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Abstract

Currently, the modelling of the railway catenaries for straight tracks is well understood and demonstrated via an accepted number of applications and an international benchmark. However, in the presence of a general track geometry, including curved tracks, challenges still remain. The fundamental difficulty is how to address the catenary layout in the presence of curves for a long railway track in which a large number of catenary sections are defined. To support the catenary design, e.g. in new electrification projects, it is fundamental to identify the track geometry, which is not always accessible to the overhead line engineers. This work purposes a novel approach for track geometry reconstruction using aerial views. The reconstruction methodology is based on the general geometry rules for track design. The track geometry obtained is then used as the reference to define the layout of the catenary. The Network Rail Series 1 overhead equipment is used here and the 3D finite element model of the catenary is built on the curved track geometry. The results of this work demonstrate, not only the advances proposed to study more realistically the pantograph-catenary interaction, but also the need to consider general curved tracks in acceptance and interoperability studies.

Keywords

Pantograph-catenary interaction realistic catenary models curved tracks current collection performance

Introduction

New rail electrification projects raise several challenges to Overhead Contact Line (OCL) engineers as the catenary layout has to accommodate the track singularities, such as level crossings, bridges and tunnels, but also has to deal with the geometry of the track. In general, the nominal track geometry consists of straight and circular segments interconnected by transition curves that ensure the continuity of the first and second derivatives of the track in the transition points. This assemblage guarantees curvature continuity, making the ride smoother for the passengers.^1,2 In a curve, the outer rail is raised to tilt the vehicles in order to minimize the lateral accelerations perceived by the passangers and balance the loads on inner and ourter wheels of the train. The difference of heights between the two rails in a horizontal banked curve is called cant. The tracks can also go up and down, e.g., in a mountain^3–5 and therefore, have also a vertical grade.^6,7 Therefore, the geometry of the track is defined by its curvature, cant and grade.⁸

In reality, the geometry of the track is not perfect. Several irregularities^9,10 arise on the track layout resulting from: (a) construction imperfections; (b) usage in service; (c) temperature rise, originating the so-called sun kink, and; (d) changes on the track substructure^11–13 resulting from, e.g., environmental effects or soil motion. The track irregularities influence the wheel-rail interaction^14–16 and are especially important when studying problems associated to the passengers’ ride comfort,^17,18 fatigue of the vehicle structural elements and the deterioration of the railway infrastructure.¹⁹ Previous works²⁰ have demonstrated that the track irregularities have a negligible effect on pantograph-catenary interaction and, therefore, the track imperfections will not be considered in the present work.

The development and application of computational methods capable of simulating the pantograph-catenary interaction is a very active field of research. The pantograph-catenary benchmark^21,22 shows how existing, state-of-the-art, numerical tools deal with pantograph-catenary dynamics in straight tracks. However, real railway tracks are not strictly straight, displaying curves with varying radius and, as such, these tools may not be capable of fully represent catenaries that sit on realistic track geometries.

The European standard EN50318:2018²³ specifies functional requirements for the validation of pantograph-OCL tools and simulation methods to ensure compliance with its results. Until recently, this standard only required dynamic analysis to be run on straight catenaries. However, the 2018 release of EN50318 improves on the previous version of the standard, by including an AC stitched catenary installed on a curved track of constant radius. Computational tools wishing to be validated against AC stitched overhead equipment must successfully model and analyze catenaries in curved tracks.

The aim of this work is to propose an advanced methodology to reconstruct realistic railway track geometries using aerial images and design rules available in the literature. This approach is able to identify the curvature and cant characteristics of the track, as well as the necessary OCL parameters needed to correctly build catenary models in curved track, such as the stagger. Once the realistic track geometry is defined, then the 3D catenary model is built using the track layout as reference and the other OCL parameters identified on the aerial images. To demonstrate the methodology proposed here, an application example is presented consisting of a complete catenary model, with a number of sections, that is built on the top of a realistic track.

Overview of the computational tool PantoCat

One of the tools presented in the pantograph-OCL benchmark^21,22 is PantoCat,²⁴ which is the pantograph-catenary dynamic interaction analysis program used here to build the 3D catenary models that sit on tracks with realistic geometries. PantoCat is able to analyse, in realistic operation conditions, models of complete overhead energy collecting systems that include all mechanical details of the pantograph components and the complete topology and structural elements of the catenary.^24–27 The development of a pantograph-catenary dynamic analysis tool is a multidisciplinary exercise and PantoCat uses the Finite Element (FE) methodology²⁸ to study the catenary²⁹ and the multibody formulation³⁰ to represent the pantograph.^31–34 These methodologies are integrated via an efficient co-simulation procedure,^35–37 where a contact model based on a penalty formulation is used to represent the pantograph-catenary interaction.^24,38 PantoCat is extensively used in research and consultancy projects for the rail industry, being certified by a notified body to the EN50318:2018²³ for all catenary types and validated against field data in several countries.

PantoCat is a fully 3D tool, enabling to model rigid and flexible catenaries with multiple sections, their overlaps³⁹ and gradients, the operation of multiple pantographs^40–42 and consider complex loads on the components, including aerodynamic effects.^20,43 A bespoke geometry pre-processor is able the spatially define the position of all FE nodes and elements of the catenary model according to track data, such as curvature, cant and grade, and also includes the OCL geometric data. This feature enables to perform the dynamic studies in tangent or curved tracks⁴⁴ with or without irregularities and perturbations.⁴⁵

Catenary modelling on straight tracks

In their simplest form, catenaries are composed by two cables, the messenger and contact wires, tensioned at the supports in the beginning and end of each section, also known as wire runs. These wires are periodically mounted on poles, and supported at each one by cantilevers. The distance between two poles is called a span. The contact wire is supported from above, via a steady arm, to ensure that there is enough clearance for the pantograph to pass safely, but also to provide the necessary stagger. In between poles, the contact wire is supported by the messenger wire via dropper cables, which allow to control the sagging and elasticity of the contact wire.⁴⁶ Due to technical limitations, such as weight, costs and ease of maintenance, sections have a finite length, usually no longer than 1500 m. To ensure contact continuity, some of the terminal spans of a section and initial spans of the following section are placed alongside each other, in what is commonly referred to as an overlapping zone.

The messenger and contact wire deflect between poles, as a consequence of their weight. They are tensioned at the beginning and end of a section to ensure the correct geometry of the catenary and proper operating conditions. One particularity of pantograph-catenary interaction is the elastic wave, which is the propagation of an upwards deflection of the contact wire position, caused by the pantograph running on it. When train speeds approach the wave propagation speed of the contact wire, also called critical velocity, the amplitude of catenary oscillations increases and contact quality is greatly reduced. Applying high axial forces (tension) on the wires, allows for higher wave propagation speeds. To ensure proper contact quality, trains speeds should not exceed 70% of the critical velocity.⁴⁷

To avoid excessive friction, heat and grooving of the pantograph contact strip, the contact wire is laterally displaced so that it sweeps a larger area of its contact strip and distributes wear. This lateral offset is commonly known as stagger. Stagger is imposed by a steady arm, at each pole, and its design takes into consideration the contact strip useable length, track geometry, operational requirements and weather-specific tolerances, imposed not only by infrastructure managers, but also by the applicable standards. Generally, stagger design is done in such a way that it allows for the longest span length possible, thus reducing the number of poles and, consequently, the catenary construction costs.

The catenary is represented here via a linear FE model, which is well suited for the small rotations and deformations characteristic of the dynamic behaviour of these overhead systems. In order to build the OCL model accurately, the following parameters must be known:

• Number of spans and their length;

• Dropper length and spacing for all spans;

• Contact and messenger wires height and stagger at each support;

• Material properties for all wires and steady arms, namely cross section, linear mass, Young’s modulus, tension (if applicable);

• Mass of all wire clamps;

• Stiffness and damping characteristics of all catenary elements.²²

Catenary modelling on curved tracks

Modelling catenaries on curved tracks requires: (i) all the data detailed before for straight tracks; (ii) the information about the track geometry, and; (iii) the wire stagger arrangement in the curves. The track geometry, in nominal conditions, is fully characterized by three parameters, which are defined as function of the track length, namely: curvature, cant and grade. Based on this information, it is possible to build the spatial layout of the track and, using the correct design rules, to realistically define the 3D model of the catenary. However, the track geometric parameters are not always accessible to the overhead line engineers. In the following, a methodology to obtain, via geometric reconstruction, the track layout is presented. Then, the realistic catenary model can be built based on that track geometry.

Track geometry reconstruction

The track geometry is property of the infrastructure managers who reserve the right of its release, and is usually unavailable to the public. Therefore, a geometric reconstruction process based on available aerial images is envisaged here to define the track layout. The limitation is that the track grade profile cannot be obtained from the aerial images and, therefore, the catenary model is built on a planar track. Nevertheless, as the contact wire heights are measured as nominal values from the top of the rails, the effect of this limitation on the accuracy of the pantograph-catenary dynamics should be negligible.

The reconstruction process of the track geometry developed here follows some basic steps:

• Identify on aerial images, obtained via Google Maps or equivalent, the area of interest;

• Identify, in the image, the $(x_{s}, y_{s})$ coordinates of representative points along the track centreline;

• Scale the track discretized geometry from screen coordinates $(x_{s}, y_{s})$ to world coordinates $(x, y)$ ;

• Fit the discretized geometry with straights, constant radius curves and transition curves to define the track curvature versus track length information;

• Based on the train service speed, compute the cant of the track along its length.

Aerial identification of railway tracks

A section of track between Bristol Parkway and Old Sodbury in the Great Western Mainline (GWML), UK, is used to demonstrate the methodology developed in this work. This section includes several curves, bridges and overpasses, and it is known by the infrastructure manager as being one of the stretches that has more demanding design requirements. The track section has approximately 6 km long and the aerial images are obtained using Google Maps.

Based on the image data, which has a dimension of approximately 20,000 x 4600 pixels, points along the track centreline are selected, with a separation of approximately 100 pixels, being their coordinates recorded. The aerial images with the identified points superimposed are depicted in Figure 1. The position of the points along the track centreline, depicted in Figure 1, is expressed in screen (pixel) coordinates. The coordinates must be scaled to global $(x_{m}, y_{m})$ coordinates, in metres. It is found, on the aerial image information, that the scaling factor is 0.3 m/pixel. Upon its application on the points representing the track, leads to the track geometry.

Figure 1.

Aerial images for selected track with superimposed selected points.

Definition of track geometry along the track length

The track geometry is, in general, defined by the curvature, cant and grade as function of the track length, generally referred to as kilometric point. It represents the linear distance between the track starting point and a given point along the rails, using the coordinates obtained previously. The track length up to any point is the sum of the travel length until the previous point, plus the distance between the previous point and the current one. Therefore, the kilometric distance of point $i$ can be computed as

\begin{array}{c} P_{k m, i} = p_{k m, i - 1} + \sqrt{{(x_{m, i} - x_{m, i - 1})}^{2} + {(y_{m, i} - y_{m, i - 1})}^{2}} \end{array}

(1)

where

(x_{m}, y_{m})

are the coordinates of the current point, identified with subscript

i

, and of the preceding point, with subscript

i - 1

, and

p_{k m, i - 1}

is the track length up to the preceding point.

Track geometries can be divided in three zones: straights, curves with constant radius, also known as circular curves, and transition curves. Transition curves are zones between a straight and a circular curve, where the track curvature and cant increase linearly to allow for a smooth transition between the curves and straights. These transition zones are curves of increasing or decreasing radius, also known as clothoides, and can be approximated as cubic parabolas.

Reconstructing the track geometry requires information about which parts of the track are straight, circular curves and transitions. To find out the straight zones, a linear regression is applied to the coordinates of the points selected. The coefficient of determination of this regression, $r^{2}$ , is equal to 1 if the points selected are perfectly fitted with a straight line.

Through an iterative process, the coefficient of regression is computed for an increasing number of selected points. If the points lead to a regression coefficient $r^{2} = 1$ within a specified tolerance, in this case defined to 0.001, then it is assumed that they sit in a straight line, which enables to define the length of the straight track zones. In this way, the initial and end points of each straight are identified, effectively restraining the curved areas to the segments between them.

Between two straights, there is a circular curve and two transition curves of equal length. The total length of the curved zone is known, by imposition of the limits of the straight zones. However, the radius of the circular curves and the length of the transition curves is not known. The complete curved zone is approximated to a single circular curve and, using methods of circle fitting,⁴⁸ an estimation for the radius of the entire curved zone is found. The estimated radius is then used to compute the length of transition curves required to meet the design rules. The transition length is defined in the track design handbook⁴⁹ as

\begin{array}{c} L = 0.09381 \frac{V^{3}}{R} \end{array}

(2)

where

V

is the train speed in km/h and

R

is the curve radius in metres. The track curvature is defined as

\begin{array}{c} c = \frac{1}{R} \end{array}

(3)

Verification of the track geometry reconstruction

To ensure that the curvature profile is computed correctly and that it accurately represents the coordinates of the points taken from the aerial images, the track, in terms of XY coordinates, is reconstructed from the estimated curvature profile. A method for this reconstruction is described in the literature.⁵⁰ The reconstructed $x$ , $y$ and $θ$ coordinates of point $i$ are given by

\begin{array}{c} θ_{i} = θ_{i - 1} + \frac{Δ l (c_{i} + c_{i - 1})}{2} \end{array}

(4)

\begin{array}{c} x_{i} = x_{i - 1} + Δ l \cos [\frac{1}{2} (θ_{i} + θ_{i - 1})] \end{array}

(5)

\begin{array}{c} y_{i} = y_{i - 1} + Δ l \sin [\frac{1}{2} (θ_{i} + θ_{i - 1})] \end{array}

(6)

where

c

is taken from the curvature profile and

Δ l

is a track step, in metres, used to reconstruct the point coordinates. The smaller the step, the finer the reconstructed geometry. The new XY coordinates are computed and plotted against the initially measured XY coordinates to check for inconsistencies in the curvature profile. Through an iterative process, the track geometry estimation can be improved by varying the radius of the curved zones and, consequently, updating the length of the transition curves.

Reconstruction of track cant

The track cant in curves is defined by four main parameters:

• Theoretical cant $E_{T}$ , the cant at which the centrifugal forces are fully offset and the train operation speed is ideal;

• Applied cant $E_{A}$ , the actual cant existing on the track;

• Cant deficiency $D = E_{T} - E_{A}$ ;

• Rate of change of cant $d E$ .

The cant geometry design rules state that, under desirable conditions of $d E$ < 35 mm/s, the theoretical cant in millimetres is given by⁴⁹

\begin{array}{c} E_{T} = 11.82 \frac{V^{2}}{R} \end{array}

(7)

where

V

is the train speed in km/h and

R

is the curve radius in metres. The regulations state that the track cant is limited to a maximum of 150 mm and the cant deficiency is limited to a maximum of 110 mm ⁵¹. In the application example considered here, if the values of theoretical cant are higher than the track rules specified limit, a value of 150 mm applied cant is considered and cant deficiency occurs.

Summary of the track geometry reconstruction

The objective of this work is to reconstruct a realistic track geometry and build a catenary model based on representative track geometry. In order to provide a complete model for the dynamic analyses, a section of straight track is added before and after the realistic geometry obtained. The full track geometry and the track length covered by each catenary section are depicted in Figure 2, where the greyed areas represent the OCL overlapping zones. The catenary sections are defined as follows:

• Section 1 (SC1), straight track;

• Sections 2 and 3, curvature changes;

• Sections 4 and 5, circular curve;

• Section 6, curvature changes;

• Section 7, straight track.

Figure 2.

Reconstructed track geometry and overall section arrangement.

All catenary sections have 23 55-m spans, resulting in a total section length of 1265 m, representing a total track length of 7975 m. The complete track geometry is described in Table 1.

Table 1.

Reconstructed track geometry and track cant for GWML.

Zone	Zone length (m)	Pkm (m)	R (m)	EA (mm)
—	—	0.00	∞	0.00
Straight track	1497.35	1497.35	∞	0.00
Transition curve	445.23	1942.58	2400.00	150.00
Circular curve	182.20	2124.78	2400.00	150.00
Transition curve	445.23	2570.01	∞	0.00
Straight track	150.68	2720.69	∞	0.00
Transition curve	227.35	2948.04	4700.00	127.32
Circular curve	2742.03	5690.07	4700.00	127.32
Transition curve	227.35	5917.42	∞	0.00
Straight track	2057.58	7975.00	∞	0.00

Stagger in curves

The stagger in the contact wire is designed, among other reasons, to ensure wear is distributed through the entire pantograph contact strip and avoid grooving. In a straight track the stagger usually takes the form of $b_{i + 1} = {- b}_{i}$ , as depicted in Figure 3 (a), i.e., alternating lateral deviations from the straight line. However, as the track curvature increases, the offset in the inner side of the curve is decreased, in order to keep the contact wire sweep inside the limits enforced by the operation safety conditions. In this case, depicted in Figure 3(b) and (c), the stagger has offsets for which $b_{i + 1} \neq {- b}_{i}$ . In the case of small radius curves it may happen that the offset in the inner side is increased to the point of $b_{i + 1} = b_{i}$ , as depicted in Figure 3(c). Eventually, there is a point where constant inner stagger is not enough to meet the design rules, and span length of the catenary must be shortened.^44,46

Figure 3.

Contact wire stagger: (a) straight track; (b) large radius curves; (c) low radius curves.

Stagger data for catenaries is usually expressed in the form of stagger tables, defined as a function of the curve radius.⁵² The stagger data for curved track for the GWML catenary is not known and, therefore, the design rules available and the best engineering judgment is used to build a stagger table for this catenary. When designing overhead contact lines, the rules for the stagger $b$ at the registration arms comply with:⁴⁶

• Span length as long as possible;

• Contact wire sweep $Δ b_{m}$ > 1.5 mm/m;

• Contact wire sweep $Δ b_{m}$ < 1.5 mm/m is allowed for up to 30% of the span length;

• Lateral (radial) force at the supports 80 N < $F_{l a t}$ < 2000 N;

• Difference of lateral forces at adjacent supports as low as possible.

The contact wire sweep of the pantograph, due to the lateral movement of the contact wire, can be defined as the derivative of contact wire position in relation to the track centreline. The contact wire position in mm, for span $j$ supported at poles $i$ and $i + 1$ , is defined as⁴⁶

\begin{array}{c} e_{C W} (x) = b_{i - 1} + x \frac{b_{i} - b_{i - 1}}{L_{j}} + x \frac{L_{j} - x}{2 R} \end{array}

(8)

where

b

is the stagger in metres,

L_{j}

is the span length in metres,

R

is the curve radius in metres, and

x

is the position along the track in metres. The sweep

Δ b_{m}

, in mm/m, is defined as

\begin{array}{c} Δ b_{m} = \frac{d e_{C W}}{d x} = \frac{b_{i} - b_{i - 1}}{L_{j}} + \frac{L_{j} - 2 x}{2 R} \end{array}

(9)

The lateral forces in the steady arm at poles $i$ and $i + 1$ , are given by⁴⁶

\begin{array}{c} F_{l a t}^{i} = H (\frac{L_{j}}{R} + 2 \frac{b_{i + 1} - b_{i}}{L_{j}}) \end{array}

(10)

\begin{array}{c} F_{l a t}^{i + 1} = H (\frac{L_{j}}{R} + 2 \frac{b_{i} - b_{i + 1}}{L_{j}}) \end{array}

(11)

where

H

is the contact wire tension expressed in N and assuming that the stagger is constant at every other pole, i.e.,

b_{i + 2} = b_{i}

. In those conditions,

F_{l a t}^{i + 2} = F_{l a t}^{i}

The sweep, $Δ b_{m}$ , and the lateral forces, $F_{l a t}$ , are computed for various radius/stagger pairs of three known catenaries from Belgium, Germany and France, here named A, B and C respectively.⁵² The stagger data, computed lateral forces and sweep rates are described in Table 2–4. The analysis of this formulation and respective results allows for a better understanding on how catenary stagger is designed. The following characteristics can be identified:

• The absolute value of lateral forces increases with curvature. For R ≥1000 m, the maximum force does not exceed three times the lateral force under straight track conditions;

• For $b_{2} = - b_{1}$ , increasing the stagger leads to higher maximum forces and lower minimum forces, effectively increasing the force difference between adjacent poles;

• Stagger $b_{1}$ in curves might be increased from its value in straight sections;

• As curvature increases, stagger $b_{2}$ increases from $- b_{1}$ up to $+ b_{1}$ ;

• Span length is reduced only when $b_{2} = b_{1}$ is not enough to meet the force and/or sweep requirements;

• Problematic span lengths (where $Δ b_{m}$ < 1.5 mm/m, nicknamed %p in the tables) are unavoidable for curves with R <5000 m. The problematic length has its maximum value for this radius and decreases as the curvature is increased.

Table 2.

Stagger data, lateral forces and rate of pantograph sweep for catenary A.

						MAX RADIUS					MIN RADIUS
R (m)		Span (m)	b1 (mm)	b2 (mm)	Tension (N)	F_lat i (N)	F_lat i+1 (N)	Δb (mm/m)			F_lat i (N)	F_lat i+1 (N)	Δb (mm/m)
≤	>	Span (m)	b1 (mm)	b2 (mm)	Tension (N)	F_lat i (N)	F_lat i+1 (N)	min	max	%p	F_lat i (N)	F_lat i+1 (N)	min	max	%p
∞	15,000	63	200	−200	9938	126.2	−126.2	6.3	6.3	0.0	167.9	−84.5	4.2	8.4	0.0
15000	10,000	63	200	−150	9938	152.2	−68.7	3.5	7.7	0.0	173.0	−47.8	2.4	8.7	0.0
10000	7000	63	200	−100	9938	157.3	−32.0	1.6	7.9	0.0	184.1	−5.2	0.3	9.3	13.8
7000	5000	63	200	0	9938	152.5	26.3	0.0	7.7	31.4	188.3	62.1	0.0	9.5	23.8
5000	4000	63	200	100	9938	156.8	93.7	0.0	7.9	23.8	188.1	125.0	0.0	9.5	19.0
4000	3500	63	210	210	9938	156.5	156.5	0.0	7.9	19.0	178.9	178.9	0.0	9.0	16.7
3500	3000	63	240	240	9938	178.9	178.9	0.0	9.0	16.7	208.7	208.7	0.0	10.5	14.3
3000	2500	63	260	260	9938	208.7	208.7	0.0	10.5	14.3	250.4	250.4	0.0	12.6	11.9
2500	2200	63	290	290	9938	250.4	250.4	0.0	12.6	11.9	284.6	284.6	0.0	14.3	10.5
2200	2000	63	320	320	9938	284.6	284.6	0.0	14.3	10.5	313.0	313.0	0.0	15.8	9.5
2000	1800	56	340	340	9938	278.3	278.3	0.0	17.5	10.7	309.2	309.2	0.0	19.4	9.6
1800	1700	56	360	360	9938	309.2	309.2	0.0	19.4	9.6	327.4	327.4	0.0	20.6	9.1
1700	1600	56	380	380	9938	327.4	327.4	0.0	20.6	9.1	347.8	347.8	0.0	21.9	8.6

Table 3.

Stagger data, lateral forces and rate of pantograph sweep for catenary B.

						MAX RADIUS					MIN RADIUS
R (m)		Span (m)	b1 (mm)	b2 (mm)	Tension (N)	F_lat i (N)	F_lat i+1 (N)	Δb (mm/m)			F_lat i (N)	F_lat i+1 (N)	Δb (mm/m)
≤	>	Span (m)	b1 (mm)	b2 (mm)	Tension (N)	F_lat i (N)	F_lat i+1 (N)	min	max	%p	F_lat i (N)	F_lat i+1 (N)	min	max	%p
∞	15,000	80	400	−400	16,000	320.0	−320.0	10.0	10.0	0.0	405.3	−234.7	7.3	12.7	0.0
15,000	10,000	80	400	−340	16,000	381.3	−210.7	6.6	11.9	0.0	424.0	−168.0	5.3	13.3	0.0
10,000	9200	80	400	−300	16,000	408.0	−152.0	4.8	12.8	0.0	419.1	−140.9	4.4	13.1	0.0
9200	8600	80	400	−290	16,000	415.1	−136.9	4.3	13.0	0.0	424.8	−127.2	4.0	13.3	0.0
8600	8100	79	400	−310	16,000	434.6	−140.6	4.3	13.6	0.0	443.6	−131.5	4.0	13.9	0.0
8100	8000	78	400	−320	16,000	449.5	−141.3	4.2	14.0	0.0	451.4	−139.4	4.1	14.1	0.0
8000	7600	78	400	−310	16,000	447.3	−135.3	4.0	14.0	0.0	455.5	−127.1	3.7	14.2	0.0
7600	7400	77	400	−330	16,000	465.5	−141.3	4.0	14.5	0.0	469.9	−136.9	3.9	14.7	0.0
7400	7100	77	400	−320	16,000	465.7	−132.7	3.7	14.6	0.0	472.7	−125.7	3.5	14.8	0.0
7100	7000	76	400	−340	16,000	482.8	−140.3	3.8	15.1	0.0	485.3	−137.9	3.7	15.2	0.0
7000	6700	76	400	−330	16,000	481.1	−133.7	3.6	15.0	0.0	488.9	−125.9	3.3	15.3	0.0
6700	6500	75	400	−350	16,000	499.1	−140.9	3.7	15.6	0.0	504.6	−135.4	3.5	15.8	0.0
6500	6300	75	400	−340	16,000	500.3	−131.1	3.3	15.6	0.0	506.2	−125.3	3.1	15.8	0.0
6300	6100	74	400	−360	16,000	516.6	−140.7	3.4	16.1	0.0	522.7	−134.6	3.2	16.3	0.0
6100	5900	74	400	−350	16,000	518.4	−130.2	3.1	16.2	0.0	525.0	−123.6	2.8	16.4	0.0
5900	5800	73	400	−370	16,000	535.5	−139.6	3.2	16.7	0.0	538.9	−136.2	3.0	16.8	0.0
5800	5600	73	400	−360	16,000	534.5	−131.8	2.9	16.7	0.0	541.7	−124.6	2.6	16.9	0.0
5600	5500	72	400	−380	16,000	552.4	−141.0	3.0	17.3	0.0	556.1	−137.2	2.8	17.4	0.0
5500	5300	72	400	−370	16,000	551.7	−132.8	2.7	17.2	0.0	559.6	−124.9	2.4	17.5	0.0
5300	5100	71	400	−390	16,000	570.4	−141.7	2.7	17.8	0.0	578.8	−133.3	2.4	18.1	0.0
5100	5000	71	400	−380	16,000	574.3	−128.8	2.3	17.9	0.0	578.7	−124.3	2.1	18.1	0.0
5000	4900	76	400	120	16,000	361.1	125.3	0.0	11.3	19.7	366.1	130.3	0.0	11.4	19.3
4900	4800	79	400	80	16,000	387.6	128.3	0.0	12.1	18.6	393.0	133.7	0.0	12.3	18.2
4800	4700	80	400	70	16,000	398.7	134.7	0.0	12.5	18.0	404.3	140.3	0.0	12.6	17.6
4700	4600	80	400	80	16,000	400.3	144.3	0.0	12.5	17.6	406.3	150.3	0.0	12.7	17.2
4600	4500	80	400	90	16,000	402.3	154.3	0.0	12.6	17.2	408.4	160.4	0.0	12.8	16.9
4500	4400	80	400	100	16,000	404.4	164.4	0.0	12.6	16.9	410.9	170.9	0.0	12.8	16.5
4400	4300	80	400	110	16,000	406.9	174.9	0.0	12.7	16.5	413.7	181.7	0.0	12.9	16.1
4300	4200	80	400	120	16,000	409.7	185.7	0.0	12.8	16.1	416.8	192.8	0.0	13.0	15.7
4200	4100	80	400	130	16,000	412.8	196.8	0.0	12.9	15.7	420.2	204.2	0.0	13.1	15.4
4100	4000	80	400	140	16,000	416.2	208.2	0.0	13.0	15.4	424.0	216.0	0.0	13.3	15.0
4000	3900	80	400	150	16,000	420.0	220.0	0.0	13.1	15.0	428.2	228.2	0.0	13.4	14.6
3900	3800	80	400	170	16,000	420.2	236.2	0.0	13.1	14.6	428.8	244.8	0.0	13.4	14.2
3800	3700	80	400	180	16,000	424.8	248.8	0.0	13.3	14.2	433.9	257.9	0.0	13.6	13.9
3700	3600	80	400	190	16,000	429.9	261.9	0.0	13.4	13.9	439.6	271.6	0.0	13.7	13.5
3600	3500	80	400	210	16,000	431.6	279.6	0.0	13.5	13.5	441.7	289.7	0.0	13.8	13.1
3500	3400	80	400	220	16,000	437.7	293.7	0.0	13.7	13.1	448.5	304.5	0.0	14.0	12.7
3400	3300	80	400	240	16,000	440.5	312.5	0.0	13.8	12.7	451.9	323.9	0.0	14.1	12.4
3300	3200	80	400	250	16,000	447.9	327.9	0.0	14.0	12.4	460.0	340.0	0.0	14.4	12.0
3200	3100	80	400	270	16,000	452.0	348.0	0.0	14.1	12.0	464.9	360.9	0.0	14.5	11.6
3100	3000	80	400	290	16,000	456.9	368.9	0.0	14.3	11.6	470.7	382.7	0.0	14.7	11.2
3000	2900	80	400	310	16,000	462.7	390.7	0.0	14.5	11.2	477.4	405.4	0.0	14.9	10.9
2900	2800	80	400	300	16,000	481.4	401.4	0.0	15.0	10.9	497.1	417.1	0.0	15.5	10.5
2800	2700	80	400	330	16,000	485.1	429.1	0.0	15.2	10.5	502.1	446.1	0.0	15.7	10.1
2700	2600	80	400	340	16,000	498.1	450.1	0.0	15.6	10.1	516.3	468.3	0.0	16.1	9.7
2600	2500	80	400	350	16,000	512.3	472.3	0.0	16.0	9.7	532.0	492.0	0.0	16.6	9.4
2500	2400	80	400	370	16,000	524.0	500.0	0.0	16.4	9.4	545.3	521.3	0.0	17.0	9.0
2400	2300	80	400	380	16,000	541.3	525.3	0.0	16.9	9.0	564.5	548.5	0.0	17.6	8.6
2300	1800	80	400	400	16,000	556.5	556.5	0.0	17.4	8.6	711.1	711.1	0.0	22.2	6.8
1800	1700	79	400	400	16,000	702.2	702.2	0.0	22.5	6.8	743.5	743.5	0.0	23.8	6.5
1700	1600	78	400	400	16,000	734.1	734.1	0.0	24.1	6.5	780.0	780.0	0.0	25.6	6.2
1600	1500	77	400	400	16,000	770.0	770.0	0.0	25.9	6.2	821.3	821.3	0.0	27.7	5.8

Table 4.

Stagger data, lateral forces and rate of pantograph sweep for catenary C.

						MAX RADIUS					MIN RADIUS
R (m)		Span (mm)	b1 (mm)	b2 (mm)	Tension (N)	F_lat i (N)	F_lat i+1 (N)	Δb (mm/m)			F_lat i (N)	F_lat i+1 (N)	Δb (mm/m)
(m)	>	Span (mm)	b1 (mm)	b2 (mm)	Tension (N)	F_lat i (N)	F_lat i+1 (N)	min	max	%p	F_lat i (N)	F_lat i+1 (N)	min	max	%p
∞	7500	63	200	−200	12,000	152.4	−152.4	6.3	6.3	0.0	253.2	−51.6	2.1	10.5	0.0
7500	5000	63	240	−50	12,000	211.3	−9.7	0.4	8.8	13.1	261.7	40.7	0.0	10.9	23.8
5000	4000	63	240	0	12,000	242.6	59.8	0.0	10.1	23.8	280.4	97.6	0.0	11.7	19.0
4000	3000	63	240	90	12,000	246.1	131.9	0.0	10.3	19.0	309.1	194.9	0.0	12.9	14.3
3000	2500	63	240	160	12,000	282.5	221.5	0.0	11.8	14.3	332.9	271.9	0.0	13.9	11.9
2500	2250	63	240	200	12,000	317.6	287.2	0.0	13.2	11.9	351.2	320.8	0.0	14.6	10.7
2250	1500	63	240	240	12,000	336.0	336.0	0.0	14.0	10.7	504.0	504.0	0.0	21.0	7.1
1500	1200	59	240	240	12,000	472.0	472.0	0.0	22.3	7.6	590.0	590.0	0.0	27.9	6.1
1200	950	54	240	240	12,000	540.0	540.0	0.0	30.0	6.7	682.1	682.1	0.0	37.9	5.3

By using a systematic search procedure, a compromise between radius ranges, span lengths and staggers is achieved for the GWML catenary. A stagger table for this catenary is presented in Table 5.

Table 5.

Stagger data, lateral forces and rate of pantograph sweep for the GWML catenary.

						MAX RADIUS					MIN RADIUS
R (m)		Span R (m)	b1 (mm)	b2 (mm)	Tension (N)	F_lat i (N)	F_lat i+1 (N)	Δb (mm/m)			F_lat i (N)	F_lat i+1 (N)	Δb (mm/m)
≤	>	Span R (m)	b1 (mm)	b2 (mm)	Tension (N)	F_lat i (N)	F_lat i+1 (N)	min	max	%p	F_lat i (N)	F_lat i+1 (N)	min	max	%p
∞	15,000	63	230	−230	16,500	241.0	−241.0	7.3	7.3	0.0	310.3	−171.7	5.2	9.4	0.0
15,000	7500	63	240	−230	16,500	315.5	−176.9	5.4	9.6	0.0	384.8	−107.6	3.3	11.7	0.0
7500	6000	63	260	−230	16,500	395.3	−118.1	3.6	12.0	0.0	429.9	−83.4	2.5	13.0	0.0
6000	5000	63	320	−230	16,500	461.3	−114.8	3.5	14.0	0.0	496.0	−80.2	2.4	15.0	0.0
5000	4000	63	320	160	16,500	291.7	124.1	0.0	8.8	23.8	343.7	176.1	0.0	10.4	19.0
4000	3300	63	320	190	16,500	328.0	191.8	0.0	9.9	19.0	383.1	246.9	0.0	11.6	15.7
3300	2800	63	320	220	16,500	367.4	262.6	0.0	11.1	15.7	423.6	318.9	0.0	12.8	13.3
2800	2500	63	320	250	16,500	407.9	334.6	0.0	12.4	13.3	452.5	379.1	0.0	13.7	11.9
2500	2250	63	320	280	16,500	436.8	394.8	0.0	13.2	11.9	483.0	441.0	0.0	14.6	10.7
2250	2000	63	320	320	16,500	462.0	462.0	0.0	14.0	10.7	519.8	519.8	0.0	15.8	9.5
2000	1800	62	320	320	16,500	511.5	511.5	0.0	16.0	9.7	568.3	568.3	0.0	17.8	8.7
1800	1600	61	320	320	16,500	559.2	559.2	0.0	18.1	8.9	629.1	629.1	0.0	20.3	7.9
1600	1400	59	320	320	16,500	608.4	608.4	0.0	20.9	8.1	695.4	695.4	0.0	23.9	7.1
1400	1300	56	320	320	16,500	660.0	660.0	0.0	25.0	7.5	710.8	710.8	0.0	26.9	7.0
1300	1200	53	320	320	16,500	672.7	672.7	0.0	28.1	7.4	728.8	728.8	0.0	30.4	6.8
1200	1100	49	320	320	16,500	673.8	673.8	0.0	32.1	7.3	735.0	735.0	0.0	35.0	6.7
1100	1000	45	320	320	16,500	675.0	675.0	0.0	36.8	7.3	742.5	742.5	0.0	40.5	6.7

Finite element model of the catenary on curved track

The Series 1 overhead system data and the reconstructed GWML geometry, summarized in Table 1 and Table 5, are used to create the necessary input files to generate the catenary model in the computational tool PantoCat. The finite element mesh for the 3D catenary model is depicted in Figures 4 and 5 from different perspective views.

Figure 4.

Catenary FE mesh, isometric view.

Figure 5.

Catenary FE mesh, side view of overlap.

The catenary is represented by a linear FE model, which is well suited for the small rotations and deformations characteristic of the dynamic behaviour of these overhead systems. All catenary wires and structural components are modelled, in the FE formulation, as a 2 node, 6 degree of freedom, Euler-Bernoulli beam elements.²⁴ The clamps that hold the various structures of the catenary together are modelled as lumped masses, and the messenger wire cantilevers are modelled as equivalent 3D spring-damper elements. Both the messenger and contact wires are constrained by fixed points at the supports at the beginning and end of the wire runs.²⁶

Conclusions

In this work, an innovative methodology to reconstruct railway track geometries from aerial images is presented. The objective is to use this realistic geometry, which is not always accessible to the overhead line engineers, to build reliable 3D catenary models. These models will be fundamental to perform pantograph-catenary dynamic studies and access with accuracy the current collection performance of different configurations, enabling to optimize the catenary design and support new rail electrification projects or OCL renewals.

The methodology proposed here is demonstrated in a 8 km track section of the Great Western Mainline in the UK. The area of interest is identified in Google Maps, discretised by the identification of a set of key points along the track centreline, and described in spatial coordinates using the position of those points. The track is subdivided into straights, transition curves and circular curves using the methods presented above. The reconstructed geometry is verified against that identified in the aerial images to ensure proper geometric correlation. The cant and stagger information are reconstructed based on the design rules and good industry practices available in the literature and shared by the infrastructure manager.

The Network Rail Series 1 overhead equipment is considered here. All construction rules of this catenary, such as wire tensions, span arrangements and material properties of components, are known and used by the computational tool PantoCat to build the 3D finite element catenary model that sits on the realistic curved track. This work demonstrates the advances proposed to study more realistically the pantograph-catenary interaction and the need to consider general curved tracks in acceptance and interoperability studies.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the FCT, through IDMEC, under LAETA, project UIDB/50022/2020.

ORCID iDs

JM Rebelo

P Antunes

J Pombo

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Track and catenary reconstruction from aerial images for the realistic analysis of rail electrification systems

Abstract

Keywords

Introduction

Overview of the computational tool PantoCat

Catenary modelling on straight tracks

Catenary modelling on curved tracks

Track geometry reconstruction

Aerial identification of railway tracks

Definition of track geometry along the track length

Verification of the track geometry reconstruction

Reconstruction of track cant

Summary of the track geometry reconstruction

Stagger in curves

Finite element model of the catenary on curved track

Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iDs

References