Crosswind Effects on Interaction Performance of High-speed Railway Pantograph-catenary System: A Case Study in Chengdu-Chongqing Passenger Special Railway

As a common disturbance to the railway pantograph-catenary system, the crosswind may deteriorate the current collection quality and threat operational safety. The main topic of this paper is to study the effect of crosswind on the interaction performance of pantograph-catenary considering the aerodynamic forces acting on both the pantograph and catenary. The pantograph-catenary system of the Chengdu-Chongqing passenger special railway is adopted as the analysis object. The absolute nodal coordinate formulation (ANCF) is employed to build the catenary model, of which the numerical accuracy is validated via the comparison with the field measurement data collected from an inspection vehicle operating at 378 km/h. A special spatial grid is defined for the pantograph-catenary system to generate the stochastic wind field based on the empirical spectrum. According to the quasi-steady theory, the wind load acting on the catenary is derived. Computational fluid dynamics (CFD) is employed to calculate the lift and drag forces acting on each component of the pantograph, which are used to derive the equivalent aerodynamic force that can be applied in the lumped-mass model. The simulation results indicate that the pantograph-catenary system of Chengdu-Chongqing passenger special railway has an acceptable performance with a crosswind speed of 20 m/s. But when the crosswind increases up to 30 m/s, some contact force statistics exceed the safety threshold with a turbulence intensity of more than 17%. Through the analysis of the operational safety, it is found that the contact wire always works within the safety range of the pantograph head with a crosswind speed of 30 m/s. But some safety issues can be seen from the maximum uplift of the pantograph head with a turbulence intensity of more than 21%.


Introduction
Due to the complexity of the modern high-speed railway system, many independent relationships exist among the vehicle [1], the infrastructure [2], the overhead system [3] and the environmental disturbance [4], which interact, depend upon and restrict each other. A significant example is the pantograph-catenary system, responsible for transmitting electric power to the electric train [5]. As shown in Figure 1, the catenary is a tensioned cable structure constructed along the railroad. The electric current is transferred from the catenary to the pantograph through a sliding contact between the contact wire and the pantograph head. The current collection quality of the electric train is determined by the interaction performance between the catenary and pantograph.
As the most vulnerable part of the traction power system [6], the catenary is a long-span structure with high flexibility, which is very susceptible to crosswind. A substantial wind load can aggravate the vibration of the pantograph-catenary system, which may increase the contact force fluctuation [7], the risk of contact loss and the occurrence of arcing [8]. These issues affect the stable transmission of the electricity [9], which definitely deteriorates the current collection quality and aggravates the wear and ablation of the contact surface [10], reducing service life. It is necessary to evaluate the interaction performance of the pantograph-catenary subjected to the crosswind load, especially in the design phase. The mathematical model of pantograph-catenary has been a primary tool to evaluate the contact force, which is the main indicator to reflect the contact quality between the contact wire and the pantograph head. The modelling technique of pantograph-catenary experiences a significant advancement in the last decades, as summarised in [9]. The catenary is normally modelled by the finite element method or finite difference method [10], which can effectively describe the wave propagation [11], the geometrical nonlinearity [12] and the dropper slackness [13]. The pantograph is normally modelled by a lumped-mass model which can reflect the first two or three critical modes [14,15]. Some multibody dynamics models of pantograph are also developed to reproduce a realistic geometry [16][17][18]. Some realistic disturbances, such as the wear [19], the irregularity [20,21], the geometrical deviation [14] and the vehicle perturbations [22], are properly modelled and included in the numerical model to evaluate the contact forces.
The wind load, which presents one of the main environmental disturbances to the long-span structure and moving vehicle [23], also attracts the interest of some scholars from the railway engineering community. The galloping of a catenary reported in [24,25] is a rare phenomenon caused by aerodynamic instability. Once it happens, the catenary vibrates with huge amplitude, and the pantograph cannot run through the contact wire. The buffeting is the most common windinduced vibration for the catenary in daily operation, which has a direct impact on the contact force.
In [26], the fluctuating wind field along the railway catenary is constructed using empirical spectrums. The wind-induced vibration and its effect on the contact forces are analysed. But the wind load on the pantograph is not considered in this work. In [27], the pantograph aerodynamics is considered to evaluate the contact force. But the stochastics of the wind load is not taken into account. In [28], the Pseudo-Excitation Method is used to evaluate the dispersion of the catenary response subjected to a crosswind. However, the geometrical nonlinearity and dropper slackness cannot be involved in a response spectrum analysis method.
These shortfalls in previous research are addressed in this paper to evaluate the effect of crosswind on the pantograph-catenary interaction performance. The analysis object of this paper is the Chengdu-Chongqing passenger special railway in China high-speed network, as shown in Figure   2. The top design speed for this railway is 380 km/h. In this work, the current collection quality under crosswind is evaluated based on a pantograph-catenary model validated at 378 km/h through an experimental test. The aerodynamic coefficients of the catenary are obtained through a wind tunnel test. A CFD model of the pantograph is built to analyse the pantograph aerodynamics with a crosswind. The stochastic wind field is constructed along the catenary, and the current collection quality is evaluated through a stochastic analysis.

Pantograph-Catenary Formulations
In order to govern the large deformation of the catenary under the impact from both the pantograph and wind load, the ANCF beam is adopted to model the contact wire, messenger wire and stitch wire [29]. The catenary is modelled based on the design parameters in Chengyu high-speed railway.
The model is validated by comparison with the field test data at 378 km/h. Figure 3. ANCF beam element

Catenary model
The ANCF is a nonlinear finite element approach that can effectively describe the flexibility of the catenary [30]. In this work, the ANCF beam element is used 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. For an ANCF beam element [31], as shown in Figure 3, the nodal degree of freedom (DOF) vector that contains the displacements and the gradients are defined as: (1) in which χ is the local coordinate in the undeformed configuration ranging from 0 to the unstrained length L0. The position vector in the deformed configuration r is interpolated using the shape function matrix N as r = Ne (2) in which N can be defined as follows:  (6) Similarly, the tangent stiffness matrices of the ANCF cable element can also be derived. It should be noted that the axial stiffness changes to zero when the dropper works in slackness.
The shape-finding procedure has been given in [21] with details.

Modelling of the pantograph
in which m1, m2 and m3 are the equivalent mass of the pantograph head, upper arm and lower arm, respectively. k1, k2 and k3 are the corresponding equivalent stiffness. c1, c2 and c3 are the corresponding equivalent damping. F0 is the uplift force. Fc is the contact force. Fair is the equivalent aerodynamic force, which is the contribution of aerodynamic forces acting on each pantograph component to the contact force. The derivation of Fair is presented in Section 4.

Modelling of contact
The contact between the pantograph collector and contact wire is described by the penalty method as follows.
Using the above equation, the equation of motion for the pantograph-catenary system can be obtained as

Validation with Experimental Test
To validate the numerical model presented above and analyse the pantograph-catenary interaction performance at super-higher speed, an instrumented pantograph (see Figure 4) is mounted on an inspection vehicle (see Figure 5), which regularly runs on China high-speed network. According to En 50317 [32], 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 pantograph head to measure the inner forces between the collector and the framework. The contact force can be seen as the sum of the inner forces, inertial forces and the aerodynamic force as follows: a f eq c inner, head, aero 11 a n n ii ii (12) in which inner,i f is the inner force. f n is the number of spring sensors. eq m is the equivalent mass of the pantograph head. a n is the number of accelerometers on the pantograph head. head,i a is the acceleration measured by each accelerometer. aero f is the aerodynamic correction part, which has been determined in a wind tunnel test. The initial configuration of the catenary is presented in Figure 6. Then the dynamic simulation is performed with a TSG-19-type pantograph. The measurement and simulation contact forces are presented in Figure. 7. 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 [32], 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 [33] 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 pantograph head 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 behaviours of the pantograph-catenary interaction. Figure 6. Initial configuration of Chengdu-Chongqing high-speed catenary Table 1. Comparison of critical indicators between simulation and measurement

Derivation of Aerodynamic Forces Acting on Pantograph-Catenary
In this section, the aerodynamic forces on the pantograph caused by the crosswind are derived based on the pantograph geometry and CFD simulation. The aerodynamic forces on the catenary are derived based on the Quasi-steady theory and spatial coordinate transformation.    It is seen that the fluctuating components u, v, w and the aerodynamic coefficients CD and CL should be obtained to determine the aerodynamic forces used in the numerical simulation. In this work, the wind tunnel test is conducted to measure the aerodynamic coefficients of a realistic contact wire subjected to a crosswind. As shown in Figure 8, a contact wire section is built with a scale ratio of 5:1. The wind tunnel test is conducted in the Fluid Mechanics Laboratory at the Department of Energy and Process Engineering, NTNU Gløshaugen. The measurement results of CD and CL are presented in Figure 9. Third-order polynomials are utilised to fit the curves of the measurement parameters, which are used in the numerical simulation to update the aerodynamic coefficients in each time step. The cross-sections for other wires (including the messenger wire, dropper and steady arm) are assumed to be circular. Therefore, the lift coefficient can be neglected. The drag coefficient CD with different Reynolds numbers can be found in [35].    where  , b ,  and  denote the rotation angles of different arms in the pantograph, as shown in Figure 10. According to the geometrical relationships between the pantograph arms, the rotation angles  , b and  can be expressed as follows when  is determined.

Wind Field Construction
The stochastic wind field is constructed by inversing the empirical spectrum to time history. The Von Karman spectrums [36] in longitudinal, lateral and vertical directions are adopted here.
Considering the spatial correlation, the spectral matrix is generated as follows. Therefore, for two arbitrary spatial points M and P, the cross-spectral density matrix for the wind components in the two points can be expressed by  24) and (25). For the analysed catenary shown in Figure 6, the spatial grid is depicted in Figure 14. It is seen that the spatial grid has four layers in the vertical direction. The top one is for the messenger wire and stitch wire. The second one is for dropper wires. The third one is for the contact wire, and the last one is for the pantograph.  Figure 14. Spatial grid of wild field for pantograph-catenary

Performance Assessment with Crosswind
In this section, the pantograph-catenary system's current collection quality and operation safety are evaluated under the crosswind with different turbulence intensities. The terrain categories primarily determine the turbulence intensity. For open terrains, the turbulence intensity usually is no more than 20%. But for some complex terrains, big turbulence may be expected. In the simulations, the turbulence intensity changes from 9% to 27%. The crosswind speeds of 20 m/s and 30 m/s are adopted in the assessment. According to the design specification [37], the current collection quality and the operation safety should be ensured in the serviceability limit state, in which the maximum wind speed is mostly 30 m/s. The train speed is set as 378 km/h, which is very close to the maximum design speed and has been validated against the measurement data in Table 1. The main indicators adopted in this assessment of current collection quality are the standard deviation, the statistical maximum and the statistical minimum of contact forces filtered within 0-20 Hz [38]. The maximum lateral deviation of contact point and maximum vertical uplift of pantograph head are analysed to evaluate the operational safety under crosswind.

Preliminary analysis of Current Collection Quality
The evaluated contact force statistics (namely the standard deviation, statistical maximum and statistical minimum) as a function of turbulence intensity with the crosswind speed of 20 m/s and 30 m/s are presented in Figure 15   From the boxplot analysis, it is seen that the pantograph-catenary has an acceptable performance at 20 m/s crosswind speed. Some issues of current collection quality can be observed at 30 m/s, which is the serviceability limit state for most pantograph-catenary systems. A probabilistic analysis is performed here to quantify the possibility of exceeding the safety threshold.
The probability density function (PDF) of the statistical maximum and minimum contact forces with a crosswind speed of 30 m/s is presented in Figure 20 (a) and (b), respectively. It is seen that the statistical maximum contact force has a 0%, 2.65%, 47.53% and 89.85% possibility to exceed the safety threshold with the turbulence intensity of 13%, 17%, 21% and 25%, respectively. The statistical minimum contact force has a 0%, 12.58%, 73.03% and 96.92% possibility to be negative with the turbulence intensity of 13%, 17%, 21% and 25%, respectively. Thus, more attention should be paid to improve the wind-resistant capability when the railway crosses a complex terrain with a turbulence intensity of more than 17%.

Saftey Assessment
The above analyses mainly focus on the assessment of current collection quality. In this section, the operation safety caused by the crosswind is analysed. Generally, there are two safety issues for the pantograph-catenary interaction under crosswind. One is that the contact wire exceeds the working range of the pantograph head, which may cause the scraping of the pantograph collector. The other is the large uplift of the pantograph head, which may damage the catenary and cause the wire

Conclusions
In this paper, the interaction performance of the pantograph-catenary is investigated under the crosswind with different turbulence intensities. The pantograph-catenary system of Chengdu-Chongqing passenger special railway is adopted as the analysis object. The absolute nodal coordinate formulation is employed to build the catenary model, which can describe the nonlinearity of geometrical deformation and dropper slackness. The field measurement data collected from an inspection vehicle operating at 378 km/h is used to validate the numerical accuracy. The stochastic wind field is constructed for the pantograph-catenary system based on the empirical spectrum. The wind load acting on the catenary is derived based on the quasi-steady theory. The CFD simulation is employed to calculate the lift and drag forces acting on each component of the pantograph. Based on multibody dynamics, the equivalent aerodynamic force used in the lumped-mass model is derived. The analysis of current collection quality indicates that the pantograph-catenary system has an acceptable performance at a crosswind of 20 m/s. But when the crosswind increases up to 30 m/s, some contact force statistics exceed the safety threshold with a turbulence intensity of more than 17%. The operational safety analysis indicates that no dewirement issues should be concerned under the crosswind of 30 m/s for the Chengdu-Chongqing passenger special railway. But the maximum uplift of pantograph head may exceed the ultimate limit with a turbulence intensity of more than 21%.