Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First...

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Rotation Speed Varibles: MG1, MG2, and Engine: ϖ m1, ϖ m2, and ϖ ice. Power Efficiencies: Inverter, axle to tire, engine to MG1, reduction gear, and engine to axle efficiencies ηinv, ηaxle, ηeng_m1, ηred, and ηaeng_axle Motor M2 Grear Reduction GR m2 2.636 := MG1 Sun Planet Ring ICE MG2 Gear 2010 Prius Split Hybrid Drive www.springerlink.com/index/DL1Q76921M22P7ML.pdf http://autos.yahoo.com/green_center-article_24/ Patent:US6005297 Description and Design of Synergy (Planetary Gear) Drive http://www.mekatro.com/pdf/Eleco05_SPHEV.pdf Modeling and Simulation of Serial Parallel Hybrid EV http://en.wikipedia.org/wiki/Toyota_Prius Basic Description of Prius T.icem2Procedure: First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and power. Define vehicle and road parameters. MG1 Starter and Controller Acceleration Protocol : MG1 is the battery initiated starter for the ICE. "Simulate" MG1's spin startup, 1init m2 and 1init ice , as a ramp from 0 to its maximum speed, along with the motor M2 and ICE, respectively, which are also driven by the Prius Controller to ramp up. Refer to Startup Curves on bottom of page 6. This is an arbitrary set of conditions just to start the generator turning. Note that this startup is not a function of time, but of corresponding rotations. This startup condition simulates a low gear mode for the ICE. Evaluate two different Prius Control Strategies: Max Acceleration and Nominal. See curves on page 4. During acceleration, the largest torque is from motor/generator MG2, which is geared to the axle. The axle determines the vehicle speed. Use the MG2 rotation, ϖ m2 , as our control variable. During acceleration, supply peak electrical power to MG2 from both the battery (peak battery power for 10 seconds) and from MG1, which is now run as a generator at it's max speed and is mechanically driven by the ICE. As MG2 speeds up and demands increasing power, throttle back it's peak drive/torque by limiting the MG2's electrical power input to the peak Inverter Output, i.e. the sum of the battery's and MG1 generator's peak power. Refer to the curve of M2 Torque vs.ϖ m2 on page 3. Peak acceleration and 0 - 60 mph is programmed by setting the MG1 max speed: max m1 Use equations for vehicle dynamics to calculate the time to 60 mph. Plot performance simulation curves. Calculate All Electric Range, AER, miles from a single charge of the battery. Calculate AER using various EPA driving modes. Compare Prius Performance to GM Volt. Macro Model of Hybrid Vehicle Performance: Prius http://www.leapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.mcd PRIUS GEN IV 1.8L PERFORMANCE SIMULATION

Transcript of Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First...

Page 1: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

Rotation Speed Varibles:MG1, MG2, and Engine: ωm1, ωm2, and ωice.

Power Efficiencies: Inverter, axle to tire, engine to MG1, reduction gear, and engine to axle efficiencies ηinv, ηaxle, ηeng_m1, ηred, and ηaeng_axle

Motor M2 Grear

Reduction

GRm2 2.636:=

MG1 Sun

Planet

Ring

ICE

MG2 Gear

2010 Prius Split Hybrid

Drive

www.springerlink.com/index/DL1Q76921M22P7ML.pdf

http://autos.yahoo.com/green_center-article_24/ Patent:US6005297Description and Design of Synergy (Planetary Gear) Drive

http://www.mekatro.com/pdf/Eleco05_SPHEV.pdfModeling and Simulation of Serial Parallel Hybrid EV

http://en.wikipedia.org/wiki/Toyota_PriusBasic Description of Prius

T.icem2Procedure:First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and power. Define vehicle and road parameters. MG1 Starter and Controller Acceleration Protocol:

MG1 is the battery initiated starter for the ICE. "Simulate" MG1's spin startup, Ω1initm2 and Ω1initice, as a ramp from 0 to its maximum speed, along with the motor M2 and ICE, respectively, which are also driven by the Prius Controller to ramp up. Refer to Startup Curves on bottom of page 6. This is an arbitrary set of conditions just to start the generator turning. Note that this startup is not a function of time, but of corresponding rotations. This startup condition simulates a low gear mode for the ICE. Evaluate two different Prius Control Strategies: Max Acceleration and Nominal. See curves on page 4.

During acceleration, the largest torque is from motor/generator MG2, which is geared to the axle. The axle determines the vehicle speed. Use the MG2 rotation, ωm2, as our control variable. During acceleration, supply peak electrical power to MG2 from both the battery (peak battery power for 10 seconds) and from MG1, which is now run as a generator at it's max speed and is mechanically driven by the ICE.

As MG2 speeds up and demands increasing power, throttle back it's peak drive/torque by limiting the MG2's electrical power input to the peak Inverter Output, i.e. the sum of the battery's and MG1 generator's peak power. Refer to the curve of M2 Torque vs.ωm2 on page 3.

Peak acceleration and 0 - 60 mph is programmed by setting the MG1 max speed: Ωmaxm1

Use equations for vehicle dynamics to calculate the time to 60 mph. Plot performance simulation curves. Calculate All Electric Range, AER, miles from a single charge of the battery. Calculate AER using various EPA driving modes.

Compare Prius Performance to GM Volt.

Macro Model of Hybrid Vehicle Performance: Prius

http://www.leapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.mcd

PRIUS GEN IV 1.8L PERFORMANCE SIMULATION

Tom
VXPhysics Signage16
Page 2: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

ωmg1 2.6 ωmg2⋅+ 3.6ωice=

kiceη ρ⋅ Pc⋅

Ω ice_max=

Tice mice kice⋅ dgas⋅=

Pice mice kice⋅ Ωshaft⋅=

J, f, and Ωshaft are the moment of inertia, friction coefficient, and speed of the shaft. Tice and Tdcg are the ICE and generator torques. The ICE pressure, Pice, results from the flow of gasoline. η is the ICE efficiency, ρ the density of gasoline, Pc the calorific value of gasoline and Ωice_max the maximum rotation speed of the engine. The control of the machine is ensured by the mice ratio, which defines the actual flow of gasoline, dgas, through a specific actuator. The relation for the shaft rotational velocities ωice, ωmg1, and ωmg2 describes the action of the Hybrid Synergy Drive Planetary Gear (shown in the above illustration).

JtΩshaft

d

d⋅ f Ωshaft⋅+ Tice Tdcg−=

Simulation of a Series Hybrid Electric Vehicle based on Energetic Macroscopic RepresentationW. Lhomme1, A. Bouscayrol1, Member, P. Barrade2

Model Equations for the Driveshaft, ICE, and Planetary Gears

0 1000 2000 3000 4000 50000

1020304050607080

Prius 1.8L Atkinson ICE Model (kW)

Engine Speed (RPM)

Pow

er (

kW)

Pice w( )

w

0 1000 2000 3000 4000 5000100

110

120

130

140

150Prius 1.8L Atkinson ICE Torque Model

Engine Speed (RPM)

Tor

que

(N m

)

TiceE w( )

w

TiceE w( ) TIII ice w( ) XT⋅:=Pice w( ) PIIIice w( ) XP⋅:=XT142

115:=XP

73

57

73

76⋅:=Scale Up to Gen IV Specs:

PIIIice w( ) if w 5200> PIIIice 5200( ), PIIIice w( ),( ):=TIII ice w( ) if w 5200> TIII ice 5200( ), TIII ice w( ),( ):=

TIII ice w( ) if w Ω idle< 0, Pidle Tslope w Ω idle−( )⋅+, 60 k⋅

2 π⋅ w⋅⋅:=PIIIice w( ) if w Ω idle< 0, Pidle Pslope w Ω idle−( )⋅+, :=

Tslope60.23 Pidle−

4000:=Pslope

57 Pidle−

4000:=Pidle Tidle

2 π⋅ Ω idle⋅

60 k⋅

⋅:=Ω idle 1000:=Tidle 85:=k 1000:=

Generation III Torque and Power Models:http://www.leapcad.com/Transportation/Toyota_Prius_Model_Specifications.pdf

Specifications for Different Prius ModelsMatch Model Parameters to:

Scale the Power and Torque Models for Toyota Atkinson Gen III 1.5L ICE to Gen IV 1.8L

Prius Gen III ICE Performance Curves

Page 3: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

km 1.08:=Rotational Inertia Coeff:

Mcurb 3042 lb⋅:=Curb Weight:RRroad 0.002:=Road Rolling Resist:

Vcw 0 mph⋅:=Average Cross Wind:ρ 1.293gm

liter⋅:=Air Density:

θ (radians):θ atan 0.0( ):=(Average 0% road grade)Cdcw 0.000014:=Cross Wind Drag Coff:

RRtire 0.007:=Rolling Resistance Per Tire:Cd 0.25:=Drag Coeff:Af 1.98 m

2=Af Afg SCF⋅:=

v_r w( )2 π⋅ rtire⋅ w⋅ RPM⋅

GR mph⋅:=velMG2_at_rpm w( )

2 π⋅ rtire⋅ w⋅ RPM⋅

GR:=M2rpm_at_mph s( )

GR s⋅2 π⋅ rtire⋅ RPM⋅( ):=

Velocity vs Shaft (MG2 Rotaional Speed) and Hybrid Synergy Gearing

Fo 60 mph⋅( ) 358.884 N=Fo 60 mph⋅( ) 358.884 N=Fo v( ) Fa v( ) Fr v( )+:=Opposing Force, Fo:

Fa 60 mph⋅( ) 230.295 N=Fa v( ) 0.5ρ⋅ Af⋅ v Vw+( )2

Cd⋅ Cdcw 0.5 v⋅ Vcw+( )2⋅+

⋅:=Aerodynamic Loss, Fa:

Fr 60 mph⋅( ) 128.589 N=Fr v( ) Mgrossg⋅ RRtire RRroad+( ) cos θ( )⋅ sin θ( )+ ⋅:=Road Resistance, Fr:

Vehicle Dynamics Equations - Find Velocity and Time for Maximum Acceleration

Mbatt 68 kg⋅:=Mgross 3.212 103× lb=Mgross Mcurb Passengers2+:=Gross Weight:

Passengers2 170 lb⋅:=Passenger Weight:

Tmaxm2 207ft lbf⋅ GRm2⋅ ηaxle⋅:=Tmaxice 142 N⋅ m⋅ ηeng_m1⋅:=Tmaxm1 207 ft⋅ lbf⋅ ηeng_m1⋅:=Max Motor Torque:

Pmaxice 93 hp=Pmaxm1 47.137 hp=

Pmaxice 73 kW⋅ ηeng_m1⋅:=Pmaxm1 37 kW⋅ ηeng_m1⋅:=Pmaxm2 60 kW⋅ ηaxle⋅:=Max Motor Power:

η red 0.95:=ηeng_m1 0.95:=ηaxle 0.95:=ηeng_axle 0.95:=η inv 0.90:=GR 3.267:=Gear Ratio and Efficiencies:Specifications for Different Prius ModelsToyoto Prius Gen IV 1.8L-Vehicle, Motor and Road Parameters:

Gen III NHW20 MG2 Motor Performance Curves

Frontal Area Corrected:SCF 0.85:=Shape Correction Factor:

Afg 2.33 m2⋅:=Frontal Area*:Vw 0 mph⋅:=Average Wind Velocity:

Chasis and Environmental Parameters

Pmaxbat 27 kW⋅:=Energybat 6.5 kW⋅ hr⋅:=Tmaxm2 702.815 N m⋅=Redline Ωmaxice min⋅:=Set Pmaxbat to 0 for Extremum

Note MG1's power is speed (6500 rpm) limitedTmaxm1 Ωmaxm1⋅ 39.993 kW=

Ωmaxice 5200 RPM⋅:=Ωmaxm1 9000 RPM⋅:=Max. Acceleration Mode==>Max Motor Speeds:

RPM min1−:=rtire 0.287 m⋅:=

Page 4: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

MG1 Motor

0 10 20 30 40 50 60 70 80 90 100110 1201000

0

1000

2000

3000

4000

5000

6000

7000

8000Control Strategy: Nom ICE Idle Profile

Vehicle Velocity (mph)

Rot

atio

nal S

peed

s (r

pm)

0 10 20 30 40 50 60 70 80 90 100110 1200

1000

2000

3000

40005000

6000

7000

8000

90001 .10

4Control Strategy: Max Acceleration

Vehicle Velocity (mph)

Rot

atio

nal S

peed

s (r

pm)

Plot Colors: MG2 Shaft Rotational Speed (Red), ICE RPM (Blue), and MG1 Rotor (Green).Note Max Accel Plot: ICE does not turn on until MG2 propels Vehicle Speed up to 15 mph

Control Stategy Plots: Max Acceleration and Nominal Cruise Profiles

Ω ice M2rpm_at_mph 120 mph⋅( )( ) 6.712 103×=

Ω iceP1 ωmg2( ) Ω1Prflm1 ωmg2( ) 2.6 ωmg2⋅+( )3.6

:= Ω iceP2 ωmg2( ) Ω1Prfl2vm1 v_r ωmg2( )( ) 2.6 ωmg2⋅+( )3.6

:=

Use these expressions Just for generating plots at bottom of page

Ω ice ωmg2( ) Ω1Prfl2vm1 v_r ωmg2( )( ) 2.6 ωmg2⋅+( )3.6

:= Ωmg2 ωice( ) root xm2ωice 3.6⋅ Ω1Prfl2vm1 v_r xm2( )( )−

2.6− xm2,

:=

Ω ice ωmg2( ) Ω1Prflm1 ωmg2( ) 2.6 ωmg2⋅+( )3.6

:= Ωmg2 ωice( ) root xm2ωice 3.6⋅ Ω1Prflm1 xm2( )−

2.6− xm2,

:=

ICE and MG2 Rotation Profiles From MG1 Profiles Ω1Ω1Ω1Ω1Prfl1m1 (Enabled) or Ω1Ω1Ω1Ω1Pffl2m1 (Disabled ):

xm2 1500:=Ω1Prfl2vm1 v( ) if v 60< 65007500

60v⋅−, 1000− v 60−( )

4000

60⋅+,

:=

Examination of above then gives us the form of Ω1Prfl2vm1 v( )

Ω1Prfl2m1 ωmg2( ) 3.6 Ω iceVel ωmg2( )⋅ 2.6ωmg2−:=

Ω iceVel 5000( ) 4.159 103×=Ω iceVel ωmg2( ) Ω ice_vel

velMG2_at_rpmωmg2( )mph

:=Calculate Control Point:Rev up ICE from Cruise at Vehicle Velocity Velrev_up

See Plot Lower Right

Ω ice_vel vel( ) if vel Velrev_up< 1800, 1800 vel Velrev_up−( ) 4000 1800−100 Velrev_up−

⋅+,

:=Velrev_up 60:=

Use just Ω ice_vel vel( ) to find resultant relation between MG1 and Velocity, i.e. MG2

Nominal Control Strategy P2: Constant ICE (rpm= 1800) to 60 mph, then ramp up. Profile 2- Ω1Prfl2m1

Ω1Prflm1 ωmg2( ) Ω1initm2 ωmg2( ):=

Ω1initice ωice( ) if ωice 1700< Pmaxbat 1 kW⋅>∧ 1000Ωmaxm1

1950

ωice

RPM⋅+, Ωmaxm1 min⋅,

:=

Ω1initm2 ωmg2( ) if ωmg2 700< Pmaxbat 1 kW⋅>∧ 1000Ωmaxm1

820

ωmg2

RPM⋅+, Ωmaxm1 min⋅,

:=

Hybrid Synergy DriveMG1 Control Stategy:

Max MG1 RotationΩmaxm1

Initial M2 Spinup ( Ωm1),

Ω1initm2 ωmg2( )and Driving Profiles, Ω1Prflm1, Ω1Prfl2m1

Max Acceleration Control Strategy P1: Turn on ICE at 15 mph (See Control Plot)Apply maxium MG1 power/speed Ωmaxm1 = 8500 RPM from Generator MG1 to MG2.

Control Strategies (Hybrid Synergy Drive Speed Relationships):Develop an MG1 Profile and then Find Speed Relationships of MG2(ICE) and ICE(MG2)

Page 5: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

Ftω ωmg2( ) Ttot ωmg2( ) GR⋅ ηred⋅

rtire:=

Tractive Road Force, Total: Ftot v( )Ttot M2rpm_at_mph v( )( ) GR⋅ ηred⋅

rtire:= PM2 ω( ) Pm2 Ωmg2 ω( )( )

kW:=

Plot Terms: Ptot v( ) Ftot v( ) v⋅:= Ptot 60 mph⋅( ) 108.955 hp=

Applying maximum motor torque, find the velocity and time starting from initial velocity = 0 mph.

End 40:=

Third Law of Motion:Given

tv t( )d

d

Ftot v t( )( ) Fo v t( )( )−

km Mgross⋅= v 0( ) 0= velocity Odesolve t End,( ):=(dv/dt is acceleration)

Pt ωm2( ) Ttot ωm2( ) 2⋅ π⋅ ωm2⋅ RPM⋅ kW1−⋅:=

accel t( )Ftot velocity t( )( ) Fo velocity t( )( )−

km Mgross⋅:=

Prius Spec is 9.8 secTime 0 sec⋅:= time v( ) root v velocity Time( )− Time,( ):= time 60 mph⋅( ) 9.171s=

Passing 40 to 60 mph: Passing time 60 mph⋅( ) time 40 mph⋅( )−:= Passing 4.495s=

Given Control Acceleration Stategy Ω1Ω1Ω1Ω1Prfl1m1: Calculate Power and Torque

Torque, ICE: Ticem2 ωmg2( ) TiceE Ω ice ωmg2( )( ) N⋅ m⋅:= Ω1Prflm1 7000( ) 9 103×=

Torque,MG1: Tm1 ωmg2( ) Ticem2 ωmg2( )− 3.61−⋅ ηeng_m1⋅:=

PinvAm2 ωmg2( ) η inv Pmaxbat Tm1 ωmg2( ) 2⋅ π⋅ Ωmaxm1⋅−( )⋅:=

PinvBm2 ωmg2( ) if PinvAm2 ωmg2( ) Pmaxm2 η inv⋅≥ Pmaxm2 η inv⋅, PinvAm2 ωmg2( ),( ):=

Max Power to Inverter: < P2max, P1max + Pbatmax and Pm1< P1max and Pice x 2.6/3.6

MG2 Inverter Power, Pinv:Pinvm2 ωmg2( ) if Tm1 Ω1initm2 ωmg2( )( ) 2⋅ π⋅ Ωmaxm1⋅( )− Pmaxm1> Pmaxbat Pmaxm1+( ) η inv⋅, PinvBm2 ωmg2( ), :=

Torque MG2 Power LimitedBreak Point, ωinv

Problem, Find ωinv such that:

(Tmaxm2) ωinv = Pinvm2(ωinv)

Guess for winv, wx: wx if Pmaxbat 1 kW⋅< 8, 800,( ):=

Solution (rpm): ωinv root Tmaxm2

Pinvm2 wx( )

2 π⋅ wx 1+( ) RPM⋅− wx,

:= ωinv 759.842=

Torque, MG2: Tm2 ωmg2( ) if ωmg2 ωinv≤ Tmaxm2,Pinvm2 ωmg2( )2 π⋅ ωmg2 RPM⋅

,

:= Tm2 ωinv( ) 2⋅ π⋅ ωinv RPM⋅ 55.923kW=

Tractive Torq, Total: Ttot ωmg2( ) Tm2 ωmg2( ) Ticem2 ωmg2( ) 2.6

3.6⋅ ηeng_axle⋅+:= Pm2 ω( ) Tm2 ω( ) 2⋅ π⋅ ω⋅ RPM⋅:=

Tractive Road Force, Total:

Page 6: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

P R I U S G E N I V P E F O R M A N C E S I M U L A T I O N C U R V E S

0 1000 2000 3000 4000 5000 60000

100

200

300

400

500

600

700

800

900Torque (Total,MG2,ICE,MG2) vs. Speed

MG2 Motor Speed (RPM)

Dyn

o M

otor

Tor

que

(Nm

), P

ower

(hp

)

Ttot ωm2( )Tm2 ωm2( )Ticem2 ωm2( )

Tm1 ωm2( )−

ωm2

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

80Velocity & g Acceleration (x 100) Curve

time (sec)

velo

city

(m

ph) velocity t( )

mph

accel t( )

g100⋅

t

MG2 Torque/Power is Limited byICE & BatteryPeak Power

0 10 20 30 40 50 60 70 80 90 100150025003500450055006500750085009500

Dyno Road Force (N) vs. Velocity (mph)

velocity (mph)

Mot

or F

orce

(N

)

Ftot v mph⋅( )

N

v

0 1000 2000 3000 4000 5000 60000

1020304050607080ICE and MG2 Power (kW) vs. ICE RPM

ICE Speed (RPM)

DY

NO

Pow

er (

kW)

PM2 ω ice( )Pice ω ice( )

Redline

ω ice

0 1000 2000 3000 4000 5000 60000

102030405060708090

100Dyno Power vs. ICE RPM

ICE Speed (RPM)

Dyn

o P

ower

(kW

) Pmax.icePt Ωmg2 ω( )( )Pm2 Ωmg2 ω( )( ) ξ⋅

Pice ω( )

Redline

ω0 1000 2000 3000 4000 5000 6000

0

100200300400500600700800900

Dyno Axle Torque vs. RPM

Axle (MG2) Speed (RPM)

DY

NO

Mot

or T

orqu

e (N

m)

Ttot ω( )

Tm2 ω( )

Ticem2 ω( )

ω

0 1000 2000 3000 4000 50000

100020003000400050006000700080009000

1 .104MG1 and MG2 Startup vs. ICE Spin

Engine Revs (RPM)

Mot

or R

evs

(RP

M)

Ω1initm2 ωmg2( )Ω1initice ω ice( )Ωmg2 ω ice( )

ωmg2 ω ice, ω ice,0 1000 2000 3000 4000 5000 6000

0

100200300400500600700800900

Dyno Torque vs. ICE RPM

ICE Speed (RPM)

Dyn

o M

otor

Tor

que

(N

m)

Ttot ω( )

Tm2 Ωmg2 ω( )( )TiceE ω( )

ω

<===These are an arbirary set of ICE and MG1 initial

conditions just to start the MG1

generator turning.

Page 7: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30Single Charge Cruise Mileage Range

velocity (mph)

Cru

ise

Ran

ge (

mile

s)

Cruise_Range v mph⋅ 50, SOCgen,( )mi

Cruise_Range v mph⋅ 100, SOCgen,( )mi

v

v 0 2, 80..:= Cruise_Range 70 mph⋅ 50, 0.5,( ) 14.31mi=

Cruise_Range 62.5 mph⋅ 50, SOCgen,( ) 16.697 mi=Velocity Range

Cruise_Range 60 mph⋅ 50, SOCgen,( ) 17.601 mi=

Cruise_Range 55 mph⋅ 50, SOCgen,( ) 19.591 mi=

Cruise_Range 50 mph⋅ 50, SOCgen,( ) 21.842 mi=

Cruise_Range 40 mph⋅ 50, SOCgen,( ) 26.844 mi=

Cruise_Range 30 mph⋅ 50, SOCgen,( ) 32.869 mi=

Single Charge Highway Cruise Range Estimate

Cruise_Range v Po, S,( )Energybat 1 S−( )⋅ NumStarts v Po, S,( ) Energyaccelv( )

TInvE GPE⋅

Regen Mgross⋅ v2⋅

2−

⋅−

v⋅

PowerdissLossv Po,( ):=

NumStarts v Po, S,( ) floor

Energybat 1 S−( )⋅ NS v Po, S,( ) Energyaccelv( )

TInvE GPE⋅

Regen Mgross⋅ v2⋅

2−

⋅−

PowerdissLossv Po,( ) Timessr⋅

:=

NS v Po, S,( )Energybat 1 S−( )⋅ NSo v( )

Energyaccelv( )

TInvE GPE⋅

Regen Mgross⋅ v2⋅

2−

⋅−

PowerdissLossv Po,( ) Timessr⋅:=NSo v( ) 2

50 mph⋅v

2

⋅:=

NSo and NS are iterative converging estimates of NumStarts

Energyaccelv( ) Pmaxm2 time v( )⋅:=

RTEff 0.92:=PowerdissLossv Po,( ) Fo v( ) v⋅TInvE GPE⋅

Po watt⋅

IPEE+:=

USABC Round Trip Battery Energy Efficiency

SOCgen 0.5:=Regen 0.69:=GPE 0.97:=IPEE 0.95:=TInvE 0.90:=Timessr 30min:=

State of Charge for generator is SOCgen. SOCgen is 50% for recharge. 201V HV battery idle power is Po. 12V battery gives Accessory Power. The Traction Inverter x motor Efficiency - TInvE, HV Power Electronics at Idle Efficiency - IPEE, and Gear Power Efficiency - GPE are 90%, 95%, and 97%, respectively. Brake Regen efficiency of kinetic energy is 69% @ deacceleration = 0.315g. Then the number of starts per hour as a function of velocity, NS, NumStarts(v, Po), is

Drive Train Power Efficiency - Battery Loss to Force Commanded Vehicle Velocity:

Driving Pattern/Profile: Given we cruise at constant speed and Time for start, stop, and regen breaking, Timessr= every 15 minutes.

Find the Single Charge (@SOC = 50%) Cruise Range for a given Velocity

Page 8: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

Cruise Range as a Function of Traction Battery Idle Power, Po

Cruise_Range 15 mph⋅ 50, 0.3,( ) 57.727 mi=

Cruise_Range 55 mph⋅ 50, 0.5,( ) 19.591 mi=

180km

hr⋅ 111.847

mi

hr=

0.5 0.25−0.5

0.5=Cruise_Range 55 mph⋅ 100, 0.25,( ) Cruise_Range 55 mph⋅ 100, 0.5,( )−

Cruise_Range 55 mph⋅ 100, 0.5,( )0.482=

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 800

10

20

30

40Single Charge Cruise Mileage Range

velocity (mph)

Cru

ise

Ran

ge (

mile

s)

Cruise_Range v mph⋅ 50, 0.3,( )

mi

Cruise_Range v mph⋅ 50, 0.4,( )

mi

Cruise_Range v mph⋅ 50, 0.5,( )

mi

Cruise_Range v mph⋅ 100, 0.3,( )

mi

Cruise_Range v mph⋅ 100, 0.4,( )

mi

Cruise_Range v mph⋅ 100, 0.5,( )

mi

v

Page 9: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

n6 0 598..:=US06 US06F1⟨ ⟩:=time US06F0⟨ ⟩:=

HWY10V submatrix HY10 0, rows HY10( ) 1−, 1, cols HY10( ) 1−,( ):=

FTP10V submatrix FP10 0, rows FP10( ) 1−, 1, cols FP10( ) 1−,( ):=

Rhwy rows HWY( ):=HWY HWYF 1⟨ ⟩:=

rows UDDS( ) 1.37 103×=UDDS UDDSF1⟨ ⟩:=

rows FTP( ) 1.875 103×=FTP FTPF1⟨ ⟩:=tx FTPF0⟨ ⟩:=

The Federal Test Procedure(FTP) is composed of the UDDS followed by the first 505 seconds of the UDDS. It is often called the EPA75. FP10 is a 10 Hz Sampling. HY10 is the 10 Hz Highway schedule.

The US06 cycle represents an 8.01 mile (12.8 km) route with an average speed of 48.4 miles/h (77.9 km/h), maximum speed 80.3 miles/h (129.2 km/h), and a duration of 596 seconds.

http://www.epa.gov/nvfel/testing/dynamometer.htmRead US06 and FTP Driving Profile Files

Find the Power to Maintain Constant Velocity

Powercruise v Po,( ) PowerdissLossv Po,( ):=Note: The generator's output is 54 kW. This allows it produce a net charge up to 80 mph cruise. Powercruise 60 mph⋅ 100,( ) 11.132 kW=

n 0 200..:= τnn

10:= w

nn

20:= vv

nn

2:=

Pcruisen

Powercruise vvn

mph⋅ 100,( )k watt⋅

:=

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

40

45Cruise Power Curve

velocity (mph)

Cru

isin

g P

ower

(kW

atts

) Powercruise vvn mph⋅ 10,( )k watt⋅

Powercruise vvn mph⋅ 50,( )k watt⋅

Powercruise vvn mph⋅ 100,( )k watt⋅

vvn

FTPF READPRN "http://www.leapcad.com/Transportation/FedTestProc.TXT"( ):=

UDDSF READPRN "http://www.leapcad.com/Transportation/uddscol.txt"( ):=

HWYF READPRN "http://www.leapcad.com/Transportation/hwycol.txt"( ):=

FP10 READPRN "http://www.leapcad.com/Transportation/FTP10Hz.TXT"( ):=

HY10 READPRN "http://www.leapcad.com/Transportation/HWY10Hz.txt"( ):=

US06F READPRN "http://www.leapcad.com/Transportation/US06PROFILE.TXT"( ):=

AER Given Three Different Driving Schedules

Page 10: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

max Distance.( ) 80.08=max Distance( ) 110.414=

Distance.rr

0

rr

rr

US06rr∑

=

10

60 60⋅⋅:=rr 0 rows US06( ) 1−..:=Distance

r0

r

r

FTPr∑

=

10

60 60⋅⋅:=r 0 rows FTP( ) 1−..:=

MPGepa 24.98=MPGepa 0.55 MPGcity⋅ 0.45 MPGhwy⋅+:=

X1

40:=MPGhwy

1

0.0013761.3466

AER HWY 1,( )+

:=MPGcity 25.992=MPGcity1

0.0032591.18053

AER FTP 1,( )+

:=

EPA 20085 Cycle MPG Fuel Economy Least Squares Fit Regression for AER to SOC = 0

AER HWY10 10,( ) =AER HWY 1,( ) 33.053=AER FTP 1,( ) 33.524=AER US06 1,( ) 22.039=

HWY10r1

HWY10Vceil

r1 1+

10

1− mod r1 10,( ),:=r1 0 rows HY10( ) 10⋅ 1−..:=

AER P Hz,( ) Ebat Ediss vold 0←←←

n 1−←

N rows P( ) 1−←

n n 1+←

t mod n N,( )←

v Pt

vavg v vold+( ) 0.5⋅←

Paccel

km Mgross v vold−( )⋅mph Hz⋅

sec⋅ vavg⋅⋅ mph

TInvE GPE⋅← v vold>if

Paccel km Mgross v vold−( )⋅mph Hz⋅

sec⋅ vavg⋅ mph⋅ REff vold v−( ) Hz⋅ Gg⋅ ⋅← otherwise

Ediss EdissPowerdissLossv mph⋅ 100,( ) Paccel+( ) sec⋅

kW hr⋅ Hz⋅+←

vold v←

Ebatn

Ediss←

EdissEnergybat

kW hr⋅<while

R

0

n

m

Pmod m N,( )

Pmod m 1+ N,( )

+( ) mph⋅ sec⋅

2 mi⋅ Hz⋅∑=

R

:=

REff g( )85

770.01⋅ 1 e

27.129− g⋅−( ) 91.235⋅ 28.408− ⋅:=Regen Efficiency Curve vs Decel (g): Ggmph

sec g⋅:=

All Electric Range, AER, for Driving Profile Velocity/Time File, P Sampling Rate, Hz, and SOC = 0

Page 11: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

SAVE PROFILES

WRITEPRN "EFTP.PRN"( ) AER FTP 1,( ) 40⋅:= EFTP READPRN "EFTP.PRN"( ):= max EFTP( ) X⋅ 27.275=

WRITEPRN "EUS06.PRN"( ) AER US06 1,( ) 40⋅:= EUS06 READPRN "EUS06.PRN"( ):= max EUS06( ) X⋅ 19.263=

WRITEPRN "EHWY.PRN"( ) AER HWY 1,( ) 40⋅:= EHWY READPRN "EHWY.PRN"( ):= max EHWY( ) X⋅ 26.8=

0 50 100 150 200 250 300 350 400 450 500 550 6000

102030405060708090

100US06 Drive Cycle: Distance, E(Bat Drain)

Time (seconds)

Spe

ed (

mph

), D

ist

(Mile

/10)

, E

(kW

*40)

US06

Distance.

EUS06

time

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 19000

10

20

30

40

50

60

70

80

90

100

110FTP Drive Cycle: Distance, E(Bat Drain)

Time (seconds)

Spe

ed (

mph

), D

ist

(Mile

/10)

, E

(kW

*40)

FTP

UDDS

Distance

EFTP

tx

Page 12: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

Pvv Volt 8⟨ ⟩:= PcruiseV Volt 9⟨ ⟩:=

ωxn

6000 n⋅200

:= Ttotn

Ttot ωxn( ) q⋅:= Tm2

nTm2 ωx

n( ) q⋅:= Ticen

Ticem2 ωxn( ) q⋅:= Vx

nvelocity tz

n( ):=

Ax Vx( )Ftot Vx mph⋅( ) Fo Vx mph⋅( )−( ) 100⋅

km Mgross⋅ g⋅:= ax

nAx Vx

n( ):= Ftn

Ftot Vxn

mph⋅( )N

:= Ptn

Ptot Vxn

mph⋅( )hp

:=

PreWtTotM2ICVA augmentωx tz, Ttot, Tm2, Tice, Vx, ax, Ft, Pt,( ):= WRITEPRN "Pre3PB0.txt"( ) PreWtTotM2ICVA:=

2010 Prius (Black) vs. Sustaining Mode Volt (Red) Acceleration Performance

Volt Power reduced from 111 kW to Generator Power x 90% = 47.7 kW in Sustaining Mode. Volt 0 to 60 mph time increases from 8.5 to 15 seconds.

Volt 40 mph to 60 mph Sustaining passing time increased from 3.3 seconds to 8.5 seconds.

0 1000 2000 3000 4000 50000

100

200

300

400

500

600

700

800

900Torque (Total,MG2,ICE,Volt) vs RPMs

Motor Rotor Speed, (RPM)

Dyn

o M

otor

Tor

que

(Nm

)

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

80Vehicle Velocity & g Acceleration (x100)

time (sec)

Veh

icle

vel

ocity

(m

ph)

and

g's

x 10

0

0 10 20 30 40 50 60 70 80 90 1001500

2500

3500

4500

5500

6500

7500

8500

9500Dyno Road Force (N) vs. Velocity (mph)

Dyno velocity (mph)

Dyn

o M

otor

For

ce (

N)

0 10 20 30 40 50 60 70 800

20

40

60

80

100

120

140

160Dyno Power (hp) vs. Velocity (mph)

Dyno velocity (mph)

Dyn

o P

ower

(hp

)

Read Charge Depletion Mode Data Read Charge Sustaining Mode Data (Disabled)

Volt READPRN "Volt_tVelwMAgTPmFP.prn"( ):= Volt READPRN "VoltSus_tVelwMAgTPmFP.prn"( ):=

Volt READPRN "VoltGenOnly_tVelwMAgTPmFP.prn"( ):= PT v( )Ptot v mph⋅( )

hp:=

cols Volt( ) 10=n 0 300..:= Ptot v( ) Ftot v mph⋅( ) v⋅ mph⋅ hp

1−⋅:=

timev Volt 0⟨ ⟩:= vv Volt 1⟨ ⟩:= ωv Volt 2⟨ ⟩ k⋅:= mphv Volt 3⟨ ⟩:= gv Volt 4⟨ ⟩:= tzn

n

10:=

q N m⋅( )1−:= Tv Volt 5⟨ ⟩:= Pv Volt 6⟨ ⟩:= Fv Volt 7⟨ ⟩:=

Page 13: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100110120

Cruise Power Demand Curve

velocity (mph)

Cru

isin

g P

ower

(kW

atts

)

Fuel_Use READPRN "http://www.leapcad.com/Transportation/Prius%201_5L%20Atkinson%20Fuel%20Use.TXT"( )T:=

0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100DYNO Power (hp) vs. Velocity (mph)

DYNO velocity (mph)

DY

NO

Pow

er (

hp)

0 10 20 30 40 50 60 70 80 90 1001500

2500

3500

4500

5500

6500

7500

8500

9500Road Force (N) vs. Velocity (mph)

velocity (mph)

Mot

or F

orce

(N

)

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

80Velocity & g Acceleration (x 100) Curve

time (sec)

velo

city

(m

ph)

and

g's

x 10

0

0 1000 2000 3000 4000 50000

100

200

300

400

500

600

700

800

900Torque (Total,MG2,ICE) vs. Speed

Motor Speed (RPM)

Mot

or T

orqu

e (N

m)

Comparison Prius No Battery (Solid) vs. Battery (Dotted)

Comparison Prius Gen III (Sold) vs. Prius Gen IV (Dotted)Agx t( )

accel t( )

g:=Pz Pre8⟨ ⟩:=Fz Pre7⟨ ⟩:=

az Pre6⟨ ⟩:=Vz Pre5⟨ ⟩:=Ticez Pre4⟨ ⟩:=Tm2z Pre3⟨ ⟩:=Ttotz Pre2⟨ ⟩:=timez Pre1⟨ ⟩:=ωz Pre0⟨ ⟩:=

PreWtTotM2ICVAPre READPRN "PreGenIII.txt"( ):=Pre READPRN "Pre3PB0.txt"( ):=

Read Comparison Files Either for GenIV No Battery Power or Full Power Gen III Prius

Page 14: Mathcad - Prius Gen IV 1.8L Simleapcad.com/Transportation/Prius_Gen_IV_1.8L_Simulation.pdf · First model the Prius Internal Combustion Engine's (ICE) and the motor's, torque and

Switch Solid and Dotted Curves

Comparison Prius No Battery (Dotted) vs. Battery (Solid)

Comparison Prius Gen III (Dotted) vs. Prius Gen IV (Solid)

0 1000 2000 3000 4000 50000

100

200

300

400

500

600

700

800

900Torque (Total,MG2,ICE) vs. Speed

Motor Speed (RPM)

Mot

or T

orqu

e (N

m)

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

80Velocity & g Acceleration (x 100) Curve

time (sec)ve

loci

ty (

mph

) an

d g'

s x

100

0 10 20 30 40 50 60 70 80 90 1001500

2500

3500

4500

5500

6500

7500

8500

9500Road Force (N) vs. Velocity (mph)

velocity (mph)

Mot

or F

orce

(N

)

0 10 20 30 40 50 60 70 80 90 1000

102030405060708090

100DYNO Power (hp) vs. Velocity (mph)

DYNO velocity (mph)

DY

NO

Pow

er (

hp)