Mathias Kliem^{1} , Daniel Johansen^{2} , Jan Høgsberg^{3}
^{1, 2, 3}Department of Mechanical Engineering, Technical University of Denmark, Lyngby, Denmark
^{1}Corresponding author
Journal of Vibroengineering, Vol. 20, Issue 6, 2018, p. 22682288.
https://doi.org/10.21595/jve.2018.19978
Received 17 May 2018; received in revised form 21 July 2018; accepted 28 July 2018; published 30 September 2018
Copyright © 2018 Mathias Kliem, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Steel lattice towers with suspended insulator strings are typically used to carry highvoltage overhead transmission lines. The installation of nonconductive power pylons made of glass fibre reinforced plastics enables a direct cablepylon connection, as the composite structure acts as an unibody insulator. At the same time, windinduced vibrations, such as the severe cable vibration phenomenon galloping, will consequently be directly transferred to the slender composite mast structure, potentially leading to extensive damage. The aim of the study is therefore to investigate the galloping behaviour of iced conductor lines with regard to different cable support conditions. Furthermore, additional damping in the composite power pylon structure is assumed to mitigate conductor line galloping and therefore reduce the risk of phase flashovers between adjacent conductor lines. A numerical galloping simulation is carried out in order to evaluate the effect of a rigid cablepylon connection with enhanced damping properties on the cable vibration amplitudes. A pyloncable system, consisting of 3×300 m spans, is investigated. It was found that the support conditions of the conductor lines have a significant influence on the galloping mode, the vibration amplitudes and the orientation of the characteristic galloping ellipse. The addition of damping to the pylon decreases the vibration amplitudes slightly and leads to a reorientation of the galloping ellipse.
Keywords: nonconductive composite power pylons, galloping conductor lines, numerical galloping simulation, direct cablepylon connection.
National electrical grid systems are facing significant transformations: The increasing demand for alternative and renewable energy solutions and, at the same time, the ageing of highvoltage infrastructure are just a few examples, demonstrating the need for renewed and expanding electrical grid systems [14]. As a result, a beautification of the 400 kV network has been decided by a national directive 2008 in Denmark [5]. The standard steel lattice towers with its long insulator strings, basically unchanged in visual their appearance for decades and dominating today’s landscape, are planned to be replaced by power pylon structures based on composite materials. The use of nonconductive composite materials, such as glass fibre reinforced plastics (GFRP) or aramid fibre reinforced plastics (AFRP), allows an innovative design and a significantly reduction in size, due to the possibility of integrating the insulators directly on the pylon arm. However, this direct and stiff connection of the conductor lines to the nonconductive composite structure may lead to an increased dynamic interaction.
Windinduced vibrations and motions, such as the cable vibration phenomenon called galloping, may therefore be directly transferred to the power pylon, potentially leading to structural damage due to excessive vibration amplitudes at resonance [6]. Conductor line galloping is a severe cable vibration phenomenon typically occurring at temperatures below the freezing point and at frequencies between 0.1 Hz and 1 Hz with large amplitudes of up to 0.03 times the span length [7, 8]. The aerodynamic instability is initiated by asymmetric ice aggregations along the conductor, when subjected to moderate or strong cross winds. The galloping motion typically describes a vertically oriented ellipse with a small horizontal component, also known as the galloping ellipse. Depending on the wind conditions, the ice aggregations and the related dynamic characteristics of the conductor lines, a single loop or a few loops of a standing wave may be represented [8] over a time period of up to 24 h. Thus, fatigue damage in the cable, the structure or the clamping may potentially be induced. The design for the severe galloping vibration phenomenon is therefore an important dynamic load case for high voltage power pylons.
Furthermore, the galloping instability is a governing factor in determining the spacing between the conductors. A mitigation in galloping amplitudes may potentially allow a decrease in spacing between adjacent phase conductors due to a reduced risk of phase flash overs. This may consequently lead to a desired downsizing of the composite power pylon, giving it a smaller footprint in its environment. A mitigation of conductor line galloping is assumed to be achieved by the direct cabletower attachment, compared to the much more flexible standard cable connection with insulator strings [9].
Additional damping in the load carrying pylon structure, potentially obtained by using materials with enhanced damping properties or by the application of damping treatments, may further lead to a mitigation of wind induced galloping vibrations in the cable. Therefore, special regard is given to the dynamic mechanical material behaviour of the nonconductive composite materials, like GFRP and AFRP, used for the power pylon structure. Due to the viscoelastic nature of the polymer resin material, the damping behaviour is up to several magnitudes larger compared to traditional engineering materials such as metals [10]. The damping properties of various polymer resin materials and composite materials have thoroughly been investigated at environmental conditions typical for conductor line galloping [11, 12]. The damping depends highly depends on the fibre direction of the laminate. The lowest damping is obtained in the direction parallel to the fibres, whereas the damping is maximized at fibre angles $\phi \approx $ 30°45°. Furthermore, the application of additional damping treatments may enhance the overall damping behaviour of the entire structure. The passive constrained layer damping treatment, suitable for highvoltage applications, was experimentally and numerically studied on generic composite mast substructures [13].
Several analytical, numerical and experimental investigations have been carried out on galloping conductor line cables over the last 90 years, when galloping was observed and documented the first time. A fundamental formulation of galloping was proposed by Den Hartog, describing the vertical oscillation mechanism of asymmetrically iced conductor lines [7]. A potential mitigation of cable galloping was studied using external devices, such as spiral wires and circular rings, locally mounted to the cables [14]. Other options to reduce galloping effects are seen in the use of twisted conductors, the reduction of the span length and the adjustment of the cable tension to a certain level, minimizing a potential lateral movement of the conductor line [15, 16].
To the best knowledge of the authors, an investigation on a potential mitigation of conductor line galloping by connecting the cable directly to the nonconductive power pylon has not been conducted yet. The aim of this study is therefore to analyse the effect of different cable support conditions and additional damping of the pylon on the vibration amplitudes of galloping conductor lines. A numerical galloping analysis of a representative 3×300 m cablepylon system is carried out in order to compare the resulting cable vibration behaviour with respect to the different support conditions and properties.
The required expansion and upgrade of national grid systems, due an ageing infrastructure or the increasing demand for renewable energies, provides the opportunity to install visually improved power pylons compared to standard steel lattice towers, which are basically unchanged in their design for the last 60 years (see Fig. 1).
Fig. 1. Visual impact of the compositebased power pylon (computer aided image) [17]
The use of the nonconductive composite materials for the design of highvoltage power pylons offers several advantages [18] compared to steel:
1) Visually improved and compact design.
2) Cost competitive due to reduced transportation effort, a monopile foundation and a shorter construction time.
3) Unibody insulator by using nonconductive composite materials like GFRP and AFRP.
4) Enhanced damping properties due to the viscoelastic nature of the polymer.
The composite pylon and the related span properties and dimensions are described in more detail in the following sections.
The composite power pylon is developed in a multimaterial design, exploiting the unique properties of each individual material. The pylon consists of a tubular and tapered steel tower, carrying a pair of cantilevered, tapered and hollow composite cylinders at which three pairs of double cable bundles are directly attached, equally distributed at each side (see Fig. 2). The composite cross arms with an inclination of 30° are designed for static and dynamic bending loads in $x$ and $y$ direction, such as the static self weight of the (iced) conductor line and the dynamic wind or galloping loads [19].
Fig. 2. Multimaterial design of the power pylon
The laminate of the cylindrical composite structure with the stacking sequence ${\text{[0\xb0,\xb120\xb0,0\xb0]}}_{n}$, shown in Fig. 3, gradually decreases in thickness from a thickwalled rootend section (${t}_{{A}_{1}}=$ 29 mm) to a thinwalled shelllike zone at the tip of the arm (${t}_{{A}_{2}}=$ 8 mm).
Fig. 3. Global coordinate system (COS) and local material COS
The pylon structure is designed for maximum stiffness in order to prevent any resonant vibrations of the structure due to galloping conductor lines at frequencies between 0.1 Hz and 1 Hz [8]. At the same time, enhanced damping properties of the composite cross arms are desired to potentially damp wind induced cable vibrations, which are directly transferred to the pylon due the rigid cabletower connection.
Furthermore, damping of the composite cross arm is assumed to mitigate conductor line galloping and thereby prevent any structural damage. The dimensions and mechanical properties of the composite power pylon structure are listed in Table 1. The material properties of the representative unidirectional (UD) GFRP lamina are given in the literature [20].
Table 1. Dimensions and properties of composite pylon
Parameter

Symbol

Unit

Value

Height tower

${h}_{T}$

m

21.6

Wall thickness tower

${t}_{T}$

m

30.0

Wall thickness cross arm

${t}_{A}$

mm

8.029.0

Length cross arm

${l}_{A}$

m

13.6

Inclination cross arm

${\alpha}_{A}$

(°)

30

Lamina orientation (cross arm)

–

(°)

${\text{[0,\xb120,0]}}_{n}$

Lamina thickness (quadriaxial woven fabric)

${t}_{L}$

mm

1.0

–

${E}_{x}$

GPa

34.9

*Smeared lamina properties

${E}_{y}$

GPa

10.8

*(based on Fig. 3)

${G}_{xy}$

GPa

5.6

–

${\nu}_{xy}$

–

0.43

The composite power pylon is designed to carry transmission lines with two 400 kV circuits on each side. The initial span length is defined by 300 m, using ACSR conductor lines of the type “Martin” [9]. The physical properties of iced conductor lines with a similar characteristic are given in Table 2. A typical sag configuration of a 300 m span at –17.7 °C (0 °F) was modelled, expecting galloping to happen at temperatures below 0°C [23].
Galloping vibrations are sensitive to specific parameters, such as wind velocity, cable sag and cable damping. Any change may affect the vibration characteristics of the conductor line significantly [24]. Several numerical sensitivity studies are reported, investigating the influence of such parameters: galloping vibrations are amplified by an increased wind speed and a decreased cable damping and cable sag [25]. However, strong winds are not necessarily a sufficient condition to initiate conductor line galloping. It is possible that even extremely strong winds may not lead to galloping, when the instability criterion is not met [26].
Table 2. Mechanical properties and dimensions of iced conductors similar to the properties of an ACSR conductor line type “Martin” [21]
Parameter

Symbol

Unit

Value

Axial stiffness

$EA$

10^{6} N

20.23

Torsional rigidity

$GI$

Nm^{2}/rad

101.0

Diameter bare conductor

${d}_{c}$

10^{3} m

28.6

Height of the Dshaped ice [22]

${h}_{ice}$

10^{3} m

63.5

Mass per unit length (incl. ice)

$m$

kg/m

2.38

Moment of inertia per unit length

$J$

10^{4} kg m

3.33

In this numerical galloping analysis, typical values for the wind velocity, the cable sag and the cable damping were chosen and kept constant throughout the numerical galloping analysis, in order to maintain comparability. The focus of the study is to analyse the effect of different cable support conditions on the vibration behaviour of galloping conductor lines, with regard to the stiffness and damping properties of the supporting structure. It is assumed that a stiff cablepylon connection in combination with additional damping, introduced for example by a passive constrained layer damping treatment, may lead to mitigation of the galloping vibration amplitudes [27].
A realistic 400 kV threespan system was modelled, containing two cable attachment points (CAP) and one representative conductor line over the whole range of 3×300 m. Two cable connections are investigated: 1) cable attachment to a flexible insulator string and 2) cable attachment directly to a stiff power pylon structure. The latter configuration is shown in Fig. 4.
Fig. 4. Representative double 400 kV threespan system of 3×300 m with a representative centerspancable exposed to aerodynamic forces $$ Displacements due to galloping are analysed at the center (1/2) of an adjacent cable [A], at the cable attachment point (CAP) of the power pylon [B], at the center (1/2) of the galloping cable [C] and at a quarter (1/4) of the galloping cable [D]
The representative conductor line is thereby rigidly connected to the composite cross arm at its outer most position. The remaining cables are represented by point masses and equivalent spring stiffnesses. The numerical galloping analysis, including geometrically nonlinear effects of the cables, was conducted using the commercially available finite element (FE) software Abaqus. A usersubroutine, written in Fortran, was used to subject the representative conductor line to aerodynamic loads only on the intermediate 300 m span, thereby initiating a galloping event. Prior to that, the numerically obtained static equilibrium of the pretensioned conductor line, calibrated to a typical sag condition, was verified by the analytical expression, calculating the cable sag:
where $m$, $g$, ${L}_{x}$ and ${H}_{x}$ are the mass per unit length of the cable, the constant of gravity, the span length and the horizontal cable tension, respectively. Furthermore, a numerical frequency analysis of the single power pylon and the entire threespan system was carried out in order to identify the first natural frequencies. Finally, the galloping analysis of the cablepylon system was conducted. The numerical analysis is based on the following steps:
1) Static: Pretension of the cable in the global $x$ direction (see Fig. 4),
2) Static: Application of gravity to the conductor lines and pylon structures in the global direction,
3) Dynamic: Galloping analysis by applying aerodynamic loads to the representative midspan section of the conductor line.
Beam elements with a hybrid formulation were used to model the cable and the pylon, specially recommended for slender structures with a high axial stiffness compared to its bending stiffness [2831]. Each conductor line with its span length of 300 m was thereby discretised into 200 elements. A reduction to 30 elements per span led to comparable results, but to instabilities in the solving process.
The schematic cross section of a conductor line with a Dshaped ice accretion, frequently used in numerical galloping studies [33, 34], is represented in Fig. 5. The related properties and dimensions are listed in Table 2. Dshapes with their vertical front facing the wind are found to behave very unstable and are therefore highly prone to galloping [8, 25].
Fig. 5. Schematic conduct or line with a Dshaped ice accretion and the resulting aerodynamic loads (${F}_{L}$, ${F}_{D}$ and $M$) due to wind loading with the velocity ${U}_{W}$ [35]
To simplify the Dshaped cross section of the conductor line for the numerical simulation, a circular cross section with equivalent properties is assumed in order to maintain the same axial stiffness $EA$, torsional rigidity $GI$, moment of inertia $J$ and the mass per unit length $m$ (see Table 3) [21].
Table 3. Equivalent properties of the presented iced conductor lines (see Table 2) used as input parameters for the numerical simulation [21]
Eq. diameter ${d}_{c\text{'}}$

Eq. density $\rho \text{'}$

Eq. Emodulus $E\text{'}$

Eq. Gmodulus $G\text{'}$

0.0335 m

2706.36 kg/m^{3}

23.02·10^{9} Pa

821.2·10^{6} Pa

Although the level of selfdamping in transmission lines is low [3638], mainly caused by friction between the strands, the damping of additional ice may influence the galloping behaviour and is therefore taken into account in the numerical simulations. Damping ratios in the wide range between 0.4 % and 6 % are typically used to simulate the galloping vibration of iced conductor lines, depending on the type of conductor line and ice accretion [3942]. An average damping ratio of ${\zeta}_{c}=$ 3.2 % was employed, applicable to Abaqus using the Rayleigh model. The mass and stiffness proportional Rayleigh damping coefficients ${\alpha}_{R}$ and ${\beta}_{R}$ are described as:
with ${\omega}_{i}$ and ${\omega}_{j}$ representing the two adjacent angular frequencies of the cable vibration mode, respectively.
The aerodynamic loads ${F}_{L}$, ${F}_{D}$ and $M$, acting on an iced transmission line (see Fig. 5), are defined in Eq. (3) by the aerodynamic coefficients ${C}_{L}$, ${C}_{D}$ and ${C}_{M}$, the density of the air ${\rho}_{air}$, the velocity of the wind ${U}_{w}$ and the height $h$ of the windexposed object:
For conductor lines with a Ushaped ice accretion, the height is equivalent to the diameter of the bare conductor ${d}_{c}$, whereas for Dshaped ice accretions the height $h$ is assumed to be the height of the ice front ${h}_{ice}$ [43] (see Fig. 5).
The wind velocity is a sensitive parameter with a significant impact on the galloping vibration amplitudes [24]. A wind velocity of ${U}_{w}=$ 8.5 m/s was used for the numerical galloping analysis, representing a typical wind speed in Denmark [44]. The aerodynamic coefficients ${C}_{L}$, ${C}_{D}$ and ${C}_{M}$, experimentally determined in wind tunnel tests for the specific Dshape of the iced conductor lines [22], are shown in Fig. 6.
Fig. 6. Aerodynamic coefficients for lift ${C}_{L}$, drag ${C}_{D}$ and moment ${C}_{M}$ as a function of the angle of attack $\alpha $, specifically determined for an iced conductor line with a characteristic Dshape [22]
The aerodynamic coefficients are independent of the wind velocity [45] and only depend on the instantaneous angle of attack $\alpha $, which is constantly changing due to the dynamics of a galloping system. The angle of attack can be approximated by:
where $R$, ${\theta}_{0}$, $\dot{\theta}$ and $\dot{V}$ represent the radius of the bare conductor, the initial angle of attack, the torsional angular velocity and the vertical velocity of the conductor line, respectively [46, 47]. The latter two properties and the nodal displacement and the torsional angle, describing the time dependent state of motion of the conductor line, are obtained for each time step by numerical time integration. A userdefined element (UEL) subroutine is therefore introduced into Abaqus [21], generating user elements without mass and stiffness to share the same nodes as the beam elements used for modelling the conductor line (see Fig. 7).
Fig. 7. Schematic beam elements and user elements
As the bending stiffness of cables is usually low, the bending degreesoffreedom (DoF) may be released (DOF 5 and 6 at Node 1; DOF 11 and 12 at Node 2). Based on a previously conducted sensitivity analysis, a release was found to only change the static deflection of the cable by 0.8 % [48], simultaneously leading to numerical instabilities. The bending DOFs were therefore not released for the conducted numerical galloping analysis.
The implementation of a userdefined element subroutine enables access on the stateofmotion parameters for each cable element at time $t$ and $t+\mathrm{\Delta}t$. This allows the calculation and prediction of the angle of attack $\alpha $ by (4) and the application of the aerodynamic loads ${F}_{L}$, ${F}_{D}$ and $M$ by (3) at time $t$ and $t+\mathrm{\Delta}t$. The HilberHughesTaylor time integration method is used in Abaqus by default [49]. The user element, introduced by the UEL, is defined to share the same nodes as its equivalent cable element, exhibiting no mass and stiffness. The required input in the Abaqusspecific matrix AMATRX for the UEL, containing the mass matrix $M$, the damping matrix $C$ and stiffness matrix $K$, can therefore be set to zero for the user element. The residual load vector $RHS$, also required as input parameter for the UEL, is defined by:
with $\nu =$ –0.05, ${F}_{L}$, ${F}_{D}$, and $M$ representing the slight numerical damping and the aerodynamic loads for lift, drag and moment at time $t$ and $t+\mathrm{\Delta}t$ [21]. The predicted aerodynamic loads at time $t+\mathrm{\Delta}t$ are applied to the two shared nodes of the beam and user element, in order to calculate the nodal “stateofmotion” properties at time $t+\mathrm{\Delta}t$.
The initial angle of attack ${\theta}_{0}$, required in Eq. (4) to calculate the angle of attack ${\alpha}_{t+\mathrm{\Delta}t}$, was set to 10°.
In order to compare the effect of different cable support conditions on the galloping vibration amplitudes, a simplified threespansystem with a standard cableinsulator string connection is modelled. The hanging insulator strings are thereby represented by linear static springs in the $x$$z$ plane [50], determined by:
where ${L}_{i}$ and ${m}_{i}$ represent the length and the weight of the insulator string, respectively. It is assumed that the insulator string and the load carrying tower structure are entirely stiff. The support conditions used in the numerical galloping simulation are schematically shown in Fig. 8.
Fig. 8. Simplified support conditions for standard conductor line connection using hanging insulator strings at static equilibrium (contour plot: deflection ${U}_{2}$)
Spatial beam elements were used to model the power pylon with its multimaterialdesign. The material properties of steel were assigned to the tapered tower section, whereas the composite cross arm is defined by the smeared lamina properties (see Table 1). The tower and the cross arm are divided into several subsections to represent the decreasing wall thickness and diameter towards each tip. In order to evaluate potential resonant vibrations due to a galloping event, a frequency analysis of the power pylon was carried out. The conductor lines are thereby represented by point masses and equivalent springs, acting to both sides of each cable attachment point (CAP) in the global $x$ direction. As the present sagtospan ratio is smaller than 1/8, the equivalent spring stiffness ${k}_{x}$ of a taut conductor line may be calculated by:
where ${k}_{e}$ and $s$ represent the horizontal stiffness of a perfectly taut cable and the cable sag, respectively [51]. The horizontal stiffness of a perfectly taut cable ${k}_{e}$ is described as:
with $AE$ and $\phi $ representing the axial stiffness of the conductor line and the angle between the chord line and a horizontal reference [51].
The numerical galloping simulation is based on a preliminary power pylon design [52], considering a variation of windiceloading scenarios [53, 54] and a maximum allowable static deflection in the vertical $y$ direction of 200 mm, defined by the architects.
In order to investigate the effect of damping on the cable vibration amplitudes due to galloping, a reasonable damping ratio is therefore assigned to the composite cross arm section of the power pylon model.
The damping properties of GFRP composite materials significantly decrease with decreasing temperature [11] and may not offer sufficient damping at gallopingprone temperatures below the freezing point. However, the application of constrained layer damping treatments, using viscoelastic materials with its peak damping in that temperature range, may considerably increase the damping of the composite structure [27]. A damping ratio of ${\zeta}_{\mathrm{}CA}=$ 0.03 is therefore assigned to the composite cross arm section in the numerical galloping analysis. The damping is implemented using the Rayleigh damping coefficients, calculated by Eq. (2), where ${\omega}_{i}$ and ${\omega}_{j}$ represent the two angular frequencies, at which the composite cross arm vibrates during the galloping event.
The numerical galloping analysis was carried out using the threespan model with different cable support conditions. Only one conductor line is fully modelled, subjected to aerodynamic loading in the center span (see section 2.2). The remaining cables are represented by point masses and equivalent horizontal springs in the global $x$ direction (see section 3.3). All relevant codes in order to reproduce the numerical galloping analysis, such as the Fortrancode for the UEL, the Pythoncode for extracting and postprocessing the displacement data from Abaqus and the input files for Abaqus, are published and freely available [55].
The results of the numerical galloping simulation of the threespan system are presented in this section. The results of the static deflection and the horizontal reaction forces are discussed before presenting the natural frequencies of the system and the power pylon with its equivalent cable springs and point masses. Furthermore, the effect of the cable support condition on the cable vibration amplitudes are investigated by comparing the direct cable attachment with the standard cableinsulator string connection. Finally, the potential mitigation of conductor line galloping by the implementation of additional damping in the composite cross arm is discussed.
The results of the static deflections of a threespan system are shown in Figs. 8 and 10. The latter figure represents the direct cablepylon connection approach, containing a set of power pylons and a representative 3×300 m long conductor line with ice accretion, positioned at the outer most CAP on the composite cross arm. The user elements (UEL) are assigned to the conductor line at midspan, representing the point of attack of the aerodynamic loads. The results of the equilibrium condition of a single, 300 m long cable span are listed in Table 4. The maximum cable sag at the center with ${s}_{num}=$ 5.727 m corresponds to the analytical solution based on Eq. (1) with ${s}_{an}=$ 5.730 m. In order to considerably reduce the computational effort for the numerical galloping simulation, the remaining 11 cables of the system are represented by equivalent springs (based on Eq. (8)) and point masses (see Fig. 9). The properties are listed in Table 4.
Fig. 9. Boundary conditions for the composite power pylon with cables represented by springs and point masses
Fig. 10. Maximum static deflection of the threespansystem with the representative conductor line subjected to aerodynamic loads (contour plot: deflection ${U}_{2}$)
Table 4. Static conditions of an iced conductor line with a span length of 300 m, based on the properties shown in Table 2 and 3
Per cable

Per cable attachment point (CAP)


–

Tension ${H}_{x}$

Sag $s$

Cable inclination ${\alpha}_{c}$

Eq. Stiffness ${k}_{x}$

Eq. Point mass ${m}_{c}$

Numerical

45840 N

5.727 m

4.01°

1.3·10^{5} N/m

714 kg

The results of the natural frequencies of the individual power pylon and the entire cablepylonsystem are discussed in the following.
The mode shapes and the natural frequencies of the individual power pylon with and without equivalent cable masses and stiffness are presented in Fig. 11.
Fig. 11. Frequency analysis of the composite power pylon without and with attached cables (represented by Eq. springs ${k}_{x}$ and point masses ${m}_{c}$ at each CAP as shown in Fig. 9)
The consideration of 12 taut cables significantly changes the vibration behaviour of the power pylon. The equivalent cable masses lead to a decrease of the first natural frequency by 55 % while maintaining its bending mode shape in the $yz$plane, compared to the power pylon without equivalent cable attachments. However, the resulting first natural frequency of the power pylon ${f}_{{p}_{1}}=$ 1.32 Hz (bending mode in the $yz$plane) is very close to the galloping induced pylon vibration ${f}_{{g}_{p}}=$ 1.23 Hz (bending mode in $xz$plane), discussed in more detail in section 4.3.1. An increase in the vertical bending stiffness by locally reinforcing the cross arm and the tower in the $yz$plane may reduce a potential resonance vibration of the pylon due to galloping. The natural frequencies of the higher modes are above 2 Hz and therefore not prone to resonance due to galloping induced vibrations.
The natural frequencies of the entire cablepylonsystem are dominated by the conductor lines. The results are listed in Table 5. The mode shape related to the lowest frequencies of the 3×300 m span system corresponds to a single, horizontally oriented loop in the $xz$plane. As conductor line galloping occurs in most of the cases as a vertically oriented single loop, the initial guess of the numerically simulated galloping frequencies was set to the corresponding mode 46 with its natural frequencies between 0.286 Hz and 0.312 Hz for the calibration of the Rayleigh damping coefficients (see Section 4.3.1 and Table 6).
Table 5. Natural frequencies of threespan system (3×300 m conductor lines, each with a sag $s=$ 5.73 m and a horizontal reaction force ${H}_{x}=$ 45.84 kN)
–

Mode

13

46

79

1012

1315

1618

1921

Natural frequencies
of the system ${f}_{{s}_{n}}$ [Hz]

min

0.231

0.286

0.459

0.461

0.687

0.690

0.908

max

0.231

0.312

0.460

0.461

0.688

0.690

0.910


Nr. loops per span (plane)

1 ($xz$)

1 ($xy$)

2 ($xy$)

2 ($xz$)

3 ($xz$)

3 ($xy$)

3 ($xzxy$)

In the following the results of the numerical galloping simulation for a threespan conductor line system with different cable boundary conditions are presented. The parameters like wind speed ${U}_{w}$, initial angle of attack $\alpha $, cable sag $s$, damping of the iced conductor line ${\zeta}_{\mathrm{}c}$ and horizontal tension force ${H}_{x}$ remain unchanged throughout the entire analysis (see section 3).
In the first section, the calibration of the Rayleigh coefficients is described, as the damping for the cable and the cross arm is implemented using Rayleigh damping. Three galloping models
• Model 1: Standard cableinsulator string connection,
• Model 2: Direct cablepylon connection (${\zeta}_{\mathrm{}CA}=$ 0),
• Model 3: Direct cablepylon connection (${\zeta}_{\mathrm{}CA}=$ 0.03),
are studied and discussed in the following sections. Each galloping simulation was carried out over a period of 1500 s, in order to reach a steady state condition. Finally, the vibration amplitudes of the three models with regard to the midspan are compared and discussed in more detail in the section “Comparison of vibration amplitudes”.
The calibration of the Rayleigh damping coefficients ${\alpha}_{R}$ and ${\beta}_{R}$ for the cable and cross arm was carried out iteratively, based on Eq. (2). The iteration is needed, as the galloping characteristics may change when introducing a change in the damping level.
1) Galloping analysis of a cable, fully constrained at each end; Calibration of the initial Rayleigh coefficients based on an expected single loop (in $xy$plane) at frequencies close to the natural frequencies of the 3×300 m cablepylon system (see Table 5),
2) Recalibration of the Rayleigh coefficients based on the resulting galloping frequencies of a cablepylonsystem (shown in Fig. 12 for Model 2), representing a threespanmodel (3×300 m) with a galloping cable in the center span (see Fig. 10),
3) Verification step of the galloping analysis, based on Rayleigh coefficients calibrated in step 2), showing galloping frequencies in the same range.
The Rayleigh coefficient for the cable, used in the numerical galloping analysis, are shown in Table 6 and are based on the galloping frequencies presented in Fig. 12.
The galloping frequencies were determined by using a Fast Fourier Transformation (FFT) of the time histories at the CAP (location B) and the galloping conductor lines (location D) at steady state in horizontal and vertical direction (see Fig. 4).
The location D along the galloping conductor line of the reference model was chosen instead of location C, as the iced cable was galloping in a 2loop constellation with its antinodes at 1/4 of the span. As expected, the CAP of the power pylon vibrates in vertical direction with mainly the same frequencies as the galloping conductor line (${f}_{1}=$ 0.61 Hz and ${f}_{2}=$ 0.84 Hz) and the horizontal vibration of the CAP (${f}_{2}=$ 1.23 Hz). The latter is basically twice the vertical galloping frequency of the cable, as the power pylon is pulled sidewards when the cable vibrates between its maximum and minimum.
Fig. 12. Results of the FFT of Model 2, based on the time histories of displacements in vertical ${u}_{y}$ and horizontal ${u}_{x}$, ${u}_{z}$ directions at location B (CAP) and D (1/4) of the galloping cable span)
The Rayleigh coefficients of the cross arm are determined with ${\alpha}_{R}^{pylon}=$ 0.152 and ${\beta}_{R}^{pylon}=$ 0.005, based on the galloping induced vibrations frequencies of the structure and its damping ratio of ${\zeta}_{\mathrm{}CA}=$ 0.03 (see section 3.3).
Table 6. Calibration of the Rayleigh coefficients ${\alpha}_{R}^{cable}$ and ${\beta}_{R}^{cable}$ for the galloping cable for Model 2 and 3
Model

Mode
(steady st.)

Damping ${\zeta}_{c}$ [–]

Galloping freq. [Hz]

Rayleigh coefficients


Cable

Cable

Tower

${\alpha}_{R}^{cable}$

${\beta}_{R}^{cable}$


Initial (1×300 m)

1Loop ($xy$)

0.032

0.290.31

–

0.060

0.017

Iteration 1 (3×300 m)

2Loop ($xy$)

0.032

0.61  0.84

0.60  1.23

0.143

0.007

Iteration 2 (3×300 m)

2Loop ($xy$)

0.032

0.52  0.61

0.57  1.01

No recalibration

The Rayleigh coefficients for the cable, used in Model 1, were determined as ${\alpha}_{R}^{cable}=$ 0.154 and ${\beta}_{R}^{cable}=$ 0.0066, based on the FFTplot represented in Fig. 13.
Fig. 13. Results of the FFT of Model 1, analysing the time histories of displacements in vertical ${u}_{y}$ and horizontal ${u}_{x}$, ${u}_{z}$ directions at location B (Insulator string connection) and D (1/4 of the galloping cable span)
The slightly increased galloping frequencies of Model 1 compared to Model 2 (see Fig. 12) are due a galloping mode with 3 loops, instead of a 2loops condition at steady state.
The spring stiffnesses used to represent the hanging insulator strings (see Fig. 8) are defined by ${k}_{{i}_{x}}=$ 8.61 kN/m and ${k}_{{i}_{z}}=$ 8.36 kN/m [9]. Time histories of the vertical and horizontal cable vibration amplitudes due to galloping are shown in Figs. 14 and 15 for different locations along the threespan system.
Fig. 14. Model 1Time history plot of vertical displacements at location A (Center, adjacent span), location B (CAP, insulator string), C (Center, mid span) and D (1/4, mid span) (see Fig. 4), representing different vibration conditions (phase III)
Fig. 15. Model 1  Time history plot of horizontal displacements at location A (Center, adjacent span), location B (CAP, insulator string), C (Center, mid span) and D (1/4, mid span) (see Fig. 4), representing different vibration conditions (phase III)
After a period of about 200 s, denoted as phase I, the galloping instability reaches steady state (phase II), in which a 3loop galloping mode is identified (see Fig. 16).
The observation is in agreement with the determined galloping frequencies, presented in Fig. 16. The natural frequencies of the system, corresponding to a vertical 3loop mode in the $xy$plane, are with ${f}_{{s}_{1618}}=$ 0.69 Hz close to the galloping frequencies of the cable with ${f}_{{g}_{c}}\approx $ 0.76 Hz (see Table 5).
The vertical amplitudes at the center and at the quarter point of the galloping midspan (Loc. C and D) are determined as 4.9 m and 3.6 m in phase II, respectively. It is clearly observed that the adjacent span (Loc. A) is excited by the galloping conductor lines at midspan to vibrate in the vertical direction, leading to amplitudes of about 1.1 m. The pronounced coupling is due to the cable connection by the hanging insulator strings, which are free to move in the $xz$plane and only restricted by the stiffness from the adjacent cables.
Fig. 16. Comparison of the scaled galloping modes of Model 1 at steady state (phase II) and Model 2 and 3 with a 3loop (phase II) and a 2loop galloping mode (phase IV)
The horizontal amplitudes in the midspan at steady state (phase II) are with maximum 0.4 m much smaller compared to the vertical amplitudes. However, an excitation of the adjacent spans to vibrate in the $z$direction is also observed. The trace of the galloping conductor line in $yz$plane at the center of the midspan (Loc. C) is presented in Fig. 17, showing the chaotic buildup phase (phase I) and the clearly visible galloping ellipse during steady state (phase II).
Fig. 17. Model 1 – Galloping ellipse traced at Loc. C (center, midspan) during 3loop galloping
The galloping behaviour of the conductor line, rigidly attached to the composite cross arm, is much more unsteady compared to Model 1, demonstrated by the vertical and horizontal time history plots in Fig. 18 and 19.
After the initial buildup phase with a chaotic vibration behaviour of the cable, a 3loop condition is observed (phase II) over a range of about 500 s. The vertical amplitudes at the center of the midspan are with 2.7 m about 30 % higher compared to the location D at onequarter of the midspan. The steady state (phase IV) is reached after about 1050 s, representing a 2loop galloping mode with a maximum vertical amplitude of 7.6 m at location D. The vertical and horizontal amplitude almost vanishes at the center of the midspan, demonstrating a wellbalanced 2loop condition (see Fig. 16).
The vibration response of the cable consists of a dominating part at the frequency ${f}_{{g}_{c}1}\approx $ 0.6 Hz and a minor portion at ${f}_{{g}_{c}1}\approx $ 0.81 Hz (see Fig. 12).
The unsteady vibration behaviour of the cable in vertical ($y$axis) and horizontal ($z$axis) direction is assumed to be due to the direct power pylon connection with its high bending stiffness, resulting in a limited displacement at the CAP in the $xy$plane. This leads to a delayed and irregular excitation of the adjacent spans, which consequently effects the vibration behaviour of the galloping cable at midspan.
Fig. 18. Model 2 – Time history plot of vertical displacements at location A (Center, adjacent span), location B (CAP, pylon), C (Center, mid span) and D (1/4, mid span) (see Fig. 4), representing different vibration conditions (phase IIV)
The observed vibration amplitudes of the adjacent spans (see Loc. A in Fig. 18 and 19) exhibit the same magnitude compared to Model 1, but only in the transient phases I and III and just over a short period of 150 s. In phase I and III of Model 2, the observed and steady vibration amplitudes in vertical direction with maximum 0.1 m, which corresponds to only 9 % of the vertical amplitudes in Model 1 with its standard insulator string connection (compare with at Loc. A in Fig. 14).
Fig. 19. Model 2 – Time history plot of horizontal displacements at location B (CAP, pylon), C (Center, mid span) and D (1/4, mid span) (see Fig. 4), representing different vibration conditions (phase IIV)
Fig. 20. Model 2 – Galloping ellipse traced at Loc. D (1/4, midspan) during 2loop galloping
The horizontal vibration amplitudes of the adjacent spans of Model 1 and 2 also vary in magnitude. In Model 1, the amplitudes are with 0.18 m about 90 % higher compared to Model 2. The variation in vibration amplitudes of the different models at midspan is discussed in more detail in section 4.3.5. In Fig. 20, the characteristic galloping ellipse at Loc. D is shown for the 2loop galloping condition at steady state.
Additional damping in the composite cross arm of the power pylon is assumed to mitigate the galloping induced cable vibration amplitudes. A damping ratio of ${\zeta}_{CA}=$ 0.03 is thereby implemented to the cross arm by Rayleigh damping, calibrated based on the vibration frequencies at Loc. B (CAP) shown in Fig. 12. The results of the time dependent cable displacements in vertical and horizontal direction are presented in Fig. 21 and 22 for different locations along the threespan system.
Fig. 21. Model 3 – Time history plot of vertical displacements at location A (Center, adjacent span), location B (CAP, Power pylon), C (Center, mid span) and D (1/4, mid span) (see Fig. 4), representing different vibration conditions (phase IIV)
Fig. 22. Model 3 – Time history plot of horizontal displacements at location B (CAP, pylon), C (Center, mid span) and D (1/4, mid span) (see Fig. 4), representing different vibration conditions (phase IIV)
The difference in the time history plots of the vertical and horizontal amplitudes of Model 2 and 3 is mainly related to a changed duration of the different phases. Whereas the durations of the transition phases I and III are increased, the period of phase II with its 3loop galloping mode is reduced.
However, the overall vibration behaviour with the 2 and 3loop mode is the same for Model 2 and 3. No significant difference in horizontal and vertical vibration amplitude of the adjacent spans is observed. A comparison of the vibration amplitudes of the midspan is discussed in more detail in the following section.
As expected, the vertical vibration amplitudes at steady state are larger during a 2loop galloping mode (see Fig. 20) compared to a 3loop mode (see Fig. 17). In order to fairly compare the effect of the cable support condition on the vibration amplitudes of the galloping cable, the vibration conditions need to be similar. Therefore, the galloping ellipse of Model 1, representing a standard cableinsulator string connection, is compared to the galloping ellipses of the models with a direct cabletower connection (Model 2 and 3) at a time of about 500 s in phase II (see Fig. 18 and 21). At this state, the cables of all three models vibrate in a 3loop mode with its local maxima in the center and close to onequarter and threequarter at midspan. The resulting galloping ellipses for Model 1, 2 and 3 at Loc. C and D are therefore shown in Fig. 23.
Fig. 23. Comparison of the galloping ellipses of Model 1 (Insulator string connection), Model 2 (direct connection without cross arm damping) and Model 3 (direct connection with cross arm damping ${\zeta}_{CA}=$ 0.03) with regard to a 3loop condition at Loc. C and D, each at about 500 s (phase II)
a)
b)
A change in cable support condition from an insulator string to a stiff pylon structure leads to a significant change in the vibration behaviour with respect to the $xz$plane in the center (Loc. C) and close to onequarter (Loc. D) at midspan. Two phenomena are thereby observed at both locations: 1) a change in length of the major axis of the galloping ellipse and 2) a rotation of the ellipse towards the $z$axis. The length of the major axis of the galloping ellipse significantly reduces from Model 1 to Model 2, resulting in a reduced vertical vibration amplitude of about 48 % due to the increased stiffness at the CAP. The application of damping to the cross arm with ${\zeta}_{CA}=$ 0.03 in Model 3 leads to an additional reduction in vertical amplitudes by 15 % for the 3loop condition at Loc. C.
At the same time, a gradually rotating galloping ellipse towards the $z$axis is observed for Model 1 to 2 by 7 % and for Model 2 to 3 by additional 2.3 %. This consequently leads to an increase in horizontal amplitude by 53 % from Model 1 to Model 2. A further increase in horizontal vibration amplitude is not observed for Model 3. Instead, the application of damping to the composite cross arm leads to a reduction of the galloping ellipse in horizontal direction by 8 %.
In summary, it can be stated that a change from a flexible cableinsulator string connection to a stiff cablepylon connection seems to considerably reduce the size of the galloping ellipse, while slightly increasing the horizontal vibration amplitude. Additional damping in the composite cross arm further reduces the vertical vibration amplitudes and, at the same time, the horizontal vibration amplitude compared to Model 2 without any damping.
Windinduced vibrations, such as the severe galloping cable vibration phenomenon, may lead to extensive damage at the cable attachment point or even for the entire mast structure. The objective of this research was to investigate a potential mitigation of conductor line galloping by rigidly connecting the conductor line to a nonconductive composite power pylon with enhanced damping properties.
A numerical galloping analysis of a representative 3×300 m span system was carried out in order to evaluate different cablesupport conditions with respect to the resulting vibration amplitudes. The outer most cable of the midspan was therefore subjected to aerodynamic loading in order to initiate conductor line galloping. The remaining cables were represented by equivalent spring stiffnesses and point masses. The main conclusions are drawn for the conducted simulations:
1) The vibration behaviour of a galloping conductor line, directly attached to a stiff pylon structure, is with a 3loop and 2loop condition much more complex compared to a flexible cableinsulator string connection with a constant 3loop condition.
2) By changing the cable support conditions from an insulator string to a stiff power pylon, the vertical vibration amplitude at the location of maximum deflection may be reduced by 48 %, while the small horizontal vibration amplitude increases by 53 %.
3) The addition of damping to the composite cross arm with ${\zeta}_{CA}=$ 0.03 leads to a reduction of vertical and horizontal vibration amplitudes by 15 % and 8 %, respectively, compared to the model without any damping in the composite cross arm.
Thanks to the Innovation Fund Denmark for the financial support of the project “Power Pylons of the Future” (PoPyFu) in collaboration with Bystrup and Tuco Marine ApS.
The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This research is supported by Innovation Fund Denmark via the project Power Pylons of the Future (PoPyFu) in collaboration with Bystrup and Tuco Marine ApS, which are gratefully acknowledged.