812IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, VOL. PAS-89, NO. 5/6, MAY/JUNE 1970 Step-by-Step Formation of Bus Admittance Matrix KASI NAGAPPAN 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. 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 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 (2) Authorized licensed use limited to: King Fahd University of Petroleum and Minerals. Downloaded on July 26, 2009 at 14:37 from IEEE Xplore. Restrictions apply. -" = Ofor panda Next the entries Ya6t of Ybus' are computed as follows: Y for . . consisting of rows and columns corresponding to the basic or independent loops [5]. Proofs for (6) and (7) are given in the Appendix. yqq a .t is the value from the bus admittance matrix before the addition of p-q. requiring more storage space and time in the digital computer. This adds a new bus to the network.(5 _Vq_ Since the networks of the power system are bilateral. is the voltage across the element p-a and vp_c =1 forp = a Vf.. to which a line p-q is added. [13] to form Zbus by adding one line or element at a time and computing the resultant matrix. Downloaded on July 26. Authorized licensed use limited to: King Fahd University of Petroleum and Minerals.q) is computed as follows: Yqa where Ya.VP-L. First Yaq (a = 1. Addition of branch p-q to network. Ya# = Y#ffa for all a and (. The matrix A has extra rows and columns corresponding to the fictitious nodes for the links. mutually coupled to the existing elements in the network. ALGORITHM TO COMPUTE Ybus The matrix Ybus is formed by step-by-step addition of a line or passive element. The matrix Ybus is a submatrix of Yaux. The size of the matrix is increased by one when a tree branch is added. Besides the method of singular transformations. [11]. and the added element v-. In order to avoid the major inversion. [6]. = Yaq = Yp-q.6 = ya. n q n (7) q[ Ibu YbusV' Vqa |Ya Yqq_]V. This is a major inversion. whereas all the entries of the matrix are modified when a link is added. Addition of Branch p-q Consider a network with n buses. (4) Fig. The performance equation of the network with the new bus q is 1 1 q 9p-q. is added.. Such nonsingular transformations of ly] and [z] result in the matrices Yaux and Zaux as follows: Yaux = At[y ]A (3) Zaux = Ct[Z]C. the primitive admittance matrix of all the existing elements. an algorithm has been developed by El-Abiad [9].n. 3.*. 2009 at 14:37 from IEEE Xplore. [11]. These matrices are also referred to as orthogonal network matrices [5]. 2. shown in Fig. ¢ is the row of [y]. there is a method of nonsingular transformation. Note that there is a dual relationship between Ybus and Z100p. This method involves inversion of small-size matrices whenever an element or line. Restrictions apply. whereas the matrix C has extra rows and columns corresponding to the open path or loops for the tree branches. as shown in Fig. The new bus admittance matrix is (n + 1) X (n + 1) given in (5). Addition of link p-q to network.NAGAPPAN: STEP-BY-STEP FORMATION OF BUS ADMITTANCE MATRIX 813 where C is the basic loop incidence matrix of size e X L and L is the number of independent or basic loops. ALGORITHM TO COMPUTE ZbuS Having formed Ybus and Zloop0 Zbus and Yloop can be obtained by inversion of the corresponding matrices. which employs augmented incidence matrices A and C [11]. 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_ . modifying the existing entries of Ybus. [6]. 0= 1. . whereas the size remains the same for the addition of a link but the entries of the existing matrix are modified. Fig.d + YaqY2d. The size of the matrix is increased by one for the addition of a tree branch that adds a new bus.= -1 for a = a a.pTVp-a (6) where p-a is the list containing all the existing elements mutually coupled to p-q and the element p-q. a v.2.2.B . 2. The line p-q has mutual coupling with some or all of the existing lines or elements in the network.. consisting of rows and columns corresponding to the buses or independent nodes. whereas Zl00p is a submatrix of Zaux. 3. P)) n..l n . 7] (1) = 1.E.iITTATTnA PnlkITTq V. (ii) + oc8 Yq rOp = Yccp Yocq Y. . n) ov) =_ ta ECG RPERENC p-Ir .... 3. CASE Or NK z cO H4 | oFOR * t ofo't| (V) yop NOT Y= 1 Y.°-pJ.0 CO1 0-4 X. =I f Vfa FOR =0 FOR f&cOr'#C ) .f6..L . Yoc Ycp CC lb C)Yco1. n ( = = 2.. (ii)ct.EReNCE r" | /A R PBRENCE RtlERENC E NOT TH!E| RFFZRZNCf 6 "STRE 8AN) H fh FyO j (I) YPc =Yc =l ii)YSc0Yp U = S* 4 X=q -P |(ii) Y BRANCH/| f= oc vp ..TABLE I 00 Pn STTMMA-Rv OR F. P =0 FOR fPd& + a |(V) Yjp = Ypp+ VP-Jc.x3...ip (W) (ii) Yocp Yxp y + Y. = z == Y t4 0 tfa fl FOR f SAME AS p | IN NO r --1 FOR Ot:O TH (J) . 2.p...2..5 n. -5 S NOT coupLeD /s A ' /5 EL COUPZLD WITH 4X/STINGeTl |Ar- ADDZD IS NOT rHE R. N/N R!PtREA1Ct = ( O = i 2.e? I yI H 0 o'ii') (1 L |v)Y = ii) YLP * Y&. -I FOR czC. 4..n) I zi (J H- (i) 'YZL =YLU= _____ LetiI YLa=Y o. 2..84vs~L. O7'I..&f (ca.2.FOR TsO FOR =o Yiii) =-. * SAMR A5 IA THE CASE OP ExceP: CO (ii) Y. 2-.p=Y pf (i')YSsy. + =Y y ocSp =235 |.3.. 3..+I Yop=yL=oYt 8VgyYp= Yi yq P xcEPr Llb YpzO YL ..o. P-5 (ii) Y(P xYp (oc= 1. 6364 0. YLL = = YP-q.1816 -5.4 0 0 2-3 Lo 0.6 I Authorized licensed use limited to: King Fahd University of Petroleum and Minerals. YaL (a = 1. 4. NUMERICAL EXAMPLES To illustrate the algorithm. 2009 at 14:37 from IEEE Xplore.25 0.4546 8.4546 2 . TABLE II Ybus (new) = Ybus'. p-a:0-1 and 1-2(1). of Ybus' in (8) are computed as follows: 0D . q = 2.25 0.4 0-1 1-2(1) Ybus = 01 1[ 10. .6364] 2 Step 4 Add the link 1-2(2) between 1 and 2..5 0. q = 3.01 [y] = 1-2(1) -1.3 Xp-q. 0-1 1-2(1) 2-3 0-1 0.NAGAPPAN: STEP-BY-STEP FORMATION OF BUS ADMITTANCE MATRIX 815 Then the performance equation of the network is 1 'IbusL Liq-LI n First. a simple network.9091 -1.L][V L YLa YLL. (10) = Then.01 1 2 0-1[21 1 2 1[ ]. the entries Ya.8182 1 3.2 0.n) is computed as follows: YLa = (8) 8 YaL where -Yp-q.1816 2_-5.25 0.p-q (pu) 0.-o (pu) 0. Downloaded on July 26.5.25 0. P-q* for a. For simplicity. i. shown in Fig.8182 0.* * .2 ya Ya.9091 -1. [z] = 0-1 [0.6364 -5 .2 0 0 0.0 0. q = 1.45463. p = 1. A summary of the equations appears in Table I.5] [y] = 0-1 Step 3 Add the branch between 2 and 3. p = 2. and a bus admittance matrix is obtained.25 0 [z] = 1-2(1) 0.5 0. ignoring the Lth row and column.eL ] . p-qVp- (9) vp_ = 1 for p = a rra = -1 for o.8182 3. 0-1 1-2(1) 2-3 1-2(2) Ybus 1 10.e.20 * Reference point is zero.YL YLL 1. Data for the problem are given in Table II.2 0.2 0 0.25 0. is considered.= VP-" = 0 for p and uNext. This is not coupled with 0-1 or 1-2(1). DATA FOR NUMERICAL EXAMPLE* The new bus admittance matrix after the addition of the link p-q is Ybus' in (8). YLL is computed as : a a. 0-1 Ybus = Step 2 Add the branch 1-2(1) between 1 and 2. p-o-: 0-1 and 1-2(1). q = 2.2. * 1 L n [Ybus' Ya. Step 1 Add the branch between 0 and 1. p = 0.0 2-3 L 0.25 0. Restrictions apply. Network for example.0 5.5 0. This has coupling with the branch 0-1.n.2. S Number 1 2 3 4 5 Line p-q 0-1 1-2(1) 2-3 1-2(2) 0.4 0 0 0 0 0 0. (12) Proofs for [9]-[12] are given in the Appendix. real numbers are assumed for the line constants and the line charging is neglected.2j 0-1 1-2(1) 2-3 0-1 F 2. -5 5_ 3L 0.25 1-2(1)_0.8182 1-2(1) -1.4 0.6364] [z] = 1-2(1) 1-2(3 1-2(2)[ 0-1 0.4546 -5. 3 [z] Y 0-l O. L = 2(2).5 with Line Coupled 1-2(1) 0-1 1-2(1) Xp-pq. This has coupling with the existing branches p = 1. (11) '6 1-2(2) Fig.5 0.6 0. 4.0 0-1 1-2(1) 0-1 [2. for this network is computed by the algorithm [9].8000 -5. 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 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.6 2.0000° VP-a = -1 for af = a The bus impedance matrix Zbus.0 2 -5.816 IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS. = -Yp q. From Fig. The second bus q from the reference.2. for a line coupled with other Y= a Io = (19) elements.0 in (14) can be written as I = It can be checked that the product of Ybus and Zbus is an identity matrix. Ybus 2 -5 . By dual relations between Ybus and Zl.l = Step 5 Add the link between 0 and 3. . If p-q is coupled with i-k or k-i only. since the addition of uncoupled p-q does not still under investigation. a separate algorithm has to be developed similar to the one developed for modifications in Zbu8 [9]-[11]. the resultant bus admittance matrix is 1 2 3 10. APPENDIX I.n.2 j 0.6 0 1-2(2) 11.4001 10.4 4.B and a before the addition of p-q. and when I. (15).4001 8. 1998 8.k-a.8000 -5. and (16) Yqa = Yaq = as a. This is Y#1a is the transfer admittance between buses .0 -5. since V1-k or )k = 0. The bus admittance matrix including the fictitious node L is Suppose p-q is coupled with a-k only. 2 Ie .0000. By definition coupled with other lines. then ipq Therefore = Yp-q.8 1 .2.? is made zero by eliminating current'q = Yqa.0 0 5. by the same reasoning ip.6 L 0 . if p-q is coupled with more than one elernent in the network.4001 5.4001 If p-q is coupled with k-a only. This has no coupling p = 0 q = 3.1998 -5. Zbus 2 3 1 1 0.6364 -5 0. Restrictions apply. However. but Va-k ipq = pu y 1 _2 bs-3 -5.0 0 2 2-- After eliminating the Lth node. 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 . *.. the same amount of current has to be subtracted. Ybus = 2 -5.2121 0. = = 1.[11] and by vp-or = Ofor pand Combining (13). 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.3010 0. ip-q = 0. while all the buses except a are short circuited with the reference node. since Vk-a = -1. The bus admittance matrix including the fictitious node L is 1 2 I ybu = bs-3 the bus q.3010 -gp-qy p-TPp- (17) 3LO 1503 0. and JI& is the extra current flowing into bus /3 due to the coupling effect of p-q with the existing elements. it is seen from (5) that Yqa = L 2 3 0 0.1503 = 2j 0. ADDITION OF BRANCH p-q By injecting 1 pu voltage from the reference node to bus a (a = 1. MAY/JUNE 1970 -y = 0-1 1-2(1) 2-3 3.2 10=. Downloaded on July 26. L = 3.q).4546 8. _k. * .' = I.2 -5. ipq can be written as (16) P g p-( Y =p//fr0gpo with p-o spanning all the coupled elements including p-q and VP-or= 1 for p = a 3L ° -5.pa. - (13) (14) (15) 1 1b = 0 -ip.YaqYqp/Yqq.6j . a-k* L 0 o -5. due to the current I.0 O 3L Iv Y. Therefore.4001 0 8. °°°° 7 .8000 -52.2j 0. Further investigation is in progress in this direction. a-kV.5. 2009 at 14:37 from IEEE Xplore.1816 After eliminating the Lth node. The currenit I. 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.4213 0.2 -2.8 0 0 5 2-32) 0 -1.4001 0 10401998 0 1 2 3 L Yp-q. n.3578j with ip-r having the coniditiorLs as above.2211 0.8 1-2(2) L0. the resultant bus admittance matrix is 2 1 3 1 0 -5.4 0 0-1 1-2(1) -2.2121 0. It may be recalled that the entries of Ybus have to be modified change the current into bus /. Authorized licensed use limited to: King Fahd University of Petroleum and Minerals.4546 5 0 -5 0 0 2.op it seems logical to visualize the possibility of computing Zloop by an algorithm. If. H.. For example. (Power Apparatus and Systems). . I: six basic reference frames. 1969.* . Mass. in the development of the short-circuit program (with mutuals)." IEEE Trans. J. pp. 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. El-Abiad.YU iALl[=n[-]. April 24-26. pp.. or any multiplications whatsoever. p-q (29) case YOa1 (a. (Power Apparatus and Systems). Therefore Ybus' is obtained by ignoring the row and column corresponding to the fictitious node L. Cleveland. Substituting (19) and (23) into (18) and combining with (14) Y = Yea + Y1*qyqa yqq VpL* (24) II. "Orthogonal networks. Hale and J. vol. 77. "Digital calculation of line-to-ground short circuits by matrix method. 1950. "Nodal representation of large complex-element networks including mutual reactances. W. Siegel and G. Ohio): Based on our experience in the Cleveland Electric Illuminating Company." AIEE Trans. A. November 1956. (31) Ybus' Vbus = Ibus because there are actually n buses. the Ybus terms can be calculated (term by term) without any need for either the formation of A or A' matrices. Ybus' 1413 (26) Discussion YLL = iqL. n. ADDITION OF LINK P-q Referring to Fig. Happ. 385-392. W.3 = 1." IEEE Trans."Special cases of orthogonal networks-tree and link. vol. 1951. A similar approach is described in [14]. pp. New York: McGraw-Hill.. Kirchmayer. p-2q (28) YLL Yp-q.. "Tensorial analysis of integrated transmission systems. 262. 74. Bills. pp. 323-332.n) is equal to zero I. for a particular value of (3 I13. YVa1 = YLL Now the fictitious node L has to be eliminated by making eL equal to 0.*. Phoenix. pp.n) is computed the same as in the of a branch p-q. i." presented at the 1st Allerton Conference on Circuits and Systems Theory. Power Apparatus and Systems. (Power Apparatus and Systems). Sato. PSCC. the equations derived for a branch p-q in Section I of the Appendix. we would like to offer the following com- Ya V1 = Ia. Computer Methods in Power System Analysis. 476-481. F. C. 1966. Ramarao (Cleveland Electric Illuminating Company. October 1960. Then Vpq becomes equal to VPL. = 1. Ward. Restrictions apply. Matrix Analysis of Electric Networks. Vq = yqa yqq (22) Substituting the value of Vq given by (22) and Va = 0 (a = 1. Kavuru A. J. H. . vol. a. 880-891. PAS-85.2. vol. Kron. 4-8. [5] H. Franklin Inst. Siemaszko. Substituting eL equal to zero in (26) YVa1 + YaLYL3 (30) 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. [6] . we form Ybus using Ybus = At[y] A. (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. Manuscript received July 7. 1285-1297."Digital computer analysis of large linear systems. [3] N. A. 79. vol. 79." AIEE Trans. 2 (Network Analysis).: Harvard University Press. M. vol. Logic and algebraic addition are used rather than multiplication or division. 1226-1229." J. . 1239-1248. [7] A. 1968. vol. 3. (Power Apparatus and Systems). ments." Proc." AIEE Trans. vol. Substitution of Iq = Yqa and Va = 0 (a = 1." AIEE Trans. August 1966. Kron. pp.NAGAPPAN: STEP-BY-STEP FORMATION OF BUS ADMITTANCE MATRIX 817 From the performance equation (5) yields (20) I'B= Y3aTVa + Y13qVg (21) Iq = ?qaV7a + Y1qV5. and L. El-Abiad. 76.. November 1963. holding good as far as YaL and YLa (a = 1. June 1960. Urbana. B. = Y# qay. Downloaded on July 26. Habermann. 719-726. El-Abiad. Glimn. Ariz. K. H. REFERENCES [14] H." [8] [9] [10] [11] AIEE Trans. 1963. "Digital computation of driving point and transfer impedances. August 1957.2.2. YLL has to be evaluated as follows: When eL is equal to 1 pu and V13 ( 1= 1. pt. The short-circuit program (with mutuals) developed in the Cleveland Electric Illuminating Company contains these simplifications and is in use. "Algorithm for direct computation and modification of solution matrices of networks including mutual impedance. G. "Digital calculation of network inverse and mesh transformation matrices. "Improved procedure for interconnecting piece-wise solutions. 70.n) in (21) Yqa = YqqV. pp. vol.n) in (20).* . L is a fictitious node created by a voltage source between L and q such that eL = Vpq - (25) [12] Then the performance equation can be written as 1 n L [13] YaL V13 (6 IV a YLL eLJ L LL 1QL EL Since p-L can be treated as a branch." presented at the 1st PICA Conference. H. Cambridge. Rept. *A. December 1955. pt. pp. "Digital calculation of network impedances. Henderson. pp. but q should be replaced by L. H. Jr. Stagg and A. 2. Ill. Le Corbeiller.n) are concerned. March 1966. 281-294. "Reduced matrix calculus application to the network computation.2. [4] G. 1958 (February 1959 sec. Power Apparatus and Systems.2. as in (1). p_eL = = YP-q. B. [2] P.e. (Power Apparatus and Systems).).. (23) REFERENCES [1] G. Authorized licensed use limited to: King Fahd University of Petroleum and Minerals. W. PAS-85.. 2009 at 14:37 from IEEE Xplore." AIEE Trans. can be easily constructed. The inversion considered here involves only small matrices.25/0. to Y2. only the upper or lower triangle is kept during calculation.8182 3.. matrix. 25 1 0. This method suffers from the drawback that the Ybranch matrix has to be calculated at each stage.2 0. Ramamoorty (Indian Institute of Technology. 2 . whereas the Zbu. then a total of n + (n . Restrictions apply. El-Abiad [9] gave a method for step-by-step construction of Zbus without going through Ybus. whereas in the Zbug matrix formation only one new column is computed. 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. For example. 1 O X O0 To change yo to its new value yi. matrix.2 -2. thus resulting in faster convergence. Z Z. The addition of a link mutually coupled with existing elements is slightly faster when forming Yb. has no special meaning in the formation of the Ybu.2 1_ O [ = 2. Ill. was that it required an inversion of Ybus.8 _ 0. 1969. Now a similar approach has been suggested by the author for Ybus construction. The use of Zbu. it requires (n3 X n2)/2 multiplications or divisions. (n X n). For example. and finally to Y4 does not require a matrix inversion at each step.5 = 0. Would the author com- 1/0.5 C.and mutual branch impedances.. multiplications or divisions to compute their primitive admittance matrices.5 0 0o 0. the addition of a branch requires modification of all the elements in the matrix (3). whereas it takes n3 to compute A-1 by other known methods.5 -0. a (3 X 3) matrix must be inverted. The product form of the matrix will take only (ni3 + n2)/2. matrix.6O y Y4 = 3. it is present.4 0 .7273 1_ - and at the end of step 4 Z4 Z0. Obtain = 0. Ill. whereas Zbus formation will require 2n + n2. Applied to the example given in the paper. 2.4 -2 . then a total of n + n2 multiplications or divisions is required for Ybu. Zbu.6 .) and H.. India): The author to be congratulated for his timely paper on Ybus matrix construction. Even though the paper claims that matrix inversion is not used. Chicago. It has been determined [16] that to obtain the inverse of a matrix A. or an iterative procedure to arrive at the fault values [13. the effect of all the other buses is considered. but only elementary transformations [16]. In the conventional Ybw. The values of estimated operations stated before assume that since the matrices Ybus and Zbus are symmetric. Usability: The Ybus matrix is not usually considered for the calculation of short circuits since its use requires an inversion routine. and Yb.1)2 multiplications or divisions is required for Ybus formation. At step 1. as described in the paper. H. 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. The Zbu. The discusser would like to know the advantages of this method. The post. is more advantageous than other methods. neglecting mutuals. In general. The reasons for this have not been discussed yet. at step 2. Two factors worthy of comment. the inverse of the (3 X 3) impedance matrix corresponding to the three coupled branches is obtained. Didriksen (Harza Engineering Company. at the end of step 4. larger matrix inversions would have been required.2 1 0. Since the primitive network is described in terms of self. for the example given in the paper it may have taken Z3 i3 -36 multiplications or divisions to compute all the required inverses. if a branch is to be added at step n.818 IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS. and Authorized licensed use limited to: King Fahd University of Petroleum and Minerals. matrix has very few zero elements and so while making any change at a particular bus during any stage. in the manner sketched above..0 0 0o O= -0. we find that prior to step 1.8182 0] 1.4 0. formation will only require n. the same property results in a low convergence rate for load flow problems using Ybus.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. One possible reason appears to be that a Ybu. Chicago. This property was used by Tinney and others to reduce the storage requirements by optimal ordering.25 0 ] = 0. 2009 at 14:37 from IEEE Xplore. Ybu. Kanpur. this is so because when forming the Ybu.25].25 0] 0. a (1 X 1) matrix must be inverted.3636 -0.. The point is that at step 1.25 0. nonetheless. = -0.. Would the author comment on those electric network problems where the use of Ybu. Had the example included a larger number of coupled branches. The difficulty with Zbu. its generation becomes a very straightforward matter. if their primitive impedance matrices are inverted at each step. 7= -0.5- then YI = X2 1 0 A s3 0 be f -771 0 O- - 2. Downloaded on July 26. In the example presented. The addition of a branch mutually coupled with existing elements is much faster when forming the Zbu.8 -1. methods. the described method requires this technique to obtain the primitive Manuscript received Julv 11. MAY/JUNE 1970 M. proceed as follows: let 6 = yo-new column Manuscript received July 1. matrix has a maximum number of zero off-diagonal elements. This is because the changes made at a particular bus are not effective at other buses. . However. Thus. involving repeated inversions of the Zbranch matrix. the author had to invert the Z matrix at steps 1.6 0. and its effect on the y matrix can be readily computed.): This paper is of theoretical interest. table 4-1] shows the following facts. and the author should be commended for his success in adding one more path to the table of formation of network matrices (Fig. for n coupled branches it takes P = n2 (n + 1)2 n3= ment on the comparison stated. Computation Efficiency: A comparison of the formulas in Table I against the Zbug formation formulas given in [11. In the discusser's opinion any extension or omission of lines from the existing network can be easily carried by the Zbus method. Y2 0.4 4. 1969.8-1. but not considered within the paper are usability and computation efficiency. is used in load flow studies. but nevertheless. constructions.and premultiplication of this matrix with singular or nonsingular matrix [11] gives rise to the Ybus matrix. author's closure]. admittance matrix [y]. a (2 X 2) matrix must be inverted. the primitive Ybranch matrix is obtained by inverting the Zbranch matrix. and 4.5 0. Pachon (UNIVAC. formation. if a link is to be added at step n. The topological classification of a line as a branch or as a link. 25 0.6364 0 L 0.9091 -1. 1).5 1 0 0 0 t 1a A similar procedure can be followed to obtain at the end of step 2 Z2 -[0. step 3 does not require an inversion algorithm since branch 2-3 is not coupled with existing elements.2_ We note that moving from yo to yi. The use of the product form would take only 18. and at step 4. the Z matrix for the coupled branches can be considered as ZO= 0 1 0 O O -1 0 O1 for which the inverse is yo changes to = Io = (3 X 3) unit matrix. but since mutual impedances are not considered within this problem. fl to range over only these buses and not a. 1969.2 -5. Since these buses are known when gps.0 .6 0.4 0. Has the author attempted multiple-line additions? It would appear that if all lines in a mutually coupled group were added simultaneously. However.5 0 0 0 0 1.4 0-1 1-2(1) 1-2(2) 0. 1963.167 0 3 -5.667 -1. Ya.23331 2 1..667 nonzero. A change in the self-impedance or the removal of a coupled line is simulated by the addition of a "new" line in parallel.NAGAPPAN: STEP-BY-STEP FORMATION OF BUS ADMITTANCE MATRIX 819 will still provide the elements of the primitive admittance matrices required at each step.4 and no couplings. especially for changes involving coupled lines. 57. It should be noted that when a single-line p-q is added to a network.0 7.167 -4. The rows -and columns that change are those corresponding to buses p.0 7.21-2(2) L 0.s are zero unless calculated to be (35) LYLL_I from which 1 LL_-1. Downloaded on July 26. Reitan (University of Wisconsin. Reitan and K. it is often difficult to choose.4 1 (32) 8. impedance matrix for svstem changes involving mutual couplings. but with no mutual couplings included.4 -2. Wilf. New York: Wiley.653 1-2(1) 2. Mathematical Methods for Digital Computers. August 1969. 3021j 2 Yb. The author might also note the column matrix up-a is a column of the bus incidence matrix for the partial network. This is not true. suppose it is desired to obtain Ybus for the same network. Hadley. Step 2.1) 0. . Mass. and use precisely a notation.8 -5.250 z'] (33) 0.0 3L 0 (37) .: AddisonWesley. would seem to be of great interest.2. Linear Programming.5 1-2(1) 0. and any buses affected through mutual couplings with line p-q. K. As an example of the problem: at one point the author states that p-a includes "all the existing elements and the added element.6 -0..9167] 2 -1. 0 1[ 3.2 0.167 9. 0-1 This result may be checked by inspection of the network since there are no couplings involved. the following illustrates such a procedure for the same changes as made above.. Authorized licensed use limited to: King Fahd University of Petroleum and Minerals.0 7. C.8333 -1.8 ] 0-1 [-1.25 0. 48-50.167 -5. 08 10.3021] 0-1 F0." These are three slightly different definitions.8'- (39) (40) [Y]new = 1-2(1) 1-2(2) = 0 --2.4 and with mutual impedances of 0. .4 -0.0 -5. K. vol. Reading. IEEE (Letters).1. q. 1432-1433. Ralston and H.8 ['Ay] [y] new (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 2 0-1 -21 2 -1 O- (42) AY= A[Ay]A = 1 1 [-4. . pp.. 1960-1967." Proc. no further changes will arise in the corresponding row and column of Ybus. Wis. 44-45. is calculated.8 -1. "Modifications of the bus Manuscript received July 10. the successive recalculating of ygp-q. Ybus = 2 .167 2 -4. [161 A. The mutual impedances of the "new" line are of the same sign and magnitude as those of the line to be changed.2 and 0.1) with self-impedance = -0. Eds. pp.02 -5. the algorithm could be made more efficient by allowing a. In writing a paper of this type.0 0 2. 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.0 3. The ability to modify Ybug easily by an algorithm.5] 2 1 3 1F 6.# = 1. Restrictions apply. If a "special algorithm" is desired in which several changes may be made simultaneously. Add line 1-2(3) whose self-impedance equals 0.0_ (44) (34) B::] __ 1 2.5333 1-2(2) _-0. As a further observation. To illustrate with the author's algorithm and numerical example.667 -5.2 0 -0. 2 -1.2 -2.5.167 9.6 0-1 1-2(1) 1-2(2) 10-1 [ 3. In the conclusion." The work in the numerical example tends to support this definition.36671 3 (43) Adding the terms in (43) to the appropriate terms in (32) gives Yb.2 2 1 1 6.1979 0. p-a would be avoided. respectively.): The author has provided an algorithmic approach to the formation and modification of the bus admittance matrix. n..0 2 -4. Ybus' = YLL [L 2.. 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 4. once all lines incident to a given bus and all lines affecting this bus through mutual couplings have been processed by the algorithm. S. REFERENCES (36) [17] D. at other places p-a is said to contain "all the existing elements mutually coupled to p-q and the element p-q. define.0 . all of which are functional so far as the algorithm is concerned. Add line 1-2(.0333 1. as the author states.25 with 1-2(2) and 0-1.667 - which agrees with (38).0 -5. the bus admittance matrix describing the network is changed only in certain rows and columns.4 0.0_3 Step 1. nor is it true in the algorithm for the bus impedance matrix [17].2 1. C.q and Yq. Kruempel.667 7.8 .3 1.2333 0.2 0 0.25 0-1 1-2(1) 1-2(2) 1'2(.167 -5. p-v = 1-2(-1)[1.25 1-2(2) 0 1-2(-1)LO. the author states that a separate algorithm is necessary in order to modify coupled lines. (38) K." In the Appendix p-a is defined to span "all the coupled elements including p-q.2500 -1. Define these matrices: 0-1 [Y]old = 1-2(1) -2.167 -4.' = 0.6 2.6667 - [Ay] = [Y] old 0-1 1-2(1) 1-2(2) 2. For the network with mutuals 1 2 3 0. REFERENCES [15] G. 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.. Madison. 2009 at 14:37 from IEEE Xplore.' = 3 0 6. pp.4 j 1-2(2) 1-2(-1) 1-2(1) Yp-q.4 and no mutual couplings. Kruempel and D. Manuscript submitted February 17. G. NO. it is restricted to small-size matrices. (c) Simulation with fundamentalfrequency source in first L-R branch. Restrictions apply. N. Since the number of coupled lines in a power system is small. While appreciating the interest shown by Mr.L (a) INTRODUCTION R1 R2 02 dc power transmission schemes being commissioned. AC HVDC System Impedance in System Studies MICHAEL F.. the size of the matrix to be inverted will not exceed six. Simulation of ac system impedance. it is hoped that accounting for mutual couplings will result in a well-conditioned bus admittance matrix to facilitate faster convergence of iterative technique. Ramamoorthy and Mr. and fault conditions. Pachon. there has been considerable investigation into better simulation (digital. and a line coupled with the two existing Manuscript received August 15. W rITH THE increasing number and size of high-voltage R3 % R 03 :T:Cnl F AC SYSTEM (b) Z Ln F Ip 1L3 I RI (i) % Rn tR2 R3 C2 C3 :T:Cn ACT SYST AC SYSTEM (c) Fig. (b) Simulation with fundamental-frequency source in series with whole ac system impedance. PAS-89. Hingorani is with the Bonneville Power Administration. Although it is conventional to neglect mutual coupling as far as load flow studies and short-circuit studies are concerned. Some of these problems are 1) design of ac filters. 1(a) ] at various frequencies be simulated correctly. 1969. 1969. 5/6. 1969. BURBERY SENIOR MEMBER. Reitan in working out an example illustrating their technique. VOL. 1. whereas the method developed in the paper will accommodate the system changes with fewer computations. the author wishes to state that although inversion is not completely eliminated. made available for printing April 14. Ramarao will not yield readily for the system changes. Paper 69 TP 632-PWR. Downloaded on July 26. for several studies better simulation of the impedancefrequency characteristic (from power frequency to a few kilohertz) of the ac system is important. (a) System for simulation. However. Burbery is with GEC-AEI Ltd. of which three are coupled. when there are 15 lines in the system. Didriksen and Mr. Didriksen and Mr. To answer the point raised by Mr. Ore. Tex. . or model) of HVDC systems. 2009 at 14:37 from IEEE Xplore. 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. 97208. The author is developing programs in order to bring about comparison of different techniques to provide more information regarding computation efficiency raised by Mr. recommended and approved by the Power System Engineering Committee of the IEEE Power Group for presentation at the IEEE Summer Power Meeting. F. 1969. L1Lp L3 S. England. 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. 2) overvoltages at the converter station resulting from various switching operations. Manchester. ac system impedance is represented by its inductance at power frequency. Authorized licensed use limited to: King Fahd University of Petroleum and Minerals. the maximum size of the matrix to be inverted is restricted to smaller sizes. lines. For the design of the HVDC system itself. blocking-unblocking. IEEE. AND Abstract-In some studies of HVDC system design.. The transformation method referred to by Mr. For given typical ac system impedance diagrams (usually obtained from ac system models and simulators). Dallas.820 IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS. Simulation of NARAIN G. Portland. MAY/JUNE 1970 Kasi Nagappan: The author appreciates the interest shown by the discussers and is grateful for the comments made by every one of them. it is important that the impedance of the ac system [Fig. 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. HINGORANI. For example. analog. Pachon with regard to inversion to obtain a primitive admittance matrix. M. June 22-27. Kruempel and Prof.