Step by Step Formation of Ybus and Zbus

812

IEEE TRANSACTIONS ON

Step-by-Step

Formation

POWER APPARATUS AND SYSTEMS, VOL. PAS-89, NO.

of

5/6, MAY/JUNE 1970

Bus Admittance Matrix

KASI NAGAPPAN

Abstract-An algorithm has been developed to form the bus admittance matrix Ybu,, by forming the network through a step-bystep addition of a line or a passive element to the system, taking

into account the mutual coupling between the elements. Two sets of formulas have been derived for the addition of a tree branch and for the addition of a link. This method eliminates the formation of incidence matrices and does not require singular or nonsingular transformations. This algorithm is very convenient for calculation in digital computers.

INTRODUCTION

POWER system analysis, like load flow studies, shortcircuit studies, and transient stability studies, has become very convenient with the advent of digital computers. More and more complex systems can now be handled by suitable mathematical models, constituting an ordered collection of system parameters in the form of matrices. These models depend on the selection of independent variables. When the voltages are selected as independent variables, the corresponding currents are dependent and the matrix relating the voltages to the currents is then in the admittance form. When these voltages and currents are referred to the buses (independent nodes), the reference is the bus frame, and the resulting equations are usual independent nodal equations. The voltages and currents, when referred to independent loops, are related by the admittance matrix in the loop frame of reference. When the currents are treated as independent variables, the matrices are impedance matrices in the respective frames of reference. It is obvious from the literature that these bus admittance and impedance 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 a given system, which require various connection or incidence matrices [1]-[6]. Algorithms for forming the bus impedance matrix and its dual, the loop admittance matrix, have been developed and are widely used in various system studies [9][111. Fig. 1 describes how various parameter matrices are formed from the primitive impedance and admittance matrices, which give the self-impedance or admittance and the mutual impedance or admittance, but not the interconnection of transmission lines. An algorithm has been developed to form the bus admittance matrix Ybu,s by building the network through a step-by-step addition of a line or passive element to the system, taking into account the mutual coupling between the elements. This method eliminates the formation of incidence matrices and does not require singular or nonsingular transformation [11, [41-[6]. Paper 69 TP 629-PWR, recommended and approved by the Power System Engineering Committee of the IEEE Power Group for 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 the bus impedance matrix, such as 1) comparatively low storage space requirements in the computer, 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 and multiplications, and 4) simpler modifications in network matrices to follow system changes. REVIEW OF TRADITIONAL METHODS Given the self-impedances and mutual impedances of transmission lines, the primitive impedance matrix, [z] can be formed selecting the order of the lines arbitrarily. The matrix [z] when inverted gives the primitive admittance matrix. Neither primitive matrix reveals the interconnection of various lines when the lines are numbered serially. The size of these matrices is e X e where e is the number of lines in the system. It has been developed [1 ]-[4], [11] that the bus admittance matrix Ybu,s can be obtained by singular transformation

Ybus

=

At[y]A

(1)

where A is the bus incidence matrix showing the incidence of lines or elements to the buses in the system and its size is e X b, with b the number of buses (b = n-1, where n is the number of nodes); At is the transpose of matrix A, [yI is the primitive admittance 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 impedance matrix, can be formed by singular transformation as follows:

Zloop= Ct[z]C

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(2)

813

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

where C is the basic loop incidence matrix of size e X L and L is the number of independent or basic loops. Note that there is a dual relationship between Ybus and Z100p. Besides the method of singular transformations, there is a method of nonsingular transformation, which employs augmented incidence matrices A and C [11]. The matrix A has extra rows and columns corresponding to the fictitious nodes for the links, whereas the matrix C has extra rows and columns corresponding to the open path or loops for the tree branches. Such nonsingular transformations of ly] and [z] result in the matrices Yaux and Zaux as follows:

Yaux = At[y ]A

(3)

Zaux

(4)

Ct[Z]C,

=

These matrices are also referred to as orthogonal network matrices [5], [6]. The matrix Ybus is a submatrix of Yaux, consisting of rows and columns corresponding to the buses or independent nodes, whereas Zl00p is a submatrix of Zaux, consisting of rows and columns corresponding to the basic or independent loops [5], [6], [11].

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

ALGORITHM TO COMPUTE ZbuS Having formed Ybus and Zloop0 Zbus and Yloop can be obtained by inversion of the corresponding matrices. This is a major inversion, requiring more storage space and time in the digital computer. In order to avoid the major inversion, an algorithm has been developed by El-Abiad [9], [11], [13] to form Zbus by adding one line or element at a time and computing the resultant matrix. The size of the matrix is increased by one when a tree branch is added, whereas all the entries of the matrix are modified when a link is added. This method involves inversion of small-size matrices whenever an element or line, mutually coupled to the existing elements in the network, is added. ALGORITHM TO COMPUTE Ybus The matrix Ybus is formed by step-by-step addition of a line or passive element. The size of the matrix is increased by one for the addition of a tree branch that adds a new bus, modifying the existing entries of Ybus, whereas the size remains the same for the addition of a link but the entries of the existing matrix are modified.

Addition of Branch p-q Consider a network with n buses, shown in Fig. 2, to which a line p-q is added. This adds a new bus to the network. The line p-q has mutual coupling with some or all of the existing lines or elements in the network. The performance equation of the network with the new bus q is 1

q[ Ibu

1

n

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 as follows: Yqa

=

Yaq

=

Yp-q,pTVp-a

9p-q, ¢ is the row of [y], the primitive admittance matrix of all the existing elements, and the added element v-, is the voltage across the element p-a and vp_c =1 forp = a Vf,= -1 for a = a a. a v,-" = Ofor panda Next the entries Ya6t of Ybus' are computed as follows: Y

q

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

q

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

(6)

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

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

...

n

(7)

where Ya,t is the value from the bus admittance matrix before the addition of p-q. Proofs for (6) and (7) are given in the Appendix. 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 source eL between node L and bus q such that eL = Vp_ - VP-L, as shown in Fig. 3.


IS NOT

FOR

A '

r"

RtlERENC

EL COUPZLD WITH 4X/STINGeTl |Ar-

rHE R,E,EReNCE

/5

U

Pn

TABLE I

F,iITTATTnA

| /A

PnlkITTq

NK

|

BRANCH/|

FOR f

Y,,L

=_

=I

FOR

f

=o

(

.... n)

X,°-pJ, 2, 3....ln

= 1,2,3,..f6,&f

+

ov)

Y=

NOT

1 Y,84vs~L,

t

Yi yq

YpzO

r

ta ECG RPERENC p-I-

SAMR A5 IA THE CASE OP

=

S*

-P

(J)

(1 L |v)Y

o'ii')

=

P xcEPr Llb

.

=

== Y

yI

p

YLa=Y o; O7'I,e?

I

|

4

=-,

ExceP:

NO r

SAME AS

(ii) Y(P

|(ii) Y

(ca.p, 2-.n) I

CC

4,2,5 n..

ii) YLP * Y&. .

ofo't| (V) yop *

C)Yco1,

LetiI

oc8

= Yy

Yoc Ycp

lb

= = 2,x3, P)) n,o,..

(W) (ii) Yocp Yxp

,

P

=0 FOR fPd& + a |(V) Yjp = Ypp+ VP-Jc, P-5

Yiii)

(1)

(i')YSsy,,+IYop=yL=oYt8VgyYp=

( O = i 2, 3, *

oFOR

--1 FOR Ot:O

tfa fl

, (ii)ct,p=Y pf

(ii)

,,

7]

ocSp =235 |,.

Y, ++

Yq rOp = Yccp Yocq Y.ip

y

)

=0 FOR f&cOr'#C

(i) 'YZL =YLU= _____

(ii)

vp

-I FOR czC- FOR TsO

f= oc

(oc= 1, 2,... n

Vfa

=

TH

CASE

Or

YL

.0

N/N R!PtREA1Ct

IN

X=q

R PBRENCE

xYp

6 "STRE

/sS NOT coupLeD E NOT TH!E| RFFZRZNCf

V.

8AN) H fh FyO j (I) YPc =Yc =l ii)YSc0Yp

ADDZD

-5

STTMMA-Rv OR

0-4

CO1

CO

z

H4 cO

0

0

z

t4

H

(J

H-

zi

00

815


Then the performance equation of the network is 1 L n [Ybus' Ya.L][V L YLa YLL- eL ] ,n) is computed as follows:

1

'IbusL Liq-LI n

First, YaL (a = 1,2, *

YaL

YLa

=

-Yp-q, p-qVp-

8

(8)

(9)

where

vp_

1 for p =

a

rra = -1 for o- =

a

=

VP-" = 0 for p and uNext, YLL is computed as

YLL

=

:

a.

YP-q, P-q*

(10)

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

Ya,YL

ya

YLL

=

for a,

1,2,* * ,n.

0D

=

(12)

Ybus'.

Proofs for [9]-[12] are given in the Appendix. A summary of the 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. For simplicity, real numbers are assumed for the line constants and the line charging is neglected. Data for the problem are given in Table II.

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

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

[y] =

Ybus

=

0-1[21 1 2 1[ ].

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

[z] Y

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

1-2(1) -1.8182 1

Ybus

TABLE II

DATA FOR NUMERICAL EXAMPLE*

S Number 1 2

Line p-q 0-1 1-2(1)

2-3 1-2(2) 0.3

3 4 5

*

Reference point is

1 10.1816 2_-5.4546

3.6364] 2

-5.45463.6364]

Xp-q,p-q (pu) 0.5 0.4 0.2 0.6 0.5

Coupled

with Line

1-2(1) 0-1 1-2(1)

Xp-pq,-o (pu) 0.25 0.25 0.20

zero.

Step 3 Add the branch between 2 and 3. This is not coupled with 0-1 or 1-2(1); p = 2, q = 3. 0-1 1-2(1) 2-3 0-1 0.5 0.25 0 [z] = 1-2(1) 0.25 0.4 0 0 2-3 Lo 0.2j 0-1 1-2(1) 2-3 0-1 F 2.9091 -1.8182 0.01 [y] = 1-2(1) -1.8182 3.6364 0.0 0.0 2-3 L 0.0 5.01

Ybus

=

3

2

1

0-1 1-2(1)

0-l O.5 0.25 1-2(1)_0.25 0.4

1-2(2)

Fig. 4. Network for example.

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

Ybus (new)

'6

.2

(11)

01 1[ 10.1816 -5.4546 2 - 5.4546 8.6364 -5 . -5 5_ 3L 0.0

Step 4 Add 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

[z]

=

1-2(1) 1-2(3 1-2(2)[

0.5 0.25 0.25 0.4

0

0

0

0.2

0

0

0.2

0

0.2

0


0

0.6

I

816

IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS,

-y

=

0-1 1-2(1) 2-3 3.2 -2.4 0 0-1 1-2(1) -2.4 4.8 0 0 5 2-32) 0 -1.6 0

1-2(2) L0.8

1-2(2)

0.8

11.6

.

2.2

j

The bus admittance matrix including the fictitious node L is

1

L 2 3 0 0.2 -5.4546 8.6364 -5 0.6j - 5.4546 5 0 -5 0 0 2.2j 0.6 L 0 .2

10=.1816

1 2 I ybu = bs-3

After eliminating the Lth node, the resultant bus admittance matrix is 1 2 3 10.1998 -5.4001 0 8.8000 -52. Ybus 2 -5 .4001 5.0 -5.0 O 3L

1b

the bus q, the same amount of current has to be subtracted. Comparing this equation with (7) will reveal that the extra current term is added when the new bus is introduced and extra 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 to bus a (a = 1,2, *, n,q), while all the buses except a are short circuited with the reference node, it is seen from (5) that Yqa

Y,pa,' = I,

,

Ie

=

Step 5 Add the link between 0 and 3. This has no coupling p = 0 q = 3, L = 3. The bus admittance matrix including the fictitious node L is

1

10401998

1 _2 bs-3

-5.4001 0

L

0

2

3

L

0 o -5.4001 8.8000 -5.0 0 5.0 2 -5.0 0 2 2--

After eliminating the Lth node, the resultant bus admittance matrix is 2 1 3 1 0 -5.4001 10. 1998 8.8000 -5.0000. Ybus = 2 -5.4001

3L

°

-5. °°°°

7

.0000°

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

Zbus

(13) (14)

Iv

=

=

1,2, * ,n. -

From Fig. 2

0

=

y

MAY/JUNE 1970

2 3 1 1 0.2211 0.2121 0.1503 = 2j 0.2121 0.4213 0.3010

3LO 1503 0.3010 0.3578j

It can be checked that the product of Ybus and Zbus is an identity matrix. CONCLUSIONS The same algorithm used to compute Ybus can be used for a removal or for a small change in the line constant of a line not

-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 or k-i only, ip-q = 0, since V1-k or )k = 0. Therefore, if p-q is coupled with more than one elernent in the network, ipq can be written as

(16) P g p-( Y0gpo =p//frwith 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

as

a.

Combining (13), (15), and (16) Yqa = Yaq =

-gp-qy p-TPp-

(17)

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 not coupled with any of the existing elements in the network, and JI& is the extra current flowing into bus /3 due to the coupling effect of p-q with the existing elements. By definition

coupled with other lines. However, for a line coupled with other Y= a Io = (19) elements, a separate algorithm has to be developed similar to the one developed for modifications in Zbu8 [9]-[11]. This is Y#1a is the transfer admittance between buses ,B and a before the addition of p-q, since the addition of uncoupled p-q does not still under investigation. It may be recalled that the entries of Ybus have to be modified change the current into bus /. The current I& can be calculated by injecting a voltage into in accordance with the following expression when the qth bus has to be eliminated: Yafi' = Yap - YaqYqp/Yqq. The second bus q from the reference, and short circuiting all the other buses term on the right-hand side corresponds to the current into bus to the reference [principle of superposition] such that the a, due to the current I; and when I,? is made zero by eliminating current'q = Yqa.


817


From the performance equation (5)

(20) I'B= Y3aTVa + Y13qVg (21) Iq = ?qaV7a + Y1qV5. 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 yqq

Vpq

-

[8] [9]

(24)

II. ADDITION OF LINK P-q Referring to Fig. 3, L is a fictitious node created by a voltage source between L and q such that eL =

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. Cambridge, Mass.: Harvard University Press, 1950. [3] N. Sato, "Digital calculation of network inverse and mesh transformation matrices," AIEE Trans. (Power Apparatus and Systems), vol. 79, pp. 719-726, October 1960. [4] G. Kron, "Improved procedure for interconnecting piece-wise solutions," J. Franklin Inst., vol. 262, pp. 385-392, November 1956. [5] H. H. Happ, "Orthogonal networks," IEEE Trans. Power Apparatus and Systems, vol. PAS-85, pp. 281-294, March 1966. [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,"

(25)

VpL*

Then the performance equation can be written as 1 n L

I,YU iALl[=n[-].

[10] [11]

[12]

[13]

AIEE Trans. (Power Apparatus and Systems), vol. 74, pp. 1285-1297, December 1955. H. W. Hale and J. B. Ward, "Digital computation of driving point and transfer impedances," AIEE Trans. (Power Apparatus and Systems), vol. 76, pp. 476-481, August 1957. A. H. El-Abiad, "Algorithm for direct computation and modification of solution matrices of networks including mutual impedance," presented at the 1st PICA Conference, Phoenix, Ariz., April 24-26, 1963. ,"Digital computer analysis of large linear systems," presented at the 1st Allerton Conference on Circuits and Systems Theory, Urbana, Ill., November 1963. G. W. Stagg and A. H. El-Abiad, Computer Methods in Power System Analysis. New York: McGraw-Hill, 1968. J. C. Siegel and G. W. Bills, "Nodal representation of large complex-element networks including mutual reactances," AIEE Trans. (Power Apparatus and Systems), vol. 77, pp. 1226-1229, 1958 (February 1959 sec.). A. H. El-Abiad, "Digital calculation of line-to-ground short circuits by matrix method," AIEE Trans. (Power Apparatus and Systems), vol. 79, pp. 323-332, June 1960.

(26)

YaL V13 (6 IV a YLL eLJ L LL 1QL EL Since p-L can be treated as a branch, the equations derived for a branch p-q in Section I of the Appendix, holding good as far as YaL and YLa (a = 1,2,* ,n) are concerned, YLL has to be evaluated as follows: When eL is equal to 1 pu and V13 ( 1=1, 2,*,n) is equal to zero

Ybus'

1413

YLL

=

iqL.

(27)

Since L is not short circuited with the reference and p is short circuited with the reference iqL = iLp From (27) and (28)

=

Yp-q, p_eL

YLL

=

=

YP-q, p-2q

(29)

Yp-q, p-q

YOa1 (a,3 = 1,2, ,n) is computed the same as in the of a branch p-q, but q should be replaced by L. YVa1

=

YVa1

+

(28)

YaLYL3

case

(30)

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

Ya V1 = Ia, a,

= 1,2,

n.

(31)

Ybus' Vbus = Ibus because there are actually n buses. Therefore Ybus' is obtained by ignoring the row and column corresponding to the fictitious node L.

Discussion

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

ments.

It is stated in the Abstract that the method described eliminates the need for the formation of the incidence matrices and does not require singular or nonsingular transformations. If, as in (1), we form Ybus using Ybus = At[y] A, the Ybus terms can be calculated (term by term) without any need for either the formation of A or A' matrices, or any multiplications whatsoever. For example, Yij is calculated by merely adding and subtracting the terms from the primitive admittance matrix y as indicated by the branch numbers connected to the nodes i and j. Logic and algebraic addition are used rather than multiplication or division. The short-circuit program (with mutuals) developed in the Cleveland Electric Illuminating Company contains these simplifications and is in use. A similar 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.


818

IEEE TRANSACTIONS ON POWER APPARATUS AND

M. Ramamoorty (Indian Institute of Technology, Kanpur, India): The author to be congratulated for his timely paper on Ybus matrix construction. The use of Zbu, and Yb,9 methods for load flow calculations has been widely discussed in the literature. It has been found that Zbu9 methods have better convergent properties as compared to the Yb,, methods. The reasons for this have not been discussed yet. One possible reason appears to be that a Ybu, matrix has a maximum number of zero off-diagonal elements. This property was used by Tinney and others to reduce the storage requirements by optimal ordering. However, the same property results in a low convergence rate for load flow problems using Ybus. This is because the changes made at a particular bus are not effective at other buses. The Zbu,, matrix has very few zero elements and so while making any change at a particular bus during any stage, the effect of all the other buses is considered, thus resulting in faster convergence. The difficulty with Zbu, was that it required an inversion of Ybus. El-Abiad [9] gave a method for step-by-step construction of Zbus without going through Ybus, Since the primitive network is described in terms of self- and mutual branch impedances, Zbu, can be easily constructed. Now a similar approach has been suggested by the author for Ybus construction. This method suffers from the drawback that the Ybranch matrix has to be calculated at each stage, involving repeated inversions of the Zbranch matrix. In the conventional Ybw, constructions, the primitive Ybranch matrix is obtained by inverting the Zbranch matrix. The post- and premultiplication of this matrix with singular or nonsingular matrix [11] gives rise to the Ybus matrix. In the discusser's opinion any extension or omission of lines from the existing network can be easily carried by the Zbus method. The discusser would like to know the advantages of this method.

admittance matrix [y]. The inversion considered here involves only small matrices, but nevertheless, it is present. We have considered this matrix inversion problem and have found that the "product form of a matrix" [ 151 appears to be a very efficient way to compute the required [y] -matrix elements at each step. In the example presented, the author had to invert the Z matrix at steps 1, 2, and 4; step 3 does not require an inversion algorithm since branch 2-3 is not coupled with existing elements, and its effect on the y matrix can be readily computed. The point is that at step 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 be inverted. Had the example included a larger number of coupled branches, 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 from that of a matrix whose inverse in known. Applied to the example given in the paper, we find that prior to step 1, the Z matrix for the coupled branches can be considered as

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.

-1

0 O-

O

O

ZO= 0 1 0 for which the inverse is yo changes to

=

Z,

=

Io

1

(3 X 3) unit matrix. At step 1, Z

=

-0.5 0 0o 0. 25 1 0. 1 O X O0

To change yo to its new value yi, proceed as follows: let 6 = yo-new column

-0.5


C. H. Didriksen (Harza Engineering Company, Chicago, Ill.) and H. Pachon (UNIVAC, Chicago, Ill.): This paper is of theoretical interest, and the author should be commended for his success in adding one more path to the table of formation of network matrices (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 closure]. Ybu. is used in load flow studies, but since mutual impedances are not considered within this problem, its generation becomes a very straightforward matter. The topological classification of a line as a branch or as a link, neglecting mutuals, has no special meaning in the formation of the Ybu, matrix. Would the author comment on those electric network problems where the use of Ybu,,, as described in the paper, 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] shows the following facts. The addition of a branch mutually coupled with existing elements is much faster when forming the Zbu. matrix; this is so because when forming the Ybu. matrix, the addition of a branch requires modification of all the elements in the matrix (3), whereas in the Zbug matrix formation only one new column is computed. For example, if a branch is to be added at step n, then a total of n + (n - 1)2 multiplications or divisions is required for Ybus formation, whereas the Zbu. formation will only require n. The addition of a link mutually coupled with existing elements is slightly faster when forming Yb,,, For example, if a link is to be added at step n, then a total of n + n2 multiplications or divisions is required for Ybu, formation, whereas Zbus formation will require 2n + n2. The values of estimated operations stated before assume that since the matrices Ybus and Zbus are symmetric, only the upper or lower triangle is kept during calculation. Would the author com-

SYSTEMS, MAY/JUNE 1970

= 0.25]. Obtain

1/0.5-

7= -0.25/0.5

then YI =

-771 0 O-

X2 1 0 A s3 0 be f

2.0 0 0o O= -0.5 1 0

-

0

0 t 1a

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

Z2

0.25 0] 0. 25 0.4 0 , Y2 0.2 1_ O

[

-[0.5 =

=

2.9091 -1.8182 0] 1.8182 3.6364 0 L 0.3636 -0.7273 1_ -

and at the end of step 4

Z4

Z0.5 0.25 0 ] = 0.25 0.4 0.2 1 0.2 0.6O

y Y4

=

3.2 -2.4 -2 .4 4.8 _ 0.8 -1.6

0.8-1.6 . 2 .2_

We note that moving from yo to yi, to Y2, and finally to Y4 does not require a matrix inversion at each step, but only elementary transformations [16]; nonetheless, at the end of step 4, the inverse of the (3 X 3) impedance matrix corresponding to the three coupled branches is obtained. It has been determined [16] that to obtain the inverse of a matrix A, (n X n), in the manner sketched above, it requires (n3 X n2)/2 multiplications or divisions, whereas it takes n3 to compute A-1 by other known methods. Thus, for the example given in the paper it may have taken Z3 i3 -36 multiplications or divisions to compute all the required inverses. The use of the product form would take only 18. In general, for n coupled branches it takes n3=P = n2 (n + 1)2

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


819


will still provide the elements of the primitive admittance matrices required at each step. We are looking into the applications of the product form of a matrix to network matrix formations and would appreciate the author's comment on the particular application described in this discussion. REFERENCES [15] G. Hadley, Linear Programming. Reading, Mass.: AddisonWesley, 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, Madison, Wis.): The author has provided an algorithmic approach to the formation and modification of the bus admittance matrix. The ability to modify Ybug easily by an algorithm, especially for changes involving 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 existing elements and the added element." The work in the numerical example tends to support this definition. However, at other places p-a is said to contain "all the existing elements mutually coupled to 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 slightly different definitions, all of which are functional so far as the algorithm is concerned. The author might also note the column matrix up-a is a column 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 is simulated by the addition of a "new" line in parallel. The mutual impedances of the "new" line are of the same sign and magnitude as those of the line to be changed; the self-impedance of the "new" line is chosen so that the parallel combination of self-impedances is 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 with no mutual couplings included. This is accomplished by removing the coupled line 1-2(1) and adding in its place line 1-2(3) with selfimpedance equal to 0.4 and no mutual couplings. For the network with mutuals 1 2 3 0. 08 10.2 -5.4 1 (32) 8.8 -5.0 . Ybus = 2 - 5.4 0.0 -5.0 7.0_3 Step 1. Add line 1-2(- 1) with self-impedance = -0.4 and with mutual impedances of 0.2 and 0.25 with 1-2(2) and 0-1, respectively. 0-1

1-2(1) 1-2(2) 1'2(- 1)

0.250 z'] (33) 0.2 0 -0.4 j 1-2(2) 1-2(-1) 1-2(1) Yp-q, p-v = 1-2(-1)[1.2500 -1.1979 0.8333 -1.3021] F0.5 1-2(1) 0.25 1-2(2) 0 1-2(-1)LO.25 0-1

0-1

0 0.2 0.6 0.2

0.25 0.4 0.2

(34)

__

B::]

1 2.9167] 2 -1.667

LYLL_I

LL_-1. 3021j

-

(35)

from which 3 0 6.667 -5.0 . 2 -1.667 7.02 -5.0 3. 0 1

Yb,,'

=

2

1[ 3.667 -1.667


(36)

Step 2. Add line 1-2(3) whose self-impedance equals 0.4 and no couplings.

Ybus'

YLL [L 2.5] 2 1 3 1F 6.167 -4.167 9.167 -5.0 2 -4.167 -5.0 7.0 3L 0

=

(37)

.

(38)

This result may be checked by inspection of the network since there are no couplings involved. If a "special algorithm" is desired in which several changes may be made simultaneously, the following illustrates such a procedure for the same changes as made above. Define these matrices:

1-2(1) 1-2(2)

0-1

[Y]old

[ 3.2 -2.4

0-1

10-1 [Y]new =

1-2(1) 1-2(2)

=

0 --2.0 0 2.5 0 0 0 0 1.6667

[y] new

[Y] old 0-1 1-2(1) 1-2(2) 2.4 -0.8 ] 0-1 [-1.2 1.653 1-2(1) 2.4 -2.3 1.6 -0.5333 1-2(2) _-0.8

['Ay]

[Ay]

0.8'-

1-2(1) -2.4 4.8 - 1.6 2.21-2(2) L 0.8 -1.6 0-1 1-2(1) 1-2(2)

=

=

(39) (40)

-

(41)

A is the bus incidence matrix for partial network including only coupled lines 1 --1 1 A = 1-2(1) 1-2(2) L 1

0-1

AY= A[Ay]A =

2 O-

-21

1 1 [-4.0333

2

(42)

-1

2

1.23331

1.2333 0.36671

(43)

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

Yb,,'

=

2 1 1 6.167 -4.167 9.167 2 -4.167 0 3 -5.0

3

-5.0 7.0_

(44)

which agrees with (38). It should be noted that when a single-line p-q is added to a network, the bus admittance matrix describing the network is changed only in certain rows and columns. The rows -and columns that change are those corresponding to buses p, q, and any buses affected through mutual couplings with line p-q. Since these buses are known when gps, ,, is calculated, the algorithm could be made more efficient by allowing a,fl to range over only these buses and not a,# = 1,2, ..., n, as the author states. Ya,q and Yq,s are zero unless calculated to be

nonzero. As a further observation, once all lines incident to a given bus and all lines affecting this bus through mutual couplings have been processed by the algorithm, no further changes will arise in the corresponding 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 simultaneously, the successive recalculating of ygp-q, p-a would be avoided. REFERENCES

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

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


820

IEEE TRANSACTIONS ON POWER APPARATUS AND

Kasi Nagappan: The author appreciates the interest shown by the discussers and is grateful for the comments made by every one of them. The transformation method referred to by Mr. Ramarao will not yield readily for the system changes, whereas the method developed in the paper will accommodate the system changes with fewer computations. To answer the point raised by Mr. Ramamoorthy and Mr. Didriksen and Mr. Pachon with regard to inversion to obtain a primitive admittance matrix, the author wishes to state that although inversion is not completely eliminated, it is restricted to small-size matrices. For example, when there are 15 lines in the system, of which three are coupled, and a line coupled with the two existing

Manuscript received August 15, 1969.

Simulation of

5/6, MAY/JUNE 1970

lines, the size of the matrix to be inverted will not exceed six. Since the number of coupled lines in a power system is small, the maximum size 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 hoped that accounting for mutual couplings will result in a well-conditioned bus admittance matrix to facilitate faster convergence of iterative technique. The author is developing programs in order to bring about comparison of different techniques to provide more information regarding computation 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, the author wishes to state that the procedure outlined is itself an algorithm to modify the matrix due to addition or removal of a coupled line and partial changes in a coupled line.

AC

HVDC NARAIN G. HINGORANI,

SYSTEMS, VOL. PAS-89, NO.

System Impedance in System Studies

SENIOR MEMBER, IEEE, AND

MICHAEL F. BURBERY

Abstract-In some studies of HVDC system design, ac system impedance is represented by its inductance at power frequency. However, for several studies better simulation of the impedancefrequency characteristic (from power frequency to a few kilohertz) of the ac system is important. This paper presents a simple approach to calculating an approximate equivalent network consisting of parallel LCR branches and having an impedance-frequency characteristic similar to that given for the ac system. For given typical ac system impedance diagrams (usually obtained from ac system models and simulators), simulation of an ac system by corresponding equivalent networks during studies of certain HVDC problems would provide a more accurate means of designing HVDC systems.

(a) L1Lp

INTRODUCTION

R1

WrITH THE increasing number and size of high-voltage

dc power transmission schemes being commissioned, there has been considerable investigation into better simulation (digital, analog, or model) of HVDC systems. For the design of the HVDC system itself, it is important that the impedance of the ac system [Fig. 1(a) ] at various frequencies be simulated correctly. 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,

02

L3

S.L

Ip

RI

F

R3 % R 03 :T:Cnl

AC SYSTEM

(i)

Paper 69 TP 632-PWR, recommended and approved by the Power System Engineering Committee of the IEEE Power Group for 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.

R2

1L3

(b) Z Ln F

% Rn tR2 R3 C2 C3 :T:Cn ACT SYST

AC SYSTEM

I

(c)

Fig. 1. Simulation of ac system impedance. (a) System for simulation. (b) Simulation with fundamental-frequency source in series with whole ac system impedance. (c) Simulation with fundamentalfrequency source in first L-R branch.


Step by Step Formation of Ybus and Zbus

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