Catenary Model Research Articles

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

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.

Study on the Length of the Effective Vibration Area of the Catenary in a Pantograph–Catenary Interaction System

The effective vibration area includes most of the catenary vibration caused by pantograph–catenary interactions and is the basis of the real-time catenary model for hardware-in-the-loop simulation. However, while the length of the effective vibration area is one of the most important parameters of the real-time catenary model, it has not been fully studied at present. In this paper, the length of the effective vibration area is first investigated. A pantograph–catenary interaction model is developed based on the modal superposition method. After the validation of the model, the vibration energy distribution of the catenary is used to determine the length of the effective vibration area based on the converged total energy. The influence of vehicle velocity and contact wire tension on the vibration energy distribution and length of the effective vibration area is investigated. The obtained appropriate length of effective vibration area is validated by a real-time catenary model and online measurement data of the contact force. The investigation results show that the energy distribution of the catenary can accurately determine the length of effective vibration area, and it increases with increasing vehicle velocity but decreases with increasing contact wire tension. The appropriate length of effective vibration area should be at least 160 m (approximately three spans) in the pantograph–catenary system.

Load characteristics of screen panels in vibrating flip-flow screens

A vibrating flip-flow screen (VFFS) is an effective solution for screening viscous and fine-grained materials. As one of the critical components, the load characteristics of the polyurethane screen panel primarily affect the fatigue life. However, more research is needed to understand the load characteristics of the screen panel. In this article, the discrete screen panel dynamic model is proposed to research the load characteristics of the screen panel based on the catenary model. The dynamic response parameters of the screen panel are obtained by the catenary model, and the finite element model of the screen panel is established to extract the dynamic characteristic parameters of the screen panel. The accuracy of the catenary model and the finite element model is verified by the experiment, and then the rationality of the dynamic theory is verified by the simulation. Moreover, the effects of hardness, amplitude, frequency, and angle on the tension and vibration strength of the screen panel are investigated. The results show that hardness, amplitude, frequency, and angle significantly affect the tension and vibration strength of the screen panel. The sensitivity of each factor on tension from high to low: amplitude > hardness > frequency > angle. Adjusting the value of each factor within a certain range is beneficial to increasing the vibration strength; meanwhile, it can reduce the tension of the screen panel. This research helps improve the service life and provides a theoretical basis for designing and optimizing the screen panel.

A comparative study on the influence of typical track failures on high-speed pantograph-catenary interaction dynamics

Different track failures are proven to have different influences on train-track interactions, but their influence on pantograph-catenary interactions has not been investigated thus far. In this work, the influences of typical track failures on pantograph-catenary interaction dynamics are first investigated. A reduced train-track-pantograph-catenary interaction model is developed to study these effects. In this model, the long catenary is reduced to a small moving area around the moving pantograph, and this area is considered and modelled as the reduced catenary model based on the arbitrary Lagrangian–Eulerian method. The reduced catenary model and moving pantograph are formulated as the reduced pantograph-catenary interaction model, and it is further unidirectionally connected with the existing reduced train-track interaction model to formulate the reduced train-track-pantograph-catenary interaction model. The present reduced pantograph-catenary model is first validated by the EN50318-2018 standard, and the influences of track irregularity, track buckling, fastening failure, and track settlement on pantograph-catenary interaction dynamics are studied. The investigation results first show that the track settlement with a long wavelength has a significant influence on the pantograph-catenary interaction dynamics and can even cause contact loss with a good train-track interaction behaviour. Thus, it should be further mentioned in the maintenance of the track structure.

Enhanced Detection of Subway Insulator Defects Based on Improved YOLOv5

Insulators, pivotal to the integrity of railway catenaries, demand impeccable functioning to prevent system failures. Their consistent assessment is vital for railway safety. Current insulator evaluations in subways predominantly involve human intervention, a method fraught with inefficiencies, inaccuracies, and oversights, exacerbated by the complex backdrop of subway tunnels and minuscule defect dimensions. This study introduces an enhanced algorithm, anchored in the YOLOv5 framework, to refine insulator defect identification. Challenges in defect detection include limited, imbalanced data samples and adaptability. Addressing this, an accurate catenary model mirrors the subway line’s architecture, facilitating the creation of synthetic instances of both intact and impaired insulators. An atomization technique augments the dataset volume, fortifying the algorithm’s resilience in reduced visibility conditions, such as fog. Tackling sample equilibrium, the study introduces an equilibrium loss function, assigning disparate weights to various sample categories during training, thereby sharpening the algorithm’s focus on positive instances, particularly those that are challenging to discern, and rectifying the disproportion in sample categories. Incorporating lightweight structures like GhostNet and the Efficient Channel Attention Network (ECA-Net) channel attention scheme not only diminishes the network’s computational demands, thereby elevating the detection capabilities, but also minimizes superfluous data processing, enhancing the accuracy in identifying smaller targets. Empirical analyses indicate substantial model optimization: a size reduction to 60 pp of its original (from 15 MB to 9 MB), a near 1.4 pp increase in mean average precision (mAP) to 96.57%, and a tripling of the detection speed (from 30 to 90 FPS). Real-world image assessments further reveal a mAP improvement of approximately 2.5 pp (reaching 98.43%), confirming the model’s suitability for real-time applications.

Tension Characteristics of Cable-Driven System Based on ALE Formulation

This paper mainly studies the tension characteristics of cables in Cable-Driven Parallel Mechanisms (CDPMs). The exact catenary model and cable model based on the Arbitrary Lagrangian Eulerian description (ALE) are used. First, the ALE cable model and the exact catenary model are used to analyze and compare the tension at each point of a single cable, which proves the calculation accuracy of the ALE cable model. Then, the cable model described by ALE is used to model the Cable-Driven Parallel Mechanisms, the end effector is enabled to do a specific helical motion, the tension characteristics of the cable are extracted and analyzed, and the mesh independence of the ALE elements and calculation convergence are verified.

A Novel Kinematics Model of Flip-Flow Screen Panel: Inclined Catenary Model

A flip-flow screen can effectively screen viscous particles, and its kinematic characteristics determine its screening performance. Since previous kinematic models have errors, a novel kinematics model of the flip-flow screen panel, namely the inclined catenary model, is developed. It is verified by comparing theoretical motion trajectory with experimental motion trajectory. Then, the kinematic characteristics, i.e., displacement, velocity and acceleration, obtained using four kinematic models, are compared. Finally, the effects of rotation speed n, eccentricity e, incline angle α and tensional amount Δl on displacement, velocity and acceleration of the midpoint are investigated. The results show that displacement, velocity and acceleration of each point in the screen panel can be calculated by using the inclined catenary model, and the inclined catenary model possesses higher prediction accuracy than the three previous kinematic models. Moreover, with the increase in n, the absolute value of velocity and acceleration increases, and the maximum absolute value of displacement remains unchanged. With the increase in e, the absolute value of displacement, velocity and acceleration increases. With the increase in α, the absolute value of transverse components of displacement, velocity and acceleration increases slowly and the absolute value of longitudinal components of displacement, velocity and acceleration decreases slightly. With the increase in Δl, the absolute value of displacement, velocity and acceleration increases. Therefore, the inclined catenary model can provide good guidance for selecting reasonable screening parameters.

Method for real-time reconstruction of a transmission line based on the LiDAR point cloud data of a partial line segment

In a light detection and ranging (LiDAR) anti-external force damage monitoring system, the LiDAR installed on an electric power tower can only collect some point cloud data from the transmission line within a few tens of meters. Consequently, its anti-break system fails to calculate the distance between the transmission line and ground objects. In this study, a real-time reconstruction method for a transmission line based on a catenary mathematical model is proposed. Based on the three-dimensional (3D) point cloud data of the local power line, a 3D point cloud is mapped to a two-dimensional (2D) plane. In the 3D plane coordinate system, the catenary equation is fitted and calculated using the catenary model, and the 2D point cloud dataset of the transmission line is calculated according to this equation. Subsequently, the transmission line of the 2D plane is restored to the 3D space, and finally, the entire transmission line is restored to the 3D point cloud data. The field test results reveal that the proposed method can be used to effectively calculate and reconstruct the 3D transmission line model in real time and solve the spatial distance calculation problem of the anti-external force damage monitoring system.

A framework for analyzing the accuracy, complexity, and long-term performance of cable-driven parallel robot models

Cable-driven parallel robots require accurate and reliable control models to satisfy the demands of their diverse applications. The models’ usefulness in practice depends on their structure, the considered effects, and the ability to determine their best parameter values. This work establishes a unified model framework, containing eleven of the most commonly used cable robot models. The framework can optimize, evaluate, and compare models based on measurement datasets that capture the behavior of the robot. The models are evaluated based on their accuracy in the platform pose and the cable forces. Their computational costs, optimization process indicators, and the influence of the dataset size are also studied. This analysis is exercised for a large-scale cable robot based on two measurement datasets that are recorded 1.5 months apart and cover different regions of the robot’s state–space. The results show that a catenary model with force offsets offers the best overall accuracy for the given cable robot. While the models’ relative behavior is consistent in both datasets, their accuracy is significantly reduced on the second dataset due to long-term effects acting on the robot.

Modelling mooring line snap loads using a high-order finite-volume approach

Wave energy converters’ mooring systems are subjected to highly dynamic motions responsible for complex loading mechanisms. Snap loads differ from other dynamic load types by the propagation of tension discontinuities along the cable, which can lead to premature mooring failures if not correctly predicted. Accurate prediction of these loads requires numerical models that can resolve non-smooth solutions. This work developed a (conservative) finite-volume formulation for solving the hyperbolic cable dynamics equation by applying the high-order non-oscillatory central-upwind scheme to calculate the numerical fluxes at the cell interfaces. The central-upwind scheme is a variant of the Godunov-type central finite-volume method and does not need a Riemann solver. This new numerical formulation was tested against analytical and experimental data for a single catenary model under snap-loading conditions. It is shown the method’s ability to simulate snap loads correctly, predicting motions and tensions. Furthermore, it exhibits second-order accuracy in space for smooth solutions. This approach is highly robust (with a numerical dissipation independent of the time step) and simpler to implement compared to other codes in the literature.

Hardware-in-the-loop simulations of a railway pantograph with a finite element periodic catenary model

The pantograph-catenary dynamic interaction problem is addressed by Hardware-in-the-Loop (HIL) tests to simulate the dynamic interaction between a numerical model (the catenary) and a physical device (the pantograph). The real-time simulation requires a computationally efficient numerical model and an on time and accurate transference of the response to the pantograph, for which a Periodic Finite Element Model (PFEM) of the catenary is considered. Firstly because of the acceptable computation time required to solve it, makes it suitable for real-time simulations and, secondly, its ability to allow for the delay caused by the transfer of the numerical model response. The catenary PFEM we used considers the non-linear behaviour of the dropper slackening, leading to highly accurate HIL test results, which were validated up to a frequency of 25 Hz.

Identifiability of linear compartmental tree models and a general formula for input-output equations

A foundational question in the theory of linear compartmental models is how to assess whether a model is structurally identifiable – that is, whether parameter values can be inferred from noiseless data – directly from the combinatorics of the model. Our main result completely answers this question for models (with one input and one output) in which the underlying graph is a bidirectional tree; moreover, identifiability of such models can be verified visually. Models of this structure include two families of models often appearing in biological applications: catenary and mammillary models. Our analysis of such models is enabled by two supporting results, which are significant in their own right. One result gives the first general formula for the coefficients of input-output equations (certain equations that can be used to determine identifiability) that allows for input and output to be in distinct compartments. In another supporting result, we prove that identifiability is preserved when a model is enlarged and altered in specific ways involving adding a new compartment with a bidirected edge to an existing compartment.

Experiments and a tension-bending catenary model for the progressive collapse resistance of concrete beam-column structures

Catenary action (CA) is an extremely vital resistance mechanism for the progressive collapse of concrete beam-column structures. However, the systematically theoretical researches on CA are rather scarce due to the complexity of structural forces in the large deformation stage. To address this dilemma, an evaluation method for distinguishing the types of structural anti-progressive collapse performance was put forward and applied in 75 existing beam-column structures, by which the conditions of potential CA were revealed. Based on our experiments and collected databases, an innovative Tension-Bending Catenary (TBC) model was proposed, which considers the eccentric tension of the beam, the deformation compatibility conditions of the overall structure, and the plastic rotation capacity of the beam. The CA peak capacity, displacement, and horizontal resultant force can be predicted by the proposed model, whose accuracy and feasibility were verified via 16 specimens. Moreover, the main influencing factors on the CA capacity were revealed via a series of parametric analyses, including the beam depth-span ratio (h/ln), the properties of beam top longitudinal reinforcements (area As, ultimate strength fu, and ultimate tensile strain εsu), and the relative axial stiffness (ra). Parametric analysis results indicate that the CA capacities positively correlate with the beam top longitudinal reinforcements (As, fu, and εsu). Nevertheless, the CAcapacity markedly declines for a relative axial stiffness of end supports of less than 1.0. The CA capacity first displays an ascendant trend and then decreases with the beam depth-span ratio. Meanwhile, the bearing capacity of CA is superior to that of compressive arch action (CAA) until a critical depth-span ratio. Furthermore, the ultimate strain of the longitudinal reinforcements εsu is recommended to be as large as possible and not less than about 0.1 because a minor εsu causes the CA capacity to drop significantly. Especially, εsu should exceed 0.05 to make CA greater than CAA. Finally, practical anti-progressive collapse strategies were summarized and proposed.

Research on reliability analysis of catenary model based on the fusion particle swarm least square support vector machine algorithm

The reliability of contact network is always an important part of the reliability analysis of traction power supply system. In this paper, combined with the failure rate data of the main parts of the contact network, the small sample data is expanded by bootstrap non-parametric regeneration sampling method using the fused particle swarm least squares support vector machine (PSOLSSVM) algorithm to provide training set data for particle swarm optimisation. Mann, Schuer and Fertig fit test and the two-parameter Weibull distribution of the model based on MATLAB were used. The key components such as load cable and insulator were selected, and the data was combined to establish a model to characterise the overall reliability of the contact network. Based on the established model, the relevant parameters of each main part are estimated separately, which accords with the actual situation.

Predictive model for sag and load on overhead transmission lines based on local deformation of transmission lines

• Based on the elastic catenary model, a novel predictive model for sag and load on overhead transmission lines is proposed in the paper. • The predictive model comprehensively considers the cross-section shrinkage and the elastic deformation after the transmission line is stretched and the temperature effect. • The model takes local deformations on the transmission line as the input parameters to obtain the sag and load on the overhead transmission line. • This paper proposes a novel monitoring method for overhead transmission lines by combining the novel model with fiber Bragg grating (FBG) sensors. Ice accretion on transmission lines (TLs) is one of the major causes of power grid outages in northern regions. Therefore, how to stably and accurately monitor the icing load and sag on overhead TLs is a top priority. Considering the shortcomings of the existing monitoring methods, a mathematical model based on the elastic catenary model is proposed in this paper, which comprehensively considers the cross-section shrinkage and the elastic deformation after the transmission line (TL) is stretched and the temperature effect. The model takes local deformations on the TL line as the input parameters to obtain the sag and load on the overhead TL. In addition, this paper simulates an engineering case by ABAQUS finite element analysis software, and compares the simulation results with the prediction results of this model and the existing model, which proves that the model has higher accuracy. Finally, this paper proposes a novel monitoring method for TLs by combining the developed model with fiber Bragg grating (FBG) sensors. Not only does this monitoring method have higher accuracy, but it can also be used on older TLs that have been completed.

Development of a High-Precision and Flexible Model for Accurate Simulation of Trolleybus Grids

The reduction of climate-changing emissions is vital, particularly in urban areas. To achieve this goal, the decarbonization of the transport sector is decisive. Trolleybus systems represent a solution towards sustainability as long as the integration of the electrical infrastructure with renewable sources as well as the introduction of full-electric fleets are actualized. To ensure a smart electrical transition the circuit modelling phase is crucial and must enable the most accurate simulations possible. The modularity of the catenary model and a versatile graphical user interface, which allows extensive topological change flexibility, distinguish the novel trolleybus network simulator presented in this study. The model completes the gaps left by earlier block-based simulation tools with its high precision and low processing effort. The suggested model’s enhancement in precision is evidenced by a graphical study of the voltage distribution examined in a section of Bologna’s trolleybus grid. Finally, albeit non-iterative and thus computationally even cheaper, the proposed model is applicable for the simulation of more complex realistic scenarios involving the non-linearity of the equivalent power grid. The article will show satisfactory results on the convergence of the simulator.

Finite element periodic catenary model to perform HIL pantograph tests considering non-linear dropper behaviour

In this paper, we propose a general approach to compute the dynamic response of periodic infinite structures subjected to a moving load. The method only considers one repetitive block of the structure which is modelled by the Finite Element Method. The problem is first shifted to the frequency domain where the periodicity condition is easily applied and then the temporal response is obtained. An infinite periodic catenary system has been chosen to illustrate the proposed formulation. The linear formulation is extended to include the non-linear behaviour of droppers. The efficiency and accuracy of the catenary model obtained makes it very suitable for use in Hardware in the Loop (HIL) pantograph tests. We propose to combine this catenary model with an iterative strategy to achieve the steady-state response of the coupled system and its performance is analysed in a virtual HIL simulation.

A disturbance observer based lumped-mass catenary model for active pantograph design and validation

ABSTRACT To understand and thus improve the dynamic behaviour of the railway pantograph-catenary system, researchers have developed a number of approaches to the catenary modelling. The Finite Element (FE) approach provides a high degree of accuracy, but at the cost of significant computing time. In contrast, researchers in active pantographs generally use relatively simple (and fast running) lumped-mass catenary models. This paper aims to improve on the models available for the design and validation of active pantographs. Firstly, the veracity of the classic lumped-mass model is assessed and a new catenary model is proposed, by augmenting the lumped-mass model with a dynamics-matched mass and a disturbance which can be found using a disturbance observer (DO). Open-loop, passive pantograph, results show that the proposed model can represent the dynamic behaviour exhibited in the FE catenary very well. Secondly, two PID controllers are designed for an active pantograph, and tested with the FE, classic and DO-based lumped-mass catenary models. It is shown that if a controller is validated on the classic lumped-mass model, the credibility will be low. In contrast, the new DO-based model is found to deliver more robust results, providing efficient simulations for active pantograph design and validation.

Hybrid Modeling of Deformable Linear Objects for Their Cooperative Transportation by Teams of Quadrotors

This paper deals with the control of a team of unmanned air vehicles (UAVs), specifically quadrotors, for which their mission is the transportation of a deformable linear object (DLO), i.e., a cable, hose or similar object in quasi-stationary state, while cruising towards destination. Such missions have strong industrial applications in the transportation of hoses or power cables to specific locations, such as the emergency power or water supply in hazard situations such as fires or earthquake damaged structures. This control must be robust to withstand strong and sudden wind disturbances and remain stable after aggressive maneuvers, i.e., sharp changes of direction or acceleration. To cope with these, we have previously developed the online adaptation of the proportional derivative (PD) controllers of the quadrotors thrusters, implemented by a fuzzy logic rule system that experienced adaptation by a stochastic gradient rule. However, sagging conditions appearing when the transporting drones are too close or too far away induce singularities in the DLO catenary models, breaking apart the control system. The paper’s main contribution is the formulation of the hybrid selective model of the DLO sections as either catenaries or parabolas, which allows us to overcome these sagging conditions. We provide the specific decision rule to shift between DLO models. Simulation results demonstrate the performance of the proposed approach under stringent conditions.

Hybrid modeling approach for the tether of an airborne wind energy system

The design of controllers for the automatic operation of airborne vehicles is challenging. Especially in the emerging research field of airborne wind energy, where tethered aircraft convert energy from wind at higher altitudes into electricity, the development of robust and safe control systems will be a critical success factor. Typically, these systems consist of one or more controlled aircraft, a ground station, and one or more tethers that transfer aerodynamic forces acting on the wings to the ground. To understand these systems and control them for real-time simulations, suitable tether models must extend the classical flight mechanical models. In this paper, a hybrid approach is presented, combining an elastic spring model with a catenary model of the tether. In particular, for real-time simulations in a model-in-the-loop environment where a static modeling approach is sufficient, this hybrid model is intended to provide an alternative to the commonly used spring models. Furthermore, such a static tether model neglects high-frequency dynamic oscillations that can occur with dynamic lumped mass models. As consequence, the focus is on primary influences such as stiffness and sag due to external loads. The developed hybrid model is analyzed and compared with the spring model for a simulated tethered flight. It can be concluded that this hybrid static model, although subject to strong assumptions such as symmetrical tether sag, represents the tether more accurately. For example, it allows a more precise design of the flight controller before conducting any flight tests. This model can be used as a reasonable alternative to the simple spring model whenever a static modeling approach of the tether is adequate.