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A method to calculate jump height of iced transmission lines after ice-shedding
A method to calculate jump height of iced transmission lines after ice-shedding Chuan Wu, Bo Yan, Liang Zhang, Bo Zhang, Qing Li PII: DOI: Reference:
S0165-232X(16)30004-0 doi: 10.1016/j.coldregions.2016.02.001 COLTEC 2235
To appear in:
Cold Regions Science and Technology
Received date: Revised date: Accepted date:
5 June 2015 20 January 2016 1 February 2016
Please cite this article as: Wu, Chuan, Yan, Bo, Zhang, Liang, Zhang, Bo, Li, Qing, A method to calculate jump height of iced transmission lines after ice-shedding, Cold Regions Science and Technology (2016), doi: 10.1016/j.coldregions.2016.02.001
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A method to calculate jump height of iced transmission lines after ice-shedding
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Chuan Wua, Bo Yana,b,*, Liang Zhanga, Bo Zhangc,d ,Qing Lic,d
a
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College of Aerospace Engineering, Chongqing University, Chongqing 400030, China
b
State Key Laboratory of Transmission & Distribution Equipment and Power System Safety and New
Technology, Chongqing University, Chongqing 400044, China Henan Electric Power Institute, State Grid Corporation of China, Zhengzhou 450052, China
d
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c
Key Laboratory of Transmission Line Galloping Simulation, State Grid Corporation of China,
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Zhengzhou 450052, China
* Corresponding author. Tel.: +86-23-6510-2561; fax: +86-23-6510-2561.
Email address: [email protected] (Bo Yan)
ACCEPTED MANUSCRIPT Abstract Based on the energy conservation, stress-sag relation, geometrical relation of spans and
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equilibrium of suspension insulators of a multi-span transmission line following
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ice-shedding, a theoretical method to calculate the maximum jump height of the line is
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presented. The jump heights of transmission lines with various parameters, including the number of spans, span length, conductor type, ice thickness and suspension insulator
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length, after ice-shedding are determined by the presented method and the finite element method, and the results obtained by both methods agree well. The method presented in
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this paper is more accurate than those previously proposed by other authors and more convenient than the finite element method. It is suggested that the presented method be
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employed to predict the jump height of a transmission line located in an ice zone after
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ice-shedding, which is a key factor in the design of electric insulation clearance.
Keywords: Iced transmission line, ice-shedding, jump height, theoretical method, finite
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element simulation.
1. Introduction
Ice-shedding from an iced transmission line in an ice zone under natural conditions, such as temperature rise and wind action, will cause vertical jump of the line and may lead to flashover if the phase-to-phase or phase-to-tower clearance is smaller than the tolerable insulation distance, which may jeopardize the safe operation and even result in large economic loss. Therefore, it is necessary to determine the maximum jump height of a transmission line after ice-shedding and to provide a reference for the design of the
ACCEPTED MANUSCRIPT transmission line. The jump heights of iced transmission lines after ice-shedding were investigated with
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experimental, numerical and theoretical methods by many authors. Jamaleddine et al. (1993) carried out modeling tests on a two-span reduced-scale model to study the
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dynamic responses of a transmission line after ice-shedding. Based on the direct
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observations that the loss of ice during the end of glaze ice episodes “unzips” along the length of the span rather than falling off in one piece, Van Dyke et al. (2007) carried out
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three ice-shedding tests to simulate the unzipping effect on the middle span of a test line.
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Kollár and Farzaneh (2012) investigated the vertical vibration and bundle rotation of a one-span twin bundle conductor utilizing a small-scale model in laboratory. In addition, Morgan and Swift (1964) and Meng et al. (2012) performed a series of tests on a
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five-span and a three-span transmission line sections respectively to investigate jump
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heights and dynamic responses of the conductor lines after releasing heavy weights, which were used to simulate ice loads on the middle span. With the improvement of computational mechanics and software, numerical simulation methods were used to study the dynamic responses and jump heights of transmission lines after ice-shedding. Roshan Fekr and McClure (1998) simulated the dynamic responses of transmission lines with different parameters, including ice thickness, span length and elevation difference, after ice-shedding by means of the finite element method (FEM). Kollár and Farzaneh numerically simulated the dynamic behavior of bundle conductors following ice-shedding from one sub-conductor (2008), and analyzed the displacement and tension
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of the conductor following ice-shedding propagation on a single conductor and on a
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circuit three conductors linked with inter-phase spacers (2013). Moreover, Mirshafiei et
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al. (2013) numerically investigated the dynamic responses of a conductor ice-shedding following the rupture of a conductor, and Yang et al. (2014) studied the jump heights,
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unbalanced tensions and vertical loads of a seven-span transmission line after
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ice-shedding. However, these numerical investigations focus on the dynamic responses of
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transmission lines after ice-shedding, and the calculation method of the jump height was rarely discussed. Recently, based on a lot of finite element simulations, Yan et al. (2013)
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presented a simple formula to determine the jump height of a transmission line after ice-shedding, which depends on the initial tension stress in the conductor and the sag difference of the conductor in the states before and after ice-shedding. List and Pochop
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(1963) proposed an approximate formula for jump height, which relies on the tension
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stress of the iced conductor, the Young's modulus and the weight per unit length of the conductor. Based on the unzipping ice-shedding tests, Van Dyke et al. (2007) proposed that the jump height above the conductor position without ice may be estimated to be equivalent to the initial deformation of the conductor due to the ice load. There is a little theoretical work on the jump height of a transmission line after ice-shedding. Oertli (1950) presented an integral formula, in which the displacements of the suspension points of the insulators and the tension variation of the conductor during motion are ignored, to determine the jump height. Lips (1952) derived a quadratic equation to determine jump height by the balance of the strain energy in the conductor
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and the work done by the adjacent spans, and the two suspension points of the span from
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which ice sheds are assumed to be fixed during motion. Based on the assumption that ice
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deposits only on one span and totally sheds from it in a multi-span section, Morgan and Swift (1964) gave out a theoretical method to calculate the jump height, in which tension
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variation in the conductor and vertical displacements of the suspension points are
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ignored.
In this paper, a new theoretical method to calculate the jump height of a transmission
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line after ice-shedding is presented, and the tension variation in the conductor, the displacements of suspension points of the insulators and the influence of the adjacent
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spans are taken into account at the same time. This method is verified by the finite element simulation, and it can be used to determine the jump height of a transmission line after ice-shedding, which is fundamental for the design of electric insulation clearance of
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a transmission line in an ice zone.
2. Calculation method of jump height for multi-span transmission lines 2.1. Basic assumptions and equations As indicated by Van Dyke et al. (2007) and Kollár et al. (2013), actual ice-shedding is typically progressive and they studied the dynamic responses of the conductor lines after progressive ice-shedding experimentally and numerically. However, it is difficult to determine the ice-shedding propagation process and velocity. In addition, Yan et al. (2013) numerically simulated the dynamic responses of a multi-span transmission line after partial ice-shedding from one span and indicated that the maximum jump heights of the
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conductor after partial ice shedding are smaller than that as ice sheds fully from the entire
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span. It is also difficult to determine what kind partial ice-shedding case is reasonable for
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the design of transmission lines. Therefore, the extreme case that ice suddenly sheds from the entire span is currently used to assess jump height of the conductors after
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ice-shedding.
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In the design of a multi-span transmission line, the jump heights of the lines with odd number of spans are usually considered (Morgan and Swift, 1964; Oertli, 1950; Lips,
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1952; Yan et al., 2013). As mentioned by Roshan Fekr and McClure (1998), because the maximum response calculated with equal-span section exceeds that of the unequal-span
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section, it is not necessary to consider unequal spans for the design purpose so long as equal spans of maximum length are checked. Yan et al. (2013) numerically investigated the dynamic responses of transmission lines with three, five and seven spans, and
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indicated that the maximum jump height arrives when ice-shedding takes place on the middle span, and the number of spans, elevation difference and wind load nearly do not
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affect the jump height for a multi-span line with more than three spans. Based on these conclusions, the jump height of a transmission line with five equal spans without elevation difference and wind load is studied firstly. The method is then extended to a line with any number of spans. The suspension shape of a conductor line can be described by the flat parabola as shown in Fig. 1. It is assumed that ice sheds from one span instantaneously; the conductor is always in tension state and only one loop vibration shape occurs in the first jumping up process after ice-shedding; the bending stiffness and damping of the conductor are ignored; and the elongation of the conductor is small and in elastic range.
ACCEPTED MANUSCRIPT y a
dx
d
fe
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f
c
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b o
l/2
x
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-l/2
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Fig. 1. Conductor shape with flat parabola.
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The flat parabola equation of a suspension cable is wx 2 2σ0
(1)
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y
where w is the weight per unit length and per unit area of the conductor, and σ0 is the initial horizontal stress in the conductor. Based on the flat parabola equation, the
f
wl 2 8σ0
(2)
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stress-sag relation of the conductor is given by
where l is the span length and f the sag of the conductor. The conductor length L is then determined by
Ll
w2l 3 24σ02
(3)
2.2. Model of a five-span line A five-span transmission line is shown in Fig.2. Each span has the equal span length l and there is no elevation difference in each span. Both ends of the line section are dead-ends, and they are set to be fixed in the model. The weight of the iced conductor,
ACCEPTED MANUSCRIPT which includes the self-weight of the conductor and the weight of the ice on it, per unit length and per unit area and horizontal stress in the conductor are respectively w0 and σ0. Span 3
Span 4
Span 5
l
l
l
l
l
Insulator 1
Insulator 2
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Span 2
Insulator 3
III
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II
Insulator 4
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Span 1
I
s12
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s23
Fig. 2. Three typical states of a five-span transmission line after ice-shedding at central
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span.
It is assumed that ice sheds from the middle span, i.e. span 3, with a rate of β, and three
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states I, II and III, which respectively correspond to the equilibrium state before ice-shedding, static equilibrium state after ice-shedding and maximum jump height state
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following ice-shedding of the conductor, are shown in Fig. 2. In state I, the strain energy of span 3 is the maximum and its kinetic energy is zero. When ice suddenly sheds from span 3, the conductor vertically accelerates and moves upward due to the inertial force. As the conductor passes through the position corresponding to state II, it moves up but decelerates until state III. The transmission line then oscillates and finally arrives at static state due to damping. It is noted that the suspension insulators at the two ends of span 3 swing during the jump process of the line.
2.3 Calculation of jump height of the five-span line
ACCEPTED MANUSCRIPT Due to the symmetries of the line section and the loads on it, the motion of span 1 and that of span 5 are the same after ice sheds from span 3, so are spans 2 and 4. The linear
written as
L (σ σ ) E 3 0
(4)
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L3
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elongation ΔL3 of span 3 at a time after ice-shedding relative to its initial state can be
where E is the Young's modulus and σ3 is the horizontal stress of span 3 at current time.
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On the other hand, the length of span 3 at a time can be determined by Eq. (3), so ΔL3 can
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be determined by the change of the conductor length relative to its initial length, i.e. (5)
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w32 (l 2s23 )3 w2l 3 w2 (l 2s23 )3 w02l 3 L3 [(l 2s23 ) ] [ l 0 2 ] 2s23 3 24σ0 24σ32 24σ02 24σ32
where w3 is the equivalent weight per unit length and per unit area of span 3 after ice-shedding, and s23 is the horizontal swing of suspension insulator 2, which increases
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the length of span 3. It is noted that subscript 3 represents variables corresponding to span 3, and subscript 23 represents variables corresponding to insulator 2 between spans 2 and
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3. Because the right sides of Eqs. (4) and (5) are equal to each other, the following relation is obtained
w2 (l 2s23 )3 w02l 3 L (σ3 σ0 ) 2s23 3 E 24σ32 24σ02
(6)
which is the so-called geometrical equation for span 3. Similarly, the geometrical equations for span 1 and span 2 can be obtained as the follows
w2 (l s12 )3 w02l 3 L (σ1 σ0 ) s12 0 E 24σ12 24σ02
(7)
w2 (l s23 s12 )3 w02l 3 L (σ2 σ0 ) s23 s12 0 E 24σ22 24σ02
(8)
ACCEPTED MANUSCRIPT A suspension insulator string and the conductor are connected with a clamp, which can be simplified as a mass point in the model. The forces on clamp 2 connected with
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insulator 2 are shown in Fig. 3. According to Newton’s second law, the dynamic
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equations in vertical and horizontal directions of the clamp can be written, respectively, as
(9)
T3 cos 2 TR 23 sin 2 T2 cos 2 m2 ax2
(10)
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TR 23 cos2 T3 sin 2 T2 sin 2 m2 g m2 a y2
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where m2 is the mass of clamp 2， ax2 and a y2 are its horizontal and vertical accelerations respectively，TR23 the tension in insulator 2，and the angles α2, β2 and θ2 are shown in
T3 sin 2
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Fig.3. From Figs. 2 and 3, the following relations can be obtained
w3 L3 wL A , T3 cos 2 3 A , T2 sin 2 0 2 A , T2 cos 2 2 A 2 2
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sin 2
2 s23 R 2 s23 , cos 2 R R
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where A is the cross-sectional area of the conductor, R is the length of the suspension insulator. Substitute above relations into Eqs. (9) and (10) and delete TR23 by these two equations, equation (10) can be written as
σ3 A
(w3 L3 +w0 L2 )s23 2
2 R2 s23
A σ 2 A m2 [ax2
( g a y2 ) s23 2 R 2 s23
]
(11)
Because the mass and the inertial force of the clamp are much smaller than those of the conductor during motion after ice-shedding, the mass can be ignored as discussed by Morgan and Swift (1964). Thus, Eq.(11) can be simplified as
ACCEPTED MANUSCRIPT σ 2 σ3
(w3 L3 +w0 L2 )s23 2 2 R2 s23
0
(12)
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It is noted that although the inertial forces on the clamps are ignored, the inertial force
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of the conductor included in the term w3L3, which is much larger than that of the clamp, is taken into account.
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Similarly, the horizontal dynamic equation of clamp 1 connected with insulator 1 is
(w0 L1 w0 L2 )s12 2 2 R2 s12
0
(13)
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σ1 σ2
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given by
During moving up of span 3 from state I to state III, as shown in Fig. 3, the energy
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conservation law can be expressed as,
VI 2W VIII Wg
(14)
where VI and VIII are the strain energies at state I and state III, respectively, 2W is the
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work done by spans 2 and 4 to span 3 and Wg the work done by the gravity of span 3.
y2
m2
β2 T2
Insulator 2
TR23 θ2
m2g
x2
R ht
α2 T3
III
f3t f0 H
I
Fig. 3. States I and III of span 3 after ice-shedding
ACCEPTED MANUSCRIPT By means of sag-stress relation (2), VI and VIII can be written as
σ02 w2l 4 LA AL 0 2E 128Ef 02
(15)
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σ32t w32t (l 2s23t )4 L3t A VIII AL 2E 3t 128Ef32t
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VI
(16)
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where the subscript t represents variables at state III. The work Wg is (17)
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2 Wg wg AL[ ( f 0 f3t ) ht ] 3
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where the first term inside the square bracket on the right side represents the equivalent sag difference between state I and state III. As shown in Fig. 1, the equivalent sag fe is defined by setting the equivalent rectangular area a-b-c-d be equal to the area enclosed
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by the flat parabola curve a-o-d, that is l
f el 2l ( f
2
l wx 2 wl 2 wx 2 wl 3 )dx 2l ( )dx 2 0 2 0 12 0 2 8 0
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By means of Eq. (2), it is known that
2 f 3
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fe
In addition, wg in Eq. (17) is the weight per unit length and per unit area of the iced conductor after ice-shedding, including the weight of the conductor and that of the residual ice on the conductor, and ht is the vertical displacement of the suspension point from state I to state III, which can be determined by the following geometrical relation 2 ht R R2 s23 t
(18)
Finally, the work W is given by W
s23t
0
σ3 Ads23
1 2
ht
0 w3 ALdh
(19)
ACCEPTED MANUSCRIPT Using Eqs. (6)-(8) and (12)-(13), σ3, w3 and h can be expressed as a function of s23,
s23t
0
F1(s23 )ds23
s23t
0
F2 (s23 )ds23
(20)
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W
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respectively. That means W can be described as
where F1 and F2 are functions of s23. Insert Eqs. (15)-(18) and (20) into Eq. (14), it is
s23t
0
F1(s23 )ds23
s23t
0
F2 (s23 )ds23 )
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w02l 4 LA 2( 128Ef 02
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obtained that
(21)
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w2 (l 2s23t )4 L3t A 2 2 ] 3t + wg AL[ ( f 0 f 3t ) R R 2 s23 t 2 3 128Ef3t
There are three unknowns, s23t，f3t and w3t in Eq. (21). In order to solve these variables,
can be derived out as
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additional equations are needed. Based on Eq. (2), the state equation of span 3 at state III
w3t (l 2s23t )2 8σ3t
(22)
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f3t =
written as
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Similar to Eqs. (6)-(8), the geometrical equations of spans 1, 2 and 3 at state III can be
w2 (l 2s23t )3 w02l 3 L (σ3t σ0 ) 2s23t 3t 0 E 24σ32t 24σ02 w2 (l s12t )3 w02l 3 L (σ1t σ0 ) s12t 0 0 E 24σ12t 24σ02
(23)
w2 (l s23t s12t )3 w02l 3 L (σ 2t σ0 ) s23t s12t 0 0 E 24σ 22t 24σ02
In addition, the simplified horizontal dynamic equations of the clamp 1 and 2 at state III are similar to Eqs. (12) and (13), and they are
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σ1t σ 2t
(w3t L3t + w0 L2t )s23t 2 2 R 2 s23 t (w0 L1t w0 L2t )s12t 2 2 R 2 s12 t
0
(24) 0
T
σ 2t σ3t
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In the seven equations of (21)-(24), there are seven unknowns, s12t, s23t, f3t, w3t, σ1t, σ2t
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and σ3t, which can be determined by solving these equations. Then the maximum jump height of the conductor after ice-shedding can be determined by
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2 f H f 0 ht f3t f 0 R R2 s23 t 3t
(25)
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It is very difficult to analytically solve the equations due to the integral terms in the nonlinear equation (21). However, it can be numerically resolved by “fsolve” solver in
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MATLAB software, in which the integral terms in Eq. (21) are solved by trapezoidal numerical integration method. To solve these equations, initial values of the seven variables have to be set. It is suggested that values of all the seven variables at state II,
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values.
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which can be determined by solving Eqs. (6)-(8) and (12)-(13), be selected as their initial
2.4. Calculation of jump height of an n-span line In this section, the method to determine the jump height of a transmission line with n spans as ice sheds from span r is investigated. As discussed in Section 2.3, the geometrical equation of span r at a time behaves the following form similar to Eq. (6) 2 3 Lr 0 wr20lr 03 wr (lr 0 s(r -1)r sr (r +1) ) (σ σ ) s(r -1)r sr (r +1) E r r0 24σr20 24σr2
(26)
where the subscript r represents the variables of span r, and 0 represents variables at the initial state. In addition, subscript (r-1)r represents variables corresponding to the
ACCEPTED MANUSCRIPT suspension insulator (r-1) between spans (r-1) and r. Because the number n can be an odd or even number, the geometrical equations of the ith span in the range of span 1 and span
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(r-1), and the jth span in the range of span (r+1) and span n may be different, and they
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can be written respectively as follows similar to Eqs. (7) and (8),
(27)
L j0 w2 l 3 w2j 0 (l j 0 s( j -1) j s j ( j +1) )3 (σ j σ j 0 ) s( j -1) j s j ( j +1) j 0 j20 E 24σ j 0 24σ 2j
(28)
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2 3 Li0 wi20li03 wi0 (li0 s(i-1)i si(i +1) ) (σ σ ) s(i-1)i si(i +1) E i i0 24σi20 24σi2
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It is noted that there are n geometrical equations corresponding to the n spans. The simplified horizontal dynamic equations of the ith clamp connected with the ith
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suspension insulator in the range of insulator 1 and insulator (r-2), the (r-1)th clamp, the rth clamp and the jth clamp connected with the jth suspension insulator in the range of insulator (r+1) and insulator (n-1) with the mass and inertial of the clamps ignored are
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obtained similar to Eqs. (12) and (13) σi σi +1
(wi0 Li +w(i +1)0 Li +1)si(i +1)
σ r 1 σ r
(w(r -1)0 Lr 1 +wr Lr )s(r -1)r
σr 1 σr
(wr Lr +w(r +1)0 Lr 1)sr (r +1)
σ j +1 σ j
2 Ri2(i +1) si2(i +1)
2 R(2r -1)r s(2r -1)r
2 Rr2(r +1) sr2(r +1)
(w j 0 L j +w( j +1)0 L j +1)s j ( j +1) 2 R2j ( j +1) s 2j ( j +1)
(29)
(30)
(31)
(32)
There are (n-1) simplified horizontal dynamic equations corresponding to (n-1) clamps.
ACCEPTED MANUSCRIPT In addition, the state equation of the ice-shedding span, i.e. span r, can be written as
wr (lr 0 s(r -1)r + sr (r +1) )2 8σr
(33)
VI Wr -1 Wr +1 VIII Wg
wr2 (lr 0 s(r -1)rt + sr (r +1)t )4 Lrt A
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VIII
wr20lr40 Lr 0 A 128Ef r20
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VI
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where
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Finally, the energy conservation equation of span r becomes
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fr
128Ef rt2
(34)
(35)
(36)
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For a transmission line with any number of spans and uneven span length, the displacements at the suspension points of insulator (r-1) and insulator r connecting with span r are also different. Thus, the work Wr-1 done by span (r-1) and Wr+1 done by span
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(r+1) to span r are different, and Wg, Wr-1, Wr+1 can be respectively given by
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2 1 Wg wg ALr 0[ ( f r 0 f rt ) (h(r -1)rt +hr (r +1)t )] 3 2 s( r-1) rt
0 s Wr +1 0 Wr -1
r ( r +1)t
1 2
h( r-1) rt
0 1 h σ r Adsr (r +1) 2 0
σ r Ads(r -1)r
wr ALr 0dh(r -1)r
r ( r +1)t
(37)
(38)
wr ALr 0dhr (r +1)
where h(r-1)rt and hr(r+1)t are respectively the vertical displacements of insulator (r-1) and insulator r at the maximum jump height state of span r. For a transmission line with n spans, n geometrical equations, (n-1) simplified horizontal dynamic equations, one energy conservation equation and one state equation are set up, and there are (2n+1) unknowns, including σit , from σ1t to σnt, s(i-1)it, from s12t to s(n-1)nt, frt and wrt, which can be determined by solving the (2n+1) equations. The
ACCEPTED MANUSCRIPT numerical method described in Section 2.3 can be used to solve these equations. After solution of the equations, the maximum jump height can then be determined by (39)
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1 H f r 0 (h(r -1)rt +hr (r +1)t ) f rt 2 1 f r 0 f rt ( R(r -1)r R(r -1)r 2 s(2r -1)rt Rr (r +1) Rr (r +1)2 sr2(r +1)t ) 2
3. Verification of calculation method for jump height
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The FEM has been employed to simulate the dynamic responses of transmission lines
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after ice-shedding by several authors (Roshan Fekr and McClure, 1998; Kollar and Farzaneh, 2008, 2013; Mirshafiei et al., 2013; Yang et al.,2014; Yan et al.,2013), and the
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method proposed by Yan et al. (2013) is utilized now to verify the method presented in this paper to calculate the jump height of a multi-span transmission line after ice-shedding.
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In the finite-element model of a transmission-line section, the conductor line is simplified as a cable, and the spatial two-node truss element in ABAQUS software are
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used to discretize the cable by setting its material property as “no compression”. Besides, the suspension insulator strings are modeled with the spatial two-node beam elements, and only three translation degrees of freedom of their upper ends are constrained, which allows them to swing freely whenever an unbalanced load existed. All degrees of freedom of the dead-end insulator strings are constrained. In addition, the weight of the conductor and the ice load acted on the conductor modeled by applying the equivalent density and equivalent acceleration. A five-span transmission line section with equal span length is studied. The length of each span is 500m, and the tension stress in the conductor under self-weight and ice load
ACCEPTED MANUSCRIPT is 49.09MPa. The conductor is JLHA1/G1A-1250/100, whose geometrical and physical parameters are listed in Table 1. The length of each suspension insulator is 10.225m, and
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its Young’s modulus and Poisson’s ratio are set to be 200GPa and 0.3 respectively. The
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jump heights of the middle span after ice-shedding from it are determined by the method presented in Section 2 and the FEM, and the cases that ice sheds from the middle span
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with ice-shedding rate of 50%, 60%, 80% and 100% are discussed.
Cross-sectional
(mm)
area (mm2)
47.85
1350.04
Young’s
Weight per unit
Poisson’s
modulus (MPa)
length (kg/m）
ratio
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Diameter
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Table 1 Parameters of conductor JLHA1/G1A-1250/100
65000.00
4.25
0.30
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Although the damping will attenuate vibration amplitude of the conductor line after ice-shedding, as mentioned by Van Dyke et al. (2007), it is difficult to accurately define it quantitatively now. Fortunately, the damping has little influence on the initial transient
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response, which decides the maximum jump height after ice-shedding as mentioned by
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McClure and Lapointe (2003). So the damping is ignored in the following discussion. Variation of the jump height of the conductor versus ice-shedding rate determined by the presented method and the FEM respectively are shown in Fig. 4. It can be seen that the jump heights determined by these two methods agree well, and the maximum relative error between them is only 3.25%. It is demonstrated that the presented method can be employed to determine the maximum jump height of a transmission line after ice-shedding.
ACCEPTED MANUSCRIPT 20
FEM Method in this paper
T
10
5
50
60
70
80
90
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0
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Jump height (m)
15
100
Ice-shedding rate (%)
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Fig. 4. Jump height of a typical transmission line versus ice-shedding rate determined by
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the method presented in this paper and the FEM.
4. Applicability of calculation method for jump height
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4.1 Limitation of existing theories and formula As mentioned above, several theoretical methods for the jump height of a transmission line after ice-shedding have been presented by some authors (Morgan and Swift, 1964;
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Oertli, 1950; Lips, 1952). However, in all of the theories, it is assumed that the ice
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deposits only on the middle span and sheds from it in a multi-span transmission line. Morgan and Swift (1964) proposed a method to determine jump height based on the energy equation, geometrical equation and equilibrium equation, ignoring the tension variation in the conductor and the displacements of the two suspension points of the span, from which ice sheds. In addition, in the Chinese design code for overhead transmission lines, the jump height of a transmission line after ice-shedding is estimated by the formula H m(2 l /1000)f
(40)
ACCEPTED MANUSCRIPT which is proposed in a reference book published in pre-Soviet Union (China Electric Power Press, 2003). In Eq. (40), m, ranging from 0.5 to 0.9, is a factor reflecting the
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effect of partial ice-shedding, and △f is the sag difference of the conductor between the
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two states before and after ice-shedding. Here the parameter m is set to be 1.0 because ice-shedding from the full span is considered in this case. The jump heights of some
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transmission lines with five-spans determined by Eq. (40), the method presented in this
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paper, the method presented by Morgan and Swift, and the FEM are shown in Fig. 5. Six scenarios with different span length l, different ice thickness d, and the same ice-shedding
the results are listed in Table 2.
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rate of 100% are discussed. Moreover, these corresponding values and relative errors of
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It can be seen that jump heights determined by the presented method are close to those by the FEM, and the maximum error is about 3.76%. However, the jump heights determined with the method presented by Morgan and Swift are much less than those
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with the FEM, and their maximum error is up to 48.27%. In addition, the jump heights
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determined with Eq. (40) are smaller than those by the FEM in most cases, which means that the formula proposed in the Chinese design code may underestimate the jump height, and this is probably the reason why a great number of flashover accidents induced by ice-shedding took place recently.
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FEM Method in this paper Method by Morgan and Swift Formula in Chinese design code
60
40
l=500m d=30mm
20 l=500m d=20mm
l=300m d=20mm l=400m d=20mm
0
10
20
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0
T
l=500m d=40mm
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Jump height (m)
l=500m d=50mm
30
Sag difference (m)
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Fig. 5. Jump heights of typical transmission lines after ice-shedding determined by
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different methods.
Table 2 Jump heights of typical transmission lines determined by different methods
Scenarios
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Jump height
Relative error with
(m)
NM
FEM (%)
MM
FC
FEM
NM
MM
FC
6.43
5.99
5.38
6.43
0.06
6.84
16.32
l=400m, d=20mm
11.33
9.36
7.00
10.92
3.76
14.27
35.94
l=500m, d=20mm
17.33
9.96
12.68
16.83
3.02
40.81
24.65
l=500m, d=30mm
31.32
16.71
22.83
30.45
2.88
45.13
25.03
l=500m, d=40mm
46.98
24.18
32.47
46.75
0.49
48.27
30.54
l=500m, d=50mm
58.04
29.93
40.38
57.12
1.61
47.60
29.31
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l=300m, d=20mm
Note: NM: method presented in this paper; MM: method presented by Morgan and Swift; FC: formula proposed in Chinese design code.
4.2 Applicability of the presented method It is known that structure and load parameters of a transmission line section, such as the span length, number of spans, conductor type, suspension insulator length, ice thickness and ice-shedding rate, may affect the jump height of the line after ice-shedding
ACCEPTED MANUSCRIPT (Yan et al., 2013). The jump heights of the transmission lines with different parameters after ice-shedding determined by the method presented in this paper and the FEM are
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shown in Fig. 6. It is noted that in all cases, the ice sheds from the middle span of the
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lines.
The jump heights of transmission line sections with three, five and seven spans
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respectively under different ice-shedding rates are investigated by these two methods.
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The span length and initial stress of each conductor are respectively 500m and 49.09MPa, and the ice thickness is 20mm. It can be seen from Fig. 6(a) that the jump heights
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determined by the method presented in this paper are very close to those by the FEM, and the jump height nearly does not change with the number of the spans for a line with more
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than three spans as discussed by Yan et al. (2013). The jump heights of the five-span transmission line sections with different span lengths of 300m, 400m, 500m and 600m, in which initial tension stresses are respectively
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52.42MPa, 50.19MPa, 49.09MPa and 48.48MPa, covered with 20mm thick ice after
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ice-shedding under different rates are studied by these two methods. It is observed from Fig. 6(b) that the jump heights determined by these two methods are very close to each other.
It is known that the stiffness and cross-sectional area of a conductor may affect its jump height after ice-shedding. The jump heights of the five-span transmission line sections with different types of conductors are studied by these two methods. The span length of each conductor is 500m and ice thickness is 20mm. It can be seen from Fig. 6(c) that the jump heights determined by these two methods are close to each other, and the
ACCEPTED MANUSCRIPT jump height decreases with the increase of the cross-sectional area of the conductor, which reflects its geometrical stiffness.
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The jump heights of the five-span transmission line sections, whose span length of
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each conductor is 500m, covered with different ice thickness of 20mm, 30mm, 40mm and 50mm after ice-shedding are investigated by these two methods, and the corresponding
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initial stresses in the conductor are respectively 49.09MPa, 34.92MPa, 26.69MPa and
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24.11MPa. It can be seen from Fig. 6(d) that the jump heights determined by these two methods agree well with each other.
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The jump heights of the five-span transmission line sections with suspension insulators of different lengths after ice-shedding are investigated by the two methods. The span
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length is 500m and ice thickness is 20mm. It can be seen from Fig. 6(e) that the jump heights determined by the two methods are very close to each other, and the suspension length nearly does not affect the jump height.
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Method in this paper with 50% FEM with 50% Method in this paper with 60% FEM with 60% Method in this paper with 80% FEM with 80% Method in this paper with 100% FEM with 100%
30
20
Jump height (m)
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Jump height (m)
30
20
Method in this paper with 50% FEM with 50% Method in this paper with 60% FEM with 60% Method in this paper with 80% FEM with 80% Method in this paper with 100% FEM with 100%
10
10
0 2
4
6
Number of spans
(a)
8
300
400
500
Span length (m)
(b)
600
ACCEPTED MANUSCRIPT
30
60
20
40
20
10
0 60
70
80
90
100
Ice-shedding rate (%)
50
(d)
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20
40
Method in this paper with =50% FEM with =50% Method in this paper with =60% FEM with =60% Method in this paper with =80% FEM with =80% Method in this paper with =100% FEM with =100%
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Jump height (m)
30
30
Ice thickness (mm)
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(c)
20
SC
50
Method in this paper with =50% FEM with =50% Method in this paper with =60% FEM with =60% Method in this paper with =80% FEM with =80% Method in this paper with =100% FEM with =100%
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Jump height (m)
40
Jump height (m)
50
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80 Method in this paper with conductor JL/G1A-500/65 FEM with conductor JL/G1A-500/65 Method in this paper with conductor JL/G1A-630/45 FEM with conductor JL/G1A-630/45 Method in this paper with conductor JL/G1A-800/55 FEM with conductor JL/G1A-800/55 Method in this paper with conductor JL/G2A-1000/80 FEM with conductor JL/G2A-1000/80 Method in this paper with conductor JL/G1A-1250/100 FEM with conductor JL/G1A-1250/100
10
10
20
30
(e)
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Suspension insulator length (m)
Fig. 6. Jump heights of transmission lines versus different parameters determined by the method presented in this paper and the FEM. (a) Number of spans. (b) Span length. (c) Conductor type. (d) Ice thickness. (e) Suspension insulator length.
As discussed above, the jump heights determined with the method presented in this paper are close to those with the FEM, and more accurate than those with the method by Morgan and Swift and the formula in the Chinese design code. With the method presented in this paper, the maximum jump height of a transmission line after ice-shedding can be determined easily and quickly, and it is more convenient than the
ACCEPTED MANUSCRIPT FEM because the finite element model of the transmission line is needed to be set up in the finite element simulation.
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Because the damping ratio is ignored in the derivation of the theoretical method in this
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paper, the maximum jump height may be overestimated. Fortunately, from safety point of
NU
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view, overestimation of the jump height is reliable in the design of transmission lines.
5. Conclusions
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A new theoretical method for prediction of jump heights of iced transmission lines after ice-shedding is presented based on the energy conservation, stress-sag relation,
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geometrical relation of spans and equilibrium of suspension insulators following ice-shedding. It is concluded that
1) the method presented in this paper is more accurate than those previously proposed
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by other authors, since the tension variation of the conductor, the displacements of the
account.
AC
suspension points and influence of the adjacent spans after ice-shedding are taken into
2) the jump heights of transmission lines after ice-shedding determined by the method presented in this paper are consistent with those by the finite element simulation under different parameters, and this method is more convenient than the FEM. 3) the formula proposed in the Chinese design code for transmission lines may underestimate the jump height, especially in the case of large span length, which is probably the reason why more flashover accidents induced by ice-shedding took place recently, and the method presented in this paper can be employed to easily and quickly
ACCEPTED MANUSCRIPT determine the jump height of a transmission line after ice-shedding in the design of
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electric insulation clearance.
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Acknowledgement
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This work was supported in part by the Natural Science Foundation Project of China (51277186) and in part by the Science and Technology Project of State Grid Corporation
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of China (521702140013).
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References
Jamaleddine, A., McClure, G., Rousselet, J., Beauchemin, R., 1993. Simulation of ice-shedding on electrical transmission lines using ADINA. Comput. Struct., 47 ( 4/5),523-536.
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Kollár, L.E., Farzaneh, M., 2008. Vibration of bundled conductors following ice shedding. IEEE Trans. Power Del., 23(2), 1097–1104.
Kollár, L.E., Farzaneh, M., 2012. Modeling sudden ice shedding from conductor bundles. IEEE
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Trans. Power Del., 28(2), 604-611.
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Kollár, L.E., Farzaneh, M., Van Dyke, P., 2013. Modeling ice shedding propagation on transmission lines with or without interphase spacers. IEEE Trans. Power Del., 28(1), 261 -267. Lips, K., 1952. Die Schnellhohe von freileitungsseilen nach abfallen von zusatzlastcn. Bull. Assoc. Suisse Elect., 43, 62. List, V., Pochop, K., 1963. Mechanical Design of Overhead Transmission Lines. Prague, Czechoslovakia: SNTL Publisher of Technical Literature. McClure, G., Lapointe, M., 2003. Modeling the structural dynamic response of overhead transmission lines. Comput. Struct., 81, 825–834. Meng, X.B., Wang, L.M., Hou, L., Fu, G.J., Sun, B.Q., MacAlpine, M., Hu, W., Chen, Y., 2012. Oscillation of conductors following ice-shedding on UHV transmission lines. Cold Reg. Sci. Technol., 30, 393-406.
ACCEPTED MANUSCRIPT Mirshafiei, F., McClure, G., Farzaneh, M., 2013. Modelling the dynamic response of iced transmission lines subjected to cable rupture and ice shedding. IEEE Trans. Power Del., 28(2),
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948-954.
of ice loads. Proc. Inst. Elect. Eng., 111(10), 1736-1746.
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Morgan, V. T., Swift, D. A., 1964. Jump height of overhead-line conductors after the sudden release
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Northeast Electric Power Design Institute of National Electric Power Corporation, 2003. Design manual of high voltage transmission lines for electric engineering, China Electric Power
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Press.,Beijing, China, pp.173-176.
Oertli, H., 1950. Oscillations de cables electriques aeriens apres la chute de surcharges. Bull. Assoc.
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Suisse Elect., 41,501.
Roshan Fekr, M., McClure, G., 1998. Numerical modelling of the dynamic response of ice-shedding
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on electric transmission lines. Atmos. Res., 46, 1-11. Van Dyke, P., Havard, D., Laneville, A., 2008. Effect of ice and snow on dynamics of transmission line conductors, in: Farzaneh, M. (Ed.), Atmospheric Icing of Power Networks. Springer, pp. 219-222.
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Yan, B., Chen, K.Q., Guo, Y.M., Liang, M., Yuan, Q., 2013. Numerical simulation study on jump
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height of iced transmission lines after ice-shedding. IEEE Trans. Power Del., 28(1), 216-225. Yang, F.L., Yang, J.B., Zhang, H.J., 2014. Analyzing loads from ice shedding conductors for UHV transmission towers in heavy icing areas. J. Cold Reg. Eng., 28, 04014004-1-18.
ACCEPTED MANUSCRIPT
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Highlights
>A theoretical method to calculate the maximum jump height of a multi-span
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transmission line after ice-shedding is developed.
>The presented method is verified by the finite element method and it is more accurate
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than previous methods proposed by other authors.
>The applicability of the presented method in the design of a transmission line in ice
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zone is discussed.
S0165-232X(16)30004-0 doi: 10.1016/j.coldregions.2016.02.001 COLTEC 2235
To appear in:
Cold Regions Science and Technology
Received date: Revised date: Accepted date:
5 June 2015 20 January 2016 1 February 2016
Please cite this article as: Wu, Chuan, Yan, Bo, Zhang, Liang, Zhang, Bo, Li, Qing, A method to calculate jump height of iced transmission lines after ice-shedding, Cold Regions Science and Technology (2016), doi: 10.1016/j.coldregions.2016.02.001
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ACCEPTED MANUSCRIPT
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A method to calculate jump height of iced transmission lines after ice-shedding
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Chuan Wua, Bo Yana,b,*, Liang Zhanga, Bo Zhangc,d ,Qing Lic,d
a
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College of Aerospace Engineering, Chongqing University, Chongqing 400030, China
b
State Key Laboratory of Transmission & Distribution Equipment and Power System Safety and New
Technology, Chongqing University, Chongqing 400044, China Henan Electric Power Institute, State Grid Corporation of China, Zhengzhou 450052, China
d
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c
Key Laboratory of Transmission Line Galloping Simulation, State Grid Corporation of China,
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Zhengzhou 450052, China
* Corresponding author. Tel.: +86-23-6510-2561; fax: +86-23-6510-2561.
Email address: [email protected] (Bo Yan)
ACCEPTED MANUSCRIPT Abstract Based on the energy conservation, stress-sag relation, geometrical relation of spans and
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equilibrium of suspension insulators of a multi-span transmission line following
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ice-shedding, a theoretical method to calculate the maximum jump height of the line is
SC
presented. The jump heights of transmission lines with various parameters, including the number of spans, span length, conductor type, ice thickness and suspension insulator
NU
length, after ice-shedding are determined by the presented method and the finite element method, and the results obtained by both methods agree well. The method presented in
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this paper is more accurate than those previously proposed by other authors and more convenient than the finite element method. It is suggested that the presented method be
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employed to predict the jump height of a transmission line located in an ice zone after
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ice-shedding, which is a key factor in the design of electric insulation clearance.
Keywords: Iced transmission line, ice-shedding, jump height, theoretical method, finite
AC
element simulation.
1. Introduction
Ice-shedding from an iced transmission line in an ice zone under natural conditions, such as temperature rise and wind action, will cause vertical jump of the line and may lead to flashover if the phase-to-phase or phase-to-tower clearance is smaller than the tolerable insulation distance, which may jeopardize the safe operation and even result in large economic loss. Therefore, it is necessary to determine the maximum jump height of a transmission line after ice-shedding and to provide a reference for the design of the
ACCEPTED MANUSCRIPT transmission line. The jump heights of iced transmission lines after ice-shedding were investigated with
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experimental, numerical and theoretical methods by many authors. Jamaleddine et al. (1993) carried out modeling tests on a two-span reduced-scale model to study the
SC
dynamic responses of a transmission line after ice-shedding. Based on the direct
NU
observations that the loss of ice during the end of glaze ice episodes “unzips” along the length of the span rather than falling off in one piece, Van Dyke et al. (2007) carried out
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three ice-shedding tests to simulate the unzipping effect on the middle span of a test line.
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Kollár and Farzaneh (2012) investigated the vertical vibration and bundle rotation of a one-span twin bundle conductor utilizing a small-scale model in laboratory. In addition, Morgan and Swift (1964) and Meng et al. (2012) performed a series of tests on a
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five-span and a three-span transmission line sections respectively to investigate jump
AC
heights and dynamic responses of the conductor lines after releasing heavy weights, which were used to simulate ice loads on the middle span. With the improvement of computational mechanics and software, numerical simulation methods were used to study the dynamic responses and jump heights of transmission lines after ice-shedding. Roshan Fekr and McClure (1998) simulated the dynamic responses of transmission lines with different parameters, including ice thickness, span length and elevation difference, after ice-shedding by means of the finite element method (FEM). Kollár and Farzaneh numerically simulated the dynamic behavior of bundle conductors following ice-shedding from one sub-conductor (2008), and analyzed the displacement and tension
ACCEPTED MANUSCRIPT
of the conductor following ice-shedding propagation on a single conductor and on a
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circuit three conductors linked with inter-phase spacers (2013). Moreover, Mirshafiei et
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al. (2013) numerically investigated the dynamic responses of a conductor ice-shedding following the rupture of a conductor, and Yang et al. (2014) studied the jump heights,
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unbalanced tensions and vertical loads of a seven-span transmission line after
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ice-shedding. However, these numerical investigations focus on the dynamic responses of
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transmission lines after ice-shedding, and the calculation method of the jump height was rarely discussed. Recently, based on a lot of finite element simulations, Yan et al. (2013)
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presented a simple formula to determine the jump height of a transmission line after ice-shedding, which depends on the initial tension stress in the conductor and the sag difference of the conductor in the states before and after ice-shedding. List and Pochop
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(1963) proposed an approximate formula for jump height, which relies on the tension
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stress of the iced conductor, the Young's modulus and the weight per unit length of the conductor. Based on the unzipping ice-shedding tests, Van Dyke et al. (2007) proposed that the jump height above the conductor position without ice may be estimated to be equivalent to the initial deformation of the conductor due to the ice load. There is a little theoretical work on the jump height of a transmission line after ice-shedding. Oertli (1950) presented an integral formula, in which the displacements of the suspension points of the insulators and the tension variation of the conductor during motion are ignored, to determine the jump height. Lips (1952) derived a quadratic equation to determine jump height by the balance of the strain energy in the conductor
ACCEPTED MANUSCRIPT
and the work done by the adjacent spans, and the two suspension points of the span from
T
which ice sheds are assumed to be fixed during motion. Based on the assumption that ice
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deposits only on one span and totally sheds from it in a multi-span section, Morgan and Swift (1964) gave out a theoretical method to calculate the jump height, in which tension
SC
variation in the conductor and vertical displacements of the suspension points are
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ignored.
In this paper, a new theoretical method to calculate the jump height of a transmission
MA
line after ice-shedding is presented, and the tension variation in the conductor, the displacements of suspension points of the insulators and the influence of the adjacent
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spans are taken into account at the same time. This method is verified by the finite element simulation, and it can be used to determine the jump height of a transmission line after ice-shedding, which is fundamental for the design of electric insulation clearance of
AC
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a transmission line in an ice zone.
2. Calculation method of jump height for multi-span transmission lines 2.1. Basic assumptions and equations As indicated by Van Dyke et al. (2007) and Kollár et al. (2013), actual ice-shedding is typically progressive and they studied the dynamic responses of the conductor lines after progressive ice-shedding experimentally and numerically. However, it is difficult to determine the ice-shedding propagation process and velocity. In addition, Yan et al. (2013) numerically simulated the dynamic responses of a multi-span transmission line after partial ice-shedding from one span and indicated that the maximum jump heights of the
ACCEPTED MANUSCRIPT
conductor after partial ice shedding are smaller than that as ice sheds fully from the entire
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span. It is also difficult to determine what kind partial ice-shedding case is reasonable for
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the design of transmission lines. Therefore, the extreme case that ice suddenly sheds from the entire span is currently used to assess jump height of the conductors after
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ice-shedding.
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In the design of a multi-span transmission line, the jump heights of the lines with odd number of spans are usually considered (Morgan and Swift, 1964; Oertli, 1950; Lips,
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1952; Yan et al., 2013). As mentioned by Roshan Fekr and McClure (1998), because the maximum response calculated with equal-span section exceeds that of the unequal-span
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section, it is not necessary to consider unequal spans for the design purpose so long as equal spans of maximum length are checked. Yan et al. (2013) numerically investigated the dynamic responses of transmission lines with three, five and seven spans, and
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indicated that the maximum jump height arrives when ice-shedding takes place on the middle span, and the number of spans, elevation difference and wind load nearly do not
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affect the jump height for a multi-span line with more than three spans. Based on these conclusions, the jump height of a transmission line with five equal spans without elevation difference and wind load is studied firstly. The method is then extended to a line with any number of spans. The suspension shape of a conductor line can be described by the flat parabola as shown in Fig. 1. It is assumed that ice sheds from one span instantaneously; the conductor is always in tension state and only one loop vibration shape occurs in the first jumping up process after ice-shedding; the bending stiffness and damping of the conductor are ignored; and the elongation of the conductor is small and in elastic range.
ACCEPTED MANUSCRIPT y a
dx
d
fe
T
f
c
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b o
l/2
x
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-l/2
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Fig. 1. Conductor shape with flat parabola.
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The flat parabola equation of a suspension cable is wx 2 2σ0
(1)
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y
where w is the weight per unit length and per unit area of the conductor, and σ0 is the initial horizontal stress in the conductor. Based on the flat parabola equation, the
f
wl 2 8σ0
(2)
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stress-sag relation of the conductor is given by
where l is the span length and f the sag of the conductor. The conductor length L is then determined by
Ll
w2l 3 24σ02
(3)
2.2. Model of a five-span line A five-span transmission line is shown in Fig.2. Each span has the equal span length l and there is no elevation difference in each span. Both ends of the line section are dead-ends, and they are set to be fixed in the model. The weight of the iced conductor,
ACCEPTED MANUSCRIPT which includes the self-weight of the conductor and the weight of the ice on it, per unit length and per unit area and horizontal stress in the conductor are respectively w0 and σ0. Span 3
Span 4
Span 5
l
l
l
l
l
Insulator 1
Insulator 2
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Span 2
Insulator 3
III
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II
Insulator 4
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Span 1
I
s12
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s23
Fig. 2. Three typical states of a five-span transmission line after ice-shedding at central
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span.
It is assumed that ice sheds from the middle span, i.e. span 3, with a rate of β, and three
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states I, II and III, which respectively correspond to the equilibrium state before ice-shedding, static equilibrium state after ice-shedding and maximum jump height state
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following ice-shedding of the conductor, are shown in Fig. 2. In state I, the strain energy of span 3 is the maximum and its kinetic energy is zero. When ice suddenly sheds from span 3, the conductor vertically accelerates and moves upward due to the inertial force. As the conductor passes through the position corresponding to state II, it moves up but decelerates until state III. The transmission line then oscillates and finally arrives at static state due to damping. It is noted that the suspension insulators at the two ends of span 3 swing during the jump process of the line.
2.3 Calculation of jump height of the five-span line
ACCEPTED MANUSCRIPT Due to the symmetries of the line section and the loads on it, the motion of span 1 and that of span 5 are the same after ice sheds from span 3, so are spans 2 and 4. The linear
written as
L (σ σ ) E 3 0
(4)
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L3
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elongation ΔL3 of span 3 at a time after ice-shedding relative to its initial state can be
where E is the Young's modulus and σ3 is the horizontal stress of span 3 at current time.
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On the other hand, the length of span 3 at a time can be determined by Eq. (3), so ΔL3 can
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be determined by the change of the conductor length relative to its initial length, i.e. (5)
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w32 (l 2s23 )3 w2l 3 w2 (l 2s23 )3 w02l 3 L3 [(l 2s23 ) ] [ l 0 2 ] 2s23 3 24σ0 24σ32 24σ02 24σ32
where w3 is the equivalent weight per unit length and per unit area of span 3 after ice-shedding, and s23 is the horizontal swing of suspension insulator 2, which increases
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the length of span 3. It is noted that subscript 3 represents variables corresponding to span 3, and subscript 23 represents variables corresponding to insulator 2 between spans 2 and
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3. Because the right sides of Eqs. (4) and (5) are equal to each other, the following relation is obtained
w2 (l 2s23 )3 w02l 3 L (σ3 σ0 ) 2s23 3 E 24σ32 24σ02
(6)
which is the so-called geometrical equation for span 3. Similarly, the geometrical equations for span 1 and span 2 can be obtained as the follows
w2 (l s12 )3 w02l 3 L (σ1 σ0 ) s12 0 E 24σ12 24σ02
(7)
w2 (l s23 s12 )3 w02l 3 L (σ2 σ0 ) s23 s12 0 E 24σ22 24σ02
(8)
ACCEPTED MANUSCRIPT A suspension insulator string and the conductor are connected with a clamp, which can be simplified as a mass point in the model. The forces on clamp 2 connected with
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insulator 2 are shown in Fig. 3. According to Newton’s second law, the dynamic
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equations in vertical and horizontal directions of the clamp can be written, respectively, as
(9)
T3 cos 2 TR 23 sin 2 T2 cos 2 m2 ax2
(10)
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TR 23 cos2 T3 sin 2 T2 sin 2 m2 g m2 a y2
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where m2 is the mass of clamp 2， ax2 and a y2 are its horizontal and vertical accelerations respectively，TR23 the tension in insulator 2，and the angles α2, β2 and θ2 are shown in
T3 sin 2
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Fig.3. From Figs. 2 and 3, the following relations can be obtained
w3 L3 wL A , T3 cos 2 3 A , T2 sin 2 0 2 A , T2 cos 2 2 A 2 2
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sin 2
2 s23 R 2 s23 , cos 2 R R
AC
where A is the cross-sectional area of the conductor, R is the length of the suspension insulator. Substitute above relations into Eqs. (9) and (10) and delete TR23 by these two equations, equation (10) can be written as
σ3 A
(w3 L3 +w0 L2 )s23 2
2 R2 s23
A σ 2 A m2 [ax2
( g a y2 ) s23 2 R 2 s23
]
(11)
Because the mass and the inertial force of the clamp are much smaller than those of the conductor during motion after ice-shedding, the mass can be ignored as discussed by Morgan and Swift (1964). Thus, Eq.(11) can be simplified as
ACCEPTED MANUSCRIPT σ 2 σ3
(w3 L3 +w0 L2 )s23 2 2 R2 s23
0
(12)
T
It is noted that although the inertial forces on the clamps are ignored, the inertial force
RI P
of the conductor included in the term w3L3, which is much larger than that of the clamp, is taken into account.
SC
Similarly, the horizontal dynamic equation of clamp 1 connected with insulator 1 is
(w0 L1 w0 L2 )s12 2 2 R2 s12
0
(13)
MA
σ1 σ2
NU
given by
During moving up of span 3 from state I to state III, as shown in Fig. 3, the energy
PT ED
conservation law can be expressed as,
VI 2W VIII Wg
(14)
where VI and VIII are the strain energies at state I and state III, respectively, 2W is the
AC
CE
work done by spans 2 and 4 to span 3 and Wg the work done by the gravity of span 3.
y2
m2
β2 T2
Insulator 2
TR23 θ2
m2g
x2
R ht
α2 T3
III
f3t f0 H
I
Fig. 3. States I and III of span 3 after ice-shedding
ACCEPTED MANUSCRIPT By means of sag-stress relation (2), VI and VIII can be written as
σ02 w2l 4 LA AL 0 2E 128Ef 02
(15)
RI P
σ32t w32t (l 2s23t )4 L3t A VIII AL 2E 3t 128Ef32t
T
VI
(16)
SC
where the subscript t represents variables at state III. The work Wg is (17)
NU
2 Wg wg AL[ ( f 0 f3t ) ht ] 3
MA
where the first term inside the square bracket on the right side represents the equivalent sag difference between state I and state III. As shown in Fig. 1, the equivalent sag fe is defined by setting the equivalent rectangular area a-b-c-d be equal to the area enclosed
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by the flat parabola curve a-o-d, that is l
f el 2l ( f
2
l wx 2 wl 2 wx 2 wl 3 )dx 2l ( )dx 2 0 2 0 12 0 2 8 0
CE
By means of Eq. (2), it is known that
2 f 3
AC
fe
In addition, wg in Eq. (17) is the weight per unit length and per unit area of the iced conductor after ice-shedding, including the weight of the conductor and that of the residual ice on the conductor, and ht is the vertical displacement of the suspension point from state I to state III, which can be determined by the following geometrical relation 2 ht R R2 s23 t
(18)
Finally, the work W is given by W
s23t
0
σ3 Ads23
1 2
ht
0 w3 ALdh
(19)
ACCEPTED MANUSCRIPT Using Eqs. (6)-(8) and (12)-(13), σ3, w3 and h can be expressed as a function of s23,
s23t
0
F1(s23 )ds23
s23t
0
F2 (s23 )ds23
(20)
RI P
W
T
respectively. That means W can be described as
where F1 and F2 are functions of s23. Insert Eqs. (15)-(18) and (20) into Eq. (14), it is
s23t
0
F1(s23 )ds23
s23t
0
F2 (s23 )ds23 )
NU
w02l 4 LA 2( 128Ef 02
SC
obtained that
(21)
MA
w2 (l 2s23t )4 L3t A 2 2 ] 3t + wg AL[ ( f 0 f 3t ) R R 2 s23 t 2 3 128Ef3t
There are three unknowns, s23t，f3t and w3t in Eq. (21). In order to solve these variables,
can be derived out as
PT ED
additional equations are needed. Based on Eq. (2), the state equation of span 3 at state III
w3t (l 2s23t )2 8σ3t
(22)
CE
f3t =
written as
AC
Similar to Eqs. (6)-(8), the geometrical equations of spans 1, 2 and 3 at state III can be
w2 (l 2s23t )3 w02l 3 L (σ3t σ0 ) 2s23t 3t 0 E 24σ32t 24σ02 w2 (l s12t )3 w02l 3 L (σ1t σ0 ) s12t 0 0 E 24σ12t 24σ02
(23)
w2 (l s23t s12t )3 w02l 3 L (σ 2t σ0 ) s23t s12t 0 0 E 24σ 22t 24σ02
In addition, the simplified horizontal dynamic equations of the clamp 1 and 2 at state III are similar to Eqs. (12) and (13), and they are
ACCEPTED MANUSCRIPT
σ1t σ 2t
(w3t L3t + w0 L2t )s23t 2 2 R 2 s23 t (w0 L1t w0 L2t )s12t 2 2 R 2 s12 t
0
(24) 0
T
σ 2t σ3t
RI P
In the seven equations of (21)-(24), there are seven unknowns, s12t, s23t, f3t, w3t, σ1t, σ2t
SC
and σ3t, which can be determined by solving these equations. Then the maximum jump height of the conductor after ice-shedding can be determined by
NU
2 f H f 0 ht f3t f 0 R R2 s23 t 3t
(25)
MA
It is very difficult to analytically solve the equations due to the integral terms in the nonlinear equation (21). However, it can be numerically resolved by “fsolve” solver in
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MATLAB software, in which the integral terms in Eq. (21) are solved by trapezoidal numerical integration method. To solve these equations, initial values of the seven variables have to be set. It is suggested that values of all the seven variables at state II,
AC
values.
CE
which can be determined by solving Eqs. (6)-(8) and (12)-(13), be selected as their initial
2.4. Calculation of jump height of an n-span line In this section, the method to determine the jump height of a transmission line with n spans as ice sheds from span r is investigated. As discussed in Section 2.3, the geometrical equation of span r at a time behaves the following form similar to Eq. (6) 2 3 Lr 0 wr20lr 03 wr (lr 0 s(r -1)r sr (r +1) ) (σ σ ) s(r -1)r sr (r +1) E r r0 24σr20 24σr2
(26)
where the subscript r represents the variables of span r, and 0 represents variables at the initial state. In addition, subscript (r-1)r represents variables corresponding to the
ACCEPTED MANUSCRIPT suspension insulator (r-1) between spans (r-1) and r. Because the number n can be an odd or even number, the geometrical equations of the ith span in the range of span 1 and span
T
(r-1), and the jth span in the range of span (r+1) and span n may be different, and they
RI P
can be written respectively as follows similar to Eqs. (7) and (8),
(27)
L j0 w2 l 3 w2j 0 (l j 0 s( j -1) j s j ( j +1) )3 (σ j σ j 0 ) s( j -1) j s j ( j +1) j 0 j20 E 24σ j 0 24σ 2j
(28)
NU
SC
2 3 Li0 wi20li03 wi0 (li0 s(i-1)i si(i +1) ) (σ σ ) s(i-1)i si(i +1) E i i0 24σi20 24σi2
MA
It is noted that there are n geometrical equations corresponding to the n spans. The simplified horizontal dynamic equations of the ith clamp connected with the ith
PT ED
suspension insulator in the range of insulator 1 and insulator (r-2), the (r-1)th clamp, the rth clamp and the jth clamp connected with the jth suspension insulator in the range of insulator (r+1) and insulator (n-1) with the mass and inertial of the clamps ignored are
AC
CE
obtained similar to Eqs. (12) and (13) σi σi +1
(wi0 Li +w(i +1)0 Li +1)si(i +1)
σ r 1 σ r
(w(r -1)0 Lr 1 +wr Lr )s(r -1)r
σr 1 σr
(wr Lr +w(r +1)0 Lr 1)sr (r +1)
σ j +1 σ j
2 Ri2(i +1) si2(i +1)
2 R(2r -1)r s(2r -1)r
2 Rr2(r +1) sr2(r +1)
(w j 0 L j +w( j +1)0 L j +1)s j ( j +1) 2 R2j ( j +1) s 2j ( j +1)
(29)
(30)
(31)
(32)
There are (n-1) simplified horizontal dynamic equations corresponding to (n-1) clamps.
ACCEPTED MANUSCRIPT In addition, the state equation of the ice-shedding span, i.e. span r, can be written as
wr (lr 0 s(r -1)r + sr (r +1) )2 8σr
(33)
VI Wr -1 Wr +1 VIII Wg
wr2 (lr 0 s(r -1)rt + sr (r +1)t )4 Lrt A
MA
VIII
wr20lr40 Lr 0 A 128Ef r20
NU
VI
SC
where
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Finally, the energy conservation equation of span r becomes
T
fr
128Ef rt2
(34)
(35)
(36)
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For a transmission line with any number of spans and uneven span length, the displacements at the suspension points of insulator (r-1) and insulator r connecting with span r are also different. Thus, the work Wr-1 done by span (r-1) and Wr+1 done by span
CE
(r+1) to span r are different, and Wg, Wr-1, Wr+1 can be respectively given by
AC
2 1 Wg wg ALr 0[ ( f r 0 f rt ) (h(r -1)rt +hr (r +1)t )] 3 2 s( r-1) rt
0 s Wr +1 0 Wr -1
r ( r +1)t
1 2
h( r-1) rt
0 1 h σ r Adsr (r +1) 2 0
σ r Ads(r -1)r
wr ALr 0dh(r -1)r
r ( r +1)t
(37)
(38)
wr ALr 0dhr (r +1)
where h(r-1)rt and hr(r+1)t are respectively the vertical displacements of insulator (r-1) and insulator r at the maximum jump height state of span r. For a transmission line with n spans, n geometrical equations, (n-1) simplified horizontal dynamic equations, one energy conservation equation and one state equation are set up, and there are (2n+1) unknowns, including σit , from σ1t to σnt, s(i-1)it, from s12t to s(n-1)nt, frt and wrt, which can be determined by solving the (2n+1) equations. The
ACCEPTED MANUSCRIPT numerical method described in Section 2.3 can be used to solve these equations. After solution of the equations, the maximum jump height can then be determined by (39)
SC
RI P
T
1 H f r 0 (h(r -1)rt +hr (r +1)t ) f rt 2 1 f r 0 f rt ( R(r -1)r R(r -1)r 2 s(2r -1)rt Rr (r +1) Rr (r +1)2 sr2(r +1)t ) 2
3. Verification of calculation method for jump height
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The FEM has been employed to simulate the dynamic responses of transmission lines
MA
after ice-shedding by several authors (Roshan Fekr and McClure, 1998; Kollar and Farzaneh, 2008, 2013; Mirshafiei et al., 2013; Yang et al.,2014; Yan et al.,2013), and the
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method proposed by Yan et al. (2013) is utilized now to verify the method presented in this paper to calculate the jump height of a multi-span transmission line after ice-shedding.
CE
In the finite-element model of a transmission-line section, the conductor line is simplified as a cable, and the spatial two-node truss element in ABAQUS software are
AC
used to discretize the cable by setting its material property as “no compression”. Besides, the suspension insulator strings are modeled with the spatial two-node beam elements, and only three translation degrees of freedom of their upper ends are constrained, which allows them to swing freely whenever an unbalanced load existed. All degrees of freedom of the dead-end insulator strings are constrained. In addition, the weight of the conductor and the ice load acted on the conductor modeled by applying the equivalent density and equivalent acceleration. A five-span transmission line section with equal span length is studied. The length of each span is 500m, and the tension stress in the conductor under self-weight and ice load
ACCEPTED MANUSCRIPT is 49.09MPa. The conductor is JLHA1/G1A-1250/100, whose geometrical and physical parameters are listed in Table 1. The length of each suspension insulator is 10.225m, and
T
its Young’s modulus and Poisson’s ratio are set to be 200GPa and 0.3 respectively. The
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jump heights of the middle span after ice-shedding from it are determined by the method presented in Section 2 and the FEM, and the cases that ice sheds from the middle span
SC
with ice-shedding rate of 50%, 60%, 80% and 100% are discussed.
Cross-sectional
(mm)
area (mm2)
47.85
1350.04
Young’s
Weight per unit
Poisson’s
modulus (MPa)
length (kg/m）
ratio
MA
Diameter
NU
Table 1 Parameters of conductor JLHA1/G1A-1250/100
65000.00
4.25
0.30
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Although the damping will attenuate vibration amplitude of the conductor line after ice-shedding, as mentioned by Van Dyke et al. (2007), it is difficult to accurately define it quantitatively now. Fortunately, the damping has little influence on the initial transient
CE
response, which decides the maximum jump height after ice-shedding as mentioned by
AC
McClure and Lapointe (2003). So the damping is ignored in the following discussion. Variation of the jump height of the conductor versus ice-shedding rate determined by the presented method and the FEM respectively are shown in Fig. 4. It can be seen that the jump heights determined by these two methods agree well, and the maximum relative error between them is only 3.25%. It is demonstrated that the presented method can be employed to determine the maximum jump height of a transmission line after ice-shedding.
ACCEPTED MANUSCRIPT 20
FEM Method in this paper
T
10
5
50
60
70
80
90
SC
0
RI P
Jump height (m)
15
100
Ice-shedding rate (%)
NU
Fig. 4. Jump height of a typical transmission line versus ice-shedding rate determined by
MA
the method presented in this paper and the FEM.
4. Applicability of calculation method for jump height
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4.1 Limitation of existing theories and formula As mentioned above, several theoretical methods for the jump height of a transmission line after ice-shedding have been presented by some authors (Morgan and Swift, 1964;
CE
Oertli, 1950; Lips, 1952). However, in all of the theories, it is assumed that the ice
AC
deposits only on the middle span and sheds from it in a multi-span transmission line. Morgan and Swift (1964) proposed a method to determine jump height based on the energy equation, geometrical equation and equilibrium equation, ignoring the tension variation in the conductor and the displacements of the two suspension points of the span, from which ice sheds. In addition, in the Chinese design code for overhead transmission lines, the jump height of a transmission line after ice-shedding is estimated by the formula H m(2 l /1000)f
(40)
ACCEPTED MANUSCRIPT which is proposed in a reference book published in pre-Soviet Union (China Electric Power Press, 2003). In Eq. (40), m, ranging from 0.5 to 0.9, is a factor reflecting the
T
effect of partial ice-shedding, and △f is the sag difference of the conductor between the
RI P
two states before and after ice-shedding. Here the parameter m is set to be 1.0 because ice-shedding from the full span is considered in this case. The jump heights of some
SC
transmission lines with five-spans determined by Eq. (40), the method presented in this
NU
paper, the method presented by Morgan and Swift, and the FEM are shown in Fig. 5. Six scenarios with different span length l, different ice thickness d, and the same ice-shedding
the results are listed in Table 2.
MA
rate of 100% are discussed. Moreover, these corresponding values and relative errors of
PT ED
It can be seen that jump heights determined by the presented method are close to those by the FEM, and the maximum error is about 3.76%. However, the jump heights determined with the method presented by Morgan and Swift are much less than those
CE
with the FEM, and their maximum error is up to 48.27%. In addition, the jump heights
AC
determined with Eq. (40) are smaller than those by the FEM in most cases, which means that the formula proposed in the Chinese design code may underestimate the jump height, and this is probably the reason why a great number of flashover accidents induced by ice-shedding took place recently.
ACCEPTED MANUSCRIPT
FEM Method in this paper Method by Morgan and Swift Formula in Chinese design code
60
40
l=500m d=30mm
20 l=500m d=20mm
l=300m d=20mm l=400m d=20mm
0
10
20
SC
0
T
l=500m d=40mm
RI P
Jump height (m)
l=500m d=50mm
30
Sag difference (m)
NU
Fig. 5. Jump heights of typical transmission lines after ice-shedding determined by
MA
different methods.
Table 2 Jump heights of typical transmission lines determined by different methods
Scenarios
PT ED
Jump height
Relative error with
(m)
NM
FEM (%)
MM
FC
FEM
NM
MM
FC
6.43
5.99
5.38
6.43
0.06
6.84
16.32
l=400m, d=20mm
11.33
9.36
7.00
10.92
3.76
14.27
35.94
l=500m, d=20mm
17.33
9.96
12.68
16.83
3.02
40.81
24.65
l=500m, d=30mm
31.32
16.71
22.83
30.45
2.88
45.13
25.03
l=500m, d=40mm
46.98
24.18
32.47
46.75
0.49
48.27
30.54
l=500m, d=50mm
58.04
29.93
40.38
57.12
1.61
47.60
29.31
AC
CE
l=300m, d=20mm
Note: NM: method presented in this paper; MM: method presented by Morgan and Swift; FC: formula proposed in Chinese design code.
4.2 Applicability of the presented method It is known that structure and load parameters of a transmission line section, such as the span length, number of spans, conductor type, suspension insulator length, ice thickness and ice-shedding rate, may affect the jump height of the line after ice-shedding
ACCEPTED MANUSCRIPT (Yan et al., 2013). The jump heights of the transmission lines with different parameters after ice-shedding determined by the method presented in this paper and the FEM are
T
shown in Fig. 6. It is noted that in all cases, the ice sheds from the middle span of the
RI P
lines.
The jump heights of transmission line sections with three, five and seven spans
SC
respectively under different ice-shedding rates are investigated by these two methods.
NU
The span length and initial stress of each conductor are respectively 500m and 49.09MPa, and the ice thickness is 20mm. It can be seen from Fig. 6(a) that the jump heights
MA
determined by the method presented in this paper are very close to those by the FEM, and the jump height nearly does not change with the number of the spans for a line with more
PT ED
than three spans as discussed by Yan et al. (2013). The jump heights of the five-span transmission line sections with different span lengths of 300m, 400m, 500m and 600m, in which initial tension stresses are respectively
CE
52.42MPa, 50.19MPa, 49.09MPa and 48.48MPa, covered with 20mm thick ice after
AC
ice-shedding under different rates are studied by these two methods. It is observed from Fig. 6(b) that the jump heights determined by these two methods are very close to each other.
It is known that the stiffness and cross-sectional area of a conductor may affect its jump height after ice-shedding. The jump heights of the five-span transmission line sections with different types of conductors are studied by these two methods. The span length of each conductor is 500m and ice thickness is 20mm. It can be seen from Fig. 6(c) that the jump heights determined by these two methods are close to each other, and the
ACCEPTED MANUSCRIPT jump height decreases with the increase of the cross-sectional area of the conductor, which reflects its geometrical stiffness.
T
The jump heights of the five-span transmission line sections, whose span length of
RI P
each conductor is 500m, covered with different ice thickness of 20mm, 30mm, 40mm and 50mm after ice-shedding are investigated by these two methods, and the corresponding
SC
initial stresses in the conductor are respectively 49.09MPa, 34.92MPa, 26.69MPa and
NU
24.11MPa. It can be seen from Fig. 6(d) that the jump heights determined by these two methods agree well with each other.
MA
The jump heights of the five-span transmission line sections with suspension insulators of different lengths after ice-shedding are investigated by the two methods. The span
PT ED
length is 500m and ice thickness is 20mm. It can be seen from Fig. 6(e) that the jump heights determined by the two methods are very close to each other, and the suspension length nearly does not affect the jump height.
CE
Method in this paper with 50% FEM with 50% Method in this paper with 60% FEM with 60% Method in this paper with 80% FEM with 80% Method in this paper with 100% FEM with 100%
30
20
Jump height (m)
AC
Jump height (m)
30
20
Method in this paper with 50% FEM with 50% Method in this paper with 60% FEM with 60% Method in this paper with 80% FEM with 80% Method in this paper with 100% FEM with 100%
10
10
0 2
4
6
Number of spans
(a)
8
300
400
500
Span length (m)
(b)
600
ACCEPTED MANUSCRIPT
30
60
20
40
20
10
0 60
70
80
90
100
Ice-shedding rate (%)
50
(d)
MA
20
40
Method in this paper with =50% FEM with =50% Method in this paper with =60% FEM with =60% Method in this paper with =80% FEM with =80% Method in this paper with =100% FEM with =100%
PT ED
Jump height (m)
30
30
Ice thickness (mm)
NU
(c)
20
SC
50
Method in this paper with =50% FEM with =50% Method in this paper with =60% FEM with =60% Method in this paper with =80% FEM with =80% Method in this paper with =100% FEM with =100%
T
Jump height (m)
40
Jump height (m)
50
RI P
80 Method in this paper with conductor JL/G1A-500/65 FEM with conductor JL/G1A-500/65 Method in this paper with conductor JL/G1A-630/45 FEM with conductor JL/G1A-630/45 Method in this paper with conductor JL/G1A-800/55 FEM with conductor JL/G1A-800/55 Method in this paper with conductor JL/G2A-1000/80 FEM with conductor JL/G2A-1000/80 Method in this paper with conductor JL/G1A-1250/100 FEM with conductor JL/G1A-1250/100
10
10
20
30
(e)
AC
CE
Suspension insulator length (m)
Fig. 6. Jump heights of transmission lines versus different parameters determined by the method presented in this paper and the FEM. (a) Number of spans. (b) Span length. (c) Conductor type. (d) Ice thickness. (e) Suspension insulator length.
As discussed above, the jump heights determined with the method presented in this paper are close to those with the FEM, and more accurate than those with the method by Morgan and Swift and the formula in the Chinese design code. With the method presented in this paper, the maximum jump height of a transmission line after ice-shedding can be determined easily and quickly, and it is more convenient than the
ACCEPTED MANUSCRIPT FEM because the finite element model of the transmission line is needed to be set up in the finite element simulation.
T
Because the damping ratio is ignored in the derivation of the theoretical method in this
RI P
paper, the maximum jump height may be overestimated. Fortunately, from safety point of
NU
SC
view, overestimation of the jump height is reliable in the design of transmission lines.
5. Conclusions
MA
A new theoretical method for prediction of jump heights of iced transmission lines after ice-shedding is presented based on the energy conservation, stress-sag relation,
PT ED
geometrical relation of spans and equilibrium of suspension insulators following ice-shedding. It is concluded that
1) the method presented in this paper is more accurate than those previously proposed
CE
by other authors, since the tension variation of the conductor, the displacements of the
account.
AC
suspension points and influence of the adjacent spans after ice-shedding are taken into
2) the jump heights of transmission lines after ice-shedding determined by the method presented in this paper are consistent with those by the finite element simulation under different parameters, and this method is more convenient than the FEM. 3) the formula proposed in the Chinese design code for transmission lines may underestimate the jump height, especially in the case of large span length, which is probably the reason why more flashover accidents induced by ice-shedding took place recently, and the method presented in this paper can be employed to easily and quickly
ACCEPTED MANUSCRIPT determine the jump height of a transmission line after ice-shedding in the design of
T
electric insulation clearance.
RI P
Acknowledgement
SC
This work was supported in part by the Natural Science Foundation Project of China (51277186) and in part by the Science and Technology Project of State Grid Corporation
NU
of China (521702140013).
MA
References
Jamaleddine, A., McClure, G., Rousselet, J., Beauchemin, R., 1993. Simulation of ice-shedding on electrical transmission lines using ADINA. Comput. Struct., 47 ( 4/5),523-536.
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Kollár, L.E., Farzaneh, M., 2008. Vibration of bundled conductors following ice shedding. IEEE Trans. Power Del., 23(2), 1097–1104.
Kollár, L.E., Farzaneh, M., 2012. Modeling sudden ice shedding from conductor bundles. IEEE
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Trans. Power Del., 28(2), 604-611.
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Kollár, L.E., Farzaneh, M., Van Dyke, P., 2013. Modeling ice shedding propagation on transmission lines with or without interphase spacers. IEEE Trans. Power Del., 28(1), 261 -267. Lips, K., 1952. Die Schnellhohe von freileitungsseilen nach abfallen von zusatzlastcn. Bull. Assoc. Suisse Elect., 43, 62. List, V., Pochop, K., 1963. Mechanical Design of Overhead Transmission Lines. Prague, Czechoslovakia: SNTL Publisher of Technical Literature. McClure, G., Lapointe, M., 2003. Modeling the structural dynamic response of overhead transmission lines. Comput. Struct., 81, 825–834. Meng, X.B., Wang, L.M., Hou, L., Fu, G.J., Sun, B.Q., MacAlpine, M., Hu, W., Chen, Y., 2012. Oscillation of conductors following ice-shedding on UHV transmission lines. Cold Reg. Sci. Technol., 30, 393-406.
ACCEPTED MANUSCRIPT Mirshafiei, F., McClure, G., Farzaneh, M., 2013. Modelling the dynamic response of iced transmission lines subjected to cable rupture and ice shedding. IEEE Trans. Power Del., 28(2),
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948-954.
of ice loads. Proc. Inst. Elect. Eng., 111(10), 1736-1746.
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Morgan, V. T., Swift, D. A., 1964. Jump height of overhead-line conductors after the sudden release
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Northeast Electric Power Design Institute of National Electric Power Corporation, 2003. Design manual of high voltage transmission lines for electric engineering, China Electric Power
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Press.,Beijing, China, pp.173-176.
Oertli, H., 1950. Oscillations de cables electriques aeriens apres la chute de surcharges. Bull. Assoc.
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Suisse Elect., 41,501.
Roshan Fekr, M., McClure, G., 1998. Numerical modelling of the dynamic response of ice-shedding
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on electric transmission lines. Atmos. Res., 46, 1-11. Van Dyke, P., Havard, D., Laneville, A., 2008. Effect of ice and snow on dynamics of transmission line conductors, in: Farzaneh, M. (Ed.), Atmospheric Icing of Power Networks. Springer, pp. 219-222.
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Yan, B., Chen, K.Q., Guo, Y.M., Liang, M., Yuan, Q., 2013. Numerical simulation study on jump
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height of iced transmission lines after ice-shedding. IEEE Trans. Power Del., 28(1), 216-225. Yang, F.L., Yang, J.B., Zhang, H.J., 2014. Analyzing loads from ice shedding conductors for UHV transmission towers in heavy icing areas. J. Cold Reg. Eng., 28, 04014004-1-18.
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Highlights
>A theoretical method to calculate the maximum jump height of a multi-span
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transmission line after ice-shedding is developed.
>The presented method is verified by the finite element method and it is more accurate
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than previous methods proposed by other authors.
>The applicability of the presented method in the design of a transmission line in ice
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zone is discussed.