Journal of Energy Storage: Petr Vyroubal, Tomá Š Kazda
Journal of Energy Storage: Petr Vyroubal, Tomá Š Kazda
Journal of Energy Storage: Petr Vyroubal, Tomá Š Kazda
A R T I C L E I N F O A B S T R A C T
Article history:
Received 1 September 2017 Numerical modelling is the method by virtually testing and verifying the functionality of a specific
Received in revised form 4 October 2017 product or component. The primary goal is to get approximate results of how the system behaves in a
Accepted 11 October 2017 given time and environment. We are able to accept a certain numerical error from a real experiment, thus
Available online xxx significantly speeding up part of the development of the device. In the field of electrochemistry of
lithium-ion accumulators, several variants of numerical models have been proposed that yield
satisfactory results in modelling certain physical fields of these batteries (electric field, temperature field,
current field).
© 2017 Elsevier Ltd. All rights reserved.
One model is the battery model with its equivalent circuit, frequency with offset of a certain value of the polarization voltage
which is represented by passive components, namely a resistor and (mostly between 1 and 10 mV) [2].
a capacitor. This is a model that describes the internal behaviour of
a battery by RC circuit. However, the problem is to find or measure 2. EIS basics
these parameters. One way to get the parameters of this spare
circuit is by means of electrochemical impedance spectroscopy The impedance value of the system is calculated using the ratio
(EIS). of alternating voltage and alternating current. This is composed of
This article deals with the application this method to obtain the amplitude and phase shift values that are dependent on the set
parameters of the equivalent battery circuit and comparing the frequency. Thanks to this, the dependence of the system
charging and discharging curves of a specific battery with the ECM impedance on the frequency is obtained. The individual chemical
model. elements that make up the measured system differ from each other
by the time constant of the polarization, which causes the
1. Introduction individual components of the impedance to vary with frequency
variations [3] and [4].
00
Electrochemical Impedance Spectroscopy (EIS) allows the These components are: Z0 which is the real impedance, jZ the
observation of the electrochemical and physical processes occur- imaginary impedance and |Z| the total impedance. The ratio of
ring in the monitored system. This is a very sensitive method in impedances "Z0 /Z00 / Z is called the loss factor tgd. EIS requires stable
which the interpretation of results is rather complex due to its high system, which is difficult in practice due to the possibility of
sensitivity. The total measurement result consists of a mixture of influencing the measurement by external noise, the temperature
responses from all the events that occur in the measured system, change and contamination. It is necessary to perform the
which often leads to ambiguity of the result [1]. measurement in the moment when the electrochemical response
For a good interpretation of the results, it is necessary to of the system is stable. The impedance is described by the formula
understand the physical and chemical essence of the measured according to [5–7]:
system. Then compare the results with other analytical methods, 00
both electrochemical and physical. The basic principle of this Z ¼ Z0 þ jZ ð1Þ
method is to set up a small sinusoidal AC voltage of a given Alternatively, using polar coordinates:
https://doi.org/10.1016/j.est.2017.10.019
2352-152X/© 2017 Elsevier Ltd. All rights reserved.
24 P. Vyroubal, T. Kazda / Journal of Energy Storage 15 (2018) 23–31
2.1. Resistor R
2.2. Capacitor C
1
Z¼ ð4Þ
jvC
2.3. Inductor L
Fig. 6. Schematic representation of Li-ion mass transfer phenomena which occur in Liion battery electrodes and their respective Nyquist plots. [13].
26 P. Vyroubal, T. Kazda / Journal of Energy Storage 15 (2018) 23–31
and
r ðs þ r’þ Þ ¼ j ð10Þ dðsocÞ IðtÞ
¼ ð17Þ
dt 3600Q Ah
resp.
For a given battery, the open circuit voltage, resistors
r ðs þ r’þ Þ ¼ ðjECh jshort Þ ð11Þ
resistances, and capacitors capacitances are functions of the
and battery state of charge (SOC) (it was measured by 0, 25, 50, 75 and
100 % SOC). These functions could be expressed in The fifth order
r ðs þ r’þ Þ ¼ j ð12Þ Polynomial form:
resp.
Rs ¼ rs0 þ rs1 ðsocÞ þ rs2 ðsocÞ2 þrs3 ðsocÞ3 þrs4 ðsocÞ4
r ðs r’ Þ ¼ ðjECh jshort Þ ð13Þ þ rs4 ðsocÞ5 ð18Þ
Cp is specific heat capacity, T is thermodynamic temperature, k is
thermal conductivity, F+ and F is the cathode- and anode-side
electric potentials, t is time, s+ a s are conductivities of positive R1 ¼ r10 þ r11 ðsocÞ þ r12 ðsocÞ2 þr13 ðsocÞ3 þr14 ðsocÞ4
and negative electrode, s þ a s are potentials of positive and þ r14 ðsocÞ5 ð19Þ
negative electrode, jECh a q_ ECh volumetric current and heat
generated by electrochemical reactions, respective jshort a q_ short
volumetric current and heat generated by internal short circuit (in R2 ¼ r20 þ r21 ðsocÞ þ r22 ðsocÞ2 þr23 ðsocÞ3 þr24 ðsocÞ4
normal conditions are these variable zero values). þ r24 ðsocÞ5 ð20Þ
Electrochemical sub-models relate the local current density to
the potential. The cell model couples sub-models to thermal and
electrical fields within the cell, integrates over multiple electrode- C1 ¼ c10 þ c11 ðsocÞ þ c12 ðsocÞ2 þc13 ðsocÞ3 þc14 ðsocÞ4
pairs [16]. þ c14 ðsocÞ5 ð21Þ
The internal model is calculated by means of the Newman,
Tiedemann, Gu, and Kim model, which is simple semi-empirical
electrochemical model. It was proposed by Kwon [17] and has been
used by [14] and [15]. C2 ¼ c20 þ c21 ðsocÞ þ c22 ðsocÞ2 þc23 ðsocÞ3 þc24 ðsocÞ4
According to these equations are the potential, temperature and þ c24 ðsocÞ5 ð22Þ
current fields calculated by means finite volume method (FVM).
In the Equivalent Circuit Model (ECM), battery electric
behaviour is represented by an electrical circuit. The numerical Voc ¼ voc0 þ voc1 ðsocÞ þ voc2 ðsocÞ2 þvoc3 ðsocÞ3 þvoc4 ðsocÞ4
model was prepared using ANSYS FLUENT system. It has adopted þ voc4 ðsocÞ5 ð23Þ
the six parameter ECM model following the work of Chen [5]. In
this model, the circuit consists of three resistors and two capacitors The energy source term comes from the contribution of Joule
(see Fig. 8). The voltage-current relation can be obtained by solving heating, electrochemical reaction heating, and the entropic
the electric circuit equations: heating:
Fig. 9. The process of simplifying the layers of a real battery into a homogeneous MSMD model [20].
Table 1
Dimensions and properties of the measured battery.
Zone Pc Pe S Ne Nc Total
d [m] 8.40E-04 2.44E-03 9.24E-04 4.28E-04 1.05E-04 8.52E-03
Density [kg m3] 2700 1500 492 2660 8900 8.04E+02
Specific Heat capacity [J (kg1 K1] 903 1260 1978 1437 385 6.94E+02
Thermal conductivity [W (m1 K1)] 238 1.48 0,334 1,04 398 1.47E+01
Electrical conductivity [S m1] 3.89E+07 1.00E-06 1.00E+04 6.33E+07 r_p 1.92E+06
r_n 3.90E+05
Fig. 12. Nyquist diagram of impedance measuring the KOKAM battery by different SOC.
Table 2 Table 3
The parameters of equivalent circuit model by discharging. The parameters of equivalent circuit model by charging.
SOC [%] Voc [V] Rs [V] R1 [V] C1 [F] R2 [V] C2 [F] SOC [%] Voc [V] Rs [V] R1 [V] C1 [F] R2 [V] C2 [F]
1 4.195754 0.01889 0.001540 1.241 0.002314 4.626 1 4.19785 0.01879 0.002278 4.6990 0.001542 1.266
0.75 3.920000 0.01889 0.002425 4.378 0.001645 1.193 0.75 3.94147 0.01878 0.002328 3.9170 0.001466 1.259
0.5 3.790000 0.01889 0.003286 4.734 0.001850 1.181 0.5 3.80990 0.01885 0.003221 4.4160 0.001751 1.189
0.25 3.678890 0.01895 0.002222 1.281 0.007209 5.312 0.25 3.72448 0.01889 0.002096 1.2520 0.006020 5.118
0 3.008480 0.01902 0.002970 1.591 0.021300 4.857 0 3.02384 0.01906 0.003355 1.5710 0.022690 4.880
Fig. 13. The polynomial functions obtained by discharging (dotted line fitted, full line measured).
P. Vyroubal, T. Kazda / Journal of Energy Storage 15 (2018) 23–31 29
Fig. 14. The polynomial functions obtained by charging (dotted line fitted, full line measured).
5. Parameters extraction were then inserted into the graphs and interleaved by the 5th order
polynomial function. In this way, we have obtained equations
The values of the equivalent circuit parameters were extracted describing the behaviour of the battery in different charging/
from EIS measurements using the fitting program that is included discharging state (Fig. 13).
in the Biologic booster. The fitted circuit was chosen according to The battery circuit parameters obtained in this way were
the mentioned equivalent circuit in Fig. 8. For the results of these inserted into the numerical model. The charging and discharge
measurements, it is important in which state of charge the battery curves of the battery were modelled and compared with the curves
is at the beginning of the measurement. measured at the laboratory measurement.
The Nyquist diagram (see Fig. 12) shows the impedance
measurement at different battery SOC using the EIS method in 6. Results and discussion
laboratories.
The obtained curves were then fitted to an equivalent circuit The actual comparison of the discharge characteristics is shown
identical to the circuit shown in Fig. 8 above. Thanks to this in Fig. 14. It is obvious that the curves have the same course and are
imitation, the parameters R and C (Tables 2 and 3) were obtained slightly offset. This 0.1 V spacing is not essential to affect too much
and then were used in Eqs. (18)–(23) above. These dependencies the properties of the simulated battery and can be considered
Fig. 15. Comparison of measured and simulated discharge curves of the battery.
30 P. Vyroubal, T. Kazda / Journal of Energy Storage 15 (2018) 23–31
Fig. 16. Comparison of measured and simulated charge curve 0.2C of the battery.
satisfactory, particularly for copying the same waveform as the this method it can be stated that, within the tolerance of a certain
measured battery. It is necessary to realize that the battery degree of inaccuracy, it has proved to be an effective tool for
represents an ideal RC circuit. Some parameters represented by the obtaining a battery model at the RC circuit level. The use is
Nernst and Butler-Volmer equations are neglected. unambiguously where we do not know the exact properties of the
Discharging by 4C can be definitely considered as fast materials in which the batteries are composed. Conventional
discharging for example: Maximum NCR18650B discharge limit methods of battery microstructure modelling are based on often
is 2C under continuous current load or KOKAM SLPB7570180 10.6 complex electrochemical models. These models are loaded with
Ah with maximal continuous discharge 3C. errors already at the time of determining some of the coefficients
The charging characteristics were measured as well as the of the given materials.
accuracy of the previous measurements. To simulate battery When modelling a particular device (electric vehicle, electric
charging, input data was used as described above. The charging train, etc.) when looking at a battery in a microstructure is not
current was set to 0.2C, which ensures a slow but still kindly charge important, replacing the battery with the model thus obtained is
of the battery (see Fig. 15). sufficient. In principle, we are able to obtain a solution for any
Numerical model allows us to display some physical fields. discharge current by approximation. However, as soon as we
Fig. 16 shown the potential distribution at the positive and exceed a certain amount of discharging current, the endothermic
negative electrode at the end of discharging (Fig. 17). reaction battery begins to run. This model is unable to capture it.
But the temperature field can be captured. Therefore, it is also
7. Conclusion important to monitor this parameter and not exceed the
manufacturers declared permissible values.
The method presented in this paper, which is suitable for the The presented process of numerical modelling was compared
extraction of lithium ion battery parameters, is called electro- with real-time measurements, which yielded satisfactory model
chemical impedance spectroscopy. In assessing the suitability of match and exact results.
Fig. 17. Potential distribution at the positive (left) and negative (right) electrode.
P. Vyroubal, T. Kazda / Journal of Energy Storage 15 (2018) 23–31 31
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