Coupling of Three-Phase Sequence Circuits Due to Line and Load Asymmetries

Transcription

1 Coupling of Three-Phae Sequence Circuit Due to Line and Load Aymmetrie DEGO BELLAN Department of Electronic nformation and Bioengineering Politecnico di Milano Piazza Leonardo da inci 01 Milano TALY Atract: - n thi paper a rigorou circuit repreentation i derived for the coupling etween poitive- and negative-equence circuit due to aymmetrical load and/or line in three-phae ytem. t i hown that each aymmetrical impedance reult in a two-port network connecting the two circuit. An effective circuit repreentation i derived in the paper for uch two-port network aed on the definition of an ideal tranformer with complex turn ratio. The propoed equivalent circuit i ueful to get deeper inight into the fundamental mechanim generating negative-equence voltage/current from ytem aymmetrie. Moreover the propoed circuit repreentation can e implemented into oftware for circuit imulation of complex ytem. Key-Word: -power ytem analyi; ymmetrical component tranformation; aymmetrical three-phae load 1 ntroduction Three-phae power ytem under teady tate condition are uually analyzed in the frequency (phaor) domain y mean of the ymmetrical component tranformation [1]. Other imilar tranformation are availale operating in the frequency or in the time domain []-[]. The main aumption underlying uch method i the ymmetry of the three-phae load and/or the line i.e. the three impedance mut e equal and with the ame coupling coefficient (if any). The reaon i clearly due to the fact that under uch aumption the tranformation mentioned aove decouple the modal circuit (e.g. the poitive negative and zero equence circuit in the ymmetrical component tranformation) allowing a much eaier and direct analyi. f the aumption of ymmetrical load/linei not met the tranformation provide coupled modal circuit leading to a much more complicated analyi. To the Author knowledge no much work ha een done to evaluate the impact of the load/line aymmetry on modal circuit [4]-[6]. Thi i an intereting iue ecaue aymmetry can e a feature of modern power ytem for everal reaon. Firt geometric arrangement of neary power ytem or component can reult in aymmetrical ehavior of line and/or load. Second the analyi of power ytem under ditorted condition i of increaing importance. At harmonic frequencie each electric component can how a different ehavior due to the effect of unpredictale paraitic element. Since equence circuit are widely ued in the analyi of three-phae ytem in thi work the impact of aymmetrie on equence circuit i invetigated. t i hown that each impedance aymmetry reult in coupling etween poitive- and negative-equence circuit which can e effectively repreented y an ideal tranformer with complex turn ratio. The equivalent circuit derived in the paper can e ueful to get deeper inight into the fundamental mechanim of aymmetry effect and for the implementation in oftware for circuit imulation of complex ytem. Background Three-phae power ytem with ymmetrical load can e conveniently analyzed y reorting to the well-known ymmetrical component tranformation. ndeed even in the cae of mutual coupling etween the phae the aumption of ymmetrical ytem reult in three uncoupled equence circuit. Solving each equence circuit i much impler than olving the ytem a a whole. The tranformation matrix in it rational form i defined a 1 α α 1 S = 1 α α (1) SBN:

2 where 1 e j π α = = + j () and. The tranformation matrix i a Hermitian matrix i.e.. The ymmetrical component tranformation when applied to phaor voltage provide + 0 = S a c () where and are the poitive negative and zero equence voltage. Of coure the ame tranformation applie to the current. Symmetrical three-phae load can e decried in term of an impedance matrix with the following tructure m m = m m (4) m m equence circuit when the tranformation i applied to the whole three-phae ytem. Aymmetrical Line/Load Let u conider an aymmetrical three-phaeload coniting in three uncoupled impedance and taking value poily different with repect to a nominal value.notice that uch model can e actually ued to repreent either a load or an aymmetrical line reulting from a non-ideal tranpoition of the conductor. We aume therefore (9a) (9) (9c) where the relative deviation and are complex quantitie where oth the real and the imaginary part can take poitive or negative value. The aymmetrical impedance matrix can e written By defining the column vector + = 0 + = 0 a = c a = (5) c the tranformed current/voltage relationhip for a ymmetrical load can e written (10) where and = SS = 1 (6) = (7) + = = m (8a) = (8) 0 + m By applying the tranformation (6) to (10) the nominal impedance matrix remain unchanged (i.e. ) ince it tructure i the ame a (4) with out-of-diagonal element equal to zero. Therefore only the tranformed verion ofmatrice and mut e analyzed. After ome algera the following expreion can e otained: (11) (1) The diagonal form of the equence impedance matrix (7) lead to the aove-mentioned uncoupled SBN:

3 (1) n thi work the analyi i performed for power ytem with three wire. Therefore the zeroequence circuit correponding to the zero-equence voltage and current in (6) i not defined. Thu y taking into account (11)-(1) the poitive- and negative-equence component of the equence impedance matrix provide the following relationhip etween poitive- and negativeequence variale: repreentation however can e derived y reorting to the definition of a new component coniting in an ideal tranformer with complex turn ratio (ee Appendix). z δ a + (14) From a circuit point of view the four additive impedance term in (14) (15a) (15) δ δ c Fig. 1. Serie connection of two-port network repreenting nominal impedance and coupling due to impedance aymmetrie. (15c) (15d) a+ δ a / a correpond to a erie connection of four two-port network(ee Fig. 1) whoe explicit circuit repreentation will e provided in the following Section. 4 Circuit Repreentation of Aymmetrie n thi Section the circuit repreentation of the four two-port network (15a)-(15d) are provided. The two-port network correponding to (15a) i imply a pair of uncoupled impedance i.e. the nominal impedance of the load. The circuit repreentation of (15) conit in an impedance connected in parallel with the two port (ee Fig. ). The two-port network (15c) and (15d) are imilar each other. Notice that ince the two matrice in (15c) and (15d) are not ymmetrical the equivalent circuit cannot e given in term of reciprocal component only. A imple Fig.. Two-port network repreenting aymmetry in a. Fig.. Two-port network repreenting aymmetry in. + c+ δ / δ c / c Fig. 4. Two-port network repreenting aymmetry in c. n fact for the two-port network (15c) it can e eaily hown that the equivalent circuit (ee Fig. ) conit in an ideal tranformer with complex turn ratio given y and an impedance α α SBN:

4 connected in parallel to one of the two port (y reporting the impedance through the ideal tranformer it value doe not change ince the caling coefficient i ). Ι n thi work a rigorou circuit repreentation of coupling etween poitive- and negative-equence circuit due to aymmetrie in three-phae line and/or load ha een derived. Simple equivalent circuit have een otained thank to the definition of the ideal tranformer with complex turn ratio. The propoed equivalent circuit allow deeper inight into the analyi of the effect of three-phae aymmetrie. Moreover the propoed equivalent circuit i uited for implementation into the oftware for circuit imulation. Future work will e devoted to numerical quantification of aymmetrie effect and to the analyi of pecific cae of practical interet. δ a / Appendix An ideal tranformer with complex turn ratio ki defined y (paive ign convention at the two port): + δ / α (A1) (A) Notice the complex conjugate of k in(a). Such definition lead to conervation of complex power: α (A) δ c / Fig. 5. Complete two-port network repreentation of coupling etween poitive- and negative-equence circuit due to aymmetrie in impedance. For the two-port network (15d) the circuit repreentation i imilar to (15c) where the ideal tranformer ha complex turn ratio intead of and the impedance of coure i (ee Fig. 4). The complete circuit repreentation of coupling etween poitive- and negative-equence circuit i hown in Fig. 5. t i worth noticing that in many cae of practical interet only one impedance deviate ignificantly from it nominal value (e.g. in an untranpoed line one conductor i geometrically aymmetrical with repect to the other two conductor). n uch cae the circuit coupling i due only to the contriution of the aymmetrical impedance wherea the other two reult in hortcircuited two-port network in Fig Concluion Moreover the caling coefficient for reporting impedance from the econdary to the primary ide i. n fact an impedance on the econdary ide reult in a primary ide equivalent impedance (A4) Reference: [1] C. L. Fortecue "Method of ymmetrical coordinate applied to the olution of polyphae network" Tran. AEE 1918 pp [] G. C. Paap "Symmetrical component in the time domain and their application to power network calculation" EEE Tran. on Power Sytem vol. 15 no. May 000 pp [] G. Chicco P. Potolache and C. Toader "Analyi of three-phae ytem with neutral under ditorted and unalanced condition in the ymmetrical component-aed framework" EEE Tran. on Power Delivery vol. no.1 Jan. 007 pp [4] P. Paranavithana S. Perera R. Koch and. Emin "Gloal voltage unalance in M network due to line aymmetrie" EEE Tran. on Power Delivery Oct. 009 pp SBN:

5 [5]. Emin and D. S. Criford "Negative phaeequence voltage on E&W tranmiion ytem" EEE Tran. on Power Delivery vol. 1 no. July 006 pp [6] Electromagnetic compatiility (EMC) limit Aement of emiionlimit for the connection of unalanced intallation to M Hand EH power ytem EC Tech. Rep SBN: