Engineering the Curve: Understanding Conductor Sag in Overhead Lines

INTRODUCTION

Sagging of cables is a crucial aspect of overhead lines, a common sight in our streets, around substations, and even within our homes. The phenomenon is not just a visual occurrence; it’s a complex interplay of engineering and mathematics. Take, for example, the simple act of hanging clothes on a rope. The curvature that forms near the middle is not random—it’s the result of precise scientific principles.

In this article, I will dissect the ‘Hanging Conductor’ from a mathematical and physical engineering standpoint. We’ll explore how sag charts are created and examine some actual printed tables from PLS-CADD exports. The journey from classical hand calculations to modern software-assisted design reflects the evolution of engineering practices. These tools are now indispensable for designers in ensuring proper clearance and safety.

THE CONDUCTOR’S SAG DETERMINATION IS DONE BY:

Temperature, Unit Weight of a conductor, Deformation of Plastic (creep, strand settlement, etc.), Tension in the conductor Span Length, The elasticity of conductor materials, Weather Patterns (effect of wind, ice)

in this article we discuss just shape of the sag how to calculate in Mathematical and some physics way.

CABLE SAG: THE MATH BEHIND THE DROOP

When discussing the sag of cables, particularly in the context of overhead power lines, three mathematical functions are commonly referenced to describe the shape of the cable under the influence of gravity:

1-Parabola Function: This function is used when the sag is relatively small compared to the span of the cable. The equation for a parabolic curve is given by: y=ax^2+bx+c

2-Hyperbolic Function: This function can be used for longer spans where the sag is more pronounced, and the cable behaves more rigidly. The hyperbolic function is represented as: (y^2/b^2)-(x^2/a^2)=1

Catenary Function: For very long spans and heavy cables, the catenary function provides the most accurate representation of the cable’s curve. The catenary curve is described by the equation: y=a*cosh(x/a)

Each of these functions serves to model the cable’s shape under different conditions and is crucial for ensuring the safety and reliability of overhead line installations. the most used or the most suitable one is Catenary function. we will break down the catenary function and it will be used in overhead line to determine sag of the cable.

CATENARY FUNCTION "TRANSMISSION LINE"

The shape of a catenary is a function of the conductor weight per unit length, w, and the horizontal component of tension, H. The sag of the conductor, D, is a function of these parameters, the span length, L, and the difference in elevation between the span support points.

1-Catenary Curve level spans

S= horizontal Span length

H= Horizontal tension force

A= Support A

B= Support B

D= Mid span Sag

TR= Tension Force Right side

TL= Tension Force Left side

W= weight of Cable (Conductor)

If the lowest point of the curve above is the origin, and the curve must be placed in the middle of "y" vertex. The following equation can also be in use to model it:

where:

x = horizontal distance from span’s lowest point (m)

y = vertical distance from the span’s lowest point

a = Catenary constant

The catenary constant is effectively the ratio of the horizontal tension in the conductor to the unit weight in the absence of wind:

where:

T = Horizontal component of tension (N)

W = distributed load on conductor (N/m)

M = unit mass of conductor (kg/m)

g = gravitational constant of 9.81 m/s2

For a level span, the low point is in the center, and the sag, D, is found by substituting x = S/2 in the preceding equations. The exact catenary and approximate parabolic equations for sag become the following:

H = Horizontal Tension

*Important note

The parabolic tension function can be used for sag calculations:

• When the sag-to-span ratio is less than approximately 1/8.
• For shorter spans, typically found in distribution lines, where the sag is not as pronounced.

You can check the formula derivatization here.

2-"Catenary" and "Parabola" curve inclined spans

Inclined spans, where the support points are at different elevations, can be analyzed using the same equations as level spans. The catenary equation remains applicable for calculating the conductor height above the low point in the span.

However, for inclined spans, the total span is conceptually divided into two separate sections: one to the right and one to the left of the lowest point (as illustrated in Figure below). It's important to note that the catenary shape itself, relative to the low point, remains unaffected by the difference in elevation between the support points (i.e., the inclination of the span).

The equation for the conductor elevation, y(x), relative to the low point, is given for each section extending outwards from the low point. (The equation would follow here).

Note that x is considered positive in either direction from the low point. The horizontal distance, X L or X R, from the left support point to the low point or visa versa the derivative of finding sag with different height support with parabola equation and catenary Equation.

• Finding Left and right Distance with Parabola equation: -

catenary constant: -

here: -??

So: -

There for: -

So: -

Finding Left and right Distance with Catenary Equation: -

We have: -

As we know the catenary equation will be shown as

The formula's second term becomes zero because the cable sag forms a catenary curve, attached to the x-axis in the certain point. Therefore, the difference (a) in the formula equals zero.

catenary constant: -

?The catenary cable will be divided in to two part left side and right side. The left part formula will be shown the height of the support to lowest part of the cable.

Known: -

We will rewrite the equation (D)?to equation (E).

There for: -

So, you can use the (F)?formula to find (X left)

SAG VS STRESS

Sag (conductor dip) and tension are inversely proportional in overhead lines. Higher tension (due to conductor weight) reduces sag, but excessively high tension can stress conductors and supports. Design considers this balance alongside temperature and weather impacts. Accurate calculations use the catenary equation, while the parabolic equation offers a simpler approximation for smaller sags.

PLS-CADD SAG CHART WITH PARAMETERS

The table in the report shows the sag and tension of the conductor for various spans and temperatures. The leftmost column shows the span length in meters. The next five columns show the sag of the conductor at different temperatures (30°C, 40°C, 50°C, 60°C, and 70°C). The rightmost column shows the vertical projection of the span.

The table below the main table shows the wave time of the conductor for various spans and temperatures. Wave time is the time it takes for a wave to travel the length of the conductor.

The bottom row of the report shows the horizontal tension of the conductor under various conditions.

Overall, this stringing chart report provides detailed information about the sag and tension of a power line under various conditions. This information is important for ensuring the safe and reliable operation of the power line.

CONCLUSION

The article discusses the importance of accurately modeling cable sag in overhead lines. It highlights the use of mathematical functions—parabolic, hyperbolic, and catenary—to represent the cable’s shape under different conditions. The catenary function is emphasized as the most effective for long spans and heavy cables, ensuring safety and reliability in line installations. This underscores the critical role of precise engineering calculations and modern software in designing overhead lines.

Author:

Seevan Abdulmajid Ahmed

(Seevan Harshami)

Resorces: -

Saric, S. (2016) Sag and tension of conductor, Academia.edu. Available at: https://www.academia.edu/25942985/Sag_and_Tension_of_Conductor (Accessed: 11 April 2024).

Argasinska, H., and Franc Jakl. SAG-Tension Calculation Methods for Overhead Lines: Task Force B2.12.3. Cigre, 2016.

Alen Hatibovic: SAG-TENSION CALCULATION METHODS FOR OVERHEAD LINES: received 21 February 2014; accepted after revision 4 April 2014