Download Modeling d.c. Stray Currents Using a Multi-Layer Model

Modeling d.c. Stray Currents Using a Multi-Layer Model
H.W.M. Smulders, M.F.P. Janssen
Movares, Utrecht, The Netherlands
Abstract
Stray currents from d.c.-powered railway and tram lines can cause significant damage to pipelines, cables
and structural works in the vicinity of those lines. The level of stray current can be reduced by a variety of
measures, like improving the insulation between tracks and earth, using stray current collection systems
and decreasing the longitudinal resistance of the tracks.
In order to assess the effectiveness of these measures, the railway system can be modeled using various
techniques. The most elementary models exist of a grid of resistances, representing the tracks, the earth
and the insulation between them (or lack thereof). Also very complex 3-D models of the soil do exist.
Although the simple models may provide sufficient accuracy for a first impression, they often lack the
possibility to include the various measures against stray currents and the accuracy to assess their
effectiveness. Another difficulty is to determine the correct values of the discrete resistances in these
simple models, based on physical parameters as rail-to-track resistance and specific earth resistance.
The complex 3-D models have the major disadvantage that the effort required is very high, and modeling
several km long structures is difficult. Therefore the practical use of these models is quite limited.
Movares (on 1 May 2006 Holland Railconsult changed the company name to Movares) has developed
the tool STARTRACK (STray current Analysis tool for Railway TRACKs) which tries to overcome these
problems. Based on basic physical properties of soil (galvanic coupling is dominant for d.c.) a set of
equations has been derived to determine the values of the discrete components in the model, such as the
resistance between the track and structural works, between structural works and earth and between
parallel pipelines and earth. The earth can be modeled with two layers, therefore providing the possibility
to take the properties of the top layer of the earth into account. Using these basic equations a semi 2-D
model of the rail infrastructure including the soil was made. The implementation into a simulation tool is
based on the existing SimspoG tool, which has it's origin in 25 kV 50 Hz power supply systems.
This new simulation tool STARTRACK is a significant improvement over many existing models. Given the
relative simplicity it is possible to model complex track situations with reasonable accuracy and effort. The
model is especially valuable in comparing various measures against stray currents and in determining
their effectiveness. The use for predicting actual stray current levels remains limited. The main reason is a
practical one: normally in real life practice it is very difficult to determine the local properties of the most
critical parameters, such as soil resistivity and rail-to-earth resistance. Instead, it quite often must be
assumed that they are homogenous and average values for these parameters have to be used. Local
deviations from this homogenous model can cause relatively large deviations in the results. Nevertheless
first impressions of the use of the model, for instance for complex situations like a light rail tunnel with
stray current collection system have given very promising results.
1. Introduction
Various modeling techniques have been developed to study the behavior of stray currents caused by
railway systems. Railway systems by nature are large, one-dimensional systems, and current distribution
in the soil is a typical 3-D phenomena, thus difficulties arise. Several approaches (Figure 1) can used:
A. Direct discretization into a circuit model;
B. Resistor network models;
C. Cylindrical layer models;
D. Full 3-D models (not shown);
E. Longitudinal sectioning, 2-D approach per section;
ρ1
ρ2
ρ3
Circuit model of 1
branch
A. Direct Circuit Models
SS
LOAD
C. Cylindrical Layer Models
SS
TR
AC
K
FA
R
EA
RT
H
Section L
VIC
TI
M
B. Resistor Network Models
E. Longitudinal sectioning, 2-D
Figure 1 Overview of various types of modeling techniques
An advantage of A [1] is that a relatively simple model can be derived, however the characterization of
the circuit elements based on the key parameters of a railway system can quite a problem. The
advantage of B is that it is relatively simple but the accuracy is limited and simulation of complex
situations is cumbersome. C, cylindrical layer models can be linked to 3-D models [2], and can be used
relatively easy, therefore large structures can be modeled. Problems however occur when truing to model
a layered soil correctly, or when multiple tracks, situations with points etc are present. Basically the
problem is to solve the Poisson equation and derive a potential for a line, which is by definition not
possible. Full-3-D models, D, already exist, and are commercially available. In railway engineering
applications the use is limited, due to the effort needed to set up the model, and discrepancy between
transversal and longitudinal distances of interest in a railway system.
In order to overcome the problems of the previous approaches a new approach E, longitudinal sectioning,
2-D, has been developed. The system is divided in longitudinal sections, for each section a transversal
equivalent circuit is developed based on a 3-D soil modeling for that section. Long complex structures can
now be modeled with relative ease. It can also be proven that the error caused by the semi 2-D approach
is relatively small compared to the error caused by the parameter variations of the soil. Normally the soil
is not known in great detail. In the following section this technique is explained.
2. Physical Models
2.1. Basics, homogeneous soil, transversal
A homogeneous soil can be described as a half-infinite space with a constant resistivity. A half spherical
earth electrode will be used to represent a conductor in contact with soil with a certain footprint. The
Voltage distribution can be determined by injecting a current [3]. Based on this the resistance αij between
different points at distance d as well as infinity (conventionally put at 0 V) αii can be determined, Figure 2.
α ii =
ρ
2πrX
α ij =
ρ
2πd
(1)
For a complete transversal section the admittance matrix be derived:
[V ] = [A] • [I ]
[B] = [A]−1
[I] = [B ] • [V ]
coefficients αii and αij of [A] given by (1). We now define [Γ] by rearranging the coefficients of [B]:
(2)
γ 1N 
γ 2 N 
γ NN 
(3)
N
γ 11 = (β 11 + β 12 + β13 + . + β 1N ) = ∑ β 1i
rx
V1,I 1,r1
Vx
V2,I 2,r2
γ12
dr
γ11
coefficients
γ1N
V3,I 3,r3
γ23
VN ,I N,rN
Section N
I
Section N+1
γ 12 = γ 21 = − β 12 = − β 21
n =1
Section N-1
 γ 11 γ 12
[Γ] =  γ 21 γ 22
γ N 1 γ N 2
γ3N
γ22
γ33
γNN
transversal
longitudinal
Figure 2 Overview of soil modeling
2.2. Longitudinal model
Galvanic coupling between various sections is taken into account for longitudinal current propagation
(Figure 2). The coupling between sections N-2, N+2, etc. need not be taken into account. These terms
can be neglected because:
• Normally section length is larger than the transversal distances studied;
• Admittance decreases linear with increasing length;
• For railway systems longitudinal conductors will always be present with a good conductivity (rails);
• The complexity of the model would increase considerably.
Also the cross-couplings can be neglected for the following reasons:
• The voltage drop between adjacent sections is small compared to the transversal direction;
• The admittance present in the transversal combined with the longitudinal one gives a galvanic
coupling in cross direction in the model.
Given the relative uncertainty of some input parameters such as soil resistivity, a more precise modeling
is not useful. A check has been made to see if results are independent from section length. When section
length was increased or decreased by a factor 3, results varied in a range of ± 20%, this is acceptable.
Similar to section 2.1. coupling resistances can be determined:
N
ri
ρ
ρ (L − 2 ∗ riL )
(4)
2
α iL =
2πriL
riL =
N
∑ ri
∗
∑r
n =1
i
R Li =
π
∗
riL ∗ (L − riL )
n =1
2.3. Coupling between transversal and longitudinal modes
The model of Figure 3, (left) has only longitudinal effects, so combining this longitudinal mode with
transversal mode is needed. The impedance Z towards remote earth in the transversal mode changes
due to the addition of the longitudinal mode. Of course in the physical reality this is not true, so a
correction for γ11 is needed. The impedance RA (Figure 3 right) must be such that RA in parallel with the
characteristic impedance of the transmission lines leads to the impedance G i (Gi = 1/γ11):
2 * Gi2 Gi
R Ai = Gi +
+
∗ 4 ∗ Gi2 + 2 ∗ RLi ∗ Gi
R Li
RLi
(5)
This gives [∆] with coefficients:
(6)
δ ij = γ ij i ≠ j
δ ii = 1
R Ai
For modeling the soil the ∆ matrix, formula (6), and coupling impedances, formula (4) will be used.
section N-1
section N
Z
section N+1
RL
RiL
RjL
RiL
RL
RA
RL
RA
RL
RA
RjL
Figure 3 Overview of longitudinal coupling
2.4. Multi layer soil
A model has been derived for a homogeneous soil, but normally soil is not homogeneous. Often layers of
soil with different electrical properties will be on top of each other. As it is difficult to obtain electrical data
of soil as a function of depth we will limit ourselves here to a two-layered soil1, but the approach is also
applicable for a multi-layered soil. Here too a half sphere earth electrode is used, Figure (4).
R
I
β
ρ1
dR
d
ρ2
dα
Figure 4 Overview of two layer soil with half spherical earth electrode
By injecting a hypothetical current in the earth the voltage distribution can be determined, and from this
the coupling admittances and the admittance towards remote earth.
ρ ∞ 1
R2 + d 2
VX = I x ∗
∗∫ 2 ∗
∗ dR
2π a R   ρ 2 − ρ1 

2
2

 ∗ d + R + d 
 ρ

1



(7)
When Φ(R) is the primitive of the integral in (7):
ρ
VX = I x ∗
∗ (Φ (∞ ) − Φ (a ))
2π
(8)
Leading to a new version of the coefficients given in (1) & (4) (homogeneous soil) in case of a 2-layer soil:
ρ
ρ
ρ
(9)
α ii = 2 ∗ Φ (ri )
α ij = 2 ∗ Φ(e )
R Li = 2 ∗ (Φ (riL ) − Φ(L − riL ))
2π
2π
π
In case of d→0 or d→∞, or ρ1=ρ2 formula (7) gives the same results as (1) & (4).
2
2
2.5. Synthesis
The complete model for the conductors in contact with the soil is constructed combining all of the above
elements and is shown in Figure 5. R ii is the (possible) impedance between the conductor and the actual
footprint on the soil. Section 3 deals with the linking of the model to key parameters of the railway system.
The modeling as presented above needs to implemented in software tool, section 4 treats this aspect.
1
Normally in The Netherlands a two layer soil is present. A top layer 1–30 m thick, of well conducting
peat or clay on top of a very thick more ancient layer of sand, which a much lower conductivity.
Rii
Rii
Rii
Rii
RLi
δij
Rii
δii
δij
δii
δii
δij
Rii
RLi
δii
δij
δij
δii
RLi
δij
δii
Figure 5 Overview of the complete model
3. Example of model parameters
The generic model is based on geometry, footprint, and soil resistivity. However a match still must be
made with the parameters of a railway system. A number of typical situations has been studied:
• Normal ballast track;
• Track with enhanced insulation (e.g. coated rail);
• Pipelines or other conductors in the soil;
• Track in or on civil structures;
• Stray current collection systems.
Here we will limit ourselves to ballast track. Key parameters are:
• Ztrack to earth
Zt→e
4 Ω/km
• Width of one rail
B
0,2 m
• Number of rails
N
2
• Specific soil resistivity
ρ
50 Ω.m
• Fraction soil contact
f
25 %
• Section length model
L
100 m
1000 ∗ Z t → e
N ∗B∗L∗ f
ρ
(10)
rx =
Rii =
−
2π
L
2π ∗ N ∗ B ∗ L ∗ F
These parameters lead to the (realistic) values for rx and R ii :
• rx
1,26 m
• Rii
33,7 Ω
4. Numerical Modeling
4.1. Overview
Since 1994 simulations and measurements on the behavior of a.c. traction power supply systems have
been made by Movares, the simulation tool SimspoG has been developed, which has proved to be a very
useful tool. Simulations on a.c. railway lines, short circuit calculations, current distributions and resonance
calculations have been made. For a.c. modeling, the equations of Carson-Pollaczek [4] are used to
calculate the impedances of all conductors. This includes the inductive coupling between all conductors
and the influence of the soil. For d.c. systems, the Carson-Pollaczek equations can not be used. As an
alternative, the physical model as described above is used for STARTRACK. The basics of the a.c. and
the d.c. model is the same.
4.2 Mathematical background
For a linear system of n conductors, the impedance of these conductors is given by a general impedance
matrix Z. For an a.c. model, the impedances on the diagonal Zii give the self impedance of the
conductors. The other values Z ij give the mutual inductive coupling between conductors i & j. For a d.c.
model, all Z ij values are zero and the Z ii values give the ohmic resistance of the conductors. The
additional conductors in the model linked to the physical conductors in contact with the ground are also
part of the matrix.
Z
 Z11 Z12 Λ
Z
Z 22
21
= 
 Μ
Ο

Z n1 Z n2 Λ
Z1n 
Z2 n 

Μ

Z nn 
(11)
Petrov [5] & Kontcha [6]), give the general equation (12) describing the system. In this equation a relation
between voltage and current on the beginning (U' and I') and the end (U'' and I'') of the system is given.
[U ′]
[U ′′]
 [I ′]  = A ∗  [I ′′] 




(12)
SS
Cat
Cat
Train
Cat
Figure 6 Division of system in sections
Taking the simple system shown in Figure 6 as an example, this system can be written as:
(13)
A = ACat ∗ ACat * ATrain ∗ ACat
Multiplying the individual transfer functions gives the overall transfer function of the system. Each of the
individual transfer functions gives a relationship between the voltages and currents on either sides of the
section. The A cat transfer function is based on the Z-matrix. Other parts of the infrastructure, like trains or
physical connections between conductors, are described by other transfer functions. Using the applicable
boundary conditions, the voltages and currents on either side can be calculated by solving (12). The
intermediate currents and voltages can be found by multiplying the voltages and currents on the
beginning of each section by its transfer function.
4.3. Validation
In the last several years, on several occasions the results of the SimspoG simulations (the a.c. model)
were compared to measurement results (Janssen [7]). In all cases, the simulation results correspond very
well with the measurements. No detailed comparison for STARTRACK with d.c. railway lines has been
done yet, but some measurements are planned for spring/summer 2006. A preliminary comparison with
data available from the RET network (municipal transport company of Rotterdam) was satisfactory.
5. Examples
5.1 Basic examples
Figure 7,left, shows the layout: 2 substations, 12 km apart, feed a double track line with a load of 1 kA at
12 km. The rails of both tracks are interconnected every 4 km. At a distance of 50 m from the tracks, a
pipeline is buried in the soil. Figure 7, right shows the current through the soil and pipeline. Positive
current flow is to the right of the model. Some of the key parameters are:
• soil resistivity: 100 Ohm.m
• depth of pipeline: 1m
• radius of pipeline: 25 cm
• resistance of the coating: 3000 Ohm.m 2
• resistance between tracks and soil: 10 Ohm.km
15
Pipeline
Soil
10
5
SS1
SS2
1kA
Track 1
interconnection
-5
-10
-15
Km 20
Km 17
-20
Km 13
Km 9,0
Km 5,0
Km 0,0
Track 2
Current [A]
0
-25
0
2
4
6
8
10
12
Position [km]
14
16
18
20
Figure 7 Basic example of dual track line, results: current through soil and pipeline
20
20
Reinforcement
Soil
15
10
10
5
Current [A]
Current [A]
5
0
-5
0
-5
-10
-10
-15
-15
-20
-20
-25
Reinforcement (top layer)
Reinforcement (bottom layer)
Soil
15
0
20
40
60
80
100
120
Position [km]
140
160
180
200
-25
0
20
40
60
80
100
120
Position [km]
140
160
180
200
Figure 8 Overview of current distribution for a system on reinforced concrete slab (left) and for a
concrete slab with two layers of reinforcement insulated from each other (right).
Next the effect of reinforcement in the concrete civil structure supporting the railway on the current
distribution is studied. In Figure 8 (left), a 200 km long model (load case from Figure 7) is used which is
constructed on a reinforced concrete slab. This may seem exceptionally long, but this gives more insight
than a short model. It can clearly be seen that the current in the soil reaches very far (tens of kilometers in
both directions), even though the substations and the load are located quite close together in the middle
of the line. Figure 8 (right) shows the effect of an additional layer of reinforcement, insulated from the top
layer. It is assumed that both layers are insulated by a plastic foil with an overall resistance of 3 Ohm for
100 m of track. The current in the soil is reduced by a factor of two. A further reduction of the current in
the soil can be realized by increasing the overall resistance between the two layers of reinforcement.
However, a quite high (possibly not realistic) resistance is needed to reduce the current substantially. For
a factor 10 reduction in stray current, an increase of ≈100 is needed for the resistance.
5.2. Simulations of a real life situation
101,0
103,0
22 km
10 tracks
107,5
110,7
113,5
115,0
yard and
workshop
3 tracks in
workshop
detail of yard
and workshop
pipeline
Figure 9 Overview of RET line with yard and workshop
The next step is to make a model of a real situation. In co-operation with RET, a model was created of a
line including a yard and a workshop. The track layout and the location of the 6 substations on the 22 km
long line is shown in Figure 9. Parallel to a part of the line is a natural gas pipeline. STARTRACK is used
to model the situation, Voltage&Current distributions for a number of load cases have been calculated.
Some details:
• The resistance between the rails and earth is assumed to be 4 Ohm.km.
• In the workshop, the tracks are cross bonded and connected to earth. The earth resistance is
1 Ohm.
• The tracks in the workshop are insulated from the outside tracks by insulating joints. These can
be bridged by switches if needed.
• The pipeline has a radius of 25 cm, no insulating flanges and a coating of 3000 Ohm.m2.
The results show that the stray currents increases more or less linearly with the total power demand of
the line, maximum stray current is 1∼1.5% of the total return current. This may seem rather low, but it is
mainly caused by the short distance between the substations. The current on the pipeline ≈10% of the
current in the soil. This seems (from a modeling point 0f view) a reasonable level for a rather poorly
insulated pipeline. For the maximum total load (4 trains on the main line, 2 trains on the yard, 1600 A per
train), the maximum rail to earth voltage is just over 60 Volt, maximum voltage between pipeline and earth
is 250 mV, both in accordance with expectations.
6. Conclusions
The tool STARTRACK which has been developed is a reasonable compromise between physical
correctness and technical usefulness. The results obtained so far have been proven to be very
satisfactory, with limited effort predictions can be made. As discussed above, it remains questionable
whether a model like this (or any other stray current model for that matter, due to lack of detailed reliable
input data) can be used to calculate stray currents a priori with high reliability. However, the differences in
stray current levels for various solutions can be calculated quite well. The tool therefore has a major use
for making various design choices.
Acknowledgements
The authors would like to thank Diederik Verheul and Gerrit Disberg of Movares for giving us the
opportunity to develop STARTRACK. We would also like to thank Richel van der Schulp and Leo
Vliegenthart of RET Rotterdam for the information about their network, and their co-operation.
References
[1] Lucca, G., M. Moro, “Conductive coupling among electrified traction lines and buried structures”,
Conference Proceedings EMC Europe 2004, Eindhoven, 2004, pp. 505-509.
[2] Hill, R.J., S. Brillante, P.J. Leonard, “Modelling electromagnetic fields in DC traction systems using
three dimensional finite-element analysis”.
[3] Koch, W., “Erdung in Wechselstromanlagen über 1 kV; Berechnung und Ausführung”, Springer
Verlag, Berlin, 1961.
[4] Pollaczek F., “Über das Feld einer unendlichen langen wechselstrom durchflossene Einfachleitung”,
Mitt. Aus dem telegraphisen Reichsamt, 1926 (in German).
[5] Petrov O., O. Grimalski, “Berechnung der elektrischen Größen in Oberleitungen und Schienen von
Wechselstrombahnen”, Elektrische Bahnen 88 (1990) 11, pp. 398 - 402 (in German).
[6] Kontcha
A.;
“Mehrpolverfahren
für
Berechnungen
in
Mehrleitersystemen
bei
Einphasenwechselstrombahnen”, Elektrische Bahnen 94 (1996) 4, pp. 97 - 109 (in German).
[7] Janssen M.F.P., H.J. van Dijk, H.W.M. Smulders, G. van Alphen, "Application of the SimspoG
simulation tool- modelling the railway infrastructure", Comprail 2002.