Study on wind tunnel test and galloping of iced quad bundle conductor

Accepted Manuscript Study on wind tunnel test and galloping of iced quad bundle conductor

Qiong Wang, Liming Wang PII: DOI: Reference:

S0165-232X(18)30082-X https://doi.org/10.1016/j.coldregions.2018.12.009 COLTEC 2711

To appear in:

Cold Regions Science and Technology

Received date: Revised date: Accepted date:

7 March 2018 27 November 2018 20 December 2018

Please cite this article as: Qiong Wang, Liming Wang , Study on wind tunnel test and galloping of iced quad bundle conductor. Coltec (2018), https://doi.org/10.1016/ j.coldregions.2018.12.009

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ACCEPTED MANUSCRIPT Study on Wind Tunnel Test and Galloping of Iced Quad Bundle Conductor Qiong Wang a*, b, Liming Wang a a

Department of Electrical Engineering, Graduate School at Shenzhen, Tsinghua University,

Shenzhen University City, Shenzhen, China State Grid Jiangsu Electric Power Co., Ltd. Economic Research Institute, Nanjing, China

*

[email protected]

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b

Abstract: Conductor galloping is a common phenome non during the ope ration of the

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trans mission line, which can easily result in electrical and mechanical accidents such as

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tower collapses and inte rphase flashover. And aerodynamic parameters are key factors to study galloping. In this pape r, the aerodynamic parameters of crescent-shaped and D-shaped conductors, which are two typical iced conductors, are studied by wind tunnel

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test. The instability analysis is carried out according to Den Hartog instability

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mechanism and the linear galloping model. The results indicate that wind velocity, ice thickness, conductor type, bundle spacing and splitting number all have a great influence on ae rodynamic parameters and conductor galloping. Then the galloping

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situation of a 500 kV quad bundle conductor are simulated according to the parameters obtained from wind tunnel test. The calculating results correspond with the actual situation. The results obtained in this pape r are important to pre dict the galloping amplitudes unde r different weathe r conditions and investigate anti-galloping technology of iced quad bundle conductor. Key words：Quad bundle conductor; wind tunnel test; aerodynamic parameters; instability analysis; conductor galloping

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ACCEPTED MANUSCRIPT 1. Introduction The iced conductor in overhead power transmission lines located in a steady airflow is subject to lift, drag and torque due to its non-circular and unsymmetrical cross-section. These forces are significant factors that induce galloping. Galloping is a low frequency (0.1~3Hz) and large amplitude (possibly > 10m) self-excited nonlinear vibration caused by wind and asymmetrical ice, which usually lasts for a long time

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(Chabart et al., 1998; Hu et al.,2012; Cai et al., 2015).

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Conductor galloping is very harmful to the transmission line due to its large amplitude and long duration. Severe galloping may cause cascading of the transmission line and disruption of the power

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supply. The dynamic tension generated in the process may lead to fatigue rupture of conductor and

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damage of tower steelwork and associated hardware (Wang, 2008; Zhang et al., 2000). Previous work shows that the galloping phenomenon in an overhead transmissio n line is a

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complicated fluid-solid coupling nonlinear time-dependent aerodynamics problem, which are affected by many factors such as tiny terrain, wind velocity, ice shape, ice thickness, as well as transmission line

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configuration. Den Hartog (DEN HARTOG, 1932) and Nigol (Nigol et al., 1974) originally proposed two famous galloping theories (the vertical oscillation mechanism and the torsional oscillation mechanism) ,

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which are generally accepted. From the study of Den Hartog and his equation, galloping mechanism may be explained as a self-excited vibration triggered when the aerodynamic forces become increasingly

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sensitive to variation of wind angle of attack. Therefore, small vibration of the conductor (or any other object) will lead to large variation of aerodynamic forces and thus large amplitude vibration (self excitation). According to N igol mechanism, galloping happens when the frequency of the conductors’ torsional vibration is close to that of the transverse wind self- vibration, the resonance will occur, which results in large-amplitude lateral vibration. Then, Yu et al. (1992) proposed inertially coupled galloping, which indicates that the torsional torque causes the conductor to twist and the aerodynamic force changes. The increase of the lift force will cause the acceleration and the inertia force to rise, leading to the increase

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ACCEPTED MANUSCRIPT of the torsional torque. The positive feedback makes the transverse amplitude of the conductor larger and larger. Aerodynamic parameters of iced conductors are the key factors to study galloping. In the aspect of aerodynamic parameter research, the fluid computation software CFD and wind tunnel test are generally used. However, the results of CFD simulation need to be verified by wind tunnel test or actual physica l flow field due to the fact that the definition of boundary conditions and physical parameters cannot be

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consistent with reality in the simulation of complex flow field by CFD technology. Besides, the accuracy of the calculation method and the post processing should be modified and supplemented with the

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corresponding experiments. In the aspect of wind tunnel test, Lou et al. (2014) studied the aerodynamic

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parameters of crescent-shaped and D-shaped bundled conductors under different ice thicknesses and ice accretion angles. Hu et al. (2012) investigated the aerodynamic parameters of crescent-shaped iced

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conductors with the ice thicknesses of 12mm, 20mm and 28mm. Nishihara et al. (2012) studied the effects

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of initial angle of ice and snow accretion growth on the aerodynamic force by wind tunnel test. In general, previous research on wind velocity mainly varies from 10m/s to 18m/s, which are easy to induce galloping. But it does not include the aerodynamic parameters of all wind velocity range, and the study of

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aerodynamic parameters under high wind velocities are also rare. However, it is of great significance to

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study the difference of aerodynamic parameters of the iced conductor under all wind velocities if we want to understand the galloping characteristics under different wind velocities. Besides, in the study of ice thickness, previous work mainly focuses on ice thicker than 10mm, and there is almost no test on iced

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conductors whose ice thickness is thinner than 5mm. However, the recent field data in galloping areas in China shows that the ice thickness is about 3mm in Anhui when conductor gallops, and 1~2mm in Hubei, and 3~8mm in Jibei. That is to say, conductors covered with thin ice are easy to induce galloping. In addition, the current research on aerodynamic parameters mostly focus es on wind velocity, ice shape, ice thickness and initial angle of ice. Other factors such as conductor type and bundle spacing need to be further studied.

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ACCEPTED MANUSCRIPT In this paper, the author made different crescent-shaped and D-shaped iced conductor section models to measure the aerodynamic parameters of the iced conductors under wind velocities from 2m/s to 30m/s by the wind tunnel test; and the aerodynamic parameters of iced conductors with various ice thickness are also tested, including thin thicknesses such as 1mm. In addition, the author also studied influence of conductor type and the bundle spacing on the aerodynamic parameters of iced conductors. In order to

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analyse shielding effect on aerodynamic parameters, this paper studies the aerodynamic parameters of both single conductor and quad bundle conductor. Based on the aerodynamic parameters of the wind tunnel test

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results, the aerodynamic behaviour of the studied conductor model is predicted according to Den Hartog

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mechanism and the linear model. Finally, the galloping situation of a 500kV quad bundle iced conductor is simulated according to the aerodynamic force, and the multi-span 4DOF (4 degree-of-freedom) nonlinear

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dynamic model for conductor- insulator is established. The results of this paper have certain guiding

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significance for early warning technology in the galloping areas. 2. Model and Equipment of Wind Tunnel Test

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2.1. Segment Model of Iced Conductor

The ice shape of the conductor is strongly affected by weather conditions. When the temperature is

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relatively low and there is gentle wind and light rain, the rain drop freezes immediately and the ice shape tends to be crescent. When the temperature rises and the rain is heavy, the rain drop cannot be condensed

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on the surface of the conductor, and the separation point is generated by the wind flow, forming an approximate D shape. The crescent-shaped and D-shaped conductors are typical and common found in China in recent years which can easily gallop. Therefore, they are chosen as ice shapes used in the wind tunnel test in this paper, as shown in Fig.1. Both the conductor model and the icing model are made of ABS material. The conductor model is selected as LGJ-630/45. In order to compare the influence of different conductor types on aerodynamic parameters, the conductor model of LGJ-400/35 and LGJ300/40 are adopted. The effective length of the segment is 0.8m. The model diameter is defined as the

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ACCEPTED MANUSCRIPT diameter of the bare sub-conductor. Thus, the low aspect ratios of the three kinds of conductors are 23.81,

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29.83 and 33.42, respectively.

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(a) Crescent-shaped iced conductor

(b) D-shaped iced conductor

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Fig.1 M odel of iced conductors in wind tunnel

Seven kinds of crescent-shaped iced conductors and three kinds of D-shaped iced conductors were

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designed. The section size of the segmental models of the crescent-shaped iced conductor with a maximum ice thickness of 2.5mm and the D-shaped iced conductor with a maximum sectional dimension of 70mm are shown in Fig.2. In order to unify the expression, the thickness of the D-shaped iced conductor is converted to an equivalent value of the thickness of crescent-shaped iced conductor. The expression of equivalent ice thickness is as follows:

b

4A D

(1)

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ACCEPTED MANUSCRIPT where b is the equivalent ice thickness; A is the sectional area of iced conductor; D is the outer diameter of the bare sub conductor. Thus, the equivalent ice thicknesses of the D-shaped iced conductor with the maximum sectional dimensions of 70mm, 80mm and 90mm are 39.32mm, 61.64mm and 86.94mm respectively. Fig.3 shows the definitions of wind angle and aerodynamic force of crescent-shaped single

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conductor and quad bundle conductor, and so are the definitions of the D-shaped iced conductor. The attack angle was taken as 0°~180° with 5° intervals. Furthermore, since the aerodynamic characteristic of

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conductor between the uniform flow and 5% turbulence intensity is the most critical to cause conductor

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galloping, this experiment adopts uniform flow (Lin, 2012; Lou et al., 2014).

70 mm

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33.6 mm

2.5 mm

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33.6 mm

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Wind

FL Fy FD

a

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Fig.2 Conductor segment model

M

d Fx

(a) Single conductor

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d,D

FL Wind

Wind angle

FD M

S

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(b) Quad bundle conductor Fig.3 Wind direction and aerodynamic force of crescent-shaped conductor

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2.2. Test Equipment

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To numerically simulate the wind flow around the iced conductor, wind tunnel tests are widely used to acquire the aerodynamic parameters. The wind tunnel test is being done in the ZD-1 boundary layer

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wind tunnel in Zhejiang University, which is a closed single return flow wind tunnel with a rectangular

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cross section, as is shown in Fig.4. The working cross-sectional dimension of the tunnel is 4m (width) × 3m (height) × 18m (length), which can generate uniform flow with the velocity ranging from 3m/s to 55m/s. The aerodynamic force of iced conductor model is measured by high frequency force measuring

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has a maximum error of 3‰.

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device with a sampling frequency of 200 Hz, and each wind direction is sampled for one minute. The force

(a) The test setup

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(b) The schematic illustration of the test setup

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Fig.4 Wind tunnel test for measuring quad bundle conductor

Fig.4 shows the setup for measuring the overall aerodynamic forces on the quad bundle conductor.

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The forces on the single bundle conductor are measured in the same way. Upper plate is connected to the

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wind tunnel top surface by three screws. The gap distance between the plate and the model is chosen to be 1mm to ensure that the wind load on the top plate would not be transferred to the conductor model. The

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bottom plate was lifted from the ground to eliminate the boundary layer effect of the wind tunnel. The force balance is placed under the centre of the bottom plate. The influence of the bottom plate on the

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measurement of aerodynamic force can be neglected. 2.3. Definition of Aerodynamic Coefficients

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The drag FD, the lift FL and the torque M can be measured by the force balance, and the aerodynamic

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coefficients can be expressed as follows:

Cd =

FD 1 N 0.5U 2 Ld

(2)

Cl 

FL 1 N 0.5U 2 Ld

(3)

Cm 

1 M N 0.5U 2 Ld 2

(4)

where ρ is the density of air; U is the test wind velocity; L is the length of the infinitesimal section of the conductor; d is the upwind size of the conductor and is simplified as the diameter of the bare sub-

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ACCEPTED MANUSCRIPT conductor; N is the number of the sub-conductors; Cd and Cl are the drag and lift coefficients. The torque coefficient Cm is not shown for there is no obvious law. 3. Effects of Wind Velocity 3.1. Aerodynamic Coefficients under Different Wind Velocities

The conductor galloping is closely related to wind velocity which strongly influences the Reynolds

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number. Thus, it is of great significance to study the aerodynamic characteristics of the conductors under

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different wind velocities. In order to make up for the shortage of aerodynamic parameters at high wind velocities, the aerodynamic parameters of crescent-shaped iced conductor with 10mm ice thickness and D-

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shaped iced conductor with 39.32mm ice thickness under wind velocities from 2m/s to 30m/s are studied 3

to .2

1

4

based on the

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in this paper, which corresponds to a Reynolds number range of 4. 4 1

diameter of the bare sub-conductor which lies within the sub-critical regime. The results of the force

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coefficients are presented in Fig.5 and Fig.6.

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Fig.5 Variation of aerodynamic coefficients for the crescent-shaped quad bundle conductor at 7 different Reynolds numbers and attack 180°

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angles ranging from 0°

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Fig. Variation of aerodynamic coefficients for the D-shaped quad bundle conductor at to 1

°

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ranging from °

different Reynolds numbers and attack angles

Figs.5 and 6 show the aerodynamic coefficients varying by attack angles at different Reynolds

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numbers for crescent-shaped and D-shaped quad bundle conductors. For crescent-shaped iced conductor, the drag coefficient shows a half sine wave between 0° and 180° but there is a decline in the curve around

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the angles of 45°, 90° and 135° due to the bundled interference effects. The lift coefficient shows a full sine wave. In addition, when Re>36321, there exist peak points in the curve of the lift coefficients of

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crescent-shaped iced conductor near the wind attack angles of 10° and 175°. For D-shaped iced conductor, there exist two extreme values near the attack angle of 75° and 135° in the lift coefficient curve. The minimum drag coefficient occurs near the attack angle of 90°. It can be seen from the figures that when Re<48429, the magnitudes of lift coefficients and drag coefficients of both crescent-shaped and D-shaped iced conductors show an increasing trend with the increase of Reynolds number. For larger Reynolds

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The Den Hartog instability criterion is widely used to predict the possibility of conductor galloping, which can be expressed as: ∂Cl +C𝑑 < ∂α

(5)

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Den=

where Den is the Den Hartog coefficient, α is the wind attack angle. The conductor will gallop if Den is

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negative.

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The Den curves of the iced conductors at different Reynolds numbers are presented in Fig.7. For crescent-shaped iced conductor, it can be seen that the increase of Reynolds number leads to an overall

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upward trend of the absolute value of the Den Hartog coefficient at most attack angles. For Re<24214, the

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Den Hartog coefficients are all positive, which means there is little chance to gallop for conductors under small Reynolds numbers. When Re>36321, the Den Hartog coefficient becomes negative near the attack

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angles of 60°, 130° and 180°, so the conductor has a good chance to gallop under higher Reynolds numbers. In addition, it should be noted that the smallest value of Den Hartog coefficient is not obtained

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when the Reynolds number is the greatest .For example, the minimum Den Hartog coefficient is -1.25 when Re equals to 48429 rather than 72644 near the attack angle of 180°. The result indicates that the

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crescent-shaped iced conductor under the condition that Re =48429 may cause the most serious potentia l incidence of galloping. For D-shaped iced conductor, there exist two unstable zones with negative Den Hartog coefficients under all Reynolds numbers. One is within the range of 60°~80°， and the other is within the range of 140°~180°. Besides, the minimum Den Hartog coefficient occurs when Re=48429 at the attack angle of 60°.

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(b) D-shaped iced conductor

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(a) Crescent-shaped iced conductor

180°

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Fig.7 Den Hartog coefficients at 7 different Reynolds numbers and attack angles ranging from 0°

3.3. The General Linear Model for Galloping

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In this section, the original linearized 1-D model is also used to analyse the stability of iced conductors. In this model, since the wind direction is normal to the axis of the conductor, the aerodynamic

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stability of galloping can be expressed as (Macdonald and Larose., 2006; Raeesi, 2015): ∂C

X a =Recos γ{ cos γ[Cd (2+ tan2 γ)+ ∂Red R e -

∂C d ∂α

∂C

∂C

tan γ]-sin γ[Cl + ∂Rel R e - ∂αl tan γ ]}

( )

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where Xa is a nondimensional parameter specifying the aerodynamic stability; γ is the direction between the oscillating plane of the conductor and the conductor-wind plane. Galloping instability may occur on

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the condition that Xa is negative. It is worth noting that for across- flow vibrations in flow normal to the

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conductor axis (i.e. γ=90°), Eq. (6) reduces to: ∂C

Xa =Re(Cd + ∂Rel

(7)

In absence of the Reynolds number, Eq. (7) is exactly the same as Eq. (5), which means that the general instability criterion for galloping in this model covers the special case of Den Hartog criterion. The aerodynamic coefficients are obtained from the wind tunnel test results and are used to determine the derivative terms, which are

∂Cd ∂Re

and

∂Cl ∂Re

. Fig.8 and Fig.9 present the values of Xa at the

studied range of Reynolds number and attack angle with different directions, including the along-wind

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ACCEPTED MANUSCRIPT direction γ=0, γ=30°, γ=45°, γ=60° and the across-wind direction γ=90°. It can be seen from Fig.8 that for crescent-shaped iced conductor, no aerodynamic instability occurs when γ=0, 30° and 45° in the studied range of Reynolds numbers and attack angles. Negative value of Xa can be observed for 4

Re 4 1

near the attack angle of 180° when γ=60°. As for across-wind direction γ=90°, there appear

three instability zones around the attack angle of 60°， 130° and 180°. Calculation of Xa function with different directions of motion covering a range of γ

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180° shows that the most critica l ° and Re=72644 in Fig.8(e),

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case in terms of the largest instability zone corresponds to γ=90° for α 1

1 4 , which is not when minimum Den Hartog coefficient occurs.

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and the minimum value of Xa is - .

For γ=90°, with the increase of Reynolds number, the range of the attack angle which could cause

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instability becomes larger.

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For D-shaped conductor, there is no chance for conductor to gallop in the case of along-wind direction (γ=0). For γ=30° and 45°, galloping instability occurs near the attack angles of 140° for the

for 4

1

4

Re 4

1

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full studied range of Reynolds numbers. There is also a small instability zone near the attack angle of 110° 4

when γ=45°. For γ=60°, the instability zone near the attack angles of 140°

4

Re 5.

observed for value of Xa value of Xa

1

4

and

<α<

1

4

Re

and 14 <α<1

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3.5 1

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becomes larger and two new instability zones are observed near the attack angle of 60 ° for

-1.

- .3

1

1

5

.2 1

4

. For γ=90 ° , similar instability zones can also be

for the full studied range of Reynolds numbers. The minimum

occurs when γ=60° for α 14 ° and Re=72644 in Fig.9(d). The minimum

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when γ=90 ° occurs when α

° and Re=48429, which is also when

minimum Den Hartog coefficient occurs.

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(b) γ=30°

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(a) γ=0

(d) γ=60°

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(c) γ=45°

(e) γ=90°

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Fig.8 Value of X a for crescent-shaped iced conductor

(b) γ=30°

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(a) γ=0

(d) γ=60°

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(c) γ=45°

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The results obtained in this part when γ=90°for both crescent-shaped and D-shaped conductors correspond with the analysis of Den Hartog instability mechanism. At the same time, some other instability zones can be seen with other directions in the general linear model.

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4. Effects of Other Factors on Aerodynamic Coefficients 4.1. Ice thickness

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Apart from wind velocity, ice thickness, conductor type, bundle spacing and bundle number will all

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have an influence on aerodynamic parameters. Since these factors are not very sensitive to the Reynolds

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number, this part will focus on how these factors themselves influence aerodynamic parameters and analyse conductor galloping according to Den Hartog instability mechanism.

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Ice thickness affects wind excitation on the conductor and has a significant effect on the aerodynamic forces. The aerodynamic parameters will be very different with the ice thickness changing.

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Fig.10 and Fig.11 show the aerodynamic parameters and Den Hartog coefficients of crescent-shaped iced

(Re=24214).

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conductors and D-shaped iced conductors in different ice thicknesses under the wind velocity of 10m/s

It can be seen from the Fig.10 (a) that the amplitude of drag coefficient of crescent-shaped iced

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conductor increases with the increase of ice thickness. In the attack angle range of 25°~155°, the drag coefficient increases with the increase of ice thickness. In addition, the thicker the ice thickness, the bigger the change of rate of drag coefficient. However, in the range of 0~25° and 155°~180°, the drag coefficient decreases with the increase of ice thickness. Fig.10 (b) shows that the absolute value of the lift coefficient increases with ice thickness increasing in the full range of attack angle. In addition, with the increase of ice thickness, peak points appear near the attack angles of 20°, 85° and 175 °. For ice thickness thinner than 20mm, the Den Hartog coefficients are all positive and the conductor remains stable. When the ice is

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full range of wind attack angle, as shown in Fig.11 (a). The lift coefficient shows a clear increasing trend with the increase of ice thickness in the attack angle of 100°~170°, as presented in Fig.11 (b). The

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minimum value of the Den Hartog coefficient decreases as the ice thickness increases, which indicates that

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the thicker ice may cause more serious potential incidence of galloping.

(b) Lift coefficient

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(a) Drag coefficient

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Fig.10 Aerodynamic curves of crescent-shaped iced conductor under different ice thicknesses

(b) Lift coefficient

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(a) Drag coefficient

(c) Den Hartog coefficient

Fig.11 Aerodynamic curves of D-shaped iced conductor under different ice thicknesses

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The increase of the ice thickness makes both crescent-shaped and D-shaped conductors easier to gallop. The study of aerodynamic parameters in different ice thicknesses is important to further analyse the influence of various ice and wind parameters on conductor galloping, which is of great significance for the research of the galloping mechanism and the improvement of anti-galloping measures. 4.2. Conductor type

The parameters of the conductors are crucial factors that influence galloping. Therefore, it is crucial

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ACCEPTED MANUSCRIPT to study aerodynamic parameters of different types of conductors for reasonable selection of conductors in order to prevent galloping. Three different conductor types (LGJ-630/45, LGJ-400/35, LGJ-300/40) are chosen to study and their parameters are set out in Table 1. The aerodynamic parameters and Den Hartog coefficients of crescent-shaped iced conductor with 10mm ice thickness under wind velocity of 20m/s are studied in this paper, as shown in Fig.12.

Mechanical Parameters of the Conductors LGJ-630/ 45

Cross-section area(mm2)

666.55

Diameter(mm)

33.6

Weight per unit length(kg/km)

2060

Rated tensile strength(kN)

148.7

Young’s modulus(MPa) Coefficient of linear expansion(10-6 )

LGJ-400/ 35

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Conductor model

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Table 1

LGJ-300/ 40 338.99

26.82

23.94

1349

1133

103.9

92.22

63000

65000

73000

20.9

20.5

19.6

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425.24

(a) Drag coefficient

(b) Lift coefficient

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(c) Den Hartog coefficient

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Fig.12 Aerodynamic parameters of different types of crescent-shaped iced conductors

When the attack angle is in the range of 50°~140°, the drag coefficient increases with the increase

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of the diameter of the conductor. When the attack angle is in the range of 0~50° or 140°~180°, the drag coefficient decreases with the increase of the diameter. For the lift coefficient, with the increase of the

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diameter of the conductor, the absolute value of the it increases for most attack angles and there appears a peak point around the attack angle of 175° for LGJ-630/45. Since the negative zone of Den Hartog

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coefficient for LGJ-630/45 appears at the same attack angle, the appearance of peak point of lift

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coefficient is significant for the potential incidence of galloping, for it vividly reduces the slope value of lift coefficient. In addition, the conductor with a larger cross section is difficult to twist after eccentric icing since it has a bigger torsional stiffness, which makes the difference of ice thickness between the

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windward and leeward surface of the conductor bigger. The eccentric icing becomes serious and the conductor is easier to gallop. Therefore, in practical engineering applications, the use of smaller cross section conductors in a reasonable range can be considered to reduce the galloping. 4.3. Bundle spacing

The sub span oscillation and the electrical characteristics are the main two aspects to be taken into account when it comes to selecting the bundle spacing according to the current standard (Rakosh, 2011).

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ACCEPTED MANUSCRIPT The sub span oscillation is serious when the ratio of bundle spacing S to sub diameter D is less than 10, which should not be used. The sub span oscillation can be avoided when S/ D is greater than 16. In the electric field, when S/ D is approximately 10, the surface electric field of the conductor is minimal and the corona is less likely to appear. In general, the value of S/ D of the bundle conductor is designed to be between 10~16. The aerodynamic parameters of the conductor under different bundle spacings are studied,

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and the new requirements for the selection of the bundle spacing are put forward from the perspective of reducing the conductor galloping. In this paper, the aerodynamic parameters of the crescent-shaped

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conductor with 10mm ice thickness and the D-shaped iced conductor with 39.32mm ice thickness are

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studied at different bundle spacings at the wind velocity of 20m/s. The results are shown in Figs.13 and 14.

(b) Lift coefficient

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(a) Drag coefficient

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Fig.13 Aerodynamic parameters of crescent-shaped iced conductors with different bundle spacings

(b) Lift coefficient

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(a) Drag coefficient

(c) Den Hartog coefficient

Fig.14 Aerodynamic parameters of D-shaped iced conductors with different bundle spacings

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For both the crescent-shaped and D-shaped iced conductor, the drag coefficient decreases as the bundle spacing increases, but not by much. The absolute values of the lift coefficients show an obvious decreasing trend. As for Den Hartog coefficient, the minimum value occurs when the bundle spacing is 425mm. And it is obvious that for D-shaped conductor, the instability zone becomes smaller with the bundle spacing increasing. Therefore, it is beneficial to increase the bundle spacing of the conductor to prevent the conductor from galloping. For instance, as for the 500kV 4×LGJ630/45 quad bundle

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33. =53 . mm

Thus, the optimum bundle spacing is 537.6mm considering galloping and the effect of corona. Since the value of bundle spacing of 500kV transmission line in China is not more than 500mm, the bundle

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spacing can be selected as 500mm in the galloping areas. 4.4. Bundle conductor

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There are great differences between single and bundle conductors in the galloping. It is generally

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acknowledged that the single conductor is prone to twist under the influence of icing eccentricity due to its small torsion stiffness, making the shape of icing symmetric and round. The bundle conductor is not easy

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to twist because of the large torsional stiffness. Therefore, the asymmetric icing leads to the fact that the

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bundle conductor is easy to gallop. In this paper, the difference of aerodynamic parameters of single and bundle conductors are obtained through wind tunnel test so as to analyse the difference of galloping

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between them from the perspective of aerodynamic excitation. The aerodynamic parameters and Den Hartog coefficients of crescent-shaped iced single conductor

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paper, as shown in Fig.15.

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and quad bundle conductor with 10mm ice thickness under the wind velocities of 20m/s are studied in this

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(b) Lift coefficient

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(a) Drag coefficient

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(c) Den Hartog coefficient

Fig.15 Aerodynamic parameters of crescent-shaped iced single conductor and quad bundle conductor

The curves of aerodynamic parameters of single and quad bundle conductors show similar overall

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trend, but there is a significant difference in the local area. Due to shielding effect, the drag coefficient of quad bundle conductor decreases in the attack angle degree of 45°，90° and 135°, especially in the 90° attack angle. The two leeward sub conductors are completely blocked by the two windward sub conductors in the 90 ° attack angle, so the drag coefficient decreases greatly. In general, the drag coefficient of the quad bundle conductor is smaller than that of the single conductor. For lift coefficient, the amplitude of the quad bundle conductor is greater than that of the single conductor. Moreover, due to

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ACCEPTED MANUSCRIPT the shielding effect, the lift coefficient of the quad bundle conductor changes more intensely. The Den Hartog coefficient curve of the single conductor shows a gentle change in the most range of attack angles, and the value of Den Hartog coefficient is always positive. That is to say, the single conductor is more difficult to gallop compared with the bundle conductor. 5. Simulation of Galloping

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Ice thickness, wind velocity, damping and conductor type all have an important influence on the

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galloping of the conductor. In this section, a multi-span 4DOF (4 degree-of-freedom) nonlinear dynamic model for conductor galloping is developed to simulate the galloping situation. Thus, it is possible to

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release the warning of galloping to the relevant departments to reduce the damage of the conductor

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galloping.

Since each line section is considered to be independent of the adjacent sections, a line section is

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adopted to study the galloping characteristics of 500kV transmission line. The test line section has four spans, of which the lengths are 550m, 455m, 481m and 479m, and the height differences are 0m, 6m, -9m

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and 0m. There are four conductors of Type LGJ-630/45 bundled together. This test line galloped seriously

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in 2008 in Henan Province, China. The situation of field galloping is as follows: the temperature is -5℃； the wind velocity is about 15m/s； the ice shape is approximately crescent, and the ice thickness is 10mm. The galloping amplitude in the vertical direction is about 8m and the frequency is about 10~12 times per

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minute, namely 0.167~0.2Hz. The results are obtained from video footage by a high-precision camera. Since the span length of an overhead transmission line is long, and the stiffness of the conductor has little effect on the shape of a cable hung between two suspension points, the conductor can be assumed as a soft chain. The distribution of the displacement and tension of the conductor can then be obtained, which is taken as the initial condition for the dynamic simulation of conductor. In this paper, the nonlinear dynamic model of the multi-span 4DOF (4 degree-of-freedom) dynamic model for conductor-insulator

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ACCEPTED MANUSCRIPT which is implemented in C++is established, as shown in Fig. 16. This model assumes the tension insulator to be at both ends of the line section. The cable is divided into four-dimensional (x, y, z, θ) elements and the weight of the conductor is concentrated at the nodes of the elements. In this work, 500 elements were used per span. As is shown in Fig. 16, the y direction is the vertical diction, and is the direction of gravity. The z direction is the direction of the axis of the transmission line, and the x direction is the direction of

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z×y. The multi-span model assumes the tension insulators to be at both ends of the line section.

o z

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x

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y

U

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Fig. 1 . M ulti-span 4DOF dynamic model for the conductor

The stiffness matrix is replaced by the tension vector and moved to the right of the equation. The

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dynamic equations of each element of the iced conductor are as follows (Fu, 2012):

where

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̈

̈

̇ θ

(8)

̇

(9)

is the mass of the element i of the iced conductor, which includes the self-weight and the weight

of covered ice; c is the damping of the translational direction;

is the moment of inertia;

θ

is the

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damping of torsional vibration; ̇ and ̈ are the velocity and the acceleration vector of i; ̇ and ̈ are the angular velocity and the angular acceleration;

is the external force vector of i, which includes the

gravity, the external forces exerted on the conductor and the aerodynamic force;

is the inertia force

vector, which is produced for the fact that in the process of galloping, ice on conductors is mostly asymmetrical, and the mass center of the conductor is not the same as its torsion centre (the centre of the

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ACCEPTED MANUSCRIPT bare conductor).

is the tension vector of i;

,

and

are the torque produced by twisting, the

external torque and the moment of inertia. The line parameters are the same as the actual line parameters mentioned above, which are shown in Table 2. The ice shape is crescent covered with 10mm-thick ice. The wind speed is 15m/s, and its direction is perpendicular to the axis of the conductor. The damping c and

θ

have obvious influence on the

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galloping amplitude. In this paper, the damping ratio ζx, ζy and ζz are all chosen as 0.012 and ζθ is chosen as 0.0054 (Fu, 2012). In the process of galloping, the external forces and torque caused by aerodynamic

(10)

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FD =0.5Cd U 2 Ld

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forces are the drag FD, the lift FL and the torque M, which can be defined as:

FL =0.5Cl U 2 Ld

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(11)

M =0.5Cm U 2 Ld 2

(12)

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Table 2

Parameters of the conductor Sectional area

External diameter

Line density

Breaking load

Modulus of elasticity

Coefficient of linear expansion

LGJ-630/45

666.55mm2

33.6mm

2020kg/km

148.7kN

63000MPa

20.9×10-6

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Conductor type

Polynomial fittings are done for lift coefficient, drag coefficient and torque coefficient. The

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expressions of the three force coefficient and attack angle after the curve fitting are as follows: n

Cd   adn n (n  1)

(13)

0

n

Cl   aln n (n  1)

(14)

0 n

Cm   amn n (n  1)

(15)

0

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ACCEPTED MANUSCRIPT where α is the attack angle; n is the order and is a positive integer; adn is the n order fitting coefficient of the drag coefficient; aln is the n order fitting coefficient of the lift coefficient; amn the n order fitting coefficient of the torque coefficient. The lift, drag and torque of the conductor in the process of galloping are calculated by the results of the corresponding wind tunnel test. The ice thickness of crescent-shaped iced conductor is 10mm and the

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wind velocity is 15m/s, which corresponds with the field weather conditions. The lift coefficient, drag

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MA

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coefficient and torque coefficient are shown in Fig.17.

Fig.1 Aerodynamic parameters of the conductor

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The fitting coefficients are calculated for aerodynamic coefficients using the selected 37 attack angles and the corresponding test values, which are shown in Table 3. The lift, drag and torque of the

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conductor under any attack angle in the process of galloping can then be obtained. Table 3 Fitting coefficients of aerodynamic curve

n

adn

aln

amn

0

0.7382

-0.0042

0.63813

1

3.3626

0.2926

-24.6672

2

-43.3503

4.0855

196.2353

3

283.1521

9.3677

-772.4825

4

-940.7520

-27.9778

1715.1664

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151.0069

-2886.1658

6

-2139.3100

-284.5593

3427.0318

7

1648.0600

295.1216

-2805.3214

8

-837.1920

-186.7797

1552.1233

9

278.4861

74.0050

-566.4008

10

-58.3703

-17.9439

130.1875

11

6.9947

2.4372

-17.0502

12

-0.3654

-0.14215

0.9692

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The lift, the drag and the torque of the conductor element are calculated according to the fitting

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coefficients above. The calculated forces of the lift, the drag and the torque are projected in the direction

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of X, Y, Z and θ. The external force component and the external torque component of each conductor unit in the process of galloping are then obtained. After that, an explicit direct integration algorithm based on a

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central difference is used to solve the problem. As a result, the whole galloping process of the conductor

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can be simulated using the self-programming software. Fig.18 shows the displacement of the midpoint of the longest span in the y direction. As shown in Fig.18, the conductor remains unstable and gallops, which

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is the same as the results of the linear models in the same condition in Part 3.2 and 3.3.

Fig.1 Vertical displacement of conductor

The simulation results and the actual observation results of the conductor galloping under the condition of 15m/s wind velocity and 10mm ice thickness are shown in Table 4.

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ACCEPTED MANUSCRIPT Table 4 Comparison of simulation and field results Galloping amplitude in the Frequency/Hz vertical direction/m Simulation results

7.8

0.143

Observation results

8.0

0.167

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As shown in Table 4, the simulation results are very close to the actual observation results, which proves that the calculation results of the software have a certain guiding significance to the prediction of

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the field transmission line galloping. 6. Conclusions

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In this paper, the wind tunnel test is done to calculate the lift, the drag and the torque coefficients of the conductor under different wind velocities, ice thicknesses and line conditions so as to analyse the

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influence of different factors on the aerodynamic coefficients. The instability incidence is identified according to Den Hartog instability mechanism and the linear galloping model. Then a simulation mode l

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for conductor galloping of 500 kV transmission lines has been developed and the conductor galloping is

follows:

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simulated, the results of which are close to the observation data. The findings may be summarized as

(1) With the increase of wind velocity (Reynolds number), the lift coefficient and the drag

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coefficient of the crescent-shaped and D-shaped iced conductors both show an obvious increasing trend. For larger Reynolds number which is closer to the critical regime, the aerodynamic coefficients decrease with the increase of wind velocity. (2) The instability results based on Den Hartog instability mechanism is a special case of the general linear model for galloping. According to the instability analysis, the crescent-shaped conductor has a good

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ACCEPTED MANUSCRIPT chance to gallop under higher Reynolds numbers. For D-shaped iced conductor, there exist two unstable zones with negative Den Hartog coefficients under all Reynolds numbers. (3) With the increase of ice thickness, the maximum value of the drag coefficient and the maximum absolute value of the lift coefficient of the crescent-shaped iced conductor increase in the whole range of attack angle. For D-shape iced conductors, the drag coefficient increases in the whole range of attack angle,

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and the lift coefficient increases in the attack angle of 100°~170°. Thicker ice may cause more serious potential incidence of galloping.

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(3) The conductor type has an impact on the galloping. For crescent-shaped iced conductor, when

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attack angle is about 50°~140°, the drag coefficient increases with the increase of the conductor diameter. The absolute value of the lift coefficient increases with the increase of the conductor diameter for most

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attack angles. As the conductor diameter increases, peak point of lift coefficient will occur and there is a

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good chance of galloping.

(4) With the increase of bundle spacing, the drag coefficients of crescent-shaped iced conductor and D-shaped iced conductor both decrease, but just by little. The absolute values of the lift coefficient of the

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conductors have a decreasing trend. For D-shaped conductor, the instability zone becomes smaller.

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(5) Because of the shielding effect, the drag coefficient of the crescent-shaped quad bundle conductor is smaller than that of the single conductor. The lift coefficient of the quad bundle conductor changes more intensely with the change of the attack angle than the single conductor. The single conductor

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is more difficult to gallop compared with the bundle conductor. Acknowledgments

The authors would like to thank for the financial support by the National Key Research and Development Program of China (No. 2017YFB0902700).

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ACCEPTED MANUSCRIPT References Cai, M., Yan, B., Lu, X., Zhou, L., 2015. Numerical simulation of aerodynamic coefficients of iced-quad bundle conductors. IEEE Transactions on Power Delivery. 30(4), 1669-1676. Chabart, O., Lilien, J. L., 1998. Galloping of electrical lines in wind tunnel facilities. Journal of Wind Engineering & Industrial Aerodynamics. 74-76(98), 967-976. DEN HARTOG, J. P., 1932. Transmission Line Vibration Due to Sleet. Transactions of the American Institute of

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Electrical Engineers. 51(4), 1074–1076.

Fu, G., “Research on galloping characteristics of ultra high voltage transmission lines and key technologies of anti

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galloping”, Ph. D. dissertation, Dept. Electron. Tsinghua Univ, Beijing, 2 12. (in Chinese)

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Hu, J., Yan, B., Zhou, S., et al., 2012. Numerical investigation on galloping of iced quad bundle conductors. IEEE Transactions on Power Delivery. 27(2), 784-792.

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Lin, W. “Wind tunnel and numerical study on aerodynamic characteristics of ice accreted transmission lines”,

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Master thesis, Zhejiang University, Hangzhou, 2012 (in Chinese). Lou, W., Lv, J., Huang, M. F., et al., 2014. Aerodynamic force characteristics and galloping analysis of iced bundled conductors. Wind and Structures, An International Journal, 18(2).

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Macdonald, J. H. G., & Larose, G. L., 2006. A unified approach to aerodynamic damping and drag/lift instabilities,

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and its application to dry inclined cable galloping. Journal of Fluids & Structures, 22(2), 229-252. Nigol, O., Clarke, G. J., 1974. Conductor galloping and control based on torsional mechanism. IEEE Transactions on Power Apparatus and Systems, PA93(6), 1729-1729.

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Nishihara, T., Shimizu, M., Matsumiya, H., 2012. Effects of initial ice and snow accretion angle on steady aerodynamic coefficients of ice and snow accreted conductor for overhead transmission lines. Kozo Kogaku Ronbunshu A, 58, 589-598.

Raeesi, A., 2015. Wind-induced response of bridge stay cables in unsteady wind. Electronic Theses and Dissertations. 5452. https://scholar.uwindsor.ca/etd/5452. Rakosh Das Begamudre. 2011. Extra High Voltage AC Transmission Engineering.

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ACCEPTED MANUSCRIPT Wang, J., 2008. Overhead Transmission Line Vibration and Galloping. 2008 International Conference on High Voltage Engineering and Application, Chongqing, pp. 120–123. Yu, P., Shah, A. H., Popplewell, N., 1992. Inertially coupled galloping of iced conductors. Journal of Applied Mechanics,59(1):140~145. Zhang, Q., Popplewell, N., Shah, A. H., 2000. Galloping of bundle conductor. Journal of Sound and Vibration.

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234(1), 115-134.

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ACCEPTED MANUSCRIPT Highlights Aerodynamic coefficients of crescent-shaped and D-shaped conductors under a wide range of wind velocities (Reynolds numbers), different ice thicknesses, conductor types, bundle spacings and bundle numbers are obtained by wind tunnel test. Effects of wind velocity (Reynolds number) on galloping are analyzed according to Den Hartog instability mechanism and the general linear model for galloping. Effects of ice thickness, conductor type, bundle spacing and bundle number are studied by Den Hartog instability mechanism.

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A multi-span 4DOF (4 degree-of-freedom) nonlinear dynamic model for conductor galloping is developed to simulate the galloping situation.

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