available-power prediction of lithium-ion batteries in electric ...

Aachener Beiträge des ISEA

Alexander Farmann

A comparative study of reduced-order equivalent circuit models for state-of- available-power prediction of lithium-ion batteries in electric vehicles

Band 124

A comparative study of reduced-order equivalent circuit models for

state-of-available-power prediction of lithium-ion batteries in

electric vehicles

Von der Fakultät für Elektrotechnik und Informationstechnik

der Rheinisch-Westfälischen Technischen Hochschule Aachen

zur Erlangung des akademischen Grades eines Doktors der

Ingenieurwissenschaften genehmigte Dissertation

vorgelegt von

Diplom-Ingenieur (FH)

Alexander Farmann, M. Sc. aus Teheran, Iran

Berichter:

Univ.-Prof. Dr. rer. nat. Dirk Uwe Sauer

Univ.-Prof. Dr. Ing. Olfa Kanoun

Tag der mündlichen Prüfung: 25. Januar 2019

Diese Dissertation ist auf den Internetseiten der Universitätsbibliothek online verfügbar.

ii

AACHENER BEITRÄGE DES ISEA Vol. 124

Editor:

Univ.-Prof. Dr. ir. h. c. Rik W. De Doncker

Director of the Institute for Power Electronics and

Electrical Drives (ISEA), RWTH Aachen University

Copyright ISEA and Alexander Farmann 2019

All rights reserved. No part of this publication may be reproduced, stored in a

retrieval system, or transmitted in any form or by any means, electronic,

mechanical, photocopying, recording, or otherwise, without prior permission of

the publisher.

ISSN 1437-675X

Institut für Stromrichtertechnik und Elektrische Antriebe (ISEA)

Jägerstr. 17/19 • 52066 Aachen • Germany

Tel: +49 (0)241 80-96920

Fax: +49 (0)241 80-92203

[email protected]

iii

Alexander Farmann

A comparative study of reduced-

order equivalent circuit models for state-of-available-power prediction of

lithium-ion batteries in electric vehicles

v

Preface When I started my diploma thesis at Forschungsstelle für Energiewirtschaft in Munich

in 2010, I was slightly sceptical at first and often honestly wondered whether I had

taken the right decision to take a specific topic dealing with battery technologies. Over

time, as I kept learning and became more familiar with the topic, I realized that one will

never stop to gain new knowledge about a battery’s behavior. However, obviously this is

an open-end story as even after nine years since those days I have to admit that there

are still many of topics to be investigated and understood that I need to investigate and

understand. During my time at ISEA, I was given the opportunity of doing research on

many aspects of battery technology/behavior, and I tried to make the best of it. This

work is the result of countless hours of my life during the time when I worked at ISEA

and afterwards.

My gratitude and appreciation go especially to professor Dirk Uwe Sauer, who made this

work possible and supervised my thesis. He always encouraged me to follow my

ambitions, he believed in me to bring this work to an end and supported me in

publishing several scientific papers in renowned journals and magazines. In addition,

special thanks go to professor Kanoun for undertaking the second examiner’s role and for the interesting discussions in advance.

I thank all my colleagues and friends at ISEA for their support during the time we

worked together. Many thanks for all the animated technical and non technical

discussions which we had together. Special thanks go to Wladislaw Waag who always

contributed his experience and technical knowledge. Performing measurements would

not have been possible without the helping hand of Holger Blanke, who continously

supported me in manufacturing and adapting the test benches with his longstanding

expertise.

I would like to express my gratitude and appreciation to my colleagues, Thomas Olbrich

and Dietmar Vogt at Continental AG, who have always offered their support and

believed in me. It is a great pleasure for me to have got to know you.

I cannot thank my family enough who made everything possible for me. My parents who

helped me through school and supporting me to come to Germany for studying.

Moreover, especial thanks goes to my grandfather and my brother who always believed

in me and supported me.

Landshut, March 2019

Alexander Farmann

vi

Abstract Lithium-ion batteries (LIBs) definitely belong to the most promising commercially

available energy storage systems (ESS) for use in electric vehicles (EVs). Their higher

specific volumetric and gravimetric energy and power density, higher cycle lifetime and

lower self-discharge rate in contrast to settled ESS (e.g., lead-acid batteries, nickel-

cadmium or nickel-metal hydride) have gained the attention of many vehicle

manufacturers, suppliers and research institutions in order to explore and improve

different LIB technologies in recent years.

Battery management systems (BMS) consisting of both hardware and hardware are

responsible for reliable and safe operation of LIBs in EVs. State-of-Charge (SoC), State-

of-Health (SoH) and State-of-Available-Power (SoAP) are the major battery states that

must be determined by means of so-called monitoring algorithms. The main focus of the

present study lies on on-board SoAP prediction of LIBs in EVs. The prediction of the

maximum power that can be applied to the battery by (dis)charging it during

acceleration, regenerative braking and gradient climbing is definitely one of the most

challenging tasks of BMS. The available battery power is limited by the safe operating

area (SOA), where SOA is defined by battery temperature, current, voltage

and SoC. Accurate SoAP prediction allows the energy management system to regulate

the power flow of the vehicle more precisely and optimize battery performance and

improve its lifetime accordingly.

In this study, LIBs at different aging states using various active materials are

investigated whereby the primary focus lies on investigating the electrical behavior of

LIBs using lithium titanium oxide, Li4Ti5O12 (LTO) anodes. In addition, other LIB

technologies such as lithium nickel cobalt manganese

oxide, Li(Ni1/3Co1/3Mn1/3)O2 (NMC) and lithium iron phosphate, LiFePO4 (LFP) are

examined. Characterization tests are performed over a wide temperature range (-

20 °C⋯+40 °C) by employing electrochemical impedance spectroscopy and current pulse

tests. Furthermore, the behavior of battery impedance parameters and open-circuit

voltage over the battery lifetime with regard to temperature, SoC is investigated

comprehensively.

The closed-loop model-based approaches using reduced order equivalent circuit models

(ECM) for battery state estimation have received increasing attention in recent scientific

publications due to their simple nature and the possibility for implementation on low-

cost embedded systems. The aforementioned techniques are often reliable and can track

the changes of impedance characteristics over the battery lifetime. However, most of the

vii

methods presented in the literature are often validated under nominal conditions using

standardized load profiles and neglect major internal and external factors, among others,

extreme temperature variation or adaptability of the applied algorithm to present

operating condition and aging state of the battery. In this study, a comparative study of

a wide range of impedance-based ECMs for on-board SoAP prediction is carried out. In

total, seven dynamic ECMs including ohmic resistance, RC-elements, ZARC elements

connected in series with a voltage source are implemented. The investigated ECMs are

verified under varying conditions (different temperatures and wide SoC range) using

real vehicle data obtained in an EV prototype and current pulse tests. Furthermore, the

current dependence of the charge transfer resistance is considered by applying the

Butler-Volmer equation. The dependence of voltage estimation and SoAP prediction

accuracy for different prediction time horizons on SoC, temperature and applied current

rate is examined.

viii

Table of contents

1 Introduction and motivation ....................................................................................... 1

2 Lithium-ion batteries and their experimental characterization ................................ 5

2.1 Fundamentals of lithium-ion batteries ...................................................................... 5

2.1.1 High performance electrode materials for lithium-ion batteries ................ 8

2.2 The lithium-ion batteries investigated in this work ................................................ 14

2.2.1 Experimental setup .................................................................................... 15

2.3 Experimental techniques commonly applied for electrical characterization of

lithium-ion batteries ................................................................................................. 15

2.3.1 Electrochemical impedance spectroscopy .................................................. 15

2.3.2 Hybrid pulse power characterization ......................................................... 19

2.4 Measurement results and discussion ....................................................................... 20

2.4.1 Characterization of the impedance characteristics ................................... 20

2.4.2 Characterization of the open-circuit voltage .............................................. 36

2.5 Validation datasets ................................................................................................... 55

2.5.1 Load profiles investigated in this work for verification reasons ............... 56

2.6 Summary ................................................................................................................... 63

3 Review of monitoring algorithms and battery models presented in the literature . 66

3.1 State-of-Charge estimation ....................................................................................... 70

3.1.1 Methods based on coulomb counting technique ......................................... 70

3.1.2 Methods based on measured or estimated OCV ........................................ 71

3.1.3 Methods based on active or passive impedance spectroscopy ................... 73

3.2 State-of-Health and capacity estimation .................................................................. 73

3.2.2 Methods based on measured or estimated OCV ........................................ 79

3.2.3 Methods based on electrochemical models................................................. 89

3.2.4 Methods based on Incremental capacity analysis and differential voltage

analysis methodologies ............................................................................... 91

3.2.5 Methods based on aging prediction models ............................................... 95

3.3 State-of-Available-Power prediction ......................................................................... 97

3.3.1 Methods based on adaptive characteristic maps ..................................... 104

3.3.2 Methods based on equivalent circuit models ........................................... 106

3.4 State-of-Safety estimation ...................................................................................... 118

3.5 Lithium-ion battery models .................................................................................... 121

3.5.1 Classification of battery models ............................................................... 121

3.5.2 Requirements on battery models employed in monitoring algorithms ... 132

3.6 Summary ................................................................................................................. 132

ix

4 The equivalent circuit models investigated in this work ....................................... 134

4.1 Equivalent circuit model using an ohmic resistance (Model 1) ............................. 136

4.2 Equivalent circuit models using an ohmic resistance connected in series with a

finite number of RC-elements (Models (2-4)) ......................................................... 136

4.3 Equivalent circuit models using an ohmic resistance connected in series with a

finite number of ZARC-elements (Models (5-7)) ..................................................... 139

4.4 Summary ................................................................................................................. 142

5 Applied method for battery state estimation ......................................................... 143

5.1 Method for impedance parameters estimation ....................................................... 143

5.1.1 A selection of simulation results for impedance parameters estimation 147

5.1.2 Sensitivity analysis of impedance parameters and voltage estimation

accuracy during an extreme temperature variation .............................................. 148

5.2 Method for battery State-of-Available-Power prediction ....................................... 150

5.2.1 Basic idea of the applied technique for SoAP prediction ......................... 151

5.2.2 Results and discussion ............................................................................. 153

5.3 On-board open-circuit voltage estimation .............................................................. 168

5.4 Summary ................................................................................................................. 169

6 Summary and outlook ............................................................................................. 170

Bibliography ..................................................................................................................... 174

List of abbreviations ........................................................................................................ 195

List of symbols .................................................................................................................. 199

Appendices ....................................................................................................................... 205

Appendix A: Measured OCVs of NMC/LTO and NMC/C-based Lithium-ion batteries at

different aging states for various temperatures ..................................................... 205

Appendix B: Step-wise illustration of on-board capacity estimation of lithium-ion

batteries .................................................................................................................. 207

Appendix C: Charge and discharge curves behavior over the battery lifetime .............. 208

Appendix D: Static charge and discharge power capability of the battery ..................... 209

Appendix E: Comparison of model performances for Cell-D-new and Cell-A-aged using

the examined driving cycles .................................................................................... 210

Appendix F: Impact of prediction time horizon on SoAP prediction accuracy for different

applied current rates and State-of-Charges using the examined driving cycles ... 211

List of publications ........................................................................................................... 213

Conference and magazine contributions ................................................................ 213

x

Peer-Reviewed journal contributions ..................................................................... 213

Introduction and motivation 1

1 Introduction and motivation

Upcoming and existing regulations offered on national or international scale in order to

reduce the vehicle local emissions present new technical and economic challenges to the

automotive industry. Partial or full electrification of the vehicle powertrain is one of the

certain unfailing measures to reach particular targets in the aforementioned sector [1].

In this regard, alternative energy storage systems (ESS) as a fundamental part of the

vehicle powertrain play a key role in achieving the desired strategic objectives. According

to already adopted regulations in the European Union (EU), the fleet average local CO2

emissions of passenger cars must be limited to 95 g/km from 2021 [2]. In order to achieve

the aforementioned target, hybridization of the vehicle powertrain in mid-term, is a

preferred solution of the vehicle manufactures where the aforementioned target can be

fulfilled with a moderate cost increase. However, in long-term (i.e., very probably from

2025), there are ambitions to further limit vehicle local CO2 emissions to

75 g/km. Thereafter, full electrification of the vehicle powertrain becomes increasingly

important. Fig 1 illustrates a selection of important requirements on ESS for use in

electric vehicles (EVs) [3].

Fig. 1: A selection of requirements on energy storage systems for use in electric vehicles


Lithium-ion batteries (LIBs) definitely belong to the most promising commercially

available ESS for use in automotive applications. The higher specific volumetric and

gravimetric energy and power density, lower weight, higher cycle lifetime and lower self-

discharge rate of LIBs [4] in comparison to settled ESS (e.g., lead-acid batteries, nickel-

cadmium or nickel-metal hydride) have gained the attention of many vehicle

manufacturers, suppliers and research institutions in order to explore and improve

different LIB technologies in recent years. Beside some technical and infrastructural

challenges, the main barrier for market entry of the EVs still remains their price that is

in turn highly dependent on battery price. Thereafter, EVs can only be successful in

achieving a commercial breakthrough if their price decreases significantly and end user

acceptance is high enough. Following the noticeable drop in LIB prices in recent years

and the forecast for continuing trend in coming years mainly as a result of increasing

production volume (e.g., in Ref. [5]), almost nothing stands in the way of successful EVs

market entry.

Nevertheless, there are still some drawbacks associated with LIBs which need to be

addressed. First, up to now the aging mechanisms of LIBs are not sufficiently

understood [6]. Second, on the one hand their lack of safety at high temperatures or in a

crash case and on the other hand their limited performance at very low temperatures

(e.g., especially challenging for hybrid electric vehicles (HEVs) with regard to cold

cranking capability) belong to the main concerns of the vehicle manufacturers [7].

Consequently, certain targeted improvements have to be made in terms of safety,

developing innovative electrode and electrolyte materials yielding higher lifetime, cycle

stability and more efficient power and capacity performance even at low temperatures

[8].

In large lithium-ion battery packs several hundred LIBs are connected in series and/or

in parallel to fulfill specific electrical and thermal requirements. The performance of

LIBs and their operation in EVs are controlled and diagnosed by means of so-called

battery management systems (BMS) consisting of both software and hardware. In other

words, among the main tasks of BMS are battery state estimation, protection against

battery over/under-voltage, over-current, over-temperature, under-temperature etc. The

main battery states of interests are often State-of-Charge (SoC), State-of-Health (SoH)

and State-of-Available-Power (SoAP). Since the aforementioned battery states are not

measureable on-board, the so-called monitoring algorithms must be used.


Various internal and external factors such as temperature distribution inside battery

pack, change of battery impedance characteristics (i.e., mainly because of battery aging

or temperature change etc.), cell-to-cell variations either in an initial state or over the

battery lifetime resulting from different cell impedances or capacities, influence the

battery behavior with regard to energy content and/or power capability. Therefore, using

robust monitoring algorithms working precisely over the battery lifetime is necessary. In

ideal case, monitoring algorithms employed for battery state estimation must work

reliable over the battery lifetime. Furthermore, a precise estimation of battery states

should be ensured in order to satisfy end consumers with regard to e.g., power and/or

energy management strategy or safety aspects etc. During the early stage of

development phase of monitoring algorithms, an especial emphasis should be put on low

computational effort, low model parameterization effort, high model fidelity over a wide

SoC and temperature range and so far possible consideration of physical equivalence.

As will be discussed later in this work, closed-loop model-based approaches using

reduced order equivalent circuit models (ECM) for battery state estimation have received

increasing attention in recent scientific publications due mainly to their simple nature

and the possibility for implementation on low-cost embedded systems. The

aforementioned techniques are often reliable and can track the changes of impedance

characteristics over the battery lifetime. However, most of the methods presented in the

literature are often validated under nominal conditions using standardized load profiles

and neglect major internal and external factors, among others, extreme temperature

variation or adaptability of the applied algorithm to present operating condition and

aging state of the battery. One of the main objectives of this work is to investigate the

influence of the employed ECMs on voltage estimation and SoAP prediction accuracy

under varying conditions. The applied algorithm is validated using real vehicle data

obtained in an EV prototype and pulse tests performed under varying conditions. In

total, seven different impedance based ECMs are implemented and dependence of

voltage estimation and SoAP prediction accuracy on temperature, SoC and applied

current rate is examined comprehensively. Moreover, dependence of SoAP prediction

accuracy on various prediction time horizons often of interest in EVs is examined.

LIBs using various active materials are commercially available on the market for

automotive and stationary applications. The primary focus of this work lies on

investigating the electrical behavior of LIBs at different aging states using lithium

titanium oxide, Li4Ti5O12 (LTO) anodes. In addition, other LIB technologies such as

lithium nickel cobalt manganese oxide, Li(Ni1/3Co1/3Mn1/3)O2 (NMC) or lithium iron


phosphate, LiFePO4 (LFP) are examined. The measurements are performed over a wide

temperature range (− °C⋯ °C). It is worth noting that the investigated ECMs and

the applied estimator can be employed independent of battery types/chemistries.

The main objectives of this work can be summarized as follows:

Discussion and analysis of various internal and external factors influencing the

impedance characteristics of the battery,

Discussion and analysis of various internal and external factors influencing the

open-circuit voltage (OCV) behavior of the battery,

A comprehensive review of available techniques for battery states/parameters

estimation (e.g., SoC, SoH and actual battery capacity, SoAP etc.) and various

classes of battery models presented in the literature,

Discussion and analysis of requirements on verification load profiles for battery

monitoring algorithms and detailed description of the ECMs investigated in this

work,

Analysis of dependence of voltage estimation and SoAP prediction accuracy of the

investigated ECMs on temperature, SoC and applied current rate.

In chapter 2, fundamentals and principal functionality of LIBs are discussed and a brief

overview of active materials commonly used in for LIBs is provided. Moreover, the

results of extensive experimental impedance and OCV measurements performed at

different aging states of the battery and over a wide temperature and SoC range are

shown and the findings are described profoundly.

A review of available techniques for on-board battery state/parameters estimation, in

particular methods for on-board SoC, actual battery capacity, SoH, State-of-Safety (SoS)

estimation and SoAP prediction is provided in chapter 3.

Chapter 4 addresses the ECMs investigated in this work and physical meaning of

impedance parameters employed in the respective ECMs is discussed comprehensively.

The applied estimator for on-board impedance parameters estimation and SoAP

prediction of the battery is described in chapter 5. In addition, the impact of battery

temperature, SoC and applied current rate on voltage estimation and SoAP prediction

accuracy is discussed. Finally, this work is concluded in chapter 6 with a summary and

discussion of the presented results.

Lithium-ion batteries and their experimental characterization 5

2 Lithium-ion batteries and their experimental characterization

In this chapter, principle functionality of LIBs and various commercially available LIB

technologies including a brief description of commonly used anode and cathode materials

are discussed profoundly. Moreover, the LIBs investigated in this work and their

technical specifications are presented in subsection 2.2. Experimental techniques

commonly used for electrical characterization of LIBs are discussed in subsection 2.3.

Subsection 2.4 presents a selection of the measurement results obtained from battery

impedance characterization tests and OCV tests performed on the investigated LIBs at

different temperatures and aging states. Moreover, in subsection 2.5, the datasets used

for validating the applied algorithm are shown and important requirements on

verification datasets for battery monitoring algorithms are discussed.

2.1 Fundamentals of lithium-ion batteries

The first primary LIB using lithium in the anode and manganese oxide in the cathode

was commercialized by the company Sanyo in 1972. After permanent improvements of

used active materials and cell structure in order to achieve for example higher voltage

level, the first commercially available secondary LIB was introduced to the market by

Sony in 1991. The particular LIB used amorphous carbon anode and lithium cobalt

dioxide, LiCoO2 (LCO) cathode [9].

Although the LIBs mainly differ in their characteristics and the used active materials

(materials used in the cathodes or the anodes), the basic functionality of the LIBs

remains always the same. A LIB consists of two electrodes that are separated by an

electronically insulating but ionically conducting separator. Using separator often made

of porous polymer membrane is necessary in order to prevent a short-circuit in a LIB.

Between both electrodes, an ion conductive electrolyte often containing lithium salts (i.e.,

LiPF6) is used where electrolyte is usually mixed with carbonates (i.e., ethylen carbonate

or dimethyl carbonate). It is worth noting that alternative electrolyte materials based on

gel, polymeric or glassy matrices are currently under development [10]. The anode of LIB

in its most conventional structure is made of graphite while the cathode is made of metal

oxides (e.g., LCO, lithium manganese oxide, LiMn2O4 (LMO) etc.). A more detailed

description of commercially available anode and cathode materials will be provided later

in following subsection. The current collector of the negative electrode often consists of

copper and that of the positive electrode consists of aluminum because of its passivation


properties. Fig. 2 shows a typical structure of a secondary LIB using graphite anode and

metal oxide cathode.

Fig. 2: Schematic illustration of a common lithium-ion battery using graphite anode and metal oxide cathode

The chemical equation representing the intercalation and deintercalation phenomenon of

lithium-ions in electrode materials in an example of a LIB using lithium metal oxide

(LiMO2, e.g., LiCoO2) cathode and graphite anode is as follows (Eq. (1)):

yC+LiMO2↔LixCy + Li(1-x)MO2 (1)

where x ≅ 0.5 and y=6 for V= 3.7 V.

LIBs operate according to the so-called “rocking chair” principle. During battery operation (i.e., charging or discharging process), the lithium-ions (Li+) are transferred

from one electrode through the electrolyte and the separator to the other electrode and

then stored in active material. In particular, during discharge Li+ move from the anode

to the cathode while electrons flow from the anode through the external circuit to the

cathode producing electrical power. The aforementioned processes are reversed when the

LIB is charged [9]. The LIBs commercially available for automotive applications can be

classified into following three formats [11]-[12]:

Cylindrical,

Pouch,


Prismatic.

The electrical and mechanical design of a lithium-ion battery pack including cell

selection is significantly influenced by type of respective EV (i.e., HEV or battery electric

vehicle (BEV)). Accordingly, an unsophisticated decision could have an enormous impact

on different factors such as power/energy requirements, costs, packaging or over/under-

sizing of lithium-ion battery pack.

The cells integrated into a lithium-ion battery pack should be protected against the

intrusion of liquids by adequate measures. Thereafter, the housing of a lithium-ion cell is

often based on flexible pouch foils or robust metal housings [13]. The three LIB formats

mentioned above can be found in different EVs currently released in series on the

market. Unfortunately, vehicle manufacturers worldwide have not been agreeing on a

standardized LIB format for EVs independent of vehicle and local market1 up to now.

The latter fact could have huge impact on cost minimizing in long-term since with

increasing sales numbers of a particular LIB format that is used in a wide range of EVs,

the cost of production significantly decrease.

The cylindrical LIBs with more than 3 billion production figures a year mainly for

consumer applications such as laptops are widespread and actually belong to the most

produced LIBs. Along with prismatic LIBs these both cell formats are based on robust

metal housing where at least one predetermined breaking point inside the cell or over-

pressure valve is implemented. The aforementioned over-pressure valve becomes active

at pressures about 10 to 15 bars. The main difference between the mentioned LIB

formats is their different heat dissipation characteristics whereby prismatic LIB format

is often the preferred choice because of its larger external surface2.

The next format of LIBs often used in EVs is called pouch format or so-called stacked

electrode design. The housing of the aforementioned LIBs is based on multi-layer film

composites with an aluminum core. As current collectors metallic foils (commonly copper

for negative electrode and aluminum for positive electrode) are used. The housing of

pouch cells deflects noticeably under gas evolution occurring inside the LIB and, thus no

over-pressure valve is implemented in such LIBs [13]. In addition to the above

mentioned safety devices, the LIBs are often equipped with additional safety devices

such as melting fuse, shutdown separators or positive temperature coefficient contact

disc [12], [14].

1 The German vehicle manufacturers defined DIN SPEC 91252:2011-01 (D) in 2011 in order to support the

standardization process at least in Germany. 2 It is assumed that the chemistry and the energy content of LIBs are the same.


In Table 1, a general comparison of automotive specified LIB-formats available on the

market is given. It is worth noting that the given values and advantages and

disadvantages are a generalization and could differ between individual cell manufacture

etc.

Table 1: Lithium-ion battery cell formats commonly used in automotive applications (adopted from Refs. [12]-[13])

Cell format Dimension

Format 1

Dimension

Format 2

Dimension

Format 3 Advantages Disadvantages

Cylindrical 18650 26650 217003

Low cost and

considerable

experience

(e.g.,

production)

Cooling and

packaging

Pouch

(LxHxT

[mm3])

- 121x243xT 330x162xT

Higher

energy

density

because of

low case

weight and

well cooling

characteristi

c; cheap;

simple

production

Low

mechanical

stability; low

cost

Prismatic

(LxHxT

[mm3])

173x115x

32 (or 45)

120x85x

12.5

V1: 175x85x21

V2:

148x91x26.5V

3:

148x125x26.5

Efficient

thermal

management

; packaging;

Mechanical

stability

High

volumetric

energy density

on battery

pack level

2.1.1 High performance electrode materials for lithium-ion batteries

In the following two subsections, a brief overview of cathode and anode materials

commonly used in LIBs for automotive application is given. It is worth mentioning that

the purpose of the aforementioned subsections is not discussing individual active

material in great detail as it is out of the scope of this work. Interested readers are

referred to the review articles given below.

Advanced cathode materials are essential to meet specific requirements regarding e.g.,

voltage level and durability of LIBs. In general, it can be stated that layered transition

3 This cell format is still under development and will in all probability be used in diverse mobile applications such as EVs

or pedelecs etc.


metal oxides based on α-NaFeO2, spinel and olivine structures are the most used cathode

materials in commercially available LIBs [15]. Among the main challenges for present

and future cathode material structures are the stabilization of crystal structure during

lithiation and delithiation and prevention of reaction with electrolyte. In addition, anodic

stability of electrolyte must be ensured since it directly influences the highest useable

potential for charging and the maximum releasable capacity from the cathode,

respectively [8]. A wide overview of available cathode materials and recent advances and

developments can be found for example in Refs. [8], [15]. A selection of the state-of-the-

art cathode materials commonly used in LIBs is listed in Table 2.

Table 2: A selection of advanced cathode materials commonly used for LIBs adopted from Refs. [8]-[11], [15]

Cathode materials

Cathode

capacity

(mAh ∙g− )

Structure

Voltage

vs.

Li|Li+

Advantages Disadvantages

LiCoO2 140-155 α-NeFeO2 3.9

High stability,

good rate

capability

Expensive;

Availability of

cobalt

LiNiO2 160-170 α-NeFeO2 3.8

Cheaper than

LiCoO2, High

current rate

capability,

higher energy

density then

LiCoO2

Expensive;

Less stable

and less

ordered than

LiCoO2,

Li(Ni1-x-y,Mnx,Coy)O2 140-180 α-NeFeO2 ≈3.8

High capacity,

operation at

high voltages

Safety

Li(Ni0.8,Co0.15,Al0.05)O2 190 α-NeFeO2 3.6

High capacity

and high power

capability,

higher

structure

stability

Safety

LiMn2O4 100-120 Spinel

structure 4.0

Lower cost and

safer than

LiCoO2, very

fast cathode

material

because of the

possibility for

multi-

directional

diffusion of

lithium-ions

Lower

capacity than

cathodes

based on

α-NeFeO2

structure,

occurrence of

phase changes

during cycling

V2O5 and

LiV3O8 ≈300

Layered

compounds 3.0

Higher capacity

than above

Low voltage

and low


compounds energy

density,

respectively

LiMnPO4 150

Olivine

structure

4.1

Higher redox

potential and

higher energy

density,

respectively

than LiFePO4

Lower

capacity and

lower rate

capability

than LiFePO4

LiFePO4 ≈ 3.45

Safe, Excellent

rate capability

and high

capacity

Low electronic

conduction,

low voltage

LiCoPO4 <135 4.8

Higher redox

potential and

higher energy

density,

respectively

than LiFePO4

Lower

capacity and

lower rate

capability

than LiFePO4

LiNiPO4 160-170 5.2

Higher redox

potential and

higher energy

density,

respectively

than LiFePO4

Moderate

ionic and

electronic

conductivity

Sulfur 1675 - ≈2.5 Higher capacity

and low cost

High volume

expansion

during

discharging

and the

possibility of

occurrence of

parasitic

reactions

Graphite with a capacity of approx. 350 – 372 mAh∙g-1 and a flat operating voltage range

between 0.05 - 0.2 V vs. Li|Li+ is definitely the most used anode material in

commercially available LIBs [8]. The low working potential vs. lithium of graphite yields

high voltage and high energy content, respectively. Among further advantages of

graphite are its low cost and promising cycle life. However, at the same time, the main

disadvantage of graphite anode is that at high discharge current rates, graphite anode

may be polarized and, thus highly reactive dendritic lithium can be formed on surface of

the electrode. At the same time, lithium metal as anode material with a theoretical

specific capacity of approx. 3800 mAh∙g-1 and a potential of approx. 0 V vs. Li|Li+

indicate the highest anode materials capacity and full cell voltage in combination with

wide range of cathode materials [16]. However, mainly because of some safety issues

such as instability in organic solvents or formation of dendrites on lithium metal surface


during charge/discharge processes, an internal short-circuit can occur. Therefore, the use

of lithium metal as anode material in LIBs is limited at least in mobile applications [17].

In general, anode materials can be classified in following three groups [16]:

Insertion/de-insertion materials such as carbon-based, porous carbon, carbon

nanotubes, graphene etc.,

Alloy/de-alloy materials or so-called group IV elements such as Si, Ge, SnO2 etc.,

Transition metal oxides (e.g., MnxOy, NiO), metal sulphides, metal nitrides etc.

In fact, it can be stated that carbon-based anode materials (hard or soft carbon) belong to

the most used anode materials in LIBs mainly because of their thermal stability, low

cost and availability. However, among their main disadvantages are their low coulomb

efficiency and high voltage hysteresis [16]. Moreover, the carbon-based anode materials

suffer from solid electrolyte interface (SEI) formation.

LIBs using LTO anodes as a kind of zero strain insertion material [16] in combination

with various cathode materials such as LMO, LFP or NMC definitely belong to the most

promising ESS in various aspects [18]. During full lithiation the LTO anode reaches a

theoretical capacity of 175 – 330 mAh∙g-1 [16]. Moreover, because of the two-phase

lithium insertion/extraction process in the electrode a flat voltage of approx. 1.55 V vs.

Li|Li+ appears while its spinel symmetry structure remains almost unaltered [19]-[20].

All of this means that LIBs using LTO anodes show a reduction in operating cell voltage,

electronic conductivity and theoretical capacity yielding less overall energy density but

high power density while the opposite is valid for graphite-based LIBs [16]. For HEVs

where high power and cycling performance in a wide temperature range are required,

LIBs using LTO anodes are proved with their excellent power performance.

Moreover, the LTO anode may not suffer from SEI and dendrite formation mainly

because of its high potential level [8]. Therefore, for LIBs using LTO anodes, the SEI

phenomenon is mainly referred to the cathode [21]. Among the further advantages of

LIBs using LTO anodes are their cyclability even at low temperatures, without the

occurrence of lithium plating [22] and exfoliation of active materials, and their promising

thermal stability at higher temperatures [16]. A selection of state-of-the-art carbon and

non-carbon-based anode materials commonly used in LIBs is given in Table 3.


Table 3: A selection of advanced anode materials commonly used for LIBs adopted from Refs. [8], [16], [23]

Anode materials

Anodes

capacity

(mAh ∙ g− )

Potential

vs. Li|Li+ Advantages Disadvantages

Carbon nanotubes 1100

< . Low cost and

safety

Low columbic

efficiency, high

voltage hysteresis

and high irreversible

capacity

Hard carbon 200-600

Graphene 780/11164

SiO >1600 0.6 High capacity, good

cycling stability, low

cost and high power

capability

High volume change ≈ %, Low

columbic efficiency

and high capacity

fade

Silicon (Si) 4200 <0.5

Germanium (Ge) ≈1600 0.12

Tin oxide(SnO2) 783 0.6

Transition metal

oxides and metal

phosphides/sulfides

500 - 2000 0.8 - 1.6

High (specific)

capacity, high

energy, low cost and

low operation

potential

Low coulomb

efficiency, high

voltage hysteresis,

short cycle life and

high production costs

Titanium oxides 175 - 330 ≈1.5

Safety, good cycle

lifetime and high

power capability,

excellent low

temperature

performance

Low capacity and low

energy density

Based on the capacity values listed in

Table 3, it can be stated that Si with

approx. 4200 mAh∙g-1 indicates the highest capacity among the discussed anode

materials. The high capacity value is mainly related to formation of intermetallic

lithium-silicon compounds. At the same time, Si is the second most abundant element in

the earth’s crust and, thus, inexpensive and environmental friendly [8].

Summarized, it can be concluded that the material combination (i.e., between the anode

and the cathode materials) should be performed considering safety aspects,

possible/required operating voltage range and high power and capacity capability. In this

regard, a selection of the active materials discussed above by giving the respective

potential vs. Li/Li+ is illustrated in Fig. 3.

4 These values can be achieved when a number of graphene sheets are considered together [16].


Fig. 3: Potential ranges of different active materials used in lithium-ion batteries adopted from Refs. [11], [13]

In order to configure a LIB that theoretically indicates the highest possible voltage,

active materials must be selected where the difference between their potentials (anode

and cathode potentials) is the highest. At the same time, capacity of the electrodes (

or ) need to be considered. The specific energy (E) of a LIB increases with increasing

capacity of electrodes and cell voltage; thereafter, the E can be determined as follows

(Eq. (2)): = + (2)

Even if the housing of LIB is not involved in the electrochemical reaction occurring in

LIB, the cell case can have an enormous impact on LIBs power and energy density,

current carrying capacity and durability. In order to fullfill the increasing power and

energy requirements on future ESS, improvement of existing and development of novel

active materials and electrolyte becomes essential. The energy density of LIBs can be

increased by either using high voltage cathode materials or high capacity anodes and

cathodes with promising cycle lifetime [13]. At the same time, developing novel

electrolyte materials such as LiFAP or fluorinated solvents that e.g., indicate more

thermal stability at high operating voltage become important. The theoretical specific

energy density of LIBs using the above discussed active materials is approx. 400 Wh∙kg-1.

When additional factors such as electrolyte, cell case, current collectors etc. are

considered, theoretical specific energy density decreases by more than 50% [13].


Consequently, from a single LIB to a large lithium-ion battery pack by considering

additional components, a maximum theoretical specific energy density of 8 – Wh ∙kg− may be reached.

2.2 The lithium-ion batteries investigated in this work

In this work, LIBs at different aging states using various active materials are

investigated. An overview of the LIBs investigated in this work and a brief description of

their previous history is given in Table 4.

Table 4: Overview of lithium-ion batteries investigated in this work

Cell name

Cell chemistry

(cathode/

anode)

Nominal

capacity

[Ah]

Actual

capacity

[Ah]

Aging history

Cell-A-new NMC/LTO 15 16.8 New cell

Cell-A-aged NMC/LTO 15 13.7

Aged battery cell after

approximately 1500 full cycles

(100% DoD) at 35 °C

Cell-B-new NMC/LTO 20 20 New cell

Cell-C-new NMC/LTO 15 16 New cell

Cell-C-aged NMC/LTO 15 12.3


approximately 3 months calendric

aging stored at 40 °C and at 2.5 V

Cell-D-new NMC/LTO 15 15.7 New cell

Cell-E-new NMC/LTO 15 16.6 New cell

Cell-E-aged NMC/LTO 15 12.4



aging stored at 40 °C and at 2.5 V

Cell-F-new NMC/C 40 40.3 New cell

Cell-F-aged NMC/C 40 35.9



aging stored at 60 °C

Cell-G-new NMC/C 40 42 New cell

Cell-H-new NMC/C 25 25 New cell

Cell-H-aged NMC/C 25 21.2

Aged battery cell after approx. 5

months calendric aging stored at

60 °C

Cell-I-new LFP/C 3.6 3.6 New cell


Cell-J-new LFP/C 8.2 8 New cell

Cell-J-aged LFP/C 8 7.2 Aged battery cell after approx. 500

full cycles at 30 °C

2.2.1 Experimental setup

The experimental measurements are carried out using test bench consisting of MK53

temperature chamber manufactured by the company BINDER GmbH in Tuttlingen,

Germany with a temperature range between -40 °C and +180 °C and Digatron BTS-600

manufactured by the company Digatron Power Electronics GmbH in Aachen, Germany.

Electrochemical impedance spectroscopy (EIS) measurements are performed in a

frequency range of kz mz. Within this work, the galvanostatic EIS

measurement technique is applied using an experimental setup (Digatron EISmeter)

developed at ISEA of RWTH Aachen and manufactured by the company Digatron Power

Electronics GmbH in Aachen, Germany. In Table 5, the specification of the used test

bench is given.

Table 5: Characteristics of the measuring test bench

Characteristic Maximum measurement

inaccuracy in a full scale

Available

measurement

range

Sampling rate

Voltage

measurement ±6 mV or ±0.1% from the end value [0…6] V

100 ms Current

measurement

±200 mA or ±0.1% from the end

value [-200…+200] A

2.3 Experimental techniques commonly applied for electrical characterization of lithium-ion batteries

EIS measurement and hybrid pulse power characterization (HPPC) techniques are often

used for electrical characterization of batteries. As a main difference between both

measurement techniques, the experimental setup and employed excitation signals may

be addressed. In the following two subsections, a brief review on basic theory of EIS

measurements and HPPC tests for parameterizing the battery models is discussed.

2.3.1 Electrochemical impedance spectroscopy

EIS is often used as a promising non-invasive technique for parameter identification and

investigation of a battery’s dynamic behavior. Generally, two techniques can be used for

EIS measurement. The first technique, so-called galvanostatic EIS measurement,

applies a small ac excitation (sinusoidal) current at various frequencies and the voltage

response is measured. The second measurement technique, a so-called potentiostatic EIS


measurement, applies a sinusoidal voltage to the LIB as an excitation signal and the

resulting current is measured. Performing EIS measurement allows deep insight into

electrochemical processes occurring inside the cell over a wide frequency range. This

yields a very simple and accurate parameter and process identification from very low

(some µHz) up to high frequencies (some kHz and even more) [24]. The complete

impedance spectrum can be obtained determining the complex impedance for each

frequency in the investigated frequency range using current and responded voltage as

follows (Eq. (3)): = (3)

Commonly, the impedance spectrum is shown in a so-called Nyquist diagram by plotting

the negative imaginary part as a function of the real part. The measured complex

impedance spectrum can be subdivided into five sections referring to various

electrochemical processes occurring in the battery (Fig. 4) [25]:

Section 1, corresponding to the inductivity of the LIB at high frequencies,

Section 2, corresponding to the ohmic resistance (R0),

Section 3, corresponding to SEI,

Section 4, corresponding to charge transfer processes,

Section 5, corresponding to diffusion, migration and remaining convection

processes.

It is worth noting that the ECM5 shown in Fig. 4 is not investigated in the later sections

of the present work and has only representative character of a possible sophisticated

battery modeling considering the above mentioned processes.

5 Constant phase element (CPE) corresponds to the generalized capacitive element forming a straight line in a complex

plane. The impedance of a CPE-element is determined as follows: C E = . where 0 < ξ < 1 and A refers to

generalized capacitance, ξ is a depression factor and is the frequency range [4].


Fig. 4: Typical Nyquist plot of LTO/NMC lithium-ion battery measured at -20 °C and 50% State-of-Charge, frequency range: 5 kHz – 1 mHz

At very high frequencies (i.e., above approx. 1000 Hz), the impedance spectrum (section 1

in Fig. 4) is mainly dominated by the inductive behavior of the battery caused by wires

and windings. However, L is often neglected in BMS since it does not contain valuable

information about the battery’s electrochemical state for monitoring algorithms. Moreover, it is only measureable when very high currents are applied to the battery

(e.g., several hundred amperes) or when the measuring cables are long.

Section 2 in Fig. 4 is used to identify the R0 of the battery. When the impedance

spectrum crosses the real axis (i.e., m = ), the corresponding point on the real

axis refers to R0 [26]. Fig. 4 illustrates the dependence of R0 of the investigated LIBs

(Cell-A-new, Cell-A-aged, Cell-B-new and Cell-G-new) on temperature and SoC.

According to the results it can be concluded that the impact of battery temperature on R0

in contrast to the impact of the SoC is more pronounced. The latter fact is mainly

referred to high resistive behavior of electrolyte differing at various temperatures.

Moreover, it may be stated that R0 increases slightly with decreasing SoC. In Ref. [26],

the authors show that dependence of R0 on SoC for NMC/C LIBs is negligible and the R0

remains almost constant over the SoC range. For Cell-A-new, Cell-A-aged, it is obvious

that the increase of the R0 over the battery lifetime is negligible and, thus the obtained

curves almost overlap.


Fig. 5: Dependence of ohmic resistance on State-of-Charge and temperature (relative to the values measured at 23 °C and 50% State-of-Charge)

The first small high frequent semi-circle (section 3 in Fig. 4) is attributed to SEI in the

anode or in the cathode [21]. However, it is worth mentioning that the SEI formation in

the anode for LIBs using graphite is generally more pronounced than the one in the

cathode [16].

Charge transfer processes are investigated considering the second large semi-circle of

the impedance spectrum (section 4 in Fig. 4). The time constant (τ) of the second semi-circle, which is an important indicator for the dynamic behavior of battery, is obtained

investigating the maximum frequency, when the negative imaginary part of the

impedance spectrum reaches its local minimum (section 3 and section 4 in Fig. 4) as

follows (Eq. (4)):

= / ∙ (4)


Based on the derived charge transfer resistance (Rct) and τ, the value of the double layer capacitance (Cdl) can be determined. Due to the slower reactions and respectively higher

impedance values, both above mentioned semi-circles can be clearly distinguished at low

temperatures or when the LIB is aged. At higher temperatures and especially for a LIB

in new state, the time constants of both processes are very similar which leads to

inseparable semi-circles indicating very high charge transfer kinetics [27]. According to

the authors in Ref. [28], the SEI-related semi-circle appears clearly at low temperatures

because of the following three reasons: First, the frequency range of the impedance of the

inductive component overlaps with that for SEI. Second, the time constants of anode and

cathode are almost identical and third, the value of the SEI-resistance is negligible.

Solid state diffusion in the active material at very low frequencies (frequencies below

approx. 1.5 Hz) can be investigated by exact exploration of section 5 in Fig. 4. This part

of Nyquist plot shows a -45° slope and is often described by means of the Warburg

impedance ( ) or a ZARC-element [29]. The latter two elements will be investigated in

great detail in later sections of this work. In Ref. [4], the authors investigate the aging

behavior of NMC/C LIBs over a wide SoC range by means of EIS measurement in a

frequency range of 5 kHz f 0.2 Hz. The impedance parameters of the employed ECM

including two ZARC-elements and have been determined through a fitting procedure

of the measured spectrum. Moreover, the increase of R0 over the battery lifetime is then

used for SoH estimation where no clear interdependence between Rct, τ and battery’s actual aging state for investigated LIBs could be identified.

2.3.2 Hybrid pulse power characterization

The second experimental approach often used to examine the impedance characteristic

and available battery power is the so-called HPPC test. Within this measurement

technique, current pulses with various amplitudes for charging and discharging cases

with durations of few seconds (e.g., 10 s or 20 s) are applied to the battery. Different

parameters and processes can then be identified by investigating the corresponding part

of the battery voltage response. As long as the battery is under load, its voltage differs

from the measured C and can be written as follows (Eq. (5)):

− C = Δ +Δ E , +Δ E , +Δ , +Δ , +Δ (5)

Accordingly, the difference between the measured voltage and the C is equal to the

sum of the following overvoltages, each referring to a specific physical phenomenon:


Ohmic overvoltage6,

SEI-related overvoltages on both electrodes,

Charge transfer overvoltages on both electrodes,

Remaining concentration-related overvoltages including diffusion, migration and

convection processes.

In the first moment directly after the beginning of the current pulse (approx. 50 ms), R0

can be determined (employing ohmic law) by observing the measured voltage drop after

discharging or voltage rise after charging. The voltage drop/rise because of R0 is mainly

caused by the resistive behavior of the battery’s electrolyte, poles and current collectors.

Kinetic processes and current dependence of internal resistance can be identified by

employing the Butler-Volmer equation (BVE). For this reason, the sum of Rct and R0,

referring to the d.c. resistance (Rdc) during the current pulse is determined as a voltage

value reached after 2 s from the beginning of the current pulse divided by the current.

Diffusion overvoltage and corresponding impedance parameters can be obtained

exploring the voltage response in a range of a few seconds (e.g., ∆t >= 5 s [30]). According

to Ref. [30], the diffusion effect becomes more pronounced during discharging than

during charging of the LIB.

2.4 Measurement results and discussion

In this subsection, the measurement procedures applied for battery impedance

characterization and OCV investigation are described. Moreover, a selection of

measurement results obtained from EIS and HPPC tests for electrical characterization

of the battery are presented. The OCV behavior of the investigated LIBs is investigated

over a wide temperature range and the obtained results and the findings are discussed

in great details.

2.4.1 Characterization of the impedance characteristics

In total, 28 characteristic measurement points (various temperatures and SoCs) have

been examined. Before each test is performed, LIBs are tempered to 23 °C for 3 h and

then charged to 100% SoC with 1C current rate (i.e., 15 A for Cell-A-new and Cell-A-aged

LIBs) and 2.75 V by applying the constant current-constant voltage (CC-CV) charging

strategy. The charging process is interrupted as soon as the applied current rate

6 Strictly speaking ohmic overvoltage (Δ ) is not an overvoltage, since it is not caused by electrochemical reactions in

the LIB. However, in this study for simplicity reasons it is included under the term battery overvoltage.


decreases below 0.01C. For EIS measurement the following test procedure is performed

for each investigated temperature point:

1. Tempering the LIB in the temperature chamber to desired temperature for at

least 3 h.

2. Performing EIS measurements at 100% SoC.

3. Pause for 15 min.

4. Discharging to the next SoCF

7 level with 0.5C.

5. Pause for 2 h.

6. Performing EIS measurements for the given SoC.

7. Repeating steps 3 to 6 for each SoC (90%, 70%, 50%, 30%, 20%, 10%).

HPPC tests with various current rate amplitudes and pulse duration of 20 s are

performed to examine the current dependence of Rdc and Cdl. For this reason, the

following current rates are employed in sequence (±0.2C, ±0.5C, ±0.7C, ±1C, ±1.5C, ±2C,

±3C, ±5C8). To ensure the consistency of each SoC step, the amount of discharged and

charged Ah during current pulses is then equalized by charging/discharging during the

so-called equalization step. In the present work, the defined measurement procedure for

HPPC tests differs slightly with regard to the current pulse duration and the relaxation

time between each applied pulses from other specifications reported for example in Ref.

[31]. In Ref. [32], the authors only investigate the battery voltage response behavior after

discharging and neglect the charging case completely. Moreover, the authors investigate

battery polarization effects after a very long current pulse duration (i.e., 720 s) that

refers mainly to solid electrode diffusion processes.

For HPPC tests the following test procedure is applied to each battery beginning from a

fully charged state for each examined temperature:

1. Tempering the LIB in the temperature chamber to desired temperature for at least

3h.

2. HPPC test:

7 In this subsection, the SoC is related to the battery’s nominal capacity (Cnom). 8 ± C is not applied to every investigated LIB at different temperatures.


2.1. Charging the LIB for 20 s with a predefined current rate9

2.2. Pause for 15 min in order to tempering the cell

2.3. Discharging the LIB for 20 s with a predefined current rate

3. Pause for 15 min.

4. Discharging to the next SoC level with 0.5C.

5. Pause for 2 h.

6. Performing the HPPC test as described in step 2.

7. Repeating steps 4 to 6 for each SoC level (90%, 70%, 50%, 30%, 20%, 10%).

In fact longer pause phases between two pulses as implemented in our measurement

procedure yield more accurate results from thermal balance point of view. Impedance

characteristics of ESS, in particular LIBs, depend mainly on the following factors [26]:

SoC,

Temperature,

Actual aging state of the battery.

Some works in the literature show results of extensive accelerated calendar and cycling

aging tests at various temperatures and Depth-of-Discharge (DoD) discussing the change

of impedance parameters over the battery lifetime and how this information can be used

for predicting the battery’s remaining useful lifetime (RUL) [33]. A detailed review on

available techniques for actual capacity and RUL estimation will be discussed later on in

chapter 3.

In Ref. [35], the authors show results of experimental investigations over a wide

temperature range using LIBs in a new state with lithium nickel cobalt aluminum oxide,

LiNi0.8Co0.15Al0.05O2 (NCA) cathodes and graphite anodes. High dependence of R0 on

temperature is shown while dependence on SoC is less pronounced. In addition, an

increase of the Rct with decreasing SoC is observed. Furthermore, the low temperature

power capability of LIBs using LCO cathodes and graphite anodes is investigated by

performing three electrode measurements and the simulation of the ECM in Ref. [36].

The main contribution of Rct and its increase with decreasing temperatures is noted as a

major source of cell polarization at low temperatures.

9 At 100% SoC only discharging current pulses are applied to the LIB. Otherwise by applying charging current pulses, the

predefined upper voltage limit would be hit.


A cycle life analysis of LIBs using NMC cathodes and LTO anodes is performed in Ref.

[34]. The authors report a higher capacity fade due to active material loss in the cathode

than the degradation occurred in the anode. For degradation analysis, the incremental

capacity analysis (ICA) technique is consulted. In Ref. [27], the same results are

obtained by investigating diverse LIBs using different cathodes (e.g., LMO and LCO

cathodes) and LTO anodes. According to the authors, for the cycle-aged LIBs, the loss of

cyclable lithium and the loss of positive electrode active material are identified as the

main aging contributor. Results indicate minor loss of negative electrode active material

and negligible aging for calendar-aged LIBs.

In Ref. [21], impedance measurement results of LIBs using LCO cathodes and LTO

anodes in a new state for various temperatures and DoDs are shown. An ECM employing

passive elements (resistances, capacitances, inductance etc.) representing the cathode

and the anode is used for parameterization of the LIBs. The main focus lies on

understanding the capacity fade mechanism on both electrodes. The results indicate that

the increase of the Rct and the capacity fade of the cathode are major degradation

mechanisms.

However, most of the researches published in the literature explore the impedance

characteristic of LIBs using graphite anodes. Up to now, few works have investigated the

aging mechanism and especially the behavior of impedance characteristics of the LIBs

using LTO anodes under different conditions. In the following, LIBs using LTO anodes

(Cell-A-new, Cell-A-aged, Cell-C-new, Cell-C-aged) at different aging states are

characterized in order to determine the influence of SoC, temperature and current rate

on impedance characteristics and battery power capability, respectively. Kinetic

processes described by the BVE with regard to the current dependence of Rdc and the

behavior of BVE parameters, in particular exchange current density ( ) and symmetry

factor (), are analyzed. A full evaluation of the Butler-Volmer behavior of the

investigated LIBs is performed over a wide temperature and SoC range at different

aging states.

In Fig. 6 and Fig. 7, selected impedance spectra of Cell-A-new and Cell-A-aged, measured

at 23 °C and -20 C are presented. While in the impedance spectrum measured at 23 °C

(Fig. 6) only one semi-circle (mainly caused by charge transfer processes as described in

subsection 2.3.1) is observable, in the impedance spectrum measured at -20 °C (Fig. 7) a

second semi-circle can be distinguished, resulting from the described SEI phenomenon.


Fig. 6: a) Impedance spectra of Cell-A-new and b) Cell-A-aged at 23 °C and various State-of-Charge (Please note the different scale of the graphs)

Fig. 7: a) Impedance spectra of Cell-A-new and b) Cell-A-aged at -20 °C and various State-of-Charge (Please note the different scale of the graphs)

In the following subsections, the impedance parameters extracted based on the

measurement procedure described beforehand and their dependence on temperature,

SoC and actual aging state of the battery are discussed.

Ohmic resistance 2.4.1.1

As discussed in subsection 2.3, the R0 of LIBs is extracted at high frequencies in the

range from approx. 500 Hz up to 900 Hz. In particular, for automotive applications,

where battery power capability in a time horizon of 1-20 s is often of interest (battery

voltage response including lower frequencies), the knowledge of R0 is not sufficient. As a

consequence, it is necessary that R0 is considered along with Rct and the Diffusion


resistance (Rd). Fig. 8 presents the behavior of the R0 measured at different SoCs and

temperatures for Cell-A-new and Cell-A-aged.

Fig. 8: Ohmic resistance of Cell-A-new and Cell-A-aged determined as described in subsection 2.3.1: a) dependence on temperature b) dependence on State-of-Charge

As shown in Fig. 8, by decreasing the battery’s SoC, R0 increases slightly which is mainly

due to the change of ionic conductivity of LTO during discharging [37]. The shape of the

dependence of the R0 on the SoC remains almost unchanged over the battery lifetime.

The results obtained from slightly aged LIB cycled up to 500 cycles approve this

assumption. As expected, the impact of temperature on R0 in comparison to impact of the

SoC is more pronounced10, which is mainly due to the high contribution of electrolyte at

different temperatures [38].

Charge transfer resistance 2.4.1.2

Rct corresponds mainly to charge transfer processes of lithium-ions between the

electrolyte and the electrodes. The Rct is obtained from the impedance spectrum when R0

is subtracted from the measured Rd (Eq. (6)), while Rd corresponds to begin of transport

10 The values of the R0 for Cell-A-aged measured at -20 °C and at SoCs < 50% are missing: In combination with advanced

aging state of the battery and capacity fade, this leads to missing SoC points at lower SoC range.


limitations in solid and liquid phases at approx. 1 Hz (see Fig. 4), which is highly

temperature dependent [26]. = − (6)

R0 or a combination of R0 with Rct and Rd is often used as an input for SoH and RUL

estimation. According to the authors in Ref. [39], Rd values (determined in a frequency

range between 0.1 Hz and 1 Hz) increase significantly over the battery lifetime. Within

the present work, the Rd values for Cell-A-new are obtained in a frequency range

between 1 Hz and 1.6 Hz and for Cell-A-aged between 1.5 Hz and 4 Hz at 23 °C. The

same tendency is also observed for other temperatures, where the respective frequency

range decreases with decreasing temperature.

Due to kinetic limitations of the processes and according to the Arrhenius law, the high

dependence of Rct on temperature is expected and actually exists as shown in

Fig. 9a. According to Ref. [40], the impact of temperature on the negative electrode

(LTO) is less pronounced compared to the positive electrode. At the same time the

dependence of Rct on SoC is surprisingly less pronounced. An increase of Rct (approx.

15%) with decreasing SoC (SoC < 50%) is detected for aged LIBs (Fig. 9b). According to

the report published in Ref. [21], the main contributor for increasing the Rct with

decreasing SoC is the cathode. However, this effect might also be referred to the change

of electronic conductivity of the LTO during discharge.

Furthermore, the slight change of Rct over the battery lifetime is approved. This fact

might be related to the absence of SEI growth in the anode. In Ref. [27], the authors

have performed cycle and calendar life tests on both full and half cells using LTO anode

material. Based on the obtained results it may be concluded that the positive electrode is

the main impedance contributor, especially for cycle-aged cells. The increase of the

impedance in the negative electrode is noticeably small so that the impedance increase

over the battery lifetime can mainly be related to the positive electrode.


Fig. 9: Charge transfer resistance determined as described in subsection 2.3.1 for Cell-A-new and Cell-A-aged. a) dependence on temperature (linear representation) b) dependence on State-of-Charge11

The high temperature dependence of Rct can be investigated more detailed, when

resistance values are plotted logarithmically. By decreasing the temperature by 10 °C

the resistance increases by a factor in the range of 1.6 - 2.2. A so-called Arrhenius plot is

presented in Fig. 10, using the extracted values of Rct for Cell-A-new and Cell-A-aged at

100%, 50% and 10% SoC. This plot illustrates the relation between Rct and temperature

according to the Arrhenius equation (Eq. (7)):

= · −∙ (7)

where A is the proportionality constant in [Ω-1], T is the temperature in Kelvin, R is the

gas constant (=8.314 J ∙ mol ∙ K− ) and Ea refers to the activation energy [41]. Higher

activation energy means higher temperature dependence and higher required lithiation

energy than delithiation energy [42]. The activation energy of lithium-ion transfer

determined for Cell-A-new is approx. 48±0.8 kJ ∙ mol− and for Cell-A-aged

is approx. 44±0.8 kJ ∙ mol− which is almost consistent with the data reported in Ref. [43].

11 An unexpected behavior of Rct for Cell-A-aged at -20 °C and 90% SoC is detected. At 90% SoC, the value of the Cell-A-

aged decreases below that of Cell-A-new, which could not be expected from a theoretical point of view and is most likely

related to inaccurate measurement.


Moreover, it can be stated that the activation energy remains almost constant over the

investigated SoC range.

Fig. 10: Arrhenius plot for charge transfer resistance of Cell-A-new and Cell-A-aged at various State-of-Charges

In Ref. [44], the authors have determined an activation energy of 44 - 49 kJ ∙ mol− for

LIBs using LTO anode and LCO cathode materials by employing resistance values at

1 Hz corresponding to lithium-ion transfer at electrode-electrolyte interface. In Ref. [26]

almost the same results have been achieved for LIBs using NMC cathodes and graphite

anodes. In Ref. [21], the authors show activation energy values of 58 kJ ∙ mol− for the

cathode (LCO) and 29 kJ ∙ mol− for the anode (LTO) for LIBs in a new state. In Ref. [40],

an activation energy of 41 kJ ∙ mol− for LIBs using NCA cathode and graphite anode

materials has been determined. Moreover, an activation energy of 40 kJ ∙ mol− for LIBs

using LFP cathodes is derived in Ref. [42]. By comparing the results obtained in this

work and the data reported in the literature, the derived values for activation energy

seem to be more related to the activation energy of the cathode.

Time constant and double layer capacitance 2.4.1.3

Time constant of the battery is often employed as an indicator for investigating the

dynamic behavior of the battery. The higher the τ the slower is the voltage response of

the battery. τ changes over the battery lifetime and is highly temperature dependent

[45]. As mentioned in subsection 2.3.1, within this work τ is determined by investigating

the maximum frequency of the impedance spectrum at the point where the local


minimum of the negative imaginary axis is reached. Fig. 11 shows the dependence of τ

on temperature for Cell-A-new and Cell-A-aged at selected SoCs in linear and

logarithmical form. τ increases by a factor of approx. 2.5 with decreasing temperature by

10 K. For both LIBs τ reaches its minimum in the middle of the SoC range for almost all

investigated temperatures. Furthermore, it may be stated that τ decreases over the

battery lifetime which is actually in agreement with results presented in Ref. [26].

Fig. 11: Dependence of the time constant on temperature at various State-of-Charges for Cell-A-new and Cell-A-aged: a) linear representation and b) semi-logarithmical representation

When τ is known, Cdl can be simply derived dividing τ by Rct (see subsection 2.3.1).

Unfortunately, in the literature the dependence of τ or Cdl on SoC or temperature has

found little attention up to now, which makes a clear comparison between the results

obtained in this work with others difficult. According to the obtained results, the Cdl

increases by a factor of 1 - 1.5 with decreasing temperature by 10 K and changes slightly

over the battery lifetime, which is in agreement with the results reported in Ref. [21].

According to the authors in the latter reference, the obtained values for Cdl are mainly

dominated by the capacitance of the cathode and are obviously one order lower than the

capacitance of the LTO anode. The dependence of Cdl on SoC is less pronounced and no


clear tendency is observed. In Ref. [46], the authors show an increase of Cdl with the

decrease of temperature and over the battery lifetime for LIBs using NMC cathodes and

graphite anodes. Moreover, the authors state that for LFP/C LIBs the Cdl decreases with

decreasing temperature and over the battery lifetime. In Ref. [37], the same tendency of

Cdl behavior with regard to temperature and actual aging state for LFP/C LIBs is shown

which is in agreement with results obtained in the present work.

According to Ref. [47], the Cdl of lead-acid batteries shows high current dependence,

whereas up to now its existence for LIBs has not been investigated extensively. The

current dependence of Cdl can be simply determined using HPPC tests as follows (Eq.

(8)): = ∆∆ (8)

where ∆ refers to the amount of charge throughput during current pulse and ∆ refers

to the voltage difference between measured battery voltage at the end of the current

pulse and the voltage before applying current pulse. Fig. 12 illustrates the current

dependence of Cdl of Cell-C-new and Cell-C-aged for different temperatures. The

measurements are performed at 50% SoC.

Fig. 12: Current dependence of the double layer capacitance determined as described in Eq. (8) for Cell-C-new and Cell-C-aged at different temperatures and 50% State-of-Charge

Based the results shown in Fig. 12, it may be stated that the current dependence of Cdl

might be neglected for the aforementioned LIBs, since the applied current rate does not

significantly influence the Cdl behavior.


Total battery resistance 2.4.1.4

In applications where the available battery power plays a major role the knowledge of

impedance values in very low frequency ranges is important. In this work the measured

total battery resistance (R0.001Hz) at 1 mHz (lowest measured frequency) is used to

investigate the impact of SoC and temperature on R0.001Hz for Cell-A-new and Cell-A-

aged. R0.001Hz can also be interpreted as a sum of all resistances obtained from the

impedance spectrum. Fig. 13 shows the behavior of R0.001Hz over the battery lifetime at

different SoCs and temperatures. R0.001Hz does not show any significant SoC dependence

but is highly temperature dependent. The R0.001Hz increases by a factor of 2 - 2.5 with

decreasing temperature by 10 K. Presented results are in agreement with the results

shown in Ref. [48] with regard to the impact of SoC on R0.001Hz. In Ref. [49], the authors

present the same results according to the behavior of R0.001Hz (in that case, resistances at

0.2 Hz) over the battery lifetime. However, the authors demonstrate an increase of

R0.001Hz with decreasing SoC.

Fig. 13: Total resistance determined as described in subsection 2.3.1 for Cell-A-new and Cell-A-aged: a) dependence on temperature b) dependence on State-of-Charge


Current dependence of the direct current resistance 2.4.1.5

HPPC tests with various charging and discharging current rates are performed on

Cell-C-new and Cell-C-aged to investigate the current dependence of the Rdc. The Rdc is

derived by examining the battery voltage drop after 2 s by employing ohmic law.

A nonlinear dynamic ECM (Fig. 14) including three impedance parameters and C is

sufficient to reproduce a nonlinear characteristic of the battery for applications such as

BMS, where computing power is limited. The employed ECM considers the nonlinearity

of Rct, which results in an additional parameter the so-called kI. As will be discussed in

great detail later in chapter 4, the latter parameter depends on SoC, actual aging state

and temperature of the battery where the impact of temperature is significantly higher

than SoC. Generally, in such simplified ECMs, battery electrodes are represented by

simple RC-elements. However, for BMS applications, it is not significantly relevant

whether the implemented RC-element refers to the cathode or anode, since monitoring

algorithms are not able to distinguish between this additional information. Thereafter,

the implemented RC-element represented by a parallel connection of Rct and Cdl

corresponds mainly to charge transfer processes occurring mainly on the interfacial

between electrode and electrolyte. This yields the share of the current through Cdl, the

so-called capacitive current (Ic) and faradaic current through Rct.

Fig. 14: Nonlinear first order lithium-ion battery model commonly employed in battery management systems

Based on the ECM shown in Fig. 14, the relation between the measured battery voltage

and the overvoltages caused by the current flow through the LIB can be described as

follows (Eqs. (9)-(10)): − C = Δ + Δ (9) − C = Δ (10)


where is the measured battery voltage, C ≈ OCV is the battery open-circuit

voltage, Δ is the voltage drop/rise due to R0 and Δ corresponds to the voltage drop

related to the charge transfer processes. Since the battery voltage drop after 2 s is

assumed to be equal to the sum of Δ and Δ , the Eq. (10) can simply be derived from

Eq. (9).

The BVE is often used to describe the kinetic processes occurring at the interface

between electrode and electrolyte: If charge transfer reactions are the only limiting

processes (which means that other processes are neglected), the concentration of

reactants might be assumed to be constant. This yields the equal rate constants of

forward and backward reactions which is concluded in a factor as presented in Eq. (11)

[50]. = ∙ exp ∙ ∙∙ ∙ Δ − exp − ∙ ∙∙ ∙ Δ (11)

where factor denotes the exchange current density, which depends on the

concentration of the reactants and is therefore often employed for the characterization of

equilibrium state, n is the number of electrons involved in the reaction (n = 1 for LIBs), T

is the absolute battery temperature in Kelvin, F is the Faraday constant, R is the

universal gas constant, and is the symmetry factor of anodic and cathodic reactions in

the electrodes. The latter parameter shows how symmetrical the dependence of both

reactions on electrode’s potential is [41].

Fig. 15 depicts Rdc determined at different temperatures and 50% SoC for Cell-C-new

and Cell-C-aged by applying the HPPC technique as described in subsection 2.3.2. As

expected, Rdc increases with decreasing temperature, where its current dependence is

interpreted by BVE. However, the current dependence is negligible for Cell-C-new for

temperatures above 0 °C. Rdc increases when the LIB is aged and its current dependence

becomes more pronounced even at temperatures above 0 °C. However, a typical Butler-

Volmer behavior shown for example in Ref. [26] is not observed which could be an

indicator for the low impact of current on Rdc for the LIBs investigated in this work.


Fig. 15: Direct current resistance for various temperatures at 50% State-of-Charge (a: Cell-C-new and b: Cell-C-aged)

The current dependence of Rdc for 0 °C at different SoCs is illustrated in Fig. 16. A strong

current dependence for all investigated SoCs can be observed. What is remarkable is the

fact that Rdc increases with decreasing discharging current at SoC of 100% but remains

almost constant at all other SoCs for discharging currents. Furthermore, it may be

pointed out that obviously the Rdc for charging case is lower than the discharging case for

the same current value. This may be advantageous for applications where the battery

has to accept a high amount of power in a short period of time, for example during

regenerative braking phases in EVs.

Fig. 16: Direct current resistance for various State-of-Charges at 0 °C (a: Cell-C-new and b: Cell-C-aged)

However, in most of the presented researches the current dependence of Rdc is neglected

and it is assumed to be linear. In order to fit the respective parameters of the BVE to the

measured values, a simple nonlinear least-square (LS) technique is employed. In Fig. 17,

based on a relation between applied current and Δ , fitting results for both parameters


of BVE, namely symmetry factors ( , assuming + = and

are illustrated.

The following conclusions may be derived from the results shown in Fig. 17:

The high influence of battery temperature on , and is observed.

The influence of SoC in comparison to the impact of temperature is less

pronounced for both parameters.

The , referring to the anodic reaction (Li-deintercalation) of the cathode

increases by a factor of 1 - 1.3 with decreasing temperature by 10 K, which

denotes a larger anodic reaction than cathodic reaction in the cathode, especially

at low temperatures.

The symmetry factors ( and ) change slightly over the battery

lifetime. However, unfortunately, no clear tendency can be observed.

Higher battery temperature yields higher values for . increases by a factor of

1.1 - 2 with increasing temperature by 10 K. The slope of this increase decreases

over the battery lifetime.

As expected, decreases over the battery lifetime and it can be concluded that

decreases with decreasing SoC.


Fig. 17: a, b: Dependence of and on temperature for selected SoCs for Cell-C-new and Cell-C-aged; c: dependence of on temperature for selected SoCs for Cell-C-new and Cell-C-aged – Please notice the different scale of the graphs

In addition to the above discussed results, measured charge and discharge curves of the

aforementioned NMC/LTO-based LIBs can be found in appendix C.

2.4.2 Characterization of the open-circuit voltage

In modern monitoring algorithms, accurate estimation of battery SoC and SoAP is often

coupled with the knowledge of nonlinear monotonic SoC-OCV correlation [51]. When the

battery is in an equilibrium state and all the side reactions are finished, the nonlinear

monotonic correlation between the OCV or strictly speaking, the electro motive force

(EMF12) and the battery SoC can be used. The OCV is always a function of SoC,

OCV = f(SoC). In fact, as a consequence of some external or internal issues such as

temperature or actual aging state of the battery, the SoC-OCV correlation may change

under varying conditions [52] and long relaxation time resides in a range of some hours

12 The term EMF refers to the battery voltage under open-circuit condition when the thermodynamic equilibrium potential

of the main and side reactions is reached. In this study, the OCV is employed as an equivalent to the EMF.


is necessary until the equilibrium state is reached [53]. A detailed description of methods

applied for on-board OCV estimation will be discussed later in great detail in chapter 4.

From electrochemical point of view, the OCV of a full cell when no current flows and the

electrode potentials are in equilibrium state, can be determined as follows (Eq. (12)): OCV = − (12)

The OCV of each electrode depends on temperature and concentration of lithium-ions in

the electrode that is normalized by the SoC of the respective electrode [54]. The OCV of

LIBs depends mainly on the following factors:

Applied current rate before current interruption [53], [55]-[56],

Battery temperature [53], [55]-[57],

Previous history [55]-[56],

Cell-to-cell variations [53], [57],

Actual aging state of the battery etc. [53], [57]-[58].

Deep understanding of the impact of the above mentioned factors on relaxation behavior

of the battery is of great interest as it can make a significant contribution to improving

the SoC estimation accuracy. Prior researches on the above mentioned factors and theirs

impact on OCV of LIBs are limited in amount and scope. Up to now, most of the

researches published in the literature have explored the OCV behavior of LIBs either

under nominal condition or when the battery was in a new state. Unfortunately, the

impact of actual aging state or temperature of the battery is often neglected or assumed

to be negligible. In Refs. [59]-[60], the authors investigate the OCV behavior of LIBs in a

new state using LFP cathodes whereby special emphasis is put on investigating the OCV

hysteresis of the aforementioned LIBs; in this regard a hysteresis model for BMS

applications is presented and verified. Moreover, in Ref. [55], the OCV behavior of LIBs

using various active materials in new state is investigated comprehensively. The authors

present measurement results with respect to determined OCV gradient, OCV hysteresis

and OCV behavior of the LIBs at different temperatures. However, in the latter

reference the impact of battery aging is not considered further. In Ref. [61], the authors

propose an accurate normalized OCV modeling technique considering the impact of

aging and temperature of the battery. According to the authors, the SoC-OCV correlation

remains unchanged over the battery lifetime for various temperatures.


The main contribution of the present subsection is to discuss the importance of precise

battery OCV characterization procedure and examine various factors influencing the

battery OCV behavior of LIBs. Moreover, special emphasis is put on understanding the

impact of temperature and aging on battery’s relaxation behavior. The qualitative discussion should give the readers an overview and possible explanation of the OCV

behavior of the investigated LIBs and potentially clarify the origin of the found

dependences. The OCV measurements are performed over a wide temperature range

(-20 °C⋯0 °C⋯+10 °C⋯+23 °C⋯+45 °C). The OCV characterization measurements are

carried out using the measuring test bench presented in Table 5. A careful selection of

experimental setup used for OCV measurements with regard to voltage measurement

accuracy and defining a standardized measurement procedure is essential before the

OCV measurements are started. Unfortunately, available test manuals (e.g., Ref. [62]),

do not provide a comprehensive and consistent test procedure to assess OCV of the

batteries. It is worth mentioning that by setting experimental setup for OCV

measurements, it is necessary to keep the temperature of the battery during

measurement procedure constant since it could influence the OCV behavior, as will be

examined later.

Within this work, the following test procedure is applied to each LIB for each

characteristic measurement point:

1. Tempering the battery cell in the temperature chamber to 23 °C for 3 h.

2. Charging the battery cell to 100% SoC by applying the CC-CV charging strategy

with 1C current rate until upper cut-off voltage is reached and the current

decreases below 0.01C.

3. Tempering the battery in the temperature chamber to desired temperature

(-20 °C⋯ +45 °C) for 8 h.

4. Discharging to the next SoC13 level (5% increment) with 0.5C followed by interim

relaxation time of ∆t=8 h.

5. Repeating step 4 for each SoC until the battery voltage reaches the lower cut-off

voltage limit. The OCV values obtained after the relaxation process build the

OCV external discharge boundary.

13 In this subsection, SoC is related to the battery’s actual capacity (Cactual) unless otherwise noticed. The Cactual is

measured at 23 °C with 1C discharge current rate immediately after performing OCV test. It is worth noting that for

SoC-OCV modeling, the battery’s actual capacity must be employed since the SoC-OCV correlation/curve

differs/changes for different capacity values employed for SoC determination.


6. Repeating steps 4 and 5 for charging case until the battery voltage reaches the

upper cut-off voltage limit. The OCV values obtained after the relaxation process

build the OCV external charge boundary.

Open-circuit voltage behavior of lithium-ion batteries under nominal condition 2.4.2.1

The short-time previous history of a battery has an enormous impact on measured

battery OCV. Depending on whether the LIB is charged or discharged, the possible

equilibrium potentials differ at the same SoC, respectively. In particular, the OCV after

discharging is lower than the OCV after charging at the same SoC under the same

operating condition. The difference between the OCV values after charging and

discharging process is called hysteresis. Thereafter, the existence of OCV hysteresis

implies that alone the knowledge of battery OCV without considering the previous

history (i.e., charge/discharge history) is not sufficient for an accurate SoC estimation.

From electrochemical point of view, the hysteresis phenomenon is known to result from

mechanical stress, microscopic misconception in active materials and different

thermodynamic equilibrium potentials at same SoC [61]. The depletion of lithium ions in

pores can be described by the path dependent shrinking-core model [63]. Moreover, it is

known that the direction, amplitude and duration of the applied current before current

interruption influence the OCV hysteresis [61]. In addition, the lithium intercalation in

graphite anode can be assumed as a further source to OCV hysteresis [64]. The

hysteresis phenomenon has been reported for a wide range of active materials such as

LCO, LiMnO2 and LFP [55], [65]. However, olivine-based LFP cathodes are known to

exhibit the maximum hysteresis in contrast to other LIB chemistries [59], [65].

In Fig. 18, the results obtained from OCV measurements performed on NMC/LTO-based

LIBs (Cell-B-new and Cell-E-new, Cell-E-aged) are illustrated. The measurements are

performed at 23 °C. In particular, measured OCV curves, OCV hysteresis and OCV

gradients determined based on the OCV rate of variation per % SoC are examined

profoundly. According to the results, it can be stated that the OCV of the Cell-E-new

changes significantly over the battery lifetime. Moreover, it is observed that OCV

hysteresis of Cell-E-aged is lower than that of Cell-E-new over a wide SoC range (i.e.,

20% - 80%). The OCV hysteresis of Cell-E-aged becomes higher than OCV hysteresis of

Cell-E-new in a SoC range above 90% SoC. A local maximum of OCV hysteresis of

approx. 15 mV at approx. 55% SoC and a local minimum of approx. 0 mV are detected in

a SoC range of approx. 65% - 75%. In contrast to Cell-E-new, Cell-E-aged, the OCV


hysteresis of Cell-B-new is almost negligible and becomes more pronounced with

decreasing temperature as will be discussed later in subsection 2.4.2.2.

Furthermore, the behavior of the OCV gradients for the aforementioned LIBs is

examined. In this regard, it can be stated that the OCV gradient of Cell-B-new is

significantly higher than the OCV gradients of Cell-E-new and Cell-E-aged. For

Cell-B-new, a local maximum of OCV rate of change per % SoC after discharging of

approx.10 mV/% is detected in a SoC range above 90% and a local minimum of approx.

2 mV/% is observed at 65% SoC. The local maximum of OCV rate of change per % SoC

for Cell-E-new is approx. 5 mV and for Cell-E-aged is approx. 4.5 mV whereby the local

maxima are detected in a SoC range of 50% - 60%. For the latter two LIBs, two local

minima are observed in a SoC range of approx. 25% - 35% and 75% - 85%. Hence, if the

measurement accuracy of test bench is 5 mV, then the SoC error obtained through OCV-

based estimation technique would be approx. 0.5% for Cell-B-new and 1% for Cell-E-new

and Cell-E-aged.

Fig. 18: OCV curves, OCV hysteresis and OCV variation per % SoC of Cell-E-new, Cell-E-aged, and Cell-B-new measured at 23 °C

Fig. 19 shows the results obtained from OCV measurements performed at 23 °C on

NMC/C-based LIBs (Cell-F-new, Cell-F-aged, Cell-H-new and Cell-H-aged). The OCV


hysteresis of Cell-F-new and Cell-F-aged, reaches the local maximum of approx. 30 mV

in a SoC range of 25% - 35%. At the same time, the corresponding OCV hysteresis of the

aforementioned LIBs decreases significantly with increasing SoC. The OCV hysteresis of

Cell-F-new and Cell-F-aged remains lower than 10 mV in a SoC range above 70%.

Summarized, it can be stated that OCV hysteresis of the aforementioned LIBs increases

gradually with decreasing SoC up to approx. 40% SoC and decreases again. Almost the

same tendency can be approved for Cell-H-new and Cell-H-aged. Moreover, the OCV

hysteresis of Cell-F-new decreases over the battery lifetime for a wide SoC range (i.e.,

SoCs below 70%) while OCV hysteresis of Cell-H-new remains almost unchanged over

the battery lifetime. However, in contrast to Cell-F-new and Cell-F-aged, lower OCV

hysteresis peaks are observed for Cell-H-new and Cell-H-aged.

The corresponding OCV rate of change per % SoC of Cell-F-new and Cell-F-aged

increases up to approx. 15 mV/% at both ends of SoCs (i.e., SoC>90% and SoC<10%) and

reaches its local minimum of approx. 5 mV/% at approx. 40% SoC. For Cell-H-new and

Cell-H-aged the OCV rate of change per % SoC with some slight exceptions is lower than

Cell-F-new and Cell-F-aged. The local minimum of approx. 2.5 mV/% is detected at 40%

SoC. The OCV gradient of Cell-H-new becomes more pronounced than the OCV gradient

of Cell-H-aged in a SoC range below 25%. The shape of the OCV gradients after charging

and discharging remains almost unchanged where the local peaks of OCV gradients of

Cell-F-aged and Cell-H-aged are shifted about 5% towards high SoCs. Thereafter, it can

be concluded that local maximum of OCV gradient peak is shifted slightly as the LIB

ages. According to Ref. [66], the two peaks (plateaus) observed in the OCV curves of the

LIBs using the same chemistry are mainly referred to staging phenomena occurring in

the anodes. The staging phenomenon occurs because of transitions between two-phase

regions in the anode.


Fig. 19: OCV curves, OCV hysteresis and OCV variation per % SoC of Cell-F-new, Cell-F-aged, Cell-H-new and Cell-H-aged measured at 23 °C

Fig. 20 shows the results obtained from OCV measurements performed at 23 °C on LFP-

based LIBs (Cell-I-new, Cell-J-new and Cell-J-aged). As expected, the measured OCV-

curves of the LFP-based LIBs are significantly smoother than the OCVs of the NMC/C-

based LIBs discussed beforehand. Based on the obtained results for Cell-J-new and

Cell-J-aged, it can be stated that their OCV curves change significantly over the battery

lifetime as the OCV curves of Cell-A-new and Cell-A-aged do. The change of the OCV

curve at high SoC range (i.e., voltage above 3.3 V) is mainly related to loss of lithium

inventory that actually cannot be used for main conversion reaction [60]. In contrast to

the aforementioned NMC/C-based LIBs, the existing plateau of the Cell-J-new at high

SoC range disappears over the battery lifetime whereas the large plateau behavior in the

middle SoC range of Cell-J-aged is shifted to high SoC range. The latter fact could be

referred to an increase of battery impedance over the battery lifetime.

The OCV hysteresis of Cell-I-new shows two local peaks; the first local maximum of OCV

hysteresis reached at 30% SoC is approx. 40 mV and the second hysteresis peak of

26 mV is detected at 65% SoC. However, for Cell-J-new and Cell-J-aged, it seems that


the OCV hysteresis does not change remarkably over the battery lifetime; a slight

difference is observed in a SoC range above 70%. For the latter two LIBs, the OCV

hysteresis reaches its local maximum of approx. 20 mV at 20% SoC. The OCV gradient of

the investigated LIBs increases significantly at very low SoC range (i.e., below 20% SoC).

The maximum corresponding OCV rate of change per % SoC reaches approx. 15 mV/% in

the mentioned SoC range whereas for remaining SoC range (i.e., above 20% SoC) the

OCV rate change per % SoC remains below 5 mV/%.

Fig. 20: OCV curves, OCV hysteresis and OCV variation per % SoC of Cell-I-new, Cell-J-new and Cell-J-aged measured at 23 °C

In total, it can be concluded that the OCV gradient of Cell-J-new changes significantly

over the battery lifetime in a SoC range above 50% SoC. An OCV rate of change per %

SoC of approx. 3 mV/% means that when voltage measurement accuracy of test bench is

5 mV, the SoC error that is obtained through OCV estimation technique for Cell-J-new

and Cell-J-aged is approx. 1.6% which is three time higher than the expected error for

the investigated LTO-based LIBs.

The monitoring algorithms presented in the literature often neglect the OCV hysteresis

and employs the mean value of the OCV after charging and discharging instead. For

LFP-based LIBs because of the mentioned flat OCV curve over a wide SoC range, a

difference of only few millivolts in measured OCV value yields significant error in SoC


estimation. However, for the investigated NMC/C-based LIBs (Cell-F-new, Cell-F-aged,

Cell-H-new and Cell-H-aged), a separate consideration of charge and discharge OCV

curves seems not to be necessary since its influence on SoC estimation accuracy is

negligible. According to Ref. [60], the slope of the OCV curves of the investigated LFP-

based LIBs in flat SoC range is approx. 0.1 mV/% ∆SoC and in steep SoC range is

1.5 mV/%∆SoC that is quite similar to the results achieved in this work. However, the

authors state that the examined LIBs show a maximum OCV hysteresis of 50 mV at

23 °C, which is quite higher than the results obtained in this work. Moreover, in Ref.

[67], the authors show results of OCV measurements performed on LIBs in new state

using various active materials (NMC/C, NMC/LTO, LMO/C and LFP/C). According to the

results, the maximum SoC rate of change per mV voltage is approx. 3% for the examined

LFP/C LIBs. Thereafter, when voltage measurement accuracy is 5 mV, the resulting SoC

estimation error is approx. 13%. Therefore, the voltage measurement accuracy of the test

bench should be at least 1 mV. However, to the best knowledge of the author, at the

present the measurement accuracy of commercially available battery test benches can

only reach approx. 5 mV.

Open-circuit voltage behavior of lithium-ion batteries at different temperatures 2.4.2.2

The OCV behavior of the examined LIBs under nominal condition has been discussed in

the previous subsection. The battery temperature plays a major role regarding the OCV

and the SoC estimation accuracy, respectively. Therefore, for accurate voltage based SoC

estimation, it is highly recommended that the impact of battery temperature on OCV is

considered in the applied algorithm [68].

Fig. 21 shows a selection of measured OCVs for Cell-E-new and Cell-E-aged at various

temperatures (i.e., in particular -20 °C and +45 °C), determined OCV difference at a

specific temperature to OCV of Cell-E-new at 45 °C14 as well as determined OCV

gradients after discharging. Moreover, OCVs measured after charging and discharging

for the mentioned temperatures are illustrated in Fig. 21a. According to the obtained

results, the OCVs change over the battery lifetime at different temperatures

significantly. The highest OCV change is observed in a SoC range of 30% - 70%.

Furthermore, in order to investigate the impact of battery aging on OCV at different

temperatures more detailed, the difference between the measured OCVs after

14 In order to ensure better visibility of the measured curves, only a selection of the obtained results are illustrated in

Fig. 21 and Fig. 22; thus giving the interested readers an opportunity to use the OCV values measured over the whole

temperature range (i.e., -20 °C, 0 °C, 10 °C, 23 °C and 45°C), a complete list of the OCV values can be found in tables

A.1 and A.2 in Appendix A.


discharging for individual temperature (i.e., -20 °C and 45°C) to the respective OCVs

after discharging of the Cell-E-new at 45 °C is determined. The respective results are

shown in Fig. 21b. According to the results obtained at -20 °C, the OCVs after

discharging change significantly over the battery lifetime whereby the aforementioned

difference reaches its local minimum of approx. -40 mV in a SoC range of 50% - 60%. The

results obtained from OCV measurements after charging indicate almost the same

tendency for the investigated LIBs. However, after analyzing all measured OCVs for

examined temperatures, it may be stated that no clear tendency regarding the

dependence of battery OCV on temperature can be observed. In total, the determined

difference between the OCV values for the investigated temperatures becomes more

pronounced in the middle SoC range.

In addition, the OCV gradients of the aforementioned LIBs are investigated for different

temperatures. In this regard, Fig. 21c illustrates the determined OCV gradients of

Cell-E-new, Cell-E-aged after discharging for -20 °C and 45 °C. The determined OCV

gradients at 45 °C remain almost unchanged over the battery lifetime whereby the local

maximum of OCV rate of change per % SoC of approx. 4 mV/% is detected at approx. 50%

SoC and a local minimum of approx. 2 mV/% is observed at 80% SoC. A local maximum

of OCV rate of change per % SoC of approx. 5 mV/% at 60% SoC and local minima of

approx. 1 mV/% at 80% SoC and 30% SoC are observed for Cell-E-aged at -20 °C. The

determined local maximum seems to be shifted about 10% to higher SoC range over the

battery lifetime. In total, we conclude that OCV gradients of the mentioned LIBs are

affected by battery temperature.


Fig. 21: Open-circuit voltage behavior of Cell-E-new and Cell-E-aged measured at 45 °C and -20 °C, a) OCV curves after charging and discharging, b) The difference between the OCVs after discharging for individual temperature to OCVs of Cell-E-new at 45 °C, c) The gradients of the OCV curves after discharging (Please note that the SoCs are related to the battery’s nominal capacity).

In addition to the above mentioned LIBs, the OCVs of Cell-F-new and Cell-F-aged are

investigated at different temperatures. Fig. 22 illustrates a selection of OCVs of the

aforementioned LIBs measured at 0 °C and 45 °C15, determined OCV difference at a

specific temperature to OCV of the Cell-F-new at 45 °C as well as determined OCV

gradients after discharging. Fig. 22a illustrates the OCVs after charging and discharging

measured at mentioned temperatures. The obtained results indicate that the OCVs of

the respective LIBs change over the battery lifetime (mainly at SoCs below 35% SoC).

For Cell-F-new and Cell-F-aged the influence of actual battery aging state on the

determined OCV difference (i.e., the difference between measured OCV at particular

temperature and OCV of Cell-F-new at 45 °C) seems to be more pronounced in a SoC

15 In order to prevent lithium plating and accelerated aging, respectively, no OCV measurements are performed on Cell-F-

new and Cell-F-aged at -20 °C.


range of 40% - 70%. According to the results obtained at 45 °C, the absolute values of

OCVs increase over the battery lifetime while the same tendency can be approved for

other temperature points.

Fig. 22: Open-circuit voltage behavior of Cell-F-new and Cell-F-aged measured at 45 °C and 0 °C, a) OCV curves after charging and discharging, b) The difference between the OCVs after discharging for individual temperature to OCVs of Cell-F-new at 45 °C, c) The gradients of the OCV curves after discharging (Please note that the SoCs are related to the battery’s nominal capacity).

Furthermore, the behavior of OCV gradients of the aforementioned LIBs is examined for

respective temperatures. In contrast to Cell-E-new, the OCV gradients of Cell-F-new do

not change remarkably over the battery lifetime for the examined temperatures. A slight

change of the determined OCV gradient over the battery lifetime is observed in a SoC

range below 50% SoC. A local minimum of OCV rate of change per % SoC of 5 mV/% is

observed in a SoC range of 30% - 50%. At the same time, the local maximum of OCV rate

of change per % SoC of approx. 12 mV/% is detected in a SoC range above 80%. In a SoC

range below 20% SoC, the OCV gradients increase extremely. As expected, the OCV


gradients of Cell-F-new and Cell-F-aged are significantly higher than the determined

OCV gradients of Cell-E-new and Cell-E-aged. In total, it can be concluded that the

impact of temperature on OCVs of NMC/LTO-based LIBs is higher than on OCVs of

NMC/C-based LIBs. The impact of battery temperature on OCV becomes more

pronounced over the battery lifetime mainly as a result of increasing battery impedance

and decreasing diffusion coefficient.

According to the authors in Ref. [55], in general, the OCV gradients of LIBs increase

with increasing temperature. Moreover, the authors show a parabola-shaped curve for

OCV hysteresis at different temperatures indicating the increase of OCV hysteresis with

increasing temperatures. Unfortunately, the presented results are not in agreement with

the results obtained in this work. In Ref. [60], no clear tendency for the OCV behavior of

LFP-based LIBs for different temperatures is observed which seems to be in agreement

with the results obtained in this work. In Ref. [68], the authors investigate the OCVs of

LFP-based LIBs in new state in a temperature range of 0 °C - 50 °C. As a result, a

significant difference between OCV curves for examined temperatures is observed in a

SoC range of 30% - 80%. Unfortunately, almost all latter references address the cases

where battery is new state and the impact of battery aging is neglected completely.

Factors influencing the relaxation behavior of lithium-ion batteries after 2.4.2.3

current interruption

Directly after current interruption, in the first few minutes up to some hours, the

relaxation behavior of the battery is dominated by fast diffusion, migration and charge

transfer processes occurring at the electrode/electrolyte interface or in the electrolyte.

Afterwards, over a longer time period, solid-state diffusion processes become more

pronounced. As postulated by Fick’s law, it is known that with increasing temperature all reaction and diffusion processes take place faster [69]. The diffusion processes

occurring in the solid electrode can be described by second Fick’s law.

In the present subsection, the results obtained from OCV measurements performed on

Cell-B-new, Cell-F-new and Cell-F-aged are presented with an especial emphasis on

their relaxation behavior after current interruption. Furthermore, the factors

influencing the relaxation behavior of the aforementioned LIBs after current

interruption are analyzed. In particular, the voltage response of LIBs after current

interruption and the needed relaxation time until all the side reactions are declined are

investigated comprehensively. From electrochemical point of view, charge transfer and

diffusion overvoltages occurring in the middle and at very low frequency region can be

attributed as main contributors to the relaxation process of the battery.


The relaxation behavior of the aforementioned LIBs at different temperatures and SoCs

is illustrated in Fig. 23. Fig. 23a.1 and 23b.1 show the battery’s voltage response over the relaxation period (i.e., I = 0 A): The battery voltage at the end of the relaxation period

( ) is subtracted from the battery voltage measured directly after current

interruption (i.e., ∆ = − . As a result, it can be concluded that the voltage

response of the battery after current interruption changes over the battery lifetime for

different temperatures; this evidence is caused mainly by the fact that internal

resistance of the battery is remarkably influenced by actual aging state and temperature

of the battery. Moreover, the battery SoC affects the shape of the voltage response; this

phenomenon is more pronounced for Cell-F-new and Cell-F-aged than for Cell-B-new

which is in agreement with the results presented in Ref. [53]. However, in the latter

reference, the impact of actual aging state of the battery on battery’s relaxation behavior is neglected.

Furthermore, in Fig. 23a.2 and Fig. 23b.2 voltage relaxation curves of Cell-B-new and

Cell-F-new after charging and discharging with 0.5C at 23 °C for different SoCs are

shown. The low pronounced OCV hysteresis of Cell-B-new in contrast to Cell-F-new is

confirmed as discussed beforehand in subsection 2.4.2.1.

Fig. 23a.3 and Fig. 23b.3 depict the determined required relaxation times of Cell-B-new,

Cell-F-new and Cell-F-aged at different temperatures over the investigated SoC range.

The needed relaxation time for individual LIB is determined as follows: The battery is in

an equilibrium state as soon as the gradient of the voltage relaxation curve over time is

approx. zero (i.e., ∆V/∆t ≅ 0 where the measurement inaccuracy of the test bench is

considered). Consequently, it may be stated that a high dependence of needed relaxation

time of the aforementioned LIBs on SoC and temperature exists. The increase of

required relaxation time with decreasing SoC might be referred to increase of time

period for homogeneous distribution of lithium ions in the electrode. The latter effect

seems to be more pronounced for Cell-F-aged than for Cell-F-new.

Since the diffusion coefficient decreases over the battery lifetime and varies for different

temperatures, the needed relaxation time is influenced by actual aging state of the

battery. Fig. 23c illustrates the required relaxation time of the aforementioned LIBs at

50% SoC and different temperatures. As expected, with decreasing temperature the

required relaxation time increases significantly. Based on the obtained results, it can be

stated that the required relaxation time depends linearly on battery’s temperature. The

knowledge of the aforementioned relationship/dependency is useful for BMS developers


in order to save costs and time effort with regard to OCV measurements during product

development phase.

Fig. 23: Impact of temperature and aging on relaxation behavior of the battery, a.1) Voltage response of Cell-B-new after charging and discharging at different temperatures and 50% SoC under open-circuit condition, a.2) Voltage response of Cell-B-new after charging and discharging at 23 °C and different SoCs under open-circuit condition, a.3) Required relaxation time of Cell-B-new after discharging at different temperatures and SoCs, b.1) Voltage response of Cell-F-new and Cell-F-aged after charging and discharging at different temperatures and 50% SoC under open-circuit condition, b.2) Voltage response of Cell-F-new after charging and discharging at 23 °C and different SoCs under open-circuit condition, b.3) Required relaxation time of Cell-F-new and Cell-F-new after discharging at different temperatures and SoCs, c) Required relaxation time of Cell-B-new, Cell-F-new and Cell-F-aged for different temperatures at 50% SoC.

In Ref. [70], the relaxation behavior of LFP-based LIBs is examined under nominal

conditions. The authors state that for the examined LIBs over relaxation period from 1

hour up to 24 hours at 45% SoC, the applied OCV-based SoC estimator could introduce


an error of approx. 4% into SoC estimation. It is worth noting that in addition to the

above mentioned factors, the impact of applied current rate on required relaxation time

until all the side reactions are finished on relaxation behavior of LIBs is enormous and

its consideration in a full OCV characterization procedure is unavoidable. In this work,

the impact of applied current rate on OCV behavior of the battery is not further

examined as it has been investigated comprehensively by other researches in the past.

According to Ref. [53], an increase of the applied current rate yields an increase of

required relaxation time for LIBs using NMC cathodes. However, in contrast to the

results obtained in the present work, the authors state that the OCV characteristics of

the investigated LIBs do not change over the battery lifetime or strictly speaking the

impact of aging is marginal. In Refs. [59]-[60], the opposite results are achieved

regarding the impact of the applied current rate on OCV behavior for LFP-based LIBs.

The authors state that with increasing applied current rate the required relaxation time

decreases.

Simplified OCV-model considering temperature correction term and actual 2.4.2.4

aging state of the battery

As discussed in the previous subsections, for a precise SoC estimation over the battery

lifetime and under varying conditions, it is necessary that dependence of OCV on

temperature and actual aging state of the battery is considered. Therefore, the drift of

the SoC dependence of OCV on battery temperature over the investigated temperature

range can be employed for an accurate OCV modeling. In addition, it is worth

mentioning that the dependence of the battery OCV on further thermodynamic

quantities such as pressure and concentration plays also an important role in the field of

accurate OCV modeling. But the latter dependences are not considered further in this

work.

Assuming that battery OCV after charging and discharging depends (approx.) linearly

on temperature, the SoC-based temperature correction term ( [V ∙ K− ]) can be simply

determined for individual SoC after charging or discharging as follows (Eqs. (13)-(14)):

, oC = C @ °C C − C @ °C C (13)

, oC = C @ °C C − C @ °C C (14)

From electrochemical point of view, the derived kth is equal to battery entropy (S), when

the faraday constant (F) and the number of electrons per reaction (n) are included in the

Eqs. (13)-(14) [71]-[72]. The entropy differs for various active materials and affects the

reversible heat generation and the output voltage response of the LIB, thus it is of great


interest that kth behavior is investigated comprehensively. In this regard, the

corresponding kth values of Cell-E-new, Cell-E-aged, Cell-F-new and Cell-F-aged are

determined based on Eqs. (13)-(14). The obtained results are illustrated in Fig. 24.

Fig. 24: Temperature correction term (kth) determined as described in Eqs. (13)-(14) for Cell-E-new, Cell-E-aged, Cell-F-new and Cell-F-aged

As discussed in subsection 2.4.2.2, the dependence of OCV on temperature becomes more

pronounced over the battery lifetime; this tendency can be approved according to the

determined kth values. For Cell-F-new and Cell-F-aged, a difference of

approx. 0.3 mV ∙ K− between the kth values after charging and discharging is observed in

a SoC range above approx. 40% SoC. As a result, it can be concluded that the impact of

kth over the battery lifetime for a wide SoC range is almost negligible and becomes more

pronounced in a SoC range below 40%. However, in contrast to Cell-F-new and

Cell-F-aged, the kth values (i.e., absolute values) of Cell-E-new and Cell-E-aged increase

with battery aging (especially in a SoC range of 40% - 60%) whereby the difference

between kth values after charging and discharging remains almost negligible which is in

agreement with the results presented beforehand.


As discussed in the previous subsections, the OCVs of LIBs may change over the battery

lifetime depending on the LIB chemistry. In this regard, for Cell-E-new, Cell-E-aged,

Cell-F-new and Cell-F-aged, the coefficient of OCV change (i.e., the difference between

OCV values at two different aging states for respective temperature and SoC) over the

battery lifetime (∆SoH), the so-called kSoH, is determined as follows Eqs. (15)-(16):

, oC = C , @ °C C,ϑ − C , @ °C C,ϑ− (15)

, oC = C , @ °C C,ϑ − C , @ °C C,ϑ− (16)

Fig. 25 shows the kSoH values of Cell-E-new, Cell-E-aged, Cell-F-new and Cell-F-aged

determined as proposed in Eqs. (15)-(16). As a result, no clear tendency regarding the

dependence of the kSoH on temperature can be observed. It is worth noting that for

realizing a reliable comparison between both LIB chemistries, the determined OCV

values of Cell-F-new and Cell-F-aged below 0 °C are not considered. For Cell-E-new,

Cell-E-aged, the corresponding maximum ∆OCV rate of change per % ∆SoH is approx. 0.5 mV/% at 50% SoC. The resulting difference between the determined kSoH values after

charging and discharging for the aforementioned LIBs is negligible. For Cell-F-new and

Cell-F-aged, the corresponding minimum ∆OCV rate of change per % ∆SoH is approx. -0.8 mV/% at 60% SoC and 23 °C. However, in contrast to Cell-E-new and Cell-E-aged,

the obtained differences between the kSoH values after charging and the kSoH values after

discharging of Cell-F-new and Cell-F-aged seem to be more pronounced.


Fig. 25: Coefficient of the OCV change over the battery lifetime (kSoH) determined as described in Eqs. (15)-(16) for Cell-E-new, Cell-E-aged, Cell-F-new and Cell-F-aged (The SoCs are related to the battery’s nominal capacity)

By employing the Eqs. (13)-(14) and considering the impact of battery aging, a simplified

equation for determining the battery OCV and SoC dependences of the OCV, can be

derived as follows (Eq. (17)): OCV oC, ϑ, o = OCV °C oC + OCV , °C oC − OCV , °C oC ∙ − o ∙ϑ − °C . oC (17)

In the Eqs. (15)-(17), the SoH must be considered in a range between 0 and 1 whereby

the zero refers to aging state of the battery at End of Life (EoL) and the one refer to

aging state of the battery at Begin of Life (BoL). In this work, the EoL criterion of

battery is defined as the decrease of battery’s capacity to 80% of its initial value.

In the present work, the OCV measurements are performed on single LIBs and, thus the

influence of cell-to-cell variations regarding the individual OCV value of the LIBs is

neglected. However, for large lithium-ion battery packs where a high amount of LIBs are

connected in series, considering an additional correction term referring to the influence

of cell-to-cell variations on OCV in Eq. (17) is highly recommended.


2.5 Validation datasets

In order to optimize the software functionalities and decrease the required development

phase and costs of embedded software for BMS applications, the Automotive Open

System Architecture (AUTOSAR)-compliant along with model-based software

development have become more and more popular in the automotive industry in recent

years [13]. The state-of-the-art product development process for BMS and large battery

systems in automotive applications is based on V-Model [73] where further automotive

standards such as ISO 26262 [74] and Automotive SPICE process assessment and

process reference model [75] that rely on V-model, are often used. For example, ISO

26262 standards has ten different parts, whereby the requirements for product

development at system level with respect to functional safety are specified in part 4.

Fig. 26 depicts schematically the product development process (i.e., systems engineering

and software engineering) of BMS according to the mentioned V-Model.

Fig. 26: Product development process according to V-Model (adopted from Ref. [73], [75])

The product development process according to V-Model begins by analyzing the

stakeholder requirements on Sys.1 level. Thereafter, on Sys.2 and Sys.3 levels, system

requirements and system architecture are defined. According to V-Model, at each

individual development phase e.g., the developed BMS software model or the developed

BMS must be verified individually. In this regard, for verification tests in individual

development phase (module, integration and system tests) the developed models etc. are


often verified by means of model-in-the-loop (MIL), software-in-the-loop (SIL) and

Hardware-in-the-loop (HIL) test environment [75]. The last stage of product verification

is then often performed on vehicle level.

In fact, load profiles and environmental conditions that the battery is exposed to in the

field differ in terms of characteristic, frequency etc. to that in laboratory environment.

Hence, for analyzing the performance of monitoring algorithms, the application of real

load profiles collected under real conditions over a long period of time (i.e., over days,

weeks or even more) is highly recommended. Unfortunately, the monitoring algorithms

presented in the literature are often verified under nominal conditions or by applying

simple pulse tests or standardized load profiles such as urban dynamometer driving

schedule (UDDS), dynamic stress test (DST) or federal urban driving schedule (FUDS)

etc.

Beside the above mentioned factors referring to the necessity of using real load profiles

(i.e., load profiles measured on-board during vehicle operation) for validating the

monitoring algorithms, collecting real vehicle data over a long period of time can provide

helpful information on battery’s aging behavior etc. In general, real vehicle data can be employed for developing new operation strategies of the vehicle and optimizing power

demand/capabilities and operation limits of the battery.

In this regard, among the main parameters to be collected on-board in EVs are [10], [76]:

Amount of charge and energy throughput,

Actual capacity and impedance of the battery referring to battery’s SoH both on cell and system-levels,

Temperature spread inside lithium-ion battery pack etc.

In the following subsection the load profiles used in this work for verification reasons are

presented.

2.5.1 Load profiles investigated in this work for verification reasons

Within this work, the following two different types of load profiles with various

characteristics are examined for validating the applied monitoring algorithms:

Real vehicle data under varying conditions,

HPPC (see subsection 2.3.2).

The aforementioned load profiles are applied to the investigated LIBs at different aging

states over a wide temperature and SoC range.


Real vehicle data obtained under varying conditions 2.5.1.1

The ECM parameters depend not only on battery characteristic itself but also on the load

profile the battery is exposed to. The latter fact is a result of following two issues: First,

the ECMs commonly used for monitoring algorithms are simplified and, thus the

employed ECMs are not able to reproduce the battery behavior (both thermal and

electrical) precisely. Second, the only signals that are usually available on-board for

monitoring algorithms are current, voltage and temperature of the battery. Therefore,

the ECM parameters can be estimated accurately when there is enough fluctuation

available in applied current and voltage of the battery [46]. Adaptive filters used for

ECM parameters identification minimize the resulting error between the measured

voltage and the modeled battery voltage. Thereafter, such filters optimize the estimated

parameters only for the applied load profile available during the parameter

identification process. Consequently, the identified ECM parameters could differ for

respective load profiles.

The main objective of the datasets investigated in this work is to understand the

performance of different ECMs over a wide range of temperature, SoC and applied

current rates. As will be discussed later in chapter 5, the investigated ECMs are further

examined with regard to voltage and SoAP prediction accuracy. An especial emphasis is

put on following factors: First, the battery behavior and performance of the applied

algorithm are investigated over a wide SoC range including varying relaxation and

charging periods (see Fig. 27). Second, the impact of “extreme” temperature variation performed between individual driving cycle or during a single driving cycle over a wide

temperature range (-20 °C⋯-5 °C⋯+10 °C⋯30 °C) on battery’s and algorithms performance is investigated. Third, for long-term robustness analysis, tests using real

vehicle data including various vehicle operating conditions (e.g., highway or city,

defensive or aggressive driving styles) are considered. For this purpose, 39 real driving

cycles from ten different drivers are recorded in a time range of one week.

Fig. 27 illustrates a selection of driving cycles investigated in this work for validation of

the applied battery state estimator. The normalized energy content released or applied

to the LIB and the temperature profile of the LIB measured on its surface are shown.

The third driving cycle always refers to the case where the vehicle is driven on the

highway and approx. 40% of the total battery energy content is released. During a

driving phase on a highway, as the LIB must mainly deliver energy to vehicles

powertrain and the recuperation phases are rarely given, the increase of battery

temperature is significantly higher than e.g., the driving cycle obtained in the city


including different stop-phases. In addition, after the last driving cycle (7th driving cycle)

a relaxation period of 24 h is inserted to the load profile where no current flows to/from

the LIB.

The negative power values refer to cases where the battery is discharged and positive

values refer to cases where the battery is charged (e.g., regenerative phases or charging

periods). The sampling rate of battery temperature is 10 s since temperature of the

battery does not change rapidly under normal/safe operating conditions as for example

current and voltage do. During vehicle parking and charging phases, battery voltage is

measured with a sampling rate of 10 s where sampling rates of current and voltage

during driving phases are decreased to 100 ms. Furthermore, in order to have

deterministic points for verifying the predicted voltage and SoAP, respectively, a current

pattern consisting of two current pulses in both charge and discharge directions with

amplitudes of 1C and 2C each for 20 s is integrated into the load profiles. It is worth

noting that normalized energy content illustrated in Fig. 27 is determined using nominal

voltage and capacity values of the examined LIBs.


Fig. 27: Examples of synthetic real driving cycles over a wide temperature and normalized energy content range; a) Cell-A-aged, b) Cell-B-new, c) Cell-D-new, d) Cell-G-new, e) Cell-J-aged.


In order to provide a better visibility of a single load profile, the second driving cycle of

the Cell-G-new is illustrated exemplary in Fig. 27. In particular, the measured power

profile and the determined State-of-Available-Energy (SoAE16) of the respective driving

cycle are investigated under real condition.

Fig. 28: Illustration of the measured power and the determined State-of-Available-Energy profile of the second driving cycle for Cell-G-new depicted in Fig. 27.

Relative distribution of specific parameters of the investigated load profiles 2.5.1.2

In the previous subsection a detailed description of the investigated driving cycles is

provided. As mentioned beforehand, the main objective is to investigating the battery

behavior over a wide SoC and temperature range. In this regard, the relative

distribution of the aforementioned parameters of the driving cycles is used as a basis for

analyzing and a explanation of possible deviations. Fig. 29 presents relative distribution

of temperature and SoC of the investigated driving cycles for selected LIBs discussed

above. Unfortunately, because of some measurement setup failures during performing

measurements on Cell-A-aged and Cell-D-new, the attached temperature sensors on

surface of the aforementioned LIBs were loose. Thus, the measured temperature profiles

illustrated in Fig. 27 refer mainly to temperature of the climate chamber and not the

real surface temperature of the aforementioned LIBs, which in turn makes an

illustration of their relative distribution values unreasonable.

16 The technique applied for SoAE determination will be presented in great detail in subsection 3.1.


Fig. 29: Relative distribution of State-of-Charge and temperature of the investigated driving cycles for selected LIBs (Cell-B-new, Cell-G-new and Cell-J-aged)

As can be seen in Fig. 29, relative temperature distribution of Cell-B-new is relatively

homogenous for the examined temperature range and lies below approx. 15%. For

Cell-G-new and Cell-J-new, the temperature distribution differs slightly as the

temperature distribution in a range of above 10 °C is significantly higher than lower

temperatures. As discussed beforehand, during a single driving cycle mainly depending

on respective load profile, temperature of the battery increases by some degrees

(5 °C⋯10 °C). Consequently, the mentioned temperature increase has a high influence

on impedance characteristic of the LIB. The SoC distribution of the investigated load

profiles show only slight differences at high SoC ranges (i.e., above 95%) while for

remaining SoC range the distribution sems to be very similar. It is worth mentioning

that the battery SOC shown in Fig. 29 is related to battery’s nominal capacity.

Hybrid pulse power characterization tests 2.5.1.3

Along with the presented real vehicle data measured under varying conditions, HPPC

tests are performed at different temperatures. A detailed description of test procedure


for HPPC tests was provided in subsection 2.3.2. In order to minimize the impact of

battery temperature on impedance parameters of the LIBs, an especial focus was put on

temperature equalization between individual charge and discharge current pulses

during a single HPPC test. Fig. 30 shows current and voltage profiles measured during

HPPC test performed at 23 °C on Cell-G-new. Moreover, the normalized energy content

determined using battery’ nominal capacity is illustrated.

Fig. 30: Hybrid pulse power test performed at 23 °C on Cell-G-new

The upper voltage limit of the LIB was set to 4.2 V and the lower voltage limit was set to

2.7 V as proposed in the datasheet. The maximum applied current is limited to ±3C.

Frequency analysis of the employed load profiled 2.5.1.4

In the previous subsections, the electrical and thermal characteristics of the investigated

load profiles and their impact on battery behavior have been discussed. In Fig. 31 the

normalized frequency spectra of current profile of the HPPC test and the second driving

cycle (see Fig. 28) both applied to Cell-G-new are illustrated. The spectra are determined

based on simple discrete-time Fourier transformation (DTFT).


Fig. 31: Normalized frequency spectra of two different current profiles for Cell-G-new. The determination of spectra is performed using discrete-time-Fourier transformation divided by root-mean square value of the current as proposed in Ref. [290]

According to the results, it can be concluded that the current amplitude of the HPPC test

is significantly higher than the current amplitude of the investigated driving cycle;

especially in a frequency range of (0.1 ⋯ 0.01) Hz. The normalized current amplitude

reaches its local maximum of approx. 0.8 of its root-mean square (RMS) value at approx.

0.5 Hz. At the same time, the amplitudes of both load profiles become almost similar

with slight exceptions at frequencies below 0.01 Hz. In general, it can be stated that in

the frequency range where the current amplitude is high or in other words where there

is enough dynamic/fluctuation available, the ECMs used in modern monitoring

algorithms are favor to reproduce the battery behavior more accurate.

2.6 Summary

In this chapter, the results of extensive electrical characterization tests performed on

NMC/LTO LIBs at different aging states are presented and compared to those presented

in the literature. First, the impedance parameters of the battery are characterized and

the impact of battery SoC and temperature on impedance characteristics of the LIBs is

investigated comprehensively. The influence of temperature on impedance

characteristics of the LIBs is investigated in a temperature range of -20 °C ⋯ +40 °C.

According to the results, the values of R0 and Rct increase with decreasing temperature

and Cdl shows high temperature dependence while its values decrease over the battery


lifetime. The R0 increases slightly with decreasing SoC while at the same time a low

impact of SoC on Rct, and Cdl is observed. Generally, it can be summarized that the

basic shape of the dependence of the R0 and Rct on SoC and temperature remains almost

unchanged over the battery lifetime, especially at high temperatures. Based on the BVE,

the behavior of and I0 is examined. The obtained results indicate the high impact of

SoC and temperature on I0 whereby the impact of temperature is significantly more

pronounced in contrast to the impact of SoC. It is also shown that the current

dependence of the Rdc at high temperatures is less pronounced. Hence, for simplicity

reasons depending on the application the current dependence might be neglected.

However, its consideration at lower temperatures (i.e., below 0 °C) and at advanced

aging state seems to be necessary.

Second, the results of extensive OCV characterization tests performed on LIBs using

various active materials are presented. The impact of actual aging state, temperature

and previous history on OCV behavior of the LIBs are investigated comprehensively.

Impact of aging of the battery on OCV behavior can be summarized as follows:

The required relaxation time until the equilibrium state of the battery is reached

increases over the battery lifetime,

The aging of the battery influences the OCV hysteresis where the dependence of

OCV hysteresis on actual aging state of the battery differs for individual LIB

chemistry,

The shape of battery’s voltage response after current interruption changes over the battery lifetime,

The battery aging influences the OCV gradients; the change of the OCV gradients

over the battery lifetime for NMC/LTO LIBs is more pronounced than other cell

chemistries investigated in this work.

Impact of temperature of the battery on OCV behavior can be summarized as follows:

The required relaxation time until all the side reactions are finished increases

with decreasing temperature whereby it can be assumed that a linear dependence

exists between the aforementioned parameters (i.e., required relaxation time and

temperature of the battery),

The temperature of the battery influences significantly the measured OCVs; in

case of OCV curves after discharging for NMC/C and NMC/LTO LIBs, regarding

the temperature dependence of the hysteresis no clear statement can be provided,


The OCV curves change over the battery lifetime at different temperatures,

The OCV gradients of the NMC/C LIBs remain almost unchanged for different

temperatures while the impact of temperature on OCV gradients of NMC/LTO

LIBs seems to be more pronounced.

Moreover, the load profiles used in this work for validating the applied monitoring

algorithms are presented. In this regard, two different load profiles with different

characteristics are used. First, driving cycles obtained in an EV prototype over a time-

period of one week are applied to the LIBs over a wide temperature and SoC range

where the battery temperature varies during/between each driving cycle. Additional

factors such as relaxation and charging periods are considered in the investigated load

profiles. Second, HPPC tests are performed on LIBs at different temperatures over a

wide SoC range where current pulses with different amplitudes are applied to the LIBs

for both charge and discharge direction. The frequency spectra analysis of the

aforementioned load profiles show that the amplitude of the applied current differ

significantly in a frequency range below 0.1 Hz mainly as a result of the higher constant

current rates applied to the LIB during HPPC test in contrast to the driving cycle used

for verification reasons.

Review of monitoring algorithms and battery models presented in the literature 66

3 Review of monitoring algorithms and battery models presented in the literature

In order to fulfill specific power and energy requirements in EVs generally specified by

vehicle manufactures, LIBs are connected in series and/or in parallel (depending on

topology of the battery pack), thus forming a large lithium-ion battery pack [77]. The

EVs mainly differ in degree of electrification of their electrical distribution system that

correlates with resulting voltage level of the lithium-ion battery pack. When d.c. voltage

of the appropriate battery system is above 60 V and a.c. voltage is above 30 V [78], the

respective battery system is defined as a high voltage (HV) battery system. At the same

time, a battery system indicating voltage level below the aforementioned levels is

handled as low voltage (LV) system [79] and requires no contact protection etc. [80]. In

Table 6, the four commonly used voltage levels for lithium-ion battery pack or vehicle

electrical distribution system available in series production or currently under

development are given.

Table 6: Comparison of different voltage levels of lithium-ion battery systems commonly used in automotive industry

Voltage level

(nominal) 12 V 48 V 400 V 800 V

Vehicle type Conventional or

micro HEVs Mild HEVs

Full EV or HEV

(including

PHEV)

Full EV

The higher the voltage level of the battery system is, the lower the power loss of the

system would be, mainly as a result of lower current rates. Accordingly, total weight of

the respective system decreases and available range for fully electrical drive “could” increase. The performance of LIBs and their operation in EVs are controlled and

diagnosed by means of BMS consisting of both software and hardware. In other words,

protection against battery over/under-voltage, over-current, over-temperature, under-

temperature and cell balancing are among the main tasks of BMS. Fig. 3217 illustrates

schematically a simplified lithium-ion battery pack including the following BMS electric

and electronic components [81]:

Cell Controller Board (CCB),

BMS Master,

LV and HV interfaces,

17 It is worth noting that thermal connections and cooling system are not illustrated in Fig. 32.


Fuses, contactors etc.

Fig. 32: Schematic illustration of simplified modular-based lithium-ion battery pack including high and low voltage interfaces, lithium-ion batteries, electric and electronic components (adopted from Refs. [10], [82])

Besides the above mentioned electric/electronic specifications; from thermal point of

view, the battery system need to be tempered (cooling or heating according to

environmental specifications) actively or passively, thus keeping the LIBs in predefined

temperature range during operation (e.g., charging or driving phases).

For a safe operation of battery pack, software-based applications implemented on

electronic control unit (ECU) are needed where the extension of the realized software

applications is influenced directly by degree of system complexity [13]. As shown in

Table 6, voltage level of lithium-ion battery pack is directly correlated with the system

complexity. The system complexity of a 12 V battery systems is significantly lower than

that of 800 V battery systems given in Table 6, whereby for example for a 12 V battery

system a particular number of diagnosis functions, thermal functions etc. are not

required. A selection of software-based functions operated by BMS can be found in Fig.

33.


Fig. 33: A selection of software-based functions operated by battery management systems

The main task of the CCB is to measure a particular number of cell voltages and

temperatures (e.g., 1-12 cells) [81]. In fact, the CCB is the only physical interface

between the BMS and the battery cells. Furthermore, cell balancing is performed by

means of balancing resistances implemented on each CCB and controlled by an

algorithm implemented on the ECU in the BMS master. Depending on topology of

lithium-ion battery pack and number of LIBs which have to be monitored, one or more

CCBs are required. Each CCB communicates with the other CCBs via Controlled Area

Network (CAN) or Isolated Communications Interface (isoSPI). Measured cell voltages

and temperatures are then submitted via CAN to the BMS Master and analyzed,

respectively. On the one hand the BMS Master is responsible for evaluating the

measured battery pack’s current and voltage as well as triggering the contactors (opening, closing) during pre-charge phases or when a crash or fault is detected, and on

the other hand it performs plausibility checks and diagnoses of the battery states and

parameters. Furthermore, monitoring algorithms implemented for battery state

estimation are running on the ECU. Submitting or receiving data to/from the vehicle

EMS as well as the electronic power supply are all performed over a designed LV

interface.

The accurate estimation of battery states such as SoC, SoAP or SoH, is still a

challenging task, keeping in mind that the implemented monitoring algorithms have to

work accurately over years in the particular application. Monitoring algorithms use on-

board measured values such as battery temperature, battery voltage and current for

state estimation. However, since a direct insight into electrochemical processes inside


the LIBs in the field is not possible, estimated battery states or parameters may differ

from real values. For example, according to Ref. [83], the difference between the

measured cell temperature on the surface and in the core may be approximately 10 K.

Keeping in mind that often one single temperature sensor is used in a battery module for

monitoring the module’s temperature18 and the measurement accuracy of implemented

temperature sensors are often low, it becomes obvious how challenging the task of

reliable and accurate battery monitoring can be by considering all uncertainties and

disturbances.

The electrical and thermal performance of a lithium-ion battery pack is mainly

influenced by the following factors:

Topology of the lithium-ion battery pack [84]-[85],

Internal and external factors, such as unequal aging behavior of the LIBs over

the battery lifetime (capacity and impedance spread between each individual LIB

in the battery module/pack) [77], [86],

Inconsistency of the SoCs [76], [85],

System losses occurring in the battery pack (e.g., power rails, contactor

resistances etc.) [76]-[77],

Limitation of the use of the battery’s maximum power and energy capacity, caused by the increase of battery impedance and decrease of available battery

capacity over the battery lifetime [85],

Distribution of the temperature in the battery pack19 etc. [38], [76].

Therefore, considering the above mentioned factors is necessary during system design

phase in order to prevent oversizing the system and reducing costs, respectively, while

specified requirements are fulfilled. In the following subsections a review of available

techniques for estimating the following battery states/parameters is provided:

SoC,

SoH and actual battery capacity,

SoAP,

SoS.

18 Each battery module consists of several LIBs connected in series or parallel. 19 Temperature differences between individual LIB must be kept in a range of 3 °C - 5 °C [288].


3.1 State-of-Charge estimation

On-board estimation of battery SoC is always a part of BMS. The Battery SoC is

equivalent to the fuel gauge used in conventional vehicles. The SoC is often employed as

a basis for reliable energy/power management strategy [87]. The SoC corresponds to the

ratio of releasable battery capacity (Cr) to the actual battery capacity (oC =∙100%). An inaccurate SoC estimation might have fatal consequences for vehicle

manufacturers with regard to customer acceptance or safety failures (i.e., occurrence of

unwanted battery overcharge or over-discharge etc.). Moreover, an inaccurate SoC

estimation leads to a change of operating SoC window that in short term has high

impact on battery power capability prediction and in long term on expected battery

lifetime [65]. Battery SoC definitely belongs to the most investigated battery states and

it has often been discussed in many researches in the past. Comprehensive reviews on

available SoC estimation techniques can be found, for example, in Refs. [82], [87]-[89]. In

the following, a brief overview of available techniques for SoC estimation is provided.

3.1.1 Methods based on coulomb counting technique

The simplest and may be the most common technique for SoC estimation is the so-called

coulomb counting (CC) technique. The CC technique is based on integrating the current

over time while it is assumed that the actual battery capacity and initial SoC value are

known or determined by other supporting algorithms (Eq. (18)) [82]:

oC = oC + ∫ (18)

where SoC(t0) refers to the SoC at the initial time, is the actual battery capacity

determined by other algorithms and represents the current applied to the battery.

Among the main advantages of the CC technique are low computing power, simple

nature and the possibility of combination with other supporting techniques. However,

since the CC technique is an open-loop estimator, the SoC estimation accuracy is affected

by measurement uncertainties (e.g., current measurement), disturbances or inaccuracies

by setting the initial SoC or the battery capacity [67]. Therefore, in order to ensure long-

term stability of the CC technique and compensate the mentioned disadvantages, the CC

technique is often combined with other supporting techniques such as OCV-based

estimation techniques.

In addition to SoC, the SoAE of the battery is a further key factor/battery state for a

precise energy optimization/management of EVs. The SoAE is often employed for an

accurate remaining driving range prediction either in combination with SoC or


completely standalone [90]. Simply speaking, with the support of an accurate SoAE

estimator, the possibility is given to figuring out how long the battery would last for a

specific power demand. However, the main challenge is to ensure high SoAE estimation

accuracy during battery operation as the measurement inaccuracies and battery

nonlinearities affect significantly estimation results. The main difference between the

SoAE and the SoC is that for SoAE estimation the measured battery voltage must be

considered in addition to the measured current [91]. The SoC is not an appropriate state

for predicting the available battery energy and remaining drive range of the vehicle since

it considers the amount of charge (As) applied to the battery whereas the SoAE accounts

for the amount of available energy (Ws) and can be employed as a direct indicator for

accurate vehicle energy optimization. The SoAE can be determined as follows [91]-[93]:

oAE = oAE + ∫ = oAE + ∫ OCV C ∙ ∙ CC (19)

where oAE is the oAE of the battery at the initial time ( ), P is the applied power at

time and is the actual energy content of the battery which mainly depends on

battery temperature and applied current. Moreover, the SoAE is highly nonlinear mainly

because of the voltage considered in the equation. Consequently, it can be stated that

among the main factors influencing the SoAE of the LIBs are temperature, applied

current and battery voltage (i.e., strictly speaking the OCV) [94]. In general, the

techniques applied for SoAE estimation of LIBs can be subdivided into following two

classes:

Model-based methods using ECMs or advanced electrochemical battery models,

Neuronal network.

Unfortunately, in contrast to SoC, the topic of SoAE estimation has found little attention

in the literature up to now and few articles have performed systematic investigation on

SoAE estimation. For more detailed description of methods for on-board SoAE

estimation of LIBs, the readers are referred to Refs. [91], [94]-[97].

3.1.2 Methods based on measured or estimated OCV

As described in the previous chapter, the OCV is always a function of SoC; i.e.,

OCV = f(SoC). Thereafter, this correlation is often employed to recalibrating the CC-

based SoC estimator and setting the initial SoC value [98]. In this regard, when the

battery is in an equilibrium state and all the side reactions are finished, the SoC-OCV

correlation is used to recalibrate the CC-based SoC estimator. Fig. 34 illustrates the

OCVs of LIBs at different aging states using various active materials measured at 23 °C.


Fig. 34: Open-circuit voltages of Lithium-ion batteries at different aging states using various active materials measured at 23 °C

As discussed in chapter 2, the SoC-OCV correlation may change over the battery

lifetime; hence, the latter fact should be considered in the applied algorithm for an

accurate SoC estimation. A detailed review of references available in the literature

dealing with the topic of OCV-based SoC estimation will be discussed later on in

subsection 3.2.2.

Alternatively, SoC can be estimated in a closed-loop form using model-based techniques

based on ECMs or advanced electrochemical models [87].Within the former technique

(i.e., ECM-based methods), the battery SoC is not directly incorporated in the ECM and

it is determined over the SoC-OCV correlation based on estimated OCV from the

implemented ECM. In an example of the ECM shown in Fig. 14, the OCV can be simply

determined by subtracting the determined overvoltages from the measured battery

voltage as follows (Eq. (20)):

C = − ∆ − ∆ (20)

Unfortunately, the ECM-based methods are not able to precisely reproduce the

electrochemical processes occurring in the LIB, mainly because of their significant

simplification. In this regard, the electrochemical battery models are the promising

solution for accurate battery modeling. The electrochemical battery models are able to

capture various electrochemical processes such as diffusion, intercalation and migration

processes. Unfortunately, as a result of their complex mathematical model structure and


required high computational effort, the aforementioned models have received less

attention in the field of battery monitoring algorithms up to now.

3.1.3 Methods based on active or passive impedance spectroscopy

The methods based on active or passive impedance spectroscopy are often applied for

battery impedance parameters and battery state estimation. Thereafter, when the

dependence of the impedance parameters on SoC is known, battery SoC can be simply

determined [99]. For this purpose, it is necessary that following two factors are

considered: First, a pronounced relationship between the battery impedance parameters

such as R0 and the SoC should be given. Second, impedance characteristics of LIBs

change over the battery lifetime and, thus the applied estimator must be able to track

the aforementioned changes, otherwise the SoC estimation accuracy decreases as the

LIB ages.

A selection of techniques applied for SoC estimation is illustrated in Fig. 35.

Fig. 35: A selection of techniques commonly applied for on-board State-of-Charge estimation

3.2 State-of-Health and capacity estimation

Battery capacity with ampere hours (Ah) or ampere seconds (As) as a unit corresponds to

the amount of charge extractable from the battery until cut-off discharge voltage limit is

reached when starting from a fully charged state. One important issue is that the


capacity is not a constant parameter, and it decays over the battery lifetime due to

internal aging processes when the battery is cycled or even if it is not being used due to

calendar aging [100], [101]. In automotive applications, battery temperature, discharging

and charging current rates, the DoD during battery operation and the SoC during rest

periods are the major degradation factors [102]. In general, it can be stated that

degradation mechanisms of LIBs are either electrochemical or mechanical where often

the source of aging mechanism has its origin in battery electrodes. From an

electrochemical point of view, the capacity loss of LIBs generally occurs because of loss of

cyclable lithium due to SEI formation in electrodes and impedance increase of the LIB.

In general, aging processes occurring inside the LIBs can be mainly related to e.g.,

structural changes, mechanical changes, defoliation of active mass, current collectors

corrosion, increase of internal resistance or contact loss etc. [6], [102]-[104]. However, a

detailed discussion of aging mechanisms of LIBs is out of the scope of this work and,

thus the readers are referred to given review articles.

The SoH is a figure of metric to evaluate the actual aging state or present condition of

the battery and, thus it must be determined by BMS. In particular, the SoH refers to

battery’s capacity fade and/or power fade [105]. When the SoH has to reflect the battery

energy capability (oC), then the actual battery capacity ( ) is of interest. At the

same time, when the SoH is used to describe the power capability (o ) of the battery,

then the actual battery impedance ( ) is of interest [87]. An accurate estimation of

the SoH is of great importance for BMS since the obtained information is often employed

for identifying ongoing battery degradation and prevents a possible battery system

failure. As mentioned above, the actual battery capacity and actual battery impedance

are the main factors for SoH estimation. Thereafter, the SoH is often defined either

as a ratio between the actual battery capacity at nominal conditions (nominal

temperature and nominal discharge current) and the battery’s nominal capacity,

i.e., oC = , or

as a ratio between the actual battery impedance value at nominal conditions

(nominal temperature and nominal discharge current) and its nominal

impedance, i.e., o = .

The unit of SoH is percent where 100% refers to battery aging state at BoL and 0% to

battery aging state at EoL. Generally, as an established rule in automotive applications,

the EoL of the battery is reached when its actual capacity has decreased to 70% - 80% of


its nominal value or in applications where the available power plays more important role

such as HEVs, the EoL is often defined as being reached when the battery impedance is

doubled [10], [106]. For plug-in hybrid electric vehicle (PHEV) where both parameters

(i.e., power and energy capability) are of interest, the oC and the o of the battery

should be considered [82]. However, even if the EoL (SoH=0%) of the battery is reached,

the LIBs can still be used for a while, for example in second-life applications etc. [107].

Fig. 36 shows a selection of possible techniques for on-board SoH estimation commonly

applied for BMS applications.

Fig. 36: A selection of techniques commonly applied for on-board State-of-Health estimation (adopted from Ref. [105])

In general, methods for SoH estimation can be subdivided into experimental and model-

based techniques. The experimental-based techniques require the fundamental

knowledge of interdependences between operation and degradation of the LIB while the

model-based estimators determine the SoH thorough specific impedance parameters of

the LIB (e.g., R0 etc.) that are sensitive to aging state of the battery [105]. A clear

distinction and description of advantages and drawbacks of the aforementioned SoH


estimators can be found in Ref. [105]. For more detailed reviews with respect to available

methods for SoH estimation and aging mechanism of LIBs, the readers are referred to

Refs. [82], [87], [103], [105], [108]. In Fig. 37, possible integration of on-board SoH and

actual battery capacity estimators in the context of simplified BMS architecture is

illustrated.

Fig. 37: Integration framework of State-of-Health and actual battery capacity estimators in the context of simplified battery management system

Generally, the methods for on-board capacity estimation can be divided into the

following four categories:

Voltage-based estimation methods using the OCV-SoC correlation during an idle

or operation time,

Electrochemical model-based methods using electrolyte’s conductivity and electrode’s porosity as an indicator for SoH estimation,

Incremental capacity analysis (ICA) and differential voltage analysis (DVA)

methods, through which information about the aging mechanism can be directly

obtained,

Aging prediction methods, in which an aging model prediction is used for

estimating the capacity and the RUL.

Discussion of different battery capacity definitions 3.2.1.1

Different definitions for battery capacity are given in the literature. Unfortunately, they

are often inconsistent and unclear. In this subsection, a proposal for consistent


definitions of battery capacity is given. The following three main capacity definitions,

illustrated in Fig. 38, may be employed [98], [109]:

Nominal Capacity (Cdatasheet),

Initial capacity (Cinitial),

Actual capacity (Cactual).

Fig. 38: Comparison of different battery capacity definitions

The nominal capacity (Cdatasheet) is the capacity of a battery defined by the manufacturer

for operation under nominal conditions, including nominal temperature (e.g., 23 °C),

nominal discharge current (e.g., one hour discharge rate (1C)), and being fully

discharged from an initially fully charged state. Often, the nominal capacity is the

capacity guaranteed by the manufacturer, and the real capacity of a given battery cell

under nominal conditions is slightly higher. This leads to possible negative values of SoC

if the nominal capacity of the LIB is used as reference for SoC estimation.

The initial capacity (Cinitial) refers to the highest amount of charge that can be extracted

from the battery in a new state and starting from a fully charged state.

The actual capacity (Cactual) is the highest amount of charge that can be extracted from

the battery in its actual aging state starting from a fully charged state. When the battery

is new, its initial capacity is the same as its actual capacity. The difference between the


initial and the actual capacity of the battery over the battery lifetime refers to capacity

loss caused by aging effects.

The Cinitial and Cactual can be defined either at nominal conditions, including nominal

temperature, nominal charging and nominal current or at conditions different from

nominal. The influence of the conditions might be quite substantial. For example, a high

battery temperature generates an increase in the capacity/energy due to a decrease of

the internal resistance and an increase in the kinetics and thermodynamic aspect of the

chemical processes involved in the cell reaction [291]. In Ref. [110], the authors have

investigated the battery capacity for a wide temperature range. An enhancement of

battery capacity by 7.7% at 45 °C in comparison to 15 °C is observed in the latter

example. At the same time, the higher the current rate is, the lower the extractable

battery capacity is [111]-[112]. Empirical derivation of battery capacity based on the

peukert-equation for automotive applications by considering different temperature

ranges and current rates is shown in Refs. [113]-[114]. It is worth noting that the battery

SoC is always defined based on the battery capacity. Depending on the employed battery

capacity (nominal, initial, actual, at nominal conditions or at other conditions) the

respective SoC value will be different.

Basic idea and challenges of on-board capacity estimation 3.2.1.2

The simplest way to determine Cactual is to discharge the battery with nominal current at

nominal temperature from a fully charged state until the battery’s cut-off voltage is

reached [108], [115]. This procedure can be simply applied in the laboratory. However, in

EVs, the use of this procedure is hardly possible, and, therefore, other methods are

required. For example, capacity can be defined as a ratio between ampere hours

charged/discharged and the difference between SoCs (∆oC )20 before and after the

charge or discharge process, as shown in the following equation (Eq. (21)) [116]-[118]: ∫ d = oC − oC (21)

where denotes the actual capacity of the LIB, and is the columbic efficiency,

defined as the ratio between the discharged and charged ampere hours. However, in

most cases, the coulomb efficiency of LIBs can be assumed as ( ≈ ) due to the relatively

low rate of side reactions in comparison to other battery technologies, such as lead-acid

or nickel metal-hydride batteries [110], [119].

It is worth noting that the only useable SoC estimation technique within this method is

a voltage-based SoC estimation method and not CC-based technique discussed in

20 SoCv refers to the voltage-based estimation technique of SoC.


previous subsection. The application of a CC-based estimator would be meaningless, as it

would lead to a circularity of dependencies in the algorithms [116], [120]. Thereafter, the

Eq. (21) can be rewritten as follows (Eq. (22)):

= ∫ C − C (22)

A possible implementation of the use of Eq. (22) for on-board capacity estimation is

shown in step-by-step manner in appendix B. In step 1, all necessary variables are

initialized. In step 2, the OCV of the battery is measured at the beginning of the driving

cycle, and the respective SoC is estimated by the predefined SoC-OCV relation. In

general, it is a good compromise to estimate the OCV(t1) using this method at the

beginning since EVs and PHEVs are often used only for a few hours a day and are

parked during the remaining time, and the battery can be relaxed for a more accurate

measurement. In step 4, OCV(t2) refers to the battery voltage (OCV) during driving or

after driving (during a key-off mode). When both SoCs are estimated, ΔoC can then be

calculated for a considered time span (t1 ⋯ t2) in step 5. For this purpose, it is

recommended that the battery is sufficiently discharged and the ΔSoC is respectively

higher than a predefined value where an accurate SoC estimation (e.g., ΔSoC > 60%) is

possible [121]. At the same time, a moderate temperature change from an initial state

has to be met; otherwise, the influence of temperature on the battery capacity, as

described in previous subsection, cannot be neglected. These conditions are checked in

step 6. Finally, in step 7, the capacity of the battery is determined as defined in Eq. (22)

where the transferred charge from step 3 is divided by the estimated ∆SoC. An accurate

current and voltage measurement of the battery in the vehicle is essential. Especially for

step 3 (determination of the transferred charge), it is important that the current is

measured as precisely as possible. Otherwise, a logged current offset will be considered

in the measurement and consequently in the model, which will result in biased battery

capacity estimation.

3.2.2 Methods based on measured or estimated OCV

As discussed in subsection 3.1, one common method for estimating the battery SoC given

in Eqs. (21)-(22)) is to use the available SoC-OCV correlation. The use of SoCV in

latter equations yields in the following two possible on-board techniques for determining

the value of the OCV (or, strictly speaking, the EMF21) and the SoC both after current

interruption or when the battery is under load:

21 As described in chapter 2, the term “OCV” is used as an equivalent to EMF.


The OCV can be measured accurately after enough periods in the idle mode when

the battery is in its equilibrium state.

The OCV and, accordingly, the SoC can be estimated under load using model-

based techniques by applying one of the state identification methods.

OCV estimation techniques during an idle mode of the battery 3.2.2.1

As shown in chapter 2, less relaxation time is needed at higher temperatures than at

lower temperatures since diffusion processes and chemical reactions taking place inside

the cell are faster at higher temperatures than at lower temperatures, as first postulated

by Fick’s law [69], [122].

Ref. [119] presents an algorithm for OCV and capacity prediction after the battery is

charged by the CC-CV method. The focus lies on the consideration of overpotentials (i.e.,

ohmic overvoltage, due to charge transfer overvoltage or due to electrolyte migration and

due to solid state diffusion) during the relaxation process and its change over the battery

lifetime. In Ref. [121], an on-board fitting algorithm of the OCV relaxation model to the

measured OCV progression with a capacity estimation accuracy of 2% is developed. This

model is based on an ECM consisting of one ZARC-element and a voltage source in

series, representing the OCV. Presented model is fully adaptive and considers the

change of the battery impedance characteristics over its lifetime. Refs. [123]-[124]

present an adaptive OCV estimation method based on LS method for a second order

ECM. A linear dependency between a time constant of a diffusion element and the OCV

is detected for LIBs using LMO and LFP cathodes. For both chemistries, OCV is

accurately estimated within 20 minutes after a current interruption.

Further BMS algorithms and empirical methods for OCV prediction by observing and

fitting the OCV-curve during a relaxation process for a predefined time range are

discussed in Refs. [57], [125]-[126]. The advantage of the OCV prediction method is that

it is not necessary to wait many hours until the battery reaches a steady state, as the

relaxed voltage value can be predicted after mere minutes. Inaccuracies can occur

because of the battery’s potential behavior, such as the hysteresis effect or a flat OCV curve over a wide SoC range. Furthermore, in order to improve the estimation accuracy,

the possible change of SoC-OCV correlation over the battery lifetime need be considered

[52]. However, depending on the application, using such a method, even if there is

potential for an evident lack of accuracy, could be beneficial regarding the capacity

estimation for applications such as taxis due to their low idle time and the lack of

opportunities to wait a defined relaxation time.


The application of the OCV prediction technique is meaningful when a pronounced

dependency between OCV and SoC is available. For example, this technique can be

employed with sufficient estimation accuracy for LIBs using NMC or NCA cathodes;

mainly because of their steep OCV curves for almost the entire SoC range. The

algorithms mentioned above are summarized in Table 7 by means of their

implementation complexity.

Table 7: A selection of open-circuit voltage-based literature sources for on-board capacity estimation

Employed Methodology Relevant possible

references

Complexity of the model

implementation

Using the SoC-OCV relation directly after

the charging and discharging process [117], [120], [126] Low

Using empirical-based models for OCV

prediction [57], [127]-[128] Medium

Using adaptive filters for capacity

estimation

[119], [121],

[125],[129] High

OCV estimation techniques during an operation mode of the battery 3.2.2.2

As mentioned previously, another possible way to estimate the battery capacity is to use

the voltage obtained through an ECM. The basic principle of this method is based on the

comparison of the measured and the simulated battery voltages where the states and

parameters of the battery model are estimated. An accurate determination of OCV from

an ECM is highly dependent on the employed battery model. The more accurate the

battery model is, the more accurate the determined battery overvoltages and the battery

OCV can be. Nevertheless, consideration of all overvoltages, especially overvoltages with

higher time constants (some hours), is hardly possible with ECMs. This means that the

extracted OCV from Eq. (20) is not the same as the OCV discussed beforehand.

Considering the limited available computing power of BMS, reducing the model

parameters to a minimum by trying to depict the electrochemical processes as accurately

as possible becomes necessary. In this regard, ECMs employed in the BMS often consist

of a C , a simple linear R0 and a number of RC-elements connected in series. The

number of applied RC-elements depends highly on the application, relevant time

constants and required estimation accuracy. A higher number of RC-elements yield

higher estimation accuracy whiles at the same time the computational effort increases

[130], [131]. An example of a possible ECM for use in monitoring algorithms is shown in


Fig. 39. Each element of the ECM refers to a certain electrochemical process during

battery operation or even in a battery steady state.

Fig. 39: An example of an equivalent circuit model employed in battery management systems considering various time constants

In order to ensure high estimation accuracy over the battery lifetime and under various

operating conditions (e.g., various temperatures and SoCs), ECM parameters need to be

adapted to the actual aging state of the LIB on-board [132]. In Ref. [86], the authors

show that the estimation accuracy could decrease over the battery lifetime even if the

parameters of the ECM are adapted to the actual aging state of the battery. Algorithms

presented in many researches in the past were often validated under nominal conditions

or when the battery was in a new state.

As a common solution for ECM parameters estimation, the battery model is represented

in a discrete-time state-space form. The methods differ in the way how the states and

parameters of the model are estimated. Generally, it can be differentiated between the

following two techniques [133]:

Adaptive joint estimation technique,

Adaptive dual estimation technique.

With the joint estimation technique an adaptive filter is applied on a single vector of

unknown parameters and states for their estimation. The dual estimation technique

uses two filters in parallel. One filter is used for the state estimation and the other one

for the parameter estimation. A selection of well-known methodologies that are often

applied for the on-board estimation of the ECM parameters is listed in Fig. 40.


Fig. 40: A selection of commonly applied techniques for ECM parameters estimation

In the following subsections, the available algorithms using adaptive filters and observer

techniques for on-board capacity estimation are discussed in more detail. A comparison

of the techniques is conducted, and the complexity of the model implementation is

presented.

3.2.2.2.1 Least squares-based methods

The underlying idea of LS-based algorithms is very simple; it is mainly based on the

minimization of the sum of squared prediction errors. The LS-based filters consider

input data, model states and parameters as deterministic signals. These filters are

simple to handle for on-board estimation purposes, especially because of their simple

implementation and low computational effort in embedded systems. Therefore, the

implementation of LS-based algorithms on microcontrollers for automotive applications

is very popular. Among the main disadvantages of LS-based algorithms are their

divergence problem and the fact that they are not suitable for application on very

complex ECMs. There are different classifications of LS-based filters known for time

varying parameters estimation, such as recursive least square (RLS), recursive extended

least square (RELS), least square with forgetting factor, weighted least square (WLS),

weighted recursive least square (WRLS) etc. [134]. However, their divergence problem

and inapplicability on very complicated nonlinear battery models can be considered

disadvantages [87].


In Refs. [118], [120], [135], on-board capacity estimation algorithms using RLS for ECM

parameters identification are proposed. Developed algorithms are evaluated for both

driving and charging mode in PHEVs. Refs. [136]-[138] present WRLS-based methods for

extracting ECM parameters. From estimated impedance parameters, based on a

combined weighted filter, SoC and SoH of the battery and, correspondingly its actual

capacity are estimated. Joint estimation techniques based on RLS and an adaptive

extended Kalman filter (AEKF) for online parameter identification and SoC estimation

employing a first order ECM are applied in Ref. [139].

Moreover, in Refs. [60], [140]-[141], the authors investigate the voltage behavior of LIBs

using LFP cathodes due to capacity degradation. Changes in the OCV-curve observed,

particularly at high SoC ranges, but only the constant current part of the charge curve is

investigated. The drawback of this cell’s behavior is that the changes of the OCV curve and the consequent capacity loss can be detected when the cell is degraded enough. The

authors propose detecting the capacity loss by defining a prediction residual between the

estimated OCV from a simplified ECM (by employing the RLS-method) and the

measured one. The residual (∆OCV) values increase over the battery lifetime while OCV

behavior of the LIBs change. This is then used for capacity estimation by comparing the

actual residual value and a predefined threshold. The proposed technique is only verified

for LFP-based LIBs, and only battery aging due to cycling and its influence on OCV

characteristics are investigated whereas the impact of calendar aging processes is

neglected.

In Ref. [142] charging curves of LIBs with different chemistries are investigated for SoH

estimation. Based on the observation of a CC-regime of a charging curve, a

transformation function is derived and model parameters are identified by employing a

nonlinear LS technique. After examination of the results, an absolute difference of

approximately 2% between the estimated and measured SoH over 1800 cycles is

achieved. Because of the simplicity and promising results of the investigated algorithm,

it is definitely one of the most promising methodologies for on-board capacity estimation.

In Ref. [143], the authors investigate the on-board capacity and SoH estimation by

investigation of charging time and current for four different LIB types. Unlike the above-

presented methods, the authors investigated both charging regimes (including the CV-

regime) of the curves. The authors found a correlation between the charging time during

the CV-regime due to the intercalation process in this phase and the capacity loss of


LIBs using LFP cathodes. However, its applicability in the field needs to be investigated

in more detail.

3.2.2.2.2 Kalman filter, similar filters and observer technique estimation based

methods

Algorithms based on the KF technique are widely spread in the field of monitoring

algorithms. KF was for the first time presented in 1960 by Kalman in Ref. [144].

However, since KF can only be applied to linear systems and batteries show a highly

nonlinear behavior (e.g., current dependence of the Rct etc.), it is often used in

combination with other extended filters (e.g., LS, Extended Kalman filter (EKF), Sigma-

Point Kalman filter (SPKF) etc.). The working principle of KF-based algorithms is very

simple, whereby the simulated battery voltage is compared with the measured voltage

and based on their resulting difference (ε) which is amplified with K (the Kalman gain),

states of a state space model are updated to minimize ε. Contrary to LS-based methods

KF accepts only stochastic inputs with zero-mean Gaussian covariance.

However, there are some disadvantages of the KF-based algorithms which need to be

addressed: a) required complicated matrix calculations (i.e., Jacobian matrix), b) the

linearization of the nonlinearity of the battery characteristics and the necessity of

knowing a priori the variable of the system noises, such as mean value, relevance and

covariance matrix. An inconvenient information matrix of the system noise may yield

estimation error and parameters divergence, respectively [67], [145]-[146].

As further improved versions of the KF methodology, the EKF, AEKF, and sigma point

KF (SPKF) (including unscented KF (UKF) and central difference KF (CDKF)) can be

addressed. In the system theory, EKF is widely used for the state estimation of

nonlinear systems where at each time step a linearization process of the system by

means of a linear time-varying (LTV) system is performed. It is assumed that the noise

processes are zero mean white Gaussian stochastic processes of covariance matrices

which refer to the uncertainty of the state estimation. As a main difference to the KF

algorithm, battery nonlinearities (nonlinear state transition function and nonlinear

measurement function) are approximated by means of first-order Taylor-series

expansions at each time step [108], [133], [147]. In this context, different approaches

using EKF as joint or dual estimator for SoAP predictions are discussed in great detail in

Refs. [148]-[149].

To overcome the disadvantages of EKF, such as linearization of the nonlinear battery

model and high computational effort, more improved algorithms, such as UKF or CDKF,


which both belong both to the SPKF class, can be used for more accurate and robust

battery state estimation without the need for system linearization or calculation of the

Jacobi matrix. The latter technique approximates the probability density of the state

estimation by using characteristic points, so-called sigma points [150]. In comparison to

EKF, SPKF algorithms do not need the linearization of the equation at each sample

point by using a Taylor-series. SPKF uses an unsteady, function evaluation for

covariance matrix estimation. In the literature, there are many different algorithms

known from this class and discussed for battery state estimation. However, an extensive

exploration of the aforementioned methods is out of the scope of this work. In the

following, some relevant methods and papers will be introduced and discussed.

Ref. [151] introduces an algorithm whereby SoC and battery capacity are estimated by

utilizing an EKF. SoC is estimated by employing EKF, while a RLS filter is used for

ECM parameter estimation. Based on the resulting SoC, capacity is derived using the

difference between the SoC increment of a fresh cell and the SoC increment at present

aging state of the battery. The more accurate the estimated SoC increment for a defined

sample data is, the more accurate the battery capacity that can be estimated by the

employed technique is. The authors show estimation accuracy of less than 4% employing

New European Driving Cycle (NEDC) as a reference profile for LIBs at different aging

states.

In comparison to the described EKF technique, the AEKF improves the estimation

accuracy by an adaptive update of the noise covariance. In Ref. [152], the authors

present an algorithm based on adaptive joint filter methodology using AEKF for state

estimation of LIBs with LFP cathode materials. Furthermore, they proposed an

algorithm for on-board capacity estimation called the iterative transferred charge

method. The basic idea of this methodology is almost the same as that presented in Eq.

(22). An absolute error of 4% for battery capacity estimation over its life-time was

achieved using this methodology.

A model-based approach for on-board battery state estimation is presented in Ref. [153].

Moreover, a battery capacity and driving range prediction methodology for EVs is

presented. For this reason, EKF and ∞-Filter for SoC estimation is applied, and the

corresponding results are compared and analyzed. Based on Eq. (22), the battery

capacity and driving range are estimated. While, for an accurate performance of KF

estimation methods, the priori noise is assumed to be zero-mean Gaussian, estimation

techniques based on a sliding mode observer or ∞-Filter actually do not require this


prerequisite. Maximum estimation error is minimized while an alternative feedback gain

for considering estimation errors and measurement noises is applied. After analyzing the

results of both models, the authors found almost the same performance and convergence

speed for EKF and ∞-filter.

Almost the same methodology has been proposed in Refs. [154]-[155]. A dual EKF, which

employs two EKFs in parallel based on a simplified ECM, is implemented. The first EKF

estimates the SoC, and the second one, the so-called weight filter, is responsible for

capacity estimation. Because of the high complexity of the proposed algorithm and the

compensation for the model error, the above filters are extended by the measurement

noise model to separate the state and weight filter. The algorithm was validated with a

maximum absolute capacity estimation error of 5%.

A dual extended Kalman filter (DEKF) combined with pattern recognition based on a

Hamming network, which is normally used for binary pattern search, is presented in

Refs. [156]-[158]. Two patterns have been investigated for SoC, capacity and,

respectively, SoH estimation at different temperatures. The investigated patterns, so-

called the capacity pattern and (dis)charging voltage pattern are predefined from

laboratory measurements, while the input-patterns are compared to find the closest

using statistical analysis. Estimation results within a range of ±5% for SoC and capacity

are achieved.

In Ref. [159], the authors investigate the combination of the EKF technique and a per-

Unit system for SoC and capacity estimation by employing a simplified ECM

(considering ∆ and diffusion overvoltages). A capacity estimation error of less than 5%

is achieved. However, due to the high computational effort of the proposed algorithms,

intensive simplifications need to be carried out for their application in the field. In Refs.

[160]-[162], an algorithm based on dual sliding-mode observer for SoC and SoH

estimation is presented. This algorithm consists of two observers, where the first fast-

paced time-varying observer is responsible for battery parameter and voltage estimation

and the second slow-paced filter for SoH estimation in terms of battery capacity and

resistance. The Lyapunov equation is applied to ensure the convergence of both

observers. During each sampling time, the state and parameter observers are updated

after each other. The state observer is updated using previous estimated parameters

from the second observer. Subsequently, the parameters are updated by using the new

estimated state values.


A dual joint estimation filter is introduced in Ref. [116], whereby SPKF is used for SoC

estimation and various total least square (TLS) algorithms for the second filter are

employed and compared for capacity estimation. After analyzing different TLS-based

filters, promising results could be achieved from the weighted TLS (WTLS) method,

whereby noises on both the accumulated charge and SoC difference are considered. The

algorithms are verified only by using simulated battery data. As a further investigation,

the adaptability of the WTLS to embedded systems with real vehicle data could be

performed.

Refs. [163]-[164] present an adaptive EKF-based algorithm for SoC and SoH estimation

on lead-acid battery example. However, due to a difference between the SoC-OCV

relationship of lead-acid batteries and LIBs and their various behaviors during the

battery lifetime, the exact implementation and use of the proposed methodology for LIBs

need to be optimized and adapted. A combined SoC and SoH estimation method using a

dual Kalman Filter in combination with a support vector machine is presented in Ref.

[165]. The presented dual KF consists of a simple KF and UKF for SoC and capacity

estimation, respectively. The support vector machine (SVM) is implemented and

additionally coupled with a dual filter to predict the RUL of the battery. Refs. [150],

[166]-[167] used a CDKF filter that affords lower parameter count than UKF using a

linear parameter varied for a simplified ECM. The advantage of the presented algorithm

is the need for only one filter that is able to estimate a battery’s internal resistance and capacity for SoH determination as a parameter and SoC as a battery state. The needed

computational effort is, accordingly, greatly decreased, and promising performance could

be observed due to the robustness of the algorithm.

An alternative to the KF is a particle filter (PF). The main difference between them is

that the PF does not need an assumption of the zero mean Gaussian distribution of

model states and noises and it can generally deal with any Probability density function

(PDF) by applying the Monte Carlo method. For this reason, a limited set of samples are

employed. In Refs. [168]-[170], SoC and capacity estimation techniques based on PF

methodology are investigated for EVs and stationary applications. After long-term

simulation tests for EVs, accurate results are achieved for capacity estimation with an

estimation error of 2% or better, whereby the authors propose the applicability of their

algorithm for on-board implementation on a microcontroller. According to the authors,

the computational effort of a PF increases linearly with the amount of defined samples.

However, employing a reasonable amount of states and samples and their determining

techniques yields a well compromise for its implementation on a microcontroller.


Moreover, in Ref. [171], the authors propose a combined model employing genetic global

optimization algorithm and Levenberg-Marquardt local optimization algorithm to

estimate the impedance parameters of the ECM for SoH estimation.

The discussed algorithms and respective literature sources for on-board capacity

estimation based on ECMs are listed and compared in Table 8.

Table 8: A selection of ECM-based literature sources for on-board capacity estimation of LIBs

Possible relevant

References

Employed

Methodology

Combination with other

adaptive filters required

Complexity of the

model

implementation

[116] LS SPKF Medium

[60], [118],[135],

[140], [141], [172] RLS No Low

[154]-[155] DEKF No Medium

[152] AEKF Iterative transferred charge Medium

[150], [166]-[167] CDKF No High

[165] Dual Filter

(KF+UKF) SVM High

[168]-[170] PF No High

[93] Genetic

algorithm

No Medium

EKF (for SoC estimation)

3.2.3 Methods based on electrochemical models

The models discussed in the previous subsection are generally preferred for

implementation on low-cost microcontrollers mainly because of their simplicity and their

high robustness. However, the main drawback of such models is the less pronounced

chemical and physical meaning of parameters. This results in increasing estimation

inaccuracy over the battery lifetime, as different aging processes cannot be tracked

accurately. As an improvement with more emphasis on the physical meaning of

parameters at the expense of computational effort, electrochemical-based models can be

noted [173]. In the following, methodologies regarding on-board capacity and SoH

estimation based on these models and their principals are discussed. More detailed

explanation regarding electrochemical battery modeling and their pros and cons will be

given in 3.5.


In Refs. [174]-[176], the authors investigate electrochemical models based on pervasively

accepted single particle model (SPM) theory. The scope of their research lied in the field

of on-board SoC and SoH estimation. For this reason, an observer technique based on

partial differential equations (PDE) has been investigated. The corresponding SoH

estimator algorithm was divided into two parts: i) the capacity estimator ii) the power

fade estimator (impedance rise). Due to the challenge of on-board parameter estimation

by the PDE technique, it is therefore combined with a first order pad 22 approximator

(for the estimation of the diffusion coefficient), a backstepping PDE estimator and a LS

technique.

In Refs. [177]-[179], the authors investigate a SPM model based on a UKF methodology

for SoC estimation. LS technique is used for estimation of battery capacity and internal

resistance rise, which is consequently connected to SoH estimation. As a parameter for

the detection of the capacity loss, the porosity of the cathode (ε, the volume fraction of the intercalation particles) was considered; for power fade detection, the decrease of the

effective electrolyte conductivity (keff) has been considered. One advantage of the

presented model is the ability to distinguish between the capacity loss due to impedance

increase and due to active material loss. Unfortunately, only the LIB degradation of the

cathode material is examined, which is possibly a limiting factor for accurate capacity

estimation. A control-oriented simplified SPM for estimating SoH of the LIB and

determining battery’s impedance and solid phase diffusion time of lithium ions in the positive electrode is introduced in Ref. [180]. The coefficients of the third-order pad approximated transfer function (used for discretization of the SPM) are related to aging

relevant parameters (capacity and impedance). The linear increase of the total resistance

is related to the increase of the Rct and contact resistance, while the increase of the

diffusion time as battery ages occurs because of the SEI layer’s growth of active particles of the positive electrode, according to the authors. Furthermore, a non-monotonic

increase in the capacity factor is observed, which can hardly be used as SoH indicator.

In Ref. [181], a comparison between SPM and a semi-empirical electrical model is

performed. Parameter identification is also performed by employing a nonlinear LS

technique. Several criteria, such as root mean square error, sum of the squared error,

and voltage prediction boundaries, are investigated for comparison of both models. Both

models require the same computational time, while the SPM model delivered more

accurate results due to voltage prediction.

22 A Pad approximant corresponds to a simple function represented by a ratio of two power series. Its main difference

from Taylor series is the possibility to match the Taylor series of the function that needs to be approximated [289].


Furthermore, Ref. [93] compare a simple second-order ECM (consisting of 6 adaptive

parameters) with an extended ECM based on the SPM model (consisting of 10 adaptive

parameters), whereby the focus of their research lie in the improvement of the voltage

estimation accuracy in the low SoC range (SoC < 20%). The concentration of lithium in

the electrode particle (describing solid-phase diffusion processes) represented by the

surface SoC in the developed model is employed in order to enhance the estimation

accuracy. According to the authors, the estimation accuracy through the improved model

(extended ECM) could be improved by 50%. A selection of literature sources dealing with

electrochemical-based models for battery capacity and SoH estimation can be found in

Table 9.

Table 9: A selection of electrochemical model-based literature sources for on-board capacity estimation of LIBs

References Employed

Methodology


adaptive filters required


implementation

[182]

PDE observer

No

High [174]-[176] LS

[177]-[179] LS UKF Medium

[181] LS No Low

[180] Gradient-based

recursive estimator LS Medium

3.2.4 Methods based on Incremental capacity analysis and differential voltage analysis methodologies

The investigation of LIBs inner chemical reactions and aging mechanisms can be carried

out by means of differential mathematic approaches, such as ICA or DVA. Generally,

these approaches are widely applied to investigate the behavior of a battery in a

laboratory environment in order to characterize the chemical and physical processes that

take place inside the cell during its operation and to examine how each of the processes

evolves over the lifetime of the battery. Recently various concepts and techniques have

been developed in order to address the use of these methods, including on-board capacity

loss estimation for EVs.

ICA has been already investigated as a reliable offline tool to study the electrochemical

behavior of a battery in Refs. [183]-[185]. The analysis is achieved by the differentiation

of the charged or discharged battery capacity in respect to the terminal voltage. The

obtained curves V – dQ/dV transform the plateau present in the cell voltage trend

(representative of electrochemical equilibrium phases during the battery operating


conditions) into peaks, with different amplitude depending on the cell chemistry and

aging conditions. Refs. [183]-[185] have demonstrated how powerful the differential

analysis is as a non-invasive tool, which does not require the physical separation of each

electrode. The investigations have shown the feasibility of using the ICA to observe how

the characteristic peaks of the curves change in terms of amplitude and position over the

battery lifetime, with the intention of detecting the different aging mechanisms that

take place in the LIB.

This evidence makes the offline SoH estimation relatively simple compared to a post-

mortem analysis, in terms of aging mechanism identification. As will be discussed

further later on, and as can already be foreseen, one of the drawbacks and limitations of

this method lies in the fact that the cell has to be charged or discharged with a constant

current in the entire voltage region where at least one of the peaks is detectable; with

the compromise that the current rate has to be limited enough to show the voltage

plateau, which is otherwise covered from the resistance voltage drop in the case of high

load current operation.

The DVA investigates the working operation of a battery through the differentiation of

the terminal voltage trend with respect to the (dis)charged capacity, whereby the

obtained trend Q – dV/dQ represents, for each peaks, the transition between two

electrochemical equilibrium conditions during the cell operation. As for ICA, also in this

case, several studies e.g., Refs. [186]-[190] have investigated how the curves obtained

through this analysis change over the battery lifetime, in qualitative and quantitative

terms. For example, Ref. [190] applies the DVA analysis to the full and single electrode

voltage curves, and further derived a model to predict the capacity loss due to aging. The

proposed model is able to detect irreversible capacity loss and useable positive and

negative active material losses during calendar and cycle life. Furthermore, the

potentiality of the DVA can also be extended to obtain information relative battery SoC,

as the position of each peak can be addressed to the quantity of lithium charges present

in the positive and the negative electrodes, and consequently to the SoC.

The possibility of using the aforementioned methodologies in a BMS for on-board SoH

estimation has been introduced in Ref. [191]. In the latter specific case, a methodology

for using the ICA method on EVs for on-board battery capacity estimation is proposed.

Firstly, the lifetime modification of the ICA curves for 8 LFP-based LIBs cycled at the

same conditions is investigated as already demonstrated in Ref. [184]; according to the

obtained results, the change in the positions and, mainly, in the height of the ICA peaks


over the lifetime for this type of LIB chemistry is significant, thus the information

obtained from the differential analysis can be used directly for the estimation of the

battery capacity. Furthermore, the authors explain how the ICA curve can be

determined on-board taking into account that for PHEVs and EVs, it is possible to

occasionally have a charge curve with constant current, they chose to apply ICA only

during the charging process and in a defined voltage range. The assumption is that one

of the peaks must be clearly visible and detectable, i.e., a complete plateau in the voltage

curve has to be measured. Examples of applications of ICA and DVA on the charging

process for SoH estimation through SVM method can be found in Refs. [34], [192]-[194].

First, an online fitting process of a piece of the charge curve − is accomplished, and

then a numerical differentiation of the curve is carried out in order to obtain the

characteristic peak curve. The height of the peak is then compared with the offline data

to obtain a first estimation of actual capacity of the battery.

Moreover, the validation of the SoH estimation is performed for different LIBs of the

same type. Nevertheless, some critiques of the method can be found by considering the

real on-board conditions that can be encountered during the normal operation of EVs.

First of all, the authors explain that the method can be applied in each moment to a

battery voltage profile that does not necessarily have to be an OCV curve. However, as

soon as ICA is applied to a charge voltage curve obtained under high current rates, the

possibility to detect the peaks of the ICA curve corresponding to phase transformation of

the cathode becomes limited, thus it is suggested to obtain ICA curve only when the

current rate is low. This is due to the different voltage polarization that a cell can

present over a wide SOC range. Moreover, unfortunately, for online application is shown:

the necessity of testing such an approach in a battery pack, where the cell variability

plays a fundamental role, cannot be neglected.

An alternative approach using differential analysis methods for on-board capacity

estimation is proposed in Ref. [195]. The aging behavior of LFP-based LIBs is examined

by applying cyclic voltammetry (CV) analysis. According to the authors, the method

gives the same qualitative results as the ICA and DVA methodologies [196]-[197].

Thereafter, the authors discuss the difficulties of applying the ICA and DVA for on-board

battery capacity estimation due to the following requirements:

The voltage samples have to be preciously measured and collected with an

adequate frequency, especially when the current rate is limited enough in the SoC

range in which the plateau is completely visible,


The collected samples have to be filtered or smoothed in order to obtain a correct

and feasible differentiation analysis as they would otherwise be unclear and full

of useless small peaks,

Fitting and a differentiation process have to be accomplished,

At the end, a comparison between offline and online data has to be performed.

This is a complex issue, as the information, in terms of the identified aging

mechanism, has to be converted into a single cell capacity value.

Due to these conditions, the authors proposed the using PDF as a simple approach that

is comparable qualitatively and quantitatively with differential analysis. The advantage

of the PDF method is not only to be restricted to the on-board SOH estimation, whereas

the PDF method realizes the discretization of the ICA/DVA method. Such a

discretization provides two conveniences in battery SOH research: First, when

processing ICA/DVA, the derivative through probability counting of the sampled voltage

using the PDF method can directly be derived without curve fitting. Second,

discretization property of the PDF can be easily applied to on-board SOH estimation.

However, the PDF method needs some calibration work before being applied for on-board

use.

In fact, when the PDF function is applied to the battery terminal voltage curve, the

result is again a curve with peaks in the correspondence values where voltage plateaus

are detected, or simply speaking, each peak expresses the number of measurements in

which the same voltage value is detected. The on-board application of the PDF is

achieved by comparing the curve measured offline (e.g., stored in the form of a simple

lookup table) with the information gathered on-board, taking into account that:

The voltage curve in which the PDF is applied has to be obtained with a constant

current process,

The (dis)charge current has to be sufficiently limited to allow for clear detection of

the voltage plateau,

The (dis)charge process has to cover at least the voltage range in which a

complete plateau can be detected.

In Ref. [192], PDF method combined with a fuzzy logic approach is proposed. The main

advantage of the fuzzy logic method is the grant of particular level of uncertainty and

ambiguity in transforming noisy data [108]. However, the applicability of this method in


a complex system (which is a single cell or a battery pack) in everyday life conditions has

not yet been proven and needs further investigations.

In Table 10, a selection of the relevant mentioned references are shown and compared.

Table 10: Incremental capacity analysis and differential voltage analysis-based methods for capacity estimation

Employed

Methodology References

Adaption of model parameters to the actual

aging state of the

battery

Complexity of the

model

implementation

SVM [191], [193] Yes Medium

PDF [192], [195] Yes Medium

3.2.5 Methods based on aging prediction models

The aging behavior of LIBs under different operating conditions has been deeply

investigated in the past and in particular, recently. The aim is generally to understand

how the energy and power capability change over the battery lifetime by modifying the

conditions under which a defined LIB operates. The main concern of this approach is

that, normally, lifetime tests are very time consuming: in fact, considering that LIBs are

able to reach more than thousand cycles, the goal is to accomplish the tests in a

reasonable time and, at the same time, to gather enough information in terms of battery

lifetime.

Generally, the test results are finalized by obtaining a model that describes the battery

capacity loss or resistance increase during the lifetime based on operating conditions

(actual voltage, current, temperature, etc.). Aging models attempt to predict the capacity

and internal resistance in terms of SoH estimation based on actual operating conditions

and past history. Several authors have already shown numerous results following this

procedure [198]-[201].

Refs. [33], [66], [202] investigate the aging behavior of LIBs using NMC cathodes in

storage and cycling tests for different temperatures and SoCs. The obtained capacity and

resistance model predictions are validated with measured data obtained from cycling

tests, with accurate approximation for capacity loss prediction. In Ref. [199], the authors

investigate a semi-empirical cycle life of a LIB with LFP as cathode material by using a

large test matrix. Experimental results show that the capacity loss is strongly influenced

by time and temperature, while, for high current rates, the charge and discharge rate is

more relevant. Similar results were obtained from Ref. [198] for capacity loss both for

storage and cycling, while Ref. [200] proposed a prediction model based on accelerated

aging tests able to describe the internal mechanism for capacity loss. The authors in Ref.


[201] carried out accelerated aging tests on LFP-based LIBs and developed a model that

correlates the effect of combined multiple stress factors on the capacity degradation, with

the goal of predicting the lifetime of the battery under dynamic operating conditions.

The necessity of building a prediction model can be generally related, on one hand, to the

lifetime prediction of the battery when it is subjected to some defined working conditions

or, on the other hand, to the possibility of using this on-board information for capacity

estimation. Another important contribution is provided in Ref. [203]. The authors use

the data obtained from accelerated aging tests as input for a SVM, which is a powerful

machine learning algorithm, in order to obtain an identification system able to work in

an on-board environment for diagnosis and prognosis by considering temperature,

current rate and SoC. The basic idea of such algorithms is to set a vector as an input of

the proposed model and another as its target values. The goal is then defined as

updating the model parameters by observing the dependency of output and input values,

which enables the possibility of making an accurate prediction of the output vector in the

future.

In Refs. [204]-[207], a machine learning methodology based on a relevance vector

machine (RVM) and a PF from a Bayesian network framework is presented. RVM, as

proposed for example, in Ref. [203], is responsible for identification of an employed

model. The main difference between RVM and SVM is the PDF as an output of the

employed models. The PF is used for estimation of the RUL of the LIB in the form of a

PDF. The main difference of the investigated model from the model mentioned in the

previous subsection is the investigation of in series with the to Rct. LIBs using NCA

cathodes are investigated and based on the mentioned techniques, the battery’s state and parameters are estimated.

An alternative novel data driven technique-based on sample entropy (SampEn) has been

recently shown by some authors. Richman and Moorman presented a SampEn algorithm

for the first time in the field of health prognosis [208]. The authors in Ref. [209]

performed HPPC tests on different LIBs using NMC cathodes, whereby SampEn of a

responding voltage during a constant current regime is used as an input for an LS-based

estimator. Ref. [210] shows an algorithm based on a determined surface temperature of

the battery during the charging process. The authors have developed a model using

information gathered during the battery’s life-time in combination with PF algorithms

for remaining battery capacity estimation. Furthermore, Ref. [211] presented a combined


model consisting of a machine learning technique based on SVM methodology and a

SampEn as their input vector regarding network training for SoH estimation.

Moreover, in Ref. [100], the authors tested LIBs in a cycling condition using two dynamic

profiles, the so-called charge depleting and charge sustaining profiles, while varying the

temperature and the average SOC. In Ref. [212], cylindrical LIBs with five different

dynamic profiles for roughly 120 days in order to investigate the aging path dependence

due to the magnitude and frequency of the power pulses and due to the variation of the

temperature during cycling are cycled. In Ref. [213], an energy-based battery model was

developed and investigated; the model consists of calendar and cycle aging prediction

parts. Different scenarios, such as various driving cycles, charging strategies and peak-

shaving, and their aging effect on battery lifetime have been simulated. In Ref. [214], the

authors designed a system based on ECMs that model the behavior of each cell electrode;

summing up the contribution of each electrode the total cell voltage can be simulated

and reproduced. The inputs of the model are the degradation mode, i.e., the type of

mechanism involved in the aging process. In this case, it is necessary to have a system

that is able to predict which type of degradation has taken place inside the cell

depending on the actual and past operating condition.

The above mentioned algorithms and references are listed and concluded in Table 11.

Table 11: Aging model based methods for capacity estimation

A selection of

relevant references

Employed

Methodology


filter required


implementation

[203] SVM No Medium

[204]-[207] RVM PF Medium

[209]-[210] SampEn PF Medium

3.3 State-of-Available-Power prediction

SoAP is a further battery state that must be determined continuously by BMS in EVs.

Up to now, different approaches for SoC and SoH estimation of LIBs have been shown

and reviewed in the literature by many authors in the past. However, the topic of SoAP

prediction has not been explored enough sufficiently yet and there are still a lot of

researches required to optimize, improve and understand this challenging task. As a

general rule, the time horizon for SoAP prediction lies between 1 s and 20 s in EVs [26].

By knowing the available battery power in addition to the actual battery SoC and SoH, a


reasonable EMS strategy can be applied and specified vehicle functions, such as vehicle

acceleration, deceleration or gradient climbing can be performed without exiting battery

SOA and affecting the lifetime of the battery or causing safety damage, respectively

[215]-[217]. Fig. 41 shows schematically the interaction between various battery states

and SoAP in the context of a simplified BMS architecture.

Fig. 41: Schematic illustration of possible interaction between different battery states/parameters and State-of-Available-Power in the context of a simplified battery management system architecture

One of the state-of-the-art approaches used for static power capability determination is

the HPPC method presented by the partnership for new generation vehicles (PNGV)

battery test manual, published by the Idaho National Engineering & Environmental

Laboratory of the U.S. Department of Energy [218]. An improved method for the

determination of power capability by means of pulse tests with some modifications

regarding applied current rates, resting time etc., was presented in subsection 2.3. In

fact, accurate results may be achieved by applying this technique in laboratory

environments. But under real conditions, such as in vehicles where available peak

current or voltage for a specified time horizon need to be known, accurate power values

are not provided and the results are mostly overestimated since only the operational

design limit of battery voltage is considered [219]-[222]. Fig. 42 shows the dependence of

battery power capability on SoC for Cell-C-new and Cell-C-aged measured at 23 °C. The


available battery power is determined based on resistance values obtained as a voltage

value reached after 20 s from the beginning of the current pulse divided by the current.

Fig. 42: Power capability of Cell-C-new and Cell-C-aged at 23 °C determined as proposed in Eqs. (23)-(24); (Vcell,min=1.5 V, Vcell,max=2.75 V)

The power fade occurred over the battery lifetime directly influences the driving

performance of the vehicle in terms of acceleration or battery charging during

regenerative phases and charging periods [223]. From the relationship between battery

SoC and power capability shown in Fig. 42, it becomes obvious that the charging power

capability decreases with the increase of SoC and vice versa. The maximum available

power of a lithium-ion battery pack is derived by investigating the battery’s VOCV and

resistance as follows [106], [215], [224]:

Pdischarge= [Vcell,min∙ ( C , - Vcell,min)/Rcell,discharge]∙ , ∙ , (23)

Pcharge= [Vcell,max∙(Vcell,max - C , )/Rcell,charge]∙ , ∙ , (24)

Based on Eqs. (23)-(24), it can be concluded that the power fade over the battery lifetime

is mainly influenced by the increase of battery impedance. The current that can be

applied to the battery can simply be derived from Eqs. (23)-(24) after some mathematical

calculations from [224]. It is worth noting that the applied technique for determination

of the battery power capability is independent of the battery chemistry used. However, it

is important to ensure the constant battery temperature and to adjust the correct SoC

level during the tests.


As discussed in subsection 3.2, the EoL criterion of the battery is mainly dominated by

actual capacity and actual impedance of the battery. In fact, in terms of available battery

power prediction the latter parameter plays a major role. In Ref [106], the authors

examine the power capability of a battery pack at different aging states using three

different driving cycles obtained from an EV. The obtained results indicate that when

the available battery power fades down to 50% (i.e., the battery impedance is doubled)

the vehicle might have trouble accommodating the discharge or charge power at low and

at high SoC ranges, respectively. Furthermore, according to the results, the decrease of

available battery power leads to the increase of acceleration time, in particular at low

SoCs.

The available battery power is limited by the SOA, where the SOA is defined by battery

temperature, current, voltage and SoC [46], [87]. However, during a single applied

current over a time horizon of several seconds (∆t <= 10 s), the battery’s temperature and SoC do not change significantly. It can be therefore concluded that in practice battery

voltage and current are the major limiting factors for SoAP prediction [87], [225].

Because of safety reasons the battery must be operated in a specified voltage window

( ); this limitation influences the maximum current which can be

drawn from or fed into the battery and it is therefore often used as an indicator for the

battery’s power capability. Consequently, the maximum power which is allowed to be applied to the battery can be determined when the predefined SOA limit of battery

current or battery voltage is reached. It is worth noting that not only the voltage of the

lithium-ion battery pack but also the voltage of individual cells must be monitored.

For this reason, by the beginning of the pulse (t=t0), battery voltage and battery current

at the end of the applied pulse (t=t0+∆t) should be predicted (Fig. 4323). As by the

beginning of the pulse it is not known whether the voltage or the current or both of them

are limiting factors, one must distinguish between the following four different cases

[225], [226]:

Case A: Charging with cell or battery system voltage limitation,

Case B: Charging with battery current limitation,

Case C: Discharging with cell or battery system voltage limitation,

Case D: Discharging with battery current limitation.

23 Fig. 43 shows cases where it is assumed that constant current pulses are applied to the battery; however, depending on

the application, it is also possible that constant power pulses are applied to the battery which means that both the

battery current and voltage would change during the applied power pulse.


In Fig. 43a and 43c, the cases are illustrated where the battery voltage at the end of the

applied charge or discharge current reaches exactly the allowed upper or lower voltage

limit (see case A and case C above). In these two cases the predefined current limit is not

exceeded. At the same time in Fig. 43b and Fig. 43d, the maximum limiting charge and

discharge current is applied to the battery while the upper or lower voltage limit is not

reached (see case B and case D above). In fact, case B is the only case among all the four

listed cases where the battery is charged with the maximum limiting current while the

absolute value of the available battery power is increasing. In the remaining cases, the

absolute value of the available battery power decreases over particular charging or

discharging periods.

Fig. 43: Schematic illustration of four cases for limiting available battery power in case of battery charging or discharging with constant current: a) applying constant charge current (Icharge<Imax) while battery voltage is the limiting factor (in this case the battery could be at very high SoC range); b) applying constant charge current (Icharge=Imax) while current is the limiting factor (SoCcase,b < SoCcase,a); c) applying constant discharge current (Idischarge>Imin) while battery voltage is the limiting factor (in this case the battery could be at very low SoC range); d) applying constant discharge current (Idischarge=Imin) while current is the limiting factor (SoCcase,d > SoCcase,c)


In an example of the ECM shown in Fig. 39, the predicted battery voltage ∆ at

the end of the applied current pulse for a certain time horizon of ∆t can be derived as

follows (Eqs. (25)-(29)): ∆ = ∙ (25) ∆ ∆ = = ∙ −∆ + ∙ ( − −∆ ) ∙ (26) ∆ ∆ = = ∙ −∆ + ∙ ( − −∆ ) ∙ (27) ∆ ∆ = = ∙ −∆ + ∙ ( − −∆ ) ∙ (28) ∆ = ∆ + ∆ ∆ + ∆ ∆ + ∆ ∆ + C oC + oC . ∆. 24 (29)

In addition to the discussed impacts of battery SoC, current and voltage on available

battery power, it is essential not to neglect the influence of battery temperature. Since

applied high charge or discharge current rates or battery temperatures may influence

the aging behavior of the battery (e.g., lithium plating25 etc.) [227], the SoAP prediction

algorithm must consider these boundary conditions as well [228]. However, as discussed

in Refs. [46], [225], the accurate prediction of the available battery power becomes more

challenging at low temperatures or when the battery is aged, as the LIB reaches the

predefined SOA and power limits, respectively, readily. Fig. 44 shows the dependence of

SoAP on temperature and applied current rate in an example of a NMC/LTO LIB in a

new state. As expected, the available battery power for both charge and discharge

direction decreases with the decrease of battery temperature, as a consequence of

increasing battery resistance. At the same time, the available battery power increases

with the increase of temperature while aging processes occurring in the LIB are faster

under such operating conditions.

24 oC [V ∙ A− ∙ s− ] represents the change of the C per As of current flow. 25 Lithium plating refers to deposition of metallic lithium in form of dendrites in the anode [6]. Lithium plating occurs

directly on locations where local anode potential drops below 0 V vs. Li|Li+.


Fig. 44: Dependence of available battery power on battery current and SoC at different temperatures in an example of Cell-C-new (Vcell,min=1.5 V, Vcell,max=2.75 V) determined as proposed in Eqs. (23)-(24)

This subsection aims to give an overview of available techniques for SoAP prediction of

LIBs in EVs. Unfortunately, it seems that there is no uniform definition available in the

literature for battery power capability. The definition used in this work, namely SoAP,

differs little from the definition of State-of-Function (SoF) estimation which is often

employed. SoF is a figure of merit that describes the battery capability to perform a

certain task [229]-[230]. In the particular case of battery power capability, it may be

concluded that SoF is a kind of logical signal that only allows yes or no response to the

required battery power for a certain function, such as cranking capability etc. At the

same time, SoAP is mainly related to the amount of power which the battery can deliver

to or accept from the vehicle powertrain over a certain time horizon (∆t) [231]-[234].

Therefore, in order to prevent misinterpretation of the used parameters, in the following

only the SoAP definition whose meaning and particular definition is described above, is

used. An alternative definition is given in Refs. [102], [235], where the authors state that

SoAP is actually defined as a ratio of available battery power to the maximum nominal

battery power.

In general, the available techniques for predicting the SoAP can generally be subdivided

into the following two groups:


Methods based on (adaptive) characteristic maps (CM),

Methods based on ECMs.

3.3.1 Methods based on adaptive characteristic maps

The methods based on CM use the known static interdependences between battery

power capability and battery parameters and states. Related battery parameters and

states generally used in this context are listed as follows [31], [236]:

SoC,

Temperature,

Duration of the power pulse,

Required power,

Battery voltage.

In order to build the CM: First, the above mentioned dependences are characterized by

various battery characterization techniques, such as HPPC tests or EIS. Second, the

extracted parameters are stored as a more dimensional matrix in a non-volatile memory

in the ECU used as initial parameter sets of the battery.

As discussed in the previous section, because of aging mechanisms occurring in the LIBs,

their impedance characteristics change over the battery lifetime. This makes the use of

adaptive techniques for tracking the CMs over the battery lifetime necessary. The basic

idea of the techniques using CMs is simple: The battery power measured on-board is

compared with the predicted value (∆P = Pmeas – Ppredict) and the corresponding reference

point in the CM is adapted if a deviation between both values is detected. This process

yields more accurate future power prediction in the corresponding area of the map.

Among the main advantages of CM-based techniques for predicting the SoAP are their

simple implementation and straight forward characteristic. However, since batteries are

nonlinear devices, there are some issues which need to be considered thoroughly. First,

battery’s characteristics (e.g., dynamic behavior etc.) depend strongly on its previous history. Various electrochemical processes, such as charge transfer processes become

pronounced when dynamic load is applied to the battery. Since these effects are not

considered in the CMs, the SoAP prediction accuracy is affected significantly. Second,

the dependences of the above mentioned parameters are stored on the memory in

multidimensional form. This means that a high amount of storage capacity is needed

depending on the amount of parameters. Unfortunately, in most cases, data storage


capability and computational resources on microcontrollers are limited. Therefore, a

specific functional extension is required in order to reduce the computational effort, so

that the dependences of the parameters are approximated by empirical functions (e.g.,

polynomial functions) [226], [237]-[239].

In Ref. [236], the authors present an adaptive CM-based (ACM) technique where the

dependences between actual SoC, temperature, duration of the applied load (1 s – 10 s)

and available power of LIBs are considered. Moreover, battery voltage and current are

measured and observed over a predefined time horizon and consequently compared with

a reference power value of the CM. If any difference is detected and it is higher than a

value specified beforehand, the respective reference point of the CM is adapted

accordingly.

In Ref. [239], a method for determining the maximum available charge or discharge

battery power depending on actual battery SoC, temperature and its actual aging state

(available discharge capacity) is discussed. A 5th order polynomial function is applied in

order to build the interrelation function between power and SoC and between power and

available discharge capacity. The interrelation function between temperature and power

is built by means of a second order polynomial function.

In Refs. [237]-[238], a self-learning technique for SoAP prediction in HEVs using NiMH

and Lead-Acid batteries is introduced. The authors introduce an algorithm employing

ACM where the second order polynomial function is used to approximate the operating

point of the CM by means of simple multidimensional regression techniques, resulting in

less consumption of the memory on the microcontroller. Otherwise, according to the

authors, about 3000 measured values would have to be stored which requires a high

amount of memory capacity. In this context, the battery behavior is represented by the

coefficients of the used polynomial function. In order to track the battery’s characteristics over its lifetime, the applied regression function is continuously updated

during operation. The dependences between voltage, SoC, temperature, pulse duration

and the absolute power value are employed to build the ACM. The implemented

algorithm is able to predict the SoAP under the above conditions by considering the

allowed voltage window.

In Ref. [240], an algorithm for estimating certain parameters such as battery discharge

power, available charge capacity, over-charge or over-discharge detection of lithium-ion

battery packs in HEVs is presented. For this reason, the authors propose to use the

determined battery internal resistance as a reference for estimating the above


mentioned battery parameters. In this regard, the proposed algorithm uses the values of

internal resistance and OCV for a given SoC from CM and determines the available

battery power based on Eqs. (23)-(24). However, the non-adaptive nature of the proposed

algorithm and neglect of the influence of temperature are its main disadvantages.

Consequently, the values of the internal resistance remain constant over the battery

lifetime although an increase of battery resistance during operation is expected.

3.3.2 Methods based on equivalent circuit models

As a second technique for predicting the battery SoAP, model-based methods employing

ECMs are discussed in this subsection. These techniques differ mainly in the used ECM

and methodology for on-board parameters identification. In order to be able to predict

the available battery power accurately, it is necessary to use a dynamic battery model

which reproduces electrochemical processes occurring in the battery as accurately as

possible. However, a feasible replica of each of the processes can rarely be obtained since

up to now many processes occurring under different conditions are not identified or

understood sufficiently. In fact, because of their high complexity and the required high

computational effort, such battery models are not suitable for battery state estimation.

As described in the previous subsection the KF-based and LS-based estimators belong

definitely to the most often employed recursive estimators in various engineering

disciplines. In both cases, the system uses a linear regression model in a generic form

[241]-[242]. In the following subsection, in the literature presented methods from the

class of KF and LS for SoAP prediction presented in the literature are discussed.

Kalman filter-based methods 3.3.2.1

In Ref. [232], a SoAP prediction technique using KF for estimating the ECM parameters

(R-RC ECM) is introduced. The main differences with regard to the definitions of SoF

and SoAP are carefully addressed and alternative possibilities are provided. The applied

algorithm considers the change of C during the applied discharge current pulse

(∂ C / ∂ ∙ ∆) which consequently improves the C convergence and estimation

accuracy of the algorithm. For verification reasons, HPPC tests and UDDS load profile

are used. Eqs. (23)-(24) are employed as a basis for SoAP prediction where battery

nonlinearities are neglected and only the discharging case is considered.

In Ref. [243], the authors present an approach based on a KF filter for SoAP prediction

in an example of lead-acid battery. An ECM (R-RC) considering the nonlinearity of

charge transfer processes by means of the BVE is applied and the results are compared

with the same model neglecting the mentioned nonlinearity. After analyzing the results,


the authors state that considering the aforementioned nonlinearity improves the

estimation accuracy significantly. However, there are some points which may be

addressed here: First, the is assumed to be equal for both anodic and cathodic

reactions (i.e., = 0.5) in this work, which is actually not applicable to lead-acid

batteries and could at best be applied to LIBs [145], [244]. Second, for both models the I0

is assumed to be constant. In fact, the I0 decreases over the battery lifetime and is highly

temperature-dependent as shown in chapter 2. This means that the accuracy of the

proposed algorithm might decrease over the battery lifetime. Real vehicle data gathered

in the field are used to verify the algorithm whereas the SoAP estimator is only

validated for cases when maximum allowable discharge current is drawn. Furthermore,

in Ref. [245], the same authors investigate the Arrhenius equation and consider

temperature as model input for the determination of resistance and overvoltage in an

example of LIBs using LFP cathode. Based on the KF technique, parameters of the ECM

are identified on-board. In contrast to the previous work, the temperature dependence of

the ECM parameters is considered in the model. Obtained results indicate the necessity

of considering the nonlinearity at very low temperatures while a significant

improvement with regard to the voltage prediction accuracy could be achieved.

In Ref. [246], an approach based on R-RC ECM for predicting the battery’s SoAP is

presented. Battery current and voltage are considered as limiting factors for the applied

algorithm. As an advantage, in the employed algorithm the difference between each

individual cell VOCV is considered to identify the weakest cell in the battery pack. VOCV is

determined based on a nonlinear SoC-OCV correlation characterized in the laboratory.

Evaluating each cell’s C by its own has on the one hand the advantage that the

weakest LIB (LIB with minimum C for discharging case or LIB with maximum C

for charging case) in the battery pack can thus be simply identified, but it requires on

the other hand a higher computational effort since in large lithium-ion battery packs

often large amounts of LIBs are connected in series and each LIB have to be supervised

individually. In this regard, the authors propose using only the information from

maximum C in case of charging and minimum VOCV in case of discharging.

Unfortunately, no information has been provided with regard to the estimation

technique applied for impedance parameters; the authors mention, however, that KF-

based algorithms could be applied.

In Ref. [247], a novel algorithm for SoAP prediction is proposed where an impedance-

based ECM using the connected in series with a simple R-RC ECM is employed. An


adaptive approach based on fractional joint KF is applied for on-board impedance

parameters and SoAP estimation. Unfortunately, the RC-element representing the

charge transfer processes is neglected and thereafter it is simplified to a linear

resistance. The authors define SoC, I and V for both charge and discharge cases as

limiting factors for the SoAP prediction algorithm. For verification reasons, constant

current pulses with durations of 10 s and 60 s are inserted to the FUDS load profile and

the predicted SoAP at the end of the pulse is analyzed. As discussed beforehand, such

algorithms may work accurately and robust under the nominal condition described by

the authors but it is definitely of interest to examine the applied technique under

conditions beyond nominal.

In Ref. [148], a model-based approach using EKF for estimating impedance parameters,

voltage across the RC-element and SoAP prediction in an example of a HEV is

presented. Minimum charging current or maximum discharging current for (∆t =1 - 10 s)

is determined after performing some simple mathematical calculations from the

employed ECM (R-RC). Based on the derived limiting current values, (dis)charge

voltages of the battery pack are calculated for the desired time horizon. The proposed

algorithm is verified under nominal conditions in a MIL environment.

Furthermore, in Ref. [149], an EKF-based algorithm for on-board impedance parameters

estimation is proposed where predefined limits for ECM-parameters depending on

battery temperature and SoC are set. A simple R-RC ECM is employed where the correct

and meaningful convergence of the impedance parameters is ensured by setting the

upper and lower limits of the impedance parameters. Moreover, in Ref. [248] the authors

present a novel technique for estimating the impedance parameters of the ECM and

instantaneous SoAP prediction. The parameters of the lithium-ion battery pack used in a

HEV are extracted by means of EIS measurements for an initial state. Based on various

filters, such as high-pass, low-pass or band-pass filters and/or a combination thereof, the

measured battery’s voltage response is split up into different appropriate frequency ranges (medium to high frequency range as well as low and steady-state part). When the

impedance parameters of the ECM for the respective frequency range are identified, the

battery’s limiting current and instantaneous power can then be simply determined. The authors propose an EKF-based algorithm for estimating the parameters of the RC-

element on-board. However, since the corresponding frequencies are influenced by many

factors such as temperature and actual aging state of the battery, the impedance

parameters estimation from each frequency range might be underlined some


inaccuracies when the battery is operated under conditions other than those initially

obtained. In addition, since commercially available hardware for performing EIS-

measurement in the vehicles is still in a primitive development stage, accurate

measurements over a wide frequency range can hardly be carried out.

In contrast to the above referenced literatures where adaptive joint estimators using

EKF for SoAP prediction are proposed, in Refs. [220], [235] the authors propose a SoAP

estimation technique based on the DEKF. The DEKF method uses two EKFs (i.e., a

Kalman state and a weight filter) that run concurrently at every time step, where one

filter is responsible for state estimation (e.g., SoC or SoAP etc.) and the other one for

parameters identification (e.g., R0, τ, Rct etc.) while both of them concurrently provide

each other with derived information and ensure a more stable independent estimation

path in this way.

In Ref. [235], an R-RC ECM using the DEKF algorithm for on-board parameters

estimation is used, where battery current and voltage are considered as limiting

constraints for SoAP estimation. Each of the limiting factors (current and voltage) is

considered separately and its impact on SoAP is discussed thoroughly. For the applied

DEKF algorithm, the flowing current through the Rct is taken into the state vector, and

the remaining parameters of the ECM (R0, τ, VOCV) with a slower rate of change are

stored in the parameter vector. The proposed algorithm is examined by performing

verification experiments on the LFP-based LIBs at different aging states and

temperatures. For verification reasons the FUDS load profile is used as a reference.

Furthermore, the maximum time that the current can be applied to the battery and

voltage limits is determined based on an ECM employed. Unfortunately, during

verification tests a narrow range of applied current (-3C - +2C) is examined. According to

the authors, promising results for SoAP prediction with a relative estimation error of

less than 5% could be achieved. However, the proposed technique neglects the battery

nonlinearities and no SoAP prediction results for a particular time horizon are provided.

In fact as discussed in chapter 2, the impact of the applied current on the voltage

estimation accuracy is not negligible; therefore, higher current rates which actually often

occur in a real field must be considered. Moreover, the obtained results indicate the

decrease of voltage estimation accuracy at very low SoC range, which occurs mainly

because of high pronounced diffusion processes [46].

Furthermore, in Ref. [215] the authors propose a static SoAP prediction (using SoC-Ri

relationship) methodology based on the EKF technique (EKF is employed for SoC


estimation). An R-RC ECM with a little deviation concerning the internal resistance is

implemented. The instantaneous available battery power is derived based on Eqs. (23)-

(24). The authors combine the ohmic and charge transfer resistance in one linear

resistance connected in series with the RC-element representing diffusion overvoltage.

The combination of both resistances may help to overcome the mentioned linearization

problem of battery’s nonlinear characteristics by EKF algorithms. However, since the impedance parameters change over the battery lifetime, it is essential to ensure their

adaption to the actual aging state of the battery.

In Refs. [220], [249]-[250], battery voltage, current and SoC limits are considered as

constraints for SoAP prediction in a lithium-ion battery pack. In Ref. [220], the

possibility to employ EKF and DEKF methods for impedance parameters and SoC

estimation of LFP-based LIBs is discussed comprehensively. Moreover, in Refs. [249]-

[250] maximum discharge and minimum charge currents are determined based on

derived battery SoC and resistance values. Two algorithms are proposed for SoAP

prediction and their performances are compared, where simple ECMs employing a

simple R0 connected in series with VOCV are implemented and the available battery

power is determined based on Eqs. (23)-(24). In order to be able to predict the available

battery current for a ∆t, the change of C over the prediction time horizon is estimated

by means of the Taylor-series expansion. In the second algorithm proposed, the ECM is

implemented in a state-space form and the limiting battery current considering the

voltage and SoC limits is estimated based on the bisection search method. However, in

both cases the employed ECMs are very simple and neglect battery nonlinearities and

polarization effects. Moreover, the available battery power is determined separately for

individual LIBs in the battery pack; this fact requires high computational effort that is

actually not given on commercial microcontrollers used in the automotive industry.

In Ref. [251], an adaptive joint estimator employing a simple R-RC ECM for SoC and

SoAP estimation is presented. LIBs at different aging states using LFP cathodes are

investigated for different temperatures. The Nernst model is used to build and adapt the

nonlinear SoC-OCV correlation over the battery lifetime and make the SoC be a part of

the developing model, respectively. Battery SoC and SoAP are estimated by means of

AEKF where the UDDS load profile is used as a reference for verification reasons.

Moreover, a simple LS estimator is used to identify the ECM parameters. In addition to

the estimated SoC, the battery’s design limits, namely voltage and current, are defined as constraints in the implemented SoAP estimator. The available battery power is


determined and validated for ∆t=0 and ∆t = 30 s. Since current and voltage limits for the

respective operating condition and the battery’s aging state are calculated on-board,

safety damaging the LIBs by either over- or underestimating the available power can be

prevented. However, for verification tests, the applied current is downscaled to ±2C,

which is actually low (especially in case of HEVs) as mentioned above and might

influence the reliability analysis of the proposed algorithm.

Furthermore, in Ref. [146], the authors present an AEKF-based six-step joint estimator

for SoC and SoAP estimation (verified for ∆t = 15 s, 30 s, 60 s) in an example of PHEV.

An R-RC ECM is applied, where HPPC tests and UDDS load profiles are applied in

order to verify the robustness and real-time applicability of the proposed algorithm.

However, since the authors use the Nernst equation to build the nonlinear SoC-OCV

correlation and the limiting battery current is determined based on the actual battery’s SoC, the estimation accuracy of the model might decrease over the battery lifetime or

when the operating temperature differs extremely from the one used to parameterize the

Nernst equation.

In Ref. [216], the authors present a multi-states joint estimator employing the RLS

technique with optimal forgetting factor (RLSF) for impedance parameters estimation

and an AEKF for SoC and SoAP estimation. In addition, battery voltage, current and

SoC are considered as limiting factors for the SoAP prediction algorithm. LFP-based

LIBs at various aging states are used for verification reasons and an R-RC ECM is

employed, where the R0 is implemented separately for charge and discharge directions.

Thereafter, in total there are four impedance parameters needed to be identified on-

board. The identified parameters are transmitted to the prior state estimator in order to

update the battery’s SoC. Since the SoC-OCV correlation is built by the Nernst equation,

the C is then updated over the equation when new SoC values are available.

Thereafter, the battery’s SoAP is predicted based on the results from the SoC and parameter identification estimators. According to the authors, promising results for

SoAP prediction with a relative estimation error of less than 1% could be achieved.

Least-square-based methods 3.3.2.2

In the following, LS-based techniques for SoAP prediction are discussed

comprehensively. In Ref. [252], the authors show the results of the applied SoAP and

voltage estimator by performing verification tests on the test bench and in HEV. The

obtained results of the predicted voltage indicate an estimation accuracy of approx. 5%

where the impedance parameters of the ECM are identified by means of the RLS


algorithm, and a predictive control algorithm based on step response, the so-called

Dynamic Matrix Control (DMC), is applied for the maximum discharge current

estimation. The applied DMC algorithm belongs to the class of filters employed in the

systems theory which is able to determine a sequence of manipulated variables for

predicting the upcoming behavior of the model. The main important difference to the

previously discussed literature lies in the applied ECM, where n RC-elements are

connected in series and the time constant of each RC-element is determined offline.

However, the developed algorithm is only evaluated for the discharging case whereas the

charging case (e.g., regenerative braking etc.) is neglected.

A novel approach for SoAP prediction considering both voltage and current as limiting

factors is presented in Ref. [253]. Within this approach, a so-called moving average (MA)

noise covering the remaining battery overvoltages (e.g., diffusion processes etc.) is

connected in series with RC-element and R0 in the ECM. Furthermore, an RELS

algorithm is applied for impedance parameters identification. The main advantage of the

RELS algorithm over the common RLS algorithm is the additional implementation of a

feedback process which allows the estimate of unmeasurable noise. The proposed

algorithm is validated by UDDS load profile and HPPC tests while the algorithm’s accuracy is analyzed for various time horizons (5 s - 120 s). The achieved results indicate

a more accurate and reliable SoAP prediction by employing the proposed ECM rather

than a simple R-RC ECM, especially for cases where the SoAP is predicted for a large

prediction time horizon. As the next step, the presented approach might be investigated

in more detail under real conditions and especially over a wide temperature range since

it is only validated under nominal condition.

In Ref. [222], the authors propose an algorithm for SoAP prediction considering SoC,

current and voltage as limiting factors. A RLSF technique is applied to estimate battery

impedance parameters. The main difference of the proposed technique (RLSF) in

comparison to the common RLS filter is that if the model parameters change

continuously but slowly, the RLS technique is rarely able to capture the occurred

changes instantaneously. Therefore, an enhanced filter is required, which gives more

weight to recent data than to an older dataset as proposed in Ref. [216]. The on-board

applicability of the proposed algorithm is then verified by means of HIL tests using the

FUDS load profile as a reference. A simple R-RC ECM is employed where the VOCV is

modeled by Nernst model. Moreover, the change of the VOCV over a prediction time

horizon is determined, which leads to higher voltage prediction accuracy. According to


the authors, promising results could be achieved with the battery voltage estimated

within an accuracy range of 1%.

In Ref. [224], the authors investigate a method for SoAP prediction in an example of a

transporter, the so-called Segway. Based on Eqs. (23)-(24), the available battery power is

simply determined. The authors propose using LS or RLS-based techniques for

estimating the battery resistance and VOCV. Furthermore, based on the estimated values

for available battery power, the maximum velocity of the transporter is determined. In

this particular case it is assumed that the current is equal to the electric-motor current

and the resistance is determined as a sum of the motor winding resistance and battery

resistance.

In addition to the methods discussed above, in Refs. [225], [254] the authors present an

adaptive technique for SoAP prediction of the lithium-ion battery pack using the varied

parameters approach (presented in Ref. [145]) for impedance parameters estimation. The

current dependence of the Rct is considered in the proposed algorithm by simplifying the

BVE. The authors employ a nonlinear R-RC ECM where the differences between SoCs

and resistances of individual LIBs are considered in an implicit way. Battery voltage,

current and SoC are considered as limiting factors for SoAP prediction. In order to

identify the weakest LIB in the battery pack, C and resistance are considered as

supporting indices. In this regard, the LIB with the highest VOCV for charging case or

lowest C for discharging case as well as the LIBs with the highest resistance is

considered as the weakest. As an improvement to the used ECM shown in references

above, its extended version is presented in Ref. [46] for SoAP prediction. For this reason,

a second RC-element capturing diffusion processes is added to the basis ECM. A RLS

filter is used for estimating the parameters of the diffusion branch. The proposed

algorithm is verified under various conditions (various SoC ranges and different

temperatures) by using real vehicle data. Unfortunately, no long-term robustness

analysis of the proposed algorithm is performed.

In Refs. [30], [255], the authors apply a direct differential (DD) algorithm to estimate

battery states (SoC, SoAP and SoH) and the WRLS algorithm to identify the impedance

parameters of an R-RctRdC ECM. The Rd is derived empirically and connected in series

with Rct. According to the authors, considering the Rd in the ECM increases the SoAP

prediction accuracy for discharge cases and for prediction time horizons in a range of

∆t >= 10 s significantly. However, the Rd is not implemented as an adaptive parameter in

the proposed algorithm, and the fact that it changes over the battery lifetime and under


different operating conditions [46] may lead to the decrease of the estimation accuracy

over the battery lifetime.

In Refs. [256]-[257], the authors present an algorithm based on the WRLS technique for

estimating the impedance parameters of an ECM that considers the mass transport

effects by incorporating the . Moreover, the current dependence of the Rct is

considered, and the applicability of the Tafel-equation for determining the Rct in case of

high current rates is proven. The applied algorithm is verified under various conditions

(different temperatures and different aging states of the battery) by investigating LIBs

using NMC and LFP cathode materials. Based on the developed algorithm, battery VOCV

and SoC, respectively, are derived from an employed ECM by using the SoC-OCV

correlation. Furthermore, the actual battery SoH using the estimated battery impedance

and SoAP is determined on-board. However, for SoAP prediction the authors consider

only the battery voltage as a limiting factor and determine the maximum current rate

that the battery can deliver or accept.

Other adaptive filters and observer methods 3.3.2.3

In Ref. [258], the authors propose an SoAP estimation algorithm putting special

emphasis on investigating time horizons much larger than in the researches presented

above (i.e., ∆t >= 60 s). Asymmetric ECM parameters (R-RC-RC) for charge and

discharge directions are considered in the applied algorithm. According to the authors,

variation of the ECM parameters is necessary when SoAP prediction is performed for

larger time horizons. The asymmetric parameter values yield higher estimation accuracy

for appropriate charge and discharge directions. In this regard, two different SoAP

prediction algorithms are investigated. First, a voltage limited-based method

extrapolating battery resistances and VOCV (so-called VLERO) is investigated. Second, a

method based on a so-called multi-step model predictive iterative (MMPI) technique.

Furthermore, battery SoC, voltage and current are considered as limiting factors for

SoAP prediction. Applied algorithms are able to calculate the maximum battery current

whereas voltage and SoC design limits are used. For verification reasons, the proposed

algorithms are evaluated by means of the UDDS load profile at low temperatures (0 °C

and -20 °C). According to the obtained results, the MMPI technique shows a higher

degree of SoAP prediction accuracy. However, since the proposed algorithms are only

evaluated in a MIL environment, it is recommended to examine their performance under

real conditions as the next step.


Moreover, in Ref. [259], the authors propose a model-based approach using the

Levenberg-Marquadt methodology for SoAP estimation in an example of a lithium-ion

battery pack (ns=20, np=4). The proposed technique is then compared with the technique

based on the bisection methodology described in Ref. [249]. LIBs using NMC and LFP

cathode materials are used to perform verification tests. The applied algorithm is

evaluated by means of FUDS, DST load profiles and HPPC tests. The authors

investigate two R-RC ECMs where the first ECM neglects the VOCV hysteresis and the

second one uses a one-state hysteresis. Battery SoC, voltage and current are considered

as limiting factors for SoAP prediction. According to the authors, the bisection technique

shows some advantages over the proposed algorithm with regard to its computational

efficiency. However, the inconsistency of individual LIBs in the lithium-ion battery pack

is neglected and it is assumed that the LIBs show a similar behavior. Moreover, battery

nonlinearities are neglected.

In Ref. [219], a model-based approach for SoAP estimation in an example of a lithium-ion

battery pack (ns=16, np=1) for HEVs is discussed. In addition to the battery’s voltage design limits, SoC and current limits are calculated and considered as further limiting

factors for SoAP prediction. An R-RC ECM is employed and based on determined

impedance parameters, the SoAP is predicted. However, the proposed SoAP prediction

algorithm is verified under nominal conditions for LIBs in new state by comparing the

results from HPPC test and DST load profile. The authors do not give information about

the applied technique for on-board impedance parameters estimation. According to the

results shown in the aforementioned paper, the discussed theory with regard to the over

and under estimation of the available battery power by means of the HPPC test in the

previous subsection is approved.

In Ref. [217], contrary to the above-mentioned approaches, the authors apply the PF

technique for estimating the SoAP where battery SoC serves as an input parameter. The

main advantage of PF-based techniques in comparison to other Bayesian filters is the

possibility of approximating the PDF even if non-Gaussian measurement uncertainty is

given. A dependence of the polarization resistance on SoC and current (R=f(SoC, I)) is

used as a basis for an accurate instantaneous SoAP prediction. Thereafter, the Gaussian

shape is chosen as membership function of the employed fuzzy set. The measured

current and the voltage estimated from the employed ECM are used for the implemented

PF algorithm. In the presented approach, the battery’s SoC is firstly estimated based on

the PF method, and the SoAP is determined by evaluating the resulting PDF function of

the estimated SoC performed by means of a simple look-up table representing the


relationship between SoAP and battery SoC. However, the authors use a simplified

“fuzzy” ECM (ECM 1 is responsible for charging case, ECM 2 is responsible for discharging case) whereas the double-layer capacitance and the temperature’s influence are neglected.

In Ref. [221], the impedance parameters of the employed ECM (R-RC) and battery SoAP

are estimated based on a genetic algorithm. The presented approach is validated by

means of the FUDS load profile for LFP-based LIBs in a new state whereupon a lithium-

ion battery pack (ns=1, np=9) is used. The difference between both charge and discharge

ohmic resistances (Rdch, Rch) is considered in the employed ECM. In the first step, the

authors apply a SOC-limited method for instantaneous SoAP prediction. Afterwards, a

voltage-limited method is applied where the change of VOCV over the prediction time

horizon is considered in order to improve the SoAP prediction accuracy. The obtained

results compared with the results of the HPPC-based method discussed beforehand

indicate a higher estimation accuracy of the proposed algorithm within an accuracy

range of 2%.

Fuzzy logic and machine learning-based methods 3.3.2.4

In addition to the techniques discussed above, there are some publications dealing with

machine learning-based algorithms for SoAP prediction. The underlying idea of such

techniques is based on the ability to learn and to adjust the relationship between input

and output signals without the need of complex mathematical functions by using human

knowledge. However, the development of machine learning methods is still in a primitive

stage and the applied techniques need to be further modified and verified under real

conditions.

In Ref. [260], the authors investigate a technique based on the BP neural network

training algorithm for SoAP prediction. The BP neural network algorithm is a forward,

mono transmission multi-layers network including an input layer (containing 2 nodes),

hidden layer (containing 4 nodes) and output layer (containing 1 node) network. It allows

considering the nonlinear battery behavior without the need of high-level hardware. The

basic idea of the proposed algorithm is simple: When the output vector does not meet the

expected values, then the output error is transformed to input layer and distributed to

units of hidden layers, respectively. Thereafter, the estimation error of each layer can be

obtained. Based on the obtained estimation error, the weight of each neuron is adjusted

until the output vector reaches the correct values. The proposed technique is verified

under nominal conditions by means of offline simulations and constant discharging


power profiles. Its applicability on real-time applications needs to be investigated in

greater detail in the future.

In Ref. [261], the authors present a method using an adaptive neuro-fuzzy-inference

system (ANFIS) for on-board SoAP prediction whereby the main focus lies on the battery

voltage prognosis. The proposed approach combines the strengths of two common

techniques, the artificial neuronal networks (ANN) and fuzzy inference system (FIS), in

such a way that the battery parameters are adapted to the actual aging state. In this

regard, according to the authors, a meaningful convergence of the parameters can be

ensured. However, the authors neglect the battery current as further limiting factor for

predicting the SoAP. The obtained SIL test results indicate high algorithm feasibility

and estimation accuracy. For a predefined time horizon the battery voltage is predicted

within a relative estimation accuracy of less than 1%. The implemented algorithm acts

more as an SoF algorithm since the model output delivers an answer as to whether a

power pulse can be performed or not. Furthermore, in Ref. [262], the authors show

results employing the same algorithm as before with the difference that the battery’s SoC is also considered as an additional input parameter of the employed model. Battery

SoC is determined by means of the robust extended Kalman filter (REKF).

In Ref. [228], the authors perform a large experimental investigation considering the

impact of temperature, SoC, resistance on SoAP using HPPC method as a basis. The

measurements have been performed on LIBs using LMO cathodes over a wide

temperature range (i.e., from -20 °C up to 50 °C) whereby the increase of available

battery power with increasing temperature is approved as shown in Fig. 44. The authors

present two models, namely parametric model considering the impact of temperature on

power capability of the battery in Eq. (23) and non-parametric model using SVM

technique based on radial basis function (RBF). The latter method (i.e., SVM) is a

popular machine learning technique for data analysis and pattern recognition,

respectively. In general, the SVM method tries to find the best compromise solution

between the model complexity and the learn ability. According to the results, with

decreasing temperature and SoC the parametric model becomes more accurate than the

non-parametric model and vice versa. Unfortunately, the available battery power is only

investigated for discharge cases and the authors neglect charging cases completely.

The algorithms mentioned above and the respective literature sources for techniques

based on equivalent circuit models are listed and compared in Table 12.


Table 12: Referenced works based on equivalent circuit models for SoAP prediction

Possible References Employed ECM Applied estimation technique

Current

dependence of

the charge

transfer

resistance is

considered?

[253] R-RC-(C(z)) RELS No

[225], [254] R-R(I)C VPA Yes

[46] R-R(I)C-RC VPA, RLS Yes

[249], [250] R KF No

[232], [246] R-RC KF No

[235] R-RC DEKF No

[256]-[257] R-R(I)C- W WRLS Yes

[146], [251] R-RC AEKF No

[148]-[149], [215], [248] R-RC EKF No

[252] R-RC RLS and DMC No

[247] R-RC- W Fractional Joint KF No

[258] R-RC-RC a) VLERO, b) MMPI No

[243], [245] R-RC KF Yes

[216] R-RC RLSF and AEKF No

[219] R-RC - No

[221] R and R-RC Genetic algorithm No

[217] R PF Yes

[261]-[262] - ANFIS, ANFIS+ REKF -

[260] - BP neural network -

[228] - SVM No

[222] R-RC RLSF No

[30], [255] R-RctRdC WRLS No

[259] R-RC Levenberg-Marquadt No

3.4 State-of-Safety estimation

The safety of LIBs is of paramount importance not only for vehicle manufacturer but

also producers of consumer electronics. Unfortunately, because of several fire incidents

in various applications (e.g., aviation or automotive industry etc.) in recent years, the

attention regarding safety state of particular applications has been increased

significantly. As discussed beforehand, in order to provide the respective system with a

specific amount of power and energy, several hundreds of LIBs are connected in series

and/or parallel in a lithium-ion battery pack. Thereafter, a failure in a single LIB could

have fatal consequences throughout the battery system.


In general, safety of the battery can be defined inversely proportional to the concept of

abuse as follows (Eq. (30)) [263]: = (30)

where refers to the state of abuse and represents the SoS whereby x

addresses all state and control variables such as temperature, current, voltage, battery

deformation etc. at the time t. Consequently, it can be stated that there is a direct

correlation between SoS, SOA and other battery states such as SoAP or SoAE;

thereafter, ignoring the aforementioned control variables ( , , ) may damage the

battery. In this regard, the SoS can be defined as a function of following

parameters/battery states (Eq. (31)): , , , = , , , oC (31)

Generally, the battery safety is classified by means of hazard levels (HL) originally

defined by European Council for Automotive Research and Development (EUCAR) [264].

The HL is defined in a range of 0 up to 7 referring to the severity of the respective

hazard where the lowest level (i.e., HL = 0) indicates that no damage is detected and the

highest level (i.e., HL = 7) refers to the case where an explosion is detected. In total, it

can be stated that an ESS is defined as safe when the HL is lower than 4.

Safety measures in a lithium-ion battery pack equipped with BMS can be subdivided

into internal or external methods. In case of former case, measures such as shut down

separators or in subsection 2.1 discussed cell-built-in current interrupt switches

activated after detection of an internal gas pressure etc., are used. External methods

mainly rely on hardware for lithium-ion battery pack protection against abuse conditions

such cell bypass techniques where the weakest cell is temporarily or permanently

bypassed. In this regard, fuses or contactors are used for protecting the battery system

against short-circuiting, over-charging and over-discharging etc.

Among further battery abuse test procedures are FreedomCAR [265] and SAE J2464

[266]. In addition to the latter standards, the SAE J2929 [267] is a logical follow-up to

J2464 establishing pass/fail criteria for lithium-ion battery packs used in EVs. However,

the aforementioned test procedures are based on the idea revealing the battery to

different abuse conditions and then observe its behavior. In fact, no information

regarding the likelihood of the hazards and the risks accused by them is provided [268].

Potential hazardous external events can be classified into the following four cases:


Electrical hazard (e.g., short-circuit, over-charge etc.),

Thermal hazard (e.g., fire),

Mechanical hazard (e.g., crush, nail penetration etc.),

System hazard (e.g., contactor fails).

In general, it can be stated that hazardous failure on cell-level is mainly referred to

electrolyte leakage, internal short-circuit and occurring thermal runaway. In Table 13,

HL ranges and corresponding descriptions and required criteria are described.

Table 13: Hazard levels adopted from Ref. [265]

Level Description Criteria

0 No effect No effect; no loss of functionality

1 Reversible loss of function No defect; no leakage; no venting or fire detected; no

explosion, temporary loss of functionality

2 Irreversible damage/ defect No leakage; no venting or fire, no explosion but

irreversibly damaged

3 Minor leakage or venting

(i.e., ∆mass < 50%)

No leakage; no venting or fire, no explosion. Weight loss

< 50% of electrolyte weight. Light smoke

4 Major leakage or venting

(i.e., ∆mass % )

No fire or flame; no rupture, no explosion. Weight loss % of electrolyte weight. Heavy smoke

5 Fire or flame No rupture, no explosion (i.e., no flying part)

6 Rupture No explosion, but flying parts.

7 Explosion Explosion (disintegration of the LIB)

According to Ref. [263], based on Eq. (30), SoS is defined in a numerical range between 0

and 1 whereby with increasing the absolute value of abuse, the value of safety decreases

to zero and vice versa. The authors propose a quadratic function for general definition of including three parameters. In this regard, the EoL value of the battery (i.e., = .8 ∙ ) is considered as a boundary condition indicating whether the

battery is at acceptable value of abuse. Within the aforementioned study, the authors

examine the abuse boundaries of LFP-based LIBs considering subfunctions such as

current, voltage (over and under voltage), thermal and mechanical limits.

In Ref. [268], the authors propose the so-called Hazard Modes and Risk Mitigation

Analysis (HMRMA) methodology considering three simple principles for reducing the

hazardous risk. First, the design process should be considered at all levels (i.e., cell,

module and battery pack). Second, the control, sensing and protection devices and the


impact of each individual component on the battery pack should be included. Third, the

hazards are identified while possible risks are analyzed and quantified, respectively. In

contrast to EUCAR, the presented methodology covers also module and battery pack

level. In the aforementioned study the hazard risk ( ) is defined as a product of hazard

severity ( ) and hazard likelihood ( ) as follows (Eq. (32)): = ∙ (32)

where HS refers to values between 0 and 7 according to hazard level, HL refers to values

between 1 and 10 based on the rate of occurrence. According to Eq. (32) the of

respective application can be reduced either by decreasing or . The correlation

between the latter parameters is represented by a hyperbolic curve.

The topic of LIB safety in various applications is still a very challenging task for

application engineers and, thus there is still a need for significant efforts to be made in

various disciplines.

3.5 Lithium-ion battery models

The electrochemical processes occurring in LIBs are influenced by many internal and

external factors such as battery temperature, external applied potential, concentration

and concentration gradient of the reaction partners etc. Subsequently, developing a

battery model that is able to consider the aforementioned factors is not a simple issue.

Experimental techniques based on e.g., EIS measurement are powerful tools for

describing and understanding electrochemical reactions occurring in LIBs and, thus can

be applied for accurate battery modeling. The main challenge for an accurate battery

modeling is always to ensure a reasonable precision of model results in accordance to the

results obtained from experimental investigations. Unfortunately, most of the

parameters of the LIBs are not measureable on-board and, thus cannot be employed for

direct comparison to the corresponding variables of the employed battery model.

Thereafter, such parameters can be often determined by comparing the model results

and the experimental data [269]. In the following subsections, a brief review on different

classes of LIB models presented in the literature is given.

3.5.1 Classification of battery models

The battery models proposed in the literature mainly vary in terms of model structure,

model complexity, required computing power and reliability of the obtained results. The

more precise the electrochemical processes occurring during battery operation can be

reproduced by the employed battery model, the more accurate the obtained results can


be. However, the latter fact can be only achieved at the expense of increasing

computational effort and required simulation time. In this regard, it is of great interest

for application engineers that a battery model is employed that satisfies the needs of the

particular application.

In general, battery models can be classified into following three groups [270]:

White box models,

Grey box models,

Black box models.

The white box models refer to the battery models that are based on complex

mathematical equations describing physical and electrochemical processes occurring in a

LIB. As a basis for developing such battery models, tools based on e.g., Finite Element

Method (FEM)26, Finite Volume Method (FVM) and Finite Difference Method (FDM) etc.

are employed [271]. However, as mentioned above the more complex the employed model

is the higher the required computing power can be. Contrary to the latter class of battery

models, the grey box battery models are widespread in BMS applications mainly because

of their simple nature and low computing power. However, from accuracy point of view

such battery models are less accurate than white box models, mainly as a result of

lacking interpretation of electrochemical processes. As mentioned in in chapter 2, grey

box models are often parametrized offline by means of EIS-measurements or HPPC

tests. Such battery models are based on ECMs while their parameters are always

determined using prior empirical knowledge and experimental data. Such battery

models simulate the voltage response of the battery for different current-based or power-

based loads applied to the battery.

In contrast to the above mentioned battery model classes, the black box battery models

refer to the category of battery models that are based on machine learning techniques

such as neuronal network, fuzzy logic or evolutionary algorithms etc. [272]. In such

battery models, the battery is often observed as a black box where a detailed

representation of electrochemical processes occurring in the LIB is neglected. Among the

main advantages of such battery models are their simple structure and possibility for

fast simulation. However, in contrast to the latter two battery model classes, the low

estimation accuracy and high parameterization effort of black box battery models can be

addressed as their main disadvantages.

26 The FEM-based tools are available from zero dimensional up to more complex systems such as 3 dimensional models.


In Ref. [269], a systematic review on various battery models from empirical models to

atomistic models is given in a wide length and time scale. Within the aforementioned

study, the authors classify the battery models into four different groups; empirical

models, electrochemical models, multi-physics models and molecular/atomistic models.

Moreover, a review article addressing safety-focused modeling of LIBs is presented in

Ref. [14]. According to the authors in the latter reference, the following three scales can

be considered for an accurate battery modeling:

Material level (Microscopic length scale) where electrochemical processes such as

charge transfer, diffusion processes are modeled,

Cell level where thermodynamic, thermal and mechanical phenomena are

described,

Battery pack level where the LIBs are often undifferentiated and the integration

of LIBs in advanced applications such as EVs is addressed.

White box battery models 3.5.1.1

As discussed beforehand, the white box battery models or well-known as electrochemical

battery models are mainly based on complex mathematical equations (e.g., methods

based on PDE or differential algebraic equations (DAE)). Within such battery models,

the focus of modeling mainly lies on representing the interaction between physical and

chemical properties of the internal components such as electrodes, electrolyte and

separator. In this regard, thermodynamic, fluid mechanics and kinetic reactions are

considered in great detail for reproducing the behavior of the battery as accurate as

possible [272]. Such battery models are parameterized by means of material properties.

Consequently, during development phase the influence of material property on battery’s electrochemical behavior can be investigated upon on the results obtained from such

models [273]. However, it is worth noting that the current/voltage behavior of battery

cannot be reproduced by the aforementioned models.

In general, white box battery models can be classified into two groups, namely P2D-

based models and SPM-based models. The P2D models are based on porous electrode,

concentrated solution theory and kinetics equations [274]-[275]. In fact, because of their

high computational effort, employing P2D models in real-time applications such as BMS

is limited. Hence, the focus has turned to simplifying such models into so-called SPM

models in recent years. Within such models ionic intercalation are defined as a dominant

dynamics part [293]. Generally, the SPM models neglect specific properties of battery’s electrolyte and the transfer processes are simplified significantly (e.g., variations in


concentration of electrolyte and potential are neglected) [271]. The latter simplifications

yield inaccurate simulation results for specific operating conditions such as when high

discharge current rates are applied to the battery or when thick electrodes need to be

simulated. The main advantages of SPM models are their comparatively simple nature,

low computing power and given possibility for implementation on embedded systems for

on-board state estimation such as SoC or SoH.

3.5.1.1.1 Pseudo-two-Dimensional battery model

Because of porous structure of battery electrodes, the interfacial area between the solid

electrode and the electrolyte solution increases and negatively affects occurring

electrochemical reactions in these areas. Such complex porous structure makes an

efficient and accurate battery model development very difficult. The first porous

electrode theory for batteries using macroscopic approach was presented in 1975 by

Newman and Tiedemann in Ref. [274]. The proposed method for battery modeling has

become the basis structure for most of the P2D models. It was only years later as the

first P2D LIB model was presented in 1993 in Refs. [275]-[276]. The aforementioned

porous electrode theory and the concentrated solution theory were combined together in

an example of lithium metal/TiS2 battery with polymeric separator. The behavior of

battery electrodes is simulated using spherical particles surrounded by electrolyte.

Another P2D-based battery model was presented for LMO/C-based LIBs using organic

electrolyte in 1994 in Ref. [277].

Fig. 45 illustrates schematically the three domains (i.e., porous anode, porous separator

and porous cathode) of a P2D battery model.


Fig. 45: Schematic illustration of pseudo-two-dimensional lithium-ion battery model

The latter references employ a 1D approach that considers the (de)intercalation

processes through the surface area of spherical particles surrounded by electrolyte. The

1D approach considers only the dynamics along one axis (x-axis as illustrated in Fig. 45)

whereby the dynamics along other dimensions (y-axis and z-axis) are neglected.

However, often the latter approximation (i.e., only consideration of x-axis) is a well

compromise for LIB structures since the length scale of remaining axes are at least

thousand times (approx. 100000 µm) higher than the length scale of x-axis (approx. 100

µm) [173]. Equations for P2D lithium-ion battery models can be explained as follows

[271]:

1. The concentration of Li+ in solid state (cs) is determined based on Fick’s law.

2. The concentration of Li+ in electrolyte (ce) and in separator is determined based

on conservation of Li+.

3. The solid-state potential ( ) of electrode is determined by ohmic law.

4. The electrolyte potential ( ) and in separator is determined based on Kirchhoff’s and ohmic laws.

5. The transport/flux of Li+ through electrolyte to electrodes is described by the

BVE.


The respective governing nonlinear equations of P2D battery models are summarized in

Table 14. In total, there are 14 nonlinear partial differential algebraic equations (PDAE)

with 14 unknown parameters [173].

Table 14: Governing equations27 of pseudo-two-dimensional lithium-ion battery models adopted from Refs. [271] and [278]

Region Governing equations Brief description of the respective

equation

Electrodes

(k=(n, p);

n=negative

electrode

and p=

positive

electrode)

, , , = , ∙ ( ∙ , , , )

Conservation of Li+ species in a

spherical solid phase particle that

is described by Fick’s law.

, , = ( , ∙ , , ) +− + ∙ ,

, , refers to electrolyte

concentration and and ,

differ for each domain (anode,

cathode and separator).

, , , = ,

, refers to effective

conductivity of the electrode.

Conservation of charge in solid

phase of each electrode represents

the potential in solid phase. − , ∙ , , − , ∙ , , + , ∙ − +∙ , =

Combination of Kirchhoff’s law with ohmic law in electrolyte

phase.

, = ( , − , )∙ ( , )∙ ( , , )∙ [ ( , , )− (− , , )]

The relationship between current

density, concentration and

overvoltage based on BVE as

described in chapter 2.

, = , , − , , − ( , ); = , , − , , − ∙

The terminal voltage is

determined based on electrode

potential at the end of two solid

electrodes and contact resistance.

Separator

(k=s) , , = ( , ∙ , , ) See above

27 where represents the effective electronic conductivity in solid electrode, is the specific surface area of electrode, I

is the current density (I=I0/A), is the effective ionic diffusion coefficient, R the universal gas constant, T the absolute temperature in Kelvin, is the porosity of the region, is the reaction rate constant of electrode , F the Faraday’s constant, C is contact resistance, kis the wall flux of Li+ on particle of the electrode, eff the effective ionic conductivity

of electrolyte in region k and + is the transference number of Li+ in electrolyte.


− , ∙ , , + , ∙ − +∙ , =

See above

3.5.1.1.2 Single particle battery model

As discussed beforehand, in order to make the P2D models suitable for real-time

applications, some simplifications must be performed. In this regard, two approaches can

be considered: First, the model order is reduced whereby the fidelity of the respective

system should be preserved. Second, some model properties are neglected whereby

information loss should be accepted [173]. A SPM model of LIBs can be simply derived

from P2D models after some simplifications where each battery electrode is idealized as

single spherical solid particles in which (de)intercalation processes occur.

In Ref. [279], the authors present a model considering the diffusion and intercalation

processes inside a single electrode particle where the anode and the cathode are

implemented with the same surface area. However, because of the mentioned

simplifications, SPM models can only be used with some limitations (i.e., low current

rates and thin electrodes [269]). In Ref. [271], the authors state that SPM is a preferred

choice rather than the P2D model at very low current rates (I<=1C) . However, more

sophisticated battery models must be employed for high discharge current rates.

Fig. 46 shows schematically a SPM model for LIBs.

Fig. 46: Schematic illustration of single particle model

A comprehensive review on simplified P2D models with especial emphasis on SPM

models can be found in Ref. [271]. Moreover, in Ref. [292], the authors propose an

extended P2D-based approach considering the cell-to-cell variations on battery pack level

under constant voltage charging and discharging. In this regard, coupled P2D model

considering aging (only capacity fade), porosity variation and temperature distribution is

presented. The authors state that the influence of temperature distribution on individual


LIB performance in the battery pack in contrast to the impact of remaining parameters

is more significant and, thus using an efficient cooling strategy is necessary.

Grey box battery models 3.5.1.2

The grey box battery models abstracts away the electrochemical processes occurring in

the LIB. Such models employ simple passive elements such as resistances, capacitances,

inductances for representing the nonlinear behavior of the battery. As discussed in

chapter 2, the aforementioned passive elements depend mainly on temperature, SoC,

applied current rate and actual aging state of the battery.

The aforementioned battery models are often based on experimental data gathered in

laboratory for a specific SoC and temperature operating range and, thus the estimation

accuracy decreases when the battery is operated beyond the specified range [269]. Beside

the possibility for current or voltage simulation by employing the aforementioned battery

models, further battery parameters/states such as SoC, SoAP, SoH or heat generation

can be determined based on the determined overvoltages and the applied current.

Generally, the main criterion for differentiating between grey box battery models and

black box battery models is the higher emphasis on physical-chemical background of

grey box battery models. For control systems or for on-board battery state estimation,

grey box battery models are often the preferred choice [173]. Reviews and descriptions of

different grey box battery models may be found in Refs. [280]-[283].

A simple impedance-based ECM describing the electrode reactions in the battery was

proposed by Randle in Ref. [284]. The proposed ECM consisting of resistances (including

R0 and Rct) and a Cdl connected in series with is still very popular in in the field of

battery modeling. Fig. 47 shows the aforementioned Randles circuit representing

electrode and cell reaction.

Fig. 47: Equivalent circuit model of LIBs adopted from Ref. [284]


The Randles circuit shown in Fig. 47 considers in addition to the in chapter 2 described

electrochemical processes, the occurring diffusion processes by means of where 28

refers to the Warburg coefficient, is the angular frequency and j is the imaginary

number √− [280]. A comprehensive description of remaining impedance parameters can

be found in chapter 2. The impedance of the ECM illustrated in Fig. 47 can be simply

determined as follows: = + + +σ∙ −√ (33)

= − = ,+ +√ ,σ ∙ − (34)

where , = ∙ and ,σ = σ .

After some simple mathematical formulations, the can be written as follows: = ,+ +√ ∙ , ∙ − (35)

where k is a parameter in a range between 0.001 and 1.

Based on the boundary conditions applied to differential equations, the can be

expressed by three various cases. For the first case, the so-called semi-infinite case with

x=0 and a constant concentration ( ), the at x→∞ can be calculated as follows

(Eq. (36)): = ∙ ∙ ∙√ ∙ −√ (36)

As discussed in chapter 2, the follows a straight line with -45° angle in the complex

plane [237]. The second solution is the so-called ideal reservoir where it is assumed that

from a finite length/distance ( ) the concentration remains constant even if diffusion

current is still flowing. Consequently, the is determined as follows (Eq. (37)): = ∙ ∙ ∙√ ∙ tanh √ (37)

In addition to the latter cases, the third and the last case is the so-called non-permeable

wall at where it is assumed that diffusion is blocked from a and no diffusion

current is flowing. Thereafter, the can be determined as follows (Eq. (38)):

28 w = ∙ ∙ ∙ ∙√ where R is the ideal gas constant, T is the temperature, m is the molarity of the reacting

species, F is the Faraday constant, D is the diffusion constant and is the effective electrode surface.


= ∙ ∙ ∙√ ∙ coth √ (38)

A sophisticated ECM consisting infinite number of RC-element connected in series to a

finite length transmissive and a simple capacitance representing charge storage in

electrodes is discussed in Ref. [285]. The impedance parameters of the proposed ECM are

identified by means of EIS measurements. In Ref. [130], the authors examine the impact

of the number of RC-elements on battery voltage estimation accuracy. According to the

results, two RC-elements are enough for precise battery voltage estimation where with

the increase of used RC-elements the estimation accuracy increases but the speed of this

increase becomes slower.

A sensitivity analysis of ECM output of a simple R-RC-RC ECM based on Morris

technique is performed in Ref. [286]. The authors explicitly investigate which of the

impedance parameters can be assumed to be constant without having a considerable

impact on voltage output. The influence of temperature on impedance parameters

behavior is neglected and only the SoC dependent parameters are examined. According

to the results, if a certain number of impedance parameters, in particular C1, C2 and

are fixed at their mean values and their variation with SoC is avoided, the output

voltage is affected minimal and the obtained error of estimated voltage in contrast to the

case where the latter parameters change with SoC, increases just slightly. In total, it can

be stated that the model output is mainly sensitive to variation of and .

In Ref. [230], the performance of ten different ECMs with respect to estimation accuracy

SoC and SoAP is investigated. The impedance parameters of the employed ECMs and

the battery states are estimated by means of DEKF algorithm. For verification of the

applied algorithm and performing a sensitivity analysis, NEDC load profile is employed

in a SoC range of 5% - 90% and LIBs in new state using NMC and LFP cathodes are

investigated. According to the presented results, the ECM using two RC-elements

connected in series to R0 and VOCV provides the most accurate result. The authors use the

determined average root-mean square error (RMSE) as a merit for SoC and SoAP

estimation accuracy. Furthermore, in Ref. [229], the same authors investigate the

behavior of the impedance-based battery model using two RC-elements. A sensitivity

analysis regarding the influence of ECM parameters on resulting battery impedance is

performed where it is shown that the R0 is the most dominant impedance parameter. As

a result, the authors state that a change of approx. 50% in the value of R0 yields a

deviation of


approx. 8 mΩ in the determined battery impedance. The impedance parameters of the employed ECM are identified by means of EIS measurements based on Levenberg-

Marquardt algorithm. Unfortunately, the impact of actual battery aging state and

temperature on impedance behavior of the investigated LFP-based LIBs is neglected.

In Ref. [131], a wide overview of ECMs often applied for battery state estimation in BMS

is given. The Employed ECMs are compared considering their robustness, model

complexity and accuracy of the estimated battery voltage. Multi-swarm particle (MPSO)

algorithm is applied for on-board estimation of the ECM parameters. On the one hand,

using a complex ECM yields a higher degree of voltage estimation accuracy, but on the

other hand the parameter identification of such ECMs requires high computing power

which is rarely given on low cost embedded systems. According to the authors, the ECM

using R0 connected in series with one RC-element yields most accurate results with

regard to the above mentioned criteria for voltage estimation.

In addition to the latter references, a comparative study of five various ECMs

considering SoC estimation accuracy and dynamic model performance is performed in

Ref. [287]. For this reason, LMO-based LIBs in a new state are investigated while the

employed ECMs are evaluated by means of HPPC tests and standardized load profiles

(i.e., DST and FUDS). According to the obtained results, the R-RC-RC ECM indicates

the highest SoC estimation accuracy and best dynamic performance. In the

aforementioned study, a method based on genetic algorithm is applied for identifying the

time constant of RC-element and a REKF approach is applied for SoC estimation.

Black box battery models 3.5.1.3

The black box battery models mainly refer to class of battery models that are based on

analytical techniques [272]. As mentioned beforehand, in such models the physical

dependencies are almost neglected and the battery behavior is modeled by simple

functional dependencies or system equations. In general, the black box battery models

can be subdivided into methods based on neuronal networks and fuzzy logic.

The basic principal of neuronal network-based models is very simple whereby the

relationships are reproduced by means of so-called neurons and nodes. The neurons and

nodes are interconnected together with various weights. Unfortunately, the

parameterization of the mentioned weights requires large amount of training and real

datasets. Moreover, the fuzzy logic-based battery models are likewise based on empirical

approach working according to Boolean logic where the battery is implemented as a

network of insufficient described states. The latter fact yields only a qualitative


statement of system state. Among the main advantages of black box battery models are

their low computational effort and simple nature. However, the aforementioned battery

models are not able to reproduce the electrochemical processes and, thus they are not

employed for an advanced battery modeling.

3.5.2 Requirements on battery models employed in monitoring algorithms

As often mentioned in the previous subsections, the battery modeling technique is highly

affected by respective application. In case of real-time applications/embedded systems

where available computing power is limited, many functions (e.g., battery state

estimation, diagnosis etc.) must work robust and accurate over years and over a wide

operating range. In general, following requirements on battery models for use in BMS

applications in EVs must be considered [272]:

Required implementation effort,

Computational effort or simply speaking algorithms efficiency,

Required (offline)-parameterization effort,

Consideration of aging phenomena and inhomogeneity’s of current and temperature distribution on battery pack level,

High accuracy over a wide SoC range; especially at very low SoCs,

Physical equivalent.

3.6 Summary

In this chapter, available techniques for major battery states/parameters (i.e, SoAP, on-

board capacity, SoH, SoS etc.) estimation as well as different classes of LIB models

presented in the literature are discussed comprehensively. A substantial number of

approaches for capacity estimation use the same simple principle, i.e., the relationship

between ampere-hours charged or discharged from the battery and the difference

between the two values of the voltage-based SoC (SoCV) relative to the initial and end

point where the ampere-hour throughput is measured. In addition, the methodologies

such as ICA and DVA, electrochemical model-based techniques and aging prediction

models are addressed. The main challenge for capacity estimation of lithium-ion battery

packs is still to apply a technique that is able to detect the distribution of capacity loss

from each single LIB over the battery lifetime and to try to optimize the operation

strategy of the battery in order to increase its lifetime and reduce further costs for end

consumers.


The SoAP refers to the amount of power that the battery can deliver to or accept from

the vehicle powertrain. The available battery power is mainly limited by battery SoC,

temperature, current and voltage. At the same time, battery power capability of the

battery is influenced significantly by battery’s aging state. The techniques that have up to now been applied for predicting the SoAP can be classified into the following two

groups: The first group is based on so-called ACM; the second group, which up to now

has received most attention, is based on ECMs.

In addition to the presented techniques for estimating the aforementioned battery states,

a brief review on different classes of battery models is provided. Generally, the battery

models can be subdivided into three various classes. The first class of battery models is

the so-called white box models whereby occurring electrochemical processes such as

mass transport, diffusion, ion transport and the temperature distribution inside the LIB

are represented in great detail. Moreover, grey box battery models neglect detailed

representation of occurring electrochemical phenomena and try to reproduce the

nonlinear behavior of LIBs by simple passive elements such as resistances and

capacitances. The parameters of ECM can be identified on-board by means of adaptive

techniques or offline using EIS measurements or HPPC tests. Techniques using adaptive

filters such as KF, LS or similar observer methods are the most promising compromise

between estimation accuracy and low computational effort for battery state/parameter

estimation. The use of adaptive methods for parameter identification is necessary as

impedance parameters of the employed ECMs change over the battery lifetime. In this

regard, the grey box battery models are very popular in the field of BMS mainly because

of their simplicity and low computational effort. Black box battery models are the third

class of battery models that are based on machine learning techniques such as neural

network and fuzzy logic. The latter models neglect physical dependencies of occurring

electrochemical reactions and consider simple functional system dependencies.

The equivalent circuit models investigated in this work 134

4 The equivalent circuit models investigated in this work

As discussed in the previous chapter, the computing power of ECUs commonly used in

BMS applications is limited. Thereafter, it is not technically feasible to apply white-box

battery models for BMS applications at least at present. In this regard, grey box battery

models are very popular approaches for use in monitoring algorithms since such battery

models are simple and can precisely reproduce the battery dynamics over a wide

frequency range. In this work, a systematic overview over impedance-based ECMs often

employed for battery state estimation is given, thus a feasibility study is performed. In

total, seven different impedance-based ECMs incorporating various passive components

such as simple linear (as described in chapter 2) up to more complex components such

as ZARC-elements are investigated. It is worth noting that the nonlinearity of the charge

transfer processes is considered by applying the BVE (Eq. (11)). The investigated ECMs,

their model equations in discrete form and a brief description of each can be found in

Table 15.

Table 15: Specification of the equivalent circuit models investigated in this work

Equivalent circuit models Model equations ECM description

Model 1:

∆ = ∙

Vsim(k)=∆VR0(k)+

VOCV(k)

The employed ECM

consists of R0 and

VOCV. The ECM

consists of one

adaptive parameter.

Model 2:

∆ + = , ∙ −∆+ ,∙ − −∆∙

Vsim(k)=∆VR0(k)+

∆Vct(k)+VOCV(k)

= ∙ refers to

time constant and

∆Vct refers to

overvoltage of the RC-

element.

The ECM consists of

R0, RC-element

representing charge

transfer processes and

VOCV.

The ECM consists of

four adaptive

parameters.


Model 3:

∆ , + = ,∙ −∆+ , ∙− −∆∙

Vsim(k)=∆VR0(k)+

∆Vct(k)+∆Vd,1(k)+VOCV(k)

= , ∙ , refers to

time constant and ∆ , refers to

overvoltage of the

second RC-element.

The ECM consists of

R0, RC-element

representing charge

transfer processes,

RC-element

representing diffusion

processes and a VOCV.

The ECM consists of

six adaptive

parameters.

Model 4:

∆ , + = ,∙ −∆+ , ∙− −∆∙

Vsim(k)=∆VR0(k)+

∆Vct(k)+

∆Vd,1(k)+∆Vd,2(k)+

VOCV(k)

= , ∙ , refers to

time constant and ∆ , refers to

overvoltage of the

third RC-element.

The employed ECM

consists of R0, three

RC-elements

representing charge

transfer and diffusion


The ECM consists of

eight adaptive

parameters.

Model 5:

∆ ZA C, += ZA C, ∙ −∆+ ZA C, ∙ − −∆∙

Vsim(k)=∆VR0(k)+

∆VZARC,1(k)+VOCV(k)

= ZA C, ∙ ZA C,

refers to time

constant and ∆ ZA C,

refers to overvoltage

of the ZARC-element.

The employed ECM

consists of R0, ZARC-

element representing

charge transfer


The ECM consists of

five adaptive

parameters.

Model 6:

∆ ZA C, += ZA C, ∙ −∆+ ZA C, ∙ − −∆∙

Vsim(k)=∆VR0(k)+

∆VZARC,1(k)+∆VZARC,2(k)+

VOCV (k)

= ZA C, ∙ZA C, refers to time


refers to overvoltage

of the second ZARC-

element.

The employed ECM

consists of R0, two

ZARC-elements

representing charge




The ECM consists of

eight adaptive

parameters.

Model 7:

∆ ZA C, += ZA C, ∙ −∆+ ZA C, ∙ − −∆∙

Vsim(k)=∆VR0(k)+

∆VZARC,1(k)+∆VZARC,2(k)+

∆VZARC,3(k)+VOCV(k)

= ZA C, ∙ ZA C, refers to time


refers to determined

overvoltage of the

third ZARC-element.

The employed ECM

consists of R0, three

ZARC-elements

representing charge



The ECM consists of

eleven adaptive

parameters.

4.1 Equivalent circuit model using an ohmic resistance (Model 1)

Each of the investigated ECM consists of R0 representing resistive contribution of the

electrolyte, current collectors etc. The R0 is highly temperature dependent and increases

over the battery lifetime while its dependence on battery SoC is more or less pronounced

depending on active materials of LIBs. The C represents the voltage when the LIB is

in steady state and all side reactions are finished (for more detailed explanation

regarding internal or external factors influencing the aforementioned parameters the

readers are referred to chapter 2). The impedance of the model 1 is determined as follows

(Eq. (39)):

, = (39)

4.2 Equivalent circuit models using an ohmic resistance connected in series with a finite number of RC-elements (Models (2-4))

In order to capture the dynamics of LIBs more precisely, a finite number of RC-elements

are connected in series with R0. Each RC-element consists of a resistance and a

capacitance. The models 2-4 differ in the number of the RC-elements. The higher the

number of RC-elements is the higher the accuracy of estimated voltage can be. However,

at the same time, the execution time or required computing power increases, thus often a

compromise between these both parameters needs to be found. The first RC-element

with as time constant is always responsible for capturing charge transfer processes


occurring at the interface between the electrode and the electrolyte. It is worth noting

that depending on ECM parameterization and relating frequency of respective processes,

the first RC-element could also refer to SEI in the anode. However, in the present work,

the impact of SEI formation is neglected for battery modeling and it is assumed that

time constant of both processes are very similar and does not lead to an inseparable

semi-circle in the measured impedance spectrum. In this regard, the impedance of the

aforementioned models can be determined as follows (Eq. (40)):

, − = + ∑ ∙ += (40)

Charge transfer processes in batteries can be divided into the following two types:

Homogenous or heterogeneous charge transfer processes [244]. Homogenous charge

transfer corresponds to charge carrier (ion) exchange in electrolyte and second type

corresponds to charge transfer between ions in electrolyte and ions in the electrode.

Charge transfer overvoltage is determined when the difference between the amount of

lithium-ions during intercalation and deintercalation at both electrodes is not zero and

the LIB is being charged or discharged. The correlation between current density and ∆

can be described by the BVE as described beforehand in chapter 2 as follows: = ∙ exp ∙ ∙∙ ∙ Δ − exp − ∙ ∙∙ ∙ Δ (41)

When the effective internal surface ( ) is considered, the multiplication of the current

density and the is equal to the current flowing into the electrodes (i.e., = ∙ ).

Accordingly, Eq. (41) can be rewritten as follows (Eq. (42)): = ∙ exp ∙ ∙∙ ∙ Δ − exp − ∙ ∙∙ ∙ Δ (42)

The definition of each parameter given in Eq. (42) was provided in chapter 2; thus no

additional description is provided here. The current dependence of the Rct is considered

in the employed ECMs by applying the BVE. Considering the current dependence of Rct

becomes essential at very low temperatures or when the LIB is aged as discussed in

chapter 2. The latter fact yields higher battery voltage prediction accuracy. For

simplicity reasons it is assumed that the of both anodic and cathodic reactions are

equal (i.e., = = . 29) and, thus the BVE can be solved analytically (Eq.

(43)): = sinh . ∆ (43)

29 For most of the LIBs, may be assumed to be approx. 0.5. However, for other battery technologies, such as lead-acid

batteries this simplification is not applicable, whereas values between 0.25 and 0.3 seem to be feasible [244].


After some simple mathematical formulations the current dependence of Rct can be

derived as follows (Eq. (44)):

= , ∙ ( . ∙ +√ . ∙ +

. ∙ )

30 (44)

As described in chapter 2, the changes for different temperatures and decreases

significantly over the battery lifetime and, thus it is of great interest for application

engineers to consider this term in an adaptive manner for monitoring algorithms.

Thereafter, within this work, an additional impedance parameter, so-called is

employed. is determined as follows (Eq. 45): = . [A− ] (45)

kI describes the behavior of that is mainly aging state and temperature dependent.

Moreover, in order to avoid singularities, the factor , is employed which is equal to Rct

at IR=0 A, which is determined by L'hopital's rule. In Fig. 48, Butler-Volmer behavior of

Rct are depicted schematically using the above mentioned equations for different current

rates and kI values.

Fig. 48: Schematic illustration of the nonlinear behavior of Butler-Volmer characteristic; a: anodic and cathodic reactions, b) normalized charge transfer resistance for various kI values where it is assumed that symmetry factor is 0.5

30 In order to decrease the computational effort for considering the current dependence of the Rct it can be implemented by

a means of simple look-up table as follows: ln + √ + .


The dynamic behavior of LIBs is influenced by ; the higher is the slower is the voltage

response. The time constant of individual RC-element must be differentiated in a range

of some (mili)seconds up to some hours. Another important impedance parameter used

in the models 2 – 4 is the Cdl. As discussed in chapter 2, the Cdl can be determined in a.c.

or d.c. domain using EIS measurement or HPPC tests. In model 3 and model 4, the

second and third RC-elements with time constants and refer to diffusion processes

occurring at very low frequencies (i.e., approx. f 1 Hz). Summarized, the electrical

behavior of RC-elements described above can be expressed as follows (Eqs. (46)-(47)):

[ ∆∆ ,∆ , ] = [ −∆ −∆ −∆] ∙ [ ,, ] + [

∙ − −∆, ∙ − −∆

, ∙ − −∆ ] ∙

[ ] (46) = ∆ + ∆ + ∆ , + ∆ , + C (47)

4.3 Equivalent circuit models using an ohmic resistance connected in series with a finite number of ZARC-elements (Models (5-7))

Unfortunately, the RC-elements described above are not able to reproduce accurately

battery dynamics for specific operating conditions and often yield unsatisfactory results;

especially when the battery voltage needs to be predicted over a large time horizon or

when the battery is operated at very low SoCs or very low temperatures. The latter issue

will be discussed in great detail in chapter 5. Moreover, in some cases the intention of

the employed ECM could be to modeling the inhomogeneous material distribution or

modeling the porosity [171]. Therefore, the so-called ZARC-element is often a preferred

choice for an accurate battery modeling in the aforementioned operating areas and

describes depressed semi-circles more precise. If a CPE-element is used instead of the

capacitance in a simple RC element, the resulting circuit is a ZARC-element with the

following impedance: ZA C = .. + ; where R refers to the resistance, A is the

generalized capacitance; ξ is the depression factor and is the frequency range [280].

Purely capacitive behavior is expected when ξ=1. Moreover, as described in chapter 2,

ξ=0.5 yields a -45° branch whereby ξ=0 refers to purely resistive behavior of the CPE-

element. The impedance of the above mentioned models can be determined as follows

(Eq. (48)):

− = + ∑ . . += (48)


Since the aforementioned parameters are in frequency domain, they must be

approximated in time domain for application in on-board monitoring algorithms. The

ZARC-element can be approximated by means of finite number of RC-elements. A well

compromise between efficiency and accuracy can be met by approximating the ZARC-

element using three or five RC-elements. In the present work the 3 RC model is applied.

For more detailed information regarding the applied approximation technique, the

readers are referred to Ref. [280]. Summarized, the electrical behavior of ZARC-elements

employed in models 5-7 can be expressed in discrete form as follows (Eqs. (49)-(50)):

[∆ ZA C,∆ ZA C,∆ ZA C, ] =[ −∆ −∆ −∆

] ∙ [ ZA C,ZA C,ZA C, ] +

[ ZA C, ∙ − −∆

ZA C, ∙ − −∆ZA C, ∙ − −∆ ]

∙ [ ] (49)

= ∆ + ∆ ZA C, + ∆ ZA C, + ∆ ZA C, + C (50)

In Fig. 49, the impedance spectrum, impedance amplitude and phase of Cell-G-new

measured at 23 °C and 50% SoC are illustrated. In addition, the above discussed ECMs

are fitted to the measured impedance spectrum and their impedance behavior is

analyzed.


Fig. 49: Impedance spectra of the investigated ECMs compared with an impedance spectrum of Cell-G-new measured at 23 °C and 50% State-of-Charge; a) Nyquist plot, b) Bode plot.

In order to compare the resulting impedances of the ECMs, the RMSE of individual ECM

is determined as follows (Eq. (51)):

REZ = √∑ | | , −| | ,− (51)

where refers to the lowest measured frequency (i.e., approx. 1 mHz) and is the

highest measured frequency before inductive tail of the impedance spectrum starts; in

other words, refers to frequency where R0 is detected (i.e., approx. 500 Hz). The results

of the obtained RMSE values are given in Table 16. According to the obtained RMSE

values, model 7 indicates the highest accuracy with regard to the determined battery

impedance.

Table 16: Comparison of impedance behavior of the investigated ECMs based on determined root-mean square error

Model number Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7

REZ [Ω] 0.325 0.098 0.0543 0.0522 0.09 0.043 0.0175


4.4 Summary

In this chapter, the ECMs investigated in this work are presented. Moreover, physical

meaning and electrochemical dependencies of impedance parameters are described for

the respective ECMs. In total, seven different ECMs are examined; resulting voltage

equations and electrical behavior of the ECMs are given. According to the obtained

results for impedance behavior of the examined ECMs, it can be concluded that model 7

is the most precise ECM with regard to battery impedance determination for the

examined frequency range. The accuracy of the determined complex battery impedance

using the aforementioned ECM is increased approx. by a factor 2 in comparison to

model 1. In the following chapter, the performance of the aforementioned ECMs will be

investigated with regard to accuracy of voltage estimation and SoAP prediction based on

load profiles presented in chapter 2.

Applied method for battery state estimation 143

5 Applied method for battery state estimation

As described in chapter 3, adaptive techniques often applied for on-board ECM

parameters estimation can be subdivided into following two groups: First, methods based

on adaptive joint estimation techniques. Second, methods based on adaptive dual

estimation techniques. In this work, a recursive joint estimator, the so-called enhanced

varied parameters approach (EVPA) is applied for on-board parameters identification of

the investigated ECMs. Combination of the implemented ECMs (see chapter 4) and the

EVPA applied for impedance parameters estimation yields very high voltage estimation

accuracy. The employed algorithm can further be used to estimate other battery states

such as for voltage-based SoC (SoCV) or SoH. As discussed in chapter 3, the main

limiting parameters for SoAP are voltage and current of the battery whereby it must be

ensured that the battery voltage is always in predefined voltage limits. An accurate

prediction of battery voltage for a certain current belongs always to a successful SoAP

prediction strategy. The more precise the predicted battery voltage, the higher is the

accuracy of the predicted SoAP.

This chapter first describes the basic idea of the applied algorithm and provides

examples of the obtained results regarding the estimated ECM parameters.

Furthermore, the basic idea of the algorithm applied for voltage and SoAP prediction is

discussed comprehensively. Thereupon, dependence of the voltage estimation and SoAP

prediction accuracy on following parameters is investigated as follows:

Dependence of voltage estimation and SoAP prediction accuracy on applied

current rate,

Dependence of voltage estimation and SoAP prediction accuracy on SoC,

Dependence of voltage estimation and SoAP prediction accuracy on battery

temperature.

Dependence of SoAP prediction accuracy on prediction time horizon (∆). 5.1 Method for impedance parameters estimation

As discussed beforehand, the impedance parameters and states of the battery can be

estimated on-board by means of adaptive methods. In this work, a model-based joint

estimator is applied for estimating ECM parameters and different overvoltages referring

to various electrochemical processes, respectively. The core algorithm of EVPA is based

on a straight forward recursive technique and it was presented for the first time in detail


in Ref. [46]. The aforementioned algorithm is able to adapt each adaptive parameter to

particular aging state and/or operating condition of the battery and works accurately

over a wide temperature range. The latter fact is a key factor for accurate voltage

estimation. Fig. 50 illustrates schematically operating principle of the employed

algorithm in context of simplified BMS architecture.

Fig. 50: Schematic illustration of operating principle of the applied algorithm for impedance parameters and voltage estimation in the context of simplified battery management system architecture

In the first step, the algorithm is initialized by ACM for given temperature and SoC. In

real application which means that when a code generation is performed and the

algorithm is implemented on microcontroller, an ACM is implemented for each adaptive

parameter which is based on a polynomial approximation (i.e., 2nd order polynomial

function). In this regard, the impedance parameters of the employed ECM can be

extracted by means of EIS measurements or HPPC tests in the laboratory.

The underlying idea of the parameter estimation algorithm is the combination of certain

values of average LIB parameters within a parameter set P for a nonlinear ECM

consisting of various impedance parameters discussed in the previous chapter (e.g., for

model 2 means that the model consists of four adaptive parameters ( , , kI, ) for

example P = = 0.8 mΩ, = 1 mΩ, = 100 F and kI = 0.01 A-1. The load profile

applied to the LIB is then divided into short evaluation periods with a duration of some

seconds each (e.g., k = 200 values at sampling frequency of 10 Hz). During each


evaluation period n different parameter sets (P1, P2, … Pn) are generated and then n

battery voltages at each sample step k during this period are determined using

measured battery current with these parameter sets.

It is assumed that during the evaluation period the OCV remains constant or its change

is negligible. The decision for starting and terminating of each of the evaluation periods

is made according to following criteria:

It is verified whether the current through the capacitance over a short period of

time is low (i.e., < , for ∆ ). The definition of this criterion is important,

since the battery voltage can only be determined on the basis of the previous ∆ .

At the same time, it is important to minimize the impact of the previous history of

the battery on resulting battery voltage. Moreover, high current fluctuation in

load profile allows obtaining more valuable information regarding the battery

impedance. Thereafter, it is necessary that the current fluctuation is evaluated.

When the conditions mentioned beforehand are fulfilled and the current profile

for a predefined time range is evaluated, the evaluation period can be terminated

and preferably the next evaluation directly begins.

The estimated voltages are then compared to the measured battery voltage. The

parameter set which results in the lowest quadratic voltage deviation or so-called cost

function ( C ) between the measured and the determined voltage (Eq. (52)) is selected

as the ideal parameter set and then used as a basis to generate parameter sets for the

next evaluation period.

C = ∑ , − ,= (52)

As discussed in the previous subsections, a compromise often has to be achieved between

an accurate battery modeling and a low computational effort in BMS applications. In

contrast to the core algorithm presented in Ref. [46], an improvement is performed

regarding the amount of the parameter loops need to be compared. The latter fact yields

almost the same grade of accuracy for impedance parameters estimation and faster

convergence of each adaptive parameter. Moreover, the simplification allows the

contribution of more complex nonlinear ECMs. The principle of searching method used

by the EVPA for an example of three adaptive parameters is shown schematically in Fig.

51.


Fig. 51: Schematic illustration of searching principle of the applied algorithm for impedance parameters estimation

The C given in Eq. (52) is determined for the initial parameter set and for parameter

sets containing each time one different value of individual parameter. Each parameter is

varied in positive and negative directions. This means that for n adaptive parameters,

only 2n calculations instead of 3n calculations as proposed in Ref. [46] need to be

performed yielding an enormous decrease of the needed computing effort and faster code

generation. For example, in case of parameter , the coefficient (∆ ) is used to calculate

two variations of this parameter value as follows: +( ∙ ∆ ) and − ∙ ∆1). The

coefficient (∆ ) is implemented as a variable, which may change during each iteration

period in order to accelerate the parameter convergence. In general, when the parameter

adaption is stable or simply speaking the correct convergence of the parameter was

successful, the value of ∆ remains constant. It is worth noting that the width of ∆ plays

a major role in order to ensure the parameter’s convergence in a correct direction. Choosing high initial values may lead to wrong results and the parameter might be

converges to the wrong direction or diverges. Therefore, it is recommended to explore

proper initial values that yield accurate results for each of parameters during offline

tests.

In Fig. 51, the blue point in the middle represents a reference parameter set where the

new parameter pairs are built as described above around this point with the width of ∆ .

Green points indicate that the value of the cost function in Eq. (53) in this direction for

these two dimensions yield lower value than value for other parameter sets illustrated


with red points. The third dimension which is marked with orange color shows no

improvement in the determined results and therefore no more investigation will be

performed in respective dimensions. Once all the dimensions are examined a new

parameter set is set for next evaluation period.

5.1.1 A selection of simulation results for impedance parameters estimation

Based on the method described in the previous subsection, the results obtained for model

6 are investigated using Cell-G-new as a reference. As discussed in chapter 4, the

aforementioned model consists of totally eight adaptive parameters

i.e., R0, RZARC,1, RZARC,2, AZarc,1, AZarc,2, , , kI. In Fig. 52, the adaption process of the

impedance parameters measured and simulated voltages are illustrated for the

examined load profile as shown in Fig. 27.

Fig. 52: Results of voltage simulation and adaption process of the impedance parameters of model 6 for Cell-G-new

According to the results, the adaption of the impedance parameters during temperature

change for respective SoC can be observed. For example, an increase of resistances or

decrease of capacitances with decreasing temperature can be clearly observed which is in

agreement with results presented in chapter 2. In addition to the results shown above,


Fig. 53 depicts measured and simulated battery voltages including determined

overvoltages (i.e., ∆ + ∆ ZA C, etc.) and C . The C is determined according to Eq.

20 by subtracting the simulated voltage from the sum of the determined overvoltages.

Fig. 53: Measured and simulated battery voltages including determined overvoltages and estimated open-circuit voltage based on model 6 for Cell-G-new

5.1.2 Sensitivity analysis of impedance parameters and voltage estimation accuracy during an extreme temperature variation

In the previous subsection, the robustness analysis of the applied algorithm is performed

and applicability of the used ECMs for BMS applications in an example of model 6 is

analyzed. According to the results, consideration of extraordinary internal or external

battery temperature change is essential for monitoring algorithms. The latter fact occurs

as a result of high impact of the battery’s temperature on battery impedance characteristic. In order to examine the impact of temperature change on the applied

algorithm regarding robustness and correct convergence of the estimated parameters, a

measurement procedure on the test bench is defined and the load profile shown in Fig.

27 (in particular the second day driving cycle) is used. The measurement was performed


on each investigated LIB while only the results obtained from Cell-B-new are discussed

in this subsection.

The defined measurement procedure is as follows: The LIB was fully charged at 23 °C

using CC-CV technique and then tempered in temperature chamber for more hours to -

20 °C at fully charged state. Thereafter, the test was performed while the temperature

was increased to +30 °C after 1 hour driving until the battery was almost fully

discharged and the driving cycle is ended. Fig. 54 shows exemplary the results of

adaption process of estimated impedance parameters of model 6 obtained for Cell-B-new.

Fig. 54: Sensitivity analysis of adaption process of the impedance parameters of model 6 during an extreme temperature variation (from -20 °C up to +30 °C) over a wide SoC range for Cell-B-new

For robustness analysis, the initial values of the impedance parameters shown in Fig. 54

are intentionally chosen incorrect. The results indicate the fast adaption (approx. below

10 minutes) of the parameters to correct values for given conditions. For example, correct

adaption of resistances can be seen while they are increasing with decreasing

temperatures and decreasing with increasing temperatures respectively.

Fig. 55 shows exemplary the results of sensitivity analysis with regard to battery voltage

estimation accuracy in an example of Cell-B-new.


Fig. 55: Sensitivity analysis and instantaneous voltage estimation during an extreme temperature variation (from -20 °C up to +30 °C) over a wide SoC range for Cell-B-new

As expected due to higher pronounced diffusion processes occurring in the battery at

very low temperatures31, the deviation between estimated and measured voltage is

higher than the calculated error when the temperature is increased. Furthermore, at

very low SoCs (i.e., SoCs 20%) the estimation accuracy decreases again due to

pronounced diffusion phenomena and rapid/extreme OCV change at very low SoCs as

discussed in chapter 2.

5.2 Method for battery State-of-Available-Power prediction

In the previous subsection, the basic idea of the applied algorithm for battery impedance

parameters estimation is presented. Moreover, the results of sensitivity analysis for

adaption process of the impedance parameters and the impact of temperature change

during a driving cycle on impedance behavior of the LIB are discussed. In the following

subsections: First, the basic idea and relating mathematical equations of the applied

technique for a precise SoAP prediction is discussed. Second, model performances (i.e.,

31 It is assumed that the impedance parameters are converged to their right values.


models 1 - 6) are analyzed and compared with regard to voltage estimation and SoAP

prediction accuracy.

5.2.1 Basic idea of the applied technique for SoAP prediction

As described in the previous chapter, the investigated ECMs are employed as a basis for

battery voltage estimation whereby, at the same time the battery voltage can be

predicted for a certain time horizon (∆). The battery voltage for charging and

discharging cases for models 2 - 7 are determined as follows (Eqs. (53)-(54)): ∆ = ∆ + = ∙ −∆ + ∙ ∙ ( − −∆ ) + = ∙ −∆ + ∙∙ − −∆ + C = + oC ∙ ∆ ∙ (53)

∆ = ∆ + = ∙ −∆ + ∙ ∙ ( − −∆ ) + = ∙ −∆ +∙ ∙ − −∆ + C = + oC ∙ ∆ ∙ (54)

where i refers to the number of RC-elements or ZARC-elements ( ∈ … ) employed in

the respective ECM given in Table 15. Description and method for determining

and oC have been described in chapters 3 and 4.

In addition to the latter equations, for determining the limiting charge/discharge current

( ) that yields exactly the upper or the lower battery voltage limits (i.e., or ),

the following equation must be solved (Eq. (55)): = ∆ + = ∙ −∆ + ∙ ∙ ( − −∆ ) + = ∙ −∆ + ∙ ∙ −−∆ + C = + oC ∙ ∆ ∙ (55)

where = and = refer to the case when the battery is charged and = and = refer to the case when the battery is discharged,

respectively. By employing the aforementioned equations, the available battery power

for charge and discharge cases (i.e., and ) for a predefined prediction

time horizon (∆) can be simply calculated as follows (Eqs. (56)-(57)): = ∙ ∆ (56) = ∙ ∆ (57)

As described in chapter 3, Eqs. (56)-(57) are valid for the case where constant current

pulses are applied to the battery (see Fig. 43). However, for the case when constant

power pulses are applied to the battery, the change of current over ∆ should additionally

be considered in the aforementioned equations.

Furthermore, it is worth mentioning that Eqs. (53)-(55) are ideal candidates for voltage

and current prediction of models 2 – 7. Unfortunately, these equations cannot be applied


to model 1 as the latter ECM consists of only one linear impedance parameter and

transient behavior of the battery is neglected completely. Therefore, available power of

the battery considering design limits; i.e., voltage, SoC and current can be predicted as

follows (Eqs. (58)-(67)) [249]-[250]: ∆ = ∆ + C + oC ∙ ∆ ∙ (58) ∆ = ∆ + C + oC ∙ ∆ ∙ (59)

, , = OCV C −+ C ∙∆ ∙ (60)

, , = OCV C −+ C ∙∆ ∙ (61)

, , C = C − C∆ (62)

, , C = C − C∆ (63)

, = max , , , , , , , C (64)

, = min , , , , , , , C (65) = ∆ ∙ , (66) = ∆ ∙ , (67)

It is worth noting that the definition of and mentioned above in Eqs.

(56)-(57) and Eqs. (66)-(67) is equal to SoAP presented in chapter 3. Please notice that

the SoAP mentioned in the following subsections is calculated based on the

aforementioned equations.

As described in chapter 3 (see Fig. 43), for the case where the battery voltage reaches its

predefined limit (i.e., or ) and the applied current is within the predefined

battery current limits (i.e., cases a and c), the prediction of SoAP is more challenging

than the remaining two cases (i.e., cases b and d). Unfortunately, the real available

SoAP is unknown and, thus the inaccuracy of the predicted SoAP cannot be derived

directly. In fact, the real maximum available battery power can only be predicted when

the exact current (Ireal,max power) that leads to ∆ = or ∆ = 32 is known or in

other words when the battery voltage hits or . However, the Ireal,max power cannot

be calculated; if so, then we have an ideal SoAP prediction.

As a basis for an accurate SoAP prediction it is necessary that battery voltage and SoAP,

respectively, are predicted precisely for a certain constant current pulse (i.e., ) over a

prediction time horizon. Hence, based on the estimated impedance parameters, battery

32 The ∆ and ∆ are determined as follows: ∆ = − and ∆ = − .


voltage is predicted for a specific ∆ and compared with voltage/power measured on test

bench. Thereafter, the relative SoAP prediction error is determined as follows (Eq. 68):

∆ = ,∆− ∆∆ (68)

5.2.2 Results and discussion

In this subsection, the results obtained from HPPC tests and load profiles shown in

Fig. 27 are presented. In this regard, first, dependence of calculated RMSE for estimated

battery voltage ( ) on SoC, temperature and applied current rate is examined for the

aforementioned load profiles. For this reason, a statistical analysis of the obtained

results is performed whereby is determined as follows (Eq. (69)):

= √∑ , − ,= (69)

The dependence of on SoC, temperature and applied current rate is investigated as

follows:

1. The accuracy analysis of the employed ECMs is performed based on determined

for each manifold 10% SoC range (0-10%, 10%-20%, 20%-30%,⋯,90%-100%).

It is worth mentioning that the SoCs are calculated using battery’s nominal

capacity.


in following temperature ranges:

a. ∆1 = -20 °C⋯-5 °C

b. ∆2 = -5 °C⋯10 °C

c. ∆3 = 10 °C⋯25 °C

d. ∆4 = 25 °C⋯40 °C


for following current rate ranges:

a. ∆ -4C

b. ∆ = -4C⋯-2C

c. ∆ =-2C⋯0

d. ∆


It is worth mentioning that because of occurring parameters convergence problem,

model 7 is not considered for further analysis in the present chapter. The main reason

for convergence problem of the estimated impedance parameters of model 7 can be

referred to high number of implemented adaptive parameters included in the respective

model. Thereafter, only the first six ECMs are investigated in the following subsections.

Dependence of voltage estimation accuracy ( ) on SoC, temperature and 5.2.2.1

applied current rate

In this subsection, the main focus lies on performance analysis of the investigated ECMs

with regard to estimation accuracy of . In Fig. 56, the results obtained from HPPC

tests performed at 23 °C on Cell-B-new, Cell-C-new, Cell-C-aged and Cell-G-new are

illustrated.


Fig. 56: Model performances based on HPPC tests performed at 23 °C on Cell-B-new, Cell-C-new, Cell-C-aged and Cell-G-new33

In fact, for Figs. 56-57 where the results obtained from the performed HPPC tests are

shown, charging pulses with the same amplitudes as illustrated discharging pulses are

available. However, in order to stay conform to the later results derived from driving

cycles where charging current pulses are limited, the investigated current range (i.e., x-

axis, respectively) is limited to 2C. According to the results illustrated in Fig. 56 it can be

concluded that the model 1 indicates the poorest performance in contrast to other ECMs

for the investigated LIBs. For Cell-G-new the differences between models 2-6 with

regard to current dependence of is almost negligible. The obtained differences are

simply too low for a convincing statement regarding the most accurate model for voltage

estimation. However, by investigating the SoC dependence of , the performance of

the models differ for various SoC ranges. In total, it may be stated that despite some

slight exceptions model 4 seems to be more accurate than other ECMs. In contrast to

Cell-G-new, the results obtained for Cell-B-new shows that performance of the models

are slightly different. The slight difference between the derived results for model 4 and

model 6 indicate that their performances depending on SoC and applied current rate are

almost similar. In addition to the latter two LIBs, the results obtained for Cell-C-new

and Cell-C-aged are analyzed. According to the obtained results, it may be stated that

model 4 indicates the most accurate results for both LIBs. As it can be seen, the models

accuracy decreases slightly when the battery is aged while the latter fact is more

pronounced for model 1. Moreover, it becomes obvious that estimation accuracy of the

investigated models decreases significantly with decreasing SoC.

The results discussed above have a representative character since the influence of

temperature is not considered and the HPPC tests are performed under nominal

condition. In chapter 2, the significant impact of temperature on battery’s impedance characteristics is discussed comprehensively. In this regard, in case of model 1 the

impact of battery temperature on voltage estimation accuracy is examined. It is worth

noting that selection of model 1 is not met according to any special reasons and should

just give an indication for possible behavior of other models at different temperatures.

33 It is worth mentioning that in the present chapter, the x-axis of figures referring to current dependence of and

SoAP are based on battery’s nominal capacity.


Fig. 57 illustrates the results obtained for the dependence of the determined on

applied current rate and SoC at various temperatures, in particular 0 °C, 23 °C and 40

°C.

Fig. 57: Comparison of model 1 performance based on HPPC test performed at different temperatures (i.e., 0 °C, 23 °C and 40 °C); a) Cell-G-new, b) Cell-C-new and c) Cell-C-aged

As expected, the performance of model 1 decreases with decreasing temperature for the

investigated LIBs. The model inaccuracy at 0 °C is approx. 5 times higher than the

model inaccuracy at 40 °C. Contrary, the SoC dependence of calculated seems to be

less pronounced at 23 °C and 40 °C. Moreover, it becomes obvious that at 0 °C voltage

estimation accuracy decreases continuously with decreasing SoC whereas this trend

cannot be approved for Cell-C-new. In the latter case estimation accuracy at 23 °C seems

to be slightly better than at 40 °C which probably has its source in measurement


uncertainty; however, unfortunately no clear explanation can be provided regarding this

unexpected behavior.

Up to now, the model performances for the performed HPPC tests are examined. In the

following, the performance of the investigated models for the examined driving cycles is

discussed comprehensively. Fig. 58 illustrates the results obtained from the investigated

driving cycles shown in Fig. 27 for different temperature, SoC and applied current

ranges.

Fig. 58: Model performances based on the examined driving cycles for Cell-B-new, Cell-G-new and Cell-J-aged34

As a general explanation for the obtained results for current dependence of determined

, it may be concluded that the differences between model accuracies are very similar

and almost negligible. The voltage estimation accuracy is improved by approx. 15% from

model 1 to model 4 or model 6. Depending on application of interest and its

requirements, the application engineers may weight the advantages and disadvantages

of the respective models for particular application.

34 The results obtained for Cell-D-new and Cell-A-aged can be found in appendix E.


Moreover, by analyzing the results obtained for SoC dependence of estimated for

Cell-G-new, two local peaks become obvious in SoC range of 30%-40% and SoC range of

60%-70%. The same behavior can also be detected in the results obtained for Cell-B-new

and Cell-J-aged whereby the obtained voltage errors are slightly lower pronounced. In

contrast to Cell-B-new and Cell-G-new, for Cell-J-aged the first peak is shifted to SoC

range of 20% - 30%. According to SoC ranges where the aforementioned peaks are

observed, these peaks may be referred to staging phenomenon occurring in the anode.

As shown in Fig. 57, temperature of the battery may affect significantly the voltage

estimation accuracy. Consequently, voltage estimation accuracy decreases with

decreasing temperature. In general, the same tendency can also be observed in Fig. 58;

however, for Cell-G-new the voltage estimation accuracy decreases significantly in a

temperature range of -5 °C ⋯ +10 °C in contrast to other temperature regions. The

reason for the mentioned observation can be found in temperature distribution of the

examined driving cycle shown in Fig. 29. Unfortunately, no clear statement can be

provided with regard to selection of the most accurate model using the obtained results

since the determined deviations for voltage estimation accuracy of the models; in

particular, model 3, model 4 and model 6 are marginal and by considering further

external uncertainty sources such as measurement errors etc. an exact answer cannot be

provided to the question which model is the most accurate one. The obtained results may

be employed for a model selection when the target system/application and the respective

operating SoC range, temperature and the applied current rates are more specified.

In the following subsections, model performances with regard to their SoAP prediction

accuracies depending on the mentioned parameters (i.e., temperature, SoC and applied

current rate) are analyzed.

Dependence of SoAP prediction accuracy on SoC, temperature and applied 5.2.2.2

current rate

In the previous subsections, model performances with regard to the determined for

different applied current rates, SoCs and temperatures are discussed. In the present

subsection, the SoAP prediction accuracy is analyzed for the investigated ECMs where

the ∆ as proposed in Eq. (68) is applied for analysis of the predicted SoAP. In the first

step, dependence of SoAP prediction accuracy on applied current rate followed by

examining its SoC and temperature dependency are investigated based on HPPC tests

performed at 23 °C and driving cycles. Moreover, in the second step, the influence of


defined prediction time horizon on SoAP prediction accuracy is examined using model 1

as an example. For this reason, five various prediction time horizons are investigated.

As a basis for precise scientific evaluation, the performance of the investigated ECMs is

compared using so-called boxplots. For better interpretation of the boxplots, a brief

description is provided as follows:

1. The tops and the bottoms of the boxplots refer to the 25th and 75th percentiles of

the samples whereby the distance between them (i.e., tops and bottoms of the

boxplots) refers to interquartile ranges.

2. The median of the samples are illustrated using a line in the middle of each

boxplot. Moreover, the skewness of the samples occurs when the median is not

centered.

3. The lines extending above and below each boxplot are the so-called whiskers. In

this work, the whiskers are determined as follows:

a. The whiskers correspond to ±2.7 and 99.3% coverage of the samples if

the data are normally distributed.

b. The illustrated whiskers extend to the most extreme data value identified

not to be an outliner.

c. In this work, the outliners are excluded in the figures and not shown

further. A data value is identified as an outliner if it is higher than

P2+w∙(P2-P1) or less than P1-w∙(P2-P1) where P1 and P2 refer to 25th and 75th

percentiles of the samples.

5.2.2.2.1 Dependence of SoAP prediction accuracy (∆) on applied current rate

The current range investigated for accuracy analysis of determined ∆ differs slightly to

the current range used beforehand for accuracy analysis of . The examined current

ranges are as follows:

a. ∆ -4C

b. ∆ =-4C⋯-2C

c. ∆ =-2C⋯-1C

d. ∆ =-1C⋯0

e. ∆ = ⋯1C


f. ∆ =1C⋯2C

g. ∆ =2C⋯4C

h. ∆ 4C

In Fig. 59, the results obtained from HPPC tests performed at 23 °C on the investigated

LIBs are shown. The illustrated boxplots address the dependence of the determined ∆

on applied current rate for a prediction time horizon of 20 s. It is worth mentioning that

the range of applied current rates (on the x-axis) for Cell-C-new and Cell-C-aged differs

to that of other LIBs. That is mainly caused by the fact that for the aforementioned two

LIBs the applied current rates were limited to ±3C. Moreover, the illustrated results

indicate much lower values for the determined ∆ (i.e., prediction accuracy is very high)

in a current rate range of ∆ . The latter fact occurs actually as a result of battery

discharge with a constant current rate of 0.5C between two various SoCs whereby no

SoAP prediction is performed in this areas (i.e., ∆ = ). Thereafter, in combination with

”correct” determined ∆ values derived based on applied current pulses in this range

(e.g., ∆ =0.5C or ∆ =0.75C etc.), resulting ∆ is low in the aforementioned ranges.

However, when the constant discharge phases are excluded, resulting model accuracies

(i.e., at least the tendency of model behavior) should be very similar to that obtained in a

current range of ∆ .

According to the results shown in Fig. 59 for HPPC tests performed at 23 °C, following

conclusions may be drawn:

1. As expected, model 1 indicates the poorest performance in contrast to other

models with regard to SoAP prediction accuracy. The model inaccuracy increases

by a factor of approx. 1.5 - 2.

2. The accuracy analysis indicates that SoAP prediction accuracy decreases

gradually with increasing (dis)charge current.

The differences between model accuracies of models 2 – 6 are very low and, thus the

derived differences between model accuracies can be neglected. Unfortunately, the

prediction accuracy is not significantly improved with increasing number of RC-elements

or by employing ZARC-elements.


Fig. 59: Dependence of the determined ∆ on applied current rates

for ∆ = based on HPPC tests performed at 23 °C for the investigated LIBs

In addition to the above discussed results for HPPC tests, the results obtained from the

investigated driving cycles are examined. As shown in Fig. 27, the investigated driving

cycles include multiple charging regimes with constant current rates

(i.e., Icharge=0.5C). Since no SoAP prediction is performed during the aforementioned

charging phases and ∆ is equal to zero, respectively, the determined ∆ values are


very low in the respective current range (∆ ) where charging regimes and remaining

applied charging currents (e.g., recuperation phases)are included. Thus, from theoretical

point of view when the charging phases are excluded from the processing data, the

obtained ∆ values should be in the same scale as it is the case for ∆ . Moreover, it is

worth noting that ∆ values calculated in a current range of 2C - +4C a small amount of

charging current pulses of +2C.

Fig. 60 illustrates the determined ∆ values obtained based on the investigated driving

cycles for the investigated LIBs.

According to the results shown in Fig. 60, following conclusions may be drawn for

current dependence of ∆ :

1. With increasing (dis)charging current rates the SoAP prediction accuracy

decreases significantly.

2. In contrast to the results shown in Fig. 59, the prediction accuracy of model 1 is

not significantly lower than other models. But nevertheless, the SoAP prediction

accuracy of model 1 is still slightly lower than other models.

In total, it may be stated that the investigated models excluding model 1 perform almost

similar for the respective applied current range. In fact, the differences between model

inaccuracies for the investigated models are negligible and, thus no clear statement can

be provided regarding the selection of the most accurate ECM for SoAP prediction.


Fig. 60: Dependence of the determined ∆ on applied current rates

for ∆ = based on driving cycles for the investigated LIBs


5.2.2.2.2 Dependence of SoAP prediction accuracy (∆) on SoC and temperature

In contrast to previous subsection where model performances are compared with regard

to current dependence of SoAP prediction error; in this subsection SoC and temperature

dependence of determined ∆ for ∆=20 s is discussed. It is worth mentioning that in

order to provide better visibility of the obtained results, only discharge cases are

considered for ∆ calculation while charging cases are neglected. The results obtained

from charging cases are almost similar and indicate the same behavior tendency.

Fig. 61 illustrates the dependence of the determined ∆ values on SoC and measured

temperature. According to the results shown in Fig. 61, following conclusions may be

drawn for SoC dependence of calculated ∆ :

1. With decreasing SoC the SoAP prediction accuracy decreases slightly.

2. The prediction accuracy of model 1 (at least for most of the investigated SoC

ranges) indicates comparative results to other models.

3. However, for SoCs below approx. 40%, model 1 seems to perform very poor and be

very inaccurate in contrast to other ECMs. Unfortunately, the main reason for

the aforementioned significant difference for the determined ∆ values between

model 1 and other models is not clear.

According to the results shown in Fig. 61, following conclusions may be drawn for

temperature dependence of calculated ∆ :

1. With decreasing temperature the SoAP prediction accuracy decreases.

2. Model 1 indicates poorest performance and yields very inaccurate results with

regard to predicted SoAP. The inaccuracy of model 1 is approx. 3 times higher

than other ECMs.

3. The obtained results for models 2 - 4 indicate a very similar performance and

quite the same degree of accuracy.


Fig. 61: Performance comparison of the investigated ECMs for ∆ = based on the driving cycles for the investigated LIBs; a) Dependence of determined ∆ on State-of-Charge; b) Dependence

of determined ∆ on temperature

5.2.2.2.3 Impact of prediction time horizon on SoAP prediction accuracy (∆)

In the previous subsections, model performances with regard to the determined ∆

depending on SoC, temperature and applied current rate are analyzed. In the

aforementioned cases, determined ∆ is based on a specific prediction time horizon of ∆ = s. In general, the tendency of model performances should not differ significantly

for other prediction time horizons (∆). However, in the present subsection, the impact of


various prediction time horizon on SoAP prediction for a wide range of common

prediction time horizons often of interest in EVs is investigated. In particular, the

investigated prediction time horizons (∆) are as follows:

∆=0.5 s,

∆=2 s,

∆=5 s,

∆=10 s,

∆=20 s.

Fig. 62 illustrates the model performances for the investigated LIBs for various

prediction time horizons depending on applied current rate. The left column (a) shows

the results obtained from HPPC tests performed at 23 °C and the right column (b) refers

to the results obtained from driving cycles. The results illustrated in the aforementioned

figure are obtained using model 1 as an example. The investigated results obtained from

other ECMs indicate the tendency and do not show any significant deviation.

According to the results shown in Fig. 63 following conclusions may be drawn:

1. As expected, with increasing prediction time horizons, the SoAP prediction

accuracy decreases significantly.

2. Moreover, it seems that the impact of the defined prediction time horizon becomes

more pronounced for the examined driving cycles than HPPC tests.

3. In total, it can be concluded that difference between SoAP prediction accuracy for

a prediction time horizon below 5 s (i.e., ∆ 5 s) is not significant and the

prediction accuracy decreases significantly for ∆ 10 s. For selected discharge

currents; especially for HPPC tests, it may be stated that prediction inaccuracy is

doubled when ∆ is increased from 10 s up to 20 s.


Fig. 62: Performance comparison of model 1 for various prediction time horizons for the investigated LIBs; a) Results obtained from HPPC tests at 23 °C; b) Results obtained from driving cycles35

35 Dependence of determined ∆ on applied current rate and SoC for various prediction time horizons for Cell-J-aged can

be found in Appendix F. Moreover, dependence of SoAP prediction accuracy on SoC for different prediction time

horizons for the investigated LIBs is illustrated in Fig. F.2


5.3 On-board open-circuit voltage estimation

In the previous subsections, model performances with regard to voltage estimation

(determined using as a basis for analysis) and SoAP prediction accuracy

(determined using ∆ as a basis for analysis) over a wide range of SoCs, temperatures

and applied current rates are discussed. In addition to the above mentioned parameters/

battery states, the on-board OCV estimation accuracy of the investigated ECMs is

examined in the present subsection. In this regard, a single driving cycle performed on

Cell-G-new is selected and the OCV is determined as proposed in Eq. 20. It is worth

noting that a simple low-pass filter is applied to the determined OCV values for each

sampling time in order to filtering away possible fluctuation in the calculated OCVs.

Fig. 63 shows the results of measured and estimated OCVs using Cell-G-new as an

example. The load profile employed for verification reasons is performed at 23 °C.

Fig. 63: Measured and determined open-circuit voltages and relative estimation error for the investigated ECMs using a single driving cycle performed at 23 °C on Cell-G-new. Please note that the SoCs are related to battery’s nominal capacity

Table 17 gives the calculated RMSE values of the estimated OCVs over the whole SoC

range. According to the results, it can be concluded that model 4 is a preferred candidate


for OCV estimation under described condition while OCV estimation accuracy of model 1

is very low; in particular, the model accuracy is approx. 3 time lower than other models.

Table 17: Root-mean-square error of the determined open-circuit voltages for the investigated ECMs

Model number Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 RE C [mV] 74.5 22 23.3 22.1 21.9 22.3

Furthermore, it can be stated that based on the results, OCV estimation accuracy

decreases significantly with decreasing SoC. The estimation accuracy decreases by a

factor 2 or higher for SoCs below 20%.

5.4 Summary

In this chapter, the basic idea of the applied method for battery impedance parameters

estimation is presented and a sensitivity analysis is performed with regard to model

behavior under varying conditions. Furthermore, model performances with regard to

voltage estimation and SoAP prediction accuracy are compared and analyzed. The

accuracy analysis is performed using the load profiles presented in chapter 2 while the

main focus lies on investigating the dependence of the aforementioned parameter/state

on SoC, temperature and applied current rates.

Summary and outlook 170

6 Summary and outlook

This work contributes to the topic of dynamic battery modeling and on-board lithium-ion

battery state estimation for BMS. In order to meet specified power and/or energy

requirements, hundreds of LIBs are connected in series and/or parallel, thus building a

large lithium-ion battery pack. Thereafter, to ensuring an efficient operation strategy

and an optimal system design for a wide range of applications, an accurate modeling and

understanding the battery’s nonlinear behavior both on cell and system levels becomes necessary. Accurate electrical and thermal modeling of LIBs is a very challenging task

since LIBs are nonlinear devices; especially when the employed battery model is

responsible for reproducing the behavior of the battery at very low temperature and SoC

range. The main objective of this work is to investigate the impact of the employed ECMs

on estimation/ prediction accuracy of battery states and parameters such as

instantaneous voltage, SoAP for a predefined prediction time horizon and OCV etc. In

particular, accuracy of the estimated voltage and the predicted SoAP for different

prediction time horizons is examined over a wide SoC, temperature and applied current

range. The aforementioned target is achieved starting from studying internal and

external factors influencing the impedance characteristics of LIBs under varying

conditions, and proceeding through a comprehensive literature review on relating topics

right down to implementation of the ECMs and accuracy analysis of the examined ECMs

with regard voltage estimation and SoAP prediction. The measurements performed on

the examined LIBs at different aging states and the results achieved are summarized as

follows:

First, a short introduction to principle operation of LIBs and a brief overview of different

LIB formats and commonly used anode and cathode materials is given. In addition, the

methods for investigation and interpretation of impedance characteristics/parameters

and OCV behavior of LIBs are discussed comprehensively. In this regard, dependence of

impedance characteristics of LIBs on internal and external factors is investigated where

a nonlinear first order ECM is parameterized for different SoCs and temperatures

(-20 °C ⋯ +45 °C). The measurements are performed using EIS and HPPC tests. For this

purpose, the following parameters are examined extensively: , double-layer

capacitance, charge transfer resistance, time constant, total resistance and OCV. In

total, it can be stated that the aforementioned impedance parameters are influenced

significantly by actual aging state of the LIB. In contrast to temperature the impact

battery SoC on impedance parameters seems to be much lower. Moreover, current


dependence of direct current resistance and double-layer capacitance is examined

comprehensively. According to the results, it can be stated that the current dependence

of double-layer capacitance of NMC/LTO-based LIBs is negligible while current

dependence of battery’s direct resistance becomes more pronounced with decreasing

temperature and over the battery lifetime. Contrary to the aforementioned impedance

parameters, it is shown that the influence of temperature on OCV is more or less

pronounced depending on the used active materials. In particular, for NMC/LTO LIBs

the impact of temperature on OCV behavior seems to be very significant whereas for

NMC/C LIBs the OCV behavior is less affected. Furthermore, it can be stated that SoC-

OCV correlation changes remarkably when the LIB ages. Based on the obtained

information from the latter facts, a simple OCV model considering temperature

correction term and aging state of the battery is proposed. Moreover, the impact of

battery temperature and SoC on required relaxation time until all side reactions are

finished and the behavior of battery voltage response after current interruption is

investigated. As expected with decreasing temperature the required relaxation time

increases significantly and it can be assumed that a linear dependence between the both

parameters exists.

Second, an extensive review of available techniques for major battery state and/or

parameters estimation/prediction such as SoAP, SoC, SoH, actual capacity and

impedance parameters presented in the literature is provided. For this purpose, approx.

300 literature sources, patents etc. are analyzed and reviewed while a special emphasis

is put on elaborating their strengths and weaknesses. In addition, the requirements on

battery monitoring algorithms and importance of considering realistic load profiles for

verifying the applied monitoring algorithms for respective application are discussed. As

pointed out, the applied monitoring algorithms should be able to consider internal and

external factors affecting battery’s electrical/thermal behavior such as extreme

temperature change and battery aging over years etc. Thereafter, the necessity of using

adaptive techniques working robust over the battery lifetime and under varying

conditions is highlighted. In addition to the presented techniques for battery state

estimation, a review on different classes of battery models presented in the literature is

given. Generally, the battery models can be subdivided into three various classes,

namely white box, grey box and black box battery models. A brief overview of relevant

literature sources is provided and the strengths and the weaknesses of individual

battery model class are addressed. The result is that grey box battery models are wide

spread in the field of on-board battery state estimation due mainly to their simple


nature, low computational effort and high accuracy over a wide frequency/operating

range. At the same time, white box battery models show very accurate results with

regard to precise representation of electrochemical processes occurring in the LIB during

operation and battery’s nonlinear behavior, respectively. The latter fact is a result of detailed consideration of electrochemical processes by of complex mathematical

equations. However, up to now, the aforementioned battery models have received little

attention in the field of monitoring algorithm mainly because of their complex structure

and required high computing power which is rarely given on embedded systems

implemented in BMS.

Third, a wide range of reduced order impedance-based ECMs is investigated

comprehensively with regard to electrical and impedance behavior of individual ECM. A

selection of the ECMs is performed based on the forecast/possibility of implementation

on target low cost embedded systems in BMS. Since computational power capability of

BMS in EVs is limited, the employed ECMs must be significantly simplified. In total,

seven different impedance-based ECMs from a simple ECM consisting of connected in

series with VOCV up to more complex ECMs consisting of so-called ZARC elements are

implemented and investigated. The current dependence of the is considered in the

employed ECMs by means of BVE. Moreover, physical meaning of the employed

impedance parameters and relating electrochemical processes occurring during LIB

operation are discussed carefully. From theoretical point of view, it can be stated that

with increasing number of ECM-elements (i.e., number of RC-elements or ZARC-

elements), voltage estimation or impedance determination accuracy “should” increase;

however, at the expense of higher computational effort. Based on the results achieved

from impedance analysis of the investigated ECMs, the ECM consisting of an and

three ZARC elements connected in series with a VOCV indicates the most accurate result

for a selected frequency range with an RMSE error of approx. 20 mΩ.

Fourth, the investigated ECMs are used as a basis for model-based battery

state/parameters estimation, in particular SoAP prediction and OCV estimation etc.

Based on the EVPA method, impedance parameters of the employed ECMs are

estimated on-board for various load profiles with different characteristics. Thereupon, in

the first step, the model accuracies are compared based on determined RMSE of battery

voltage for the investigated LIBs. Subsequently, the impact of the investigated ECMs on

SoAP prediction accuracy is analyzed based on the calculated relative prediction error

using the deviation between measured and predicted power. For both latter parameters,

the accuracy analysis is performed concerning the dependence of the determined voltage


error and SoAP on temperature, SoC and applied current range. In total, it can be stated

that the calculated model error differences between model 2 - 6 are negligible and, thus

no clear statement can be provided regarding the most sufficient and accurate ECM.

However, as expected, model 1 consisting of an connected in series with a voltage

source indicates the poorest performance in contrast to other examined ECMs. It is

pointed out that it is always recommendable to functions developer to studying the

requirements of target application carefully before development process is started. In

this regard, temperature profile/distribution, SoC operating area of target system and

current rates which the battery will be exposed to under real conditions in EVs are very

important factors for precise battery modeling and selection of most appropriate battery

model class. It is also shown that when for particular target application the

requirements on voltage estimation or SoAP prediction accuracy are not high, the

application of model 1 may be sufficient.

The methods and the obtained results discussed in this work give a general overview

over necessity of considering different approaches/requirements during product

development phase. Concerning the impact of ECM on battery states and parameters

estimation accuracy for various operating conditions, following battery

states/parameters may be of interest for further investigation in future works:

Remaining charging time prediction,

SoAE determination,

Actual capacity and SoH estimation etc.

Moreover, in addition to the ECMs investigated in this work, examining other advanced

ECMs consisting of e.g., Warburg impedance in various combinations such as a Randles

circuit discussed in chapter 3 or more advanced ECM where a Warburg impedance is

connected in series with ZARC elements etc. may be helpful.

An especial focus could lie on further development of white box battery models or in

particular SPMs with an especial emphasis on computational efficiency for use in BMS.

Thereafter, occurring electrochemical processes should be reproduced computationally

efficient and very precise so far as possible. Furthermore, the application of white box

battery models can be investigated with regard to SoAP prediction accuracy.

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List of abbreviations

ACM adaptive characteristic maps

AEKF Adaptive extended Kalman filter

ANFIS Adaptive neuro-fuzzy-inference system

AUTOSAR Automotive open system architecture

BEV Battery electric vehicle

BMS Battery management system

BoL Begin of Life

BSE Battery state estimation

BVE Butler-Volmer equation

CAN Controller area network

CC-CV Constant current-constant voltage


CDKF Central difference Kalman filter

CPE Constant phase element

CV Cyclic volumetric

DAE Differential algebraic equation

DEKF Dual extended Kalman filter

DTFT Discrete-time Fourier transformation

DD Direct differential

DMC Dynamic matrix control

DoD Depth-of-Discharge

DST Dynamic stress test

DVA Differential voltage analysis

E Specific energy

ECU Electronic control unit

ECM Equivalent circuit model

EES Energy storage system

EoL End of life

EKF Extended Kalman filter

EMF Electro motive force

EMS Energy management system

EU European Union

EUCAR European Council for Automotive Research and Development

EV Electric vehicle

EU European Union

EVPA Enhanced varied parameters approach

FDM Finite difference method

FEM Finite element method

FIS Fuzzy inference system


FUDRLS Fast upper-diagonal recursive least square

FUDS Federal urban driving schedules

FVM Finite volume method

HR Hazard risk

HL Hazard likelihood

HS Hazard severity

HEV Hybrid electric vehicle

HIL Hardware-in-the-Loop

HMRMA Hazard Modes and Risk Mitigation Analysis

HL Hazard level

HV High voltage

ICA Incremental capacity analysis

isoSPI Isolated communications interface

KF Kalman filter

LIB Lithium-ion battery

LFP Positive electrode active materials with a common formula LiFePO4

LMO Positive electrode active materials with a common formula LiMn2O4

LTV linear time-varying

LS Least square

LTO Negative electrode active materials with a common formula Li4Ti5O12

LTV linear time-varying

LV Low voltage

MA Moving Average

MMPI multi-step model predictive iterative

MIL Model-in-the-Loop

NEDC New European Driving Cycle

NMC Positive electrode active materials with a common formula of Lix(NixMnyCoz)O2


NCA Positive electrode active materials with a common formula of Lix(NixCoyAlz)O2

OCV Open-circuit voltage

PDE Partial differential equation

PDF Probability density function

PHEV Plug-in hybrid electric vehicle

P2D pseudo-two-dimensional

RBF Radial basis function

REKF Robust extended Kalman filter

RELS Recursive extended least square

RMSE Root-mean square error

RLS Recursive least square

RLSF Recursive least square with optimal forgetting factor

RUL Remaining useful lifetime

RVM Relevance vector machine

SampEn Sample entropy

SIL Software-in-the-loop

SoA Safe operating area

SoAE State-of-Available Energy

SoH State-of-Health

SoC State-of-Charge

SoF State-of-Function

SoAP State-of-Available Power

SoS State-of-Safety

SPKF Sigma point Kalman filter

SPM Single particle model

SVM Support vector machine

TLS Total least square

List of symbols 199

UDDS Urban dynamometer driving schedule

UKF Unscented Kalman filter

V2G Vehicle to grid

WLS Weighted least squares

WRLS Weighted recursive least squares

WTLS Weighted total least square

List of symbols

Symbol Unit Meaning

Ai F∙sξ-1 Generalized capacitance – Parameter of the CPE-element where i ∈1 … 3 refers to the number of ZARC element

m2/m3 Table 4: Specific surface area of electrode

A m2 Surface area of electrode

Aeff m2 Chapter 4: Effective internal electrode surface

Ah Actual battery capacity

Ah Nominal battery capacity

Ah Initial battery capacity

Cr Ah Releasable battery capacity

cs,k mol/m3 Table 4: Solid-state concentration

ce,k mol/m3 Table 4: Electrolyte concentration

F Double-layer capacitance – Parameter of the ECM

D m2/s Diffusion coefficient

m2/s Table 4: Effective ionic diffusion coefficient

E Wh Chapter 2.1: Specific energy

Wh Chapter 3.1: Actual energy content of the battery

kJ ∙ mol− Activation energy

EMF V Electromotive force

List of symbols 200

F As∙mol-1 Table 4: Faraday constant

f Hz Frequency

Hz Maximum frequency when the imaginary part of the impedance

spectrum reaches its local minimum

- State of safety of the battery

- State of abuse of the battery

Im Ω Imaginary part of impedance

HS - Hazard severity

HL - Hazard likelihood

HR - Hazard risk

A Current

A Table 15: Current

I A/m2 Chapter 2 & 4: current density

IC A Chapter 2 & 5: Capacitive current flowing through double-layer

capacitance

IC,max A Chapter 5: Predefined value of the current on the capacitance of

the employed ECM

Icharge A Chapter 3: Applied charge current

Idischarge A Chapter 3: Applied discharge current

Imax A Chapter 3: Maximum charge current

Imin A Chapter 3: Minimum discharge current

, , A Chapter 5: Maximum charge current whereby the upper voltage

limit is considered as a constrain

, , A Chapter 5: Maximum discharge current whereby the lower voltage

limit is considered as a constraint

, , C A Chapter 5: Maximum charge current whereby the upper SoC limit

is considered as a constraint

, , C A Chapter 5: Maximum discharge current whereby the lower SoC

limit is considered as a constraint

List of symbols 201

A/m2 Table 4: Exchange current density

A Chapter 2 & 4: Exchange current density

Jk mol/s.m2 Table 4: Wall flux of Li+ on electrodes particle

j - Complex variable: j=√−

m2.5/s.mol0.5 Table 4: Reaction rate constant of electrode

V ∙ K− Temperature correction term of

the OCV model

V/% Coefficient of OCV change over the battery lifetime

kI A-1 A parameter of the employed ECMs

Lk m Table 4: Thickness of the region

ldiff m Pore length

n - Number of electrons involved in the reaction (n = 1 for LIBs)

ns - Chapter 3: Number of cells connected in series

np - Chapter 3: Number of cells connected in parallel , - Chapter 3: Number of cells connected in series , - Chapter 3: Number of modules connected in parallel

OCV V Open-circuit voltage

Pn - Chapter 5: Parameter set of the employed ECM where n refers to

the number of parameter sets

W Chapter 3: Discharge power

W Chapter 3: Charge power

Pmeas W Chapter 3: Measured battery power

Ppredict W Chapter 3: Predicted battery power Δ As Amount of charge throughput

R J ∙ mol ∙ K− Universal gas constant

Ω Actual battery impedance

C Ω Contact resistance

List of symbols 202

Ω Ohmic resistance

Rcell,discharge Ω Chapter 3: Discharge resistance of the battery

Rcell,charge Ω Chapter 3: Charge resistance of the battery

Ω Charge transfer resistance

, Ω Chapter 4 & 5: Parameter of the ECM referring to diffusion

resistance

Ω Chapter 2: Resistance corresponding to begin of transport

limitations in solid and liquid phases

Ω Direct current resistance

R0.001Hz Ω Chapter 2: Total battery resistance Re Ω Real part of impedance Z

S J ∙ K− Entropy

SoCV % Voltage-based Stage-of-Charge

T or Kelvin Temperature + - transference number of Li+ in electrolyte s Initial time

∆t s Chapter 2: Interim relaxation time

V Battery voltage

V Simulated battery voltage

V Measured battery voltage ∆ V Chapter 5: Predicted discharge voltage of the battery ∆ V Chapter 5: Predicted charge voltage of the battery

Vcell,min V Chapter 3: Minimum cell voltage

Vcell,max V Chapter 3: Maximum cell voltage

, V Table 15: Voltage on the first (double-layer) capacitance of the

employed ECM

, V Table 15: Voltage on capacitances of the employed ECMs where i ∈1 … 2

List of symbols 203

V Chapter 5: Maximum voltage of the battery

V Chapter 5: Minimum voltage of the battery

V Chapter 3: Operating voltage of the battery

V Chapter 5: Limiting battery voltage

C V Open-circuit voltage of the battery V C , V Chapter 3: Open-circuit voltage after discharging V C , V Chapter 3: Open-circuit voltage after charging

V Equilibrium potential of the positive electrode

V Equilibrium potential of the negative electrode

V ∆ V Chapter 3: Predicted voltage of the battery for a predefined time

horizon

VZA C, V Table 15: Voltage on the ZARC-element of the employed ECM

where i ∈1 … 3 refers to the number of ZARC elements

Greek , - Symmetry factor of anodic reactions used for BVE - Symmetry factor of cathodic reactions used for BVE S/m Effective electronic conductivity of electrodes solid phase Ω/√s Warburg coefficient - Porosity of the region

ε V Difference between measured and simulated voltage - Coloumb efficiency V Solid-phase potential of electrode V Electrolyte potential s Time constant oC V ∙ A− ∙ s− The change of open-circuit voltage per 1 As current flow

Δ V Ohmic overvoltage or simply speaking voltage drop due to ohmic

resistance

List of symbols 204

Δ E , V SEI-related overvoltage on the anode Δ E , V SEI-related overvoltage on the cathode Δ , V Charge transfer overvoltage on the anode Δ , V Charge transfer overvoltage on the cathode

Δ V Concentration-related overvoltages including diffusion, migration

and convection processes

Δ V Table 15: Determined overvoltage of the RC-element representing

charge transfer processes

Δ , V Table 15: Determined overvoltage of the second RC-element

representing diffusion processes

Δ , V Table 15: Determined overvoltage of the third RC-element

representing diffusion processes

∆ - Chapter 5: A coefficient used for determination of two variants of

the parameter

Δ ZA C, V Table 15: Determined overvoltage of respective ZARC-element of

the employed ECM ∆ s Predefined prediction time horizon

∆tC s Chapter 5: Time period where the current through capacitance of

the employed ECM is lower than a predefined value ∆ W Chapter 3: Difference between measured and predicted power S/m Effective ionic conductivity of electrolyte

1/s Angular frequency

ξi - Chapter 4: Depression factor used for ZARC-elements employed in

the ECMs where i ∈1 … 3 refers to the number of ZARC element

Ω Complex impedance

Ω Warburg impedance

Appendices 205

Appendices

Appendix A: Measured OCVs of NMC/LTO and NMC/C-based Lithium-ion batteries at different aging states for various temperatures

Appendix A presents the open-circuit voltages of NMC/LTO and NMC/C-based LIBs at

different aging states for different temperatures. A detailed description of performed

measurement strategy is given in chapter 2.

Table A.1: Measured OCVs of Cell-E-new and Cell-E-aged at different temperatures

Cell name State-of-

Charge [%]

Open-circuit voltage after

discharging [V]


charging [V]

Temperature [°C]

Cell-E-new

-20 0 23 45 -20 0 23 45

90% 2.601 2.604 2.612 2.606 NaN NaN 2.623 2.615

80% 2.584 2.587 2.592 2.588 NaN 2.599 5.599 2.599

70% 2.571 2.578 2.57 2.575 NaN 2.587 2.587 2.584

60% 2.525 2.559 2.538 2.55 2.56 2.576 2.552 2.569

50% 2.499 2.511 2.489 2.504 2.522 2.545 2.506 2.533

40% 2.464 2.47 2.46 2.47 2.475 2.491 2.471 2.487

30% 2.448 2.45 2.447 2.452 2.454 2.46 2.454 2.461

20% 2.437 2.439 2.436 2.438 2.441 2.445 2.441 2.445

10% 2.421 2.429 2.419 2.418 2.424 2.433 2.425 2.427

Cell-E-aged

90% 2.588 2.598 2.599 2.604 NaN 2.599 2.592 2.618

80% 2.575 2.583 2.579 2.586 NaN 2.587 2.579 2.595

70% 2.568 2.569 2.546 2.571 2.579 2.576 2.554 2.582

60% 2.522 2.526 2.494 2.538 2.565 2.544 2.507 2.562

50% 2.454 2.476 2.46 2.49 2.498 2.493 2.469 2.521

40% 2.444 2.453 2.446 2.461 2.454 2.461 2.449 2.477

30% 2.43 2.438 2.425 2.449 2.438 2.448 2.431 2.465

20% 2.412 2.421 2.399 2.43 2.414 2.427 2.403 2.439

10% NaN 2.399 NaN 2.412 NaN 2.403 NaN 2.418

Appendices 206

Table A.2: Measured OCVs of Cell-F-new and Cell-F-aged at different temperatures

Cell name State-of-Charge

[%]


discharging [V]


charging [V]

Temperature [°C]

Cell-F-New

0 10 23 45 0 10 23 45

90% 4.058 4.057 4.059 4.062 NaN 4.071 4.063 4.066

80% 3.941 3.947 3.948 3.954 3.962 3.962 3.957 3.96

70% 3.822 3.825 3.825 3.834 3.843 3.842 3.83 3.845

60% 3.734 3.736 3.73 3.747 3.753 3.753 3.75 3.758

50% 3.673 3.676 3.67 3.685 3.691 3.69 3.688 3.695

40% 3.63 3.634 3.63 3.642 3.65 3.65 3.649 3.655

30% 3.586 3.589 3.584 3.583 3.611 3.614 3.614 3.613

20% 3.528 3.533 3.53 3.53 3.55 3.556 3.556 3.554

10% 3.465 3.47 3.47 3.472 3.471 3.482 3.486 3.484

Cell-F-aged

90% 4.072 4 4.076 4.077 NaN 4.007 4.077 4.076

80% 3.955 3.887 3.965 3.966 3.959 3.898 3.967 3.966

70% 3.854 3.801 3.864 3.867 3.869 3.812 3.87 3.867

60% 3.771 3.699 3.745 3.785 3.787 3.739 3.753 3.79

50% 3.704 3.65 3.687 3.718 3.72 3.688 3.697 3.724

40% 3.656 3.596 3.633 3.662 3.672 3.646 3.655 3.675

30% 3.609 3.54 3.582 3.602 3.628 3.592 3.603 3.626

20% 3.551 3.509 3.495 3.55 3.567 3.523 3.505 3.565

10% 3.492 3.166 3.287 3.496 3.498 3.423 3.3 3.502

Appendices 207

Appendix B: Step-wise illustration of on-board capacity estimation of lithium-ion batteries

Appendix B shows schematically a widespread technique for actual capacity estimation

of LIBs in EVs. A detailed description of boundary conditions and available methods for

on-board battery capacity estimation is given in chapter 3.

Fig. B.1: Step-wise presentation of on-board capacity estimation based on the OCV-SoC correlation

START

Initialization of the variables(Ahinitial = 0)

Determination of the OCV(t1) and

SoC(t1)

Ah Counter

ΔSoC = SoC(t2)-SoC(t1)

SoC

QC

Step 2

Step 3

Step 4

Step 5

Step 7

2

1)(

t

tdttiQ

Border conditions fullfilled?(Δυ< Δυmax) && (ΔSoC > ΔSoCmin)

No

Step 6

Determination of the OCV(t2) and

SoC(t2)

Yes

END

Step 1

Appendices 208

Appendix C: Charge and discharge curves behavior over the battery lifetime

Appendix C describes the impact of current and aging state on (dis)charge curves and

actual capacity of NMC/LTO-based LIBs comprehensively discussed in chapter 2.

Fig. C.1: Discharge curves of Cell-C-new over the battery lifetime discharged from a fully charged state with 1C (≅15 A) current rate measured at 23 °C

Fig. C.2: Discharge curves of Cell-C-aged at End of Life discharged with various current rates from a fully charged state measured at 23 °C

Appendices 209

Fig. C.3: Charge curves of Cell-A-new and Cell-A-aged with various current rates measured at 23 °C

Appendix D: Static charge and discharge power capability of the battery

In appendix D, power capability of Cell-G-new determined based on Eqs. (23)-(24)

proposed in chapter 3 is given.

Fig. D.1: Power capability of Cell-G-new at 23 °C determined as proposed in Eqs. (23)-(24); the voltage boundaries are predefined as follows: Vcell,min=2.7 V and Vcell,max=4.2 V.

Appendices 210

Appendix E: Comparison of model performances for Cell-D-new and Cell-A-aged using the examined driving cycles

Appendix E addresses the results of the determined and SoAP prediction accuracy

for Cell-D-new and Cell-A-aged based on the examined driving cycles.

Fig. E.1: Model performances based on the examined driving cycles for Cell-D-new and Cell-A-aged

Appendices 211

Appendix F: Impact of prediction time horizon on SoAP prediction accuracy for different applied current rates and State-of-Charges using the examined driving cycles

The results obtained for Cell-J-aged based on the examined driving cycles concerning the

impact of prediction time horizon on SoAP prediction accuracy are shown in Fig. F.1. In

this regard, model 1 is employed for accuracy analysis of predicted SoAP depending on

applied current rate and SoC for various prediction time horizons. Moreover, the results

obtained for dependence of SoAP prediction accuracy of the investigated ECMs on SoC is

illustrated in Fig. F.2.

Fig. F.1: Performance analysis for various prediction time horizons for Cell-J-aged using model 1 as a reference

Appendices 212

Fig. F.2: Performance comparison of model 1 for the investigated LIBs for various prediction time horizons based on the investigated driving cycles

Appendices 213

List of publications

Conference and magazine contributions

A. Farmann, D. U. Sauer (2018): High-voltage Lithium-ion Batteries –

Methods for On-board State Estimation – ATZ elektronik worldwide, Vol. 13, pp 58- 63

A. Farmann, D. U. Sauer (2018): HV-Lithium-Ionen-Batterien — Monitoring-

Algorithmen für die Onboard-Zustandserkennung – ATZelektronik, Vol. 13, pp 58- 63

A. Farmann, C. Vogel, D. U. Sauer (2015): On-board impedance parameters estimation

of SLI batteries in micro-hybrid vehicles: Part I. Requirements and Modeling, Kraftwerk

Batterie

A. Farmann, C. Vogel, D. U. Sauer (2015): On-board impedance parameters estimation

of SLI batteries in micro-hybrid vehicles: Part II. Parameter Identification and cranking

capability, Kraftwerk Batterie

A. Farmann, A. Gitis, H. S. Luong, D. U. Sauer (2014): Investigation of battery

monitoring algorithms for on-line power prediction in stationary applications, Kraftwerk

Batterie

Peer-Reviewed journal contributions

A. Farmann and D. U. Sauer: Comparative study of reduced order equivalent circuit

models for on-board state-of-available-power prediction of lithium-ion batteries in

electric vehicles, Journal of Applied Energy, 2018, vol. 225, pp. 1102 - 1122.

A. Farmann and D. U. Sauer: A study on the dependency of the open-circuit voltage on

temperature and actual aging state of lithium-ion batteries, Journal of Power Sources,

2017, vol. 347, no. 0, pp. 1 - 13.

A. Farmann and D. U. Sauer: A comprehensive review of on-board State-of-Available-

Power prediction techniques for lithium-ion batteries in electric vehicles, Journal of

Power Sources, 2016, vol. 329, no. 0, pp. 123 - 137.

A. Farmann, W. Waag and D. U. Sauer: Application-specific electrical characterization of

high power batteries with lithium titanate anodes for electric vehicles, Energy, 2016, vol.

112C, no. 0, pp. 294 - 306.

Appendices 214

A. Farmann, W. Waag and D. U. Sauer: Adaptive approach for on-board impedance

parameters and voltage estimation of lithium-ion batteries in electric vehicles, Journal

of Power Sources, vol. 299, 2015 , no. 0, pp. 176 - 188.

A. Farmann, W. Waag, A. Marongiu and D. U. Sauer: Critical review of on-board

capacity estimation techniques for lithium-Ion batteries in electric and hybrid electric

vehicles, Journal of Power Sources, 2015 , vol. 281, no. 0, pp. 114 - 130.


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ABISEA Band 74 Baumhöfer, T. Statistische Betrachtung experimenteller Alterungs-untersuchungen an Lithium-Ionen Batterien 1. Auflage 2015, 174 Seiten ISBN 978-3-8440-3423-3 ABISEA Band 75 Andre, D. Systematic Characterization of Ageing Factors for High- Energy Lithium-Ion Cells and Approaches for Lifetime Modelling Regarding an Optimized Operating Strategy in Automotive Applications 1. Auflage 2015, 210 Seiten ISBN 978-3-8440-3587-2 ABISEA Band 76 Merei, G. Optimization of off-grid hybrid PV-wind-diesel power supplies with multi-technology battery systems taking into account battery aging 1. Auflage 2015, 194 Seiten ISBN 978-3-8440-4148-4 ABISEA Band 77 Schulte, D. Modellierung und experi-mentelle Validierung der Alterung von Blei-Säure Batterien durch inhomogene Stromverteilung und Säureschichtung 1. Auflage 2016, 168 Seiten ISBN 978-3-8440-4216-0 ABISEA Band 78 Schenk, M. Simulative Untersuchung der Wicklungsverluste in Geschalteten Reluktanz-maschinen 1. Auflage 2016, 142 Seiten ISBN 978-3-8440-4282-5

ABISEA Band 79 Wang, Y. Development of Dynamic Models with Spatial Resolution for Electro- chemical Energy Converters as Basis for Control and Management Strategies 1. Auflage 2016, 198 Seiten ISBN 978-3-8440-4303-7 ABISEA Band 80 Ecker, M. Lithium Plating in Lithium-Ion Batteries: An Experimental and Simulation Approach 1. Auflage 2016, 170 Seiten ISBN 978-3-8440-4525-3 ABISEA Band 81 Zhou, W. Modellbasierte Auslegungs-methode von Tempe-rierungssystemen für Hochvolt-Batterien in Personenkraftfahrzeugen 1. Auflage 2016, 192 Seiten ISBN 978-3-8440-4589-5 ABISEA Band 82 Lunz, B. Deutschlands Stromversor-gung im Jahr 2050 Ein szenariobasiertes Verfahren zur vergleich-enden Bewertung von Systemvarianten und Flexibilitätsoptionen 1. Auflage 2016, 196 Seiten ISBN 978-3-8440-4627-4 ABISEA Band 83 Hofmann, A. Direct Instantaneous Force Control Key to Low-Noise Switched Reluctance Traction Drives 1. Auflage 2016, 244 Seiten ISBN 978-3-8440-4715-8

ABISEA Band 84 Budde-Meiwes, H. Dynamic Charge Acceptance of Lead-Acid Batteries for Micro-Hybrid Automotive Applications 1. Auflage 2016, 168 Seiten ISBN 978-3-8440-4733-2 ABISEA Band 85 EngeI, S. P. Thyristor-Based High-Power On-Load Tap Changers Control under Harsh Load Conditions 1. Auflage 2016, 170 Seiten ISBN 978-3-8440-4986-2 ABISEA Band 86 Van Hoek, H. Design and Operation Considerations of Three-Phase Dual Active Bridge Converters for Low-Power Applications with Wide Voltage Ranges 1. Auflage 2017, 242 Seiten ISBN 978-3-8440-5011-0 ABISEA Band 87 Diekhans, T. Wireless Charging of Electric Vehicles - a Pareto-Based Comparison of Power Electronic Topologies 1. Auflage 2017, 156 Seiten ISBN 978-3-8440-5048-6 ABISEA Band 88 Lehner, S. Reliability Assessment of Lithium-Ion Battery Systems with Special Emphasis on Cell Performance Distribution 1. Auflage 2017, 202 Seiten ISBN 978-3-8440-5090-5


ABISEA Band 89 Käbitz, S. Untersuchung der Alterung von Lithium-Ionen-Batterien mittels Elektroanalytik und elektrochemischer Impedanzspektroskopie 1. Auflage 2016, 258 Seiten DOI: 10.18154/RWTH-2016-12094 ABISEA Band 90 Witzenhausen, H. Elektrische Batteriespeichermodelle: Modellbildung, Parameteridentifikation und Modellreduktion 1. Auflage 2017, 286 Seiten DOI: 10.18154/RWTH-2017-03437 ABISEA Band 91 Münnix, J. Einfluss von Stromstärke und Zyklentiefe auf graphitische Anoden 1. Auflage 2017, 178 Seiten DOI: 10.18154/RWTH-2017- 01915 ABISEA Band 92 Pilatowicz, G. Failure Detection and Battery Management Systems of Lead-Acid Batteries for Micro- Hybrid Vehicles 1. Auflage 2017, 212 Seiten DOI: 10.18154/RWTH-2017-09156 ABISEA Band 93 Drillkens, J. Aging in Electrochemical Double Layer Capacitors: An Experimental and Modeling Approach 1. Aufl. 2017, 179 Seiten DOI: 10.18154/RWTH-2018-223434

ABISEA Band 94 Magnor, D. Globale Optimierung netz-gekoppelter PV-Batterie-systeme unter besonderer Berücksichtigung der Batteriealterung 1. Auflage 2017, 210 Seiten DOI: 10.18154/RWTH-2017-06592 ABISEA Band 95 Iliksu, M. Elucidation and Comparison of the Effects of Lithium Salts on Discharge Chemistry of Nonaqueous Li-O2 Batteries 1. Aufl. 2018, 160 Seiten DOI: 10.18154/RWTH-2018-223782 ABISEA Band 96 Schmalstieg, J. Physikalisch-elektrochemische Simulation von Lithium- Ionen-Batterien: Implementierung, Parametrierung und Anwendung 1. Aufl. 2017, 176 Seiten DOI: 10.18154/RWTH-2017- 04693 ABISEA Band 97 Soltau, N. High-Power Medium-Voltage DC-DC Converters: Design, Control and Demonstration 1. Aufl. 2017, 176 Seiten DOI: 10.18154/RWTH-2017-04084 ABISEA Band 98 Stieneker, M. Analysis of Medium-Voltage Direct-Current Collector Grids in Offshore Wind Parks 1. Aufl. 2017, 144 Seiten DOI: 10.18154/RWTH-2017-04667

ABISEA Band 99 Masomtob, M. A New Conceptual Design of Battery Cell with an Internal Cooling Channel 1. Aufl. 2017, 167 Seiten DOI: 10.18154/RWTH-2018-223281 ABISEA Band 100 Marongiu, A. Performance and Aging Diagnostic on Lithium Iron Phosphate Batteries for Electric Vehicles and Vehicle-to-Grid Strategies 1. Aufl. 2017, 222 Seiten DOI: 10.18154/RWTH-2017-09944 ABISEA Band 101 Gitis, A. Flaw detection in the coating process of lithium-ion battery electrodes with acoustic guided waves 1. Aufl. 2017, 132 Seiten DOI: 10.18154/RWTH-2017-099519 ABISEA Band 102 Neeb, C. Packaging Technologies for Power Electronics in Automotive Applications 1. Aufl. 2017, 132 Seiten DOI: 10.18154/RWTH-2018-224569 ABISEA Band 103 Adler, F. S. A Digital Hardware Platform for Distributed Real-Time Simulation of Power Electronic Systems 1. Aufl. 2017, 156 Seiten DOI: 10.18154/RWTH-2017-10761


ABISEA Band 104 Becker, J. Flexible Dimensionierung und Optimierung hybrider Lithium-Ionenbatteriespeichersysteme mit verschiedenen Auslegungszielen 1. Aufl., 2017, 157 Seiten DOI: 10.18154/RWTH-2017-09278 ABISEA Band 105 Warnecke, A. Degradation Mechanisms in NMC Based Lithium-Ion Batteries 1. Aufl. 2017, 158 Seiten DOI: 10.18154/RWTH-2017-09646 ABISEA Band 106 Taraborrelli, S. Bidirectional Dual Active Bridge Converter using a Tap Changer for Extended Voltage Ranges 1. Aufl. 2017 ABISEA Band 107 Sarriegi, G. SiC and GaN Semiconductors – The Future Enablers of Compact and Efficient Converters for Electromobility 1. Aufl. 2017 ABISEA Band 108 Senol, M. Drivetrain Integrated Dc-Dc Converters utilizing Zero Sequence Currents 1. Aufl. 2017 ABISEA Band 109 Koijma, T. Efficiency Optimized Control of Switched Reluctance Machines 1. Aufl. 2017

ABISEA Band 110 Lewerenz, M. Dissection and Quantitative Description of Aging of Lithium-Ion Batteries Using Non-Destructive Methods Validated by Post-Mortem-Analyses 1. Aufl. 2018 ABISEA Band 111 Büngeler, J. Optimierung der Verfüg-barkeit und der Lebens-dauer von Traktionsbatterien für den Einsatz in Flurförder-fahrzeugen 1. Aufl. 2018 ABISEA Band 112 Wegmann, R. Betriebsstrategien und Potentialbewertung hybrider Batteriespeichersysteme in Elektrofahrzeugen 1. Auflage 2018 ABISEA Band 113 Nordmann, H. Batteriemanagementsysteme unter besonderer Berück-sichtigung von Fehlererken-nung und Peripherieanalyse 1. Aufl. 2018 ABISEA Band 114 Engelmann, G. Reducing Device Stress and Switching Losses Using Active Gate Drivers and Improved Switching Cell Design 1. Aufl. 2018 ABISEA Band 115 Klein-Heßling, A. Active DC-Power Filters for Switched Reluctance Drives during Single-Pulse Operation 1. Aufl. 2018

ABISEA Band 116 Burkhart, Bernhard Switched Reluctance Generator for Range Extender Applications - Design, Control and Evaluation 1. Aufl. 2018 ABISEA Band 117 Biskoping, Matthias Discrete Modeling and Control of a versatile Power Electronic Test Bench with Special Focus on Central Photovoltaic Inverter Testing 1. Aufl. 2018 ABISEA Band 118 Schubert, Michael High-Precision Torque Control of Inverter-Fed Induction Machines with Instantaneous Phase Voltage Sensing 1. Aufl. 2018 ABISEA Band 119 Van der Broeck, Christoph Methodology for Thermal Modeling, Monitoring and Control of Power Electronic Modules 1. Aufl. 2019 ABISEA Band 120 Hust, Friedrich Emanuel Physico-chemically motivated parameterization and modelling of real-time capable lithium-ion battery models – a case study on the Tesla Model S battery 1. Aufl. 2019 ABISEA Band 121 Ralev, Iliya Accurate Torque Control of Position Sensorless Switched Reluctance Drives 1. Aufl. 2019


ABISEA Band 122 Sarah Paul Ayeng’o Optimization of number of PV cells connected in series for a direct-coupled PV system with lead-acid and lithium-ion batteries 1. Aufl. 2019 ABISEA Band 123 Stefan Andreas Koschik Permanenterregte Synchron-maschinen mit verteilter Einzelzahnsteuerung - Regelkonzepte und Betriebs-strategien für hochintegrierte Antriebssysteme 1. Aufl. 2019

ISSN 1437-675X

In this study, LIBs at diferent aging states using various active materials are investigated whereby the primary focus lies on investigating the electrical behavior of LIBs using LTO anodes. In addition, other LIB technologies such as NMC/C and LFP/C are examined. Characterization tests are performed over a wide temperature range by employing electrochemical impedance spectroscopy and current pulse tests. Furthermore, the behavior of battery impedance parameters and open-circuit voltage over the battery lifetime for various temperatures and SoCs is investigated. Model based approaches using reduced-order equivalent circuit models (ECM) for battery state estimation have received increasing attention in recent publications mainly due to their simple nature and the possibility for implementation on low-cost embedded systems. The aforementioned techniques are often reliable and can track the changes of impedance characteristics over the battery lifetime. In this study, a comparative study of a wide range of impedance-based ECMs for on-board SoAP prediction is carried out. In total, seven dynamic ECMs including ohmic resistance, RC-elements, ZARC elements connected in series with a voltage source are implemen-

ted. The investigated ECMs are veriied under varying conditions using real vehicle data obtained in an EV prototype and current pulse tests. Furthermore, the current dependence of the charge transfer resistance is considered. The dependence of voltage estimation and SoAP prediction accuracy for diferent prediction time horizons on SoC, temperature and applied current rate is examined.

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