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 of low-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 transmission 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 information 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 notebook-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 transmission 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 transmission 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 points are (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 = c

i=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(xi3 A) - Yil

+ [aft+; 4y) (13)

Therefore, the Newton-Raphson iteration to search for I such that g(n) = 0 is

R - 48 n+1-

X1=: (f(xi2 4) - Yi)af(Xi, n,)ian - n 1 CL ( a%,4 I 1 822 (ftxi, &I - Yi) +

(af EiT 4>‘)

(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, A will 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,, OIL are 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 = BAi cash ((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 convergence/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)I 2 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 approximating 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 transmission 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

Number of nonorigin points 10

Convergence tolerancea 0.00000000010 Relative noise amplitude 0.199999988 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 Maximum absolute residual 0.09877181986 Mean absolute residual 0.04790142076 STD DEV of residual vector 0.03072120388

aTh~s tolerance was reached in six iterations.

It should be further noted that the same rapid convergence 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

Number of nonorigin points 10 Convergence tolerancea 0.00000001000 Relative noise amplitude 0.0 Computed value of a 124.00728267588 Computed value of b -9.00139688372 Computed value of c 855.68602456075 Maximum absolute residual 0.02391056077 Mean absolute residual 0.00922825148 STD DEV of residual vector 0.00668603082

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 approximation 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 121 I l.Oe - 6 then il:= 1.0

appear after the assignment statement corresponding to equation (A6).


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