A Simple, Effective Lead-Acid Battery Modeling Process for Electrical System Component Selection _ SAE-2007!01!0778-Battery-Modeling-Process

7/28/2019 A Simple, Effective Lead-Acid Battery Modeling Process for Electrical System Component Selection _ SAE-2007!01!

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2007-01-0778

A Simple, Effective Lead-Acid Battery Modeling Process forElectrical System Component Selection

Robyn A. JackeyThe MathWorks, Inc

Copyright 2007 The MathWorks, Inc.

ABSTRACT

Electrical system capacity determination for conventionalvehicles can be expensive due to repetitive empiricalvehicle-level testing. Electrical system modeling andsimulation have been proposed to reduce the amount ofphysical testing necessary for component selection [1,

2].

To add value to electrical system component selection,the electrical system simulation models must regard theelectrical system as a whole [1]. Electrical systemsimulations are heavily dependent on the battery sub-model, which is the most complex component tosimulate. Methods for modeling the battery are typicallyunclear, difficult, time consuming, and expensive.

A simple, fast, and effective equivalent circuit modelstructure for lead-acid batteries was implemented tofacilitate the battery model part of the system model.

The equivalent circuit model has been described indetail. Additionally, tools and processes for estimatingthe battery parameters from laboratory data wereimplemented. After estimating parameters fromlaboratory data, the parameterized battery model wasused for electrical system simulation. The battery modelwas capable of providing accurate simulation results andvery fast simulation speed.

INTRODUCTION

Modeling and simulation are important for electricalsystem capacity determination and optimum component

selection. The battery sub-model is a very importantpart of an electrical system simulation, and the batterymodel needs to be high-fidelity to achieve meaningfulsimulation results. Current lead-acid battery models canbe expensive, difficult to parameterize, and timeconsuming to set up. In this paper, an alternative lead-acid battery system model has been proposed, whichprovided drive cycle simulation accuracy of batteryvoltage within 3.2%, and simulation speed of up to10,000 times real-time on a typical PC.

In Figure 1, a conventional design process is contrastedwith Model-Based Design for electrical systemcomponent selection. The conventional design processfor component selection, shown in Figure 1a, involves acostly, time-consuming, iterative process of building atest vehicle, evaluating performance, and then modifyingthe electrical system components. Using Model-Based

Design, Figure 1b, introduces additional steps that makethe overall design process more efficient. Model-BasedDesign requires only one or two iterations of modifyingthe test vehicle and re-verifying the electrical systemdesign.

Figure 1 [6]: Component Selection Processes

Build TestVehicle

FinalizeVehicle

EvaluateResults

1a. Conventional

Design

1b. Model-Based

Design

SpecifyElectrical

Architecture

ModifyTest

Vehicle

SpecifyElectrical

Architecture

TestComponents

DevelopSystem Model

Build TestVehicle

ValidateSystem Model(Vehicle Test)

Optimize Designthrough Simulation

Specify OptimalComponents

Modify TestVehicle

Verify SystemPerformance(Vehicle Test)

ValidateSystem Model(Vehicle Test)


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3

Cell Temp

2

SOC

1

Voltage

theta_a

Pstheta

Thermal Model

ns

Gain

Tau1

SOC

ImR2

Compute R3

DOC R1

Compute R1

SOC R0

Compute R0

Vpn

thetaIp

Compute Ip

SOC

thetaEm

Compute Em

theta

Im

DOC

SOC

Charge and Capacity

I

R1

C1

Em

R0

R2

Ip

V

Ps

Im

Vpn

Battery Circuit Equations

u y

1/(Taup.s+1)

2

Ambient

Temp

1

Current

Ps

Ps

Vpn/(Ts +1)Ip (A)

SOC

SOC

SOC

SOC

SOC

DOC

DOC

DOC

Tau1

Vpn

Vpn

Vpn

Im

Im

Im

Im

Im

Im

theta (C)

theta (C)

theta (C)

theta (C)

theta (C)

R0 (Ohm)

C1 (F)

R2 (Ohm)

R1 (Ohm)

Em (V)

V (V)

A parameterization method has also been proposed,with testing requirements of standard discharge andcharge curves. The battery model was used in electricalsystem simulations to study component sizing andselection for application to various vehicle electricalsystem configurations. The actual data collected duringthe battery modeling study were customer proprietary,and therefore were not included in this paper.

BATTERY MODEL

BATTERY MODEL STRUCTURE

A physical system lead-acid battery model was created1.

The battery model was designed to accept inputs forcurrent and ambient temperature, as shown in Figure 2.

The outputs were voltage, SOC, and electrolytetemperature.

1Modeled Using Simulink

Current

Ambient Temp

Voltage

SOC

Cell Temp

Figure 2 [6]: Battery Model

A diagram of the overall battery model structure is

shown in Figure 3, which contains three major parts: athermal model, a charge and capacity model, and anequivalent circuit model. The thermal model trackselectrolyte temperature and depends on thermaproperties and losses in the battery. The charge andcapacity model tracks the batterys state of charge(SOC), depth of charge remaining with respect todischarge current (DOC), and the batterys capacity

The charge and capacity model depends on temperatureand discharge current. The battery circuit equationsmodel simulates a battery equivalent circuit. Theequivalent circuit depends on battery current and severanonlinear circuit elements.

Figure 3 [6]: Simulink Model Structure


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EQUIVALENT CIRCUIT

The structure of the battery circuit equations in Figure 3was a simple nonlinear equivalent circuit [4], which isshown in Figure 4. The structure did not model theinternal chemistry of the lead-acid battery directly; theequivalent circuit empirically approximated the behaviorseen at the battery terminals. The structure consisted oftwo main parts: a main branch which approximated thebattery dynamics under most conditions, and a parasiticbranch which accounted for the battery behavior at theend of a charge.

P

N

1

2

1V

[ParasiticBranch]

[Main Branch]

Rp

R2

R1

R0

Ep

Em

C1

Figure 4 [6]: Battery Equivalent Circuit

The battery equivalent circuit represented one cell of thebattery. The output voltage was multiplied by six, thenumber of series cells, to model a 12 volt automotivebattery. In Figure 3, the number of series cells wasentered into the Gain block with parameter value ns.

The voltage multiplication by six assumed that each cellbehaved identically. Figure 4 shows the electrical circuitdiagram containing elements that were used to createthe battery circuit equations.

Each equivalent circuit element was based on nonlinearequations. The nonlinear equations includedparameters and states. The parameters of the equationswere dependent on empirically determined constants.

The states included electrolyte temperature, storedcharge, and circuit node voltages and currents. Theequations were as follows:

Main Branch Voltage

Equation 1 approximated the internal electro-motiveforce (emf), or open-circuit voltage of one cell. Thecomputation was performed inside the Compute Emblock in Figure 3. The emf value was assumed to beconstant when the battery was fully charged. The emfvaried with temperature and state of charge (SOC).

( )( )SOCKEE Emm += 12730 (1

where:

Em was the open-circuit voltage (EMF) in voltsEm0 was the open-circuit voltage at full charge in voltsKE was a constant in volts / C

was electrolyte temperature in CSOC was battery state of charge

Terminal Resistance

Equation 2 approximated a resistance seen at thebattery terminals, and it was calculated inside theCompute R0 block in Figure 3. The resistance wasassumed constant at all temperatures, and varied withstate of charge.

( )[ ]SOCARR += 11 0000 (2

where:

R0

was a resistance in OhmsR00 was the value of R0 at SOC=1 in Ohms

A0 was a constantSOC was battery state of charge

Main Branch Resistance 1

Equation 3 approximated a resistance in the mainbranch of the battery. The computation was performedinside the Compute R1 block in Figure 3. Theresistance varied with depth of charge, a measure of thebatterys charge adjusted for the discharge current. Theresistance increased exponentially as the batterybecame exhausted during a discharge.

( )DOCRR ln101 = (3

where:

R1 was a main branch resistance in OhmsR10 was a constant in OhmsDOC was battery depth of charge

Main Branch Capacitance 1

Equation 4 approximated a capacitance (or time delay

in the main branch. The computation was performedinside the Compute C1 block in Figure 3. The timeconstant modeled a voltage delay when battery currenchanged.

111 RC = (4

where:

C1 was a main branch capacitance in Farads

1 was a main branch time constant in secondsR1 was a main branch resistance in Ohms


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Main Branch Resistance 2

Equation 5 approximated a main branch resistance. Thecomputation was performed inside the Compute R2block in Figure 3. The resistance increasedexponentially as the battery state of charge increased.

The resistance also varied with the current flowing

through the main branch. The resistance primarilyaffected the battery during charging. The resistancebecame relatively insignificant for discharge currents.

( )[ ]( )+

=

IIA

SOCARR

m22

21202

exp1

1exp(5)

where:

R2 was a main branch resistance in OhmsR20 was a constant in Ohms

A21 was a constantA22 was a constantEm was the open-circuit voltage (EMF) in voltsSOC was battery state of chargeIm was the main branch current in AmpsI* was the a nominal battery current in Amps

Parasitic Branch Current

Equation 6 approximated the parasitic loss current whichoccurred when the battery was being charged. Thecomputation was performed inside the Compute Ipblock in Figure 3. The current was dependent on theelectrolyte temperature and the voltage at the parasiticbranch. The current was very small under most

conditions, except during charge at high SOC. Note thatwhile the constant Gpo was measured in units ofseconds, the magnitude of Gpo was very small, on theorder of 10

-12seconds.

( )

+

+=

f

p

P

pPN

pPNp AV

sVGVI

1

1exp

0

0(6)

where:

Ip was the current loss in the parasitic branchVPN was the voltage at the parasitic branchGp0 was a constant in seconds

p was a parasitic branch time constant in secondsVP0 was a constant in volts

Ap was a constant

was electrolyte temperature in Cfwas electrolyte freezing temperature in C

CHARGE AND CAPACITY

The Charge and Capacity block in Figure 3 tracked thebatterys capacity, state of charge, and depth of charge.Capacity measured the maximum amount of charge that

the battery could hold. State of charge (SOC) measuredthe ratio of the batterys available charge to its fulcapacity. Depth-of-charge (DOC) measured the fractionof the batterys charge to usable capacity, becauseusable capacity deceased with increasing dischargecurrent. The equations that tracked capacity, SOC, andDOC were as follows:

Extracted Charge

Equation 7 tracked the amount of charge extracted fromthe battery. The charge extracted from the battery wasa simple integration of the current flowing into or out othe main branch. The initial value of extracted chargewas necessary for simulation purposes.

( ) ( ) dIQtQ mt

initee += 0_ (7

where:

Qe was the extracted charge in Amp-secondsQe_init was the initial extracted charge in Amp-seconds

Im was the main branch current in Amps was an integration time variablet was the simulation time in seconds

Total Capacity

Equation 8 approximated the capacity of the batterybased on discharge current and electrolyte temperatureHowever, the capacity dependence on current was onlyfor discharge. During charge, the discharge current wasset equal to zero in Equation 8 for the purposes ocalculating total capacity.

Automotive batteries were tested throughout a largeambient temperature range. Lab data across the entiretested current range showed that battery capacity beganto diminish at temperatures above approximately 60C

The look-up table (LUT) variable Kt in Equation 8 wasused to empirically model the temperature dependenceof battery capacity.

( )( )( )

( )

LUTKIIK

KCKIC t

c

toc =+

= ,*11

, * (8

where:

Kc was a constantC0* was the no-load capacity at 0C in Amp-secondsKt was a temperature dependent look-up table

was electrolyte temperature in CI was the discharge current in AmpsI* was the a nominal battery current in Amps

was a constant


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State of Charge and Depth of Charge

Equation 9 calculated the SOC and DOC as a fraction ofavailable charge to the batterys total capacity. State ofcharge measured the fraction of charge remaining in thebattery. Depth of charge measured the fraction ofusable charge remaining, given the average dischargecurrent. Larger discharge currents caused the batteryscharge to expire more prematurely, thus DOC wasalways less than or equal to SOC.

( ) ( ) ,1,

,01

avg

ee

IC

QDOC

C

QSOC == (9)

where:

SOC was battery state of chargeDOC was battery depth of chargeQe was the batterys charge in Amp-secondsC was the batterys capacity in Amp-seconds

was electrolyte temperature in CIavg was the mean discharge current in Amps

Estimate of Average Current

The average battery current was estimated as follows inEquation 10.

( )11 +=

s

II mavg

(10)

where:

Iavg was the mean discharge current in AmpsIm was the main branch current in Amps

1 was a main branch time constant in seconds

THERMAL MODEL

Electrolyte Temperature

The Thermal Model block in Figure 3 tracked thebatterys electrolyte temperature. Equation 11 wasmodeled to estimate the change in electrolytetemperature, due to internal resistive losses and due toambient temperature. The thermal model consists of afirst order differential equation, with parameters for

thermal resistance and capacitance.

( )

( )

+=t

as

init dC

RP

t0

(11)

where:

was the batterys temperature in C

a was the ambient temperature in C

init was the batterys initial temperature in C, assumed to beequal to the surrounding ambient temperature

Ps was the I2R power loss of R0 and R2 in Watts

R was the thermal resistance in C / WattsC was the thermal capacitance in Joules / C was an integration time variablet was the simulation time in seconds

BATTERY PARAMETER IDENTIFICATION

Parameterization of a battery model, such as theprocedure proposed in [1], can be complex and difficult

The parameterization process in [1] requires difficultnonstandard test procedures. A more automatedapproach was studied in detail and implemented. Theautomated approach used an optimization routine toadjust the battery model parameters, using dischargeand charge test data.

DATA REQUIREMENTS

The parameterization method required standard lab tesdata. For discharge, full batteries were discharged aconstant currents and temperatures. For chargebatteries were charged at constant current, until theterminal voltage approached the gassing voltage [5] fothe battery. Then, the charges were continued aconstant voltage, until the batteries reached a fulcharge. Several typical currents and ambientemperatures were used for the testing procedure. Thequality and consistency of the lab data were veryimportant in achieving a good fit with the lab data. The

data showed some significant variability between testsbut the majority of the variability observed was at opencircuit (no current). An example is shown in Figure 5.

10.5

11

11.5

12

12.5

13

Voltage

Lead-Acid Battery Discharge Curves

0 5 10010

0

Current

Time (hours)

Figure 5 [6]: Variability of Measured DischargeCurves


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AUTOMATIC PARAMETER TUNING

Due to the variability observed in battery dischargecurves, a set of average discharge curves was used forthe parameter tuning process. The average curves werecreated by calculating the mean voltage of all dischargetests taken at a unique combination of operatingconditions (test runs at the same temperature anddischarge current). An example is shown in Figure 6.

0 0.5 1 1.5 2 2.510.5

11

11.5

12

12.5

13

Time (hours)

Voltage

Average of Battery Curves

AverageCurve

StopHere

Figure 6 [6]: Mean-Value Discharge Curves

The overall estimation process involved several steps:

1. Optimizing capacity model2. Optimizing discharge parameters3. Optimizing charge and discharge parameters

Step 1: Capacity Model Optimization

The capacity was calculated from discharge test data foreach tested temperature and discharge current. Themean-value discharge curves were used. Then, anoptimization routine

2was used to adjust the nonlinear

capacity parameters to best fit the capacity equations tothe measured mean capacities. These optimizedparameter values were the final capacity parameters.

Step 2: Discharge Optimization

Next, the battery parameters were optimized to minimize

the error between measured and simulated dischargecurves. Only parameters that affected the dischargesimulation were tuned

3. The tuned parameters were

given an initial value and min/max constraints.

The selection of the parameters and constraints foroptimization was not a trivial process. Determiningwhich parameters affected only the discharge cycleswas not immediately straightforward. The parameter

2Using Optimization Toolbox

3Using Simulink Parameter Estimation

selections were the result of a significant amount of triaand error testing. Only parameters that were welexercised in the discharge curve data could beoptimized, because of the risk of throwing off parameterswhich were dominant under other battery conditionssuch as charging.

The optimization routine also adjusted the initial SOC oeach discharge simulation, within reasonableconstraints. By changing initial SOC, the entiresimulated discharge curve effectively shifted down andto the left, or up and to the right as shown in Figure 7Shifting the initial SOC allowed the simulated dischargecurve to better align with the test data. Initial SOC was anecessary degree of freedom, because the batterieswere not consistently fully charged before the tesbegan.

0 1 2 3 4 510.5

11

11.5

12

12.5

13

13.5

Time (hours)

Voltage

Initial SOC Variation

SOCinit

=100%

SOCinit

=90%

Figure 7 [6]: Ini tial SOC Variation

The last 25% of the measured discharge data wasignored for the initial optimization. The data wasignored because a large voltage error would occur ithere was a slight error in the capacity, causing thesimulated battery to fully discharge and the voltage todrop off before the tested battery voltage did. Thevoltage error would throw off the optimized parameterssignificantly.

After the optimization, each of the results was plotted foanalysis. The curve fitting was considered to bereasonable. Some typical fitting variability undedifferent battery conditions is shown in Figure 8 and

Figure 9. Some variability was typical at the beginningand end of the discharge cycle, including differences inthe batterys capacity. Some of the differences wereattributed to using different physical batteries for eachlab test.


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0 2 4 6 811

11.5

12

12.5

13

13.5

Voltage

Battery Discharge Result

Time (hours)

SimulatedMeasured

Figure 8 [6]: Discharge Curve Example 1

0 2 4 6 811

11.5

12

12.5

13

Voltage

Battery Discharge Result

Time (hours)

SimulatedMeasured

Figure 9 [6]: Discharge Curve Example 2

Step 3: Combined Charge and Discharge Optimization

After the battery discharge parameters were fullyoptimized, then the charge curves were optimized.However, while some parameters primarily affected thebattery charging condition, other parameters affectedboth the discharge and charge. Consequently, bothdischarge and charge test data and parameters wereused in the final optimization. Simulated voltage andcurrent both needed to be optimized for charging curvesonly, because neither was held constant throughout the

entire tests. Weight coefficients were used to balancethe optimization of the current vs. voltage errors,because the raw current and voltage values had somemagnitude difference.

The results were reasonable with respect to the batterytest data. Many of the simulated and measured resultslined up very closely. Note that the discharge testresults did worsen slightly during the combined chargeand discharge optimization (step 3), but the curves werestill very close. Some of the worst-case resultsobserved are shown in Figure 10 and Figure 11.

0 1 2 3 4

12.5

13

13.5

14

Voltage

0 1 2 3 40

10

20

30

Current

Example Charge Tuning Result

0 1 2 3 40

0.5

1

Time (hours)

SOC/

DOC

Simulated

Measured

Simulated

Measured

SOC

DOC

Figure 10 [6]: Final Charge Curve - Worst CaseExample Observed

0 0.5 1 1.5 2 2.5 311

11.5

12

12.5

13

13.5

Voltage

Example Discharge Tuning Result

0.5 1 1.5 2 2.5 30

0.5

1

SOC/

DOC

Time (hours)

SOC

DOC

Simulated

Measured

Figure 11 [6]: Final Discharge Curve - Worst Case

Example Observed

Additional Tuning

After the major optimizations, the batterys thermamodel was adjusted manually. The thermal resistanceand capacitance were adjusted based on data from theapplication of the battery in a vehicle test. Theadjustment was made because the thermal propertiesdepend heavily on the installation of the battery in avehicle.


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The parameters R1 and 1 affected the end of dischargeslope and time constant. The parameters were tuned

4

with a few discharge curves that included the after-discharge settling time, similar to the curves shown inFigure 5. The parameters were not well-exercised inthe rest of the battery parameter estimation process, sothey had to be tuned separately. Once the parametervalues were determined, the entire fitting process wasre-run with the new values to ensure an optimal fit to thelab data.

USE IN ELECTRICAL SYSTEM SIMULATIONS

The final battery model block, shown in Figure 12, wasused within an electrical system simulation model.Multiple battery sizes were parameterized to facilitateelectrical system component selection. The batteryblock was used in drive cycle simulations with differentparameters for the different battery sizes.

2

SOC

1

Temp

Ambient Temp SOC

Cell Temp

+

-

Ceraolo Battery

Batt Harness

1

Env Temp

Figure 12 [6]: Battery Model for Electrical SystemSimulation

Speed and Accuracy

The battery model was capable of simulating one hour ofsimulation time in less than half a second of real time.

The simulation was performed on a typical PC using aRunge-Kutta (4, 5) variable step solver. Open-loop inputcurrent data was used at a 1 second sample time.

To validate the battery model, a test vehicle was runthrough several drive cycles to gather actual RPM,temperature, and battery current and voltage data tocompare to the simulation. An example is shown inFigure 13. The battery model was simulated usingRPM, temperature, and current inputs. On a 1-hourstop-and-go cycle, the accuracy of the simulated batterymodel voltage was within 3.2% (0.42 Volts) throughout.

On a 1-hour idling simulation with transmission in park,the voltage was within 1.2% (0.15 Volts).

4Using Simulink Parameter Estimation

850 900 950 1000 1050 1100 1150

12.2

12.4

12.6

12.8

13

13.2

13.4

time (s)

Voltage

Measured vs. Simulated Voltage

Measured

Simulated

Figure 13 [6]: Battery Simulation Voltage

FUTURE WORK

Battery modeling is a difficult and time-consuming task

Given additional time, many additions and changescould have been made to improve the resultsImprovements have a point of diminishing returns whenthe error experienced in the battery model becomessmaller than the variability experienced with reabatteries. However, there were a few improvements thacould have been investigated to further improve thebattery parameterization process.

Battery Aging and Typical Performance

Undesired aging effects can occur when test batteriesare fully charged and discharged repeatedly. In this

paper, a new battery was tested for only four dischargeand charge cycles, to minimize aging effects. Howeverthe possibility of instead obtaining parameterization datawith weaker, broken-in batteries should be studiedUsing slightly aged batteries might improve the batteryparameterization, allowing simulation of more typicaelectrical system performance.

Simpler Application to Other Batteries

Approximating parameters for a battery of the samechemistry but different capacity should be possiblewithout repeating the entire data collection and

optimization process. In this paper, two batterycapacities were parameterized, however, a direcrelationship could not be found between the optimizedparameters of each of the two batteries. The lack of adirect relationship between parameters of two batterieshaving the same chemistry but different capacity couldhave been caused by noise in the battery data or bysome parameters not being sensitive. The parameterelationship for different capacities would be a veryuseful topic for further study, because a capacityrelationship could provide a cost savings if a few knownadjustments could be made to the parameters to create


9/9

additional battery sizes without repeating the entire testprocess for each new battery capacity.

Measuring Battery Capacity

Battery capacity was very difficult to estimate correctly.One reason for the difficulty was battery variability.Another reason for the difficulty was that ensuring thebattery was fully charged before discharge testing wasnot easy. Fully charging the battery was more of anissue at higher temperatures where charging losses aresignificant, and thus achieving a full charge becomesdifficult. The charging difficulties at higher temperaturesshould be taken into consideration during the lab testingprocess. The batteries should be as completely chargedas possible before discharge tests begin.

CONCLUSION

A lead-acid battery model was developed, along withtools to parameterize the model from laboratory data.Construction of an equivalent circuit model has beendescribed. A semi-automated process was used toestimate parameters for the battery model fromlaboratory data. The completed battery model simulatedat approximately 10,000 times real-time. The accuracyof the simulated battery model voltage was within 3.2%in comparison to vehicle drive cycle measurements.

ACKNOWLEDGEMENTS

Special thanks to Andrew Bennett of The MathWorks forproviding the original battery model structure and forsupporting the model development. Thanks also to BoraEryilmaz of The MathWorks for supporting the batteryparameterization with Simulink Parameter Estimation,and to Peter Maloney of The MathWorks for modelingsupport and for reviewing this technical paper. Also,data and application feedback provided by Nick Chungand J K Min of Hyundai Motor Company wereinstrumental in the projects success.

REFERENCES

1. Schttle, R., W. Mller, H. Meyer, and E. SchochMethods for the Efficient Development andOptimization of Automotive Electrical Systems,970301, SAE, Warrendale, PA, 1997.

2. Wootaik Lee, Hyunjin Park, Myoungho SunwooByoungsoo Kim, Dongho Kim. Development of aVehicle Electric Power Simulator for Optimizing the

Electric Charging System, 2000-01-0451, SAEWarrendale, PA, 2000.

3. Ceraolo, Massimo and Stefano Barsali. DynamicaModels of Lead-Acid Batteries: ImplementationIssues, IEEE Transactions on Energy ConversionVol. 17, No. 1, IEEE, March 2002.

4. Ceraolo,. New Dynamical Models of Lead-AcidBatteries, IEEE Transactions on Power SystemsVol. 15, No. 4, IEEE, November 2000.

5. D. Linden and T. B. Reddy (editors), Handbook oBatteries, 3rd edition, McGraw-Hill, New York, NY2001.

6. The MathWorks, Inc. retains all copyrights in the

figures and excerpts of code provided in this article.These figures and excerpts of code are used withpermission from The MathWorks, Inc. All rightsreserved.

CONTACT

Robyn A. J ackeyTechnical Consultant, The MathWorks, Inc.39555 Orchard Hill Place, Suite 280Novi, MI 48375Email: [email protected]

*The MathWorks, Inc. retains all copyrights in the figures and excerpts ocode provided in this article. These figures and excerpts of code are used withpermission from The MathWorks, Inc. All rights reserved.

1994-2007 by The MathWorks, Inc.

MATLAB, Simulink, Stateflow, Handle Graphics, Real-Time Workshop, andxPC TargetBox are registered trademarks and SimBiology, SimEvents, andSimHydraulics are trademarks of The MathWorks, Inc. Other product or brandnames are trademarks or registered trademarks of their respective holders.

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A Simple, Effective Lead-Acid Battery Modeling Process for Electrical System Component Selection _ SAE-2007!01!0778-Battery-Modeling-Process