .times. C c 1 / R c .times. C c 1
/ R c .times. C s - ( 1 R u .times. C s + 1 R c .times. C s ) ] ,
.times. B = ( 0 1 / R u .times. C s ) , D = ( 1 / C s 0 ) , C = [ 0
, 1 ] , u = T f ##EQU00010##
[0082] In the design of an interval observer for the above thermal
model, it is assumed that the thermal parameters R.sub.c, C.sub.c,
R.sub.u, and C.sub.s are constant, i.e., no parametric
uncertainties in the model. Here, it is the internal heat
generation term, i.e. {dot over (Q)}(t), that is treated as the
uncertainty entering the model. From Eq. (43), since I(t) is
measured, the max and min of voltages (y and y) are accessible, and
the upper and lower bounds of z are inherited from the electrical
model interval observer, the upper and lower bounds for {dot over
(Q)}(t) can be computed. For example, if current is positive
(I(t)>0), then
{dot over (Q)}(t)==I(t)[y(t)-OCV(z(t))], {dot over
(Q)}(t)=I(t)[y(t)-OCV(z(t))] (47)
The thermal model (45)-(46) can be reformulated as
dx/dt=Ax+Bu+D{dot over
(Q)}(t)+K(y.sub.T-Hx)=(A-KC)x+Bu+Ddq/dt+Ly.sub.T (48)
[0083] First, K=[K.sub.1 K.sub.2].sup.T needs to be chosen such
that the matrix (A-KC) is Hurwitz and Metzler. Since
( A - K .times. C ) = [ - 1 / R c .times. C c 1 R c .times. C c - K
1 - 1 / R c .times. C s - ( 1 R u .times. C s + 1 R c .times. C s )
- K 2 ] , ( 49 ) ##EQU00011##
[0084] L1.ltoreq.1/R.sub.cC.sub.c is required. In addition, since
matrix A is itself Hurwitz, the choice of a positively small
K.sub.1 and a positively large K.sub.2 would enforce the matrix
(A-KC) to be Hurwitz and Metzler.
[0085] The interval observer is then
dx/dt=(A-KC)x+Bu+D{dot over (Q)}(t)+LyT (50)
dx/dt=(A-KC)x+Bu+D{dot over (Q)}(t)+Ly.sub.T (51)
[0086] The dynamics for the estimation error =x-x and e=x-x are
given by
d /dt=(A-KC) +D({dot over (Q)}(t)-{dot over (Q)}(t))+K(yT-y.sub.T),
(52)
de/dt=(A-KC)e+D({dot over (Q)}(t)-{dot over
(Q)}(t))+K(y.sub.T-y.sub.T) (53)
[0087] where .gtoreq.0 and e.gtoreq.0 according to monotonic system
theory.
[0088] Thus, an interval observer for core temperature can then be
designed, by treating the heat generation {dot over (Q)}.sub.k(t)
uncertain, and only using the max and min surface temperature
information. As shown in FIG. 5, the interval observer for the
battery pack thermal model is used to estimate the range of core
and surface temperature uncertainties T and T of the battery cells,
then enhance the interval observer for SOC.
[0089] Results
[0090] In order to validate the interval observer design, numerical
studies are carried out on NMC battery cells modeled with a lumped
electrical-thermal model (1)-(10). The state-dependent electrical
model parameters are taken from Xinfan Lin et al. "A lumped
parameter electro-thermal model for cylindrical batteries", J. of
Power Sources, 257:1-11, 2014. The total current fed to the battery
is a UDDS driving cycle. The interval observer is used to estimate
the lower and upper bounds on the internal states from only total
current and voltage measurements. Three scenarios are considered.
First, the state estimation of a single battery cell is tested,
which accounts for uncertainties linked to SOC and temperature
dependent parameters. Then, the same observer is used to estimate
the state interval for a parallel arrangement of five cells, which
involves uncertainty due to cell heterogeneity as well as SOC and
temperature dependent parameters. Last, a temperature-enhanced
interval observer is used to estimate the state interval for a
series arrangement of five cells, which involves uncertainty due to
cell heterogeneity as well as SOC and temperature dependent
parameters.
[0091] Interval Observer for Single Battery Cell
[0092] A single cell and design of the interval observer according
to the previous section is considered. As shown in FIG. 3B, the
initial value for the SOC trajectory simulated from the plant model
is 30%, and the initial values on the interval observers (lower and
upper bounds) are 20% and 40%. The observer gains are chosen to be
L=[10-0.1].sup.T and L =[10-0.1].sup.T. FIG. 3A shows the applied
measured current, and FIG. 3B shows the estimation performance for
SOC. The solid curve is the plant model simulated SOC and the
dashed curves correspond to the estimation, with upper bounds and
lower bounds. From these plots, the lower and upper bound estimates
computed from the interval observer are able to recover quickly
(less than c.a. 20 s) from large initial errors and always enclose
the true SOC of the battery. These results confirm the stability
and inclusion properties of the designed interval observer, given
uncertain initial conditions and state-dependent parameters.
[0093] Internal Observer for Battery Cells in Parallel
[0094] A parallel arrangement of five cells, which differ in their
initial SOCs and model parameters is considered. The interval
observer is designed according to the previous sections. The
initial SOCs are 20%, 30%, 34%, 37%, and 49%, and the initial
bounds (interval observer) on SOCs are 15% and 54%. FIG. 3C shows
the estimation performance for SOC, and FIG. 3A shows the measured
total current, which is the summation of all the cell local
currents. The solid curves represent the true SOC of each cell in
the parallel arrangement, and dashed curves are the lower and upper
bound estimates. These plots show that the bound estimates provided
by the interval observer are close to the minimum and maximum
values of the state during its temporal evolution. They also
envelop the state distribution across the five cells. Therefore,
the results show that cell heterogeneity can be included as unknown
but bounded uncertainties, which can be exploited to develop an
interval observer that provides reliable bound estimates for the
system state. Moreover, stability and inclusion of the observer are
guaranteed.
[0095] Temperature Enhanced Interval Observer for Battery Cells in
Series
[0096] The interval observers described above (electrical model
enhanced by the thermal model) were used for determining the SOC
and core temperature of five sells in series. FIG. 4A shows the
estimation performance for SOC for the five cells. Curves z.sub.1
to z.sub.5 represent the true SOC of each cell in the series
arrangement, and dashed curves z and z are the lower and upper
bound estimates. These plots show that the bound estimates provided
by the interval observer are close to the minimum and maximum
values of the state during its temporal evolution. They also
envelop the state distribution across the five cells. FIG. 4B shows
the estimation performance for the core temperature of the five
cells. The solid curve represents the true T.sub.c of each cell in
the series arrangement, and dashed curves are the lower and upper
bound estimates of the T.sub.c.
CONCLUSIONS
[0097] An interval observer based on an equivalent circuit-thermal
model for lithium-ion batteries has been presented. The
SOC-temperature-dependent parameters are considered as unknown but
bounded uncertainties in the single cell model. Then, both parallel
and series arrangements of five cells are used for observer design,
where cell heterogeneity is now accounted for through the
uncertainty bounding functions. Given that the reduced nominal
battery model is locally observable, the original uncertain model
can be transformed into a partial linear form for monotone systems,
which enables interval estimation. By properly choosing the
observer gains, the state matrix of the estimation error is Hurwitz
and Metzler, which guarantees stability and inclusion of the state
bound estimates. Furthermore, a temperature-enhanced interval
observer is designed to simultaneously estimate the SOC and the
core temperature of five battery cells. A major feature of the
present estimation approach is its scalability, since the number of
states of interval observers is independent of the number of cells.
Simulation results showcase the effectiveness of the interval
observer design.
[0098] FIG. 6 shows a method for operating batteries in line with
the above-described approach. For example, in S601, a state of
charge of the batteries is estimated; in S602, a surface
temperature and a core temperature of the batteries are estimated;
and in S603, charging of the batteries is adjusted based on the
state of charge, the surface temperate, and the core
temperature.
[0099] Numerous modifications and variations of the embodiments
presented herein are possible in light of the above teachings. It
is therefore to be understood that within the scope of the claims,
the disclosure may be practiced otherwise than as specifically
described herein.
* * * * *