Localization Method and Finite Element Modelling of the Mid-Point Anchor of High-Speed Railway Distributed in Long Straight Line with Large Slope

Abstract

In order to ensure the safe and reliable operation of a high-speed railway, the precise positioning of the mid-point anchor in the catenary is very important. In view of the two problems in the calculation of the mid-point anchor position of the catenary in a long ramp section, the calculation accuracy is low, and the calculation of the central anchor clamp position is lacking. In this study, the predetermined location of the mid-point anchor is chosen based on the mid-point anchor location principle and the line condition, and the range of the allowable error of the mid-point anchor setting is determined according to the predetermined position of the mid-point anchor. Secondly, by considering the impact of the line ramp and using the measured span length, the tension difference of the clue in the direction of the line is calculated. Then, the theoretical location of the mid-point anchor clamp is determined using the downhill component and the tension difference. Finally, the theoretical position of the clamp is corrected according to the setting of the dropper to obtain the corrected position of the clamp. An FEM (finite element method) of the catenary is established in ANSYS software to calculate the height difference between the messenger cable and the contact wire at the point of the mid-point anchor setting, and then the length of the mid-point anchor rope is obtained. Through the calculation of actual case data, the maximum value of the relative error of the location of the mid-point anchor obtained by this proposed method is very small compared with the actual position, which verifies the effectiveness of this method.

1. Introduction

High-speed rail is a project with great energy consumption, and it has great practical significance to reduce its energy loss under the goal of “carbon neutrality and carbon peaking” [1]. When the train is running, the electrical energy needs to be continuously provided as its power, and the catenary can provide it with the required electrical energy. The mid-point anchor is an important part of the catenary. The mid-point anchor of the catenary can balance the tension of the left and right half-anchor clues, prevent the clues from moving, and narrow down the scope of accidents. An inaccurate location of the mid-point anchor can lead to the tension of the left and right half-anchors being unbalanced, which increases the degree of wear of the contact wires and the risk of breakage of the contact wires [2] On the one hand, it is possible to create a hard point at the mid-point anchor, which will worsen the environment in which the pantograph receives the current, resulting in a large amount of energy loss. For example, the installed power of the Fuxing EMU (electric multiple units) CR400 produced in China is 10 MW, and the energy loss caused by hard points is about 0.1–0.15% of the total power [3]. On the other hand, if a mid-point anchor rope is loose, that can also lead to faults of the pantograph and OCS (overhead contact system) [4]. In addition, in the downhill section, inaccurate positioning of the mid-point anchors will cause the catenary to wear out, thus affecting the recovery and utilization of regenerative braking energy [5,6]. Therefore, it is of great practical significance to study the location method of the mid-point anchor of a catenary in the slope section.

At present, the location method of the mid-point anchor is mainly carried out by calculating the tension difference of the clue in combination with the line conditions [3]. This method mainly determines the position of the mid-point anchor by the tension deviation of clues, calculated based on the offset of the dropper and the positioning device when temperature changes. However, the main inadequacy of this method is as follows: During the calculation, the influence of small ramps is ignored and the span length uses the average, which is inconsistent with reality. Additionally, when determining the position of the mid-point anchor, it is usually placed in the center of the span, and the corresponding theoretical analysis and calculation are lacking. As a result, it has a poor calculation precision of the mid-point anchor location, especially when the road conditions are more complicated, such as a long straight line with large slop. Due to the lack of corresponding theoretical guidance, when installing the mid-point anchor, it only can be adjusted repeatedly on site, which not only increases the workload of workers, but also causes numerous quality problems.

Before the calculation of the length of the mid-point anchor rope, the height difference between the messenger cable and the contact wire at the position of the mid-point anchor needs to be calculated. For the problem of catenary modeling, Rajasekaran et al. [7] proposed a mathematical method to accurately determine the mechanical and geometric elements of the catenary when the supports were at different heights. Wu et al. [8] proposed the parabola method which determined the length of the dropper by simplifying the initial shape of the messenger cable to a quadratic parabola. However, this method did not consider the messenger wire sag and the mechanical balance relationship between the catenary structure, and the calculation accuracy was insufficient. Fang et al. [9] proposed the dividing module method. First, the shape of the contact wire was used to determine the dropper tension, then this tension of the dropper was applied to obtain the displacement of the messenger wire. This method obtained sufficient accuracy; however, it decomposed the catenary into several subsystems, which was not conducive to the subsequent dynamic innovation. Zhou Ning et al. [10] proposed the negative sag method. By modifying the initial value calculated iteratively, the initial equilibrium state of the catenary was obtained. Li [11] compared the dividing module method with the negative sag method. Hou et al. [12,13,14] adopted the finite element method to discretize a catenary and used nonlinear iterative solutions to establish a three-dimensional model of the catenary. Tur et al. [15] developed a method to obtain an FEM of the catenary which could be used to analyze its dynamic characteristics at different load types. Liu et al. [16] proposed a multi-objective constraint and a nonlinear finite element program structure search method for the problem of low accuracy to solve the initial equilibrium state of a catenary. Song et al. [17] proposed a catenary model based on truss elements and nonlinear cable that could fully describe the initial configuration and nonlinearity of each wire. Then, Song et al. [18] proposed an iterative procedure based on the equilibrium equations of a flexible cable element to address the unbalanced forces caused by remeshing a contact wire. According to the static balance equation, Benet [19] et al. proposed a method to study the force of a catenary. This method determined the initial configuration of the catenary parameters by calculating the length of the dropper. Cho [20] studied the influence of pre-sag on the dynamic interaction between pantograph and a catenary. Kim [21] studied the sag and tension of a catenary structure with consideration of the elastic deformation. Antonio et al. [22] proposed a simplified analytical model for the dynamic characteristics of a catenary, and they regarded the speed and distance of the pantograph as free parameters to study their influence on the dynamics of the catenary. Ruan et al. [23] used the FEM to establish a calculation model for a complete anchor section, and they proposed a correction method for different line conditions such as flat curves and transition curves. Chen [24] established the FEM of a catenary in ANSYS software to simulate the installation and construction processes of the contact wire, the messenger cable, and the dropper. Jinfa et al. [25] used an FEM to establish a dynamic simulation model of a pantograph and a catenary. Menicanti et al. [26] used the finite element modeling to establish a dynamic simulation model of a pantograph catenary system.

By introducing the slope of the line, the actual span length was used in the calculation, and the position of the mid-point anchor of the catenary was calculated by using the tension difference of the clue generated in the direction of the line when the dropper and the positioning device were offset, also considering the component of the gravity of the clues along the downhill direction of the line. An FEM of the catenary was established and the height difference between the messenger cable and the contact wire was determined, so the length of the mid-point anchor rope at the location of the mid-point anchor clamp was calculated in the model. Furthermore, the actual data of a high-speed railway line is used as an example to verify the validity of this method.

2. Calculation of Central Anchor Position

2.1. Determination of the Permissible Deviation of the Mid-Point Anchor Setting

The basic principle of the mid-point anchor setting is to make the tension of the clues on both sides of the mid-point anchor as equal as possible. In the straight line, the mid-point anchors are generally set in the middle of anchor segments, and the section of the curved line should be set on the side with many curves and small radius [3].

According to the above principles and the actual situation of the line, the preset position of the mid-point anchor was determined.

2.2. Tension Deviation Resulting from the Offset of the Dropper

The n-th span was selected as the study object, which was between the span of the mid-point anchor located and the compensating device, and then all droppers of this span were equivalent to one dropper. The droppers were in plumb position at average temperature. While the temperature changed, the catenary wires expanded or shrunk, leading to the deviation of the dropper, and the suspension position of the dropper moved from point A to B. Letting ∆l_n be the amount of expansion and contraction of the trolley wire of the n-th span when the temperature changed, the dropper’s stress analysis is shown in Figure 1.

(1)$P_n\cos {\theta }_n+T_{jn+1}\sin \theta -T_{jn}\sin \theta -g_j{l}_n=0$

(2)$\cot {\theta }_n=\frac{\sqrt{c^2-{(\Delta l_n\cos \theta )}^2}}{\Delta l_n\cos \theta }\approx \frac{c}{\Delta l_n\cos \theta }$

(3)$\cot {\theta }_n=\frac{\sqrt{c^2-{(\Delta l_n\cos \theta )}^2}}{\Delta l_n\cos \theta }\approx \frac{c}{\Delta l_n\cos \theta }$

(4)$\Delta l_n=\alpha \Delta t{\displaystyle \sum _{i=1}^n{l}_i}$

Therefore, the tension deviation ΔT_jn resulting from the offset of the dropper was

(5)$\Delta T_{jn}=T_{jn+1}-T_{jn}=\frac{g_j{l}_n}{\cos \theta \cot {\theta }_n+\sin \theta }$

where g_j is the unit weight of the trolley wire, in N/m, l_n is the n-th span length, in m, θ is the slope angle of the line, c is the dropper’s average length, and α is the linear expansion coefficient of the contact wire.

Next, taking a span as the research object, analysis of the tension deviation resulting from the offset of the dropper in the line direction is shown in Figure 2. The tension deviation ΔT_jn resulting from the dropper offset is decomposed along the direction of the line, so as to obtain the tension deviation ΔT_jnx generated in the direction of the line when the dropper offset.

(6)$\sin {\phi }_1=\frac{L}{\sqrt{L^2+{(a_{\mathrm{A}}-a_{\mathrm{G}})}^2+{(L{t}_1)}^2}}$

(7)$\Delta T_{jnx}=\Delta T_{jn}\sin {\phi }_1$

where L is the span length, in m, a_A and a_G are the pull-off values of point A and G, respectively, in m, and t₁ is the slope (the slope angle is γ).

Suppose that there are m spans between the mid-point anchor and the compensator, then the total tension deviation ΔT_jdx along the direction of the line is generated when all the droppers of the m spans are offset.

(8)$\Delta T_{jdx}={\displaystyle \sum _{n=1}^m\Delta T_{jnx}}$

2.3. Tension Deviation Resulting from the Offset of the Positioning Device

The catenary is zigzag distributed in the straight line, and the horizontal force on a wire within a straight-line segment is called the zigzag force [3,27]. Temperature changes cause the wire’s elongation or shortening, positioning device shift, and generation of a tension deviation of the wire.

2.3.1. Calculation of the Zigzag Force

The heights of the suspension of the trolley wire section between suspension point of the first dropper and its adjacent anchor point can be regarded as unequal. Taking the trolley wire section between any positioning point C and the suspension point of the first dropper on its left or right as the research object, firstly, the tension T_c1 and T_c2 of the contact wire segment on both sides of the point C were calculated. Then, components of the zigzag force T_z1 and T_z2 at the point C were calculated, respectively. Finally, the zigzag force T_z at the point C was obtained.

As shown in Figure 3, the trolley wire section IC was on the left of the positioning point C, then we calculated the component of the zigzag force T_z1 as follows.

(9)$\overline{EJ}=\sqrt{L_1{}^2+{(a_1-a_2)}^2}-L_0$

Then, according to the principle of $\Delta EJI\backsim \Delta EGC$,

(10)$\overline{JI}=\frac{(\sqrt{L_1{}^2+{(a_1-a_2)}^2}-L_0)L_1{t}_1}{\sqrt{L_1{}^2+{(a_1-a_2)}^2}}$

where L₁ is the span length; a₁ and a₂ are the stagger values at the positioning point E and point C, respectively; L₀ is the distance between suspension point I of the first dropper and its adjacent positioning point C.

The equation of the unequal height suspension is

(11)$y=\frac{g_j}{2T}{x}^2+C_{1}x+C_2$

where C₁ and C₂ are real constants; T is the trolley wire rated tension, in N.

According to Equations (9) and (10), the coordinates of point I can be obtained. Then, substituting the coordinates of points C and I into Equation (11), the values of C₁ and C₂ can be obtained. Further, by deriving the curve equation and substituting the coordinates of point C, the inclination angle θ₁ of the curve at point C can be obtained. The tension T_c1 of the contact wire at point C was

(12)$T_{\mathrm{c}1}=\frac{T}{\cos {\theta }_1}$

As shown in Figure 3, the intersection of the tangent of the curve IC at point C and the projection EG of the contact wire on the horizontal plane was K. Hence, the distance between points K and G was

(13)$\overline{KG}=\overline{GC}\cot {\theta }_1=L_1{t}_1\cot {\theta }_1$

According to the principle of $\Delta GKL\backsim \Delta GEF$,

(14)$\overline{GL}=\frac{\overline{GK}}{\overline{GE}}\overline{GF}$

The tension T_c1 was decomposed along the direction of the straight line CM, and the zigzag force component T_z1 of the contact wire segment CI at point C could be obtained using Equation (15):

(15)$\{\begin{array}{l}\overline{CK}=\frac{\overline{GK}}{\cos {\theta }_1}\hfill \\ T_{\mathrm{z}1}=T_{\mathrm{c}}\frac{\overline{GL}}{\overline{CK}}\hfill \end{array}$

As shown in Figure 4, the trolley wire section CI′ was on the right of the positioning point C, then we calculated the component of the zigzag force T_z2 as follows.

The distance between C and A′ is

(16)$\overline{C{A}^{\prime }}=\sqrt{L_2{}^2+{(a_2-a_3)}^2}$

According to the principle of $\Delta C{I}^{\prime }{J}^{\prime }\backsim \Delta C{E}^{\prime }{A}^{\prime }$,

(17)$\overline{I^{\prime }{J}^{\prime }}=\frac{\overline{C{J}^{\prime }}}{\overline{C{A}^{\prime }}}\overline{E^{\prime }{A}^{\prime }}$

where a₂ and a₃ are the stagger value at the positioning point C and the point A′, respectively, m; L₂ is the span length, m.

The coordinates of point I′ can be obtained according to Equations (16) and (17). Now, substituting the coordinates of points C and I′ into Equation (11), the values of C₁ and C₂ can be obtained. Further, deriving the curve equation and substituting the coordinates of point C, the inclination angle θ₂ of the curve at point C can be obtained. The tension T_c2 of the contact wire at point C can be calculated by Equation (18):

(18)$\{\begin{array}{l}\overline{I^{\prime }{J}^{\prime }}=\frac{\overline{C{J}^{\prime }}}{\overline{C{A}^{\prime }}}\overline{E^{\prime }{A}^{\prime }}\hfill \\ T_{\mathrm{c}2}=\frac{T}{\cos {\theta }_2}\hfill \end{array}$

According to the principle of $\Delta C{L}^{\prime }{J}^{\prime }\backsim \Delta C{B}^{\prime }{A}^{\prime }$,

(19)$\overline{C{L}^{\prime }}=\frac{\overline{C{J}^{\prime }}}{\overline{C{A}^{\prime }}}\overline{C{B}^{\prime }}$

The tension in the contact wire segment CI′ at point C is decomposed along the direction of the straight line CB′, and the zigzag force component T_z2 of the contact wire segment CI′ at point C can be calculated by Equation (20):

(20)$\{\begin{array}{l}\overline{C{K}^{\prime }}=\frac{\overline{C{J}^{\prime }}}{\cos {\theta }_2}\hfill \\ T_{\mathrm{z}2}=T_{\mathrm{c}2}\frac{\overline{C{L}^{\prime }}}{\overline{C{K}^{\prime }}}\hfill \end{array}$

Therefore, the zigzag force of the positioning point C is

(21)$T_{\mathrm{zc}}=T_{\mathrm{z}1}+T_{\mathrm{z}2}$

2.3.2. Calculation of the Positioning Device Offset

For a semi-inclined catenary system, because the messenger cables are arranged in the central direction of the line, the offset of the positioning device refers to the steady arm offset. For a straight catenary system, because all the contact wires and messenger cables are arranged along the center of the line, they can be treated as a whole, and the positioning device offset refers to the cantilever offset.

Suppose that there were (m + 1) spans from the span where the mid-point anchor was located to the compensator, there should be m positioning devices. The m-th positioning device offset is

(22)$\Delta l_m=L_m{\alpha }_p\Delta t$

where α_p is the trolley wire’s or the messenger cable’s linear expansion coefficient in a semi-inclined catenary system or a straight catenary system; L_m is the distance from the m-th positioning device to the mid-point anchor, m; $\Delta t$ is the amount of temperature changed.

2.3.3. Tension Deviation Resulting from the Offset of the Positioning Device

The tension deviation ΔT_jwi of the clubs resulting from the offset of the i-th positioning device is calculated according to the i-th positioning device offset Δl_i and its length d_i, and the zigzag force T_zi at the position of the i-th positioning device between the span where the mid-point anchor located and the compensator. As shown in Figure 5, the resulting tension deviation is calculated as follows:

For the small angle ${\phi }_2$, there is $\tan {\phi }_2\approx \sin {\phi }_2$.

(23)$\Delta T_{jwi}=\frac{T_{\mathrm{zi}}}{d_{\mathrm{i}}}\Delta l_{\mathrm{i}}$

Thus, the total tension deviation ΔT_jw resulting from all the positioning devices of m spans which are between the span mid-point anchor located and the compensator is

(24)$\Delta T_{jw}={\displaystyle \sum _{i=1}^m\Delta T_{jw}{}_i}$

2.4. Downslope Component of Wire Gravity

When the catenary is in a ramp section, the weight of the thread itself will produce a downslope component in the downslope direction along the line. Among them, when the catenary is a semi-inclined catenary suspension, because the messenger cable is arranged along the center of the line, the contact wires are arranged in zigzags. The clues here refer to the contact lines; when the catenary is a vertical catenary suspension, the clue refers to both the messenger cable and trolley wire.

(25)$\Delta G_n=\sin (\arctan (t_1))g_{\mathrm{w}}{l}_n$

where ΔG_n is the downslope component of the gravitational force along the line between the mid-point anchor and the compensator, in N.

2.5. Determination of Accurate Location of the Mid-Point Anchor

The tension deviation of the half anchor section resulting from the offset of the dropper, the offset of the positioning device, and the downslope component of the wire gravity was determined as follows:

(26)$\Delta T_j=\Delta G_n+\Delta T_{jdx}+\Delta T_{jw}$

While the mid-point anchor is located on each span within its allowable margin of error in turn, tension deviations of the contact wires of the left and right half-anchor ΔT_jL and ΔT_jR were calculated according to Formulas (1)–(26). The tension deviation of contact wire ΔT_j0 between the right and the left half-anchor was calculated as follows:

(27)$\Delta T_j{}_0=\left|\left|\Delta T_{jL}\right|-\left|\Delta T_{jR}\right|\right|$

For the one-span mid-point anchor, when ΔT_j0 takes the minimum value, the span where the mid-point anchor is located is the accurate location of the mid-point anchor. For the two-span mid-point anchor, when ΔT_j0 takes the minimum or the secondary minimum value, the exact location of the mid-point anchor is at the corresponding two spans.

3. Calculation of the Mid-Point Anchor Clamp’s Position

3.1. Calculation of the Location of the Mid-Point Anchor Clamp

The theoretical location of the clamp of the mid-point anchor of the trolley wire was determined according to the tension deviation of the wires between both sides’ half anchors when the mid-point anchor clamp moved, and the location of the mid-point anchor clamp would be corrected depending on the arrangement of the dropper in the span.

3.1.1. Calculation of Theoretical Position of Mid-Point Anchor Clamp

When the clamp of the mid-point anchor of the trolley wire moved to the span of the accurate location of the mid-point anchor with a certain step in a certain direction, the tension deviation of the wire ΔT_j0(i) between the right and left half-anchors was calculated according to Formulas (1)–(27), where i is the position number when the clamp moves. When ΔT_j0(i) is equal to its average value, the location of the clamp is the theoretical location of the mid-point anchor clip of the trolley wire.

3.1.2. Correction of the Mid-Point Anchor Clamp’s Position

It is noticeable from Figure 6 that when the mid-point anchor is located at the theoretical position, the dropper in the span may interfere with the setting of the mid-point anchor rope; thus, the theoretical location of the mid-point anchor clamp of the contact wire needs to be corrected.

For the dropper and the mid-point anchor in positions (a) and (b), the location of the mid-point anchor clamp can be corrected as follows:

(28)$\{\begin{array}{ll}\begin{array}{l}{x}_{j\mathrm{a}}=x_i+\Delta L_j\\ x_{j\mathrm{b}}=x_i-\Delta L_j\end{array}\hfill & \Delta T_{jL}-\Delta T_{jR}>0\hfill \\ \begin{array}{l}{x}_{j\mathrm{a}}=x_{i+1}+\Delta L_j\\ x_{j\mathrm{b}}=x_{i+1}-\Delta L_j\end{array}\hfill & \Delta T_{jL}-\Delta T_{jR}<0\hfill \end{array}$

For the dropper and the mid-point anchor in positions (c) and (d), the position of the mid-point anchor clamp can be corrected as follows:

(29)$\{\begin{array}{c}{x}_{j\mathrm{c}}=x_{i+1}+\Delta L_j\\ x_{j\mathrm{d}}=x_{i+1}-\Delta L_j\end{array}$

where x_ja, x_jb, x_jc, and x_jd are the corrected positions of the mid-point anchor clamp in different situations, m; x_i, x_i+1 are positions of the suspension points of droppers on both sides of the clamp, m; ΔL_j is half of the length of the anchor clamp in the center of the contact line, m.

3.2. Calculation of the Length of the Mid-Point Anchor Rope

As Figure 7 shows, the mid-point anchor rope length was determined according to the design regulations.

(30)$L_m=5h+\Delta L_{\mathrm{c}}+\Delta L_j$

where L_m is the length of the mid-point anchor rope, in m; ΔL_c is the length of the mid-point anchor clamp of the messenger cable, in m; and h is the height difference between the messenger cable and the trolley wire at the mid-point anchor clip junction at the trolley wire, in m.

Ignoring the influence of the mid-point anchor rope and taking the allowable deviation range of position of the mid-point anchor as the research object [16], the finite element model of the catenary was established through the following steps.

• The geometric model was established according to the catenary parameters.

• As Figure 8 shows, the gravity of both the wire and constraints were applied to estimate the deformation of the catenary, the displacements obtained with the analysis were added to the nodes of the FEM, and the geometry of the FEM was updated.

• By shortening the length of the dropper to raise the contact wire, the process was repeated until the catenary was suspended to a position where it met all of the design requirements under its own weight.

The local coordinate system was adopted to convert the catenary model into a planar model within each span. During the processing of the constraints, the nodal coordinate system was adopted to release the axial constraints of the messenger cable and the trolley wire.

Using the finite element model of the catenary, the height difference between the messenger cable and the contact wire at the position of the mid-point anchor clamp of the contact wire can be obtained, and then according to Formula (30), the length of the mid-point anchor rope can be obtained.

4. Validation

We took the data of 44# anchoring section of Xicheng Passenger Dedicated Line for example to verify the effectiveness of this method. This anchoring segment is in the straight ramp section; it has a total of 28 spans, a total length of 1336.73 m, and semi-inclined catenary suspension. According to the proposed method of setting the mid-point anchor position, the 14th span was chosen as the preset location, so the spans from 12th to 16th could be selected as the range of the allowable deviation of the mid-point anchor setting position.

Since the messenger cable of the 44# anchoring section of the Xicheng Line were of the same material as the contact wire, when the temperature changed, the dropper would not be offset, so it would not cause a tension difference at the contact wire. Since the anchoring section was in a straight segment, and it used the semi-inclined catenary suspension, the contact wire was zigzag arranged, and the messenger cable was arranged along the center of the line, so the messenger cable tension difference caused by the rotation of the wrist arm could be ignored when the temperature changed. The positioning device is here referred to the localizer, and the messenger cable tension difference is referred to the contact wire tension difference. Thus, the location of the mid-point anchor was obtained by calculating the contact wire tension deviation resulting from the deviation of the localizer and the downslope component of the trolly wire gravity along the direction of the line.

Then, after substituting the parameters of the catenary of the 44# anchoring section of the Xicheng Line into Formulas (9)–(21), the zigzag force at the localization point of the trolly wire was calculated. The corresponding result is presented in Figure 9.

It was evident that the zigzag force calculation result obtained with the proposed method was consistent with that of the design institute.

When the contact wire mid-point anchor was located at the 14th span, the offset values of the 12# localizers of the 1st–13th spans with the changing temperature and the contact wire tension difference resulting from the localizer offset along the line direction were calculated by Formulas (22) and (23) and are shown in Figure 10. Since the contact wire was anchored at both ends, a deflection angle of approximately 15 degrees existed between the transfer pillar and the anchor pillar, resulting in a significant increase in the zigzag force at the 2# localizer. Therefore, the contact wire tension deviation resulting from the deviation of the 2# localizer became significantly larger.

When the contact wire mid-point anchor was located at the 14th span, the total contact wire tension difference along the direction of the line caused by the localizer offset of the 1st–13th spans was 801.900 N (calculated with Formula (24)). The downslope component caused by the contact wire’s gravity of the 1st–13th spans was 193.356 N, which was calculated with Formula (25).

Similarly, the offset values of the 13 localizers of the 15–28th spans with the changing temperature and the contact wire tension difference resulting from the localizer deviation in the direction of the line were obtained, as shown in Figure 11. For the same reason, the 27# localizer had a significantly larger contact wire tension difference than the other localizers.

According to the above algorithm, when the contact wire mid-point anchor was in the 12th to 16th spans, in turn, the following forces were obtained: the tension difference of the contact wires (ΔT_jwL and ΔT_jwR) resulting from the offset of the localizers of the left and right half-anchor, the downslope component of the gravity (ΔG_L and ΔG_R) of the left and right half-anchor contact wires, the contact wire tension deviation (ΔT_j) between both sides of the mid-point anchor, and these results are shown in Figure 12.

Since the mid-point anchor of the 44# anchoring segment of the Xicheng Passenger Dedicated Line was two-span installation type, the 14th and 15th spans were selected as the setting position of the contact wire mid-point anchor; this is shown in Figure 12. According to the design specifications, the mid-point anchor of this anchoring segment was located at the 14th and 15th spans.

It was assumed that both sides of wire clamps of the mid-point anchor rope were moved from left to right simultaneously in the same step size at the 14th and 15th spans, respectively. According to Formula (25), the downslope component of the gravity of the left and right half-anchor contact wires can be obtained. The trolley wire tension deviation of both sides’ anchors were determined with Formula (26). The trolley wire tension deviation between both sides’ anchors was obtained with Formula (27). Among them, the maximum tension deviation of the trolley wire of half an anchoring section was 1140 N, which met the regulation of the design specifications that a half-anchoring section contact wire tension difference was not more than 15% of its rated tension.

When the mid-point anchor clip moved, some forces were obtained, according to Formulas (26) and (27), as follows: the trolley wire tension deviation of the left half-anchor and the right half-anchor (ΔT_jL, ΔT_jR), and the trolley wire tension deviation (ΔT_j0) between both sides of the anchor. The results are shown in Figure 13.

According to the intension deviation of the trolley wire between both half-anchors when the clip moved, its average value can be obtained: 78.05 N. When the trolley wire tension deviation between both half-anchors was equal to 78.05 N, the location of the mid-point anchor clip was its theoretical position.

As shown in Figure 14, the corrected location of the mid-point anchor clip was determined by correcting the theoretical position of the mid-point anchor clamp according to Formulas (28) and (29). The theoretical position of the left clamp was 20.950 m away from point A. Because the length of the clamp was 0.2 m, the left clamp was installed at a distance of 20.75 m from point A. The distance between point A and the suspension point of the dropper was 20.908 m. The corrected position of the left clamp was obtained with Formula (29). It was detected that the corrected position of the left clamp was 21.108 m away from point A. The theoretical position of the right clamp was 29.710 m away from point B, and the distance between point B and the suspension point of the dropper was 21.264 m. The corrected position of the right clamp was obtained with Formula (28). It was detected that the corrected position of the right clamp was 29.196 m away from point B. Therefore, the error between the corrected location of the mid-point anchor clamp and the actual location of the clamp was obtained, and the maximum relative error was 1.401%, which met the accuracy requirement. The specific comparison results are displayed in Table 1.

The catenary parameters of the catenary where the central anchor was located are presented in Table 2.

The FEM of the catenary is displayed in Figure 15. Ignoring the influence of the anchor rope of the contact wire, the FEM of catenary model was established. Figure 15 shows the displacement in the Y-direction of the actual suspension height of the trolley wire and the design height, and the different colors represent different intervals of the Y-direction displacement.

The height deviation between the actual suspension height of the trolley wire and the design height is shown in Figure 16; it met the design specification of ±30 mm, and the deviation between the adjacent dropper suspension points on the contact wire did not exceed 10 mm. The length of the dropper in the model was calculated and compared with its actual length. The maximum error of the calculated dropper length was 8.2 mm, as shown in Figure 17, which met the accuracy requirement. This indicated that the established model was accurate and that the model could reflect the actual state of the catenary more realistically.

In this model, the heights between the trolley wire at both the left and right wire clamps and the messenger wire were 1.020 m and 1.014 m, respectively. According to Equation (30), the mid-point anchor rope length on the 14th and 15th spans was calculated as 5.500 m and 5.470 m, respectively. The total length of the mid-point anchor rope obtained by this method was 10.97 m, which was 31.7% less than the total length of the mid-point anchor rope (16.06 m) given by a certain design institute.

5. Conclusions

• The proposed method can be used for the position calculation of both two-span and one-span mid-point anchors.

• The total length of the mid-point anchor rope obtained by the proposed method was reduced by 31.7% compared with the existing method, which reduces the construction cost and improves the economic benefit.

• The effectiveness of the proposed method was verified by the data of the Xicheng Passenger Dedicated Line. Compared with the actual position of the mid-point anchor clamp, the maximum relative error of this method was 1.40%, which meets the engineering requirements. This shows that the proposed method can provide reference and basis for the position calculation of the mid-point anchor during design of the catenary, and at the same time provide theoretical support for the installation of the mid-point anchor in the field.

• The application of the finite element method in the localization method of the mid-point anchor of a high-speed railway distributed in long straight line with large slope was effective.

• According to the ideas of this article, we can further study the positioning method of the mid-point anchor of the curve segment in the later stage.

Author Contributions

Conceptualization, X.W.; methodology, X.W., K.Y. and Y.M.; software, K.Y.; validation, X.W. and K.Y.; formal analysis, X.W. and Y.M.; investigation, K.Y.; resources, Y.M.; data curation, X.W.; writing—original draft preparation, X.W. and K.Y.; writing—review and editing, X.W., K.Y. and Y.W.; visualization, K.Y.; supervision, Y.M.; project administration, Y.M.; funding acquisition, Y.M. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

This work is supported by Foundation of One Hundred Youth Talents Training Program of Lanzhou Jiaotong University.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. Stress analysis of the dropper offset.

Figure 2. Tension deviation resulting from the dropper offset along the line direction.

Figure 3. Analysis of the zigzag of the trolley wire on the left of the point C.

Figure 4. Analysis of the zigzag of the trolley wire on the right of the point C.

Figure 5. Analysis of tension deviation of the contact wire resulting from the offset of the positioning device.

Figure 6. Correction diagram of the position of the mid-point anchor clip. (Subgraphs (a–d) represent four different relationships between the theoretical position of the dropper and the mid-point anchor clip within the span, respectively).

Figure 7. Schematic diagram of mid-point anchor rope installation.

Figure 8. FEM of the catenary.

Figure 9. Zigzag force calculation result.

Figure 10. Localizer offset values of the 1st–13th spans and the contact wire tension difference.

Figure 11. Localizer offset values of the 15–28th spans and contact wire tension difference.

Figure 12. Changes in contact wire tension difference and the downslope component of the trolley wire gravity at different positions of the mid-point anchor.

Figure 13. Changes of the trolley wire tension deviation when the mid-point anchor clip was at different positions.

Figure 14. Correction diagram of two-span mid-point anchor clip position.

Figure 15. FEM of the railway catenary (the span where the central anchor was located).

Figure 16. The partial results of the height deviation of the contact wire.

Figure 17. The partial results of the dropper length comparison.

Table 1

Results of the position of the mid-point anchor clip.

	Theoretical Position (m)	Correct Position (m)	Actual Position (m)	Relative Error
Left clip	20.950	21.108	21.408	1.401%
Right clip	29.710	29.196	28.896	0.592%

Table 2

Parameters of the span where the central anchor was located.

	Contact Wire	Messenger Wire		Stitched Wire		Dropper
Wire type	CTS150	JTMH120		JTMH-35		JTMH-10
Cross-sectional area (m2)	1.50 × 10−4	1.20 × 10−4		3.50 × 10−5		1.00 × 10−5
Elastic modulus (MPa)	1.24 × 1011	1.24 × 1011		1.24 × 1011		1.24 × 1011
Wire density (kg/m)	1.35	1.065		0.303		0.089
Poisson’s ratio	0.33	0.33		0.33		0.33
Tension (N)	25,000	20,000		3500		/
Span length (m)	49.95	44.84	49.77		50.66	48.1
Interval of droppers (m)	4/7.954/7.954/7.954/7.954/7.954/4
	4/7.168/7.168/7.168/7.168/7.168/5
	5/7.954/7.954/7.954/7.954/7.954/5
	5/8.132/8.132/8.132/8.132/8.132/5
	5/7.620/7.620/7.620/7.620/7.620/5
Structure height (m)	1.540/1.529	1.529/1.520	1.520/1.513		1.513/1.539	1.539/1.528
Line slope	0.025
Stagger value (m)	−0.258/−0.253/0.263/−0.252/0.275/−0.242

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