Estimate of Forces Exerted Against Spacers When Faulty Condition Occurs

IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS VOL. PAS-892 NO. 7, SEPTEMBER/OCTOBER 1970

Estimate of Forces Exerted Against Spacers When

Faulty Condition OccursHIROYUKI HOSHINO

Abstract-Certain problems exist concerning spacers in bundletransmission lines. The issues involve forces caused by fault current,

hybrid fluttering (galloping) of the bundle, vibration, etc.; otherproblems such as the wear, fatigue, and relaxation of the materialsused over a period also arise due to these forces. Because the existingliterature on forces exerted during a fault event are concerned onlywith specific conditions, it is difficult to predict the force applied foran entirely different condition. The summarized analysis describedhere has been developed to predict forces exerted against spacers

caused by fault conditions by using common basic calculations;experimental data, including those of the author, are provided forcomparison to support the estimates made.

INTRODUCTION

AMONG THE FORCES exerted against the spacer of a

Abundled transmission line, force resulting from a faultcurrent is the largest in a normal stringing condition. Althoughcertain experimental data exist concerning force created in theevent of fault current passing through the bundle conductor[1]-[5], it is difficult to predict this force under any differentstringing conditions. This is because the method of extendingsuch data to other stringing conditions might not have de-veloped formerly.For future development of bundled transmission lines under

new stringing conditions, the author has analyzed the quantita-tive relations between conditions such as conductor tension, con-

ductor weight per unit length, amperage and duration of faultcurrent, spacer frame elasticity, and force generated against thespacer.

For convenience in applying the new method of calculatingforce exerted against spacers over a wide range of conditions, theauthor has abbreviated and nondimensionalized the differentialequation of conductor motion and obtained a solution for thenondimensionalized equation. Consequently, a force exerted on a

spacer for a given stringing condition can be calculated byapplying a certain coefficient to nondimensionalized variables.

LIST OF SYMBOLS

CT coefficient for obtaining total exerted force; CT =

1.0,V 3, and 3.0 for two, three, and four bundles,respectively

c wavespeed on subconductor, m/sd diameter of subconductor, metersE nondimensionalized constant of spacer self-elasticity

Paper 70 TP 89-PWR, recommended and approved by theTransmission and Distribution Committee of the IEEE PowerGroup for presentation at the IEEE Winter Power Meeting, NewYork, N. Y., January 25-30, 1970. Manuscript submitted Septem-ber 4, 1969; made available for printing December 9, 1969.The author is with Hitachi Cable Ltd., Tokyo, Japan.

Ep practical value of spacer self-elasticity, kg/mFa attraction force between subconductors per unit

length, kg/mg gravity constant, 9.8 M/s2I subconductor current, amperes

n number of subconductorsp nondimensionalized variable corresponding to distance

z on line directionr nondimensionalized variable corresponding to time trCut.ff value of r corresponding to fault current cutoffs nondimensionalized variable corresponding to spacing

Tttow

x

XsXm ax

yz

Xssubconductor tension, kgtime, secondstime to conductor collision, secondsweight of subconductor per unit length, kg/mcoordinate in direction of conductor arrangementspacing of conductors at arbitrary point, metersspacing at point of spacer, meterscoordinate in direction upright to X and z

distance in direction of line, meters.

EQUATION FOR STRUNG CONDUCTOR AFFECTED BY CURRENT

In cases of bundled conductor lines, equal currents are carriedby each subconductor, causing forces of attraction between thesubconductors. These forces are given by the following formulaas the force between parallel currents:

(1)I2

Fa = 2.04 X 10-8 X X12

To obtain total force exerted against a subconductor, Fashall be vectorially summed up in a combination of all sub-conductors, such as the quad bundle shown in Fig. 1. The strungconductor equation can be simplified as explained below fora horizontal pair of subconductors.For subconductor I, illustrated in Fig. 2, the conductor

equation effected by current is given as

(2)T

2XI W d2XI 2.04 X 10-8 X 12T - =

CJZ2 g 4qt2 xs

where subscript I implies the position of subconductor I. Theright-hand term in (2) is force exerted by current; the firstterm on the left is reactive force exerted by subconductor tension,and the second term is subconductor inertia.

For subconductor II

T 92X11 W d XII 2.04 X 10-8 X 120Z2 g at2 XS

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IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, SEPTEMBER/OCTOBER 1970

aoFXs o I

CTxAC a AC -- AB + ADCT 3.0 r.o

Fig. 1. Vector summation of electromagnetic force.

III

Fig. 3. Solution of (8).

S0.3.

r cut off

Fig. 2. Illustration of horizontal arrangement.

Equation (3) is like (2) except for the direction of force in theright term.As for vertical coordinate y

T YI2 - WA/1 + (ayI/aZ)2 (4)

T 2_ = WV1 + (ay`II/Z)2. (5)

Because of the horizontal arrangement, Y1 = YII throughoutthe subconductors.By subtracting (3) from (2) and observing that X$ = XI -

XII

T al2Xs W a2X_ 2.04 X 10-8 X (6)2 dZ2 2g at2 XS

Fig. 4. Various values of collision time and r,,Xtoff.

CONDUCTOR MOTION IN CENTER OF SUBSPAN

Assuming that the subconductors are parallel and have in-finite length, the equation of subconductor motion due tocurrent is expressed as

Wa2X- 2.04X 10-8 X 122g at2 xs (7)

because all parts of the subconductors move identically, re-taining a parallel state, so that the first term on the left of (6)becomes zero. Then letting

itr =-/(4.08 CTg X 10-8/W) Xa

xmax

X's

Xs =Xmax

Sheer gravity force exerted against the conductor is activethroughout the entire duration of the short circuit. Therefore,calculation can be effected by aiming at the sagged plane onwhich the subconductors are strung.

Thus, the term due to gravity can be abbreviated. By similarcalculation, the gravity term is also abbreviated in the equationfor a vertical pair of subconductors, permitting calculation byaiming only at the spacing X8 between the subconductors, withcoefficient CT taking the total force.

when t = 0, and (7) is nondimensionalized. Then it becomes

a2sI 1ar2 s

(8)

Putting s = 1 and as/ar = 0 when r = 0, an integratedsolution is obtained as shown in Fig. 3. Making use of Fig. 3,the collision of conductors may be investigated. Although it wasan assumption to quote the subconductor length as infinite, the

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HOSHINO: FORCES EXERTED AGAINST SPACERS

solution is applicable as long as subspan length is sufficientlylong and the effect of the bundle spacer is not transmitted to thefull subspan before collision. In other words, the result in Fig. 3should not be used in a close spacer range up to length ct0 wherec = VITg/W is the wavespeed on a subconductor and to is timeuntil collision occurs. If the current is continued to collision, tois calculated as

to = X aV/ (4.08 CTg X 10-8) (9)

where the coefficient 1.25 is the value of r on collision.If the current is cut off before collision, the curve on Fig. 3

should be straightened at the cutoff point because inertia causesthe subconductors to move with constant speed without ac-celeration by attraction forces after cutting off the current.Thus, to is longer than in the case of continuous current. Thevalues of r where these straightened lines reach zero are shownin Fig. 4, with the variation of r,ut.ff.

TRANSIENT FORCE EXERTED AGAINST SPACERS

The above analysis described the movement of a subconductorfar away from a spacer; however, movement near a spacerwould be suppressed by the spacer, since a transient force wouldact on the spacer as a reaction of suppression. The motion equa-tion in a fault condition is expressed by (6) with a coefficientCT for summing up the total combined force. Letting

P= V/(4.08 CT X 10-8)/T XZXmax

r = /(4.08C gX 10-8)/W XmXmax

Xss=

Xmax

Xs =Xm ax

when t = 0, and the nondimensionalized equation for conductormovement is derived as

Is_ s = - while fault current flowsaP2 Or2 8

= 0 after current is cut off. (10)

For an initial condition, s = 1, Os/Op = 0 when r = 0. For aboundary condition, s = 1 where p = 0. The integrated solutionof a condition where the faulty current is continued is shown inFig. 5. In this integration, note that, due to collision, spacingof the subconductors never decreases less than the diameter;i.e., the values of s never fall below d/Xmax in the integratingprocess of (10).

In the nondimensionalizing operation of (10), the variables corresponds to conductor spacing X, p corresponds to thedistance in the direction of conductor z, and r corresponds totime t. Thus, subconductor motion can be visualized everymoment by relating Fig. 5 to these corresponding values. Afault current is usually cut off by a circuit breaker within afew cycles. Conductor motion in a condition involving shortduration of fault current can be calculated by supplying a

= 0.05

- 0.s- 0.6- 0.7

pFig. 5. Solution of (10).

7dgdp

6

S

4

3

2

0 0.S

Lo1.0 K 0.0o5

K

LS

0.6

1.0 L5 2.0 2.5 3.0' 3.5 A

r cuotf-0.4

1.0 4.5

Fig. 6. Relation of r and ds/dp with various values of rcutOff.

conductor constant speed motion in the middle subspan aftercutting off fault current.From the results of these calculations, force exerted against a

spacer can be derived as a product of conductor tension andbending gradient of the conductor in the vicinity of a spacer.Therefore, values of ds/dp relate to the gradient and proportionalto dXs/dz are shown in Fig. 6, with the variation of r and r,,t0ff.

DIFFERENCES OF FORCE EXERTED AGAINST SPACERS DUETO CONDUCTOR DIAMETER

Using a digital computer to obtain results of forces exertedagainst spacers, a certain difference was noted between theseforces that exists due to a difference among conductor diameters.The figures given in the former description were based on acondition that the ratio of d/Xmax is assumed to be 0.05. Sincethe maximum value throughout occurrence of a fault event ismost important for spacer strength design, these maximumvalues for a different condition of d/Xmax are shown in Fig. 7,with variations of rcu1off.The maximum value of ds/dp for d/Xmax = 0.07 illustrated

in Fig. 7 is less than a condition of 0.05. In a condition of 0.09,it is even smaller. This difference is explicit in the region ofrcutoff over 1.6. Consequently, a considerable difference ofmaximum force is caused when a fault current is continued untilmaximum force is generated at a point r z 2.0.

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1.0 1.5r cut of f

Fig. 7. Relation of routoff and (ds/dp)max with various valuesof d/Xmax.

EFFECT OF SPACER ELASTICITY

Some spacers are produced with a supporting frame thatpossesses a certain elasticity and is deformable in the event of afault current. After the fault current passes over, the framerecovers through its self-elastic property. Some experimentshave shown that force exerted against spacers can be reduced byutilizing such elastic spacers [3], [4]. Force exerted againstthese elastic spacers can be calculated by the modifying boundarycondition in the integration process of (10).The boundary condition of former integration for rigid spacers

is s = 1 at the point p = 0 because the spacing never changedat that spacer point. Regarding the elastic spacer, a force bal-ance would be maintained at the spacer point because a magneticforce is generated proportional to ds/dp, causing a reduction ofs. Thus, spacing s at the spacer point can be written as

s(1-s)E (11)dp

or by using a practical value,

T dXs - (Xmax - Xs)Ep Xmax(1 -s)Ep. (12)dz

From the relation of nondimensionalization

dsdp _ (1-s)E _ 1

T dXs Xmax(1 -s)Ep >K 108TCpdz

Therefore,

E = EpXmax (13)IA/4.08 X 10-8TCT

The relation of (11) must be kept throughout the integration;this is easily accomplished when programming by digital com-puter. A result of this calculation is shown in Fig. 8.

dpmax EIId,

d757 :E0 005 5 O s104Xmax 331XI 1--I 0I'

6

5-

4

0~~~~~~~~~~1

0O 0.5 1.0 1.5 .r cut oft

Fig. 8. Relation of rCt.ffand (ds/dp)max with variation of elasticity.

EXAMPLE OF CALCULATING FORCE EXERTEDAGAINST SPACERS

An example of force generated against spacers utilizing theabove results is shown below.

In this case T = 4030 kg, WV = 2.94 kg/m, Xmax. = 0.457meter, I = 12 000 amperes, tcutoff = 0.083 second, and n = 2

p V/(4.08 X 10-8)/4.030. 12 000 =- 0.0835z0.457

r= V1/(9.8 X 4.08 X 108)/2.94 = 9.66t0.457

r1utoff 0.083 X 9.66 = 0.8

s = s- = 2.18X3.0.457

Referring to Fig. 7, (ds/dp)max for rcutoff = 0.8 is 0.9.Therefore, maximum force exerted against a rigid spacer is

dXs 0.0835T - 4030 X 0.9 X 21 = 140 (kg).dz 2.18

This calculation, related to the case in Table I, was reported in[3]. If the spacer has an elasticity of 17 300 kg/m (i.e., 200 lb/in),the nondimensionalized constant E becomes

17 300 X 0.457

12 000/4.08)10- X430

Referring to Fig. 8, (ds/dp)max corresponding to rcutoff = 0.8is 0.75 for E = 1.0 X 102 and 0.28 for E = 3.0 X 101. Therefore,the force exerted against the spacer for E = 1.0 X 102 is

T dXs = 4030 X 0.75 X 0X0835 = 116 (kg)dz 2.18

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TABLE IFAULT CURRENT TEST RESULTS AND CALCULATED VALUES

Experimental CalculatedSpacer Impact Estimated Spacer Impact

Short-Circuit Force Time rms Current Force TimeCondition* (kg) (seconds) (kA X cycles) (kg) (seconds)

Experiments in [11]Two 954-kemil ACSR Spacers (kA X cycles)

A) Tension 7.7 43.2 10.8 31.52950-3180 kg 14.2 X 4.5 93.0 19.9 114

Conductor weight 15.5 150.0 21.7 1501.650 kg//m 17.5 191.0 24.5 215

Spacing 18.5 195.0 25.9 2650.406 meter 19.2 218.0 26.8 295

Experiments in 12]Two 795-kemil ACSR Spacers (kA)

B) Tension 36.4 543 6.5 6702090 kg 24.3 X 5 257

Conductor weight 4 1461.530 kg/m 3 95

Spacing0.406 meter

Experiments in 13]Two 91-Strand 5005 AAAC 1.657-inch

Diameter Spacers (A. s rms)C) Tension 1188 44.5 14.3 34

2380 kg 1202 45.0 14.5 37Conductor weight 1785 93.2 21.5 852.940 kg/m 1995 117.0 24.0 104

Spacing 1120 0.37 13.5 0.430.457 meter 1350 0.345 16.3 0.30

1650 0.205 19.9 0.221300 0.17 21.7 0.182520 0.11 30.3 0.112550 0.115 30.7 0.11

D) Tension 1188 49.2 14.3 464030 kg 1202 51.0 14.5 49

Conductor weight 1785 147.0 21.5 1082.940 kg/m 1995 183.0 24.0 140

Spacing 1250 54.6 15.0 500.457 meter 1700 109.0 20.4 95

2100 270.0 25.2 1581120 0.37 13.5 0.431350 0.365 16.3 0.301650 0.215 19.9 0.221800 0.17 21.7 0.182520 0.11 30.3 0.11

Experiments in 141(kA)

E) Tension 12.5 456 0.177 12.5 510 0.1763250kg 19.0 690 0.11 19.0 775 0.115

Conductor weight 29.0 1020 0.08 29.0 1180 0.0761.952 kg/m

Spacing0.4 meter

* A)-60 Hz asymmetric.B)-60 Hz asymmetric; no description for duration.C)-duration 0.083 second.D)-duration 0.083 second.E)-cutoff after maximum force.

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TABLE I (Cont'd)FAULT CURRENT TEST RESULTS AND CALCULATED VALUES

Experimental CalculatedSpacer Impact Estimated Spacer Impact

Short-Circuit Force Time rms Current Force TimeCondition* (kg) (seconds) (kA + cycles) (kg) (seconds)

Experiments in [6](kA peak X cycles)

F) Tension 13.7 X 8.8 75 9.7 522000kg 13.8 X 8.7 73.5 9.8 52

Conductor weight 14.0 X 8.9 75.0 9.9 561.320 kg/m 13.8 X 87 73.0 9.8 52

Spacing2 bundles, 0.4 meter

G) Tension 15.0 X 10 52 10.6 942000 kg 14.7 X 9.5 57 10.4 36

Conductor weight 14.8 X 9.5 55 10.5 391.320kg/m 15.1 X 10 48 10.7 41

Spacing 14.9 X 10 42 10.6 433 bundles, 0.4 meter,Xma. = 0.462 meter

H) Tension1600 kg 13.8 X 11 27.2 9.8 31

Conductor weight 16.6 X 10 33.2 11.8 411.110kg/m 18.3 X 9 51.8 13.0 45

Spacing 17.1 X 11 65.2 12.1 524 bundles, 0.4 meter,Xmax = 0.566 meter

F)-62.5 Hz.G)-62.5 Hz.

and the force exerted for E = 3.0 X 101 is

dXs 0.0835T s = 4030 X 0.28 X = 48.5 (kg).dz 2.18

The value for E = 52 must be between these two values.Consequently the amount of reduction from the force exertedagainst a rigid spacer due to its self-elastic characteristic isestimated at more than 24 kg but less than 96.5 kg-i.e., be-tween 17 and 69 percent of the force exerted against the rigidspacer. The experimental value for this condition reported in[3] was 30.4 percent.

COMPARISON OF CALCULATED AND MEASURED VALUES

For a further assessment of these calculations, a comparisonof calculated values against experimental values was made(Table I). Most of the experimental values compared wereextracted from data appearing in published papers. (Experi-ments on two-, three- and four-bundle conductors [5] in additionto tests reported in [1]-[4] are covered in the table.) In thiscomparison the most important point is the estimation of rmsvalue of the fault current. However, this was obscure in somepublications because the fault current had a dc component orasymmetry. In [3] this was shown as a product of ampere-seconds. In this case, the currents were derived by dividing bythe duration.The conditions in Table I are quite different for each paper,

conductor tension, weight of conductor, current amperage,duration-and, of course-the spacer force; however, thecalculations and experiments agree quite well despite such dif-ference in conditions.

CONCLUSIONS

The quantitative relation between transmission line condi-tions and force exerted against spacers explained herein can besummarized as follows.

1) The time to collision of conductors when a fault currentarises is proportional to the subconductor spacing and the squareroot of the conductor weight, inversely proportional to the faultcurrent. This calculation is applicable in the center of the sub-span where the effect of maintaining spacing by the spacers doesnot arise before collision occurs.

2) The effect of the spacer is transferred by the wavespeed ofthe conductor; thus the length transferred before collision is theproduct of wavespeed and time to collision.

3) In case the current is cut off before collision, the time tocollision is longer than the case of continued current because,rather than the conductor being accelerated, it moves by self-inertia after the current is cut off.

4) Maximum transient forces exerted against spacers areinfluenced by the duration of continued fault current. If thecurrent is cut off before a collision of conductors, the exertedforce-versus-time curve is an oscillogram displaying a flat topand a trapezoid-shaped base (shown in Fig. 6).

5) If the current flows for a longer time after collision, theexerted force curve demonstrates a sharp rise minus the flat top(Fig. 6) [4]. This causes a much greater force to be exertedagainst the spacer.

6) These results agree with experimental data not only re-garding maximum force but also concerning the shape of force-time curve in a wide variety of line conditions.

7) The influence of the conductor diameter on the forceexerted against the spacer is to some extent in the region ofcurrent continued longer than collision time.

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8) The proper self-elasticity of the spacer reduces the forceexerted against the spacer; however, the value of its elasticitymufst be designed appropriately. Larger elasticity is effective forshorter current duration, lesser elasticity is effective only forlonger duration. An elastic spacer must be so designed that it iscapable of recovering its original shape after expiration of thefaulty current.

REFERENCES[1] A. L. Mvalmstrom, L. G. Gifford, and J. 0. Smith, "Short-circuit

tests oni bundle conductors," Elec. Engrg., vol. 77, pp. 724-727,August 1958.

[2] R. L. Retallack, T. R. Fry, and C. A. Popeck, "Fault and loadcurrent testing of a bundle conductor spacer," IEEE Trans.Power Apparatus and Systems, vol. 82, pp. 646-652, October1963.

[3] J. R. Ruhlman, R. A. Eucker, and R. L. Swart, "Electrodynamicstudies of bundled conductor spacers," IEEE Trans. PowerApparoatus and Systems, vol. 82, pp. 750-760, October 1963.

[4] C. Manuzio, "An investigation of the forces on bundle conductorspacers under fault conditions," IEEE Trans. Power Apparatusand Systems, vol. PAS-86, pp. 166-184, February 1967.

[5] K. Hayashi, Y. Suzuki, S. Yamamoto, H. Hoshino, and S.Yamazaki, "Shortcircuit and twist characteristics of multi-conductor transmission lines," Hitachi Rev., vol. 8, p. 24,July 1959.

The mechanical impulse is given by

Fadt = - t04s

(15)

and the mechanical impulse is equal to the change in momentum:

K12 t = mv,4s

(16)

where m is the mass per unit length and v, is the peak velocity ofan element.Assuming the subconductor motion to have the form x = x,

sin ct, the maximum velocity at any point along the length of thesubspan can be expressed in terms of the peak transverse deflectionat that location:

Vz = X = X2 j vT/rm (17)

where x, is the peak transverse deflection of any element, co is thefundamental frequency in radians per second, L is the length of thesubspan, and T is the subconductor tension.

Since the peak deflection will vary from 0 at the rigid spacer to amaximum at the center of subspan in the form x = Xo sin 7rz/L, themomentum of the entire system can be written in terms of the peaktransverse deflection at z = L/2, the midpoint of a subspan:

LKl t = L Xom -TM4s ir L

(18)

where Xo is the peak deflection at L/2, the midpoint of the subspan.Rearranging terms gives

X = KI2 tr 1 L-VmiT.2s 2m -r (19)

James C. Poffenberger (Preformed Line Products Company,Cleveland, Ohio): My compliments to Dr. Hoshino on an interestingand informative paper regarding the effect of fault currents on

bundled conductor spacers. Dr. Hoshino's use of charts simplifiesthe calculation of spacer forces for a wide range of line parameters,spacer rigidities, and fault current conditions. For a special caserelated to United States practice and rigid spacers, the spacer forcecalculation can be further simplified.

In the United States, the general practice on EHV lines is toattempt to clear faults rapidly. This rapid clearing leads to a simpli-fication for the special case of rigid spacers, twin conductors, andshort fault durations. Briefly, the approximate analysis assumesthat the electrodynamic forces of attraction are applied as a singleimpulse at the initial separation distance and that the subconductorsaccelerate almost instantaneously to maximum velocity. In actu-ality, of course, the subconductors start from rest and acceleratetoward each other, with the force of attraction increasing as theseparation distance decreases.With the force of attraction assumed constant during the fault,

the velocity of the conductor at the end of the impulse can be calcu-lated directly from the electrodynamic input. Assuming that theforce impulse excites a single-loop sinusoidal motion over the sub-span, the shape of the deflection curve can be found in terms of lineparameters. This leads to the force reaction of the conductor on thespacer.That is, for two parallel subconductors, and using Dr. Hoshino's

notation wherever possible, the force of attraction at the initialseparation is

KI2F. = (14)4s

where I is the total rms current flowing in the two subconductors.K is a constant of proportionality, and s is the initial separationbetween subconductors.

Manuscript received February 20, 1970.

The transverse reaction at a spacer from the subspan underconsideration is the product of the conductor tension and the slopeof the conductor at the spacer, dx/dz = Xo 7r/L, when the transversedeflection is at a peak:

(20)2 = X0 L T

where P is the spacer reaction.Consequently the total spacer reaction may be found by sub-

stituting (19) into (20) and multiplying by 2 to account for thereaction from the adjacent subspan:

w,rKI2ti1LV/Tp 4\1KwL T T.4 s m r L

(21)

If the pound/foot/second system of reference is used, K = 4.5 X10-8 and with m = WVg, the spacer force reaction becomes

(22)p 4.50 X 10 8

4 s

If symmetric faults are imposed, then the rms value can be useddirectly. In the case of asymmetric faults, the effective value ofcurrent and time may be determined from oscillographic recordsand noted as an electrical impulse in terms of amperes and seconds.This was the procedure followed by my colleagues in [3]. Whereelectrical impulse is used and not reduced to an equivalent rms value,then time must be entered in the denominator as shown in (23). Ifthe accelerations due to gravity and other constants are combinednumerically, the resulting equation provides a method for directcalculation of spacer forces

(23)P = 0.20(14 -VT/Were) s t

where.le is the electrical impulse in ampere-seconds.

Discussion

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TABLE IIFORCES IN TWO-CONDUCTOR RIGID SPACERS FROM SHORT-DURATION FAULTS

CurrentSubspan Peak Asym- Effective FOrCe.Length metric rms Time Impulse Calculated Measured

Spacers (feet) (kA) (kA) (seconds) (A - s) (pounds) (pounds)Experiments in [s], PLP

Tension 150 9.50 5.780 0.100 578 24 155250 pounds 150 51.0 28.650 0.058 1662 349 320

Weight 300 37.0 18.0 0.095 1710 226 1811.946 lb/ft 300 22.0 12.89 0.098 1263 119 106

Spacing 250 21.0 12.88 0.100 1288 122 1091.417 feet 250 20.0 12.90 0.082 1058 100 95

250 27.0 18.88 0.082 1524 208 181250 30.0 19.90 0.084 1l672 244 212250 45.0 27.0 0.084 2268 449 431250 42.0 26.4 0.083 2191 424 385

Tension 150 37.0 18.0 0.095 1710 294 2448900 pounds 150 22.0 12.89 0.098 1263 155 113

Weight 250 21.0 12.88 0.100 1288 158 1541.946 lb/ft 250 20.0 12.9 0.082 1058 130 104

Spacing 250 27.0 18.58 0.082 1524 270 2731.417 feet 250 30.0 19.9 0.084 1672 318 347

250 45.0 27.0 0.084 2268 585 652250 42.0 26.4 0.083 2191 552 575300 42.0 25.62 0.083 2126 520 595

Experiments in [1], Detroit EdisonTension 200 19.7 8.85 0.111 981 103 94

6750 pounds 200 31.2 15.25 0.079 1204 218 208Weight 200 36.7 16.4 0.077 1261 245 331

1.075 lb/ft 200 55.2 24.9 0.068 1693 501 484Spacing 200 49.7 23.0 0.069 1587 434 448

1.333 feet 200 41.3 18.95 0.080 1517 342 430

Experiments in [51, HitachiTension 9.7 0.1405 1365 143 1654400 pounds 9.8 0.139 1361 143 162

Weight 9.9 0.142 1410 150 1650.885 lb/ft 9.8 0.139 1361 143 161

Spacing1.310 feet

Thus, for short-duration faults on two-conductor bundles theforce in rigid spacers is a function of current magnitude and timeduration, subconductor tension and weight, and initial separationbetween subconductors. The length of subspan does not enter intothe relationship.

Spacer force calculations based on (23) are compared to previouslyacquired test data in Table II and Figs. 9-13, and there is generallygood agreement between experiment and theory. Experimental datafrom my colleagues' Preformed Line Products Company (PLP)work [31 were obtained during asymmetric faults and electricalimpulses levels were determined from the oscillographic records.The Detroit Edison tests [11 also included asymmetric faults, andthe authors were good enough to make the oscillograms of theirpioneering experiments available to my colleagues for detailedexamination some years ago. The electrical impulses shown for theHitachi data [5] were based on the time duration and effectiverms values shown in the present paper by Dr. Hoshino.

Fig. 9 compares experiment and theory from the PLP work at aconductor tension of 5250 pounds. Time is included in the ordinate,since there was variation from one test to another in fault duration.The separation distance was 1.417 feet, or 17 inches. (In the PLPwork, we made force measurements at 17-inch separation and wetested prototype commercial spacers empirically at 18-inch sep-aration.) Fig. 10 shows data obtained in PLP testing at a tension of8900 pounds and,a separation of 1.417 feet. Fig. 11 compares DetroitEdison experiment with theory at a separation of 1.33 feet or 16inches. The tension used for calculation was 6750 pounds.As shown in Fig. 12, a family of curves is generated by the simpli-

fied theory-a different curve for each significant variation in theparameters of subconductor separation, weight, or tension. Tocompare all the experimental data against a single curve, it is neces-sary to include these parameters, as well as fault duration, in the

ordinate of a composite display graph; this was done in plottingFig. 13.

Fig. 13 includes the four pertinent test data from the presentpaper, and the composite display shows the relation between experi-ment and theory for a representative range of conductor configura-tions and short time faults.For short-duration faults and line parameters generally used in

the United States, the simplified formulation can be used for rapidcalculation of compressive force in rigid spacers. The tensile force inrebound will be of approximately the same magnitude. In the case offlexible spacers, the forces will be smaller, as shown by my colleaguesand others; how much smaller depends on the degree of spacerflexibility. He-re a more exact design analysis must follow the rigorouissolutions, as in the case of long fault durations.

If (23) is modified to account for the division of current in three-and four-conductor bundles and if Dr. Hoshino's values for CTand Xmax are used, calculated values of spacer force follow the sametrend as shown by the experimental data in the paper. However,the values calculated from the simplified theory are higher than themeasured forces, averaging 11 percent greater for the three-conductordata and 30 percent greater for the four-conductor results. Becausefault times werelong, up to 11 cycles, the calculated values should havebeen lower than the measured forces, but there may have been otherfactors involved. For example, the first and fifth tests on three-conductor bundles show the same fault intensity, 10 cycles at 10 600amperes rms, but the experimental values vary by 24 percent, 52to 42 kg. The corresponding theoretical forces shown in the paperhave an even greater variation. Would Dr. Hoshino be good enoughto clarify my understanding of these variations for what wereapparently identical test conditions.

Returning now to United States practice and field experience, formy part I have not heard of any spacer difficulties in the United

1482

1483HOSHINO: FORCES EXERTED AGAINST SPACERS

U

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T =5250 LB60 W = 1I.95 LB /FT L -

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THEORY20

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iv 0 500 1000 1500 2000 2500 3000ELECTRICAL IMPULSE (AMP-SEC)

Fig. 9. Experimental data acquired on 600-foot outdoor spanwith asymmetric faults ranging in duration from 0.058 to 0.100second.

6 DETROIT EDISON

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Fig. 11. Data from tests in [11 with 200-foot subspans andasymmetric faults to 55 200 amperes peak.

zt 6.0o.0 MPOI

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Tests conducted with subspan lengths of 150, 250,and 300 feet.

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C.)0-O 500 1000 1500 2000 2500 3000ELECTRICAL IMPULSE (AMP-SEC)

Fig. 12. Spacer force is function of subconductor separation,tension, and weight, as well as fault intensity.

Fig. 13. Experimental data from three laboratories in generalagreement with simplified theory for two-conductor bundles,rigid spacers, and short fault duration.

IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, SEPTEMBER/OCTOBER, 1970

States arising from fault currents. This includes rigid spacers,flexible spacers, and spacer dampers. Within our experience, me-chanical problems pose a much more serious threat to spacer per-formance, not only in the United States but throughout the world.These problems summarized by Dr. Hoshino in his abstract-hybridfluttering, vibration, and wear, fatigue, and relaxation of materials-arise primarily from wind-induced phenomena. Although the faultcurrent-spacer force mechanism is now well defined, thanks tothe contributions by Dr. Hoshino and other investigators referencedin his paper, subconductor oscillation and methods to suppress it arenot clearly understood at this time. It is hoped that the work nowbeing done will clarify the wind-induced problems to the sameextent as the electrical problem.

H. Hoshino: The author would like to show that (23) agrees withhis calculation in the limited region of fault current duration.

In a case of a two-bundle conductor, the spacer force P is calcu-lated as

P = Td? S = A/4.08 X 10-8 X T I (s

where (ds/dp)ma:x relates the maximum of nondimensionalizedgradient of subconductor at a spacer.For an approximation of (ds/dp)max, the relation of rcutff to

(ds/dp)max shown in Fig. 7 can be equated as

(ds)mx Vcutoff = \/4.08g X 10-8/W Itutaff\dp)max Xmax

in the region of 0 < r,ut,ff < 1.2Therefore

T dXs = V/4.08 X 10-8T I /4.08g X 10-8/W Itectoffdz Xmax

= 4.08 X 10-8 Tg/W utoffX max

This equation is similar to (22) and can be rewritten by the dis-cusser's notation of electrical impulse:

P = 4.08 X 10-8 (I) 1 V\T/wVWS t

which is similar to (23) except for the proportional constant due tothe differences entailed by the metric system and the pound/foot/second system.

Manuscript received April 24, 1970.

Consecutively, the author would like to describe the standpointand the progress on the development of this theoretical calculationof spacer forces.

In 1957, a fault current test was conducted by Hitachi CableLtd., in Japan and the experimenters obtained the principal data inthe fault event. The author then tried to calculate the spacer forcestheoretically for the various stringing conditions and fault condi-tions by a simplified calculation that took the impulsive momentumin calculation as a product of the current duration and the electro-magnetic force at a fixed value of subconductor spacing. This methodcould achieve results similar to those of the discussion. However, adevelopment of this method was given up because the bases did notagree with the facts that had been shown experimentally.The principal points that did not agree with the experiments are:1) The subconductors in the middle part of subspan retain a

parallel state at a time before collision. The subconductors nearthe point of spacer form a curve. By observation of a photographicrecord, the moving subconductors in a fault event are seen to be agroup of parallel straight lines with the curved part in both ends.The length of the curved part grows with the wavespeed on the sub-conductor.

2) The subconductors collide with each other even if the faultcurrent has been cutoff before collision.

3) The parallel part of the subconductor collides at the same timein the middle part of the subspan except in the case of a short sub-span length.

4) Consequently, the shape of a moving subconductor is not asinusoidal or parabolic curve.

5) The subconductors move a considerable distance within thefault current duration even if the current is cut off by fast operationof the breaker. Consequently, there are some theoretical difficultieswhen the impulse is calculated from. a constant value of spacing. Thespacing should change with time in the calculation.

6) The waveforms calculated theoretically shall agree, or at leastbe similar to, the experimental results, which are a trapezoid if thecurrent is cut off before the subconductor collision and a sharptriangle waveform if the current flows for a longer time.

In light of factors 1)-6), the author intended to consider a newmethod based on the partial differential equation. This was difficultbecause the numerical solution of such an equation is possible onlywith a digital computer having a great memory. With the develop-ment of such computers, the solution for a given condition hasbecome possible by numerically solving partial differential equation.These computer results showed considerable agreement with experi-mental results, especially with respect to waveforms.

Subsequently, the author improved this calculation by utilizingdimensional analysis techniques for comprehensive use of this result.The discusser's equation of spacer force could be used in the

limited region 0 < rcuwff < 1.2 so that the current is cut off beforethe conductors collide.

If the current flows longer time, the value of (ds/dp)max shall notbe equated as proportional to r,0utfff. Therefore, the simplifiedequation is in error.The principal advantages of this paper are its indications of a

wide region of application and of a capability of showing the wave-form of spacer force.

1484

Documents

Estimate of Forces Exerted Against Spacers When Faulty Condition Occurs