# Validation and Analysis on Numerical Response of Super-High-Speed Railway Pantograph-Catenary Interaction Based on Experimental Test

Authors: Jia Yang; Yang Song; Xiaobing Lu; Fuchuan Duan; Zhigang Liu; Ke Chen
Journal: Shock and Vibration
 (2021)

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## Abstract

The numerical tools can be used to facilitate the design of the railway pantograph-catenary system. The validation of the current numerical results is mostly performed at a speed slower than 350 km/h. This paper aims at the validation and analysis of the numerical results at a super-high-speed. The catenary model is constructed based on a nonlinear finite element approach employing the absolute nodal coordinate formulation. A multibody dynamics model is adopted to represent the pantograph. The measurement data are collected by an inspection vehicle equipped with an instrumented pantograph operating at 378 km/h in Chengdu-Chongqing high-speed line. Comparing the numerical simulation and the field test shows that the present pantograph-catenary model can provide reliable numerical results at 378 km/h. The numerical analysis of pantograph-catenary interaction at super-high-speed shows that the trailing pantograph performance does not comply with the assessment standard at 378 km/h. The adjustment of double-pantograph interval and messenger wire tension can effectively improve the trailing pantograph performance.


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## Body

## 1. Introduction

In the modern high-speed railway system, the pantograph-catenary system is responsible for powering electric trains. The pantograph is usually installed on the train roof to contact the catenary’s contact wire, constructed along the railroad, as shown in Figure1. Generally, the current collection quality is directly determined by the interaction performance of the pantograph and catenary. The ever-increasing of train speed results in significant technical challenges [1]. One of them is to keep a stable contact between the pantograph and the catenary and ensure the electric railway’s safe and reliable operation without traffic disruptions.Figure 1
Realistic pantograph and catenary systems: (a) high-speed catenary; (b) TSG19-type pantograph.
(a)(b)


### 1.1. Problem Description

Due to the high cost of a field test, numerical modelling has been the most popular approach to study the interaction performance of pantograph-catenary, to guarantee numerical accuracy has been an urgent issue to both the scientific community and the practical industry. That is why Bruni et al. [2] organized a comparison among ten mainstream software devices to set up a benchmark for the validation of numerical results. Based on the world benchmark, the European standard community announced the latest standard, En 50318, in 2018 [3]. However, the maximum validation speed only reaches 320 km/h. In some countries, the super-high-speed railway has experienced a distinct advancement in the last several decades. Especially in China, the operating speed for most high-speed railway has reached 350 km/h. The design speed for some newly built railways has reached 380 km/h or even 400 km/h. In the UK, the design speed for the HS2 project will be 360 km/h. The newly built Honam high-speed line in South Korea has a design speed of 400 km/h [4]. However, no mature standard can currently be used to validate the numerical results at more than 350 km/h. The validation at super-high-speed has been a topic of interest as it is crucial for understanding the dynamic behavior and facilitating the design and optimization of a pantograph-catenary system at super-high-speed.


### 1.2. Literature Review

The pantograph-catenary has been recognized as the most vulnerable part of the traction power system [5]. Therefore, the interaction performance has attracted the attention of many scholars and engineers. The numerical modelling technique of pantograph-catenary has experienced rapid development as the ever-increasing demand for the speed upgrade [6]. A lumped mass-spring system was usually used to represent the catenary in the early era [7]. Such a simple model can only describe the stiffness distribution along the span length and cannot describe the wave propagation in the contact wire [8]. With the speed upgrade, the wave propagation was proven to have an important implication on pantograph-catenary dynamic performance [9]. Therefore, many continuous modelling approaches were developed based on the finite element method (FEM) [10–12], the absolute nodal coordinate formulation (ANCF) [13, 14], the modal superposition method [15], and the finite difference method (FDM) [16]. All of them were compared and summarized in [2] to establish a benchmark to validate numerical results. The pantograph was typically modelled by a lumped mass model with two or three degrees of freedom (DOFs) [17], which can represent two or three primary modes for a realistic one [18]. This simplification is widely accepted as it can accurately describe the response at 0–20 Hz, which is specified to be the frequency of interest in En 50367 [19]. The advanced multibody pantograph model has been developed by some scholars [20, 21] to represent the physical configuration, which is of great importance to optimize the parameters [22], develop the controllers [23], and test passive dampers [24] for a better current collection quality. Apart from the simulation in ideal working conditions, the environmental perturbations and the pantograph-catenary errors are appropriately modelled to investigate their effect on the current collection quality. The wind field along the catenary was constructed [25, 26], and the buffeting behavior was analyzed in [27]. The traditional vortex-induced vibration in bridge engineering [28] was not the main issue for railway catenary. The aerodynamic instability has been found to have an incredible effect on the catenary [29, 30], which may cause a large amplitude vibration and make it impossible for the pantograph to operate [31]. The vehicle-track vibration was included as the vertical and spatial perturbations to assess pantograph-catenary interaction performance [32–34]. The catenary model’s ice-coating effect was included to evaluate the initial configuration and the contact force [35]. The catenary’s equilibrium state was calculated with a defective dropper, and its effect on the contact force was evaluated [36, 37]. The irregularities were included in the catenary to investigate their effect on the pantograph-catenary interaction [38, 39]. The dispersion of the results can be significantly observed with the random disturbance [40]. The detection methods for the irregularities are developed based on Ensemble Empirical Mode Decomposition (EEMD) [41] and time-frequency representations [42].To ensure numerical accuracy, some scholars devoted their attention to the field test and attempted to validate the numerical model. In [43], the simulation results were compared with several experimental tests to validate the contact force standard deviation. In [44], the catenary model was validated according to the laboratory test. However, most of these validations were performed at a speed slower than 350 km/h. The numerical accuracy at a higher speed cannot be guaranteed.


### 1.3. Contribution of This Paper

It is seen from the literature review that the numerical simulations of pantograph-catenary were seldom performed and validated at a super-high-speed. This shortcoming is tackled in this paper. A nonlinear finite element approach is utilized to model the catenary. The train speed is set up to 378 km/h to reproduce the physical behavior of pantograph-catenary interaction. An instrumented pantograph with several accelerometers is installed on the train, which runs on Chengdu-Chongqing high-speed line to collect the dynamic measurement data. The results obtained from the numerical simulation and the field test are compared. An optimization approach is proposed to improve the current collection quality based on a validated numerical model.


## 1.1. Problem Description

Due to the high cost of a field test, numerical modelling has been the most popular approach to study the interaction performance of pantograph-catenary, to guarantee numerical accuracy has been an urgent issue to both the scientific community and the practical industry. That is why Bruni et al. [2] organized a comparison among ten mainstream software devices to set up a benchmark for the validation of numerical results. Based on the world benchmark, the European standard community announced the latest standard, En 50318, in 2018 [3]. However, the maximum validation speed only reaches 320 km/h. In some countries, the super-high-speed railway has experienced a distinct advancement in the last several decades. Especially in China, the operating speed for most high-speed railway has reached 350 km/h. The design speed for some newly built railways has reached 380 km/h or even 400 km/h. In the UK, the design speed for the HS2 project will be 360 km/h. The newly built Honam high-speed line in South Korea has a design speed of 400 km/h [4]. However, no mature standard can currently be used to validate the numerical results at more than 350 km/h. The validation at super-high-speed has been a topic of interest as it is crucial for understanding the dynamic behavior and facilitating the design and optimization of a pantograph-catenary system at super-high-speed.


## 1.2. Literature Review

The pantograph-catenary has been recognized as the most vulnerable part of the traction power system [5]. Therefore, the interaction performance has attracted the attention of many scholars and engineers. The numerical modelling technique of pantograph-catenary has experienced rapid development as the ever-increasing demand for the speed upgrade [6]. A lumped mass-spring system was usually used to represent the catenary in the early era [7]. Such a simple model can only describe the stiffness distribution along the span length and cannot describe the wave propagation in the contact wire [8]. With the speed upgrade, the wave propagation was proven to have an important implication on pantograph-catenary dynamic performance [9]. Therefore, many continuous modelling approaches were developed based on the finite element method (FEM) [10–12], the absolute nodal coordinate formulation (ANCF) [13, 14], the modal superposition method [15], and the finite difference method (FDM) [16]. All of them were compared and summarized in [2] to establish a benchmark to validate numerical results. The pantograph was typically modelled by a lumped mass model with two or three degrees of freedom (DOFs) [17], which can represent two or three primary modes for a realistic one [18]. This simplification is widely accepted as it can accurately describe the response at 0–20 Hz, which is specified to be the frequency of interest in En 50367 [19]. The advanced multibody pantograph model has been developed by some scholars [20, 21] to represent the physical configuration, which is of great importance to optimize the parameters [22], develop the controllers [23], and test passive dampers [24] for a better current collection quality. Apart from the simulation in ideal working conditions, the environmental perturbations and the pantograph-catenary errors are appropriately modelled to investigate their effect on the current collection quality. The wind field along the catenary was constructed [25, 26], and the buffeting behavior was analyzed in [27]. The traditional vortex-induced vibration in bridge engineering [28] was not the main issue for railway catenary. The aerodynamic instability has been found to have an incredible effect on the catenary [29, 30], which may cause a large amplitude vibration and make it impossible for the pantograph to operate [31]. The vehicle-track vibration was included as the vertical and spatial perturbations to assess pantograph-catenary interaction performance [32–34]. The catenary model’s ice-coating effect was included to evaluate the initial configuration and the contact force [35]. The catenary’s equilibrium state was calculated with a defective dropper, and its effect on the contact force was evaluated [36, 37]. The irregularities were included in the catenary to investigate their effect on the pantograph-catenary interaction [38, 39]. The dispersion of the results can be significantly observed with the random disturbance [40]. The detection methods for the irregularities are developed based on Ensemble Empirical Mode Decomposition (EEMD) [41] and time-frequency representations [42].To ensure numerical accuracy, some scholars devoted their attention to the field test and attempted to validate the numerical model. In [43], the simulation results were compared with several experimental tests to validate the contact force standard deviation. In [44], the catenary model was validated according to the laboratory test. However, most of these validations were performed at a speed slower than 350 km/h. The numerical accuracy at a higher speed cannot be guaranteed.


## 1.3. Contribution of This Paper

It is seen from the literature review that the numerical simulations of pantograph-catenary were seldom performed and validated at a super-high-speed. This shortcoming is tackled in this paper. A nonlinear finite element approach is utilized to model the catenary. The train speed is set up to 378 km/h to reproduce the physical behavior of pantograph-catenary interaction. An instrumented pantograph with several accelerometers is installed on the train, which runs on Chengdu-Chongqing high-speed line to collect the dynamic measurement data. The results obtained from the numerical simulation and the field test are compared. An optimization approach is proposed to improve the current collection quality based on a validated numerical model.


## 2. Numerical Modelling of Pantograph-Catenary

A mathematical model of the pantograph-catenary system is built based on the ANCF and multibody dynamics in this section. A preliminary validation is performed to validate the numerical accuracy at 320 km/h operating speed.


### 2.1. Nonlinear Modelling of Catenary

The ANCF is a nonlinear finite element approach to describe the geometrical nonlinearity of large deformation. The ANCF beam element is utilized to model the tensioned wires (including contact wire, messenger wire, and stitch wire). The ANCF cable element is adopted to model the dropper wire. The steady arm is modelled by the truss element. The claws and clamps on the wire are assumed as lumped masses. The tangent stiffness matrices of the ANCF beam and cable elements have been derived in [13]. It should be noted that the axial stiffness of the cable element changes to zero when the dropper works in compression. In this way, the dropper behavior in slackness can be described.The mass matrixMCG and the stiffness matrix KCG are assembled by the element matrix of each catenary component as follows:(1)MCG=∑ncwMcw,ne+∑nmwMmw,ne+∑ndrMdr,ne+∑ncsMsa,ne+∑nclMcl,ne,KCG=∑ncwKcw,ne+∑nmwKmw,ne+∑ndrKdr,ne+∑ncsKsa,ne+∑nmsKms,ne,in which Mcw,ne, Mmw,ne, Mdr,ne, Msa,ne and Mcl,ne are the element mass matrix of contact wire, messenger wire, dropper, steady arm, and claws, respectively. Kcw,ne, Kmw,ne, Kdr,ne, Ksa,ne and Kms,ne are the corresponding stiffness matrices. The initial configuration can be iteratively calculated through a TCUD-ANCF combined method reported in [13], in which the unstrained length of each element is taken as unknown to solve the equilibrium equation. Employing a Rayleigh damping matrix CCG, the equation of motion for the catenary can be written by(2)MCGU¨Ct+CCGU˙Ct+KCGUCt=FCGt,in which UCt and FCGt are the vectors of DOF and external force. The stiffness matrix KCG is updated in each iterative step to account for the dropper slackness and geometrical nonlinearity. The Rayleigh damping coefficients identified from a realistic China high-speed railway catenary [45] are adopted in the numerical model.


### 2.2. Modelling of the Pantograph

A two-dimensional geometry of a pantograph is presented in Figure2, in which F, G, and P are the centers of mass of the lower link, lower arm, and upper arm, respectively. m1, m2, and m3 are their respective masses; J1, J2, and J3 are the corresponding moments of inertia; l1, l2, and l3 are the corresponding lengths for each part; l4 is the length of CD; l5, l6, and l7 are the lengths of FA, GB, and PC respectively; CA, CB, CC and CD are the damping of revolute joints A, B, C, and D, respectively; α is the rising angle of the pantograph; M0 is the static uplift moment; zEt and zht are the displacement of the upper framework and the pantograph collector, respectively. mh, kh, and ch are the mass, stiffness, and damping of the pantograph collector, respectively. le and lf are, respectively, the vertical and horizontal distances between the two joints A and B on the base. δ is the fixed angle from the upper arm to arm CD. fc is the contact force on the pantograph collector.Figure 2
Pantograph geometry configuration.Lagrange’s equations of motion for the pantograph configuration in terms of the rising angleα and the vertical displacement of the pantograph collector zht can be written by(3a)ddt∂L∂α˙−∂L∂α=Q1,(3b)ddt∂L∂z˙h−∂L∂zh=Q2,in which L is the difference between the kinetic energy and the potential energy. Q1 and Q2 are the generalized forces acting on the two degrees-of-freedom α and zht, respectively, which can be derived from the principle of virtual work. The equations of motion for the multibody pantograph can be written as follows:(4a)h1αα¨+h2αα˙2+h3αα˙+h4α+h5α,α˙,zht,z˙ht×Δα−M0=0,(4b)mhz¨ht−h5α,α˙,zht,z˙ht+fct=0,in which the definitions of h1α, h2α, h3α, h4α, h5α,α˙,zht,z˙ht and Δα are given as follows:(5a)h1α=m1l52+J1+m2q12l62+q12J2+m3l12+m3q22l72+2m3q2l1l7cosα+γ+q22J3+mh−l3q2sinγ+l1sinα2,(5b)h2α=m2q1l62∂q1∂α+q1J2∂q1∂α+m3q2l72∂q2∂α+m3l1l7cosα+γ∂q2∂α−q21+q2m3l1l7sinα+γ+q2J3∂q2∂α+mh−l3q2sinγ+l1sinα−l3∂q2∂αsinγ−l3q22cosγ+l1cosα,(5c)h3α=CA+CBq12+CC1+q22+CDq1+q22,(5d)h4α=−khzht−l1sinα−l3sinγl1cosα+q2l3cosγ+m1gl5cosα+q1m2gl6cosβ+m3gl1cosα+m3q2gl7cosγ,(5e)h5α,α˙,zht,z˙ht=−chz˙ht−l1cosα+l3q2cosγα˙−kHzht−l1sinα−l3sinγ,(5f)Δα=l1cosα+q2l3cosγ,where(6a)β=arctanl1sinα−lel1cosα+lf+arccosl22+lt2−l422l2lt,(6b)γ=arccosl42+lt2−l222l4lt−arctanl1sinα−lel1cosα+lf−δ,(6c)q1=l12+lfl1cosα−lel1sinαl12+le2+lf2+2lfl1cosα−2lel1sinα+lt2+l42−l22lt4lt2l22−lt2+l22−l422l1ltlfsinα+lecosα,(6d)q2=−l12+lfl1cosα−lel1sinαl12+le2+lf2+2lfl1cosα−2lel1sinα+lt2+l22−l42lt4lt2l22−lt2+l42−l222l1ltlfsinα+lecosα(6e)lt=l12+le2+lf2+2lfl1cosα−2lel1sinα.


### 2.3. Modelling of Contact

The contact between the pantograph collector and contact wire is described by the penalty function method shown in equation (15).(7)fc=kcyp−yc,yp≥yc,0,yp<yc.In this work, a very large contact stiffness of 20 0000 N/m and a high sampling frequency of 2000 Hz is adopted in the numerical simulation to ensure numerical stability. The element size in the contact wire is set as 0.25 m to ensure that numerical error cannot affect the results in the frequency range of interest. Using equation (15), the equation of motion for the pantograph-catenary system can be obtained as(8)MGU¨t+CGU˙t+KGtUt=FGt,in which MG, CG and KGt are the mass, damping, and stiffness matrices for the whole pantograph-catenary system, respectively. FGt is the external force vector. A Newmark integration scheme is adopted to solve equation (16). The stiffness matrix KGt is updated each time step to adequately describe the nonlinearity from the wire deformation and the dropper slackness.


### 2.4. Preliminary Verification

The work in [2] provides a benchmark for both static and dynamic validations of a pantograph-catenary model. Based on the reference model parameters in the benchmark, the pre-sag and elasticity of the contact wire calculated by the present model are compared with the reference values in Table 1. It shows that the pre-sag obtained by the present method is precisely the same as the benchmark. The maximum error of the elasticity is just 6.52%. The dynamic validation is implemented by introducing a lumped mass pantograph model operating at 320 km/h. The contact force statistics are compared with the benchmark in Table 2. The most significant error only reaches 6.9% for the minimum contact force, which is still smaller than the threshold.Table 1
Static validation of the present model against the benchmark.
Dropper No.Pre-sagElasticityBenchmark (mm)Present (mm)Error (%)Benchmark (mm/N)Present (mm/N)Error (%)Sup0000.2060.192576.5210000.1650.156475.172242400.2730.267741.933414100.3450.32685.284525200.3880.368325.075555500.40.375096.236525200.3880.368325.077414100.3450.32685.288242400.2730.267741.9390000.1650.156475.17Sup0000.2060.192576.52Table 2
Dynamic validation of the present model against the benchmark.
BenchmarkPresent modelError (%)Fm (N)169169.150.09σ (0–20 Hz) (N)53.9152.592.45σ (0–2 Hz) (N)38.2738.250.05σ (0–5 Hz) (N)41.0441.000.10σ (5–20 Hz) (N)34.8032.995.20Fmax (N)313.22305.852.35Fmin (N)60.4056.226.9


## 2.1. Nonlinear Modelling of Catenary

The ANCF is a nonlinear finite element approach to describe the geometrical nonlinearity of large deformation. The ANCF beam element is utilized to model the tensioned wires (including contact wire, messenger wire, and stitch wire). The ANCF cable element is adopted to model the dropper wire. The steady arm is modelled by the truss element. The claws and clamps on the wire are assumed as lumped masses. The tangent stiffness matrices of the ANCF beam and cable elements have been derived in [13]. It should be noted that the axial stiffness of the cable element changes to zero when the dropper works in compression. In this way, the dropper behavior in slackness can be described.The mass matrixMCG and the stiffness matrix KCG are assembled by the element matrix of each catenary component as follows:(1)MCG=∑ncwMcw,ne+∑nmwMmw,ne+∑ndrMdr,ne+∑ncsMsa,ne+∑nclMcl,ne,KCG=∑ncwKcw,ne+∑nmwKmw,ne+∑ndrKdr,ne+∑ncsKsa,ne+∑nmsKms,ne,in which Mcw,ne, Mmw,ne, Mdr,ne, Msa,ne and Mcl,ne are the element mass matrix of contact wire, messenger wire, dropper, steady arm, and claws, respectively. Kcw,ne, Kmw,ne, Kdr,ne, Ksa,ne and Kms,ne are the corresponding stiffness matrices. The initial configuration can be iteratively calculated through a TCUD-ANCF combined method reported in [13], in which the unstrained length of each element is taken as unknown to solve the equilibrium equation. Employing a Rayleigh damping matrix CCG, the equation of motion for the catenary can be written by(2)MCGU¨Ct+CCGU˙Ct+KCGUCt=FCGt,in which UCt and FCGt are the vectors of DOF and external force. The stiffness matrix KCG is updated in each iterative step to account for the dropper slackness and geometrical nonlinearity. The Rayleigh damping coefficients identified from a realistic China high-speed railway catenary [45] are adopted in the numerical model.


## 2.2. Modelling of the Pantograph

A two-dimensional geometry of a pantograph is presented in Figure2, in which F, G, and P are the centers of mass of the lower link, lower arm, and upper arm, respectively. m1, m2, and m3 are their respective masses; J1, J2, and J3 are the corresponding moments of inertia; l1, l2, and l3 are the corresponding lengths for each part; l4 is the length of CD; l5, l6, and l7 are the lengths of FA, GB, and PC respectively; CA, CB, CC and CD are the damping of revolute joints A, B, C, and D, respectively; α is the rising angle of the pantograph; M0 is the static uplift moment; zEt and zht are the displacement of the upper framework and the pantograph collector, respectively. mh, kh, and ch are the mass, stiffness, and damping of the pantograph collector, respectively. le and lf are, respectively, the vertical and horizontal distances between the two joints A and B on the base. δ is the fixed angle from the upper arm to arm CD. fc is the contact force on the pantograph collector.Figure 2
Pantograph geometry configuration.Lagrange’s equations of motion for the pantograph configuration in terms of the rising angleα and the vertical displacement of the pantograph collector zht can be written by(3a)ddt∂L∂α˙−∂L∂α=Q1,(3b)ddt∂L∂z˙h−∂L∂zh=Q2,in which L is the difference between the kinetic energy and the potential energy. Q1 and Q2 are the generalized forces acting on the two degrees-of-freedom α and zht, respectively, which can be derived from the principle of virtual work. The equations of motion for the multibody pantograph can be written as follows:(4a)h1αα¨+h2αα˙2+h3αα˙+h4α+h5α,α˙,zht,z˙ht×Δα−M0=0,(4b)mhz¨ht−h5α,α˙,zht,z˙ht+fct=0,in which the definitions of h1α, h2α, h3α, h4α, h5α,α˙,zht,z˙ht and Δα are given as follows:(5a)h1α=m1l52+J1+m2q12l62+q12J2+m3l12+m3q22l72+2m3q2l1l7cosα+γ+q22J3+mh−l3q2sinγ+l1sinα2,(5b)h2α=m2q1l62∂q1∂α+q1J2∂q1∂α+m3q2l72∂q2∂α+m3l1l7cosα+γ∂q2∂α−q21+q2m3l1l7sinα+γ+q2J3∂q2∂α+mh−l3q2sinγ+l1sinα−l3∂q2∂αsinγ−l3q22cosγ+l1cosα,(5c)h3α=CA+CBq12+CC1+q22+CDq1+q22,(5d)h4α=−khzht−l1sinα−l3sinγl1cosα+q2l3cosγ+m1gl5cosα+q1m2gl6cosβ+m3gl1cosα+m3q2gl7cosγ,(5e)h5α,α˙,zht,z˙ht=−chz˙ht−l1cosα+l3q2cosγα˙−kHzht−l1sinα−l3sinγ,(5f)Δα=l1cosα+q2l3cosγ,where(6a)β=arctanl1sinα−lel1cosα+lf+arccosl22+lt2−l422l2lt,(6b)γ=arccosl42+lt2−l222l4lt−arctanl1sinα−lel1cosα+lf−δ,(6c)q1=l12+lfl1cosα−lel1sinαl12+le2+lf2+2lfl1cosα−2lel1sinα+lt2+l42−l22lt4lt2l22−lt2+l22−l422l1ltlfsinα+lecosα,(6d)q2=−l12+lfl1cosα−lel1sinαl12+le2+lf2+2lfl1cosα−2lel1sinα+lt2+l22−l42lt4lt2l22−lt2+l42−l222l1ltlfsinα+lecosα(6e)lt=l12+le2+lf2+2lfl1cosα−2lel1sinα.


## 2.3. Modelling of Contact

The contact between the pantograph collector and contact wire is described by the penalty function method shown in equation (15).(7)fc=kcyp−yc,yp≥yc,0,yp<yc.In this work, a very large contact stiffness of 20 0000 N/m and a high sampling frequency of 2000 Hz is adopted in the numerical simulation to ensure numerical stability. The element size in the contact wire is set as 0.25 m to ensure that numerical error cannot affect the results in the frequency range of interest. Using equation (15), the equation of motion for the pantograph-catenary system can be obtained as(8)MGU¨t+CGU˙t+KGtUt=FGt,in which MG, CG and KGt are the mass, damping, and stiffness matrices for the whole pantograph-catenary system, respectively. FGt is the external force vector. A Newmark integration scheme is adopted to solve equation (16). The stiffness matrix KGt is updated each time step to adequately describe the nonlinearity from the wire deformation and the dropper slackness.


## 2.4. Preliminary Verification

The work in [2] provides a benchmark for both static and dynamic validations of a pantograph-catenary model. Based on the reference model parameters in the benchmark, the pre-sag and elasticity of the contact wire calculated by the present model are compared with the reference values in Table 1. It shows that the pre-sag obtained by the present method is precisely the same as the benchmark. The maximum error of the elasticity is just 6.52%. The dynamic validation is implemented by introducing a lumped mass pantograph model operating at 320 km/h. The contact force statistics are compared with the benchmark in Table 2. The most significant error only reaches 6.9% for the minimum contact force, which is still smaller than the threshold.Table 1
Static validation of the present model against the benchmark.
Dropper No.Pre-sagElasticityBenchmark (mm)Present (mm)Error (%)Benchmark (mm/N)Present (mm/N)Error (%)Sup0000.2060.192576.5210000.1650.156475.172242400.2730.267741.933414100.3450.32685.284525200.3880.368325.075555500.40.375096.236525200.3880.368325.077414100.3450.32685.288242400.2730.267741.9390000.1650.156475.17Sup0000.2060.192576.52Table 2
Dynamic validation of the present model against the benchmark.
BenchmarkPresent modelError (%)Fm (N)169169.150.09σ (0–20 Hz) (N)53.9152.592.45σ (0–2 Hz) (N)38.2738.250.05σ (0–5 Hz) (N)41.0441.000.10σ (5–20 Hz) (N)34.8032.995.20Fmax (N)313.22305.852.35Fmin (N)60.4056.226.9


## 3. Validation with Experimental Test

To validate the numerical model presented above and analyze the pantograph-catenary interaction performance at super-higher-speed, an instrumented pantograph (see Figure3) is mounted on an inspection vehicle (see Figure 4), which is regularly running on China high-speed network. According to En 50317 [46], the instrumented pantograph is equipped with four accelerometers on its pantograph collector, collecting the inertial part of the contact force. Two spring sensors are placed under the panhead to measure the inner forces between the panhead and the framework. The contact force can be seen as the sum of the inner forces, inertial forces, and the aerodynamic force as follows:(9)fc=∑i=1nffinner,i+meqna∑i=1naahead,i+faero,in which finner,i is the inner force measured by sensors 10 and 11. nf is the number of spring sensors. meq is the equivalent mass of the pantograph head. na is the number of accelerometers on the pantograph head. ahead,i is the acceleration measured by each accelerometer. faero is the aerodynamic correction part, which has been determined in a wind tunnel test.Figure 3
Instrumented pantograph to measure contact force.Figure 4
High-speed inspection vehicle.The measurement contact force is collected from a super-high-speed test in Chengdu-Chongqing high-speed line. The design speed for this line is 350 km/h. The running speed of the inspection vehicle reaches 378 km/h in this super-high-speed test. The two tensile sections (from 83.093 km to 85.981 km) are taken as the analysis object. According to the design data, the catenary model with two tensile sections is constructed using the aforementioned method. The initial configuration of the catenary is presented in Figure5. Then, the dynamic simulation is performed with a TSG-19-type pantograph. The measurement and simulation contact forces are presented in Figure 6. It is seen that the fluctuation range of the simulation contact force shows a good agreement with the measurement contact force. According to En 50317 [46], the measurement data has an up to 10% inevitable error due to the limitation of the measurement equipment. Therefore, the contact force waveform cannot be directly used for comparison. Some statistics of the contact force and uplift specified in En 50318 [3] are typically used to validate the numerical model. The comparison of these statistics is presented in Table 3. It is seen that the most important indicator, contact force standard deviation evaluated by the present model, only has a 4.17% error against the measurement data, which is much smaller than the threshold of 20%. The uplifts of the panhead and the support are almost identical to the measurement values. Even though the actual maximum and minimum contact forces are not included in the validation in En 50318, the most significant difference of these values against the measurement data is still smaller than 20%. Through the comparison, it can be demonstrated that the present model has good performance to evaluate the comprehensive and local behaviors of the pantograph-catenary interaction.Figure 5
Catenary initial configuration using the design data of Chengdu-Chongqing high-speed line: (a) front view; (b) side view; (c) top view.
(a)(b)(c)Figure 6
Contact force time history: (a) measurement; (b) simulation.
(a)(b)Table 3
Comparison of critical indicators between simulation and measurement.
MeasurementSimulationErrorThresholdSpeed (km/h)378378——Mean (N)189.24 N189.96 N0.72 N±2.5 NStandard deviation (N)35.26 N36.73 N4.17%±20%Statistical maximum (N)288.8 N300.15 N3.9%—Statistical minimum (N)87.5 N79.77 N8.83%—Range of panhead vibration68 mm68.3 mm0.3 mm±20 mmSupport uplift90 mm90.82 mm0.82 mm−10 mm; +20 mmActual maximum315.00 N298.68 N5.18%—Actual minimum85.00 N70.30 N17.29%—


## 4. Assessment with Numerical Results

The current assessment standard En 50367 [19] stipulates that the statistics of the contact force and uplift should satisfy the criteria. When the train speed is over 200 km/h, the mean contact force should be over 0.00047v2+60N, but smaller than 0.00097v2+70N. The statistical minimum contact force should be positive, while the actual minimum should be over 0 N for Chine high-speed network. The statistical and actual maximum contact force should be smaller than 350 N at up to 350 km/h. There is no current safety threshold for the maximum contact force at super-high-speed. It is expected that this threshold value can be increased with the increase of the mean contact force. In this section, the assessments are performed according to the standard [19] based on the numerical results. Both single-pantograph and double-pantograph operations in practice are considered.


### 4.1. Assessment with Single Pantograph

In the daily operation of China high-speed network, most electric motor units (EMUs) are equipped with one pantograph. When a single pantograph is used, the assessment indicators are presented in Table4. It can be seen that all the indicators satisfy the criteria of the standard. The pantograph-catenary system shows an acceptable performance when running at 378 km/h.Table 4
Assessment of simulation results against En 50367 (single pantograph).
Simulation resultAcceptanceMean (N)189.96 N>127.16 N and < 208.60Statistical maximum (N)300.15 N<350 N (up to 350 km/h)Statistical minimum (N)79.77 N>0 NSupport uplift90.82 mm<150 mmActual maximum298.68 N<350 N (up to 350 km/h)Actual minimum70.30 N>10 N


### 4.2. Assessment with Double Pantographs

However, some EMUs are equipped with double pantographs to improve the carrying capacity. When double pantographs are used, the trailing pantograph performance is often deteriorated by wave propagation in the catenary [47]. A standard distance of 210 m between the two pantographs is used in the simulation. The contact forces of leading and trailing pantographs are presented in Figure 7. It is seen that the leading pantograph has a significantly better performance than the trailing one. The fluctuation of the trailing pantograph contact force is much more violent than the leading one. Even some contact loss can be observed. Table 5 collects the assessment indicators for both leading and trailing pantographs. It can be seen that the behavior of the leading pantograph is very similar to a single pantograph. However, almost all the indicators of the trailing pantograph do not satisfy the criteria. When the train is operating at 378 km/h, the trailing pantograph performance cannot pass the acceptance. In Section 4.3, some suggestions are proposed to improve the trailing pantograph performance at a super-high-speed.Figure 7
Simulation contact force with double pantographs.Table 5
Assessment of simulation results against En 50367 (double pantograph).
Leading pantographTrailing pantographAcceptanceMean (N)189.45 N189.58>127.16 N and < 208.60Statistical maximum (N)300.43 N385.90 N<350 N (up to 350 km/h)Statistical minimum (N)78.47 N−6.74 N>0 NSupport uplift92.4 mm101.2 mm<150 mmActual maximum295.81 N393.82 N<350 N (up to 350 km/h)Actual minimum86.51 N0 N>0 NUnsatisfactory results are in bold.


### 4.3. Suggestions for Improving Trailing Pantograph Performance at Super-High-Speed

The most effective measure to improve the trailing pantograph performance is to increase the tension in the contact wire. The current tension in the contact wire has reached 30 kN. Due to the material limitation, the extra tension will increase the risk of the contact wire breakage. Therefore, this section investigates the potential measure to improve the trailing pantograph performance at a given tension class. According to the authors’ previous research [48], the trailing pantograph performance depends on the match between the catenary mode excited by the pantograph and the double pantograph’s interval. If the pantographs’ interval is consistent with the structural wavelength (which can be extracted from the mode), a resonance will happen, and the trailing pantograph’s bad performance is expected. Based on this theory, the pantograph interval or messenger wire tension may be adjusted to avoid the resonance and achieve good performance for the trailing pantograph.


#### 4.3.1. Adjustment of Pantograph Interval

The nominal pantograph interval is 210 m. According to China railway standard, the minimum and maximum pantograph intervals are 200 m and 215 m, respectively. In this analysis, the pantograph interval is defined (from 190 m to 230 m) slightly over the standard range to investigate the potential performance. The dynamic simulations of double pantographs-catenary interaction are performed with different messenger wire tensions. The contact force statistics of the leading pantograph are presented in Figure8. It is seen that the change of the pantograph interval does not have a distinct effect on the leading pantograph performance. The contact force statistics of the trailing pantograph are presented in Figure 9. Unlike the leading pantograph, the trailing pantograph behavior is significantly changed by slightly adjusting the pantograph interval. When the interval is from 190 m to 205 m, a positive statistical minimum contact force can be obtained. Especially, when the interval is 205 m, both the statistical and actual minimum contact forces are positive. It should be noted that that the maximum contact forces of the trailing pantograph are always more extensive than the threshold 350 N. But this threshold is defined for the operating speed lower than 350 km/h and should be increased at super-high-speed as the increase of the mean contact force. Therefore, the numerical result indicates an acceptable performance of the trailing pantograph at the pantograph interval of 205 m.Figure 8
Contact force statistics of the leading pantograph with different pantograph intervals.Figure 9
Contact force statistics of the trailing pantograph with different pantograph intervals.Figure 10
Contact force statistics of the leading pantograph versus messenger wire tension.


#### 4.3.2. Adjustment of Messenger Wire Tension

The nominal messenger wire tension for the analysis catenary is 21 kN. In the numerical simulations, the messenger wire tension is changed from 19 kN to 29 kN, which does not exceed the contact wire tension. The contact force statistics of the leading pantograph with different messenger wire tensions are presented in Figure10. It is seen that the change of the messenger wire tension has some effect on the leading pantograph performance. Nevertheless, all the statistics are still within the acceptance range. When the messenger wire tension is 29 kN, the leading pantograph has the best performance. The contact force statistics of the trailing pantograph are presented in Figure 11. It is seen that, for the messenger wire tensions from 19 kN to 25 kN, the minimum contact forces are smaller than 0 N. Only when the messenger wire tensions are 27 kN and 29 kN, both the statistical and actual minimum contact forces are positive. The actual and statistical maximum contact forces are also relatively lower than in other cases.Figure 11
Contact force statistics of the trailing pantograph versus messenger wire tension.


#### 4.3.3. Adjustment of Both Pantograph Interval and Messenger Wire Tension

The actual and statistical minimum contact force of the trailing pantograph versus messenger wire tension and pantograph interval are presented in Figures12 and 13, respectively. It is seen that when the messenger wire tension increases to 27 kN and 29 kN, the trailing pantograph generally has a more acceptable performance. With a messenger wire tension of 27 kN, both actual and statistical minimum contact forces are positive at a pantograph interval from 190 to 215 m. With a messenger wire tension of 29 kN, both actual and statistical minimum contact forces are positive for most pantograph intervals except 205 m, 220 m, and 230 m. For other messenger wire tensions, only two acceptable cases can be observed.Figure 12
Actual minimum contact force of the trailing pantograph versus messenger wire tension and pantograph interval.Figure 13
Statistical minimum contact force of the trailing pantograph versus messenger wire tension and pantograph interval.


## 4.1. Assessment with Single Pantograph

In the daily operation of China high-speed network, most electric motor units (EMUs) are equipped with one pantograph. When a single pantograph is used, the assessment indicators are presented in Table4. It can be seen that all the indicators satisfy the criteria of the standard. The pantograph-catenary system shows an acceptable performance when running at 378 km/h.Table 4
Assessment of simulation results against En 50367 (single pantograph).
Simulation resultAcceptanceMean (N)189.96 N>127.16 N and < 208.60Statistical maximum (N)300.15 N<350 N (up to 350 km/h)Statistical minimum (N)79.77 N>0 NSupport uplift90.82 mm<150 mmActual maximum298.68 N<350 N (up to 350 km/h)Actual minimum70.30 N>10 N


## 4.2. Assessment with Double Pantographs

However, some EMUs are equipped with double pantographs to improve the carrying capacity. When double pantographs are used, the trailing pantograph performance is often deteriorated by wave propagation in the catenary [47]. A standard distance of 210 m between the two pantographs is used in the simulation. The contact forces of leading and trailing pantographs are presented in Figure 7. It is seen that the leading pantograph has a significantly better performance than the trailing one. The fluctuation of the trailing pantograph contact force is much more violent than the leading one. Even some contact loss can be observed. Table 5 collects the assessment indicators for both leading and trailing pantographs. It can be seen that the behavior of the leading pantograph is very similar to a single pantograph. However, almost all the indicators of the trailing pantograph do not satisfy the criteria. When the train is operating at 378 km/h, the trailing pantograph performance cannot pass the acceptance. In Section 4.3, some suggestions are proposed to improve the trailing pantograph performance at a super-high-speed.Figure 7
Simulation contact force with double pantographs.Table 5
Assessment of simulation results against En 50367 (double pantograph).
Leading pantographTrailing pantographAcceptanceMean (N)189.45 N189.58>127.16 N and < 208.60Statistical maximum (N)300.43 N385.90 N<350 N (up to 350 km/h)Statistical minimum (N)78.47 N−6.74 N>0 NSupport uplift92.4 mm101.2 mm<150 mmActual maximum295.81 N393.82 N<350 N (up to 350 km/h)Actual minimum86.51 N0 N>0 NUnsatisfactory results are in bold.


## 4.3. Suggestions for Improving Trailing Pantograph Performance at Super-High-Speed

The most effective measure to improve the trailing pantograph performance is to increase the tension in the contact wire. The current tension in the contact wire has reached 30 kN. Due to the material limitation, the extra tension will increase the risk of the contact wire breakage. Therefore, this section investigates the potential measure to improve the trailing pantograph performance at a given tension class. According to the authors’ previous research [48], the trailing pantograph performance depends on the match between the catenary mode excited by the pantograph and the double pantograph’s interval. If the pantographs’ interval is consistent with the structural wavelength (which can be extracted from the mode), a resonance will happen, and the trailing pantograph’s bad performance is expected. Based on this theory, the pantograph interval or messenger wire tension may be adjusted to avoid the resonance and achieve good performance for the trailing pantograph.


### 4.3.1. Adjustment of Pantograph Interval

The nominal pantograph interval is 210 m. According to China railway standard, the minimum and maximum pantograph intervals are 200 m and 215 m, respectively. In this analysis, the pantograph interval is defined (from 190 m to 230 m) slightly over the standard range to investigate the potential performance. The dynamic simulations of double pantographs-catenary interaction are performed with different messenger wire tensions. The contact force statistics of the leading pantograph are presented in Figure8. It is seen that the change of the pantograph interval does not have a distinct effect on the leading pantograph performance. The contact force statistics of the trailing pantograph are presented in Figure 9. Unlike the leading pantograph, the trailing pantograph behavior is significantly changed by slightly adjusting the pantograph interval. When the interval is from 190 m to 205 m, a positive statistical minimum contact force can be obtained. Especially, when the interval is 205 m, both the statistical and actual minimum contact forces are positive. It should be noted that that the maximum contact forces of the trailing pantograph are always more extensive than the threshold 350 N. But this threshold is defined for the operating speed lower than 350 km/h and should be increased at super-high-speed as the increase of the mean contact force. Therefore, the numerical result indicates an acceptable performance of the trailing pantograph at the pantograph interval of 205 m.Figure 8
Contact force statistics of the leading pantograph with different pantograph intervals.Figure 9
Contact force statistics of the trailing pantograph with different pantograph intervals.Figure 10
Contact force statistics of the leading pantograph versus messenger wire tension.


### 4.3.2. Adjustment of Messenger Wire Tension

The nominal messenger wire tension for the analysis catenary is 21 kN. In the numerical simulations, the messenger wire tension is changed from 19 kN to 29 kN, which does not exceed the contact wire tension. The contact force statistics of the leading pantograph with different messenger wire tensions are presented in Figure10. It is seen that the change of the messenger wire tension has some effect on the leading pantograph performance. Nevertheless, all the statistics are still within the acceptance range. When the messenger wire tension is 29 kN, the leading pantograph has the best performance. The contact force statistics of the trailing pantograph are presented in Figure 11. It is seen that, for the messenger wire tensions from 19 kN to 25 kN, the minimum contact forces are smaller than 0 N. Only when the messenger wire tensions are 27 kN and 29 kN, both the statistical and actual minimum contact forces are positive. The actual and statistical maximum contact forces are also relatively lower than in other cases.Figure 11
Contact force statistics of the trailing pantograph versus messenger wire tension.


### 4.3.3. Adjustment of Both Pantograph Interval and Messenger Wire Tension

The actual and statistical minimum contact force of the trailing pantograph versus messenger wire tension and pantograph interval are presented in Figures12 and 13, respectively. It is seen that when the messenger wire tension increases to 27 kN and 29 kN, the trailing pantograph generally has a more acceptable performance. With a messenger wire tension of 27 kN, both actual and statistical minimum contact forces are positive at a pantograph interval from 190 to 215 m. With a messenger wire tension of 29 kN, both actual and statistical minimum contact forces are positive for most pantograph intervals except 205 m, 220 m, and 230 m. For other messenger wire tensions, only two acceptable cases can be observed.Figure 12
Actual minimum contact force of the trailing pantograph versus messenger wire tension and pantograph interval.Figure 13
Statistical minimum contact force of the trailing pantograph versus messenger wire tension and pantograph interval.


## 4.3.1. Adjustment of Pantograph Interval

The nominal pantograph interval is 210 m. According to China railway standard, the minimum and maximum pantograph intervals are 200 m and 215 m, respectively. In this analysis, the pantograph interval is defined (from 190 m to 230 m) slightly over the standard range to investigate the potential performance. The dynamic simulations of double pantographs-catenary interaction are performed with different messenger wire tensions. The contact force statistics of the leading pantograph are presented in Figure8. It is seen that the change of the pantograph interval does not have a distinct effect on the leading pantograph performance. The contact force statistics of the trailing pantograph are presented in Figure 9. Unlike the leading pantograph, the trailing pantograph behavior is significantly changed by slightly adjusting the pantograph interval. When the interval is from 190 m to 205 m, a positive statistical minimum contact force can be obtained. Especially, when the interval is 205 m, both the statistical and actual minimum contact forces are positive. It should be noted that that the maximum contact forces of the trailing pantograph are always more extensive than the threshold 350 N. But this threshold is defined for the operating speed lower than 350 km/h and should be increased at super-high-speed as the increase of the mean contact force. Therefore, the numerical result indicates an acceptable performance of the trailing pantograph at the pantograph interval of 205 m.Figure 8
Contact force statistics of the leading pantograph with different pantograph intervals.Figure 9
Contact force statistics of the trailing pantograph with different pantograph intervals.Figure 10
Contact force statistics of the leading pantograph versus messenger wire tension.


## 4.3.2. Adjustment of Messenger Wire Tension

The nominal messenger wire tension for the analysis catenary is 21 kN. In the numerical simulations, the messenger wire tension is changed from 19 kN to 29 kN, which does not exceed the contact wire tension. The contact force statistics of the leading pantograph with different messenger wire tensions are presented in Figure10. It is seen that the change of the messenger wire tension has some effect on the leading pantograph performance. Nevertheless, all the statistics are still within the acceptance range. When the messenger wire tension is 29 kN, the leading pantograph has the best performance. The contact force statistics of the trailing pantograph are presented in Figure 11. It is seen that, for the messenger wire tensions from 19 kN to 25 kN, the minimum contact forces are smaller than 0 N. Only when the messenger wire tensions are 27 kN and 29 kN, both the statistical and actual minimum contact forces are positive. The actual and statistical maximum contact forces are also relatively lower than in other cases.Figure 11
Contact force statistics of the trailing pantograph versus messenger wire tension.


## 4.3.3. Adjustment of Both Pantograph Interval and Messenger Wire Tension

The actual and statistical minimum contact force of the trailing pantograph versus messenger wire tension and pantograph interval are presented in Figures12 and 13, respectively. It is seen that when the messenger wire tension increases to 27 kN and 29 kN, the trailing pantograph generally has a more acceptable performance. With a messenger wire tension of 27 kN, both actual and statistical minimum contact forces are positive at a pantograph interval from 190 to 215 m. With a messenger wire tension of 29 kN, both actual and statistical minimum contact forces are positive for most pantograph intervals except 205 m, 220 m, and 230 m. For other messenger wire tensions, only two acceptable cases can be observed.Figure 12
Actual minimum contact force of the trailing pantograph versus messenger wire tension and pantograph interval.Figure 13
Statistical minimum contact force of the trailing pantograph versus messenger wire tension and pantograph interval.


## 5. Conclusions

Most current numerical simulations of pantograph-catenary are mostly performed and validated at a speed of no more than 350 km/h. In this paper, an instrumented pantograph is mounted on a high-speed inspection vehicle operating at 378 km/h in China high-speed network. A numerical model of pantograph-catenary is constructed based on ANCF and multibody dynamics. Comparing the contact force and dynamic uplift between the numerical simulation and the field test is performed. Some conclusions can be drawn as follows:(1)
The proposed pantograph-catenary model can provide reliable numerical results at 378 km/h. The statistics of contact force and uplift have a high similarity with the measurement data.(2)
When a single pantograph is used, the pantograph-catenary interaction performance still satisfies the assessment standard at 378 km/h. However, when the double pantographs are used, the trailing pantograph performance is no longer acceptable.(3)
The adjustment of double-pantograph interval and messenger wire tension can effectively improve the trailing pantograph performance. These measures are promising in practice as they do not need to increase the catenary tension class.It should be noted that the assessment in this paper uses the contact force with a 20 Hz cut-off frequency. This frequency range may not be enough to describe the interaction behaviors of the pantograph-catenary and deserves to be improved. But it will entail some necessary fundamental research in the future, including the development of measurement equipment for high-frequency data, the numerical scheme for simulating high-frequency behaviors, and the appropriate modelling of the disturbances, which may affect high-frequency performance. The analysis of this paper also indicates that the pantograph interval or catenary parameters may be adjusted to achieve good performance for the trailing pantograph. But it is not easy to give an analytical explanation on the selection of pantographs’ interval or catenary parameters. Firstly, the catenary has infinite modes mathematically. One or several dominant modes are excited by moving pantographs with a given speed. An appropriate solution is desired to explicitly determine which mode is the dominant one at a given speed. Secondly, the catenary mode is time-varying due to the strong nonlinearity, including the geometrical nonlinearity and dropper slackness. Reasonable solutions are expected to quantify the effect of the time-varying mode on the interaction performance.



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*Source: 9922404-2021-08-17.xml*