Fault Current Characteristics of the DFIG under Asymmetrical Fault Conditions

Transcription

1 Eergie 05, 8, ; doi:0.3390/e80097 Article OPEN ACCESS eergie ISSN Fault Curret Characteritic of the DFIG uder Aymmetrical Fault Coditio Fa Xiao, Zhe Zhag ad Xiagge Yi State Key Laboratory of Advaced Electromagetic Egieerig ad Techology, Huazhog Uiverity of Sciece ad Techology, Wuha , Chia; (Z.Z.); (X.Y.) Author to whom correpodece hould be addreed; Tel.: Academic Editor: Frede Blaabjerg Received: 0 July 05 / Accepted: 6 September 05 / Publihed: 30 September 05 Abtract: Durig o-evere fault coditio, crowbar protectio i ot activated ad the rotor widig of a doubly-fed iductio geerator (DFIG) are excited by the AC/DC/AC coverter. Meawhile, uder aymmetrical fault coditio, the electrical variable ocillate at twice the grid frequecy i ychroou dq frame. I the egieerig practice, otch filter are uually ued to extract the poitive ad egative equece compoet. I thee cae, the dyamic repoe of a rotor-ide coverter (RSC) ad the otch filter have a large ifluece o the fault curret characteritic of the DFIG. I thi paper, the ifluece of the otch filter o the proportioal itegral (PI) parameter i dicued ad the implified calculatio model of the rotor curret are etablihed. The, the dyamic performace of the tator flux likage uder aymmetrical fault coditio i alo aalyzed. Baed o thi, the fault characteritic of the tator curret uder aymmetrical fault coditio are tudied ad the correpodig aalytical expreio of the tator fault curret are obtaied. Fially, digital imulatio reult validate the aalytical reult. The reearch reult are helpful to meet the requiremet of a practical hort-circuit calculatio ad the cotructio of a relayig protectio ytem for the power grid with peetratio of DFIG. Keyword: wid power; doubly-fed iductio geerator (DFIG); fault curret characteritic; o-evere fault; aymmetrical fault

2 Eergie 05, Itroductio Sice wid eergy i expected to play a importat role i the future eergy idutry, wid power geeratio techology ha received world-wide attetio. Wid power geeratio baed o the doubly-fed iductio geerator (DFIG) ha gaied icreaig popularity due to everal advatage, icludig maller coverter ratig aroud 30% of the geerator ratig, variable peed ad four-quadrat active ad reactive power operatio capabilitie, lower coverter cot, ad power loe compared with the fixed-peed iductio geerator or ychroou geerator with full-ized coverter [,]. A the power peetratio from the DFIG-baed wid turbie ito the grid ha icreaed teadily i recet year, the grid code require that the wid power geerator hould ot be tripped from the grid but tay coected durig the fault coditio [3], which mea that the wid turbie have the capability of low voltage ride through (LVRT) [4 6]. The operatio characteritic of DFIG uder LVRT coditio have a large ifluece o the fault characteritic of curret [7]. Thi mea that the DFIG brig may ew problem ad challege to the traditioal relayig protectio of the power grid. Therefore, i order to etablih a olid bai for the relay protectio tudy of the power grid with DFIG, the fault curret characteritic of the DFIG hould be tudied. Durig evere fault coditio, i order to eure the afety of the DFIG, the crowbar protectio [8] will be activated to hort circuit the rotor widig ad divert the urge curret from the rotor-ide coverter (RSC). Up to ow, the fault curret characteritic of the DFIG uder evere fault coditio have bee dicued widely [9 ]. Wherea the crowbar protectio will ot be activated ad the rotor widig are till excited by the AC/DC/AC coverter uder o-evere fault coditio. The fault curret characteritic of the DFIG uder o-evere fault coditio, uch a traiet compoet ad dampig time cotat, are much differet from the evere fault coditio. For o-evere fault coditio, the fault curret of the DFIG i tudied baed o the aumptio that the excitatio curret will keep cotat before ad after the fault occurrece [] or rie rapidly to the maximum value ad the keep cotat durig the grid fault [3]. However, ice the repoe of the AC/DC/AC coverter i very fat ad complicated durig the grid fault, the aforemetioed aumptio do ot eem to work. I [4,5], the dyamic behavior of the fault curret of the DFIG uder o-evere fault coditio ha bee dicued. Neverthele, the reearch reult are baed o ymmetric fault coditio, ad there are o aalytical expreio of the fault curret uder ubalaced fault coditio. Uder ubalaced fault coditio, the covetioal phae locked loop (PLL) techique caot preet accurate phae agle etimatio becaue the ecod harmoic produced by the egative-equece compoet of the grid voltage will propagate through the PLL ytem ad will be reflected i the extracted phae agle [6]. I the egieerig practice, otch filter are uually ued to extract the poitive- ad egative-equece compoet [7,8]. Thi mea that the traiet behavior of the DFIG are chaged by the otch filter ad the reearch reult preeted i [4,5], do ot agree with reality very well. Hece, further reearch work hould be implemeted to tudy the fault curret characteritic of the DFIG uder ubalaced fault coditio. I order to fill thi gap, a theoretical aalyi method for the fault curret characteritic of the DFIG uder ubalaced fault coditio i propoed. Sectio itroduce the DFIG equatio i pace vector otatio. Sectio 3 aalyze the ifluece of the otch filter o the proportioal itegral (PI)

3 Eergie 05, parameter ad the implified calculatio model of the rotor curret are etablihed. Sectio 4 aalyze the dyamic performace of the tator flux likage. Baed o thi, the characteritic of the tator fault curret uder ubalaced fault coditio are tudied i Sectio 5. Simulatio reult give i Sectio 6 validate the theoretical aalyi reult.. Doubly-Fed Iductio Geerator (DFIG) Equatio i Space Vector Notatio The poitive (dq) + ad egative (dq) ychroou rotatig referece frame are choe to model the DFIG baed o the fifth-order two-axi repreetatio, ad the model of DFIG i commoly kow a the Park model [9]. Uig geerator covetio for the tator widig ad motor covetio for the rotor widig, the voltage equatio ad flux likage equatio are expreed a: u Ri p u Ri p u Ri p u Ri p d + d + q + d + q + q + d + q + rd r rd r rq+ rd+ rq r rq r rd + rq + L i L i d + d m rd L i L i q q m rq + L i Li rd + m d r rd L i Li rq m q r rq + u Ri p d d q d u Ri p q q d q u Ri p rd r rd r rq rd u Ri p rq r rq r rd rq Li L i d d m rd Li L i q q m rq Li Li rd m d r rd Li Li rq m q r rq where u i the voltage, i i the curret, R i the reitace, L i the iductace, ad ψ i the flux likage. The ubcript d ad q idicate the direct ad quadrature axi compoet of the correpodig electrical quatity. The ubcript ad r idicate the tator ad rotor quatitie, repectively. L m i the magetizig iductace. ω i the ychroou agular velocity, ad ω = ω. ω ( ω ω) r r i the rotor lip i the poitive dq frame, ad ω ( ω ω r r). The ubcript + ad idicate poitive-equece ad egative-equece compoet, repectively. The upercript + ad idicate poitive (dq) + ad egative (dq) ychroou rotatig referece frame, repectively. p d / dt i the differetial operator. Figure depict the equivalet circuit model of the DFIG. () () (3) (4)

4 Eergie 05, I d + R L Lr Rr I rd I R d L Lr Rr I rd U d d L m rd U rd U d d L m rd U rd I q j q R L Lr j( r) rq Rr I rq I q j q R L Lr j( r) rq Rr I rq U q q L m rq U rq U q q L m rq U rq j d j( r) rd j d (a) j( r) rd (b) Figure. Equivalet circuit model of the doubly-fed iductio geerator (DFIG). (a) Equivalet circuit model of the DFIG i the poitive dq referece frame rotatig at the ychroou peed ω ; ad (b) equivalet circuit model of the DFIG i the egative dq referece frame rotatig at the ychroou peed ω. 3. Dyamic Behavior of the Rotor Curret Durig the fault traiet period, the meauremet of the active ad reactive power are iaccurate. Moreover, it i ureaoable to make the DFIG operate with a uity power factor ad capture the maximum wid power. Therefore, a typical hadlig method i to hut dow the exteral power cotrol loop whe a fault i detected [0]. Uder thi coditio, oly the operatio characteritic of the ier rotor curret cotrol loop are take ito coideratio for the fault traiet aalyi. I additio, the rotatio peed i coidered cotat for the followig aalyi. 3.. Modelig of the Rotor-Side Coverter (RSC) Accordig to the equatio of Equatio () ad (4), the d-axi ad q-axi compoet of the rotor flux likage i (dq) + ad (dq) referece frame ca be expreed i Equatio (5) baed o the tator flux likage ad the rotor curret: L ( Li ψ ) /L Li rd m m rd d r rd L ( Li ) /L Li rq m m rq q r rq L ( Li ) /L Li rd m m rd d r rd L ( Li ) /L Li rq m m rq q r rq Meawhile, the tator reitace i o mall that it ca be eglected. Uder thi coditio, ubtitutig Equatio (5) ad the firt two equatio of Equatio () ad (3) ito the lat two equatio of Equatio () ad (3), the d-axi ad q-axi compoet of the rotor voltage i (dq) + ad (dq) referece frame ca be expreed: (5)

5 Eergie 05, u ( Ri σl pi ) Li ( U ) L / L u ( Ri L pi ) ω Li L / L rq r rq r rq r r rd lip d m u ( Ri L pi ) σ Li ( U ) L / L u ( Ri Lpi ) Li L /L rq r rq r rq r r rd lip d m rd r rd r rd r r rq lip q m rd r rd r rd r r rq lip q m (6) where σ = L / LL m r, =, ad = lip r lip r. It ca be ee from Equatio (6) that the d-axi ad the q-axi rotor voltage i (dq) + ad (dq) referece frame exert a effect o each other. Hece, i order to decouple the couplig term r Li r rq,, σω Li r Li r rd r r rq, ad σω Li r r rd i both traiet ad teady-tate coditio, the d-axi ad the q-axi rotor curret ca be eparately cotrolled uig feed forward compeatio []. Moreover, it ca be foud from (6) that the trackig error of the rotor curret i caued by the tator curret ad tator flux. Hece, the ifluece the term U L / L ad U L / Lare doe by the feed forward r m r - m compeatio ad eglectig the derivative of the tator flux likage. I additio, due to the egative compoet ocillatig at twice the grid frequecy ( ) i the ychroou dq frame while the poitive compoet are realized a DC quatitie. Therefore, otch filter N() are uually applied to filter out the ifluece of the 00-Hz frequecy compoet, it tructure i deiged a: N () (7) where i the otch frequecy which i the ame a that of, ad i the cutoff frequecy of the otch filter. Hece, uder ubalaced fault coditio, the otch filter are applied to filter out the ifluece of the 00-Hz frequecy compoet of the d-axi ad q-axi compoet of the rotor curret i (dq) + ad (dq) referece frame. Moreover, the time delay of the pule-width modulatio (PWM) cotrol i diregarded becaue PWM cotrol ha a peed that i markedly fater tha that of the electromagetic dyamic [0]. Hece, takig the cotrol diagram of d-axi ad q-axi compoet i the (dq) + referece frame, for example, the implified block diagram of the rotor curret cotrol loop of the RSC i how i Figure. I Figure, G () i the trafer fuctio of the PI cotroller. The ' u ', u ', u ', ad u PI rd rq rd rq are cotrolled by the error of the rotor curret. (a) (b) Figure. Simplified block diagram of the ier rotor curret cotrol loop of the RSC: (a) implified cotrol diagram of the d-axi compoet i (dq) + referece frame; ad (b) implified cotrol diagram of the q-axi compoet i (dq) + referece frame. RSC: rotor-ide coverter.

6 Eergie 05, I Figure, both the rotor curret referece ad the tator flux likage ifluece the dyamic repoe characteritic of the rotor curret. I fact, the dyamic repoe characteritic of the rotor curret are eetially a fuctio of tator voltage becaue the iput of the tator flux likage are determied by the voltage. The rotor voltage referece are ued to geerate the PWM wave to cotrol RSC witchig. Hece, the rotor voltage of the DFIG ca be expreed i Equatio (8): ' u u Li U L / L rd rd r r rq m ' u u Li rq rq r r rd ' u u Li + U L / L (8) rd rd r r rq m ' u u Li rq rq r r rd Aumig that the rotor voltage ca alway track the referece by pace vector pule width modulatio, the rotor voltage referece are obtaied by: ' u k ( i i ) k ( i i )dt rd irp + rd rd iri + rd rd ' u k ( i i ) k ( i i )dt rq irp + rq rq iri + rq rq ' u k ( i i ) k ( i i )dt rd irp rd rd iri rd rd ' u k ( i i ) k ( i i )d t rq irp rq rq iri rq rq where k irp + ad k are the proportioal ad itegral cotat of the ier cotroller i the (dq) + iri + referece frame, repectively. k irp ad kiri are the proportioal ad itegral cotat of the ier cotroller i (dq) referece frame, repectively. The i ad i rd rq are the referece igal of the d-axi ad q-axi compoet of the rotor curret i (dq) + referece frame, repectively. The i rd ad i rq are the referece igal of the d-axi ad q-axi compoet of the rotor curret i (dq) referece frame, repectively. 3.. Rage of the Croover Frequecy ad Phae Margi It ca be ee from the Figure, the ope-loop trafer fuctio of the cotrol ytem without time delay ca be preeted a: k iri G () G () N() (k + )( )( ) R L PI r irp r r where G () i the trafer fuctio of a DFIG motor body. r Aume that the referece igal of the expected phae margi ad expected croover frequecy are, repectively, ad. Accordig to the cotrol theory, the magitude repoe of the ope-loop cr m trafer fuctio at the croover frequecy hould be 0 db; thu, the followig equatio ca be cr etablihed a: G ( j ) G ( j ) N( j ) PI cr r cr cr (9) (0) k k irp cr iri cr = R ( L ) ( ) (εω ω ) cr r cr r cr cr () The phae agle of thi forward path loop gai at the cro over frequecy i give, i radia, by: cr

7 Eergie 05, { G ( j ) G ( j ) N( j )}= PI cr r cr cr m k j R j L j k cr irp L r cr cr =arcta arcta arcta k R iri cr = (k + )( )( ) irp cr r cr r cr cr iri r cr () Accordig to Equatio () ad (), the followig cotroller parameter are foud: k AB A irp cr k B A iri with A ta arcta (σlω R ) arcta εω ω (ω ω ) m r cr r cr cr ad B ( L ) R ( ) (εω ω ) cr r cr r. cr cr cr I order to eure the ability to rapidly chage the igal of the curret loop ad reduce the igal ditortio, the badwidth cb of the cloed loop ytem will take more tha the maximum frequecy of the effective igal. Moreover, i order to avoid the effect of the witchig harmoic, the cmax witchig harmoic wat to exit i the high frequecy bad of the ier rotor curret cotrol loop. Hece, the witchig frequecy hould be higher tha 0 cb. I egieerig, the deired cloed-loop badwidth of the ier rotor curret cotrol loop i uually take value betwee. ad.4 time of the croover frequecy of ope loop ytem []. Accordigly, the rage of the expected croover frequecy i the cloed loop ytem ca cr be obtaied: k cmax cr ad k (.,. 4) 0k T (4) where = p 60 cmax max ad T i the cotrol period of the rotor iverter. p i the umber of pole pair ad max i the maximum peed of the rotor. From Equatio (), the phae margi ca be obtaied a how i Equatio (5) if the cotroller time cotat i deiged a k k = L R irp iri r r : (3) cr = arcta m cr (5) Whe the phae margi i higher tha the value of m i Equatio (5), the effect of the itegratig elemet will drop teeply ad teady trackig performace will be woreed. Therefore, the value of m i Equatio (5), i et a the maximum phae margi of the PI cotroller i the egieerig applicatio. Accordigly, the rage of the expected phae margi ca be obtaied: m arcta cr m cr (6) 3.3. Traiet Characteritic of Rotor-Side Coverter (RSC) Accordig to the Figure a, the d-axi compoet of the rotor curret i (dq) + referece frame i: where: i () G () i () G () p () rd ird rd ird d + (7)

8 Eergie 05, G ird G () N() () R L G N PI () () (8) r r PI ad: G ird N () ()= R+ L+ G () N () (9) r r PI I order to get the maximum phae margi for the ier rotor curret cotroller, ubtitute k k = L R irp iri r r ito Equatio (3) to obtai the expreio of kiri a give i Equatio (0): k = R ( ) ( ) ( ) iri r c cr cr cr (0) Accordig to the rage of the expected croover frequecy i Equatio (4), the rage of the cr itegral cotat kiri ca be obtaied: 4. R k 806. R r iri r () with R r Accordig to the rage of the itegral cotat k iri, ubtitute k k = L R irp iri r r ito Equatio (0), the root locu diagram of the ope-loop trafer fuctio i how i Figure 3. From Figure 3, it ca be oberved that the eigevalue of the ytem are located i the left-half-plae which mea that the cotrol ytem i table. Moreover, it ca be oberved that the rage of the imagiary part for the domiat pole i betwee 60 ad 640. Thi mea that the repoe compoet of the domiat pole of the ytem will be limited by the otch filter. Hece, G (). ird (a) (b) Figure 3. Root locu diagram of the ope-loop trafer fuctio. (a) root locu with a chagig ; ad (b) the domiat pole of the of the ope-loop trafer fuctio with a cr chagig. cr Accordigly, by ubtitutig Equatio (7) ad k k = L R irp iri r r i the equatio of Equatio (9), the cloed-loop trafer fuctio to the diturbace of differetial flux durig the fault traiet period i: G 3 + ()= (R + L )[ +( k R ) + k R ] ird 3 r r iri r iri r ()

9 Eergie 05, The bode diagram of G () i how i Figure 4. ird From Figure 4, it ca be oberved that the maximum gai of the cloed loop trafer fuctio to the diturbace compoet at i early 54.9 db, which mea that the harmoic compoet i p will be greatly damped. Hece: d i = G () p 0 rd ird d + (3) Figure 4. Bode diagram of the ier curret cloed-loop trafer fuctio to the diturbace d + compoet (uch a p ) durig the fault traiet period. Hece, durig the fault traiet period, the d-axi ad q-axi compoet of the rotor curret i (dq) + referece frame are: i i rd rq i i rd rq (4) Accordig to the [3], the total referece of rotor curret reult from the additio of two term. The firt term i the referece igal of teady-tate compoet i rf. The ecod term i the demagetizig curret that help to reduce or elimiate the electromotive force iduced i the rotor. It i calculated from the atural flux i r. Hece, the referece igal of the rotor curret i (dq) + referece frame ca be expreed: + i = i i i K rd rdf rd rdf d d + + i = i i i K (5) rq rqf rq rqf d q with K L ( LL ). ad are the atural compoet of the tator flux i the (dq) + referece d m r d+ q+ frame, repectively. Likewie, the expreio a how i Equatio (6) ca be obtaied:

10 Eergie 05, where ad d q i i = i i i K i i = i i i K rd rd rdf rd rdf d d rq rq rqf rq rqf d q are the atural compoet of the tator flux i the (dq) referece frame, repectively. (6) 4. Dyamic Performace of Stator Flux Likage Accordig to the ymmetrical compoet theory, if the grid upply i ubalaced, ay three-phae quatity, e.g., voltage, curret, or flux, deoted a a vector U ca be eparated ito the poitive ad S egative equece compoet a: j t j t U e U e S S S U (7) where U S ad U are the poitive- ad egative-equece voltage compoet i the tator, repectively. S - The otch filter N() are applied to filter out the ifluece of the 00-Hz frequecy compoet i the tator voltage i the (dq) + ad (dq) referece frame. Meawhile, aumig that the d-axi of the referece frame i aliged with the tator voltage vector [4] ad applyig Laplace traformatio to them, it ca be obtaied: u () NU () ()+ U () N ()/( ), u () 0 d + S S q + u () NU () ()+ U () N ()/( ), u () 0 (8) d S S q By ubtitutig Equatio (5) ad (6) ito the firt two Equatio of () ad (4), the d-axi ad q-axi compoet of the tator curret ca be expreed i Equatio (9): i () L i ()/ L [ L K ()+ ()]/ L d + m rdf m d d + d + i () L i ()/ L [ L K ()+ ()]/ L q + m rqf m d q + q + i () L i ()/ L [ L K ()+ ()]/ L d m rdf m d d d i () L i ()/ L [ L K ()+ ( )]/ L q m rqf m d q q (9) Uder table operatig coditio, the tator flux likage ca be obtaied from the Equatio () ad (3) uch that U /, U /, ad 0. Hece, the expreio of the atural q + q d + d flux ca be obtaied a: d+ d + U q q d d U q q I fact, the repoe of the ier rotor curret cotrol loop i much fater tha the dyamic chage of the tator flux likage [5]. Therefore, the term R Li S m rdf, R Li, R Li S m rqf S m rdf, ad R Li S m rqf ca be eglected. Moreover, the amplitude of the term RLKU L ad RLKU L are much maller S m d S m d tha U ad U, repectively. Hece, the term RLKU L ad RLKU L ca alo be S m d S m d eglected. Baed o thi, ubtitute Equatio (9) ad (30) ito the firt two Equatio of () ad (3) ad the voltage equatio of the tator ca be obtaied a follow: u ( ) R ()/ L () () d + d + q + d + 0 R ()/ L+ () () U / q + d + q + (30) (3)

11 Eergie 05, u ( ) R ()/ L () () d d q d 0 R ()/ L+ () () (3) q d q where U i the amplitude of the tator voltage vector uder ormal coditio ad L L L L. m r Accordig to the Equatio (8) ad (3), takig the d-axi compoet of the tator flux i the (dq) + referece frame, for example, it ca be expreed a: R L U ( R L) U U S S () N() d+ ( R L) D () ( R L) (33) d+ where D ()=[ ( + R / L) ] ( ). Aumig the D d+ d+ () pole ca be obtaied uch that: i equal to 0, the cloed-loop R L j, (34) j 34, where i much greater tha R / L. Hece, the, i the domiat pole of the trafer fuctio D. Meawhile, the amplitude of ( R L ) / D ( ) d +, i o mall that the ifluece of () d + egative-equece voltage ca be eglected. Therefore, the cloed-loop trafer fuctio repreeted i the form for the d-axi compoet of the tator flux i the (dq) + referece frame durig the fault traiet period ca be rewritte a: U R L U S () N() d + ( R L) ( R L) U A AB AB (35) d = U S ( R L) ( + ( R L) + ) The expreio for the coefficiet A d, A, A, B, ad B ca be foud i Appedix A. Likewie, expreio a how i Equatio (36) ca be obtaied: A U R L U q AB A B S () q+ ( R L) ( + ( R L) + ) (36) The expreio for the coefficiet A q, A 3, A 4, B 3, ad B 4 ca be foud i Appedix B. Sice oe of the dampig time cotat = /. 5m, thi mea that the traiet period i approximately 0.0 m, which ca be eglected. Meawhile, the term R / L i o mall compared with the term ad. Hece, the term R /L ca be eglected. Uder thi coditio, the approximate value of the term A A d, B, A q, A 3, ad 3 B ca be obtaied. Accordigly, A 0, A 6 7, B 9 7, ad A, A 9 7 d q 3, ad B Baed o thi, the d-axi ad q-axi compoet of the tator flux i the (dq) + referece frame ca be expreed a: Rt/L Ke i ( t ) d+ Rt/L U K e co ( t ) q + with K ( 9U - 7U ) 7 U (7 ) ad arcta 6 U ( 7U 9U + ). Likewie, the expreio of the d-axi ad q-axi compoet of the tator flux likage i the (dq) referece frame ca be obtaied: (37)

12 Eergie 05, Ke i ( t + ) Rt/L d Rt/L + co + q U K e ( t ) (38) with K 3U ( 7 ) ad arcta( 3). 5. Fault Curret Characteritic of the Doubly-Fed Iductio Geerator (DFIG) By ubtitutig the expreio of tator flux likage (,,, ad ) ad the expreio d + q + d q of rotor curret ( i rd, i, i rq rd, ad i rq ) ito the equatio of Equatio (), the time domai expreio of the d-axi ad q-axi compoet of the tator curret i the (dq) + ad (dq) referece frame ca be obtaied a follow: Rt/L i() t L i L + K (L K ) e i( t+ ) L d m rdf m d i() t ( L i U ) L + K (L K ) e co( t+ ) L Rt/L q m rqf m d Rt/L i() t L i L + K (L K ) e i( t ) L d m rdf m d i() t ( L i U ) L K (L K ) e co( t ) L Rt/L q m rqf m d (39) (40) The coordiate traformatio i applied to Equatio (39), the poitive-equece compoet of three-phae tator fault curret ca be derived a: i ( t) I co( t+ + )+ K ( L K )e i( - ) L i t I K L K L i t I K L K L Rt/L a a m d a Rt/L ( ) co( t+ + )+ ( )e i( - ) b b m d b Rt/L ( ) co( t+ + )+ ( )e i( - ) c c m d c where =, = / 3, = / 3, ad a 0 b 0 c 0 0 i the agle betwee the tator voltage vector ad + + tator curret vector at the momet that the fault occur. Meawhile, I = ( Li ) +( Li U ) ( L) m rdf + m rqf + + ad γ arcta[( Li U ) Li ]. + m rqf + m rdf Likewie, the egative-equece compoet of three-phae tator curret ca be derived a: i () t I co( t+ + )+ K ( L K )e i( + ) L i t I K L K L i t I K L K L Rt/L a a m d a Rt/L ( ) co( t+ + )+ ( )e i( + ) b b m d b Rt/L () co( t+ + )+ ( )e i( + ) c c m d c where =, = + / 3, = / 3. Meawhile, I = ( Li ) +( Li U ) ( L) ad a 0 b 0 c 0 m rdf m rqf arcta[( L i U ) L i ]. m rq m rd Accordigly: i () t i ()+ t i () t a a + a i () t i ()+ t i () t b b+ b i () t i ()+ t i () t c c + c (4) (4) (43)

13 Eergie 05, Simulatio Study Uder ubalaced fault coditio, there are variou cotrol trategie available for the LVRT requiremet of the DFIG [6 33]. Therefore, the effect of the cotrol trategy o the fault curret characteritic of the DFIG mut be coidered. 6.. Headig Steady-tate Referece Sigal of Rotor Curret I thi paper, i order to eure balaced heatig o the three-phae tator widig, the cotrol target for balaced tator curret of the DFIG i ued [8]. Therefore, uder table operatig coditio: i d 0, i 0 q (44) Moreover, i order to provide ufficiet reactive curret to meet the requiremet of the grid code [6], the reactive curret referece igal deoted by I T durig grid fault hould be: I T 0 >0.9 ( 09. ) (45) where all parameter are give i per uit ytem, ad repreet the magitude of the poitive-equece grid voltage after a fault occur. Due to the q-axi compoet of the tator curret i the (dq) referece frame beig zero, the q-axi compoet of the tator curret i the (dq) + referece frame uder table operatig coditio ca be expreed a: + i ( 09. u ) q + d (46) I additio, to eure the afety of the RSC, the maximum rotor curret I ret, which i withi the admiible overload capability of the RSC, hould alo be provided. The typical value of I ret i. per uit (p.u.) [9]. By ubtitutig Equatio (44) ad (46) ito the Equatio () (4), the teady-tate referece igal of the rotor curret i the (dq) + ad (dq) referece frame uder ubalaced fault coditio ca be expreed a follow: + i I i i i rdf + ret rqf + rdf - rqf i = u L -L 0. 9-u L i 0 rdf - - i u L rqf d - m rqf + d + m d + m (47) Accordig to Equatio (47), o coditio that the cotrol target for balaced tator curret i applied i the DFIG, the expreio of the poitive- ad egative-equece compoet of the three-phae tator fault curret ca be obtaied. 6.. Simulatio Aalyi I order to validate the previou theoretical aalyi reult, a imulatio model with the DFIG i built i the PSCAD/EMTDC oftware eviromet. Amog them, the imulatio model of the DFIG cotrol ytem i derived from referece [7] ad the cotrol target for balaced tator curret of the DFIG i derived from referece [6].

14 Eergie 05, It hould be oted that the improved cotrol trategy of the GSC preeted i adopted to limit the fluctuatio of the DC-lik voltage [3]. Additioally, the rotate peed i kept cotat durig grid fault. The detail of the 0.5 kv, 50 Hz tet ytem how i Figure 5 are a follow. I Figure 5, the lie parameter are r() = r() = 0.7 Ω/km ad x() = x() = Ω/km. The Zero equece parameter of the lie are r(0) = 0.9 Ω/km ad x(0) =0.53 Ω/km. The total legth of lie L i 5 km. The rated capacity of the T i.5/.5 MVA, the tur ratio i 0.69 kv/0.5 kv, the widig type i Y/D, ad the leakage reactace i 0.06 p.u. The voltage magitude (L-L) of the ychroou geerator G i 0.5 kv, the phae agle i 0, ad the iteral reitace i 0.5 Ω. The parameter of the.5-mw rated DFIG are: U 690V, f 50HZ, L L 39p.u.., L 767p.u., R p.u., R p.u., 7, r m r c p, =800r/mi. max Figure 5. Tet ytem of the DFIG. For the followig imulatio example, the output active power of the DFIG i.0 p.u. ad the grid voltage i.0 p.u. before the fault occur. The fault curret characteritic of the DFIG are tudied i a variety of fault coditio, icludig differet fault type ad fault poit. Amog them, the fault happe to be 5 km away from the DFIG i the lie L at time t = 5.0 ad reult i Phae-C groud fault that ha bee ued for the imulatio example. Uder thi fault coditio, the poitive-ad egative-equece compoet of the tator voltage are 0.57 p.u. ad 0.5 p.u., repectively Stator Flux Likage Figure 6 how the imulatio reult of the d-axi ad q-axi compoet of the tator flux likage (,,, ad d + q + d q ) i the (dq) + ad (dq) referece frame. I Figure 6, Y repreet the flux likage. pp ad repreet the poitive-ad egative-equece compoet, repectively. The -ap repreet the imulatio reult. Aother waveform repreet the theoretical aalyi reult. Notice that all value i the theoretical aalyi reult ad imulatio reult are the per-uit value (p.u.). It ca be oberved that there are damped fudametal frequecy compoet i the q-axi ad d-axi compoet of the tator flux likage i the (dq) + ad (dq) referece frame. Meawhile, there are o DC compoet i ad. However, there are DC compoet i ad, whoe d + d q + q amplitude are approximately proportioal to the amplitude of poitive-ad egative-equece compoet of the grid voltage, repectively.

15 Eergie 05, Figure 6. d-axi ad q-axi compoet of the tator flux likage i the (dq) + ad (dq) referece frame Rotor Curret ad Stator Curret Figure 7 how the compario betwee the theoretical aalyi reult ad imulatio reult of the d-axi ad q-axi compoet of the rotor curret i the (dq) + ad (dq) referece frame. Figure 7. d-axi ad q-axi compoet of the rotor curret i the (dq) + ad (dq) referece frame. It ca be ee from Figure 7 that the differece betwee the imulatio reult ad theoretical aalyi reult of the rotor curret are very mall. Whe the cotrol target for the balaced tator curret i applied i the DFIG, there are damped fudametal frequecy compoet ad DC compoet i the rotor curret ( rd, i rq, ad i rq- ), but oly damped fudametal frequecy i

16 Eergie 05, compoet i the d-axi fault compoet of the rotor curret i rd i the (dq) referece frame. Additioally, the amplitude of the DC compoet i ot damped. I additio, it ca be ee i Figure 7 that there i a mall deviatio from the calculated rotor curret at iitial coditio (about / cycle after fault iitiatio) ad the imulated rotor curret. Thi i due to the derivatio of the aalytical expreio for rotor curret beig baed o the aumed coditio that the voltage fluctuatio durig the grid fault coditio are eglected. I fact, whe a fault occur i the lie L, there are ocillatio i the termial voltage of the DFIG due to the imbalace of the itataeou power durig the iitial tage. Thi i the reao why the iitial coditio of the calculated rotor curret are differet for the imulated rotor curret. Figure 8 how the compario betwee the theoretical aalyi reult ad imulatio reult of the d-axi ad q-axi compoet of the tator curret i the (dq) + ad (dq) referece frame. Figure 9 illutrate the amplitude of the poitive- ad egative-equece fudametal frequecy compoet of the tator curret ( AMP Ia pp ad AMP Ia ), the phae agle of poitive-equece fudametal frequecy compoet ( PH Ia pp ), ad the damped DC compoet ( AMP ). Ia dc Figure 8. d-axi ad q-axi compoet of the tator curret i the (dq) + ad (dq) referece frame. It ca be oberved from Figure 8 ad Figure 9 that there are damped DC compoet ad teady-tate fudametal frequecy compoet i the poitive-equece compoet of the tator curret ( i d ad i q ). The amplitude ad phae agle of the poitive-equece fudametal frequecy compoet i the tator curret are related to the correpodig cotrol target of the DFIG ad the amplitude of the poitive-equece tator voltage compoet. The amplitude of the damped DC compoet i ot oly related to the correpodig cotrol target of the DFIG ad the amplitude of the poitive-equece tator voltage compoet, but alo affected by the phae agle of the pre-fault curret ad the amplitude of the egative-equece tator voltage compoet.

17 Eergie 05, Figure 9. The A phae curret uder coditio of a Phae-C groud fault. The compoitio of the egative-equece tator curret compoet i related to the cotrol target of the DFIG. Whe the cotrol target for the balaced tator curret i applied i the DFIG, there i oly a damped fudametal frequecy compoet ad without the DC compoet i the egative-equece tator curret compoet ( i d ad i ). However, there i ot oly the DC q- compoet, but alo the damped fudametal frequecy compoet of the egative-equece tator curret compoet i the (dq) referece frame o the coditio that the cotat electromagetic torque or balaced rotor curret of the cotrol target i applied i the DFIG [8]. I additio, it ca be ee from Figure 8 that the amplitude of the damped fudametal frequecy compoet i oly related to the amplitude of the egative-equece tator voltage compoet. However, the iitial phae agle of the damped fudametal frequecy compoet i cotat ad it i idepedet of the egativeequece tator voltage compoet. Moreover, the amplitude of the damped fudametal frequecy compoet i proportioal to the amplitude of the egative-equece tator voltage compoet. I Figure 0, the imulatio reult of the Phae-C tator curret o the differet coditio, of the DFIG without the otch filter ( I SC ap ) ad with the otch filter ( I SC ap ), ad the theoretical aalyi reult of the Phae-C tator curret o the coditio that the DFIG with the otch filter ( I SC ), are compared. Due to lack of pace, oly the compario of the tator curret (with or without otch filter) ha bee how i thi paper. I fact, the AC compoet with frequecy of exit i the tator flux likage, the rotor curret, the tator curret, ad the tator voltage o the DFIG, which the phae agle caot be a accurate etimatio by the covetioal PLL techique ad without the otch filter.

18 Eergie 05, Figure 0. The C phae curret uder coditio of a Phae-C groud fault. It ca be oberved form the Figure 0 that the AC compoet with frequecy of i the tator curret ha bee elimiated by the otch filter ad there exit the AC compoet with a frequecy of i the tator curret without the otch filter. Moreover, the differece betwee the imulatio reult ad theoretical aalyi reult of the Phae-C curret i very mall, which validate the effectivee of the theoretical aalyi. I additio, there are oly damped DC compoet ad teady-tate fudametal frequecy compoet i the tator curret baed o the aforemetioed aalyi. For the fudametal frequecy compoet, there are o-liear relatiohip betwee the fault voltage ad the fault curret of the DFIG. Hece, o the coditio that the cotrol target for balaced tator curret i applied i the DFIG, the DFIG ca be repreeted by a cotrolled poitive-equece curret ource. O the coditio that the cotrol target for balaced rotor curret or cotat electromagetic torque i applied i the DFIG, the DFIG ca be repreeted by a cotrolled poitive-equece curret ource ad a cotrolled egative-equece curret ource. 7. Cocluio The fault curret characteritic of the DFIG are tudied i thi paper o coditio that a ubalaced fault occur ad the rotor widig are excited by the AC/DC/AC coverter. Uder ubalaced fault coditio, the electrical variable ocillate at twice the grid frequecy i the ychroou dq frame. I the egieerig practice, otch filter are uually ued to extract the poitive ad egative equece compoet i the ychroou dq frame. Hece, the otch filter have a large ifluece o the fault curret characteritic of the DFIG. I thi paper, the dyamic performace of the tator flux likage ad the rotor curret are aalyzed. Baed o thi, the tator fault curret characteritic of the DFIG uder ubalaced fault coditio are tudied. Thi tudy i of great igificace for afe ad table operatio of the power grid with peetratio of DFIG. The obtaied cocluio ca be draw a follow: () The amplitude ad phae agle of the tator flux likage i the DFIG are chaged by the otch filter. Moreover, the rage of the PI parameter ad the dyamic performace of the RSC are alo iflueced by the otch filter.

19 Eergie 05, () The traiet behavior of the DFIG are aalyzed, ad imple expreio of the DFIG traiet electrical variable are propoed. The acquired traiet characteritic provide good referece to improve DFIG cotrol durig ubalaced grid fault. (3) Uder ubalaced fault coditio, the tator fault curret characteritic of the DFIG are differet from thoe of a covetioal ychroou geerator. There i oly a damped DC compoet ad teady-tate fudametal frequecy compoet i the tator fault curret. The amplitude of the damped DC compoet of the tator fault curret i related to the depth of the grid voltage dip ad the phae agle of the pre-fault curret. The teady-tate fudametal frequecy compoet i ot oly related to the depth of the grid voltage dip, but alo affected by the correpodig cotrol target of the DFIG. (4) For the fudametal frequecy compoet, o the coditio that the cotrol target for balaced tator curret i applied i the DFIG, the DFIG ca be repreeted by a cotrolled poitive-equece curret ource. O the coditio that the cotrol target for balaced rotor curret or cotat electromagetic torque i applied i the DFIG, the DFIG ca be repreeted by a cotrolled poitive-equece curret ource ad a cotrolled egative-equece curret ource. Thi i helpful to meet the requiremet of a practical hort-circuit calculatio ad the cotructio of a relayig protectio ytem for the power grid with peetratio of DFIG. Ackowledgmet Thi work wa upported by the Natioal Natural Sciece Foudatio of Chia (No ad No ). Author Cotributio Fa Xiao ad Zhe Zhag deiged the model ad the experimet. Fa Xiao carried out the experimet. Fa Xiao ad Xiagge Yi aalyzed ad iterpreted data. Fa Xiao, Zhe Zhag ad Xiagge Yi cocluded the cietific fidig ad prepared the maucript. Coflict of Iteret The author declare o coflict of iteret. Appedix A K (R L + )K K ( K R K LK ) R L (R L ) R L ( ) (48) + R L R / L (R L ) R L R L A A d R L R L (49) B K 3 B R L (50)

20 Eergie 05, B A K A (5) K (R L )K K + R L 3 A R L (R L ) R / L (R L ) (5) R L K R L (53) K R L ( 4 R L + R L + )A d (54) (55) K A R L A ( R L ) 3 d d K ( R L )A 4 d (56) Appedix B A 3 K (R L ) K K K R K L K R / L (R L ) R L R L R L R L R / L (R L ) A (57) q R L (58) B K 8 B 3 4 R L (59) B A K A (60) K K R L K R L (6) A 4 3 R L R L K 5 R L (6) K (R L )A 6 q (63) K A 4 AR L ( R L )A 7 q q q (64) K 8 R / L (R L ) A. q (65) Referece. Muller, S.; Deicke, M.; de Docker, R.W. Doubly fed iductio geerator ytem for wid turbie. IEEE Id. Appl. Mag. 00, 8, Iwaki, G.; Koczara, W. DFIG-baed power geeratio ytem with UPS fuctio for variable-peed applicatio. IEEE Tra. Id. Electro. 008, 55, Jauch, C.; Matevoya, J.; Ackerma, T.; Bolik, S. Iteratioal compario of requiremet for coectio of wid turbie to power ytem. Wid Eergy 005, 8, The grid Code 3. Available olie: http// (acceed o Jauary 006).

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