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6th International Symposium on

SHORT-CIRCUIT CURRENTS IN POWER SYSTEMS LIÈGE (BELGIUM) 6.-8. SEPTEMBER 1994

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

Gerhard Herold

University of Erlangen-Nürnberg Federal Republic of Germany

Report 2.1

Elmar Zeitler

2.1 SHORT-CIRCUIT CURRENTS – CALCULATION OF EFFECTS THE SECOND EDITION OF IEC PUBLICATION 865 Wolfgang Meyer Gerhard Herold Elmar Zeitler University of Erlangen-Nürnberg Federal Republic of Germany In the case of flexible conductors, IEC 865 (1986) was restricted 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.

Abstract In 1993 the second edition of the IEC Publication 865-1 „Short-circuit currents – Calculation of effects” was published. 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 conductors 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 forces in flexible conductors and the swing out. The comparison with test results is shown.

On the basis of extensive tests the methods known from literature [2, 4-13] have been developed further for calculating 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].

INTRODUCTION In 1986 the Technical Committee 73 of the International Electrotechnical Commission IEC published the first international standard „Calculation of the effects of shortcircuit 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 investigations have been done in Study Committee 23 „Effects of high currents” Working Group 02 „Substations” of the International Conference on Large High Voltage Electric Systems CIGRÉ 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 September 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 Standardization CENELEC decided to publish it as European Standard EN 60865-1:1994.

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

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

During or after line-to-line or three-phase short circuits, two main conductors swing away from each other, for example L1 and L2. Figure 1 [2] shows three typical conductor movements. Thereby the maximum swing-out angle δm is the decisive parameter: – δm < 70°: The span is displaced until the first reversal point is reached at δm and then returns to the steadystate 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 δm. – 70º ≤ δm < 180°: The span is displaced and drops down from the position indicated by δm 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 maximum at the time tt. – 180° ≤ δm: 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.

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. Detailed information is given in [2].

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-

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 η It applies to a.c. single-phase and three-phase systems for rated voltages up to and including 420 kV.

2.1.1

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. Consequently Ft, Ff and bh are calculated for a line-to-line short circuit. In the case of short circuits far from genera′′ = 3/2 I k3 ′′ [18]. In contrast, Fpi is to be calculated tor, I k2 in a single-phase system for the line-to-line short-circuit current I k2 ′′ and in a three-phase system for the three′′ . phase short-circuit current I k3 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 physical pendulum, which obeys the differential equation Jδ + mg s sin δ = M (t ) (1) n

where m = ∫ dm ≈ nms′l

∫ y( x) d m

s=

m

2 b 3 c

distance to center of gravity

8

∫ y d m ≈ 15 nms′ lbc = ∫ F ′( x ) y ( x ) cos δ d x

J = M

≈

mass

2

2

moment of inertia 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 b J ≈ 2π 0,8 c mgn s gn

T = 2π

The double integration of (1) can only be done numerically. 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 independent of time. If the moment is calculated by a substitute force on the center of gravity averaged over the swing-out angle ~ + n~ ) k ls cosδ , (2) M = F ′( m

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

d

(1) can be integrated analytically twice. F′ is the constant force per unit length caused by the initial short-circuit current I k2 ′′ :

MAXIMUM SWING-OUT ANGLE δm

F′ =

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 vertical axis is the swing-out angle δ (t). The longitudinal and transversal conductor waves are excluded.

1 1 + Sl nEs As

′′ ) µ 0 ( I k2

2π

2

a

(3)

After the breaking of the current at δ = δk, the moment M ~ und n~ from [3], here marked by a is zero. The factors m tilde to distinguish them from other variables used, consider the a.c. and d.c. components of the short-circuit current and kd the averaged influence of the angle δ. Integration of (1) with d δ2 /d δ = 2δ leads to: − for 0 ≤ δ ≤ δk with the initial values δ = δ = 0

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 under the stiffness norm N =

(*23)

δ =

8π T

(*25)

2.1.2

2

2

( rres sin δ + cosδ − 1)

(4)

− for δk < δ with the initial values δ = δ k and δ = δk 8π

δ =

T

2

2

( cosδ − cosδ k ) + δk2

(5). Inserting (4) with δ = δk leads to the maximum swingout angle, which is only dependent on rers and δk:

δ m = arccos (1 − rers sin δ k ) = arccos χ

(5)

The course of δm(δk) reaches its maximum at δ k = 90° and then decreases. Because Tk ≤ Tk1, the worst case δ k = 90° is also decisive. So it follows that:

where rres is the ratio of electromagnetic force per unit length to gravitational force per unit length G′ = nms′ gn rers =

~ + n~ ) k F ′( m d

(6)

G′

ξ =

1 k

sin

δ*

k = sin

,

8π T

2 2

1 + rers

2

2

( 1 − k 2ξ 2 )(1 − ξ 2 )

Separation of variables gives an elliptic integral of the first kind; the result is the Jacobis elliptic sinus amplitudinis function sn [19]. Taking the initial value δ * = −δ 1, one gets after substituting back:

π ⎛ δ1 ⎞ 1− ⎜ ⎟ 64 ⎝ 90° ⎠

2

=

radial

(*24)

tangential

for 0 ≤

Tk1 Tres

for 0,5 <

≤ 0,5

Tk1

for

-0,985 ≤ χ < 0,766

for

χ < -0,985

(*31)

R′ = F ′ sin δ + G ′ cos δ + m′yδ2 nms′ yδ = F ′ cos δ − G ′ sin δ

R′ is the radial force per unit length, which changes very slowly compared with the eigenvalue of the conductor as a swinging cord. The integration of the second equation with the initial value δ(t = 0) = 0 and insertion in to the first one leads to

Therefore, the angle at the end of the short-circuit current flow is with t = Tk1: ⎧ ⎡ ⎛ ⎞⎤ ⎪δ ⎢1 − cos⎜ 2π Tk1 ⎟ ⎥ 1 ⎜ ⎟ ⎪⎪ ⎢⎣ ⎝ Tres ⎠ ⎥⎦ δk = ⎨ ⎪ ⎪ 2δ 1 ⎪⎩

0,766 ≤ χ ≤ 1

Figure 3 shows the sectional view of a span. The forces per unit length are acting on a conductor element:

T ⎛ π 2 ⎛ δ ⎞2 ⎞ 1 4 1 + r 2 ⎜1 − ⎜ ⎟ ⎟⎟ res ⎜ ⎝ 64 ⎝ 90° ⎠ ⎠

for

SHORT-CIRCUIT TENSILE FORCE Ft

with the resulting period during short-circuit current flow: 2

(*30)

For short-circuit durations greater than 0,1 s the influence ~ can be disregarded. of the d.c. component is low and m ~ For distribution networks usually n = 1. The calculation of factor kd is not easy. As 0,9 < kd < 1 applies for usual spans, kd = 1 can be used [2, Figure 4.2.6]. Accordingly, (6) becomes: F′ rers ≈ r = (*20) G′

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 amplitudedependent periodicity of the sn-function by approximation: ⎡ ⎛ t ⎞⎤ ⎟⎟ ⎥ δ ( t ) = δ 1 ⎢1 − cos⎜⎜ 2π ⎢⎣ ⎝ Tres ⎠ ⎥⎦

Tpend

for 90°< δ k

⎧1,25arccos χ ⎪ δ m = ⎨10°+ arccos χ ⎪ ⎩180°

⎡ ⎛ ⎞⎤ t π ⎢ ⎜ δ = δ 1 + 2 arcsin k sn ⎜ 2 π − , k ⎟⎥ ⎢ T 2 ⎟⎠⎥⎦ pend ⎝ ⎣

Tres =

⎩1 − rers

A comparison with test results [11, 14] shows that δm is too low when calculated by (7). This is due to the simple model of a constant force per unit length in (2) on the one hand, and the pendulum movement superposed by longitudinal oscillations of the conductor on the other hand. Therefore, the following corrections are made in (7): Small swing-out angles are adjusted by a constant factor of 1,25 and large angles with the addition of 10°, which gives continuation at δm = 50° and 180°. Hence

by which the span swings around the steady state position δ1 = arctan rers :

ξ =

for 0 ≤ δ k ≤ 90°

The value δm =180° is obtained at χ = −1. If χ < −1, the span rotates.

δ1

2

⎧1 − rers sin δ k

χ =⎨

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

δ * = δ − δ1 ,

(7)

(*29)

Tres

When Tk1/Tres > 0,5, the maximum value of δk for Tk1/Tres = 0,5 is to be inserted, because the actual shortcircuit duration Tk can be lower than the short-circuit duration 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 δ = δm it is δ = 0 in

Figure 3 Swing out of the conductor in midspan and forces during current flow

2.1.3

R′ = 3r sin δ + 3 cos δ − 2 G′

(8)

R′ reaches its maximum at

δ1 = arctan r

Figure 4 Conductor movement in midspan for calculating the tensile force Ff

(*21)

if δk ≥ δ1, i.e. Tk1 ≥ Tres/4, otherwise at δ = δk, if Tk1 < Tres/4. The change in conductor length by increasing the radial force from Rst′ = G′ to Rt′ = G ′( 1 + ϕ ) becomes with the change-of-state equation for a conductor: lt − lst = Nl ( Ft − Fst )

The upper limit follows with δf = 0: Ff = Fst 1 + 4 ζ (1 − cos δ m )

(9)

and with the approximation 1 - cos δm ≈ (2/π) δm:

The approximate lengths of the conductor with ν = t,st 3 ⎛ R′ l ⎞ l ⎛ Rν′ ⎜⎜ lν = 2 sinh ⎜⎜ ν ⎟⎟ ≈ l + G′ 24 ⎝ Fν ⎝ Fν 2 ⎠

Fν

Ff = Fst 1 + 8ζ

⎞2 ⎟⎟ ⎠

(

)

Ft = Fst ( 1 + ϕψ )

Ff = 1,2 ⋅ Fst 1 + 8ζ m 180°

(*34)

When 0,6 < r < 2, the approximation

⎧ ⎛ ⎪ 3⎜ 1 + r 2 − 1⎞⎟ ⎠ ϕ= +1= ⎨ ⎝ G′ ⎪⎩ 3( r sin δ + cosδ − 1) Rmax ′

for δ k ≥ δ 1 for δ k < δ 1

δ m = arccos (1 − r sin δ k ) ≈ arccos (1 − r ) applies, which with (11) gives us (22) in IEC 865 (1986); δm=180° in (12) results in (23) there.

(*32)

HORIZONTAL SPAN DISPLACEMENT

( n ms′lgn ) 2 =

Into the change-of-state equation in the case of elastic and thermal elongation

(*28)

24 Fst3 N

2 2 ⎡ ⎛ G ′ ⎞2 ⎤ l ⎢⎛ Rt′ ⎞ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥ = ε ela + ε th 24 ⎢⎣⎝ Ft ⎠ ⎝ Fst ⎠ ⎥⎦

(10) becomes

ϕ ψ + ϕ ( 2 + ζ )ψ + ( 1 + 2 ζ )ψ − ζ ( 2 + ϕ ) = 0 2

3

2

(*33)

with which ψ can be determined by 0 ≤ ψ ≤ 1 or taken from Figure *7.

the sags in the midspan 2 2 G ′l R ′l , bc = bct = t 8 Fst 8 Ft are inserted and solved for bct:

DROP FORCE Ff At the highest point of the conductor movement δm according to Figure 4, the maximum potential energy is stored: Epot = (1 − cos δ m ) nms′lgn s

2 3⎛ l ⎞ bct = bc 1 + ⎜⎜ ⎟⎟ ε ela + ε th = CD bc 8 ⎝ bc ⎠

(

During the fall the energy is partially or completely converted into elongation energy depending on height and movement of fall: 1 2 2 E ela = Nl Ff − Fst 2

(

(*35)

The drop force only needs to be calculated if δm ≥ 70°, which occures if r > 0,6.

ϕ follows from (8)

ζ

(12)

180°

δ

(10)

Hence the short-circuit tensile force Ft is estimated as:

With the stress factor

δm

In general the velocity and hence the potential energy of the conductor are not zero and drop forces are observed in tests which are greater than those calculated with (12). This can be taken into considered by a factor of 1,2:

are inserted in (9) 2 3 ⎡ ⎛ R ′ ⎞2 ⎤ l ⎢⎛ Rt′ ⎞ ⎜⎜ ⎟⎟ − ⎜⎜ st ⎟⎟ ⎥ = Nl Ft − Fst ⎢ 24 ⎣⎝ Ft ⎠ ⎝ Fst ⎠ ⎥⎦

(11)

ε ela = N ( Ft − Fst )

(*36)

and the thermal elongation due to the heating by the shortcircuit current

From the height of fall one obtains the balance of the complete energy conversion:

(

(11)

CD corresponds to (*38). The elastic elongation is generated due to the change of the tensile force from Fst to Ft

)

( cos δ f − cos δ m ) nms′ lgn s = 21 Nl Ff2 − Fst2

)

ε th =

)

2 α th ⎛ I k2 ′′ ⎞

2 ⎛ I k2 ′′ ⎞ ⎜ ⎟ T = cth ⎜ ⎟ T ⎝ A ⎠ κcρ ⎝ A ⎠

(*37)

with the material constant cth according sub-clause *2.3.2.4. The maximum sag bct for Tk1 ≥ Tres/4 is reached at the angle δ1 and for Tk1 < Tres/4 at the end of the current flow at δk . Thus T is:

and from this: Ff = Fst 1 + 4ζ ( cos δ f − cos δ m )

2.1.4

⎧ Tres / 4 T=⎨ ⎩ Tk1

for Tk1 ≥ Tres / 4 for Tk1 < Tres / 4

THE PINCH EFFECT OF BUNDLED CONDUCTORS (*37)

Figure 5 shows a contracted regular twin-bundle. Regular, here, means that the position of the cross sectional centres of the n sub-conductors of the bundle are marked by the corners of a regular polygon.

Because the shape already differs from the parabola with low values of r and approaches a triangle with high values of r, the actual sag is higher than that estimated with (13). The correction is made using the form factor CF with linear interpolation in the range 0,8 < r < 1,8: ⎧1,05 ⎪ CF = ⎨0,97 + 0,1 r ⎪ , ⎩115

Sub-conductors are considered to clash effectively if the clearance as between the midpoints of adjacent sub-conductors, as well as the distance ls between two adjacent spacers fulfill either (*43) or (*44)

for r ≤ 0,8 for 0,8 < r < 0,8

(*39)

as/ds ≤ 2,0 and ls ≥ 50 as as/ds ≤ 2,5 and ls ≥ 70 as

for r ≥ 1,8

A calculation of the bundle force then does not have to be done; in this case it is sufficient to add 10 % to the tensile force Ft in (*34) [20].

CF is derived from tests, because it is only possible numerically to get the form factor for each span. The maximum horizontal displacement is the projection of the sag onto the horizontal axis − with slack conductors: for δ m ≥ 90°

⎧ CF CDbc bh = ⎨ ⎩ CF CDbc sin δ m

for δ m < 90°

The calculations for contraction of the bundle described in the literature up to now can only be solved numerically. Hence, the parabola model is introduced below, based on equations and diagrams which can be solved analytically for the most occurring cases of clashing sub-conductors and by diagrams or numerically for non-clashing subconductors.

(*40)

− with strained conductors: ⎧ CF CDbc sin δ 1 bh = ⎨ ⎩ CF CDbc sin δ m

for δ m ≥ δ 1 for δ m < δ 1

Physical model for regular n-conductor bundles The sub-conductors touch each other in the range 2xP - ls < x < 0. When considering the bundle in its static steady state of contraction and assuming the shape of the curve of the non-clashing section a parabola, the equation for the first quadrant is: ⎛ x ⎞2 y ( x ) = yA + yP ⎜⎜ ⎟⎟ (14) ⎝ xP ⎠ The flexural stiffness of the conductor which acts above all near the spacers is ignored, so that for the ratio of the transversal and the longitudinal component of the tensile force between two adjacent spacers the following holds

(*41)

because they swing out almost horizontally [13], so that bh is a maximum at about δ1. EXTENSION ON STRAINED CONDUCTORS The methods written above are only valid for slack conductors, which are connected to support insulators, and hence represent a homogenious arrangement. In long spans, the mass of the insulator strings is a multiple of the conductor mass and is essentially decisive for the sag. The period T according to (*23) contains only the sag. To take the insulator strings into account, an equivalent static conductor sag at midspan is defined: bc =

G′l

Fy ( x) dy y x = =2 P dx Fx xP xP

2

which is – for slack conductors, the actual static sag, – for strained conductors, the sag, when the actual arrangement is replaced by an arrangement of conductors with the same conductor type and static tension, but without insolator strings. As the electromagnetic force acts only on the conductor, the difference between the conductor length lc and the span length l is to be taken into account in (3): F′ =

where lc = l lc = l − 2li

( )

′′ µ 0 I k2 2π

a

2

lc l

(15)

The longitudinal component Fx is local independent. On the other hand, the transversal component per unit length Fy′ is local independent with respect to (15)

(*22)

8 Fst

(*43) (*44)

2

Fx

d y dx

2

= Fy′ ( x ) = 2

yP 2

xP

Fx

(16)

At the location xP of the spacer, (15) becomes Fy ( xP ) = 2

yP F xP x

(17)

and according to (16) yields Fy ( xP ) = xP Fy′ ( xP )

(*19)

(18)

for slack conductors, for strained conductors, where li is the length of one insulator chain.

With both assumptions a good concurrence is achived with test results [13]. Figure 5 Idealized geometry of a clashed twin-bundle 2.1.5

Fy′ ( xP ) is the electromagnetic force per unit length acting

Electromagnetic force To determine Fy′( x P ) , the electromagnetic force on a parabolic arc of the length xP through the n - 1 sub-conductors of the bundle is to be calculated:

on a parabolic arc caused by the other n-1 sub-conductors of the bundle. The tensile force on the bundle clamp for n sub-conductors consists of the static tensile force Fst and the tensile force Fela brought about by the lengthening of the span and the bending of the dead end structure ⎛ 1 ∆l ⎞ ⎟⎟ = Fst ⎜⎜1 + ⎝ NFst l ⎠

Fpi = nFx = Fst + Fela

2

µ ⎛ I ⎞ xP Fy ( x P ) = ( n − 1) 0 ⎜ ⎟ 2π ⎝ n ⎠ asw

with the effective short-circuit current I of the main conductor and the effective centre-line distance asw between sub-conductors. With (15) the transversal component becomes 2 µ ⎛ I ⎞ xP xP Fx = ( n − 1) 0 ⎜ ⎟ (24) 2 π ⎝ n ⎠ asw 2 y P

(19)

With k equidistant spacers in the span one obtains by the approximation for the curve lengthening of the parabolic arcs ∆l = ( k + 1)

2

4 yP

The effective distance asw between the sub-conductors depends on yP and can be approximated with the equation of a parabola. The electromagnetic force is estimated at the location xP with the distance 2y(x) of the parabolic arcs according to (14) x µ ⎛ I ⎞2 P dx Fy ( xP ) = ( n − 1) 0 ⎜ ⎟ ∫ 2π ⎝ n ⎠ 0 2 y ( x )

(20)

3 xP

By substituting equations (16) and (20) in equation (19) the equation for determining xP or yP is: nFy′ x P − Fst x P y P − 2 3

k +1 4 Nl 3

3

yP = 0

(21)

If xP or yP are related to their maximum values and abbreviated by 1/ξ and η 1

The integration and the comparision with (24) leads to:

xP

l lc ≤ 1 with xP,max = s = 2( k + 1) 2 ξ xP,max yP as − ds ≤ 1 with yP,max = η = 2 sin( 180°/ n) yP,max =

asw =

yP = yPmax =

(22)

ε st =

3

(

2 a −d s s

)

2

⎛ 180° ⎞ ⎜ sin ⎟ ⎝ n ⎠

2

3 3 Fν ls N ⎛ 180° ⎞ ε pi = n ⎜ sin ⎟ 8 a −d 3⎝ n ⎠

3

(

s

s

)

(*47)

In any case, fη is thus equal to the quotient of the effective distances and depends on the conductor geometry:

and the substitution fη =

(23)

For the solution of (22) one must decide whether or not the sub-conductors are clashing or only reducing their distance. If the sub-conductors touch at one point, then ξ = η = fη. In this case εpi = 1 + εst follows from (22). The quantity j=

indicates: j≥1 j

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SHORT-CIRCUIT CURRENTS IN POWER SYSTEMS LIÈGE (BELGIUM) 6.-8. SEPTEMBER 1994

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

Gerhard Herold

University of Erlangen-Nürnberg Federal Republic of Germany

Report 2.1

Elmar Zeitler

2.1 SHORT-CIRCUIT CURRENTS – CALCULATION OF EFFECTS THE SECOND EDITION OF IEC PUBLICATION 865 Wolfgang Meyer Gerhard Herold Elmar Zeitler University of Erlangen-Nürnberg Federal Republic of Germany In the case of flexible conductors, IEC 865 (1986) was restricted 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.

Abstract In 1993 the second edition of the IEC Publication 865-1 „Short-circuit currents – Calculation of effects” was published. 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 conductors 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 forces in flexible conductors and the swing out. The comparison with test results is shown.

On the basis of extensive tests the methods known from literature [2, 4-13] have been developed further for calculating 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].

INTRODUCTION In 1986 the Technical Committee 73 of the International Electrotechnical Commission IEC published the first international standard „Calculation of the effects of shortcircuit 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 investigations have been done in Study Committee 23 „Effects of high currents” Working Group 02 „Substations” of the International Conference on Large High Voltage Electric Systems CIGRÉ 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 September 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 Standardization CENELEC decided to publish it as European Standard EN 60865-1:1994.

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

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

During or after line-to-line or three-phase short circuits, two main conductors swing away from each other, for example L1 and L2. Figure 1 [2] shows three typical conductor movements. Thereby the maximum swing-out angle δm is the decisive parameter: – δm < 70°: The span is displaced until the first reversal point is reached at δm and then returns to the steadystate 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 δm. – 70º ≤ δm < 180°: The span is displaced and drops down from the position indicated by δm 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 maximum at the time tt. – 180° ≤ δm: 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.

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. Detailed information is given in [2].

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-

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 η It applies to a.c. single-phase and three-phase systems for rated voltages up to and including 420 kV.

2.1.1

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. Consequently Ft, Ff and bh are calculated for a line-to-line short circuit. In the case of short circuits far from genera′′ = 3/2 I k3 ′′ [18]. In contrast, Fpi is to be calculated tor, I k2 in a single-phase system for the line-to-line short-circuit current I k2 ′′ and in a three-phase system for the three′′ . phase short-circuit current I k3 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 physical pendulum, which obeys the differential equation Jδ + mg s sin δ = M (t ) (1) n

where m = ∫ dm ≈ nms′l

∫ y( x) d m

s=

m

2 b 3 c

distance to center of gravity

8

∫ y d m ≈ 15 nms′ lbc = ∫ F ′( x ) y ( x ) cos δ d x

J = M

≈

mass

2

2

moment of inertia 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 b J ≈ 2π 0,8 c mgn s gn

T = 2π

The double integration of (1) can only be done numerically. 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 independent of time. If the moment is calculated by a substitute force on the center of gravity averaged over the swing-out angle ~ + n~ ) k ls cosδ , (2) M = F ′( m

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

d

(1) can be integrated analytically twice. F′ is the constant force per unit length caused by the initial short-circuit current I k2 ′′ :

MAXIMUM SWING-OUT ANGLE δm

F′ =

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 vertical axis is the swing-out angle δ (t). The longitudinal and transversal conductor waves are excluded.

1 1 + Sl nEs As

′′ ) µ 0 ( I k2

2π

2

a

(3)

After the breaking of the current at δ = δk, the moment M ~ und n~ from [3], here marked by a is zero. The factors m tilde to distinguish them from other variables used, consider the a.c. and d.c. components of the short-circuit current and kd the averaged influence of the angle δ. Integration of (1) with d δ2 /d δ = 2δ leads to: − for 0 ≤ δ ≤ δk with the initial values δ = δ = 0

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 under the stiffness norm N =

(*23)

δ =

8π T

(*25)

2.1.2

2

2

( rres sin δ + cosδ − 1)

(4)

− for δk < δ with the initial values δ = δ k and δ = δk 8π

δ =

T

2

2

( cosδ − cosδ k ) + δk2

(5). Inserting (4) with δ = δk leads to the maximum swingout angle, which is only dependent on rers and δk:

δ m = arccos (1 − rers sin δ k ) = arccos χ

(5)

The course of δm(δk) reaches its maximum at δ k = 90° and then decreases. Because Tk ≤ Tk1, the worst case δ k = 90° is also decisive. So it follows that:

where rres is the ratio of electromagnetic force per unit length to gravitational force per unit length G′ = nms′ gn rers =

~ + n~ ) k F ′( m d

(6)

G′

ξ =

1 k

sin

δ*

k = sin

,

8π T

2 2

1 + rers

2

2

( 1 − k 2ξ 2 )(1 − ξ 2 )

Separation of variables gives an elliptic integral of the first kind; the result is the Jacobis elliptic sinus amplitudinis function sn [19]. Taking the initial value δ * = −δ 1, one gets after substituting back:

π ⎛ δ1 ⎞ 1− ⎜ ⎟ 64 ⎝ 90° ⎠

2

=

radial

(*24)

tangential

for 0 ≤

Tk1 Tres

for 0,5 <

≤ 0,5

Tk1

for

-0,985 ≤ χ < 0,766

for

χ < -0,985

(*31)

R′ = F ′ sin δ + G ′ cos δ + m′yδ2 nms′ yδ = F ′ cos δ − G ′ sin δ

R′ is the radial force per unit length, which changes very slowly compared with the eigenvalue of the conductor as a swinging cord. The integration of the second equation with the initial value δ(t = 0) = 0 and insertion in to the first one leads to

Therefore, the angle at the end of the short-circuit current flow is with t = Tk1: ⎧ ⎡ ⎛ ⎞⎤ ⎪δ ⎢1 − cos⎜ 2π Tk1 ⎟ ⎥ 1 ⎜ ⎟ ⎪⎪ ⎢⎣ ⎝ Tres ⎠ ⎥⎦ δk = ⎨ ⎪ ⎪ 2δ 1 ⎪⎩

0,766 ≤ χ ≤ 1

Figure 3 shows the sectional view of a span. The forces per unit length are acting on a conductor element:

T ⎛ π 2 ⎛ δ ⎞2 ⎞ 1 4 1 + r 2 ⎜1 − ⎜ ⎟ ⎟⎟ res ⎜ ⎝ 64 ⎝ 90° ⎠ ⎠

for

SHORT-CIRCUIT TENSILE FORCE Ft

with the resulting period during short-circuit current flow: 2

(*30)

For short-circuit durations greater than 0,1 s the influence ~ can be disregarded. of the d.c. component is low and m ~ For distribution networks usually n = 1. The calculation of factor kd is not easy. As 0,9 < kd < 1 applies for usual spans, kd = 1 can be used [2, Figure 4.2.6]. Accordingly, (6) becomes: F′ rers ≈ r = (*20) G′

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 amplitudedependent periodicity of the sn-function by approximation: ⎡ ⎛ t ⎞⎤ ⎟⎟ ⎥ δ ( t ) = δ 1 ⎢1 − cos⎜⎜ 2π ⎢⎣ ⎝ Tres ⎠ ⎥⎦

Tpend

for 90°< δ k

⎧1,25arccos χ ⎪ δ m = ⎨10°+ arccos χ ⎪ ⎩180°

⎡ ⎛ ⎞⎤ t π ⎢ ⎜ δ = δ 1 + 2 arcsin k sn ⎜ 2 π − , k ⎟⎥ ⎢ T 2 ⎟⎠⎥⎦ pend ⎝ ⎣

Tres =

⎩1 − rers

A comparison with test results [11, 14] shows that δm is too low when calculated by (7). This is due to the simple model of a constant force per unit length in (2) on the one hand, and the pendulum movement superposed by longitudinal oscillations of the conductor on the other hand. Therefore, the following corrections are made in (7): Small swing-out angles are adjusted by a constant factor of 1,25 and large angles with the addition of 10°, which gives continuation at δm = 50° and 180°. Hence

by which the span swings around the steady state position δ1 = arctan rers :

ξ =

for 0 ≤ δ k ≤ 90°

The value δm =180° is obtained at χ = −1. If χ < −1, the span rotates.

δ1

2

⎧1 − rers sin δ k

χ =⎨

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

δ * = δ − δ1 ,

(7)

(*29)

Tres

When Tk1/Tres > 0,5, the maximum value of δk for Tk1/Tres = 0,5 is to be inserted, because the actual shortcircuit duration Tk can be lower than the short-circuit duration 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 δ = δm it is δ = 0 in

Figure 3 Swing out of the conductor in midspan and forces during current flow

2.1.3

R′ = 3r sin δ + 3 cos δ − 2 G′

(8)

R′ reaches its maximum at

δ1 = arctan r

Figure 4 Conductor movement in midspan for calculating the tensile force Ff

(*21)

if δk ≥ δ1, i.e. Tk1 ≥ Tres/4, otherwise at δ = δk, if Tk1 < Tres/4. The change in conductor length by increasing the radial force from Rst′ = G′ to Rt′ = G ′( 1 + ϕ ) becomes with the change-of-state equation for a conductor: lt − lst = Nl ( Ft − Fst )

The upper limit follows with δf = 0: Ff = Fst 1 + 4 ζ (1 − cos δ m )

(9)

and with the approximation 1 - cos δm ≈ (2/π) δm:

The approximate lengths of the conductor with ν = t,st 3 ⎛ R′ l ⎞ l ⎛ Rν′ ⎜⎜ lν = 2 sinh ⎜⎜ ν ⎟⎟ ≈ l + G′ 24 ⎝ Fν ⎝ Fν 2 ⎠

Fν

Ff = Fst 1 + 8ζ

⎞2 ⎟⎟ ⎠

(

)

Ft = Fst ( 1 + ϕψ )

Ff = 1,2 ⋅ Fst 1 + 8ζ m 180°

(*34)

When 0,6 < r < 2, the approximation

⎧ ⎛ ⎪ 3⎜ 1 + r 2 − 1⎞⎟ ⎠ ϕ= +1= ⎨ ⎝ G′ ⎪⎩ 3( r sin δ + cosδ − 1) Rmax ′

for δ k ≥ δ 1 for δ k < δ 1

δ m = arccos (1 − r sin δ k ) ≈ arccos (1 − r ) applies, which with (11) gives us (22) in IEC 865 (1986); δm=180° in (12) results in (23) there.

(*32)

HORIZONTAL SPAN DISPLACEMENT

( n ms′lgn ) 2 =

Into the change-of-state equation in the case of elastic and thermal elongation

(*28)

24 Fst3 N

2 2 ⎡ ⎛ G ′ ⎞2 ⎤ l ⎢⎛ Rt′ ⎞ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥ = ε ela + ε th 24 ⎢⎣⎝ Ft ⎠ ⎝ Fst ⎠ ⎥⎦

(10) becomes

ϕ ψ + ϕ ( 2 + ζ )ψ + ( 1 + 2 ζ )ψ − ζ ( 2 + ϕ ) = 0 2

3

2

(*33)

with which ψ can be determined by 0 ≤ ψ ≤ 1 or taken from Figure *7.

the sags in the midspan 2 2 G ′l R ′l , bc = bct = t 8 Fst 8 Ft are inserted and solved for bct:

DROP FORCE Ff At the highest point of the conductor movement δm according to Figure 4, the maximum potential energy is stored: Epot = (1 − cos δ m ) nms′lgn s

2 3⎛ l ⎞ bct = bc 1 + ⎜⎜ ⎟⎟ ε ela + ε th = CD bc 8 ⎝ bc ⎠

(

During the fall the energy is partially or completely converted into elongation energy depending on height and movement of fall: 1 2 2 E ela = Nl Ff − Fst 2

(

(*35)

The drop force only needs to be calculated if δm ≥ 70°, which occures if r > 0,6.

ϕ follows from (8)

ζ

(12)

180°

δ

(10)

Hence the short-circuit tensile force Ft is estimated as:

With the stress factor

δm

In general the velocity and hence the potential energy of the conductor are not zero and drop forces are observed in tests which are greater than those calculated with (12). This can be taken into considered by a factor of 1,2:

are inserted in (9) 2 3 ⎡ ⎛ R ′ ⎞2 ⎤ l ⎢⎛ Rt′ ⎞ ⎜⎜ ⎟⎟ − ⎜⎜ st ⎟⎟ ⎥ = Nl Ft − Fst ⎢ 24 ⎣⎝ Ft ⎠ ⎝ Fst ⎠ ⎥⎦

(11)

ε ela = N ( Ft − Fst )

(*36)

and the thermal elongation due to the heating by the shortcircuit current

From the height of fall one obtains the balance of the complete energy conversion:

(

(11)

CD corresponds to (*38). The elastic elongation is generated due to the change of the tensile force from Fst to Ft

)

( cos δ f − cos δ m ) nms′ lgn s = 21 Nl Ff2 − Fst2

)

ε th =

)

2 α th ⎛ I k2 ′′ ⎞

2 ⎛ I k2 ′′ ⎞ ⎜ ⎟ T = cth ⎜ ⎟ T ⎝ A ⎠ κcρ ⎝ A ⎠

(*37)

with the material constant cth according sub-clause *2.3.2.4. The maximum sag bct for Tk1 ≥ Tres/4 is reached at the angle δ1 and for Tk1 < Tres/4 at the end of the current flow at δk . Thus T is:

and from this: Ff = Fst 1 + 4ζ ( cos δ f − cos δ m )

2.1.4

⎧ Tres / 4 T=⎨ ⎩ Tk1

for Tk1 ≥ Tres / 4 for Tk1 < Tres / 4

THE PINCH EFFECT OF BUNDLED CONDUCTORS (*37)

Figure 5 shows a contracted regular twin-bundle. Regular, here, means that the position of the cross sectional centres of the n sub-conductors of the bundle are marked by the corners of a regular polygon.

Because the shape already differs from the parabola with low values of r and approaches a triangle with high values of r, the actual sag is higher than that estimated with (13). The correction is made using the form factor CF with linear interpolation in the range 0,8 < r < 1,8: ⎧1,05 ⎪ CF = ⎨0,97 + 0,1 r ⎪ , ⎩115

Sub-conductors are considered to clash effectively if the clearance as between the midpoints of adjacent sub-conductors, as well as the distance ls between two adjacent spacers fulfill either (*43) or (*44)

for r ≤ 0,8 for 0,8 < r < 0,8

(*39)

as/ds ≤ 2,0 and ls ≥ 50 as as/ds ≤ 2,5 and ls ≥ 70 as

for r ≥ 1,8

A calculation of the bundle force then does not have to be done; in this case it is sufficient to add 10 % to the tensile force Ft in (*34) [20].

CF is derived from tests, because it is only possible numerically to get the form factor for each span. The maximum horizontal displacement is the projection of the sag onto the horizontal axis − with slack conductors: for δ m ≥ 90°

⎧ CF CDbc bh = ⎨ ⎩ CF CDbc sin δ m

for δ m < 90°

The calculations for contraction of the bundle described in the literature up to now can only be solved numerically. Hence, the parabola model is introduced below, based on equations and diagrams which can be solved analytically for the most occurring cases of clashing sub-conductors and by diagrams or numerically for non-clashing subconductors.

(*40)

− with strained conductors: ⎧ CF CDbc sin δ 1 bh = ⎨ ⎩ CF CDbc sin δ m

for δ m ≥ δ 1 for δ m < δ 1

Physical model for regular n-conductor bundles The sub-conductors touch each other in the range 2xP - ls < x < 0. When considering the bundle in its static steady state of contraction and assuming the shape of the curve of the non-clashing section a parabola, the equation for the first quadrant is: ⎛ x ⎞2 y ( x ) = yA + yP ⎜⎜ ⎟⎟ (14) ⎝ xP ⎠ The flexural stiffness of the conductor which acts above all near the spacers is ignored, so that for the ratio of the transversal and the longitudinal component of the tensile force between two adjacent spacers the following holds

(*41)

because they swing out almost horizontally [13], so that bh is a maximum at about δ1. EXTENSION ON STRAINED CONDUCTORS The methods written above are only valid for slack conductors, which are connected to support insulators, and hence represent a homogenious arrangement. In long spans, the mass of the insulator strings is a multiple of the conductor mass and is essentially decisive for the sag. The period T according to (*23) contains only the sag. To take the insulator strings into account, an equivalent static conductor sag at midspan is defined: bc =

G′l

Fy ( x) dy y x = =2 P dx Fx xP xP

2

which is – for slack conductors, the actual static sag, – for strained conductors, the sag, when the actual arrangement is replaced by an arrangement of conductors with the same conductor type and static tension, but without insolator strings. As the electromagnetic force acts only on the conductor, the difference between the conductor length lc and the span length l is to be taken into account in (3): F′ =

where lc = l lc = l − 2li

( )

′′ µ 0 I k2 2π

a

2

lc l

(15)

The longitudinal component Fx is local independent. On the other hand, the transversal component per unit length Fy′ is local independent with respect to (15)

(*22)

8 Fst

(*43) (*44)

2

Fx

d y dx

2

= Fy′ ( x ) = 2

yP 2

xP

Fx

(16)

At the location xP of the spacer, (15) becomes Fy ( xP ) = 2

yP F xP x

(17)

and according to (16) yields Fy ( xP ) = xP Fy′ ( xP )

(*19)

(18)

for slack conductors, for strained conductors, where li is the length of one insulator chain.

With both assumptions a good concurrence is achived with test results [13]. Figure 5 Idealized geometry of a clashed twin-bundle 2.1.5

Fy′ ( xP ) is the electromagnetic force per unit length acting

Electromagnetic force To determine Fy′( x P ) , the electromagnetic force on a parabolic arc of the length xP through the n - 1 sub-conductors of the bundle is to be calculated:

on a parabolic arc caused by the other n-1 sub-conductors of the bundle. The tensile force on the bundle clamp for n sub-conductors consists of the static tensile force Fst and the tensile force Fela brought about by the lengthening of the span and the bending of the dead end structure ⎛ 1 ∆l ⎞ ⎟⎟ = Fst ⎜⎜1 + ⎝ NFst l ⎠

Fpi = nFx = Fst + Fela

2

µ ⎛ I ⎞ xP Fy ( x P ) = ( n − 1) 0 ⎜ ⎟ 2π ⎝ n ⎠ asw

with the effective short-circuit current I of the main conductor and the effective centre-line distance asw between sub-conductors. With (15) the transversal component becomes 2 µ ⎛ I ⎞ xP xP Fx = ( n − 1) 0 ⎜ ⎟ (24) 2 π ⎝ n ⎠ asw 2 y P

(19)

With k equidistant spacers in the span one obtains by the approximation for the curve lengthening of the parabolic arcs ∆l = ( k + 1)

2

4 yP

The effective distance asw between the sub-conductors depends on yP and can be approximated with the equation of a parabola. The electromagnetic force is estimated at the location xP with the distance 2y(x) of the parabolic arcs according to (14) x µ ⎛ I ⎞2 P dx Fy ( xP ) = ( n − 1) 0 ⎜ ⎟ ∫ 2π ⎝ n ⎠ 0 2 y ( x )

(20)

3 xP

By substituting equations (16) and (20) in equation (19) the equation for determining xP or yP is: nFy′ x P − Fst x P y P − 2 3

k +1 4 Nl 3

3

yP = 0

(21)

If xP or yP are related to their maximum values and abbreviated by 1/ξ and η 1

The integration and the comparision with (24) leads to:

xP

l lc ≤ 1 with xP,max = s = 2( k + 1) 2 ξ xP,max yP as − ds ≤ 1 with yP,max = η = 2 sin( 180°/ n) yP,max =

asw =

yP = yPmax =

(22)

ε st =

3

(

2 a −d s s

)

2

⎛ 180° ⎞ ⎜ sin ⎟ ⎝ n ⎠

2

3 3 Fν ls N ⎛ 180° ⎞ ε pi = n ⎜ sin ⎟ 8 a −d 3⎝ n ⎠

3

(

s

s

)

(*47)

In any case, fη is thus equal to the quotient of the effective distances and depends on the conductor geometry:

and the substitution fη =

(23)

For the solution of (22) one must decide whether or not the sub-conductors are clashing or only reducing their distance. If the sub-conductors touch at one point, then ξ = η = fη. In this case εpi = 1 + εst follows from (22). The quantity j=

indicates: j≥1 j

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