Step by Step Formation of Ybus and Zbus

IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, VOL. PAS-89, NO. 5/6, MAY/JUNE 1970

Step-by-Step Formation of Bus Admittance MatrixKASI NAGAPPAN

Abstract-An algorithm has been developed to form the busadmittance matrix Ybu,, by forming the network through a step-by-step addition of a line or a passive element to the system, takinginto account the mutual coupling between the elements. Two sets offormulas have been derived for the addition of a tree branch andfor the addition of a link. This method eliminates the formation ofincidence matrices and does not require singular or nonsingulartransformations. This algorithm is very convenient for calculationin digital computers.

INTRODUCTION

POWER system analysis, like load flow studies, short-circuit studies, and transient stability studies, has become

very convenient with the advent of digital computers. Moreand more complex systems can now be handled by suitablemathematical models, constituting an ordered collection ofsystem parameters in the form of matrices. These models dependon the selection of independent variables. When the voltages areselected as independent variables, the corresponding currentsare dependent and the matrix relating the voltages to the cur-rents is then in the admittance form. When these voltages andcurrents are referred to the buses (independent nodes), thereference is the bus frame, and the resulting equations are usualindependent nodal equations. The voltages and currents, whenreferred to independent loops, are related by the admittancematrix in the loop frame of reference. When the currents aretreated as independent variables, the matrices are impedancematrices in the respective frames of reference.

It is obvious from the literature that these bus admittance andimpedance matrices, as well as loop admittance and impedances,have been widely used for various power system calculations.There are traditional methods of forming these matrices for agiven system, which require various connection or incidencematrices [1]-[6]. Algorithms for forming the bus impedancematrix and its dual, the loop admittance matrix, have beendeveloped and are widely used in various system studies [9]-[111. Fig. 1 describes how various parameter matrices areformed from the primitive impedance and admittance matrices,which give the self-impedance or admittance and the mutualimpedance or admittance, but not the interconnection of trans-mission lines.An algorithm has been developed to form the bus admittance

matrix Ybu,s by building the network through a step-by-stepaddition of a line or passive element to the system, taking intoaccount the mutual coupling between the elements. This methodeliminates the formation of incidence matrices and does notrequire singular or nonsingular transformation [11, [41-[6].

Paper 69 TP 629-PWR, recommended and approved by thePower System Engineering Committee of the IEEE Power Groupfor presentation at the IEEE Summer Power Meeting, Dallas,Tex., June 22-27, 1969. Manuscript submitted December 16, 1968;made available for printing April 14, 1969.The author is with the Thiagarajar College of Engineering,

Madurai-15, Madras, India.

Fig. 1. Formation of network matrices from primitive matrices.

This method has the same advantages as the algorithm for thebus impedance matrix, such as

1) comparatively low storage space requirements in the com-puter,

2) less time (due to elimination of large matrix multiplications)required by transformation and major inversions of matrices,

3) greater accuracy, by avoiding matrix inversions andmultiplications, and

4) simpler modifications in network matrices to follow systemchanges.

REVIEW OF TRADITIONAL METHODSGiven the self-impedances and mutual impedances of trans-

mission lines, the primitive impedance matrix, [z] can be formedselecting the order of the lines arbitrarily. The matrix [z] wheninverted gives the primitive admittance matrix. Neither primitivematrix reveals the interconnection of various lines when thelines are numbered serially. The size of these matrices is e X ewhere e is the number of lines in the system. It has been de-veloped [1 ]-[4], [11] that the bus admittance matrix Ybu,s can beobtained by singular transformation

Ybus = At[y]A (1)where A is the bus incidence matrix showing the incidence oflines or elements to the buses in the system and its size is e Xb, with b the number of buses (b = n-1, where n is the numberof nodes); At is the transpose of matrix A, [yI is the primitiveadmittance matrix, and Ybus is the bus admittance matrix,whose size is b X b.

It has also been shown [1]-[41, [11 ] that Zloop, the loop imped-ance matrix, can be formed by singular transformation as fol-lows:

Zloop= Ct[z]C (2)

812

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NAGAPPAN: STEP-BY-STEP FORMATION OF BUS ADMITTANCE MATRIX

where C is the basic loop incidence matrix of size e X L and L isthe number of independent or basic loops. Note that there is adual relationship between Ybus and Z100p.

Besides the method of singular transformations, there is amethod of nonsingular transformation, which employs aug-mented incidence matrices A and C [11]. The matrix A has extrarows and columns corresponding to the fictitious nodes for thelinks, whereas the matrix C has extra rows and columns corre-sponding to the open path or loops for the tree branches. Suchnonsingular transformations of ly] and [z] result in the matricesYaux and Zaux as follows:

Yaux = At[y]A (3)

Zaux = Ct[Z]C, (4)These matrices are also referred to as orthogonal networkmatrices [5], [6]. The matrix Ybus is a submatrix of Yaux,consisting of rows and columns corresponding to the buses orindependent nodes, whereas Zl00p is a submatrix of Zaux, con-sisting of rows and columns corresponding to the basic orindependent loops [5], [6], [11].

ALGORITHM TO COMPUTE ZbuSHaving formed Ybus and Zloop0 Zbus and Yloop can be ob-

tained by inversion of the corresponding matrices. This is amajor inversion, requiring more storage space and time in thedigital computer. In order to avoid the major inversion, analgorithm has been developed by El-Abiad [9], [11], [13]to form Zbus by adding one line or element at a time andcomputing the resultant matrix. The size of the matrix is in-creased by one when a tree branch is added, whereas all the entriesof the matrix are modified when a link is added. This methodinvolves inversion of small-size matrices whenever an element orline, mutually coupled to the existing elements in the network, isadded.

ALGORITHM TO COMPUTE YbusThe matrix Ybus is formed by step-by-step addition of a line

or passive element. The size of the matrix is increased by onefor the addition of a tree branch that adds a new bus, modifyingthe existing entries of Ybus, whereas the size remains the samefor the addition of a link but the entries of the existing matrixare modified.

Addition of Branch p-q

Consider a network with n buses, shown in Fig. 2, to which aline p-q is added. This adds a new bus to the network. Theline p-q has mutual coupling with some or all of the existinglines or elements in the network. The performance equation ofthe network with the new bus q is

1 1 n q

q[ Ibu YbusV' |Ya ]V.(5q Vqa Yqq_ _Vq_

Since the networks of the power system are bilateral, Ya# =Y#ffa for all a and (. First Yaq (a = 1,2,*.. ,n,q) is computed asfollows:

Yqa = Yaq = Yp-q,pTVp-a (6)

where p-a is the list containing all the existing elements mutuallycoupled to p-q and the element p-q.

Fig. 2. Addition of branch p-q to network.

Fig. 3. Addition of link p-q to network.

9p-q, ¢ is the row of [y], the primitive admittance matrix ofall the existing elements, and the added element v-, is thevoltage across the element p-a and

vp_c =1 forp = a

Vf,= -1 for a = a

v,-" = Ofor pandaa a.

Next the entries Ya6t of Ybus' are computed as follows:

Y ,6 = ya.d + YaqY2d, for a ,B0= 1,2, ... nyqq

where Ya,t is the value from the bus admittance matrix beforethe addition of p-q. Proofs for (6) and (7) are given in the Ap-pendix. The new bus admittance matrix is (n + 1) X (n + 1)given in (5).

Addition of Link p-q

A fictitious node L is created by inserting a voltage sourceeL between node L and bus q such that eL = Vp_ - VP-L,as shown in Fig. 3.

813

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Then the performance equation of the network is

1 1 n Ln 'Ibus- [Ybus' Ya.L][V 8LLiq-LI L YLa YLL- eL ] (8)

First, YaL (a = 1,2, * ,n) is computed as follows:

YaL YLa = -Yp-q, p-qVp- (9)

where

vp_ = 1 for p = a

rra = -1 for o- = a

VP-" = 0 for p and u- : a.

Next, YLL is computed as

YLL = YP-q, P-q* (10)

Then, the entries Ya. of Ybus' in (8) are computed as follows:

= ya Ya,YL for a, = 1,2,* * ,n. (11)YLL

The new bus admittance matrix after the addition of thelink p-q is Ybus' in (8), ignoring the Lth row and column, i.e.,

Ybus (new) = Ybus'. (12)

Proofs for [9]-[12] are given in the Appendix. A summary ofthe equations appears in Table I.

NUMERICAL EXAMPLES

To illustrate the algorithm, a simple network, shown in Fig. 4,is considered, and a bus admittance matrix is obtained. Forsimplicity, real numbers are assumed for the line constants andthe line charging is neglected. Data for the problem are given inTable II.

Step 1

Add the branch between 0 and 1, p = 0, q = 1.

0-1[z] = 0-1 [0.5]

0-1[y] = 0-1[21

1Ybus = 1[ 2 ].

Step 2

Add the branch 1-2(1) between 1 and 2. This has couplingwith the branch 0-1, p = 1, q = 2, p-o-: 0-1 and 1-2(1).

0-1 1-2(1)

[z] 0-l O.5 0.251-2(1)_0.25 0.4

0-1 1-2(1)0-1 [2.9091 -1.8182

Y 1-2(1) -1.8182 3.6364]1 2

Ybus1 10.1816 -5.4546-2_-5.4546 3.6364]

0D.2 '6 1-2(2)

Fig. 4. Network for example.

TABLE IIDATA FOR NUMERICAL EXAMPLE*

S Xp-q,p-q Coupled Xp-pq,-oNumber Line p-q (pu) with Line (pu)

1 0-1 0.5 1-2(1) 0.252 1-2(1) 0.4 0-1 0.253 2-3 0.24 1-2(2) 0.6 1-2(1) 0.205 0.3 0.5

* Reference point is zero.

Step 3

Add the branch between 2 and 3. This is not coupled with0-1 or 1-2(1); p = 2, q = 3.

0-1 1-2(1) 2-30-1 0.5 0.25 0

[z] = 1-2(1) 0.25 0.4 02-3 Lo 0 0.2j

0-1 1-2(1) 2-30-1 F 2.9091 -1.8182 0.01

[y] = 1-2(1) -1.8182 3.6364 0.02-3 L 0.0 0.0 5.01

1 2 31[ 10.1816 -5.4546 01

Ybus = 2 - 5.4546 8.6364 -5 .3L 0.0 -5 5_

Step 4Add the link 1-2(2) between 1 and 2. This has coupling with

the existing branches p = 1, q = 2,L = 2(2), p-a:0-1 and 1-2(1).

0-1 1-2(1) 2-3 1-2(2)0-1 0.5 0.25

[z] = 1-2(1) 0.25 0.41-2(3 0 01-2(2)[ 0 0.2

0

0

0.20

0

0.20

0.6 I

815


IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, MAY/JUNE 1970

0-10-1 3.2

-y= 1-2(1) -2.42-32) 01-2(2)L0.8

1-2(1)-2.44.80

-1.6

2-30050

1-2(2)0.811.6 .

2.2 j

The bus admittance matrix including the fictitious node L is

11 10=.1816

ybu I=

2 - 5.4546bs-3 0

L 0 .2

2-5.45468.6364

-50.6

3 L0 0.2

-5 0.6j5 00 2.2j

the bus q, the same amount of current has to be subtracted.Comparing this equation with (7) will reveal that the extracurrent term is added when the new bus is introduced andextra current flows into the new bus q.By dual relations between Ybus and Zl,,op it seems logical to

visualize the possibility of computing Zloop by an algorithm.Further investigation is in progress in this direction.

APPENDIX

I. ADDITION OF BRANCH p-q

By injecting 1 pu voltage from the reference node tobus a (a = 1,2, *, n,q), while all the buses except a areshort circuited with the reference node, it is seen from (5) that

After eliminating the Lth node, the resultant bus admittancematrix is

1 2 3

1b 10.1998 -5.4001 00

Ybus = 2 -5 .4001 8.8000 -52.

3L O -5.0 5.0

Step 5

Add the link between 0 and 3. This has no coupling p = 0q = 3, L = 3. The bus admittance matrix including the fictitiousnode L is

11 10401998

y _2 -5.4001bs-3 0

L 0

2-5.40018.8000

-5.00

3 L0 o

-5.0 05.0 22 2--

After eliminating the Lth node, the resultant bus admittancematrix is

1 2 310. 1998 -5.4001 0 1

Ybus = 2 -5.4001 8.8000 -5.0000.3L ° -5. °°°° 7 .0000°

The bus impedance matrix Zbus, for this network is computedby the algorithm [9]- [11] and by

Yqa = Iv

Y,pa,'= I, , = 1,2, *- ,n.

(13)

(14)

From Fig. 2

I e = -ip.l (15)

Suppose p-q is coupled with a-k only; then

ipq = Yp-q, a-kV, _k, but Va-k = 1 pu

Therefore

ipq = Yp-q, a-k*

If p-q is coupled with k-a only, by the same reasoning ip, =-Yp q,k-a, since Vk-a = -1. If p-q is coupled with i-k ork-i only, ip-q = 0, since V1-k or )k = 0. Therefore, if p-q iscoupled with more than one elernent in the network, ipq canbe written as

=p//fr-g p-( YP0gpo (16)

with p-o spanning all the coupled elements including p-q and

VP-or= 1 for p = a

VP-a = -1 for af = a

vp-or = Ofor pand a s a.

Combining (13), (15), and (16)1 2

1 0.2211 0.2121Zbus = 2j 0.2121 0.4213

3LO 1503 0.3010

30.15030.30100.3578j

It can be checked that the product of Ybus and Zbus is anidentity matrix.

CONCLUSIONS

The same algorithm used to compute Ybus can be used for aremoval or for a small change in the line constant of a line notcoupled with other lines. However, for a line coupled with otherelements, a separate algorithm has to be developed similar tothe one developed for modifications in Zbu8 [9]-[11]. This isstill under investigation.

It may be recalled that the entries of Ybus have to be modifiedin accordance with the following expression when the qth bushas to be eliminated: Yafi' = Yap - YaqYqp/Yqq. The secondterm on the right-hand side corresponds to the current into busa, due to the current I; and when I,? is made zero by eliminating

(17)Yqa = Yaq = -gp-qy p-TPp-

with ip-r having the coniditiorLs as above.The currenit I,0 in (14) can be written as

I = lo0 + I'fi (18)

where Igo is the current into bus : when the element p-q is notcoupled with any of the existing elements in the network, andJI& is the extra current flowing into bus /3 due to the couplingeffect of p-q with the existing elements. By definition

Io =Y= a (19)

Y#1a is the transfer admittance between buses ,B and a before theaddition of p-q, since the addition of uncoupled p-q does notchange the current into bus /.The current I& can be calculated by injecting a voltage into

bus q from the reference, and short circuiting all the other busesto the reference [principle of superposition] such that thecurrent'q = Yqa.

816



From the performance equation (5)

I'B= Y3aTVa + Y13qVg (20)

Iq = ?qaV7a + Y1qV5. (21)

Substitution of Iq = Yqa and Va = 0 (a = 1,2,.* ,n) in (21)yields

Yqa = YqqV, i.e.,

Vq = yqa (22)yqq

Substituting the value of Vq given by (22) and Va = 0 (a =1,2, *A,n) in (20), for a particular value of (3

I13. = Y# qay. (23)

Substituting (19) and (23) into (18) and combining with(14)

Y = Yea + Y1*qyqa (24)yqqII. ADDITION OF LINK P-q

Referring to Fig. 3, L is a fictitious node created by a voltagesource between L and q such that

eL = Vpq - VpL* (25)

Then the performance equation can be written as

1 n L

Ybus' YaL V13 IV a (6I,YU iALl[ =n[-]. (26)L 1413YLL ELeLJ LL 1QL

Since p-L can be treated as a branch, the equations derivedfor a branch p-q in Section I of the Appendix, holding good asfar as YaL and YLa (a = 1,2,* ,n) are concerned, YLL hasto be evaluated as follows: When eL is equal to 1 pu and V13( 1=1,2,*,n) is equal to zero

YLL = iqL. (27)

Since L is not short circuited with the reference and p is shortcircuited with the reference

iqL = iLp = Yp-q, p_eL = YP-q, p-2q (28)

From (27) and (28)

YLL = Yp-q, p-q (29)

YOa1 (a,3 = 1,2, ,n) is computed the same as in the case

of a branch p-q, but q should be replaced by L.

YVa1 = YVa1 + YaLYL3 (30)YLL

Now the fictitious node L has to be eliminated by making eLequal to 0. Then Vpq becomes equal to VPL. Substituting eLequal to zero in (26)

Ya V1 = Ia, a, = 1,2, n. (31)

Ybus' Vbus = Ibus because there are actually n buses. There-fore Ybus' is obtained by ignoring the row and column corre-sponding to the fictitious node L.

REFERENCES[1] G. Kron, "Tensorial analysis of integrated transmission

systems, pt. I: six basic reference frames," AIEE Trans.,vol. 70, pp. 1239-1248, 1951.

[2] P. Le Corbeiller, Matrix Analysis of Electric Networks. Cam-bridge, Mass.: Harvard University Press, 1950.

[3] N. Sato, "Digital calculation of network inverse and meshtransformation matrices," AIEE Trans. (Power Apparatus andSystems), vol. 79, pp. 719-726, October 1960.

[4] G. Kron, "Improved procedure for interconnecting piece-wisesolutions," J. Franklin Inst., vol. 262, pp. 385-392, Novem-ber 1956.

[5] H. H. Happ, "Orthogonal networks," IEEE Trans. PowerApparatus and Systems, vol. PAS-85, pp. 281-294, March1966.

[6] ,"Special cases of orthogonal networks-tree and link,"IEEE Trans. Power Apparatus and Systems, vol. PAS-85, pp.880-891, August 1966.

[7] A. F. Glimn, B. Habermann, Jr., J. M. Henderson, and L. K.Kirchmayer, "Digital calculation of network impedances,"AIEE Trans. (Power Apparatus and Systems), vol. 74, pp.1285-1297, December 1955.

[8] H. W. Hale and J. B. Ward, "Digital computation of drivingpoint and transfer impedances," AIEE Trans. (Power Ap-paratus and Systems), vol. 76, pp. 476-481, August 1957.

[9] A. H. El-Abiad, "Algorithm for direct computation andmodification of solution matrices of networks including mutualimpedance," presented at the 1st PICA Conference, Phoenix,Ariz., April 24-26, 1963.

[10] ,"Digital computer analysis of large linear systems,"presented at the 1st Allerton Conference on Circuits andSystems Theory, Urbana, Ill., November 1963.

[11] G. W. Stagg and A. H. El-Abiad, Computer Methods in PowerSystem Analysis. New York: McGraw-Hill, 1968.

[12] J. C. Siegel and G. W. Bills, "Nodal representation of largecomplex-element networks including mutual reactances,"AIEE Trans. (Power Apparatus and Systems), vol. 77, pp.1226-1229, 1958 (February 1959 sec.).

[13] A. H. El-Abiad, "Digital calculation of line-to-ground shortcircuits by matrix method," AIEE Trans. (Power Apparatusand Systems), vol. 79, pp. 323-332, June 1960.

Discussion

Kavuru A. Ramarao (Cleveland Electric Illuminating Company,Cleveland, Ohio): Based on our experience in the Cleveland ElectricIlluminating Company, in the development of the short-circuitprogram (with mutuals), we would like to offer the following com-ments.

It is stated in the Abstract that the method described eliminatesthe need for the formation of the incidence matrices and does notrequire singular or nonsingular transformations. If, as in (1), weform Ybus using Ybus = At[y]A, the Ybus terms can be calculated(term by term) without any need for either the formation of A orA' matrices, or any multiplications whatsoever. For example, Yij iscalculated by merely adding and subtracting the terms from theprimitive admittance matrix y as indicated by the branch numbersconnected to the nodes i and j. Logic and algebraic addition areused rather than multiplication or division. The short-circuit pro-gram (with mutuals) developed in the Cleveland Electric Illumi-nating Company contains these simplifications and is in use. Asimilar approach is described in [14].

REFERENCES[14] H. Siemaszko, "Reduced matrix calculus application to the

network computation," Proc. PSCC, pt. 2 (Network Analysis),Rept. 4-8, 1966.

Manuscript received July 7, 1969.

817


IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, MAY/JUNE 1970

M. Ramamoorty (Indian Institute of Technology, Kanpur, India):The author to be congratulated for his timely paper on Ybus matrixconstruction. The use of Zbu, and Yb,9 methods for load flow calcula-tions has been widely discussed in the literature. It has been foundthat Zbu9 methods have better convergent properties as comparedto the Yb,, methods. The reasons for this have not been discussed yet.One possible reason appears to be that a Ybu, matrix has a maximumnumber of zero off-diagonal elements. This property was used byTinney and others to reduce the storage requirements by optimalordering. However, the same property results in a low convergencerate for load flow problems using Ybus. This is because the changesmade at a particular bus are not effective at other buses. The Zbu,,matrix has very few zero elements and so while making any changeat a particular bus during any stage, the effect of all the other busesis considered, thus resulting in faster convergence. The difficultywith Zbu, was that it required an inversion of Ybus. El-Abiad [9]gave a method for step-by-step construction of Zbus without goingthrough Ybus,

Since the primitive network is described in terms of self- andmutual branch impedances, Zbu, can be easily constructed. Now asimilar approach has been suggested by the author for Ybus con-struction. This method suffers from the drawback that the Ybranchmatrix has to be calculated at each stage, involving repeated in-versions of the Zbranch matrix. In the conventional Ybw, construc-tions, the primitive Ybranch matrix is obtained by inverting theZbranch matrix. The post- and premultiplication of this matrixwith singular or nonsingular matrix [11] gives rise to the Ybusmatrix.

In the discusser's opinion any extension or omission of linesfrom the existing network can be easily carried by the Zbus method.The discusser would like to know the advantages of this method.


C. H. Didriksen (Harza Engineering Company, Chicago, Ill.)and H. Pachon (UNIVAC, Chicago, Ill.): This paper is of theore-tical interest, and the author should be commended for his successin adding one more path to the table of formation of networkmatrices (Fig. 1).Two factors worthy of comment, but not considered within the

paper are usability and computation efficiency.Usability: The Ybus matrix is not usually considered for the

calculation of short circuits since its use requires an inversion routine,or an iterative procedure to arrive at the fault values [13, author's clo-sure]. Ybu. is used in load flow studies, but since mutual impedancesare not considered within this problem, its generation becomes a verystraightforward matter. The topological classification of a line as abranch or as a link, neglecting mutuals, has no special meaning in theformation of the Ybu, matrix. Would the author comment on thoseelectric network problems where the use of Ybu,,, as described in thepaper, is more advantageous than other methods.Computation Efficiency: A comparison of the formulas in Table I

against the Zbug formation formulas given in [11, table 4-1] showsthe following facts. The addition of a branch mutually coupled withexisting elements is much faster when forming the Zbu. matrix;this is so because when forming the Ybu. matrix, the addition of abranch requires modification of all the elements in the matrix(3), whereas in the Zbug matrix formation only one newcolumn is computed. For example, if a branch is to be added at stepn, then a total of n + (n - 1)2 multiplications or divisions is requiredfor Ybus formation, whereas the Zbu. formation will only require n.The addition of a link mutually coupled with existing elements isslightly faster when forming Yb,,, For example, if a link is to beadded at step n, then a total of n + n2 multiplications or divisions isrequired for Ybu, formation, whereas Zbus formation will require2n + n2. The values of estimated operations stated before assumethat since the matrices Ybus and Zbus are symmetric, only the upperor lower triangle is kept during calculation. Would the author com-ment on the comparison stated.Even though the paper claims that matrix inversion is not used,

the described method requires this technique to obtain the primitive

Manuscript received Julv 11, 1969.

admittance matrix [y]. The inversion considered here involves onlysmall matrices, but nevertheless, it is present. We have consideredthis matrix inversion problem and have found that the "productform of a matrix" [ 151 appears to be a very efficient way to computethe required [y] -matrix elements at each step.

In the example presented, the author had to invert the Z matrix atsteps 1, 2, and 4; step 3 does not require an inversion algorithmsince branch 2-3 is not coupled with existing elements, and itseffect on the y matrix can be readily computed. The point is that atstep 1, a (1 X 1) matrix must be inverted; at step 2, a (2 X 2)matrix must be inverted; and at step 4, a (3 X 3) matrix must beinverted. Had the example included a larger number of coupledbranches, larger matrix inversions would have been required.The product form of a matrix deals with the problem of computing

the inverse of a matrix for which only one column is different fromthat of a matrix whose inverse in known. Applied to the examplegiven in the paper, we find that prior to step 1, the Z matrix for thecoupled branches can be considered as

-1 0 O-ZO= 0 1 0

O O 1

for which the inverse is yo = Io = (3 X 3) unit matrix. At step 1, Zchanges to

-0.5 0 0oZ, = 0.25 1 0.

O0 O1XTo change yo to its new value yi, proceed as follows: let

6 = yo-new column

-0.5= 0.25].

Obtain

1/0.5-7= -0.25/0.5

then

-771 0 O- - 2.0 0 0oYI = X2 1 0 O= -0.5 1 0

A s3 0 be f 0 01at

A similar procedure can be followed to obtain at the end of step 2

-[0.5Z2 = 0.25

O

0.25 0]0.4 0 ,0.2 1_

[ 2.9091Y2 = - 1.8182

L 0.3636

-1.8182 0]3.6364 0

-0.7273 1_

and at the end of step 4

Z0.5Z4 = 0.25

O

0.25 0 ]0.4 0.210.2 0.6-

y 3.2Y4 = -2 .4

_ 0.8

-2.44.8

-1.6

0.8--1.6 .2 .2_

We note that moving from yo to yi, to Y2, and finally to Y4 doesnot require a matrix inversion at each step, but only elementarytransformations [16]; nonetheless, at the end of step 4, the inverseof the (3 X 3) impedance matrix corresponding to the three coupledbranches is obtained. It has been determined [16] that to obtain theinverse of a matrix A, (n X n), in the manner sketched above, itrequires (n3 X n2)/2 multiplications or divisions, whereas it takes n3to compute A-1 by other known methods. Thus, for the examplegiven in the paper it may have taken Z3 i3 -36 multiplications ordivisions to compute all the required inverses. The use of the productform would take only 18. In general, for n coupled branches it takes

n3 = n2 (n + 1)2P=

multiplications or divisions to compute their primitive admittancematrices, if their primitive impedance matrices are inverted at eachstep. The product form of the matrix will take only (ni3 + n2)/2, and

818



will still provide the elements of the primitive admittance matricesrequired at each step. We are looking into the applications of theproduct form of a matrix to network matrix formations and wouldappreciate the author's comment on the particular applicationdescribed in this discussion.

REFERENCES[15] G. Hadley, Linear Programming. Reading, Mass.: Addison-

Wesley, 1963, pp. 48-50.[161 A. Ralston and H. S. Wilf, Eds., Mathematical Methods for

Digital Computers. New York: Wiley, 1960-1967, pp. 44-45.

K. C. Kruempel and D. K. Reitan (University of Wisconsin, Madi-son, Wis.): The author has provided an algorithmic approach to theformation and modification of the bus admittance matrix. Theability to modify Ybug easily by an algorithm, especially for changesinvolving coupled lines, would seem to be of great interest.

In writing a paper of this type, it is often difficult to choose,define, and use precisely a notation. As an example of the problem:at one point the author states that p-a includes "all the existingelements and the added element." The work in the numericalexample tends to support this definition. However, at other placesp-a is said to contain "all the existing elements mutually coupledto p-q and the element p-q." In the Appendix p-a is defined to span"all the coupled elements including p-q." These are three slightlydifferent definitions, all of which are functional so far as the algorithmis concerned. The author might also note the column matrix up-a is acolumn of the bus incidence matrix for the partial network.In the conclusion, the author states that a separate algorithm

is necessary in order to modify coupled lines. This is not true,nor is it true in the algorithm for the bus impedance matrix [17].A change in the self-impedance or the removal of a coupled line issimulated by the addition of a "new" line in parallel. The mutualimpedances of the "new" line are of the same sign and magnitude asthose of the line to be changed; the self-impedance of the "new"line is chosen so that the parallel combination of self-impedancesis the desired value of the "modified" line.To illustrate with the author's algorithm and numerical example,

suppose it is desired to obtain Ybus for the same network, but withno mutual couplings included. This is accomplished by removing thecoupled line 1-2(1) and adding in its place line 1-2(3) with self-impedance equal to 0.4 and no mutual couplings.For the network with mutuals

1 21 10.2 -5.4

Ybus = 2 - 5.4 8.83 0.0 -5.0

30.08

-5.0 .

7.0_-(32)

Step 1. Add line 1-2(- 1) with self-impedance = -0.4 and withmutual impedances of 0.2 and 0.25 with 1-2(2) and 0-1, respectively.

0-10-1 F0.5

z'] 1-2(1) 0.251-2(2) 01-2(-1)LO.25

1-2(1)0.250.40.20

1-2(2) 1'2(- 1)0 0.25-0.2 00.6 0.20.2 -0.4 j

0-1 1-2(1) 1-2(2) 1-2(-1)Yp-q, p-v = 1-2(-1)[1.2500 -1.1979 0.8333 -1.3021]

__ 1- 2.9167]

B::] 2 -1.667

LYLL_I LL_-1. 3021jfrom which

1 2 31[ 3.667 -1.667 0

Yb,,' = 2 -1.667 6.667 -5.0 .3. 0 -5.0 7.02

Step 2. Add line 1-2(3) whose self-impedance equals 0.4 and nocouplings.

YLL [L 2.5]1 2

1F 6.167 -4.167Ybus' = 2 -4.167 9.167

3L 0 -5.0

(37)

(38)

3

-5.0 .

7.0

This result may be checked by inspection of the network since thereare no couplings involved.

If a "special algorithm" is desired in which several changes maybe made simultaneously, the following illustrates such a procedurefor the same changes as made above. Define these matrices:

0-10-1 [ 3.2

[Y]old = 1-2(1) -2.41-2(2) L 0.8

1-2(1)-2.44.8

-1.6

1-2(2)0.8'-

-1.62.2-

(39)

(40)

0-1 1-2(1) 1-2(2)10-1 --2.0 0 0

[Y]new = 1-2(1) 0 2.5 01-2(2) 0 0 1.6667

['Ay] = [y] new - [Y] old

0-1 1-2(1) 1-2(2)0-1 [-1.2 2.4 -0.8 ]

[Ay] = 1-2(1) 2.4 -2.3 1.6531-2(2) _-0.8 1.6 -0.5333

(41)

A is the bus incidence matrix for partial network including onlycoupled lines

1 20-1 --1 O-

A = 1-2(1) 1 -1 (42)1-2(2) L 1 -21

AY= A[Ay]A =

1 21 [-4.0333 1.23331

2 1.2333 0.36671(43)

Adding the terms in (43) to the appropriate terms in (32) gives

1 21 6.167 -4.167

Yb,,' = 2 -4.167 9.1673 0 -5.0

3

-5.07.0_

(44)

which agrees with (38).It should be noted that when a single-line p-q is added to a net-

(33) work, the bus admittance matrix describing the network is changedonly in certain rows and columns. The rows -and columns that changeare those corresponding to buses p, q, and any buses affected throughmutual couplings with line p-q. Since these buses are known whengps, ,, is calculated, the algorithm could be made more efficient byallowing a,fl to range over only these buses and not a,# = 1,2, ...,

(34) n, as the author states. Ya,q and Yq,s are zero unless calculated to benonzero.As a further observation, once all lines incident to a given bus

(35) and all lines affecting this bus through mutual couplings have beenprocessed by the algorithm, no further changes will arise in thecorresponding row and column of Ybus,Has the author attempted multiple-line additions? It would appear

that if all lines in a mutually coupled group were added simultane-ously, the successive recalculating of ygp-q, p-a would be avoided.

(36)


REFERENCES[17] D. K. Reitan and K. C. Kruempel, "Modifications of the bus

impedance matrix for svstem changes involving mutualcouplings," Proc. IEEE (Letters), vol. 57, pp. 1432-1433,August 1969.

819


IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, VOL. PAS-89, NO. 5/6, MAY/JUNE 1970

Kasi Nagappan: The author appreciates the interest shown by thediscussers and is grateful for the comments made by every one ofthem.The transformation method referred to by Mr. Ramarao will not

yield readily for the system changes, whereas the method developedin the paper will accommodate the system changes with fewercomputations.To answer the point raised by Mr. Ramamoorthy and Mr. Didriksen

and Mr. Pachon with regard to inversion to obtain a primitiveadmittance matrix, the author wishes to state that although in-version is not completely eliminated, it is restricted to small-sizematrices. For example, when there are 15 lines in the system, ofwhich three are coupled, and a line coupled with the two existing

Manuscript received August 15, 1969.

lines, the size of the matrix to be inverted will not exceed six. Sincethe number of coupled lines in a power system is small, the maximumsize of the matrix to be inverted is restricted to smaller sizes.Although it is conventional to neglect mutual coupling as far as

load flow studies and short-circuit studies are concerned, it is hopedthat accounting for mutual couplings will result in a well-conditionedbus admittance matrix to facilitate faster convergence of iterativetechnique.The author is developing programs in order to bring about com-

parison of different techniques to provide more information regardingcomputation efficiency raised by Mr. Didriksen and Mr. Pachon.While appreciating the interest shown by Mr. Kruempel and Prof.

Reitan in working out an example illustrating their technique, theauthor wishes to state that the procedure outlined is itself an al-gorithm to modify the matrix due to addition or removal of acoupled line and partial changes in a coupled line.

Simulation of AC System Impedance in

HVDC System StudiesNARAIN G. HINGORANI, SENIOR MEMBER, IEEE, AND MICHAEL F. BURBERY

Abstract-In some studies of HVDC system design, ac systemimpedance is represented by its inductance at power frequency.However, for several studies better simulation of the impedance-frequency characteristic (from power frequency to a few kilohertz)of the ac system is important. This paper presents a simple ap-proach to calculating an approximate equivalent network consistingof parallel LCR branches and having an impedance-frequencycharacteristic similar to that given for the ac system. For given typicalac system impedance diagrams (usually obtained from ac systemmodels and simulators), simulation of an ac system by correspondingequivalent networks during studies of certain HVDC problems wouldprovide a more accurate means of designing HVDC systems.

INTRODUCTION

WrITH THE increasing number and size of high-voltagedc power transmission schemes being commissioned,

there has been considerable investigation into better simulation(digital, analog, or model) of HVDC systems. For the design ofthe HVDC system itself, it is important that the impedanceof the ac system [Fig. 1(a) ] at various frequencies be simulatedcorrectly. Some of these problems are

1) design of ac filters,2) overvoltages at the converter station resulting from various

switching operations, blocking-unblocking, and fault conditions,

Paper 69 TP 632-PWR, recommended and approved by thePower System Engineering Committee of the IEEE Power Groupfor presentation at the IEEE Summer Power Meeting, Dallas,Tex., June 22-27, 1969. Manuscript submitted February 17, 1969;made available for printing April 14, 1969.N. G. Hingorani is with the Bonneville Power Administration,

Portland, Ore. 97208.M. F. Burbery is with GEC-AEI Ltd., Manchester, England.

(a)

L1Lp L3 S.L

R 1 R2 R3 % R F

02 03 :T:Cnl

AC SYSTEM(b)

Ip 1L3 Z Ln

RI tR2 R3 % Rn

(i) C2 C3 :T:CnACTSYST

AC SYSTEM

FI

(c)Fig. 1. Simulation of ac system impedance. (a) System for simula-

tion. (b) Simulation with fundamental-frequency source in serieswith whole ac system impedance. (c) Simulation with fundamental-frequency source in first L-R branch.

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Step by Step Formation of Ybus and Zbus

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