Short Circuit Currents Calculation of Effects the Second Edition of Iec Publication 865

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 Elmar Zeitler University of Erlangen-Nürnberg 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-Nürnberg 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 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 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 η 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 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 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 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 maxi- mum 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. 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 δm 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 δ (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 N Sl nE A = +1 1 s 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 ( )δ δ+ =n (1) where m dm nm l= ≈ ′∫ s mass s y x m m b= ≈∫ ( ) d 2 3 c distance to center of gravity J y m nm lb= ≈ ′∫ 2 8 15 d s c 2 moment of inertia M F x y x x= ′∫ ( ) ( ) cos dδ 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 T J mg s b g = ≈2 2 0 8π π 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 δ , (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 µ0 2 2π k2 (3) After the breaking of the current at δ = δk, 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 δ. Integration of (1) with d � /d ��δ δ δ2 2= leads to: − for 0 ≤ δ ≤ δk with the initial values δ δ= =� 0 ( )� sin cosδ δ δ= + −8 122πT rres (4) 2.1.3 − for δk < δ with the initial values δ δ= k and � �δ δ= k ( )� cos cos �δ δ δ δ= − +8 22πT k k2 (5) where rres is the ratio of electromagnetic force per unit length to gravitational force per unit length ′ = ′G nm gs n ( ) r F m n k Gers d= ′ +′ ~ ~ (6) For the second integration of (4), the following substitu- tions are inserted to calculate the angle δk at the end of the short-circuit current: δ δ δ ξ δ δ* , sin * , sin= − = =1 11 2 2k k by which the span swings around the steady state position δ1 = arctan rers: ( )( )�ξ ξ ξ= + − −8 1 1 122 2 2 2 2πT 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 δ δ* = − 1, one gets after substituting back: δ δ= + − ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥1 2 2 2arcsin sn ,k t T kπ π 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: δ δ( ) cost t T = − ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥1 1 2π res with the resulting period during short-circuit current flow: T T T r res pend res = − ° ⎛ ⎝⎜ ⎞ ⎠⎟ = + − ° ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟1 64 90 1 1 64 90 2 1 2 24 2 1 2π πδ δ (*24) Therefore, the angle at the end of the short-circuit current flow is with t = Tk1: δ δ δ k k1 res k1 res 1 k1 res for 2 for = − ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ≤ ≤ < ⎧ ⎨ ⎪ ⎪⎪ ⎩ ⎪ ⎪⎪ 1 1 2 0 0,5 0,5 cos π T T T T T T (*29) When Tk1/Tres > 0,5, the maximum value of δk 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 δ = δm it is �δ = 0 in (5). Inserting (4) with δ = δk leads to the maximum swing- out angle, which is only dependent on rers and δk: ( )δ δ χm ers k= − =arccos sin arccos1 r (7) 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: χ δ δ δ= − ≤ ≤ ° − °< ⎧ ⎨ ⎩ 1 0 90 1 90 r r ers k k ers k for for sin (*30) The value δm =180° is obtained at χ = −1. If χ < −1, the span rotates. 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 longi- tudinal 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 δ χ χ χ χ χ m for 0,766 1 for -0,985 0,766 for -0,985 = ≤ ≤ °+ ≤ < ° < ⎧ ⎨⎪ ⎩⎪ 1 10 180 ,25arccos arccos (*31) For short-circuit durations greater than 0,1 s the influence of the d.c. component is low and ~m can be disregarded. 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: r r F Gers ≈ = ′′ (*20) SHORT-CIRCUIT TENSILE FORCE Ft Figure 3 shows the sectional view of a span. The forces per unit length are acting on a conductor element: radial tangential s ′ = ′ + ′ + ′ ′ = ′ − ′ R F G m y nm y F G sin cos � �� cos sin δ δ δ δ δ δ 2 ′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 Figure 3 Swing out of the conductor in midspan and forces during current flow 2.1.4 ′ ′ = + − R G r3 3 2sin cosδ δ (8) ′R reaches its maximum at δ1 = arctan r (*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 ′ = ′R Gst to ( )′ = ′ +R Gt 1 ϕ becomes with the change-of-state equation for a conductor: ( )l l Nl F Ft st t st− = − (9) The approximate lengths of the conductor with ν = t,st l F G R F l l l R Fν ν ν ν ν ν = ′ ′⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ≈ + ′⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟2 2 24 3 2 sinh are inserted in (9) ( )l R F R F Nl F F 3 2 2 24 ′⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ − ′⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − t t st st t st (10) Hence the short-circuit tensile force Ft is estimated as: ( )F Ft st= +1 ϕψ (*34) ϕ follows from (8) ( )ϕ δ δ δ δ δ δ = ′ ′ + = + −⎛⎝⎜ ⎞ ⎠⎟ ≥ + − < ⎧ ⎨⎪ ⎩⎪ R G r r max sin cos 1 3 1 1 3 1 2 1 1 for for k k (*32) With the stress factor ( )ζ = ′n m lg F N s n st 2 324 (*28) (10) becomes ( ) ( ) ( )ϕ ψ ϕ ζ ψ ζ ψ ζ ϕ2 3 22 1 2 2 0+ + + + − + = (*33) with which ψ can be determined by 0 ≤ ψ ≤ 1 or taken from Figure *7. DROP FORCE Ff At the highest point of the conductor movement δm ac- cording to Figure 4, the maximum potential energy is stored: ( )E nm lg spot m s ncos= − ′1 δ During the fall the energy is partially or completely con- verted into elongation energy depending on height and movement of fall: ( )E Nl F Fela f st2= −12 2 From the height of fall one obtains the balance of the complete energy conversion: ( ) ( )cos cosδ δf m s n f st− ′ = −nm lg s Nl F F12 2 2 and from this: ( )F Ff st f m= + −1 4ζ δ δcos cos Figure 4 Conductor movement in midspan for calculating the tensile force Ff The upper limit follows with δf = 0: ( )F Ff st m= + −1 4 1ζ δcos (11) and with the approximation 1 - cos δm ≈ (2/π) δm: F Ff st m= + °1 8 180ζ δ (12) 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: F F tf s m= ⋅ + °12 1 8 180, ζ δ (*35) The drop force only needs to be calculated if δm ≥ 70°, which occures if r > 0,6. When 0,6 < r < 2, the approximation ( ) ( )δ δm = − ≈ −arccos sin arccos1 1r rk applies, which with (11) gives us (22) in IEC 865 (1986); δm=180° in (12) results in (23) there. HORIZONTAL SPAN DISPLACEMENT Into the change-of-state equation in the case of elastic and thermal elongation l R F G F 2 2 2 24 ′⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ − ′⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = + t t st ela thε ε the sags in the midspan b G l F b R l Fc st ct t t = ′ = ′ 2 2 8 8 , are inserted and solved for bct: ( )b b l b C bct c c ela th D c= + ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ + =1 3 8 2 ε ε (11) CD corresponds to (*38). The elastic elongation is gener- ated due to the change of the tensile force from Fst to Ft ( )εela t st= −N F F (*36) and the thermal elongation due to the heating by the short- circuit current ε ακ ρth th k2 th k2= ′′⎛⎝⎜ ⎞ ⎠⎟ = ′′⎛ ⎝⎜ ⎞ ⎠⎟c I A T c I A T 2 2 (*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 cur- rent flow at δk . Thus T is: 2.1.5 T T T T T T T = ≥ < ⎧ ⎨ ⎩ res k1 res k1 k1 res for for / / / 4 4 4 (*37) 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 lin- ear interpolation in the range 0,8 < r < 1,8: C r r r r F for for 0,8 < for = ≤ + < ≥ ⎧ ⎨⎪ ⎩⎪ 1 05 0 8 0 97 0 1 0 8 115 18 , , , , , , , (*39) CF is derived from tests, because it is only possible nu- merically 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: b C C b C C bh F D c m F D c m m for for = ≥ ° < ° ⎧ ⎨ ⎩ δ δ δ 90 90sin (*40) − with strained conductors: b C C b C C bh F D c 1 m 1 F D c m m 1 for for = ≥ < ⎧ ⎨ ⎩ sin sin δ δ δ δ δ δ (*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 con- ductors, 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 con- ductor sag at midspan is defined: b G l Fc st = ′ 2 8 (*22) which is – for slack conductors, the actual static sag, – for strained conductors, the sag, when the actual ar- rangement 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 I a l l µ π 0 2 2 k2 c (*19) where l lc = for slack conductors, l l lc i= − 2 for strained conductors, where li is the length of one insulator chain. With both assumptions a good concurrence is achived with test results [13]. THE PINCH EFFECT OF BUNDLED CONDUCTORS 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. Sub-conductors are considered to clash effectively if the clearance as between the midpoints of adjacent sub-con- ductors, as well as the distance ls between two adjacent spacers fulfill either (*43) or (*44) as/ds ≤ 2,0 and ls ≥ 50 as (*43) as/ds ≤ 2,5 and ls ≥ 70 as (*44) 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]. 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 sub- conductors. 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: y x y y x x ( ) = + ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟A P P 2 (14) 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 d d ( )y x F x F y x x x y x = = 2 P P P (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) F y x F x y x Fx y x d d ( ) 2 2 22= ′ = P P (16) At the location xP of the spacer, (15) becomes F x y x Fy x( )P P P = 2 (17) and according to (16) yields F x x F xy y( ) ( )P P P= ′ (18) Figure 5 Idealized geometry of a clashed twin-bundle 2.1.6 ′F xy ( )P is the electromagnetic force per unit length acting 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-conduc- tors 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 F nF F F F NF l lxpi st ela st st = = + = + ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟1 1 ∆ (19) With k equidistant spacers in the span one obtains by the approximation for the curve lengthening of the parabolic arcs ( )∆l k y x = +1 4 3 2 P P (20) By substituting equations (16) and (20) in equation (19) the equation for determining xP or yP is: nF x F x y k Nl yy′ − − + =P st P P P3 32 1 4 3 0 (21) If xP or yP are related to their maximum values and abbre- viated by 1/ξ and η ( ) ( ) 1 1 2 1 1 ξ η = ≤ = = + = ≤ = − ° x x x l l k y y y a d n P P,max P,max s c P P,max P,max s s with 2 with 2 180 /sin then (21) becomes ξ η ε ξ η ε η3 3 2 0+ − =st pi f (22) with the strain coefficients ( )εst st s s s = − °⎛ ⎝⎜ ⎞ ⎠⎟ 3 2 1802 2 2F l N a d n sin (*47) ( )ε ν pi s s s = − °⎛ ⎝⎜ ⎞ ⎠⎟ 3 8 1803 3 3 n F l N a d n sin (*48) the electromagnetic force per unit length also related to the maximum value f F F y y η = ′ ′ ≤,max 1 and the substitution F F lyν = ′,max s (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 = + ε ε pi st1 (*49) indicates: j ≥ 1 the sub-conductors are clashing, j < 1 the sub-conductors have reduced their dis- tance, but are not clashing. Electromagnetic force To determine ′F xy ( )P , the electromagnetic force on a parabolic arc of the length xP through the n - 1 sub-con- ductors of the bundle is to be calculated: ( )F x n I n x ay ( )P P sw = − ⎛⎝⎜ ⎞ ⎠⎟1 2 0 2µ π with the effective short-circuit current I of the main con- ductor and the effective centre-line distance asw between sub-conductors. With (15) the transversal component be- comes ( )F n I n x a x yx = − ⎛⎝⎜ ⎞ ⎠⎟1 2 2 0 2µ π P sw P P (24) The effective distance asw between the sub-conductors de- pends 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) ( )F x n I n x y xy x ( )P d 2 ( ) P = − ⎛⎝⎜ ⎞ ⎠⎟ ∫1 2 0 2 0 µ π The integration and the comparision with (24) leads to: a y y y y ysw A P A P A = 2 / arctan / (25) According to (18) and (24), the local independent trans- versal force per unit length ′F xy ( )P is maximum, if the sub-conductors clash in one point, i.e. if ( )y y a d nP Pmax s s= = − °2 180sin / then ( )[ ]y d nA s= °/ sin /2 180 , and asw becomes its maxi- mum: ( )a d n a d a d asw,max s s s s s s= ° − − =sin / / arctan /180 1 1 3 ν (*A.7) with the abbreviation ν3, and gives the electromagnetic force according to (23) ( )F n I n l aν µ π ν= − ⎛ ⎝⎜ ⎞ ⎠⎟1 2 0 2 3 s s (26) In any case, fη is thus equal to the quotient of the effective distances and depends on the conductor geometry: f Fy Fy a y a yη = ′ ′ = ≤, ( ) ( )max sw P,max sw P 1 asw(yP) and asw(yP,max) are determined with (25). The effective short-circuit current has been inserted in (24) without stating the functional connection between I and the initial short-circuit current ′′Ik . The time elapsed until the contraction of the bundle is complete, Tpi may be assessed from the boundary value problem for the middle of the bundled conductor. It is far easier to establish analytically, assuming a close bundle and the usual short- circuit currents, by stating a constant force. The results are in good accordance as long as the assumptions are observed. In this case with the formula for a mass element 2.1.7 ′ms accelerated by constant electromagnetic force per unit length ( )′ = ′ − − ⎛⎝⎜ ⎞ ⎠⎟ ° m y a m n I n n a ts s s s2 1 2 2 1800 2 2µ π sin / Tpi becomes for y = ds: ( ) T n a d m I n n a f I Ipi s s s s 1 k= ° − ′ ⎛ ⎝⎜ ⎞ ⎠⎟ − = ′′1 180 2 10 2 sin µ π ν (27) with the abbreviation ( )ν1 0 2 1 180 2 1 = ° − ′ ′′⎛ ⎝⎜ ⎞ ⎠⎟ −f n a d m I n n a sin s s s k s µ π (*46) f means the frequency of the power system. The natural frequency of the bundled conductors as oscillating strings is in the range of a few cycles and therefore well below the excitation frequency of 50 Hz and 100 Hz, respec- tively. Thus, the time-averaged values of the force per unit length as well as of the short-circuit current are rele- vant for the pinch force. Therefore: I T i t t T 2 2 0 1= ∫ pi d pi ( ) (29) For long contraction periods, like those occuring with wide bundles, the squared current may be approximated by using the the factors ~m and ~n taken from [3]. For short contraction periods, however, it is absolutely neces- sary to resolve (29) exactly. With ( )[ ]i t I tk t( ) sin e sin/= ′′ − + −2 ω γ γτ for short circuits far from generator, where τ is the time constant of the network and γ = arctan(2πfτ) [4]. fTpi is the solution of (27), here abbreviated by: ν ν1 2= fTpi (*A.5) with ( )ν2 2= ′′I I/ k , where Tpi is dependent on I. Figure *8 shows ν2 as a function of ν1, which converges more and more to 1 + ~m according to Figure *12a for increasing ν1. Substituting I in (26) gives the electromagnetic force which is to be inserted into (*48): ( )F n I n l aν ν ν= − ′′⎛ ⎝⎜ ⎞ ⎠⎟1 2 0 2 2 3 µ π k s s (*45) Clashing of sub-conductors If the sub-conductors are clashing, then fη = 1 and (22) becomes with η = 1: ξ ε ξ ε3 2 0+ − =st pi (*51) It contains constant coefficients. ξ is analytically calcula- ble, dependent on j with the parameter εst, whereby j2/3 ≤ ξ ≤ j holds. From (19) the tensile force in the bundle clamp can be given F Fpi st e st = + ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟1 ν ε ξ (*50) Here, an overswing factor νe is introduced, which ac- counts for the dynamic process in the clashing of bundled conductors and has still to be determined more exactly. Non-clashing of sub-conductors If the sub-conductors reduce their distance without clash- ing, then xP,max = ls/2 and yA = as/(2sin180°/n) - yP ac- cording to Figure 4. (22) becomes with ξ = 1: η ε η ε η3 0+ − =st pi f (*62) Here, the nonlinear factor fη must be considered, too, which is a function of as/ds. Therefore, the quantity η can only be numerically determined dependent on j and the parameters εst and as/ds, whereby 0 < η < 1. Instead of de- termining η(j) by (*62), an analytical solution j(η) can be obtained from (*49) with the inverse function: ( )j f= + + η ε η ε η 3 1 st st (*61) From (19) the tensile force in the clamp can be given with the overswing factor νe: F Fpi st e st = + ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟1 2 ν ε η (*54) Dynamic overswing factor The calculation of the pinch force Fpi has been based on quasi-static considerations of the geometry and forces ob- served with the final state of the contracted bundle. In practice, however, the sub-conductors are accelerated by the electromagnetic force acting upon them. The kinetic energy thereby absorbed is converted into elongation work of the system. The elongation work required to reach the final state of contraction, however, is lower than the kinetic energy absorbed until the final static position has been reached for the first time. The balance is converted into collision, deformation, and frictional en- ergy, as the sub-conductors oscillate around their final position. Thereby, the first overswing movement leads to the actual maximum of the pinch force. So far, the above situation has been accounted for by a constant factor, which in practice, however, will be dependent upon the geometry of the bundle and the boundary conditions of the contraction. On a differential portion of a sub-conductor, the electro- magnetic force performs the differential work dWm. Tak- ing into account the number of sub-conductors and spac- ers, and assuming the shape of the curve of the clashed conductors to be a parabola, the summation of these dif- ferential works along the non-clashing section gives the work performed on the whole bundle, as: ( ) ( ) ( )W n k I n n y y x x y y xx m ( 0 P,max P d d= + ⎡⎣⎢ ⎤ ⎦⎥ −∫∫2 1 2 1 20 2µ π ) (30) with y(x) according (14). With the energy conservation law and neglecting the bending stiffness of the conductor as well as the friction between the individual conductors it is: W W Wm kin spring+ + = 0 (31) This means that the electromagnetic work performed has been converted into elongation work and kinetic energy at 2.1.8 that point in time when the final static position has been reached for the first time. The elongation work of the spring is: W N l l lspring = 1 2 1 ∆ ∆ (32) Subsequently, the kinetic energy causes a further elonga- tion of the equivalent spring of the overall system and thus an additional variation in length of the sub-conduc- tors, ∆l+: W N l l lkin = + + 1 2 1 ∆ ∆ (33) At the time of maximum contraction of the bundle and in analogy to (19) the tensile force becomes F nF F NF l l l lpi x st st = = + +⎛⎝⎜ ⎞ ⎠⎟ ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ +1 1 1∆ ∆∆ and comparison with (*50) and (*54), respectively, results in the overswing factor νe, which is weighted by a factor of 1/2 in order to account for energy dissipation due to de- formation, friction, emission of sound, etc.: νe = + ⎛ ⎝⎜ ⎞ ⎠⎟ +1 2 1 ∆l l From (31) with (30), (32) and (33) it is found ( ) ( ) ∆ ∆ µ π l l n n Nl k I n x y y y y y + = −+ ⎛ ⎝⎜ ⎞ ⎠⎟ − ⎧ ⎨⎪ ⎩⎪ ⎫ ⎬⎪ ⎭⎪ −9 1 4 1 2 1 10 2 P 3 P 4 A P A P arctan Hence the different overswing factors νe for the cases of clashing and non-clashing sub-conductors can be calcu- lated, i. e. (*52) and (*55). YOUNG'S MODULUS FOR STRANDED CONDUCTORS Young's modulus for aluminium-steel stranded conductors is extremely non-linear in the range of relatively low stress (< 50 N/mm2), as is usually found with outdoor substations, and may even take on values far below those given by standards or furnished by manufacturers, respec- tively. Since on the one hand, Young's modulus is an im- portant parameter for calculating the tensile forces, but on the other hand, an exact establishment of the modulus by expensive tensile testing is neither possible nor desirable. As a result of extensive tests, an empirical expression for the approximate calculation of Young's modulus is given: E E F nA F n As st s fin st s finfor= + ° ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ≤0,3 0,7 90sin σ σ (*26) where σfin = 5·107 N/m2 is the lowest value of σ, when Young's modulus becomes constant. The final Young's modulus E for stranded conductors shall be used. DESIGN LOAD For designing the substations, the following should par- ticularly be noted: – In most cases, the tensile forces Ft during a short circuit are maximum for the coldest conductor. – The higher the span temperature the higher the span displacements and the tensile forces Ff after a short cir- cuit. Therefore, in most cases, the tensile forces Ff after a short circuit are maximum for high temperatures. – The tensile forces Fpi caused by the pinch effect are al- so maximum for the coldest conductor. To be shure to get the worst case, it is necessary to do the calculations on the basis of the static tensile force Fst – at the local minimum winter temperature, e.g. -20°C, – at the maximum operating temperature, e.g. 60°C. For each tensile force, the worst case shall be taken into account. COMPARISON BETWEEN CALCULATIONS AND TESTS Extensive tests have been carried out to verify the calcu- lation methods given above, e.g. see references in [2]. In the case of flexible strained conductors, Figure 6 com- pares the calculations according IEC 865-1 (1993) with the results of more than 110 tests described in [2, 13-17, 21] with 123 kV and 420 kV arrangements. In addition, the location of the relative error of -25 %, 0 % and +25 % is depicted as broken lines. As can be seen, the relative er- rors, as they are not on save side, lie within the technical limit of about 25 % and are sufficient small. The outcome of these tests confirm the calculation methods. OUTLOOK TC 73 is going to publish a Report 865-2 „Short-circuit currents – Examples of calculation of the effects of short- circuit currents”, in which the calculation procedures for 6 examples with rigid and flexible conductors are shown. A standard for calculation of the effects of short-circuit currents in d.c. auxiliary installations is in preparation. For flexible conductor arrangements the effects on drop- pers, the stiffness and dynamic behaviour of portals and foundations are to be investigated. The extension of the methods for voltages above 420 kV is to be checked by comparing with tests as well as the use for spans with lengths more than 60 m. REFERENCES [1] IEC Publication 865, Calculation of the effects of short-circuit currents, Genève: IEC, 1986. [2] IEC TC 73/CIGRÉ SC 23 WG 02: The mechanical effects of short-circuit currents in open air substa- tions, Genève: IEC, Paris: CIGRÉ, 1987. [3] IEC Publication 865-1, Short-circuit currents – Cal- culation of effects, Genève: IEC, 1993. [4] W. Lehmann, P. Sieber, „Mechanische Kurzschluß- beanspruchung durch Leitungsseile in Schaltanla- gen“, VDE-Fachberichte, vol.24, pp. 149-153, 1966. [5] L. Möcks, „Die Teilfeldlänge in Bündelleitern“, Bull. SEV, vol. 59, pp. 726-723, 1968. [6] G. Hosemann, „Berechnung der Kurzschlußströme und -kräfte in der Normung“, etz-A, vol. 97, no. 5, pp. 298-303, May 1976. [7] B. Engel, „Mechanische Beanspruchung von Leiter- seilen in Anlagen nach einem Kurzschluß“, Elektri- zitätswirtschaft, vol. 78, pp. 186-189, 1979. [8] B. Engel, „Vereinfachte Berechnung der Seilaus- lenkungen durch Kurzschlußkräfte bei Gerätever- bindungen in Schaltanlagen“. Archiv für Elektro- technik, vol. 61, no. 6, pp. 287-292, 1979. 2.1.9 Figure 6 Results of tests versus calculation a) Maximum Fmax of tensil forces Ft and Ff b) Pinch force Fpi c) Horizontal displacement bh The forces are related to the respective static tensile forces. Indices: m measured; c calculated [9] L. Gauffin, J.L. Lilien, „Mechanical effects of short- circuit currents in substations with strain bus systems – Medium complexity calculation methods“, in CIGRÉ Symposium 06-85, Bruxelles, Report 330-01, pp. 1-6, 1985. [10] J.L. Lilien, „The pendulum model“, CIGRÉ SC 23- 81 (WG 02) 11-IWD, 1981, unpublished. [11] M. Waeber, „Bestimmung der Ausschwingbewe- gung von schwach gespannten Leiterseilen und des Fallseilzugs in Schaltanlagen bei Kurzschlüssen“, etzArchiv, vol. 5, pp. 103-107, 1983. [12] B. Olszowsky, „Computation of jumper swings of EHV-substations under short circuit currents“, CI- GRÉ SC 23-85 (WG 02) 36-IWD, 1985, unpublished. [13] B. Herrmann, N. Stein, G. Kießling, „Short-circuit effects in substations with strained conductors – Sys- tematic full scale tests and a simple calculation method“, IEEE Trans. Power Delivery, vol. 4, pp. 1021-1028, 1989. [14] G. Kießling, „Das Seilspannfeld als physikalisches Pendel – eine analytische Lösung der Kurz- schlußvorgänge“, Archiv für Elektrotechnik, vol. 70, pp. 273-281, 1987. [15] G. Kießling, „Kurzschlußkräfte bei Zweierbündeln – Messungen und analytische Lösung mit dem Pa- rabelmodell“, etzArchiv, vol. 10, pp. 53-60, 1988. [16] G. Hosemann, I. Landin, W. Meyer, „Progress in in- ternational standardization: mechanical effects of short-circuit currents in substations with flexible conductors“, in 4th International Symposium on Short-Circuits in Power Systems, Liège 3-4 Sep- tember 1990, Report 3.3, pp. 3.3.1-3.3.8. [17] E. Zeitler, „Contraction of bundled conductors in outdoor substations determined by the parabola model“, ETEP, vol. 3, pp. 285-291, 1993. [18] IEC Publication 909, Short-circuit current calcula- tion in three-phase a.c. systems, Genève: IEC, 1988. [19] G.A. Korn, T.M. Korn, Mathematical handbook for scientists and engineers, New York; McGraw-Hill, 1968. [20] M. Mathejczyk, N. Stein, „Kurzschlußseilzüge eng- gebündelter Doppelseile in Schaltanlagen“, etz-A, vol. 97, pp. 232–328, 1976. [21] Y. Serizawa, „Behaviour of dead-end suspension double-conductor bus during short-circuit”, JIEE of Japon, vol. 11, pp. 2269-2278, 1967