__________________________________________
* Poznan University of Technology.
Krzysztof BUDNIK*
Wojciech MACHCZYŃSKI*
Jan SZYMENDERSKI*
SIMULATION OF STRAY CURRENTS GENERATED BY DC ELECTRIC TRACTION
The paper presents two methods of the 3D simulation of the primary scalar potential of the electric flow field produced in the earth by d.c. traction stray currents. In the first method the equivalent rail is considered as an earth return circuit with distributed parameters, whereas in the second method the rail is treated as a circuit with lumped parameters. It is assumed in the paper that the system considered is linear, that the earth is homogeneous medium of finite conductivity and that the effects of currents in nearby underground metal installations on the potential generated in the earth by track currents (primary earth potential) can be disregarded. An extensive parametric analysis to examine the roles of various factors, which affect the primary earth potential caused by stray currents, may be performed using simulation program developed. The technical application of the method presented, which can be useful at design stage e.g. of metal structures buried in the stray currents area, is illustrated by examples of computer simulation.
KEYWORDS: stray currents, d.c. traction, earth return circuit, earth scalar potential, simulation
1. INTRODUCTION
According to the definition (EN 50122-2: 2004) stray current means the part of a current which follows paths other than the intended paths. They deviate from their intended path primarily because the resistance of the unintended path is lower than that of the intended path, or the parallel combination of the two allows part of the current to take the unintended path.
Engineering practice often deals with problems connected with harmful effects that direct current sources have on nearby earth-return circuits (pipelines, cables, etc.). The stray currents from the d.c. rail-return circuit may flow into the earth and into the underground structure (earth return circuit), returning to the rails or negative feeder taps in the vicinity of the substation or power plant. The general nature of the stray current problem is illustrated schematically in Fig. 1.
Stray currents when entering or discharging metallic buried structures can cause damage in the form of structure coating disbonding in the area where the current
enters and electrolytic corrosion in the area where it discharges. The extent of electrolytic damage is a function of the character of the metal and the amount of current and of time and soil conditions. Railway systems conventionally have the substation negative grounded to the rails. The system corrodes the rails remotely from the substation, whereas the foreign structures corrode (anodic zone) near the substation [1 – 5].
Substation
Rail current return path Current in overhead conductor
Soil current return path Pipeline Anodic zone
Electric engine
Fig. 1. Earth-return circuit in stray current area
The objective of the paper is to summarize the problems of the modeling of d.c.
stray currents generated in the vicinity of the electrified railways. The special concern will be however given to the 3D simulation of the primary scalar potential of the electric flow field produced in the earth by traction stray currents.
Two methods are presented. An analytical method of calculation is based on the complete field method of solution of the transmission-line problem. The analysis given is applicable to any d.c. railway system in which tracks can be represented by a single earth return circuit with distributed parameters (equivalent rail) with current (shunt) energization [2 – 8].
In the approximate method, the equivalent rail with current energization is modeled as a large multinode electrical equivalent circuit with lumped parameters. The circuit is a chain of basic circuits, which are equivalents of homogenous sections of the rail [9, 10].
The use of the methods presented permits to calculate such parameters as longitudinal rail current, rail leakage current, rail potential and primary earth potential. It is assumed in the paper that the system considered is linear, that the earth is isotropic, homogeneous medium of finite conductivity and that the effects of currents in nearby underground metal installations on the potential generated in the earth by track currents (primary earth potential) can be disregarded.
The analysis described in the paper may be useful in understanding effects on metal installation buried in the stray current area. The simulation model presented can be especially useful in the design stage of new earth return circuit buried in the stray current area, when frequent alterations are made as the design progresses. The efficiency of the simulation program developed is demonstrated by illustrative calculations.
2. CURRENT AND POTENTIAL OF THE EQUIVALENT RAIL OF THE D.C. RAILWAY SYSTEM
2.1. Solution for a rail modeled as a circuit with distributed parameters – exact method
The d.c. railway system shown in Fig. 1 can be applied directly by superposition in building up electrified railway system as in Fig. 2. In this system tracks are represented by a single conductor – equivalent to a rail continuously in contact with the earth through the track ballast. The conductor is energized with the currents I0 and (– I0) by a substation and an electric engine at points x = x0 and x = xL, respectively [2, 3].
I0
Ir(x) x = x0
I0
Ir(x) x = xL
+
Fig. 2. Current (shunt) energization of the equivalent rail of the d.c. railway system shown in Fig. 1
It should be noted that the basic model can be applied directly by superposition if there are a number of loads to be considered.
The starting point for the analytical solution for current and potential along an equivalent rail located along the x –axis of the Cartesian co-ordinate system is, according to the multi-conductor line theory, the system of linear differential equations:
) x ( j ) x ( dx YV
) x ( dI
) x ( f ) x ( dx ZI
) x ( dV
r r
r r
(1)
where Vr denotes the rail potential, Ir – the rail current, Z – the longitudinal impedance per unit length (p.u.l.), Y – the p.u.l. shunt admittance, and f and j are the p.u.l. external sources (longitudinal and shunt, respectively) driving the homogeneous line.
If the equivalent rail is infinite in the length and energized with the current I0
by a substation at point x = x0, the solution of the equations (1) for the current along the rail, taking into account the boundary conditions:
) 2 ( 0 I0
x
Ir , ) 2
( 0 I0
x
Ir (2)
where Ir(x0) and Ir(x0) denote the left-hand and right-hand limits of the function Ir(x) when x approaches to x0, is given in the form [2]
| x x 0 | 0
r e 0
2 )I x x ( sign )
x (
I (3)
where is the propagation constant.
Potential along the equivalent rail can be calculate from the relationship:
dx ) x ( dI Y ) 1 x (
Vr r (4)
thus taking into account formulas (2) and (3)
| x x 0 | 0
r e 0
2 Z I ) x (
V (5)
where: Y – the shunt admittance per unit length and Z0 – characteristic impedance of the equivalent rail, respectively. The details of the parameters of the rails and the equivalent rail can be found in literature, e.g. [1–5].
For the case of current energization of the rail with I0 at the point x = xL (Fig. 2), currents and potentials are calculated from the equations (3) and (4) with I0 = -I0 and x0 = xL, respectively.
Consider next the case of a finite rail extending from x = x1 to x = x2. The rail is energized with the current I0 at x = x0 and is open circuited on both ends. The current along the rail can be now determined from the following expression [7]:
x x x
x
r I e Ae Be
x x sign x
I 0 0 |0| ) 2
( )
( (6)
where A and B are constants which are to determined from the boundary conditions.
Taking into account that:
0 ) ( )
(x1 I x2
Ir r (7)
the constants A and B become:
) 1
( 2
0 2
0 e x
L sh
x x ch
A I
, ( ) 2
2
1 0
0 e x
L sh
x x ch
B I
(8)
where L = x2 x1 denotes the rail length.
It should be pointed out, that for the case of other kind of the boundary conditions, e.g. defined by impedances of finite value at rail both ends, the constants can be evaluated in similar way.
2.2. Solution for a rail modeled as a circuit with lumped parameters – approximate method
Assuming a segment of the length l of the equivalent rail to be homogeneous (e.g. Z, Y = const.), it is possible to model the circuit by a π – two port, as shown in Figure 3 [9, 10], with the series impedance
) (
0sinh l Z
Z (9)
and the shunt admittance
0
tanh 2 2
Z l Y
(10)
Z
l a)
Z
l I0
b)
2 Y
2 Y 2
Y 2
Y
Fig. 3. a) π- two port model of an elementary homogeneous segment of rail, b) π- two port model of a rail with a current source representing the current energization
The whole rail length can be subdivided into elementary cells which may have different lengths or different specific parameters.
If the rail is subjected to the external sources, the passive model (Fig. 3a) has to be completed by the active elements. This leads to a circuit representation for the current/shunt energization (substation and loads) of the equivalent rail, Fig. 3b.
After being divided into sections the equivalent rail can be composed of such basic two-ports which define the nodes and branches of the network model, which is well suited for computer - aided circuit analysis using simulation programs. The number of subdivisions of the rail can theoretically be as large as required, according to the wanted degree of discrimination in the potential and current computation.
3. SCALAR POTENTIAL IN THE EARTH DUE TO CURRENT IN THE EQUIVALENT RAIL
The knowledge of the earth potential of the electric flow field in the vicinity of the tracks is required for the evaluation of stray currents on nearby structures.
The potential (primary potential) can be obtained by the technique used in the earth return circuit theory, when the conductor with earth return carries a longitudinal current [2, 7]. The basic circuit for the calculation of the earth potential is shown in Fig. 4.
The equivalent rail is placed on the earth surface and is carrying the longitudinal current Ir(x) which flows in the positive direction of the x axis lying along the rail. The rail can be regarded as a set of current elements of length dτ.
From each element an elementary leakage current (-dIr(τ)/dτ) flows into the earth
with the conductivity γ, producing the elementary scalar potential. In the observation point P the scalar potential can be determined from the expression:
r d dI P
dV
r
e
2
) ( )
(
(11)
where r is the distance from the current element (source point) to the observation point.
y
Ir(τ) x
τ dτ
P(x,y,z) rail
soil
Fig. 4. Equivalent rail with longitudinal current flow on the earth surface
Solution for a rail modeled as a circuit with distributed parameters
For the case of infinitely long equivalent rail, located in the xy plane (y = 0, z = 0), the scalar potential in the earth becomes:
x y z d
d dI P
V
r
e 2 2 2
) (
) ( 2
) 1 (
(12)
If the equivalent rail is energized with the current I0 by a substation at point x = x0 , the scalar potential, taking into account the relationship (3), takes the form
0 0 0
0
2 2 2 2
2 2 0
) ( )
4 ( ) (
x x x
x
e d
z y x
e e d z y x
e e P I
V
(13) If the equivalent rail is energized with the current (– I0) by an electric engine at point x=xL , the scalar potential, taking into account the relationship (3), takes the form
L L L
L
x x x
x
e d
z y x
e e d z y x
e e P I
V
2 2 2 2
2 2 0
) ( )
4 ( )
( (14)
The total earth potential in the observation point P results from the superposition of the expressions (13) and (14) and can be numerically solved.
Solution for a rail modeled as a circuit with lumped parameters
The approximate method based on the equivalent circuit of the rail with current energization as shown in Fig.3b. enables one to calculate node potentials if the value of the passive and active elements of the circuit are known. The
potential in the earth in the observation point P due to leakage current flowing from the k-node to the earth can be calculated from eqn.(12) taking into account, that the leakage current density equals to VkGk and Gk denotes the per unit length shunt conductance.
Thus
c k k k
e x y z
G d P V
V ( ) 2 ( )2 2 2
(15)
where c is the integration path lying along the longitudinal branch of the π- two port.
The total earth potential in the observation point P results from the superposition
N
k k e
e x y z V x y z
V
1
) , , ( )
, ,
( (16)
where N is the number of the π- two ports modeling the equivalent rail.
It should be noted that the models of the equivalent rail with current energization and the concept of superposition allow one to consider more complicated d.c. railway systems using a segmental approximation of the complex railway route and taking into account greater number of substations and loads at any location. The earth potential in the observation point has to be evaluated for each rail segment with leakage current applying each time a new co-ordinate system and transforming appropriately boundary conditions and co- ordinates of energization points [5].
4. EXAMPLES OF CALCULATION
The usefulness and efficiency of the computation algorithm developed shall be demonstrated by an example of the calculation of the rail current, rail potential and earth potential in the vicinity of the d.c. single track traction. The equivalent rail is treated as an infinitely long earth return circuit with distributed parameters R = 0.02 Ω/km and G = 0.76 S/km and is energized at points x0 = -2.5 km (substation) and xL = 2.5 km (vehicle) with an unit-current. The earth potential due to stray currents of the d.c. electrified railway system has been calculated at the depth 1m in the soil with the conductivity γ = 0.01 S/km. The results of calculations based on the exact method are shown in Fig. 5 and Fig. 6.
Figure 7 shows a potential profile calculated below the equivalent rail at depth z = -1 m obtained by use of exact and approximate method respectively.
The parameters used in the computations are the same as in previous examples.
The length of the π- two ports of the circuit with lumped parameters lk = 100 m and N = 200. The terminating impedances at points x = ±10 km are equals to the characteristic impedance of the equivalent rail. Both curves on the Fig. 7 are nearly identical with the exception ends of curve obtained by use of the
approximate method. The reason of the discrepancy is due to the fact, that the approximate method does not take into account the influence of the leakage currents at points x > 10 km.
Fig. 5. Current and potential distributions along the equivalent rail
Fig. 6. 3D distribution of the earth potential in the stray current area
Fig. 7. Earth potential profiles according to the exact and approximate methods
5. FINAL REMARKS
The paper presents two methods of the 3D simulation of the primary scalar potential of the electric flow field produced in the earth by d.c. traction stray currents. In the first method the equivalent rail is considered as an earth return circuit with distributed parameters, whereas in the second method the rail is treated as a circuit with lumped parameters. It is assumed in the paper that the system considered is linear, that the earth is isotropic, homogeneous medium of finite conductivity and that the effects of currents in nearby underground metal installations on the potential generated in the earth by track currents (primary earth potential) can be disregarded.
The direct analytical approach used enables any physical interpretation of the phenomena being simulated. The formulas obtained in the paper require numerical integration which can be performed by the use of freely available tools.
Commercially available electrical circuit simulation packages enable modeling and simulation of complex d.c. railway systems modeled as circuits with lumped parameters to be carried out without the need to make simplifications and assumptions that may render the simulation unrealistic.
The models of the equivalent rail with current energization and the concept of superposition allow one to consider more complicated d.c. railway systems using a segmental approximation of the complex railway route and taking into account greater number of substations and loads at any location. The earth potential in the observation point has to be evaluated for each rail segment with leakage current applying each time a new co-ordinate system and transforming appropriately boundary conditions and co-ordinates of energization points.
The analysis described in the paper may be useful in understanding effects on metal installation buried in the stray current area. The simulation models presented can be especially useful in the design stage of new earth return circuit buried in the stray current area, when frequent alterations are made as the design progresses.
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