On the Dynamics of Bundled Conductors in Overhead Transmission Lines

Himanshu Verma∗1, Ashish M. Dighe2, andPeter Hagedorn1

1 Institut fur Mechanik, Technische Universtat Darmstadt, Darmstadt, Germany.2 Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai, India.

The Energy Balance Principle (EBP) is well established for estimating the vibration levels of wind induced oscillations ofsingle overhead transmission lines. The mathematical model, wherein a conductor is treated as a continuous system, resultsin a transcendental eigenvalue problem (EVP), which gives numerical difficulties in the case of bundled conductors. In thispaper, different approaches for solving transcendental EVP and their relative merits are discussed. A new method namedcontinuous spectrum approachprovides a good engineering solution. Results from different approaches are compared.

1 Introduction

In the vibration analysis of a continuous system, one comes across the transcendental eigenvalue problem. For example, thevibration analysis of bundled conductors with attached spacer dampers leads to [1, 2]

J(s)a = 0 (1)

where the elements of the system matrixJ are transcendental functions of the complex eigenvalues (= −δ + iω). There is anon-trivial solution to Eqn.(1) iff

det (J (s)) = 0. (2)

The characteristic equation (2), being transcendental, has an infinite number of roots. For energy balance [1, 2, 3] it issometimes required to calculate the first thousand eigenvalues or so (in the frequency range of interest) and the correspondingeigenvectors of the transmission line system.

2 Determinant search method

This is the most common approach for solving the transcendental eigenvalue problem in which the roots of the characteristicpolynomial are searched in the complex domain, so as to satisfy Eqn.(2). The nonlinear optimization functionfsolveofMATLAB r 6.5 is used to this end.

For bigger problems the elements of system matrixJ(s) have a large difference in the order of their magnitudes. Thedifficulty with this approach is that for bigger problems the function does not converge to zero, rather it sticks to minima.Hence, it requires additional checks to confirm whether the obtained values are eigenvalues of the system or not. Moreover, itis difficult to obtain all of the eigenvalues (even if good initial guesses are given) of bundled conductors with a large numberof spacer dampers, because of the optimization criterion being poor conditioned.

3 Alternate approach

In this approach the numerical behavior is improved by avoiding the direct solution of the homogeneous problem of Eqn.(1).The homogeneous set of the equations is transformed into a non-homogeneous one, which satisfies Eqn.(2) implicitly. Theproposed algorithm is as follows :

1. Eqn.(1) is a system ofP - homogeneous linear-simultaneous equations, which can also be written as

P∑

k=1

Jj,k(s)ak = 0, j = 1, 2, 3, . . . , P. (3)

2. Substitutear = C; (1 ≤ r ≤ P ) in any (P − 1) equations and solve the resulting system of non-homogeneous linearsimultaneous equations forak (k 6= r). HereC is some constant, e.g. equal to1.

∗ Corresponding author: e-mail:himanshu@mechanik.tu-darmstadt.de, Phone: +49 6151 16 2486, Fax: +49 6151 16 4125

PAMM · Proc. Appl. Math. Mech. 4, 115–116 (2004) / DOI 10.1002/pamm.200410039

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

3. Use this solution and substitute in the remaining equation, say thelth equation, and defining the error asP∑

k=1

Jl,k(s)ak = ε. (4)

An optimization is now performed finding values fors, in complex domain, for which|ε| equals (or tends to) zero. Thisapproach leads to a procedure which has an improved numerical behavior, converging faster than the determinant search.

4 Continuous spectrum approach

In problems where the frequency spectrum is very dense, as is the case of the bundled conductors, one may substitute thecontinuous frequency spectrum by a discrete one [4]. This means that corresponding to eachω there exits an eigenvalue,satisfying Eqn.(2). For a given circular frequency it is still important to know the factorδ, representing the damping of thecorresponding mode. Therefore, the problem now is to find the correspondingδ for a givenω. This simplification converts our2-D optimization problem (in complex domain) to a 1-D (in real domain) one. It should be noted here that the correspondingeigenvectors are still needed for the energy balance.

5 Results

To compare the effectiveness of the described approaches, a representative example problem of a bundled conductor has beenconsidered. A quad bundled transmission line of span length 140 m has been taken. It carries two quad spacer dampers,which are attached respectively at 40 m and 90 m, from the left end. Both the spacer dampers are taken of the same type. Thediameter of the conductor is 17.5 mm and its mass per unit length is 0.981 kg/m. The tension in all four conductors is 50 kN.

The transcendental EVP was solved with the above described approaches and the eigenvalues were searched in the fre-quency range of 20 to 30 Hz. Fig.1 compares the obtained eigenvalues from these three approaches. As the system eigen-frequencies are very closely spaced, for a better illustration only the eigenvalues in the frequency range of 24.5 to 26.5 Hzare shown in Fig.1. In the figure, the damping ratios are plotted over the frequency in Hz. Because of their small values, thedamping ratios are plotted on a log-scale. It can be observed that there are many eigenvalues, which were either missed by thedeterminant search methodor by thealternate approach. The continuous curve, which has been obtained by thecontinuousspectrum approach, passes through all the eigenvalues found from the other two approaches. It was later observed that all thepeaks in the continuous curve represent the system eigenvalues, as can also be seen from Fig.1. On the basis of this fact, onecan easily conclude that thecontinuous spectrum approachis a useful tool in finding the eigenvalues (and eigenvectors) inbundled conductors.

24.5 25 25.5 26 26.5

10−4

10−3

Determinant search methodAlternate approachContinuous spectrum approach

Dam

ping

ratio

( =δ

√δ2+

ω2

)

Frequency in Hz(= ω

2π

)

Fig. 1 Eigenvalues obtained by different methods (24.5 to 26.5 Hz)

References

[1] Verma, H., and Hagedorn, P., 2004. “Wind Induced Vibrations of Long Span Electrical Overhead Transmission Lines: A ModifiedApproach”.Submitted to Wind and Structures.

[2] Hagedorn, P., Mitra, N., and Hadulla, T., 2002. “Vortex-Excited Vibrations in Bundled Conductors: A Mathematical Model”.Journalof Fluids and Structures,16(7), pp. 843–854.

[3] Anderson, K., and Hagedorn, P., 1995. “On the Energy Dissipation in Spacer-Dampers in Bundled Conductors of Overhead Transmis-sion Lines”.Journal of Sound and Vibration,180(4), pp. 539–556.

[4] Hagedorn, P., 1982. “On the Computation of Damped Wind Excited Vibrations of Overhead Transmission Lines”.Journal of Soundand Vibration,83(2), pp. 253–271.

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© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim