Wind induced coupling effects of transmission tower-line system at microtopography

The coupled and uncoupled Dynamic Finite Element Models (DFEMs) of the transmission tower-line system consisting of two towers and three spans of conductors and ground wires were built at the windward and crosswind slope of a cosinoidal hill. The acceleration ratios of mean wind profiles of cosinoidal mountains were summarized from previous studies, then the mean wind profile over the windward, crosswind and leeward slopes were revised. The wind loads acting on the transmission tower-line system were calculated based on the revised profile. The wind-induced dynamic responses of the coupled and uncoupled DFEMs were detected, and the coupling effects on reaction force of hanging points, displacements and accelerations at the tower top, axial forces of main rods and wind-induced vibration coefficients were analysed. The results show that due to the lack of the TMD-like effect, the uncoupled system is more sensitive to wind load and has significant wind-induced vibration compared to the coupled system, making the three-dimensional force of the hanging point, displacements and accelerations at the tower top, axial forces of main rods increasing, but the wind-induced vibration coefficients change little. Generally, the coupling effects of the tower-line system are significant. Ignoring it will make the structural design of transmission towers conservative.


Introduction
The transmission tower-line system is one of the most prevalent structures in mountainous areas.It has a great impact on the safety of the transmission tower-line system if the wind field in the mountainous area changes.The vibration of conductors and ground wires and transmission tower interact, making the coupling effects between them exist [1][2][3] .The wind-induced responses will be overestimated or underestimated if the coupling effects are ignored.Therefore, it is essential and necessary to investigate the impact of the coupling effects on wind-induced responses of a transmission tower-line system.
Microtopography defines as special terrains located in a large terrain, such as hills, canyons and narrow passes between mountains, which influence the wind velocity profiles.Airflow from open terrain has a phenomenon of "solitary bypass" when it passes through a mountain, then the horizontal and vertical flow is generated.Due to the "solitary bypass", airflow at both sides and top of a mountain will be speeded up [4 , leading to an increase of wind pressure acting on structures and buildings.
Yao [4] found that airflow waked at the leeward slope of the mountain because of the occlusion effect, then the wind velocity at the leeward mountain foot was slowed down, i.e., the wind speed acceleration ratio was less than 1.Li et al. [5] investigated the features of wind field around a mountain using a wind tunnel test and Computational Fluid Dynamics simulation (CFD), summarized the wind velocity profiles at the mountain foot and side, and fitted the revised equations based on the profiles.Sun et al. [6] modified the equation proposed by Walmsley et al. [7] using the wind tunnel test results of the wind velocity field of micro-terrain, and proposed the revised equation of wind speed profile at the leeward mountain foot.Li et al. [8] found that the wind velocity acceleration ratio at the mountain top exponentially increased along with the height.Liu et al. [9] numerically simulated the wind flow field around a mountain in Wenzhou where a transmission tower-line system was located, and found that the wind velocity profile in a complex mountain area was different from the exponential profile defined by the Chinese standard GB 50009-2012.
The wind induced dynamic responses of transmission tower-line systems in mountain areas have been studied in previous research.Yao et al. [10] numerically simulated the wind velocity field around a 3-dimensional cosine mountain and compared the numerical results to several standards, then classified the mountain into 4 areas, i.e., significantly increased, increased, lightly increased and reduced areas according to the wind-induced bending moment at the tower foot.Liu et al. [11] took the terrain factor of wind velocity profile into account when calculating the wind-induced responses of a transmission tower on a mountain, and found that the mean displacement at the tower top increased, while the standard deviations of displacement and acceleration did not.Li et al. [12][13] pointed out that the wind induced responses of a transmission tower in mountain terrain increased by 23.0%~59.6%compared to the tower in open terrain.Okamura et al. [14] measured the wind induced responses of a tower at the leeward slope of the mountain, and found that the vertical component had a significant impact on the responses.
Generally, the revised equation of wind velocity profile defined in the Chinese standard GB 5009-2012 [15] is adopted when the wind induced responses of the transmission tower are calculated in Chinese engineering practice.However, the revised equation is quite conservative, and cannot be in good agreement with the acceleration ratio of wind velocity profiles simulated by CFD or wind tunnel test.Meanwhile, the coupling effects of the transmission tower and line are not clear yet.Therefore, firstly, the acceleration ratios of mean wind profiles of cosinoidal mountains were summarized from previous studies, then the mean wind profile over the windward, crosswind and leeward slopes were revised.Then the coupled and uncoupled DFEMs of the transmission tower-line system are established, and their wind-induced responses when located at the windward and crosswind slopes are analysed and compared.Finally, the coupling effects of the transmission tower-line system are investigated.

Acceleration ratio of wind velocity
The acceleration ratio of mean wind velocity is defined as: where 0 ( ) U z is the mean wind speed at the height of z, is the mean wind speed at the mountain top or foot, ( ) S is the acceleration ratio at the height of z, and the subscripts t and f represent for mountain top and foot.

Revised equation
The acceleration ratio of mean wind velocity at a certain spot on the mountain slope is defined by the acceleration ratio at the mountain top and foot in several standards [15][16][17][18][19] .However, it has been pointed out in many studies that it is more reasonable to calculate the acceleration ratio at the mountain slope by vertical linear interpolation of acceleration ratio at the mountain top and foot than horizontal linear interpolation.The interpolation equation can be written as: where S is the acceleration ratio at mountain slop, H is the mountain height, h is distance above the ground at mountain slop.According to Li et al.'s work [8] , the acceleration ratio t S at mountain top is: where At  and C is the correction factors of mountain shape, As  and Ah  are correction factors for mountain height and slop, respectively, L is the horizontal distance from mountain top to half mountain slop, Bs  , Bh  and Bt  are correction factors of slop, height and terrain, and The acceleration ratio at the windward, crosswind and leeward slope is different, and can be calculated as: where K and a K are conversion factors for mountain height and slop, and G H is gradient wind height.

Mountain model
A 3-D cosine mountain is used, as shown in Figure 1.The mountain shape can be determined by Equation (9).
where mountain height H =100 m, and mountain radius 1 L =400 m.A two-tower and three-span line system is adopted, as shown in Figure 2. Two layouts of the system are arranged at the windward and crosswind slopes, respectively.In Case 1, towers are located at the mountain top and foot of the windward slope, while in Case 2, towers are located at the mountain top and foot of the crosswind slopes.

Transmission tower-line system model
The tower is 115 m in height, and has three crossarms.The main rods of this tower are made of steel tubes with a strength of Q345.The conductors' type is JL1-G3A-1250/70, while the ground wire's type is JLB20A-240.Each span of the system is 800 m.In the coupled and uncoupled DFEMs, towers are modeled by Beam 188, conductors and ground wires are modeled by Link 10, and insulators are modeled by Link 8.The initial position of conductors and ground wires are calculated as [20][21] : where 0 L is the span, Q is the transmission line weight per meter, T is the initial tension of transmission line, c is altitude difference between two hanging points of lines, and x and z are coordinates.The harmony superposition method was used to simulate the wind field around the mountain.The exponential mean wind velocity profile and Davenport spectrum were adopted to simulate the wind field.The related parameters in the simulation are listed in Table 1.The simulated history and spectrum of wind speed are shown in Figure 5.It is shown that the simulated spectrum is in agreement with the target Davenport spectrum.TMD-like damper.When the system was located at the windward slope, the wind-induced vibrations of transmission lines are smaller, making the system stiffness enhanced.However, when the system was located at the crosswind slope, the wind-induced vibrations of lines are larger, making the system stiffness weakened.Table 2. Three-way force at the pegging point of coupled and uncoupled windward slope systems No.

Gust factors of reaction forces at hanging points
The gust factors of reaction forces at hanging points are defined as [22] : where max s f are peaks of s f , and 2.5 = g are peak factor.The calculated gust factors coupled and uncoupled system at the windward and crosswind slopes are listed in Tables 4 and 5.The gust factors of conductors hanging point of the non-coupled system on the windward slope are greater than the values of the coupled system, but the gust factors of the ground line are less than that of the coupled system.On the crosswind slope, the gust factors of the crosswind hanging point of the non-coupled system conductors and ground lines are greater than those of the coupled system.The time histories of along wind displacement and acceleration at the tower top are shown in Figures 7  and 8.As shown in Figure 7, the displacement of the uncoupled system is larger than that of the coupled system, and the displacement of the tower at the crosswind slope is larger than that at the windward slope.The features of accelerations in Figure 8 are similar to displacement.It indicates that the uncoupled system is more sensitive to wind load.

Displacements of tower body
The means and standard deviations of displacements of tower bodies in couple and uncoupled systems are shown in Figure 9.The displacements of the system at the crosswind slope are larger than those at the windward slope.And the displacements of the uncoupled system are larger than those of the coupled system.9. Means and standard deviations of tower displacement on windward and crosswind slopes for coupled and uncoupled systems

Accelerations of tower body
The standard deviations of accelerations of tower bodies in couple and uncoupled systems are shown in Figure 10.Similar to displacement, the accelerations of uncoupled systems are larger than that of coupled systems.

Conclusion
The wind-induced dynamic response of the coupled and uncoupled systems of a transmission tower-line system on the windward and crosswind slope is analyzed using the DFEMs, and the coupling effect on reaction forces of hanging point, displacements, accelerations, and gust factors are discussed.The following conclusions can be obtained from the above study.
(1) The wind-induced responses of the system at the windward slope are larger than those at the crosswind slope, due to the larger area where wind loads act.
(2) The wind-induced responses of the uncoupled system are larger than those of the coupled system, due to the lack of a TME-like damper generated by transmission lines.
(3) Generally, it is more conservative to design the transmission tower using an uncoupled system than a coupled system.

Figure 1 .
Figure 1.Configuration of the Cosine Mountain Figure 2. Configuration of tower-line system (a) Coupled model (b) uncoupled model Figure 3. Finite element model with two towers three wires system: (a) couple and (b) uncoupled system The coupled and uncoupled DFEMs of the transmission tower-line system are shown in Figure 3.In the coupled DFEM, conductors and ground wires are joined with insulators.However, in the uncoupled DFEM, transmission lines and towers are built separately.Transmission lines are jointed at the fixed supports.The reaction forces at the supports are calculated when wind loads act on transmission lines.Then the reaction forces are put on the hanging points at towers.This procedure is shown in Figure 4. (a) Wind load of conductors and ground wires (b) Node force applied at the hanging point Figure 4. Application of wind loads in a schematic illustration on transmission lines for a non-coupled system speed at a height of 10 m /(m/s) 32 Analog circular frequency (rad/s) [0, 4π] Fraction of frequency 1024 Time step for simulation /s 0.25 The simulation's total duration /s 512 (a) Time history of wind speed (b) Power spectrum of wind speed Figure 5. Simulated wind speed at the tower top on the windward slope (z=115 m) 4. Elegant hanging point force 4.1.Three-way hanging point force The numbers of hanging points of insulators are shown in Figure 6.Due to the symmetry, the reaction forces of points 1, 3, 5, 7, 9, 11, 13, and 14 are analyzed hereby.

Figure 6 .
Figure 6.Number of transmission towers on the guide line hanging point

Figure 7 .Figure 8 .
Figure 7. Time histories of tower top displacement on windward and crosswind slopes for coupled and uncoupled systems

Figure 10 .
Figure 10.Standard deviations of acceleration comparison on windward slope and crosswind for coupled and uncoupled systems 5.4.Gust factors of tower The gust factors of the tower based on the responses of coupled and uncoupled systems and Chinese standard GB 5009-2012 are listed in Table6.The gust factors of coupled systems at windward slope are larger than those of uncoupled systems, while it is the opposite for gust factors of coupled systems at crosswind slope.However, the gust factors based on coupled and uncoupled systems are smaller than the values based on Chinese standards.Table6.Transmission tower wind vibration coefficients on windward and sidewind sides

Table 3 .
Three-way force at the pegging point of coupled and uncoupled lateral ward slope systems z f (10 5 N) x  (10 3 N) y  (10 3 N)

Table 4 .
Wind vibration coefficients for systems with pegging point forces that are both coupled and uncoupled to the windward

Table 5 .
Wind vibration coefficients for systems with pegging point forces that are both coupled and uncoupled to the lateral ward

Table 6 .
The gust factors of coupled systems at windward slope are larger than those of uncoupled systems, while it is the opposite for gust factors of coupled systems at crosswind slope.However, the gust factors based on coupled and uncoupled systems are smaller than the values based on Chinese standards.Table6.Transmission tower wind vibration coefficients on windward and sidewind sides