In this section, the analysis is done for the conductor static profile in the form of the deformed shape of a completely flexible conductor structure suspended between two towers and defined as a catenary. This catenary configuration of power line conductors when strung on towers can be treated as the deformation of slender structures. This analysis is imperative because the Sag-tension calculations predict the behaviour of conductors based on recommended tension limits under varying loading conditions. These tension limits specify at a certain percentages of the conductor’s rated breaking strength that are not to be exceeded during installation in order to guarantee long life for the line and also conform to regulations. To accurately determine the sag limits for stringing the power line is very essential during the line design process. The sag of conductors is used to select support point heights and span lengths so that the minimum clearances will be maintained over the life of the line.
As explained in [14], because the conductor is the most expensive component of any power line, from an economic perspective, it is disadvantageous to employ low conductor tensions. Based on specification, a minimum clearance is required between the ground and the lowest point of an overhead conductor. Therefore, if the span length is fixed, the height required for a transmission- line tower increases as the conductor tension decreases. Alternatively, if the amount of conductor sag is fixed, the span length required decreases as the conductor tension decreases; in this case, the more number of towers required, not the tower height, must increase to transmit electricity over a specified distance. Both scenarios result in greater transmission-line costs. It has been shown that significant cost reduction is possible if higher conductor tensions could be used safely.
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The conductor’s size, tension, and span length, together, primarily affect a line’s susceptibility to Aeolian vibration [2]. The amount of energy imparted on a conductor varies directly with the span length. The longer the span, the more wind-induced energy is absorbed. Conductors tend to vibrate more readily at higher tensions because of their low self-damping capability (the frictional interaction between strands) to reduce the imparted energy. In order to identify the axial tension limit to employ, the condition at which the conductor damping capability balances the energy imparted on it by wind loading is of importance in this study. The analysis of various sags and the conductor slack is used as the bases of ascertaining the cost benefit of stringing a line using the span length as the reference with regards to percentage of its ultimate braking point. This can be used to determine the cost saved by power utilities in terms of cost contributed by the conductor in constructing the power line.
The single span of a transmission line conductor static profile as shown in figure (3.1), is considered as a continuous structure and can be described by a set of parabolic or hyperbolic functions i.e. parabola or catenary curve. This deformed shape curves under the static condition, the conductor deforms due to gravity. It assumes a catenary profile by sagging along the span.
Figure 3.1: The conductor static profile
In the analysis, a parabolic curve is the shape that is formed by a conductor supporting an evenly distributed horizontal weight, whereas a catenary is the shape that is formed by hanging the conductor whose weight is constant per unit of the arc length.
Though the hyperbolic or the catenary curve equations are more accurate, the mathematical formulae which are used for the derivations of conductor as a parabola are much simpler with very good results in comparison. Detail derivation of equations for both hyperbolic and parabolic
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equation can be found in [74]. The equations for the power line curves for cases were the power line passes non-level and mountainous terrain can be found in [75].
Consider a conductor attached to two fixed points A and B, supporting it weight as shown in figure (3.2). Assume the conductor between AB carries its weight that is uniformly distributed along the horizontal. If wL denote the weight per unit length.
Figure 3.2: The Conductor parabola or catenary curve.
For the conductor with a span-length LC, weight 𝑤L, and horizontal tension 𝐻, the maximum sag distance LD (the vertical distance between the point of attachment and the cable, at the lowest point in the span) is described by the parabolic function. Apply the equation of a parabola to a span of the same elevation, with a vertical axis and its vertex are at the origin of coordinates located at point A.
H x wL
tan …………..….……. (3.1)
The equations of the parabola for this curve are given as:
H x x w
y L
) 2 (
2
…………..….……. (3.2a)
H L LD wL S
8
2
…………..….……. (3.2b)
2 22
1 24 H w L L
LC S S L …………..….……. (3.2c)
Where LD = mid-span sag (m), wL = conductor weight (N/m), LC = horizontal span length (AB), (m), H = conductor tension (N).
For the case in which the single span of the transmission line is represented by a set of hyperbolic functions which is similar to the description of catenary curve. The equations describing the conductor as a hyperbolic function curve are given as:
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cosh 1
)
( H
x w w
x H
y L …………..….……. (3.3a)
1
cosh 2
2
H L w w
LD H L C …………..….……. (3.3b)
H wL w
LC H S
sinh 2
2 …………..….……. (3.3c)
The slack which is the difference between the conductor length and the span length can be evaluated by
2 3 2
24H L L w
L
L C S L S
…………..….……. (3.4)
It is pertinent to note, the main motivation for this study has to do with determining conductor damping, but the overall aim was to determine at least the sag that can effectively damped out the imposed energy on the conductor. This directly determines minimum conductor chord length required for a particular span length. If this length is adequately determined and effectively implemented, on a long distance stretch of a power line, it saves the power utilities huge savings as the cost of the conductor is the most expensive component of any power line. Analysis of the sag as a function of different stringing tensions for single span is done in appendix B