Modeling and Analysis of the Technical Performance of DC-MotorElectric Bicycle Drives Based on Bicycle Road Test Data

Annette Muetze, Ying Chin TanDept. of Electrical and Computer Engineering

University of Wisconsin–Madison, Madison, WI 53706 USAmuetze@engr.wisc.edu, yctan@wisc.edu

Abstract— Electric motor powered bicycles have been makingtheir way into the U.S. market for about two decades. Custom-designed electric bicycles allow common issues such as high costand weight to be overcome. To this aim, customized modelingtools are required. The paper discusses the modeling of a direct-drive dc-motor electric bicycle drive system for technical per-formance evaluation, where the operating cycle profiles used arebased on actual road tests. It is explained how the measurementdata can be processed and coupled with the “usual” modelof a direct-drive dc-drive system, thereby extending commonmodeling approaches. Then, the different riding profiles areanalyzed with the developed tool. The results both illustrate theability of such low-cost drives to serve for commuting purposeswith moderate driving styles and their limits to support rathersporty rides.

Index Terms— Power-assisted bicycle, direct-drive, efficiency,modeling, performance evaluation.

I. INTRODUCTION

Electric motor powered bicycles have been making theirway into the U.S. market for about two decades. Such electricbicycles can be used for a large variety of purposes, includingserving as a vehicle for police or law enforcers, a guide bikeduring races, and for leisurely rides and commuting (e.g. [1]–[8]). When designing electric bicycles (including their electricdrive systems) and trying to overcome common issues suchas high bicycle cost and weight, it is important that the drivebe most efficient over a given operating cycle; this leads tocustom-designed electric bicycles such as “city bicycles,” “hillbicycles,” “distance bicycles,” and “speedy bicycles.” In thispaper, a model of an electric bicycle drive is presented thatcan be used to evaluate the technical performance of a givendrive system both instantaneously and over a whole operatingcycle. A very unique characteristic of the model is that theoperating cycle profiles used as input are based on actual roadtests. Special emphasis is based on the coupling techniques ofthese experimental data to the different modules of the drivemodel. For example, both the command and the load torque arerequired in the model, but only one measured value per timestep is available. Furthermore, the speed has been measured,which, in the “conventional” model, is a function of the nettorque available at the shaft. As far as the authors know, thisis the first time that such an approach on analysis and designof electric bicycle drives is reported on in the literature. Themodel allows investigation of:

1) Instantaneous drive parameters (e.g. currents, voltages,torques, battery loading, efficiencies).

2) Overall technical performance of the drive over a givendriving cycle (e.g. efficiency, energy consumption).

3) Influence of the parameters of the different drive com-ponents on the technical performance and other outputparameters.

First, in the style of a concise review, the different elementsof the “conventional” model are presented (Section II). Then,the drive simulation technique is discussed. This includes thetest vehicle and data recording, as well as the data processingand model extension to analyze the technical performanceof a drive using such measured riding profiles (Section III).Here, special emphasis is put on the different ways of usingthe measurement data as input into the model. In the nextsection, following a short overview of the four measuredriding profiles, the performances of different dc-motor drivesis analyzed using the presented model (Section IV). Boththe advantages and limits of the drives as well as of themodeling approach are shown. The findings are summarizedand prospects of future work are given at the end (Section V).

II. MODEL STRUCTURE AND IMPLEMENTATION

A. Objectives

The model is designed for the investigation of both theinstantaneous and the overall performance of direct-drive dc-motor electric bicycle drive systems under different ridingconditions. To this aim, it allows:

1) Investigation of various instantaneous drive parameters,such as motor current, voltage, torque, remaining batteryenergy, and system efficiency.

2) Overall performance evaluation of the drive over a givendriving cycle, such as system efficiency and total powerconsumption.

3) Investigation of the influence of controller, battery, andmotor parameters on the different drive parameters. Forexample, battery internal resistance, motor inductanceand resistance. Thereby, modifications to better meetthe demands of custom-designed electric bicycles canbe identified and verified.

Here, only direct-drive systems with brushed dc-motors offully-powered electric bicycles are considered. However, themodel can be extended at a later stage to include brushlessdc motors and/or the additional control of the output powerdue to the required human-to-motor power ratio of “pedelec”-type electric bicycles. With “pedelec”-type electric bicycles,only a certain pre-determined, speed-dependent fraction of thebicycle propulsion power is delivered by the drive.

15741-4244-0743-5/07/$20.00 ©2007 IEEE

B. Structure

The overall model is organized in submodels that areimplemented as individual submodules. It consists of the fourtoday well-known main components of such drive systems(Sections II-C through II-E):

1) Electric motor.2) Controller, itself consisting of three submodules:

a) Transformation of the user demand into the corre-sponding duty cycle.

b) Control of the switching based on the duty cycle.c) A capacitor to provide a stiff voltage at the switch

throw terminal.3) Battery.4) Electromechanical system that relates net torque and

acceleration, using a simplified single-wheel represen-tation.

and of a series of technical performance modules (Section II-F). A sketch of the electric components, 1) to 3), and theirmain interactions is shown in Fig. 1, the equivalent circuit inFig. 2.

Battery MotorController

Batteryvoltage

Batterycurrent

Demandvoltage

Feedbackcurrent

Dem

and

Throttle

Fig. 1. Overview of the three electric modules and their main interactions.

emf= kv

Rb

VbattCVbatt

VcIc

Ra La

B J

Tnet = ktIa - Tload

Ia

Ibatt Ibatt-IcBattery

DC Motor

Controller

Fig. 2. Equivalent circuit of the system.

The model is implemented using the commercially availablesoftware package MATLAB R© Simulink R©. All parameters aredefined globally.

C. Motor model including mechanical equation

The motor including the mechanical equation are modeledin the today well established way consisting of:

1) The electrical equivalent circuit ((1) - (3))

Va = RaIa + LadIa

dt+ kvω (1)

EMF = kvω (2)

Te = ktIa (3)

with the armature voltage, current, resistance, and induc-tance Va, Ia, Ra, and La, the back-EMF voltage given by

the product of back-EMF constant kv and rotor speed ω,the electrical torque Te and torque constant ke.and

2) The mechanical equivalent circuit (4):

Tnet = Te − Tload = Jdω

dt+ Bω (4)

where Tnet, J , and B are the net torque, inertia, anddamping coefficient respectively.

The corresponding block is shown in Fig. 3.

Electrical

1Ra+sLa

1B+sJ

kt

Electro-mechanical

Torqueconstant

Ia Te Speed

Tload

kv

Back-emf constant

Vcmd

V-limiter

Va

Fig. 3. Motor model block including mechanical equation (constant motorflux).

D. Controller model

The controller controls the power flow from the batterysource to the motor. The controller model module consistsof three sub-modules (Fig. 4):

1) Transformation of the user demand into the switchcontrol (Fig. 4(a)):The controller controls the switching of the switch thatconnects the motor terminals either to the battery voltageVbatt or to zero voltage. The torque command givenby the rider Tcmd is transformed into the correspondingcurrent and compared to the armature current. The erroris controlled to become zero with a Proportional Integral(PI) PWM controller. The output D of the controllertakes on the values 0 or 1.

2) Control of the switch (Fig. 4(b)):The instantaneous throw voltage Vcmd and current Icmd

that result from the torque command Tcmd and the actualpoint of operation of the motor are obtained from theoutput of the controller D, the armature voltage Va, andthe armature current Ia. The command voltage Vcmd isused to control the motor (Fig. 3), whereas the currentIcmd is required to calculate the voltage across thevoltage stiffening capacitor.

3) A capacitor to provide a stiff voltage at the switch throwterminal (Fig. 4(c)):The capacitor voltage Vc is obtained by integration ofthe capacitor current which is the difference betweenthe battery and throw currents Ibatt and Icmd.

E. Battery model

The battery model block comprises the battery voltage Vbatt

and the battery internal resistance Rb. Temperature and loaddependence are not considered. The battery voltage interactswith the capacitor voltage, and the battery current is obtainedfrom the voltage drop across the battery internal resistance(Fig. 5).

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Tcmd1kt

Ra

Ia Ra

PI-Controller

DComparator0/1

(a) Processing of user demand.

D

Vc

0

Vcmd D

Ia

0

Icmd

(b) Switch.

Ibatt

Icmd

1C

Vc

Vbatt

1sx0

(c) Voltage stiffening capacitor.

Fig. 4. Controller model block including processing of the user demand.

Vbatt

Vc

1Rb

Ibatt

Fig. 5. Battery model block.

F. Technical performance modules

The following parameters are investigated with the technicalperformance modules:

1) Instantaneous electrical input power Pin.2) Instantaneous mechanical output power Pmech.3) Input energy (battery energy output) Win (accumulated

since the beginning of the driving cycle).4) Battery capacity Wbatt0.5) Battery remaining energy Wbatt(t).6) Instantaneous drive efficiency η.7) Average drive efficiency over driving cycle ηave.The equations implementing these seven aspects are simple

and well-established. Therefore, not all technical performancemodules are shown in Fig. 6.

VbattPin

Ibatt

TePmech

Pin Win1

3600

1s

Pmech

Pin

Pmech

Pin~Pin

100

if execution

1) 2)

3)

6)

Fig. 6. Selected technical performance modules.

For the realization of the technical performance blocks,several if-execution blocks are used. This is required to preventthe occurrence of undefined numbers caused by divisions byzero. For example, at the beginning of a simulation (t = 0),unless initialized otherwise, both the electrical input powerPin and the mechanical output power Pmech are zero, andcalculation of the efficiency would result in an undefinednumber. Using the if-execution block, this is avoided as

follows (nominator N and denominator D):The input D is processed as follows to become D:

1) N = 0,D = 0 : D is set to be a non-zero constant sothat N/D = 0.

2) N = 0,D = 0 : D takes the value D(t) = D(t − ∆t)so that N/D = inf .

3) Other than above: D = D remain unchanged.

For the example of the instantaneous drive efficiency η (case6) it is N = Pmech,D = Pin, and D = Pin.

III. DRIVE SCENARIO SIMULATION TECHNIQUE

A. Test vehicle description

For the experimental investigation, an electric bicycle witha brushed dc-motor installed in the front hub, a controller, athumb throttle, and a battery pack is used (Fig. 7). This bicycleis a commercially available bicycle that has been available inthe laboratory. All experiments were carried out using this testvehicle. The electric hub-motor in the front wheel is not usedduring the measurements, yet, using this bicycle, the actual setup of an electric bicycle is represented.

Fig. 7. Electric bicycle test set-up used for the experimental investigation.

The torque and speed are directly measured in the hub ofthe rear wheel of the test bicycle, using a Power Tap R© hub(Fig. 8) [9]. The measurement information is transmitted tothe CPU through the receiver. For all measurements, the tirepressure was kept at 50,...,60 psi, which is typical for bicyclesthat are used for leisure and commuting and that are commonlynot re-inflated before each ride. The anemometer that can beseen in Fig. 7 is not used for the measurements discussed inthis paper.

B. Data recording

The riding profiles were recorded (measured) in terms ofpower, PPT, torque, TPT, and ground speed, ωPT. The samplinginterval is set at its minimum time, ts = 1.26s. Table I showsa sample data set.

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Fig. 8. Power Tap R© hub [9] used for the experimental investigation.

TABLE I

SAMPLE DATA SET FROM POWER TAP R©

Time Torque Speed Power DistanceMinutes Nm km/h W km0 0 0 0 00.021 6.1 4.1 21 0.0060.042 4.5 6.5 24 0.0080.063 3.7 6.5 20 0.01

C. Data processing and model extension

The measurements of the riding interval profiles as de-scribed above (Section III-B) are inputs to the model using theSimulink R© signal builder block. In addition to the torque TPT

and the speed ωPT, the speed difference ∆ωPT and acceleration(dω/dt)PT = ∆ωPT/ts between two sampling intervals areused for this purpose.

1) Command torque and load torque: The model containsthe load and command torques Tload and Tcmd as inputs (Figs. 3and 4). However, only the torque “produced” by the rider isavailable in the form of TPT, along with the speed ωPT. Asthis torque shall be produced by the electric motor, it is usedas input for the torque command, Tcmd = TPT = Tcmd,PT.

The load torque Tload needs to be derived from the measure-ment data at each sampling interval. Two different approachesare taken:

1) Delay of Tload when compared with Tcommand:The load torque at time t, Tload(t), is approximatedby the command torque of the previous time interval,Tload(t + ts) = Tcmd(t), or, Tload(t) = Tcmd(t − ts).

2) Approximation of Tload using the mechanical equation:The load torque Tload is derived using the mechanicalequation (4) that becomes

Tload,PT = Tcmd,PT − BωPT + J∆ωPT

ts. (5)

Both approaches have been implemented. For this purpose,the motor and controller models (Figs. 3 and 4) have beenmodified to use the measured data Tcmd,PT and, in the caseof approach no. 2, ωPT and (dω/dt)PT as inputs. From these,

the new parameters Tload,PT∗,DPT∗, Vcmd,PT∗, and Icmd,PT∗ arederived. Here, the “∗” signifies that a values is obtainedthrough processing of the measured data TPT and ωPT. Fig. 9shows selected elements of the modified controller modelblock for method 2.

1kt

Ra

Ia Ra

Tcmd,PT DPT*Vcmd,PT*

PTddt J

Icmd,PT*

Tload,PT*

Fig. 9. Modified controller model block (selected) with the command andload torques derived from the experimentally obtained riding profiles; the “∗”signifies that a value is derived from recorded (measured) data.

Approach 1 completely ignores the outside drive parametersthat contribute to the riding profile, whereas approach 2includes some of these through the simplified single-wheelmodel. Even though the slopes of the route are not consideredexplicitly, some information is included in the accelerationthat is taken from the recorded (measured) data (dω/dt)PT.Therefore, we are using approach 2 in the following.

2) Speed: The recorded (measured) speed ωPT can be usedas input into the model to compute:

1) The back-EMF.2) The rolling friction feedback in the mechanical equation.Both approaches are implemented so that the parameters

that are derived from the motor speed are computed from theproper speed, even at operating points when the motor cannotproduce the commanded torque due to limitations imposed bythe stator current or back-EMF. The modified model of thedc-motor and the electromechanical system with ωPT used asinput for both computations is shown in Fig. 10. With thisapproach, the command signals that the drive receives arecompletely decoupled from the ability of the drive to producethe command torque at a given speed as well as from errorsintroduced by the single-wheel model behind the mechanicalequation.

kv

Vcmd,PT*

PT

Te0

1sx0

1La

1Ra

1kt

ktIa Te Tnet

0

1sx0

1J

B

Tload,PT*

sim

Fig. 10. Modified model of dc-motor and electromechanical system (constantmotor flux) with the command voltage, load torque, and the motor speed forthe computation of the back-EMF and the rolling friction feedback in themechanical equation derived from the experimentally obtained riding profiles;the “∗” signifies that a value is derived from measured data. Compare withFig. 3.

3) Efficiency calculations: With the measured speed ωPT

available and being used as input to one or both feedbackloops, it can be of interest to only concentrate on behavior andefficiency of the motor “before” the simplified single-wheelmodel of the mechanic system (mechanical equation). Thiscan be notably of interest with configurations and operating

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points where the drive cannot deliver the required torque ata given speed. Therefore, a second module to calculate themechanical power Pmech is implemented that uses the electricalmotor torque Te and the recorded (measured) speed ωPT asinput (Fig. 11).

Te

sim

PmechPmech

Te

PT

~

Fig. 11. Modification of Pmech module to be used when simulated speed,ωsim, and recorded (measured) speed, ωPT, are not in agreement.

IV. RESULTS

A. Riding interval profiles

The test-bicycle described above (Section III-A) was usedto obtain riding interval profiles as follows: The measure-ments were designed to obtain road data of “real-life appli-cations.” To this aim, riding interval profiles, profiles no. 1through 4, of four different riders and with intervals of 15 to 25minutes were recorded, where the bicycle was used for a shortleisurely ride, grocery shopping, or commuting (Table II).

TABLE II

CHARACTERISTICS OF RECORDED RIDING PROFILES

Profile Rider Riding Pmech,max Pmech,ave Tmax Taveno. weight interval

kg min W W Nm Nm1 50 18 204.0 35.6 27.9 4.72 75 16 389.1 133.9 40.8 8.23 85 22 368.6 66.3 26.4 5.94 95 25 857.0 179.0 50.2 9.9

Exemplarily, the measured torque versus time and powerversus time characteristics of riding profiles 1 and 4 areshown in Figs. 12–15. They illustrate the spread of the profilecharacteristics, and thus requirements on the drive, as ridingprofiles 1 and 4 are those with the lowest and highest torqueand power demands respectively (It should be noted that themaximum speed of the riding profile 4 exceeds the speed limitfor low-speed electric bicycles according to U.S. law, whichis 20mph.). The characteristics of riding profiles 2 and 3 arein-between these two extremes.

B. Simulation results

1) Parameters: The four rides were analyzed using thedeveloped model and example-case values of a dc-drive sys-tem. The base example-case values were adapted from thecommercially available electric bicycle that was also used astest vehicle for the riding profile measurements (Section III-A,Fig. 7). The bicycle has a 24V brushed dc-hub motor and abattery system of two 12V, 12Ahr lead acid batteries.

In this work, we seek to identify the limits of the drivesand the simulation techniques and not to optimize one singledrive configuration for one given riding profile. Therefore, weselect by intention a comparatively large value of the arma-ture resistance to account for low-cost motors and additionalresistance of the connections. The parameters of the motor

0

10

20

30

40

50

0 5 10 15 20 25Time [min]

Torque [Nm]

Fig. 12. Measured torque versus time of riding profile no. 1 (same scalesas Fig. 14 by intention), Tmax = 27.9Nm, Tave = 4.7Nm.

0

100

200

300

400

500

600

700

800

900

0 5 10 15 20 25Time [min]

Power [W]

Fig. 13. Measured power versus time of riding profile no. 1 (same scales asFig. 15 by intention), Pmax = 204.0W, Pave = 35.6W.

0

10

20

30

40

50

0 5 10 15 20 25Time [min]

Torque [Nm]

Fig. 14. Measured torque versus time of riding profile no. 4, Tmax =50.2Nm, Tave = 9.9Nm.

are set to Ra = 1Ω, La = 1mH, kv = kt = 1Nm/A, andRbatt = 2 · 12mΩ. The parameters of the PI-controller weredesigned to achieve 0.06s rise time, 5% overshoot,0.2s settlingtime, and 30rad/sec bandwidth (Kp = 100,Ki = 0.3). We setJ = 10kg/m2 according to [10].

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0

100

200

300

400

500

600

700

800

900

0 5 10 15 20 25Time [min]

Power [W]

Fig. 15. Measured power versus time of riding profile no. 4, Pmax = 857.0W,Pave = 179.0W.

Using these base example-case values of the differentparameters as starting point, we vary selected parametersthroughout the analysis. When doing so, the abilities of themodeling technique are analyzed using the performance ofthe low-cost example-case drive for illustration. For reasonsof comparison, only the first 840s (14min) of the rides aresimulated. The results illustrate well how the performances ofa given drive system depends on the characteristic of a givenriding profile.

2) Speed: A comparison of the simulated and the recorded(measured) speed shows the limitations given by the simplifiedsingle-wheel model of the mechanical system: Exemplarily,Fig. 16 shows an extract of riding profile no. 1. The simplifiedmodel does not reproduce the speed (dynamic behavior) ofthe drive correctly, notably for fast speed slopes, but the driveitself is able to produce the required torque as can be seenfrom Fig. 17.

It is important to recall that–as a result of the way therecorded data are used as inputs to the model–the commandsignals the motor receives are completely decoupled frominaccuracies of the speed simulation (Section III-C.2). Corre-sponding simulations were also carried out for the other ridingprofiles, with different values of the lumped inertia as smallas J = 0.1kgm/s2, and of the controller, leading to similarresults. In order to correct these inaccuracies, more complexmodels of the mechanical system, such as those suggested in[10] or [11], would need to be implemented and adapted towork with the measurement data as inputs.

As a result of the limits of the speed simulation, the outputefficiency needs to be calculated via the modified performancesubmodule, using the recorded (measured) speed ωPT and theelectric torque Te (Fig. 11).

3) Torque: Operating points can occur where the drivecannot produce the command torque due limitations imposedby the current or the back-EMF. Whenever the commandtorque exceeds the torque limit of the drive at its currentspeed, deep speed dips can be seen in the simulated speedωsim when compared with the recorded (measured) one, ωPT.For illustration, the simulation of riding profile no. 1 is

50 100 150 2000

5

10

15

Time [s]

Spee

d [r

ad/s

]

wPT

wsim

Fig. 16. Recorded and simulated speed wPT and wsim of riding profile no. 1(40s-230s): The simplified model does not reproduce the speed (dynamicbehavior) of the drive correctly, notably for fast speed slopes, but the driveitself is able to produce the required torque as can be seen from Fig. 17.

50 100 150 200−10

−5

0

5

10

15

20

25

30

Time [s]

Tor

que

[Nm

]

Tcmd,PT

Te

Fig. 17. Command and electrical torques Tcmd,PT and Te of riding pro-file no. 1 (40s-230s): The drive is able to produce the required torque.

shown in Fig. 18, where several of such dips can be seen (atapproximately 30s, 250s, 400s, 440s, 480s, 615s, and 720s).The inability of the drive to produce the command torque canfurther be seen in Fig. 19, where the electric torque does notreach the command torque whenever a speed dip can be seenin Fig. 18.

These limits of the drive are even more evident with thesimulations of the riding profile no. 4. (Figs. 20 and 21).

If not the recorded (measured) speed ωPT but the simulatedone ωsim were used as inputs into the feedback loops, and in a“real” case, such decreases in speed would cause a decrease ofback-EMF and thus increase of current and torque. However,as the motor does not receive the simulated, but the higherrecorded (measured) speed as inputs in the simulations, thesimulated torque remains low, further decreasing the simulatedspeed. Therefore, drawing incorrect conclusions from theserapid drops in the simulated speed drops must be avoided.

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0 200 400 600 8000

5

10

15

Time [s]

Spee

d [r

ad/s

]

wPT

wsim

Fig. 18. Recorded and simulated speed wPT and wsim of riding profile no. 1(first 840s); at approximately 30s, 250s, 400s, 440s, 480s, 615s, and 720s, thecommand torque exceeds the torque limit of the drive at the current speed.

0 200 400 600 800−10

−5

0

5

10

15

20

25

30

Time [s]

Tor

que

[Nm

]

Tcmd,PT

Te

Fig. 19. Command and electrical torques Tcmd,PT and Te of riding pro-file no. 1 (first 840s); at approximately 30s, 250s, 400s, 440s, 480s, 615s, and720s, the command torque exceeds the torque limit of the drive at the currentspeed.

Of course, such speed simulation results with speed dipswhere the motor cannot produce the command torque can evenless be used for energy consumption calculations than thosediscussed above (Section IV-B.2). Therefore, the modifiedperformance submodule (Fig. 11) needs to be used.

The orders of magnitudes are illustrated with the followingsimple approximation: With torque and back-EMF constant ofkt = kv = 1Nm/A, the back-EMF is 10V at 10rad/s speed.Neglecting a possible current limit of the machine, 14V are leftfor the voltage drop at the resistance(s), giving approximately14Nm torque.

4) Efficiencies: First, we consider the following two com-puted values for each of the four riding profiles: (i) The batteryenergy output, which is the energy input to the drive, Win,and is calculated from the battery voltage and current Vbatt

and Ibatt (Fig. 6.1). (ii) The drive output energy Wout,∗ whichis calculated from the recorded (measured) speed ωPT and theelectric torque Te (Fig. 11) (Table III).

0 50 100 150 200

−30

−20

−10

0

10

20

Time [s]

Spee

d [r

ad/s

]

wPT

wsim

Fig. 20. Recorded and simulated speed wPT and wsim of riding profile no. 4(first 200s): The drive is not able to produce the required torque. Thediscrepancy is that large that the simulated speed becomes negative.

0 200 400 600 800−10

0

10

20

30

40

50

Time [s]

Tor

que

[Nm

]

Tcmd,PT

Te

Fig. 21. Command and electrical torques Tcmd,PT and Te of riding pro-file no. 4 (first 840s): The drive is not able to produce the required torque.

TABLE III

RESULTS OF DRIVE SCENARIO SIMULATIONS

Profile Win Wout,∗ Wout,∗/Win Wprofile Wout,∗/Wprofileno. [Wh] [Wh] [Wh]1 29.36 9.36 0.31 12.03 0.782 34.48 15.45 0.45 16.36 0.953 37.03 22.37 0.60 24.93 0.904 30.00 17.29 0.58 54.76 0.32

From these two values, the ratio Wout,∗/Win can be calcu-lated, which has the form of an “equivalent efficiency.” Thisquantity does not consider if the drive is able to produce therequired torque, but it only considers the torque the motor candeliver. As both the input and the output energy are derivedfrom the recorded speed, these computations are not affectedby the inaccuracies of the simulated speed. Here, relative largeamounts of time during which the drive is operated at highspeed translates into higher values of this ratio for the differentriding profiles.

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Next, the energy requirement of the riding profiles, Wprofile,each for the first 840s, are considered (Table III), from whicha second ratio, Wout,∗/Wprofile, can be derived. From thisquantity conclusions on how much of the riding profile canbe produced by the drive can be drawn.

In the model, the armature and battery currents Ia and Ibatt

can be both positive and negative. As a result, the simulationsinclude the possibility of generating mode. This becomesnotably obvious with riding profile no. 4. At operating pointswhere the back-EMF that is computed from the recorded (mea-sured) speed ωPT exceeds 24rad/s, the back-EMF is larger thanthe maximum command voltage, the armature current becomesnegative (Fig. 10), eventually translating into a negative batterycurrent Ibatt (Figs. 4 and 5) that recharges the battery.

By comparing Wout∗ and Wprofile, conclusions on how muchof the riding profile can be produced by the drive can be drawn.

These results do not change with a change of J , as thedifferent energy computations and the command signals aredecoupled from the simulated speed. Furthermore, the simu-lations with different values of the controller, Kp = 2 andKi = 0.5, were carried out, leading to the same results.

Regarding the energy requirements of the different ridingprofiles in a more general way, most profiles could be sup-plied from one or 1.5 laptop-size batteries, when assuming aconservative estimate of 30% overall efficiency.

V. CONCLUSIONS

Different technical performance criteria of electric bicycledrive systems for given operating cycles can be evaluatedvia a model implementation that uses riding profiles basedon actual road tests. Such analysis can contribute to de-signing better, custom-designed electric bicycles and therebyovercoming common issues such as high cost and weight.The implementation shows how the characteristic of a givendrive depends on the characteristic of the riding profile. Thepresented tool can be used to develop more general answers tooperating areas of drives with respect to the different torque-speed combinations and their derivations with respect to timeas they can occur with electric bicycles.

APPENDIX

Appendix A: List of abbreviations

Acronym DefinitionDC Direct CurrentEMF Electro-Magnetic ForcePI Proportional-IntegralPWM Pulse-Width Modulation

Appendix B: List of symbols

Name DescriptionB rolling resistanceIa armature currentIbatt battery currentIcmd current resulting from command dutyIcmd,PT current resulting from command duty,

derived from measured valuesJ inertia

List of symbols continued

Name Descriptionkv back-EMF constantkt torque constantLa armature inductancePin input powerPmech mechanical powert timeRa armature resistanceRb battery resistanceTcmd command torqueTcmd,PT command torque, derived from recorded valuesTPT recorded (measured) torqueTe electric torqueTload load torqueTload,∗ load torque, derived from recorded valuesTnet net torqueTPT measured torqueVbatt battery voltageVc dc-link voltageVcmd command voltageVcmd,PT command voltage, derived from recorded valuesWbatt battery remaining energyWbatt0 battery capacityWin energy inputWmech mechanical energyω speedωPT recorded (measured) speed

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[9] Power Tap R© is by Graber Products, Inc., 5253 Verona Road, Madison,WI USA, http://www.cycle-ops.com.

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