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IEEE TRANSACTIONS ON ENERGY CONVERSION 1

Time-Domain Parameter Extraction Method for

Thévenin-Equivalent Circuit Battery Models
Ari Hentunen, Member, IEEE, Teemu Lehmuspelto, Member, IEEE, and Jussi Suomela

Abstract—This paper presents an analytical time-domain-based validate the component selection and sizing as well as to pro-
parameter identification method for Thévenin-equivalent circuit- vide information for vehicle’s control algorithms. Accurate and
based lithium-ion battery models. The method is based on the computationally light battery models are needed for these sim-
analysis of voltage-relaxation characteristics of pulse discharge
and pulse charge experiments, and the method can be used for ulations. Accurate battery models accompanied with parameter
both discharge and charge operation with any number of parallel adaptation algorithms are also needed in certain real-time ap-
resistor-capacitor branches. The use of the method is demonstrated plications, e.g., in battery management systems (BMSs), to es-
for a second-order model and validated with a real-world duty timate important quantities such as state of charge (SOC), state
cycle. Experimental results for a commercial lithium-ion battery of health, and available power [2].
module are presented.
Modeling of electrochemical batteries is challenging due to
Index Terms—Batteries, battery management systems (BMSs), their high level of nonlinearity. The characteristics such as
electric vehicles (EVs), equivalent circuits, parameter extraction. impedance are, in general, nonlinear multivariable functions of
the SOC, temperature, aging, current direction, and rate. There
I. INTRODUCTION are several different kinds of modeling approaches, which can
be generally divided into electrochemical [3], mathematical [4],
HE interest in vehicle electrification has been rising
T steadily as we are moving toward sustainable transporta-
tion. Due to increased demand from customers and matur-
and electrical [5]–[7] modeling. Lately, also models that com-
bine mathematical and electrical models have been introduced
(e.g., [8]). For system-level simulations of EVs, Thévenin-based
ing technology, automobile manufacturers are now introduc- [6] or impedance-based [7] electrical models are commonly
ing electric vehicles (EVs), such as battery EVs, hybrid EVs, used, because they are fast to execute, simple and intuitive to
and plug-in hybrid EVs in increasing pace. Also, heavy vehicle analyze, and provide accurate SOC, open-circuit voltage (OCV),
and non-road mobile machinery industries have shown rising and terminal voltage prediction under dynamic load current pro-
interest to electrify the drive train. The driving forces are the re- file [9]. The model can be augmented to predict also the tem-
markably tightening legislation and regulations for the exhaust perature [10], which is often needed in the simulations. Due to
emissions of the diesel engines used in NRMM, the desire to inherent simplicity of electrical models, they can also be trans-
improve performance, safety, and operator comfortability, and formed into online models, which run inside a BMS [11].
the desire to decrease operating cost. Common for all EVs is The main difference between the impedance-based and
that a large electrochemical battery is used as an energy storage Thévenin-based modeling methods is that the parameters of
to power the vehicle. Lithium-ion (Li-ion) chemistries offer su- impedance-based models are extracted based on electrochemi-
perb properties such as high power rating, high energy density, cal impedance spectroscopy measurements in frequency domain
and high cycle life, and therefore, they are likely to be largely [7], while Thévenin-based models are parameterized typically
adopted by EV manufacturers [1]. by (CP) experiments in time domain [12], i.e., experimental
In the early stages of a vehicle electrification development, current and voltage time-series data. Thévenin-based models
simulations are usually used as a tool to evaluate concept studies are attractive, because no impedance measurements need to be
and to validate early design goals. As the development process done. In addition, also battery modules and packs can be char-
goes on, more accurate models of subsystems are needed to acterized and modeled directly based on the data from a battery
cycler during performance tests of a battery module or pack.
Manuscript received November 13, 2012; revised March 5, 2014; accepted The basic experiments at the typical temperature and rate can
April 10, 2014. This work was carried out in the HybLab and ECV projects
funded by the Multidisciplinary Institute of Digitalization and Energy (MIDE) be done in a couple of days [13]. If also the temperature and
of Aalto University and Finnish Funding Agency for Technology and Innovation current-rate effects need to be extracted, the duration of the tests
(Tekes). Paper no. TEC-00592-2012. increases accordingly.
A. Hentunen is with the Department of Electrical Engineering and Automa-
tion, School of Electrical Engineering, Aalto University, 02015 Espoo, Finland The Thévenin-based electrical model of [6] is commonly used
(e-mail: ari.hentunen@aalto.fi). for Li-ion battery modeling. The voltage response to current ex-
T. Lehmuspelto is with the Department of Engineering Design and Produc- citation is modeled as a Thévenin equivalent circuit, which is
tion School of Engineering, Aalto University, 02015 Espoo, Finland (e-mail:
teemu.lehmuspelto@aalto.fi). represented in Fig. 1, where sQ is the SOC, uo c is the OCV,
J. Suomela is with Hybria Oy, 02150 Espoo, Finland (e-mail: jussi.suomela@ ub is the terminal voltage, ib is the terminal current, R0 is
hybria.fi). the ohmic resistance, R1 , . . . , Rn are the dynamic resistances,
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org and C1 , . . . , Cn are the corresponding dynamic capacitances.
Digital Object Identifier 10.1109/TEC.2014.2318205 All resistances and capacitances are functions of the SOC,

0885-8969 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
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2 IEEE TRANSACTIONS ON ENERGY CONVERSION

Fig. 1. Electrical equivalent circuit.

Fig. 2. Illustration of a PD experiment for a battery module.

temperature, rate, current direction, calendar life, and cycle life.
The voltage source provides the predicted OCV as a function of
the SOC. Generally, also temperature dependence and hysteresis can be used for both discharge and charge operation with any
effect can be included into the OCV prediction. number of parallel RC branches. The computational burden of
The number of resistor-capacitor (RC) branches depends on the proposed method remains low even if long rest times and
the desired accuracy and bandwidth of the model [14]. One to high sampling frequency were used in the experiments. With the
three branches are typically used. One RC branch is enough presented method, the parameter extraction can be done system-
for early simulations of EVs. It is, however, often found that atically and it can be easily automated. The use of the method
more branches are needed for more detailed simulations, ve- is demonstrated and validated with experiments with a com-
hicle software development, or BMS development [10], [11], mercial 40 Ah Li-ion battery module. The presented parameter
[15]–[17]. The parameterization of one RC branch model is extraction method is not limited to automotive applications, but
straightforward. However, the lack of systematic offline param- it can be applied to other applications and battery chemistries
eter identification methods for multiple RC branches has yield as well.
into use of ad hoc methods and iterative numerical optimization
methods to parameterize the model.
II. MODEL EXTRACTION
The parameters for a first-order model can be predicted an-
alytically from a short CP [12] or from the voltage relaxation The proposed RC parallel circuit identification method is
after a CP has ended [13]. With some restrictions, the CP method based on the analysis of the voltage relaxation characteristics
[12] can also be applied for two RC-circuits. However, due to during the rest-times of the experimental PD and PC charac-
short duration of the CP, the shorter time constant is in the or- terization tests. These PD and PC tests consist of a sequence
der of less than a second and the longer time constant is in the of constant CPs with a rest time between pulses. The exper-
order of tens of seconds. These are too short time constants iments should cover the typical rates of the anticipated load
for system-level simulation of many battery systems, including profile to provide accurate results. The duration of rest times
EV applications. The analytical method of Hu et al. [17] can and the number of pulses are both compromises of the desired
be applied for multiple-order models. However, it is limited for accuracy and test duration. Rest times between 5 and 60 min
discharge operation. In addition, if the parameters are identified and pulse rates between 2% and 20% are typically used. Fig. 2
from the data during the load current, as in [12] and [17], then illustrates a PD sequence at 1C rate, 10% pulse rate, and 30-min
the SOC and OCV do not stay constant during the identification, rest time. Each PD or PC test sequence characterizes the whole
which causes error to the parameters. SOC range systematically. The starting times and ending times
Most of the presented parameter identification methods for of the CPs are first identified, and the data are split accordingly
Thévenin-based models are based either on online parameter into sections. Each rest time is analyzed separately. The rest
identification (e.g., [10], [11], [16], [18], [19]) or on iterative period under investigation is then partitioned into separate time
numerical optimization (e.g., [8], [9], [20], [21]). Online meth- windows, which are used to identify the corresponding model
ods are not directly applicable for offline simulations. Iterative parameters.
methods such as genetic algorithms and nonlinear least squares The parameters are typically scheduled for the SOC, tem-
typically need initial guesses and have a high number of param- perature, and rate. Two parameter sets are identified, one for
eters to be identified, and thus, those methods are very sensitive discharging, i.e., for positive current, and one for charging, i.e.,
for initial conditions, they may end up in a local minimum, and for negative current. The resulting parameter mappings can be
they may have numerical problems with convergence. They also implemented, e.g., as mathematical functions or multidimen-
need excessive iterative simulations to identify the parameters sional look-up table.
accurately and may need a lot of effort to find a good set of The SOC must be tracked in order to make the SOC-
parameters. Thus, the identification process of iterative brute scheduling. The SOC is defined as the ratio of the available
force methods may be very tedious and time intensive. charge to the usable charge capacity:
A simple and fast analytical time-domain-based method for Q
RC circuit parameter extraction of Thévenin equivalent circuit sQ = (1)
Qus
battery models is presented in this paper. The method is based
on the analysis of voltage relaxation characteristics of pulse dis- where sQ is the SOC, Q is the available charge in the battery,
charge (PD) and pulse charge (PC) experiments, and the method and Qus is the usable charge capacity of the battery. In (1), the
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HENTUNEN et al.: TIME-DOMAIN PARAMETER EXTRACTION METHOD FOR THÉVENIN-EQUIVALENT CIRCUIT BATTERY MODELS 3

variable for the SOC is dimensionless and has a range of 0–1.

It is also common to express it in percents. The SOC can be
tracked, e.g., by the coulomb-counting method [6]:

1
sQ = sQ 0 − ib dt (2)
Qus
where sQ 0 is the initial SOC, and ib is the battery current. It
is recommended to first measure the real capacity and use it
instead of the nominal capacity, if the values differ noticeably.
The ohmic resistance R0 can be easily identified at the time-
instants when a CP is beginning or ending:
  Fig. 3. Voltage relaxation characteristics during a PD test. Example time
 ΔU 
R0 =   (3) instants for two time windows are also shown. Time axis has been shifted to
ΔI  have origin at the beginning of rest time, i.e., t0 .

where ΔU is the instantaneous voltage difference during the

current step, and ΔI is the amplitude of the current step.
For a model with nth order, the following time-domain equa- as in Fig. 1. The beginning time of the rest period should be
tion represents the terminal voltage during a rest time after a CP extracted to be the time instant when the current reaches zero,
has ended: i.e., right after the instantaneous voltage step caused by the
ohmic resistance. The ending time instant should be extracted

n
− τt to be the time instant right before the next CP begins. The
ub = uo c − Ui e i (4)
parameters Ui and τi can then be estimated from the transient
i=1
circuit voltage uτ data.
where Ui is the initial voltage of the ith RC parallel branch and Let us use two RC circuits as a minimal example to demon-
τi = Ri Ci is the corresponding time constant. The RC parallel strate the full capabilities of the model extraction method. One
branches model the activation polarization or charge-transfer RC circuit is then a special case of the presented method. For
overvoltage and concentration polarization at anode and cath- two RC branches, (6) becomes
ode [22]. During rest time, the terminal voltage of a battery
uτ (t) = U1 e− τ 1 + U2 e− τ 2 .
t t
approaches OCV as the polarization decreases, finally reaching (7)
equilibrium. In (4), the RC branches voltages decrease with de-
cay rates determined by the time constants τi , and eventually Next, the rest period is divided into two partitions, which are
the battery voltage equals the OCV. used for parameter extraction of short and long time-constant
During a rest period of a PD or PC test, the transient circuit branches, respectively. If more RC branches were used in the
voltage, i.e., the total voltage over the resistor-capacitor network model structure, the rest period would need to be divided into
of the (EEC) of Fig. 1, can be expressed as as many partitions as there were RC circuits. Let us first define
the time instants:
uτ (t) = Uo c − ub (t) (5) 1) tcp e is the time instant when the CP ends and current starts
to fall.
where Uo c is the OCV, which stays constant during the rest
2) t0 is the time instant when the current has fallen back to
period, and ub (t) is the battery voltage as a function of time.
zero, i.e., the beginning of rest time. In the following, this
Assuming that the rest time is sufficiently long so that the voltage
will be referred as t = 0 s.
reaches steady state, the OCV can be identified by taking the
3) t11 is the starting time of the first time window, typically
final voltage value at the end of each rest time. If shorter rest
t11 = t0 .
times are employed, other methods for the OCV-identification
4) t12 is the ending time of the first time window.
are recommended. Rapid methods are presented in [13].
5) t21 is the starting time of the second time window.
Even though the transient circuit voltage of (5) is not a real
6) t22 is the ending time of the second time window.
measurable quantity and cannot be measured directly during
7) tend is the ending time of the rest period.
the experiment, it can be obtained a posteriori from the data.
The partitioning of one rest time is illustrated in Fig. 3. Time
By assuming the model structure of Fig. 1, the transient circuit
windows should have a time gap between them, during which the
voltage during a rest period can be represented as
voltage of the shorter time constant branch should fall very close

n
to zero. This will be illustrated later in an example. The natural
Ui e− τ
t
uτ (t) = i (6) starting time for the first time window is t0 , i.e., t = 0 s. The
i=1
starting time for the second time window should be later than
where Ui is the initial voltage of the ith RC branch and τi is the three times the shorter time constant. This condition confirms
corresponding time constant. For PD and PC experiments, the that the voltage of the shorter time-constant branch has dropped
transient circuit voltage uτ and all of the initial voltages Ui have below 5% of its initial value, and thus, it has only a negligible
always the same sign, i.e., positive for discharge and negative for effect on the parameter extraction of the long time-constant
charge when discharge is defined as positive current direction branch. However, because the predicted time constants cannot
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4 IEEE TRANSACTIONS ON ENERGY CONVERSION

be known exactly before the parameter extraction, this condition The initial voltage of the first branch at time instant t0 can then
should be checked afterward. be obtained as follows:
Let us start with the latter partition and choose the time in- t11
Û1 = uτ (t11 ) e τ̂ 1 . (15)
stants t21 and t22 as its starting and ending time instants. This
time window is now used to extract the long time-constant char- The predicted voltage of the first RC branch can be written as
acteristics. It is first assumed that the two time constants are of
û1 (t) = Uˆ1 e− τ̂ 1
t

different time scales, i.e., one is in the order of tens of seconds for t ≥ 0. (16)
and the other is in the order of minutes or tens of minutes. It is The total predicted transient circuit voltage can now be ex-
also assumed that at the beginning of the second time window pressed as
the fast time-constant voltage has dropped to zero. Then, the
ûτ = Û1 e− τ̂ 1 + Û2 e− τ̂ 2 .
t t
following expression can be written: (17)
t −t
− τ 21 For a model with nth order, generic expressions for the tran-
uτ (t) = uτ (t21 ) e 2 for t ≥ t21 , (8)
sient circuit voltage uτ i (t) as well as the time-constant τ̂i , initial-
where the uτ (t21 ) is the voltage at the starting time of the second voltage Ûi , and voltage ûi (t) predictions of the ith RC branch
time window. By setting the time as the ending time of the time can be formulated as follows:
window, i.e., t = t22 , the time constant τ2 can be solved: ⎧
⎨ Uo c − ub (t), n
⎪ if i = n
t22 − t21 
τ̂2 =   for uτ = 0 (9) uτ i (t) = U − u (t) − (18)
uτ (t21 ) ⎪
⎩ oc b ûi+1 (t), if i < n
ln i+1
uτ (t22 )
ti2 − ti1
where a hat is used to denote predicted quantities to distinct τ̂i = u τ i (t i 1 )
, for uτ i = 0 (19)
them from the measured quantities. ln u τ i (t i 2 )
The above equation works for both current directions, i.e., for ti1
both the PD and PC experiments, because the possible minus Ûi = uτ i (ti1 ) e τ̂ i (20)
signs of the voltages cancel each other. It should be noted, − τ̂t
however, that after a very long rest time, the transient circuit ûi (t) = Ûi e i , for t ≥ 0. (21)
voltage approaches zero, which is not allowed. Therefore, the The resistance Ri and capacitance Ci values can be extracted
ending-time for the determination of the time constant should with the knowledge of the preceding CP amplitude Icp and
be selected appropriately so that the transient circuit voltage is duration tcp , as follows:
not very close to zero. The initial voltage of the second branch
at time instant t0 can then be obtained: Ûi
Ri = tcp (22)
−
Û2 = uτ (t21 ) e
t21
τ2
. (10) Icp 1 − e τ̂ i

The predicted voltage of the second RC branch can then be τ̂i

Ci = . (23)
expressed as Ri
û2 (t) = Uˆ2 e− τ̂ 2
t
for t ≥ 0. (11) In the above equations, the resistance and the time constant
are assumed to be constant during the CP, which is obviously
After U2 and τ2 have been obtained, the first time window can not true in general. However, the resulting inaccuracy in the
be used to extract the parameters of the short time-constant RC parameter values causes only a negligible error in the voltage
branch. Those parameters can be extracted the same way as for prediction. This issue can also be partly rounded by having a
the second time window, but the data must be first preprocessed high number of CPs, e.g., 20, in the characterization experiment.
by subtracting the predicted long time-constant branchs voltage However, the duration of the experiment increases, respectively,
from the experimental data: to the number of CPs and the duration of the rest period.
uτ (t) = uτ (t) − û2 (t) = Uo c − ub (t) − û2 (t) (12) This parameter extraction method can be generalized to any
number of RC branches as long as their time constants are
where an apostrophe denotes altered voltage after the subtraction well separated. The procedure goes always from the longest
of the predicted long time-constant voltage. The altered transient time-constant identification towards the shortest. There should
circuit voltage can be expressed as also be a sufficiently long time gap between consecutive time
t −t 1 1 windows to let the shorter time-constant branches voltages to
uτ (t) = uτ (t11 ) e− τ1
for t ≥ t11 (13) drop into the vicinity of zero before the beginning of the next
where the uτ (t11 ) is the voltage of the altered transient circuit time window.
voltage at the starting time of the first time window. Then, the
time constant can be solved: III. EXPERIMENTAL
t12 − t11 A module-level test environment for battery characterization
τ̂1 =    for uτ = 0. (14)
uτ (t11 ) has been built in the laboratory (see Fig. 4). A commercial
ln
uτ (t12 ) Li-ion polymer battery module from Kokam is investigated in
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HENTUNEN et al.: TIME-DOMAIN PARAMETER EXTRACTION METHOD FOR THÉVENIN-EQUIVALENT CIRCUIT BATTERY MODELS 5

Models are made with MATLAB/Simulink/Stateflow. The ECU

is connected to the host computer via high-speed link. All mea-
surements as well as other signals can be monitored online from
the host computer through dSPACE ControlDesk software.
All relevant data are logged with 50-ms sampling interval.
Data from ControlDesk and power analyzer are merged and
postprocessed afterward to make a unified data structure that
has all relevant information for battery characterization. For
characterization and model validation, the power analyzer is
considered to be more accurate and reliable source of data than
the others, and hence, the power analyzer is used as the source
of current, voltage, and ampere-hour data. Temperature data are
obtained from ControlDesk.
Prior to starting the characterization, the battery was charged
to its maximum voltage and the BMS balanced the cells to
be in equal SOC. After the balancing was finished, the BMS
was disabled, i.e., the balancing was set off and the BMS was
not able to control the charger, the load, nor the contactors.
Everything were controlled by the MABX and monitored from
the host computer. The experiment was stopped when the first
cell reached its cutoff voltage, i.e., 2.7 V.
Characterization of a battery module is more complicated
Fig. 4. Battery test setup. Also a power analyzer was used to measure battery than that of a single cell, because the cells are not identical but
terminal voltage, current, and ampere-hours. have slightly different capacities and resistances, and they may
evolve differently when aging. Due to capacity differences, the
capacity of a battery is determined by the cell with the lowest
TABLE I
SPECIFICATION OF THE BATTERY CELL AND MODULE capacity. If all cells are fully charged at the beginning of a
discharge test, only the weakest cell reaches the cutoff voltage.
Property Unit Cell Module
The average voltage curve does not reach the cutoff voltage.
Also the shape of the voltage curve may differ from the ideal
Nominal capacity Ah 40 40 curve, because near the end-of-test (EOT) condition the cells are
Nominal voltage V 3.7 25.9
Max voltage V 4.2 29.4
not in the same SOC. From a modeling perspective, this yields
Cutoff voltage V 2.7 18.9 to a situation where the shape of the OCV curve of a battery
Max charge current A 80 80 pack may differ slightly from that of a single cell.
Cont. discharge current A 200 200
Peak discharge current A 400 400
For high-fidelity modeling of a multicell battery, every cell
should be modeled separately [19], [23]. The model of a bat-
tery can then be formed by connecting the cell models in ap-
propriate series and parallel configuration [23]. However, this
the experiments. The battery consists of seven series-connected approach increases significantly the computational complexity
SLPB 100216216H lithium-ion polymer pouch-type cells. The of the model as well as parameter extraction effort, as the bat-
positive electrode material is lithium-nickel-manganese-cobalt- tery pack may consists of tens or hundreds battery cells in series
oxide and the negative electrode material is graphite. The spec- and parallel connections. For system-level simulations, simplic-
ification of the battery is shown in Table I. ity and light computational burden is often preferred. Hence,
Load current is made with a water-cooled programmable dc in this paper, the battery module was modeled as a single bat-
electronic load, model PLW12K-120-1200 from Amrel, which tery cell with the characteristics of the whole battery module.
has maximum current, voltage, and power ratings of 1 200 A, The resulting model characterizes the behavior of the module,
120 V, and 12 kW, respectively. A Powerfinn PAP3200 is used as but does not reveal accurate information about any of the cells.
a power supply during charging. The power supply can be used The module voltage can be divided by the number of cells to
as a controlled voltage or current source with output voltage area achieve the average value for cell voltage, OCV, and resistance
of 0–36 V and current area of 0–127 A. Its maximum output and capacitance values.
power is 3.2 kW at 24 V. A Hioki 3390 power analyzer with a
Hioki 9278 current clamp is used to measure current, voltage,
and ampere-hours. IV. RESULTS
All equipment except the power analyzer is controlled with All experiments were made at room temperature. Inherent
a dSPACE MicroAutoBox (MABX) DS 1401/1505/1507 rapid heating due to thermal losses was evident. The average temper-
control prototyping electronic control unit (ECU). Model-based ature (Tavg ) and temperature change (ΔT ) of the experiments
software development is utilized to produce code for the ECU. are presented in the end of this section together with the results
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6 IEEE TRANSACTIONS ON ENERGY CONVERSION

Fig. 5. Parallel RC circuit parameter extraction. Measured and estimated tran-

sient circuit voltage u τ during the rest time. The vertical gray line marks the
starting time of the second time window.

of the model validation. Unless otherwise stated, all temperature

and error measures are for the SOC range of 10–100%.

A. Model Extraction
The capacity of the battery was measured to be approximately
45 Ah. For simplicity, the aging effects and self-discharge were
ignored. A PD experiment with 1C rate, 10% pulse rate, and
30 min rest time was made to characterize the OCV and to
extract base values for the ohmic resistance and RC-network
parameters for discharge. Two RC circuits were used in the
EEC, and the data were divided accordingly into sections. For
simplicity, constant time windows were used: t11 = 0 s, t12 =
12 s, t21 = 240 s, and t22 = 600 s. However, below 10% SOC,
the first time window was time-shifted for 5 s and lengthened to
15 s, i.e. t11 = 5 s, t12 = 20 s, to ensure correct behavior. This
issue is further discussed later in this section. The resulting mean
values of the time constants were 22 and 570 s. The parameter
maps were upsampled by a factor of 10 with shape-preserving
piecewise cubic interpolation method. This resulted in smoother
parameter maps. Linear interpolation between data points were
used in the simulations.
Fig. 5 illustrates the extraction procedure with an example.
The figure shows the experimental transient circuit voltage data
as well as the transient circuit voltage estimate of a model with
two RC parallel branches during a rest time of 30 min. Also
the predicted voltage of each RC branch as well as the starting
time instant of the second time window are shown in the figure.
The subtraction of the battery voltage from the OCV removes
the final offset and inverts the voltage curve. Hence, the transient
circuit voltage always decays from the initial voltage towards
zero. It can be seen from the figure that the estimated voltage
characterized the real behavior very well and that the condition
for the separation of time windows was fulfilled.
After the PD test, a PC test was made to identify the param-
eters for charging. Then, PD and PC tests were made at 2C
rate as well as at 4C rate for discharging. Two experiments (1C
charge and 2C discharge) were made with 10-min rest time,
while the others were made with 30-min rest time. The same
time windows were used for all experiments. The resulting pa-
rameter mappings are presented in Fig. 6. No experiments at Fig. 6. Parameter mappings. (a) u o c . (b) R 0 . (c) R 1 . (d) R 2 . (e) C 1 . (f) C 2 .
very low rates were done, because due to low current even a
moderate error in parameter values causes only a minor error
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HENTUNEN et al.: TIME-DOMAIN PARAMETER EXTRACTION METHOD FOR THÉVENIN-EQUIVALENT CIRCUIT BATTERY MODELS 7

Fig. 9. PC experiment at 2C rate, 10% pulse rate, and 30 min rest time.
Fig. 7. PD experiment at 1C rate, 10% pulse rate, and 30 min rest time. (a) Measured and simulated voltage on the left axis and measured current on
(a) Measured and simulated voltage on the left axis and measured current on the right axis. (b) Voltage error.
the right axis. (b) Voltage error.

anymore characterize the real behavior, but a third RC circuit

would be needed to represent the second-scale time constant.
This second-scale resistance increases very rapidly and cap-
tures a high voltage. With only two RC branches, this resistance
should not be included in neither of the RC branches. This can be
ensured in the model extraction algorithm by shifting the time
windows at low SOC further from the beginning of the rest pe-
riod, i.e., by selecting t11 to be something like 5 s, and if needed,
the other time instants time-shifted as well. It would be possible
to include the second-scale resistance partly or fully into the
ohmic resistance R0 . However, then, the ohmic resistance R0 of
the model would not represent a pure ohmic resistance anymore.
Therefore, for clarity, the second-scale resistance was ignored.
It can also be seen from the figure that the OCV at the EOT
is inaccurate, which is caused by the inaccuracies in the SOC
prediction and OCV mapping. Even small inaccuracies get visi-
Fig. 8. Part of a PD experiment at 1C rate, 10% pulse rate, and 30 min rest ble below 10% SOC, because of the high nonlinearity and steep
time. (a) Measured and simulated voltage. (b) Voltage error. MAPE = 0.07%,
RMSPE = 0.07%. slope of the OCV-curve.
The results of a PC experiment at 2C rate, PD experiment at
4C rate, and a constant-current discharge (CCD) experiment at
1C rate are shown in Figs. 9–11, respectively.
into the voltage prediction. It was also confirmed from the val-
idation experiments that the voltage error during low-current
periods remained very low, indicating that the parameter values B. Model Validation
are close to 1C-rate values. The model was validated using a measured power profile
The results of a PD experiment at 1C rate are shown in Figs. 7 of an underground mining load-haul-dump (LHD) loader [24].
and 8. The simulation results match very well with the experi- The power profile is shown in Fig. 12. The validation current
mental results. The mean absolute percentage error (MAPE) and profile has short-time current peaks with different amplitudes as
root mean square percentage error (RMSPE) were 0.10% and well as constant-current periods, and thus, is a good profile for
0.12%, respectively. The maximum absolute percentage error validation purposes in general.
(APE) was 0.4%. The average power of the cycle was 30 kW. A series-hybrid
Below 10% SOC, the error during the transient increased to topology with a battery in the dc link was considered. The
several percents, the maximum APE being 5.7%. The main rea- internal combustion engine (ICE) operates with constant power
son for such a high error lies in the rapidly slowing dynamics of of 25 kW, and thus, the power profile of the battery was obtained
a battery when approaching fully discharged state. This starts to by moving the power profile of Fig. 12 downwards 25 kW. Since
happen at SOC level of 10–20% [13]. Two RC circuits cannot the ICE power was less than the average power of the duty cycle,
 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

8 IEEE TRANSACTIONS ON ENERGY CONVERSION

Fig. 10. PD experiment at 4C rate, 10% pulse rate, and 30 min rest time.
(a) Measured and simulated voltage on the left axis and measured current on
the right axis. (b) Voltage error.
Fig. 13. Model validation, LHD cycle repeated until cell cutoff voltage is
reached. (a) Measured and simulated voltage on the left axis and measured
temperature on the right axis. (b) Voltage error.

Fig. 11. CCD experiment at 1C rate. Measured and simulated voltage on the
left axis and measured current on the right axis.

Fig. 12. Duty cycle of an underground mining LHD loader. Fig. 14. Model validation, approximately one LHD cycle in the middle of the
experiment. (a) Measured and simulated voltage on the left axis and measured
current on the right axis. (b) Voltage error. MAPE = 0.18%, RMSPE = 0.23%.

the considered cycle represented a charge depleting cycle with

approximately 2% decrement in SOC per cycle. The power temperature rose in the beginning and in the end and had a
was scaled down with a factor of 28 to represent the power plateau in the middle of the experiment, where the temperature
of a single battery module. The battery current was calculated stayed within 5 ◦ C for a length of over 50% of the full capacity.
in real-time in the MABX ECU based on the measured battery The results show that with a second-order model less than 0.3%
module voltage. The result of the validation experiment is shown MAPE and 0.4% RMSPE for voltage prediction at SOC range
in Fig. 13 and a part of the full test is shown in Fig. 14 to show of 10–100% was achieved despite the non-isothermal condi-
details during one cycle. tions. It can be deduced that the presented parameter extraction
The average temperature, temperature change, and error mea- method performs well. Temperature scheduling and better ther-
sures of the experiments are shown in Table II. Typically the mal stability would further increase the accuracy.
 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

HENTUNEN et al.: TIME-DOMAIN PARAMETER EXTRACTION METHOD FOR THÉVENIN-EQUIVALENT CIRCUIT BATTERY MODELS 9

TABLE II [13] S. Abu-Sharkh, and D. Doerffel, “Rapid test and non-linear model charac-
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PD 2C / 10 min 34 ◦ C 14 ◦ C 0.8% 0.18% 0.24% accurate battery model applicable for EV and HEV simulation,” in Proc.
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