DFIG SteadyState
Transcript of DFIG SteadyState
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J. McCalley
Double-fed electric machines
steady state analysis
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Four congurations
2
We will study
only this one,
the DFIG.
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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.
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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%
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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!
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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
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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
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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.
$%
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eferring !uantities
9
33
33
3333
Rr&s5sL@r
rs+sss
Rs &5sL@sIs Ir
Bs Br
=ow write the rotorside 'oltage euation $re#erred to stator%:
rrsrsr ILjsREsV )( +=
Di'ide 4y srrs
rs
r ILjs
RE
s
V)( +=
and we get the #ollowing !ir!uit:
33
33
3333
Rr9s&5sL@r
ss
Rs &5sL@sIs Ir
Bs Br9s
2he 'oltage on 4oth sides
o# the #"r is the sa"e,
there#ore, we "ay eli"inate
the #"r. .We re)resent a
"agneti0ing indu!tan!e
&5sL"in its )la!e.
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eferring !uantities
10
33
3333
Rr9s&5s
L@r
&5sL"s
Rs &5s
L@s
Is
Ir
Bs Br9s
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Po"er relations
11
33
3333
Rr9s&5s
L@r
&5sL"s
Rs &5s
L@s
Is
Ir
Bs Br9s
s
sRR
s
sRR
s
sR
s
sRsRR
s
R rr
rrrrrrr )1( +=
+=+
=
s
sVV
s
sVV
s
sV
s
sVsVV
s
V rr
rrrrrrr )1( +=
+=+
=
Change the !ir!uit a!!ordinglyE.
We "odi#y the a4o'e !ir!uit slightly in order to !learly se)arate sli)de)endent ter"s
#ro" loss ter"s:
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Po"er relations
12
33
3333
Rr&5s
L@r
&5sL"s
Rs &5sL@sIs Ir
Bs Br
It is )ossi4le to )ro'e that the "e!hani!al )ower out o# the "a!hine is the )ower
asso!iated with the sli)de)endent ter"s R$?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$?s%9s
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Po"er relations
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33
3333
Rr&5s
L@r
&5sL"s
Rs &5sL@sIs Ir
Bs Br
( ) { }
{ }*2
*2
*2
Re1
3)1(
3
1Re3
)1(3
Re33
rrr
r
rrr
r
reqeqrmech
IVs
s
s
sRI
Is
sV
s
sRI
IVRIP
=
=
=
Re+
Rr$?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 $see
4elow% as "otor "ode i")lies Reter" should
4e added and Be ter" should 4e su4tra!ted.
P"e!h
I# 6s6?
Reter" is )ositi'e
Beter" is )ositi'e
u))lying P to !!tI# 7s7?Reter" is negati'eBeter" is negati'eConsu"ing P #ro" !!t.
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rs or!uee$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
$): 1 o# )ole
)airs%
Re!all #ro" slide K: ;s
rs
=
s
msm ss
== 1)1(
2here#ore:
{ }
{ }*2
*2
Re33
Re33
rr
rr
rr
rr
r
m
mr
mrr
m
em
IVpRIp
IVpR
Ip
T
=
=
r
m
r
s
s
m
s
s
==1
and
( )ivrrrr
rr
em IV
pRIp
T = cos33
2
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secon e!u &a en or!uee$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 +=
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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$?s%9s
Rotor )ower:
( )
( )
( ) ( ) msrsrsrrr
msrsrsr
rr
msrsrsrrrrr
LjsIILjsRIV
LjIILjs
RI
s
V
LjIILjs
sRRI
s
sVV
+++=
++
+=
++
++=+
11
Rotor 'oltage:
u4stitute Brinto Pr: { } ( ) ( )( ){ }
( ){ }( ){ }*222
****
**
Re3
Re3
Re3Re3
rsmsrmsrrsrr
rsmsrrmsrrrsrrr
rmsrsrsrrrrr
IILjsILjsILjsIR
IILjsIILjsIILjsIIR
ILjsIILjsRIIVP
+++=+++=
+++==
2he "iddle two ter"s are )urely i"aginary, there#ore:
( ){ }*2Re3 rsmsrrr IILjsIRP +=
First ter" is )urely real, only the se!ond ter" !ontains real and i"aginary, there#ore:{ }*2 Re33 rsmsrrr IILjsIRP +=
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e$pression
17
{ }*2 Re33 rsmsrrr IILjsIRP +=
=ow su4stitute Psand Printo the )ower 4alan!e euation:
rlossslossrsmech PPPPP ,, +=
{ }*2 Re33 srmssss IILjIRP +=
{ } { } 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"$a%Observe thatRe(a! " # Im(a!
2here#ore:
{ } { }
**Im3Im3
rsmssrmsmech IILsIILP
=
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e$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$! " # Im(a!
2here#ore:
{ } { }{ } { }
{ } { }[ ]{ }[ ]sIILIIsIIL
IILsIIL
IILsIILP
rsms
rsrsms
rsmsrsms
rsmssrmsmech
==
=
=
1Im3
ImIm3
Im3Im3
Im3)(Im3
*
**
**
***
Re!all: )1( ssm =
{ }*Im3 rsmmmech IILP =2here#ore:
mmechem
p
PT =Re!all:
{ }*Im3 rsmem IIpLT =
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e$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.
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e$pressions
21
Fro" stator winding euation:
s
rmss
L
ILI
=
Fro" rotor winding euation:
r
smrr
L
ILI
=
u4stitute into torue e)ression 1E.
{ }*Im3 rsmem IIpLT =
{ }
{ }
{ }*
2*
**
*
Im3
Im3
Im3
Im3
rs
s
m
rmrs
s
m
rrmrs
s
m
r
s
rmsmem
IL
Lp
ILILLp
IILIL
Lp
IL
ILpLT
=
=
=
=
Msing stator winding euation:
Purely
real
{ }
{ }
{ }*
2*
**
*
Im3
Im3
Im3
Im3
rs
r
m
smrs
r
m
ssmrs
r
m
r
smrsmem
IL
Lp
ILIL
Lp
IILIL
Lp
L
ILIpLT
=
=
=
=
Msing rotor winding euation:
Purely
real
#i d li
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#irgap and slip po"erOn s%ides &' and &) *e derived the fo%%o*in+ re%ations for the po*er into the
stator and rotor respe,tive%-:
{ }
*2 Re33srmssss
IILjIRP +=
{ }*2 Re33 rsmsrrr IILjsIRP +=S.btra,tin+ %osses from both sides) *e obtain:
{ }*2 Re33 srmssss IILjIRP = { }*2 Re33 rsmsrrr IILjsIRP =
This /.antit- is the po*er that f%o*s
from the stator termina%s to the rotor
(ne+ative for +enerator operation!0 Inother *ords) it is the po*er a,ross
the air+ap0 Therefore:
{ }*2 Re33 srmssssairgap IILjIRPP ==
This /.antit- is the po*er that is
transferred from the +rid to the rotor
thro.+h the ,onverter (ne+ative*hen it is into the +rid!0 It is ,a%%ed
the s%ip po*er0 Therefore:
{ }*2 Re33 rsmsrrrslip IILjsIRPP ==1rin+ o.t front the 2s3 in the s%ip po*er e4pression and .se Re5a6"#Im(a!
(both!:Use Im(a$! " #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
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#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==
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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
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#cti&e po"er relations - summary
25
sr s =
E4a,t Appro4imate1oth
{ }*2 Re33 srmssss IILjIRP +=
{ }*
2 Re33 rsmsrrr IILjsIRP +={ }*2 Re33 srmssssairgap IILjIRPP ==
{ }*2 Re33 rsmsrrrslip IILjsIRPP ==
airgapslip sPP =
rlossslossrsmech PPPPP ,, +=
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
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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
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)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)
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)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
$>0motor
%>0
&
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7$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$ample Problem
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7$ample Problem
43
(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
(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
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$ample Problem
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7$ample Problem
44
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:
(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
(h! Rotor rea% po*er
(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
rlossslossrsmech PPPPP ,, +=
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$ample Problem
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7$ample Problem
45
(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
(! Tem { }*
Im3 rss
mem I
L
LpT =
( ){ } &NmTem 9.125.164.18079028.1Im10587.2
105.223
*
3
3
=
=
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:
8ind turbine control le&els
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 $su))lying%.
{ }
*
Im3 rss
m
em IL
L
pT =
Fied torue i")lies #ied rotor
!urrent i# stator #lu is #ied.
(e!ause 2e"+P"e!h)95", P"e!h
"ust de!rease as 5"in!reases.
e&el 9 control
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e&el 9 control
59
1oth rotor ,.rrent and stator ,.rrent e/.ations have rea% part determined b- Qs
and ima+inar- part determined b- Tem(Vsis at ^@ so rea% part of ,.rrents is in
/.adrat.re *ith Vs!
6Ts63Bs9Ls5s$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
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e e co o
60
?i4ed Qs"@ ?i4ed Tem"#&
* 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
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61
?i4ed Qs"@ ?i4ed Tem"#&
* 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
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62
?i4ed Qs"@ ?i4ed Tem"#&
* ##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
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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)
-
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*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)
-
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*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)
-
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*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)
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*C5) &s DF5)
78
(or!ue e!uation for *C5)
-
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79
33
3333
Rr&5sL@r
&5sL"
Rs &5sL@sIs Ir
BsB"
I"
SKIG
Rr$?s%9s
ssRIP rrmech
)1(3 2 = =ote that the [s\ on the deno"inator )ro'idesthat P"e!his )ositi'e #or s7, "otor a!tion, andnegati'e #or s6, generator a!tion.
s
RI
p
s
sRI
s
p
s
sRI
pP
pPT
rr
s
rr
s
rr
m
mech
m
mech
m
em
22
2
3)1(
)1(3
)1(3
1
=
=
==
=
Jo* to obtain IrB 0 (ne4t s%ide!
(or!ue e!uation for *C5)
-
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80
33
3333
Rr&5sL@r
"+&5sL"
s+Rs&Y@sIs Ir
BsB"
I"
Rr$?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
-
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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. $net slide%
u4syn!hronous u)ersyn!hronous
(or!ue e!uation for DF5)
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82
33
Re
&5sLe+&Ye9s33
3333
Rr&5sL@r
&5sL"
Rs &5sL@sIs Ir
Bs Br9sB"
I"Re$?s%9s
Rr$?s%9s
+=
s
sRRIP
eqr
rmech
)1)((3 2
+=
+
=
+==
=
s
RRI
p
s
sRRI
s
p
s
sRRI
pP
pPT
eqr
r
s
eqr
r
s
eqr
r
m
mech
m
mech
m
em
22
2
3)1)((
)1(3
)1)((3
1
Jo* to obtain IrB 0 (ne4t s%ide!
*C5) &s DF5)
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*C5) &s DF5)
83
33
Re
&5sLe+&Ye9s33
3333
Rr&5sL@rRs &5sL@sIs Ir
Bs Br9sB"
I"Re$?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
-
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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 $slides ---O%:
22
2
2 /)(3
3
+++
++
+=
+=
s
XXX
s
RRR
sRRpV
s
RRI
pT
eq
rth
eqr
th
seqrtheqr
r
s
em
(or!ue-slip characteristic for DF5)
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85
2he sign o# Reand Ye
are #or rotor !urrent
dire!tion de#ined out o#
the rotor. 2hese signs
re'erse #or rotor !urrent
dire!tion into the rotor
as we ha'e done.
7=ciencyC id /W i d # ?OK d i #
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86
Consider our /W assign"ent, at a s)eed o# ?OK r)" and unity )ower #a!tor.
Co")ute the e##i!ien!y o# the DFIG.
&W
ssRRIP reqrmech
1500
)1667.0/()01667.01)(00263.005375.0()6.1125(3
/)1)((3
2
2
=++=
+=
At ?OK, the sli) is s+$?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
-
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87
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
RIP eqrr
29.204
)05375.0()6.1125(3
3
2
2
=
=
=2he rotor )ower is
33
Re
&5sLe+&Ye9s
33
3333
&5sL@rRs &5sL@sIs Ir
Bs Br9sB"
I"Re$?s%9s
Rr$?s%9s
"+&5sL"
2his )ower is negati'e $4e!ause Reis negati'e%J it is su)ersyn!hronous, there#ore it
is #lowing out o# the rotor to the RC.
Fro" your ho"ework, you should !o")ute that Is+?-N. a")eresIr+??K.- a")eres, Re+.K3OK oh"s, Ye+.OK? oh"s.
7=ciencyC id /W i t t d # ?OK d it # t
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88
Consider our /W assign"ent, at a s)eed o# ?OK r)" and unity )ower #a!tor.
Co")ute the e##i!ien!y o# the DFIG.
&W
RIP rrrlosses
0.10)00263.0()6.1125(3
3
2
2
,
==
=
2he rotor and stator winding losses are
33
Re
&5sLe+&Ye9s
33
3333
&5sL@rRs &5sL@sIs Ir
Bs Br9sB"
I"Re$?s%9s
Rr$?s%9s
"+&5sL"
Fro" your ho"ework, you should !o")ute that Is+?-N. a")eresIr+??K.- a")eres, Re+.K3OK oh"s, Ye+.OK? oh"s.
&W
RIP ssslosses
07.9)00265.0()2.1068(3
3
2
2
,
==
=
7=ciencyC id /W i t t d # ?OK d it # t
-
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89
Consider our /W assign"ent, at a s)eed o# ?OK r)" and unity )ower #a!tor.
Co")ute the e##i!ien!y o# the DFIG.
2he stator a!ti'e )ower is
33
Re
&5sLe+&Ye9s
33
3333
&5sL@rRs &5sL@sIs Ir
Bs Br9sB"
I"Re$?s%9s
Rr$?s%9s
"+&5sL"
Fro" your ho"ework, you should !o")ute that Is+?-N. a")eresIr+??K.- a")eres, Re+.K3OK oh"s, Ye+.OK? oh"s.
&WIVP ssss 64.1276)180cos(2.1068
3
690cos3 ===
7=ciencyC id /W i t t d # ?OK d it # t
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90
Consider our /W assign"ent, at a s)eed o# ?OK r)" and unity )ower #a!tor.
Co")ute the e##i!ien!y o# the DFIG.
2he total )ower deli'ered to the grid is
33
Re
&5sLe+&Ye9s
33
3333
&5sL@rRs &5sL@sIs Ir
Bs Br9sB"
I"Re$?s%9s
Rr$?s%9s
"+&5sL"
Fro" your ho"ework, you should !o")ute that Is+?-N. a")eresIr+??K.- a")eres, Re+.K3OK oh"s, Ye+.OK? oh"s.
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
-
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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 $)#% lagging to .K)# leading at the !olle!tor 4us 4ased on the wind #ar" aggregated 8W out)ut.\
DF5) for non-unity po"er factor
-
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92
. Ca"" and C. dwards, [Rea!ti'e Co")ensationyste"s #or Large Wind Far"s,\ I 2rans"ission
and Distri4ution Con#eren!e and )osition, N.
DF5) for non-unity po"er factor[Along with the e'olution o# wind tur4ine te!hnology te!hni!al standards o# wind
-
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93
Along with the e'olution o# wind tur4ine te!hnology, te!hni!al standards o# wind
generation inter!onne!tions 4e!o"e "ore restri!ti'e. For ea")le, unity )ower #a!tor
has 4een reuired #or wind generation inter!onne!tions in "any utilities or !ontrol areas
in earlier years. Re!ently, the "ore restri!t reuire"ent with .K lead and lag )ower
#a!tor has 4een under dis!ussion sin!e the DFIG and #ull !on'erter wind tur4ine
te!hnology has 4e!o"e "ainstrea" o# wind generation inter!onne!tion reuests.\
I. Green and Z. hang, [Cali#ornia IH e)erien!e with wind #ar"
"odeling,\ I Power and nergy o!iety General 8eeting, ??.
DF5) for non-unity po"er factor
-
7/23/2019 DFIG SteadyState
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94
33
Re
&5sLe+&Ye9s
33
3333
Rr&5sL@r
&5sL"
Rs &5sL@sIs Ir
Bs Br9sB"
I"Re$?s%9s
Rr$?s%9s
cos3cos3
s
sssss
V
PIIVP ==
De#ine: as )ower #a!tor angle: ?N66?N
Identi#y the !urrent )hasor as
)s!"(cos jII ss =
2here#ore: )cos
s!"1(3)s!"(coscos3
jV
PjV
PI
s
s
s
ss ==
Re!alling 222 cos1s!"1coss!" ==+ , we "ay write
)cos
cos11(
3
2
= j
V
PI
s
ss
Psis negati'e #or genJ then !os is also negati'eJ
Psis )ositi'e #or "otorJ then !os is also )ositi'eJ
so Isis always )ositi'e.
We ha'e &ust "ade the nu"erator
)ositi'e #or all 'alues o# .
-
7/23/2019 DFIG SteadyState
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DF5) for non-unity po"er factor
-
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96
33
Re
&5sLe+&Ye9s
33
3333
Rr&5sL@r
&5sL"
Rs &5sL@sIs Ir
Bs Br9sB"
I"Re$?s%9s
Rr$?s%9s
De#ine the "agneti0ing !urrent #a!tor: ratedsmmrateds
mm IKI
I
IK ,
,
==
Fro" the !ir!uit,
-
7/23/2019 DFIG SteadyState
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97
33
Re
&5sLe+&Ye9s
33
3333
Rr&5sL@r
&5sL"
Rs &5sL@sIs Ir
Bs Br9sB"
I"Re$?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%:
DF5) for non-unity po"er factor
-
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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
DF5) for non-unity po"er factor
-
7/23/2019 DFIG SteadyState
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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. $net slide%
DF5) for non-unity po"er factor
-
7/23/2019 DFIG SteadyState
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100
33
Re
&5sLe+&Ye9s
33
3333
Rr&5sL@r
&5sL"
Rs &5sL@sIs Ir
Bs Br9sB"
I"Re$?s%9s
Rr$?s%9s
Rotor !urrent #or rated statora!ti'e )ower and rea!ti'e )ower
generation
2
2,
cos
cos113
++=
m
s
rateds
rK
V
PI
It is interesting to see the relati'e "agnitude
4etween Irand Is. Again, #ro" slide Q: cos3cos3,
,
s
rateds
rateds
s
ss
V
PI
V
PI ==
22
,
,, cos
cos11
3
cos3
++=
m
s
rateds
rateds
s
rateds
rrs
K
V
P
P
V
I
IK
22
cos
cos11cos
++=
mrs
KK
DF5) for non-unity po"er factor
-
7/23/2019 DFIG SteadyState
101/104
101
33
Re
&5sLe+&Ye9s
33
3333
Rr&5sL@r
&5sL"
Rs &5sL@sIs Ir
Bs Br9sB"
I"Re$?s%9s
Rr$?s%9s
Rotor !urrent #or rated statora!ti'e )ower and rea!ti'e )ower
generation
2
2,
cos
cos113
++=
m
s
rateds
rK
V
PI
Rotor !urrent #or rated stator
a!ti'e )ower and stator unity
)ower #a!tor
2,0.1 13
m
s
ratedspf
r KV
PI +==
Ratio o# rotor !urrent reuired #or
a gi'en stator )ower #a!tor when
su))lying T and that reuired #or
unity stator )ower #a!tor, all at
rated stator a!ti'e )ower
2
22
0.1 1
cos
cos11
m
m
pf
r
r
K
K
I
I
+
++
==
DF5) for non-unity po"er factor
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102/104
102
33
Re
&5sLe+&Ye9s
33
3333
Rr&5sL@r
&5sL"
Rs &5sL@sIs Ir
Bs Br9sB"
I"Re$?s%9s
Rr$?s%9s
-
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103
-
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104/104
Ratio o#