Vector Control of Induction Machines

dq3

2 IM 32 dq

su si

qq

abab

Introduction

• The traditional way to control the speed of induction motors is the V/Hz-control

• Low dynamic performance• In applications like servo drives and rolling

mills quick torque response is required.• Desire to replace dc drives led to vector

control• Braunschweig, Leonhard, Blaschke, Hasse,

late 70-ies

What is vector control?

• Vector control implies that an ac motor is forced to behave dynamically as a dc motor by the use of feedback control.

• Always consider the stator frequency to be a variable quantity.

• Think in synchronous coordinates.

Basic blocks of a vector controlled drive

dq3

2 IM 32 dq

su si

qq

abab

Addition of a block for calculation of the transformation angle

dq3

2 IM 32 dq

su si

qq

abab

T ra n s fo rm a tio na n g le

c a lc u la tio n

rq

The current is controlled in the d- and q-directions

jref ref refs sd sqi i i

magnetization

torque production

Vector controller

dq3

2 IM 32 dq

su si

qq

abab

T ra n s fo rm a tio na n g le

c a lc u la tio n

rq

C u rre n tc o n tro lle r

refsi

+-

Stator and rotor of an induction machine

Magnetization current from the stator

The flux

1

r

The rotation

2

View from the rotor

v

2

dl e v B

v

B

Induced voltage and current

2

F

Torque production

2

Ampere-turn balance

Rotor flux orientation

• Difficult to find the transformation angle since the direction of the flux must be known

• Flux measurement is required• Flux sensors (and fitting) are expensive and

unreliable• Rotor position measurement does not tell the

flux position• The solution is flux estimation

Rotor flux orientation using measured flux

Original method suggested by Blaschke•Requires flux sensors•Flux coordinates: aligned with the

rotor flux linkage

arctan r

r

b

a

a

b

r *

jes

f s srsr

y y y

Rotor flux orientation

From Chapter 4

ssu

ssi

smi

sri

j sr r

rR

sR slL rlL

mL

s

dR (stator)

d

ss sss st

u i

r

dj R (rotor)

d

ss sr

r r rt i

Transformation to flux coordinates

j j j js

de j e e R e

d

ff f fss s st

u i

j j j jr

de j e j e R e

d

ff f frr r r rt

i

1 s

dj R

d

ff f fss s st

u i

2 r

dj R

d

ff frr rt

i

2 1 r

The flux coordinate system is ”synchronous” only at steady-state. During transients the speed of the rotor flux and the stator voltage may differ considerably.

The rotor equation (5.9)

2 r

dj R

d

ff frr rt

i

m

r r

1 L

L Lf f fr r s i i

r m r2

r r

d R L Rj

d L L

ff f frr r st

i

Split into real and imaginary parts

0frq d

0d

frq

t

r m r

r r

d R L R

d L L

ff frdrd sdit

m r2

r

L R0

Lf frd sqi

Rotor flux dynamics are slow

rr

r

LT

R

f frd0 m sd0ψ L i

Torque control

*m

r

3 Lp Im

2 Lf fr sT i

m

r

3 Lp

2 Lf frd sqT i

mL

refref rsdi

Rotor flux orientation using estimated flux

• The rotor flux vector cannot be measured, only the airgap flux.

• Flux sensors reduce the reliability• Flux sensors increase the cost• Therefore, it is better to estimate the rotor

flux.

The "current model" in the stator reference frame

(Direct Field Orientation)

r

dj R

d

ss sr

r r rt i

m

r r

1 L

L Ls s sr r s i i

m

r r

ˆd L1ˆj

d T T

ss sr

r r st

i

The current model

C u rren tco n tro l

C u rren tm o d e l

IMd riv e

f

fs

s

refrd

refT

fsi s

si

ssu

fsu

r

The "current model" in synchronous coordinates (Indirect Field Orientation)

m r m2 2

r r

0fsqf f

rd sq frd

iL R Li

L T

2r

1 sq

sd

i

T i f f

rd0 m sd0L i

Transformation angle

1 d tq

1 2r

1

Tsq

r rsd

i

i

Remarks on indirect field orientation

• Does not directly involve flux estimation (superscript f dropped)

• Not ”flux coordinates” but ”synchronous coordinates”

• Since the slip relation is used instead of flux estimation, the method is called indirect field orientation

Indirect field orientation based on the current model

C u rren tco n tro l

s lipre la tio n

IMd riv esy

s

s

refrd

refT

fsi s

si

ssu

fsu

r

sy

q

q

Feedforward rotor flux orientation

1r

1refsq

r refsd

i

T i

•Significantly reduced noise in the transformation angle•Fast current control is assumed (ref.value=measured value)•No state feedback => completely linear

The voltage model

•The current model needs accurate values of the rotor time constant and rotor speed•The trend is to remove sensors for cost and reliability reasons•Simulate the stator voltage equation instead of the rotor voltage equation

s

ˆd

d

ss sss sR

t

u i

Solve for the rotor current and insert in

s ms s rL L i i

r mr r sL L i i

rs m

m

s s s sr s s s

LL L

L i i

m r/L L

2m m

sr r

s s sr s s

L LL

L L

i

Multiplication by

yields

2m

s σr

LL L

L

r

m

Lˆ ˆ L

Ls s sr s s i

Solve forsr

Direct field orientation using the voltage model

C u rren tco n tro l

V o ltag em o d e l

IMd riv e

f

fs

s

refrd

refT

fsi s

si

ssu

fsu

ˆ frd

Stator flux orientation

1

ˆd nominal

d

sssst

u

m

r

ˆ ˆs s ss r s

LL

L i

At low frequencies the current model can be used together with:

"Direct self-control" (DSC) schemes first suggested by Depenbrock, Takahashi, and Noguchi in the 1980s.

Field weakening

M a x im u mto rq u e ra n g e

F ie ld w e a k e n in g ra n g e= > R e d u c e d to rq u e

refsdi

maxr refr

Current control

R L

++su se

si

dL R

d

ss s s

t

iu i e

j j j jdL e e R e e

d tq q q q i u i e

j j j j jdL e jω e e R e e

d tq q q q q

i+ i u i e

dL jω R

d t

i+ i u i e

d+ j

dL R Lt

iu i e

d

dd

d d q d

iL u Ri Li e

t

d

dq

q q d q

iL u Ri Li et

Transfer function and block diagram of a three-phase load

( )

1

js

s L R

G

+-

( )sG

e

ui

Review of methods for current control

• Hysteresis control• Stator frame PI control• Synchronous frame PI control

Hysteresis control(Tolerance band control)

• Measure each line current and subtract from the reference. The result is fed to a comparator with hysteresis.

• Pulse width modulation is achieved directly by the current control

• The switching frequency is chosen by means of the width of the tolerance band.

• No tuning is required.• Very quick response

Drawbacks of hysteresis control

• The switching frequency is not constant.• The actual tolerance band is twice the chosen

one.• Sometimes a series of fast switchings occur.• Suitable for analog implementation. Digital

implementation requires a very high sampling frequency.

Stator frame PI control

• Two controllers: one for the real axis and one for the imaginary axis

• Cannot achieve zero steady-state error• Tracking a sinusoid means that steady-state is

never reached in a true sense• Integral action is useless except at zero

frequency

Synchronous frame PI control

• In a synchronous reference frame the current is a dc quantity at steady-state.

• Zero steay state error is possible.• Coordinate transformations necessary• Easily implemented on a DSP• Usually the best choice!

Design of synchronous frame PI controllers

Remove cross-coupling

j L u u i

d+ j

dL R Lt

iu i e

dL R

d t

iu i e

( )1

ss L R

G

+

+

-

refi ( )sF ( )sG

( )sG

u

e

i+

-

j L

Desired closed-loop system

( )ss

aa

cG

ln(9)rta

( )i

s p

kk

s F

( ) ( )( )

( ) ( )1

s ss

s s

c

F GG

F G

Choice of controller parameters

-1( ) ( )

RL R L+s s s

s s s

a a aa F G

Lpk a

Rik a

Speed control

• Applications: pumps and fans in the process industry, paper and steel mills, robotics and packaging, electric vehicles

• Very different dynamic requirements• Most drives have low to medium high

requirements on dynamics. These drives are considered here.

• Cascade control is sufficient

Block diagram of a speed-controlled drive system

Currentcontroller refuI

irefi

Speedcontroller

Inverter

refm

m

Electricmotor

The mechanical system

d

dm

e l mT T J bt

1/ J s bm

eT

lT

The speed controller

• The task of the speed controller is to provide a reference value for the torque (or current) which makes the mechanical system respond to the speed reference with a specified rise time.

ln(9)rt a

Block diagram with speed controller

1/ J s bm

eT

lT

Inner loopSpeed

controller

TF c refsqi

refm

1/ Tc

132

Trefmrd

r

cL

pL

1refm m mF

J s b

1oF G

J s b

1o

co

GG

G s

aa

oG s

a

Choice of controller parameters1

FJ s b s

a

PI

bF J s b J

s s

a aa

Realistic choice of bandwidth

• Care must be taken that the bandwidth of the speed controller is not unnecessarily high.

• In fact this should be decided during the first steps in the design process of a drive system

• The bandwidth is directly connected to the current rating of the inverter.

A change in the speed reference

qpi J a

max max mbaseC

How large steps should be foreseen?

qp nomi I max With and

maxnom mbaseI C J a

max

nom

mbase

I

C Ja

Check if the current controlleris sufficiently fast.