Dynamic Model of Lead Acid Battery Charging

Md. Navid Akbar, Md. Maidul Islam, Asif Al Hye and S. Shahnawaz Ahmed

Department of Electrical and Electronic Engineering, Bangladesh University of Engineering and Technology
Dhaka 1000, Bangladesh

akbar.navid@hotmail.com

Abstract— In smart grid environment, batteries charged fromMatlab/Simulink

grids which is difficult to implement. In this
during off peak hours for use either in electric vehicles or as emergency
paper, an easy to implement dynamic model of battery
power source play an important role as Distributed Energy Resources charging with specified constant current has been developed
(DERs) at the demand side. It is important to estimate the support available
so that the OCV versus time and OCV versus SOC of a
from such a resource, for energy management during a blackout in the grid
battery charged from a grid system can be determined from
system. In this paper, a dynamic (time varying) model of charging with
constant current has been developed for lead acid batteries, which the derived mathematical expressions without requiring an
are the
most commonly used and the cheapest energy storage unit. Thisexperimental
model execution. The results obtained from the model
provides the open circuit voltage (OCV) at a given time and applied
the on a battery have been compared with a practical test
corresponding state of charge (SOC) without requiring the test data. of the
battery by an experimental setup. A few sets of data obtained from
practical tests of different sizes of batteries have been used to determine
The model can be implemented on the consumer side smart
model parameters. Then the model has been applied on another battery andso that estimated battery SOC and voltage can be sent to
meter
the output compared with its available test data for validation.
utility server in smart grid environment.
Index Terms— Smart grid, Battery charging, Dynamic
modelling, State of charge, Open circuit voltage. II. DEVELOPMENT OF CHARGING MODEL
A typical charging profile [7] of a lead acid battery at
constant current as in Fig. 1 can be divided into two sections.
I. INTRODUCTION
The first zone can be approximated as linear zone up to 70
Lead acid batteries are the most commonly used and the percent of the total charging time. The OCV in this phase
cheapest energy storage units. In smart grid environment, lead increases linearly with time. In the exponential zone, the rate
acid batteries play an important role as Distributed Energy of OCV increase is higher, which settles to a constant value at
Resources (DERs) at the consumer or demand side. Energy the end of charging. The linear zone has been represented by a
storing systems with lead acid batteries are gaining simple combination of a constant capacitor C b and internal
widespread application in transportation systems as well as in resistance Rb. The exponential zone includes a time dependent
the electric utility sector. capacitor Cexp and Rb .The charging curve suggests that Cexp (t)
should decrease with time at the beginning of the exponential
A variety of models with a varying degree of complexity zone, later increase and become constant at the end of
and accuracy currently exist that predict battery behavior. A charging.
Thevenin model of lead acid battery considering state of
charge (SOC) as a linear function of open circuit voltage
(OCV) has been developed in both [1] and [5]. However,
OCV was estimated using state space model or Kalman
filtering technique which needs parameters that may not be
easy to obtain for any battery system in general. A
mathematical approach to predict battery charging behavior
developed in MATLAB/SIMULINK platform [2] considers
memory effect showed by lead acid batteries. The model
presents Voc, Ibat and SOC graphs taking into account the
gassing effect, self-discharge, diffusion capacitance and
double layer capacitance. However, aassumptions of
capacitor values are not supported by experimental data. A
battery model [3] considering the charging period, discharging
period and the rest period is developed to determine the
battery voltage output. The battery parameters vary in the
different phases. But the model is not continuous i.e. valid for
every few minutes interval only and gives SOC from open
circuit voltage for a specific time interval. The work [4]
presents a method for battery parameter estimation using Fig.1: Typical charging profile of a nominal 12V battery
 The proposed constant current charging model of a lead Linear zone ends at t1. Exponential zone starts from here. By
acid battery is shown in Fig. 2. The zener diodes and switches similar method voltage across Cexp,
here are used for modelling purposes and they represent the I charge × t '
switching between the linear and the exponential zone. The V2 = (3)
switch in the S1 position at time t<t1 represents the linear Cexp (t ')
phase with a combination of Cb and Rb .When t≥t1, the circuit The total mathematical expression for Voc
switches to S2 position and Cexp is in effect. The linear zone
capacitance is replaced by a constant voltage and is in series I charge I charge × t '
with Cexp. At the end of charging, C exp is also replaced by Voc (t) = V0 + [t×u(t)-t ' . u(t ' ) ] + u(t ' )
Cb Cexp (t ')
voltage Vexp. The final battery voltage is the sum of initial open
(4)
circuit voltage (V0), linear zone voltage (VL) and the
(Here, t is in hours, t' = t-t1)
exponential zone voltage (Vexp).

B. Parameter Determination through OCV Analysis

Data from standard battery tests [6] were employed into
the battery for extracting its different models parameters. The
tests include OCV versus time and SOC versus time curves
for nominal 12V batteries of three different Ampere-hour (Ah)
ratings (40, 70, 100 Ah). Charging rates for all the batteries
Ah
are given by ( ) A. The OCV vs. time curves are as in Fig.
10
3.

(a)
Fig. 2: Battery equivalent circuit

Cb linear zone capacitance [F]

Cexp exponential zone capacitance [F]
Ibat battery current [A]
Icharging/Ichrg constant charging current [A]
Rb battery internal resistance [Ω]
t time in hour [h]
Vbat battery terminal voltage [V]
Voc OCV of battery [V]
Vo OCV of uncharged battery [V] (b)
VL linear zone voltage limiter [V]
Vexp exponential zone voltage limiter [V]

A. Mathematical Equation Formation

In the linear zone, internal resistance and a fixed capacitor
is considered. So current through Cb is I charge.
dV 1
I charge = Cb (1)
dt (c)
I charge × dt
dv 1 = Fig. 3: OCV vs. time for (a) 40, (b) 70 and (c) 100 Ah battery
Cb
Upon integration the voltage across Cb,
I charge × t
V1 = (2)
Cb
 ¿ When a battery is completely uncharged, its SOC is zero
Using the equation Icharge = CbdVoc (t )¿
dt , capaitance Cb percent. At the final and highest steady OCV, the SOC should
is estimated from the linear segments of the corresponding be 100%. Specific Gravity (SG) or Acid Density (AD) gives
charging profiles. an accurate and direct indication [8] of SOC against time.

By examining the different values of C b, it is found to be SOC = {1-(ADFull-ADMeasured)×7.1}×100% (8)

proportional to the respective Ah of the batteries. The SOC depends on the acid density between the two plates.
proportionality factor is found averaging all the three Thus, SOC is related to the difference of acid density at any
proportionality factors corresponding to the three profiles in time from that in fully charged condition as:
Fig. 3 to be 0.539. So, for any ‘X’ Ah battery:
SOC = a1×( ADFull-ADMeasured) + a0 (9)
Cb= X × 0.539 (5)
For the exponential segments of the battery charging AD versus time is therefore analogous to SOC versus time.
profiles, capacitance Cexp is estimated from a similar relation
¿
of Icharge= Cexp (t')dVoc (t )¿
dt .

Using the three sets of Voc (t) and time curves, a set of
discrete Cexp (t') values is calculated. Using these C exp (t')
values, the following solid-line graphs in Fig. 4 are obtained.
On the same axes, programmatic curve fitting of C exp (t') in
MATLAB gives the dashed-linegraphs.The curve fitting is
done with terms of the form:

Cexp (t') = A + B×t' + C×exp(-t') (6) (a) (b)

Fig.5: SOC vs. time for (a) 40 and (b) 70 Ah

The SOC versus time graphs can also be approximated as 2

straight lines as done in Fig. 5 for a 40 Ah and a 70 Ah
battery. These two regions again correspond to the linear and
exponential zones. From the approximating lines, a general
equation governing SOC versus time is found. For any ‘X’ Ah
battery:

SOC (t) = {(7.845e-3)[t×u(t)- t'×u(t')] + (32e-3)×u(t')} ×7.1 (9)

(a) (b)
III. EXPERIMENTAL VALIDATION
The equivalent circuit model of a lead acid battery and the
coefficients used in its associated equations were developed in
this paper using a set of practical data. This model can now
determine the characteristics of a different battery not used in
the determination process of model parameters.

Fig. 6 compares the OCV versus time curve obtained

through MATLAB simulation of the developed model with
the curve plotted from test data [6] of a 12V, 80Ah battery,
and shows close agreement.
(c)

Fig. 4: Cexp (t') vs. time for (a) 40, (b) 70 and (c) 100 Ah battery

Comparing the coefficients of Cexp (t') for the test batteries, it

can established that for any ‘X’ Ah Battery:

Cexp (t') = -0.1X + (0.063X)×t' + [-0.019X^2+2.64X

-43.20]×exp(-t') (7)

C. SOC versus Time Analysis

 REFERENCES
[1] Chiasson, John, and Baskar Vairamohana. "Estimating the state of
charge of a battery." Control Systems Technology, IEEE Transactions
on 13.3 (2005): 465-470
[2] Saiju, R., & Heier, S. (2008, April). “Performance analysis of lead acid
battery model for hybrid power system.” In Transmission and
Distribution Conference and Exposition, 2008. T&D. IEEE/PES (pp. 1-
6).
[3] Mischie, Septimiu, and Dan Stoiciu. "A new and improved model of a
lead acid battery.” Facta universitatis-series: Electronics and Energetics
20.2 (2007): 187-202
[4] Daowd, Mohamed, et al. "Battery Models Parameter Estimation based
on MATLAB/Simulink®." EVS-25 Shenzhen, China (2010).
[5] S. Pang, J. Farrell, J. Du, and M. Barth. “Battery state-of-charge
estima-tion,” in Proc. Amer. Control Conf., vol. 2, Jun. 2001, pp.
1644–1649
[6] Practical data obtained from battery charging tests, courtesy of Power
Electronics Lab, Bangladesh University of Engineering & Technology
(BUET), April 2014
Figure 6: Comparison of simulated Voc vs. t profile with experimentally [7] Battery data, courtesy of North West Energy Storage Company. URL
Ah as of 17th June, 2014:
obtained [6] test data using Ichrg =( ) = 8A http://www.scubaengineer.com/documents/
10 lead_acid_battery_charging_graphs.pdf
[8] S. Shahnawaz Ahmed: "Battery for Solar System" in Solar Home
System, a book published by GIZ GmbH, (a German Govt. concern for
The OCV versus SOC graph has also been simulated as in collaboration in international development activities), 2013, Dhaka,
pp.55-75
Fig. 7 for the same test battery.

Figure 7: Simulated Voc vs. SOC curve for the 80 Ah battery using
Ah
Ichrg = ( ) = 8A
10

IV. CONCLUSION
A dynamic (time dependent) and simple mathematical model
of a lead acid battery charging is derived and tested with
experimental data to validate the model. Mathematical
expressions for OCV and SOC as functions of time have been
formulated. The parameters of this model depend on the
charging current magnitude. The developed model will be
helpful in determining SOC of battery without testing it in an
experimental rig. The model can be implemented on the
consumer side smart meter, so that estimated battery SOC and
voltage can be sent to utility server in smart grid environment
for energy management using such batteries as DERS in the
event of blackout in the grid.

This work can be further extended to model battery

discharging. Charging data of the same battery under different
charging currents might also be used to enhance the model.