Two-Terminal Algorithm Analysis for Unsymmetrical Fault Location on 110 kV Lines

ene gies A icle Two-Te minal Algo i hm Analysis o Unsymme ical Faul Loca ion on 110 kV Lines Zuzana Buk iso a *, Ja osla a O sago a , Da id Topolanek and Pe Toman Depa men o Elec ical Powe Enginee ing, B no Uni e si y o Technology, Technicka 12, 61600 B no, Czech Republic; [email p o ec ed].cz (J.O.); opolanek@ u b .cz (D.T.); [email p o ec ed].cz (P.T.) *Co espondence: [email p o ec ed].cz; Tel.: +420-606-202-347 Recei ed: 5 Ma ch 2019; Accep ed: 25 Ma ch 2019; Published: 27 Ma ch 2019   Abs ac : This wo k analyses a wo- e minal algo i hm designed o loca e unsymme ical aul s on 110 kV powe ansmission lines. The algo i hm p ocesses synch onized ol age and cu en da a ob ained om bo h ends o he p o ec ed ansmission line and calcula es he dis ance o he aul . I is based on decomposing he equi alen ci cui in o he posi i e-, nega i e- and ze o-sequence componen s and inding he poin whe e he ou pu ol ages o he igh and he le side o he ansmission line a e equal. Compa ed o he con en ional dis ance elay loca o , he accu acy o his me hod is highe and less in luenced by he aul esis ance, he pa allel-ope a ed line e ec and line asymme y, as discussed in his wo k. I is, howe e , e y sensi i e o he synch oniza ion accu acy. The ma hema ical model o he powe sys em was c ea ed in he PSCAD (Powe Sys ems Compu e Aided Design) en i onmen and he compu a ional algo i hm was implemen ed in Ma hema ica so wa e. Keywo ds: unsymme ical aul loca ion; wo- e minal algo i hm; sequence componen s 1. In oduc ion To loca e a aul du ing single-phase aul s in a 110 kV line, a aul loca o , which is one o he unc ions o dis ance p o ec ion, is cu en ly used. The basic pa ame e s o a p o ec ed line a e se in he elay and used o compu e he aul dis ance and usually o ini ia e he loca o unc ion as well. Basically, a e he pick up o he ip o he dis ance p o ec ion, a sho -ci cui loop is de e mined and he cu en s and ol ages measu ed in his loop a e used o he sho -ci cui loop eac ance and esis ance calcula ion. A e ha , a aul loop and i s impedance ZL h ough he measu ed phase cu en s and ol ages a e de e mined. I a g ound aul loca ion is calcula ed, i is necessa y o ake accoun o alues o he esidual compensa ion ac o s R E /R 1 ,X E /X 1 as well. The compu a ional algo i hm used o he aul impedance ZL de e mina ion is based on he solu ion o he sho -ci cui loop using he alue o he ol age measu ed a he p o ec ion connec ion poin , he cu en in he o wa d di ec ion ( o he aul ) and he cu en in he backwa d di ec ion (back o he poin o measu emen ). In he case o a single-phase aul in phase i(i= 1, 2, 3) i will be: ZL =ULi ILi−kEIE . (1) The aul dis ance is e alua ed om he eac ance o he aul XL , which is he imagina y componen o he aul impedance ZL . Thus, he co ec e alua ion o he single-phase aul dis ance is always s ongly dependen on he eac ance pe kilome e XL and esidual compensa ion ac o kE o he p o ec ed line se in he dis ance elay. Since mos o he 110 kV lines a e no ully ansposed, aul loca o e o s a e o en caused by unequal impedance alues o he phase-g ound loops and mu ual impedances o hese loops. Ano he Ene gies 2019,12, 1193; doi:10.3390/en12071193 www.mdpi.com/jou nal/ene gies Ene gies 2019,12, 1193 2 o 14 e o can be caused by non-homogenei y o he ansmission line. Non-homogenei y means ha he line pa ame e s a e no cons an o he whole line leng h bu a y in indi idual sec ions. Typical examples a e changes in c oss-sec ion o di e en g ound wi e conduc i i y. Nega i e e ec o he aul esis ance on aul loca ion is desc ibed in [ 1 , 2 ]. In sys ems wi h pa allel-ope a ed lines, he accu acy o he aul loca ion is also g ea ly in luenced by mu ual impedances. In gene al, he g ound wi e conduc i i y and he sho -ci cui con ibu ion o he pa allel line powe supply will ha e he majo in luence on he loca o e o a e. Many esea che s ha e ocused hei a en ion on possible e o elimina ion. The p oposed algo i hms di e depending on he inpu da a hey ha e a ailable and he me hod o calcula ion. The me hods can be di ided in o h ee main g oups: •T a elling wa e-based me hods, •A i icial in elligence-based me hods, •Impedance-based me hods. 1.1. T a elling Wa e- and A i icial In elligence-Based Me hods T a elling wa e-based compu a ional me hods a e usually based on he co ela ion be ween he o wa d and backwa d wa es ha a el alongside he ansmission line. When a aul occu s, he wa es a el om he poin o he aul o bo h ends o he ansmission line. The aul is hen iden i ied using he ansien -s a e analysis. Some s udies [ 3 , 4 ] sugges c ea ing a da abase o a ious aul scena ios o make he ansien analysis easie . These me hods can wo k wi h one- o mul iple- e minal da a. The accu acy o hese algo i hms is mainly dependen on he da a sampling a e. Fo he aul de ec ion, he wa ele ans o m p o es o be e y eliable. In elligen echniques help o imp o e e iciency o he aul de ec ion and classi ica ion. The ad an age o using neu al ne wo k is i s abili y o ecognize a pa e n and ca ego ize he da a. Acco ding o [5], he mos used echniques based on he a i icial in elligence a e: •Expe Sys em Techniques, •A i icial Neu al Ne wo ks, •Fuzzy Logic Sys ems. Recen ly, a use o ime- ime (TT) ans o m in signal p ocessing has been discussed [ 6 ]. Re e ence [ 7 ] p oposes o apply TT- ans o m o se ies-compensa ed lines, which p o ed o be e icien e en when p ocessing a signal in luenced by high noise. In [ 8 ], he TT- ans o m and a aul classi ica ion based on suppo ec o machine (SVM) is used o loca e aul on a hyb id line. Al hough his echnique p ocesses ansien ol age signal ob ained om only one end, he aul was iden i ied and loca ed wi h a high accu acy. 1.2. Impedance-Based Me hods The basic p inciple o he impedance-based algo i hms o he aul loca ion calcula ion is simple. The p o ec ion elay uses posi i e and ze o sequence impedances and measu ed ol ages and cu en s o de e mine he dis ance o he aul by calcula ing he impedance, as desc ibed by Equa ion (1). These me hods a e simple and commonly used and hei accu acy can be signi ican ly imp o ed i da a om mul iple e minals is acqui ed. Algo i hms ha use only local measu emen da a a e called single-ended (o one- e minal) algo i hms. These me hods a e o en implemen ed, e.g., in mic op ocesso -based p o ec i e elays [9,10] . Thei ad an ages a e simplici y due o hei lack o communica ion equi emen s. Howe e , he accu acy o hese me hods is g ea ly in luenced by aul esis ance, load low and sou ce impedances. In [ 11 ], a single-ended echnique, which is based on using he cu en and ol age measu emen s a local e minal and an es ima ed sho -ci cui capaci y o he emo e sys em, is p oposed. Al hough his me hod is no una ec ed by he p oblems men ioned abo e, he e o s we e wi hin accep able limi s. Ene gies 2019,12, 1193 3 o 14 Two- e minal algo i hms wo k wi h he measu emen s om bo h ends o he ansmission line. These algo i hms can be u he di ided in o me hods using synch onized o unsynch onized measu emen s. I is also possible o p ocess da a om mul iple e minals o imp o e he accu acy o he calcula ion. Using he synch onized wo- e minal ol age and cu en phaso s, he calcula ion o he aul loca ion is signi ican ly imp o ed [ 1 ]. The da a collec ed om digi al eco de s is e alua ed in a cen al compu e using specialized so wa e. Wi h he elecommunica ions de elopmen , he synch oniza ion and as and eliable measu ed phaso da a exchange a e becoming easie and he e o e i is possible o implemen hese algo i hms di ec ly in o he p o ec ion. The ad an ages o hese me hods a e: • Elimina ion o aul loca ion e o s caused by inaccu acies in esidual compensa ion ac o s de e mina ion, •Supp ession o an e o caused by he aul esis ance, •Reduc ion o he e ec s o mu ual coupling and line asymme y. The impedance-based me hods a e cu en ly he mos widely used me hods o aul loca ion and wi h he de elopmen and ins alla ion o he phaso measu emen uni s (PMUs) o digi al elays wi h global posi ioning sys ems, he echniques based on he undamen al powe equency componen s can be imp o ed. The bigges ad an age o hese echniques, p oposed e.g., in [ 12 – 14 ], is ha he e o caused by a ia ions in he sou ce and aul impedances is elimina ed. Ex ensi e placing o he PMUs is, howe e , e y limi ed by he high ins alla ion cos s and he e o e, some wo ks ocus on de eloping an op imal PMU placemen s a egy [ 12 ]. Mo eo e , he de eloped echniques a e o en in luenced by o he ac o s, such as need o high da a sampling a e [14]. Re e ence [ 13 ] p oposes a me hod ha u ilizes synch onized ol age and cu en da a de i ed a bo h ends o he ansmission line and exp esses he ol age ac oss he aul in e ms o he measu ed da a. This wo k is based on simila p inciple and i will be discussed la e in his ex . Compa ed o [ 13 ], howe e , he exac e alua ion o he aul dis ance is no pe o med. Ins ead, he accu acy o he analyzed algo i hm is imp o ed by using no only posi i e-, bu nega i e-sequence componen s as well. Then, he leas squa e me hod is applied o ind he poin o he aul . Some pape s [ 15 – 17 ] y o emo e he need o ob aining synch onized da a. Re e ence [ 15 ] sugges s modi ying he echnique in oduced in [ 13 ] by conside ing only he magni udes o he ol age a he aul poin . This assump ion allows o use unsynch onized phaso s o measu ed ol ages and cu en s and emo e he e o caused by he unsynch onized da a. Me hods desc ibed in [ 16 , 17 ] a e based on a simple assump ion ha a aul impedance is pu ely esis i e. Finding a solu ion o his condi ion is hen used o ind he synch oniza ion angle. Al hough hese algo i hms enable o use he unsynch onized measu emen s, which is a big imp o emen , he main aim o his wo k is o ind a usable me hod o he aul loca ion on 110 kV lines in he Czech Republic, which a e sho and o en pa allel-ope a ed. Me hods p oposed in [ 15 – 17 ] do no deal wi h he e ec o he pa allel line. In his wo k, he e ec o mu ual coupling is ully conside ed. The es o he pape is o ganized as ollows: Sec ion 2desc ibes he basic p inciple o he algo i hm. Sec ion 3discusses he model used o es ing he me hod and esul s o a ious scena ios. Sec ion 4gi es he conclusions. 2. Desc ip ion o he Analyzed Algo i hm Fi s ly, a me hod using he one- e minal app oach will be desc ibed o ou line a basic p inciple o he p oposed impedance-based algo i hm and he e o s ha can occu . The pa allel-ope a ed line e ec is desc ibed he e as well. La e in his sec ion, an analyzed wo- e minal algo i hm will be discussed. Ene gies 2019,12, 1193 4 o 14 2.1. Basic Desc ip ion o he One-Te minal Algo i hm Figu e 1displays a si ua ion when a single-phase aul wi h a aul esis ance R occu s in a ansmission line. The aul is loca ed in he dis ance dmeasu ed om he loca o a he x-poin , he o al line leng h is l. Using his example, an idea o one- e minal algo i hm e o s can be gi en. Ene gies 2018, 11, x 4 o 15 2.1. Basic Desc ip ion o he One-Te minal Algo i hm Figu e 1 displays a si ua ion when a single-phase aul wi h a aul esis ance R occu s in a ansmission line. The aul is loca ed in he dis ance d measu ed om he loca o a he x-poin , he o al line leng h is l. Using his example, an idea o one- e minal algo i hm e o s can be gi en. Figu e 1. A simpli ied aul y ansmission line scheme. An equi alen ci cui is o med by he in e connec ion o impedance sequence componen ci cui s and he aul esis ance—see E o ! Re e ence sou ce no ound.. The indexes o he posi i e, nega i e and ze o-sequence componen s a e 1, 2 and 0, espec i ely. I he Ki chho ’s ci cui laws a e applied in indi idual loops o he equi alen ci cui , an e alua ion o he aul dis ance d using he cu en x Iand ol age x U a io used by he loca o placed a he x-poin can be done: () 1 2 0 1x 1L 2x 1L 0x 0L 0x 1L 0x 1L 1x 2x 0x x x 1x2x0x 1x2x0x U U U dIZ IZ IZ IZ IZ UU U U IIII III +++ + + + − ++ == ++ ++ , (2) ()() 1 2 0 1L 1x 2x 0x 0x 0L 1L x x1x2x0x ..UU U dZI I I IdZ Z U IIII +++ ++ + − =++ . (3) I 0L 1L E 1L 3 Z ZkZ−= (whe e 0L Z and 1L Z a e line impedances pe uni leng h o a ze o- and posi i e-sequence componen , espec i ely, and E k is a esidual compensa ion ac o ) is assumed, hen: 0x E 1L 1 2 0 1 2 0 xEx 1L 1L E xx1x2x0x x1x2x0x 31 IkZ U U U U U U UI dZ dZ k I I II I I II I   ++ ++ =+ + = + +   ++ ++   , (4) whe e he ol age a he aul poin can be w i en as: ()()()() 1 2 0 1x 1y 2x 2y 0x 0y Ex Ey 33 3U U U RI I RI I RI I RI I++= += + = + = + . (5) Applying hese equa ions, he impedance measu ed by he loca o a he x-poin is: Ex Ey xEx 1L E xxx .1 II UI dZ k R III +  =++   (6) and simila ly, o he loca o placed a he y-poin : () yEyExEy 1L E yyy .1 UIII ldZ k R III  + =− + +    . (7) I he line pa ame e s 1L Z and E k, aul esis ance R and he ea h cu en s Ey I and Ex I acqui ed om bo h ends a e known, he Equa ions (6) and (7) can be used o p ecisely calcula e he aul dis ance. I hese alues a e no ob ained, he loca o algo i hm co ec ion me hod will always be jus an es ima ion and he e o e po en ially able o ac ually inc ease he loca o e o . E en mo e complica ed si ua ion occu s o he pa allel-ope a ed lines, when he equi alen ci cui con ains also he pa allel-line sequence impedances—in he E o ! Re e ence sou ce no ound. d awn in ed. Figu e 1. A simpli ied aul y ansmission line scheme. An equi alen ci cui is o med by he in e connec ion o impedance sequence componen ci cui s and he aul esis ance—see Figu e 2. The indexes o he posi i e, nega i e and ze o-sequence componen s a e 1, 2 and 0, espec i ely. I he Ki chho ’s ci cui laws a e applied in indi idual loops o he equi alen ci cui , an e alua ion o he aul dis ance dusing he cu en Ix and ol age Ux a io used by he loca o placed a he x-poin can be done: Ux Ix =U1x +U2x +U0x I1x +I2x +I0x =U1 +U2 +U0 +dI1xZ1L +I2xZ1L +I0xZ0L +I0xZ1L −I0xZ1L I1x +I2x +I0x , (2) Ux Ix =U1 +U2 +U0 +d.Z1L.I1x +I2x +I0x+I0xdZ0L −Z1L I1x +I2x +I0x . (3) I Z0L −Z1L = 3 kEZ1L (whe e Z0L and Z1L a e line impedances pe uni leng h o a ze o- and posi i e-sequence componen , espec i ely, and kE is a esidual compensa ion ac o ) is assumed, hen: Ux Ix =d Z1L +I0x3kEZ1L Ix!+U1 +U2 +U0 I1x +I2x +I0x =dZ1L1+kE IEx Ix+U1 +U2 +U0 I1x +I2x +I0x , (4) whe e he ol age a he aul poin can be w i en as: U1 +U2 +U0 =3R I1x +I1y=3R I2x +I2y=3R I0x +I0y=R IEx +IEy. (5) Applying hese equa ions, he impedance measu ed by he loca o a he x-poin is: Ux Ix =d.Z1L1+kE IEx Ix+R IEx +IEy Ix (6) and simila ly, o he loca o placed a he y-poin : Uy Iy =(l−d).Z1L 1+kE IEy Iy!+R IEx +IEy Iy . (7) I he line pa ame e s Z1L and kE , aul esis ance R and he ea h cu en s IEy and IEx acqui ed om bo h ends a e known, he Equa ions (6) and (7) can be used o p ecisely calcula e he aul dis ance. I hese alues a e no ob ained, he loca o algo i hm co ec ion me hod will always be jus an es ima ion and he e o e po en ially able o ac ually inc ease he loca o e o . E en mo e Ene gies 2019,12, 1193 5 o 14 complica ed si ua ion occu s o he pa allel-ope a ed lines, when he equi alen ci cui con ains also he pa allel-line sequence impedances—in he Figu e 2d awn in ed. Ene gies 2018, 11, x 5 o 15 Figu e 2. The equi alen ci cui o pa allel-ope a ed ansmission lines. F om he ci cui diag am i is clea ha he posi i e and he nega i e sequence componen s o he impedance measu ed by he loca o emain he same, bu he ze o sequence does no . This is caused by he common pa h o he ea h cu en o bo h ansmission lines o med by he g ound and he g ound wi e. The mu ual coupling is symbolized in he ze o-sequence componen by ou addi ional induced ol age sou ces. Two o hem a e pa o he ze o-sequence componen o he aul ed sec ion. These ol ages a e induced by he ze o-sequence cu en 0x pa I o he pa allel line: 0A 0M 0x pa ..UdZI=, (8) () 0B 0M 0x pa ..UldZI=− . (9) The emaining wo ol age sou ces a e loca ed in he ze o-sequence componen o he heal hy line and a e induced by he ze o-sequence cu en 0x I o he aul y line: 0C 0M 0x ..UdZI=, (10) () () 0D 0M 0x ..UldZII=− − . (11) Vol ages can be subs i u ed o p oduc o cu en s ha induce hese ol ages and mu ual ze o sequence impedance 0M Z acco ding o Equa ions (8)–(11). Then, he equi alen ci cui o he ze o- sequence impedance can be modi ied, as shown in Figu e 3. The loca o placed a he x-poin calcula es he impedance as () 1x 2x 0xx x1x2x0x 1 2 0 1x 1L 2x 1L 0x 0L 0x 1L 0x 1L 0x pa 0M 0y 0L 1x 2x 0x UU U U IIII U U U dIZ dIZ dIZ dIZ dIZ lI Z l dIZ II I ++ =++ +++ + + + − + −− =++ , (12) () xE yE 0x pa 0M 0y 0L xEx 1L E xxxxx .1 () II I ZIZ UI dZ k R l l d I III I +  =++ + −−   . (13) Figu e 2. The equi alen ci cui o pa allel-ope a ed ansmission lines. F om he ci cui diag am i is clea ha he posi i e and he nega i e sequence componen s o he impedance measu ed by he loca o emain he same, bu he ze o sequence does no . This is caused by he common pa h o he ea h cu en o bo h ansmission lines o med by he g ound and he g ound wi e. The mu ual coupling is symbolized in he ze o-sequence componen by ou addi ional induced ol age sou ces. Two o hem a e pa o he ze o-sequence componen o he aul ed sec ion. These ol ages a e induced by he ze o-sequence cu en I0x pa o he pa allel line: U0A =d.Z0M.I0x pa , (8) U0B =(l−d).Z0M.I0x pa . (9) The emaining wo ol age sou ces a e loca ed in he ze o-sequence componen o he heal hy line and a e induced by he ze o-sequence cu en I0x o he aul y line: U0C =d.Z0M.I0x, (10) U0D =(l−d).Z0M.I0x −I . (11) Vol ages can be subs i u ed o p oduc o cu en s ha induce hese ol ages and mu ual ze o sequence impedance Z0M acco ding o Equa ions (8)–(11). Then, he equi alen ci cui o he ze o-sequence impedance can be modi ied, as shown in Figu e 3. Ene gies 2019,12, 1193 6 o 14 The loca o placed a he x-poin calcula es he impedance as Ux Ix=U1x+U2x+U0x I1x+I2x+I0x =U1 +U2 +U0 +dI1xZ1L+dI2xZ1L+dI0xZ0L+dI0xZ1L−dI0xZ1L+lI0x pa Z0M−(l−d)I0yZ0L I1x+I2x+I0x , (12) Ux Ix =d.Z1L1+kE IEx Ix+R IxE +IyE Ix +lI0x pa Z0M Ix −(l−d)I0y Z0L Ix . (13) Fo he loca o placed a he y-poin can be w i en as: Uy Iy =(l−d).Z1L 1+kE IEy Iy!+R IEx +IEy Iy +lI0y pa Z0M Iy −(l−d)I0x Z0L Iy . (14) I is ob ious ha due o he numbe o quan i ies ha in luence he impedance calcula ion, i is much mo e e ec i e o use wo- e minal algo i hms. Ene gies 2018, 11, x 6 o 15 Fo he loca o placed a he y-poin can be w i en as: () () Ex Ey yEy0y pa 0M 0x 0L 1L E yyyyy .1 () II UI IZ IZ ldZ k R l ld IIIII +  =− + + + −−    . (14) I is ob ious ha due o he numbe o quan i ies ha in luence he impedance calcula ion, i is much mo e e ec i e o use wo- e minal algo i hms. . Figu e 3. A ze o-sequence componen equi alen ci cui o pa allel-ope a ed ansmission lines. 2.2. Two-Te minal Algo i hm The equi alen ci cui used o wo- e minal algo i hms is displayed in E o ! Re e ence sou ce no ound.. The ansmission line be ween he sou ce and he aul poin is subs i u ed by he dis ibu ed pa ame e line model wi h se ies impedances and shun admi ances. The comple e equi alen ci cui consis s o indi idual sequence componen ci cui s connec ed o he aul esis ance R . Figu e 4. An equi alen ci cui o a single-phase aul . To calcula e he aul loca ion, indi idual sequence componen ol ages a he aul poin (whe e i = 1, 2, 0) a e de e mined. These ol ages ep esen he ou pu ol ages o he espec i e ansmission line model and can be calcula ed using he inpu alues o cu en s in xi I and ol ages in xi U a he x-poin : ou in in x x x cosh( . ) sinh( . ) ii i i ii i UU U dlIZ dl γγ == − , (15) o using he inpu alues o cu en s in yi I and ol ages in yi U a he y-poin : Figu e 3. A ze o-sequence componen equi alen ci cui o pa allel-ope a ed ansmission lines. 2.2. Two-Te minal Algo i hm The equi alen ci cui used o wo- e minal algo i hms is displayed in Figu e 4. The ansmission line be ween he sou ce and he aul poin is subs i u ed by he dis ibu ed pa ame e line model wi h se ies impedances and shun admi ances. The comple e equi alen ci cui consis s o indi idual sequence componen ci cui s connec ed o he aul esis ance R . To calcula e he aul loca ion, indi idual sequence componen ol ages a he aul poin (whe e i= 1, 2, 0 ) a e de e mined. These ol ages ep esen he ou pu ol ages o he espec i e ansmission line model and can be calcula ed using he inpu alues o cu en s Iin ix and ol ages Uin ix a he x-poin : Ui =Uou ix=Uin ixcosh(γid.l)−Iin ixZi sinh(γid.l), (15) o using he inpu alues o cu en s Iin iyand ol ages Uin iya he y-poin : Ui =Uou iy=Uin iycosh(γi(1−d).l)−Iin iyZi sinh(γi(1−d).l), (16) whe e γi=pZiL.YiL is he p opaga ion cons an and Zi = ZiL YiL is he cha ac e is ic impedance o he p o ec ed ansmission line (i= 1, 2, 0 o he posi i e-, nega i e- and ze o-sequence componen o he line impedance pe uni leng h ZiLand he line admi ance pe uni leng h YiL). I Ui =Uou ix=Uou iy , hen he aul loca ion dcan be de e mined. Fo his pu pose, he posi i e-sequence componen can be used because i s pa ame e s γ1=pZ1L.Y1L and Z1 = Z1L Y1L a e known and hei calcula ion is no in luenced by he soil esis i i y. Ene gies 2019,12, 1193 7 o 14 Ene gies 2018, 11, x 6 o 15 Fo he loca o placed a he y-poin can be w i en as: () () Ex Ey yEy0y pa 0M 0x 0L 1L E yyyyy .1 () II UI IZ IZ ldZ k R l ld IIIII +  =− + + + −−    . (14) I is ob ious ha due o he numbe o quan i ies ha in luence he impedance calcula ion, i is much mo e e ec i e o use wo- e minal algo i hms. . Figu e 3. A ze o-sequence componen equi alen ci cui o pa allel-ope a ed ansmission lines. 2.2. Two-Te minal Algo i hm The equi alen ci cui used o wo- e minal algo i hms is displayed in E o ! Re e ence sou ce no ound.. The ansmission line be ween he sou ce and he aul poin is subs i u ed by he dis ibu ed pa ame e line model wi h se ies impedances and shun admi ances. The comple e equi alen ci cui consis s o indi idual sequence componen ci cui s connec ed o he aul esis ance R . Figu e 4. An equi alen ci cui o a single-phase aul . To calcula e he aul loca ion, indi idual sequence componen ol ages a he aul poin (whe e i = 1, 2, 0) a e de e mined. These ol ages ep esen he ou pu ol ages o he espec i e ansmission line model and can be calcula ed using he inpu alues o cu en s in xi I and ol ages in xi U a he x-poin : ou in in x x x cosh( . ) sinh( . ) ii i i ii i UU U dlIZ dl γγ == − , (15) o using he inpu alues o cu en s in yi I and ol ages in yi U a he y-poin : Figu e 4. An equi alen ci cui o a single-phase aul . I he igh sides o he Equa ions (15) and (16) a e equal, hen i is possible o use only he equa ions o he posi i e-sequence ol age a he aul poin Uin 1x cosh(γ1d.l)−Iin 1xZ1 sinh(γ1d.l)−Uin 1y cosh(γ1(1−d).l) + Iin 1yZ1 sinh(γ1(1−d).l) = 0. (17) Assuming γ1=γ2=pZ1L.Y1L and Z1 =Z2 = Z1L Y1L , o inc ease he calcula ion eliabili y, he nega i e-sequence ol age is used as well Uin 2x cosh(γ1d.l)−Iin 2xZ1 sinh(γ1d.l)−Uin 2y cosh(γ1(1−d).l) + Iin 2Z1 sinh(γ1(1−d).l) = 0. (18) Fo each equa ion, a de ia ion om ze o o a ious alues o dcan be de e mined ε1(d)=Uou 1x −Uou 1y , (19) ε2(d)=Uou 2x −Uou 2y . (20) A e ob aining he de ia ions o posi i e- and nega i e-sequence componen and applying he leas squa e me hod, a simple unc ion F is c ea ed. This unc ion eaches i s minimum in he dis ance d, which co esponds o he aul poin : F(d) = |ε1(d)|2+|ε2(d)|2. (21) The ou pu o his algo i hm is he alue o dis ance dwi h he minimal F(d) alue. 2.3. Synch oniza ion To co ec ly calcula e he dis ance o he aul dusing his algo i hm, i is necessa y o measu e he inpu cu en s and ol ages a x- and y-poin s synch onously. This, howe e , is mos ly impossible in cu en 110 kV dis ibu ion ne wo ks. The dis ance p o ec ion loca o s do no use mu ual ime synch oniza ion; he e o e, he da a eco ds mus be addi ionally synch onized be o e pe o ming he calcula ions. This can be done wi h use o he ansien da a eco ds in he ime domain, e.g., Ene gies 2019,12, 1193 8 o 14 cap u ing he momen o he ansien incep ion (a su icien da a sampling a e equi ed). Ano he me hod is based on he phaso co ec ion by es ima ing he inpu quan i ies mu ual phase shi . Using his es ima ed phase shi alue, he co esponding quan i ies a e shi ed by angle δ -synch oniza ion ope a o . I Iin xis he synch onizing quan i y, hen: •Phaso co ec ion a he x-poin : Iin x=Iin x∠(α−α),Uin x=Uin x∠(β−α), •Phaso co ec ion a he y-poin : Iin y=Iin y∠(ϕ−ϕ+δ),Uin y=Uin y∠(γ−ϕ+δ), whe e α , β,γ , ϕ a e he angles o he unsynch onously eco ded phaso s and δ is he synch oniza ion ope a o de e mined by he phase shi o he cu en s Iin xand Iin y. Conside ing he 110 kV dis ibu ion ne wo ks ope a ion, he phase shi be ween he inpu cu en s can be de e mined using he sho -ci cui impedances Zxs and Zys ep esen ing he p o ec ed line load, as shown in a simpli ied diag am in Figu e 5. Acco ding o [ 1 ], o es ima e he synch oniza ion ope a o δ , he inpu cu en and ol age da a cap u ed du ing he no mal ope a ion o du ing he aul (o hei combina ion) can be used. The same p inciple o he synch oniza ion in [13] is used. Ene gies 2018, 11, x 8 o 15 Figu e 5. A simpli ied equi alen ci cui o a single-phase aul . Ano he me hod is, as men ioned abo e, based on ansien obse a ion. To analyze he equency componen s o he signal, he Fou ie ans o m is commonly used. The p oblem wi h using Fou ie ans o m is ha i is no capable o de e mine when he pa icula equency changes. To acqui e in o ma ion abou bo h ime and equency, a sho - ime Fou ie ans o m, which uses a sliding window, can be applied. This echnique, howe e , limi s he equency esolu ion. Be e solu ion is o use he wa ele ans o m [18]. The wa ele ans o m decomposes he signal in o unc ions loca ed bo h in Fou ie and he eal- ime space. I is basically an in ini e se o a ious ans o ms. The p oblem wi h eco ds synch oniza ion using he ansien s udy is he unequal ime ha he ansien needs o ge om he aul poin o he poin o measu emen . The ansien a i es soone a he close e minal, so cap u ing he momen o he ansien incep ion a bo h e minals and subsequen compa ison can be qui e inaccu a e. Fu u e esea ch will be ocused on his opic. The synch oniza ion me hod should be chosen based on he da a a ailabili y. Howe e , he p e e able one is he ansien analysis me hod. I he equi ed da a a e no a ailable, he synch oniza ion ope a o needs o be calcula ed. 3. Analysis o he Two-Te minal Algo i hm Tes ing 3.1. Ma hema ical Model o a 110 kV Line The p esen ed wo- e minal algo i hm was implemen ed in he Ma hema ica so wa e ( e sion 11.1.1.0, Wol am Resea ch, Champaign, IL, USA) o p ocess he ol ages and cu en s measu ed a bo h ends o he 110 kV line modelled in PSCAD so wa e ( e sion 4.6.0.0, Mani oba HVDC Resea ch Cen e, Winnipeg, MB, Canada). The ol age and cu en da a a e ob ained synch onously, which means ha quan i ies mu ual ime shi is ze o. The c ea ed model consis s o wo non- ansposed pa allel-ope a ed 110 kV lines wi h six conduc o s and one g ound wi e, as in Figu e 6. One side o he owe ca ies he conduc o s o he i s line, he o he side ca ies he second line. The ma hema ical model o his line is buil acco ding o he eal dis ibu ion pa allel-ope a ed 110 kV line. The ansmission line is modelled using he equency dependen line model. The soil esis i i y is 50 Ωm, he leng h o he line is 27.88 km. The e a e six 240 AlFe4 conduc o s and a combined g ound wi e wi h 48 ibe s (0.2 Ω DC esis ance and 18 mm diame e ). Line pa ame e s we e de i ed om se ies impedance and shun admi ance ma ixes in PSCAD line cons an p og am ou pu ile. The esul s a e lis ed in Table 1. Figu e 5. A simpli ied equi alen ci cui o a single-phase aul . Ano he me hod is, as men ioned abo e, based on ansien obse a ion. To analyze he equency componen s o he signal, he Fou ie ans o m is commonly used. The p oblem wi h using Fou ie ans o m is ha i is no capable o de e mine when he pa icula equency changes. To acqui e in o ma ion abou bo h ime and equency, a sho - ime Fou ie ans o m, which uses a sliding window, can be applied. This echnique, howe e , limi s he equency esolu ion. Be e solu ion is o use he wa ele ans o m [ 18 ]. The wa ele ans o m decomposes he signal in o unc ions loca ed bo h in Fou ie and he eal- ime space. I is basically an in ini e se o a ious ans o ms. The p oblem wi h eco ds synch oniza ion using he ansien s udy is he unequal ime ha he ansien needs o ge om he aul poin o he poin o measu emen . The ansien a i es soone a he close e minal, so cap u ing he momen o he ansien incep ion a bo h e minals and subsequen compa ison can be qui e inaccu a e. Fu u e esea ch will be ocused on his opic. The synch oniza ion me hod should be chosen based on he da a a ailabili y. Howe e , he p e e able one is he ansien analysis me hod. I he equi ed da a a e no a ailable, he synch oniza ion ope a o needs o be calcula ed. 3. Analysis o he Two-Te minal Algo i hm Tes ing 3.1. Ma hema ical Model o a 110 kV Line The p esen ed wo- e minal algo i hm was implemen ed in he Ma hema ica so wa e ( e sion 11.1.1.0, Wol am Resea ch, Champaign, IL, USA) o p ocess he ol ages and cu en s measu ed Ene gies 2019,12, 1193 9 o 14 a bo h ends o he 110 kV line modelled in PSCAD so wa e ( e sion 4.6.0.0, Mani oba HVDC Resea ch Cen e, Winnipeg, MB, Canada). The ol age and cu en da a a e ob ained synch onously, which means ha quan i ies mu ual ime shi is ze o. The c ea ed model consis s o wo non- ansposed pa allel-ope a ed 110 kV lines wi h six conduc o s and one g ound wi e, as in Figu e 6. One side o he owe ca ies he conduc o s o he i s line, he o he side ca ies he second line. The ma hema ical model o his line is buil acco ding o he eal dis ibu ion pa allel-ope a ed 110 kV line. Ene gies 2018, 11, x 9 o 15 Figu e 6. T ansmission line conduc o a angemen . Table 1. Line pa ame e s. Posi i e- and nega i e-sequence impedance () 121 1 j 3.513 j 11.096ZZ R X==+ = + Ω Pe kilome e : () 1L 2L 1L 1L j 0.126 j 0.398 /kmZZ R X==+ = + Ω Ze o-sequence impedance () 00 0 j 7.165 j 32.369ZR X=+ = + Ω Pe kilome e : () 0L 0L 0L j 0.257 j 1.161 /kmZR X=+ = + Ω Posi i e- and nega i e-sequence admi ance () 75 12 1 1 jB 2.788.10 j 8.263.10 SYY G −− ==− = − Pe kilome e : () 86 1L 2L 1L 1L j 10 j 2.964.10 S/kmYY G B −− ==− = − Ze o-sequence admi ance () 75 00 0 j 2.788.10 j 4.851.10YG B −− =− = − Ω Pe kilome e : () 86 0L 0L 0L j 10 j 1.740.10 S/kmYG B −− =− = − Residual compensa ion ac o 0L 1L EREXE 1L j 0.346 j 0.639 3. ZZ kkk Z − ==+=+ Using he c ea ed PSCAD model, i is also possible o calcula e he aul dis ance. To do his, he Equa ion (1) is used simila ly o he common aul loca o . The compu a ional algo i hm is buil wi h logical blocks and unc ions implemen ed in he PSCAD lib a y. I u ilizes ob ained cu en and ol age da a p ocessed using he Fas Fou ie T ans o m. The ou pu o his calcula ion can be compa ed o he ou pu o he p oposed algo i hm. The simula ion da a a e s o ed in a Com ade ile o ma . The PSCAD so wa e con ains a componen called RTP/COMTRADE Reco de , which is able o eco d he simula ed da a a sa e hem as Com ade. The e o e, he da a can be u he iewed in a so wa e ha uses his o ma . 3.2. Impac o he Faul Resis ance Se e al es s we e pe o med o assess he impac o he aul esis ance alue on he p oposed algo i hm. The aul occu ed in he L1 phase in he dis ance o 4.49 km om he e minal a he x- poin . The aul esis ance was changed up o 100 Ω, which appea s e y a ely in he 110 kV ne wo k and e en a aul wi h such high esis ance was success ully localized, as seen in E o ! Re e ence sou ce no ound., whe e he minimum o he unc ion [ ] ln F( )d deno es he calcula ed aul loca ion. Figu e 6. T ansmission line conduc o a angemen . The ansmission line is modelled using he equency dependen line model. The soil esis i i y is 50 Ω m, he leng h o he line is 27.88 km. The e a e six 240 AlFe4 conduc o s and a combined g ound wi e wi h 48 ibe s (0.2 Ω DC esis ance and 18 mm diame e ). Line pa ame e s we e de i ed om se ies impedance and shun admi ance ma ixes in PSCAD line cons an p og am ou pu ile. The esul s a e lis ed in Table 1. Table 1. Line pa ame e s. Posi i e- and nega i e-sequence impedance Z1=Z2=R1+jX1=(3.513 +j 11.096)Ω Pe kilome e : Z1L =Z2L =R1L +jX1L =(0.126 +j 0.398)Ω/km Ze o-sequence impedance Z0=R0+jX0=(7.165 +j 32.369)Ω Pe kilome e : Z0L =R0L +jX0L =(0.257 +j 1.161)Ω/km Posi i e- and nega i e-sequence admi ance Y1=Y2=G1−jB1=2.788.10−7−j 8.263.10−5S Pe kilome e : Y1L =Y2L =G1L −jB1L =10−8−j 2.964.10−6S/km Ze o-sequence admi ance Y0=G0−jB0=2.788.10−7−j 4.851.10−5Ω Pe kilome e : Y0L =G0L −jB0L =10−8−j 1.740.10−6S/km Residual compensa ion ac o kE=Z0L−Z1L 3.Z1L =kRE +jkXE =0.346 +j 0.639 Using he c ea ed PSCAD model, i is also possible o calcula e he aul dis ance. To do his, he Equa ion (1) is used simila ly o he common aul loca o . The compu a ional algo i hm is buil wi h logical blocks and unc ions implemen ed in he PSCAD lib a y. I u ilizes ob ained cu en and ol age da a p ocessed using he Fas Fou ie T ans o m. The ou pu o his calcula ion can be compa ed o he ou pu o he p oposed algo i hm.