Reduction and equivalencing of equations of electrical network based on matrix annihilators Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

Reduction and Equivalencing of Equations of Electrical Network based on Matrix Annihilators

Magomed Gadzhiev*, Misrikhan Misrikhanov, Vladimir Ryabchenko and Yury Sharov

Institute of Electrical Power Engineering, National research university MPEI, Moscow, Russia

Abstract— This paper proposes an original method for the reduction of node voltage equations, aimed at equivalencing electrical network. The method is based on equivalent matrix transformations using matrix annihilators. The offered method, in comparison with the traditional one, makes it possible to improve the conditionality of the solved equations by an order of magnitude or more. This has a positive effect on the numerical stability of the resulting electrical network equivalents. The results of reduction of a small, large, and very large system of node voltage equations are presented.

Index Terms — electrical network, nodal stress equations, reduction, matrix annihilator, equivalencing, conditionality, large and very large matrices.

I. Introduction

It is well known that the calculation of electric power systems (EPS) conditions is multivariate. Solving the complete systems of equations covering all the nodes and connections of large power systems, even with simplified models, poses a serious problem, because of the lack of reliable information on all elements of the network. The problem is exacerbated by the need to accumulate and store large amounts of information [1] and the corresponding increase in the requirements for speed of computers [2].

The need to reduce the computation time arises mainly in multivariate and multimode calculations in operational control, and in EPS operation planning [3, 4]. Equivalencing makes it possible both to reduce the time of solving the node voltage equations (NVE), and to reconcile the amount of information and its error.

Equivalencing of EPS is the transformation (reduction) of a complex mathematical model into a simpler one while preserving the most important (required) properties within a given accuracy. This approach is widely used with the toolkit of Krylov subspaces [5]. There is one more

* Corresponding author.

E-mail: [email protected]

http://dx.doi.org/10.25729/esr.2018.01.0011

Received: January 30, 2018. Revised: April 20, 2018.

Accepted: April 21, 2018. Available online: April 25, 2018.

© 2018 ESI SB RAS and authors. All rights reserved.

approach to the calculation of EPS conditions, where equivalencing is reduced to the transformation of an equivalent circuit and its parameters to the one having a smaller number of nodes and branches and suitable for modeling of the initial EPS conditions.

The paper proposes an original NVE reduction method for equivalencing an electrical network. The method is based on the NVE matrix transformations with the help of algebraic objects, called annihilators of matrices.

This method, in comparison with the traditional algorithm based on the Schur complement, makes it possible to improve significantly the conditionality of the solved equations, especially for large (up to 10 000 equations) and very large (up to 100 000 equations) systems of NVE. It has a significant effect on the accuracy of the obtained electrical network equivalents and on the numerical correctness of numerical models.

II. THE TRADITIONAL APPROACH TO NVE

Equivalencing

The procedure of equivalencing is given later as an example. Excluded nodes (set M) and nodes stored in equivalent units (set N) are given in [2].

The nodes were renumbered so that the first nodes were from the set N and the rest of the nodes were from the set M. In this case, the NVE were structurally divided into blocks (block matrices and subvectors)

ynn ymn un in

ymn ymm um im

Here, UN, UM - the voltage subvectors; I

N ' M

current sub-vectors in the vector

n

(1)

IM

Submatrix YNN

in the conductivity matrix

Y Y nn Y ymn

Y ymn Y ymm

(2)

(3)

is square and covers only the constraints on the set of nodes N, and the square submatrix YMM corresponds to the

connections on the set M. Note that YNM and YMN are

rectangular matrices.

Expanding equation (1) up to the selected blocks

Y TT A-Y TT — T

innljn ^ nm m — 1n>

Y TT A-Y TT = T

mn n mm m 1m>

(4)

and expressing the voltage vector of the excluded nodes UM from the second equation (4), we obtain the equation

Y -Y Y~l Y TT = I —

*nn 1nm1mm1mn 17 n n

Y Y~l T

1nm1mm1 m

which with the introduction of new notations

nn

'ww

Y Y~ Y

inmimmimn'

annihilator; rank(•) - rank of the matrix; size(•) -dimensions of the matrix (vector dimension); null ( • ) -basis of the null space of the matrix; cond ( • ) - condition number of the matrix; 11 ■ 11 -given vector norm [6,7].

In this paper, the so-called left annihilator of matrices is used, which is called the annihilator. Recall [8, 9], that the maximal rank matrix annihilator of to x n matrix M of

rank r is called matrix M1, and MM = 0,

(n — r)xm '

(5)

with rank M

= n — r .

T3 =T —Y F 1 /

jn jn inmimm1m'

is reduced to the equivalent form

Y IT = I

Thus, the equivalencing procedure considered above, for the given excluded nodes (set M) preserves only the nodes of the set N in the resulting equivalent solution (equivalent).

From a formal point of view, the described equivalencing procedure is based on the well-known Schur complement algorithm [6, 7]. One of the drawbacks of this approach is the complexity and, often, the impossibility of preconditioning ("regulation" of the conditionally) of the solved equations.

The next section of the paper describes an original method. This method is an alternative to the Schur complement algorithm, and the equivalencing method is constructed on its basis.

This method, as was said earlier, makes it possible to effectively affect the NVE condition, and, as a consequence, reduce the computational errors and increase the correctness of the equivalent solutions obtained.

III. Mathematical Justification Of The Alternative Method

The following notations and definitions will be used: 0 ^ -n x to zero matrix; E - nxnunitmatrix; ( )T

71x771 tl v /

-transposed matrix; (•)+ -pseudo-inverse matrix according to Moore-Penrose; (•) - maximal rank matrix

For simplicity, we shall assume that the annihilators of zero satisfy the orthogonality condition

M±MJ-T = E

n—r

Well-developed methods for computing the null space null(M) of a matrix M can be used for calculation of

annihilator matrices [6, 10]. In this case M1 = null (MT)

We will consider the NVE in the following block

decomposition: (6)

1 ! A,

where A1, A - are rectangular submatrices of

size Ay = n x riy,

A

= n x ra„

(7)

(8)

in this case r^ + n,2 = n . The decomposition (7), (8) is

clearly shown in Fig. 1.

The statement: the solution of linear equation (7) for

invertible block matrix A equivalent formulas [9]

A2 is determined by the

xi — b '

-l

x2 —

X1 = b'

= a£ b-A

(9)

(10)

1*1

here Ay , A2 are the left annihilators of zero of the maximal rank submatrices A{ . A2, respectively, A{ , A^ are the pseudoinverse matrices of the submatrices /1

A2 •

э

x

l

b

2

2

n, + «,

Figure 1. Block partition of matrices and vectors in the matrix equation.

IV. Reduction Of NVE Based On Matrix Annihilators

Considering NVE (1) in the division into blocks, as it is done in equation (7), we introduce the notations

yn =

nn

mn

Y —

IM ~

mn

1 mm

(11)

and write (1), taking into account (11). The obtained equation is

ln

lM

N jn\

Kl 1m

(12)

Annihilator Y,, is introduced and it satisfies the

m

following conditions:

Y Y = 0

1m1m mxm'

F1F1T— F

Then, according to the first equation (10), from the theorem proved earlier we can write

y-Ly Tj _ y1-

iminljn — im

Let us introduce new notations

'n

lm

Y3

inn

Y Y

m n"1

p 1n

lM

ln

m

Y3 U =r

znnun ân'

then, equation (14) can be rewritten in a generalized form

(16)

Equations (6) and (16) are different, but they have the

same solution UN. The principal difference of equation

(16) is that it allows solving the problem of preconditioning in order to minimize computational errors simultaneously with equivalencing.

It is well known [6, 11, 12] that in order to reduce the influence of errors in the initial data, to increase the accuracy of the solution, and to accelerate the convergence

of the iterative methods, various algorithms are used that usually consist of elementary transformations of rows (columns) of matrices in equation (7): scaling, regularization, balancing, change of conditioning (preconditioning, use of spectrally equivalent operators), etc.

With respect to matrix equation (16), the problem of reducing errors will consist in minimizing the ratio [11]

(17)

However, the direct determination of the value rin terms of the coefficients of the matrices of the original equation is difficult due to the nonlinearity of the valuation operation. Therefore, it is preferable to use a qualitative characteristic called the matrix condition number [6, 10, 11]. In the considered case, this number is

||At/iv| p 1n

INI' AI3 n

cond YNN -

nn

-1

nn

(18)

and it satisfies the inequality

Ai/

N\\

< cond Y

U

3

nn -

<

AJ3

n

AM

p 1N

(19)

Given (15), the ratio (19) is transformed to the form

(13)

||AUn lluj

U cond (y^n )<

A

( I ^

1 m

V Im y y

( I \ 1 m

V Im y

(20)

(14)

(15)

The larger the condition number (18), the greater the impact of the original data errors on the solution to NVE.

Reduction of the condition number (18) can be achieved by further transformation of the NVE equivalent system (16) by introducing a new matrix D , which should be [12]

as close as possible to Y,, Y

easily computable and

easily invertible. In this case, the NVE will be replaced by the equation

where

BY3 U = D I3 1jinnun u1n

cond DY3nn < cond Y3nn

V. Reduction Of A Small NVE System

Let us consider the computational example [9]. We analyze the electric network shown in Fig. 2 [2]. For

3

3

±

m

1

convenience of calculation, we take all the line resistances to be the same r — 10 Ohm ( V =0,1 S), except for the

y y , f

two lines ?î ,¡=20 Ohm ( K|:, = 0.05 S)

r24 = 5 Ohm

(YM =0,2S).

The basic mode corresponds to the NVE system

-0,25 0,1

0,1 -0,4

0,05 0,1

0 0,2

0,05 0 0,1 0,2 -0,25 0 0 -0,2

V» -8,5

UÏ 1

-7

K -2

(21)

'n

Y —

IM ~

-0,25 0,1 0,1 -0,4 0,05 0,1 0 0,2

0,05 0 0,1 0,2 -0,25 0 0 -0,2

(22)

considering that

Y^ -

null

0,05 0

0,1 0,2

-0,25 0

0 -0,2

0,9646 0,0915 0,2295 0,0915 -0,1837 0,6752 0,2333 0,6752

we have

Y Y 1M1M

0,9646 0,0915 0,2295 0,0915 -0,1837 0,6752 0,2333 0,6752 0,05 0

0 0

0,1 0,2 -0,25 0 0 -0,2

0 0

Performing further calculations, we obtain

-0,2205 0,1011 0,1251 -0,1301'

Y3 =Y^Y =

inn m n

Il = F,"1

lN

M

jn -9,8979

jm -0,7473

II = Y3 ^ n inn

-1

p =

85,0 87,5

which exactly corresponds to the values indicated in Fig. 2.

It is worth noting, that the condition number of matrix (3) and the conductivity number of original matrix (22) is

cond Y

n

Y,, — 15,9373 for the original matrix, and

cond Y^n — cond

-0,2205 0,1011 0,1251 -0,1301

= 5,5204,

for the equivalent, which is less by almost 3 times. This number can be further reduced if V ,, is

The results of the calculation are shown in Fig. 2. Let us suppose that the set M of excluded nodes is 3 and 4. Then, matrix (3) has the following block decompositions:

*M

m

6,6150 4,7836 3,2365 4,7836 3,5836 8,6518 4,1774 8,6518

The use of annihilator (23) in the calculations provides -0,2205 0,1011 0,1251 -0,1301

Y3 -

inn ~

(23)

(24)

cond Y^n = 1,2444, which is by more than an order of

magnitude less than cond | Kv matrix

-0,24 0,12

YM . In this case, the

Y —

inn ~

0,12 -0,56

calculated by the traditional method, has a 3,5 times greater condition number than matrix (24).

Figure 2. The scheme of the electrical network in the basic mode.

VI. Reduction Of A Large NVE System

Let us consider the reduction in a large NVE system, the basic mode of which corresponds to a matrix

t

Y Y nn Y ymn

Y ymn Y ymm

size

and vector of

=1000x1000

size

= 1000.

Matrix (3) and vector (2) for the basic mode are dense and their elements vary within the following limits: -4,2; ...;5,7. The total number of non-zero elements of

the matrix from (3) is ~ 9,91 • 105.

Let the set M of excluded nodes be equal to 995, and, accordingly, the number of nodes N = 1000 - 995 = 5 .

Calculation by formulas (11) - (16) in Matlab, using orthogonal annihilators leads to the following results:

F3 =

1NN

1,8571 -0,7800 1,4434 0,4049 -0,4329 -0,7242 0,6592 1,5200 0,5827 1,6713 0,2350 -0,3492 -0,6982 0,9702 -1,3692 -0,6484-0,6965 0,0061 -1,1241 0,5262 -0,8512 0,2244 0,1312 1,1808 1,2175

TT _ y3 I3 -

I3N = [0,7807; - 0,1045; - 1,4519; - 0,2478; - 0,1782J,

' 1,6907 0,4469 -0,8643 . -0,3758 1,4110

In this case, the condition number of matrix (25) is 7,8019, and the Euclidean error rate of solution (26) with respect to the exact value of the solution vector is 3,825310-13.

Calculations using conventional methods allow us to obtain the following matrix and vector

F3 -

NN ~

41,7122 19,8072 -2,6691 84,1475 -29,5413 -55,6173-48,4439 9,7584 -51,1095 43,1407 -84,5763-19,1018 14,5854 -42,1711 75,9152 87,8123 63,1374 -21,0482 19,9958 -59,6218 49,3494 -8,4865 36,5263 64,3591 -27,9839

8,3722 -44,0346 -41,1712 , 103,2302 -15,6018

with more than 3,5 times the Euclidean norm of the error

I3 -

with respect to the exact value of vector 1,465110-12. The condition number of matrix (27) is 39,4146 and it is almost 5 times higher than the condition number of matrix (25).

VII. Reduction In A Very Large NVE System

Let us suppose that a very large NVE system is given, whose dimension is 104. Matrix (3) has no zero elements, thus the number of non-zero elements is 108, that is one hundred million (100 • 106). In this case, the elements in the matrix and vector (2) vary in the range from -10,7 to 11,5 . Suppose that the set of excluded nodes M is equal to 9 -103, which means that the number of nodes left is N = 104 - 9 -103 = 100 .

As a result of the calculations, the 100 x 100 matrix was obtained and it had the condition number

cond Y3n

47,3464, while the condition number of

(25)

matrix

cond Y,

nn

with the same size,

was

nn

10390,4112 , that is more than 200 times

higher.

We note that it is not possible to directly use the Schur complement algorithm for such a large matrix. In this case, the authors used the parallelization of the computation process. (26)

VIII. Conclusion

An original equivalencing algebraic method is proposed to reduce the equations of electric network steady-state conditions on the basis of matrix annihilators. The method allows transformations of the equivalent circuit and its parameters to a form having a significantly smaller number of nodes and branches. Numerical procedures for computing matrix annihilators are well developed for large (100 < n < 1000) and very large (103 < n < 105) matrices, while the variation in the annihilators properties makes it possible to significantly (ten times and more) improve the ^§onditionality of the resulting NVE equivalents and thereby reduce computational errors and improve the correctness of the solution.

Acknowledgment

This work was supported financially by the Russian Science Foundation (Project №18-19-00225).

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Magomed Gadzhiev graduated from the Electric Power Department (EPD) of Moscow Power Engineering Institute (MPEI) in 2006. In 2012 he defended his PhD thesis "Improving the accuracy of accounting for power losses to the crown during optimization of EPS operating conditions" in MPEI. Currently he is the Head of the Research Laboratory of the Department of Electric Power Systems at MPEI.

Misrikhan Misrikhanov graduated from the Electric Power Department (EPD) of Moscow Power Engineering Institute (MPEI) in 1974. In 2010 he defended his doctoral dissertation «Fundamentals of the theory of invariant control and its application to the creation of technical systems». Currently he is a Professor at Ivanovo State University.

Vladimir Ryabchenko graduated from Irkutsk High Military Aviation Engineering School in 1983 with a specialty in «Aeronautical Instrument and Electrical Equipment». In 2002 he defended his doctoral dissertation in Air Force Academy named after Professor N.E. Zhukovsky. Leading researcher of the Research Laboratory «Automation of Electrical Distribution Networks» National Research University MPEI.

Yuriy Sharov graduated from the EEF MPEI in 1986. In 1994 he defended his PhD thesis "Development of methods and tools for assessing the effectiveness of the management of electric power systems under large disturbances" in MPEI. Currently he is the Head of the Department of Electric Power Systems

w »

X

V I

at MPEI.