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Difference between force and constant power control

applied to electrical bike

Arnaud Sivert, IEEE Reviewer E-mail : arnaud.sivert@iut.u-picardie.fr

Franck Betin, E-mail : franck.betin@u-picardie.fr IEEE Member,

Sebastien Carrière, E-mail : sebastien.carriere@iut.u-picardie.fr

Laboratoire des Technologies innovantes (L.T.I)

Institut Universitaire de Technologie de l’Aisne, I.U.T, 13 av. F.Mitterrand, 02880 Cuffies, France

Abstract:

This paper outlines the benefits of the constant power control compared to the constant force control.

The constant power control allows to cancel the intensity peaks supplied by the battery and have better

dynamic speed. Indeed, for the same energy consumption during acceleration, displacement is larger

with the constant power control. However, this control strategy causes a current peak motor. Therefore,

there are trade off which exist between the constant power and constant driving force to control a

motor. Many curves present in theory and practice the two control strategies.

The power constant control is obviously applicable to any electric vehicle. We applied the two

commands to 1500W brushless electric bikes from a test bench. These bikes reach 60 km/h with a

difficult compromise between weight, power, autonomy and price. The constant power control is the

most suitable because it increases the life of batteries which represent 35% of price. The constant

power control is achieved through regulation of the battery current and not of regulation motor.

However, a limitation of motor current priority must be made for low speed values. In addition, the

battery current control makes it easy to limit the current to 1C during deceleration or downhill runs

(regeneration).

Keywords: control strategy, electrical bike, torque control, power control.

1. Introduction

The paper deals demonstrate the benefits of the

constant power control compared to the constant

force control. The constant power control is used

to cancel the intensity peaks supplied by the

battery and to have better dynamic speed [1]. The

benefits of power control are obviously

applicable to any electric vehicle. In the paper,

the both strategies control will apply to a

brushless motor of 1500W for bike. It is possible

to use these bike engines 60 km/h by taking

insurance for lower engine power inferior at

4000W. These e-bikes have a difficult

compromise between weight, power, autonomy

and price. These bikes were made possible thanks

to new battery Li-po or Li-ion. But to increase

the lifetime of batteries which represents 35% of

the price cycle, we applied the constant power

control which allows limiting the intensity peaks

during acceleration.

2. E-bikes

Our bikes can go 60 km/h on flat road for the

most powerful. The acceleration is 4 seconds to

reach 36 km/h. The maximum intensities for

battery and for motor, the maximum speed and

acceleration time can be configured in the

controller. In 2010, the cost of our bikes was €

1.400 with the instrumentation for a power of

1500 W. In 2011, the cost decreased to € 1.000.

These bikes do not need pedal assistance but only

800 W, 42 Km/h,

a handful throttle accelerator [3].

Liion 48V 12 A.H,

1500 W, 60Km/h,

Lipo 54V 16 A.H c

500 W, 36 Km/h,

Lipo 48V 5 A.H,

Fig 1: Our electric bikes from 500W to 1500W.

1

Bicycle DC Motors are brushless wheel motor

that have very high specific power rate but it is

possible to use classical motors outrunner of

4000W. The controller (1500W, 60V, 40A max,

2400W max) can brake and reload the battery on

the road downhill. The charger of battery reloads

until to 10A and can balance Li-po to 5A. We

will now mathematically quantify the electric

bike to know these features and understand its

control. For the sake of simplicity, we will not go

into details of the mechanical losses of the motor,

or control (regulation speed and current), internal

resistance of batteries. But, the reader can

download the detailed study carried out by our

students on the website: http://aisne02geii.ekart.fr/.

Now, we will see the force and power required

by e-bike in steady state.

3. Forces and power in steady state speed

In a steady state speed, the motor force is equal to

the resistive force. This force depends on the

bearings, the tires, the road slope, and the air

resistance.

Their respective equations are:

(1)

FP ( N) = M (kg ) ⋅ g ⋅ slope(%) with g=9.81 (2)

Fresistive ( N) = FRoad + FP + FA

FA ( N) = f ⋅ [V(Km / h ) + Vwind ]2

(3)

The Froad depends of the pavement and driver

weight. It is negligible compared to the air

resistance FA. The power needed can be observed

in a steady state speed on figure 2.

The power corresponds in steady state speed to

the following equation (4):

Presistive ( W ) = Fresistive ( N ) ⋅

V(km / h )

= Phumane + Pelec

3,6

The average human power is setting from 150W

to 300W for a pedaling rate from 10 to 100 rpm.

The cyclist is always adjusting the gear ratio to

the relief in order to obtain the same power and a

constant pedaling rate due to the resistance

power. Now that the power of resistance is

known, the accelerating force to start the vehicle

must be studied. The motor are often controlled

using force or constant torque strategy. We will

see the dynamics of these types of speed control.

4. Motor force control

We will use the constant force to accelerate and

decelerate the vehicle. These forces are limited

by the values of motor intensity which is

configured in the controller.

The cyclist fixes the motor reference with the

throttle handle. The electro mechanicals relations

of the engine are:

v(m.s-1)=Um /k = α. UBatt /k

Fm ( N) = I m ⋅ k ⋅ ηmotor

(5)

(6)

Where Um and Im are the motor voltage and

current. The α coefficient varies from 0% to

100%. It’s the PWM duty cycle delivered by the

controller.

The mechanical and electrical power is

determined by the following equation:

P ( W) = Fresitive ⋅ v( t ) = α ⋅ U Batt ⋅ I Batt ⋅ η motor

Power (Watt)

slope de 5%

0%

slope road -5%

Speed (km/h)

Figure 2: Motive power vs speed for

different slopes [ M=100kg, f=0,26 N/(m.s-1)2 ]

(7)

With Ubatt and Ibatt the batteries voltage and

current, η the efficiency.

For simplicity in steady state speed, the resisting

force will be considered constant at 30 N, the

mass of bike and rider is 100 kg. It can be seen in

Figure 3 that the intensity limit is set to start at 56

A. So the driving force of 280N will start because

k is equal to 5.

The dynamics of the speed is imposed by the

fundamental mechanical equation following:

Fm = M

dv

+ FRe sistive

dt

(8)

2

We can observe on the fig 3 that during the motor

force is to 280 N and during the deceleration, it

decreases to -220N.

The speed dynamic is determined by the

differential equation (8):

v( m / s) =

( Fm − Fresistive )

⋅ t + v ( t = 0)

M

Intensity motor(A)

56A

6A

(9)

-44 A

The acceleration and deceleration time will be 4

seconds to reach 10 m/s.

On Figure 3, motor power and energy

consumption can be seen. This energy is

composed of course of the kinetic energy and of

the energy required by the resisting force. The

energy corresponds to the following equation

(10):

Speed (km/h)

Forces electric motor (N)

36 km/h

280 N

∫

E(W.H) = E kinetic + Eforce resistive = Fm (t) ⋅ v(t) ⋅ dt

30 N

For example, during acceleration, the energy

required is equal to:

1

V t2

2

E ( W.H) = M ⋅ V + Fresistive ⋅

⋅

2

t acc 2

(11)

We can see that the energy recovered during

braking is almost equal to the energy of

acceleration recess near the resisting force. Note

also the peak intensity to be provided by the

batteries during acceleration and advanced

regeneration during deceleration. However, all

batteries are limited in rate of discharge and

charge currents in such a way to do not destroy

them. So there are trade off between the desired

dynamics and the maximum currents allowed by

the battery.

The following table shows that price depends on

the rate of discharge for an accumulator 48V

10A.H.

It can also be observed with this table that the

weight and volume increases when a current peak

battery is required [4]. In addition, it can be seen

that the current deceleration must be set by the rate

of battery charge.

To cancel the current peak of the outgoing and

incoming batteries, we will see that the constant

power control is suitable.

tRE=10s

tacc=4s

deplacement

20

tdec=4s

-220 N

120

time

140 m

Power motor (W)

Intensity battery(A)

2800 W

56 A

300 W

6A

-44 A

-2200

Energy (W.H)

2.4 W.H

Energy

recover

1.5 W.H

1.16 W.H

Fig 3: force, power, energy and intensity according

to trapezoidal speed profile for a battery of 50V.

Kinds of battery

li-po

10 A.H 12S

13 A.H 12S

12 A.H 12S

Table 1 : Comparison of batteries different 48V

Size & Volume cm3

Mass

Price

charge rate

kg

2011

max

(1*10.6*10.2) 1300

2.5 kg

420 €

10 A= 1C

(0.6*20.8*13) 2000

3.9 kg

620 €

20 A= 1.5C

(0.8*20.8*13) 2600

4.3 kg

800 €

24 A= 2 C

discharge rate

max

30 A = 3C

104 A = 8C

180 A = 15 C

R

1 mΩ

? mΩ

? mΩ

3

5. Motor control with constant power.

When the battery power is limited, the bike runs

at constant engine power Pmlimit [1] [5].

To know the speed dynamics, we have to solve

the following differential equation:

Pm lim it

V

=M

V (m / s) =

D( m) =

dV

+ Fresistive

dt

2 ⋅ Pm lim it ⋅ t

M

2 ⋅ Pm lim it

M

+ V ( t = 0) 2

Speed (m/s)

constant power

(12)

constant force

(13)

movement (m)

2

⋅ ⋅ t3/ 2

3

27 m

(14)

By neglecting the resistive force, the equation

(12) can be solved and gives the dynamics of

velocity and displacement corresponding to

equations (13) and (14)

If the resisting force cannot be neglected, the

differential equation is not resolvable then it will

be simulated as shown in Figure 4 to compare the

two control strategies.

If the power is limited to 1400W corresponding

to the average power (Figure 3) during the

acceleration, it will also take 4 seconds to reach

the speed of 36 km/h (10 m/s). The energy will

be the same for the two control strategies. But

with the constant power control, the distance will

be 27 m instead of 20m. Indeed, the dynamic

speed at constant power is higher than that of the

driving at force constant as it can be seen in

Figure 4. In addition to the constant power, the

battery intensity is constant equal to 28A during

acceleration and does not reach 56A.

Nevertheless, the motor intensity is very

important for the low speeds. To not oversize the

switches of controller, the motor intensity will be

limited to 150A. The dynamics with this

limitation can be seen on the figure 5.

constant power

20 m

constant force

time (s)

Fig 4 : Energy, speed, distance for control power

and constant force with a load of 30 N.

When the motor current is limited, this causes a

ramp on the current battery until it reaches 28A

corresponding to the constant power. So the

dynamics of speed will be slightly lower

compared to Figure 4. But if the constant power

to 1470 W is increase, the speed reached 10 m/s

in 4 s.

limitation motor intensity

Motor intensity

Battery intensity

Limitation intensity battery 28A

Speed (m/s)

Energy of battery(j)

Without limitation motor intensity

constant power control

With limitation motor intensity

constant force control

Fig 5 : Motor intensity and battery for constant

power control with limitation intensity motor.

With a constant power control, there is never

likely to exceed the maximum intensity of the

batteries. But, there is a large motor current for

4

low speeds. Therefore, a thermal relay is required

to protect the motor.

With the constant force control, the motor

intensity limit is 3 to 4 times the rated current. So

it is possible to exceed the maximum motor

power when the slope of the road is important

and destroy it and the batteries. A thermal relay is

also necessary to protect the engine, but use more

temperature sensors to monitor the batteries.

6.

A test bench for e-bikes

The choice of rear motor is motivated by a way

to reload the battery thanks to the human muscle

strength. A pinch roller is installed on the device.

A generator is connected on the pinch roller to

test the motor wheel (see on figure 6). Some

embedded instruments measure the voltage, the

intensity, the wheel speed and calculate the

power and the energy of the battery. A LCD

monitor displays all parameters values:

7. Control strategies for motor brushless

of e-bike

To control the motor of an electric bike without

sensor assistance, there are several strategies such

as:

- Intensity limiting motor only (motor control

with constant force during the start)

- controlling the speed and limiting the intensity

motor,

- limiting the intensity from the battery (motor

control with constant power during the start).

The second strategy is interesting because it can

use a security sensor that has to be installed on

the chain ring with the following features (fig 7) :

- If the pedaling frequency vanishes or is equal to

0.1 rd/s, the motor runs as a freewheel and the

speed set-point is 0 km/h, whatever any action on

the twist handle throttle

- If the pedaling frequency is lower than 0.15

rd/s, the speed set-point equals 13km/h even if

the throttle is getting up to 100%,

- If the pedalling frequency is greater than 0.15

rd/s, the speed set-point matches with a ratio of the

twist throttle,

- An electrical breaking will occur when the

throttle is at its start position when the bike is

getting over 13 km/h. Below this speed the motor

operates at a free wheel.

Imax=15A

Speed 38 km/h

LCD instrumentation

13 km/h

Drive

roller

Generator DC

Figure 6: e-bike and test bed

An oscilloscope and a wattmeter recorder are

used to measure the dynamic of the intensity,

voltage and speed of the e-bike.

Pedaling

throttle

0%

Braking intensity

-5A throttle 0%

I=4.7A

throttle à 100%

freewheel

Fig 7 Second strategie : regulation speed and

intensity battery (FR 23N, wheel mass 7kg)

We have found again approximately the

acceleration and deceleration times given by the

equation (5) and (7) using the test bench.

Fmotor = K • I motor = 5 • (15A) = 75 N

(V

− Vinit ) ⋅ M (38 − 13) ⋅ 7kg

t accel = final

=

=0,9s

3,6 ⋅ (Fm + FR )

3,6 ⋅ (75 − 23)

t decel =

(Vfinal − Vinit ) ⋅ M (13 − 38) ⋅ 7kg

=

=0,8s

3,6 ⋅ (−25 − 23)

3,6 ⋅ (Fm + FR )

5

With the test bench, the mass corresponds to the

7kg of motor and not the rider (100kg) as shown

in Figure 3 & 4.

We can observe in Figure 8 that, using the third

strategy without pedal sensor, the current battery

is limited to 40A for 0.3s max at startup in such a

way to have a good acceleration and after the

current is limited to 20 A.

This application allows us to observe the

regulation of current and the steady state current

of 15A when the speed reached 64 km/h with a

battery of 50V. Figure 8 allows us to observe the

speed and the acceleration during the 2 phases to

40A (33m.s-2) and 20A (5m.s-2). After, it can

observe the stop freewheel without regeneration.

Imax40A

Speed

max

64 km/h

Steady state speed

I limiting 20A

intensity battery 15 A

Started

Freewhell stop

Fig 8 third strategie :

regulation intensity battery and speed

Conclusion

We have shown that using a constant power

control can be eliminated the peak current of the

battery but there are not protection from overload

the motor. This control is achieved through

regulation intensity of battery power and not on

the motor. But a limitation of motor intensity

priority has to be made for low speed values. In

addition during deceleration or downhill, the

battery intensity regulation allows to limit the

current to load rate maximum. This constant

power control allows having better control over

acceleration constant force for the same power

consumption. The profits of constant power

control are obviously applicable to all electrical

vehicles. Moreover, it has long been used for

electric traction railway [5].

However, many manufacturers offer variable

speed control with constant torque and never

constant power control.

References

[1] Donoghue, John F.; Burghart, James H « Constant

Power Acceleration Profiles for Electric Vehicles »

Industrial Electronics, IEEE Transactions on Digital, 1987 ,

Page(s): 188–191 : 10.1109/TIE.1987.350953

[2] Benoit Rozel, Wilfried Frelin, Emmanuel Hoang, Gilles

Feld, “ charge simulator for Home Trainer”, CETSIS'2005,

Nancy, 25-27 octobre 2005

[3] A.Sivert, F.Betin, J.Becar “An Electrical Bike For

Project Based Learning Platform”, EVER ecologic vehicles

& renewable energies de MONACO, Avril 2011.

[4] Vandana, R.; Fernandes, B.G.; “Optimal sizing of motor

Battery system for in wheel electric vehicles” IECON 2010

36th Annual Conference on IEEE Industrial Electronics ,

2510-2515, 10.1109/IECON.2010.5675157.

[5] Popa, G “Determining the Optimal Operating Regime of

the Traction Motors for Constant Power Operation of the

Vehicle” Automation, Quality and Testing, Robotics, 2006

IEEE International Conference on , 205-208 , Identifier:

10.1109/AQTR.2006.254526

[6] Chang-Hua Lin; Hom-Wei Liu; Chien-Ming Wang;

“Design and implementation of a bi-directional power

converter for electric bike with charging feature” Industrial

Electronics and Applications (ICIEA), 2010 the 5th IEEE

Conference ,538–543, 10.1109/ICIEA.2010.5517092

Arnaud.sivert was born in France.

He received the Ph.D. degree from the University of

Picardie Jules Verne, Soissons, France, in 2000.

In 1994, he joined an Institute University of

Technology in the Department of Electrical

Engineering, as an Assistant Professor. His major

research interest is the control of electrical machines

I.U.T has produced many prototypes electric vehicles

since 2008 and participates in the French national

challenge of electric kart. In 2011, he participated in

the first challenge of French National electric bike.

The e-bike as a teaching support is used in technical

field activity as electrical engineering or mechanical

engineering and also in theoretical field activity as

physics and mathematics. The use of this teaching

support is also adequate with the syllabus of

undergraduate students and bachelor of technology

students. The e-bike teaching tool turns all mechanical

or human parameters such as forces and powers into

their electrical analogy representation. The e-bike

allows understanding some facts.

6

100 words

We will demonstrate the benefits from a constant

power control compared a constant force control.

We will show that the constant power control

allow to cancel the intensity peaks supplied by

the battery and to have better dynamic speed.

Indeed, for the same energy expenditure during

acceleration, displacement is larger with a

constant power control.

However, this command causes a current peak

motor. Therefore, there is a compromise between

the constant power and constant driving force to

control a motor. Many curves are presented in

theory and practice the two commands.

7