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PARAMETER IDENTIFICATION OF THE LEAD-ACID BATTERY MODEL
Nazih Moubayed
1
,
Janine Kouta
1
,
Ali EI-AIi
2
,
Hala Dernayka
2
and Rachid Outbib
2
1 Department
of
Electrical Engineering
Faculty
of
Engineering 1 - Lebanese University - Lebanon
2
Laboratory
of
Sciences in Information and Systems (LSIS)
Aix-Marseille III University, Marseille - France
ABSTRACT
The lead-acid battery, although known since strong a long
time, are today even studied in an intensive way because
of
their economic interest bound to their use in the
automotive and the renewable energies sectors.
In
this
paper, the principle
of
the lead-acid battery is presented. A
simple, fast, and effective equivalent circuit model
structure for lead-acid batteries was implemented. The
identification
of
the parameters
of
the proposed lead-acid
battery model is treated. This battery model is validated by
simulation using the Matlab/Simulink Software.
INTRODUCTION
Lead-acid batteries, invented in 1859 by French physicist
Gaston Plante, are the oldest type of rechargeable battery.
In 1880, Camille Faure finalizes a technique facilitating the
manufacturing of the lead-acid battery. Since, the technical
development didn't stop progressing (properties of the
alloys, additives of the active matters, etc.)
[1).
Despite having the second lowest energy-to-weight ratio
(next to the nickel-iron battery) and a correspondingly low
energy-to-volume ratio, their ability to supply high surge
currents means that the cells maintain a relatively large
power-to-weight ratio.
In
addition, the lead-acid batteries
are important thanks to the availability
of
the used
materials and the possibility
of
their recycling
[2).
These
features, along with their low cost, make them attractive
for use in cars, as they can provide the high current
required by automobile starter motors. They are also used
in vehicles such as forklifts, in which the low energy-to
weight ratio may
in
fact be considered a benefit since the
battery can be used as a counterweight. Large arrays
of
lead-acid cells are used as standby power sources for
telecommunications facilities, generating stations, and
computer data centers. They are also used to power the
electric motors in diesel-electric (conventional) submarines
[3). The lead-acid battery is also used for storage energy
which is delivered by a renewable energy system (solar
energy system, and/or wind energy system .... ) [4).
Today, more
of
the third
of
the world production
of
lead are
used by the manufacture
of
batteries (60% to 65%
of
the
market
of
the batteries concern the sale
of
lead-acid
batteries).
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2008 IEEE
Modelling and simulation are important for electrical
system capacity determination and optimum component
selection. The battery model is a very important part
of
an
electrical system simulation, and this model needs to be
high-fidelity to achieve meaningful simulation results. This
paper treats the case
of
the lead-acid battery. For it, an
introduction to lead-acid battery is presented. The
modelling
of
this battery is illustrated in two different
models. The parameter identification
of
the studied model
is also discussed. This identification is followed by a
validation
of
the treated model by simulation using the
Matlab/Simulink software. Finally, a conclusion about the
obtained results are presented and discussed.
THE LEAD-ACID BATTERY
A lead-acid battery is an electrical storage device that
uses a reversible chemical reaction to store energy. It
uses a combination of lead plates or grids and an
electrolyte consisting of a diluted sulphuric acid to convert
electrical energy into potential chemical energy and back
again
[5).
Each cell contains (in the charged state)
electrodes of lead metal (Pb) and lead (IV) oxide (Pb02) in
an electrolyte of about 37% wlw (5.99 Molar) sulfuric acid
(H2S04).
In
the discharged state both electrodes tum into
lead(lI) sulfate (PbS04) and the electrolyte loses its
dissolved sulfuric acid and becomes primarily water. Due
to the freezing-point depression
of
water, as the battery
discharges and the concentration of sulfuric acid
decreases, the electrolyte is more likely to freeze.
Because
of
the open cells with liquid electrolyte in most
lead-acid batteries, overcharging with excessive charging
voltages will generate oxygen and hydrogen gas by
electrolysis
of
water, forming an explosive mix. This should
be avoided. Caution must also be observed because
of
the extremely corrosive nature
of
sulfuric acid.
Lead-acid batteries have lead plates for the two
electrodes. Separators are used between the positive and
negative plates
of
a lead acid battery to prevent
short-circuit through physical contact, mostly through
dendrites ('treeing'), but also through shedding of the
active material. Separators obstruct the flow
of
ions
between the plates and increase the internal resistance
of
the cell (Fig.
1).
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&&
tL
I
.
I-
-
( I
Reec\a ;lh
Reacts
IoI1th
0
sulfuric ootd
sulfate ;ons
.c
.c
'form 1 ...
to form
load
e.
e.
s
ulfato.
Must
sulfate. Pb
s uppl y lootrons
supplios
Iw
end
1
Ion
H
2
SO
4
poslt1ve
positive
chergno
end
H
2
O
1
...
eloctrode
-- -
1. left
---
IIOQ8tive
Figure 1: Lead-acid battery [6].
MODELING OF THE LEAD-ACID BATTERY
The lead-acid battery represents a fundamental and main
element in the renewable energy systems and in the
hybrid vehicles. Therefore, it is necessary to study the
modeling of this type of batteries.
In
fact, very big
quantities of models exist, from the simplest, containing
impedance placed in series with a voltage source, to the
most complex. In general, these models represent the
battery like an electric circuit composed of resistances,
capacities and other elements, constant or variable
(function of the temperature or the State Of Charge SOC
that gives an idea on the quantity of active substance)
[7],[8].
The
simplified
model
The simplest model of a lead-acid battery is composed of
a voltage source placed
in
series with impedance (Fig. 2).
,...L
r------
Figure
2:
Lead-acid battery simplest model.
The main problem of this model is that the two elements
E(p) and Z(p) must be at least function of the State Of
Charge (SOC) and of the battery's temperature e [9,10].
The improvement of the simple model takes place while
adding a parasitic branch in parallel (Figure 3).
I,.
.. C1SOC )
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2008 IEEE
Figure 3: Lead-acid battery general model.
In
fact, the parasitic branch represents the irreversible
reactions that take place in the battery as for example the
electrolysis of water that occurs at the end of the charging
process, especially in the case of overcharge. In this
branch an Ip current circulates. The charge stocked in the
battery is only joined to
1m
(current of the main branch, in
amperes). A part
of
the total current
I,
which is the
Ip
current, is a lost current and cannot be restored.
The third order
model
[11]
The model is consisted of two main parts: a main branch
which approximated the battery dynamics under most
conditions, and a parasitic branch which accounted for the
battery behavior at the end of a charge. The main branch
is formed of a R/C block placed in series with a resistance
(Figure 4). All elements of figure 4 are functions of the
State
Of
Charge (SOC), the charging/discharging currents
and the temperature of the electrolyte 9.
RO
,...L
+
Em
v
N
Figure 4: Lead-acid battery third order model.
where:
Em was the main branch voltage,
R1
was the main branch resistance,
C1
was the main branch capacitance,
R2 was the main branch resistance,
I
01pn)
was the Parasitic branch current,
Ro was the Terminal resistance.
Main branch voltage (Em)
Equation 1 approximated the internal electro-motive force
(emf), or open-circuit voltage of one cell. The emf value
was assumed to be constant when the battery was fully
charged. The emf varied with temperature and state of
charge (SOC):
Em
=EmO
-
KE
.(273 + 9)(1- SOC) (1)
where:
Em was the open-circuit voltage (EMF) in volts,
Emo
was the open-circuit voltage at full charge in volts,
KE was a constant in volts 1
DC,
9 was electrolyte temperature in
DC,
SOC was battery state of charge.
Main branch resistance R1
Equation 2 approximated a resistance in the main branch
of the battery. The resistance varied with depth of charge,
a measure of the battery's charge adjusted for the
discharge current. The resistance increased exponentially
as the battery became exhausted during a discharge.
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(2)
where:
R1 was a main branch resistance in Ohms,
R10
was a constant in Ohms,
DOC was battery depth of charge.
Main branch capacitance
C1
Equation 3 approximated a capacitance (or time delay) in
the main branch. The time constant modeled a voltage
delay when battery current changed.
C
1
=l (3)
RI
where:
C1 was a main branch capacitance in Farads,
T1 was a main branch time constant in seconds,
R1
was a main branch resistance in Ohms.
Main branch resistance R2
Equation 4 approximated a main branch resistance. The
resistance increased exponentially as the battery state
of
charge increased.
The resistance also varied with the current flowing through
the main branch. The resistance primarily affected the
battery during charging. The resistance became relatively
insignificant for discharge currents:
(4)
where:
R2 was a main branch resistance in Ohms,
R20 was a constant in Ohms,
A21 was a constant,
A22 was a constant,
Em was the open-circuit voltage (EMF) in volts,
SOC was the battery state of charge,
1m was the main branch current in Amps,
1* was the nominal battery current in Amps.
Terminal resistance RO
Equation 5 approximated a resistance seen a t the battery
terminals. The resistance was assumed constant at all
temperatures, and varied with the state of charge:
Ro
=Roo [1
+
Ao(I-S0C)]
(5)
where:
Ro was a resistance in Ohms
Roo was the value of RO at SOC=1 in Ohms
Ao
was a constant
SOC was the battery state of charge
Parasitic branch current Ip
Equation 6 approximated the parasitic loss current which
occurred when the battery was being charged. The current
was dependent on the electrolyte temperature and the
voltage at the parasitic branch. The current was very small
under most conditions, except during charge at high SOC.
978-1-4244-1641-7/08/ 25.00
2008 IEEE
Note that while the constant
Gpo
was measured in units of
seconds, the magnitude of Gpo was very small, on the
order of 10-
12
seconds.
I
=V G [V
PN
/( t
p
.s+l)
A ( 1 - ~ ) l
PN. poexp + p (6)
Vpo Sf
where:
Ip
was the current loss in the parasitic branch,
VPN
was the voltage at the parasitic branch,
GpO was a constant in seconds,
Tp was a parasitic branch time constant in seconds,
Vpo was a constant in volts,
Ap was a constant,
8 was the electrolyte temperature in DC,
8t was the electrolyte freezing temperature in DC.
Some definitions
Extracted charge Qe
Equation 7 tracked the amount of charge extracted from
the battery. The charge extracted from the battery was a
simple integration of the current flowing into or out of the
main branch. The initial value of extracted charge was
necessary for simulation purposes.
t
Qe(t) =Qe_init
+
f-Im(t).dt
o
Total capacity
C
(7)
Equation 8 approximated the capacity of the battery based
on discharge current and electrolyte temperature.
However, the capacity dependence on current was only for
discharge. During charge, the discharge current was set
equal to zero in Equation 8 for the purposes of calculating
total capacity.
C(I,9)
= K,.C,' ,
{ l - ~ )
1 + ( K c - l ~ I ~ ) Sf
(8)
where:
Kc
was a constant,
Co*
was the no-load capacity at OC in Amp-seconds,
8 was the electrolyte temperature in
DC,
I was the discharge current in Amps,
I
was the nominal battery current in Amps,
and E were a constant.
State
Of
Charge
SOC) and
Depth
Of
Charge
DOC)
Equations 9 and 10 calculated the SOC and DOC as a
fraction of available charge to the battery's total capacity.
State of charge measured the fraction of charge remaining
in the battery:
SOC =1- Q
e
C(O,S)
(9)
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Depth of charge measured the fraction of usable charge
remaining, given the average discharge current. Larger
discharge currents caused the battery's charge to expire
more prematurely, thus DOC was always less than or
equal to SOC.
(10)
where:
SOC was battery state of charge,
DOC was battery depth of charge,
Q
e
was the battery's charge in Amp-seconds,
C was the battery's capacity in Amp-seconds,
a was the electrolyte temperature in c,
lavg was the mean discharge current in Amps.
Estimate
of
Average
urrent
The average battery current was estimated as follows in
Equation 11.
lavg
=
1m
(11)
(t
l
s+l)
where:
lavg
was the mean discharge current in Amps,
1m
was the main branch current in Amps,
T1 was a main branch time constant in seconds.
Thermal model
Equation 12 was modeled to estimate the change in
electrolyte temperature, due to intemal resistive losses
and due to ambient temperature. The thermal model
consists of a first order differential equation, with
parameters for thermal resistance and capacitance.
(12)
Where:
a was the battery's temperature in c,
aa was the ambient temperature in c,
aini was the battery's initial temperature in c, assumed
to be equal to the surrounding ambient temperature,
P
s
was the
12R
power loss of Ro and R2 in Watts,
Re
was the thermal resistance in c 1Watts,
Ce was the thermal capacitance in Joules 1C,
T was an integration time variable,
t was the simulation time in seconds.
PARAMETERS IDENTIFICATION
The mentioned equations of the lead-acid third order
model contain constants that must be determined
experimentally by tests in the laboratory. These constants
or parameters can be divided in four categories:
- The main branch parameters used in equations 1 to 5:
EmO,KE
,RIO,R20,A21'
A22 ,Roo,A
o
- The parasitic branch parameters used in equation 6:
978-1-4244-1641-7/08/ 25.00
2008 IEEE
Gpo, Vpo,Ap.
- The capacitance parameters used in equation 8:
Kc,Co,E,O.
- The thermal parameters used in equation 12:
Ca,R
a
Main branch parameters identification
All parameters are calculated experimentally through very
appropriate tests. The most adequate test is illustrated in
figure
5.
"'0 J)
J1
J'oltop
14
r
ummt
"'3
1
Figure 5: Test serving in determining the parameters
of
the
main branch
of
the third order lead-acid model.
To identify Emo and KE, one needs two equations, these
equations are obtained while measuring the voltage in the
beginning and at the end
of
the test,
Vo
and
V1
(they are
equal to the emf at the beginning and at the end). For The
values
of
the load state, SOCbeginning and
SOCend,
they can
be known easily.
It is sufficient one equation to identify R10. This equation
was obtained by making the following difference, (V1-V4),
which is due to the presence
of
the resistance R1.
The main branch resistance is neglected
R2.
Same test is applied as for the emf parameters. Roo and Ao
are identified while measuring the instantaneous drop
voltage following the application
of
the current I.
Parasitic branch parameters identifi cation
The identification of the constants GpO, Vpo and Ap is
obtained by making tests when the battery is completely
full. In this case, 1m is supposed to be neglected and the
temperature of the electrolyte can be estimated from the
ambient temperature.
Capacitance parameters identification
This identification needs four equations. To do that, two
methods can be used. The first one is based on the data
given by the manufacturer and the second one is based on
the experimental test.
Thermal parameters identificat ion
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The proposed thermal model is very simple. It is formed of
thermal resistance Re and
of
thermal capacitance Ceo
These two parameters are determined experimentally or
are given by the manufacturers
of
batteries.
It should be noted that, contrary to all others parameters,
the thermal resistance depends on the site where the
battery is placed.
SIMULATION
The presented third order model
of
the lead-acid battery
using its identified parameters is used
in
Matlab/Simulink
software in order to validate its functioning. The linearity
of
the model is due to the omission
of
the parasitic branch in
the general model.
Charging state
To simplify the modeling
of
the chosen accumulator, the
temperature
of
the electrolyte is supposed equal to the
ambient temperature.
In
addition:
- The accumulator is supposed to be empty,
- The initial extracted charge
is
negligible (Qe_init
=
),
- The ambient temperature is supposed equal to 25C,
- The initial values
of
the SOC and DOC are equal to 0.2.
The model functioning
in
the charging state is illustrated
in
figure
6. In
fact, before the beginning
of
this phenomenon,
the current
in
the model was zero, the voltage is equal to
1.95 V and the SOC is set to be 0.2. The charging
of
the
module of the studied accumulator takes place with
constant current equal to 20
A.
The duration
of
the
transient state
is
about 5000 seconds. During this period,
the voltage across the model terminals increases
in
a
linear way as far as reaching its maximal value Erno which
is equal to 2.22
V.
Same, the
SOC
increases linearly. After
the accumulator's charging, the voltage becomes equal to
2.15 V and the SOC approaches to 0.8. This means that
the accumulator will
be
able to continue charging as the
SOC didn't reach the unity value.
Figure 6: Battery charging
Discharging state
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2008 IEEE
With regard to the discharging phase
of
the accumulator,
several initial conditions are taken into consideration.
In
fact:
- The accumulator is supposed to be completely charged,
- The initial charge extracted is zero (Qe_init
=
),
- The ambient temperature is supposed equal to 25C,
- The initial values
of
SOC and DOC are equal to 0.8.
The phase
of
the discharge
is
presented
in
figure
7.
Cum'
o
.1 : .: : :'. ': :I ':::: ::: :, : : : : . i : I ~ : : : : : : : : :
. :::.:::
: : ~ . ' . ' . : : : ' . : : : ' . - ' : ' [ : 1 : : : : : : : : : : : : : : i
25L-_-L-_--- -
L-_--'-_--- '--_-'-_---'-_----'
Voltage
Figure 7: Battery discharging.
In
general, before the accumulator's connection with a
load, the voltage across its terminals is equal to 2.15
V.
When the load is placed, the accumulator begins to
provide current. This one is supposed constant. The
duration
of
this phase is supposed to be equal to 5000
seconds. During this period, the voltage across the model
terminal decreases in a linear way as far as reaching its
minimal value. In the same way, the SOC decreases
linearly. After the accumulator's discharge, the voltage
becomes equal to 1.95 V and the SOC approaches to 0.2.
CONCLUSION
The electric lead-acid batteries are devices that provide
the electric energy from chemical one. These are electro
chemical generators. They store the energy that they
restore according to the needs. They can
be
recharged
when one reverses the chemical reaction; it is what
differentiates them from the electric batteries.
These accumulators are used
in
several applications, for
example, they serve to supply electrically the cars, the
heavy weights, the planes, etc.. One uses them like
stationary batteries, assuring the lighting and the working
of
the embarked devices.
Seen their interests
in
the daily life, the electric lead-acid
batteries are studied in
this paper. The principle
of
working
and the battery's modeling are discussed.
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Several lead-acid battery models are conceived, for
example, the mathematical model
and
the parallel branch
model. But the third order model is the simplest one to
identify.
As
conclusion,
all
parameters of this model, which is
studied
in
this paper,
can be
identified
by
laboratory tests
or taken from the manufacturer's data. The third order
model of the lead-acid
has been
validated
by
simulation
on
the software Matlab/Simulink.
REFEREN ES
[1]
D.
Linden et
T.
B.
Reddy,
"Handbook of Batteries",
3rd
edition, McGraw-Hili, New York,
NY,
2001.
[2] Ceraolo, "New Dynamical Models of Lead-Acid
Batteries", IEEE Transactions
on
Power Systems,
vol.
15,
No.4,
IEEE,
November 2000.
[3]
Robyn
A. Jackey, "A Simple, Effective Lead-Acid
Battery Modeling Process for Electrical System
Component Selection", The MathWorks, Inc., Janvier
2007.
[4] Wootaik Lee, Hyunjin
Park,
Myoungho Sunwoo,
Byoungsoo Kim and Dongho Kim. "Development of a
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on
Power
Systems,
VOL. 15,
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[6] http://hyperphysics.phy-astr.gsu.edu/Hbase/electricl
leadacid.html
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Stefano Barsali and Massimo Ceraolo, "Dynamical
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VOL. 17,
NO.1, Mars 2002.
[8]
Ziyad
M.
Salameh, Margaret,A. Casacca William
and
A.
Lynch, "A Mathematical
Model
for Lead-Acid
Batteries", Departement of Electrical Engineering,
University of Lowell,
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[9]
Michel
F.
de
Koning and
Andre Veltman, "modeling
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annual
IEEE
Power Electronics Specialists Conference,
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[10]
Sabine Piller, Marion perrin
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"Methods for state of charge determination
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Jackey,
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978-1-4244-1641-7/08/ 25.00
2008 IEEE