A Comparative Study on the Wind Deflection of Railway Overhead Contact Line Based on Empirical Formula and Finite Element Approach

The assessment of wind deﬂection of the railway overhead contact line (OCL) is of great importance to ensure the safe and reliable operation of the electriﬁed railway. This paper performs a comparative study of the railway OCL wind deﬂection based on the traditional empirical formula and ﬁnite element method (FEM). Considering the geometrical nonlinearity, a nonlinear ﬁnite element approach is adopted to model the OCL. The crosswind load is applied to the OCL through an iterative procedure. The comparative study indicates that the maximum wind deﬂection estimated by the empirical formula is generally the same as the FEM result when the turbulence intensity is 10%. However, with more than 10% of the turbulence intensity, the empirical formula may give a risky result. Then, some coeﬃcients are introduced to modify the empirical formula according to the FEM results. The numerical examples are presented with diﬀerent OCL tension classes to verify the validation of the modiﬁed empirical formula.


Introduction
e OCL is widely used in modern electrified railway systems and serves as the only source of power for electric trains [1]. As shown in Figure 1, the OCL is a tensioned cable structure constructed along the railway track, which transmits the electric current to the locomotive through the sliding contact with the pantograph installed on the carbody's roof.
e quality of the current collection of the electric train is directly affected by the dynamic performance of the pantograph-OCL interaction. Typically, the OCL is the most vulnerable part of the electrified railway system, as it suffers multiple impacts from the vehicle-track vibration [2], the temperature variation [3], the irregularities in the contact line [4], and the crosswind load [5]. Often, the OCL has a large span length to reduce the construction cost and also has high flexibility to decrease the unevenness of the elasticity along the contact line. ese features make the OCL very susceptible to the crosswind. Generally, wind is comprised of even wind and fluctuating wind. e former is able to cause deviation of the OCL with respect to its original configuration. e latter excites a drastic forced vibration of the OCL, which directly disturbs its contact with pantographs. More precisely, fluctuating winds may increase the fluctuation of the contact force and the possibility of the occurrence of contact loss, which creates more wear [6,7], vibration [8], fatigue [9], arcing and sparking [10,11], and more severe accidents like dewirement and line breakage in some extreme cases. Recently, the negative effect of the wind-induced vibration of the OCL has been frequently reported from Chinese high-speed networks, especially the scarping of the pantograph head caused by the wind deviation of the contact line.
In general, the contact force between the registration strip of the pantograph and the contact line of the OCL is a direct reflection of the quality of the current collection [12]. An excessive contact force can aggravate the wear of the contact interface. In contrast, an inadequate contact force may cause the separation of the pantograph collector from the OCL. In order to ensure a stable contact force, the pantograph-OCL system should be appropriately designed. Many numerical models of the OCL have been developed to reproduce the realistic behaviour [13][14][15] and explore the implications of design parameters on the dynamic performance [16][17][18]. Based on these simulation tools, several optimisation strategies are proposed to reduce the elasticity unevenness of OCL [19], decrease the wave propagation interference [20], and avoid the resonance with pantographs [21]. e research studies mentioned above seldom consider the effect of crosswind load. Generally, there are three types of vibration for long-span structures caused by wind load. One is galloping [22], which is a kind of instability induced by negative aerodynamic damping [23]. For the OCL, galloping is rare to be observed in real life [24], except in some extreme conditions [25]. e second is vortex-induced vibration, which is common in bridge deck [26,27] and power transmission line [28]. e vortex-induced vibration of OCL has not been reported in any existing literature. Apart from these two types, the buffeting of the OCL caused by fluctuating wind is the most common one, which challenges the current collection quality and the safe operation. In order to understand the aerodynamics of the OCL, the CFD model of contact line cross section was established by Sanchez-Rebollo et al. [29] with different working conditions. In order to explore the buffeting behaviour of the OCL, Song et al. [30] derived the fluctuating forces acting on the contact line and analysed the wind-induced response of OCL based on a nonlinear finite element approach. To ensure a good interaction performance of pantograph-OCL subjected to a strong wind field, a control strategy was proposed based on the instantaneous response of the pantograph [31]. Considering the crosswind acting on the pantograph and the OCL simultaneously, Pombo et al. [32] conducted a simulation of the pantograph-OCL interaction with wind load. en, the wind load was introduced in the simulation of multiple pantograph-OCL interactions to study its effect on the contact forces on both pantograph heads [33].
In the current industry [34], the wind deflection of the contact line is evaluated using empirical formulas, which only gives an empirical result in the design phase of the OCL. Some dynamic effects caused by the fluctuating wind, especially with high turbulence intensity, cannot be considered in the empirical equation. e work in [35] has indicated the difference in the results between the empirical equation and the response spectrum analysis. In this paper, a nonlinear finite element method is used to model the OCL. e crosswind load acting on the OCL is derived according to the fluid-induced vibration theory. e results of wind deflection evaluated by both the FEM and empirical formula are compared to indicate the shortfall of the empirical formula.
en, some modifications are suggested to improve the empirical equation. e background and current research state are introduced in Section 1. In Section 2, the empirical formula to estimate the wind deflection is derived. In Section 3, a nonlinear finite element of the OCL is built based on FEM. In Section 4, the crosswind field along the OCL is constructed.
e aerodynamic forces acting on the OCL are derived in Section 5. In Section 6, a preliminary comparative study is presented. e modification is recommended in Section 7. Some discussions and conclusions are drawn in Section 8.

Estimation of Wind Deviation Using Empirical Formula
is section presents the derivation procedure to obtain the empirical formula [34]. For the contact line, the wind load p acting on the per-unit contact line can be estimated by in which V 2 R is the design wind speed at the height of 10 m above ground, β q is the wind pressure non-uniformity coefficient, μ s is the wind load profile coefficient, μ z is the wind speed height variation coefficient, and d is the diameter of the contact line.
Considering a contact line subjected to the crosswind, as shown in Figure 2, the wind deflection y w (x) of the contact line can be estimated by in which x is the distance of the reference point with respect to the left steady arm, l is the span length, and T is the contact line tension. If no wind load is applied, the lateral displacement of the contact line with respect to the track centre can be obtained by in which s i is the stagger value of the left steady arm and s i+1 is the stagger value of the right one. erefore, the total deviation under crosswind can be obtained by e position of the maximum wind deflection can be estimated from equation (4) e explicit formula of the position of maximum wind deflection can be obtained as erefore, the maximum wind deflection can be obtained by substituting equation (6) into equation (4), which can be written as follows: In most cases, s i � s i+1 � s. us, equation (7) can be simplified to

Finite Element Model of OCL
In order to describe the geometrical nonlinearity in large deformation of the OCL caused by the crosswind, a nonlinear finite element approach is adopted to model the OCL. e ANCF (absolute nodal coordinate formulation) is a commonly used nonlinear finite element method to deal with large deformations in various engineering backgrounds [36]. As shown in Figure 3, the ANCF beam is utilised to model the contact and messenger lines. e ANCF cable is employed to model the dropper and the steady arm. e claws and clamps on the dropper and steady arm are assumed as lumped masses. As for an ANCF beam element, the nodal degree of freedom (DOF) vector can be expressed by in which χ is the local longitudinal coordinate in the undeformed configuration from 0 to the element length L 0 . e position vector in the deformed configuration r can be interpolated using the shape function matrix S as r � Se. (10) According to [37], S is defined using a classic form of shape function for beam element as follows: e strain energy of the beam element can be calculated by the summation of the energies in axial and bending deformation as in which E is the elastic modulus, A is the section area, I is the inertial moment of the beam, ε l is the longitudinal strain, and κ is the curvature. Differentiating equation (12) with respect to the element DOF vector e yields the generalised elastic force vector Q as e secant stiffness matrix K e for the given element can be determined according to equation (13). In an integration algorithm, the tangent stiffness matrix K T is mostly used to calculate the incremental nodal DOF vector Δe. Notably, in the shape-finding procedure, the tangent stiffness matrix K L related to the incremental unstrained length ΔL 0 is necessary to determine the unstrained size of each element. us, the tangent stiffness matrices K T and K L are obtained by differentiating both sides of equation (13).
e stiffness matrix of the ANCF cable element can be obtained through a similar derivation procedure. It should be noted that the cable element used to describe the nonlinear behaviour of droppers cannot withstand compression. e axial stiffness is set to zero when the dropper works in compression. Assembling the element matrices by the FEM yields the global incremental equilibrium equation for the whole OCL as follows: where ΔF G is the global unbalanced force vector and K G T and K G L are the global stiffness matrices related to the incremental nodal displacement vector ΔU C and the incremental unstrained length vector ΔL 0 , respectively. Additional constraint conditions for the design specification can be provided to reduce the number of unknowns in equation (15) and ensure a unique solution. en, the Newton-Raphson iteration can be used to solve the equilibrium state of the OCL with the gravity and tensions [38]. Including a consistent mass matrix M G C and a Rayleigh damping matrix C G C , the equation of motion for the OCL is written by in which K G C (t) is the global stiffness matrix of the OCL and F G C (t) is the external force vector applied on the OCL. Table 1 presents the main parameters of a conventional-speed railway OCL. e initial configuration of the OCL can be calculated according to the above approach, of which the central two spans are taken as the analysis object in the following analysis.

Crosswind Field of OCL
Generally, the natural wind can be divided into even wind and fluctuating wind. For a given spatial point M, the wind speed at this point can be expressed by in which V(M, t) is the even wind speed and V ′ (M, t) is the fluctuating wind speed. ey can be expressed by the following two vectors: in which U(M) is the steady wind speed in the longitudinal direction and u(M, t), w(M, t), and v(M, t) denote the fluctuating wind speed in the longitudinal, lateral, and vertical directions, respectively. In this section, the spatial wind field is constructed for the OCL. e empirical spectrum is a standard way to describe the stochastics of the fluctuating wind speed. In this work, the von Karman spectrums [39] in longitudinal, lateral, and vertical directions are adopted to generate the wind speed time histories. For two arbitrary spatial points M and P, the power spectral density matrix for this point can be expressed by in which coh M,P εη represents the correlation function between two points M and P in ε and η (ε, η � u, w, v) directions. e power spectral density matrix for one point M can be expressed by After determining an appropriate spatial grid, the global power spectral density matrix S V (ω) can be obtained by assembling equations (19) and (20). e spatial grid depicted in Figure 4 is adopted. It is seen that the spatial grid has two layers in the vertical direction. e top one is for the messenger/upper half of the dropper line. e second one is for the contact/lower half of the dropper line. e spatial interval of the grid in the X direction is 10 m. After obtaining the spectrum matrix, the time history of fluctuating wind speed can be obtained through the inverse Fourier transform.

Aerodynamic Forces
After obtaining the fluctuating wind speed, the aerodynamic forces are derived in this section. As shown in Figure 5, the aerodynamic forces acting on the contact line are derived as follows. e aerodynamic forces on the messenger line, dropper, and steady arm can be obtained through a similar derivation procedure.
e contact line has a certain displacement in the lateral direction, and the contact line is not always perpendicular to U. erefore, the wind speed vectors should be converted to the local reference system for each element. For each element, the even wind U e and fluctuating wind components u e , v e , and w e in the element local reference system can be obtained by the spatial coordinate transformation [40]. e contact line cross section subjected to U e , u e , and w e is shown in Figure 5. As U e is parallel to the y-axis, "y-o-z" is called an absolute wind-axis reference system. e drag F e D and lift F e L are the aerodynamic forces acting on the contact line section caused by the wind load. A dynamic wind angle β is induced by the movement of the contact line subjected to the wind load, which can change the angle of attack. β can be estimated by β � arctan w e − _ z e cr U e + u e − _ y e cr , (21) in which _ z e cr and _ y e cr are the vertical and lateral velocities of the contact line in the reference system (called relative windaxis reference system) defined by the dynamic angle b, lift F e Lr , and drag F e Dr . _ z e cr and _ y e cr can be obtained by transferring the vertical and lateral velocities _ z e c and _ y e c in the absolute wind-axis reference system to relative wind-axis reference system. F e Lr and F e Dr can be expressed by in which ρ air is the air density, D is the diameter of the contact line cross section, C L (β) and C D (β) are the lift and drag coefficients at the angle of attack β, and U e r is the effective wind velocity, which can be expressed by According to the geometrical relationship, the drag F e D and lift F e L in the absolute wind-axis coordinate system can be obtained by

⎧ ⎨ ⎩ (24)
e spatial coordinate transformation is performed again to transfer F e D and F e L to the global reference system, which can be directly exerted on the finite element model. e aerodynamic coefficients C D and C L usually are dependant on the cross-sectional profile and the Reynolds number. In this section, the wind tunnel test results obtained in the authors' previous work [41] are adopted in the following simulations.

Comparative Analysis
According to the technical specification, the design wind speed for the railway OCL is 30 m/s, which is defined in the following analyses. e numerical simulations of wind deflection with different turbulence intensities from 9% to 25% are performed. Taking the positions where the maximum wind deflection occurs as an analysis object, Figure 6 shows maximum wind deflection with different turbulence intensities. e red dash line denotes the result given by equation (8). e blue dash line represents the maximum allowed deflection specified in the standard [42]. It is seen that the increase of the turbulence intensity significantly increases the maximum wind deflection. However, the variation of the wind deflection versus the turbulence intensity cannot be considered in the empirical formula. e result given by the empirical formula is generally the same as the FEM result with 10% turbulence intensity. For the turbulence intensity smaller than 10%, the empirical equation only gives conservative results. In contrast, when the turbulence intensity is bigger than 10%, the empirical formula may give dangerous results. Mainly, when the turbulence intensity is over 15%, the maximum wind deflection is over the safety threshold, which may result in fatal accidents. Figures 7 and 8 present the time history of the contact line wind deflection in the position where the maximum wind deflection occurs with the 9% and 11% turbulence intensities, respectively. It is seen that all the displacements with 9% turbulence intensity are smaller than Shock and Vibration 5 the value given by the empirical formula. Some outliers can be seen when the turbulence intensity moves up to 11%. Assuming that the contact line wind deflection follows the normal distribution, the probability density functions (PDFs) of contact line displacement at the maximum deflection position are presented in Figure 9. It is seen that the contact line displacement has a 0% possibility to exceed the empirical value with 9% turbulence intensity. However, this possibility increases to 0.19%, 0.88%, 2.15%, and 3.31% with the turbulence intensities of 11%, 13%, 15%, and 17%. With 25% turbulence intensity, the contact line displacement has a 9.65% possibility to exceed the empirical value, which will definitely give risky results.

Modification of Empirical Formula
In this section, a modified empirical formula is proposed to make it consider the change of the turbulence intensity. e modified form of the empirical formula is given by introducing the modification coefficients α 1 , α 2 , α 3 , and α 4 as follows: in which I represents the turbulence intensity × 100. Using the simulation results in Figure 6, the modified coefficients α 1 , α 2 , α 3 , and α 4 can be obtained by the curve fitting, which is indicated in Figure 10. e fitted coefficients α 1 , α 2 , α 3 , and α 4 are presented in Table 2.
In order to validate the proposed modified empirical formula, several simulations with different OCL systems are performed with 15% turbulence intensity. In these simulations, the main parameter of the OCL, tension, is changed to be 0.8-2 times the original value. e resulting maximum wind deflections evaluated by the FEM, traditional empirical formula, and modified empirical formula are presented in Figure 11. It is seen that the accuracy of the empirical  formula to estimate the maximum wind deflection is significantly improved after the modification. For most cases, the evaluated results of the modified empirical formula are slightly larger than the FEM results, which demonstrates that the modified empirical formula gives conservative results in the design phase to ensure operational safety.

Conclusions
In this paper, a comparative study of the railway OCL wind deflection is performed based on the traditional empirical formula and FEM. A nonlinear finite element approach is presented to model the OCL. e aerodynamic forces acting on the OCL caused by the crosswind load are derived. e simulation results indicate that the empirical formula result can only be effective when the turbulence intensity is 10%. Risky results may be given by the empirical formula with a turbulence intensity of more than 10%. e empirical formula is modified by introducing some modification coefficients. According to the FEM results, a modified empirical formula is obtained. Some numerical examples are presented with different tension classes to verify the validation of the present modified empirical formula.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Acknowledgments
is study was supported in part by the National Natural Science Foundation of China (U1734202) and the funding of Chengdu Guojia Electrical Engineering Co. Ltd. (no. NEEC-2019-B09).   Shock and Vibration 7