Calculation of remote effects of stray currents on rail voltages in DC railways systems

www.ietdl.org

Published in IET Electrical Systems in TransportationReceived on 14th June 2012Revised on 28th February 2013Accepted on 18th March 2013doi: 10.1049/iet-est.2012.0022

ISSN 2042-9738

Calculation of remote effects of stray currents on railvoltages in DC railways systemsJorge Valero Rodríguez, Javier Sanz Feito

Electrical Engineering Department, Carlos III University of Madrid, Av. de la Universidad, 30. 28911 Leganés (Madrid),

Spain

E-mail: [email protected] m.es

Abstract: This study focuses on the analysis of the remote effects of stray currents and rail potentials along the rail line in DCtraction systems, although the results can be extrapolated to AC systems. This study is essential to ensure the equipment andpassenger safety, as well as the maintenance of rail infrastructures, subjected to corrosion in DC railway traction systems. Themathematical model developed in this study allows to obtain stray current and rail voltage distributions at the point where thetrain or the short circuit is located, x, and what is more important, it lets also to calculate its remote effects at any other pointalong the rail, y. Most of the railway electric models published in the literature do not calculate the remote effects of straycurrents and rail potentials. Calculations are often done just at the specific point where the current is injected into the rail.However, the train and short-circuit currents actually generate remote effects as a voltage wave-front that propagates along therails. Another contribution of this article is that the problem is defined as a function of the power demanded by the vehicle ateach point, P(x), so the consumed current, I(x), is obtained by means of a power flow algorithm. Nevertheless the vastmajority of references assume an already known input current, which, may only be obtained by power flow methods. Withthe aim of simplifying the lecture of this document, Table 1 provides the list of symbols used in this study.

1 Introduction

About half of the worldwide railway systems use DC tractionsubstations. One of its main features is the high-powerconsumption (MW) which at the supply voltage, 750–3600V, causes the circulation of high currents. Therefore in anormal DC traction system there will always be asignificant percentage of stray currents which flow throughthe earth out of the rails, because the isolation level of therailway lane is quite high, but not infinite [1–11]. Straycurrents leak into the earth until they finally emerge at theearthing terminal of the substation, fulfilling Kirchhoff’sfirst law [10–15].The main effect of rails potential and the leakage currents

in DC systems is the degradation of metallic infrastructuresbecause of electrolytic corrosion processes [1–15], with theconsequent economic cost of maintenance that they entail.For this specific reason, in DC traction systems there mustbe a strict separation between the feedback circuit andearthing systems [3, 4, 6].Owing to this high isolation level imposed by the railway

standards, stray currents may result in significant railpotentials which can damage the equipment and harmpeople, so it makes it necessary to study and analyse itsnature in order to limit its values according to the existingregulations [10, 11].Near the point where the train is, or near the point where

the short circuit occurs, the current leaks into earth,emerging back to the rail as we approach the negative

IET Electr. Syst. Transp., 2013, Vol. 3, Iss. 2, pp. 31–40doi: 10.1049/iet-est.2012.0022

terminal of the substation. In both cases, the sum ofcurrents flowing through rail, land and any otherconductive surface must be equal to the current leavingthe injection point and the substation, according toKirchhoff’s first law. Rail potentials reach its maximumvalue at the point of current injection and at thesubstation rail connection, because at both locationscurrents flowing throughout the rail are maximum and soare stray currents and rail voltages.Some authors [1], obtained a similar mathematical model

by means of applying the uniformly distributed parametermodel used for conductor transmission lines, being the onlydifference that solutions were obtained in terms of thecurrents instead of their power consumption at every point,P(x), as it occurs in reality (see Table 1). However, we gofurther in this paper, since apart from obtaining the straycurrents and the rail voltage distribution at the point wherethe train or short circuit is placed, x, we show the remotevoltage wave-front because of the movement of the train orshort circuit.Other authors [2], concentrate on the mathematical solution

of a large-scale multi-branched DC traction power networks.This, however, is beyond the scope of this paper, since wefocus on a method to obtain stray currents, rail voltages andvoltage wave-fronts at any kilometrical point, y, with agamma, Γ, topology, Fig. 1.Some other authors [3, 4], focus on the DC traction power

system grounding methods and their relationship toequipment and personal safety.

31& The Institution of Engineering and Technology 2013

www.ietdl.org

In our mathematical model, we used an ungrounded(floating) system, Fig. 1. This system provides the leaststray current, however, it can be dangerous since it mayappear high DC voltage with respect to earth, especiallyduring ground faults. We have chosen this groundingsystem with the aim of easily proving the evolution of straycurrents and rail voltage distributions.In this paper, stray current and rail potential distributions

are studied, and a model to calculate voltage wave-fronts asa remote effect of trains circulations and short circuits ispresented. Its main contributions, in comparison to some ofthe existing models, are the following:

Fig. 1 Γ-Topology traction system in normal conditions

Table 1 List of symbols

Symbols Meaning

a(x) acceleration of the vehicle, m/s2

Rav average resistance of the rail, ΩR’, R’R catenary and rails linear resistances, Ω/kmi(x) current along the rail, AI(x) current consumed by the train, Ar(x) curvature radius, kmFi(x) inertia effort, daNG’RE linear conductance, S/kmUSS nominal voltage of the collateral substation, VY plane (km) which contains the point where the

measurement is doneX plane (km) which contains the point where the

current, I(x), is injectedy point where effects are measured, kmx point where the current, I(x), is injected in the

rail and train position, kmP(x) = I(x)U(x) power consumption, MWL power supply section length, kmdx rail differential length, kmB rail gauge, mM rail propagation constant, km–1

LC = 1/M rail surge or characteristic length, kmZC rail surge or characteristics impedance, ΩURE(x) rail voltage, VA, B, C resistance coefficients, daN, daN h/km,

daN h2/km2, respectivelyCmg rotating-mass coefficientdi(x) stray current, Adi(x)/dx stray current linear density, mA/mR0 substation output resistance, Ωs(x) terrain slope, daN/TmFt(x) traction effort, daNv(x) train speed, km/hM vehicle mass, TmU(x) voltage between catenary and rails, V


† Our model uses a method to characterise traction vehiclesas dynamic loads defined by their power consumption at eachpoint, P(x). Therefore the load current, I(x), as well as thevoltage between pantograph and rails, U(x), are calculatedby means of power flow methods, in order to achievemaximum accuracy.† Apart from obtaining rail potential values, URE(x), at thepoint where the short circuit or the train inject current intothe rail, x, the main advantage of this model is that it allowsus to obtain rail potential wave-fronts at any point along therailway track, y, because of the train or short-circuitmovement.† At the same time, it is possible to calculate rail potentialevolution at a fixed point, y, where the observer is placed,as a result of the displacement of the train or short circuitalong the railway track.

In this paper, we present the most basic stray current modelwith the following assumptions:

† There is only a single train in the power supply section.† We consider a problem with uniform conditions, that is,constant train speed, v(x) = v0, constant curvature radius, r(x) = r0 and constant terrain slope, s(x) = s0, in the powersupply section. Therefore the power consumption is alsoconstant. Nevertheless, it should be noted that as a trainmoves along the line, it demands a variable power profileover time, P(x). With the aim of simplifying this paper, wearbitrarily consider that the power consumption is equal toP(x) = 1 MW at each point.† We work with a gamma, Γ, topology with an ungrounded(floating) system, in which the power supply section is onlyfed by the left collateral substation [11], Fig. 1.† This rail model is also feasible to be used in AC railwaysystems, 1 × 25 kV and 2 × 25 kV, since the model showsan almost only resistive behaviour, and the inductive andcapacitive effects can be neglected [10].

This paper is organised as follows. In Section 2, a DCtraction system is described. In Section 3, the rail model isobtained taking into account the physical phenomenon ofleakage current. In Sections 4 and 5, simulations in normaloperating conditions and under short-circuit conditions arepresented.

2 Traction system description

In the rest of the paper, we assume a gamma, Γ, topology,consisting of a substation that feeds a unique power sectionfrom the left side [11]. Fig. 1 shows Γ-topology tractionsystem in normal conditions with its most relevantmagnitudes, whose values are presented in Table 2 [11].In this paper, we will also assume that the locomotive

draws a known power, P(x), which implies the circulationof a current I(x) through the catenary, resulting in a voltage

Table 2 Γ-Topology magnitudes

Constants Variables depending on x

USS = 3600 V I(x)L = 10 km U(x)R0 = 0.2 ΩR’ = 0.03 Ω/kmR’R = 0.01 Ω/kmP(x) = 1 MW


www.ietdl.org
drop, U (x), between the pantograph and rail, as shown inFig. 1. Equation (1) describes the circuit shown in Fig. 1.
USS = R0 + R′ + R′R

( )x

[ ]I(x)+ U (x)

P(x) = I (x)U (x)(1)

Contrary to what it is usually found in technical literature, it ismore accurate to work with the power consumption of thevehicle, P(x), so we must calculate the current demanded bythe train, I(x), and the catenary-to-rail voltage, U(x), from aload power flow method, instead of using the directabsorbed current by the train fed by nominal voltage.That is because railway specifications are given in terms of

train speed, v(x) and railway layout as terrain slope, s(x) andcurvature radius, r(x), which are related to the powerconsumption, P(x), according to the well-knownexpressions (2)

P(x) = Fi(x)+ Ft(x)[ ]

v(x)

360

Fi(x) = Cmga(x)m10−2

Ft(x) = A+ Bv(x)+ Cv2(x)[ ]+ m

500b

r(x)+ s(x)

[ ] (2)

Using P(x) instead of I(x), allows us to have a better definedproblem and work with absolute precision at every point, x,where technical specifications vary, that is, in railway trackslopes, in section with small curvature radius and where thetrain accelerates. In addition, we have to take into accountthat we are working with a time-variable geometry circuit,and therefore there will be different electrical circuitconfigurations to be solved, according to the train orshort-circuit position, x. This scenario will become evenmore complicated when several trains circulate through thesame section.

3 Uniformly distributed sleepers rail model

The return circuit, composed of rails, sleepers and ballast,must be designed to carry traction, regenerative braking andshort-circuit currents back to the substations through a lowimpedance path, in order to limit voltage drops along therail, and keep rail-to-earth voltages within safe values [1–11].To ensure the safety of people, rail potentials during normal

and fault conditions must not exceed the maximum allowablevalues listed in the standards EN 50122-1 and EN 50122-2.At the same time, steady-state rail voltages will not exceed120 V, except in workshops where the limit is 60 V [10,11]. The electrical coupling between rails and earth dependson many factors [2, 6, 7], the most important being:

† Type of superstructure, sleepers, ballast and rail insulationlevel.† Characteristics of the sleeper joint on the ground: sandballast, gravel and concrete or soil.† Level of soil contaminants, moisture, temperature, rain, iceand other variables that affect G’RE.

Table 3 shows the average values of G’RE depending on theearth features.Measurements carried out in normal and short-circuit

conditions show that rail-to-earth complex impedance inAC systems have an angle usually comprised between


1° and 3°, which indicates that the reactive component canbe practically ignored compared with the resistive one [10].This means that we can consider an almost only galvaniccoupling between rails and earth, so the rail behaviour isthe same for both, DC and AC systems. As a result, thereturn circuit can be characterised by the specific railresistance, R’R, with a parallel distributed rail-to-earthconductance, G’RE, for both technologies.The acceptance of an almost only resistive impedance

confirms the universal validity of the mathematical modeldeveloped in this paper, which can be used to analyse DCand AC traction systems [10]. As a result, to study thephysical phenomenon of current dissipation a mathematicalmodel of uniformly distributed parameters has beendeveloped for a Γ-topology scheme. Nevertheless, themethod can be easily generalised in the case of existingseveral substations and trains by applying the superpositionprinciple.Distribution of stray currents and rail voltages generated in

two different situations are studied:

† In normal operating conditions, the traction circuit iscomposed of the rectifier substation, the catenary, thelocomotive and the rails, Fig. 1.† In fault conditions, the traction circuit is similar to that inFig. 1, but replacing the vehicle by a short circuit betweencatenary and rails, so that U(x) = 0.

In this way, we can consider the substation and the point ofcurrent injection as current sources, whose current values, I(x), have to be obtained by means of the power flowmethod depending on the power consumption, P(x).Therefore we can work with the following equivalenttraction circuit shown in Fig. 2.Since the isolation of the sleepers is not perfect, there will

always be leakage currents, di(x). At a certain distance from

Fig. 2 Equivalent traction circuit

Table 3 Rail to earth linear conductance, G’RE

Concrete sleepers on sand ballast with stone paving 5 S/kmconcrete sleepers on gravel ballast with stone paving 1 S/km


www.ietdl.org
the substation and because of its influence, such leakagecurrents reverse their direction, that is, the current is no longerlost to the earth, but emerges from it and returns to thesubstation via rolling rails [11–15]. Stray currents can belimited in different ways, including electrical circuits based onpower electronic semi-conductors [3–5, 12–14], differentearthing schemes [3, 4, 9, 13], reducing specific railresistance, R′R, increasing rail-to-ground specific conductance,G’RE [8, 13, 14] and reducing substation separation [8, 11].To develop the proposed rail model we will work with the
mathematic traction circuit shown in Fig. 3, which models thegalvanic coupling between rail and earth by means of R′R andG’RE[3, 11].The traction circuit is divided into three sections according

to the current behaviour in each of them, as can be observed inFig. 2.

† Upstream from the substation (Section 1) and downstreamfrom the point of current injection (Section 3), the length ofthe rail is long enough to consider the current to be zero atits ends, as a result of the current leakages.† Just at the point where the current, I(x), is injected to therail, it splits into two directions. One component is directedtowards the substation, Ia and the other one, Ib, is dissipatedalong the rail in the opposite direction (Section 3), so it willreturn to the substation from the earth.† The whole current that returns to the substation, I(x), ismade up of two components, Ic and Id, which have thesame value as Ia and Ib, respectively.† When performing the simulations, we will only focus onSections 2 and 3, because they are the only ones involvedin our electrical analysis in Γ-topology.

From these considerations, we take for granted that

† In Section 1, as the distance from the substation decreases,the rail current increases because of the return from earth,until it reaches the value Ib at the point where the substationis connected to the rail. Therefore the rail potential must benegative in this section.† At the end of Section 2, current Ia is injected, whichdissipates along the rail as we approach the substation. Inthe midpoint of the section, this current begins to emerge


on the rail from the earth, until its value is completelyrestored at the substation connection point to the rail. As aresult, there will be a positive rail voltage distribution nearthe current injection point, and a negative distribution in thevicinity of the substation.† Finally, the injected current, Ib, in Section 3 is dissipatedalong the rail, until it reaches a null value at its end.Therefore the rail voltage in this section has a positive sign.

By applying Kirchhoff’s first law at any differential meshon the circuit shown above, we obtain the differentialequation, (3), which defines the physical phenomenon ofcurrent dissipation in a steel rail supported on an uniformlydistributed set of sleepers

d2i(y)

d2y−M2 i(y) = 0

M = 1/LC =��R′R G

′RE

√(3)

Where i(y) is the rail current at any point, y, M is the railpropagation constant and LC is the rail surge orcharacteristic length. This equation has a hyperbolic sinesolution, according to expression (4)

i(y) = C1eyM + C2e

−yM (4)

The value of the constants C1 and C2 is determined from theboundary conditions [11]. Expressions of rail current, i(y), atany point, y, as a function of injected current, I(x), are shownin Table 4.Where x is the point where the train or short-circuit current,

I(x), is injected and y is the position where the rail current, i(y), and rail voltage, URE(y), are measured.Rail voltages at every point, URE(y), are obtained by taking

into account the relation between voltage and stray current, di(y), shown in (5), [11]

URE(y) = − 1

G′RE

di(y)

dy(5)

Where di(y)/dy is the stray current linear density (mA/m).Therefore with this equation it is possible to obtain an

Fig. 3 Mathematical model of the rail


www.ietdl.org

accurate value of the linear stray current, according toEN-50122–2, provided that the calculation takes intoaccount the average density of rail traffic on the railwayline. It is important to point out that according to standardEN 50122–2, the average permissible value, during a year,for not obtaining an appreciable corrosion on the rail in 25years, is 2.5 mA/m. Not only does this value depend on thecurrent injected into the railway lane by each train, but alsoon the train frequency, according to the traffic grid. Thecurrent injected at each specific abscissa y is defined by (6)

di(y)

dy= −G′

REURE(y) (6)

After deriving, we obtain the final equations for rail voltage,URE(x), shown in Table 5.Where ZC, is the rail surge or characteristics impedance

ZC = M

G′RE

=��R′R

G′RE

√(7)

It is important to take into account that the magnitude ofvoltage equation in Section 2, shown in Table 5 isessentially the same as that listed in the standard EN-50122,with the opposite sign criteria, equation (C.1).

URE(y) = −ZCI(x)

21− e

−x

LC

⎡⎣

⎤⎦ (8)

However, this standard applies a simplified expression since itonly allows to obtain the rail voltage at the point where thecurrent is injected, x. Nevertheless, our method is more

Table 4 Rail current expressions

Section 1, y ∈ (−∞, 0)

i(y) = − I(x)2

exM + e−xM − 2exM − 1

[ ]e−yM

Section 2, y ∈ (0, x)

i(y) = − I(x)2

1− e−xM( )

eyM + exM − 1( )

e−yM

exM − 1

⎡⎣

⎤⎦

Section 3, y ∈ (x,∞)

i(y) = I(x)2


[ ]e−yM

Table 5 Rail voltage expressions

Section 1, y ∈ (−∞, 0)

URE(y) = −ZCI(x)2


[ ]e−yM

Section 2, y ∈ (0, x)

URE(y) = ZCI(x)2

1− e−xM( )

eyM − exM − 1( )

e−yM

exM − 1

⎡⎣

⎤⎦

Section 3, y ∈ (x,∞)

URE(y) = ZCI(x)2


[ ]e−yM


general, since it allows to calculate the voltage available ina generic position, y, because of a train or short circuitinjecting current at position x, therefore the mathematicalexpressions are more complicated.If we want to obtain the rail voltage at the point of current

injection, we have to impose y = x in the voltage equation ofSection 2, so we obtain the following expression

URE(x) = ZCI(x)

2

exM + e−xM − 2

exM − 1

[ ]

= ZCI(x)

21+ e−xM − 1

exM − 1

[ ](9)

For high values of x, we obtain the same expression asequation (C.1) of standard EN-50122–2

URE(x) = ZCI(x)

21+ e−xM − 1

exM − 1

[ ]= ZC

I(x)

21+ −1

exM

[ ]

= ZCI(x)

21− e−xM[ ]

(10)

Nevertheless, there is a loss of very important information inthe proximity of the substation, this is to say, when x→ 0.The expressions developed by us coincide with those of the

standard EN 50122–2 when taking into account the specificconditions of application of Annex C.

4 Simulations in normal conditions

When carrying out simulations, we always considersteady-state conditions, since the rail model does notinclude dynamic phenomena. This is equivalent to acceptthat the speed of the train is very low in comparison withthe speed propagation of electromagnetic phenomena.The conditions under which simulations are carried out are

shown in Tables 2 and 3. Besides, the power required by thetrain, P(x), is considered to be constant, and it is arbitrarily setto1 MW. The nominal voltage for DC long-distance mainrailway lines in Spain is 3000 V, so the maximum internalvoltage of the feeding substation, Uss, has been set to 3600V, according to the Spanish specifications.

4.1 Rail voltages

Maximum rail voltages for different railway features areshown in Table 6. It is important to highlight that now weare only calculating rail voltages at the point where the trainis placed, y = x at Section 2 according to Table 5, (11).Where y = x is the point with the maximum rail voltage,because the current in the rail, i(y), is also maximum at thispoint

URE(y) = ZCI(x)

2

eyM + e−yM − 2

eyM − 1

[ ](11)

The current consumed by the train at each point, I(x), has beenobtained applying a power flow method, using (1) and P(x) =1 MW.It can be observed that rail voltage increases when R’R, or

ZC, increases because current dissipated into the earth goes upas well. On the other hand, rail voltage increases when G’RE


www.ietdl.org

decreases because of the improvement in the rail insulationlevel, Fig. 4.According to Fig. 4, it can be proved that rail voltage is

always 0 V when the train is placed at substation, x = 0,since all the current injected in the rail circulates throughthe substation, according to first Kirchhoff’s law, and thereare not any stray currents along the rail. We can prove itmathematically, applying L’Hopital’s rule to (11). Thereforewe obtain that URE(y = 0) = 0.According to EN-50122–1, Section 9.3.2.1, in the steady

state, accessible voltage cannot exceed the limit set by thestandard, 120 V. Accessible voltages in Fig. 4 does notexceed this limit because operation currents, I(x), are not sohigh as those found in short-circuit conditions. However,when several vehicles are travelling along the same feedsection, the rail potentials will increase because of thesuperposition of remote effects, and therefore high railvoltages will appear [14].

4.2 Spatial rail voltage distribution

It is essential to study the remote effects caused by the traincurrent, that is, the rail voltage at any point, y, differentfrom that one, x, where the current has been injected. Notonly does this method show voltage levels at the pointwhere the vehicle is placed, x, but it lets us quantify itsremote effects upstream and downstream as well. For CaseII, the final result is a spatial distribution of rail voltagesshown in Fig. 5. Where the X-axis indicates the point wherethe vehicle is placed, and the Y-axis shows the point atwhich the rail voltage is measured. Z-axis registers railvoltage magnitude.

Table 6 Voltage magnitudes for different railway features, innormal conditions

Case Constants Maximumvoltage

Point

I R’R = 0.01 Ω/km, G’RE = 5 S/km,M = 0.224 km− 1, ZC = 0.045 Ω

5.8 V 10 km

II R’R = 0.01 Ω/km, G’RE = 1 S/km,M = 0.1 km− 1, ZC = 0.1 Ω

9.2 V 10 km

III R’R = 0.02 Ω/km, G’RE = 1 S/km,M = 0.14 km− 1, ZC = 0.14 Ω

15.8 V 10 km


When cutting this three-dimensional (3D) distribution withthe vertical plane X = Y we obtain the rail voltage distributionat the point where the train is located, x, that is, we obtainexactly the function shown in Fig. 4, Case II.If we suppose that the train is placed at x = 3 km, the rail

voltage distribution along the railway line, y, is obtained bycutting the spatial distribution through the plane X = 3 km,Fig. 6.The dotted line shows the wave-front rail voltage

distribution along the rail as a result of the remote effectcaused by the train located at kilometer x = 3 km. Thiscurve is divided into two different sections. Sectionupstream vehicle location is Section 2 in Fig. 2, while thedownstream rail portion is Section 3, which has atheoretically infinite length.† Section 2 (upstream from the train).At the point where the train is placed, rail voltage reaches

the value of 3.7 V. As we approach the substation, y = 0 km,rail potential decreases to a value of zero at y = x/2 = 1.5 km.From that point, rail voltage is negative, reaching –3.7 V aty = 0 km.† Section 3 (downstream from the train).The voltage tends to zero along a decaying exponential

function as a result of leakage current. At the end, the railpotential is zero, but it is always positive, Table 5.The solid curve is the envelope of maxima of the dotted

one, and coincides with the Case II curve shown in Fig. 4.To obtain the solid curve we have to impose that y = x inthe URE(y) equation in Section 2 in Table 4, that is to say,we obtain expression (11).If we cut the surface shown in Fig. 5 using different planes,

X = x, we obtain a succession of rail voltage distributionsalong the supply section, y, because of the train movement.

4.3 Rail voltage evolution at a fixed point

The model can also determine the evolution of the rail voltageat a fixed point, y, when the train moves along the railwayline. If we want to measure voltage at y = 3 km when thetrain is moving from x = 0 km to x = 10 km, we have tocut the surface on Fig. 5 by the plane Y = 3 km, obtainingFig. 7.Voltage at y = 3 km is zero when the train is placed at the

substation, x = 0 km, because all the current flows throughthe substation, and there is no current in the rail. As

Fig. 4 Rail voltages in normal conditions


www.ietdl.org

Fig. 5 3D rail voltage distribution

Fig. 6 Bi-dimensional distribution of rail voltage because of a train placed in x = 3 km

Fig. 7 Rail voltage at the fixed point y = 3 km

IET Electr. Syst. Transp., 2013, Vol. 3, Iss. 2, pp. 31–40 37doi: 10.1049/iet-est.2012.0022 & The Institution of Engineering and Technology 2013

www.ietdl.org
expected, when the train is at y = 3 km, voltage reaches3.7 V.
5 Simulations under short-circuit conditions

In this case, simulations are obtained moving the point wherethe short circuit occurs.

5.1 Rail voltages

Maximum rail voltages for different railway features areshown in Table 7. It is important to point out that we areonly calculating rail voltages at the point where the shortcircuit takes place, y = x, at Section 2 according to Table 5,(11). Where y = x is the point with the maximum railvoltage, because the current in the rail, i(y), is alsomaximum at this point.In this case, I(x), is the short-circuit current obtained from

(1), imposing that the voltage between the catenary and therails is zero, U(x) = 0 V.According to railway features shown in Table 7, we obtain

the rail voltages shown in Fig. 8.According to EN-50122, Section 7.3.1, Table 4, in the

steady state (t > 0.5 s), contact voltages cannot exceed thelimit set by the standard, that is, 395 V. As we can see inFig. 8 contact voltages do not exceed this limit, whichproves the validity of the developed model. However, thesesustained conditions do not appear in practice, because theelectrical protections at the substation will trip and clear thefault in a very short time.

5.2 Spatial rail voltage distribution

In the 3D rail voltage distribution, X-axis indicates the pointwhere the short circuit takes place, and Y-axis shows thepoint where the rail voltage is measured. Z-axis registers themagnitude of rail voltage. Fig. 9 shows the 3D rail potentialdistribution for Case II.When cutting the spatial distribution through the plane

X = Y we obtain the rail voltage at the point where theshort circuit occurs, x, that is, we obtain the curve shown inFig. 8, for Case II.


If a short circuit takes place at x = 3 km, rail voltagedistribution along the railway line, y, is shown inFig. 10. In this case, we have extended the analysis for alength of 50 km to fully appreciate the overall voltageprofile.The dotted line shows the wave-front rail voltage

distribution along the rail as a result of remote effectscaused by the short circuit. This curve is divided into twodifferent sections, as we previously mentioned for normalconditions. The solid curve is the envelope of the maximaof the dotted one, and it coincides with Case II shown inFig. 8, but with the abscissa axis increased up to the pointx = 50 km.To obtain the solid curve we have to impose that y = x in

the URE(x) equation at Section 2 in Table 5, that is to say,we obtain expression (11).When cutting the surface shown in Fig. 9 by means of

different planes X = x, we obtain a succession of railvoltage distributions along the supply section because of theshort-circuit displacement.

5.3 Rail voltage evolution at a fixed point

To calculate rail voltage at kilometer y = 3 km, when theshort circuit moves along the railway line, we cut thesurface of Fig. 9 through the plane Y = 3 km, Fig. 11.The rail voltage at y = 3 km is zero when the short circuit

occurs at the negative terminal connection of the substation tothe rail, x = 0 km, because all the current flows through thesubstation, and there is no current in the rail. As expected,

Table 7 Voltage magnitudes in short-circuit conditions

Case Constants Maximumvoltage

Point

I R’R = 0.01 Ω/km, G’RE = 5 S/km,M = 0.224 km− 1, ZC = 0.045 Ω

291.5 V 6.4 km

II R’R = 0.01 Ω/km, G’RE = 1 S/km,M = 0.1 km− 1, ZC = 0.1 Ω

190.9 V 8.6 km

III R’R = 0.02 Ω/km, G’RE = 1 S/km,M = 0.14 km− 1, ZC = 0.14 Ω

135.7 V 5.4 km

Fig. 8 Rail voltages in short-circuit conditions


www.ietdl.org

Fig. 9 Three-dimension rail voltage distribution

Fig. 11 Rail voltage at the fixed point y = 3 km

Fig. 10 Bi-dimensional distribution of rail voltage because of a short circuit in x = 3 km

IET Electr. Syst. Transp., 2013, Vol. 3, Iss. 2, pp. 31–40 39doi: 10.1049/iet-est.2012.0022 & The Institution of Engineering and Technology 2013

www.ietdl.org
when the short circuit occurs at x = 3 km, the voltage reachesa positive value of 146 V.
6 Conclusions

The main problem in rail traction systems is to feed loads thatpresent a variable-geometry topology. The load current, I(x),and pantograph-to-rail voltage, U(x), have to be obtained as afunction of the power drawn by the vehicle, P(x), using powerflow methods. It is not correct to assume a constant loadcurrent, I, since it depends on many factors as we pointedout in this paper. Therefore this is the most accuratemethod, instead of using the direct absorbed current by thetrain fed by nominal voltage, and probably the most straightforward method to include multiple trains fed by the samesection (multiple train study).Once our uniformly distributed sleeper model has been

developed, it is possible to obtain stray currents, di(x), straycurrents linear density, di(x)/dx, as well as rail potentialdistributions, URE(x). The wave-front rail voltage can beobtained, from the 3D rail voltage distributions, as a remoteeffect of a moving train or a short circuit being produced atdifferent locations along the rail. This information allows usto design railway traction tracks and substation spacing,under normal and fault system operation conditions, takinginto account rail voltage and stray currents linear densitylimits. At the same time, 3D rail voltage distribution allowsvoltage evolution at a fixed point because of remote effectsto be obtained. Therefore we are able to obtain theinstantaneous rail voltage distribution along any section,and for any position of the train or short circuit.Another important feature of this model is that it has been

developed as a part of a railway traction software, so we cancalculate the equations with real parameters which vary at anypoint of the railway lane, x, and even with the time, t.

7 Acknowledgment

This work has been funded and supported by the Spanishcompany ELECTREN S.A. (www.electren.es) as part of theactivities of the ‘ELECTREN Chair for R&D in ElectricTraction Systems’ at Carlos III University of Madrid.


8 References

1 Yu, J.G., Goodman, C.J.: ‘Modelling of rail potential rise and leakagecurrent of DC rail transit systems’. IEE Colloquium, Stray CurrentEffects of DC Railways and Tramways, October 1990, pp. 221–226

2 Cai, Y., Irving, M.R., Case, S.H.: ‘Modeling and numerical solution ofmultibranched DC rail traction power systems’, IEE Proc. Electr. PowerAppl., 1995, 142, (5), pp. 323–328

3 Paul, D.: ‘DC traction power system grounding’, IEEE Trans. Ind. Appl.,2002, 38, (3), pp. 818–824

4 Lee, C.H., Lu, C.J.: ‘Assessment of grounding schemes on rail potentialand stray currents in a DC transit systems’, IEEE Trans. Power Deliv.,2006, 21, (4), pp. 1941–1947

5 Alamuti, M.M., Nouri, H., Jamali, S.: ‘Effects of earthing systems onstray current for corrosion and safety behaviour in practical metrosystems’, IET Electr. Syst. Trans., 2011, 1, pp. 69–79

6 Charalambous, C., Cotton, I.: ‘Influence of soil structures on corrosionperformance of floating-DC transit systems’, IET Electr. Power Appl.,2007, 1, pp. 9–16

7 Brenna, M., Dolara, A., Leva, S., Zaninelli, D.: ‘Effects of the DC straycurrents on subway tunnel structures evaluated by FEM Analysis’. IEEEPower and Energy Society General Meeting, 25–29 July 2010, pp. 1–7

8 Cotton, I., Charalambous, C., Aylott, P., Ernst, P.: ‘Stray current controlin DC mass transit systems’, IEEE Trans. Veh. Technol., 2005, 52,pp. 722–730

9 Jamali, S., Alamuti, M.M., Savaghebi, M.: ‘Effects of different earthingschemes on the stray current in rail transit systems’. Proc. 43rd Int.Universities Power Engineering Conf. (UPEC 2008), 43rdInternational, September 2008, pp. 1–5

10 Puschmann, K., Schneider, S., ‘Contact lines for electric railways’(SIEMENS, 2009, 2nd edn.)

11 Rodríguez, J.V., ‘Analysis and simulation of rail potentials in DCrailway systems’. Master thesis dissertation. Electrical EngineeringDepartment, Carlos III University of Madrid, November 2011.Available at http://www.hdl.handle.net/10016/15944

12 Fotouhi, R., Farshad, S.: ‘A new novel power electronic circuit to reducestray current and rail potential in DC railway’. Proc. 13th PowerElectronics and Motion Control Conf. 2008, (EPE-PEMC 2008), 1–3September 2008, pp. 1575–1580

13 Niasati, M., Gholami, A.: ‘Overview of stray current control in DCrailway systems’, Railway Engineering-Challenges for RailwayTransportation in Information Age, 2008, (ICRE 2008), 25–28 March2008, pp. 1–6

14 Tzeng, Y.-S., Lee, C.-H.: ‘Analysis of rail potential and stray currents ina direct-current transit system’, IEEE Trans. Power Deliv., 2010, 25,pp. 1516–1525

15 Pham, K.D., Thomas, R.S., Stinger, W.E.: ‘Analysis of stray current,track-to-earth potentials and substation negative grounding in DCtraction electrification system’, Railroad Conf., 2001. Proc. 2001IEEE/ASME Joint, 2001, pp. 141–160


Date post:	24-Dec-2016
Category:	Documents
Upload:	javier-sanz
View:	228 times
Download:	2 times

Calculation of remote effects of stray currents on rail voltages in DC railways systems

Documents