1. Bibliographic Information

1.1. Title

The central topic of this paper is "Effective Measures to Improve Current Collection Quality for Double Pantographs and Catenary Based on Wave Propagation Analysis."

1.2. Authors

The authors are Zhao Xu (Student Member, IEEE), Yang Song (Member, IEEE), and Zhigang Liu (Senior Member, IEEE).

• Zhao Xu: Received B.S. in Engineering Mechanics and is pursuing a Ph.D. in Electrical Engineering at Southwest Jiaotong University, China. His research focuses on finite element modeling, structural dynamics, and their applications in railway pantograph-catenary systems.
• Yang Song: Received Ph.D. in Electrical Engineering from Southwest Jiaotong University, China. He was a Research Fellow at the University of Huddersfield, UK, and is currently a Postdoctoral Fellow at the Norwegian University of Technology, Norway. His interests include pantograph-catenary interaction assessment, wind-induced vibration of railway structures, and coupling dynamics in railway engineering.
• Zhigang Liu: Received Ph.D. in Power System and its Automation from Southwest Jiaotong University, China, where he is currently a Full Professor. His research interests lie in the electrical relationship of EMUs and traction, as well as detection and assessment of pantograph-catenary in high-speed railways. He is a Fellow of The Institution of Engineering and Technology (IET) and an Associate Editor for several IEEE Transactions.

1.3. Journal/Conference

This paper was published in a journal affiliated with the Institute of Electrical and Electronics Engineers (IEEE), a globally recognized professional association for electronic engineering and electrical engineering. Publication in an IEEE journal signifies a high standard of peer review and relevance in the field of electrical engineering and related applications, especially in transportation systems.

1.4. Publication Year

The paper was published on 2020-04-06T00:00:00.000Z (April 6, 2020 UTC).

1.5. Abstract

The abstract introduces the problem of current collection quality deterioration, particularly for the trailing pantograph, in double-pantograph Electrical Multiple Unit (EMU) trains operating on catenary systems. This deterioration is caused by mechanical waves excited by the leading pantograph propagating along the contact wire. The paper aims to propose effective measures to improve this quality. It establishes a double pantographs-catenary model using the Finite Element Method (FEM) to analyze wave propagation. Through analyzing the contact wire uplift response from a single moving force, the paper discusses optimal intervals for double pantographs. It finds that bad intervals correspond to velocity peaks of the contact wire uplift, while good intervals align with valley values of the uplift velocity. Based on this, a formula for the optimal interval is proposed and validated using parameters from European and Chinese high-speed networks. Additionally, the paper investigates the effect of introducing a damper to the steady arm to reduce wave intensity, showing that slight steady arm damping positively impacts the double pantographs-catenary interaction.

1.6. Original Source Link

The original source link is /files/papers/692d816c421ae58817ac7111/paper.pdf. This indicates that the paper is an officially published work.

2. Executive Summary

2.1. Background & Motivation

The core problem the paper addresses is the deterioration of current collection quality for the trailing pantograph in double-pantograph high-speed train operations.

• Importance: High-speed railways, especially in countries like China, face immense pressure to increase carrying capacity (the number of passengers or cargo a train can transport). A common solution is to connect multiple EMUs (Electrical Multiple Unit trains), requiring multiple pantographs to collect sufficient electrical energy from the catenary (the overhead line system supplying power). Double-pantograph operation is becoming mainstream in newly built high-speed networks.
• Specific Challenges/Gaps: When multiple pantographs interact simultaneously with the catenary, the mechanical wave generated by the leading pantograph propagates along the contact wire and significantly disturbs the trailing pantograph. This wave propagation can lead to intermittent contact, arcing, and ultimately, a decrease in current collection quality and potential damage to components. Prior research has attempted to find optimal pantograph intervals but, as the paper points out, existing formulas (e.g., from Zhang et al. [29] and Liu et al. [30]) are not perfectly consistent with simulation results or have limitations regarding practical interval ranges. The effectiveness of introducing damping components to mitigate wave intensity has also not been thoroughly investigated in published literature.
• Paper's Entry Point/Innovative Idea: The paper's innovative idea is to conduct a deeper wave propagation analysis by examining the contact wire uplift response caused by a single moving force. It seeks to precisely identify the relationship between the contact wire's velocity (specifically, its peaks and valleys) and the optimal/bad placement of the trailing pantograph. This allows for the development of a more accurate formula for optimal pantograph interval. Furthermore, it introduces a novel measure: integrating a damper into the steady arm of the catenary to actively reduce wave intensity.

2.2. Main Contributions / Findings

The paper makes several primary contributions to improving current collection quality in double-pantograph systems:

• Revealing Deterioration Mechanism: Through wave propagation analysis and studying the contact wire uplift response under a single moving force, the paper identifies that bad intervals for the trailing pantograph occur when it interacts with the velocity peak of the contact wire uplift caused by the leading pantograph. Conversely, good intervals occur at the valley value of the contact wire uplift velocity, where vibrations are offset and attenuated.

• New Optimal Interval Formula: A novel formula for calculating the optimal interval of double pantographs is proposed. This formula, derived from the contact wire uplift response and its velocity, considers parameters such as train speed ($v$), the length of the passing stage ($L_c$), and the fluctuation frequency of the post-passage stage ($f_c$). It explicitly defines both good and bad intervals and demonstrates improved accuracy compared to previous methods.

• Validation of the Proposed Formula: The validity of the new formula is rigorously verified through extensive numerical simulations using parameters from both the Beijing-Tianjin high-speed line in China and a reference model from the European high-speed network (French high-speed network benchmark).

• Investigating Steady Arm Damping: The paper introduces and investigates the effect of adding a damper to the steady arm of the catenary. It finds that slight steady arm damping (e.g., 50 Ns/m in their specific simulation) has a positive effect, significantly reducing the contact force standard deviation for both leading and trailing pantographs and preventing loss of contact. However, excessive damping can create hard spots and negatively impact performance.

• Robust Simulation Model: A double pantographs-catenary model is established using a FEM approach, and its accuracy is verified against the EN 50318:2018 standard, providing a reliable tool for further analysis.

  These findings address the challenge of trailing pantograph performance by offering both passive (optimal spacing) and active (damping) measures, leading to improved current collection quality and enhanced reliability for high-speed multiple pantograph operations.

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

To fully understand this paper, a novice reader should be familiar with several fundamental concepts related to railway electrification, structural dynamics, and numerical methods.

• Pantograph-Catenary System: This is the primary subject of the paper.
  • Pantograph: A device mounted on the roof of an electric train (or tram) that collects electrical current from overhead lines. It consists of a collector strip that slides along the contact wire and a mechanical structure (e.g., lower frame, upper frame) with springs and dampers to maintain constant contact force.
  • Catenary: The overhead line system that supplies electrical energy to the train via the pantograph. It is a complex structure typically comprising:
    • Contact Wire (CW): The lowest wire, directly touched by the pantograph's collector strip. It's designed for smooth sliding contact.
    • Messenger Wire (MW): A stronger, higher-tension wire that supports the contact wire.
    • Droppers: Vertical or inclined wires that connect the messenger wire to the contact wire, distributing the load and maintaining the desired height and sag of the contact wire.
    • Steady Arm: A horizontal or angled arm that holds the contact wire at a specific lateral position relative to the track, especially at supports, to account for stagger (lateral offset).
    • Support Points: Structures (e.g., masts, gantries) from which the entire catenary system is suspended.
    • Span: The distance between two consecutive support points.
• Electrical Multiple Unit (EMU) Train: A type of multiple-unit train consisting of self-propelled carriages, using electricity as the motive power. EMUs do not require a separate locomotive. Multiple pantographs are often used on EMUs when multiple units are coupled together to increase the total power collection capacity.
• Current Collection Quality: A critical performance indicator for pantograph-catenary systems. It refers to the effectiveness and consistency of electrical contact between the pantograph and the contact wire. Good current collection quality means minimal arcing, low contact force variations, and no loss of contact (momentary separation). It's often quantified by metrics like mean contact force, standard deviation of contact force, and number of loss of contact events.
• Wave Propagation (Mechanical Waves): In this context, when a pantograph slides along the contact wire, it exerts a dynamic force that generates mechanical waves (vibrations) that travel along the wire.
  • Forward Wave: Travels in the same direction as the train.
  • Backward Wave: Travels in the opposite direction to the train.
  • Doppler Effect: A change in the frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. In pantograph-catenary interaction, it means the forward wave has a different frequency/amplitude than the backward wave relative to a stationary point on the wire, often making the backward wave more impactful.
• Finite Element Method (FEM): A numerical technique for finding approximate solutions to boundary value problems for partial differential equations. It involves dividing a complex structure (like a catenary) into many small, simple parts called finite elements. The behavior of each element is described by simple equations, and these are then assembled to model the behavior of the entire structure. FEM is particularly useful for structures with complex geometries and non-linear behavior (e.g., large displacements, non-linear material properties, contact mechanics).
  • Euler-Bernoulli Beam: A simplified model for beams, often used for contact wire and messenger wire, that assumes plane sections remain plane and perpendicular to the neutral axis, suitable for slender beams.
  • Truss Element: A simplified structural element that can only carry axial tension or compression, often used for droppers and steady arms where bending stiffness is negligible or intentionally ignored.
• Lumped-Mass Model: A common simplification for dynamic systems (like a pantograph). Instead of modeling every component in detail, the mass, stiffness, and damping characteristics of a system are concentrated into a few discrete points (lumped masses) connected by springs and dampers. This reduces computational complexity while still capturing essential dynamic behavior.
• Proportional Damping (Rayleigh Damping): A method to define the damping matrix $\mathbf{C}$ in structural dynamics. It assumes $\mathbf{C}$ is a linear combination of the mass matrix $\mathbf{M}$ and the stiffness matrix $\mathbf{K}$, i.e., $$\mathbf{C} = \alpha \mathbf{M} + \beta \mathbf{K}$$. $\alpha$ and $\beta$ are constants typically derived from experimental measurements. This simplification is widely used because it preserves the orthogonality of modes, simplifying modal analysis.
• Penalty Function Method: A technique used in FEM to enforce contact constraints. Instead of explicitly modeling the contact area, a "penalty" force is introduced when contact is detected (i.e., when two bodies interpenetrate). This force is proportional to the penetration depth and a contact stiffness (k_c), pushing the bodies apart. If there is no penetration, the force is zero.
• Standard Deviation ($\sigma$): A statistical measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In pantograph-catenary interaction, a low standard deviation of contact force indicates good current collection quality.

3.2. Previous Works

The paper frames its contribution by addressing shortcomings in previous attempts to define optimal pantograph intervals for double-pantograph operations.

• Zhang et al. [29]: This work proposed an optimal interval based on the idea that good performance is achieved when the mechanical wave phase of the leading pantograph is opposite to that of the trailing pantograph. The formula provided is: $ L_{\mathrm{p}} = \left{ \begin{array}{ll} (2k + 1) \frac{Lu}{\alpha \sqrt{T/\rho}} , & k = 1, 2, 3 \ldots \mathrm{ ~ ~ { ( Optimal ~ interval ) } } \ 2k \frac{Lu}{\alpha \sqrt{T/\rho}} , & k = 1, 2, 3 \ldots \mathrm{ ~ ~ { ( Bad ~ interval ) } } \end{array} \right. $ where:
  • $L_{\mathrm{p}}$ is the interval between pantographs.
  • $L$ is the span length of the catenary.
  • $u$ is the velocity of the train.
  • $\alpha$ is a correction factor.
  • $T$ is the tension of the contact wire.
  • $\rho$ is the linear density of the contact wire.
  • $k$ is an integer (1, 2, 3...) representing the order of the interval. The paper criticizes this formula because its underlying assumption that wave frequency is simply calculated by wave speed and span length is simplistic. It also assumes stable contact wire vibration, which isn't always true during forced vibration phases. As shown in Figure 4, the optimal intervals predicted by this formula (red dash lines) often do not align with the simulation's observed optimal intervals.
• Liu et al. [30]: This study proposed an optimal interval based on wave interference theory, expressed as: $ L_{\mathrm{p}} = \left( \frac{3}{2} + k \right) \frac{(C - u)L}{C} , k = 1, 2, 3 \ldots $ where:
  • $L_{\mathrm{p}}$ is the interval between pantographs.
  • $C$ is the wave propagation speed.
  • $u$ is the velocity of the train.
  • $L$ is the span length of the catenary.
  • $k$ is an integer (1, 2, 3...). This formula is primarily focused on very short pantograph intervals, where the leading pantograph acts as an auxiliary to minimize negative effects. The paper notes that while some of its predicted good intervals (black dash lines in Figure 4) might coincide with local minima in simulation, it often predicts many non-optimal intervals and is less applicable to the larger intervals common in China's high-speed networks.

Beyond these specific formulas, the paper also refers to broader research in pantograph-catenary interaction, including:

• Simulation Tools: [1], [2], [5] highlight the importance of numerical simulation (e.g., FEM for catenary [6], multibody or lump-mass models for pantograph [7]) for studying dynamic performance.
• Computational Efficiency: Moving mesh methods [9], [10] and modal coordinate models [11] are used to improve the speed of simulations.
• Damping: The importance of damping in improving structural performance [12] and methods for identifying damping properties [13], [14] are acknowledged.
• Disturbances: External disturbances (wind load [15], [16], aerodynamic instability [17], locomotive excitation [18], electromagnetic interference [19], temperature variation [20]) and internal disturbances (wave propagation [21], component anomalies [22], wire wear/irregularities [23]) affect performance.
• Control Strategies: Various control strategies [24]-[26] are proposed to mitigate negative effects.
• Multiple Pantographs: Other works [27], [28] have also indicated the significant impact of mechanical waves from leading pantographs on trailing pantograph performance.

3.3. Technological Evolution

The evolution of pantograph-catenary technology has been driven by increasing train speeds and demand for higher carrying capacity.

• Single Pantograph Era: Early electric trains typically used a single pantograph per locomotive or EMU.
• Higher Speeds & Carrying Capacity: As train speeds increased, the dynamic interaction became more critical. Maintaining stable contact at hundreds of kilometers per hour is a major engineering challenge. The need for greater power collection for longer trains or coupled EMUs led to the adoption of multiple pantographs.
• Challenges of Multiple Pantographs: The introduction of multiple pantographs brought new challenges, primarily the wave propagation phenomenon. The mechanical waves generated by the first pantograph drastically affect the contact quality of subsequent pantographs, requiring solutions for optimal pantograph interval and wave mitigation.
• Advanced Modeling and Simulation: The field has moved from simplified analytical models to sophisticated Finite Element Method (FEM) models and multi-body dynamics simulations that can capture complex non-linear behaviors, validated by international standards like EN 50318.
• Focus on Optimization and Control: Current research, including this paper, focuses on optimizing system parameters (like pantograph interval) and incorporating active or passive control elements (like dampers) to enhance performance and reliability. This paper's work fits within this technological timeline by addressing a critical operational challenge of multiple pantograph high-speed trains through advanced wave propagation analysis and proposing practical solutions.

3.4. Differentiation Analysis

Compared to previous studies, this paper offers several core differences and innovations:

• Refined Wave Propagation Analysis: Unlike Zhang et al. [29], which simplified wave frequency calculations and assumed stable vibration, this paper conducts a more detailed analysis of the contact wire uplift response and its velocity profiles under a single moving force. It precisely links the deterioration mechanism to velocity peaks and valleys, providing a more nuanced understanding of resonance and attenuation.
• Improved Optimal Interval Formula: The proposed formula directly incorporates parameters derived from the single pantograph interaction, namely the length of the passing stage ($L_c$) and the frequency of the post-passage stage ($f_c$). This makes the formula more responsive to the actual dynamic behavior of the catenary and train speed, overcoming the inconsistencies observed with previous formulas (as shown in Figure 4).
• Broader Applicability for Intervals: While Liu et al. [30] focused on very short intervals (where the leading pantograph might act as an auxiliary), this paper's formula is designed to be applicable and validated across a wider range of pantograph intervals, which is more relevant for high-speed networks in China and Europe.
• Novel Damping Solution: The paper introduces and quantifies the effect of adding a damper to the steady arm as an active wave intensity reduction measure. Previous research primarily discussed structural damping in general. This specific application of a steady arm damper is a novel contribution, offering a practical engineering solution beyond just pantograph interval optimization.
• Rigorous Validation: The model and proposed formula are thoroughly validated against the EN 50318:2018 standard and parameters from real-world high-speed lines (Beijing-Tianjin, French high-speed network), lending strong credibility to its findings.

4. Methodology

4.1. Principles

The core idea of the method used in this paper is to analyze the dynamic mechanical wave propagation within the contact wire of a catenary system. Specifically, it focuses on how the vibrations (or uplift response) caused by a leading pantograph affect a trailing pantograph. The theoretical basis is that the current collection quality of the trailing pantograph is largely determined by the phase and magnitude of the mechanical wave it encounters. If the trailing pantograph is positioned where the contact wire is undergoing a velocity peak due to the leading pantograph's wave, it can lead to resonance and poor contact. Conversely, if it's placed where the contact wire's velocity is at a valley (or is being attenuated), the interaction can be improved. This principle allows for:

1. Optimizing Pantograph Interval: By understanding the characteristic wave patterns (specifically uplift velocity peaks and valleys) generated by a single pantograph, a formula can be derived to position the trailing pantograph at optimal intervals where it experiences minimal disturbance.
2. Mitigating Wave Intensity: Recognizing that wave intensity is detrimental, the paper also explores introducing damping elements into the catenary structure (specifically, the steady arm) to absorb wave energy and reduce its impact on trailing pantographs.

4.2. Core Methodology In-depth (Layer by Layer)

The methodology involves establishing a detailed Finite Element Method (FEM) model of the double pantograph-catenary system, analyzing wave propagation dynamics, deriving an optimal interval formula, and investigating damping effects.

4.2.1. Modeling of Catenary

The catenary system, depicted in Figure 2, is a complex structure comprising contact wire, messenger wire, steady arm, messenger wire support, droppers, and clamps. To capture its geometrical nonlinearity and dynamic behavior, the FEM approach is used. The following figure (Figure 2 from the original paper) illustrates the components of a catenary:

该图像是示意图，展示了接触网的结构，其中包括夹具、信号线、接触线、稳臂和下垂线等组件。接触线的适当设置对于双受电弓的电流采集质量至关重要。

The equation of motion for the catenary, representing its dynamic behavior, is given by: $ \mathbf { M } _ { \mathrm { C } } \ddot { \mathbf { U } } _ { \mathrm { C } } + \mathbf { C } _ { \mathrm { C } } \dot { \mathbf { U } } _ { \mathrm { C } } + \mathbf { K } _ { \mathrm { C } } \mathbf { U } _ { \mathrm { C } } = \mathbf { F } _ { \mathrm { C } } $ where:

• $\mathbf{M}_{\mathrm{C}}$ is the global mass matrix of the catenary.

• $\mathbf{C}_{\mathrm{C}}$ is the global damping matrix of the catenary.

• $\mathbf{K}_{\mathrm{C}}$ is the global stiffness matrix of the catenary.

• $\ddot{\mathbf{U}}_{\mathrm{C}}$ is the global acceleration vector of the catenary.

• $\dot{\mathbf{U}}_{\mathrm{C}}$ is the global velocity vector of the catenary.

• $\mathbf{U}_{\mathrm{C}}$ is the global displacement vector of the catenary.

• $\mathbf{F}_{\mathrm{C}}$ is the external force vector applied to the catenary (e.g., from pantographs).

  The mass matrix $\mathbf{M}_{\mathrm{C}}$ and stiffness matrix $\mathbf{K}_{\mathrm{C}}$ are assembled by summing the element matrix of each catenary component. The mass matrix $\mathbf{M}_{\mathrm{C}}$ is assembled as: $ \mathbf { M } _ { \mathrm { C } } = \sum _ { n _ { \mathrm { c w } } } \mathbf { M } _ { \mathrm { c w } , n } ^ { \mathrm { e } } + \sum _ { n _ { \mathrm { m w } } } \mathbf { M } _ { \mathrm { m w } , n } ^ { \mathrm { e } } + \sum _ { n _ { \mathrm { d r } } } \mathbf { M } _ { \mathrm { d r } , n } ^ { \mathrm { e } } + \sum _ { n _ { \mathrm { c s } } } \mathbf { M } _ { \mathrm { c s } , n } ^ { \mathrm { e } } + \sum _ { n _ { \mathrm { c l } } } \mathbf { M } _ { \mathrm { c l } , n } ^ { \mathrm { e } } $ where:

• $$\mathbf{M}_{\mathrm{cw}, n}^{\mathrm{e}}$$ is the element mass matrix for the $n$-th contact wire (cw) element.

• $$\mathbf{M}_{\mathrm{mw}, n}^{\mathrm{e}}$$ is the element mass matrix for the $n$-th messenger wire (mw) element.

• $$\mathbf{M}_{\mathrm{dr}, n}^{\mathrm{e}}$$ is the element mass matrix for the $n$-th dropper (dr) element.

• $$\mathbf{M}_{\mathrm{cs}, n}^{\mathrm{e}}$$ is the element mass matrix for the $n$-th steady arm (cs) element.

• $$\mathbf{M}_{\mathrm{cl}, n}^{\mathrm{e}}$$ is the element mass matrix for the $n$-th clamp (cl) element.

  The Euler-Bernoulli beam model is used for the contact wire, messenger wire, and steady arm. For a contact wire element, its element mass matrix $$\mathbf{M}_{\mathrm{cw}, n}^{\mathrm{e}}$$ is expressed by: $ \begin{array} { l } { \displaystyle \mathbf { M } _ { c w , n } ^ { \mathrm { e } } = \sum \rho _ { c w } A _ { c w } \int _ { 0 } ^ { l _ { \mathrm { e } } } \mathbf { N } _ { x } ^ { \mathrm { r } } \mathbf { N } _ { x } ^ { \mathrm { r } } \mathrm { d } x + \rho _ { c w } A _ { c w } } \ { \displaystyle \qquad \times \int _ { 0 } ^ { l _ { \mathrm { e } } } \mathbf { N } _ { y } ^ { \mathrm { r } } \mathbf { N } _ { y } ^ { \mathrm { r } } \mathrm { d } x + \rho _ { c w } A _ { c w } \int _ { 0 } ^ { l _ { \mathrm { e } } } \mathbf { N } _ { z } ^ { \mathrm { r } } \mathbf { N } _ { z } ^ { \mathrm { r } } \mathrm { d } x + \rho _ { c w } W _ { \mathrm { r } } } \ { \displaystyle \qquad \times \int _ { 0 } ^ { l _ { \mathrm { e } } } \mathbf { N } _ { \theta x } ^ { \mathrm { r } } \mathbf { T } _ { \theta x } ^ { \mathrm { r } } \mathrm { d } x } \end{array} $ where:

• $\rho_{\mathrm{cw}}$ is the linear density of the contact wire.

• $A_{\mathrm{cw}}$ is the sectional area of the contact wire.

• $W_{\mathrm{r}}$ is the polar moment of inertia of the contact wire cross-section.

• $l_{\mathrm{e}}$ is the element length of the contact wire.

• $\mathbf{N}_{x}^{\mathrm{r}}$, $\mathbf{N}_{y}^{\mathrm{r}}$, $\mathbf{N}_{z}^{\mathrm{r}}$ are the shape functions of the Euler-Bernoulli beam along the X, Y, and Z axes, respectively.

• $$\mathbf{N}_{\theta x}^{\mathrm{r}}$$ is the shape function around the X axis.

• The terms $$\mathbf{M}_{\mathrm{mw}, n}^{\mathrm{e}}$$ (messenger wire) and $$\mathbf{M}_{\mathrm{cs}, n}^{\mathrm{e}}$$ (steady arm) have similar forms, with subscripts changed to mw and cs respectively.

  The droppers are modeled using nonlinear truss elements. The element mass matrix $$\mathbf{M}_{\mathrm{dr}, n}^{\mathrm{e}}$$ for a dropper is: $ \begin{array} { r } { \mathbf { M } _ { d r , n } ^ { \mathrm { e } } = \sum \rho _ { d r } A _ { d r } \int _ { 0 } ^ { l _ { \mathrm { e } } } \mathbf { N } _ { x } ^ { \mathrm { d } } { } ^ { \mathrm { T } } \mathbf { N } _ { x } ^ { \mathrm { d } } \mathrm { d } x + \rho _ { d r } A _ { d r } } \ { \int _ { 0 } ^ { l _ { \mathrm { e } } } \mathbf { N } _ { y } ^ { \mathrm { d } } \mathbf { N } _ { y } ^ { \mathrm { d } } \mathrm { d } x + \rho _ { d r } A _ { d r } \int _ { 0 } ^ { l _ { \mathrm { e } } } \mathbf { N } _ { z } ^ { \mathrm { d } } \mathbf { N } _ { z } ^ { \mathrm { d } } \mathrm { d } x } \end{array} $ where:

• $\rho_{\mathrm{dr}}$ is the linear density of the dropper.

• $A_{\mathrm{dr}}$ is the sectional area of the dropper.

• $\mathbf{N}_{x}^{\mathrm{d}}$, $\mathbf{N}_{y}^{\mathrm{d}}$, $\mathbf{N}_{z}^{\mathrm{d}}$ are the shape functions of the truss element along the X, Y, and Z axes, respectively. The clamps (\mathbf{M}_{\mathrm{cl}, n}^{\mathrm{e}}) are represented as lumped masses (diagonal matrix).

Similarly, the stiffness matrix $\mathbf{K}_{\mathrm{C}}$ is assembled as: $ \mathbf { K } _ { \mathrm { C } } = \sum _ { n _ { \mathrm { c w } } } \mathbf { K } _ { c w , n } ^ { \mathrm { e } } + \sum _ { n _ { \mathrm { m w } } } \mathbf { K } _ { m w , n } ^ { \mathrm { e } } + \sum _ { n _ { \mathrm { d r } } } \mathbf { K } _ { { \mathrm { d r } } , n } ^ { \mathrm { e } } + \sum _ { n _ { \mathrm { c s } } } \mathbf { K } _ { c s , n } ^ { \mathrm { e } } + \sum _ { n _ { \mathrm { m s } } } \mathbf { K } _ { m s , n } ^ { \mathrm { e } } $ where:

• $$\mathbf{K}_{\mathrm{cw}, n}^{\mathrm{e}}$$ is the element stiffness matrix for the $n$-th contact wire element.

• $$\mathbf{K}_{\mathrm{mw}, n}^{\mathrm{e}}$$ is the element stiffness matrix for the $n$-th messenger wire element.

• $$\mathbf{K}_{\mathrm{dr}, n}^{\mathrm{e}}$$ is the element stiffness matrix for the $n$-th dropper element.

• $$\mathbf{K}_{\mathrm{cs}, n}^{\mathrm{e}}$$ is the element stiffness matrix for the $n$-th steady arm element.

• $$\mathbf{K}_{\mathrm{ms}, n}^{\mathrm{e}}$$ is the element stiffness matrix for the $n$-th messenger wire support (ms) element.

  The element stiffness matrix $$\mathbf{K}_{\mathrm{cw}, n}^{\mathrm{e}}$$ for the contact wire is written by: $ { \begin{array} { l l } { \mathbf { K } _ { \mathrm { c w } , n } ^ { \mathrm { e } } = \mathbf { K } _ { \alpha } + \mathbf { K } _ { \beta } + \mathbf { K } _ { \chi } + \mathbf { K } _ { \delta } } \ { \mathbf { K } _ { \alpha } = \sum E _ { \mathrm { c w } } A _ { \mathrm { c w } } \int _ { 0 } ^ { L _ { \mathrm { c } } } ( \frac { \partial \mathbf { N } _ { \mathrm { c s } } ^ { \mathrm { r } } } { \partial x } ) ^ { \mathrm { T } } \frac { \partial \mathbf { N } _ { \mathrm { c s } } ^ { \mathrm { r } } } { \partial x } \mathrm { d } x } \ { \mathbf { K } _ { \beta } = E _ { \mathrm { c w } } I _ { \mathrm { c w y } } \int _ { 0 } ^ { l _ { \mathrm { c } } } ( \frac { \partial ^ { 2 } \mathbf { N } _ { \mathrm { c s } } ^ { \mathrm { r } } } { \partial x ^ { 2 } } ) ^ { \mathrm { T } } \frac { \partial ^ { 2 } \mathbf { N } _ { \mathrm { r s } } ^ { \mathrm { r } } } { \partial x ^ { 2 } } \mathrm { d } x } \ { \mathbf { K } _ { \chi } = E _ { \mathrm { r w } } I _ { \mathrm { c w z } } \int _ { 0 } ^ { l _ { \mathrm { c } } } ( \frac { \partial ^ { 2 } \mathbf { N } _ { \mathrm { y } } ^ { \mathrm { r } } } { \partial x ^ { 2 } } ) ^ { \mathrm { T } } \frac { \partial ^ { 2 } \mathbf { N } _ { \mathrm { y } } ^ { \mathrm { r } } } { \partial x ^ { 2 } } \mathrm { d } x } \ \mathbf { K } _ { \delta } = G _ { \mathrm { c w } } I _ { \mathrm { c w } } \int _ { 0 } ^ { l _ { \mathrm { c } } } ( \frac { \partial ^ { 2 } \mathbf { N } _ { \mathrm { g } , x } ^ { \mathrm { g } } } { \partial x ^ { 2 } } ) ^ { \mathrm { T } } \frac { \partial ^ { 2 } \mathbf { N } _ { \mathrm { g } , x } ^ { \mathrm { g } } } { \partial x ^ { 2 } } \mathrm { d } x \end{array} $ where:

• $E_{\mathrm{cw}}$ is the Young's modulus of the contact wire.

• $I_{\mathrm{cwx}}$, $I_{\mathrm{cwy}}$, and $I_{\mathrm{cwz}}$ are the moments of inertia of the contact wire with respect to the X, Y, and Z axes, respectively.

• $G_{\mathrm{cw}}$ is the shear modulus (rigid modulus) of the contact wire.

• The various $\mathbf{K}$ terms ($$\mathbf{K}_{\alpha}`,`\mathbf{K}_{\beta}`,`\mathbf{K}_{\chi}`,`\mathbf{K}_{\delta}$$) represent contributions from axial, bending (in Y and Z planes), and torsional stiffness, respectively. The messenger wire (\mathbf{K}_{\mathrm{mw}, n}^{\mathrm{e}}) has a similar form.

For the dropper (truss element), the element stiffness matrix $$\mathbf{K}_{\mathrm{dr}, n}^{\mathrm{e}}$$ is expressed by: $ \mathbf { K } _ { \mathrm { d r } , n } ^ { \mathrm { e } } = \sum E _ { \mathrm { d r } } A _ { \mathrm { d r } } \int _ { 0 } ^ { l _ { \mathrm { e } } } \left( \frac { \partial \mathbf { N } _ { x } ^ { \mathrm { d } } } { \partial x } \right) ^ { \mathrm { T } } \frac { \partial \mathbf { N } _ { x } ^ { \mathrm { d } } } { \partial x } \mathrm { d } x $ where:

• $E_{\mathrm{dr}}$ is the Young's modulus of the dropper. Importantly, droppers are modeled to only withstand tension (pulling force) and not compression (pushing force), meaning their stiffness becomes $0 N/m$ under compression. The steady arm (\mathbf{K}{\mathrm{cs}, n}^{\mathrm{e}}) uses the same truss element form. The messenger wire support (\mathbf{K}{\mathrm{ms}, n}^{\mathrm{e}}) is treated as a virtual support point with equivalent lumped stiffness.

The structural damping matrix $\mathbf{C}_{\mathrm{C}}$ is introduced based on the proportional damping assumption (also known as Rayleigh damping), which simplifies the damping calculation: $ \mathbf { C } _ { \mathrm { C } } = \alpha \mathbf { M } _ { \mathrm { C } } + \beta \mathbf { K } _ { \mathrm { C } } $ where:

• $\alpha$ and $\beta$ are two constant values, known as Rayleigh damping coefficients, derived from experimental measurements.

4.2.2. Modeling of Pantograph

The pantograph is represented by a widely used lumped-mass model, as shown in Figure 3. This model simplifies the complex physical structure of a pantograph into equivalent masses, stiffnesses, and dampings obtained from bench tests (controlled laboratory experiments). The following figure (Figure 3 from the original paper) shows the pantograph model:

该图像是一个示意图，展示了双机械系统的模型，其中包含两个质量 $m_1$ 和 $m_2$，各自通过弹簧 $k_1$ 和 $k_2$ 相连，同时附带阻尼器 $c_m$ 和 $c_d$。该模型用于分析波动传播对轨道和供电系统的影响。

The contact force (f_c) between the pantograph collector and the contact wire is described using the penalty function method, which avoids complex explicit contact detection by introducing a force when penetration occurs: $ f _ { \mathrm { c } } = \left{ \begin{array} { l l } { k _ { \mathrm { c } } \left( y _ { \mathrm { p } } - y _ { \mathrm { c } } \right) } & { y _ { \mathrm { p } } \geq y _ { \mathrm { c } } } \ { 0 } & { y _ { \mathrm { p } } < y _ { \mathrm { c } } } \end{array} \right. $ where:

• $f_{\mathrm{c}}$ is the contact force between the contact wire and pantograph collector.
• $y_{\mathrm{p}}$ is the vertical displacement of the pantograph collector.
• $y_{\mathrm{c}}$ is the vertical displacement of the contact point in the contact wire.
• $k_{\mathrm{c}}$ is the contact stiffness, representing the resistance to penetration. This equation means that a force is generated only when the pantograph collector (y_p) tries to penetrate below the contact wire (y_c), and the force magnitude is proportional to the penetration depth.

4.2.3. Model Verification

To ensure the reliability of the simulation results, the double pantographs-catenary model is verified against the latest European standard EN 50318:2018. This standard provides a reference model and acceptance ranges for key performance indicators, ensuring consistency and accuracy across different simulation tools. The verification involves comparing static configuration (e.g., contact wire pre-sag) and dynamic results (e.g., contact force metrics) with the standard's requirements.

4.2.4. Analysis of Wave Propagation and Deterioration Mechanism

The core of the proposed solution stems from a detailed analysis of wave propagation in the contact wire excited by a single moving force (representing a leading pantograph).

• Single Moving Force Simulation: A constant moving force (e.g., 150 N) is applied to the catenary model at various speeds.
• Contact Wire Uplift Response: The vertical displacement (uplift) and velocity of the contact wire at specific points (e.g., support points) are observed and analyzed over time.
• Identification of $L_c$ and $f_c$:
  • The uplift velocity response is divided into three stages:
    1. Pre-passage stage: Slight oscillations from the forward wave.
    2. Passing stage (red background in Figure 7(c)): The period when the moving force is directly passing the observation point. The length of this region along the track is denoted as $\mathbf{L_c}$.
    3. Post-passage stage (green background in Figure 7(c)): The contact wire is affected by the backward wave, showing periodic fluctuations in velocity. The frequency of this fluctuation is denoted as $\mathbf{f_c}$.
• Correlation with Current Collection Quality: The simulation results showed a strong correlation:

  • Bad intervals for double pantographs appear when the trailing pantograph encounters the velocity peak of the contact wire uplift caused by the leading pantograph (Figure 7(b)). This condition can lead to resonance and poor current collection quality.

  • Good intervals appear when the trailing pantograph encounters a valley value of the contact wire uplift velocity (Figure 7(b)). At these points, the vibrations from the leading pantograph are attenuated or offset, leading to improved contact.

    The following figure (Figure 7 from the original paper) illustrates the uplift and velocity response of the contact wire:

    该图像是示意图，展示了接触线的抬升响应及其速度变化。图(a)显示了接触线的抬升量与位置的关系，标示了“坏区间”和“好区间”。图(b)展示了不同位置处的速度，强调了相同的区间。图(c)则明显标示了位置分界点 $L_c$ ，并注释了对应频率 $f_c$。这些结果有助于理解双受流器与接触线之间的相互作用。

4.2.5. Proposed Optimal Interval Formula

Based on the mechanism that bad intervals occur at velocity peaks and good intervals at velocity valleys of the post-passage stage, a new formula for optimal pantograph intervals (L_p) is proposed. The relationship for a bad interval is initially derived as: $ \frac { v } { \relax _ { p } - \relax { L } _ { \mathrm { { c } } } } = \frac { 2 } { 2 k - 1 } f _ { \mathrm { { c } } } $ where:

• $v$ is the train speed.

• $L_p$ is the interval of pantographs.

• $L_c$ is the length of the passing stage (determined by the contact wire's response to a single pantograph, as defined above).

• $f_c$ is the fluctuation frequency of the contact wire uplift in the post-passage stage.

• $k$ is an integer (1, 2, 3...).

  From this, the formulas for both optimal and bad intervals (L_p) are summarized as: $ \begin{array} { r l } & { L _ { \mathrm { p } } } \ & { = \left{ \begin{array} { l l } { \frac { 2 k - 1 } { 2 } \cdot \frac { v } { f _ { \mathrm { c } } } + L _ { \mathrm { c } } } & { k = 1 , 2 , 3 . . . . . . \mathrm{ ~ ~ { ( Optimal ~ interval ) } } } \ { k \cdot \frac { v } { f _ { \mathrm { c } } } + L _ { \mathrm { c } } } & { k = 1 , 2 , 3 . . . . . . . \mathrm{ ~ ~ { ( Bad ~ interval ) } } } \end{array} \right. } \end{array} $ where:

• $L_p$ is the interval between pantographs.

• $v$ is the train speed.

• $f_c$ is the fluctuation frequency of the contact wire uplift in the post-passage stage (obtained from frequency analysis of the uplift response, e.g., Figure 8).

• $L_c$ is the length of the passing stage (obtained from single pantograph simulation, e.g., Figure 7(c)).

• $k$ is an integer (1, 2, 3...) representing the order of the interval.

  The values of $L_c$ and $f_c$ are crucial and are determined from single pantograph simulations for specific speeds and catenary structures. $f_c$ is found to be one of the natural frequencies of the catenary, excited by the moving load.

4.2.6. Introduction of Damper to Steady Arm

As an alternative or complementary measure, the paper investigates adding a damper to the steady arm of the catenary. This aims to actively reduce the intensity of the mechanical wave propagating from the leading pantograph, thereby mitigating its negative effect on the trailing pantograph. The damper is modeled as a component that dissipates vibrational energy. The effectiveness is evaluated by simulating double pantograph-catenary interaction with varying damping coefficients for the steady arm.

The following figure (Figure 13 from the original paper) shows the cantilever system with a damper on the steady arm:

该图像是图13，展示了带有阻尼器的悬臂系统。图中标示了阻尼器的位置，旨在有效降低波动强度，以改善双弓架-接触网的交互效果。

5. Experimental Setup

5.1. Datasets

The experimental validation in this paper primarily uses two sets of catenary and pantograph parameters:

1. Reference Model from EN 50318:2018 (European High-Speed Network): This is a standardized model used for benchmarking and validating pantograph-catenary simulation tools. It represents a realistic line from the French high-speed network.

   • Source: European standard EN 50318:2018.

   • Characteristics: Detailed geometrical and material properties for contact wire, messenger wire, droppers, steady arm, and MW support. Includes specific values for line density, tension, Young's modulus, cross-section, dropper rigidity, span length, stagger value, etc.

   • Usage: Used for initial model verification (Section II.C) and later for validation of the optimal interval formula (Section V.B). The following are the results from Table I of the original paper:

     Catenary material property
     Contact wire(CW)	Line density: 1.35 kg/m; Tension: 22 kN; Young's modulus: 100 kN/mm²; Cross section; 150 mm²
     Messenger wire(MW)	Line density: 1.08 kg/m; Tension: 16 kN; Young's modulus: 97 kN/mm²; Cross section; 120 mm²
     Dropper	Line density: 0.117 kg/m; Clamp mass; 195 g (on CW), 165 g (on MW) Tensile rigidity from dropper1 to 9 (kN/m): 197; 223; 247; 264; 269; 264; 247; 223; 197; Line density: 0.739 kg/m
     Steady arm	Line density: 0.739 kg/m
     MW support	Stiffness: 500 kN/m; Damping: 1000 Ns/m
     Catenary geometrical property
     Encumbrance: 1.2 m; Interval of droppers: 6.25 m; Number of droppers: 9; Number of span: 29; Length of span: 50 m; Stagger value: 0.2 m; Steady arm length: 1.2 m;

2. Beijing-Tianjin High-Speed Line (China High-Speed Network): This represents an operational high-speed line in China.

   • Source: Parameters for the Beijing-Tianjin railway.

   • Characteristics: Different material and geometrical properties for contact wire, messenger wire, droppers, MW support, and steady arm stiffness. It also includes specific pantograph properties (masses, stiffnesses, dampings, static force).

   • Usage: Primary object for investigating the shortfalls of previous solutions (Section III) and for verification of the proposed optimal interval formula (Section V.A) and the damping study (Section VI). The following are the results from Table IV of the original paper:

     Catenary material property
     Contact wire(CW)	Line density: 1.082 kg/m; Tension: 27 kN; Young's modulus: 100 kN/mm²; Cross section; 120 mm²
     Messenger wire(MW)	Line density: 1.068 kg/m; Tension: 21 kN; Young's modulus: 97 kN/mm²; Cross section; 120 mm²
     Dropper	Tensile rigidity: 105 N/m
     MW support	Fixed Catenary geometrical property
     Encumbrance:1.6 m; Interval of droppers: 10 m; Number of droppers: 5;
     Number of span: 29;Length of span 50 m; Stagger value: 0.3 m; Steady arm stiffness: 1.25× 10⁷ N/m;
     Pantograph property
     m1: 5 kg; m2: 18.98 kg; k1: 6000 N/m; k2: 0.5 N/m; cm: 5 Ns/m; cd: 350 Ns/m; kp1: 0.006 Ns²/m²; kp2: 0.006 Ns²/m²;Fst: 70 N.

These datasets were chosen because they represent both a standardized benchmark for simulation tool validation and a real-world operating high-speed line, ensuring the proposed methods are applicable and robust for practical scenarios.

5.2. Evaluation Metrics

The paper uses several metrics to evaluate current collection quality and dynamic interaction performance:

1. Contact Force Standard Deviation ($\sigma$):

   • Conceptual Definition: The standard deviation of the contact force measures the fluctuation or dispersion of the force around its mean value. A lower standard deviation indicates more stable contact and better current collection quality. This metric directly reflects how smooth and consistent the interaction is.
   • Mathematical Formula: $ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (F_i - \bar{F})^2} $
   • Symbol Explanation:
     • $\sigma$: Standard deviation of the contact force.
     • $N$: Total number of contact force data points recorded during the simulation.
     • $F_i$: The $i$-th individual contact force measurement.
     • $\bar{F}$: The mean contact force over the entire measurement period.

2. Mean Contact Force ($F_m$):

   • Conceptual Definition: The average force maintained between the pantograph collector and the contact wire. It indicates the nominal pressure exerted by the pantograph to ensure electrical contact.
   • Mathematical Formula: $ F_m = \frac{1}{N} \sum_{i=1}^N F_i $
   • Symbol Explanation:
     • $F_m$: Mean contact force.
     • $N$: Total number of contact force data points.
     • $F_i$: The $i$-th individual contact force measurement.

3. Maximum Contact Force ($F_{max}$):

   • Conceptual Definition: The highest instantaneous contact force recorded during the interaction. Excessive maximum forces can lead to increased wear on both pantograph and catenary, and potential damage.
   • Mathematical Formula: $F_{max} = \max(F_1, F_2, \ldots, F_N)$
   • Symbol Explanation:
     • $F_{max}$: Maximum contact force.
     • $F_i$: The $i$-th individual contact force measurement.

4. Minimum Contact Force ($F_{min}$):

   • Conceptual Definition: The lowest instantaneous contact force recorded. A minimum force that approaches zero (or becomes zero) indicates loss of contact, which can lead to arcing and power supply interruptions.
   • Mathematical Formula: $F_{min} = \min(F_1, F_2, \ldots, F_N)$
   • Symbol Explanation:
     • $F_{min}$: Minimum contact force.
     • $F_i$: The $i$-th individual contact force measurement.

5. Loss of Contact (Y/N):

   • Conceptual Definition: A binary indicator (Yes/No) that signifies whether the pantograph momentarily loses electrical contact with the catenary. This occurs when the contact force drops to zero. Loss of contact is highly undesirable as it causes arcing, interferes with power supply, and damages equipment.
   • No specific mathematical formula is provided in the paper as it is a qualitative indicator derived from $F_{min}$ values. If $F_{min}$ is 0 or less, loss of contact occurs.

5.3. Baselines

The paper primarily compares its proposed optimal interval formula against two existing formulas from prior research:

• Zhang et al. [29]'s formula (Eq. 9): Based on mechanical wave phase being opposite.
• Liu et al. [30]'s formula (Eq. 10): Based on wave interference theory. These are chosen because they represent the main previous attempts to mathematically define optimal intervals for double pantographs, making them direct and relevant baselines for comparison. For the steady arm damping study, the baseline is implicitly the system with zero damping in the steady arm.

6. Results & Analysis

6.1. Core Results Analysis

The experimental results are presented in several tables and figures, systematically validating the model, identifying shortcomings in previous work, demonstrating the proposed solution's effectiveness, and exploring the impact of steady arm damping.

6.1.1. Model Verification Results

The accuracy of the FEM model for the catenary is verified using the EN 50318:2018 standard. The following are the results from Table II of the original paper:

Table II presents the static configuration results (pre-sag and equivalent elasticity). The maximum percentage error for pre-sag is 0.96% (at dropper 4 and 6), and for equivalent elasticity, it is 3.22% (at dropper 2 and 8). These errors are significantly below the 10% threshold specified by the standard, indicating high accuracy for the static model.

The following are the results from Table III of the original paper:

Table III shows the dynamic simulation results for contact force metrics ($F_m$, $\sigma$, $F_{max}$, $F_{min}$) at speeds of $275 km/h$ and $320 km/h$. All simulated values fall within the range of acceptance defined by EN 50318. This comprehensive verification confirms that the developed FEM model is accurate and reliable for further analysis of pantograph-catenary interaction.

6.1.2. Shortcomings of Previous Optimal Interval Formulas

The paper evaluates the effectiveness of existing optimal interval formulas (Zhang et al. [29] and Liu et al. [30]) against its own FEM simulation results for the Beijing-Tianjin high-speed line. The following figure (Figure 4 from the original paper) compares previous optimal intervals and simulation results:

该图像是图表，展示了双受电弓间隔（米）与标准偏差（N）之间的关系。图中的蓝线表示标准偏差的变化，红色虚线和黑色虚线分别代表Zhang等人和Liu等人的结果，为不同间隔下的标准偏差提供了参考。

Figure 4 plots the contact force standard deviation (\sigma) of the trailing pantograph against pantograph interval at $350 km/h$. The blue line represents the simulation results, showing a clear periodic relationship. The red dashed lines represent optimal intervals calculated by Zhang et al.'s formula (Eq. 9), and black dashed lines by Liu et al.'s formula (Eq. 10).

• Zhang et al.'s Formula (Eq. 9): Most red dashed lines do not align with the local minima of $\sigma$. In fact, they appear closer to local maxima or non-optimal positions, indicating that this formula is not consistent with the simulation.
• Liu et al.'s Formula (Eq. 10): While some black dashed lines coincide with local minima of $\sigma$, the formula also predicts many non-optimal intervals, especially for larger spacings, making it less useful for the pantograph intervals typically used in China's high-speed network. This analysis clearly demonstrates the shortcomings of previous solutions and highlights the need for a more accurate method.

6.1.3. Deterioration Mechanism and Proposed Formula

To understand the deterioration mechanism, the contact wire uplift response and velocity caused by a single moving force are analyzed. The following figure (Figure 6 from the original paper) illustrates the vibration of the contact wire at 350 km/h:

该图像是图 6，展示了在 $350 ext{ km/h}$ 速度下接触网的振动情况。图中标示了前向波、后向波、受力路径、好的间隔与坏的间隔，分析了不同时间点位置的接触网起伏变化。

Figure 6 shows the contact wire uplift contour over position and time at $350 km/h$. The red line is the moving force trajectory. The yellow and white lines represent forward and backward wave propagation, respectively. The backward wave (white line) has a significantly larger amplitude, explaining its greater influence on the trailing pantograph. Specific bad interval (135 m, yellow dashed line) and optimal interval (177 m, red dashed line) are highlighted from Figure 4.

Referring back to Figure 7 (in Methodology section), the analysis of contact wire uplift and velocity at a support point confirms that the bad interval (yellow dashed line) appears at a velocity peak of the contact wire uplift, while the optimal interval (red dashed line) appears at a valley value of the uplift velocity. This supports the idea that resonance at velocity peaks causes deterioration, and attenuation at velocity valleys improves performance. The post-passage stage fluctuation frequency (f_c) and the length of the passing stage (L_c) are extracted from these responses.

The following figure (Figure 8 from the original paper) shows the frequency domain of contact wire uplift:

该图像是图8，展示了接触线抬升的频域特性。横坐标为频率（Hz），纵坐标为功率，图中标示了频率峰值 $f_{c2} = 1.424$ Hz，表现出在特定频率下的功率变化趋势。

Figure 8 shows the frequency domain of the contact wire uplift at $350 km/h$, revealing a peak frequency (f_c) of 1.424 Hz in the post-passage stage. Using this $f_c$ and the derived $L_c$ in the proposed formula (Eq. 12), new optimal intervals for $350 km/h$ are calculated as 116.3 m, 184.5 m, 252.8 m, and 321.1 m.

The following figure (Figure 9 from the original paper) compares optimal intervals and calculation results:

该图像是一个图表，展示了不同双一单元间隔下电流采集质量的标准差（单位：N）。红色虚线标记了良好间隔的位置，数据表明在170m到250m的间隔内，标准差有明显的波动，体现了双单元操作对电流采集质量的影响。

Figure 9 compares these calculated optimal intervals (red dashed lines) with the simulation results for the Beijing-Tianjin line. The red dashed lines align very well with the local minima of the contact force standard deviation (\sigma), demonstrating a strong consistency between the proposed formula and the FEM simulation. This validates the new formula's accuracy and its ability to predict optimal intervals.

6.1.4. Verification with Beijing-Tianjin High-Speed Line at Various Speeds

The proposed formula (Eq. 12) is further validated for the Beijing-Tianjin high-speed line across a range of speeds ($300 km/h$ to $380 km/h$). The following are the results from Table V of the original paper:

Table V shows the extracted $f_c$ and $L_c$ values, and the calculated optimal intervals for various speeds. $f_c$ slightly increases with speed, and is found to correlate with one of the catenary's natural frequencies (Figure 10). The following figure (Figure 10 from the original paper) compares $f_c$ and natural frequency:

该图像是图10，展示了自然频率与模态阶次的关系。图中标记了在不同速度（300km/h, 320km/h, 340km/h, 350km/h, 360km/h, 380km/h）下的情况，数据点与模态阶次之间的对应关系清晰可见。

The following figure (Figure 11 from the original paper) compares optimal intervals calculation results at different speed:

该图像是一个图表，展示了在不同速度下双边斜杆间隔的标准偏差计算结果。图中包含五个子图，分别对应速度为300 km/h (a)、320 km/h (b)、340 km/h (c)、360 km/h (d)和380 km/h (e)的情况。在每个图中，X轴表示斜杆之间的间隔（米），Y轴表示标准偏差（N），红色虚线标识了“良好间隔”位置。

Figure 11 presents five subplots, each comparing the calculated optimal intervals (red dashed lines) with the simulation results for $\sigma$ at speeds from $300 km/h$ to $380 km/h$. In all cases, the red dashed lines consistently align with the local minima of the standard deviation, further solidifying the validation and accuracy of the proposed formula across different operating conditions.

6.1.5. Verification with Reference Model in Benchmark (European Network)

The proposed formula is also validated using the EN 50318 reference model (representing a French high-speed network). The following are the results from Table VI of the original paper:

Table VI shows $f_c$, $L_c$, and the calculated optimal and bad intervals for this model at $320 km/h$. The following figure (Figure 12 from the original paper) compares optimal interval calculation results:

该图像是一个图表，展示了双解析图之间的间距与标准差之间的关系。横轴为解析图间距（米），纵轴为标准差（牛顿），并标示了良好间距（红色虚线）和不良间距（黑色虚线）。

Figure 12 compares these calculated optimal (red dashed lines) and bad (black dashed lines) intervals with the simulation results for the EN 50318 reference model. Both sets of lines show excellent consistency with the local minima and maxima of the standard deviation, respectively. This multi-system validation confirms the robustness and general applicability of the proposed formula.

6.1.6. Effect of Steady Arm Damping

The paper investigates the impact of adding a damper to the steady arm on the current collection quality. The simulations use the Beijing-Tianjin line parameters, with a pantograph interval of 200 m and speed of $350 km/h$. The following are the results from Table VII of the original paper:

Table VII presents standard deviations (\sigma) for both leading and trailing pantographs, along with loss of contact occurrences, for steady arm damping values from $0 Ns/m$ to $250 Ns/m$.

• Optimal Damping: A damping value of $50 Ns/m$ yields the lowest $\sigma$ for the trailing pantograph (61.15 N) and also for the leading pantograph when compared to $0 Ns/m$. Significantly, it's the first damping value where loss of contact is avoided (N). This suggests that slight steady arm damping has a positive effect, reducing wave intensity and improving overall current collection quality.
• Excessive Damping: As damping increases beyond $50 Ns/m$, the $\sigma$ for the leading pantograph starts to increase, and for the trailing pantograph, it eventually increases as well. At $225 Ns/m$ and $250 Ns/m$, loss of contact re-occurs, and $\sigma$ values are higher than with no damping. This indicates that excessive steady arm damping behaves like a hard spot in the catenary, detrimentally affecting the pantograph-catenary interaction.

6.2. Ablation Studies / Parameter Analysis

The paper implicitly conducts parameter analysis through:

• Varying Pantograph Interval: The relationship between pantograph interval and contact force standard deviation is extensively explored (Figures 4, 9, 11, 12) to identify periodic patterns and validate the optimal interval formula.
• Varying Train Speed: The optimal interval formula is tested and validated across a range of speeds ($300 km/h$ to $380 km/h$, see Figure 11 and Table V). This demonstrates the formula's consistency under different operating speeds, showing how $f_c$ and $L_c$ change with speed and how this affects optimal spacing.
• Varying Steady Arm Damping: Different damping coefficients for the steady arm are simulated (Table VII) to quantify its impact on current collection quality. This acts as an ablation study to understand the optimal range and negative effects of excessive damping, effectively verifying the positive impact of slight steady arm damping.

7. Conclusion & Reflections

7.1. Conclusion Summary

This paper rigorously investigates effective measures to enhance current collection quality for double pantograph-catenary systems in high-speed railway applications. Utilizing a Finite Element Method (FEM) model validated against the EN 50318:2018 standard, the authors delve into the wave propagation mechanism responsible for trailing pantograph performance degradation. The key finding is that bad intervals correspond to velocity peaks of the contact wire uplift caused by the leading pantograph, while good intervals occur at velocity valleys. Building upon this insight, a novel formula for determining optimal double-pantograph intervals is proposed, incorporating train speed ($v$), the length of the passing stage ($L_c$), and the frequency of the post-passage stage ($f_c$). This formula demonstrates superior accuracy and consistency compared to previous methods, validated across both Chinese and European high-speed networks. Furthermore, the study introduces and evaluates the effect of steady arm damping, revealing that slight steady arm damping positively improves current collection quality for both leading and trailing pantographs and helps prevent loss of contact.

7.2. Limitations & Future Work

The authors acknowledge several limitations and suggest future research directions:

• Dynamic Nature of Optimal Interval: The proposed optimal interval formula is directly dependent on train speed ($v$) and catenary structural parameters (which influence $L_c$ and $f_c$). In reality, trains do not always operate at a single speed, and catenary structures can vary significantly across different spans of a railway line. Therefore, a single calculated optimal interval may not be consistently ideal for an entire real-world line, implying a need for more adaptive solutions.

• Unclear Relationship between Velocity and $f_c$: While the paper notes that the post-passage fluctuation frequency (f_c) correlates with one of the catenary's natural frequencies and changes slightly with moving velocity, the explicit relationship between velocity and which natural frequency is excited remains unclear. This warrants further investigation to provide a more fundamental understanding and potentially improve the predictive power of the formula.

• Practical Implementation of Optimal Interval: The paper highlights the difficulty of applying a fixed optimal interval in a real-world scenario with varying speeds and catenary configurations.

  Future work could focus on:

• Developing adaptive strategies for pantograph interval adjustment based on real-time speed and catenary characteristics.

• Further fundamental research into the exact relationship between moving load velocity and the excitation of catenary natural frequencies.

• Exploring active control mechanisms for pantographs or catenary components that can dynamically mitigate wave propagation effects, rather than relying solely on fixed optimal intervals or passive damping.

7.3. Personal Insights & Critique

This paper provides a robust and insightful analysis of a critical problem in high-speed rail electrification. The approach of dissecting the contact wire uplift response and correlating velocity peaks and valleys with bad and good intervals is a logical and effective way to understand the underlying wave propagation mechanism. This mechanistic understanding is a significant improvement over purely empirical or simplified wave-based formulas.

One particular strength is the rigorous model validation against the EN 50318 standard, which lends high credibility to the FEM model and subsequent simulation results. The comprehensive verification of the proposed formula across different speeds and two distinct high-speed networks (China and Europe) strongly supports its general applicability.

From a practical engineering perspective, the investigation into steady arm damping is highly valuable. While optimal pantograph interval is a design consideration, incorporating damping elements is a tangible modification that can be implemented in existing or new catenary systems. The finding that slight damping is beneficial but excessive damping creates hard spots provides crucial guidance for system designers.

Potential Issues/Areas for Improvement:

1. Real-time Adaptability: As the authors noted, a fixed optimal interval is not always practical. Future work could explore how to dynamically adjust pantograph spacing in coupled EMUs (if mechanically feasible) or how to implement active pantographs that can counteract wave disturbances based on real-time measurements.

2. Uncertainty Quantification: The paper relies on deterministic simulations. In reality, catenary properties can have variations due to manufacturing tolerances, wear, or environmental factors (e.g., temperature). Incorporating uncertainty quantification into the model could provide a more robust assessment of performance and optimal intervals.

3. Energy Efficiency of Damping: While damping improves current collection quality, it also dissipates energy. A more detailed analysis of the energy trade-offs involved with steady arm damping might be beneficial for optimizing system efficiency.

4. Multi-Pantograph Systems: The paper focuses on double pantographs. Extending the analysis to three or more pantographs (e.g., for very long trains) would be a natural progression, as the wave interference patterns become significantly more complex.

   Overall, this paper offers significant contributions to the field, providing both a deeper scientific understanding of pantograph-catenary dynamics and practical, validated measures for improving current collection quality in high-speed rail systems. Its methods and conclusions could potentially be transferred to other domains involving moving loads on flexible structures where wave propagation is a concern.