be found unnecessary to take this action.The authors' proposals for using filters,

with a bandwidth equal to the criticalbandwidth of the ear, to measure theambient level excluding the transformerharmonics, and others with narrow bandcharacteristics for measuring the individualcomponents of the hum, are most attractivefrom both a technical and practical view-point, and should be developed.The discusser feels the paper would be

of greater value, particularly to those con-cerned with transformers but not wellacquainted with acoustics, if definitionscould be given of the many terms used inthe paper such as "flat-weighted octaveband values" and "critical bandwidth ofthe ear in decibels" etc. It would also bevaluable to have further explanations andcomparisons of the various methods offrequency analysis referred to in the paper,such as A-weighted spectrum level-db/cps.

M. W. Schulz, Jr., and R. J. Ringlee: Mr.Edwards has pointed out that the methodof predicting audibility by comparing theA-weighted sound level of the ambient andthe total transformer hum is generallysatisfactory and is easier to apply, par-ticularly for those who are not wellacquainted with acoustical measurements.This is true; however, we anticipate thatas people gain more experience in measur-

ing transformer noise they will acquireadditional equipment and knowledge whichwill permit the more complex measurementand analysis of sound spectra. With theadditional information thus provided, amore accurate prediction of public reactionshould be possible.

It may never be practical to attempt topredict the response of a particular indi-vidual, but we believe that public responseis predictable in a statistical sense. Itshould be possible to obtain sufficient datafrom a proper sample of substations andplot it in Fig. 2. A determination of thefraction of points representing complaintsin each zone A, B, C, etc., would then yielda useful method for predicting the risk orprobability of complaint in a given situation.

It may be possible to devise a table ofcorrection numbers similar to those givenby Stevens.8 There is already a correctionfor previous exposure. Corrections mightalso be made for property zoning, land-scaping, overhead transmission lines, sizeof substation, etc. Until enough datahas been accumulated we agree with Mr.Edwards that the practical procedure shouldbe to make provision in the design of asubstation for reducing the noise andproceed with the required measures onlyafter it has been demonstrated that theyare needed.

Although it was implied in Figs. 6(A)and (B) and in the discussion of them in

the paper, the greater variability of theindividual harmonics compared to totalsound deserves further emphasis. Theharmonics are more variable both withtime and space particularly in the vicinityof a reflecting structure, hence the samplingprocedures must be carefully examined.Tests must be repeated and more micro-phone locations must be used. The authorsdo not have enough data to give a firmdecision, but we estimate that approxi-mately four times as many microphonestations should be used to provide the sameaccuracy as that for total sound levelmeasurements.One way to systematize harmonic meas-

urements at a measuring station is to takereadings at the station and at four positionsspaced approximately 30 inches radiallyfrom the station; two positions on thediameter along the line of sight to the source,and two on a diameter at right angles.We are considering mounting the micro-phone on a rotating boom and electricallyaveraging over the path traversed.Mr. Edwards and others have suggested

that acoustical terminology used in thepaper may not be familiar to all who areinterested in the subject of transformernoise. Although a satisfactory explanationof all the terms is not feasible here, Ap-pendix I, giving definitions of some ofthe more important terms, has been addedwith the hope that it will prove helpful.

Digital Calculation of Line-to-GroundShort Circuits by Matrix Method

AHMED H. EL-ABIADASSOCIATE MEMBER AIEE

METHODSof calculating short-circuitcurrents in power systems with the

aid of digital computers have been de-scribed in a number of technical papers. 1-4One of the more complicated aspects hasbeen the treatment of mutual impedancesin the zero-sequence network for line-to-ground faults. The purpose of this paperis to present an analytical basis to includethe effects of mutual impedances in a newmatrix method and digital computer pro-gram described in an unpublished paperand which has been called "the short-circuit matrix method." The authorprefers to call it the "bus impedance ma-trix method" because the matrix usedis actually the open-circuit driving pointand transfer impedance matrix.

In developing this technique, particularemphasis was placed on developing amethod applicable to the calculations ofshort circuits for systems with largenumbers of mutual impedances. In addi-tion, careful consideration was given tocomputer logic required to develop the

prograuii to carry out the actual calcula-tions on a high-spSed digital computer.The main advantages of the new ap-

proach are given in the following.1. Considerable savings in computer stor-age requirements and in computing timehave been realized because neither a trans-formation matrix nor matrix inversion isused.462. Greater accuracy can be achieved thanin an iterative method in which precision isquestionable, especially where impedancesare low.

3. The bus impedance matrices for bothpositive and zero sequence are calculatedonly once for any system and retained forlater use.

4. The impedance matrix can be modifiedeasily for transmission system and capacitychanges even when mutual impedances areaffected.5. System subdivisions can be handledeasily by utilizing parts of the main pro-gram, thus increasing the maximum size ofthe power system that can be studied onany computer.

The power system is represented by the

familiar symmetrical components' se-quence networks. The required data arethe positive- and zero-sequence networksassuming that the negative-sequence net-work is identical to the positive-sequencenetwork. The positive-sequence gen-erator busses and zero-sequence sourcebusses are chosen to be the referencebusses for the positive-sequence and zero-sequence networks respectively. Theopen-circuit driving point and transferimpedance matrix of each sequence iscomputed by forming the network, add-ing one branch at a time and closingthe loops as they occur, The main differ,ence between the positive-sequence andzero-sequence networks is in the mutualimpedances.

Fig. 1 shows the procedure for comput-ing the bus impedance matrix. Addinga new branch to the partial networkadds a new row and column to the partialmatrix which corresponds to a new bus,if the new branch adds a new bus; orcorresponds to a new loop, if the newbranch closes a new loop. If the newbranch is coupled to the partial network(in the case of the zero-sequence network),

Paper 60-180, recommended by the AIEE SystemEngineering Committee and approved by the AIEETechnical Operations Department for presenta-tion at the AIEE Winter General Meeting, NewYork, N. Y., January 31-February 5, 1960. Man-uscript submitted November 2, 1959; made avail-able for printing December 18,1959.

AHMED H. EL-ABIAD is with Purdue University,Lafayette, Ind.

El-A biad-Digital Calculation of Line-to-Ground Short CircuitsJUNE, 1960 323

Fig. I (above). Pro-cedure for comput-ing bus impedance

matrix

Fig. 2 (left). Meth-od oF modifying busimpedance matrixFor system changes

each element of the new row and columnis corrected for the coupling. The cor-rections take into consideration the directcoupling between the new branch andbranches of the partial network as wellas the coupling between these branchesand other branches in the partial network,and so on. In general, the number ofsuch involved branches is relatively small.A new row and column corresponding toa loop is then eliminated7 before addingthe next branch.

Fig. 2 shows the procedure for modify-ing the bus impedance matrix for systemchanges by correcting one element at atime.

Fault currents, line contributions, andbus voltages are directly computed fromthe appropriate matrix elements as givenby equations 14 through 22.

Description of Methodof Calculation

The following sections give a detaileddescription of the procedure to be followedin the treatment of mutual impedances.The symbols used in this description aregiven in the Nomenclature. Lower caseletters represent branch quantities; cap-ital letters represent bus or nodal quanti-ties; a bar over a letter represents amatrix; unprimed bus number subscriptsmay take any value, i.e., correspond to awhole matrix; and primed bus numbersubscripts can only take values of thosebusses which belong to a group of coupledbranches, i.e., correspond to a submatrix.

NOMENCLATURE

QuantitiesE,e = voltagel,i= currentZ,z = impedanceY,y = admittanceN total number of busses not counting the

ground (reference) busB1= total number of branchesU=unit matrix

Subscriptsp,q= bus numbers for branch quantities;

p is always a sending end and q isalways a receiving end

m,n = bus numbers for bus quantitiesL=loop or linkr=newly added row and column, i.e., r=q

if q is a new bus and r=L if q is nota new bus

Zpq,ypq = branch impedance or admittance,two subscripts: self-quantity, i.e., adiagonal element of a matrix

ZP'ql'.P22'q2' = branch impedance or admit-YP1'Ql',P2'q2'

tance, four subscripts: mutual quan-tity, i.e., off-diagonal element of amatrix

(a)=phase (a) quantities(1) = positive-sequence quantities

El-A biad-Digital Calculation of Line-to-Ground Short Circuits JUNE, 1960324

(2) = negative-sequence quantities(0) = zero-sequence quantities

COMPUTATION OF Bus IMPEDANCEMATRIX

The following are the equations to beused in the computation of the bus im-pedance matrix (Fig. 1).

For a grounded branch without coupling(P= 0):1. If q is a new bus (i.e., Oq is a treebranch), then,

P2 q2 psrq',pl'qlj zp,'v, l' Zpi'qo,p,q21 I Zp1tq1,p3'Q2'

Zp'q',p'q' =p2 q2 -Ptp2'q',Pl'q' zP2Q02' Zps',08

p3'qs' ZPa'Q,',Po'qo' Zp8'Pj'pt'qg I ZpgI'l

=branch impedance submatrix which represents the involved group of coupledbranches P1'1'q,P2'q2,.-. .

In general, some of the mutual impedances will be zeros.Invert the above branch impedance submatrix to give:

ypo'qo' p',q'",PV',po'go' Yv'a'.P,q

YP,= [zp'qiJ 1 Po,Io

where Yp,,Q= branch admittance submatrixypo'qo'= self-admittance of the newly added branch po'qo', a scalar

Zqq °oq (2)

2. If q is not a new bus (i.e., Oq is a

link branch), then,

ZLnS -Zqn

ZLL:=ZLq+ZoQ

(3)

(4)

For an ungrounded branch withoutcoupling:

1. If q is a new bus, then,

Z4n = Zpn

and

Zqq = Zpq +Zpq

2. If q is not a new bus, then,

ZLn Zpn Zn

ZLL = ZLP ZLq +Zpq

(5)

Pi'ql' P3gat

YpO'qo,p'q= Po qo I Ypo',o',lql' I YpoI'qo''qIYPO'O',PA'Iq ' I

= row submatrix of mutual admittances between new branch Po'qo' and coupledbranches pl'ql', P2q2', *

Po 'q. 'pl'ql' Ypl'ql',Vg'qo'

Yp'q',po'o.' = P2'g2' Y ,t'q2',po'qo'p3'q3a YPI,Q3PO'qOI

- the transpose of yz,.,',',Ipl' because of symmetryand

Pa'q3'

(6)

(7)

(8)

If the newly added branch is coupledto a group of branches already considered,p'q' (i.e., pl'ql', P2'q2', .), thebranch impedance submatrix is formed.The new branch in this case is calledPo'qo'.

Pooqo' Pqo'

Z o, PO 'qo |ZP°'°°', PO'aON P'Fqp q 1 2p'q', po'qo' ,'_PYQ, PSa I

where

2p'q- branch impedance submatrix of all

coupled branches affected by and in-cluding the newly added line

zQo'qo' self-impedance of the newly addedbranch po'qo'

Po'q1 YPi'#1' YPo'1q',vP'Xq' YPI'QI'v,PSrYP'O',P'Q' =P2'q2' YP2'Q2'.P1'Ql' YP2'q0' YPv'o'sP'|s1G

Po'qa' YP&'Oq'.Pv'q' I YPVjq,p'q2', YPs'qs'

-branch admittance submatrix of the involved coupled branches pl'ql', p2'q2',Also form the following column matrices:

n

Pl'ql' Zpl n -ZqltnZp'q',n =P2 q2 Zpt'mnZ,nI

p3qa'q Zps'n-Z'n

where: Zp,ln, Zq1 'n, Zq2'n,. .. are elements from the partial bus matrix; n takes allvalues of the partial bus list including the newly added r.The equations for coupling correction are:

Zpo' Gol' q'0 p'Y q', n'p'Zrn(corrected) =Zrn(uncorrected) _ s

I1-2PO 20 ,P Q YP Q ,P.W 00(9)

[Zpo0'c0, p'qyp'q',vq'p'q'prJ -[Zp*'q'# Zpg'qg,p'qep'q,p,'*'e]

Zrr(corrected) = Zrr(uncorrected) - --1 -2pO f,q"p Q yp ,Q"p0G(

( 10)

I I go'ZPOGO,P'q' P=o'qo' ZP0'q*',1P0fl' I ZPO'qO',Pt'q2' I Z01qo',p2'qg'

=row submatrix which is composed of the mutual impedances of the new branchpl'qo' with the involved group of coupled branches pl'ql', P2'q2', . . .

pI 12I

4'q' ,Po qo' " p2fq2lP3"q3 I

= the transpose of zPO'QO',P'q'because of symmetry

and

Notice that: 1. r= qo' if the newlyadded branch is a tree branch, and r=Lif the newly added branch is a link branch;2. the value of Zrr (uncorrected) is com-

puted from the corrected values of Zrnusing the appropriate equations 2, 4, 6, or

8.

To eliminate the newly added loop row

and column, each element of the partialbus impedance matrix is to be correctedaccording to the following equation:

Zmn (after elimination)= Zmn (before

elimination)-LMZLn (11)ZLL

MODIFICATION OF Bus IMPEDANCEMATRIX FOR SYSTEM CHANGES

If, after computing the bus imped-ance matrix, a change is made in the

El-Abiad-Digital Calculation of Line-to-Ground Short Circuits

ZqnO=

and

(1)

pliqll

P21921

pA'l2f

JUNE 1 960 325

system, a direct method for correctingeach element of the matrix is used, ratherthan starting to compute a completelynew matrix. If the changes in the systemare extensive, however, it may be easierto start a new matrix, but such changesare unusual. In general, the sys-tem changes include adding a few newbranches or changing the impedances of afew branches. Opening a line is equiv-alent to increasing its self-impedanceto a very large value and decreasing itsmutual impedances to zeros.New branches can be added to a system

by the same method used to compute theoriginal bus impedance matrix.Changes in impedance of a few branches

modify the bus impedance matrix accord-ing to the following equation (Fig. 2):

Zmn (new) = Zmn (old) +Zm,p'q'YP'q',P'O'XZp'q',n (12)

where Zmn=any element of the bus im-pedance matrix.

pl'ql'

find the closing-in fault current, there is noneed to compute all the elements of Zmn(new)compute only those elements necessary forthe case.

4. Zm,pyn= thetransposeof Zv'g' ,m. There-fore, since m and n take the values 1, 2,N, it is only necessary to calculate either2m,p'q' for different m's or Zp'Q',n for differ-ent n's. One is found from the other bytransposinig.

CALCULATION OF LINE-TO-GROUND FAULT

Applying the usual approximations andassumptions made for short-circuit stud-ies, the required fault quantities are com-puted from the positive-sequence andzero-sequence bus impedance matrices.For a line-to-ground fault on phase (a)

at bus (n), i.e., En(a)-=O

- 3In (a) =

-

- ( 14)2Znn(l) +Znn(o)

(0) 2Znn(i) +Znn(o)

Palq2tZm,vPf' =m Zmpl,-Zmql' | Zmp2'-Zmq2' I Zmpa'-Zma,

n

Pi "qi Zpltn-Zqi'nZp'P',n =P2q2a Zpt'n Zqs'n

P3 q$8 Zpx'n-Zgs'n

Ypta2, -= 17-(yp5-yp'q'(new))Xzp e 'P'q'(]p'q' Yp'Q(new))

YP'a'= [Zp'5']-

branch admittance submatrix, whichincludes the changed branches, beforethe change

Yp'Q'(new) = [ p' '(new) 1I=same branch admittance submatrixafter the change

and

Ema 1 -Zmn(lE(1)' 1 2Znn(l)+Znn(o)

EM - Zmn(2)2Znn(j) +Znn(o)

-Zmn(o)E

1 2Znn(l) +Znn(o)2Zrn(l) +Zmn(o)

Em(a) 2Znn(l) +fZnn(o)

zp((l) [2Znn([) +Znn(o)-1

','"(0) =2Z.n(i)+ Znn()XpP'Si' ,p'q'(o)0p)ZP(o),n I

pilgl ' Yz '=P2'q2' y '

p31q3' yz,

YIl,PI qIOpl~qa'7 lt

P2'q2' pa'qs'Pa'Bl' I )Pl'2l'(o I ypl'sPt'ql'(o) I Y'l' Q,' I

andn

p1qfL Zpilnf(0)- Zo,n(o)Zp'q'(0),n = P2 q2 Z 2'n(0)- Zq'n(,)

p3qI9 Zp,nl(0) ZQf(0)

CALCULATION OF MATRIX FOR LARGESYSTEMSIf the system is too big to be studied

as one network, it is divided into twoor more subnetworks. Each of these isstudied in turn, taking into considerationthe effects of each of the other subnet-works. The procedure would be as fol-lows for a system to be divided into twosubnetworks, A and B. The same stepswould be performed for a system dividedinto three or more subnetworks.

1. Divide the original network into the re-quired number of subdivisions so that thesize of each is approximately the same;the network is cut only at busses (cut-lineimpedances are zeros); and the numberof cut points between any two divisionsis kept to a minimum.2. Calculate the bus impedance matrix foreach division in the normal way.3. Include the effects of subnetwork B onsubnetwork A as follows:

(a). Compare the bus numbers of thebus impedance matrices of the two sub-networks and determine the commonbusses (or cut points).(b). Construct an equivalent matrix forsubnetwork B. Its axes are the common

p21q3tplZql'2p,2-Zp,q1' I-Zql,'P,' +ZQl,q,' Zpl,p,'-Zt,'a'-Zq,'p,' +ZQl'q,' ZP,'ps'-ZPl'Qs'-Z2l'p,' +Zql'q3'

IZP'a', l'Q'q=P292I iPt,P -ZP2,ql, -Zq2,'p, +Zqs'g,' 2pi,12' ZPat'qt' -Zq'P2' +Zqt'aS' ZPt'PS'-Zps'qat'-Zq'ap'±+Zq2' q,'PA q3' 2ps,l,P1-Zp3',Ql'-2a'Pl' +ZQ3'1 ZP3'Vl'-ZP&'Qt'-Z9I'P2' +Zg392IS 2P,wPS,-ZP$'q3-Zq3,p, Zs'qg'I

Notice that:

1. If the changed lines are uncoupled, equa-tion 12 reduces to:

Zmn(new) Zmn(old) +

v(Zpm-ZQm)(Zpn Zqn)(Ypq Ypq(new))Z J-(yp- ypq(new))(Zpp +ZQQ- 2Zpq)Pe

and-1

iPxt471(a) 2- X2Znn(l)+ Znn(0)2[Zvin(Pa')Ze,'n(,) (+

3pl Ql (I)

YPI al ptP g O)ZP'f'(o),nt (22)

(13) where

2. Both equations 12 and 13 compute onenew element of the bus impedance matrixfor simultaneous branch changes.3. If the change in the system is permanent,every element of the new bus impedancematrix must be computed. If the changeis only temporary, e.g., opening a line to

YP'q'(O) = IZP q'(0)] 1

busses and its elements are the corre-sponding elements from the bus imped-ance matrix of subnetwork B. Thisequivalent matrix is equal to the opencircuit driving point and transfer impe-dance matrix measured for subnetworkB at the cut busses.(c). Consider this equivalent matrix asa group of new coupled branches (equiv-alent network) to be added to subnet-work A. Each branch of this equivalentnetwork adds a new loop (link) to the bus

pilqll _ P2'qt, P3'3fP1'ql' ,YPl,','(O) YP,'qi' P2' 03' Ypl,'ql',P3Q '((o)

= P2Iq2' YPs',q'2,vp ql'(O) YPl'Q'(O) Yp'a'2' Pa'S,' o)Pa'q3' Ypa'qa',p,'Iq'(o) YPa'qa',ps'q2'(o) YP2'qa'(0)

El-A biad-Digital Calculation of Line-to-Ground Short Circuits

D

r

r

r

I

I

P2'ql

326 JUNE: 1960

impedance matrix of A, which is then elim-inated. After considering all branchesof the equivalent network, the resulting REbus impedance matrix of A will be thesame size, but will represent both A and Bsubnetworks.

4. Repeat step 3 to find the bus impedancematrix for B including the effect of A.

Given a certain memory size in a com- P3 3

puter for storing the bus impedance --matrix and assuming that it is to be stored epfas a triangular matrix since it is sym-

n 3 3

metric, the maximum size of the networkthat can be handled will be approximatelyproportional to the number of sub- p /q 2divsons raised to the power of three --halves, e.g.,

Numberofsubdivisions: 1,2,3,4 2 2Relative size of maximum network: 1 , 2.8,5.1, 8

Conclusions J

The bus impedance matrix method has plq \ pbeen proved at this time to be most con-venient for short-circuit studies. Givencomplete branch information (branch im-pedances and branch interconnections) fora power system, efficient techniques havebeen developed for: 1. computing the bus (A)impedance matrix, 2. directly modifyingthe bus impedance matrix for systemchanges, and 3. easily subdividing a largenetwork in order to study it on a relativelysmall computer.The bus impedance matrix completely

describes the construction of the powersystem and may be called its mathe-matical model. It is equivalent to a com- / P1 p343 /pletely plugged and set network analyzer.The success of this method of solving

complex problems has pointed to the need / 3

for more research into the understandingof mathematical models, such as bettermethods of derivation, modification,transformation, etc., in order to realize 2 2 2

more fully the value of the high-speeddigital computer.

Appendix 1. Derivation ofEquations p 1p;q /^

Adding a Tree Branch tFig. 3 shows a tree branch po'go' being / P/

added to a partial network which includesa group of lines pl'ql', p2'q2', . . which aredirectly or indirectly coupled to branchpo'qo'. After adding the new branch, theperfonnance equation of the partial net- I,work is:

[F ZiZ-. ...Z Z'qo' 1 (B)

En Zni Znn Zngo' X In Fig. 3. Adding tree branch po'qo' which is coupled tc

:jj

.,.:- A-Method of cdlculating Z,.'nE#°0 ZBO'1 ZO'n ! Zg0'Q*' IBO' B-Method of calculting Z0'qO,'(23)

O pl'ql , P2'q2', Pa'qs'

El-Abiad-Digital Calculation of Line-to-Ground Short CircuitsJUNE 1960 327

where Zl. .., Zin, .. , Znl *, Znn, ... ZQo',o' = Zp0'-0' ±ZpO*qg' -are known from previous calculations, and --pI

ZQO, * * ., Zgonp . . ., Zq0'q0o are to be found. ZPolQolptyp gp1. In order to find Zqo'n, make all cur- ZPaoqO1zPO1q01,P'q'YP'q',PO'q0Orents equal 0, except In= 1, then: 1- pqO1,p Q yp'q' ,pO9

ZLU = Zpn-Znand if p is a ground bus (p=0):

ZL = -Zqn

Ei -=Zn

En = Znn

Ego' = ZQo'nReferring to Fig. 3(A):

Ego, = Ep*' -epo'qoand

If the newly added bianch is not coupled(24) to the rest of the partial network, i.e.,

p'g' vanishes and the added branch is calledpq, then equation 33 becomes:

Z'7q = Zpq+zPJrzp(6)(25) and if p is a ground bus, i.e., p = 0, then:

Zqq Zoq (2)

2. To find ZLL, make all currents in equa-

tion 34 equal to zero, except IL=1, andfrom equations 28, 31, 32, and 35:

ZLL = ZPo'L ZQo'L +ZpVo0'qZPOQOI,PaqYp 'P o'g [ZP'L-ZQ'I

Zp9'qg'iVO'qO',p'q'Yp'q',pg' O'J1PwoSpo'qo' ,P'q'YP'q',P Vo'Qo

(37)

which, for no coupling, reduces to:Po'qo' p'q'

Po'qo' 1Cep'q* | Po'qo' | I|

p'q' | | p'q' |p,'.po,'qo'LzP'Q',v'q | P q

or

potgot pIgr

p I ipjP4rI

P r2wPS |yttswt || Pa

(26)(8)ZLLZPL -ZQL +Zpq

and if p is a ground bus (p=0):

ZLL= ZQL+ZOq (4)

3. Since EL is normally equal to zero,

equation 34 can be reduced to:7(27)

also

p'q' e P'qo' E (28)

From equations 24, 25, 26, 27, and 28:

Zqoo',n = Zp01n'-

(29)

If the newly added branch is notcoupled to the rest of the partial network,p'q' vanishes and the added branch is simplycalled pg.

Equation 29 then becomes:

Zen Zpn (5)

and if p is a ground (reference) bus, then:

Zqn=° (1)

2. To find Zqo'q.o', make all currents inequation 23 equal to zero except IqO' =1,then:

E1 = Ziqo'

En =Znqo (30)

E Ro = Zroino o

Referrng to Fig. 3(B):

(25)Eg Ep, -e,,,q,

and

Adding a Link BranchFig. 4 shows a link branch, po'q0', being

added to a partial network which includesa group of lines, pl'ql', P2'q2', ..., whichare directly or indirectly coupled to branchpo'qo'. After adding the new link, the per-forinance equation of the partial network is

ElZi... Z15 ... ZIL II1

En zni ZXX. . ZnL In

L ZLI .ZL,.. .ZLL LL

(34)

where

EL=voltage source of link po'q0' which isnormally equal to zero

IL= link current --ivoo

Z,1, ... Zini, .. Znl...., Znno, ..areknown from previouscalculations; ZLp.L..ZLn, * ., and ZLL are to be found.

Referring to Fig. 4-

EL -= EO, - Eq0o - ep,'oq' (35)1. To find ZLn, make all currents in equa-tion 34 equal to zero, except In= 1, and fromequations 26, 27, 28, and 35:

ZL-= Zpo'n ZO2'nzpo'qo',P'q'5P'p'Pq'q' [Zp'n Zq'n]

1 po'qo',p'q'y,q'Y pq.o(36)

which, for no coupling, reduces to:

Po'qo' p'q'

po'go' eI v|.,g Po'qo' FZP8T'qT'O,l,v PQ

-PqI-P'Q' | P q | SP'Q',Po'ao' zP'Q'PIQIt I |. 'I

or

Po'qo' p'q'

po'qo' po'qo' Fv7T".7v' ' ep.,'P'q' I p'q' Ipp'lQJ,po v'Y P'.,'Q ||, |o.Z

From equations 25, 28, 30, 31 and 32:

(31)

(32)

Am= [Zmn7ZnLZlL ZL,n llnor

Zmn (after elimination) = Zmn (before

ZmLzZLnelimination) -ZLL

ZLL

(38)

(11)

Modifying Bus Impedance Matrixfor System ChangesGiven a system whose bus impedance

matrix has been calculated, its performanceequation is:

Em= ZmnIn (39)

Introducing a change in a group of coupledbranches pl'ql', P2'92Q' the new perform-ance equation would be:

Rm(nev) = Zmn(new)jn

or

Am(new) =Zmn (In+Aln)

(40)

(41)

where AIn are fictitious currents to be super-imposed on the unchanged system such thatthe new bus voltages would be the same as

those resulting on the changed system byIn.

Referring to Fig. 5:

In= 1

In = 0

Aln= Aip'q'AIn= - ipgq'andA1n =O

at n=nOat n p noat n=p'at n=q'

at n#p' or q'

Substituting equation 42 into 41:

Em(new) = Zmno +(Zmp'- Zmq' )Aip'q'

Z-mno+Zm ,p'q'Aip'q'

where

zm,p'q' (Zmp'-Zmq')

but

Aip,, = OP(Yp'-yp'q(new))ep'q'(oew)

and

(42)

(43)

(44)

(45)

El-A biad-Digital Calculation of Line-to-Ground Short Circuits

(7)

(3)

JUNE 1960328

e1p'Q(nefw) = Ep,(new) - Eel (new) (46)

From equations 42, 45, and 46:

ep'q'(new) - (Zp'no-Zq'no)+(Zp'p'-Zvpqt-Zq'P' +7Zq'q')Aip'q'

Zp'q',, +'Zp'q',pfq'iKp'lq (47)

where

-7P'f',nzo-- (Zp'no-Z q no) (48)and

p'Q', ''=(7p'p'-Zp'7Q' p'+

7Zq'q') (49)

Solving equations 45 and 47 for Aip'q':

Aip q = YC- 9p'q'-YV'q'(new) .Ve'Q' 1-1{Yv'q'Y'P'(new) }Zv'q',no

= Yp'q' .p'q' Zp'q',nO (50)

where

YP'Q,Y.P'g-[ Yp'q'-Yp'q(new) IZpIQe,pY I3 IYPql-9pYq(new)} (51)

Substituting equation 50 into 43:

Em(ne )'4= Zminno +Zm,p'pq'Yp'q'I,p'I'Qp'' ,nO

However, from equation 40 for I,, =0 exceptat n,, I,o=l:

Em(new) = Zmno(ne-)

and since no can be any bus n, then:

Zmn(new)-= Zmn(old) +Zm ,p'g' Yv'q p'q'7p'g'n(12)

If there is no mutual coupling among thechanged branches at all, then equation 12reduces to:

Zmn(uew) = Zmn(old) +

(Zpm -Zqm)(Zpn-zfqn)(..L )wo~~~~~~~~zZq Opq(new)

-- )(zpp-2Zpq+Zaq)zPq Zpg(new)(13)

Appendix 11. NumericalExample

Given:

From Fig. 6:

Branch Self Coupled MutualStep p-q Impedance p-q Impedance

p ip qf q/ la 1.-..1 0.1

2.....0-2....0.5

3. 0-3.. 0.5

4 1-2 0.4. 1-4.... 0.1

E 3-4 -0.2

'L.q' 5. 1-4 0.2 1-2 0.1

6........ 3-4 .0.... 0.3...... 1-2 ......-0 .2

(B)Computing Bus Impedance Matrix

Fig. 4. Adding link branch po'qc' which is coupled to pl'ql', P2'q2', p3'qsA-Method of calculating ZL.; EL is such that IL=0 Step 1. Add first branch: 0-1. Check for

B-Method of calculating ZLL; EL is such that lL=1 ground connection. If 1 is a new bus,B-Method of cdlculdting ZLL. EL iS such thdt IL-1 add it to bus list. Use equation 2:

El-A biad-Digital Calculation, of Line-to-Ground Short Circuits 329

EL epfqiI .

(A)

JUNE, 1960

Netwrk

Fig. 5. Method oF calculating Zmn(new) For a change in p'q'i change in Zp'q' substituted bysuperimposing Ip' and Iq'I ip'q'(now)neW branch current from p' to q' after change in Zp'q'

Use equation 11 to eliminate L:

1[

Step 2. Add next branch: 0-2. Check forground connection. If 2 is a new bus, addit to bus list. Use equations 1 and 2.

1 2

2 5

Step 3. Add next branch: 0-3. Check forground connection. If 3 is a new bus, addit to bus list. Use equations 1 and 2.

1 2 3

Step 4. Add next branch: 1-2. Cheek forground connection. If 2 is not a new bus,this branch closes a loop. Use equations7 and 8. There is no coupling with apartial network.

1 2 3 L1 0.1 0.12 0.5 -0.53 0.5 0

LI 0. 1 -0.5 0 1.0

0.2 0.30.1 0.2

0.4

0.1 055©

1 2 3

1 0.09 0.051 0I2 10.05 0.25 013 l0 l0 l0.5l

Step 5. Add next branch: 1-4. Check forground connection. If 4 is a new bus, addit to bus list. Use equation 5. Correct forcoupling with partial network (with 1-2)using equation 9, then use equation 6 andcorrect for coupling using equation 10.

1-4 1-2

1-4 [0.2 l0.1]1-2 0.1 4

1-4 1-21-4 5.71 -1.43

Y 1-2 - 1.43 2.86

Z41 = 0.09_(°(0.1)-(2.86)(0.09-0.05) = 0.08Z41=0.09-1-(0.1)(-1.43)In the same way Z42= 0.1 and Z43= 0.

Z= 0.08+0.2-(0.1 )(2.86)(0.08- 0.1)- (0.2)(0.1)(-1.43)

1-(0.-)(-1.43)= 0.26

1 2 3 4

1 0.09 0.05 0 0.082 0.05 0.25 o 0.13 0 0 0.5 0

4 0.08 0.1 0 0.26

Step 6. Add next branch: 3-4. Check forground connection. If 4 is not a new bus,this branch closes a loop. Use equation 7.Correct for coupling with partial networkusing equation 9, then use equation 8 andcorrect for coupling using equation 10.

3-4 1-2 1-4

3-4 0.3_ -0.2 _02p'q,=1-2 -0.2 0.4 0.1

1-4 0 0.1

3-4 1-2 1-4

3-4 5 .4 3.08 -1.54Ypv'q' 1-2 3.08 4.62 -2.31

1-4 -1.54 -2.31 6.15

ZLI=0-O.08-

1-0.2, 0] 4.61, -2.311F0.09-0.051' L -2.31, 6.16JL0.09-0.08J

1- [-0.02, 0]3 08

_=-0.06Similarly, ZL2 -0.2, ZL3= 0.5, and ZL4=

-0.22.

ZLL 0.5-(-0.22)+0.3-

[-0.2, 0l[_.2, -2.366X

-0.06-(-0.2)0

_-0.06-(-0.22) -0.3[-0.2, 0]Xr3.08--1 .54

1- f-0.2, ]301-l54]=0.944

1 2 3 4 L

1 0.09 0.05 0 0.08 -0.062 0.05 0.25 0 0.10 -0.202 0 0 0.5 0 0.54 0.08 0.10 0 0.26 -0.22L -0.06 -0.20 0.50 -0.2210.944

Eliminate L using equation 11,'therefore:1 2 3 4

1 0.086 0.037 0.032 0.0662 0.037 0.208 0.106 0.053

Zmn3 0.032 0.106 0.234 0.117

4 0.066 0.053 0.117 0.209

Modifying Z.n for System Changes

Change Z3-4 from 0.3 to 0.24; and 23-4, 1-from -0.2 to 0.

1-2 1-4 3-4

1-2 0.4 0.1 -0.2

Zp'l'= l1-A .1 0.2 0

3-4 -0.2 0 0.3

and

1-2 1-4 3-4

1-2 4.615[ -2308 3.077yp'p'= 1-4 -263086.154 -1.538

3-4 3.077 - 1.538 1 5

1-2 1-4 3-4

1-2 0.1 0.1 0

Zp'q'(new) 1A4 0.1 0.2 0

3-4 0 0 1 0.24 1

El-Abiad-Digital Calculation of Line-to-Ground Short Circuits

Fig. 6. Sample syste

REFERENCE BUS

m

JUNE 1960330

and

1-2 1-4 3-4

1-2 2.857 - 1.429 0

Ypt'(new)-=1-4 -1.429 5.714 03-4 0 0 4.

1.758 _

-0. 8793.077 _

1-4 3-4

-0.879 3.0770.440 -1.538

-1.538 1 .218

REFERENCE BUS | @1

Fig. 7. Dividing sample system into sub-networks A and B

A few questions arose as the paperwas studied:

1. In the list of main advantages, it isstated ".. . -neither a transformation matrixnor matrix inversion is used." In the"Computation of Bus Impedance Matrix"section, however, it is stated "Invert theabove branch impedance submatrix togive:... " Does this mean that only smallordered matrices have to be inverted?2. When an impedance matrix has to bemodified, doesn't it require considerablymore change than for a nodal representation?

1-2 1-4 3-4

1-2 0.219 0.036 -0.087Zp'q,p' =1L4 0.036 I0.162 0.057

3-4 z-0.087 0.057 0.028(The following'should be a symmetricmatrix.)

1-2 1-4 3-4

1-2 3.995 -1.997 4.775

Yp'2,D'q'= 1-4-41.991 0.995 -2.3763-4 4.7751-2.387 3.475

1 2 3 4

1 0.083 0.042 0.040 0.059Zmn(new) = 2 0.042 0.240 0.049 0.074

3 0.040 0.049 0.250 0.1304 0.059 0.074 0.130 0. 192

Network Subdivision

Dividing the sample system into two sub-networks, A and B as shown in Fig. 7, theresults of the computation proceed asfollows:

0-1 0-2

0-1 r 1 IZA=

0II .I

0-3 3-4 4-1 1-2

0-3 0.53-4 0.3 -0.24-1 ____ 0.2 -0.11 2 --0.2 -0.1 0.4

1 2

7A 1 1 °2 I 0.5

1

,Z 2ZB= 3

4

1 2 3 4

1.0 0.7 0.5 0.80.7 0.8 0.5 0.60.5 0.5 0.5 0.50.8 0.6 l0.5 0.8

0-1ZA-equivalent-- 2

0-1ZB-equivalent= 0-2

0-1 0-2

[0.1 °l]I0.0 ().51

0-1 0-2

1.0 0.7l0.7 0.8l

ZA+ B-

ZB+ A-

1 2 3. It appears that the method used in the

1 0.086 0.037 paper automatically selects the correctequivalent 2 0 037 O.207 number of loops for describing a network by

the mesh method. Has the author en-

countered any cases where the networkequivalent topology gave trouble?

2 3 4. For what computers have short-circuit1 0.086 0.037 0.032 0.066

=2 0.037 0.208 0.106 0.0533 0.032 0.106 0.234 0.1174 0.066 0.053 0.117 0.208

ReFerences1. DIGITAL CALCULATION OF SHORT-CIRCUIT CUR-RENTS IN LARGE COMPLEX IMPEDANCE NEBTWORKS,L. W. Coombe, D. G. Lewis. AIEE Transactions,pt. III (Power Apparatus and Systems), vol. 74,1955(Feb. 1956 section), pp. 1394-97.

2. DIGITAL SHORT-CIRCUIT SOLUTION OF POWERSYSTEM NBTWORKS INCLUDING MUTUAL IMPED-ANCE, Martin J. Lantz. Ibid., vol. 76, 1957 (Feb.1958 section), pp. 1230-35.3. NODAL RIPRESENTATION oF LARGE COMPLEX-ELEDMENT NETWORKS INCLUDING MUTUAL REACT-ANCE, J. W. Siegel, G. W. Bills. Ibid., vol. 77, 1958(Feb. 1959 section), pp. 1226-29.4. DIGITAL CALCULATION OF POWER SYSTEM NET-WORKS UNDER FAULTED CONDITIONs, R. T. Byerly,R. W. Long, C. J. Baldwin, Jr., C. W. King. Ibid.,pp. 1296-1307.5. DIGITAL CALCULATION OF NETWORK IM-PEDANCES, A. F. Glinin, R. Habermann, Jr., J. M.Henderson, L. K. Kirchmayer. Ibid., vol. 74, Dec.1955, pp. 1285-95.6. DIGITAL COMPUTATION OF DRIVING POINTAND TRANSPBE IMPBDANCE, H. W. Hale, J. B. Ward.Ibid., vol. 76, Aug. 1957, pp. 476-81.7. TENSOR ANALYSIS OF NTrwoRKs (book),Gabriel Kron. John Wiley & Sons, Inc., NewYork, N. Y., 1949.

8. DIGITAL SHORT-CIRCUIT SOLUTION OP POWERSYSTEM NETWORKS INCLUDING MUTUAL IMPED-ANCE, A. H. EIl-Abiad. Report no. 605, AmericanElectric Power Service Corporation, New York,N. Y., Aug. 1959.9. TE3NSORS EXTBND NETWORK ANALYSIS FAULTSTUDIES, R. Bruce Shipley. Electric Light andPower, Chicago, Ill., Dec. 1955, p. 94.

DiscussionG. W. Bills (North American Aviation, Inc.,Downey, Calif.): This paper is of consider-able interest to power-system engineers whomust analyze networks containing mutualinductances. If network solutions can beobtained without the employment of itera-tion or matrix inversion techniques, suchdifficulties as convergence or roundoff andtruncation errors will disappear or be mini-mized.

programs been made?5. Could the author give the actual ma-chine computing times for a typical studyand specify the size of the network?6. Has a study been made which includesresistance as well as inductance in theelements?

This method has the merit of making acomputer perform the labor of setting upthe network equations and then solving forthe currents. It also makes use of GabrielKron's excellent tensorial methods, which,combined with digital techniques and digitalcomputers, make a very powerful tool forthe analysis of power-system networks.Soon the various programs which havebeen published for determining short-circuit currents with digital computersshould be compared on a large machine suchas an International Business Machines Corp-oration 709 to find the most efficient tech-nique.The author's method is certainly one that

should be included in such a comparison andI commend him for his contribution.

V. Caleca (American Electric Power ServiceCorporation, New York, N. Y.): Speakingas a relay engineer, I think I am voicingthe sentiments of most of my colleagueswhen I say that we are most happy to seethe apparently successful outcome of thedigital computer program for short-circuitcalculations. Charged as we are with theresponsibility of protecting our systemsagainst the effects of all types of faults,those due to the perversity of nature as wellas the malpractices of man, an accurateknowledge of short-circuit capabilities isbasic to our work.

For many years the d-c network analyzerhas been a satisfactory tool. More recentlyhowever, the increasing size and complexityof power systems and their interconnections.have made its shortcomings more acute.In particular, the inability to represent zero-sequence coupling and negative impedances.often means that special techniques haveto be used, or the results of a study supple-mented by time-consuming longhand cal-culations. The a-c analyzer, of course,.has also been available, but in many in-stances other practical considerations havemade it inaccesible to the relay engineerat a given time.

El-A biad-Digital Calculation of Line-to-Ground Short Circuits

(Yp'q' -Yp'q(new) )1-2

1-2= 1-43-4

--

JUNE, 1960 331