Electrical Battery Model for Use in Dynamic Electric Vehicle ...

Thevenin equivalents and impedances, or provide runtimemodels. Combinations are often used. Theveninmodels [11]-[13] assume the open-circuit voltage isconstant and use a series resistor and an RC parallelnetwork to track battery response to transient loads [8].An increase in the number of parallel RC networks canincrease the accuracy of the predicted battery response.However, between the SOC and time-constantdependence on cycle number and temperature, predictionerrors for estimating run time and SOC tend to be high.Impedance-based models, like Thevenin models, areaccurate only for a fixed SOC and temperature setting;hence their accuracy when predicting dc response andbattery runtime is limited [14].Impedance spectroscopy can be used to fit acomplicated equivalent network to measured impedancespectra in order to validate the time constants found in theThevenin models. Runtime-based electrical models usecontinuous or discrete-time implementations to simulatebattery runtime and dc voltage response in SPICEcompatiblesimulations for constant current discharges.Inaccuracy increases as the load currents vary [15].Combinations of these circuit models (in particular theThevenin and run-time models) can take advantage of thepositive attributes of each [8], [16], [17] such thataccurate SOC prediction, transient response, run-time,and temperature effects can be obtained.III. MODELThe model in [8] is capable of predicting run time and I-V performance for portable electronics, but is notaccurate for the transient response to short-duration loads(less than 1 s). As a result, it does not predict accuratelythe SOC throughout drive cycles for HEV simulations.Fast time constants of Li-ion batteries have been shownin [13], [16], [17] and are necessary to determine thelosses within a battery pack during vehicle drive cycles.Accurate determination of the discharge capacity, whichis a function of the discharge rate it (), temperaturef [ ( )] 2Tt , and cycle number f [ 3ncycle] [16], must alsoinclude a rate factor, f [()] it 1. The rate factor accountsfor a decrease in capacity due to unwanted side reactions[8], [18] as the current increases. The proposed model forpredicting SOC, terminal voltage, and power losses ofLi-ion, NiMH, and lead-acid batteries is shown in Figure1. It is governed by equations representing SOC and:V ter min alSOC[(), i t T(), t ncycle,] t SOCinitial f1[()]i tt02 3terminal bat int(1)f [ T( t)] f [ n ] i( t)dtcycleV Voc( SOC, T) i ( t) R ( SOC,T ) (2a)Rtransienti () t R ( , . )battransients m h1tR ( SOC) C ( SOC)s s ( R ( SOC) e (2b)s1tR ( SOC) C ( SOC)m mR ( SOC) e m1tR ( SOC) C ( SOC)h h R ( SOC) e )hEq. (1) for SOC can be modeled as the left circuit in Fig.1 when temperature and cycle number are given (e.g.short vehicle drive cycles at a low speed and grade).Here (1) is normalized to battery capacity, so V SOCliesbetween 0 and 100% and represents the SOC of thebattery [8]. I batteryis the battery load, modeled as acurrent source, while Rself dischargeand R cap _ fadeareparalleled to C capacityand represent normalized selfdischargeand battery capacity correction factor [16].V ter min alrelates to the parameters shown in the rightcircuit of Figure 1, a multiple time-constant approach( sec, min, and hour) for modeling the transientbehavior of the terminal voltage where each parameter isa function of the SOC [8]. This battery model has beenimplemented in the Simulink environment.IV. DATAMeasurements of the circuit model parameters of Fig. 1for different batteries are found using a battery testingapparatus which consists of data loggers, an electronicload, power supply, and other equipment. Eachcomponent is controlled via Labview 7.1, which is usedto program test sequences and analyze the recordedFigure 1 Proposed electrical battery model for use in HEV simulation at a constant temperature1337Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply.

V OCmeasurements. The open circuit voltage ( ) andterminal voltage versus SOC at room temperature wastested for five Panasonic CGR18650A 3.7 volt,2200mAh cylindrical Li-ion batteries using a constantcurrent discharge profile at several different charge rates.Each of the parameters in the model of Fig. 1 is anonlinear function of the various effects. For purposesof the model, each is represented as a polynomialfunction of SOC up to sixth order, with coefficientsgiven by:20 + 1* 2* ...Parameter a a SOC a SOC (3)The values in Table 1, when inserted into (3), formulatethe open-circuit voltage as the SOC changes. Theresulting expressions can be used within modelingprograms for efficient computation of the open-circuitvoltage.Estimates of min(seconds/minutes range), hour,and the corresponding resistive and capacitive circuitparameters are shown in [8] and [13]. Both the secandmintime constants have been measured here usingcharging and discharging currents at intervals of 5%SOC. The series resistance and resistive and capacitivecomponents used to model secand minare shown as afunction of SOC for discharging and charging currentswhen the values in Table 3 are inserted into equation 3.(D) and (C) in Table 3 denote time constants measuredduring discharging and charging current profiles,respectively. Determination of the longest timeconstant, hour, and its resistive and capacitivecomponents is a time-consuming process, hence thevalues found for discharging currents will be used tomodel hourfor charging currents as well. The measuredResistance (ohms)0.10.080.060.040.02R seriesR secondsR minutesR hours00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1SOCFigure 2 Resistance values versusSOC for alldischarging current resistive circuit parametersvalues for Rseries , Rt_s, Rt_m, and R t_hduringdischarge currents are shown in Figure 2. Thefunctions of the resistive components versus SOC areinterpolated values between the measured data points;model simulations may use either the interpolated valuesbetween the measured points (i.e. a look-up table) or thebest-fit polynomial functions from the data in Table 1.To check for time constants smaller than sec,which would be expected to have effects on dynamics ofvehicle power electronics, a buck converter switching at10 kHz was placed between a single PanasonicCGR18650A cell and a load. Figure 3 depicts themeasured battery voltage, current, converter output, andswitching function for a load of 1.5 . The waveformshows a voltage drop corresponding to the switchingfrequency, caused by the R seriescomponent of thebattery model. The voltage drops do not show evidenceof exponential behavior on this time scale. Hence timescales faster than secdo not impact the model.TABLE 1 CIRCUIT PARAMETER FUNCTION VALUES FOR THE OPEN-CIRCUIT VOLTAGE AND TIME CONSTANTSParameter a0a1a2a3a4a5a6Voc3 13.433 -90.038 284.33 -453.64 355.88 -108.97Rseries (D) 0.0482 0.144 -0.4577 0.4965 -0.1297 -0.049 0R (D) 0.1457 -1.0586 4.867 -10.131 9.5077 -3.3024 0t_sC (D) 0.7472 9.0675 -30.712 32.551 -7.358 -3.4621 0t_sR (D) 1.7423 -33.837 228.96 -712.75 1118.1 -859.2 -.12t_mC (D) 239.49 -14755 264236 -825025 947250 -371150 0t_mRseries (C) 0.051 0.2078 -1.1148 2.1656 -1.7766 0.5251 0R (C) 0.1134 0.9721 5.0929 -11.278 10.914 -3.845 0t_sC (C) 0.9936 14.521 -91.976 199 -179.67 -57.923 0t_sR (C) 1.6561 -35.02 246.62 -78559 1250.7 -970.92 292.72t_mC (C) 216.52 28308 -166348 431168 -491497 199631 0t_mR 0.1484 -1.1978 3.5946 -4.3618 1.8335 0 0t_hC 493923 3E+07 -1E+08 2E+08 -9E+07 0 0t_h1338Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply.

Open Circuit Voltage (volts)4.243.83.63.43.2362.5 degrees Celcius27 degrees Celcius3 degrees CelciusFigure 3 Battery voltage (500 mV/ div), load current(500 mA/ div), converter dc output (5 V/div), and f switch(10 kHz), respectively.Determination of the rate factor, f [()] it 1, is shownin Figure 3 for the 2C discharge rate. A reference curvefor the 2C discharge rate is shown as the dotted line inFigure 4, and represents V OCwith included batteryinternal voltage drops due to the internal resistance anddischarge rate, 2 Cref Voc Rint (i2 C iC/ 25).Variables a and b correspond to the SOC at which thevoltage reaches the minimum (specified bymanufacturer) for 2Crefand 2Cdischarge curves,respectively. The ratio of b to a is equal to f [()] 1it forthe corresponding discharge current [16] and must betaken into account to accurately predict SOC duringdrive cycles with large load transients. Tables 2 and 3include the measured values of the rate factors fordischarging and charging currents, respectively, at ratesof C/25, C/5, C/2, and C.Trials of the Li-ion battery open circuit voltage at atemperature of 3°C, 27°C, and 62.5°C are shown inFigure 5. The temperature coefficient ( f [ Tt ( )])can2be found the same way the rate factor ( f [()] it 1 ) wasVoltage [volts]4.44.243.83.63.43.2C/25 DischargeCurve withresistive lossesBattery Voltage Discharge CurvesC/25C/52CC/2 C30 0.2 0.4 0.6 0.8 1SOCbFigure 4 Determination of rate factor, ( f [()] it 1 ), frommeasurements of voltage vs. SOC at different discharge rates.a2.810.80.6 0.4SOC0.2Figure 5 Open circuit voltage versus SOC at constant temperaturesof 3°C, 27°C and 62.5°C.found. At a higher temperature, there is very little alteration ofthe open-circuit voltage as compared to the room temperaturecase; a drop in open-circuit voltage can be seen for the 3°Ccase, and accurate results at low temperatures can be achievedby including a voltage drop equation within the model. As thetemperature changes, a voltage versus SOC equation will beused to determine the voltage drop at a certain temperature andSOC.TABLE 2 RATE FACTORS FOR VARIOUS DISCHARGE CURRENTSiload0.0808 0.4389 1.0886 2.1603f [ ] 1iload1 0.9946 0.96495 0.9232TABLE 3 RATE FACTORS FOR VARIOUS CHARGE CURRENTSiload-0.0838 -0.4386 -1.0988 -2.202f [ ] 1iload1 0.99324 0.9803 0.9665V. RESULTSTests have been performed in Simulink to ensureaccurate model performance during driving schedules.Fig. 6 depicts the subcomponents of the vehiclesimulator, including new battery storage model, as wellas the flow of variables between each subcomponent.Simulations for four different HEV configurations,corresponding to plug-in hybrids with Li-ion pack sizesof 25, 50, 75, and 100 kg were performed on the FederalUrban Driving Schedule (FUDS). TCL PL-383562 850-mAh polymer Li-ion batteries were used in a series andparallel configuration to model the battery pack.Equations for all battery parameters (except sec) weresimulated within the model and taken from [8]. Resultsof the four simulations are shown in Fig. 7. Fig. 7aconfirms that the model vehicle (see [3] for modeldetails) was able to follow the FUDS schedule during thesimulations. The decrease in SOC (from an initial SOCof 90%) is shown in Fig. 7b for each vehicle. The resultsare optimistic because [8] does not include a rate factor.Fig. 7c shows the resistive losses within each01339Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply.

Figure 6 High-level model of EV and PHEV simulators in the MATLAB environmentbattery pack, and Fig. 7d shows losses of all fourvehicles compared with the road power. The proposedbattery model successfully represents Li-ion batterieswithin a PHEV throughout driving schedules. Dataextracted from the results provide insight on the effectsof various driving behaviors and the short-term transientson losses within a battery pack. Simulations oncomputers with Intel’s Core 2 Duo processor takeapproximately 2 min per 1 s of driving schedule; this isten times longer than driving simulation times with theprevious Thevenin-based lead-acid battery model [2].This increase is caused by algebraic loops introducedinto the Simulink model from differential equationsdescribing the three time constants.speed [m/s]30252015105a. Speed profileVEH 1FUDS 151000 500 1000 1500time [s]SOC [%]10.90.80.7VEH 1VEH 2VEH 3b. SOCVEH 40.60 200 400 600 800 1000 1200 1400Time [s]Power [W]15001000500c. Battery lossesVEH 1VEH 2VEH 3VEH 4Power [kW]x 10 4 d. VEH 1 road power and losses2Road Pow erBattery Losses1.510.500 20 40 60 80 100 120Time [s]00 20 40 60 80 100 120Time [s]Figure 7 (a-d) Simulation of VEH 1, 2, 3, and 4 on the FUDS with battery pack mass as the variable1340Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply.

Results of verification of the Simulink model against theactual response of the tested Li-ion batteries are shownin Figures 8-11. Figure 8 depicts the simulated batteryterminal voltage and measured voltage fluctuationsversus time when both the model and batteries are putthrough a simulated city/highway driving schedule. Afterthe 38 min. driving schedule, the error between the actualand simulated terminal voltage is 0.199%. Thiscorrelates to a SOC deviation of the simulated model tothe actual of 1.00%. Figure 9 is the first 500 s of thesimulated driving schedule where the battery modelterminal voltage tracks the measured LI-ion terminalvoltage. Discrepancies between the terminal voltage ofthe simulated and measured batteries near the upperpeaks of the voltage during charging periods (simulatedregenerative breaking) are inconclusive. Although itwould appear that the resistive constants during cyclecharging may be inaccurate, the measured data pointsoccur at approximately every second, while thesimulated driving schedule has the current chargingpeaks at below one second time scales. Hence, themeasuring apparatus is too slow to pick up all details offluctuations of the real battery during the drivingschedule. The lower terminal voltage peaks match upwell since the driving schedule maintains the dischargingcurrent peaks at a rate above one second.Figures 10 and 11 display the simulated road powerand measured Li-ion output power during a trial usingthe 22 min.Federal Urban Driving Schedule. TheSimulink Li-ion battery model, along with the measuredmodel circuit parameters shown previously was placedinto the University of Illinois Electric Vehicle Simulatorto run the driving schedule [3]. Figure 10 depicts theroad power divided by the number of batteries in theentire pack (in this case 6831 batteries) for both thesimulated and measured power into and out of thebattery. Tracking of the measured battery output poweragainst the Simulink model road power per battery isaccurate. Figure 11 is a close-up of the output powerfrom 4000 to 5000 s of the FUDS and just as with Figure8 peak output power has a limited time fidelity.Terminal Voltage (Volts)4.254.24.154.14.05Simulated VoltageMeasured Voltage40 100 200 300 400 500Time (s)Figure 9 First 500 seconds of simulated and measured battery terminalvoltage versus time for city/highway driving scheduleThe simulated peak road power per battery occurs duringintervals less than a second, hence the measurementapparatus will not pick up thosepeaks during the measured Li-battery trial.VI. CONCLUSIONThe proposed battery model will accurately representlithium-ion battery behavior within a dynamic HEVsimulator. Experimental parameters found from benchtests characterize Li-ion cells well at a constanttemperature and cycle life, and the effects on thoseparameters caused by fluctuating temperature can easilybe modeled in a dynamic simulator through open circuitvoltage characterization at a few temperatures. Simulinkmodels of battery pack/cell response to simulatingdriving schedules have been verified using a hardwarein-the-loopbattery cycle testing apparatus; the simulatedSOC, terminal voltage and output power response to theFUDS and another formulated city/highway drivingschedule tracks the measured responses. Although boththe University of Illinois HEV simulator and the4.2514Terminal Voltage (Volts)4.24.154.14.0543.95Simulated VoltageMeasured VoltageRoad Power (Watts)121086420-2Simulated Road PowerMeasured Road Power3.9-43.850 500 1000 1500 2000 2500Time (s)Figure 8 Simulated and measured battery terminalvoltage versus time for city/highway driving schedule-60 200 400 600 800 1000 1200 1400Time (seconds)Figure 10 Simulated and measured road power perbattery versus time for FUDS1341Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply.

Road Power (Watts)543210-1-2Simulated Road PowerMeasured Road Power-3800 850 900 950 1000Time (seconds)Figure 11 200 seconds of simulated and measured roadpower versus time for the FUDSproposed battery model are capable of providingsimulations which include self-discharge and batterycyclingdegradation, the simulation times necessary toget results over these effects on various battery types arecurrently too long to validate the measured affects. Theaccurate response of the battery model now allows forextensive tests over the efficiency and losses within thebattery pack during various driving schedules andalteration in vehicle parameters.ACKNOWLEDGEMENTThis work was supported in part by the Power AffiliatesProgram at the University of Illinois.REFERENCES[1] K. B. Wipke, M. R. Cuddy, S. D. Burch, “ADVISOR 2.1: a userfriendlyadvanced powertrain simulation using a combined backward/forwardapproach,” IEEE Trans. Vehicular Tech., vol. 48, no.6, pp. 1751-1761, November 1999.[2] D. L. Logue, P. T. Krein, “Dynamic hybrid electric vehiclesimulation, version 1.0,” University of Illinois, Technical ReportUILUENG-98-0409, December 1998.[3] M. Amrhein, P. T. Krein, “Dynamic simulation for analysis ofhybrid electric vehicle system and subsystem interactions, includingpower electronics,” IEEE Trans. Vehicular Tech., vol. 54, no. 3, pp.825-836, May 2005.[4] D. W. Dennis, V. S. Battaglia, and A. Belanger, “Electrochemicalmodeling of lithium polymer batteries,” J. Power Sources, vol. 110,no. 2, pp. 310-320, 2002.[5] L. Song and J. W. Evens, “Electrochemical-thermal model of lithiumpolymer batteries,” J. Eletrochem. Soc., vol. 147, pp. 2086-2095,2000.[6] J. Newan, K. E. Thomas, H. Hafezi, and D. R. Wheeler, “Modelingof lithium-ion batteries,” J. Power Sources vol. 119-121, pp. 838-843, Jun. 2003.[7] P. M. Gomadam, J. W. Weidner, R. A. Dougal, and R. E.White, “Mathematical modeling of lithium-ion and nickel batterysystems,”J. Power Sources, vol. 110, no. 2, pp. 267-284, Aug. 2002.[8] M. Chen and G. A. Rincon-Mora, “Accurate electrical battery modelcapable of predicting runtime and I-V performance,” IEEE Trans.Energy Conversion, vol. 21m no. 2, pp. 504-511. Jun. 2006.[9] P. Rong and M. Pedram, “An analytical model for predicting theremaining battery capacity of lithium-ion batteries,” in Proc. Design,Automation, and Test in Europe Conf., 2003, pp. 1148-1149.[10] M. Pedram and Q. Wu, “Design considerations for battery-poweredelectronics,” in Proc. 1999 Des. Autom. Conf., pp. 861-866.[11] B. Y. Liaw, G. Nagasubramanian, R. G. Jungst, D. H. Doughty,“Modeling of lithium ion cells- a simple equivalent-circuit modelapproach,” Solid State Ionics, vol. 175, pp. 835-839, 2004.[12] C. J. Zhan, X. G. Kromlidis, V. K. Ramachandaramurphy, M.Barnes,N. Jenkins and A. J. Ruddell, “Two electrical model of the leadacidbattery used in a dynamic voltage restorer;” in Proc. IEEEGeneration, Transmission, and Distribution, vol. 150, no. 2, pp.175-182, Mar. 2000.[13] B. Schweighofer, K. M. Raab, and G. Brasseur, “Modeling of highpower automotive batteries by the use of an automated test system,”IEEE Trans. Instrum. Meas., vol. 52, no. 4, pp. 1087-1091, Aug.2003.[14] S. Buller, M. Thele, R.W. D. Doncker, E. Karden, “Impedancebasedsimulation models of supercapacitors and li-ion batteries forpower electronics applications,” in Rec. 2003 Ind. Appl. Soc. Ann.Meeting, vol. 3, pp 1596-1600.[15] S. Gold, “A PSPICE macromodel for lithium-ion batteries,” inProc. 12 th Annu. Battery Conf. Applications and Advances, 1997,pp. 215-222.[16] L. Gao, S. Liu, and R. A. Dougal, “Dynamic lithium-ion batterymodel for system simulation,” IEEE Trans. Components and PackagingTech., vol. 25, no. 3, pp. 495-505, Sept. 2002.[17] S. Abu-Sharkh and D. Doerffel, “Rapid test and non-linear modelcharacterization of solid-state lithium-ion batteries,” J. PowerSources, vol. 130, pp. 266-274, 2004.[18] R. Rao, S. Vrudhula, and D. N. Rakhmatov, “Battery modeling for1342Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply.