A Fast Simulation Model of Pantograph–Stitched-Catenary Interaction in Long-Distance Travel

Abstract

The increasing operation speed of high-speed trains allows the pantograph to continuously interact with the catenary over a long distance in a short time, and many new methods have been developed to efficiently calculate its dynamics. However, the existing methods only consider simple catenary systems, which limits their application in high-speed railway systems. In this work, a reduced pantograph–stitched-catenary interaction model is developed to simulate pantograph–stitched-catenary interactions during long-distance travel. Based on the existing reduced catenary model, the stitched catenary system is first considered, where the stitched wire is simplified into a part of the messenger wire supported by two spring-damping elements. The present model is validated by test results and the EN 50318:2018 standard, and it is subsequently used to study the dynamic performance of the pantograph–stitched-catenary system at an overdesigned speed in Sweden. The results show that the proposed model can be seven times faster than the traditional modal superposition method with the same accuracy in a stitched catenary system, and the existing catenary system cannot be operated at an overdesigned speed without increasing the contact wire tension. The present model gives an efficient solution to pantograph–stitched-catenary interaction problems.

1. Introduction

The pantograph–catenary system is one of the most important parts in the electrical railway system, and its working performance directly influences the current collecting quality of the trains [1]. Unstable dynamic behavior of the pantograph–catenary system can cause abnormal wear of both the contact strip and contact wire and damage these two key components [2,3], which further limits the operational speed and increases the maintenance cost [4,5]. With the increasing speed of high-speed train operation, the dynamic characteristics of the pantograph–catenary system have become much more complex, and the pantograph continuously interacts with the catenary over a long distance in a short time, as shown in Figure 1, where the “…” means the long catenary with the same structure. Note that the simulation time of the pantograph–catenary system should be long enough to present the dynamic characteristics of the whole system, and this makes the catenary model in the pantograph–catenary system become longer and longer. To better simulate the pantograph–catenary interaction dynamics and optimize its dynamic performance, it is necessary to consider the long catenary model in the pantograph–catenary interaction model.

Many scholars have contributed to pantograph–catenary dynamic studies with low vehicle operation speed and a short catenary model, including the basic pantograph–catenary interaction modeling and simulation [6,7,8,9], the influence of multi-pantograph operation on system dynamic responses [10,11,12,13], pantograph and catenary fatigue problem [14,15,16], and the system’s stability under environmental influences (like wind and ice) [4,17,18]. In these studies, the finite element method (FEM) [19,20,21,22,23] and modal superposition method (MSM) [1,24,25,26,27] are mainly used, and they are proven to be accurate in studying these problems. However, as mentioned before, in the high-speed operation case, the catenary can be very long, and both the MSM and FEM must model this very long catenary system. This will greatly increase the degrees of freedom (DOF) of the whole model and decrease the computational efficiency. In 2017, Facchinetti and Bruni [1] found that the vibration of the catenary structure caused by the pantograph–catenary interaction mainly concentrates in about 2–3 spans due to structural damping, and they developed a real-time catenary model to efficiently simulate the pantograph–catenary interaction in the hardware-in-the-loop case. In this model, the long catenary is reduced to 3–5 spans around the pantograph, and this small reduced catenary becomes the basis of the real-time catenary model. Xu et al. [28] also developed a reduced catenary model to efficiently simulate the pantograph–catenary interaction, where the long catenary systems are reduced to small regions around the moving pantograph. The pantograph and the small area around the pantograph where most of the vibration is located are moving, and the catenary itself keeps the longitude static. Thus, there is a relative movement between the small certain area and the catenary structure, and the arbitrary Lagrange–Eulerian method (ALE method) is used to describe the movement of the catenary in this small moving area. The MSM is further used to model this small area, and the reduced catenary model is finally formulated. Based on this model, the influence of track structure maintenance on pantograph–catenary interaction dynamics is analyzed. Jimenez-Octavio et al. [29] further developed a mesh moving method to efficiently solve the pantograph–catenary interaction dynamic responses, where only the abovementioned small area of the catenary around the pantograph needs to be meshed in detail and coupled with a coarser fixed mesh domain of the whole catenary structure. Pan et al. [30] further investigated the length of the effective vibration area of the catenary to decide the length of the reduced catenary model, where the traditional MSM was used to model the pantograph–catenary interaction system, and the vibration energy of the catenary was calculated to determine the length of the effective vibration area.

The abovementioned new methods greatly decrease the number of DOFs of the catenary model and have proven to be highly efficient and accurate in solving pantograph–catenary interaction dynamic responses with long catenary structures. However, the existing methods only consider simple catenary systems, and stitched catenary systems are ignored. This is mainly caused by the application of the MSM, where the stitch wire connects with the messenger wire at the end of every two neighboring spans, and the shape function of this structure is difficult to express. While the FEM can be used to model a stitched catenary system, it is difficult to combine it with the abovementioned new modeling methods. Because the stitched catenary system is widely applied in high-speed railway systems [31,32], the application of the existing new modeling methods with only simple catenary structures is limited.

In this study, a new reduced pantograph–stitched-catenary interaction model is developed to accurately and efficiently simulate pantograph–stitched-catenary interactions with a long catenary structure. In this model, based on the existing reduced catenary model and MSM-based pantograph–catenary model [28,30], the long stitched catenary structure is reduced and modeled using the ALE method and MSM. The stitch wire and catenary suspension are simplified into a part of the messenger wire supported by a couple of spring-damper elements, where the distance of these two spring-damper elements and the stiffness and damping of each spring-damping element are chosen to ensure that the stiffness of the stitch wire model is equal to the original stiffness. Because only the reduced stitched catenary model needs to be modeled and the stitch wire is simplified, the number of DOFs of the whole pantograph–catenary system decreases, and the dynamic responses of the pantograph–catenary interaction can be efficiently calculated. This greatly benefits pantograph–stitched-catenary interaction simulations in high-speed railway systems with long catenary structures, where the dynamic characteristics of pantograph–catenary systems can be more accurately investigated. The present model can consider several pantographs with a longer reduced catenary model. After the formulation of the present model, it is validated by measurement data from an on-track test of the Swedish Regina train and the EN 50318:2018 standard [33]. Then, the present model is used to study the dynamic performance of the pantograph–catenary interaction system at an overdesigned speed in Sweden. The remainder of this paper is organized as follows. The proposed reduced pantograph–stitched-catenary interaction model is formulated in Section 2, and it is validated in Section 3. The dynamics of the pantograph–catenary interaction with operational speeds higher than the designed speed is investigated based on the present model in Section 4. Finally, conclusions based on this study are presented in Section 5.

2. Reduced Pantograph–Stitched-Catenary Interaction Model

In the stitched catenary system, the stitch wire is designed to optimize the stiffness of the catenary structure near the registration arms, which can further optimize the dynamic performance of the pantograph–catenary interaction. The stitch wire is located above the registration arms and connects to the messenger wire and dropper, as shown in Figure 2a. Note that the function of the stitch wire is similar to that of the messenger wire, which also has tension and carries the contact wire through the dropper. The stitch wire is also supported by the messenger wire through the clamp. Therefore, the stitch wire can be simplified to a part of the messenger wire at the same location to carry the contact wire through the dropper. The catenary suspension is then combined with the original messenger wire above the stitch wire and modeled by a couple of spring-damping elements to further support this part of the messenger wire, as shown in Figure 2b. This part of the messenger wire and spring-damping elements form the stitch wire model. Because the stitch wire is simplified, the existing reduced catenary model based on the MSM can be further extended to the stitched catenary system. The detailed modeling processes are shown below.

2.1. Reduced Stitched Catenary Model

Based on the reduced catenary model, the reduced stitched catenary model is formulated first. The small area around the moving pantograph is chosen, and the ALE method and MSM are used to model the catenary located in this small area [1]. The contact and messenger wire are considered as Euler‒Bernoulli beams, and the stitch wire is considered as a part of the messenger wire with a couple of spring-damping elements supporting this part, which makes the structure of the present stitch catenary model familiar to the simple catenary system. Thus, the derivation process of the dynamic equation of the contact and messenger wire in the reduced stitched catenary model is just the same as that shown in [28], and it is not repeated here. The dynamic equations of the contact and messenger wire in the reduced stitched catenary model are

$$M_m{\ddot{q}}_m+C_m{\dot{q}}_m+K_m{q}_m=Q_{flowm}+Q_{Fm}$$

(1)

$$M_c{\ddot{q}}_c+C_c{\dot{q}}_c+K_c{q}_c=Q_{flowc}+Q_{Fc}$$

(2)

where M_m, C_m and K_m are the mass, damping and stiffness matrices of the messenger wire, and M_c, C_c and K_c are the mass, damping and stiffness matrices of the contact wire, respectively. Rayleigh damping is considered as the numerical damping for convergence, and the damping coefficient is chosen as 0.02 [10]. Q_flow is the additional generalized force vector generated by the relative motion between the reduced stitched catenary model and the material of both the contact and messenger wire, and Q_F is the generalized force vector caused by the force act on the contact wire, including the pantograph–catenary interaction force and gravity. The expression of these matrices can be found in [28].

The stitch wire model is then expressed here. As mentioned above, the stitch wire is simplified to a part of the messenger wire, and the catenary suspension and the original messenger wire are modeled as a couple of spring-damping elements. The distance between these two spring-damper elements is d_S, and the stiffness and damping of these two elements are $K_S^{\prime }$ and $C_S^{\prime }$, respectively. These values are chosen to ensure that the stiffness of the catenary at the stitch wire part is equal to the original stiffness. The dynamic force of the spring-damping elements can be calculated by

$$F_S=K_S^{\prime }{d}_S+C_S^{\prime }{\dot{d}}_S$$

(3)

where $d_S$ is the displacement of the messenger wire at the location of the spring-damping element. Note that the location of these spring-damping elements also changes as the pantograph travels along the catenary based on the theory of the reduced catenary model. Let $x_{S0i}$ be the initial position of the ith stitch wire model at o-xyz, and the position of the ith dropper $x_{Si}$ at any time can be expressed as

$$x_{Si}=x_{S0i}-Vt$$

(4)

where V is the operation velocity of the vehicle.

The dropper and registration arm are then modeled. Based on the reduced catenary model, the dropper is modeled as bilinear spring elements, and the registration arm (attachment points) is modeled as a lumped mass $m_S$ attached to the contact wire, which can present the shock to the pantograph and catenary. Details of these models are shown in [28].

Based on the dynamic equations of the messenger and contact wires, the stitch wire model, and the model of the dropper and registration arm, the reduced stitched catenary model is finally formulated as

$$\left[\begin{array}{cc}{M}_m& \\ & M_c\end{array}\right]\left[\begin{array}{c}{\ddot{q}}_m\\ {\ddot{q}}_c\end{array}\right]+\left[\begin{array}{cc}{C}_m& \\ & C_c\end{array}\right]\left[\begin{array}{c}{\dot{q}}_m\\ {\dot{q}}_c\end{array}\right]+\left[\begin{array}{cc}{K}_m& \\ & K_c\end{array}\right]\left[\begin{array}{c}{q}_m\\ q_c\end{array}\right]=\left[\begin{array}{c}{Q}_{flowm}+Q_{Fm}\\ Q_{flowc}+Q_{Fc}\end{array}\right]$$

(5)

Note that the stagger forces are applied on every registration arm acting point to form the stagger of the catenary.

In Equation (5), the initial shape of the messenger and contact wires are initially straight, which is different from the real static shape presented in Figure 2a. To accurately calculate the dynamic responses of the catenary, its static shape should be accurately calculated. In the present model, the sag distance at the dropper-connection position of the contact wire should first be determined based on the design parameters of the catenary system; then, the static shape of the catenary is calculated from Equation (5). The calculation results $q_m$ and $q_c$ are then used as the initial conditions of Equation (6), which correspond to their static shape. Note that the shape of the contact wire is the same as the designed shape, but the shape of the messenger wire at the stitch wire is different due to the simplification of the stitch wire. This difference has little influence on the calculation accuracy, as shown in the validation results in Section 3.

2.2. Modeling of the Pantograph

The pantograph is modeled as a lumped mass system model, which consists of a contact strip, panhead, frame, and spring-damper elements. Based on [31], the contact strip, panhead, and frame are simplified as lumped masses, and the contact strip and panhead are combined and connected with the frame through spring-damper elements, as shown in Figure 2b. The frame is connected to the car body through spring-damper elements as well. The contact strip, panhead, and frame mass of the pantograph are m₃, m₂, and m₁, respectively. The stiffness and damping of the spring-damper elements between the panhead and frame are K₂ and C₂, respectively, and those of the spring-damper elements between the frame and car body are K₁ and C₁, respectively. The parameters of the pantograph can be found in [31], which result in the same frequency responses from the model as those from a real pantograph. The dynamic equation of the pantograph is expressed as

$$M_P{\ddot{q}}_P+C_P{\dot{q}}_P+K_P{q}_P=Q_{cP}+Q_P$$

(6)

where $M_P$, $C_P$, and $K_P$ denote the spring, damping, and stiffness matrices of the pantograph, respectively. $q_P$ is the generalized coordinate vector, $Q_{cP}$ is the generalized force vector caused by the pantograph–catenary interaction, and $Q_P$ is the generalized force vector generated by the pantograph itself, which includes gravity, uplift forces, and aerodynamics. The uplift force of the present pantograph consists of a constant uplift force and an aerodynamic force. The aerodynamic force is applied on m₂ (panhead), and the constant uplift force is applied on m₁ (frame). Details of these matrices are shown in [1]. Note that the present model is developed based on Sweden’s high-speed pantograph, and it only has one contact rail (contact strip). In addition, the main target of the present work is developing a fast simulation model of pantograph–stitched-catenary interaction. Thus, the influence of contact rails’ distance on the stability of the pantograph–catenary interaction is not considered in the present investigation.

2.3. The Reduced Pantograph–Stitched-Catenary Interaction Model

Based on the reduced stitched catenary model and pantograph model, the reduced pantograph–stitched-catenary interaction model can be finally formulated, which is

$$\begin{array}{l}\left[\begin{array}{ccc}{M}_m& & \\ & M_c& \\ & & M_P\end{array}\right]\left[\begin{array}{c}{\ddot{q}}_m\\ {\ddot{q}}_c\\ {\ddot{q}}_P\end{array}\right]+\left[\begin{array}{ccc}{C}_m& & \\ & C_c& \\ & & C_P\end{array}\right]\left[\begin{array}{c}{\dot{q}}_m\\ {\dot{q}}_c\\ {\dot{q}}_P\end{array}\right]\\ +\left[\begin{array}{ccc}{K}_m& & \\ & K_c& \\ & & K_P\end{array}\right]\left[\begin{array}{c}{\dot{q}}_m\\ {\dot{q}}_c\\ {\dot{q}}_P\end{array}\right]=\left[\begin{array}{c}{Q}_{flowm}+Q_{Fm}\\ Q_{flowc}+Q_{Fc}\\ Q_{cP}+Q_P\end{array}\right]\end{array}$$

(7)

In Equation (7), $Q_{Pc}$ means the generalized force vector caused by the pantograph–catenary contact force $F_c$, which is calculated by [10]:

$$F_c=\left\{\begin{array}{c}{k}_n{d}_r,d_r\ge 0\\ 0,d_r<0\end{array}\right.$$

(8)

where $k_n$ is the contact stiffness, and $d_r$ is the gap between the contact wire and contact strip at the contact point. The $k_n$ in the present model is 50,000 N/m [10]. Note that the pantograph–catenary contact forces are calculated in the same way as those shown in [30]. This is caused by the fact that the same stitched catenary system is considered in [30], and the present model and the traditional pantograph–catenary contact model are used in these two models.

3. Validation and Discussion

3.1. Validation

3.1.1. Validation Based on the Measurement Data

The test results are first compared to validate the present model. In 2020, Trafikverket (The Swedish Transport Administration) initiated a project to test whether the SYT7.0/9.8 catenary system (designed train speed 200 km/h, messenger wire tension 7 kN, contact wire tension 9.8 kN) can support high-speed trains running at a maximum speed of 300 km/h, as shown in Figure 3. The railway between Falköping and Töreboda in Sweden was chosen, and a Swedish Regina train (top speed 300 km/h) with a high-speed pantograph was used for the field test. The test results indicated that the SYT7.0/9.8 catenary system can run the high-speed train at a maximum speed of 302 km/h for only a short time (time of the field test) while maintaining the resulting contact forces at acceptable levels.

The measurement data from this test are used for validation purposes, acquired from the Schunk group in Sweden. The date sample frequency is 500 Hz, and the measurement data are filtered in 0–20 Hz. Parameters of the pantograph–catenary interaction model are listed in [27]. Structural damping is also considered, and the damping ratio of the catenary is 0.02, as used in [27]. Based on the pre-analysis method [28], the reduced stitched catenary model is 180 m long with three spans, d_S is 12 m, and $K_S^{\prime }$ is $3\times {10}^5$ N/m. The constant operational speed of 222 km/h is taken from the on-track test for model validation, which corresponds to the designed operation speed of SYT7.0/9.8. Based on the test conditions, the train is considered to run on a straight track. The first 120 modes of the messenger and contact wires are considered, and the number of DOFs of the entire pantograph–catenary interaction model is 722. Based on the key parameters from the experiment, the pantograph uplift force considered in the pantograph model is calculated by

$$F_{up}=50+0.097V^2$$

(9)

The key dynamic characteristic of the pantograph–catenary system is the contact force [11]; thus, it is calculated here to validate the reduced catenary model. The time histories of the contact forces of the pantograph–catenary system and their corresponding spectral analyses at 222 km/h are shown in Figure 4, where the calculation results from the present model are compared with those from the experiment. The measurement data of the contact force are obtained based on the acceleration of the contact strips on the pantograph, and the contact force from both the present model and the measurement data are filtered in a 0–20 Hz span. As shown in Figure 4, the calculation results from the present model are similar to the measurement data at 222 km/h, and the amplitudes of the contact force from the present model are close to those of the measurement data. The spectral analysis results show that the peak frequencies of the spectrum from both the measurement data and the reduced stitched catenary model are in good agreement with each other in the three frequency spans considered (0–2 Hz, 0–5 Hz and 5–20 Hz). The spectral analysis results are further sorted and presented in

Table 1. Contact force sorted according to frequency span from different models and train speeds.

	Frequency Span	0–2 Hz		0–5 Hz		5–20 Hz
	Train Operation Speed	Present Model	Test Results	Present Model	Test Results	Present Model	Test Results
Contact force (N)	222 km/h	91.24	90.85	97.22	97.85	7.32	6.81
Standard deviation	222 km/h	11.43	11.98	16.19	16.76	23.62	24.21

. Table 1 shows that the spectral analysis results from the present model are in good agreement at both low frequencies (0–5 Hz) and high frequencies (5–20 Hz).

3.1.2. Validation Based on the EN50318-2018 Standard and FEM

The present model is then validated based on the EN50318:2018 standard [30], which is a European standard for pantograph–catenary interaction simulation validation of the AC stitched catenary and its corresponding pantograph. Note that the present model is firstly validated by the measurement data from SYT7.0/9.8, which was developed before 2018, and the original EN50318:2018 standard is more suitable. In addition, the recent publications about the pantograph–catenary interaction modeling still use the traditional EN50318:2018 standard to validate the model. Thus, the original EN50318:2018 standard is used for validation. In the present validation process, the double pantograph is considered, the length of the reduced catenary model is 360 m, d_S is 19 m, and $K_S^{\prime }$ is $7\times {10}^5$ N/m. These values are decided by the pre-analysis method. The number of modes considered in the modeling is n = 360. Other parameters of both the pantograph and catenary in the reduced pantograph–stitched-catenary model are set based on the requirement of EN50318:2018 (like tension and dropper distance). The time histories of the pantograph–catenary contact forces based on the EN50318:2018 standard are shown in Figure 5, where a similar model is also calculated, and the same results are presented in [30]. The corresponding statistical results are shown in

Table 2. Statistical results of the pantograph–catenary interaction with respect to different pantographs and speeds.

Speed [km/h]	275		320
Pantograph	Leading	Trailing	Leading	Trailing
Fm [N]	143.1(143–144)	143.7(142–144)	169(169)	169(169)
σ [N]	24.1(20.2–24.7)	28.9(24.4–36.2)	23.2(20.5–24.7)	36.9(30.4–38.3)
σ (0–5 Hz) [N]	14.8(11.7–15.2)	17.9(17.0–18.2)	13.0(11.8–13.3)	22.9(20.4–24.2)
σ (5–20 Hz) [N]	18.5(16.5–19)	25.5(16.4–27.4)	19.1(15.2–20.9)	28.8(21.5–29.8)
Actual maximum of contact force [N]	198.5(185–199)	225.2(203–252)	229.2(210–232)	242.3(239–255)
Actual minimum of contact force [N]	95.6(92–102)	86.2(56–88)	121.4(105–128)	74.4(43–78)
Range of vertical position of the point of contact [mm]	22.4(18–25)	32.5(26–36)	21.6(13–23)	59.8(38–63)
Maximum uplift at support [mm]	68.2(55–79)	71.3(51–79)	88.6(74–95)	91.1(69–95)
Percentage of loss of contact [%]	0(0)	0(0)	0(0)	0(0)

. As shown in Figure 5 and Table 2, the results from the present model are reasonable, and its corresponding statistical results meet the demand of Table A.7 in EN50318-2018. Based on these results, the accuracy of the reduced pantograph–stitched-catenary interaction model can be validated, and the present stitch wire model can accurately present the dynamic behavior of the stitch wire parts. Note that the results shown in Figure 5 are similar to those shown in [30]. This is caused by the fact that the MSM-based pantograph–catenary model in [30] is validated by the same standard EN50318:2018 with the same pantograph and catenary parameters. Thus, the measurement data are further used in Section 3.1.1 to validate the present model.

The FEM model developed by Liu et al. [10] is further considered to validate the present model. The static shape of the contact wire is considered, and both the FEM model and the present model are used to calculate it. The results are shown in Figure 6, where the results from both models are compared with each other. It can be seen from Figure 1 that the results from the present model are also in good agreement with those from the FEM model, and the maximum relative difference between these two results is no more than 2.7%, which further validates the accuracy of the present model.

3.1.3. Validation of the Calculation Efficiency with a Long Vehicle Travel Distance

After validating the accuracy of the reduced pantograph–stitched-catenary model, the calculation efficiency of the reduced pantograph–stitched-catenary interaction model with a long vehicle travel distance is further validated with the traditional model. In the traditional model, the pantograph–catenary interaction is developed based on the MSM and achieved in MATLAB 2022a. The pantograph–catenary interaction case used in Section 3.1 is also considered, where the train operation speed is V = 200 km/h. The long vehicle travel distance is considered, where the train operation time is chosen from 5 s, 10 s and 20 s, and their corresponding catenary lengths are 300, 600 and 1200 m, respectively. In the traditional model, the full length of the catenary needs to be modeled; however, the length of the reduced stitched catenary model is set at 180 m.

The DOFs and CPU times of the traditional MSM model and the proposed reduced pantograph–stitched-catenary model at various train moving times are shown in Figure 7. While the number of DOFs of the traditional pantograph–stitched-catenary model increases with the train running distance, the number of DOFs of the reduced pantograph–stitched-catenary model is the same. Therefore, the calculation time of the traditional pantograph–catenary model increases from 2452 s to 21,206 s, whereas the calculation time of the reduced catenary model only increases from 1159 s to 3108 s. The relative difference between the present and traditional model calculation times also increases from 439% to 687%. This means that the calculation time of the present model is only 1/7 of that of the traditional MSM, and it can further increase with increasing train operation time. Therefore, the present reduced pantograph–stitched-catenary interaction model is highly computationally efficient in solving the dynamic responses of pantograph–stitched-catenary interactions during long-distance travel. This will greatly benefit the investigation of long-term pantograph–catenary interaction dynamics. For example, when investigating the high-speed pantograph contact strip wear problem, a much longer catenary structure can be considered, and the wear rate of the contact strip under different conditions can be more efficiently and accurately obtained based on the present model. The comparison of the present model, FEM model, and MSM model is shown in

Table 3. Comparison of reduced pantograph–stitched-catenary model and other models.

Models	Reduced Pantograph–Stitched- Catenary Model	FEM Model (Liu et al. [10])	MSM Model (Zhang et al. [18])
Characteristics	The long stitched catenary is reduced to a small region around the moving pantograph and modeled using the modal superposition method and ALE method, and only the reduced area of the catenary can be modeled.	The whole long catenary structure is considered and modeled using the finite element method, and most of the components are detail modeled.	The whole long catenary structure is considered and modeled using the modal superposition method, only the main parts of the catenary can be modeled.
Accuracy	Accurate in main pantograph–catenary interaction dynamics but can not simulate component’s dynamic responses	Accurate in both the main pantograph–catenary interaction dynamics and component’s dynamic responses	Accurate in main pantograph–catenary interaction dynamics but can not simulate component’s dynamic responses
Efficiency	Fastest	Slowest	Middle

, which further describes the advantages of different models.

3.2. Discussion

According to the modeling process, the accuracy of the proposed model is strongly influenced by three factors, including the length of the reduced stitched catenary model, the distance between every two spring-damping elements, and the stiffness of the spring-damping elements in the stitched wire model. These three aspects are discussed to further understand the present model.

The influence of the length of the reduced stitched catenary model L on the calculation accuracy in different cases is firstly discussed here. The mean contact force and standard deviation of the pantograph–catenary contact force with respect to different L in both the test case and the EN50318-2018 case are shown in Figure 8, where L changes from 60 m (1 span) to 240 m (4 spans) in the original case and 240 m (4 spans) to 420 m (7 spans) in the EN50318-2018 case. It can be seen from Figure 8 that both the mean contact force and standard deviation approach the test results and standard range with increasing L, but different cases have different corresponding convergence values. In the original case, when L is greater than 180 m, the corresponding mean contact force and standard deviation converge to the test results. However, the mean contact force and standard deviation in the EN50318-2018 case converge when L is larger than 360 m. This means that different pantograph–catenary-interaction cases have different suitable L values, which should be decided by the pre-analysis method before investigation.

The influence of the distance between every two spring-damping elements d_S in the stitch wire model on the calculation accuracy is then discussed. Based on the same case, the mean contact force and standard deviation of the contact force at the lead pantograph with respect to different d_S are calculated and shown in Figure 9. Figure 9 shows that both the mean contact force and standard deviation converge to standard values at d_S = 19 m, and they quickly increase with increasing and decreasing d_S. Note that the original length of the stitch wire is 18 m in the EN50318-2018 standard, which means that the d_S should be slightly longer than the original stitch wire to ensure the accuracy of the present model. In addition, because different stitched catenary systems have different stitch wire lengths, the d_S should be decided in different cases where the pre-analysis method can also be used.

Finally, the influence of the spring-damping element stiffness $K_S^{\prime }$ in the stitch wire model on the calculation accuracy is discussed. The EN50318:2018 validation case is further considered, and the mean contact force and standard deviation of the contact force at the lead pantograph with respect to different $K_S^{\prime }$ are calculated and shown in Figure 10. Figure 10 shows that both the mean contact force and standard deviation of the contact force decrease with increasing $K_S^{\prime }$ and converge to the standard value when $K_S^{\prime }$ is larger than $7\times {10}^5$ N/m, which means that $K_S^{\prime }$ should be no less than $7\times {10}^5$ N/m to maintain the accuracy of the present model. This value should also be decided by the pre-analysis method in different stitched catenary systems. In addition, unlike the original reduced catenary model, the parameters of the stitch wire model in the present reduced stitched catenary model further influence the calculation accuracy, which means that the stitch wire influences the dynamic responses of the whole catenary, and the stitch wire should be carefully modeled with proper parameters.

4. Influence of Train Overdesigned-Speed Operation on Pantograph–Catenary Interaction Dynamics

After the validation and discussion of the present model, it is further applied to investigate the influence of train overdesign-speed operation on pantograph–catenary interaction dynamics, where the pantograph continuously interacting with the long stitched catenary in a short time is considered. The abovementioned experiment is chosen, where the vehicle operation speed (302 km/h) is higher than the designed top speed of the SYT7.0/9.8 catenary system (200 km/h). The operation time is considered to be 10 s, which corresponds to a 900 m long stitched catenary structure.

Based on the present model, the pantograph–catenary interaction at 302 km/h is simulated, and the time histories of the pantograph–catenary contact forces and their corresponding spectral analysis results are shown in Figure 10. The simulation results are also compared with those from the measurement data. Both the measurement data and simulation results are adjusted through a low-pass filter at 0–20 Hz. The statistical results are also compared with those obtained at V = 200 km/h, as shown in

Table 4. Comparison of contact force statistics at different speeds.

Speed	200 km/h	302 km/h
Mean value	84.2 N	104.9 N
Standard deviation (0–20 Hz)	15.7	23.2
Maximum value	109.7 N	152.6 N
Minimum value	46.7 N	65.4 N

. Figure 11 and Table 4 show that the pantograph–catenary interaction performance at V = 302 km/h remains sufficient. Although the present speed is almost 1.5 times greater than the catenary design speed and considering the influence of train–track interaction dynamics, both the simulation results and measurement data of the contact force are acceptable. The positive offset from measurement data is caused by the fact that the catenary in the real line is not idealized and the local irregularity of the catenary causes this offset. The statistical results of the contact force at V = 302 km/h show only a small increase compared to those at V = 200 km/h. The actual minimum contact force of the present results is slightly higher than that in standard EN 50318 at V = 300 km/h (59.2 N with [30 N, 55 N]); however, the actual maximum contact force of 151.2 N is much smaller than that in the standard, which should be [190 N, 225 N]. This is caused by the small tension of the contact wire, which decreases the stiffness of the catenary system. The spectral analysis results show that the contact force is mainly dominated by low-frequency components below 10 Hz.

To better understand the stability of the present pantograph–catenary system in higher operation velocity, the influence of different vehicle velocities on the pantograph–catenary interaction above its design speed is then investigated. The time histories of the pantograph–catenary contact forces with respect to different train operation speeds and their minimum contact forces are shown in Figure 11, where the vehicle velocity V changes from 250 to 320 km/h. Figure 12 shows that the minimum contact forces quickly decrease with increasing V after V = 275 km/h. The minimum contact force even decreases from 59.2 N to 0 N with V increasing from 300 km/h to 312 km/h. This means that the stability of the present pantograph–catenary interaction system is decreased with V = 300 km/h, and it reaches a critical state at V = 312 km/h. At the critical state, the pantograph loses contact with the catenary, and the current collection performance obviously decreases. In addition, because V = 300 km/h is close to 312 km/h and the pantograph–catenary interaction system is highly nonlinear, a small change in the pantograph and catenary parameters can cause unstable dynamic behavior at V = 300 km/h, such as contact loss. Because the parameters of the pantograph–catenary interaction system cannot remain constant at any time and the present system at V = 300 km/h is close to its critical state, it is impossible to let the train operate at 300 km/h in the existing SYT7.0/9.8 catenary system in the long term. This also supports the decision of Trafikverket that the existing catenary system will not operate high-speed trains with overdesigned speeds in the long term.

As mentioned in [28], to increase the maximum operation speed of the existing catenary system, the most efficient way is to increase the tension of the contact wire. Based on the present model, the influence of increased contact wire tension on pantograph–catenary interaction dynamics is investigated. The time histories of the pantograph–catenary contact force with respect to different contact wire tensions and vehicle velocities are shown in Figure 13, and the minimum contact force with respect to different contact wire tensions and vehicle velocities is shown in Figure 14. As shown in these two figures, when the vehicle velocity is 320 km/h, the tension of the contact wire only needs to increase to 12.8 kN to stabilize the system, and no contact loss will occur. However, when the vehicle velocity is 350 km/h, the contact wire tension should be increased to 15.8 kN to stabilize the system. Note that the catenary system should have enough design margin to ensure its current collection performance at a high vehicle operation velocity. Therefore, the tension of the contact wire should be increased to 15.8 kN to allow the present SYT7.0/9.8 catenary system to operate at V = 300 km/h.

5. Conclusions

In this study, a reduced pantograph–stitched-catenary interaction model is developed based on the existing reduced catenary model to efficiently investigate the dynamic characteristics of pantograph–stitched-catenary interactions during long-distance travel. Note that the function of the stitch wire is similar to that of the messenger wire, which also has tension and carries the contact wire through the dropper. The stitch wire is also supported by the messenger wire through the clamp. Therefore, in the present model, the stitch wire and catenary suspension are simplified into a part of the messenger wire supported by a couple of spring-damper elements, where the distance of these two spring-damper elements and the stiffness and damping of each spring-damping element are chosen to ensure that the stiffness of the stitch wire model is equal to the original stiffness. Because only a small area instead of the whole long stitched catenary needs to be modeled, the number of DOFs of the pantograph–catenary interaction system decreases, and the dynamic responses of the pantograph–catenary interaction can be efficiently calculated. The present model is validated with measurement data from an on-track test of the Swedish Regina train and EN 50318 standard, and then, it is used to investigate the dynamic response of the existing pantograph–catenary interaction system under train overdesigned-speed operation.

Based on the obtained calculation results, the following conclusions are drawn:

(1)

The proposed reduced pantograph–stitched-catenary model can efficiently and accurately investigate the dynamic behavior of pantograph–stitched-catenary interactions. When calculating the pantograph–catenary-interaction dynamic responses, the calculation time of the present model is only 1/7 of that of the MSM with a train operation distance of 1200 m. This greatly benefits the investigation of the long-term service evolution of the pantograph–stitched-catenary system, such as the wear rate calculation in pantograph contact strip wear investigations.

(2)

When the train operation speed is higher than the designed top speed of the SYT7.0/9.8 catenary system and up to 300 km/h, the pantograph–catenary interaction dynamic performance of the present pantograph–catenary system is still acceptable, and the risk of contact loss is small. However, when V = 300 km/h, the present pantograph–catenary system is not stable and is close to its critical state, which means that it can result in contact loss with a small change in the pantograph–catenary system parameters. Thus, it cannot operate at V = 300 km/h in the long term. To increase the maximum operation speed of the existing SYT7.0/9.8 catenary system to 300 km/h, the contact wire tension should be increased to 15.8 kN.

Finally, the limitations of the proposed dynamic model are that only the main dynamic characteristics of the pantograph–catenary interaction can be simulated and investigated, and the detailed dynamic responses of the components in the catenary system cannot be fully presented, especially those of the stitch wire. In our future works, a FEM-based reduced catenary model will be formulated to accurately and efficiently investigate both the main dynamic characteristics of the pantograph–catenary interaction and the dynamics of key components.