DFIG SteadyState

7/23/2019 DFIG SteadyState

1/104

J. McCalley

Double-fed electric machines

steady state analysis


2/104

Four congurations

2

We will study

only this one,

the DFIG.


3/104

Basic concepts

3

AC

DC

DC

AC

DFIG

Rotor

Power

Grid

DC Link

Rotor is wound: it has 3 windings.tator has three windings.

Indu!tion "a!hine looks like a trans#or"er with a rotating se!ondary $rotor%.

In DFIG, we will in&e!t a 'oltage !ontrol signal 'ia that !on'erter.


4/104

Basic Concepts

4

p

fn ss

60=(alan!ed 'oltages a))lied to stator windings )ro'ides a

rotating "agneti! #ield o# s)eed

whi!h indu!es an e"# in the rotor windings a!!ording to

*eind+indu!ed e"# in one !ondu!tor o# rotor

*'+'elo!ity o# !ondu!tor relati'e to stator #lu rotation*(+stator "agneti! #lu density 'e!tor*L+length o# !ondu!tor in dire!tion o# wire

LBveind = )(

rotor

$#s: - /0,

): 1 o# )ole

)airs%


5/104

Basic concepts

5

rpm60

;

;rad/sec377;

p

fn

n

nn

psslip

ss

s

ms

mms

s

ms

=

=

==

==

We !an "ani)ulate to get:

)1(

)1(

s

snn

sm

sm

=

=

2he indu!ed rotor 'oltages ha'e #reuen!y o# :msr =

u4stitution into sli) e)ression a4o'e yields: srsrs

r sffss ===

Observe three modes of operation:

5"6 5s5r7s7u4syn!hronous o)eration

5"+ 5s5r+s+yn!hronous o)eration

5"75s5r6s6u)ersyn!hronous o)eration

8e!hani!al

rad9se!


6/104

model

6

sssss IjXREV )( +=STATOR VOLTAGE EQUATION: at #s

rrrrsr IXjREV

+

=

)(

ROTOR VOLTAGE EQUATION:at #

r

+stator 'oltage with #reuen!y #ssV

sEsI

sR

sX

+ e"# in the stator windings with #reuen!y #s+ stator !urrent with #reuen!y #s

+stator resistan!e

+stator leakage rea!tan!e

+rotor 'oltage with #reuen!y #rrV

rsE

rI

rR

rX

+indu!ed e"# in the rotor windings with #reuen!y #r

+indu!ed rotor !urrent with #reuen!y #s

+rotor resistan!e

+rotor leakage rea!tan!e+

2hese uantitiesare re#erred to

rotor side,

indi!ated 4y

)ri"e notation.

2hese uantities

are re#erred tostator side.

rrL


7/104

eferring !uantities

7

ol'e 4oth relations #or "and euate:

(ut re!all:

A))li!ation o# Faraday;s Law allows the stator 4a!k e"# and the indu!ed rotor

'oltage to 4e e)ressed as:

mrrrrs

mssss

fNKE

fNKE

2

2

== rotor uantities, res)e!ti'ely

": "agneti0ing #lu

rrr

sss

rs

s

rrr

rs

sss

s

m fNK

fNK

E

E

fNK

E

fNK

E=

==

22

sr sff =sNK

NK

sfNK

fNK

E

E

rr

ss

srr

sss

rs

s ==

2he ratio


8/104

eferring !uantities

8

We &ust deri'ed that:a

EsE srs=

At a lo!ked rotor !ondition $s+?%, the de'i!e is si")ly a stati! trans#or"er,and we ha'e:

aII

aEE

rr

rsrs

/==

rsss

rs EaEa

EE ==

2his tells us it we want to "o'e a 'oltage #ro" rotor side to stator side, we "ulti)ly it

4y a+=s9=r. We !an o4tain si"ilar relationshi)s #or !urrents and i")edan!es, and so

we de#ine the rotor uantities re#erred to the stator a!!ording to:

2

2

aLL

aRR

rr

rr

==

33

33

3333

Rr&5rL@r

rss

Rs &5sL@sIs Ir

Bs Br

Rotor uantities are re#erred

to the statorside, indi!ated

4y un)ri"ed uantities.

2his is lo!ked rotor !ondition

$s+?%, there#ore 5r+5sand

rs+s

We !an a!!ount #or other

sli) !onditions using 5r+s5s

and #ro" $%, a;rs+ss.

$%


9/104

eferring !uantities

9

33

33

3333

Rr&s5sL@r

rs+sss

Rs &5sL@sIs Ir

Bs Br

=ow write the rotorside 'oltage euation $re#erred to stator%:

rrsrsr ILjsREsV )( +=

Di'ide 4y srrs

rs

r ILjs

RE

s

V)( +=

and we get the #ollowing !ir!uit:

33

33

3333

Rr9s&5sL@r

ss

Rs &5sL@sIs Ir

Bs Br9s

2he 'oltage on 4oth sides

o# the #"r is the sa"e,

there#ore, we "ay eli"inate

the #"r. .We re)resent a

"agneti0ing indu!tan!e

&5sL"in its )la!e.


10/104

eferring !uantities

10

33

3333

Rr9s&5s

L@r

&5sL"s

Rs &5s

L@s

Is

Ir

Bs Br9s


11/104

Po"er relations

11

33

3333

Rr9s&5s

L@r

&5sL"s

Rs &5s

L@s

Is

Ir

Bs Br9s

s

sRR

s

sRR

s

sR

s

sRsRR

s

R rr

rrrrrrr )1( +=

+=+

=

s

sVV

s

sVV

s

sV

s

sVsVV

s

V rr

rrrrrrr )1( +=

+=+

=

Change the !ir!uit a!!ordinglyE.

We "odi#y the a4o'e !ir!uit slightly in order to !learly se)arate sli)de)endent ter"s

#ro" loss ter"s:


12/104

Po"er relations

12

33

3333

Rr&5s

L@r

&5sL"s

Rs &5sL@sIs Ir

Bs Br

It is )ossi4le to )ro'e that the "e!hani!al )ower out o# the "a!hine is the )ower

asso!iated with the sli)de)endent ter"s R$?s%9s and Br$?s%9s. 2o do so, use:

rlossslossrsmech PPPPP ,, +=

where Psand Prare )owers entering the "a!hine through the stator > rotor windings,res)e!ti'ely, and Ploss,sand Ploss,rare the stator and rotor winding losses, res)e!ti'ely.

)ressing the righthandter"s o# the )ower 4alan!e relation in ter"s o# the a4o'e

!ir!uit )ara"eters leads one to identi#y the sli)de)endent ter"s as P"e!h.

Rr$?s%9s

Br$?s%9s


13/104

Po"er relations

13

33

3333

Rr&5s

L@r

&5sL"s

Rs &5sL@sIs Ir

Bs Br

( ) { }

{ }*2

*2

*2

Re1

3)1(

3

1Re3

)1(3

Re33

rrr

r

rrr

r

reqeqrmech

IVs

s

s

sRI

Is

sV

s

sRI

IVRIP

=

=

=

Re+

Rr$?s%9s

Be+

Br$?s%9s

I# P"e!h7the "a!hine is deli'ering )ower through the sha#t: 8H2HR

I# P"e!h6the "a!hine is re!ei'ing )ower through the sha#t: G=

Rotor !urrent $Ir% dire!tion is out o# )ositi'e side

o# 'oltage sour!eJ there#ore it su))lies )ower to

!ir!uit. (ut a nor"al $)ositi'e% resistan!e Realways !onsu"es )ower. o these two ter"s

should 4e o))osite sign. De#ining P"e!h

7 $see

4elow% as "otor "ode i")lies Reter" should

4e added and Be ter" should 4e su4tra!ted.

P"e!h

I# 6s6?

Reter" is )ositi'e

Beter" is )ositi'e

u))lying P to !!tI# 7s7?Reter" is negati'eBeter" is negati'eConsu"ing P #ro" !!t.


14/104

rs or!uee$pression

14

{ }*2 Re13)1(3 rrrrmech IVs

s

s

sRIP

=

{ }*2 Re13)1(3 rrm

rr

mm

mechemm

emmemmech IVs

sp

s

sRI

ppPT

pTTP

====

33

3333

Rr&5sL@r

&5sL"s

Rs &5sL@sIs Ir

Bs Br

Re+

Rr$?s%9s

Be+

Br$?s%9s

$): 1 o# )ole

)airs%

Re!all #ro" slide K: ;s

rs

=

s

msm ss

== 1)1(

2here#ore:

{ }

{ }*2

*2

Re33

Re33

rr

rr

rr

rr

r

m

mr

mrr

m

em

IVpRIp

IVpR

Ip

T

=

=

r

m

r

s

s

m

s

s

==1

and

( )ivrrrr

rr

em IV

pRIp

T = cos33

2


15/104

secon e!u &a en or!uee$pression

15

{ }*Re3 sss IVP=

33

3333

Rr&5sL@r

&5sL"s

Rs &5sL@sIs Ir

Bs Br

Re+

Rr$?s%9s

Be+

Br$?s%9s

tator )ower:

( ) ( ) msrssssss LjIILjRIV +++=tator 'oltage:

u4stitute Bsinto Ps: { } ( ) ( )( ){ }

( ){ }( ){ }*222****

**

Re3

Re3

Re3Re3

srmssmssssss

srmsssmssssssss

smsrssssssss

IILjILjILjIR

IILjIILjIILjIIR

ILjIILjRIIVP

+++=+++=

+++==

2he "iddle two ter"s are )urely i"aginary, there#ore:

( ){ }*2Re3 srmssss IILjIRP +=First ter" is )urely real, only the se!ond ter" !ontains real and i"aginary, there#ore:

{ }*2 Re33 srmssss IILjIRP +=


16/104

e$pression

16

{ }*Re3 rrr IVP=

33

3333

Rr&5sL@r

&5sL"s

Rs &5sL@sIs Ir

Bs Br

Re+

Rr$?s%9s

Be+

Br$?s%9s

Rotor )ower:

( )

( )

( ) ( ) msrsrsrrr

msrsrsr

rr

msrsrsrrrrr

LjsIILjsRIV

LjIILjs

RI

s

V

LjIILjs

sRRI

s

sVV

+++=

++

+=

++

++=+

11

Rotor 'oltage:

u4stitute Brinto Pr: { } ( ) ( )( ){ }

( ){ }( ){ }*222

****

**

Re3

Re3

Re3Re3

rsmsrmsrrsrr

rsmsrrmsrrrsrrr

rmsrsrsrrrrr

IILjsILjsILjsIR

IILjsIILjsIILjsIIR

ILjsIILjsRIIVP

+++=+++=

+++==

2he "iddle two ter"s are )urely i"aginary, there#ore:

( ){ }*2Re3 rsmsrrr IILjsIRP +=

First ter" is )urely real, only the se!ond ter" !ontains real and i"aginary, there#ore:{ }*2 Re33 rsmsrrr IILjsIRP +=


17/104

e$pression

17

{ }*2 Re33 rsmsrrr IILjsIRP +=

=ow su4stitute Psand Printo the )ower 4alan!e euation:



{ } { } rlossslossrsmsrrsrmsssmech PPIILjsIRIILjIRP ,,*2*2 Re33Re33 +++=

H4ser'e we ha'e loss ter"s added and su4tra!ted in the a4o'e, so they go away.

{ } { }** Re3Re3 rsmssrmsmech IILjsIILjP +==ow !onsider what ha))ens when you take the real )art o# a 'e!tor "ulti)lied 4y &

$or rotated 4y degrees%:

Re$&a%

a

&a

I"$a%Observe thatRe(a! " # Im(a!

2here#ore:

{ } { }

**Im3Im3

rsmssrmsmech IILsIILP

=


18/104

e$pression

18

Let;s !onsider another 'e!tor identity: taking i"aginary )art o# a !on&ugated 'e!tor:

{ } { }** Im3Im3 rsmssrmsmech IILsIILP =

I"$a%

a

a

I"$a%

Observe that

Im(a$! " # Im(a!

2here#ore:

{ } { }{ } { }

{ } { }[ ]{ }[ ]sIILIIsIIL

IILsIIL

IILsIILP

rsms

rsrsms

rsmsrsms

rsmssrmsmech

==

=

=

1Im3

ImIm3

Im3Im3

Im3)(Im3

*

**

**

***

Re!all: )1( ssm =

{ }*Im3 rsmmmech IILP =2here#ore:

mmechem

p

PT =Re!all:

{ }*Im3 rsmem IIpLT =


19/104

e$pressions

19

{ }*Im3 rsmem IIpLT =( )ivrrrr

rr

em IVpRIp

T

= cos33

2

2orue e)ression 1?: =eed rotor

s)eed, rotor 'oltage and rotor !urrent2orue e)ression 1: =eed stator

!urrent and rotor !urrent

A third set o# eui'alent torue e)ressions #ollowE.


20/104


21/104

e$pressions

21

Fro" stator winding euation:

s

rmss

L

ILI

=

Fro" rotor winding euation:

r

smrr

L

ILI

=

u4stitute into torue e)ression 1E.

{ }*Im3 rsmem IIpLT =

{ }

{ }

{ }*

2*

**

*

Im3

Im3

Im3

Im3

rs

s

m

rmrs

s

m

rrmrs

s

m

r

s

rmsmem

IL

Lp

ILILLp

IILIL

Lp

IL

ILpLT

=

=

=

=

Msing stator winding euation:

Purely

real

{ }

{ }

{ }*

2*

**

*

Im3

Im3

Im3

Im3

rs

r

m

smrs

r

m

ssmrs

r

m

r

smrsmem

IL

Lp

ILIL

Lp

IILIL

Lp

L

ILIpLT

=

=

=

=

Msing rotor winding euation:

Purely

real

#i d li


22/104

#irgap and slip po"erOn s%ides &' and &) *e derived the fo%%o*in+ re%ations for the po*er into the

stator and rotor respe,tive%-:

{ }

*2 Re33srmssss

IILjIRP +=

{ }*2 Re33 rsmsrrr IILjsIRP +=S.btra,tin+ %osses from both sides) *e obtain:

{ }*2 Re33 srmssss IILjIRP = { }*2 Re33 rsmsrrr IILjsIRP =

This /.antit- is the po*er that f%o*s

from the stator termina%s to the rotor

(ne+ative for +enerator operation!0 Inother *ords) it is the po*er a,ross

the air+ap0 Therefore:

{ }*2 Re33 srmssssairgap IILjIRPP ==

This /.antit- is the po*er that is

transferred from the +rid to the rotor

thro.+h the ,onverter (ne+ative*hen it is into the +rid!0 It is ,a%%ed

the s%ip po*er0 Therefore:

{ }*2 Re33 rsmsrrrslip IILjsIRPP ==1rin+ o.t front the 2s3 in the s%ip po*er e4pression and .se Re5a6"#Im(a!

(both!:Use Im(a$! " #Im(a! on s%ip e4pression:

22

{ }*2

Im33 srmssssairgap IILIRPP == { }*2

Im33 rsmsrrrslip IILsIRPP ==

{ }rsmsrrrslip IILsIRPP *2

Im33 ==

The term 7Im56 in the s%ip po*er e4pression is 8air+ap0 Therefore:

{ }*2 Im33 srmssssairgap IILIRPP ==

airgapslip sPP =

#i d li


23/104

#irgap and slip po"er

23

So *e .st proved that:

O.r po*er ba%an,e re%ation states:

Re,a%%:s

ms

=1

airgap

s

mmech PP

=

airgapsm

airgaps

m

mmechem P

pp

P

p

PT

===slip

s

em Ps

pT

=

s

rs

= slip

sr

sem P

pT

=

slip

r

em P

p

T

=

airgapslip sPP = *here

{ }*2 Re33 srmssssairgap IILjIRPP == { }*2

Re33 rsmsrrrslip IILjsIRPP ==

slipairgap P

rlossr

P

slosssrlossslossrsmech PPPPPPPPP ,,,, +=+=

Therefore: slipairgapmech PPP +=

S.bstit.tin+ airgapslip sPP = *e obtain ( ) airgapairgapairgapmech PssPPP == 1

S.bstit.tin+:slipairgapairgapslip P

sPsPP

1==


24/104

acti&e po"ers

24

On s%ides &' and &) *e derived the fo%%o*in+ re%ations for the po*er into the

stator and rotor respe,tive%-:

{ }*2 Re33srmssss

IILjIRP += { }*2 Re33 rsmsrrr IILjsIRP +=

If *e ne+%e,t the stator %osses (7RSIs9! and rotor %osses (7RrIr

9!:

{ }*Re3 srmss IILjP = { }*

Re3 rsmsr IILjsP =1rin+ o.t front the 2s3 in the rotor po*er e4pression and .se Re5a6"#Im(a! (both!:

{ }*Im3 srmss IILP = { }*

Im3 rsmsr IILsP =Use Im(a$! " # Im(a! on the rotor po*er e4pression

{ }*Im3 srmss IILP = { }rsmsr IILsP *

Im3 =

The term 7Im56 in the rotor po*er e4pression is 8S0 Therefore: sr sPP =Re,a%% the po*er ba%an,e re%ation: rlossslossrsmech PPPPP ,, +=

Ne+%e,tin+ %osses:rsmech PPP +=

S.bstit.tin+ 8re4pression: sssmech PssPPP )1( ==

Re,a%%:s

ms

=1

s

s

mmech PP

=

s

sm

s

s

m

m

mechem P

pp

P

p

PT

===

r

s

em Ps

pT

=

s

rs

= r

sr

sem P

pT

=

r

r

em P

p

T

=

# ti l ti


25/104

#cti&e po"er relations - summary

25

sr s =

E4a,t Appro4imate1oth


{ }*

2 Re33 rsmsrrr IILjsIRP +={ }*2 Re33 srmssssairgap IILjIRPP ==

{ }*2 Re33 rsmsrrrslip IILjsIRPP ==

airgapslip sPP =


slipairgapmech PPP +=

( ) airgapmech PsP = 1

s

ms

=1

airgap

s

mmech PP

=

airgap

s

em Pp

T

=

m

mechem

pPT

=

slipr

em P

p

T

=

{ }*Re3 srmss IILjP =

{ }*

Re3 rsmsr IILjsP ={ }*Re3 srmssairgap IILjPP ==

{ }*Re3 rsmsrslip IILjsPP ==

sr sPP =

rsmech PPP +=

smech PsP )1( =

s

s

mmech PP

=

s

s

em Pp

T

=

r

r

em P

p

T

=

P b l


26/104

Po"er balance

26

Ps

Ploss,s

Pairga) Psli) Pr

Ploss,r

slipairgap P

rlossr

P

slosssrlossslossrsmech PPPPPPPPP ,,,, +=+=

2hese #igures assu"e )ro)er sign !on'ention

$)ower #lowing to the rotor is )ositi'e%.

Pgrid PsPairga) Psli) PrPgrid

ith %osses itho.t %osses

P"e!h P"e!h

) t d


27/104

)enerator modes

27

ems

s Tp

P = em

rr T

pP

=

Mode *lip andspeed Pmech Ps Pr

1. Motor(Tem>0)

ss(suprsynchrnsm)

>0 (mchdelivers mechpwr)

>0 (mchreceivespower viastator)

>0 (mchreceivespower viarotor)

2. enerator(Tem0 (mchdelivers mechpwr)

>0 (mchreceivespower viastator)


28/104

)enerator modes

28

sm >

sm 0motor

%>0

&>0a"sor"in'

>0

Observe that ,.rrent an+%e is a%*a-s

ne+ative of impedan,e an+%e) Yi"#Y

ealp"r

eacti&ep"r

$>0motor

%>0

&


42/104

7$ample Problem

42

(a! S-n,hrono.s speed(b! Line#to#ne.tra% vo%ta+e

(,! Line ,.rrent

(d! Stator f%.4

(e! Rotor ,.rrent(f! Rotor f%.4

(+! Rotor vo%ta+e

(h! Rotor rea% po*er

(i! Rotor rea,tive po*er(! Tota% rea% po*er +enerated

(! Tem

(b! Line#to#ne.tra% vo%ta+e: 'o%s04.39803

690==sV

(,! Line ,.rrent: amps1804.167304.3983

102

330

*6

*

*

=

=

==+ ss

ssss V

P

IIVjP

(d! Stator f%.4

( )eers9028.1

16.314

106.2)1804.1673(04.398)( 3=

=

=

jj

RIV

s

ssss

sssss jRIV +=

(a! S-n,hrono.s speed: rad/sec16.314)50(22 === ss f

A%ternative%-) the s-n,hrono.s speed *as +iven as &'@@ rpm) therefore:

sec/08.157sec60

m!"2

m!"

1500rad

rev

radrevs ==

sec/16.314)08.157(2 radp ss ===

The 9 ; >?IG +iven b- the data on s%ide 7 is de%iverin+) from the stator) rated

%oad (9 ;! at rated vo%ta+e *ith Fero stator rea,tive po*er in a '@ JF +rid0 The

s%ip is s"#@09' (s.per#s-n,hrono.s!0 Komp.te:

7$ample Problem


43/104

7$ample Problem

43


(,! Line ,.rrent

(d! Stator f%.4


(+! Rotor vo%ta+e



(! Tem

(e! Rotor ,.rrent

(f! Rotor f%.4

(+! Rotor vo%ta+e

rmsss ILIL +=

amps5.164.1807105.2

)1808.1673(10587.29028.13

3

=

=

=

m

sss

r L

IL

I

This is the referred rotor ,.rrentZ

e ,an obtain the a,t.a% rotor ,.rrent from a (or .! "@07P:

amps5.165.6145.164.1807)34.0( === rr IaIThis phasor is at the rotor

fre/.en,-) of

fr"sfs"#@09'('@!"#&90' JFrrsmr ILIL +=eer4.77358.15.164.180710587.21808.1673105.2 33 =+= r

'o%s9.1652.102)4.77358.1)(16.314)(25.0(109.2)5.164.1807( 3 =+= jVr

rsrrr jsRIV +=

A,t.a% rotor vo%ta+e: =

== 9.1656.30034.0

9.1652.102

a

VV rr




7$ample Problem


44/104

7$ample Problem

44





(,! Line ,.rrent

(d! Stator f%.4


(+! Rotor vo%ta+e



(! Tem


(i! Rotor rea,tive po*er

(! Tota% rea% po*er +enerated

{ }*Re3 rrr IVP=

{ } MW55.0)5.164.1807(9.1652.102Re3 * ==rP

{ }*Im3 rrr IV% =

{ } +,-R4.23)5.164.1807(9.1652.102Im3 * ==r%

MW55.255.02 ==+ rs PP


Komments:&08mm.st be %ar+er in ma+nit.de to s.pp%- %osses

90 This *ind t.rbines ratin+ sho.%d be 90'' ;0

70 The >?IG stator *indin+ is rated for 9;0

7$ample Problem


45/104

7$ample Problem

45


(,! Line ,.rrent

(d! Stator f%.4


(+! Rotor vo%ta+e



(! Tem

(! Tem { }*

Im3 rss

mem I

L

LpT =

( ){ } &NmTem 9.125.164.18079028.1Im10587.2

105.223

*

3

3

=

=




8ind turbine control le&els


46/104

8ind turbine control le&els

46

Leve% I: Re+.%ates po*er f%o*

bet*een +rid and +enerator0

Leve% II: Kontro%s the amo.nt

of ener+- e4tra,ted from the

*ind b- *ind t.rbine rotor0

Leve% III: Responds to *ind#

farm or +rid#,entra% ,ontro%

,ommands for ; dispat,h)

vo%ta+e) fre/.en,-) or inertia%,ontro%0

Rotor#side ,onverter (RSK! is

,ontro%%ed so that it provides

independent ,ontro% of Temand Qs0 Lets st.d- the

stead-#state a,tions of this

parti,.%ar ,ontro% f.n,tion0

e&el 9 control


47/104

e&el 9 control

47

This (open#

%oop! ,ontro%

not heavi%-

.sed for >?IGs

Ass.me >K b.s vo%ta+e is

,ontro%%ed b- +rid#side

,onverter (GSK! to a pre#determined va%.e for

proper operation of both

GSK and RSK0

e a,hieve

,ontro% obe,tivesb- ,ontro%%in+

rotor#side

vo%ta+e0

e ,ontro% rotor

vo%ta+e to a,hieve a

spe,ified tor/.e and

stator rea,tive po*er0

e&el 9 control


48/104

e&el 9 control

48

O.r obe,tive here is) for a fi4ed stator vo%ta+e (fi4ed b- the +rid!) and a

desired tor/.e Tem)refand a desired stator rea,tive po*er Qs)ref) *e *ant to

determine the rotor vo%ta+e to mae it so0 e are a%so interested in the

stator f%.4) stator ,.rrent) rotor ,.rrent) and rotor f%.4) and stator rea%po*er) as sho*n in the dia+ram be%o*0

e&el 9 control


49/104

e&el 9 control

49

e dra* the phasor dia+ram *ith stator f%.4 as the referen,e (@ de+rees!0 Jere)

the stator f%.4) denoted b- [s(instead of s!) is spe,ified as the referen,e0 e

have identified parti,.%ar an+%es in this phasor dia+ram0 It is operatin+ as a

motor (,.rrent is a%most in phase *ith vo%ta+e!) and the stator is absorbin+rea,tive po*er (Ishas a ne+ative an+%e re%ative to Vs) so motor"VsHIshas a

positive an+%e) indi,atin+ it is ind.,tive and therefore absorbin+0

e&el 9 control4 + e!uation


50/104

e&el 9 control4 +se!uation

50

?rom vo%ta+e e/.ation (s%ide 7'!: sssss jRIV +=

If *e ne+%e,t drop a,ross the stator resistan,e (it is t-pi,a%%- ver- sma%%!) then:

sss jV S.bstit.te into the stator rea,tive po*er e/.ation: { } { }** Im3Im3 ssssss IjIV% ==Use Im(a!"Re(a!: { }*Re3 ssss I% =?rom previo.s s%ide) note that :iis the an+%e b- *hi,h Is%eads s) i0e0)

issss II == ;0S.bstit.tin+:

{ } { }

{ } isssiisss

isssissss

IjI

II%

cos3s!"cosRe3

Re30Re3

====

?ina% e/.ation for Qs: issss I% cos3=

e&el 9 control4 ( e!uation


51/104

e&el 9 control4 (eme!uation

51

?rom J7 (see s%ide 7\!:

A+ain (from phasor dia+ram!) note that :iis the an+%e b- *hi,h Is%eads s) i0e0)

issss II == ;0S.bstit.tin+:

?ina% tor/.e e/.ation:

{ }ssem IpT ,Im3 *=

{ } { }

{ } issiiss

ississem

IpjIp

IpIpT

s!"3s!"cosIm3

Im30Im3

=+===

issem IpT s!"3=

e&el 9 control4 5s e!uation


52/104

e&el 9 control4 5se!uation

52

?rom phasor dia+ram:

1.t re,a%% o.r Qsand Teme/.ations:

S.bstit.tin+ into Ise/.ation:

isiss jIII s!"cos +=

issem IpT s!"3=

issss I% cos3= sss

is

%I

3

cos =

s

emis

p

TI

3s!" =

s

em

ss

ss

p

Tj

%I

33+=

Re,a%% from s%ide '@: sss jV sssV

S.bstit.tin+ into Ise/.ation:

s

ems

s

s

s pV

Tj

V

%I

33

+=

e&el 9 control4 ;r e!uation


53/104

e&el 9 control4 ;re!uation

53

Usin+ these re%ations) to+ether *ith:

?rom s%ide 9@:

rmsss

ILIL +=

rrsmr ILIL +=

r

rs

ms

s

sLL

L

LI

=

1

r

r

s

rs

mr

LLL

LI

1+= rs

m

LL

L2

1=

sss jV s

ems

s

ss

pV

Tj

V

%I

33

+=

*e ma- derive:

=

m

rs

s

ems

m

rs

s

s

m

r

s

sr

L

LL

pV

Tj

L

LL

V

%

L

LV

33

=

m

s

s

ems

m

s

s

s

ms

sr

L

L

pV

Tj

L

L

V

%

L

VI

33

1

e&el 9 control4 ;r e!uation


54/104

e&el 9 control4 ;re!uation

54

+

=

m

rs

s

s

m

r

s

sr

m

rs

s

emsrr

LLL

V%

LLVj

LLL

pVTV

33

Ne+%e,tin+ the vo%ta+e drop in the rotor resistan,e) *e ma- derive:

No* .se the rotor f%.4 e/.ation derived on the previo.s s%ide

to+ether *ith the rotor vo%ta+e e/.ation (s%ide 7'!:

=

m

rs

s

ems

m

rs

s

s

m

r

s

sr

LLL

pVTj

LLL

V%

LLV

33 rsrrrjsRIV +=

e&el 9 control4 summary


55/104

e&el 9 control4 summary

55

+

=

m

rs

s

s

m

r

s

sr

m

rs

s

emsrr

L

LL

V

%

L

LVj

L

LL

pV

TV

33

=

m

rs

s

ems

m

rs

s

s

m

r

s

s

r L

LL

pV

Tj

L

LL

V

%

L

LV

33

=

m

s

s

ems

m

s

s

s

ms

sr

L

L

pV

Tj

L

L

V

%

L

VI

33

1

s

ems

s

ss

pV

Tj

V

%I

33

+=

A%so) *e have stator and rotor po*ers as a f.n,tion of Tem:

s

ss

V

=

ems

s Tp

P = em

rr T

p

P

=

e&el 9 control4 magnitudes


56/104

e&el 9 control4 magnitudes

56

2

2

2

22

33

+

=

m

rs

s

s

m

r

s

sr

m

rs

s

emsrr

L

LL

V

%

L

LV

L

LL

pV

TV

22

2

33

+

=

m

rs

s

ems

m

rs

s

s

m

r

s

sr

LLL

pVT

LLL

V%

LLV

22

2

33

1

+

=

m

s

s

ems

m

s

s

s

ms

sr

L

L

pV

T

L

L

V

%

L

VI

22

2

33

+

=

s

ems

s

ss

pV

T

V

%I

And this sho*s that these terms are f.n,tions of o.r desired referen,e /.antities0

),,( emssIs T%Vf=

),,( emssr T%Vf=

),,( emssIr T%Vf=

),,,( remssVr T%Vf =

;a+nit.des are attra,tive be,a.se then *e ,an p%ot them0

The above re%ations are +iven as a f.n,tion of r) b.t it ma- be more int.itive

to p%ot them as a f.n,tion of rotor speed) m) *here *e ,an ,omp.te

r "smH(s!0 ]o. ,an thin of the rotor speed as m"(s! s*hi,h sho*s

that for %o* positive s%ips) rotor speed is .st be%o* s-n,hrono.s speed) and

s

ss

V

=

)( ss Vf=

emr

r Tp

P = ),(r remTf =

e&el 9 control


57/104

e&el 9 control

57

?i4ed Qs"@ ?i4ed Tem"#&

22

2

33

+

=

s

ems

s

ss

pV

T

V

%I

* Isis inde)endent o# 5"4ut in!reases with S2e"S and with STsS

* Isis the sa"e inde)endent o# whether "a!hine is a4sor4ing or su))lying 'ars.

* A4o'e euation indi!ates Isshould 4e the sa"e #or 2e"+?, 2e"+?. /owe'er,

a4o'e euation negle!ted stator resistan!e Rs. Assu"ing #ied Bs, in "otor

"ode $2e"+?%, Rs!auses 'oltage a!ross rotor !ir!uit to 4e less, and so Ir"ust

4e greater to deli'er sa"e torue. In gen "ode, Rs!auses 'oltage a!ross

e&el 9 control


58/104

e&el 9 control

58

22

2

33

1

+

=

m

s

s

ems

m

s

s

s

ms

sr

L

L

pV

T

L

L

V

%

L

VI

?i4ed Tem"#&

* Iris inde)endent o# 5"#or #ied torue 4ut in!reases as Ts"o'es #ro" $a4sor4ing%

to U $su))lying%.

{ }

*

Im3 rss

m

em IL

L

pT =

Fied torue i")lies #ied rotor

!urrent i# stator #lu is #ied.

(e!ause 2e"+P"e!h)95", P"e!h

"ust de!rease as 5"in!reases.

e&el 9 control


59/104

e&el 9 control

59

1oth rotor ,.rrent and stator ,.rrent e/.ations have rea% part determined b- Qs

and ima+inar- part determined b- Tem(Vsis at ^@ so rea% part of ,.rrents is in

/.adrat.re *ith Vs!

6Ts63Bs9Ls5s$rea!ti'e )ower into stator, a4s%

;a+netiFed from rotor ,.rrentTs+ $no stator rea!ti'e )ower%:

s

s

V

%

3 ;a+netiFed from stator ,.rrent0

;a+netiFed from both ,.rrents0

s

ems

s

ss

pV

TjV

%I 33

+=

= m

s

s

ems

m

s

s

s

ms

sr

L

L

pV

T

jL

L

V

%

L

V

I 33

1

+

+=

++=+=

m

s

s

ems

s

ems

m

s

s

s

ms

s

s

s

m

s

s

ems

m

s

s

s

ms

s

s

ems

s

srsm

L

L

pV

T

pV

Tj

L

L

V

%

L

V

V

%

L

L

pV

Tj

L

L

V

%

L

V

pV

Tj

V

%III

333

1

3

33

1

33

Bery !lose to 0ero sin!e LsVL".8agneti0ing !o")onent.

ms

s

L

V 1

Ts6 $rea!ti'e )ower #ro" stator, su)%:

Ts+3Bs9Ls5s$rea!ti'e )ower into stator, a4s%

;a+netiFed from both ,.rrents0

Add the" to

o4tain

"agneti0ing

!urrent

e&el 9 control


60/104

e e co o

60


* Prlinearly de!reases w9 5"#or U2e"$gen% and linearly in!reases w9 5"#or 2e"$"ot%.

* Pris inde)endent o# whether "a!hine is a4sor4ing or su))lying 'ars.

emr

r Tp

P =

Re"e"4er: m"(s!s)

r"ss0

e&el 9 control


61/104

61


* Bris linearly de!reasing with 5"to 5"+5sand then linearly in!reasing with 5".* Brde)ends "ainly on s)eed o# "a!hine.

* Brdoes not !hange "u!h with 2e"or with Ts4e!ause BsLr95sL"tends to do"inate.

2

2

2

22

33

+

=

m

rs

s

s

m

r

s

sr

m

rs

s

emsrr

L

LL

V

%

L

LV

L

LL

pV

TV

Re"e"4er: m"(s!s)

r"ss0

e&el 9 control


62/104

62


* ##i!ien!y in!reases with 5"under all !onditions $see net slide%:

* In the su4syn!hronous "ode, stator windings !arry SP"e!hSSPrS.

* In the su)ersyn!hronous "ode, stator windings !arry SP"e!hSSPrS.

* ##i!ien!y de!reases as STsS in!reases $"ost e##i!ient #or unity )ower #a!tor%.

* 8ore e##i!ient when a4sor4ing $"agneti0ed #ro" stator% than su))lying

$"agneti0ed #ro" rotor%

)enerator modes


63/104

63

sm >

sm ?IG

The differen,e bet*een the ma,hines in terms of stead-#state mode%s isthe abi%it- to e%e,tri,a%%- absorb or s.pp%- ,omp%e4 po*er S via the rotor0

here do *e see rotor %osses in these ,ir,.itsB (ne4t s%ide!

*C5) &s DF5)


75/104

*C5) &s DF5)

75

33

Re

&5sLe+&Ye9s

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"

$Ye+ 5rLe%

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

BsB"

I"

SKIG

>?IG

Sp%it .p the RHs terms in ea,h ,ir,.it as RMR(s!Hsand the rotor %osses be,ome immediate%- apparent0

here do *e see me,hani,a% po*er in these ,ir,.itsB (ne4t s%ide!

Rr$?s%9s

Re$?s%9s

Rr$?s%9s

*C5) &s DF5)


76/104

*C5) &s DF5)

76

33

Re

&5sLe+&Ye9s

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"

$Ye+ 5rLe%

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

BsB"

I"

SKIG

>?IG

The me,hani,a% po*er is representedb- the s%ip#dependent resistan,es0

1.t *hat do the other t*o terms in the >?IG ,ir,.it representB (ne4t s%ide!

Rr$?s%9s

Re$?s%9s

Rr$?s%9s

*C5) &s DF5)


77/104

*C5) &s DF5)

77

33

Re

&5sLe+&Ye9s

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"

$Ye+ 5rLe%

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

BsB"

I"

SKIG

>?IG

These terms represent the rea% and rea,tive po*er e4,han+e bet*een the rotorand the RSK0 As *e sa* on s%ide ) these terms) Re/and We/,an be pos (rotor

transfers po*er to RSK! or ne+ (RSK transfers po*er to rotor!0

Jo* to ,omp.te tor/.e in for these ma,hinesB (ne4t t*o s%ides!

Rr$?s%9s

Re$?s%9s

Rr$?s%9s

*C5) &s DF5)


78/104

*C5) &s DF5)

78

(or!ue e!uation for *C5)


79/104

79

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

BsB"

I"

SKIG

Rr$?s%9s

ssRIP rrmech

)1(3 2 = =ote that the [s\ on the deno"inator )ro'idesthat P"e!his )ositi'e #or s7, "otor a!tion, andnegati'e #or s6, generator a!tion.

s

RI

p

s

sRI

s

p

s

sRI

pP

pPT

rr

s

rr

s

rr

m

mech

m

mech

m

em

22

2

3)1(

)1(3

)1(3

1

=

=

==

=

Jo* to obtain IrB 0 (ne4t s%ide!

(or!ue e!uation for *C5)


80/104

80

33

3333

Rr&5sL@r

"+&5sL"

s+Rs&Y@sIs Ir

BsB"

I"

Rr$?s%9s

Find 2he'enin looking in here.ms

msth

##

#VV

+=

ms

msth

##

###

+=

rrth

thr

jXsR#

VI

++

=)/(

3333

Rr&5sL@rthIs Ir

Bth

Rr$?s%9s

( ) 22

2 /33

rthr

th

srthrr

s

em

XX

s

RR

sRpV

s

RI

pT

++

+

==

Co""ent: "77, so

Bth]Bs, th+sis not a

4ad a))roi"ation.

*C5) (or!ue-slip characteristicZou "ay re!all #ro" 33 or your undergraduate !ourse on ele!tri! "a!hines that


81/104

81

Zou "ay re!all, #ro" 33 or your undergraduate !ourse on ele!tri! "a!hines that

the toruesli) !hara!teristi! o# the suirrel!age indu!tion generator $CIG% a))ears

as 4elow. Hne o4ser'es that the CIG o)erates as a generator only when it is in

su)ersyn!hronous "ode and a "otor only when it is in su4syn!hronous "ode.

8otoring

Generating

=ow let;s take a look at the torues)eed !ur'es #or the DFIGE. $net slide%

u4syn!hronous u)ersyn!hronous

(or!ue e!uation for DF5)


82/104

82

33

Re

&5sLe+&Ye9s33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

+=

s

sRRIP

eqr

rmech

)1)((3 2

+=

+

=

+==

=

s

RRI

p

s

sRRI

s

p

s

sRRI

pP

pPT

eqr

r

s

eqr

r

s

eqr

r

m

mech

m

mech

m

em

22

2

3)1)((

)1(3

)1)((3

1

Jo* to obtain IrB 0 (ne4t s%ide!

*C5) &s DF5)


83/104

*C5) &s DF5)

83

33

Re

&5sLe+&Ye9s33

3333

Rr&5sL@rRs &5sL@sIs Ir

Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

Find 2he'enin looking in here.

33

Re

&5sLe+&Ye9s

3333

Rr&5sL@rIs Ir

Bth Br9s

Re$?s%9s

Rr$?s%9sth

"+&5sL"

ms

msth

##

###

+=

ms

msth

##

#VV

+=

)/(/)( sXXjsRR#

VI

eqreqrth

thr ++++

=

22

2

2 /)(3

3

+++

++

+=

+=

s

XXX

s

RRR

sRRpV

s

RRI

pT

eq

rth

eqr

th

seqrtheqr

r

s

em

Co""ent: "77, soBth]Bs, th+sis not a

4ad a))roi"ation.

(or!ue-slip characteristic for DF5) h d 4t i th t li h t i ti # th DFIG


84/104

84

o how do we o4tain the toruesli) !hara!teristi! #or the DFIG

?. De'elo) 'alues o# e#or 'arious 'alues o# torues)eed !ontrol )oint $slides ---O%:

s

em

ss

ss

sR

Tp

R

VVI

2

3

42

= ( )sssssm LjRIVV +=

( )s

ms

ssssssmr I

Lj

LjRIVIII

+==

( )

++==+

r

rsrrm

r

reqseqeq

I

LjsRIVs

I

VLjsR#

Aside: 2he a4o'e )oints result #ro" the tur4ine !ontrol !hara!teristi!. 2his !hara!teristi!originates #ro" the "ai"u" )ower etra!ted #ro" the wind, whi!h is gi'en 4y the )ower

!ur'e, des!ri4ed 4y P"e!hV5"3.(ut P"e!h+2e"5"there#ore 2e"V 5"

.

. For ea!h 'alue o# e, e)ress 2e"as a #un!tion o# s $or 5"+ 5s$?s%% #or 'arious

'alues o# s. torues)eed !ontrol )oint $slides ---O%:

22

2

2 /)(3

3

+++

++

+=

+=

s

XXX

s

RRR

sRRpV

s

RRI

pT

eq

rth

eqr

th

seqrtheqr

r

s

em

(or!ue-slip characteristic for DF5)


85/104

85

2he sign o# Reand Ye

are #or rotor !urrent

dire!tion de#ined out o#

the rotor. 2hese signs

re'erse #or rotor !urrent

dire!tion into the rotor

as we ha'e done.

7=ciencyC id /W i d # ?OK d i #


86/104

86

Consider our /W assign"ent, at a s)eed o# ?OK r)" and unity )ower #a!tor.

Co")ute the e##i!ien!y o# the DFIG.

&W

ssRRIP reqrmech

1500

)1667.0/()01667.01)(00263.005375.0()6.1125(3

/)1)((3

2

2

=++=

+=

At ?OK, the sli) is s+$?K?OK%9?K+.?--O

2he "e!hani!al )ower su))lied to the generator

33

Re

&5sLe+&Ye9s

33

3333

&5sL@rRs &5sL@sIs Ir

Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

"+&5sL"

Fro" your ho"ework, you should !o")ute that Is+?-N. a")eres

Ir+??K.- a")eres, Re+.K3OK oh"s, Ye+.OK? oh"s.

7=ciencyC id /W i t t d # ?OK d it # t


87/104

87



&W

RIP eqrr

29.204

)05375.0()6.1125(3

3

2

2

=

=

=2he rotor )ower is

33

Re

&5sLe+&Ye9s

33

3333


Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

"+&5sL"

2his )ower is negati'e $4e!ause Reis negati'e%J it is su)ersyn!hronous, there#ore it

is #lowing out o# the rotor to the RC.

Fro" your ho"ework, you should !o")ute that Is+?-N. a")eresIr+??K.- a")eres, Re+.K3OK oh"s, Ye+.OK? oh"s.



88/104

88



&W

RIP rrrlosses

0.10)00263.0()6.1125(3

3

2

2

,

==

=

2he rotor and stator winding losses are

33

Re

&5sLe+&Ye9s

33

3333


Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

"+&5sL"


&W

RIP ssslosses

07.9)00265.0()2.1068(3

3

2

2

,

==

=



89/104

89



2he stator a!ti'e )ower is

33

Re

&5sLe+&Ye9s

33

3333


Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

"+&5sL"


&WIVP ssss 64.1276)180cos(2.1068

3

690cos3 ===



90/104

90



2he total )ower deli'ered to the grid is

33

Re

&5sLe+&Ye9s

33

3333


Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

"+&5sL"


93.148029.20464.1276 =+=+= rsg PPP

2he di##eren!e 4etween P"and Pgis the losses on the stator and rotor windings:

07.1993.14801500 ,, ==+= slossesrlossesgm PPPP

##i!ien!y is:

7.981500

93.1480===

m

g

P

P

DF5) for non-unity po"er factor[FRC --? A ^?_ s)e!i#ies that large ind #ar"s " st "aintain a )o er #a!tor ithin the


91/104

91

[FRC --?A ^?_ s)e!i#ies that large wind #ar"s "ust "aintain a )ower #a!tor within the

range o# .K leading to .K lagging, "easured at the PHI as de#ined in the Large

Generator Inter!onne!t Agree"ent $LGIA% i# the 2rans"ission Pro'ider shows, in the

syste" i")a!t study that they are needed to ensure the sa#ety or relia4ility o# the

trans"ission syste"..\^?_ Order for Wind Energy, Hrder =o. --?A, ?N CFR Part 3K $De!e"4er ?, K%. ee also Interconnection for Wind Energy,

Hrder =o. --?, O FR 3Q3 $`une ?-, K%, FRC tats. > Regs. 3?,?N- $K% $Final Rule%J see also Hrder Granting

tension o# ##e!ti'e Date and tending Co")lian!e Date, O FR QO3 $Aug. ?, K%, ?? FRC -?,?O3 $K%.

. Ca"" and C. dwards, [Rea!ti'e Co")ensation yste"s #or Large Wind Far"s,\ I 2rans"ission and Distri4utionCon#eren!e and )osition, N.

[2he le!tri!al yste" H)erator $IH% o# Hntario essentially reuires rea!ti'e )ower

!a)a4ilities #or large wind #ar"s that are eui'alent to that #or syn!hronous generators,

taking into !onsideration an eui'alent i")edan!e 4etween the generator ter"inals andthe PHI ^_. 2he reuire"ents in!lude:E u))lying #ull a!ti'e )ower !ontinuously while

o)erating at a generator ter"inal 'oltage ranging #ro" .K )u to ?.K )u o# the

generator;s rated ter"inal 'oltage.\

[2he Al4erta le!tri! yste" H)erator;s reuire"ents ^Q_ in!lude: 2he wind #ar";s

!ontinuous rea!ti'e !a)a4ility shall "eet or e!eed . )ower #a!tor $)#% lagging to .K)# leading at the !olle!tor 4us 4ased on the wind #ar" aggregated 8W out)ut.\

DF5) for non-unity po"er factor


92/104

92

. Ca"" and C. dwards, [Rea!ti'e Co")ensationyste"s #or Large Wind Far"s,\ I 2rans"ission

and Distri4ution Con#eren!e and )osition, N.

DF5) for non-unity po"er factor[Along with the e'olution o# wind tur4ine te!hnology te!hni!al standards o# wind


93/104

93

Along with the e'olution o# wind tur4ine te!hnology, te!hni!al standards o# wind

generation inter!onne!tions 4e!o"e "ore restri!ti'e. For ea")le, unity )ower #a!tor

has 4een reuired #or wind generation inter!onne!tions in "any utilities or !ontrol areas

in earlier years. Re!ently, the "ore restri!t reuire"ent with .K lead and lag )ower

#a!tor has 4een under dis!ussion sin!e the DFIG and #ull !on'erter wind tur4ine

te!hnology has 4e!o"e "ainstrea" o# wind generation inter!onne!tion reuests.\

I. Green and Z. hang, [Cali#ornia IH e)erien!e with wind #ar"

"odeling,\ I Power and nergy o!iety General 8eeting, ??.



94/104

94

33

Re

&5sLe+&Ye9s

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

cos3cos3

s

sssss

V

PIIVP ==

De#ine: as )ower #a!tor angle: ?N66?N

Identi#y the !urrent )hasor as

)s!"(cos jII ss =

2here#ore: )cos

s!"1(3)s!"(coscos3

jV

PjV

PI

s

s

s

ss ==

Re!alling 222 cos1s!"1coss!" ==+ , we "ay write

)cos

cos11(

3

2

= j

V

PI

s

ss

Psis negati'e #or genJ then !os is also negati'eJ

Psis )ositi'e #or "otorJ then !os is also )ositi'eJ

so Isis always )ositi'e.

We ha'e &ust "ade the nu"erator

)ositi'e #or all 'alues o# .


95/104



96/104

96

33

Re

&5sLe+&Ye9s

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

De#ine the "agneti0ing !urrent #a!tor: ratedsmmrateds

mm IKI

I

IK ,

,

==

Fro" the !ir!uit,


97/104

97

33

Re

&5sLe+&Ye9s

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

)

cos

cos11(

3

2

,

+= j

V

PIjKI

s

sratedsmr

Assu"e the "a!hine is o)erated at rated )ower, Ps,rated, and re!all

cos3cos3

,

,

s

rateds

rateds

s

ss

V

PI

V

PI ==Re!all #ro" slide Q:

)cos

cos11(

3

2,

,

+= j

V

PIjKI

s

rateds

ratedsmr

and the su4stitute into )re'ious e)ression :

)cos

cos11(

3cos3

2,,

+= j

V

P

V

PjKI

s

rateds

s

rateds

mr

Fa!tor out the Ps,rated93BsE.$net slide%:



98/104

98

33

Re

&5sLe+&Ye9s

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

)cos

cos1

1(3cos3

2,,

+= jVP

V

P

jKIs

rateds

s

rateds

mr

Fa!tor out the Ps,rated93Bs

++

= )

cos

cos11(

cos3

2,

j

jK

V

PI m

s

rateds

r

Co"4ine ter"s with [&\

++=

cos

cos1

cos1

3

2, m

s

rateds

r

Kj

V

PI

i")li#y

=

cos

cos11

3

2, m

s

rateds

r

Kj

V

PI



99/104

99

33

Re

&5sLe+&Ye9s

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

o this is #or a4sor4ing$undere!ited o)eration%

=

coscos11

3

2

, m

s

ratedsr

KjV

PI

I# we re)eat the eer!ise #or

su))lying $o'ere!ited o)eration%,

we will o4tain this:

+

=

cos

cos11

3

2, m

s

rateds

r

Kj

V

PI

2he di##eren!e in sign on the suare root ter" indi!ates higher rotor !urrent is reuired

#or o'ere!ited o)eration than #or undere!ited o)eration. =o 4ig sur)rise there

And so the rotor winding should 4e rated #or the o'ere!ited o)eration, at

rated stator a!ti'e )ower out)ut. 2his would 4eE. $net slide%



100/104

100

33

Re

&5sLe+&Ye9s

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

Rotor !urrent #or rated statora!ti'e )ower and rea!ti'e )ower

generation

2

2,

cos

cos113

++=

m

s

rateds

rK

V

PI

It is interesting to see the relati'e "agnitude

4etween Irand Is. Again, #ro" slide Q: cos3cos3,

,

s

rateds

rateds

s

ss

V

PI

V

PI ==

22

,

,, cos

cos11

3

cos3

++=

m

s

rateds

rateds

s

rateds

rrs

K

V

P

P

V

I

IK

22

cos

cos11cos

++=

mrs

KK



101/104

101

33

Re

&5sLe+&Ye9s

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s

Rotor !urrent #or rated statora!ti'e )ower and rea!ti'e )ower

generation

2

2,

cos

cos113

++=

m

s

rateds

rK

V

PI

Rotor !urrent #or rated stator

a!ti'e )ower and stator unity

)ower #a!tor

2,0.1 13

m

s

ratedspf

r KV

PI +==

Ratio o# rotor !urrent reuired #or

a gi'en stator )ower #a!tor when

su))lying T and that reuired #or

unity stator )ower #a!tor, all at

rated stator a!ti'e )ower

2

22

0.1 1

cos

cos11

m

m

pf

r

r

K

K

I

I

+

++

==



102/104

102

33

Re

&5sLe+&Ye9s

33

3333

Rr&5sL@r

&5sL"

Rs &5sL@sIs Ir

Bs Br9sB"

I"Re$?s%9s

Rr$?s%9s


103/104

103


104/104

Ratio o#

DFIG SteadyState

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