Consideration of the influence of supports in modeling the electromagnetic fields of 25 kV traction networks under emergency conditions

0 Introduction

Electromagnetic fields (EMFs) are among the main factors that determine the electromagnetic safety conditions at electric power facilities,including transport [1,2].They can generate interference that disrupts the normal functioning of electrical and electronic devices [2],cause the ignition of flammable substances,and lead to serious accidents involving personnel working on disconnected power and communication lines,owing to the effect of induced voltages.

EMFs also have an adverse effect on humans [1],which inhibits processes in the central nervous system and causes headaches,lethargy,and fatigue.Changes in the blood composition and pressure and an increase in heart rate can also occur.Capacitance currents at high EMF intensities can change metabolic processes.Power frequency fields have a considerable influence on people because with an increase in frequency,there is an effect of inertia in the opening of cell membranes,and with its decrease,induced and capacitance currents decrease.

Traction networks (TNs) of electrified AC railways are among serious EMF sources owing to electromagnetic imbalance.The electromagnetic interference of the TN causes a large voltage in adjacent devices,which can cause severe equipment damage and electrical-related injury.

Many studies have focused on modeling the EMFs of power transmission lines and TNs.COMSOL Multiphysics software modules were used to calculate the EMFs generated near high-voltage power lines [3].The focus of[4] was the analysis of the EMFs along the route of a highvoltage transmission line and the results of the analysis of the influence of key variables on the EMF level.The findings of a study on the EMF between a power line and railway are presented in [5].In [6],the authors addressed the EMFs of 132 kV transmission lines,which were calculated using the Biot-Savart law and Maxwell’s equations.The superposition method was used to simplify the calculation of the magnetic field.In [7],the magnitude of the EMF intensity is shown to depend on the distance between the supports.Additionally,the results of theoretical studies related to the absorption of electromagnetic energy and an evaluation of the effectiveness of measures to protect personnel from EMF effects are presented.This article [8]presents data that characterize the EMF levels under 500 kV lines.The simulation was performed for horizontal and vertical conductor arrangements under balanced and unbalanced conditions,respectively.The results of the EMF distribution analysis for high-voltage substations are provided in [9].The authors of [10] present a model for examining the power-frequency EMF created by a highvoltage transmission line.The distribution of the electric field along the route of a 400 kV power transmission line was analyzed in [11].The influence of the electric field caused by high-voltage power lines on humans was assessed in [12].The results of modeling the electric field of 330 kV power transmission lines located near buildings are presented in [13].In [14],a technique was developed to model the EMF of a TN near a catenary system and rails.A method for calculating the low-frequency EMF around 15 kV power lines is discussed in [15].The calculation results for the EMFs of overhead transmission lines are provided in[16].The EMFs around the power lines of various designs are compared in [17].The findings of research on the EMF at the Xijiang traction substation are given in [18].The effects of the EMF produced by an electrified railway section are analyzed in [19].Methods for predicting EMFs in high-voltage substations based on fuzzy models are described in [20].The EMFs of substations are analyzed in [21,22].The problems related to electromagnetic compatibility and safety on the routes of electrified railways are considered in [23] and [24].The EMFs produced by high-speed transport systems are analyzed in [25].The aspects of modeling and measuring the parameters that determine the conditions for electromagnetic compatibility and safety at railway electrical substations are examined in [26].The results of the simulation and analysis of the electromagnetic environment of TNs are presented in [27].

The analysis of the described publications suggests that they address critical aspects of determining the EMF produced by power lines and TNs,and analyze electromagnetic safety conditions.However,the discussed publications do not provide a method for modeling the EMF near the metal supports of catenary systems under emergency conditions.Such a method can be implemented based on the algorithms given in [28] and the Fazonord software [29].

The methodology and algorithms for determining the EMF created by 25 kV TNs under emergency conditions are presented in [30].Based on the obtained results,it was concluded that the currents of the catenary system during a short circuit are approximately five times greater than the currents of the load,and the magnetic field intensities maintained the same ratio.Importantly,this conclusion is valid for a plane-parallel field created by currents that flow through wires and rails.Near the support on which a short circuit occurred,the situation is much more complicated owing to the influence of currents that flow through the support.The field becomes three-dimensional and its calculation becomes much more challenging.The findings of studies that address this problem are presented below.

1 Problem statement

Despite the large number of studies concerned with modeling the EMF of TNs [5,14,19,25] and power lines[3,4,6-17],the problem of determining three-dimensional electromagnetic fields near metal supports when a shortcircuit current flows through them remains unsolved.In this case,the field has a complex spatial structure that greatly complicates intensity calculations.Below are the methods,algorithms,software,and digital models proposed to determine the EMF in the described emergency scenarios.During modeling,the objects were represented by segments of thin wires [28] to examine the distribution of the electric charge and calculate the intensities of the electric and magnetic fields.This approach was implemented in the Fazonord software [29].

There are two main types of emergency conditions in 25 kV TNs: contact wire-to-rail short circuits when the supports are grounded to the rail track;and contact wire-to-ground short circuits in sections with supports disconnected from the rails.Grounding the supports of the contact network on the rail track reduces the reliability of track circuits and complicates track maintenance without removing the voltage because of the need to disconnect the grounding leads from the rails [30,31].If the spreading resistance of the support foundation is less than 100 Ω,the rail should be connected through an earth arrester,which requires a large amount of work to check and replace faulty arresters.Reference [31] showed that for the majority of practically significant conditions,the probability of hazardous situations in sections with supports disconnected from the rails was considerably lower than the same indicator for sections with supports connected to them.

A three-dimensional electromagnetic field near a metal support with a short-circuit current flowing through it is characterized by a complex spatial structure,which significantly complicates the intensity calculations.The EMF near the supports under emergency conditions can be determined using the approach proposed in [28] and implemented using the Fazonord software [29].In this case,the current-carrying parts are represented by segments of thin wires to calculate the distribution of the electric charge and the intensities of the electric and magnetic fields.

The proposed approach has the following features that distinguish it from the aforementioned methods designed to determine the EMFs of power transmission lines and TNs.

• EMFs can be calculated for TNs of various designs,for example,25,2 × 25 kV,TNs equipped with shielding wires,line feeders,and suction transformers.

• The system approach to EMF modeling is implemented because EMF determination involves calculating the operating parameters of a complex electric power system or traction power supply system.

• This method allows for the determination of the technical efficiency of devices used to reduce EMF levels,such as shielding wires.

Below is a brief description of the algorithms

2 Modeling method

The method proposed for modeling the EMFs created by current-carrying parts of finite length is based on algorithms for determining the EMF levels described in [2].The main difference from the methods of analyzing electromagnetic safety conditions [3,27] determined by the EMF levels is that these parameters are calculated based on the operating parameters of complex systems in the phase coordinates [29,32,35-38].

Digital models for determining EMF have the form of a system of nonlinear equations

where F represents a nonlinear vector function,and S, V are the adjustable and non-adjustable operating parameters,respectively.

The parameters S found from these equations are used to determine the horizontal and vertical electric and magnetic field intensities [2] for a given set of spatial coordinates х,y;additionally,the amplitude values are calculated:

The amplitudes of the EMF intensity were determined using the method proposed in [2] as follows:

This approach provides seamless information exchange,which is crucial in the implementation of cyberphysical systems for traction power supplies [36].The procedure provides an adequate determination of the EMF generated by the electric rolling stock and power lines of various designs,which is confirmed by comparison with instrumental measurements.The data for such a comparison are presented in [2],which shows that the maximum intensity values are almost equal to the calculated values.In addition,the adequacy of the model was verified by the high values of the correlation coefficients between the experimental and calculated data (Fig.1).The following section outlines the development of this procedure for modeling the EMF generated by conductors of a limited length.

Fig.1 Calculated values and experimental measurements of the electric field level of the catenary system

The EMF of short wires can be modeled by representing such objects as a series of parts connected successively to calculate the electric charge distribution and further determine the intensities of the electric and magnetic fields[28,37,38].First,it is necessary to calculate the power flow of the electrical system as it determines the current and voltage of the wires.

Modeling is based on the following main propositions:

• The objects under consideration are segments of straight thin wires arbitrarily located in space.

• Some current-carrying parts (e.g.,cables) can be buried.

• The size of the set of objects must be limited to operate with the concepts of electrical circuits and the equations of the quasi-stationary zone.For a frequency of 50 Hz and significant harmonics,these dimensions should not exceed the first hundreds of meters.

A careful analysis of the problem raises the difficult question of whether it is possible to use the concepts of self-inductance and the mutual inductance of short-wire segments.These concepts imply the presence of loops that correspond to the magnetic fluxes.The problem of determining such loops for short segments is not trivial because the magnetic field in this situation ceases to be plane-parallel.However,mutual inductive effects can be excluded from consideration for current-carrying parts with lengths in the order of several tens of meters.

This approach is justified as follows: The voltage induced by mutually inductive couplings can be estimated using mathematical expressions [34,35] for long and parallel wires.If we consider extreme cases with singlephase short-circuit currents reaching 50 kA,with distances between the wires of 20 m,and a length of the affected wire of 50 m,then the induced electromotive force of magnetic influence is 610 V.This value conservatively places upper limits on the possible induced voltage of the magnetic influence;a decrease in the influencing current and length of the affected wire proportionally reduces the induced electromotive force.The induced voltage at operating currents in the order of 1 kA is limited from above by a value of approximately 12 V,which makes it possible not to factor in mutually inductive couplings between individual short wires.Symmetrical short circuits and edge effects of short current-carrying parts lead to much lower values of the induced voltage.Under normal load conditions,the effects of mutually inductive coupling can be neglected.There was also no induced voltage in the case of a mutually perpendicular arrangement of the wires.This means that the potential of grounded objects in the absence of working or emergency currents can be considered to be zero;nevertheless,these objects determine the structure of the electric field.

Following the logic of the Fazonord software [29],the EMF calculation for short wires assumes that the result of the power flow calculation for the system to which they belong is known.Because of the small influence of short wires on power flow,the calculation of the latter involves modeling similar to that of long current-carrying parts.The voltage and current of the short wires were determined by calculating the power flow,which are necessary to determine the levels of the electric and magnetic fields.The calculation formulas are applied further,and the algorithm for determining the EMF levels is as follows:

Fig.1 shows the coordinate system and a single short wire.The location of the X0Z plane was chosen such that it coincides with the plane of Earth’s surface.The current system assumes Nw short wires,each of which operates or emergency current flows.In addition,earthed conductive objects without current are considered.Each short wire i has a length L and is divided into a number ni of line elements,each of which has a length Δli=Li/ni.In some cases,the length of a line element is denoted by Δlij,with index j indicating the location of the element on wire i;the numbering of the line elements starts with unity from the beginning of wire 1,as shown in Fig.2.The magnitude Δlij does not depend on j.

Fig.2 Calculating the electric field of a line element of the wire

At the observation point М,having coordinates (x, y, z),wire i creates potential determined by the formula:

where rij denotes the distance from the middle of segment Δlij to the observation point,denotes the distance from the middle of the mirror image of line element Δlij to the observation point,represents the charge complex of line element j of wire i,and nI indicates the number of line elements in the wire.When choosing the equivalent charges in the form of point charges,the values of ri j andare determined using the given coordinates of the beginning xi1,yi1,zi1 and end xi2,yi2,zi2 of the short wire:

The choice of an equivalent charge located on the wire axis is more efficient than the representation using point charges.The potential created at the observation point by the axis of the line element j with length Δli of wire i with charge densityand the reflection of the line element in the ground (Fig.3) is equal to

Fig.3 Scheme of the contribution of the line element Δlij to the potential of the segment Δlkl

wheredenote the distances from the beginning and end of line element j and from the beginning and end of its reflection to observation point M,defined as follows:

The system of equations can be derived using the method of equivalent charges to calculate the complex valueof wire i of line element j:

Potential coefficients are calculated using the following formula:

where αij ≠ αji can be attributed to the possible differences in the lengths of the line elements of different wires.

With line elements j and l located on different wires with numbers i and k,the distances are determined by the following expressions:

In the event that line elements are located within the same wire,j ≠l,and the index of the element,near which the observation point M is located on the surface of the wire (Fig.4),is indicated by symbol l,and then,assuming the proximity of point M to the wire axis,at a line element length equal to at least several wire radii,we can obtain the following expression:

Fig.4 Line elements within one wire

When observation point M is located within the same wire,the contribution determined by the reflection charges can be calculated using the point-charge formula:

If j =l (Fig.5),then (6) is used,and the distances within a line element are determined according to the following expression:

Fig.5 Determining self-potential coefficient

where ri denotes the radius of wire i.

The magnetic field intensity of the short wire system(Fig.6) was calculated using the Biot-Savart formula after calculating the operating parameters:

Fig.6 Magnetic field of a line element of the wire

The positive current direction in (10) corresponds to the direction from start node 1 to end node 2.The vector product has the following projections onto the coordinate axis:

However,it is easier to calculate the magnetic field of a short wire using the following formula (Fig.7):

Fig.7 Magnetic field of the short wire

To correctly determine the signs,the cosines of the angles α1 and α2 are calculated using the scalar products of the vectors:

The coordinates of the beginning (xα,yα,zα) of the perpendicular ri are determined by the equations of a straight line that passes through the beginning and end of the wire and a plane perpendicular to this line passing through the observation point M with coordinates (x,y,z).

The coordinates of the point where the wire is divided perpendicularly from the observation point at x21 ≠0 are

There can be the following options:

Once the total values of the field intensity complexes are calculated using (9) and (16),the projections on the coordinate axes can be determined [28].In particular,for an electric field,

The square of the instantaneous value is

Extreme points E2(t) are determined by the zeros of the derivative

The distinguishing feature of the described computer technology for modeling the EMF of short wires in comparison with the calculations of fields of multiwire systems is that all single-wire objects in the calculated scheme constitute a single group of components that create the field.The EMF is determined based on calculations of the operating parameters of the electrical system or TN.

The developed technology was implemented in the Fazonord software to determine EMFs at objects that include live parts of limited length,such as metal supports.

3 Modeling results

The Fazonord software was used for modeling.EMF was calculated for two cases:

1) Short circuit of the contact wire to the rail;

2) Short circuit through the self-grounding resistance of the support.

Fig.8 shows the model diagrams for modeling threedimensional EMFs for the first and second cases.The diagrams include models of the following components: a 220 kV power transmission line,two traction substations with 40 MVA transformers,and a TN of the intersubstation zone with a length of 50 km.Additionally,models of the support of a catenary system and a system of short wires that correspond to the contact wire and rails are presented to model the EMF according to the method proposed in [28].The conductivity of Earth was considered as 0.01 S/m.The support was represented by four rods 10 m high with the following x and z coordinates: -5.5 m,-0.15 m;-5.2 m,-0.15 m;-5.5 m,0.15 m;-5.2 m;0.15 m (Table.1).Selfgrounding resistance RS of a metal support is taken as 20 Ω.

Table 1 Coordinates of the location of the support rods

Fig.8 Model diagram for a contact wire-to-rail short circuit

The modeling yielded the х-and z-coordinate dependences of the amplitudes for the electric and magnetic fields at a height of 1.8 m for contact wire-to-rail short circuit (Figs.9-12),and for short circuit through the selfgrounding resistance of the support (Figs.13-16).The spatial structure of the distribution of these parameters is shown in Fig.17 for the contact wire-to-rail short circuit and in Fig.18 for the short circuit through the self-grounding resistance of the support.

Fig.9 Magnetic field amplitude Hmax(х) at a height of 1.8 min the case of contact wire-to-rail short circuit.A variation range of х: -10…10 m;at: I -z=0 m,II -z=4 m,III -z=8 m

Fig.10 Magnetic field amplitude Hmax(z) at a height of 1.8 m in the case of contact wire-to-rail short circuit.A variation range of z: -4 ...6 m;at: I -x=-5 m,II -x=-4 m,III -x =-3 m,IV -x=-7 m,V -x=-6 m

Fig.11 Electric field amplitude Emax(х) at a height of 1.8 m in the case of contact wire-to-rail short circuit.A variation range of x: -15…15 m;at: I-z =0 m,II-z =8 m,III-z =16 m,IV -z =20 m

Fig.12 Electric field amplitude Emax(z) at a height of 1.8 m in the case of contact wire-to-rail short circuit.A variation range of z: -3…3 m;at: I-x=-5 m,II-x=-7 m,III -x=-6 m

Fig.13 Electric field amplitude Emax(х) at a height of 1.8 m for a short circuit through the self-grounding resistance of the support.A variation range of х: -15…15 m;at: I -z=0 m,II -z =2 m,III -z=4 m,IV -z=8 m

Fig.14 Magnetic field amplitude Hmax(x) at a height of 1.8 m for a short circuit through the self-grounding resistance of the support.A
variation range of х: -10…10 m;at: I -z =0 m,II -z=4 m

Fig.15 Electric field amplitude Emax(z) at a height of 1.8 m for a short circuit through the self-grounding resistance of the support.A variation range of z: -5…5 m;at: I -x=-7 m,II - x=-6 m,III - x=-5 m,IV -x=-4 m,V - x=-3 m

Fig.16 Magnetic field amplitude Hmax(z) at a height of 1.8 m for a short circuit through the self-grounding resistance of the support.A variation range of x: -5…5 m;I -x =-7 m,II - x=-6 m,III - x=-5 m,IV -x=-4 m,V -x=-3 m

Fig.17 Spatial structure of intensity distribution for the electric (a) and magnetic (b) fields in the case of the contact wire-to-rail short circuit

Fig.18 Spatial structure of intensity distribution for the electric (a) and magnetic (b) fields in the case of a short circuit through the self-grounding resistance of the support

Fig.9 Magnetic field amplitude Hmax(х) at a height of 1.8 m in the case of contact wire-to-rail short circuit.A variation range of х: -10…10 m;at: I -z=0 m,II -z=4 m,III -z=8 m

4 Discussion of modeling results

The data that capture the modeling results are shown in the findings of the research indicate that:

-The maximum value of the magnetic field amplitude in the case of contact wire short circuit to the rail reaches 2.3 kA/m near the support,which can cause the malfunction of electronic equipment;the same parameter for the electric field is 0.84 kV/m and does not exceed the allowable values;

-The three-dimensional EMF in the case of a contactwire short circuit through the support to the rail and a short circuit through the self-grounding resistance of the support has a complex spatial structure.

There is a significant increase in the EMF level near the support;for example,the magnetic field intensity in the case of a short circuit through the self-grounding resistance of the support reaches 683 A/m,and the electric field intensity for the same short circuit is 27 kV/m.

As the distance from the support increased,the intensity levels decreased rapidly.

To verify the adequacy of the model,EMFs were calculated without considering the support of the two options.The method described above was first used to calculate the EMF of the short conductors.The second option suggests the use of repeatedly tested algorithms for modeling plane-parallel EMFs,which were validated by comparing with the results of instrumental measurements[2].Discrepancies in the amplitudes that did not exceed 3%indicate the adequacy of the method for determining the EMF of conductors with a limited length.

5 Conclusions

This paper presented an approach developed to correctly determine the influence of supports in modeling the electromagnetic fields of TNs of mainline railways.Its main feature is that all the single-wire objects in the model diagram (support rods and a system of short wires that correspond to the contact wires of the catenary system and rails) constitute a single group of components that create a field,and the EMF is determined based on the calculation of the operating parameters in the phase coordinates.

This approach was implemented using the commercial software Fazonord.It is universal and can be used to determine the EMFs at any electrical power facility,including current-carrying parts of limited length.It can be used to

• Determine the EMFs near the supports of overhead power lines and TNs of various designs.

• Calculate the EMF intensities at substations.

• Consider the influence of metal structures located near TNs,such as pedestrian crossings at railway stations.

The proposed approach differs significantly from the methods used for determining the EMFs of power transmission lines and TNs because it allows the calculation of EMFs for TNs.Power transmission lines of various designs rely on the systems approach to EMF modeling,making it possible to determine the EMF levels based on the calculation of the operating parameters of a complex electric network,and enables the evaluation of the technical performance of devices used to reduce intensities.

Owing to the difficulty of organizing EMF intensity measurements under emergency conditions during short circuits of contact wires to the rail or ground,the adequacy of the model was verified via an indirect method,that is,by comparing the results of EMF calculations based on the method using the concept of conductors with limited length and the method using repeatedly tested algorithms for calculating EMF in a plane-parallel setting.The discrepancy in the results amounted to a few percentages,indicating the adequacy of the results presented in this study.

Currently,research is being conducted to develop a proposed methodology to calculate the EMF intensities for traction substations of AC railways.

Acknowledgments

The research was carried out within the framework of the State Assignment “Conducting applied scientific research”on the topic “Development of methods,algorithms,and software for modeling the operating conditions of traction power supply systems for DC.”

Declaration of Competing Interest

We declare that we have no conflict of interest.