Introduction to vector control of PMSM v3.ppt [Mode … · 10 Principle of functioning of a PMSM -...

[email protected]

Introduction to Vector Control

of Permanent Magnet Synchronous Machinesusing

Energetic Macroscopic Representation

[email protected] Professor in Electrical Engineering

PhD - HdR

Arts et Métiers ParisTech – Lille - FranceL2EP – Control Team

ERASMUS MUNDUS MASTER STEPS

02&03/12/2013

3Pedagogical Outcomes

At the end of the lecture, students have to be able to:

Use a methodological approach to represent models ofelectrical systems in order to help the deduction of dedicatedcontrol schemes

Understand what are the variables of a Permanent MagnetSynchronous Machine (PMSM) to be controlled

Know, with the help of SIMULINK simulations, how to control aPMSM in the abc reference frame and what are the inherentlimitations

Know, with the help of SIMULINK simulations, how to control aPMSM in the dq reference frame and what are the necessarymathematical transformations to implement

4Agenda

• Models and Representations for the control of electrical systems• Modelling and Energetic Macroscopic Representation (EMR) of

Permanent Magnet Synchronous Machines (PMSM)• Principles of Inversion Based Control• Inversion Based Control of Permanent Magnet Synchronous

Machines (PMSM) – abc reference frame– Simulation with SIMULINK

• Inversion Based Control of Permanent Magnet SynchronousMachines (PMSM) – dq reference frame– Simulation with SIMULINK

5Models and Representations

What are the quantities to be controlled ?

fdtdJTT resmotor

dtd

Equations are usually written with differential operators, however:

tttt

dttd

t

)()(lim0

Physical causality is integral

Motor

Load

The torque at time t cannot dependon the speed at time t+t Physically impossible!



Motor

Load

fdtdJTT resmotor

dtd

The position at time t depends on the speed up to t The speed at time t depends on the torque up to t

There are natural (physical) causalities

JsfTT resmotor

s

If equations are written in integral form, then:

t

dt0



Motor Torque (Tmotor) -> Rotation of the load ()

Action implies Reaction

Motor

Load

s

Motor Torque (Tmotor) x Rotation of the load () = Power

Action x Reaction = Power

JsfTT resmotor

8

resout Tp

motorin Tp

Energetic Macroscopic Representation makes possible to represent a model in order to:• To respect the natural (integral) causality • To highlight actions and reactions and power flows

Motor

Load

Motor Env

*motorT

motorT

resT

""storedlossesoutin pppp

)1(s

)2(JsfTT resmotor

1

2

Losses + storage

Source

Source

Models and Representations

9Modelling and EMR of a PMSM

Constitution of a PMSM- Stator composed of three windings- Rotor composed of surface mounted or buried magnets

10

Principle of functioning of a PMSM- Stator creates a rotating magnetic field (equivalent to a rotating

magnet although stator windings are at stand still)

Modelling and EMR of a PMSM

Each winding produces a magnetic field with a magnitudeproportional to the coil current and a direction corresponding tothe coil axis

If the three coils are supplied with a 3-phase AC sine currentsystem, the resultant magnetic field is rotating

http://en.wikipedia.org/wiki/Rotating_magnetic_field

11

Principle of functioning of a PMSM- Stator creates a rotating magnetic field (equivalent to a rotating

magnet although stator windings are at stand still)- Rotor magnets are attracted by the rotating field and rotate at the

same speed (in steady state)

Bss

Brr

+ C

N

S

ssrr BBkC

Torque Control requires a VECTOR CONTROL


12

Modelling of a PMSM

d 3’

2 3

1

2’

x

d

q

N

S

1

q

2

3

y

1’

F

v1

i1 vF

iF

i2

v2

i3

v3

y

x

Only one pole pair for the sake of simplicity.The permanent magnet is equivalent to a winding supplied

with a DC current


13

Modelling of a PMSM

r

r

r

s

s

s

3

2

1

3

2

1

3

2

1

Stator flux linkage equations are

with

3

2

1

3

2

1

iii

LMMMLMMML

sss

sss

sss

s

s

s

and

3/4cos3/2cos

cos

max

3

2

1

r

r

r


FF iMmax in case of wound-rotor

cstemax in case of rotor with permanent magnets

14

Modelling of a PMSM

r

r

r

s

s

s

ss dtd

dtd

iii

Rdtd

iii

Rvvv

3

2

1

3

2

1

3

2

1

3

2

1

3

2

1

3

2

1

Stator voltage equations are then

3

2

1

3

2

1

3

2

1

3

2

1

eee

iii

dtd

LMMMLMMML

iii

Rvvv

sss

sss

sss

s


15

Modelling of a PMSM

Or in a more concise way (matrix notation)

edt

idLiRv ss

with

3

2

1

vvv

v

3

2

1

iii

i

Voltage vector

Current vector

3

2

1

eee

e EMF vector

sR Stator resistance

sL Stator inductance matrix


16

Modelling of a PMSM

emmaglossesin pppp The power balance is

With

332211. ivivivivp tin

iLi

dtdi

dtidLp st

tsmag .

21

23

22

21

2. iRiRiRiRp sssslosses


17

Modelling of a PMSM

and

emtem Tieieieiep 332211

then

332211 ieieieiepT tem

em

As

dd

dd

dtd

dtde rrr

then id

dT tr

em

Torque Control requires a VECTOR CONTROL


18

Modelling of a PMSM

If

3/4cos3/2cos

cos

max

3

2

1

r

r

r

3/4sin3/2sin

sin

max

3

2

1

3

2

1

r

r

r

dd

eee

and

then

3/4sin3/2sinsin 321max iiiTem


19

Modelling of a PMSM

Question: Find the current references that lead to a constant torque under minimum copper losses


20

Modelling of a PMSM

Solution:

d

dTd

d

ddTi remr

r

em

2max

*

2

**

23

2. iRp slosses is minimum if is minimum iFor a given , *

emT is minimum if is collinear to i

ddkii r*

i

dd r

i.e.

Then 2

*

ddk

ddk

dd

T rrtrem


21

Modelling of a PMSM

Solution:

3/4sin3/2sin

sin

32

max

*

emTi

Finally

Current vector is collinear to EMF vector


22

Modelling of a PMSM

If the stator windings are star coupled:

13u

23u

1v

2v 3v

Only two voltages and two currents are sufficient to model the machine:

2i

1i

3i

0321 iii

3

2

1

3

2

1

3

2

1

3

2

1

eee

iii

dtd

LMMMLMMML

iii

Rvvv

sss

sss

sss

s

Question: Rewrite the voltage equation withas variables

231321 ,,, uuii


23

Modelling of a PMSM

Solution:

32

31

2

1

2

1

32

31

23

13

2112

2112

eeee

ii

dtdL

ii

Rvvvv

uu

cs

ssc MLL is called the stator cyclic inductance

Remark: if we add the same quantity vZ (Zero-sequence) to v1, v2and v3, u13 and u23 remain the same.


24

EMR of a PMSMThe VSI (Voltage Source Inverter) is considered as an amplifier:

Mono-physical conversion (elecelec)

Multi-physical conversion (elecméca)


11, kbuskk sVsv

32

31

2

1

2

1

32

31

23

13

2112

2112

eeee

ii

dtdL

ii

Rvvvv

uu

cs

emTieeiee 232131

fdtdJTT resem

0321 vvv0321 iii

Bus Env

DC bus VSI Star coupling LoadPMSM

Vbus

ibus

[s]=[s*]

[v]

[i123]

[u]

[i]

[i]

[e]

Tem

Tres

VSI:

Star coupling:

PMSM:

Load:

25

From EMR to Simulink…


VSI Star coupling PMSM+Loadf+J

PMSMR+L

PMSMEM conversion

Load

Env

DC Source

Bus Vbus [v ] [u]

[i123]ibus

[i]

[i]

Tem Omega Tres

Omega[e]

[s*]

26Principles of Inversion Based Control

(.)input output

u(t) y(t)

-1(.)desiredoutput

Controlling a system for output tracking can be interpreted as inverting the system

yref(t) Control

… if we can implement a good approximation of the system’s inverse

System

uref(t)

inputreference

Open loop and closed-loop controls

27

Open loop controls

v(t)

-1(.)

Let’s take a simple example: current control of a resistance

Control

System

)(~)(:.~ **1 tiRtv

R1:.

*v R

i(t)i

v*(t) i*(t)Con

trol

syst

em

*iv

In case of acausal relationship (no accumulation), open loop control is possible (No need of controllers).The inversion of the element is said “direct”.

Principles of Inversion Based Control

28

Open loop controls


Direct Control with EMR formalism

y2

u2

Objective: to control y2

u1

y1

y2 = f(u1 )+g(u2 )

Compensate u2Manipulate u1

y2*u1*u1*=f-1(y2* )+g(u2 )~

u1

y1 k

y2 = k x u1y2

u2

y1 = (1/k)x u2

y2*u1~

k*

k* = y2 */u1~

29

Open loop controls


Direct Control with EMR formalism

u1

y1

u21

no measurement no controller

p weighting variables

y2pu2p

u1

y2p-ref

y21-ref

kW1…kWp

refpWprefW ykyku 2'

21'

11 ...

Implement a compromise or

prioritize outputs.

VDC

v1=VDC

i2v2=VDC

i1VDC

icoup

refWrefWDC vkvkV 21 )1(

v1ref

kW

v2ref

Example: current node

0 ≤ kW ≤ 1

30

Closed-loop controls


v(t)

-1(.)

Let’s take a simple example: current control of a R+L circuit

Control

System LsR

1:.

*vR

i(t)i

v*(t) i*(t)Con

trol

syst

em

*iv

In case of causal relationship (accumulation of energy), open loop control is not possible Need of controllers

L

)(~~)(:.~ **1 sisLRsv No derivative in real-time

31



v(t)

Controller

Let’s take a simple example: current control of a R+L circuit

Control

System*v

Ri(t)

i

v*(t)i*(t)

Con

trol

syst

em

*iv

In case of causal relationship (accumulation of energy), closed-loop control is mandatory

Need of measurements and controllers (Indirect control)

L

)()()(: ** sisisCsvController

LsR

1:.

32



u2

i

i

u1

Indirect Control with EMR formalism(System with accumulation of energy)

U2(s)

U1(s)

-

+(s)

U(s) I(s)

+

U1(s)

+I*(s)U*(s)

C(s)+-

iref

u2

i

u1*

~

1u

2u

i

33

Inversion based-control of the PMSM in the abc reference frame

Inversion Based Control of PMSM (abc frame)

emTieeiee 232131

32

31

2

1

2

1

32

31

23

13

2112

2112

eeee

ii

dtdL

ii

Rvvvv

uu

cs

The VSI (Voltage Source Inverter) is considered as an amplifier:

fdtdJTT resem

11, kbuskk sVsv0321 vvv0321 iii

Energysource

Mono-physical

conversionelement

Accumulationelement

Couplingelement

Direct inversion

Controller(Indirect

Inversion)

Strategy

EMR legend

Measurement

Multi-physical

conversionelement

34

From EMR and IBC to Simulink…


VSI

TorqueStrategy

Tem *

ParkTransformation

PMSM+Loadf+J

PMSM R+L

PMSMEM conversion

ModulationStrategy

Load

MS

Inversionstar coupling 1

InversionVSI Inversion

EM conversion

DC Source

ES

Current Control

Vbus [v ] [vdq ]

[i123]ibus

[idq]

[idq]

Tem Omega Tres

[edq]

thetaOmega

[vdq *]

vz *

[v *][idq*]

[kdq]

Tem*

35



[s*]

1

Vbus

2

[v*]

1

[v*]

1

R2* u

L2* u Kp_i[i*]

3

[i]

2

[e]

1

[(d/dt)*] [i*]

1

[k]

2

Tem *

1

[v*]

12/3

2/3

1/3

1/3

vz*

2

[u*]

1

u13*

u23*

vz*

1 0

[k]

1 e2

sin(u-2*pi /3)

e1

sin(u)

1s-2/(3*phi _max)

Omega

1theta

36



Questions: Using the Simulink file PMSM_abc- Analyse the simulation (Model-Control-Strategy)- Comment the effect of the knowledge of the system’s parameters

and the gain Kp_i on the overall performances- Conclude

Solution:To attain good performances, very good knowledge of the system’s parameters is necessary (which is difficult) and a high value for Kp_igain is required to perfectly track the references (which can lead to instability) Another type of control is required

37

Modelling of a PMSM in the dq reference frame

Inversion Based Control of PMSM (dq frame)

The control of the currents is difficult because the system in the abcreference frame is highly coupled

32

31

2

1

2

1

23

13

2112

2112

eeee

ii

dtd

ii

Ruu

ss

By using the following change of variables (called Concordiatransformation):

ii

ii

Cii

21

61

032

2

1

38



The system becomes :

32

31

23

13

2112

2112

eeee

Cii

dtdLCC

ii

CRCuu

C tc

ts

tt

Considering:

23

13

uu

Cvv t

23

13

ee

Cee t

It comes:

ee

ii

dtd

LL

ii

RR

vv

c

c

s

s

00

00

The real three phase machine is equivalent to a fictitious twoindependent phase machine (All matrices are diagonal).

39



However, to obtain a constant torque, stator currents have still to besinusoidal (since the rotor rotates!) and then difficult to track.

d

1

q

2

3

y

F

v1

i1 vF

iF

i2

v2

i3

v3

x

d

q

y

F

v

i vF

iF

i

v

x

s r

40



If the stator windings rotate at the same speed as the rotor, statorcurrents will become constant!

By using the following change of variables (called Rotation):

q

d

q

d

ii

ii

Rii

cossinsincos

It comes:

q

d

q

dc

q

dc

q

ds

q

d

ee

ii

Lii

dtdL

ii

Rvv

0110

Although the system is (weakly) recoupled, currents in steady state areconstant and then easy to control.

41



d

q

y

F

v

i vF

iF

i

v

x

d

q

q

y

F

vF

iF

iq

vq

x

d id

vd

s r

Moreover, flux vector depends only on iq. Then id can be kept equalto zero (ed=0).

Equivalence with a DC Brushless machine

s

Vector control is easy to implement in dq reference frame.

42

d

q

y

F

v

i vF

iF

i

v

x



Concordia Rotation

Park

d

q

q

y

F

vF

iF

iq

vq

x

d id

vd

d

1

q

2

3

y

F

v1

i1 vF

iF

i2

v2

i3

v3

x

43

Modelling of a PMSM in the dqreference frame


vA

vB

vC

iA

iB

iC

vd

vq

PARK

iq

iA

iB

iC

id

Park-1

Park

44

EMR of a PMSM in the dq reference frame


dqdqcdq

cdqsdq eiLdtid

LiRv

0110

23

13

uu

CRvRv tdq

2

11

ii

CRiRidq

45Inversion Based Control of PMSM (dq frame)

Questions: Using the Simulink file PMSM_dq_students- Simulate the control system using the EMR- Conclude


46References[1] “Control Strategies for Open-End Winding Drives Operating in the Flux-Weakening Region”. IEEETransactions on Power Electronics, 9-2013, Alexandru-Paul SANDULESCU, Fabien MEINGUET, XavierKESTELYN, Eric SEMAIL, Antoine BRUYERE[2] “Model-based decoupling control method for dual-drive gantry stages: A case study with experimentalvalidations”. Control Engineering Practice, Vol. 21, N°. 3, pages. pp. 298-307, 3-2013, Ivan MauricioGARCIAHERREROS, Xavier KESTELYN, Julien GOMAND, Ralph COLEMAN, Pierre-Jean BARRE[3] “A Vectorial Approach for Generation of Optimal Current References for Multiphase PermanentMagnet Synchronous Machines in Real-time”. IEEE Transactions on Industrial Electronics, Vol. 58, N°.11, pages. 5057 - 5065, ISBN0278-00462-2011, Xavier KESTELYN, Eric SEMAIL[4] “Control of a Symmetrical Dual-drive Gantry System using Energetic Macroscopic Representation”.Solid State Phenomena (SSP), Vol. 144, pages. 181-185, ISBN3-908451-60-42-2009, XavierKESTELYN, Julien GOMAND, Alain BOUSCAYROL, Pierre-Jean BARRE[5] “FPGA Implementation of a General Space Vector Approach on a 6-Leg Voltage Source Inverter”.IECON 2011 - IEEE International Conference On Industrial Applications of Electronics, N°. 37, pages.3482-3487, ISBN978-1-61284-969-09-2011, Alexandru-Paul SANDULESCU, Lahoucine IDKHAJINE,Sebastien CENSE, Frédéric COLAS, Xavier KESTELYN, Eric SEMAIL, Antoine BRUYERE[6] “Teaching drive control using Energetic Macroscopic Representation - expert level”. EPE’09,Barcelona, Spain, 7-2009. Alain BOUSCAYROL, Philippe DELARUE, Frédéric GIRAUD, XavierGUILLAUD, Xavier KESTELYN, Betty LEMAIRE-SEMAIL, Walter LHOMME[7] “Vectorial Modeling and Control of Multiphase Machines with Non-salient Poles Supplied by anInverter”. Chapter 5 of "Control of Non-conventional Synchronous Motors. ISTE Ltd and John Wiley &Sons Inc, pages. 448-470, ISBN978-1-84821-331-912-2011, Xavier KESTELYN, Eric SEMAIL[8] “Multiphase Voltage source Inverters”. Chapter 8 of "Power electronic Converters - PWM Strategiesand current control techniques". ISTE Ltd and John Wiley & Sons Inc ., pages. 203-242, ISBN978-1-84821-195-73-2011, Xavier KESTELYN, Eric SEMAIL

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