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 23ton axle load cars, which will be exacerbated by 27ton 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 NSGAII 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 32track 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 hardmid 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 hardmid 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 hardmid rolling order has a large enough time interval to pass the friction retarders and turnouts.
The popularly used pointcontinued speed control system in China has an interval braking point which can adjust the interval time for the hardmid 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 AlgorithmII (NSGAII) is used to get the Pareto solution set in this paper.
The arithmetic flow chart is shown in Figure
NSGAII 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 



Highspeed rolling 



Deceleration 

Range of slopes.
Section type  Acceleration  Highspeed 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·JK2A(50), and TJK3A(50). Their technical parameters are shown in Table
Friction retarder technical parameters.
Type  T·JK2  T·JK2A(50)  T·JK  T·JK3A(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 36track (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 

AG_{1}  5.167  250  11.280  22.544  12.122  34.666 
AG_{2}  5.833  200  10.190  20.362  103.107  123.469 
AG_{3}  6.750  200  11.795  23.562  141.69  165.252 
AG_{4}  1.750  200  3.054  6.109  217.355  223.464 
AG_{5}  0.500  200  0.873  1.745  243.291  245.036 
AG_{6}  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 m^{2} facade area and 30 t total weight. The middle rolling car is a gondola car with 5.94 m^{2} 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 NSGAII 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/s^{2})
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/s^{2})
Standard gravity acceleration (m/s^{2})
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 (m^{2})
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).