ELE B7 Power Systems Egieerig Symmetrical Compoets
Aalysis of Ubalaced Systems Except for the balaced three-phase fault, faults result i a ubalaced system. The most commo types of faults are sigle liegroud (SLG) ad lie-lie (LL). Other types are double lie-groud (DLG), ope coductor, ad balaced three phase. The easiest method to aalyze ubalaced system operatio due to faults is through the use of symmetrical compoets Slide # 1
Symmetrical Compoets The key idea of symmetrical compoet aalysis is to decompose the ubalaced system ito three sequece of balaced etworks. The etworks are the coupled oly at the poit of the ubalace (i.e., the fault) The three sequece etworks are kow as the positive sequece (this is the oe we ve bee usig) egative sequece zero sequece Slide # 2
Symmetrical Compoets Ubalace Currets Balace Systems Sequece Currets zero sequece Usymmetrical Fault Ubalace System I C I A Symmetrical compoets Zero Sequece Positive Sequece Three balaced Systems positive sequece I B Negative Sequece egative sequece Slide # 3
Slide # 4 Assumig three ubalace voltage phasors, A, B ad C havig a positive sequece (abc). Usig symmetrical compoets it is possible to represet each phasor voltage as: C C C C B B B B A A A A Where the symmetrical compoets are: Positive Sequece Compoet Negative Sequece Compoet Zero Sequece Compoet Symmetrical Compoets
Symmetrical Compoets, The Positive Sequece Compoets ( A B C ) Three phasors Equal i magitude Displaced by 12 o i phase Havig the same sequece as the origial phasors (abc), C o 12 o o 12 12 B A,, The Negative Sequece Compoets ( ) A B C Three phasors Equal i magitude Displaced by 12 o i phase Havig the opposite sequece as the origial phasors (acb) B o 12 o o 12 12 C A,, C The zero Sequece Compoets ( A B ) Three phasors Equal i magitude Havig the same phase shift ( i phase) B C A Slide # 5
Example A Zero Sequece A B C A B C A B C A B C A B C A A C Positive Sequece C o 12 o 12 A A A B C A B Ubalace oltage Negative Sequece B o 12 o 12 C B Sythesis Usymmetrical phasors usig symmetrical compoets Slide # 6
Sequece Set Represetatio Ay arbitrary set of three phasors, say I a, I b, I c ca be represeted as a sum of the three sequece sets I I I I a a a a b b b b c c c c I I I I I I I I where a b c a b c a b c I, I, I is the zero sequece set I, I, I is the positive sequece set I, I, I is the egative sequece set Slide # 7
Coversio Sequece to Phase Oly three of the sequece values are uique, a a a I, I, I ; the others are determied as follows: a b c 2 3 3 = 112 1 I I I (sice by defiitio they are all equal) 2 b a c a b a c I I I I I I I 2 Ia I 1 1 1 1 a 1 1 I a + 2 2 I b I a 1Ia Ia 1 I a I 1 2 2 c 1 Ia Slide # 8
Coversio Sequece to Phase Defie the symmetrical compoets trasformatio matrix 1 1 1 2 A 1 2 1 The I A A AI Ia Ia I I b Ia I s I c Ia I Slide # 9
Coversio Phase to Sequece By takig the iverse we ca covert from the phase values to the sequece values s 1 I A I 1 1 1 1 1 2 with A 1 3 1 2 Sequece sets ca be used with voltages as well as with currets Slide # 1
Example If the values of the fault currets i a three phase system are: 1545 2515 13 I A I B Fid the symmetrical compoets? Solutio: I C O Slide # 11
Example If the values of the sequece voltages i a three phase system are: o 1 26 112 Fid the three phase voltages Solutio: 26 A 112 1 A 36 B 124( 26 ) 112( 112 ) 1 B 3 6 C 112( 26 ) 124( 112 ) 1 C Slide # 12
Use of Symmetrical Compoets 1. The Sequece circuits for Wye ad Delta coected loads Cosider the followig Y-coected load: I I I I a b c I Z I Z ag a y ( Z Z ) I Z I Z I ag Y a b c Z I ( Z Z ) I Z I bg a Y b c Z I Z I ( Z Z ) I cg a b Y c I a I b I c ag Zy Z Z Z Ia Z Z Z Z I bg y b I cg Z Z Zy Z c ab bc Z Y ca I I ao Z Y Z Y 3 a Z Slide # 13
Use of Symmetrical Compoets ag Zy Z Z Z Ia Z Z Z Z I bg y b I cg Z Z Zy Z c ZI, A, I AI 1 1 s s s s A ZAI A ZAI A Zy 3Z Z A Z y Z y s s Slide # 14
Networks are Now Decoupled Zy 3Z I Zy I Z y I Systems are decoupled ( y 3 ) y Z Z I Z I I ao Z I Z Y y I a Z Y I a Z Y ao Z o 3Z a Z a Z Zero Sequece Circuit Positive Sequece Circuit Negative Sequece Circuit Slide # 15
Y-coected load (Isolated Neutral): I a I b I c ab bc ca Z Y If the eutral poit of a Y-coected load is ot grouded, therefore, o zero sequece curret ca flow, ad Z Symmetrical circuits for Y-coected load with eutral poit is ot coected to groud are preseted as show: Z Y Z Y a ao Zero Sequece Circuit I a Z Y a Z Positive Sequece Circuit I a Z Y a I ao Zo Z Z Y Z Negative Sequece Circuit Slide # 16
Delta coected load: The Delta circuit ca ot provide a path through eutral. Therefore for a Delta coected load or its equivalet Y-coected ca ot cotai ay zero sequece compoets. Z ab I ab, bc Z I bc, The summatio of the lie-to-lie voltages or phase currets are always zero 1 3 ( ab bc ca ) ab ca Z ad Therefore, for a Delta-coected loads without sources or mutual couplig there will be o zero sequece currets at the lies (There are some cases where a circulatig currets may circulate iside a delta load ad ot see at the termials of the zero sequece circuit). I ao Z I a I ca / Z 3 1 3 ( I ab bc ab I ca bc I a I b I c I I a I ab ca ) Z I / Z ab 3 I bc Z Z I ca ao Zero Sequece Circuit a Positive Sequece Circuit a Negative Sequece Circuit Slide # 17
Sequece diagrams for lies Similar to what we did for loads, we ca develop sequece models for other power system devices, such as lies, trasformers ad geerators For trasmissio lies, assume we have the followig, with mutual impedaces a b I a I b Z aa Z aa Z ab a b c I c Z a Z aa c I Z Slide # 18
Sequece diagrams for lies, cot d Assume the phase relatioships are a Zs Zm ZmIa b Zm Zs Z m I b c Zm Zm Zs Ic where Z Z s m self impedace of the phase mutual impedace betwee the phases Writig i matrix form we have ZI Slide # 19
Sequece diagrams for lies, cot d Similar to what we did for the loads, we ca covert these relatioships to a sequece represetatio ZI A I AI 1 1 s s s s A Z AI A Z AI A Z A Z s 2Z m Z Z s m s Z s Z m s Slide # 2
Sequece diagrams for lies, cot d Therefore, Where, s Z Z 2Z Z Z o aa s m Z s Z m Z s Z m Z Z Z 2Z m ab Z Z Z 2Z The groud wires (above overhead TL) combied with the earth works as a eutral coductor with impedace parameters that effects the zero sequece compoets. Havig a good groudig (depeds o the soil resistively), the the voltages to the eutral ca be cosidered as the voltages to groud. a a a I ao Z o a a a a a a a I a I a Z Z a a a a Slide # 21
Sequece diagrams for geerators Key poit: geerators oly produce positive sequece voltages; therefore oly the positive sequece has a voltage source I a I ao E a Z a Z a Z go ao 3Z Durig a fault Z + Z X d. The zero sequece impedace is usually substatially smaller. The value of Z depeds o whether the geerator is grouded Slide # 22
Sequece diagrams for Trasformers The positive ad egative sequece diagrams for trasformers are similar to those for trasmissio lies. The zero sequece etwork depeds upo both how the trasformer is grouded ad its type of coectio. The easiest to uderstad is a double grouded wyewye Z + Z - Z Referece Bus Referece Bus Referece Bus Slide # 23
Trasformer Sequece Diagrams Slide # 24