Statistical Analysis of Wind-Induced Dynamic Response of Power Towers and Four-Circuit Transmission Tower-Line System

Only one wind field model loading the transmission tower or the tower-line system was investigated in the previous studies, while the influence of two different wind field models was not considered. In addition, only one sample of the wind speed random process was used in the past numerical simulations, and the multiple dynamic response statistical analysis should be carried out. In this paper, statistical analysis of the wind-induced dynamic response of single towers and the transmission tower-line system is performed with the improved accuracy. A finite element model of the transmission tower-line system (the tower consisted of both steel tubes and angel steels) is established by ANSYS software. The analysis was performed by three statistical methods. The effects of the length of the time history and of the number of samples were investigated. The frequency histograms of samples follow the Gaussian distribution. The characteristic statistical parameters of samples were random. The displacements and the axial forces of the low tower are larger than those of the high tower. Two wind field models were applied to simulate the wind speed time history. In field 1 model, Davenport wind speed spectrum and Shiotani coherence function were applied, while in field 2 model Kaimal wind speed spectrum and Davenport coherence function were used. The results indicate that wind field 1 is calmer than wind field 2. The displacements and the axial forces of the tower-line system are less than those of single towers, which indicate damping of wind-induced vibrations by the transmission line. An extended dynamic response statistical analysis should be carried out for the transmission tower-line system.


Introduction
The power transmission tower-line system is a complex spatial coupled vibration system.Vibration is the main cause of damage to the power circuits [1][2][3][4].Because of the high flexibility of power towers and the geometrical nonlinearity of the conductor and insulator strings, it is difficult to simulate precisely the dynamic response of a tower-line system.Usually, the transmission line and the towers were analyzed independently, the coupling effect of the line and tower was ignored; thus, the calculation analysis was inaccurate [5][6][7][8].
The wind response of the power transmission towerline systems has been considered in a series of studies.Mara and Hong [9] performed a numerical simulation to investigate the effect of wind direction on the response and the surface capacity of a lattice transmission tower under wind loads.The inelastic response parameters and the capacity curve for a single tower were calculated in the paper.Yang et al. [10,11] developed the three-dimensional finite element models of UHV single anchor tower and the tower-line system and analyzed the responses of the anchor tower and tower-line system under a wind load.The results show that the displacement, the axial force, and the main column counteracting force have a nonlinear change trend with the initial stress increase.Under a wind load, the mean displacement and the maximum axial force on anchor tower are much higher than those under the equivalent static load.Chen et al. [12] and Aboshosha and El Damatty [13] emphasized that the influence of dynamic interaction of the conductor and ground wires on towers needs to be investigated.Zhou et al. [14,15] introduced the theory of rain wind-induced vibration on the stay cable in transmission line.Hamada et al. [16][17][18][19][20] developed a nonlinear three-dimensional four-nodded cable element and the nonlinear threedimensional numerical model to simulate the transmission line and towers behaviour.They studied the progressive failure of the lattice transmission tower-line systems under tornado wind loads and proved the importance of the transmission lines in the analysis of transmission tower-line systems under tornado loads.Tian and Zeng [21] carried out a parametric study of tuned mass dampers for a long span transmission tower-line system under wind loads.Barbosa et a1.[22] proposed a methodology for the reduction of faults in transmission tower-line system under high-speed wind events and gusts, including a safety analysis and recommendations for corrective maintenance.The above researchers mainly focused on either angle steel tower or the steel tube tower, while towers consisted of both steel tubes and angel steels were less investigated.Therefore, the coupling effect of the tower-line system on the wind vibration response should also be considered for the case of the tower which consisted of both steel tubes and angel steels.It should be noted that only one wind field model loading the transmission tower or the tower-line system was investigated in the previous studies, while the influence of two different wind field models was not considered.So far, two wind field models must be investigated.
In addition, only one sample of the wind speed random process was used in the past numerical simulations.Taking into account actual complicated wind field, the multiple dynamic response statistical analysis should be carried out.According to a Japanese standard [23], the response of a single mass system on wind acting 160 times in 10 min was calculated, and the results showed that the interval analysis should be performed at least four times to make the results 90% credible.The multinode system deviations may be larger.Gui-niu [24] analyzed the wind vibration of seven typical high-rise buildings, made statistical analysis of the results, and concluded that the internal forces of the structure, the displacement, and the acceleration responses to the wind on the top are all relatively consistent, verified the correctness and applicability of the method, and provided a reasonable statistical method to assess the wind vibration time history analysis result.
This paper is based on the four-circuit electrical power line.We designed a finite element model of the transmission tower-line system using ANSYS software.The effects of the length of the time history and of the number of samples were investigated.To improve the credibility of the wind vibration response analysis, we selected 600 s sample length of time history and the total number of samples 10 for the further analysis.Based on the engineering practice, we considered 90 ∘ wind angle under two wind field models.Then, we carried out statistics and comparative analysis of the wind vibration response of single towers and the transmission tower-line system.Finally, the results obtained with two different wind field models have been compared.This study supplies an advanced statistical analysis of the vibration response of transmission tower-line system.

Structural Model
. .Project Profile.In this paper, we investigated a 220 kV transmission line of four circuits as the engineering background, adopting the method of strain tower, tangent tower, tangent tower, and strain tower, which spans on 280 m, 653 m, and 259 m.The tangent tower is a four-circuit tower consisted of steel tubes and angel steels, the nominal height of the high tower is 72 m, and its total height is 97.3 m with the root of 22 m; the nominal height of the low tower is 45 m, its total height is 70.3 m, and the root is 11 m.The conductor line is a double split type JLHA2/LB14-630/45, and the upper two ground lines are JLHA2/LB14-95/55.The tower advocate is made of Q345 steel tube, and the other parts are made of Q235 equilateral angle steel.The type and parameters of the lines are listed in Table 1.
. .e Finite Element Model.The model of the transmission tower-line system was designed by ANSYS software.We employed Beam 188 unit for simulations; the quality of steel tube and angle steel elements, the nodal plate, auxiliary materials, and fittings is considered by adjusting the density of the material.The elasticity modulus and the Poisson ratio of the steel for Q235 equilateral angle steel and Q345 steel tube were 206 GPa and 0.3, respectively.The tower-line system was built for four circuits.The conductor line of the tower is vertically arranged and consists of four layers.The top one is the ground line, and the remaining layers are four pairs of double split conductors.Each pair of double split conductors and the ground line were simulated with cables.According to the search theory of the conductor line, we used Link 10 to model the line and applied the initial strain for the initial

Numerical Simulation of the Wind Speed
. .ree-Dimensional Wind Field Parameters.In the wind speed time history curve, the instantaneous wind speed consists of two parts: one is a rather long period of more than 10 min, the mean wind that does not change with time; the other one is the fluctuating wind with the period of a few seconds.The wind speed in the structure can be expressed by the mean wind speed and the fluctuating wind speed.See the following formula:  (, , , ) = V () + V (, , , ) . ( . . .e Mean Wind.Davenport [25] proposed an exponential function to describe the mean wind speed; it changes with the height.In the wind-resistant design of civil engineering, the exponential function is usually adopted as follows: where () is the mean wind speed at the height of ; (  ) is the mean wind speed at the height of   ;  is the height;   is the standard reference height;  is the ground roughness exponent.
. . .e Fluctuating Wind.The turbulent characteristics of the fluctuating wind within the frequency domain are described by the longitudinal fluctuating wind speed spectrum and the coherence function of the fluctuating wind speed.
The specification in China [26] adopts the Canadian Davenport wind speed spectrum [25].Davenport obtained more than 90 strong wind records based on different places and heights in the world and established the empirical mathematical expression: where () is the fluctuation wind speed spectrum;  is the frequency;  is the height; () is the mean wind speed at the height of ;  10 is the mean wind speed at the standard height of 10 m;  is the ground roughness exponent;  is the terrain rough factor.The transmission tower is a high structure, suitable for the use of the Kaimal wind speed spectrum [27] as the target power spectrum to simulate the wind speed.Its mathematical expressions are the following: where (, ) is the fluctuation wind speed spectrum;  * is the longitudinal friction velocity;  V is the root variance of the pulsating wind speed; the rest of the symbolic parameters are the same as in (3).
According to the specification in China [26], the threedimensional frequency-independent spatial coherence function was adopted by Shiotani and Arai [28].Its expression is as follows: where Coh , () is the square root of the coherent function;  = , ,  are the coordinates;   = 50,   = 50,   = 60.
The coherence function proposed by Davenport [25] is related to the frequency, and its expression is where Coh , (, ) is the square root of the coherent function;  is the frequency;  = , ,  are the coordinates; (  ), (  ) are the mean wind speed at the heights of   and   ;   ,   ,   are the attenuation coefficient,   = 16,   = 8,   = 10.
. . .Method of Numerical Simulation of the Fluctuating Wind Speed.Many researchers in China and abroad observed and studied the fluctuating wind, and it is generally believed that the fluctuating wind can be approximated as a stationary random process with a zero mean value.The simulation method of the stationary random process is divided into two kinds: linear filtration and harmonic synthesis.In recent years, the Autoregressive (AR) model of the linear filtering method is widely used in studies of random vibrations and the time domain analysis.It possesses a small amount of calculations and fast.After the linear filtering, the white noise random process (zero mean value) becomes a stationary random process with the characteristic spectrum.In this paper, the numerical simulation of the fluctuating wind speed was carried out by the Autoregressive model of the linear filtering method (AR [29]).
The AR model of  points associated with the fluctuating wind (, , , ) is expressed as follows: where are the coordinates of number  point,  = 1, 2, . . ., ;  is the order number of AR model; Δ is the time step;   is  ×  matrix, the Autoregressive coefficient matrix of AR model;  = 1, 2, . . ., ; () is the independent random process vector.
. .Simulation of the Wind Speed Time History.The main parameters of the wind speed simulation are shown in Table 2, according to the specification in China [26] and the engineering practice.The specification in China states that the fluctuating wind field is composed of Davenport wind speed spectrum and Shiotani coherence function.Therefore, this paper considers Davenport wind speed spectrum and Shiotani coherence function as the wind field 1, but Davenport wind speed spectrum does not change with altitude, while Kaimal wind speed spectrum changes with altitude.The transmission tower is a high structure, suitable for the use of the Kaimal wind speed spectrum as the target power spectrum to simulate the wind speed.In addition, Shiotani coherence function does not take into account the frequency, However, Davenport coherence function is related to the frequency.Therefore, this paper considers the Kaimal wind speed spectrum and Davenport coherence function as the wind field 2.
Due to the large node of the transmission tower, we simplified the simulation area in this paper.Figure 3 shows the subsection schematic of single towers and lines.We used the wind speed simulation at a centre point as the representative of each area.In each area, the wind speed time history at all nodes is the same as the wind speed time history of the representative point.Using MATLAB software to initiate the fluctuating wind speed time history simulation The initial frequency is 0.01 Hz and the end frequency is 10 Hz program, we simulated the wind speed for the single towers and the lines, respectively.We used AR model of the linear filtering method (AR [29]) to simulate the wind speed time history at 90 ∘ wind angle.Figure 4 shows the simulation results of the fluctuating wind speed time history for the transmission tower-line system at 61.05 m, sample 5, under the influence of wind fields 1 and 2. Figure 5 shows the comparison of the fluctuating wind speed spectrum with the target spectrum at 61.05 m, sample 5, under wind fields 1 and 2.
It can be seen from Figure 5 that the spectral line trend of the simulated spectrum is consistent with that of the target spectrum.Using the Autoregressive model of the linear filtering method (AR model) to simulate the wind field, we achieve good accuracy.The simulation results of the wind speed time history show that the fluctuating wind under Davenport wind speed spectrum possesses more broad fluctuation range.
. .Calculation of the Wind Load Time History.According to the manual description in China [30], the wind load on the transmission tower can be expressed as follows: where   is the wind load for transmission tower;  is the wind load shape coefficient;   is the wind area of the element in the transmission tower;  is the wind speed.
When the wind load is applied to the towers, after working out each section of the wind load, the load values were distributed into four nodes in this section of the whole body of the tower to calculate the internal forces of the structure.
The wind load of the line can be expressed as follows: where   is the standard value of horizontal wind load for vertical line action;  is the uneven factor of the wind speed;   is the wind load adjustment coefficient of the line;   is the wind load shape coefficient of the line diameter < 17 mm or ice cover (no matter the diameter of the wire),   = 1.2, the wire diameter ⩾ 17 mm, and the wire is not ice covered,   = 1.1;   is the wind pressure height coefficient;  is the diameter of the line or the mean outer diameter of the line under ice cover (for bundled conductor, the outer diameter of the subconductor). ℎ is horizontal span;  is the angle between the wind and the line;  0 is the wind pressure,  0 =  2 /1600.
When the wind load is applied to the lines, it is also applied to nodes of the sections.

Statistical Analysis of the Wind-Induced Dynamic Response
Figure 6 shows the location map of the maximum displacement nodes and the maximum axial force elements.
. .Total Length and Total Number of Samples Selection.In this paper, the total time history length and a total number of samples are selected for the high tower (without the load of the lines).The length is 100 s, 200 s, 600 s, 1000 s, and 2000 s; the total number of samples is 1, 3, 5, 8, 10.To analyze the results of the average dynamic response, we removed the first 10 s of the unstable stage of structural vibration.The statistical comparison and analysis results of the calculation of the Ux displacement (the displacement along the wind direction) on high tower top node 35# are presented in Tables 3, 4, and 5.
It can be seen from Table 3 that the extreme displacement value increases gradually with the increase of the sample length; the reason of this phenomenon is that there will be such result in the dynamic analysis, and the numerical is very big but the frequency is very low, relating to the sample length.With an increase in the sample length, the error of this method also increases as the statistical result of dynamic response.It can be concluded from Table 3 that the extreme  displacement value is not consistent with one sample.The relative error of the statistical results for the number of more than one sample is no more than 2%, and the deviation is small.The relative error of the mean value of the displacement response (Table 4) is no more than 3%; it does not change with the sample length and the number of samples.The average wind that is equivalent to static forces is the main factor that causes the average result; thus, the statistical results of the time history are calculated on average.
From Table 5, it can be seen that the root variance of the displacement response tends to a certain fixed value with an increase in the sample length and the number of samples.This value is about 14.50 mm.The effect of sample length variation is small, and the relative error is about 8%, which can be reduced by an increase in the number of samples.To sum up, we resume that high accuracy of the calculation results of the wind-induced nonlinear dynamic response statistics may be provided by the length of samples of 600 s and the total number of samples of 10.The time step was 0.1 s, excluding the initial 10 s of the unstable stage in structural vibration, and the total number of statistical points was 59,000.

. . Dynamic Response Statistical Analysis. Using ANSYS and
Origin software, we analyzed the nonlinear dynamic response of single towers and the tower-line system for the selected 600 s length 10 samples under two kinds of the wind field.Figures 7-14 show the time history curves and the frequency histograms of the single tower and tower in the tower-line system under wind fields 1 and 2.
The frequency histograms show that the frequency distribution of displacement and axial force statistical results possess Gaussian distribution (thin lines are the Gaussian curve fittings by Origin).
Tables 6-21 present statistics of 10 samples parameters of single towers and towers in the system under two kinds of the wind field.
In Tables 6-21, one may see the statistical description of the samples from the central tendency, the dispersion degree, the Skewness, and Kurtosis.The Skewness and Kurtosis describe particular characteristics of an individual sample.Skewness characterizes the distribution of symmetry.Kurtosis is a measure of the degree of Kurtosis of a set of data.As can be seen from Tables 6-21, the values of Skewness and Kurtosis are basically around 0 that is in accordance with the Gaussian distribution.
We used three statistical methods in this paper [31].Method 1, the maximum average: the extreme value in the analysis of the wind vibration is not most representative, but sometimes it cannot be ignored.Considering the maximum average results as a statistical result is relatively conservative.Method 2, "3  rule": we add three times root variance to the mean of samples.And then we calculate the average of the sum: in the simulation, the wind speed is assumed to be a stationary Gaussian random process; the time history analysis results basically have normal distribution, also following "3  rule" of statistical analysis.Method 3, the root mean squares average (RMS): because the average of the fluctuating wind is near zero, the wind-induced dynamic response of displacement and internal force response of root mean square (RMS) value is approximately equal to the average, so it relates to the response caused by the average wind.
The dynamic response displacements and axial force values of single towers and the transmission tower-line system were obtained for the sample length of 600 s and 10 samples by the three statistical methods mentioned above, and the results are listed in Tables 22-25.

Conclusion
Based on the numerical simulation method, the dynamic response of the transmission tower-line system is obtained within the sample length of 600 s for 10 samples.The conclusions are the following: (1) The Autoregressive model of linear filtering method (AR model) provides good accuracy of the wind field simulation.The simulation results of the wind speed time history show that the fluctuating wind under Davenport wind speed spectrum has broad fluctuation range.The wind speed fluctuation range simulated under Kaimal wind speed spectrum is less broad.(2) The sample length of 600 s, 10 samples, and the number of statistical points 59,000 were selected to satisfy the accuracy of calculation results of the displacement and the axial force of the single tower and tower-line system.The frequency histograms of samples basically follow the Gaussian distribution.The characteristic statistical parameters of the samples are random.
(3) The results of the dynamic response of single towers and the transmission tower-line system show that the data obtained by Method 3 are closer to the average of the mean value.In the other two methods, for the single high tower under wind field 1, the displacements are 58%-70% larger than the average of mean, and the axial forces are 23%-29% larger than the average of mean values.Under the wind field 2, the displacements were 31%-35% larger than the average of mean, and the axial forces were 14%-15% larger as well.
In the case of the single low tower under wind field 1, the displacements are 61%-75% larger than the average of mean, while the axial forces are 24%-29% larger.Under the wind field 2, the displacements were 35%-41% larger, and the axial forces were 16%-18% larger than the average of mean values.
(4) For the high tower in the system under wind field 1, the displacements are 28%-35% larger, and the axial forces are 13%-15% larger than their average of mean values.Under the wind field 2, the displacements were 22%-28% larger than the average of mean, and the axial forces exceeded on 11%-13% the average of mean values.For the low tower in the system, under wind field 1, the displacements are 30%-40% larger, while the axial forces are 14%-20% larger than the average of mean.And under the wind field 2, the displacements were 24%-34% larger than the average of mean; the axial forces were 11%-20% larger than the average of mean.
(5) The effect of statistical results of the low tower is greater than those of the high tower.The values obtained with wind field 1 is slightly larger than those under wind field 2, indicating that wind field 1 is more conservative than wind field 2. The results of the tower-line system are smaller than those of the single towers, which shows that the transmission line damps the transmission towerline vibrations.The extended dynamic response statistical analysis should be carried out for the transmission tower-line system.

Figure 1 :Figure 2 :
Figure 1: Schematic of the transmission tower-line system crossing a watercourse.

Figure 3 :
Figure 3: Subsection schematic of single towers and lines.

Figure 4 :Figure 5 :
Figure 4: The fluctuating wind speed time history under wind fields 1 and 2.

Figure 6 :
Figure 6: The location map of the maximum displacement nodes and the maximum axial force elements.

Figure 7 :Figure 8 :
Figure 7: Time history curve of single high tower under wind field 1.

Figure 9 :Figure 10 :
Figure 9: Time history curve of single low tower under wind field 2.

Figure 11 :Figure 12 :
Figure 11: Time history curve of high tower in the tower-line system under wind field 2.

Figure 13 :Figure 14 :
Figure 13: Time history curve of low tower in the tower-line system under wind field 1.

Table 1 :
Types and parameters of lines.

Table 2 :
The main parameters of wind speed simulation.

Table 3 :
The average of extremum Ux displacement on tower top node 35# (mm).

Table 4 :
The average of mean Ux displacement on tower top node 35# (mm).

Table 5 :
The average of variance Ux displacement on tower top node 35# (mm).

Table 6 :
Statistics of 10 samples parameters of the single high tower top node 35# Ux displacement time history under wind field 1 (m).

Table 7 :
Statistics of 10 samples parameters of the single high tower element 1972# axial force time history under wind field 1 (N).

Table 8 :
Statistics of 10 samples parameters of the single high tower top node 35# Ux displacement time history under wind field 2 (m).

Table 9 :
Statistics of 10 samples parameters of the single high tower element 1972# axial force time history under wind field 2 (N).

Table 10 :
Statistics of 10 samples parameters of the single low tower top node 874# Ux displacement time history under wind field 1 (m).

Table 11 :
Statistics of 10 samples parameters of the single low tower element 3986# axial force time history under wind field 1 (N).

Table 13 :
Statistics of 10 samples parameters of the single low tower element 3986# axial force time history under wind field 2 (N).

Table 14 :
Statistics of 10 samples parameters of high tower in the system top node 35# Ux displacement time history under wind field 1 (m).

Table 23 :
Comparison of statistical results on single low tower node 874# Ux displacement and element 3986# axial force.

Table 24 :
Comparison of statistical results on high tower in the system node 35# Ux displacement and element 1972# axial force.

Table 25 :
Comparison of statistical results on low tower in the system node 874# Ux displacement and element 3986# axial force.