HUNGARIAN JOURNAL OF
INDUSTRY AND CHEMISTRY

Vol. 45(1) pp. 67–71 (2017)

hjic.mk.uni-pannon.hu

DOI: 10.1515/hjic-2017-0010

INITIAL ELECTRICAL PARAMETER VALIDATION IN LEAD-ACID
BATTERY MODEL USED FOR STATE ESTIMATION

BENCE CSOMÓS, DÉNES FODOR,
*
AND GÁBOR KOHLRUSZ

Department of Automotive Mechatronics, Institute of Mechanical Engineering, University of
Pannonia, Egyetem u. 10., Veszprém, H-8200, HUNGARY

The paper presents a current impulse-based excitation method for lead-acid batteries in order to define the ini-
tial electrical parameters for model-based online estimators. The presented technique has the capability to track
the SoC (State of Charge) of a battery, however, it is not intended to be used for online SoC estimations. The
method is based on the battery’s electrical equivalent Randles’ model [1]. Load current impulse excitation was
applied to the battery clamps during discharge while the voltage and current was logged. Based on the Randles’
model, a model function and a fit function were implemented and used by exponential regression based on the
measured data. The diffusion-related non-linear characteristic of the battery was approximated by a capacitor-
like linear voltage function for speed and simplicity. The initial capacitance of this bulk capacitor was estimated
by linear regression on measurements recorded in the laboratory. Then, the RC parameters of the equivalent
battery model were derived from exponential regression on transients during each current impulse cycle. The
battery model with initial RC parameters is suitable for model-based online observers.

Keywords: Battery, SoC, Exponential regression, Randles’ model, Load current impulse

1. Introduction

In our daily lives, the number of mobile devices and
utilities that can operate without grid connections is
increasing. Even though lithium batteries possess better
performance properties and energy indicators, lead-acid
batteries are still cheaper, significantly present in
commercial applications and almost fully recyclable.
Therefore, any developments in lead-acid battery
systems are still of interest.

According to Ref. [1], several methods exist to
estimate a battery’s State of Charge (SoC) and State of
Health (SoH) but model-based prediction is the most
widespread because of its reliablity and robustness.
Model-based methods, as the name suggests, need a
valid, properly detailed electric battery model. The
Randles’ model as a standard battery model is very
popular in the contexts of lead-acid and lithium-ion
batteries because of its cost-effectiveness and the
similarities of both types. By similarity it is meant that
the same model can be reasonably used for the
parameter estimation of both battery types [2-3].

Some additions to the standard Randles’ model can
be made if more details in electrochemistry are required
such as diffusion in the bulk and porosity amongst
others.

The model requires values of initial resistance (R)
and capacitance (C). The more accurate the initial

*Correspondence: fodor@almos.vein.hu

parameters of the model, the faster and more reliable the
convergence of a model-based predictor to the actual
state, that is, the actual SoC.

The scope of the present work is to identify the
initial values of RC components (parameters) by
evaluating the voltage impulse responses excited by
load currents in the time domain.

2. Battery model for impulse excitation

In this paper, a standard Randles’ model [7-12] was
analyzed that consists of charge-transfer resistance, Rct,
battery serial resistance, Rs, double-layer capacitance,
Cdl, and bulk capacitance, Cb (Fig.1). The voltage
references of the capacitors and currents in Fig.1 were
set for discharge.

Figure 1. Randles’ battery model applied to a
discharging battery pack

Rct

Cdl Cb

Udl

ibi(t)

Uocv(t)

ict

idl

CSOMÓS, FODOR, AND KOHLRUSZ

Hungarian Journal of Industry and Chemistry

2.1. State-space model and model function

By neglecting the intermediate mathematical steps and
rearranging the state-space model, the system can be
written in the following form:

U
RC





















































dl
ctdl

0
1

(1)

iR
U

U
U

s10

ocv
b

dl

















 . (2)

By solving the output in Eq.(2) in terms of the time
domain using a current impulse as the system input, the
output function in Eq.(2) can be expressed by Eq.(3)
that can be considered as the model function of the
system. This form of the output equation serves as a
basis for creating a fit function on measured voltage
data and then, derive R and C values from the fit
parameters.

For clarity, each of the terms of the output
equation are grouped by alphabetical letters:

DBtAetU
t



)(ocv (3)

where tags A, B and C can be written according to

)(ctdl 0 tiRUA  (4)

ti
B 0b

)(
 (5)

)()( cts tiRRD  (6)

where Uocv is the battery’s open-circuit voltage,  is the
system’s time constant, t is the measurement time, and
i(t) is the impulse load current. Since the proposed
method is based on load and relaxation cycles that
follow each other during the analysis, Ub0 represents the
initial voltage of the bulk capacitor while Udl0 is the
initial voltage of the double-layer capacitor at the
beginning of each impulse cycle.

It should be noted that changes in current during
each impulse cycle can be neglected as a result of
working in the short time-constant region of the
discharge curve, thus the current can be considered as a
constant.

The battery model shown in Fig.1 is prepared for
short-time transient analysis. Even though the model
can be used and remains valid for modelling discharge
processes that last for several hours, that is, for long-
time transients, the accuracy of the model becomes poor
under such circumstances. The reason for this is that the
battery model presented excludes the diffusion effect
that can even be observed by the initial valley-like
voltage drop and later as a circle-like voltage response
on the long-time discharge voltage curve (Fig.2). Such
an exclusion was made because the scope of the current

work is to focus on short-time voltage responses to
avoid excessive measurement time intervals and
computational resources. Consequently, the battery
model was optimized for fast SoC detection by short-
time battery checks.

2.2. Determining the initial capacity of Cb

In Eq.(1), the Ub voltage represents the equilibrium
voltage of the battery, therefore, it is related to the
battery’s main charge and as a result its SoC. If the
battery is excited by small C/10 - C/30 currents, Ub and
Uocv can be considered equal. Instead of Ub, Uocv can be
measured at the battery terminals. Therefore, the
relationship between Uocv and SoC is important and can
be determined from laboratory OCV measurements.

A small discharge current was applied to the
battery terminals under controlled conditions until the
battery’s OCV reached the factory’s minimum voltage
threshold from a fully charged state. The voltage,
current and temperature were logged while the SoC
could be calculated by the simple Coulomb-counting
method during the process and saved in a lookup table.
Then, the lookup table could be used to determine a
discrete relationship betwen Uocv and SoC in itself. The
Uocv - SoC characterisation method can be extended by
the regression on lookup data in order to create a
continous Uocv - SoC relationship. A linear function,
such as a capacitor-like regression of Uocv - SoC
characteristics can be legitimate if the battery’s
excitation current is small, i.e. between C/10 and C/15
and is not discharged under 20-25% of SoC. This could
be the case when low-power devices are considered as
loads.

In the case of plain discharge, the SoC changes can
be basically tracked by the basis of the B term like in
Eq.(5) as conducted in the Coulomb-counting method
[1]. In the presented battery model, the B term provides
information on the long-term state of the battery and
requires initial parameters such as Cb and Ub0. The
former represents the battery’s initial capacity, the latter
is related to the battery’s initial voltage at the beginning
of each impulse cycle. Right before the first current
impulse, Ub0 is equal to the battery’s equilibrium
voltage since a relaxation time of between 30 minutes
and one hour is sufficient for chemical processes to
decay.

Figure 2. Discharge characteristics of a 15Ah AGM
battery excited by different load currents

INITIAL ELECTRICAL PARAMETER VALIDATION IN LEAD-ACID BATTERY MODEL

45(1) pp. 67–71 (2017)

The initial value of Cb is crucial because it has a
strong influence on both the initial voltage drop and the
gradient of the long-time discharge voltage curve. In the
proposed model, a simplified, that is, capacitor-like
formula was used for initial battery capacity estimation
so it approximates the battery non-linear discharge
characteristic linearly. The introduction of a regression
error of a few percent, however, can lead to an easier
and faster determination of the initial capacity Cb.

The calculation of the battery’s initial capacity was
realized according to [4]. The fully charged battery at
room temperature needed to be slowly discharged by a
C/15 current until its OCV voltage reached the factory
recommended minimum voltage threshold. Once the
discharge had finished, 2 hours of relaxation had to be
observed in order to return the battery back to its almost
equilibrium state. Then, a C/15 slow charge had to
proceed until the battery’s OCV voltage reached the
factory recommended maximum voltage threshold. In
both cases, the battery’s OCV voltage, current and
temperature were recorded (Fig.3/a).

After averaging the discharge and charge-voltage
measurements, linear regression was conducted on it
according to

0bb
b

15/C|
)(

1
)( UtQ

C
tU ocv  (7)

where Qb(t) can be estimated by Qb(t) = i0 t.
Eq.(7) can be identified by the standard form of

the linear curve y=mx+b. In Eq.(7), Cb is the battery’s
initial capacity, Qb(t) is the actual charge of the battery
and Ub0 is the actual voltage of Cb at the beginning of
the impulses. By rearranging Eq.(7), Cb can be
determined. Even though charge/discharge currents and
SoC levels are constrained, the ambient temperature
should be controlled as well to provide a constant
temperature during the test periods. The value of Cb was
calculated as 37766 F at 22°C using C/15 load currents.

3. Exponential regression to derive RC
model parameters

The evaluation of measurement data and the
comparision of measurements and simulations were
realized using Matlab. The RC parameters in the model
shown in Fig.1 and later in an OrCAD circuit were
derived by an exponential regression using an
appropriate fit function on the measured voltage data.
The regression error between the measured and
modelled characteristics can be minimized if the fit
function follows the form of the model function, that is,
both of them implement similar dynamics and the
physical background of the inspected system. Therefore,
the fit function can be written as

BeAtU
t

ˆˆ)( ˆocv 




(8)

in a form that is similar to Eq.(3). The hat sign means
that the form of Eq.(8) is similar to Eq.(3), but uses a
different reference system.

When using Eq.(8) attention must be paid to the
determination of RC parameters. Eq.(8) gives voltage
references with respect to the ground, that is the x-axis,
while Eq.(3) yields the references Udl and Us which
correspond to Ub. References used by Eqs.(3) and (8)
must be matched to derive correct RC parameters
(Fig.3/b).

It can be seen that Eq.(8) neglects the linear term
from Eq.(3). Since the effect of continuous slow
discharge of the battery, which is linked to the Cb bulk
capacitance in the battery model, possesses a time-
constant several orders of magnitude higher than the
Cdl-Rct subsystem, it cannot be observed during the short
impulse cycles. However, it should be considered
during the whole discharge process so Eqs.(3) and (8)
need to be used together to estimate the SoC during
long-term discharge processes.

According to Refs. [5-6], it is practical to evaluate
only the discharge component of the voltage responses
during every load cycle. This method is also supported
by the fact that the dynamic behaviour of the battery for
both load and relaxation states can be described by the
same battery model since both responses originate from
the same battery structure. Because load curves are only
needed, the measured voltage should be separated from
global data and then, concatenated right after each other.
Since voltage data has been prepared in this way,

Figure 3. (a) Linear regression on the average of C/15
discharge and charge voltage data to derive initial
value of Cb. Initial cut-off has been filtered.
(b) References of model function and fit function

CSOMÓS, FODOR, AND KOHLRUSZ

Hungarian Journal of Industry and Chemistry

regression can be made during every discharge impulse
cycle simultaneously.

The relaxation time during each impulse was set to
provide enough time for the battery’s double-layer
capacitance to almost fully discharge. It is practical
because it simplifies Eq.(8) by reducing the effect of the
Udl0 term in Eq.(4).

The load time was set taking into consideration the
time constant of the Cdl - Rct subsystem. According to
experiments, it is within the range of 1 - 4s. The

calculation method of the term B̂ in (8) is based on the
calculation of the limit of the exponential term in
Eq.(8). Practical experiments have proven that
maintaining a load cycle of 4 in length can speed up
and correct the regression.

The RC values change during discharge. Due to
offline voltage response evaluation, it is not possible to
optimise these parameters according to load conditions.
Therefore, the average RC values can be used for the
whole period of discharge though this introduces a
slight misalignment between the measured and
modelled voltage characteristics.

By applying Eq.(8) on the prepared measurement
data, the average RC parameters shown in Table 1 were
derived.

4. Model implementation and validation of
the impulse excitation method in OrCAD

The aim of the OrCAD implementation was to validate
the proposed battery model and the applicability of the
impulse excitation method for RC parameter
determination. The equivalent circuit model shown in
Fig.1 was transformed into an electric circuit. The
duration of the simulation was set to 1 hour and the time
step was equal to the sample time of the real
measurement, namely 100 ms.

4.1. Initialization of the OrCAD simulator and
RC elements

The initial voltages of the capacitors and the proper
directions of OrCAD elements should be set carefully.
OrCAD uses a reference system of its own that
influences the current references of each component.
This should be considered while placing an element in
the circuit editor and when comparing the electric
circuit references to the Randles’ model.

The initial voltages of the capacitors, Udl and Ub,
were set to describe the battery’s discharge process and

thus also follow the voltage references of the Randles’
model, shown in Fig.1. The values of initial capacitor
voltages were derived from the chemical background
and assumptions. Cdl can be considered fully discharged
through Rct after the battery had relaxed (no load) for 2
hours so its initial voltage, Udl, was set to 0. The bulk
capacitance of the battery, Cb, reflects the lengthy time
constant as well as diffusion-related processes and is
related to the main charge storage capability of the
battery. If the relaxation time is sufficiently long, the
battery reaches an equilibrium state and its OCV
becomes equal to Ub. An optimal relaxation time that is
sufficiently long was derived from preceding
measurements of discharge/charge cycles by applying
different load currents and relaxation times. Based on
experiments, a relaxation time of 2 hours was applied
and as a result the OCV could be considered equal to
Ub. Using these assumptions, the initial voltage, Ub, was
set to 12.7 V and Udl to 0 V.

The introduction of the load current as an impulse
excitation can be achieved through a switched power
resistor element which is setup in a similar way to the
real arrangement (Fig.3/b). The switching routine of the
OrCAD element was tuned in accordance with real
load-relaxation cycles that were applied to the real
testbench. The resistor sets the load current that
discharges the battery.

During this work, a load current of 3A was used
with a load of 5s and relaxation cycles that lasted 10s.
All of the initial values are summarized in Table 2.

4.2. Comparing the simulation with the battery
model

In Fig.4/a, the results of measured and simulated
voltage responses are shown. The validity of the model
was analyzed by the comparison of measured and
simulated voltages. According to the setup, the
comparison is performed within a time frame of 5,600 s.
The blue curve that represents the simulated data has a
longer tail than the red one because filtering needed to
be performed on the measured data to cut the
initialization process at the beginning of the testbench.

The zoomed-in segment shows the fitness of the
simulated voltage response. The difference between the
curves is relatively small, around 0.05 V to be precise.
This error occurred because average RC values were
used in the model instead of online tuned ones.

Table 2. Initial values of simulation parameters that
need to be set before a run

Simulation parameter Initial value

Simulation step time 100 ms

Simulation duration 1 hour

Udl0 0 V

Ub0 12.7 V

Relaxation time / cycles 10 s

Load time / cycles 5 s

Load current 3 A (with a 4 Ohm load)

Table 1. Derived RC values of the battery model from
regression

Derived parameter name Average value

Rct 0.032 Ohm

Cdl 92 F

Rs 0.056 Ohm

SoC after discharge 87.9 %

INITIAL ELECTRICAL PARAMETER VALIDATION IN LEAD-ACID BATTERY MODEL

45(1) pp. 67–71 (2017)

5. Conclusions

This work shows a current impulse-based excitation
method that can be used to either track SoC changes
during moderate discharge or find proper RC model
values for model-based algorithms. The method is
founded on an equivalent model-based approach that
can implement the dynamic behaviour of a battery
without using excessive chemical equations.

The technique uses offline analysis of a battery,
therefore, it is not able to reasonably track SoC changes
in real-time. This technique is not intended to estimate
the SoC of batteries by itself, indeed, it estimates good
set points for online estimators such as Kalman filters or
other model-based observers.

The advantages of the presented approach are its
rapid nature and simplicity while minimising the error
of SoC estimation. The disadvantages of this method are
its offline nature and related consequences.

This technique could potentially be applied to
embedded systems and commercially.

6. Acknowledgement

This research was supported by the EFOP-3.6.1-16-
2016-00015 project. The project was supported by the
Hungarian Government and co-financed by the
European Social Fund. The project was supported by
the European Union and co-financed by the European
Social Fund (EFOP- 3.6.2-16-2017-00002).

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Figure 4. (a) Validation of modelled (OrCAD) and
measured voltage data in Matlab environment
(b) Battery testbench used during the analyzis