2020 (Mattgias Kuipers) An Algorithm for an Online Electrochemical Impedance Spectroscopy and Battery Parameter Estimation Development, Verification and Validation

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Journal of Energy Storage
journal homepage: www.elsevier.com/locate/est
An Algorithm for an Online Electrochemical Impedance Spectroscopy and
Battery Parameter Estimation: Development, Verification and Validation
Matthias Kuipersa,d, Philipp Schröera,d, Thomas Nemetha,d, Hendrik Zappena,d,e,
Alexander Blömekea,d, Dirk Uwe Sauera,b,c,d
a Chair for Electrochemical Energy Conversion and Storage Systems, Institute for Power Electronics and Electrical Drives (ISEA), RWTH Aachen University, Jägerstrasse 17-
19, 52066 Aachen, Germany
bHelmholtz Institute Münster (HI MS), IEK 12, Forschungszentrum Jülich, 52425 Jülich, Germany
c Institute for Power Generation and Storage Systems (PGS), E.ON ERC, RWTH Aachen University, Mathieustrasse 10, 52074 Aachen, Germany
d Jülich Aachen Research Alliance, JARA-Energy, Templergraben 55, 52056 Aachen, Germany
e Safion GmbH, Hüttenstraße 7, 52068 Aachen, Germany
A R T I C L E I N F O
Keywords:
Li-ion battery
Battery management system
Electrochemical impedance spectroscopy
battery diagnostics
Electrical equivalent circuit model
A B S T R A C T
More advanced battery diagnostic approaches are required for a safer and more reliable operation of today's and
future battery technologies. For the purpose of evaluating a battery's internal conditions, Electrochemical im-
pedance spectroscopy has proven to be a powerful tool, but nowadays it is only used in laboratory setups. It
could provide valuable information about the battery's internal states, if instead it is applied online in an actual
battery application. Therefore, we have developed an efficient algorithm, which is designed to run on a battery
management system continuously carrying out measurements of the electrochemical impedance by iteratively
evaluating measurements of battery current and voltage. Furthermore, the algorithm adapts the parameters of an
equivalent circuit model to best match the battery's impedance, hence providing characteristic measures of a
battery's internal conditions. The algorithm is implemented in a generic form, which severs as a baseline that can
be adjusted to more specific requirements and circumstances in corresponding battery applications. The scope of
this work focuses on an introduction of the generic algorithm as well as a proof-of-concept. The later was carried
out in two steps. First, the algorithm was run against a battery model. Subsequently, the algorithm was validated
utilizing measurements from a real battery cell. In this case, laboratory electrochemical impedance spectroscopy
measurements served as reference. The algorithm was able to estimate impedance with a high accuracy in both
tests. A high accuracy was also achieved for the parameter estimation; however, its accuracy decreases with
large superposed DC current rates.
1. Introduction
The electrification of mobility as well as the utilization of stationary
storage systems is thriving thanks to recent developments in battery
technologies. Energy density and power capability keep increasing for
state-of-the-art and future battery technologies. Nevertheless, these
improvements commonly come along as a trade-off with the batteries’
lifetime and safety. Consequently, more intense observations of the
batteries’ safety as well as aging status and aging mechanisms become
more crucial. Based on such continuous online diagnostics an adapta-
tion of operating strategies, such as the modification of charging rates
and voltage thresholds, will become equally important and will ensure
longer lifetimes and safer operation. The operating strategy itself re-
quires more accurate estimations of the batteries’ internal states and
parameters. Hence, researchers are developing more advanced diag-
nostic mechanisms as well as new sensor techniques. A measurement
technique which is strictly speaking not a new sensor technique since it
is still based on collecting measurements of current and voltage is the
electrochemical impedance spectroscopy (EIS). It works on the prin-
ciple of exciting a battery in a precisely defined manner and applies
advanced processing techniques in order to obtain accurate measure-
ments of the battery's impedance spectrum. It is a powerful, yet ex-
pensive tool commonly used in laboratory scale. Besides investigating
the electrical behavior alone, it is utilized for the identification of
processes within the battery, such as aging and/or safety related pro-
cesses. Since laboratory EIS measurements require a lot of processing,
the data is commonly buffered before it is processed, urging the need
for comparably large amount of data storage space [1]. Amongst others,
this is a reason why EIS measurements are commonly performed offline.
Nevertheless, it would be very beneficial to perform EIS measurements
online whilst the battery is utilized in its applications. In [1] the authors
proposed to use system noise as excitation signal required for EIS. The
results look promising, yet this work focuses on an experimental setup
and uses offline Fourier transform method for calculating the
https://doi.org/10.1016/j.est.2020.101517
Received 23 February 2020; Received in revised form 11 April 2020; Accepted 1 May 2020
Journal of Energy Storage 30 (2020) 101517
Available online 25 May 2020
2352-152X/ © 2020 Elsevier Ltd. All rights reserved.
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impedance spectrum. The work also does not contain any interpretation
of the impedance spectra. In [2] and [3] the authors implement a
mathematical calculation of equivalent circuit parameters from im-
pedance values at only three frequencies. If one of these measurements
is slightly inaccurate, a large inaccuracy needs to be expected for the
equivalent circuit parameters. A more robust approach would be based
on an overdetermined equation system. In [4] a review is given re-
garding the estimation of battery internal temperature. Online EIS is
mentioned as a promising technique, yet based on partial components
of battery impedance only, such as real part, imaginary part, phase
change or intercept frequency [11–13]. In contrast, evaluating battery
impedance for multiple frequencies would essentially provide a more
robust estimation as well as quantitatively more information.
Within the scope of our work we have developed an algorithm for
an online electrochemical impedance spectroscopy (OEIS), aiming to
provide a continuous measurement of the high-frequency electrical
behavior of a battery in operation. In contrast to ordinary EIS mea-
surement techniques, the algorithm is designed in a way it requires only
little storage space and performs its calculations in an iterative ap-
proach, distributing calculation effort evenly over time. Furthermore,
the OEIS algorithm estimates multiple values of battery impedance by
running these iterative calculations in parallel, minimizing the time it
takes to obtain a spectrum. In subsequent steps, the OEIS can adapt an
equivalent circuit model onto the spectrum. Hence, it does not only
provide continuous estimations of multiple values of battery im-
pedance, moreover it provides continuous estimations of characteristic
internal parameters. It therefore provides an alternative approach to
state of the art estimation techniques, such as Kalman filters, which
estimate battery impedance iteratively in time domain [9]. The OEIS
algorithm on the other hand performs its parameter estimation in fre-
quency domain after evaluating a large amount of measurements of
voltage and current.
The fundamental assumption for the online electrochemical im-
pedance spectroscopy (OEIS) algorithm to work is the availability of
current ripples on the battery under test. In some battery applications,
ripples are inherently available,because of loads containing electric
motors. Alternatively, they are caused by loads containing PWM-based
converters [10,12]. These kinds of ripples are commonly interpreted as
noise and therefore are undesired, yet they can provide valuable in-
formation about the battery's electrical behavior. In case no inherent
current ripples are available in a feasible frequency range or their ex-
citation is too low in comparison to expected inaccuracies in current
and voltage measurements, it is conceivable that additional ripples are
generated artificially. The simplest approach is to switch loads on and
off in a desired manner. In case OEIS is applied onto Lithium Ion bat-
teries, there will also be a balancing system available. If the utilized
hardware allows an intrusion into the control of the balancing switches,
their corresponding resistors can be switched onto the battery cells in a
desired fashion. Nevertheless, in most BMS a 4-point-measurement1 is
not available. Hence, the sense wires will have an effect on the mea-
surement itself. This issue is nonexistent if the already available power
electronics, such as dc-dc converters in hybrid- and electric vehicles, are
utilized for generating artificial noise. In this case, the converter's
controls need to be manipulated in a corresponding way. In [5] such a
concept is introduced. A similar approach is proposed in [6], except the
fact that the power electronics of the charger are utilized in this sce-
nario. The charger is able to induce a frequency sweep onto the battery
as soon as it is fully charged. Alternatively, the ripple excitation can be
integrated directly into the battery management system (BMS) as dis-
cussed in [7]. In this case, a simple version of a switched mode am-
plifier is integrated into the BMS. Unfortunately, this approach requires
additional hardware components and therefore increases costs.
Since in many applications it will be undesired to cause additional
ripples, it is also possible to just apply these ripples for only a short
amount of time. However, this concept drives the question, for how
long the ripples need to be applied so the OEIS algorithm can perform
an estimation. Unfortunately, it is impossible to give a general answer
as it depends on different factors, such as the frequency range under
investigation. Yet in comparison to laboratory EIS measurement de-
vices, which commonly apply a frequency sweep2 the OEIS algorithm
evaluates multiple frequencies at the same time, thus reducing the re-
quired time frame significantly. In order to give a rough idea of what
time frame is to be expected, it shall be noted, a time frame of less than
30 s was enough throughout the investigations presented within the
scope of this work.
This study is aiming to provide a systematic proof of concept for the
novel OEIS algorithm rather than investigating the availability of ex-
citation signals and how they are generated. It is therefore assumed that
current ripples are available for the desired frequencies and with suf-
ficient excitation. Accordingly, the utilized current profiles were de-
signed specifically to test the algorithm in certain ways, leaving aside
boundary conditions, which in the future may be set by whichever
application the OEIS is applied to.
2. Working principle of the online electrochemical impedance
spectroscopy (OEIS) algorithm
Most diagnostic approaches, such as recursive least square algo-
rithms, Kalman Filters, particle filters etc. are carried out in time do-
main. The general idea of the OEIS algorithm is to adapt the battery
impedance in frequency domain rather than in time domain.
Accordingly, the process consists of several steps. Figure 1 presents a
schematic illustration of the OEIS algorithm's setup together with their
most fundamental interface signals. The current implementation of
OEIS algorithm consists of five submodules, namely:
Goertzel Algorithm (GZ) – Corresponding to its name, the funda-
mental principle of this algorithm is based on Goertzel algorithm [8], a
mathematical principle to investigate a periodic signal of a desired
frequency within an input signal. This input signal may contain many
other components besides the one, which shall be observed. An algo-
rithm was developed based on the GZ, yet including further calculations
for estimating battery impedance. Its working principle will be in-
troduced in Section 2.1.
Passive Electrochemical Impedance Spectroscopy (PEIS) – A basic
version of the PEIS algorithm was introduced in [11–13]. It is based on
the observation of a frequency band in order to calculate a value of
impedance. This part of the OEIS algorithm will be introduced in
Section 2.2.
Varied Parameters Approach (VPA) – This estimator is based on a
kind of pattern search algorithm and was first applied for battery di-
agnostics in [14]. In contrast to its original purpose, the implementa-
tion of VPA within OEIS is carrying out calculations in frequency do-
main rather than in time domain. Hence, it was redesigned entirely. A
more detailed insight is granted in Section 2.4.
Equivalent Circuit Model (ECM) – The VPA requires a model to be
fitted onto the impedance values in frequency domain. All sorts of
different ECMs can be used for this purpose. In Section 2.3 an advanced
ECM is introduced which is used within the scope of this work.
Adaptive Characteristic Maps (ACM) and prediction – This part of1 A “4-point measurement” refers to the general practice of attaching four
instead of two wires to an electrical device under investigation. Two of the four
wires can be used for applying electrical loads to the device, whereas the other
two are used for measuring voltage. Thanks to this sort of separation, the vol-
tage measurement is not afflicted by contact resistances and the resistance of
the sense wires themselves.
2 A “frequency sweep” refers to the process of applying a ripple of only one
frequency at a time. An entire measurement hence consists of several individual
measurements, whilst increasing or decreasing frequency step by step
M. Kuipers, et al. Journal of Energy Storage 30 (2020) 101517
2
the algorithm is not further discussed within this manuscript. Yet it
represents further processing of the output values, such as the im-
plementation of adaptive characteristic maps (ACMs). ACMs describe a
parameter across a specific operating range eg. across the battery's state
of charge (SoC) and its temperature. They are adaptable, hence can
serve as a tool for a continuous tracking of a parameter at a desired
operating point, which then again can be utilized for further inter-
pretations and predictions. Amongst others, predictions of power cap-
ability, estimations of the dominating aging effects as well as estima-
tions of internal battery temperatures can be carried out on the basis of
the OEIS algorithm.
As mentioned above, the OEIS algorithm performs its adaptation of
battery impedance in frequency domain rather than in time domain.
Accordingly, the measurements of battery voltage and -current, which
are always performed in time domain, need to be transformed in one
way or the other. GZ- and PEIS algorithm were developed for this
purpose. As presented at the point marked with the letter A in Figure 1,
they both consume measurements of current and voltage over time and
provide discrete values of battery impedance Zi for corresponding
frequencies fi as output. Accordingly, the outputs of both algorithms can
be expressed as parameter triplets pti containing values of frequency,
real- and imaginary part of the impedance according to equations (1)
and (2).
=pt f Z f Z f{ , ( ), ( )}i i i i i i (1)
= +Z f Z f Z f( ) ( ) j· ( )i i i i i i (2)
These intermediate results can be visualized in a Nyquist diagram as
depicted in Figure 1 within the plot marked by the letter B. It becomes
evident, that Goertzel- and PEIS algorithms perform the same task,
estimating close to identical values of impedance. In an actual appli-
cation it would not be required to implementboth algorithms, yet for
the purpose of development, it is beneficial to investigate alternative
approaches, especially as the two algorithms have different ground
laying concepts. Accordingly, both algorithms were implemented and
investigated within the scope of this work. If one algorithm outperforms
the other, a profound recommendation can be expressed. Nevertheless,
as both algorithms prove to perform very similar it is up to individual
preferences or dedicated investigations in future work. Disregarding the
comparison of the two algorithms, triplets pti from only one of the two
algorithms are passed on to the subsequent part of the OEIS. The un-
derlying choices are described in the sections introducing verification
and validation, respectively.
Within the next step of the OEIS algorithm, the sets of frequencies
and impedance values are handed over to the VPA as presented in
Figure 1. The VPA's job is to perform a fitting of an ECM onto the
measured impedance data. For this purpose, the VPA provides fre-
quencies to the ECM, which then again hands back its impedance for
the given frequencies. VPA uses these values to calculate a root mean
squared error (RMSE). The interconnection between VPA and ECM is
implemented in a generic way enabling the possibility to use different
ECMs or even to switch between ECMs during operation, if desired. This
can be of great benefit if the battery's electrical behavior changes sig-
nificantly over the desired operating range, for example for extreme
SoC or temperature conditions. The ECM utilized by the current im-
plementation of the OEIS will be introduced in Section 2.3. As output,
the VPA provides those parameters of the ECM, which match the esti-
mated values of impedance best. In Figure 1, another Nyquist diagram -
marked with the letter C - is visualizing the intermediate results at this
step. Together with the ECM, the parameters from VPA now provide a
continuous description of the battery's impedance.
Finally, the adapted parameter values can be handed over from VPA
to adaptive characteristic maps (ACM) together with predictors such as
power prediction algorithms, none of which are further introduced
within this work. There are several additional applications conceivable
for the acquired parameter sets. They provide a robust and consistent
adaptation and differentiation between the electrical parameters of a
battery such as ohmic resistance and charge transfer resistance. This
information can provide additional insight into which distinguished
aging mechanisms are predominant at the current aging status [15].
Alternatively, an estimation of the internal battery cell temperature can
be carried out based on the outputs of the OEIS algorithm [16].
2.1. Goertzel based Calculation of Impedance
The Goertzel algorithm is a rather well-known method for in-
vestigating an input signal regarding its frequency contents for a pre-
defined frequency at a time [8]. First, within an evaluation period,
information are collected from the input signal gk using an iterative
calculation presented in (3)
= +x g
f
f
x x2·cos
2· ·
·k k k k
target
sample
1 2
(3)
The frequency for which the investigation is carried out is called
ftarget, whilst fsample denotes the sampling frequency of the input signal
gk. Index k refers to a discrete time step. Within the first part of this
work fsample is assumed to be equal 1 kHz. Formula (3) is utilized to
Figure 1. Schematic illustration of submodules and most fundamental signal flows within OEIS algorithm.
M. Kuipers, et al. Journal of Energy Storage 30 (2020) 101517
3
calculate intermediate signals xk, xk 1 and xk 2. According to Goertzel
algorithm xk and xk 1 are required for the calculation of the complex
frequency component G f( )target as presented in (4). It contains the am-
plitude and phase of that frequency component within the input signal,
which refers to the target frequency [8]. As the original Goertzel al-
gorithm is a mathematical tool, its signals G f( )target , xk, xk 1 and xk 2 all
inherit their unit from the input signal gk.
= +G f x x
f
f
( ) ·( 1)·exp j·
2· ·
k ktarget 1
target
sample (4)
Equation (4) could be carried out every time step in order to provide
an even more continuous calculation of G f( )target . Nevertheless, within
this work it is only carried out once after each evaluation period, hence
reducing calculation effort. Additionally, evaluating a large number of
input values prior to the calculation of G f( )target ensures a better signal
to noise ratio and a conversion of G f( )target towards the true value. For
the purpose of this work the evaluation period was set to 10 s, thus 10k
samples were processed using equation (3) before a calculation of
G f( )target is carried out. Furthermore, the algorithm is implemented in a
way it does not need to buffer measurement values. It merely needs to
update the intermediate signals in an iterative fashion. Accordingly, the
algorithm needs little storage.
For the purpose of measuring battery impedance, Goertzel algo-
rithm needs to be carried out in parallel for the input gk equals battery
current i(t) and battery voltage u(t), respectively. Consequently, six
intermediate signals are calculated from the two Goertzel algorithms
running in parallel, namely xku, xku 1, xku 2 for processing input voltage
and xki , xki 1, xki 2 for processing input current. These can be used for
calculating outputs I f( )target and U f( )target in frequency domain, which
then again could be used to calculate the complex battery impedance
according to equation (5).
=Z f
U f
I f
( )
( )
( )target
target
target (5)
Nevertheless, complex calculations cannot be carried out on low
cost micro controllers. Therefore, the equation for the complex im-
pedance needs to be split into real and imaginary part. For this purpose,
equation (6) is taken as baseline.
= =
+
+
Z f
U f
I f
x x
x x
( )
( )
( )
·( 1)·exp j·
·( 1)·exp j·
k
u
k
u f
f
k
i
k
i f
f
target
target
target
1
2· ·
1
2· ·
target
sample
target
sample (6)
The final results are shown in (7) and (8), whilst a constant KGZ was
introduced according to (9). A more detailed derivation can be found in
Appendix A.
= +
+
Z x x x x x x x x
x x x x
Re{ } · · ·cos(K ) · ·cos(K ) ·
( ) 2· · ·cos(K ) ( )
k
u
k
i
k
u
k
i
k
u
k
i
k
u
k
i
k
i
k
i
k
i
k
i
1 GZ 1 GZ 1 1
2
1 GZ 1
2 (7)
=
+
Z x x x x
x x x x
Im{ } · ·sin(K ) · ·sin(K )
( ) 2· · ·cos(K ) ( )
k
u
k
i
k
u
k
i
k
i
k
i
k
i
k
i
1 GZ 1 GZ
2
1 GZ 1
2 (8)
=
f
f
K
2· ·
GZ
target
sample (9)
Within the current version of our OEIS algorithm, six different
target frequencies are to be investigated. The calculations introduced
above equations (3), ((7) and (8)) need to be carried out individually
for every single one of these frequencies. It is accordingly simple to
adjust the algorithm to a different amount of frequencies, if desired.
Theoretically, this concludes the basic version of our GZ algorithm.
However, there are mechanisms, which may corrupt the calculations of
the algorithm, such as the unavailability of sufficient excitation, non-
linearities etc. Therefore, a set of countermeasures have been
implemented in order to make the algorithm more trustworthy and
robust.
By implementing minor modifications to the GZ algorithm the ab-
solute values I f| ( )|target and U f| ( )|target can be calculated additionally.
The absolute values are a measure for the degree of excitation and
system response. Comparing these measures to wisely chosen threshold
values can provide a mechanism to evaluate the availability of suffi-
cient excitation as well as sufficient system response. Hence, if one of
these criteria fails to pass, the OEIS algorithm will skip processing the
output values for the corresponding evaluation period.
A fundamental assumption for the GZ algorithm is that the battery
fulfills the criteria of a linear and time invariant (LTI) system.
Unfortunately, batteries tend to have a nonlinear electrical behavior.Yet, if the operating point does not change significantly, the battery is
in a quasi-linear state. For example, when using an evaluation period of
10 s, the battery's SoC and temperature will not change noticeably.
Assuming a current rate of 5C, the SoC would only change by around
1.4 %. At a heat capacity of around 1 kJkg K· on cell level [10] and joule
heating of 100 W, a 0.5 kg would experience an increase in temperature
by 2 K only.
A more critical nonlinearity is caused by the charge transfer process
or “reaction over-potential”, which depends on current rate. In order to
avoid incorrect adaptations in times during which the battery is bur-
dened with strongly fluctuating current rates and therefore is in a
nonlinear state, we have implemented rudimentary countermeasures.
The maximum and minimum current values of the corresponding eva-
luation period are tracked. If the difference between these values
crosses a predefined threshold, it must be assumed the battery was not
operated in a linear range. Accordingly, OEIS algorithm will skip fur-
ther processing of the output values of the afflicted evaluation phase.
Besides tracking the maximum and minimum current, it is also required
to track the average current rate. This value will later be an input to the
nonlinear ECM introduced in Section 2.3.
The GZ algorithm works best on a periodic input signal.
Unfortunately, this criterion is rarely met. In most cases, there will be a
certain drift in current and especially in the voltage signals across the
evaluation period of 10 s. In other words: the signal is corrupted by low
frequency components. There are different countermeasures, which can
correct this signal drift effect. We have decided to implement an infinite
impulse response (IIR) Butterworth high-pass filter of 2nd order, which
is responsible for eliminating the low frequency components in the
signal. This filter is preprocessing both the measured current and vol-
tage signals before they are provided to the GZ algorithm.
2.2. Passive Electrochemical Impedance Spectroscopy (PEIS)
The PEIS algorithm was introduced in [11–13]. In these cases, PEIS
was utilized for measuring ohmic resistance. Hence, only one high
frequency band was observed and only the real part of the impedance
was processed. For the purpose of OEIS, multiple frequency bands are
required and imaginary parts need to be processed as well.
Similar to our design of the GZ algorithm, it consists of an evalua-
tion phase to collect and process a large amount of measurement data,
followed by the actual calculation of the impedance itself. There are
two steps, which are carried out within the evaluation phase. First, a
band-pass filter is applied onto current i(t) and voltage u(t) measure-
ments, to filter out a desired frequency range, obtaining iBPF(t) and uBPF
(t), respectively. In our version of the PEIS algorithm, an infinite im-
pulse response (IIR) Butterworth high-pass filter of 4th order was im-
plemented. The cut-off frequencies were set at ± 20 % around the
target frequency ftarget. Next, the root mean square values of current
IRMS and voltage URMS are calculated according to equations (10) and
(11) as well as the average effective power P̄ according to equa-
tion (12).
M. Kuipers, et al. Journal of Energy Storage 30 (2020) 101517
4
=T U u t dt· ( )RMS
T
BPF
2
0
2
(10)
=T I i t dt· ( )RMS
T
BPF
2
0
2
(11)
=T P u t i t dt· ¯ ( )· ( )
T
BPF BPF
0 (12)
For all these calculations, T is the duration of the evaluation phase.
Analog to the GZ algorithm, the PEIS implementation does not require
to buffer measurement values. The required amount of storage is lim-
ited to just a few unit delay blocks within the band-pass filter as well as
for the integrals.
As soon as a desired amount of measurement values has been pro-
cessed within the evaluation phase, the corresponding impedance is
calculated. First, the absolute value and the real part of the impedance
are calculated according to equations (13) and (14).
=Z f T U
T I
( ) ·
·target
RMS
RMS
2
2
2 (13)
=Z f T P
T I
( ) ·
¯
·target RMS2 (14)
Afterwards, the imaginary part is calculated according to equation (15)
=Z f Z f Z f( ) ( ) ( )target target target2 2 (15)
Identical to the implementation of GZ algorithm PEIS is designed to
investigate six different target frequencies or rather frequency bands.
The calculations need to be carried out individually for every single one
of these target frequency-bands.
The PEIS algorithm can also experience issues in specific cases.
Hence, it is equipped with almost the same countermeasures, which
were used for the GZ algorithm. An additional high-pass filter is not
required since the band-pass filters should provide sufficient filtering.
2.3. Introduction of an electrical equivalent circuit model (ECM)
The VPA is implemented in a way it can adapt different amounts of
parameters. Accordingly, it is possible to vary the ECM in a plug and play
fashion. If desired, the OEIS algorithm would even be able to cope with
switching in between ECMs online. This feature can be helpful if battery
behavior changes towards more extreme operating conditions and at the
same time the application requires a wide range of operating conditions in
terms of SoC and/or temperature. Yet, within this study, we restrict our-
selves to just one ECM. The utilized ECM is depicted in Figure 2. It consists
of an ohmic resistance R0 in series to a ZARC3 element, which in this case is
displayed in its representation as a resistor R1 parallel to a constant phase
element4 CPE1. R1 and CPE1 represent the charge transfer resistance and
the non-uniform double layer capacitance, respectively. Furthermore, the
charge transfer resistance is a nonlinear resistance assumed to follow Butler
Volmer equation (also see appendix C).
The utilized ECM presented in Figure 2 is rather complex in
comparison to commonly used ECMs in battery management systems,
which mainly consist of an ohmic resistance in series with one or more
RC-branches (parallel connection of a resistor and a capacitor). The
reason for using a ZARC element instead of an RC-branch is an increase
in accuracy. An RC-branch only models one time constant whereas a
ZARC element represents an infinite distribution of time constants,
which should model the battery's nonideal electrical behavior more
accurately. On the other hand, it is also the reason why modelling ZARC
elements in time domain can only be done via a simplification, for
example by approximating it using 3 or 5 RC-branches for each ZARC
element. This is one of the main reasons why diagnostic algorithms,
which are running online in time domain, try to avoid ZARC elements.
Yet, since the OEIS algorithm is performing its estimations in frequency
domain rather than in time domain, it can easily cope with using more
complex ECM-elements, such as ZARC elements.
The complex impedance Z ( ) of the ECM depicted in Figure 2 can
be described in frequency domain by using the following equation:
= + = +
+
Z Z j Z R R
A R
( ) ( ) · ( )
1 (j ) · ·ECM ECM ECM 0
1
1 1 (16)
As mentioned above R0 is the ohmic resistance, R1 is the charge transfer
resistance and the parameters A0, and γ describe the constant phase ele-
ment CPE1. In order to minimize the number of parameters required to
fully describe the ECM, we decided to set = 0.9. We assume this sim-
plification provides sufficiently good results even if the true value of γ
changes during operation and equals anything between 0.8 and 1. There
may be batteries, which do not fulfill this criterion. In these cases, the
operator can parameterize a desired value or may include γ as a into the
adapted parameter set. Nevertheless, a reduced number of degrees of
freedom increases in general the robustness of such algorithms.
For avoiding complex computations, the real and imaginary part of
the complex impedance must be derived. A detailed derivation is pre-
sented in Appendix B. The results are stated below:
= = +
+
+ +
( )
( )Z Z R
R AR
A R A R
Re{ ( )} ( )
· · ·cos ·
1 2· · · ·cos · ( · · )
ECM ECM 0
1 1 1
2
2
1 1 2 1 1
2
(17)
= =
+ +
( )
( )Z Z
A R
A R A R
Im{ ( )} ( )
( 1)· · · ·sin ·
1 2· · · ·cos · ( · · )
ECM ECM
1 1 2
1 1 2 1 1
2
(18)
The current dependency of R1 is based on the Butler Volmer
Equation [14]:
=I i V V· exp ·z·F
R·T
· exp ( 1)· (1 )·z·F
R·T
·R C C0
(19)
It describes the nonlinear relationship between IR, which is the
Figure 2. Utilized equivalent circuit model.
3 A ZARC element is an electrical circuit element often used for modelling
batteries’ electrical behavior. It consists of a parallel connection of a resistor
and a constant phase element. It behaves somewhat similar to a parallel con-
nection of a resistor and a capacitor, yet instead of providing a single main time
constant, the ZARC elements provides a distribution of infinite time constants
around a main time constant. Its impedance is calculated according to:
= +Z ( )ZARC
R
A R
1
1 (j ) · 1· 1
4 A constant phase element is an electrical circuit element often used for
modelling ZARC elements. It may be compared to a capacitor with a non-uni-
form surface area. Its impedance is calculated according to: =Z ( )CPE A
1
(j ) · 1
M. Kuipers, et al. Journal of Energy Storage 30 (2020) 101517
5
current through the charge transfer resistance R1 and the reaction
overpotential VC, which refers to the voltage across the double layer
capacitor. Furthermore, it includes constants such as the Faraday's
constant F and the universal gas constant R. The value of z refers to the
number of electrons, which participate in the electrochemical reaction,
and is therefore a constant value for the utilized battery technology. It is
fair to assume that temperature T is available through a direct mea-
surement or a thermal model. Furthermore, the current IR is similar to
the average current Ī calculated by GZ or PEIS, as long as the criterion
for nonlinearity is not violated. Accordingly, there are only two para-
meters to describe current dependency, namely the symmetry factor of
anodic and cathodic reaction α and the exchange current i0. In [14, 17]
and [18] a symmetry factor of = 0.5 was assumed to be feasibly for
most lithium ion batteries under common operating conditions. Based
on this assumption the charge transfer resistance can be calculated
using equation (26), which includes constant KBV as defined in (21).
= =
+
R I R I T
I i
( ) ( ) K · · 1
(2· )
R R
R
1 ct,SS BV 2
0
2 (20)
=K R
·z·FBV (21)
A detailed derivation of these equations is presented in Appendix C.
It shall be noted equation (20) describes the small signal resistance
instead of large signal resistance. In frequency domain small signal
values need to be used since the information about the DC-component
of the load current gets lost along the way. Nevertheless, as long as the
exchange current i0 is adapted correctly using the equation of small
signal resistance in frequency domain, it can also be used in time do-
main using the equation for the large signal resistance. The latter is also
derived in Appendix C.
In summary, we use a parameter set consisting of three parameters
to fully describe the battery's electrical behavior, namely:
=p R i A{ , , }0 0 1 (22)
These values are calculated via VPA and they are provided as output
of the OEIS.
2.4. Estimation of ECM parameters using a varied parameters approach
(VPA)
GZ and PEIS provide triplets of frequency, real- and imaginary part
of the corresponding impedance according to (1). In the current im-
plementation of the OEIS algorithm six frequencies are investigated,
hence six triplets are passed onto the VPA either from GZ or from PEIS.
The VPA's task is to fit an ECM onto the input values. The basic concept
of the VPA for the purpose of battery diagnostics was published in [14].
It performs an estimation in time domain by processing measurements
of voltage and current. At the same time, it is designed to process many
samples (>> 1000 samples) within a single evaluation phase, hence a
single estimation. In our case the estimation shall be carried out in
frequency domain and it only consist of six triplets as input values.
Therefore, we decided to redesign the VPA entirely in order to optimize
its performance for the given setup. In the following, our version of the
VPA is presented as it is implemented within the OEIS algorithm.
VPA is triggered as soon as a calculation of impedance is carried out
by GZ and PEIS and none of the criteria for sufficient excitation and
linearity are violated. It is closely coupled to an ECM, which provides
its impedance Z ECM split into real and imaginary part for a given input
frequency f and a parameter set p according to:
= +Z f p Z f p j Z f p( , ) ( , ) · ( , )ECM ECM ECM (23)
The parameter set contains a list of adjustable values for all com-
ponents describing a certain ECM. A generic representation of this
parameter set is depicted in (24)
= …p p p p{ , , , }n1 2 (24)
Within its first iteration, VPA triggers the ECM utilizing an initial
parameter set pinit.
= = …p p p p p{ , , , }init init init init1, 2, n, (25)
In a subordinate loop the input frequency f is iterated through the
target frequencies stored in the triplets which were passed from GZ or
PEIS. At the same time, a root mean squared error RMSE is summed up
according to:
= +RMSE p
n
Z Z f p Z Z f p( ) 1 ( ( , )) ( ( , ))
i
i ECM i i ECM i
2 2
(26)
The RMSE compares the n values of impedance from GZ or PEIS
according to (2) with those of the ECM by considering both, the errors
in real part and imaginary part. It is valid for a specific parameter set p.
Within the following iterations VPA is modifying the parameter set p,
whilst recalculating a corresponding RMSE every single time. The
modification is designed in a way, it only changes one parameter at a
time. The rows of the following matrix show the parameter sets, which
the VPA is iterating through:
…
+ …
…
+ …
…
… +
…
p p p
p k p p
p k p p
p p k p
p p k p
p p p k
p p p k
·(1 )
·(1 )
·(1 )
·(1 )
·(1 )
·(1 )
n
n
n
n
n
n
n
1 2
1 2
1 2
1 2
1 2
1 2
1 2 (27)
Besides the initial parameter set (first row), the parameters are in-
creased and decreased one after the other by a fraction k. The value of k
correlates to a relative scatter width and therefore has to fulfill.
< <k0 1 (28)
In our implementation, it is initialized equal 0.2 in order to start
searching in a relatively wide range. After the RMSE has been calcu-
lated for all parameter sets, the parameter set with the lowest value of
RMSE is set to be the new initial parameter set, completing an adap-
tation loop. Yet, within an overlying loop the process can be repeated in
order to get even closer to the mathematical minimum which fits the
model's impedance best onto the measured impedance values from GZ
or PEIS. The new initial parameter set serves as a baseline for the next
iteration of this overlying loop. In case the new initial parameter set is
identical to the old one, it is evidently close to the minimum, which is
why the value of k is reduced. Within the current version of our VPA k is
halved any time the initial set provides the lowest deviation. The pro-
cess stops as soon as a maximum number of iterations is reached. The
final parameter set is given as output and its RMSE serves as a quality
criterion.
3. Verification against a battery model
In order to verify the functionality of the OEIS algorithm a ver-
ification test was carried out. Instead of applying measurements of
current and voltage from a real battery cell, a battery model is used in
this case. Accordingly, the true ECM parameters are known and hence
serve as an ideal reference. The ECM utilized for simulating battery
behavior is presented in Figure 3.
The left hand side of the ECM in Figure 3 is identical to the ECM
introduced in Section 2.3. Consequently, the OEIS algorithm should be
able to adapt to the corresponding parameter values. In order to si-
mulate the battery's low frequency behavior moreaccurately, an ad-
ditional RC-branch R2||C2 was introduced into the ECM. The RC-branch
is supposed to model slow effects, such as diffusion. Its time constant is
comparably larger than the main time constant of the ZARC element.
M. Kuipers, et al. Journal of Energy Storage 30 (2020) 101517
6
Accordingly, the discrepancy between the ECM utilized in the OEIS
algorithm and the ECM used for simulating battery behavior is low for
most of the comparably high frequencies.
The battery simulation is carried out utilizing a current load profile
and calculating voltage response based on the ECM presented in
Figure 3. Both timelines are presented in Figure 4. The current profile
consists of three parts categorized according to their DC component.
For the first 70 s, no DC current is applied. Subsequently, a DC current
of -10 A is applied for 50 s followed by a DC current of -40 A for another
50 s.
Additionally, an AC ripple is applied onto the battery. It starts at
t = 20 s and consists of two sets of superposed sinusoidal signals of
different frequencies. First, a set of six frequencies is imposed, which
later is evaluated by the algorithm:
=f Hz Hz Hz Hz Hz Hz{1 , 2 , 5 , 10 , 40 , 150 }i (29)
Each of these signals is applied with an amplitude of 0.2 A. In order
to challenge the algorithm a second set of frequencies is imposed, ser-
ving as noise:
=f Hz Hz Hz Hz{0.1 , 3.5 , 20 , 100 }noise i, (30)
Their amplitude is 0.15 A. The algorithm will not be parameterized
to evaluate these frequencies, nor shall it be disturbed by their pre-
sence. It shall be noted, the frequencies in both sets (29) and (30) were
in general chosen arbitrarily, yet with a tendency to a logarithmic
distribution. In addition to noise at specific frequencies, the current
signal was further superposed by a small white gaussian noise with a
standard deviation of 5 mA.
The overall contents of the AC ripple are visualized in the plot on
the bottom of Figure 4. It shows the result of Fourier Transforms carried
out across the current load profile utilizing windows of 2 s each. The
plot also visualizes disturbances at t = 70 s and t = 120 s caused by
steep current ramps whilst switching between DC currents. These dis-
turbances are a challenge to the algorithm, especially due to their large
amplitudes. Their amplitude exceeds the current ripples’ amplitudes by
far, which is why the scale of the current ripple amplitude is saturated
at 0.3 A. Otherwise, the AC signals would barely be visible.
In Figure 4 there are three 10 s time frames highlighted and num-
bered, (1), (2) and (3). These time frames represent evaluation periods
of GZ and PEIS and their subsequent execution of VPA and will be used
for further investigations. All of these time frames have in common that
sufficient time (~30 s) has passed for the algorithms to settle, yet they
differ in prevailing DC current.
First, the impedance outputs from GZ and PEIS shall be compared.
Figure 5 shows these intermediate signals separated into real- and
imaginary part. For the purpose of an improved visualization, the
imaginary part is presented across a logarithmic scale.
Since there is no AC excitation within the first 20 s, there are no
feasible results from GZ and PEIS at these times. At t = 20 s AC ex-
citation is applied. Accordingly, it takes the algorithms a duration of
one evaluation period (10 s) to collect data and subsequently to pro-
viding the first reasonable results around t = 30 s. Nevertheless, it takes
another evaluation period for GZ and PEIS to swing in onto almost
identical values. The same effect takes place after the disturbances
through switching between DC currents, whereas PEIS is influenced less
than GZ. In all cases it does not take longer than two evaluation periods
or 20 s for the algorithm to provide feasible results, assuming the ex-
citation signal is good. Accordingly, it seems GZ and PEIS both provide
reasonable values of impedance at all of the three highlighted time
frames. Figure 6 substantiates this claim. It shows the results of GZ and
PEIS at the three time frames in frequency domain. Furthermore, it
presents the results from VPA and the true values calculated from the
input data to the simulated battery model. All the data is in good
alignment with the impedance of the battery model. The RMSE calcu-
lated by the algorithm is used for comparing GZ and VPA impedance. It
does not exceed a value of 0.008 mΩ. Hence, the algorithm is working
correctly. Even if a large DC current is superposed, the impedance is
met with high accuracy, as can be seen in time frame (3). It seems the
algorithm is able to cope well with the nonlinearities of the ECM.
Another effect in Figure 6, which needs to be addressed, is the
shrinking size of the first semi-circle. This behavior was expected due to
the nonlinear charge transfer resistance. Nevertheless, it shall be noted
Figure 3. Equivalent circuit model utilized for the purpose of simulating a
battery's voltage response.
Figure 4. Test profile for the verification of the OEIS algorithm.
Figure 5. Comparison of the impedance outputs from GZ and PEIS within the
scope of the verification test carried out on a battery model.
M. Kuipers, et al. Journal of Energy Storage 30 (2020) 101517
7
the main time constant of the ZARC element shrinks too, as a result of
this effect. In Figure 6 this phenomenon can be identified in a shift of
the GZ and PEIS impedance values further to the right of the semi-
circle. In the same way the peak of the imaginary part within the Bode
plot, moves towards higher frequencies. On one hand this should make
it easier to distinguish the first semi-circle from the diffusion effects, but
on the other hand, even small impacts from diffusion effects onto the
impedance of the first semi-circle, can have a more significant impact, if
the impedance of the semi-circle itself is small.
Next, a closer look is taken at the VPA output parameters. Figure 7
shows the three parameters, which are adapted by VPA over time. The
true values fed into the simulation model are presented alongside the
VPA's outputs. Within this run of the algorithm, impedance values from
GZ were utilized as input to VPA. The values from PEIS have proven to
be more stable against disturbances, nevertheless in steady state GZ and
PEIS provide similar values.
At time t = 0 s it can be identified that the algorithm was initialized
with parameter values equal 50 % of the true values. The first adap-
tation is carried out around t = 40 s, thus as soon as the results from GZ
and PEIS have sufficient quality. VPA converges quickly, adapting its
parameters close to the true values. At time frame (1) the parameters
have reached a good accuracy. The subsequent steps in superposed DC
currents briefly disturb the algorithm, but it converges quickly after.
The disturbance could have been avoided by applying harsher thresh-
olds for the countermeasure against nonlinearity, as implemented in GZ
and PEIS. For time frames (2) and (3) the accuracy of R0 and A1 remains
good. For the exchange current i0 on the other hand, a decreasing trend
is identified. This effect can be explained when taking a closer look at
Butler Volmer equation. In Figure 8 the charge transfer resistance is
shown according to the Butler Volmer relation (α = 0.5), whereas the
three blue curves are presenting large signal resistances for different
exchange current rates, namely 5 A, 10 A, 15 A. The same curves are
shown for small signal resistances in red.
For the adaptation of the charge transfer resistance in frequency
domain the expression of the small signal resistance must be utilized.
Taking a look at current rates with a large absolute value e.g.
=I A40R , it becomes evident that the curves for very different ex-
change current rates get close to each other. Hence, even if the charge
transfer resistance is estimated with only a small error, the corre-
sponding uncertainty in the exchange current is large. This phenom-
enon explains the poor accuracy ofthe exchange current in time frame
(3). The small errors required for such a deviation can be explained by
numerical errors and the influence of the diffusion effect on the im-
pedance of the first semi-circle. The effect is less severe for the curves
representing large signal resistances. In summary, it shall be noted that
the OEIS algorithm shows good performance as long as the superposed
DC current is moderate. Based on Figure 8, a relation of IR ≤ i0 would
provide a conservative definition of such a moderate current rate.
4. Validation against measured data
Since the proof-of-concept was successful with a battery model, a
validation test against a real battery cell is carried out next. For this
Figure 6. Comparison of impedance spectra and - values in frequency domain
within the scope of the verification test carried out on a battery model; (top) for
time frame (1); (middle) for time frame (2); (bottom) for time frame (3).
Figure 7. ECM parameters as output from VPA within the scope of the ver-
ification test carried out on a battery model.
Figure 8. Nonlinear behavior of the charge transfer resistance as large- and
small signal resistance and its respective depedency on the exchange current.
M. Kuipers, et al. Journal of Energy Storage 30 (2020) 101517
8
purpose, a dedicated electrical test was conducted on a lithium ion
battery cell. The cell under investigation is a 2.9 Ah prismatic cell with
a lithium titanate spinel anode and a lithium manganese spinel / li-
thium cobalt dioxide blend on the cathode. Even though its nominal
voltage is only 2.4 V it achieves a power density of more than 2 kW/kg
thanks to charging and discharging rates beyond 50 C. The battery cell's
impedance is accordingly small, which is a challenge to the OEIS al-
gorithm.
The measurement has been conducted using a current generator fed
with a multi sinusoidal signal while the cell's voltage response and load
current were recorded simultaneously with a sample rate of 8 kHz. At
the same time, the battery cell was placed in a temperature chamber for
precise tempering to, in this case, 0°C. Since the cell's electrical para-
meters are unknown and hence cannot be utilized as reference, a la-
boratory EIS measuring device was connected to the cell as well.
Accordingly, laboratory EIS-measurements were carried out prior to the
multi sinusoidal signal test and were later taken as reference. The test
itself was carried out for a duration of 30 s and at a state of charge of
90 %. The load profile did not contain any DC current but a superposed
set of six sinusoidal signals, each with an amplitude of roughly 0.45 A.
Accordingly, the current profile looks like a zero-mean noise similar to
the signal presented in Figure 1 in the plot marked with the letter A.
The signal was applied after a delay of around 1.5 s into the mea-
surement. The frequencies of the six signals were chosen in a way that
they were manually distributed across the first semi-circle of the bat-
tery's impedance, namely:
=f Hz Hz Hz Hz Hz Hz{4 , 12 , 24 , 60 , 188 , 800 }i (31)
In an actual application of the OEIS algorithm these frequencies
have to be parameterized according to the utilized battery and oper-
ating conditions. Alternatively, they can be adjusted online via a
dedicated addition to the algorithm.
Both GZ and PEIS were adjusted to the chosen frequencies.
Furthermore, the duration of the evaluation period was reduced from
10 s to 2.5 s, because the slowest frequency of 4 Hz is four times slower
than the slowest frequency used in the verification test. The results of
GZ and PEIS are presented in Figure 9.
Excluding the first evaluation period, because of the absence of the
sinusoidal signals up to t = 1.5 s, PEIS mostly settles after two eva-
luation periods. GZ on the other hand takes one more evaluation period
to settle. Both algorithms provide consistent results for the remainder of
the load profile. Furthermore, both algorithms converge to almost
identical values of real and imaginary part throughout all the selected
frequencies. Thanks to the fast convergence, under the given circum-
stances it would be sufficient to provide a ripple for only ten seconds, in
order to get a first feasible estimation of impedance. Thus, saving time
and efforts in an actual application.
Even though GZ and PEIS seem to provide close to identical results,
both scenarios have been run, in which either GZ or PEIS were chosen
as input to VPA. As expected, the results from VPA were almost iden-
tical for both cases. The VPA's results with GZ as input are shown in
Figure 10. It visualizes a snapshot of the results, which were calculated
from an evaluation period that stretched from around t = 7.5 s until
t = 10 s. These results are output subsequent to the evaluation period.
In Figure 10, a laboratory EIS measurement is presented alongside the
OEIS results and serves as reference.
The impedance values from GZ and PEIS are in good agreement
with the laboratory EIS measurement. The laboratory measurement
itself reveals an additional small semi-circle towards the highest fre-
quencies. Since the VPA's model only contains one semi-circle, it tries to
provide a reasonable estimation, but will never be able to reproduce the
impedance perfectly. The RMSE equal 0.082 mΩ with respect to im-
pedance values from GZ. Using PEIS as input to VPA results in a RMSE
of 0.083 mΩ. Considering the absolute values of battery impedance
between roughly 1 mΩ and 4 mΩ, it is fair to assume an error of
<< 10 % can be achieved in this particular example. Evidently, the
OEIS algorithm provides an accurate and robust estimation of the
battery's impedance. Increasing the complexity of the utilized ECM
would surely increase the accuracy even further, yet it might require
more frequencies to be observed in order to retain its robustness.
Regarding the comparison of GZ and PEIS, both algorithms work
well. Furthermore, there are plenty of parameters within each of the
two algorithms for tuning different aspects of its performance. None of
the two algorithms outperforms the other, hence it will be up to the
applications requirements to decide which algorithm appears more
suitable. If, for example, the frequencies of the excitation signals are not
precisely known or may vary slightly around a given value, PEIS might
be a more suitable approach, since it considers frequency bands rather
than individual frequency values. GZ algorithm, on the other hand, is
less complex in terms of its mathematical calculations and hence
computational effort is lower in comparison to PEIS. If a wide range of
operating conditions is utilized or if more complex ECMs are utilized, it
may become necessary to implement a dedicated algorithm for
adapting target frequencies online. Adjusting GZ algorithm to different
frequencies is simple, yet for PEIS it would require recalculating the
filter coefficients of its band-pass filters, further increasing its compu-
tational effort. In the end it depends on the circumstances of the ap-
plications to decide on which algorithm is to be used.
5. Summary and outlook
Within the scope of this manuscript, we have introduced an algo-
rithm for an online electrochemical impedance spectroscopy (OEIS). It
is implemented in an efficient way and can therefore be implemented
on a BMS with little effort. A first proof-of-concept was carried out
against a battery model. The OEIS algorithm estimated the model's
impedance with high accuracy. Its parameter estimation was also ac-
curate yet proved to have weaknesses for large superposed DC current
rates at strong nonlinear behavior. Within a second step, the OEIS al-
gorithm was validated based on measurement data from a real battery
cell with a comparably low impedance. The impedance outputs were
compared to a laboratory EIS measurement and showed good agree-
ment even though the battery's electrical behavior exceeds the com-
plexity of the utilized equivalent circuit model (<< 10 %). Besides its
accuracy the algorithm also showed fastconvergence (< 10 s), thus
reducing the time frame in which an excitation signal needs to be
present. Since in some applications it is expected that an artificial ex-
citation signal must be generated, the intrusion into the system can be
reduced to a minimum, triggering the excitation and the OEIS algo-
rithm only at distinct operating conditions.
The algorithm is designed in a way it can easily cope with other
equivalent circuit models. It would be interesting to widen the fre-
quency range and to implement more complex models, which are es-
timated for a large operating range. However, this step would require a
sophisticated mechanism to adapt the investigated frequencies online.
The mechanism would have to consider the battery's operating condi-
tions and electrical behavior, in order to ensure feasible impedance
values are estimated before carrying out the parameter fitting.
CRediT authorship contribution statement
Matthias Kuipers: Conceptualization, Methodology, Software,
Validation, Formal analysis, Investigation, Data curation, Writing -
original draft, Visualization, Supervision. Philipp Schröer: Validation,
Investigation, Writing - review & editing. Thomas Nemeth: Writing -
review & editing. Hendrik Zappen: Writing - review & editing.
Alexander Blömeke: Writing - review & editing. Dirk Uwe Sauer:
Resources, Writing - review & editing, Supervision.
M. Kuipers, et al. Journal of Energy Storage 30 (2020) 101517
9
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influ-
ence the work reported in this paper.
Acknowledgement
This work has received funding from the European Unions Horizon
2020 research and innovation program under the grant ”Electric
Vehicle Enhanced Range, Lifetime And Safety Through INGenious
battery management” (EVERLASTING 713771).
Supplementary materials
Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.est.2020.101517.
Appendix A
The complex impedance Z can be calculated for a desired target frequency ftarget from the corresponding values of complex voltage U and
complex current I , which are determined by Goertzel algorithm running once for voltage and once for current.
Figure 9. Comparison of the impedance outputs from GZ and PEIS within the scope of the verification test carried out on a real battery cell.
Figure 10. Comparison of impedance spectra and - values in frequency domain within the scope of the verification test carried out on a real battery cell.
M. Kuipers, et al. Journal of Energy Storage 30 (2020) 101517
10
https://doi.org/10.1016/j.est.2020.101517
=Z
U
I
(f )
(f )
(f )target
target
target (A1)
Applying the equations from the original Goertzel algorithm (3), (4) leads to:
= =
+
+
( )
( )Z
U
I
x x
x x
(f )
(f )
(f )
·( 1)·exp j·
·( 1)·exp j·
k
u
k
u
k
i
k
i
target
target
target
1
2· ·f
f
1
2· ·f
f
target
sample
target
sample (A2)
Next, a constant KGZ can be defined for one particular ftarget according to:
=K
2· ·f
fGZ
target
sample (A3)
Furthermore, the complex expressions of voltage and current are transferred into Cartesian coordinates using Euler's formula.
= + +
+ +
Z x x x
x x x
·( 1)·cos( K ) j· ·( 1)·sin( K )
·( 1)·cos( K ) j· ·( 1)·sin( K )
k
u
k
u
k
u
k
i
k
i
k
i
1 GZ 1 GZ
1 GZ 1 GZ (A4)
Real and imaginary part of the impedance are calculated next:
= + +
+ +
Re Z x x x x x x x x
x x x x x
{ } · · ·cos(K ) · ·cos(K ) · ·[cos (K ) sin (K )]
( ) ( ) ·cos (K ) 2· · ·cos(K ) ( ) ·sin (K )
k
u
k
i
k
u
k
i
k
u
k
i
k
u
k
i
k
i
k
i
k
i
k
i
k
i
1 GZ 1 GZ 1 1
2
GZ
2
GZ
2
1
2 2
GZ 1 GZ 1
2 2
GZ (A5)
= +
+ +
Im Z x x x x x x x x
x x x x x
{ } · ·sin(K ) · ·sin(K )·cos(K ) · ·sin(K ) · ·sin(K )·cos(K )
( ) ( ) ·cos (K ) 2· · ·cos(K ) ( ) ·sin (K )
k
u
k
i
k
u
k
i
k
u
k
i
k
u
k
i
k
i
k
i
k
i
k
i
k
i
1 GZ 1 1 GZ GZ 1 GZ 1 1 GZ GZ
2
1
2 2
GZ 1 GZ 1
2 2
GZ (A6)
Applying the trigonometric identity
+ =x xcos ( ) sin ( ) 12 2 (A7)
together with some final simplifications, the equations for determining impedance are put into a final form as presented in (A8), (A9) and (A10).
= +
+
+
+
Z x x x x x x x x
x x x x
x x x x
x x x x
· · ·cos(K ) · ·cos(K ) ·
( ) 2· · ·cos(K ) ( )
j· · ·sin(K ) · ·sin(K )
( ) 2· · ·cos(K ) ( )
k
u
k
i
k
u
k
i
k
u
k
i
k
u
k
i
k
i
k
i
k
i
k
i
k
u
k
i
k
u
k
i
k
i
k
i
k
i
k
i
1 GZ 1 GZ 1 1
2
1 GZ 1
2
1 GZ 1 GZ
2
1 GZ 1
2 (A8)
= +
+
Re Z x x x x x x x x
x x x x
{ } · · ·cos(K ) · ·cos(K ) ·
( ) 2· · ·cos(K ) ( )
k
u
k
i
k
u
k
i
k
u
k
i
k
u
k
i
k
i
k
i
k
i
k
i
1 GZ 1 GZ 1 1
2
1 GZ 1
2 (A9)
=
+
Im Z x x x x
x x x x
{ } · ·sin(K ) · ·sin(K )
( ) 2· · ·cos(K ) ( )
k
u
k
i
k
u
k
i
k
i
k
i
k
i
k
i
1 GZ 1 GZ
2
1 GZ 1
2 (A10)
For the purpose of minimizing calculation effort cos(KGZ) and sin(KGZ) are calculated and initialized as constants either offline or during in-
itialization. Accordingly, they do not have to be calculated every time the impedance is determined.
Appendix B
The impedance of an equivalent circuit model consisting of an ohmic resistance in series with a ZARC element can be calculated according to:
= +
+
Z R R
A R
( )
1 (j ) · ·0
1
1 1 (B1)
Applying Euler's formula the denominator is split into its real and imaginary part.
= +
+ +( ) ( )Z R
R
A R A R
( )
1 · · ·cos · j· · · ·sin ·
ECM 0
1
1 1 2 1 1 2 (B2)
In order to avoid a complex value in the dominator, both sides of the fraction are multiplied with the denominator's conjugated complex value,
resulting in:
= +
+
+ + +
( ) ( )
( ) ( ) ( )Z R
R A R A R
A R A R A R
( )
· · ·cos · j· · · ·sin ·
1 2· ·· · ·cos · ( · · ) ·cos · ( · · ) ·sin ·
ECM 0
1 1 1
2
2 1 1
2
2
1 1 2 1 1
2 2
2 1 1
2 2
2 (B3)
Applying the trigonometric identity of:
+ =cos
2
· sin
2
· 12 2
(B4)
the equation for the battery model's impedance is reduced to:
= +
+
+ +
( ) ( )
( )Z R
R A R A R
A R A R
( )
· · ·cos · j· · · ·sin ·
1 2· · · ·cos · ( · · )
ECM 0
1 1 1
2
2 1 1
2
2
1 1 2 1 1
2
(B5)
M. Kuipers, et al. Journal of Energy Storage 30 (2020) 101517
11
Finally, it can also be denoted according to its real and imaginary part:
= = +
+
+ +
( )
( )Z Z R
R A R
A R A R
Re{ ( )} ( )
· · ·cos ·
1 2· · · ·cos · ( · · )
ECM ECM 0
1 1 1
2
2
1 1 2 1 1
2
(B6)
= =
+ +
( )
( )Z Z
A R
A R A R
Im{ ( )} ( )
( 1)· · · ·sin ·
1 2· · · ·cos · ( · · )
ECM ECM
1 1
2
2
1 1 2 1 1
2
(B7)
Appendix C
In order to derive the equation for the small signal charge transfer resistance, Butler Volmer equation is required:
=I i V V· exp ·z·F
R·T
· exp ( 1)· (1 )·z·F
R·T
·R C C0
(C1)
In [14] it is shown that = 0.5, describing a synchronous behavior between anodic and cathodic reaction, gives a sufficiently accurate estimation
for the behavior of a lithium ion battery, even if Butler Volmer equation is applied onto a cell measurement rather than a measurement on individual
electrodes. As long as this simplification is applicable according to [14] the charge transfer resistance can be denoted as:
= =
+ +
R I R I
k I k I
k I
( ) ( 0)·
ln[ · ( · ) 1 ]
·ct R ct R
I R I R
I R
2
(C2)
Utilizing substitutions:
= =R I
i
( 0) R·T
2· · ·z·Fct R 0 (C3)
=k
i
1
2·I 0 (C4)
An algebraic expression for the over potential VC can be denoted as:
= = + +V R I I I
i
I
i
( )· R·T
·z·F
· ln
2· 2·
1C ct R R R R
0 0
2
(C5)
Based on (C5) an algebraic formula can be derived for the charge transfer resistance Rct in terms of both: large (Rct,LS) and small signal (Rct,SS)
resistance according to (C6) and (C7) respectively.
= =
+ +( )
R I V
I
T
I
( ) K · ·
ln 1
ct LS R
C
R
I
i
I
i
R
, BV
2· 2·
2R R
0 0
(C6)
= = + +R I dV
dI
d
dI
T I
i
I
i
( ) K · · ln
2· 2·
1ct SS R C
R R
R R
, BV
0 0
2
(C7)
For the purpose of minimizing calculation effort constant KBV is introduced, combining all constant values.
=K R
·z·FBV (C8)
Calculating the derivativeof equation (C7) results in expression:
=
+ +
+
+( ) ( )
R I T
i
( ) K · · 1
1
· 1
2·
2·
2· 1
ct SS R
I
i
I
i
I
i
I
i
, BV
2· 2·
2 0
(2· )
2·
2R R
R
R
0 0
0 2
0 (C9)
The equation can be simplified into:
=
+
R I T
I i
( ) K · · 1
(2· )
ct SS R
R
, BV 2
0
2 (C10)
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	An Algorithm for an Online Electrochemical Impedance Spectroscopy and Battery Parameter Estimation: Development, Verification and Validation
	Introduction
	Working principle of the online electrochemical impedance spectroscopy (OEIS) algorithm
	Goertzel based Calculation of Impedance
	Passive Electrochemical Impedance Spectroscopy (PEIS)
	Introduction of an electrical equivalent circuit model (ECM)
	Estimation of ECM parameters using a varied parameters approach (VPA)
	Verification against a battery model
	Validation against measured data
	Summary and outlook
	CRediT authorship contribution statement
	Declaration of Competing Interest
	Acknowledgement
	Supplementary materials
	Appendix A
	Appendix B
	Appendix C
	References