Dynamic modeling for electric vehicle land speedrecord performance prediction

Matilde D’Arpino∗, Martin Villing†, Jeffrey P. Chrstos∗, Giorgio Rizzoni∗

∗The Ohio State University - Center for Automotive Research, Columbus, OH 43212 USA†University of Stuttgart - 70174 Stuttgart, Germany

Email: darpino.2@osu.edu

Abstract—This paper presents a non-linear time-domain dy-namic model to support the design, optimization, and character-ization of electric vehicles, especially for racing application. Thevehicle system is described by means of accurate mathematicalmodels using standard simulation platforms: MATLAB® andCarSim®. The approach is based on co-simulation architectureoffering flexibility, accurate dynamic performance, and fast com-putation, which are suitable for real-time testing. To demonstratethe benefits of the proposed platform, the Venturi Buckeye Bulletland speed record electric vehicle has been modeled and analyzedas case study.

Index Terms—dynamic modeling, electric vehicle, land speedrecord.

I. INTRODUCTION

Limits of vehicular technologies are constantly pushed bymotorsports with the aim to achieve higher performance.Electric vehicles (EVs) are gaining an important role in wheel-driven categories for their capability to challenges conven-tional vehicles and promote emission reduction. As is wellknown, EVs have several advantages compared to conventionalvehicles, such as less mechanical complexity, high energyefficiency, high torque/power density of electric motors, widespeed range with constant torque, and constant power opera-tion [1]. The control of an electric vehicle represents anotherdistinct advantage due to the accurate, high bandwidth torquegeneration. This means that EVs offer a great opportunity forperformance improvements of motorsports in traction capabil-ity, stability, and safety. However, a systematic and optimizeddesign process is needed to realize their full potential.

For racing application, modeling and simulation represent aunique and indispensable tool to design, predict, optimize thevehicle overall performance, and calibrate the control strategy.Several tools have been previously presented in literature forEV simulations, such as ADVISOR and PSAT [2]. However,modeling vehicles for motorsports application presents severalrequirements, such as the necessity of:• precise aerodynamic model due to the high operating

speed;• accurate dynamic model of components for the evaluation

of mechanical stress;• proper dynamic and thermal model of the powertrain

working in extreme load requirement;

Fig. 1. The Venturi Buckeye Bullet 3 (VBB3).

• embedded software for evaluating vehicle control andstability strategies.

The scope of this paper is to present the tool developed forthe study of the performance of the Venturi Buckeye Bullet3 (VBB3), the World’s fastest electric vehicle depicted in fig.1. An accurate simulation tool (afterwards named VBB3sim)has been developed to evaluate vehicle stability, shift strategy,and new hardware-installation impact on performance. Due tothe high complexity of the vehicle dynamic model and pow-ertrain structure, VBB3sim has been designed and developedusing a co-simulation approach, based on both CarSIM® andMATLAB-Simulink®. The former programming environmentrepresents a modular framework for an efficient implemen-tation of 3D vehicle dynamic modeling to accurately predictthe vehicle performance in response of driver, roadway, andenvironment inputs. However, modeling advanced powertrainand control technique requires an extremely flexible approachthat can be successfully provided by the graphical program-ming language of MATLAB-Simulink®. Therefore, the co-simulation approach enhances the modularity and flexibility ofthe simulation in modeling components and control algorithmallowing for system-level and components-level analysis. Si-multaneously, a reduction of the computational requirement isachieved, allowing for real-time simulations, such as driver-in-the-loop (DIL), hardware-in-the-loop (HIL), and software-in-the-loop (SIL).

This paper presents the VBB3 structure in section II, fol-lowed by a detailed mathematical model description (section

978-1-5386-2894-2/17/$31.00 © 2017 IEEE

III). Section IV includes the implementation of the proposedsimulation approach.

II. VEHICLE STRUCTURE

The Ohio State University (OSU) has a long tradition of rac-ing EVs. Since 2001, several high-speed all-electric vehicles,the Buckeye Bullets (BBs), have been designed and built [3],[4] achieving several national and international speed records.The cars are characterized by a streamlined aerodynamic shapeoptimized for LSR, having a long narrow body with fourwheels enclosed inside the bodywork. VBB3 is the most recentbattery-powered EV at OSU. The vehicle has been designedto reach the ambitious speed of 400 mph, and gain the titleas World’s Fastest Electric Vehicle [4]. VBB3 Land SpeedRecord (LSR) attempts are performed at the Bonneville SaltFlats, Utah, on an 8-9 miles straight and flat course. Duringsummer 2016, VBB3 achieved a World LSR of 341 mph inthe FIA category A-VIII-8, with a peak speed of 358 mph.VBB3 is a four-wheel driven EV powered by two independentpowertrains, one per axle, with a total peak power of 1.5MW. The vehicle structure and design is widely analyzed in[4]. Each powertrain is composed by two interior permanentmagnet synchronous machines (IPMSMs) connected througha friction clutch to a custom two-speed transaxle with anauxiliary shaft for emergency braking. Each motor has anominal power of 200 kW and an overload capability of 1.885;it is controlled by a proper voltage source inverter poweredby two 825-V battery packs. The vehicle is equipped with atotal of 8 battery packs (92.4 kWh - 2.8 MW) based on 14-Ah lithium ion phosphate cell. Two cooling circuits separatelyremove heat from the motors (oil cooled) and inverters (watercooled). The vehicle is equipped with a supervisory controland a data acquisition system based on MATLAB-Simulink®

and implemented on ETAS® ES910 prototyping unit. Thecontroller is connected to inverters and battery BMSs, aswell as custom designed control and acquisition boards byseveral CAN networks. The shifting of the two gear-boxes iselectrically actuated.

The logistical complexity of the record attempt requiresadditional tools for the vehicle analysis and testing. VBB3powertrain is tested on a 1.5 MW dynamometer at the OSUCenter for Automotive Research and low speed vehicle testing(up to 150 mph) can be performed at the TransportationResearch Center (TRC, [5]) on a 7-mile loop track. Thesimulation approach provides, thus, a very effective way toevaluate and optimize the vehicle performance.

III. ADVANCED VEHICLE MODEL

VBB3sim is based on a causality approach (forward-facingmodel) and accepts as input driver data (accelerator andbrake pedal position, gear selection, steering angle), trackinformation (friction coefficient, road shape and grade), windspeed and direction, and initial state of the vehicle, as shownin Fig. 2. The vehicle components have been described withmodels having the reasonable level of accuracy to predict the

Fig. 2. VBB3 vehicle simulator model.

vehicle performance. Steady-state maps, obtained by experi-mental data, have been used to define the electrical modelsof battery and electrical drive. Electrical dynamics can beneglected since they are at least one order of magnitude fasterthan mechanical ones. Physics-based model approach has beenused to model the vehicle mechanical components. Due to thespecific application, the following model discharges brakingaction and road grade, however the description can be easilyextended.

A. Powertrain model

Standard battery-powered EVs are usually equipped withone powertrain coupled to one of the two axles by means of amechanical transmission. VBB3 has two independent power-trains, one per axle, as described in section II. Thus, to ensurethe required flexibility, the mathematical model of the vehiclepowertrains has been implemented in MATLAB-Simulink®

environment. The dynamic of the powertrain is described witha free-body diagram technique, as shown in fig. 3(a). Themodel is based on the connection of rotating inertia and aspring-damper-system representing the dry clutch behavior [6].The other elements has been considered having negligiblespring-damper coefficients. The two powertrain models aretreated separately until connected though the vehicle dynamicmodel (section III-B).A torque reference is derived from the driver’s acceleratorpedal position and the vehicle control provides the torque ref-erence to each electrical drive. Defining Tm as the motor torqueat the shaft (sum of the produced torque of two motors), theangular acceleration ω̇m is obtained by the dynamic equationat the inertial load (Jm + Jc,in):

ω̇m =Tm − Tc,in

Jm + Jc,in(1)

where Tc,in is the reaction torque of the clutch. The frictionclutch is then described by a spring-damper system using the

(a)

(b)

Fig. 3. Powertrain mechanical model (a), and electrical model (b).

Karnopp approach [7]. The spring and damping characteristicsof the clutch are described by Ts and Td, as follows:

Ts = k(ϕs) · (ϕc − ϕt)Td = bc · (ωc − ωt)

(2)

where the angular speed of clutch disc and transmissioninput shaft are ωc and ωt, while ϕc and ϕt are the relatedangular displacements. k(ϕs) and bc are the spring constantand damper coefficient. The clutch is normally closed (stickingmode, ωc = ωm), however, if properly actuated, can enable thetransmission shaft to rotate independently from the propulsionone (slipping mode, ωc 6= ωm). Thus, the reported mathemat-ical model is based on two states in sticking mode and threestates in slipping mode. When the clutch is the slipping mode,it can be described according to eq. (1) and the following setof equations:

ω̇c =Tc,in − Tc,out

Jc,outTc,in = Fn · µcl ·Ra · sign(ωm − ωc)

Tc,out = Ts + Td

(3)

The clutch transmitted torque Tc,in is function of the axialclamping force (Fn), the friction coefficient (µcl), and activeradius (Ra) of the clutch plates. This characteristics is usuallyprovided by the clutch manufacturer as function of the releasetravel of the diaphragm spring. Tc,out represents the outputtorque of the clutch. When the clutch is in sticking mode, it isrigidly coupled to the electrical machines, eq. (1) is connectedto the following equations:{

Tc,in = Tc,out = Ts + Td ≤ Tmaxc

Tmaxc = Fn · µstick ·Ra · sign(Tc,in)

(4)

where Tmaxc represents the maximum transmittable torque as

function of the friction coefficient in sticking mode (µstick).The clutch is then coupled with a gear-box (gear ratio ngear)and a final drive (gear ratio nfd):

ω̇t =Tc,out − Tt,in

Jt

Tt,out =Tt,in

ngear

Ttr,1 + Ttr,2 =Tt,out

nfd

(5)

The traction torque at the wheel (Ttr,i with i=1,2) is obtainedsplitting the output torque of the transmission (Tt,out) consid-ering eventual differential slip effect (steering). No differentialcoupling is usually adopted in LSR vehicle.

1) Electrical drive: The electrical drive model is reportedin fig. 3(b). The torque reference, obtained from the driveraccelerator pedal position, is controlled to not exceed themotor current limits and vehicle acceleration limit, it is deratedin case of drives reaching overtemperature, and it is limitedin case of vehicle stability issues. The two IPMSMs on eachvehicle axle are independently controlled by traction invertersusing a Maximum Torque Per Ampere (MTPA) algorithm[8]. The d − q current control in the machine synchronousreference frame (id, iq) is assumed to be instantaneous, thus themotors are modeled with maps as function of dc-link voltage,torque reference, and motor speed. The motor maps, suchus torque-speed curve, AC current and voltage, power factor,are obtained by experimental data. Efficiency maps for bothelectrical machine (ηm) and traction inverter (ηvsi) have beenalso included in the simulation. Thus, the mechanical powerof the electrical machines (Pm), and electrical powers of bothelectrical machines and inverters (Pe, Pdc) are known at everyinstant. The power balance at the DC-link allows evaluatingthe voltage VDC as function of the currents and state of chargeof the battery packs.

2) Electrical power source: The VBB3 battery packconsists of Lithium-Iron-Phosphate cells [9]. A first-ordertemperature-dependent model has been used to estimate dc-link voltage and state of charge (SoC) starting from the currentof each battery pack, Ipack:

VDC = E0(SoC)−Rpack(SoC, Tcells)Ipack

SoC = SoCinit −∫Ipackdt

Qpack

Tpack = T0 +

∫I2

packRpack(SoC, Tpack)dt

Cpack

(6)

where E0 is the open circuit voltage as function of the SoC,Rpack is the temperature and SoC dependent series resistance,Qpack is the total battery capacity. Tpack is the pack temperatureevaluated as deviation from the initial temperature T0 due tothermal effect of the pack resistance. Cpack is the thermal massof the pack.

B. Vehicle dynamic modelThe vehicle dynamic model of VBB3sim uses 15 degrees

of freedom for the VBB3 model: six for the dynamic of the

sprung mass (body x-y-z directions, roll, pitch, yaw), twofor each unsprung mass (wheel spin and jounce), and onedegree is related to the steering angle. The CarSIM® modelis described in [10]. For simplicity the following descriptionis mostly related to the longitudinal dynamic of the vehicle,however VBB3sim takes care of all the additional components.The considered reference frame is reported in fig. 1.A wheel subject to a traction force Ftr produces a tractiveforce Fd at the tire-ground contact patch. When the vehicle ismoving forward, the friction force Fd causes the slip betweenthe tire and the ground, and contributes to the motion of thevehicle [11]. A slip ratio κ can be defined on the longitudinalaxle as the ratio between the linear Vx and the angular speedof the wheel ωw:

κ =ωwRe − Vx

|Vx|(7)

where Re is the effective tire radius. It is possible to define theforce Fmax

d as maximum tractive force that can be transmittedto the ground before the wheelspin will occur (fig. 4). Thus,

Fmaxd = µ(λ)Fz ≥ Ftr (8)

where Fz is the normal force at each wheel and µ(λ) isthe friction coefficient between the tire and the ground. Thiscoefficient depends on tire characteristics, road conditions,contact patch area, and it is function of the slip ratio. TheBonneville Salts Flat surface can exhibit great variability ofµ(λ). Coefficient of friction of 0.25-0.3 represents the saltsurface when granular like sand, and 0.45-0.5 in the best caseof prepared course (hard crust). Wheelspin is likely to occurat lower gears and high torque request. Traction control andmodulation of the torque become crucial to success at Bon-neville. Fast and accurate control reaction, refined suspensionsettings, and driver expertise can limit tire slipping, optimizethe vehicle acceleration, especially at low speed, and improvethe lateral stability. As matter of fact, the top speed of mostvery high power vehicles racing at Bonneville is limited by themaximum traction rather than available engine/motor power.The equations of motion can be expressed applying Newton’sSecond Law to the vehicle of mass m in the x-direction andto the wheel, considering the two independent axle dynamics:

V̇x =Fd − Faero,x

m

˙ωw,F =Ttr,Fi

−Re,FiFd,Fi

− Trr,Fi

Jwi = 1, 2

˙ωw,R =Ttr,Ri

−Re,RiFd,Ri

− Trr,Ri

Jwi = 1, 2

Fd =

2∑i=1

(Fd,Fi + Fd,Ri −

Trr,Fi

Re,Ri

− Trr,Ri

Re,Ri

) (9)

where Faero,x is the aerodynamic drag, ωw is the wheel speed,Jw is the wheel inertia, Trr is the rolling resistance torque atthe wheel. The subscripts F and R indicate the front and rearpowertrain, while i represents the wheel index.

Fig. 4. Wheel tractive force as function of the normal load.

1) Axle normal force model: The normal force Fz is evalu-ated at each wheel as the static corner weight plus the weighttransfer contribute due to acceleration/braking, aerodynamicdrag, and lift forces. When the vehicle is in motion, additionaldynamic forces are exerted by the suspension and irregularitiesin the track surface. Considering the acceleration effect, theload force Fz,F and Fz,R can be calculated as:

Fz,F =msp(gδR − V̇xh)

δF + δRFz,R = (msp · g)− Fz,F

(10)

where msp is the vehicle sprung mass, g is the gravitationalacceleration, h represents the height of CG, δF and δR are thedistance of the front and rear axle with respect to the CG.

2) Aerodynamic model: The aerodynamic drag takes intoaccount the air resistance against the motion of the vehicle.This results in a drag force directed opposite to the vehiclemotion (x-axis), a side force (y-axis), and a lift force (z-axis) applied to the vehicle center of pressure. At low speed,aerodynamic drag and lift forces are small, but they become ofprimary importance at high speed (more than 250 mph) [12].The magnitude of these forces depends on the air density (ρ),the aerodynamic cross-section area Af, and the speed of theair, as follow:

Faerox = −Cx(β) ·1

2ρAfV

2

Faeroy = −Cy(β) ·1

2ρAfV

2

Faeroz = Cz(β) ·1

2ρAfV

2

(11)

where Cx, Cy, Cz respectively represent the aerodynamic forcecoefficients for drag, side, and lift depending on the vehicleshape and size. These parameters are function of β thatrepresents the angle from the vehicle heading (vehicle localx direction) to the relative wind. They can be rigorouslydetermined (as well as the vehicle center of pressure) withCFD analysis [13], and estimated with wind tunnel and coastdown tests. In the same way it is possible to define the roll,

pitch, and yaw moments.Taerox = Cyaw(β) ·

1

2ρAf(δF + δR)V

2

Taeroy = −Croll(β) ·1

2ρAf(δF + δR)V

2

Taeroz = −Cpitch(β) ·1

2ρAf(δF + δR)V

2

(12)

An important consideration about the vehicle stability is therelative location of center of pressure, center of mass, and tirecontact patches. LSR wheel-drive vehicles may be stronglyconstrained about ground stability due to the high force thevehicle is subject at high speed. In general, the center ofpressure is required to be located behind the center of mass.Thus, if the vehicle develops a yaw moment, the aerodynamicforces acting on the center of pressure will tend to decreasethe moment itself thanks to the self-balancing action.

3) Rolling Resistance model: Tire rolling resistance repre-sents the force necessary to move the vehicle wheels due tothe tire losses on the contact patches. Rolling resistance ateach wheel Trr is modeled by two components. The first onedepends on the vertical load of the wheel (Fz), a constant (Rrc )and a speed depended coefficient (Rrv ). The second componentis due to the application of the traction force Ftr,

Mrr = Fz,wKrsurfRe · (Rrc +RrvVx) + Ftr · (Re −Rl) (13)

where Krsurf describes the road surface effect.4) Tire model: The tire performance affects the vehicle

behavior for several reasons. Tires support the entire ver-tical load, and the contact patches between tire and roadsupport acceleration, braking, and lateral forces, responsiblefor directional stability [14]. However, tires generate rollingresistance, drag in the moving direction, and steering torque.Inflation pressure, temperature, and rotational speed affect themaximum friction coefficient of the tire-road system, thus, thecapability of the vehicle to accelerate [15]. In LSR applicationtires are subject to a high variability of operating conditionsduring the whole race; due to the high speed and the roadfriction, tires increase pressure, temperature, stiffness (Kl), andeffective radius (Re); the aerodynamic drag affects the vehicleweight distribution resulting in a different loaded radius (Rl)[11]. Bonneville tires are narrow, with little tread, roundedprofile, and they run at high inflation pressure (usually bias-ply tires). This results in a very small contact patch betweenthe tire and the ground. The considered model is based on aapproximated function depending on the vehicle speed, wheelspeed, and normal force:Rl =

Fz

Kl0

(1 + cr · Vx)

Kl = Kl0(1 + ck · Vx)

Re = k6ω3w + k5ω

2w + k4Fzωw + k3Fz + k2ωw + k1

(14)

where cr and ck represent velocity coefficient for loadedtire radius and stiffness, Kl0 is the tire initial stiffness. Thedescription of the effective tire radius growth is more complexdue to the combined effect of Fz and ωw, as shown in fig. 5.To accurately represent the tire growth, a polynomial model

Fig. 5. Radial and bias-ply tire growth comparison [14].

has been approached based on experimental data (up to 200mph). VBB3 tires have experienced up to 3cm of growth indiameter at 300 mph.

C. Track and driver model

Roughness and irregularity of the racing surface has a largeimpact on vehicle performance. Surface waviness can increasedrag due to unexpected turbulence in the air flow. The terraincondition is obtained by embedded model of CarSim. Complexroad geometry and surface parameters can be setup to emulatethe LSR salt track with eventual irregularity (like bumps).The driver is modeled as a look-ahead PI loop following thetargeted vehicle path compensating for wind and track effects.

IV. NUMERICAL RESULTS

The VBB3sim models have been calibrated using data fromracing and testing events of the actual VBB3 and previousvehicles [4], [13], [16], [17]. VBB3 is a well instrumentedvehicle, providing accurate data for calibration and validationof the mathematical model. Accelerator position, wheel speed,gear position, GPS data, motor resolver, currents and voltages,brake fluid pressure, cooling temperature, and flow rate, aresome of the acquired sizes.Figures 6-12 report the numerical results obtained with theVBB3sim in case of vehicle driving on a straight 6-milecourse. The driver’s torque request is combined by the VBB3control with the shifting request to obtain the p.u. torquereference of front and rear powertrains, as shown in fig. 6.The VBB3 structure allows for a power-shifting since thegearboxes of the two powertrains can be independently shifted.In detail, a stability analysis has been performed to definethe shifting strategy; as results the rear powertrain is shiftedfirst to reduce the torque level when the vehicle is rear drivenand the front shifting procedure is actuated (fig. 7). Figure8 shows the vehicle speed and the position on the track.The simulation is validating the VBB3 design requirement ofachieving a speed of 400 mph in 90s. The vehicle accelerationis limited by the losses reported in fig. 9, such us rollingresistance, driveline, and aerodynamic losses. It is possible tonotice how the aerodynamic effect increases quadratically withthe vehicle speed. Figure 10 reports the load of the front and

rear axle showing the effect of the weight distribution due tovehicle weight and aerodynamic forces at the increasing of thevehicle speed. The tire model allows estimating the effectiveand loaded radii, as reported in fig. 11. The effective radius ishighly affected by the speed increase. Finally, figure 12 showsthe voltage of the battery packs when the vehicle starts withan SoC=99%. At the end of the race the battery packs havea residual SoC of 56%.

V. CONCLUSION

In this paper, a co-simulation approach is presented forthe development of the dynamic model for the performanceevaluation of the VBB3 land speed record electric vehicle.Experimental results demonstrate that the proposed approachcan accurately model the performance of a LSR EV, and can beused for performance optimization, such as shifting strategy.

ACKNOWLEDGMENT

The authors would like to thank the Buckeye Bullet studentteam and the sponsors for the commitment and dedication tothe project.

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Fig. 6. Driver’s and control’s throttle signals.

Fig. 7. Traction torque at the wheels.

Fig. 8. Vehicle speed.

Fig. 9. Vehicle power losses.

Fig. 10. Vehicle axle loads.

Fig. 11. Effective and loaded radius of the tires.

Fig. 12. Battery voltages.