ELECTROCHEMICAL-BASED ITHIUM Tham_Tran_Thesis.pdf · temperature =10ºC. [18] Reprinted from "Experimental investigation of the lithium-ion battery impedance characteristic at various

ON THE DEVELOPMENT OF

ELECTROCHEMICAL-BASED LITHIUM-

ION BATTERY MODELS FOR BATTERY

MANAGEMENT SYSTEMS

Ngoc Tham Tran

BEng, MEng

Submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Electrical Engineering and Computer Science

Faculty of Science and Engineering

Queensland University of Technology

2019

On The Development of Electrochemical-Based Lithium-Ion Battery Models For Battery Management

Systems i

Keywords

Battery management systems;

Grid-connected battery storage systems;

Lithium ion batteries;

Battery modelling;

Reduced order models electrochemical models;

Single particle models;

Extended single particle model;

Pseudo-two-dimension model;

Padé approximate model;

Thermal models;

Degradation models;

ii On The Development of Electrochemical-Based Lithium-Ion Battery Models For Battery Management

Systems

Abstract

Renewable energy sources such as wind and photovoltaics have been broadly

integrated on the power network to reduce the reliance on fossil fuels and achieve

desired carbon emissions targets. However, these power generation technologies are

inherently variable. This variability presents a major challenge for grid stability. Grid-

connected battery storage systems have been widely accepted as an effective solution

to this problem. The battery storage systems are capable of providing a rapid response

to counteract the fluctuations and filter out the variabilities associated with renewable

generation and therefore stabilize grid performance and maximize system security

benefits.

Typical battery management systems are used to control the charge and

discharge of the battery systems within a safe operation window. These battery

management systems often employ battery equivalent circuit models which have

limited ability to accurately represent battery dynamics, which can lead to situations

in which batteries are not fully utilized, or used optimally. This motivates the need for

a next generation of battery management systems with advanced features. These

advanced management systems employ electrochemical models which have greater

capabilities in terms of predicting battery dynamics, states and degradation. They offer

the promise of better battery resource utilization and prolonged system life. There are

existing works on these management systems, however, major issues related to

electrochemical-based state estimation and the development of sophisticated, whilst

computationally efficient, electrochemical models embodying coupled

electrochemical-thermal-degradation have not yet been addressed.

The main objective of this thesis is to construct high-performance, reduced order,

electrochemical models incorporating thermal and degradation phenomenon that can

be used in advanced battery management systems. These models have been developed

to satisfy the requirement of computationally simple, whilst maintaining their ability


Systems iii

to accurately represent the major electrochemical and thermodynamical degradation

mechanisms of lithium ion batteries. The performance of these models is validated

numerically by comparing their output with solutions of a more sophisticated (and

computationally expensive) pseudo-two-dimensional electrochemical model.

Furthermore, an electrochemical-based adaptive state estimation algorithm is proposed

and validated via experiment.

iv On The Development of Electrochemical-Based Lithium-Ion Battery Models For Battery

Management Systems

Table of Contents

Keywords ........................................................................................................................ i

Abstract .......................................................................................................................... ii

Table of Contents .......................................................................................................... iv

List of Figures .............................................................................................................. vii

List of Tables ............................................................................................................... xii

List of Abbreviations .................................................................................................. xiii

Statement of Original Authorship ................................................................................ xv

Acknowledgements ..................................................................................................... xvi

Chapter 1: Introduction ................................................................................ 1

1.1 Background .......................................................................................................... 1

1.2 Research objectives.............................................................................................. 4

1.3 Original contributions .......................................................................................... 4

1.4 Thesis outline ....................................................................................................... 7

Chapter 2: Literature Review ....................................................................... 9

2.1 Empirical equivalent circuit models .................................................................... 9

2.2 Pseudo-two-dimensional electrochemical model .............................................. 14

2.3 Reduced order models of P2D model ................................................................ 18

2.4 Physics-based equivalent circuit models for lithium ion batteries ..................... 22

2.5 Thermal model ................................................................................................... 26

2.6 Degradation model ............................................................................................. 27

2.7 Battery state estimation techniques .................................................................... 29

2.8 Conclusion and research gaps ............................................................................ 30

Chapter 3: Single Particle Models and State Estimation by Sigma-point

Kalman Filter 33


Systems v

3.1 Introduction ........................................................................................................33

3.2 Extended SPM and a practical implementation scheme for electrochemical

battery models .........................................................................................................................35

3.3 Extended single particle model and sigma-point Kalman filter .........................44

3.4 Experimental result and discussion ....................................................................51

3.5 Conclusions ........................................................................................................57

Chapter 4: Padé Approximation of Linearized P2D Model .................... 58

4.1 Introduction ........................................................................................................58

4.2 Analytic Laplace domain transfer functions of linearized P2D model ...............59

4.3 Padé approximation model .................................................................................63

4.4 Result and discussion .........................................................................................69

4.5 Conclusions ........................................................................................................79

Chapter 5: Coupled electrochemical-thermal model for small-format

lithium ion cells 81

5.1 Introduction ........................................................................................................81

5.2 Simplified thermal model for small-format cylindrical cells .............................82

5.3 Coupled electrochemical thermal model for small-format cell ..........................83

5.4 Result and discussion .........................................................................................85

5.5 Conclusions ........................................................................................................91

Chapter 6: A Computationally Efficient Electrochemical-thermal-

degradation Model for Large-format Lithium Ion Cells ..................................... 92

6.1 Introduction ........................................................................................................92

6.2 Model domain and reduced orders electrochemical model in each wind ...........93

6.3 Thermal approximation model ...........................................................................98

6.4 Coupled degradation-electrochemical-thermal models ....................................107

6.5 Results and discussion ......................................................................................109

6.6 Conclusions ......................................................................................................117

vi On The Development of Electrochemical-Based Lithium-Ion Battery Models For Battery

Management Systems

Chapter 7: Conclusions.............................................................................. 118

7.1 Summary and discussion ................................................................................. 118

7.2 Directions for further research ......................................................................... 120

Bibliography .................................................................................................. 123

Appendices ..................................................................................................... 131

Appendix A ................................................................................................................ 131

Appendix B ................................................................................................................ 133


Systems vii

List of Figures

Figure 2.1 (a) Current pulse technique, (b) EIS technique, (c) A widely used

equivalent circuit model for lithium ion battery [18]. Reprinted from

"Experimental investigation of the lithium-ion battery impedance

characteristic at various conditions and aging states and its influence on

the application," by W. Waag, S. Käbitz, and D. U. Sauer, Applied

Energy, vol. 102, pp. 885-897, Copyright (2013), with permission from

Elsevier. ....................................................................................................... 10

Figure 2.2 Dependence of ohmic resistance (a), charge transfer resistance (b) and

double layer capacitance (c) on temperature, SOC and battery aging

[18]. Reprinted from "Experimental investigation of the lithium-ion

battery impedance characteristic at various conditions and aging states

and its influence on the application," by W. Waag, S. Käbitz, and D. U.

Sauer, Applied Energy, vol. 102, pp. 885-897, Copyright (2013), with

permission from Elsevier. ............................................................................ 12

Figure 2.3 Dependence of direct current resistance rDCR (rDCR = r0 + rct ) on

temperature, SOC and battery aging. (a) New cell, SOC =50%. (b) Aged

cell, SOC =50%. (c) New cell, temperature =10ºC. (b) Aged cell,

temperature =10ºC. [18] Reprinted from "Experimental investigation of

the lithium-ion battery impedance characteristic at various conditions

and aging states and its influence on the application," by W. Waag, S.

Käbitz, and D. U. Sauer, Applied Energy, vol. 102, pp. 885-897,

Copyright (2013), with permission from Elsevier. ...................................... 13

Figure 2.4 Schematic of one-dimensional electrochemical cell model of a lithium

ion cell [28] .................................................................................................. 15

Figure 2.5 Single Particle Model of a Li-ion battery [4]............................................ 19

Figure 2.6 The 1D cell diagram with transfer functions in each domain [7] ............. 21

viii On The Development of Electrochemical-Based Lithium-Ion Battery Models For Battery

Management Systems

Figure 2.7 Equivalent circuit for a porous electrode comprised of three finite

regions, adapted from [44] ........................................................................... 23

Figure 2.8 P2D equivalent circuit network implementation [46]. Reprinted from

" A physically meaningful equivalent circuit network model of a

lithium-ion battery accounting for local electrochemical and thermal

behaviour, variable double layer capacitance and degradation," by M.-

T. von Srbik, M. Marinescu, R. F. Martinez-Botas, and G. J. Offer,

Journal of Power Sources, vol. 325, pp. 171-184, Copyright (2016),

with permission from Elsevier ..................................................................... 24

Figure 2.9 Cell level scheme of the simplified physics-based equivalent circuit

model [47]. Reprinted from " An easy-to-parameterise physics-

informed battery model and its application towards lithium-ion battery

cell design, diagnosis, and degradation," by Y. Merla, B. Wu, V. Yufit,

R. F. Martinez-Botas, and G. J. Offer, Journal of Power Sources, vol.

384, pp. 66-79, Copyright (2018), with permission from Elsevier .............. 25

Figure 2.10 The list of commonly reported degradation mechanisms, mode,

cause and effect, adapted from [63] ............................................................. 28

Figure 2.11 General principle of Kalman filter algorithm. This figure is adapted

from [74] ...................................................................................................... 30

Figure 3.1 Simulink model of a lithium ion cell connected to a load ........................ 39

Figure 3.2 Equivalent circuit of the SPM-3P for lithium ion cell .............................. 41

Figure 3.3 Cell voltage response during 1C discharge rate ........................................ 42



Figure 3.6 Cell voltage response during 10C discharge rate ...................................... 43

Figure 3.7 Electrolyte concentration in the separator and positive electrode

during the dynamic charge/discharge current profile. ................................. 46

Figure 3.8 Diagram of the SPKF algorithm ............................................................... 50


Systems ix

Figure 3.9 Experiment setup to test the half-cell ....................................................... 53

Figure 3.10 (a) Current rate, (b) comparison between the reference SOC and

SOC reference and (c) SOC estimation error .............................................. 54

Figure 3.11 (a) Comparison of the voltage estimation and the measured voltage

and (b) the voltage estimation error. ............................................................ 55

Figure 3.12 OCP-SOC relationship of LiFePO4 positive electrode. .......................... 56

Figure 4.1 Schematic diagram of a lithium ion cell embodying the P2D model

taken from [4] and transfer function of each variable in each domain

based on [7]. ................................................................................................. 60

Figure 4.2 A low order Padé approximation model of the linear P2D model. .......... 66

Figure 4.3 Frequency response of the Padé approximants and transcendental

transfer functions of the four variables within the negative electrode. ........ 67

Figure 4.4 Current input and corresponding frequency content of a battery

storage system that is being used to smooth the power generated from

a wind farm connected to a grid. Current data sourced from [86]. .............. 69

Figure 4.5 Comparison between the Padé approximation model and the P2D

model for a pulse current profile .................................................................. 71

Figure 4.6 Comparison between the Padé approximation model and the P2D

model for the UDDS applied current profile. .............................................. 72

Figure 4.7 Comparison between the Padé approximation model with the full

(nonlinear) P2D model for the applied current profile associated with

the wind farm application, as depicted in Figure 4.4(a). ............................. 73

Figure 4.8 Comparison between two charging methods (a). Cell voltage during

charging. (b). The over potential s e at negx L ..................................... 76

Figure 4.9 Higher order Padé approximation model of the linear P2D model .......... 77

Figure 4.10 Cell voltage error obtained from low order and higher order Padé

approximation models when compared to the predictions of the P2D

model............................................................................................................ 79

x On The Development of Electrochemical-Based Lithium-Ion Battery Models For Battery Management

Systems

Figure 5.1 A schematic representation of the proposed cell model ........................... 84

Figure 5.2 (a) Cross-section view of a small-format cylindrical cell which shows

negligible temperature variation along the cell’s radius, (b) Cross-

section view of a large-format cylindrical cell considered in [94] which

shows significant temperature derivation along the cell’s radius ................ 85

Figure 5.3 Comparison between the proposed cell model and the rigorous 1D

radial PDE model in Comsol using1C charging/discharging of a small-

format cell (18650 cell) ................................................................................ 88

Figure 5.4 Comparison between the proposed cell model and the rigorous 1D

radial PDE model in Comsol using UDDS current profile of a small-

format cell (18650 cell) ................................................................................ 89

Figure 5.5 Temperature error between the proposed cell model and the Comsol

1D radial PDE model: 1C charging/discharging of a large-format cell. ...... 90

Figure 5.6 Temperature error between the proposed cell model and the rigorous

1D radial PDE model in Comsol using UDDS current profile of a large-

format cell. ................................................................................................... 90

Figure 6.1 (a) Illustration of a cylindrical lithium ion battery with spirally wound

design and its cross-sectional view, (b) the component layers in each

wind and (c) the domains that constitute half of each wind and the

corresponding Padé model transfer functions [59]. ..................................... 95

Figure 6.2 Illustration of cross section of the cylindrical lithium ion battery with

boundary conditions ................................................................................... 102

Figure 6.3 Parallel connection of Padé models ........................................................ 108

Figure 6.4 A schematic representation of the DET cell ........................................... 108

Figure 6.5 Simulation result of temperature distribution at different convection

coefficients from DET model during a 1C charge-rest-discharge-rest

cycle. .......................................................................................................... 110


Systems xi


coefficients from the DET model during UDDS cycles. ........................... 111

Figure 6.7 Comparison of temperature variation between DET model and the

full P2D model in COMSOL at aRr (outer wind) and

0Rr (inner

wind) at different convection coefficients during a 1C charge-rest-

discharge-rest cycle. ................................................................................... 112

Figure 6.8 Comparison of temperature variation between DET model and the

full P2D model in COMSOL at ar R (outer wind) and

0r R (inner

wind) at different convection coefficients during UDDS cycles. .............. 113

Figure 6.9 Evolution SEI resistance filmR and maximum concentration of lithium

ion s,maxc in solid phase in negative electrode of inner and outer winds (

2100W m Kh ). ........................................................................................ 116

xii On The Development of Electrochemical-Based Lithium-Ion Battery Models For Battery

Management Systems

List of Tables

Table 2.1 The matrix showing the tests used in [18]. Reprinted from

"Experimental investigation of the lithium-ion battery impedance

characteristic at various conditions and aging states and its influence on

the application," by W. Waag, S. Käbitz, and D. U. Sauer, Applied

Energy, vol. 102, pp. 885-897, Copyright (2013), with permission from

Elsevier. ........................................................................................................ 11

Table 3.1 FreedomCAR Cell Model parameters [27] ................................................ 39

Table 3.2 Half-Cell Model Parameters ...................................................................... 48

Table 4.1 Cell parameters used in the simulation ...................................................... 70

Table 4.2 Comparison of simulation time of the proposed approximation model

to compute all cell variables at all discrete spatial locations ....................... 74

Table 4.3 Comparison of simulation time of the proposed approximation model

to compute only cell voltage ........................................................................ 75

Table 4.4 Comparison of RMS error of cell voltage and the computational

workload between low order and higher order Padé approximation

models .......................................................................................................... 78

Table 5.1 Comparison of simulation time .................................................................. 86

Table 6.1 Comparison of simulation time ................................................................ 114


Systems xiii

List of Abbreviations

SOC State of charge

BMS Battery management system

ABMS Advanced battery management system

UDDS Urban dynamometer driving scheduler

P2D Pseudo-two-dimensional electrochemical model

BESS Battery energy storage system

PDE Partial differential equations

SEI Solid electrolyte interphase

OCV Open circuit voltage

OCP Open circuit potential

SPKF Sigma-point Kalman filter

EIS Electrochemical impedance spectroscopy

SPM Single particle model

ESPM Extended single particle model

DRA Discrete-time realization

FVM Finite volume method

EKF Extended Kalman filter

FEM Finite element method

CC Constant current

KF Kalman filter

CV Constant voltage

MD-MS Multi-dimensional, multi-scale

xiv On The Development of Electrochemical-Based Lithium-Ion Battery Models For Battery

Management Systems

DET Degradation-electrochemical-thermal model

NMPC Nonlinear model predictive control

DAE Differential algebraic equations


Systems xv

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the best

of my knowledge and belief, the thesis contains no material previously published or

written by another person except where due reference is made.

Signature:

Date: ___21/6/2019____________

QUT Verified Signature

xvi On The Development of Electrochemical-Based Lithium-Ion Battery Models For Battery

Management Systems

Acknowledgements

I would like to express my deepest appreciation and gratitude to my supervisors,

Prof. Mahinda Vilathgamuwa, and Prof. Troy Farrell. This dissertation would not have

been possible without their expertise, continual guidance, encouragement and support.

I would like to thank them for giving me the opportunity to pursue a PhD and work on

this innovative and advanced research project. I am extremely thankful for this

incredible experience.

I would like to extend my sincere gratitude to my external advisor Prof. San

Shing Choi for his persistent guidance and valuable advises he provided for the

improvement of this dissertation and other published articles.

I would like to acknowledge the organisations that funded my research, namely,

the Queensland University of Technology for QUT Postgraduate Research Award,

QUT Higher Degree Research Tuition Fee Sponsorship, QUT Excellence Top Up

Scholarship, QUT HDR travel grant and other resources, IEEE for student travel

grants, and the Australian Research Council.

I would like to thank Prof. Peter Talbot, Prof. Jose Alarco, Dr. Jawahar Nerkar,

Assoc. Prof. Hongxia Wang and Dr. Teng Wang for their support with the

experimental results. I would like to thank QUT School of Electrical Engineering and

Computer Science, Power engineering discipline, School of Mathematics, and High

Performance Computing group for their administrative and technical support.

I am grateful to Dr. Yang Li, Joseph Teague, Prof. Colin Please, Assoc. Prof.

Paul Corry, Dr. Shawn Neilsen and Assoc. Prof. Geoff Walker for their in-depth

discussions and contributions to improve the quality of this dissertation and other

publications.

I wish to thank committee members of my confirmation and final seminars

including Prof. Gerard Ledwich, Prof. Jose Alarco, Dr Yateendra Mishra and Assoc.

Prof. Hongxia Wang for their insightful comments and encouragement.


Systems xvii

Last but not least, I would like to thank my parents and my brother for their love

and encouragement during all my educational endeavours. To my partner Quynh

Phuong Lai thank you for her love and patience thorough this journey. I dedicate this

dissertation to you all, my beloved ones.

Chapter 1: Introduction 1

Chapter 1: Introduction

1.1 BACKGROUND

Lithium ion batteries are now commonly used in conjunction with renewable

generation systems, such as wind farms and photovoltaic stations. The role of the

batteries is to alleviate negative impacts the renewable generators may have on power

systems due to the inherent uncertainties and intermittency of the renewable sources

[1]. A typical example is the Hornsdale Power Reserve battery storage system

(129MWh capacity and rated at 100MW of discharge power) which has been effective

in providing network security services to South Australian grid system since 2017 [2].

A Battery Management System (BMS) is essential in order to utilize battery

storage capacity optimally, achieve safe charging and discharging of the battery system

and prolong the battery lifetime [3]. In this connection, optimal control algorithms in

advanced BMSs will necessitate models that can provide insight into cell internal

electrochemical variables, thermal dynamics and degradation [4]. An accurate battery

model that can represent the complex and nonlinear internal characteristics of batteries

is therefore desirable.

Depending on the application, the capacity of grid-connected battery energy

storage systems (BESSs) can vary from kWh to MWh [1]. Given that a single cell has

typically only 7.5Wh energy capacity [5], the BESSs are usually constructed from

many battery packs, with each pack consisting of large numbers of series- and parallel-

connected lithium ion cells to achieve the required voltage and capacity level [6]. As

a result, not only is a highly accurate battery model desirable but the model itself must

also be able to be evaluated quickly, using highly efficient computational algorithm.

Current BMSs rely significantly on equivalent circuit models of the battery due

to the simplicity of the models and the inherently low computational burden required

to implement them [7]. However, the models give only limited information on the

electrochemical characteristics and degradation response of the battery [8]. In addition,

2 Chapter 1: Introduction

equivalent circuit models require extensive battery testing in order to validate them in

a particular operational range in which the battery circuit parameters have been

identified [8]. As an alternative to equivalent circuit models, the pseudo 2-dimensional

electrochemical (P2D) model, first developed by Doyle, Fuller, and Newman [9, 10],

does provide detailed insights on these characteristics [8]. However, the P2D model

consists of partial differential equations (PDEs). To solve these, spatial discretization

methods are applied to yield a system of differential algebraic equations (DAEs) in the

time domain [4]. These methods are computationally expensive and are usually

implemented on a desktop computer, often using commercial software. Therefore, in

industrial applications, the methods are not suitable for embedded BMSs which require

much lower computational workloads. For example, as reported by Lee, Chemistruck

and Plett [7], the P2D model implemented in COMSOL Multiphysics [11] requires

about 13 min on a desktop computer, to produce 25 min of simulated voltage response

to the Urban Dynamometer Driving Scheduler (UDDS) current profile [12]. Such

computational requirements render a direct implementation of the P2D model

infeasible for use in a BMS.

On thermal management, information on heat generation is fundamentally

important for managing thermal issues such as thermal runaway, electrical cell

unbalance within the battery pack and poor performance at low temperatures [13].

However, heat generation inside a cell is a complex process that requires the

knowledge of the physical characteristics of the cell during its operation. Total heat

generation is due to irreversible and reversible processes, Joule heating in the solid and

electrolyte and heating from the electrode/current collector contact resistance [8]. It is

accompanied by changes in the electrochemical properties of the cell, such as entropy,

solid and electrolyte concentrations due to chemical and electrochemical reactions and

changes in the solid and electrolyte potentials.

Another important issue is the degradation of battery performance over time and

usage. The degradation is mainly caused by the formation and growth of the solid

electrolyte interphase layer (SEI), which scavenges active lithium ions and electrolyte

materials and increases battery resistance, leading to capacity and power fades,

respectively [14]. SEI growth is also coupled with the thermal behaviour of the cell as


higher temperatures increase the SEI growth rate which in turn causes higher SEI

resistance and ohmic heat generation, which elevates the temperature [15].

Thermal and degradation issues become more significant in large format

cylindrical batteries because non-uniformities in temperature and degradation occur

when the batteries operate [16]. Such non-uniformities exacerbate further degradation

and temperature gradients within the batteries. Current BMSs, which rely on

equivalent circuit models, have limited insight on the electrochemical characteristics

of a battery and, given the close coupling between the two, equivalent circuit models

are not able to model battery thermal behaviour and degradation mechanism precisely

[8].

In addition, information about battery state of charge (SOC) is often used by

BMSs to predict the available energy remaining in the batteries [4]. SOC can be

tracked by observing the ratio of the average lithium ion concentration to the maximum

concentration of the solid phase. This immeasurable quantity is a standard input for

control algorithms of the BMSs. A precise SOC information of the battery is required

since it can decide the effectiveness of the control algorithms. Studies on online SOC

estimation algorithms based on equivalent circuit model are immensely popular, those

based on physics first principle battery models have not yet matured and remains a

challenge to be considered.

The main goal of this thesis is to develop new electrochemical reduced order

battery models while taking into consideration the thermal and degradation dynamics

within the cells. Furthermore, a new adaptive electrochemical model-based SOC

estimation algorithm is also proposed. The developed models and estimation algorithm

have to be accurate and computationally efficient for application in grid-connected

battery storage systems. Given this, the research objectives of this work, aimed at

facilitating this goal, are outlined in the next section.


1.2 RESEARCH OBJECTIVES

To achieve the aforementioned goal, this thesis will be focused on the following

four objectives:

Objective 1: Develop a new SOC estimation algorithm using an electrochemical

model and an adaptive observer. This objective leads to the contribution 1 which is

described in Section 1.3.

Objective 2: Derive a new simplified model of Li-ion battery based on the

linearized pseudo-two-dimensional electrochemical model (P2D) using Padé

approximation method. This objective leads to the contribution 2 which is described

in Section 1.3.

Objective 3: Develop a new computationally efficient electrochemical-thermal

model for small-format lithium ion cells by incorporating a simplified battery thermal

model into the Padé approximation model. This objective leads to the contribution 3

which is described in Section 1.3.

Objective 4: Develop a new sophisticated electrochemical-thermal-degradation

model for large-format lithium ion cells by coupling together the Padé approximation

model, an approximated radial thermal model and degradation model. This objective

leads to the contribution 4 which is described in Section 1.3.

1.3 ORIGINAL CONTRIBUTIONS

Working toward achieving the objective of the study, the following original

contributions have been made:

Contribution 1:

The first contribution of the thesis is the development of a new adaptive state

estimation algorithm using the combination of the extended single particle and sigma-

point Kalman filter and the development of a practical scheme to implement the single

particle model in Simulink. The feasibility of the proposed model is validated by

comparing the estimated cell voltage and state of charge of the cell obtained with that


from test measurements. For this contribution, the associated published conference

papers are:

N. T. Tran, M. Vilathgamuwa, T. Farrell, and S. S. Choi, "Matlab simulation

of lithium ion cell using electrochemical single particle model," in 2016 IEEE 2nd

Annual Southern Power Electronics Conference, p. 210. (IEEE student travel grant

prize)

N. T. Tran, M. Vilathgamuwa, Y. Li, T. Farrell, S. S. Choi, and J. Teague, "State

of charge estimation of lithium ion batteries using an extended single particle model

and sigma-point Kalman filter," in 2017 IEEE 3rd Annual Southern Power Electronics

Conference, pp. 1-6. (IEEE student travel grant prize)

Contribution 2:

The second contribution of the thesis is the development of a new reduced cell

model by applying Padé approximations to simplify the complicated transcendental

transfer functions of the linearized P2D model to rational polynomial transfer

functions that are amenable to rapid computation for embedded BMS applications. The

reduced model output includes the spatial and temporal variation of all of the state

variables of the P2D model. Consequently, it can be used to determine any of the state

variables at any specific spatial location. There is no need to compute the state

variables at all of the spatial locations, as is the case when using the spatial

discretization methods to solve the P2D model. This proposed model is the first

attempt at using the Padé approximation method to simplify the linearized P2D model

whilst preserving physical meaning of the model. The model overcomes the limitation

of single particle models when applied to thick electrodes high-energy lithium ion

batteries used in grid-connected applications. The proposed model is validated by

comparing to the result obtained with that based on the P2D model solved by finite

volume method and implemented using an open source code. For this contribution, the

associated published journal paper is:

N. T. Tran, M. Vilathgamuwa, T. Farrell, S. S. Choi, Y. Li, and J. Teague, "A

Padé Approximate Model of Lithium Ion Batteries," Journal of the Electrochemical

Society, vol. 165, no. 7, pp. A1409-A1421, January 1, 2018. (Q1)


Contribution 3:

The third contribution of the thesis is the development of a new computationally

efficient electrochemical-thermal model of small-format cylindrical lithium ion cells.

This proposed model takes advantage of the aforementioned Padé approximation

model by computing only electrochemical variables at specific spatial locations. The

Padé electrochemical model is then combined with a simplified thermal model of

small-format cylindrical lithium ion cells to form a new model which requires low

computational burden. The proposed model is validated by comparing to the results

obtained with those using the full, 1-dimensional, radial implementation of the P2D

model implemented in Comsol Multiphysics®. For this contribution, the associated

accepted conference paper is:

N. T. Tran, M. Vilathgamuwa, T. Farrell, S. S. Choi, Y. Li, and J. Teague, "A

Computationally-Efficient Electrochemical-Thermal Model for Small-Format

Cylindrical Lithium Ion Batteries," the 2018 IEEE 4th Annual Southern Power

Electronics Conference, Singapore. (IEEE student travel grant prize)

Contribution 4:

The fourth contribution of the thesis is the development of a new and

sophisticated electrochemical-thermal-degradation model for large-format spirally

wound, cylindrical lithium ion cells. It takes into account the non-uniform degradation

and thermal behaviour within the cells. This proposed model is able to provide insight

into the variation and evolution of the local temperature and degradation rate of each

individual wind along the cell radius. The performance of the proposed model is

validated by comparing to the full, 1-dimensional, radial implementation of the P2D

model implemented in Comsol Multiphysics®. For this contribution, the associated

manuscript to be submitted to a journal is:

N. T. Tran, M. Vilathgamuwa, T. Farrell, S. S. Choi, and Y. Li, "A

Computationally Efficient Electrochemical-Thermal Model for Large Format

Cylindrical Lithium Ion Batteries," 2019. (Under review, the Journal of the

Electrochemical Society, Q1)


1.4 THESIS OUTLINE

The remainder of this thesis is organized as follows.

In Chapter 2, the key literature related to battery empirical equivalent circuit

models, electrochemical models, degradation, thermal models and battery state

estimation techniques are critically analysed. Areas where further research are needed

are identified at the end of this chapter.

Chapter 3 is aimed at developing an adaptive electrochemical model-based

battery state estimation algorithm. A practical scheme to implement the

electrochemical model in Simulink is also developed. This scheme is then used in

sophisticated models proposed in Chapters 5 and 6 where thermal and degradation

models are coupled with reduced order electrochemical models. Chapter 3 addresses

the Objective 1 of the thesis.

Chapter 4 focuses on simplifying the complex transcendental transfer functions

relating the electrochemical variables of the linearized P2D model into rational transfer

functions. Padé approximation is used to substantially reduce the complexity of the

P2D model whilst still maintaining a reasonable level of accuracy. This approximation

model overcomes limitations of single particle models in high energy cells of thick

electrodes where the current density is non-uniform. This chapter addresses the

Objective 2 of the thesis.

In Chapter 5, a computationally efficient electrochemical-thermal model for

small-format lithium ion cells is developed. This model is based on the Padé

approximation model proposed in Chapter 4, but incorporates a simplified thermal

model of the cells. Temperature variations within the small-formatted cylindrical cells

are ignored. The practical scheme proposed in Chapter 3 is also used to implement the

proposed model in Simulink. Chapter 5 addresses the Objective 3 of the thesis.

Chapter 6 provides the details on the development of a novel electrochemical-

thermal-degradation model for large-format lithium ion cells. The Padé approximation

model proposed in Chapter 4 is coupled with a battery degradation model and a

simplified radial thermal model. This is in order to efficiently represent the dynamics

and the non-uniform temperature and degradation in the cylindrical cells. The practical


scheme of Chapter 3 is also used here to implement this sophisticated model in

Simulink. Chapter 6 addresses the Objective 4 of the thesis.

Chapter 7 concludes the main findings of this thesis and presents suggestions

for future works.

Chapter 2: Literature Review 9

Chapter 2: Literature Review

This chapter provides an overview of published works on lithium ion battery

modelling. Sections 2.1 to 2.7 will review topics in the literature related to the

modelling of lithium ion batteries. In Section 2.8 conclusions from this review will be

drawn noting the research gaps and discussing their connection to the objectives of

this thesis.

2.1 EMPIRICAL EQUIVALENT CIRCUIT MODELS

Empirical equivalent circuit battery models have been used extensively in BMS

due to their simplicity and low computational requirement. The battery models can be

represented by an Open Circuit Voltage (OCV) in series with battery impedance

parameters including an ohmic resistance 0r , a parallel circuit consisting of the charge

transfer resistance ctr and a double layer capacitance

dlC , as depicted in Figure 2.1(c)

[17]. The battery output voltage can be calculated by the OCV and the voltage drop

across the battery impedance when an input current is applied. The parametric values

of these battery impedance parameters depend significantly on the battery SOC,

operating current rates, state of cell degradation and temperature [18]. These

dependencies can be determined from tests using such techniques as the current pulse

test [18] and/or the Electrochemical Impedance Spectroscopy (EIS) test [18-20], as

depicted in Figure 2.1(a) and Figure 2.1(b), respectively. In the current pulse technique

[18], the battery voltage drop 0V at the moment of applying the input current occurs

due to the ohmic resistance. The subsequent voltage decay 1V occurs across the

parallel ctr and

dlC circuit of the charge transfer resistance. In the EIS test [18], a small

signal excitation current ( )i f at a given frequency f is applied to the battery and the

voltage response of the battery ( )v f is measured. The complex impedance of the

battery at the frequency f can be expressed as ( )( )

( )v f

z fi f

. By applying a range of

frequencies, the battery impedance spectrum can be obtained. As shown in Figure

10 Chapter 2: Literature Review

2.1(b), 0r is determined at the point where Im ( ) 0z f . The resistance

dr is

determined at the minimum local point of Im ( )z f . The resistance ctr is the

difference between dr and

0r (ct d 0r r r ). The time constant τ (

ct dlr C ) is determined

by the maximum local point of Im ( )z f ( max2 f ). Based on these techniques,

the information of the battery impedance dependency on different input current rates,

SOC, degradation and temperature can be obtained [18].

Figure 2.1 (a) Current pulse technique, (b) EIS technique, (c) A widely used

equivalent circuit model for lithium ion battery [18]. Reprinted from "Experimental

investigation of the lithium-ion battery impedance characteristic at various conditions and

aging states and its influence on the application," by W. Waag, S. Käbitz, and D. U. Sauer,

Applied Energy, vol. 102, pp. 885-897, Copyright (2013), with permission from Elsevier.

The work of W. Waag et al. [18] showed an experimental investigation of battery

impedance dependency on various operating conditions and degradation levels by

using both current pulse technique (CP) and EIS technique (IS). The test method and

results reported by W. Waag et al. [18] are shown in Table 2.1, Figure 2.1, Figure 2.2

and Figure 2.3. Table 2.1 shows the matrix of CP and IS tests at different temperature


and SOCs. In each CP test, the test is repeated with ten different current rates ranging

from 0.25C to 4.25C. After testing the battery, the battery impedance is extracted from

the measured current and voltage data. Figure 2.2 and Figure 2.3 show the result of

these tests in which the battery impedance dependency on current rates, SOC,

degradation and temperature is obtained. By using these tests, full characterization of

the battery impedance can be parameterized and stored in look-up tables for battery

modelling and state estimation in the BMS, as did numerous works [21-23]. However,

the accuracy of the empirical equivalent circuit models strongly relies on operating

conditions in which the batteries have been tested and parametrized [8]. In other words,

the batteries need to be tested in various operation conditions to determine the change

in battery impedance as can be seen in the mentioned example. Thus, many tests which

are costly and time consuming need to be conducted in order to broaden the model

validity range [8]. Furthermore, the same tests need to be repeated if a different battery

type is to be modelled.

Table 2.1 The matrix showing the tests used in [18]. Reprinted from "Experimental

investigation of the lithium-ion battery impedance characteristic at various conditions and

aging states and its influence on the application," by W. Waag, S. Käbitz, and D. U. Sauer,

Applied Energy, vol. 102, pp. 885-897, Copyright (2013), with permission from Elsevier.

10% 20% 30% 50% 70% 90% 100%

-10°C IS IS IS IS IS IS IS

0°C IS, CP IS IS, CP IS, CP IS, CP IS, CP IS



40°C IS IS IS IS IS IS IS


Figure 2.2 Dependence of ohmic resistance (a), charge transfer resistance (b) and

double layer capacitance (c) on temperature, SOC and battery aging [18]. Reprinted from

"Experimental investigation of the lithium-ion battery impedance characteristic at various

conditions and aging states and its influence on the application," by W. Waag, S. Käbitz, and

D. U. Sauer, Applied Energy, vol. 102, pp. 885-897, Copyright (2013), with permission from

Elsevier.


Figure 2.3 Dependence of direct current resistance rDCR (rDCR = r0 + rct ) on

temperature, SOC and battery aging. (a) New cell, SOC =50%. (b) Aged cell, SOC =50%.

(c) New cell, temperature =10ºC. (b) Aged cell, temperature =10ºC. [18] Reprinted from

"Experimental investigation of the lithium-ion battery impedance characteristic at various

conditions and aging states and its influence on the application," by W. Waag, S. Käbitz, and

D. U. Sauer, Applied Energy, vol. 102, pp. 885-897, Copyright (2013), with permission from

Elsevier.

Alternatively, battery impedance can be estimated online by using parameter

estimation algorithms such as weighted recursive least squares [24, 25] and Dual

Extended Kalman filter [26]. These methods use measurable quantities such as battery

current as an input and battery voltage as a reference to adaptively update the battery

impedance values so that the errors between the battery voltages calculated by the

model and the measured voltages are minimized. Battery SOC can also be estimated

using the same approach and will be discussed in detail in Section 2.7. However, the


parameter estimation algorithm becomes complicated and its complexity level could

be as high as the electrochemical model itself (the model which will be reviewed in

the later part) [22]. Importantly, the empirical equivalent circuit model does not

provide insight into internal electrochemical variables of the batteries which are the

key to explain the battery degradation [7].

2.2 PSEUDO-TWO-DIMENSIONAL ELECTROCHEMICAL MODEL

The pseudo-2 dimensional electrochemical model (P2D) for lithium ion

batteries, first developed by Doyle, Fuller, and Newman [9, 10], does provide insight

into battery electrochemical behaviour. The P2D model is applied to four domain

elements of a lithium ion battery, namely, the negative electrode, the separator, the

electrolyte and the positive electrode [4]. During discharge, lithium ions in the negative

electrode diffuse from the interior to the surface of the spherical solid particles that

constitute the porous electrode, where they undergo an electrochemical reaction and

are transferred into the electrolyte phase. Thereafter, the ions travel through the

electrolyte solution to the positive electrode where they again react at the surfaces of

the solid particles (constituting that porous electrode) and are intercalated into these

particles through the solid electrolyte interphase (SEI) between the electrolyte solution

and the solid particles. These processes are shown schematically in Figure 2.4 along

with the definition of the modelling domain in which the negative electrode (neg) is

defined from 0x to negx L ; the separator (sep) is defined from negx L to

neg sepx L L and the positive electrode (pos) is defined from neg sepx L L to totx L .

The reverse of the depicted discharge process occurs in the charge process, whereby

lithium ions flow from the solid phase of the positive electrode to the solid phase of

the negative electrode. As the separator forms an electrically insulated barrier between

the electrodes, the flow of electrons associated with the charge and discharge processes

occurs via an external circuit connecting the two compartments, doing useful work as

they do so [27].


Figure 2.4 Schematic of one-dimensional electrochemical cell model of a lithium ion

cell [28]

The P2D model that describes the above processes consists of five state variables

including the electric potential ,s x t in the solid electrode, the electric potential

,e x t in the electrolyte, the lithium concentration of the active material , ,sc x r t of

the positive and negative electrodes, the lithium concentration ,ec x t in the

electrolyte, and the molar fluxes ,j x t of the charge that flows between the active

material in each electrode and the electrolyte. The governing equations of the model

are given as follows [4, 7] with the definitions of the symbols shown in the equations

being given in the Appendix B.

The potential in the solid phase of each (negative and positive) electrode is derived

from the principle of conservation of charge and is given by (for i neg, pos ),

i

seff,i

app e

,,i

x tI i x t

x

, (2.1)

with the boundary conditions

i i

s seff,i eff,i0, ,tot

app

t L tI

x x

.

i i

s s2

s2

, , , ,1c x r t c x r tD r

t r rr

negi posi sepi

i i

s s2

s2

, , , ,1c x r t c x r tD r

t r rr

e¯appI

Li

Li

e¯

appI

Li

Li

i

seff,i

e

,,i

app

x tI i x t

x

i

seff,i

e

,,i

app

x tI i x t

x

e i i

s

,,

ii x ta F j x t

x

i i ie e eeff,i 0

e e

, , 1,i

c x t c x tD t i x t

t x x F x

SEI

Electrolyte

Solid

D

i

e eeff,i eff,i

e

, ln ,,

i

ix t c x t

i x tx x

e ,ii x t

Negative electrode Separator Positive electrode

0x neg sepx L L neg sep pos totx L L L L negx L

tot

cell s s, 0,V t L t t


The electrolyte current is defined along the whole cell domain and is given by,

(for i neg, sep,pos and sep , 0j x t ):

e i i

s

,,

ii x ta F j x t

x

(2.1)

The electrolyte potential is defined along the whole cell domain and is given by

(for i neg, sep,pos ),

D

i

e eeff,i eff,i

e

, ln ,,

i

ix t c x t

i x tx x

, (2.2)

where,

eff,i eff,i 0 /D i

e

ln21 1

ln

c ad fRTt

F d c

,

with the boundary conditions

i i

e e0, ,0

tott L t

x x

.

Lithium ions are transported in the solid particles of each electrode by diffusion and

their concentration is given by (for i neg, pos ),

i i

s s2

s2

, , , ,1c x r t c x r tD r

t r r r

, (2.3)

where the boundary conditions and initial condition are given by

i

s

s

,0,0

c x tD

r

,

i

s s

s

, ,,i

c x R tD j x t

r

and i i

s s,0, ,0c x r c .

The equation for the lithium ion concentration in the electrolyte phase (across the

whole cell domain) is derived from the principle of conservation of mass in which the

change of electrolyte concentration is due to the diffusion and migration of the ions and

their transfer at the solid/solution interfaces (for i neg, sep,pos and sep , 0j x t ),

namely,

i i ie e eeff,i 0

e e

, , 1,i

c x t c x tD t i x t

t x x F x

. (2.4)


The boundary conditions and initial condition are

i i

e e0, ,0

totc t c L t

x x

and i i

e e,0,0c x c , respectively.

The transfer of charge at the solid/electrolyte interfaces in each electrode is given by

(for i neg, pos ),

11i eff,i i i i i

e s,max s,e s,e

i i

, , , ,

1exp , exp ,

j x t r c x t c c x t c x t

F Fx t x t

RT RT

, (2.5)

where, i

s,ec is the surface concentration of lithium in a spherical electrode particle.

The overpotential of the intercalation reactions in each electrode is given by (for

i neg, pos ),

i i i i i i i

s e OCP s,e film, , , , ,x t x t x t U c x t F R j x t . (2.6)

The potential difference between the positive and the negative current collectors

yields the cell voltage, namely,

tot

cell s s, 0,V t L t t . (2.7)

Equations (2.1) – (2.7) constitute a system of coupled PDEs and algebraic

equations that represent the P2D model for a lithium ion battery. Under galvanostatic

conditions, the input of the model is the applied current appI t and the output of the

model is the cell voltage cellV t . In the separator ( neg neg sepL x L L ), the model

involves two coupled PDEs which are the electrolyte concentration given by Eq. (2.4)

and the electrolyte potential given by Eq. (2.2). In the electrodes ( neg0 x L ,

neg sep totL L x L ), the model involves the coupled PDEs for electrolyte concentration,

given by Eq. (2.4), electrolyte potential, given by Eq. (2.2) and solid phase potential,

given by Eq. (2.1) on the macro (cell) scale (x) and Eq. (2.3) on the micro (particle)

scale (r) (which exists at every x in the electrode domains) [29]. The nonlinear nature

of the P2D model means that it must be solved numerically. As exemplified by the work

of Farrell and coworkers [30, 31], this is generally achieved using a desktop computer


running finite volume or finite element software. As noted earlier, this complexity

prevents the implementation of the P2D model in embedded real-time applications

such as BMSs.

2.3 REDUCED ORDER MODELS OF P2D MODEL

The P2D model introduced above is an accurate model to describe battery

operation [9, 10, 32]. With this model, the battery terminal voltage can be accurately

predicted. However, the computational cost associated with solving the coupled, non-

linear partial differential equations that define the model is very high in comparison

with the battery equivalent circuit models. A simplified electrochemical model that

has lower computational overheads, whilst maintaining its precision in a specific range

of battery operation, would therefore be ideal in order to facilitate the accurate, real-

time resolution. Indeed, there are several reduced-order electrochemical models

reported in the literature [4, 29].

2.3.1 Single particle models

A common approach, which results in the development of single particle model

(SPM), is one of the practical solutions used to reduce the complexity of the pseudo-

2D electrochemical model from a system of coupled five PDEs and one algebraic

equation into a single PDE and an algebraic equation [33]. It is based on the

assumptions that the applied current to the battery is small and the conductivity of the

electrolyte is large. Each electrode compartment is represented by a single particle, as

depicted in Figure 2.5. The electrolyte concentration is assumed to be constant and that

the current in the electrolyte does not vary spatially [4]. By applying these

assumptions, the lithium ion cell model is simplified from one consisting of the PDEs

in Eqs. (2.1), (2.3), (2.4) and the algebraic equation (2.5) to the following two PDEs

(one for each electrode in Eq. (2.9)) and the algebraic equation (2.13) [4].


Figure 2.5 Single Particle Model of a Li-ion battery [4]

The governing equation of lithium concentration in the solid particle of each

electrode is (for i neg, pos ):

2

( , ) ( , )10

i i

s ss

c r t c r tD

t r r r

(2.8)

Initial condition and the boundary on the particles is given by:

0,( ,0) ( )i

s sc r c r (2.9)

0

0i

s

r

c

r

(2.10)

( )

( )

p

iis

i i

r R

c I tj t

r Fa L

(2.11)

Output voltage of the lithium ion cell is:

0 0

( ) θ ( ) θ ( ) ( )pos neg

s s collectorV t t t I t R (2.12)

where:


0 0

,max

( )2 ( )θ ( ) asinh ( ( ))

2 ( )( ( ) ( ))

i

fi i i

s ss i ii i i i i

eff e ss s ss

R I tRT I tt U c t

F a La L r c c t c t c t

(2.13)

and the Open Circuit Potential, ( ( ))i i

ssU c t , is a function of surface concentration in

the solid phase ( )i

ssc t .

The advantage of SPM is that it can be evaluated more rapidly, which is essential in

real-time applications. However, this model has inherent limitation due to its low

current rate validity range [34].

2.3.2 Discrete-time realization method (DRA) and reduce-order state space

model

An alternative to the SPM is reported by Lee et al. [7] wherein the authors applied

truncated Taylor series expansions about a set point to each of the variables of the P2D

model to obtain a linear form of the model, around the desired point

*

s,e OCP s,0 s,e s,0 e e e ,0{ ( ), , ( , ) (0, ), }ep U c c c x t t c c [7]. Transcendental transfer

functions of each variable in each domain are then formed by Laplace transform. Figure

2.6 depicts a set of internal electrochemical variables of a lithium ion cell in each

domain. The electric potential s ( , )z t in the solid electrode, the lithium concentration

on the surface of the active material s,e ( , )c z t , the molar ion fluxes n ( , )j z t between the

active materials in the electrodes and the electric potential s ( , )z t in the solid electrode

are functions of the dimensionless spatial variable z [7]. The variable z is defined only

in the electrodes and equal to / negx L in the negative electrode and equal to

( ) /tot posL x L in the positive electrode. The electric potential e ( , )x t in the electrolyte,

the lithium concentration e ( , )c x t are not scaled with z as is the case with the other

parameters. e ( , )x t and

e ( , )c x t are the function of the spatial location x across the cell.


Figure 2.6 The 1D cell diagram with transfer functions in each domain [7]

Lee et al. [35] then introduced a discrete-time realization algorithm (DRA) to reduce

those transcendental transfer functions of the cell electrochemical variables into an

optimal reduced order state space form. The DRA contains the approximation of the

discrete-time impulse response for each transfer function H(s) of the electrochemical

variables [35]. Ho-Kalman algorithm is used to produce a state space form of the model

to yield an optimized, reduced-order, discrete-time, state-space model. It reduces the

computation time requirements by a factor of more than 5000, compared to the P2D

model. However, the parameters of such state space models have no direct physical

meaning and the practical use of the DRA algorithm in real-time applications, where

rapid updating of the parametric values of the parameters is required, is still

questionable.

2.3.3 Padé approximations for SPMs

Padé approximations provide a way of greatly simplifying complicated

transcendental transfer functions in order to produce simpler rational polynomial

transfer functions that can be used in embedded applications [36]. Previously, Padé

approximations have been applied to the transcendental transfer functions that model


lithium ion diffusion in a single, solid, spherical particle in the SPM [37, 38] for lithium

ion batteries. Marcicki et al [39], Zhang et al [40] and Yuan et al [41, 42] used Padé

approximations for an extended version of the SPM, which incorporates concentration

change within the electrolyte solution while maintaining the uniform current density

assumption of the SPM. These authors report that their Padé approximate model is

computationally more than 50 times faster computationally than the P2D model and

attains reasonable accuracy in comparison to the SPM PDE models [42]. However, the

applications of Padé approximations reported in the existing literature are limited to

the SPMs only. Therefore, related research gaps in reduced order electrochemical

models will be further discussed and dealt with in the following chapters.

2.4 PHYSICS-BASED EQUIVALENT CIRCUIT MODELS FOR LITHIUM

ION BATTERIES

A number of reported works proposed in the literature to transform the P2D

model into equivalent circuit models while retaining the physical meaning of the

electrical elements associated with the models. Newman [43] reported a physics-based

equivalent circuit model constructed using a network of resistors and voltage sources.

In this network, high resistance values represent the low conductivity of the electrolyte

and separator and low resistance values represent the high conductivity in the solid

phase. Farrell et al. [44] proposed a physics-based equivalent circuit model that

comprised of a number of finite circuit loops as shown in Figure 2.7. Each circuit loop

includes resistances which represent the conductivities of the solid and solution phase

of the electrodes. Kirchhoff’s current and voltage laws are applying to the circuit

yielding a set of linear algebraic equations for the currents in the circuit. The members

of the coefficient matrix in the linear algebraic equations are determined from the

ohmic resistances corresponding to the conductivity of the solid and solution phase of

the electrodes and electrolyte, respectively [44]. This model provides a useful physical

interpretation of the differential equations associated with the electrochemical model

to investigate the current and reaction distributions within the electrodes in the first

moment after applying the current to the battery.


Figure 2.7 Equivalent circuit for a porous electrode comprised of three finite regions,

adapted from [44]

S. Raël et al. [45] used equivalent circuit model to represent mass and charge

transport in lithium ion batteries. M.-T. von Srbik et al. [46] developed a detailed

physics-based equivalent circuit model which translate all variables of the P2D model

into the electrical circuit components shown in Figure 2.8. In addition, M.-T. von Srbik

et al. [46] considered the double layer capacitor, which represents the electrochemical

charge transfer process takes place at the interface between solid particles and the

solution. The degradation and thermal models are also coupled in this equivalent

circuit model. This detailed model provides a full physical interpreting of the P2D

model into electrical circuit analogy which can be easily implemented and used in

optimal control algorithms. Yu Merla et al. [47] developed a simpler version of the

detailed model proposed by M.-T. von Srbik et al. [46]. This simplified equivalent

circuit keeps most of the features of the aforementioned detailed model such as the

transport in the solid phase, double layer capacitance, degradation as shown in Figure

2.8. However, the lithium ion transport in solution phase is modelled by only resistors

and the diffusion phenomenon occurred in this phase and thermal model are omitted.

This simplified model is also easily implemented and modified due to its modular

structure as shown in Figure 2.9(a) and (b).


Figure 2.8 P2D equivalent circuit network implementation [46]. Reprinted from " A

physically meaningful equivalent circuit network model of a lithium-ion battery accounting

for local electrochemical and thermal behaviour, variable double layer capacitance and

degradation," by M.-T. von Srbik, M. Marinescu, R. F. Martinez-Botas, and G. J. Offer,

Journal of Power Sources, vol. 325, pp. 171-184, Copyright (2016), with permission from

Elsevier


Figure 2.9 Cell level scheme of the simplified physics-based equivalent circuit model

[47]. Reprinted from " An easy-to-parameterise physics-informed battery model and its

application towards lithium-ion battery cell design, diagnosis, and degradation," by Y.

Merla, B. Wu, V. Yufit, R. F. Martinez-Botas, and G. J. Offer, Journal of Power Sources,

vol. 384, pp. 66-79, Copyright (2018), with permission from Elsevier

Overall, these models have the advantages of easier implementation in well-

accepted software package such as Simulink® which provides set of solvers to deal

with the ordinary differential equations in the model. However, these physics-based

equivalent circuit models are developed based on finite volume or finite element

methods which may require expensive computation load when a high number of

discretised node points is chosen.


2.5 THERMAL MODEL

Battery thermal management is a critical function of a BMS in order to ensure

safe operation of the battery safety and to enhance battery life time [48]. Coupled

thermal and electrochemical models have been presented previously in the literature

[49, 50]. Pals and Newman [51] presented a one-dimensional, coupled thermal-

electrochemical model that calculates the heat generation rate and temperature of a

single cell and then in a second paper [52] they used their model to determine the

temperature distribution within a cell stack. Wang et al [53] modelled the thermal

performance of a battery pack with various cell configurations in the pack and with

different cooling strategies. However, due to their complexity and computational cost

these aforementioned models are implemented using powerful commercial software,

embodying integrated finite element and computational fluid dynamical packages [54-

56] such as COMSOL Multiphysics® [57] or ANSYS® Fluent [58]. Guo and White

[12] reported that a 2-dimensional, coupled, thermal-electrochemical model of a

cylindrical cell required approximately 20 hours to simulate a 30-minnute constant

discharge profile. In the same work, a proposed 1-dimensional, radial model of a

spirally wound cell required 20 minutes to simulate the same discharge profile. It is

noted that those reported simulation times are in the case of a constant discharge

current. For dynamic current profiles such as UDDS cycles or grid-connected battery

currents, much higher computational time per sample are required [59]. Thus, these

methods are not suitable for embedded BMSs, which require much lower

computational workload. An approximate thermal-electrochemical model is therefore

required for real-time applications.

A number of approximate thermal-electrochemical models have been reported

in literature [50, 60]. For example, the lumped thermal model, which is based on the

assumption that the whole interior temperature of the cell is uniform [60], has been

used in battery thermal management [8]. This model is simple and requires low

computational overhead and is useful in the thermal management of battery packs

where individual cells within the pack are considered as simple heat sources. However,

this approach is not able to compute non-uniform temperature distributions such as

those that occur in large format cylindrical cells, especially in the radial direction [61].


Kim et al. [62] proposed a second order polynomial approximation to the temperature

of the cell in the radial direction and in which the temperature at the center of the cell

is at the maximum. However, due to the hollow core in spirally wound cells the

maximum will not occur at the center of the cell and in large format cylindrical cells

this approximation is likely to be unreasonable. In addition, previous works use only

a single value of material density [62], specific heat and thermal conductivity for the

whole cell and incorporate a fitting approach to estimate these parameters and

incorporate a Kalman filter to estimate the cell’s temperature. The computational

requirements of such an approach are likely to be relatively high. High efficient

thermal modelling for cylindrical cells is examined further in Chapter 5 and 6 where

the inclusion of thermal model on electrochemical models are discussed.

2.6 DEGRADATION MODEL

Degradation in the lithium ion battery is affected by a variety of factors such as

cycling numbers, storage time, current rates, SOC and mechanical stress. A list of

commonly reported degradation mechanisms, mode, cause and effect is as shown in

Figure 2.10 [63]. Three main degradation modes are classified in [63]. Degradation

mechanisms that link to the loss of lithium inventory include Solid Electrolyte

Interphase (SEI) growth, SEI decomposition, electrolyte decomposition and lithium

plating [63]. Degradation mechanisms that link to the loss of active material in

negative and positive electrode include binder decomposition, structural disordering,

loss of electrode contact, electrode particle cracking and corrosion of current collectors

[7]. The end result is battery capacity fade and power fade. In order to prevent

degradation and optimally operate the battery using health-conscious optimal control

schemes, physics-based degradation models are required.


Figure 2.10 The list of commonly reported degradation mechanisms, mode, cause and

effect, adapted from [63]

Various works have been reported on the modelling of battery degradation. These

works mainly focus on the most dominant degradation mechanisms such as the growth

of SEI layer [63, 64]. SEI growth is attributed to 50% of the battery degradation [63].

Up to this time, SEI is still considered as “the most important but least understood

solid electrolyte in rechargeable Li batteries” due to the complexity of chemical and

electrochemical reactions involved and its physical properties [14, 65]. Ramadass et

al. [66] proposed a model of SEI layer growth which is represented by a continuous

film formation over the surface of the solid particles. It in turn increases the surface

resistance of the particles and the irreversible capacity loss. Safari et al. [67] developed

an isothermal, multimodal, aging model for SEI growth at the carbonaceous anode

material by considering the solvent decomposition kinetics and solvent diffusion

through the SEI layer. Henrik and G¨oran [68] proposed a degradation model based on

linear combination of continuous SEI layers growth in microscopic scale and SEI

layers cracks in macroscopic scale due to the expansion of the solid particles that are

covered by the SEI layers.

An important aspect should be noted is that the degradation rates are strongly

affected by cell’s temperature as higher temperature increases the SEI film growth rate


leading to higher loss of active material and higher SEI resistance [63]. This material

loss and resistance increment in turn induces higher temperature. Moreover, during

operation, non-uniformities in local temperature and degradation exacerbate high

degradation and temperature gradients within large format cylindrical batteries [16].

Unfortunately, coupled electrochemical-thermal-degradation models considering non-

uniformities have not yet been well-studied in the literature due to the complexity

associated with these sophisticated models. An additional challenge is that under

embedded applications there is the need to solve the coupled problem in a fast

computational manner whilst still being able to produce an accurate outcome. These

challenges are examined further in Chapter 6 where the thermal and degradation

models are efficiently included on electrochemical models for large format cylindrical

batteries.

2.7 BATTERY STATE ESTIMATION TECHNIQUES

Precise and reliable battery SOC estimation is important in order to optimize and

ensure safe battery operation and extended battery lifetime. Battery SOC can be also

determined using Coulomb counting method which is based on integration of

measured battery current [69]. In real applications, the SOC initial value is unknown

and current measurement errors exist. This method is also not suitable and therefore

an adaptive observer is required to predict the battery SOC.

A number of model-based approaches have been used to adaptively estimate

battery SOC and generally divided into two categories: equivalent circuit-based and

physics-based estimation algorithms. Equivalent circuit based estimation algorithms

rely exclusively on equivalent circuit models of the battery due to the simplicity of

these models implementation [70]. However, as mentioned above, equivalent circuit

models do not provide insights into the electrochemical characteristics of the battery

during its operation [28]. A limited number of reported works use a physics-based

battery model, like P2D or SPM, to determine SOC.

Among these works, Kalman filters (KF) are attractive approaches and

favourably used in combination with the electrochemical model(s) for state estimation


[70]. Generally, KF algorithms include two main steps in each measurement interval:

state update and measurement update. In the state update step, the present state, the

output of the model are predicted. Then, in the measurement step, the model state is

corrected using a weighted correction factor. This factor is the product of an adaptive

Kalman gain Lk and the error between the measured output of the real system and

estimated one of the model as shown in Figure 2.11. A standard KF with a reduced

form of the P2D model is used to estimate SOC in [71]. In comparison with the

standard KF, extended Kalman filter (EKF) is a highly efficient adaptive filter and is

preferable for estimating the state of the battery [72]. In the EKF, the nonlinear

equations of the system need to be linearized at each time step by using the first order

Taylor series expansion. The linear approximation can reduce the accuracy and result

in an unstable filter [72]. Sigma-point Kalman filter (SPKF) is an alternative algorithm

for state estimation and has some advantages such as its ability to operate without

linearizing the original system functions and the need to calculate their derivatives [72,

73]. SPKF algorithm is discussed in greater detail in Chapter 3 where this algorithm is

adapted to the SOC estimation study using simplified electrochemical models.

Figure 2.11 General principle of Kalman filter algorithm. This figure is adapted from

[74]

2.8 CONCLUSION AND RESEARCH GAPS

Based on this literature review, an essential requirement of advanced BMS is

that it should be able to precisely provide insight into battery electrochemical

dynamics. This function is necessary in order to control the battery operation to


achieve overall technical and economic benefits. In large scale battery storage systems,

another requirement is that the battery model needs to be computationally highly

efficient. This allows the battery model to operate effectively in embedded real time

application. Battery equivalent circuit models have been widely used to represent the

battery dynamic in real time application due to its simplicity and robustness. However,

this model is not able to give insight into battery electrochemical characteristic due to

its limitation on representing physical phenomena occurring inside the battery [7]. The

pseudo-2D battery model can overcome this problem by providing full information of

battery electrochemical characteristic accurately [4]. However, high complexity of the

model prevents its implementation in embedded real-time application. Since high

computational workload is required to solve coupled PDEs, pseudo-2D battery model

is often solved with the aid of a desktop computer and powerful commercial finite

element software. Many works have been reported to reduce the order of the high

fidelity P2D model, however, they still have some limitations discussed above and

need to be improved. High performance coupled electrochemical-thermal-degradation

models considering battery geometry are not yet well developed. Furthermore,

estimating battery SOC in real time is mandatory in BMS. In real applications, the

SOC initial value is unknown and current measurement errors exist. Consequently, the

research problem at hand is to develop a computationally efficient electrochemical

battery model and an adaptive electrochemical model-based estimation algorithm

suitable for a modern BMS of a large battery storage system. The developed scheme

shall precisely and efficiently evaluate battery electrochemical variables such as

lithium ion concentrations in solid phase and solution phase, cell state of degradation,

temperature and be able to predict the battery state of charge. In order to address this

research challenge, there are several gaps from this literature review that are

considered for this thesis as follows.

The combination of the ESPM model and the SPKF technique for battery SOC

estimation has not yet been reported. The ESPM accounts for the electrolyte

concentration and electrolyte potential which are not considered in SPM models

leading to a more accurate electrochemical model compared with SPM. Concurrently,

the SPKF theoretically outperforms EKF and have been demonstrated in previous


works related to battery SOC estimation. This research challenge will be addressed in

Chapter 3 to achieve the Objective 1 of the thesis.

The Padé approximation method has not yet been applied to approximate the

pseudo-2D electrochemical model. This method can reduce the complicated

transcendental transfer functions corresponding to lithium ion electrochemical

parameters to rational transfer functions. These transfer functions will be simpler and

applicable for the embedded BMS application. This research gap will be fulfilled in

Chapter 4 to achieve the Objective 2 of the thesis.

Coupled electrochemical-thermal models of lithium ion batteries are often

associated with high complexity and computational cost. They are implemented using

powerful commercial software, embodying integrated finite element and

computational fluid dynamical packages. A highly computationally efficient,

simplified electrochemical-thermal model is essential in real-time applications of

small-format cylindrical lithium ion cells where temperature variation along cell’s

radius can be negligible. This research challenge will be addressed in Chapter 5 to

achieve the Objective 3 of the thesis.

For large format lithium ion cells, a more sophisticated electrochemical-thermal-

degradation model would be more suitable as it accounts for non-uniform temperature

and degradation distributions such as those that occur in these cells, especially in the

radial direction. Such type of model has not been available in the literature. This

research gap will be closed in Chapter 6 to achieve the Objective 4 of the thesis.

Advanced control of the battery storage systems requires the insightful

information of battery electrochemical variables, degradation and thermodynamics. In

this thesis, a systematic approach to derive high-performance reduced-order

electrochemical models will be presented in Chapters 3-6. These models are capable

of efficiently computing the variations of cell electrochemical variables, temperature

and the degradation of grid-connected lithium-ion battery banks.

Chapter 3: Single Particle Models and State Estimation by Sigma-point Kalman Filter 33

Chapter 3: Single Particle Models and State

Estimation by Sigma-point

Kalman Filter

3.1 INTRODUCTION

From Chapter 2, it has been explained that large-scale lithium ion battery energy

storage systems are widely used in conjunction with renewable resources such as

electrical power generated from wind and solar. The batteries are to act as energy

buffers so as to alleviate the inherent intermittency of the renewables. The advantages

of batteries are their low self-discharge rates, long life and high power and energy

density [3]. In conjunction with these applications, it is essential to employ a suitable

BMS to ensure the batteries operate optimally with respect to safety and service life. A

battery model that yields accurate SOC estimation is an important requirement in the

design of advanced control strategies in the BMSs [4, 28]. Most estimation algorithms

require an accurate battery model that can precisely describe the voltage response of

the battery during the battery charge/discharge process. Therefore, battery modelling

has become an integral part in the development of an effective BMS.

Simple equivalent circuit models are normally applied in model-based methods.

In such connection and as described in Section 2.1, the battery is represented by an

Open Circuit Voltage (OCV) source connected in series with an internal resistance and

a parallel circuit that includes a charge transfer resistance and a double layer

capacitance [17]. This circuit model is very simple and it requires minimal

computation cost to solve. However, batteries have highly nonlinear electrochemical

characteristics that depend significantly on external and internal conditions. Simple

circuit models have only limited capability in accurately accounting for the dynamic

electrochemical behaviour of lithium ion batteries.

34 Chapter 3: Single Particle Models and State Estimation by Sigma-point Kalman Filter

The electrochemical model, which accounts for phenomena such as the transport

of charge within the cell, the intercalation processes that occur within the solid phase

electrode materials and the electrochemical kinetics occurring at the charge transfer

interfaces inside the battery during its operation, is an alternative to battery modelling

[9, 10, 32]. With this model the battery voltage can be accurately predicted. However,

as explained in Section 2.2, the computational cost associated with solving the

coupled, non-linear partial differential equations that define the model is very high in

comparison with equivalent circuit models.

A simplified electrochemical model that has low computational overheads,

whilst maintaining its precision in a specific range of battery operation, would

therefore be ideal in order to facilitate the accurate, real-time resolution and control of

individual cells within a battery energy storage system. There are several reduced

electrochemical models reported in the literature [4, 29]. The SPM that is reviewed in

Section 2.3.1 is one of the practical solutions to reduce the complexity of the P2D

model. An improvement of SPM is the extended SPM (ESPM) proposed in [75, 76],

which takes into account the variations in electrolyte concentration and potential. The

ESPM in [75] approximates the distribution of the electrolyte concentration and

potential within the electrodes with cubic functions of time and space whereas a

quadratic function is used to describe the electrolyte concentration and electrolyte

potential in the separator. The ESPM in [76] approximates the lithium concentration

in the solid and electrolyte phases by quadratic functions of time and space while the

electrolyte potential is solved analytically based on the assumption that the applied

current passed through the electrode is distributed uniformly over the electrode.

Regarding SOC, in order to ensure safe battery operation and to extend the battery

lifetime, precise and stable battery SOC estimation is essential [22, 28]. Examples

include different optimal control strategies suggested in [8] [77] to improve the battery

lifetime by charging/discharging the battery optimally based on the SOC information.

A number of model-based SOC estimation techniques has been reviewed in the

Chapter 2. Among these techniques, Sigma-point Kalman filter (SPKF) outperforms

standard Kalman filter and extended Kalman filter (EKF) algorithm. Therefore, SPKF

algorithm is chosen in this study for SOC estimation algorithm.


In the first part of this chapter, a simple electrochemical model based on the SPM

[33] is developed and in which the governing PDE is approximated by two differential

algebraic equations and one algebraic equation. This is done in a manner similar to that

given in [29] where the lithium concentration in the single particle that comprises each

electrode can be expressed by a polynomial function of the particle radius. The model

is implemented in Simulink in such a way that it can be used in all types of system

simulations. The feasibility of the proposed model is validated by comparing the

predicted cell voltage obtained from this model with that of the SPM. It is found that

the proposed model significantly reduces the computation time for simulation whilst

producing a reasonable approximation to the SPM output. Hence the first part of this

chapter is mainly to introduce a practical scheme that is used to convert an

electrochemical model into a Simulink cell block.

In the second part of this chapter, the accuracy of the simplified SPM model is

improved by taking into account the evaluation of the electrolyte concentration

distribution as did Han et al. [76]. This improvement yields an extended SPM, so called

ESPM. A new SPKF-ESPM approach is then proposed, which is a combination of the

ESPM and the use of the SPKF technique to estimate the battery SOC. The feasibility

of the proposed model is validated by comparing the predicted cell voltage and SOC

from this algorithm with data obtained from laboratory experiments. The proposed

method is shown to be capable of estimating the SOC of the battery with high accuracy

and computational efficiency.

3.2 EXTENDED SPM AND A PRACTICAL IMPLEMENTATION SCHEME

FOR ELECTROCHEMICAL BATTERY MODELS

3.2.1 Three parameter approximation model for the lithium solid concentration

A highly efficient approximation for the SPM can be achieved by reducing the

PDE in Eq. (2.8) of the lithium diffusion in particles (see Section 2.3.1) to differential

algebraic equations. Two- and three-parameter models that do this are presented in

[29], with the three-parameter model found to be more accurate at representing the

solutions of the PDE. Here the same three- parameter model reduction approach as


given in [29] is applied to Eq. (2.8) of the SPM presented in Section 2.3.1. A summary

of the model reduction approach is as follows [29].

The lithium concentration is assumed to be a polynomial function of the single

particle radius and given as,

2 4

s

p p

( , ) ( ) ( ) ( )r r

c r t a t b t d tR R

. (3.1)

By using this assumption, the first boundary condition ( 0r ) in Eq. (2.10) is

satisfied and the second boundary condition ( pr R ) in Eq. (2.11) is changed into Eq.

(3.2):

s sn

p p

2 4( ) ( ) ( )

D Db t d t j t

R R . (3.2)

The volume-averaged concentration and volume-averaged concentration flux

are expressed in Eq. (3.3) and Eq. (3.4), respectively:

p 2

s s2

p p0

( ) 3 ( , )

R

r

r rc t c r t d

R R

, (3.3)

p 2

s s2

p p0

( ) 3 ( , )

R

r

r d rq t c r t d

R dr R

. (3.4)

Substituting Eq. (3.1) into Eq. (3.3) and Eq. (3.4) and evaluating, one obtains

3 3

( ) ( ) ( ) ( )5 7

sc t a t b t d t , (3.5)

3 2

( ) ( ) ( ) ( )2

s

p p

q t a t b t d tR R

. (3.6)

Evaluating Eq. (3.1) at the surface condition ( pr R ) yields

( ) ( ) ( ) ( )sc t a t b t d t . (3.7)

The three coefficients ( )a t , ( )b t and ( )d t can be obtained by manipulating

equations Eq. (3.8), Eq. (3.9) and Eq. (3.10) to obtain:

39 35

( ) ( ) ( ) 3 ( )4 4

ss s s pa t c t c t q t R , (3.8)


( ) 35 ( ) 35 ( ) 10 ( )ss s s pb t c t c t q t R , (3.9)

105 105

( ) ( ) ( ) 7 ( )4 4

ss s s pd t c t c t q t R . (3.10)

Substituting Eq. (3.8), Eq. (3.9) and Eq. (3.10) into Eq. (3.1),

2

4

39 35( , ) ( ) ( ) 3 ( ) ( 35 ( ) 35 ( ) 10 ( ) )

4 4

105 105( ) ( ) 7 ( )

4 4

s ss s s p ss s s p

p

ss s s p

p

rc r t c t c t q t R c t c t q t R

R

rc t c t q t R

R

. (3.11)

Volume averaging the governing equation (2.8) produces

2

2 2

0

( , ) ( , )13 0

pR

s ss

p pr

c r t c r tr rD d

R t r r r R

. (3.12)

Substituting Eq. (3.11) into Eq. (3.12), the volume-averaged concentration then

becomes:

( )

( ) 3 ns

p

j tdc t

dt R . (3.13)

Differentiating Eq. (2.8) and volume averaging yields

22

2

0

( , ) ( , )1

3 0

p

s sR s

p pr

c r t c r tD

t r r rr rd

R r R

. (3.14)

Substituting Eq. (3.11) into Eq. (3.14), the volume-averaged concentration flux

is then governed by:

2 2

( )45( ) 30 ( ) 0

2

s ns s

p p

D j tdq t q t

dt R R . (3.15)

The surface concentration of lithium in the particles ( )ssc t is then obtained by

substituting Eq. (3.9) and Eq. (3.10) into Eq. (3.2) to give:

8

( ) ( ) ( ) ( )35

p p

ss s s n

s

R Rc t c t q t j t

D . (3.16)

Equations (3.13), (3.15) and (3.16) represent the SPM-Three parameter model

for the lithium concentration at the surface of the single particle that constitutes each


electrode in the SPM presented above. As stated above, one has reduced a single PDE

in Eq. (2.8) into an algebraic equation (3.16) and two ordinary differential equations

(Eq. (3.13) and Eq. (3.15)). The initial conditions for Eq. (3.13) and Eq. (3.15) are

given by the initial state of the cell. For a single discharge of a new cell these values

will be 0,( ,0) ( )s sc r c r and (0) 0sq . To obtain the cell voltage using this model, the

lithium surface concentration ( )ssc t from Eq. (3.16) is substituted into Eq. (2.13), which

are then substituted into Eq. (2.12).

3.2.2 Implementation of SPM-Three parameter model for lithium ion cell in

Simulink

In this investigation, the SPM-Three parameter model (henceforth, labelled as

SPM-3P model) that is presented in the previous section (Eq.s (2.12), (3.13), (3.15)

and (3.16)) has been implemented in the Matlab/Simulink environment. In Simulink,

it represents a “lithium ion cell block” which can be applied to various applications in

power electronics including an electrochemical cell within a large battery pack. Since

the SPM-3P model consists of three differential algebraic equations, the computation

requirement is dramatically reduced from that of the SPM and full electrochemical

models. Consequently, the cell block can be connected in series and parallel to achieve

the required battery pack voltage and capacity of large battery energy storage system.

Figure 3.1 shows an example of the cell model in a simple application in

Simulink. The cell is connected to a Controlled Current Source that represents a

constant current load. The lithium ion cell is continuously discharged by different

constant current rates ranging from 1C to 10C. The parameters used in the model are

those from the FreedomCar cell and are given in [27] and listed in Table 3.1.


Figure 3.1 Simulink model of a lithium ion cell connected to a load

Table 3.1 FreedomCAR Cell Model parameters [27]

Symbol Units Negative

Electrode Separator

Positive

electrode

410 cm 50 25.4 36.4

pR 410 cm 1 - 1

,maxsc

3 -310 molcm

16.1 - 23.9

,0ec

3 -310 molcm

1.2 1.2 1.2

0i

3 -210 Acm

3.6 - 2.6

collectorR mΩ 1.9 - 1.9

sD

2 -112 cm1 s0

2.0 - 3.7

fR mΩ 0 - 0

L 410 cm 50 25 36.4

eD

2 -16 cm1 s0

2.6 2.6 2.6

s - 0.58 - 0.50

0% , 0%

- 0.126 - 0.936

100% ,

100%

- 0.676 - 0.442



Electrode Separator

Positive

electrode

s - 0.58 - 0.50

The Open Circuit Potential of the negative and positive electrodes are functions

of stoichiometry and are adapted from [27] as follows:

neg neg neg neg 1/2 4 neg 1

5 neg 3/2 neg

neg

( ) 8.00229 5.0647 12.578( ) 8.6322 10 ( )

2.1765 10 ( ) 0.46016exp[15.0(0.06 )]

0.55364exp[( 0.92)]

U x

x

(3.17)

pos pos pos 6 pos 5 pos 4 pos 3

pos 2 pos pos 115

( ) 85.681( ) 357.7( ) 613.89( ) 555.65( )

281.06( ) 76.648 0.30987exp(5.657( ) )

13.1983

U

(3.18)

where, stoichiometry ss s,max( ) / ( )c t c t is the ratio between the lithium ion

concentration at the surface of the solid particle and the maximum lithium ion

concentration.

Figure 3.2 shows the detailed equivalent circuit of a designed electrochemical

cell named Electro_cell_SPM3. The cell model consists a SPM-3P model block, a

Controlled Voltage Source block, a resistance and a current sensor. The SPM-3P

model block contains Matlab Functions blocks and Integrator blocks used to calculate

the cell voltage based on Eq.s (2.11), (2.12), (3.13), (3.15), and (3.16). The input of

the SPM-3P model block is the charge/discharge current from the current measurement

block. In order to avoid the algebraic loop error, an initial condition block (IC block)

is added into the SPM-3P model block to set the initial condition of the current input.

The output of the SPM-3P model block then forms an input of the Controlled Voltage

Source block which converts the input signal into an equivalent voltage source. The

resistor collectorR represents the cell current collector contact resistance.


Figure 3.2 Equivalent circuit of the SPM-3P for lithium ion cell

Figure 3.3, Figure 3.4, Figure 3.5, and Figure 3.6 depict the comparison between

the voltage response of the SPM-3P model, a Two-parameter version of the SPM

(labelled as SPM-2P, details not given here) and the SPM (solved by using the “pdepe”

solver in Matlab) for discharge current rates of 1C, 5C, 7C and 10C. SPM-2P is built

using similar approach used for SPM-3P. The difference is that the lithium ion

concentration in the solid phase is approximated using a quadratic function of the

single particle radius. It is observed that the difference between the SPM-2P model

and SPM-3P model increases as the current rate increases. Furthermore, the SPM-3P

model and the SPM give very similar results.


Figure 3.3 Cell voltage response during 1C discharge rate





The SPM-3P model requires 0.011 second to simulate the 3600 second discharge

profile on a desktop computer, which is about 3.06 µsec/sample. On the other hand,


the “pdepe’’ solver in Matlab requires 0.478 second to simulate the same profile on

the same desktop computer, which is about 0.133 msec/sample. Thus, the SPM-3P

model is approximately 40 times faster than using the SPM with one governing PDE.

Consequently, the SPM-3P model is a possible candidate for real-time control

applications since it requires low computation while maintaining high accuracy

compared to the SPM.

3.3 EXTENDED SINGLE PARTICLE MODEL AND SIGMA-POINT

KALMAN FILTER

3.3.1 Extended single particle model ESPM

The ESPM is an approach to improve the accuracy of the SPM [75]. It is based on

the assumptions that each electrode compartment is represented by a single particle

and the electrolyte concentration in it varies with time and space. The concentration is

computed as shown below.

The output voltage of the lithium ion cell is expressed in Eq. 3.19 using ESPM

approach. The detailed derivation can be found in [78, 79].

pos

f1

pos pospos pos pos 0 pos pos pos

eff e ss s,max ss

neg

f1

neg negneg neg neg 0 neg neg neg

eff e ss s,max ss

pos p

e tot e col ss

2sinh

2

2sinh

2

, 0,

I t R I tRTV t

F a La L r c c c t c t

I t R I tRT

F a La L r c c c t c t

L t t I t R OCP c

os neg neg

sst OCP c t

. (3.19)

In Eq. (3.19), the electrolyte potential difference ttL ,0, etote is approximated as in

Eq. (3.20), based on the assumption that the molar flux is the same in each electrode

[78]. The difference between the electrolyte potential at the positive electrode-current

collector boundary and negative electrode-current collector boundary is

0

neg sep pos e tot

e tot e neg sep pos

eff eff eff e

2 12 ,, 0, ln

2 0,

RT tL L LI t c L tL t t

A F c t

. (3.20)


The governing equations of electrolyte concentration txc ,e in the negative electrode,

separator and the positive electrode are expressed as in Eq. 3.21 [75].

neg

e e eeff 0

e,neg s neg

sep

e e eeff

e,sep neg nm

pos

e e eeff 0

e,pos s nm tot

, ,1 , ,0

, ,,

, ,1 , ,

c x t c x tD t a j x t x L

x x x

c x t c x tD L x L

x x x

c x t c x tD t a j x t L x L

x x x

. (3.21)

In the ESPM, the electrolyte concentration is then approximated as quadratic

functions of position x along the electrodes and separator [76]:

2

1 2 3 neg

2

4 5 6 neg nm

2

7 8 9 nm tot

, 0

,

,

e

a t x a t x a t x L

c a t x a t x a t L x L

a t x a t x a t L x L

. (3.22)

Using six boundary conditions and three equations of total amount of lithium ion as

derived in [76], the nine coefficients 91 aa governing the electrolyte concentration can

be obtained through solving the simultaneous equation Eq. (3.23).

7 tot 8

2

neg 3 neg

e 1 neg 3 neg e

sep 3 3 2 2 sep

e 4 nm neg 5 nm neg 6 sep e

pos 3 3 2 2 pos

e 7 tot nm 8 tot nm 9 pos e

2 2

1 neg 3 4 neg 5 neg 6

2

4 nm 5 nm

2 0

0

2 6 6

2 3 6 6

2 3 6 6

a t L a t

a t

a t L a t L Q t

a t L L a t L L a t L Q t

a t L L a t L L a t L Q t

a t L a t a t L a t L a t

a t L a t L

2

6 7 nm 8 nm 9

neg eff sep eff sep eff

1 e e,neg neg 4 e e,sep neg 5 e e,sep

sep eff sep eff pos eff pos eff

4 e e,sep nm 5 e e,sep 7 e e,pos nm 8 e e,pos

2 2

2 2

a t a t L a t L a t

a t D L a t D L a t D

a t D L a t D a t D L a t D

. (3.23)

Figure 3.7 depicts the electrolyte concentration distribution in the separator and the

positive electrode during a particular charge-discharge process of a half-cell

constructed in the Queensland University of Technology’s (QUT) battery research

facility. The current profile is shown in Figure 3.10(a). The negative electrode of the

half-cell is a lithium foil and so the electrolyte concentration at the boundary between

the foil and the separator is considered constant throughout the charge-discharge


process. On the other hand, in the separator and in the positive electrode, the electrolyte

concentration is shown to have non-uniform profiles as the dynamic charge-discharge

proceeds. The electrolyte concentration varies spatially and temporally in these two

domains which clearly invalidates the constant electrolyte concentration assumption

made in developing the SPM. This is one of the reasons why the results obtained from

the SPM deviate from that obtained from the P2D model [75]. Therefore, by

introducing the variation of electrolyte concentration in the ESPM, one can obtain a

more accurate determination of the cell terminal voltage.

Figure 3.7 Electrolyte concentration in the separator and positive electrode during the

dynamic charge/discharge current profile.

3.3.2 Sigma-point Kalman filter algorithm

In order to estimate the SOC of the battery with high precision and stability, the

SPKF algorithm reported in [73] is used in this work. For the problem at hand, the cell

ESPM can be represented in the discrete-time state space form:

, , 1 11 1 1 1

,max ,max ,max

, , , 1 3s k s k k

k k k k k

s s s p

c c j tx f x u w k w

c c c R

. (3.24)

where, , , ,x u w k are state of the system, input current, process noise and time

step, respectively.

The cell SOC can be calculated using the transformation shown in Eq. (3.25). It

is noted that the SOC given in Eq. (3.25) is calculated in terms of the physical


parameters of the cell such as average lithium concentration ,s kc and the maximum

lithium concentration in the solid particle ,maxsc . SOC is calculated based on the ratio

of the average lithium concentration in the solid phase to the maximum possible

concentration in the electrode.

,

0%

,max

100% 0%

s k

s

k

c

cSOC

. (3.25)

where, 0% 100%, are stoichiometric limit at SOC 0% and SOC 100%,

respectively.

The model output cell voltage is given as Eq. (3.26).

1

pos pos pos 0 pos pos pos

eff e ss s,max ss

pos

f1

pos posneg neg neg 0 neg neg neg

eff e ss s,max ss

neg sep pos

neg sep

eff eff eff

2, , , sinh

2

2sinh

2

2

2

k k k k

I tRTy g x u v k

F a L r c c c c t

I t R I tRT

F a La L r c c c c t

L L LI t

A

0 neg

e tot f

pos neg neg

e

pos pos neg neg

ss ss col

2 1 ,ln

0,

k

RT t c L t R I t

F c t a L

OCP c t OCP c t I t R v

(3.26)

Within the first term on the RHS of Eq. (3.26), the reaction rate effr is fitted as an

empirical function of the input current and the normalized value of surface

concentration in the solid phase. This fitting of the reaction rate contributes to

compensate the weakness of the kinetic expression used in the Butler-Volmer equation

which cannot represent correctly the specific phenomenon related to a phase transition

electrode material [80]. For the particular half-cell used in this study, the reaction rate

pos

effr in Eq. (3.26) is fitted using measured voltage and current data from experiment

and shown in Eq. (3.27). The detail of the experiment is discussed in the next section.


pos 8 6 6 2 3

eff

2pos pos pos

ss ss ss6 8 8

pos pos pos

s,max s,max s,max

pos pos

ss ss5 2 6

pos

s,max s

2.025 10 1.156 10 7.141 10 0.01484

2.048 10 2.462 10 3.059 10

3.546 10 3.772 10

r I t I t I t

c t c t c tI t

c c c

c t c tI t I t

c c

2

pos

,max

. (3.27)

Regarding the second term on the RHS of Eq. (3.26), the Open Circuit Potential

(OCP) of the negative and positive electrodes are functions of stoichiometry . The

analytical expression to describe the function can be obtained by fitting the curve

relating the measured OCP against the stoichiometry or a lookup table can be used.

This OCP-stoichiometry relationship and the form as well as the coefficients of the

function depend on the material that is used in each electrode. In the case of the

particular half-cell that is used in experiments carried out in this study, the OCP of the

negative electrode which is made by lithium foil is considered to be zero and the

positive electrode (LiFePO4) is a function of stoichiometry as adopted from [81] as

follows:

neg neg 0OCP , (3.28)

pos pos pos 1.3198

6 pos 3.8003

6 pos 3.7995

3.4323 0.8428 exp 80.2493 1

3.247 10 exp 20.2645 1

3.2482 10 exp 20.2646 1

OCP

. (3.29)

Of the remaining terms on the RHS of Eq. (3.26), the values of the various

parameters of the half-cell used in this study are listed in Table 3.2.

Table 3.2 Half-Cell Model Parameters


Electrode Separator

Positive

electrode

pR 910 m - - 53

s,maxc -3molm - - 21,900



Electrode Separator

Positive

electrode

e,0c -3molm 1,000 1,000 1,000

colR 2m 0.001 - 0.001

sD 18 2 110 m s

- - 8

fR 0 - 0

L 610 m 750 25 30

eD 10 2 110 m s

0.22 0.22 0.22

s - - - 0.48

0% , 0%

- - - 0.99

100% , 100% - - - 0.01

The schematic diagram of the ESPM-SPKF algorithm is depicted in Figure 3.8.

The lithium ion cell used in this investigation has its applied current (input) and its

terminal voltage (output) measured at every time step. The unmeasurable battery state

of the cell (either the cell SOC or its average lithium ion concentration in the solid

particle) is unknown and needs to be estimated. Thus, the battery model and the ESPM-

SPKF estimator will have the same battery current as input, and the estimated output

voltage ˆkV as the output of the estimator. The estimated state of the model is processed

by the SPKF within each time interval via two steps: state update and measurement

update. The output of the state update step ˆkx

is calculated based on the prior estimated

state 1ˆ

kx

of the previous time step. ˆkx

is then used as an input to compute the estimated

voltage output of the model. The error between the output of the battery and the model

is multiplied by the SPKF gain and the resulting signal is then used to update the

estimated state again in the measurement update step. This final updated state estimate


value ˆkx at each time step is exported to the BMS and is used to calculate the state

estimate of the next time interval. In this way, the ESPM-SPKF is used to estimate the

SOC of the half-cell.

State update

Lithium ion cell

Measurement update

Z-1 SPKF gain

+

++ +

k-1 k-1 k-1f(x ,u ,w ,k - 1), , )k k kg(x ,u v k

ˆk-1

x

1ku

ˆk

x

ˆk

x

kvkw

cellVcellI

errorV

ˆkV

Real system

Battery model and estimation by ESPM-SPKF

Figure 3.8 Diagram of the SPKF algorithm

A summary of the relevant equations in the SPKF algorithm adopted from [73] is as

follows:

; ;a T T T

k k k kx x w v ,

; ;T

a x w T v T

k k k k

,

2 dim a

kp x ,

Initialization at k=0:

0 0x̂ x Ε , T

, , ,

,0 0 0 0 0 ,0ˆ ˆ diag , ,a a a a a

x x w vx x x x

Ε ,

T

,0 0 0 0 0ˆ ˆ

x x x x x

Ε , , T

0 0 0ˆ ˆ ; ;a ax x x w v

Ε ,

For k = 1, 2, …

Step 1: State update


, , , , , ,

1 1 1 , 1 1 , 1ˆ ˆ ˆ, ,a a a a a a

k k k x k k x kx x x

,

, , ,

, 1, 1 1,, , , 1x x w

k i k i k k if u k

,

( ) ,

0 ,ˆ p m x

k i i k ix

,

T

( ) , ,

, 0 , ,ˆ ˆp c x x

x k i i k i k k i kx x

,

Step 2: Output estimate

, ,

, , 1 1,, , ,x v

k i k i k k iY f u k

,

( )

0 ,ˆ p m

k i i k iy Y ,

Step 3: Estimator gain matrix

T( )

, 0 , ,ˆ ˆp c

y k i i k i k k i kY y Y y ,

T( ) ,

, 0 , ,ˆ ˆp c x

xy k i i k i k k i kx Y y

,

, ,k xy k y kL ,

Step 4: Measurement update

ˆ ˆ ˆk k k k kx x L y y ,

T

, , ,x k x k k y k kL L .

where, the detailed definition of the symbols shown in these steps is given in

[73].

3.4 EXPERIMENTAL RESULT AND DISCUSSION

In order to verify the proposed method for estimating the battery SOC using

ESPM-SPKF, the half-cell alluded to earlier is assembled and used for an experimental

study. Since the half-cell is built in the QUT’s battery research facility, it is possible

to obtain cell’s physical parameters such as electrode thickness, electrode plate area,

and active material volume fraction from measurements. The half-cell contains a

LiFePO4 porous electrode, a lithium foil as a reference electrode, a porous separator


and electrolyte. The nominal capacity of the half-cell is 1.679 mAh. The thickness of

the lithium foil used in this half-cell is 0.75mm which is 25 times thicker than the

thickness of the positive electrode which is 0.03mm. Thus, the lithium foil can present

itself as an “inexhaustible lithium source” to the positive electrode. Therefore, the

lithium concentration at the separator and lithium foil boundary can be maintained

constant during the half-cell charge/discharge operations.

After the assembly process, the half-cell is subjected to a preconditioning

charge-discharge operation at low current rate (C/10). This is to allow the formation

of a stable solid electrolyte interphase (SEI) layer. After this preconditioning step, the

half-cell is ready for the main test which will also be emulated in the simulation study.

The half-cell is subjected to a sequence of dynamic current pulses, with the current

rate varying from 1C to 5C, as is shown in Figure 3.10(a). The duration for each

charge/discharge pulse is two minutes. This current pulse profile used in this

experiment will be helpful for verifying the ESPM model for predicting the cell

voltage. The discharge amount is higher than the charge amount in this profile. It

reflects the operation of the battery where the overall discharging is higher than

charging when the battery is to power electric vehicles on the road.

A VMP-300 battery test station from BioLogic Science Instruments, running

EC-Lab Software was used to cycle (charge and discharge) the half-cell. The Galvano

Profile Importation (GPI) function in the EC-Lab Software enables the user to import

the charge-discharge patterns designed with dynamic current pulses as a text file. The

half-cell is kept in a Coin Cell Holder and connected to one channel of the VMP-300

battery test station via a low current probe. A computer controls the charge/discharge

process based on the imported current profile and automatically records cell current

and voltage every second. The experiment setup is shown in Figure 3.9. The proposed

algorithm is implemented in a Matlab program. The measured data is imported into

this program whereby the current data is used as an input to the model and the voltage

is used as the reference for the state estimator. SOC is also calculated by the Coulomb

counting method which integrates the measured current over the time [69]. Since the

initial SOC is known and the error accumulated over the short operation period is

assumed to be negligible, the outcome of the Coulomb counting method can be used


as the reference or the so-called true SOC. In the simulation, incorrect initial SOC

value (60%) is intentionally chosen as the initial value for the SOC estimation to check

whether the proposed algorithm is able to converge to the true SOC value.

Figure 3.9 Experiment setup to test the half-cell

Figure 3.10(b) gives the progress of the SOC reference and the estimated SOC

values during the validation experiment, plotted over time. The estimated SOC by the

ESPM-SPKF algorithm starting at SOC 60% can converge to the SOC reference within

180s and thereafter able to track the SOC reference. This can be seen in Figure 3.10(c),

where the estimation error is shown. Similarly, in the case of estimating the battery

voltage, the voltage estimate starting with a wrong initial voltage value can (due to

wrong initial SOC) converge and track the measured voltage, as illustrated in Figure

3.11(a). The voltage estimated error is shown in Figure 3.11(b).


Figure 3.10 (a) Current rate, (b) comparison between the reference SOC and SOC

reference and (c) SOC estimation error


Figure 3.11 (a) Comparison of the voltage estimation and the measured voltage and

(b) the voltage estimation error.

The open-circuit potential (OCP) vs SOC characteristic of LiFePO4 based

battery exhibits a flat profile over a wide range of SOC, as shows in Figure 3.12. The

OCP-SOC relationship of LiFePO4-based positive electrode presents nearly constant

OCP over the range 20% to 90% of the SOC. OCP of positive electrode is a dominant

part of the full cell OCP. Theoretically it should be straightforward to infer from a

given SOC the corresponding OCP value. However, in practice, the shallow gradient

in the OCP-SOC characteristic introduces difficulty in estimating the SOC of the

battery from a given the OCP value. Traditional method reliant on OCP-based

estimation is thus unsatisfactory for this type of battery material. For example, when

the controller initializes the system or after the system is reset, the terminal voltage of

the battery which is in equilibrium after a resting period is considered as the OCP of

the battery. This value is used to deduce the SOC value from OCP-SOC relationship.


For LiFePO4-based battery, this method is not feasible. A sustained SOC error can be

introduced, without the help of the SPKF algorithm. The proposed method is therefore

most useful for this type of battery material.

Figure 3.12 OCP-SOC relationship of LiFePO4 positive electrode.

Importantly, the computation time of the ESPM-SPKF algorithm is small. The

algorithm is implemented in Matlab and is computed in a desktop computer with Intel

Core i7-6700 CPU running at 3.4GHz and 16GB RAM. The total computational time

required is 2.977s to complete 8,900s of the study period, which is about 0.33ms per

sample. If the sampling time is 1s in real-time application, the proposed ESPM-EKF

algorithm is sufficiently fast to compute for 3,000 cells within one sampling time. In

practice, the cells can be connected in series or parallel to form a battery module. A

battery pack can be configured by many modules and the ESPM-SPKF algorithm can

be used to estimate the SOC of each module instead of each cell, with the assumption

that every cell in the battery module is identical. By doing so, the computational speed

of the proposed algorithm can be increased sufficiently for use in a large battery bank.

Therefore, there is scope for the ESPM-SPKF algorithm to be adopted in real-time

BMS applications for grid-connected battery energy storage systems.


3.5 CONCLUSIONS

This chapter presents a simple electrochemical SPM-3P model, its extension

ESPM and the incorporation of it into a state estimation scheme based on the sigma-

point Kalman Filter (SPKF) algorithm for lithium ion cells. Specifically, the

implementation of the SPM-3P model as a Simulink block has been carried out. This

Simulink block could be a useful tool for the simulations of battery systems such as

those used in electric vehicles or renewable energy applications. Through ESPM, the

accuracy of the SPM-3P is improved by taking into account the variation of electrolyte

potential and current. Finally, a state estimation algorithm for estimating battery SOC

has been proposed by combining the ESPM model with the SPKF estimation scheme.

The results shows that the proposed ESPM-SPKF algorithm can effectively track the

reference SOC and voltage whilst enjoying low computational overheads.

This contribution closes the research gap and meets the requirement from

industry on the management of modern electrical distribution networks which have

high penetration of renewables and battery storage systems. Such an application

requires low computational-burden battery models which can at the same time provide

insightful information of battery SOC and electrochemical variables, as was explained

in Chapter 2.

58 Chapter 4: Padé Approximation of Linearized P2D Model

Chapter 4: Padé Approximation of Linearized

P2D Model

4.1 INTRODUCTION

The batteries used in storage systems for grid applications are designed to have

a high energy density; as opposed to cells that are specifically designed to deliver high

power (like those used in electric vehicle applications [4]). Such high energy density

cells are generally made with thicker electrodes [4], which results in non-uniform,

lithium ion concentration and current density profiles across the electrode [4].

However, in both the SPM and extended SPM the current density is assumed to be

constant, spatially as discussed in Chapter 3. Thus, these models can yield inaccurate

predictions when applied to the thick electrodes of high energy density cells [4].

Accordingly, this chapter describes the attempt to improve on the SPM and

ESPM using on Padé approximation technique. To do this, the transcendental transfer

functions of the linearized P2D model and the corresponding corrections for nonlinear

behaviour given by Lee, Chemistruck and Plett [7] are adopted, with the view to

improve on the accuracy of the linearized P2D model. Padé approximations to these

complicated transcendental transfer functions are applied to develop simpler, rational

polynomial transfer functions, which form a simpler model of cell operation that is

amenable to rapid computation and therefore embedded BMS applications. It is

believed that this is the first attempt at using the Padé approximation method to

simplify all transcendental transfer functions of the linearized P2D model. The

proposed reduced model output includes the spatial and temporal variation of all of the

variables of the P2D model. In addition, it can be used to determine any variable at

any specific spatial location without the need to compute variables at all of the

discretized spatial locations; as is the case when using spatial discretization methods

to solve these models. Importantly, the physical meaning of the approximation model

can be preserved since the coefficients of these rational polynomial transfer functions

Chapter 4: Padé Approximation of Linearized P2D Model 59

directly link to the physical parameters of the cell such as, SEI layer resistance, particle

size of the active material, lithium ion diffusivity and solid and solution phase volume

fractions. If the operating conditions of the cell change (for example, due to cell aging),

which in turn changes the values of the physical parameters, these coefficients are

readily updated as they themselves are in computationally simple algebraic forms.

4.2 ANALYTIC LAPLACE DOMAIN TRANSFER FUNCTIONS OF

LINEARIZED P2D MODEL

Truncated Taylor series expansions about a set point can be applied to each of

the variables of the P2D model to obtain a linear form of the model. The details of this

are given by Lee, Chemistruck and Plett [7]. The set-point, p*, used by these authors is

given by the initial conditions of the cell, defined as

*

, ,0 , ,0 ,0p , , , 0s e OCP s s e s e eU c c c c c j , where s,e s e . These authors also

define a set of transfer functions for the cell, shown in their respective domains in

Figure 4.1, where the dimensionless distance z is related to the spatial variable x via the

transformations, neg/z x L ( neg0 x L ) and tot pos/z L x L ( neg sep totL L x L ).

Subsequently, the “~” notation is appended to the variables , , ,0s e s e OCP sU c , and

, , ,0s e s e sc c c . The transcendental transfer functions are formed by first linearizing the

governing equations of the P2D model around the set point p* and then taking Laplace

transforms of these linearized equations. The details of these transfer functions are as

follows.


Figure 4.1 Schematic diagram of a lithium ion cell embodying the P2D model taken

from [4] and transfer function of each variable in each domain based on [7].

For the phase potential difference neg

s,e in the negative electrode:

neg

negneg negeffs,e

neg negnegapp

neg

eff

1cosh

,

1sinhcosh 1

s zz s L

I s A s ss z

, (4.1)

where:

s,0

12

neg neg negneg neg negs s OCP s

totneg neg neg

eff eff s,e s

tanh

tanhc

a a U Rs L R

c FD

, (4.2)

neg

s neg

s

sR

D (4.3)

and s is the frequency parameter in the Laplace domain.

For the molar flux in the negative electrode:


neg neg

neg neg neg negapp s eff eff

neg neg neg neg

eff eff

neg

,

cosh cosh 1

sinh

J z s s

I s a FAL

s z s z

s

. (4.4)

For the over potential in the negative electrode:

neg neg

neg

ct

app app

, ,z s J z sFR

I s I s

. (4.5)

For the solid phase Li+ concentration on the surface of the particles in the

negative electrode:

neg neg neg

s,e s

neg neg neg neg negapp s s eff eff

neg neg neg neg

eff eff

neg

, tanh

tanh

cosh cosh 1

sinh

C z s s R

I s a FAL D

s z s z

s

, (4.6)

For the potential in solid phase of the negative electrode:

neg neg neg negnegeffs

neg neg neg neg negapp eff eff eff

neg neg neg neg neg

eff

neg neg neg neg neg

eff eff eff

cosh cosh 1,

sinh

1 cosh sinh

sinh

L s s zz s

I s A s s

L s z s s

A s s

. (4.7)

The transfer functions for the electrolyte potential in the positive electrode

pos

s,e app, /z s I s , the molar flux in the positive electrode app, /posJ z s I s , the solid

concentration on the surface of the particle in the positive electrode pos

s,e app, /C z s I s

and the potential in the solid phase of the positive electrode pos

s app, /z s I s , are given

by multiplying the corresponding transfer functions in the negative electrode by (-1)

and substituting the corresponding parameters of the positive electrode.

For the Li+ concentration in the electrolyte:

*

,

1app app

,;

Ke ke

k

k

C sC x sx

I s I s

, (4.8)

where,


* ** neg pos

e,k k k

app app app

1

k

C s J s J s

I s s I s I s

, (4.9)

Here k is the kth eigenvalue, and ; kx the corresponding eigenfunction, for

app, /eC x s I s and K is the order of the expansion [7]. The transfer functions,

*neg

k app/J s I s and *pos

k app/J s I s as well as those for the electrolyte potential in

each domain are given in the Appendix A.

The nonlinear corrections for the internal variables are computed as follows [7]:

ct app

final

s

app

11 1eff

e s,max s,e s,e

, ,

2

2 , , ,

R I tx t z t

a LA

I tRT

F c x t c c x t c x t

, (4.10)

, , ,0, ,s e s e sc x t c z t c , (4.11)

,0, ,e s ec x t c z t c , (4.12)

s,e s,e ,, , ,OCP s ex t z t U c x t , (4.13)

e e e1 1, , 0,x t x t t , (4.14)

0

e

e e 1e

2 1 ,, , ln

0,

RT t c x tx t x t

F c t

. (4.15)

Finally, the cell voltage is calculated from:

pos tot neg

cell film film

pos tot neg

OCP , OCP s,e

tot tot

e

, 0,

, 0,

, , 0,

s e

final final

V t F R j L t R j t

U c L t U c t

L t L t t

. (4.16)

The advantage of these transfer functions is that they model the P2D variables at

any spatial location in the cell domain based only on the applied current appI s . The

transfer functions given in Eq.s (4.1) – (4.9) and Eq.s (7.1) – (7.5) are complicated and

computationally expensive to calculate which, like the P2D model from which they

are derived, makes them not amenable to real time application. The Discrete Time


Realization algorithm (DRA) proposed by Lee, Chemistruck and Plett [35] can reduce

the transcendental transfer functions into an optimized, reduced order, discrete-time,

state-space model. In the DRA, Lee et al. first compute the discrete-time pulse response

of the transcendental, continuous-time, transfer functions (Eq.s (4.1) – (4.9) and Eq.s

(34) – (38)). They then apply the Ho-Kalman algorithm to produce the state-space

realization from this discrete-time impulse response. These authors show that the

resulting simple, state-space model produced from the DRA can reduce the

computation time requirements by more than 5000 times, compared to the P2D model.

However, the DRA requires about 11 min on a desktop computer to produce the

reduced state-space model from the transcendental transfer functions for a given set of

parameters (e.g. filmR , effr ,

sD , eD ,

e and s ). There have been two major

improvements to the classical DRA in terms of reducing the computation overhead, the

first by Gopalakrishnan et al. [82] who implemented an improved singular value

decomposition scheme on a virtual Hankel matrix to speed up the DRA. This resulted

in a reduced order model that was approximately 100 times faster than that using the

classical DRA. The second was by Rodríguez et al. [83] who applied variation of

parameters to speed up the computation of the electrolyte concentration transfer

function. In this case the modified transfer function can be computed 3800 times faster

than the previous electrolyte concentration transfer function given by Lee, Chemistruck

and Plett [7]. The speed up of an overall reduced order model using this modified

transfer function is not considered by Rodríguez et al [83], however. Regardless of these

improvements however, when the values of the parameters change, for example due to

cell aging, the DRA needs to be re-run to regenerate another state-space model. This

results in an increased run time for DRA based reduced order models that may prevent

their practical use in real-time applications where rapid parameter updating is required.

4.3 PADÉ APPROXIMATION MODEL

In order to simplify the above transfer functions, Padé approximations is applied.

The aim of this approach is to reduce the transcendental nature of the above transfer

functions to ones that contain only rational functions of simple polynomials in the


Laplace domain [36]. It is noted that an (M, N) order Padé approximation of a given

transfer function, iG s , centered at 0s has the form [36],

0

0

Mi m

mi m

Ni n

n

n

a s

P s

b s

, (4.17)

where the superscript i associates the quantity with the corresponding transfer

function, namely, phase potential, molar flux, overpotential, lithium concentration and

solid phase potential as given by Eq.s (4.1) – (4.9) above. In Eq. (4.17) the numerator

has order M with M coefficients i

ma and the denominator has order N with N

coefficients i

nb .

The M+N+1 equations used to solve for the coefficients i

ma and i

nb , of iP s can be

determined from the polynomial equation [36],

2

0 0 0

0N M N M

i k i n i m

k n m

k n m

s b s a s

, (4.18)

where the coefficients i

k can be determined from the power series expansion of

the transfer function ( )iG s , namely,

0

1( )

!

ki i

k k

s s

dG s

k ds

. (4.19)

Equation (4.19) generates a polynomial of order N(N+M+2) in s [36]. Its right hand

side equals zero for all s, therefore, the coefficients of s equal zero [36]. Consequently,

a set of M+N+1 linear equations given in Eq. (4.20) are created, which includes N

equations dependent on i

nb and M+1 equations dependent on i

ma and i

nb , namely,

0

0

0 , 1, ...,

, 0, ...,

Ni i

n M j k

n

ki i i

m k m k

m

b k N

b a k M

. (4.20)


where the zeroth order term, 0

ib , is assumed to be equal to 1 to normalize the

solutions [36]. Solving the equations in Eq. (4.20), the M+1 coefficients,

, ( 0,..., )i

ma m M , and N coefficients, , ( 1,..., )i

nb n N can be obtained [37, 84].

There are several readily available routines that automate the above methodology.

In this chapter the 0PadeApproximant , , ,iG s s M N function in Wolfram

Mathematica [85] is used to determine the Padé approximations to the transfer

functions.

As a specific example it is noted that the transfer function of the molar flux in the

negative electrode, app, /negJ z s I s , given by Eq. (4.4), can be approximated to first

order by the Padé approximation,

neg neg

neg neg

neg

1 0

app 1 0

, J J

J J

J z s a s a

I s b s b

, (4.21)

where,

neg

neg

s,0neg neg neg

eff s s totneg

s,e1 s

neg neg neg neg neg neg 2

eff eff eff eff eff eff

2neg2

effneg neg neg 2

s s sneg neg

eff eff

( )4 5

3

2 6 3

4 5 23

13 10 4 1 5 2

J

neg

OCP cR D FR

ca R

z z

z zR a D FL

z z

z z

, (4.22)

neg

neg

,0 neg neg neg neg neg neg neg neg 2

0 s eff eff eff eff eff eff eff

,

( )180 2 6 3

sJ

s e

OCP ca D z z

c

, (4.23)

neg

neg

,0neg neg neg

eff s s totneg neg

,1 s s

neg neg neg neg neg neg 2

eff eff eff eff eff eff

n

effneg neg neg neg 2 neg neg

s s s s eff eff

( )12 5

2 6 3

s

J negs e

neg neg

OCP cR D FR

cb Aa FL R

z z

Aa FL R a D FL

2eg 2

neg 2 2

eff

8 15 2

7 15 2

z z

z z

, (4.24)

neg

neg negnegeff effs,0neg neg neg neg neg

0 s s eff eff neg neg neg 2

s,e eff eff eff

2( )180

6 3

JOCP c

b Aa D F Lc z z

. (4.25)


Similarly, the remaining transcendental transfer functions, introduced above, can

be approximated by their rational polynomial, Padé approximants, given in Figure 4.2.

In this way a system of transfer functions is developed that represent the variables of

the P2D model for a lithium ion battery and which have the advantage of being easy

to implement and compute in a real-time application. In addition, the physical meaning

of the approximation model can be preserved since the coefficients of these rational

polynomial transfer functions directly link to the physical parameters of the cell such

as filmR , effr ,

sD , eD ,

e and s . If the operating conditions of the cell and hence the

values of the physical parameters change, these coefficients are readily updated as they

themselves have computationally simple algebraic forms as shown in Eq.s (4.22) –

(4.25). This is in distinct contrast to the DRA approach discussed above. It is noted

that the outputs of the Padé approximants are used within their corresponding

nonlinear correction term, given by Eq.s (4.10) – (4.15), in order to improve the

accuracy of the approximations to the purely linear P2D model [7].

Figure 4.2 A low order Padé approximation model of the linear P2D model.

Figure 4.3 compares the frequency response of the Padé approximant with their

transcendental transfer function counterparts for the four internal electrochemical

variables within the negative electrode. It is noted that the frequency response of each


set of transfer functions (where a set consists of a number of curves corresponding to

the range of z values between z=0 and z=1) depicted in Figure 4.3 is for a particular

set-point p*. A range of p* values for each set is considered, resulting in a family of

sets, with each member of this family corresponding to a different p*. From this family

of sets, the actual p* value that was used to produce a given set shown in Figure 4.3

was the value that maximized the root mean squared error (when compared to the

linearized P2D model). This ensures that the frequency responses of the Padé

approximations shown in Figure 4.3 are the worst case scenarios (i.e. the ones with

maximum error when compared to the linearized P2D model).

Figure 4.3 Frequency response of the Padé approximants and transcendental transfer

functions of the four variables within the negative electrode.

It can be observed that the error between the Padé approximants and the

transcendental transfer functions increases as the input frequency is increased.

However, it is noted that in grid applications low frequencies dominate the battery


input current profile as shown in Figure 4.4. Figure 4.4(a) depicts the current profile,

reported by Li et al. [86], of a battery storage system that is being used to smooth the

power generated from a wind farm connected to a grid. Figure 4.4(b) shows the fast

Fourier transform of the current signal in Figure 4.4(a). Figure 4.4(c) plots the Power

Spectrum Density (PSD) and identifies 99% occupied bandwidth [87], which is

calculated by determining where the integrated power crosses 0.5% and 99.5% of the

total power in the spectrum [87]. As shown in the Figure 4.4(c), 99% of the occupied

bandwidth of the power signal have frequencies less than 8.225 mHz. Comparing the

Padé approximants with the linearized P2D model outputs in the mHz frequency in

Figure 4.3 it is observed that the two responses show an acceptable match, suggesting

that it is reasonable to apply the low order Padé approximants for grid scale

applications.


Figure 4.4 Current input and corresponding frequency content of a battery storage

system that is being used to smooth the power generated from a wind farm connected to a

grid. Current data sourced from [86].

4.4 RESULT AND DISCUSSION

Comparisons of Padé approximants of various cell variables with those obtained

from the full P2D model under various applied current conditions are given in Figure

4.5, Figure 4.6 and Figure 4.7. The P2D model has been implemented using the open

source code supplied by Torchio et al. [87]. The cell parameters used for these

simulations were obtained from [7] and are listed in Table 4.1.


Table 4.1 Cell parameters used in the simulation


electrode Separator

Positive

electrode

Li μm 128 76 190

sR μm 12.5 _ 8.5

A 2m 1 1 1

1Sm 100 _ 3.8

s _ 0.471 _ 0.297

e _ 0.357 0.724 0.444

brug _ 1.5 1.5 1.5

max

sc 3molm

26,390 _ 22,860

,0ec 3molm

2000 2000 2000

,mini _ 0.05 _ 0.78

,maxi _ 0.53 _ 0.17

sD 2 1m s 3.9E-14 _ 1.0E-13

eD 2 1m s 7.5E-11 7.5E-11 7.5E-11

0t _ 0.363 0.363 0.363

k 1/2 5/2 1mol m s 1.94E-11 _ 2.16E-11

_ 0.5 0.5 0.5

filmR 2m 0 _ 0

eff eff brug eff brug

e e e, ,s eD D

2 4 7 2 10 3 14 4

e e e e4.1253x10 5.007x10 4.7212 10 1.5094 10 1.6018 10ec c c c c

neg

OCP 0.16 1.32exp 3 10.0exp 2000U

pos

OCP 0.4924656

6

14.19829 0.0565661 tanh 14.5546 8.60942 0.0275479 1.90111

0.998432

0.157123 exp 0.04738 0.810239 exp 40 0.133875

U

Figure 4.5 compares the electrolyte concentration ,e totc L t , 0,ec t and the surface

concentration in the solid phase , ,s e totc L t , , 0,s ec t and cell voltage for a pulse current

profile. Figure 4.5 shows a similar comparison when the Urban Dynamometer Driving

Schedule (UDDS) current profile is applied. Figure 4.6 shows the comparison when


the input current is that of the wind farm, grid application obtained from [86], as

depicted in Figure 4.4(a).

Figure 4.5 Comparison between the Padé approximation model and the P2D model for

a pulse current profile


Figure 4.6 Comparison between the Padé approximation model and the P2D model for

the UDDS applied current profile.


Figure 4.7 Comparison between the Padé approximation model with the full

(nonlinear) P2D model for the applied current profile associated with the wind farm

application, as depicted in Figure 4.4(a).

In all of these cases the Padé approximations compare very well to the P2D model

outputs. The computation times (in seconds) for each of these applied current profiles

is shown in Table 4.2. It is noted that all simulations were implemented in MATLAB

R2016b and run on the same desktop computer with an Intel Core i7-6700 CPU

running at 3.4GHz and 16GB RAM. In addition, the Padé approximants are

determined at the same discrete spatial locations as given by the FVM grid used to


solve the P2D model. It is observed that the Padé approximations require

approximately 9.6s to simulate the pulse current profile in comparison to the P2D

model, which takes approximately 12s. The time savings of the Padé approximations

are more evident in the cases of the UDDS and grid current profiles, with the Padé

approximation model solving in approximately 7 and 42.5s, respectively and the P2D

model taking approximately 565 and 22,494s, respectively. It is this feature of the

reduced model coupled with the fact that the form of the Padé approximations, once

obtained, only require relatively simple rational functions to be evaluated, which leads

to the significant computational savings observed in Table 4.2.

Table 4.2 Comparison of simulation time of the proposed approximation model to

compute all cell variables at all discrete spatial locations

Simulation time (s) Pulse current

profile UDDS

Grid current

profile

Proposed

approximation model 9.546 s 7.198 s 42.519 s

P2D model 15.316 s 565.823 s 22494.1 s

Like the P2D model, the Padé approximation model can be used to predict how

important cell variables will evolve both spatially and temporally. However, another

significant advantage of the Padé approximation approach is that variables can be

evaluated at discrete locations within the domain, without the need to compute all

values of the variable at all discrete locations, as is the case with the FVM

implementation of the P2D model. This can result in further significant time savings.

For instance, as shown in Eq. 4.16, the transfer functions of

, e, , , , , , ,s e ej x t c x t x t c x t and ,x t at only two locations, totx L and 0x , are

required to compute the cell voltage. If only the cell voltage is calculated then, as

shown in Table 4.3, the computation time of the Padé approximation model reduces to

approximately 4.7s for the pulse current profile, and approximately 4.5 and 29.1s for

the UDDS and grid-scale current profiles, respectively.


Table 4.3 Comparison of simulation time of the proposed approximation model to

compute only cell voltage

Simulation time (s) Pulse current

profile UDDS Grid current profile

Proposed

approximation model

4.726 s 4.5322 s 29.063 s

P2D model 15.316 s 565.823 s 22494.1 s

The application of the proposed reduced model to battery charging is now

considered. The traditional constant current-constant voltage (CC-CV) charging

method is widely used to charge lithium batteries [88]. As the name suggests, this

method applies constant current (CC) to charge the battery to a preset voltage at which

time the charging mode switches to a constant voltage (CV) regime until the current

reduces to a preset value. The CV regime is often lengthy resulting in a long charging

time and in addition, as the battery ages, the preset voltage limit for the CC regime

often results in the cell being overcharged [4]. When the cell is being overcharged, it

can be damaged and the risk of explosion is potentially increased [89]. A faster and

safer method of charging has been reported by Chaturvedi et al. [89], and is based on

observing the value of the potential difference s e at the negative

electrode/separator boundary ( negx L in Figure 4.1). It is noted that at this point,

s e , will be a minimum within the negative electrode during charging [90]. Thus,

when s e at negx L is negligible (point B shown on Figure 4.8(a)), charging should

be stopped to avoid overcharge and this is the reason for monitoring this point in the

approach presented by Chaturvedi et al [89]. Figure 4.8(a) depicts the cell voltage and

Figure 4.8 (b) depicts the over potential, s e , as given by the Padé and P2D models

for a single charge step at a rate of 1C. In the voltage limit method, the cell voltage is

monitored and the charging process is switched from constant current charging to a

constant voltage mode when the cell voltage reaches 4.2V. From Figure 4.8(a), at the

point of switching (point A, 4.2V) this corresponds to a state of charge (SOC) of

93.2%. This SOC will continue to increase under the constant voltage regime, but will


do so very slowly. From Figure 4.8(b), however, at the point of switching the potential

difference s e at negx L is greater than 0V. Therefore, under the method proposed

by Chaturvedi et al. the charger would continue to charge at a constant current until

0,s e yielding a 6.8% extra SOC. The additional time required for gaining 6.8%

extra SOC in this constant current regime is 243s, which is much faster than the time

required to gain the same amount of charge if the charger had switched to a constant

voltage regime (which would take on the order of hours). The pivotal point here is that

the values of s e at negx L predicted by the Padé approximation model compare

very favorably to those predicted by the P2D model but have the advantage of being

easily computed. An accurate and fast prediction of the s e values at negx L , which

cannot be easily measured empirically, means that the Padé approximation method

could be useful for control purposes in fast charging.

Figure 4.8 Comparison between two charging methods (a). Cell voltage during

charging. (b). The over potential s e at negx L


Finally, it is noted that the accuracy of the proposed Padé approximation model

can be further improved by increasing the order of the rational functions used in the

approximations. However, higher order models result in higher computational

requirements. In order to investigate this tradeoff, the approximation order is increased

as shown in Figure 4.9.

Figure 4.9 Higher order Padé approximation model of the linear P2D model

Figure 4.10 depicts the voltage error (when compared with the P2D model

predictions) for the low order approximations given in Figure 4.2, and the higher order

approximations given in Figure 4.9 for the three previously applied current profiles

(i.e. the pulse current profile, UDDS, grid-scale application). Table 4.4 shows the root

mean square value of the voltage error over the length of each current profile. From

these figures it can be observed that the accuracy of the higher order approximation

model increases by some 5.4% and 1.1% for the pulse current profile and the UDDS

profile, respectively. In the pulse current profiles and the grid-scale profile, there are

negligibly small differences between the low and higher order approximations models.

However, as can also be seen in Table 4.3, the computational time requirement of the

higher order approximation model increases by approximately 20% for the pulse and

UDDS current profiles and 7.7% for the grid current profile. It can be observed that


the higher order approximation model does not result in significant improvements for

any of these applied current profiles and therefore, increasing the approximation order

is unnecessary.

Table 4.4 Comparison of RMS error of cell voltage and the computational workload

between low order and higher order Padé approximation models

Pulse current

profile

UDDS current

profile

Grid current

profile

Order selection 1 (low

order)

0.0098 V 0.0091 V 0.00822 V

Order selection 2

(higher order)

0.0093 V 0.0090 V 0.00824 V

Voltage error reduce 5.4% 1.1 % -0.24%

Order selection 1 (low

order)

4.726 s 4.532 s 29.063 s

Order selection 2

(higher order)

6.053 s 5.5 s 31.486 s

Computation increase 21.9 % 17.6 % 7.7 %


Figure 4.10 Cell voltage error obtained from low order and higher order Padé

approximation models when compared to the predictions of the P2D model.

4.5 CONCLUSIONS

In this chapter, a reduced order model for a lithium ion battery is presented in

which Padé approximants were used to simplify complex, transcendental, transfer

functions associated with the linearized, pseudo 2-dimensional (P2D) electrochemical


model of the battery. The resulting transfer functions take the form of simple, rational

polynomial functions, which can be used to compute any variable at any location

within a one-dimensional representation of the battery domain. Corrections for

nonlinear behaviour are also incorporated into the reduced model. The proposed model

substantially reduces the complexity of the P2D model whilst maintaining the

functionality of predicting all of the variables of that model. The form of the Padé

approximations, once resolved, only require relatively simple rational functions to be

evaluated, which leads to significant computational savings when solving the reduced

model. The results demonstrate that the reduced model predictions match very closely

with those obtained from the full (nonlinear) P2D model but in a fraction of the

computational time. Importantly, variables can be evaluated at specific discrete

locations within the domain, without the need to compute all values of the variable at

all discrete locations, as is the case with the FVM implementation of the P2D model.

This can result in further significant time savings and enhance the flexibility of the

model for a variety of applications. This contribution closes the research gap and fulfils

the requirement on grid-connected battery storage systems. The developed

computationally efficient models provide insightful information of electrochemical

variables which therefore fulfils the need of industry. In the next two chapters, this

proposed model and its advantages will be adapted to couple with thermal and aging

effects.

Chapter 5: Coupled electrochemical-thermal model for small-format lithium ion cells 81

Chapter 5: Coupled electrochemical-thermal

model for small-format lithium ion

cells

5.1 INTRODUCTION

One of critical functions of a battery management system is the thermal

management. Information of evolution of cell temperature is essential for handling the

thermal issues including electrical cell unbalance within a battery pack due to

temperature difference and thermal runaway [13].

Previous chapters are concerned with the modelling of the electrochemical processes

within the battery. Thermal issues have not been considered there. In this chapter, the

focus is on developing a simpler electrochemical-thermal model for small-format

cylindrical cells. This model takes advantage of the Padé approximation model that was

developed in Chapter 4. This model can compute the cell voltage by using only rational

polynomial transfer functions at specific locations in the cell’s domain, without the need

to compute all the variables at all discrete locations, as in the case with the finite volume

method implementation of the P2D model [59]. Due to the uniform temperature over

the radius of small-format cylindrical lithium ion cells, in particular the popular type

18650 lithium ion cell, the spatial temperature dependence can be assumed to be

negligible [8]. The simplified thermal model, which neglects the spatial distribution of

temperature along the cell radius, is then coupled with the Padé approximation model

developed in Chapter 4. Consequently, the resulting model has low computation burden

whilst it can perform reasonably well in predicting cell voltage, specific

electrochemical variables at specific cell locations and the evolution of temperature

over the duration of the cell operation. The proposed model is verified by comparing

the results obtained based on this model with that obtained from solving the 1D radial

electrochemical-thermal battery model contained in Comsol.

82 Chapter 5: Coupled electrochemical-thermal model for small-format lithium ion cells

5.2 SIMPLIFIED THERMAL MODEL FOR SMALL-FORMAT

CYLINDRICAL CELLS

The one dimensional radial equation governing temperature distribution along

the cell radius is obtained from [91] as:

( , ) 1 ( , )

( , )p gen

T r t T r tc r q r t

t r r r

, (5.1)

where, 3[ ]kg m is the material density, pc [ ]J kgK is the specific heat and

[ ]W mK is the thermal conductivity. The left hand side of Eq. (5.1) models the

accumulation of thermal energy per unit volume. The first term on the right hand side

of Eq. (5.1) models the heat flux due to the thermal conductivity of the material.

The last term genq is the heat generated per unit volume calculated using Eq. (5.2) [92],

respectively.

gen chem cq q q , (5.2)

where,

cellchem cell cell

c c

OCVI Iq V OCV T

V V T

,

and,

2

collectorc

s c

R Iq

A V .

Assume a

( , )0 , 0

T r tr R

r

(

aR is the outer radius of the cylindrical cell) as

the temperature distribution along the radius in small-format cylindrical cell is

negligible [8]. This yields the simplified thermal model Eq. (5.3).

( , )

( ) ( )p gen conv

T r tc q t q t

t

, (5.3)

where, the rate of convective heat removal per unit volume is given as

c amb

conv

c

hA T Tq

V

. (5.4)


During battery operation, the temperature changes and this in turn impacts the

model parameters such as solid phase diffusion coefficient, reaction rate, electrolyte

diffusion coefficient and the conductivity [8]. The changes in these physicochemical

properties can be calculated based on the Arrhenius’s law on temperature dependency

Eq. (5.5) [8].

1 1

exp aref ref

E

R T T

, (5.5)

where, represents the changing parameter at the temperature T. ref is the

reference value of that parameter at the reference temperature refT . For the purpose of

this study, the estimated values of these parameters can be adopted from [93].

The temperature dependence open circuit potential (OCP) of the battery is

approximated by a first-order Taylor series expansion with respect to temperature, as

given by Eq. (5.6) [91]:

ref

ref

OCPOCP OCP T T

T

. (5.6)

5.3 COUPLED ELECTROCHEMICAL THERMAL MODEL FOR SMALL-

FORMAT CELL

The proposed model consists of the aforementioned reduced order

electrochemical model using the Padé approximation method and the simplified

thermal model. The interplay between the electrochemical and thermal models is as

shown in Figure 5.1: the reduced order electrochemical model calculates the variations

of the total heat generation which are inputted to the simplified thermal model.

Rational transfer functions of electrochemical variables at totx L and 0x are

computed to determine the cell voltage and the open circuit voltage OCV . The heat

generation rate is then calculated using Eq. (5.2). Simultaneously, the thermal model,

which is coupled with the electrochemical model, computes the evolution of the cell

temperature which is in turn fed back to the reduced-order electrochemical model. The

model is then implemented in Simulink using a similar approach proposed in Chapter


3. It is noted that the present model is simpler than the one that will be discussed in the

next chapter for the large-format cell geometry depicted in Figure 5.2(b). In that work,

each wind is represented by an electrochemical model.

Figure 5.1 A schematic representation of the proposed cell model

Neg Sep Pos

app

,iJ z s

I s

app

,i z s

I s

s,e

app

,iC z s

I s

e

app

,iC z s

I s

q t

T t( )

( )p

T tc q t

t

FET cell model

Applied current Iapp,

Convection coefficient h,

Ambient temperature Tamb

Variation of specific internal

electrochemical variables and

temperature with time

e

app

,i z s

I s


Figure 5.2 (a) Cross-section view of a small-format cylindrical cell which shows

negligible temperature variation along the cell’s radius, (b) Cross-section view of a large-

format cylindrical cell considered in [94] which shows significant temperature derivation

along the cell’s radius

5.4 RESULT AND DISCUSSION

In order to verify the accuracy of the model, the results obtained using the

proposed model are compared with that using the 1D radial electrochemical thermal

PDE model implemented in Comsol Multiphysics software. Figure 5.3 and Figure 5.4

show the outcome of the comparison on the computed temperature variation, the

voltage and other specific variables such as lithium concentration in the solid phase

and electrolyte phase for 18650 cell. The test cases correspond to that during a 1C

charge/discharge cycle and Urban Dynamometer Driving Schedule (UDDS) cycle

[12]. The time-step used in the simulation is 1s.


In both cases, the results obtained using the proposed model compare favorably

with that using the Comsol 1D radial electrochemical-thermal PDE model. All the

simulations were carried out on the same desktop computer with an Intel Core i7-6700

CPU running at 3.4GHz and 16GB RAM. The computation times for each of the test

cases are shown in Table 5.1. It is observed that the proposed model requires

approximately 2s to simulate a charge/discharge cycle, in comparison to the Comsol

model which takes approximately 6 minutes. The time saving of the proposed model

is even more significant in the case of the UDDS test whereby the proposed model

took approximately 2s to complete the simulation while the Comsol model required

some 40 minutes. It is also noted that the reported computation time of the Comsol

model shown in Table 5.1 is based on extremely coarse physics-controlled meshes,

with built-in parameter sets in Comsol. The computational time will be even higher in

the case of finer meshes. It is this feature of the proposed model which combines the

reduced order electrochemical model with the Padé approximations and the simplified

thermal model which leads to the significant computational savings. In addition, as

shown in Figure 5.3 and Figure 5.4, it is observed that the relative small temperature

differences radially under different convection coefficient h values (h=10W/m2/K

represents natural convection and higher values represent forced convection conditions

[95]). This observation confirms that the assumption of uniform temperature is

reasonable. Therefore the proposed model can be used for small-format cylindrical

cells.

Table 5.1 Comparison of simulation time

Simulation time (s) Proposed

model

1D radial P2D

model

1C discharge-rest-charge h=5 W/m2/K 2.21s 6min 14s

h=10 W/m2/K 2.44s 6min 23s

h=100 W/m2/K 2.16s 6min 26s

UDDS h=5 W/m2/K 2.48s 39min 37s


Simulation time (s) Proposed

model

1D radial P2D

model

h=10 W/m2/K 2.43s 33min 17s

h=100 W/m2/K 2.29s 40min 33s

In the case of a large-format cell such as that with the geometry reported in [94]

where the cell diameter is 45mm, the present study shows that the proposed model

fails to predict the cell temperature accurately. High convection coefficients induces a

large divergence of temperature between the inner and outer winds of the cell [94], as

illustrated in Figure 5.2(b). This fact contradicts the assumption made in developing

the proposed model in that a uniform temperature profile is assumed along the cell

radius. Therefore, large error in the predicted temperature can be observed in case of

large format cell, as shown in Figure 5.5 and Figure 5.6. Consequently, for small-

format cylindrical cell, particularly the 18650 cell, the proposed model would be useful

as it provides favourably accurate temperature prediction whilst requires low

computation burden. For large format cell, a more sophisticated electrochemical-

thermal model would be more suitable.


Figure 5.3 Comparison between the proposed cell model and the rigorous 1D radial

PDE model in Comsol using1C charging/discharging of a small-format cell (18650 cell)


Figure 5.4 Comparison between the proposed cell model and the rigorous 1D radial

PDE model in Comsol using UDDS current profile of a small-format cell (18650 cell)


Figure 5.5 Temperature error between the proposed cell model and the Comsol 1D

radial PDE model: 1C charging/discharging of a large-format cell.

Figure 5.6 Temperature error between the proposed cell model and the rigorous 1D

radial PDE model in Comsol using UDDS current profile of a large-format cell.


5.5 CONCLUSIONS

This chapter presents a computationally efficient electrochemical-thermal model

developed for small-format cylindrical lithium ion cells. It is shown that the proposed

model is able to predict the behaviour of the cells satisfactorily. In comparison to the

1-D radial PDE model, the proposed model enjoys a much reduced computational

burden which enhances its applicability for a variety of real-time battery management

applications. It is noted that the application of the proposed model in this chapter is

limited for small-format cells due to the assumption of uniform temperature over the

cell radius. In large-format cylindrical lithium ion cells, the temperature profiles are

no longer uniform and hence, the assumption used in this chapter is no longer valid.

It is also noted that the main contribution of this chapter is to develop a low

burden computation electrochemical-thermal model for small-formatted cells. This

model can allow users to avoid the use of a more complex model that is developed and

discussed in the next chapter. In that chapter, a sophisticated electrochemical-thermal-

degradation model will be developed, which accounts for the variation of the

temperature and degradation along the radius of large-format cylindrical cells.

92 Chapter 6: A Computationally Efficient Electrochemical-thermal-degradation Model for

Large-format Lithium Ion Cells

Chapter 6: A Computationally Efficient

Electrochemical-thermal-degradation

Model for Large-format Lithium Ion

Cells

6.1 INTRODUCTION

Chapter 5 discussed a simple model for small-format cylindrical cells, however,

it is not suitable for large format cylindrical batteries in which non-uniformities in

temperature and degradation occur within the cells [16]. Furthermore, such non-

uniformities exacerbate further degradation and temperature gradients within the

battery. Therefore, a sophisticated model is required to represent these dynamics.

Coupled thermal, degradation and electrochemical models are uncommon,

however, Tanim et al. [15] do present a SPM coupled with thermal and SEI models.

The result is a 1D spatial model that ignores non-uniform distributions of temperature

and degradation along the radius of the cell, which, given their dimensions, does not

seem reasonable for large-format cylindrical cells. Smith et al. [16], present an

empirical degradation model coupled with a multi-dimensional, multi-scale (MD-MS)

cell model for large format cylindrical batteries. However, PDE-based models are

computationally expensive and are used in battery design rather than real-time, optimal

control situations. These models are too complex to be implemented in real time for

most control algorithms [96].

In this chapter, an efficient 1-dimensional, radial, coupled degradation-

electrochemical-thermal model of a spirally wound, cylindrical lithium ion battery,

noted hereafter as the DET model, is proposed. The DET model couples previous

reduced order, electrochemical model proposed in Chapter 4 with an approximate

Chapter 6: A Computationally Efficient Electrochemical-thermal-degradation Model for Large-format

Lithium Ion Cells 93

thermal profile for the cell, based on the steady state solution of the one-dimensional,

radial, heat equation, and an SEI growth model [66]. The model accounts for the central,

electrolyte filled, hollow core and the spirally wound material. The proposed DET

model is computationally efficient whilst maintaining a high degree of accuracy

(compared with that obtained when using the P2D model) in simulating the radial

temperature and degradation distributions within the cell over time. It is a predictive

modelling tool that can be applied to a wide variety of battery applications. The model

is novel in that it applies to large-format, spirally wound, cylindrical batteries and

accounts for non-uniform degradation and thermal behavior coupled to electrochemical

phenomena. The proposed model is able to provide insight into the variation and

evolution of the local temperature and degradation rate of each individual wind along

the cell radius. For the first time, this insight information is computed using the

underlying physics of degradation, rather than the fitting of empirical models.

This chapter is organized in the following way. The next section introduces the

model structure and briefly describe the P2D model and the previous reduced order,

electrochemical model. In the third section, the approximation for the radial

temperature distribution in the cell is developed. Then, the fourth section shows how to

couple this approximation to the reduced order model of each wind in the cell. Finally,

results are presented in fifth section where the proposed DET model is compared to the

full, one-dimensional, radial implementation of the P2D model implemented in Comsol

Multiphysics®.

6.2 MODEL DOMAIN AND REDUCED ORDERS ELECTROCHEMICAL

MODEL IN EACH WIND

6.2.1 Model domain

A schematic of a cylindrical lithium ion cell and its cross-section is given in Figure

6.1(a). The cell has an outer radius aR and inner radius

0R . The cell structure includes

a centrally located hollow core, which is filled with electrolyte. The wound (or “jelly

roll”) structure of the cell consists of repeated electrode, separator and current collector



layers [61] as shown in Figure 6.1(b). A single “wind” is also defined in Figure 6.1(b) and

consists of a sandwich of 8 layers beginning with the negative current collector and

ending with the negative electrode.

+

- +

neg

s

app

( , )

( )

z s

I s

neg

s,e

app

( , )

( )

C z s

I s

neg

app

( , )

( )

J z s

I s

sep

e

app

( , )

( )

C z s

I s

sep

e

app

( , )

( )

z s

I s

pos

s

app

( , )

( )

z s

I s

pos

s,e

app

( , )

( )

C z s

I s

pos ( , )

( )app

J z s

I s

Neg Pos Pos Neg Neg Pos Pos Neg

neg

e

app

( , )

( )

C z s

I s

neg

e

app

( , )

( )

z s

I s

pos

e

app

( , )

( )

z s

I s

pos

e

app

( , )

( )

C z s

I s

wind ith

Neg Sep Posz=0

...

Ra

R0

r

+- - +

SEI layer

,iT r t ,i kT r t

neg ( , )iq r tsep ( , )iq r t pos ( , )iq r tcollector ( , )iq r t

collector ( , )iq r t

( , ) 1 ( , )( , )p


t r r r

(a)

(b)

(c)

z=1 z=0 z=1 z=1z=0

wind ith+k

Padé model



Figure 6.1 (a) Illustration of a cylindrical lithium ion battery with spirally wound

design and its cross-sectional view, (b) the component layers in each wind and (c) the

domains that constitute half of each wind and the corresponding Padé model transfer

functions [59].

The proposed model framework is multi-scale and includes a macroscopic

domain, characterized by the radial variable, r (a0 Rr ), in which the temperature

distribution is solved and a microscopic domain, characterized by z ( 0 1z ), in which

the coupled electrochemical and degradation models are solved. In the negative

electrode domain, 0z at the electrode-current collector boundary and 1z at the

electrode-separator boundary. In the separator domain, 0z at the negative electrode-

separator boundary and 1z at the separator-positive electrode boundary. In the

positive electrode domain, 1z at the separator-electrode boundary and 0z at the

electrode-current collector boundary.

In this chapter, temperature, ,T r t , is assumed to vary along the radial

coordinate, r, of the cell in such a way that the temperature within each wind (at the

microscale) is uniform in space, however, the temperature across winds (at the

macroscale) can vary. As the temperature in a wind is assumed to be uniform in space,

and given that the applied current density is identically symmetric over each half

domain of the wind, the electrochemical process occurring inside the three main

domains in the first half of the wind (namely, the negative electrode, the separator and

the positive electrode as shown in Figure 6.1(c)) will be identical to the corresponding

ones in three “mirrored” domains in the other half of the wind (namely, the positive

electrode, the separator and the negative electrode). Therefore, it is only required to

solve for the electrochemical variables in one half of the wind as these values are then

“mirrored” in the remaining half of the wind.

For each wind, an electrochemical model and degradation model in sub-domain

z (the microscopic domain) associated with the corresponding ,T r t is solved.

Specifically, the input of each wind is the temperature ,T r t , which affects the

electrochemical processes including the transport of ions, reaction kinetics and SEI



growth. The output of the electrochemical model are a series of heat generation rates,

which in turn form the input of the thermal model in the domain r (macroscopic

domain).

The electrochemical characteristics within each wind is solved using a Padé

approximation based, reduced order, P2D model that was previously developed in

Chapter 4 for five state variables; including the electric potential s ,z t in the solid

electrode, the electric potential e ,z t in the electrolyte, the lithium concentration of

the active material s p, ,c z r t of the positive and negative electrodes, the lithium

concentration e ,c z t in the electrolyte, and the molar fluxes ,j z t of the charge that

flows between the active material in each electrode and electrolyte. In this model Padé

approximations are used to simplify the complicated transcendental transfer functions

that result from the Laplace transform solution of the linearized P2D model [7, 97].

Corrections for nonlinear behavior, as given by Lee, Chemistruck and Plett [7], are

incorporated in the model. This approach reduces the complex nature of the above

transfer functions to ones that contain only rational functions of simple polynomials in

the Laplace domain [36]. In this way a system of transfer functions, that represent the

variables of the P2D model for each wind, is developed. These transfer functions have

the advantage in that they can be easily implemented and are computationally efficient.

Here, they are used to calculate the voltage of each wind and the volume average heat

generation rate in each of the domains shown in Figure 6.1(c). The coupling of these

transfer functions with the temperature and degradation models will be discussed in

Section 6.4, but first these models are introduced in the following sections.

6.2.2 Degradation model for each wind

A SEI layer on the surface of the particles that form the negative electrode is one

of the major causes of capacity loss and impedance rise for lithium ion battery [14]. In

this chapter cell degradation is accounted for by including model equations, based on

those developed previously by Ramadass et al. [66], which describe the growth of the

SEI layer during charging.



The irreversible side reaction, which forms a film at the solid-electrolyte interface

of the negative electrode, is given by,

S 2Li 2e P , (6.1)

where, S is the solvent of the electrolyte and P is the “product” that forms the SEI layer.

The ohmic resistance in the SEI film is related to the thickness of the film,

film ,z t , namely,

film

film film

,, ,0

p

z tR z t R z

. (6.2)

where,

film

film

,,

p

p

Mz tJ z t

t

, (6.3)

and pM is the molecular weight of the SEI film, p is the density of the film, p is the

conductivity of the film.

Here, film ,J z t denotes the molar flux of the irreversible side reaction Eq. (6.1)

and is expressed as,

0,film filmfilm film, exp ,

i FJ z t z t

F RT

. (6.4)

where 0,filmi is the exchange current density for the side reaction Eq. (6.1), R is the

universal gas constant and F is the Faraday’s constant. The overpotential, film ( , )z t ,

of the irreversible side reaction Eq. (6.1) is calculated by,

film s e neg SEI

film film

, , , ,

, , ,

z t z t z t OCP z t OCP

FR z t J z t J z t

. (6.5)

In this chapter, this degradation model is solved at the microscopic scale and is

coupled with the Padé model and the thermal model. Details of this coupling will be

discussed in Section 6.4.



6.3 THERMAL APPROXIMATION MODEL

In this section an approximation for the radial temperature distribution within the

cell depicted in Figure 6.1 is derived. In Section 6.3.1, the governing equation of heat

conduction, suitable for a 1-dimenasional radial domain, r , is recalled and the heat

generation terms in each microscopic domain, z , are briefly describe. These generation

terms are calculated using the electrochemical variables mentioned in Section 6.2.1. In

Section 6.3.2, an average thermal conductivity for the wound material within the cell is

introduced. This conductivity is used to approximate the governing thermal equation as

shown in Section 6.3.3.

6.3.1 Governing equation of the thermal model and heat generation terms

It is known that temperature variations occurring along the radius of cylindrical

cells are caused by heat conduction in the radial and azimuthal directions [60]. Chen et

al [95] reported that the heat flow in the azimuthal direction of cylindrical lithium ion

cells that have a high number (20 or more) of winds can be negligible [95]. They

showed that when the number of winds is high, heat flow is predominantly radial and

therefore a 1-dimensional, radial model can be used in preference to a 2-dimensional

model to accurately represent the temperature distribution of such cells [95]. In this

chapter, that finding is adopted and a 1-dimensional, radial, thermal model is used to

calculate the temperature distribution within the cell.

The governing equation for the radial temperature distribution within the cell is

given by [91],

( , ) 1 ( , )

( , )p


t r r r

, (6.6)

where, 3[kg m ] is the material density, pc [ J kgK] is the specific heat, [ W mK]

is the thermal conductivity. Boundary conditions for Eq. (6.6) are shown in Figure 6.2

as BC1, BC2 and BC3.

The average heat generation rate per unit volume ( , )q r t 3[ W m ] in each domain

of each wind consists of reversible and irreversible heating, Joule heating in the solid

and solution and resistive heating at the electrode/current collector interface. These



heating terms are calculated using the electrochemical variables that have been solved

using Padé model at the microscopic scale as described in the Section 6.2.1.

Irreversible heating, iq , is due to electrochemical chemical reactions and is given

by [8],

i s ( , ) ( , )q a Fj z t z t . (6.7)

where,sa is the specific interfacial area.

Reversible heating, rq , results from a change in entropy, namely [8],

r s ( , )OCP

q a Fj z t TT

. (6.8)

where OCP is the Open Circuit Potential of the of the electrode.

Joule heating in the solid, s

Jq , is a result of ohmic resistances and is given by [8],

2eff

s

2

( , )s

Ji

z tq

zL

. (6.9)

where, eff

s is the effective solid conductivity, is the solid conductivity,s is

the volume fraction of the solid phase, iL is the thickness of the electrode, and

neg,posi .

Joule heating in the electrolyte, e

Jq , is ohmically related to the variation of

electrolyte concentration and potential, namely [8],

eff

2effe De e eJ 2 2

e

( , ) ( , ) ( , )1 1

( , )i i

z t c z t z tq

z c z t z zL L

. (6.10)

where, eff Brug

e is the effective electrolyte conductivity, is the electrolyte ionic

conductivity, e is the electrolyte volume fraction, Brug is the Bruggeman coefficient,

eff

D is defined as eff

D eff 0

e

d ln1 1

d ln

fRTt

F c

, 0t is the transference number, f is the

mean molar activity coefficient, and neg, sep, posi .

The heat generation rate per unit volume, collectorq , from current collector is [8],



2

collectorcollector

c c

( )( , )i

I t Rq r t

AV . (6.11)

where, collectorR is current collector resistance,

cA is the cell surface area, and cV is the

cell volume.

Given the above equations the average heat generation term in each electrode is

calculated as,

1

neg neg neg s, neg e, neg

i

0

( , ) di r J Jq r t q q q q z , (6.12)

1

pos pos pos s, pos e, pos

i

0

( , ) di r J Jq r t q q q q z . (6.13)

whilst the average heat generation term in the separator is calculated only from the

Joule heating in the electrolyte, namely,

1

sep e, sep

0

( , ) di Jq r t q z . (6.14)

The average heat generation rate per unit volume of a wind, q , in Eq. (6.6) is

calculated by summing the products of the average (volumetric) heat generation in

each domain (collectorq for current collector, posq for positive electrode, negq for negative

electrode or sepq for separator) with the volume of that particular domain and dividing

this sum by the total volume of each wind.

6.3.2 Average thermal conductivity for the wound material

In order to simplify Eq. (6.6) in Section 6.3.3, an average thermal conductivity wound

[ W mK] for the wound material (0 aR Rr ) is first introduced as shown in Figure 6.2.

The average conductivity, wound , is expressed in Eq. (6.15) and is based on the concept

of discrete thermal resistances (in each domain of Figure 6.1(c)) in series [98].

Given the temperature difference between the points at layer

i jr r and ir as shown

in Figure 6.2 , the heat transfer rate can be expressed as [98]:



layer

transfer

layer

cell

layer

2

ln

i i j

i

j

i j

i

T r T r rq

H

r r

r

, (6.15)

where, cellH is the cell height, layer

j is the thermal conductivity of each of the domains,

ir is the radius of each wind and layer

jr is the thickness of each of the domains as shown

in Figure 6.2.

Expression (6.15) has a similar form to Ohm’s law in which the potential

difference ( layer

i i jT r T r r ) across a conductor (layer

layer

cell2 lni j

j

i

r rH

r

) between

two points ( layer

i jr r and ir ) is directly proportional to the current ( transfer

iq ) through it.

Therefore, the heat transfer problem in a composite cylinder, which consists of

multiple domains, each with a different thermal conductivity layer

j , can be represented

as a system of thermal resistances connected in series as shown in Figure 6.2. The

thermal resistance of a single domain in the composite cylinder is given as [98]:

layer

wound

layer

ln

2

i j

ii

cell j

r r

rR

H

. (6.16)

The total resistance of the wound material, woundR , can then be expressed as [98]:

layer

wound

layer1

ln

2

i j

ni

i cell j

r r

rR

H

. (6.17)

Treating the whole wound material region as a single layer as shown in Figure

6.2 the average thermal conductivity, wound , can be defined as:

a

wound 0

wound

cell

Rln

R

2 H R

. (6.18)

The average thermal conductivity wound can therefore be obtain as:



a

wound 0

layer

layer1

Rln

R

1ln

ni j

i j i

r r

r

. (6.19)

Figure 6.2 Illustration of cross section of the cylindrical lithium ion battery with

boundary conditions

6.3.3 Approximation of the thermal model

In order to solve the governing PDE in Eq. (6.6), some methods such as finite Fourier

transforms [99] and spatial discretization methods [57, 58] can be used; however, they

0

0 ,T r t

R0Ra

00

0

( , )BC1: 0

r

T r t

r

a

woundwound wound

a

( , )BC2: ( , ) amb

r R

T r th T R t T

r

woundwound0 0 0

0

wound

0 0 0

( , ) ( , )

BC3:

( , ) ( , )

T R t T R t

r r

T R t T R t

wound ,T r t

layer

layer

cell

ln

2

i j

ii

j

r r

rR

H

wound

ir

layer

i jr r

layer

i jT r r

iT r



require high computational workloads [4]. For the feasibility of using DET model in

control design, an approximation profile for Eq. (6.6) is derived, which is based on its

steady state solution.

At steady state ( ( , ) 0T r t t ) and assuming a uniform constant source term q , Eq.

(6.6) becomes,

1 ( )T r

r qr r r

, (6.20)

which has the general solution [98],

2

1 2 0 a( , ) ( )ln ( ), R R4

qT r t r C t r C t r

. (6.21)

Noting Eq. (6.21) an approximation of the temperature within the cell is proposed

and given by,

0 0

wound 2 2

a a 0 a

( , ) ( ), 0 R( , )

( , ) ( ) ( ) R ( ) ln ln R , R R

amb

amb

T r t T D t rT r t

T r t T A t B t r C t r r

. (6.22)

where the initial temperature along the cell radius is equal to the ambient temperature,

ambT , namely,

amb( ,0)T r T . (6.23)

It is noted that the approximation given in Eq. (6.22) satisfies a symmetry boundary

condition at 0r (BC1), namely,

00

0

( , )0

r

T r t

r

. (6.24)

The boundary condition at aRr (BC2) describes the convective transfer of heat at

the outer surface of the cylindrical battery, namely,

a

woundwound wound

a amb

( , )(R , )

r R

T r th T t T

r

, (6.25)

where h 2[W m K ] is the convection coefficient.



Thermal continuity is assumed at the boundary 0Rr (BC3 in Figure 6.2), thus,

wound

wound0 0 00

(R , ) (R , )T t T t

r r

, (6.26)

and

wound

0 0 0(R , ) (R , )T t T t . (6.27)

Substitution of the approximate form of ,T r t given in Eq. (6.22) into the boundary

conditions Eq. (6.25) to Eq. (6.27) gives,

0

0

2 2

0 a 0 a

1 a

a

10 ( )2R ( )

R

( ) ( ) ( ) R R ( ) ln R ln R

1( )2R ( ) ( )

R

B t C t

D t A t B t C t

B t C t hA t

(6.28)

Solving for ( )B t , ( )C t and ( )D t in terms of ( )A t , yields,

2

0

2 2

0 a 0 a

wound a

2

0 a

( ) ( ) ( )2

( ) 1 R R ln R ln R ( ) ( )

( ) ( ) ( )R 1

R R

B t A t A tR

D t A t A t

hC t A t A t

(6.29)

Now, integrating both sides of Eq. (6.6) gives,

0 a 0

0

a a

0 0

R wound

0 00

0 0

woundwound

( , ) ( , )( , ) 1

1 ( , )

R R

p p

R

R R

domain

R R

T r t T r tT r tc rdr c rdr r rdr

t t r r r

T r tr rdr q rdr

r r r

. (6.30)

Substituting Eq. (6.22) and Eq. (6.29) into Eq. (6.30) and integrating, namely,



0

0

0

0

0

2 2

0 ,0 , a a

0

2 2 2N M20

0 ,0 , a a

1 1

( , ) ( , )LHS of Eq.(6.30)

( ) ( )1 ln ln

( ) ( )1 ln ln

2 2 2 2

a

a

j

j

i j

R R

p p

R

R R

p j p

R

p j p

i jr L

T r t T r tc rdr c rdr

t t

dA t dA tc rdr c r R r R rdr

dt dt

RdA t dA t r rc c R r R

dt dt

1

2 2

1 1 2

, a 1 a

N M

0 ,0 0 2 21 1

1 2

, a 1 a

1 ln ln2 2 2

( ),

1 ln ln2 2 2

i j

j

j

r L

i j i j

j p i j

p

i ji j i j

j p i j

r L r Lc R r L R

dA tc R

dt r L r Lc R r L R

0 a a

0 00

2 2N M

1

1 a a

1 1a

1 ( , ) 1 ( , )RHS of Eq.(6.30)

1( ) 2 , .

2 2

R R R

domain

R R

i j i jdomain

j i

i j

T r t T r tr rdr r rdr q rdr

r r r r r r

r L r LA t R R q r t

R

Now defining , , and as:

2 2

1 1 2

, a 1 a

N M

0 ,0 0 2 21 1

1 2

, a 1 a

1 ln ln2 2 2

1 ln ln2 2 2

j

j

i j i j

j p i j

p

i ji j i j

j p i j

r L r Lc R r L R

c R

r L r Lc R r L R

,

wound

a a

a

12R R

R

,

and

2 2N M

1

1 1

,2 2

i j i jdomain

i

i j

r L r Lq r t

, gives,

( )

( )dA t

A tdt

, (6.31)

Here domainq is the heat generation rate of each layer in each wind and is calculated

in Eq. (6.11) - Eq. (6.14), collector, neg, sep, posdomain , N is the number of winds,

neg neg neg neg neg sep pos pos pos sep neg

1 2 collector 3 collector 9 collector collector0, , , ....,L L L L L L L L L L L L L L L ,

M is the number of layers in a wind, neg neg sep pos pos

collector collector, , , , andL L L L L are the thickness of



the negative current collector, negative electrode, separator, positive electrode and

positive current collector, respectively.

Equation (6.31) is the first order ordinary differential equation (ODE) for ( )A t . Solving

Eq. (6.31) for ( )A t then allows for ( )B t , ( )C t and ( )D t to be determined from Eq. (6.29),

which in turn determines the radial temperature distribution within the battery

according to Eq. (6.22). The proposed approach considers the thermal properties such

as material density, the specific heat and the thermal conductivity of different layer in

the wounded material; however, it requires low computational workload since there is

only one ODE function of ( )A t needed to be solved.

The parameters within the Padé model are temperature dependent. These

parameters include the solid phase diffusion coefficient sD , the reaction rate , the

electrolyte diffusion coefficient eD and the conductivity . These temperature

dependencies can be calculated using an Arrhenius law [8], namely,

1 1

exp aref ref

E

R T T

, (6.32)

where, represents one of the parameterssD , ,

eD , and , at temperature, T, and

ref is the reference value of that parameter at the reference temperature refT [93].

The temperature dependence in the open circuit potential OCP of the electrode is

approximated by a first-order Taylor series expansion, namely [91],

ref ref OCPOCP OCP T T

T

, (6.33)

where, refOCP is the OCP of the electrode at the reference temperature refT .



6.4 COUPLED DEGRADATION-ELECTROCHEMICAL-THERMAL

MODELS

This section shows how to couple the thermal approximation to the Padé model

and degradation model of each wind in the cell. To illustrate how the DET model is

developed, the cell dimension of the large-format cell reported by Lee et al. in [61] is

used. The cylindrical cell has the inner radius 0R 4mm and the outer radius

aR 22.5mm . The parameters are obtained from [7] and are given in Table 4.1. The

radial geometry of the cylindrical battery consisting of 20 winds with 8 layers in each

wind as illustrated in Figure 6.1. As mentioned in Section 6.2, due to the assumption

that temperature within one wind is uniform in space, electrochemical process in each

wind is represented by one Padé model. Since the negative electrodes and positive

electrodes of the 20 winds connect to the same negative current collector and positive

current collector, respectively, this is modelled by connecting 20 Padé models in

parallel as shown in Figure 6.3. The output voltage and the input current of each Padé

model must satisfy Eq. (6.34) according to Kirchhoff’ circuit laws, namely,

app, app, all

1

cell,1 cell,2 cell,N

( ) ( )

( ) ( ) ... ( )

N

i

i

I t I t

V t V t V t

. (6.34)

The coupling of the degradation, Padé and thermal models is shown

schematically in Figure 6.4. It can be seen that the degradation model at each point z

of each wind is coupled with the Padé model. The heat generation rates collectorq , posq ,

negq and sepq calculated from the electrochemical variables of the Padé model are used

to simulate the temperature variation ,T r t at the macroscopic scale r . In turn the

temperature ,T r t in each wind coupled with the Padé model and the degradation

model. In the large format cell, consisting of 20 winds, there will be 41 coupled “sub-

systems” of equations, 20 of which are Padé sub-systems, 20 are degradation sub-

systems and one of which is a thermal sub-system as shown in Figure 6.4.



Figure 6.3 Parallel connection of Padé models

Figure 6.4 A schematic representation of the DET cell

Wind 1 Wind 2 Wind nth

Iapp,1

Iapp,all

T1 T2 Tn

Vcell,NVcell,1 Vcell,2

Padé model

for wind 1

Padé model

for wind 2

Padé model

for wind nth ......Vcell,1 Vcell,2 Vcell,N

Iapp,all

Iapp,2 Iapp,N

Iapp,1 Iapp,2 Iapp,N

Neg Sep Pos

1

.

q

1T

.

nq

nT

( , ) 1 ( , )p

T r t T r tc r q

t r r r

…

…Wind 1 Wind nth

Neg Sep Pos

DET model

film

film film

film

film,

,, ,0

,,

p

p

k

p

z tR z t R z

Mz tJ z t

t

film,1 1,J T

Degradation model wind 1

Thermal model

Padé approximation model 1 Padé approximation model nth

app

,iJ z s

I s

app

,i z s

I s

s,e

app

,iC z s

I s

e

app

,iC z s

I s

e

app

,i z s

I s

app

,iJ z s

I s

app

,i z s

I s

s,e

app

,iC z s

I s

e

app

,iC z s

I s

e

app

,i z s

I s

film,1 ,max,1, sR c

… …

Degradation model wind nth

film

film film

film

film,

,, ,0

,,

p

p

k

p

z tR z t R z

Mz tJ z t

t

film, ,max,,n s nR cfilm, ,n nJ T



6.5 RESULTS AND DISCUSSION

This section demonstrates the performance of the DET model by comparing its

outcomes to those of a full P2D model that has been modified to include degradation

and thermal behavior. In the full P2D model, instead of approximating the PDE

governing equations as presented in the Sections 2.2, and 6.3.1, the coupled governing

PDE equations are directly solved using COMSOL Multiphysics® [57]. Specifically,

the P2D model coupled with the degradation model mentioned in Section 2.2 is solved

to simulate the electrochemical variables on the microscopic scale. Simultaneously,

Eq. (6.6), which governs the temperature on the macroscopic scale, is solved. Same

parameters, that are used for the DET model simulation implemented in

MATLAB/Simulink® [100], are used in the COMSOL simulation. These parameters

are adapted from [7] and are listed in Table 4.1.

Figure 6.5 and Figure 6.6 show simulation results from the DET model, for the

temperature distribution along the cell radius (here expressed in terms of the wind

number) for different values of the convection coefficient h , plotted over time. Figure

6.5 is for an input current of 1C for a single charge-rest-discharge-rest cycle and Figure

6.6 is for an input current associated with a UDDS (Urbane Dynamometer Driving

Schedule) profile [12]. Both of these current profiles are given explicitly in [59, 101].

From each figure it is observed that larger convection coefficient values h , result in

more non-uniform temperature profiles, spatially. In Figure 6.5 it can be seen that the

temperature difference between the outer wind and the inner wind can be as high as

4°C. Alternatively, low values of h , result in higher cell temperatures over time,

however, the temperature distribution along the cell radius is nearly uniform in these

cases.




coefficients from DET model during a 1C charge-rest-discharge-rest cycle.




coefficients from the DET model during UDDS cycles.

Figure 6.7 and Figure 6.8 show the temperatures at 0Rr and aRr and the cell

voltage over time resulting from the charge-rest-discharge-rest cycle and the UDDS

cycle, respectively, for the DET and P2D models. Curves for different convection

coefficient values are shown. It can be seen that in all of these cases the DET model

results compares favorably to the full P2D model results.

The computation times for each of these applied current profiles is shown in

Table 6.1. All simulations were carried out on the same desktop computer with an Intel

Core i7-6700 CPU running at 3.4GHz and 16GB RAM. I can be observed that the

DET model requires approximately three minutes to simulate the 1C charge-rest-

discharge-rest cycle profile in comparison to the reference model, which takes

approximately eight minutes. The time savings of the DET model are more evident in



the case of the UDDS profiles, with the DET model solving in approximately 90

seconds, and the reference model taking approximately two hours.

Figure 6.7 Comparison of temperature variation between DET model and the full P2D

model in COMSOL at aRr (outer wind) and

0Rr (inner wind) at different

convection coefficients during a 1C charge-rest-discharge-rest cycle.



Figure 6.8 Comparison of temperature variation between DET model and the full P2D

model in COMSOL at ar R (outer wind) and

0r R (inner wind) at different

convection coefficients during UDDS cycles.



Table 6.1 Comparison of simulation time

Simulation time (s) Proposed model 1D radial P2D model

1C discharge-rest-

charge

h=5 W/m2/K 179 s 526 s

h=10 W/m2/K 179 s 526 s

h=50 W/m2/K 187 s 527 s

h=100 W/m2/K 187 s 534 s

UDDS cycles h=5 W/m2/K 91.6 s 2h 12m 8s

h=10 W/m2/K 90.31 s 2h 11m 42s

h=50 W/m2/K 88.6 s 2h15m 6s

h=100 W/m2/K 90.6 s 2h 10m 18s

Figure 6.9 shows the evolution of the degradation at the inner and outer winds

for a periodic 1C charge-rest-discharge-rest current profile and 2100W m Kh . Figure

6.9(c) shows the SEI resistance at 0Rr and

aRr , which can be seen increase over

time. It is also noted that the SEI resistance of the inner wind is higher than that at the

outer wind. This is due to the non-uniform temperature distribution across the cell. The

higher temperatures at the inner radii facilitate the degradation reaction leading to

thicker SEI layers and thus, higher resistances. Figure 6.9(d) shows the maximum

lithium ion concentrations at 0Rr and

aRr , which decrease over time as lithium

ions are removed by the degradation reaction. Furthermore, consistent with

observations for Figure 6.9(c), the maximum concentration in the inner wind decreases

faster than that for the outer wind. Over extended charge cycles it is observed that both

the SEI film resistance and the maximum lithium concentration of the inner wind and

outer wind diverge significantly.

The DET model has the capability of predicting spatial and temporal temperature

and degradation distributions for a spiral wound cell. Based on this information,



optimal control algorithms that minimize, over time, distributions that degrade the

cell’s state-of-health, can be designed. Examples of such algorithms have been given

in the literature for situations where cell temperature is spatially uniform. Offline

trajectory optimization is reported in [102] and online Nonlinear Model Predictive

Control reported in [77, 103] for these simpler cases. The development of such

algorithms for large format cells is beyond the scope of this thesis but will be addressed

in the future works.



Figure 6.9 Evolution SEI resistance filmR and maximum concentration of lithium ion

s,maxc in solid phase in negative electrode of inner and outer winds ( 2100W m Kh ).



6.6 CONCLUSIONS

This chapter presents a new and computationally efficient one-dimensional, radial,

coupled degradation-electrochemical-thermal model of a large format, spirally wound

cylindrical cell. The Padé approximant model proposed in Chapter 4 is employed to

compute the variation of electrochemical variables and heat generation terms in each

wind and couple it with a degradation model. An approximate model for the radial

temperature distribution of the cell is proposed that is, in turn, coupled with the Padé

and degradation models. The model is able to return accurate predictions when

compared to those of a full P2D model, but in a fraction of the computation time.

The development in this chapter helps to fulfil the requirement on battery models

used in designing the management schemes of battery storage systems. The developed

models are seen to be computationally efficient and can provide insightful information

of battery degradation, thermodynamics and electrochemical variables.

118 Chapter 7: Conclusions

Chapter 7: Conclusions

This chapter provides a summary of the thesis, followed by possible research

directions for future work based on the outcomes of this thesis.

7.1 SUMMARY AND DISCUSSION

In this section, the objectives of this thesis, that are outlined in Chapter 1, are

restated and detail provided on how they were achieved throughout the thesis.

Objective 1: Develop a new adaptive electrochemical model-based adaptive

estimation algorithm.

Chapter 3 has described a practical scheme to implement the three-parameter

SPM in Simulink and an adaptive state estimation algorithm. The scheme also has

provided a novel way to implement other reduced order models and sophisticated

coupled models in Simulink as demonstrated in Chapter 5 and 6. For the first time the

ESPM-SPKF algorithm was proposed for adaptive SOC estimation which is a

combination of ESPM and SPKF. An experiment using a half-cell that was built by

the QUT battery lab with lithium foil and LiFePO4 positive electrode was used to

verify the proposed ESPM-SPKF algorithm. The results shown that the ESPM-SPKF

algorithm was able to converge to the reference SOC within a reasonable time and

tracked the reference SOC thereafter. Both predicted cell voltage and SOC from this

algorithm show excellent agreement with the measured data from the experiments.

These contributions fulfilled the research gaps on implementing electrochemical

models in Simulink and using them in battery state estimation.

Objective 2: Develop a new approximate model of the linearized P2D model.

Chapter 4 has developed a new reduced cell model for a lithium ion battery in

which Padé approximants are used to simplify all complex, transcendental, transfer

functions associated with the linearized, pseudo 2-dimensional (P2D) electrochemical

model of the battery. The resulting transfer functions took the form of simple, rational

polynomial functions, which can be used to compute any variable at any location

Chapter 7: Conclusions 119

within a one-dimensional representation of the battery domain. Corrections for

nonlinear behaviour were also incorporated into the reduced model. This proposed

model was validated by comparing results based on the developed model with those

obtained from the full (nonlinear) P2D model solved by finite volume method and

implemented using an open source code reported by Torchio et al. [87]. The results

obtained using the reduced model compare favourably to those from the P2D model

and the computational time required to produce these results was significantly reduced.

Importantly, the form of the reduced model means that variables can be evaluated at

specific discrete locations within the cell domain, without the need to compute all

values of the variable at all discrete locations, as is the case with the spatial

discretization methods most commonly used to implement the P2D model. This results

in further significant time savings and enhances the suitability of the model for a

variety of applications.

Objective 3: Develop a new computationally efficient coupled

electrochemical-thermal model for small-format lithium ion cells.

A computationally efficient coupled electrochemical-thermal model for small-

format cylindrical lithium ion cells has been developed in Chapter 5. The special

feature of the aforementioned Padé approximation model proposed in Chapter 4 was

taken advantage of by coupling it to the thermal model intended for small-format

cylindrical lithium ion cells in which temperature was assumed to be uniform along

the cell radius. This proposed model was validated by comparing the results based on

this model with those obtained using the full, 1-dimensional, radial P2D model

implemented in Comsol Multiphysics®. The results demonstrated that the objective of

developing a fast computational model while maintaining a reasonable level of

accuracy has been achieved. The application of the proposed model is limited for

small-format cylindrical cells, however.

Objective 4: Develop a new sophisticated coupled electrochemical-thermal-

degradation model for large-format lithium ion cells

For the first time a sophisticated coupled electrochemical-thermal-degradation

model for large-format cylindrical lithium ion cells has been developed in Chapter 6.

The model accounts for non-uniform degradation and thermal behaviour coupled to


the electrochemical phenomena. The Padé approximation model proposed in Chapter

4 was used in this chapter to represent the dynamics of electrochemical variables in

each wind. The SEI film growth model reported in [66] was adapted to represent the

degradation process in each wind. A new approximation of the thermal model was also

proposed. This approach considers the thermal properties of different layers in the

wound material. However, there is only one ODE function which needs to be solved,

resulting in a significant computational saving. The practical scheme proposed in

Chapter 3 was used in this chapter to implement the model in Simulink which is

essential for computationally scaling up the cell model to a battery pack system.

Consequently, the proposed coupled electrochemical-thermal-degradation model was

able to provide insight into the variation and evolution of the local temperature and

degradation rate of each individual wind along the cell radius. The results from the

proposed model matched very closely with those obtained from the full, 1-

dimensional, radial implementation of the P2D model implemented in Comsol

Multiphysics® but only require a fraction of its computational time.

7.2 DIRECTIONS FOR FURTHER RESEARCH

Notwithstanding the encouraging progress made so far, there are a number of

fruitful areas for further works.

Direction 1: Identify parameters for electrochemical models

Battery manufacturers do not disclose information of battery physical

parameters for battery users [104]. Using cell tear down and physical tests to measure

these parameters is notoriously difficult and costly. Importantly, specific subsets of

parameters such as health related or kinetic parameters gradually changed as batteries

getting degraded. Therefore, online estimating these parameters presents an important

advance in the field. The identified parameters can be used to update the battery

electrochemical model characteristics based on the real-time operating conditions.

Such these ‘adaptive’ battery electrochemical models can be used to develop robust

fault diagnosis, degradation models which are essential in health-consciously optimal

controls. Future works on identifying specific subsets of parameters, such as health

Chapter 7: Conclusions 121

related or kinetic parameters can be focussed and explored. Hyper-heuristics employed

Particle Swarm Optimization [105] and Simulated Annealing [106] are possible

approaches to identify cell parameters. The cost function of the parameter

identification algorithm is the error between the outputs of the electrochemical model

and the measurable cell input current, output voltage, temperature and impedance.

Using these real-time, easily measurable quantities is a feasible way to implement this

proposed identification algorithm in online applications.

Direction 2: Virtually parallel-like connection for serial-connected batteries

In a battery pack, lithium ion cells are connected in series and parallel to achieve

desired voltage and power levels. Cell-to-cell unbalancing can occur due to various

factors such as cell manufacturing and assembly process, parameter variation during

battery operation, and inhomogeneous aging stress factors [107]. Unbalanced serial-

connected cells in a string without appropriate control schemes can exacerbate further

imbalance and inhomogeneous degradation when the same current rate is applied to

every cell. A novel virtually parallel-like connection can be used to control the current

rate for each serial-connected cell in a string. In a virtually parallel-like connection of

series connected batteries, the series batteries are mathematically modelled as if they

are connected in a parallel connection. These simulated current rates are then used to

control the output currents of cell-level converters so that the lithium ion cells receive

the same current rates that they may receive if they are connected in parallel. By

virtually connecting these serial-connected cells in parallel, their ability to stay

balanced throughout the life of the battery pack can be enhanced. Therefore, the usable

capacity of the battery pack can be fully utilized. It is noted that the high computational

burden may be imposed when dealing with large-capacity battery packs in the

centralized control scheme of the traditional BMS in which a master controller is used

for sensing and computing all cells in the pack. However, this issue can be addressed

by using the decentralized control scheme that currently attracts interest in research

and industry [108, 109]. The impact of this decentralized control scheme is that the

cost may increase as additional Cell Management Units comprising of sensors and

microcontrollers for individual cells are added to the systems. However, benefits can


be gained from the improvement the life time and safety of the battery pack. This trade-

off can be also investigated as part of the future work.

Direction 3: Apply Padé approximation model in fast charging applications

One of significant advantages of the reduced order model using Padé

approximation approach proposed in Chapter 4 is that variables can be evaluated at

discrete locations within the domain. This feature can be used in fast charging

applications since the dynamics of interested variables can be computed and monitored

with low computational burden without the need to compute all values of the variable

at all discrete locations. Future investigation on fast charging scheme can be carried

out to minimize the charging time while ensuring the potential voltage 0s e . This

approach is expected to reduce significantly charging time in comparison to traditional

CC-CV charge method whilst ensuring safe operation of the batteries. In this thesis,

the degradation is modelled by the SEI growth during charging. Fast discharging has

yet to be considered. One challenge in the fast discharging application will be the

higher computational requirement, as a more comprehensive degradation model needs

to be used. This challenge may be addressed in further research.

Direction 4: Optimally control large-format cylindrical cells

The proposed model in Chapter 6 can be used in optimal control scheme such as

optimally controlling of large format batteries in which the cell’s hot spots can be

observed. This is because the proposed model is able to provide information of the

radial distribution of cell temperature and degradation which is essential for such those

applications. By optimally controlling the battery based on the information of the cell

hot spots, the batteries life can be extended and safety of the battery during its

operation can be ensured. Offline trajectory optimization reported in [102] and online

nonlinear control algorithms such as Nonlinear Model Predictive Control reported in

[77, 103] can be possible approaches. Furthermore, the implementation structure of

the model in form of coupled sub-systems makes it feasible to add additional

mechanism such as mechanical stress on solid particles which can also be developed

in this future work.

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Appendices 131

Appendices

Appendix A

The electrolyte concentration transfer functions

The transfer functions used to compute the electrolyte concentration is given as

[7],

* 0 * * neg neg neg negneg1 neg neg eff effk

2 2* neg neg neg neg

appneg eff eff

20 neg neg * neg

1 eff eff neg

2 2* neg neg neg

neg eff eff

1 sin cosh

sinh

1 cos

k t L L s sJ s

I s FA L s s

k t L s

FA L s

, (7.1)

and,

* 0 * * pos pos pos pospos6 pos tot eff effk

2 2* pos pos pos pos

apppos eff eff

0 * * pos neg pos pos

5 pos nm eff eff


pos eff eff

0

6

1 cos cosh

( ) sinh

1 sin cosh

sinh

1

k t L L s sJ s

I s FA L s s

k t L L s s

FA L s s

k t L

* * pos neg pos pos

pos nm eff eff


pos eff eff

0 * * pos pos pos pos

5 pos tot eff eff


pos eff eff

0 pos * pos

5 eff tot eff

cos cosh

sinh

1 sin cosh

sinh

1 cos c

L s s

FA L s s

k t L L s s

FA L s s

k t L

2* pos

nm

2 2* pos pos pos

pos eff eff

20 pos * pos * pos

6 eff tot eff nm

2 2* pos pos pos

pos eff eff

os

1 sin sin

L s

FA L s

k t L L s

FA L s

, (7.2)

where, *

neg sep /n k nL L D , *

pos pos /p k pL L D , *

nm neg sep /p k pL L L D ,

*

tot tot /p k pL L D .

For the electrolyte potential in the negative electrode ( neg0 x L ):

132 Appendices

negnegneg neg negeff

neg neg nege eff1

neg neg neg negapp eff eff

neg neg

neg neg

neg

neg neg neg neg

eff eff

1 cosh sinh,

sinh

cosh cosh

sinh

x sL x s s

x s L

I s A s s

L x sL s

L

A s s

, (7.3)

For the electrolyte potential in the separator ( neg neg sepL x L L ):

negnegneg negeff

sep negsepe eff1

sep neg neg negapp eff eff eff

1 tanh, 2

sL s

x s L x

I s A A s

, (7.4)

For the electrolyte potential in the positive electrode ( neg sep totL L x L ):

negnegneg negeff

pos negsepe eff1

sep neg neg negapp eff eff eff

negpos negeff

neg

eff

neg neg neg neg

eff eff

neg seppos

po

1 tanh, 2

1 cosh

sinh

cosh

sL s

x s L

I s A A s

L s

A s s

L L xL

L

neg

s

neg neg neg neg

eff eff

neg totpos negeff

neg pos

eff

neg neg neg neg

eff eff

neg sep

neg neg

eff eff

sinh

cosh

sinh

s

A s s

L xL s

L

A s s

L L x

A

. (7.5)

Appendices 133

Appendix B

List of symbols

iL Thickness of the layer (m)

sR Particle radius (m)

sa Specific interfacial area ( -1m )

maxs,c Maximum concentration of lithium in the solid phase ( -3mmol )

s,0c Initial concentration of lithium in the solid phase ( -3mmol )

e,0c Steady-state concentration of lithium in the electrolyte phase ( -3mmol )

es,c Surface concentration of lithium in a spherical electrode particle ( -3mmol )

effeD Effective electrolyte diffusivity ( -12 sm )

sD Solid diffusivity ( -12 sm )

f Mean molar activity coefficient

filmR Solid electrolyte interphase layer film resistance ( -1m )

R Universal gas constant (8.314 -1Jmol )

0t Transference number for the anion

sD Solid phase lithium diffusion coefficient ( 12sm )

eD Electrolyte phase lithum ion diffusion coefficient ( 12sm )

eQ Total amount of lithium ion in each domain ( m ol )

pr Radial coordinate across the solid particle radius (m)

effr Reaction rate constant ( 5.1-11-2 smmolsmmol )

A Electrode plate area ( 2m )

T Temperature (K)

F Faraday’s constant (96,478 -1Cmol )

s , e Active material, electrolyte phase volume fraction

Solid phase conductivity ( -1Sm )

Electrolyte phase ionic conductivity ( -1Sm )

134 Appendices

eff Effective electrolyte conductivity ( -1Sm )

a , c Charge transfer coefficients

s s,maxc c Stoichiometry of electrode

min,i Stoichiometry of electrode at 0% state of charge

max,i Stoichiometry of electrode at 100% state of charge

pos/neg/sep Pertaining to the positive/negative electrode/separator

N Number of winds

M Number of layers in a wind

The material density ( 3kg m )

pc The specific heat ( J kgK )

The thermal conductivity [ W mK]

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ELECTROCHEMICAL-BASED ITHIUM Tham_Tran_Thesis.pdf · temperature =10ºC. [18] Reprinted from "Experimental investigation of the lithium-ion battery impedance characteristic at various