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Transmission tower-line systems are designed using static loads specified in various codes. This paper compares the dynamic response of a test transmission line with the response due to static loads given by Eurocode. Finite element design software SAP2000 was used to model the towers and lines. Non-linear dynamic analysis including the large displacement effects was carried out. Macroscopic aspects of wind coherence along element length and integration time step were investigated. An approach is presented to compare the probabilistic dynamic response due to 7 different stochastically simulated wind fields with the response according to EN-50341. The developed model will be used to study the response recorded on a test line due to the actual wind speed time history recorded. It was found that static load from EN overestimated the strength of conductor cables. The response of coupled system considering towers and cables was found to be different from response of only cables with fixed supports.

Collapse of transmission tower-line systems is not a well understood phenomenon. These systems are subjected to various loads like wind, snow, icing and earthquake. Comparatively, wind loads are more complex for these structures due to high geometric non-linearity of cables and randomness of wind turbulence. This thesis is aimed at understanding the dynamic behavior of transmission tower-line systems under fluctuating wind loads. Present code recommendations are based on static loading. In this paper, [

Coupled transmission tower-line systems are highly complex in their behavior due to the interaction between non-linear conductors and stiff towers which results in closely spaced frequencies. [

A 3D non-linear analysis including the large displacements was carried out to study the dynamic response of conductors. Effects of parameters like coherence along element length and integration time step were considered. The response of cables and insulators was found to be non-Gaussian. Methods lately published for calculating extreme values of a non-Gaussian process were used. These extreme values were then compared to the response due to wind pressure recommended in EN-50341. The chosen transmission line is in Rostock, Germany. It has 2 end towers and 2 suspension towers. Each of the 3 spans is about 400 m. A 3D finite element model of a real transmission line was modeled in finite element software SAP2000. Two models were studied: only conductors and conductors coupled with towers. The results were compared to show the importance of conductor-tower interaction.

The test line is a 380 kV line with 2 circuits and 3 phases. The conductors are 2 × 3 quad bundle made of aluminum conductor with steel core (ACSR). There are 2 end towers referred as WA15 and WA18 and 2 supporting towers referred as T17 and T16 (

SAP2000v15 OAPI (Open Application Programming Interface) was used to create the geometry of towers. A typical transmission tower can have close to 1500 members and 500 nodes. To recreate the towers with varied slenderness ratios can be time consuming. The developed VB code solved this problem. Parameters like slope of main columns, base and top width, height of the broader part of tower and total height were taken as input.

In past, [

The created tower geometry was checked for disjointed nodes. The towers have been created in SAP2000 using open application programming interface (OAPI) by taking required height as input. Each member length is numerically calculated and geometry is created in SAP2000 v15. This sometimes results in common nodes falling out of a member while it was created with other members. These occurrences are very less and can be pin pointed by dead load analysis. After rectifying the disjointed nodes the sectional and material properties were defined for each member. Each member was discretized into 3 parts to ensure adequate accuracy. A dead load analysis was performed on these tower models. Characteristics of the 3 tower types are given in

Each tower is at a different ground level and this has been accounted for. Distances and sag in each span is given in

Tower | Nodes | Members | Height (m) | Dead load (kN) |
---|---|---|---|---|

WA18/WA15 | 895 | 2125 | 51.4 | 372.4 |

T17 | 724 | 1393 | 64.7 | 274.3 |

T16 | 735 | 1313 | 57.2 | 221.9 |

Span | Distance [m] | Vertical sag at middle (m) | Relative height: left to right (m) |
---|---|---|---|

WA15-T16 | 393.5 | 17 | 0 - 0 |

T16-T17 | 406.5 | 17 | 0 - 10.27 |

T17-WA18 | 439 | 17 | 10.27 - 15.67 |

The conductors were modeled as tension-only linear elastic material. The non-linearity of these flexible cables was taken into account for dynamic analysis. Quad bundle of conductors was assumed as one conductor with 4 times cross sectional area. In this study, each conductor was divided into 20 parts for application of wind load time histories. The sectional and material properties for conductors are given in

The sectional and material properties for towers, insulator strings and conductors are as per the technical drawings. Realistic modeling of aerodynamic damping is very complex for such systems. Aerodynamic damping affects the conductors in a varied way. Aerodynamic damping is an aeroelastic phenomenon and opposes the action of cables depending on the direction of motion. This cannot be modeled in SAP2000v15 however a satisfactory model has been recently presented by [

In most of the numerical research works, either the insulator strings have not been modeled or have been assumed to be a beam element while towers have been neglected. [

The processing time depends on the computing platform however with best commercially available platform also the software and degrees of freedom can be a restriction. The analysis time for complete SAP2000 model was about 60 - 75 hours due to large number of degrees of freedom. To reduce the analysis time, the towers were reduced to equivalent beams. As presented by [

Two models were made to compare the effect of interaction between towers and conductors. The coupled model with reduced tower is shown in

Property | Value |
---|---|

Cross sectional area | 297.8 mm^{2} × 4 = 1191.2 mm^{2 } |

Overall diameter | 22.4 mm (of single conductor) |

Weight | 998 Kg/km × 4 = 3992 Kg/km |

Modulus of elasticity | 74,000 N/mm^{2} |

Length of insulators | 5.3 m |

Tower | Modal stiffness (kN/m) | ||
---|---|---|---|

Mode I | Mode II | Mode III | |

T16 | 0.051 | 0.054 | 0.111 |

Reduced | 0.050 | 0.052 | 0.117 |

T17 | 0.043 | 0.045 | 0.102 |

Reduced | 0.043 | 0.046 | 0.068 |

WA | 0.092 | 0.100 | 0.125 |

Reduced | 0.091 | 0.102 | 0.152 |

Separate time histories of 300 seconds were generated for 19 points on each conductor. Distance between each point on the conductor is about 20 m. The mean wind speed at respective heights for generating the time history is from DIN-1055-4:2005-03 [

where C is the decay factor taken as 11 (

Mean wind speed was then added to the turbulence generated for each point. The drag force coefficient (C_{d}) for conductors was taken to be 1 (EN-50341). A force time history was generated for each point using

There are 16 conductors in two spans and each divided into 20 parts, hence 304 time histories were simulated in one wind field. Seven such fields were simulated for probabilistic analysis. Each time history was applied to the model which was a time consuming task. To reduce the efforts of generating the loads and applying the time history, OAPI was used. Hilber-Hughes-Taylor time integration method (1977) was used for direct integration initially with α = −0.33. Integration time step was taken as 0.1 s at first to get results. To study the effect of integration time step a comparative study was done.

though 0.01 was the right time step, due to constraints of SAP2000’s memory usage, a time step of 0.02 s was used. With an integration time step of 0.01 s the whole time history could not be solved due to inadequate system memory. The system being used is a dual core processor with 64 bit OS and 24 GB RAM. We are of the opinion that the multi-threaded solver in SAP2000 could not recognize the 64 bit system so as to enable it to use the whole system memory (more than 4 GB). This error may be attributed to the .NET communication between SAP2000 and the 64 bit OS. However, a detailed investigation of this issue is still under process.

A separate static model was created on which static loads were applied. The design loads for overhead electrical lines exceeding 45 kV are taken from EN 50341. The variation of wind pressure along height is given as per Equations (2) and (3).

where h is the height above ground level in meters and ^{2}. The wind loads on conductors, insulators and towers were calculated as per Equations (4), (5) and (6) respectively.

where q is the wind pressure as per Equations (2) and (3), G is the dynamic response factor given by 0.45 + 60/L for spans greater than 200 m and as 0.75 for spans lesser than 200 m, d is the diameter of the conductor, L is the length of conductor exposed,

In time history analysis due to wind loads, extreme value of the response is generally found out. This is due to the fact that the applied wind loads are generated from a random process and the extreme value of the response will vary with each time history. In probabilistic analysis this is called as mean extreme value

where ν is the mean frequency of occurrence of zero crossings with positive slopes only and is given by Equation (8).

and

here

It has been shown by [

here

Equation (9) is valid only for a Gaussian process. However, it was observed that the response of the structure is Gaussian (

The skewness and kurtosis for the 4 response parameters for the 7 wind fields can be seen in

Kareem and Kwon [

where

Parameter | Moments | Peak Factors | |||
---|---|---|---|---|---|

Skewness | Kurtosis | Davenport | Kwon & Kareem | Huang et al. | |

Cable Disp. (m) | |||||

TH1 | −0.30 | 3.13 | 3.06 | 2.75 | 2.67 |

TH2 | 0.07 | 2.57 | 2.89 | 2.59 | 2.98 |

TH3 | −0.49 | 3.26 | 2.99 | 2.55 | 2.37 |

TH4 | −0.30 | 2.65 | 3.12 | 2.21 | 2.73 |

TH5 | −0.30 | 3.13 | 3.06 | 2.75 | 2.67 |

TH6 | 0.07 | 2.57 | 2.89 | 2.59 | 2.98 |

TH7 | −0.001 | 2.59 | 2.91 | 2.52 | 2.92 |

Cable Tension (kN) | |||||

TH1 | 0.27 | 3.09 | 3.23 | 3.75 | 3.71 |

TH2 | −0.01 | 2.62 | 3.24 | 2.71 | 3.27 |

TH3 | 0.38 | 2.64 | 3.19 | 3.39 | 3.82 |

TH4 | 0.70 | 3.24 | 3.31 | 4.63 | 4.53 |

TH5 | 0.27 | 3.10 | 3.23 | 3.75 | 3.71 |

TH6 | −0.01 | 2.62 | 3.24 | 2.71 | 3.27 |

TH7 | 0.21 | 2.59 | 3.24 | 3.06 | 3.62 |

Insulator Disp. (m) | |||||

TH1 | −0.48 | 3.31 | 3.18 | 2.75 | 2.50 |

TH2 | 0.112 | 2.35 | 3.19 | 2.28 | 3.40 |

TH3 | −0.41 | 2.93 | 2.98 | 2.35 | 2.47 |

TH4 | −0.05 | 2.55 | 3.11 | 2.48 | 3.08 |

TH5 | −0.48 | 3.31 | 3.18 | 2.75 | 2.50 |

TH6 | 0.112 | 2.35 | 3.19 | 2.28 | 3.40 |

TH7 | −0.67 | 3.17 | 3.05 | 2.26 | 2.16 |

Insulator Tension (kN) | |||||

TH1 | 0.35 | 2.98 | 3.24 | 3.78 | 3.85 |

TH2 | 0.16 | 2.63 | 3.24 | 3.04 | 3.55 |

TH3 | 0.48 | 3.04 | 3.20 | 3.96 | 3.98 |

TH4 | 0.72 | 3.41 | 3.37 | 4.91 | 4.67 |

TH5 | 0.35 | 2.98 | 3.24 | 3.78 | 3.85 |

TH6 | 0.16 | 2.63 | 3.24 | 3.04 | 3.55 |

TH7 | 0.31 | 2.80 | 3.31 | 3.60 | 3.89 |

[

where the variables have the same meaning as in Equation (10).

Peak factors for these parameters using the above methods are shown in

Extreme values based on the peak factors from [

value from the set of extreme values obtained. The last column of the table shows these values that have been considered for comparison with the response due to static wind loads given by EN-50341.

In

There is a difference of 12-30% between the two responses for various parameters. To investigate the effect of coupling between the towers and the cables, the responses of the two models were compared. The results are shown in

There is a difference in the results from two models. Hence, it is more accurate to consider the coupled model for analysis of such structures. At the same time the coupled model increases the analysis time considerably even with the reduced towers. In addition the accuracy of peak factors is less as the time step used was 0.02 s.

In the interest of dynamic response of power transmission lines, the following main conclusions were made from this study:

Aerodynamic damping is an influential parameter in the response of transmission lines. For this study an equivalent viscous damping suggested on the basis of experimental results gave satisfactory results. However recently [

The effect of wind coherence along the element length in the simulated stochastic wind field is of great importance. It was found that if the element length is larger than the eddy size, the effective wind load on that element can be up to 45% larger. Two options are suggested based on this study: firstly, considering an element length which is smaller than the eddies with higher frequencies; or secondly, reducing the force on an element based on the joint acceptance function.

Parameter | Extreme value | Considered value | ||||||
---|---|---|---|---|---|---|---|---|

TH1 | TH2 | TH3 | TH4 | TH5 | TH6 | TH7 | ||

Cable disp. (m) | 23.1 | 25.4 | 23.7 | 22.6 | 23.1 | 25.4 | 24.3 | 25.4 |

Cable tension (kN) | 97.2 | 93.6 | 103.9 | 105.2 | 97.2 | 93.6 | 97.9 | 105.2 |

Insulator disp. (m) | 4.6 | 5.0 | 4.75 | 4.9 | 4.6 | 5.0 | 4.4 | 5.0 |

Insulator tension (kN) | 40.0 | 38.8 | 42.1 | 42.9 | 40.1 | 38.8 | 40.5 | 42.9 |

Parameter | Design load | Dynamic analysis | Difference, % |
---|---|---|---|

Cable displacement [m] | 19.4 | 25.4 | 30.9 |

Cable tension [kN] | 93.9 | 105.2 | 12.0 |

Insulator displacement [m] | 4.2 | 5.0 | 19 |

Insulator tension [kN] | 35.8 | 42.9 | 19.8 |

Parameter | Only cables | Coupled system | Difference % |
---|---|---|---|

Cable displacement [m] | 22.1 | 25.4 | 15.0 |

Cable tension [kN] | 100.6 | 105.2 | 4.6 |

Insulator displacement [m] | 4.5 | 5.0 | 12.4 |

Insulator tension [kN] | 36.8 | 42.9 | 16.7 |

Response of a transmission line to gust wind is non-Gaussian in nature. An appropriate method to calculate the peak factor for a non-Gaussian random process gave results that were higher than the davenport’s peak factors. A numerical verification of these peak factors can be envisaged as a future research task.

Static wind loads specified in EN-50341 overestimate the cable strength. Extreme values of 4 response parameters were found to be greater than the static design response.

The response for a coupled model is different from the response when only the cables are considered. The results from the model with only the cables gave wrong estimations for the displacements of lines and insulators. The swing angle of insulators can be larger in coupled model thereby resulting in flashovers. To account for the difference in displacement values, it is recommended to use an equivalent stiffness at supports, instead of the towers, in the model with only cables. This can reduce the analysis time to 3 - 4 hours and also satisfactorily account for the difference in the response as compared to the coupled model.

As a future scope for the project, it would be interesting to input the wind speed records from the test line and compare the numerical response with the recoded response. The insulator swing angle can be conveniently calculated from the numerical model and the same parameter is being recorded at the test line.

Alok Dua,Mathias Clobes,Thomas Höbbel,Vasant Matsagar, (2015) Dynamic Analysis of Overhead Transmission Lines under Turbulent Wind Loading. Open Journal of Civil Engineering,05,359-371. doi: 10.4236/ojce.2015.54036