State of Charge Estimation for Advanced Batteries:
Reduced Order Electrochemical Modeling with Error Compensation
by
Kyle David Malinowski
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
December 12, 2011
Keywords: state of charge, Lipolymer, coulomb counting,
equivalent circuit model, reduced order electrochemical model
Copyright 2011 by Kyle David Malinowski
Approved by
Songyul Choe, Chair, Associate Professor of Mechanical Engineering
Jeffrey W. Fergus, Professor of Materials Engineering
David G. Beale, Professor of Mechanical Engineering
ii
Abstract
Advanced battery management systems are essential for efficient electrification of
vehicles. Correct state of charge (SOC) estimates will maintain a battery within its charge limits
and extend its usable life. Existing SOC algorithms for highpower Lipolymer batteries are not
capable of accuracies greater than 95% in realistic operating conditions. This thesis proposes the
use of a reduced order electrochemical model with error correction to achieve that accuracy
target in the current range of 15C with various load profiles and ambient temperature (Tamb)
range of 040?C. Hybrid electric vehicle (HEV) and electric vehicle (EV) drivers deserve to
precisely know their remaining driving range and should not have to worry about premature
battery pack exhaustion or failure.
Coulomb counting, equivalent circuit model, and electrochemical model approaches are
evaluated based on their deviation from a standard open circuit voltage curve. The equivalent
circuit is a 2nd order Randle circuit and the electrochemical model is a single particle model.
Order reduction was accomplished by assuming that reaction current is not influenced by the
electrolyte concentration, the ButlerVolmer equation may be linearized, and ion conductivity is
constant.
Two methods are proposed; one based on empirical data and one based on a closedloop
feedback system. In the first method, error rates are calculated depending on operating current
rate and ambient temperature and are then applied to the reduced order electrochemical model
SOC estimates during constant current and pulse loading. In the second method, model SOC
iii
estimates are updated continuously by comparing the model terminal voltage estimate to battery
terminal voltage measurements with a variable gain. Performance during chargedepleting and
chargesustaining modes is investigated.
This thesis is the first to document SOC estimation performance of an equivalent circuit
model and electrochemical model at ambient temperatures other than 25?C or room temperature.
Also pioneered is the exploration of high current rates, charging efficiency computations as a
function of temperature, and use of an open circuit voltage curve as a comparison tool rather than
coulomb counting. The model error characterization and minimization techniques employed are
novel as is the ROM feedback error compensation technique.
iv
Acknowledgments
I would like to express my gratitude to my advisor, Dr. Songyul Choe, for generously
offering me this research opportunity. Under his guidance I was able to develop a strong problem
solving skill set that will serve me well the rest of my life. I would also like to acknowledge
committee member Dr. Jeffrey W. Fergus for providing interdisciplinary insight. Special thanks
to John Marcell for always providing assistance with a smile on his face and Steven Moore for
maintaining our experimental equipment.
Many thanks to my colleagues Xueyan Li, Rujian Fu, Victor Agubra, Meng Xiao, Trey
Zarecor, Farzana Rahman, and Yinyin Zhao. They taught me more than any textbook or journal
article ever could. Additional thanks to Xueyan Li for brainstorming with me on a daily basis. I
hope to meet with them each again in the professional world.
Finally, I would like to convey my deep appreciation to my parents, David and Patricia
Malinowski, for their unwavering support.
v
Table of Contents
Abstract ........................................................................................................................................... ii
Acknowledgments.......................................................................................................................... iv
List of Tables ................................................................................................................................. ix
List of Figures ................................................................................................................................. x
List of Abbreviations ................................................................................................................... xiii
Chapter 1: Introduction ................................................................................................................... 1
1.1 Problem Statement ................................................................................................................ 1
1.2 Definition of SOC ................................................................................................................. 2
Chapter 2: Battery Fundamentals.................................................................................................... 4
Chapter 3: Mathematical Modeling Fundamentals ....................................................................... 21
Chapter 4: Literature Review ........................................................................................................ 26
4.1 Error Correction Techniques ............................................................................................... 27
4.2 SOC Estimation without a Model ....................................................................................... 31
4.2.1 Literature Rundown ...................................................................................................... 37
4.3 SOC Estimation with a Model ............................................................................................ 42
4.3.1 Literature Rundown ...................................................................................................... 43
4.4 Patent Review ...................................................................................................................... 56
4.5 Summary ............................................................................................................................. 65
Chapter 5: Experimental Setup ..................................................................................................... 68
vi
5.1 Test Equipment ................................................................................................................... 68
5.2 Software Programming ....................................................................................................... 71
5.3 LiPolymer Battery Specifications ...................................................................................... 73
Chapter 6: Battery Modeling ........................................................................................................ 75
6.1 Minimum Rest Time ........................................................................................................... 75
6.2 OCV Curve .......................................................................................................................... 76
6.3 Equivalent Circuit Model (ECM) ........................................................................................ 79
6.3.1 RC Circuit ..................................................................................................................... 79
6.3.2 Parameter Extraction .................................................................................................... 80
6.4 Full Order Thermal Electrochemical Model (FOM) ........................................................... 82
6.5 Reduced Order Thermal Electrochemical Model (ROM) ................................................... 85
6.6 Thermal Component of ROM ............................................................................................. 89
Chapter 7: Existing Method Error Characterization ..................................................................... 90
7.1 Experimental Test Matrix.................................................................................................... 91
7.2 Coulomb Counting .............................................................................................................. 93
7.3 ECM .................................................................................................................................... 95
7.4 ROM .................................................................................................................................... 96
7.5 Summary ............................................................................................................................. 98
Chapter 8: SOC Estimation Error Compensation of ROM Using an Empirical Method ........... 100
8.1 ROM Error Rate ................................................................................................................ 101
8.2 2D Interpolation ................................................................................................................ 104
8.3 Single Discharge ............................................................................................................... 105
8.4 Single Cycle ...................................................................................................................... 107
vii
8.5 HPPC Test ......................................................................................................................... 107
8.6 5 Cycles ............................................................................................................................. 108
Chapter 9: SOC Estimation Error Compensation of ROM Using a Feedback Method.............. 110
9.1 Feedback Design ............................................................................................................... 111
9.2 Gain Optimization ............................................................................................................. 113
9.3 Single Discharge ............................................................................................................... 114
9.3.1 Effect of Feedback ...................................................................................................... 115
9.4 Single Cycle ...................................................................................................................... 116
9.4.1 Effect of Feedback ...................................................................................................... 117
9.5 HPPC Test ......................................................................................................................... 117
9.5.1 Effect of Feedback ...................................................................................................... 118
9.6 5 Cycles ............................................................................................................................. 118
9.6.1 Effect of Feedback ...................................................................................................... 119
Chapter 10: Error Analysis ......................................................................................................... 120
10.1 ChargeDepleting Mode .................................................................................................. 120
10.2 ChargeSustaining Mode ................................................................................................. 122
Chapter 11: Conclusions and Future Work ................................................................................. 124
References ................................................................................................................................... 126
Appendix 1: Tree Diagram of SOC Estimation Methods and Principle Authors ....................... 133
Appendix 2: Electrochemical Model Parameters ....................................................................... 134
Appendix 3: Experimental Test Matrix ...................................................................................... 135
Appendix 4: Single Discharge Average SOC Error ................................................................... 136
Appendix 5: Single Cycle Final SOC Error ................................................................................ 137
viii
Appendix 6: HPPC Test Average SOC Error ............................................................................. 138
Appendix 7: 5 Cycle Final SOC Error ........................................................................................ 139
ix
List of Tables
Table 1: Advantages and drawbacks of existing SOC estimation techniques .............................. 66
Table 2: Highpower test equipment............................................................................................. 69
Table 3: SOC Tamb correction factors ............................................................................................ 79
Table 4: ECM extracted parameters ............................................................................................. 82
Table 5: Efficiency coefficients .................................................................................................... 91
Table 6: Proposed investigation range .......................................................................................... 92
Table 7: ROM single discharge final SOC errors ....................................................................... 101
Table 8: ROM discharge error rates ............................................................................................ 102
Table 9: ROM charge error rates ................................................................................................ 103
Table 10: Total discharge times .................................................................................................. 106
x
List of Figures
Figure 1: Microcell diagram ........................................................................................................... 5
Figure 2: Discharge kinetics ........................................................................................................... 6
Figure 3: SEI layer structure ........................................................................................................... 8
Figure 4: Sources of battery aging .................................................................................................. 9
Figure 5: Pouch type Cobased cathode microcell ....................................................................... 20
Figure 6: Second order Randle circuit .......................................................................................... 21
Figure 7: Modified Thevenin circuit ............................................................................................. 21
Figure 8: Fuzzy logic characterization example ........................................................................... 27
Figure 9: Artificial neural network layers ..................................................................................... 28
Figure 10: State observer feedback loop ....................................................................................... 29
Figure 11: Kalman filter layers ..................................................................................................... 30
Figure 12: Typical OCVSOC curve ............................................................................................ 35
Figure 13: EIS shape analysis ....................................................................................................... 36
Figure 14: Patent categorization ................................................................................................... 65
Figure 15: CT VI calibration plot ................................................................................................ 69
Figure 16: Test station wiring diagram ......................................................................................... 70
Figure 17: Test station photograph ............................................................................................... 71
Figure 18: LabVIEW front panel screenshot ................................................................................ 72
Figure 19: LabVIEW current control block diagram .................................................................... 72
xi
Figure 20: LabVIEW data recording block diagram .................................................................... 73
Figure 21: Terminal voltage relaxation......................................................................................... 75
Figure 22: Minimum rest time ...................................................................................................... 76
Figure 23: OCV generator pulse sample ....................................................................................... 77
Figure 24: OCV curve ................................................................................................................... 78
Figure 25: Effect of Tamb on 1C full discharge ............................................................................. 78
Figure 26: ECM RC circuit ........................................................................................................... 80
Figure 27: ECM parameter extraction pulse ................................................................................. 81
Figure 28: Curve fitting example .................................................................................................. 82
Figure 29: Electrochemical model schematic ............................................................................... 83
Figure 30: Electrochemical model calculation scheme ................................................................ 85
Figure 31: HPPC test profile ......................................................................................................... 92
Figure 32: Coulomb counting error characterization .................................................................... 94
Figure 33: Coulomb counting SOC estimation error summary .................................................... 95
Figure 34: ECM error characterization ......................................................................................... 96
Figure 35: ROM error characterization ......................................................................................... 97
Figure 36: ROM SOC estimation error summary ......................................................................... 98
Figure 37: Empirical method schematic block diagram ............................................................. 100
Figure 38: Single charge data at 1C, 0?C .................................................................................... 103
Figure 39: ROM discharge error rate as a function of Tamb ........................................................ 104
Figure 40: Constant discharge  empirical .................................................................................. 105
Figure 41: Single cycle data at 1C, 0?C ...................................................................................... 107
Figure 42: HPPC cycle data at 1C, 0?C ...................................................................................... 108
xii
Figure 43: 5 cycle data at 1C, 0?C .............................................................................................. 109
Figure 44: Feedback method schematic block diagram .............................................................. 111
Figure 45: Constant discharge  feedback ................................................................................... 115
Figure 46: Effect of feedback  single discharge ........................................................................ 116
Figure 47: Effect of feedback  single cycle ............................................................................... 117
Figure 48: Effect of feedback  HPPC ........................................................................................ 118
Figure 49: Effect of feedback  5 cycles ..................................................................................... 119
Figure 50: Chargedepleting cycle analysis at 0?C ..................................................................... 120
Figure 51: Chargedepleting cycle analysis at 25?C ................................................................... 121
Figure 52: Chargedepleting cycle analysis at 40?C ................................................................... 121
Figure 53: Chargesustaining cycle analysis at 0?C ................................................................... 122
Figure 54: Chargesustaining cycle analysis at 25?C ................................................................. 123
Figure 55: Chargesustaining cycle analysis at 40?C ................................................................. 123
xiii
List of Abbreviations
a Activity of species (units: mol?cm3)
as Interfacial surface area (units: cm1)
A Electrode crosssectional area (units: cm2)
C Reactant surface concentration (units: mol?cm2)
Cp Specific heat capacity (units: J?kg1?K1)
ce Electrolyte concentration (units: mol?cm3); Electron concentration (units: mol?cm3)
ci Ion concentration (units: mol?cm3)
cs Solid phase concentration (units: mol?cm3)
cs,ave Volumeaveraged solid phase concentration (units: mol?cm3)
cs,max Maximum solid phase concentration (units: mol?cm3)
cs,surf Solid phase surface concentration (units: mol?cm3)
d Depth of battery (units: m)
Deeff Effective diffusion coefficient (units: cm2?s1)
Di Ion diffusion coefficient (units: cm2?s1)
Ds Solid phase diffusion coefficient (units: cm2?s1)
e or Q0 Elementary charge (?1.6022?1019 C)
Ecell Cell potential (units: V)
E?cell Standard cell potential (units: V)
Eeq Equilibrium potential (units: V)
xiv
En Potential without current (units: V)
F Faraday constant (96485 C?mol1)
h Planck constant (6.626?1034 J?s); Heat transfer coefficient (units: W?m2?K1)
i Current density (units: A?cm2)
Icutoff Full charge cutoff current, specified by manufacturer (units: A)
i? Side reaction current (units: A)
In Exchange current (units: A)
i0 Exchange current density (units: A?cm2)
j Reactant flux (units: mol?cm2?s1)
ji Flux density (units: mol?cm2?s1)
jLi Lithiumion current density (units: A?cm3)
k Reaction rate coefficient (units: s1); Thermal conductivity (units: W?m1?K1)
kB Boltzmann constant (1.381?1023 J?K1)
Keq Reaction quotient (dimensionless)
L Thickness of negative electrode (units: cm)
n Number of electrons transferred (dimensionless)
Ni Ion flux density (units: mol?cm2?s1)
p Bruggeman porosity exponent (dimensionless)
q Heat transfer rate between the cell and its surroundings (units: W?m3)
qave Volumeaveraged concentration flux (units: mol?cm2?s1)
Qcharge Amount of charge input to battery during a full charge (units: Ah)
Qdischarge Amount of charge output from battery during a full discharge (units: Ah)
Qgen Heat generation rate per unit volume (units: W?m3)
xv
Qmax Maximum charge capacity of a battery (units: Ah)
Qrated Rated charge capacity of a battery as defined by manufacturer (units: Ah)
Qreleasable Amount of charge remaining in a battery that may be discharged until Vmin is reached
(units: Ah)
r Radius of electrode particle (units: cm)
Rs Maximum radius of electrode particle (units: cm)
S0 Stoichiometry values at 0% SOC (dimensionless)
S100 Stoichiometry values at 100% SOC (dimensionless)
SOCCC SOC estimate from coulomb counting (units: %)
SOCROM SOC estimate from the ROM (units: %)
SOCOCV SOC estimate from OCV curve (units: %)
t+0 Li+ transference number (dimensionless)
Tamb Ambient temperature (units: ?C)
ti Transference number (dimensionless)
ttotal Total time of a full discharge at a specified current and ambient temperature (units: s)
U Equilibrium potential (units: V)
ui Ion mobility (units: cm2?s1?V1)
v Stoichiometric coefficient (dimensionless)
V Bulk fluid velocity (units: cm?s1)
ve Electron velocity (units: cm?s1)
Vmax Maximum terminal voltage of battery, specified by manufacturer (units: V)
Vmin Minimum terminal voltage of battery, specified by manufacturer (units: V)
VT Terminal voltage (units: V)
xvi
ze Electron count (dimensionless)
zi Ion charge number (dimensionless)
? Transfer coefficient (dimensionless)
?a Anodic transfer coefficient (dimensionless)
?c Cathodic transfer coefficient (dimensionless)
? Electrode width (units: cm); SEI film thickness (units: cm)
? Permittivity (units: F?cm1)
?e Electrolyte phase volume fraction (dimensionless); Active material volume fraction
(dimensionless); Porosity (dimensionless)
?f Conductive filler volume fraction (dimensionless)
?p Polymer phase volume fraction (dimensionless)
? Total overpotential (units: V); Efficiency coefficient (dimensionless)
?act Activation overpotential (units: V)
?conc Concentration overpotential (units: V)
?ohm Ohmic overpotential (units: V)
?eff Effective ionic conductivity (units: S?cm1)
? Resistivity (units: ??cm); Density (units: kg?m3)
? Conductivity (units: S?cm1)
?2 SOC estimation variance (units: %2)
?2ave Average SOC estimation variance (units: %2)
?eff Effective solid conductivity (units: S?cm1)
?cc Coulomb counting SOC estimation error (units: %)
?ECM ECM SOC estimation error (units: %)
xvii
?ROM ROM SOC estimation error (units: %)
ROM?? Coulomb counting SOC estimation error rate (units: %/hr)
?e Electrolyte potential (units: V)
?s Solid phase potential (units: V)
?se Phase potential difference (units: V)
1
Chapter 1: Introduction
The electrification of automobiles is becoming a popular field of study as consumer
demand increases for vehicle efficiency and manufacturers face strict emission and fuel economy
requirements. Batteries provide safe, durable, and efficient electrical energy storage, some types
more than others. Lithiumion polymer cells have highest energy density of any battery on the
market today, provide the greatest electrochemical potential, are relatively light in weight,
require no maintenance, exhibit negligible hysteresis, have long cycle lives, demonstrate little
selfdischarge, and are environmentally safe to dispose. All these factors make them desirable to
OEMs and explain their relatively recent implementation in hybrid electric vehicles (HEV) and
electric vehicles (EV). This thesis will focus on a fundamental issue common to all highpower
batteries.
1.1 Problem Statement
Accurately predicting the remaining driving range in a hybrid electric or full electric
vehicle is essential. As HEVs/EVs grow in popularity and complexity, new monitoring methods
are needed to better track performance. Drivers need to know how much further they can travel
before their vehicle batteries require a recharge. The state of charge (SOC) of a battery or pack of
batteries is analogous to a fuel gauge in an internal combustion vehicle. The remaining charge
calculation must be correct in order to utilize a battery?s full capability. This must be
accomplished within charge limitations of the battery so as to not cause unwanted degradation in
the form of permanent charge capacity loss. Precision of SOC evaluation is instrumental in
alleviating EV range anxiety. Most OEMs have a SOC estimation error target of ? 5 % under all
2
realistic driving conditions, yet no existing estimation techniques are capable of this. The
objective of this research is to devise a new method to meet the target of ? 95% accuracy in the
current range of 15C with various load profiles and Tamb range of 040?C. The selected
investigation range covers the vast majority of realistic HEV/EV battery pack operating
conditions.
1.2 Definition of SOC
SOC is defined as ratio of the charge capacity in a cell (Qreleasable) to the maximum charge
capacity at full charge (Qmax). The preferred SOC reference should be Qmax rather than the rated
charge capacity of a new cell (Qrated) because Qmax decreases as the cell ages. With this approach,
the SOC range will always be 0100%. The equation for SOC may be generalized as
%100m a x ?? QQS O C r e le a s a b le
where Q has units of amperehours (Ah). Units of charge capacity in highpower batteries are
typically Ah. Qreleasable and Qmax are a measure of electrons flowing from a battery, therefore it
does not represent battery internal dynamics. The standard Tamb and Crate that these variables
are evaluated at is 25?C and 1C, respectively.
A more precise definition considers the components that contribute to charge capacity,
which are the lithiumions inside a battery. SOC is an average of the ratio of the concentration of
lithiumions in the negative electrode to the maximum concentration in the negative electrode,
exemplified by
? ??? ?? ???
?
?
? 0 0100m a x, 0m a x,, )(1 dxSSc SccS O C s sa v es
3
where ? is the width of the negative electrode (cm), cs,ave is the volumeaveraged solid phase
concentration (mol?cm3), cs,max is the maximum solid phase concentration (mol?cm3), and S0 and
S100 are the lithium stoichiometry values at 0% and 100% SOC, respectively. This definition
accounts for physical electrode limitations; therefore, it will be adopted as the formal SOC
definition.
4
Chapter 2: Battery Fundamentals
Batteries are electrochemical energy storage devices. They are categorized into two
groups, primary and secondary. Primary batteries are those that may only be discharged once and
do not possess the ability to be charged. Charge is permanently depleted during use, such as with
most small portable electronics. Secondary batteries are those that may be discharged and
charged multiple times. Charge may be replenished, such as with most batteries found in laptops
and automobiles. A power conversion cycle exists whereby chemical potential is converted to
electric power during discharge and electric power is converted back to chemical potential
during charge.
A few key electrochemical terms are integral for complete understanding of a battery.
The terms battery and cell are used interchangeably. The amount of energy stored in a battery is
known as charge (Q). It is defined as the quantity of electricity and has units of coulombs (C).
Charge can be positive or negative depending on whether the matter exhibits attraction or
repulsion in the presence of other matter. The elementary charge (e or Q0) is defined to be the
electric charge carried by a single proton or electron and is equal to ?1.6022?1019 C; positive for
a proton and negative for an electron. Within matter, there exist two types of charge, fixed and
mobile. Mobile charges consist of ions and electrons found in conductors while fixed charges are
those found in insulative matter. Mobile charge material is the foundation for battery internal
construction. Examples of electronic conductors include copper and carbon, and an example of
an ionic conductor is battery electrolyte.
Batteries are manufactured with multiple layers of electrodes, each consisting of a
combination of electronic and ionic conductors. Desired performance may be enhanced by
5
altering the electrode chemistry, which determines the amount of lithiumions initially
intercalated, or bound between the electrode compounds. Ions are conserved during operation,
meaning that the amount is the same whether it has been cycled zero times or one thousand.
Lithiumion (Li+) polymer intercalationtype batteries are made of many layers of
microcells. A typical microcell diagram is shown in Figure 1.
Figure 1: Microcell diagram [47]
The term intercalationtype implies that the electrodes have a lattice structure.
Charging/discharging the cell causes the Liions to leave the positive/negative electrode and
enter the lattice structure of the negative/positive electrode, respectively. Discharge kinetics are
shown in Figure 2.
6
Figure 2: Discharge kinetics [61]
This process of ions being moved in and out of an interstitial site in the lattice is called
intercalation. Lipolymer batteries evolved from lithiumion batteries with the difference being
that the electrolyte is held in a solid polymer composite rather than an organic solvent [71]. The
electrolyte consists of lithium salts such as LiPF6, LiBF4, or LiClO4 [33],[36],[59]. One microcell
has five components including a negative current collector, negative electrode, separator with
infused electrolyte, positive electrode, and positive current collector. Both electrodes are also
infused with electrolyte. A binder material, such as polyvinylidene fluoride (PVDF), appends the
separator to the electrode. The typical main ingredient of a negative electrode is carbon while the
typical for the positive electrode is a metal oxide consisting of manganese, nickel, and cobalt.
The voltage of a lithiumion polymer cells typically varies from 2.5 to 4.2 V.
The microcell layer order is crucial to battery operation. During discharge, Li+ ions
diffuse to the surface of carbon particles in the negative electrode where they deintercalate and
transfer to an electrolyte solution. The positively charged ions travel via diffusion and migration
through the electrolyte solution, where they again react and diffuse into metal oxide active
7
material particles [8],[33],[36],[39],[40],[41],[47],[54],[55],[59],[68],[80],[90]. Electrons,
produced in the negative electrode reaction and consumed in the positive electrode reaction, are
blocked by the electronically insulating separator and instead must travel through an external
circuit. During discharge, these electrons provide power to an external load. A battery is
considered fully charged/discharged when the Liion concentration in the negative electrode
reaches a maximum/minimum, respectively. The manufacturer determines these limits so as not
to damage a battery.
Ions migrate through channels in the separator material between the electrodes. These
active molecules attract Li+ ions. Driven by the electromotive force, the Li+ ions bond with active
molecules in succession until they reach the opposite electrode. Also, the concentration gradient
will drive Li+ ions to disperse evenly in the electrolyte when there is no load on the battery.
Transport modes will be discussed in detail later on.
Ion and electron transport may be inhibited by temperature or resistive layer growth.
High temperature may cause the electrolytic polymer to decompose resulting in active material
becoming inactive. Low temperature may cause the Li+ ions to plate out in solid metal form. This
results in a loss of Li+ ions which decreases the electrolyte conductivity since it is a function of
Li+ ion concentration. The resistive layer between the anode particles and surrounding electrolyte
is known as the solid electrolyte interface (SEI), formed initially by the decomposition of organic
solvents in the electrolyte upon primary charging. It is deliberately formed by the manufacturer
in order to provide a barrier that restricts reactions between the carbon anode and the electrolyte.
Figure 3 shows the SEI layer structure.
8
Figure 3: SEI layer structure [24]
During the first activation cycles, a part of Li+ ions will remain in the negative electrode
and another part will be consumed in the SEI, which is an irreversible process [54]. This layer
grows with cycling as organic solvents and Li+ ions continue to plate the anode surface [58].
Significantly more cathode material than anode material, electrode overlap and subsequent
shorting during manufacturing, and local nonuniformities in polarization augment the
probability of Li metal plating. In addition, transition metals from the cathode may dissolve and
become part of the SEI [60]. The growth of the SEI is related to the rate of the side reactions by
FL Mit ??? ????
where ? is the film thickness (cm) composed of lithium products and i? is the rate of the side
reaction (A). The other parameters are all constants for a given reaction and a given battery
chemistry [55],[56].
Many researchers consider SEI growth to be the major source of battery aging. Other
contributors may be seen in Figure 4, with high and low temperatures being detrimental for
different reasons.
9
Figure 4: Sources of battery aging [60]
The fundamentals of electrochemistry will now be discussed. Current (I) is defined as the
rate of flow of electric charge. It is the measure of flow of coulombs per second, with units of
amperes (A). Current flows in the opposite direction of electron flow. When considering current
to or from a battery, the current direction during discharge is defined as positive. It may be
written mathematically as a function of the Faraday constant F (96485 C?mol1), the cross
sectional area A (cm2), the electron velocity ve (cm?s1), and the electron concentration ce
(mol?cm3) in the form of
ee cFAvdtdQI ???
where F, defined as the magnitude of electric charge per mole of electrons, equals Avogadro?s
number NAv, times the elementary charge Q0, is calculated to be
1191230 96485)106022.1)(100221.6( ??? ?????? m o lCCm o lQNF Av
Current density (i) is the measure of the density of flow of charge. It is defined as the
current per unit area with units of amperes per square centimeter (A?cm2), and calculated as
AIi?
10
Current, however, is defined with different variable values for electrons and ions. For
electrons,
eee cFAvzI ?
where ze is the number of electrons.
For ions,
?? i iii cvzFAI
where zi is the charge number of the ion.
Flux density, ji, is defined as the moles per area per second (mol?cm2?s1) and calculated
as
iii cvj ?
??? i ii jzFi
Electric potential (?) is defined as the electric potential energy divided by the charge. It is
a scalar quantity measured in volts (V) or joules per coulomb (J/C). Within a battery, voltage is
equal to the difference in electrical potential between two materials. There will be zero voltage
between identical electrodes since they share the same potential. Lithium has different potentials
when placed in an interstitial lattice site of the solid active materials of the anode or cathode [15].
Standard potential is calculated as the difference between the potential of a material and
that of a standard hydrogen electrode. Hydrogen gas at one bar pressure and 25?C is bubbled
over an inert platinum electrode in an aqueous solution of 1M concentration of H+(aq). The
standard cell potential (E?cell) is the difference between the cathode and anode potentials at
standard conditions, measured in volts. There is a tendency for reduction to occur at one of the
electrodes. E?cell may be positive or negative depending on the reduction halfreaction. Tables
11
have been created that document the standard electrode potentials of many known solutions. Li+
ions have one of the greatest negative standard electrode potentials, 3.04 V.
Electrical conductivity (? or ?) is the measure of a material?s ability to conduct current. It
is defined as the ratio of current density ( ji) to the magnitude of the electric field (E). It is the
inverse of resistivity and has the units of siemens per centimeter (S?cm1), with one Siemen equal
to one ampere per volt. Materials with higher electron conductivity, such as copper, allow for
easier current flow while electron insulating materials, such as water, are much more resistive to
flow. Different materials have different charge carriers which might include electrons or
specified anions or cations.
Electrical resistivity (?) is the measure of a material?s ability to oppose electric current.
According to Ohm?s law, it is defined as the ratio of voltage to the current and has units of ohm
centimeters (??cm). Ohmic resistance is due to the current flow through the various metallic
components while ionic resistance is due to voltage drop resulting from the flow of ions across
the separator [19]. Capacitance (C) is the ability of a body to hold an electrical charge. It is
defined as the ratio of charge to the voltage and has the units of farads (F). Charge can be written
as
ELAECQ s o u r c e ???????????? ?
where C is capacitance, ? is permittivity (F?cm1), and E is the potential.
There are three modes of ion transport within most batteries, which includes migration,
diffusion, and convection. Convection does not occur in lithiumion polymer batteries due to
their solid electrolyte. Migration is potentialdriven. It is directly related to the solid phase
surface concentration gradient [58]. An electric field across a solution creates a driving force.
Double layer capacity is a buildup of charge just outside of the two electrodes due to polarized
12
molecules attracted to either electrode. This charge separation creates a voltage drop that drives
lithiumion transfer [19]. It follows the equation
???? iiim ig r a tio ni FcuzN ,
where ui is the ion mobility (cm2?s1?V1) and ci is ion concentration (mol?cm3). In each of the
three ion transport equations, Ni is the flux density (mol?cm2?s1).
Diffusion is concentrationdriven. Ions diffuse from a higher concentration to a lower
concentration. It follows the equation
iid iffu sio ni cDN ???,
where Di is the ion diffusion coefficient (cm2?s1).
Convection is velocitydriven. Ions are carried along with the bulk movement of a fluid.
This does not affect lithiumion polymer batteries since the electrolyte is solid. It follows the
equation
VcN ico n vectio ni ??,
where V is the bulk fluid velocity (cm?s1).
Since batteries are electrochemical devices, both electrical and chemical terminology
must be understood. Current requires the transfer of electrons or ions. Chemical reactions known
as oxidation and reduction make this possible. Oxidation occurs when a material releases
electrons, thereby releasing cations or accepting anions, while reduction is the acceptance of
electrons, thereby reducing the number of cations or increasing the number of anions. In any
oxidationreduction (redox) reaction, there are two halfreactions. One species undergoes
oxidation and another undergoes reduction. Basically, electrons are transfers from one to the
other.
13
A standard change in Gibbs free energy, ?G?, is proportional to the standard potential, E?,
as
?? nFEG ???
Using the Gibbs function,
STHG ?????
the free energy change for the reaction can be calculated where the change in entropy of a battery
is calculated as
TEnFS ????
where n is the number of electrons transferred in the reaction, F is the Faraday constant, and E is
the cell potential.
Redox chemical equations must be balanced with respect to mass and charge. In an
oxidation halfreaction equation, a number of neutralcharged molecules will be split into a
number of ions plus a number of electrons. Likewise, in a reduction halfreaction equation, a
number of ions and a number of electrons combine to form a number of neutralcharged
molecules. The same number of electrons should appear in each halfreaction equation, in order
to follow conservation of charge laws.
The two halfreaction equations may be combined to obtain an overall reaction equation.
Typical cathode materials used in lithiumion polymer batteries are manganese, nickel, and
cobalt. The following chemical equations show the discharge reactions with a manganesebased
cathode [55]
221 Li M n Oxex LiM n OLi x ??? ??? (Reduction in cathode)
Cxex L iLiC x 66 ??? ?? (Oxidation in anode)
14
CLi M n OLiCM n OLi xx 62621 ???? (Net balanced equation)
A halfcell consists of an electrode immersed in a solution of ions. Two of these joined
by a salt bridge form a voltaic (galvanic) cell. The salt bridge allows for ion transport and
inhibits any electron flow. It is needed however for electron flow through the outer loop from
electrode to electrode. The electrons may only travel between electrodes through an external load
while the ions may only travel from halfcell solution to the other. The electrode at which
oxidation occurs is referred to as the anode and the electrode at which reduction occurs is
referred to as the cathode. The cell potential (Ecell) is the driving force that moves electrons from
the anode to the cathode. The electrons flow in this direction during discharging of a battery.
When electrons move in the opposite direction it is called charging. A simplified voltaic cell
diagram may be written with the anode halfcell reaction on the left, the cathode halfcell
reaction on the right, and a double vertical line in between indicating the salt bridge. It would
look like
)()()()( 2 sMnaqMnaqLisLi ??
As mentioned before, standard electrode potentials (E?) are derived in comparison to a
hydrogen electrode under standard conditions. Ecell is equal to the electric potential of the
cathode minus the electric potential of the anode. If this is a positive value, then the reaction
from the left side of the voltaic cell representation shown above is considered to be spontaneous
(galvanic cell). If negative, it is nonspontaneous (electrolyte cell). The system is considered to
be at equilibrium when Ecell is equal to zero, which means that the battery is dead. Electricity
may only be produced when the cell reaction is not at equilibrium.
15
The reaction quotient (Keq) is a function of the concentrations of chemical species
involved in the chemical reaction. Standard Gibbs free energy change and standard cell potential
may be written as
eqKRTG ln??? ? and eqcell KnFRTE ln??
The Nernst equation relates the nonstandard cell potential to the standard cell potential.
The concentrations of species in a cell change with current and will continue to change until
equilibrium is reached. The concentration of species may be determined by measuring the
nonstandard cell potential by using
)(lo g303.2 10 anFRTEE c e llc e ll ?? ?
where a is the activity of species (mol?cm3), related to Keq by
?
?
?
i
v
ti
j
v
tj
eq ij
j
a
a
K
}{
}{
in which the subscript j refers to reactants, the subscript i refers to products, and v is the
stoichiometric coefficient.
A battery stores electrical energy until connected to a load when it releases that energy.
The anode experiences oxidation while the cathode experiences reduction. During this process,
the change in Gibbs free energy is negative, which means that the battery has less free energy
after releasing the energy than it did at the beginning.
Overpotentials are a source of voltage drop and power loss. There are three types
including ohmic, activation, and concentration. They increase with current and temperature [45].
They may be added together to calculate the total overpotential (?) as a function of applied
16
current. This represents the change in potential that a charged species would go through as it
passes through the spherical particle into the electrolyte [15]. At each electrode,
Ues ??? ???
where ?s is the solid phase potential, ?e is the electrolyte potential, and U is the equilibrium
potential.
Ohmic overpotential (?ohm) also known as resistance polarization, is equal to the
difference in potential across a length of solution. It is linked to the finite conductivity of the
nonelectroactive cell constituents. It is a function of current and resistance, considered as a
voltage drop, and may be calculated as [55]
IRAILAdxIdxidL LLLohm ????????? ??? ???????
000
)()0(
Electrical work (Wel) can be calculated in numerous ways including
Gn F EQEW el ?????
nF STHnF GE )( ?????????
where Q is charge, E is potential, n is the number of electrons, and F is the Faraday constant.
Current density, from Faraday?s law, is
)()( rdrdoxoxrdox CkCknFjjnFn F ji ?????
where ?
????? ??? RT GhTkk B exp
,
j is the reactant flux (mol?cm2?s1), k is the reaction rate coefficient (s1) as a function of Gibbs
free energy, C is the reactant surface concentration (mol?cm2), kB is the Boltzmann constant
(1.381?1023 J?K1), and h is the Planck constant (6.626?1034 J?s).
17
The change in Gibbs free energy may also be written as a function of chemical and
electrical terms. In general,
FEGG a c tiv a tio n ?????
where ? is the transfer coefficient, defined as the number of electrons transferred during
multistep reactions.
The reaction rate coefficients for reduction and oxidation may now be written in terms of
? and ?G as
?????? ??? RT FEkk rdrdrd ?e x p,0
?????? ??? RT FEkk oxoxox ?e x p,0
where ?
????? ???? RTGhTkk a c t i v a t i o nB e x p0
.
If the reaction rate coefficients are then plugged back into the current density equations,
the net current density becomes
?????? ?????? ????????? ??? rdrdrdoxoxox CRT FEkCRT FEknFi ?? e x pe x p ,0,0
Exchange current density (i0) is a measure of how active an electrode surface is. The
higher it is, the easier it is for charge to flow. When the net current density is equal to zero the
system is said to be in equilibrium, the term is then called exchange current density, calculated as
rdeqrdrdoxeqoxox CRT
FEn F kCRT FEn F ki ??
?
???
?
? ?????
?
???
?
? ??? ?? e x pe x p ,0,00
where Eeq is the equilibrium potential.
18
The ButlerVolmer equation goes a step further in the net current density equation by
replacing nFk0C with i0, so
? ? ? ?
????
?
????
? ??
?
???
?
? ?????
?
???
?
? ??? RT EEFRT EEFii eqrdeqox ?? e x pe x p
0
Activation overpotential refers to the potential needed to overcome the large initial
potential step in a multistep reaction. It is linked to the thermodynamic irreversibility of some
chemical processes. Using the ButlerVolmer and Nernst equations, the activation overpotential
(?act) can be derived in [55]
rdoxa c tna c tn IIRT FIRT FII ???????? ????????? ??? ???? e x p)1(e x p
where In is the exchange current. The equation presented matches that mentioned at the
beginning.
Concentration overpotential refers to conditions of limits of diffusion. It is linked to the
transient nonhomogeneity in the spatial repartition of the electroactive species. If lithiumion
concentrations at an electrode surface are too high or too low, the mass transport of the species
can affect the cell potential. When the electrode is represented as a collection of spheres, the ratio
of Li+ surface concentration over the maximum concentration represents the amount of readily
available power. This is because the surface lithium is easily removed, whereas the ions near the
center of the particle are limited by the rate of diffusion to reach the surface [22],[23],[33],[39].
Based on Nernst?s law, the concentration overpotential (?conc) can be calculated as [55]
bOsR
sObR
nc o n c cc ccnFRTEE ln????
where En is the potential when no current is applied, R and O represent reduction and oxidation
species, and s and b represent surface and bulk, respectively.
19
There are documented examples of hysteresis occurring in Liion batteries observed
during the intercalation of lithiumions into carbon. It is believed that the lithium atoms may
bond to hydrogenterminated edges of hexagonal carbon. The capacity for the insertion of
lithium is strongly dependent on the hydrogen content of the carbon materials. If the inserted
lithium bonds to a carbon atom, which also is bonded with a hydrogen atom, a corresponding
change to the carboncarbon bond from (sp)2 to (sp)3 may occur, contributing to a large
hysteresis during lithium insertion [54]. Another possible cause for hysteresis is phase transitions
in the positive electrode.
There are multiple sources of heat generation within a cell. These include Joule heating,
exothermic/endothermic chemical reactions, heat of mixing, and overpotentials. Joule heating is
created by the flow of electrons in the current collectors, and to a much smaller extent the flow
of electrons in the electrode and flow of ions in the electrolyte, and is proportional to the square
of the current. Chemical reactions which occur during intercalation and deintercalation either
expel or absorb heat as new molecules are formed, proportional to charge. Heat of mixing is
caused by ion flow in the electrode and is proportional to the gradient of ion concentration.
Overpotential, specifically activation overpotential, causes heat generation at the interface
between electrode and electrolyte. Irreversible thermodynamic processes create entropy,
observed as heat, proportional to the current. As cell temperature decreases, slow kinetics require
an increase in overpotentials to drive reactions. Slow diffusion results in increased solid and
electrolyte phase concentration gradients and also results in decreased ionic conductivity of the
electrolyte, which leads to larger electrolyte phase potential gradients [58].
There are three main types of lithiumion polymer batteries, which are pouch, cylindrical,
and prismatic. Pouchtype cells consist of multilayer microcells stacked atop one another in a
20
rectangular shape with a soft outer shell. Cylindricaltype cells consist of one long microcell
wrapped tightly into a cylinder shape with a hard shell. Prismatictype cells are similar to
cylindricaltype only they are pressed into a cubical shape. An example of a pouchtype cell with
a cobaltbased cathode is shown in Figure 5.
Figure 5: Pouch type Cobased cathode microcell [24]
Applications, causes of degradation, and quick charging techniques are key focuses of
research presently. Various processes may contribute to battery aging such as electrolyte
decomposition, the formation of surface films on both electrodes, compromised interparticle
contact at the cathode, or formation of Co3O4 [46]. Due to this battery?s inherent high energy
density, it has great potential for various energy storage roles in the future.
21
Chapter 3: Mathematical Modeling Fundamentals
An equivalent circuit model (ECM) may represent a cell [2],[3],[4],[7],
[10],[17],[19],[20],[29],[34],[37],[38],[42],[43],[48],[50],[51],[65],[67],[69],[70],[78],[97],[98].
For lithiumion cells, a Randle or modified Thevenin circuit may be used. These are composed
of resistances signifying cell internal resistances such as ohmic, charge transfer, and film. It also
has capacitances signifying cell internal capacitances such as double layer and film. A second
order Randle model is illustrated in Figure 6 followed by an example of a modified Thevenin
circuit in Figure 7.
Figure 6: Second order Randle circuit [8]
Figure 7: Modified Thevenin circuit [2]
These have been shown to represent a battery?s internal dynamics and impedance
spectrum well. This model is a simplification of electrochemical processes and may not work for
all cells. The validity of equivalent circuits when modeling highenergy cells remains
questionable [18],[20],[56]. The parameters must be updated periodically in order to account for
aging effects [48]. They are considered to be accurate in the short term [4].
22
An ECM may be continuous or discrete. Continuous models are based on RC circuits like
the ones pictured above. They derive OCV from VT measurements using the equation
?? OCV ?VT ??
where ? is the total cell overpotential and VT is the terminal voltage. The discrete
form, also known as state space, is a discretized version of an RC circuit. It takes the general
form of
)()()( )()()( tDutCxty tButAxtx ?? ???
Typical discrete time state space cell models take on the form
kkkk
kkkk vuxgy wuxfx ?? ??? ),( ),(1
where wk is process noise and vk is measurement noise.
Electrochemical models may also represent a cell
[2],[4],[8],[13],[15],[16],[17],[18],[19],[20],[23],[25],[26],[28],[29],[32],[33],[36],[39],[40],[41],
[46],[47],[49],[56],[58],[63],[100],[8],[10]; however, they are typically of greater complexity and
require greater computational power. These consist of mathematical equations meant to
accurately depict cell behavior. Some authors prefer an electrochemical model over an
equivalent circuit model because it is able to predict physical cell limitations, such as lithium
concentration depletion and saturation [23],[41],[57],[8]. Common approaches to the
electrochemical model of a battery include a rigorous 2D model, a pseudo2D model, and a
single particle model. Model reduction is an important area of interest for battery management
system (BMS) users. Resulting equations for solid and electrolyte concentrations (cs and ce) and
solid and electrolyte potentials (?s and ?e) are generalized below
[8],[18],[23],[32],[33],[39],[40],[41],[47],[58],[82],[8].
23
Fick?s law of diffusion: ?
????? ??????? r txrcrrrDt txrc sss ),,(),,( 22
with boundary conditions: 0
0
???
?r
stc and Fa jtcD
s
Li
Rr
ss
s
????
?
Electrolyte ion concentration distribution: ),(1),(),( 0 txj
F ttxcxDxt txc Liee ffeee ????????? ??????? ?
with boundary conditions: 0
0
??????
?? Lx
e
x
e tctc
Charge conservation derived from Ohm?s law: 0),(),( ???
????? ???? txjtxxx Lise f f ??
0),(),(ln),( ???????? ??????????? ???? txjtxcxxtxxx Liee ffDee ff ???
with boundary conditions: 0
0 ??
????
?? Lxexe xx ??
ButlerVolmer equation: ?
????? ?????????????? ???? RT FRT Ftxiatxj casLi e x pe x p),(),( 0
Overpotential: Ues ??? ???
where r is the radius of the electrode particle, Ds is the solid phase diffusion coefficient (cm2?s1),
?e is the electrolyte phase volume fraction, Deeff is the effective diffusion coefficient (cm2?s1), t+0
is the transference number defined as the fraction of the current carried by that ion in a solution
of uniform composition, F is Faraday?s constant, jLi is the lithiumion current density (A?cm3),
is the solid phase potential, ?eff is the effective solid conductivity (S?cm1), ?eff is the effective
ionic conductivity (S?cm1), ce is the lithiumion concentration in the electrolyte phase, i0 is the
exchange current density (A?cm3), ?a is the anodic transfer coefficient, ?c is the cathodic transfer
coefficient, ? is the overpotential, and as is the interfacial surface area (cm1) calculated by [47]
s?
24
s
ss Ra ?3?
where ?e is the active material volume fraction and Rs is the maximum electrode particle radius
(cm). To create a precise battery model that mimics performance in real operating conditions the
underlying principles must be understood.
Different approaches to physical modeling of a battery have unique advantage and
disadvantages. A single particle model represents each electrode with a single spherical particle
whose surface area is equivalent to that of the active area of the solid phase in the porous
electrode. Only the distance from the center of the particle to the surface, r, is considered. This
model neglects the solution phase of the cell. A pseudo2D model, also known as a porous
electrode model, represents each electrode as multiple particles. It considers concentrations and
potentials in the r and y directions. A rigorous 2D model considers solution phase concentration
and potentials in the x, y, and r directions inside an electrode which is made up of many spheres
or particles of active material.
There are different mathematical model reduction strategies such as approximating
concentrations within each particle with a polynomial rather than PDEs. A polynomial is used to
approximate the parabolic shape, thereby simplifying the calculation [56]. Three common
assumptions made in model order reduction are 1) reaction currents are not influenced by the
electrolyte concentration, 2) the ButlerVolmer equation may be linearized, and 3) ion
conductivity is constant. Each have proven to significantly reduce algorithm calculation time.
All equivalent circuit or mathematical models require great computing power to reduce
calculation time. Accuracy is dependent on how well the model represents the true physics. As
always, algorithm robustness depends on the developer. Computational efficiency suffers as
model complexity increases. Numerical stability is dependent on the type of numerical method
25
used to solve the partial differential equations. Cost can be very high for algorithm development
and hardware to run the model.
26
Chapter 4: Literature Review
Several SOC estimation methods have been investigated [170]. Each method has unique
advantages and disadvantages. The evaluation criteria are based on accuracy, algorithm
robustness, computational efficiency, numerical stability, and cost.
Accuracy is essential. Battery and EV/HEV manufacturers state that the SOC error target
is less than or equal to 5% when compared to coulomb counting, a method that will be discussed
later. It must remain accurate under varying operating conditions such as ambient temperature,
current rate, vibration, humidity, and cell age. High temperatures, large SOC variance, and
strenuous load profiles may accelerate battery aging, thereby changing its maximum charge
capacity, Qmax [28]. The process must also be robust enough to handle slight variations in cell
construction, attain numerical convergence, and be as inexpensive to implement and operate as
possible.
SOC estimation methods can be arranged into two main categories; with or without a
mathematical model. Those without are based solely on measurement. These can be arranged
into three subcategories including coulomb counting, open circuit voltage (OCV), and
resistance/impedance. Others, like quantum magnetism and neutron imaging, are rather new with
no documented publications other than two patents. Methods with a model can be arranged into
two subcategories including equivalent circuit modeling and electrochemical modeling. It is
commonplace to combine SOC estimation methods with error correction techniques, such as
with fuzzy logic, a Kalman filter, a state observer, or a neural network in order to increase
accuracy.
27
4.1 Error Correction Techniques
Fuzzy logic is a manyvalued logic that is fluid rather than fixed. There is a truth value
that exists between 0 and 1 that corresponds to a statement being 100% false to 100% true,
similar to human decision making. This allows conclusions to be drawn from ambiguous data.
Information is given a subjective description, such as fast or slow. A schematic representation is
shown in Figure 8.
Figure 8: Fuzzy logic characterization example [70]
The input and output variables are written as ?if?then? statements, also known as rules.
For example, ?if current is high and temperature is warm, then output is high.? Rules may be
designed by an expert or inferred by previous data. A qualitative data set is collected and then
mapped into a numerical range, thereby converting it to a quantitative data set that can be used to
characterize a cell [10],[17],[19],[26],[34],[37],[39],[50],[51],[56],[59],[70].
In general, the fuzzy logic method is accurate if proper numerical ranges are chosen,
algorithm robustness depends on the developer, computational efficiency is high since it relies on
ranges of data, numerical stability is not relevant since this is not a numerical approximation, and
cost is low.
Artificial neural networks act like the human brain in that they sort out patterns and
assign appropriate weighting factors; essentially, they ?learn.? There exists an input layer, a
neutron layer, and an output layer. The neutrons accept inputs, manipulate them with weighting
28
factors, and produce outputs, as shown in Figure 9.
Figure 9: Artificial neural network layers [73]
It is considered a network when many neutron layers are in series. Weighting factors are
continuously updated, bringing the final output closer to the true value. A key advantage to
neural networks is that they can represent linear and nonlinear systems. They also can be refined
with data previously gathered from a similar cell. This technique requires highly empirical
components and the error may never converge to the global minimum. Training data and the
training method strongly influence the error
[2],[7],[10],[17],[26],[34],[37],[43],[48],[50],[51],[59],[68],[71],[73],[100].
Generally, the neural network method is accurate if weighting factors are accurate.
Computational efficiency may be low if many values are sought, numerical stability if high since
the weighting factors, new or from previous data, can remove any model instability, and cost
may be high depending if there is a model incorporated and how complex it is.
A state observer estimates the internal states of a battery by relying on a mathematical
model coupled with measurements, usually of the form shown in Figure 10.
29
Figure 10: State observer feedback loop
with the following equations
)()(?)(? )()](?)([)(?)1(? kDukxCky kBkykyLkxAkx ?? ?????
where, at time k, x(k) is the plant's state, u(k) is its inputs, y(k) is its outputs, and L is the observer
gain. The hat above a variable denotes that it is an observer variable. The observer error can then
be calculated as
?? e(k ) ? ? x (k ) ? x (k ) ? 0
when ?? k?? . Choosing the appropriate gain is crucial.
A Kalman filter (KF) can take continuous, noisy, inaccurate data and provide data that is
closer to the true values. The purpose is to remove unwanted noise. It uses input values derived
from physical models and sensor measurements. First, a new state is predicted along with an
associated uncertainty. Then a new measurement is taken and given an uncertainty value. A KF
usually takes on the form of , where Fk is the state transition model, Bk
is the controlinput model, and wk is the process noise. It may take on the model form shown in
Figure 11.
kkkkkk wuBxFx ??? ?1
30
Figure 11: Kalman filter layers [72]
A weighted average scheme, where the value with the least uncertainty gets the greatest
weight, produces a new state value. The estimates tend to be closer to the true values than the
original measurements because the weighted average has a better estimated uncertainty than
either of the values that went into the weighted average [72]. An extended Kalman filter (EKF) is
a state estimator for a nonlinear system. These algorithms are often complex because there are
many state variables to consider. There must be a good statistical model of the model error
variance. In the flat region on the OCV curve, the SOC estimate is very sensitive to a poorly
chosen model error variance. Since the model contains components that are not directly
measured, selecting the error variance is difficult. If improperly chosen, the SOC variable will be
noisy and may diverge. A sigmapoint Kalman filter (SPKF) may correct EKF assumptions
including the output random variable mean and the output random variable covariance. Rather
than using Taylorseries expansions to approximate covariance matrices, a number of function
evaluations are performed. Then the results are used to compute a covariance matrix. Other
Kalman filter techniques include unscented (UKF), adaptive (AEKF), adaptive unscented
(AUKF), and central difference (CDKF). All Kalman filters typically require significant
computational power
[3],[4],[7],[10],[13],[16],[17],[19],[20],[26],[33],[34],[36],[37],[38],[43],[48],[50],[51],[52],[59],
[62],[63],[64],[68],[71],[72],[100].
31
The Kalman filter method is usually accurate if the feedback gain, physical model, and
Gaussian distribution of the external noise is accurate. Computational efficiency may be low if
many values are sought, numerical stability is high since the weighting factors can remove any
model instability, and cost may be high depending on the complexity of the model.
4.2 SOC Estimation without a Model
Included in this section are SOC estimation methods that rely on direct measurement.
There is no complex mathematical manipulation of the input or output data. Most methods
presented here require a lookup table to be generated before implementation for reference
purposes.
Coulomb counting calculates SOC by integrating the measured current with time,
resulting in units of ampereseconds or coulombs, which are then converted to Ah. Current
sensing techniques include the use a shunt resistor, Hall effect transducer, or giant
magnetoresistance (GMR) sensor. Industry favors Hall effect transducers for their reliable
operation and low cost, even though they are susceptible to noise. Ah are inserted into a cell
during charging and extracted during discharging. The calculation is as follows
%100)0()(
m a x
0 ?
??
?? ? Q
dti
S O CtS O C
t ?
where ? is the efficiency coefficient defined as
ech
edischQQ
arg
arg??
where Qdischarge and Qcharge are the measured charge output during a full discharge and input
during a full charge, respectively.
32
An inherent problem is that ? is different for charging and discharging. A lithiumion
polymer battery will deliver less charge than the amount input during charging. This is due to
charge losses accrued from lithiumions plating out on the anode during charging
[23],[30],[45],[50]. As a result, the amount of electrochemically active Li+ ions will be reduced
[46]. Typical values for ? are 1.0 during charge and 0.981.0 during discharge
[13],[42],[50],[63],[71].
Coulomb counting has several drawbacks. It is not possible to determine the initial SOC
by measurement. The initial SOC may be estimated by recalling the previous coulomb counting
history, which may already have an offset error [2],[26],[61],[81]. Also, since this method is
based on current measurement any sensor offset errors will accumulate and further compromise
accuracy
[2],[3],[7],[10],[13],[15],[16],[26],[30],[35],[37],[42],[43],[48],[49],[50],[51],[55],[59],[61],[68],
[71],[74],[75],[76],[78],[79],[81],[84],[92],[98],[100],[101],[104],[105],[13],[10]. The initial
SOCs are defined to be zero or one hundred percent when fully discharged or charged,
respectively. Internal side reactions and selfdischarging cause errors that accumulate with time
[20],[56],[59],[61],[74]. Recalibration is required to remove accumulated error
[2],[10],[16],[30],[34],[35],[42],[48],[51],[68],[71],[74],[75]. To obtain an exact updated charge
capacity a cell would have to undergo a full charge/discharge. This requires a long time, subjects
the battery to possible over/undercharge, and shortens a cell?s life each time
[13],[16],[17],[23],[30],[35],[61],[63],[68],[84],[100]. Another factor to consider is that current
integration in a real world application will not be continuous. This means that the sampling rate
much be fast enough to capture any sudden load change [26],[41],[61],[98].
33
The relation between charge capacity and charge extracted is nonlinear due to the
multitude of factors that may affect it. These include, but are not limited to, the cell voltage and
temperature, the rate at which cells have been charged and discharged, the chemical composition
of the various active chemicals and any doping which has been used, the possibility and effects
of contamination, the shape and length of the physical current paths within the cell, the volume
of electrolyte, the thickness of the electrolyte and the separator, the resistivity of the components,
the rate of mass transfer of the ions through the electrolyte, the rate of chemical action at the
surface of the electrodes or the rate of absorption of the ions into the intercalation layers, the
actual surface area of the electrodes, the effective surface area of the electrodes taking into
account the particle sizes of the chemicals, the effect of passivation on the electrode surface, the
ambient temperature, the Joule heating effect, the self discharge rate of the cells, and the time
between charges
[2],[4],[13],[23],[27],[30],[31],[34],[45],[48],[50],[55],[59],[61],[65],[68],[71],[74],[84],[98],[10
6],[107].
Qmax depends on temperature and age of the battery
[2],[27],[29],[35],[45],[55],[59],[61],[71],[74]. A cell loses active material and consequently full
charge capacity as it experiences cycling or harsh environments [23]. Examples of degradation
modes include loss of active material sites in either electrode, growth of a solid electrolyte
interface (SEI) layer on the anode, and material structural changes causing increased diffusion
impedance at the cathode [23]. Ohmic and transport limitations reduce available capacity at high
rates [47]. The fact that temperature and battery age affect SOC is common for all methods in
this study.
34
Coulomb counting is accurate as long as the initial SOC is known and the current sensor
error is small. It is a direct measurement method so there is no need to assess algorithm
robustness, computational efficiency, or numerical stability. Cost of implementation is relatively
low, but may increase with the number and accuracy of sensors.
OCV is a lookup table approach. It requires an OCV curve be generated initially by
charging/discharging a cell at low current for finite times with rest periods between loadings
until it is fully charged/discharged [42]. The current is integrated during loading using the
coulomb counting process and a corresponding SOC is linked to a measured rest period cell
voltage that is considered to be the OCV [2],[13]. These values are stored in a lookup table and
referenced during battery operation.
OCV is inaccurate when there is external current flowing through a cell
[16],[20],[34],[38],[43],[46],[50],[54],[55],[59],[61],[63],[67],[68],[71],[74],[79],[83],[85],[88],[
90],[93],[94],[96],[100],[101],[103],[13],[10]. The measured terminal voltage (VT) differs from
the OCV due to overpotentials inside the cell and any external voltage losses
[16],[20],[34],[55],[61].
Overpotentials include activation, concentration, and ohmic. Lithiumion polymer cells
exhibit a nearly flat OCVSOC relationship slope from approximately 2080% SOC so a highly
accurate voltage sensor would be needed [20],[26],[34],[50],[66],[71],[79],[80],[90],[99],[13].
Another factor is that identical batteries may have slight variations in their OCV curves [38]. A
battery?s OCV curve may change nonlinearly with the ambient temperature and degradation
from age [2],[7],[13],[26],[34],[45],[59],[65],[71],[77]. It has been demonstrated that the OCV
SOC relationship does not change during cycling of the battery, if SOC is expressed in relative
capacity [45],[54],[59]. Multiple lookup tables would have to be generated in order to cover the
35
range of expected temperatures. Figure 12 shows a typical lithiumion polymer OCVSOC curve,
with the solid line representing an aged battery.
Figure 12: Typical OCVSOC curve [46]
In general, the OCV method is accurate only when the battery has been at rest for an
extended period of time. OCV without a model is a direct measurement method so there is no
need to assess algorithm robustness, computational efficiency, or numerical stability. Cost is
very cheap since a voltage sensor and stored lookup tables are the only components.
The resistance/impedance is another lookup table approach. Active material in the cell
and corresponding resistance/impedance changes as the anode and cathode alter charge state.
Resistance (R) is voltage divided by a DC current, written as
IVR? .
Impedance (Z) is voltage
divided by an AC current, written as
IVZ? ,
where
)( VtjeVV ????
and
)( ItjeII ????
. This is
similar to resistance only it includes the phase angle. Impedance measurement entails imposing a
frequency sweep on the battery, typically 10kHz1mHz, offering the opportunity to reach
conclusions about internal processes with unique time constraints [1],[7],[29],[34],[61],[69]. This
is accomplished with the use of an electrochemical impedance spectroscopy (EIS) machine.
36
There are two EIS operation modes: galvanostatic and potentiostatic. The former refers to
the manner of applying an AC current while measuring the cell voltage response and the latter
refers to the manner of applying an AC voltage while measuring the current response. Voltage
fluctuation must be kept less than or equal to 10mV in order to ensure impedance measurements
are conducted under linear conditions [59]. When results are presented in the form of a Nyquist
plot with the negative imaginary part on the yaxis and the positive real part on the xaxis,
semicircles followed by a linear slope may be seen, as depicted in Figure 13.
Figure 13: EIS shape analysis [7]
Xaxis intersection points, maximum semicircle heights, and linear section slopes each
represent a specific internal component. It has been discovered that the resistive components
represent the current collectors, electrolyte, bulk of the electrode materials, and transfer reactions
at the electrode surfaces. Capacitive components represent the doublelayer capacitance.
Inductive components represent the porous nature of the electrodes [69]. When these values are
fit to a battery equivalent circuit model, particular parameters may be extracted. R and Z are
measured using an ohmmeter and impedance spectroscope, respectively, and correlated to a SOC
in a lookup table.
37
Similar to OCV, R is nonlinear with respect to charge/discharge rate, ambient
temperature, and age [1],[27],[34],[35],[42],[59],[61],[68],[69],[71],[100]. Z measurements have
the same disadvantages; only certain aspects of the frequency spectrum are not proportional to
the available capacity of the cell [35]. An additional drawback is that the R/Z method requires a
constant DC/AC current, respectively, making measurements inaccurate under real battery
demand conditions [16],[51],[55],[63],[68],[71],[105]. Both of these methods are indicative only,
meaning that a lookup table would have to be generated for various operating conditions. Also,
identical batteries may have slight variations in R/Z [28].
Classically, the R/Z method is only accurate when the current and ambient temperature
are constant. It is a direct method so there is no need to assess algorithm robustness,
computational efficiency, or numerical stability. Cost of measuring internal resistance is very
low, but cost of an impedance spectrometer can be high.
4.2.1 Literature Rundown
An enhanced coulomb counting method has been conceived in which the initial SOC is
read from memory while making use of the efficiency factor ?, Ah are counted during loading,
and recalibration occurs every time the battery reaches 0 or 100% SOC as stated by the
manufacturer, where 100
arg
arg ??
ech
ed ischQQ? . Also, there is a selfdischarge correction for batteries
being stored for extended durations. All experiments have been at room temperature. Linear
equations to determine SOC are derived during charging at 1C by plotting the voltage versus
SOC in constant current (CC) mode and current versus SOC in constant voltage (CV) mode.
Multiple lookup tables would have to be generated at various currents, ambient temperatures,
and cycleages. Experiments showed that batteries have a higher charging efficiency factor at
38
lower SOC. The opposite holds true for the discharge efficiency factor. By periodically updating
the efficiency factors, longterm error can be minimized [30].
A twopulse method has been suggested by which a battery at rest is subjected to a 10
second pulse of constant load current of known magnitude. The voltage drop is recorded during
the pulse. Then the battery rests for ten seconds. At the end of the rest period, the terminal
voltage is measured (Vmax). A second load current pulse, identical to the first, is applied for
another ten seconds while the voltage drop is again measured (?V2). Vmax is assumed to be equal
to OCV. The SOC is deduced from Vmax by recalling an OCV curve for the battery at a specified
temperature. There was proven to be a linear relationship between the Crate used and ?V2. C
rate is deduced from previously generated tables. Using the equation
maxQ
ICrate ? , a new value
for Qmax can be calculated. It has been confirmed that the second pulse gives consistent results as
a battery ages. This study was conducted on valveregulated leadacid batteries at room
temperature, so future work should expand this concept to other types and batteries and ambient
temperatures [35]. The same authors previously mentioned wrote an earlier paper describing an
estimation SOC method using a combination of terminal voltage, current, and impedance
measurements. A complex equivalent circuit model, similar to a 6th order Randle model, was
implemented. 1kHz was the frequency used for impedance measurement. The impedance was
normalized and used in conjunction with the change in voltage to estimate SOC in the hyperbolic
OCVSOC regions near 0 and 100% SOC. Two equations are given;
01 )()( nomRT ZCVO C V ?? ?? for the linear OCVSOC region and
3102 )()( ??? nomnomRT ZZCVO C V ???? for the hyperbolic regions, where
maxQ
ICrate ? and ?1,
?2, and ?3 are found through experimentation. The normalized impedance at fully charged or
39
depleted SOC is periodically updated to help account for aging. Future work would be to update
Qmax and analyze ambient temperature effects [42].
A paper has described how to model battery behavior. The OCVSOC curve is generated
by applying a small current from 100% SOC to 0% SOC. The small current is intended to
remove overpotentials; however, they still exist. A better approach would be to discharge with a
small current for a short amount of time, rest for at least 2 hours, then record the terminal voltage
at the end of rest and consider that to be the OCV since all overpotentials will have been
completely removed. The OCVSOC shifts upward with decreasing ambient temperature,
although negative temperatures have not been studied. Overpotentials increase with current rate
and cell age. The overpotential model includes ohmic, kinetic, and diffusion overpotential, and
considers temperature. The OCV technique is only applicable when the battery is at rest, so it
was recommended to use coulomb counting with overpotential prediction [54].
It has been stated that there are three reasons why a battery?s measured voltage does not
equal its OCV: ohmic voltage loss, voltage loss caused by charge transfer, and voltage loss
caused by diffusion. Ohmic voltage loss does not have a time delay and is constant. Voltage loss
caused by charge transfer has a time delay and is then constant. Voltage loss caused by diffusion
has a time delay and then increases with time. Rest time after loading before taking an EIS
reading is considered arbitrary. There are five sections of the Nyquist plot: Section 1) at very
high frequencies the spectrum shows inductive behavior caused by inductive reactances of
metallic elements in the cell and wires, 2) ohmic resistance (R?) of the cell at the intersection
with the real axis, sum of the resistances of current collectors, active material, electrolyte, and
separator, 3) first semicircle associated typically with the SEI and is formed during cycling on
the surface of the anode, 4) second semicircle representing the double layer capacity and charge
40
transfer resistance at the electrodes, and 5) linear slope represents the diffusion processes in the
active material of the electrodes at very low frequencies. SEI resistance and double layer
resistance increase at lower temperatures. R? increases with temperature, yet does not change
with SOC making it alone a poor indication of SOC for lithiumion cells [1].
It has been stated that the first xaxis intersection point in a Nyquist plot corresponds to
the R?, the high frequency semicircle shows components of passivation, the large semicircle
shows the charge transfer reaction, and the low frequency linear region shows diffusion. The
radius of the large semicircle decreases from 20% to 80% SOC and then increases from 80% to
100% SOC. A polynomial of degree three was then curve fit to the time constant of two
components of the cell at various SOCs. The time constants of those specific components,
namely the left and right intercepts of the large semicircle, can be continuously monitored and
compared to values in a lookup table [7].
Phase has been deemed an accurate SOC estimation indicator. A constant 40Hz AC
current was applied to two different cointype cells while they were being cycled. Both cells
have exhibited phase increases while charging and phase decreases while discharging. A simple
model of two resistors and a constant phase element has been proposed to describe the behavior.
When the phase variance was plotted versus SOC, a nearlinear trend appears [27].
AC impedance alone has been used to determine SOC of a battery. Multiple equivalent
circuit models have been fitted to battery impedance data with multiple conclusions drawn as to
which component of the Nyquist plot corresponded to which component of the model. Many
different battery types have been examined. The resulting Nyquist plot for a lithium cell shows a
high frequency inductive loop and two lower frequency capacitive semicircles. The inductive
loop is attributed to the electrode porosity and the capacitive semicircles are attributed to the
41
electrode process. The equivalent series capacitance at 25mHz varies linearly with SOC, so it
was the most promising parameter to monitor [69].
Neutron radiography has been used to physically measure the lithium concentration
across battery electrodes at low current rates. A neutron beam is attenuated as it passed through a
cell, producing a 2D projection. The number of neutrons counted is given by a Poisson
distribution and stroboscopic averaging is used to account for limitations of the imaging
equipment. The change in beam intensity is proportional to the Li concentration in the solid
electrodes. Observed blurring in the neutron images are attributed to inplane roughness or
curvature of the battery. The effects of high current behavior remain unknown. While this may
provide accurate results in a lab setting, it is impractical for use in a HEV/EV [22].
A battery state flowchart has been produced. Five states are mentioned including initial,
standby, charge, transitional, and discharge; each defined by the measured current and voltage of
the battery. When there is no external current flow, OCV is used to determine SOC. When there
is external current flow, coulomb counting is used. Aging is considered by updating Qmax using
chsisf QS O CS O CQ ?? 100m a x
, where SOCsi and SOCsf denote the initial and final SOC in the
standby state, respectively, and Qch denotes the amount of charge flowing into the battery during
the charge state. A disadvantage to this approach is that is requires two separate rest periods to
calculate SOCsi and SOCsf. An SOC error is 1.2% was observed when compared to coulomb
counting alone [31]. The same authors wrote an earlier paper describing the concept of the
multiplestate model. Overpotentials are modeled as ohmic, kinetic, and diffusion resistances.
The effects of aging on the total overpotential may be determined by f
ch
achf
dad ???? ? where a
represents aged, f represents fresh, d represents discharge, and ch represents charge [45].
42
A battery mathematical model has been presented that predicts OCV using experimental
aging data. OCV is shown to decrease at constant SOC, noticeably after hundreds of cycles.
OCV and coulomb counting methods estimate SOC separately. Those estimates are given weight
factors determined empirically at different current rates and ambient temperatures. The final
SOC is a combination of weighted OCV and coulomb counting estimates. High current rates
were not considered [55].
A method has been presented that uses OCV to determine SOC when a battery has been
at rest for a while. This eliminated effects of selfdischarge. Coulomb counting was then used
during load operation. Impedance measurements were found to suffer from transient effects and
cell aging, so it was continuously updated in order to predict cell terminal voltage [61].
Yet another lookup table approach is quantum magnetism. A cell is polarized by external
means and the resultant magnetic field attractive force is measured, following the equation
)( BVqF ?? . Intercalation/deintercalation alters an electrode?s structure thereby modifying its
magnetic field susceptibility [80],[90]. The resultant change in attractive force is sensed and
correlated to a SOC in a lookup table. The magnetic sensor accuracy must be great since the
change in attractive force is on the order of mN. This method is fairly new and has not yet had
any published journal articles describing its merit. The effects of polarizing a cell remain
unknown. This method also has the drawback that it might not be possible to put windings
around each individual battery inside a battery pack without significant interference issues.
4.3 SOC Estimation with a Model
The methods discussed in this section all use mathematical strategies to improve SOC
estimates. They may take the form of an equivalent circuit model or electrochemical model.
43
4.3.1 Literature Rundown
A hybrid method has been developed wherein OCV is used to estimate SOC when a
battery has been at rest for an appropriate length of time and coulomb counting is used when
there is ion flow. An equivalent circuit model with a reducedorder state space observer
accurately predicted SOC, while considering aging effects by magnifying the battery?s nominal
impedance value by 2. The observer was of the form
xCy yxCGBuxAx ?? )?(?? ? ????
? [2].
A modified Thevenin equivalent circuit has modeled a battery. The components include
ohmic resistance, electrochemical polarization resistance, concentration polarization resistance,
electrochemical polarization capacitance, concentration polarization capacitance, and OCV.
OCV is defined as the average of OCVs recorded during charging and discharging in order to
eliminate the effect of hysteresis. The ohmic resistance is found to be independent of the load
current, but rather proportional to SOC. An EKF algorithm is implemented and circuit
parameters identified. Coulombic efficiency was found to decrease as current rate increases,
although it was always greater for discharging. The adaptive EKF monitors changes in the
dynamics of the target, treating them as noise or modifying the model parameters. The AEKF
has superior accuracy when compared to an EKF [3].
A screening process combined with an extended Kalman filter has been suggested. The
screening process first decides which group of cells has similar characteristics within a battery
pack and selected an appropriate equivalent circuit model to represent them. Then the extended
Kalman filter uses the selected model along with measurements to estimate SOC. Error of ? 5%
was documented when this method was compared to coulomb counting [4].
44
A reduced order model has been developed based on porous electrode theory. This means
that the electrodes are represented as multiple spheres spanning the thickness. The Liions
diffuse into and out of these spheres. Multiple microcells are linked to mimic a single pouch type
cell. Three main assumptions are made including the reaction current not being influenced by the
electrolyte concentration, the ButlerVolmer equation may be linearized, and the ion
conductivity is held constant. The nonlinear ion concentration PDEs are replaced with ODEs
using a polynomial approach and the definition of SOC is altered to include electrode Li
concentrations even at 0 or 100% SOC. Calculation time is drastically reduced when compared
to the full order model [8]. The same authors compared terminal voltage and SOC results of the
reduced order model with the full order model, equivalent circuit model, and experimental data.
Two current conditions are tested, constant and pulse. The proposed ROM achieves an SOC
estimation error of less than 1% during a full discharge. It also demonstrates a significant drop in
calculation time; from 79 seconds with the full order model to 8.7 seconds at 1C. Ongoing
research will focus on the ROM response to high current rates, up to 10C, as well as heat
generation at various ambient temperatures. Implementation of a feedback control loop may
further increase accuracy [8].
ECM parameters have been derived using an algebraic method. It has been proven that a
1st order Randle model poorly depicts battery dynamics, a 2nd order model improves that
response, and a 3rd order model almost perfectly matches experimental data. A 2nd order model
provides the greatest efficiency, calculated as a ratio of the relative accuracy to calculation time
[4].
The Nernst equation has been used to relate OCV with SOC in conjunction with a first
order Randle ECM. A recursive least squares method extracts parameters by weighting recent
45
data heavier than old data. SOC estimates are used to update the model and then the model is
used to update SOC. The model performs well under hybrid pulse power characterization (HPPC)
and dynamics stress test (DST) simulations when compared to experimental data, with a
maximum SOC error of 2.1%. The ECM parameters will need to be updated as the battery ages
[4].
A paper has focused on the model order reduction process. Six techniques for converting
diffusion PDEs to ODEs are investigated, including the Analytical Method (AM), Integral
Method Approximation (IMA), Pad? Approximation Method (PAM), Finite Element Method
(FEM), Finite Difference Method (FDM), and Ritz Method (RM). The goal is to determine
which method most quickly provides diffusion dynamics results within an error threshold of a
porous electrode electrochemical model. The PAM proves to offer the best convergence [10].
An adaptive algorithm has been developed that identifies model parameters for a second
order Randle ECM. By computing an observer gain and estimation error, new battery parameters
can be extracted. SOC is calculated as a weighted average of estimates from coulomb counting
and the ECM. The SOC estimation error is less than 3%. The inherent coulomb counting
problem of accumulation of error remains an issue [13].
A state space model has been presented in which a nonlinear Luenberger observer with a
KF is applied to an ECM. Simulations results of the ECM match an existing nonmodel SOC
algorithm. The proposed ECM was not validated experimentally, and ambient temperature and
battery age were not considered [10].
An adaptive unscented Kalman filter approach has been presented based on a zerostate
hysteresis battery model. The AUKF adaptively adjusts the covariance values of the process and
measurement noise based on the output voltage residual sequence. It was shown to have superior
46
performance when compared to EKF, AEKF, and UKF. This method has a relatively low
computational power requirement. Future work should be on replacing the zerostate hysteresis
model with a model that more accurately predicts battery dynamics, especially during relaxation
[10].
A radial basis function (RBF) neural network was used in conjunction with an extended
Kalman filter to estimate SOC. The neural network is trained by using coulomb counting data
from a previously cycled battery. Inputs to the neural network are the terminal voltage at the
previous time step, the SOC at the present time step, and the current at the present time step. The
RMS error is calculated to be 3%. The initial training of the neural network will become obsolete
as the battery ages. It is suggested that lookup tables be created for range of ambient
temperatures since this work focused only on room temperature [13].
A single particle model has been used to model a cell. Accuracy is lost at current rates
above 1C because the model assumes spatial averages in concentration over the electrodes even
though in actuality the concentration is nonuniform [15].
A discrete state space model based on an equivalent circuit has been used. The estimator
is designed in such a way as to eliminate the need for a recursive gain calculation. A linear
parameter varying (LPV) technique is utilized for its low computational power requirement,
being at most half that needed for a Kalman filter. A RMS error of 4% is observed, yet only
room temperature is considered [16].
An adaptive Luenberger observer based on an equivalent RC circuit has been used to
estimate SOC of a battery pack. The gain is continuously adjusted to minimize error between the
model estimated voltage and the measured terminal voltage. The Hybrid Pulse Power
Characterization (HPPC) test is performed on a cell while a numerically nonlinear least squares
47
algorithm determines the unknown model parameters. The circuit model is not capable of
accounting for heat generation in a cell [17].
A reduced order partial differential algebraic equations (PDAE) mathematical model has
been created with the assumptions that the lithium concentration in the electrolyte is uniform.
Also, the mass transfer in the solid material is approximated in the radial direction in terms of
volume averaged concentration, particle surface concentration, and averaged concentration flux.
There are two observer gains since the positive electrode is made of two materials. The model
has difficulty tracking true values when high current rates with subsequent high heat generation
rates are experienced. Validation of the estimated overpotentials requires reliable measurements
of the potential distribution across the electrodes, which is not easy to obtain [18].
A 1D electrochemical model has been developed with the assumption that the reactions
occur on the surface of the electrodes. Another is that the OCV depends on the lithium
concentration in the electrolyte adjacent to the electrode. An EKF is implemented using Euler?s
method to discretize the model. The method proves an increase in accuracy over traditional
EKFs and slidingmode observers, yet it does not account for lithiumion concentration
distribution within the electrodes [19].
A porous electrode based model including high current rate limitations has been created
together with an unscented filter. These limitations include diffusion restraints in the electrolyte
and solid phase transport. The unscented filter uses sigmapoints chosen based on the variance of
the measured variable to map the nonlinearities of the measured variables to that in the state
variables. The sigmapoints are the eigenvalues of the Cholesky decomposition of the covariance
matrix. It does not require linearization of the system equations, thereby reducing computational
power requirements. Each electrode is modeled as spherical particles arranged along the
48
thickness. To make the model quicker, diffusion and kinetic parameters are held constant and
thermal effects are assumed to be negligible. Capacity fade was not considered, but may be
deduced from model parameters upon further investigation [20].
Modeling constraint management during pulse operation has been investigated. An
electrochemical model with a linear filter is used; however, the linear filter performs poorly
when the SOC was outside the range of 3070%. It is stated that the ratio of Li concentration in
the anode to the maximum allowable in the anode must always be between zero and one. The
model order reduction procedure groups together modes that have similar eigenvalues, and are
still able to reproduce varying frequency response of diffusion. A Kalman filter is added to
reduce process and measurement noise. Electrode surface stoichiometries are found to change
much faster than SOC since they are more closely coupled to the recent charge/discharge history.
Model convergence time could be decreased with less noisy current sensors. Controls should be
implemented that allow pulse charging of a battery beyond its manufacturerspecified maximum
voltage without significant lithium plating side reactions on the anode surface. It has been
observed that the positive electrode shows a much stronger dependency on surface stoichiometry
than does the negative electrode. It is suggested that future work focus on model parameter
adaptation as a battery ages [23]. The same authors? previous work was designing a linear
Kalman filter based on a reduced order electrochemical model. There are six states in the state
space model including five that describe positive and negative electrode solidstate concentration
gradient dynamics and one that describes electrolyte concentration gradient dynamics. This is
much easier to solve than the author?s previous 313th order nonlinear finitevolume model.
Reduction of the model from 5 to 3 states does not have much effect of SOC estimation error.
The linear KF performs well only when SOC remains in the range 3070%. Future work should
49
focus on identifying battery degradation [40]. An earlier paper by the same authors presents a 1D
electrochemical model under the assumptions of quasilinear behavior and decouples the local
reaction current from the electrolyte concentration. A detailed description is given for the
electrochemical model and reduced order support vector machine (SVM) state space transfer
functions. The bandwidth range is altered to be 010Hz. The higher the current rate, the greater
the order of the model needs to be to capture electrode surface saturation/depletion nonlinearities.
The electrode surface lithium concentrations are used to estimate SOC. A 313th order nonlinear
CFD model is reduced to a 12th order state variable model and accuracy remains within 1%. The
model still requires high computational power [47].
A thesis by K. A. Smith has incorporated a solidstate diffusion model into a 1D
electrochemical model and then reduced its order. This limits the high current rate performance
by protecting the electrodes from lithium saturation and depletion. Model parameters are
gathered experimentally since he was not allowed to disassemble a cell. Surface concentrations
are found to change much quicker than bulk concentrations. A potential difference limit, rather
than a manufacturer terminal voltage limit, is introduced which allowed more efficient, faster
charging of a battery. More heat is found to be generated at lower ambient temperatures.
Electrode surface concentration is taken as an average and electrolyte concentration is taken on a
distributed basis. A KF is implemented which demonstrated improved computational efficiency.
Future work would include studying the effects of ambient temperature on the model, upgrading
the KF to an EKF, further order reduction if possible, and model validation through the lifetime
of a battery [58].
EIS has been combined with fuzzy logic methodology. A Sugeno fuzzy model and
centerofgravity defuzzification method are used. It has been demonstrated that the real value of
50
resistance grows as SOC decreases. It is also shown that the resistance decreases as the DC offset
current increases [25].
A mixed algorithm combining coulomb counting with a second order Randle equivalent
circuit mathematical model has been presented. A low accuracy Hall effect current sensor and
voltage sensor are used to better simulate realistic onboard EV/HEV systems. The model is able
to compute OCV from current and voltage measurements. The coulomb counting component of
the mixed algorithm captures quick changes in SOC while the model component corrects the
longterm coulomb counting accumulated errors. The two components are coalesced into a single
closed loop scheme, with gains chosen to minimize steady state and dynamic error. The
algorithm has proven to be more stable than coulomb counting alone [26]. The same authors go
into more detail of their mixed algorithm in another paper. SOC estimations based on coulomb
counting using a highly accurate current sensor proves to diverge considerably from those using
a standard current sensor. A closeloop controller is used. The equivalent circuit cell model
represents the plant of the control system, SOC based on coulomb counting is the input variable,
and voltage is the output. The reference output is the measured cell voltage. The measured
current is a disturbance. An integral gain is incorporated. Future work is to test the algorithm at
temperatures other than room temperature [34].
A discretized equivalent circuit model with a dual extended Kalman filter has been
developed in order to estimate the battery?s ohmic resistance, which is monitored experimentally
for hundreds of cycles and then correlated to SOH. Ohmic resistance variation increases with the
number of cycles [28].
A remapped Randle equivalent circuit has been presented. The model shows advantages
for the estimation of the components associated with the electrochemical transient effects within
51
the battery. This is due to having two states in parallel rather than in series as in the Randle
model. Error is reduced when an Utkin observer and Kalman filter are applied. The main charge
store capacitor (Cb) in the model is shown to vary proportionally with Qmax. By monitoring Cb, it
is proven possible to update SOC so that its range always remains 0100% [29].
Two reduced order particlebased models have been suggested. The electrode averaged
model (EAM) assumes that the electrolyte concentration is held constant and neglects both the
solid concentration distribution along the electrode and the spatial dependence of the Butler
Volmer current. A finite difference method converts the PDEs to ODEs. The state values model
(SVM) divides the governing equations into submodels in order to derive analytical transfer
functions. The EAM presents accurate SOC estimation but is highly sensitive to nonlinear model
parameters. The SVM is stable but provides poor SOC estimation [32]. The same authors also
describe an electrochemical model of a cell has been simplified by holding the electrolyte
concentration constant and neglecting the solid concentration distribution along the electrode.
Also, the spatial dependence of the ButlerVolmer current is ignored and is instead an average
across the electrode, thereby representing the solid material as a single particle. An extended
Kalman filter is applied with good results [33]. The same authors have a third paper describing
the same model mentioned previously [36]. A fourth paper by the same authors describes the
original electrochemical model, in which the parameters are obtained empirically. SOC is
defined by the positive electrode stoichiometry as follows
%0%100
%0)( ?? ?? ???tS O C [41].
An equivalent circuit model has been coupled with a slidingmode observer. Nonlinear
unknown disturbance is added to the model in order to increase accuracy. Models parameters are
extracted from multiple tests using an equivalent circuit model. The results prove that the
observer provides true SOC estimations even when model parameters have errors. Total SOC
52
error is within 3% when compared to coulomb counting alone. Ambient temperatures other than
25?C and high currents have yet to be evaluated [37].
A new OCVSOC relationship has been presented. Since similar batteries may have
slight differences in their OCV curves, the curves need to be normalized in order to use the same
OCVSOC curve for each battery. Normalization is achieved by arbitrarily choosing a cutoff
OCV and associated SOC. A first order Randle equivalent circuit is used to model the battery. A
dual EKF is utilized, one being a state filter for SOC and the other a weight filter for capacity. A
modified SOC is derived from the normalized OCVSOC curve. The conventionally defined
SOC becomes
c u t o f f
i f i e dalc o n v e n t i o n S O CS O CS O C ?? 1 m o d [38].
A porous electrode model has been provided which includes high rate limitations with an
unscented filter. The unscented filter is ideal due to the nonlinearity of the physical equations at
high current rates. A single particle model assumes that the concentration and potential changes
in the solution phase are negligible. A porous electrode model includes the spatial distribution of
concentration and potential along an electrode as well as voltage drop across the electrolyte. It is
stated that an EKF does not accurately capture nonlinearities in the battery response during high
currents. An unscented filter using sigma points is used instead. A single particle model is shown
to only be accurate when there is little process noise. SOC estimation error with the unscented
filter is < 3% while still providing physical insight of the battery [39].
A reduced order equivalent circuit model with an extended Kalman filter has been
presented. The proposed model is a modified version of the first order Randle model, in which
the series resistance is replaced with a parallel path of two identical resistances with diodes. It is
claimed to better represent hysteresis by allowing the current direction to determine the internal
resistance value. Reduction of the model comes at the cost of omitting the fast dynamics.
53
Measurement noise and data rejection are included in the EKF model, bringing its SOC
estimation error to within 2%. Ambient temperatures other than 25?C and high currents have yet
to be evaluated [43].
An EMF model has been described. Lithiumion cell hysteresis, although small,
maintains a constant form during aging. This is found to be true at all temperatures. EMF or
OCV, however, changes with aging and can lead to SOC estimation errors of more than 10% if
only a fresh cell SOCOCV curve is used for measurement. The battery OCV drop is directly
linked to the positive electrode decomposition. This adaptive EMF model is used in future
electrochemical models to reduce SOC estimation error [46].
A combination RC and hysteresis model has been offered. Hysteresis is minimal in
lithiumion batteries so it may be neglected. The equivalent circuit model has three resistors and
two capacitors, the values of which are determined empirically. SOC is determined via an EKF
from terminal voltage and current measurements. Error is found to be within 2% when compared
to coulomb counting. This will increase with time unless the equivalent circuit parameters are
periodically updated, which would require a significant amount of time for the battery to be out
of regular operation. Low temperatures and high current rates cause error to grow beyond 2%
[48].
An EKF with a parameter estimator has been developed. It is designed for NiMH
batteries but would also be applicable to lithiumion. Diffusion is found to have the most
profound effect on SOC. Great computational power is required to run the EKF and only room
temperature was considered in the study [49].
A modelbased Kalman filter SOC estimation technique has been proposed. The model is
an equivalent circuit model with 3 resistors and 2 capacitors, which is said to identify cell
54
parameters easier than previously proposed models. The KF is expanded to an EKF to account
for nonlinearities in the model, with a resulting error of less than 3%. The EKF helps correct
errors caused by model order reduction but it requires significant computational power [50].
A slidingmode observer SOC estimation method has been presented for a lithiumion
polymer battery. It is chosen for its simple control structure and robust tracking performance. A
first order Randle model is used to capture the battery dynamics. Model parameters are extracted
experimentally; however, it is noted that resistances will change with SOC. It has been proven
that they will also change with current rate and ambient temperature, making the equivalent
circuit model inaccurate. The Lyapunov inequality equation prove that the observer will
converge and error is found to be < 3% when compared to coulomb counting [51].
Advanced algorithms using sequential probabilistic inference or optimal estimation
theory have been studied. Sigmapoint Kalman filters (SPKF) have outperformed EKFs in SOC
estimation. State vectors such as SOC or current changed quickly while parameter vectors such
as resistance increases change over a long period of time. A discretetime state space model is
used, of the form
kkk
kktk vS O CO C Vy wS O CeS O C e ff ?? ?? ??? )(1 ?
where model parameters are estimated in real time to account for any manufacturing differences
between cells and to track the effects of aging. SOC estimation error is found to be less with the
SPKF than with other types of filters [52]. The same author described a squareroot SPKF in
another paper and proved that it is more numerically efficient. SOC estimation errors are reduced
to less than 1% [53]. An earlier paper by the same author describes several state space cell
models that incorporate Kalman filters. These models are able to demonstrate the effects of cell
relaxation, internal resistance, and hysteresis [66].
55
A review of lithiumion battery models has been conducted. It is stated that the porous
electrode approach models a battery well but is time consuming to calculate while the single
particle approach requires less time but neglects solution phase diffusion limitations. Using
polynomial approximations with the porous electrode approach reduces calculation time while
maintaining model accuracy [56]. The same authors present an EKF model that uses terminal
voltage measurements to estimate SOC. The change in Qmax with aging is not considered. SOC is
based on the ratio of the concentration of Liions in the negative electrode to the maximum in the
negative electrode [57].
A state space model with an EKF has been described in 3 papers. The first provides the
theory of linear KFs and EKFs. The second gives the theory of battery modeling. SOC is a state
in the model, which means its value is given as well as its uncertainty. Knowing the uncertainty
gives a user a precise range of remaining charge. Many different models are investigated and
evaluated based on RMS error under various temperatures. Adding hysteresis to the models
increased accuracy at the cost of increased computational power required. The third paper
outlines the primary applications of EKF to the estimation requirements for battery management.
The discrete time state space model takes on the form
kkk
kktk vS O CO C Vy wS O CeS O C e ff ?? ?? ??? )(1 ?
where toff is the time that the battery is at rest. The model is tested using a pulse test and by
applying the Urban Dynamometer Driving Schedule (UDDS). The EKF always demonstrates
convergence. A dual EKF is introduced whereby states and parameters are estimated separately
with the outcome of updating parameters as the cell ages. SOC estimation error is found to be
less than or equal to 5% in the temperature range 050?C. Future work should include the
evaluation of the dual EKF state space model at negative ambient temperatures [62],[63],[64].
56
An adaptive digital filter for lithiumion battery SOC estimation has been presented. A
first order Randle equivalent circuit LPV model is used. It is incorrectly assumed that age and
temperature do not affect the OCVSOC curve. A least squares algorithm recursively identifies
model parameters [65].
A Thevenin equivalent circuit has been modified to estimate SOC. The input is terminal
voltage, which is then transformed into state space with other parameters to account for
nonlinearities. It demonstrates the ability to predict SOC but would become more inaccurate
under high currents, changes in temperature, or cell degradation. The model presented is
intended for leadacid batteries and may not be appropriate for lithiumion batteries. Also, real
time results are not attainable [67].
Impedance measurements have been combined with fuzzy logic inference. An equivalent
circuit is fitted to the EIS data in order to extract parameters. Error is high at 10%, meaning that
the model chosen does not appropriately demonstrate the battery dynamics. When fuzzy logic is
combined with coulomb counting the error is reduced to 5%. Lithiumion type batteries and
temperatures other than room temperature are not investigated [70].
4.4 Patent Review
Several patents have been evaluated [74109]. The majority of patents were assigned by
GM Global Technologies, Inc, with Nissan Motor Co., Ltd. and Panasonic EV Energy Co., Ltd.
tied for second most. The OCV method accounts for nearly twothirds of the patent market share.
Not all are written for lithiumion polymer type batteries; however, many concepts still apply. It
may be noted that many patents have basic ideas in common with only minor differentiation
between documentation. For the sake of the reader, each patent was summarized separately.
57
A method has been presented wherein OCV is used to estimate SOC when there is no
external current flow and coulomb counting is used when there is a load. This concept is similar
to ordinary coulomb counting only it has many recalibration points, rather than just fully charged
or fully discharged. The error depends on battery rest intervals; the longer the period of use
between long rests, the greater the accumulated error. Many lookup tables to account for
different ambient temperatures would have to be stored in memory. SOC determined from OVC
is not calculated by linear interpolation, but rather by a curvefit equation. Qmax is updated
periodically by means of cycling the battery. A predictive OCV equation is discussed that
determines remaining run time based on OCV and estimated overpotentials [74].
The accuracy of SOC estimation has been increased by updating Qmax by monitoring a
battery?s internal resistance during discharge. It is said to be a linear relationship [75].
It has been suggested to use OCV under no load and coulomb counting with multiple
current sensors under load. Three different current sensors are averaged to provide one SOC
number. Each sensor takes readings at a different frequency, with sensor accuracy decreasing as
its frequency increased [76].
A concept has been introduced in which OCV is used to estimate SOC when there is no
load on the battery. Multiple OCV curves are stored in lookup tables for multiple ambient
temperatures. Coulomb counting is used when there was a load on the battery. An equation for
Qmax as a function of cycle number is determined from experimental data. The updated Qmax is
then used to update SOC during coulomb counting [77].
An equivalent circuit model has been used to extract battery parameters. Terminal
voltage and current are input into model equations to determine the double layer voltage (Vdl),
which is then correlated to OCV and consequently SOC [78].
58
A SOC estimation algorithm has been developed in which OCV, coulomb counting, and
transient response are weighted and then averaged. Component weights are calculated as the
integral of each change with SOC. The transient response, or dominant pole position, is found to
change significantly with SOC when compared to OCV so it is given a relatively larger weight
[79].
Magnetic response has estimated SOC. A battery is wound externally with wires in which
current was passed creating a magnetic field. That magnetic field induces a second magnetic
field within the battery. A resultant magnetic field is sensed as the difference between the first
and the second. The resultant field, or change of attractive force, is found to be linearly
proportional to the SOC in the range 10100% [80]. The same authors have an almost identical
earlier patent, only it combined magnetic response detection with coulomb counting. Weight
factors are given to each based on the change in Ah [90].
SOC has been estimated by the application of a constant voltage for a specified length of
time. The battery must be in equilibrium before beginning, making this concept unrealistic. The
resulting current is measured and correlated to a previously generated lookup table [81].
An electrochemical model has been used to estimate lithium concentration diffusion in
the electrode active material. Based on the voltage and temperature, physicsbased equations
express the SOC. Concentration is correlated to OCV, once all overpotentials were removed by
calculation, and diffusion limitations are accounted for. The model is comprised of a few
separate estimation units including diffusion, OCV, and current density. Current density relies on
terminal voltage measurements while concentration distribution and OCV are purely
mathematical estimates. All are generated from the mathematical equations described earlier.
Great accuracy is achieved at the cost of great required computation power [82].
59
An OCV method has been divulged in which the relationship between voltage and
current is plotted, with current as the xaxis and voltage as the yaxis. The point where the trend
line crosses the voltage axis is considered to be the OCV. That value is then correlated to a
stored OCVSOC lookup table. This method does not consider different ambient temperatures or
battery aging [83].
A hybrid OCV/coulomb counting method has been presented where both are used
simultaneously. Rest is defined as a specified period of time where the external current equal to
or less than a preset value. The SOC estimations from both methods are assigned weights
depending on how well the rest condition is met. Theoretically, the OCV method should
periodically recalibrate the SOC ascertained from coulomb counting, yet the drawbacks of each
during loading are not eliminated [84].
A multistep state vector approach has been developed. First, a time interval for which
the battery has been at rest is determined. Previous state vectors are recalled from memory. A
second state vector is predicted based on the previous history and rest time. Once the battery
starts to charge/discharge a terminal voltage measurement, or first state vector, is obtained. A
second terminal voltage is estimated based on the predicted second state vector. A voltage error
value is calculated as the difference between the first and second terminal voltage values. Then a
third state vector is estimated based on the second predicted state vector and the voltage error
value [85]. The same author has an earlier patent in which an EKF is fitted to an electrochemical
model with temperature taken into consideration. SOC is an internal system state so the
uncertainty of estimation is an output. The work follows the author?s journal articles mentioned
previously. The estimation algorithm would require significant computational power [100].
60
Similar to [83], a battery?s VI relationship has been plotted during charging and
discharging. The point where the trend line crosses the voltage axis is assumed to be the OCV.
That OCV is then correlated to a SOC in a previously generated lookup table [86].
An OCV method has been introduced that reduces errors from hysteresis. An SOC range
is specified as the range in which corrections need to be made based on hysteresis being so
influential there, such as 2080%. A stored map of correction factors based on current and
temperature is to be kept in memory. If the SOC, or corresponding OCV, range is within the
range specified then a correction factor is applied in order to provide a more accurate SOC [87].
A patent has been suggested whereby terminal voltages V1 and V2 are measured at times
T1 and T2. Measurements are taken during rest periods making this method inaccurate during
charging or discharging. Voltages are then plotted versus the inverses of the square roots of the
times. The linear relationship is extrapolated to find the voltage axis intersection value, which is
assumed to be the OCV. SOC is estimated based on the OCV. This method could be used to
estimate OCV after a load has been removed but before equilibrium is reached; however, it is not
applicable in dynamic loading situations [88].
A coulomb counting method has been pioneered in which there are two current sensors.
The first has a measurement range that covers the expected continuous current range. It is located
inside a junction box connected directly to a battery. The second is a sensor that detects
motor/generator current based on a motor control signal. It is located inside a DC/DC converter.
When the current range of the first sensor is exceeded, the second is used thereby increasing
accuracy [89].
SOC has been estimated from temperature, current, and voltage measurements. Within
the hysteresis a range of voltages and SOCs are defined. SOC is then calculated based on the
61
average SOCs and voltages with a given equation. Lithiumion batteries do not exhibit strong
hysteresis making this method not applicable [91]. The same author holds another patent in
which OCV estimated SOC. Charge/discharge swings are predefined to meet current flow and
duration requirements. If a charge swing is followed by a discharge swing, or vice versa, then the
measured terminal voltage is said to be equal to OCV. Also, OCV will be used when the battery
has been at rest longer than a predetermined time [95].
A SOC estimation technique has been presented in which a straightline approximation
using a least squares recursive method is used to find the voltage intercept in a VI plot from
experimental data. Voltage and current are measured. The change in Ah is a function of the
change in terminal voltage with variable associated adjustment coefficients, change in
electromotive force constant, and polarization voltage generation constant based on the physical
properties of the battery [92]. The same inventors have another patent that employs the OCV
technique. Once a battery is disconnected from a load, terminal voltage is measured and a stored
polarization voltage from a lookup table is subtracted from it. The calculation gave OCV which
was then linked to a SOC. When there is external current, a noload voltage is determined by
straightline regression analysis from terminal voltage and current measurements. Coulomb
counting kept track of Ah charged or discharged. A polarization voltage is calculated based on
the Ah charged or discharged. Then the polarization voltage is subtracted from the noload
voltage to obtain OCV. It is stated that this method is not practical since it takes about a month
for all polarization effects to leave a cell once it is disconnected [96].
An OCV method has been adduced whereby OCV is plotted and an equation fitted to the
terminal voltage measurements as it relaxes after a charge or discharge. The equation is also a
function of age, temperature, and approximate SOC. Each equation is based on 20 hours of data.
62
Every time the external current flow to a battery is interrupted one of the predetermined
equations could be used to estimate the steadystate voltage, or OCV, which is then correlated to
SOC [93].
It has been alleged that coulomb counting performs more accurately under constant
current conditions, so that is when it is used; otherwise OCV. An adaptive digital filter with a
battery model determines OCV from terminal voltage and current measurements. Even though
the current may be constant, the current sensor errors will continue to accumulate [94].
SOC has been estimated by the voltage drop measured as the difference between the
terminal voltage when charging and the terminal voltage afterward at equilibrium. A plurality of
these values is stored in memory. This method does not consider that impedance of a cell will
change as it ages. Also, it will not be accurate under high current rates such as those seen in
HEVs/EVs [97].
An equivalent circuit has been patented in which voltage, current, and temperature are
continuously input. Depending on the operating point of the battery, state and parameter
variables are corrected with weighting factors. The values are then used to estimate OCV, and
therefore SOC. The equations basically combined OCV with coulomb counting, weighting each
depending on the current load. No considerations are made for temperature or aging [98].
Pulsed loads have been applied to a battery to obtain SOC. Current is passed from the
battery through a known resistive load while the terminal voltage was measured. The measured
voltage is then compared to a stored lookup discharge voltage profile and a SOC is deduced. The
voltage profile will change as the battery ages or under different temperatures making this
technique invalid [99].
63
A patent has exemplified use of an EKF with a state vector model. The EKF uses the
measured terminal current to predict terminal voltage. A difference is calculated between the
predicted and measured terminal voltages, which are then used to correct the state variables.
Ambient temperature and charging/discharging efficiency are included. The technique is very
computationally intensive [101]. The same inventors have another patent that combines coulomb
counting and OCV techniques. A weighted average is taken to determine the SOC. The voltage
based SOC is based on a modified Nernst equation while the Ahbased SOC is based on simple
coulomb counting. The weights are derived by piecewise functions based on how the terminal
voltage changes with SOC as a function of time [102].
A battery?s current limits have been used to estimate SOC. Voltage and current
measurements are taken at two different times while the battery is under load. A trend line is
drawn between those points on a VI plot. The minimum voltage is identified on the plot and a
line drawn. The intersection of the trend line and Vmin is noted. A line is then drawn from the
intersection to the current axis depicting Ilimit. Imin is specified by the battery manufacturer.
Remaining charge is calculated as the difference of Ilimit and Imin divided by the maximum
limiting current slope. The VI plots will change with aging and temperature [103].
A coulomb counting method with efficiency factors has been presented. The charging
efficiency factor is a function of measured and maximum applied current while the discharging
efficiency factor is a function of Ah removed under standard and present operation conditions.
These values are stored in multiple lookup tables with different temperatures being accounted for.
A user must know SOC, temperature, and current to get an appropriate efficiency factor. These
tables will change as the battery ages; for example, efficiencies will drop [104].
64
A multistep approach with a pseudoSOC has been patented. Coulomb counting is used
initially to give a pseudoSOC. An OCV is estimated based on the pseudoSOC. Terminal
voltage is measured and the overpotentials are removed by physicsbased equations. The
estimated and calculated OCVs are compared and assigned a correction factor from previously
generated lookup tables. The correction factor tables will need to be periodically updated to
account for battery degradation [105]. Similarly, a patent has been presented in which coulomb
counting error is reduced with a correction factor, although the deterioration in Qmax is
periodically updated by fully cycling the battery [107].
The Peukert equation has been used to calculate Qmax, of the form tIQ k?max where k is
the Peukert constant. Coulomb counting measures the Ah charged or discharged with the use of
two current sensors. One takes samples very often while the other takes samples occasionally. A
weighted average is computed and input into the coulomb counting SOC estimation equation
[106].
SOC has been estimated based on internal resistance from terminal voltage measurements
during a disconnected state. When current is applied, the R and I measurements are referenced to
a previously generated RI plot with various curves representing different SOCs. A temperature
correction is applied in order to standardize the effects of temperature. Cell resistance will
increase as the battery ages, so the RI curves will have to be updated [108].
An OCV concept has been described wherein terminal voltage measurement values are
converted to OCV values with current measurements and a stored internal resistance value,
which can be decremented with cycling. Basically, ohmic overpotentials are removed from the
terminal voltage measurements. It remains unclear how the internal resistance value will be
accurately decremented [109].
65
Summary pie charts shown in Figure 14 depict the percentages of patents (2008present)
by assignee (left, blue) and by method (right, green).
Figure 14: Patent categorization
4.5 Summary
A tree diagram may be seen in Appendix 1 that gives an overall view of all existing SOC
estimation methods. The methods have dark backgrounds while the principle author(s) have
white backgrounds. The diagram illustrates commonalities between literatures reviewed and
categorizes all journal articles, conference papers, and patents into those with and those without a
mathematical model. Then subcategories provide further categorization. The majority of
literature employs a model. Table 1 provides an overview of the advantages and drawbacks to all
existing SOC estimation methods.
66
Table 1: Advantages and drawbacks of existing SOC estimation techniques
Technique Advantages Drawbacks
Coulomb counting ? Easy implementation
? Low cost
? Measurement errors accumulate
? Requires periodic re
calibration
? Unable to estimate initial SOC
? Selfdischarge not accounted for
OCV
No model ? Easy implementation
? Low cost
? Battery must be at
electrochemical equilibrium
Equivalent
circuit model
? Simple to construct RC
circuit
? Relatively low
computational power
required
? Unable to describe physical cell
limitations or heat generation rate
? Model parameters are
temperature, current, and age
dependent
Thermal
electrochemical
model
? Describes physical cell
limitations and heat
generation rate
? Order may be reduced to
decrease computation time
? Relatively high computational
power required
Impedance ? Provides insight into
dynamics of multiple
internal components
? Highly temperature, current, and
age sensitive
? Requires constant AC amplitude
Not a single article or patent reviewed documents ECM or ROM results at any ambient
temperature other than 25?C or room temperature, nor does any reflect the effects of higher C
rates or duration of loading on SOC estimation. Also, none provide data for efficiency coefficient
calculations and non have used the OCV curve as a ?true? value. Rather, coulomb counting is
used with a highprecision CT which maintains the flaw of error accumulation.
In fact, only one paper describes a concept of correcting coulomb counting errors with a
model [34], in which the difference between an ECM terminal voltage estimate and the measured
terminal voltage are multiplied by an integral gain and subtracted from the coulomb counting
SOC estimate. Basically, it regulates the model output voltage at the value of the actual
67
measured voltage. This concept must rely on coulomb counting history for initial SOC and does
not account for CT measurement error accumulation. The authors assume a highprecision CT to
be the ?true? value. It also does not account for various ambient temperatures, various Crates, or
loading for longer than a single discharge. There exist many SOC estimation concepts, but none
have proven the ability to retain great accuracy in the gamut of realistic operating conditions.
68
Chapter 5: Experimental Setup
The author designed and constructed several battery test stations at Auburn University
capable of charging/discharging a battery with any desired current profile and ambient
temperature.
5.1 Test Equipment
A LEM IT 400S high performance current transducer (CT) was used to measure current.
It has a range of 0400A and rated accuracy of 0.0044% and is comparable to what might be
found inside a HEV/EV. The CT transforms the current signal passing through it to a much
smaller current at a ratio of 2000:1. A 2.5? precision resistor with a tolerance of 0.01% was
connected across the smaller current carrying wires and a voltage was recorded across the
resistor in order to determine current a battery was experiencing. The conversion factor from
measured voltage to battery current remained constant, even under variations in ambient
temperature of ?5?C. AC and DC signals may be recorded. The CT was calibrated through the
use of an Agilent 34410A digital multimeter with an associated accuracy of 0.17mV and a
Tektronix AM503B current probe with an associated error of <0.2%. A VI characteristic plot
was generated by recording voltage from the CT with the multimeter and current of the wire
traveling through the CT with the current probe. The data was then input into the LabVIEW
calibration software, for the voltage range 0.5 to 0.5V, where it was saved for all future
measurements. The voltage range selected corresponds to approximately 130 to 130A, which
covers the maximum current of 5C (78.5A). Figure 15 shows that there was negligible error
between the CT manufacturer?s VI plot and the experimental VI plot.
69
Figure 15: CT VI calibration plot
The sample rate for single cycle tests was 10Hz while that for longer term tests was 1Hz
in order to conserve computer memory. A summary of power supply and electronic load
specifications is given in Table 2. All tests in this thesis were conducted with a Sorensen DCS 8
125E power supply and a Kikusui PLZ1004W electronic load.
Table 2: Highpower test equipment
Type Model Rated Voltage (V) Rated Current (A)
Power Supply Sorensen DCS 8125E 08 0125
Electronic Load Kikusui PLZ1004W 1.5150 0200
Ambient temperature was manipulated by a Thermotron 8200 with a Watlow Series 922
microprocessorbased ramping control. This unit allows experiments to be conducted in the
temperature range 70?C to 150?C, which covers the realistic range a HEV/EV will experience.
Ambient temperature was recorded using an Omega Ktype thermocouple with an associated
uncertainty of measurement of 0.1%. The thermocouple was calibrated in the temperature range
of 50?C to 100?C using an Omega Ice Point Calibration Reference Chamber capable of
maintaining a constant temperature of 0?C.
70
Terminal voltage measurements are taken directly from the battery terminal tabs using
dedicated short, twisted wires so as to minimize voltage drop. Terminal voltage measurement
was calibrated using an Agilent 34410A digital multimeter with an accuracy of 0.17mV.
A test station wiring diagram is shown in Figure 16. Figure 17 shows a photograph of the
test station. The equipment is organized within a metal rack with one shelf for the computer, one
for the electronic load, one for the power supply, and one for the data acquisition board, with the
computer peripherals on top. This compact design allowed for optimum use of space within the
laboratory and improved the ease of transportation.
Figure 16: Test station wiring diagram
71
Figure 17: Test station photograph
5.2 Software Programming
National Instruments LabVIEW software allows a user to define a program. National
Instruments PCI cards, connector blocks, and modules allow a user to output analog or digital
voltage control signals while recording input measurement voltage signals.
A screenshot showing the front control panel of a program is shown in Figure 18,
followed by a screenshot of the discharge current control section of the rear block diagram in
Figure 19. The easily manipulated controls may be placed on the front panel at user?s discretion.
All block diagrams have the same basic sequence structure: initialization, accept input, produce
output, repeat until a stop condition is met, then clear input and close.
72
Figure 18: LabVIEW front panel screenshot
Figure 19: LabVIEW current control block diagram
73
Feedback control of the sink current is continuously optimized based on CT readings to match
the current setpoint. The loop ceased to iterate once a preset stop condition, such as Vmin, is
reached. Any desired current profile may be programmed.
Data is recorded in TDMS format which can be read in Excel after downloading a TDMS
Importer addin. Figure 20 shows a block diagram of the data recording section of a LabVIEW
program.
Figure 20: LabVIEW data recording block diagram
5.3 LiPolymer Battery Specifications
The batteries being used for experimental validation are pouch type with a cathode
consisting of a mixture of manganese, nickel, and cobalt oxides. Each has a nominal capacity of
15.7Ah. Their maximum temperature allowable before thermal runaway occurs is 75?C. Due to a
confidentiality agreement with the supplier, no additional information will be given.
74
All cells are initially fully discharged and then fully charged 5 times before any testing.
This is to provide adequate charge transfer for the SEI layer to stabilize. If not stable, Qmax may
vary significantly between cycles. For the specific battery being investigated, a full charge is
defined as reaching 4.15V at 1C in CC mode then reaching the cutoff current of 800mA in CV
mode. A full discharge is defined as reaching 2.5V at 1C.
75
Chapter 6: Battery Modeling
The fundamentals presented in Chapter 2 will now be applied. This section offers a
detailed description of how the ECM, FOM, and ROM are formulated.
6.1 Minimum Rest Time
A standard rest time of 30 minutes was adapted for all experiments based on battery
relaxation characteristics. Figure 21 depicts unfiltered terminal voltage relaxation after a 5
minute 1C charge at 25?C with an arbitrary initial SOC. Rest time is practically identical for
other Crates and ambient temperatures.
Figure 21: Terminal voltage relaxation
The OCV method remains an accurate SOC estimation technique for a battery at
electrochemical equilibrium. A minimum time to reach equilibrium may be determined from
studying the battery?s voltage relaxation characteristics; the shorter the required rest time, the
more coulomb counting recalibration opportunities. An allowable SOC change rate may differ
between OEMs, but the author might suggest a target of ?0.01% SOC per minute. Based on the
76
voltage relaxation plot as shown in Figure 22, terminal voltage changes by ?0.072mV per minute,
or ?0.01% SOC per minute, after 18 minutes. Figure 22 is a zoomedin image of Figure 21 with
a lowpass filter applied. Every time a HEV or EV rests its battery pack for at least 18 minutes
then the accumulated time (t) should be reset to zero and a terminal voltage measurement could
be used to estimate SOC based on the OCV curve. The figure proves that 30 minutes per rest
period is more than sufficient.
Figure 22: Minimum rest time
6.2 OCV Curve
A fully charged battery was discharged at a 1Crate for 30 seconds and then rested for 30
minutes while Tamb remains a constant 25?C. This process was repeated until the terminal voltage
reached the manufacturer?s Vmin of 2.5V. After 30 minutes, terminal voltage changes by less than
1mV per 10 minutes corresponding to less than 0.014% SOC per minute so it was deemed an
appropriate rest time.
The discharge duration of 30 seconds was chosen in order to provide approximately 120
data points. The measured voltage at the end of a 30 minute rest, before load was applied, is
77
considered to be equal to OCV since it is considered ample time for the cell to reach chemical
equilibrium. A sample of the terminal voltage profile undergoing OCV discharge pulses is shown
in Figure 23. The rest periods between pulses are not shown.
Figure 23: OCV generator pulse sample
Current was measured and integrated during the discharges. The coulomb counting
method was applied with an initial SOC of 100% and a Qmax measured experimentally
beforehand. CT measurement errors are assumed to be negligible due to the short load times and
rest periods between each data point. OCVs and SOCs were then linked to generate Figure 24.
78
Figure 24: OCV curve
SOCs corresponding to the battery?s Vmax and Vmin were interpolated. That is why the chart has
values below 0% SOC and above 100% SOC.
The effect of Tamb on terminal voltage after a full discharge at 1C can be seen in Figure
25.
0 600 1200 1800 2400 3000 3600 4200 4800
2 .6
2 .8
3 .0
3 .2
3 .4
3 .6
3 .8
4 .0
4 .2
T
e
rmi
n
a
l
Vo
lt
a
g
e
(V)
T i m e (s)
0 ? C
2 5 ? C
4 0 ? C
Figure 25: Effect of Tamb on 1C full discharge
79
The OCV curve standard is 25?C; therefore, if the same OCV curve is applied at 0?C the
SOC will be underestimated and at 40?C overestimated. Table 3 shows the SOC correction
factors at 0% SOC. These values are considered to be the ?true, calibrated? values to which all
calculations of error will be based.
Table 3: SOC Tamb correction factors
0?C 25?C 40?C
Final terminal voltage 3.4798 3.0233 2.7864
Final SOC 7.94 0.58 2.08
Correction factor at 0% +8.52 N/A 1.50
6.3 Equivalent Circuit Model (ECM)
6.3.1 RC Circuit
A second order Randle model is the most common equivalent circuit model in literature
[3],[4],[8],[13],[17], [19], [34], [43], [51],[65], [70], [98]. With its 3 resistors and 2 capacitors it is
able to accurately portray battery dynamics. First and third order models were attempted, with
poor results. The components include ohmic resistance (R1), electrochemical polarization
resistance (R2), concentration polarization resistance (R3), electrochemical polarization
capacitance (C2), and concentration polarization capacitance (C3) [3], as can be seen in Figure 26.
80
Figure 26: ECM RC circuit
OCV is equal to the sum of the other voltages, given by
Taa VRIRIRIO C V ???? 332211 where the currents are those flowing across the resistors.
2
22 RVI a ? and the current across the capacitor C2 is given by
2
2121222 RVIIIdtdVCI ab ????? . This simplifies to
2
1
22
22 CICRVdtdV ?? . With the boundary
conditions of 0)0(2 ?V and 212 )( RIV ?? , the solution becomes )1()( 22212 CR teRItV ??? . When
the approach is applied to all components in the circuit the solution becomes
TCR
tCR t VeReRRIO C V ?????? ?? )]1()1([
3322 3211 . The estimated OCV is then input to a lookup
table to estimate SOC.
6.3.2 Parameter Extraction
To determine to correct values for R1, R2, R3, C2, and C3 a series of pulses were applied to
a battery to measure dynamic response. Current and terminal voltage are measured during 1C, 1
minute discharges with subsequent 30 minute rest periods. A single pulse depicting terminal
voltage change is shown in Figure 27.
81
Figure 27: ECM parameter extraction pulse
The immediate voltage change after current interruption is equal to the ohmic
overpotential (V1) since it does not include a capacitor that would slow voltage relaxation. V1 is
divided the current, 15.7A, to extract R1, according to Ohm?s law. The remaining voltage change
after the initial change until the end of rest is equal to the concentration and activation
overpotentials (V2 + V3). The MATLAB curve fitting tool was implemented using a nonlinear
least squares exponential equation of the form ?a*exp(b*x)+c*exp(d*x)? since that matches the
form of V2(t) and V3(t) determined previously. Figure 28 shows an example curve fitting, in
which the blue dots are data points and the red line is the curve fit.
82
Figure 28: Curve fitting example
The data from multiple separate discharge pulses were averaged, with an average R
squared value above 0.99, to provide final values, as seen in Table 4.
Table 4: ECM extracted parameters
Parameter Value @ 0?C Value @ 25?C Value @ 40?C
R1 0.010613 0.005752 0.002181
R2 0.001772 0.001149 0.000653
R3 0.000214 0.000154 0.000173
C2 32860.11 22777.05 71809.62
C3 4318320 2237890 8202080
The parameters vary appreciably with Tamb. They are also expected to vary between cells
considering minute manufacturing difference as well as with age as material degradation leads to
internal resistance increases.
6.4 Full Order Thermal Electrochemical Model (FOM)
The model for electrodes is built based on the theory that it consists of multiple spherical
particles, as shown in Figure 29
83
Figure 29: Electrochemical model schematic [9]
where lithiumion concentration, cs, in a single particle is described by Fick?s law of diffusion
?????? ??????? rcrrrDtc sss 22 (1)
with boundary conditions
0
0
???
?r
src and Fa jrcD
s
Li
Rr
ss
s
????
?
(2)
where r is the radius of electrode particle, Rs is the maximum electrode particle radius, as is the
interfacial surface area, and F is the Faraday constant.
Distribution of ion concentrations in the electrolyte can be described by
Liee f feee jF tcxDxtc 01)( ????????? ??????? ? (3)
with boundary conditions
0
0
??????
?? Lx
e
x
e tctc (4)
where L is the thickness of the microcell, ce is the lithiumion concentration in the electrolyte
phase, ?e is the porosity, t+0 is the transference number, and Deeff is the effective diffusion
coefficient calculated using the equation
84
peeeffe DD ??? (5)
The governing equation of potential is based on the principle of charge conservation
derived from Ohm?s law
0???????? ???? Lise ff jxx ?? (6)
with boundary conditions
AIxx xse f fxse f f ?????? ???? ?? ????
(7)
00 ?????? ?? Lxsxs xx ?? (8)
where ? is the thickness of the anode (cm) and ?+ is the thickness of the cathode (cm).
Charge conservation in the electrolyte yields
0ln ???????? ??????????? ???? Liee ffDee ff jcxkxxkx ? (9)
with boundary conditions
00 ?????? ?? Lxexe xx ?? (10)
Current density is obtained from the ButlerVolmer electrochemical kinetic equation
?????? ?????????????? ???? RT FRT Fiaj casLi e x pe x p0
(11)
where ? is the overpotential, defined as the difference between the solid and electrolyte potential
minus the equilibrium potential
Ues ??? ??? (12)
85
Figure 30 depicts the calculation scheme for a single cell model. Governing equations in
the microcells are interconnected with a single cell.
Figure 30: Electrochemical model calculation scheme [9]
All the PDEs are discretized by finite difference methods, implicit method in the time
domain and CrankNicolson method in the space domain. The implicit method is unconditionally
stable and first order accurate while the CrankNicolson method is stable although oscillatory
and is second order accurate. Nonlinear terms are linearized by Taylor series expansion. The
model is solved in discretized state space form. Model parameters may be found in Appendix 2.
6.5 Reduced Order Thermal Electrochemical Model (ROM)
The equations above are PDEs and functions of three variables: x, r, and t that represent
the location of a particle along the thickness of an electrode, the radial coordinate within a
particle, and time, respectively. PDEs can be solved numerically by discretization. When the
86
domain is discretized with 100 nodes in the xdirection and 50 nodes in the rdirection, the total
number of differential algebraic equations becomes 100?50.
According to the suggestion made by V. Subramanian et al. [20], ion concentration PDEs
in the solid electrodes can be replaced with a polynomial under the assumption that ion
concentrations inside the spherical particles are a function of the radius of the sphere
????????????????????? 4422 )()()(),( sss RrtdRrtbtatrc (13)
where the coefficients, a(t), b(t), and d(t) are a function of time.
The coefficients are solved using cs,ave, volumeaveraged concentration fluxes (qave), and
surface concentrations (cs,surf), given by
a v essa v es u r fs
a v essa v es u r fs
a v essa v es u r fs
cRqctc
cRqctb
cRqcta
,,
,,
,,
4
1057
4
105)(
351035)(
4
353
4
39)(
???
????
???
(14)
where the volumeaveraged concentration, volumeaveraged concentration flux, and surface
concentration are defined as
??
?
?
??
?
?
????
?
?
??
?
?
???
?
?
?
?
?
?
?
?
?
?
?
?
4
4
2
2
,
0
3
2
0
3
2
,
)()()(),(
),(
3
),(
3
ss
s u r fs
R
s
s
a v e
R
s
s
a v es
R
r
td
R
r
tbtatrc
drtrc
rR
r
q
drtrc
R
r
c
s
s
(15)
By combination with Fick?s law and the boundary conditions, the three above equations
become
87
Fa
j
qDcc
R
D
FaR
j
q
R
D
q
dt
d
FaR
j
c
dt
d
s
Li
a v esa v ess u r fs
s
s
ss
Li
a v e
s
s
a v e
ss
Li
a v es
????
???
??
8)(35
0
2
45
30
03
,,
22
,
(16)
Thus, the ion concentration in equation (1) is completely replaced by equations (14), (15),
and (16).
The equations above for the potentials are further simplified under following assumptions:
1. Reaction current is not influenced by the electrolyte concentration
0ln ??????? ???? eeffD cxkx (17)
2. The ButlerVolmer equation may be linearized
)(0 URT Fiaj sesLi ?? ? (18)
3. Ion conductivity (keff) is constant
?????? ??????????? ???? ee f fee f f xxkxkx ?? (19)
The phase potential difference, ?se, is defined as
esse ??? ?? (20)
So, the potential equations (6) and (9) can then be simplified in the following manner
Liee ffLiee ffDee ff jxkxjcxkxxkx ??????? ????????????? ??????????? ???? ?? 0ln ? 0???????? ????? e ffLie kjxx ?
0???????? ???? Lise ff jxx ?? ? 0???????? ???? effLis jxx ??
88
?????? ???????? ????? e f fe f fLise kjxx 11?? (21)
The capacity of a cell can be expressed as a function of lithiumion concentration in the
electrodes as
?? ???? ? ???0 0m a x,, ASFcA d xFcQ ssa v essr e le a s a b le (22)
???? ?? ASSFcQ ss )( 0100m a x,m a x (23)
where ?s is the active material volume fraction. The first term in equation (22) can be understood
as the average lithiumion concentration integrated across the anode thickness. It has units of
coulombs and only considers the active material. The second term in equation (22) accounts for
the charges remaining in the cell at 0% SOC, which cannot be released. Equation (23) is similar
only S100 is included rather than an integral to account for releasable charges at 100% SOC.
SOC can be expressed as a ratio of the average lithium concentration to the maximum
lithium concentration in the negative electrode, calculated as
?
?
?
?
?
?
???
??
?? ??
??
?
?
??
???
??
)()( 0100m a x,
0
0m a x,,
0100m a x,
0
0m a x,,
m a x SSc
Scdxc
ASSFc
ASFcA d xFc
Q
QS O C
s
sa v es
ss
ssa v ess
r e le a s a b le
? ?
?? ?
?
??
?
? ?? ?
?? ?
??
?? ?
)()( 0100m a x,
0
0m a x,,
0100m a x,
0 0
0m a x,,
SSc
dxScc
SSc
dxScdxc
s
sa v es
s
sa v es
? ??? ?? ???
?
?
? 0 0100m a x, 0m a x,, )(1 dxSSc SccS O C s sa v es
(24)
The FOM and ROM PDEs are solved using the same mathematical techniques. Source
code will not be presented in this thesis as it is proprietary to Dr. Choe?s battery research group
at Auburn University.
89
6.6 Thermal Component of ROM
Heat transfer is described by
qQzTkzyTkyxTkxtTC g e nzyxp ???????? ????????????? ??????????? ????????
where ? is density (kg?m3), Cp is specific heat capacity (J?kg1?K1), k is thermal conductivity
(W?m1?K1), Qgen is the heat generation rate per unit volume (W?m3), and q is the heat transfer
rate between the cell and its surroundings (W?m3) defined by
)( ambTTdhq ??
where h is the heat transfer coefficient (W?m2?K1) and d is the depth of the battery (m).
Qgen includes both reversible and irreversible heat generation and is described by
?????? ?????? TO C VTVO C VIQ Tg e n
90
Chapter 7: Existing Method Error Characterization
SOC estimates based on the OCV curve (SOCOCV) are considered the ?true? value.
Coulomb counting, ECM, and ROM SOC estimation data will be compared to that of the OCV
curve after a rest period of at least 30 minutes has passed. Inevitably, this means that realtime
dynamic model validation is not possible. Given that the initial SOC is 100%, the final SOC can
be inferred by measuring terminal voltage after rest, and the current during discharge is constant,
a linear OCV line may be interpolated. The OCV lines at various discharge currents and ambient
temperatures then become the basis for error characterization.
A battery is initially fully charged to its Vmax and Icutoff, which are for these batteries
4.15V and 800mA, respectively. It is then fully discharged to its Vmin of 2.5V. During the
discharge the current is integrated to calculate a unique Qmax for that specific battery at a specific
Tamb. This is a standard procedure for all batteries since Qmax is essential to estimate SOC based
on the equation
%100)0()(
m a x
0 ?
??
?? ? Q
dti
S O CtS O C
t ?
The efficiency coefficient, ?, is always 1.0 during charge but is a function of Tamb during
discharge, as seen in Table 5, based on the equation
ech
edischQQ
arg
arg??
At low temperatures reaction kinetics are slowed to a point in which ion migration from
the anode to the cathode is decreased due to the reduction of active channels in the polymer
separator.
91
Table 5: Efficiency coefficients
Temperature (?C) 0 25 40
? 0.975 1.0 1.0
These values are expected to change as a battery ages; however, due to the limited number of
cycles each battery will experience these values are assumed to be constant.
As a test begins, a battery is fully charged again to 4.15V and 800mA corresponding to
an initial SOC of 100%. Then the battery will be cycled a specified number of times while the
current is integrated to determine SOC. Full charge is redefined as the condition whereby the
charge (Q) is equal to the previously determined Qmax, Q=Qmax. Full discharge is redefined as
Q=0. After cycling, the battery rests for 30 minutes. Finally, the battery terminal voltage is
measured, which is assumed to be OCV at this point. The SOC based on the OCV curve
(SOCOCV) is compared to the SOC based on coulomb counting (SOCcc). Recorded current data
will be the input for the ECM and ROM models to create modelbased SOC estimates (SOCmodel)
enabling comparisons to SOCOCV.
Qmax is considered to be accurate even though it employs the coulomb counting technique
since the measurement conditions are low current, constant rate, 25?C, and only 1 cycle. Also,
Qmax is not expected to change by more than 1% during any single test. This was confirmed by
conducting 5 cycles on a battery at 1C and 25?C while measuring Qmax during each cycle.
7.1 Experimental Test Matrix
Table 6 shows the variables that will be investigated. These have each been proven to
affect SOC estimation results. The objective is to determine errors as a function of temperature,
current, and time.
92
Table 6: Proposed investigation range
Current 1, 2, 5C
Mode Constant, Hybrid Pulse Power Characterization (HPPC)
Tamb 0, 25, 40?C
Duration full discharge, 1 cycle, 5 cycles
For clarification, 1C = 15.7A and the HPPC battery test profile was developed by the U.S.
Department of Energy (DOE) at the Idaho National Laboratory [110]. Figure 31 charts the
relative current profile.
Figure 31: HPPC test profile
It was chosen because it is based on technical targets instituted for energy storage to meet
DOE battery performance and life cycle standards. It is applied to a battery for 1 hour so as to
not fully discharge a battery. Relative currents utilized are 1C, 2C, and 5C.
In the test matrix context, a ?cycle? refers to a discharge followed by a charge in the SOC
range 2080%. These SOCs are calculated by coulomb counting as fractions of the measured
Qmax. This corresponds to a typical HEV charge sustaining mode SOC range [44]. Appendix 3
contains the complete list of tests conducted. Three batteries were used in total with their roles
93
being rotated for each group of tests so that they each experienced similar times at each Tamb so
as to not cause uneven degradation.
7.2 Coulomb Counting
As can be seen in Figure 32, coulomb counting error is a linear function of time. These
are plots at 1, 2, and 5C at 0, 25, and 40?C. All plots show a linear trend; linear because of
constant error accumulation with constant current. Coulomb counting error (?cc) becomes more
positive as Tamb increases, where
O C Vcccc S O CS O C ??? ,
attributed to the effect of temperature on maximum charge capacity.
Coulomb counting follows
%100)0()(
m a x
0 ?
??
?? ? Q
dti
S O CtS O C
t ?
where Qmax is always measured at a standard 25?C. Qmax would be different if the Tamb were
altered. High temperatures induce greater reaction kinetics, which leads to greater ion migration
between Vmax and Vmin, and a greater maximum charge capacity. The opposite occurs at low
temperatures. Temperature change may be a consequence of internal heat generation due to ion
current, too, based on the ButlerVolmer equation. Since Qmax and the maximum charge capacity
at operating temperature are different, coulomb counting estimates will diverge from OCV curve
estimates. This is exemplified in the fact that the coulomb counting SOC estimates at 1C and
25?C had the least error.
94
Figure 32: Coulomb counting error characterization
Figure 33 shows the SOC estimation error of coulomb counting when compared to the
OCV line. All slopes are linear. The 1C, 25?C line has an error even though the OCV curve was
generated by the CT with the same conditions because Qmax is slightly different from cell to cell
depending on manufacturing tolerances and loading history. Maximum charge capacity is larger
for the OCV curve due to its pulsed current profile whereby a battery is given time to rest
between load pulses during which its terminal voltage rises. Upon initiation of the succeeding
pulse, activation overpotential must be overcome once again to reach the same voltage it was at
before which consumes amphours falsely included in the SOC calculation. A higher precision
95
CT would reveal the same flaw since divergence is a function of battery voltage swings relative
to Tamb.
Figure 33: Coulomb counting SOC estimation error summary
7.3 ECM
As can be seen in Figure 34, ECM error (?ECM) is not a function of time, where
O C VE C ME C M S O CS O C ???
where SOCECM is a function of the ECMestimated OCV. These are plots at 1, 2, and 5C at 0, 25,
and 40?C. Immense error is attributed to model parameters. ?ECM increases significantly with
current due to internal heat generation altering the model parameters, which are assumed to be
constant. They are very sensitive to variations from cell to cell, age, and temperature. There is no
observable trend between ?ECM and Tamb.
96
Figure 34: ECM error characterization
7.4 ROM
As can be seen in Figure 35, ROM error (?ROM) is a function of time, where
O C VR O MR O M S O CS O C ???
These are plots at 1, 2, and 5C at 0, 25, and 40?C. ?ROM becomes more positive as Tamb increases
and is greatest at 0?C. These observations are analogous to those with coulomb counting only the
error accumulation is less per time step. The ROM calculates SOC based on
97
? ??? ?? ???
?
?
? 0 0100m a x, 0m a x,, )(1 dxSSc SccS O C s sa v es
which is based on a standard 25?C.
Figure 35: ROM error characterization
Figure 36 shows the SOC estimation error of the ROM when compared to the OCV line.
Note that all slopes are linear. There is a small error accumulation with each time step since no
model is able to perfectly represent a physical phenomenon. The average anode lithium
concentration must be less at low temperatures and greater at high temperatures since it is the
only nonconstant in the SOC equation. This is typified in the negative slope at 0?C and the
98
positive slopes at 25?C and 40?C. As expected, the discharge at 1C and 25?C had the least
average error.
Figure 36: ROM SOC estimation error summary
7.5 Summary
The ROM exhibited less error than coulomb counting in all cases. Coulomb counting
offers no connection between measured current and battery internal dynamics. It can provide Ah
calculations but cannot relate those to the state of a battery since Qmax changes with Tamb and age.
The ROM can relate current and temperature to SOC by accounting for their effects on lithium
concentration in the anode. The model input current can be a setpoint, to eliminate CT
measurement error, or simply a CT measurement. If taken as a setpoint, there may be a small
initialization error not captured by the model as power equipment takes time to ramp up to a
setpoint, however, this is on the scale of <0.1% SOC so it could be neglected. CT measurements
were the input for all models in this thesis.
99
Regardless of the rated accuracy of a CT, there will always exist coulomb counting
measurement error that will accumulate with time. From gathered data, it appears that the ROM
error also accumulates with time since no model is perfect. Small imperfections accumulate with
each time step. Also, the ROM may rely on the CT for current input, adding to the linear error
accumulation. External disturbance, such as a strong random magnetic field pulse, may lead to
false readings. A higher precision CT may improve SOC estimation but still would be
susceptible to the drawbacks previously mentioned.
The ROM also exhibited less error than the ECM in all cases. This is due to the inability
of the ECM to describe physical cell limitations, such as maximum/minimum lithiumion
concentration and maximum lithiumion surface concentration or diffusion rates in the anode, or
heat generation rate. Also, its current input is based on an imperfect CT current measurement.
Model parameters are highly sensitive to temperature and current. This implies that the
parameters would have to be periodically updated and may vary from cell to cell based on age or
manufacturing variations. The ROM continuously updates for temperature and current and is
capable of including lithiumion diffusion depletion and saturation limitations.
Even though the ECM calculation time may be 10+ times faster, increasing SOC
estimation speed is secondary in this thesis. SOC estimation accuracy augmentation is the
primary goal. The ROM calculation time is approximately a factor of 10 less than that for the
FOM with no evident difference in SOC estimates. Therefore, only the ROM will be further
evaluated for inclusion in a new method.
100
Chapter 8: SOC Estimation Error Compensation of ROM Using an Empirical Method
A new SOC estimation algorithm is proposed that corrects the error associated with ROM
estimates. Theoretically, the linear error accumulation previously documented may be removed
by subtracting a constant correction rate term dependent on I and Tamb. An algorithm loop has
been conceptualized containing a 2D interpolator and an error correction path, as shown in
Figure 37, designed to minimize SOC estimation error. The current input is a desired value or
setpoint, not a measured value so as to eliminate CT measurement offset error. Terminal voltage
input from the battery is used only to set initial SOC.
Figure 37: Empirical method schematic block diagram
101
Combined method concepts using coulomb counting with a model such as a weighted
average scheme, a piecewise method with smoothing, and a Kalman filter approach have been
evaluated. The mathematicallysimplest approach, error characterization and removal, proved to
perform well due to the linear nature of the ROM error.
The OCV method remains valid if a battery has been at rest for a specified amount of
time. It may only be used to estimate initial and final SOCs if the rest condition has been
satisfied.
8.1 ROM Error Rate
ROM error (?ROM) accumulates linearly with time. It is calculated as
)( O C VR O MR O M S O CS O C ???
where the SOCs used in the equation are single point values at the end of a full discharge, as
shown in Table 7. ROM final SOC error increases with Crate but not always with Tamb.
Table 7: ROM single discharge final SOC errors
Current (Crate) Tamb (?C) Final SOC Error (%)
1
0 3.00
25 1.92
40 3.17
2
0 2.72
25 2.10
40 3.15
5
0 5.14
25 2.15
40 3.27
Since error is a linear function of time due to model imperfections accruing with each
time step, a constant discharge error rate ( ROM?? ) may be calculated as
102
t o t a l
O C VR O M
t o t a l
R O MR O M t S O CS O Ct )( ??? ???
where ttotal is the total discharge time (s) at a specific Crate and Tamb.
Table 8 shows the experimental results for ROM?? . The ROM discharge error rate
increases with Crate. At similar Crates, the error rate tends to have a maximum value at 0?C
and a minimum at 25?C.
Table 8: ROM discharge error rates
Current (Crate) Tamb (?C) Error Rate (?104) (%/s)
1
0 9.5
25 5.4
40 8.8
2
0 17.8
25 11.9
40 17.5
5
0 103.2
25 31.4
40 45.9
Charge error rates were calculated the same way. They differ significantly from discharge
error rates. Unfortunately, it is not possible to compare coulomb counting or ROM SOC
estimates to the OCV curve during a full charge since it consists of CC and CV modes. SOC
does not linearly increase but rather tapers off in CV mode as the applied current decreases.
Charge error rate is assumed to be constant during CC and CV modes. Example charge data at
1C and 0?C is shown in Figure 38. Time (t) used in error rate calculated is reset to zero whenever
current is interrupted or polarity switched.
103
Figure 38: Single charge data at 1C, 0?C
It is also not possible to backinterpolate an OCV SOC estimate from the beginning and end of
charge since there is not rest period between discharging and charging phases. Charge error rate
results are shown in Table 9. These values were tuned so as to minimize final SOC error after a
full charge.
Table 9: ROM charge error rates
Current (Crate) Tamb (?C) Error Rate (?104) (%/s)
1
0 5.0
25 4.2
40 7.7
2
0 7.1
25 7.2
40 13.0
5
0 79.2
25 9.7
40 2.3
104
8.2 2D Interpolation
Rarely will real world operating conditions exactly match a Crate and Tamb evaluated.
Interpolation is required to calculate ROM error rates in the Crate range of 15C and the
temperature range of 040?C.
First, ROM?? is interpolated as a function of Tamb. A 2nd order polynomial method was
chosen due to the nonlinear nature of the relationship. ROM?? as a function of Crate is nearly
linear. Each equation for discharge, which is shown in Figure 39 with dotted lines representing
an interpolation curve, has an associated coefficient of determination (R2) = 1.
Figure 39: ROM discharge error rate as a function of Tamb
The calculated value is then linearly interpolated with respect to Crate. This process is
also applicable to charge error rates. An example calculation for a battery being discharged at 3C
and 30?C is provided below:
42
42
10780.14)30(18.10315.81106.0)(:5@ 10753.41)30(773.176962.10204.0)(:2@ ?
?
??????? ??????? R O MR O M R O MR O M TTTC TTTC ?? ?? ??
??
sR O M %10762.2310753.41)10753.4110780.14()25( )23( 4444 ???? ?????????????
105
8.3 Single Discharge
Error correction is applied to the single discharge data previously shown for the ROM, as
portrayed in Figure 40. There is a significant improvement in SOC estimation accuracy in all
cases, so much so that any differences between the model with error correction and the OCV line
are indiscernible in a plot. The maximum SOC error is < 0.003% under all investigated operating
conditions.
Figure 40: Constant discharge  empirical
Variances (?2i) at each time step are calculated using the equation
22 )( O C Vii S O CS O C ???
106
where i may represent coulomb counting, the ECM, or the ROM. Variances eliminate any
negative values and are able to better show error distribution, compared to taking the absolute
value of the differences, when integrated.
Average variances (?2i,ave) are then calculated by
dtS O CS O Ct to ta lt O C Vi
t o t a la v ei ?
??
0
22, )(1?
where ttotal is the total time (s) of a full discharge at a specified current and Tamb, as shown in
Table 10. Total discharge time increases with temperature and decreases with Crate.
Table 10: Total discharge times
Current (Crate) Tamb (?C) ttotal (s)
1
0 3157.4
25 3569.0
40 3617.4
2
0 1532.1
25 1762.3
40 1801.2
5
0 498.5
25 683.9
40 713.5
Average errors (?i,ave) are then calculated by
tdS O CS O Ct to ta lt O C Vi
to ta la v eia v ei ?
???
0
22,, )(1??
This process is similar to taking the 2norm of ?i. Appendix 4 shows the results for each method.
The ROM with error correction based on empirical data has negligible error in every case for a
single discharge.
107
8.4 Single Cycle
The proposed SOC estimation algorithm is applied during a single full cycle, 100% to 0%
to 100% SOC, in order to evaluate its accuracy during discharging and charging. Example cycle
data at 1C and 0?C is shown in Figure 41.
Figure 41: Single cycle data at 1C, 0?C
The table in Appendix 5 shows the final SOC errors when compared to the final OCV for
coulomb counting, the ROM, and the ROM with error correction. The ROM with empirical error
correction presents considerably more desirable results than the other methods since it was tuned
to minimize final SOC error.
8.5 HPPC Test
The HPPC test profile is applied for 1 hour to determine accuracy under pulse load
conditions. Example cycle data at 1C and 0?C is shown in Figure 42. The ECM is omitted due to
poor performance.
108
Figure 42: HPPC cycle data at 1C, 0?C
Using the same average error calculation described beforehand, Appendix 6 shows
average errors for each method during HPPC tests. The ROM with empirical correction proved
to be more accurate at 25?C and 40?C but less accurate at 0?C. This is due to greater changes in
temperature at low ambient temperatures with similar internal heat generation rates. These
temperature changes affect model parameters. The error rate calculations were based on constant
current so they do not aptly capture pulse behavior.
8.6 5 Cycles
Longterm tests are conducted to prove that the ROM error correction is effective.
Similar to a single cycle, only final SOC estimates of the ROM and OCV curve may be
compared. Example cycle data at 1C and 0?C is shown in Figure 43. The ECM has spikes in
SOC after current polarity changes due to time being so great that the exponential terms from its
109
OCV equation become nearly zero. Basically, concentration and activation overpotentials are
ignored.
Figure 43: 5 cycle data at 1C, 0?C
Appendix 7 illustrates the final SOC error after 5 cycles at various ambient temperatures
and Crates. The ROM with empirical correction proved far less accurate than the original ROM
in every case. Obviously, the charge error rates are not appropriate for cycling conditions since
they were calculated based on CC/CV charging modes rather than just CC mode as is the
circumstance for the 5 cycle tests.
110
Chapter 9: SOC Estimation Error Compensation of ROM Using a Feedback Method
The empirical method proves to have immaculate accuracy during a single discharge and
single cycle loading, but pulse and longterm accuracy is questionable. The error rates are not
adjustable so if the dynamics of a battery differ much from those used for error rate calculations
then error will grow radically. To counter this, a second SOC estimation algorithm is proposed
that corrects the error associated with ROM estimates using a feedback loop. Figure 44 shows
the schematic block diagram. Theoretically, this approach should eliminate accumulating CT and
ROM errors by comparison to battery terminal voltage measurements.
111
Figure 44: Feedback method schematic block diagram
9.1 Feedback Design
A typical state space observer model takes the form of
DuCxy e r r o rLBuAxx kk
kk
?? ???? ??
?
11
1
where L is an observer gain.
With the proposed method, the difference in terminal voltage (?VT) between the ROM
and the battery is treated as the error term, so the state space model is of the form
112
DuSCV VLBuSAS kkT T
kk
??? ?????? ??
?
11
1
where S is the stoichiometry number, which is related to cs,ave and subsequently SOC, to be
discussed later. This representation demonstrates that L must be a vector. First, an appropriate
correction term insertion point must be determined. The model function containing cs,ave is
03, ?? FaR jcdtd ss Lia v es
After discretization the equation becomes
FaR jt cc ss
Lik a v esk a v es 3,1, ??
??
?
Then k and k+1 terms are separated as shown
tFaR jcc ss Lik a v esk a v es ????? 3,1,
in order to solve all k+1 terms simultaneously with the accompaniment of all the other
unaffected equations in the model.
Since cs,ave is nonlinear through the thickness of an electrode, an average value of all
particles across the electrode thickness is used to calculate SOC using S in the equation
0100
0SS SSSOC ???
where
max,
,
s
avesccS?
Also, jLi can be averaged as
?? ? ????? A Ij
Li
113
If the previous equation for cs,ave is altered by substituting in the average values of cs,ave
and jLi then dividing through by cs,max the equation for the anode, with inclusion of error
correction, becomes
tVLtFAaR IcSSVLBuSAS TssskkTkk ????????????????? ?? ?31m a x,11
where new coefficients for the A and B matrices are shown.
The state space output is derived after linearization of S as follows
??? ??????? SCSfO C VV T )(
So the C and D matrix coefficients remain unchanged in
DuSCVO C VV kkTkT ?????? ??? 111 ?
9.2 Gain Optimization
The observer gain vector (L) consists of 3 sections to account for the anode, cathode, and
separator. In accordance with charge conservation laws, whatever value is subtracted from the
anode must be added to the cathode typified in the equation
? ? ? ?c a t h o d esanodes ???? ?????? ??
where A and F were on both sides but cancelled out and L in the separator region is 0. When the
model parameters are plugged in the equation becomes
5.0104.3658.01050 44 ???????? ?? c a t h o d eanode ??
Then separate unknown variables from the given numbers and multiply numerator and
denominator by a common multiplier (? ) to attain
5.0104.36 4 ????? ??? anode and 58.01050 4 ???? ??? c a th o d e
So L takes the form of
114
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
??
?
?
?
?
?
?
?
?
?
?
???
???
?
?
?
?
?
c at hod e
s e par at or
anode
L
58.01050
58.01050
0
0
5.0104.36
5.0104.36
4
4
4
4
where ? is optimized by pole placement using the MATLAB function ?place()?. Desired poles
are optimized based on state space matrices A and C to provide a constant optimized gain. The
desired pole must be negative for system stability. Performance of pole placement was
determined based a trial and error method considering many different poles by analyzing the
resultant SOC estimation error. The resulting desired pole was 0.5 with an associated gain of ? =
0.0442.
9.3 Single Discharge
The algorithm with feedback is applied to the single discharge data previously shown for
the ROM, as portrayed in Figure 45. There is a significant improvement in SOC estimation
accuracy in all cases. The maximum SOC error is > 10% near the end of a 5C, 0?C discharge
suggesting improper low temperature model parameters, such as k or h. Errors are greatest in the
SOC range 020% due to a rapid measured terminal voltage dropoff relative to model estimates.
ROM overpotential estimates are erroneous near the end of discharge. In HEV/EV application
this is of little concern since a battery pack will never experience < 20% SOC. Errors at the
beginning of discharge are associated with improper initial SOC estimation. Since the gain is
fairly small, the model requires time to adjust to large initial ?VT?s.
115
Figure 45: Constant discharge  feedback
Average errors (?i,ave) are calculated the same way as before. Results are included in
Appendix 4. At low temperature the ROM with feedback proves advantageous over the regular
ROM, but the opposite is true at high temperature with the difference being less than 1% SOC.
Its average error was < 5% SOC in every case.
9.3.1 Effect of Feedback
The ROM with feedback error compensation will be compared to the ROM without error
compensation and coulomb counting to determine how the SOC estimate is corrected during
116
each type of test. All plots are considering data recorded during experiments at 1C and 25?C. All
other Crates and ambient temperatures demonstrated similar trends. Figure 46 shows results
during a single constant current full discharge. The deviation is almost identical to that from the
OCV line with the maximum being 2.9% SOC.
Figure 46: Effect of feedback  single discharge
9.4 Single Cycle
The proposed SOC estimation algorithm is applied during a single full cycle, 100% to 0%
to 100% SOC, in order to evaluate its accuracy during discharging and charging. Final SOC error
results are included in Appendix 5. The ROM with feedback revealed < 2% SOC error in every
case.
117
9.4.1 Effect of Feedback
Figure 47 shows results during a single cycle. During discharge, the feedback correction
shows little change from the original ROM. During charge, SOC is greater due to correction
from terminal voltage measurements leading to a more accurate final SOC value. SOC does not
reach 0% with the ROM since reaching the Vmin of 2.5V at 1C does not indicate full depletion of
lithiumions, referring to S0, in the anode. Residual charge could be extracted if a lower current
was applied.
Figure 47: Effect of feedback  single cycle
9.5 HPPC Test
The HPPC test profile is applied for 1 hour to determine accuracy under pulse load
conditions. Average SOC errors of the ROM with feedback for various operating conditions are
included in Appendix 6. Average error is < 5% SOC in every test.
118
9.5.1 Effect of Feedback
Figure 48 shows results during one hour of HPPC testing. The ROM with feedback error
compensation has a slightly greater SOC estimate in the range of 10098.5% and slightly lower
SOC estimate in the range 9695% which leads to less average error overall.
Figure 48: Effect of feedback  HPPC
9.6 5 Cycles
Longterm tests are conducted to prove that the ROM error correction is effective.
Similar to a single cycle, only final SOC estimates of the ROM and OCV curve may be
compared. Results are included in Appendix 7. The ROM with feedback revealed < 3.5% SOC
error in every case.
119
9.6.1 Effect of Feedback
Figure 49 shows results during 5 cycles. The SOC estimates from the ROM with
feedback error compensation never reach 20% or 80% due to terminal voltage measurements
greater than or less than, respectively, the ROM terminal voltage estimates.
Figure 49: Effect of feedback  5 cycles
120
Chapter 10: Error Analysis
SOC estimates of the proposed methods are compared to classical methods during
various charge modes. An error analysis determines which method performs best under which
operating conditions.
10.1 ChargeDepleting Mode
Single discharge and HPPC test SOC estimation method errors are analyzed in this
section. Figures 5052 show the data in Tables 11 and 13 graphically. The black dashed line
represents the typical OEM SOC error estimate maximum of 5%. ECM estimates were removed
so as not to skew the plot scales since its error was notably more for each instance. Some errors
are too small to see in the plots.
Figure 50: Chargedepleting cycle analysis at 0?C
121
Figure 51: Chargedepleting cycle analysis at 25?C
Figure 52: Chargedepleting cycle analysis at 40?C
The ROM with feedback correction is the only method that achieves the SOC accuracy
target of > 95% in all situations. Average errors were the least at 25?C since the OCV curve
standard temperature is 25?C, increased with Crate due to increases in heat generation rate that
are not perfectly accounted for in the thermal component of the model, and were greater during
HPPC testing since it constitutes a pulsed current profile. The charge error rates are based on
122
CC/CV charging yet the HPPC testing is only CC charging. The proposed feedback method
proves to be the best option for the range of investigated currents and ambient temperatures.
10.2 ChargeSustaining Mode
Single full cycle and 5 partial cycles SOC estimation methods errors are analyzed in this
section. This charge mode has the disadvantage that results may not be dynamically compared to
the OCV curve since the data is not monotonically decreasing or increasing with rest periods
before and after. Figures 5355 show SOC error data for all methods after a single 1000100%
cycle and after 5 802080% cycles. Note that the yaxes are absolute values of the final SOC
errors. Same as before, some errors are too small to see in the plots.
Figure 53: Chargesustaining cycle analysis at 0?C
123
Figure 54: Chargesustaining cycle analysis at 25?C
Figure 55: Chargesustaining cycle analysis at 40?C
The ROM with error correction performs best and achieves the target of less than 5%
final SOC error for all investigated operating conditions, as do most other methods. The results
from the ROM with empirical correction have significant error suggesting that the charge error
rates chosen are not appropriate. In general, greatest accuracy is attained at 25?C with low
current for the same reasons mentioned in the previous section.
124
Chapter 11: Conclusions and Future Work
Battery state of charge estimation methods were evaluated to determine which has the
greatest accuracy under various loading conditions. The ROM proved superior to coulomb
counting and the ECM in each test when the results were compared to an OCV curve.
Improvements were made to the ROM and two novel SOC estimation techniques were evaluated.
The first method uses empirical data to tune SOC estimates. ROM error during a single
discharge was found to be linear due to small flaws in model parameters that accumulate with
each time step. Due to this linearity, error rates were calculated and applied to the ROM
estimates, drastically reducing error. Of the 36 examined cases, 3 did not meet the
aforementioned target of > 95% accuracy; however, during a single discharge or single cycle the
accuracy is impeccable. Unfortunately, a HEV or EV charge demand profile will realistically
never be constant.
The second method attempts to counteract the inherent drawbacks of assuming constant
error rates by including the ROM in a closedloop feedback system whereby model SOC
estimates are updated continuously by comparing the model terminal voltage estimate to battery
terminal voltage measurements. A variable gain is implemented to maintain the state space poles
in the desired location. Of the 36 examined cases, all met the aforementioned target of > 95%
accuracy, making this algorithm ideal for potential use in all HEVs and EVs.
Theoretically, the proposed SOC estimation method would perform well with other
advanced battery types as long as the model parameters were updated to reflect changes in
battery chemistry. All of the physicsbased equations used in the ROM remain applicable. A
125
future improvement might be to update the thermal component of model so as to better capture
internal heat generation at high Crates.
Other equivalent circuit models may be considered such as a Randle of order greater than
3 or a modified Thevenin, even though it has been stated that a 2nd order Randle model provides
the greatest efficiency [4]. Also, other electrochemical models could be assessed such as full
order single particle or porous electrode models. These offer marginal accuracy gains at the price
of excessive computational power required for calculation. Studying how each ROM model
parameter varies with temperature and age is a worthwhile venture.
Future work should include the investigation of the accuracy of this SOC estimation
algorithm outside the range of 040?C and 15C. There is little documentation on subzero
temperature as well as high Crate effects on Lipolymer batteries. Also, longer cycling times
would increase confidence in the proposed method. Being a relatively new technology, there is
much to explore.
State of health estimation is the next evolutionary phase. Tracking changes in maximum
lithium concentration would enable SOC to always maintain the range of 0100% in
electrochemical models. The ROM with feedback correction would be ideal if it could be
updated periodically to account for cell degradation and maintain accuracy in all realistic
operating conditions.
126
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133
Appendix 1: Tree Diagram of SOC Estimation Methods and Principle Authors
134
Appendix 2: Electrochemical Model Parameters
Parameter Negative electrode Separator Positive electrode unit
Design
specifications
(geometry
and volume
fractions)
Thickness, ? 50*104 25.4*104 36.4*104 cm
Particle radius, Rs 1*104 1*104 cm
Active material volume fraction, ?s 0.58 0.5
Polymer phase volume fraction, ?p 0.048 0.5 0.11
Conductive filler volume fraction, ?f 0.04 0.06
Porosity, ?e 0.332 0.5 0.33
Solid and
electrolyte
phase Li+
concentration
Maximum solid phase concentration,
cs, max 35.4*10
3 52.6*103 mol?cm3
Stoichiometry at 0% SOC: x0%, y0% 0.1279 0.9878
Stoichiometry at 100% SOC: x100%,
y100% 0.7512 0.3181
Average electrolyte concentration, ce 1.2*103 1.2*103 1.2*103 mol?cm

3
Kinetic and
transport
properties
Exchange current density
coefficient, ki0 12.9 6.28 A?cm
2
Chargetransfer coefficient, ?a, ?c 0.5, 0.5 0.5, 0.5
Solid phase Li diffusion coefficient,
Ds 2.0*10
12 3.7*1012 cm2?s1
Solid phase conductivity, ? 1 0.1 S?cm1
Electrolyte phase Li+ diffusion
coefficient, De 2.6*10
6 2.6*106 2.6*106 cm2?s1
Bruggeman?s porosity exponent, p 1.5 1.5 1.5
Electrolyte phase ionic conductivity,
?
15.8ce
exp(
13472ce1.4)
15.8ce
exp(
13472ce1.4)
S?cm1
Li+ transference number, t+0 0.363 0.363 0.363
135
Appendix 3: Experimental Test Matrix
Test Number Temperature (?C) Current (Crate) Mode Duration
1 0
1
constant 1 discharge
2
5
2 25
1
2
5
3 40
1
2
5
4 0
1
constant 1 full cycle
2
5
5 25
1
2
5
6 40
1
2
5
7 0
1
HPPC 1 hour
2
5
8 25
1
2
5
9 40
1
2
5
10 0
1
constant 5 partial cycles
2
5
11 25
1
2
5
12 40
1
2
5
136
Appendix 4: Single Discharge Average SOC Error
Current
(Crate) Tamb (?C)
Coulomb
counting
(%)
ECM
(%)
ROM
(%)
ROM with
empirical
correction
(%)
ROM with
feedback
correction
(%)
1
0 2.22 4.63 3.19 < 0.01 3.09
25 0.85 4.02 1.11 < 0.01 1.74
40 3.82 1.37 3.59 < 0.01 1.72
2
0 1.87 4.79 3.35 < 0.01 2.89
25 0.98 9.29 1.21 < 0.01 1.63
40 3.98 3.85 3.73 < 0.01 2.96
5
0 2.26 37.97 1.95 < 0.01 1.84
25 1.04 24.24 1.24 < 0.01 1.20
40 5.15 6.95 4.86 < 0.01 3.58
137
Appendix 5: Single Cycle Final SOC Error
Current
(Crate) Tamb (?C)
Coulomb
counting
(%)
ECM
(%)
ROM
(%)
ROM with
empirical
correction
(%)
ROM with
feedback
correction
(%)
1
0 1.86 0.08 0.34 0.02 1.51
25 0.14 0.54 0.01 0.02 0.24
40 0.10 0.87 0.04 0.01 0.13
2
0 2.97 0.25 0.23 0.02 1.40
25 0.03 0.53 0.05 0.01 0.26
40 0.16 0.77 0.04 0.01 0.09
5
0 4.15 0.57 0.82 0.02 1.28
25 2.31 0.66 0.11 0.01 0.24
40 1.33 0.93 0.24 0.03 0.10
138
Appendix 6: HPPC Test Average SOC Error
Current
(Crate) Tamb (?C)
Coulomb
counting
(%)
ECM
(%)
ROM
(%)
ROM with
empirical
correction
(%)
ROM with
feedback
correction
(%)
1
0 2.03 6.36 2.83 2.96 1.84
25 1.01 4.42 1.03 1.00 0.52
40 1.11 1.74 1.11 1.08 0.52
2
0 5.20 14.10 6.52 6.84 2.61
25 3.16 10.28 3.03 2.88 2.23
40 4.62 3.10 4.42 4.27 1.37
5
0 7.93 31.79 11.41 12.35 4.95
25 7.57 22.41 7.12 6.39 3.47
40 8.76 9.25 8.02 7.19 3.42
139
Appendix 7: 5 Cycle Final SOC Error
Current
(Crate) Tamb (?C)
Coulomb
counting
(%)
ECM
(%)
ROM
(%)
ROM with
empirical
correction
(%)
ROM with
feedback
correction
(%)
1
0 2.01 0.28 2.36 1.91 1.65
25 1.32 6.71 1.53 2.79 0.85
40 1.32 3.49 1.45 2.59 0.79
2
0 0.10 2.05 0.35 4.38 0.33
25 0.25 13.71 0.58 3.10 0.70
40 0.24 5.91 0.80 3.21 1.48
5
0 6.11 5.07 1.07 7.55 3.15
25 0.45 14.36 0.09 4.42 0.89
40 0.27 5.76 0.40 4.77 1.79