CHAPTER 7 SYMMETRICAL COMPONENTS AND REPRESENTATION OF FAULTED NETWORKS

Transcription

1 HAPTER 7 SMMETRAL OMPOETS AD REPRESETATO OF FAULTED ETWORKS A uled three-phe yte e reolved ito three led yte i the iuoidl tedy tte. Thi ethod of reolvig uled yte ito three led phor yte h ee propoed y. L. Forteue. Thi ethod i lled reolvig yetril opoet of the origil phor or iply yetril opoet. thi hpter we hll diu yetril opoet trfortio d the will preet how uled opoet like - or -oeted lod, trforer, geertor d triio lie e reolved ito yetril opoet. We the oie ll thee opoet together to for wht re lled equee etwork. 7. SMMETRAL OMPOETS A yte of three uled phor e reolved i the followig three yetril opoet: Poitive Sequee: A led three-phe yte with the e phe equee the origil equee. egtive equee: A led three-phe yte with the oppoite phe equee the origil equee. ero Sequee: Three phor tht re equl i gitude d phe. Fig. 7. depit et of three uled phor tht re reolved ito the three equee opoet etioed ove. thi the origil et of three phor re deoted y, d, while their poitive, egtive d zero equee opoet re deoted y the uript, d repetively. Thi iplie tht the poitive, egtive d zero equee opoet of phe- re deoted y, d repetively. ote tht ut like the voltge phor give i Fig. 7. we lo reolve three uled urret phor ito three yetril opoet. Fig. 7. Repreettio of () uled etwork, it () poitive equee, () egtive equee d (d) zero equee.

2 . 7.. Syetril opoet Trfortio Before we diu the yetril opoet trfortio, let u firt defie the - opertor. Thi h ee give i (.4) d i reprodued elow. e (7.) ote tht for the ove opertor the followig reltio hold d o o e e e e e e e e (7.) Alo ote tht we hve (7.) Uig the -opertor we write fro Fig. 7. () d (7.4) Siilrly fro Fig. 7. () we get d (7.5) Filly fro Fig. 7. (d) we get (7.6) The yetril opoet trfortio trix i the give y (7.7) Defiig the vetor d,

3 .4 we write (7.4) (7.8) where i the yetril opoet trfortio trix d i give y (7.9) The origil phor opoet lo e otied fro the ivere yetril opoet trfortio, i.e., (7.) vertig the trix give i (7.9) d oiig with (7.) we get (7.) Fro (7.) we write (7.) (7.) (7.4) Filly, if we defie et of uled urret phor d their yetril opoet, we the defie (7.5) Exple 7.: Let u oider et of led voltge give i per uit y.,. d. Thee iply d The fro (7.7) we get ( )

4 .5 ( ). pu 4 ( ) We the ee tht for led yte the zero d egtive equee voltge re zero. Alo the poitive equee voltge i the e the origil yte, i.e.,, d Exple 7.: All the qutitie give i thi exple re i per uit. Let u ow oider the followig et of three uled voltge.,. d.9 f we reolve the uig (7.4) we the hve Therefore we hve ,.96 6., Furtherore ote tht Rel d Retive Power The three-phe power i the origil uled yte i give y P * * * T Q (7.6) where i the oplex ougte of the vetor. ow fro (7.) d (7.5) we get P (7.7) T T Q

5 .6 Fro (7.) we get T Therefore fro (7.7) we get ( ) P Q (7.8) We the fid tht the oplex power i three tie the utio of the oplex power of the three phe equee. Exple 7.: Let u oider the voltge give i Exple 7.. Let u further ue tht thee voltge re lie-to-eutrl voltge d they upply led - oeted lod whoe per phe ipede i..8 per uit. The the per uit urret i the three phe re pu pu pu The the rel d retive power oued y the lod i give y P Q ( ) o( ).9559 pu ( ) i( ).85 pu ow uig the trfortio (7.5) we get pu Fro the reult give i Exple 7. d fro the ove vlue we opute the zero equee oplex power Q.8 P.55 pu The poitive equee oplex power i

6 .7 Q.954 P.745 pu Filly the egtive equee oplex power i Q.67 P.69 pu Addig the three oplex power together we get the totl oplex power oued y the lod P ( P P P ) ( Q Q Q ) Q pu 7.. Orthogol Trfortio ted of the trfortio trix give i (7.9), let u ited ue the trfortio trix (7.9) We the hve (7.) ote fro (7.9) d (7.) tht ( T ). We therefore tte ( T ), where i ( ) idetity trix. Therefore the trfortio trie give i (7.9) d (7.) re orthogol. ow ie T T ( ) we write fro (7.7) P (7.) Q We hll ow diu how differet eleet of power yte re repreeted i ter of their equee opoet. ft we hll how tht eh eleet i repreeted y three equivlet iruit, oe for eh yetril opoet equee.

7 .8 7. SEQUEE RUTS FOR LOADS thi etio we hll otrut equee iruit for oth d -oeted lod eprtely. 7.. Sequee iruit for -oeted Lod oider the led -oeted lod tht i how i Fig. 7.. The eutrl poit () of the widig re grouded through ipede. The lod i eh phe i deoted y. Let u oider phe- of the lod. The voltge etwee lie d groud i deoted y, the lie-to-eutrl voltge i deoted y d voltge etwee the eutrl d groud i deoted y. The eutrl urret i the ( ) ( ) (7.) Therefore there will ot e y poitive or egtive equee urret flowig out of the eutrl poit. Fig. 7. Sheti digr of led -oeted lod. The voltge drop etwee the eutrl d groud i (7.) ow (7.4) We write iilr expreio for the other two phe. We therefore write (7.5) Pre-ultiplyig oth ide of the ove equtio y the trix d uig (7.8) we get

8 .9 (7.6) ow ie We get fro (7.6) (7.7) We the fid tht the zero, poitive d egtive equee voltge oly deped o their repetive equee opoet urret. The equee opoet equivlet iruit re how i Fig. 7.. While the poitive d egtive equee ipede re oth equl to, the zero equee ipede i equl to (7.8) f the eutrl i grouded diretly (i.e., ), the. O the other hd, if the eutrl i kept flotig (i.e., ), the there will ot e y zero equee urret flowig i the iruit t ll. Fig. 7. Sequee iruit of -oeted lod: () poitive, () egtive d () zero equee. 7.. Sequee iruit for -oeted Lod oider the led -oeted lod how i Fig. 7.4 i whih the lod i eh phe i deoted y. The lie-to-lie voltge re give y (7.9) Addig thee three voltge we get

9 . Fig. 7.4 Sheti digr of led -oeted lod. ( ) (7.) Deotig the zero equee opoet, d d tht of, d we rewrite (7.) (7.) Agi ie we fid fro (7.). Hee -oeted lod with o utul ouplig h ot y zero equee irultig urret. ote tht the poitive d egtive equee ipede for thi lod will e equl to. Exple 7.4: oider the iruit how i Fig. 7.5 i whih -oeted lod i oeted i prllel with -oeted lod. The eutrl poit of the -oeted lod i grouded through ipede. Applyig Kirhoff urret lw t the poit P i the iruit we get ( ) The ove expreio e writte i ter of the vetor [ ] Sie the lod i led we write

10 . Fig. 7.5 Prllel oetio of led d -oeted lod. Pre-ultiplyig oth ide of the ove expreio y the trfortio trix we get ow ie we get Seprtig the three opoet, we write fro the ove equtio Suppoe ow if we overt the -oeted lod ito equivlet, the the opoite lod will e prllel oitio of two -oeted iruit oe with ipede of d the other with ipede of /. Therefore the poitive d the egtive equee ipede re give y the prllel oitio of thee two ipede. The poitive d egtive equee ipede i the give y

11 . ow refer to Fig The voltge i give y ( ) Fro Fig. 7.5 we lo write. Therefore Thi iplie tht d hee. We the rewrite the zero equee urret expreio t e ee tht the ter i et fro the zero equee ipede. 7. SEQUEE RUTS FOR SHROOUS GEERATOR The three-phe equivlet iruit of yhroou geertor i how i Fig..6. Thi i redrw i Fig. 7.6 with the eutrl poit grouded through retor with ipede. The eutrl urret i the give y (7.) Fig. 7.6 Equivlet iruit of yhroou geertor with grouded eutrl.

12 . The derivtio of Setio. ue led opertio whih iplie. A per (7.) thi uptio i ot vlid y ore. Therefore with repet to thi figure we write for phe- voltge ( ) ( ) ( ) ( ) E M M L R E M L R ω ω ω ω ω (7.) Siilr expreio lo e writte for the other two phe. We therefore hve ( ) [ ] E E E M M L R ω ω (7.4) Pre-ultiplyig oth ide of (7.4) y the trfortio trix we get ( ) [ ] E E E M M L R ω ω (7.5) Sie the yhroou geertor i operted to upply oly led voltge we ue tht E E d E E. We therefore odify (7.5) ( ) [ ] E M M L R ω ω (7.6) We eprte the ter of (7.6) ( ) [ ] g M L R ω (7.7) ( ) [ ] E E M L R ω (7.8) ( ) [ ] M L R ω (7.9) Furtherore we hve ee for -oeted lod tht, ie the eutrl urret doe ot ffet thee voltge. However. Alo we kow tht. We therefore rewrite (7.7) ( ) g (7.4) The equee digr for yhroou geertor re how i Fig. 7.7.

13 .4 Fig. 7.7 Sequee iruit of yhroou geertor: () poitive, () egtive d () zero equee. 7.4 SEQUEE RUTS FOR SMMETRAL TRASMSSO LE The heti digr of triio lie i how i Fig thi digr the elf ipede of the three phe re deoted y, d while tht of the eutrl wire i deoted y. Let u ue tht the elf ipede of the odutor to e the e, i.e., Sie the triio lie i ued to e yetri, we further ue tht the utul idute etwee the odutor re the e d o re the utul idute etwee the odutor d the eutrl, i.e., The diretio of the urret flowig through the lie re idited i Fig. 7.8 d the voltge etwee the differet odutor re idited. Fig. 7.8 Luped preter repreettio of yetril triio lie. Applyig Kirhoff voltge lw we get (7.4) Agi ( ) ( ) (7.4) (7.4)

14 .5 Sutitutig (7.4) d (7.4) i (7.4) we get ( ) ( )( ) ( ) (7.44) Sie the eutrl provide retur pth for the urret, d, we write ( ) (7.45) Therefore utitutig (7.45) i (7.44) we get the followig equtio for phe- of the iruit Deotig ( ) ( )( ) (7.46) d (7.46) e rewritte ( ) (7.47) Sie (7.47) doe ot expliitly ilude the eutrl odutor we defie the voltge drop ro the phe- odutor (7.48) oiig (7.47) d (7.48) we get ( ) (7.49) Siilr expreio lo e writte for the other two phe. We therefore get (7.5) Pre-ultiplyig oth ide of (7.5) y the trfortio trix we get ow (7.5)

15 .6 ( ) ( ) ( ) ( ) Hee ( ) ( ) ( ) ( ) 6 Therefore fro (7.5) we get (7.5) The poitive, egtive d zero equee equivlet iruit of the triio lie re how i Fig. 7.9 where the equee ipede re 6 Fig. 7.9 Sequee iruit of yetril triio lie: () poitive, () egtive d () zero equee. 7.5 SEQUEE RUTS FOR TRASFORMERS thi etio we hll diu the equee iruit of trforer. A we hve ee erlier tht the equee iruit re differet for - d -oeted lod, the equee iruit re lo differet for d oeted trforer. We hll therefore tret differet trforer oetio eprtely.

16 oeted Trforer Fig. 7. how the heti digr of - oeted trforer i whih oth the eutrl re grouded. The priry d eodry ide qutitie re deoted y uript i uppere letter d lowere letter repetively. The tur rtio of the trforer i give y α :. Fig. 7. Sheti digr of grouded eutrl - oeted trforer. The voltge of phe- of the priry ide i A A A A Expdig A d A i ter of their poitive, egtive d zero equee opoet, the ove equtio e rewritte (7.5) A A A A A A A otig tht the diretio of the eutrl urret i oppoite to tht of, we write equtio iilr to tht of (7.5) for the eodry ide (7.54) ow ie the tur rtio of the trforer i α : we write Sutitutig i (7.54) we get α A A α A α α ( A A A ) α A Multiplyig oth ide of the ove equtio y α reult i α ( ) A A A α A (7.55) A

17 .8 Filly oiig (7.5) with (7.55) we get ( ) A A A ( α ) A α (7.56) Seprtig out the poitive, egtive d zero equee opoet we write α (7.57) A α (7.58) A [ ( ) ] α (7.59) A A Fig. 7. ero equee equivlet iruit of grouded eutrl - oeted trforer. Fro (7.57) d (7.58) we ee tht the poitive d egtive equee reltio re the e tht we hve ued for repreetig trforer iruit give i Fig..8. Hee the poitive d egtive equee ipede re the e the trforer lekge ipede. The zero equee equivlet iruit i how i Fig. 7.. The totl zero equee ipede i give y ( ) (7.6) The zero equee digr of the grouded eutrl - oeted trforer i how i Fig. 7. () i whih the ipede i give i (7.6). f oth the eutrl re olidly grouded, i.e.,, the i equl to. The igle lie digr i till the e tht how i Fig. 7. (). f however oe of the two eutrl or oth eutrl re ugrouded, the we hve either or or oth. The zero equee digr i the how i Fig. 7. () where the vlue of will deped o whih eutrl i kept ugrouded. Fig. 7. ero equee digr of () grouded eutrl d () ugrouded eutrl - oeted trforer.

18 oeted Trforer The heti digr of - oeted trforer i how i Fig. 7.. ow we hve AB A B A A A B B B AB AB (7.6) Agi AB α Therefore fro (7.6) we get AB Fig. 7. Sheti digr of - oeted trforer. ( ) α (7.6) AB AB The equee opoet of the lie-to-lie voltge AB e writte i ter of the equee opoet of the lie-to-eutrl voltge (7.6) AB A (7.64) AB A Therefore oiig (7.6)-(7.64) we get Hee we get ( ) A A α (7.65) A α d α (7.66) A Thu the poitive d egtive equee equivlet iruit re repreeted y erie ipede tht i equl to the lekge ipede of the trforer. Sie the -oeted widig doe ot provide y pth for the zero equee urret to flow we hve A

19 . However the zero equee urret oetie irulte withi the widig. We the drw the zero equee equivlet iruit how i Fig oeted Trforer Fig. 7.4 ero equee digr of - oeted trforer. The heti digr of - oeted trforer i how i Fig t i ued tht the -oeted ide i grouded with the ipede. Eve though the zero equee urret i the priry -oeted ide h pth to the groud, the zero equee urret flowig i the -oeted eodry widig h o pth to flow i the lie. Hee we hve. However the irultig zero equee urret i the widig getilly le the zero equee urret of the priry widig. Fig. 7.5 Sheti digr of - oeted trforer. The voltge i phe- of oth ide of the trforer i relted y A α Alo we kow tht A A We therefore hve A A A α A A A A ( ) A (7.67) Seprtig zero, poitive d egtive equee opoet we write

20 . α (7.68) A A A α α (7.69) A α α (7.7) The poitive equee equivlet iruit i how i Fig. 7.6 (). The egtive equee iruit i the e tht of the poitive equee iruit exept for the phe hift i the idued ef. Thi i how i Fig. 7.6 (). The zero equee equivlet iruit i how i Fig. 7.6 () where. ote tht the priry d the eodry ide re ot oeted d hee there i ope iruit etwee the. However ie the zero equee urret flow through priry widig, retur pth i provided through the groud. f however, the eutrl i the priry ide i ot grouded, i.e.,, the the zero equee urret ot flow i the priry ide well. The equee digr i the how i Fig. 7.6 (d) where. Fig. 7.6 Sequee digr of - oeted trforer: () poitive equee, () egtive equee, () zero equee with grouded -oetio d (d) zero equee with ugrouded -oetio. 7.6 SEQUEE ETWORKS The equee iruit developed i the previou etio re oied to for the equee etwork. The equee etwork for the poitive, egtive d zero equee re fored eprtely y oiig the equee iruit of ll the idividul eleet. erti uptio re de while forig the equee etwork. Thee re lited elow.. Aprt fro yhroou hie, the etwork i de of tti eleet.. The voltge drop ued y the urret i prtiulr equee deped oly o the ipede of tht prt of the etwork.. The poitive d egtive equee ipede re equl for ll tti iruit opoet, while the zero equee opoet eed ot e the e the. Furtherore utriet poitive d egtive equee ipede of yhroou hie re equl. 4. oltge oure re oeted to the poitive equee iruit of the rottig hie. 5. o poitive or egtive equee urret flow etwee eutrl d groud.

21 . Exple 7.5: Let u oider the etwork how i Fig 7.7 whih i eetilly the e tht diued i Exple.. The vlue of the vriou rete re ot iportt here d hee re ot give i thi figure. However vriou poit of the iruit re deoted y the letter A to G. Thi h ee doe to idetify the ipede of vriou iruit eleet. For exple, the lekge rete of the trforer T i pled etwee the poit A d B d tht of trforer T i pled etwee D d E. The poitive equee etwork i how i Fig Thi i eetilly e tht how i Fig..4. The egtive equee digr, how i Fig. 7.9, i lot idetil to the poitive equee digr exept tht the voltge oure re et i thi iruit. The zero equee etwork i how i Fig. 7.. The eutrl poit of geertor G i grouded. Hee pth fro poit A to the groud i provided through the zero equee rete of the geertor. The priry ide of the trforer T i -oeted d hee there i diotiuity i the iruit fter poit A. Siilr oetio re lo de for geertor G d trforer T. The triio lie ipede re pled etwee the poit B, D d F. The eodry ide of trforer T i ugrouded d hee there i rek i the iruit fter the poit F. However the priry ide of T i grouded d o i the eutrl poit of geertor G. Hee the zero equee opoet of thee two pprtu re oeted to the groud. Fig. 7.7 Sigle-lie digr of -hie power yte. Fig. 7.8 Poitive equee etwork of the power yte of Fig. 7.7.

22 . Fig. 7.9 egtive equee etwork of the power yte of Fig Fig. 7. ero equee etwork of the power yte of Fig. 7.7.