Energy an Power Engineering, 00,, 8-47 oi:0.46/epe.00.405 Publise Online November 00 (ttp://www.scirp.org/journal/epe) Te Effects of Mutual Coupling an Transformer Connection Type on Frequency Response of Unbalance Tree Pase Electrical Distribution System Abstract Omer Gül, Anan Kaypmaz Electrical Engineering Department, Istanbul Tecnical University, Maslak, Turkey E-mail: gulomer@itu.eu.tr, kaypmaz@itu.eu.tr Receive June 4, 00; revise August, 00; accepte September, 00 In tis paper, a novel armonic moeling tecnique by utilizing te concept of multi-terminal components is presente an applie to frequency scan analysis in multipase istribution system. Te propose moeling tecnique is base on gatering te same pase busses an elements as a separate group (pase grouping tecnique, PGT) an uses multi-terminal components to moel tree-pase istribution system. Using multiterminal component an PGT, istribution system elements, particularly, lines an transformers can effectively be moele even in armonic omain. Te propose moeling tecnique is applie to a test system for frequency scan analysis in orer to sow te frequency response of te test system in single an tree-pase conitions. Consequently, te effects of mutual coupling an transformer connection types on tree-pase frequency scan responses are analyze for symmetrical an asymmetrical line configurations. Keywors: Harmonic Resonance, Unbalance Distribution System, Frequency Scan. Introuction Harmonic stuies ave become an important aspect of electrical istribution system analysis an esign in recent years largely ue to te increasing presence of solistate electronic power converters. Moreover, sunt capacitors are extensively use in electrical istribution systems (EDS) for power factor correction. Due to te proliferation of nonlinear loas, awareness of armonic effects as been increasing [,]. It is terefore tat te possibility of resonance because of sunt capacitor soul ten be analyze by te utilities [-5]. Te first ecision to make in any armonic stuy of istribution system is weter a tree pase moel is require or a single pase moel will be sufficient. Tree-pase istribution systems are generally unbalance an asymmetrical. Hence, asymmetrical treepase istribution systems must be moele by pase co-orinations an teir analysis can be performe eiter uner sinusoial or non-sinusoial conitions [,6-9]. Compare to te single-pase analysis, problem size increases tree-times in pase coorinate base moeling an analysis of EDS. In aition, wen armonics are present in te system, te moels must be realize for eac armonic component, wic requires new metos in tree-pase armonic analysis of istribution systems in orer to ecrease computation time an memory requirement [7]. A number of ifferent stuies relate to armonic moeling an analysis of EDS ave been given in te literature (e.g. [,6]). Grainger [0] applie te matrix factorization tecnique (MFT) to armonic stuies to acieve a significant saving in computational effort. In te paper, only te require columns of te bus impeance matrix wic represent tose busses supplying nonlinear loas are obtaine instea of performing a full inverse. As one of te most common an simple armonic analysis tecnique, frequency scan meto is use to ientify te frequency response of EDS. However tis is not an easy task for some cases as sown in [5,6]. Firstly, bus amittance matrix of EDS becomes bot complicate an large-scale base on te number of busses an tree-pase system components. Seconly, te maximum armonic orer to be consiere is of importance in terms of storage an computational effort for frequency scan base armonic analysis. If te maximum armonic orer to be consiere is as ig as tat of te number of Copyrigt 00 SciRes.
O. GÜL ET AL. 9 busses in te EDS to be analyze a new approac to solve suc systems is neee in orer to ecrease computation time an memory requirements. In tis paper, te following improvements are acieve in moeling an computation tecniques. Multiterminal component concept is use to fin te matematical moel of tree-pase asymmetric EDS in armonic omain []. As for te matematical moels of EDS, pase grouping tecnique, PGT is use []. Te tecnique is base on te separation of same pase buses an components into ifferent groups (PGT) so tat more unerstanable moels can be constitute an savings in memory use can be obtaine. Moreover, MFT is preferre in tis stuy to etermine te frequency scan of te EDS. Differing from Grainger, only te require element of te bus impeance matrix on iagonal wic represent tose busses supplying nonlinear loas are obtaine instea of performing a full inverse of te bus amittance matrix [0]. In tis paper, te aforementione ieas are combine to fin a solution for multi-pase frequency scan of asymmetric EDS.. Multipase Distribution System Moeling Obtaining te general moel of electric circuits wit te ai of multi-terminal element is given in etail in moern circuit teory. General form of algoritms given for multi-terminal elements becomes more simple an unerstanable wen it is use for matematical moeling of power systems. Grap an terminal equations associate wit multi-terminal elements represent te matematical moel of multi-terminal element an sow te wole features of it. In tis section, multi-terminal component moels of a istribution system, wic is use in obtaining te armonic epenent moeling of EDS, is given togeter wit PGT. Harmonic epenent moels are use in orer to fin te frequency response of te network... Basics of Multi-Terminal Approac for Matematical Moeling To obtain te require moels for power system analysis, all buses in te system is generally esire to be sown in te moel an pase to groun voltages are neee for power system moeling. As a result of tis, te grap of electric power systems tat can be represente as a multiterminal element becomes oriente grap of wic common noe enotes te groun an terminals of te grap represent te buses of te system as sown in Figure. Oriente grap of Figure (b) togeter wit Equation () gives te matematical moel of multiterminal component. Multi-terminal component moeling tecnique can be use in moeling of EDS for various aspects suc as single-pase, symmetrical components an pase coorinate moels of EDS witout limitations [,]. Equation () gives te terminal equations. I Y Y Y V (,) (,) (, n) I Y(,) Y(,) Y (, n) V. I Y Y Y V n ( n,) ( n,) ( n, n) nxn n nx.. Matematical Moel of Electrical Distribution System Eac element in an electric istribution system can be represente as a multi-terminal component wit its matematical moel, explaine in etail above. It is terefore an electric power network itself tat can be moele as a multi terminal component as sown in Figure 7 troug te combination of multi-terminal elements, wic is performe by using te parallel connection meto of multi-terminal components. Te terminal equation of tree-pase electric power network in armonic omain is given by Equation (). a a ab ac a I bus Ybus Ybus Y bus V bus b ba b bc b Ibus Ybus Ybus Ybus. Vbus c ca cb c c I bus Ybus Ybus Ybus Vbus (n+)-terminal component r (reference) n r (reference)... n () () Figure. Multi-terminal moeling of n-bus EDS (a) an its grap representation (b). a a a. a n b b b... bn (n+)-terminal component c c c. c n a a reference (groun) a an b b b bn reference c c Figure. (a) Multi-terminal representation of a tree-pase electric power network; (b) Oriente grap of an electric power network. cn c Copyrigt 00 SciRes.
40 O. GÜL ET AL.. Propose Approac Te propose tecnique is base on bot separation of same pase buses an oter power system components into ifferent groups, i.e., eac pase group contains same pase busses an elements, wic can be represente as in te matematical moel of a tree-pase line moel in Subsection.. Since electric power networks are compose of multi-terminal components connecte to eac oter, components moels are firstly presente. Hence, an electric power network itself can be moele as a multi-terminal structure wic is a combination of its constituents. Base on te topology of te system, te combination proceure of multi-terminal elements representing system constituents is carrie out ere troug parallel connection meto of multi-terminal elements, wic is well-known in moern circuit teory []. Since te most common elements in electric power networks are lines an transformers, te moels of tese elements are given only in tis paper. Yet, one can get te oters by following te proceure wic is given in te next section. After getting te element moels in te form of multi-terminal component, te matematical moels are stemme from te proceure as explaine in te following sections... Te Line Moel In general, te lines are represente as -equivalent circuit in most applications. Te series impeance an sunt amittance lumpe- moel representation of te tree-pase line is sown in Figure [,7]. To obtain a symmetrical moel of funamental components, te lines are generally transpose so as to eliminate te effect of long lines. However, tis aim can not be reace wen te system ave armonic components. Furtermore, long line effect takes place in relatively sort istances, if te lines carry signals wit ig frequencies. Due to tese facts, it is a must to use pase coorinate moels in armonic epenent line moeling. In tis case, tree-pase representation of lines as multi-terminal component an its oriente grap are given accoring to PGT in Figure. Accoring to propose approac, te following proceure is given for obtaining te matematical moel of lumpe- moel. Step Neglecting skin effects, armonic epenent series impeance matrix of a line or cable in Figure is given as in Equation () z r jx () were, r an x are funamental series re- a b c y ac Z ab Z bc Z a Z b Z c Z ac y ac y a b c y a y y y b y c y ab y bc y ab y bc Figure. Lumpe- moel representation of te treepase line. sistance an reactance matrices of a line or cable for te pases a, b, c respectively an is te armonic orer. Harmonic epenent sunt amittance matrices of pases a, b an c are given as below: y Re (/ ) Im y j y (4) y Re y j(/ ) Im y Real parts of Equation (4) are neglecte for line an cables. Step By inversion of primitive impeance matrix for Figure, series primitive amittance matrix in armonic omain are obtaine as in Equation (5) y y y y z y y y a ab ac ba b bc ca cb c y y y By gatering te primitive amittance values of same pase elements in one group, te primitive amittance matrix of Figure an its sort form are given as Equation (6) an (7), respectively. a ab ac y 0 0 y 0 0 y 0 0 a ab ac 0 y 0 0 y 0 0 y 0 a ab ac 0 0 y 0 0 y 0 0 y ba b bc y 0 0 y 0 0 y 0 0 bc b bc y 0 y 0 0 y 0 0 y 0 bc b bc 0 0 y 0 0 y 0 0 y ca cb c y 0 0 y 0 0 y 0 0 ca cb c 0 y 0 0 y 0 0 y 0 ca cb c 0 0 y 0 0 y 0 0 y (6) a ab ac y y y ba b bc y y y y (7) ca cb c y y y a b c (5) Copyrigt 00 SciRes.
O. GÜL ET AL. 4 a b c were y, y an y are te sub amittance matrices of pase elements wic are groupe base on k te pases a, b, c. y is a mutual amittance matrix between k an (k = a, b, c; k ; = a, b, c). Step Eac system element of Figure (b) is first assume to be excite by a voltage source in orer to obtain te terminal equation of tree-pase line, wic is represente in te form of multi terminal component in Figure (a). In tis case, te close loop equations for oriente grap of tree-pase line wose components ave been excite by ifferent voltage sources can be arrange by gatering te same pase terminals togeter as in Equation (8). In te same way, basic cut-set equations can be written as Equation (9). a ( V ) b ( ) a B 0 0 U 0 0 V c ( ) b V 0 B 0 0 U 0 a 0 (8) c V 0 0 B 0 0 U b V c V a ( I ) b ( ) a U 0 0 Q 0 0 I c ( ) b I 0 U 0 0 Q 0 a 0 c 0 0 U 0 0 Q I b I c I B, B, B are basic loop matrices relate to pases, a, b an c, an Q, Q, Q are a b c basic cut-set matrices relate to pases, a, b an c, respectively. Moreover, as known, te expression B Q T is vali for all pases. Step 4 By using te Equation (7), Equation (8) an Equation (9), terminal equations in armonic omain can be given as Equation (0), wic is well known in circuit teory [6]. Oriente grap of Figure (b) togeter wit Equation (0) gives te armonic epenent matematical moel of multi-terminal line component. Were a b c a a I V a a I a ab ac V Y b bus Ybus Y bus b I ba b bc V. b Ybus Ybus Ybus b I ca cb c V c Ybus Ybus Ybus c I V I V c c (9) (0) Pase sub-amittance an sub-mutual amittance matrices in Equation (0) are given as Equation () an () for t -armonic orer, respectively. T bus B B, k a, b, c k k k k Y y () T k k k Ybus B y B, k a, b, c; (k ); a, b, c () Equation () an () represent 9 ifferent sub-matrices for eac armonic orer, or rater tese matrices simply form te terminal equation of armonic epenent tree-pase line in te newly propose meto. Tese 9 matrices for eac armonic orer can be calculate inepenently an for tis reason te matrix calculations can be one at te same time by more tan one computer in parallel processing. Simplifie assumptions. ) a) As sown in Equation (), te primitive armonic amittance matrices associate wit eac pase can be assume to be equal on te conition tat te pase caracteristics suc as conuctor size, line lengt, pase material an te number of component belong to eac pase are same. y y y y () a b c pase b) As sown in Equation (4), te mutual amittance matrices of t -orer can be assume to be equal on te conitions tat long line effect is neglecte an te lines are transpose, wic leas to symmetrical mutual coupling between lines. y y y y ab ac ba bc ca cb m y y y (4) As a consequence, wen assumptions are mae relate to Equations () an (4) it is enoug to form te primitive amittance matrix representing te wole treepase line (as in Equation (0)) by just using bot onepase primitive amittance matrix an a mutual amittance matrix. ) As sown in Equation (5), basic loop matrices are assume to be equal wen te topology of pases is ientical. a b c B B B B (5) Instea of using te Equations ()-(), wic contain 9 ifferent equations for eac armonic, only te Equations (6) an (7) can be use in orer to form te armonic epenent terminal equation of te line on te conitions tat te performance equations of a primitive network are as Equations ()-(4) an te topology of pases are ientical. Copyrigt 00 SciRes.
4 O. GÜL ET AL. a b c T pase Y = Y = Y = B y B (6) bus bus bus b b b b c Y = Y = Y = Y ab ac ba bc bus bus bus bus ca cb T m = Y bus = Y bus = B y B (7) a a 7-terminal component c c a a c Te algoritm given above for lines is also vali for power networks compose of more tan one line. In tat case, if a mutual coupling between ifferent tree-pase lines exists, all tese lines must be moele as a single multi-terminal component. For tis reason, te formation of matematical moel is realize as explaine above. Consequently, as one migt expect tat te propose algoritm can be applie to a system wose bus numbers are ifferent at ifferent pases, wic sow anoter merit of te propose moeling tecnique... Tree-Pase Transformer Moel Magnetizing current in transformers leas to armonic currents ue to its saturate core. Due to te fact tat te transformers soul be moelle in armonic omain so tat armonic currents are require to take place in te moel [,7]. However, armonic currents is not inclue in te transformer moel since our interest in tis stuy is to etermine te frequency response of EDS. Matematical moel associate wit any of transformer can be obtaine by utilizing te concept of multi-terminal component. However, te most common transformers in use, i.e., Y- connecte tree-pase transformers are preferre ere to sow te potential application of te propose meto. In respect of te propose meto, a transformer moel is given ere in te case tat te mutual coupling between pases of primary an seconary winings is not neglecte. Y- connecte tree-pase transformer is illustrate in Figure 4. Step Y- connecte tree-pase transformer is represente as a multi-terminal component in Figure 5. Wen it is esire to form te matematical moel of te transformers wit isolate neutral point, one soul take te neutral point into consieration. As a result, te terminal number in multi-terminal representation of transformer is increase from 7 to 8, wic leas to increase imension in oriente grap an terminal equations. Step In multi-pase system representation, te power transformer are represente by reactance an resistance matrices for eac pair of winings. Accoring to propose PGT, te primitive amittance matrix of te transformer t armonic orer is given by Equation (8). r Figure 4. (a) Tree-pase representation of a line in te form of multi-terminal component; (b) Oriente grap of a tree-pase line. V a I a I V b b V c I c I a I b I c r V a V b V c Figure 5. Y- connecte tree-pase transformer wit single core. y ym ym y m y m ym ym y ym ym ym ym y y y y y y m m m m m y y m y m y m y ym ym y y y y y y y y y y y y m m m m m m m m m (8) were, y is te amittance at primary sie y is te amittance at seconary sie ym is te mutual amittance between primary an seconary sie at same pases ' y m is te mutual amittance between primary coils. '' ym is te mutual amittance between primary an seconary coil on ifferent cores. ''' y m is te mutual amittance between seconary coil. As a result of PGT, te primitive amittance matrix in Equation (8) as a simple an symmetrical structure, wic can easily be sown in its sort form as in Equation (9). pase m m y y y m pase m y y y y (9) m m pase y y y Copyrigt 00 SciRes.
O. GÜL ET AL. 4 m y is generally taken as zero for eac armonic. It is terefore tat te primitive matrix y becomes equal to te primitive matrix of tree single-pase transformers connecte to eac oter, base on teir connection group. Step Transformers performance equations in real values are converte to expressions in p.u. by using te turns ratio of transformers. Moreover, pase sifting originate from connection group of tree-pase transformer (Delta-Wye) soul be inclue in matematical moel of te transformers. Furtermore, instea of moelling te voltage regulators iniviually, pase sifting of voltage regulators can also be inclue in transformers matematical moel on te conition tat te moel of voltage regulator itself is not neee. In tis stuy, transformers performance equations are assume to be given in p.u. an te equivalent circuit of Figure 6 is use in orer to inclue bot magnitue an pase sifting in te matematical moel. Apart from tese, te pase sifting operations in pase sifting tree-pase transformers are inclue in te matematical moel in te same way. Wen te aforementione conitions, wic represent te very general form of transformer moel, are taken into consieration, voltage ratio for eac pase in tree -pase transformer are given as Equation (0) wit te assumption tat pases ave ifferent complex turns ratio. As for te current turns ratio of te transformer, it is given in Equation () as complex conjugate of voltage turns ratio. k k αv 0 δ v, k a, b, c k 0 βv k k * (0) δi δ v, k a, b, c () Wit tese assumptions, sub-matrices of primitive amittance matrix of t armonics are given in general form as Equations () an (). a a kk k pase k y δ i. y. δ v, k a, b, c () b b 8-terminal component r c c a a b b c Figure 6. (a) Multi-terminal representation of Y- connecte tree-pase transformer; (b) oriente grap of Y- connecte transformer. r c I Y V V Figure 7. Basic equivalent circuit in p.u. for coupling between primary an seconary coils wit bot primary an seconary off-nominal tap ratios of an. k k m y δ i. y. δ v, k a, b, c; (k ); a, b, c () Since pase sifts in voltage an current equations ue to transformer connection type are same for all pases, te primitive amittance matrices can be given as Equations (4) an (5). As a result of -connection of te transformer, canges in pase angles an magnitues in tese equations are same. For tis reason, te structure of primitive amittance matrix as become simple an powerful in computation. I *.. y y y δ y δ (4) a b c pase v v y y y y y y δ y δ ab ac ba bc ca * m.. cb v v (5) In tis stuy, pase sifting ue to transformers connection group is inclue in te matematical moel troug above equations. Step 4 To form te terminal equations (or matematical moel) of tree pase transformer, close loop equations for voltage soul be obtaine as Equation (6). * V B U 0 (6) V As sown in Equation (7), let B be ivie into two sub-matrices as B an B, wic are sub-loop matrix relate to pases an basic loop matrix associate wit te noes of star connection, respectively. B B B (7) Te matrix B exist wen te star connection noe is not groune. As for te matrix B, it can be written as Equation (8) for te six most common connections of tree-pase transformers (Yg-Yg, Yg-Y, Yg-, Y-Y, Y-, -). U B O B O U B O U B (8) Copyrigt 00 SciRes.
44 were, te sub-matrix B given as below is a matrix associate wit -connection of tree-pase transformers. ( ) 0 B 0 ( ) (9) In Equation (9), te coefficients an take te values or 0 base on te transformers connection type. Wilst is wen te primary winings of transformer is wye, is 0 wen te primary sie of te transformer is elta. As for te coefficients, it takes te value wen seconary sie is wye, an takes te value 0 wen seconary sie is elta. Furtermore, te sub-matrices U an O are given as below 0 0 0 U an O 0 0 0 (0) Basic sub-loop matrix B in Equation (7) can be given as Equation (), if te star connection noes are not groune. T B b b b () If star connection noe in one of te primary an seconary winings is not groune, te sub-matrix b in Equation () can be given as Equation (). b λ γ () If te star connection noes in bot sies are not groune, ten, te sub-matrix b is given by Equation () λ 0 b 0 γ () Step 5 As a consequence, bus amittance matrix of a transformer is given as a function of primitive amittance matrix an loop equations as Equation (4). T Y bus B.. B y (4).. Oter Components Loaing soul be inclue in te system representation because of its amping effect near resonant frequencies. However, an accurate moel for te system loa is ifficult to etermine because te frequency-epenent caracteristics are usually unknown an te loa itself is canging continuously. In general, if a loa is linear, te loa is represente as an amittance using te CIGRE loa moel at te intereste frequencies [4]. If loa is nonlinear, te loa is represente as an open circuit in frequency scan stuy. O. GÜL ET AL. Capacitors are often place in istribution networks to regulate voltage levels an reuce real power loss. Capacitor bank size an locations are te most important factors in etermining te response of istribution system to a armonic source. For accurate representation of capacitors, a sunt capacitor can be moelle as wye connecte or elta connecte constant amittance [,,7]. For armonic stuies of EDS, it is usually sufficient to represent te entire transmission system by its 50 Hz sort-circuit equivalent resistance an inuctance at te ig sie of te substation transformer [7]. 4. Tree Pase Frequency Scan Analysis An electric energy system simply consists of te resistances (R), inuctances (L) an capacitances (C). All circuits containing bot capacitance an inuctance ave one or more natural resonant frequencies [5,4-7]. Normally electric energy systems are esigne to operate at frequencies of 50 Hz so as not to be uner resonance for funamental frequency. However, certain types of loas prouce currents an voltages wit frequencies tat are integer multiples of te 50 Hz funamental frequency. Tese iger frequencies are a form of electrical istortion known as power system armonics. Wen one of te natural frequencies correspons to an exciting frequency prouce by non-linear loas, armonic resonance can occur. Voltage an current will be ominate by te resonant frequency an can be igly istorte. It is terefore tat for all effective armonic frequencies, te system soul be analyze for weter a resonance is going to occur or not. Frequency scan analysis is te most common an effective meto to etect te armonic conition in a network. Te simplest way to etermine te frequency response of a network is to implement frequency or impeance scan stuy. Te process of frequency scan stuy can be performe by solving te network equation for te frequencies of.f o. Here, f 0 is te funamental frequency an is te armonic orer. V Z I (6) were V is te noal vector, I is te current vector. Te aim of te frequency scan stuy is to etermine impeance as a function of frequency. Te frequency scan tecnique basically involves following steps: A current injection, wic is a scan of sine waves of magnitue one, is firstly applie to te bus of interest. Seconly a resultant voltage of tat bus is measure. V I Z V 0 Z Z (7) For large-scale systems, Z is simply erive from bus impeance matrix, wic is efine by taking te Copyrigt 00 SciRes.
O. GÜL ET AL. 45 inverse of bus amittance matrix as below: Z - Y (8) an te expane form of noal impeance matrix at any frequency is Z Z Zn Z Z Zn Z (9) Zn Zn Znn were Z ij = transfer impeance between noes i an j Z ii = riving point impeance at noe i, r. Te effects of, te transformer connection types an te mutual coupling between lines on frequency response are examine on a tree-bus inustrial test system sown in Figure 8 by using te frequency scan tecnique. Te test system consists of tree busses utility, IND an IND. IND an IND busses are connecte troug a sort tree-pase an four-wire line. Te system is supplie by te utility troug 69/.8 kv transformer. Wile a motor an linear loa are connecte on bus IND, a armonic proucing nonlinear loa an a linear loa are connecte on bus IND. Harmonic currents of te nonlinear loa are given in te Table. Since zero sequence armonics are not foun an since only one armonic source is present in te test system, te system can be assume to be balance an symmetric. Tat is wy single pase analysis can be use to solve tis system. Te values on Table are calculate in pu system. Te selecte base quantities are 0.000 kva an.8 kv. Te ata an calculations are available on te web site ttp://www.ee.ualberta.ca/pwrsys/ieee/ownloa.tml. Te following assumptions are mae in te analysis. ) Te loa points are supplie from an infinite bus system. ) Te linear loas are moele wit its series resistance an reactance. ) For te motor loas, a locke rotor impeance are use [,,]. In tis stuy, tree-pase moels are use an following four-cases are consiere in te frequency scan simulation of te test system. Te frequency responses wit tree-pase moels are given in comparison to single-pase moels in te Figures 9- for eac of te following cases. Firstly, te transformer connection type is selecte as wye-groune/wye-groune. In te secon case, te mutual coupling between lines is taken into consieration for te transformer connection type as wye-groune/wye-groune an te value of mutual impeance is taken as one-tir of pase impeance. Te system is still symmetric in tis case. So, only one-pase frequency response analysis is enoug for te system. 69kV.8kV IND IND Figure 8. Te consiere tree-bus inustrial test system. Figure 9. Frequency responses of single an tree-pase EDS (transformer connection type is wye-groune/wyegroune). Figure 0. Frequency responses of single an tree-pase EDS (transformer connection type is wye-groune/wyegroune an tere is a mutual coupling between lines). Figure. Frequency responses of single an tree-pase EDS. Copyrigt 00 SciRes.
46 O. GÜL ET AL. Table. Harmonic current spectrum of nonlinear loa at bus IND. 5 7 7 9 5 %Ic 0. 0.4 0.09 0.077 0.059 0.05 0.04 0.04 Ic 0.9 0.085 0.054 0.046 0.05 0.0 0.06 0.04 0 0 0 0 9 5 7 4 4 47 49.5.5 Z (p.u.).5 0.5 pase, a %Ic 0.04 0.0 0.09 0.07 0.04 0.0 0.0 0.0 Ic 0.00 0.09 0.07 0.06 0.04 0.04 0.0 0.0 0 0 0 0 In te tir case, te transformer connection type is selecte as wye-groune/elta an it is sown in Figure tat elta connection of transformer as an effect on frequency response similar to tat of case-. Finally we a te system moifie so as to ave an asymmetric tree-pase network. Te asymmetry in te fourt case is obtaine by canging te compensation capacitors values for te pases a, b an c as a XC 0.455j p.u, X b C 0.068j p.u. an c XC 0.9j p.u respectively. As in te secon case, mutual impeance is taken as one-tir of pase impeance an te transformer connection type is wye-groune/wye-groune too. It is sown in Figure tat frequency responses of all pases are ifferent from eac oter. Hence, frequency responses of eac pase in asymmetric networks soul be etermine iniviually. For large-scale istribution systems moele an analyze by pase coorinates, te moifie MFT will reuce te number of computation for large-scale networks since te MFT is capable of computing only te necessary elements in impeance matrix instea of taking a full inverse of te bus amittance matrix Because of te asymmetric compensation capacitors for te pases, tree-pase frequency response becomes ifferent from single pase response on te conition tat coupling between lines an/or transformer connection types are taken into consieration in tree-pase moeling. Te frequency responses of all pases in asymmetric networks are ifferent from eac oter. 5. Conclusions In tis paper, to provie savings in storage an computation time, frequency scan analysis in multipase asymmetric istribution system is realize eiter by combining PGT an MFT or iniviually. Te solution algoritms are base on PGT an MFT. Wilst te PGT uses 4 Z (p.u.) 7 6 0 5 0 5 0 5 0 5 40 45 50 7 6 5 5 Z (p.u.) 4 (a) pase, b 0 5 0 5 0 5 0 5 40 45 50 (b) pase, c 0 5 0 5 0 5 0 5 40 45 50 (c) Figure. Frequency responses of asymmetric tree-pase network (a) frequency response of pase a; (b) frequency response of pase b; (c) frequency response of pase c. te concept of multi-terminal component moeling tecnique to obtain te armonic epenent moel of EDS in terms of frequency scan, te MFT uses only te require element of te bus impeance matrix on iagonal to etermine te frequency scan of te EDS. Terefore an MFT base algoritm is given to fin te latter. In orer to sow te accuracy of te propose tecnique, a symmetric tree-pase inustrial system wit -bus is preferre for simplicity. As a result, single-pase an tree-pase frequency responses of EDS are ob- Copyrigt 00 SciRes.
O. GÜL ET AL. 47 taine. Te results sow tat tree-pase frequency response becomes ifferent from single pase response on te conition tat coupling between lines an/or transformer connection types are taken into consieration in tree-pase moeling. In aition, frequency responses of all pases in asymmetric networks are ifferent from eac oter. Consequently, besie in asymmetrical moeling, one can easily extract tat a pase-coorinate base moel must be use to etect te frequency response of EDS, even in te symmetrical moeling. 6. References [] IEEE St. 59-99, IEEE Recommene Practice an Requirements for Harmonic Control in Electric Power Systems, IEEE Press, New York, 99. [] J. Arrillaga, B. C. Smit an N. R. Watson, Power System Harmonics Analysis, Jon Wiley an Sons, New York, 997. [] IEEE Guie for Application of Sunt Power Capacitors, IEEE Stanar 06-99. [4] W. Xu, X. Liu an Y. Liu, Assessment of Harmonic Resonance Potential for Sunt Capacitor, Electric Power System Researc, Vol. 57, No., 00, pp. 97-04. [5] R. A. W. Ryckaert an A. L. Jozef, Harmonic Mitigation Potential of Sunt Harmonic Impeances, Electric Power Systems Researc, Vol. 65, No., 00, pp. 6-69. [6] M. F. McGranagan, R. C. Dugan an W. L. Sponsler, Digital Simulation of Distribution System Frequency- Response Caracteristics, IEEE Transaction on Power Apparatus an Systems, Vol. PAS-00, No., 98, pp. 85-855 [7] IEEE Power Engineering Society, Tutorial on Harmonic Moelling an Simulation, IEEE Catalogue Number: 98TP5-0, Piscataway, 998. [8] M. A. Laugton, Analysis of Unbalance Polypase Networks by te Meto of Pase Coorinates, Part System Represantation in Pase Frame of Reference, Proceeing of IEE, Vol. 5, No. 8, 968, pp. 6-7. [9] W. E. Dillon an M.-S. Cen, Power System Moelling, Proceeing of IEE, Vol. 6, No. 7, July 974, pp. 90-95. [0] L. G. Grainger an R. C. Spencer, Resiual Harmonic in Voltage Unbalance Power System, IEEE Transactions on Inustry Applications, Vol. 0, No. 5, 994, pp. 98-405 [] O. Gül, Harmonic Analysis of Tree Pase Distribution Networks by Utilizing Concept of Multi-terminal Components an Pase Coorinates, P.D Dissertation, Istanbul Tecnical University, Istanbul, 00. [] O. Gül, A. Kaypmaz an M. Tanrıöven, A Novel Approac for Tree Pase Power System Moelling by Utilizing te Concept of Multi-Terminal, International Review on Moelling an Simulations, Vol., No., February 00, pp. 90-00. [] D. E. Jonson, J. L. Hilburn an R. J. Jonson, Basic Electric Circuit Analysis, Prentice Hall, Englewoo Cliffs, 990. [4] CIGRE Working Group 6-05, Harmonics, Caracteristics Parameter, Metos of Stuy, Estimates of Existing Values in te Network, Electra, No. 77, 98, 5-54. [5] Z. Huang, W. Xu an V. R. Dinavai, A Practical Harmonic Resonance Guieline for Sunt Capacitor Application IEEE Transactions on Power Systems, Vol., No. 0, p. 64. [6] T. A. Haskew, J. Ray an B. Horn, Harmonic Filter Design an Installation: A Case Stuy wit Resonance, Electrical Power System Researc, Vol. 40, No., 997, pp. -5. [7] CIGRE 6.05, WO CCO Report Guie for Assessing te Network Harmonic Impeances. List of te Symbols an Abbreviations EDS Electrical Distribution Systemb PGT Pase Grouping Tecnique MFT Matrix Factorization Tecnique α Primary off-nominal tap ratings β Seconary off-nominal tap ratings V Voltage pasor I Current pasor Y Amittance pasor Z Impeance pasor Y Primitive amittance Z Primitive impeance B Basic loop matrices N Bus numbers Star point inex Delta connection inex a, b, c Line inexes M Mutual coupling armonic inex * Complex conjugate Branc inex j Imaginary unit V Voltage inex I Current inex Matrix Matrix transpose T Copyrigt 00 SciRes.