4. UNBALANCED 3 FAULTS
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1 4. UNBALANCED AULTS So fr: we hve tudied lned fult ut unlned fult re more ommon. Need: to nlye unlned ytem. Could: nlye three-wire ytem V n V n V n Mot ommon fult type = ingle-phe to ground i.e. write node eqution in term of Vn, Vn, Vn,,, Alterntive: ue Symmetril omponent expre: eh unlned equene,, where i voltge or urrent : um of lned equene (lled ymmetril omponent) then: poitive phe equene (--) negtive phe equene (--) zero phe equene (ll phor in phe) nlye eprte lned network Reltion n e implified y uing the opertor. 4. Ue of the opertor Define: Thu: Alo: j e.5 j j j.866 * j j.866 j or lned ytem with phe equene --: V V n n Vn n V V V V V V V n n n ELEC46: Anlyi of power ytem fult p. ELEC46: Anlyi of power ytem fult p.4
2 4. ntrodution to ymmetril omponent Exmple: Now onider n unlned equene,, + + nd the ymmetril omponent equene: Poitive equene,, -- Negtive equene,, -- Zero equene,, = ELEC46: Anlyi of power ytem fult p.5 ELEC46: Anlyi of power ytem fult p.6
3 Unlned equene: 6 degree of freedom,,,,, Symmetril omponent equene: 6 degree of freedom,,,,, Will now how tht, given ny unlned - equene,,, n find +ve, -ve nd zero equene,,,,,, nd,, whih um to it: i.e. tht i in mtrix form: Proof: n exerie, how tht thu ymmetril omponent given y will um to,, qed tht i the ymmetril omponent for,, re given y: ELEC46: Anlyi of power ytem fult p.7 ELEC46: Anlyi of power ytem fult p.8
4 Exmple:.,, lned with -equene -- i.e. nd o tht 4.,, V V V line-line voltge. Then zero eq. volt. V V V V (y KVL) ELEC46: Anlyi of power ytem fult p.9 i.e. no zero-equene omponent in line-line voltge (ut n exit in line-neutrl voltge).,, line urrent into Y-onneted lod. Then zero-equene urrent i: onlude: n N () neutrl urrent = x zero-equene urrent () iolted neutrl zero-equene urrent = 4. open iruit in one line: ELEC46: Anlyi of power ytem fult p. Suppoe: 7.. A o tht: A nd:. A. A Symmetril omponent vlue for urrent in eh phe re thu: Phe Phe Phe ( eq.)... (+ve eq.) 5 9 (-ve eq.) 5 9 Note: ut i.e. n hve non-zero ymmetril omponent of phe vrile whih i zero.
5 Power in ymmetril omponent V n V n V n Complex - power i: * * * n n n S P jq V V V unlned - ytem: Vn, Vn, Vn nd,, * * T * T * Vn Vn Vn Vp. p AV A * where: Vp phe voltge vetor; p phe urrent vetor; V equene voltge vetor; equene urrent vetor. V n A ; V V n ; V n thu: * * i.e. T T S V A A * * Vn Vn Vn * * * * n n n S V V V * * * n n n V V V S 4. Applition of ymmetril omponent t i neery to determine ymmetril omponent of urrent in ll prt of network nd then omine to get true urrent. Thi require determintion of the tul network in whih equene urrent flow: poitive equene: network i norml lned equivlent. for negtive equene: for poitive equene exept for genertor. No voltge oure. for zero equene: equivlent network depend on the tul onfigurtion. Sequene network: for lned-y impedne lod g Z y Z y Z y Z n V Z Z Z Z Z Z Z Z g y n n y n y n n n V Z Z Z Z g n y n n V Z Z Z Z g n n y n Z=R+j ELEC46: Anlyi of power ytem fult p. ELEC46: Anlyi of power ytem fult p.
6 or: V Z Z Z Z g y n n n Vg Zn Zy Zn Z n V g Zn Zn Zy Z n or: Vp Zp. p ut: Vp AV. nd p A. V where: A ; V V V Hene: AV. Z. A. where: V Z p Z. nd A Zp A Sequene impedne mtrix: Thu: y n V Z Z Z V Zy Z V Z Z y Z i zero equene impedne Z i poitive equene impedne Z i negtive equene impedne ; A Zy Zn Z Zy Z y Thu for the Y iruit, the equene network re: Zero: Poitive: Negtive: V V V Z y Z n Z y Z y Z Z Z Z Z y Z y Z y when there i no neutrl onnetion; Zn nd thu there i no zero equene urrent. when olidly erthed; Zn or onnetion: onvert into equivlent Y iruit (uing the -Y trnformtion) wherey ZY Z /. Thu the equene network re: Zero: Poitive: Negtive: = Z Z Z Z Z Z n ELEC46: Anlyi of power ytem fult p. ELEC46: Anlyi of power ytem fult p.4
7 Repreenttion of plnt item in equene network. Synhronou mhine: // Poitive equene impedne d Negtive equene i not = +ve equene // Typilly:.5 to.5.5 to.5.4 to.4 Only voltge ville re poitive equene one.. Line nd le: Poitive nd negtive equene impedne vlue re equl. Zero equene impedne depend on nture of return pth through the erth if no fourth wire ville.. Trnformer: Poitive nd negtive equene vlue re equl nd re norml lned vlue. Zero equene omponent vry widely, depending on trnformer onnetion. Zero equene impedne for vriou t/f onnetion. or eh of equene network, we need to redue them to Thévenin equivlent een from the fult, i.e. Z Z Z V V V V _ zero eq. +ve eq. -ve eq. ELEC46: Anlyi of power ytem fult p.5 ELEC46: Anlyi of power ytem fult p.6
8 Exmple 9. [ee textook] Note tht for the line: Z V Z.5 pu 9.4 B Bline SB 6 Thévenin equivlent: Ue MVA,.8kV e (genertor zone). Prefult voltge i V.5 o. Drw equene digrm nd redue to Thévenin equivlent for fult t u (motor terminl). zero eq. +ve eq. Z j.5 Z j.89 V.5 + _ V + V _ j. j.5 j.455// j. -ve eq. Z j V _ j.475// j. Ce : lned fult t u Ue only poitive equene iruit V.5 j7.558 pu Z j.89 Here: ; V ; V j7.558 ; V Uing: p A ELEC46: Anlyi of power ytem fult p.7 ELEC46: Anlyi of power ytem fult p.8
9 // // where: p nd j7.558 // // // thu j // pu Ce : A ingle-phe ground fult ult ondition (expreed in phe domin) re: ; V Z if ring fult (r impedne Z ) or V if olted fult ( Z ) But: A p () V V Vp AV V V V V V V V V () Generl -phe u () nteronneted equene network ig. 9.7 (Glover et l) Thu: V VV V Z Z () Condition () nd () n e tified y interonneting the equene network hown in () ove. rom thi: V Z ZZ Z or uing: p A. V Z ZZ Z () (4) (5) Uing previou exmple: Single-phe ground fult on u with Z ELEC46: Anlyi of power ytem fult p.9 ELEC46: Anlyi of power ytem fult p.4
10 Ce : A line-line fult () Generl -phe u ig. 9.9 (Glover et l).5 j j.964 j5.89 pu pu () nteronneted equene network Conider line-line fult from phe to. ult ondition, expreed in phe domin, re: (6) (7) V V Z (8) Trnform thee into the equene domin: But: V V V V V V V V So eq.8 eome: V V V V V V Z (9) But from eq.9, nd o: V V Z Hene: V V Z Thu, fult ondition in the equene domin: V V Z Thee ondition n e tified y onneting the poitive- nd negtive-equene network in prllel t the fult terminl through the fult impedne Z. rom thi: V Z Z Z Trnform into the phe domin: ELEC46: Anlyi of power ytem fult p.4 ELEC46: Anlyi of power ytem fult p.4
11 j j V () ZZ Z Alo: () () Ce 4: A doule line-to-ground fult () Generl -phe u ig. 9. (Glover et l) ELEC46: Anlyi of power ytem fult p.4 () nteronneted equene network Conider fult from phe to through fult impedne Z to ground. ult ondition, expreed in phe domin, re: () V V (4) V Z (5) Trnform thee into the equene domin. Note tht: nd thu from eq.: But: V V V V (6) V V V V So eq.4 eome: V V V V V V or: V V Sutitute into eq.6: V V V V V (7) But: = Hene, from eq.5 nd eq.: V V Z Z Thu, fult ondition in the equene domin: V V V V Z Thee ondition n e tified y onneting the zero-, poitive- nd negtive-equene network in prllel t the fult terminl. n ddition, Z i inluded in erie with the zero-equene network. rom thi: ELEC46: Anlyi of power ytem fult p.44
12 V ZZ // Z Z Uing urrent diviion: Z Z Z Z Z Z Z Z Z Thee n e trnformed into the phe domin vi: Ce 5: Effet of trnformer phe hift Note tht we ignored effet of -Y trnformer phe hift in the previou exmple. ult urrent nd ontriution to fult urrent on the fult ide of -Y trnformer re not ffeted y -Y phe hift; ontriution to the fult from the other ide re ffeted. ELEC46: Anlyi of power ytem fult p.45
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