2 APPLICATION OF A CATENARY MODEL .1 Introduction
2.4 Sag Equation and its Use
The conductor sag is the distance measured vertically from the conductor to the straight line (chord) joining two support points of a span. Actually the sag varies on the interval of the span, i.e. increases from zero to maximum, then decreases to zero, going from the left–hand side support to the right–hand side one. It can be appropriately described by the equation for the sag, D(x), as the function of x, where x varies from zero to the span length, x[0,S]. The curve of D(x) is called here a sag curve and is shown in Fig. 2.4. The sag value at some point within the span is a vertical distance from the x–axis to the sag curve.
Distance
Sag
D(x)
xC S
0
(xc;Dmax ) Dmax
Fig. 2.4: Sag curve
Differently to the conductor curve, which has the low point, the sag curve has the maximum point. The coordinates of the latter are xc and Dmax.
Some literature identifies the sag curve with the conductor curve even though they have own separate equations. It is mathematically incorrect according to the explanation given above.
However, the vertical distance measured from the straight line to the conductor curve at some point of the span is equal with the vertical distance measured from the x–axis to the sag curve at the same point of the span. The actual use of the sag equation is a calculation of the sag at an arbitrary point of the span. It is necessary for example, to obtain the clearance over the conductors at some points within the span.
2.4.1 Derivation of the Sag Equation
The equation for the sag curve (shortly called as sag equation), D(x), can be derived by the use of the equation for the conductor curve. For this purpose, firstly the equation for the straight line, yline(x), passing through the support points A and B has to be defined on the interval [0,S], then subtract (2.34) according to (2.36). The result provided is the sag equation, which is usable for the sag calculation at any point of the span.
S maximum sag in a span, and also for defining special formulas for the following characteristic sags of the catenary conductor curve: Maximum sag Dmax
Mid–span sag D(S/2)
Low point sag D(xMIN), where 0 ≤ xMIN ≤ S.
2.4.2 Location of the Maximum Sag in a Span
Finding the first derivative of (2.37) and considering 2sinhx·coshx=sinh2x [52], then solving equation (2.39), the location of the maximum sag in a span is obtained and given by (2.42).
From (2.42) it is obvious that the maximum sag of the catenary in an inclined span is not located at a mid–span, but it is moved toward one of the two support points. Now there is a question whether it is moved toward the higher or the lower one. The answer to this question is given below without the use of numerical examples, but strictly analytically.
Denoting the second summand in (2.42) with q yields expression (2.43):
S q xC
2 (2.43)
Now let us assume that the maximum sag is moved from the mid–span toward the higher support point and that the right–hand side support point is higher than the left–hand side one, i.e. assume that relation (2.44) is valid and then check it mathematically step by step.
If h1h2 q0 (2.44) The initial conditions are given: S>0, c>0, h1>0, h2>0. The main steps for checking the validity of the assumption given by (2.44) are shown in the following lines:
/2
0sinh arcsinh 2
arcsinh 2 1 2 1
c S c
h h S
h
c h (2.45)
S c
c
h h S
h h
2 / sinh arcsinh 2
arcsinh 2 1 2 1
(2.46)
The inverse hyperbolic sine (see Fig. 2.5) is a monotonic, strictly increasing function [57], so if x2x1 arcsinh(x2)arcsinh(x1) (2.47)
x arcsinh(x)
0
Fig. 2.5: Curve of arcsinh(x)
Applying (2.47) in (2.46) gives (2.48), which can deduce (2.50)
S c
c
h h S
h h
2 / sinh 2
1 2 1
2
(2.48)
S c
c
S 2 sinh /2 1 1
(2.49)
S c
c
S/2 sinh /2 (2.50)
Taking into consideration relations (2.51) and (2.52), it can be stated that the previous one is valid. This way the validity of the assumption (2.44) is also proved. The same process applied for cases h1>h2 and h1=h2 gives further two relations:
if h1h2 q0 (2.53) if h1 h2 q0 (2.54) Thus, the above question of the movement of Dmax has been satisfactory answered.
Relation (2.54) refers to a level span when there is no movement of Dmax. Summarizing (2.44), (2.53) and (2.54) the final conclusion in connection with the location of Dmax related to the mid–span, proved analytically here, is the following:
The maximum sag of the catenary conductor curve in a level span is located at a mid–span, but in an inclined span it is moved from a mid–span toward a higher suspension point.
This is an essential difference in comparison to the parabola, since the maximum sag of the parabolic conductor curve is always located at a mid–span, in level and inclined spans as well.
This feature effectively simplifies the parabola based algorithms for overhead line design.
2.4.3 Characteristic Sags
Since xC is obtained, it can be used to determine the maximum sag (2.57). The main steps of the deduction are the following:
) characteristic sags can be defined as follows.
Mid–span sag:
Low point sag (sag at the lowest point of the conductor):
S vertex point have then the same location. In these cases the low point sag is in fact identical with the vertex point sag. The term low point sag is frequent in literature and that is why it is also used here, instead of the term vertex point sag. The adequate clarification can be done with the following five relations taking into consideration that in some cases xMIN as x–
coordinate of the vertex point may be outside the span.
If xMIN 0 D