Large amplitude vibrations of long-span transmission lines withbundled conductors in gusty wind

Pham Viet Hung a, Hiroki Yamaguchi a,n, Masanori Isozaki b, Jawad Hussain Gull a

a Graduate School of Science and Engineering, Saitama University, Saitama, Japanb R&D Center, Tokyo Electric Power Co., Yokohama, Japan

a r t i c l e i n f o

Article history:Received 27 May 2012Received in revised form28 November 2013Accepted 13 January 2014Available online 5 February 2014

Keywords:Transmission lineBundled conductorsWind-induced oscillationField dataEigenvalue analysisGust response

a b s t r a c t

Large amplitude wind-induced vibrations of ice-accreted/unaccreted conductors in overhead transmis-sion lines are frequently observed in the field. Damage due to the large vibrations is costly and affectsmany aspects of modern society. In this study, an attempt is made to identify the large amplitude gustresponses and to distinguish them from the unstable phenomena of galloping in field-observedvibrations of long-span-overhead transmission lines that have bundled conductors. An extensive methodof combining field-measured data analysis, eigenvalue analysis and gust response analysis is applied. Thefield-measured wind and vibration characteristics and their relations are first discussed to studypreliminarily the types of field-measured responses. Next, the natural frequencies and mode shapes ofthe transmission lines are estimated by eigenvalue analysis using reliably created finite element modelsto verify the field-measured response characteristics in the frequency domain. Gust response analysis isfinally conducted to interpret intensively the large-amplitude gust responses of overhead conductors,and results in good agreement with field-measured vibrations. Through this extensive study, it isconcluded that most of the field-measured responses are gust-type vibrations and that a gust responsecan be sufficiently large to cause damage in the overhead transmission lines, regardless of their type.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

When considering the causes of large-amplitude wind-inducedvibrations, people often think that such vibrations in overheadtransmission lines are caused by a galloping unstable phenom-enon, which was first identified and explained by Den Hartog(1932). This galloping phenomenon, which is characterised as alow-frequency, large-amplitude, wind-induced vibration with aself-excitation mechanism, usually occurs in moderately strongand steady wind, while its incidence is rather infrequent andunpredictable under conditions in which there is an interaction ofthe wind and asymmetrical ice or wet snow accreted conductors.The galloping phenomenon is well recognised as one of the majorwind-induced vibrations that causes damage in transmission lines.Phase-to-phase flashovers can occur and lead to widespreadelectrical power outages. Large amplitude vibrations of conductorscan cause overload on towers and fatigue damage to hardware aswell as insulators and conductors. The galloping phenomenon isone of the classical problems in overhead transmission lines undercertain climatic conditions. It has been studied by many research-ers through field observations (Yukino et al., 1995; Diana et al.,

1990; Rawlins, 1981), wind tunnel experiments (Keutgen andLilien, 2000; Nakamura, 1980; Novak and Tanaka, 1974; Nigoland Buchan, 1981a), and numerical analyses (Yamaguchi et al.1995; Desai et al., 1995; Ohkuma and Marukawa, 1999). Further-more, there has been extensive research on the identification ofthe galloping mechanism (Nigol and Buchan, 1981b; Wang andLilien, 1998; Blevins and Iwan, 1974; Nakamura, 1980) and itsprevention through different devices (Havard and Pohlman, 1979;Hunt and Richards, 1969; Richardson, 1965). An excellent survey ofthe state-of-the-art on galloping unstable phenomena is given inthe technical brochure (CIGRÉ, 2005).

Since Den Hartog's finding of the galloping phenomenon, anylarge amplitude wind-induced vibration is generally thought to becaused by the galloping unstable mechanism. However, a so-calledgust response that is a randomly forced vibration in gusty windcan be another source of large amplitude wind-induced vibrationsin overhead transmission line conductors (Yamaguchi et al., 1995).Because of the high level of flexibility in the transmission lines, thepossibility of a large amplitude random response due to atmo-spheric gusty wind cannot be overlooked; this phenomenon isshown in the relevant numerical analysis (Ohkuma andMarukawa, 1999). It is also confirmed by the analysis of field-measured data (Gurung et al., 2003; Gull et al., 2011) which showsthat besides the galloping, the occurrence of a large-amplitudegust response is not only in the horizontal direction but also in the

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Journal of Wind Engineeringand Industrial Aerodynamics

0167-6105/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jweia.2014.01.002

n Corresponding author. Tel./fax: þ81 48 858 3552.E-mail address: hiroki@mail.saitama-u.ac.jp (H. Yamaguchi).

J. Wind Eng. Ind. Aerodyn. 126 (2014) 48–59

vertical direction. With regard to the computational modellingof gusty wind-transmission line conductor interactions, a refinedcomputational model of the conductor in the surrounding movingair is used to investigate the dynamic interaction between the windand the conductor motion (Keyhan et al., 2013). In the computationalmodel, wind load time histories are used as input for nonlineardynamic analysis, with the direct time integration of incrementalequations of motion. In consideration of the nonlinear dynamic res-ponse of transmission line conductors that were subjected to gustywind in arbitrary directions, a formulation and computing procedurehas been proposed and introduced. In the computational modelling,the exact static equilibrium configuration according to the elasticcatenary equation under self-weight, non-levelled supports, arbitrarysag, large displacement and deformation fields have been considered(Impollonia et al., 2011; Miguel et al., 2012; Keyhan et al., 2013). Mostof the references emphasised the computational methods for thetransmission line-imposed wind force interaction in time domain,without interpreting the characteristics of the large amplitude gustywind-induced vibration and validating the methods by full-scalevibration measurements.

Despite such numerous field observations, studies and applicationson the large amplitude wind-induced vibrations for more than a halfcentury, a practical protection method that is recognised as fullyreliable has not yet been developed. Aminimisation or control methodfor wind-induced vibrations of transmission lines still depends on afield trial-and-error procedure (Ohkuma and Marukawa, 1999; CIGRÉ,2005). Accidents such as the loosening of bolts and the breaking ofinsulator attachments, spacers, and porcelain plates due to the largeamplitude vibrations of iced/un-iced transmission line conductors ingusty winds have been observed even recently by Tokyo ElectricPower Company (TEPCO). For the rational design of overhead trans-mission lines and their smooth operation, some measure for control-ling both the galloping and the gust response is necessary. However,because the characteristics of the galloping and gust response areentirely different, the methods for minimising or controlling themwould be different. It is, therefore, indispensable to identify gallopingand gust responses separately.

In this study, an extensive method that consists of field-measured data analysis, eigenvalue analysis and gust responseanalysis based on the finite element (FE) model is applied to fullyidentify and interpret the characteristics of large amplitude gustresponses that were observed in long-span, overhead transmissionlines that have bundled conductors. The field-measured wind anddisplacement characteristics, such as the mean wind velocity, winddirection, turbulence intensity, root mean square (RMS) and powerspectral density (PSD) of the wind velocity and response displace-ment, are first discussed, to have an initial idea of the type of field-measured vibrations. Next, eigenvalue analysis is performed to eval-uate the response spectral peaks followed by the gust response ana-lysis in the frequency domain. From the gust response analysis, theresponse PSDs and the RMS responses are obtained for the measuredwind characteristics and are compared with the field-measured PSDsand RMS responses, to ascertain the type of field-observed vibrationsin different types of transmission lines.

2. Outlines of transmission lines and field measurements

TEPCO has been recording wind-induced vibrations of itsoverhead transmission lines with multiple bundled conductors,both in iced and un-iced conditions. In the present study, thevibration data with the wind data recorded by TEPCO is analysedfor three different types of long-span transmission lines, whichwere selected because some damage was reported in differentcomponents of these transmission lines due to large wind-inducedvibrations. The maximum peak-to-peak amplitudes (MPPAs) are

observed, for example, at approximately 7 m in the horizontaldirection, 5 m in the vertical direction and 701 in the torsionaldirection. Such large span vibrations can result in the dynamicresponse of components such as insulators, spacers, and jumpers,which can lead to damage. Therefore, objectives for the fieldmeasurements were set to identify clearly the causes of largeamplitude vibrations that could have caused damage in differentcomponents of the transmission lines.

Fig. 1 (a), (b) and (c) show the geometries of the studiedtransmission lines, Line A, Line B and Line C, respectively, with theinstrumentation for the field measurements. Line A has eightbundled conductors in a single dead-end span of 615 m betweentwo anchoring towers, No. 58 and No. 59, with a 40 m difference intheir levels. Line B has four bundled conductors in two spans: a624 m span between the towers No. 3 and No. 4, and a 407 m spanbetween towers No. 4 and No. 5. The level differences of twoanchoring points in the first and second spans are 137 m and51.6 m, respectively. The two spans are not aligned, and they havean acute angle of 8o580, as shown in Fig. 1(b). It should be emphasisedthat Line B is anchored at the intermediate tower and is connectedthrough a jumper line, which can change the dynamic characteristicsof the transmission line system. Line C has two bundled conductorsin three spans: a 249 m span between towers No. 36 and No. 37, a439 m span between towers No. 37 and No. 38, and a 421 m spanbetween towers No. 38 and No. 39. The level differences of theanchoring/suspending points in the first, second and third spans are82.7 m, 59.7 m and 109.8 m, respectively. In Line C, suspension-typeinsulators are used for the intermediate towers. The specifications oflines A, B, and C are given in Table 1.

The accelerations in all three translational directions and theangular velocity were measured by using accelerometers and angularvelocimeters, respectively, at the quarter- and mid-spans, as shownin Fig. 1, for observing wind-induced vibrations that were dominatedby symmetric and anti-symmetric lower-frequency modes in up tothree-loop modes. For higher-frequency anti-symmetric modes,

Fig. 1. Geometries of the transmission lines and cross sections of bundle con-ductors: (a) Line A; (b) Line B; and (c) Line C.

P.V. Hung et al. / J. Wind Eng. Ind. Aerodyn. 126 (2014) 48–59 49

more measured locations should be considered. In the case of Line B,in addition, the motions of a jumper pipe are measured by theaccelerometer that is located at its midpoint. The sampling frequencyof each record is 10 Hz, and the record length is 10 min. For themeasurement of the wind velocity, a three-dimensional ultrasonicanemometer was placed at the top of the tower in each transmissionline, with a sampling frequency of 50 Hz.

The measurement data used in the present study comprise allof the measured vibration events that involve wind data fromDecember 2008 to July 2009. It should be noted that the wirelesssensor that was developed by TEPCO to measure the vibrationacceleration of the conductor bundle in the three directions, wasmounted on the spacer frame of the conductor bundle. The windand acceleration measurements were synchronised for real time,but different sampling frequencies and/or different trigger levelswere used for each type of data. The wind velocity data wererecorded continuously for four months. Therefore, 10-min-lengthwind records that correspond to each 10-min-length accelerationdata can be extracted appropriately from the continuous windvelocity data, in which a 10-min-length wind record is commonlyassumed to be stationary in practice. The summary of the fieldmeasurements is given in Table 2, and it indicates the number ofdatasets of all of the measured vibration events and the vibrationevents that have wind data.

3. Preliminary study on field-measured data

To have insight into the characteristics of gusty wind anddynamic responses in the field, the mean wind direction, mean

wind velocity, wind turbulence intensity, RMS and MPPA of thedynamic displacement responses were first studied. It is notedthat the measured acceleration is numerically integrated twice toobtain the time series for the displacement. Before each integra-tion step in the acceleration record, the data are filtered to removethe low-frequency components of noise by using the Butterworthhigh-pass filter with an appropriately selected cut-off frequency. Inthe current analysis, the cut-off frequency of a high-pass filter isset to be half of the first natural frequency, 0.05 Hz for the data ofLines A and B, and 0.07 Hz for the data of Line C.

3.1. Mean wind velocity, wind direction and turbulence intensity

Fig. 2 shows the variation in the direction and magnitude of themean wind velocity during the measured vibrations along withthe alignment of the transmission lines. As seen in the figure, thedirection of the mean wind velocity is not normal to the transmis-sion line, and the variation in the wind direction is large, especiallyin the case of Line B. This arrangement indicates the importance ofhaving a skew correction when calculating the component of themean wind velocity that is normal to the transmission line.

The horizontal/vertical turbulence intensity that is mentionedin this section is the ratio of the standard deviation of thefluctuation wind velocity in the along-wind direction/fluctuationwind velocity in the across-wind vertical direction per mean windvelocity. The magnitude of the turbulence intensity is plotted withrespect to the normal component of the mean wind velocity inFig. 3. All of the transmission lines are located on mountainousterrain and, therefore, the turbulence intensities are relativelyhigh, especially in the case of Line B. It is confirmed in the figurethat the turbulence intensity decreases with an increase in themean wind velocity and that the vertical component is almostone-half of the horizontal component. Both the high turbulenceintensities and the skew of the wind direction could preclude thepossibility of vortex formation or the aerodynamic effect of oneconductor on the other in the bundle of conductors.

3.2. RMS and MPPA of wind-induced vibrations

The RMS of the horizontal, vertical, and torsional responses inLine A, Line B, and Line C are plotted with respect to the normalcomponent of the mean wind velocity in Fig. 4. It is noted that theRMS responses appeared to be more correlative with the normal

Table 1Specification of the transmission lines: Line A, Line B, and Line C.

Line A Line B Line C

Trans–mission lines No. conductors 8 4 2No. of spans 1 2 3Span length (m) 615 624 439Sag/span ratio 0.057 0.051 0.023

Conductor Elastic modulus (109 N/m2) 69.9 90.6 90.6Diameter (mm) 40.3 22.4 24.5Spacing (m) 0.4 0.4 0.4Weight (kg/m) 3.056 1.11 1.328

Spacer Weight (kg) 28 9.5 27Elastic modulus (109 N/m2) 210 210 210

Tension Length (m) 13.91 7.43 3.6insulator Weight (kg) 14,330 4076 275

Elastic modulus (109 N/m2) 210 210 210Suspension Length (m) 3.5insulator Weight (kg) 290

Elastic modulus (109 N/m2) 210Jumper Diameter (mm) 120pipe Elastic modulus (109 N/m2) 63Jumper Diameter (mm) 40hanger Elastic modulus (109 N/m2) 210

Table 2Number of field data sets from December 2008 to July 2009.

Lines Datasets Left Span Jumper Right Span

L/2 3L/4 L/4 L/2

Line A All vibration data – – – 44 191Vibration with wind data – – – 44 191

Line B All vibration data 104 220 121 137 157Vibration with wind data 88 166 116 135 152

Line C All vibration data 10 61 – 52 1Vibration with wind data 4 22 – 27 1

P.V. Hung et al. / J. Wind Eng. Ind. Aerodyn. 126 (2014) 48–5950

wind component than the mean wind component. As shown inFig. 4, the trend of the RMS response versus the normal compo-nent of the wind is parabolic in almost all of the cases, except forthe vertical vibration of Line C. While there might be a possibilityof having an aerodynamically unstable phenomena, it is difficult toascertain the type or the mechanism of the vibration in Line C witha small amount of data. It is also understood from Fig. 4 that theRMS responses of Line B are larger and more scattered than thoseof Line A and Line C, which is mainly due to the difference in boththe magnitude and dispersion of the turbulence intensity of thewind, as shown in Fig. 3. Except for Line C, furthermore, thehorizontal responses are larger than the vertical responses, whichis partly because the turbulence intensities in the horizontaldirection are higher than those in the vertical direction. Almostall of the above discussion on the RMS response characteristicssuggests that most of the field-measured responses could be gustresponses regardless of the types of transmission line. Tthedifferences in the RMS values for the different measured points,are highly related to the dominant modes in the wind-inducedresponses, which will be discussed later.

Fig. 5 shows the variation in the MPPA with respect to thenormal component of the wind velocity for Line B, which is anexample, that has the largest MPPA among the three transmissionlines. The parabolic trend of MPPA in the figure is almost similar tothe trend of the RMS response, which is also one of the character-istics of the gust response because the peak response can beobtained by multiplying the RMS response by a peak factor. Themaximum values of the MPPAs for the three lines are summarisedin Table 3. While the RMS responses are relatively small for Line Aand Line C, their maximum MPPAs are large enough, up to 1.62 mand 4.48 m, respectively. There is an event that shows very largeresponses with horizontal, vertical, and torsional MPPAs of 7.65 m,4, 83 m and 71.01, respectively, at the middle point of the 407 mspan of Line B. This type of event should be carefully identified as agust response or a galloping phenomenon in further analysis. It isnoted that the large horizontal MPPA of 1.65 m at the midpoint ofthe jumper is observed, which can induce some damage in the

jumper line. The mechanism of this jumper vibration will bediscussed later in relation to the dynamic characteristics of Line B.

4. Frequency domain characterisation of responses witheigenvalue analysis

To evaluate the dynamic characteristics of transmission linesystems, eigenvalue analysis is performed. The results of eigenva-lue analysis not only are used to interpret field-observed vibra-tions but are also used in conjunction with modelled wind forces,to conduct the gust response analysis. The FE models of transmis-sion lines are created and analysed by using the general-purposecomputing programme FEMAP/NX Nastran.

4.1. FE models and static equilibrium configuration analysis

The details of FE models for different components in threetransmission lines are indicated in Fig. 6. The modelling of electricconductors is most essential in the FE modelling of transmissionline systems, and three degree-of-freedom, two-node tube ele-ments with zero bending stiffness are applied to the conductors.

The spacers in the transmission line systems have the functionof maintaining the distance between conductors in a bundle ofconductors and each conductor is clamped at the end of a spacerarm. The tangential rotation of each conductor can be free or fixeddepending on the tightness of the clamp, which thereby affects onthe torsional stiffness of the conductor bundle. However, it hasbeen found by preliminarily checking different types of spacermodels that the spacer type does not change the torsionalfrequencies significantly. In this study, therefore, the spacers aremodelled by two-node rigid tube elements that have a rigidconnexion with the conductors.

The insulators consist of small links or ball-socket joints andhave very low bending stiffness. Therefore, both tension andsuspension insulators are modelled by three degree-of-freedom,two-node tube elements with zero bending stiffness. In the cases

Fig. 2. Direction and magnitude of the mean wind velocity: (a) Line A; (b) Line B; and (c) Line C.

Fig. 3. Turbulence intensity of the wind versus the normal component of the mean wind velocity: (a) Line A; (b) Line B; and (c) Line C.

P.V. Hung et al. / J. Wind Eng. Ind. Aerodyn. 126 (2014) 48–59 51

of Line A and Line B, the connections between the insulators andconductors are established by a complex assembly of yoke plates,which affects the torsional frequencies of the bundled conductors.In this analysis, the yoke plates are modelled by rigid plateelements.

With respect to the jumper line in the case of Line B, its mainpart is a jumper pipe, which establishes a path for the conductorsthrough the tower. The jumper pipes are connected to theinsulators through the flexible jumper hangers at both ends bypivotal connections. In this analysis, the jumper pipes are mod-elled by two-node rigid tube elements, while three degree-of-

freedom, two-node tube elements are applied to the flexiblejumper hangers.

FEMAP with NX Nastran is different from other commercialsoftware. It does not address the cable element. For a modellingcable property, a one-dimension element that can carry onlytension and torsion has been used. In creating the FE models oftransmission line systems, the conductor profile under its body-load has been derived carefully due to the nonlinear characteristicsof the conductor and the significant effect of the sag-to-span ratioon the natural frequencies of the vertical symmetric modes asshown in Fig. 7 (Ramberg and Griffin, 1977 and Irvine, 1981).

Fig. 4. RMS response versus the normal component of the mean wind velocity: (a) Line A; (b) Line B; and (c) Line C.

Fig. 5. MPPA response versus the normal component of the mean wind velocity for Line B.

P.V. Hung et al. / J. Wind Eng. Ind. Aerodyn. 126 (2014) 48–5952

In fact, in the cases of Line A and Line B, their sag-to-span ratiosare located close to a so-called cross-over point at which thenatural frequencies of vertical symmetric and anti-symmetricmodes coincide with each other, and slight errors in setting thesag-to-span ratios cause relatively large changes in the naturalfrequencies.

The static equilibrium configuration analysis that is associatedwith a pre-stiffening in the transmission lines under their bodyloads for inducing the tension inside the conductors is a nonlinearproblem, which is solved by an iterative algorithm that employsthe sag as a convergence-controlled parameter. The iterativealgorithm in this study is summarised in four steps. In the firststep, inelastic catenary configurations are theoretically calculatedas input data. Then, the theory-based models are analysed byusing the static nonlinear solution of FEMAP/NX Nastran to obtainthe initial sag in the second step. In the third step, the difference

between the current sag and the real target sag is checked. If theerror is within the convergence tolerance, then the calculationgoes to the fourth step; otherwise, the calculation returns to thefirst step and has a modified sag. Finally, in the fourth step, theiteration programme is stopped with appropriate transmissionline configurations.

4.2. Natural frequencies and mode shapes of transmission linesystems

The eigenvalue analyses are then conducted for the FE modelsof transmission lines with appropriate static equilibrium config-urations, and the results of the eigenvalue analyses are sum-marised in Table 4. In the case of having a horizontal mode in eachtransmission line, it is confirmed that the lowest natural frequencycorresponds to one loop-per-span mode and that the highernatural frequencies are nearly equal to the integral multiple ofthe lowest frequency. With respect to the vertical and torsionalmodes, on the other hand, these trends are somewhat perturbedby the sag effect, especially for Line A and Line B, as discussedpreviously for Fig. 7.

In the case of Line A, for example, the lowest frequency for thevertical one-loop mode is very close to the second naturalfrequency of the vertical two-loop mode, and the one-loop modeshape is modified from the half-sinusoidal waveform. It is alsonoted that the torsional natural frequency is very close to thevertical frequency for each loop-per-span mode except for the one

Table 3Maximum of MPPAs of Line A, Line B, and Line C.

Lines Span Measured point Horizon (m) Vertical (m) Torsion (1)

Line A L/2 1.42 1.29 13.2L/4 1.62 1.53 9.81

Line B Left L/2 4.62 3.40 31.43L/4 3.44 3.74 26.3

Jumper L/2 1.65 0.42 33.9Right L/4 5.86 3.98 37.7

L/2 7.65 4.83 71.0Line C Left L/2 2.36 2.38 20.1

3L/4 1.31 3.39 16.9Right L/4 2.54 2.24 20.53

L/2 2.97 1.99 20.7

Fig. 6. FE models of the transmission lines: (a) Line A; (b) Line B; and (c) Line C.

Fig. 7. Changes in the natural frequencies with respect to the sag-tospan ratio andthe positions of Lines A, B and C.

Table 4Natural frequencies and mode shapes of the transmission lines.

Line Horizontal (Hz) Vertical (Hz) Torsion (Hz)

Line A 0.096 0.180 0.148

0.192 0.190 0.201

0.287 0.293 0.2980.383 0.382 0.394

Line B 0.099 0.145 0.1310.156 0.190 0.1830.197 0.195 0.2120.296 0.298 0.3130.310 0.312 0.337

Line C 0.168 0.177 0.181

0.178 0.186 0.189

0.300 0.309 0.312

0.337 0.344 0.348

0.357 0.359 0.364

P.V. Hung et al. / J. Wind Eng. Ind. Aerodyn. 126 (2014) 48–59 53

loop-per-span mode. This result occurs because the vertical andtorsional modes are in-phase and out-of-phase motions of con-ductors in the bundle, respectively.

In the case of Line B, the jumper is also modelled to investigateits effect on the dynamic characteristics of the transmission lines.Fig. 8 shows a coupling of a jumper vibration with a two-looptorsional mode of the 407 m span in Line B. The correspondingnatural frequency is 0.34 Hz. The figure shows clearly the couplingof the horizontal and torsional vibrations of the jumper with thetorsional mode of the shorter span. Such a coupling mode isimportant in the sense that a large vibration in the jumper can beinduced by a wind-induced span-vibration, which could result inits being damaged, as observed by TEPCO.

4.3. Characteristics of field-measured vibrations in the frequencydomain

The characteristics of field-measured vibrations can be under-stood by the analytically evaluated natural frequencies and modeshapes. The spectral analysis is first conducted to investigate thetime-averaged characteristics of the wind-induced responses andthe wind velocities in the frequency domain. The dominantfrequencies are then identified based on the peak responsespectrum with the analytical natural frequencies and modeshapes.

Example PSDs of field-measured, wind-induced responses inthe three transmission lines are depicted in Fig. 9. All of thedominant peak frequencies in the PSD are very close to theanalytical natural frequencies, as shown in the figure, which alsoindicates the analytical natural frequencies and the associatedmode shapes.

In the case of Line A, there are two equal-level dominant peaksin the horizontal PSD of the quarter span response, whichcorrespond to one-loop and two-loop modes, resulting in nearlyequal RMS values of horizontal responses at the mid-span andquarter-span of Line A in Fig. 4(a). In the figure, on the other hand,the vertical RMSs are also on the same level, which is caused bythe fact that there exist almost identical dominant peaks for themid-span and quarter-span vertical responses in Fig. 9(a); thesepeaks correspond to the closely spaced natural frequencies of theone-loop and two-loop vertical modes.

In the case of Line B, the dominant peak of the quarter-spanresponse, which corresponds to the two-loop mode in the longerspan, is the most significant in the vertical PSD, as shown in Fig. 9(b). This finding can explain clearly why the RMS of the quarter-span response is larger than that of the others in Fig. 4 (b).Furthermore, the relatively large dominant peak of the jumperresponse is recognised in its lateral PSD and is associated with the

torsional dominant peak of the jumper as well as the torsionaldominant peak of the quarter-point of the 407 m span, which havealmost the same frequency of approximately 0.34 Hz. This field-observed phenomenon is explained clearly by the previouslydiscussed coupling mode of the jumper and span shown in Fig. 8.

In the case of Line C, the PSDs of two spans are similar not onlyin their aspects but also in the levels of their dominant peaks. Thisphenomenon can be seen in all three-direction responses, asshown in Fig. 9(c). The horizontal PSD, for example, showsdominant peaks of two spans at almost the same frequencies,approximately 0.17 Hz for the mid-span and 0.35 Hz for thequarter-span. This finding is caused by the fact that the two spanshave similar structural geometries and hence they can be coupledin their vibration modes, as shown in the figure and in Table 4.

5. Interpretation of the field-measured responses based on thegust response analysis

5.1. Gust response analysis of transmission lines

Davenport's approach for determining the resonant dynamicresponses of structures in specific modes under gusty wind(Nicholas, 2012; Davenport, 1962), summarised in Fig. 10, isapplied to the wind-induced vibrations of transmission lines. Eq.(1) is derived by neglecting both the aerodynamic interactionamong conductors in a conductor bundle and the nearly zero liftforce in an individual conductor because of the circle-cross sectionof a conductor. The first assumption is plausible because theseparation between two conductors is larger than 10 times theconductor diameter (Zhang et al., 2000), and the second assump-tion can be justified by the conductor's right angle to the winddirection and the circular shape in the absence of ice.

Srðf Þ ¼ Suðf Þ � ½ρCDχðf iÞdcnclU�2 ∑N

i ¼ 1

φ2i

M2i

Jðf iÞj2�� ��Hðf iÞj2 ð1Þ

where Sr , Su are the power spectral densities of the dynamicresponse and wind velocity fluctuation, ρ is the air density, CD isthe drag force coefficient for a single conductor, jχ iðf Þj2is theaerodynamic admittance, dc is the diameter of the conductor, nc

is the number of conductors in a bundle, l is the span of thetransmission line, U is the mean wind velocity normal to thetransmission line, φi is the mode shape vector of the ithmode, andMi is the mass normalising coefficient, which is expressed asfollows:

Mi ¼mZ l

0φ2

i ðxÞdx ð2Þ

where m is the mass per unit length of a bundle of conductors.jJiðf Þj2 in Eq. (1) is the joint acceptance function and can be

thought of as a weighting function for wind loads that is depen-dent on the frequency-dependent correlation of the wind velocityacross the structure and can be written as follows:

jJiðf Þj2 ¼Z l

0

Z l

0exp �kf

Ux1�x2

!φiðx1Þφiðx2Þdx1dx2

����������

ð3Þ

where x is the position along the length of the transmission line,expð�ðkf =U Þjx1�x2jÞ represents the frequency-dependent corre-lation of the wind velocity, and k is an empirical constant.

Furthermore, jHiðf Þj2 is the mechanical admittance of thetransmission line and can be expressed as follows:

jHiðf Þj2 ¼1

ð2πf iÞ41

f1�ðf =f iÞ2g2þð2ξif =f iÞ2ð4Þ

Fig. 8. Coupling of the jumper motion with the two-loop torsional mode of the407 m span of Line B.

P.V. Hung et al. / J. Wind Eng. Ind. Aerodyn. 126 (2014) 48–5954

where f i is the natural frequency of the ithmode, and ξi is the sumof the structural and aerodynamic damping ratios for the ithmode.

Once the PSD of the response is obtained by Eq. (1), the RMS ofthe response can be calculated by the following equation:

RMSr ¼Z 1

0Srðf Þdf

� �0:5ð5Þ

5.2. Assumptions about the parameters in the gust response analysis

The measured wind data are used to calculate the gustresponses of three transmission lines, while there are manyuncertain parameters that are assumed in the gust responseanalysis. Tables 5 and 6 summarise the assumed values of thoseuncertain variables in the present analysis, and the following arebrief explanations and discussions on their appropriateness.

The air density depends on the temperature, pressure, andpresence of moisture and is 1.342 kg/m3 at �10 1C and 1.204 kg/m3 at 20 1C under the standard atmospheric pressure and in theabsence of moisture. In this analysis, its value is taken as 1.204 kg/m3 because the possible change in its value might not affect theresults of the analysis significantly.

Fig. 9. Examples of the PSD of the field-measured responses: (a) Line A; (b) Line B; and (c) Line C.

Fig. 10. Davenport's approach for the wind-induced resonant response Rawlins (1981).

Table 5Uncertain parameters assumed in the gust response analysis.

Variable Description Value

ρ Air density (kg/m3) 1.204CD Drag coefficient single conductor 1.07χ Aerodynamic admittance 1k Empirical constant in frequency dependant

correlation of wind speed15

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The drag coefficients for the different conductors are providedby TEPCO, and their values for an un-iced single conductornormally range from 1.25 to 0.89, which can be affected by theturbulence intensity of the wind. In this study, an average value of1.07 is assumed, while the RMS response that is calculated byusing this averaged value can have a 16% difference from theresponse that is calculated by using the extreme value of the dragcoefficient.

The aerodynamic admittance is introduced to account for theeffect of the flow modification due to the motion of the structureon the force coefficients. The transmission line conductors, beingvery slender and moving with a very small velocity compared tothe velocity of the oncoming wind, do not modify the flowsignificantly. Therefore, the aerodynamic admittance for thetransmission lines is assumed to be one.

An appropriate value of k, which is the empirical constant inthe frequency-dependent correlation of the wind velocity, can bedetermined only through experimentation, but Holmes discussesits general range as being from 10 to 20 (Holme, 2001). Thepossible change in its value might not affect the results signifi-cantly and, therefore, the value of k is taken to be 15.

The damping in the transmission lines is dominated by theaerodynamic damping because their structural damping is negligiblysmall. Matheson and Holmes (1981) and Holmes (2001) noted thatthe resonant response of the transmission line conductor is largelydamped out because of the very large aerodynamic damping ofapproximately 20% to 25% of the critical damping for a high wind

velocity. Regarding the structural damping of the transmission lineconductor, McClure and Lapointe (2003) selected 2% of the criticaldamping for the bare cable. By referring to the quasi-steady theory ofaerodynamic damping (Stroman, 2006), appropriate values for damp-ing ratios, which give the appropriate sharpness and width of theresonant peak, are assumed in the current analysis, as given in Table 6.

The objective of the gust response analysis in this section is tointerpret and characterise the field-measured responses as gustresponses but not to investigate the effects of varying-uncertainparameters on the gust responses. Furthermore, it is not feasible torun a series of values for studying the propagation of uncertaintiesthat correspond to such an enormous dataset of field-measuredresponses.

5.3. Results of the gust response analysis and comparison with thefield measured responses

The analytically evaluated RMS responses were compared withall of the measured-vibration data at all of the measured points forthree transmission lines, and it was found that almost all of thecases show good agreement. In the following, therefore, onlyimportant examples of RMS comparisons are discussed. To under-stand completely the field-measured vibrations, beside the casesof good agreement in the RMS comparisons, some other casesalong with their specific characteristics are also selected fordiscussion.

One example of a good agreement in the RMS comparisons isthe case of Line A at the mid-point of the 615 m span, as shown inFig. 11. It is found that all of the three response components arereproduced very well by the theory-based gust response analysis.A good agreement can be seen by looking at the PSD comparisonof each event, and the PSD of a typical event in the gust responseanalysis of Line A is depicted in Fig. 12, in which the analytical andexperimental responses show a very small discrepancy.

As mentioned previously in the results of eigenvalue analysis,the horizontal and torsional motions of the jumper are coupledwith the torsional motion of the span at a natural frequency of

Fig. 11. RMS comparison at L/2 of the 615 m span of Line A.

Fig. 12. PSD comparison at L/2 of the 615 m span of Line A.

Table 6Aerodynamic damping ratios assumed in the gust response analysis.

Mode Line A (%) Line B (%) Line (%) C

Hor. Vert. Tor. Hor. Vert. Tor. Hor. Vert. Tor.

1 6 4 2 5 2 2 4 2 12 7 3 2 4 2 1 3 3 13 5 3 1 4 2 14 4 3 1

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0.34 Hz in the case of Line B. This characteristic was reflected inthe gust response analysis and results in a good agreement in theRMS comparison in Fig. 13, which is supported by the smalldiscrepancy in the PSD comparison in Fig. 14. The couplingcharacteristics can also be seen in these figures. The RMSresponses of the jumper in Fig. 13(a, b) show a similar tendencyas the RMS response of the quarter-point of the 407 m span inFig. 13(c), and all of the PSD comparisons in Fig. 14 show the samedominant peaks around their coupling frequency of 0.34 Hz.

Another special case of Line B is the vibration responses at themid-point of the 407 m span, which has very large responseamplitudes compared to all of the other cases in the three lines,as shown in Fig. 5 and Table 3. Their RMS comparisons betweenthe field measurement and the gust response analysis are shownin Fig. 15. In the figures, the various RMS values are small and havevery good agreement in the horizontal and vertical RMS compar-isons, while a relatively large discrepancy is seen in the torsionalRMS comparison. Fig. 16 shows the time series of the response intwo events selected in Fig. 15, in which Event no. 1 and Event no.2 are located at the same mean wind velocity and have good

agreement and weak agreement in the torsional RMS, respectively.The time series of two events suggests that both the horizontaland vertical responses are possibly gust responses because ofsuddenly developed large amplitudes over a short period of time,but that the torsional response could not be a gust responsebecause of a gradually developed large amplitude of approxi-mately 500 s in Event no. 1. These suggestions are supportedpartly by the PSD comparison having a good agreement in thehorizontal and vertical responses but a significant discrepancy inthe torsional response in Event no. 1, as shown in Fig. 17.

One more example of a large discrepancy in the RMS compar-ison is the vertical response at L/4 of the 439 m span in Line C inFig. 18(a), where Event no. 3 and no. 4 are selected for the cases ofweak agreement and good agreement, respectively. Their PSDcomparisons are shown in Fig. 18(b, c), which indicates a verysignificant discrepancy in the case of Event no. 3. This findingsuggests that Event no. 3 could not be a gust response and thatmore study that provides detailed analysis is necessary.

Based on the above results and discussion, it is concluded thatthe gust response analysis can reproduce well the field-measured

Fig. 13. RMS Comparison of the coupled jumper and the span responses in Line B: (a) Horizontal RMS of the jumper; (b) Torsional RMS of the jumper; and (c) Torsional RMSof the quarter-point of the 407 m span.

Fig. 14. PSD Comparison of the jumper in Line B. (a) Horizontal PSD of the jumper; (b) Torsional PSD of the jumper; and (c) Torsional PSD of the quarter-point of the407 m span.

Fig. 15. RMS comparison at L/2 of the 407 m span in Line B.

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vibrations in three different transmission lines and that most ofthe field-measured large amplitude vibrations are possibly gust-type vibrations, regardless of the transmission line type, except forin certain cases.

6. Conclusions

Full-scale measurements of large-amplitude vibrations andtheir characterisation by the extensive analysis of enormousnumber of datasets are conducted for different types of long-span bundled-conductor overhead transmission lines. The essen-tial conclusions can be summarised as follows:

Detailed field-measurements of wind-induced responses andwind characteristics in three different transmission lines enablethe characterisation of their wind-induced vibrations. The para-bolic pattern of the RMS response plotted with the mean windvelocity indicates the possibility that observed vibrations are gustresponses.

The results of eigenvalues and gust response analyses provideunderstandable information in the identification of field-measuredevents, in which the proper FE models of transmission lines with

their accurate static-equilibrium-configurations play an importantrole.

Furthermore, reasonable agreement between the gust theory-based and field-observed RMS responses as well as their PSDsconfirm that most of the field-measured vibrations could be gustresponses, while there might be a possibility of galloping phe-nomena in some cases, and a more vivid interpretation foridentifying the galloping is required in the future.

Through this extensive study of combining field-measured dataanalysis, eigenvalue analysis and gust response analysis, it isconcluded that gust responses can be sufficiently large to causedamages in overhead transmission lines, regardless of their type.

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