ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 60 (2012), nr. 3
22
Parabola and Catenary Equations for Conductor Height
Calculation
Alen HATIBOVIC ∗
Abstract
This paper presents new equations for conductor height calculation based on the known maximal sag. The
equations can be directly applied for plotting the conductor curve after a completed sag-tension calculation for
planning and designing overhead lines. The basic differences between the parabola and the catenary curves
are also discussed. The validity of the shown formulas has also been proved by some numerical examples.
Keywords: transcendental functions, algebraic functions, overhead lines, leveled span, inclined span, sag
1. Introduction
The origin of the x-y coordinate system for
sag-tension calculation is generally put at the
top of the conductor curve [1]. However, it is
more advantageous to set the origin to the
bottom of the left-hand side support of the
span for defining the equation for the
conductor height. By this way the ycoordinate presents the height of the
conductor curve related to x-axis. The
distance of conductor’s arbitrary point from
the y-axis is then the distance from the lefthand side support. This paper shows both
the catenary and the parabola equations
under this condition.
The actual conductor curve can be
described by a catenary function, i.e. by the
hyperbolic cosine function, which belongs to
the group of the transcendental functions.
The parabola curve can be described by a
quadratic function, which belongs to the
group of the algebraic functions. The basic
difference between the algebraic and
transcendental functions is in their exponent.
While the exponent of the algebraic functions
is permanent, it is varying in the case of
transcendental functions. Despite the fact
that the parabola and the catenary functions
are mathematically quite different, their
curves can be very similar. Therefore, when
planning overhead electrical lines the
catenary is often approximated by the
parabola, since it results in a significant
simplification of the calculation. It is
acceptable, because in most of the cases the
difference between the catenary and the
parabola is negligible [2]. It is a generally
accepted fact in the literature that the
conductor curve can be approximated by a
parabola for spans up to about 400 metres.
For longer spans the exact catanary based
calculation shall be used, because the
difference between the catenary and the
parabola curves cannot be ignored.
2. Catenary equation
The top of the catenary curve is located at
the point (0,c) as it is shown in Figure 1, see
curve y1.
y
y1=c·cosh(x/c)
(0; c)
y2=c·cosh(x/c) - c
x
Figure 1. Graphs of catenary curves
Its basic equation is the following [3]:
∗
Alen HATIBOVIC, Senior engineer for electrical network
development, EDF DÉMÁSZ Hálózat, Szeged, Hungary,
[email protected]
y = c ⋅ ch(x / c )
(1)
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 60 (2012), nr. 3
For the determination of the catenary
equation it is necessary to know its
parameter c (c >0). It can be obtained by
using the sag-tension calculation for planning
overhead lines. When the origin of the
coordinate system is at the top of the
catenary curve, the equation becomes:
y = c ⋅ ch( x / c ) - c
(2)
After this step the origin shall be moved to
the bottom of the left-hand side support of
the span according to Figure 2.
y
MIN
y MIN
x MIN
x
Figure 2. Catenary curve in an inclined span
This figure shows an inclined span with a
vertex at point MIN.The equation for
conductor height will be defined by the
following data:
c – parameter of the catenary curve
xMIN – x coordinate of the vertex point
yMIN – y coordinate of the vertex point
The final catenary equation for conductor
height is (3). Its exponential form is given by
(4). The interval is always x ∈ [0,a], so it is not
necessary to be written each time.
y = c ⋅ ch
c
y =  e
2
x − x MIN
− c + y MIN
c
x − x MIN
c
+e
−
x − x MIN
c

 − c + y MIN


(3)
(4)
Both of the two equations are universal,
since these are valid in the case of any type
of inclined span (h1<h2 or h1>h2) and in the
case of leveled span (h1=h2=h), too. So by (3)
or (4) it is possible to calculate the conductor
height at any point of the span.
It can be seen in (4) that the x variable is
located at the exponent, which is an
important feature of transcendental functions.
In the case of leveled spans the equations
23
(3) and (4) get simpler forms given in (5) and
(6) on the basis of the known maximal sag
bmax. The height of the two supports is
denoted with h.
y = c ⋅ ch
x − a/2
− c + h − bmax
c
x −a / 2
−
c
y =  e c + e
2
x−a / 2
c

 − c + h − bmax


(5)
(6)
3. Parabola equation
There is a very important defference
between the parabola and the catenary
concerning the maximal sag of the
conductor.
Since the maximal sag of the parabola is
always located at a mid-span, both in the
case of leveled and inclined spans, the
maximal sag of the catenary in an inclined
span is slightly moved toward the higher
suspension point of the conductor. This is
one of the reasons of the simplicity of the
parabola based calculation in comparison to
the catenary based calculation. When the
maximal sag of the parabola is known it is
possible to obtain the parabola equation for
the conductor height, since the parabola is
defined by any three points of its curve.
In case of the catenary it is more difficult,
because it is necessary to know both the
parameter of the catenary c and the
coordinates of the vertex point according to
the base equation (3). So the knowledge of
the maximal sag is not enough for the
determination of the catenary.
The standard equation of the parabola
is (7),
y = Ax 2 + Bx + C
(7)
where A, B and C are the coefficients of the
parabola. The coefficient A defines the shape
of the parabola curve. If A>0, the curve has
the minimum [4], [5], and if A<0, the curve
has the maximum. It will be mathematically
proved later that in the case of the equation
for the conductor height the coefficient A is
positive.
3.1. Parabola equation by three points
Figure 3 demonstrates the applied method
for an inclined span with h1<h2. Points A and
B are the suspension points of the conductor,
while C is the conductor point at a mid-span.
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 60 (2012), nr. 3
24
The parameters shown in Figure 3 are the
following:
a – span length
h1 – height of the left-hand side suspension
point
h2 – height of the right-hand side susp. point
xMIN – x coordinate of the vertex point
yMIN – y coordinate of the vertex point
A=
h1
0
1
h2
h1 + h 2
− b max
2
a
a
2
1
0
a2
2
a
 
2
1
1
B(a ;h 2)
h2 y
h 1 A(0;h 1)
C
MIN
yc
y MIN
B =
0
a2
2
a
 
2
0
a
a
2
2
x MIN
0
a /2
a
Figure 3. Parabola curve in an inclined span
with h1<h2
Since the curve is a parabola, the
maximal sag bmax is located at x=a/2. The lefthand side and the right-hand side
suspension points A(0;h1) and B(a;h2) are
always known points. The third necessary
point C is obtained by the known maximal
sag (8). The value of the maximal sag can be
obtained from the sag-tension calculation.
a h +h

C  ; 1 2 − bmax 
2

2
(8)
Based on the three points of the parabola
the system of three algebraic equations (9)(11) can be written by utilizing of the
standard equation of the parabola (8):
C = h1
(9)
Aa 2 + Ba + C = h2
(10)
2
A(a / 2) + B (a / 2) +C = (h1 + h2 ) / 2 −bmax
(11)
Writing these equations in the matrix form
P·x=Q and using the Cramer’s rule [6], [7] to
find the solution, the unknown coefficients of
the parabola A and B can be obtined by:
x j = det(P j ) / det(P)
(j=1, 2, 3 )
(12)
=
a
2
a
 
2
h − h 1 − 4 b max
= 2
a
4 b max
a2
(13)
1
h1
h2
h1 + h 2
− b max
2
0
x
1
0
1
a
a
2
1
1
1
1
=
(14)
1
After the substitution of the coefficients A,
B, C into (7), the equation for the conductor
height gets its final form (15):
y=
4bmax 2 h2 − h1 − 4bmax
x +
x + h1
a2
a
(15)
This is a universal equation of the
parabola, since it is usable for both leveled
and inclined spans. In the case of the leveled
span the equation (15) changes into (16) and
the coefficients change into (17).
y=
A=
4bmax 2 4bmax
x −
x+h
a2
a
4bmax
a2
B=
− 4bmax
a
(16)
C=h
(17)
There is a very important consequence
from (15) and (16): beside the same span
length and maximal sag both in leveled and
inclined spans, the coefficient A of the
parabola does not change.
The validity and usability of (15) and (16)
will be proved in the following three
examples, one for a leveled and two for
inclined spans.
Example 1. Leveled span (h1=h2=h)
a=200m; h=18m; bmax=8m; y(x)=?
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 60 (2012), nr. 3
y [m ]
y = 8 ⋅ 10 −4 x 2 − 16 ⋅ 10 −2 x + 18
(18)
18
16
14
12
10
8
6
4
2
0
0
25
50
75
100
125
150
175
200
x [m]
Figure 4. Parabola curve from example 1. (leveled
span)
Example 2. Inclined span (h1<h2)
a=200m; h1=18m; h2=34m; bmax=8m; y(x)=?
y = 8 ⋅ 10 −4 x 2 − 8 ⋅ 10 −2 x + 18
(19)
32
28
y [m]
24
20
16
25
examples it has been proved that the
equation for the conductor height (15) is
correct and universal, since it can be used in
each type of the span. Therefore, we do not
pay attention to either h1<h2 or h1>h2,
because the input data is the same in each
case of the tasks. We have also seen that in
the case of the equation for the conductor
height, the coefficient A of the parabola is
always positive, since the span length and
the maximal sag of the parabola are also
positive. Therefore, the conductor curve has
the minimum. Let us mention that the vertex
of the parabola and the lowest point of the
conductor are generally the same point (point
MIN) and xMIN ∈ [0,a], like in each of the
previous figures. Figure 7. shows one rare
case of the inclined span when the vertex
point MIN is out of the span, i.e. xMIN ∉ [0,a].
In this case the lowest point of the conductor
(point M) is equal with the lower suspension
point of the span. For the appropriate
presentation of this rare case [8] the parabola
curve is shown in an interval x ∈ [0, 2xMIN ].
12
8
4
0
0
25
50
75
100
125
150
175
200
x [m]
Figure 5. Parabola curve from example 2. (inclined
span with h1<h2)
Example 3. Inclined span (h1>h2)
a=200m; h1=34m; h2=18m; bmax=8m; y(x)=?
y = 8 ⋅ 10 −4 x 2 − 24 ⋅ 10 −2 x + 34
(20)
Figure 7. Inclined span with M ≠ MIN
32
28
3.2. Parabola equation in the vertex form
y [m ]
24
Beside the standard equation of the
parabola (7) its vertex form is also often used
(21). Equation (7) can be transformed into
form (21) by using (22):
20
16
12
8
4
2
y = A(x − x MIN ) + y MIN
0
0
25
50
75
100
125
150
175
(21)
200
x [m]
Figure 6. Parabola curve from example 3. (inclined
span with h1>h2)
The vertex of the curve is shown in each
last three figures and they will be needed in
the next paragraph 3.2.
With the help of the previous three
2
4 AC − B 2
B 

y = A x +
 +
2A
4A

(22)
According to (22) the coordinates of the
vertex point can be defined as (23) and (24):
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 60 (2012), nr. 3
26
x MIN = −
B
a  h − h1 

= 1 − 2
2A 2 
4bmax 
(23)
2
y MIN
 h −h 
4 AC − B 2
=
= h1 − bmax 1 − 2 1  (24)
4A
4bmax 

Substituting (23) and (24) into (21), the
vertex form of the equation for the conductor
height (25) is obtained:
2
y=
4bmax
a2

a  h2 − h1 
 +
 x − 1 −
2
4bmax 

 h − h1 

+ h1 − bmax 1 − 2
b
4
max 

2
(25)
In the case of the leveled span the vertex
coordinates become xMIN=a/2 and yMIN=h–bmax,
so the previous equation changes into (26):
instead of the maximal sag bmax.
Knowing the mathematical connection
(30) between the coefficient A of the
parabola and its parameter p, the identity
(31) is obtained.
p=
1
2A
4bmax
1
=
2
a
2p
⇒
A=
⇒
1
2p
bmax
(30)
a2
=
8p
(31)
The parameter p always has a positive
sign. Generally, the coefficient A of the
parabola can be positive or negative, but
since in our case it is always positive, there
are not any problems with signs in (30).
Using identity (31), the equation for the
conductor height can be written both in a
standard form (32) and in a vertex form (33)
of the parabola equation:
2
4b 
a
y = max
 x −  + h − bmax
2
a 
2
(26)
By using equation (25) it is easy to
calculate the coordinates of the vertex point.
So, by equations (26) and (25) we can now
write the equations in the vertex form of the
parabola from examples nr. 1, 2 and 3, as it
is shown below:
Example 1. Leveled span (h1=h2=h)
Vertex point MIN (100;10)
2
y = 8 ⋅ 10 −4 ( x − 100) + 10
(27)
Example 2. Inclined span (h1<h2)
Vertex point MIN (50;16)
2
y = 8 ⋅ 10 −4 ( x − 50 ) + 16
2
a 
1 2  h2 − h1
x + 
−
 x + h1
2p
2 p 
 a
(32)
2
1 
a  2p

y=
 x − 1 − 2 (h2 − h1 )  + h1 −
2p 
2 a

−
1 a  2 p

 1 − 2 (h2 − h1 ) 
2p 2  a

2
(33)
In a special case when the suspension points
on two supports are on the same elevation, the
previous two equations get simple forms as (34)
and (35):
y=
1 2 a
x −
x+h
2p
2p
y=
1 
a
a2
x
h
−
+
−


2p 
2
8p
(28)
(34)
2
Example 3. Inclined span (h1>h2)
Vertex point MIN (150;16)
y = 8 ⋅ 10 −4 ( x − 150) + 16
y=
(29)
Since the vertex point and its coordinates
are already shown in Figures 4., 5. and 6., it
is easy to check the results of the numerical
calculation for defining xMIN and yMIN. It can be
concluded that the results are correct.
3.3. Parabola equation with a parameter p
Hereunder shall be shown how the
equation for the conductor height shall be
defined by the parameter of the parabola p
(35)
4. Usage of the parabola equations for
the conductor height
Any of the above presented equations are
useful for calculating the conductor height
and for plotting the graph of the conductor
line as well. For solving other tasks the
appropriate one has to be chosen from the
shown equations depending on the actual
task.
Equation (15) as the standard form of the
parabola equation is very practical for a quick
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 60 (2012), nr. 3
finding of the first derivative y’(x) and the
second derivative y”(x) of a function [9]. The
x coordinate of the vertex can be obtained by
solving y’(x)=0. After that it is easy to define
the y coordinate of the vertex by solving
yMIN=y(xMIN). This method will be shown
numerically in equation (20) from the
example 3. We already know that the vertex
point is MIN(150;16), but it can be checked
with the method below.
y ( x ) = 8 ⋅ 10 −4 x 2 − 24 ⋅ 10 −2 x + 34
y ' ( x ) = 16 ⋅ 10 −4 x − 24 ⋅ 10 −2
y' ( x) = 0
⇒
x MIN
16 ⋅ 10 −4 x MIN − 24 ⋅ 10 −2 = 0
x MIN =
24 ⋅ 10 −2
= 150
16 ⋅ 10 −4
⇒
y MIN = y ( x MIN )
y MIN = 8 ⋅ 10 −4 ⋅ 150 2 − 24 ⋅ 10 −2 ⋅ 150 + 34 = 16
MIN ( x MIN ; y MIN ) = MIN (150;16)
As it can be seen the same results prove
the validity of the shown method for using
equation (15) for defining the vertex point.
Equation (25) as the vertex form of the
parabola equation has an even bigger
usability than the equation (15). Beside the
determination of the vertex point and the
coefficient A, it can be used to replace the
conductor curve within the x-y coordinate
system. A concrete example of such an
application is the following formula [10] for
the determination of the conductor length in
inclined spans on the basis of the known
maximal sag of the parabola.
L=
a2
16bmax
+ arsh
8bmax

arsh a 2 (a − xMIN ) +
8bmax
xMIN +
a2
2
8b

(a − xMIN ) 1 +  8bmax
(a − xMIN )2 +
+ max
2
2 
a
 a 
8b
 8b

xMIN 1 +  max
+ max
xMIN 
2
2
a
 a

2
(36)



For deriving a formula (36) the curve had
27
to be appropriately replaced within the x-y
coordinate system in order to make possible
the integral calculus for the conductor length.
In the case of the leveled span the
previous formula has a much simpler form:
4b
4b
a2 
 4b 
L=
arsh max + max 1 +  max 

8bmax 
a
a
 a 
2

 (37)

5. Conclusions
The equation for conductor height defined
by the parabola equation has more variations
than the one obtained by the catenary
equation. The simplicity of defining the
parabola equation is partly due to the fact
that the maximal sag of the parabola is
always located at a mid-span, either it is a
leveled or an inclined span. In the case of a
catenary this rule is not valid. As it is shown,
the parabola gives a possibility to find a
solution in different ways, so it ensures an
effective way of checking the results. These
are some of the reasons for the frequent
usage of the parabola, when the difference
between the parabola and the catenary is
insignificant.
Through the numerical examples we have
seen the way of obtaining the equation for
the conductor height in the case of any type
of the span. It has also been shown how to
define the coordinate of the vertex point. The
vertex is very often equal with the lowest
point of the conductor and in the case of the
inclined span it is one critical point of the
conductor, so its checking is highly
recommended. The special case of the
inclined span has also been discussed, when
the vertex point is out of the span.
Using different variations of the parabola
equation for the conductor height the validity
of shown equations has been proved through
numerical examples.
A very important result of the shown 3
examples is that in the case of the same
span length a and the value of the maximal
sag bmax, both in leveled and inclined span,
the coefficient A of the parabola does not
change. This paper highlighted this important
characteristic of the parabola.
6. Acknowledgment
This paper presents the main part of EDF
DÉMÁSZ
project
called
Designer
Programme, which entered the 19.
28
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 60 (2012), nr. 3
Hungarian Innovation Award Competition.
The Designer Programme as a recognised
innovation got into the Innovation Award
2010 book under number 12.
7. References
[1] CIGRÉ 324, Sag-tension calculation methods
for overhead lines, CIGRÉ 2007
[2] HATIBOVIC
A.,
Usage
of
Parabola
Calculation for Planning of Electrical
Overhead Network, ENELKO conference,
Kolozsvár 2011
[3] PANSINI
A.,
Electrical
Distribution
Engineering, The Fairmont Press, Inc., 2007
[4] GUSTAFSON D., FRISK P. and HUGHES J.,
College Algebra, CENGAGE Learning 2010
[5] OBÁDOVICS GY., Matematika, SCOLAR
Budapest 2012
[6] TURKINGTON D. A., Mátrix Calculus & ZeroOne Matrices, CAMBRIDGE 2005
[7] GENTLE J. E., Matrix Algebra, Springer 2007
[8] HATIBOVIC A., TOMIC M., Determination of
the lowest point of conductor for inclined
spans, CIGRÉ conference, Sarajevo 2011
[9] OBÁDOVICS GY., Felsőbb matematikai
feladat-gyűjtemény, SCOLAR Budapest 2002
[10]HATIBOVIC A., Integral Calculus Usage for
Conductor Length Determination on the Basis
of Known Maximal Sag of a Parabola,
Periodica Polytechnica Electrical Engineering,
Budapest University of Technology and
Economics 2012
8. Biography
Alen HATIBOVIČ was born in
Tuzla (Bosnia-Herzegovina), on
October 1, 1966. He graduated
from University of Tuzla, Faculty of
Electrical Engineering in Tuzla
(Bosnia-Herzegovina), in 1992.
He is a senior engineer for electrical network
development and works for EDF DÉMÁSZ in
Szeged (Hungary). His research interests
concern: distribution of electric energy,
planning and designing electrical network
(especially overhead lines).