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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.

The power transmission tower-line system is a complex spatial coupled vibration system. Vibration is the main cause of damage to the power circuits [

The wind response of the power transmission tower-line systems has been considered in a series of studies. Mara and Hong [

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 [

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.

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

Types and parameters of lines.

Conductor line | Ground line | |
---|---|---|

Type | JLHA2/LB14-630/45 | JLHA2/LB14-95/55 |

| 666.6 | 152.8 |

| 33.6 | 16 |

| 2030 | 670 |

| 58116 | 26120 |

| 77488 | 32650 |

| 62400 | 97400 |

| | |

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 stress of the line with the basic length of the element of 20 m. The suspension insulator string was modeled with Link 8. The foot of the tower was fixed with constraints. The stiffness of the tension tower is large, and both ends of the line have fixing constraints. The finite element model of the tower-line system consists of 2450 nodes and 5084 elements. It is shown in Figures

Schematic of the transmission tower-line system crossing a watercourse.

Finite element model of the transmission tower-line system.

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:

Davenport [

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 [

The transmission tower is a high structure, suitable for the use of the Kaimal wind speed spectrum [

According to the specification in China [

The coherence function proposed by Davenport [

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 [

The AR model of

The main parameters of the wind speed simulation are shown in Table

The main parameters of wind speed simulation.

Parameters | Values |
---|---|

Surface roughness class | A type ( |

Mean wind profile | Exponential function |

| 35 m/s |

Fluctuation wind speed spectrum | Davenport/Kaimal |

Coherence function | Shiotani/Davenport |

Simulation method | The Autoregressive model of linear filtering method (AR) |

Length of time history | 100 s, 200 s, 600 s, 1000 s, 2000 s |

Number of samples | 1, 3, 5, 8, 10 |

Time step | 0.1 s |

Frequency range | The initial frequency is 0.01 Hz and the end frequency is 10 Hz |

Due to the large node of the transmission tower, we simplified the simulation area in this paper. Figure

Subsection schematic of single towers and lines.

We used AR model of the linear filtering method (AR [

The fluctuating wind speed time history under wind fields 1 and 2.

Comparison of the pulsating wind speed spectrum with the target spectrum under wind fields 1 and 2.

It can be seen from Figure

According to the manual description in China [

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:

When the wind load is applied to the lines, it is also applied to nodes of the sections.

Figure

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

High tower

Low tower

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

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

The length of samples (s) | ||||||
---|---|---|---|---|---|---|

100 | 200 | 600 | 1000 | 2000 | ||

The total number of samples | 1 | 170.52 | 176.07 | 172.02 | 176.38 | 185.61 |

3 | 167.29 | 179.30 | 172.84 | 177.68 | 181.66 | |

5 | 168.59 | 176.76 | 171.85 | 176.58 | 185.29 | |

8 | 169.94 | 173.83 | 170.77 | 180.29 | 184.28 | |

10 | 169.77 | 173.44 | 171.71 | 179.58 | 183.38 |

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

The length of samples (s) | ||||||
---|---|---|---|---|---|---|

100 | 200 | 600 | 1000 | 2000 | ||

The total number of samples | 1 | 124.27 | 126.19 | 123.76 | 123.11 | 124.46 |

3 | 121.85 | 124.86 | 123.23 | 124.78 | 124.63 | |

5 | 124.37 | 124.21 | 123.78 | 124.53 | 124.95 | |

8 | 124.73 | 123.41 | 123.27 | 124.67 | 125.00 | |

10 | 124.44 | 123.63 | 123.45 | 124.59 | 124.92 |

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

The length of samples (s) | ||||||
---|---|---|---|---|---|---|

100 | 200 | 600 | 1000 | 2000 | ||

The total number of samples | 1 | 14.54 | 14.06 | 13.60 | 15.48 | 14.14 |

3 | 13.88 | 15.03 | 13.65 | 14.83 | 14.23 | |

5 | 14.41 | 14.97 | 14.04 | 14.64 | 14.35 | |

8 | 14.89 | 14.61 | 14.33 | 14.76 | 14.47 | |

10 | 14.71 | 14.50 | 14.32 | 14.73 | 14.51 |

It can be seen from Table

The relative error of the mean value of the displacement response (Table

From Table

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.

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

Time history curve of single high tower under wind field 1.

Frequency histogram of single high tower under wind field 1.

Time history curve of single low tower under wind field 2.

Frequency histogram of single low tower under wind field 2.

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

Frequency histogram of high tower in the tower-line system under wind field 2.

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

Frequency histogram of low tower in the tower-line system under wind field 1.

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

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

Displacement | Mean | SD | Skewness | Kurtosis | Minimum | Maximum |
---|---|---|---|---|---|---|

Sample 1 | 0.35442 | 0.06831 | 0.13752 | 0.09248 | 0.12564 | 0.62239 |

Sample 2 | 0.35890 | 0.07097 | 0.05107 | −0.15181 | 0.11660 | 0.57983 |

Sample 3 | 0.35444 | 0.06559 | 0.05681 | 0.34004 | 0.12688 | 0.63887 |

Sample 4 | 0.35050 | 0.06803 | −0.02753 | 0.13739 | 0.08682 | 0.59494 |

Sample 5 | 0.35621 | 0.06778 | −0.13817 | −0.06441 | 0.07765 | 0.58402 |

Sample 6 | 0.34745 | 0.06941 | 0.17547 | 0.12119 | 0.11916 | 0.61964 |

Sample 7 | 0.35154 | 0.07096 | 0.05388 | 0.09499 | 0.10151 | 0.58926 |

Sample 8 | 0.35595 | 0.06740 | 0.02192 | 0.19480 | 0.10866 | 0.63173 |

Sample 9 | 0.35450 | 0.06794 | 0.01347 | −0.19614 | 0.14479 | 0.56227 |

Sample 10 | 0.35138 | 0.06998 | −0.03027 | −0.12573 | 0.12174 | 0.5984 |

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

Axial force | Mean | SD | Skewness | Kurtosis | Minimum | maximum |
---|---|---|---|---|---|---|

Sample 1 | −2949190 | 225007 | −0.17856 | 0.07085 | −3822520 | −2119440 |

Sample 2 | −2950810 | 232705 | −0.03571 | −0.15095 | −3660450 | −2131130 |

Sample 3 | −2937700 | 219395 | −0.07871 | 0.40531 | −3985950 | −2135060 |

Sample 4 | −2940790 | 225055 | 0.03573 | 0.15023 | −3730720 | −2032810 |

Sample 5 | −2941240 | 225111 | 0.14216 | 0.02959 | −3762320 | −1956890 |

Sample 6 | −2931170 | 232844 | −0.20734 | 0.12401 | −3910300 | −2142180 |

Sample 7 | −2935120 | 232508 | −0.04941 | 0.05375 | −3698390 | −2099490 |

Sample 8 | −2936460 | 227044 | 0.00731 | 0.03378 | −3816710 | −2145960 |

Sample 9 | −2927360 | 228612 | −0.01472 | −0.13648 | −3705960 | −2211190 |

Sample 10 | −2938370 | 230554 | 0.0376 | −0.13684 | −3739450 | −2212520 |

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

Displacement | Mean | SD | Skewness | Kurtosis | Minimum | Maximum |
---|---|---|---|---|---|---|

Sample 1 | 0.35656 | 0.03693 | −0.18703 | −0.03583 | 0.21504 | 0.47104 |

Sample 2 | 0.35073 | 0.03515 | 0.23975 | 0.21275 | 0.25089 | 0.49667 |

Sample 3 | 0.34378 | 0.03559 | −0.06313 | −0.01272 | 0.21787 | 0.45803 |

Sample 4 | 0.35171 | 0.03376 | −0.04804 | −0.07481 | 0.23818 | 0.47092 |

Sample 5 | 0.35240 | 0.03495 | −0.15822 | 0.04742 | 0.22354 | 0.46427 |

Sample 6 | 0.35231 | 0.03791 | −0.19994 | 0.46160 | 0.19914 | 0.46968 |

Sample 7 | 0.34313 | 0.03985 | −0.03358 | −0.19496 | 0.19995 | 0.47154 |

Sample 8 | 0.34723 | 0.03761 | −0.02709 | −0.04106 | 0.22602 | 0.46813 |

Sample 9 | 0.34644 | 0.03561 | −0.13752 | −0.25035 | 0.21269 | 0.45174 |

Sample 10 | 0.35075 | 0.03550 | −0.11431 | 0.00106 | 0.21375 | 0.49495 |

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

Axial force | Mean | SD | Skewness | Kurtosis | Minimum | maximum |
---|---|---|---|---|---|---|

Sample 1 | −2948080 | 140919 | 0.17295 | −0.17540 | −3370600 | −2444340 |

Sample 2 | −2935270 | 132886 | −0.26298 | 0.19441 | −3477780 | −2534960 |

Sample 3 | −2905090 | 134097 | 0.16187 | 0.05996 | −3339770 | −2400450 |

Sample 4 | −2930700 | 127225 | 0.02708 | −0.10905 | −3376020 | −2549530 |

Sample 5 | −2942440 | 130176 | 0.15989 | −0.01083 | −3319490 | −2430920 |

Sample 6 | −2923830 | 143796 | 0.16689 | 0.32566 | −3368890 | −2332000 |

Sample 7 | −2902400 | 153641 | 0.08227 | −0.21682 | −3376560 | −2331160 |

Sample 8 | −2921840 | 142735 | −0.04108 | −0.19449 | −3413370 | −2492820 |

Sample 9 | −2907490 | 132675 | 0.13648 | −0.30934 | −3263360 | −2526720 |

Sample 10 | −2932580 | 134586 | 0.1526 | −0.14411 | −3447680 | −2437930 |

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

Displacement | Mean | SD | Skewness | Kurtosis | Minimum | Maximum |
---|---|---|---|---|---|---|

Sample 1 | 0.22218 | 0.04530 | −0.02671 | 0.38976 | 0.02268 | 0.42222 |

Sample 2 | 0.22022 | 0.04435 | −0.1314 | 0.18416 | 0.03907 | 0.3918 |

Sample 3 | 0.22132 | 0.04658 | −0.0305 | 0.02998 | 0.06550 | 0.38712 |

Sample 4 | 0.21970 | 0.04688 | −0.01637 | −0.01960 | 0.04685 | 0.38675 |

Sample 5 | 0.22132 | 0.04325 | −0.03241 | −0.19524 | 0.07749 | 0.35659 |

Sample 6 | 0.21745 | 0.04478 | 0.05446 | −0.00820 | 0.06996 | 0.3807 |

Sample 7 | 0.22377 | 0.04309 | 0.0077 | −0.25533 | 0.08035 | 0.38412 |

Sample 8 | 0.22365 | 0.04780 | 0.1368 | 0.17935 | 0.05807 | 0.40093 |

Sample 9 | 0.22044 | 0.04361 | 0.0464 | 0.16881 | 0.06084 | 0.39196 |

Sample 10 | 0.21944 | 0.04352 | 0.00647 | −0.02461 | 0.05496 | 0.37344 |

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

Axial force | Mean | SD | Skewness | Kurtosis | Minimum | maximum |
---|---|---|---|---|---|---|

Sample 1 | −2082860 | 169524 | 0.01765 | 0.34298 | −2832540 | −1319350 |

Sample 2 | −2073850 | 162592 | 0.13724 | 0.13390 | −2656770 | −1448890 |

Sample 3 | −2077750 | 174300 | 0.05555 | −0.04310 | −2655310 | −1522670 |

Sample 4 | −2068890 | 173793 | 0.02291 | −0.07825 | −2666350 | −1421670 |

Sample 5 | −2078850 | 161427 | −0.00686 | −0.24323 | −2584010 | −1584550 |

Sample 6 | −2060190 | 166659 | −0.0369 | −0.03360 | −2654160 | −1465200 |

Sample 7 | −2088820 | 158783 | −0.00452 | −0.17713 | −2714900 | −1553110 |

Sample 8 | −2083710 | 177654 | −0.19925 | 0.19016 | −2777500 | −1513340 |

Sample 9 | −2075190 | 159751 | −0.02932 | 0.13758 | −2664530 | −1492870 |

Sample 10 | −2069550 | 160157 | −0.02257 | −0.00476 | −2633630 | −1525840 |

Statistics of 10 samples parameters of the single low tower top node 874# Ux displacement time history under wind field 2 (m).

Displacement | Mean | SD | Skewness | Kurtosis | Minimum | Maximum |
---|---|---|---|---|---|---|

Sample 1 | 0.22514 | 0.02572 | −0.03497 | 0.35147 | 0.10754 | 0.31219 |

Sample 2 | 0.22209 | 0.02837 | −0.02827 | −0.22073 | 0.14054 | 0.32388 |

Sample 3 | 0.22374 | 0.02627 | 0.01650 | −0.23694 | 0.12845 | 0.31262 |

Sample 4 | 0.21937 | 0.02918 | 0.09423 | −0.10346 | 0.12755 | 0.31399 |

Sample 5 | 0.22691 | 0.02537 | 0.17305 | 0.16969 | 0.11514 | 0.31096 |

Sample 6 | 0.21709 | 0.02526 | −0.02530 | −0.15620 | 0.12656 | 0.30634 |

Sample 7 | 0.22686 | 0.02407 | −0.09220 | 0.15697 | 0.13991 | 0.31507 |

Sample 8 | 0.21504 | 0.02781 | 0.01384 | −0.01940 | 0.12086 | 0.30812 |

Sample 9 | 0.21722 | 0.02517 | −0.11976 | 0.66410 | 0.12086 | 0.32400 |

Sample 10 | 0.22322 | 0.02428 | 0.09103 | −0.01395 | 0.13545 | 0.30322 |

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

Axial force | Mean | SD | Skewness | Kurtosis | Minimum | maximum |
---|---|---|---|---|---|---|

Sample 1 | −2094690 | 107707 | 0.0077 | 0.22676 | −2452950 | −1616830 |

Sample 2 | −2082530 | 118333 | 0.02739 | −0.24416 | −2500100 | −1732870 |

Sample 3 | −2090200 | 110671 | −0.000196 | −0.25047 | −2453230 | −1702140 |

Sample 4 | −2068760 | 122979 | −0.09519 | −0.16871 | −2454330 | −1708950 |

Sample 5 | −2101960 | 108367 | −0.20278 | 0.15694 | −2453410 | −1628760 |

Sample 6 | −2059300 | 104250 | 0.06863 | −0.17650 | −2425230 | −1687330 |

Sample 7 | −2104580 | 101206 | 0.06531 | 0.18746 | −2500850 | −1750660 |

Sample 8 | −2049980 | 117069 | −0.01751 | −0.04761 | −2429990 | −1668020 |

Sample 9 | −2060060 | 106047 | 0.23684 | 0.86047 | −2494290 | −1629370 |

Sample 10 | −2087080 | 101083 | −0.17578 | 0.14326 | −2451000 | −1745540 |

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

Displacement | Mean | SD | Skewness | Kurtosis | Minimum | Maximum |
---|---|---|---|---|---|---|

Sample 1 | 0.32330 | 0.02974 | 0.10779 | 0.18315 | 0.22458 | 0.44661 |

Sample 2 | 0.31841 | 0.03038 | −0.02766 | −0.00132 | 0.21733 | 0.43928 |

Sample 3 | 0.31864 | 0.02934 | 0.08242 | −0.11575 | 0.22185 | 0.4174 |

Sample 4 | 0.32026 | 0.03065 | −0.00738 | 0.08926 | 0.20795 | 0.42238 |

Sample 5 | 0.32299 | 0.03063 | 0.04103 | 0.20988 | 0.18418 | 0.44216 |

Sample 6 | 0.32308 | 0.02990 | −0.0116 | 0.30768 | 0.21552 | 0.43896 |

Sample 7 | 0.32486 | 0.03049 | 0.03309 | 0.00446 | 0.22536 | 0.43633 |

Sample 8 | 0.32136 | 0.03042 | 0.03473 | 0.10439 | 0.20734 | 0.44905 |

Sample 9 | 0.32192 | 0.03045 | −0.10315 | 0.08275 | 0.21565 | 0.41596 |

Sample 10 | 0.32186 | 0.03094 | 0.13139 | −0.14000 | 0.21494 | 0.43481 |

Statistics of 10 samples parameters of high tower in the system element 1972# axial force time history under wind field 1 (N).

Axial force | Mean | SD | Skewness | Kurtosis | Minimum | maximum |
---|---|---|---|---|---|---|

Sample 1 | −2764420 | 117829 | −0.08952 | −0.06287 | −3153370 | −2368250 |

Sample 2 | −2745900 | 115359 | 0.0534 | −0.04482 | −3155580 | −2354910 |

Sample 3 | −2751510 | 114232 | −0.05331 | −0.00765 | −3169110 | −2340380 |

Sample 4 | −2756110 | 120891 | −0.04072 | −0.08704 | −3177180 | −2372540 |

Sample 5 | −2766420 | 119284 | −0.03748 | 0.11486 | −3245510 | −2305030 |

Sample 6 | −2767990 | 116249 | −0.01359 | 0.18738 | −3171830 | −2374290 |

Sample 7 | −2774830 | 119638 | 0.11258 | 0.01836 | −3141770 | −2336580 |

Sample 8 | −2761040 | 118749 | 0.0004 | −0.01720 | −3176570 | −2347060 |

Sample 9 | −2762820 | 120316 | 0.05717 | 0.05883 | −3188790 | −2362880 |

Sample 10 | −2766160 | 121813 | −0.056 | −0.19360 | −3139350 | −2348950 |

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

Displacement | Mean | SD | Skewness | Kurtosis | Minimum | Maximum |
---|---|---|---|---|---|---|

Sample 1 | 0.32390 | 0.01979 | 0.02170 | −0.10203 | 0.26349 | 0.40088 |

Sample 2 | 0.31469 | 0.02212 | −0.20609 | 0.23266 | 0.23122 | 0.37960 |

Sample 3 | 0.31516 | 0.02067 | −0.08654 | 0.50949 | 0.24704 | 0.41275 |

Sample 4 | 0.31720 | 0.02374 | 0.20045 | −0.21313 | 0.23982 | 0.40715 |

Sample 5 | 0.32499 | 0.02618 | 0.34043 | 0.14454 | 0.25832 | 0.41941 |

Sample 6 | 0.32549 | 0.02616 | −0.10295 | 0.17649 | 0.23848 | 0.41391 |

Sample 7 | 0.32417 | 0.02269 | 0.33877 | 0.12758 | 0.25590 | 0.41618 |

Sample 8 | 0.31876 | 0.02523 | 0.07881 | 0.05582 | 0.24046 | 0.42604 |

Sample 9 | 0.31902 | 0.02232 | 0.10203 | 0.06224 | 0.25438 | 0.40586 |

Sample 10 | 0.32502 | 0.02393 | −0.08207 | −0.09111 | 0.24802 | 0.41339 |

Statistics of 10 samples parameters of high tower in the system element 1972# axial force time history under wind field 2 (N).

Axial force | Mean | SD | Skewness | Kurtosis | Minimum | maximum |
---|---|---|---|---|---|---|

Sample 1 | −2766730 | 80424 | 0.09442 | 0.04115 | −3040150 | −2484060 |

Sample 2 | −2729260 | 89952 | 0.36791 | 0.23231 | −3025500 | −2366020 |

Sample 3 | −2734140 | 85454 | 0.1637 | 0.15528 | −3136020 | −2459260 |

Sample 4 | −2740050 | 102842 | −0.26335 | −0.33218 | −3106210 | −2456440 |

Sample 5 | −2773030 | 109547 | −0.38259 | 0.24823 | −3155320 | −2454760 |

Sample 6 | −2777580 | 109056 | −0.07278 | 0.36029 | −3137000 | −2430630 |

Sample 7 | −2774270 | 90187 | −0.21247 | 0.00087 | −3089440 | −2480530 |

Sample 8 | −2746430 | 109665 | −0.08952 | 0.00632 | −3094720 | −2375430 |

Sample 9 | −2750570 | 94028 | −0.07622 | −0.06608 | −3214130 | −2479610 |

Sample 10 | −2779340 | 98622 | 0.21257 | 0.12578 | −3099170 | −2434880 |

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

Displacement | Mean | SD | Skewness | Kurtosis | Minimum | Maximum |
---|---|---|---|---|---|---|

Sample 1 | 0.20172 | 0.02069 | 0.17222 | 0.23026 | 0.13554 | 0.30662 |

Sample 2 | 0.20006 | 0.02114 | 0.08235 | 0.00699 | 0.12600 | 0.27096 |

Sample 3 | 0.20069 | 0.02073 | 0.06948 | 0.10717 | 0.12750 | 0.2815 |

Sample 4 | 0.19951 | 0.01945 | 0.1794 | 0.27326 | 0.12948 | 0.28295 |

Sample 5 | 0.20105 | 0.02043 | 0.05912 | 0.05919 | 0.13795 | 0.27355 |

Sample 6 | 0.20208 | 0.02008 | 0.13388 | 0.04497 | 0.13019 | 0.26971 |

Sample 7 | 0.20135 | 0.02076 | 0.14814 | 0.25605 | 0.13445 | 0.29724 |

Sample 8 | 0.20067 | 0.01950 | 0.12126 | 0.02237 | 0.12363 | 0.27096 |

Sample 9 | 0.20280 | 0.01959 | 0.16538 | 0.24519 | 0.13675 | 0.29595 |

Sample 10 | 0.19928 | 0.02025 | 0.0133 | 0.05149 | 0.11540 | 0.27521 |

Statistics of 10 samples parameters of low tower in the system element 3986# axial force time history under wind field 1 (N).

Axial force | Mean | SD | Skewness | Kurtosis | Minimum | maximum |
---|---|---|---|---|---|---|

Sample 1 | −1989210 | 91975 | −0.23888 | 0.53071 | −2486860 | −1723230 |

Sample 2 | −1982100 | 93565 | −0.10102 | 0.15964 | −2302060 | −1637630 |

Sample 3 | −1985410 | 92553 | −0.0233 | 0.12298 | −2325000 | −1632300 |

Sample 4 | −1980010 | 86823 | −0.28028 | 0.75853 | −2412110 | −1657230 |

Sample 5 | −1986780 | 90776 | −0.05874 | 0.20562 | −2316480 | −1633210 |

Sample 6 | −1993140 | 89244 | −0.16064 | 0.17712 | −2380800 | −1677460 |

Sample 7 | −1987410 | 91865 | −0.28757 | 0.89180 | −2553190 | −1685600 |

Sample 8 | −1985370 | 85843 | −0.13573 | 0.38890 | −2362630 | −1538990 |

Sample 9 | −1995940 | 87631 | −0.23869 | 0.65880 | −2479220 | −1652330 |

Sample 10 | −1977210 | 89291 | −0.05338 | 0.23054 | −2303600 | −1516040 |

Statistics of 10 samples parameters of low tower in the system top node 874# Ux displacement time history under wind field 2 (m).

Displacement | Mean | SD | Skewness | Kurtosis | Minimum | Maximum |
---|---|---|---|---|---|---|

Sample 1 | 0.20474 | 0.01716 | 0.21285 | −0.25034 | 0.15718 | 0.26938 |

Sample 2 | 0.19786 | 0.01478 | −0.18439 | −0.02441 | 0.14613 | 0.24476 |

Sample 3 | 0.19756 | 0.01429 | 0.6389 | 1.95256 | 0.15665 | 0.29188 |

Sample 4 | 0.19713 | 0.01639 | 0.25773 | 0.50871 | 0.14560 | 0.27491 |

Sample 5 | 0.20114 | 0.01573 | 0.23138 | −0.07892 | 0.15503 | 0.26071 |

Sample 6 | 0.20166 | 0.01705 | 0.24128 | 0.67868 | 0.15298 | 0.28411 |

Sample 7 | 0.20188 | 0.01491 | 0.16281 | 0.67360 | 0.15510 | 0.27966 |

Sample 8 | 0.20029 | 0.01518 | 0.06761 | 0.09526 | 0.13861 | 0.26583 |

Sample 9 | 0.20143 | 0.01667 | −0.09939 | −0.02418 | 0.15240 | 0.2647 |

Sample 10 | 0.20109 | 0.01560 | −0.08412 | −0.13405 | 0.13951 | 0.24841 |

Statistics of 10 samples parameters of low tower in the system element 3986# axial force time history under wind field 2 (N).

Axial force | Mean | SD | Skewness | Kurtosis | Minimum | maximum |
---|---|---|---|---|---|---|

Sample 1 | −2003970 | 81613 | −0.25174 | −0.12699 | −2393590 | −1773560 |

Sample 2 | −1970750 | 70321 | 0.21342 | 0.26403 | −2201780 | −1631920 |

Sample 3 | −1970020 | 67161 | −0.72856 | 2.71910 | −2458630 | −1755420 |

Sample 4 | −1968010 | 76148 | −0.3753 | 1.31182 | −2458310 | −1690060 |

Sample 5 | −1989460 | 75171 | −0.19104 | 0.12478 | −2306520 | −1698410 |

Sample 6 | −1991600 | 78931 | −0.28651 | 1.12501 | −2457960 | −1753990 |

Sample 7 | −1990110 | 69899 | −0.30044 | 1.64027 | −2464100 | −1734350 |

Sample 8 | −1983460 | 70168 | −0.06092 | 0.77690 | −2364690 | −1583980 |

Sample 9 | −1989120 | 78892 | 0.02409 | 0.28657 | −2373170 | −1728910 |

Sample 10 | −1985380 | 74337 | 0.10206 | 0.12704 | −2228420 | −1593240 |

In Tables

We used three statistical methods in this paper [

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

Comparison of statistical results on single high tower node 35# Ux displacement and element 1972# axial force.

Number | Statistical methods | Single high tower | |||
---|---|---|---|---|---|

Wind field 1 | Wind field 2 | ||||

Displacement (m) | Axial force (N) | Displacement (m) | Axial force (N) | ||

1 | Method 1 | 0.60214 | −3783277 | 0.47170 | −3375352 |

2 | Method 2 | 0.55944 | −3622471 | 0.45836 | −3336793 |

3 | Method 3 | 0.36013 | −2947645 | 0.35139 | −2928202 |

4 | The average of mean | 0.35353 | −2938821 | 0.34950 | −2924127 |

5 | Item 1/Item 4 | 1.703 | 1.287 | 1.350 | 1.154 |

6 | Item 2/Item 4 | 1.582 | 1.233 | 1.311 | 1.141 |

7 | Item 3/Item 4 | 1.019 | 1.003 | 1.005 | 1.001 |

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

Number | Statistical methods | Single low tower | |||
---|---|---|---|---|---|

Wind field 1 | Wind field 2 | ||||

Displacement (m) | Axial force (N) | Displacement (m) | Axial force (N) | ||

1 | Method 1 | 0.38756 | −2683970 | 0.31304 | −2461538 |

2 | Method 2 | 0.35570 | −2575358 | 0.30012 | −2409227 |

3 | Method 3 | 0.22547 | −2082638 | 0.22321 | −2082823 |

4 | The average of mean | 0.22095 | −2075966 | 0.22167 | −2079914 |

5 | Item 1/Item 4 | 1.754 | 1.293 | 1.412 | 1.183 |

6 | Item 2/Item 4 | 1.610 | 1.241 | 1.354 | 1.158 |

7 | Item 3/Item 4 | 1.020 | 1.003 | 1.007 | 1.001 |

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

Number | Statistical methods | High tower in transmission tower-line system | |||
---|---|---|---|---|---|

Wind field 1 | Wind field 2 | ||||

Displacement (m) | Axial force (N) | Displacement (m) | Axial force (N) | ||

1 | Method 1 | 0.43429 | −3171906 | 0.40952 | −3109766 |

2 | Method 2 | 0.41255 | −3117028 | 0.39069 | −3048073 |

3 | Method 3 | 0.32309 | −2764258 | 0.32169 | −2758864 |

4 | The average of mean | 0.32167 | −2761720 | 0.32084 | −2754673 |

5 | Item 1/Item 4 | 1.35 | 1.15 | 1.28 | 1.13 |

6 | Item 2/Item 4 | 1.28 | 1.13 | 1.22 | 1.11 |

7 | Item 3/Item 4 | 1.00 | 1.001 | 1.00 | 1.00 |

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

Number | Statistical methods | Low tower in transmission tower-line system | |||
---|---|---|---|---|---|

Wind field 1 | Wind field 2 | ||||

Displacement (m) | Axial force (N) | Displacement (m) | Axial force (N) | ||

1 | Method 1 | 0.28247 | −2392195 | 0.26844 | −2370717 |

2 | Method 2 | 0.26171 | −2256128 | 0.24781 | −2206980 |

3 | Method 3 | 0.20194 | −1988296 | 0.20110 | −1985582 |

4 | The average of mean | 0.20092 | −1986258 | 0.20048 | −1984188 |

5 | Item 1/Item 4 | 1.40 | 1.20 | 1.34 | 1.20 |

6 | Item 2/Item 4 | 1.30 | 1.14 | 1.24 | 1.11 |

7 | Item 3/Item 4 | 1.00 | 1.00 | 1.00 | 1.001 |

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:

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors gratefully acknowledge the financial support from Northeast Electric Power University (BSJXM-201521), Jilin City Science and Technology Bureau (20166012), and Natural Science Foundation of China (51378095).