Application of the Sherman–Morrison formula to short-circuit analysis of transmission networks with phase-shifting transformers

Electric Power Systems Research 155 (2018) 289–295

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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

Application of the Sherman–Morrison formula to short-circuit analysis of transmission networks with phase-shifting transformers Piotr Kacejko a,∗ , Jan Machowski b a b

Electrical Engineering and Computer Science Faculty, Lublin University of Technology, Poland Faculty of Electrical Engineering, Warsaw University of Technology, Poland

a r t i c l e

i n f o

Article history: Received 21 March 2017 Received in revised form 6 September 2017 Accepted 23 October 2017 Keywords: Power system Short circuit calculations Phase-shifting transformers Sherman–Morrison formula

a b s t r a c t For transmission network with in-phase transformers only, the nodal admittance matrix is symmetric. For such networks the short-circuit calculations can be done using sparsity-oriented factorisation of large complex symmetric matrices. In some transmission networks the phase-shifting transformers are used in order to control the real power flows. The series branch of the ␲-port network modelling such transformer has directional property. When such anisotropic branches are included in the network model the admittance matrix becomes asymmetric. Typically the number of phase-shifting transformers installed in transmission network is very small when compared to the whole number of transformers. It means that very few asymmetric element of the matrix imposes a larger computational effort especially for factorisation, which is the most time consuming and memory using process. In this article it is shown that (in spite of the asymmetric elements) the short-circuit calculations can be done by factorisation performed by procedure applicable to symmetric matrices. A new calculation method is proposed where the admittance model of the phase-shifting transformer is divided into the isotropic ␲-port network and a series anisotropic branch. The network admittance matrix is formed only from the isotropic branches and is factorised by procedure for symmetric matrices. The anisotropic branches are taken into account by modification of the impedance matrix elements using the formulas derived in this paper on the basis of the Sherman–Morrison formula. Proposed method can be easily implemented in already existing computer programs originally developed for transmission networks without the phase-shifting transformers. © 2017 Elsevier B.V. All rights reserved.

1. Problem definition High voltage transmission networks use transformers with onload tap changers [1–4,8]. Transformers regulating only the voltage magnitude are referred to as the in-phase transformers. They are used to control voltage and reactive power. Flows of the real power can be controlled by phase-shifting transformers (PST) regulating the voltage phase. In power system analyses such as the load flow, power system stability, short-circuit calculations both types of transformers can be modelled with the use of the following models:

(a) Admittance model i.e. the ␲-port network, which admittances of series and shunt branches depend on the transformation ratio of the transformer.

∗ Corresponding author. https://doi.org/10.1016/j.epsr.2017.10.025 0378-7796/© 2017 Elsevier B.V. All rights reserved.

(b) Current injection model i.e. the ␲-port network with constant admittances and additional nodal currents depending on the transformation ratio and nodal voltages. Each of these models has its own advantages and disadvantages. Generally, choice of model depends on the type of analysis and numerical method used. In optimal load flow computer programmes (where iterative methods are used and transformation ratios are treated as variables) the current injection models are preferred. In stability studies and short-circuit calculations (where transformation ratios are treated as predetermined parameters and network is modelled by nodal admittance matrix) the admittance models of transformers are predominantly used. In this article the admittance transformer models used to the short-circuit calculations are considered. Generally, in existing transmission networks the number of the phase-shifting transformers is very small if compared to the inphase transformers. A vast majority of transformers used in practice are the in-phase transformers. In some transmission networks the phase-shifting transformers are not used.

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P. Kacejko, J. Machowski / Electric Power Systems Research 155 (2018) 289–295

Therefore, many existing short-circuit computer programs do not have ability to model the phase-shifting transformers. In such computer programs the admittance models (␲-port networks) of the in-phase transformers are simply added to the admittance network model. Nodal admittance matrix of such network model is symmetrical, what simplifies all procedures. Usually the sparse-matrix technique is used to memorise and factorise the symmetrical admittance matrix. The impedance matrix (inversion of the admittance matrix) is not sparse. Therefore the impedance matrix of the whole transmission network is not computed and is not memorised. Instead of this the columns of the impedance matrix necessary for the short-circuit analysis are computed step by step for each node of the interest. In the case of the phase-shifting transformer the series branch of the ␲-port network has directional properties i.e. admittance from the first node to the second node is not equal to the admittance from the second node to the first node [1,2]. In this article such branch is referred to as the anisotropic branch. When anisotropic branches are inserted to the network model, then the network admittance matrix becomes asymmetric. The simple way is to use factorisation method designed to asymmetric matrices and then perform all further calculations on the basis of the asymmetric impedance matrix. However, such formal approach seems to be irrational. There are usually very few phase-shifting transformers in the transmission network and the number of asymmetric elements in the admittance matrix is very small. Such very small amount of asymmetric elements imposes larger computational effort especially in factorisation, which is the most time consuming part of all computations. This article shows that (in spite of the asymmetric elements) in short-circuit calculations the factorisation of large matrix can be performed for the symmetric matrix. In the proposed method the asymmetric admittance matrix is replaced by the sum of the symmetric matrix and an additional (corrective) matrix. Symmetric matrix is factorised and then the required part of the impedance matrix is calculated using formulas resulting from the Sherman–Morrison formula concerning the inverse of a sum of matrices [9,10]. In terms of computer memory capacity required, duration of computations and software structure, the method proposed here is more efficient than the method consisting in direct application of procedures for factorisation of asymmetric matrices. Another important advantage of the proposed method is the fact that models of phase-shifting transformers can be easily introduced into already existing short-circuit computer programs based on symmetric matrices without the need for uncomfortable replacement of computational procedures that deeply interfere with the structure of a computer program tried and tested for many years. This has been exactly the main reason for looking for a solution discussed in this article.

2. Short-circuit calculations The basis for short-circuit calculations is a network model shown in Fig. 1. This model is obtained by using the superposition method in the following way. Short-circuit through impedance Z F is assumed to occur in any node k. To this impedance two opposing voltage sources +V 0k and −V 0k are connected, where V 0k corresponds to voltage at node k in the pre-fault state. Such modified network model is replaced with a sum of two models:

(a) a model for the pre-fault state, (b) a supplementary model being a difference between models for the short-circuit state and pre-fault state.

Fig. 1. Circuit diagrams used for short-circuit calculations. (a) supplementary model (b) Thevenin equivalent circuit

Fig. 1a illustrates only the supplementary model. In this model, the sources of voltage (electromotive forces of generators) are short-circuited to the reference bus N (left side of the diagram). Voltages relative to the reference bus have values of (V k − V 0k ), (V i − V 0i ), (V j − V 0j ), etc., respectively, where superscript 0 refers to the pre-fault state. In the part of the diagram inside the dotted line, only the passive network occurs. After this part is replaced with a single impedance as seen from the short-circuit node, a simple equivalent circuit is obtained as in Fig. 1b. This circuit diagram is also consistent with Thevenin’s theorem. Based on this equivalent circuit, short-circuit current and voltage at node k can be calculated using the following formulas: Ik =

V ok ; Z Th + Z F

V k = V ok − Z Th I k

(1)

The part of the circuit diagram in Fig. 1a within the dotted line can be described with the following nodal admittance equation:

⎡

0

⎤

⎡

..

.

.. .

.. .

⎤⎡

.. .

.. .

⎤

⎥⎢ ⎥ ⎢ .. ⎥ ⎢ . . . Y ii Y ij Y ii . . . ⎥ ⎢ V i − V oi ⎥ ⎢ . ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥⎢ o ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎢ ⎥ V . . . Y Y Y . . . − V ji jj jk j ⎥ ⎢ ⎥= ⎥⎢ j ⎥ ⎢ −I ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ k ⎥ ⎢ . . . Y ki Y kj Y kk · · · ⎥ ⎢ V k − V ok ⎥ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ . ..

.. .

.. .

.. .

..

.

(2)

.. .

Design of the admittance nodal matrix is simple, because the off-diagonal elements are equal to the admittances of the branches taken with the opposite sign, i.e. Y ij = −yij = −1/ , where  , yij ij

ij

are, respectively, impedance and admittance of the branch connecting nodes i, j, and Y ij is a off-diagonal element of the matrix. Diagonal elements are equal to the sum of admittances  of the branches connected to the respective node, i.e. Y ii = yi0 +

yij ,

j= / i

where yi0 = −1/ is admittance of the shunt branch linking the i0 respective node to the reference bus N (Fig. 1a). Such defined matrix is diagonally dominant and is well conditioned.

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In Eq. (2), nodal current is different from zero only in node k and is equal to −I k , i.e. to the negative value of the short-circuit current. In the remaining nodes, nodal currents are equal to zero. In Eq. (2), the unknown voltages V i , V j , etc. occur on the right side of the equation. For this reason, in the subsequent calculations, the following equation is used

⎡

⎤

.. .

⎡

..

.. .

.

.. .

⎤⎡

.. .

0

⎤

⎢ ⎥ ⎢ ⎥ .. ⎥ ⎢ V i − V oi ⎥ ⎢ . . . Z ii Z ij Z ik . . . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎢ ⎥⎢ o ⎥ ⎢ V j − V j ⎥ = ⎢ . . . Z ji Z jj Z jk . . . ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎢ V − Vo ⎥ ⎢ . . . Z Z Z ⎥ −I k ⎥ ⎢ ⎥ ··· ⎥ ⎢ k ⎢ ki kj kk k⎥ ⎣ ⎦ ⎣ ⎦⎣ . ⎦ .. .

.. .

.. .

.. .

..

.

(3)

..

in which the impedance matrix is equal to the inverted admittance matrix under Eq. (2), i.e. Z = Y−1 . From that equation, the following is obtained: V k = V ok − Z kk I k

(4)

Comparison of Eqs. (1) and (4) shows that Z Th = Z kk . This means that nodal impedance matrix for the model in Fig. 1a has a very useful property consisting in that diagonal elements of that matrix are equal to Thevenin’s impedances of the corresponding nodes of the network in question. Therefore, the knowledge of that matrix allows one to easily, using Formula (1), to calculate short-circuit currents in all network nodes. In addition, using Eq. (3), voltage at any node can be calculated, in the short-circuit state in the selected node: V i = V oi − Z ik I k

(5)

V j = V oj − Z jk I k

(6)

The Ohm’s law says that the current flowing through a branch with impedance  can be calculated using the following formula: ij

I i−j =

Vi − Vj 

=

Vi0 − Vj0 

ij

ij

−

Z ik − Z jk 

Ik

(7)

ij

The first component on the right hand side of (7) corresponds to the current flowing through the branch in the pre-fault state. Therefore, one can state as follows: I i−j = I 0i−j −

Z ik − Z jk 

Ik

(8)

ij

where  is the impedance of the branch, and Z ik , Z jk are elements ij

of the nodal impedance matrix. It should be emphasised here, that in order to compute the shortcircuit current in a selected node k and the currents flowing through all branches (7) it is sufficient to know only one k-th column of the impedance matrix. Owing to this fact in sparse matrix technique the full impedance matrix is not memorised. Short circuits are analysed node by node and for each node only one column is computed on the basis of the factorised admittance matrix and by using fast backward and forward substitution. 3. Phase-shifting transformer model If short-circuit calculations do not take into account internal short-circuits in the transformer and only short-circuits in the network are analysed, phase-shifting transformers (PST) may be modelled using a complex transformation ratio  and equivalent admittance y seen from transformer terminals [11–13]. Such model is illustrated in Fig. 2a.

Fig. 2. Equivalent circuit diagrams of phase-shifting transformers (a) circuit diagram with complex transformation ratio, (b) equivalent ␲-port network with anisotropic branch, (c) current injection model.

Fig. 3. Illustration of definition of in-phase and quadrature transformation ratios.

Obviously, the equivalent admittance seen from transformer terminals depends on its transformation ratio. For example, report [11] shows that, for a transformer with quadrature regulation, equivalent reactance seen from transformer terminals can be described by the following formula:

X = X() = Xmin + [Xmax − Xmin ]

tg tgmax

2 (9)

where Xmin = X( = 0) and Xmax = X( =  max ) are equivalent reactances for zero value and maximum value of the phase shift , respectively. The definition of complex transformation ratio  is illustrated in the phasor diagram shown in Fig. 3. Assuming that the booster (series) transformer connected between nodes a,c is fed by the excitation transformer from node a (Fig, 2a), the following relationships can be written: V P = V a ; V Q = ˇV a

(10)

where  and ˇ are useful parameters closely related to the complex transformation ratio. Such transformation ratio is defined as follows: V = c = Va and





V a + ˇV a + iV a Va



tan  = / 1 + ˇ







= 1 + ˇ + i

(11)

(12)

where angle  is phase shift, (1 + ˇ) is in-phase transformation ratio,  is quadrature transformation ratio, i2 = − 1 is imaginary number.

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Fig. 4. Equivalent circuit used for solving the problem.

The right part of the circuit diagram in Fig. 2a corresponding to admittance y can be described with the following nodal admittance equation:

Fig. 5. Network model with anisotropic elements.

(13) In accordance with the definition of complex transformation ratio for the circuit diagram in Fig. 2a, the following can be stated: I c = I a /∗ and V c = V a . By inserting these relationships to equation (13), the following is obtained after simple transformations:

(14) The nodal admittance matrix under (14) describes a ␲-port equivalent network (Fig. 2b) with the following parameters: yab =  ∗ y;



yba = y



ya0 =  ∗  − 1 y;





yb0 = 1 −  y

(15a) (15b)

In the general case of a complex transformation ratio  from (15a) it results that yab = / yba , which means that the series branch of the ␲-port network is directionally dependent (is anisotropic). After inserting into Formulas (15a) and (15b) the values of  and ∗ resulting from (11), the following is obtained: yab =  ∗ y = (1 + ˇ)y − iy = y␤ − y

(16a)

yba = y = (1 + ˇ)y + iy = y␤ + y

(16b)

y␤ = (1 + ˇ)y and y = i␥y

(16c)

(dotted line in Fig. 5) is not included in the network model prior to the calculation of the required part of the nodal impedance matrix. This branch will be taken into account in the correction of the impedance matrix elements as described below. Formulas for the correction of the impedance matrix elements will be derived based on the formula for inverse of a sum of matrices (A + A), where A is the invertible matrix, and A is a corrective matrix. It is assumed that the corrective matrix has the form A = u × vT , where u, v are column matrices. With such assumptions, under the Sherman–Morrison formula, the following is true [9,10]:

(A + A)

−1

= A−1 −

A−1 u · v T A−1 1 + v T A−1 u

(18)

Obviously, expression (1 + vT A−1 u) in the denominator of this formula is a scalar. The Formula (18) can be used after demonstrating that the correction of the admittance matrix corresponding to the inclusion / yba is expressible by the prodof the anisotropic branch yab = uct of the column and row matrices. For further consideration it is assumed here that the nodal admittance matrix has the following form:

It is easy to prove that model from Fig. 2b can be replaced by current injection model shown in Fig. 2c, while both additional nodal currents are given by the following formula I a = I b = y(V a − V b ) = iy(V a − V b )

(17)

In such model the phase shift introduced by the phase-shifting transformer is replaced by two injection currents depending on quadrature transformation ratio  and voltages V a , V b at both sides of the transmission link. This model can be used in iterative computation methods similar to that proposed in Ref. [2]. However, the iterative methods are not the scope of this article. 4. Proposed method Based on Formulas (16a) and (16b), the circuit diagram shown in Fig. 2b can be presented as in Fig. 4. In this circuit diagram, the lower series branch and the shunt branches are isotropic, i.e. they are not directionally dependent. The upper branch is anisotropic (dependent on the quadrature transformation ratio ). In accordance with the assumptions of the proposed method, only the isotropic part of PST model is initially inserted to the network model. This is illustrated in Fig. 5. The anisotropic branch

(19) The Formulas (16a) and (16b) and diagrams presented in Figs. 4 and 5 show that, by including an anisotropic branch to the network model, the following corrections should be entered to the admittance matrix: Y aa

new

= Y aa + yab = Y aa − y

(20a)

Y ab

new

= Y ab − yab = Y aa + y

(20b)

Y bb

new

= Y bb + yba = Y aa + y

(20c)

Y ba

new

= Y ba + yba = Y aa + y

(20d)

P. Kacejko, J. Machowski / Electric Power Systems Research 155 (2018) 289–295

It means that, upon inclusion the anisotropic branch, the new nodal admittance matrix is described by formula Ynew = Y + Y, whereas:

(21) As is readily demonstrable, such matrix can be expressed as

⎡ . ⎤ . ⎢ . ⎥ ⎢ 0 ⎥ ⎢ ⎥  ⎢ ⎥ Y = u · vT = ⎢ 0 ⎥ · . . . 0 0 0 −1 +1 ⎢ 0 ⎥ ⎢ ⎥ ⎣ y ⎦

(22)

293

(1) Calculate parameters of all ␲-port networks with isotropic branches modelling all network elements. For phase-sifting transformers memorise parameters of the anisotropic branches (Fig. 4). (2) Use sparse matrix technique to create the symmetric admittance matrix. (3) Factorise the symmetric admittance matrix. (4) Use fast backward and forward substitution to compute the columns of the impedance matrix relating to the nodes being the terminals of the phase-sifting transformers. Memorise these columns. (5) Select subsequent node in the area of interest for which the short-circuit currents must be found. (6) Use fast backward and forward substitution to compute the column of the impedance matrix relating to this node. (7) Compute short-circuit current, voltages at selected nodes and current flows in selected branches. (8) If all nodes from the area of interest have been considered then finish. Else go to (5).

y which is consistent with the assumption made in Formula (18). It means that the modified impedance matrix Znew = Y−1 new can be calculated using the following formula: Znew = (Y + Y)

−1

Zu · vT Z =Z− 1 + vT Zu

(23)

where matrices u, v are as in Eq. (22). Performance of transformation under this formula requires, in editorial terms, an area wider than a single column. For this reason, the applicable mathematical transformations are presented in Appendix at the end of this paper. Eq. (A5) in the Appendix shows that elements (i, j) and (j, i) of matrix Zu · vT Z are expressed by the following formulas: y (Z ia + Z ib )(Z bj − Z aj ) for (i, j)

(24a)

y (Z ja + Z jb )(Z bi − Z ai ) for (j, i)

(24b)

Taking into account Eqs. (24a), (24b) and (A5), it may be stated based on Formula (23) that, as a result of inclusion of an anisotropic branch, elements of the impedance matrix are modified using the following formulas: (a) upper part of matrix Zij(new) = Zij −

y (Z ia + Z ib )(Z bj − Z aj ) 1 + y (Z bb − Z aa + Z ba − Z ab )

for i = 1, ..., n and j ≥ i

(25a)

(b) lower part of matrix Zji(new) = Zji −

y (Z ja + Z jb )(Z bi − Z ai ) 1 + y (Z bb − Z aa + Z ba − Z ab )

for i = 1, ..., n and j < i

(25b)

/ Zji(nowe) , which means that, after the As is evident, Zij(nowe) = inclusion of a corrective anisotropic branch, the nodal impedance matrix becomes asymmetric. Obviously, for diagonal elements, i.e. for j = i, both Formulas (25a) and (25b) provide an identical value. Formulas (25a) and (25b) apply to the inclusion of a single anisotropic branch. In deriving these formulas, no assumption was made as to the symmetry of modified matrices. Therefore, having included a single anisotropic branch only, using the same formulas, the impedance matrix can be further modified, taking into account subsequent anisotropic branches occurring in the models of the remaining phase-shifting transformers. 5. Sparsity-oriented algorithm The algorithm of the short-circuit computer program using the proposed method and sparsity of the admittance matrix consist of the following steps:

Methods to memorise and factorise the large sparse matrices are well described in the literature, for instance in [5,18,19]. 6. Inclusion of isotropic branch If a network contains only isotropic branches, the admittance and impedance matrices are symmetric. Inclusion of an additional isotropic branch with impedance z = 1/y into such network results in correction of the admittance matrix Y which can be expressed in the following way:

⎡ ⎢ ⎣

..

.. .

.

.. .

⎤

⎡

⎥

⎢

.. .

⎤

⎥

Y = ⎢ · · ·

+y

⎢ ⎥ −y ⎥ ⎦ = ⎣ +y ⎦ . . .

···

−y

+y

+1

−1



(26)

−y

In such case, Formula (23) leads to the following correction of the impedance matrix: Zij(new) = Zij −

(Z ia − Z ib )(Z aj − Z bj )

(27)

z + (Z aa − 2Z ab + Z bb )

This formula is identical to the one obtained for isotropic networks by C.H. El-Abiad using the laws of electricity only [6,7]. Note, however, that no mathematical proof of that formula has been known in literature so far. Also Formulas (25a) and (25b), concerning the inclusion of anisotropic branches derived in this article, have not been known. 7. Example The procedures based on Formulas (25a) and (25b) have been implemented in the program SHORT used for analysis of power systems [14–17]. Below are described calculations for a small test system (Fig. 6) which can be performed and verified using generally available mathematical tools, such as MATLAB. The network of the test system consists of six transmission lines with nominal voltage 220 kV and one phase-shifting transformer with control of the quadrature transformation ratio. This transformer is connected in series to line L4. The system has two generating units G1, G2 and an equivalent generator G3 which is an equivalent source for the remaining part of the system. For example calculation it is assumed that ˇ = 0 and  = 30◦ . For such √ assumption from Formula (12) it results that  = tan 30o = 1/ 3. Table 1 presents impedances of network elements  = Rij + iXij ij

and admittances yij = 1/ . The value of impedance of branch L4 ij

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The admittance nodal matrix taking into account the correction branch is described by matrix Ynew = Y + Y, where correction matrix Y is given by Formula (21). Hence, taking into account the value given by (29) and matrix (A5) matrix (A7) is obtained as shown in the Appendix. The right lower submatrix of that matrix is asymmetric. Using the asymmetric matrix inversion, matrix (A8) provided in the Appendix is obtained. Now it is necessary to check if the same matrix is obtained by proposed method based on the Sherman–Morrison Formula (23). For considered example and value from (29) the following column matrices are obtained from (22): Fig. 6. Circuit diagram of three-machine test system with one phase-shifting transformer (PST).

Table 1 Data of the test system from Fig. 6. Branch

Nodes

L1 L2 L3 L5 L6 L4 G1 G2 G3

B1 B2 B2 B3 B4 B5 B1 B6 B3

B2 B3 B6 B5 B5 B6 N N N

Impedance

Admittance

6.0 + 59.5i 10.7 + 90.0i 3.5 + 30.8i 4.2 + 47.0i 3.5 + 30.8i 5.3 + 56.0i 3.5 + 70.2i 2.2 + 40.3i 0.4 + 6.1i

(1.68–16.64i) × 10−3 (1.30–10.96i) × 10−3 (3.64–32.05i) × 10−3 (1.89–21.11i) × 10−3 (3.64–32.05i) × 10−3 (1.68–17.70i) × 10−3 (0.71–14.21i) × 10−3 (1.35–24.74i) × 10−3 (10.70–163.23i) × 10−3

takes into account the transformer reactance calculated by Formula (9) for  = 30◦ . As shown in Fig. 1a, generator reactances are shunt branches connected to the appropriate generator node and reference bus N. Impedances of branches G1, G2 contain in them subtransient reactances of generators G1, G2 and reactances of step-up transformers. Impedance of the G3 branch is calculated based on short-circuit power of the part of the system replaced by that source. All impedances are recalculated to the level of network nominal voltage, i.e. 220 kV. transformation ratio is obtained: For assumed √ data the following √ ˇ = 0,  = 1/ 3 or  = 1 + i/ 3. Parameters of the ␲-port equivalent network with isotropic branch are determined by Formulas (15a), (15b) and (16c): y␤ = (1 + ˇ)y = y = 10−3 (1.68 − 17.70i)

(28a)

 √ ya0 =  ∗  − 1 y = 10−3 (1/3 + i/ 3) y = 10−3 (0.33 + 0.58i)(1.68 − 17.70i)

(28b)

= 10−3 (10.77 − 4.93i)

⎡

0

⎤

⎡

0

⎤

⎢ ⎥ ⎢ 0 ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎥; v = ⎢ 0 ⎥ u = 10−3 ⎢ ⎢ ⎥ ⎢ 0 ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎣ 10.21 + 0.97i ⎦ ⎣ −1 ⎦ 10.21 + 0.97i

(30)

+1

Taking into account these columns and matrix given by (A6) matrix (A9) shown in the Appendix is obtained from the Sherman–Morrison formula. As it is easy to see both matrices (A8) and (A9) are identical which means that the Sherman–Morrison formula provides the same result as the inversion of the asymmetric matrix. The same matrix is obtained stepwise using Formulas (25a) and (25b).

8. Conclusions Admittance models of phase-shifting transformers introduce to the network model series branches with directional (anisotropic) properties. For such models, short-circuit calculations can be performed using procedure of calculation of the inversion of large asymmetric sparse complex matrix. Compared to the overall number of network branches, the number of phase-shifting transformers is very small. This leads to a situation where, due to a small number of asymmetric elements, procedures for factorisation of large asymmetric sparse matrices are necessary. This is computationally inefficient. This paper proposes different approach consisting of procedure for factorisation of asymmetric nodal admittance matrix is used and the resulting nodal impedance matrix is modified using the Sherman–Morrison formula. Proposed method is computationally efficient and, what is even more important, existing short-circuit computer program based on the symmetric matrices can be still used with slight modification only.

 √ yb0 = 1 −  y = − (i/ 3) y = −0.58i(1.68 − 17.70i) 10−3

(28c)

= 10−3 (−10.21 − 0.97i)

Appendix A. Matrix operations using Sherman–Morrison formula (A1) Matrix operations using Sherman–Morrison formula

where subscripts a, b correspond to nodes B5, B6. Taking into account the data provided in Table 1 and the values in Formulas (28a)–(28c), a nodal admittance matrix (A5) is obtained as provided for in the Appendix. The rows and columns in that matrix are ordered in accordance with numbers of the nodes in the network on Fig. 6. This matrix is symmetric. Its inversion is given by matrix (A6) shown in the Appendix. Admittance of the correction branch is determined by Formula (16c) and has the following complex value: √ y = (i/ 3)y = 0.58i(1.68 − 17.70i) = 10−3 (10.21 + 0.97i) (29)

(A1) (A2) 1 + vT Zu = 1 + y(Z ba − Z aa ) + y(Z bb − Z ab ) = 1 + y(Z bb − Z aa + Z ba − Z ab ) (A3)

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(A4) (A5) Admittance and impedance matrices in the example

(A5)

(A6)

(A7)

(A8)

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