喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== 喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== Proceedings of the ASME 2010 International Design Engineering Technical Conferences &Computers and Information in Engineering ConferenceIDETC/CIE 2010August 15-18, 2010, Montreal, Quebec, CanadaDETC2010-28814SYMBOLIC MATH-BASED BATTERY MODELINGFOR ELECTRIC VEHICLE SIMULATIONAden N. SeamanDepartment of Systems Design EngineeringUniversity of WaterlooWaterloo, Ontario, Canada. N2L 3G1Email: anseamanreal.uwaterloo.caJohn McPheeDepartment of Systems Design EngineeringUniversity of WaterlooWaterloo, Ontario, Canada. N2L 3G1Email: mcpheereal.uwaterloo.caABSTRACTWe present results of a math-based model of a battery elec-tric vehicle (BEV) designed in MapleSim1. This model has thebenefits of being described in a physically consistent way us-ing acausal system components. We used a battery model byChenandRin con-Moratodevelopamath-basedmodelof acom-plete battery pack, and developed simple power controller, mo-tor/generator, terrain, and drive-cycle models to test the vehicleunder various conditions. The resulting differential equationsare simplified symbolically and then simulated numerically togive results that are physically consistent and clearly show thetight coupling between the battery and longitudinal vehicle dy-namics.1IntroductionVehiclemodelingisacomplicatedandchallengingtask. Au-tomotive companies release several new vehicles each year, andall of these need to be simulated and tested before they are actu-ally manufactured.With the push towards cleaner and more energy-efficientvehicles, powertrains are incorporating motors, generators,continuously-variable transmissions, energy storage devicessuch as batteries and fuel-cells, and traditional internal combus-tion engines (ICEs).One of the techniques that can ease the growing complex-ity of vehiclemodelingis acausal math-basedmodelingin which1Maple and MapleSim are trademarks of MapleSoftthe system is described using the physics-based equations thatgovern the behaviour of its components. These mathematicalequations are processed symbolically before finally being solvednumerically to generate output data. This approach makes it eas-ier for designers to specify component behaviour, and constrainsthemtodescribecomponentsinamorephysically-consistentlan-guage. This makes it easier to swap or modify components andsimplifies the description of the system 1.The Modelica 2 description language has been used bymany authors 37 to model hybrid electric vehicle systemsacausally, mostly using the Dymola 8 simulation environment.We have chosen to use MapleSim 9 from MapleSoft asour simulation environment, as this allows us to access the un-derlying mathematical equations which govern the system beingsimulated.Thisapproachyieldsasimplifiedequation-baseddescriptionof the system which can be simulated efficiently. The equationscan also be used in real-time simulationfor hardware-in-the-loop(HIL) applications, and can be used in sensitivity analysis andsystem optimization 10,11.In this paper we present the results of a battery electric ve-hicle (BEV) we have modeled using math-based modeling tech-niquesinMapleSim. See Fig.1forablockdiagramoftheoverallBEV system. This is the beginning of a more complex math-based hybrid electric vehicle (HEV) model we aim to developusing symbolic mathematics.We have incorporated a lithium-ion electric-circuit batterymodel by Chen and Rin con-Mora 12 into the BEV system. We1Copyright c ? 2010 by ASMEProceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2010 August 15-18, 2010, Montreal, Quebec, Canada DETC2010-? Downloaded 25 Jun 2011 to 113.204.33.35. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmmodified the battery equations to simulate a battery pack com-posed ofseries and parallel combinationsof single cells. Inorderto connect the battery pack to a motor we had to develop a powercontroller model as part of the system integration. We furtherincorporateda simple one-dimensionalvehicle model that driveson an inclined plane, a terrain model that controls the incline,and a drive cycle model that controls the vehicles desired speed.By varying the drive cycle and terrain model, we tested theBEV under various driving conditions.FIGURE 1.BLOCK DIAGRAM OF OVERALL BEV MODEL2System Modeling and SimulationThe technique we decided to use was math-based model-ing using MapleSim as the simulation environment, which has agraphical interface for interconnecting system components. Thesystem model is then processed by the Maple mathematics en-gine, and finally the differential-algebraic equations (DAEs) de-scribing the system are simulated numerically to produce out-put data. For 3D multibody simulation it uses the DynaFlex-Pro engine, which uses linear graph-theory for system simula-tion 1,11.2.1BatteryOne of the most importantcomponentsof an electric vehicle either BEV or HEV is the battery. There are many ways ofmodeling different battery chemistries depending on the fidelityneeded and the battery parameters of interest. See the article byRao et al. 13 for an overview of some of the techniques. Gen-erally, with increasing model accuracy comes increased compu-tational requirements.Some modeling techniquesthat we reviewed were: the lead-acid model of Salameh et al. 14; the mathematical lithium-ionmodel of Rong and Pedram 15 that incorporates state-of-healthand temperature effects; the lumped-parameter model in section3.1 of the Partnership for a New Generation of Vehicles (PNGV)Battery Test Manual 16; the Kalman filtering techniques ofPiller et al. 17; the electrical circuit modelofChen and Rin con-Mora12; andthe impedancemodelof Nelson et al. 18. Thesedifferenttechniqueshavetheirstrengthsandweaknessesandlim-ited ranges of application.There is a great interest in using lithium-ion batteries inelectric vehicles, as they are light and have a higher power-to-weight and power-to-size ratio than Lead-Acid or Nickel-basedbatteries. Great demands are placed on vehicle batteries as thedriver accelerates and brakes regeneratively,putting the batteriesthrough periods of high current draw and recharge. Dependingon the driving environment,the batteries can also be subjected tolarge temperature variations, which can have a significant effecton the batterys performance and lifetime.Thus we needed to model a lithium-ion battery chemistryover a wide state-of-charge (SOC) range, under widely-varyingcurrents, for various temperatures. Since we would eventuallylike to model this vehicle in a hardware-in-the-loop (HIL) sys-tem, we neededa model that was not computationallyexpensive,and we did not require a high-fidelity model.These requirements led us to the electrical circuit model ofChen and Rin con-Mora. We implemented their components inMapleSimandusedacustomfunctionblocktorepresentthenon-linear relationship between the state of charge and the electricalcomponents (Equations 2 to 6 in their paper). See Fig. 2 for ablock diagram of the battery.FIGURE 2.BLOCK DIAGRAM OF SINGLE-CELL BATTERYMODELSince their model is of a single cell, we modified their equa-tions to simulate a battery of cells in parallel and series. TheChen and Rin con-Mora battery can be divided into two linearcircuits with a non-linear coupling between them. See Fig. 2 forlabels of these different circuits. One circuit is a large capaci-tor in parallel with a resistor that models the charge state of thebattery and self-discharge. This can be called the “capacity cir-cuit”. Anothercircuit is a voltage source in series with a resistor-2Copyright c ? 2010 by ASMEDownloaded 25 Jun 2011 to 113.204.33.35. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmcapacitor network that models the time response of the battery.This can be called the “time response circuit”.To adapt their single cell model to simulate an entire bat-tery pack, let Nparallelbe the number of cells in a parallel pack,and let Nseriesbe the number of parallel packs placed in seriesto make the whole battery. The open circuit voltage in the timeresponse circuit is multiplied by Nseries. The current flowing inthe time response circuit is divided by Nparallelwhen it flows inthe capacity circuit. The resistors in the time response circuit aremultipliedbyNseries/NparallelandthecapacitorsaremultipliedbyNparallel/Nseries.A singlecell ofthebatterymodelhas anopen-circuitvoltageof 3.3 V and a capacity of 837.5 mAh at a 1 A discharge ratestarting at 100% state of charge. By placing 8 cells in parallel,and 74 of these parallel packs in series, a 244.2 V, 6.7 Ah batterypack was created. This pack is comparable to that in a 2007Toyota Camry hybrid 19.The Chen and Rin con-Mora battery model is simple enoughto simulate in a short amount of time while being complexenough to provide the following: variations in the open circuitvoltage with SOC; transient effects of charge depletion and re-coveryandtheir dependenceonSOC; andthe variationin batterycapacity with discharge current. Furthermore, since it is an elec-tricalcircuitmodelitcaneasilybeincorporatedintotheelectricalsystem of the BEV model and is amenable to being representedusing math-based modeling techniques.Oneofthedownsidesofthis modelis thatnotemperatureef-fects of any kind are modeled, although Chen and Rin con-Morastate it would not be difficult to include them. In an electric ve-hicle the temperature will vary with external environmental con-ditions, with heating of the battery due to internal losses, andwith endo- and exothermic chemical reactions. The only modelwe encountered that explicitly included temperature dependencewas the mathematical model of Rong and Pedram 15, but theirmodel assumes a constant discharge current and thus is not suit-able for our BEV system.The Chen and Rin con-Mora model can also be overchargedand does not consider the increasing resistance of the battery asit nears a full charge. Furthermore the variations of the batterysstate-of-health (SOH) with time and charge cycles is not mod-eled. These downsides are acceptable given that in future mod-eling the vehicles control system will limit maximum batterycharge,andalthoughinthis paperwe are notinterested inmodel-ingtemperatureorstate-of-health,theyshouldnotbetoodifficultto incorporate.2.2Power ControllerThe next important component of an electric vehicle is apower converter that acts as an interface between the battery andthe drive motor/generator. This component controls the amountof power going to the motor during driving, and the amount ofpower going back into the battery during regenerative braking.Generally, boost or buck converters are used depending onwhether the output voltage is higher or lower, respectively, thanthe input voltage 20. By varying the duty cycle of a high-frequency switching circuit, the output voltage and thus currentand power can be controlled.Insteadof modelingthe highfrequencycircuit in MapleSim,we decided to use a simple approximation that can serve as botha boost or buck converter with power flowing from the batteryto the motor, or vice-versa. Figure 3 is a picture of the powercontrollerblock diagram. Althoughthe currentmodelhas a fixedconverter efficiency of 100%, a more realistic efficiency modelsuch as the one used by Hellgren 3 can be incorporated.FIGURE 3.BLOCK DIAGRAM OF POWER CONTROLLERMODELUsing a signal-driven current source in the output loop, theoutput voltage is measured and the output power is calculated.The input currentis adjusted by a PID controller so that the inputpower matches the output power. This circuit works both forpositive or negative current, which determines the direction ofpower flow. This model avoids the divide-by-zero problem of asimple algebraic power converter when the output voltage andcurrent goes to zero, and adapts to changing input and outputimpedance. However it does not take into consideration physicallimitations of componentssuch as the batterys maximumchargeor discharge rates, and voltage and current limits of the motor,wires, or power electronics.2.3Electrical MotorThe electrical motor used in the vehicle model is the Model-ica DC permanent magnet motor, which includes internal resis-tance, inductance, and rotor inertia 21.Its mechanical and electrical behaviour are modelled byEquations 1 and 2 where Jais the armature inertia,(t) is thearmaturerotationangle,Vnom, Inom, and fnomare the nominalmo-tor voltage, current, and rotational frequency, respectively.(t)is theshafttorque,andLaandRaare thearmatureinductanceand3Copyright c ? 2010 by ASMEDownloaded 25 Jun 2011 to 113.204.33.35. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmresistance, respectively. FinallyV(t), andI(t) arethe voltageandcurrent at the motor terminals, respectively.Ja(t)30(VnomRaInom)I(t)fnom(t) = 0(1)LaI(t)+RaI(t)V(t)+30(VnomRaInom)(t)fnom= 0(2)We chose to use the physical parameters of the LEM-200ModelD127DCpermanentmagnetmotorfromL.M.C.Ltd22.However we modified the rated current and voltage of the motorto be more compatible with our battery voltage. This would ef-fectively require re-winding the motor with different wire andchanging its magnets.The parameters used for the motor are presented in Table 1.Note that the peak current and power of the motor are twice therated value.TABLE 1.MOTOR MODEL PARAMETERSNameValueResistance0.0175 Inductance13HInertia0.0236 kgm2RPMrated3600 rpmVrated150 VIrated96 APrated12.56 kW2.4Vehicle DynamicsThe vehicle model we used was very simple.Its physi-cal parameters were based on the 2007 Toyota Camry hybrid.Since we were concerned only with the performance of the pow-ertrain components, we did not concern ourselves with vehiclesuspension or steering. We used a one-dimensional model of africtionless cart on an incline under the force of gravity. Thedrive motor is connected to one of the slipless wheels of the cartthrough a fixed transmission with a ratio of 9 motor revolutionsper wheel revolution. The wheels have the same diameter as theP215/60VR16.0 tires on the Camry.Equation 3 describes the relationship between the rotationand torqueof the motor shaft.(t) is the torqueseen at the motorshaft, m is the vehicles mass, R is the drive tire radius,is thegear ratio from the motor to the tire,(t) is the motor shaftsrotational displacement, g is the gravitational constant, and(t)is the terrain inclination angle.Table 2 lists the values used for these parameters.(t) =mR?Rd2dt2(t)+gsin(t)?(3)TABLE 2.VEHICLE MODEL PARAMETERSNameSymbolValuemassm1613 kgtire radiusR32.25 cmgear ratio9gravityg9.8 m/s2The only type of braking included in this model is regener-ative braking where the current to the motor is reversed and thebattery is charged with the kinetic energy of the vehicle. We didnot take into consideration recharge current limits of the battery.To this vehicle model we attached a simple terrain model.A time-dependent lookup table controlled the inclination of theterrainonwhich thevehicletraveled. This allowed us to simulatethe vehicles performance on flat and hilly terrain.Thedrivecycle is a time-dependentlookuptable of the vehi-cles desired speed. A PID controllercomparesthe desired speedto the actual speed and drives the input of the power controller totransfer power to the motor, or to extract power from the motoruntil the vehicles speed matches the desired speed.See Fig. 1 for the block diagram of the overall BEV model.2.5Numerical SimulationAfter MapleSim converts the vehicle model into differentialequations, it simplifies and reduces the system of equations sym-bolically. Then using this reduced equation set it solves themnumerically to produce the final output data.MapleSim simulated our system with its non-stiff solver,which uses a Fehlberg fourth-fifth order Runge-Kutta method4Copyright c ? 2010 by ASMEDownloaded 25 Jun 2011 to 113.204.33.35. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmwith degree four interpolant.We used an adaptive time-stepwith absolute and relative error tolerances of 1e-7, and turned onMapleSims native code generation ability which runs the sim-ulation faster. The model was simulated on a 3 GHz Intel Core2 Duo using MapleSim version 3 for Linux. It was set to simu-late over a 30 second time interval, and took 10 seconds of actualtime to complete.3Results0500010000150002000025000Time (s)33.23.43.63.844.2Voltage (V)ModeledActualFIGURE 4.MODELED-VS-ACTUAL 12 BATTERY UNDERPULSED CONSTANT-CURRENT DISCHARGEFigure 4 is a comparison between the MapleSim model andan actual cell for a pulsed current discharge of a single batterycell. The actual cell data was extracted from Fig. 5 of Chenand Rin con-Moras paper. Like the model in their paper, ourmodel does not include a self-discharge resistor. An initial SOCof 98% gives a close match to the experimental results, trackingthem very well until the battery capacity is almost exhausted.Our model requires one discharge cycle more than the actual tosee a rapid collapse in the battery terminal voltage.Using our vehicle model we performedtwo simple and intu-itive tests. Table 3 lists the parameters used in the drive cycles.3.1AccelerationsThe first test we did was to simulate the vehicle driving un-der hard and gentle accelerations on flat terrain. Battery and in-ternal combustion engine vehicles are more efficient if gentle ac-celeration is used compared to hard acceleration, due to internallosses. The initial accelerations of the hard and gentle cycles aredifferent,but the maximumspeed and rate of decelerationare thesame. See the hard and gentle curves of Fig. 5 for a plot of thedrive cycle speed with time.TABLE 3.DRIVE CYCLE AND TERRAIN MODEL PARAME-TERSNameValueVmax9 m/sahard1.607 m/s2agentle0.968 m/s2hill height8.67 mhill angle8Figure 6 plots the batterys state of charge versus time. Re-call that this model is without rollingresistance. One can see thatthe hard acceleration drive cycle ends up with a lower final stateof charge than the gentle cycle. This difference is due to ohmiclosses in the motor windings and chemical losses in the battery.3.2HillsThe second test we did was to drive the vehicle up and downa hill. The battery should lose energy going uphill as the vehi-cle gains gravitational potential energy, and gain energy goingdownhill as the vehicle loses potential energy. See the hill cyclecurve of Fig. 5 for a plot of the drive cycle speed with time. Theterrain cycle is very simple: at t=9.5 s the vehicle encounters thehill, then it drives up or down an 8incline before returning toflat terrain at t=20.5 s.Figure 7 plots the batterys state of charge versus time forthis test. In both cases the battery loses energy as it accelerates051015202530Time (s)0123456789Velocity (m/s)Hard & Hill cyclesGentle cycleFIGURE 5.DRIVE CYCLE: SPEED-VS-TIME FOR HARD, GEN-TLE, AND HILL CYCLES5Copyright c ? 2010 by ASMEDownloaded 25 Jun 2011 to 113.204.33.35. Redistribution subject to ASME license or copyright; see http:/www.asme.org/terms/Terms_Use.cfmthe vehicle, transferring energy from the battery to the vehicleskinetic energy.In the uphill case the state of charge decreases. The drivecontrollerappliesmore powerto the motorto match the vehiclesspeed tothe desiredspeed, and thebatterys energyis put intothevehicles gravitational potential energy.In the downhill case the state of charge increases. The drivecontroller applies the regenerative “brakes” to keep the vehiclesspeed constant, and the vehicles gravitational potentialenergyistransferred to the battery.Finally the vehicle encounters a flat spot and uses regenera-tive braking to come to a halt, transferring the vehicles kineticenergy to the battery.051015202530Time (s)78.87979.279.479.679.880State of Charge (%)Gentle startHard startFIGURE 6.STATE OF CHARGE FOR HARD AND GENTLE AC-CELERATIONS ON FLAT TERRAIN051015202530Time (s)76777879808182State Of Charge (%)DownhillUphillFIGURE 7.S