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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 3, MAY/JUNE 2005
Impedance-Based Simulation Models of
Supercapacitors and Li-Ion Batteries
for Power Electronic Applications
Stephan Buller, Member, IEEE, Marc Thele, Rik W. A. A. De Doncker, Fellow, IEEE, and
Eckhard Karden, Member, IEEE
Abstract—To predict performance of modern power electronic
systems, simulation-based design methods are used. This work
employs the method of electrochemical impedance spectroscopy
to find new equivalent-circuit models for supercapacitors and
Lithium-ion batteries.
Index Terms—Lithium-ion (Li-ion) batteries, simulation models,
supercapacitors (SCs).
I. INTRODUCTION
S
IMULATION-BASED development methods are increasingly employed to cope with the complexity of modern
power electronic systems. For these methods, suitable submodels of all system components are mandatory. However,
compared to the submodels of most electric and electronic
components, accurate dynamic models of electrochemical
energy storage devices are rare. Therefore, this paper employs
the method of electrochemical impedance spectroscopy (EIS)
to extent the physics-based, nonlinear equivalent circuit models
of supercapacitors (SCs) [2] to describe Lithium-ion (Li-ion)
batteries.
The following section briefly introduces the method of electrochemical impedance spectroscopy and presents measured
impedance spectra. From these spectra, appropriate equivalent-circuit models are deduced. After this, the Matlab/Simulink
implementation of the new simulation models is discussed and
simulation results as well as verification measurements are provided. Finally, conclusions are drawn and future perspectives
of the new impedance-based modeling approach are outlined.
Paper IPCSD-05-006, presented at the 2003 Industry Applications Society
Annual Meeting, Salt Lake City, UT, October 12–16, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Power
Electronics Devices and Components Committee of the IEEE Industry Applications Society. Manuscript submitted for review July 1, 2003 and released for
publication March 3, 2005.
S. Buller was with the Institute for Power Electronics and Electrical Drives
(ISEA), Aachen University of Technology (RWTH-Aachen), D-52066 Aachen,
Germany (e-mail: Stephan.Buller@gmx.de).
M. Thele and R. W. A. A. De Doncker are with the Institute for
Power Electronics and Electrical Drives (ISEA), Aachen University
of Technology (RWTH-Aachen), D-52066 Aachen, Germany (e-mail:
te@isea.rwth-aachen.de; dedoncker@isea.rwth-aachen.de).
E. Karden is with Energy Management, Ford Research Center Aachen (FFA),
D-52072 Aachen, Germany (e-mail: ekarden@ford.com).
Digital Object Identifier 10.1109/TIA.2005.847280
II. IMPEDANCE SPECTRA OF SCS AND LI-ION BATTERIES
Electrochemical impedance spectroscopy can be performed
either in a galvanostatic or in a potentiostatic mode. Following the first approach, a small ac current flows through the
storage device under investigation and its ac voltage response
is measured. From the ac current and the measured ac voltage
response, the storage impedance is determined online using
discrete Fourier transforms (DFTs). Superimposed with the ac
excitation signal, a dc current (charging or discharging) defines
the overall working point of the cell. Due to the pronounced
nonlinearity of most electrochemical storage systems, espeis
cially of batteries, the differential impedance
usually not equal to the quotient
. In these cases, modeling
the large-signal behavior of an energy storage device requires
impedance measurements at several working points followed
by integration of the differential impedance with respect to
. In addition, the impedance of
current, i.e.,
storage devices usually depends on temperature and state of
charge. Therefore, sets of impedance spectra have to be analyzed systematically [3]–[5]. Due to mass transport phenomena,
dynamic battery performance during continuous discharging
or charging of batteries differs significantly from that during
dynamic microcycling with frequent changes between charging
and discharging. As the latter is typical for many practical
battery applications (e.g., hybrid-electric vehicles or stop/start
vehicles), EIS on Li-ion batteries has been performed using
a specific microcycle technique [5].
During the investigation of the SCs, impedance spectra have
been recorded at four different voltages and five temperatures.
As an example, Fig. 1 shows the complex-plane representation
V in a
of impedance spectra of a 1400-F SC at
frequency range from 70 Hz down to 160 mHz. This frequency
range corresponds to typical time constants in most high-power
applications, e.g., cranking of a vehicle with a combustion
engine.
Hz , the SCs show inIn the high-frequency range
ductive behavior. Then, at approximately
m ,
the impedance plots intersect the real axis. For intermediate frequencies, the complex-plane plots form an angle of approximately 45 with the real axis. This angle is explained by the
limited current penetration into the porous structure of the electrodes (which has been discussed in [2]). For lower frequencies,
the spectra approach a nearly vertical line in the complex plane,
which is typical of ideal capacitors.
0093-9994/$20.00 © 2005 IEEE
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BULLER et al.: IMPEDANCE-BASED SIMULATION MODELS OF SCs AND Li-ION BATTERIES
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Fig. 3. Equivalent-circuit model of the Li-ion battery.
Fig. 1. Complex-plane diagram of impedance spectra of a 1400-F SC
manufactured by Montena Components SA, U = 1:25 V.
Fig. 4. Complex-plane impedance diagram of measured and modeled
impedance data of the Li-ion battery at # = 25 C, 80% SOC, I = 0 A and
I = 1 A charge.
III. EQUIVALENT-CIRCUIT MODELS
Fig. 2.
Impedance spectra of an Li-ion battery (I
= 0 A,
# = 25
C) .
Fig. 2 shows measured impedance spectra of a Li-ion battery
(Saft LM 176065, 3.6 V/5 Ah) at room temperature for different
states of charge (in this case with zero dc current). Impedance
data have been recorded for eight frequencies per frequency
decade starting at 6 kHz. For all spectra, some characteristic
,
frequencies are given. At approximately
the real axis intersection of the impedance spectra is observed.
For lower frequencies, all spectra show two capacitive semicircles. The first semicircle is comparably small and slightly depressed, whereas the second one is larger, nearly nondepressed,
and grows remarkably with decreasing state of charge. Finally,
at the low-frequency end of the depicted spectra, diffusion becomes visible. The diffusion impedance shows a 45 slope,
which is typical of a so-called Warburg impedance. Due to the
boundary condition for diffusion of Li ions in the electrodes,
the diffusion branch of the spectrum approaches a capacitor-like
impedance spectrum 90 for even lower frequencies.
Impedance spectra of valve-regulated lead-acid batteries
(VRLA) with different superimposed dc currents have also
been measured at several state of changes (SOCs) and temperatures. These results are beyond the scope of this paper but can
be found in [1], and [4].
The discussion of the model topology and the general modeling principle in this section concentrates on the Li-ion battery technology. In the case of SCs, excellent agreement with
the measured impedance spectra was achieved using a ladder
network model, consisting of the resistance of the pore electrolyte and the nonlinear double-layer capacitance of the phase
boundary electrode/electrolyte. Detailed results as well as the
lumped-element representation of the ladder network are reported in [1] and [2].
To model the recorded impedance spectra, suitable equivalent-circuit topologies have to be defined. Based on the underlying physical processes, the equivalent circuits should allow
an optimum representation of the measured spectra with a minimum set of model parameters. In a second step, the model
parameters have to be calculated. To minimize the deviations
between modeled data and measured spectra, a least-square
fitting algorithm is employed.
In Fig. 3, the electric equivalent circuit of an Li-ion battery
is depicted. This circuit consists of an inductance , an ohmic
representing a deresistance , a so-called element
pressed semicircle in the complex-plane [1], [3], a nonlinear
and
) as well as of a Warburg impedance
RC circuit (
. Using the depicted model topology, the observed ac behavior of an Li-ion battery can be described accurately. The
following adaptation of the model to the measured impedance
spectra shows that, despite several simplifications, all relevant
processes including porosity, charge transfer and diffusion are
modeled with sufficient precision.
Fig. 4 compares measured and calculated impedance data of
the Li-ion battery at 25 C and 80% SOC. For all frequencies
and both depicted dc currents, the corresponding curves show
nearly perfect agreement.
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Fig. 5. Nonlinearity of the resistance R
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 3, MAY/JUNE 2005
(# = 25 C, 50% SOC).
Fig. 6. Open-circuit voltage as a function of the state of charge (Li-ion).
Obviously, the diameter of the low-frequency semicircle, i.e.,
, strongly depends on the dc current that is suparameter
perimposed during the impedance measurement. The nonlinwith dc current is depicted in Fig. 5. Apart from
earity of
the data points, which have been determined from the measured
impedance spectra, Fig. 5 also shows a calculated curve which
.
models the current dependency of
The relation between the dc current and the corresponding
can be described
overvoltage at the impedance element
by a Boltzmann-type equation. In electrochemistry, this type of
equation is known as Butler–Volmer equation (1). The constants
are the exchange current , the number of transferred elemen, and the
tary charges , the symmetry coefficient
thermal voltage
(
mV if
K).
(1)
The nonlinear charge transfer resistance
can be calculated
as
(2)
Equation (2) can be solved analytically for the special cases
,
(irreversible reactions), and
(symmetric kinetics). In all other cases, numeric calculation is required. Therefore, to determine the Butler–Volmer parameters
, , and from the data points in Fig. 5, a second fitting algorithm is employed.
The best approximation of the data points is obtained for
A,
, and
. For classic redox reactions, represents the change in oxidation number of the reaction ions. Hence, integer values are expected. However, for
the Li-ion battery, a noninteger value for was found to be
best. This result might be explained by the specific nature of the
Li intercalation reaction or might be due to the simplifications
that are necessary to allow a parameterization of the simulation
model without reference electrode measurements. An electrochemical investigation of this finding is not required for the further development of the dynamic battery model and is therefore
considered beyond the scope of this work.
as
Finally, Fig. 6 shows the open-circuit battery voltage
function of the state of charge. The Li-ion battery was partly
Fig. 7.
Approximation of a ZARC element by RC circuits.
discharged and the open-circuit voltage was measured after a
minimum rest period of five hours. The curve in Fig. 6 can be
stored into a lookup table.
IV. MODEL IMPLEMENTATION
So far, the dynamic behavior of the modeled energy storage
devices is still described in the frequency domain. The time-domain behavior of the equivalent-circuit model can be calculated
by solving a set of ordinary differential equations. For this calculation, simulation tools like Matlab/Simulink can be employed.
However, not all complex impedance elements (e.g., ZARC elements and Warburg impedances) can directly be implemented
in a common circuit simulation tool. For these elements, appropriate approximations by means of RC circuits or RC ladder network topologies have to be found first [1].
As an example, the basic idea for the representation of a
ZARC element, employed to model a depressed, capacitive
semicircle in the complex-plane diagram, is depicted in Fig. 7.
The approximation is based on a series connection of nonlinear
RC circuits. All RC circuits are fully determined by the parameters of the ZARC element. Thus, the number of experimental
parameters remains constant. With an increasing number of
RC circuits, the approximation of the ZARC elements becomes
more and more precise. However, the calculating time increases. Thus, an appropriate compromise between simulation
accuracy and computation effort has to be found. This question
is thoroughly discussed in [1].
The specific model parameters which are needed for the simulation of a certain battery are stored in a separate file. To allow
a linear adaptation of a parameterized battery model to differently sized batteries of the same technology, all parameters are
defined with respect to the battery’s nominal current and the
number of battery cells connected in series. Furthermore, due to
the nonlinearity of some impedance elements, the original battery current in the simulation model is replaced by the relative
.
current
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BULLER et al.: IMPEDANCE-BASED SIMULATION MODELS OF SCs AND Li-ION BATTERIES
Fig. 8. Current profile for the verification of the SC model.
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Fig. 10. Comparison of the measured voltage response and the data obtained
from different simulation models.
TABLE I
COMPARISON OF MEASURED AND SIMULATED EFFICIENCY DATA. WORKING
POINT: 1.5 V, ROOM TEMPERATURE, CYCLE DEPTH
615% Q
Fig. 9. Measured and simulated voltage response to the current profile.
V. VERIFICATION AND APPLICATION OF THE MODELS
As a final step, the results of the simulation models are compared with measured data in the time domain. Both, the SC
model as well as the model of the Li-ion battery have been verified in detail [1]. In this section, some examples of these verification measurements are given.
For the verification of the SC model, the current profile depicted in Fig. 8 has been employed [2]. The imposed charging
and discharging pulses model a highly dynamic load at the beginning as well as deeper charging and discharging periods at
the end of the evaluation. The corresponding voltage curves are
depicted in Fig. 9. The measured and the calculated data show
excellent agreement.
The influence of the porous structure of the SC electrodes
can be illustrated by means of a comparison of the full simulation model with the voltage response of the simplified model
which only consists of a series connection of the ohmic resisof the SC. For this comparison,
tance and the capacitance
Fig. 10 provides an enlarged view of the first current pulses of
the verification profile. Once more, the excellent agreement of
the measured and simulated voltage data becomes obvious. In
addition, remarkable deviations due to the neglect of porosity
are observed for the simplified model. For low frequencies, i.e.,
for comparably long relaxation times, these deviations could be
by the larger dc
overcome by replacing the ohmic resistance
resistance
with
being the resistance of the
electrolyte in the pores of the electrodes [2]. In this case however, the fast voltage transients would not be well represented
anymore.
One advantage of SCs used as energy storage devices is their
good energy efficiency. This efficiency is also influenced by
the porous structure of the electrodes which means that the increasing real part of the impedance with decreasing frequency
has to be taken into account.
To compare measured and simulated efficiency data, an SC
is partly charged and discharged with constant dc currents of
various amplitudes. The cycle depth is chosen to be 540 A s
which corresponds approximately to
. For each current amplitude, the charge/discharge cycle is repeated ten times
but only the last five cycles, which start and finish at the same
internal conditions (quasi-stationary), are used for the efficiency
calculation. In a second step, the same current profile is simulated by means of the newly developed capacitor model. Measured and simulated efficiency data are compared in Table I.
Again, very good agreement is observed.
Next, a simulation example of the Li-ion battery model is
presented. For the model verification, the dynamic discharge
current profile depicted in Fig. 11 has been selected. The
comparison of the simulated and the measured voltage response
to this current profile at a state of charge of 77.5% and
room temperature is shown in Fig. 12. Excellent agreement
of the measured and the simulated voltage curves is found.
The outstanding accuracy of the simulation model is due to
the exact representation of the complex battery impedance
including all important nonlinearities.
An important precondition for the high quality of the simulation results is the nearly perfect reproducibility of the battery
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Fig. 11.
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 3, MAY/JUNE 2005
Current profile for the verification of the Li-ion battery model.
storage technologies, e.g., NiMH batteries or even fuel-cell
stacks in the future. This versatility will allow the combined
simulation of different energy storage devices for the evaluation
of new storage-hybridization concepts. Furthermore, by means
of additional submodels, e.g., describing mass transport in
VRLA batteries, the validity range of the existing models can
be further enlarged. By this, the simulation of long-lasting
constant-current charging or discharging periods will become
possible.
Another interesting future application of the impedancebased simulation models is a detailed thermal battery design.
All mechanisms of heat generation of SCs or batteries can be
precisely represented. Combined with the mechanisms of heat
transport and dissipation, the thermal behavior of batteries can
be simulated. Consequently, the influence of different future
cooling concepts, for example, on life-cycle costs, could be
evaluated.
ACKNOWLEDGMENT
The authors are grateful to the Ford Research Center Aachen
(FFA), especially to Dr. D. Kok and Dr. L. Gaedt for supporting
this research project.
REFERENCES
Fig. 12. Measured and simulated voltage response of the Li-ion battery (77.5%
SOC, 25 C).
behavior during operation as well as the lack of parasitic reactions. From this point of view, Li-ion batteries are especially
suited for any kind of model-based description. Compared to
Li-ion batteries, the simulation of lead acid batteries turns out
much more difficult. Nevertheless, the described simulation
approach could also be successfully adapted to this battery
technology [1], [4].
VI. CONCLUSION AND FUTURE PERSPECTIVES
This paper has shown that nonlinear, lumped-element equivalent-circuit models meet the accuracy requirements for simulation models of energy storage devices. To demonstrate the
power of this modeling concept, Li-ion batteries and SCs were
selected.
For the determination of suitable equivalent-circuit topologies as well as for the parameterization of these models, the
method of EIS was employed. After the implementation of the
models, the simulation results were compared to test-bench
data. Excellent agreement of simulated and measured voltage
data was found.
Due to the versatility the impedance-based modeling approach, the described concept can also be employed for other
[1] S. Buller, “Impedance-based simulation models for energy storage devices in advanced automotive power systems,” Ph.D. dissertation, ISEA,
RWTH Aachen, Aachen, Germany, 2003.
[2] S. Buller, E. Karden, D. Kok, and R. W. De Doncker, “Modeling the
dynamic behavior of supercapacitors using impedance-spectroskopy,”
IEEE Trans. Ind. Appl., vol. 38, no. 6, pp. 1622–1626, Nov./Dec. 2002.
[3] E. Karden, “Using low-frequency impedance spectroscopy for characterization, monitoring, and modeling of industrial batteries,” Ph.D. dissertation, ISEA, RWTH Aachen, Aachen, Germany, 2001.
[4] S. Buller, M. Thele, E. Karden, and R. W. De Doncker, “Impedancebased nonlinear dynamic battery modeling for automotive applications,”
J. Power Sources, vol. 113, pp. 422–430, 2003.
[5] E. Karden, S. Buller, and R. W. De Doncker, “A method for measurement and interpretation of impedance spectra for industrial batteries,” J.
Power Sources, vol. 85, pp. 72–78, 2000.
Stephan Buller (M’97) received the Ph.D. degree from the Institute for Power Electronics and
Electrical Drives (ISEA), Aachen University of
Technology (RWTH-Aachen), Aachen, Germany, in
2002.
He joined ISEA in 1997, spending five years as a
Research Associate. From 2002 until 2004, he was
a Chief Engineer at ISEA. His research activities
were mainly in the area of batteries and other energy
storage systems. In January 2005, he joined an
international consulting company.
Marc Thele received the Diploma in Electrical
Engineering from the Institute for Power Electronics
and Electrical Drives (ISEA), Aachen University of
Technology (RWTH-Aachen), Aachen, Germany, in
2002.
In June 2002, he joined ISEA as a Research Associate. His research activities are in the area of battery
simulation models of different technologies.
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BULLER et al.: IMPEDANCE-BASED SIMULATION MODELS OF SCs AND Li-ION BATTERIES
Rik W. A. A. De Doncker (M’87–SM’99–F’01)
received the Doctor of Electrical Engineering degree
from the Katholieke Universiteit Leuven, Leuven,
Belgium, in 1986.
During 1987, he was appointed as a Visiting
Associate Professor at the University of Wisconsin,
Madison. In December 1988, he joined the General
Electric Company Corporate R&D Center, Schenectady, NY. In 1994, he joined Silicon Power
Corporation as Vice President. In October 1996,
he became a Professor at Aachen University of
Technology (RWTH-Aachen), Aachen, Germany, and Head of the Institute for
Power Electronics and Electrical Drives (ISEA).
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Eckhard Karden (M’00) received the Ph.D.
degree from Aachen University of Technology
(RWTH-Aachen), Aachen, Germany.
He is a Research Engineer for storage systems in
the Energy Management Group of the Ford Research
Center Aachen (FFA), Aachen, Germany. Before he
joined Ford in 2002, he was Chief Engineer at the
Institute for Power Electronics and Electrical Drives
(ISEA), RWTH-Aachen.
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