Rational modeling of overhead transmission lines considering finite length impedance and admittance

expressions

Rodolfo A. R. Moura Universidade Federal de

São João Del Rei, São João Del Rei, MG, Brasil

Marco A. O. Schroeder Universidade Federal de

São João Del Rei, São João Del Rei, MG, Brasil

Antonio C. S. Lima Universidade Federal do

Rio de Janeiro, Rio de Janeiro, RJ, Brasil

M. Teresa Correia de Barros Instituto Superior Técnico Universidade de Lisboa,

Lisboa, Portugal

Abstract—— An upcoming challenge in overhead lines is wide river crossing as the span between towers may be over 2 km. One challenge in this configuration is to accurately represent the line behavior. The expressions for the evaluation of impedance and admittance matrices in overhead lines are based on the assumption that the total line length must be at least a few order of magnitude higher that the highest conductor average height. However, in case of a wide line span such hypothesis is not strictly valid and the per-unit of length impedance and admittance must consider the finite length of the circuit. The cascading of short line in time-domain is inefficient, however, if a rational approximation of the equivalent admittance is used a more efficient line model is obtained. A test case based on actual line configuration used to cross the Amazon River is considered and the results indicate that a more accurate frequency domain behavior is obtained leading to a more suitable rational approximation, which is not only more accurate but also is less prone to the residue perturbation in the case of passivity enforcement.

Index Terms-- Transmission lines, rational modeling, network synthesis, Time-Domain.

I. INTRODUCTION

As power system grow more complex and interconnected there is a premium for the correct evaluation and assessment of its performance in steady-state and during transient phenomena. Another point is the ever increasing demand of environmental issues which stress the importance for a suitable model of any given transmission network. In some developing countries such as Brazil, China and India there is the challenge of power transmission through wide river crossings and through forests.

A key aspect in the evaluation of power system transients is the parameter determination of the electrical network involved in the analysis [1]. For the assessment of overhead lines transient behavior, the assumption related to quasi-TEM (transverse electromagnetic) is usually acceptable leading to impedance and admittance matrices that can be assembled in a suitable computer model [2][3].

For the modeling of overhead lines there are a number of simplifying hypothesis, such as constant height along the line length, and the assumption that for the per unit length parameter determination that the line is infinite in length. These assumptions were considered for the determination of ground return impedances of underground cables and overhead lines [4][5][6] and are also used when simplified expressions such as the complex ground plane [7] or asymptotic approximations [8] are considered. It is worth mentioning that even when the quasi-TEM propagation is waived, conductors are assumed at a constant height and the line length is supposed infinite [9]. More recently, in [10] it is proposed a modification in the per unit length expressions for the impedance and admittance of overhead lines to take into account its finite length.

In this work, we evaluate the rational modeling of a nonuniform line (NuL) obtained by a cascading of several uniform lines. Each uniform line is represented using finite length expressions for impedance and admittance matrices. To asses this proposition we consider the case of an actual wide river crossing where the line span is over 2 km. It is based on the Amazon river crossing. This paper is structured as follows: Section II presents the basics of the NuL modeling assuming cascade of uniform lines. Section III shows the rational approximation of the characteristic admittance and propagation function associated with a finite length circuit. For the frequency domain fitting, we use the method known as Vector Fitting [11][12] which is a rational formulation of the methodology presented in [13]. Impulse response tests used to evaluate the time-domain responses are presented in Section IV. The main conclusions of this paper are presented in Section V.

II. NONUNIFORM LINE (NUL) MODELING

An overhead line can be represented by a nonuniform line (NuL) whenever is not possible to assume that the per unit length parameters are constant along the line. This is the case when there is a large difference in conductor heights along the span or a wide span where the sag is very pronounced. To

This work received a partial financial support from “Conselho Nacional deDesenvolvimento Científico e Tecnológico” - CNPq, “Coordenação deAperfeiçoamento de Pessoal de Nível Superior” - CAPES and “InstitutoNacional de Energia Elétrica” - INERGE and “Fundação de Amparo àPesquisa do estado de Minas Gerais” - FAPEMIG. Paper submitted to the Power Systems Computation Conference (PSCC2016) in Genoa, Italy, June 20-24, 2016.

illustrate this Fig. 1 depicts the phase conductors and ground-wire sags for a wide river cross. In this case, the span is of 2.1 km and it is used to cross the Amazon river in Brazil. For this river crossing the tower are higher than 340 m.

Fig. 1. Sag for a 2.1 km span used for the crossing of the Amazon river.

The equations that define the line behavior in the

frequency domain are given in (1)

( )( , ) ( )

( )( , ) ( )

d s dx s s

dx dxd s d

x s sdx dx

= −

= −

VZ I

IY V

(1)

where both series impedance per unit length, Z(x,s) and shunt admittance also per unit length Y(x,s) are functions of both frequency s=jω and the position x along the line. Unlike the uniform case, where both Z and Y do not depend on x, there is no closed-form solution for the voltage at the line terminals [14]. An approximate solution consists in assuming that for a given length, ( , ) ( )x s s≈Z Z and ( , ) ( )x s s≈Y Y

which leads to an uniform line. If the admittance matrix associated with each uniform line

is transformed into a transfer matrix relating the voltage and current at the uniform line segment is possible to obtain an equivalent transfer matrix or chain matrix [15] for the whole NuL which can then be converted back to an admittance matrix. The nodal admittance matrix associated with a NuL is given by a block matrix with the following structure, (using the notation proposed in [16])

RR SRn T

SR SS

=

Y YY

Y Y (2)

where SS RR≠Y Y . In the case of a uniform line segment,

system where both diagonal block matrices are equal and SRY

is symmetric. To illustrate this consider a case where a given line span

is divided in three segments, each with a matrix nKY where

k=1,2, 3 and has a structure similar to (2). The transfer matrix associated with each k segment is given by

1 1

1 1

k k k

k k k k k k

RS RR RSk kk

k k SR SS RS RR SS RS

− −

− −

− − = = −

Y Y YA BQ

C D Y Y Y Y Y Y (3)

The transfer matrix eqQ is obtained by the dot product of

each kQ and the equivalent nodal admittance matrix, eqY , of

a NuL is given by

( ) ( )

( ) ( )

1 1

1 1

eq eq eq eq eq eq

eq

eq eq eq

− −

− −

− = − −

D B C D B AY

B B A (4)

The implementation of (4) in a frequency domain program is straightforward as the main issue is related to the inversion of some of the block matrices, see [15] for details. On the other hand, the time-domain modeling of such system is rather cumbersome, as one need to represent each uniform line segment, which implies in a loss of computational efficiency and a limitation regarding the maximum time-step possible in the simulation. Using a time-domain model, one must use a time-step that is at least smaller than the fastest travelling time found in the uniform line sectioning.

An alternative formulation for the time-domain analysis of a NuL was proposed in [17]. Unfortunately, this approach was prone to numerical oscillations in both voltage and current at the NuL terminals. A distinct approach for the time-domain was proposed in [18] that recognizes that a NuL is a particular case of a frequency dependent network equivalent (FDNE) which in turn can be approximated using a rational modeling. The main drawback in this proposition is that it fails to assess whether or not the smallest eigenvalues at the lower frequency range are accurately represented in the rational approximation.

Another important issue in NuL modeling is the length of the uniform line segmentation. In the technical literature there are number of propositions such as the Courant-Friedrichs-Levy (CFL) criterion as in [19] or to use a finite-difference approximation of (1)[20] non unlike the procedure used to model electric towers for power system transients. If either one of these approaches is applied, the length of the uniform line section is around 30 m.

If this type of line segmentation procedure is applied to the wide river crossing, shown schematically in Fig. 1, there will be several line segments where the length would be one considerably smaller than the average height of phase conductors. Such scenario contradicts one of the hypothesis of the uniform line modeling. A possibility to improve this scenario is to use the so-called finite length line as proposed in [10]. In appendix A, we briefly present the impedances and admittances per unit length expressions for such a case.

III. RATIONAL APPROXMATION OF WIDE-RIVER CROSSING

To illustrate the proposed methodology, we consider the case of a wide-river crossing that it is being presently installed to cross the Amazon River and has a 2.1 km span as shown in Fig. 2. The conductor data is shown in Table I. The span is segmented in 70 uniform lines using the expressions for finite length conductors to obtain the per unit length

-1.0 -0.5 0.0 0.5 1.00

50

100

150

200

250

300

Phase Wire

Ground Wire

[km]

[m]

matrices, i.e., Z and Y. Each line segmented is converted to a transfer matrix Q as show in section II. All transfer matrices are cascaded to obtain the chain matrix which is then converted to an equivalent nodal admittance matrix as shown in (4). Then, we subject this matrix to the pole relocating algorithm known as Vector Fitting (VF), thus

11

0n

q

N

ne

n

ss p=

−= + ++ R

Y R R (5)

where s jω= , np is a set of poles which are real or come in

complex conjugate poles, nR is the residue matrix in which

elements are real or come in complex conjugate pairs, 0R

and 1−R are real matrices. Although, only stable poles are

used in (5), the overall approximation might not be passive. Thus, a post-processing is needed to enforce passivity [21][22].

(a) Actual tower

(b) Bundle conductors and ground wire arrangement

Fig. 2. Actual tower and conductor arrangement for a river crossing of the Amazon River.

TABLE I—NUL PARAMETERS FOR THE 500 KV DOUBLE CIRCUIT

Operating Voltage 500 kV Number of conductors/phase 4

Number of cable grounded-wires 2 Distance among subconductors 0,457 m

Height phase A 313.2 m Height phase B 323.2 m Height phase C 313.2 m

Height grounded-wires 332.7 m Phase conductors diameter 29.591 mm

Ground-wires diameter 9.14 mm Horizontal distance between phases 5 m

Horizontal distance between ground-wires 17.6 m

One limitation of the rational approximation of the nodal admittance matrix lies is that poorly observable eigenvalues, i.e., eigenvalues associated with the open-circuit behavior are not correctly identified using VF. This loss of accuracy is important whenever open-circuit responses are of interest. To overcome this limitation was proposed to use the Modal Vector Fitting [23]. However, in this approach the system to be solved to identify the poles is no longer sparse. Recently, it was proposed to use a similarity transformation known as

Mode Revealing Transformation that preserves the passivity of the original model and increases the observability of the small eigenvalues [24].

In Fig. 3 is depicted the results of the rational approximation of the admittance matrix of the river crossing using the NuL approach. In Fig. 4, it is presented the fitting results if a single uniform line is used to represent the wide river crossing. In both scenarios 50 poles were considered and although a very good match can be observed, but as it will be shown in the paper, there are some inaccuracies in the lower frequencies in what concern the quality of fitted eigenvalues. It is worth mentioning that the modeling of the single span as a NuL presented a frequency response quite different from the one obtained using a uniform line. These differences will also be presented in the eigenvalues and very distinct time responses are to be expected.

Fig. 3. Nodal admittance matrix and the rational model for the River Crossing. Rational Modeling using 50 poles (NuL approach).

Fig. 4. Nodal admittance matrix and the rational model for the River Crossing. Rational Modeling using 50 poles (conventional approach).

Figure 5 depicts the behavior of the eigenvalues obtained from the rational approximation using 50 poles as well as the eigenvalues of the original data. There are large deviations in the eigenvalues associated with the open-circuit response at lower frequencies. Initially, one may think that an increase in the number of poles would provide further accuracy

102

104

10-6

10-4

10-2

100

102

Frequency [Hz]

Mag

nitu

de [

p.u.

]

Original

FRVF

102 10410-6

10-4

10-2

100

102

Frequency [Hz]

Mag

nitu

de [p

.u.]

OriginalFRVF

eliminating these low frequencies mismatches. Unfortunately, this is not the case. Figure 6 depicts the comparison of the eigenvalues when 80 poles are considered in the rational model. Although, some improvement can be observed, this rather large increase in the number of poles was not sufficient to eliminate the loss of accuracy.

Fig. 5. Eigenvalues of the equivalent nodal admittance matrix and the rational model for the River Crossing without MRT. Rational Modeling using 50 poles (NuL approach).

Fig. 6. Eigenvalues of the equivalent nodal admittance matrix and the rational model for the River Crossing without MRT. Rational Modeling using 80 poles (NuL approach).

The eigenvalues from the original nodal admittance matrix obtained after the chain matrix, are presented in Fig. 5 together with the ones obtained using the rational approximation using 50 poles. In Fig. 6 is depicted the results considering 80 poles. It can be seen that the main difference lies in the representation of the smallest eigenvalues. An increase in the number of poles seems to increase the accuracy at lower frequencies, however it is not possible to eliminate the mismatches. The results may indicate that a further increase in the number of poles would improve the rational modeling. However, it was found that this was not the case and furthermore, as the number of poles increased

there is a nonlinear gain in the overall time needed for the passivity enforcement. If some cases, for instance considering 120 poles, it was not possible to find a passive approximation and in other cases, even though the system is now passive, the model perturbation was such that a great loss of accuracy was found. Table II presents the passivity enforcement time as a function of the number of poles in an off-shelf computer using intel i7 (I7-3770) with 12 Gb of RAM.

Fig. 7. Eigenvalues of the equivalent nodal admittance matrix and the rational model for the River Crossing using MRT. Rational Modeling using 80 poles, (NuL approach).

If on the other hand, the poles and residues are identified

using a Mode Revealing Transformation (MRT), it is possible to improve the quality of the smallest eigenvalues within a reasonable order. Figure 7 compares the eigenvalues of the original system with the ones obtained using the rational approximation. A sensible increase in accuracy at lower frequencies is found.

TABLE II—PASSIVITY ENFORCEMENT OF RATIONAL APPROXIMATION

Number of poles Time [minutes] 40 3.3 50 4.5 60 5.5 70 7.2 80 23.7

It is interesting to compare the results of the NuL

modeling with the ones we obtain assuming a uniform line, i.e., using a single line segment with the average height along the span. Figures 8 and 9 compares the behavior of the eigenvalues obtained using a uniform line and its rational approximation. Again, it can be observed that the MRT improves the low frequency accuracy of the smallest eigenvalues. It also is worth mentioning that if we compares these set of eigenvalues with the ones obtained using the NuL approach there are some noticeable differences in both set of eigenvalues. A transmission line has two set of eigenvalues, one associated the the short-circuit response and other with the open circuit response. At least one of each set of

102 10410-8

10-6

10-4

10-2

100

102

Frequency [Hz]

Eig

enva

lues

of Y

[S]

Original DataFitted

102 10410-8

10-6

10-4

10-2

100

102

Frequency [Hz]

Eig

enva

lues

of Y

[S]

Original DataFitted

102 10410-8

10-6

10-4

10-2

100

102

Frequency [Hz] E

igen

valu

es o

f Y [S

]

Original DataFitted

eigenvalues present a very distinct behavior when compared with the results associated with the NuL.

Fig. 8. Eigenvalues of the equivalent nodal admittance matrix and the rational model for the River Crossing without MRT. Rational Modeling using 80 poles, (Uniform line approach).

Fig. 9. Eigenvalues of the equivalent nodal admittance matrix and the rational model for the River Crossing using MRT. Rational Modeling using 80 poles, (Uniform line approach).

IV. TIME-DOMAIN RESPONSES

For the evaluation of the time-domain response, we consider the circuit configuration shown in Fig. 10 where the voltage at the first conductor is a double exponential. Here, we considered a 1.2/50 µs voltage source.

Fig. 10. Schematic for the double exponential response test.

To assess the impact of the proposed methodology we

consider two scenarios. In the first one, the whole span is represented as a uniform line, where the average heights of each conductor is used. In the second one, we use the approach described in the previous sections.

Figure 11 depicts the time-domain responses at terminal 2 using both approaches. It can be noticed that there are very noticeable differences between each approach. The use of the NuL with nonuniform line is richer and presents more spikes than the uniform line approximation. The same behavior is found in the other nodes, as shown in Fig. 12.

Fig. 11. Voltage at the first phase of terminal #2 for a double exponential voltage (1.2/50 µs) at terminal #1.

Fig. 12. Voltage at the second phase of terminal #2 for a double exponential voltage (1.2/50 µs) at terminal #1

V. CONCLUSIONS

This work has presented a rational modeling of a nonuniform line (NuL) suitable for time-domain simulations. A case of a wide-river cross was used to test the modeling. As the chain matrix was used to obtain the equivalent nodal admittance matrix, the line span need to divided in smaller uniform segments

One peculiarity of such case is that given the tower elevate height the conductors’ sag is so pronounced that the uniform line assumption of a constant height cannot be considered.

102 10410-8

10-6

10-4

10-2

100

102

Frequency [Hz]

Eig

enva

lues

of Y

[S]

Original DataFitted

102 10410-8

10-6

10-4

10-2

100

102

Frequency [Hz]

Eig

enva

lues

of Y

[S]

Original DataFitted

Futhermore, given the span length, the common assumption in overhead lines that the circuit length is at least a few orders of magnitude higher than the highest average height does not hold.

To overcome these two limitations, it is possible to divide the line in smaller uniform line segments, where a constant height hypothesis can hold and given the even shorter line length, the expressions for the impedances and admittance per unit length must be corrected using the so-called finite length expressions.

Using the chain matrix is possible to derive a matrix transfer function relating voltage and currents at the towers which can then be converted to an equivalent nodal admittance matrix. This procedure was quite common in the past when only frequency domain simulation were considered for the analysis of power system transients. If a rational approximation of the equivalent nodal admittance matrix is possible, a more efficient time-domain simulation is attained as this avoids the common practice of dealing with very short line segments implying in an unnecessary small time-step.

To illustrate the proposed approach, an actual case of a wide river crossing was considered. The results in both time and frequency domain indicated that the NuL approach provide a greater insight in the behavior of the line span without a sensible increase in simulation complexity or overall computation time. In fact, although not addressed here in detail, this approach was around 10 times faster when compared with the actual representation of each line segmentation. Therefore, the authors strongly recommend that for study of fast transients in unusual transmission line such as the one under analysis, to take into account the information presented in this paper.

Future work will deal with the lightning performance of such peculiar line configuration and an assessment of the impact of the NuL in the overall performance of the whole circuit.

APENDIX A – LINE PARAMETERS AND FORMULATION

OF NODAL ADMITTANCE

Consider a case of two overhead conductors over a homogeneous and uniform soil. Conductor i has radius ri and coordinates 0, xi, height hi and length s1 , conductor j has radius rj and coordinates xj1, xj2, height hj and length s2. Both lengths are across the x-axis, yij is the distance between conductors i and j along the y-axis. The expressions for the mutual impedance per unit length between conductor i and j is given by

2

1

0 1 2 1 2

2 2 2 20 1 2 1 2

4 (s s ) (s s )

ji

j

xx

ij

x ij ij

j ds dds ds s

dZ

S

ωμπ

−− +

= − +

(6)

where 2 2( )ij i j ijh hd y= − + , 2

22 sij i j ijh h y

jS

ρωμ

= + + +

,

the ground resistivity is sρ . For the mutual admittance per

unit length the expression is 1

02jω πε −=Y P (7)

where the matrix 0P has elements Pij given by

2

1

1 2 1 2

2 2 2 20 1 2 1 2

1

2 (s s ) (s s )

ji

j

xx

ij

x ij ij

ds ds

d S

ds dsP

= −

− + − +

. (8)

The analytical solutions for (6) and (7) ) [10] were used to determineboth Z and Y matrices. The characteristic admittance, Yc, and the propagation matrix, H, are obtained as follow

( )1 expc− ⋅ ⋅ − ⋅= =Y Z Z Y H Z Y (9)

where 1 2s s= = is the length of the uniform section. To

derive the line model, we must assume that all the conductors involved have the same length. Eigenvalues/eigenvector or Schur decompositions can be used to obtain the matrix square root in (8). Each finite length uniform line segment can be assembled in a nodal admittance matrix as show below

( )( ) ( )

( ) ( ) ( )

1 12 2 2

1 12 2 2

2

2

c c

n

c c

− −

− −

+ − − = −

−

− + −

Y I H I H Y H I HY

Y H I H Y I H I H (10)

where I is an identity matrix with the same dimensions as the number of conductors involved.

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