Railway freight trains consist of many cars heading to different destinations. Hump is the special equipment that distributes cars with different destinations to different tracks in a marshalling station. In recent years, with the development of Chinese freight car technology, the axle load has risen from 21 ton to 23 ton and will rise to 27 ton in the future. Many rolling problems appear in the hump distributing zone with the application of 23-ton axle load cars, which will be exacerbated by 27-ton axle load cars. This paper proposes a multiobjective optimization model based on the angle of the hump profile design with minimizing weighted accumulating rolling time (WART) and hump height as optimization goals and uses the improved genetic algorithm NSGA-II to determine a solution. In case study, Pareto solution set is obtained, and the contrast analysis with traditional method is made.
Marshalling station is the station that disassembles, sorts, and reassembles freight trains to different destinations and the hump is the disassembling equipment in marshalling station (shown in Figure
Composition of marshalling station.
In China, most humps were built or rebuilt in the 1980s and 1990s. The design of the cars has an 18 t and 21 t axle load with sliding bearings [
In the field of hump operation, Adlbrecht et al. [
In hump design fields, Zhang et al. [
In the field of multiobjective optimization, Deb et al. [
Above all, the hump distributing zone design has not been solved perfectly. This paper aims to provide an optimization method for a hump distributing zone design for the upcoming 27 t axle load car in China. The paper is organized as follows. The proposed model, solution algorithm and coefficient calibration are described in Section
The hump distributing zone is located between the hump crest and entry to the braking position in the classification yard. The profile design of the hump distributing zone is the vertical design of hard rolling track that has the maximum total resistance in all shunting tracks.
Before modeling, for calculation convenience, a coordinate system needs to be established. The gradient change point of the hump crest and acceleration gradient is set as the origin of the coordinate system, with the vertical downward direction as the
Coordinate system of hump distributing zone.
The hump is used for car classification. Making the cars pass through the distributing zone as quickly as possible and meeting the speed demands of the control system is the goal of optimization, which can be divided into two optimization targets:
Rolling time is an important index for distributing zone evaluation. With branching turnout as a dividing point, different sections connect different numbers of tracks and have different traffic volumes. Taking a 32-track classification yard, for example, half of the range is shown in Figure
Sketch map of rolling time weight in distributing zone (half range).
The distributing zone is composed of many differently slope sections. The basic rolling resistance and air resistance are changing along with speed variation when the car is rolling. To calculate the WART under dynamic accelerating conditions, the distributing zone is divided into a large number of small equal parts. In each small part, the basic rolling resistance and air resistance are assumed constant. With the speed, acceleration, and distance formula, the rolling time of each small part can be computed as follows:
With the recursion calculation, the rolling time of each section can be obtained, and the WART of profile design calculating car can be computed as follows:
To get the acceleration
Force analysis of rolling car.
In Figure
Usually,
The value of
The total resistance consists of 4 parts: the rolling resistance, the air resistance, the turnout resistance, and the curve resistance, which can be calculated as follows:
The model for calculating specific rolling resistance can be obtained in [
The specific air resistance can be calculated as follows:
According to parameters in the “Code for Design on Hump and Marshalling Yard of Railway” (“the Code” for short) and assuming the resistance force of turnout and curve is constant,
Taking formulas (
The meaning of each symbol is similar to the former.
Making rolling cars pass through the distributing zone as quickly as possible requires the gradients be as large as possible. However, this will make the height of the distributing zone too high, which will increase the difficulty of hump pushing for the locomotive and increase energy consumption. Therefore, the other optimization objective is to make the hump height as low as possible:
According to the Code, the gradient change point should be kept away from the friction retarder, switch rails, and the frog. The shortest distance is along the tangent of vertical curve:
Suppose that friction or turnout lies in front of the gradient change point:
With a 1.4 m/s humping speed, under advantageous rolling condition, the easy rolling car’s entry speed to turnout and the friction retarder is lower than the specified speed.
To make the car continue rolling if it is clamped to stop, the minimum slope where the friction retarder is located needs to be limited:
Referring to the Code, the maximum and minimum gradient of the acceleration section should be limited:
According to the Code, with a 1.4 m/s humping speed and disadvantageous rolling conditions, the speed of the hard rolling car should not fall below a certain value at the end of distributing zone.
The hard-mid rolling order means that the front rolling car is the hard rolling car, and the car behind is the middle rolling car. For the middle rolling car runs faster than the hard rolling car, the interval time of the hard-mid rolling order will gradually be reduced, which leads to the risk of collision. The turnouts or friction does not have enough time to change the working status of the middle rolling car. The Code gives the limits that under disadvantageous rolling conditions and 1.4 m/s humping speed, the hard-mid rolling order has a large enough time interval to pass the friction retarders and turnouts.
The popularly used point-continued speed control system in China has an interval braking point which can adjust the interval time for the hard-mid car rolling order, so the key section of track is the one from the hump crest to the first turnout and interval braking point:
For convenience of maintenance work, the minimum slope length needs to be limited.
The profile design optimization model of the hump distributing zone can be summarized as follows:
The meaning of each symbol is the same to the former.
The model belongs to the Multiobjective Optimization Problem (MOP). Its solution is the Pareto set with no better solution than where one goal is optimized while another goal is degraded. The Nondominated Sorting Genetic Algorithm-II (NSGA-II) is used to get the Pareto solution set in this paper.
The arithmetic flow chart is shown in Figure
NSGA-II algorithm flow.
Sketch map of plane outspread drawing of hard rolling track.
In Figure
According to practical experience, if the first acceleration slope is less than 28 m, the interval time will not be long enough and this will lead to a shunting accident. The length of switch rail and frog of 6# turnout are 7.437 m and 9.994 m. The distributing zone can be roughly divided into three main sections with friction retarder as cutting points. The ranges of slope changing points can be limited as in Table
Ranges of slope changing points.
Section type | Ranges of changing point |
---|---|
Acceleration |
|
|
|
High-speed rolling |
|
|
|
Deceleration |
|
Range of slopes.
Section type | Acceleration | High-speed rolling | Deceleration |
---|---|---|---|
Gradient ( |
35 |
5 |
−1 |
The range of the gradient is −1
The gradient convert formula is
The binary code length can be calculated as follows:
With regard to the length of the slope, for maintenance work convenience, the slopes are integers except the last one. The accuracy of encoding is 1 m, and its range of lengths is 15~200 m, the binary code length is 8.
In China, the popular friction retarder types are the T·JK, T·JK2-A(50), and TJK3-A(50). Their technical parameters are shown in Table
Friction retarder technical parameters.
Type | T·JK2 | T·JK2-A(50) | T·JK | T·JK3-A(50) |
---|---|---|---|---|
Braking action time | ||||
Braking(s) | 0.6 | 0.6 | 0.6~1.0 | 0.8 |
Full braking(s) | 1.4 | |||
Release action time | ||||
Release(s) | 0.5 | 0.4 | 0.8~1.23 | 0.4 |
Full release(s) | 0.9 | 0.6 | 1.94 | 0.8 |
In China, the main turnout switch equipment is an electropneumatic switch machine in the distributing zone. Taking the ZK3 for reference, the turnout switching time is less than 0.6 s.
The parameters of temperature, wind speed and calculation car depend on the specific hump.
Taking a 36-track (34 tracks are used for rolling) hump for example. The plane outspread drawing of hard rolling track of the distributing zone is shown in Figure
The
Number |
|
---|---|
|
0.000 |
|
12.122 |
|
34.666 |
|
44.386 |
|
74.386 |
|
84.707 |
|
100.227 |
|
103.107 |
|
123.469 |
|
131.301 |
|
141.690 |
|
165.252 |
|
168.252 |
|
190.772 |
|
207.351 |
|
217.355 |
|
223.464 |
|
231.041 |
|
243.291 |
|
245.036 |
|
254.731 |
|
266.322 |
|
367.751 |
|
397.760 |
|
414.560 |
|
518.660 |
Plane outspread drawing of hard rolling track (sketch map).
Curve parameters of hard rolling track are shown in Table
Curve parameters of hard rolling track.
Number | Angle (°) | Radius (m) | Tangent length (m) | Curve length (m) | Start |
End |
---|---|---|---|---|---|---|
AG1 | 5.167 | 250 | 11.280 | 22.544 | 12.122 | 34.666 |
AG2 | 5.833 | 200 | 10.190 | 20.362 | 103.107 | 123.469 |
AG3 | 6.750 | 200 | 11.795 | 23.562 | 141.69 | 165.252 |
AG4 | 1.750 | 200 | 3.054 | 6.109 | 217.355 | 223.464 |
AG5 | 0.500 | 200 | 0.873 | 1.745 | 243.291 | 245.036 |
AG6 | 29.057 | 200 | 51.773 | 101.429 | 266.322 | 367.751 |
Consider too many slopes can affect the car rolling and increase the amount of hump maintenance work. Referring the Code, 6 slopes are set in this paper.
The climate conditions are shown in Table
Climate conditions of hump design.
Condition | Temperature (°C) | Wind speed (m/s) | Wind direction (°) |
---|---|---|---|
Disadvantageous | −5 | 4 | 0 |
Advantageous | 27 | −3 | 0 |
According to the data of Table
Reasonable range of slope changing point.
Slope change point | Ranges |
---|---|
|
|
|
|
|
|
|
|
|
|
|
|
Given the small ranges of
For the WART calculation, the connect track number and the
Connect track number and the
Branching turnout | Connect tracks |
|
Value |
---|---|---|---|
|
34 |
|
54.380 |
|
17 |
|
141.295 |
|
5 |
|
217.345 |
|
2 |
|
264.725 |
Equilateral turnout has lead curve, the hump uses 6# equilateral turnout, and its radius and angle are 180 m and 4.731°. The
Turnouts
Number | Start |
End |
---|---|---|
TO1 | 36.949 | 54.38 |
TO2 | 64.392 | 81.823 |
TO3 | 123.864 | 141.295 |
TO4 | 199.914 | 217.345 |
TO5 | 223.604 | 241.035 |
TO6 | 247.294 | 264.725 |
The hard rolling car is the P70 with 9.82 m2 facade area and 30 t total weight. The middle rolling car is a gondola car with 5.94 m2 facade area and 70 t total weight. The max entry speed of turnout and friction retarder is 6.5 m/s. The minimum slope of friction retarder is 2
Solution set with two methods of hump profile design of distributing zone.
No. |
|
|
|
|
|
|
|
|
|
|
|
|
HH | WART |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
54.7 | 18.4 | 8.7 | 8.0 | 5.5 | 1.5 | 30 | 83 | 110 | 196 | 310 | 393.66 | 4.255 | 778.827 |
|
54.8 | 17.8 | 6.7 | 2.7 | 9.0 | 0.6 | 30 | 83 | 164 | 198 | 302 | 393.66 | 4.213 | 780.720 |
|
54.9 | 18.2 | 5.0 | 5.0 | 6.1 | 3.6 | 30 | 83 | 114 | 193 | 298 | 393.66 | 4.147 | 783.875 |
|
53.2 | 19.1 | 7.9 | 7.9 | 4.0 | 0.4 | 30 | 83 | 146 | 197 | 348 | 393.66 | 4.097 | 786.410 |
|
54.5 | 16.7 | 9.4 | 7.4 | 2.5 | 3.3 | 30 | 83 | 144 | 198 | 307 | 393.66 | 4.014 | 787.948 |
|
53.4 | 18.9 | 3.3 | 9.3 | 4.2 | 2.5 | 30 | 83 | 144 | 195 | 313 | 393.66 | 3.977 | 791.551 |
|
54.0 | 19.9 | 3.8 | 3.8 | 6.7 | −0.2 | 28 | 83 | 113 | 194 | 324 | 393.66 | 3.885 | 794.150 |
|
54.9 | 17.2 | 1.7 | 7.9 | 3.8 | 2.0 | 29 | 83 | 113 | 198 | 325 | 393.66 | 3.863 | 795.772 |
|
54.1 | 16.3 | 9.9 | 2.1 | 4.2 | 1.8 | 28 | 83 | 159 | 199 | 274 | 393.66 | 3.778 | 797.065 |
|
54.5 | 18.8 | 4.5 | 2.0 | 3.8 | 3.5 | 28 | 83 | 163 | 194 | 352 | 393.66 | 3.728 | 798.674 |
|
54.7 | 16.2 | 2.1 | 6.2 | 3.4 | 2.8 | 30 | 83 | 116 | 197 | 295 | 393.66 | 3.681 | 800.504 |
|
54.2 | 18.1 | 2.9 | 3.6 | 4.8 | 1.1 | 29 | 83 | 160 | 199 | 335 | 393.66 | 3.630 | 803.904 |
|
54.8 | 15.8 | 4.6 | 5.1 | 3.6 | 1.9 | 29 | 83 | 164 | 195 | 332 | 393.66 | 3.584 | 806.544 |
|
52.4 | 19.4 | 3.9 | 6.1 | 2.1 | 2.5 | 28 | 83 | 154 | 198 | 346 | 393.66 | 3.510 | 809.348 |
|
51.7 | 18.4 | 4.4 | 4.0 | 4.4 | 0.9 | 29 | 83 | 145 | 196 | 292 | 393.66 | 3.484 | 813.706 |
|
54.8 | 15.9 | 0.0 | 4.5 | 3.6 | 2.3 | 29 | 83 | 112 | 198 | 288 | 393.66 | 3.402 | 816.304 |
|
53.0 | 12.2 | 8.1 | 9.3 | 1.4 | −0.1 | 28 | 83 | 144 | 197 | 340 | 393.66 | 3.337 | 829.300 |
|
52.0 | 13.7 | 5.9 | 9.3 | 5.8 | −0.9 | 28 | 83 | 160 | 192 | 268 | 393.66 | 3.289 | 833.284 |
|
51.6 | 12.5 | 6.2 | 6.7 | 2.0 | 1.5 | 29 | 83 | 112 | 199 | 280 | 393.66 | 3.267 | 840.509 |
|
54.1 | 10.0 | 6.9 | 4.6 | 2.5 | 1.9 | 28 | 83 | 165 | 198 | 352 | 393.66 | 3.247 | 843.870 |
|
50 | 16 | 8 | 6 | 3 | - | 28 | 83 | 168 | 185 | 393.66 | - | 3.717 | 827.582 |
As seen in Table
This paper focuses on the hump profile optimization problem. Based on the analysis of the hump distributing zone and the importance of each part, a model was established with the smallest WART and the lowest of hump height selected as optimization objectives, with the gradients change points, Maximum Entry Speed, and so on, as constraints. The NSGA-II algorithm was imported for model solution. In the case study, a comparison is made between the method proposed in this paper and traditional manual method, and the optimization effect is significant.
Length of the small part (m)
Initial speed in the small part (humping speed is the original speed) (m/s)
Acceleration (m/s2)
Rolling time of small part
Total number of small parts
Gravity (N)
Components of the forces along the slope (N)
Total rolling resistance (N)
Gradient (downgrade of rolling direction is positive; upgrade is negative) (
Car weight (kg)
Number of axles
Specific air resistance (N/kN)
Specific curve resistance (N/kN)
Coefficient of turnout (1 when car rolls in the range of turnout, otherwise 0)
Curve coefficient (1 when car rolls in the range of curve, otherwise 0)
Rolling speed (m/s)
Wind speed (negative when the same as the rolling direction and positive when opposite) (m/s)
Included angle of the wind and rolling direction (°)
Slope gradient of slope
Radius of the vertical curve (m)
Rolling speed of the easy car under advantageous rolling conditions (m/s)
Max allowed entry speed of turnout (m/s)
Minimum slope for hard rolling car automatic rolling (
Limited minimum gradient (
Design gradient (
Lengths of the hard rolling car (m)
Humping speed (m/s)
Rolling time of hard rolling car from the top of hump to the first turnout (s)
Rolling time of the hard rolling car from the top of hump to the first friction retarder (s)
Friction retarder working status change time (s)
The
Binary code length of the gradient
Rolling time of car in small part (s)
End speed in the small part and also the initial speed of next small part (m/s)
The WART (s)
Weighting coefficient of small part
Gradient angle of the slope (°)
Normal supporting force (N)
Components of the forces in the normal direction (N)
Acceleration of
Acceleration that takes into account rotational inertia (m/s2)
Standard gravity acceleration (m/s2)
Specific rolling resistance (N/kN)
Specific turnout resistance (N/kN)
Coefficient of air resistance
Direction coefficient of turnout (1 for reverse turnout, 0.5 for forward turnout)
Front projected facade area of the rolling car (m2)
Height of the hump distributing zone (m)
Included angle of the compound direction (wind and rolling direction) and the rolling direction (°)
Starting
Length along the tangent (m)
Value of the adjacent slope change (
Number of slopes
Max allowed entry speed of the friction retarder (m/s)
Slope at the friction retarder (
Speed of the hard rolling car at the end of the distributing zone (m/s)
Limited maximum gradient (
Specified end speed (m/s)
Lengths of middle rolling car (m)
Turnout working status change time (s)
Rolling time of middle rolling car from the top of hump to the first turnout (s)
Rolling time of middle rolling car from the top of hump to the first friction retarder (s)
Minimum slope length (m)
Converted slope gradient (
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research is supported by the National Natural Science Foundation of China Project (51308029) and the Fundamental Research Funds for the Central Universities (2017JBM036).