Elektrotehniški vestnik 85(5): 286-293, 2018 Original scientific paper Calculation of the insulator-set displacement induced by the ice and/or wind loads using 3D catenary equations Borut Zemljaric Design Engineering Department, Elektro Gorenjska Company, Ulica Mirka Vadnova 3a, 4000 Kranj, Slovenia E-posta: borut.zemljaric@elektro-gorenjska.si Abstract. This paper presents a mathematical method to calculate the displacement of overhead-line conductors and suspension insulator sets during changeable conditions caused by ice and/or wind. The method which is consistent and mathematically transparent is primarily intended to be used in a 3D space to determine the initial conditions for a later more advanced dynamic calculation based on the finite element method. The method is based on the 3D catenary differential equations which for appropriate boundary conditions and insulator-set motion constraints form a mathematical system of algebraic-differential equations that are solved numerically. A numerical example shows that the method can be used with confidence as a simple alternative to other static methods or commercial softwares. Using the method, a wide spectrum of practical design problems associated with the overhead-line and ice and/or wind loads can be analyzed. Keywords: insulators, displacements, transmission, overhead line, catenary, numerical method Izračun pomikov izolatorjev s tridimenzionalnimi enačbami verižnice ob delovanju žlednih in vetrnih obtežb Članek predstavlja matematično metodo za izračun pomikov izolatorskih sestavov in stanj verižnice visokonapetostnih daljnovodov, nastalih zaradi delovanja zunanjih vremenskih dejavnikov, predvsem žledne ali vetrne obtežbe na posamezne elemente daljnovoda. Predstavljena metoda je matematično transparentna in je namenjena predvsem za uporabo pri 3-D prostorskih problemih, ko moramo določiti začetne pogoje za reševanje zahtevnejših dinamične problemov. Ti po navadi temeljijo na metodi končnih elementov. Za ustrezno reševanje dinamičnih numeričnih integracij so natančni začetni pogoji nujni pogoj za doseganje konvergenc rešitev. Predstavljena matematična metoda temelji na tridimenzionalnih diferencialnih enačbah verižnice, ki skupaj z ustreznimi robnimi pogoji in enačbami mehanskih omejitev gibanja izolatorskih sestavov skupaj tvorijo sistem algebrajsko-diferencialnih enačb, ki ga moramo rešiti. Predstavljeni numerični rezultati kažejo, da je predstavljena metoda zanesljiva, in jo je mogoče uporabiti kot preprosto alternativo drugim metodam oziroma plačljivim komercialnim programskim paketom.. Ključne besede: izolatorji, pomiki, prenos, verižnica, numerične metode Nomenclature A- Conductor cross-section a -Span length b0 -Perpendicular distance CE- Euler-transformation matrix Cw-Wind-transformation matrix Cx- Vee-transformation matrix Cat, Qn, Q&- Non-dimensional drag coefficients d0- Conductor sag at a/2 Dc -Conductor diameter Da - Ice-load thickness on conductor E - Young modulus of elasticity F- Force vector Fti, Fni, Fbi- Wind-force components g- Gravitational constant h -Vertical height distance H- Horizontal conductor force Hl , Hr - Horizontal force in the left- and right-hand side suspension point ]k - I insulator-set weight )x, ]z - Vee insulator-set weight components L0- Conductor stressed length Lu- Conductor unstressed length Lk, Lp, Lb - Hanging I, post and brace insulator lengths mfc, mp, mb- Hanging I, post and brace insulator masses mc -Conductor mass per length PL PR - - Perpendicular force in left- and right-hand side suspension point rx, rz - Vee insulator gravity center components s- Conductor unstressed material point Td - Drag force vector Tl- Conductor tension vector Th - Conductor tension tangential component t,n,b- Local conductor frame of reference vectors Received 13 August 2018 Accepted 5 October 2018 CALCULATION OF THE INSULATOR-SET DISPLACEMENT INDUCED BY THE ICE AND/OR WIND LOADS USING 3D. 287 U -Absolute wind velocity vr -Wind velocity vector vrt, vm, vrb - Wind components' relative velocity V- Vertical force , Vr - Vertical force at left- or right-hand side suspension point wL -Ice weight x, y, z-Coordinates in global frame of reference XYZ- Global frame of reference X1Y1Z1- Local frame of reference Vee insulator-set a-Angle between wind velocity and y,z plane A- Displacements vector S, £, tj -Components of displacements vector p- Air density 9, (p, Euler angles k - Vector of geometric span data 1 Introduction The development of new compact overhead powertransmission lines (OL) or reconstruction of the existing ones to decrease their visual impact on the landscape remains to be a very important aspect of the design process. One of the possible solutions involves the use of different types of insulator systems. Speaking in terms of the mechanical strength, it is important that when analyzing and designing such OLs, conductor sags, internal and external safety electrical distances and insulator shape, determined by using an appropriate design tool are considered in the calculation process. While the OL static design is well known and supported by commercial software, to enable OL dynamic analysis, OL dynamic models must in most cases be developed and transformed to the software. Dynamic models based on the finite element methods (FEM) enabling a dynamic simulation, the initial conditions must be implemented. Usually, the commercial software is a closed black box from which the initial conditions needed for a dynamic model cannot be directly obtained. The paper presents a new numerical-mathematical approach to solving this issue. The method provides a tool to consider most of the used OL designs needing no special commercial software to establish the initial conditions to be used in dynamic simulations and analyses. This paper deals with the problem of calculating the suspension-insulator sets displacement, caused by ice or wind loads or a combination of both. These loads can affect one or more spans of the high-voltage OL. An external load affecting a conductor changes its tensile force manifested as a displacement of a suspension-insulator set. When on insulator-set moves in a span with an increased conductor tensile force, the conductor force tends to reduce the span length and increases it in the adjacent spans. From its neutral position, the insulator set rotates to a new position, thus causing a conductor clip-in point displacement and giving rise to a new conductor-insulator force equilibrium inside the OL tension field. No matter how simple is a vertically hanging insulator set, the calculation of a relatively simple problem is difficult as in addition to the ice load, the wind load too, must be taken into account. The impact of the wind load on the conductor turns a 2D problem into a three-dimensional (3D) problem where the catenary equation is not fully solved. The paper presents a consistent and mathematically transparent method to calculate the conductor displacements and sags induced by different ice or wind for an arbitrary shape of the insulator set. Today, the usual approach to the problem solution and providing a corresponding software tool is using a method based on the conductor analytical catenary equation solved in 2D. The method is well presented by Kiesling et al. in [1]. They provide a physical understanding of the problem without considering the wind load in the mathematical model. Though not directly connected to our problem here, classical approaches to considering wind loads on a conductor are well presented by Peyrot et.al. [2]. Here in the development of a cable element, the solved catenary equation in 2D is still used. In the model, the conductor is virtually segmented into several elements and the wind pressure assumed to affect each element is constant and equal to that at the mid-length of the effected element. This concept is used in a common software tool [3]. The third concept comes from the field of dynamic problems where the commercial software, is based on the classical finite-element method (FEM). It used by Yan et al. [4] and McClure et al [5]. The conductor is divided into finite elements presented as beam elements by adding external ice and wind forces. Unlike the existing methods which use an analytical conductor catenary solution in a 2D space or divide conductors into smaller elements, our method is directly based on differential cable equations in a 3D space. In searching for a numerical solution, we use the today's standard PCs and their numerical capabilities. We developed a relatively simple mathematical method using single-span differential equations developed by Bliek [6]. For a multi-span conductor we developed a system of algebraic differential equations (ADEs) that mathematically describe the OL tension field. Another advantage of this method is that it can be used for any arbitrary insulator-set geometry. The paper is organized as follows. Following this introduction, section 2 presents the span-displacement equilibrium equations used as the boundary conditions to solve the ADEs. Section 3 provides the conductor 3D governing static differential equations and includes the wind and ice loads as part of the conductor governing equations. Section 4 introduces the movement constraints expressed with displacement equations for a particular insulator set. Equations for the two most common insulator sets are presented. The first one is the classical vertical hanging I insulator and the second one 288 ZEMLJARIC is the rotating Vee brace insulator set [7, 8]. Section 5 presents a numerical example and draws conclusions. 2 Span-displacement equilibrium equations Let us consider that the OL consists of n spans. The conductor tension field is divided into n spans with suspension sets i = 1,2, ..,n-l, mounted on supporting towers. The first and the last tower are tension towers with a tension set. A single span with index i has a span length of a0i, span height difference of h0i, and span perpendicular distance of b0i, as presented in Fig. 1. A global XYZ frame of reference is used. The subscript 0 indicates the OL initial state. This means that there are no external forces relating due to the ice or wind loads and the conductor horizontal force which is equal for all n-spans is H0. The initial span length and the vertical and perpendicular distances between the suspension points in the i-th span, written as a vector, are K0i = Ki h0i boi]T . (1) Consequently, at no ice or wind loads, all the insulator strings inside the tension field lie in the initial position. This state is taken as a reference position with the displacements equaling zero. Note that the conductors are clipped-in in the insulator sets. In Fig. 1, the initial states are shawn with a continuous catenary line. Figure 1. OL profile in the initial (continuous line) and displaced (dash line) state. accuracy, the parabola equation [9]. In the initial state, besides to the conductor horizontal tension force, only the gravitational force, determines the catenary shape inside individual spans. For OLs where the catenary profile is usually flat, the sag-to-span ratio is 1:8 or less, so for maximum conductor sag d0 at the span midpoint and conductor length L0i in single span i the following two equations apply d0i = mcga-li /QHoi (2) L0i = Jh0i + {aoi[1 + (8/3)(doi/aoiy]}2. (3) In (3), L0i is the actual conductor stretch length for any initial temperature. As our focus is on the conductor-state changes caused by wind and/or ice forces, it is assumed that the temperature is constant. So, the conductor elongation and/or contraction due to temperature variations are omitted in the mathematical model. Besides the initial span data (1), by knowing unstressed conductor length Lui as a supplement to horizontal tension force H0i, the catenary equation is fully determined. Assuming that the tension in the conductor is constant throughout the span length, the unstressed conductor length gives the following equation adopted from [2] Lu^ = Lo^(l-HlS). (4) At this point, we define force vector F acting on the insulator-set suspension point with the components defined in the global frame of reference. Inside the i-th span on the left-hand side end suspension point and equally on the right-hand side and suspension point, the forces are FLi = [Hu VLi PU]T , Fm = [HRi VRi PRi]T. (5) For a moment, we must assume that force vectors (5) are known. However, when the conductor condition is changed by a wind or ice load, the conductor tensile forces change in all the spans. If the conductor suspension points in a span move, the higher tensile force moves the suspension points in these directions and consequently changes the span data in the adjacent spans as shown in Fig. 2 and illustrated with a dotted catenary line in Fig. 1. In most cases the initial position for the classical I insulator-set is in the vertical Z direction. The neutral position of the Vee brace insulator is in the X direction perpendicular to OL. The initial OL data and the initial conductor tension force allow us to calculate the conductor span length and sag using the classical catenary equation [1], [2], or, with an acceptable CALCULATION OF THE INSULATOR-SET DISPLACEMENT INDUCED BY THE ICE AND/OR WIND LOADS USING 3D. 289 i+1 Figure 2. Displacements of the I insulator from its initial position. The newly shaped catenary data with the displaced insulator-set for an z-th span can be written in the following vector form K = [at h b]T. (6) Displacements vector Aj of an z-th insulator suspension set with horizontal 8t, vertical ei; and perpendicular ^ components in the global frame of reference for a single suspension set is A = £î VîV (7) Fi = [Ht V PîY = FRi_1-FL (8) The displacement vector (7) in fact describes a an enforced insulator-set movement and is given by the kinematics of the chosen suspension-set type. The displacement equations can be derived for any suspension-set geometry, as we will show later in Section 4. Besides the given material and the geometrical insulator-set data (weight, insulator dimensions, constraints) that are constants, the displacement is only a function of the resultant forces (8) in the suspension point. So At = A;(FL,FR). Note that the conductor force at the conductor end (5) and the catenary data (6) are also determined and can be viewed as a function of K = k(Fl). Referring to Fig. 2, the following vector equation holds for all spans Kqî + (Ai+1 -Ai)-Ki = Oi = 1,2, ... n. (9) Equation (9) which describes the boundary conditions to be met when searching for the algorithm, is only a function of the conductor tension forces. By determining the conductor forces in an individual span that satisfies equation (9), the displacement in a static configuration can be calculated by taking into account the wind and/or ice loads for an arbitrary span. 3 3D Conductor static equations In this section, equations are given to calculate the new span data (6) and the adjacent span resultant forces (8) in the suspension point. At this point our approach differs significantly from the existing methods. The forces are calculated directly with differential 3D catenary equations developed in [6]. They are extracted a static solution of the original dynamical equation describing 3D conductor motions in a single span. Using them under a static condition, the single-span catenary is fully determined with seven ADEs written in the conductor natural frame of reference fixed on the conductor and expressed in terms of the Euler angles. After minor adaptations to better fit our problem and adding the ice load and ice thickness variables, the equations are summarized as the governing equations J^ + Fti- (mc9 + WlÙ sin