6th International Symposium on

SHORT-CIRCUIT CURRENTS IN POWER SYSTEMS

LIGE (BELGIUM)

6.-8. SEPTEMBER 1994

SHORT-CIRCUIT CURRENTS CALCULATION OF EFFECTS THE SECOND EDITION OF IEC PUBLICATION 865

Wolfgang Meyer Gerhard Herold Elmar Zeitler

University of Erlangen-Nrnberg Federal Republic of Germany

Report 2.1

2.1.1

SHORT-CIRCUIT CURRENTS CALCULATION OF EFFECTS THE SECOND EDITION OF IEC PUBLICATION 865

Wolfgang Meyer Gerhard Herold Elmar Zeitler

University of Erlangen-Nrnberg Federal Republic of Germany

Abstract In 1993 the second edition of the IEC Publication 865-1 Short-circuit currents Calculation of effects was pub-lished. In contrast to the first edition, the new standard is valid for nominal voltages up to 420 kV. The methods for calculation of the tensile forces in flexible conductors is completely revised and now applicable on slack conduc-tors as well as on strained conductors which are connected to portals with a span length of up to 60 m. This paper gives a short description of the new standard and derives and explains the equations for calculating the tensile for-ces in flexible conductors and the swing out. The compa-rison with test results is shown.

INTRODUCTION

In 1986 the Technical Committee 73 of the International Electrotechnical Commission IEC published the first in-ternational standard Calculation of the effects of short-circuit currents [1]. It was only valid for rated voltages up to and including 72,5 kV and an extension to higher voltages should be done if simple methods are available for slack and strained flexible conductors. The investiga-tions have been done in Study Committee 23 Effects of high currents Working Group 02 Substations of the In-ternational Conference on Large High Voltage Electric Systems CIGR and published along with IEC TC 73 [2]. Immediately after the printing of the first edition, TC 73 decided to extend its work on higher voltages. In Septem-ber 1993 the second edition of IEC Publication 865-1 Short-circuit currents Calculation of effects [3] came out after intense preparation and with the agreement of 19 National Committees. In parallel voting the members of the European Committee for Electrotechnical Standardi-zation CENELEC decided to publish it as European Stan-dard EN 60865-1:1994.

IEC PUBLICATION 865-1 (1993)

The IEC Publication 865-1 (1993) is clearly organized Section 1 - General Section 2 - The electromagnetic effect on rigid conductors

and flexible conductors Section 3 - The thermal effect on bare conductors and

electrical equipment Annex A - Equations for calculation of diagrams Annex B - Iteration procedure for calculation of factor It applies to a.c. single-phase and three-phase systems for rated voltages up to and including 420 kV.

Section 1 contains the normative references, the lists of symbols and units, and the definitions for Sections 2 and 3.

In Section 2, the method for calculating the forces and stresses in arrangements with rigid conductors was taken over from IEC 865 (1986) with some clarifications. De-tailed information is given in [2].

In the case of flexible conductors, IEC 865 (1986) was re-stricted to the calculation of swing-out and fall-of-span tensile forces in single conductors of up to 20 m span length which are connected to support insulators without consideration of the short-circuit duration. In high-voltage arrangements, the flexible conductors are often connected to portals with tension insolator strings and bundle conductors are predominant. Besides this, the conductor movement is of interest because minimum clearances to the neighbouring phase conductors and earthed parts of the structure have to be taken into consideration. The short-circuit duration has a great influence on the forces.

On the basis of extensive tests the methods known from literature [2, 4-13] have been developed further for calcu-lating the swing-out and fall-of-span tensile forces and the horizontal displacements [13, 14] and a new method for estimation of the tensile force due to the contraction of the bundle conductors is introduced [15-17].

Section 3 was taken over from IEC 865 (1986) with some clarifications.

In the following, the methods for calculation of the forces and conductor displacement in arrangements with flexible conductors are derived. References to clauses, equations and figures of IEC 865-1 [3] are marked by an asterix (*).

CONDUCTOR MOVEMENT AND TENSILE FORCES

During or after line-to-line or three-phase short circuits, two main conductors swing away from each other, for ex-ample L1 and L2. Figure 1 [2] shows three typical con-ductor movements. Thereby the maximum swing-out an-gle m is the decisive parameter: m < 70: The span is displaced until the first reversal

point is reached at m and then returns to the steady-state position with damped oscillations. During or at the end of the short circuit, the short-circuit tensile force Ft is at its maximum and the sag bct at m.

70 m < 180: The span is displaced and drops down from the position indicated by m in the direction of the suspension points. At the end of the first fall, the drop force Ff is at its maximum. The sag reaches its maxi-mum at the time tt.

180 m: The electromagnetic forces accelerate the span so much, that it rotates once or several times, until the stored energy is exhausted. At the bottom of the curves, tensile-force peaks of approximately the same order as the maximum drop force Ff may occure at approximately equal intervals corresponding to the natural frequency of the span. The sag reaches its maximum at the time tt.

The stresses occuring in line-to-line short-circuits and balanced three-phase short-circuits are approximately equal. However, for line-to-line short-circuits, conductor swing out typically results in decreasing minimum clear-

2.1

2.1.2

ances when the adjacent conductors carrying short-circuit current move towards one another after the short circuit. In the case of a balanced three-phase short-circuit, the center conductor moves only slightly because of its inertia and the alternating bidirectional forces acting on it. Con-sequently Ft, Ff and bh are calculated for a line-to-line short circuit. In the case of short circuits far from genera-tor, = I Ik2 k33 2/ [18]. In contrast, Fpi is to be calculated in a single-phase system for the line-to-line short-circuit current Ik2 and in a three-phase system for the three-phase short-circuit current Ik3.

Figure 1 Short-circuit location curves in the middle of the span for two adjacent phases in the case of a line-to-line short circuit in L1 and L2 [2]

left side: movement of the conductor right side: surface area required

locus during current flow locus after current flow

MAXIMUM SWING-OUT ANGLE m In the following, the sag of the conductor is assumed to be a parabola and the shape remains in a plane during swing out, Figure 2. The angle between the plane and the verti-cal axis is the swing-out angle (t). The longitudinal and transversal conductor waves are excluded.

Each main conductor consists of n sub-conductors with the mass per unit length ms, the cross section As, and the actual Young's modulus Es. The spring constant of both supports S and the Young's modulus Es are subsumed un-der the stiffness norm

NSl nE A

= +1 1s s

(*25)

Figure 1 Spans L1, L2 side-by-side during a line-to-line short circuit

For calculation of the maximum swing-out angle ###m the movement of the span is described by the nonlinear physi-cal pendulum, which obeys the differential equation

J mg s M t sin ( ) + =n (1) where m dm nm l= s mass s

y x mm

b= ( ) d 23 c

distance to center of gravity

J y m nm lb= 2 815

d s c2 moment of inertia

M F x y x x= ( ) ( ) cos d exciting moment The approximations apply to usual spans where bc l for the static sag holds. The period of conductor oscillation for small angles ### and constant sag bc is given by

TJ

mg sbg

= 2 2 0 8 n

c

n, (*23)

The double integration of (1) can only be done numeri-cally. Because the period of oscillation of the span is long compared with the period of the short-circuit current, the time history of the current can be substituted by the initial short-circuit current, whereby the force becomes inde-pendent of time. If the moment is calculated by a substi-tute force on the center of gravity averaged over the swing-out angle

( )M F m n k ls= +~ ~ cosd , (2) (1) can be integrated analytically twice. F is the constant force per unit length caused by the initial short-circuit current Ik2 :

( ) = F I

a

02

2k2 (3)

After the breaking of the current at = k, the moment M is zero. The factors ~m und ~n from [3], here marked by a tilde to distinguish them from other variables used, con-sider the a.c. and d.c. components of the short-circuit cur-rent and kd the averaged influence of the angle . Integration of (1) with d /d 2 2= leads to: for 0 k with the initial values = = 0

( ) sin cos = + 8 122T rres (4)

2.1.3

for k < with the initial values = k and = k

( ) cos cos = +8 22T k k2 (5) where rres is the ratio of electromagnetic force per unit length to gravitational force per unit length = G nm gs n

( )

rF m n k

Gersd= +

~ ~ (6)

For the second integration of (4), the following substitu-tions are inserted to calculate the angle k at the end of the short-circuit current:

* , sin * , sin= = =1 11

2 2kk

by which the span swings around the steady state position 1 = arctan rers:

( )( ) = + 8 1 1 122 2 2 2 2T r kers Separation of variables gives an elliptic integral of the first kind; the result is the Jacobis elliptic sinus amplitu-dinis function sn [19]. Taking the initial value * = 1, one gets after substituting back:

= +

1 2 2 2arcsin sn ,k

t

Tk

pend

For the usual range 0 < rres < 10 a simple approximation can be given, which is derived from the linearization of equation (1) for small angles and contains the amplitude-dependent periodicity of the sn-function by approxima-tion:

( ) cost tT

=

1 1 2 res

with the resulting period during short-circuit current flow:

TT T

rres

pend

res

=

=+

1 64 90 1 1 64 90

21

224

21

2 (*24)

Therefore, the angle at the end of the short-circuit current flow is with t = Tk1:

k

k1

res

k1

res

1k1

res

for

2 for

=

0,5, the maximum value of k for Tk1/Tres = 0,5 is to be inserted, because the actual short-circuit duration Tk can be lower than the short-circuit du-ration Tk1 given by the protection concept, and the worst case shall be considered for design purposes.

The maximum swing-out angle is obtained as follows: In the reversal point of the movement = m it is = 0 in

(5). Inserting (4) with = k leads to the maximum swing-out angle, which is only dependent on rers and k: ( ) m ers k= =arccos sin arccos1 r (7) The course of m(k) reaches its maximum at k = 90 and then decreases. Because Tk Tk1, the worst case k = 90 is also decisive. So it follows that:

=