DOI: 10.1002/ente.201600154
Analysis of Lithium-Ion Battery Models Based on
Electrochemical Impedance Spectroscopy
Uwe Westerhoff,*[a, c] Kerstin Kurbach,[a, c] Frank Lienesch,[b] and Michael Kurrat[a, c]
This work is an overview of various equivalent circuits (ECs)
containing various degrees of detail. The ECs are evaluated
in terms of model accuracy and parameterization time for
the systematic assignment of an equivalent circuit to application fields. For this purpose, impedance spectra were measured using electrochemical impedance spectroscopy at different states of charge, health and temperatures. Then the
parameters of the EC were extracted using the least-squares
method and the Levenberg–Marquardt algorithm. After
comparing the simulated to the measured impedance spec-
trum, a review and assignment of equivalent circuits for potential applications is given. Simple equivalent circuits with
a series resistor and a maximum of two resistance–capacitance (RC) elements are ideal for simulations with lower dynamics. Equivalent circuits with up to five RC elements or
even a constant-phase element (CPE) are promising for simulating highly dynamic processes. By using RCPE elements
the impedance spectrum can be modeled with the highest accuracy, which is why this type of model should be used for diagnostic purposes.
Introduction
In the literature three different approaches of modeling Liion batteries are typically proposed: theoretical quantitative
models (white box),[1] qualitative models with experiment
(gray box),[1] and experimental quantitative models.[1, 2] Many
parameters are required for the calculation of differential
equations, which are obtained from the literature or identified by using complex measurement methods for the quantitative theoretical models of the white box method.[1] Therefore measurements of the conductivity of the electrolyte or
electrode, porosity, particle radius distribution, tortuosity, or
diffusion coefficients of the individual materials should be
conducted and their dependency on temperature and aging
determined.[3] So-called black modeling is based on purely
mathematical models.[4] In this case the input and output variables are interconnected using control structures and empirical data. The quality of the models improves the more data
(so-called training data) it receives.[5] Thus the models learn
to simulate the electrical behavior without needing the physical information of the battery. The use of electrical equivalent circuits is firmly established and belongs to the gray box
category, the qualitative models using experimental data. If
the correct model assumptions and order are fulfilled, the
equivalent circuits can be a quantitative model. In the literature, a great variety of options for modeling Li-ion batteries
with electrical equivalent circuits has been presented.[6] The
choice of the equivalent circuit depends strongly on the cell
chemistry and the detailed characteristics. Therefore no standard model can be used for every battery type,[7, 8] as the
probability that the model is over- or under-modeled is high.
To obtain a model with an optimal reproduction of the cell
characteristics, the basic structure of a battery has to be
known. Additionally, an approach is needed for the estimation of the parameters of the equivalent circuit. In Figure 1,
an approach is shown that describes the individual components of a battery cell with electrical and electrochemical elements. The series connection of the equivalent circuit elements yields the entire model for a battery cell.
Another approach to determine the optimum equivalent
circuit configuration is the density function of the distribution of relaxation times (DRT).[9, 10] Using this method, the
high-intensity characteristic frequencies of a measured impedance spectrum are determined to derive the number of RCelements. A simplified version to determine the equivalent
circuit configuration is curve sketching to identify the
minima, maxima, and inflection points in the Nyquist plot.[11]
The frequencies of the distinctive points are also called process frequencies as they show in which frequency range of
[a] U. Westerhoff, K. Kurbach, Prof. Dr. M. Kurrat
Institute of High Voltage Technology and Electrical Power Systems (elenia)
Technische Universit-t Braunschweig
Schleinitzstraße 23, 38106 Braunschweig (Germany)
E-mail: u.westerhoff@tu-bs.de
[b] Dr. F. Lienesch
Physikalisch-Technische Bundesanstalt
Bundesallee 100, 38116 Braunschweig (Germany)
[c] U. Westerhoff, K. Kurbach, Prof. Dr. M. Kurrat
Battery LabFactory Braunschweig
Technische Universit-t Braunschweig
Langer Kamp 19, 38106 Braunschweig (Germany)
The ORCID identification number(s) for the author(s) of this article can
be found under http://dx.doi.org/10.1002/ente.201600154.
T 2016 The Authors. Published by Wiley-VCH Verlag GmbH & Co. KGaA.
This is an open access article under the terms of the Creative Commons
Attribution Non-Commercial License, which permits use, distribution and
reproduction in any medium, provided the original work is properly cited,
and is not used for commercial purposes.
Part of a Special Issue on “Li-Ion Batteries”. To view the complete issue,
visit: http://dx.doi.org/10.1002/ente.v4.12
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the impedance spectrum the individual electrochemical processes occur.[12]
Modeling
In various publications different equivalent circuits are selected as the model approaches for simulating the battery
characteristics.[5] The quality of an equivalent circuit depends
on the application and differs essentially by the speed of parameterization and the accuracy. In Figure 2, the applied
equivalent circuit models are shown, which differ in terms of
the two aforementioned criteria. The examined equivalent
circuit models can be used to simulate the impedance spectra
of circuits consisting of only resistors, inductors, and capacitors. Most battery systems, such as those in full-vehicle simulations, stationary storage in the home, or large-scale electrical grid storage, can already be described with sufficient accuracy using these models.[13–15] For more detailed considerations of the cell characteristics and the dependence on the
state of charge (SOC), the temperature (T) or the state of
health (SOH), further equivalent circuit elements can be
used. This is implemented by using constant-phase elements
(CPEs, Figure 3). With CPEs it is considered that the real
electrodes are not plates and have no uniform boundary because they are produced with a high porosity to intercalate
the lithium ions.[16] The diffusion behavior of the electrodes
is highly dependent on the particle size distribution.[32] In addition, the diffusion behavior (especially the solid-state diffusion) should be modeled properly to consider the slow processes occurring in the battery; therefore a Warburg element
is used. For diffusion processes, the impedance spectrum Nyquist diagram extends with a 458 angle upward slope. This
behavior can also be modeled using a CPE element; however, the physicochemical description of Li-ion batteries is located at the CPE for the porosity and particle radius distribution, whereas the physicochemical origin at a Warburg element describes the diffusion behavior of the carriers. The
equivalent circuit configurations in Figure 3 were derived
from accumulated experience with impedance measurements.
Parameter estimation
The parameters for the equivalent circuit are, as mentioned
above, elements determined from the measured impedance
spectra. For this purpose, as a first step curve sketching is
used to estimate the parameters of the components. The estimated parameters are the initial values for the minimization
function, which is based on the least-squares method in
Equation (1).[11] It is necessary to set appropriate initial conditions for the algorithm so that it can locate the global minimum and does not get stuck in a local minimum. Using the
Figure 2. Equivalent circuit models with resistance, inductor, and a variation in the the number of RC elements; these are the RC models.
Energy Technol. 2016, 4, 1620 – 1630
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Figure 1. Description of individual cell components with equivalent circuit elements.
least-squares method, the parameters are adjusted so that
the sum of squared deviation (residuals) is minimized.
min f ð xÞ ¼
x
min
R;C;L;CPE;W
fmax
X
@
ZEC ðR; C; L; CPE; W; fn Þ @ Zmeasure;n
n¼fmin
ð1Þ
For variation of the parameters in the direction of the global
minimum, there are a number of algorithms. The following
two examples of these algorithms are discussed in more
detail:[17–19]
· trust-region algorithm
· Levenberg–Marquardt algorithm
With the trust-region expansion the iteration increment is
increased. In a defined radius around the iterate (the confidence interval) the algorithm searches for a greater minimum as it is defined by the normal increment. This allows
the process to converge very fast toward the steepest descent. The Levenberg–Marquardt algorithm is based on a similar principle. The iteration step size is increased by a factor
that is recalculated from one step to another. This provides
a robust algorithm that converges despite having poor start
parameters in the direction of the steepest descent. It combines the advantages of the steepest descent method and the
Gauss–Newton method. The difference between the two
methods is that, in the trust-region method, the radius is determined directly whereas, in the Levenberg–Marquardt
method, it is determined implicitly, through the use of
a damping parameter.[17, 18]
Electrochemical Impedance Spectroscopy
Electrochemical impedance spectroscopy has long been used
to characterize the condition of a battery and for the description of the electrochemical characteristics and processes in
the cell.[2, 20–22] Before the electrochemical impedance spectroscopy measurement is applied, it is important to take into
account that a battery is a nonlinear, time-invariant system,
and therefore a long rest time is needed to ensure that the
battery is in electrochemical equilibrium.[23] Information on
Energy Technol. 2016, 4, 1620 – 1630
>2
the function of electrochemical impedance spectroscopy and
the measurement setup are described in the Experimental
Section. For a description of the physical and electrochemical
effects that can be identified in the impedance spectrum, the
impedance spectrum is divided into individual frequency
ranges.[24, 20] In the very high frequency range > 20 kHz, an inductive behavior is measured in addition to the real part of
the impedance. The behavior is mainly caused by the measurement setup such as the connecting lines and the type of
cable wiring. That the battery cell does not have an inductive
component cannot be excluded, however this was not explicitly observed in these measurements. Other processes are, for
example, the charge-transport processes in the electrolyte,
the solid–electrolyte interphase (SEI), and in the active material (including the anode and cathode). The charge-transfer
processes, both from the electrolyte into the SEI and from
the SEI into the active material of the anode/ cathode, are
located in the middle frequency range (typically 1 kHz–
10 mHz) and are represented in the impedance spectrum in
the form of semicircular arches.[25] However, the transitions
of the processes are often so subtle, that a discretization of
impedance ranges for the individual transport and transfer
processes is very difficult. The separator is produced as
a porous structure so that the lithium ions can run through it.
Therefore, the separator is shown as a capacitor in parallel
with a resistor. However the capacity effect of the separator
is so small compared to the other components that this RC
element can usually be neglected. In the low-frequency
range (< 10 kHz) diffusion processes dominate in the anode
and cathode. The course of the impedance spectrum corresponds to a 458 rising line in the Nyquist plot. However, not
only electrochemical processes can be derived from the impedance spectrum but also physical quantities such as the
conductivity of lithium-ions and electrons. The only directly
measured parameter for an equivalent circuit is a purely
ohmic resistance at abscissa zero crossing for the imaginary
part of the impedance. This resistance is dominated by the
conductivity of the electrolyte and thus constitutes a resistance for the lithium-ion transport.
When recording impedance spectra as a function of various experimental states (i.e., SOC, T, SOH) the selected step
sizes must be sufficiently large so that a change in the impedance spectrum is caused. Only with the right selection of
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Figure 3. Equivalent circuits with resistance, inductance, capacitance, constant-phase-elements, and Warburg elements; these are the CPE models.
Results and Discussion
To evaluate the equivalent circuits, first the characteristics of
a Li-ion battery cell were determined by means of electrochemical impedance spectroscopy in various metrological
studies. For this purpose, various states of the battery were
prepared to investigate the changes of the characteristics in
the impedance spectrum. For the metrological tests, the variables with the greatest influence on the battery performance
and characteristics have been varied. These variables are the
ambient temperature, the change in the state of charge, and
the continuous state of aging. The investigations were performed in the following areas:
This dependence changes the entire characteristic curve
behavior, which also becomes visible in all parameters of the
equivalent circuit. The value of the abscissa zero crossing
and thus the value of the internal resistance increases greatly
with decreasing temperature. Furthermore, the first semicircle increases slightly, whereas the second semicircle is
spreading much stronger. The internal resistance increases
even further with these two expansions of the semicircle. If
the temperature rises again, the impedance spectrum reverses back to its original state, and thus the temperature behavior is a reversible process. The requirement is that the cell is
operated within the prescribed temperature range, which was
the case. In Figure 5, the impedance spectrum expands with
decreasing state of charge in the measured mean frequency
range of 1 Hz > f > 50 Hz. This area is often associated in the
literature with the charge-transfer process, which describes
the intercalation process of the lithium ions into the active
material of the electrodes.
· temperature range: 50 to @20 8C
· state of charge range: 100 to 0 %
· state of health range: 100 to 86 %
The different states of health have been set by the decrease in capacity due to a 1 C cyclization of cells. In the laboratory, the battery cells were charged at a depth of discharge (DOD) of 100 % with constant current–constant voltage (CC–CV) and discharged with CC. The percentage decrease in capacity relative to the initial capacity defines the
SOH. The temperature was first investigated at @40 8C.
However, the frequency range was not changed during the
experiment, which proved to be disadvantageous afterwards.
In Figure 4 an abscissas zero crossing could not be measured
at a temperature of @20 8C because of the shift of the process
frequencies which will be discussed in Figure 7. Nevertheless,
the temperature-dependent behavior in the impedance spectrum is clearly visible.
Figure 4. Change of the impedance spectrum showing dependence on the
temperature at SOH = 100 %, SOC = 50 %, frequency range 100 kHz–10 mHz,
and AC amplitude 1/20 C.
Energy Technol. 2016, 4, 1620 – 1630
Figure 5. Change of the impedance spectrum showing the dependence on the
state of charge at T = 25 8C, SOH = 100 %, frequency range 100 kHz–5 mHz,
and AC amplitude 1/20 C.
With decreasing state of charge, the resistive parts of the
impedance rise correspond to an increase in the internal resistance. Due to the increase of the internal resistance,
a higher voltage drop arises, and the temperature of the cell
increases at the same current. This behavior is observable influences the open-circuit voltage characteristics of a Li-ion
battery cell, as the battery voltage has a very strong nonlinear decrease shortly before reaching the final discharge voltage. The change in the state of charge is a reversible process
when the prescribed voltage range is maintained. The impedance spectra of the aged cell in Figure 6 also enlarge with
time, at constant temperature and state of charge. In addition, the internal resistance of the cell increases almost continuously. These aging processes are irreversible, and therefore attention was paid to a careful treatment of the cells
throughout the investigations.[26]
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step sizes can the electrochemical processes be considered
separately and expressed in response to the experimental
conditions.
All of the influencing variables (temperature, state of
charge, aging) result in the expansion of the second semicircle and thus lead to an increase in charge-transfer resistance.
To be able to state which of the three factors is responsible
for the increase, the other parameters have to be included as
well. For this reason, the process frequencies were also determined to deduce the original influence factor. Figure 7 a illustrates in which process parts these special frequencies have
to be determined in the impedance spectrum. At the zero
crossing of the imaginary part of the impedance is the frequency fZIM,0, which indicates the frequency point of a purely
ohmic resistance. The semicircle in the middle of the impedance spectrum is dominated the most by the charge-transfer
process and has its characteristic frequency fZIM,max at the
maximum of the imaginary part of the impedance. From the
frequency fZIM,min the diffusion processes start in the battery.
This frequency is determined at the minimum of the imaginary part of the impedance. For the range of the frequency,
the following boundary condition applies: fZIM,0 @ fZIM,max >
fZIM,min.
In Figure 7 b it is shown that the frequencies in the course
of the discharge increase only slightly and then fall continuously. Upon reducing the state of charge, the frequency
changes, which describes significantly the charge-transfer
process. This shows that an impedance measurement at
a fixed frequency (e.g., 1 kHz for AC resistance measurement) cannot always measure a specific process frequency. In
the course of aging (Figure 7 c) there is a continuous, permanent shift in the process frequencies in addition to the
change in frequency caused by the state of charge and temperature. With continuous aging, the impedance of a cell is
increased by depletion of the electrolyte of free lithium ions,
which results in growth of the SEI layer. Thus, the entire impedance spectrum shifts further into the capacitive range of
impedance. The result is an increase of the zero crossing frequency. At low temperatures, the process frequencies shift
Energy Technol. 2016, 4, 1620 – 1630
Figure 7. Shifts in the process frequencies depending on various factors of an
impedance spectrum: a) impedance spectra, b) state of charge, c) state of
health, d) temperature.
with a nonlinear, continuous behavior to lower frequencies,
as shown in Figure 7 d. Only the frequency fZIM,0 is greater,
which simply indicates an increase in the internal resistance.
Before the evaluation of the equivalent circuits it was examined which algorithm is the most suitable for parameter
estimation. Only the five equivalent circuits of 1 RC up to
5 RC were used to be able to eliminate the influence of different equivalent circuit elements within the CPE. The
equivalent circuit equation of the impedance, generally written, forms Equation (2).
ZEC;iRC ðR; C; f Þ ¼ R0 þ
5
X
i¼1
Ri
1 þ jw ? Ri ? Ci
ð2Þ
The parameters for these equivalent circuits were determined by the least-squares method in Equation (1) and the
Levenberg–Marquardt algorithm and trust-region algorithm.
The numerically calculated parameters for the respective
equivalent circuit elements were parameterized to simulate
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Figure 6. Change of the impedance spectrum showing dependence on the
state of health at T = 25 8C, SOC = 100 %, frequency range 100 kHz–5 mHz,
and AC amplitude 1/20 C.
rRE ð f Þ ¼
ZRE;xRC ð f Þ
@ 1;
ZRE;measure ð f Þ
rIM ð f Þ ¼
ZIM;xRC ð f Þ
@1
ZIM;measure ð f Þ
ð3Þ
From the relative deviation of the complex impedance
components, the relative deviation of the total impedance is
calculated by means of the absolute value function in Equation (4).
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðrRE Þ2 ?ðrIM Þ2
rZ ð f Þ ¼
ð4Þ
As rz is still a function of frequency, this value is further evaluated to derive a characteristic value for the quality of the
whole impedance spectrum. The quality criteria are the average deviation [Eq. (5)] and the standard deviation [Eq. (6)],
which are determined by the relative deviation of the modeled to the measured impedance spectrum. At the average
deviation, the sum of all relative deviations is divided by the
number of frequencies:
r¼
1
Nmeasure
?
X
Nmeasure
ð5Þ
rZ;i ð f Þ
i¼1
For the calculation of the standard deviation, the average deviation is also used. These two deviations are the quality parameters in this investigation.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
NX
measure
u
E
C2
1
s¼t
?
rZ;i ð f Þ @ rZ
Nmeasure @ 1 i¼1
Evaluating the equivalent circuit models
The reproduction of a measured impedance spectrum with
the examined equivalent circuits is presented graphically in
Figure 9 and as data values in Table 1. In Figure 9 only the
frequency range 20 kHz < f < 20 mHz is shown because the
diffusion section and the inductive area of the impedance
spectrum are not shown in the Nyquist diagram. But in the
Bode diagrams, Figure 10–Figure 12, the complete frequency
range can be observed.[27] This is particularly true as the
middle frequency range has the decisive influence on the
quality of the equivalent circuit.[28]
The impedance spectrum in Figure 9 was measured at
a charge state of 10 %, an ambient temperature of 25 8C, and
a state of health of 98 %. With the simplest equivalent circuit
ð6Þ
In Figure 8, the results of the average deviation and standard deviation are shown by the selected equivalent circuits
from 1 RC to 5 RC. As shown in Figure 8 the blue and red
dashed lines deviate only slightly from each other, and therefore the difference between the algorithms is difficult to see.
Only for the equivalent circuits with four and five RC elements was the Levenberg–Marquardt algorithm 0.1 % better
than the trust-region algorithm. The standard deviation
shows a much greater dependence on the selected algorithm.
With a difference of not more than 0.14 %, the difference is
relatively large compared to the deviation of the average deviation. Consequently the Levenberg–Marquardt algorithm
has been selected for further parameterization of the equivalent circuit models. The Levenberg–Marquardt algorithm is
particularly robust towards the choice of a poor start parameter.
Energy Technol. 2016, 4, 1620 – 1630
Figure 8. Deviations of the simulated from the measured impedance spectrum at a state of charge of 50 %, temperature of 25 8C, and no aging: L–
M: Levenberg–Marquardt, T-R: trust-region, (solid line) standard deviation,
(dashed) average deviation.
Figure 9. Comparison of measured impedance spectrum (red) at 25 8C,
SOC = 10 %, and SOH = 98 %, with all simulated equivalent circuits of this investigation (blue), RC-EC’s: with marking, CPE EC’s: without marking.
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the impedance spectrum in a function of the previously measured frequencies. This was followed by the comparison of
the measured with the simulated impedance spectrum. The
relative deviation of the individual impedances was separately calculated for the real and imaginary parts of the impedance in Equation (3).
EC-component/ EC
R
1 RC
2 RC
3 RC
4 RC
5 RC
1 RCPE
2 RCPE
3 RCPE
R0 [W]
L0 [mH]
R1 [W]
C1 j CPE1[a]
R2 [W]
C2 j CPE2[a]
R3 [W]
C3 j CPE3[a]
R4 [W]
C4 j CPE4[a]
R5 [W]
C5 j CPE5[a]
0.846
–
–
–
–
–
–
–
–
–
–
–
0.399
0.148
1.295
0.040
–
–
–
–
–
–
–
–
0.399
0.166
0.980
0.012
9.157
10.580
–
–
–
–
–
–
0.399
0.173
0.272
0.002
0.694
0.091
8.077
9.983
–
–
–
–
0.399
0.174
0.154
0.001
0.256
0.018
0.602
0.190
8.910
10.160
–
–
0.399
0.175
0.108
0.001
0.205
0.008
0.483
0.099
0.324
2.538
11.640
10.750
0.399
0.176
0.087
0.002
1.041
0.243/0.509
–
0.782/0.879
–
–
–
–
0.399
0.176
0.107
0.009
0.098
0.004/0.829
0.819
0.212/0.664
–
6.975/0.850
–
–
0.399
0.176
0.043
0.002/0.959
0.186
0.017/0.788
0.789
0.217/0.680
–
6.923/0.848
–
–
[F]
[F]
[F]
[F]
[F]
[a] If the field has one value, it is a capacitor. When the field has two values, the first value is the CPE parameter for the imperfect capacitor with the unit
Farad, and the second value is the CPE-parameter for the exponent, which sets the phase angle.[11]
Figure 11. Magnitude (blue) and phase response (red) of the measured impedance spectrum at 25 8C, SOC = 10 %, and SOH = 98 %, compared to the
magnitude and phase responses of the simulated RCPE-equivalent circuit
models (gray).
Figure 10. Magnitude (a) and phase (b) responses of the measured impedance spectrum a = blue; b = red at 25 8C, SOC = 10 %, and SOH = 98 %, and
the simulated RC equivalent circuit models (gray).
Figure 12. Standard deviation (left) and average deviation (right) of the relative deviation from the simulated to the measured impedance spectra.
Energy Technol. 2016, 4, 1620 – 1630
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Table 1. Estimated parameter values of the equivalent circuit components of the least-squares method and the Levenberg–Marquardt algorithm to a measured impedance spectrum at 25 8C, SOC = 10 % and SOH = 98 %
Energy Technol. 2016, 4, 1620 – 1630
is due to the calculations of the relative deviation of the total
impedance according to Equations (3) and (4). The relative
deviation of rIM is constant at 100 % due to the missing imaginary part of the equivalent circuit, whereas the value of the
real part rRE at the maximum is 160 %. Thus, the statistical
spread of the deviation is very low, which is reflected in the
standard deviation. The average deviation in Figure 12
(right) decreases with an average of D(r = 0.12 % with increasing complexity of the equivalent circuit model. Both quality
parameters reveal a difference between the 5 RC and
1 RCPE < 1 %. Moreover, both quality parameters result in
an increasing accuracy at low states of charge. This is due to
the maximum expansion of the second semicircle at low
states of charge. The superimposed electrochemical processes
in the semicircles can be better modeled because the individual physicochemical processes are discernible.
As the third and final quality characteristic of the equivalent circuit models, the number of iteration steps which are
needed to achieve the global optimum of the parameter estimation has been counted. The results are shown in Figure 13.
The exact duration and therefore the parameterization speed
depends on the respective computing performance of the
simulation computer. As described the least-squares method
with Levenberg–Marquardt algorithm was used for the parameter estimation.
Figure 13. Number of iterations until the global optimum of the parameter
estimation.
Conclusions
The standard deviation, the average deviation, and the
number of iterations were used as quality criteria for assessing the quality of an equivalent circuit simulation for battery
modeling. If all criteria are considered together, the following statements about the quality of the equivalent circuits by
the examining the impedance spectra in Figure 9–11 are described in this work. The use of an equivalent circuit of R is
completely unsuitable for simulating an impedance spectrum.
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R, only one point on the real axis can be generated. This
equivalent circuit can represent any capacitive or inductive
effects and is also not dependent on the frequency. The 1 RC
equivalent circuit is essentially based on fZIM,0 and fZIM,max because it does not require fZIM,min, and an element for simulating the diffusion behavior does not exist in this equivalent
circuit. The situation is different for the 2 RC equivalent circuit, which is based on fZIM,0 and fZIM,min and thus can simulate the diffusion behavior. For that case, fZIM,max cannot be
considered. The simplest and most suitable RC equivalent
circuit is the 3 RC model, which is based on all three process
frequencies. With the equivalent circuits 4 RC and 5 RC,
good reproductions of the impedance spectra were also produced. The focus in the models lies on the consideration of
fZIM,0 and fZIM,min, which is the same for the 2 RC model. In
the magnitude response of the Bode diagram in Figure 10 a,
it was observed that the process frequencies mark the area
of the inflection point in the measured profile. In the three
equivalent circuits 3 RC, 4 RC, and 5 RC the inflection points
are in the immediate proximity of the process frequencies.
In Figure 10 b, the measured phase response of the Bode
diagram is plotted in red. It is noticeable that no RC equivalent circuit simulates the phase at 08 very accurately. Furthermore, the 5 RC equivalent circuit exhibits an unusual drop in
phase at the process frequency fZIM,min. Because the parameters are derived from a numerical estimate and not from
a physical computation with material constants, this phenomenon has to be caused by the estimation method. As shown
in Figure 10 b, the profile of the phase response cannot be simulated with any RC equivalent circuit very well. Based on
the Nyquist diagram in Figure 9, the capacitor is essential for
the RC models and not the resistor. A capacitor is not appropriate for modeling the compression of the semicircle, so
that equivalent circuits with CPE elements show an advantage by modeling this behavior, which is shown in Figure 11.
The magnitude responses show only very small differences
between the simulated and measured values. Deviations
occur only in the very high-frequency range > 20 kHz. This is
the inductive area of the impedance spectrum, and due to
the influence of the measurement setup they are not further
investigated. The phase responses of the CPE models show
the same deviation as the RC models at a phase of 0 8C. Furthermore, the phase response from 1 RCPE in the middle
frequency range is similar to the 5 RC. This is also the reason
why a constant phase element is often converted into a series
circuit of a multitude of RC elements.[29] The difference between 2 RCPE and 3 RCPE is very small over the entire frequency range. As a function of state of charge, the simulated
impedance spectra are analyzed in terms of the standard deviation and the average deviation to indicate a quality characteristic in Figure 12. These deviations are calculated from
the relative deviation of the simulated impedance of an
equivalent circuit to the measured impedance versus the frequency [see Eq. (2)–(5)].
The standard deviation in Figure 12 (left) is reduced by an
average of Ds = 0.14 % with increasing complexity of the
equivalent circuit. The very low standard deviation for R-EC
Energy Technol. 2016, 4, 1620 – 1630
Figure 14. Recommended assignment of equivalent circuit models for simulations of different application scenarios of Li-ion batteries.
tion), the model can be simplified to establish any network
need. Should the focus be more on the derivation of physicochemical processes to identify safety-critical features in real
time, which adjust themselves due to battery aging, at least
3rd-order models are necessary for the home storage and
electric-mobility models with time constants. With these
equivalent circuit models, it is possible to obtain information
on the SEI growth, increase in the charge transfer resistance,
and diffusion behavior. In the cell production field, a variety
of R k CPE elements should be used additionally to determine the contact resistance between the arrester and active
material as well as the impact of calendering.
Due to the long duration of parameterization when using
CPE elements, the simulation speed will be greatly reduced
with CPE models for a battery cell simulation. It is still a current topic of investigation as to what effect this would have
on a simulation model of a battery cell. The created simulation model is used here to predict the battery status in varying load scenarios and does not reflect the value of impedance, but rather the battery voltage and many other sizes as
an output value. As this model is an addition to the equivalent circuit block, other blocks are required such as a temperature, open circuit, and lifetime models to achieve a complete
battery model. The other mentioned blocks are only meant
as an outlook and could be incorporated into future work.
Experimental Section
For setting up a battery cell model, measurements on real cells
were necessary to incorporate the characteristics of the model.
The measured pouch battery cells in Figure 16 were self-made in
the Battery LabFactory Braunschweig (BLB). They consist of
LiNi1/3Mn1/3Co1/3O2 cathodes with an electrode formulation of
4 %:2 %:4 %:90 % (binder/additive/carbon black/active material)
coated onto an aluminum foil and a graphite anode with an electrode formulation of 4 %:1 %:2 %:93 % coated on a copper foil.
The electrodes had a size of 50 X 50 mm2, and the cell had a nominal capacity of approximately 50 mAh. The anode was contacted
with a welded nickel conductor and the cathode with an aluminum conductor. Between the two electrodes a ceramic-coated
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However, with the element1 RC the relevant factors of the
charge-transfer resistance can be simulated, such as temperature, state of charge and state of health. Along with 2 RC,
which can also represent the range of the diffusion processes
because of the additional RC element, these two equivalent
circuit models are very good for larger simulation models because it requires only a small effort for parameterization and
selection of the equivalent circuit parameters. This could be,
for example, a battery model with a high number of cells,
which is integrated into a distribution grid simulation (or
smart grid) or in a battery management system (BMS) for
photovoltaic storage. The accurate simulation of battery
characteristics is not necessary for either simulation models
in dynamic supply or load cases. Perhaps, the application of
the R model would be suitable in a simulation with a very
large number of cells as is the case for MWh-scale storage
that is integrated in the high-voltage grid because the assessment of the power loss and the sudden change in voltage are
essentially important. Considering the process frequencies,
the 3 RC model is the simplest RC equivalent circuit model
that simulates an impedance spectrum accurately and requires only a low number of iteration steps until the determination of the optimal parameters. When using equivalent circuits with a higher number of RC elements, the deviations
are minimized, but the number of required iterations is
greatly increased. Due to the higher accuracy and the consideration of different time constants, these models can be used
for simulations of mobility applications. Thus, the rates of
current change for a traction battery of a hybrid electric vehicle (HEV) or a battery electric vehicle (BEV) are highly
dynamic and rarely continuous. When comparing the 5 RC
and 1 RCPE circuits, the discrepancies between the two
models are hardly visible. But they differ significantly in the
number of necessary iterations, which is why the 1 RCPE
should be chosen. Small differences are observed in the Nyquist plot (Figure 9) and Bode diagram (Figure 11) between
the most complex models 2 RCPE and 3 RCPE. Only the
standard deviation and average deviation parameters show
the trend that the 2 RCPE has a lower deviation average of
2.4 % and will only use one third of the number iterations for
optimal parameter estimation. These two models should not
be used as part of a higher-level simulation, but as a diagnostic tool for deriving the physical factors that influence the
state of function of a battery cell. By convention, a higher
number of equivalent circuit elements causes no improved
significance. Using too many elements does not allow for the
assignment of additional electrochemical processes. Figure 14
summarizes the evaluation to derive a recommendation for
the use of equivalent circuit models in the simulations of different applications.
A battery system that is made up of many individual cells
can be modeled by an equivalent circuit network with x series equivalent circuits. For MWh-scale energy storage, which
consists of thousands of individual cells, it is recommended
to use the simplest possible equivalent circuit to keep the
simulation effort low. By multiplication or division of the
equivalent circuit parameters (depending on the interconnec-
To record the impedance spectrum, the battery was connected
using a four-wire measuring system to a galvanostat, and then
the measuring program was started. The cell was first charged to
an initial charge level of 100 % and then discharged gradually.
Each short discharge was followed by a one-hour break so that
the voltage could relax, and the cell was set in electrochemical
equilibrium before the impedance spectrum was measured. The
galvanostat generated a sinusoidal alternating current of a small
amplitude (1/20 C) and measured the sinusoidal voltage response. From the difference between the magnitude and phase,
the frequency-dependent impedance could be determined by
using the complex alternating current calculation. The impedance spectrum was determined for using 10 frequency measurements per decade with a stepwise variation of the frequency
starting at the highest frequency of 500 kHz to the lowest frequency of 5 mHz. During the measurement, the position of the
cell was not changed and no mechanical stress was exerted onto
the cell. The cells are very sensitive to mechanical stress, which
has an effect on the result of the impedance spectroscopy measurement.[30, 31] For this reason an apparatus was constructed that
fixed the cells and enabled a reproducible measurement. During
the measurement the battery was placed in a climate-controlled
chamber to adjust constant ambient conditions (Figure 15). The
list of equipment used in the measurement setup is presented in
Table 2 and shown schematically in Figure 16.
Figure 15. Left: measurement setup consisting of a climate chamber, the Galvanostat, and a power booster to improve performance. Right: self-built laboratory Pouch cell with a nominal capacity of approximately 50 mAh.
Table 2. List of the important devices at measurement setup.
equipment
label
galvanostat VersaSTAT 3
shielded sensor cable to the front-end
current cable to the back-end
PC with the VersaStudio software
climate chamber VT 4030
M1
L1
L2
P1
T1
Figure 16. Schematic of the test setup for the electrochemical impedance
spectroscopy measurements.
List of Symbols
AC
BEV
BLB
BMS
DRT
EC
HEV
L–M
LiMxOx
LiNiMnCoO2
LiPF6
RC
SEI
SOC
SOH
T-R
Cx [F]
CPEx [F]
f [Hz]
fmax [Hz]
fmin [Hz]
fZIM,0 [Hz]
fZIM,max [Hz]
fZIM,min [Hz]
IB [A]
L0 [H]
Nmeasure [1]
w [1/s]
fZ [8]
rIM [%]
rRE [%]
rZ [%]
(rz [%]
R0 [W]
Rx [W]
s [%]
t [s]
Energy Technol. 2016, 4, 1620 – 1630
alternating current
battery electric vehicle
Battery LabFactory Braunschweig
battery management system
distribution of relaxation time
equivalent circuit
hybrid electric vehicle
Levenberg–Marquardt algorithm
lithium–metal oxide
lithium–nickel–manganese–cobalt dioxide
lithium hexafluorophosphate
parallel circuit resistor and capacitor
solid–electrolyte interface
state of charge
state of health
trust-region algorithm
capacitor of the equivalent circuit
constant-phase element
frequency
maximum frequency
minimum frequency
frequency at ZIM = 0
frequency at ZIM = ZIM,max
frequency at ZIM = ZIM,min
battery current
inductivity in the impedance spectrum
number of measured values
angular frequency
phase response of the impedance spectrum
relative deviation of the imaginary part of impedance
relative deviation of the real part of impedance
relative deviation of the impedance
average deviation of the impedance
purely ohmic resistance in the impedance
spectrum
resistance of the equivalent circuit
standard deviation of the impedance
time
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separator was inserted for electrical insulation of the electrodes
from each other. The entire electrode assembly was packed in
a laminated aluminum bag with the addition of an electrolyte
based upon LiPF6 and then vacuum sealed.
ambient temperature
battery voltage
open-circuit voltage
Warburg element of the equivalent circuit
magnitude of the complex impedance Z
complex impedance of the equivalent circuit
imaginary part of the impedance
measured impedance
real part of the impedance
Acknowledgements
The authors thank the Nds. Ministerium fgr Wissenschaft und
Kultur of the State of Lower Saxony for the financial support
of this work with Graduiertenkolleg Energiespeicher und
Elektromobilit-t Niedersachsen (GEENI). We thank colleagues of the Battery LabFactory Braunschweig (BLB) for
the excellent cooperation and the use of equipment infrastructure for cell production.
Keywords: electrochemical impedance spectroscopy · energy
storage · equivalent circuits · lithium-ion batteries · modeling
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21944296, 2016, 12, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/ente.201600154 by South Korea National Provision, Wiley Online Library on [03/08/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
T [8C]
UB [V]
UOC [V]
Wx [W/s@1/2]
jZj [W]
jZjEC [W]
ZIM [W]
jZjmeasure [W]
ZRE [W]