Transactions on the Built Environment vol 18, 1996 WIT Press, ISSN

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1 Modelling electromagnetic fields in DC traction systems using three-dimensional finite-element analysis RJ. Hill, S. Brillante, P.J. Leonard School of Electronic and Electrical Engineering, University of Bath, Down, ##& &42 7,47, Abstract The use of the finite-element (FE) method to calculate the potentials and fields surrounding the track in railway traction systems has concentrated on twodimensional modelling exploiting longitudinal track symmetry. This paper introduces a three-dimensional FE track model, involving solution of the diffusion equation to calculate ground flow in a DC traction system. It is shown that difficulties concerned with data accuracy, numerical convergence and verification need to be solved. 1 Introduction The problem addressed in this paper is the development of modelling for the calculation of electromagnetic (EM) fields around railway track using threedimensional (3D) finite elements (FE). The main difficulties are in reducing the model complexity which results from the removal of longitudinal uniformity and the inclusion of conduction s within the ground. Knowledge of the EM fields around railway track is required to: calculate remote magnetic potentials, fields and electric stresses assess induced voltages or corrosion s on parallel pipes and cables determine the screening effect of metal structures such as bridges derive equivalent circuit parameters for track and traction line modelling. A 3D FE track model represents a more elegant solution compared to both 2D FE and existing analytical bulk parameter models, which cannot deal with complex geometries and the effect of the presence of trains on the track. Specific applications of 3D modelling are concerned with "real-world" conditions and their objective is to improve on the simple analytical models used by railway engineers which are accurate only to within about ±15%. A main

2 424 Computers in Railways motivation is to enhance safety by gaining an understanding of the conditions of interaction between traction and track circuit s. In this respect, there is a specific need to model: signalling track circuit termination areas, such as around blockjoints and electrically tuned separation joints, to quantify longitudinal feedthrough and lateral crosstalk interference between track circuits earthing arrangements near power substations, to determine ground flow between grounding grids and rail connections field distortion near critical points such as track cross bonding. The novelty of the work relates to the provision of a numerate, rational approach for problems involved with the calculation of 3D electromagnetic fields in traction systems. The paper commences by reviewing existing 2D FE work. This is followed by a report on progress in setting up a 3D model of a DC traction system. Emphasis is given on measures to improve model correctness with a discussion of verification issues. 2 Review of two-dimensional FE track modelling FE can model systems comprised of bulk media with variable electrical material properties, and has proved very useful on linear and nonlinear systems with transient and steady DC or AC excitation. The longitudinal symmetry of railroad track has encouraged the development of 2D FE modelling for a wide variety of applications concerned with potential and field values, coupling coefficients and equivalent impedances. Examples include assessment of crosstalk between parallel track circuits [1] and analytical modelling to obtain equivalent track transmission line parameters required in electric circuit modelling [2]. 2.1 Principle of finite elements and user procedure FE packages solve an appropriate set of Maxwell's equations over a problem space by calculating the scalar or vector potential values at defined mesh nodes for the minimum energy condition through the minimization of a functional. By interpolation with shape (weight) functions over the space of each finite element, the complete field distribution within the model is then known. The exact algorithms employed are embedded in software and are transparent to the user. The software is designed for solution stability with minimization of storage requirements and run time. The user constructs a physical representation of the device or system being modelled, defining each material region in terms of its electrical conductivity, permeability and permittivity. Each region is then divided into triangular or quadrilateral elements for 2D, or 6-sided bricks or triangular prisms for 3D, with size depending on the field changes expected. The formulation, or equation set to solved, is then specified, the model boundary conditions defined (usually in terms of the behaviour of the electric and magnetic field vectors), and the excitation conditions specified (usually total for magnetic solutions and voltage for electrostatic solutions). In addition to the EM field variables, post-processing routines can calculate conductor s and voltages by integration. External ports are available which behave as physical circuit terminals connected to conductors within the model. They may be used as inputs to apply voltages to, or inject in, the

3 Computers in Railways 425 model, or as outputs to test for induced voltages or s for specific excitation conditions. 2.2 Validity of 2D track models To reduce the complexity of 2D FE models with respect to the computer resources available, simplifications must be made to the system geometry and problem formulation. First, consider railway track geometry. Track is a 3D structure laid on the surface of the ground whose weakly conducting behaviour affects the EM field distribution. Fields from the s flowing along the rails and other parallel cables spread into the air and ground and give rise to energy flow predicted by the E x H Poynting vector. For 2D modelling, the energy flow must be in the z-direction (Fig. 1). In practical traction and signalling circuits, the electric and magnetic fields have the largest intensity in the space between the catenary and rails and between the rails, respectively. Most energy thus flows in the vicinity of the track, but since the fields and hence energy can extend far from the track, the boundary of the model must be considered carefully. 2D "Slice" at position of longitudinal symmetry Catenary Track circuit power source Track substructure and ground Figure 1: Rail track traction system model Ex, Ey, ddx-dt, ddy/dt

4 426 Computers in Railways The directions of the E and H vectors determine whether the FE model may be simplified to 2D. The track longitudinal symmetry requirement implies that the fields must not change in the z-direction. This implies that: the direction of conductive flow through the rails and ground at the position of the model "slice" must be normal to the x-y plane (this applies only away from points of common-mode injection near substations) the change of voltage along the rails due to series impedance must be small the change of rail due to inter-rail admittance must be small. The resultant electric field must be in the z-direction which implies that the field arising from eddy s and z-direction conduction s must be much greater than that due to inter-rail and rail-earth diffusion and loss s. The last two observations enable the magnetic (eddy ) and electrostatic conditions to be decoupled. This implies that radiative effects cannot occur, so the transmission line wavelength should be long compared with the section length being modelled. The absence of waves implies that and voltage variations throughout the unit length section are small. For rail track to satisfy this criterion, the FE model validity is restricted to frequencies less than about 30 khz, with a corresponding wavelength of several hundred metres. 2.3 Problem formulation Decoupling the electrostatic and magnetic parts of a solution removes the need to simultaneously consider eddy s and displacement s in the same formulation. The equations for the eddy part of the solution can be written in terms of the vector potential A which is defined by B = VxA (1) It can be shown [3] that the problem may be solved using the symmetric set of equations at a (2) -U, (3) where A% is the z-component of the vector potential, E% =JE% dt is the time integral of the applied electric field and I^t is the total. The displacement and conductive problem is solved by a different set of Maxwell's equations. For AC conditions, an electrostatic problem exists with the conducting rails given prescribed voltages, laid over the ground which is given specific electrical conductivity and permittivity. In the solution, the scalar potential V is the variable of interest and is defined by E = -VV (4) The equation to be solved is derived from Maxwell's equations as

5 Computers in Railways 427 at at (5) 3 Three-dimensional FE modelling of traction systems 3.1 Problem formulation To reduce the computer resources required for large-scale 3D work, further investigations are required to simplify the modelling process. Since the track is no longer longitudinally symmetric, the process of decoupling the two solutions must be examined. Because excitation along straight rails is in the longitudinal z-direction only, and some of the leaves the rails and flows through the earth, the ground will have transverse x and y components in addition to a longitudinal z component. The eddy density from the ground will be much less in magnitude than that from the concentrated rail, so it can be neglected. The EM conditions may thus still be modelled in 2D, albeit as a series of "slices" as shown in Fig. 1 due to the variable rail with longitudinal direction. The required excitation in the z-direction in rail and ground is an initial condition which must be determined from a separate prior 3D diffusion model. 3.2 DC traction system: practical tests and analytical model A feature of DC traction systems is the relatively large proportion of traction entering or leaving the ground from the rails between the locomotive wheels and substation feed. To explore this phenomenon and to generate a benchmark for subsequent modelling, practical measurement data obtained from the SPOORNET (South Africa) 3 kv DC traction system were examined. Fig. 2 shows the test conditions. Current was fed from a substation to a locomotive along a straight, level track over uniform ground. The distribution in the feed (catenary, contact wire and parallel feeder) and return (rails and earth wire) were measured at a test location 1168 m from the substation as the position and power demand of the locomotive were varied. Measuring point Catenary Figure 2: Practical traction return test The exact measurement conditions were imprecise, with in particular the locomotive position in error by ±5%. The variation of propagation constant with (Table 1) could be due to change of ground conditions during the

6 428 Computers in Railways course of the tests. Nevertheless, interpretation using classic single-phase transmission line theory assuming a uniform (single-layer) ground conductivity yields an effective propagation constant which decreases with increasing measurement section length. This implies that, because for longer lengths the ground flows at greater depth, the effective rail-to-ground conductance and ground conductivity reduce with depth. From Table 1, the largest and smallest effective rail-to-ground conductances range from 2.93 ms/m to 12.6 ms/m, assuming an effective rail return resistance of 50 mq/m. The next task was to devise an analytical model which could reproduce the above results and which could subsequently be used to verify FE methodology. It is clear from the practical tests that an essential feature of any 3D traction system model is a realistic ground conductivity representation. Physically, a multi-layer ground model is necessary with a geometry amenable to mathematical analytical modelling. Table 1: DC traction system rail practical measurements Sbstn.l RaiP Earth Dist. to locom. (m) Prop %g cons tant (m-1 ) Sbstn.l Rail? Earth Dist. to locom. (m) Propag. constan t (m-1) Substation = feeder + catenary + contact wire 2 Rail includes earth wire In the scheme adopted, shown in Figure 3A, the rails are represented by an equivalent semi-cylinder set into the ground, which is divided into layered semicylindrical segments. The model deviates from reality in that: the rail-ground contact is a continuous cylindrical surface in uniform contact with the inner soil layer, instead of through the rail fasteners, ties and ballast the ground conductivity stratifications are cylindrical in the model rather than horizontal in practice the boundary conditions in the model are that no horizontal enters or leaves the line at the points of injection; this does not occur in practice. The model geometry enables analytical calculation of the equivalent series resistance and shunt conductance values of each ground layer, which is necessary for the formation of an equivalent multiconductor transmission line (MTL) as shown in Fig. 3B. The MTL can then be solved for given excitation and boundary conditions by a modal calculation routine within the MATLAB environment (Fig. 3C). The results can also be verified by circuit simulation using a grid of equivalent resistances and conductances from Fig. 3B. The model was set up with the dimensions and conductivities of the cable and ground layers chosen to produce the same order of magnitudes for the track s and voltages as were found in the practical measurements of Table 1. A three-layer ground conductivity profile was then imposed on the model to

7 Computers in Railways 429 improve the agreement with the practical results. The reverse problem of selecting ground conductivities was solved by intuition and trial-and-error with knowledge of practical ground conditions [4]. The result (Table 2) is an improvement compared with the constant ground conductivity model implicit in the propagation constant calculations of Table 1. A. Semi-cylindrical track model B. Equivalent transmission line Rails Ground layers C. Modal transmission line -CD Ground layers Figure 3: 3D DC traction system model Table 2: Atialytical model conditions L^yer Material Conductivity m 0-1 m 1-20m m Cable Ground 1 Ground 2 Grounds S/m S/m S/m S/m 3.3 DC traction system model: FE modelling A 4 km length model, with feed and return points spaced 2 km apart symmetrically in the centre of the section, was created with the same geometry as for the analytical model. The following practical considerations arose: Model size As the mesh of Fig. 4 shows, even geometrically simple models can require very large numbers of elements. There is a trade-off between model complexity, computational speed and temporary storage requirements. Tests with models of 20k to 80k elements revealed that with a compromise number of 55k, the computation time required was about 10 hours using a HP Apollo Series 715 workstation with 128 Kbytes storage.

8 430 Computers in Railways Figure 4: FE mesh for 3D DC traction system model Element size For a well-conditioned model, two features must be minimised. The first is the volume difference between the largest and smallest elements. This can be considerable since when the number of elements is increased, more are inserted close to the cable where the field changes rapidly. It is also desirable to limit the aspect ratio of the brick or prism sides to near 1:1:1. This is inherently difficult with long, thin geometries. The compromise model was given 400 elements along its length (representing 10 m) and the worst conditioned brick had sides in the ratio 0.17:137:6.75. Boundary condition The previous 2D representation had the far boundary condition set at zero potential to represent remote earth. However, to model longitudinal conductive flow, a physically more realistic condition would be an insulating boundary. Both insulating and conducting far boundary conditions were investigated, as shown in Fig. 5. Boundary distance From Fig. 5, it appears that the problem is to find the smallest model size which gives accurate results irrespective of the boundary condition. Testing for zero flow along a conducting boundary will confirm that the model size is correct. Models with various boundaries were investigated, and it was found that boundaries at 100 km, 10 km and 1 km contained 100%, > 99.9% and < 90% of the in the earth, respectively. A 10 km boundary was chosen as the optimum. Solution convergence Great care should be taken in setting up physically correct models with realistic data, and to check solutions for reasonableness. One factor found to affect computational accuracy and convergence is setting too great a difference between conductivities either side of a material boundary (the rail/ground conductivity ratio in the present model can be Figure 5: Track centreline equipotentials with 100 V applied between feed (top left) and centre of symmetry (top right ofpicture) Left: Insulated boundary Right: Conducting boundary with V = 0

9 Computers in Railways 431 Rail, ground density and ground equipotential contours were computed in the FE model and verified by comparison with the analytical model. Fig. 6 shows the longitudinal density along the rail and the vertical leaving the rail in the model as a function of distance along the track. As expected, enters the ground from the rail on both sides of the point of injection. The rail is forced to zero at the end of the model 1 km from the injection point. Fig. 7 shows the ground density in the horizontal (z) and vertical (y) directions as functions of depth at the point of injection (z = 0) and a point near to the central line of symmetry (z = 500 m). The vertical is largest at the injection point, and almost zero at the centre due to model symmetry. The horizontal density, however, is larger at the track centre by a factor of three compared with the injection point. This flow of is consistent with the equi potential plots of Fig. 5. Figure 6: Rail density (Aim*) as a function of distance A, /^ Longitudinal (z-direction) B, 1% Vertical (y-direction) Figure 7: Vertical (Jy) and horizontal (Jz) components of ground density at point of injection and L/2 as functions of depth (scale 1 km) A, Jy at z=0 (scale A/m2) C, Jy atz=l/2 (scale A/m2) B, Jz at z=0 (scale 4.0 A/m2) D, Jz at z=l/2 (scale 1.4 AJm2)

10 432 Computers in Railways 4 Conclusions The finite element method is a very powerful modelling tool used for a wide variety of engineering devices and systems. Its use for large, unbounded models, such as traction systems, works well provided care is taken in setting up and specifying the model. 2D FE has become an established technique to calculate field magnitude and gradients, and to derive equivalent transmission line parameters for circuit modelling. Present work is focused on calibrating the transmission line models using practical testing to define their range of validity. Future work is planned on additional excitation conditions such as combined DC and AC signals. 3D FE represents a new technique that will provide information about ground flow near substations, conditions at track circuit terminations and aspects of electromagnetic interference. The computer resources required for 3D models of traction systems are very large and much more work is necessary to improve model accuracy. 5 Acknowledgements The author has received financial support from UK EPSRC through research grant #GR/J/ Benny Stein of SPOORNET, Johannesburg, South Africa, is acknowledged for provision of traction measurement data. 6 References 1. Carpenter D.C. and Hill R.J.: Railroad track electrical impedance and adjacent track crosstalk modelling using the finite-element method of electromagnetic systems analysis, IEEE Transactions on Vehicular Technology, v. 42, n. 4, November 1993, pp Hill R.J. and Carpenter D.C.: Rail track transmission line distributed impedance and admittance: theoretical modeling and experimental results, IEEE Transactions on Vehicular Technology, v. 42, n. 2, May 1993, pp Carpenter D.C. and Hill R.J.: Finite-element method modelling of overhead catenary rail track impedance and admittance. Computers in Railways III, Vol. 2: Technology: 3rd International Conference on Computer Aided Design, Manufacture and Operation in the Railway and other Advanced Mass Transit Systems: COMPRAIL 92 (Southampton: Computational Mechanics Publications 1992), Washington DC, August 1992, pp Carpenter D.C. and Hill R.J.: A continuous-function ground conductivity model for the determination of electric railway earth conductance, IEEE Transactions on Geoscience & Remote Sensing, v. 31, n. 5, September 1983, pp Yu, J.G. and Goodman, C.J.: Stray design parameters for DC railways, pp , Proc. Joint ASME/IEEE Railroad Conf., Atlanta, IEEE, New York, Wedepohl L.M.: Application of matrix methods to the solution of travelling wave phenomena in polyphase systems. IEE Transactions, v. 110, n. 12, Dec 1973, pp