Aerodynamic damping in cables of overhead transmission lines subjected to wind loads

Abstract

Traditionally, the analysis of cables under the influence of wind is performed using an equivalent static analysis. However, the occurrence of countless accidents in structural systems where cables are involved, even in cases where the project speed limit has not been reached, indicates that the collapse may have been caused by dynamic actions or by underestimated values of wind speed. The aim of this article is to introduce a dynamic analysis methodology for cables under the influence of wind and its forces. The validation procedure proposed is performed through a comparison among the results obtained, results from other researches, and wind tunnel experimental results. In parallel, comparison of the results is obtained in both static and dynamic analysis.

Keywords

Dynamic analysis in cables cables under wind forces aerodynamic damping transmission lines structures

Introduction

In Brazil, a growing demand of electric energy and the presence of water resources in abundance indicate the need for a distribution network based on transmission lines usually supported by metallic latticed steel towers. In the project of these towers, once they are slender and lightweight, the wind represents the major responsible for the actions to be considered (Holmes, 2015). However, an increase in the number of accidents involving electric transmission towers has been observed, many of these related to the occurrence of high intensity winds (Blessmann, 2005; Henriques et al., 2015). Therefore, an adequate evaluation of wind effects on structures is of fundamental importance.

Papers on the analysis of structural failure in transmission lines have been published, among them Lam and Tin (2011), Albermania and Kitipornchai (2003), and Albermania et al. (2009). Aiming at observing structures submitted to wind response, studies in a wind tunnel were developed by Wang et al. (2015), Yang et al. (2015), Loredo-souza and Davenport (2003), and Stengel et al. (2017).

Currently, in Brazil, there are two engineering standards from the Brazilian Association of Technical Standards (ABNT). Both are from the 1980s, presenting guidelines for determining forces due to wind: the ABNT NBR 5422:1985 (1985) that determines the necessary conditions for the complete project of electric transmission lines and provides the specific procedures for the determination of wind forces, which act in latticed structures, insulating strings and cables, and ABNT NBR 6123:1988 (1988) which determines the required conditions for the consideration of dynamic and static wind forces in the design of buildings. In few words, the first standard is specific to the project of electric transmission lines and provides, additionally, information about the forces caused by wind movement through the transmission line. The second standard provides guidelines for determining forces due to wind in a wider context, with the possibility of applications in less specific cases, for example, latticed structures and cables.

The ABNT NBR 5422:1985 (1985) considers only the equivalent static wind actions, admitting that these actions do not produce significant inertial force in the system to be considered. Using this simplification, the conducting cable movement is ignored and only the static traction will be considered. However, a significative number of accidents occur even for speeds lower than those recommended in the project, and in most cases, even without the rupture of the conductors, which will only fall because of the tower collapse. This behavior pattern indicates that the dynamic effects of atmospheric turbulence can affect these structural systems.

This article presents a methodology for the dynamic analysis of transmission line cables under the influence of wind, considering the aerodynamic damping.

Developed methodology

In order to assess the assumption of using wind loading as equivalent static force, a dynamic analysis was developed in a tridimensional model of an isolated cable, considering the geometric nonlinearities and aerodynamic damping. Battista et al. (2003) conducted a similar study; however, in this case, aerodynamic damping was not considered. Other authors have evaluated the importance of taking into consideration aerodynamic damping in the dynamic behavior of suspended electric cables and developed experimental studies (Stengel et al., 2015).

A procedure aerodynamic damping, a procedure based on the dynamic analysis in the time domain (“time-history”) was proposed which considers the aerodynamic pressure calculus according to relative speeds between cable and wind (Carvalho, 2015; Carvalho et al., 2017). Force loading caused by wind was modeled according to its statistic properties which resulted in a random process. Since the dynamic effects are considerable, it is necessary that the cables have speed values close to the values of wind speed.

Seeking the validation of the proposed procedure, which considers aerodynamic damping; comparisons were made with the results obtained by Davenport (1988) and Vickery (1992) and the wind tunnel values obtained by Loredo-Souza (1996). The effect of taking into account aerodynamic damping was evaluated by the comparison between the dynamic analysis with and without taking into consideration damping.

Comparisons and evaluations were made based on a virtual model of a real transmission line composed of a simple conductor and two supports separated by 400 m at the same level. The supports were considered fixed. This simplification was assessed and considered acceptable (Carvalho, 2015), based on comparisons between the results obtained in this study and dynamic analysis conducted by Oliveira (2006), with flexible cable support.

The maximum values of dynamic response were compared with those obtained from a static analysis performed in agreement to the prescriptions of the ABNT NBR 6123:1988 (1988). Equivalent forces were used as well.

In order to use the experimental results obtained by Loredo-Souza (1996), a virtual model similar to reality was designed, although all technical features remained the same, and in this case, supports were separated by 150 m.

Numerical procedures for dynamic analysis in cables

General description

A pendant cable, in the same way as an electric transmission line, assumes the geometric shape of a catenary. In the case where supports are at the same level, the catenary is symmetric to the central axis that passes through the vertex. The catenary vertex is related with cable weight, span length, temperature, and tension.

For the development of this research, virtual cable modeling was performed using the non-linear truss element (link 10) in the commercial program ANSYS^® (version 12.1, 2009), the key option set to zero. This configuration imposes the elements to work with tension forces.

The formulation of large displacement and initial deformation values for the truss elements must be considered into the displacement calculation. Fixed supports were used at both ends. The dynamic analysis involves the next stages:

First stage. Gravitational forces are gradually implemented. The final configuration of the cable is obtained from a non-linear static analysis (dynamic effects are disabled in this structural loading stage)

Second stage. The aerodynamic forces, which correspond to the average portion of the wind speed, are implemented in the cable as nodal forces. At this stage, the analysis is already dynamic; this fact implies some additional observations. The loads must be slowly introduced, in small increments, in a way that the cable speed, at this stage, is not expressive, and, therefore, does not interfere in the results of the next stage.

Third stage. Wind forces, composed by average and random component, are included, as an arbitrary function of time, for every cable node. The dynamic analysis is processed in a transient regime.

Aerodynamic damping

Aerodynamic damping is defined as a retardant force, which is derived from the relative movement between the structure and wind. For the calculation of the damping, in the case of prismatic structures as in the case of cables, in a uniform flow with the wind direction (aerodynamic drag), a formulation was purposed by Davenport (1988) and Vickery (1992), as is presented in the next equation.

ζ_{aj} = (\frac{C_{D}}{4 π}) (\frac{ρ_{a} d^{2}}{m}) (\frac{V}{f_{j} d})

(1)

where $ζ_{aj}$ is aerodynamic damping in the jth mode, $C_{D}$ is drag coefficient, $ρ_{a}$ is air density, $d$ is cable diameter, $m$ is mass per length unit, $V$ is wind speed, and $f_{j}$ is jth cable natural frequency in Hz.

The formulation for aerodynamic damping as proposed in the study is directly considered in the wind pressure calculation, with the use of the relative speed between wind and structure. The basic formulation for wind pressure and relative speed calculus is presented in the next equation

q_{wind} = 0.613 V_{R}^{2}

(2)

V_{R} = (V (t) - V_{str})

(3)

V (t) = {\bar{V}}_{z} + v (t)

(4)

{\bar{V}}_{z} = {\bar{V}}_{10} {(z / 10)}^{p}

(5)

where $q_{wind}$ is wind dynamic pressure; $V_{R}$ is relative speed between wind and structure, in the node considered; $V (t)$ is wind speed; $V_{str}$ is structure speed, in wind direction, in the considered node; $v (t)$ is fluctuant component of speed; ${\bar{V}}_{z}$ is average longitudinal speed component, calculated in 10 min; ${\bar{V}}_{10}$ is project average speed at 10 meters from the ground, calculated in 10 min; $z$ is a height from the ground; and $p$ is exponential ground rugosity coefficient.

In this research, the recommendations of the ABNT NBR 6123:1988 (1988) are adopted. They suggest, for the execution of the dynamic analysis, the utilization of the average speed calculated in the 10-min interval, as described by the following equation

{\bar{V}}_{10} = 0.69 V_{0} S_{1} S_{3}

(6)

where $V_{0}$ is wind gust speed, calculated in a 3-s interval; $S_{1}$ is topographic factor according to the ABNT NBR 6123:1988; $S_{3}$ is statistical factor associated to the destruction probability, according to the ABNT NBR 6123:1988.

With this formulation, the procedure for the aerodynamic damping calculation becomes generic, independent from the dynamic structure characteristics, able to be applied, basically, to any kind of structure under wind influence.

It must be observed that the aerodynamic drag coefficient, for the wind forces calculation, needs to be adequate for the kind of structure and wind position. Moreover, special considerations are necessary when a structure point has high speed in a transversal wind direction.

Some studies, among others Nagao et al. (2003), assessed the spatial correlation between aerodynamic pressures and revealed that the correlations for the “longitudinal speed fluctuation” did not coincide with the “aerodynamic pressure fluctuation” process. However, in this study, it is assumed that the pressures, which act on the structure, are direct functions of the speed, as in the classic model from Davenport adopted in the ABNT NBR 6123:1988 (1988), spectral density and crossing correlation functions are not considered for the pressure fluctuation.

As the values of force are depend on the speed assumed by the structure, the transient dynamic solution has to be finalized at each increment of time, so it is possible to store the speed values at each node and recalculate the wind force modulus. The solution must end at each step once the acquisition of the structure instant speed values is done through a post-processing module. Once an iteration is completed and the solution is stored, a new solution is computed from the preserved conditions from the previous iteration (speed, accelerations, internal forces, etc.), using the new aerodynamic pressures.

The procedure proposed in this study attends, as well, to the dynamic analysis with no consideration of aerodynamic damping. In that case, the instant cable speed must be canceled (equal to zero) at each time-step.

Simulation, in time, of the fluctuant component of wind speed

For the accomplishment of a non-deterministic dynamic analysis in time domain, the creation of time functions for the floating portion of the longitudinal wind speed is necessary. To create an aleatory sign with a null average, from a given energy spectrum, a Fourier series is used. The function $v (t)$ can be generated from the following equation (Pfeil, 1993)

v (t) = \sqrt{2} \sum_{i = 1}^{N} \sqrt{S^{V} (f_{i}) Δ f} \cos (2 π f_{i} t + θ_{i})

(7)

where $S^{v} (f_{i})$ is spectral density function; $N$ is the number of frequency intervals Δf considered in the spectrum; $f_{i}$ is frequency in Hz; $t$ is time in seconds; $Δ f$ is frequency increment, in Hz; and $θ_{i}$ is aleatory lag angle, between 0 and 2π.

Proceeding the spectrum division, natural frequencies of the structure must be included in the frequencies $f_{i}$ so that the result is not an underestimate of the real one. The current model is extremely expensive, in computational means, once that for each time interval, a new spectrum division is considered and a summation of the last equation is done.

Turbulence power spectrum

The major application of the power spectrum is for determining composition, in frequency, of an aleatory process. For the determination of the spectral density function $S^{V}$ , a formulation proposed by Kaimal is used, shown in the equation (Blessmann, 2005)

\frac{{fS}^{V} (z, f)}{u_{*}^{2}} = \frac{200 x}{{(1 + 50 x)}^{5 / 3}}; x (z, f) = \frac{z f}{{\bar{V}}_{z}}

(8)

Where $f$ is frequency, in Hz; $u_{*}$ is friction speed, in m/s; and $z$ is height above ground, in meters.

Friction speed can be described as

u_{*} = \frac{k {\bar{V}}_{z}}{\ln (z / z_{0})}

(9)

where $k$ is Kárman’s constant, approximately 0.4 and $z_{0}$ is terrain rugosity.

Statistic characteristics of the interdependence between aleatory processes

For large structures, not only temporal series but also multiple series correlated in space are necessary.

According to Davenport (1979), the probabilistic distribution of wind speed is considered a normal or Gaussian distribution. Taking $v_{1}$ and $v_{2}$ , two aleatory processes, representing the wind speed fluctuation at two points of a structure, it is possible to measure their interdependence using the cross-spectral density and cross-correlation functions, presented in the next equations

S^{v_{1}, v_{2}} (f) = \int_{- \infty}^{+ \infty} C^{v_{1}, v_{2}} (τ) e^{- i 2 π f τ} d τ

(10)

C^{v_{1}, v_{2}} (τ) = \int_{- \infty}^{+ \infty} S^{v} (f) e^{- F} e^{i 2 π f τ} df

(11)

Where $S^{v_{1}, v_{2}} (f)$ is cross-spectral density function for points 1 and 2; $C^{v_{1}, v_{2}} (τ)$ is cross-correlation for points 1 and 2 and $τ$ is arbitrary time interval.

Function $F$ is expressed by

F = \frac{f {[C_{1 x}^{2} (x_{1} - x_{2}) + C_{1 z}^{2} (z_{1} - z_{2})]}^{1 / 2}}{\bar{V} (10)}

(12)

where $x_{1}, x_{2} {and z}_{1} {, z}_{2}$ are horizontal and vertical coordinates from points 1 and 2; $C_{1 x} and C_{1 z}$ are decay coefficient in the vertical and transversal direction.

Wind tunnel testing indicates that the values of decay coefficient depend on several factors, among them, average speed, roughness of the terrain, and height above surface. Simiu and Scanlan (1986) suggest values of $C_{1 x} = 16$ and $C_{1 z} = 10$ for the usual practice in projects.

Adopting two temporal series ${(v}_{1} (t) and v_{2} (t))$ , simultaneously, at points 1 and 2, means that τ = 0, a cross-correlation function C1 is obtained

C_{1} = C^{v_{1}, v_{2}} (0) = \int_{- \infty}^{+ \infty} S^{v} (f) e^{- F} df

(13)

Calculating the value of C1 for different performance ranges $(Δ l = (x_{1} - x_{2}) ou {(z}_{1} - z_{2}))$ , it is possible to build a graphic that correlates the coefficients (C1) obtained with the performance ranges (ΔL).

The autocorrelation function of processes (at the same point) is given by

C^{v} (τ) = \int_{- \infty}^{+ \infty} S^{v} (f) e^{i 2 π f τ} df = \int_{- \infty}^{+ \infty} S^{v} (f) \cos (2 π f τ) df

(14)

Knowing the value of the autocorrelation function, it is possible to find time $τ_{1}$ for which the autocorrelation is equal to the calculated cross-correlation value considering null τ. Thereby, the temporal functions at points 1 and 2, especially correlated, can be expressed by the same temporal series, with a time lag equal to $τ_{1}$ .

The following list is a brief description of the stages which need to be followed in order to obtain a spatial correlation between neighboring temporal series:

Define bandwidth ΔL for the temporal series;

Define the cross-correlation value C1;

Define the time interval $τ_{1}$ ;

Creation of the temporal series according to the subclause 3.3, separated by a $τ_{1}$ time interval.

Results

Proposed procedure versus Davenport and Vickery’s procedure

Figures 1 to 3 present the temporal evolution of the cable nodal displacement, nodal velocity, and the support reactions in the wind direction for the methodology of this study, the methodology proposed by Davenport (1988) and Vickery (1992) and the simulation without aerodynamic damping (no structure–fluid interaction), for a 32 m/s wind gust. The x-axis is longitudinal to the cable direction, the y-axis is the horizontal transversal one, and the z-axis is the vertical one.

Figure 1.

Temporal evolution of the central node displacement with and without taking into consideration aerodynamic damping, by different methodologies.

Figure 2.

Temporal evolution of the central node velocity with and without taking into consideration aerodynamic damping, by different methodologies.

Figure 3.

Temporal evolution of the transversal reactions with and without taking into consideration aerodynamic damping, by different methodologies.

Proposed procedure versus static analysis according to the ABNT NBR 6123:1988

Table 1 presents the comparison of the maximum cable reactions for a 50 m/s wind gust obtained by the proposed procedure, which considers aerodynamic damping, and the results obtained by the static analysis with equivalent forces according to the ABNT NBR 6123:1988 (1988).

Table 1.

Comparison between reactions obtained by the proposed procedure and the procedures proposed by the ABNT NBR 6123:1988.

Reaction (kN)	Proposed procedure	ABNT NBR 6123:1988 (1988)
Fx	55.51	48.52
Fy	7.82	7.34
Fz	1.26	1.35

Proposed procedure versus Loredo-Souza’s experimental results

Figure 4 presents the temporal evolution of the transversal cable reaction, in the wind direction, for the proposed procedure, applied to the wind tunnel tested model by Loredo-Souza (1996). Table 2 presents the comparison between the values obtained in simulations and the wind tunnel results.

Figure 4.

Transversal reaction temporal evolution obtained by the proposed procedure.

Table 2.

Comparison between cable transversal reactions obtained by proposed procedure and Loredo-Souza (1996).

Fy (kN)	Loredo-Souza (1996)	Proposed procedure	Error (%)
Minimum	1.7	1.5	11.8
Mean	3.0	2.9	3.3
Maximum	4.5	4.3	4.4

Conclusion

A procedure for the analysis of cables under the dynamic influence of wind, which takes into consideration aerodynamic damping, was proposed. This procedure is based on the calculation of the aerodynamic pressures starting from the relative speed between cables and wind. The obtained results from the utilization of this procedure were compared with results obtained by the formulation proposed by Davenport (1988) and Vickery (1992) and wind tunnel testing performed by Loredo-Souza (1996). Both comparisons were considered satisfactory.

The cable responses were evaluated in two situations, taking and not taking into consideration aerodynamic damping. The results from the second situation were smaller. This observation indicates that dynamic analysis of cables under the influence of wind must not be performed without taking into consideration aerodynamic damping, once those results are too conservatives.

Even with the reduction in the results due to aerodynamic damping, the maximum reaction values in the horizontal directions are slightly above the results obtained from the static analysis with equivalent forces, according to the ABNT NBR 6123:1988 (1988). For this reason, for the design of support structures, such as the electric transmission metallic lattice tower, it is suggested to review the usual practice considering the dynamic analysis of the structural set composed by towers, insulator strings and cables.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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