It is indicated how the stray current influences the power grid in this figure. Grounding points, including electric poles or the ground grid of the substations along the subway, form a DC current loop. The direct current flows between transformers, transmission lines and lightning lines. When subway compartment lies between substation A and station B, an un-uniform potential distribution along the line is generated by stray currents is1 and is2. The grounding points at transformer T1 and transformer T2 have a potential difference, which causes a ground loop to form through the AC grid and the earth, causing DC intrusion into the grid.
The neutral point current of transformer is affected by the operation of subway. The current in the neutral point of the main transformer has obvious macro periodicity. During the subway operation time segment—the period from 5 a.m. to 12 a.m.–the amplitude significantly increases.
2.1. The Formation Principle Of Grounding DC Loop
In this paper, a mathematical method for calculating the potential distribution near the subway is proposed. Two methods of power supply to the subway mode are widely used, including single-side feeding and two-side feeding. The impact of subway operation is larger in the case of the single-side power supply mode. A diagram of the single side power supply mode of a subway is presented in
Figure 3. As shown in
Figure 3,
I represents the load current of the locomotive and
L represents the distance between the locomotive and substation. The traction station is assumed at point O; the current positive direction is set the same as the direction of the subway compartment; and the current i(x) flowing through rail is a function of
x calculated by the formula i
s(
x) = I
i(
x).
The potential distribution along the subway is affected by many factors, including the soil structure, the power supply mode of the subway, and the driving position of the locomotive. To simplify the analysis to find the effect of each factor in the mathematical model, the following assumptions are made:
- (1)
There is no leakage current of the traction grounding grid. It is assumed that all the stray current is caused by the operation of the subway.
- (2)
The equivalent resistance of the rails, the steel mesh in the track bed and the resistivity of soil are well regulated equivalently. The subway tunnel is regarded as a line type.
- (3)
The material of the catenary is viewed as a good conductor without resistance.
There are applicable models for the general operation of a subway. The mathematical model is shown in
Figure 4, including a carriage, rail, drainage facilities and nearby regional power system.
There are two main equivalent models of locomotives widely used, including the current source model and constant voltage source model. The current source model was selected for simulation and calculation in this paper. A time-varying current source is selected to represent the total current source in the equivalent DC loop, which is numerically equal to the sum of the traction current flowing in the rail and the stray current in the soil. The traction current of the engine is represented by current source I, which is a time-varying current source.
The longitudinal equivalent resistance of two rails (Rz, in Ω/km) and the transition resistance between the rail with the buried metal (Rg, in Ω/km) can be obtained from the Engineering Design Manual. r is the resistance of the catenary in Ω/km, Rgl is the transition resistance from the buried metal to ground in Ω/km, and RM is the longitudinal resistance of the buried metal structure in Ω/km.
The voltage
u(x) at any point
x on the rail is a function of the position
x, in V. The stray current
is (x) at the position of
x is a function of
x, in A. On the foundation of Kirchhoff’s voltage principle, combined with the calculation of the node voltage and branch current of the calculation model,
u(x) and
is (x) are given as follows:
where
Differential form of formula (1) is given as formula (2), and general solution of formula (2) is given as Equation (3):
A, B are undetermined coefficients, where
The current
i(x) is presented as:
Border conditions are assumed as
i(x = 0) = I and
i(x = L) = I, then A and B are obtained as:
The expressions of voltage
u(x) and current
i(x) are obtained as:
From formula (7), when the concentration parameter is used, the expression of the stray current from the rail to the soil is:
For each position on the rail, the stray current flowing into the earth is negative while the current flowing from the earth to the rail is positive. Then, the distributed stray current density function of the soil is obtained by Equation (8) as follows:
In the model, rails are assumed as ideal conductors with evenly distributed resistance. It is supposed that the current is I, and each point of the rail from the point (0,0,h) to the point (L,0,h) has leakage of current. The Green’s function expression in the horizontal two-layer soil by the equivalent complex image method is used to solve the ground potential distribution function in the horizontal plane along the line: u(x, y, z) = f(L, x, y, z, h, H, I, ρ1, ρ2).
Since the structure of soil is mostly double-layered, it was focused on the study of two-layer soil distribution in this paper. The rail is divided into numerous parts by the discrete sampling method. Every part is seen as a current source. The vector of sources is
dis = [
dis1 dis2 dis3...disn], where
n→∞. According to the superposition theorem combined with the horizontal two layers of soil Green’s function expression, the potential distribution caused by the stray current into the ground is given as
Ug(x,y,z) as follows:
where
rn (n = 1,2,3,4) in the formula are as follows:
, , , , , k is the reflection coefficient given as .
2.2. Grid Grounding Model of Loop DC Distribution Network
The operation of the subway affects the potential distribution of regions along the subway. A DC loop is formed in the power grid through a ground potential difference at grounding points in different positions. There are neutral grounding points in the voltage level of 110 kV and above power system. So, in our research, the DC intrusion generated by the subway operation is considered only in the AC grid of 110 kV and above. A double-column transformer is widely used in a system with 110 kV or higher and all the transformers are divided into single-phase transformers and three-phase transformers. For a double-column transformer, either a single-phase transformer or a three-phase transformer can be calculated using the transformer winding DC model in
Figure 5.
For a single-phase situation, the resistance of transformer winding is given by formula (11), where R
T is the DC resistance (Ω) of the single-phase winding, P
k is the copper loss (kW) of the single-phase transformer, U
N is the rated voltage of single-phase (kV), and S
N is the rated capacity of single-phase (MVA). All these parameters are available on the transformer nameplate.
In the equivalent DC loop model of the power grid, three-phase transmission lines are regarded as three resistance branches. The transmission model is shown in
Figure 6.
The DC resistance is given by formula (12).
where
RL is the single-phase resistance (Ω) of the transmission line, ρ is the resistivity of transmission line (Ω•m),
S is the cross-sectional area (m
2) of the single-phase line, and
l is the transmission line length (m) which connects bus lines of two substations.
There is a lightning line above the transmission line in long distance transmission systems. The overhead ground wire is directly grounded through the pole tower. When there exists a tower-lightning line, the line acts as another DC intrusion loop due to ground potential difference.
Figure 7 shows a DC model of the tower-lightning line system. The calculation method is consistent with the method of the transmission line.
In the typical model of a railway system, the traction power supply system model of the subway can be divided into three or four layers. A four-layer model, as shown in
Figure 8, was chosen in this paper, consisting of rails, an electric drainage net, structural reinforcement and grounding. The longitudinal resistance of the running rails shall be low. Where
RLij is resistance of transmission line between substation
i and the substation
j. The equivalent resistance of neutral point of the transformer is
Rg. According to the real interconnected system of the regional power grid, the DC model can be extended with grounding points. Other substations nearby are also expressed in the equivalent circuit as
RTi+1 and
RTj+1, as shown in the
Figure 8. The minimum module of the equivalent model is double π unit to simplify the model.
According to the analysis above, the DC current I
ij between two substations with grounding points can be obtained by the following formula (13):
where
Vi and
Vj are, respectively, the potential of the grounding sites of the two substations,
RTi and
RTj are the equivalent resistance of the three-column winding of the two grounding transformers in parallel,
Rgi and
Rgj are the neutral point resistances of the two transformers,
RLij is the single-column equivalent DC resistance of the transmission line between the two stations,
n is the number of times that the transmission lines are returned.
The railway is usually located in the center of the city. There are interactions between multiple transformer grounding points to form a complex DC loop. The grounding resistance is not the same at every position. In this simulation, we set the grounding resistance of each grounding position as the maximum value of normal operation for the convenience of calculation. The minimum DC current in the power grid during the operation of the locomotive can be calculated by formula (14):
where
V is the voltage vector of all ground nodes in the grid,
Y is the node conductance matrix in the grid, and
I is the current of neutral points of transformers.
Y is a global quantity. In the calculation of this paper, we used a two-layer distribution model of soil to carry out the calculation of the ground potential distribution. The magnitude of the substation grounding resistance is mainly depended on the resistivity distribution at the grounding position. DC intrusion into power system can be calculated by the method we have introduced.