HRVATSKI OGRANAK MEĐUNARODNE ELEKTRODISTRIBUCIJSKE KONFERENCIJE - HO CIRED 4. (10.) savjetovanje Trogir/Seget Donji, 11. - 14. svibnja 2014.

SO1 – 14

Alen Hatibović, electrical engineer (PhD student) Electrical network designer, Baja (Hungary) [email protected]

ALGORITHM FOR THE CONDUCTOR LENGTH CALCULATION IN INCLINED AND LEVELLED SPANS BASED ON THE PARABOLA MODEL SUMMARY This paper shows a complete mathematical solution for calculation of the conductor length using integral calculus. The presented method is based on the parabola model and is applicable for spans approximately up to 400 metres, i.e. in those cases when the difference between the catenary and the parabola is negligible, and so the catenary can be approximated by a parabola. The algorithm is prepared for inclined spans, but it is also usable in levelled spans. Since the conductor length changes with temperature, each calculation refers to a chosen temperature of the conductor, using the maximum sag concerned to that temperature, beside the chosen tension and conductor type. Hence, the maximum sag of the parabola is one of the input data for the calculation. Beside the conductor length calculation in a full span the provided algorithm also gives a possibility to calculate the conductor length in an arbitrary part of the span. Thus, it can solve classical tasks in practice, but also some unconventional ones. The use of the algorithm and its universal formula are presented in different types of the span. Key words: inclined span, sag, parabola, overhead lines, vertex point

ALGORITAM ZA IZRAČUN DULJINE VODA U KOSOM I RAVNOM RASPONU ZA MODEL PARABOLE SAŽETAK Referat prikazuje cjelovito matematičko rješenje za izračun duljine voda uporabom integralnog računa. Prikazana metoda se zasniva na modelu parabole i može se primjenjivati za raspone do oko 400 metara, tj. u onim slučajevima kada je odstupanje između lančanice i parabole zanemarljivo, te se lančanica može aproksimirati parabolom. Algoritam je izrađen za kose raspone, ali se takođe može primjenjivati i za ravne raspone. Obzirom da se duljina voda mijenja sa temperaturom, svaki izračun se odnosi na izabranu temperaturu voda, uz uporabu najvećeg provjesa za tu temperaturu, izabrano naprezanje i tip vodiča. Stoga, najveći provjes parabole je jedan od ulaznih podataka za proračun. Pored izračuna duljine voda u cijelom rasponu izrađeni algoritam takođe pruža mogućnost izračuna duljine voda u prizvoljno izabranom dijelu raspona. To znači da riješava klasične, ali takođe i neke nekonvencionalne zadatke u praksi. Uporaba algoritma i njegove univerzalne formule je prikazana u različitim vrstama raspona. Ključne riječi: kosi raspon, provjes, parabola, nadzemni vodovi, tjemena točka

1

1.

INTRODUCTION

Due to the sag of the overhead line (OHL), the conductor within the span is always longer than the span itself. Thus, the conductor length calculation is important for constructing overhead lines. In case of the parabola based calculation for an OHL design the existing scientific literatures generally give a solution for the conductor length in levelled spans only. The same length formula is frequently in use also in inclined spans despite the fact that it makes errors in calculations. This is the reason for providing the universal algorithm for calculation of the conductor length, which ensures correct results in each case, i.e. in levelled and inclined spans as well. Such a complex task can be effectively solved by the application of the integral calculus. In addition, this approach also solves the conductor length calculation in any part of the span. The maximum sag of the parabola can be obtained by a sag–tension calculation or taken from the available sag–tension–temperature tables. The determination of the maximum sag is not the task of this paper. It is discussed in details in [1], [2]. The conductor sag is well seen in Figure 1.

Fig. 1. Overhead line For providing a universal algorithm for the conductor length calculation, which is based on the parabola model, Figure 2 will be used. It contains all applied symbols and shows a parabolic conductor curve in an inclined span. The equation y(x) of the curve is essential for the deduction and it can be defined by the application of the three known points A, B and C of the parabolic curve [3]. Using the coordinate system shown in Figure 2 the necessary data for all calculations are the following in this paper: the span length, the heights of the suspension points with respect to the x–axis and the maximum sag. The additional input data ( x1 and x2 ) are necessary only for the conductor length calculation in part of the span. The following symbols are used in Figure 2:

A (0;h1) – left–hand side suspension point, B (S;h2) – right–hand side suspension point MIN (xMIN ; yMIN) – vertex point C (xC; yC) = C (S/2; (h1 + h2)/2 – Dmax ) – conductor’s point with a maximum sag S – span length Dmax – maximum sag

y (x) – parabolic conductor curve ψ – angle of the span inclination

x1, x2 – start and end points of part of a span, x1 < x2 E (x1; y (x1)) – start point of the conductor in part of a span F (x2; y (x2)) – end point of the conductor in part of a span 2

B

h2

Height

F

h1

y(x)

D max

ψ

A E

yC

y MIN

C

MIN

0

x1

x MIN x C = S / 2

x2

S

Distance Fig. 2. Parabolic conductor curve in an inclined span with h1 < h2 The equation for the parabolic conductor curve in Figure 2 is (1). It is given in the vertex form of the parabola equation according to the basic expression (2) [4], [5] where xMIN (4) and yMIN (5) present the coordinates of the parabola's vertex point. The a (3) is a coefficient of the parabola and it defines its shape. Let us mention that this coefficient is the same in both vertex and general forms of the parabola equation [6]. Being a quadratic function, the parabola belongs to the group of algebraic functions. 2

 h −h  4 Dmax  S  h2 − h1   + h1 − Dmax 1 − 2 1  y( x) =  x − 1 − 2 S  2 4 Dmax  4 Dmax  

y ( x ) = a ( x − x MIN ) + y MIN 2

a= x MIN =

y MIN

2

x ∈ [0, S ]

(1) (2)

4 Dmax S2

(3)

S  h2 − h1  1 −  2  4 Dmax 

 h −h  = h1 − Dmax  1 − 2 1  4 Dmax  

(4) 2

(5)

In order to simplify the deduction, equation (2) will be applied for providing the algorithm, but (3) and (4) will be used at the end of the deduction. The special final formulas for the characteristic tasks in connection with the conductor length calculation will be separately defined.

2.

CALCULATIONS IN INCLINED SPANS

2.1.

Conductor length in part of an inclined span

The length of the parabola (2) on the interval following mathematical formula:

Lx1 x2 =

x2

∫

x1

[x1, x2], shown in Figure 2, can be determined by the 2

 dy  1 +   dx  dx 

(6)

3

Using the basic derivation rule (7), the first derivative of (2) is (8). Squaring it results in (9):

d (cx n ) = ncx n−1 dx

(7)

dy = 2a ( x − x MIN ) dx

(8)

2

 dy  2   = [2a ( x − x MIN )]  dx 

(9)

Inserting (9) into (6) and evaluating the integral by the application of the substitution method are shown step by step in the following lines:

Lx1 x2 =

x2

1 + [2a ( x − x MIN )] dx

∫

2

(10)

x1

2a ( x − x MIN ) = sht

(11)

2adx = cht ⋅ dt

(12)

1 cht ⋅ dt 2a

(13)

dx =

t = arsh (2a (x − x MIN ))

x = x1

⇒

x = x2

⇒

(14)

t1 = arsh (2a ( x1 − x MIN ))

(15)

t 2 = arsh (2a (x 2 − x MIN )) t2

Lt1 t2 = ∫ 1 + sh 2 t t1

1 cht ⋅ dt 2a

(16)

t

1 2 2 ch t ⋅ dt 2a t∫

(17)

1 2 1 + ch 2t ⋅ dt 2a t∫ 2

(18)

Lt1 t2 =

1

t

Lt1 t2 =

1

t

Lt1 t2 =

1 2 (1 + ch 2t ) ⋅ dt 4a t∫

(19)

1

t

1  sh 2t  2 t +  4a  2  t1

(20)

1 (t + sht ⋅ cht )tt21 4a

(21)

1 [arsh (2a (x − x MIN )) + sh (arsh (2a (x − x MIN ))) ⋅ ch (arsh (2a (x − x MIN )))]xx21 4a

(22)

x2 1  2 arsh (2a ( x − x MIN )) + 2a ( x − x MIN ) ⋅ 1 + (2a ( x − x MIN ))   x 4a  1

(23)

Lt1 t2 = Lt1 t2 = Lx1 x2 =

Lx1 x2 =

x

2 1 1 2  Lx1 x2 =  arsh (2a ( x − x MIN )) + (x − x MIN ) ⋅ 1 + (2a ( x − x MIN ))  2  4a  x1

4

(24)

1 1 arsh(2a ( x 2 − x MIN )) − arsh (2a (x1 − x MIN )) + 4a 4a 1 1 2 2 + ( x 2 − x MIN ) ⋅ 1 + (2a (x 2 − x MIN )) − ( x1 − x MIN ) ⋅ 1 + (2a ( x1 − x MIN )) 2 2

Lx1x2 =

(25)

Substituting (3) and (4) into previous expression yields (26):

Lx1x2 =

 8D S2 arsh max  S2 16 Dmax 

 8Dmax  h − h1    S S2   x 2 − 1 − 2   − arsh    16 D  S2 2 4 D max max     

 h − h1    S  x1 − 1 − 2   +   2 4 D max      2

 8D 1 S h − h1     ⋅ 1 +  max +  x 2 − 1 − 2  S2 2 2 4 Dmax   

 S h − h1     x 2 − 1 − 2  −   2 4 Dmax    

 8D 1 S h − h1     ⋅ 1 +  max −  x1 − 1 − 2   S2 2 2 4 Dmax   

 S h − h1     x1 − 1 − 2   2 4 Dmax    

(26)

2

Formula (26) is a universal one for the conductor length calculation based on the parabola model since it can be directly used for deriving the final formulas for calculations in a full inclined span, but also in a levelled span (full or its part). 2.2.

Conductor length in a full inclined span

In order to obtain the formula for the conductor length calculation in a full inclined span expression (24) can be used from the previous section, but the integral limits have to be changed into: x1=0 and x2=S. In fact, this is the special case of the span–part when the integral limits present the x–coordinates of the two suspension points of the conductor in the given span. The main steps for the determination of the final formula are the following: 2

S

 dy  L = ∫ 1 +   dx  dx  0

(27) S

1 1 2 L =  arsh(2a ( x − x MIN )) + (x − x MIN ) ⋅ 1 + (2a (x − x MIN ))  4 a 2  0

(28)

1 1 arsh (2a (S − x MIN )) − arsh (2a (0 − x MIN )) + 4a 4a 1 1 2 2 + (S − x MIN ) ⋅ 1 + (2a (S − x MIN )) − (0 − x MIN ) ⋅ 1 + (2a (0 − x MIN )) 2 2

(29)

1 1 arsh (2a (S − x MIN )) + arsh (2ax MIN ) + 4a 4a 1 1 2 2 + (S − x MIN ) ⋅ 1 + (2a (S − x MIN )) + x MIN ⋅ 1 + (2ax MIN ) 2 2

(30)

L=

L=

After substituting (3) and (4) into the previous expression it becomes (31):

L= +

 4D S2 arsh  max 16 Dmax  S

 4D  h − h1   S2 1 + 2   + arsh  max  4 Dmax   16 Dmax   S

 4D h − h1  S  1 + 2  ⋅ 1 +  max  S 4 4 Dmax  

2

 h − h1   1 − 2   +  4 D max   

 4D  h − h1   h − h1  S 1 + 2   + 1 − 2  ⋅ 1 +  max   S 4 Dmax   4 4 Dmax   

(31)

 h − h1   1 − 2  4 Dmax   

2

Equation (31) is the final formula for the conductor length calculation within the whole span. It is needed more frequently than the adequate formula for the conductor length in the part of the span given by (26). Let us mention that both (26) and (31) are obtained by the application of the same algorithm, with the use of the appropriate integral limits in two different cases. 5

2.3.

Application of 1/cosψ multiplier and its impact on the conductor length calculation

The same parabola curve in levelled and inclined spans mathematically has the same maximum sag [6], [7]. In other words if the span length and the coefficient a remain unchanged, then the change of the span inclination does not cause the change of the maximum sag of the parabola. It is different in the case of the catenary. Its maximum sag increases when the span inclination increases. This contradiction is partly compensated using 1/cosψ multiplier [7] to increase the parabola's sag in an inclined span in comparison to its sag in a levelled span. The following formula is in use for the maximum sag:

Dmax ψ =

1 ⋅ Dmax cosψ

(32)

After the application of 1/cosψ, the maximum sag in an inclined span is denoted as Dmax ψ. The multiplier 1/cosψ can be obtained by (33):

1 h −h  = 1 + tg 2 ψ = 1 +  2 1  cosψ  S 

2

(33)

The following expression is the equivalent of the previous one and thus gives equal results. However, it is not so frequent in use, due to its complexity.

 1  h − h  = ch (arsh(tgψ )) = ch  arsh 2 1   cosψ  S  

(34)

The impact of 1/cosψ can be well seen in Figure 3, which shows three conductor curves in an inclined span, i.e. the catenary and its parabolic approximation with and without the application of 1/cosψ multiplier. The three curves are denoted as:

• ycat(x) – catenary • ypar(x) – parabolic approximation of ycat(x) without the use of 1/cosψ • ypar ψ(x) – parabolic approximation of ycat(x) with the use of 1/cosψ

h2

MIN par ψ

ψ

h1 Height

MIN par MIN cat MIN par – vertex of y par ( x ) MIN par ψ – vertex of y par ψ( x ) MIN cat – vertex of y cat ( x )

y par ( x ) y par ψ ( x ) y cat ( x ) 0

S

Distance Fig. 3. Conductor curves in an inclined span

The lengths of the curves in Figure 3 can be appropriately denoted in the following way:

• Lcat – length of ycat(x) on the interval [0,S] • Lpar – length of ypar(x) on the interval [0,S] • Lpar ψ – length of ypar ψ(x) on the interval [0,S] According to Figure 3, the application of 1/cosψ ensures better parabolic approximation of the catenary. As a consequence, the length calculation is also more accurate. Two following relations concern to inclined spans with low inclination. Note that in steep spans the parabola is not used, only the catenary.

y par ( x ) > y par ψ ( x ) > ycat ( x ) 6

∀

0