modeling, hybridization, and optimal charging of
TRANSCRIPT
Clemson UniversityTigerPrints
All Dissertations Dissertations
8-2016
Modeling, Hybridization, and Optimal Charging ofElectrical Energy Storage SystemsYasha ParviniClemson University
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MODELING, HYBRIDIZATION, AND OPTIMAL CHARGINGOF ELECTRICAL ENERGY STORAGE SYSTEMS
A DissertationPresented to
the Graduate School ofClemson University
In Partial Fulfillmentof the Requirements for the Degree
Doctor of PhilosophyMechanical Engineering
byYasha ParviniAugust 2016
Accepted by:Dr. Ardalan Vahidi, Committee Chair
Dr. John R. WagnerDr. Mohammed Daqaq
Dr. Simona Onori
Abstract
The rising rate of global energy demand alongside the dwindling fossil fuel resources has
motivated research for alternative and sustainable solutions. Within this area of research, electrical
energy storage systems are pivotal in applications including electrified vehicles, renewable power
generation, and electronic devices. The approach of this dissertation is to elucidate the bottlenecks
of integrating supercapacitors and batteries in energy systems and propose solutions by the means
of modeling, control, and experimental techniques.
In the first step, the supercapacitor cell is modeled in order to gain fundamental under-
standing of its electrical and thermal dynamics. The dependence of electrical parameters on state
of charge (SOC), current direction and magnitude (20-200 A), and temperatures ranging from -
40 °C to 60 °C was embedded in this computationally efficient model. The coupled electro-thermal
model was parameterized using specifically designed temporal experiments and then validated by
the application of real world duty cycles.
Driving range is one of the major challenges of electric vehicles compared to combustion
vehicles. In order to shed light on the benefits of hybridizing a lead-acid driven electric vehicle via
supercapacitors, a model was parameterized for the lead-acid battery and combined with the model
already developed for the supercapacitor, to build the hybrid battery-supercapacitor model. A hard-
ware in the loop (HIL) setup consisting of a custom built DC/DC converter, micro-controller (µC)
to implement the power management strategy, 12 V lead-acid battery, and a 16.2 V supercapacitor
module was built to perform the validation experiments.
Charging electrical energy storage systems in an efficient and quick manner, motivated
to solve an optimal control problem with the objective of maximizing the charging efficiency for
ii
supercapacitors, lead-acid, and lithium ion batteries. Pontryagins minimum principle was used to
solve the problems analytically. Efficiency analysis for constant power (CP) and optimal charging
strategies under different charging times (slow and fast) was performed. In case of the lithium
ion battery, the model included the electronic as well as polarization resistance. Furthermore, in
order to investigate the influence of temperature on the internal resistance of the lithium ion battery,
the optimal charging problem for a three state electro-thermal model was solved using dynamic
programming (DP).
The ability to charge electric vehicles is a pace equivalent to fueling a gasoline car will be
a game changer in the widespread acceptability and feasibility of the electric vehicles. Motivated
by the knowledge gained from the optimal charging study, the challenges facing the fast charging
of lithium ion batteries are investigated. In this context, the suitable models for the study of fast
charging, high rate anode materials, and different charging strategies are studied. The side effects
of fast charging such as lithium plating and mechanical failure are also discussed.
This dissertation has targeted some of the most challenging questions in the field of elec-
trical energy storage systems and the reported results are applicable to a wide range of applications
such as in electronic gadgets, medical devices, electricity grid, and electric vehicles.
iii
Dedication
I dedicate this dissertation to my beloved family. Thanks for your endless love and support
in all stages of my life, including my Ph.D. studies.
iv
Acknowledgments
In the course of my Ph.D. studies, I had the privilege to interact, collaborate, and learn from
top notch scientists, researchers, and engineers. This indeed is the true value of my spectacular
Clemson experience.
I would like to begin by expressing my gratitude to my advisor, Dr. Ardalan Vahidi. The
ability to think freely as a researcher to come up with authentic ideas, as well as taking the lead in my
research, are the outstanding qualities that I gained while working under Dr. Vahidi’s supervision.
We had a plan from the first day to prepare me for a job in academia which is realized today, thanks to
the opportunities created by four years of funded research and the privilege to experience teaching,
during my Ph.D. studies. Dr. Vahidi’s student-centered approach in mentoring is the reason of my
great learning experience, I can’t thank you enough.
The motivation to conduct high quality research was boosted by the team work nature of our
group. Thanks to Mr. Alireza Fayazi who contributed to the work on optimal charging of lithium
ion batteries as well as Dr. Nianfeng Wan for all the smart feedbacks and fruitful discussions. The
guidelines I received in my Ph.D. studies from our groups alumni, Dr. Hoseinali Borhan and Dr.
Chen Zhang is fully appreciated.
I would like to recognize the insight and input provided to my research by the Ph.D. com-
mittee members, Dr. Mohammed Daqaq, Dr. Simona Onori, and Dr. John Wagner.
Next, I would like to thank Dr. Anna Stefanopoulou for providing the opportunity to con-
duct collaborative research under her supervision at the Battery Lab during my one year visit at the
University of Michigan. I am grateful to Dr. Jason Siegel from the Battery Lab, for his genuine
support and effective feedbacks. Many thanks to the members of the Battery Lab, Dr. Youngki
v
Kim, Dr. Xinfan Lin, and Mr. Shankar Mohan for all our productive dialogues.
Thanks to Dr. Levi Thompson and Dr. Saemin Choi of Chemical Engineering Department
at the University of Michigan for their collaboration on the hybrid battery supercapacitor work.
I would like to acknowledge the technical support of our government and industry partners,
Dr. Vasilios Tsourapas and Dr. Yi Ding.
Last but not the least, I would like to thank Dr. Gholamreza Vossoughi who mentored
me during my early research career. I was first acquainted to research by late Professor Vahab
Pirouzpanah. His memory will last forever.
vi
Table of Contents
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Supercapacitor Electrical and Thermal Modeling, Identification, and Validation fora Wide Range of Temperature and Power Applications . . . . . . . . . . . . . . . . 82.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Electrical Model of the Supercapacitor . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Frequency Analysis Using Electrochemical Impedance Spectroscopy Measurements 252.5 Thermal Model of the Supercapacitor . . . . . . . . . . . . . . . . . . . . . . . . 292.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Range Extension of a Lead-Acid Battery Driven Vehicle using Supercapacitors: Mod-eling and Hardware In the Loop Validation . . . . . . . . . . . . . . . . . . . . . . . 413.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Energy Storage Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Hybrid Battery-Supercapacitor Energy Storage . . . . . . . . . . . . . . . . . . . 543.4 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 573.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Heuristic versus Optimal Charging of Supercapacitors, Lithium-Ion, and Lead-AcidBatteries: An Efficiency Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . 654.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Charging of the Supercapacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
vii
4.3 Efficiency Analysis for the Supercapacitor . . . . . . . . . . . . . . . . . . . . . . 744.4 Charging of the Lead-Acid Battery . . . . . . . . . . . . . . . . . . . . . . . . . . 764.5 Efficiency Analysis for the Lead-Acid Battery . . . . . . . . . . . . . . . . . . . . 824.6 Charging of the Li-ion Battery . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.7 Efficiency Analysis for the Li-ion Battery . . . . . . . . . . . . . . . . . . . . . . 914.8 Effect of Temperature On Optimal Charging of the Li-ion Battery . . . . . . . . . 924.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5 Fast Charging Challenges in Lithium-Ion Batteries . . . . . . . . . . . . . . . . . . . 1025.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.2 Side Effects of Fast Charging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.3 Suitable Models for Fast Charging . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.4 Controls in Fast Charging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.5 Anode Materials for Fast Charging . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6 Dissertation Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2 Discussions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.3 Dissemination of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
viii
List of Tables
2.1 Supercapacitor cell specification . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The coefficients of the fitted second order polynomial for different temperatures and
current of 191 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 RMS error between model and experiment . . . . . . . . . . . . . . . . . . . . . . 242.4 Physical parameters of the cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.5 Identified thermal parameters at 25 °Cand -20 °Cusing UAC . . . . . . . . . . . . . 39
3.1 Lead acid battery specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2 Supercapacitor specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Estimated parameters for the SC cell . . . . . . . . . . . . . . . . . . . . . . . . . 543.4 Scooter specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5 Range results for the stand-alone battery and HES system . . . . . . . . . . . . . 63
4.1 Supercapacitor cell specification . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Efficiency comparison for the supercapacitor . . . . . . . . . . . . . . . . . . . . 764.3 Lead acid module specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.4 Polynomial coefficients for the OCV and R as a function of SOC . . . . . . . . . 784.5 Efficiency comparison for the lead acid battery . . . . . . . . . . . . . . . . . . . 824.6 Li-ion cell specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.7 Li-ion physical and thermal parameters . . . . . . . . . . . . . . . . . . . . . . . 944.8 Polynomial coefficients for R as a function of Tc . . . . . . . . . . . . . . . . . . 944.9 Unconstrained CC and optimal charging in 10 minutes . . . . . . . . . . . . . . . 984.10 CC and optimal charging in 10 minutes considering voltage and temperature con-
straints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
ix
List of Figures
2.1 The supercapacitor cell connected to the power supply with the thermocouple at-tached to the surface and placed in the thermal chamber. . . . . . . . . . . . . . . 14
2.2 Schematic of the equivalent electric circuit model with “n” number of RC branches. 152.3 Ideal and real OCV for low and high currents versus SOC. . . . . . . . . . . . . . 182.4 Sample pulse-relaxation test at 25 °Cand 191 A current. . . . . . . . . . . . . . . . 192.5 Rs as a function of SOC for all temperatures at each pulse current level during
discharging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6 Rs as a function of SOC for all currents at each temperature during discharging. . 212.7 Supercapacitor cell surface temperature at different ambient temperatures and pulse
current rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.8 Measured average Rs as a function of temperature and current. . . . . . . . . . . . 232.9 Comparison between terminal voltage from experiments and the OCV −Rs −RC
model for the sample test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.10 Estimated R1 as a function of temperature and current. . . . . . . . . . . . . . . . 252.11 Estimated C1 as a function of temperature and current. . . . . . . . . . . . . . . . 262.12 Identified single RC electrical model. . . . . . . . . . . . . . . . . . . . . . . . . 272.13 Nyquist plot of the EIS measurements at -20 °Cand 25 °Cand three SOC levels of
0, 50, and 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.14 Bode plot of the single RC electrical model versus EIS measurements. . . . . . . . 292.15 Given the assumption of uniform heat generation with a convective cooling bound-
ary condition at the surface, the radial temperature distribution can be approximatedas a 4th order polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.16 Sample test for the study of temperature dynamics of the supercapacitor. . . . . . . 342.17 Reversible and irreversible heat generation rates for a test at 25 °C, 140 A, 90 second
rest, and full SOC range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.18 Thermal pulse tests at -20 °Cand 140 A for different SOC ranges and rest periods. . 352.19 Coupling of the electrical and thermal models. . . . . . . . . . . . . . . . . . . . . 372.20 Thermal model parameterization result using the urban assault driving cycle (UAC)
at (a) 25 °Cand (b) -20 °C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.21 Electro-thermal model validation using scaled ECC current profile at (a) 25 °Cand
(b) -20 °C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 Schematic of the active HES with one converter (a) Direct connection of battery tothe load, (b) Direct connection of SC to the load. . . . . . . . . . . . . . . . . . . 45
3.2 Schematic of the equivalent electric circuit model with “n” number of RC branches. 46
x
3.3 Experimental current applied to the module and the corresponding terminal voltagefor obtaining the OCV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 OCV versus SOC for the lead-acid battery. . . . . . . . . . . . . . . . . . . . . . 493.5 Experimental current applied to the lead-acid module and the corresponding termi-
nal voltage for model parameterization. . . . . . . . . . . . . . . . . . . . . . . . 503.6 Parameterization results for the lead-acid battery. . . . . . . . . . . . . . . . . . . 513.7 Parameterization results for the SC cell. . . . . . . . . . . . . . . . . . . . . . . . 533.8 Schematic of the hybrid battery-SC configuration. . . . . . . . . . . . . . . . . . . 553.9 New York City driving cycle and the extracted power profile using the scooter plat-
form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.10 Hybrid battery-SC experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . 593.11 Current and terminal voltage of the stand-alone battery from the HIL experiment. . 603.12 Current and terminal voltage of the battery and SC in the HES system with 0%
regeneration rate from the HIL experiment. . . . . . . . . . . . . . . . . . . . . . 63
4.1 Schematic of the SC model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2 Charging the SC with constant power from zero to full charge in 6 minutes. . . . . 724.3 Charging the SC with constant power from zero to full charge in 30 seconds. . . . . 724.4 The effect of charging time, initial SOC, and range of SOC on SC charging efficiency. 754.5 R versus SOC for the lead-acid battery. . . . . . . . . . . . . . . . . . . . . . . . 774.6 Measured OCV versus SOC for the lead-acid battery. . . . . . . . . . . . . . . . 774.7 Charging the lead-acid battery with constant power from zero to full charge in 1 hour. 794.8 Charging the lead-acid battery with constant power from zero to full charge in 6
minutes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.9 Optimal charging current, SOC, and λ2 profiles for charging the lead-acid battery
from zero to full charge in 1 hour. . . . . . . . . . . . . . . . . . . . . . . . . . . 814.10 Schematic of the single R-C model for the Li-ion battery. . . . . . . . . . . . . . . 834.11 Actual and approximated OCV versus SOC for the Li-ion battery. . . . . . . . . . 844.12 Charging the Li-ion battery with constant power from zero to full charge in 6 minutes. 854.13 Optimal charging current, I1, and SOC for charging the Li-ion battery form zero
charge to full charge in 1 hour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.14 Optimal charging current, I1, and SOC for charging the Li-ion battery form zero
charge to full charge in 6 minutes. . . . . . . . . . . . . . . . . . . . . . . . . . . 904.15 Internal resistance R as a function of core temperature for the Li-ion battery. . . . . 954.16 Illustration of the DP grid and sample transitions. . . . . . . . . . . . . . . . . . . 964.17 CC charging results for charging from 0% to 90% in 10 minutes considering the
temperature and voltage constraints. . . . . . . . . . . . . . . . . . . . . . . . . . 994.18 Optimal charging results via DP for charging from 0% to 90% in 10 minutes con-
sidering the temperature and voltage constraints. . . . . . . . . . . . . . . . . . . . 100
xi
Chapter 1
Introduction
1.1 Overview
The focus of this dissertation is on modeling, hybridization, and optimal charging of elec-
trical energy storage systems. The electrical energy storage systems studied are supercapacitors
as the prominent storage devices in terms of power density, Lithium-ion batteries as the superior
battery chemistry in terms of energy density, and last but not least lead-acid batteries with the no-
ticeable cost advantages. Energy Storage devices play an important role in our everyday life through
facilitating the development of areas such as renewable power generation [1–5], transportation elec-
trification [2, 3], and portable medical and consumer electronic devices. In order to facilitate the
integration of these devices in power applications, we model the electrical and thermal dynamics
of them in a wide range of temperature and loads. Increasing the degree of freedom by integrating
the supercapacitor and lead-acid battery in a hybrid configuration, motivates the study of extending
the vehicle’s driving range which is a well known challenge for electric vehicle commercialization.
Charging accounts for half the life span of storage systems and is an important factor in determining
the lifetime and recharging speeds. Utilizing optimal control theory we shed light on the achievable
charging efficiencies while considering long and short charging times. The above-mentioned three
projects are the building blocks of the completed research in the dissertation.
In the first study a validated lumped and computationally efficient electrical and thermal
1
model for a cylindrical supercapacitor cell is developed. The electrical model is a two state equiva-
lent electric circuit model with three parameters that are identified using temporal experiments. The
dependence of parameters on state of charge (SOC), current direction and magnitude (20-200 A),
and temperatures ranging from -40 °C to 60 °C is incorporated in the model. The thermal model is a
linear 1-D model with two states. The reversible heat generation which is significant in double layer
capacitors is included in the thermal model. The coupling of the two models enables tuning of the
temperature dependent parameters of the electrical model in real time. The coupled electro-thermal
model is validated using real world duty cycles at sub-zero and room temperatures with root mean
square error of (20-87 mV) and (0.17-0.21 °C) for terminal voltage and temperature, respectively.
The second project evolves around studying the circumstances under which, the hybridiza-
tion of a lead-acid powered vehicle using supercapacitors will result in extended range. In the first
step the lead-acid battery equivalent electric circuit model was parameterized using experimental
data and validation was performed. The second step was to integrate the battery model with a
DC/DC converter and an existing supercapacitor model to build the hybrid energy storage (HES)
model. In the hybrid topology the supercapacitor is responsible for providing the power during trac-
tion and capturing the regenerative energy during braking. The battery which is decoupled from the
high frequency and high power dynamics of the load functions as an on-board charger for the su-
percapacitor module. The custom built full-bridge DC/DC converter is located between the battery
and supercapacitor in order to regulate the charging current from the battery to the supercapacitor
according to the power management strategy. Simulations were performed based on a 48 V electric
scooter using city driving cycle and the results were compared with experiments using a hardware
in the loop (HIL) setup specifically built for this study. The hardware in the HIL setup consists
of a 12 volt lead-acid battery, DC/DC converter, micro-controller (µC), and a 16 V supercapacitor
module. The load which includes the vehicle dynamics and the driving cycle is simulated for the
HIL experiments. The results show that by regenerating 20% of the available braking energy the
range will be improved by 10%.
The third research emphasizes on different charging strategies for stand-alone supercapaci-
tors (SC), lithium-ion (Li-ion), and lead-acid batteries. Constant power (CP) and optimal charging
2
strategies are formulated and the corresponding charging currents are obtained. Efficiency analy-
sis for different charging strategies and charging times (slow and fast) is performed. The identical
objective function for all modules is to minimize the resistive losses during a given charging time
by utilizing Pontryagin’s minimum principle. An analytical solution exists for the SC case which is
constant current (CC) charging. In case of the Li-ion battery the model includes the electronic as
well as polarization resistance. The variation of the total internal resistance with SOC in lead-acid
chemistry is considerable and the optimal charging problem results in a two point boundary value
problem with an initial and a final condition to be satisfied. Furthermore, a thermal model is coupled
with the electrical model of the Li-ion battery to study the effect of voltage, current, and temperature
constraints on the charging process.
The fourth study focuses on fast charging challenges and bottlenecks in lithium ion bat-
teries. Suitable models for fast charging, high rate anode materials, and control strategies that
emphasize on reducing the charging time are discussed. The side effects of high current charging
such as lithium plating and mechanical failure are specifically summarized.
Further details of the mentioned research directions are described in the following sections.
1.1.1 Electro-Thermal Modeling of Cylindrical Supercapacitors
High power density and long cycle life beside other features such as wide temperature
operation range, durability in harsh environments, fast and efficient cycling due to low internal re-
sistance, and low maintenance costs makes supercapacitors an attractive electrical energy storage
system for various applications [6]. A model of a supercapacitor is required for any application in
order to simulate the performance of the system while satisfying electrical and thermal constraints.
An equivalent electric circuit model was proposed for the electrical model of the supercapacitor. The
model was parameterized using pulse-relaxation data collected from the experiments conducted on
the cell. The model is valid from -40 °C to 60 °C considering the dependency of the parameters
on temperature, SOC, current direction, and current magnitude. Electrochemical impedance spec-
troscopy (EIS) measurements were used to study the frequency response of the supercapacitor and
to compare it with the electrical model obtained using pulse-relaxation experiments. The thermal
3
model included the reversible (entropic) as well as the irreversible heat generation. The thermal
model consisted of four parameters which were identified as constants using actual current pro-
files. The coupling between the electrical and the thermal model was done by feeding the total heat
generation calculated from the electrical model into the thermal model and tuning the temperature
dependent electrical model parameters according to temperature variation obtained from the ther-
mal model. Finally the electro-thermal model was validated using real world driving cycles. The
comprehensive and validated electrical model developed in this study is used to build the hybrid
supercapacitor-battery model in the next research.
1.1.2 Hybridization of Lead-Acid Battery using Supercapacitors for Range Exten-
sion of a Battery Driven Vehicle
Achieving satisfactory driving range in battery electric vehicles has been a challenge for
scientists and engineers from its early days in nineteenth century. Efforts on improving the energy
density of batteries has led to various secondary type of batteries for traction purposes in electric
vehicles (EV) and hybrid electric vehicles (HEV). Lead-acid, nickel metal hydride, and lithium-ion
batteries are the major chemistries that have been commercialized. The following challenges limit
the wider consumer acceptance of EVs:
• Accelerated battery aging when used in power applications
• Limited charging rate which curtails the amount of regenerative braking energy that can be
captured
• Battery packs for EV and HEVs typically need to be oversized to satisfy the power require-
ments
• Limited vehicle driving range for EVs
In this research, firstly the lead-acid battery’s equivalent electric circuit model is parame-
terized and validated using experimental data. In the second step the battery model is integrated
with the SC model from the above-mentioned research, to build the hybrid energy storage (HES)
4
model. In the HES topology, the SC is responsible for providing all the power during traction and
also capture the regenerated energy from braking. The battery which is decoupled from the high
power dynamics of the load, functions as an on-board charger for the SC module. The full-bridge
DC/DC converter regulates the charging current according to the power management strategy. Sim-
ulations are performed based on a duty cycle derived using an electric scooter platform and a city
driving cycle and the results are compared with experiments using a hardware in the loop (HIL)
setup specifically built for this study. The HIL setup consists of a 12 V lead-acid battery, DC/DC
converter, micro-controller (µC), and a 16.2 V SC module. The results show that by regenerating
20% of the available braking energy the range will improve by 10%.
Regenerative braking as a charging event and the importance of charging efficiency mo-
tivates the third research direction. The developed models from the first and second studies are
directly used in the third project.
1.1.3 Achievable Charging Efficiencies in Supercapacitors and Batteries using Ana-
lytical and Numerical Methods in Optimal Control
Much research and development is spurred towards monitoring and control of the cell charg-
ing method, current rate (C-rate) [7,8], temperature [9–11], number of charging/discharging that are
important factors in studying the efficiency and cycle life of batteries and supercapacitors [12–18].
In the third research the objective is to maximize the charging efficiency by minimizing the resistive
losses in a given charging time and a specified range of SOC. A major performance bottleneck is
the heat generated due to resistive losses, during charge and discharge cycles, reducing the overall
efficiency and shortening the life span [19–21]. Studies that investigate the optimization of effi-
ciency over the entire driving cycle, mask the fundamental efficiency bottlenecks of stand-alone
components such a the electrical energy storage module. This reseach, evaluates different charging
strategies for stand-alone supercapacitors (SC), lithium-ion (Li-ion), and lead-acid batteries. Con-
stant power (CP) and optimal charging strategies are formulated and the corresponding charging
currents are obtained. Efficiency analysis for different charging strategies and charging times (slow
and fast) is performed. The identical objective function for all modules is to minimize the resistive
5
losses during a given charging time by utilizing Pontryagin’s minimum principle. An analytical
solution exists for the SC case which is constant current (CC) charging. The variation of the total
internal resistance with SOC in lead-acid chemistry is considerable and the optimal charging prob-
lem results in a two point boundary value problem. In case of the Li-ion battery the model includes
the electronic as well as polarization resistance. Furthermore, in order to investigate the influence
of temperature on the internal resistance of the Li-ion battery, the optimal charging problem for a
three state electro-thermal model is solved using dynamic programming (DP).
1.1.4 Fast Charging Challenges in Lithium-Ion Batteries
Among the wide variety of energy storage systems the lithium ion battery in the transporta-
tion sector and the pumped hydro storage (PHS) in the electricity grid are the dominant technologies.
In 2014, the rated capacity of the U.S. grid-scale energy storage was 29 GW, where the cumulative
share of all kind of battery storage systems was 542 MW, accounting for only 2% of the total opera-
tional capacity (Source: The U.S. Department of Energy’s Global Energy Storage Database). Within
the battery technologies lithium ion is a promising storage system for use in grid-scale applications
due to its unique technical advantages such as high energy density, high cycling efficiency, low
self-discharge, relatively good cycle life as well as the projected decrease in its cost [22]. However,
the challenges facing the deployment of electrical energy storage technologies such as lithium ion
batteries in various applications, has motivated research and development in both academia and in-
dustry. A number of these barriers are the accelerated aging of lithium ion batteries in high power
applications, low temperature performance limitations, low energy density for applications such as
electric vehicles which results in range anxiety, and finally the limited charging rate that results in
prolonged charging times. The problem of faster charging times is an imperative factor in feasibility
of the technology and user acceptability. This chapter encompasses the research that has focused on
increasing the charging rate of lithium-ion batteries by investigating the pros of using detailed elec-
trochemical models, various control strategies, and high rate electrode materials. The negative side
effects of rapid charging such as lithium plating and mechanical failure are also discussed. Mod-
els consisting of electrode, electrolyte and thermal dynamics, anode materials with nanostructure
6
morphologies, and control strategies based on detailed models and subject to electrical and thermal
constraints; are promising approaches in addressing the fast charging problem.
1.2 Dissertation Outline
The remainder of this dissertation is organized in the following order. In Chapter 2, the
electrothermal modeling, identification, and experimental validation of the SC is presented. Chap-
ter 3 covers the driving range extension study, by introducing the hybrid battery/supercapacitor
model, configuration, power management strategy, and HIL experiments. Chapter 4, presents the
research on optimal charging of SC, lead-acid, and lithium ion batteries using analytical and numer-
ical methods. Chapter 5 includes the review on the fast charging of lithium ion batteries. Chapter 6,
summarizes this dissertation with the conclusion remarks.
7
Chapter 2
Supercapacitor Electrical and Thermal
Modeling, Identification, and Validation
for a Wide Range of Temperature and
Power Applications
2.1 Literature Review
Within different electrical energy storage technologies capacitors and supercapacitors are
interesting because of their unique characteristics and broad application spectrum. Conventional
capacitors are constructed by separating two metallic electrodes by an insulating material called
dielectric. In order to keep these capacitors small for electronic devices the only practical way to in-
crease the energy density is to use an electrical insulator (dry or electrolytic) with a higher dielectric
constant [23–25]. However the capacitance of these capacitors remain in the order of pico to milli
Farads which is not suitable for high energy and power applications. The key to achieve higher
energy density is by increasing the surface area using porous electrodes with an extremely large
internal effective surface [26]. These efforts led to the development of supercapacitors also known
8
as “ultracapacitors” or “electric double layer capacitors” (EDLC). The first EDLC was patented by
H.I. Becker of General Electric [27]. The first commercialized supercapacitor working with double
layer principle was based on the efforts of Robert A. Rightmire and his fellow researcher Donald L.
Boos from the Standard Oil Company of Ohio (SOHIO) [28, 29]. The capacitance of these devices
can be orders of magnitude larger than those of conventional dry and electrolytic capacitors [30].
Carbon is one of the materials of choice for electrode as it combines unique properties such as
high surface-area, conductivity, and temperature stability as well as corrosion resistance with rela-
tively low cost [31]. Metal oxides and polymers are also used as electrode materials [32–34]. The
electrodes are immersed inside the electrolyte which is a liquid providing pure ionic conductivity be-
tween the positive and negative electrodes. The electrolyte could be either aqueous or non-aqueous
depending on the materials and application. Aqueous H2SO4 or KOH electrolytes are common in
supercapacitors because of their higher electrical conductivity. Organic electrolyte such as tetraethy-
lammonium tetrafluoroborate (TEA-BF4) in acetonitrile (AN) or propylene carbonate (PC) are also
used which allows higher cell voltage up to 2.7 V compared to the 1 V in aqueous electrolytes [35].
Separators are used to prevent electric shorting between electrodes. Separators must be porous so
that the ions can pass through them and also stay inert in the supercapacitor cell environment [30].
Supercapacitors are manufactured in cylindrical or prismatic shapes. In the cylindrical design the
positive electrode, negative electrode, and the separator are packed in a spiral wound shape. The
supercapacitor used in this research is a cylindrical cell with activated carbon electrodes and ace-
tonitrile based electrolyte and the charge storage mechanism is based on the electric double layer
established at the electrode-electrolyte interface. Electric double layer capacitors are characterized
by high power density and exceptional cycle life. The high power capability arises from the low
internal resistance as a result of high electrolyte conductivity, low electronic resistance of the elec-
trode, and the interface between electrodes and current collectors [31]. Formation of electric double
layers is due to electrostatic forces and as minimum redox reactions or phase changes occur, the
process is highly reversible providing virtually unlimited cycle life [36].
High power density and long cycle life beside other features such as wide temperature
operation range, durability in harsh environments, fast and efficient cycling due to low internal re-
9
sistance, and low maintenance costs makes supercapacitors an attractive electrical energy storage
system for various applications [6]. Supercapacitors have been utilized in wind power generation for
smoothing fast wind-induced power variations, power leveling of wind farms, and facilitating low
voltage operation of doubly fed induction generator (DFIG) [37–44]. Examples of utilizing super-
capacitors as energy buffers in solar power generation via photovoltaic panels and Stirling engines
are studied in the literature [45–47]. The application of supercapacitors in power system protec-
tion as uninterruptible power supply (UPS) in different fields such as telecommunications is also
reported [48–52]. In [53], Bakhoum et al. introduce a tunable supercapacitor with variable capac-
itance suitable for applications such as ultra-wide-band radio transceivers and electro-mechanical
transducers with high sensitivity. Luo et al. presents supercapacitors as a promising energy storage
for applications in power systems such as transmission and distribution stabilization, voltage reg-
ulation and control and motor stating [54]. In bioengineering and medical devices supercapacitors
are utilized as energy storage units in devices including magnetic resonance imaging (MRI) and as
power supply for laser-based breast cancer detector [55, 56]. In cars, aircraft, and railway vehicles
supercapacitors have been studied as stand-alone storage modules or in combination with batter-
ies or fuel cells [18, 57–69]. Improvement in fuel economy up to 15% is reported for a full-size
SUV where the internal combustion engine is integrated with supercapacitors in a mild hybrid elec-
tric configuration [70]. Having the high efficiency and fast charging capability of supercapacitors
in mind, an optimal charging current is obtained considering the dynamics of the vehicle and the
electric motor during regenerative braking [71]. Utilizing supercapacitors in a hybrid supercapac-
itor/battery configuration and with the goal of increasing battery lifetime, a decrease in life cycle
cost of 23-50% is reported compared to the battery-only electric city bus [72]. Another interesting
application of supercapacitors is boosting the poor low temperature performance of batteries and
also helping the cold start of diesel engines [73–75].
A model of supercapacitor is required in all the above-mentioned applications in order to
simulate the performance of the system while satisfying electrical and thermal constraints. Electrical
models of supercapacitors can be categorized in two groups: (i) models that attempt to mimic all
the physical and chemical phenomenon of charging and discharging which are accurate but not
10
computationally efficient. Continuum models based on Poisson-Nernst-Planck equations, atomistic
models based on molecular dynamics, and quantum models based on electron density functional
theory (DFT) are in this category [76], (ii) models that are suitable for system level studies which are
accurate enough and also suitable for on board and real time applications. Equivalent electric circuit
models that mainly capture the terminal voltage dynamics is an example for this group. In this study
we are interested in introducing a validated model appropriate for real time integration in a system
level. A number of studies have focused on modeling the electrical behaver of supercapacitors in
time and frequency domains by proposing equivalent electric circuit models and their identification
procedure [77–81]. Musolino et al. propose a full frequency range model that captures the self-
discharge and redistribution phenomena [82]. Torregrossa et al. improves the model presented
in [77] that captures the diffusion of the SC residual charges during charging/discharging and rest
phases [83]. In [13], Rizoug et al. use frequency analysis to identify the resistive parameters and
a time domain approach for capacitance characterization. These proposed models are accurate for
fixed temperature operations as the dependence of model parameters on temperature are not taken
into account. The importance of the variation of electrical model parameters with temperature is
studied in [84–86]. Buller et al. uses electrochemical impedance spectroscopy to identify model
parameters considering dependence on four voltage levels and temperatures from -30 °C to 50 °C
[87].
The proposed equivalent circuit model in this dissertation has the following characteristics:
• Terminal voltage dynamics is captured with high accuracy (20-87 mV) suitable for all power
system applications.
• Computationally efficient with only 3 parameters to be identified.
• Carefully designed temporal experiments (pulse-relaxation) are utilized for identification pur-
poses.
• The dependence of model parameters on temperature (range:-40 °C to 60 °C), state of charge
(SOC) (range:0-100), current direction (charge/discharge), and also for the first time, current
magnitude (range: 20-200 A) is investigated.
11
• EIS measurements in the frequency domain at -20 °C and 25 °C are used to validate the model
parameterized in time domain.
• A second validation method is performed by tracking the model performance under dynami-
cally rich and real-world duty cycles rather that simple pulse tests often utilized in the litera-
ture.
On the other hand having the knowledge of how the temperature of a supercapacitor cell
varies helps predict thermal run-aways and its aging behavior [88]. It also enables obtaining models
for cooling management purposes at the stack level [89] and real time tuning of the temperature
dependent parameters of the electrical model [90–93]. The thermal dynamics can be predicted by
numerically solving the governing partial differential equations (PDE) as investigated in [94, 95].
However these complex first principle based thermodynamic models are computationally expensive
and therefore not suitable for real time applications. Utilizing a reduced order model with sufficient
accuracy for power applications is of interest in this study. In [96], Berrueta at al. propose reduced
order electrical and thermal models for a 48 V supercapacitor module and the application of the
electrical model is shown in a micro-grid case study. However the thermal model is over simplified
by neglecting the reversible heat generation effect which is significant in EDLCs [97] and consid-
ering the pack as a whole body (zero-dimensional model) which results in a high reported RMS
error for the thermal model (2.229 °C) [96]. In [98, 99] electrical and thermal models are proposed
and efforts have been made on including the reversible heat generation according to [97], however
clear results on capturing the exothermic effect during charging and endothermic behavior during
discharging are not observable in the papers.
The proposed reduced order thermal model in this study has the following characteristics:
• The thermal model is a linear 1-D model with 2 states.
• Temperature dynamics is captured with high accuracy (0.17-0.21 °C) suitable for thermal
management systems.
• Computationally efficient with only 4 parameters to be identified.
12
• Both the reversible (entropic effect) and irreversible heat generation (joule heating) are inte-
grated in the model.
• Real world duty cycles are used to parametrize the model.
• The thermal model is coupled with the electrical model to capture the changes in the param-
eters of the electrical model that depend on temperature.
• The coupled electro-thermal model is validated using a practical driving cycle rather than
simple constant current cycles often used in the literature.
This chapter is organized in the following order. In section 2.2 the experimental setup
which the parameterization and validation experiments are performed and also the EIS equipment
is described. In section 2.3 the electrical model of the supercapacitor and its parameterization is
presented. Section 2.4 covers the frequency domain analysis using EIS measurements. Section
2.5 describes the thermal model, integration of the reversible heat generation in the model, cou-
pling of the electrical and thermal models, identification results of the model, and the experimental
validation. Finally, section 2.6 concludes the chapter.
2.2 Experimental Setup
Experiments have been conducted on a cylindrical Maxwell BCAP3000 cell with activated
carbon as electrodes. The cell contains non-aqueous electrolyte allowing the maximum rated voltage
of 2.7 V. The specifications of the cell are listed in Table 2.1.
Table 2.1: Supercapacitor cell specification
Parameter ValueNominal Voltage (V) 2.7
Nominal Capacitance (F) 3000Mass (kg) 0.5
Specific Power (Wkg−1) 5900Specific Energy (Whkg−1) 6
13
All the pulse-relaxation experiments related to parametrization and validation of both the
electrical and thermal models are conducted using the following set of equipments:
• Power supply: Bitrode FTV1-200/50/2-60 cycler which is capable of supplying up to 200 A
suitable for running the pulse-relaxation and also the driving cycle validation tests.
• Thermal chamber: Cincinnati sub-zero ZPHS16-3.5-SCT/AC capable of controlling the am-
bient temperatures as low as -40 °C and up to 150 °C.
• Temperature sensor: OMEGA T-type thermocouple which its accuracy is the maximum of
0.5 °C or 0.4% according to technical information provided by the manufacturer.
Fig. 2.1 shows the cell connected to the power supply and placed in the thermal chamber
with the thermocouple attached to the surface of the cell. The cell is suspended to allow uniform air
flow around the cell for a better identification of the convective heat coefficient.
The EIS measurements are obtained using Gamry series G-750 potentiostat/galvanostat
equipment along with the thermal chamber where the cell was located for controlled ambient tem-
perature.
Figure 2.1: The supercapacitor cell connected to the power supply with the thermocouple attached to thesurface and placed in the thermal chamber.
14
2.3 Electrical Model of the Supercapacitor
The equivalent electric circuit model, identifying cell capacitance and open circuit voltage
(OCV ), and also electric model parameter estimation results are presented in this section.
2.3.1 Equivalent Electric Circuit Model
In this study the galvanostatic approach is used in the whole modeling procedure where
current is the input to the system and the output of interest is the terminal voltage. This approach
is also widely used in the battery literature for parameterizing equivalent electric circuit model pa-
rameters [100–102] and using the model in control oriented research such as optimal charging of
lithium-ion and lead-acid batteries [103]. Fig. 2.2 shows the schematic of the proposed equivalent
electric circuit model. It consists of a resistance Rs connected in series to the RC branches. Con-
Figure 2.2: Schematic of the equivalent electric circuit model with “n” number of RC branches.
sidering positive sign for the charging and negative sign for discharging and applying Kirchhoff’s
voltage law (KVL) to the circuit shown in Fig. 2.2, the equation governing the terminal voltage VT,
could be written as follows:
VT = OCV (SOC) + IRs +n∑j=1
VRC,j (2.1)
In equation (3.1), OCV is a linear function of SOC for an ideal capacitor. The SOC is
15
determined by coulomb counting by the following state equation:
dSOC
dt=
I
CVmax(2.2)
where C and Vmax are the nominal capacitance in Farads and the maximum voltage across the cell
at full charge. The second term in (3.1), is the voltage drop over the series resistance and the last
part is the sum of voltage drops across RC branches. Dynamics of each RC pair is described as:
dVRC,j
dt= − 1
RjCjVRC,j +
I
Cj(2.3)
where Rj and Cj are the corresponding resistance and capacitance of each RC branch ,respectively.
2.3.2 Cell Capacitance and Open Circuit Voltage
The OCV and capacitance (C), are the primary modeling parameters to be identified. Ca-
pacitance and capacity are the terms used to determine the amount of electric charge stored in
electrical energy storage systems. Capacitance is the term used for supercapacitors with the unit of
Farads. Equivalently the term capacity is used in the battery literature with the unit of ampere hours
(Ah). The relationship between capacity in Ah and capacitance (C) in Farads is:
capacity =CVmax
3600(2.4)
The capacitance of the cell used in this study is 3000 F which is equivalent to 2.25 Ah. The
term C-rate is the rate at which charging (discharging) is performed. For example charging a cell
with a capacity of 2.25 Ah from zero to full charge with a C-rate of one means, supplying a current
of 2.25 A that results in a charging time of one hour. Similarly the charging current values used
in this study which are 22.5 A, 67.5 A, 135 A, and 191 A are equivalent to C-rates of 10C, 30C,
60C, and 85C respectively. The power supply utilized in this research has a current limit of 200 A,
which is the reason of choosing 191 A as the maximum pulse current applied in the experiments.
The capacitance is obtained by charging the cell from zero to maximum voltage applying a small
16
constant current. The reason for using small current is to minimize the effect of resistive losses
and measure the capacitance more accurately. Constant current assumption allows to integrate (3.2)
using the boundary conditions SOC(0) = SOCi and SOC(tf) = SOCf and to find the capacitance:
C =tfI(t)
Vmax(SOCf − SOCi)(2.5)
where tf is the charging time. SOCf and SOCi are not measured and by definition are one at full
charge (VT = Vmax = 2.7 V) and zero at the empty state (VT = Vmin = 0 V). The charging current used
in the experiment was 0.112 A (equivalent to a C-rate of C20 ) and the time recorded to fully charge
the empty cell was 21.05 hours which according to (3.4) results in a capacitance of 3143 F. Similarly
a constant current of 0.1178 A was applied to discharge the fully charged cell to zero. The recorded
time was 18.73 hours resulting in a capacitance of 2934 F. The reported nominal capacitance by the
manufacturer is 3000 F which is almost the average of the measured capacitance for charging and
discharging. The OCV as a function of SOC is obtained by charging the cell from V0 = 0 V to
Vmax = 2.7 V with a small constant current of 0.45 A (equivalent to a C-rate of C5 ). By applying a
small constant current, the recorded terminal voltage at each time corresponds to the OCV at that
time. The obtained OCV profile is almost identical for a C-rate equal to C5 compared to C
20 , so we
present the results based on the C5 rate, which is a shorter test to run. Using the coulomb counting
method governed by (3.2), SOC is obtained at each time. This provides a profile for OCV as a
function of SOC to be integrated in the supercapacitor model. The difference between charging
and discharging OCV is small due to low current and small equivalent series resistance.
The OCV as a function of SOC is shown in Fig. 2.3. In this figure the ideal linear OCV ,
the measured terminal voltage with a high constant current of 140 A , and the OCV used in this
study are shown. The effect of high current is shown to compare it with the low current measure-
ments. As it can be observed, the high current results in a loss of capacity of 6% as the cell reaches
its maximum allowed voltage in a shorter time. This is the reason of using low current for identify-
ing the capacity and the OCV of the cell. The OCV of the supercapacitor under investigation has a
nonlinear relationship with SOC as shown in experiment-2 in Fig. 2.3. A fourth order polynomial
17
is fitted to the OCV versus SOC data and integrated in the model as follows:
OCV (SOC) = − 0.18(SOC)4 + 0.59(SOC)3 − 1.2(SOC)2
+ 3.5(SOC)− 1.9× 10−4(2.6)
The nonlinearity in the OCV profile is due to the small pseudo-capacity behavior of the cell [104].
Figure 2.3: Ideal and real OCV for low and high currents versus SOC.
2.3.3 Equivalent Electric Circuit Model Identification
The unknown parameters of the equivalent circuit are Rs, Rj, and Cj. In order to record a
rich set of data for identification and also to investigate the dependence of parameters on SOC, cur-
rent magnitude and direction and also temperature, the following set of pulse-relaxation experiments
are performed:
• Experiments are conducted at six temperature levels (-40, -20, 0, 25, 40, 60 °C). The lower
and upper limits for the temperature are the actual limits reported by the manufacturer.
• At each temperature level four pulse current rates (191, 135, 67.5 and 22.5 A) are applied.
• Starting at a fully discharged state the cell is charged by the pulse current to 5% SOC then
18
followed by a 20 second relaxation period. This procedure is repeated for each 5% SOC
interval until the cell is fully charged.
• Similar pulse-relaxation procedure is repeated for discharging immediately at the end of the
charging process.
Fig. 2.4 is one example of the total of 24 different pulse relaxation tests. This test will be
called the “sample test” throughout the dissertation and will be used to present the identification
process and model accuracy investigation. In this specific test the temperature is 25 °C and the
current is 191 A.
Figure 2.4: Sample pulse-relaxation test at 25 °C and 191 A current.
The relaxation period at each SOC level contains the information needed to estimate the
equivalent electric circuit model parameters. The relaxation or rest phase consists of two segments
as depicted by the insets in Fig. 2.4. The first part is a sudden change in terminal voltage at
the moment the current is set to zero. This change is observed by an instant drop in voltage while
charging and a jump in voltage during discharge and is captured byRs in the model. The second part
in the relaxation stage is the exponential behavior in voltage and is modeled by the RC branches.
Rs is obtained by dividing the instant voltage change by the pulse current at each 5% SOC level.
Fig. 2.5 illustrates the variation of Rs during discharge with respect to SOC and temperature at
19
each current level. This figure shows that as the temperature increasesRs decreases regardless of the
current magnitude except for 60 °C. In supercapacitors, the electronic resistance of the electrode and
electrolyte and also interfacial resistance between the electrode and the current-collector contribute
to the amount of Rs [105, 106]. An increase in electrolyte resistance or an increase in the interface
resistance between the electrode and the current collector could contribute to the increase in Rs
at 60 °C. The dependence of Rs on SOC is small according to Fig. 2.5 for all temperatures and
currents except the high current case (191 A). The dependence in this case can be approximated by
fitting a second order polynomial to the data in the following general form:
Rs(SOC) = a(SOC)2 + b(SOC) + c (2.7)
Table 2.2 shows the coefficients for the fitted second order polynomials to the data at dif-
ferent temperatures and current of 191 A.
Table 2.2: The coefficients of the fitted second order polynomial for different temperatures and current of191 A
Temperature (°C) a b c
-40 6.7E-5 -6.5E-5 4.4E-4
-20 1.9E-4 -1.8E-4 4.1E-4
0 1.1E-4 -1.2E-4 3.8E-4
25 1.2E-4 -1.3E-4 3.7E-4
40 5.7E-5 -6.1E-5 3.6E-4
60 4.9E-5 -6.4E-5 4.2E-4
Another representation of Fig. 2.5 is plotted in Fig. 2.6 to be able to observe the dependence
of Rs with respect to the magnitude of pulse current at different temperatures and during discharge
period. The figure indicates that the variation of Rs with the current magnitude is negligible regard-
less of the ambient temperature.
The temperature measurements form the thermocouple attached to the surface of the cell for
all 24 set of pulse-relaxation tests are depicted in Fig. 2.7. This figure shows that applying higher
20
Figure 2.5: Rs as a function of SOC for all temperatures at each pulse current level during discharging.
0 0.2 0.4 0.6 0.8 12
3
4
5x 10
−4
SOC
Rs
Temperature = −40C
0 0.2 0.4 0.6 0.8 12
3
4
5x 10
−4
SOC
Rs
Temperature = −20C
0 0.2 0.4 0.6 0.8 12
3
4
5x 10
−4
SOC
Rs
Temperature = 0C
0 0.2 0.4 0.6 0.8 12
3
4
5x 10
−4
SOC
Rs
Temperature = 25C
191 A
135 A
67.5 A
22.5 A0 0.2 0.4 0.6 0.8 1
2
3
4
5x 10
−4
SOC
Rs
Temperature = 40C
0 0.2 0.4 0.6 0.8 12
3
4
5x 10
−4
SOC
Rs
Temperature = 60C
Figure 2.6: Rs as a function of SOC for all currents at each temperature during discharging.
21
pulse current results in a higher surface temperature at the end of the charging period regardless
of the ambient temperature. The highest observed increase in surface temperature is 1.5 °C at the
highest ambient temperature of 60 °C and at the highest current. According to Fig. 2.5 a 23-30 %
increase in Rs occurs from 40 to -40 °C. This indicates that the effect of a small change in tempera-
ture during the pulse-relaxation tests on Rs is negligible. Another interesting observation from Fig.
2.7 is that during discharging the surface temperature of the cell decreases. This phenomena and
the reasoning will be addressed in the thermal modeling section of the dissertation.
Figure 2.7: Supercapacitor cell surface temperature at different ambient temperatures and pulse current rates.
At this point the knowledge of small variation of Rs with SOC and negligible temperature
effect during each pulse-relaxation test allows to consider a constant value for Rs at each temper-
ature and current level. This is done by taking the average of the Rs with respect to SOC for the
charging and discharging sections separately. The final Rs is obtained by averaging the values ob-
tained from the charging and discharging sections for each pulse relaxation test. Fig. 2.8 shows the
variation of the constant averaged Rs with temperature and current magnitude. This figure indicates
that Rs is highest at -40 °C and that the change of the average Rs with respect to temperature is
higher than that due to current magnitude.
The next step is to identify the values of resistance and capacitance of the RC branches
with the assumption that the parameters are constant and not a function of SOC. Identification of
Rj andCj is performed by minimizing the square error between the measured and simulated terminal
22
Figure 2.8: Measured average Rs as a function of temperature and current.
voltages for each pulse-relaxation experiment to obtain the estimated values for each temperature
level and current magnitude. The cost function to be minimized is:
J =∑k
(Vm(k)− VT(k))2 (2.8)
where Vm and VT are the measured and simulated terminal voltages, respectively. The number of
RC branches will be determined based on the accuracy of the estimation results.
Firstly a simple first order model named OCV − Rs which consists of resistance Rs con-
nected in series to the terminals of the supercapacitor is considered. The single parameter in this
model is Rs which is already identified. The root mean square error (RMSE) between the terminal
voltage from the model and sample test experiment considering the ideal versus the real OCV is
obtained. The RMSE numbers in Table 2.3 show that the real nonlinear OCV profile should be
integrated in the model as the results for the sample test indicate a 50% decrease in RMS error. The
drop in RMSE is also significant for other tests using the nonlinear OCV profile.
In the next step a single RC branch is added in series to the OCV − Rs to build the
OCV −Rs−RC model. Fig. 2.9 compares the terminal voltage results from the OCV −Rs−RC
model and the experimental data from the sample test.
23
Figure 2.9: Comparison between terminal voltage from experiments and the OCV − Rs − RC model forthe sample test.
The result shows that the electrical model with the estimated R1 and C1 accurately predicts
the dynamics of the terminal voltage with a RMSE of 20 mV as listed in Table 2.3. This result shows
that the OCV −Rs −RC model is accurate enough and will be used as the final electrical model.
Table 2.3: RMS error between model and experiment
Model RMSE (mV)
Linear-OCV -Rs 100
Nonlinear-OCV -Rs 50
OCV -Rs-RC 20
By performing a similar procedure used to estimate the electric model parameters for the
sample test, R1 and C1 parameters were identified for the other 23 pulse-relaxation experiments.
Fig. 2.10 and 2.11 summarize the estimated parameters R1 and C1 as a function of all temperature
and current levels, respectively. Similar to Rs that reaches its maximum value at low temperatures,
R1 also shows such a behavior. The resistance in the RC branch represents the polarization re-
sistance which in general is due to kinetic reactions and also diffusion process. In the case of a
double layer supercapacitor with carbon electrodes and organic electrolyte the kinetic reactions are
24
minimum. The charge transfer is mostly based on electrostatic diffusion of ions in the pores of the
electrode material. Similar toRs that reaches its maximum value at low temperatures,R1 also shows
such a behavior. The C1 value corresponds to the double layer capacitance of the supercapacitor.
The value of C1 depends on the surface area of the activated carbon, electrical conductivity of the
electrolyte, and the double-layer effective thickness [84,107]. The variation of any of the mentioned
parameters contribute to the observed higher value of C1 at the current of 135 A. One direction for
future work is to use a more complicated model and also in situ measurements to explain the current
dependency of model parameters in detail.
Figure 2.10: Estimated R1 as a function of temperature and current.
2.4 Frequency Analysis Using Electrochemical Impedance Spectroscopy
Measurements
Fig. 2.12 shows the electric model proposed and identified using pulse-relaxation experi-
ments in the previous section. Considering a linear relationship for theOCV as a function of SOC,
(3.1) can be re-written for the final electrical model as follows:
VT =
∫Idt
C+ IRs + V1 (2.9)
25
Figure 2.11: Estimated C1 as a function of temperature and current.
By taking Laplace transform of both sides of (2.9) and obtaining the ratio of terminal voltage (VT)
as the output to current (I) as the input, the transfer function of the electrical model of the superca-
pacitor is derived as follows:
Z(S) =1
CS+Rs +
R1
R1C1S + 1(2.10)
Substituting S = jω in (2.10), the impedance of the supercapacitor is obtained as a function
of the current frequency (ω). An informative way of representing the impedance dynamics is the
so-called Nyquist diagram which is the imaginary part of the impedance plotted versus the real part,
for a known range of frequencies. The intersection of the Nyquist plot with the real axis represents
Rs in the model. The resistances contributing to the amount ofRs are the electronic resistance of the
electrode material, resistance between the electrode and the current-collector, electrolyte resistance
and the ionic resistance of ions moving through the separator [31]. The dependence of Rs on the
electrode and electrolyte resistance is investigated in [105] showing that the lower amount of active
carbon material used in the electrode and also higher electrolyte conductivity will result in a smaller
Rs. The effect of resistance between the electrode material and the current collector is studied
in [106] illustrating that treating the current collector before applying the coating of active carbon
26
will result is a smaller Rs. The resistance in the RC branch represents the polarization resistance
which in general is due to kinetic reactions and also diffusion process. In the case of a double layer
supercapacitor with carbon electrodes and organic electrolyte the kinetic reactions are minimum.
The charge transfer is mostly based on electrostatic diffusion of ions in the pores of the electrode
material. The C1 value corresponds to the double layer capacitance of the supercapacitor.
Figure 2.12: Identified single RC electrical model.
In order to study the frequency response of the supercapacitor electrochemical impedance
spectroscopy (EIS) experiments were performed. Using the Gamry series G-750 potentiostat/galvanostat
equipment, experiments with the following specifications were conducted:
• Two temperature levels of -20 °C and 25 °C.
• Small input currents of 0.4 A for -20 °C and 0.2 A for 25 °C.
• Frequency window of 50 mHz to 100 kHz.
• The sampling rate is 20 points per each decade of frequency.
• Data is gathered at 0, 20, 40, 50, 60, 80, and 100% SOC at each temperature level.
Fig. 2.13 shows the Nyquist plot for two temperature levels. It shows that Rs is less at
higher temperatures. Also in average the value for Rs from EIS measurements is higher than the
values from pulse-relaxation experiments which is consistent with [13]. The figure also shows that
27
−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Re(Z) (Ω)
-Im(Z)(Ω
)
SOC=0,Temp=25°C
SOC=50,Temp=25°C
SOC=100,Temp=25°C
SOC=0,Temp= −20°C
SOC=50,Temp= −20°C
SOC=100,Temp= −20°C
5 5.5 6 6.5
x 10−4
0
10
20
x 10−6
Figure 2.13: Nyquist plot of the EIS measurements at -20 °C and 25 °C and three SOC levels of 0, 50, and100.
at high frequencies (points closer to the origin) supercapacitor has a pure resistive behavior and at
very low frequencies a mostly capacitive behavior.
In order to depict the dependency of the magnitude and phase of the complex impedance
on frequency explicitly, the Bode plots are presented. Fig. 2.14 compares the magnitude and phase
between the single RC model and the EIS measurements at 25 °C.
Considering the estimated values for Rs, R1, and C1 from the pulse-relaxation sample test the
transfer function for the supercapacitor is:
Z(S) =(S + 0.0005)(S + 0.9989)
S(S + 0.00056)(2.11)
As it can be observed from the phase plot of the model the system begins from a -90 degree
of phase at low frequencies which represents the integrator term in the transfer function. The term
(S + 0.9989) in the numerator of the transfer function affects the system at the corner frequency of
about (ω = 1) causing the phase to approach to zero degree. The effect of the other two terms in the
numerator and denominator of the transfer function is not observable in the bode plot as they almost
cancel out each other and do not effect the impedance. The Bode plots for the EIS measurements
differ from the model at frequencies above 100 rad/s. The increase of the phase angle from zero
28
degrees at 100 rad/s to +90 degrees at higher frequencies indicates an inductive behavior. This
behavior is not captured by the model as the frequency of the pulse-relaxation experiments is less
than 100 rad/s. This is acceptable, as in practical applications the frequency of the current as the
input to the system is much lower than 100 rad/s.
100
102
104
−70
−60
−50
−40
Mag
nitu
de (
dB)
Frequency (rad/s)
Single R−C ModelEIS Measurment
100
102
104
−90
−45
0
45
90
Pha
se (
deg)
Frequency (rad/s)
Figure 2.14: Bode plot of the single RC electrical model versus EIS measurements.
After developing the electrical model and analyzing it in the frequency domain the next
steps are to develop the thermal model of the supercapacitor and to validate the coupled electro-
thermal model.
2.5 Thermal Model of the Supercapacitor
A computationally efficient thermal model developed for cylindrical batteries [108] is mod-
ified and adopted for the thermal modeling of the supercapacitor. In the beginning of this section
the two-state thermal model will be described. In the consecutive sections entropic heat generation
29
and electro-thermal coupling will be introduced. Finally parameterization results are presented.
2.5.0.1 Two State Thermal Model
The model is based on one dimensional heat transfer along the radial direction of a cylinder
with convective heat transfer boundary conditions as illustrated in Fig. 2.15.
Figure 2.15: Given the assumption of uniform heat generation with a convective cooling boundary conditionat the surface, the radial temperature distribution can be approximated as a 4th order polynomial.
A cylindrical supercapacitor, so-called a jelly-roll, is fabricated by rolling a stack of cath-
ode/separator/anode layers. Assuming a symmetric cylinder, constant lumped thermal properties
such as cell density, conduction heat transfer, and specific heat coefficient are used [108]. Uniform
heat generation along the radial direction is a reasonable assumption according to [109]. The tem-
perature distribution in the axial direction is more uniform than the radial due to higher thermal
conductivity [110]. The radial 1-D temperature distribution is governed by the following PDE:
ρcp∂T (r, t)
∂t= kt
∂2T (r, t)
∂r2+kt
r
∂T (r, t)
∂r+Q(t)
Vcell(2.12)
with boundary conditions:
∂T (r, t)
∂r|r=0= 0 (2.13)
30
∂T (r, t)
∂r|r=R= − h
kt(T (R, t)− T∞) (2.14)
where t, ρ, cp and kt are time, volume-averaged density, specific heat, and conduction heat transfer
coefficients, respectively. The heat generation rate inside the cell is Q, The cell volume is Vcell and
R is the radius of the cell. The first boundary condition in (2.13) is to satisfy the symmetric structure
of the cell around the core. Convective heat transfer at the surface of the cell forms the boundary
condition in (2.14). Here T∞ is the ambient air temperature and h is the heat transfer coefficient for
convective cooling.
With uniform heat generation distribution, the solution to (2.12) is assumed to satisfy the
following polynomial temperature distribution as proposed in [111]:
T (r, t) = α1(t) + α2(t)( rR
)2+ α3(t)
( rR
)4(2.15)
The volume-averaged temperature T , and volume averaged temperature gradient γ are cho-
sen as the states of the thermal model. These quantities can be related to the temperature distribution
as follows:
T =2
R2
[ ∫ R
0rT dr
](2.16)
γ =2
R2
[ ∫ R
0r(∂T
∂r) dr
](2.17)
Using (2.15) the surface Ts and core Tc temperature of the cell are expressed by:
Ts = α1(t) + α2(t) + α3(t), Tc = α1(t) (2.18)
The temperature distribution T (r, t) can be written as a function of Ts, T , and γ after some
algebraic manipulation. The original PDE (2.12), can be reduced to a set of two linear ordinary
31
differential equations (ODE) with the state space representation of:
x = Ax+Bu, y = Cx+Du (2.19)
where x = [T γ]>, u = [Q T∞]>, and y = [Tc Ts]> are state, input, and output vectors
respectively. The parameter β = ktρcp
is the thermal diffusivity. Finally the linear system matrices A,
B, C, and D are:
A =
−48βh
r(24kt+rh)−15βh24kt+rh
−320βhr2(24kt+rh)
−120β(4kt+rh)r2(24kt+rh)
B =
β
ktVcell
48βhr(24kt+rh
)
0 320βhr2(24kt+rh)
C =
24kt−3rh24kt+rh
−120rkt+15r2h8(24kt+rh)
24kt24kt+rh
15rkt48kt+2rh
D =
0 4rh
24kt+rh
0 rh24kt+rh
This linear two state model is relatively easier to parameterize compared to the detailed PDE models,
as shown in the following sections.
32
2.5.0.2 Thermal Test Procedure
In order to investigate the dependence of temperature dynamics on current magnitude, depth
of discharge (SOC range) and also relaxation period (rest time), the following set of repeated cy-
cling experiments are performed:
• Experiments are conducted at two temperature levels (-20 and 25 °C).
• At each temperature level three current levels (140, 100, and 50 A) are applied.
• Two SOC ranges (0-100% and 50-100%) and two resting times (90 and zero seconds) are
studied.
• In each set of experiments a fully discharged or a half charged cell undergoes cycles of charge-
rest-discharge until the surface temperature of the cell reaches steady state, followed by a long
rest period till the surface temperature relaxes to its initial value.
Fig. 2.16 shows the surface temperature and voltage evolution under a pulse test with
current of 140 A. In this test, the cell is cycled in the upper half of the voltage range (50-100%
SOC) with an ambient temperature of -20 °C and 90 seconds of rest between every charge and
discharge period. The surface temperature reaches steady state after approximately 1 hour.
2.5.1 Irreversible and Reversible Heat Generation
The total heat generation consists of two parts. The first contributor to the total heat gener-
ation is the ohmic losses. These losses are due to the internal resistance of the cell. This irreversible
joule heating effect is associated with the losses in Rs and R1 as follows:
Qjoule = RsI2 +
V 21R1
(2.20)
where V1 is the voltage across R1 in the single RC branch.
The inclusion of reversible heat generation in the model which is the second contributor
to the total heat generation is required to accurately predict the dynamic temperature response for
33
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−20
−19
−18
−17
−16
−15
Cel
l Tem
pera
ture
(C
)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−200
0
200
Cur
rent
(A)
TemperatureCurrent
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100001
1.5
2
2.5
3
Time (s)
Vol
tage
(V
)
Figure 2.16: Sample test for the study of temperature dynamics of the supercapacitor.
cycling over different SOC ranges at both low and high currents and temperatures. The reversible
(entropic) heat generation rate is governed by [97]:
Qrev = δT I(t) (2.21)
The reversible heat generation rate is proportional to current and the volume average tem-
perature T with the unit of Kelvin (K) [97]. The constant of proportionality δ is related to the
physical properties of the cell and will be estimated from the temperature measurements. As an
example, Fig. 2.17 shows the reversible and irreversible heat generation profiles for the specific test
at 140 A, 25 °C with a 90 second rest period and charging the cell from zero to full charge. The first
two subplots show the applied current profiles and the measured surface temperature. The inset in
the heat generation plot at the steady state region shows that the reversible heat generation rate is
proportional to the magnitude and direction of the current while the irreversible heat generation rate
is always positive and almost a linear function of time. Integrating the heat generation rates during
charging and discharging at the steady state region, the reversible heat generation is calculated to be
616 joules compared to the 392 joules of the irreversible heat generation.
34
Figure 2.17: Reversible and irreversible heat generation rates for a test at 25 °C, 140 A, 90 second rest, andfull SOC range.
0 2000 4000 6000 8000 10000 12000 14000 16000 18000−20
−18
−16
−14
−12
−10
−8
Time (S)
Tem
pera
ture
( o C
)
SOC(50−100),Rest=90sSOC(0−100),Rest=90sSOC(50−100),Rest=0sSOC(0−100),Rest=0s
50505100515052005250
−9.2
−9
−8.8
−8.6
Figure 2.18: Thermal pulse tests at -20 °C and 140 A for different SOC ranges and rest periods.
35
Fig. 2.18 shows a set of four experiments at -20 °C and pulse current of 140 A that highlights
the effect of rest period between the charge and discharge cycle and depth of discharge on the
dynamics of temperature rise. All profiles show an overall increase in the cell temperature during
the test which is due to the irreversible heat generation. The effect of entropy on the temperature
rise is only observable when there is a rest. In this case the RMS current is lower and therefore
Qjoule is smaller. While in the case without any rest between charging and discharging the average
steady state temperature is not a function of SOC range. A closer look at the inset in Fig. 2.18
will also reveal that there is an obvious non-linearity in the temperature ripple. This is due to the
reversible heat generation which is a result of entropy change during charge and discharge period.
Considering the ions in the supercapacitor as the system of interest, the change of entropy of this
system from state 1 to state 2 is [112]:
∆S = −∫ 2
1
dQrev
T= −Cpln(
T2
T1) (2.22)
where Cp is the heat capacitance of the double layer supercapacitor. Entropy can be interpreted as
a measure for disorder in a system. This means that the higher the level of disorder in a system, the
higher the entropy. During charging as the ions move to the surface of the electrodes the disorder
of the system of ions is decreasing, therefore the entropy decreases. According to (2.22), for ∆S
to be negative (decreasing entropy) T2 should be greater that T1 which explains the increase in
temperature during charging. While discharging, the disorder level of ions is increasing as they
spread out in the electrolyte randomly similar to an ideal gas [97]. This results in an increase in the
entropy (∆S > 0) which dictates a decrease in temperature (T2 < T1) according to (2.22) clarifying
the observed cooling effect during discharge.
2.5.2 Coupling of the Electrical and Thermal Models
The electrical and thermal models are coupled to form the complete system model. The total
heat generation rate is calculated from the equivalent circuit model, and the temperature (which is
the output of the thermal model) feeds back into the resistance of the equivalent circuit. Fig. 2.19 is
36
the schematic of the coupled electro-thermal model.
Figure 2.19: Coupling of the electrical and thermal models.
2.5.3 Parameterization Results for the Thermal Model
The convective h and conductive kt heat coefficients, specific heat coefficient Cp, and the
parameter in (2.21) associated with reversible heat generation (δ) are the parameters to be identified
for the thermal model. The urban assault cycle (UAC) which is used to generate the battery current
profile [113] for a heavy vehicle (13.4 ton) and also utilized in parametrization of battery thermal
models in [108] and [100] was scaled up by a factor of six to generate an input current profile with
sufficient excitation for supercapacitor applications. Also a relaxation period of one hour is added to
the data set which results in useful temperature relaxation data for parameterizing the heat capacity
and coefficient of convective cooling in the model. The physical parameters of the cell that are
measurable are summarized in Table 3.3.
Table 2.4: Physical parameters of the cell
Mass (kg) Length (m) Radius (m) Volume (m3) Density (kgm−3)
0.51 0.138 0.0304 4E-4 1277
Parameter estimation is performed by minimizing the square error between the measured
37
(Tm) and simulated (Ts) surface temperatures. The cost function to be minimized is:
J =∑k
(Tm(k)− Ts(k))2 (2.23)
Fig. 2.20 compares the modeled surface temperature with experimental measurements per-
formed on the cell using the UAC at 25 °C and sub-zero temperature of -20 °C. The histogram of
the temperature error is also shown for both temperatures. The RMS error is 0.13 °C and 0.11 °C
for 25 °C , and -20 °C respectively, which is an indicator of the estimation accuracy.
Figure 2.20: Thermal model parameterization result using the urban assault driving cycle (UAC) at (a) 25 °Cand (b) -20 °C.
Table 3.4 shows the values of the identified thermal model parameters for both 25 °C and
-20 °C. The value estimated for h is in the range of forced convective heat transfer coefficient for air
which is between 10 to 200 W/(m2K). The thermal chamber consists of a fan inside it which helps
regulate the temperature to the preset value. The estimated value of h will depend on the fan being
on or off during the experiment which is the reason for different values for h at two temperatures
of 25 °C and -20 °C. The specific heat coefficient values is close to the amount of the cell’s organic
based electrolyte (Acetonitrile cp=1863 J/kgK at 25 °C). The thermal conductivity values are a result
of the combined thermal conductivity of activated carbon, electrolyte, separator and the aluminum
current collectors formed in a jelly roll shape. The value of 0.0004 for δ at 25 °C is comparable to
38
0.00033 JCoulomb−1K−1 reported in [97] for a 2.7 V, 5000 F prismatic cell with organic electrolyte
at room temperature.
Table 2.5: Identified thermal parameters at 25 °C and -20 °C using UAC
h cp kt δ
(Wm−2K−1) (Jkg−1K−1) (Wm−1K−1) (JCoulomb−1K−1)
T∞
25 157 1259 0.49 4E−4
-20 26 1480 0.74 2.3E−4
2.5.4 Electro-Thermal Model Validation
The supercapacitor was tested under a different current profile, the escort convoy cycle
(ECC). ECC which is a current profile for batteries [108], was scaled up by a factor of four and used
as the second driving cycle to validate the thermal model and also the electrical model. Under this
current profile at 25 °C and -20 °C the terminal voltage and the surface temperature of the cell are
measured. The identified parameters obtained from the parameterization procedures are fixed and
the terminal voltage and surface temperature from the model are compared to the actual measure-
ments. Fig. 2.21 shows that both the electrical and thermal models mimic the actual measurements
of voltage and temperature with good accuracy at both 25 °C and -20 °C. The histogram of the volt-
age and temperature errors are also shown. The RMS error for the terminal voltage are 82 mV and
87 mV for 25 °C and -20 °C, respectively. The surface temperature RMS error is 0.17 °C for 25 °C
and 0.21 °C for -20 °C.
2.6 Conclusions
In this dissertation an equivalent electric circuit model was proposed for the electrical model
of a supercapacitor. The model was parameterized using pulse-relaxation data from the experiments
conducted on the cell. The model is valid from -40 °C to 60 °C considering the dependency of the
parameters on temperature, SOC, current direction, and current magnitude. The final electrical
39
Figure 2.21: Electro-thermal model validation using scaled ECC current profile at (a) 25 °C and (b) -20 °C.
model consists of three parameters and the analysis shows that the parameters are more a function
of temperature than SOC or current magnitude and direction. EIS measurements were used to
study the frequency response of the supercapacitor and to compare it with the electrical model
obtained using pulse-relaxation experiments. The thermal model included the reversible (entropic)
as well as the irreversible heat generation. The thermal model consisted of four parameters which
were identified as constants using actual current profiles. The coupling between the electrical and
the thermal model was done by feeding the total heat generation calculated from the electrical
model into the thermal model and tuning the temperature dependent electrical model parameters
according to temperature variation obtained from the thermal model. Finally the electro-thermal
model was validated using real world driving cycles. The validation results show the high accuracy
of the proposed electro-thermal model which is suitable for real time implementations in all kinds
of power systems and also thermal management of supercapacitor packs.
40
Chapter 3
Range Extension of a Lead-Acid Battery
Driven Vehicle using Supercapacitors:
Modeling and Hardware In the Loop
Validation
3.1 Literature Review
There is an extensive body of literature discussing battery-SC hybrids in different applica-
tions [41, 64, 114]. The added degree of freedom via hybridizing has its own pros and cons. The
main advantage is the possibility of combining the high energy density batteries with the superior
power density of SCs. The high power density of the SC allows shaving the peak load on the bat-
tery and increases the specific power of the electrical energy storage system [62, 115, 116]. Other
research shows the benefits of the battery-SC hybrids in terms of battery life extension [117–121].
Hybrid battery-SCs are also used in stationary applications such as microgrid and uninterruptible
power supplies (UPS) [49, 122]. In [123], SCs are utilized to capture the regenerative braking en-
ergy in a race car. The inclusion of SCs resulted in a smaller battery pack, lower resistive losses,
41
and lower peak current which extends the service life of the electronic components.
The two basic challenges of hybridizing are choosing the configuration and developing a
power management strategy. On the configuration side, the topologies are divided into two major
categories, passive and active. Passive hybrid configurations include direct parallel connection of
the battery and SC, or connection with diodes [124]. In passive designs the power split is primarily
controlled by the internal resistance of the modules and voltage of the battery. In the active structure
power electronic devices are utilized as the hardware that facilitates the power split between the
battery and the SC [125–128]. The design with two bi-directional converters allows full control
of the power flow between battery, SC, and to the load. However, using one converter is cheaper,
easier to control, and most importantly has smaller losses compared to two converters [129–131].
Two potential single converter architectures for HES are shown in Fig. 3.1. In Fig. 3.1(a), the battery
is directly connected to the load which has the advantage of less voltage fluctuation compared to
SCs. In this topology however the battery suffers from high currents during high power traction and
braking events. The internal resistance of SCs is less than batteries so connecting the SC directly
to the load as in Fig. 3.1(b), increases the traction efficiency. Also SCs can undergo high charging
currents and are much more efficient during charging compared to batteries, which increases the
regenerative braking efficiency. The problem with this topology is that SCs terminal voltage varies
in a larger range than batteries and such a large voltage window is not favorable on the load side
[132]. One solution is to add a second converter between the SC and the load to regulate the voltage
to the DC bus voltage [133].
On the other hand, there are studies focusing on different control strategies to split the power
demand between the battery and the SC [61, 134, 135]. These studies are successful in controlling
the power flow in the HES while considering the limits on battery and SC current, voltage, and state
of charge (SOC). However most of them ignore one or more of the following real world scenarios:
(i) component efficiency (specifically converter efficiency), (ii) battery and SC model accuracy,
(iii) experimental validation using real world duty cycles, and (iv) comparison with a battery-only
platform considering some kind of performance metric.
The objective of increasing the mileage of EVs, has been of interest to many researchers
42
[136, 137]. Fujimoto et al. in [138], optimizes the front and rear driving-braking force distribu-
tions considering the motor losses and the slip ratio of the wheels, to improve the cruise range of
EVs. In a similar study, Wang et al. in [139], report improvement in range of an EV by effective
torque distribution between the wheels and no other additional hardware. Efforts have been made
to improve EV’s range by adding range extender modules such a fuel cell to the battery driven ve-
hicles [140, 141]. Hybrid battery-SC energy storage systems have also been considered in range
improvement studies. Yang et al. in [142], improve the range of a scooter by 20% utilizing re-
generative braking and electronic gearshifting between the battery and SC. Moreno et al., show
that a range improvement of 3.3-28.7% in a hybrid electric vehicle with lead-acid battery and SC
energy storage is achieved, depending on whether if the battery can or cannot capture regenera-
tive braking [143]. However this study does not compare the range results to the battery-only case
which masks the efficiency of added components such as the converters. Carter et al., report small
improvement in range for a lead-acid electric kit car equipped with SC for practical regeneration
rates and a 6.6-9.4% higher range for high regeneration rates (70-76%), depending on the driving
cycle. In the aforementioned study, the component models are based on commercial software and
the results are not validated by experiments [144].
In this research we present the simulated and experimentally validated range extension re-
sults based on a new HES topology combined with a widely applicable control strategy. The high-
lights of this study are:
• Models for lead-acid and SC modules are parameterized, validated, and used to build the HES
model.
• In the proposed topology the battery is completely decoupled from the load and functions as
an on-board charger for the SC.
• The topology includes only one uni-directional DC/DC converter which is placed between
the battery and the SC as shown in Fig. 3.1(b). The voltage variation on the SC side is kept
small by an appropriate power management strategy instead of using a second converter.
43
• The simplicity of the control strategy allows it to be implemented using inexpensive commer-
cial micro controllers.
• The converter is specifically built for this study which enables monitoring the real world
efficiency bottlenecks of the HES.
• The HES model is fully scalable. However for simulation and experimental implementation,
an electric scooter platform is used along with the New York City driving cycle to derive a
real world duty cycle.
• A 12 V HIL setup is used to implement the control strategy and extract the range results.
• The range results for the HES is compared to the battery-only platform.
The Remainder of this chapter is organized in the following order. In Section 3.2 the battery
and SC models are described. In Section 3.3 the HES topology, converter design, and the power
management strategy are described. Section 3.4 presents the HIL setup and the simulation and
experimental results for the stand-alone battery platform and the HES system. Finally Section 3.5
presents the concluding remarks.
3.2 Energy Storage Modeling
In this section the electric models for both the battery and SC are presented. Equivalent
electric circuit models are proposed and parameterized for both the battery and SC with the follow-
ing characteristics:
• Fewer parameters to be identified compared to the electrochemical models.
• Computationally efficient and appropriate for control oriented studies.
• Accurate terminal voltage prediction.
These validated models will be used to build the HES model.
44
Figure 3.1: Schematic of the active HES with one converter (a) Direct connection of battery to the load, (b)Direct connection of SC to the load.
3.2.1 Equivalent Electric Circuit Model
Fig. 3.2 shows the schematic of the equivalent electric circuit model. It consists of an
equivalent series resistance which represents the internal resistance of the cell and alsoRC branches
that capture the voltage behavior during relaxation. The current is the input to the model and the
output is the terminal voltage. Positive current is chosen to represent charging and the terminal
voltage is obtained by applying the Kirchhoff’s voltage law (KVL) to the circuit shown in Fig. 3.2:
VT = Vocv(SOC) + IRs +
n∑j=1
VRC,j (3.1)
In (3.1), Vocv is the open circuit voltage (OCV ) which is a nonlinear function of SOC for
both lead-acid battery and the double layer SC. The SOC is determined using coulomb counting,
governed by the following state equation:
dSOC
dt=
I
qmax(3.2)
45
where qmax is the nominal capacity in Coulombs. The second part in (3.1), is the voltage drop
over the ohmic resistance and the last part is the sum of voltage drops across parallel RC branches
connected in series. The voltage dynamics of each RC pair is described as:
dVRC,j
dt= − 1
RjCjVRC,j +
I
Cj(3.3)
where Rj and Cj are the equivalent resistance in Ohms and capacitance in Farads for each branch,
respectively.
Figure 3.2: Schematic of the equivalent electric circuit model with “n” number of RC branches.
3.2.2 Battery Capacity and Open Circuit Voltage Identification
The battery used in this study is an AP-12220EV lead-acid module with the specifications
listed in Table 3.1.
The first step in modeling is to measure the capacity and also obtain the OCV as a function
Table 3.1: Lead acid battery specification
Voltage (V) 12
Upper Voltage Limit (V) 14.7
Lower Voltage Limit (V) 10.5(at 1.1A)
Nominal Capacity (Ah) 22
Specific Power (Wkg−1) 180
Specific Energy (Whkg−1) 42
46
of SOC. The actual capacity is measured by discharging the battery from its upper voltage limit to
its lower voltage limit under a small constant current. Knowing that the applied current is constant
one can integrate (3.2) and use the boundary conditions SOC(0) = SOCi and SOC(tf) = SOCf
to find the discharging (charging) time:
tf =qmax(SOCf − SOCi)
I(t)(3.4)
The current fed into the module for capacity test is 0.55 A. This current according to (3.4)
will discharge the battery with nominal capacity of 22 Ah from SOCi = 1 to SOCf = 0 in 40 hours.
Such a small current minimizes the Peukert’s effect and hence, the measured capacity is more accu-
rate. The actual capacity measured for the module is 19.7 Ah at 100% depth of discharge (DOD).
In lead-acid batteries with an increase in DOD during cycling, positive active mass degradation is
accelerated which reduces the cycle life of the battery [145]. In order to decelerate the aging process
the battery is limited to 35% DOD (or to keep it above 65% SOC) as recommended by the battery
manufacturer. The model developed for the lead-acid battery in this study is only applicable for
discharging cycles.
The OCV as a function of SOC is obtained by running the following experiment similar
to the method used in [146]:
• Fully charge the battery using the constant current-constant voltage (CC-CV) protocol recom-
mended by the manufacturer, then allow the module to relax for at least 3 hours. The relaxed
voltage for the fully charge module is 13.04 V.
• Discharge the fully charged module at a low current of 0.55 A for 2 hours which is equivalent
to a 5% change in SOC according to (3.4) followed by a relaxation period of 3 hours.
• Repeat this 5% pulse-relaxation until the battery reaches 65% state of charge.
Fig. 3.3 illustrates the mentioned experiment to obtain the OCV profile. The measured
terminal voltage at the end of every relaxation period at every 5% SOC level, corresponds to the
OCV at that SOC level. Using the coulomb counting method shown in (3.2) the SOC of the
47
Figure 3.3: Experimental current applied to the module and the corresponding terminal voltage for obtainingthe OCV .
48
battery is obtained at each time. This provides a look-up table for OCV as a function of SOC to be
integrated in the battery model. Fig. 3.4 shows the OCV obtained as a function of the SOC from
65% to 100%.
Figure 3.4: OCV versus SOC for the lead-acid battery.
3.2.3 Lead-Acid Battery Modeling and Parameterization
The choice of the equivalent electric circuit in this study is a first order model with a single
RC branch connected in series to Rs. The parameters to be identified are Rs, R1, and C1. In
order to gather a rich data set for identification purposes the following pulse-relaxation experiment
is performed:
• Constant current-constant voltage (CC-CV) charging to 100% SOC, with 0.05C cut-off.
• Discharge the fully charged module by applying a 22 A of current for 180 seconds. This will
result in a 5% drop in SOC from 100% to 95% according to (3.4), then allow the battery to
relax for 3 hours.
• Repeat step 2 until the battery reaches 65% state of charge.
Fig. 3.5 illustrates the mentioned pulse relaxation experiment. The free response of the
system (i.e zero input) is when the pulse current indicated in Fig. 3.5 is cut off. This corresponds
49
to the so called relaxation period. The voltage response during relaxation consists of two parts as
follows:
• The first part is the instant voltage jump as shown in Fig. 3.5 which occurs right at the moment
the discharging current is cut off. This behavior is due to the internal resistance of the module.
• The second part is the exponential behavior in voltage which is captured by the RC branches
in the model.
Figure 3.5: Experimental current applied to the lead-acid module and the corresponding terminal voltage formodel parameterization.
The series resistance Rs, is calculated by dividing the instant voltage jump by the pulse
current. Rs as a function of SOC is obtained by repeating this calculation for each SOC level.
Using this data, a 2nd order polynomial in the range of 65% to 100% SOC, is fitted to Rs values as
follows:
Rs = 8.8(SOC)2 − 17(SOC) + 19 (3.5)
The second part of the relaxation period contains the dynamics to identify R1 and C1. The
identification is performed by minimizing the square error between the measured and simulated
50
terminal voltages at each time instant. The cost function is:
J =∑k
(Vm(k)− VT(k))2 (3.6)
where Vm and VT are the measured and simulated terminal voltages, respectively. The parameter-
ization results are shown in Fig. 3.6. The root mean square (RMS) error between the model and
the experimental terminal voltage is 83 mV. The estimated C1 has a constant value of 5086 F. The
identified values for R1 is used to fit a 2nd order polynomial as a function of SOC in the range of
65% to 100%:
R1 = 16(SOC)2 − 1.3(SOC) + 8.2 (3.7)
Figure 3.6: Parameterization results for the lead-acid battery.
3.2.4 Supercapacitor Modeling and Parameterization
The SC utilized in this research is a Maxwell BCAP3000 cylindrical cell with activated
carbon as electrodes. The cell specifications are listed in Table 3.2.
The cell contains non-aqueous electrolyte allowing the maximum rated voltage of 2.7 V.
51
Six of these cells are connected in series to build the SC module with a nominal voltage of 16.2 V.
Similar to the battery a first order model with a single RC branch is chosen and parameterized. The
OCV , capacity, series resistance, resistance and capacitance of the RC pairs are the parameters to
be identified based on the method used in [147]. The actual capacity and the OCV are obtained
by charging the SC from zero to its maximum nominal voltage under a low current of 0.225 A.
The SC is then discharged to zero with a similar current rate, to obtain the capacity and OCV for
discharging. The difference between charging and discharging OCV is small due to low current
and small internal resistance. Despite the linear behavior in an ideal capacitor the OCV profile
as a function of SOC is slightly nonlinear for the chosen SC and could be approximated by the
following 4th order polynomial:
Vocv = − 0.18(SOC)4 + 0.59(SOC)3 − 1.2(SOC)2
+ 3.5(SOC)− 1.9× 10−4(3.8)
Specific pulse-relation experiments as proposed in [147], are used to identify Rs, R1, and
C1. The following set of experiments at room temperature is designed for the identification of the
SC model:
• Charge the empty cell by applying a 135 A of constant current for 3 seconds which is equiva-
lent to a 5% increase in SOC according to (3.4).
• Relax the cell for 20 seconds then repeat the pulse relaxation till 100% SOC.
• Repeat a similar procedure for the discharging cycle.
The identification is performed similar to the battery by minimizing the square error be-
Table 3.2: Supercapacitor specification
Voltage (V) 2.7
Nominal Capacity (F) 3000
Specific Power (Wkg−1) 5900
Specific Energy (Whkg−1) 6
52
Figure 3.7: Parameterization results for the SC cell.
53
tween the measured and simulated terminal voltages at each time instant. Fig. 3.7 shows the param-
eterization results for the SC cell for the mentioned set of pulse relaxation experiment. The RMS
error for the voltage is 17.5 mV which indicates the model and experiment are in agreement. Table
3.3 summarizes the estimations for Rs, R1, and C1. The variation of the parameters with SOC for
the SC is negligible. In case of large temperature variations and in order to consider the effect of
temperature on model parameters, refer to the complete model developed in [147].
Table 3.3: Estimated parameters for the SC cell
Rs (mΩ) 0.46
R1 (mΩ) 8.3
C1 (F) 104417
3.3 Hybrid Battery-Supercapacitor Energy Storage
In this section the HES topology is described first. Next, the digital converter design and
control is presented. Finally, the power management governing the power flow between the battery
and SC, and its advantages are explained.
3.3.1 HES Topology
Fig. 3.8 shows a schematic of the hybrid configuration. This architecture is inspired by the
series hybrid electric powertrain. In the series hybrid the engine is decoupled from the wheels. It
only charges the batteries while operating in its most efficient region and the battery supplies all the
load demand. In the HES topology proposed in this study, the battery, identical to the engine in the
series hybrid electric vehicle, is decoupled from the load and the SC satisfies all the load demand.
The battery in this topology functions as an on-board charger for the SC module. In order to apply
this idea the topology in Fig. 3.1(b) is chosen. The DC/DC converter is located between the battery
module and the SC.
54
Figure 3.8: Schematic of the hybrid battery-SC configuration.
3.3.2 Digital Converter Design and Control
In this study the single unidirectional DC/DC converter is located between the 12 V battery
module and the 16.2 V bank of SCs. The phase-shift modulated (PSM) full-bridge converter can
achieve high efficiency and power density in many applications [148]. Unlike the traditional pulse
width modulated (PWM) bridge converter, this topology uses a fixed duty cycle close to 50%.
Furthermore a phase between the bridge columns is introduced to cause an overlap in conduction
time between the top and bottom switches. These characteristics along with the delay in PWM
of each column, result in an oscillation of the metal oxide semiconductor field-effect transistor
(MOSFET) voltage due to resonance of the MOSFET’s capacitances and the leakage inductance
of the transformer. Hence the voltage goes to zero before the next turn-on period. In this way
zero voltage switching (ZVS) is achieved resulting in almost no turn-on switching loss and hence a
significant improvement in efficiency. Higher frequency operation also enables the use of smaller
magnetic components. A 32-bit micro-controller (µC) is used to control the power electronics.
Two single input single output (SISO) control loops are implemented. The fast inner control loop
manages the switch currents for each PWM cycle. The second control loop regulates the battery
input current with respect to a desired set-point. The µC also manages the balancing between the
SC cells as well as the over-current and over-voltage circuit protection. The µC communicates with
the host personal computer (PC) through universal serial bus (USB), where new set-point commands
55
are received. Also via the same connection the measured data is transfered, to be logged by the PC
for future analysis. The overall function of the converter is to regulate and supply a predefined
amount of constant current from the battery to the SC.
3.3.3 Power Management Strategy
The control strategy that is integrated inside the µC is responsible for the power manage-
ment. In this control strategy the battery is commanded to continuously supply a preset amount
of constant current. Constant current charging is shown to be the optimal charging strategy for
SCs [71]. The amount of the constant charging current is chosen such that: (i) the voltage variation
on the SC side is minimized which waives the need for an extra hardware for SC voltage regu-
lation, (ii) minimize the DC/DC converter’s output voltage ripple which increases the converter’s
efficiency. This bang-bang control strategy charges the SC with a constant current regulated by the
DC/DC converter at all times subject to the following constraints:
• The charging current is cut off when the SC module reaches its upper voltage limit of 16.2 V.
• A hysteresis band on SC voltage is applied in order to decide when to restart the charging of
the SC. The hysteresis band in this study is 100 mV. In other words, the SC terminal voltage
should drop by at least 100 mV from the upper voltage limit until the charging command is
resent.
• The charging will stop if SOC of the lead-acid battery reaches 65% or the lower voltage limit
of the SC is reached.
The combination of the architecture and control strategy allows to minimize the stress on
the battery by decoupling it from the load and letting the low impedance SC to provide all the
positive (traction) and negative (braking) power demands. In case of the battery driven vehicle
due to the low charging rate capability of lead-acid and also life cycle considerations, the acceptable
regeneration rate is almost zero. One important advantage of this structure is the ability to efficiently
regenerate the braking energy with the SC being connected to the load which otherwise would be
56
wasted in friction braking. In this strategy the power requirement from the battery is much lower,
which allows the usage of cheaper battery chemistries with lower power densities. Operating the
battery at a low discharge rate, extends the battery life and also helps overcome the shortcomings
of the battery operation at low temperatures. This computationally efficient and effective control
strategy also allows to minimize the number of converters and their power rating. Also since the
SC has a much lower internal resistance compared to different battery chemistries the efficiency of
charging and discharging is improved on the load side.
3.4 Simulation and Experimental Results
3.4.1 Load Simulation
In order to obtain a real world duty cycle to be used in simulations and experiments, a power
cycle is extracted considering the following:
• A 48 V electric scooter with the characteristics listed in Table 3.4.
• The velocity profile of the New York City driving cycle which consists of frequent stop and
start sequences as shown in Fig. 3.9(a). This cycle is 1.9 km (1.18 miles) with a maximum
speed of 44.6 kph (27.7 mph). This data is gathered by the United States environmental pro-
tection agency (EPA).
However, the developed HES model is scalable, and any other vehicle platform could be
integrated and studied. The velocity of New York City driving cycle at each time step is used as
the input to the vehicle dynamics to calculate the power requirement of the cycle. The longitudinal
vehicle dynamics at time step t assuming zero road incline is governed by:
F (t) = Ma(t)± Crrmg ±1
2ρACdV
2(t) (3.9)
where the positive and negative signs are during traction (V (t+1) ≥ V (t)) and braking (V (t+1) <
V (t)) modes, respectively. The first term in (3.9) is due to the longitudinal and rotational inertias.
57
The mass ism and the equivalent rotational mass is chosen to be 5% ofm. The total mass is thenM .
The acceleration (deceleration) at each time step a(t), is calculated with the knowledge of velocity
of the vehicle at each time step and assuming a constant acceleration (deceleration) at each time
interval. The second term in (3.9) is due to the rolling resistance where Crr is the corresponding
coefficient. The last term in the equation is the contribution of drag resistance, where ρ, A, and Cd
are the air density, vehicle frontal area, and drag coefficient respectively.
Figure 3.9: New York City driving cycle and the extracted power profile using the scooter platform.
The distance traveled at each time step with the assumption of constant acceleration (decel-
eration) is:
∆x = x(t+ 1)− x(t) = v(t)∆t+1
2a(t)∆t2 (3.10)
Knowing the force required and the distance traveled at each time step the amount of re-
quired energy and consequently the power is derived as:
E(t) = F (t)∆x =⇒ P (t) =E(t)
∆t(3.11)
58
The final power profile (duty cycle) obtained using the 48 V scooter platform and the New
York City driving cycle is shown in Fig. 3.9(b). This single duty cycle is stacked back to back
to construct the load profile used in this study. The power profile calculated from the scooter is
associated with a 48 V system, however, the HIL setup is based on a 12 V system. Consequently,
the developed power profile is scaled down by a factor of four to match the power rating of the
experimental setup.
3.4.2 Hardware In the Loop (HIL) Setup
All the experiments related to parametrization and validation of the electrical models for
the battery, SC, and also HIL experiments on the hybrid setup are conducted using Bitrode FTV1-
200/50/2-60 cycler which is capable of supplying currents up to 200 A, which is suitable for running
the pulse-relaxation and also the driving cycle validation tests. Fig. 3.10 shows the real setup with
the components used in the experiments.
Figure 3.10: Hybrid battery-SC experimental setup.
59
Figure 3.11: Current and terminal voltage of the stand-alone battery from the HIL experiment.
60
Table 3.4: Scooter specifications
Vehicle + Passenger Mass (m) 108+70 (kg)
Drag Coefficient (Cd) 0.75
Frontal Area (A) 0.6 (m2)
Rolling Resistance (Crr) 0.0007
Air Density (ρ) 1.125 (kgm−3)
Wheel Radius (R) 0.21 (m)
3.4.3 Range Results for the Stand-Alone Battery
The first step in simulation and experiments is to run the HIL test for the stand-alone battery
to evaluate the range of the battery driven vehicle. In this case, the amount of regeneration during
braking is considered to be zero. This is due to high internal resistance of lead-acid batteries which
leads to thermal and physical stresses resulting in a lower life span for the battery, under high
charging currents. The maximum allowable charging current for the module used in this study is
6.6 A which is much smaller than the peak regeneration currents of 75 A. Fig. 3.11 depicts the
current applied to the battery and the terminal voltage dynamics of the module both for simulation
and experiments. The results show that the modeled battery terminal voltage is in good agreement
with experimental measurements. The range and the RMS error results are shown is Table 3.5. The
simulation is stopped and the range is calculated at the point where the SOC of the battery reaches
65%. This is because of the 35% DOD, recommended by the manufacturer.
3.4.4 Range Results for the HES
In the HES configuration the SC is responsible of providing the total amount of the re-
quested load. In this case regenerating the braking energy is feasible as the SC is capable of being
charged at high rates due to its low internal resistance. At the beginning of every test on the HIL
setup, the battery and the SC are fully charged. Depending on the load demand the SC is discharged
during traction and charged during regenerative braking. However the constraints are set to avoid
over-charge or over-discharge of the module. The only role of the battery is to charge the SC with
61
a constant current. In case the SC voltage reaches the upper voltage limit the charging current from
the battery is cut off. Simulation results are compared to three experimental measurements at 0%,
10%, and 20% regeneration rates. The results for the 0% regeneration case is shown in Fig. 3.12.
This result shows that the SC satisfies the total requested load while the measured terminal voltage
dynamics of the SC is followed by the model. According to the power management strategy the
battery provides a constant current of 5.5 A (also shown in Fig. 3.12) at all times unless the SOC
of the SC is equal to one with an active hysteresis band of 100 mV on the SC terminal voltage. The
amount of the constant charging current is chosen to be the lowest current possible based on the
hardware limitations. Lower currents than 5.5 A resulted in unacceptable ripple on the converter’s
output voltage and consequently lower converter efficiency. The SOC of the SC bank varies be-
tween 100% and 94%. This narrow voltage window close to the higher end of SOC is where the SC
operates more efficiently [71]. Furthermore this small voltage window on the SC side resolves the
drawback of such a topology where usually there is a large operating voltage window not favorable
for the inverter efficiency between the SC and the electric motor. The modeled terminal voltage
behavior of the battery is also in agreement with measurements with a RMS error of 41 mV. Similar
to the stand-alone case the experiment and simulation is stopped when the SOC of the battery hits
the preset lower limit of 65%.
The range of the hybrid battery-SC in the case of 0% regeneration is 11.5% less than the
stand-alone battery case. The DC/DC converter efficiency is governed by:
ηconverter =Pout
Pin(3.12)
where Pin is the input power from the battery and Pout is the output power of the converter. The effi-
ciency of the converter during 0% regeneration operation is around 86%. The hybrid structure used
in this study reduces the total impedance on the load side, by directly connecting the low impedance
SC to the load. However even this configuration could not compensate for the losses of the con-
verter. This confirms that the converter efficiency is the bottleneck of active hybrid topologies. In
order to compensate for the losses of the converter we make advantage of the configuration of the
62
Table 3.5: Range results for the stand-alone battery and HES system
Stand-Alone Battery Range HES Range
(Miles) (Miles)
0% 10% 20% 0% 10% 20%
11.19 NA NA 9.9 10.7 12.4
hybrid setup and run experiments at 10% and 20% regeneration. Table 3.5 summarizes the range
results for 0%, 10%, and 20% regeneration tests. The results show that a regeneration rate of 20%
improves the total range by 10%. Higher amounts of regeneration will result in even better range.
The available amount of regeneration in heavier vehicle platforms is higher and such a design will
result in even better range extension in such applications.
Figure 3.12: Current and terminal voltage of the battery and SC in the HES system with 0% regenerationrate from the HIL experiment.
63
3.5 Conclusions
This study investigated the circumstances under which, the hybridization of a lead-acid
powered vehicle using SCs, will result in extended range. In the first step, the lead-acid battery
model is developed and integrated with a validated SC model to form the HES model. The combi-
nation of the topology and proposed power management strategy allowed a simple implementation,
relaxed battery operation, small SC voltage variation, and regenerative braking capability. The re-
sults show that the efficiency of the converter is the bottleneck of the HES system. Capturing 20%
or higher amounts of the regenerative braking energy by connecting the SC to the load improves the
range of the hybrid system compared to the stand-alone battery. The improvement in range could
be even higher in heavier vehicles with a larger amount of available rate of regeneration.
64
Chapter 4
Heuristic versus Optimal Charging of
Supercapacitors, Lithium-Ion, and
Lead-Acid Batteries: An Efficiency
Point of View
4.1 Literature Review
Batteries have become an indispensable part of our daily life. They can be found almost ev-
erywhere from powering our electronic gadgets, computers, and phones to electrifying our vehicles
and also form a critical part of the modern centralized and distributed power grids. Supercapacitors
on the other hand are the premier energy storage devices in terms of power density, long cycle life,
and the ability to operate at extreme temperatures. Much research and development is spurred to-
wards studying important factors influencing the efficiency and cycle life of batteries and SCs, such
as monitoring and control of the cell charging method, current rate, number of charging/discharging
cycles, and temperature [8, 10, 12, 21, 149–153].
Studies that investigate the optimization of efficiency over the entire driving cycle, mask
65
the fundamental bottlenecks of efficiency in the electrical energy storage systems. In stand-alone
operation and during discharge, the cycle is often imposed by the required load and therefore there is
little that can be done in reducing resistive losses. During charging however, there is the opportunity
to choose the charging time and profile such that resistive losses are reduced. Battery manufacturers
often have a recommended charging profile which may be sub-optimal.
There are numerous studies focusing on different charging methods to achieve objectives
such as decreasing the charging time, life span and efficiency maximization, or cost minimization.
Optimal charging of Li-ion batteries is studied in [154], where minimizing the charging time while
satisfying specific physical and thermal constraints is considered. In [71], an optimal charging
problem is solved for a SC during regenerative braking with the objective of minimizing ohmic
losses. Suthar et al. in [155], use a single-particle model and aim to find the optimal current profile,
with the objective of maximizing the charge stored in the cell in a given time and with the constraint
of minimal damage to the electrode particles during intercalation. Bashash et al. in [156], focus
on optimizing the timing and charging rate of a plug-in hybrid electric vehicle from the power grid
where the goal is to simultaneously minimize the total cost of fuel and electricity and the total battery
health degradation. Optimizing the battery charging power in photovoltaic battery systems is studied
in [157] where different objectives such as charging time, battery life time, and cost of charging are
considered. Inoa et al. in [158] suggest optimal charging profiles in order to minimize charging
losses and reach a preset temperature at the end of the charging time for Li-ion batteries. More
recently, [159] and [160] have solved the optimal charging problem for Li-ion batteries considering
the trade-off between charging time and energy loss in the objective function; however neither of
these papers consider the transient effect of temperature on electrical model parameters of the Li-ion
battery. One important output of the mentioned studies is the charging/discharging current profile
that satisfies the specific objective functions. For example there are studies with the objective of
reducing the charging time which result in different types of constant current (CC) charging methods
such as multistage charging [161–165], impedance compensation [166–168] and pulse charging
[169]. Constant voltage (CV) charging has been used to improve charging speed by combining the
battery pack and charger models and the results are compared to the CC method [170]. Constant
66
power (CP) charging and discharging is also of interest to researchers as it corresponds to real world
operating conditions such as in hybrid and electric vehicles. Modeling the thermal behavior of Li-
ion batteries under CP charging and discharging cycles is investigated in [171]. Under constant
power, the relationship between the available energy in a battery and charging power has been
investigated in [172]. The aging of SCs under CP charging condition is studied in [19].
In this research we shed light on achievable charging efficiencies for the supercapacitors,
lead-acid, and Li-ion batteries. The common aspects of the problem formulation and analysis are:
• In all optimal charging formulations the objective function is to maximize the charging effi-
ciency by minimizing the resistive losses in a given charging time and for a specified range of
state of charge (SOC).
• All problems are formulated using Pontryagin’s minimum principle with the intention to solve
them analytically. In cases where analytical solution is not feasible, the problem is solved
using numerical methods.
• The constant power and optimal charging current profiles and efficiencies are obtained and
compared for all modules.
• Both slow and fast charging times are investigated.
We begin with lumped models for each module with the upper and lower bounds on SOC
being the only constraint considered in the optimal control problem formulations. This provides a
basic understanding on the effect of charging time of the charging current and efficiencies under
different charging strategies. Later on, we solve the optimal charging problem for the Li-ion battery
by coupling a reduced order, two state thermal model to the electrical model. In the electrical
models used in this study, Rs indicates the ionic and electronic resistance of electrolyte and also the
electronic resistance of the electrode. The other two parameters are the charge-transfer resistance
R1 which is in parallel with the double layer capacitance C1 formed at the interface between the
electrode and electrolyte [173]. At steady state, the sum of Rs and R1 is called the total internal
resistance (R) in this chapter.
67
In case of the SC, the electrical dynamics is modeled using a constant total internal resis-
tance (R). The open circuit voltage (OCV ) is assumed to have a linear relationship with SOC.
Another assumption is constant ambient temperature of 25 C during charging. Due to high power
density of SCs the fast and slow charging times are chosen to be 30 seconds and 6 minutes, re-
spectively. The optimal charging current and efficiency for the SC have analytical solutions and are
compared to the CC and CV strategies. For the lead-acid battery, similar to the SC, the module is
modeled using a total constant internal resistance (R) at a constant ambient temperature of 25 C.
The difference in the problem formulation for the lead-acid battery is the strong dependence of R
on SOC which is integrated into the electrical model. Both OCV and R are approximated by 2nd
order polynomials as a function of SOC. The fast and slow charging times are chosen to be 6
minutes and 1 hour, respectively. Considering these conditions for the lead-acid battery, the optimal
charging formulation results in a two point boundary value problem, which is solved numerically.
The obtained optimal charging current and efficiency are compared to CP and CC methods. For
the Li-ion battery in the first step only the electronic resistance (Rs) with a constant value at a con-
stant charging temperature of 25 C is considered. In the second step, the effect of charge transfer
resistance (R1) and double layer capacitance (C1) are added to the model. The dependence of Rs
on SOC is negligible at room temperature and the variation of R1 and C1 on SOC has not been
considered. Similar to the lead acid battery the charging times studied are 6 minutes and 1 hour for
rapid and slow charging strategies, respectively. Finally, in order to investigate the effect of temper-
ature on the electrical model parameters, a validated 3 state electro-thermal model is utilized and
the optimal charging problem is solved using the numerical method of dynamic programming. The
optimal solution in this case is obtained subject to SOC, voltage, and temperature constraints. The
unconstrained optimal charging scenarios were partially reported in [71] and [103]. The direct con-
tribution of this study is considering the effect of temperature dependent model parameters on the
optimal charging current along with the CP charging derivation and efficiency analysis for different
charging methods.
The remainder of this chapter is organized in the succeeding order. In Section 4.2 the charg-
ing of the SC, including the model used, CV, CP, and optimal charging strategies are presented.
68
Section 4.3 covers the efficiency analysis of the SC. Sections 4.4 and 4.5 explain the charging and
efficiency analysis of the lead-acid battery. Sections 4.6 and 4.7 describe the charging and effi-
ciency analysis of the Li-ion battery. Section 4.8 presents the electro-thermal model and the optimal
charging current profile considering the temperature effect as well as the voltage and temperature
constraints for the Li-ion battery. Section 4.9 presents the conclusions.
4.2 Charging of the Supercapacitor
4.2.1 Supercapacitor Model and Specifications
The SC utilized in this research is a Maxwell BCAP3000 cylindrical double layer cell. The
specifications of the cell are listed in Table 4.1.
Table 4.1: Supercapacitor cell specification
Nominal Voltage (V) 2.7
Nominal Capacitance (F) 3000
Mass (kg) 0.5
Specific Power (W/kg) 5900
Specific Energy (Wh/kg) 6
The nominal energy capacity of the SC according to Table 4.1 is 3 Wh. The equivalent
electric circuit model for this cell is identified using pulse-relaxation experiments for a wide range
of temperatures from -40 to 60 C in [147, 174]. The model indicates that the dependence of the
model parameters such as R in Fig. 4.1 on SOC and also the current magnitude is negligible. Also
as mentioned in the introduction, we do not include the thermal constraints in the charging problem
formulation and efficiency analysis for the SC. The value of constant electronic resistance (R) is
2.97 mΩ for this cell at 25 C. A linear relationship between the OCV and the SOC is considered.
69
4.2.2 Constant Voltage Charging of the Supercapacitor
It is well known that charging a capacitor and similarly a SC, from zero charge to full
charge, with a constant voltage source results in 50% energy loss irrespective of the internal and
line resistances. This can be easily shown by writing the differential equation governing the SC’s
stored charge q(t), for the circuit shown in Fig. 4.1:
R.q +
q
C= Vdc (4.1)
where C is the nominal capacitance, R is the total internal resistance, and Vdc is the charging
Figure 4.1: Schematic of the SC model.
voltage. If Vdc remains constant over time, the solution to the above differential equation from a
zero initial charge condition can be obtained to be:
q(t) = CVdc
[1− e−t/RC
](4.2)
and the constant voltage charging current Ichg = Icv is then:
Icv(t) =Vdc
Re−t/RC (4.3)
The resistive energy loss is obtained by integrating the resistive power loss RI2cv over the entire
70
charging interval [0,+∞) as:
Eloss,cv =V 2
dcR
∫ ∞0
e−2t/RCdt =1
2CV 2
dc (4.4)
This amount is equal to the total energy stored in the SC. In other words, the efficiency of charging
an empty SC with a constant voltage source is 50%, independent of resistance R. Note that the
charging efficiency depends on both the initial and final state of charge. For example charging a
SC from half to full charge with constant voltage has an efficiency of 75%. Using the definition of
efficiency, charging a SC from an initial state of charge (SOCi) to a final state of charge (SOCf)
with a constant voltage source is:
ρcv =1
1 + (1−SOCi)2
SOC2f −SOC
2i
(4.5)
4.2.3 Constant Power Charging of the Supercapacitor
Consider charging the SC with the model in Fig. 4.1 by replacing the constant voltage
source with a constant power source of P0. Applying Kirchhoff’s voltage law to the circuit the
stored charge dynamics is:
R.q +
q
C=P0.q→
.q= Icp =
− qC +
√( qC )2 + 4RP0
2R(4.6)
where Icp is the constant power charging current. Equation (4.6), which is a nonlinear differential
equation, can be solved numerically to find the charge and current. Consider charging the cell from
zero to full charge. Figs. 4.2 and 4.3 show the charge stored in the SC and the constant power
charging current for charging in 6 minutes and 30 seconds, respectively. The maximum charge
storable in the SC is 3000×2.7 = 8100 Coulombs. The constant power for charging the cell in
6 minutes and 30 seconds is 32 W and 595 W, respectively. The maximum current that the cell
undergoes in the slow charging (6 minutes) and fast charging (30 seconds) is 104.5 A and 447.6 A,
respectively. The efficiency analysis for the constant power charging will be presented in Section
71
4.3.
Figure 4.2: Charging the SC with constant power from zero to full charge in 6 minutes.
Figure 4.3: Charging the SC with constant power from zero to full charge in 30 seconds.
4.2.4 Optimal Charging of the Supercapacitor
The next natural question to ask is what charging current profile maximizes the charging
efficiency. That is the current that would charge the SC to a desired level of charge with minimum
resistive losses. Let’s choose the optimization variable to be the charging current, u(t) = Ichg(t).
The SC state of charge (SOC = q(t)qmax
) quantifies the amount of charge stored in the SC bank
normalized by the maximum charge it can accept qmax. The dynamics of SOC as the single state x1
72
of the problem is derived by coulomb counting using the current u(t), fed into the SC as follows:
d
dtx1(t) =
u(t)
qmax=
u(t)
CVmax(4.7)
where Vmax is the voltage across the SC at maximum charge. Let’s assume the SC is initially free
of charge x1(0) = SOCi = 0 and in tf units of time is charged to its final desired state of charge
SOCf; therefore x1(tf) = SOCf. The optimal input u(t) is one that minimizes the resistive losses
in the time period [0, tf] characterized by the subsequent cost function:
J =
∫ tf
0Ru2(t)dt (4.8)
This is an optimal control problem and can be solved using Pontryagin’s minimum principle [175].
First form the Hamiltonian:
H(x1, u, t) = Ru2(t) + λ1(t)u(t)
CVmax(4.9)
where λ1 is a co-state. The optimal co-state should satisfy the subsequent dynamic equation:
d
dtλ1(t) = −∂H
∂x1= 0 (4.10)
implying that in this specific problem the optimal λ1 must be a constant. The unconstrained optimal
solution will also need to satisfy the condition:
∂H
∂u= 0→ u(t) = −1
2
1
RCVmaxλ1(t) (4.11)
showing that the optimal input (charging current) must be a constant. At this point, we can integrate
(4.7) and use the boundary conditions x1(0) = SOCi and x1(tf) = SOCf to find the value of this
optimal and constant input:
uopt(t) = Iopt,SC =CVmax(SOCf − SOCi)
tf(4.12)
73
where the subscript “opt” denotes the optimal solution. This is in fact the minimizing solution since
∂2H∂u2
> 0. Given a specific charging time, the most efficient way to charge the SC will be applying
a constant current equal to (4.12). The optimal charging current is a constant current of 30.8 A and
270 A for charging the SC from zero charge to full charge in 6 minutes and 30 seconds, respectively.
4.3 Efficiency Analysis for the Supercapacitor
In order to obtain the charging efficiency of the SC, the total storable energy is required.
By the integration of power over the entire charging time and using the definition of SOC, the total
energy stored in the SC is obtained as:
Esc =1
2CV 2
max
[SOC2
f − SOC2i
](4.13)
The energy loss in the SC during optimal charging is already known and is equal to∫ tf0 Ru2
opt(t)dt.
Then the optimal charging efficiency is:
ρopt =12CV
2max(SOC2
f − SOC2i )∫ tf
0 Ru2opt(t)dt+ 1
2CV2
max(SOC2f − SOC2
i ),
Substituting for uopt from (4.12) yields:
ρopt,SC =1
1 + (2RCtf )(SOCf−SOCiSOCf+SOCi
)(4.14)
Fig. 4.4 shows the SC charging efficiency as a function of initial state of charge and tfRC . This
figure implies the expected result that longer charging times and/or smaller RC values improve the
charging efficiency. When tf approaches infinity the charging efficiency approaches 1, which is
a 100% improvement over the case with CV charging. Another observation is that beginning the
charging from a higher initial SOC and also charging within a narrower range of SOC results in
improved efficiency. Furthermore (4.14) and the first two profiles that are on top of each other in
Fig. 4.4, show that when starting from zero initial SOC, the charging efficiency is independent of
74
the final SOC and it is only a function of charging time. The optimal charging current from zero
to full charge for slow (6 minutes) and fast (30 seconds) charging methods are 22.5 A and 270 A,
respectively. The maximum peak current of the Maxwell SC cell used in this study is 2165 A and the
maximum continuous current is 147 A. Although with these upper limits the fast charging scenario
is not practical, it is insightful to compare the corresponding charging currents and efficiencies with
the slow charging example. Table 4.2 shows the dependence of SC charging efficiency for three
SOC levels, two charging times, and for CV, CP, and optimal charging strategies. Optimal charging
the SC from zero to full charge in 6 minutes is 1.53% and 45.28% more efficient than CP and CV
charging, respectively. The efficiency of charging in the range of 50 to 100% SOC (which consists
of 75% of the total energy of the SC) is almost equal for CP and CC charging methods. Constant
current is the favorable charging strategy as it is not only more efficient but also easier to supply
than the nonlinear CP charging current.
Figure 4.4: The effect of charging time, initial SOC, and range of SOC on SC charging efficiency.
75
Table 4.2: Efficiency comparison for the supercapacitor
Slow Charging Fast Charging
(6 Minutes) (30 Seconds )
CP Optimal CP Optimal CV
SOC
0-100 93.4 95.28 61.2 62.73 50
0-50 93.4 95.28 61.2 62.73 20
50-100 98.32 98.37 83.27 83.47 75
4.4 Charging of the Lead-Acid Battery
4.4.1 Lead-Acid Battery Model and Specifications
The lead-acid battery used in this study is a AP-12220EV-NB module. The module specifi-
cations are listed in Table 4.3.
Table 4.3: Lead acid module specification
Nominal Voltage (V) 12
Nominal Capacity (Ah) 22
Mass (kg) 6.4
Specific Energy (Wh/kg) 42
The real capacity of the module is obtained by discharging the fully charged module with a
low current of 0.55 A from upper voltage limit to the lower voltage limit. This measured capacity
is 19.7 Ah. Specifically designed pulse-relaxation tests such as the method used in [176], is utilized
to estimate the total internal resistance R of the cell as a function of SOC. Fig. 4.5 shows that the
internal resistance of lead-acid battery strongly depends on SOC.
The open circuit voltage of the lead-acid battery is obtained by applying a small current
of 0.05 A to charge the battery from zero to full charge. The recorded OCV for this battery as a
function of SOC is shown in Fig. 4.6.
76
Figure 4.5: R versus SOC for the lead-acid battery.
Figure 4.6: Measured OCV versus SOC for the lead-acid battery.
77
4.4.2 Constant Power Charging of the Lead-Acid Battery
The relationship between the total internal resistance and SOC of the lead-acid battery is
approximated by fitting a second order polynomial to the profile in Fig. 4.5 as follows:
R = a1(SOC)2 + a2(SOC) + a3 (4.15)
A second order polynomial is fitted to the OCV as a function of SOC profile in Fig. 4.6 as
follows:
OCV = b1(SOC)2 + b2(SOC) + b3 (4.16)
The polynomial coefficients are listed in Table 4.4.
Table 4.4: Polynomial coefficients for the OCV and R as a function of SOC
a1 a2 a3 b1 b2 b3
0.098 -0.12 0.061 -0.56 2.2 11
As the lead-acid battery is modeled with a total internal resistance ofR, applying Kirchoff’s
voltage law to the circuit that is charging the lead-acid battery with a constant power source (P0)
results in:
Icp =−(OCV ) +
√(OCV )2 + 4RP0
2R(4.17)
Substituting (4.15) and (4.16) in (4.17) and solving the nonlinear differential equation, the
charge and constant power charging current is obtained. Consider charging the empty lead-acid
battery to full charge in an hour where P0 = 248 W. The maximum charge storable in this lead-
acid battery is 19.7 Ah×3600 = 70920 Coulombs. Fig. 4.7 shows the charge and constant power
charging current profiles for charging in one hour.
For the fast charging case consider charging the same lead-acid module with a constant
78
Figure 4.7: Charging the lead-acid battery with constant power from zero to full charge in 1 hour.
power of 3660 W which is equivalent to charging the module in 6 minutes. Fig. 4.8 shows that for
the lead-acid battery both the magnitude and shape of the constant power charging current vary for
the slow and fast charging cases.
Figure 4.8: Charging the lead-acid battery with constant power from zero to full charge in 6 minutes.
4.4.3 Optimal Charging of the Lead-Acid Battery
The lead-acid battery is modeled by a single internal resistance and the only state is the
SOC of the battery governed by (4.7). The objective is to minimize the losses associated with the
79
total internal resistance as in 4.8. Therefore the Hamiltonian is:
H(x, u, t) = R(x1)u2(t) + λ2(t)
u(t)
qmax(4.18)
where R is the total internal resistance and λ2(t) is the co-state. R(x1) also shown in Fig. 4.5 is
approximated by the second order polynomial in (4.15). The necessary conditions to be satisfied
are:
−∂H∂x1
= −dR(x1)
dx1u2(t) =
d
dtλ2(t) (4.19)
∂H
∂u= 2R(x1)u(t) +
λ2(t)
qmax= 0 (4.20)
Solving for u(t) in (4.20), the optimal charging current is obtained as follows:
uopt(t) = − 1
2qmax
1
R(x1)λ2(t) (4.21)
Substituting uopt(t) from (4.21) in (4.7) and (4.19), the consecutive set of two coupled nonlinear
ordinary differential equations (ODE) are obtained:
dx1(t)
dt= − 1
2q2max
1
R(x1)λ2(t) (4.22)
dλ2(t)
dt= − 1
4q2max
dR(x1)
dx1
1
R2(x1)λ22(t) (4.23)
Charging the lead-acid battery in tf units of time from zero to full charge requires the initial
and final conditions to be satisfied:
x1(0) = SOCi, x1(tf) = SOCf (4.24)
The system of two nonlinear ODEs with one initial and another final condition forms a two
point boundary value problem which could only be solved using numerical methods. One way to
80
solve this system of ODEs is to specify the initial condition for the SOC and iteratively guess the
initial condition for λ2 until SOC reaches the final specified value. Consider the case of charging
the lead-acid battery module from zero to full charge in 1 hour. Fig. 4.9 shows the variation of
optimal charging current, SOC, and λ2 with time.
Figure 4.9: Optimal charging current, SOC, and λ2 profiles for charging the lead-acid battery from zero tofull charge in 1 hour.
As shown in the numerical results, the optimal charging current for lead-acid battery, unlike
the SC, is not constant. In order to compare the constant current charging with the optimal charging
strategy, the energy losses in both methods are calculated. For the case of charging the lead-acid
battery from zero to full charge in an hour, the resistive losses in the optimal charging strategy is
46.18 kJ compared to 48.9 kJ for constant current charging. This is a 5.5% of less energy converted
to heat which could be significant in thermal management of battery packs.
81
Table 4.5: Efficiency comparison for the lead acid battery
Slow Charging Fast Charging
(1 hour) (6 min)
CC CP Optimal CC CP Optimal
95.19 95.34 95.44 66.43 67.46 67.70
4.5 Efficiency Analysis for the Lead-Acid Battery
The efficiency for three charging strategies including CP, CC, and optimal charging is cal-
culated based on the definition of efficiency:
ρ =Ebattery
Ebattery + Eloss(4.25)
According to the lead-acid battery specification listed in Table 4.3, Ebattery is 268.8 Wh
for this module. The energy loss (Eloss) is obtained by numerically integrating the power loss
(RI2) during the charging time. Table 4.5 compares the efficiency values for charging the lead-
acid battery from zero to full charge for three strategies and two charging times. The efficiency of
optimal charging is slightly higher than the other two strategies as expected. If factors such as cost
of supplying a non-constant current is considered, one may prefer to charge the lead-acid battery
with constant current and neglect the effect of the slightly lower charging efficiency. The efficiency
of constant power charging is higher than constant current charging for the lead-acid battery.
4.6 Charging of the Li-ion Battery
4.6.1 Li-ion Battery Model and Specifications
The Li-ion battery used in this research is a A123-26650 cell with LiFePO4 chemistry.
The cell specifications are listed in Table 4.6. The cell model is developed using pulse-relaxation
experiments to identify equivalent electric circuit model parameters. The estimation is performed
82
by minimizing the least square error between the experimental and modeled terminal voltage [100].
Table 4.6: Li-ion cell specification
Nominal Voltage (V) 3.3
Nominal Capacity (Ah) 2.5
Mass (kg) 0.076
Specific Power (W/kg) 2600
For the lithium ion battery two models are studied. First only electronic resistance (Rs) is
considered and in the next step the effect of polarization resistance (R1) is also included by adding
a single R-C branch to the model as shown in Fig. 4.10. The dependence of model parameters Rs,
R1, and C1 on SOC can be neglected at room temperature [100]. The values for Rs, R1, and C1
are 0.01 Ω, 0.016 Ω, and 2200 F, respectively.
Figure 4.10: Schematic of the single R-C model for the Li-ion battery.
The OCV of the Li-ion battery as a function of SOC is depicted in Fig. 4.11. As shown
in the figure, the OCV can be approximated by a linear function using the OCV data in the range
of 10% to 95% SOC. This linear fit makes the analytical efficiency analysis possible. The linear
approximation is governed by:
OCV (t) = aSOC(t) + b (4.26)
The relationship between OCV (t) and the charge stored q(t), in the battery is obtained by substi-
83
tuting the definition of SOC in (4.26) as follows:
OCV (t) =a
qmaxq(t) + b (4.27)
where a and b for this specific battery are 0.156 and 3.226, respectively.
Figure 4.11: Actual and approximated OCV versus SOC for the Li-ion battery.
4.6.2 Constant Power Charging of the Li-ion Battery
Consider charging the Li-ion battery modeled with only the electronic resistance repre-
sented with constant Rs in Fig. 4.10 with a constant power source of P0 and the linear OCV as-
sumption of Fig. 4.11. Applying Kirchhoff’s voltage law to the circuit, the stored charge dynamics
is:
Rs.q +
[a
qmaxq(t) + b
]=P0.q
(4.28)
Solving for the constant power charging current:
.q= Icp =
−[
aqmax
q(t) + b
]+
√[aqmax
q(t) + b
]2+ 4RsP0
2Rs(4.29)
84
The nonlinear differential equation in (4.29) is solved numerically. For the cell used in this
study the maximum storable charge is 2.5 Ah×3600 = 9000 Coulombs. Fig. 4.12 shows charge
and constant power charging current profiles for charging the battery from zero to full charge in
6 minutes. The constant power charging current is linear with small variation during the charging
time and a peak amount of 25.4 A. The manufacturer recommends a fast charging of 12 minutes
with a C-rate of 4 (10 A) for this cell. However, in the literature there are reported Li-ion batteries
with nano-particles of LiFePO4 (LFP) and Li4Ti5O12 (LTO) for the positive and negative electrodes
that can undergo charging from zero to full charge with C-rates equal to 10C (6 minutes) and 15C
(4 minutes) safely [177]. The efficiency analysis for the constant power charging will be presented
in section (4.7).
0 60 120 180 240 300 36023
23.5
24
24.5
25
25.5
26
Time (s)
ChargingCurren
t(A
)
0 60 120 180 240 300 3600
1350
2700
4050
5400
6750
8100
Charge(C
oulomb)
Charging CurrentCharge
Figure 4.12: Charging the Li-ion battery with constant power from zero to full charge in 6 minutes.
4.6.3 Optimal Charging of the Li-ion Battery
Two scenarios are considered in the optimal charging of the Li-ion battery.
1) First scenario: In this step only Rs as depicted in Fig. 4.10 is considered. The value of Rs is
constant and equal to 0.01 Ω. The single state is the SOC of the battery x1, governed by (4.7). The
objective is minimizing the ohmic losses associated with Rs during the given charging time tf as
85
follows:
J1 =
∫ tf
0Rsu
2(t)dt (4.30)
The optimal charging problem is formulated using Pontryagin’s minimum principle. The
result of the optimal charging strategy in this case is also constant current. Similar to the SC case,
the value of this optimal charging current for the Li-ion battery is:
uopt(t) = Iopt =qmax(SOCf − SOCi)
tf(4.31)
Similar to the constant power case, consider charging the Li-ion battery from zero charge to
full charge in 6 minutes. According to (4.31) the optimal charging current is constant and equal to
25 A. In this scenario the constant power and optimal charging current profiles are almost identical.
2) Second scenario: In this scenario, the R-C branch is added to the model to include the effect
of the polarization resistance R1. The value of R1 is assumed to be constant and not a function
of temperature or SOC. The value for R1 is 0.016 Ω which is an average value over the SOC
range [100].
Assume I1 and I2 are the currents passing through R1 and C1, respectively (see Fig. 4.10).
Applying the Kirchoff’s current and voltage laws to the R-C branch, the second state equation
governing the dynamics of I1 is obtained. The problem in this case is to solve for the optimal
charging current for a second order system governed by the state equations:
d
dtx1(t) =
u(t)
qmax,
d
dtx2(t) =
1
R1C1
[u(t)− x2(t)
](4.32)
where the two states x1 and x2 are the SOC of the battery and the current (I1) passing through the
polarization resistance (R1). The objective, similar to the first scenario, is to maximize the charging
efficiency. The difference is that the contribution of the polarization resistance to the total ohmic
86
losses should also be considered. Therefore, the cost function to be minimized is:
J2 =
∫ tf
0
[Rsu
2(t) +R1x22(t)
]dt (4.33)
The Hamiltonian in this case is given by the subsequent equation:
H(x, u, t) = Rsu(t) +R1x22(t) + .........................
....................λ3(t)u(t)
qmax+λ4(t)
R1C1
[u(t)− x2(t)
] (4.34)
where λ3 and λ4 are the co-states. The necessary conditions for optimality are:
−∂H∂x1
=d
dtλ3(t), −∂H
∂x2=
d
dtλ4(t),
∂H
∂u= 0 (4.35)
From the first two conditions in (4.35) the dynamics of the co-states are derived and the
solution to the third condition is the optimal input as follows:
uopt(t) =
[−1
2Rsqmax
]λ3(t) +
[−1
2RsR1C1
]λ4(t) (4.36)
Substituting the optimal input into the state equations of (4.32), the optimal state dynamics
are derived. The result is a set of four linear first order ODEs:
d
dtx1(t) = a1λ3(t) + a2λ4(t)
d
dtx2(t) = b1x2(t) + b2λ3(t) + b3λ4(t)
d
dtλ3(t) = 0
d
dtλ4(t) = c1x2(t) + c2λ4(t)
(4.37)
87
where a1, a2, b1, b2, b3, c1, and c2 are constant parameters equal to:
a1 =−1
2Rsq2max, a2 =
−1
2RsR1C1qmax,
b1 =−1
R1C1, b2 =
−1
2RsR1C1qmax,
b3 =−1
2RsR21C
21
, c1 = −2R1, c2 =1
R1C1
(4.38)
Solving this system of coupled linear ODEs simultaneously, results in four algebraic equa-
tions with four unknowns. The unknown constants are obtained by applying the boundary condi-
tions specific to this problem which consist of two initial and two final conditions. The initial and
final condition for x1 = SOC are similar to (4.24). On the other hand, In this specific problem the
charging time is specified and fixed while the values of the second state at the initial and final time
are free. This results in the succeeding equations for the remaining two boundary conditions [175]:
∂h
∂x2(x2(t0)) = λ4(t0) = 0→ Initial condition for x2
∂h
∂x2(x2(tf)) = λ4(tf) = 0→ Final condition for x2
(4.39)
In general h(x(tf), tf) is the term involving the final states and final time in the cost function
which in this study is zero. Given all boundary conditions, one can solve for the states and co-states
and thus the optimal input is obtained. Consider charging a battery cell from zero charge SOCi =
x1(0) = 0 to full charge SOCf = x1(tf) = 1 in one hour. The result for this example is depicted in
Fig. 4.13.
The optimal charging current for this scenario is slightly different from the result of the first
scenario. The optimal input in this case is almost a constant current equal to 2.5 A, in the majority
of times. It may be insightful to also show the result for a fast charging case. Fig. 4.14 shows the
optimal charging current and the two states of the system when the cell is charged from zero to full
charge in 6 minutes.
This charging strategy may not be practical due to thermal and physical constraints plus
safety and lifetime issues. However it may be interesting to observe that by reducing the charging
88
Figure 4.13: Optimal charging current, I1, and SOC for charging the Li-ion battery form zero charge to fullcharge in 1 hour.
89
Figure 4.14: Optimal charging current, I1, and SOC for charging the Li-ion battery form zero charge to fullcharge in 6 minutes.
90
time, the optimal profile differs from the constant current result observed in the first scenario and
also long charging times in the second scenario.
4.7 Efficiency Analysis for the Li-ion Battery
In order to find the optimal charging efficiency, the total energy stored in the battery and
energy loss is required. Assuming a constant total internal resistance (R = Rs + R1 = 0.026 Ω)
makes the analytical efficiency analysis possible. The energy loss in the battery during optimal
charging is already known and is equal to∫ tf0 Ru2
opt(t)dt. The total energy stored in the battery is:
Ebattery =
∫ tf
0V Idt =
∫ tf
0Vdq(t)
dtdt =
∫ qf
qi
V dq (4.40)
where I , V , and q are the battery current, OCV , and the charge in Ah, respectively. Using the
linear relationship between OCV and the charge stored in the battery from (4.27) and the definition
of SOC, the maximum energy stored in the battery is obtained as follows:
Ebattery = qmax(SOCf − SOCi)
[a
2(SOCf + SOCi) + b)
](4.41)
where the unit for energy is Watt-hours (Wh). The real maximum amount of energy which the
battery can store is obtained by integrating the original OCV -q profile which results in 8.2 Wh for
the cell used in this study. Using (4.41) and charging the battery from 0 to 100% the maximum
battery energy calculated is 8.26 Wh. This illustrates that the linear approximation of OCV for
Li-ion battery is an effective approach to perform analytical efficiency analysis. Substituting the
expressions for Eloss and Ebattery in (4.25), the optimal charging efficiency for Li-ion battery is
obtained as follows:
ρopt =1
1 + Rqmax(SOCf−SOCi)
tf(12a(SOCf+SOCi)+b)
(4.42)
where tf is the charging time in hours and R is the total internal resistance in ohms. Similar to the
SC, starting the charging from a higher initial SOC results in better efficiency. Also faster charging
91
will result in higher currents and lower efficiency. For the Li-ion battery, the constant power and
the two optimal charging scenarios are almost identical in terms of the charging current profiles and
also the efficiency values.
4.8 Effect of Temperature On Optimal Charging of the Li-ion Battery
In this section, unlike the unconstrained cases solved in the previous sections, the optimal
problem is solved subject to voltage and temperature constraints. This approach represents a more
realistic solution of the optimal charging of the Li-ion battery.
4.8.1 Electro-Thermal Model of the Li-ion Battery
In order to investigate the effect of temperature on the optimal charging current a thermal
model needs to be coupled with the electrical model and integrated in the optimal charging formula-
tion. The electrical model considered for this section is a single resistance model which represents
the total internal resistance as the sum of the electronic and polarization resistances (R = Rs +R1).
The dynamics of the electrical model is governed by the single state equation (4.7) with current
being the input. On the other hand, the thermal model is a reduced order model represented by two
states. For further details on reducing the governing PDE to two linear ODEs, please refer to [108].
The thermal model is identified and validated for the A123-26650 cell with specifications listed in
Table 4.6. The state space representation of the thermal model is:
x = Ax+Bu, y = Cx+Du (4.43)
where x = [T γ]>, u = [Q T∞]>, and y = [Tc Ts]> are state, input, and output vectors
respectively. The states of the thermal model are the volume-averaged temperature T in Kelvin (K)
and the volume-averaged temperature gradient γ in (K/m). The inputs are the ambient temperature
T∞ in Kelvin and the total heat generation rateQ. The outputs of the model are the battery’s surface
temperature Ts and core temperature Tc both in Kelvin. The linear system matrices A, B, C, and D
are:
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A =
−48βh
r(24kt+rh)−15βh24kt+rh
−320βhr2(24kt+rh)
−120β(4kt+rh)r2(24kt+rh)
B =
β
ktVcell
48βhr(24kt+rh
)
0 320βhr2(24kt+rh)
C =
24kt−3rh24kt+rh
−120rkt+15r2h8(24kt+rh)
24kt24kt+rh
15rkt48kt+2rh
D =
0 4rh
24kt+rh
0 rh24kt+rh
where r, Vcell, ρ, cp, kt, and h are radius, volume of the cell, volume-averaged density, specific
heat, conduction coefficient, and convective heat transfer coefficients respectively. The parameter
β = ktρcp
is the thermal diffusivity. The values for these measured and estimated parameters for the
cell used in this research are summarized in Table 4.7.
The total heat generation rate obtained from the electrical model is equal to RI2 and is fed
into the thermal model. To complete the coupling and form the electro-thermal model, core temper-
ature as an output of the thermal model is fed into the electrical model to tuneR. The variation ofR
with core temperature is estimated by pulse-relaxation experiments at different temperatures rang-
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Table 4.7: Li-ion physical and thermal parameters
Parameter Symbol Value Unit
Radius r 12.93e-3 m
Volume Vcell 3.4219e-5 m3
Density ρ 2047 kgm3
Specific Heat cp 1109.2 JkgK
Thermal Conductivity kt 0.61 WmK
Convection Coefficient h 58.6 Wm2K
ing from -20 C (253 K) to 40 C (313 K) according to [100]. Fig. 4.15 illustrates the estimated R
as a function of core temperature beside a function fitted to the estimated values which will be used
in the simulations. The fitted function is a 6th order polynomial:
R(Tc) =c1Z6 + c2Z
5 + c3Z4 + c4Z
3 + c5Z2 + c6Z + c7
where Z = (Tc + d1)/d2
(4.44)
The coefficients of the polynomial are listed in Table 4.8.
Table 4.8: Polynomial coefficients for R as a function of Tc
c1 0.0063599 c4 0.0071517 c7 0.035803
c2 -0.011979 c5 0.0094349 d1 -284.77
c3 -0.0017172 c6 -0.023463 d2 22.165
4.8.2 Optimal Fast Charging Considering The Temperature Effect for the Li-ion
Battery
The described electro-thermal model consists of three states. Formulating the efficiency
maximization problem using Pontryagin’s minimum principle will result in three state equations
and three co-state dynamic equations with three unknown initial conditions that need to be guessed.
This makes the problem tedious to solve using normal nonlinear ODE solvers. Dynamic program-
ming (DP) is used instead, to solve this optimal control problem relying on Bellman’s principal of
94
Figure 4.15: Internal resistance R as a function of core temperature for the Li-ion battery.
optimality. We are interested in investigating fast charging scenarios to be able to trigger the voltage
and temperature limits, in order to study their effect on efficiency and the optimal charging current
profile. For this reason charging in 10 minutes is chosen where the initial charging temperature
is T∞= 25 C. The resolution for the time in the DP code is 20 seconds. At each time instant ti,
all three states of the electro-thermal model are quantized and represented by (SOCi, T i, γi). Fig.
4.16 demonstrates the implementation of DP. As shown in this figure, each grid represents the three
states (SOCi, T i, γi) moving from time ti-1 to ti with multiple possible transitions. However, only
the transitions that satisfy the state equations are acceptable which are named the admissible tran-
sitions in this study. The first phase of the DP algorithm is solved backwards in time. During the
backward phase and for every admissible transition, the minimum cost-to-go is obtained for each
time instant to the final time along with all the corresponding optimal control inputs.
During the forward phase of DP with the knowledge of the initial conditions for the states,
the optimal control input which in this study is the optimal charging current is obtained.
At first similar to the other sections in the chapter, the constraints are relaxed and the effi-
ciencies obtained from DP and CC charging from zero to full charge in 10 minutes are compared.
This result is summarized in Table 4.9 where the optimal charging efficiency is slightly higher than
CC charging.
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Figure 4.16: Illustration of the DP grid and sample transitions.
A real world implementation would require constraints on the terminal voltage and core
temperature. In the second step, the similar optimal charging problem is solved subject to a voltage
constraint of 2 V ≤ VT ≤ 3.6 V for both DP and CC charging. In the final step, keeping the
terminal voltage constraint, the optimal trajectory and efficiency is obtained by adding the core
temperature constraint of Tc ≤ 39 C to both the DP and CC charging methods. Neither CC nor DP
directly apply a limit on the maximum charging current as the voltage and temperature constraints
indirectly result in limited charging current when they become active. The amounts for the voltage
and temperature limits in the constraints are chosen from the manufacturers recommended operating
conditions. In all these cases the battery is charged from 0% to 90% since charging the battery
fully is not possible due to the aforementioned constraints as well as the short charging time of 10
minutes. The odd columns of Table 4.10 and Fig. 4.17 present the results for the CC charging. For
comparison reasons, the unconstrained CC and DP charging scenarios are re-simulated for charging
from 0% to 90%. Constant current constant voltage (CCCV) charging is the CC charging with the
voltage constraint imposed. Also CCCV∗ is the notation for CC charging with both voltage and
temperature constraints active. During CCCV charging as the inset in the first subplot in Fig. 4.17
shows, the current drops at the end of charging as the terminal voltage hits the limit of 3.6 V. During
the CCCV∗ charging the amount of initial current (20 A) is chosen such that the battery could be
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charged to 90% SOC. According to Fig. 4.17 at about 30 and 200 seconds into the CCCV∗
charging, the voltage and temperature limits are reached respectively, which results in a stepwise
decrease in current. Comparing the efficiencies listed in Table 4.10 for CC, CCCV, and CCCV∗
shows that imposing constraints on the CC charging results in smaller values for efficiency.
The even columns of Table 4.10 and Fig. 4.18 show the results for the optimal charging
using DP. The unconstrained DP, voltage constrained, and the double voltage/ temperature con-
strained problems are indicated as DP1, DP2, and DP3, respectively. The results obtained using DP
as the optimal charging currents have a slightly higher efficiency compared to their equivalent CC
charging scenarios. The general trajectory of all the three DP cases shows a warm up period at the
beginning of charging followed by a constant current charging section. The physical reason behind
this behavior is that the higher current at the beginning of charging results in a rapid increase in core
temperature which consequently decreases the total internal resistance. The lower total internal re-
sistance is in favor of minimizing the charging losses according to the objective function in (4.8).
The larger current at the beginning of charging results in terminal voltage reaching the upper limit
which results in a decrease in current for DP2 and DP3 scenarios. For DP3 as also shown via the in-
set in the second subplot in Fig. 4.18, the temperature limit is activated towards the end of charging,
which results in an instant decrease in current to avoid overheating. The DP results are based on a
20 second resolution on time which maybe the reason for the ripples in current specifically obvious
in the DP2 case.
These optimal solutions are computed using Clemson University’s Palmetto cluster because
of the high memory requirements of the DP implementation. In this particular problem, the three
state variable of SOC, T , and γ are quantized to nSOC = 200, n T = 25, and nγ = 25, respectively.
This requires each grid of Fig. 4.16 to accommodate nSOC×nT×nγ = 125000 cells. Moreover each
grid is stored by 125000×125000 cells in the implementation as each cell at time instant ti-1 has also
125000 possible transitions to the next cell at time instant (ti). In the implemented DP code, 10 vari-
ables are involved in DP backward computation with mixed single/double precision (6 bytes on av-
erage). This would mean that the minimum required memory allocation is 10×125000×125000×6
bytes or ≈ 870 GB. The involved variables are the two inputs to the objective function (R and I),
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Table 4.9: Unconstrained CC and optimal charg-ing in 10 minutes
Unconstrained
CC DP
Charging Current (A) 15 20 max.
Core Temperature (°C) 43 max. 42 max.
Terminal Voltage (V) 3.8 max. 3.7 max.
State-of-Charge @tf 100% 100%
Energy Loss (J) 2682 2638
Efficiency 91.72% 91.84%
CC: Constant CurrentDP: Dynamic Programming
Table 4.10: CC and optimal charging in 10 minutes considering volt-age and temperature constraints
Temperature-constrained
Unconstrained Voltage-constrained Voltage-constrained
CC DP1 CCCV DP2 CCCV∗ DP3
Charging Current (A) 13.5 19 max. 13.6 max. 19 max. 20 max. 18 max.
Core Temperature (°C) 41 max. 39 max. 41 max. 40 max. 39 max. 39 max.
Terminal Voltage (V) 3.6 max. 3.7 max. 3.6 max. 3.6 max. 3.6 max. 3.6 max.
State-of-Charge @tf 90% 90% 90% 90% 90% 90%
Energy Loss (J) 2254 2231 2280 2236 2325 2228
Efficiency 92.95% 93.01% 92.87% 93.00% 92.74% 93.02%
CC: Constant CurrentDP: Dynamic ProgrammingCCCV: Constant Current Constant VoltageVoltage-constrained: 2V ≤ VT ≤ 3.6VTemperature-constrained: Tc ≤ 39 °C
98
0 100 200 300 400 500 6000
10
20
Current(A
)
0 100 200 300 400 500 60025
30
35
40
Tc(C)
0 100 200 300 400 500 6003
3.2
3.4
3.6
VT(V
)
0 100 200 300 400 500 6000
1000
2000
Energy
Loss(J)
Time (s)
CC CCCV CCCV*
570 585 600
13.4
13.6
Figure 4.17: CC charging results for charging from 0% to 90% in 10 minutes considering the temperatureand voltage constraints.
99
0 100 200 300 400 500 6000
10
20
Current(A
)
DP1 DP2 DP3
0 100 200 300 400 500 60025
30
35
40
Tc(C)
0 100 200 300 400 500 6003
3.2
3.4
3.6
VT(V
)
0 100 200 300 400 500 6000
1000
2000
Energy
Loss(J)
Time (s)
570 585 60038
39
40
Figure 4.18: Optimal charging results via DP for charging from 0% to 90% in 10 minutes considering thetemperature and voltage constraints.
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three state variables (SOC, T , γ), one combined constraint, and four cost functions (cost-to-go
functions between ti, ti-1, and tf).
4.9 Conclusions
This study solved the optimal charging problem for different energy storage systems con-
sidering different levels of model complexity and also compared the optimal charging current with
constant power, constant current, and constant voltage charging strategies. Efficiency analysis was
also performed to compare different charging strategies. For the supercapacitor and the first scenario
of the Li-ion battery, the optimal charging strategy is constant current. In the supercapacitor case,
the constant power charging current has peaks that may make it less favorable to apply as it is also
less efficient. In the Li-ion case, there is not much difference in applying constant power or constant
current considering the efficiency and shape of the current profile. For the lead-Acid battery, the op-
timal strategy has a better efficiency than constant power charging, while constant current charging
has the least efficiency. Finally, the effect of temperature dependent model parameters as well as the
voltage and temperature limits on the optimal fast charging of the Li-ion battery was investigated.
This constrained optimization problem was solved using dynamic programming and was compared
to constant current charging. The results show that the charging efficiency of the optimal scenario
is slightly higher than constant current charging.
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Chapter 5
Fast Charging Challenges in
Lithium-Ion Batteries
5.1 Literature Review
The ever-growing world energy demand as well as dwindling fossil fuel resources motivates
exploring alternative solutions to the global energy problem. The transportation sector powered by
combustion engines, the industrial plants and residential buildings running on electricity produced
by non-renewable sources, are all gradually stepping towards a sustainable and low carbon model
with a zero carbon footprint within the horizon. Higher degrees of renewable penetration in the grid
alongside the electrification of ground vehicles will set the stage for a sustainable development in
both transportation and power generation sectors. Within this area of research, energy storage has
a pivotal role in electric vehicles. It is also a key component in grid applications helping improve
efficiency and system performance by allowing better supply and demand management particularly
for grids with higher renewable shares. The need to reduce the time required to charge a module res-
onates with the needs of the end-users and the acceptability of the technology in many applications.
for example, in public transportation sector there is a tendency to explore electric driven buses. The
predefined and available information regarding the route and the stops during passenger boarding
are the driving force to explore the possibilities of fast charging. Justino et al., have designed ultra
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fast charging stations for ultracapacitor driven electric buses in order to charge in a time interval
of 10-30 seconds with a required power rating of 400 kW [178]. The reason for such a high rate
capability is the high power capacity of supercapacitors [174]. In another study, Mapelli et al., study
the fast charging of an urban electric bus powered by a hybrid battery/supercapacitor energy storage
system [179]. In both of these examples the supercapacitor module is charged during stops and not
the batteries. The reason that batteries can not be used in such fast charging scenarios (under 20
minutes) is that, the electrochemical, mechanical, and thermal constraints limit the rate of charging
of the cell. The efforts to improve the charging time of lithium ion batteries can be categorized in
the following directions:
• Modeling the cell using lumped or detailed electrochemical models to deeply understand the
electrical and thermal dynamics of the battery. This approach allows to push the charging rate
to the feasible limits of the cell without violating critical constraints.
• Exercising different control strategies to charge the battery in a shorter amount of time com-
pared to the standard charging protocols currently in use.
• Seeking for novel materials for the cell components such as electrodes and the electrolyte
with higher power capacity which allow charging at higher rates.
Understanding the electrochemical, thermal, and mechanical dynamics as well as limita-
tions of lithium ion batteries, provides a powerful tool to analyze the operation bottlenecks and
therefore push the limits in order to achieve performance goals such as faster charging times. Within
this context, an accurate model will serve the purpose in the case of fast charging. The knowledge
of different ionic and electronic resistances such as charge transfer, diffusion, and electrolyte re-
sistances beside the electronic resistance of the current collector will facilitate the development of
cells where the current flows in and out in higher rates which in return results in faster charging
times. This study will review the models capable of capturing the dynamics of the cell during fast
charging.
The standard charging protocol for lithium ion batteries is the constant current-constant
voltage (CC-CV) scheme [180–182]. It consists of two phases. At the beginning the battery is
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charged by a constant current till it reaches a predefined upper voltage limit. The next step is
maintaining a constant terminal voltage which results in a tapered decreasing current. The charging
is terminated when the current drops below a certain threshold. This charging mechanism is believed
to be conservative while satisfying the safety standard during charging. Constant current charging
without the constant voltage phase in another option. However, the drawback is the limited capacity
utilization unless applying a very low charging current which substantially prolongs the charging
time. There are other numerous charging algorithms applied to lithium ion batteries with different
objectives that are not necessarily aiming to reduce the time of charging. Strategies that replace the
CV stage of a CC-CV charging with pulse of multi-step CC charging are reported [163, 183, 184].
Some studies use pulse currents for the whole duration of charging [185–187]. The term C-rate is
used in the battery literature to define a relationship between the capacity of the battery in ampere
hours (Ah) and the charging (discharging) current as well as time. Consider “C” to stand for capacity
of the cell and “n”to be a real number. In this context, a C-rate equal to “nC” is defined as the
amount of current with a magnitude of “n.C”, that is able to charge the cell in 1n hour. With this
definition it is clear that applying a higher current will result in a shorter charging time. Having
this in mind, boost charging which features higher CC charging intervals compared to the charging
rate of a CC-CV protocol, results in a shorter charging time [188, 189]. More recently, Keil and
Jossen have experimentally investigated the effect of different charging protocols on the cycle life
of three lithium ion chemistries where they conclude that the influence of the charging rate on
the deterioration of the cell is higher than the effect of the discharging current [190]. In the later
sections, the control strategies applied to the problem of charging with the objective of reducing the
charging time will be discussed.
The materials used to fabricate a lithium ion cell is another area where the power capacity
and rate performance can be improved. A battery cell consists of two circuits. The internal circuit
is the pathway for the ionic transmission, the external circuit is a passage for the electronic charge.
These two circuits guarantee the charge conservation. The internal circuit in case of the lithium ion
battery is the corridor for the transmission of lithium ions where they move from one electrode to
another through the electrolyte and the separator. The function of separator is to insulate and prevent
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electronic short circuit between the electrodes. The most commonly use material for the commer-
cialized lithium ion battery anode is graphite. Among the wide spectrum of material for cathode,
LiMO2 (M: Co, Mn) are the favorites with LiCoO2 being the most prevailing [191, 192]. During
discharging, the lithium atom that was stored in the bulk of the anode material during charging, is
extracted in the form of a lithium ion by losing a single electron (oxidation reaction) and migrating
through the electrolyte toward the cathode. The lithium ion is then intercalated into the bulk of
the cathode material by gaining an electron through a reduction reaction. Research to increase the
rate of discharge in lithium ion battery by the means of new materials often focus on the cathode
composition [193–201]. On the other hand, the charging process which is the interest of this study,
consists of the redox reactions in the opposite direction of discharging. The review of the anode
materials that are capable of undergoing high current for fast charging purposes will be discussed in
the coming sections.
On the charger hardware and the infrastructure side, there are three charging levels [202].
The level 1 charging is slow and the charger is an on-board single phase charger with a power rating
of 1.4-1.9 kW used to charge an electric vehicle (16-50 kWh) over night (11-36 h). The semifast,
level 2 charger is also on-board, often single phase, with a power of 4-19.2 kW which is capable of
charging the electric vehicle (3-50 kWh) in 2-3 hours. The third level which is the fast charging case,
is a 3-phase, off-board charger and suitable to charge an electric vehicle in 10-30 minutes [203].
According to studies such as the one conducted by Botsford et al., the expectation of the stop time
for charging is even less than what the level 3 charger can provide. The aim is to compete with
the 3 minute stop time of a regular gasoline driven vehicle [204]. To achieve such a goal all the
subsystems involved, from the battery to the charging infrastructure need to dramatically improve.
In this research, we will focus on the battery as the system of interest and review the literature to
report the current research and development efforts on the modeling, control, and material selection
that focus on improving the charging time of the lithium ion batteries.
The remainder of this chapter is organized in the following order. In Section 5.2, the un-
wanted side effects of high rate charging are reviewed. In Section 5.3, the models facilitating the
study of fast charging are described. Section 5.4, covers the control strategies with the objective
105
to minimize the charging time. Sections 5.5 and 5.6, discuss the anode materials suitable for fast
charging and conclusion remarks, respectively.
5.2 Side Effects of Fast Charging
5.2.1 Lithium Plating
This phenomena is referred to the deposition of reduced lithium cations on the surface of
the carbonaceous anode material during charging events. This process is opposed to the lithium
intercalation that is suppose to normally occur in the negative electrode of a lithium ion battery. The
major macroscopic factors that cause lithium plating are high current rates as in fast charging as well
as low temperature conditions. Lithium metal deposition on the graphite is accountable for severe
battery performance, health, and safety degradation. This is the motivation for a vast body of liter-
ature to detect the occurrence of lithium plating by the means of experiments as well as modeling
approaches [205]. Using a wide spectrum of models, researchers have elucidated that the concen-
tration saturation of lithium ions in the vicinity of the anode is an important factor that triggers the
lithium deposition [206]. On the other hand, it has been proven that the overpotential of the lithium
deposition reaction vs. Li/Li+ being less than zero is another indicator of the onset of lithium plat-
ing [207, 208]. Perkins et al., have developed a reduced order model that captures the overcharge
lithium depositions by tracking the resulting capacity loss and resistance rise [209]. While these
studies at large focus on the intercalation kinetics in the anode, the electrolyte composition is also
of significant importance in order to determine lithium plating [210]. On the qualitative side and by
the means of scanning electron microscope (SEM) and optical in-situ microscopy, the reactions on
the anode have been studied under high charging currents. In one of the earlier studies, Orsini et al.,
correlated the morphology of the lithium deposit with the current density for three types of anode
(Li, Cu, and graphite) and a LiMn2O4 cathode. The SEM micrographs showed lithium dendrites
formed on the Li and Cu electrodes under high currents, and mossy shape lithium under low cur-
rents. No special morphology was observed for the deposited lithium on the surface of the graphite
electrode [211]. More recently, Uhlmann et al., induced plating on the anode in graphite half cell by
106
applying pulse-relaxation currents up to 10C. Terminal voltage and changes in anode morphology
were analyzed during pulses and the subsequent relaxations. The pulse-relaxation measurements
indicated a specific bending in the voltage transient during charging followed by a potential plateau
in the relaxation period in case of a plating event. Furthermore, the lithium plating was confirmed
by both SEM and In-situ optical observations [212].
Efforts have been made to detect lithium plating by monitoring the variation of coulombic
efficiency [213], change in thermal heat flow via electrochemical microcalorimetry [214], and in-
crease in cell thickness [215,216]. The other operation condition that makes the anode susceptible to
lithium plating is low temperature [217, 218]. Zinth et al., utilize in situ neutron diffraction method
to study lithium deposition on the anode in a NMC/graphite cell, in a temperature of -20 °C and
under charging currents of C/5 and C/30. Results show that the rate of plating is higher under the
higher current of C/5. Also the percentage of the deposited lithium in a 20 hour relaxation period
is 17% while this rate is 19% for a discharging event right after a C/5 charging rate. The authors
also conclude that the length of the plateau observed in the discharging voltage profile is a measure
of the amount of plated lithium. Also unlike other methods they conclude that the neutron diffrac-
tion method shows that a huge fraction of the lithium plating is reversible [219]. In another study,
Schindler et al., look at the relaxation period right after charging the cell to a certain capacity at sub-
zero temperatures and combine voltage differential and electrochemical impedance spectroscopy
methods to identify the lithium deposition by tracking any changes in the cell impedance [220].
5.2.2 Mechanical Failure
The insertion/extraction of lithium ion into the volume of the anode and cathode induces
mechanical stress in the host material. The stress occurs when the battery is charged at a rate higher
than the rate that lithium can homogenize in the active material by diffusion [221]. This lithiation-
induced stress appears in the form of fracture or structural deformation in the majority of active
materials such as graphite, silicon, LiFePO4, LiMn2O4, and LiCoO2 [222–230]. One approach has
been to use mathematical modeling and fracture mechanics, to simulate the failure mechanisms in
the battery [231–237]. Christensen et al., developed a mathematical model to calculate the volume
107
expansion/contraction and the associated stress in a spherical particle of the electrode material. The
model is based on the fact that by lithium insertion a strain differential is induced between the in-
ner and outer regions of the sphere. This strain differential causes a stress that increases with the
rate of intercalation and will result in particle fracture, if surpasses the yield stress of the material.
The particle size, diffusion coefficient, and the charging rate have been considered in the model.
The model predicts that the carbon based materials will fracture at high power applications [238].
Applying a similar concept, the same authors modeled the fracture in LiMn2O4 electrodes, consid-
ering solid-state diffusion in a one-phase material and moving-boundary insertion in a two-phase
material. The results show that in a discharging event it is important to avoid over-discharge as
it results in immediate fracture [239]. In another study, Zhang et al., numerically simulate 1D
spherical as well as 3D ellipsoidal particles for a LiMn2O4 cathode. The results suggest to syn-
thesize the electrode particles to have smaller size and larger aspect ratio, in order to reduce the
intercalation stress. Furthermore, to decrease the resistive heat generation rate and the intercalation-
induced stress, the authors show that ellipsoidal particles with larger aspect ratios are preferred over
spherical particles [240, 241]. The conclusion that smaller particle size will result in less lithiation-
induces stress is also true for electrodes with different silicon material geometries such as nanowires,
spheres, and cylinders [242–246]. More recently Chen et al., combine the fracture formation and
impedance model to investigate the effect of mechanical degradation mainly caused by cracks, on
the impedance behavior of the cell. Results indicate that the microcracks have a larger influence on
the impedance response during delithiation compared to the lithiation process. Furthermore, porous
electrode impedance response suggests that the diffusion induced damage at low temperature and
high discharge/charge currents, has a larger influence on the resistance. The higher concentration
gradient explains the reason for this behavior [247]. Laresgoiti et al., propose a model based on
spherical graphite particle with solid electrolyte interphase (SEI) layer. The paper focuses on the
stress in the graphite and the SEI layer, induced by diffusion and the graphite expansion, respec-
tively. The paper further assumes that the SEI layer is more susceptible to fracture than graphite
and concludes that unlike the active material, stress in the SEI is not influenced by the current rate,
diffusion coefficient, and the particle size [248]. Barai et al., utilize a stochastic approach based on
108
the lattice spring model to come up with a fracture phase map to identify the safe/unsafe operating
conditions in terms of charging rate and particle size [249]. Researchers have also elucidated the
damage under high currents, experimentally. Choe et al., used acoustic emission (AE) technique
to study the damage taking place in a coin-type battery with lithium cobalt oxide/carbon electrodes
under accelerated cycle tests. The short duration and high amplitude of the AE signals indicated
cracking of the lithium cobalt oxide while the long duration and low amplitude signals explained
the bubbling due to the SEI layer formation on the anode [250]. In another study Kostecki et al., use
different experimental techniques such as atomic force microscopy, scanning electron microscopy,
gas chromatography, Raman spectroscopy, and synchrotron IR microscopy with the goal of iden-
tifying detrimental indicators under high current operation and storage at elevated temperatures.
The results suggest that the SEI deterioration on the anode, morphology change, and increase of
impedance in the cathode contribute to the loss of power in the cell [251].
5.3 Suitable Models for Fast Charging
The terminal voltage, current, and temperature are the only measurable parameters in a
battery cell. In order to obtain a comprehensive understanding of the electrochemical and thermal
dynamics of a cell, having a reliable and accurate model is inevitable. In general, there are two
categories of models: first principle full-order models and the reduced order models. The deciding
factor in the level of model complexity, is the application and operating conditions of the battery.
Depending on the purpose of the application, the importance of model accuracy versus computa-
tional efficiency is weighted. In the full-order models, all the underlying physics in a battery cell is
taken into account. This macroscopic approach is formulated by considering the porous electrode
and concentrated solutions theory. Such models include partial differential equations governing
the lithium ion concentration in the solid and electrolyte phases, solid and electrolyte potentials,
electrolyte ionic current, and the intercalation kinetics [252–255]. The large computational mem-
ory and power required to numerically solve the coupled system of PDEs is a burden in real time
applications. This has motivated a large community of scientists to seek techniques of reducing
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the complexity of the full order models [209, 256–261]. One direction pursued in the literature,
is using numerical methods such as volume averaging and parabolic approximation of the solid-
phase concentration [111,262–264], residue grouping [265], Pade approximation [266], and proper
orthogonal decomposition method [267]. These group of models with be referred as numerically
reduced order models (NROM) within this paper. Another direction in model reduction that has led
to a lower fidelity but widely used model, is the single particle model (SPM) [268–271]. It acquires
its name due to modeling each electrode with a single sphere particle with a surface area equivalent
to the active material area of the porous electrode. The simplification in this model results by as-
suming that the lithium concentration in the electrolyte is constant in both space and time. Another
popular set of reduced order models are the equivalent electric circuit models [100, 103, 272, 273].
This group of models are computationally efficient but are unable to fully capture electrochemical
states and parameters of interest. The model suitable for fast charging scenarios, should be accurate
enough to facilitate studying the effect of high currents on the electrochemical and thermal dynam-
ics of the cell. Among the models discussed the full order or NROM electrochemical models are
the most promising in simulating the galvanostatic charging behavior in fast charging events. These
group of models are capable of capturing the ionic transfer in both solid (electrode) and liquid (elec-
trolyte) phases. The SPM model lacks the complete dynamics of the lithium ion concentration. As
this factor is important at high C-rates [271], therefore the SPM model is not suitable for fast charg-
ing studies. Lumped equivalent electric circuit models lack the level of accuracy required to mimic
the electrochemical processes that limit the performance of lithium ion batteries at high currents.
On the other hand, the thermal dynamics of the battery cell will also be of significant importance
due to large temperature variation caused by high currents. The coupling of full order or NROM
electrochemical models with a thermal model through the total heat generation and the temperature
dependent parameters [274–276], will facilitate studies that focus on high charging rates. Hasan
et al., utilized a coupled 1D electrochemical-thermal model reported in [277], and the software
AutoLion-1D, to study the performance of lithium ion batteries under different charging protocols,
electrode designs, and the basic mechanisms of fast charging at -20 °C (low), 25 °C (moderate), and
45 °C (high) temperatures. The results elaborate that the electrolyte resistance dominates the fast
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charging process at moderate and high temperatures, while the charge transfer resistance is the dom-
inant factor at subzero temperatures. It is shown that at subzero temperatures, adiabatic conditions
compared to isothermal conditions, result in a lower charging time and higher coulombic efficiency.
The reason is the elevated temperature of the cell under adiabatic conditions due to the trapped heat,
generated inside the cell, which suppresses the charge transfer limitations and the electrolyte resis-
tance [278]. In another study, Ye et al., investigate the effect of thermal contact resistances during
fast charging in prismatic battery packs by taking a close look at temperature, temperature gradient,
and the spatial non-uniformity of current and state of charge (SOC) in the cross plane direction. The
model utilized in this study was a two dimensional electrochemical thermal model [279, 280]. The
results show that during fast charging neither constant current nor pulse charging are effective charg-
ing methods from a thermal point of view, as even intensifying the external cooling in not adequate
to avoid thermal runaways. Authors propose optimized battery design by reducing the cell thickness
as a solution to this thermal management problem [281]. Smith et al., study the charging of lithium
ion battery with a capacity of 6 Ah and nominal voltage of 2.7 V with a upper voltage limit of 3.9 V,
during charging. The authors conclude that the lithium deposition does not occur when charging
the battery from different initial SOCs to the upper voltage limit of 3.9 V, with currents rated at
approximately 15C. They suggest that setting the difference in solid and electrolyte potential rather
than the upper voltage limit as an indicator of the lithium plating occurrence increased the allowable
charging rates by 50%. The model used in this study is also a 1D electrochemical-lumped thermal
model which has allowed such detailed analysis [282]. Zou et al., discretized a 1D electrochemical
coupled with a thermal model and tune the level of discretization and order of polynomial approxi-
mations as a function of the charging rate to achieve higher model accuracy. The results show that in
order to achieve a normalized root-mean-square of 1% in the terminal voltage, compared to the full
order model, the single particle model will no longer be accurate for charging rates above 1C [283].
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5.4 Controls in Fast Charging
Unlike the discharging event that depends on the requested load, the charging process is
controllable. Certain parameters and constraints are taken into account during charging to ensure
safety, reliability, and the feasibility of the process. Considering safety from a macroscopic point of
view, the terminal voltage and temperature of the cell should be bounded in a certain range to avoid
over-potential and thermal runaways. Having cost in mind, in the long run it is favorable to extend
the battery cycle life and maintain its state of health. On the other hand, the efficiency of charging
has an important role in determining the charging strategy as it is directly linked to the cost, aging,
and safety of the cell. Last but not the least, is the charging time that is a game changer in terms
of enhancing the feasibility and acceptability of the lithium ion technology. Within this domain,
control engineering facilitates the implementation and realization of the various objectives. The
controls literature consists of studies that shed light on the charging of lithium ion batteries with the
battery cycle life in mind [165,190,284–290]. Many researchers have also investigated the charging
problem with the objective to minimize losses and improve the charging efficiency [103, 160, 291].
The main focus of this section will be on reviewing papers that emphasize on the minimum time
problem by using different control strategies [188, 189, 292]. Torchio et al., apply a quadratic dy-
namic matrix control (QDMC) approach to a pseudo two-dimensional (P2D) model of a lithium
ion battery with the objective of minimizing the charging time. To overcome the computational
burden in solving this optimal control problem, the authors have developed an approximation for
the P2D model using the least square approach. This frame work which satisfies the constraints
on output voltage, input current, and battery temperature results in charging times as low as 1200
seconds [293]. In [294], Klein et al., also use a macro-homogeneous 1-D electrochemical model of
the lithium cell with the aim to challenge the conservative constraints on current and voltage usually
used in CC-CV charging scheme. The constrained optimal control problem with the objective of
achieving the fastest charging time is formulated using nonlinear model predictive control (NMPC).
The results show a 50% decrease in the charging time compared to the CC-CV method. In another
study Moura et al., use the electrochemical states with the assumption of full state measurements
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along with a reference governor (RG) algorithm to impose constraints such as side reaction over-
potential, rather than the conventional upper voltage limit. The charging time of the RG method
is 24 min vs 38 min for CC-CV [295]. Liu et al., in [163], introduced a multi-stage constant cur-
rent charging profile by the means of Taguchi-based algorithm and orthogonal arrays that resulted
in charging the lithium ion battery to 95% in 130 minutes which was 11.2% less than the CC-CV
charging time.
5.5 Anode Materials for Fast Charging
There is an exuberant stream of literature delving into the different materials to be used in
lithium ion batteries as electrodes and electrolyte [296]. In this section the focus is on the materials
used for the negative electrode of lithium ion batteries due to its importance during the charging
process. Beside graphite that is the most widely used anode material many other materials have
also been studied. Goriparti et al., categorize the anode materials based on the chemical reaction
mechanisms [297]. First is the carbonaceous materials, where the intercalation/de-intercalation is
the dominant reaction. Porous carbon [298], carbon nanotubes [299,300], graphene [301–303], and
carbon nanofibers [304, 305] are in this category. The second group, includes materials with the
alloy/de-alloy mechanism such as silicon [306, 307], silicon oxides [308–310], germanium [311–
314], and tin [315]. The last category are the conversion materials such as metal oxide, nitride,
phosphide, and sulfides [316–325]. To be able to summarize the efforts on the material side, in
this study we focus on studies of anode materials with a nano-scale particle size for the active
material, which report a charging rate of 3C (20 minutes) or faster. Yang et al., examine a set of
three hard carbons with different nano-sized porous structures. With a high C-rate of 5C the cell
demonstrates a capacity of 332.8 mAhg−1. The reason that the cell can undergo higher charging
rates, is lower resistance and shorter pathways as a result of the nano-sized porous texture which also
contributes to the larger surface area for additional intercalation sites for the lithium ion [326]. Lee
and Park in [327], utilize a specific process to synthesize 3D porous SiO (silicon monoxide) similar
to approaches in [328–330], without altering the chemical and physical properties of SiO. The final
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carbon-coated silicon monoxide demonstrated a high capacity of 1490 mAhg−1 after 50 cycles and
a high rate capability of 74% at 3C compared to 0.1C rate. Seng et al., compare the capacity of
GeO2/Ge/C, GeO2/C, nanosized GeO2, and bulk GeO2 as anode materials. The GeO2/Ge/C sample
showed the highest rate capability compared to others where the capacity was 1750 mAhg−1 and
1680 mAhg−1 at the 5C and 10C rates, respectively. The capacity for the 10C rate was only 10%
less than the recorded capacity of 1850 mAhg−1 at a rate of 0.1C [331]. In another study, Wang et
al., directly deposit Ge on the surface of pre-synthesized Cu nanowire arrays via an rf-sputtering
method and use it as the electrode material. The Cu-Ge nanowire arrays display a capacity of
734 mAhg−1 after 80 cycles and under 60C current rate [332]. In another example of a germanium
based anode, Yuan et al., utilized specifically synthesized germanium nanowires that can undergo
currents as high as 11C with 555 mAhg−1 reported capacity [333]. Titanium oxide (TiO2) based
anodes also exhibit high rate characteristics. Liu et al., have synthesized microspheres of TiO2-B
with a pore size on 12nm. This material has showed promising results in terms of being a high power
anode material where it has a capacity of 120 mAhg−1 at a charging rate of 60C [334]. Li4Ti5O12
(LTO) is another anode material in lithium ion batteries. Prakash et al., reported using LTO as high
power anode materials with capacities of 140 mAhg−1 and 70 mAhg−1 at 10C and 100C current
rates, respectively. The nano and highly porous morphology of the electrode contributes to the high
rate capability by shortening diffusion paths [335]. Spindle-like porous α-Fe2O3 has been the main
focus of Xu and his colleagues in [336], as an anode material. The simple synthesize process beside
the smaller size of α-Fe2O3 nanoparticles cause shorter distance for transportation of Li ions, which
is critical to the rate capability. The reported capacity for such an anode was 911 mAhg−1 after 50
cycles with a rate of 0.2C. Cycling under a rate of 10C, the achieved capacity was 424 mAhg−1.
Cobalt oxides are another material used as anodes in lithium ion batteries. For example, Guan et al.,
fabricated CoO octahedral nanocages, where it was tested as anode material and exhibited good rate
capability. Under a current rate of 5C, the anode could deliver a reversible capacity of 474 mAhg−1
[337]. Lu et al., have successfully practiced fabricating a composite Ni5P4/C material for use as
anode material in lithium ion batteries. The carbon shell enhances the conductivity and limits the
aggregation of active particles which in return results in structural stability and reversibility during
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cycling. This composite material can retain a capacity of 644.1 and 357.1 mAhg−1 under currents
rates of 0.1C and 3C, respectively [338].
5.6 Conclusions
The potential to recharge an electric vehicle in an equivalent time to refuel a gasoline vehicle
as well as charging electronic devices in a fraction of the time that takes to charge them nowadays,
is the motivation for many researchers to address the problem of fast charging. A large decrease in
charging time is often associated with high currents. High current induces different types of stress
on the battery. Thermal runaway, lithium plating, mechanical failure are a few of the unwanted
circumstances of high current. A search through a wide spectrum of models for lithium ion batteries,
reveals that due to the sensitivity of the electrochemical and thermal processes to high currents, the
development and utilization of detailed first principle models provides the edge to implement more
effective fast charging strategies. On the other hand, it is identified that there is a need for advanced
control strategies that rely on the electrochemical-thermal modeling of lithium ion batteries, as
well as taking different electrical and thermal constraints into account while aiming to reduce the
charging time. Within the material science and engineering community there is a vibrant search
for feasible cost effective and high rate materials to address the charging time problem. Having
the charging process in mind, different anode materials such as carbonaceous, alloy/de-alloy, and
conversion materials with nanostructure morphologies show promise in high rate applications.
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Chapter 6
Dissertation Summary
6.1 Contributions
The major contribution of this dissertation is the proposition and validation of models and
control strategies that overcome imperative problems such as limited driving range of electric vehi-
cles and long charging times in batteries. The summary of each study is presented in this chapter.
In chapter 2, an electro-thermal model for a supercapacitor is derived and experimentally
validated. The contributions are:
1. The dependence of the electrical model parameters on temperature (range:-40 °C to 60 °C),
state of charge (SOC) (range:0-100), current direction (charge/discharge), and also for the
first time, current magnitude (range: 20-200 A) is embodied in the model.
2. The experimentally validated results show that the terminal voltage dynamics is captured with
high accuracy (20-87 mV), suitable for all kind of power system applications.
3. The computationally efficient thermal model captures temperature dynamics with high accu-
racy (0.17-0.21 °C) suitable for real-time thermal management systems.
4. The coupled electro-thermal model is validated using a practical driving cycle rather than
simple constant current cycles often used in the literature.
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Chapter 3, describes the benefit analysis on driving range extension, for a lead-acid driven
vehicle hybridized via supercapacitors. The contributions are:
1. The control strategy is simply implementable using inexpensive commercial micro controllers
and the converter is specifically built for this study, which enables monitoring the real world
efficiency bottlenecks of the hybrid energy storage (HES).
2. The developed HES model is fully scalable. For validation purposes, a 12 V hardware in
the loop (HIL) setup is used to implement the control strategy and extract the driving range
results.
In chapter 4, specifically focuses on improving the charging efficiency and duration of su-
percapacitor, lead-acid, and Li-ion batteries by the means of analytical and numerical methods in
optimal control. The contributions are:
1. The comparison of different charging strategies (constant voltage, constant current, and con-
stant power) and the associated efficiencies with their optimal charging counterpart. In case of
the lithium ion battery, the effect of temperature dependent model parameters on the optimal
charging current is investigated, by the means of using a validate electro-thermal model and
dynamic programming as the numerical technique.
2. The intention to solve this optimal control problem analytically and the gradual increase in the
model complexity, gives the study a tutorial touch which can be used as a basis for research
focusing on charging and in a larger scale, studies choosing to analytically solve optimal
control problems.
In chapter 5, the focus is on bottlenecks of fast charging in lithium ion batteries. The
contribution is through providing guidelines from the modeling, control, and material science and
engineering fields, that contribute to decreasing the time required to charge lithium ion batteries.
117
6.2 Discussions and Future Work
Beside the implementable results reported in this dissertation there is a huge potential to
expand beyond the presented studies. In the transportation sector, the electric vehicle technology is
in its infancy. Despite the noticeable progress in the past couple of decades in increasing the energy
density of batteries, the limited range of these vehicles are still an ongoing challenge. One direction
in addressing this problem is to seek for new materials that improve the energy density while at
least maintain the current performance requirements as well as competing with the decreasing cost
of the current technology. Another venue to investigate is the implementation of alternative energy
storage technologies in either stand-alone scenarios or in combination with batteries. In this regard,
supercapacitor with its high power density was studied as the alternative storage technology in this
dissertation. With the decreasing cost of supercapacitors in the coming years, their utilization in
different applications will grow. The driving range extension study in the third chapter, is only
one example of integrating supercapacitors in a vehicle level. Another interesting area that is also
covered in this dissertation is the issue of charging time. Due to the high currents applied in order
to reduce the charging time, the batteries undergo extreme electrochemical and mechanical stresses.
Reduced order electrochemical modeling and control is a area of research with a high potential
to facilitate the realization of fast charging. Such models can accurately capture the dynamics of
the cell and help impose realistic constraints on different parameters of the battery while ensuring
safety, acceptable cycle life, and feasibility to operate in real world applications.
As reported by the Energy Information Administration (EIA), the worldwide total installed
capacity of the renewable electricity generation by the year 2014, accounted for 24% (1700 GW) of
all electricity generated. This number for the U.S. is only 15.5% of the total electricity generated
in the country (180 GW). The main challenges in exploiting these energy sources, are cost and
availability. Wind and solar energies are not always available where and when needed which results
in intermittent generation. Storage technologies are a viable solution to such problems. Within the
wide variety of energy storage systems, the retired electric vehicle batteries, and also the redox flow
batteries show a lot of potential. The architecture of flow batteries makes it possible to independently
118
design the storage capacity and the power output. This allows optimal sizing of the energy storage,
unlike other battery technologies that are usually oversized due to the coupled energy and power
densities. This feature, beside utilization of cheaper organic electrolytes facilitates a lower cost per
kWh. On the other hand, batteries retired from operation in electric and hybrid electric vehicles have
a decent amount of capacity left to be used in a secondary application such as in power generation.
6.3 Dissemination of the Dissertation
6.3.1 Journal Publications
1. Yasha Parvini, Jason B Siegel, Anna G Stefanopoulou, and Ardalan Vahidi, “Supercapacitor
Electrical and Thermal Modeling, Identification, and Validation for a Wide Range of Temper-
ature and Power Applications,” IEEE Trans. Ind. Electron., 63(3):15741585, 2016.
2. Yasha Parvini, Ardalan Vahidi, S. Alireza Fayazi, “Heuristic versus Optimal Charging of
Supercapacitors, Lithium-Ion, and Lead-Acid Batteries: An Efficiency Point of View,” IEEE
Transactions on Control Systems Technology (Under Review).
3. Yasha Parvini and Ardalan Vahidi, “A Review on Fast Charging of Lithium-Ion Batteries,” to
be submitted to Journal of Power Sources.
4. Yasha Parvini, Ernesto G. Urdaneta, Jason B. Siegel, Saemin Choi, Ardalan Vahidi, Levi
Thompson, “Range Extension of a Lead-acid Battery Driven Vehicle using Supercapacitors:
Modeling and Hardware In the Loop Validation,” to be submitted to IEEE Transactions on
Vehicular Technology.
6.3.2 Peer Reviewed Conferences
1. Yasha Parvini and Ardalan Vahidi, “Maximizing Charging Efficiency of Lithium Ion and
Lead-Acid Batteries using Optimal Control Theory,” in IEEE Proc. Amer. Control Conf.,
2015, pp. 317322.
119
2. Yasha Parvini, Jason B Siegel, Anna G Stefanopoulou, and Ardalan Vahidi, “ Preliminary
Results on Identification of an Electro-Thermal Model for Low Temperature and High Power
Operation of Cylindrical Double Layer Ultracapacitors,” In P. Amer. Contr. Conf., pages
242247, 2014.
3. Yasha Parvini and Ardalan Vahidi, “ Optimal Charging of Ultracapacitors During Regenera-
tive Braking,” In IEEE Inter. Elec. Veh. Conf., pages 16, 2012.
6.3.3 Patent
1. Yasha Parvini, Saemin Choi, Jason Siegel, Ernesto Urdaneta, “Hybrid Energy Storage,” non-
provisional application submitted.
120
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