Earth-return path impedances of underground cables. Part 1: Numerical integration of infinite integrals

T.T. Nguyen

Indexing terms: Underground cables. Impeclcinces, Infinite integrals

Abstract: Although it is generally acknowledged that there is a need to closely formulate the earth- return path impedances of underground cables, research published so far has mainly concentrated on developing approximate formulations, especially for the infinite integrals which arise. Instead of seeking approximations, the paper develops a rigorous method based on direct numerical integration to evaluate the infinite integrals of underground cable earth-return path impedances. The method has immediate practical use in underground cable parameter calculations. It also provides a benchmark by which the accuracy of any proposed approximate method can be assessed. Further, the method can be used to provide training data by means of which infinite integrals can be evaluated using an array of neural networks.

List of principal symbols

co angular frequency pe earth resistivity

,uo free-space permeability (4x x 10-7H/m)

~ClOlPe distance between the conductor of phase cable j and that of phase cable k

4,<. distance between the conductor of phase cable; and the image of the conductor of phase cable k

hj, hk depth below the earth plane of cables J and k, respectively

horizontal separation between the conductors of cable; and of cable k

h, (hi + hk) /2

xjk

1 Introduction

It seems correct to say that parameter evaluation meth- ods for overhead transmission lines have been devel- oped to a greater extent than those for underground

0 IEE, 1998 IEE Proceedings online no. 19982353 Paper received 19 January 1998 The author is with the Energy Systems Centre, Department of Electrical and Electronic Engineering, The University of Westem Australia, Ned- lands, Perth, Westem Australia, 6907

IEE Proc.-Gener. Trunsni Distrib., Vol 145, No. 6, Novenibrr 1998

cable systems. A key part in the evaluation of parame- ters of both overhead transmission lines and under- ground cables is that of calculating the earth-return path contributions which are expressed in terms of infi- nite integrals [I , 21.

For both overhead lines and underground cables, the complete field system is that due to the actual conduc- tors and their images. For overhead lines, earth-path contributions derive from the images of the conductors because the images are in the body of the earth. In underground cables by comparison, earth-path contri- butions come from the conductors themselves. The images of underground cable conductors are in the air. A direct consequence is that the infinite integrals of overhead lines are nonoscillatory. This regular form allows approximate functions to be fitted to it [3]. For underground cables, on the other hand, the infinite integrals are always oscillatory. They do not lend them- selves to easy approximation.

The contributions of the earth path to underground cable parameters are significant. The earth-return path self impedances can compare up to about 90% of the total self impedances of individual cables. The mutual impedances between phases of underground cables are contributed solely by the earth paths. This is very dif- ferent from the case of overhead transmission lines where the mutual impedances between phase conduc- tors comprise components due to both the air paths and the earth paths. The mutual impedances can con- stitute about 70% of the self impedances of under- ground cables. It is therefore essential to evaluate accurately the earth-path contribution to the parame- ters of underground cables.

However, at present, approximations based on closed-form functions which are applicable for only a limited range of frequencies, earth resistivity and cable configuration are widely used [2, 4-61. There has even been a proposal to use the rapidly convergent series derived by Carson [ 11 for overhead transmission lines to calculate the infinite integrals for underground cables [5]. All of these approximate methods have low computing time requirements. However, the errors encountered in using them can be significant, particu- larly in the high frequency range.

Instead of seeking approximations, it is now feasible to evaluate the infinite integrals that arise directly, using numerical methods. In the past, the computing overheads of direct numerical integration may have been prohibitive. But nowadays, it is quite feasible. The purpose of the present paper is that of reporting recent work in which the infinite integrals of earth-return path

62 I

impedances of underground cables are evaluated directly, and the experience that has been gained in using this procedure. As far as the author is aware, there has only been a passing reference to this possibil- ity [ 2 ] .

In addition to its immediate practical use in under- ground cable parameter evaluations, direct numerical integration provides a highly accurate procedure which can be used as a benchmark against which to assess the numerous approximate methods that have been previ- ously proposed or others that might be derived. The procedure developed in this paper also provides train- ing data for a neural network approach of evaluating the infinite integrals [7].

2 Series-path impedances of underground cables

2. I The general case of an underground single-core cable with sheath and armour is shown in Fig. 1.

Self impedances of a single cable

core insulation 1 sheath insulation 2 armour

insulation 2

r6 Fig. 1 Cross-section of u cable with sheath und urmoui

I vc * - - - - -1 - -- I

V l l I I loop 1 I 1.1

4 loop 3

earth Fig.2 An elenieniul cable section

The starting point in forming the series-path parame- ters is to define three loops. These are the conductor- sheath loop, sheath-armour loop and armour-earth loop (Fig. 2). Loop parameters in the relationship

622

between loop voltages and loop currents are defined first. Transformation is then carried out to establish the final relationship between conductor-to-earth voltages and conductor currents.

Starting from the loop equations has the advantage that there is only one loop parameter involving the earth-return path impedance. After transforming from loop variables to conductor variables, each and every element of the series-path impedance matrix has a com- ponent that involves the earth-return path.

Using Fig. 2, the loop voltages in vector V, are formed from the differences between successive pairs of conductor-to-earth voltages in vector V N where

V L = C V V N (1)

('4 ( 3 )

In eqn. 1

vi, = (VI, v2, V 3 )

vt, = (VC, V " , V a )

c s a 1 1 -1

3 0 0 In eqns. 2 and 3 , superscripts 1 , 2 and 3 identify loops, and superscripts c, s and U denote core, sheath and armour, respectively.

The corresponding relationship between the loop cur- rents in vector I, and conductor currents in vector I N is

C , = 2 [' 1 3 (4)

IL = CIIN (5)

1; = ( 1 1 , 1 2 , 1 3 ) (6)

Ih = ( I C , I " , I " ) ( 7 )

In eqn. 5

c s a

C I A [: ; ':I (8) 3 1 1 1

In eqn 6, Z', I2 and Z3 are the currents in loops 1 , 2 and 3, respectively. In eqn. 7, IC, Is and I" are the cur- rents in the core, sheath and armour, respectively.

If Z, is the loop impedance matrix per unit length, the elemental loop voltage vector, AVL, for length Ax is

AVL = -zr,ILAx (9) Following transformation to conductor variables, eqn. 9 becomes

AVN = -ZNINAX (10)

(11)

In eqn. 10 Z N =c, 1 ZLCI

Eqn. 10 gives the relationship between conductor-to- earth voltages and conductor currents. This is the rela- tionship required in all power system analyses that involve underground cables. Matrix Z, is the matrix of series-path impedances of a single-core cable.

For the loop-impedance matrix, Z,, the general form for a single-core cable is

1 2 3 1 2 1 1 2 1 2 2 1 3

z L = ; [;; 2; h] (12)

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From Fig. 2, the components of the individual elements of Z L are summarised as (i) loop- 1 self impedance 2,

In eqn. 13 z1 = internal impedance of core z2 = impedance due to time-varying magnetic field in the insulation between core and sheath z3 = internal impedance of the inner surface of the tubular sheath (ii) loop-2 self impedance Z2,

In eqn. 14 z4 = internal impedance of the outer surface of the tubular sheath z5 = internal impedance of the inner surface of the tubular armour z6 = impedance due to time-varying magnetic field in the insulation between sheath and armour (iii) loop-3 self impedance Z3,

In eqn. 15 z7 = internal impedance of the outer surface of the tubular armour zg 5 impedance due to time-varying magnetic field in the insulation between armour and earth zg = self impedance of the earth-return path (iv) mutual impedance between loops 1 and 2

211 = 21 + 2% + 23 (13)

2 2 2 = 24 + 2 5 + 26 (14)

2 3 3 27 + z8 + 29 (15)

212 = 221 = 210

2 2 3 = 2 3 2 = 211

2 1 3 = 2 3 1 = 0 Because of the shielding effect of the sheath between the core and the armour, there is no magnetic coupling between loop 1 and loop 3.

have beeen derived and given elsewhere [2, 51. Impedances zl, z2, ..., zs and also zl0 and z l l are straightforward to evaluate. The expression for zg contains an infinite inte- gral [2].

In the loop impedance matrix in eqn. 12, only ele- ment Z,, contains earth-return path impedance. Fol- lowing the transformation to conductor variables, the series-path impedance matrix is given in eqn. 11. The general form of the series-path impedance matrix is

(16)

(17)

(18)

(v) mutual impedance between loops 2 and 3

(vi) mutual impedance between loops 1 and 3

Expressions for individual impedances zl-zI

c s a C 2,s z,,

a z a c 2 ,s z a a Z N = s [ 2 , s z s a ] (19)

By multiplying out the matrix products in eqn. 11 and using eqns. 12-18, elements of Z N are expressed in terms of impedances z1-zI1. Matrix Z N is the sum of two matrixes: one due to the conductor system of core, sheath and armour, and the other due to the earth path

Matrix ZNc in eqn. 20 is that which arises from the Z N = Z N C + Z N E (20)

IEE Proc -Gener. Transin Distrih , Val 145, No 6, November 1998

conductor system of core, sheath and armour. It has the general form

c s a

c 'Lc 2:s ' L a

a Z L ZL Z N C = s [zit 2Ls ZL.1 (21)

Matrix Z N E arises from the earth-return path and is given by

c s a c 29 29

Elements of Z N and Z N c are given in the following: (a) Core self impedance

z,, = 2Lc + zg (23) 8

z;c = zi + 2210 + 2211 (24) i= I

(b) Sheath self impedance

8

z:, = 2; + 2211

i=4

(c) Armour self impedance

2La = 27 + 2 8

(4 Mutual impedances between core and sheath

(29) z,, = z,, = 2LS + zg

8

Z;, = z:, 2, + ZIO + 2211 (30) 2 = 4

(e) Mutual impedances between core and armour

Z c a = Z a c = ZLa + 29

2LU = 2Lc = 27 + 2 8 + 211

Z,, = 2,s = ZLU + zg

ZLu = zhs = 27 f 28 + 211

(31)

(32)

(33)

(34)

(f) Mutual impedances between sheath and armour

The earth-return path impedance z9 contributes to each and every element of the series-path impedance matrix Z N in eqn. 19. The earth-path contribution to the total self impedances is significant; it can be up to about 90% of the total impedance.

2.2 Mutual impedances between cables The mutual loop impedance matrix Zf between the cable in phase j and the cable in phase k is given by

IC1 k2 k3

ZL, jk -!; -.? [ I 0" : ] (35) j 3 0 Z j k

In eqn. 35, j l , j 2 and j 3 identify loops 1, 2 and 3 of cable j , and k l , k2 and k3 identify loops 1, 2 and 3 of cable k, respectively.

Due to the shielding effect of the sheath and armour, there is mutual magnetic coupling between the outer-

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most loops (i.e. loop 3) only of cables j and k. The cou- pling is entirely via the earth path. In eqn. 35, z;k is the mutual earth-return path impedance between cables j and k. Like z9, the expression of .?;k also contains an infinite integral [2].

Transforming from loop variables to conductor vari- ables gives the mutual impedance matrix in the series path, Z$ , between cables j and k

z$ = C,lZ;kCr ( 3 6 ) or

In eqn. 37, j c , js and j u identify core, sheath and armour of cable j , and kc, ks and ka identify core, sheath and armour of cable k, respectively. Matrix Z$ is a full matrix each element of which is equal to

Unlike the case of overhead transmission lines, the mutual impedances of underground cables are totally contributed by the earth path only. In the case of over- head lines, the mutual impedances between phase con- ductors are the sum of external mutual impedances due to magnetic fields in the air paths and those arising from earth paths. Therefore, accurate evaluation of the mutual earth-return path impedances of underground cables is essential in representing the mutual magnetic coupling between cables. This is more so for under- ground cables than for overhead lines. In terms of magnitude, the earth-return path mutual impedances of underground cables can be in the range 35 - 70% of the self impedances, depending on frequency.

' j k .

2.3 Complete series-path impedance matrix The complete series-path impedance matrix of any underground cable system comprises submatrixes of the form given in eqns. 19 and 37.

3 cables

Earth-return path impedances of underground

In the derivation of the series-path impedance matrix for underground cables, separate symbols have been used to distinguish between the earth-return path self impedance and mutual impedance. They are z9 and z;k.

In the following general development, the notation zjk is used to denote both the mutual impedance and self impedance. For self-impedance, j = k.

The earth-return path mutual impedance associated with cablesj and k is given in [2]

where

J o L J

x exp { -2h,&/=} cos (xjk&u) du

(39) 624

In eqns. 38 and 39 KO is the modified complex Bessel function of the sec- ond kind with zero order chk = distance between cable j and cable k as shown in Fig. 3 dik3 = distance between cable j and the image of cable k as shown in Fig. 3 .xjk = horizontal distance between cable j and cable k as shown in Fig. 3

hj + h k 12, = ~

2 where hi and hk are the depths below the earth plane of cables j and k, respectively, and

WPO

P e

a = -

In eqn. 41, w is the angular frequency, pc is the earth resistivity and po = 4n x 10-7H/m.

* 'jk

Fig. 3 System of two unduground ccihles

The last term on the RHS of eqn. 38 has been re- arranged so that the integral J( j , k ) in eqn. 39 is finite and tends to zero when a tends to zero, i.e. when the angular frequency w tends to zero. This re-arrangement helps to evaluate JG, k ) numerically. To evaluate the self impedance, hi is set equal to I lk , and x;k is the outer radius of the cable.

4

The Bessel functions Ko(d,,~dba]) and K0(4kdba]) in eqn. 38 can be evaluated straightforwardly, using a standard library function. The key aspect of earth- return path impedance evaluation is to calculate the infinite integral J( j , k ) in eqn. 39, where the exponent in the exponential function in the integrand of JU, k ) is complex and is given by -2h,,,dd[u2 + 11.

In the case of an overhead transmission line, the exponent is real and equal to -211,,~dau. This is the key difference between the infinite integral for the under- ground cable and that for the overhead transmission line. In consequence, the real and imaginary parts of the transmission line infinite integral are monotonic and nonoscillatory functions [ l ] and those of the underground cable infinite integral are oscillatory func- tions of h, da for a given xik/hIll ratio. Fig. 4 shows a typical oscillatory variation 'of the real part of J( j , k ) with h,,Ya when xiklhIl7 = 0.3. The second property of the infinite integrd is that it approaches zero for large values of hnl\/a, irrespective of xjkda. This is confirmed

Properties of the infinite integral

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by very many numerical evaluations of the infinite inte- gral for a wide range of Iz,,,da and xj,<da.

o.2 1

h,

Representative variation o j the real part ojthe infinite integral Fig. 4

The monotonic and nonoscillatory infinite integral for the transmission line has led to the derivation of infinite series which converge rapidly to represent the integral [l]. The rapidly convergent series allows an accurate numerical evaluation of the infinite integral for the overhead transmission line by retaining only a few terms of the series. In the case of oscillatory infi- nite integrals for underground cables, such rapidly con- vergent series are not available.

5

When a tends to zero, the infinite integral JG, k ) in eqn. 39 tends to zero. There is, therefore, no need to evaluate J(j , k) at a = 0 (i.e. at zero frequency). When a z 0, the upper limit of the integral in eqn. 39 can be replaced with a finite value on the basis of a rapidly decreasing value of the exponential term for an increas- ing value of U. For example, when u = IOi(lz,da), the magnitude of the exponential term is less than about 2

By using the trapezoidal rule of integration, the infi- nite integral in the continuous form of eqn. 39 is replaced by the following recursive sequence:

Numerical integration of the infinite integral

10-9.

n = 1 , 2 , . . . , N (42)

J o b , I C ) = 0 (43) In eqn. 42, n denotes the step in the numerical integra- tion, N is the upper limit of n, Au is the step length and flu) is the integrand of the infinite integral in eqn. 39

f ( u ) = h,& [ d m - U ]

(44)

The upper limit in u is

The recursive sequence in eqn. 42 leads to the summa- tion form

U,,, = NAu (45)

Approximately

Extensive investigations with different step lengths and upper limits have led to the following recommenda- tions: (i) step length Au = and upper limit = 101 (/zmdu) when 0 < hmdu 5 0.5; (ii) step length Au = 142 x lo3 hlnda) and upper limit = 10/(h,,2da) when k,du > 0.5.

Evaluation by numerical integration gives accurate solutions. However, the computing time requirement is substantial. The number of samples, N , in the summa- tion in eqn. 46 can be more than 2 x lo4 on the basis of the scheme in (i) and (ii). Evaluation of the infinite integrals in the earth-return path self and mutual impedances of a three-phase underground cable for 600 frequency points requires about 2 h of computing time on a SUN SPARC workstation.

Different numerical integration methods such as Simpson’s rule and Romberg’s rule have been tried with the aim of increasing step length and achieving the same accuracy. However, the improvement is not sub- stantial, and there are cases where Simpson’s rule gives an inferior performance to that offered by the trapezoi- dal rule.

J ( j , I C ) 2 J N ( j , (47)

6 Comparison study

In the study, the results of the rigorous procedure based on direct numerical integration are used to assess the accuracy of those obtained from a widely used approximation using a closed-form function [2]. In this approximation, the earth-return path mutual imped- ance is given in [2]

In eqn. 48, d jk is the distance between cable j and cable k as shown in Fig. 3 and y is Euler’s constant.

Due to its simplicity, the approximation in eqn. 48 is widely used. On the basis of numerical integration that offers accurate results, it is now proposed to quantify the errors arising from using eqn. 48. For the purpose of evaluation, a three-phase cable is used. It is l m below the ground. Individual phase cables are in horizontal formation, and have a separation of 0.5m.

Table 1: Earth-return path mutual impedances

Numerical integration Approximation based on eqn. 48

Frequency, Hz Impedance, Q/m Impedance, Q/m

50.0 0.500 x + j0.335 x 0.499 x + j0.335 x

1000.0 0.103 x IO-* + j0.478 x IO-* 0.103 x IO-* + j0.478 x IO-* 10000.0 0.0108 + j0.0324 0.01 14 + j0.0323 100000.0 0.108 + j0.158 0.146 + j0.146

1000000.0 0.576 + 10.145 2.476 -j1.002 ~~ ~~

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Table 1 compares the earth-return path mutual impedances between two outer phase cables evaluated by numerical integration and those calculated using the expression in eqn. 48 for an earth resistivity of 5Qm.

At low frequency, the approximation is accurate. However, substantial errors occur when the frequency increases. At 105Hz, the error in resistance is about 35%. At lMHz, the results from the approximate method are obviously unacceptable. Thus the approxi- mation based on eqn. 48 is of limited application.

7 Conclusions

Cable parameters are usually evaluated once, as each cable is encountered, and held in a database for subse- quent use in any of the areas of analysis and design which require underground cable parameters. The direct numerical integration procedure developed in this paper avoids the approximations previously pro- posed and the uncertainties intrinsic to them. In the past, it was felt that the computing time overheads of direct integration were prohibitive. That may well have been the case then, but it is now feasible to use the method developed in this paper for routine evaluations of underground cable parameters.

8 Acknowledgments

The author wishes to thank the University of Western Australia for permission to publish this paper. He also wishes to thank Professor W.D. Humpage for discus- sions relating to the developments of the paper, and for his many contributions to its preparation.

References

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SAAD, O., GABA, G., and GIROUX, M.: ‘A closed-form approximation for ground return impedance of underground cables’, IEEE Trans., 1996, PWD-11, ( 3 ) , pp. 1536-1545 DOMM EL, H.W.: ‘EMTP Reference Manual’ (Bonneville Power Administration, 1986), vol. 3 SEMLYEN, A.: ‘Discussion on ‘Overhead line parameters from handbook formulas and computer programs’ by Dommel, H.W., IEEE Trans., 1985, PAS-104, (2), pp. 366-372 NGUYEN, T.T.: ‘Earth-return path impedances of underground cables. Part 2: evaluations using neural networks’, Proc IEE., pp.

1981, PAS-100, (8), pp. 3686-3693

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