Thevenin Equivalent Circuit Estimation and Application for Power System Monitoring and Protection

Transcription

1 University o Kentucky UKnowledge University o Kentucky Mster's Theses Grdute School 8 Thevenin Equivlent Circuit Estimtion nd Appliction or Power System Monitoring nd Protection Mohmmd M. tkhr University o Kentucky Click here to let us know how ccess to this document beneits you. Recommended Cittion tkhr, Mohmmd M., "Thevenin Equivlent Circuit Estimtion nd Appliction or Power System Monitoring nd Protection" (8). University o Kentucky Mster's Theses This Thesis is brought to you or ree nd open ccess by the Grdute School t UKnowledge. t hs been ccepted or inclusion in University o Kentucky Mster's Theses by n uthorized dministrtor o UKnowledge. For more inormtion, plese contct UKnowledge@lsv.uky.edu.

2 ABSTRACT OF THESS Thevenin Equivlent Circuit Estimtion nd Appliction or Power System Monitoring nd Protection The Estimtion o Thevenin Equivlent Prmeters is useul or System Monitoring nd Protection. We studied method or estimting the Thevenin equivlent circuits. We then studied two pplictions including voltge stbility nd ult loction. A study o the concepts o oltge Stbility is done in the initil prt o this thesis. A Six Bus Power System Model ws simulted using MATLAB SMULNK. Subsequently, the Thevenin Prmeters were clculted. The results were then used or two purposes, to clculte the Mximum Power tht cn be delivered nd or Fult Loction. KEYWORDS: Thevenin Equivlent Circuit, oltge Stbility, Rotor Angle Stbility, Fult Loction, Power System Monitoring Mohmmd M tkhr December 3 st 8

3 Thevenin Equivlent Circuit Estimtion nd Appliction or Power System Monitoring nd Protection By Mohmmd Museb tkhr (Director o Thesis) (Director o Grdute Studies) (Dte)

4 RULES FOR THE USE OF THESS Unpublished thesis submitted or the Mster s degree nd deposited in the University o Kentucky Librry re s rule open or inspection, but re to be used only with due regrd to the rights o the uthors. Bibliogrphicl reerences my be noted, but quottions or summries o prts my be published only with the permission o the uthor, nd with the usul scholrly cknowledgments. Extensive copying or publiction o the disserttion in whole or in prt lso requires the consent o the Den o the Grdute School o the University o Kentucky. A librry tht borrows this disserttion or use by its ptrons is expected to secure the signture o ech user. Nme Dte

5 THESS Mohmmd Museb tkhr The Grdute School University o Kentucky 9

6 Thevenin Equivlent Circuit Estimtion nd Appliction or Power System Monitoring nd Protection THESS A thesis submitted in prtil ulillment o the requirements or the degree o Mster o Science in the College o Engineering t the University o Kentucky By Mohmmd Museb tkhr Lexington, Kentucky Director: Dr. Yun Lio, Deprtment o Electricl nd Computer Engineering Lexington, Kentucky 9 Copyright Mohmmd Museb tkhr 9

7 Dedicted to My Prents, Brothers nd Sister

8 ACKNOWLEDGEMENTS would like to tke this opportunity to express my sincere thnks nd hertelt grtitude to my cdemic dvisor nd thesis chir Dr. Yun Lio or his guidnce nd support throughout my thesis. m very thnkul or his constnt encourgement during the thesis. Without him the thesis would hve never tken its present shpe. m gretly indebted or his support. My prents nd my siblings hve been gret sources o support throughout my studies. My riends hve given me lot o love without which this work would not hve been possible. lso would like to extend my thnks to Dr. Pul A Dollo nd Dr. Jimmy J Cthey or serving on my thesis committee nd providing me with invluble comments nd suggestions or improving this thesis. iii

9 Tble o Contents ACKNOWLEDGEMENTS... LST OF TABLES... LST OF FGURES.... NTRODUCTON.... BACKGROUND.... PURPOSE OF THE THESS....3 OEREW OF SYSTEM STABLTY....4 EQUATON OF MOTON OF A ROTATNG MACHNE STEADY STATE STABLTY METHODS OF MPRONG STEADY STATE STABLTY LMT TRANSENT STABLTY LMT EQUAL AREA CRTERON FACTORS AFFECTNG TRANSENT STABLTY POWER SYSTEM OLTAGE STABLTY ANALYSS.... DEFNTON AND CLASSFCATON OF POWER SYSTEM STABLTY.... CLASSFCATON OF POWER SYSTEM STABLTY....3 OLTAGE STABLTY....4 OLTAGE STABLTY ANALYSS....5 P- CURES Q CHARACTERSTCS SOME SGNFCANT RESULTS AND CRTERA N OLTAGE STABLTY POWER SYSTEM MODELNG AND THEENN EQUALENT CRCUT PARAMETERS ESTMATON TRANSMSSON LNE DATA GENERATOR DATA LOAD DATA ALGORTHM FOR THEENN EQUALENT CRCUT ESTMATON EQUATON FOR MAXMUM POWER DELERED OLTAGE AND CURRENT WAEFORMS AT LOAD BUS L WAEFORMS FOR OLTAGE AND CURRENT AT THE LOAD BUS L FAULT ANALYSS AND ESTMATON OF FAULT LOCATON UNSYMMETRCAL FAULTS SYMMETRCAL COMPONENT ANALYSS OF UNSYMMETRCAL FAULTS ANALYSS OF SNGLE LNE TO GROUND FAULT ANALYSS OF LNE TO LNE FAULT DOUBLE LNE TO GROUND FAULT ANALYSS FAULT LOCATON ALGORTHM... 5 iv

10 4.7 MPEDANCE BASED ALGORTHM 4.8 OLTAGE AND CURRENT WAEFORMS FOR DFFERENT FAULT LOCATONS CONCLUSON BBLOGRAPHY... 6 TA... 6 v

11 List o Tbles TABLE 3. GENERATORS BLOCK PARAMETER ALUES... TABLE 3. LOAD BLOCK PARAMETER ALUES... 3 TABLE 3.3 OLTAGE AND CURRENTS AT BUS L TABLE 3.4 OLTAGE AND CURRENT AT LOAD BUS L TABLE 3.5 THEENN PARAMETERS FOR 4, 5 AND 6 SETS OF MEASUREMENTS AT LOAD BUS L TABLE 3.6 THEENN PARAMETERS FOR 4, 5 AND 6 SETS OF MEASUREMENTS AT LOAD BUS L TABLE 3.7 POWER DELERED AT THE BUS L3 FOR DFFERENT POWER FACTOR ANGLES. 8 TABLE 3.8 POWER DELERED AT THE BUS L5 FOR DFFERENT POWER FACTOR ANGLES. 8 TABLE 4. FAULT LOCATON ESTMATON vi

12 List o Figures FGURE. MACHNE CONNECTED TO NFNTE BUS FGURE. POWER ANGLE CURE FGURE.3 CURE SHOWNG THE EQUAL AREA CRTERON FGURE. SMPLE RADAL SYSTEM FOR OLTAGE STABLTY ANALYSS FGURE. REACTE END OLTAGES, POWER AND CURRENT AS A FUNCTON OF LOAD FGURE.4 POWER OLTAGE CHARACTERSTCS FOR DFFERENT LOAD POWER FACTORS 7 FGURE.5 SMPLE RADAL TWO BUS SYSTEM... 8 FGURE.6 -Q CHARACTERSTCS OF THE SYSTEM N FGURE FGURE 3. SX BUS POWER SYSTEM MODEL... FGURE 3. THEENN EQUALENT CRCUT... 4 DEMAND FGURE.3 POWER OLTAGE CHARACTERSTCS FOR THE SYSTEM OF FGURE. FGURE 3.3 EQUALENT POWER SYSTEM MODEL FOR CALCULATNG MAXMUM POWER DELERED... 8 FGURE 3.4(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO DEGREES FGURE 3.4(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO DEGREES FGURE 3.5(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO DEGREES... 3 FGURE 3.5(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO DEGREES... 3 FGURE 3.6(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO 3 DEGREES... 3 FGURE 3.6(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO 3 DEGREES... 3 FGURE 3.7(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO 4 DEGREES... 3 FGURE 3.7(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO 4 DEGREES... 3 FGURE 3.8(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO 6 DEGREES FGURE 3.8(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO 6 DEGREES FGURE 3.9(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO DEGREES FGURE 3.9(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR ANGLES SET TO DEGREES FGURE 3.(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO DEGREES FGURE 3.(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO DEGREES vii

13 FGURE 3.(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO DEGREES FGURE 3.(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO DEGREES FGURE 3.(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO DEGREES FGURE 3.(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO DEGREES FGURE 3.3(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO 3 DEGREES FGURE 3.3(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO 3 DEGREES FGURE 3.4(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO 4 DEGREES FGURE 3.4(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO 4 DEGREES FGURE 3.5(A) OLTAGE SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO 6 DEGREES... 4 FGURE 3.5(B) CURRENT SGNALS FOR THE CASE WTH GENERATOR 3 ANGLES SET TO 6 DEGREES... 4 FGURE 4. A GENERAL POWER NETWORK... 4 FGURE 4. (A) POSTE SEQUENCE NETWORK AS SEEN FROM THE FAULT PONT... 4 FGURE 4. (B) NEGATE SEQUENCE NETWORK AS SEEN FROM THE FAULT PONT FGURE 4.(C) ERO SEQUENCE NETWORK AS SEEN FROM THE FAULT PONT FGURE 4. (D) THEENN EQUALENT OF POSTE SEQUENCE NETWORK AS SEEN FROM F FGURE 4. (E) THEENN EQUALENT OF NEGATE SEQUENCE NETWORK AS SEEN FROM F FGURE 4. (F) THEENN EQUALENT OF ERO SEQUENCE NETWORK AS SEEN FROM F. 44 FGURE 4.3(A) SNGLE LNE TO GROUND FAULT AT F FGURE 4.3(B) CONNECTON OF SEQUENCE NETWORKS FOR SNGLE LNE TO GROUND FAULT FGURE 4.4(A) LNE TO LNE FAULT THROUGH MPEDANCE FGURE 4.4(B) POSTE AND NEGATE SEQUENCE CONNECTONS FOR A LNE TO LNE FAULT FGURE 4.4(C) THEENN EQUALENT FOR CONNECTON OF SEQUENCE NETWORKS FOR L- L FAULT... 5 FGURE 4.5(A) DOUBLE LNE TO GROUND FAULT THROUGH MPEDANCE... 5 FGURE 4.5(B) CONNECTON OF SEQUENCE NETWORKS FOR A DOUBLE LNE TO GROUND FAULT... 5 FGURE 4.5(C) THEENN EQUALENT FOR THE SEQUENCE NETWORK CONNECTONS FOR A LLG FAULT... 5 FGURE 4.6 TRANSMSSON LNE CONSDERED FOR THE ALGORTHM [5] FGURE 4.7 NEGATE SEQUENCE NETWORK DURNG THE FAULT NEGLECTNG SHUNT CAPACTANCE [5] viii

14 FGURE 4.8(A) OLTAGE WAEFORMS FOR A PHASE A TO GROUND FAULT WTH A FAULT LOCATON OF. P.U FGURE 4.8(B) CURRENT WAEFORMS FOR A PHASE A TO GROUND FAULT WTH A FAULT LOCATON OF. P.U FGURE 4.9(A) OLTAGE WAEFORMS FOR A PHASE A TO GROUND FAULT WTH A FAULT LOCATON OF.4 P.U FGURE 4.9(B) CURRENT WAEFORMS FOR A PHASE A TO GROUND FAULT WTH A FAULT LOCATON OF.4 P.U FGURE 4.(A) OLTAGE WAEFORMS FOR A PHASE A TO GROUND FAULT WTH A FAULT LOCATON OF.6 P.U FGURE 4.(B) CURRENT WAEFORMS FOR A PHASE A TO GROUND FAULT WTH A FAULT LOCATON OF.6 P.U FGURE 4.(A) OLTAGE WAEFORMS FOR A PHASE A TO GROUND FAULT WTH A FAULT LOCATON OF.7 P.U FGURE 4.(B) CURRENT WAEFORMS FOR A PHASE A TO GROUND FAULT WTH A FAULT LOCATON OF.7 P.U ix

15 . ntroduction. Bckground The pressure on the power trnsmission network hs been incresing in recent times. Some o the resons or this which hve been mentioned in [] re A deregulted energy mrket. Environmentl constrints. Limited investment in trnsmission system reinorcement. An incresed competition in order to yield greter outputs. Hence, the Power System is orced to operte closer to the stbility limit. A Mjor problem rising out o this is voltge instbility or collpse, which cuses stedy stte security problem. When the loding o Power system pproches the mximum permissible loding, t some locl bus in the power trnsmission network, the mgnitude o the voltge tends to decrese. But only by knowing the voltge mgnitude o locl buses, we cnnot exctly ssess the impending voltge collpse. The voltge mgnitude decreses becuse o indequte locl rective power support to meet locl demnd nd losses. Lrge mounts o rective power rom other buses in the network will deteriorte the voltge proile which my led to voltge collpse. n recent yers, voltge instbility hs been responsible or mjor blckouts. The ollowing re some exmples [7]: North Est blckout, August 4, 3. Texs blckout, September 3, 8. New York Power Pool Disturbnces o September, 97 Florid System Disturbnce o December 8, 98 French System Disturbnces o December 9, 978 nd Jnury, 987 Northern Belgium System Disturbnce o August 4, 98 Swedish System Disturbnce o December 7, 983 Jpnese System Disturbnce o July 3, 987. Thus voltge stbility studies hve become o more importnce thn ever.

16 Another importnt thing to consider in this thesis is ult loction. Fult loction studies re very importnt or the trnsient stbility limit o the system. The incresed complexities o modern power systems hve rised the importnce o ult loction reserch studies []. Accurte nd st ult loction helps in reducing the mintennce nd restortion times, reduce the outge times nd thus improve the power system relibility [].. Purpose o the Thesis To operte the power system with n dequte security mrgin, it is essentil to estimte the mximum permissible loding [].The mximum power tht cn be trnserred to the lod bus in power system cn be eectively studied by estimting the Thevenin equivlent circuit o the power system Model. Thus, in one prt o my Thesis, will be clculting the Thevenin prmeters o six bus power system model. This would provide me with considerble results to clculte the mximum power tht cn be delivered. The Thevenin equivlent circuit prmeters re useul in the pplictions or power system monitoring nd protection. The Thevenin prmeters tht obtin in the irst prt re used or ult loction bsed on voltge mesurements. The ult loction lgorithm is tken rom [5], which re described in detil in Chpter 4..3 Overview o System Stbility The stbility o system o interconnected dynmic components is its bility to return to norml or stble opertion ter hving been subjected to some orm o disturbnce [8]. n power system, we typiclly del with two orms o instbility: The loss o synchronism between synchronous mchines nd voltge instbility. Synchronous stbility cn be clssiied s stedy stte nd trnsient stbility nd re studied in this chpter. The voltge stbility is studied in the next chpter. The equtions nd igures in the subsequent sections hve minly been obtined rom [4] nd [8]. As deined in [8], stedy stte stbility is the bility o the power system, when operting under given lod conditions, to retin synchronism when subjected to smll disturbnces

17 such s the continul chnges in lod or genertion nd the switching out o lines. This is lso known s dynmic stbility. Trnsient stbility dels with sudden nd lrge chnges in the system. One exmple is ults in Power system. During ult conditions, the stbility limit is less thn the stedy stte condition. Beore we mke detiled study o stedy stte nd trnsient stbility, it is importnt to study the eqution o motion o rotting mchine..4 Eqution o Motion o Rotting Mchine n this section, we will be studying the Eqution o Motion o Rotting Mchine nd deriving the swing eqution. The equtions in this section hve ll been obtined rom [8]. Let the moment o inerti o the rotor be nd the ngulr ccelertion isα. net torque pplied on the rotor. ω is the synchronous speed o the rotor (rdins/second). T is the The kinetic energy bsorbed by the rotor is given by ω Joules. The ngulr momentum is M = ω Joules-Seconds per rdin. An inerti constnt, H cn be deined s the stored energy t synchronous speed per volt-mpere o the rting o the mchine [8]. As we know tht the unit o energy used in power systems nlysis is Kilojoules or Meg joules nd i we consider the rting o the mchine to be G Meg- olt-amperes, then by multiplying G with the inerti constnt we get the kinetic energy o the mchine. GH = ω = Mω is the Kinetic Energy or the stored Energy. (.) ω = 36 Electricl Degrees per second where is the system requency in Hz. (.) Substituting (.) in (.) GH = M (36) (.3) M = GH / 8 Meg joule-seconds/electricl degree (.4) T = Mechnicl Torque nput- Electricl Torque Output d δ = (.5) dt 3

18 ( T ) d δ T ω ω = = (.6) dt xω / Here P. ω = (.7) xk. E. P = P mech P (.8) electricl By using Eqution (.) in (.7), we cn write d dt δ P = M (.9) There is n increse in the vlue o δ when there is negtive chnge in the Power output in Eqution (.9). Electricl Power output. An increse in input is ssumed to be constnt. d dt δ P = P mech P is sometimes considered s the chnge in electricl Pelectricl will increse the vlue oδ. The Power P d = or M δ + P = (.) M dt Eqution (.) is known s Swing Eqution. Now tht we hve studied bout the Eqution o Motion o Rotting Mchine, we cn nlyze Stedy Stte nd Trnsient Stbility o System bsed on this..5 Stedy Stte Stbility n this section we will be studying the stedy stte nlysis or power system. The equtions in this section hve been obtined rom [4]. The stedy stte stbility limit o prticulr circuit o power system is deined s the mximum power tht cn be trnsmitted to the receiving end without loss o synchronism [4].Figure.[4] represents simple system or the purpose o nlysis. The dynmics o this system re described by the equtions (.) to (.3) d M dt δ = P m P e (.) H M = in the Per Unit System (.) π 4

19 E P e = sinδ = Pmx sinδ (.3) X d X d is the direct xis rectnce. The plot or eqution (.3) lso known s the power ngle curve is represented in Figure.[4] X ' d X e ninite Bus P e ' E + δ Figure. Mchine Connected to ninite Bus The system hs stedy power trnser P eo = Pm nd the torque ngle is o δ s shown in Figure.. For smll increment constnt, the torque ngle chnges to ( δ + δ ) Q, δ ) [4], we get ( P o eo o P in the electric power with the input P m being o. Linerizing bout the operting point Pe P e = δ (.4) δ Rewriting Eqution (.) in the current nlysis, d δ M = P dt m ( Peo + Pe ) = Pe Using Eqution (.4) in (.5) (.5) d δ Pe M + = δ (.6) dt δ P Mk + e δ = (.7) δ 5

20 Where k = d dt The stbility o the system or smll chnges is determined by the chrcteristic eqution P [4] Mk + e = (.8) δ The roots o Eqution (.8) re given by k = ± ( P / δ ) e M (.9) P e P + eo P e P mx Genertor Q o P eo 8 9 δ o 9 δ + δ o 8 δ Motor Figure. Power Angle Curve Now the system behvior depends on the vlue o ( P e / δ ). ( P e / δ ) is positive, the roots re imginry nd conjugte. The system behvior is oscilltory boutδ o. But in our nlysis the mchine dmper windings line resistnce hd 6

21 been neglected. These cuse the system oscilltions to decy nd hence the system is stble or smll increment in power. ( P e / δ ) is negtive, the roots re rel. One is positive nd the other is negtive. Though, they re equl in mgnitude. For smll increment in power, the system is unstble s the synchronism is lost due to increse in torque ngle with increse in power. From Eqution (.3) ssuming E, to remin constnt, the system is unstble iδ > 9. (.) o The mximum power trnser without loss o stbility occurs or The mximum power trnserred is thereore given by δ o = 9 (.) E P mx = (.) X But in the nlysis we hd ssumed tht the internl mchine voltge remins constnt. n such cse s the loding is incresed, the terminl voltge dips hevily which is prcticlly not cceptble. n prctice, we must consider the stedy stte stbility limit by ssuming tht the excittion is djusted or every lod increse to keep the terminl voltge constnt. The eects o governor nd excittion control were not considered in the nlysis. Stedy stte stbility limit is very importnt s it should be tken cre tht system cn operte bove trnsient stbility but not bove stedy stte stbility limit. The trnsient stbility limit cn be mde to closely pproch the stedy stte limit currently with incresed speeds in ult clering..6 Methods o mproving Stedy Stte stbility Limit The ollowing methods cn be used depending on the conditions in order to improve the stedy stte stbility limit [4] From Eqution (.), we cn sy tht the stedy stte stbility limit cn be improved by reducing X or by incresing either E or or both. For trnsmission lines o high rectnce, the stbility limit cn be incresed by using two prllel lines. 7

22 Use o series cpcitors in the lines to get better voltge regultion rises the stbility limits by decresing the line rectnce. Employing quick excittion systems nd higher excittion voltges..7 Trnsient Stbility Limit Trnsient stbility limit is the mximum possible power tht cn be trnsmitted through point in the system when the system is operting with stbility during trnsient disturbnces [5]. The type o disturbnce nd the durtion o disturbnce ect the trnsient stbility limit. The durtion o ult determines the mount o power tht cn be trnsmitted rom one mchine to nother mchine in two mchine system without loss o synchronism. The power limit is determined using the Equl Are Criterion. This is studied in section.8..8 Equl Are Criterion As we hd considered one inite mchine system or nlysis or stedy stte stbility, we will study the Equl Are Criterion or one inite mchine swinging with n ininite bus in this section. The equtions in this section hve been minly obtined rom [5]. The detiled study o Equl Are Criterion or system with two inite mchines swinging with respect to ech other is discussed in [4] nd [5]. The Swing Eqution o inite mchine swinging with respect to n ininite mchine is d δ given by M = Pm Pe = P dt Multiplying both sides o Eqution (.3) by d δ / dt nd rerrnging, we get (.3) d δ dδ P dδ = (.4) dt dt M dt d dδ P d = δ (.5) dt dt M dt Upon integrting Eqution (.5) with respect to time, we get 8

23 dδ dt = M δ δ o P dδ (.6) dδ = ω = dt M δ δ o P dδ (.7) ω = when the mchine comes to rest with respect to the ininite mchine. The condition required or the stbility o single mchine system connected to ininite bus is δ δ o P dδ = (.8) The integrl in Eqution (.8) cn be represented s the re between the curve P m versus δ nd the curve P e versusδ. This is shown in Figure.3[5]. For the re to be ero, A P > P ) = A ( P < P ) [9]. Hence this method is clled Equl Are criterion. ( m e m e P e P m A A P δ o δ δ Figure.3 Curve Showing the Equl Are criterion.9 Fctors Aecting Trnsient Stbility The ctors ecting trnsient stbility limit mentioned in [5] re s ollows: nerti Constnt Type o Disturbnce Fult clering time 9

24 Loction o the ult nitil operting Condition o the system The wy in which the ult is clered. Thus, in this chpter we hve presented the purpose o the thesis reserch tht hs been crried out nd n overview o the bsic concepts relted to the re o my reserch. An dvnced study o these concepts cn be mde through the reerences tht hve been mentioned.

25 . Power System oltge Stbility Anlysis n chpter, gve n overview bout the importnce o voltge stbility studies s voltge instbility or voltge collpse my led to blckout. n this chpter we will be mking detiled study o the relevnt concepts tht could help us mke the understnding o voltge stbility better. Beore we study voltge stbility in prticulr, we deine nd clssiy power system stbility in generl. This is studied in the initil sections o this chpter. n the lter sections we discuss voltge stbility. These include the deinitions, concepts o mthemticl ormultion o the voltge stbility problems nd some signiicnt criteri o voltge stbility studies. The equtions nd igures hve been minly obtined rom [] nd [4].. Deinition nd Clssiiction o Power System stbility n brod terminology, power system stbility my be deined s tht property o the power system tht enbles it to remin in stte o operting equilibrium under norml operting conditions nd to regin n cceptble stte o equilibrium ter being subjected to disturbnce []. The deinitions o power system stbility though hve not been precise nd do not include ll prcticl instbility scenrios [3]. A proposl is presented in [3] which ttempts to deine power system stbility more precisely which includes ll orms o system instbility. Power system stbility is the bility o n electric power system, or given initil operting condition, to regin stte o operting equilibrium ter being subjected to physicl disturbnce with most system vribles bounded so tht prcticlly the entire system remins intct [3].. Clssiiction o Power System Stbility Power system stbility is clssiied bsed on the ollowing considertions [] The physicl nture o the instbility The size o the disturbnce The time spn

26 Bsed on the physicl nture o the instbility, it cn be clssiied s rotor ngle stbility nd voltge stbility. Bsed on the size o the disturbnce, it is clssiied s lrge disturbnce nd smll disturbnce stbility. Bsed on the time spn, it cn be clssiied s long term nd short term stbility..3 oltge Stbility oltge stbility is the bility o power system to mintin stedy cceptble voltges t ll buses in the system under norml operting conditions nd ter being subjected to disturbnce []. There is voltge instbility when there is voltge drop in the system or t bus due to severl resons which include generl disturbnce, chnge in system condition or due to luctuting lods. But the min reson or voltge instbility is the indequcy o rective power demnds to be met by the system. Rective power is injected into the system to meet the incresing demnds. For voltge stble system, s the rective power is injected into the buses in the system, the voltge mgnitude should increse. But, i the voltge mgnitude decreses even t one bus in the system or increse in rective power, the System is sid to be voltge unstble. oltge instbility is locl phenomenon but its consequences my hve widespred impct []. oltge instbility leds to low voltge proile in the system nd it cn hve cumultive eect ultimtely leding to voltge collpse..4 oltge Stbility Anlysis n this section we study voltge study nlysis or simple two terminl network. Figure.[] represents simple rdil system. This igure nd the equtions in this section re tken rom [].

27 LN θ ~ ~ R P R + jq R ~ E s LD φ Figure. Simple rdil System or oltge Stbility Anlysis E s is the oltge Source LD is the lod impednce LN is the series impednce o the system. The Current ~ cn be expressed s ~ LN ~ s = (.) ~ ~ E + LD ~ nd ~ E re the phsors o current nd source voltge. s The series impednce nd the lod impednce phsors cn be expressed s ~ = LN θ nd ~ = LD φ respectively. (.) LN LD The mgnitude o the current cn then be expressed s s = (.3) ( cosθ + cosφ) + ( sinθ + sinφ) LN LD Simpliying Eqution (.3) E LN LD 3

28 E s = (.4) ' LN Where ' LD LD = + + cos( θ φ) (.5) LN LN The Mgnitude o oltge t the receiving end cn be expressed s R = LD (.6) Substituting the vlue o current rom Eq. (.4) into Eq. (.6) = (.7) ' LD Es LN Now, clculting the power delivered t is given by P = R R cosφ (.8) = LD ' E s LN cosφ Figure.[] shows the plots o, R nd (.9) P R s unction o LN / LD. P R increses rpidly s the lod demnd i.e. LN / LD is incresed. This is done by decresing LD. P R reches mximum vlue nd then begins to decrese. The mximum vlue o P R indictes the mximum vlue o ctive power tht cn be trnsmitted through n impednce rom constnt voltge source. This power trnsmitted is mximum when the voltge drop in the line is equl in mgnitude to R tht is when LN / LD = []. With grdul decrese in LD there is n increse in nd decrese in R. The reson or rpid increse in P R initilly is due to the dominnt increse o in comprison to the decrese in R t high vlues o LD. As LD pproches LN this eect is not so dominnt nd hence there is grdul chnge rther thn shrp increse nd decrese in the vlues o nd R respectively. As domintes the increse in nd hence there is decrese in P R. LD goes below LN, the decrese in R 4

29 / sc P R / P Mx Norml Opertion Criticl lue R / E S Abnorml opertion LN / LD Figure. Rective End oltges, Power nd Current s Function o Lod Demnd The norml opertion tkes plce till the criticl vlue is reched. This corresponds to the mximum power in the Figure.. As the lod demnd increses i.e. s / is incresing the control o power would be unstble. This mens tht s the lod impednce LD is decresed the power is lso reduced. The lod chrcteristics determine i the voltge collpse tkes plce or not. For constnt impednce sttic lod chrcteristics, the system stbilizes t power nd voltge levels lower thn the desired vlues where s or constnt power lod chrcteristic the system becomes unstble through collpse o lod bus voltge []. Thus it is importnt to nlyze the reltionship between P R nd R or the purpose o voltge stbility studies. LN LD 5

30 .5 P- Curves The reltionship between P R nd R is shown in Figure.3 [] or prticulr power ctor vlue. But the voltge drop in the trnsmission lines is unction o both the ctive nd the rective power trnser s seen in Equtions (.7) nd (.9). Thus the lod power ctor ects the power voltge chrcteristics o the system Figure.4[] represents the curves or P R nd R or dierent lod power ctor vlues. R / E S Criticl oltge Figure.3 Power oltge Chrcteristics or the System o Figure. P R / P RMAX The dotted lines represent the locus o criticl operting points. This mens tht operting points bove the criticl vlues represent stisctory opertion. A sudden reduction in power ctor, which cuses n increse in the rective power delivered, cn cuse the system to chnge rom stble operting condition to n unstble condition s shown in the lower prt o the curves in Figure.3 nd Figure.4. 6

31 / E R S.9 Lg.95 Lg..95 led.9 Led Locus o Criticl Points P R / P RMx Figure.4 Power oltge chrcteristics or Dierent Lod Power Fctors.6 -Q Chrcteristics For purpose o nlysis let us consider simple rdil system s shown in Figure.5[4]. The system lod end voltge cn be expressed in terms o ( QX E ) 4X ( P + ) / P, Q s given in [4] QX + E = ± Q (.) For Rective Power Flow i.e. φ 9 X >> R E Q = cosδ [4] (.) X X E cosδ + QX = [4] (.) dq E cosδ = [4] (.3) d X 7

32 AC E R + jx P + jq LOAD Figure.5 Simple Rdil Two Bus System Q R / P RMAX..9 Locus o criticl Points Figure.6 -Q Chrcteristics o the system in Figure. R / E S Figure.6 represents the Q chrcteristics with dierent P R / PRMx rtios o R R simple two-terminl system. The system is voltge stble in the region where dq / d is positive [4]. The locus o criticl operting points is shown in the Figure.6 with dotted 8

33 lines. The criticl operting point is where the voltge stbility limit is reched i.e. dq / d =. At voltge stbility limit, the limiting rective power is given by Q lim = cos δ [4] (.4) X The prts o the curve to the right o the minim represent stble opertion. There is unstble opertion when dq / d <. But stble opertion in the region when dq / d < is cn lso be done. This is by using regulted rective power compenstion hving suicient control rnge nd high Q / gin with n opposite polrity []. The nlysis we hve presented in this section is limited to rdil systems in order to mke the understnding o dierent power system stbility concepts much esy. But in complex power systems there re mny ctors tht contribute to the system instbility nd cn be studied t higher level..7 Some Signiicnt Results nd Criteri in oltge Stbility n this section we present summry o certin other signiicnt criteri nd results in voltge stbility studies which hve not been mentioned in the previous sections. These concepts hve been minly obtined rom [4]. S oltge stbility limit is reched when = [4]. (.4) * Y S is the complex power t lod bus, Y LL is the lod bus dmittnce nd is the voltge t the lod bus. The limit o mximum loding o trnsmission line cn be given by [4] S = cri / X cri (.5) LL Where X cri is the criticl rectnce o the system ter which voltge instbility E occurs. t is expressed s [4] X cri = ( tnφ + secφ) (.6) P de Criterion [4]. The voltge stbility limit is reched when d dq dp E cos δ + + sinδ = (.7) d X d X 9

34 d d Criterion [4]. The vlue o criticl impednce beyond which voltge instbility occurs cn ound rom this criteri. oltge instbility occurs when d d = n this chpter we hve tried to present simple nd cler understnding o the voltge stbility concepts. An dvnced level o nlyticl methods or voltge stbility nd rotor ngle stbility, which oten go hnd in hnd, hs been described in [].

35 3. Power System Modeling nd Thevenin Equivlent Circuit Prmeters Estimtion n this Chpter, we will be estimting the Thevenin prmeters or the power system model tht hs been built in MATLAB nd SimPower system. The six bus power system model is represented in Figure 3.[4]. The vrious SMULNK blocks tht hve been used in the model hve been studied in the initil sections o this Chpter. The block prmeters hve been represented in tbles or ech block. The power system model is simulted nd the results re tbulted which re used or clculting the Thevenin prmeters or the power system model built or this thesis reserch. The mximum power trnserred is lso clculted. The simultion results re lso represented in terms o voltge nd current signl wveorms t the end o this chpter G3 L4 L5 L3 6 G G L L6 L Figure 3. Six Bus Power System Model 3. Trnsmission Line Dt The trnsmission lines in the power system model hve been represented by the three phse mutul inductnce block in the SimPower system. t is to be noted tht the block prmeters in this SimPower system block re resistnce nd inductnce wheres the dt used hs resistnce nd rectnce vlues. So, by using, X = πl (3.)

36 For system requency o 6 Hz nd individul rectnce vlues s mentioned in the tble, the corresponding inductnce vlues hve been clculted. 3. Genertor Dt The genertors hve been represented by three phse source block o SimPower system which is three phse voltge source in series with RL brnch. The block prmeters o this block re: () Phse-to-Phse rms voltge () (b) Phse Angle o Phse A ( degrees) (c) Frequency ( Hz) (d) Source Resistnce (Ohms ) nd Source nductnce ( Henry) The lues or the bove prmeters hve been set s show in Tble 3.[]. Tble 3. Genertors Block Prmeter lues Prmeter lue Phse-to-Phse rms oltge (v) 3 Phse Angle o Phse A ( degrees) ried rom to 6 or G ; or G nd 3 or G3 Frequency(Hz) 6 Source Resistnce (Ohms ) nd Source nductnce ( Henry). (Ohms) nd./377(henry) Respectively. The phse ngle t Genertor is vried rom to 6 degrees nd the three phse current nd voltge wveorms hve been plotted, which is shown lter in this chpter. The three voltge sources re connected in Wye with neutrl connection tht hs been internlly grounded s obtined rom []. The source impednce vlues o the genertors re represented in the tble 3.. Eqution 3. is used to convert the rectnce vlues into the inductnce nd subsequently entered in the prmeter block o the SimPower system model.

37 3.3 Lod Dt A three-phse series RLC lod block rom the SimPower system hs been used. RLC elements re combined in series to implement the three phse blnced lod []. The prmeters in this block hve been set s shown in Tble 3. Tble 3. Lod Block Prmeter lues Prmeter lue Nominl Phse to Phse oltge rms () Nominl Frequency (Hz) 6 Active Power P (W).5 nductive Rective Power(r). Cpcitive Rective Power(r) The conigurtion o the block is set to Wye (loting). Thus, the connection o the three phses is in Wye with neutrl inccessible. 3.4 Algorithm or Thevenin Equivlent Circuit Estimtion The six bus power system model in Figure 3. cn be represented by its Thevenin equivlent circuit s reerred rom lod bus L3 or L5 s shown in Figure 3.. The Thevenin equivlent prmeters cn be obtined by writing the network equtions s seen rom lod bus L3 or L5. E = (3.) S S K k Eqution (3.) represents the Network Eqution or Figure 3.. E S is the Source oltge, S is the Source mpednce. 3

38 E S - + S k k L Figure 3. Thevenin Equivlent Circuit At bus k, the oltge nd Current is represented by k nd k respectively. Eqution (3.) cn be written or vrious vlues o currents nd voltges t the bus k. This is done by vrying the phse ngle t one o the genertors. n the power system model shown in Figure 3., the genertor phse ngles t G re vried irst in order to get dierent vlues o currents nd voltges t the bus k. So, Eqution (3.) cn be urther written s E = (3.3) S S S S K k E = (3.4) K k For n vlues o genertor ngles, the Eqution (3.) cn give n vlues o currents nd voltges t the bus k. Expressing it in the orm AX = B, k k k. E s.. =. s... kn kn (3.5) 4

39 A is n (nx) Mtrix. X is ( x ) Mtrix nd B is n nx Mtrix. The solution or Eqution (3.5) is obtined using the Lest Men Squres technique. X = ( A T A) ( A T B) Thus the Thevenin equivlent prmeters re clculted. n the power system model represented in Figure 3., the phse ngles t G re vried rom to 6 degrees to get the Thevenin equivlent voltge nd current t the lod, L3. The results re tbulted s shown in Tble 3.3 Tble 3.3 oltge nd Currents t Bus L3 Angle t oltge t bus 3 Current t Active Rective Genertor (p.u.) bus 3 (p.u.) Power P ( Power Q ( ( Mgnitude(ngle p.u.) p.u. ) Degrees) in degree).5 (.6).97() (3.).836(3.5) (3.63).76(3.5) (4.7).567 (.3) (4.7).444 (4.8) (5.75).38 (-3) At lod bus L5, the currents nd voltges re tbulted by vrying the phse ngle o the genertor G. This is represented in Tble 3.4 5

40 Tble 3.4oltge nd Current t Lod Bus L5 Angle t oltge t bus 5 Current t bus Active Rective Genertor (p.u.) 5 (p.u.) Power P ( Power Q ( ( Mgnitude(ngle p.u.) p.u. ) Degrees) in degree).3(-5.4).963(.45) (-.684).588(3.34) (.47).67(-6.839) (5.76).56(- 5.) 4.94(9.44).4544(- 35.) (6.6).9(-34.3) Using these vlues o voltge nd current t the buses, the Thevenin prmeters hve been clculted. The Thevenin prmeters or dierent sets o mesurements re tbulted in Tbles 3.5 nd 3.6 Tble 3.5Thevenin Prmeters or 4, 5 nd 6 sets o mesurements t Lod bus L3 Number o Sets o Mesurements E S S E SR E SM R S X S

41 Tble 3.6Thevenin Prmeters or 4, 5 nd 6 sets o mesurements t Lod bus L5 Number o Sets o Mesurements E S S E SR E SM R S X S E is the Source oltge. is the Source mpednce. S S E SR nd E SM represent the Thevenin oltge rel nd imginry prts respectively. R S nd X S represent the Thevenin Resistnce nd Rectnce respectively. 3.5 Eqution or Mximum Power Delivered From circuit theory, mximum power is delivered to lod when the source impednce vlue is equl to the lod impednce [9]. L = S, where L is the impednce t the lod nd S is the source impednce. Suppose lod power ctor ngle isα. Then, α. L = L From the Network in Figure 3.3[9], we cn write = S E s + L The Power delivered is given by S = P + jq = (. ). L * = L = L s E s + L The power delivered t the lod buses L3 nd L5 or the system which we modeled is shown in Tbles 3.7 nd 3.8 7

42 E S - + S k P Q k L = S Figure 3.3 Equivlent Power System Model or Clculting Mximum Power Delivered Tble 3.7 Power Delivered t the Bus L3 or Dierent Power Fctor ngles Angle P (p.u) Q (p.u) Tble 3.8 Power Delivered t the Bus L5 or Dierent Power Fctor ngles Angle P (p.u) Q (p.u)

43 3.6 oltge nd Current Wveorms t Lod Bus L3 The voltge nd current wveorms re plotted by vrying the phse ngles t G nd G3. This is represented in the igures in this section.5 A B c.5 voltge smples Figure 3.4() oltge signls or the cse with genertor ngles set to degrees..8.6 A B C.4. current smples Figure 3.4(b) Current signls or the cse with genertor ngles set to degrees. 9

44 ..5 A B c.5 voltge smples Figure 3.5() oltge signls or the cse with genertor ngles set to degrees.8.6 A B C.4. current smples Figure 3.5(b) Current signls or the cse with genertor ngles set to degrees 3

45 .5 A B c.5 voltge smples Figure 3.6() oltge signls or the cse with genertor ngles set to 3 degrees.6.4 A B C. current smples Figure 3.6(b) Current signls or the cse with genertor ngles set to 3 degrees 3

46 .5 A B c.5 voltge smples Figure 3.7() oltge signls or the cse with genertor ngles set to 4 degrees A B C.. current smples Figure 3.7(b) Current signls or the cse with genertor ngles set to 4 degrees 3

47 .5 A B c.5 voltge smples Figure 3.8() oltge signls or the cse with genertor ngles set to 6 degrees.4.3 A B C.. current smples Figure 3.8(b) Current signls or the cse with genertor ngles set to 6 degrees 33

48 .5 A B c.5 voltge smples Figure 3.9() oltge signls or the cse with genertor ngles set to degrees.8.6 A B C.4. current smples Figure 3.9(b) Current signls or the cse with genertor ngles set to degrees 34

49 3.7 Wveorms or oltge nd Current t the Lod bus L5.5 A B c.5 voltge smples Figure 3.() oltge signls or the cse with genertor 3 ngles set to degrees.8.6 A B C.4. current smples Figure 3.(b) Current signls or the cse with genertor 3 ngles set to degrees 35

50 .5 A B c.5 voltge smples Figure 3.() oltge signls or the cse with genertor 3 ngles set to degrees.6.4 A B C. current smples Figure 3.(b) Current signls or the cse with genertor 3 ngles set to degrees 36

51 .5 A B c.5 voltge smples Figure 3.() oltge signls or the cse with genertor 3 ngles set to degrees.6.4 A B C. current smples Figure 3.(b) Current signls or the cse with genertor 3 ngles set to degrees 37

52 .5 A B c.5 voltge smples Figure 3.3() oltge signls or the cse with genertor 3 ngles set to 3 degrees..5 A B C..5 current smples Figure 3.3(b) Current signls or the cse with genertor 3 ngles set to 3 degrees 38

53 .5 A B c.5 voltge smples Figure 3.4() oltge signls or the cse with genertor 3 ngles set to 4 degrees A B C.. current smples Figure 3.4(b) Current signls or the cse with genertor 3 ngles set to 4 degrees 39

54 .5 A B c.5 voltge smples Figure 3.5() oltge signls or the cse with genertor 3 ngles set to 6 degrees.5 A B C.5 current smples Figure 3.5(b) Current signls or the cse with genertor 3 ngles set to 6 degrees 4

55 4. Fult Anlysis nd Estimtion o Fult Loction n this Chpter, we will be studying the concepts o unsymmetricl ult nlysis nd lso the lgorithms tht hve been proposed in Dr. Lio s work, Fult loction utilizing unsynchronized voltge mesurements during ult, Electric Power Components & Systems, vol. 34, no., December 6, pp One o the lgorithms hs been used to estimte the ult loction or my power system model. The shunt cpcitnces hve been neglected in the lgorithm used or estimtion in order to hve simplicity nd computtionl eiciency. The Thevenin prmeter vlues obtined or the power system model o Chpter 3 hve been used in the ult loction lgorithm in [5] to clculte the ult loction or the model used in this thesis. The current nd voltge signl wveorms or the dierent vlues o ult loction hve been presented t the end o this chpter. n the initil sections o this chpter, the bsic concepts o unsymmetricl ult nlysis hve been presented. The equtions nd igures o unsymmetricl ult nlysis in sections (4.-4.5) hve ll been obtined rom [4] nd we explin it s much s possible. 4. Unsymmetricl Fults The unsymmetricl ults cn be clssiied s shunt type ults nd series type ults. Shunt type ults cn gin be clssiied s [4]: () Single Line to Ground ult (b) Line to Line Fult (c) Double Line to Ground ult Beore we study in detil bout the shunt type ults, it is importnt to study the symmetricl component nlysis o unsymmetricl ults. 4. Symmetricl Component Anlysis o Unsymmetricl Fults n this section, we will nlyze how power network which is under ult condition cn be represented in terms o positive, negtive nd zero sequence networks s seen rom the point where the ult is occurring. 4

56 Figure 4. represents generl power network [4]. F is the point o ult occurrence. When the ult occurs in the system, the currents re represented by, b, c s shown, which low out o the system. The voltges o lines, b, c with respect to the ground re,, b c respectively. b c F b c b c Figure 4. A Generl Power Network Beore the ult occurs, the positive sequence voltges o ll synchronous mchines in the network re given by E. This is the pre ult voltge t F. The positive, negtive nd zero sequence networks ter the occurrence o the ult s seen rom F re represented in Figures 4. (), (b) nd (c) [4]. F Figure 4. () Positive sequence network s seen rom the ult point 4

57 F Figure 4. (b) Negtive Sequence Network s seen rom the ult point F Figure 4.(c) ero sequence Network s seen rom the ult point The Thevenin equivlents o the sequence networks re represented in Figure 4. (d), (e), () [4] E + F Figure 4. (d) Thevenin Equivlent o Positive sequence Network s seen rom F 43

58 F Figure 4. (e) Thevenin Equivlent o Negtive sequence Network s seen rom F F Figure 4. () Thevenin Equivlent o ero sequence Network s seen rom F We hve mentioned, E is present only in the positive sequence network. We cn express the sequence voltges t F s shown in Eqution (4.) [4].,, E =, re the Positive, negtive nd ero Sequence oltges respectively., re the Positive, negtive nd ero sequence Currents respectively.,, re the Positive, negtive nd ero sequence mpednces respectively. (4.) Now, we cn nlyze the dierent types o shunt ults bsed on the concepts presented in section 4.. The expressions or ult currents nd voltges in the lines re derived subsequently. 44

59 4.3 Anlysis o Single Line to Ground Fult Figure 4.3() shows the power network when single line to ground ult occurs t point F [4]. The ult occurs on Phse. b c F b = c = Figure 4.3() Single Line to Ground Fult t F The equtions or currents nd the line to ground voltges re represented s ollows [4]: = (4.) b = (4.3) c = (4.4) Using the symmetricl components method, we get the ult currents expressed in terms o positive, negtive nd zero sequence components s shown in Eqution (4.5)[4]. = 3 α α α Thereore, α = = (4.5) = (4.6) 3 From Eq. (4.4) nd (4.6), we get 3 = + + = (4.7) 45

60 46 From the Equtions (4.6) nd (4.7), it cn be nlyzed tht there is series connection o sequence networks represented by Figure 4.3(b) [4]. The sequence components o ult current nd the voltges b nd c cn be ound by using the Thevenin equivlents o the sequence networks which is shown in Figure 4.3(b). These re represented in Eqution (4.8)-Eqution (4.5)[4]. E 3 ( ) = (4.8) The ult current cn thus be given by E 3 ) ( = = (4.9) Also by Using Eq. (4.), the ult current cn be obtined. 3 ) ( ) ( ) ( E = + + (4.) E = ] 3 ) [( (4.) E 3 ( ) = (4.) Now, mking use o the method o symmetricl components nd inding the voltge o line b to ground under ult conditions. b + + = α α (4.3) Using the vlues o,, rom Eq. (4.) in (4.3), we get + + = b E α α (4.4) Using Eqution (4.9) in Eqution (4.4) b E 3 ) ( ) ( ) ( = α α α α (4.5)

61 E + F 3 F 3 F = = = = 3 F = F = = = = 3 F = Figure 4.3(b) Connection o sequence networks or single Line to Ground Fult 4.4 Anlysis o Line to Line Fult n the Figure 4.4 line to line ult through ult impednce is indicted s shown. The igures nd equtions in this section hve ll been obtined rom [4]. The currents re expressed s [4] = (4.6) = (4.7) b + c = (4.8) c b 47

62 48 F = c b c b Figure 4.4() Line to Line Fult through mpednce The voltge reltionship between b nd c is expressed s [4] b c b = (4.9) The positive, negtive nd zero sequence components o the ult currents re expressed s [4] = b b 3 α α α α (4.) On solving Eqution (4.), we get = (4.) = (4.) Now, the symmetricl components o voltges under ult t F re expressed s [4] = b b b 3 α α α α (4.3) From the Eqution (4.3) expressing nd b b ) ( 3 α α α + + = (4.4) b b α α α + + = ) ( 3 (4.5)

63 Solving Eqution (4.4) nd (4.5) 3( 3 ) = ( α α ) b = j b (4.6) Using Equtions (4.) nd (4.) in (4.) b = α α (4.7) ( ) = j 3 Substituting Eqution (4.7) in Eqution (4.6) = (4.8) From Equtions (4.) nd (4.8), we cn drw prllel connection o positive nd negtive sequence networks through s series impednce s shown in Figure 4.4(b)[4]. The zero sequence network is not connected s =. ts Thevenin equivlent is lso represented in 4.4 (c)[4]. F F Figure 4.4(b) Positive nd Negtive Sequence Connections or Line to Line Fult 49

64 E + F F Figure 4.4(c) Thevenin Equivlent or connection o Sequence Networks or L-L Fult By using the Thevenin equivlent, we cn write the expressions or E = (4.9) + + b = c = j + 3E + (4.3) The voltges t ult cn be ound out by knowing nd rom. 4.5 Double Line to Ground Fult Anlysis. This cn be clculted n this section, we will mke the nlysis o double line to ground ult. The igures nd equtions hve ll been obtined rom [4].Figure 4.5() shows the double line to ground ult in power system t point F [4]. The ult current or double line to ground ult is expressed s [4] + + (4.3) = = The voltge to ground t ult conditions re expressed s [4] = = = (4.3) b c ( b + c ) 3 5

65 b c F = c b 3 Figure 4.5() Double Line to Ground Fult through mpednce The symmetricl components o voltges under ult t F re expressed s [4] Thus, = 3 α α α α b b [ + ( α + ] = α (4.33) = ) b (4.34) 3 = ( + b ) (4.35) 3 From Equtions (4.34) nd (4.35) = ( α α ) b = b = 3 [4] (4.36) 3 = + (4.37) 3 Thus, the connection or the positive, negtive nd zero sequence networks cn be drwn bsed on the equtions obtined in this section. Figure 4.5(b) represents the connection o sequence networks or double line to ground ult [4]. 5

66 F F F 3 Figure 4.5(b) connection o sequence networks or double line to ground ult E + F F F 3 Figure 4.5(c) Thevenin Equivlent or the sequence network connections or LLG ult From Figure 4.5 (c) the ollowing expressions cn be written [4] E = (4.38) ) /( ) ( 4.6 Fult Loction Algorithm n this section we study ult loction lgorithm tht hs been presented in [5]. The impednce bsed lgorithm is studied nd mde use in order to clculte the ult loction. This lgorithm is pplicble or ll kind o unsymmetricl ults [5]. For studying the lgorithm we consider the trnsmission line represented in Figure

67 EG Gbc P bc Q Hbc EH Figure 4.6 Trnsmission Line Considered or the Algorithm [5] 4.7 mpednce Bsed Algorithm E G, Gbc re the Thevenin equivlent voltge source nd impednce respectively t terminl P. E H nd respectively t terminl Q. Hbc re the Thevenin equivlent voltge source nd impednce bc represents the line impednce. Assuming tht n unsymmetricl ult occurs, we cn mke use o the symmetricl components theory nd negtive sequence network is represented s shown in Figure 4.7 G P m ( m) Q H p q Figure 4.7 Negtive Sequence Network during the ult neglecting Shunt cpcitnce [5] We hve p p / G = (4.39) q q / H = (4.4) 53

68 ] jδ p m p = [ q ( m) q e (4.4) Where, p, p q, q Negtive sequence voltge nd current during the ult t terminl P; Negtive sequence voltge nd current during the ult t terminl Q; G, H Negtive sequence source impednce t terminl P nd Q; δ Synchroniztion ngle between mesurements t terminl P nd Q; m Per unit ult distnce rom terminl P; Totl negtive sequence impednce o the line. n the bove equtions, p nd q re known bsed on the mesurements. Then p nd q cn be obtined using Equtions (4.39) nd (4.4). Solving Eqution (4.4) will result in the ult loction. The Detiled method is reerred to the originl work presented in [5]. The lgorithm neglecting shunt cpcitnces, s presented bove, hs dvntges o simplicity nd computtionl eiciency. However, neglecting shunt cpcitnces my led to certin errors or long lines. Considertion o shunt cpcitnces is discussed in [5]. The ollowing tble lists the estimted ult loction or phse A to ground ults with ult resistnce 5 ohms. The ult resistnce vlue does not ect the ult loction. Tble 4.Fult loction estimtion Actul ult loction (p.u.) Estimted ult loction (p.u.) t is shown rom the tble tht quite ccurte estimtes hve been obtined. The errors my be due to inccurcy o Thevenin prmeter estimtes since the voltge nd current phsors utilized re not precise. 54