Parameter Identification of the Lead-Acid Battery Model

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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).

978-1-4244-1641-7/08/ 25.00

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.



3/6

(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



5/6

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

978-1-4244-1641-7/08/ 25.00

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.



6/6

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

Vehicle Electric Power Simulator for Optimizing the

Electric Charging System", SAE, Warrendale, PA,

2000.

[5] Massimo Ceraolo, "New Dynamical Models of Lead

Acid Batteries", IEEE Transactions

on

Power

Systems,

VOL. 15,

NO.4, Novembre 2000.

[6] http://hyperphysics.phy-astr.gsu.edu/Hbase/electricl

leadacid.html

[7]

Stefano Barsali and Massimo Ceraolo, "Dynamical

Models of Lead-Acid Batteries: Implementation

Issues", IEEE Transactions on Energy Conversion,

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,

1992.

[9]

Michel

F.

de

Koning and

Andre Veltman, "modeling

battery efficiency with parallel branches",

35th

annual

IEEE

Power Electronics Specialists Conference,

2004.

[10]

Sabine Piller, Marion perrin

and

Andreas Jossen,

"Methods for state of charge determination

and

their

applications", Centre for solar Energy

and

Hydrogen

Research, Joumal of power sources

96,

2001.

[11] Robyn A.

Jackey,

"A

Simple, Effective Lead-Acid

Battery Modeling Process for Electrical System

Component Selection", 2007-01-0778, The

MathWorks,

Inc.

978-1-4244-1641-7/08/ 25.00

2008 IEEE

Parameter Identification of the Lead-Acid Battery Model

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