Localization Method for Insulation Degradation Area of the Metro Rail-to-Ground Based on Monitor Information

Abstract

Since rail-to-ground insulation decreases, large-level direct currents (DCs) leak from railways and form metro stray currents, corroding the buried metal. To locate the rail-to-ground insulation deterioration area, a location method is proposed based on parameter identification methods and the monitored information including the station rail potentials, currents at the traction power substations (TPSs), and train traction currents and train positions. According to the monitoring information of two adjacent TPSs, the section location model of the metro line is proposed, in which the rail-to-ground conductances of the test section are equivalent to the lumped parameters. Using the rail resistivity and traction currents as the known information, the rail-to-ground conductances are calculated with the least square method (LSM). The rail-to-ground insulation deterioration sections are identified by comparing the calculated conductances with thresholds determined by the standard requirements and section lengths. Then, according to the section location results, a detailed location model of the degradation section is proposed, considering the location distance accuracy. Using the genetic algorithm (GA) to calculate the rail-to-ground conductances, degradation positions are located by comparing the threshold calculated with the standard requirements and location distance accuracy. The location method is verified by comparing the calculation results under different degradation conditions. Moreover, the applications of the proposed method to different degradation lengths and different numbers of degradation sections are analyzed. The results show that the proposed method can locate rail-to-ground insulation deterioration areas.

1. Introduction

Since the metro rail is incompletely insulated to the ground, direct current (DC) may leak from the rail, corroding metal [1,2,3,4,5]. Thus, to control the metro stray current level, rail-to-ground conductance per track is requested to be smaller than 0.5 S/km per track of the metro line [6,7,8,9]. However, in the field, affected by waterlogging, deterioration of rail fasteners, and accumulation of iron scraps and dust, rail-to-ground insulation decreases, which can cause an increase in the leakage of metro stray currents from rail tracks [10]. Large amounts of metro stray currents can accelerate the corrosion of buried metal, raise ground potential, interfere with communications, and flow into urban power systems, leading transformers to work in DC-biased statuses [3,11,12,13]. To decrease metro stray currents, many mitigation methods such as painting insulating coatings on the track, replacing deteriorated fasteners, and changing insulated sleepers have all been used in the field [14]. However, due to the unknown location of rail-to-ground insulation degradation, these mitigation methods are blind, ineffective, and costly. Thus, it is necessary to locate rail-to-ground insulation deterioration areas to further serve the implementation of stray current mitigation.

Rail-to-ground conductance indicates the rail-to-ground insulation level. Thus, to directly obtain rail-to-ground conductance in the field, many offline test methods have been provided in standards and published studies [7,8,15,16]. In metro tunnel lines, the conductance per length of rail-to-reinforced structures is tested to indicate the rail-to-ground conductance level. To add a DC source between the rail and the structural metal, use Hall sensors and voltage sensors to measure the currents and voltage between the rail and structural metal at a certain distance [7,8]. According to the test data, rail-to-ground resistivity is calculated based on Ohm’s theorem. In metro lines without civil structures, a DC source is added between two sides of the rail insulation joint. Through testing rail-to-ground potentials, rail-to-ground conductance is calculated using Ohm’s theorem based on the DC value and potentials under the on and off DC source. Furthermore, a high-frequency current is used as an injected current source, and a decrease in rail-to-ground conductance is located based on the traveling wave theory [15,16]. However, since the test methods must not affect the normal operation of metro trains, the measurements for rail-to-ground conductance must be applied during the stoppage time of a metro system. Moreover, because the test methods must add other sensors and current sources on the track, the test is inefficient; they can only obtain rail-to-ground insulation results within a test distance. Therefore, using the above-mentioned test methods, rail-to-ground insulation degradation areas are hard to obtain in a timely manner.

At present, no method can locate rail-to-ground insulation degradation sections directly and quickly. However, the effect of rail-to-ground conductance on the rail potential, return current, and stray current has already been fully analyzed based on calculation models, simulation models, and finite element models [1,10,12,17]. The analyses show that electrical information changes significantly with the difference in rail-to-ground conductance. When rail-to-ground conductance decreases, the rail potential level at the degradation section decreases. Then, through the insulation degradation position, the leakage current and stray current increase, leading to the return current in the rail track to decrease simultaneously. Moreover, the rail potential and return current have been monitored in metro systems at stations and traction power substations (TPSs). Thus, considering obtaining rail-to-ground insulation degradation in time, monitoring information may be useful.

To locate rail-to-ground insulation degradation, a new location method is proposed based on the monitoring information in a metro system. In the proposed method, a section location model is established between two adjacent TPSs, in which rail-to-ground conductances of the section are equivalent to the unknown lumped parameters. Based on the monitoring information of feeder currents, the rail resistivity, and traction currents, a calculation process with the least square method (LSM) is proposed to calculate the unknown conductances. Based on the conductance results, the rail-to-ground insulation deterioration section is identified by comparing the results with the threshold. Then, according to the section identification result, a detailed location model of the degradation section is established, considering the location distance accuracy. When using the genetic algorithm (GA) to calculate rail-to-ground conductances in the detailed location model, the degradation positions are located by comparing the threshold calculated with the standard requirements and the location distance accuracy. By comparing the calculation result with the different degradation conditions, the location method is verified. Moreover, the applications of the proposed method to degradation lengths and multiple degradation sections are analyzed.

The remainder of this paper is organized as follows. Section 2 describes the monitoring information and proposes the section location method. Section 3 proposes the detail location model and location method principle. The proposed method is verified and analyzed in Section 4. Section 5 concludes this paper.

2. Section Location Method

2.1. Monitored Information of Metro Systems

In the metro system, to monitor the safety of devices and trains, many online transducers and sensors are installed in stations, TPSs, and trains to test the rail voltage, current, power, and train speed and positions [13,18,19,20,21]. In the station, through an over-voltage protection device (OVPD), the rail-to-ground potential is monitored. Moreover, in a station with TPS, the traction current, feeder currents, and return currents are monitored by the rectifier, feeder device, and negative device, respectively. The monitored information of metro systems is shown in Figure 1 at the point where a train operates on the metro line.

In Figure 1, the marked electrical and positional parameters are all the known information tested and saved by monitoring devices and systems. Stations $\#1$ and $\#2$, including TPSs, are located at positions $S_1$ and $S_2$. $V_{s1}$ and $V_{s2}$ are the rail potentials at stations tested by OVPDs. Since the OVPD connects the metro rail and the grounding system, the rail potential is also the rail-to-ground voltage (i.e., rail voltage). $I_{01}$, $I_{02}$, $I_{03}$, $I_{04}$, $I_{11}$, $I_{12}$, $I_{13}$, and $I_{14}$ are the feeder currents for the catenary provided by the TPSs monitored by feeder devices. $I_{s1}$ and $I_{s2}$ are the traction currents provided by rectifiers in TPSs. $I_{05}$, $I_{06}$, $I_{15}$ and $I_{16}$ are the return currents tested by negative devices in TPSs. $P_T$ and $I_T$ are the train positions and traction currents monitored by the train automation traction system. Moreover, when the train operates, the train traction current is provided by all TPSs along the metro line.

Based on the metro monitoring devices and systems, the rail potentials, traction currents, return currents, and feeder currents in stations along the metro line, as well as the train traction currents and positions, can be obtained synchronously. Thus, when the train operates on the metro line, the test section between two adjacent TPSs can be modeled as a DC resistance network model.

2.2. Section Location Model

Based on the monitoring information of two adjacent TPSs, a section location model can be established using a DC resistance network modeling method [2]. In the model, the rail resistances between two TPSs are equivalent to the lumped conductances, and the rail-to-ground conductances are modeled as lumped conductances. The section location model of the metro line is shown in Figure 2.

In the model, the white parameters $Y_2$ and $Y_3$ are the rail conductances, which are the known parameters calculated based on per-length rail resistance and rail lengths. The rail length is determined by the distance between the train and TPS. The gray parameters $Y_1$, $Y_4$, $Y_5$, $Y_6$, $Y_7$, and $Y_8$ are the unknown parameters. $Y_1$ and $Y_4$ are the rail conductances of the rest metro line. $Y_5$ and $Y_8$ express the rail-to-ground conductances of the rest metro line. $Y_6$ and $Y_7$ are the rail-to-ground conductances in the test section.

To identify whether there is insulation degradation in a section, rail-to-ground conductances are set as the circuital parameters. Thus, the important process of the section location method is to calculate the unknown parameters, as shown in Figure 2.

2.3. Unknown Parameter Calculation and Section Identification

The section location model is a DC resistance circuit network model, which satisfies Kirchhoff’s theorem and Ohm’s theorem, as follows:

$$\begin{array}{c}\hfill \mathbf{I}=\mathbf{YU}=\left[\begin{array}{c}{I}_1\\ I_2\\ I_3\\ I_4\\ I_5\end{array}\right]=\left[\begin{array}{ccccc}{Y}_{11}& Y_{12}& 0& 0& 0\\ Y_{12}& Y_{22}& Y_{23}& 0& 0\\ 0& Y_{23}& Y_{33}& Y_{34}& 0\\ 0& 0& Y_{34}& Y_{44}& Y_{45}\\ 0& 0& 0& Y_{45}& Y_{55}\end{array}\right]\left[\begin{array}{c}{U}_1\\ U_2\\ U_3\\ U_4\\ U_5\end{array}\right]\end{array}$$

(1)

where $\mathbf{I}$ is the node current matrix; $\mathbf{Y}$ is the node conductance matrix; $\mathbf{U}$ is the node voltage matrix. In the equation, all node currents are the known information calculated by monitoring currents at TPSs and trains in the test section. The node currents at $S_1$ and $S_2$ are the traction currents of rectifiers. The node current at the train position is the train traction current $I_T$. The node currents at the rest lines are calculated by the feeder currents at $S_1$ and $S_2$. Thus, Equation (1) can be described as follows:

$$\begin{array}{c}\hfill \left[\begin{array}{c}-(I_{01}+I_{02})\\ -I_{s1}\\ I_T\\ -I_{s2}\\ -(I_{13}+I_{14})\end{array}\right]=\left[\begin{array}{ccccc}{Y}_1+Y_5& -Y_1& 0& 0& 0\\ -Y_1& Y_1+Y_2+Y_6& -Y_2& 0& 0\\ 0& -Y_2& Y_2+Y_3& -Y_3& 0\\ 0& 0& -Y_3& Y_3+Y_4+Y_7& -Y_4\\ 0& 0& 0& -Y_4& Y_4+Y_8\end{array}\right]\left[\begin{array}{c}{U}_1\\ U_2\\ U_3\\ U_4\\ U_5\end{array}\right]\end{array}$$

(2)

where $U_2$ and $U_4$ are the test rail potentials $V_{S1}$ and $V_{S2}$ connected via the OVPD at end stations of the test section. $U_3$ can be calculated by $U_2$, $U_4$, and $I_T$ as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{U}_2=V_{S1}\hfill \\ U_4=V_{S2}\hfill \\ U_3=(Y_2{U}_2+Y_3{U}_4+I_T)/(Y_2+Y_3)\hfill \end{array}\right.\end{array}$$

(3)

In Equation (2), there are 6 unknown conductance and 2 unknown node voltages. The number of unknown parameters is greater than the number of equations. Thus, rail-to-ground conductances cannot be directly calculated by Kirchhoff’s theorem and Ohm’s theorem. To solve the large number of unknown parameters, the least square method (LSM) is used to calculate rail-to-ground conductances.

The objective function of the LSM is to minimize the error in calculation and test results of station voltages, as follows:

$$\begin{array}{c}\hfill \begin{array}{c}minf={\parallel \mathbf{u}-\mathbf{V}\parallel }_2^2\end{array}\end{array}$$

(4)

where $\mathbf{u}$ is the calculation results of the station rail voltage $U_2$ and $U_4$ in Equation (2); $\mathbf{V}$ is the test results of the station rail voltage $V_{S1}$ and $V_{S2}$. Rail voltages $U_2$ and $U_4$ can be expressed via conductances, as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{U}_2={\displaystyle \frac{J_0+J_1+J_2+I_2{Y}_3{Y}_5{Y}_7{Y}_8+I_2{Y}_3{Y}_4{Y}_5{Y}_8+(Y_7+Y_8)I_2{Y}_2{Y}_3{Y}_5+(I_3+I_4)(Y_2{Y}_3{Y}_5{Y}_8+Y_1{Y}_2{Y}_3{Y}_8)}{(Y_5+Y_6)X_1+(Y_7+Y_8)X_2+Y_5{Y}_8(Y_2{Y}_3{Y}_6+Y_2{Y}_3{Y}_7+Y_2{Y}_6{Y}_7+Y_3{Y}_6{Y}_7)+Y_1{Y}_2{Y}_3{Y}_7{Y}_8+Y_2{Y}_3{Y}_4{Y}_5{Y}_6}}\hfill \\ U_4={\displaystyle \frac{J_0+J_3+J_4+(I_4{Y}_2{Y}_8+(I_4+I_5)Y_2{Y}_4)(Y_1{Y}_5+Y_1{Y}_6+Y_5{Y}_6)}{(Y_5+Y_6)X_1+(Y_7+Y_8)X_2+Y_5{Y}_8(Y_2{Y}_3{Y}_6+Y_2{Y}_3{Y}_7+Y_2{Y}_6{Y}_7+Y_3{Y}_6{Y}_7)+Y_1{Y}_2{Y}_3{Y}_7{Y}_8+Y_2{Y}_3{Y}_4{Y}_5{Y}_6}}\hfill \end{array}\right.\end{array}$$

(5)

where $J_0$, $J_1$, $J_2$, $J_3$, $J_4$, $X_1$, and $X_2$ are as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{J}_0=\hfill & (I_1+I_2+I_3+I_4+I_5)Y_1{Y}_2{Y}_3{Y}_4+(I_2+I_3+I_4+I_5)Y_2{Y}_3{Y}_4{Y}_5\hfill \\ J_1=\hfill & (I_1+I_2+I_3)((Y_7+Y_8)Y_1{Y}_2{Y}_4+Y_1{Y}_2{Y}_7{Y}_8)+(I_2+I_4)Y_2{Y}_5{Y}_7{Y}_8\hfill \\ J_2=\hfill & (I_1+I_2)((Y_7+Y_8)Y_1{Y}_3{Y}_4+Y_1{Y}_3{Y}_7{Y}_8+Y_1{Y}_2{Y}_3{Y}_8)+(I_2+I_3)\left((Y_7+Y_8)Y_2{Y}_4{Y}_5\right)\hfill \\ J_3=\hfill & (I_1+I_2+I_3+I_4)Y_1{Y}_2{Y}_3{Y}_8+(I_2+I_3+I_4)Y_2{Y}_3{Y}_5{Y}_8\hfill \\ J_4=\hfill & ((I_3+I_4+I_5)Y_3{Y}_4+(I_4+I_5)Y_3{Y}_8)(Y_1{Y}_5+Y_1{Y}_6+Y_5{Y}_6)\hfill \\ X_1=\hfill & Y_1{Y}_2{Y}_3{Y}_4+Y_1{Y}_2{Y}_3{Y}_8+Y_1{Y}_3{Y}_7{Y}_8+Y_1{Y}_2{Y}_7{Y}_8\hfill \\ X_2=\hfill & Y_1{Y}_2{Y}_3{Y}_4+Y_1{Y}_2{Y}_4{Y}_5+Y_1{Y}_2{Y}_4{Y}_6+Y_1{Y}_3{Y}_4{Y}_5+Y_1{Y}_3{Y}_4{Y}_6+Y_2{Y}_3{Y}_4{Y}_5+\hfill \\ & Y_3{Y}_4{Y}_5{Y}_6+Y_2{Y}_4{Y}_5{Y}_6\hfill \end{array}\right.\end{array}$$

(6)

The constraints are that all unknown conductance parameters are greater than 0. The judgment termination is that the error f is small enough $\left({10}^{-3}\right)$. The calculation process of unknown conductances with the LSM is shown in Figure 3.

Using the LSM, the unknown conductances can be calculated in each train operation condition. Then, by collecting all the results of unknown conductances of each train operation condition, the average values are selected as the final results. The average values of $Y_6$ and $Y_7$ are the rail-to-ground conductances in the test section. Then, by making a comparison with the conductance threshold, we can determine whether the test section contains the possibility of insulation degradation.

The conductance threshold $G_{th}$ is calculated by the rail-to-ground conductivity required by the standard and the test section length, as follows:

$$\begin{array}{c}\hfill G_{th}=G_{sd}L\end{array}$$

(7)

where $G_{sd}$ is the rail-to-ground conductivity. L is the length of the test section between two adjacent TPSs. If $Y_6$ or $Y_7$ is smaller than the threshold, the test section may contain the rail-to-ground insulation degradation positions.

According to the monitoring data of rail potentials, traction currents, and feeder currents at the two adjacent TPSs, a section location model is established based on the proposed modeling method. Then, using the LSM, the rail-to-ground conductances of the section are calculated. By comparing the threshold, we can identify whether the section contains the insulation degradation area. If the section contains the degradation area, a further detailed location is needed.

3. Detailed Location Method

3.1. Detailed Location Model and Solution

The section location method can identify the rail-to-ground insulation degradation sections between two adjacent TPSs. To locate the detailed rail-to-ground insulation degradation position, a more detailed model of the rail-to-ground conductance is necessary. After the rail-to-ground insulation degradation section is located, the detailed location model of the metro line is established, as shown in Figure 4.

In the model, the insulation degradation section is divided into multiple series of parallel structures with lumped conductances. l is the location distance accuracy, impacting the number of rail resistances and rail-to-ground conductances. The rail-to-ground conductances in the section are reconstructed as $G_2$ with unknown values. The rail conductances in the insulation degradation section are reconstructed as g, calculated by the rail conductivity and location distance accuracy l. The rail conductances $Y_1$ and $Y_4$ and rail-to-ground conductances $Y_5$ and $Y_8$ are the known parameters calculated by the section location model with LSM. Moreover, the node currents of the model are all known as the section location model.

The detailed section model is a DC circuit network. Thus, based on Kirchhoff’s theorem, the node voltages of the detailed section model can be calculated as follows:

$$\begin{array}{c}\hfill {\mathbf{U}}_d={\mathbf{G}}_d^{-1}{\mathbf{I}}_d\end{array}$$

(8)

where ${\mathbf{U}}_d$ is the node voltage matrix; ${\mathbf{I}}_d$ is the node current matrix; ${\mathbf{G}}_d$ is the node conductance matrix. In the detailed model, node currents are calculated with the monitoring information of traction currents and feeder currents. However, the node voltages and conductances are still incompletely known. In the node voltage matrix, only information on the station voltages is known. The rail potential at the train position is unknown. Meanwhile, the node conductance matrix includes the known and large unknown parameters.

Affected by the unknown parameters in the voltage matrix, the unknown rail-to-ground conductances cannot be calculated directly. Thus, the GA is used to identify rail-to-ground conductances in the detailed section location model. The GA is used to calculate the conductances; moreover, the population size of the GA is the number of rail-to-ground conductances, which is $N+1$. N represents the rail conductance number in the insulation degradation section. The genes of each population are the unknown rail-to-ground conductances, which are greater than zero. The crossover rate of the population is 0.7, and the mutation rate is 0.1. The objective of the GA is to minimize errors in calculations and test results of station voltages, as follows:

$$\begin{array}{c}\hfill min{f}_d=\sum _{i=1}^N{\parallel \widehat{V}\left(i\right)-V\left(i\right)\parallel }_2^2\end{array}$$

(9)

where N is the number of stations in the insulation degradation section; $\widehat{V}$ is the calculation result of the station rail potential; V is the monitor data of the station rail potential by OVPDs. The calculation process of the detailed location model is the same as that of the section location model shown in Figure 5.

In the calculation process with the GA, the train position and train traction current are sampled based on the monitoring data. Based on the train’s information, the traction currents of TPSs are sampled at the same time. The node current matrix is obtained based on all the traction currents. Then, initialize rail-to-ground conductances, and calculate the node conductance matrix. According to Kirchhoff’s theorem and circuit node relation, the rail potentials of stations are solved based on node voltage results. Considering the objective, the individual fitness function is used to calculate $f_d$ in Function (9). The termination condition is to reach an objective number of iterations. The output results are the rail-to-ground conductances related to the minimal fitness result.

Using the proposed calculation process, the rail-to-ground conductances for each train operation condition are calculated and collected. Then, to decrease the calculation error, the outliers in the calculation results are removed, which are identified by the Z-score method [22]. The average value of the rest rail-to-ground conductances under different train operation conditions is calculated and used for the final output.

According to the rail-to-ground conductance output, the insulation degradation sections are identified by comparing them with the conductance thresholds of each section. The thresholds of each conductance $G_{th}$ are calculated based on the rail-to-ground conductivity required in the standards, as follows:

$$\begin{array}{c}\hfill G_{loc}=G_{sd}l\end{array}$$

(10)

where $G_{sd}$ is the rail-to-ground conductivity required in the standard; l is the location distance accuracy in the insulation degradation section. If the rail-to-ground conductance is larger than the threshold, the insulation degradation area lies around the related station. Finally, the degradation area is located between the large rail-to-ground conductances.

In the detailed section location method, the rail-to-ground conductances are reconstructed, considering the location distance accuracy. Then, using monitoring information and the GA, the rail-to-ground conductances are calculated. The rail-to-ground insulation degradation area is located by comparing the thresholds.

3.2. Location Method Principle

Combining the section location and detailed location processes, the rail-to-ground insulation degradation detail areas along the metro line are located. The flow chart of the location principle is shown in Figure 6.

Collecting the station and the train monitoring information is the basis for the proposed location method. In stations, the information includes the station positions, rail potentials, and traction currents supported by TPSs. In the train, the train operation position and synchronized traction current are the critical monitoring information. Then, using the position information, the metro line is equivalent to a DC resistance section model, in which the sections between stations are modeled as lumped parameters, and the rail-to-ground conductances are unknown. Using the LSM to identify the unknown parameters, the rail-to-ground insulation degradation sections are located by comparing the results with thresholds. Collecting the insulation degradation sections, the rail-to-ground conductances of each insulation degradation section are remodeled in more detail, considering the location distance accuracy. Moreover, the rail-to-ground detailed conductances are calculated with the GA, and the insulation degradation positions are identified by comparing them with the thresholds.

4. Method Verification and Analysis

4.1. Insulation Degradation Conditions

To verify the proposed methods, a metro line model is established based on the published research using CDEGS 14.0 software [12,13]. The station positions of the metro line are shown in Figure 7.

There are 13 stations, including 5 TPSs. Using the simulation method of the metro line with CDEGS, the metal structures such as rails, catenary, structure steels, and ground wire are equivalent to conductors. The rail fastener is equivalent to a rail coating with a thickness of 10 mm. Thus, the rail-to-ground insulation property is impacted by three parts, including rail coating resistivity, coating thicknes, and soil resistivity. The main parameters of the simulation model without insulation degradation are shown in

Table 1. Main parameters of the simulation model.

Model Structure	Parameter	Value
Conductor	Catenary resistance	0.008 $\Omega$/km
	Rail resistance	0.02 $\Omega$/km
	Structure steel resistance	0.06 $\Omega$/km
	Ground wire resistance	0.05 $\Omega$/km
Coating	Rail coating resistivity	1000 $\Omega ·$km
	Rail coating thickness	10 mm
Soil	soil resistivity	100 $\Omega ·$m

.

Rail-to-ground insulation degradation is simulated by decreasing the rail coating resistivity. Moreover, in this research, 1 S/km is selected as the threshold of rail-to-ground conductivity [7]. When the train operates on the metro line with different coating resistivity, the per-length conductances are calculated as the conductivity, as shown in

Table 2. Rail-to-ground conductances with different rail coating resistivity.

Conductivity	Coating Resistivity ($\mathbf{\Omega }·$km)				Conductivity	Coating Resistivity ($\mathbf{\Omega }·$km)
(S/km)	1000	100	10	1	(S/km)	1000	100	10	1
S1	0.036	0.38	3.64	39.6	S8	0.038	0.38	3.85	38.6
S2	0.038	0.38	3.85	38.6	S9	0.038	0.38	3.85	38.6
S3	0.038	0.38	3.85	38.6	S10	0.038	0.38	3.85	38.6
S4	0.038	0.38	3.85	38.6	S11	0.038	0.38	3.85	38.6
S5	0.038	0.38	3.85	38.6	S12	0.038	0.38	3.86	39.0
S6	0.036	0.38	3.85	38.6	S13	0.036	0.38	3.86	38.6
S7	0.038	0.38	3.85	38.6	Average	0.038	0.38	3.83	38.7

.

When the rail coating resistivity is 1000 $\Omega ·$km, the average conductivity is about 0.036 S/km, which satisfies the requirements of the standard. When the resistivity decreases to 100 $\Omega ·$km, the average conductivity is 0.38 S/km, which is near the threshold of 1 S/km. When the resistivity is 10 $\Omega ·$km and 1 $\Omega ·$km, the average conductivities are 3.85 S/km and 38.6 S/km larger than the threshold of 1 S/km.

Thus, 1000 $\Omega ·$km is selected as the insulation perfection conditions. Ten $\Omega ·$km and one $\Omega ·$km are selected as the insulation degradation conditions. Ten $\Omega ·$km indicates the insulation degradation, and one $\Omega ·$km indicates serious insulation degradation.

4.2. Method Verification

To verify the proposed location method, four scenarios of a single degradation position with different degradation levels are simulated based on the metro line model. The rail-to-ground insulation degradation conditions are shown in

Table 3. Rail-to-ground insulation degradation conditions.

Insulation Degradation Parameters	Scenario
	$\#\mathbf{1}$	$\#\mathbf{2}$	$\#\mathbf{3}$	$\#\mathbf{4}$
Rail coating resistivity ($\Omega ·$km)	10	1	10	1
Insulation degradation area (m)	[3800, 4300]	[3800, 4300]	[8600, 9100]	[8600, 9100]

.

The length of the insulation degradation area is 500 m. In Scenarios $\#1$ and $\#2$, the degradation area ranges from 3800 m to 4300 m in the section between S4 to S7. In Scenarios $\#3$ and $\#4$, the degradation area ranges from 8600 m to 9100 m in the section between S7 and S10. Using the section location method, the rail-to-ground conductances of each section are calculated. Moreover, the average values of the conductances are shown in

Table 4. Rail-to-ground conductance results under different sections.

Sections	Average Conductance (S)				Threshold
	$\#\mathbf{1}$	$\#\mathbf{2}$	$\#\mathbf{3}$	$\#\mathbf{4}$	(S)
S1–S4	1.52	1.54	1.71	1.77	3.2
S4–S7	7.12	18.54	1.72	1.92	2.9
S7–S10	1.83	2.12	5.46	16.72	3.5
S10–S13	1.72	1.75	1.82	1.86	2.9

.

In the results, the thresholds of each conductance are calculated based on Function (7). In Scenarios $\#1$ and $\#2$, the average conductance values in section between S4 and S7 are larger than the threshold. Thus, in Scenarios $\#1$ and $\#2$, the degradation section are from S4 to S7. In Scenarios $\#3$ and $\#4$, the average conductance values in section between S7 and S10 are larger than the threshold. Thus, in Scenarios $\#3$ and $\#4$, the degradation section are from S7 to S10. The section location results are consistent with the deterioration conditions, which verify the accuracy of the section location method.

Based on the section location results, detailed location models of the insulation degradation section are established with a location distance accuracy of 100 m. The conductance threshold is 0.1 S, which is calculated based on Function (10). The rail-to-ground conductances in the degradation sections are calculated and shown in Figure 8.

In Scenarios $\#1$ and $\#2$, the section distance from $S4$ to $S7$ is 2.9 km, and the number of rail-to-ground conductances in this section is 30, with a location distance accuracy of 100 m. In Scenarios $\#3$ and $\#4$, the section distance from $S7$ to $S10$ is 3.5 km, and the number of rail-to-ground conductances is 36, with a location distance accuracy of 100 m. The insulation degradation positions are located and shown in

Table 5. Location results with different location distance accuracies.

Scenarios	Degradation Area (m)
	Degradation Conditions	Location Results
$\#1$	[3700, 4300]	[3700, 4400]
$\#2$	[3700, 4300]	[3700, 4700]
$\#3$	[8600, 9100]	[8500, 9300]
$\#4$	[8600, 9100]	[8500, 9400]

.

Comparing the degradation conditions with the location results, the degradation areas are included in the location results. The proposed method can be used to locate the rail-to-ground insulation degradation area.

4.3. Method Analysis

Based on the single degradation section conditions, the proposed method is verified under different degradation areas and degradation degree levels. In this section, the application effect of the proposed methods under different degradation length conditions and multiple degradation section conditions is analyzed.

4.3.1. Analysis of Degradation Length

Three scenarios are set with different degradation lengths. In the three scenarios, the rail-to-ground insulation degradation is simulated with the rail coating resistivity of 1 $\Omega ·$km. The degradation area and lengths of each scenario are shown in

Table 6. Scenarios of rail-to-ground insulation degradation.

Scenarios	Rail Coating	Degradation Condition	Length (m)
	Resistivity ($\Omega ·$km)	(m)
$\#5$	10	[3800, 4300]	500
$\#6$	10	[3800, 3900]	100
$\#7$	10	[3800, 3850]	50

.

Using the proposed method to locate the degradation positions, the distance accuracy is set to 50 m. The section location and detailed location results of the three scenarios are shown in

Table 7. Location results of different degradation length scenarios.

Scenarios	Location Sections (m)	Location Positions (m)
$\#5$	S4–S7	[3700, 4400]
$\#6$	S4–S7	[3750, 3950]
$\#7$	S4–S7	[3750, 3850]

.

Comparing the location results with the degradation scenarios, the results indicate that the proposed method can locate degradation sections with different lengths. Using the proposed method to locate the detail degradation positions, the location length is larger than the scenario. However, all the degradation positions are included in the location results. Thus, the proposed method can be used to locate the different length degradation conditions.

4.3.2. Analysis of Multiple Degradation Sections

Based on the simulation model, the multiple sections degradation conditions are analyzed. Two scenarios with different numbers of degradation sections are shown in

Table 8. Scenarios of rail-to-ground insulation degradation.

Scenarios	Rail Coating	Degradation Positions
	Resistivity ($\Omega ·$ km)	(m)
$\#8$	10	[3800, 4300], [8600, 9100]
$\#9$	10	[3800, 4300], [8600, 9100] and [11,800, 12,300]

.

Using the proposed method to locate the degradation positions, the distance accuracy is set to 50 m. The section location and detailed location results of the two scenarios are shown in

Table 9. Location results of different degradation section scenarios.

Scenarios	Location Sections (m)	Location Positions (m)
$\#8$	S4–S7, S7–S10	[3700, 4400], [8600, 9350]
$\#9$	S4–S7, S7–S10, S10–S13	[3700, 4400], [8600, 9300], [11,700, 12,450]

.

Comparing the location results with the degradation scenarios, the results indicate that the proposed method can locate the degradation section. Using the proposed method to locate the detail degradation positions, the location length is larger than the scenario. However, all the degradation positions are included in the location results. Thus, the proposed method can be used to locate multiple degradation section conditions.

By analyzing the effect of the degradation length and section number, the proposed method can locate the insulation degradation sections. Based on the analysis of the comparative results, the increase in the leakage number does not lead to an increase in the number of stray current locations. Moreover, the rail-to-ground insulation sections can be identified using the section location method. Moreover, the degradation positions are included in the detailed location position results.

According to all the analysis results, the location range exceeds the actual deterioration area. The degradation positions are included in the detailed location position results. Thus, the rail-to-ground positions can be roughly determined by the detail location method. Additionally, the positioning accuracy needs to be further improved.

5. Conclusions

To locate the rail-to-ground insulation deterioration, a location method was proposed, including two steps. The first step was to locate the degradation sections, and the second step was to locate the detail positions. In the two steps, the parameter identification methods of the LSM and the GA were used to calculate rail-to-ground conductances based on monitoring information. By comparing the calculation results with the thresholds, the degradation sections and detail areas were located. To verify the proposed method, a metro line model and various scenarios were set with different degradation areas. The location results showed that the proposed method can locate the degradation area. Then, through setting various scenarios with different lengths and degradation area numbers, the applications of the proposed method were analyzed. The results showed that the proposed method can locate rail-to-ground insulation deterioration sections. Moreover, the degradation areas were included in the detailed location results. Thus, the rail-to-ground positions can be roughly determined by the detailed location method. In the detailed location method, there is still a great possibility for improvement. In the future, we will carry out more research to improve the model’s detailed location accuracy.