parameter identification of the lead-acid battery model

PARAMETER IDENTIFICATION OF THE LEAD-ACID BATTERY MODELNazih Moubayed 1 , Janine Kouta 1 , Ali EI-AIi 2 , Hala Dernayka 2 and Rachid Outbib 21 Department of Electrical EngineeringFaculty of Engineering 1 - Lebanese University - Lebanon2 Laboratory of Sciences in Information and Systems (LSIS)Aix-Marseille III University, Marseille - FranceABSTRACTThe lead-acid battery, although known since strong a longtime, are today even studied in an intensive way becauseof their economic interest bound to their use in theautomotive and the renewable energies sectors. In thispaper, the principle of the lead-acid battery is presented. Asimple, fast, and effective equivalent circuit modelstructure for lead-acid batteries was implemented. Theidentification of the parameters of the proposed lead-acidbattery model is treated. This battery model is validated bysimulation using the Matlab/Simulink Software.INTRODUCTIONLead-acid batteries, invented in 1859 by French physicistGaston Plante, are the oldest type of rechargeable battery.In 1880, Camille Faure finalizes a technique facilitating themanufacturing of the lead-acid battery. Since, the technicaldevelopment didn't stop progressing (properties of thealloys, 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 lowenergy-to-volume ratio, their ability to supply high surgecurrents means that the cells maintain a relatively largepower-to-weight ratio. In addition, the lead-acid batteriesare important thanks to the availability of the usedmaterials and the possibility of their recycling [2). Thesefeatures, along with their low cost, make them attractivefor use in cars, as they can provide the high currentrequired by automobile starter motors. They are also usedin vehicles such as forklifts, in which the low energy-toweightratio may in fact be considered a benefit since thebattery can be used as a counterweight. Large arrays oflead-acid cells are used as standby power sources fortelecommunications facilities, generating stations, andcomputer data centers. They are also used to power theelectric motors in diesel-electric (conventional) submarines[3). The lead-acid battery is also used for storage energywhich is delivered by a renewable energy system (solarenergy system, and/or wind energy system .... ) [4).Today, more of the third of the world production of lead areused by the manufacture of batteries (60% to 65% of themarket of the batteries concern the sale of lead-acidbatteries).Modelling and simulation are important for electricalsystem capacity determination and optimum componentselection. The battery model is a very important part of anelectrical system simulation, and this model needs to behigh-fidelity to achieve meaningful simulation results. Thispaper treats the case of the lead-acid battery. For it, anintroduction to lead-acid battery is presented. Themodelling of this battery is illustrated in two differentmodels. The parameter identification of the studied modelis also discussed. This identification is followed by avalidation of the treated model by simulation using theMatlab/Simulink software. Finally, a conclusion about theobtained results are presented and discussed.THE LEAD-ACID BATTERYA lead-acid battery is an electrical storage device thatuses a reversible chemical reaction to store energy. Ituses a combination of lead plates or grids and anelectrolyte consisting of a diluted sulphuric acid to convertelectrical energy into potential chemical energy and backagain [5). Each cell contains (in the charged state)electrodes of lead metal (Pb) and lead (IV) oxide (Pb02) inan electrolyte of about 37% wlw (5.99 Molar) sulfuric acid(H2S04). In the discharged state both electrodes tum intolead(lI) sulfate (PbS04) and the electrolyte loses itsdissolved sulfuric acid and becomes primarily water. Dueto the freezing-point depression of water, as the batterydischarges and the concentration of sulfuric aciddecreases, the electrolyte is more likely to freeze.Because of the open cells with liquid electrolyte in mostlead-acid batteries, overcharging with excessive chargingvoltages will generate oxygen and hydrogen gas byelectrolysis of water, forming an explosive mix. This shouldbe avoided. Caution must also be observed because ofthe extremely corrosive nature of sulfuric acid.Lead-acid batteries have lead plates for the twoelectrodes. Separators are used between the positive andnegative plates of a lead acid battery to preventshort-circuit through physical contact, mostly throughdendrites ('treeing'), but also through shedding of theactive material. Separators obstruct the flow of ionsbetween the plates and increase the internal resistance ofthe cell (Fig. 1).978-1-4244-1641-7/08/$25.00 ©2008 IEEEAuthorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:54 from IEEE Xplore. Restrictions apply.

tLI.&&~I- -('I Reec\a"";lhReacts IoI1th0 sulfuric ootd sulfate ;ons.c .c"'form 1 .... to form loade.e. s ulfato. Must sulfate. Pbs uppl y .lootronssupplios Iwend 10 Ionposlt1vepositive H 2SO 4 chergno end1 .... eloctrode""--- H2O 1. left ""---IIOQ8tiveFigure 1: Lead-acid battery [6].MODELING OF THE LEAD-ACID BATTERYThe lead-acid battery represents a fundamental and mainelement in the renewable energy systems and in thehybrid vehicles. Therefore, it is necessary to study themodeling of this type of batteries. In fact, very bigquantities of models exist, from the simplest, containingimpedance placed in series with a voltage source, to themost complex. In general, these models represent thebattery like an electric circuit composed of resistances,capacities and other elements, constant or variable(function of the temperature or the State Of Charge SOCthat gives an idea on the quantity of active substance)[7],[8].The simplified modelThe simplest model of a lead-acid battery is composed ofa voltage source placed in series with impedance (Fig. 2).,...Lr------Figure 2: Lead-acid battery simplest model.The main problem of this model is that the two elementsE(p) and Z(p) must be at least function of the State OfCharge (SOC) and of the battery's temperature e [9,10].The improvement of the simple model takes place whileadding a parasitic branch in parallel (Figure 3).I,... (C1SOC')Figure 3: Lead-acid battery general model.In fact, the parasitic branch represents the irreversiblereactions that take place in the battery as for example theelectrolysis of water that occurs at the end of the chargingprocess, especially in the case of overcharge. In thisbranch an Ip current circulates. The charge stocked in thebattery is only joined to 1m (current of the main branch, inamperes). A part of the total current I, which is the Ipcurrent, is a lost current and cannot be restored.The third order model [11]The model is consisted of two main parts: a main branchwhich approximated the battery dynamics under mostconditions, and a parasitic branch which accounted for thebattery behavior at the end of a charge. The main branchis formed of a R/C block placed in series with a resistance(Figure 4). All elements of figure 4 are functions of theState Of Charge (SOC), the charging/discharging currentsand the temperature of the electrolyte 9.EmNFigure 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.RO,...L+Main branch voltage (Em)Equation 1 approximated the internal electro-motive force(emf), or open-circuit voltage of one cell. The emf valuewas assumed to be constant when the battery was fullycharged. The emf varied with temperature and state ofcharge (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 R1Equation 2 approximated a resistance in the main branchof the battery. The resistance varied with depth of charge,a measure of the battery's charge adjusted for thedischarge current. The resistance increased exponentiallyas the battery became exhausted during a discharge.v978-1-4244-1641-7/08/$25.00 ©2008 IEEEAuthorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:54 from IEEE Xplore. Restrictions apply.

where:• R1 was a main branch resistance in Ohms,• R10 was a constant in Ohms,• DOC was battery depth of charge.(2)Main branch capacitance C1Equation 3 approximated a capacitance (or time delay) inthe main branch. The time constant modeled a voltagedelay when battery current changed.C 1 =l (3)RIwhere:• 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 R2Equation 4 approximated a main branch resistance. Theresistance increased exponentially as the battery state ofcharge increased.The resistance also varied with the current flowing throughthe main branch. The resistance primarily affected thebattery during charging. The resistance became relativelyinsignificant for discharge currents:Note that while the constant Gpo was measured in units ofseconds, the magnitude of Gpo was very small, on theorder of 10-12 seconds.I p =V PN. G po·exp + p (6)VpoSf[VPN /(t p .s+l) A (1-~)lwhere:• 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 definitionsExtracted charge QeEquation 7 tracked the amount of charge extracted fromthe battery. The charge extracted from the battery was asimple integration of the current flowing into or out of themain branch. The initial value of extracted charge wasnecessary for simulation purposes.tQe(t) = Qe_init + f-Im(t).dt(7)owhere:• 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.(4)Total capacity CEquation 8 approximated the capacity of the battery basedon discharge current and electrolyte temperature.However, the capacity dependence on current was only fordischarge. During charge, the discharge current was setequal to zero in Equation 8 for the purposes of calculatingtotal capacity.C(I,9) = K,.C,' , {l-~)'1+(Kc-l~I~) Sf(8)Terminal resistance ROEquation 5 approximated a resistanceseen at the batteryterminals. The resistance was assumed constant at alltemperatures, 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 chargeParasitic branch current IpEquation 6 approximated the parasitic loss current whichoccurred when the battery was being charged. The currentwas dependent on the electrolyte temperature and thevoltage at the parasitic branch. The current was very smallunder most conditions, except during charge at high SOC.where:• Kc was a constant,• Co* was the no-load capacity at O°C 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 afraction of available charge to the battery's total capacity.State of charge measured the fraction of charge remainingin the battery:SOC =1-Q eC(O,S)(9)978-1-4244-1641-7/08/$25.00 ©2008 IEEEAuthorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:54 from IEEE Xplore. Restrictions apply.

Depth of charge measured the fraction of usable chargeremaining, given the average discharge current. Largerdischarge currents caused the battery's charge to expiremore prematurely, thus DOC was always less than orequal to SOC.where:• SOC was battery state of charge,• DOC was battery depth of charge,• Qe 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 CurrentThe average battery current was estimated as follows inEquation 11.(10)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 modelSEquation 12 was modeled to estimate the change inelectrolyte temperature, due to intemal resistive lossesand due to ambient temperature. The thermal modelconsists of a first order differential equation, withparameters for thermal resistance and capacitance.Gpo, Vpo,Ap.- The capacitance parameters used in equation 8:Kc,Co,E,O.- The thermal parameters used in equation 12:Ca,Ra ·Main branch parameters identificationAll parameters are calculated experimentally through veryappropriate tests. The most adequate test is illustrated infigure 5."'0 J)J'oltopJ114Cummtr "'31Figure 5: Test serving in determining the parameters of themain branch of the third order lead-acid model.To identify Emo and KE, one needs two equations, theseequations are obtained while measuring the voltage in thebeginning and at the end of the test, Vo and V1 (they areequal to the emf at the beginning and at the end). For Thevalues of the load state, SOCbeginning and SOCend, they canbe known easily.It is sufficient one equation to identify R10. This equationwas 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.Where:• a was the battery's temperature in °c,• aa was the ambient temperature in °c,• aini! was the battery's initial temperatureto 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 1 Watts,• Ce was the thermal capacitance in Joules 1°C,• T was an integration time variable,• t was the simulation time in seconds.(12)in °c, assumedSame test is applied as for the emf parameters. Roo and Aoare identified while measuring the instantaneous dropvoltage following the application of the current I.Parasitic branch parameters identificationThe identification of the constants GpO, Vpo and Ap isobtained by making tests when the battery is completelyfull. In this case, 1m is supposed to be neglected and thetemperature of the electrolyte can be estimated from theambient temperature.PARAMETERS IDENTIFICATIONThe mentioned equations of the lead-acid third ordermodel contain constants that must be determinedexperimentally by tests in the laboratory. These constantsor parameters can be divided in four categories:- The main branch parameters used in equations 1 to 5:EmO,KE ,RIO,R20,A21' A22 ,Roo,Ao·- The parasitic branch parameters used in equation 6:Capacitance parameters identificationThis identification needs four equations. To do that, twomethods can be used. The first one is based on the datagiven by the manufacturer and the second one is based onthe experimental test.Thermal parameters identification978-1-4244-1641-7/08/$25.00 ©2008 IEEEAuthorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:54 from IEEE Xplore. Restrictions apply.

The proposed thermal model is very simple. It is formed ofthermal resistance Re and of thermal capacitance CeoThese two parameters are determined experimentally orare given by the manufacturers of batteries.It should be noted that, contrary to all others parameters,the thermal resistance depends on the site where thebattery is placed.SIMULATIONThe presented third order model of the lead-acid batteryusing its identified parameters is used in Matlab/Simulinksoftware in order to validate its functioning. The linearity ofthe model is due to the omission of the parasitic branch inthe general model.Charging stateWith regard to the discharging phase of the accumulator,several initial conditions are taken into consideration. Infact:- The accumulator is supposed to be completely charged,- The initial charge extracted is zero (Qe_init = 0),- The ambient temperature is supposed equal to 25°C,- The initial values of SOC and DOC are equal to 0.8.The phase of the discharge is presented in figure 7.oCum'".1': ~ .: :! :'.!': :I!'::::.... :::....: ::~.'.'.:::'.:::'.-""""""':""""""'["""·······················:············1··:::······:::····:::··:::····::i·25L-_-L-_---'-__-L-_--'-_---"'--_-'-_---'-_----'Voltage•. ,:::: :.i:I~:::::::·:: ~.!.. :::.::: :'.~To simplify the modeling of the chosen accumulator, thetemperature of the electrolyte is supposed equal to theambient temperature. In addition:- The accumulator is supposed to be empty,- The initial extracted charge is negligible (Qe_init = 0),- The ambient temperature is supposed equal to 25°C,- The initial values of the SOC and DOC are equal to 0.2.The model functioning in the charging state is illustrated infigure 6. In fact, before the beginning of this phenomenon,the current in the model was zero, the voltage is equal to1.95 V and the SOC is set to be 0.2. The charging of themodule of the studied accumulator takes place withconstant current equal to 20 A. The duration of thetransient state is about 5000 seconds. During this period,the voltage across the model terminals increases in alinear way as far as reaching its maximal value Erno whichis equal to 2.22 V. Same, the SOC increases linearly. Afterthe accumulator's charging, the voltage becomes equal to2.15 V and the SOC approaches to 0.8. This means thatthe accumulator will be able to continue charging as theSOC didn't reach the unity value.Figure 7: Battery discharging.In general, before the accumulator's connection with aload, the voltage across its terminals is equal to 2.15 V.When the load is placed, the accumulator begins toprovide current. This one is supposed constant. Theduration of this phase is supposed to be equal to 5000seconds. During this period, the voltage across the modelterminal decreases in a linear way as far as reaching itsminimal value. In the same way, the SOC decreaseslinearly. After the accumulator's discharge, the voltagebecomes equal to 1.95 V and the SOC approaches to 0.2.CONCLUSIONThe electric lead-acid batteries are devices that providethe electric energy from chemical one. These are electrochemicalgenerators. They store the energy that theyrestore according to the needs. They can be rechargedwhen one reverses the chemical reaction; it is whatdifferentiates them from the electric batteries.Discharging stateFigure 6: Battery chargingThese accumulators are used in several applications, forexample, they serve to supply electrically the cars, theheavy weights, the planes, etc.. One uses them likestationary batteries, assuring the lighting and the workingof the embarked devices.Seen their interests in the daily life, the electric lead-acidbatteries are studied in this paper. The principle of workingand the battery's modeling are discussed.978-1-4244-1641-7/08/$25.00 ©2008 IEEEAuthorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:54 from IEEE Xplore. Restrictions apply.

Several lead-acid battery models are conceived, forexample, the mathematical model and the parallel branchmodel. But the third order model is the simplest one toidentify.As conclusion, all parameters of this model, which isstudied in this paper, can be identified by laboratory testsor taken from the manufacturer's data. The third ordermodel of the lead-acid has been validated by simulation onthe software Matlab/Simulink.REFERENCES[1] D. Linden et T. B. Reddy, "Handbook of Batteries", 3rdedition, McGraw-Hili, New York, NY, 2001.[2] Ceraolo, "New Dynamical Models of Lead-AcidBatteries", IEEE Transactions on Power Systems, vol.15, No.4, IEEE, November 2000.[3] Robyn A. Jackey, "A Simple, Effective Lead-AcidBattery Modeling Process for Electrical SystemComponent Selection", The MathWorks, Inc., Janvier2007.[4] Wootaik Lee, Hyunjin Park, Myoungho Sunwoo,Byoungsoo Kim and Dongho Kim. "Development of aVehicle Electric Power Simulator for Optimizing theElectric Charging System", SAE, Warrendale, PA,2000.[5] Massimo Ceraolo, "New Dynamical Models of Lead­Acid Batteries", IEEE Transactions on PowerSystems, VOL. 15, NO.4, Novembre 2000.[6] http://hyperphysics.phy-astr.gsu.edu/Hbase/electriclleadacid.html[7] Stefano Barsali and Massimo Ceraolo, "DynamicalModels of Lead-Acid Batteries: ImplementationIssues", IEEE Transactions on Energy Conversion,VOL. 17, NO.1, Mars 2002.[8] Ziyad M. Salameh, Margaret,A. Casacca William andA. Lynch, "A Mathematical Model for Lead-AcidBatteries", Departement of Electrical Engineering,University of Lowell, 1992.[9] Michel F. de Koning and Andre Veltman, "modelingbattery efficiency with parallel branches", 35th annualIEEE Power Electronics Specialists Conference,2004.[10] Sabine Piller, Marion perrin and Andreas Jossen,"Methods for state of charge determination and theirapplications", Centre for solar Energy and HydrogenResearch, Joumal of power sources 96, 2001.[11] Robyn A. Jackey, "A Simple, Effective Lead-AcidBattery Modeling Process for Electrical SystemComponent Selection", 2007-01-0778, TheMathWorks, Inc.978-1-4244-1641-7/08/$25.00 ©2008 IEEEAuthorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:54 from IEEE Xplore. Restrictions apply.