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Three-Phase

An electrical network under short-circuit conditions can be considered anetwork supplied by several sources (generators) with a single load con-nected to the system at the node subjected to the short circuit. The normalcustomer load currents ar e usually ignored, since they ar e small co m par edto the short-circuit current. Generally this simplification does not imparethe accuracy of the short-circuit study . T his is equivalent to the stru cturalanalysis of a bridge supported by several piers and subjected to a singleconcentrated load, with the weight of the individua l mem bers of thestructure being ignored. The re~nainderof this text concentrates on theanalysis of electrical network problems, but one should remember that thetechniques developed here apply e q ~ ~ a l l yell to structures.For a treatment of the reverse approach see Ref. 6. In the introduction


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Three-Phase Short-Circuit CalculationsDESCRIPTION OF TI1E Z-BUS MATRIXT h e 2 - b u s matrix contains the driving point imped ance of every nod e withrespect to a reference node that has been chosen arbitrarily. The drivingpoint im ped anc e of a nod e is the equivalent im ped anc e between i t and thereference. The 2-bus matrix also contains the transfer impedance betweeneach bus of the system and every other bus with respect to the referencebus. T he t ransfer impedances are determined by co ~ np ut in g he voltagesth at exist o n each of the othe r buses of the system, with respect to thereference, when a particular bus of the system is driven by an injectioncu rre nt of unity (see Fig. 3.1).T h e matrix equation relating the 2 -b u s matrix, the currents injected intothe nodes, and the node voltages is

It was recognized very early that i f the 2 - b u s matrix with the reference buschosen as the common bus behind the generator transient reactances wasavailable, the complete short-circuit analysis of the network could bereadily obtained with a small a m o u n t of additional computat ion. Re-membering, a s it was indicated earlier, that a network under fault condi-tions could b e considered to have a single node current, on e can write thematrix equation as


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Description of the 2-Bus Matrix

. Reference node ( R )

1, = 1.0Fig. 3.1. Driving point impedance: Ek - r = IZkk ; Ik = 1.O; Zkk= E k - r . Transferimpedance: lk= 1 .O; Eir= l Z i k ;Zik= Eir.

elements Z ik are the t ransfer impedance between the other buses and busk. In short-circuit calculations it is custom ary to assu m e that a ll generato rsconnected to the network are operating with 1.0 per unit voltage behindtheir internal reactances. This common point behind the generator reac-tances is used as reference. Th e network ca n therefore be considered to b esupplied by a single com m on source (see Fig. 3.2).*


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Three-Phase Short-Circuit CalculationsThe use of 1.0 pu voltage behind internal machine reactance can be

justified by applying the Helmholtz-Thevenin theorem. The prefault opencircuit voltage at the point of fault is approximately 1.0 pu and maytherefo re be assum ed to be that value. T h e short-circuit imp edan ce of thenetwork from the point of fault is determined by the impedance of thenetwork elements (including the internal machine impedances) with theinternal source voltages all short-circuited. The fault current is calculatedas the superposition of two sources: the current due to the internal sourcevoltages, which are zero; and the current due to a superposed voltagesource which will reduce the fault point voltage to zero. This voltage isobviously equal to the negative of the prefault voltage and is thus equal to- 1.0 pu fro m gro un d to the point of fault. But this gives the sam e curren ta s + 1.0 pu voltage from ground to a common bus behind transientreactances.When any node is short-circuited, it is connected to ground. Full voltageis therefore applied between the reference node and the node subjected tothe fault condition. Fo r example, for a fault on nod e 6 , the diagram can bedraw n as shown in Fig. 3.3.Since the matrix elemen ts of the Z-matrix of equa tion 3.2 are the drivingpoint impedances (diagonal e lements) and t ransfer impedances (off-diagonal elements) with respect to the reference bus, the voltages ofequat ion 3.2 will all be measured with respect to the reference node behindthe generator transient reactances. The reference node is therefore at zeropotential with respect to itself, but is at full voltage with respect to groundin the actual network. The voltage obtained from equation 3.2 for the bus


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Description of the 2-Bus M atrixunder fault condition is the full voltage of the generators with respect tothe reference. In the actual system the bus that is short-circuited is a t zeropotential with respect to ground. This difference in voltage, depending o nthe point of reference, should cause n o difficulty but must be taken intoconsideration in expressing the results of a calculation.

where E; is the voltage of bus p with ground as the reference, as it wouldbe measured in the actua l system. T h e E," is the voltage obtaine d fro m thematrix calculation of equation 3.2 and is measured with respect to thereference bus behind the generator transient reactances.When bus k is in short-circuit condition, the constraint that full voltageis applied to bus k will be satisfied by injecting a current Ik that isdetermined by equation 3.3.

See also equation 3.2.The total fault current for any bus is therefore obtained by taking thereciprocal of the corresponding diagonal element of the 2-matrix. Thevoltages that ap pe ar on the othe r buses of the system, wh en b us k is infault condition, depend on the transfer impedances as given by the offdiagonal elements of column k of the 2-bus matr ix. For example, thevoltage with respect to the reference on bus p for a short circuit on bus k


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Three-Phase Short-Circuit CalculationsT H E Z-MATRIX BUILDING ALGORITHMT o com pute the driving point an d t ransfer impedan ce matrix of a com -pletely assembled transmission system would be utterly impossible. How-ever, it is possible, by rather simple means, to modify the Z-matrix of asystem for the add ition of a single line. In this way the system c an b eassembled by starting with a system of a single transmission line, addingon e line a t a t ime, modifying the matrix fo r each line addition, a n dassembling the desired system and the matrix that corresponds to thesystem 131.DATA PREPARATION

A system d iagram is drawn. Th e junct ion points, where two or m oretransmission lines, transformers, or generator impedances are connected,ar e assigned a unique bus (node) num ber. Th e nu m ber zero is reserved forthe reference bus. In short-circuit studies the reference bus is selected asthe common point behind all generator reactances. (In other studies thereference bu s ma y be selected a s gro un d or a bus of the system. SeeChapte r 6.)Data are prepared by describing each element of the transmissionsystem by the two buses at the ends of the line an d i ts impe dan ce on acom m on per uni t base . These data a re sequenced by a n a lgor ithm from arandom ordering to a sequence such that as each l ine is selected from thedata l ist for processing, i t can be connected to the system that has beenassem bled. T h e first line in the list m ust b e on e from the reference to s om e


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(A dd ition of a Radial Line to a New BusCurrent injected into the new bus k which is connected by a radial lineto the reference, will produce no voltage on the other buses of the system

(see Fig. 3.4).Injection of curre nt into any bus of the system th at had been assem bledwill prod uce n o voltage on th e new bus k . All off diagonal elements of thenew row an d c o l ~ ~ m nre therefore zero.

II T he driving point im ped anc e of the new bus is the impe dan ce of the newline being add ed . T h e diagonal element of a new matrix axis correspondingto bus k is given by equation 3.7.

For the addition of a radial line from the reference to a new bus,augment the matr ix by a row and column of zeros. The diagonal e lementof this new axis is the imp ed an ce of the new line being a dd ed . T h e bu snumber k is added to the list of buses that comprise the system.

line


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Three-Phase Short-Circuit Calculationsto the voltages that would be produced i f current was injected into bus p(see Fig. 3.5).

The driving point impedance of bus q is equal to the driving pointimpedance of bus p plus the impedance of the line being added (see Fig.3.5).

A new axis is added to the matrix corresponding to the new bus q. T heoff-diagonal elem ents of the new row an d colum n ar e the same as theelements of the row and column of bus p of the existing system. T hediagonal element is obtained from equation 3.9. Bus q is added to thesystem bus list.

Reference bus

/ I \ \Network


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Addition of a Loop Closing LineInjection of a current of unity into bus p causes voltages to appear atevery bus of the system tha t are identical to the 2-m atrix elements of

column p of the matrix (see Fig. 3.6). Injection of a current into bus q of- 1.0 produces voltages on the buses of the system equal to the Z-m atrixelements of column q but of opposite sign. A loop current of unity can beconsidered to be a current of I, = 1.0 and I, = - 1.0 acting in concert. Thevoltages appearing on the system buses are the difference of the columnscorresponding to busesp and q as given in equa tion 3.10.- Z I qz,,

. . . .

Reference bus

Network


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Three-Phase Short-Circuit CalculationsReference bus

7 - 3

Fig. 3.7. Determination of the drivingpoint impedance of the loop.

A loop axis is ad de d to the 2-m atrix of equa tion 3.10 in whichZi-loop Zip- Ziq i# oopZ 0 Z,, - Z i# loop

T he dia gona l element is obtained from 3.12. T he loop axis is eliminatedfrom the matrix by a Kron reduction [4] by application of equation 3.15.All elements not in the loop row or column are modified. The loop rowand column are erased. The list of system buses remains unchanged.


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Addition of a Loop Closing LineII elimination. Kron reduction is given by

Z ;= Z,- Z , Z ~2,The validity of the reduction 3.15 can be proved by considering thematrix equation 3.16

[a: a:][; :]-[a:in which A, , A,, A,, A, ca n be thought of as matrices o r single coefficientsa n d X ,, X,, B , an d B, ar e vectors or single variables, respectively.Eq ua tion 3.16 in the expa nde d form is

Rewriting (3.18) givesA4X2= B 2 - A 3 X l

Premultiplication by A;' gives


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'Three-Phase Short-Circuit Calculations

A N ILLUSTRATIVE EXAMPLE OF THE BUILDING ALGORITHMConsider the network shown in Fig. 3.9 and the following data.

System DataLine x PU)

o- I 0.01o0-2 0.0 151-2 0.0840-3 0.0052-3 0.1222-4 0.0843-5 0.0371-6 O. 1266-7 0.1684-7 0.0845-8 0.0377-8 O. 140

O ornmon voltage sourcebus--


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An Illustrative Example of the Building Algorithm

Addition of the First Liiie The first line must always be a line connected tothe reference. Thus the line 0-1 can be the first line processed. At this pointno network, no matrix, and no entries in the bus list describe the system.The buses O and 1 are examined and compared with the system bus list todetermine the type of line and the algorithm to be used. The line is foundto be a line connected to the reference and to bus I. Bus 1 is comparedwith the list of buses in the system. At this point there are no buses in thenetwork. 'The line is, therefore, a degenerate case of the addition of a linefrom the reference to a new bus (see Fig. 3.4).

It is impossible to add a row and column of zeroes to the matrix, sincethere is no matrix at this point. The diagonal element of the new axis is theimpedance of the line being added. The new bus is added to the bus list.After addirig this first line we have:

1ma trix = 1[0.011 bus list= 1

The system diagram is shown below.


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36The system diagram is shown below.

Three-Phase Short-Circuit Calculations

Reference

0.015

1

The matrix shows us that injection of unit current into bus 1 and out atthe reference will cause a voltage of 0.01 to appear at bus 1 and a voltageof zero to appear on bus 2. Injection of unit current into bus 2 will producea voltage of 0.015 on bus 2 and a voltage of zero on bus I .Additiori of the Tliird Line The next line (1-2) is selected for processing.Examination of the bus numbers shows that this line is not a reference busline. Comparison of the bus numbers of the line with the system busnumbers verify that the line is a type 3 (loop closing) line.The matrix is augmented by a loop row and column by taking thedifference of the rows (and columns) corresponding to buses 1 and 2,equation 3.13. The diagonal element is obtained by equation 3.12.

loop1 0.0 1 o


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An Illustrative Example of the Building Algorithm1/Z, and a matrix inversion is not required. The modification of theelements not in the loop row and column can be most easily carried outelement by element rather than by applying equation 3.15 as a matrixequ a tion.

It can be easily verif ied that the modification of an element; Z, issimply :

~ OO P * Z l ~ ~ p - l ~ ~ pFig. 3.11 Matrix elements used in modifying element 2, by the Kron reduction.

Application of equation 3.23 is used to modify a11 elements not in theloo p axis of the matrix. T h e loo p axis is then erased.


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Three-Phase Short-Circuit CalculationsThe Matrix Reduction-A Delta-Star Coriversion The matrix reductioncan be viewed as a delta-star rediiction of the network. The network ofFig. 3.10 can be converted to the equivalent star by the standard delta-starconversion.

If tlie network of Fig. 3.12 is driven by a current of unity into bus I , thedriving point impedance Z , , is the sum of Z , and Z,.

ReferenceTI


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An Illustrative Example of the Building Algorithmof the new axis is the impedance of the line, 0.005. The new bus is added tothe bus list.

1 2 31 0.00908257 0.00137615 bus list 1,2,3matrix 2 0.00137615 0.012935793 O O 0.005

Addition of the Fifth Line The next line in the data list (2-3) is a loopclosing line, since both buses are in the system bus list. The loop axis is thedifference of the columns corresponding to the buses 2 and 3. Thediagonal element is obtained from equation 3.12.

The augmented matrix is

1

loop


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40

T h e system diagram is shown below.Three-Phase Short-Circuit Calculations

Reference

1

Addition of the Sixth Line T h e line 2-4 is identified to be a type 2 line, aline from the existing bus 2 to a new bus 4. A new axis is added to thematrix. The diagonal element is determined by use of equation 3.9.

T he off diagonal elements are obtained from equation 3.8 in which q = 4and y = 2. T h e row co rrespondiiig to bus 4 is identical to th e row of bus 2.

matrix 23

The system diagram is shown below. bus list 1,2,3,4


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An Illustrative Example of the Building AlgorithmColumn 5 is a duplicate of column 3.matrix

1

0.04182135 1bus list 1,2,3,4,5

Addition of the Eighth Line Addition to the system of line 1-6 wh ich is aline from an existing bus 1 to a new bus 6 is carried out as indicated forline 3-5 in step 7 (type 2 line).Addition of the Ninth Line The line 6-7 is also a type 2 line in which theline is from existing bus 6 to a new bus 7. The process is illustrated in step7. A fte r add itio n of these two lines the matrix is

1 2 3 4


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Three-Phase Short-Circuit CalculationsAdditiori of tlie Teiitli Liiie The line 4-7 is a loop closing liiie, since bothbuses are already in the system. The loop row and column are obtained bytaking the difference of rows 4 and 7 and the columns 4 and 7. T h ediagonal element of this new axis is obtained from equation 3.12.

The loop axis is eliininated as illustrated in steps 3 and 5.Addition of the Eleventb Liiie T he line 5-8 is determine d to be a type 2 lineand its addition has been illustrated in step 6.Addition of the Last line Line 7-8 is loop closing line. T h e method hasbeen illustrated in step 3 . I t is readily seen that regardless of the complex-ity of the network it can be assembled by this simple means of adding oneline a t a time. T h e completed matrix of the sample system is given forreference purposes.

1 2 3 4


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Voltages on Buses During Fault ConditionFAULT ANALYSIS OF A SYSTEMA complete analysis of the system is possible once the 2-matrix of thesystem is completed. Using values that are in the matrix of the samplenetwork for illustrative purposes, consider node 3 to be in fault condition.The matrix element Z,, =0.00475959 means that, if a voltage of that valueis applied between bus 3 and the reference, a total current of 1.0 will flowthrough the network. The full voltage of the generator will cause a currentthat can be determined by considering

in which I = 1.0 when E=0.00475959. It is desired to know I' whenE '= 1.0.

This is the result that is obtained by consideration of the matrix 3.2 andthe equation 3.3.The total fault value can be obtained either in amperes, pu, or MVA bydividing the base amperes, unity, or pu MVA base of the line data by thecorresponding diagonal element of the matrix (see equation 3.3).

The contribution to the fault by a line is computed using equation 3.5.The contribution from bus 2 to the fault is


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Three-Phase Short-Circuit Calculationsbut the voltage of bus 3 under fault condition is zero. Therefore, thevoltage of bus 2=0.8835.

OPENING A LINE DURING A STUDYA line of a system may be opened or removed by adding a line in parallelwitb the existing line. The impedance of the new line to be added is thenegative of the original line. The loop closing equations 3.12 and 3.13, andthe elimination of the loop by a Kron reduction 3.15 are used.

In the course of a complete fault study i t is often desirable to open eachline connected to a faulted bus, one at a time, and to obtain the new totalfault and the contribution of the remaining lines. It is undesirable tomodify the matrix of the total system because a great deal of computationwould be done unnecessarily on elements that are not required in theanalysis. Furthcrmore, i t is undesirable (because of rounding) to remove aline, add it back, remove another, add it back, and so on. Errors wouldaccumulate in the Z-matrix elements because of the repetative modifica-tion of the matrix.The best method is to extract a small matrix, from tlie total matrix, thatincludes the driving point and transfer impedances of the bus to be faultedand its immediate neighbors.

For example, if bus 3 is to be faulted, extract the small matrix


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Opening a Line During a StudyThe loop column is obtained by subtracting column 5 from column 3.

325

loop

5 loop0.004256 13 0.00050346 1

In the interest of efficiency the Kron reduction is used to modify onlyrow 3, since a fault on 3 can be completely analyzed with only thesevalues.

The modified row vector that reflects the opening of the line 3-5 is

The iiew total fault isbase -- 1Z3, 0.00482099 =207.43 pu

The contribution from bus 2 is found to be 7.4 pu. The flow from bus 5to 3 over the line X = 0.037 is the same magnitude as the flow over the line


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1 39+ auli here

Fig. 3.13. Fault transferred to open end of l ine.

BUS TIE BREAKERSCertain power systems have bus tie breakers that may be closed or openedfor operating reasons. For example, a generating station may have twogenerators connected to a bus system as shown in Fig. 3.14.

If either generator is out of service for overhaul, or for some operatingreason, i t may be necessary to close the bus tie breaker. The matrix can bemodified to represent the condition of the closed tie breaker by adding aline of zero impedance between bus P and Q. This is a perfectly satis-factory method of modeling the system for the closed bus tie breaker, butit would be impossible to open the closed breaker later in the study. Thisbecomes evident by considering the addition of a line of -0.0 impedancei? an %te\pt . t c w p n the line of impedance of 0.0 corresponding to thec osed rea er en the breaiter 1s cioseu, t n e i w u ~ u i u 1 i i i i 31 l L y


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Limitation of the 2-Matrix Methodbecome identical in value, since there is zero impedance connected be-tween the buses. The diagonal element for the addition of the -0.0impedance element, in parallel with the 0.0 impedance element, in anattempt to open the breaker becomes

when the buses are tied and Z,,, ,- ,= 0.Substitution of this value in equation 3.23 is not permitted. Therefore,

the bus tie breaker can not be opened if this representation is used.BUSTIEBREAKERTHATCANBEOPENEDTo tie two buses with a bus tie impedance of zero that can be opened oneintroduces a fictitious bus between the buses to be tied and adds a linefrom P to T with an impedance of 2 , and a line from T to Q with animpedance of - 2 , (see Fig. 3.15).

The impedance from bus P to bus Q is zero but now the breaker may beopened by adding a line from bus T to bus Q with impedance +2 ,.Thisremoves one side of the bus tie breaker.

Bus P


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Three-Phase Short-Circuit Calculationscom puted using X only withou t seriously affecting the acc ura cy of a study.Furthermore, the matrix is symmetric and can therefore be stored in uppertr iangular form. A system of 100 buses will require a matrix of 5050elements N (N + 1)/2. A system of 200 buses has 20100 elements. As late a s1958 fa uit studies of major power systems were m ade on analog modelswith fewer than 100 buses. The introduction of the digital computerprogram s increased very rapidly the num ber o f buses used in the represen-tation. It became necessary to devise methods to increase th e program size.This aspect of the problem is discussed in Chapter 4.

References1. J. B. Ward and H. W. Hale, Digital solu tion of power flow problem s, Trans. AIEE, Vol.75, Part 111, (1956), pp. 398-404.2. L. W. Coombe aod D. G. Lewis, Digital calculations of short circuit currents in largecomplex-impedance networks, Trans. AIEE, Vol. 75, Part. 111, (1956), pp. 1394-1397.3. H. E. Brown, C. E. Person, L. K. Kirchmayer, aud G. W. Stagg, Digital calculation oftbree-phase short circuits by matrix method, Trans. AIEE , Vol. 79, Part 111, (1960), pp.1277-1281.4. G . Kron, Tensor Analysis o/ Nelworks, Wiley, 1939.5 . A. H. El-Abiad, Digital calculation of line-to-ground sho rt circuits by matrix methods,Tram. AIEE , Vol. 79, Part 111, (1960), p. 323.6. R. A. Frazer, W. J. Duncan, and A. R. Collar, Elementaty Matrices, Cambridge Press,-- ----------

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