Description:TECHNICAL FIELD

The present disclosure, in general, relates to the field of electric railway systems. Particularly, but not exclusively, the present disclosure relates to overhead equipment and pantograph systems of the electric railway system. Further, embodiments of the present disclosure discloses a method and a system to determine nonlinear mechanical interactions of pantographs and overhead cables in electric railway systems.

BACKGROUND OF THE DISCLOSURE

The information in this section merely provides background information related to the present disclosure and may not constitute prior art(s) for the present disclosure.

Electric railway systems are widely adopted in high-speed and urban transit networks. Figure 1 illustrates a simple catenary cable arrangement of electric railway systems. In these systems, electrical power is transmitted through overhead wires, which are suspended above the tracks and provide a continuous source of electricity. Further, Figure 2 illustrates a pantograph in accordance with state of the art. The pantograph, mounted on top of the train, maintains consistent contact with the overhead contact wire as the train moves, allowing for efficient power transfer to the train’s electric motors. This system is favoured for its ability to deliver high power efficiently, making it suitable for high-speed rail, freight trains, and metro systems. This method of power delivery eliminates the need for onboard fuel (e.g., coal or diesel), reduces emissions, and allows trains to operate with greater energy efficiency, making it a cornerstone of modern, sustainable railway systems.
The principle behind arranging overhead cables in electric railway systems is to ensure continuous, reliable power transmission to moving trains while accommodating various mechanical and environmental factors. A simple catenary cable arrangement (refer Figure 1) typically includes a messenger wire and a contact wire suspended in parallel, with vertical droppers connecting them. The messenger wire bears most of the mechanical load, while the contact wire directly interacts with the train’s pantograph. The cable arrangement allows for optimal tension distribution, ensuring the contact wire remains taut and at a consistent height over the track. The natural sag of the cables, due to their self-weight, follows a catenary curve. The cable system is commonly tensioned using counterweights or spring-loaded systems. This tensioned, multi-wire arrangement is designed to minimize oscillations and prevent loss of contact between the pantograph and the wire, even when trains move at high speeds or the track passes through curves or gradients. The overall goal is to provide a stable, uninterrupted electrical connection with minimal wear and energy loss.
The structures supporting overhead cables are designed to ensure precise alignment, tension, and stability of the wires. These support structures typically consist of masts and poles, which are strategically spaced along the railway line to provide robust anchorage for the overhead equipment. Extending horizontally from these masts are cantilever arms, which hold the contact wire in place. The cantilevers are equipped with adjustable registration arms, which maintain the correct stagger and vertical spacing between the wires and the track, ensuring the contact wire is precisely positioned for optimal pantograph interaction.
The pantograph is designed to ensure durability, reliability, and minimal maintenance while maintaining efficient contact with the overhead contact wire. The pantograph comprises a lower base frame attached to the train's roof and an articulated arm system, usually in a diamond or single-arm configuration. The arms are connected by hinged joints that allow the pantograph to extend upward and adjust passively to the changing height of the overhead wire. The top of the pantograph features a contact strip, typically made of a carbon or graphite composite chosen for its electrical conductivity, low friction, and wear resistance. This strip ensures smooth, continuous contact with the contact wire while minimizing both electrical resistance and mechanical wear. Springs or pneumatic systems are integrated into the pantograph's arm structure to provide the upward force required to press the contact strip against the wire. These springs are calibrated to balance the pressure applied to the wire, ensuring it is neither too weak (causing loss of contact) nor too strong (causing excessive wear). The entire structure is designed to be aerodynamic to reduce drag, particularly at high speeds, and can often be folded down when not in use.
To overcome the aforementioned challenge, the present disclosure provides a system and a method to determine nonlinear mechanical interactions of pantographs and overhead cables in railway systems. The present disclosure is directed to overcome one or more limitations stated above or any other limitations associated with the prior art.

SUMMARY OF THE DISCLOSURE
One or more shortcomings of the prior art are overcome by a method and a system as claimed and additional advantages are provided through the method and the system as claimed in the present disclosure. Additional features and advantages are realized through the techniques of the present disclosure. Other embodiments and aspects of the disclosure are described in detail herein and are considered a part of the claimed disclosure.

In one non-limiting embodiment of the present disclosure, a method for determining nonlinear static and dynamic mechanical interactions between a pantograph current collector and an overhead cable network of an electrified railway line is disclosed [simply referred to as ‘the method’ hereinafter]. The method includes generating, by a processor, a first model for a plurality of cables of the overhead cable network. The plurality of cables includes a messenger cable and a contact cable. The method further includes generating, by the processor, a second model of a plurality of dropper cables. Further, each of the plurality of dropper cables have a length adjusted to achieve a target sag value of the contact cable. The plurality of dropper cables are modelled by employing a nonlinear force-deflection response configured to represent slackening of the plurality of dropper cables, upon being subjected to compression. The method further incudes generating, by the processor, a third model for support structures supporting the overhead cable network. The support structures include one or more registration arms configured to support the contact cable at the ends of each span. The method further includes generating, by the processor, a fourth model for the pantograph current collector. The fourth model includes one or more of a first vertical spring, a first vertical damper, a first mass, and a first uplift force. The method further includes defining, by the processor, a fifth model for mechanical contact between the pantograph current collector and the contact cable, to determine intermittent unilateral contact interactions. The method further includes determining, by the processor, nonlinear static and dynamic mechanical interactions between the pantograph current collector and the overhead cable network.
In an embodiment, the method of generating the first model of the plurality of cables of the overhead cable network includes the following steps. The method includes defining, material and geometric parameters of the plurality of cables. The method further includes defining, mechanical behaviour of the plurality of cables. The plurality of cables are modelled as Euler-Bernoulli beams with inertia, elastic, and damping properties being set in accordance with the defined material and geometric parameters. The method further includes incorporating, in the first model, characteristics of cable pretensions in the plurality of cables. The method further includes incorporating, in the first model, characteristics of forces exerted by gravity; incorporating characteristics of interactions with the plurality of droppers cables, and incorporating characteristics of interactions with the pantograph current collector, and the support structures.
In an embodiment, the method of generating the second model of the plurality of dropper cables includes the following steps. The method includes, defining, by the processor, a nominal length for each dropper cable of the plurality of dropper cables in the span. The method further includes, defining, by the processor, attachment points along the span, for each dropper cable of the plurality of dropper cables to be attached with the messenger cable and the contact cable. The method further includes, defining, a target sag value of the contact cable at each attachment point of the plurality of dropper cables. The method further includes determining, an actual sag value of the contact cable at each attachment point of the plurality of dropper cables. The method further includes, adjusting iteratively, the nominal length of each of the plurality of dropper cables, corresponding to the difference between the target sag value and the actual sag value of the contact cable at the attachment point. The method further includes determining, an adjusted length for each of the plurality of dropper cables. Here, the actual sag value of the contact cable for said adjusted length corresponds with the target sag value of the contact cable, within a predefined tolerance at each attachment point of the plurality of dropper cables.

In an embodiment, the method of generating the second model of the plurality of dropper cables further includes the following steps. The method includes defining, material and geometric parameters of the plurality of dropper cables with the adjusted length. The method further includes determining, inertial and damping properties of the plurality of dropper cables based on the defined material and geometric parameters. The method further includes defining, a continuously-differentiable nonlinear relationship between a change in length of each cable of the plurality of dropper cables and an elastic force response. The method further includes adjusting parameters of the continuously-differentiable nonlinear relationship defining the elastic force response to be within a predefined tolerance, for a linear response in tension. The method further includes adjusting parameters of the continuously-differentiable nonlinear relationship defining the elastic force response to correspond within a predefined tolerance, with slackening of the cable in compression. The method further includes incorporating, in the second model, characteristics of forces exerted by gravity and by interactions with the messenger cable and the contact cable.

In an embodiment, the method of generating the third model for support structures supporting the overhead cable network includes the following steps. The method includes defining, material and geometric parameters of the one or more registration arms. The method further includes defining a linear relationship between deflection of the registration arm and corresponding force response of the one or more registration arm. The method further includes determining, inertial, elastic, and damping properties of the one or more registration arm, based on the defined material and geometric parameters. The method further includes incorporating, in the third model, characteristics of forces exerted by the contact cable on the one or more registration arms.

In an embodiment, the method of generating the fourth model of the pantograph current collector includes the following steps. The method includes defining, a multi-degree-of-freedom and a multi-level spring-mass-dashpot representation of the pantograph current collector. The method further includes defining, a plurality of uplift forces, inertias, spring stiffnesses and damping parameters. The method further includes incorporating, in the fourth model, characteristics of contact forces exerted by the contact cable on a head of the pantograph current collector.

In an embodiment, the method of defining the fifth model for mechanical contact between the contact cable and the pantograph current collector includes the following steps. The method includes defining a gap function to represent a standoff distance between the head of the pantograph current collector and the contact cable. The method further includes defining, a nonlinear, continuously-differentiable relationship between a contact force and the gap function, the contact force including force exerted by the head of the pantograph current collector on the contact cable. The method further includes determining, parameters of the nonlinear, continuously-differentiable relationship between the gap function and the contact force, the parameters configured to maintain unilateral contact constraints between the head of the pantograph current collector and the contact cable within a predefined tolerance.

In another non-limiting embodiment of the present disclosure, a system for determining nonlinear static and dynamic mechanical interactions between a pantograph current collector and an overhead cable network of an electrified railway line is disclosed [simply referred to as ‘the system’ hereinafter]. The system includes a processor and a memory. The processor is configured to generate a first model for a plurality of cables of the overhead cable network. The plurality of cables includes a messenger cable and a contact cable. The processor is further is configured to generate a second model of a plurality of dropper cables. Further, each of the plurality of dropper cables have a length adjusted to complement a target sag value of the contact cable. The plurality of dropper cables are modelled by employing a nonlinear force-deflection response configured to represent slackening of the plurality of dropper cables, upon being subjected to compression. The processor is further configured to generate a third model for support structures supporting the overhead cable network. The support structures include one or more registration arms configured to support the contact cable at ends of each span. The processor is further configured to generate a fourth model for the pantograph current collector. The fourth model includes one or more of a first vertical spring, a first vertical damper, a first mass, and a first uplift force. The processor is further configured to define a fifth model for mechanical contact between the pantograph current collector and the contact cable, to determine intermittent unilateral contact interactions. The processor is further configured to determine nonlinear static and dynamic mechanical interactions between the pantograph current collector and the overhead cable network.

In an embodiment, the processor is configured to define material and geometric parameters of the plurality of cables. The processor is further configured to define mechanical behaviour of the plurality of cables. The plurality of cables is modelled as Euler-Bernoulli beams with inertia, elastic, and damping properties being set in accordance with the defined material and geometric parameters. The processor is configured to incorporate, in the first model, characteristics of cable pretensions in the plurality of cables. The processor is configured to incorporate, in the first model, characteristics of forces exerted by gravity and incorporating characteristics of interactions with the plurality of dropper cables , and incorporating characteristics of interactions with the pantograph current collector, and the support structures.
In an embodiment, the processor is configured to define a nominal length for each dropper cable of the plurality of dropper cables in the span. The processor is further configured to define attachment points along the span, for each dropper cable of the plurality of dropper cables to be attached with the messenger cable and the contact cable. The processor is further configured to define a target sag value of the contact cable at each attachment point of the plurality of dropper cables. The processor is further configured to determine an actual sag value of the contact cable at each attachment point of the plurality of dropper cables. The processor is further configured to adjust iteratively the nominal length of each of the plurality of dropper cables, corresponding to the difference between the target sag value and the actual sag value of the contact cable at the attachment point. The processor is further configured to determine an adjusted length for each of the plurality of dropper cables. Here, the actual sag value of the contact cable for said adjusted length corresponds with the target sag value of the contact cable, within a predefined tolerance at each attachment point of the plurality of dropper cables.

In an embodiment, the processor is configured to define material and geometric parameters of the plurality of dropper cables with the adjusted length. The processor is further configured to determine inertial and damping properties of the plurality of dropper cables based on the defined material and geometric parameters. The processor is further configured to define a continuously-differentiable nonlinear relationship between a change in length of each cable of the plurality of dropper cables and an elastic force response. The processor is further configured to adjust parameters of the continuously-differentiable nonlinear relationship defining the elastic force response to be within a predefined tolerance, for a linear response in tension. The processor is further configured to adjust parameters of the continuously-differentiable nonlinear relationship defining the elastic force response to be within a predefined tolerance, for slackening of the cable in compression. The processor is further configured to incorporate in the second model, characteristics of forces exerted by gravity and by interactions with the messenger cable and the contact cable.

In an embodiment, the processor is configured to define material and geometric parameters of the one or more registration arms. The processor is further configured to define a linear relationship between deflection of the registration arm and corresponding force response of the one or more registration arm. The processor is further configured to determine inertial, elastic, and damping properties of the one or more registration arm, based on the defined material and geometric parameters. The processor is further configured to incorporate in the third model, characteristics of forces exerted by the contact cable on the one or more registration arms.
In an embodiment, the processor is configured to define a multi-degree-of-freedom and a multi-level spring-mass-dashpot representation of the pantograph current collector. The processor is further configured to define a plurality of uplift forces, inertias, spring stiffnesses and damping parameters. The processor is further configured to incorporate, in the fourth model, characteristics of contact forces exerted by the contact cable on a head of the pantograph current collector [also referred to as the ‘pantograph head’ in the present disclosure].
In an embodiment, the processor is configured to define a gap function to represent a standoff distance between the head of the pantograph current collector and the contact cable. The processor is further configured to define a nonlinear, continuously-differentiable relationship between a contact force and the gap function. The contact force includes force exerted by the head of the pantograph current collector on the contact cable. The processor is further configured to determine parameters of the nonlinear, continuously-differentiable relationship between the gap function and the contact force. The parameters are configured to maintain unilateral contact constraints between the head of the pantograph current collector and the contact cable within a predefined tolerance.

It is to be understood that the aspects and embodiments of the disclosure described above may be used in any combination with each other. Several of the aspects and embodiments may be combined together to form a further embodiment of the disclosure.

The foregoing summary is illustrative only and is not intended to be in any way limiting. In addition to the illustrative aspects, embodiments, and features described above, further aspects, embodiments, and features will become apparent by reference to the drawings and the following detailed description.

BRIEF DESCRIPTION OF THE ACCOMPANYING DRAWINGS
The novel features and characteristics of the disclosure are set forth in the appended claims. The disclosure itself, however, as well as a preferred mode of use, further objectives, and advantages thereof, will best be understood by reference to the following detailed description of an illustrative embodiment when read in conjunction with the accompanying figures. One or more embodiments are now described, by way of example only, with reference to the accompanying figures wherein like reference numerals represent like elements and in which:
Figure 1 illustrates a simple catenary arrangement of an overhead equipment (OHE) in accordance with an existing art;
Figure 2 illustrates a pantograph and its components, in accordance with a conventional art;
Figure 3a is a flowchart illustrating a method for determining nonlinear static and dynamic mechanical interactions between a pantograph current collector and an overhead cable network of an electrified railway line, in accordance with an embodiment of the present disclosure;
Figure 3b is a flowchart illustrating steps included in generating a first model of the plurality of cables of the overhead cable network, in accordance with an embodiment of the present disclosure;
Figure 3c [including Figures 3c-1 and 3c-2] is a flowchart illustrating steps included in generating a second model of the plurality of dropper cables, in accordance with an embodiment of the present disclosure;
Figure 3d is a flowchart illustrating steps included in generating a third model for support structures supporting the overhead cable network, in accordance with an embodiment of the present disclosure;
Figure 3e is a flowchart illustrating steps included in generating a fourth model of the pantograph current collector, in accordance with an embodiment of the present disclosure;
Figure 3f is a flowchart illustrating steps included in generating a fifth model for mechanical contact between the contact cable and the pantograph current collector, in accordance with an embodiment of the present disclosure;
Figure 4 illustrates a system, employed for implementation of the method of the present disclosure;
Figures 5a illustrates an idealized representation of a OHE-pantograph system for the purpose of predicting their mechanical interactions, in accordance to one implementation of the present disclosure;
Figure 5b illustrates a lumped mass model of the pantograph of the Figure 5a;
Figure 6a is a plot depicting a dropper response F_d (δ)=k_d max{0,δ}, with stiffness equal to k_d in tension and zero in compression, in accordance with one implementation of the present disclosure;
Figure 6b is a plot depicting idealized dependence of the contact force f_c on the gap function g, in accordance with one implementation of the present disclosure;
Figure 7a is a plot illustrating a 1-parameter family of smooth functions {ϕ_α }_α defined in eq. (16) of the present disclosure, approximating x↦max{0,x}, in accordance with one implementation of the present disclosure; The consequent approximations of derivatives are shown on the Figure 7b;
Figure 8a to 8e are comparative plots illustrating accuracy of the method and the system of the present disclosure in comparison with the conventional approaches.

The figures depict embodiments of the disclosure for purposes of illustration only. One skilled in the art will readily recognize from the following description that alternative embodiments of the method, system, device and apparatus illustrated herein may be employed without departing from the principles of the disclosure described herein.

GENERAL DESCRIPTION

While the embodiments in the disclosure are subject to various modifications and alternative forms, specific embodiments thereof have been shown by way of example in the figures and will be described below. It should be understood, however that it is not intended to limit the disclosure to the particular forms disclosed, but on the contrary, the disclosure is to cover all modifications, equivalents, and alternative falling within the scope of the disclosure.

The terms “comprises”, “comprising”, or any other variations thereof used in the disclosure, are intended to cover a non-exclusive inclusion, such that a device, assembly, mechanism, system, method that comprises a list of components does not include only those components but may include other components not expressly listed or inherent to such system, or assembly, or device. In other words, one or more elements in a system proceeded by “comprises… a” does not, without more constraints, preclude the existence of other elements or additional elements in the system or mechanism.

In the present disclosure, a system (300) and method (200) for determining nonlinear static and dynamic mechanical interactions between pantograph and overhead cables (101) in electric railway systems is disclosed. The method (200) includes finite element and time integration methods to determine all kinetic and kinematic aspects of pantograph-overhead equipment (OHE) interactions. The aspect of determining nonlinear static and dynamic mechanical interactions includes determining at least one of the following:
static deflections of overhead cable (101) networks by accounting for pre-tension, gravity loads, and cable (104) elasticity.
temporal and spatial displacements of the pantograph head, the catenary and the messenger cables (104a),
vibrations and wave propagation in the overhead cables (101) caused by the motion of the pantograph,
time histories of contact forces between pantographs and overhead cables (101),
slackening of dropper cables (105) between the catenary and messenger wires,
mechanical stresses and strains in cables (104),
the influence of train speed on mechanical interactions,
statistical parameters related to displacements, velocities, accelerations and forces, and
failure scenarios (e.g., breakages in droppers)

Embodiments of the present disclosure relate to a method (200) for determining nonlinear static and dynamic mechanical interactions between a pantograph current collector (100) and an overhead cable (101) network of an electrified railway line [simply referred to as ‘the method (200)’ hereinafter]. The term ‘overhead cable (101) network’ as used in the present disclosure refers to cables (104) or wires spanning over railway tracks, and includes one or more contact cable(s) (104b) along with which the head of the pantograph makes contact. The term ‘catenary cables’ as used in the present disclosure refers to one or more contact cable(s) (104b) along with which the head of the pantograph makes contact.
A general description of steps and sub-steps included in the method (200) is described in the following paragraphs. The order in which the method (200) is described is not intended to be construed as a limitation, and any number of the described method blocks may be combined in any order to implement the method (200). Additionally, individual blocks may be deleted from the method (200) without departing from the scope of the subject matter described herein.
Reference is made to Figure 3a, illustrating a flowchart of a method (200) for determining nonlinear static and dynamic mechanical interactions between a pantograph current collector (100) and an overhead cable (101) network of an electrified railway line [also referred to as ‘the method (200)’ hereinafter]. The method (200) may be implemented by a system (300) including a processor (102) and a memory (106), as depicted in Figure 4. The method (200) includes steps [201] to [206] as depicted in Figure 3a, and is explained as follows. The method (200) includes the step [201] of generating, by a processor (102), a first model (103a) for a plurality of cables (104) of the overhead cable (101) network. The method (200) further incudes the step [202] of generating, by the processor (102), a second model (103b) of a plurality of dropper cables (105). The method (200) further incudes the step [203] generating, by the processor (102), a third model (103c) for support structures supporting the overhead cable (101) network. The method (200) further includes the step [204] of generating, by the processor (102), a fourth model (103d) for the pantograph current collector (100). The method (200) further includes the step [205] of defining, by the processor (102), a fifth model (103e) for mechanical contact between the pantograph current collector (100) and the contact cable (104b). The method (200) further includes the step [206] of determining, by the processor (102), nonlinear static and dynamic mechanical interactions between the pantograph current collector (100) and the overhead cable (101) network.
The term ‘plurality of cables (104)’ as used in the present disclosure includes a messenger cable (104a) and a contact cable (104b). The term ‘plurality of cables (104)’ and the term ‘cables (104)’ are used interchangeably with the term ‘plurality of wires’ and the term ‘wires’, respectively, in the present disclosure. Accordingly, the term ‘messenger cable (104a)’ is also referred to as the ‘messenger wire’ in the present disclosure. Further, the term ‘contact cable (104b)’ is used interchangeably with the term ‘contact wire’ in the present disclosure.
Figure 3b is a flowchart illustrating steps included in generating the first model (103a) for the plurality of cables (104) of the overhead cable (101) network. Generating the first model (103a) includes the step [201a] of defining, material and geometric parameters of the plurality of cables (104). The method (200) further includes the step [201b] of defining mechanical behaviour of the plurality of cables (104). The method (200) further includes the step [201c] of incorporating, in the first model (103a), characteristics of cable (104) pretensions in the plurality of cables (104). The method (200) further includes the step [201d] of incorporating characteristics of forces exerted by gravity, incorporating characteristics of interactions with the plurality of dropper cables (105), incorporating characteristics of interactions with the plurality of pantograph current collectors (100), and the support structures.

Further, the step [202] of generating, by the processor (102), the second model (103b) of the plurality of dropper cables (105), is illustrated in detail in the flowchart of the Figures 3c-1 and 3c-2 [collectively referred to as the ‘Figure 3c’ in the present disclosure]. While Figure 3c-1 illustrates the steps [202a] to [202f], the Figure 3c-2 illustrates the steps [202g] to [202l]. Aspects included in the step [202] of generating the second model (103b) are now described with reference to the Figure 3c-1 and 3c-2. Generating the second model (103b) includes the step [202a] of defining, by the processor (102), a nominal length for each dropper cable (105) of the plurality of dropper cables (105) in the span. The method (200) further includes the step [202b] of defining, by the processor (102), a target sag value of the contact cable (104b). The method (200) further includes step [202c] of determining, an actual sag value of the contact cable (104b). The method (200) further includes step [202d] of determining an adjusted length for each of the plurality of dropper cables (105). The method (200) further includes step [202e] of adjusting iteratively the nominal length of each of the plurality of dropper cables (105). The method (200) further includes step [202f] of determining an adjusted length for each of the plurality of dropper cables (105) such that the actual sag matches the target sag within a predefined tolerance.
Referring now to the Figure 3c-2, the method (200) further includes the step [202g] of defining material and geometric parameters of the plurality of dropper cables (105). The method (200) further includes the step [202h] of defining length of the plurality of dropper cables (105). The method (200) further includes the step [202i] of determining inertial and damping properties of the plurality of dropper cables (105), based on the steps [202g] and [202h]. The method (200) further includes the step [202j] of defining a continuously-differentiable nonlinear relationship between a change in length of each cable (104) of the plurality of dropper cables (105) and an elastic force response. The method (200) further includes the step [202k] of adjusting parameters of the continuously-differentiable nonlinear relationship defining the elastic force response to be within a predefined tolerance, for a linear response in tension, based on step [202j]. The method (200) further includes the step [202l] of adjusting parameters of the continuously-differentiable nonlinear relationship defining the elastic force response to be within a predefined tolerance, for slackening of the cable (104) in compression, based on step [202j]. The method (200) further includes the step [202m] of incorporating characteristics of forces exerted by gravity. The method (200) further includes the step [202n] of defining attachment points along the span for each dropper cable (105). The method (200) further includes the step [202o] of defining interactions with the messenger cable (104a) and the contact cable (104b) at the attachment points. The method (200) finally includes the step [202] of generating a second model (103b) of a plurality of dropper cables (105), based on the steps [202i], [202k], [202l], [202m] and [202o].
Figure 3d illustrates the steps [203a] to [203d] comprised in generating the third model (103c). The method (200) includes the step [203a] of defining material and geometric parameters of the one or more registration arms. The method (200) further includes the step [203b] of determining inertial, elastic, and damping properties of the one or more registration arm, based on the step [203a]. The method (200) further includes the step [203c] of defining a linear relationship between deflection of the registration arm and corresponding force response of the one or more registration arm. The method (200) further includes the step [203d] of incorporating characteristics of forces exerted by the contact cable (104b) on the one or more registration arms. The method (200) finally includes the step [203] of generating a third model (103c) for support structures supporting the overhead cable (101) network, based on steps [203b], [203c], and [203d].
Figure 3e illustrates the steps comprised in generating the fourth model (103d) of the pantograph current collector (100). The method (200) includes the step [204a] of defining a multi-degree-of-freedom and a multi-level spring-mass-dashpot representation of the pantograph current collector (100). The method (200) further includes the step [204b] of defining a plurality of uplift forces Fu, inertias, spring stiffnesses and damping parameters. The method (200) further includes the step [204c] of incorporating, in the fourth model (103d), characteristics of contact forces fc exerted by the contact cable (104b) on a head of the pantograph current collector (100). Based on the steps [204a] to [204c], the fourth model (103d) of the pantograph current collector (100) is generated.
Figure 3f illustrates the steps comprised in defining the fifth model (103e) for mechanical contact between the contact cable (104b) and the pantograph current collector (100). The method (200) includes the step [205a] of defining a gap function g to represent a standoff distance between the head of the pantograph current collector (100) and the contact cable (104b). The method (200) further includes the step [205b] of defining a nonlinear, continuously-differentiable relationship between a contact force fc and the gap function g. The method (200) further includes the step [205c] of determining, parameters of the nonlinear, continuously-differentiable relationship between the gap function g and the contact force fc, to approximate unilateral contact interaction within a predefined tolerance. The step [205c] is based on [205b]. The method (200) finally includes defining a fifth model (103e) for mechanical contact between the pantograph current collector (100) and a contact cable (104b), based on steps [205a] and [205c].

OVERVIEW OF THE STATE OF THE ART
Figure 5a shows a representative image of overhead cables (101) and a pantograph. The base of the pantograph is mounted on the carriage, while the carbon/graphite strip slides along the contact wire (also referred to as the catenary). The contact wire is suspended from the messenger wire by regularly-spaced dropper cables (105). This arrangement is commonly referred to as the simple catenary. Each span of the overhead cable (101) network consists of a pair of steady arms, a set of dropper cables (105) and specific lengths of the messenger wires (104a) and contact wires (104b). The extent of a span varies across railway systems. The messenger wire (104a), the dropper cables (105), supporting steady arms and the pre-tensions in the messenger wires (104a) and contact wires (104b) together serve to reduce the sag of the contact wire (104b) caused by its self-weight. It is desirable that the pantograph head slide at nearly constant elevation above the carriage and that the elastic stiffness of the catenary be uniform across the extent of a span. These two factors ensure that the contact force is maintained at a nominal (nearly constant) value. Alternate cable (104) arrangements, such as the stitched and compound networks have also been designed to better address these requirements, albeit at the expense of more complex installation and maintenance.
Some existing codes: Owing to the significance of predicting cable-pantograph interactions, a significant number of prediction codes have been developed by academic institutions and industrial agencies across many countries. These include:
PrOSA by DB Systemtechnik GmbH (PrOSA)
PantoCat by Instituto Superior Tecnico Lisboa
SPOPS by Korea Railroad Research Institute
CaPaSIM by Royal Institute of Technology
PCaDA by Politecnico di Milano
Gasen-do FEM by Railway Technical Research Institute
OSCAR by Societé Nationale des Chemins de Fer
PCRUN by Southwest Jiaotong University
CANDY by Universidad Pontificia Comillas de Madrid
PACDIN by Universitat Politècnica de València & TALGO S.L.
The codes mentioned above serve both government and commercial agencies. For example, a notable use case of the code OSCAR was in enabling the French TGV set a record speed of 574.8" " km/h. A majority of codes currently in use are qualified based on railway standards, most notably the European standard EN50318. While the standards prescribe acceptable ranges of predictions, they do not specify methods to adopt. Hence, codes adopt varying methodologies, resulting in noticeable differences in their predictions despite adhering to the standards.
Representation of mechanical elements and their interactions in accordance with the state of the art
The Figure 5a shows an idealized representation of a cable (104) pantograph system for the purpose of predicting their mechanical interactions. The figure depicts a two-dimensional representation. Further, Figure 5b illustrates a lumped mass model of the pantograph in Figure 5a. In principle, railway tracks can be curved, and consequently overhead networks are three dimensional. Nevertheless, two-dimensional representations provide accurate predictions as well. Three-dimensional studies requiring far greater time and effort, are usually reserved to examine specialized scenarios, such as the influence of stagger, track geometry, aerodynamic forces in tunnels or wear in the contact wire (104b) and the pantograph current collector (100).
Cables (104): The messenger cables (104a) and catenary cables are commonly represented as cable (104) elements or as Euler-Bernoulli beams. While both account for transverse stiffness, the latter accounts for bending resistance of cables (104) due to their finite thickness and is therefore better suited to high-speed networks. Beam representations need to explicitly incorporate pre-tension in the cables (104) and may additionally account for axial deformations. The inertia of the cable (104) is represented as a uniform line density and the cable’s (104) self-weight as a uniformly distributed transverse load.
Droppers: Vertical dropper cables (105) suspend the catenary from the messenger wire (104a). Small clamps at either end of the dropper cables (105) help secure them to the catenary and messenger wires (104a). One of the main sources of complexity in predicting pantograph-OHE interactions stems from the fact that transverse displacements of catenary and messenger cables (104a) result in axial deformations of the dropper cables (105). Consequently, dropper cables (105) may slacken or buckle when subjected to compressive forces. In effect, dropper cables (105) behave as springs resisting tensile forces while offering little resistance in compression.
Indeed, slackening of dropper cables (105) is one of the most crucial factors in pantograph-OHE interactions and the main sources of nonlinearity in predictions. Codes differ widely in accounting for these nonlinear effects. Proposed strategies include representing dropper cables (105) as:
bilinear springs elements with constant stiffness in tension and zero stiffness in compression, see Figure 6a. Such a representation assumes that dropper cables (105) do not transmit any compressive force. This is a reasonable approximation in practice, since the buckling loads are expected to be small for actual geometries of dropper cables (105).
nonlinear springs whose force-deflection response is measured experimentally.
beams (rather than springs) thus accounting for both axial deformation and buckling behaviour but at greater computational expense.
linear springs in which slackening is accounted for approximately by the addition of compensating elements.
The inertia of dropper cable (105) elements consists of contributions from cable (104) itself, and from the clamps at its ends. The former can be lumped along with the clamp masses, or be considered as a line density distributed along the length of the dropper.
Registration arm: Registration arms, also referred to as steady arms, support the catenary at the ends of each span. Both the geometry and support of the registration arm affect its stiffness when resisting displacements caused by the motion of the contact wire (104b). Here again, codes vary in the level of detail adopted in the mechanical idealization of the arm. The arm can be represented as Euler-Bernoulli beams with accurate geometry and damping. Others prefer a simplified representation as a damped linear springs whose stiffness is determined to match that of a detailed representation. Some codes even adopt representations for the arm as nonlinear bars. The mass of the arm may be accounted for either by lumping or as a uniformly distributed line density.
Pantograph: The representation of the pantograph is among the most important aspects of determining its mechanical interaction with the OHE. The detailed construction of a pantograph closely resembling a four-bar mechanism is shown in Figure 2. Such a level of detail, however, is not generally deemed necessary. Instead, effective linear spring-mass-dashpot representations are preferred. The EN50318 standards recommends using the simplified representation with three translational degrees of freedom, as depicted in the Figure 5a. More detailed representations include additional rotational degrees of freedom at the pantograph head and may also incorporate torsion springs. The uplift force specified in the pantograph is provided by the manufacturer and is an important tuneable parameter to enable higher running speeds (higher speeds typically require larger uplift forces).
Sliding contact: Mechanical sliding contact interactions between the pantograph and OHE underlies electrical connectivity. These contact interactions are a critical source of complexity and nonlinearity in predicting OHE-pantograph interactions. The unilateral contact interaction is characterized by the contact force. The sliding elements are deemed to be in contact if the magnitude of the force exceeds a nominal threshold, in which case, the transverse displacements of the pantograph head and the contact point on the catenary coincide. Otherwise, the elements lose contact, the contact force reduces to zero and the displacements of the pantograph and the contact wire (104b) are no longer coupled. The EN50318 standard recommends using a penalty method to enforce unilateral contact, wherein a fictitious stiff spring is added at the contact interface. Most codes adopt this strategy. The alternative method of Lagrange multipliers is also feasible but is less commonly used.
Computational methodologies: Complementary to the idealized representations of components in a pantograph-OHE system (300) are the set of computational methodologies to determine mechanical interactions. Finite element methods are the most commonly used, thanks to their versatility for approximating behaviours of linear and nonlinear mechanical components. Finite elements approximating Euler-Bernoulli beams, linear and nonlinear springs or bar elements are routine in the literature. Finite element methods are well-suited to predict static deflections of OHE cables (104) by accounting for the gravitational loading, cable (104) pretensions and cable (104) elasticity. Both explicit and implicit time integration methods/procedures (e.g., Newmark-beta method, HHT-alpha method, BDF-2, Runge-Kutta schemes) can be coupled with spatial finite element discretizations to determine dynamic deformations. Penalty and Lagrange multiplier techniques can be implemented in finite element methods to robustly predict dynamic mechanical interactions of pantograph-OHE interactions. Finite element methods are capable of accurately predicting wave propagation in the cables (104), wave reflections are droppers and masts, and vibrations of the pantograph head, all of which could significantly affect contact interactions in high-speed railways. Advanced moving-mesh techniques incorporated with finite element methods can improve the accuracy of predictions by incorporating grids over the contact wire (104b) that move with the pantograph.
It is highlighted that a dichotomy exists in the choice of methods to couple the dynamics of the pantograph with the OHE. This stems from the fact that the former is composed of rigid body elements while the latter of deformable elements. Hence, co-simulation procedures that couple distinct computational techniques implemented for rigid and deformable body dynamics are routinely used. Monolithic approaches that treat both systems together exist as well.
A second dichotomy arises from considerations of linear and nonlinear effects/interactions in the pantograph-OHE system (300). Nonlinearities most notably arise from slackening of dropper cables (105) and from the possibility of intermittent contact between the pantograph and the catenary. Modal decomposition methods are an efficient choice for linear systems but are ill-suited to account for nonlinearities as well as to account for moving contact loads. Here again, finite elements are the method of choice, even permitting the representation of geometrically nonlinear effects caused by large deflections of cables (104) in high-speed networks.

DETAILED DESCRIPTION
The pantograph-OHE system (300)
Throughout the article, a simple catenary arrangement for the OHE is considered. These systems are well studied in the literature, and several benchmark results are available for comparisons in the present disclosure. Furthermore, Indian railway networks employ simple catenary systems, thus making this work relevant to their design and analysis.
The simple catenary arrangement
Figure 5a illustrates an idealized two-dimensional representation of a pantograph and a simple catenary arrangement. The base of the pantograph is mounted on the carriage while its head slides along the contact wire (104b). Regularly spaced dropper cables (105) suspend the contact wire (104b) from the messenger wire (104a). Each span of the OHE of length l includes a pair of registration arms at the ends, a set of n_d droppers of lengths l_1,…,l_(n_d ), and the pair of pre-tensioned messenger wires (104a) and contact wires (104b) measuring length l. The extent of a span varies across railway systems (e.g., l=55" " m in the present study). The vertical offset (l_e ) of the messenger wire (104a) from the contact wire (104b) represents the encumbrance. Parameters related to each component are indicated in the figure, where tensions, bending stiffnesses, extensional stiffnesses, line densities, spring constants and masses are denoted by T, EI, EA, ρ, k and m, respectively, with subscripts m,c,d,r and p used to identify the messenger wire (104a), contact wire (104b), dropper cables (105), registration arm and the pantograph, respectively.
The messenger wire (104a), the dropper cables (105), supporting structures, and the cable (104) pre-tensions together reduce the contact wire's (104b) sag caused by its self-weight. It is desirable that the pantograph head slide at a nearly constant elevation above the carriage and that the transverse elastic stiffness of the catenary is uniform across the extent of a span. These two factors ensure the contact force f_c is maintained at a nominal (nearly constant) value. However, neither is achieved in practice, which is the point of departure for mechanistic investigations.
Mechanical models of components and interactions
In the interest of notational simplicity, the ensuing discussions are restricted to a system consisting of a single span of the OHE and one pantograph. Referring to the coordinate system indicated in figure 5a, the goal is to determine transverse (i.e., along y ) displacements (x,t)↦u_c (x,t) of the contact wire, (x,t)↦u_m (x,t) of the messenger wire, and t↦u_1 (t),u_2 (t),u_3 (t) of the lumped masses of the pantograph, as well as the time history t↦f_c (t) of the normal contact force between the pantograph and the contact wire (104b). Velocities and accelerations of cables (104) and the pantograph follows naturally. Internal forces in the OHE network, such as in the dropper cables (105) or those exerted by the registration arms can be deduced as well.
The OHE network: Span lengths are far greater than the dimensions of cable (104) cross-sections. This warrants idealizing all cables (104) in the OHE as one-dimensional structures. The messenger cables (104a) and contact cables (104b) are modelled as pre-tensioned Euler Bernoulli beams, the supports for the messenger wire (104a) at either end as linear springs having stiffness k_m, and the registration arm supporting the contact wire as a linear spring with stiffness k_r. Each dropper cable (105) is idealized as a two-dof spring-mass system. Hence, each dropper cable (105) is represented by a pair of point masses positioned at its ends and connected by a (nonlinearly) elastic spring. The mass of the dropper cable (105) is lumped with those of the clips m_cm and m_cc, attaching it with the messenger wires (104a) and contact wires (104b). Momentum balance for the messenger wire (104a) follows from examining the balance of transverse forces:

where Δl_i=l_e-l_i and familiar notations have been used for the Dirac distribution as δ(⋅) and the Macaulay bracket ⟨y⟩≜max{y,0}. The first row of eq. (1) represents the inertias of the messenger cable (104a) and the lumped masses of dropper cables (105) that are coupled with the messenger wire (104a) at discrete attachment locations {x_i }_i. The second row represents contributions from the bending elasticity of the cable (104), the pretension, and the suspension springs supporting the cable (104) at either end of the span. These are followed by the elastic resistance of the dropper cables (105). Therein, the response of the i-th dropper cable (105) is idealized as that of an elastic bar having stiffness k_i=EA_d/l_i in tension, but that slackens in compression, which is illustrated in figure 6a. The latter presumes a trivial buckling load and negligible post-buckling stiffness- both conservative but practical idealizations. The final row of eq. (1) accounts for gravitational loading, with g denoting the acceleration due to gravity.
An analogous balance law follows for the contact wire (104b):

Besides employing parameters corresponding to the contact wire (104b), eq. (2) differs from eq. (1) in the inertia of the registration arm in the first row of terms, the sign of the dropper resistance in the third, and the addition of the contact force f_c exerted by the pantograph head at its location x=x_p (t) along the span in the last row.
The pantograph: As recommended by the EN50318 standard, the pantograph is represented as a lumped three-dof spring-mass system, ignoring damping contributions from dashpots for now. Then, examining figure 5b, force balance for the three dofs follows as
█(⏟([■(m_1&0&0@0&〖" " m〗_2&0@0&0&〖" " m〗_3 )] )┬(M_p ) [■(u ¨_1@u ¨_2@u ¨_3 )]+⏟([■(k_1+k_2&-k_2&0@-k_2&k_2+k_3&-k_3@0&-k_3&k_3 )] )┬(K_p ) [■(u_1@u_2@u_3 )]=[■(f_u@0@-f_c (t))]#(3))
The interaction of the point mass m_3 at the head of the pantograph, as it slides along the contact wire (104b), is conveyed by the contact force -f_c (t), having an opposite sign compared to the force exerted on the contact wire (104b) in eq. (2). The travel speed of the pantograph is not pertinent in eq. (3) but manifests in eq. (2) as the location along the span at which the force f_c (t) acts on the contact wire (104b). For a pantograph traveling at constant speed v_p along the span while starting from an initial position x=x ‾, the simple relation x_p (t)=x ‾+v_p t is obtained. The prescribed uplift force F_u imposed on the mass m_1 is a critical operating parameter that can be tuned to help avoid contact loss, especially at high travel speeds.
Contact interaction: The time history of the contact force f_c (t) as the pantograph slides on the contact wire (104b) is a crucial unknown of interest. The impenetrability of the pantograph head and the contact wire (104b) requires that the pantograph head travel at or below the elevation of the contact wire (104b) and is conveyed by introducing the gap function
█(g(t)≜u_c (x_p (t),t)-u_3 (t)≥0#(4))
The unilateral nature of contact forbids the possibility of g<0. The force f_c≥0 can be interpreted as the Lagrange multiplier imposing the constraint in eq. (4). In particular, we have
█({■(f_c (t)>0&⟺&g(t)=0@f_c (t)=0&⟺&g(t)>0)┤#(5))
from where the well-known complementarity condition f_c g=0 ∀t, follows. During operation, it is desirable that the pantograph remain in persistent contact with the contact wire (104b). In such a scenario, g(t)=0 and f_c (t)>0 at all times. Otherwise, the contact is intermittent, with contact loss occurring at times when g(t)>0 and f_c (t)=0. In fact, the fraction of time for which g>0 is an important indicator of current collection quality.
A few remarks on the component-level models represented by eqs. (1) to (4) are noted next.
(i) Both eqs. (1) and (2) consider only transverse displacements while neglecting displacements along the span based on the expectation that cable (104) deformations are bending-dominated. This approximation is justified by the validations shown in subsequent description.
(ii) Unlike the beam models underlying eqs. (1) and (2), cable (104) models can be adopted as well. The conceptual simplicity of the latter is especially appealing at low pantograph speeds and in OHE systems with large cable (104) pre-tensions T_c and T_m. Noticeable differences between predictions of cable (104) and beam models become apparent with increasing pantograph speeds when wave propagation effects become significant.
(iii) The registration arm's stiffness k_r can be deduced from its geometry and composition, as done in our studies. The representation adopted here precludes including the influence of the arm's stagger.
(iv) The balance equations for the messenger wires (104a) and contact wires (104b) are coupled only by the dropper cable (105) responses ±k_i ⟨u_m (x_i,t)-┤ ├ u_c (x_i,t)⟩, thus conveying the purpose of dropper cables (105) in moderating the contact cable (104b) sag by anchoring them to the messenger wire (104a).
(v) The piecewise nature of dropper cable (105) responses and the implicit dependence of the contact force on cable (104) and pantograph displacements renders eqs. (1) and (2) nonlinear.
(vi) A notable omission in the description thus far is the influence of damping- in the cables (104), support structures and the pantograph. This is done in the interest of brevity, so that eqs. (1) and (2) do not get cluttered by additional damping-related contributions. In all our studies, we employ standard dashpot models for the pantograph and the registration arm and Rayleigh damping for the cables (104). These contributions are tersely represented as a damping matrix in the finite element discretization discussed next.
(vii) More elaborate pantograph models may accurately represent the kinematics of linkages in the mechanism or add rotational dofs and are generally used in specialized scenarios to examine the wear of the pantograph head or the effects of aerodynamic forces. Irrespective of the pantograph model, the fact that its coupling with the OHE is limited to just a few parameters enables co-simulation techniques, an alternative to the monolithic approach.
Finite element discretization
To study the predictions of the model, a finite element method for spatial discretization and implicit schemes for time integration are employed. Noting the fourth-order derivatives appearing in the bending terms in eqs. (1) and (2), cubic Hermite basis functions are adopted to approximate the messenger wire (104a) and contact wire (104b) displacements. For convenience, the span [0,l] with a set of n grid points is discretized for both the messenger wires (104a) and contact wires (104b) while ensuring that dropper cables (105) attachment locations {x ‾_i }_i coincide with grid points. The resulting 2n-dimensional finite element space is spanned by a set of Hermite basis functions {H_a }_(a=1)^2n. Denote the messenger wires (104a) and contact wires (104b) dofs by U_m (t),U_c (t)∈R^2n, which define the approximations of their displacement fields
u_m^h (x,t)=∑_(a=1)^2n▒  U_ma (t)H_a (x)□( ) " and " □( ) u_c^h (x,t)=∑_(a=1)^2n▒  U_ca (t)H_a (x)
Grouping the dofs of the pantograph as U_p (t)∈R^3, the composite system of unknowns is denoted by U(t)=[U_m (t),U_c (t),U_p (t)]∈ R^(4n+3).
The solution-independent contributions to the global mass matrix, stiffness matrix and load vector have the form
■(&M=diag⁡[⏟(M_m )┬(2n×2n),⏟(M_c )┬(2n×2n),⏟(M_p )┬(3×3)]@&F=[⏟(F_m )┬(2n×1),⏟(F_c )┬(2n×1),0_(3×1)]+[0_((4n+2)×1),〖" " F〗_u ]≜F_"ohe " +F_p )
where the subscripts m,c and p refer to contributions from the messenger wire (104a), contact wire (104b) and the pantograph. The pantograph matrices M_p and K^p are as defined in eq. (3). Entries of the remaining matrices and vectors follow from eqs. (1) and (2) as
■(M_(m,ab)& =ρ_m (H_a,H_b )+├ ∑_(i=1)^(n_d)▒  (m_cm+1/2 ρ_d l_i ) H_a H_b ┤|_(x ‾_i )@M_(c,ab)& =ρ_c (H_a,H_b )+├ ∑_(i=1)^(n_d)▒  (m_cc+1/2 ρ_d l_i ) H_a H_b ┤|_(x ‾_i )+m_r (├ H_a H_b ┤|_0+├ H_a H_b ┤|_l )@K_(m,ab)& =EI_m (H_a^'',H_b^'' )+T_m (H_a^' H_b^' )+k_m (├ H_a H_b ┤|_0+├ H_a H_b ┤|_l )@K_(c,ab)& =EI_c (H_a^'',H_b^'' )+T_c (H_a^' H_b^' )+k_r (├ H_a H_b ┤|_0+├ H_a H_b ┤|_l )@F_(m,a)& =-(ρ_m g,H_a )-├ ∑_(i=1)^(n_d)▒  (m_cm+1/2 ρ_d l_i ) gH_a ┤|_(x ‾_i )@〖" " F〗_(c,a)& =-(ρ_c g,H_a )-├ ∑_(i=1)^(n_d)▒  (m_cc+1/2 ρ_d l_i ) gH_a ┤|_(x ‾_i )@〖" " F〗_(p,a)& =f_u δ_a1 )
where δ_ab is the Kronecker-delta symbol and (f,g)≜∫_0^l f(x)g(x)dx represents the usual L^2-inner product.
The elastic responses of dropper cables (105) are not included in the stiffness matrices K_m and K_c, but contribute to a solution dependent residual vector R_d [U]=∑_(i=1)^(n_d ) R_i [U] that is expressed as a sum of dropper-wise residuals. Setting H(x) to be the 2n-dimensional vector of basis functions evaluated at x,h_i=[├ H┤|_(x ‾_i ),├ H┤|_(x ‾_i ),0_(3×1) ], and h ̅_i=[├ H┤|_(x ‾_i ),-├ H┤|_(x ‾_i ),0_(3×1) ], dropper residuals follow as
█(R_i [U]=k_i ⟨h ̅_i^T U-Δl_i ⟩ h_i={■(⏟(k_i h_i h ̅_i^T )┬(K_i ) U-⏟(k_i Δl_i h_i )┬(F_i )&" if " h ̅_i^T U≥Δl_i@0&" otherwise " )┤#(7))
The contribution of the i-th dropper cable (105) is nontrivial only under tension, in which case we R_i [U]=K_i U-F_i.
Similarly, the contact force contributions are solution-dependent and hence not included in the matrices and vectors above. The discretized counterpart of the gap function g is denoted by
█(g_h (t)=u_c^h (x_p (t),t)-u_3 (t)=r^T (t)U(t)," where " r(t)=[0_(2n×1),├ H┤|_(x_p (t)),0,0,-1]#(8))
The unilateral contact condition in eq. (9) now has the form
f_c^h (t)r^T (t)U(t)=0" at each " t
where, f_c^h denotes the approximation of the contact force by f_c.
Finally, denote the damping matrix by
C=diag⁡[⏟(C_"ohe " )┬(4n×4n),⏟(C_p )┬(3×3)]
which is composed of contributions C_"ohe " from the messenger wires (104a) and contact wires (104b), dropper cables (105) and the registration arm, and C_p from the pantograph.
Given initial conditions ├ U┤|_(t=0)=U_0 and ├ U ˙ ┤|_(t=0) V_0, the semi-discrete form of the model can now be stated as the problem of finding t↦(U(t),f_c^h (t)) satisfying
█({■(MU ¨(t)+CU ˙(t)+KU(t)+R_d [U(t)]=F+f_c^h r(t)@f_c^h≥0,r^T U≥0" and " f_c^h r^T U=0)┤#(9))
The initial conditions U_0 and V_0 follow from a static problem which is discussed next. In the present study, the implicit Newmark and generalized-alpha schemes are used to integrate the system of differential equations in eq. (9).
Static scenarios
The initial conditions for eq. (9) correspond to a system with the pantograph at its initial position x=x ‾ and engaged with the contact wire. Hence, the initial velocity V_0=0, while determining the initial displacement U_0 requires computing an equilibrium state. The latter follows as a specialization of eq. (9) in which accelerations and velocities are set to zero:
█(" Find " U_0∈R^(4n+3) " such that " {■(KU_0+R_d [U_0 ]=F+f_c^0 r_0 " and " @r_0^T U_0=0)┤#(10))
where r_0=├ r┤|_(t=0). Eq. (10) is a system of nonlinear algebraic equations that can be resolved using a Newton method by linearizing the residual R_d at solution iterates.
Two other static scenarios, inspecting contact wire sag and transverse stiffness, are routinely examined to evaluate OHE designs. Both problems can be conveniently derived as special cases of eq. (10). To compute the gravity-induced network sag, we seek an equilibrium configuration U ˜_0 with the pantograph disengaged. To this end, we simply set f_c^0 and the pantograph dofs of U ˜_0 to zero in eq. 10 :
█(" Find " U ˜_0∈R^4n×{0,0,0}" such that " KU ˜_0+R_d [U ˜_0 ]=F_"ohe " #(11))
For any reasonable set of design parameters, dropper cables (105) are taut when the pantograph is disengaged. In such a scenario, R_d [U ˜_0 ]=K_d U ˜_0-F_d as noted in eq. (7). Then, eq. (11) simplifies to a linear problem, and its solution defines the contact wire (104b) sag x↦u ˜_c^h (x).
Contact wire (104b) sag results in the pantograph head traveling at a nonuniform elevation. Hence, in addition to the sag, the transverse stiffness of the contact wire (104b) plays a critical role in determining the contact force history. The stiffness of the contact wire (104b) at a location x=x ‾ in response to applying a transverse load F_0 is determined in an incremental sense- as the ratio of F_0 and the contact wire's (104b) uplift at x_0 due to the application of the force. Two aspects of this stiffness are significant. First, it is non-homogeneous along the span - the contact cable (104b) is most compliant midspan and stiffer closer to the registration arm. Second, the stiffness is force-dependent. When subjected to a low transverse force, dropper cables (105) remain in tension and behave linearly. Hence, the contact wire's (104b) stiffness is unaffected by the magnitude of the force, provided that dropper cables (105) remain taut. However, dropper cables (105) may slacken at higher transverse forces, resulting in a more compliant response of the contact wire (104b). Specifically, let U ̅_0 denote the solution of the problem
█(" Find " U ̅_0∈R^4n×{0,0,0}" such that " KU ̅_0+R_d [U ̅_0 ]=F_"ohe " +F_0 [0_(2n×1),├ H┤|_x ‾ ,0_(3×1) ]#(12) )
which yields the contact wire (104b) displacement x↦u ‾_c^h (x). Then, the contact wire's (104b) transverse stiffness at x ‾ follows as
█(k ‾(x ‾)=F_0/(u ‾_c^h (x ‾)-u ˜_c^h (x ‾))#(13))
Note that determining the stiffness distribution along the span for a given force F_0 requires resolving a sequence of static problems eq. (12), one for each choice of x ‾.
Dropper cable (105) length adjustment
OHE design is generally guided by a target sag of the contact wire (104b), i.e., values of the sag {y_i }_(i=1)^(n_d ) at the locations of dropper cables (105) attachments. These typically follow a parabolic distribution. It is possible to concurrently adjust multiple parameters, including the cable (104) tensions and the dropper cables (105) locations, by posing an optimization problem to achieve the desired sag. Instead, an iterative method is proposed, which is described in the above illustrated procedure that achieves a prescribed sag by solely adjusting the set of dropper cables (105) lengths. Starting from a nominal set of dropper cables (105) lengths {l_i^0 }_i, the j-th iteration of said procedure (see below) resolves eq. (11) with the current estimate of dropper cables (105) lengths {l_i^j }_i. Notice that altering dropper cables (105) lengths changes their mass and stiffness, thus requiring recomputing the vectors F_"ohe " and R_d. Dropper cables (105) lengths for the next iteration are updated by the mismatch between the achieved and target sag values. The procedure terminates when the sag is matched within a prescribed tolerance ϵ_"tol " .

Dropper cable (105) slack and intermittent contact
Nonlinearities in the dynamic problem eq. (9) and its static counterparts as discussed stem from dropper cable (105) responses and contact constraints. Dropper cables (105) lose elastic resistance when compressed. The contact force is positive and set-valued when the gap function is zero but reduces to zero when the gap becomes positive, see figure 6b. Both sources of nonlinearity share common features - non-smoothness and an abrupt loss of load-transmission. The latter, in particular, manifests as problems of identifying the set of active dropper cables (105) at each instant and the time intervals of active contact for each pantograph. These requirements transform the prediction into a nonlinear program whose complexity increases combinatorially with the number of dropper cables (105) and pantographs.
Regularization
The main idea to address these challenges consists in regularizing the dependence of the elastic response of a dropper cable (105) on the deflection and the dependence of the contact force on the gap function. The latter is closely related to the penalty and barrier methods in constrained optimization problems.
Dropper cable (105) responses. The two-dof lumped model for dropper cables (105) as discussed previously presumes a force-displacement relationship
█(F_d (δ)=k_d max{0,δ}#(14))
where δ denotes the relative displacement of the ends. Equation (14) represents a linear response with stiffness k_d in tension and slackening of the cable (104) in compression. The relationship is reminiscent of simple damage models, in which the material abruptly loses stiffness when an internal damage parameter attains a critical value. However, an important distinction in the case of eq. (14) is that dropper cable (105) responses are fully reversible. We also note that it is straightforward to revise eq. (14) to induce slackening at a critical compressive load (e.g., the Euler buckling load) if required.
Penalty contact: The EN50318 standards suggests imposing contact constraints between the pantograph and the contact wire (104b) using a penalty method. Adopting a quadratic penalty potential, for instance, yields an approximation of the contact force as
█(f_c^pen (g)=βmax{0,-g}#(15))
where β>0 is a large parameter commonly referred to as a penalty stiffness. As illustrated in figure 6b, the dependence of f_c^pen on the gap g is no longer set-valued, but is piecewise linear nonetheless. It is helpful to observe the similarities between eq. (14) and eq. (15), with the deflection in the former replaced by the gap function in the latter. The contact force in the penalty formulation eq. (15) suffers from a non-differentiable dependence on the gap function, just as dropper forces do on the displacements in eq. (14). Similarly, the loss of load-transmission for positive values of the gap in eq. (15) is analogous to dropper cable (105) slackening under compression in eq. (14).
Smooth approximations of max{0,x}: The regularizations proposed for eqs. (14) and (15) are based on introducing smooth approximations of the function x↦max{0,x}. To this end, a 1-parameter family of smooth functions {ϕ_α }_α for α>0 is considered, defined as
█(ϕ_α (x)=1/2α(αx+log⁡(2cosh⁡(αx)))#(16))
The sequence {ϕ_α }_α approximates max {0,x} pointwise:
█(lim┬(α→∞) ϕ_α (x)=max{0,x}" for " x∈R#(17))
The convergence in eq. (17) can be intuitively justified by the observations
A formal proof follows similar reasoning. The convergence in eq. (17) is, in fact, uniform. In turn, this implies convergence in measure as well, i.e.,
█(∀ε>0,□( ) lim┬(α→∞) μ({x∈R:|ϕ_α (x)-max{0,x}|>ε})=0#(18))where μ is the usual Lebesgue measure over R. Eqs. eqs. (17) and 18) clarify the sense in which {ϕ_α }_α defines a 1-parameter family of smooth approximations of max{0,x}. Furthermore, log⁡(exp⁡(x)+exp⁡(-x))≥x implies ϕ_α≥max{0,x}, showing that ϕ_α approximates max{0,x} from above in eq. (17). Subsequently, we will also use the symmetry
█(ϕ_α (x)-ϕ_α (-x)=x#(19))
reflecting the analogous property of max{0,x}.

Besides eqs. (17) and (18), the choice of ϕ_α can also be justified by examining the approximation of derivatives. The weak derivative of max{0,x} is the unit step function H(x), set to 1 for x≥0 and to 0 otherwise, while
█(ϕ_α^' (x)=(1/2)(1+tanh⁡(αx))#(20))
It can be verified that sequence ϕ_α^' converges to H uniformly, in measure, and from above.
Figure 7 depicts the approximations of max{0,x} and H(x) by ϕ_α and ϕ_α^', respectively. The 1-parameter family of smooth functions {ϕ_α }_α defined in eq. 16 approximating x↦max{0,x} is shown in the Figure 7a. These functions help regularize dropper cable (105) responses δ↦F ˜_d (δ) and contact forces g↦f ˜_c (g) in a unified manner, see eqs. (21) and (22). The consequent approximations of derivatives are shown in the Figure 7b. Reproducing the unit step function with desired accuracy enables accounting for dropper cable (105) slackening and contact constraints in a seamless and non-programmatic manner.
The approximations are systematically improved by choosing progressively larger values of α, as quantified by the observation
∫_(-∞)^∞▒  |ϕ_α (x)-max{0,x}|dx=π^2/(24α^2 )
From eq. (19), we see that ϕ_α^' (x)+ϕ_α^' (-x)=1, thus reproducing the analogous symmetry of H(x).
Regularized residuals and linearization: The properties of ϕ_α motivate replacing the non-smooth relationship δ↦ F_d (δ) in eq. (14) with the smooth approximation
█(F ˜_d (δ)=k_d ϕ_α (δ)#(21))
and g↦f_c^"pen " (g) in eq. (15) by
█(f ˜_c (g)=βϕ_η (-g)=-β(g-ϕ_η (g))#(22))
where we have invoked property eq. (19). For generality, we have used different regularization parameters α and η in eqs. (21) and (22), respectively.
Employing eq. (21) in place of eq. (14) for dropper cable (105) responses yields an approximation of the dropper residuals R_i [U] as
█(R ˜_i [U]=k_i ϕ_α (δ_i ) h_i," where " δ_i=h ̅_i^T U-Δl_i#(23))
Similarly, replacing f_c^h in eq. (7) by eq. (22) yields a residual for the contact force as
█(R_c [U(t)]=β(g_h-ϕ_η (g_h ))r(t)" with " g_h=r^T U#(24))
Noting eqs. (23) and 25 and setting R ˜_d [U]=∑_(i=1)^(n_d ) R_i [U], the dynamic problem eq. (9) is now transformed to that of finding t↦U(t) such that
█(MU ¨(t)+CU ˙(t)+KU(t)+R ˜_d [U(t)]+R ˜_c [U(t)]=F#(25))
subject to initial conditions ├ U┤|_(t=0)=U_0 and ├ U ˙ ┤|_(t=0)=0. The static scenarios analogous to those discussed previously follow from eq. (25). Eq. (25) defines a solution for each triplet of parameters α,β,η>0. When chosen to be sufficiently large, we expect that solutions eq. (25) will approximate those of eq. (9) well.
The regularized problem in eq. (25) provides critical advantages over eq. (9). First, by replacing the piecewise definitions of dropper residuals and contact forces inherent in eq. (9), predicting the result of eq. (25) is no longer programmatic. It is no longer necessary to identify sets of slackened dropper cables (105), or examine whether contact constraints are active. These are determined automatically, with accuracy dictated by the choices of the parameters α and η. Second, eq. (25) is an unconstrained problem- it is no longer appended with contact inequalities. The contact force is determined a posteriori from eq. (22) rather than solved simultaneously with the displacements. Third, the residuals R ˜_c and R ˜_d depend smoothly on U. Consequently, their linearizations required for computing configuration updates in time-stepping schemes follow easily. For the dropper residuals, we have
R ˜_d [U+ΔU]≈R ˜_d [U]+(∑_(i=1)^(n_d)▒  K ˜_i [U])ΔU," where " {■(K ˜_i [U]=k_i ϕ_α^' (δ_i ) h_i h ̅_i^T," and " @δ_i=h ̅_i^T U-Δl_i )┤
while the linearization of the contact residual is given by
R ˜_c [U+ΔU]≈R ˜_c [U]+K ˜_con [U]ΔU," where " K ˜_con [U]=β(1-ϕ_η^' (g_h )) 〖rr〗^T
The smooth dependences of K ˜_d and K ˜_c on U is crucial for optimal convergence of Newton-type procedures to compute configuration updates in time integration schemes. In particular ϕ_α^' approximates H without oscillations around x=0, see figures 7b. This distinctive feature of eq. (19) prevents convergence difficulties in iterative solvers.
Representative results demonstrating performance.
The predictions of the invented method for the case of a pantograph interacting with a simple catenary system have been validated with the EN50318 standards. The comparisons of predictions are illustrated in figure 8, with state-of-the-art codes using available as a set of benchmark studies in the literature. The system parameters for all components, including geometric dimensions and material properties, are chosen to be identical to those in literature. The gray shaded regions in the plots represent the envelope of predictions of the other codes listed previously.
The table shown below reports the nominal dropper cable (105) lengths and the dropper cable (105) lengths optimized using the procedure described for dropper length adjustment. Since the dropper cables (105) arrangement is symmetric along the span, data of just the first five dropper cables (105) are reported.
index nominal
length optimized
length
1 1.017 1.0171
2 0.896 0.8968
3 0.810 0.8100
4 0.758 0.7599
5 0.741 0.7416

Figure 8a shows the deflected configuration of the catenary with the optimized dropper cable (105) lengths. Here, it is to be noticed that the target sag is achieved exactly. Further, the Figures 8b and 8c report the transverse stiffness of the catenary with two different uplift forces, namely, 100N and 200N. For ideal operation, it is desirable that the transverse stiffness be uniform as the pantograph slides along the span of the cable (104). The stiffness is sensitive to the uplift force because dropper cables (105) slacken at higher uplift forces. The Figure 8d reports the deflection of the central three spans of a 10-span catenary as the pantograph slides at a speed of 320 km/hr. The Figure 8e reports the time history of the contact force of the central three spans of a 10-span catenary as the pantograph slides at a speed of 320 km/hr. Observe that although the contact force oscillates, it never decreases to zero. Hence, the prediction correctly shows that the pantograph never loses contact with the catenary.
EQUIVALENTS

With respect to the use of substantially any plural and/or singular terms herein, those having skill in the art can translate from the plural to the singular and/or from the singular to the plural as is appropriate to the context and/or application. The various singular/plural permutations may be expressly set forth herein for sake of clarity.

It will be understood by those within the art that, in general, terms used herein, and especially in the appended claims (e.g., bodies of the appended claims) are generally intended as “open” terms (e.g., the term “including” should be interpreted as “including but not limited to,” the term “having” should be interpreted as “having at least,” the term “includes” should be interpreted as “includes but is not limited to,” etc.). It will be further understood by those within the art that if a specific number of an introduced claim recitation is intended, such an intent will be explicitly recited in the claim, and in the absence of such recitation no such intent is present. For example, as an aid to understanding, the following appended claims may contain usage of the introductory phrases “at least one” and “one or more” to introduce claim recitations. However, the use of such phrases should not be construed to imply that the introduction of a claim recitation by the indefinite articles “a” or “an” limits any particular claim containing such introduced claim recitation to inventions containing only one such recitation, even when the same claim includes the introductory phrases “one or more” or “at least one” and indefinite articles such as “a” or “an” (e.g., “a” and/or “an” should typically be interpreted to mean “at least one” or “one or more”); the same holds true for the use of definite articles used to introduce claim recitations. In addition, even if a specific number of an introduced claim recitation is explicitly recited, those skilled in the art will recognize that such recitation should typically be interpreted to mean at least the recited number (e.g., the bare recitation of “two recitations,” without other modifiers, typically means at least two recitations, or two or more recitations). Furthermore, in those instances where a convention analogous to “at least one of A, B, and C, etc.” is used, in general such a construction is intended in the sense one having skill in the art would understand the convention (e.g., “a system having at least one of A, B, and C” would include but not be limited to systems that have A alone, B alone, C alone, A and B together, A and C together, B and C together, and/or A, B, and C together, etc.). In those instances where a convention analogous to “at least one of A, B, or C, etc.” is used, in general such a construction is intended in the sense one having skill in the art would understand the convention (e.g., “a system having at least one of A, B, or C” would include but not be limited to systems that have A alone, B alone, C alone, A and B together, A and C together, B and C together, and/or A, B, and C together, etc.). It will be further understood by those within the art that virtually any disjunctive word and/or phrase presenting two or more alternative terms, whether in the description, claims, or drawings, should be understood to contemplate the possibilities of including one of the terms, either of the terms, or both terms. For example, the phrase “A or B” will be understood to include the possibilities of “A” or “B” or “A and B.”

In addition, where features or aspects of the disclosure are described in terms of Markush groups, those skilled in the art will recognize that the disclosure is also thereby described in terms of any individual member or subgroup of members of the Markush group.

While various aspects and embodiments have been disclosed herein, other aspects and embodiments will be apparent to those skilled in the art. The various aspects and embodiments disclosed herein are for purposes of illustration and are not intended to be limiting, with the true scope being indicated by the following claims.
 

REFERENCE NUMERALS
Particulars Referral numeral
Pantograph current collector 100
Overhead cable 101
Processor 102
First model 103a
Second model 103b
Third model 103c
Fourth model 103d
Fifth model 103e
Cables 104
Messenger cable 104a
Contact cable 104b
Dropper cables 105
Memory 106
Method 200
System 300 , Claims:1. A method (200) for determining nonlinear static and dynamic mechanical interactions between a pantograph current collector (100) and an overhead cable (101) network of an electrified railway line, the method (200) comprising:
generating, by a processor (102), a first model (103a) for a plurality of cables (104) of the overhead cable (101) network, the plurality of cables (104) including a messenger cable (104a) and a contact cable (104b);
generating, by the processor (102), a second model (103b) of a plurality of dropper cables (105), wherein each of the plurality of dropper cables (105) having a length adjusted to complement a target sag value of the contact cable (104b), and wherein the plurality of dropper cables (105) being modelled by employing a nonlinear force-deflection response configured to represent slackening of the plurality of dropper cables (105), upon being subjected to compression;
generating, by the processor (102), a third model (103c) for support structures supporting the overhead cable (101) network, the support structures including one or more registration arms configured to support the contact cable (104b) at ends of each span;
generating, by a processor (102), a fourth model (103d) for the pantograph current collector (100), the fourth model (103d) including one or more of a first vertical spring, a first vertical damper, a first mass, and a first uplift force;
defining, by the processor (102), a fifth model (103e) for mechanical contact between the pantograph current collector (100) and the contact cable (104b), to determine intermittent unilateral contact interactions;
determining, by the processor (102), nonlinear static and dynamic mechanical interactions between the pantograph current collector (100) and the overhead cable (101) network.

2. The method (200) as claimed in claim 1, wherein generating the first model (103a) of the plurality of cables (104) of the overhead cable (101) network comprises:
defining, material and geometric parameters of the plurality of cables (104);
defining, mechanical behaviour of the plurality of cables (104), wherein the plurality of cables (104) are modelled as Euler-Bernoulli beams with inertia, elastic, and damping properties are being set in accordance with the defined material and geometric parameters;
incorporating, in the first model (103a), characteristics of cable (104) pretensions in the plurality of cables (104); and
incorporating, in the first model (103a), characteristics of forces exerted by gravity, and characteristics of interactions with the plurality dropper cables (105), the pantograph current collector (100), and the support structures.
3. The method (200) as claimed in claim 1, wherein generating the second model (103b) of the plurality of dropper cables (105), comprises:
defining, by the processor (102), a nominal length for each dropper cable (105) of the plurality of dropper cables (105) in the span;
defining, by the processor (102), attachment points along the span, for each dropper cable (105) of the plurality of dropper cables (105) to be attached with the messenger cable (104a) and the contact cable (104b);
defining, a target sag value of the contact cable (104b) at each attachment point of the plurality of dropper cables (105);
determining, an actual sag value of the contact cable (104b) at each attachment point of the plurality of dropper cables (105);
adjusting iteratively, the nominal length of each of the plurality of dropper cables (105), corresponding to the difference between the target sag value and the actual sag value of the contact cable (104b) at the attachment point; and
determining, an adjusted length for each of the plurality of dropper cables (105), wherein the actual sag value of the contact cable (104b) for said adjusted length corresponds with the target sag value of the contact cable (104b), within a predefined tolerance at each attachment point of the plurality of dropper cables (105).

4. The method (200) as claimed in 3, wherein generating the second model (103b) of the plurality of dropper cables (105), comprises:
defining, material and geometric parameters of the plurality of dropper cables (105) with the adjusted length;
determining, inertial and damping properties of the plurality of dropper cables (105) based on the defined material and geometric parameters;
defining, a continuously-differentiable nonlinear relationship between a change in length of each cable (104) of the plurality of dropper cables (105) and an elastic force response;
adjusting, parameters of the continuously-differentiable nonlinear relationship defining the elastic force response to be within a predefined tolerance, for a linear response in tension; and
adjusting, parameters of the continuously-differentiable nonlinear relationship defining the elastic force response to be within a predefined tolerance, for slackening of the cable (104) in compression; and
incorporating, in the second model (103b), characteristics of forces exerted by gravity and by interactions with the messenger cable (104a) and the contact cable (104b).

5. The method (200) as claimed in claim 1, wherein generating the third model (103c) for support structures supporting the overhead cable (101) network, comprises:
defining, material and geometric parameters of the one or more registration arms;
defining, a linear relationship between deflection of the registration arm and corresponding force response of the one or more registration arm;
determining, inertial, elastic, and damping properties of the one or more registration arm, based on the defined material and geometric parameters; and
incorporating, in the third model (103c), characteristics of forces exerted by the contact cable (104b) on the one or more registration arms.

6. The method (200) as claimed in claim 1, wherein generating the fourth model (103d) of the pantograph current collector (100) comprises:
defining, a multi-degree-of-freedom and a multi-level spring-mass-dashpot representation of the pantograph current collector (100);
defining, a plurality of uplift forces, inertias, spring stiffnesses and damping parameters; and
incorporating, in the fourth model (103d), characteristics of contact forces exerted by the contact cable (104b) on a head of the pantograph current collector (100).
7. The method (200) as claimed in claim 1, wherein defining the fifth model (103e) for mechanical contact between the contact cable (104b) and the pantograph current collector (100) comprises:
defining, a gap function to represent a standoff distance between the head of the pantograph current collector (100) and the contact cable (104b);
defining, a nonlinear, continuously-differentiable relationship between a contact force and the gap function, the contact force including force exerted by the head of the pantograph current collector (100) on the contact cable (104b); and
determining, parameters of the nonlinear, continuously-differentiable relationship between the gap function and the contact force, the parameters configured to maintain unilateral contact constraints between the head of the pantograph current collector (100) and the contact cable (104b) within a predefined tolerance.

8. A system (300) for determining nonlinear static and dynamic mechanical interactions between a pantograph current collector (100) and an overhead cable (101) network of an electrified railway line, the system (300) comprising:
a processor (102) and a memory (106), wherein the processor (102) is configured to:
generate a first model (103a) for a plurality of cables (104) of the overhead cable (101) network, the plurality of cables (104) including a messenger cable (104a) and a contact cable (104b);
generate a second model (103b) of a plurality of dropper cables (105), wherein each of the plurality of dropper cables (105) having a length adjusted to complement a target sag value of the contact cable (104b), and wherein the plurality of dropper cables (105) being modelled by employing a nonlinear force-deflection response configured to represent slackening of the plurality of dropper cables (105), upon being subjected to compression;
generate a third model (103c) for support structures supporting the overhead cable (101) network, the support structures including one or more registration arms configured to support the contact cable (104b) at ends of each span;
generate a fourth model (103d) for the pantograph current collector (100), the fourth model (103d) including one or more of a first vertical spring, a first vertical damper, a first mass, and a first uplift force;
define a fifth model (103e) for mechanical contact between the pantograph current collector (100) and the contact cable (104b), to determine intermittent unilateral contact interactions;
determine nonlinear static and dynamic mechanical interactions between the pantograph current collector (100) and the overhead cable (101) network.

9. The system (300) as claimed in claim 8, wherein the processor (102) generating the first model (103a) of the plurality of cables (104) of the overhead cable (101) network, is configured to:
define material and geometric parameters of the plurality of cables (104);
define mechanical behaviour of the plurality of cables (104), wherein the plurality of cables (104) are modelled as Euler-Bernoulli beams with inertia, elastic, and damping properties are being set in accordance with the defined material and geometric parameters;
incorporate, in the first model (103a), characteristics of cable (104) pretensions in the plurality of cables (104); and
incorporate, in the first model (103a), characteristics of forces exerted by gravity and by characteristics of interactions of the plurality of dropper cables (105), the pantograph current collector (100), and the support structures.
10. The system (300) as claimed in claim 8, wherein the processor (102) generating the second model (103b) of the plurality of dropper cables (105), is configured to:
define a nominal length for each dropper cable (105) of the plurality of dropper cables (105) in the span;
define attachment points along the span, for each dropper cable (105) of the plurality of dropper cables (105) to be attached with the messenger cable (104a) and the contact cable (104b);
define a target sag value of the contact cable (104b) at each attachment point of the plurality of dropper cables (105);
determine an actual sag value of the contact cable (104b) at each attachment point of the plurality of dropper cables (105);
adjust iteratively, the nominal length of each of the plurality of dropper cables (105), corresponding to the difference between the target sag value and the actual sag value of the contact cable (104b) at the attachment point; and
determine an adjusted length for each of the plurality of dropper cables (105), wherein the actual sag value of the contact cable (104b) for said adjusted length corresponds with the target sag value of the contact cable (104b), within a predefined tolerance at each attachment point of the plurality of dropper cables (105).

11. The system (300) as claimed in claim 10, wherein the processor (102) generating the second model (103b) of the plurality of dropper cables (105), is configured to:
define material and geometric parameters of the plurality of dropper cables (105) with the adjusted length;
determine inertial and damping properties of the plurality of dropper cables (105) based on the defined material and geometric parameters;
define a continuously-differentiable nonlinear relationship between a change in length of each cable (104) of the plurality of dropper cables (105) and an elastic force response;
adjust parameters of the continuously-differentiable nonlinear relationship defining the elastic force response to be within a predefined tolerance, for a linear response in tension; and
adjust parameters of the continuously-differentiable nonlinear relationship defining the elastic force response to be within a predefined tolerance, for slackening of the cable (104) in compression; and
incorporate in the second model (103b), characteristics of forces exerted by gravity and by interactions with the messenger cable (104a) and the contact cable (104b).

12. The system (300) as claimed in claim 8, wherein the processor (102) generating the third model (103c) for support structures supporting the overhead cable (101) network, is configured to:
define material and geometric parameters of the one or more registration arms;
define a linear relationship between deflection of the registration arm and corresponding force response of the one or more registration arm;
determine inertial, elastic, and damping properties of the one or more registration arm, based on the defined material and geometric parameters; and
incorporate in the third model (103c), characteristics of forces exerted by the contact cable (104b) on the one or more registration arms.
13. The system (300) as claimed in claim 8, wherein the processor (102) generating the fourth model (103d) of the pantograph current collector (100), is configured to:
define a multi-degree-of-freedom and a multi-level spring-mass-dashpot representation of the pantograph current collector (100);
define a plurality of uplift forces, inertias, spring stiffnesses and damping parameters; and
incorporate, in the fourth model (103d), characteristics of contact forces exerted by the contact cable (104b) on a head of the pantograph current collector (100).
14. The system (300) as claimed in claim 8, wherein the processor (102) defining the fifth model (103e) for mechanical contact between the contact cable (104b) and the pantograph current collector (100), is configured to:
define a gap function to represent a standoff distance between the head of the pantograph current collector (100) and the contact cable (104b); and
define a nonlinear, continuously-differentiable relationship between a contact force and the gap function, the contact force including force exerted by the head of the pantograph current collector (100) on the contact cable (104b).