Simulation of the dynamic

response of transmission lines

in strong winds
M. J. M a t h e s o n and J. D. H o l m e s
Department of Civil and Systems Engineering, James Cook University of North
Queensland, Townsville, Queensland, 4811, Australia
(Received March 1980; revised October 1980j

A numerical simulation procedure for predicting the response of a single

span transmission line to strong turbulent winds is described. The wind
velocities are generated using a 'Monte Carlo' technique based on an
inverse fast Fourier transform; the equations of motion of the line are then
solved numerically using a finite difference method. Results obtained
using the method were compared with those from linearized random
vibration theory. Effects due to the mean swing angle of the line, and due
to the excitation of an in-plane mode of vibration were apparent.
Introduction basis. In this way, all nonlinear effects are included. The
The loading on transmission lines and their supporting simulated wind records are generated using a 'Monte Carlo'
towers due to wind is an extremely important factor in simulation procedure which recognizes the random nature
their design. A more accurate knowledge of the oscillatory of natural turbulence and allows for a realistic representa-
behaviour of transmission lines in strong winds, and of the tion of the correlation properties of the turbulent fluctua-
loads transmitted to the towers, is very desirable to produce tions along the length of the span, drawing on accumulated
more economical designs. knowledge of actual measurements of these properties in
The traditional design method has been based on the use the natural wind. The method of wind record generation
of a peak gust wind speed, i.e. the largest recorded or pre- has been described fully by Holmes, 6 and has been pre-
dicted wind speed expected to occur once in a specified viously applied to compute the response of a cable stayed
return period. However, it was recognized many years ago bridge .7
that to use the total drag force, calculated on the assump- A brief summary of the method of wind record genera-
tion that the design gust speed acted simultaneously along tion is given in the next section. Then the partial differential
a 300-500 m span, would be extremely conservative. 'Span equations of motion of the line and their numerical solution
reduction factors' have been applied to allow for this effect. by finite difference methods are discussed. The interfacing
The factors presently used have often been empirically of the two numerical procedures and the implementation
derived from the actual measurements on test lines - to compute the response of a typical transmission line span
usually, however, at wind speeds much lower than those is discussed, followed by some results of the computations,
representative of design situations. and a comparison is made with equivalent calculations using
In recent years, a more rational approach, using random random vibration theory.
vibration theory, has been advocated by several authors. 1-5
In this type of analysis, which has also been applied to other
S i m u l a t i o n o f wind velocities
structures, the reduction effect due to the tack of correlation
of wind gusts along the span, and the main features of the The basis of the numerical simulation of the wind velocity
dynamic response of the line, are taken account of in a records is that the random fluctuating or turbulent com-
rational manner. However, the method involves lineariza- ponent of the wind velocity at any point can be represented
tion both in the aerodynamics and in the line dynamic as a summation of cosine waves of N equally spaced fre-
behaviour. A linear relationship between the fluctuating quencies, with phase angles randomly distributed between 0
drag forces applied to the line and the upwind velocity and 2n radians. The amplitude of each cosine function is
fluctuations is assumed, and a linearization of the equations chosen so that its contribution to the total mean square
of motion of the line, is also made. In order to achieve the value is equivalent to the area under a specified target
latter, it is assumed that the line tension remains constant spectral density curve within a chosen frequency interval,
during the motion - this assumption avoids any considera- An. In the present case the target spectum was the yon
tion of the extensibility of the line. These assumptions have Karman-Harris wind velocity spectrum:
been questioned particularly at high wind speeds.
1.04 XuI2
In the present paper, a simulation procedure is described
in which representative wind velocities, forces, line response Su(n)- [2+ 3 (nXu~]J (1)
and lateral tower loads are computed on a step by step

0141-0296[81/020105-06/$02.00
© 1981 IPC Business Press
Eng. Struct., 1981, Vol. 3, April 105
Dgnamic response of transmission lines: M. d. Matheson and J. D. Holmes

where: Xu is a specified wavelength in metres at which In this case, the reference state was taken to be the hori-
nSu(n) is a maximum I~ is the intensity of turbulence and zontal straight line between the two supports.
a is the mean wind velocity. Whe re:
A very efficient numerical procedure is obtained by using
e = strain relative to reference state
an inverse fast Fourier algorithm to carry out the summa-
tion described above. In this case, the number, N, of s = so(1 + e) (5)
generated values in the simulated wind velocity sequence is P = po/(l + e) (6)
equal to the number of discrete frequencies into which the
specified frequency range is divided. The time increment as
-- (1 + e) (7)
between values, At, is equal to the reciprocal of the chosen 3So
maximum frequency.
Simulated random wind velocity records sampled along Equation (4) can be expanded into cartesian coordinates
a horizontal line normal to the mean flow direction are x, y, z, as follows:
generated by summing a number of complex number
vectors before performing the inverse fast Fourier transform. Po at z (1 + e)
Each vector is required to provide the correct correlation
relationship with previously generated records.
The form of the normalized co-spectrum suggested by aZy a ( T ~ o ) + f ,
Peat 2-3s0 (l+e)
(9)
Davenport 8 is used:

Po32z
~at =
~So( (1 T+ e) 32o) +fz (10)

These are the three partial differential equations of

where d is the separation distance and c is the decay con-
motion. To provide the basis for a solution algorithm,
stant. For a full exposition of the simulation method, see
further equations relating tension to strain and strain to
Holmes. 6
displacements are necessary. The strain in an element rela-
tive to the reference state:
C o m p u t a t i o n o f line r e s p o n s e
5s - 5So
Partial differential equations e (11)
8So
The mathematical equations of motion for an extensible,
perfectly flexible, suspended line are presented below. These by definition, and:
are derived by applying Newton's second law of motion to T =AE(e eo)+Tl (12)
a small element of line, 5s, as in Figure I. The basic
Where: A = area of cross-section; E = nrodulus of elasticity;
equation is:
eo = strain in an element in static state relative to reference
state and T~ = tension in an element in static state.
--=-- r +f (3)
P 3t 2 as The following assumptions have been lnade to imple-
ment the random vibration analysis mentioned ill the
where x = displacement vector from origin; p = mass/length; introduction.3
t = time; s = curvilinear distance along line; T = tension and
(a), the tension T is assumed constant throughout the cable
f = force/unit length of line.
Alternatively, by establishing a reference state, denoted and equal to the horizontal component under no wind
by a subscript '0', equation (1) can be expressed as loading, H.
(b), deflections in the x and z direction are neglected.
follows:
(c), the term 3/3s is approximated by 3lax.
Hence equation (3~ is reduced to:
po at 2 (1 + e) aso/ f(1 + e) (4)
32Y=H32Y+ f~ (13)
P ~at ~
ax '
A linear equation, like (1 3), can be solved by random
vibration theory.

Numerical model
The finite difference equations are discretized versions
of equations (8)-(12), given above, for a single span, fixed
ended line. Two numerical algorithms have been studied,
Lin~, element
one being a direct explicit approach 9 and the other a semi-
implicit-explicit approach, lo,n The latter model allows a
z~Cz -x

L: reduction of computation time by a factor of about ten.

Both sets of finite difference equations are very similar,
and a list of those corresponding to Watts' method are
presented below:
X?++I 1 = Xf + 1 + { [h(l + ein++,):)]2
_[y/n++, y?+I]2 _ [Z~++: _Zfl+t]2}u2 (14)
Figure I Suspended line and element

106 Eng. Struct., 1981, Vol. 3, April

 Dynamic response o f transmission lines: M. J. Matheson and J. D. Holmes

y/n+] = 2Y/n - Y / n - ' + (Ar/h)2{R~+'/2Y~+I Implementation

- [Ri" + '/2 + Rj" - ½] Y/n + R jn- ½ Y ] - In} The numerical procedures described in the previous two
+ (Fy)/n Ar 2 (15) sections were implemented to calculate the response of a
particular transmission line on a DEC-10 machine. Table 1
z/n +' = 2 z / n - Z / n - I lists the wind and transmission line properties used.
n tl n l~
- [Ri+,/2 + R / _ ½]Z) + R i _ ½ Z ] _ 1}
It The instantaneous wind velocity, generated as described
in the second on the simulation of wind velocities was con-
+ (Fzff Ar 2 (16) .verted to a result drag force per unit length using equation
Rn+l
i+1/2_ { [X/n+1_ 2X/n + X/n-'](h[Ar) 2
-
(19):
+ [Xln + 1 re'n+llon--lltrvn+l
--Ai_IIIXj_½SILAj+ 1 --A i
vn+l]
J
(17) fr(t) = ½PairCaDu2(t) (19)
cn+l where ur(t) is the instantaneous relative wind velocity
i+'/2 = [(AE/TI)(1 + Co)- 1]/[(AE/T1) 2 - R 7 ~ 2 ] - 1
between the air and transmission line. Thus, aerodynamic
(18)
damping effects were implicitly included in the calculations.
where: The air density Pair, was taken to be 1.20 kg/m 3 and the
drag coefficient, Ca, was assumed to be 1.0.
l = span
X, Y, Z = x/l, y/l, z/l The time increment for the wind simulation programe
was dictated by the stability requirements of the f'mite
Fy = dimensionless distribution wind force on an
difference program. Stability is ensured when:
element = f yl/Tl
Fz = dimensionless gravitational force = fzl/T1 T x/2 At
h = dimensionless space step = Axfl |/ --<1 (Watts 1°) (20)
Ax
Az = dimensionless time step = At(Tl/p)t/2[l
R = T/[T,(1 + e)] For the results described in this paper, a space step, Ax,
s.n = time n At, node j of 10 m, and a time step of 0.057 s were used. Thus the
transmission line span was divided into 30 elements. Wind
The method outlined by Watts~°has been adopted in
velocities and forces were generated at 15 points along the
this study, due to its superior computational efficiency. span. This number was dictated by computing time limita-
tions; however, an increase in the number of points where
Testing
wind forces were applied, was found to produce no signifi-
To clarify any doubts about the accuracy of the finite cant change in the response parameters. Due to limitations
difference algorithm, extensive tests were performed. Un- on computer core space, the number of generated points
fortunately, it is difficult to establish complete validity for in the velocity and response sequences was restricted to
the finite difference equations. The partial differential 4096; this gave a total record length of 233 s. However,
equations are nonlinear, and hence numerical stability the random errors in the response calculations were mini-
cannot be proved through analytical means alone. However, mized by ensemble averaging over a number of runs, each
by running the program and producing results for known one using a different sequence of random numbers in the
solutions, rigorous comparisons can be made. wind record generation.
Tensions and displacement along a particular line when
subjected to a constant load, e.g. a constant steady wind, Results
can be compared with exact solutions. A number of such
tests were undertaken and the results showed excellent Response characteristics of most interest here, are loads
agreement with the exact solutions. An indication of the exerted on supports and line deflections. Time-history plots
performance and accuracy of the system in a dynamic of a typical single wind record, corresponding centre span
situation was achieved by a free vibration analysis. Natural sway angle response 0, and lateral support reaction R, are
frequencies obtained from numerous tests were compared given in Figure 2.
with those predicted analytically by Irvine and Caughey 12 Significant features from Figure 2 are: the sympathetic
and again showed good agreement. line response to the excitation force, contribution of
Roussel 9 conducted a number of tests to compare the particular frequencies to response; and the degree of
response given by his numerical model with those from fluctuations in response. The sway angle appears to be
measurements. The agreement was good. dominated by the low wind frequencies and the first sway
Further verification of the method outlined in this mode frequency. However, the support reaction appears
paper was achieved by comparing some responses with more wide band, similar to the wind velocity record and
those calculated by Roussel's method; the computed shows higher frequency components than the sway angle.
responses were essentially identical.
Mean and r.m.s, fluctuating response
Test runs using the simulation model were run for mean
Table 1 Wind and line properties
wind velocities ranging from 10 to 20 m/s. Mean and r.m.s.
Intensity of turbulence,/u 0.15 responses recorded in Table 2, represent the ensemble
Decay constant, c 10 average of four runs at each mean wind velocity.
Peak wavelength, Ku 700 m The r.m.s, lateral support reaction is plotted against
Mass density of line, p 1.69 kg/m mean wind velocity in Figure 3. Simulation points represent
Line diameter, D 29.3 mm
averaged values, while the bar-lines indicate the maximum
Span, I 300 m
Horizontal tension (no wind), H 27.5 kN and minimum values obtained. Comparison is made with
Young's modulus, E 68 GPa the solution given by random vibration theory. Straight
lines on this graph represent power law relationships.

Eng. Struct., 1981, Vol. 3, April 107

 Dynamic response of transmission lines." M. J. Matheson and J. D. Hohnes
100r-

20

o so~

a 40 l Simulation
# v model DOi
c"
U3 o_ 0 2 0 --

/
20 I I I I
l
4
O_ o
*j 9
o
2
010-
o
&
co 0
u~

0 50 100 150 200 E

L
Time, (s)
Figure 2 Typical simulated response records vibration theory
0 05 -

Table 2 Mean and r.m.s, fluctuating responses

0 (m/s) /~ (kN) ,d~7~(kN) 6 (deg) ~ ' ~ 5 (deg)

10 0.261 0.0492 6.19 1.04

20 1.04 0.201 23.4 3.89
30 2.35 0.458 43.9 5.81 002 / I L___ Jt . . . . .
5 10 20 30 50
Mean velocity, ( m / s )
Figure 3 r.m.s, support reaction
Differences between the two solutions increase with
higher mean wind speeds, varying from 6% at ti = 10 m/s, 5
to 9% at ti = 30 m/s. These differences can be explained by
the contribution from fluctuations in line tension not
accounted for in random vibration theory, which increases
with higher mean velocity. Linearization of forces in random
vibration analysis also may contribute to the underestima- 4
tion of support loads.
The slope of the random vibration theory line was calcu-
lated as equal to 2.01, whereas a line drawn through the
simulation model points exhibits a slope of 2.03. If dynamic
behaviour of the line was ignored (e.g. a rigid bar instead of 3
a line) then the support reaction would be directly propor- 7
tional to velocity squared, i.e. the slope would be 2.00. z
These results appear to indicate that the exclusion of reso- ~z
nant dynamic response when estimating the support
reaction R , may be justified. ~
v
2

Sp ec tra
The spectrum of the support reaction is plotted in
Figure 4 for fi = 30 m/s, for both the simulation results and
for the linearized random vibration theory.
Immediately obvious from this figure, is the close agree-
ment between the spectra at low frequencies, and the 1st sway 1st in-plane
small contribution of the resonant frequencies to the total mode mode
mean square value. The latter observation is attributed to
the higher aerodynamic damping at high mean wind speeds
0 01 02 03 04 05 6
(20-25% of critical for ti = 30 m/s). Small resonant peaks
Frequency, ( Hz )
occur at the first sway mode and first in-plane mode, the Figure 4 Spectral density of support reaction (O = 30 m/s).
latter appearing in the simulation model spectrum only. (-- --G-- --), simulation (nonlinear); ( ), random vibration theory
However, at low velocities the resonant frequencies are (linear)

108 Eng. Struct., 1981, Vol. 3, April

 Dynamic response of transmission lines: M. J. Matheson and J. D. Holmes

more pronounced in the spectra because the aerodynamic Table 3 Span reduction and gust factors
damping drops. Although the inplane mode contribution
Random vibration
is significant in amplitude relative to the random vibration
Simulation model theory
theory value at this frequency, it is quite apparent that 0
overall it makes little difference to the total mean square (m/s) a G e G
and r.m.s, values. This is also indicated in Figure 3.
A comparison of line deflection response between the 10 0.662 1.55 0.626 1,46
20 0.665 1.57 0.628 1.48
two methods is of limited use. The use of random vibration
30 0.685 1.62 0.630 1.49
theory to predict deflections as expounded by Manuzio
and Paris 3 does not allow for the mean deflection of the
line under wind. The latter has two effects: the moment
arm of the horizontal wind loading is reduced and the simulation model are an ensemble average of four runs.
moment arms of the restoring gravitational forces are The span reduction factors, in both cases, have been
increased. The two effects combine to produce reduced calculated using a maximum velocity Umax, derived using
sway angles and lateral deflections at high wind speeds. equation (i) and the method of Davenport, 13 with a cut off
The spectrum of sway angle from the simulation method is frequency of 8.8 Hz and a time duration of 233 s. These
shown in Figure 5. The prominence of the first sway values correspond to the values used in the simulation
mode frequency can be seen in this figure. method. The value of Umax/~ so determined was close to
1.53 for all mean wind velocities.
Design factors In all cases, the factors given by the simulation model
The span reduction factor tx, for support reaction is a higher than the corresponding ones from the random
defined as the ratio of the maximum expected reaction, R, vibration theory. For both solutions, the factors vary little
to the unmodified reaction calculated from the peak gust with increases in velocity. Comparison between the two
velocity, i.e. : methods shows that gust factors and reduction factors
differ by about 7%.
Rmax Limited tests were conducted with an intensity of turbu-
o~ = I 2 (21)
~PairCaDumaxl/2 lence of 0.30. The simulation model results gave 20-25%
increase in G.
AS mentioned earlier in the paper, this reduction factor
accounts for the reduction in correlation of wind with
distance along the line, as well as dynamic effects. Conclusions
The peak velocity, Umax, in the denominator is difficult (1) The simulation allows the nonlinear behaviour of trans-
to define, as it depends upon the frequency response of the mission lines to be treated, and highlights the significance
instrument used to measure it, and on the duration of the of in-plane modes in the response. Excitation of in-plane
record. This results in estimates of e having considerable modes is not achieved in random vibration theory because
variability.
linearizing assumptions suppress these modes.
The gust factor, G, is defined as in (21), except t~ is used (2) Conventional linearized random vibration theory
instead of Umax, and represents the ratio of peak to mean slightly underestimates the r.m.s, support reaction.
response. Table 3 shows reduction and gust factors obtained (3) Results have supported the conclusions from random
for various velocities, where again results presented from the
vibration theory that at high velocities, dynamic response is
not dominant due to the high aerodynamic damping.
020
(4) Conventional random vibration theory needs to be
modified to consider the fluctuating behaviour about the
mean deflected position, if it is to predict the line deflec-
tions accurately. The simulation method described in this
paper can be used to predict deflections realistically.
0.15 (5) Span reduction factors and gust factors are somewhat
higher than those currently used in design. An increase in
I G of 20-25% occurs with a doubling of wind turbulent
intensity.
Future developments should include the incorporation
% o~o of vertical velocity components and hence vertical aero-
dynamic forces.
Although random vibration theory has shown itself
acceptable for determining support reactions, it shows
° / significant deficiencies when estimating line deflections.
005- Therefore, the simulation approach could prove most
1st swoy useful in studying the clashing of parallel lines in strong
mode
winds.

Acknowledgements
I I ~ 91 I
0 01 0.203 04 05 06 The support and encouragement of Mr B. J. Bulman of the
Fnzquency, (Hz) Queensland Electricity Generating Board for the work
Figure 5 Spectral density of centre-span sway angle. (--o--), described in this paper is gratefully acknowledged by the
simulation. 0 = 30 m/s authors.

Eng. Struct., 1981, Vol. 3, April 109

Dynamic response o f transmission lines." M. J. Matheson and J. D. Holmes

References f force/unit length

F dimensionless force
1 Davenport, A. G. 'The response of slender line-like structures
G gust factor
to a gusty wind',Proc. LC.E. 1962, 23,389
2 Davenport, A. G. 'Gust response factors for transmission line h dimensionless space step = A x / l
loading', Proc. 5thlnt. Conf. WindEng., Fort Collins, July 1979 H horizontal tension in static state
3 Manuzio, C. and Paris, L. 'Statistical determination of wind ]u longitudinal turbulence intensity
loading effects on overhead line conductors', C.L G.R.E. Rep. / node n u m b e r
231, 1964 l
4 Castanheta, M. N. 'Dynamic behaviour of overhead power span
lines subject to the action of wind', C.LG.R.I£ Rep. 2208, n (1) frequency, (2) n u m b e r o f time step
1970 N total n u m b e r o f time steps per record
5 Armitt, J. et al. 'Calculation of wind loadings on components R (1) dimensionless line tension, (2) lateral com-
of overhead lines',Proc. LE.E. 1975, 122, 1247
p o n e n t o f support reaction
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wind records using the Inverse Fast Fourier Transform', Cir. s distance along line
Eng. Trans.. LE. Aust. 1978, CE20, 67 S(n) spectral density
7 Holmes, J. D. 'Monte Carlo simulation of the wind-induced t time
response of a cable-stayed bridge', Proc. 3rd hit. Conf AppL T line tension
Stat. Probability Soil Strucr. Eng. Sydney, January 1979 T~
8 Davenport, A. G. 'The spectrum of horizontal gustiness near tension in static state
the ground in high winds', Quart. Z Roy. Met. Soc. 1961,87, u velocity
194 x position vector
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of cables', Comp. Meth. Appl. Mech. Eng. 1976, 9, 65
10 Watts, A. M. 'Efficient numerical solution of the dynanuc
equations of cables', Australian Appl. Math. ConJL, Katoomba,
1979
Y
z J
X, Y, Z
Cartesian coordinate system

nondimensional coordinates
11 Frith, R. J. and Watts, A. M. 'Dynamic behaviour of trans- span reduction factor
mission line conductors under the influence of wind', Ann. AT dimensionless time step
Eng. Conf., I.E. Aust, Adelaide, 1980 AX element length (in reference c o n d i t i o n )
12 Irvine, H. M. and Caughey, T. K. 'The linear theory of free
vibrations of a suspended cable', Proc. Roy. Soc. A. 1974, 341, ~ku peak wavelength ( e q u a t i o n (1))
299 P mass/unit length
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of a random function with application to zust loading', Proc. air density
Pair
1. C.E. 1964, 28, 187 0 sway angle at centre span
strain
Nomenclature
Subscripts:
A cross-sectional area o f line 0 reference state
c decay constant (equation (2)) r relative value
C~ drag coefficient
d separation distance ( e q u a t i o n (2)) Superscripts:
D line diameter mean value
E Young's m o d u l u s ' fluctuating value

110 Eng. Struct., 1981, Vol. 3, April