Investigation into the Galloping Characteristics of Multispan Iced covered Conductors

A full multispan two-degree-of-freedom (2-DOF) iced covered transmission line model is presented. Taking multispan conductors as the research object, based on the Hamiltonian principle, a nonlinear galloping model of multispan conductors considering the influence of insulator strings is established, and its galloping equation is derived. Based on the theoretical formula, the modes and frequencies in-plane and out-of-plane of the multispan conductor plane were obtained. After the partial differential equation was transformed into an ordinary differential equation by the Galerkin discrete method, the multiple scale method was used to solve the dynamic response of the multispan conductors. Through numerical calculations, this study is the first to systematically analyze the effects of parameters such as span number, span length, and tension on the in-plane and out-of-plane motion characteristics of multi-span conductors from a theoretical perspective. The research results in this paper have certain reference value for preventing multispan conductor galloping and interphase flashover.


I. INTRODUCTION
The galloping of the ice covered conductors will cause hardware wear and damage transmission lines, seriously hindering the normal operation of the line and bringing huge losses to the national economy [1]. The investigation of ice covered conductor galloping has important practical engineering value for the development of galloping prevention and galloping suppression technology.
According to different types of galloping phenomena, Den Hartog proposed a vertical galloping mechanism [2], Nigol proposed a torsional galloping mechanism [3], [4], and Yu proposed an eccentric inertial coupling mechanism [5], [6] on the basis of the former. These three galloping mechanisms can explain the causes of galloping well, so it is generally accepted. On this basis, scholars from all over the world have put forward more galloping models of iced covered conductors. The main research methods include experiments, theoretical derivations and finite element methods. The aerodynamic characteristics of conductors are the premise of research on galloping [7]. The wind tunnel experiment conducted in the literature [8] considered the aerodynamic parameters of single conductors at different wind speeds. In the wind tunnel experiment conducted in reference [9], the influence of turbulence was added to measure the aerodynamic parameters of single span conductors with different ice-covered shapes. In the wind tunnel experiment described in the literature [10], the aerodynamic coefficients of each subconductor of the foursplit conductors were measured. The study found that the wake effect had a great influence on the aerodynamic characteristics. The conductor mode and frequency are the keys to the theoretical study of ice covered conductor galloping. Reference [11] proposed the theory of linear free vibration of suspension cable with a small sag, and the normal symmetric mode of cable depends on a dimensionless parameter. Reference [12] extends the theory of suspension cables to stay cables. Reference [13] revises the formula of mode and frequency of stay cables proposed by Irvine, considering the influence of cable tilt angle on mode and frequency. According to the observation of extensive field data, the galloping of the ice-covered conductors is actually the galloping of the conductors in the tension section, so it is of great significance to study the galloping characteristics of the multispan conductors. With the maturity of computer software, the finite element method has become an effective method to study the galloping characteristics of multispan conductors. This is because theoretical studies have focused on single-span conductors. However, the existing single-span conductor theory fails to consider the influence of the interaction between insulator strings and conductors, so the galloping characteristics of multispan conductors cannot be studied. In references [14], [15], the finite element model of 3-DOF multispan conductors was established by using a three-node isoparammetric cable element. The study found that the amplitude of three-span conductors was much greater than that of single-span conductors, so the interaction between conductors could not be ignored.
A review of the research of the above scholars shows that few scholars have adopted the theoretical derivation method to study the galloping of multispan conductors at present. Reference [16] carried out a simple theoretical study on parallel length multispan conductors and believed that only the mode and frequency of the conductors could be obtained before the theoretical study on multispan conductor galloping could be carried out. Reference [17] established two independent coordinate systems to focus on the frequency and mode of the two-span conductors but ignored the axial motion of the conductors, and the dynamic tension changes caused by galloping cannot be ignored. Reference [18] considered the oscillating problem of insulators and the axial movement of multispan conductors and proposed the theoretical formula for calculating the frequency and mode of multispan conductors for the first time.
Based on the literature [18], the galloping characteristics of multispan conductors are theoretically studied for the first time. A simplified galloping model of multispan conductors considering the influence of insulators is established, and its corresponding galloping equation is derived. The fourthorder Runge-Kutta method was used to solve the equation of galloping and compared with the finite element numerical simulation results and multiscale method of reference [14] to verify the rationality of the model. The calculation efficiency of the multispan transmission line galloping calculation method proposed in this paper is higher than that of the finite element method, and considering the nonlinear effects of transmission lines, it is more reasonable than methods found in the literature [14].
According to the calculation method proposed in this paper, the effects of parameters such as span number, span length and tension on the galloping characteristics of multispan conductors are systematically analyzed for the first time from a theoretical point of view.

II. THE EQUATION OF MOTION OF ICED COVERED CONTINUOUS CONDUCTORS
A simplified mechanical model of iced covered conductors hinged at both ends without a height difference was established. The Cartesian coordinate system A-x-y-z was established with the left suspension point as the origin. The connection between the two hinged points was the x-axis, the positive direction of the y-axis was perpendicular to the downward x-axis, and the positive direction of the z-axis was perpendicular to the plane determined by x and y. The configuration of continuous span iced covered conductors can be divided into two types, namely, the static configuration under the natural state represented by the dotted line and the dynamic configuration under the wind load represented by the solid line, as shown in Fig. 1. In Fig. 1, x, y, and z represent static displacement, and u1, u2, and u3 represent dynamic displacement. The displacement in the y direction is vertical vibration, and the displacement in the z direction is horizontal vibration. li is the length of the i th span conductor, di is the sag of the i th span conductor, and the subscript i represents the i th span conductor.
To simplify the calculation, the following assumptions are made according to the actual situation of the transmission lines.
(1) Since the span is much larger than the diameter of the conductors, the bending stiffness and shear stiffness of the conductors are ignored. Since the axial displacement of the conductors is much smaller than the lateral displacement and the longitudinal vibration frequency is much higher than the transverse vibration frequency, the longitudinal displacement of the conductors is ignored.
(2) Assuming the sag ratio of any span conductor di/li<1/8, the parabolic configuration can be used to describe the static configuration of the conductors. The static configuration yi of any span conductor can be expressed as: In Equation (1), q is the weight of the conductors per unit length, and H is the static horizontal tension of the conductors. The initial horizontal tension of each spanning conductor is considered to be equal. Δx1 l2 Static configuration Dynamic configuration This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. (3) Assuming that the ice is evenly distributed on the conductors, the influence of sectional eccentricity on galloping is not considered.
(4) The interaction between the tower and conductors is ignored.
The vibration equation of any span conductor can be obtained by the Hamiltonian variational principle: In Equation (2),  represents the first-order variation, k v represents the kinetic energy,  represents the potential energy, and W represents the work done by gravity, damping force and external force. The expressions of kinetic energy and potential energy are as follows: In Equation (3), (˙) represents the first-order derivative of time, m is the mass per unit length of conductors, EA is the tensile stiffness, GIp is the torsional stiffness of conductor in the axial direction, θ is the torsional angle of conductor in the axial direction, ε is the total dynamic strain in the tension section, and its expression is as follows: In Equation (4), N is the number of spans, and le is the total length of the conductors in a tension section.
According to Equation (2), the nonlinear vibration equations of any span conductor of the multispan iced covered conductors are obtained: In Equation (5), μ2 and μ3 are damping coefficients of inplane and out-of-plane, respectively, and py and pz are aerodynamic forces of in-plane and out-of-plane, respectively.
The transverse and torsional displacements are represented through separate variables as follows: The product of ψyj(x), ψzj(x)and mode functions qyj(t), qzj(t) is separated by the mode truncation method.
In Equation (6), the subscript j represents the mode order, ψyj(x) is the in-plane modal function, ψzj(x) is the out-plane modal function, ψθj(x) is the tortional modal function, and qyj(t), qzj(t) and qθj(t) are the amplitudes of the discretizing modal functions.
The displacement function (6) is substituted into the nonlinear vibration Equation (5) of the iced covered conductors, and the nonlinear dynamic partial differential Equation (5) are transformed into nonlinear dynamic ordinary differential equations by the Galerkin method as follows: See the appendix for the expressions of coefficients in Equation (7).

III. THEORETICAL FORMULAE FOR MODES AND FREQUENCIES OF MULTISPAN CONDUCTORS
In this paper, the Galerkin method is adopted to transform the partial differential Equation (5) of the galloping transmission line into the ordinary differential Equation (7). However, the specific expression of the mode is not yet known, so Equation(7) cannot be solved. At present, the method to solve the conductor mode is divided into the finite element method and the theoretical method, but the finite element method using discrete elements to solve the conductor mode is not very accurate. In reference [18], in-plane and out-of-plane modes and frequencies of arbitrary multispan conductors were obtained based on the dynamic stiffness theory, and it was pointed out that the calculation method of anti-symmetric modes of multispan conductors was consistent with the calculation method of anti-symmetric modes of Irvine single span conductors. For symmetric modes, the multispan conductors are decomposed into several substructures, and the shape function of each substructure corresponds to the modal function of a single span conductor. This paper uses the mode superposition method to study the galloping characteristics of multispan conductors on the basis of reference [18].

A. IN-PLANE MODES AND FREQUENCIES IN MULTISPAN CONDUCTORS
Equation (7) is also suitable for solving single span conductors. According to reference [11], the symmetrical mode and frequency formula of single-span conductors can be obtained:  [18], the in-plane modes and frequencies of multispan conductors are derived based on Irvine's single span suspension cable vibration theory. The in-plane mode expression in any N-span conductor plane is as follows: The corresponding frequency is obtained by: In Equation (9), the subscript i of ψyij indicates that the mode is dominated by the i-th span, and the subscript j represents the number of mode half waves of a single span conductor. ω i kj is the dimensionless frequency obtained by the parameter of the i-th span conductor. The coefficient  is used to form the mode vectors into a normalized set of mode shapes.

B. OUT-OF-PLANE MODES AND FREQUENCIES OF MULTISPAN CONDUCTORS
The formulas for solving the out-of-plane mode and frequency of the multispan conductors are related to the span number. Since this paper mainly studies the galloping of multispan conductors with two or three spans, only the formulas of these two cases are listed. The mechanical model of the two-span conductors is proposed (see Fig. 2). According to substructure theory, reference [18] proposed the out-of-plane mode expression and frequency calculation formula of two-span conductors: In Equation (11), z is an arbitrary value. As the rotational inertia J of the insulator in Equation (12) is small, J 2 /a can be ignored for simplified calculation and solution. The mechanical model of the three-span conductors is shown in Fig. 3.  actual observations, in an external environment with low temperature and little rain, crescent-shaped ice is easily formed on the surface of the conductors, which is a common ice type in the galloping of iced covered conductors [19]. Based on the galloping mechanism of Nigol, this paper establishes a galloping model of crescent-shaped iced covered conductors, studies the effect of lift, resistance and torque on the conductors under wind load, and makes the following assumptions.

A. ESTABLISH AERODYNAMIC LOAD MODEL
(1) The steady wind blows from left to right along the zaxis horizontally. (2) The thickness and shape remain constant along the conductors. (3) Based on quasi-steady theory, the influence of conductor motion on aerodynamic coefficients is not considered [20]. In Fig. 4, Pd and Pl are the drag force and lift force for the iced covered conductor in stable wind. The Pd direction is consistent with the relative wind direction, and the Pl direction is orthogonal to the Pd direction. U is the average wind speed, Ur is the relative wind speed, α is the wind attack angle, and α0 is the initial wind attack angle. Obtaining the aerodynamic load of the iced covered conductors is important for studying galloping and its prevention and control technology. In this paper, the dimensionless aerodynamic coefficients of the iced covered conductors are defined according to aerodynamic theory as follows: In Equation (13), Cy, Cz and Cθ are load coefficients, py, pz and M are the lift force, resistance and torque, respectively, ρ is the density of air, and D is the diameter of the bare conductors.
When the average wind blows horizontally toward the iced covered conductors, an initial velocity will be generated due to an initial disturbance on the y-axis of the iced covered conductors, and then there is an angle between the relative wind speed and the real wind speed. The change in the angle of attack during the galloping of crescent-shaped iced covered conductors can be expressed as: In Equation (14), θ is the dynamic angle.

B. AERODYNAMIC COEFFICIENT
The third-order Taylor expansion of Cy, Cz, and Cθ can be obtained: See the appendix for the expressions of coefficients in Equation (15).
According to reference [14], the aerodynamic coefficient can be obtained at wind speed U=8.5 m/s and wind attack angle α=180 。 as follows: Substitute Equation (14) into Equation (15) to obtain the lift and drag in the y and z directions: This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.
Substitute Equation (17) into Equation (7) to obtain the ordinary differential equations of the coupled vibration of three degrees of freedom: The expressions of coefficients in Equation (18) are given in the Appendix.

V. MULTIPLE SCALE METHOD FOR SOLVING NONLINEAR EQUATIONS
The multiple scale method is one of the most important methods to study nonlinear problems [21]. It is widely used not only for strict periodic motion but also for attenuated vibration of dissipative systems and many other occasions. The galloping problem of iced covered conductors belongs to a weak nonlinear vibration problem. For the analysis of weak nonlinear problems, the steady-state response and transient response can be calculated by the multiple scale method.
Before solving nonlinear differential equations with the multiple scale method, this paper simply changes the form of the two-degree-of-freedom nonlinear coupled vibration Equation (18) as follows: See the appendix for the expressions of coefficients in Equation (19).
The nonlinear term in the finishing Equation (20) can be obtained as follows: In Equation (20), qy and qz represent the displacement in the y-direction and z-direction, respectively. ωy and ωz are the circular frequencies in the y-direction and z-direction, respectively. ε is a small dimensionless parameter.
The displacement can be expressed as the following function: The variables T0=t and T1=εt of the time scale are introduced into Equation (21).
The velocity and acceleration of iced covered conductors can be obtained by taking the first-order and second-order derivatives of Equation (21) Substituting Equation (22) into Equation (19) and equating the coefficients of the similar terms of ε, we can obtain the following set of linear differential equations: The periodic solution adopted as the solution to Equation (23) is as follows: Eliminating the secular terms in qy1 and qz1 gives: Assume that Ay(T1) and Az(T1) are expressed as follows: In Equation (28), ay and az are the amplitudes ofgalloping, and θy and θz are the phases.
We substitute Equation (28) into Equation (27) If the right side of each Equation (29) is zero, the steadystate amplitude of the iced covered conductors can be obtained.
Integrating Equation (29) In Equation (30) In Equation (31), Equation (31) is the analytical solution of the conductors. It is faster to use the analytical solution to obtain the displacement time-history curve of the conductors than the numerical solution.

VI. NUMERICAL EXAMPLES AND RESULT ANALYSIS
The galloping characteristics of multispan ice-covered conductors are mainly related to the length of span, the initial tension and the number of spans. When the number of spans is second or third, the multispan conductor model can accurately reflect the influence of adjacent spans on the overall galloping of the conductors in a tension section. This paper studies the galloping characteristics of ice-covered conductors under changes in these structural parameters.
Since it is very difficult to solve the analytical solution of dynamic Equation (21) according to the theory of current nonlinear dynamics, in this paper, the fourth-order Runge-Kutta method is used to solve Equation (21). To verify the accuracy of the model in this paper, the same structural parameters as those in reference [14] were selected. The specific physical parameters of the conductors are shown in Table I, with a length of 125 m, an initial tension of 15 kN This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.

A. VERIFY THE PROPOSED GALLOPING MODEL
According to actual field observations of galloping, there is torsional motion in conductor galloping [23], so some studies add the influence of torsional freedom on conductor galloping [24], [25], but some studies only establish a twodegree-of-freedom model [26], [27]. These two studies based on the Niol galloping mechanism and Den Hartog galloping mechanism have been accepted, but increasing the degree of freedom will increase the amount of calculation. Therefore, this paper first carries out numerical simulation of icecovered single span conductors (initial tension 15 kN) with 3-DOF and 2-DOF lower span spacing of 125 m and compares the influence of different degrees of freedom on the galloping characteristics of single span conductors. Equation (8) is used to obtain the modal and frequency of the single span conductors. The parameters and modal in Table I are put into Equation (19), and the time-history curves of the displacement of the single span conductors with 3-DOF and 2-DOF can be obtained by using the fourth order Runge-Kutta method, as shown in   5 shows that the in-plane amplitude of the 3-DOF singlespan conductors is 0.294 m and that of the 2-DOF single-span conductors is 0.301 m by the fourth order Runge-Kutta method. The out-of-plane amplitude was 0.009 m for 3-DOF and 0.009 m for 2-DOF. Therefore, joint torsional degrees of freedom can reduce the conductor wave amplitude, but with three degrees of freedom, calculating conductors galloping for a long time, and the two-degreeoffreedom model can also be a very good reaction adjacent to spans, leading to multiple lead wave characteristics. The influence of torsion energy does not occur through the insulator to the adjacent spans, so this is does not consider the effect of torsion.
The time-history curves of in-plane displacement within 10 s after the single span conductors tend to be stable in the literature [14]. Fig. 6 shows the calculation results of the finite element method, the multiscale method and the 2 degrees of freedom of this paper.
As shown in Fig. 6, the conductor frequency in reference [13] is 0.70 Hz, and the conductor frequency in this study is 0.68 Hz. The frequencies obtained by these two calculation methods are very close. However, Desai used finite element software to establish a single-file model and found that the in-plane amplitude was approximately 0.2 m [14]. There may be several reasons for this difference. First, Desai adopted a modal decomposition method based on linear assumptions to solve the dynamic equation, ignoring the geometric nonlinearity caused by the large galloping of the conductors, while the Runge-Kutta method was used to calculate nonlinear problems with higher accuracy. Second, the finite element method used the damping ratio as a whole mode, but multiple modes would be excited, and the change in damping ratio had a great influence on the amplitude. Finally, some errors may also occur when reading the data of the timehistory curves of single conductor displacements in the literature [14].
In this study, the in-plane amplitude of the 2-DOF single span conductors is 0.288 m by using the multiple scale method. The results obtained by the multiple scale method are very close to those obtained by the fourth order Runge-Kutta method, which also shows that the results obtained by the model in this paper are accurate. This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and

B. ANALYSIS OF GALLOPING CHARACTERISTICS OF SINGLE CONDUCTORS
In this paper, single-span conductors with 100 m span length and 15 kN initial tension are first selected to study the difference between single-span conductors and multispan conductors and the influence of the span number on the galloping characteristics of the conductors. Other parameters are shown in Table I. To verify the accuracy of the theoretical formula calculated in this paper, Abaqus finite element software was used to establish the corresponding model to calculate the frequency and mode. A cable element was used to simulate the cable, and a rod element was used to simulate the insulator string. The number of grids was 100. The first three order symmetric modes and frequencies of the single span conductors plane calculated by theoretical Equation (8) and finite element software are shown in Table  II. Table II shows that the results obtained by the two calculation methods are very close, so the conductor model established by Abaqus in this paper is also accurate. The results of the first-order in-plane symmetric modal frequency calculated by the theoretical formula in Table II are substituted into Equation (19), and the displacement timehistory curves of a single-span conductor with a length of 100 m (initial tension 15 kN) are obtained, as shown in Fig. 7.   Fig. 7 shows that the in-plane amplitude of the single-span conductors is 0.154 m and the out-of-plane amplitude is 0. 006 m. According to the frequency analysis of the characteristics of in-plane galloping of the single-span conductors, the frequency is 0.762 Hz before the single-span conductors tends to be stable and 0.755 Hz after it tends to be stable. The frequency calculated by the fourth-order Runge-Kutta method is very close to the first-order frequency of 0.764 Hz calculated by the finite element method. Therefore, the frequency of the single-span conductors does not change greatly from the beginning of galloping to stabilizing.
To compare the difference between the galloping characteristics of multispan conductors and single-span conductors after the initial tension increases, the displacement time-history curves of single-span conductors after the initial tension gradually increases are given in Fig. 8. Fig. 8 shows that the amplitude of the single-span conductors increases first and then decreases with increasing initial tension.

1) EFFECT OF SPAN LENGTH ON THE GALLOPING OF TWO-SPAN CONDUCTORS WITH UNEQUAL SPAN LENGTH
In this study, the length of the first-span conductors is kept constant, and the length of the second-span conductors is gradually increased to analyze the influence of the length between adjacent spans on the galloping characteristics of the two spans. The length of the two conductors is set to 100 m-50 m, 100 m-80 m, 100 m-150 m, and 100 m-200 m, and the initial tension is 15 kN; see Table I for other parameters. The frequency and mode of the two-span conductors obtained by theoretical Equations (10) and (12) and the Abaqus solution are shown in Table III.  Table III shows that the results of the theoretical solution and finite element solution are basically the same, and the existing errors are within a reasonable range, thus proving the accuracy and precision of the calculation results by using the theoretical formula. The in-plane and out-of-plane frequencies of the two-span conductors decrease gradually when the length between adjacent spans increases gradually, and the frequency of the two-span conductors is obviously lower than that of the single-span conductors. The modes drawn using theoretical Equation (9) Fig. 9 shows that the modal shapes of the first span conductors are basically the same. With the increase in the length of the second-span conductors, the amplitude of the mode of the second-span conductors gradually increases. Limited by space, this paper only gives the conductor modal diagram calculated by the finite element method when the length of the two-span conductors is 100 m-200 m, as shown in Fig. 10.  Fig. 10 shows that the modal shapes obtained by Abaqus are basically consistent with Fig. 9, so the modes in Fig. 9 are also accurate.
Substituting the first-order normalized symmetric modal frequency in Table III into Equation (19), the fourth-order Runge-Kutta method was used to solve, and the displacement time-history curves of unequal length two-span conductor galloping were obtained, as shown in Fig. 11.   084 m by using the multiscale method. The amplitudes of the multispan conductors obtained by the two methods are relatively close, so the model for calculating the two-span conductors in this paper is also accurate.
The amplitude of the two-span conductors with lengths of 100 m-50 m is significantly greater than that of the singlespan conductors with lengths of 100 m. This is because the energy of the two-span conductors will be transmitted to the adjacent conductors through the insulator string, resulting in a greater amplitude. Therefore, it is necessary to study the galloping of multispan conductors in practical engineering. In addition, in-plane galloping is mainly the galloping of multispans conductors [2]. To more clearly compare the effect of increasing the length of adjacent spans on amplitude, the time-history curves of displacement within 10 s are selected when the galloping of two-span conductors is stable, as shown in Fig. 12.  A   , the greater the conductor circular frequency is, the smaller the amplitude (where A is the amplitude, μ is related to wind speed and the conductor aerodynamic parameter parameters, and 0 is the circular frequency of the conductors). These results are consistent with the conclusion of the formula in this paper.
Studying the galloping trace of ice-covered conductors can provide an important basis for galloping mechanisms and antigalloping measures. The galloping traces of the two-span conductors with unequal span lengths are shown in Fig. 13. Fig. 13 shows that galloping is mainly in the vertical direction, and the trajectory is approximately oval. This is consistent with the galloping form described by the galloping mechanism widely recognized in the current galloping research field; that is, galloping is a process of gradual formation. At the initial span, the conductors oscillate with a small amplitude near the equilibrium position. Due to the accumulation of wind energy and the negative damping of air, the horizontal amplitude gradually decreases. At the same time, the vertical amplitude increases, forming an ever-larger elliptic galloping trajectory, which eventually tends to be stable due to the influence of system damping. This also proves the correctness of the numerical simulation. The circular frequency of the single-span conductors is almost unchanged before and after galloping is stable. Now, the displacement time-history curves of the two-span conductors with lengths of 100 m-200 m are selected for 10 s before and after galloping is stable, as shown in Fig. 14. This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and Fig. 14 shows that the frequencies of the two-span conductors before and after galloping tend to be stable are 0. 389 Hz and 0.449 Hz, respectively. The frequency obtained by using Abaqus is 0.385 Hz, which is very close to the frequency obtained by the fourth-order Runge-Kuta method before conductor galloping tends to be stable. However, with increasing conductor galloping time, the frequency will increase after conductor galloping tends to be stable. This is because the influence of the nonlinear term causes the inplane frequency of the two-span conductors to increase when they are galloping.

2) EFFECT OF SPAN LENGTH ON THE GALLOPING OF TWO-SPAN CONDUCTORS WITH EQUAL SPAN LENGTH
It is a special case that the length of each conductor in two spans is the same because the conductor frequency is the same. In this paper, two-span conductors with equal span lengths are selected for study. The frequency after the change in the length of the two-span conductors is shown in Table  IV, and the other parameters remain unchanged. Table IV shows that the frequencies of the two-span conductors with equal span lengths are also very accurate. As the span length of the two-span conductors with equal span length increases, the frequency gradually decreases. The modes drawn using theoretical Equation (9) 15 shows that when the conductor spacing is equal, it has the same shape. With the increase in the second conductor length, there are two downward half waves in the conductor mode because the sag increases with increasing length.
Substitute the first-order in-plane symmetric modal frequency results calculated by the theoretical formula in Table IV into Equation (19) and obtain the time-history curves of the two-span conductor displacement at different spans (initial tension 15 kN), as shown in Fig. 16.   When the total span of the two-span conductors is 300 m, the in-plane amplitude of the two conductors increases by 27.45% after the nonequal two-span conductors with distances of 100 m-200 m are changed into equal two-step conductors with distances of 150 m-150 m. Although the total span distance does not change, the amplitude will continue to increase when the frequencies of the two conductors are close.
To more clearly compare the effect of increasing the conductor spacing on the vertical amplitude, the displacement time-history curves within 10 s are selected when the two-span conductors with equal span length are galloping steadily, as shown in Fig. 17.   Fig. 17 shows that the amplitude of the two spans with equal span length increases with increasing conductor length. Therefore, regardless of whether the child traverse length is the same, the amplitude of the two spans will increase when the length of one of the two spans increases or the length of the two spans increases simultaneously.
The galloping tracks of the two-span conductors with equal span lengths are shown in Fig. 18. Fig. 18 shows that when the length between the conductors increases simultaneously, the vertical amplitude of the twospan conductors increases significantly, while the horizontal amplitude does not change significantly.

3) EFFECT OF INITIAL TENSION ON THE GALLOPING OF TWO-SPAN CONDUCTORS WITH EQUAL SPAN LENGTH
The mode and frequency of the conductors will also change after the initial tension of the conductors change. Next, two kinds of common initial tension of the conductors are selected in this paper to study the influence of different initial tensions on the galloping characteristics of the two-span conductors with span lengths of 100 m-100 m. The frequencies of the twospan conductors under the two initial tensions are shown in Table VI.  Table VI shows that the results calculated using the theoretical formula are accurate. The frequency of the equalspan two-span conductors increases with increasing initial tension. The modes drawn using theoretical Formula (9) are shown in Fig. 19. Fig. 19 shows that the modal shapes of the two-span conductors with equal length are basically the same, and the increase in initial tension has little influence on the modal of the two-span conductors.
Different initial tensions span two displacement time history curves of the conductors, as shown in Fig. 19, and when the initial tension is 40 kN and above, the two conductors do not gallop because when the initial tension increases to more than 40 kN with a smaller sag, the damping ratio remains unchanged, causing the aerodynamic damping term to be positive and unable to absorb wind energy. Fig. 20 shows that the time for the equal-span two-span conductors to gallop to stabilize gradually decreases with increasing initial tension. The amplitude of the two-span conductors with span lengths of 100 m-100 m (initial tension 20 kN) is 1.113 m, and that of the two-span conductors withspan lengths of 100 m-100 m (initial tension 30 kN) is 0.680 m. To more clearly compare the influence of the increase in initial tension on the opposite internal amplitude, the displacement time-history curves within 10 s are selected when the two conductors are galloping steadily, as shown in   21 shows that when the initial tension increases gradually, the amplitude of the equal-span two-span conductors decreases gradually, while when the initial tension is too large, the equal-span two-span conductors will not gallop. Compared with Fig. 8, as the initial tension increases from 15 kN to 20 kN, the amplitude of the twospan conductors decreases by 19.23%, while that of the single-span conductors increases by 12.32%. When the initial tension increases from 20 kN to 30 kN, the amplitude of the two-span conductors decreases by 38.90%, while that of the single-span conductors decreases by 177.69%.
The galloping traces of the two-span conductors with equal span lengths under different initial tensions are shown in Fig. 22.  Fig. 22 shows that when the initial tension increases, the galloping trace of the two-span conductors with lengths of 100 m to 100 m begins to decrease.

1) EFFECT OF SPAN LENGTH ON THE GALLOPING OF THREE-SPAN CONDUCTORS WITH UNEQUAL SPAN LENGTH
In this paper, when studying the influence of the change in adjacent span length on the overall galloping of the threespan conductors, the middle conductor length should be kept unchanged, and the conductor length on both sides should be consistent and gradually increased. Theoretical Equation (9) has the highest accuracy in calculating the inplane frequency of the multispan conductors when the length between the spans is not equal. Therefore, this paper uses the theoretical formula to solve the in-plane frequency of the three-span conductors to avoid the same value. However, when using the theoretical Equation (14) to solve the out-of-plane frequency, the value of the length between the spans can be directly substituted to ensure the highest accuracy. To verify the accuracy of the theoretical formula, Abaqus was used to establish the corresponding finite element model. The calculation results of the two algorithms are shown in Table VII. Table VII shows that the theoretical formula is close to the frequency calculated by Abaqus, so the frequency calculated by using the theoretical formula is accurate. The modes drawn using theoretical Equation (9) Fig. 23 shows that the modal shapes of the middle conductors are basically the same. As the length of the sub conductors on both sides increases at the same time, the amplitude of the conductor modes on both sides increases first and then decreases. Limited by space, this paper only gives the conductor modal diagram calculated by the finite element method when the length of the three-span conductors is 200 m-100 m-200 m, as shown in Fig. 24. Fig. 24 shows that the modal shapes obtained by Abaqus are basically consistent with Fig. 23, so the modes in Fig. 23 are also accurate.
By substituting the frequency of the first-order mode in Table VII into Equation (19), the displacement time-history curves of the three-span conductors are obtained, as shown in Fig. 25. To more clearly compare the influence of the increase in the length of adjacent spans on the amplitude of the nonequal-length three-span conductors, the displacement time-history curves of the three-span conductors within 10 s after it gallops steadily are shown in Fig. 26.    As shown in Fig. 26, when the length of the two conductors on the edge is smaller than the length of the middle conductors, the amplitude of the three-span conductors increases as the length of the two conductors on the edge increases at the same time. When the length of the two conductors on the edge is greater than the length of the middle conductors, the amplitude of the three-span conductors increases with increasing length of the two conductors on the edge. The galloping traces of the three-span conductors with unequal span lengths are shown in Fig. 27. Fig. 27 shows that the galloping traces of the three-span conductors with unequal span lengths are basically elliptical. Because there are many conductors hanging on the tension tower at present, as shown in Fig. 28, if the conductor distance is too close, it will cause accidents such as interphase flashover. Studying the amplitude of conductors is helpful for the design of lines in actual projects and provides a reference for preventing and controlling the interphase flashover of conductors.

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This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.

2) EFFECT OF SPAN LENGTH ON THE GALLOPING OF THREE-SPAN CONDUCTORS WITH EQUAL SPAN LENGTH
In this paper, the influence of the galloping characteristics of three-span conductors with the length of the conductors is studied. Four three-span conductors with a gradual increase in span length are selected, and their frequencies are shown in Table VIII. Table VIII shows that the frequency calculated by the theoretical formula is accurate. The model obtained using theoretical Equation (9) is shown in Fig. 29. As shown in Fig. 29, when the three conductors are spaced relatively close, the modal shapes of the three conductors are basically the same. However, when the three conductors are spaced relatively far apart, two downward half waves will appear in each conductor. This is due to the increase in the spans, and the conductor sag is too large.
After the frequency is obtained, the time-history curves of the displacement of the three-span conductors with equal length are calculated by the fourth-order Runge-Kutta method, as shown in Fig. 30. Fig. 30 shows that the galloping time of the equal-length three-span conductors tends to stabilize and gradually decrease with the simultaneous increase in the length of the three conductors.  Fig. 28 (a)-(c) is also because the in-plane frequency of the conductors is approximately twice the out-of-plane frequency when it gallops. To more clearly compare the influence of the amplitude of the third span conductors with the increase of the length between the secondary conductors and its counterpart, the displacement timehistory curves of the three-span conductors within 10 s after it gallops steadily are shown in Fig. 31.
It can be seen from Fig. 31 shows that the amplitude of the three-span conductors with equal spacing increases first, then decreases, and then continues to increase.
The galloping traces of the three-span conductors with equal span lengths are shown in Fig. 32 below.

3) EFFECT OF INITIAL TENSION ON THE GALLOPING OF THREE-SPAN CONDUCTORS WITH EQUAL SPAN LENGTH
In this paper, the influence of initial tension on the galloping characteristics of parallel and three-span conductors is studied. Two common initial tensions of transmission lines are selected, and their frequencies are shown in Table IX.  Table VIII shows that the frequency calculated by the theoretical formula is accurate. The model obtained using theoretical Equation (9) is shown in Fig. 33. Fig. 33 shows that the shape of the three-span conductors is basically the same under two different initial tensions, and the initial tension has little influence on the modal shape of the three-span conductors.
After obtaining the frequency, the fourth-order Runge-Kutta method is used to calculate the time-history curves of the displacement of the three-span conductors with equal length under different initial tensions, as shown in Fig. 34. This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and    Therefore, when the initial tension increases, the amplitude of the single-span conductors first increases slowly and then decreases greatly, the amplitude of the equal-pitch two-span conductors first decreases slightly and then decreases greatly, and the amplitude of the equal-pitch three-span conductors first decreases significantly and then decreases slightly.
The galloping traces of the three-span conductors with equal span lengths under different initial tensions are shown in Fig. 36.  1) The amplitude of the multispan conductors is obviously greater than that of the single-span conductors, so the interaction between adjacent spans has a great influence on the galloping characteristics of ice-covered conductors. The galloping amplitude increases first and then decreases with increasing span number. 2) The galloping amplitude of the two-span conductors with nonequal spans increases with increasing length between adjacent conductors, and the galloping amplitude of the two-span conductors with equal spans also increases with increasing length between each span of conductors. Keeping the span of the two-span conductors unequal reduces the amplitude. As the initial tension increases gradually, the galloping amplitude of the two-span conductors with equal spans decreases. When the initial tension is large enough, galloping will not occur.
3) The galloping amplitude of the three-span conductors with nonequal span or equal span increases first, then decreases and then continues to increase with the increase in the length between adjacent spans. Keeping the span of the three-span conductors consistent can reduce the amplitude. With increasing initial tension, the galloping amplitude of the three-span conductors with equal spans decreases.