L21: Induction machine control - IEA

L21: Induction machine control

L21: 7-MAY-2019

Lund University / LTH / IEA / AR / EIEN25 / 2019-05-07 2

Outlook

• Induction machine (IM) construction– Squirrel cage

– Rotor winding and slip-rings

• IM model & characteristics– Steinmetz equivalent circuit

– Vector representation and flux models

• Torque control– Feedback and observers

– Direct and deadbeat current control


AC machines

• Synchronous machines– Rotor field is created by

excitation – permanent magnets or/and externally supplied currents

• Asynchronous machines– Rotor field is created by

induction – rotor current is supplied by transformer action

Stator electrical frequency, f [1/s] [Hz]

Number of pole pairs, pp=p/2

Rotor synchronous speed, ns=f/pp*60 [1/min] [rpm]

Rotor asynchronous speed, nr=ns*(1-slip) [1/min] [rpm]

is

ir

ψsσ

ψrσ

ψm stator

rotor


IM operation principle• Rotating magnetic field by 3φ supply• Rotor winding cuts the field change• Emf is induced in the rotor conductors• Current carrying conductors create

counteracting field• Magnetic reaction forces cause a

motion1

2

31. Stator magnetizes rotor2. Stator field is changed3. Rotor conserves magnetization


Squirrel cage IM


Wound rotor IM


IM connections & control

• Terminations of interest– Y-Δ connection, 2-speed

motors – Dahlander winding– Slip control by external

resistance

• Control– Line connected– Inverter-fed

• f/V ramp• Field Oriented Control

FOC• Direct torque/flux control

DTC– doubly-fed electric machine

power electronic converter

induction

machine

machine

Udc

dc-link

is*

reference


Steinmetz equivalent circuit

• Induction machine is an electrical transformer– the magnetizing circuit is seen from no load test (NLT)

– leakage inductances are found from locked rotor test (LRT)

• Load resistance Rr’/s consists of equivalent electromechanical load resistance Rr’(1-s)/s and actual winding resistance Rr’

s

RIsT rr

em

'2'3, ψδ

ψr

ψs

s

ss


-1500 -1000 -500 0 500 1000 1500 2000 2500 3000-50

-40

-30

-20

-10

0

10

20

30

40

50

torq

ue,

T [

Nm

]

speed, n [rpm]

2 1.5 1 0.5 0 -0.5 -1

-10

-8

-6

-4

-2

0

2

4

6

8

10

pow

er,

P [

kW]

slip, s [-]

IM characteristics @ f=const

• Simplified equivalent circuit

• Single frequency– 50 Hz

• Pull out torque – Tmax=mU1

2/(2Xws)– m=3 phases– p=4 poles– Total leakage

reactance X=6.28Ω– Slip at pull out

s=R2’/X

• Braking (s>1), motoring, generating (s<0)

ss

ss

TT

m

m

max2


IM characteristics @ f=variable

Speed, n [rpm]

Torque, T [Nm]

Pull out torque

nbase nBD

High speedtorque region

Constant power region

Constant torque region

Ψ constant Ψω constant


Circuits & phasors• Magnetisation

– Resistance R1 and inductance L1of primary coil

– Leakage and mutual inductance Lσ+Lm=Ls

– Core losses Rm

– Complex current Io and magnetising flux Ψm

• Magnetically coupled– Secondary circuit

– EMF E2=jωΨm

• Electrically loaded– Max P2 power I2=E2/(Z2+R)

– Primary current I1=I2+Io

Io

IoR

IoLΨm

E2I2

I1

201 nIII 222111122222 jXRIjXRnInUjXRIEU


Mathematical model

• Stator

• Rotor

ri

si

r

r

s

s

su

su

dt

d s

ss iR

dt

diRu

dt

diRu

rrrr

ssss


IM equations

ksk

ksk

ssk

s jdt

diRu

k

rk

krk

rrk

r jdt

diRu

krm

kss

ks iLiL

k

rrk

smk

r iLiL

loadk

sk

rem Tdt

dJippT

• Voltage equations: stator vs rotor

• Flux linkage equations: stator vs rotor

• Electromagnetic torque and kinetic equation


IM equations in component form

sqksd

sdssd dt

diRu

sdksq

sqssq dt

diRu

rqkrd

rdrrd dt

diRu

rdkrq

rqrrq dt

diRu

sdsqsqsde iippT

rdmsdssd iLiL

rqmsqssq iLiL rdrsdmrd iLiL

rqrsqmrq iLiL

rdrqrqrde iippT


2

wm

1

is

p/2

1

ur

Tld

wm

is

psis

us ir

Tm

pos

psir

p/2

Step

~=

Psiis

PsiTe

Psi ir

[0 -1 ; 1 0]* u

b

1/J

1s

1s

Lm/Lr

RrRs 1

s

1s

2

load

1

us

IM model

rdsqrqsdmrs

me

mrs

sqmrqsrq

mrs

sdmrdsrd

mrs

rqmsqrsq

mrs

rdmsdrsd

LLL

LT

LLL

LLi

LLL

LLi

LLL

LLi

LLL

LLi

2

2

2

2

2


IM – 2SIE 100L4A

• Parameters– Rotor and stator resistances

Rr=1.5Ω & Rs=1.5Ω

– Rotor, magnetizing and stator inductances Lr=0.21H, Lm=0.19H & Ls=0.21H

0.0070kgm2Inertia J

2.8-Knee torque Tkn/Tn

7.3-Start current Ia/In2.4-Start torque Ta/Tn

1440rpmSpeed nn, @50Hz

4.5 (7 max)ACurrent In, @400V

84.7,85.5,84.6%Efficiency η,

Stator size Do/Di-H mm 155/95-100

Power Pn, @50Hz W 2200

Power factor cosφ - 0.83

Weight w kg 25.5

Cost* SEK 6840


Line start and switch off UL=230V

• Speed dependent load b=10 Nm @ 1500 rpm

0 0.1 0.2 0.3 0.4 0.5 0.6-30

-20

-10

0

10

20

30

time, t [s]

curr

ent,

I [

A]

i1-200

0

200

400

600

800

1000

1200

1400

1600

sppe

d, n

[rp

m]

-200

-150

-100

-50

0

50

100

150

200

term

inal

vol

tage

, U

[V

]

0 0.1 0.2 0.3 0.4 0.5 0.6-10

-5

0

5

10

15

20

torq

ue,

T [

Nm

]


Line start and switch off UL=230V

• Ud aligned to the direct axis on the field reference frame

0 0.1 0.2 0.3 0.4 0.5 0.6-40

-30

-20

-10

0

10

20

30

time, t [s]

curr

ent,

I [

A]

isdisqirdirq

0 0.1 0.2 0.3 0.4 0.5 0.6-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

flux

linka

ge,

[V

s]

sdsqrdrq


Line start and off UL=400V

• Speed dependent load b=10 Nm @ 1500 rpm

0 0.1 0.2 0.3 0.4 0.5 0.6-30

-20

-10

0

10

20

30

40

50

time, t [s]

curr

ent,

I [

A]

i1-200

0

200

400

600

800

1000

1200

1400

1600

sppe

d, n

[rp

m]

-400

-300

-200

-100

0

100

200

300

400

term

inal

vol

tage

, U

[V

]

0 0.1 0.2 0.3 0.4 0.5 0.6-20

-10

0

10

20

30

40

50

torq

ue,

T [

Nm

]


Line start and off UL=400V

• Ud aligned to the direct axis on the field reference frame

0 0.1 0.2 0.3 0.4 0.5 0.6-60

-40

-20

0

20

40

60

time, t [s]

curr

ent,

I [

A]

isdisqirdirq

0 0.1 0.2 0.3 0.4 0.5 0.6-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

flux

linka

ge,

[V

s]

sdsqrdrq


Control frame and torque

Rotor flux orientationStator flux orientation

ssrr

mrs

s

mrrss ii

L

Li

L

LiiT

α

β

stator

xy

ω s

field

(1±s)ωs

rotor

d

q

isd+jisqusr ur

rRotor

isx+jisyusk ur

kField

isα+jisβuss ur

sStator

Space current

VoltagesCoordinate

system Control

dq

αβ

αβ

3φ Drive

dq

αβ

αβ3φ

*dqu *

u

dqi

i

abci

*abcu

rje

rje


From IM equations to IM control (A)

rr

rrr

rr

ss

sss

ss

dt

diRu

dt

diRu

rr

rr

jssm

rrr

rr

jssm

rrr

rr

rsm

rrr

rrrr

rr

jrrm

sss

jrr

ssm

sss

sr

ssm

sss

ssss

ss

eiLiLieiLiL

iiLiLiL

eiLiLeiiLiL

iiLiLiL


From IM equations to IM control (B)

dt

tidLeti

dt

dLtiR

tiLetiLdt

dtiRt

dt

dtiRtu

etidt

dL

dt

tidLtiR

etiLtiLdt

dtiRt

dt

dtiRtu

rr

rjs

smr

rr

rrr

jssm

rrr

rr

rrr

rr

jrrm

ss

ss

ss

jrrm

sss

sss

ss

sss

ss

r

r

r

r

0

rjss

r

m

r

rrr

r etiL

L

L

tti

Substitute


From IM equations to IM control (C)

rrrrr

rrr

jss

r

m

r

r

rjs

smjs

sr

m

r

r

rrr

jjss

r

m

r

r

m

ss

ss

ssss

etiL

L

L

t

dt

dLeti

dt

dLeti

L

L

L

tRtu

eetiL

L

L

t

dt

dL

dt

tidLtiRtu

0

0

2

0

1

rrrrr

r

r

r

m

jssm

jssm

rjs

s

r

r

m

r

r

r

jr

r

ms

s

rss

sss

ss

etidt

dLeti

dt

dL

dt

tdeti

RLL

RL

t

etdt

d

L

L

dt

tid

LL

LLtiRtu

dt

tdeti

Lt

etdt

d

L

L

dt

tidLtiRtu

rjs

sr

m

r

r

jr

r

ms

srs

sss

ss

rrr

r

r

0


tjdt

tdti

Lt

tL

LtiLj

dt

td

L

L

dt

tidLtiRtu

tejdt

tdeeti

Let

teL

LtieLj

dt

tde

L

L

dt

tideLetiRetu

etdt

deeti

Let

eetdt

d

L

Leti

dt

dLetiRetu

dqsl

dqdq

sr

m

r

dq

dq

r

mdqsrss

dq

r

mdq

srs

dqss

dqs

dqjsl

dqjjdq

sr

m

r

jdq

dqj

r

mdqs

jrss

dqj

r

mdq

sjrs

jdqss

jdqs

jdq

e

jjdqs

r

m

r

jdq

e

jjdq

r

mjdqsrs

jdqss

jdqs

r

rr

r

r

r

slrslsl

sl

r

r

ss

rssss

sl

r

slj

rs

sl

r

sj

rsl

r

sss

0

0

0

From IM equations to IM control (D)


From IM equations to IM control (E)

tL

LtiLtiRtu

L

tL

LtiL

dt

td

L

LtiRtu

Ldt

tdisq

tiLdt

td

L

LtiRtu

L

tL

LtiL

dt

td

L

LtiRtu

Ldt

tdisd

t

tiLt

dt

td

t

tiL

dt

drq

tti

Lt

tti

L

dt

tdrd

rdr

msdrsssqssq

rs

rdr

msdrss

rq

r

msqssq

rs

sq

sqrssrd

r

msdssd

rs

rqr

mssqrss

rd

r

msdssd

rs

sd

rdr

sdm

r

framedqin

rqrq

rdr

sdmsl

sl

r

rdsd

r

mrqsl

r

rdsd

r

mrd

1

1

1

1

0

0

00

0


Rotor flux estimation• 2 vector equations -> 4

equations.

• Rewriting a lot gives:

rsi

r

s l

x

y

s

dq

drd

dt = Lm

r isd - 1

r rd

dsldt

= Lm isq

r rd disddt

= usd - Rs isd - LmLr

drd

dt + s Ls isq 1

Ls disq

dt = usq - Rs isq - s Ls isd + Lm

Lr rd 1

Lssqrd

r

m iL

LT


Rotor flux control

r

Sin Cos Power-

amp.

2

3 r

+

+

Torque

and

flux-

control

i s

is

i sd

i sq

sl

r

s s

L m

L m

r

Controlsignal

AM

T * r*

e- js

i a

i b

i c

r

rr R

L


si

r

sl

x

y

s

dq

s

Simplified Stator Flux Model

us = R is + ds

dt = R is-is0 + R is0 +

ds

dt

ds

dt = - R is0 + us - R is-is0 =

no load running

is = is0 = s0

Ls

= - Rs0

Lsus = -

s0

sus

• Control frame is coupled to stator flux

• Anchored to no-load magnetization (Is0)


Expression for stator flux

dsd

dt

Ls

risd

1

rsd Ls

disd

dtsl Ls isq

usd Rs isd

dsl

dt

Ls isq Ls r disq

dtr sd Ls r isd

usq Rs isq

sdr

• Estimation error increasing with load and degreasing frequency


Flux observer

pk

wmek-> >wel

Psiis

Psi_hat Ð> is_hat

Psi ir

Psi _hatÐ> ir_hat

K*u

1s

Integrator1

1s

Integrator

1-kk

Rr

Rs

3

wmek

2

is

1

us

• Observer = (inverse) machine model

• Fed by both voltage, current and speed– Eventually, if dynamics allows, look for sensorless solutions

• Confidence check k between measurements and estimations

• Recalculate system matrix & update model parameters


FOC: Torque producing current

usq = Rs isq + r sd + Ls isq + Ls r

disq

dtr sd - Ls r isd

sd

sdrsq

rssqrs

sdrsq

ssqrs

sdr

sqsdrs

sd

sdsqrsdrsqs

rr

rs

r

ssdsds

dt

diLLiRR

dt

diLiRR

dt

diLiR

iR

RL

RLLandiLassume

sqsd iT


Stator flux oriented vector control

• Synchronously rotating reference frame

– Reference values constant at steady state

• Direct flux vector control on d-axis and torquecontrol in q axis

)(

)(ˆ)(

2

)(ˆ)(2

)(*

1

0

**

k

niniT

RRLL

Tkiki

RR

T

LL

ku

sdr

kn

nsqsq

s

rs

rs

ssqsq

rs

s

rs

sq

s

sdsd

s

sds

sdssd

s

sdsdsdssd

sdsdssd

T

kk

L

kR

tystationariiniL

T

kkkkiRku

dt

tdtiRtu

)()()(

)()()1,()(

)()()(

*

**


Stator flux oriented vector control

ysqref

ysdref

Rough rotor speed replacement

ws

1

us*

Demux

Ü>P isd &P isq_old

Ü> isq_ref

Demux

Ü> isd & isq

delta

rect

mod

angle

rec_to_pol

old psis

old angle

v ect

angle

Out

alfa/beta Ü> d/q1

v ect

angle

Out

alfa/beta Ü> d/q

1

10e-3s+1

Transfer Fcn

ref

act

e

y

PIE

Torque control

ref

act

e

y

PIE

Stator flux control

Scope1

Scope

Saturation

Mux

Mux1

Ground

1/Ts

2/p

-1

v ect

angle

Out

d/q > alfa/beta

4

Tref

3

is

2

PsiR

1

Psiref


Field Oriented Control


increase/decrease iq

rest

is5

vect

4 isdq

3

sc

2

sb

1

sa

f(u)

f(u)

vector

f(u)

f(u)

sector

rect

mod

angle

emf

rec_to_pol

In1 Out1

low speed?

increase/decrease id

v ect

angle

Out

alfa/beta Ü> d/q

XY Graph

Scope3

Scope2Scope1

Memory

1

-1

2-D T[k]

em

[1 1 1]

[-1 -1 -1]

[1 -1 -1]

[1 1 -1]

[1 -1 1]

[-1 -1 1]

[-1 1 1]

7

8

[-1 1 -1]

|u|

Demux

-> isd & isq

4

Psis

3

T*

2

i

1

Psis*

Direct torque control

• Similar to hysteresis DCC but instead of isd is ψsd

and torque replaces isq


ABB DTC

• Transistor switching is matched to machine capability to satisfyload condition and range

– Parameter identification


SimPowSys DTC IM Drive


SimPowSys DTC IM Drive

• Example from MathWorks AC drive library

– 150kVA IM load sequence

• Start-up peaks: DC bus 870V, IM current 1300A

• Speed references: 500rpm @ 0s and 0rpm at 1.5s. Speed follows acceleration ramps

• 792Nm load is applied at 0.5s and reversed to -792Nm at 1.5s


Exercises on induction machine

• Reexamine “all” machine related questions and replace DCM and PMSM by IM

– PE ExercisesWithSolutions2019b vers 190206

• IM machine construction

• IM machine equations

• IM machine parameters and performance

• IM machine control and parameters

• Electric drive system specification for IM machine

Documents

L21: Induction machine control - IEA