Power System Modeling-Mo-Shing Chen

7/22/2019 Power System Modeling-Mo-Shing Chen

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PROCEEDINGS OF THE IEEE, VOL 62, O. 7, JULY 1974 901

,121 R. Billinton, P o w n S y s t e mReliabilityEvaluation. New York:Sys t . , vol. 82, pp. 726-735, Oct. 1963.

[3] R. Billinton and A . V. Jain, The effect of rapid start and hot eserveGordon and Breach, 1970.units in spinning reserve studies, ZEEE Trans . Power App . Sys t . ,vol. PAS-91, pp. 511-516, Mar./Apr. 1972.[4] -, Spinning reserve allocation in a complex power system,presented at the IEEE Winter Power Meeting, New York, N Y .1973, Paper C 73 097-3.[ 5 ] Reliab le loading of generating units for system operation,In 1973 Proc. Power Indus try Com pute Ap pl ica t ion Cmf . , pp. 221-229.(61 -, Interconnectedystem spinning reserve requirements,I E E E T r a n s .Power A pp . Sys t .. vol. PAS-91, pp. 517-525, Mar ./Apr.1972.[7]A. D. Patton,Short erm reliabilitycalculation, I E E ET r a n s .Power App . S ys t . ,vol. PAS-89, pp. 509-513, Apr. 1970.[a] -, A probability method for bulk power system security assess-ment, I-Basic concepts, Z E E ETrans . Power Ap p . Sys t . , vol.PAS-91, pp. 54-61, Jan./Feb . 1972.[9] J. A. Bubenko and M.Anders on, Probabilistic evaluation of the

Computer Appl ica tion Cm f . , pp. 240-249.operation reserve in a power aystem, in 1973 Proc. Power Industry[lo] A. Przyluski and 2. Reszaynska, Method of availability aesess-4th Power Systems ComputationC m f . , 1972.ment of power generating capacity in short term lanning, in Proc.1111 C.Singh and R. Billinton, A frequency and dura tion approach t o

short erm reliability evaluation, presented a t th e IEE E WinterPower Meeting, New York, N . Y., 1973, Paper T 73 094-0.[12]B. S. BiggerstaiT and T. M. Jackson, The Markov procesn as ameans of determining generating-unit st at e probabilities for use inspinning reserve applications, IEEE Tra ns . Power App . S ys t . , vol.PAS-88, pp. 423-430, Apr. 1969.[13] R. Billinton and A. V. Jain, Uni t dera ting evels in spinning reservestudies, Z E E ETrans . Power App . Syr t . , vol. PAS-90, July/Aug.1971.[14] A. D. Pa tton , A probability method for bulk power system securityassessment-111-Models for stand -by generators and field da tacollection and analysis,ZEEE Trans . Power A pp . Sys t ., vol. PA S91, pp. 2486-2493, Nov./Dec. 1972.[15] L.T. Anstine et al., Use of outage stati stics for operating and spin-

[la ] A. D. Patton, A probability method for bulk power system securityning reserve assessments, Paper 71 CP 701-PWR, Sept. 1971.assessment-11-Development of probability models f o r normally-operating components,ZEEE Trans .Power A pp . Sys t ., vol. PAS-91,[17]A. DiMarco,Asemi-Markov model of a hree-stategeneratingpp. 2480-2485, Nov./Dec. 1972.unit, Z E E E T r a n s. P o w n A p p . S y s t . , vol. PAS-91, pp. 2154-2160,Sept./Oct. 1972.[la1 C. Singh. R. Billinton, and S. Y. Lee Reliability modeling using the.deviceof stages, in 1973 Proc. Pow& Industry Computer Applicatio1191 A. D. Patton, discussion of [9], I E E E T r a n s . P ow e r A p p . S y s t . ,vol.Conf., pp. 22-30.PAS-93, pp. 17-18, Jan./Feb. 1974.

Power System Model ineMO-SHING CHEN, SENIORMEbfBER, IEEE, AND WILLIAM E. DILLON, MEMBER, IEEE

Invited paper

Absfraci-A dimmion of the philosophy of modeling of three-phase trannminnion lines, three-phase transformers, three-phasegenerators, and power system loads is presented. Although the topicis very basic, the material covered is not all conventional. Single-phase representation of a three-phase power system is discussed indetail. Assumptions usually employed in the power ndustryarestated. Also di scnssed is the mathematical representation of a non-symmetrical three-phase power systeniin which the symmetrical-component method isnot applied.An important aspect isthe study ofthe models used in present-day problems aswell as he models thatmay be requiredin the near future.

A INTRODUCTIONTH RE E- PH AS E power system consists mainly of theinterconnection of generators, transformers, transmis-sion lines, and loads. The elements in power systemare relatively simple; however, the networks which we ar efacing are ruly arge scaled. Power system engineers haveused the widely knownsymmetricalcomponentmethod osimplify many of the problems inpower system analysis. I t isthe intention of the authors to discuss the fundamental con-

Manuscript received December 27,1973.The authora are with theEnergy Systems Research Center, TheUniversity of Texaa at Arlington, Arlington, Tex. 76010.

cepts of symmetrical components in power system modelingin this paper. Although the topic is very basic, the materialcovered is not al l conventional. I t is hoped that this paperwill bring a clear picture of the single-phase representat ion ofa power system.

The models we discuss in this paper are intended for sys-tem studies. Fundamental concepts are emphasized.We havetr ied to explain the concepts of power system modeling todayin sucha way tha t itprovides room for possible modificationsfor future works by the user.

THREE-PHASERANSMISSIONINESThe most common elementf a three-phase network is th

transmission line. The interconnec tionof these elements formthe major partof the power system network. An understand-ing of the representation of a three-phase ine snotonlynecessary in the modeling itself, bu t also important for theanalysis of the whole power system. In steady-state roblems,three-phase ransmission ines are represented by lumped-*networks; series resistances and nductances between busesare lumped in hemiddle,andshuntcapacitances of thetransmission lines are divided into two halves and lumpedtbuses connecting the lines.


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902 PROCEEDINGSOF THE IEEE, JVLY 1974Series Impedance of a Transmission Line

Fig. 1 shows a three-phase ransmission inewith woground wiresw and v in which only series impedances re con-sidered. Let us write the network equation for the phase :

Similar equations maybe written for b, G, w , and v lines, or inmatrix form:

I 1

The subscript g indicates he ground return. The physicalmeanings and the methodof calculation of all the elementsfthe impedance matrix are given in [l] . A power system in-volves a connection of.many transmission lines. Since the de-signs of the lines are different (for instance, three-phase linesmay have a different number of ground conductors, rans-mission ines mayusebundledconductors,etc.), it is veryimportant for the power system engineer to replace the actualline by an equivalent three-conductor lineor system analysis.This procedure is acceptable because we are interested onlyin the performance of phase conductors. The calculation ofthe system problems is greatly simplified when all the linescan be represented by an equivalent three-conductor system.

Expressing (1) in partitioned notation,B-RB (2)Expanding (2) and acknowledging that the ground-wire volt-ages are zero,:

Avh = Z AIO ~C Z B I ~ . (3)

where

w - WV LJ Va I I

bC

b 1 C IC-

vc VC'//// ./// / //.-,= 1 I I.+ I, I

Fig.1. Three-phaaz trammissioncircuitwitht w o ground conducted.

equivalent three-conductor circuit by the impedance reduc-tion of ( 7 ) . This technique s applicable, to any number ofcircuits with any number of ground wires or bundled con-ductors. The procedure for calculating the equivalent three-conductor line is outlined in detail in 2] and [3]. When morethan one circuit is on the same right of way, the size of t heimpedance matrix should equal the number of circuits multi-plied by 3. Table I shows the impedance matrix of a typical345-kV double-circuit ( u n t r a n s p d ) 2-795 steel-reinforcedaluminum cable (ACSR) bundle, #-in ground wire. The towerand conductor configuration is shown in Fig. 2.Symmetrical Components Transfm matw n

I t has been shown that the impedance matrix of a three-phase line canbe found as

2 2I ca-g I 4 *-JEquation (8) gives the impedance of a typical transmissionline. If the line is completery transposed or canbe assumed tobe completely transposed, the impedance matrix will be re-duced to

Although few transmission lines currently in use are actuallytransposed, power system engineers customarilyuse the sym-metrical- line impedance matrix of ( 9 ) for every line. This is areasonable assumption for allbut the longestines, and allowsthe eduction of the hree-phasenetwork o hree single-phase networks. We shall begin the analysis of the symm etri -cal problem by finding the eigenvalues of the matrix in 9) :

r = Z + 2 M7 = Z My = Z - M .

Th e first eigenvalue gives the following eigenvector:

x - i or aTh e five-conductor configuration has been reduced to an whereK s any arbitrary number.


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CHEN A N D DILLON: POWER SYSTEM MODELING 903TABLE I

SERIES IMPEDANCE MATRIXFOR THREE-PHASE DOUBLE-CIRCUITRANsMISION LINEA B C a b C

0.1904 t 0.1245 +.1294 +0.1260 t.1246 t.1295 +A j0.4599O.45050.5261 j0.5423O.51841.09610.1295 + 0.1278 +.1333 +.1294 +.1280 +.1975 +

jO.5189 j0.4152 jO.4510 j0.4599 j0.51471.0903C

=a =AEC

b

and

mation (similar ity transformation) can be used to diagonalizethe completely transposed line byB ml[B F i [A 2 dV

IC 1 . . r2

where a= 1/120, u * = 1/240= 1/- 120, and-Fig. 2. Double-arcuitransmiasion line. [i] [;I and [ ITh e second and hird eigenvalues require an y choice ofXI, XI, and x such hat x1+xz+xs=ossatisfied. For are theeigenvectors of the mpedancematrix.example, 13), and (14),

I t is nteresting to note that forallcompletely ransposedsystems, the common eigenvectors are the same although theeigenvalues are different in each system. If the original phaseimpedancematrix Z h isnotperfectly ymmetrical, i.e.,Z-ZZ,,, etc., the eigenvalues and eigenvectors are differ-ent in each case.

Th e series voltage-drop equation of a transposed line is

1 (12) (19)Avc E in whichZ 2M, 2- M , and Z M are theeigenvalues of heEquations (10) and (11) suggest tha t the following transfor- system, The system described by (18) represents an uncoupled


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904 PROCEEDINGS OF THE IEEE, JULY 974

Fig. 3. (a) Original three-phase magneticallycoupled line. (b) Equiva-lent network consisting of three uncoupled single-phaee arcuits .

TABLE I1THPBE-PHAS~OUBL~-CXRCUITRANSUSSION'L~ESY~~~~ETRICAL-COMPONENTMPEDANCE MATRIX OP

0 + - 0 + -jZ.1456O.00159.0102l.34450.4468 + -0.0103 + 0.0081 + 0.3819 +0.0081 + 0.0647 + 0.0109 + -0.0170 +

j0.01020.5697-0.0101 j0.0503-0.0103 + -0.01U + 0.06W + 0.0150 +j0.00S j-0.00990.5697 ] jO.0417-

1.3445 j0.04170.05032.14560.3819 + 0 . OW -0.0170 + 0.4468 +

j0.0417 j0.05430.0150 + -0.0170 +

jO.0lse

j0.Oleeo.0062 o.oO02 +0 . 0 0 ~+jo.0059

0.0002 + O.O(n6 +

0.0103 + 0.0081 +j0.00lS j0.01020.0647 + 0.0109 +

j0.56970.0647 +0.OU3 +

j-0.0101

j-0.0099 j0.5697

system. Physically, the original three-phase circuit has beenreplaced by three uncoupled single-phase circuits as shownin Fig. 3.

The equations describing these relationships constitute thesymmetrical-component transformation:

and

where VO, I, nd z ar e called the zero-sequence, positive-sequence, and negative-sequence voltages; IO, 1 and IS arecalled the zero-sequence,positive-sequence, andnegative-sequencecurrents;and ZO Z1, and ZS re called the zero-sequence, positive-sequence, and negative-sequence imped-ances. By using the method of symmetrical components, thevariables are separated from one another, and a l l quantitiesare expressed in terms of quantities of components only. Aspointed out earlier, the symmetrical componentsill decoupleall transposed hree-conductor lines. The choice of the T .matrix is such that itwill, at the same time, benefit the gen-erator and the transformer modeling.

For a system with more than one three-phase circuit inparallel, the voltage-drop equations can be written as

ZERO - U E l . ~ 0 s . - ET.EG. -UET .o o 2 7 1 Zl

21100

Fig. 4. Uncoupled ir uit equivalent for double-circuittransmission ine.

where

ab6

xABC

Let

v - [P .

v*

I . [=A

A B C

Vp = TVI p = T I

where

Substituting (21) and (22) into (20),v = Z I

where 'Z TIZpT.The ymmetrical-component mpedancematr ix of the

double circui t of Tab le I and Fig. 2 is shown in Table 11. Asyou can see, all the off-diagonal values are small except theterms which represent the mutual coupling between th e zero-sequence line on one circuit and the ero-sequence line on theother circuit. This double-circuit transmission line is repre-sented in most of the analyses today as shown in Fig. 4. Thesmall mutuals between the symmetrical-component sequencenetworks are neglected. I t should be noted th at the mut ual


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CHEN AND DILLON: OWER SYSTEM MODELING 905coupling between the two zero-sequence networks will existeven if complete transposi tion of the lines is achieved.

Shunt Capacitance of Three-phase Transmission LinesA Capacitancematrix,elated to phase oltages and

charges as shown in (23), s calculated by inverting a potentialcoefficient matrix.

For a system with two ground wires, an equation may bewritten for the potential of the systemof conductors.

Expressing (23) in partitioned notation corresponding to t hepartitioning lines shown,

Expanding (24) and acknowledging t ha t ground-wire poten-tials are zero, then

V& = ( K - LN-'M)Q,,ac = P L Q h (25)where

Here P& is a 3 x 3 matrix which includes the effects of theground wires. The capacitance matrix of the system is foundby

Physically, we have eplaced he ransmission ineswithground wires by the equiva lent lines, withou t ground wires.This technique is similar to the series-line constant calcula-tion. A more detailed discussion on bundled conductors andmultiple circuits is given in [2] and 131. The elements in thecapacitancematrixhavea physicalmeaning in he hree -phase model as shown in Fig. 5(a).

If the line is perfectly symmetrical, all off-diagonal t ermsof C are he same and all diagonal erms of C are thesame , i.e.,

c & c Hc' -c' cTh e eigenvalues of this matrix areC- 2C', C+ C', and C+ C'.The symmetrical-component transformation may be used to

( b )Fig. 5. (a) Physical equivalents of t he elements of t he C k matrix. (b)Positive-, egative-, and zero-sequencenetworka for the diagonalizesystem of COB.

diagonalize the capacitance matrix. Rewriting (25) for a per-fectly symmetrical system,

Qoac = C&VoacQOIZ = Ta-'CetwTaVOlt

= olzvolzwhere

inwhich C-2C', C+ C , and C+C' are called the zerosequence,positive-sequence, and negative-sequencecapaci-tances of the system. The system described by this equationrepresents an uncoupled system. Physically, theriginal threephase circuit has beenreplaced by hree uncoupled singlephase circuits as shown in Fig. 5(b).

If t he original phase matrix is not perfectly symmetrical,the use of symmetrical-component transformation will resulwith mutual terms in the transformed matrix. The magnof the mutua l terms depend upon the symmetry of the original matrix. In t his case,

The abc and symmetrical-component shunt capacitance ma-trices of the double circuit of Tabl e I and Fig. 2 are shown iTables I11 and IV.

SYNCHRONOUSENERATORSCylindrical-Rotor Machine

I t is the purpose of this paper to develop basic models aswell as fundamental concepts of a three-phase synchronousgenerator. Synchronous machines have been a topic of muchstudy in the past. O ur discussion will be limited to the symmetrical-component representation of the m achine in the stem simulation. We shall begin with a machine of cylindricarotor under a steady-state condition. Salient pole machinesand the effects of rotor circuits will be discussed later. Withconstant field current, the voltage equation of the four cou


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906 PROCEEDINGS OF THE IEEE, JULY 1974TABLE I11

SHUNT ADMITTANCE MATRIXFOR DOUBLE-CIRCUITTRANWISSIONINESHOWN N FIG.2(Admittance values may be found by multiplying the correspondingelements of the capacitance matrix byjo.)

TABLE IVSWETRICA L-CObW ONENT SHUNT ADMITTANCE MATRIX ORDOUBLE-CIRCUITRANSMISSIONINESHOWN IN FIG.

0 + 0 +0.0 +O j-0.2430 I j-0.24304.5399o.1036o.10361.0140 0.0 + -0.1695 + 0.1695 +A B C A B C

0.0 + 0.0 +.0 +.0 +.0 +.0 +I-0.3918 j-0.9711 j6.2823-0.9204 j-0.1692-0.5036- 0.0 +.0 +.0 +.0 +.0 +.0 +j-0.2294 I j-0.1692 1 j-0.0789 11 j-1.0576 j-0.9711 I j6.5106

pling coil sys tem of Fig. 6 can be written asdXaat

V , = - - Lr , + Vn

dat

= La f os (e 20 ' ) 1~ L i . i b - bci ]- brb + Vn

dat= - L M I f in e 200) +- , i , i b - ~ i ]

- Gn+ Vnd x Catdat

cy,+ VnV , = ,r, + V

= L Q f os e - 40'11~ - L,i. Lsib - ~ , i , ]

dat= - wLi fI fsin e 400) + ,i, b - L&]

crc+ Vn (2 6 )where B=wt+Bo is heanglebetween he axis of the fieldwinding and theaxis of the phase a armature winding, an d ois synchronous speed of the machine.

For a cylindrical-rotor machine,L a = = LC,= L , ?Q = rb = =L a = L , = & = -L'Vn = nzn = - Ia + Ib + IC)&*

I I+ -0.0288 +0.0000 +.16950.1451 +.0 +.0263 +j0.1036 j-0.1016 j-0.-0,2430 j0.1216 j7.4891- 0.0000 +.0288 +0.1695 +0.0 +.1451 +0.0263 +

j0.1036 j-0.2237-0.1016-0.24307 . 4 B l0.1216

j-0.2430 -0.oooo.1451 +0.0263 +0.0000 +.0288 +0.1695 +- j7.4891O.12160.1036-0.2237,-0.1016

Fig. 6. Equivalent arcuit for the cylindrical-rotor machine.

Equation ( 2 6 ) may be written in phasor form

whereWLQfIf

4 2Ea =- , where + = 6 +900L.fIfa = - +1200Applying symmetrical-component transformation,

Ts [ [ jlFjs 1 TsY E L' -I L I I2


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C H E N A N D DILLON: POWER SYSTEM MODELING 907where

Eo = 0

E2 = 0Zo = r + wL0 = r + j w ( L L ')2 = r + 0 L l = r + w ( L + L')

Equation (28) may be written as

where 20, 1,and 20 re the zero-, positive-, and negative-sequence impedances f the machine. They are the impedanceswhich can be realized when only zero-, positive-, or negative-sequence current flows in the machine. Equation(30) leads tothe equivalent circuits shown in Fig. 7.I t is interesting to note that this coupled circuit device,in this case a synchronous machine, may also be reduced tothree uncoupled circuits by using the symmetric-componenttransformation. The selection of

L l

as two eigenvectors of the symmetrical-component transfor-mation matrix was arbitrary in the case of transmission linesbu t it s necessary in he synchronous machinemodeling. Thischoice made i t possible for only positive-sequence voltage tobe generated. In steady-state normal balanced operation, bothnegative-sequence and zero-sequencecircuits emainunex-cited; the positive-sequence circuit is therefore the only onewe are concerned with.

Fig. 8 gives the phasor diagrams of the positive-sequencenetwork sf a generator operating a t normal, overexcited, andunderexcited conditions. The real power ou tputs of the threecases are the same. At normal excitation, the reactive outputof the generator is zero. At overexcitation (theield cur ren t isgreater han henormalexcitation), hegeneratordeliverslagging reactive power to the system. At underexcitation (thefield curr ent is less than normal excitation), the generator de-livers leading reactivepower to the syste m r receives laggingreactive power from hesystem. It is clear that he fieldexcitation controls the reactive power outp ut of a generator.The phase angle of El represents the rotor posit ion of a gen-eratorwith espect o an arbitrary ynchronous eferenceframe. I t is well known t h a t the real power output of a syn-chronous machine is controlled mainly by the change of rela-tive position of the rotorsof the machines in the system.

Actually, the machine is more complicated than we havediscussed, mainly because of the effect of the rotor circuits. I tis well known t ha t positive-sequence armature currents in athree-phase stator winding produce a constant rotating mag-netic field in the air gap similar to the field produced by thedc field winding. Since the field structure is driven mechani-

Fig. 7. Zero-, positive-, and negative-aequenceequivalentcircuits far the cylindrical-rotormachine.

Fig. 8. Phasor diagrams representing three possible modes of machinoperation. Real power delivered by the machine s the same in eacc a s e reactive power is different.

cally at synchronous speed, the rotating magnetic ield of tharma ture winding is stationary with respect to the ield winding and any other rotor circuits of the machine. In steadystate operation, the positive-sequence impedance of t he machine is not affected by the existence of the rotor windingsThe effect of the rotor slots causes he steady-state positive-sequence impedance to vary from direct-axis reactanceXd toquadrature-axis reactance-Xq;, s slightly ess than X, haxis of the field winding is the direct axisof the machine. Thquadrature axis of th e machine s 90' out of phase of thdirect axis. Both the direct and the quadratu re axes are rotating with the rotor.

However, when a sudden disturbance in the armature circuits occurs , the changes f t he positive-sequence cur ren t wiinduce curr ent in the field circuit such th at the field flux iinitiallymaintainedonstant.Therefore,heositive-se-quence mpedance of the machine n the trans ient state greatly affected by the field winding of the machine. Actuallthe solidsteel rotor itself also nduceseddycurrents.Fre-quen tly, the effect of the solid steel rotor may be considereas three additional short-circuited windings in the rotor. Thd-axis winding and one of the q-axis windings have veryshortime constants, while the time constan t of the second q-axwinding, like th at of the field, is larger. When the effect othose windings is also t o be considered, the positive-sequencimpedance is said to be in the subtransient state. There aretwo ransient eactancesassociatedwith he ynchronousmachine: hedirect-axis ransient eactance X* and hequadrature-axis transient reactance Xi When the armaturrotating M M F wave is in line with the field winding of th


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908 PROCEEDINGS OF THE IEEE, JULY 1974

1Fig. 9. Illustratinga'method of conceptuallyfindingthesteady-statepositive-sequence mpedance of acylindrical-rotormachine.Fieldcoil is not energized and, since the source is balanced, may be eitherclosed or left open.

machine, the positive-sequence reactance will be I . Whenthe armature rotating MMF wave is in line with the quadra-tur e axis of the machine, the positive-sequence reactance willbe X i - X p ) s mainly due to thelow decay component of theeddy current of the solid steel rotor . The subtransient re-actance of the synchronous machine likewise has direct-axissubtransient reactanceXd and quadrature-axis subtransientreactance X, . Armature res istance is small and is neglectedin this paper. Let us reduce the dc voltage source E of thefield winding in Fig. 6 ozero. The rotor is continueda t con-stant speed w . A positive-sequence source is applied to theterminal. From this discussion we can visualize the differentpositive-sequence impedances as shown in Figs. 9, 10, and 11.

I n many power system analysis problems we try to usethesimplecircuits of Fig. 7 to epresent hesynchronousgenerator. If thephenomena we are concerned withoccurimmediatelyafter a disturbance, we may use subtransientreactance in the positive-sequence representation. If t he phe-nomenon we are concerned with is transient in nature and theeffect (the fast decay component)of the solid steel rotor (or itsequivalent) can be neglected due to the ast decay character-istics, we may use the ransientreactance n he positive-sequence representation. If the phenomena are of a steady-sta te nature , we may use the steady-state reactance n hepositive-sequence representation. Actually, the armature cur-rent shouldbe resolved into two components : one in theaxisand the other in the p axis, so that the proper d and p reac-tances can be used. However, for he cylindrical -rotor ma-chine, the value f the d irect-axis reactances very close to thequadrature-axis reactance. Usually, we assume that they areequal in order that the simple representationf Fig. 7 can beused.

I t is not our intention toiscuss the detailed mathematicalexpression of the var ious positive-sequence impedances of asynchronousmachine in thispaper. References [SI and [ 6 ]give the expression of those impedance terms of some funda-mental parameters of a machine.

The negative sequence reactance of the machine can bevisualized as in Fig. 12.The system shown in ig. 12 is similarto Fig. 9 except that the rotor is drivenn the opposite direc-tion. In this case, the armatu re rotating MMF wave is ro-

I4=Ei --

Fig. 10. Findingheransient eactance of themachine.Switchsc l o d after the machine s running and the initial transients over.

Fig. 11. Finding the subtransient (fast transient) reactance ofthe machine. Switch is l o d after the machine is running.

Q1 = z

'2Fig. 12. Finding the negative-sequence reactance of the machine.

tating backwards a t twice synchronous speed with respect tothe rotor, ando currents of twice rated frequency are inducedin al l rotor circuits. Therefore, the reactance for the statorcurrent is simila r to that of the subtr ansient case. This reac-


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CHEN AND DILLON: POWER SYSTEM MODELING

DIRECT A X I S

909

U Q o Eb

Fig. 13. Finding the zero-eequence reactance of the machine.tance varies from Xd to x, and usually akes he meanvalue

Th e zero-sequence reactance of the maphine canbe visual-ized as in Fig. 13. Since th e zero-sequence current s are nphase in all three phases, the resultantlux is almost canceled.Th e reactance is small [equal to L-2L' in (29)] and is af-fected very little'by the motion f the rotor. This reactance isdue to leakage lux in the slot and end winding.Salient-Pole Machine

In a salient-pole machine there are more obvious directand qua drature axes on the rotor as shown in Fig. 14. Mostsalient-pole machines have amortisseur or damper windingson the rotor. Theeffect of the amortisseur winding is similarto the eddy currents. f a solid steel round rotor. Usually, weuse one circuit in the direct axis and another circuit in thequadrature axis to represent heamortisseurwindings asshown in Fig. 15.

This discussion concerning the reactances of the round-rotor machine applies to the salient-pole machine[SI and [6]give the typical values of different machines).

Since the direct-axis reactance is quite d ifferent from thequadrature-axis reactancefor a'salient-pole machine, machinerepresentation is more complicated than for the cylindrical-rotor machine. Generally speaking, we must use additionalassumptions for this particular problem. Sometimes t is moreadvantageous to represent the machine ina different kind oftransformationother hansymmetriccomponents,such asd-q-0 components or alpha-beta-zero components.Refer-ences [l] , [SI, and [6] include a detailed discussion of thosetopics.

TRANSFORMERSAlthough transformers are one of the most common and

mechanicallysimplecomponents of modernelectricpowersystems, transformer modeling is often not highly developedin system studies. Indeed the nonlinear effects of core satu ra-tion and the transformer's response toower system transientsdo present a formidable modeling problem. Th e scope of th ispaper, however, is to show the basis of the transformer models

AXIS OF

\AXIS B F P H A S E c

I l l l A D R A TUR f A X I SFig. 14. Salient-pole rotating machine.

0t

Fig. 15. Representationofdamper windings.

presently used in steady-state analysisof balanced power systems and to demonstrate the extensionof these methods intoanalysis of unbalanced three-phase systems.Conventional Modeling of Three-Phase Transfo lm~~s

Normally the three-phase transformer is odeled in termof its symmetrical components under the assumption that powersystem is sufficiently balancedo warrant this. Tableshows theypicalsymmetrical-component models of thetransformers for he six mostcommon hree-phaseconnec-tions. Later, thesemodels will be derived from basic considetions and certain significant exceptions to these componentmodels will be explained in order tha t imp rov ements maybemade ncertain specialapplications. Th e impedances ofTable V are assumed to be the per-unit leakage impedancesobtainedby hestandardshort-circuit tests on he rans-formers. Magnetizing impedances are large shunt-connectedimpedances which are justifi ably neglected in most steady-sta te calculations. Hence they are not shown in Table.

Several things should be pointed ou t concerning the sym-metrical-component equence models shownnTable V.First, the assumption that the zero-sequence short-circuit pedanceequals the positive- and negative-sequenceshort-circuit impedances is true only for a bank of three single-phase transformers. In either the shell-type or the core-typethree-phase transformer, the zero-sequence impedance is lessthan the positive- or negative-sequence impedances: the rea-sons for this will be developed later n hissection of thepaper. Second, there is an inherent phasehift in the positiveand negative-sequence ransfer mpedances in the Wye G-DeltaandWye-Deltaconnections.Positive-andnegative-sequence voltages are shifted in opposite directions, and thedegree of phase shift is dependent n the nomenclature of th ephases, i.e., the designation of phases on primary and secondary sides of the ransformer determines whether he phase


10/15

910 PROCEEDINGSOF THE IEEE, JULY 974TABLE V

TYPICALY ElaICAL.-COMPoNENT MODELS OR THE SIX MOST COMMON CONNEcTIONS OF THREE-PHASERANWORMERSU S P U S a

WYe G WYe G

WYe G

WYe

D e l t a

D e l t a

WVe

D e l t

Del t

P O S S E aP

I I

///;.' 1,

shift is 30 or 90' and so on. However, this inherent phaseshift is usually ignored as superfluous in most system studiessince the voltage magnitudes and balanced power flows areno t affected by the shift; one simply needs to mentally con-sider the phase shift when contemplating the results of thestudy. Finally, TableV does notconsider the effects of neutralimpedances in Wye-connected windings. This affects only thezero-sequence models in heWyeG;Wye G andWye G-Delta connections, however, and one simply adds three timesthe neutral impedance in series with 2, in these cases. .ThusTable V, with judicious application of these conditions, sum-marizes themanner inwhich three-phase ransformersareusually modeled in conventional steady-state balanced systemanalyses.

Transformer Modeling ParametersBefore one can apply transformer modeling to power sys-tem analysis, there must be some basic understanding of howthe modeling parameters are obtained. Much haseen writtenabout modeling the basic two-winding single-phase trans-former. Even so , a brief review will be given here so that theseconcepts may be expanded by means of elementary circuittheory into themore complicated systems in which there maybe unbalanced three-phase voltages and currents, unbalancedthree-phaseransformeranks, or combinationshereof.Furthermore, an elementary extentionf these principles willlead to the theoretical developmentf the conceptsof Table Vand illustra te a viable method of performing unbalanced sys-tem studies.

Z E R O S E O

I Iig. 16. Primitivenetwork of a pair of magnetically coupled arcuits.

First, consider the single pair f magnetically coupled coilsin Fig. 16. This is a four-terminal network which can theo-retically be described in terms of the open-circuit impedanceframe as per (31), or in terms of the short-circuit admittanceframe as per (32).

In practi ce, however, th e coefficients of coupling are o high ina practical power t ransformer that the inversion process re-lating (31) and (32) is numerically unstable, i.e., in the per-unit system z1 and z2 are only slightly larger than . There-fore, neither the simple open-circuit parameter tests nor thesimple short-circuit parameter tests ill adequately model thepower transformer. Therefore, a hybrid set of measurements


11/15

C H E N A N D DILLON: P O W E R SYSTEM MODELING

il Y s c - t Y oc il911

- -Fig. 17. s-equivalent ransformermodel.

is made in which standard open-circuit and short-circuit testsare made. As modeled in Table V, it is this short-circuit im-pedance in per-unitform hat smost mportant n rans-former modeling since he open-circuit impedance is primarilyused to determine the exciting current for the transformer.Furthermore, the exciting current is rich in harmonics andtbestonlyapproximatesanequivalentfundamentalcompo-nent rms impedance. This approximation will be adhered toin the paper, however, since only steady-state conditions arebeing considered in the electrical model.Use of t h e Three- Terminal A f ipoximatwn

Infurtherdevelopment of the ransformer model, it isnecessary to have a good approximation of the short-ci rcui tadmittance parametersas per (32). These can be obtained byinserting the open-circuit impedance, and the short-circuitimpedance 2 into an approximate three-terminal representa-tion of the coupled circuit pair in Fig. 6. Since the admit tanceframe is sought, a r-equivalent circuit as per Fig. 17 is em-ployed. In th is figure Yo,s the reciprocal of 2, and U, is thereciprocal of 2,. Also, the spl itting of the open-circuit admit-tance into two equal parts is purely arbitrary. Both open- andshort-circuit tests would have to be made on each terminalpair of the transformer to justi fy deviation from this policy.In the per-unit system, this equal distribution should intro-duce negligible, if any , error.

Thus a good approximation for thesteady-state rans-former short-circuit parameters as per (2) is

y1 = y2E Y , (33)and

(34)Itmust be emphasizedhere that he foregoing three-

terminal model of the single-phase transformer is inadequatefor use as a building block for modeling three-phase banks.This will become evident when one at tem pts tomode1 a Wye-Delta bank from three components as per Fig. 17. From hereon, he primitive admittance mathematical modelbasedont h e four-terminalnetworkmust be mployed. Th e three-terminal model is intended only for single-phase representa-tion and to help approximate ransformer parameters fromfamiliar terminology.Parameters of Three-Phuse Transformers

Th e preceding sections on obtaining the primitive admit-tance parameters for four-terminal magnetically coupled cir-cuits are essential but incomplete inasmuch as many three-phase transformers are common-core or shell-type integrated

Fig. 18. A 12-terminalcoupledprimitivenetwork.

three-phase devices. Thus they become 12- or even 18-termi-nal coupled circuits in the primitive sense. Here the param-eters become fa r more difficult to obtain. As an illustration,consider the three-leggedcore-type ransformer of Fig. 18.For simplicity, no tertiary winding will be considered so tha ttheprimitivenetwork is onlya12-terminal ullycoupledcircuit.

The short-circuit primitive admittance matrixor this net-work is as follows:

I

In a rigorous sense, oneould have to make 1 separate short-circuit measurements to fill in the the values f the symmetr i-cal admittance matrix n (35). Furthermore, allowances forflux leakage paths through the steel tank containing the coreitself complicate matters. However, the purpose of this dis-cussion is to understand the subtle differences between threephase banks ,of single-phase ransformers and common-coretransformerharacteristics.Therefore,omelgebraicallysimplifyingassumptionssuchasperfectlysymmetricalfluxdistribution willbe made. Th e previously mentioned sho rt-circuit measurements could be carried out for any justifiedspecial case, however. Assuming flux symmetry, (35) wouldappear as (36). Also, the proper signs of the short -circuit ad-mittance values are written in 36), where coils 1, 3, and 5 areconsidered primary windings and coils 2, 4, and 6 are con-sidered the secondary windings.

Note t h a t Fig. 18 does not yet commit the three-phase trans-former it represents to any particular connection. Th e termi-nalpairscan still be connected n any of the six standardthree-phase connections as per Table V. This will be coveredlater in the iscussion of connection matrices. For comparisonconsider heprimitive dmittancematrix or hree nde-pendentingle-phaseransformersassumeddenticaloralgebraic oherency). Th e absence of the primedmutual


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91 2 PXOCEEDINGS OF THE IEEE, JULY 974

Fig. 19. A grounded Wye-Delta transformer.

primitive admittances in the three-phase bank will be shownto play a significant role in the final three-phase transformerconnection models.

AUse of the Connection Ma tr ix

In he previous hree ections, heparameters of theprimitive (unconnected) admitta nce matrices have been dis-cussed. Now the connection and its application to three-phasetransformermodelingcan bedeveloped.For he ake ofbrevity, only one example will be shown in this paper. Thiswill be the Wye G-Delta connection because it is the mostinteresting of the six ,most common three-phase connections.However, the eadercan eadilyapply heseprinciples toobtainany of theotherconfigurationseven ncludingun-balanced banks suchas the V connection sometimes found ondistribution circuits.

Consider now the Wye G D e l t a connection shown in Fig.19 and assume it represents the common-core device shownunconnected nFig. 18. In Fig. 19 the nodevoltages aredesignated by capital V s where the Wye side has lower-casephase subscripts and the Delta side has upper-case phase sub-scripts. All node voltages are with respect to ground as refer-ence. All the primitive branch voltages are denoted by thelower-case u's with numerical subscripts.

According to the connections of Fig. 19, the simple rela-tionship between the primitive branch voltages and the nodevoltages is denoted in (38a) in which the matrix of ones andzeroscompletelydetermines the physicalconnection n amathematical sense. This matrix is the connection matrix.

Now the objective is to apply the theory of Kron's con-nection matrix N to the primitive admittance matrix n orderto ob tain the needed node admittance matrix for the coupledcircuit. This is obtained by (39) where [ y p r h ] s the primitiveadmittance matrix of (36) and N L s the transpose of N .

YNODE= N ' y p r h N . (39)When hisstraightforwardmatrixmultiplication is carriedout, the resulting node admittance matrix for the groundedWye-Delta ransformation of Fig. 19 is obtained nphasequantities. This is shown in (40).

By comparing the primitive admittance matrices of (36)and (37) i t is readily seen tha t the primed ym s vanish whenthe primitive admittance matrix f a three-phase bank is sub-sti tuted into (39). Although the primed values of the ym s arenumerically smaller than the unprimed values, their ex istencehas a profound effect on the symmetr ical-component circuitmodels of the transformer as well as on the response of thethree-phase transformer in the power system. Considerationsof these symmetrical-component effects will be deferred t o alater section.Power System Three-p hase Node Admittance Matrix

Th e node admittance matrix of (40) is not yet ready foruse in a hree-phase system model. Since the primitive ad-mittances wereconsidered to beon aper-unit basiswhereboth primary and secondary voltages were nominally1.0perunit,anyWye-Delta ransformer model so obtainedmustconsider an effective turns ratio of di n order that bothWye and Delta node voltages are still a t 1.0 per unit. There-fore, both the upper right and the lower left quadrants mustbe divided by 4 3 while the lower right quadrant is dividedby 3. Then the submatrices can e used in forming the systemthree-phase node admittance matrix for study of unbalancedpower systems. In this example, the upper left quadrant reprsentshehree-phaseelf-admittanceubmatrix of thegrounded Wye side of the transformer while the lower rightquadrant is the three-phase self-admittance submatrix of theDelta winding. If there is no transmission-line coupling, thesesubmatrices are used directly as building blocks in a manneranalogous to the formation of the positive-sequence node ad-mittance matrix used for power flow models. If parallel three-phase ransmission inesdopresentconsiderablecoupling,however, then more sophisticatedmeans of deploying thetransformersubmatrices nto hesystemnodeadmittancematrix must be considered. Th e former method has provedto besuccessful in modeling all types of three-phase rans-former connections in unbalanced load flow studies. TableVI

(38a) illustrateshe basic submatr ices used inhree-phasenodeadmit tance formation for the six common three-phase con-nections for three-phase banks. Thus t he primal values of y,,,do not appear n the submatrices of Table VI. If they did, theelements of these submatr ices would be algebraically cumber-some but would, of course,presentnoproblem in formingEquation (38a) is abbreviated as

ubrsnch = [ N ] VNODE]. (38b)umericalxamples.ormallyheserimedalues of y are


13/15

CHEN AND DILLON: P O W E R SYST13.Y MODELING 913TABLE V I

BASICSusaularc~s SED N NODEADMITTANCE FORMULATIONFOR TEE SIXMOST COMMON ONNECTTONSFTHREE-PHASEUNWORMERS

Transforner Connection ?lutuaf Admittanceelf AdsittanceSUtmatciCes Submatrix

considerably smaller in magnitude than the unprimed values,so the haracteri stics of common-corehree-phaserans-formers are not radically different from those of three-phasebanks.Infact, henumericalvalues of y yp and y,,, (un-primed) are approximately equal. Therefore, we shall refer t othem as t , the per-unit leakage admittancef the transformerthat would be obtained by the short-circuit test.

For purposes of forming Table VI, the three basic typesfsubmatrices for the various connections of three-phase trans-former banks will be defined as follows:

K]- - -Yt 2Yt -YtI1 3-Y t -Yt

(43)

Here, of course, i t is assumed that ll three transformers ofthebankare dentical. Now TableVI s ormed or heseconditions.

If no better information is known than the short-circuitadmitt ances of the ransformer, henTableVI s hebestapproach omodeling hebank in a n unbalancedsystem.With more complete information such as was assumed in themodel in (40), the subtl e differences in magnitude betweenyp , y8 , nd y,,, can be exploited to the benefit of accuracy. In(40) the existence of t he primed values of y leads to the im-proved model of a common-core three-phase transformer.Symmetrical Components of Three-phase Transformers

In the vast majority f cases it is justifiable to assume thatthe hree-phase power system sbalanced.Therefore,suchdetail as was developed in the preceding section is usually notemployed in power system analysis. The ta sk a t hand now isto reduce hese hree-phase ransformernode admittancemodels to their more simplified symmetrical-component mod-els. Th e Wy e GD el ta common-core transformer as modeledin (40) willbe examined in deta il. Then appli cation of theprocedure to the remaining three-phase connections is merelya simple exercise in matrix algebra.

Considerirst theelf-admittanceubmatrix of thegrounded Wye sideof the transformerof Fig. 19. This is trans-formed into symmetrical components according to the rela-tion :

which results in

Note tha t the zero-sequence self-admittance is not equal tothe positive- and negative-sequenceself-admittances n thethree-phase transformer. If a three-phase bank had been used,ym' would have been zero and a l l three component self-admit-tances would have been equal.

Now apply the transformation to the self-admittance submatrix for he Delta side of the ransformer of Fig. 19 re-membering that the effective turns atio of d j has beenapplied.

0 I . I10 0 0

yU .. (47)12 IY, - Y i00 I1Y ; - Y;')

Here i t is verified tha t t he re is no zero-sequence self-admit-tance in the Delta windingf a balanced transformer and thathe self-admittances of the positive-andnegative-sequencenetworks are reduced by the amou nt of y, '.

Finally, t he mutual admittance submatrix of (40), modi-fied foreffective turns ratio, is transformed to find the se-quence ransfer admittances inking he Wye G and Deltasides of the transformer models in the sequence networks.

Thus

0 I I10 0 l o

Notice he forward 30 phase shift n he positive-sequencenetwork ransfer admittance while there is a backward 30'phase hift n henegative-sequence ransfer dmittance.These are , f course, expected results forhe Wye-Delta transformer. Also it is app aren t that ince the zero-sequence trans-fer admittance iszero,nozero-sequencecurrentscanpassbetween the Wye and Delta sides of a balanced transformer.Again, this is obvious from simple observation of t he three -phase connection. What was not obvious, however, is tha t the


14/15

914 PROCEEDINGS F THE IEEE, JULY1974rp+1V . MI

Fig. 20. Zero-sequence node admittance model o r aCommon-coregrounded Wye-Delta transformer. Fig. 24. Three-phase load shown aa a balancedWye-connected mpedance.

Fig. 21. Pcdtive-seguence node admittance model for acommon-core grounded Wpe-Delta traasformer.

Fig. 22. Negativesequence node admittance model for acommon-core grounded Wye-Delta transformer.

011111P -Fig. 23. Positivesequencemodel or thetransformerwith phase shift ignored.

existence ofy (the effects of the%ominon-core coupling) in-creases the transformer transfer admittance in the positive-and negative-sequence networks.

Now the results of (4 9 , (47), and (49) will be used to formthe three symmetrical-component circuitmodels of the three-phase common-core transformer (Figs. 20-22).The awkwardness of the positive- and negative-sequenceequivalent Wye G D e l t a transformer models can be avoidedif the 30 phase shifts inherent in Wye-Delta transformers areignored. As previously stated, engineersusually gnore thephase shifts in the model and mentally consider them in t heresults. If this is to be done, and we may furt her assume thaty P y , ~ y , ' (which are zero in the three-phase bank),then figure G becomes Fig. 23.

Then Fig. 22 would, of course, be the same since the direc-tion of phase shift was the only difference between the posi-tive-and henegative-sequencenetworks.Furthermore, y pmis a very small admittance since p s only slightly largerthan ym. Thus it may be justifiable t o consider yp-ym andy,-ym as open circuits. With all of these gross assumptions,we have duplicated the sequence model for the Wye GD e l t a

11VO I )Fig. 25. r e , pcei tie, andnegative-sequenceequivalentcircuits for the balanced load.

transformer of Table V except for the effects of the common-core: 1)Zero-sequence impedance is lower on the Wye G sideof a common-coreWye G D e l t a transformer han for anequivalent hree-phasebank. 2) Positive- and negative-se-quence transfer impedances are lower for the common-coretransformer than for the equivalent three-phase bank.3) Thethree sequence impedances are not equal on a common-coretransformer.

Considerable effort has been taken here to detai l the de-velopment of the Wye G D e l t a transformer whose windingsare a l l wound on a common piece of iron. This illustrates thesubt le differences between this ubiquitous transformer typeand three-phase banks. Also, it provided a quasi-justificatiorfor the simplisticmodels of Table V which most engineers usein system analysis while outlining a method-that of Kron'sconnection matrices-for developing as detailed a model ofany three-phase transformer connection that may be neededfor some special application.

Certainly, it is not the intention of the aut hors to ecom-mend detailed models for all applications. Whenever one canget by with themodels of Table V, he should do o. However,recent interest in unbalanced three-phase phenomena has in-spired the aut hors to provide a tool, previously unavailable,for including he effects of three-phase ransformers in thesystem. The only drawback to this ool is obtaining the datafor the primitive admittance matrix or a multiwinding three-phase transformer. Even so, the theoretical manipulation ofthese data as presented in this paper contributeso the under-standing of three-phase transformer modeling.

POWER YSTEMOADIn a power sys tem, it s impossible to represent every load

individually.For hisreason, oadsconsidered nasystemstudy are representationsf composite system loads.

At each substation, the substation demand P and Q) canbe obtained from the recorded readingsof the demand meterslocated at the substation sites. However, individual customerdemands at random times are not usually known. The meterat the customers' locations are watthour meters that recordthe otal energy consumption . These readings are used orcustomer billing. They must be converted into demands be


15/15

CHEN AND P O W E R SYSTEM MODELING 915fore being used in system analysis. Nevertheless, informa tionabo ut load characteristics in apower system is limitedn mostcases. Inadynamicstudy , we use constant mpedance orconstant current tomodel the realor reactive power consumedby a composite load in the positive sequence network. Thereis more uncertainty involved in modeling loads in the nega-tive- and zero- sequence networks.

Fig. 24 shows a three-phase loadwhich is represented by abalanced Wye-connected impedance.

Application of the ymmetrical-component ransformationgives

7;F]s 1V 12 31*a;i

. zThe symmetrical-component equivalent circuits are shown inFig. 25.

CONCLUSIONThe authors have at tempted to present discussion of the

conceptsunderlying hedevelopment of symmetrical com-ponentsand heirapplication o he modeling of a three-phase power system. Although symmetrical components havebeen widely discussed in he ite rature , he author s believethat this paper will add to the under standing of the founda-tions of power system modeling, part icularly for the non-specialist.

REFERENCES(11 E. Clarke, Circuit AnaZysis of A C Pmue7 Systnrrc, vol. 1. NewYork: Wiky, 1956.[Z]M. H. H e Electromagnetic and electrostatic transmiasion-lineparameters by digital computer, ZEEE T r a n s . P m u n A p p . S y s t . ,[3]M . S,. Chen, The philosophy of three-phase network transforma-vol. PAS82 , pp. 282-291, June 1963.tion, presented at the IEEE Winter Power Meeting, New York,N. Y., Jan. 1969.[4]A . E. Fitzgerald and C. Kingsley, Electric Machinery. New York:McGraw-Hill, 1961.[SI C. Concordia, S y n c h r o ~ u s Machine s, heory and Performance .New York: Wiley, 1951.[a]E. W. Kimbark, P o w n System Stability, vol. 111. New York: Wiley,1956.(71 G. Kron, T m o r A na ly si s f Networks. London, England: Mc-[ 8 ] Electrical Transmission and DisfributMn Reference Book, Westing-Donald, 1965.(91 W. E. Dillon and M. S. Chen, Transformer modeling in unbalancedhouse Corp., 1964.three-phase networks, presented at the IEEE Summer Power[lo]Modern Concepts of P a n y ste m D yn am ic s, IEEE Power Engineer-Meeting, Vancouver, B. C., Canada, July 1972,Paper C 72 460-4.ing Education Committee, 1970.

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Power System Modeling-Mo-Shing Chen