00b7d526edbdcf1507000000

Construction
survey data
Stephen J. Sugden
of transmission line catenary from
School of Information Technology, Bond University, Gold Coast, Australia
Precise survey data from observations of actual overhead transmission lines are commonly used to confirm
theoretical predictions of sag-tension calculations, based on the standard catenary model. An algorithm is
presented for determining the catenary of best fit to a set of data points in the usual Cartesian coordinates.
The method described employs the least-squares criterion for curve fitting, and uses the iterative
Newton-Raphson algorithm to solve for the required catenary parameters. A FORTRAN 77 subroutine
implementation of the algorithm is used in a conductor profile program by the South East Queensland
Electricity Board, and a PC version coded in Borland Pascal is available from the author. The program uses
the subroutine to generate the equation of a transmission line catenary from survey data, thus allowing rapid
calculation oflow-point (vertex) coordinates and conductor relative levels at arbitrary points along the span.
Experimental results are presented, which indicate typical accuracy of computed conductor height to within
approximately a conductor diameter.
Keywords: transmission line, catenary, survey data, Newton-Raphson, curve-fitting, least squares
1. Introduction
Survey departments associated with electricity transmis-
sion and distribution authorities are often given the task
of accurately determining the position of an overhead
conductor. The reasons for acquiring this information
are usually related to the requirement of maintaining a
nominated statutory clearance either from ground level
or some structure located near a conductor. Line
tensions are calculated at the design stage, and this infor-
mation is usually translated into a sag at some specific
ambient conditionPusually a series of sags for different
ambient temperatures is given. A simple parabolic or
more accurate catenary approximation to the line is used.
Errors introduced by inelastic deformation of the
conductor will result in further inaccuracies in the
determination of the vertex or any other point of the
catenary. In addition to these variations, daily variations
in load current, wind velocity, and solar radiation make
the accurate heighting of a conductor from design data
very difficult.
From the line design engineer’s point of view, the
general theory of transmission line design is well
developed and the standard methods based on the
approximate catenary or parabola give adequate results
for the customary sag-tension calculations. Many
authors rightly focus attention on the economical and
Address reprint requests to Dr. Sugden at the School of Information
Technology, Bond University, Gold Coast, Australia.
Received 30 June 1993; revised 5 October 1993; accepted 25 October
1993
reliable design of power transmission structures, with
emphasis on mathematical formulas of practical
computational value, rather than inordinate emphasis on
theoretical exactness. Lummis and Fischer’ point out
that the basic assumptions used to mathematically derive
the catenary shape (the present equation (1)) for a
suspended cable are, in fact, only approximately correct.
Nevertheless, the hyperbolic function so obtained gives
results that are sufficiently accurate for sag-tension
calculations, and in many cases even the quadratic
approximation to the hyperbolic form is adequate for
engineering design purposes (see, for example, Boyse and
Simpson’) and certainly within the required accuracy for
line construction.
1.1 Generality of approach
We are concerned here with an algorithm to fit survey
data to a standard catenary which, it is assumed,
reasonably approximates the shape of a transmission line
under normal steady-state operating conditions. The
method is independent of any physical model of
sag-tension, which of necessity must take into account
such quantities as electrical load, mechanical load,
conductor weight, temperature, and perhaps other
non-steady-state variables such as wind velocity and
direction. Simply stated, the present method is an
exercise in nonlinear least-squares curve fitting, and the
only assumption made is that the transmission line has
a shape reasonably approximating that of a catenary.
In view of the above, the method gives important
feedback on any catenary model of the transmission line.
274 Appl. Math. Modelling, 1994, Vol. 18, May 0 1994 Butterworth-Heinemann

Construction of transmission line catenary from survey data: S. J. Sugden
For a reasonably accurate survey, the present algorithm
generates the catenary of best fit, which then enables
residuals to be calculated. These are just the differences
between observed and predicted ordinates. Large
discrepancies will then indicate either (a) poor survey or
(b) poor mathematical model of the transmission line.
In either case, useful information is gained. The
method frees the surveyor from accurate observation of
low point (vertex), and frees the surveyor or line design
engineer from manual interpolation to find the low point
or any other point along the span. It gives the surveyor
an idea of the accuracy of the observations, and field
units can calculate the catenary on site and bring to light
some doubtful observations. These may be redone on
the same day, thus reducing or eliminating the need for
further field trips. Because the algorithm is easily
implemented on small hand-held calculators or note-
book-style portable computers, this suggestion is entirely
practical.
It should be noted that no claim is made that the
mathematics of the method is new; however a rather
extensive computer-based literature search failed to find
any application of catenary interpolation of survey data
for transmission lines. The technique described in this
paper provides useful information to both surveyor and
transmission line engineer, so that a description of the
details of the algorithm and a SEQEB case study form
the bulk of this paper. The approach to be described is
intended to alleviate some of the tedium associated with
converting survey data for an overhead conductor profile
into something useful. Once the catenary is generated,
the line engineer can then relate it to any theoretical
model of choice.
It could be said that the approach described herein
borrows from traditional models of overhead transmis-
sion lines only in the sense that it assumes that the
transmission line shape is sufficiently close to that of a
mathematical catenary to justify using this family of
curves as approximants in a least-squares fit of the
observed data; however, the interpolation technique is
otherwise independent of any such model.
1.2 Motivation
Motivation for development of the method came from
a SEQEB requirement for a conductor profile program
to generate the remainder of the curve on being given
observed points on an arc formed by a single span of a
transmission line catenary. Cases in which precise (to
within 1 cm) determination of overhead conductor
position is especially important include (a) costly or
specialized redesign projects and (b) those in which
confirmation of conductor clearances over navigable
waterways is required.
In recent SEQEB instances of (a), involving transmis-
sion lines energized at 110 kV or 132 kV, a system
upgrade program was associated with significant
increase in electrical load and therefore greater line sag.
This increased sag often exceeded the previously allowed
design limits, thus necessitating line retensioning. In
other cases existing tower or pole structures or their line
attachment hardware needed to be either shifted
vertically or relocated to a new position entirely.
Precise surveys help to determine the actual conductor
tension and therefore the degree to which the initially
specified tension has been maintained. It has been noted
above that theoretical predictions of conductor profile
are not in complete accord with physical reality, so that,
especially in cases of close approach to statutory
clearances, it is essential for the design engineer to
obtain confirmation of sag-tension calculations from an
accurate field survey. The present algorithm allows rapid
calculation of a catenary from field data, thus allowing
immediate calculation of the low point of the span.
Recent SEQEB examples of (b) have a generally
involved 11 kV to 132 kV lines above navigable
waterways in South East Queensland. Here, specific
statutory clearances determined by the Department of
Harbours and Marine must be maintained. In particular,
for litigation cases involving boating accidents and
high-voltage lines, it is clearly essential that the electricity
authority have highly accurate information as to the
precise location of its overhead lines under all conditions
of weather, tide levels, and electrical and mechanical
load.
Finally, from the surveyor’s point of view, the method
presented here provides a useful means of determining
several important quantities. Having observed the two
attachment points plus a number of others along the
span usually concentrated around the estimated vertex,
the surveyor needs to do the following:
1. determine the general accuracy of observations;
2. locate and eliminate bad observations and gross
errors;
3. calculate the (x, y) coordinates of the low-point
(vertex) and hence determine clearance of this point
with respect to a specified level datum.
All of these requirements are met by the present
algorithm/program, which has been coded in FORTRAN
77 for SEQEB and also in Borland Pascal for a PC. It
will be a simple matter to recode it in say, BASIC or c, if
this is desired (both languages being commonly available
on personal computers). It is recommended that
notebook-style PCs be used in the field so that the
desired catenary can be quickly computed. Gross errors
would then become obvious and further verifiable
observations could be made on the same job.
2. Mathematical theory
2.1 General
Mathematically, the requirement is for a curve-fitting
algorithm that will apply standard techniques to search
for a member of the known family of catenaries that in
some sense best fits the observed data. The best-fit
criterion to be used will be that of least squares. It is
recognized of course that experimental error is present
in all observations, and the algorithm takes this into
account; however, the two points of attachment of the
transmission line are assumed to be without error. This
Appl. Math. Modelling, 1994, Vol. 18, May 275

means geometrically that they lie exactly on the final
catenary. Such an assumption is reasonable, because the
points of attachment are almost always accessible and
can be determined very accurately with modern
laser-based theodolites. On the other hand, the interior
points need not be accessible (e.g., the transmission line
may span a river or gorge).
2.2 Statement of problem
Given the set {(xi, Y,): i = 1 . . . II - l}, it is necessary
to choose, from among those curves described by
equations (6) and (7), one which most closely fits these
data points. As outlined earlier, the least-squares
criterion is to be used to implicitly determine a value of
/z to satisfy our requirement. More precisely, defining
For a canonical catenary with vertex I/ at (a, b) and
parameter c, we have
Y-,=c(coshre)-1)
We consider the left-hand point of attachment to be
the (x, Y) origin, i.e., the curve described by (1) passes
through (0,O). This leads to
Y=2csinh(g)sinh(y) (2)
Our data pointsare (O,O),(xl, YA, (x2, y2), . . . (x,,, y,)
where (0,O) is the left point of attachment and (x,, Y,) is
the right point of attachment. Comparison of equation
(2) with equation (1) reveals that, by a suitable choice of
origin, we have now reduced a three-parameter family
(a, b, c) of catenaries to a two-parameter (a, c) family. It
is also possible to eliminate the parameter a by requiring
that the curve pass through the right-hand point of
attachment, which we have designated (x,, y,) in our
coordinate system.
Because equation (2) contains only one reference to
the parameter a, we solve it for a when (x,, y,) is
substituted for (x, Y). This process yields
Writing
i3 = sinh-’
Yll
2c sinh (xJ2c) >
and
(3)
we obtain for the equation of the catenary
;ly = sinh (Lx) sinh (R(x - x,) + 8)
where 8 is given by
It will be noted that 8 is a function of the catenary shape
parameter I only, because the supplied parameters x,, y,
are regarded as constants. In equations (6) and (7) we
now have a one-parameter (1) family of catenaries, all
passing through (0,O) and (x,, y,)-the two points of
cable attachment. Further interior data points, from the
set {(x, yi): i = 1 . . . n - 11, are also supplied from
survey. We are now in a position to formulate a precise
statement of the problem.
f(x, 1) =
sinh (Lx) sinh (2(x - x,) + 8
/z
(8)
and
n-l
s(n) = C (ftxi, i) - YJ2 (9)
i=l
we seek 3, which minimizes S(1).
2.3 Algorithm development
In order to minimize S(1) we solve dS/dL = 0 because
S(n) is a positive definite continuous function of 1 and is
only zero if every data point lies precisely on the curve.
Differentiation of (9) with respect to I gives
dS@) n- 1 df(x,, 2)
---=2 c
dA i=l
7 wi, 4 - Yi)
Defining
n- l af(xi, A)
s(4 = ci=l
ai, (ftxi2 A)- Yi)
(10)
(11)
it is required to solve
s(l) = 0 (12)
The Newton-Raphson iterative method is used to solve
(12). For this, we need g’(A):
n- 1
g’(l) = C
i
a2f(xi,A)
i=l
d12 Cf(xi3A)- Yil
+ [aft+; 4y)
(13)
Therefore, the Newton-Raphson iteration to search for
I such that g(n) = 0 is
R - 48n+1-
X1=:(f(xi2 4) - Yi)af(Xi, n,)ian-
n 1
CL (a%,4
I 1 822 (ftxi, &I - Yi) +
(af EiT4>‘)
(14)
and the partial derivatives are evaluated at 2 = i,.
2.4 Derivation of expressions for the partial derivatives
The Newton-Raphson iteration defined by (14)
requires the values of af(x,, n)/an and a2f(xi, A)/aL’ ViE
{L&3,. . . , n - l}. These values must of course be
recomputed at each iteration because, although the xi
276 Appl. Math. Modelling, 1994, Vol. 18, May

are fixed, Awill change at each step. In this section, closed
expressions obtained by the usual rules of differentiation
are derived for df(x,, A)/82 and a2f(xi, /2)/aA2. Because
differentiation of sums is easier than differentiation of
products, we convert equation (8), which defines f(x, A),
to the equivalent sum of hyperbolic cosines using a
standard identity:
21f(x, 2) = cash ((2x - x,)1 + 0) - cash (x,2 - 0)
(15)
Partial differentiation of (15) with respect to 2 yields
2f + 2J_fA= (2x - x, + 0,) sinh ((2x - x,)2 + 0)
- (x, - 0,) sinh (x,2 - 0) (16)
where the subscript I denotes a/al, i.e., af/aJ. = fL, and
the arguments of f have been omitted for simplicity. In
like fashion we denote the second partial derivative with
respect to 1 by subscript 11. Differentiation of (16) with
respect to 1 gives
2&, + 4fL = (2x - x, + Q2 cash ((2x - x,)2 + 0)
+ 28,, sinh (x/z) cash ((x - x,)3. + 0)
- (x, - 8J2 cash (x,i - 0) (17)
From a computational point of view, because x = xi and
x,, 1 are supplied, it will be seen from equations (8), (16),
and (17) that the required partial derivatives fL, fni may
be computed as soon as the values of 6,, OILare available.
These are obtained by implicit differentiation of a
rearranged form of equation (8).
Ly, = sinh (6) sinh (x, 1.) (18)
Partial differentiation of (18) with respect to 1. and then
solving for Bn yields
8
A^
= y, - x, sinh (0) cash (x,2)
cash (0) sinh (1,x)
(19)
Further differentiation gives
0 = BAicash ((3)sinh (x,1) + (fl: + x,‘) sinh (0)
x sinh (x,2) + 2x,8, cash (0) cash (x,1) (20)
It is evident that 8,, 0,, are now computable from
equations (7), (19), and (20).
3. Development of the computer algorithm
3.1 Preconditions assumed
We require that n 2 2, that all data points be distinct,
and further that no two data points share the same value
of x (abscissa). In addition, for the initial value
approximation of Appendix B to be most useful, it is
desirable that the data points are stored in order of
increasing abscissa. This will of course also be necessary
for any external tabulation of the data points. Finally,
no Xi is permitted to be zero because (0,O) is the assumed
extreme left-most point of the catenary arc. It is clear
that some of the foregoing conditions are subsumed by
others, and in fact their conjunction is equivalent to the
following two requirements:
1. n22,
2. 0 < X1 < X2 < ‘.’ < Xi < Xi+l < “’ < X,
i.e., that {xi} is a strictly positive, strictly increasing
sequence of at least two terms. These conditions are
checked and enforced by the FORTRAN subroutine by
returning an error flag, indicating the nature of any
violation. This flag is also used to indicate con-
vergence/divergence of the iteration scheme.
3.2 Details of the algorithm
The algorithm is straightforward, and may be
expressed in a Pascal-like pseudocode in its top-level
form as follows. Note that the initial approximation to
i is that obtained in Appendix B. The functions g(A) and
g’(1) are defined by equations (11) and (13), respectively.
Newton search for catenary parameter 2
read(e)
read(MaxIts)
read(n)
for i := 1 to n do read (xi, yi) endfor
NumIts := 0
A:= (Y,& - x,.Y,)l(x1x?l(x, - %I))
while Ig(/L)I2 E and (Numlts I Maxlts) do
compute sinh (x,n) cash (x,2) from data
compute sinh 8 from (7)
compute cash 8 using cash’ = 1+sinh2
compute 6 using sinh- ’
compute 8, from (19)
compute 8,, from (20)
g:=o
g’:= 0
fori:= 1 ton- 1 do
compute f(xi, 2) from (15)
compute fJxi, 2) from (16)
compute &(xi, 2) from (17)
compute g,:= g + fn(Xi, )*)(f(Xi, 3.) - Yi)
c+o;I~(“~ ,“, ‘= 9’ + _hl(xiT /2)(f(xi, A) - Yi)
AX,> ’
endfor
i := A- g(A)/g’(A)
inc(NumIts)
endwhile
if Ig(A)l < E then indicate success else indicate
failure endif
4. Limitations of the algorithm
The method is completely general, but is subject of course
to the standard conditions for convergence of the
Newton-Raphson algorithm, namely the existence of a
neighborhood of the solution in which the derivative of
the objective function is bounded away from zero. The
input data are also assumed to satisfy the conditions
of 3.1. As stated in the conclusion, no case of divergence
has yet been found; rapid quadratic convergence for 2
being observed in all test cases. Some analysis of the
denominator term g’(i) appearing in equation (14)
Appt. Math. Modelling, 1994, Vol. 18, May 277

may show divergence to be unlikely or even impossible
for some comparatively mild assumption about the input
data. This does not seem unreasonable, because the term
(dflan)2 is of course non-negative, and the term
a2fla12 is small for data that are not too wild. It
must be remembered of course that a least-squares
curve-fitting algorithm will generate a curve for any set
of data with distinct abscissas, so that examination of
calculated residuals is always advisable.
5. Conclusion
A general method has been presented for least-squares
fitting of (x, y) data points to a mathematical catenary.
The two endpoints are assumed to lie precisely on the
curve and the interior points-of which there must be at
least one-are, according to the usual assumptions,
taken to be without error in abscissa, but subject to
experimental error in the ordinate. It is anticipated that
the method will be of most value to engineering studies
such as transmission line profile calculations. The
algorithm has been implemented as a FORTRAN
subroutine, and this, along with driver program and test
data, is available from the author.
The algorithm has been observed to converge very
rapidly for all test cases used; no case of zero or small
derivative leading to divergence or floating division error
having yet been found. The reader is referred to the
previous section for limitations of the method. Experi-
mental results indicate accuracy of conductor levels
computed from the generated catenary to be within a
conductor diameter of the levels obtained from survey.
Numerical details are to be found in Appendix B.
As stated elsewhere in this paper, the least-squares
method will fit virtually any set of data, however
unreasonably, to a member of the family of approxima-
ting curves-in this case, catenaries. The physical
validity of such an approximation, however, remains a
matter for careful determination by an experienced
professional, i.e., a line design engineer or perhaps a
surveyor with appropriate experience. Examination of
the magnitude of residuals at each of the observed points,
with due regard for the physics of the problem, is usually
the most effective means of carrying out this task.
6. Acknowledgment
Thanks are extended to Steven Pryor of the Surveying
Department, South East Queensland Electricity Board,
as the person who first indicated the need for a computer
program to solve the interpolation problem, for his
advice on the practical surveying aspects of the problem,
and also for his many helpful suggestions throughout the
development of the method. His colleague, Robert Battle,
also provided some very useful assistance. The author is
indebted to John Gudgeon of the School of Mathematics,
Queensland University of Technology, for his useful
comments regarding the general nature of the algorithm
to be employed, and also to Charles Williamson of Mains
Development Department, The South East Queensland
Electricity Board for his helpful advice on the practical
engineering issues mentioned in the introduction. Thanks
also to Dr Wilson Sy, School of Mathematical Sciences,
University of Technology, Sydney, and to Dr Bernard
Duszczyk, School of Information Technology, Bond
University, for their useful comments on various drafts
of this paper.
References
Lummis, J. and Fischer, H. D. Practical Application of Sag and
Tension Calculations to Transmission Line Design. Trans. Amer.
Inst. Electrical Eng. Part III (Power Apparatus & Systems) 1955,
74, 402-4 16.
Boyse, C. 0. and Simpson, N. G. The Problem of Conductor
Sagging on Overhead Transmission Lines. J. Inst. Electrical Eng.
1944, 91, Part 2, in press.
Appendix A: Test data and results
Three test cases are given below. Case 1 contains data
points generated by the computer from a perfect
catenary. The points in Case 2 are as for those of Case
1, except that a sinusoidal noise component has been
superimposed on the generated ordinates. It was
considered desirable to observe the behavior of the
subroutine by simulating errors in the y values. The
relative amplitude chosen (20%) is rather high, but it can
be safely assumed that the subprogram will perform at
least as well on bona fide field data obtained from survey.
Such data are unlikely to contain errors quite as gross
as this. In any event, field data obtained from a South
East Queensland Electricity Board survey are supplied
as Case 3, and the results are very encouraging.
It is to be expected that residuals, i.e., the discrepancies
between ordinates computed from the generated
catenary and those observed, will be of the order of a
few centimeters in typical cases of observation of trans-
mission lines such as that reported in the present Case
3. Residuals of this order have in fact been observed, as
will be noted from the results below. Typical error in
vertical level is of the order of 1 cm, with the maximum
being approximately 2.4 cm or 1 inch. These accuracies
are of the order of a cable diameter, as claimed in the
abstract. All quantities are expressed in meters in the
computer output.
Table 1. Results part A for perfect catenary (case 1)
i x(i)” Y(i)* ycomp(i)c residual(i)d
1 0.2
2 0.4
3 0.6
4 0.8
5 1 .o
6 1.2
7 1.4
8 1.6
9 1.8
10 2.0
-0.205646
-0.357615
- 0.462008
-0.523014
-0.543081
- 0.523014
-0.462008
-0.357615
- 0.205646
0
- 0.205646
- 0.357615
- 0.462008
-0.523014
-0.523081
-0.523014
- 0.462008
-0.357615
- 0.205646
0
0
0
0
0
0
0
0
0
0
0
‘Empirical abscissa.
b Empirical ordinate.
‘Computed ordinate.
d Difference between empirical and computed ordinates

Construction of transmission line carenary from survey data: S. J. Sugden
Table 2. Results part B for perfect catenary (case 1)
Quantity Value
Number of nonorigin points 10
Convergence tolerance= 0.00000000010
Relative noise amplitude 0.0
Exact value of a 1 .ooooooooooo
Exact value of b -0.54308063482
Exact value of c 1 .ooooooooooo
Computed value of a 1.ooooooooooo
Computed value of b - 0.54308063482
Computed value of c 1 .ooooooooooo
Maximum absolute residual 0.00000000000
Mean absolute residual 0.00000000000
STD DEV of residual vector 0.00000000000
aTh~s tolerance was reached in five iterations.
Table 5. Results part A for SEQEB survey data (case 3)
Table 3. Results part A for perfect catenary + noise (case 2)
i x(i)” y(i)b ycomp(i)c residual(i)d
1 0.2 -0.240255 -0.203442 -0.0368131
2 0.4 -0.422651 - 0.353908 -0.0687431
3 0.6 - 0.475048 -0.457335 -0.0177127
4 0.8 -0.44385 -0.517803 0.0739531
5 1 -0.438926 -0.537698 0.0987718
6 1.2 -0.493786 -0.517803 0.0240171
7 1.4 -0.522715 - 0.457335 -0.0653796
8 1.6 -0.428377 - 0.353908 -0.0744693
9 1.8 -0.222596 -0.203442 -0.0744693
10 2 0 0 0
a Empirical abscissa.
b Empirical ordinate.
c Computed ordinate.
d Dtfference between empirical and computed ordinates.
Table 4. Results part B for perfect catenary + noise (case 2)
Quantity Value
Convergence tolerancea 0.00000000010
Exact value of a 1 .ooooooooooo
Exact value of b -0.54308063482
Exact value of c 1 .ooooooooooo
Computed value of a 1 .ooooooooooo
Computed value of b -0.53769782012
Computed value of c 1.00860489830
aTh~s tolerance was reached in six iterations.
It should be further noted that the same rapid conver-
gence is obtained with actual experimental data as with
machine-generated dummy data.
Appendix B: An initial estimate for the Newton
iteration
The Newton-Raphson iterative scheme, which forms the
heart of the catenary curve-fitting algorithm, requires a
reasonable starting value for I. in order to converge.
i x(i)” y(i)b ycomp( i)” residual(i)d
1 0.355 -0.057 -0.051553
2 48.593 - 5.688 - 5.676
3 84.894 -8.119 -8.10731
4 124.875 -8.999 -9.00096
5 151 ,157 -8.575 -8.57065
6 184.349 -6.849 - 6.87291
7 205.295 -5.131 -5.13744
8 234.87 - 1.822 ~ 1.80966
9 265.057 2.636 2.65014
10 265.406 2.708 2.708
- 0.005447
- 0.0120029
- 0.01169
0.00195692
- 0.0043505
0.0239106
0.00644239
- 0.0123438
-0.0141385
0
a Empirical abscissa.
‘Empirical ordinate.
‘Computed ordinate.
d Difference between empirical and computed ordinates.
Table 6. Results part B for SEQEB survey data (case 3)
Quantity Value
Convergence tolerancea 0.00000001000
Computed value of a 124.00728267588
Computed value of b -9.00139688372
Computed value of c 855.68602456075
aTh~s tolerance was reached in four Iterations
Experimental results with the algorithm have shown that
simple linear approximations used in equations (6) and
(7) give good starting values. The details follow.
Equations (6) and (7) are, respectively
2y = sinh (2x) sinh (2(x - x,) + 0) (Al)
sinh (0) = ‘“”
smh (Ax,)
Using sinh (t) zz t, we obtain
642)
(j z sinh (0) z “Y, = !A!
2,x, x,
(A3)
Applying the same approximation to (Al) and setting
. ”
A = A,, the initial value, we arrive at:
2,y = i,x
(/l,(x - xn) + Jf!!
X, >
Finally:
j. _ YX” - XY,
0-
xX,(x - xn)
W
for some (x, y).
Assuming that the data points are reasonably uni-
formly spaced in x, we choose (x, y) to be a point near
the middle of the span. Tests have consistently shown
that such a choice, which allows the linear approxima-
tion to be based essentially on the full span of the
catenary, gives the best results.
Appl. Math. Modelling, 1994, Vol. 18, May 279

Accordingly, our final expression for & is
j
‘0
= YUG - X,Y,
~,X,~X, - xn)
(‘46)
where CI= Ln/2_J.The author is indebted to an anony-
mous referee for pointing out a flaw in the original
scheme for estimating 2,.
In view of the preconditions imposed on the data-see
3.1 and Appendix A-it is apparent that Lo is always well
defined. In computing parlance, nopoating divide error
is possible as all xi are distinct, and none is zero. These
restrictions guarantee that a starting value, I,, is always
provided by equation (A6); however it should be realized
that a starting value of &, = 0 is disastrous for the
iterative algorithm. From the equation it is clear that
A0 = 0 is not permitted; indeed it corresponds to c = cc
for the catenary, i.e., physically infinite tension or zero
weight of suspended cable, and therefore a linear arc for
the span.
From equation (A6), it is clear that A0 will be zero if
and only if
Y, Yn-=- (A7)
x, XII
As noted above, this corresponds to the case of an
infinitely tensioned cable and cannot occur in practice.
What can occur, however, is that poor experimental data
may contain such a pair of points (x,, y,) and (x,, y,).
For the application motivating the present work, i.e.,
construction of transmission line catenary from survey
data, this is virtually impossible. However, in the general
case of fitting arbitrary points to a catenary, this situation
could occur, and it is recommended that in the computer
implementation, a statement such as
if 121I l.Oe - 6 then il:= 1.0
appear after the assignment statement corresponding
to equation (A6).

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