MODELING OF ULTRACAPACITOR SHORT-TERM AND LONG-TERM
DYNAMIC BEHAVIOR
A Thesis
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Yang Wang
August, 2008
MODELING OF ULTRACAPACITOR SHORT-TERM AND LONG-TERM
DYNAMIC BEHAVIOR
Yang Wang
Thesis
Approved:
Accepted:
Co-Advisor
Dr. Joan Carletta
Dean of the College
Dr. George K. Haritos
Co-Advisor
Dr. Robert Veillette
Dean of the Graduate School
Dr. George R. Newkome
Co-Advisor
Dr. Tom T. Hartley
Date
Department Chair
Dr. Alex De Abreu Garcia
ii
ABSTRACT
In this thesis several short-term models and a long-term model have been developed
for a NESSCAP3500P ultracapacitor. For the short-term ultracapacitor models, first-,
second-, third- and fourth-order transfer functions consistent with an RC ladder model are
assumed. The transfer function coefficients are identified by a least squares algorithm
based on experimental data consisting of time-varying current excitations and the
resulting terminal voltage responses.
A long-term model with six RC branches is developed by fitting the terminal voltage
transient response to an impulse charging current. Hundreds of thousands of terminal
voltage data points are recorded and least squares identification is employed to determine
the optimal values of the unknown parameters in the long-term model.
From the ultracapacitor models derived, terminal voltages under different current
profiles can be determined accurately over the time frame of one hour with an error less
than 0.1 V, the impulse charging and discharging response over a time frame of two
months can be simulated with an error less than 0.08 V, and the instantaneous power
available can be calculated.
iii
ACKNOWLEDGEMENTS
First I would like to take this opportunity to express my sincere appreciation to my
advisor Dr. Joan Carletta for her valuable guidance and encouragement throughout this
thesis. I am always illuminated from talking with Dr. Joan Carletta about solving
problems encountered in my research.
The helpful and patient advice from my co-advisors Dr. Robert Veillette and Dr. Tom
T. Hartley are gratefully acknowledged. Throughout my Masters project, I encountered a
lot of problems about how to get a good short term model to represent the dynamic
behavior of the ultracapacitor and how to do a good fit for an extremely long time data.
Each time Dr. Veillette and Dr. Hartley spend lots of time to give me necessary
background and give me wise idea to solve them. Through their help, not only I enlarge
my knowledge but also learn the techniques for how to analyze a certain problem and
how to solve it.
Special thanks are given to Dr. James Grover for his patient help to debug hardware
problems with the Microprocessor Interface Board.
My deepest gratitude goes to my family who provides love and support more than I
could ever expect.
iv
TABLE OF CONTENTS
Page
LIST OF TABLES........................................................................................................... viii
LIST OF FIGURES ........................................................................................................... ix
CHAPTER
I. INTRODUCTION ........................................................................................................... 1
1.1 Advantages of Ultracapacitors.................................................................................. 1
1.2 Disadvantage of Ultracapacitors ............................................................................... 5
1.3 The Need for Ultracapacitor Models ........................................................................ 6
1.4 Contributions of Research......................................................................................... 8
1.5 Thesis Outline ........................................................................................................... 8
II. BACKGROUND AND RELATED WORK................................................................ 10
2.1 Structure of Ultracapacitors .................................................................................... 10
2.1.1 Composition..................................................................................................... 10
2.1.2 Electric double layer structure ......................................................................... 13
2.2 The Physical Model of Ultracapacitor .................................................................... 15
2.2.1 Theoretical lumped parameter model .............................................................. 15
2.2.2 One-branch model............................................................................................ 16
2.2.3 Three-branch linear model............................................................................... 19
2.2.4 Three-branch non-linear model........................................................................ 21
v
2.2.5 Transmission line model .................................................................................. 23
2.2.6 Models based on frequency response data....................................................... 25
2.3 Summary ................................................................................................................. 28
III. EXPERIMENTAL SET-UP ....................................................................................... 30
3.1 Test Circuit.............................................................................................................. 30
3.2 Data Acquisition ..................................................................................................... 32
3.3 Flow Chart .............................................................................................................. 35
IV. SHORT-TERM MODEL............................................................................................ 39
4.1 Procedure for Fitting a Model of Given Order ....................................................... 40
4.2 Example Model Fitting: the Third-order Model .................................................... 44
4.3 The First-order Model............................................................................................. 52
4.4 The Second-order Model ........................................................................................ 55
4.5 The Third-order Model ........................................................................................... 58
4.6 The Fourth-order Model ......................................................................................... 61
4.7 Comparison of Models............................................................................................ 64
4.8 Conclusions............................................................................................................. 68
V. LONG-TERM MODEL............................................................................................... 69
5.1 Experiments for the Long-term Behavior ............................................................... 69
5.2 Long-term Model Parameter Identification ............................................................ 74
5.3 Validation................................................................................................................ 79
5.4 Instantaneous Power Discussion............................................................................. 81
5.5 Conclusions............................................................................................................. 84
VI. CONCLUSIONS ........................................................................................................ 85
vi
BIBLIOGRAPHY............................................................................................................. 88
vii
LIST OF TABLES
Table
Page
1.1 NESSCAP3500P specification ..................................................................................... 8
4.1 Poles for the four short-term models of different orders ............................................ 65
4.2 Zeros for the four short-term models of different orders............................................ 65
4.3 Integral squared voltage errors in V 2 s for three different datasets ............................ 67
viii
LIST OF FIGURES
Figure
Page
1.1 Temperature dependence of ultracapacitor parameters, from [9]................................. 4
2.1 Ultracapacitor structure............................................................................................... 11
2.2 The structure of activated carbon, from [24] .............................................................. 11
2.3 Porous structure of activated carbon, from [25] ......................................................... 12
2.4 Stern’s electrical double layer (EDL) model, from [21]............................................. 14
2.5 Ultracapacitor theoretical model................................................................................. 15
2.6 One-branch model without parallel resistor................................................................ 16
2.7 One-branch model with parallel resistor..................................................................... 17
2.8 Three-branch linear model.......................................................................................... 19
2.9 Three-branch non-linear model................................................................................... 21
2.10 Two-branch non-linear model................................................................................... 22
2.11 Transmission line model ........................................................................................... 23
2.12 Modified transmission line model ............................................................................ 24
2.13 Equivalent circuit of ultracapacitor, from [36] ......................................................... 25
2.14 Comparison of measured data and modeled data in frequency
domain, from [36] ..................................................................................................... 26
2.15 Approximation of Z p through N RC circuits, from [36] .......................................... 27
2.16 Frequency, temperature and terminal voltage model, from [35] .............................. 28
ix
3.1 Test circuit set-up........................................................................................................ 30
3.2 Analog to digital converter quality test....................................................................... 33
3.3 Flow chart to capture current and voltage data, package them
and send them out from UART................................................................................... 35
4.1 Ultracapacitor model as RC transmission line ............................................................ 39
4.2 Voltage and current data used to identify the coefficients in the
transfer function .......................................................................................................... 41
4.3 Measured voltage and current for the first validation test .......................................... 43
4.4 Measured voltage and current for the second validation test...................................... 43
4.5 Measured voltage and current for the third validation test ......................................... 44
4.6 Third-order circuit model............................................................................................ 44
4.7 Third-order modified circuit model ............................................................................ 50
4.8 Terminal voltage as a function of time for the experiment to
determine the effective parallel resistor...................................................................... 51
4.9 Finding coefficient k in Eq. (4.15) .............................................................................. 52
4.10 First-order circuit model ........................................................................................... 53
4.11 Transfer function coefficient identification for first-order model ............................ 53
4.12 Comparison of simulated and measured voltages of the first
validation for the first-order modified model ........................................................... 54
4.13 Comparison of simulated and measured voltages of the second
validation for the first-order modified model ........................................................... 54
4.14 Comparison of simulated and measured voltages of the third
validation for the first-order modified model ........................................................... 55
4.15 Second-order circuit model....................................................................................... 55
4.16 Transfer function coefficient identification for second-order model........................ 56
4.17 Comparison of simulated and measured voltages of the first
x
validation for the second-order modified model....................................................... 57
4.18 Comparison of simulated and measured voltages of the second
validation for the second-order modified model....................................................... 57
4.19 Comparison of simulated and measured voltages of the third
validation for the second-order modified model....................................................... 58
4.20 Transfer function coefficient identification for the third-order model ..................... 59
4.21 Comparison of simulated and measured voltages of the first
validation for the third-order modified model .......................................................... 60
4.22 Comparison of simulated and measured voltages of the second
validation for the third-order modified model .......................................................... 60
4.23 Comparison of simulated and measured voltages of the third
validation for the third-order modified model .......................................................... 61
4.24 Fourth-order circuit model........................................................................................ 61
4.25 Transfer function coefficient identification for the fourth-order model ................... 62
4.26 Comparison of simulated and measured voltages of the first validation
for the fourth-order modified model ......................................................................... 63
4.27 Comparison of simulated and measured voltages of the second
validation for the fourth-order modified model ........................................................ 63
4.28 Comparison of simulated and measured voltages of the third
validation for the fourth-order modified model ........................................................ 64
4.29 Comparison of the errors generated during the first validation
test by the first-, second-, third- and fourth-order modified models......................... 65
4.30 Comparison of the errors generated during the second validation
test by the first-, second-, third- and fourth-order modified models......................... 66
4.31 Comparison of the errors generated during the third validation
test by the first-, second-, third- and fourth-order modified models......................... 66
5.1 Charging current profile used at beginning of the charging-then-relaxing test .......... 70
5.2 Terminal voltage of the ultracapacitor during the charging-then-relaxing test .......... 71
xi
5.3 Discharging current profile used at beginning of the discharging-thenrelaxing test................................................................................................................. 72
5.4 Terminal voltage of the ultracapacitor during the discharging-then-relaxing test...... 73
5.5 Using the summation of exponential functions to curve fit the terminal
voltage from charging-then-relaxing test, and comparing the error
between them .............................................................................................................. 78
5.6 Validating the long term model by exciting it with the charging current profile ....... 79
5.7 Using Equation (5.6) multiplied by the discharging current area to
approximate the voltage transients from discharging-then-relaxing
test, and comparing the error between them ............................................................... 80
5.8 Validating the long term model by exciting it with discharging current profile ........ 81
5.9 First Foster Form derived from Equation (5.11)......................................................... 82
5.10 Corrected Foster Form with ESR.............................................................................. 82
5.11 First Cauer Form derived from the corrected Foster Form....................................... 83
5.12 Instantaneous energy calculation with load based on First Cauer Form .................. 83
xii
CHAPTER I
INTRODUCTION
Ultracapacitors were first used in military projects to start the engines of battle tanks
and submarines and to replace batteries in missiles. With the maturity of the
manufacturing and nano-material technology, the cost of ultracapacitors has fallen, and
the nominal capacitances have increased significantly. As a result, ultracapacitors have
begun to appear in more applications, such as diesel engine starting, railroad locomotives,
actuators and memory backup. More recently, ultracapacitors have become a topic of
some interest in the green energy world, where their ability to soak up energy quickly
makes them particularly suitable for regenerative braking applications [1,2,3]; in contrast,
batteries have difficulty in this application due to their lower rated charging current and
shorter cycle life.
1.1 Advantages of Ultracapacitors
Ultracapacitors, also known as electrochemical double layer capacitors (EDLC) or
supercapacitors, are new energy storage devices that have advantages over other energy
storage devices. In terms of energy density, existing commercial ultracapacitors range
from 1 to 10 Wh/kg [4]. Power density for ultracapacitors may typically range from 1000
to 5000 W/kg [4], and some newer ultracapacitors have higher power density. In contrast,
1
the energy density for the bipolar lead-acid battery is typically from 24 to 27 Wh/kg and
the power density is around 450 W/kg [5], the energy density for modern lithium-ion
batteries is from 150 to 200 Wh/kg and the power density is from 300 to 1500 W/kg [6],
and for automobile applications gasoline has an energy density around 12,000 Wh/kg.
Although existing ultracapacitors have energy densities that are only 1/10 those of some
batteries, their power densities are generally ten to one hundred times greater than those
of batteries. This special feature makes ultracapacitors a unique fit for applications that
require pulse power, such as burst-mode communication for wireless systems, writing to
disk and LCD operation for digital cameras, and starting vehicles. As a result,
ultracapacitors are becoming more widely used as energy storage devices. In addition to
their high power densities, ultracapacitors have several other advantages over other
energy storage devices. They have high efficiency, can operate with high currents and
over wide temperature ranges, have long cycle life and are environmentally friendly.
Each of the advantages is described in more detail next.
Coulombic efficiency is defined as the ratio of the number of electrons discharged to
the number of electrons that need to be recharged in order to bring an energy storage
device back to its original state of charge (SOC). Coulombic efficiency of ultracapacitors
is as high as 99% [7]. In addition, ultracapacitors have high round trip efficiency. The
round trip efficiency is defined as the ratio of the electrical energy produced after
charging and discharging the storage system to the electrical energy required from the
charging source. At a five-second rate (discharging to half rated voltage in five seconds,
and recharging at the same rate until the ultracapacitor is fully charged), the round trip
efficiency is greater than 70% and at a ten-second rate, it is greater than 80% [7]; this
2
round-trip efficiency is just as high as that of batteries [8]. In contrast, the round trip
efficiency of a regenerative fuel cell is about 50% [8]. The ultracapacitor’s high round
trip efficiency implies that an ultracapacitor-based energy storage system needs less
cooling capacity than most other alternative technologies, since ultracapacitors dissipate
much less energy in heat.
Since the equivalent series resistance (ESR) in ultracapacitors is extremely low, an
ultracapacitor can be charged with a very high current; this is not possible in energy
storage devices like batteries that have higher ESR, because in those devices current must
be limited to avoid overheating. In addition, no chemical reactions are involved in the
storage and release of energy from ultracapacitors. This means that charging and
discharging can be done with the same high rated current. This feature makes the
ultracapacitor a good fit for regenerative braking applications; to successfully absorb
energy from braking requires a very high charging current profile. In contrast, batterybased energy systems are not able to successfully absorb as much of the braking energy
because their charging current must be limited to avoid damage to the batteries.
Generally, ultracapacitors can operate over a wide range of temperatures. The range of
operating temperatures for ultracapacitors is determined by the electrolyte. If the
temperature is low, the mobility of the ions in the electrolyte will be low; near the
freezing point of the electrolyte the mobility of the ions will be affected dramatically. In
modern ultracapacitors, an organic solution that has a very low freezing point is
employed as the electrolyte. As a result, a typical ultracapacitor can be operated at
3
temperatures as low as -45℃. They can be operated at temperatures as high as 60℃ [9].
Figure 1.1 Temperature dependence of ultracapacitor parameters, from [9]
Throughout the range of operating temperatures, ESR and capacitance do not vary much,
as shown in Figure 1.1 [9]. In contrast, lead-acid and lithium-ion batteries, which of all
the battery types are the most tolerant of temperature changes, can be operated only from
-20℃ to 45℃ [10]. Further, for some kinds of batteries such as lithium-ion cells,
performance drastically decreases at temperatures below 0℃.
Industry standards specify that an ultracapacitor’s useful life ends when its capacitance
decreases by 20% or its ESR increases by 200%. As an ultracapacitor is used, its
performance continually degrades, and its end of life is when its performance will no
longer satisfy the application requirements. The ultracapacitor will have unlimited shelf
life if it is stored in a discharged state [9]. The ultracapacitor is good for several hundred
thousand charge/discharge cycles; this is many more than can be achieved with batteries,
4
some of which are good for only several hundred cycles. In addition, because
ultracapacitor operation involves no chemical reaction, its operation produces no
environmental pollution. Thus using ultracapacitors in hybrid electric vehicles can
improve the fuel economy and decrease vehicle emissions throughout the vehicle life.
1.2 Disadvantage of Ultracapacitors
The main disadvantage of ultracapacitors is that they can withstand only a low rated
voltage. That means that if a high terminal voltage is required, such as the 42-V modules
[11] used in some new automotive electrical systems, individual ultracapacitors must be
connected in series to form an ultracapacitor bank. Even if a bank uses all the same kind
of ultracapacitors, there will be differences in the individual capacitances; the
manufacturing tolerances on the nominal capacitance can be as high as ± 20% [12]. This
variation in capacitance places significant limitations on how the ultracapacitor bank is
controlled and used. For a series string of ultracapacitors, the current into each
ultracapacitor is the same. Assuming a simple capacitive model, the voltage v across the
ultracapacitor is governed by the equation
i=C
dv
,
dt
(1.1)
where C is the nominal capacitance of a particular ultracapacitor in the bank and i is the
current flow through the ultracapacitor bank. Mismatch in the nominal capacitances
means that ultracapacitors with smaller capacitances will have larger terminal voltage
changes dv ; this can cause terminal voltages for some cells to go beyond the rated
voltage more quickly than others when charging the ultracapacitor bank. Similarly, when
discharging a bank, different ultracapacitors will have different terminal voltages, and
5
some may even have potentially negative terminal voltages. It is dangerous to operate
cells at negative terminal voltages or at terminal voltages higher than their rated
maximum; accordingly, care must be taken in control of series-connected ultracapacitor
banks.
In order to solve the problem, different kinds of balancing circuits made up of passive
resistors, switched resistors, DC/DC converters or other components [13,14,15] may be
connected in parallel with each ultracapacitor’s terminals. Their function is to bypass
current around a cell whenever that cell’s terminal voltage exceeds a preset voltage; this
prevents the ultracapacitor from overcharging. More complicated balancing circuits
produce better control results and dissipate less energy. In [16,17], a bypassing circuit is
used so that all the ultracapacitors in a bank can be charged to the same upper voltage;
this means that when the bank is fully charged, the cells are balanced. Although
differences in cell capacitances will result in voltage imbalance at lower bank voltages,
there imbalances will not affect safety and ultracapacitor life as long as the ultracapacitor
bank is never discharged too far. For this scheme, the bank must be brought up to its
upper voltage periodically to reestablish balance for each cell; this is to avoid the
superposition of the imbalance voltages from each cell that can occur after many
charging and discharging cycles, which could eventually cause individual cells to have
negative terminal voltages.
1.3 The Need for Ultracapacitor Models
For controllable bypass circuits, a control scheme must be designed to turn the bypass
circuits on or off to balance each cell in the ultracapacitor bank while dissipating the least
6
energy. Knowing the terminal voltage of each cell is a minimum requirement for the
design of effective control strategies. To meet more stringent design requirements, the
terminal voltage should be predicted under some known current profiles. Thus it is
important to develop accurate ultracapacitor models. Ultracapacitor models can provide
detailed information useful for calculating the required volume of ultracapacitors in an
energy storage system, for designing sophisticated control strategies, and even for
extending the voltage operating range of ultracapacitors.
A simple first-order RC model (a large capacitor in series with a small resistor) can be
used to model the behavior of an ultracapacitor and simulate fast charging and
discharging in order to determine the instantaneous power available that could be stored
into or released from an ultracapacitor bank. Although this model may be sufficient for
many applications, an ultracapacitor cell behaves more as a distributed capacitance;
accordingly, a first-order model cannot account for long-term behavior nor give any
indication of how much of the stored charge should be considered available to do work in
a given interval. In addition, to better control ultracapacitors, it is not sufficient to have
models that are accurate only in short time frames. A long-term model is also needed to
account for the charge redistribution phenomenon, which can happen over time frames of
a couple of months. This phenomenon affects the instantaneous power available, since
the terminal voltage of the ultracapacitor will change gradually over long time frames.
The development of a suitably accurate high-order model of the ultracapacitor cell can:
(1) Accurately predict the terminal voltage under different current profiles;
(2) More closely model the slow transient due to charge redistribution to better
account for true stored charge and to calculate the instantaneous power available;
7
(3) Allow for a more sophisticated energy storage system control strategy;
(4) Improve pack balancing strategies;
(5) Extend the voltage operating range in some particular cases.
1.4 Contributions of Research
In this thesis we develop two kinds of ultracapacitor models for the NESSCAP3500P
ultracapacitor, one for short-term behavior and the other for long-term behavior. The
specifications for a NESSCAP3500P ultracapacitor are listed in Table 1.1. From the
short-term model the terminal voltage can be predicted under different current profiles
over time frames of one hour with the error less than 0.1 V. From the long-term model
the slow transient due to charge redistribution is simulated over time frames of two
months with the error less than 0.08 V.
Table 1.1 NESSCAP3500P specification
Rated Capacitance
3500 F
Capacitance Tolerance
-10% to 20%
Rated Voltage
2.7 V
Rated Current
700 A
ESR
0.5 mΩ
Rated Energy Density
5.29 Wh/kg
Rated Power Density
5.44 kW/kg
Temperature Range
-40℃ to 60℃
Weight
670 g
500,000 cycles
Cycle Life (25℃)
Life Time
10 years
1.5 Thesis Outline
The research work is presented as a thesis in six chapters. Chapter I gives introductory
material motivating the problem addressed and presenting the goals of the research.
8
Chapter II provides background, related work and an overview of ultracapacitor models
and parameter identification methods. Chapter III describes the test circuit and the data
acquisition method used to obtain the experimental data on which the developed models
are based. Short-term models, good for time frames of about one hour, are developed in
Chapter IV. First-, second-, third- and fourth-order models are presented, as well as the
least squares identification method used to find the model coefficients. Simulations of the
models are compared to experimental data to obtain the error of each model. In Chapter
V, a long-term model, good for time frames of about two months, is developed. The
charge redistribution phenomenon through two months is observed experimentally, and a
long-term model is derived to fit the observed slow transient. The coefficients for the
long-term model are found by least squares identification. Further, the long-term model is
tested by comparing simulation to experimental observations using impulse-like currents
over long time frames. Chapter VI draws conclusions and makes recommendations for
future work in this area.
9
CHAPTER II
BACKGROUND AND RELATED WORK
Theoretical lumped parameter models may be developed based on ultracapacitors’
physical structure. Although this kind of model represents the ultracapacitor structure, it
may include a large number of parameters as well as non-linear characteristics that make
it difficult to implement in practice. As a result, a lot of published work to model
ultracapacitors tries to derive simple models to represent the dynamic behavior and
voltage dependence of ultracapacitors.
2.1 Structure of Ultracapacitors
Ultracapacitors employ an electric double layer structure and use activated carbon as
the electrodes; these give ultracapacitors their extremely high capacitance.
2.1.1 Composition
Ultracapacitors are composed of three parts: positive and negative electrodes,
electrolyte and separator, as shown in Figure 2.1. Ultracapacitors employ activated
carbon whose surface area approaches 2000 square meters per gram [9] as the electrodes.
The reason that activated carbon has a high surface area per unit weight is that it is a
powder made up of extremely small and very rough particles, which in bulk form a low
density volume of particles with pores between them that resembles a sponge. The huge
10
Figure 2.1 Ultracapacitor structure
number of pores increases the surface area, which allows many more electrons to be
stored in the electrodes. The number of electrons stored in the electrodes is proportional
to the ultracapacitor’s capacitance. Figure 2.2, taken from [18], is a cartoon illustrating
the structure of activated carbon material.
Figure 2.2 The structure of activated carbon, from [18]
11
Since the size of pores is not uniform in the activated carbon, the capacitance is not
independent of frequency. The largest pores are “macropores” with a diameter bigger
than 25 nm. Inside the macropores there are still smaller “mesopores” with diameter
between 1 and 25 nm. Inside the mesopores there are “micropores” with diameter
between 4 and 10 angstroms and inside them there are “submicropores” with diameters
less than 4 angstroms. Figure 2.3, taken from [19], illustrates the different size pores.
Thus, ions in the electrolyte cannot charge the entire surface area of the device at all
frequencies. Qualitatively, the ions can charge big pores such as the macropores at high
frequency but can charge only small pores such as mesopores and micropores at low
frequency; this is because the ions encounter more resistance on the way to the smaller
pores, and so need a long time to reach them. Thus, the ultracapacitor will present
different capacitances at different frequencies.
Figure 2.3 Porous structure of activated carbon, from [19]
12
The electrolyte can be either an aqueous or an organic solution. Each one has its own
advantages and shortcomings. For the aqueous electrolytes, the solute is the salt and the
solvent is the water. Two example aqueous electrolytes are sulfuric acid (H2SO4) and
sodium hydroxide (NaOH). They result in a lower ESR and higher power densities than
those of non-aqueous double layer capacitors (DLC). However, aqueous DLCs can
sustain only relatively low operating voltages and have low energy densities. On the other
hand, non-aqueous electrolytes have higher energy densities [20], and higher operation
voltages, but also higher ESR and lower power densities.
The separator, placed in the electrolyte between the positive and negative electrodes, is
an electrically insulating membrane through which only ions can pass. When the ions
attempt to pass the ion-permeable membrane, they will encounter resistance from the
separator. This is one of the sources of ESR.
2.1.2 Electric double layer structure
Ultracapacitors employ a special structure called the electric double layer (EDL) that
arises at the interfaces between the porous activated carbon and the electrolyte.
Capacitance is created at this interface. It is the capacitance of this double layer that
accounts for the capacitance of the device. The double layer includes a compact layer and
a diffused layer [21,22,23], as shown in Figure 2.4. The compact layer is formed at the
interface where the solid electrode and electrolyte contacts. There are two kinds of
surface charge distribution at this interface; the first is of an electronic nature on the
electrode side, and the other is of an ionic nature with opposite sign on the electrolyte
side. The capacitance formed by the compact layer is proportional to the dielectric
13
permittivity of the electrolyte and inversely proportional to the charge separation distance
which is determined by the diameter, less than 10 angstroms [9], of the ions in the
electrolyte.
The diffused layer represents the ionic charge distribution in the electrolyte which
results from the random thermal motion. From Figure 2.4, in electrolyte, the potential of
the diffused layer closer to the compact layer is higher than the potential further from the
compact layer. It is the capacitance formed by the diffused layer that reflects the voltage
dependence of the ultracapacitor’s capacitance.
The total capacitance of the electric double layer structure is equal to the capacitance
formed by the compact layer in series with capacitance formed by the diffused layer. The
capacitance resulting from the diffused layer is voltage dependent: at low potential levels,
the capacitance formed by the diffused layer contributes significantly to the total
capacitance of the electric double-layer, while its capacitance becomes negligible at high
potential levels.
Figure 2.4 Stern’s electrical double layer (EDL) model, from [21]
14
2.2 The Physical Model of Ultracapacitor
Traditional capacitors such as electrostatic capacitors are often modeled by a single
ideal capacitor and a single ideal resistor connected in series, where the resistor
represents the ESR that prevents the modeled capacitor from acting like an ideal
capacitor. Such a model is not sufficient to model an ultracapacitor except for some
particular cases. This section describes various other models.
2.2.1 Theoretical lumped parameter model
Ideally, a model for an ultracapacitor should be based on its physical structure. In [24],
the authors presented the lumped parameter equivalent circuit for an ultracapacitor shown
in Figure 2.5. The model is constituted of an infinite number of RC branches with
voltage-dependent capacitances to mimic the activated carbon fibers in the positive and
negative electrodes, and resistances corresponding to the electrode material, the
Ranode
C p1
C p2
C p3
C pn
R p1
Rp 2
Rp3
R pn
Cn1
Cn 2
Cn 3
Cnn
Rn1
Rn 2
Rn3
Rnn
Rmembrane
Rcathode
Figure 2.5 Ultracapacitor theoretical model
15
electrolyte material, the membrane material and the various sizes of pores.
Although this model reflects the true physical structure of ultracapacitors and accounts
for physical phenomena such as charge diffusion and voltage-dependent behavior, it is
difficult to implement the model in practice because the model has so many parameters to
be identified that there is no practical method to obtain all of them. Other related work
presents several simplified models, as well as methods for parameter identification for
those models.
2.2.2 One-branch model
Figure 2.6 shows the classical equivalent one-branch circuit model for an
ultracapacitor used in [25,13], comprised of an ideal resistor and an ideal capacitor.
Although it cannot be used to reflect the long-term behavior of the ultracapacitor, it can
be used to simulate fast charging and discharging behavior. The charge redistribution
phenomenon cannot be reflected with this kind of RC model, since the model has only
i (t )
+
R
v(t )
C
−
Figure 2.6 One-branch model without parallel resistor
16
one time constant; however, the charge redistribution does not dominate in relatively
short time frames. In [13], the Challenge X team at the University of Akron employed
this model to estimate the instant available energy from the ultracapacitor bank after fast
charging and discharging. They also developed a technique for least squares
identification for the parameters in the model. The model parameters vary in time; in
practice, in time the capacitance will decrease and the resistance will increase. By taking
advantage of the simple model and their efficient identification algorithm, the work in [13]
periodically updates the parameters in the one-branch model based on recorded current
and voltage data in order to get an accurate estimation of instant available energy
information from the ultracapacitor bank. Through in-vehicle system-level tests, the
model was proven sufficient for simulating fast charging and discharging.
Another classic model of an ultracapacitor is shown in Figure 2.7. It is composed of
three ideal circuit elements: a capacitor C, a series resistor RESR which simulates energy
loss during capacitor charging and discharging, and a parallel resistor REPR which
simulates energy loss due to capacitor self-discharge. In [26] the author introduced an
i (t )
+
RESR
v(t )
C
REPR
−
Figure 2.7 One-branch model with parallel resistor
17
experimental way to identify the parameters in this model by using standard laboratory
instruments such as oscilloscopes and voltmeters. First the ultracapacitor is slowly
charged to a certain voltage; then charging is stopped and the ultracapacitor under test is
relaxed for three hours. The equivalent parallel resistor REPR is computed as
REPR =
− 10800s
,
v(t )
ln(
)C
v(0)
(2.1)
where v(0) is the terminal voltage at the point that charging is stopped, and v(t) is the
terminal voltage after three hours.
In order to get RESR , voltage and current changes after 50 milliseconds when charging
or discharging the ultracapacitor under test are recorded, then the value of RESR is
computed as
RESR =
ΔV
.
ΔI
(2.2)
The charge change in the ultracapacitor under test and the associated change in its
terminal voltage are measured to calculate the capacitance as
C=
ΔQ
.
ΔV
(2.3)
Although this method is easier than others that require specialized equipment such as a
spectrum analyzer or a controllable constant current supply, there are still some
shortcomings of the method. Firstly, the REPR value derived in this way is not accurate.
The ultracapacitor has very complex physical structure that is composed of a large
number of RC branches. When the charging current through the ultracapacitor is stopped,
the charge in the fast branch will redistribute to lower branches. It takes several weeks or
18
even longer for the ultracapacitor to reach a steady state. As a result, the decrease in
voltage seen after three hours is not due solely to self-discharge; energy redistribution
also causes a decrease. Secondly, the method to derive RESR needs to measure instant
terminal voltage change in a short time such as 50 milliseconds, and during this time
there will be a tiny change in the terminal voltage, and even worse the tiny voltage
change may be buried in the environmental noise. So the RESR derived in this way may be
not accurate. We did our own experiments based on this method. We found the value of
RESR derived from a charging test may be four to five times bigger than the one derived
from a discharging test. And in fact, the charging and discharging process should involve
the same resistance.
2.2.3 Three-branch linear model
The simplified three-branch linear model shown in Figure 2.8 was developed in [27].
Based on this model the ultracapacitor was assumed to have three different time constants,
that is, a fast branch, a medium branch and a slow branch. These three branches dominate
the behavior of an ultracapacitor in short-term time frames.
Rf
Rm
Rs
Cf
Cm
Cs
Figure 2.8 Three-branch linear model
19
In [27] a constant current excitation method was used to extract the unknown
parameters in the three-branch linear model. Time constants of the fast, medium and slow
branches are assumed to be quite different. First, the ultracapacitor under test was excited
with a constant current over a time interval shorter than the time constant of the fast
branch. During this period all charges were assumed to enter only into the fast branch; the
other branches are assumed unaffected during this time. In other words, the voltages of
the capacitors in the other branches were assumed to keep their initial values. The
terminal voltage was observed to increase. Both terminal voltage and current values were
recorded during this time, and used to identify the parameters R f and C f for the fast
branch. In addition the total charge delivered was calculated for the next parameter
identification for medium and slow branches. After that, the constant current supply was
shut down so there was no additional charge going to the ultracapacitor and the terminal
voltage was observed to decrease due to charge redistribution. The time constant of the
slow branch was assumed much longer than the time constant of the medium branch, so
that during the charge redistribution to Cm over a time interval close to the time constant
of medium branch, any charge redistribution to C s was neglected. In this stage, terminal
voltage values of the ultracapacitor were recorded. The voltage, the total charge and the
assumed time constant of medium branch were used to identify the parameters Rm and Cm
for the medium branch. After an interval longer than the time constant of the medium
branch, the capacitor in the slow branch began to be charged from its initial condition by
the fast and medium branches. The terminal voltage was observed to decrease again for
the charge redistribution from the fast and medium branches to the slow branch, and
20
terminal voltage values during this time were recorded. The voltage, the total charge and
the assumed time constant of the slow branch were used to identify the parameters Rs and
C s for the slow branch.
2.2.4 Three-branch non-linear model
Figure 2.9 shows a model proposed in [28] to reflect the behavior of an ultracapacitor
within a 30-minute time frame. It is composed of three RC branches and a parallel
resistor. The first RC branch with the elements R f , C f 0 and voltage-dependent
capacitance C f 1 × Vcf models the behavior of the ultracapacitor in the time frame of
seconds; non-linear behavior of the ultracapacitor is simplified and has only been
assigned to the fast-branch. The second branch with parameters Rm and Cm is the medium
branch modeling the behavior in the time frame of minutes; and the third branch with
parameters Rs and C s is the slow branch modeling for the behavior in the time frames of
tens of minutes. The parallel resistor Repr models the self-discharge phenomenon.
The general approach in [28] to identify the parameters in the medium and slow
branches is the same as in the last section. As far as the parameters in the fast branch are
Rf
Rm
Rs
+
Vcf
−
Repr
Cf 0
C f 1 × Vcf
Cm
Cs
Figure 2.9 Three-branch non-linear model
21
concerned, the definition of differential capacitance Cdiff , that is, the change in charge at
a given voltage, is introduced as
C diff (V ) =
dQ
|V .
dV
(2.4)
In their case,
Cdiff (V ) = C f 0 + C f 1 × Vcf .
(2.5)
In order to identify C f 0 and C f 1 , a small amount of charge was injected to the
ultracapacitor and the resulting change in voltage was measured and recorded at different
voltage levels.
A simple two-branch non-linear model was presented in [29]. The new model is shown
in Figure 2.10. In this model the first branch is composed of a constant resistance R0 , a
constant capacitance C0 and a variable capacitor with capacitance proportional to the
voltage at the device terminals. The second branch represents the voltage redistribution
phenomena inside the device. Since this model has only two branches, it is easy to do
parameter identification based on a constant current excitation lasting for a short period
+
R2
R0
Repr
V
C0
Kv ×V
C2
−
Figure 2.10 Two-branch non-linear model
22
of time. For the fast branch with the variable capacitance, the authors set up a secondorder equation to curve fit the voltage data observed from experiments in a period of time
less than the time constant of the fast branch. In this stage, they assume no change in the
capacitor charge in the slow branch. Then, they use the same technique described in
Section 2.2.3 to identify the parameters in the slow branch.
The parameter identification for the models in Figure 2.8, Figure 2.9 and Figure 2.10 is
straightforward. However, the accuracy of the models depends on the assumed time
constants for the medium and slow branches, which are chosen without prior knowledge
of the dynamic behavior of the ultracapacitor.
2.2.5 Transmission line model
Another ultracapacitor model based on the physical structure of ultracapacitors has
been proposed in [30]. The model is shown in Figure 2.11. It is composed of two parts.
The first is a non-linear transmission line connected to the terminals through an access
R1
Rn
R2
C2
Figure 2.11 Transmission line model
23
Cn
resistor R1 . This part replaces the fast branch in the three-branch model. The second part
is an n-branch RC ladder that mimics the long-term behavior of the ultracapacitor. The
sections are organized such that the shortest time constant branch is close to the
transmission line block and the time constants get longer and longer for sections farther
from the transmission line. Not only can the model reflect the non-linear behavior of the
ultracapacitor, but it is also flexible: if a short-term model is needed, the RC branches can
be truncated, but if a long-term model is needed, additional RC branches can be added to
satisfy the requirements. On the other hand, the transmission line is difficult to simulate
because it has a complex expression. This also makes the parameter identification
difficult, so implementation of the model in practice is not efficient.
Figure 2.12 shows a proposed simplification of the short-term part of the transmission
line model [31]. Four identical RC branches are used to represent the transmission line.
An access capacitor Ca was added to improve the short-term behavior of the model and a
series inductor was introduced to describe the high-frequency behavior. The behavior of
the inductor can be ignored if only the slower terminal voltage terms are of importance.
Ls
Ra
R/4
R/4
R/4
C (V ) / 4
C (V ) / 4
C (V ) / 4
R/4
Ca
Figure 2.12 Modified transmission line model
24
C (V ) / 4
Although there are five branches in this model, the last four branches have the same time
constant, so the long-term charge redistribution cannot be reflected very well; thus, this
model focuses on the short-term behavior. In [31], both constant-current tests and
frequency analyses were used to identify the parameters, taking advantage of the fact that
frequency analysis is good for determining the dynamic behavior and constant-current
analysis is good for determining the voltage dependence of the ultracapacitor.
2.2.6 Models based on frequency response data
Much of the published work on ultracapacitor modeling is based on frequency
response data obtained, for example, using electrochemical impedance spectroscopy (EIS)
[32,33,34]. The advantage of using the frequency response to identify parameters is that
the dynamic behavior of the ultracapacitor can be modeled well. Further, the parameter
identification software is embedded in some EIS systems. The disadvantage is that since
the excitation signal is usually a small AC signal with very low or fixed DC bias, the
derived model cannot accurately predict the voltage-dependent behavior. Thus, for EIS
methods different DC biases should be considered. [33] presented a new approach to
model the dynamic behavior of ultracapacitors using EIS. The device under test was
modeled by an inductor L, a resistor Ri and an impedance Z p , as shown in Figure 2.13.
L
Ri
Zp
Figure 2.13 Equivalent circuit of ultracapacitor, from [33]
25
In tests, four different DC voltages and six different temperature profiles were considered.
Take one of the tests for example. The device under test was excited with a small AC
current with a known, fixed DC bias. The terminal impedance is plotted as a function of
frequency in Figure 2.14 [33].
The plot in Figure 2.14 can be divided into two parts. One is the near vertical
impedance plot at low frequencies, the behavior of which is like an ideal capacitor. The
other is the - 45o part that forms a - 45o line at intermediate frequencies. In this model,
Z p is responsible for the - 45o degree slope. The impedance Z p models the porosity of
the ultracapacitor’s electrodes. Due to this porosity, the real part of the impedance
increases with decreasing frequency and the full capacitance of the ultracapacitor will be
seen at DC conditions. The parameter L can be identified by the impedance from very
high frequency. The parameter Ri can be identified from the intersection between the
Figure 2.14 Comparison of measured data and modeled data in
frequency domain, from [33]
26
impedance plot and the real axis. The mathematical expression used for Z p is:
Z p ( jω ) =
τ × coth( jωτ )
C × jωτ
(2.6)
There are two independent parameters in (2.6). A large number of impedance
measurements are used to find a best fit for the unknown parameters τ and C. When all
the frequency domain model parameters have been identified, the model is transferred to
time domain model by expanding the impedance Z p into the RC circuit shown in Figure
2.15. The experiments in [33] show that models with ten RC branches are in good
agreement with the measured impedance.
Another model with parameter identification done using EIS was recommended in [32].
This model, shown in Figure 2.16, has 14 RLC components. The model is used to
simulate the behavior of an ultracapacitor as a function of frequency, voltage and
temperature together. Rv and Cv elements are utilized to represent the voltage-dependent
behavior of the ultracapacitor at low frequency. Circuit 1 in Figure 2.16 is introduced to
consider the electrolyte ionic resistance in the low frequency range. Circuit 2 is
Figure 2.15 Approximation of Z p through N RC circuits, from [33]
27
Ci
Re
L
Rp 2
Ri
Rv
R p1
Ri
Rl
Ca
C p2
C p1
Cv = k × v
CR
Figure 2.16 Frequency, temperature and terminal voltage model, from [32]
used to modify the medium-frequency behavior and Circuit 3 describes the charge
redistribution and leakage current. In addition Re denotes the electronic resistance and
the inductance L is employed to describe the high-frequency behavior.
2.3 Summary
Generally speaking, there are two kinds of ultracapacitor models in related work: linear
models, and non-linear models. For the parameter identification methods, there are still
two major methods for different ultracapacitor models. One is based on the constantcurrent response; the other is based on the frequency response. Although the constantcurrent method can represent the voltage-dependent behavior of an ultracapacitor, quite
different time constants of different branches have to be assumed. In contrast, the
frequency response method can represent the dynamic behavior of an ultracapacitor, but
has difficulty reflecting the voltage dependence. Thus, there are some related work that
identify the parameters in their models using both methods.
28
The present work considers a transfer function that corresponds to an RC ladder model
of an ultracapacitor cell. A least squares identification is used to find the best transfer
function coefficients from experimental data gathered using only ordinary laboratory
instruments such as multimeters, oscilloscopes and constant voltage supplies; this is in
contrast to the controllable constant voltage supplies or EIS used in other techniques. The
identification process is done only with time-domain data, and is based on timedependent current profiles generated by manually controlling charging and discharging
switches at fifteen-second intervals. Unlike some other techniques, the process requires
no initial assumptions about the time constants.
29
CHAPTER III
EXPERIMENTAL SET-UP
Determining the parameters of an ultracapacitor model requires first that current and
voltage information be captured from the ultracapacitor under test, and then that least
squares identification be done using that current and voltage data. In this chapter, the
experimental set-up used to capture the data is described. This includes circuitry to
control the charging and discharging of the ultracapacitor, as well as the program to
measure and record the voltage and current information.
3.1 Test Circuit
Figure 3.1 Test circuit set-up
30
The test circuit, shown in Figure 3.1, consists of three NESSCAP3500P ultracapacitors
in series excited by a 5-V, 1100-W power supply. A 50-A fuse is connected in the test
circuit to protect the power supply. The rated voltage for each ultracapacitor cell is 2.7 V.
To avoid exceeding the rated terminal voltages, three identical ultracapacitor cells are
connected in series and their initial voltages are made equal. Two kinds of MOSFET
switches are employed so that the charging and discharging processes can be controlled
separately. Five p-channel MOSFETs, each of which can pass a maximum of 13 A of
current, are employed in parallel as the charging switch; when charging, their gates are
connected together to the preset -5.5-V constant voltage supply through a three-position
switch. Two n-channel MOSFETs, each of which can pass a maximum of 24 A of current,
are employed in parallel as the discharging switch; when discharging, their gates are
connected to a preset 10.5-V constant voltage supply through the same three-position
switch. Two power resistors (0.1 Ω , 1%, 250W) are used as the charging and discharging
loads, respectively.
In order to make the test circuit safe and reliable, a 10-k Ω resistor is connected
between the gate terminal of the p-channel MOSFETs and the positive terminal of the
power supply and another 10-k Ω resistor is connected between the gate terminal of the
n-channel MOSFETs and the ground. These two 10-k Ω resistors serve to prevent the
gate signals of the n-channel and p-channel MOSFETs from floating when the threeposition switch is between positions. When the switch is in the charging position, the
gates of both the n-channel and p-channel MOSFETs are at -5.5 V, so that the n-channel
MOSFETs are OFF and the p-channel MOSFETs are ON; similarly, when the switch is
31
in the discharging position, the gates of both the n-channel and p-channel MOSFETs are
at 10.5 V, so that the n-channel MOSFETs are ON and the p-channel MOSFETs are OFF.
In order to observe the transient characteristics of the ultracapacitor, the MOSFET
switches are controlled to open and close manually by means of the three-position switch
approximately every fifteen seconds during the test; this way a dynamic current profile is
generated in the test circuit.
3.2 Data Acquisition
A microcontroller board (dsPICDEM2) was employed for the terminal voltage and
current measurements. The charging current is measured indirectly by measuring the
voltage across the power resistor in series with the charging source; similarly, the
discharging current is measured indirectly by measuring the voltage across the power
resistor that serves as the discharging load. This is done by measuring the potentials of
both terminals of the power resistor with respect to the ground, subtracting the potentials
to find the voltage difference across the resistor, and converting to current through the
resistor by dividing the voltage by the resistance.
The microcontroller board has twelve-bit analog-to-digital converters (ADC). The
reference of the ADC is set to 5 V. The MPI board and the test circuit share a common
ground. Imperfections in the linearity of the ADC are a possible source of error, and so it
is important to investigate the quality of the conversion. In order to do so, known
voltages from 0 V to 5 V produced using a power supply were given to the ADC channel
of the MPI board, and then the converted values were translated back to voltages in code
32
Figure 3.2 Analog to digital converter quality test
in the microcontroller. Figure 3.2 plots the converted voltages versus the known input
voltages. In Figure 3.2, the conversion is nearly linear for input voltage values less than
3.5 V. Over 3.5 V the conversion error starts to increase. At 5.0 V the error increases to a
maximum of 0.03 V. In the experimental setup, the ultracapacitor voltage measurement is
made with one ADC terminal grounded and the other terminal potential always less than
2.0 V. Within this range the ADC error is less than 0.01 V.
For the ultracapacitor current measurements there are two sources of error. One is from
the accuracy of the power resistor, whose actual resistance may vary by as much as 1%
from its nominal resistance. The other is from non-linearity of the ADC. When charging
the ultracapacitors, one terminal of the power resistor is connected to 5 V power through
the p-channel MOSFETs and the other terminal ranges from 0 V to 4.5 V in the course of
the experiment; note that this range includes the more non-linear part of the ADC curve.
33
When discharging the ultracapacitors, one terminal of the power resistor is connected to
ground through the n-channel MOSFETs and the other terminal potential ranges from 0
V to 4.5 V in the course of the experiment; this also includes the non-linear part of the
ADC curve. The worst case for the current error is 0.33 A, which occurs at a measured
current of about 35 A.
The microcontroller board is connected to a standard RS-232 serial communication
port of the workstation through the UART circuitry on the microcontroller board. The
data passes through the analog-to-digital channels to the series port of the workstation,
and a Hyper Terminal program run on the workstation is employed to store the data
received on the series port to the hard disk. In order to facilitate Matlab processing of the
captured data, the data is stored in a fixed format. Each data set is a triplet of values, and
each value is sent in the format “#.##”, where # is a decimal digit. The first value is the
terminal voltage of the ultracapacitor under test. The second and third values are the
voltages across the charging and discharging loads, respectively. Since the load
resistances are 0.1 Ω , the charging and discharging currents can be found by multiplying
the charging and discharging resistor voltages by ten. Each data triplet is sent as a
separate line of the transmitted file, and one triplet is sent each second. In this way, as
much data as is needed can be accumulated. Later, for development of a short-term model
of the ultracapacitor, one hour of data will be used; for development of a long-term model
of the ultracapacitor, two months of data taken less frequently will be used.
34
3.3 Flow Chart
Figure 3.3 shows the flow chart for the program in the dsPIC30F4013 on the
dsPICDEM2 MPI board that samples, measures and records data. The function of each
block is described below.
Main Program
Start
Program execution
transitions
Interrupttriggered
transitions
Configure
microcontroller
pins
Initialize UART
ADC ISR
(triggered approximate
every 125 ms)
Initialize and
trigger ADC
Calculate and
store data triplet
Delay
Clear AD interrupt
flag
Convert data
triplet to character
string
UART ISR
(triggered approximate
every one second)
Set pointer to
beginning of
character string
Increment pointer
Is pointer
pointing to null
char?
Write pointed-at
character to UART
transmit register
Yes
Stop
No
Write pointed-at
character to UART
transmit register
Figure 3.3 Flow chart to capture current and voltage data, package them
and send them out from UART
35
Configure microcontroller pins: A hardware reset configures all port pins multiplexed
with the ADC as analog input pins. The four pins associated to ADC channels two, three,
four and five are used to monitor the four potentials in the test circuit.
Initialize UART: the rate of the UART communication is set to 9600 bits/sec, and the
UART is configured to use one stop bit and no parity.
Initialize and trigger ADC: the ADC is configured for automatic sampling and automatic
conversion. It is configured to begin sampling immediately after the last conversion
completes and to trigger the ADC interrupt service routine after four samples are filled in
the ADC buffer. The reference voltage for the ADC is set to 5 V by programming. The
total sample time is composed of the acquisition time and the conversion time. The
acquisition time is set to 31Tad and the conversion time is set to 14Tad , where Tad denotes
the ADC clock period which is set to 20.5Tcy by programming, where Tcy is instruction
cycle time. This makes the total sampling time 45Tad , that is, 124.5 microseconds; the
resulting sampling frequency is about 8 kHz.
Delay: This block, which is part of the main loop, delays for a fixed amount of time.
When developing the short-term model, it is programmed to delay for one second; when
developing the long-term model, it is programmed to delay for ten seconds.
36
Convert data triplet to character string: Each of the three global variables (one each for
terminal voltage, charging current and discharging current) is converted to a string of
unsigned chars with the format of #.##, where # is a unsigned char denoting a decimal
number, i.e., one of the characters “0” through “9”. Then, the three strings are packaged
into the triplet format referred to earlier to facilitate future MATLAB processing. The
triplet values are stored in a fifteen-byte buffer.
Set pointer to beginning of character string: set a pointer to point at the beginning of the
fifteen-byte buffer.
Write pointed-at character to UART transmit register: The first byte content in the
fifteen-byte buffer is written to the UART transmit register; doing this triggers interruptdriven UART communication of the entire contents of the buffer.
ADC ISR: this interrupt service routine copies the A/D conversion results from the ADC
buffer to local variables. Then it calculates the charging and discharging current based on
the captured voltage stored in the local variables. After that, the terminal voltage,
charging current and discharging current are copied to global variables. At the end of
ADC ISR, the interrupt flag is cleared for the next ADC.
UART ISR: increase the pointer to point at the next byte in the fifteen-byte buffer unless
it encounters a null character. The pointed-at byte is written to UART transmit register.
37
In summary, the program measures and records the voltage and current data of the
ultracapacitor under test and communicates with a desktop computer to store data to the
hard disk for future MATLAB processing. In the following chapters, the short term and
long term ultracapacitor models are derived to fit the voltage data under the excitation of
the current data. The parameters in each model are calculated by least squares
identification based on these data.
38
CHAPTER IV
SHORT-TERM MODEL
A simple first-order RC model was used to estimate the available energy in the
ultracapacitor bank that serves as the main electrical energy storage system for the
University of Akron’s Challenge X vehicle. A higher-order model would allow for better
control. In developing higher-order models for the ultracapacitor, we start with the idea,
used in [35], that the ultracapacitor ultimately acts like the infinite RC ladder circuit
shown in Figure 4.1, where the sections are organized such that the fastest time constant
is associated with the R and C closest to the terminals, and the time constants get longer
and longer for sections further from the terminals. This accurately models the behavior
seen on the bench: when an ultracapacitor is charged, the terminal voltage rises quickly,
but once the charging current is cut off, the terminal voltage slowly drops, as charge
i (t )
R1
v(t )
R2
C1
R3
Rn
C3
C2
Figure 4.1 Ultracapacitor model as RC transmission line
39
Cn
redistributes itself from the capacitor closest to the terminals to those that are further
away.
In this chapter, we develop four different short-term ultracapacitor models of first-,
second-, third- and fourth-order. The model coefficients are chosen to fit the behavior
over an interval of around one hour.
4.1 Procedure for Fitting a Model of Given Order
In order to derive a circuit model for the ultracapacitor that has a reasonably small
number of parameters, we chose to use a finite RC circuit model to approximate the
infinite circuit. The underlying assumption is that while a more realistic model will have
many more (and possible an infinite number of) sections, the time constants grow quickly,
and the time constants associated with the unmodeled sections can be neglected. It is
important to recognize that the model derived by neglecting sections will be good only in
time frames during which the neglected time constants do not play a significant role.
For a given finite circuit model, the form of the associated transfer function is fixed.
Deriving the model is a matter of identifying each coefficient in the transfer function.
From the physical circuit point of view, the transfer function coefficients are related to
the R and C circuit parameters, and a circuit with resistances and capacitances that are
positive and real corresponds to a transfer function that has real poles and zeros. Much of
the related work derives models by identifying real and positive R and C circuit
parameters. The approach taken here is different; it is to identify the transfer function
coefficients without regard to the circuit parameters. Thus, the resulting transfer function
is not constrained to have real poles and zeros.
40
Transforming the transfer function derived from the circuit model to a differential
equation and using Forward Euler approximations for the derivatives in the differential
equation yields a difference equation. The difference equation is arranged in matrix form
by writing it in terms of the first differences of voltage and current. Then a least squares
identification is implemented using Matlab to determine the unknown coefficients based
on thousands of experimentally captured voltage and current data points.
Once a model of a particular order is derived, it is used to simulate the response of the
ultracapacitor to the current profile used to identify the model. Based on the simulation
result, the model is modified by including one additional resistance across the terminals.
It will be seen that the addition of this resistor results in a much better fit to the
ultracapacitor behavior seen in experiments.
Measurement
1
Voltage(V)
0.8
0.6
0.4
0.2
0
0
500
1000
1500
0
500
1000
1500
2000
2500
3000
3500
4000
2000
2500
Time(seconds)
3000
3500
4000
30
Current(A)
20
10
0
-10
-20
-30
Figure 4.2 Voltage and current data used to identify the coefficients in the
transfer function
41
All four models derived in this chapter are based on the same experimental data; the
voltage and current data used to identify the coefficients in each transfer function are
plotted in Figure 4.2. Charging and discharging of the ultracapacitor under test is
controlled manually at time intervals of around fifteen seconds. During the test, the
terminal voltage of the ultracapacitor climbs from 0.1 V to 0.8 V in 500 seconds; the
terminal voltage oscillates around 0.9 V for the rest of the test.
Once all the coefficients of a transfer function are determined, several new current
profiles are used in order to validate the different order models. For each current profile,
the voltages predicted by the model are compared to those seen experimentally. The first
validation test uses the measured voltage and current plotted in Figure 4.3. The first test
lasted for 3900 seconds. In the first phase of the first test, from 0 to 550 seconds, the total
charging time was longer than the total discharging time so the terminal voltage reached
to approximate 0.9 V; In the second phase, there was no extra charge input to the
ultracapacitor and the terminal voltage oscillated around 0.9 V; In the final phase, the
total discharging time was longer than the total charging time, so the ultracapacitor was
finally discharged to 0.2 V. For the second test, whose measured voltage and current are
plotted in Figure 4.4, the terminal voltage climbed to 1.5 V gradually over 4700 seconds.
The third test ran for a much longer time, around two hours, and tested a variety of
terminal voltages from 0.5 V to almost 1.5 V; its measured voltages and current are
plotted in Figure 4.5.
Before the identification experiment and each of the three validation tests, the
ultracapacitor was discharged to zero slowly and then allowed to relax for at least two
weeks so that all tests would start with similar initial conditions.
42
Measurement
1
Voltage(V)
0.8
0.6
0.4
0.2
0
0
500
1000
1500
0
500
1000
1500
2000
2500
3000
3500
4000
2000
2500
Time(seconds)
3000
3500
4000
30
Current(A)
20
10
0
-10
-20
-30
Figure 4.3 Measured voltage and current for the first validation test
Measurement
Voltage(V)
1.5
1
0.5
0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0
500
1000
1500
2000 2500
3000
Time(seconds)
3500
4000
4500
5000
40
Current(A)
20
0
-20
-40
Figure 4.4 Measured voltage and current for the second validation test
43
Measurement
Voltage(V)
1.5
1
0.5
0
0
1000
2000
3000
4000
5000
6000
7000
0
1000
2000
3000
4000
Time(seconds)
5000
6000
7000
40
Current(A)
20
0
-20
-40
Figure 4.5 Measured voltage and current for the third validation test
4.2 Example Model Fitting: the Third-order Model
The third-order circuit model shown in Figure 4.6 is used to illustrate the procedure
followed for model identification and validation. The circuit model in Figure 4.6 gives
the transfer function
I (s )
R1
V (s )
R3
R2
C2
C1
Figure 4.6 Third-order circuit model
44
C3
V ( s ) As 3 + Bs 2 + Cs + 1
=
,
I ( s)
Ds 3 + Es 2 + Fs
(4.1)
where A = C1C2C3 R1 R2 R3 ,
B = R2 R3C2C3 + C1C3 R1 R3 + C1C2 R1 R2 + C1C3 R1 R2 + C2C3 R1 R3 ,
C = C3 R3 + C2 R2 + C1 R1 + C3 R2 + C2 R1 + C3 R1 ,
D = C1C2C3 R2 R3 ,
E = C1C3 R3 + C1C2 R2 + C1C3 R2 + C 2C3 R3 , and
F = C1 + C2 + C3 .
Rather than identifying the circuit parameters C1 , C2 , C3 , R1 , R2 and R3 , our process
identifies the transfer function coefficients A through F. The transfer function (4.1) is
equivalent to the differential equation
D&v&& + Ev&& + Fv& = A&i&& + B&i& + Ci& + i .
(4.2)
Approximating the derivatives in (4.2) as a Forward Euler difference with a time step T
yields a difference equation. The Forward Euler approximations for the first, second and
third derivatives are
v&n =
vn+1 − v n
T
(4.3)
v&&n =
v&n +1 − v&n vn + 2 − 2v n +1 + v n
=
T
T2
(4.4)
&v&&n =
v&&n +1 − v&&n v n +3 − 3v n + 2 + 3v n +1 − v n
.
=
T
T3
(4.5)
45
Substituting (4.3), (4.4) and (4.5) into (4.2) yields a difference equation, which may be
rewritten in terms of the first differences of v and i. The difference equation takes the
form
D∇ vn +3 + ( ET − 2 D )∇ vn + 2 + ( D − ET + FT 2 )∇ vn +1
= A∇in+3 + ( BT − 2 A)∇in+ 2 + ( A − BT + CT 2 )∇in+1 + T 3in+3
(4.6)
where ∇vn+i denotes vn+i − vn+i-1, ∇in+i denotes in+i − in+i-1 and T is the time step. This
form in terms of the differences ∇vn+i and ∇in+i was found to produce more accurate
results than the standard difference equation in v and i. Furthermore (4.6) can be arranged
in matrix form as
(∇vn+3
∇v n + 2
∇v n +1
∇i n + 3
∇i n + 2
D
⎛
⎜
T3
⎜
⎜ ET − 2 D
⎜
T3
⎜ D − ET + FT 2
⎜
T3
∇in +1 )⎜
−
A
⎜
⎜
T3
⎜ 2 A − BT
⎜
T3
⎜
2
⎜ BT − A − CT
⎜
⎝
T3
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟ = (in +3 ).
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(4.7)
Over a given time interval, N samples of voltage and of current are recorded. As Equation
(4.7) includes three values of the time index, we can calculate N-2 backward differences.
This gives us N-2 equations, each one an application of Equation (4.7). In the case of our
identification data, there is one hour of data captured at one-second intervals, for a total
of 3600 samples. Thus there are 3598 equations. Arranging these equations in matrix
form, we obtain
M×X = W ,
where
46
(4.8)
⎛ ∇v 3
⎜
⎜ ∇v
M =⎜ 4
M
⎜
⎜ ∇v
⎝ N
∇v 2
∇v1
∇i 3
∇i 2
∇v 3
∇v 2
∇i 4
∇i 3
M
∇i N
M
∇i N −1
M
M
∇v N −1 ∇v N − 2
∇i1
⎞
⎟
∇i 2 ⎟
⎟
M
⎟
∇i N − 2 ⎟⎠
is the matrix of differenced data,
D
⎛
⎜
T3
⎜
⎜ ET − 2 D
⎜
T3
⎜ D − ET + FT 2
⎜
T3
X=⎜
−A
⎜
⎜
T3
⎜ 2 A − BT
⎜
T3
⎜
⎜ BT − A − CT 2
⎜
T3
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
is the parameter vector to be identified, and
⎛ i3 ⎞
⎜ ⎟
⎜i ⎟
W=⎜ 4 ⎟.
M
⎜ ⎟
⎜i ⎟
⎝ N⎠
The values of the matrix X that best fit the data can be found by using the least squares
identification. The least squares identification procedure derives from the error between
measurement and estimation
e = M×X− W .
(4.9)
In Equation (4.9), e is the error function, and J = e T × e is defined as the cost function.
The main idea of least squares is to minimize the cost function J to find optimal
parameters.
Using (4.9) the cost function is written as
J = eT e = ( X T M T − W T )(MX − W) .
47
Taking the derivative with respect to the matrix X gives
∂J
∂
( XT M T − W T )(MX − W )
=
∂X ∂X
∂
( XT M T MX − XT M T W − W T MX + W T W )
=
∂X
= (2M T MX − 2M T W ).
Set
∂J
= 0 to find the optimal point. Then we get
∂X
M T MX − M T W = 0 ,
X = (M T M ) −1 M T W .
(4.10)
With the known matrix X, A through F can be determined. Since the equations for
calculating A through F are all linear equations, the transfer function coefficients can be
calculated without numerical difficulties. The parameters determined in this way are the
best fit to the measured data in a least squares sense.
Once the transfer function is determined, simulation based on it must be done to test it.
First it is transferred to a state space representation; then, the Forward Euler formula is
used for the simulation. The state space representation is written as
⎛ E
⎛ x&1 ⎞ ⎜ −
⎜ ⎟ ⎜ D
⎜ x&2 ⎟ = ⎜ 1
⎜ x& ⎟ ⎜ 0
⎝ 3⎠ ⎜
⎝
⎛ B AE
v=⎜ − 2
⎝D D
⎞
0 ⎟⎛ x1 ⎞ ⎛ 1 ⎞
⎟⎜ ⎟ ⎜ ⎟
0 ⎟⎜ x2 ⎟ + ⎜ 0 ⎟i ≡ Ax + Bi
0 ⎟⎟⎜⎝ x3 ⎟⎠ ⎜⎝ 0 ⎟⎠
⎠
⎛ x1 ⎞
C AF 1 ⎞⎜ ⎟ A
−
⎟⎜ x2 ⎟ + i ≡ Cx + Di
D D 2 D ⎠⎜ ⎟ D
⎝ x3 ⎠
−
F
D
0
1
(4.11)
where x1 , x2 and x3 are the three state variables, v is the output voltage and i is the input
current.
48
The Forward Euler formula used for the simulation is
xn +1 = xn + Tf n ,
(4.12)
where f n ≡ ( Ax) n + (Bi ) n . The s-plane stability region of the Forward Euler method is the
interior of a circle centered at (-1,0) with a radius of one unit [36]. For a stable simulation,
the eigenvalues of the matrix A multiplied by the time step must lie in the stability region.
To ensure this condition, the simulation time step may be chosen as short as necessary.
For our example, the simulation time step is set to T=0.1 second. Since the current profile
used to excite the model is captured at one second intervals, we use a linear interpolation
function to interpolate 10 simulated current excitation values for each current data sample.
It will be seen that the simulation time step T=0.1 second is short enough to give a stable
simulation for the identified models.
From the simulation results it will be seen that the model is not a good match to the
measured data. The simulated terminal voltage based on the measured input current
profiles continues to climb after the measured voltage flattens out. Further analysis shows
the shortcomings of the model in Equation (4.1):
(1) The model does not accurately account for the redistribution of charge that causes the
terminal voltage to relax when there is no charging current. This causes the error to
increase gradually over time when there is no charging current.
(2) The DC level of the model in Equation (4.1) is infinite; this is not the case for a real
ultracapacitor device.
As a result, a modified ultracapacitor model is developed by first adding a constant
parameter w to reflect the charge redistribution in the test time; the modified model is
49
V ( s ) As 3 + Bs 2 + Cs + 1
.
=
I ( s ) Ds 3 + Es 2 + Fs + w
(4.13)
The parameter w can be thought of as the reciprocal of a parallel resistance connected at
the terminals of the circuit model shown in Figure 4.6. The revised RC model with the
added resistor Repr is shown in Figure 4.7. This resistance can be used to approximate the
effect of charge redistribution only after a charging current is applied.
In an experiment aimed at finding an appropriate value for w in (4.13), the
ultracapacitor was first charged to a certain value; then, the test circuit was opened to let
the ultracapacitor relax. The terminal voltage values are recorded; then, the value w can
be calculated by
w(t ) =
1
, with Repr (t ) =
Repr (t )
−t
v (t )
)×C
ln(
v ( 0)
.
(4.14)
where v(0) is the initial voltage at the beginning of the test and the value C can be either
the nominal capacitance from the datasheet or the calculated capacitance by some other
algorithm.
In our case, the ultracapacitor was first charged to 1.6 V; then, the terminal voltage of
I (s )
Rf
V (s )
Repr
Rm
Cf
Rs
Cm
Figure 4.7 Third-order modified circuit model
50
Cs
Diffusion Test
Voltage(V)
1.6
1.4
1.2
1
0
1
2
3
4
5
Time(Day)
6
7
8
9
Figure 4.8 Terminal voltage as a function of time for the
experiment to determine the effective parallel resistor
nine days’ test was measured, recorded at ten second intervals and plotted in Figure 4.8.
In fact, the effective parallel resistance and the parameter w are time-varying parameters.
The parallel resistance increases as time goes on, from an initial value of 2 Ω to about
3000 Ω after one week. Since, in our case, we want to represent w as a constant, and our
aim is to develop a short-term model for the one-hour behavior of the ultracapacitor, we
used the value at t = 1 hour. This fixed Repr to 5 Ω and therefore w to 0.2. This value of
w was applied in all of the models, and produced good results.
Once w is fixed, another parameter k is added to the above modified model to bring the
DC level of the simulation closer to that of the experiment. The new modified model is
V (s)
As 3 + Bs 2 + Cs + 1
.
=k 3
I ( s)
Ds + Es 2 + Fs + w
(4.15)
The DC gain correction factor k in (4.15) can be determined using the following method.
In Figure 4.9 H1(s) is the real system that needs to be modeled, and H2(s) is the model to
represent the system H1(s). Suppose both of them are excited by the same signal F(s),
producing outputs Y1(s), Y2(s), respectively. If considering only the DC parts of the
system and the model,
51
Figure 4.9 Finding coefficient k in Eq. (4.15)
∞
Y (0)
=
H 1 (0) = 1
F (0)
∫ y (t )dt
1
0
∞
∫ f (t )dt
0
∞
Y (0)
=
H 2 (0) = 2
F (0)
∫ y (t )dt
2
0
∞
∫ f (t )dt
0
∞
H (0)
=
k= 1
H 2 (0)
∫ y (t )dt
1
0
∞
∫ y (t )dt
2
0
where k is the DC gain ratio of the real system to the model. Thus, multiplying the model
by the factor k ensures that the DC levels of the simulated and measured data match. In
our particular case, the value k is found by calculating the ratio of the area under the
curve of measured voltages to the area under the curve of the simulated voltages.
4.3 The First-order Model
The first-order circuit model is shown in Figure 4.10. Its transfer function is
V ( s ) As + 1
=
.
I ( s)
Bs
52
I (s )
R
V (s )
C
Figure 4.10 First-order circuit model
Figure 4.11 shows each coefficient in the first-order transfer function as derived using a
least squares fit, as a function of the number of samples used in doing the fitting. The
point at 3500 samples was chosen because it represents a balance between the transient
and steady-state parts of the data. The first-order model is found to be
V ( s ) 3.49 s + 1
=
.
I (s)
3377 s
Then, the first-order modified model is found to be
3.49 s + 1
V ( s)
= 0.839
.
3377 s + 0.2
I ( s)
The validation results are shown in Figure 4.12, Figure 4.13 and Figure 4.14.
Parameter "A"
4
3.8
X: 3502
Y: 3.485
3.6
3.4
3.2
3
2.8
0
500
1000
1500
2000
2500
3000
3500
4000
Parameter "B"
3600
3400
X: 3502
Y: 3377
3200
3000
2800
2600
0
500
1000
1500
2000
2500
3000
Number of Samples Involved in the Least Squares Fit
3500
4000
Figure 4.11 Transfer function coefficient identification for first-order model
53
0
Voltage(V)
-20
1
0.75
0.5
0.25
0
Voltage(V)
Current(A)
Measurement
20
1
0.75
0.5
0.25
0
0
500
1000
1500
2000
2500
Measurement
3000
3500
4000
0
500
1000
1500
2000
Simulation
2500
3000
3500
4000
0
500
1000
1500
2000
Error
2500
3000
3500
4000
0
500
1000
1500
2000
2500
Time(Second)
3000
3500
4000
Error(V)
0.05
0
-0.05
-0.1
Figure 4.12 Comparison of simulated and measured voltages of the first validation
for the first-order modified model
Current(A)
Measurement
20
0
-20
Voltage(V)
-40
500
1000
1500
2000 2500 3000
Measurement
3500
4000
4500
5000
0
500
1000
1500
2000
3500
4000
4500
5000
0
500
1000
1500
2000
3000
3500
4000
4500
5000
0
500
1000
1500
2000 2500 3000
Time(Second)
3500
4000
4500
5000
1.5
1
0.5
0
Voltage(V)
0
1.5
1
0.5
0
0.05
Error(V)
2500 3000
Simulation
2500
Error
0
-0.05
-0.1
Figure 4.13 Comparison of simulated and measured voltages of the second validation
for the first-order modified model
54
Current(A)
Measurement
20
0
-20
Voltage(V)
-40
1000
2000
3000
4000
Measurement
5000
6000
7000
0
1000
2000
3000
4000
Simulation
5000
6000
7000
0
1000
2000
3000
4000
5000
6000
7000
0
1000
2000
3000
4000
Time(Second)
5000
6000
7000
1
0.5
0
Voltage(V)
0
1.5
1.5
1
0.5
0
Error
Error(V)
0.15
0.1
0.05
0
-0.05
Figure 4.14 Comparison of simulated and measured voltages of the third validation
for the first-order modified model
4.4 The Second-order Model
The second-order circuit model is shown in Figure 4.15. Its transfer function is
V ( s ) As 2 + Bs + 1
=
.
I ( s)
Cs 2 + Ds
I (s)
R1
V (s )
R2
C1
C2
Figure 4.15 Second-order circuit model
55
Figure 4.16 shows each coefficient in the second-order transfer function as derived using
a least squares fit, as a function of the number of samples used to do the fitting. The point
at 3500 samples was chosen because it represents a balance between the transient and
steady-state parts of the data. The second-order model is found to be
V ( s ) 1.784 s 2 + 3.951s + 1
=
.
I ( s ) 1258.85s 2 + 2516 s
Then the second-order modified model is found to be
V (s)
1.784 s 2 + 3.951s + 1
= 0.661
.
I (s)
1258.85s 2 + 2516 s + 0.2
The validation results are shown in Figure 4.17, Figure 4.18 and Figure 4.19.
Parameter "A"
1.9
Parameter "B"
4.5
1.8
X: 3500
Y: 1.784
1.7
4
X: 3500
Y: 3.951
1.6
1.5
0
1000
2000
3000
Parameter "C"
1400
0
1000
2000
3000
4000
Parameter "D"
2800
2700
1300
X: 3500
Y: 1259
1200
2600
2500
X: 3500
Y: 2516
2400
1100
1000
3.5
4000
2300
0
1000
2000
3000
4000
Number of Samples Involved
in the Least Squares Fit
0
1000
2000
3000
4000
Number of Samples Involved
in the Least Squares Fit
Figure 4.16 Transfer function coefficient identification for second-order model
56
Measurement
Current(A)
20
10
0
-10
-20
0
500
1000
1500
0
500
1000
1500
2000
Simulation
0
500
1000
1500
2000
Error
0
500
1000
1500
Voltage(V)
1
2000
2500
Measurement
3000
3500
4000
2500
3000
3500
4000
2500
3000
3500
4000
2000
2500
Time(seconds)
3000
3500
4000
0.5
0
Voltage(V)
1
0.5
0
Voltage(V)
0.1
0.05
0
-0.05
-0.1
Current(A)
Figure 4.17 Comparison of simulated and measured voltages of the first validation
for the second-order modified model
0
Voltage(V)
Voltage(V)
-50
0
500
1000
1500
2000 2500
3000
Measurement
3500
4000
4500
5000
0
500
1000
1500
2000
3500
4000
4500
5000
0
500
1000
1500
2000
3000
3500
4000
4500
5000
0
500
1000
1500
2000 2500
3000
Time(seconds)
3500
4000
4500
5000
1.5
1
0.5
0
1.5
2500
3000
Simulation
1
0.5
0
0.2
Voltage(V)
Measurement
50
2500
Error
0
-0.2
Figure 4.18 Comparison of simulated and measured voltages of the second validation
for the second-order modified model
57
Current(A)
Measurement
20
0
-20
-40
Voltage(V)
1.5
1000
2000
3000
4000
Measurement
5000
6000
7000
0
1000
2000
3000
4000
Simulation
5000
6000
7000
0
1000
2000
3000
4000
5000
6000
7000
0
1000
2000
3000
4000
Time(seconds)
5000
6000
7000
1
0.5
0
1.5
Voltage(V)
0
1
0.5
0
Error
Voltage(V)
0.15
0.1
0.05
0
-0.05
Figure 4.19 Comparison of simulated and measured voltages of the third validation
for the second-order modified model
4.5 The Third-order Model
The third-order circuit model is shown in Figure 4.6. Its transfer function is
V ( s ) As 3 + Bs 2 + Cs + 1
=
.
I ( s)
Ds 3 + Es 2 + Fs
Figure 4.20 shows each coefficient in the third-order transfer function as derived using a
least squares fit, as a function of the number of samples used in doing the fitting. The
point at 3500 samples was chosen because it represents a balance between the transient
and steady-state parts of the data. The third-order model is found to be
V ( s) 1.254 s 3 + 4.381s 2 + 4.937 s + 1
=
.
I (s)
873.2s 3 + 2922s 2 + 2924s
Then the third-order modified model is found to be
58
V (s)
1.254s 3 + 4.381s 2 + 4.937 s + 1
= 0.746
.
I (s)
873.2s 3 + 2922s 2 + 2924s + 0.2
The validation results are shown in Figure 4.21, Figure 4.22 and Figure 4.23.
Parameter "A"
1.5
X: 3500
Y: 4.381
4
X: 3500
Y: 1.254
1
Parameter "B"
4.5
3.5
0.5
0
1000
2000
3
4000
Parameter "C"
5.5
0
1000
2000
3000
800
4000
2000
3000
4000
Parameter "D"
X: 3500
Y: 873.2
0
1000
2000
3000
4000
Parameter "F"
3400
3000
3200
X: 3500
Y: 2922
2800
2600
1000
1000
Parameter "E"
3200
0
1200
X: 3500
Y: 4.937
5
4.5
3000
0
X: 3500
Y: 2924
3000
2800
1000
2000
3000
4000
Number of Samples Involved
in the Least Squares Fit
0
1000
2000
3000
4000
Number of Samples Involved
in the Least Squares Fit
Figure 4.20 Transfer function coefficient identification for the third-order model
59
Voltage(V)
Voltage(V)
Voltage(V)
Current(A)
Measurement
20
0
-20
0
500
1000
1500
1
0.75
0.5
0.25
0
0
500
1000
1500
1
0.75
0.5
0.25
0
0
2000
Simulation
500
1000
1500
2000
Error
500
1000
1500
2000
2500
Measurement
3000
3500
4000
2500
3000
3500
4000
2500
3000
3500
4000
2000
2500
Time(Second)
3000
3500
4000
0.05
0
-0.05
0
Figure 4.21 Comparison of simulated and measured voltages of the first validation
for the third-order modified model
Current(A)
Measurement
20
0
-20
Voltage(V)
-40
Voltage(V)
500
1000
1500
2000 2500
3000
Measurement
3500
4000
4500
5000
0
500
1000
1500
2000
3500
4000
4500
5000
0
500
1000
1500
2000
3000
3500
4000
4500
5000
0
500
1000
1500
2000 2500
3000
Time(Second)
3500
4000
4500
5000
1.5
1
0.5
0
1.5
2500
3000
Simulation
1
0.5
0
Voltage(V)
0
0.05
2500
Error
0
-0.05
-0.1
Figure 4.22 Comparison of simulated and measured voltages of the second validation
for the third-order modified model
60
Current(A)
Measurement
20
0
-20
0
1000
2000
3000
4000
Measurement
5000
6000
7000
0
1000
2000
3000
4000
Simulation
5000
6000
7000
0
1000
2000
3000
4000
5000
6000
7000
0
1000
2000
3000
4000
Time(Second)
5000
6000
7000
1.5
1
0.5
0
1.5
1
0.5
0
Error
Voltage(V)
Voltage(V)
Voltage(V)
-40
0.1
0.05
0
-0.05
Figure 4.23 Comparison of simulated and measured voltages of the third validation
for the third-order modified model
4.6 The Fourth-order Model
The fourth-order circuit model is shown in Figure 4.24. Its transfer function is
V ( s ) As 4 + Bs 3 + Cs 2 + Ds + 1
=
.
I (s)
Es 4 + Fs 3 + Gs 2 + Hs
I (s )
V (s )
R3
R2
R1
C1
C2
R4
C3
Figure 4.24 Fourth-order circuit model
61
C4
Figure 4.25 shows each coefficient in the fourth-order transfer function as derived using a
least squares fit, as a function of the number of samples used in doing the fitting. The
point at 3500 samples was chosen because it represents a balance between the transient
and steady-state parts of the data. The fourth-order model is found to be
V ( s ) 0.8765s 4 + 4.105s 3 + 7.29s 2 + 5.63s + 1
=
.
I (s)
605s 4 + 2747 s 3 + 4615.3s 2 + 3083.2 s
Then, the fourth-order modified model is found to be
V ( s)
0.8765s 4 + 4.105s 3 + 7.29 s 2 + 5.63s + 1
= 0.78
.
I ( s)
605s 4 + 2747 s 3 + 4615.3s 2 + 3083.2s + 0.2
The validation results are shown in Figure 4.26, Figure 4.27 and Figure 4.28.
Parameter "A"
1
0.8
0
8
7.5
7
0
700
650
600
0
5500
5000
4500
0
5
X: 3500
Y: 0.8765
1000
2000
3000
Parameter "C"
3
0
4000
5.5
5
0
4000
1000
2500
0
3500
3000
2500
0
4000
4000
2000
3000
Parameter "F"
4000
X: 3500
Y: 2747
3000
4000
2000
3000
Parameter "D"
X: 3500
Y: 5.632
3500
X: 3500
Y: 4615
1000
2000
3000
Number of Samples Involved
in the Least Squares Fit
1000
6
X: 3500
Y: 605
1000
2000
3000
Parameter "G"
X: 3500
Y: 4.105
4
X: 3500
Y: 7.293
1000
2000
3000
Parameter "E"
Parameter "B"
1000
2000
3000
Parameter "H"
4000
X: 3500
Y: 3083
1000
2000
3000
Number of Samples Involved
in the Least Squares Fit
4000
Figure 4.25 Transfer function coefficient identification for the fourth-order model
62
Measurement
Current(A)
20
0
-20
500
1000
1500
2000
2500
Measurement
3000
3500
4000
0
500
1000
1500
2000
Simulation
2500
3000
3500
4000
0
500
1000
1500
2000
Error
2500
3000
3500
4000
0
500
1000
1500
2000
2500
Time(Second)
3000
3500
4000
Error(V)
Voltage(V)
Voltage(V)
1
0.75
0.5
0.25
0
0
1
0.75
0.5
0.25
0
0.05
0
-0.05
Figure 4.26 Comparison of simulated and measured voltages of the first validation
for the fourth-order modified model
Current(A)
Measurement
20
0
-20
Voltage(V)
Voltage(V)
-40
0
500
1000
1500
2000 2500
3000
Measurement
3500
4000
4500
5000
0
500
1000
1500
2000
3500
4000
4500
5000
0
500
1000
1500
2000
3000
3500
4000
4500
5000
0
500
1000
1500
2000 2500
3000
Time(Second)
3500
4000
4500
5000
1.5
1
0.5
0
1.5
1
0.5
0
0.05
Error(V)
2500
3000
Simulation
2500
Error
0
-0.05
-0.1
Figure 4.27 Comparison of simulated and measured voltages of the second validation
for the fourth-order modified model
63
Voltage(V)
Voltage(V)
Current(A)
Measurement
20
0
-20
-40
0
1000
2000
3000
4000
Measurement
5000
6000
7000
0
1000
2000
3000
4000
Simulation
5000
6000
7000
0
1000
2000
3000
4000
5000
6000
7000
0
1000
2000
3000
4000
Time(Second)
5000
6000
7000
1.5
1
0.5
0
1.5
1
0.5
0
Error(V)
0.1
0.05
0
-0.05
Error
Figure 4.28 Comparison of simulated and measured voltages of the third validation
for the fourth-order modified model
4.7 Comparison of Models
The poles and zeros of the different order models are listed in Table 4.1 and Table 4.2.
The first-order transfer function has one real pole and no zero, the second-order transfer
function has two real poles and one real zero, the third- and fourth-order transfer
functions have both real and complex conjugate poles and zeros.
The error from the first-, second-, third- and fourth-order models have been compared
in this section. For the comparison purpose all the models are excited with the same
dynamic current profile. The results are shown in Figure 4.29, 4.30 and Figure 4.31. And
the integral squared error from different models is listed in Table 4.3.
64
Order
Table 4.1 Poles for the four short-term models of different orders
Poles
First
− 5.9 × 10 −5
__
__
__
Second
− 7.9 × 10 −5
-2
__
__
Third
− 6.8 × 10 −5
__
-1.67+0.74i
-1.67-0.74i
Fourth
− 6.5 × 10 −5
-2
-1.27+0.97i
-1.27-0.97i
Table 4.2 Zeros for the four short-term models of different orders
Order
Zeros
First
__
__
__
__
Second
-0.29
-1.92
__
__
Third
-0.26
__
-1.62+0.7i
-1.62-0.7i
Fourth
-0.25
-1.96
-1.24+0.91i
-1.24-0.91i
0.08
First Order Model
Second Order Model
Third Order Model
Fourth Order Model
0.06
0.04
Error(V)
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
500
1000
1500
2000
2500
Time(seconds)
3000
3500
4000
Figure 4.29 Comparison of the errors generated during the first validation test by the
first-, second-, third- and fourth-order modified models
65
0.06
First Order Model
Second Order Model
Third Order Model
Fourth Order Model
0.04
0.02
0
Error(V)
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
0
500
1000
1500
2000 2500
3000
Time(seconds)
3500
4000
4500
5000
Figure 4.30 Comparison of the errors generated during the second validation test by
the first-, second-, third- and fourth-order modified models
0.15
First Order Model
Second Order Model
Third Order Model
Fourth Order Model
Error(V)
0.1
0.05
0
-0.05
0
1000
2000
3000
4000
Time(seconds)
5000
6000
7000
Figure 4.31 Comparison of the errors generated during the third validation test by the
first-, second-, third- and fourth-order modified models
66
Table 4.3 Integral squared voltage errors in V 2 s for three different datasets
1st order
2nd order
3rd order
4th order
Dataset 1
7.7
7.8
7.1
7.3
Dataset 2
24.1
33.4
26.8
25.6
Dataset 3
16.4
21.3
15.7
13.9
From the three tests, we can see each model is a good fit to the measured data in some
time frame. For example, the first-order model gives a good fit for the first test from 800
seconds to 3000 seconds but it is not a good fit for the time beyond 3200 seconds in the
same test. The second-order model has the least error for time frames less than 500
seconds in the first test, while it has the largest error throughout the second test.
Generally speaking, the third- and fourth-order models are a better fit than the lowerorder models. However, the fourth-order model is not a significant improvement over the
third-order model. Considering that the parameter identification for the fourth-order
model is significantly more complicated, the third-order model is the best trade-off
between accuracy and complexity.
For the third test, the error from different order models are shown in Figure 4.31. The
accuracy of the model deteriorates after 5000 seconds; the error between the measured
terminal voltage and the simulated voltage increases continually starting at that point in
time. The reason is the constant value w=0.2 calculated from (4.14) to model the charge
redistribution in the neglected sections is no longer a good fit for that time frame; as time
goes on, the parallel resistance increases, and therefore the appropriate value for w
decreases. If w is set to a smaller value, the behavior of the ultracapacitor after 5000
67
seconds could be corrected, but the behavior before 5000 seconds would not be as
expected. Thus, this model can only reflect behavior of the ultracapacitor for time frames
around one hour.
4.8 Conclusions
Short term first-, second-, third- and fourth-order models for a NESSCAP3500P
ultracapacitor cell have been presented. They were developed by fitting transfer function
coefficients. The behavior of the ultracapacitor within a time frame around one hour was
observed and least squares identification was employed for the parameter identification
based on dynamic current profiles; as a result, the model is a good fit to the transient
behavior of the ultracapacitor. To fit longer-term behavior we need to keep more RC
branches in Figure 4.1; that is, a higher-order model must be considered to handle the
very slow long-term charge redistribution phenomenon such as the one shown in Figure
4.9. However, this would make the identification even more complicated. Instead,
another efficient algorithm to derive the long-term model has been developed, and is
presented in the next chapter. It is expected that the accurate dynamic short-term models
developed in this chapter will allow for improved ultracapacitor management in energy
storage systems.
68
CHAPTER V
LONG-TERM MODEL
Chapter IV has presented short-term models for the NESSCAP3500P ultracapacitor.
These models fit the dynamic behavior of the ultracapacitor over a time frame of around
one hour. While the short-term models are sufficient for simulating charging and
discharging of the ultracapacitor, which are fast phenomena, they cannot be used to
accurately simulate long-term behavior. An understanding of long-term behavior is
important, for example, to estimate the terminal voltage of the ultracapacitor or the
instantaneous available energy from the ultracapacitor after it has been relaxed for two
months. Thus, long-term models are needed for implementation of control strategies for
some practical cases. In this chapter, we derive an accurate model for the slow transients
of the NESSCAP3500P ultracapacitor over a time frame of about two months.
5.1 Experiments for the Long-term Behavior
To obtain data to which to fit a long-term model, two tests were devised. The first,
denoted hereafter as the charging-then-relaxing test, started with an ultracapacitor with a
terminal voltage of 0.24 V that had been at rest (i.e. neither charged nor discharged) for a
couple of weeks. In the test, the ultracapacitor was first charged to 1.6 V in thirteen
minutes; then, the test circuit was opened to let the ultracapacitor relax for 56 days. The
second test, denoted hereafter as the discharging-then-relaxing test, used the same
69
Figure 5.1 Charging current profile used at beginning of the charging-thenrelaxing test
ultracapacitor at the end of the charging-then-relaxing test. It was discharged to 0.5 V in
two minutes; then the test circuit was opened to let the ultracapacitor relax for 58 days.
During the total of 114 days of the test, the terminal voltage of the ultracapacitor was
measured and recorded. During the first few days in each test, the terminal voltage was
recorded by the microcontroller board every ten seconds; after that a multimeter was used
to measure the terminal voltage at one day intervals. From the obtained data, the charge
redistribution characteristic could be clearly seen.
The charging current profile used to excite the ultracapacitor at the beginning of the
charging-then-relaxing test is shown in Figure 5.1. The maximum charging current was
26.5 A and the average current was 6.7 A. The charging lasted for 13 minutes. After that,
the ultracapacitor was removed from the test circuit; this was done to prevent leakage of
70
1.6
1.5
1.4
Voltage(V)
1.3
1.2
1.1
1
0.9
0
10
20
30
Time(day)
40
50
60
Figure 5.2 Terminal voltage of the ultracapacitor during the charging-thenrelaxing test
current through the MOSFET switches. Then, the ultracapacitor was relaxed for
approximately two months. The terminal voltage of the ultracapacitor under test is shown
in Figure 5.2. From Figure 5.2 we can see that, when the charging ceased, the terminal
voltage decreased exponentially with a short time constant in the first several hours; then,
the terminal voltage decreased exponentially with an intermediate time constant for
several days, and finally decreased exponentially with the largest time constant for
several weeks. This behavior is consistent with the circuit model of ultracapacitors in
Figure 4.1, which is composed of infinite RC branches with lower time constant branches
closer to the terminals and longer time constant branches further from the terminals. The
first several hours of data shown in Figure 5.2 corresponds to energy redistribution
71
between RC branches with short time constants. The next several days of data shown
corresponds to energy redistribution to the branches with relatively longer time constants.
The final weeks of data shown correspond to energy redistribution to the branches with
very long time constants. Thus, we can use several exponential functions with different
time constants to fit the experimental data. From the transfer function point of view we
need multiple poles distributed over a wide range in the s plane to account for the various
time constants.
At the end of the charging-then-relaxing test the same ultracapacitor was discharged to
0
-2
Discharging Current(A)
-4
-6
-8
-10
-12
-14
-16
0
20
40
60
80
Time(seconds)
100
120
140
Figure 5.3 Discharging current profile used at beginning of the dischargingthen-relaxing test
72
0.7
0.68
0.66
0.64
Voltage(V)
0.62
0.6
0.58
0.56
0.54
0.52
0.5
0
10
20
30
Time(day)
40
50
60
Figure 5.4 Terminal voltage of the ultracapacitor during the discharging-thenrelaxing test
0.5 V. The discharging current profile is shown in Figure 5.3. The maximum discharging
current was 16 A and the average current was 12 A. The discharging lasted for 2 minutes.
After that, the ultracapacitor was removed from the test circuit to prevent leakage of
current through the MOSFET switches. Then, the ultracapacitor was relaxed for two
months. The terminal voltage during the test is shown in Figure 5.4. At the beginning, the
terminal voltage increased rapidly then slowly and went to flat gradually. After that, the
terminal voltage began to decrease very slowly, so that a difference in the measurement
could be seen every two or three days.
Again, the response can be approximated as the sum of several exponential
components with different time constants. It is noted that the terminal voltage is
decreasing slowly at the end of the discharging-then-relaxing test. This is because of the
initial condition set up by the previous test (charging-then-relaxing), in which the
73
intermediate branches had not finished redistributing their charge to the slower branches
farther from the terminals.
5.2 Long-term Model Parameter Identification
Although the charging and discharging current profiles shown in Figure 5.2 and Figure
5.3 are exponential-like, their durations were extremely short compared to the resulting
transients (the entire test lasted for 114 days; in contrast, the charging current profile
lasted for only 800 seconds, and the discharging current profile lasted for 120 seconds).
As a consequence, the charging and discharging current profile could be seen as impulse
signals and the two slow transients due to charge redistribution after charging and
discharging could be seen as the positive impulse response and negative impulse response,
respectively.
The long-term model of the ultracapacitor was derived from the charging-then-relaxing
test. We approximate the charging current profile shown in Figure 5.1 as an impulse
signal weighted by its area. The area under the charging current curve was calculated as
800
∫ i(t )dt .
S charge =
(5.1)
t =0
The input current signal can be expressed as
i (t ) = S chargeδ (t ) .
(5.2)
Taking the Laplace transform of Equation (5.2) yields
I ( s ) = S charge .
(5.3)
Denote H (s ) as the impedance transfer function and V(s) as the s-domain output voltage
of the ultracapacitor. Then
74
V ( s ) = I ( s ) H ( s ) = S charge H ( s ) .
(5.4)
Taking the inverse Laplace transform of Equation (5.4) yields the time-domain response
v(t ) = S charge h(t ) ,
(5.5)
where h(t) is the impulse response of the ultracapacitor. From observing the terminal
voltage behavior in the charging-then-relaxing test, the summation of several exponential
functions can approximate the time response curve. Thus, we set
h(t ) = ae − bt + ce − dt + ee − ft + ge − ht + ke − lt + w ,
(5.6)
and choose Equation (5.5) as the curve fitting function to the experimental data. Taking
the Laplace transform of Equation (5.6) yields
H (s) =
a
c
e
g
k
w
+
+
+
+
+ .
s+b s+d s+ f s+h s+l s
(5.7)
Parameters b, d, f, h, l, in (5.7) are five of six poles that need to be identified (the other
pole is zero) and a, c, e, g, k, w are weights in each exponential function that need to be
identified too. Since the unknown parameters in (5.7) may be very small, we keep each
term separate (all first order) rather than combining them together for a high-order
transfer function, which would lead to round-off errors.
Equation (5.5) can be arranged in matrix form as
(e
−bt
e −dt
e − ft
e −ht
e −lt
75
⎛a⎞
⎜ ⎟
⎜c⎟
⎜ e ⎟ ⎛ v(t ) ⎞
⎟
1⎜ ⎟ =⎜
⎜ g ⎟ ⎜⎝ S charge ⎟⎠
⎜k⎟
⎜ ⎟
⎜ w⎟
⎝ ⎠
)
(5.8)
Equation (5.8) represents the terminal voltage transients of the ultracapacitor due to the
impulse charging current. In order to identify all of the parameters, different terminal
voltages and the corresponding time instants need to be measured and recorded. This will
give us as many equations as needed, each one an application of Equation (5.8). For our
identification data, we recorded the terminal voltages at ten-second intervals for the first
nine days, then reduced the sampling rate to one sample per day since there is no
measurable change in the terminal voltage at faster rates. We then get hundreds of
thousands of equations. Arranging these equations in matrix form, we obtain
M×X = V ,
(5.9)
where
⎛ e − bt1
⎜ −bt
⎜e 2
M=⎜
⎜ M
⎜ e −btn
⎝
e − dt1
e −dt2
e − ft1
e − ft2
M
e
−bt n
e − ht1
e −ht1
M
e
−bt n
M
e
−btn
⎛a⎞
⎜ ⎟
⎜c⎟
⎜e⎟
X=⎜ ⎟,
⎜g⎟
⎜k⎟
⎜ ⎟
⎜ w⎟
⎝ ⎠
and
⎛ v(t1 ) ⎞
⎜
⎟
⎜ S charge ⎟
⎜ v(t 2 ) ⎟
V = ⎜ S charge ⎟ .
⎜
⎟
⎜ M ⎟
⎜ v(t n ) ⎟
⎜S
⎟
⎝ charge ⎠
76
e − lt1
e −lt1
M
e
−bt n
1⎞
⎟
1⎟
⎟,
M⎟
1⎟⎠
We use least squares identification to search for the optimal coefficients. Based on the
least squares identification analysis in Chapter IV, we can obtain
X = (M T M ) −1 M T V .
(5.10)
In Equation (5.9) the unknown coefficients are present in both matrix M and matrix X.
Before we use Equation (5.10) to derive X we must fix coefficients b, d, f, h, l in matrix
M. Following the analysis in Section 5.1, once the pole (-b) is fixed, the pole (-d) is
chosen five times smaller than it, the pole (-f) is chosen five times smaller than the pole
(-d), the pole (-h) is chosen ten times smaller than pole (-f) and the pole (-l) is chosen ten
times smaller than the pole (-h). Based on the preset poles we get coefficients a, c, e, g, k,
w by Equation (5.10). Then, the error between the simulation and the measurement is
calculated. For the next step, we make a small deviation to the preset poles, then get the
coefficients and error again. If the error is bigger than before, we change the deviation to
the opposite sign. If the error is smaller we give another smaller deviation to the poles
again. By this way we can find the least squares error based on certain optimal poles.
Finally the optimal poles are found to be
b = 6 × 10 −3 ,
d = 1.2 × 10 −3 ,
f = 2.5 × 10 −4 ,
h = 2.5 ×10−5 ,
l = 1.5 × 10 −6 .
77
Measurement and Curve Fitting
1.6
Curve Fitting
Measured one
Voltage(V)
1.4
1.2
1
0.8
0
10
20
30
40
50
60
40
50
60
Curve Fitting Error
0.02
Error(V)
0.01
0
-0.01
-0.02
0
10
20
30
Time(day)
Figure 5.5 Using the summation of exponential functions to curve fitting the
terminal voltage from charging-then-relaxing test, and comparing the error
between them
Based on the optimal poles the parameters a, c, e, g, k, w can be determined by
Equation (5.10). The long-term model is given as
H ( s) =
3.009 ×10 −4 2.085 ×10 −4 2.386 ×10 −4 2.717 ×10 −4 4.045 ×10 −4 1.421×10 −3 . (5.11)
+
+
+
+
+
s + 6 ×10 −3 s + 1.2 ×10 −3 s + 2.5 ×10 −4 s + 2.5 ×10 −5 s + 1.5 ×10 −6
s
The curve fitting result versus the measurement, as well as the error, is plotted in Figure
5.5. The curve fitting function is designed to fit two-month data. From Figure 5.5 we can
see how good the curve fitting function is!
In order to test the model, we excite the long-term model in Equation (5.11) with the
actual charging current profiles. The simulation is shown in Figure 5.6 along with the
error between measurement and simulation. From Figure 5.6, we can see the simulation
error is between -0.05 V and 0.02 V for 56 days.
78
Measurement
Voltage(V)
1.5
1
0.5
0
0
10
20
30
Simulation
40
50
60
0
10
20
30
Error
40
50
60
0
10
20
30
Time(day)
40
50
60
Voltage(V)
1.5
1
0.5
0
Voltage(V)
0.02
0
-0.02
-0.05
Figure 5.6 Validating the long term model by exciting it with the charging current
profile
5.3 Validation
We derive the long-term model based on the charging-then-relaxing data. The model
should also fit the discharging-then-relaxing data since for an ultracapacitor the charging
and discharging process should share the same model. In order to validate the long-term
model we use the discharging-then-relaxing data to test it. First we show how well the
Equation (5.11) fits the discharging-then-relaxing data assuming the current is an impulse
as before. Then the discharging current profiles were input to the model to see the
simulation results.
The plot to show how well the Equation (5.6) derived from curve fitting the chargingthen-relaxing test data fits the discharging-then-relaxing test data in the time-domain is
shown in Figure 5.7. The error in 58 days’ test is less than 0.04 V. The discharging-then79
relaxing simulation assumes a model initial state that results from the past application of
the charging-then-relaxing test. This is not the same as assuming a relaxed initial
condition at the given terminal voltages, because the time constant from the slowest
branch of the ultracapacitor model may be more than two months. The Equation (5.6) is
weighted by the area under the discharging current profile. The simulation represents the
response to an appropriately weighted impulse current input.
Then, we repeated the validation test using the actual discharging current profile
shown in Figure 5.3 applied to the long-term model derived from the charging-thenrelaxing test.
Measurement and Fitting
0.7
Voltage(V)
0.65
0.6
0.55
0.5
0.45
Measured One
Fitting
0
10
20
30
40
50
60
40
50
60
Error
0.04
Error(V)
0.03
0.02
0.01
0
0
10
20
30
Time(day)
Figure 5.7 Using Equation (5.6) multiplied by the discharging current area to
approximate the voltage transients from discharging-then-relaxing test, and
comparing the error between them
80
Again, the simulation considers the effect of the charging-then-relaxing test on the
terminal voltage of the ultracapacitor during the discharging-then-relaxing test. The
validation plot is shown in Figure 5.8. From the results, the error between measurement
and the predicted voltage transients in 58 days’ test is less than 0.08 V.
5.4 Instantaneous Power Discussion
The long-term transfer function shown in Equation (5.11) represents the response to an
impulse current. It has the units of voltage divided by current, so it also represents the
impedance of the ultracapacitor. Since Equation (5.1) is already in a partial-fraction
expansion format, we could derive the first Foster Form [37] shown in Figure 5.9 directly
Voltage(V)
Measurement
0.65
0.6
0.55
0.5
Voltage(V)
0.45
0
10
20
30
Simulation
40
50
60
0
10
20
30
Error
40
50
60
0
10
20
30
Time(day)
40
50
60
0.65
0.6
0.55
0.5
0.45
Voltage(V)
0.08
0.06
0.04
0.02
0
Figure 5.8 Validating the long term model by exciting it with discharging current
profile
81
Figure 5.9 First Foster Form derived from Equation (5.11)
from it. The first Foster Form describes the RC driving-point impedance of the
ultracapacitor. However, in Figure 5.9 there is no term to represent the equivalent series
resistance (ESR) for the ultracapacitor. So in Figure 5.10 we add the ESR derived from
the first-order short-term model to the first Foster Form.
The first Foster Form can be transferred to the Cauer Form from which we can
calculate the instantaneous available energy more easily. The Cauer Form as well as the
R and C values for the first branch with the shortest time constant are shown in Figure
5.11. The Cauer Form consists of six RC branches with the shortest time constant closest
to the terminals and longer time constants farther from the terminals. The instantaneous
energy available can be captured using only the first branch in the Cauer Form; since this
branch has the shortest time constant, it represents the part that can release the energy.
The energy from branches with longer time constants will be distributed to the first
branch over a much longer time. Thus the instantaneous energy available
Figure 5.10 Corrected Foster Form with ESR
82
R3
R2
C2
C6
C5
C4
C3
R6
R5
R4
Figure 5.11 First Cauer Form derived from the corrected Foster Form
is only related to the parameters in the first branch over the time frame within the shortest
time constant, and during this time we can suppose there is no current in the other
branches. With this assumption, we can get Figure 5.12 with a load connected to the
terminals to calculate the instantaneous energy available.
The maximum instantaneous power happens when Rload = R1 = 1.03 mΩ. Let us denote
vc1 as the voltage across the capacitor with the capacitance of 351 F in the first branch.
Then, the energy E stored in the first branch is calculated as
1
2
E = × 351× vc1 ,
2
(5.12)
and the instantaneous power Pmax is calculated as
2
Pmax = i
R1 = 1 . 03 m Ω
Rload
2
max
× Rload
R2
C2
v
v
= ( c1 ) 2 × R1 = c1 .
2 R1
4 R1
R3
C3
R4
C4
(5.13)
R6
R5
C5
C6
Figure 5.12 Instantaneous energy calculation with load based on First Cauer
Form
83
In addition, the instantaneous power P for any load Rload is calculated as
vc1 × Rload
.
( R1 + Rload ) 2
2
P=
(5.14)
5.5 Conclusions
A long-term model with six RC branches for a NESSCAP3500P ultracapacitor cell has
been presented in this chapter. It is developed by curve fitting the terminal voltage
transient due to impulse charging current. Hundreds of thousands of terminal voltage
points are measured and recorded and least squares identification is employed to search
for the optimal value of each parameter in the long-term model. As a result, the long-term
model is a good fit to the positive and negative impulse current response over a time
frame of two months.
Since we choose real poles and zeros for the long-term model, the first Foster Form
and Cauer Form can be derived. From the Cauer Form the instantaneous power is
calculated by using the R and C values in the first branch that has the shortest time
constant.
Based on experimental observations throughout the test, and the long-term model
derived, the time constant for the last branch in the multiple RC-branch circuit model for
a NESSCAP3500P ultracapacitor cell is more than two months. This means the last RC
branch will retain the charge information for a very long time. From this point, we get the
idea that for more accurate results we must relax the ultracapacitor for a much longer
time to bring it to a known initial condition before conducting further tests on it.
84
CHAPTER VI
CONCLUSIONS
Ultracapacitors are attracting more and more people’s attention as efficient and
environmentally friendly energy storage devices. In this thesis we present short-term and
long-term linear models to predict the dynamic voltage and current response of an
ultracapacitor cell. The models could be useful in the design of a control strategy for
ultracapacitors as an energy storage system.
For the short-term model we use a third-order transfer function to represent the
dynamic behavior of ultracapacitor within a time frame of around one hour. The data for
the model identification are produced using a dynamic current excitation and measured
using only ordinary laboratory instruments such as an oscilloscope and multimeters. The
parameter identification makes use of differenced voltage and current data, producing
more accurate results than identification using the standard difference equation. The
transfer function is not restricted to have real poles and zeros, and the introduced
complex conjugate poles and zeros allow a more accurate fit to the collected data. Also
we derived the first-, second- and fourth-order models by the same method and compared
their errors. It was found that increasing the order to four does not produce significant
improvement for the model. The third-order model produces more accurate results than
the first- and second-order models, and the procedure to derive unknown parameters
85
is simpler than the fourth-order model, so we prefer the third-order model as the shortterm model.
As far as the long-term model is concerned, we want to know the terminal voltage and
instantaneous available power information for the ultracapacitor if we settle it for a long
time. We use six RC branches to describe the behavior of the ultracapacitor within a time
frame of around two months. In order to make it possible to calculate the instantaneous
power we design the long-term model to have real poles and zeros; as for the short-term
models, this time we use least squares identification to search for the optimal coefficients,
based on two months of data. Simulations show that the long-term model fits the 114
days’ impulse charging and discharging transients very well. Based on the long-term
model we derived the First Foster Form and Cauer Form circuit models. From the Cauer
Form the energy stored in the fastest branch could be calculated, as well as the
instantaneous power available from the ultracapacitor during test time. In addition we
also find the ultracapacitor retains some memory about what happened to it at least two
months ago from the long-term model analysis. The memory is from the slowest RC
branch in the model since it has a very long time constant and retains the information for
a long time. From this point we get the idea that we must settle down the ultracapacitor at
least several months to bring it to a known initial condition before conducting further
tests. The longer you settle the ultracapacitor before testing, the more accurate the test
results.
From the short-term and long-term linear models we know a lot about the
ultracapacitor cell characteristics. The charge redistribution behavior, the dynamic
behavior and the instantaneous available energy are quite clear. Based on the models,
86
complicated control strategies could be employed to support both battery-ultracapacitor
energy storage systems and ultracapacitor-only energy storage systems, as well as
improved pack balancing strategies.
There is room to improve the models. All of the tests were conducted at room
temperature. We know that ultracapacitors can be operated over a wide temperature range
but that their parameters will change, so a better model would include temperature related
information. In addition for short-term and long-term models we consider only linear
models; we could add some voltage-dependent parameter to the fast branch to reflect the
non-linear behavior for better fitting.
87
BIBLIOGRAPHY
1.
J.W. Dixon and M.E. Ortuzar, “Ultracapacitors + DC-DC converters in regenerative
braking system,” IEEE Aerospace and Electronic Systems Magazine, vol. 17, pp.
16-21, Aug. 2002.
2.
C. Ashtiani, R. Wright and G. Hunt, “Ultracapacitors for automotive applications,”
Journal of Power Sources, vol. 154, pp. 561-566, Mar. 21, 2006.
3.
J.M. Miller, P.J. McCleer and M. Cohen, “Ultra-capacitors as energy buffers in a
multiple zone electrical distribution system,” Global Powertrain Conference and
Exposition, Sep. 23-26, 2003.
4.
L. Gao, R.A. Dougal and S. Liu, “Power enhancement of an actively controlled
battery/ultracapacitor hybrid,” IEEE Trans. on Power Electronics, vol. 20, pp. 236243, Jan. 2005.
5.
D.W. Kassekert, A.O. Isenberg and J.T. Brown, “High power density bipolar leadacid battery for electric vehicle propulsion,” Intersociety Energy Conversion
Engineering Conference, vol. 1, pp. 411-417, Sep. 12-17, 1976.
6.
P.B. Balbuena and Y.X. Wang, eds., Lithium ion batteries: Solid Electrolyte
Interphase, Imperial College Press, 2004.
7.
Maxwell Technologies, “Top 10 reasons for using ultracapacitors in your system
designs,” Apr. 2006, available online at
http://www.electricdrive.org/index.php?tg=fileman&idx=get&id=7&gr=Y&path=&
file=Maxwell-top+10+reasons.pdf
8.
F. Barbir, T. Molter and L. Dalton, “Regenerative fuel cells for energy storage:
efficiency and weight trade-offs,” IEEE Aerospace and Electronic System
Magazine, vol. 20, pp. 35-40, Mar. 2005.
9.
Maxwell Technologies, “Ultracapacitor product guide,” 2001, available online at
www.maxwell.com/ultracapacitors/manuals/ultracap_product_guide.pdf
88
10. M. Perrin, P. Malbranche and E. Lemaire-Potteau, “Temperature behavior:
Comparison for nine storage technologies results from the INVESTIRE network,”
Journal of Power Sources, vol. 154, pp. 545-549, Dec. 2005.
11. D.Y. Jung, Y.H. Kim, S.W. Kim and S.-H. Lee, “Development of ultracapacitor
modules for 42-V automotive electrical systems,” Journal of Power Sources, vol.
114, pp. 366-373, Oct. 2002.
12. Nesscap Ultracapacitor Datasheet, 2003, available online at
www.nesscap.com/data_nesscap/Download%20full%20data%20sheet.pdf
13. J.A. Hicks, R. Gruich, A. Oldja, D. Myers, et al., “Ultracapacitor energy
management and controller development for a series-parallel 2-by-2 hybrid electric
vehicle,” 2007 Vehicle Power and Propulsion Conf, 2007.
14. D. Linzen, S. Buller and E. Karden “Analysis and evaluation of charge-balancing
circuits on performance, reliability, and lifetime of supercapacitor systems,” IEEE
Trans. on Industry Applications, vol. 41, pp. 1135-1141, Sep.-Oct. 2005.
15. J.M. Miller and M. Everett, “Vehicle electrical system power budget optimization
using ultracapacitor distributed modules,” 2003, available online at
http://www.ansoft.com/news/articles/VPP-Sym-paper41_JMM-ME_.pdf
16. M. Okamura, Proceedings of the 13th International Seminar on Double Layer
Capacitors and Hybrid Energy Storage Devices, Deerfield Beach, Florida, pp. 205,
2005.
17. R. Kotz, J.-C. Sauter, P. Ruch and P. Dietrich, “Voltage balancing: long-term
experience with the 250 V supercapacitor module of the hybrid fuel cell vehicle
HY-LIGHT,” Journal of Power Sources, pp. 264-271, Sep. 2007.
18. K.M. Do, “A dynamic electro-thermal model of double layer supercapacitors for
HEV powertrain applications,” MS Thesis, Dept. of Mechanical Engineering, The
Ohio State University, 2004.
19. K.-D. Henning and S. Schafer, “Impregnated activated carbon for environmental
protection,” Gas Separation and Purification, vol. 7, pp. 235-240, 1993.
20. G.L. Bullard, H.B. Sierra-Alcazar, H.L. Lee and J.L. Morris, “Operating principles
of the ultracapacitor,” IEEE Trans. on Magnetics, vol. 25, pp. 102-106, Jan. 1989.
21. F. Belhachemi, S. Rael, and B. Davat, “A physical based model of power electric
double-layer supercapacitors,” IEEE Industry Applications Conference, vol. 5, pp.
3069-3076, Oct. 8-12, 2000.
89
22. K.C. Roh, J.B. Park, C.-T. Lee and C.W. Park, “Study on synthesis of low surface
area activated carbons using multi-step activation for use in electric double layer
capacitor,” J. Industrial and Engineering Chemistry. To be published.
23. B.E. Conway, “Electrochemical capacitors: their nature, function and applications,”
Electrochemistry
Encyclopedia,
Mar.
2003,
available
online
at
http://electrochem.cwru.edu/ed/encycl/art-c03-elchem-cap.htm
24. J.-S. Lai, S. Levy and M.F. Rose, “High energy density double-layer capacitors for
energy storage applications,” IEEE Aerospace and Electronic Systems Magazine,
vol. 7, pp. 14-19, Apr. 1992.
25. E. Ozatay, B. Zile, J. Anstrom and S. Brennan, “Power distribution control
coordinating ultracapacitors and batteries for electric vehicles,” 2004 American
Control Conference, vol. 5, pp. 4716-4721, June 30-July 2, 2004.
26. R.L. Spyker and R.M. Nelms, “Classical equivalent circuit parameters for a doublelayer capacitor,” IEEE Trans. on Aerospace and Electronic Systems, vol. 36, pp.
829-836, July 2000.
27. D. New and A.J. Kassakian, “Double layer capacitors: automotive applications and
modeling,” MS Thesis, Dept. of Electrical Engineering, Massachusetts Institute of
Technology, 2002.
28. L. Zubieta and R. Bonert, “Characterization of double-layer capacitor for power
electronics application,” IEEE Trans. on Industry Applications, vol. 36, pp. 199-205,
Jan./Feb. 2000.
29. R. Faranda, M. Gallina and D.T. Son, “A new simplified model of double-layer
capacitors,” Intl. Conf. on Clean Electrical Power, pp. 706-710, May 21-23, 2007.
30. F. Belhachemi, S. Rael and B. Davat, “Supercapacitors electrical behavior for
power electronics applications,” EPE-PEMC, Kosice, Slovak Republic, Sep. 5-7,
2000.
31. W. Lajnef, J.-M. Vinassa, S. Azzopardi and O. Briat, “Ultracapacitors modeling
improvement using an experimental characterization based on step and frequency
responses,” IEEE Power Electronics Specialists Conference, vol. 1, pp. 131-134,
June 2004.
32. F. Rafik, H. Gualous, R. Gallay, A. Crausaz and A. Berthon, “Frequency, thermal
and voltage supercapacitor characterization and modeling,” Journal of Power
Sources, vol. 165, no. 2, pp. 928-934, Mar. 2007.
90
33. S. Buller, E. Karden, D. Kok and R.W. De Doncker, “Modeling the dynamic
behavior of supercapacitors using impedance spectroscopy,” IEEE Trans. on
Industry Applications, vol. 38, no. 6, pp. 1622-1626, Nov./Dec. 2002.
34. R.M. Nelms, D.R. Cahela and B.J. Tatarchuk, “Modeling double-layer capacitor
behavior using ladder circuits,” IEEE Trans. on Aerospace and Electronic Systems,
vol. 39, no. 2, pp. 430-438, Apr. 2003.
35. J.R. Miller, “Battery-capacitor power source for digital communication applications:
simulations using advanced electrochemical capacitors,” Electrochem. Soc. Proc.,
pp. 246-254, 1995.
36. T.T. Hartley, G.O. Beale, and S.P. Chicatelli, Digital Simulation: A Control System
Approach, Prentice-Hall, Mar. 1994.
37. F.E. Terman, W.W. Harman and J.G. Truxal, Automatic Feedback Control System
Synthesis, McGraw-Hill, 1964.
91