MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2017/9318025 9318025 Research Article Multiobjective Optimization Model for Profile Design of Hump Distributing Zone http://orcid.org/0000-0002-0958-5944 Zhang Hongliang 1 http://orcid.org/0000-0002-8865-234X Yang Jing 2 http://orcid.org/0000-0001-9065-1525 Yang Taoyuan 1 Chicano Francisco 1 School of Traffic and Transportation Beijing Jiaotong University Beijing 100044 China njtu.edu.cn 2 School of Civil Engineering and Transportation Beijing University of Civil Engineering and Architecture Beijing 100044 China bucea.edu.cn 2017 31122017 2017 07 07 2017 07 11 2017 11 12 2017 31122017 2017 Copyright © 2017 Hongliang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

National Natural Science Foundation of China 51308029 Fundamental Research Funds for the Central Universities 2017JBM036
1. Introduction

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 1). Trains are pushed gradually up a raised portion of track called the “hump.” Wagons are uncoupled before arriving at the crest and then coast to the desired tracks in the classification yard under the force of gravity. The distributing zone of the hump is the bottle neck of the marshalling station and has an important influence on operational efficiency and safety.

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 [1, 2]. With the development of railway wagon technology, the sliding bearing has been replaced by the rolling bearing, which has better rolling performance , and the axle load of the car has increased to 23 t. The 18 t and 21 t axle load cars are not the main transport cars any more. With the application of the 23 t axle load car, problems such as over rolling speed and insufficient braking power have arisen . Although problems can be temporarily relieved by reducing humping speed and adding more piston retarders, the humping capacity will decrease and the situation will be exacerbated with the application of 27 t axle load cars in the future . In essence, it is a problem that the hump distributing profile does not match wagon development trends, which is a problem better solved by design optimization.

2. Literature Review

In the field of hump operation, Adlbrecht et al.  propose a Mixed Integer Programme (MIP) under consideration of single-block (a block refers to rail cars that share the same destination) trains. Then conduct a series of numerical experiments and show that the solutions of the MIP improve by 10% on average. Shi and Zhou  present a time-expanded multilayer network flow model to describe the connection between different layers of yard operations and develop a mixed integer programming model to optimize the overall performance by jointly considering tightly interconnected facilities. Li et al.  propose sequencing and scheduling model considering multiple engines and inspection groups and got solution by existing commercial optimization solvers for one typical planning horizon. Bektaş et al.  propose methodology performs dynamic reassignments of empty cars through a fast and efficient solution procedure based on the assignment algorithm. Dick and Dirnberger  introduce a hump simulation system (named HYSS) as a research tool to examine how its various outputs can be used to quantify terminal performance and described the planned research program aimed at advancing the science of hump yard design and operations. Yagar et al.  proposed a computer model for sequencing trains into the rail yard humping process. The model establishes an efficient hump sequence by promoting compatibility between inbound trains and departing trains, as well as operator service priorities. Dynamic programming is used to minimize overall rail yard throughput costs for a given interval of operation. Marton et al.  combine an integer programming approach and a computer simulation tool to develop and verify an improved classification schedule for a real-world train classification instance. Lin and Cheng  introduce a simulation model which depicts typical operations in a railroad hump yard and present key performance measurements that are used to gauge the efficiency of yard operations and infrastructure. Boysen et al.  introduce and discuss the train makeup problem, analyze its complexity status, and develop suitable exact and heuristic solution procedures that are tested in a comprehensive computational study.

In hump design fields, Zhang et al.  propose a multiobjective optimization model for coupling area in marshalling yard with rolling distance of middle car under adverse condition and hard rolling car under favorable condition which are taken as optimization objectives. Zhang et al.  analyze the shortage of monthly meteorological data usage of the Code for Design on Hump and Marshalling Yard of Railway and the effect on hump design and propose the suggestion that at least daily meteorological data should be used for hump design. Huang  and Zhang  both establish multiobjective models and provide the solution algorithms. However, their optimization objectives need to be studied. In Huang’s paper, the optimization objectives are hump height and humping speed. In actual situations, the humping speed is not a suitable optimization objective. Since the uncoupling work is performed manually, increasing the humping speed can make the operator fall behind, which will lead to accidents in the shunting operation. In China, the humping speeds are usually 0.83 m/s, 1.39 m/s, and 2.22 m/s for a single car, small group of cars, and large blocks of cars, respectively, and the design speed is 1.39 m/s. As a result, the optimization objective is not suitable. In Zhang’s paper, the optimization objectives are interval time between rolling cars, hump height and amount of braking equipment. The interval time cannot be reduced infinitely, because the turnouts and the friction retarders need a certain time to change their working state for the continuous decoupling of wagons. Therefore, it is also not suitable to be used for optimization.

In the field of multiobjective optimization, Deb et al.  propose a nondominated sorting based multiobjective evolutionary algorithm (NSGA-II). Zhang and Ma  establish a multiobjective optimal model which simultaneously considers the two objectives of minimizing the production cost and minimizing the operation cost for the dry-type air-core reactor problem and then propose a memetic evolutionary algorithm which combines an elitist nondominated sorting genetic algorithm version II (NSGA-II) with a local search strategy based on the covariance matrix adaptation evolution strategy (CMA-ES). Li et al.  propose a multiobjective reverse logistics network optimization model that considers the objectives of the cost, the total tardiness of the cycle time, and the coverage of customer zones. The NSGA-II is employed for solving it.

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 3. A case study which validates the effectiveness of the proposed approach is described in Section 4. Finally, Section 5 discusses conclusions.

3. Methodology

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 y-axis direction and the rolling direction as the x-axis direction. The x-coordinate of a gradient change point indicates horizontal distance, and the y-coordinate indicates hump height. As shown in Figure 2, the black points are the gradients change points, xk indicates the horizontal distance of point k from the hump crest, and yk indicates the vertical distance from the hump crest. vk indicates the speed of a coasting wagon at point k.

Coordinate system of hump distributing zone.

3.1. Optimization Targets

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:

3.1.1. Weighted Accumulated Rolling Time (WART) of Profile Design Calculating Car

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 3, all cars to be classified will pass through the red area, in the yellow area about 1/2 of rolling cars pass through (to maximum use of the capacity of shunting yard, tracks with few traffic volumes will share with large ones nearby, so the traffic volume of each track is relatively balanced), with about 1/4 for orange, about 1/8 for blue, about 1/16 for gray, and about 1/32 for blue. Therefore, the red area is the busiest, followed by the yellow area, then the orange area, blue area, and gray area. It can be concluded that different sections have different importance, and weights can be input for the rolling time calculation. In this paper, the weights are given for final connect tracks.

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:(1)Δt=-v0+v02+2aΔsa,vt=v0+aΔt,where Δs is the length of the small part (m); Δt is rolling time of car in small part (s); v0 is the initial speed in the small part (humping speed is the original speed)  (m/s); vt is end speed in the small part and also the initial speed of next small part (m/s); a is acceleration (m/s2).

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:(2)Tw=j=1NspΔtj×cwj,where Tw is the WART (s); Δtj is the rolling time of small part j(s); cwj is the weighting coefficient of small part j; Nsp is the total number of small parts.

To get the acceleration a, an analysis of the forces is needed, as shown in Figure 4.

Force analysis of rolling car.

In Figure 4, α is gradient angle of the slope (°); Gc is gravity (N); fp and fg are the components of the forces in the normal direction and along the slope (N); fs is the normal supporting force (N); fr is total rolling resistance (N).

Usually, i is used to express the gradient, and the max gradient is usually less than 55 in China. Therefore, the equation for i can be got as follows:(3)i=tgαsinα,where i is the gradient (downgrade of rolling direction is positive, and upgrade is negative) ().

The value of fg and its acceleration can be calculated as follows:(4)fg=M×g×sinαMgi×10-3,(5)ag=fgM=gi×10-3,where ag is acceleration of fg  (m/s2), M is car weight (kg), and g is the acceleration that takes into account rotational inertia (m/s2).(6)g=gs1+0.42naxl/M×10-3,where gs is standard gravity acceleration (m/s2) and naxl is the number of axles.

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:(7)fr=wrol+wair+ktur×kturd×wtur+kcur×wcur×M×g×10-3,where the wrol is the specific rolling resistance (N/kN); wair is the specific air resistance (N/kN); wtur is the specific turnout resistance (N/kN); wcur is the specific curve resistance (N/kN); ktur is the coefficient of turnout (1 when car rolls in the range of turnout, otherwise 0); kturd is the direction coefficient of turnout (1 for reverse turnout, 0.5 for forward turnout); kcur is the curve coefficient (1 when car rolls in the range of curve, otherwise 0).

The model for calculating specific rolling resistance can be obtained in .

The specific air resistance can be calculated as follows:(8)wair=0.063sfCx1/Cx0M×10-3cos2αwinvcar±vwincosβwin2,where sf is front projected facade area of the rolling car (m2); Cx1/Cx0 is the coefficient of air resistance; vcar is rolling speed (m/s); vwin is wind speed (negative when the same as the rolling direction, and positive when opposite) (m/s); βwin is the included angle of the wind and rolling direction (°); αwin is the included angle of the compound direction (wind and rolling direction) and the rolling direction (°).

α w i n can be calculated as follows:(9)αwin=tan-1vwinsinβvcar±vwincosβ.

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, wtur is 1.375 N/kN, and wcur can be computed as follows:(10)wcur=1440π·Rcur,where Rcur is the radius of the curve (m).

Taking formulas (5)~(10) together, according to Newtonian mechanics, the acceleration can be obtained:(11)a=ag-wrol+wair+1.375×ktur×kturd+1440πRcurkcur×g×10-3.

The meaning of each symbol is similar to the former.

3.1.2. Height of Distributing Zone

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:(12)Hh=j=1Nslxj-xj-1ij×10-3,where Hh is the height of hump distributing zone (m); xj is the x-coordinate of the jth gradient change point; xj-1 is the x-coordinate of the j-1th gradient change point; and Nsl is the number of slopes; ij is the gradient of slope j ().

3.2. Model Constraints 3.2.1. Gradient Change Point

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:(13)Lvta=RvcΔi2×1000,where Lvta is length along the tangent (m); Rvc is the radius of the vertical curve (250 m in the Code) (m); and Δi is the value of the adjacent slope change ().

Suppose that friction or turnout lies in front of the gradient change point:(14)xgcxstart-Lvta,where xgc is the x-coordinate of the gradient change point and xstart is the starting x-coordinate of the gradient change point of friction or turnout.

3.2.2. Max Entry Speed

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.(15)veradvfrmax,veradvturmax,where verad is the rolling speed of the easy car under advantageous rolling conditions (m/s); vfrmax is the max allowed entry speed of friction retarder (m/s); vturmax is the max allowed entry speed of turnout (m/s).

3.2.3. Minimum Slope Where Friction Retarder Is Located

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:(16)IretIrolmin,where Iret is the slope at the friction retarder (); Irolmin is the minimum slope for hard rolling car automatic rolling ().

3.2.4. Maximum and Minimum Gradient of Acceleration Section

Referring to the Code, the maximum and minimum gradient of the acceleration section should be limited:(17)IaccminIaccIaccmax,where Iaccmin and Iaccmax are the limited minimum and maximum gradient (); Iacc is the design gradient ().

3.2.5. End Speed Limit

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.(18)vendhrdvendspe,where vendhrd is the speed of the hard rolling car at the end of the distributing zone (m/s); vendspe is the specified end speed (m/s).

3.2.6. Interval Time Limit of Hard-Mid Rolling Order

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:(19)Lhrc+Lmrc2vhum-thrhtt-tmrhttttur,Lhrc+Lmrc2vhum-thrhtr-tmrhtrtretinv,where Lhrc and Lmrc are length of hard rolling car and middle rolling car (m); vhum is humping speed (m/s); thrhtt and tmrhtt are the rolling time of hard rolling car and middle rolling car from the top of hump to the first turnout (s); thrhtr and tmrhtr are the rolling time of hard rolling car and middle rolling car from the hump crest to the first friction retarder (s). ttur is the turnout working status change time (s); tretinv is the friction retarder working status change time (s).

3.2.7. Slope Length Limit

For convenience of maintenance work, the minimum slope length needs to be limited.(20)xfp-xbp>lming,where xfp and xbp are the x-coordinate of front slope change point and the back slope change point; lmingis minimum slope length (m).

The profile design optimization model of the hump distributing zone can be summarized as follows:(21)Obj.minTw=j=1Nsptj×cwj,minHh=j=1Nslxj-xj-1ij×10-3,S.txgcxstart-Lvta,veradvfrmax,veradvturmax,IretIrolmin,IaccminIaccIaccmax,vendhrdvendspe,Lhrc+Lmrc2vhum-thrhtt-tmrhttttur,Lhrc+Lmrc2vhum-thrhtr-tmrhtrtretinv,xfp-xbp>lming.

The meaning of each symbol is the same to the former.

3.3. Model Solution

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.

3.3.1. Algorithm Flow

The arithmetic flow chart is shown in Figure 5.

NSGA-II algorithm flow.

3.3.2. Range of Decision Variable and Coding Rules

( 1) x -Coordinate of Gradient Change Point. The plane outspread drawing of hard rolling track is needed, and shown in Figure 6.

Sketch map of plane outspread drawing of hard rolling track.

In Figure 6, P1 is start point of hump profile design; P2, P8, P11, P16, P19, and P22 are the start points of curves; P3, P9, P12, P17, P20, and P23 are the end points of curves; P4, P5, P10, P15, P18, and P21 are the center coordinates of turnouts. P6; P13 and P24 are start points of friction retarders; P7, P14, and P25 are end points of friction retarders.

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 1.

Ranges of slope changing points.

Section type Ranges of changing point
Acceleration [ 28 , x p 2 - L v t a ] [ x p 2 + L v t a , x p 3 - L v t a ] [ x p 3 + L v t a , xp4-7.437-Lvta][xp5+7.437+Lvta, xp6-Lvta]

High-speed rolling [ x p 7 + L v t a , xp8-Lvta][xp8+Lvta, xp9-Lvta][xp9+Lvta, xp10-7.437-Lvta][xp10+9.994+Lvta, xp11-Lvta][xp11+Lvta, xp12-Lvta][xp12+Lvta, xp13-Lvta]

Deceleration [ x p 14 + L v t a , xp15-7.437-Lvta][xp15+9.994+Lvta, xp16-Lvta][xp16+Lvta, xp17-Lvta][xp17+Lvta, xp18-7.437-Lvta][xp18+9.994+Lvta, xp19-Lvta][xp19+Lvta, xp20-Lvta][xp20+Lvta, xp21-7.437-Lvta][xp21+9.994+Lvta, xp22-Lvta][xp22+Lvta, xp23-Lvta][xp23+Lvta, xp24-Lvta]

( 2) Range of Slopes. According to the Code and practical experience, the range of slopes can be limited and are shown in Table 2.

Range of slopes.

Section type Acceleration High-speed rolling Deceleration

( 3) Encoding Rules. Binary encoding is used in this paper. The encoding form is(22)uk=i1,i2,,iNsl,x1,x2,,xNsl,where uk is the kth chromosome.

The range of the gradient is −1~55, and the accuracy is 0.1. Considering the model calculation, the range is converted to [0, 560].

The gradient convert formula is(23)i=i+1×10,where i is the converted slope gradient ().

The binary code length can be calculated as follows:(24)2λi=560-01+1=561,λi=10,where λi is the binary code length of the gradient.

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.

3.4. Calculation of Parameters 3.4.1. Friction Retarder

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 3.

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
3.4.2. Turnout Parameters

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.

3.4.3. Other Parameters

The parameters of temperature, wind speed and calculation car depend on the specific hump.

4. Case Study

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 7. The x-coordinate of each node is shown in Table 4.

The x-coordinate of each node.

Number x -coordinate
P 1 0.000
P 2 12.122
P 3 34.666
P 4 44.386
P 5 74.386
P 6 84.707
P 7 100.227
P 8 103.107
P 9 123.469
P 10 131.301
P 11 141.690
P 12 165.252
P 13 168.252
P 14 190.772
P 15 207.351
P 16 217.355
P 17 223.464
P 18 231.041
P 19 243.291
P 20 245.036
P 21 254.731
P 22 266.322
P 23 367.751
P 24 397.760
P 25 414.560
P 26 518.660

Plane outspread drawing of hard rolling track (sketch map).

Curve parameters of hard rolling track are shown in Table 5.

Curve parameters of hard rolling track.

Number Angle (°) Radius (m) Tangent length (m) Curve length (m) Start x-coordinate End x-coordinate
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 6.

Climate conditions of hump design.

Condition Temperature (°C) Wind speed (m/s) Wind direction (°)

According to the data of Table 4, the range of the gradient change point, and the number of slopes, the range of the slope changing point can be got as shown in Table 7.

Reasonable range of slope changing point.

Slope change point Ranges
x 1 [ 28.000,34.666 - L v t a ]
x 2 [ 81.823 + L v t a , 84.707 - L v t a ]
x 3 [ 103.107 + L v t a , 123.469 - L v t a ] [ 141.690 + L v t a , 165.252 - L v t a ]
x 4 [ 190.772 + L v t a , 199.914 - L v t a ]
x 5 [ 266.322 + L v t a , 367.751 - L v t a ]
x 6 397.760 - L v t a

Given the small ranges of x2 and x6, the coordinates of x2 and x6 are given fixed values of 83.000 m and 393.660 m.

For the WART calculation, the connect track number and the x-coordinate of branching turnout are given as Table 8.

Connect track number and the x-coordinate of branching turnout.

Branching turnout Connect tracks x -coordinate Value
x t 1 34 x p 4 + 9.994 54.380
x t 2 17 x p 10 + 9.994 141.295
x t 3 5 x p 15 + 9.994 217.345
x t 4 2 x p 21 + 9.994 264.725

Equilateral turnout has lead curve, the hump uses 6# equilateral turnout, and its radius and angle are 180 m and 4.731°. The x-coordinates of each turnout are given in Table 9.

Turnouts x-coordinates.

Number Start x-coordinate End x-coordinate
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. The end speed of the optimization area is 3.6 m/s. The selected and sorted typical Pareto solution set and traditional manual method result with 5 slopes are shown in Table 10.

Solution set with two methods of hump profile design of distributing zone.

No. S1(10−3) S2(10−3) S3(10−3) S4(10−3) S5(10−3) S6(10−3) X 1 X 2 X 3 X 4 X 5 X 6 HH WART
( 1 ) 54.7 18.4 8.7 8.0 5.5 1.5 30 83 110 196 310 393.66 4.255 778.827
( 2 ) 54.8 17.8 6.7 2.7 9.0 0.6 30 83 164 198 302 393.66 4.213 780.720
( 3 ) 54.9 18.2 5.0 5.0 6.1 3.6 30 83 114 193 298 393.66 4.147 783.875
( 4 ) 53.2 19.1 7.9 7.9 4.0 0.4 30 83 146 197 348 393.66 4.097 786.410
( 5 ) 54.5 16.7 9.4 7.4 2.5 3.3 30 83 144 198 307 393.66 4.014 787.948
( 6 ) 53.4 18.9 3.3 9.3 4.2 2.5 30 83 144 195 313 393.66 3.977 791.551
( 7 ) 54.0 19.9 3.8 3.8 6.7 −0.2 28 83 113 194 324 393.66 3.885 794.150
( 8 ) 54.9 17.2 1.7 7.9 3.8 2.0 29 83 113 198 325 393.66 3.863 795.772
( 9 ) 54.1 16.3 9.9 2.1 4.2 1.8 28 83 159 199 274 393.66 3.778 797.065
( 10 ) 54.5 18.8 4.5 2.0 3.8 3.5 28 83 163 194 352 393.66 3.728 798.674
( 11 ) 54.7 16.2 2.1 6.2 3.4 2.8 30 83 116 197 295 393.66 3.681 800.504
( 12 ) 54.2 18.1 2.9 3.6 4.8 1.1 29 83 160 199 335 393.66 3.630 803.904
( 13 ) 54.8 15.8 4.6 5.1 3.6 1.9 29 83 164 195 332 393.66 3.584 806.544
( 14 ) 52.4 19.4 3.9 6.1 2.1 2.5 28 83 154 198 346 393.66 3.510 809.348
( 15 ) 51.7 18.4 4.4 4.0 4.4 0.9 29 83 145 196 292 393.66 3.484 813.706
( 16 ) 54.8 15.9 0.0 4.5 3.6 2.3 29 83 112 198 288 393.66 3.402 816.304
( 17 ) 53.0 12.2 8.1 9.3 1.4 −0.1 28 83 144 197 340 393.66 3.337 829.300
( 18 ) 52.0 13.7 5.9 9.3 5.8 −0.9 28 83 160 192 268 393.66 3.289 833.284
( 19 ) 51.6 12.5 6.2 6.7 2.0 1.5 29 83 112 199 280 393.66 3.267 840.509
( 20 ) 54.1 10.0 6.9 4.6 2.5 1.9 28 83 165 198 352 393.66 3.247 843.870
( 21 ) 50 16 8 6 3 - 28 83 168 185 393.66 - 3.717 827.582

As seen in Table 10, the Pareto solution set is numbers (1)~(20), and the traditional manual method is number (21). The hump height of number (10) is almost the same with number (21), but the Pareto solution has a low WART value with 798.674 compared with 827.582. So, the optimization effect is obvious because the WART has a large cardinal number. Also, compared with traditional manual method, the method proposed in this paper can be worked out by computer, which is more convenient and save much time.

5. Conclusions

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.

Symbols Δ s :

Length of the small part (m)

v 0 :

Initial speed in the small part (humping speed is the original speed) (m/s)

a :

Acceleration (m/s2)

Δ t j :

Rolling time of small part j (s)

N s p :

Total number of small parts

G c :

Gravity (N)

f g :

Components of the forces along the slope (N)

f r :

Total rolling resistance (N)

i :

M :

Car weight (kg)

n a x l :

Number of axles

w a i r :

Specific air resistance (N/kN)

w c u r :

Specific curve resistance (N/kN)

k t u r :

Coefficient of turnout (1 when car rolls in the range of turnout, otherwise 0)

k c u r :

Curve coefficient (1 when car rolls in the range of curve, otherwise 0)

v c a r :

Rolling speed (m/s)

v w i n :

Wind speed (negative when the same as the rolling direction and positive when opposite) (m/s)

β w i n :

Included angle of the wind and rolling direction (°)

x j - 1 :

x -coordinate of the j-1th gradient change point

i j :

Slope gradient of slope j ()

R v c :

Radius of the vertical curve (m)

x g c :

x -coordinate of the gradient change point

v e r a d :

Rolling speed of the easy car under advantageous rolling conditions (m/s)

v t u r m a x :

Max allowed entry speed of turnout (m/s)

I r o l m i n :

Minimum slope for hard rolling car automatic rolling ()

I a c c m i n :

I a c c :

L h r c :

Lengths of the hard rolling car (m)

v h u m :

Humping speed (m/s)

t h r h t t :

Rolling time of hard rolling car from the top of hump to the first turnout (s)

t h r h t r :

Rolling time of the hard rolling car from the top of hump to the first friction retarder (s)

t r e t i n v :

Friction retarder working status change time (s)

x b p :

x -coordinate of back slope change point

u k :

The kth chromosome

λ i :

Binary code length of the gradient

Δ t :

Rolling time of car in small part (s)

v t :

End speed in the small part and also the initial speed of next small part (m/s)

T w :

The WART (s)

c w j :

Weighting coefficient of small part j

α :

Gradient angle of the slope (°)

f s :

Normal supporting force (N)

f p :

Components of the forces in the normal direction (N)

a g :

Acceleration of fg (m/s2)

g :

Acceleration that takes into account rotational inertia (m/s2)

g s :

Standard gravity acceleration (m/s2)

w r o l :

Specific rolling resistance (N/kN)

w t u r :

Specific turnout resistance (N/kN)

C x 1 / C x 0 :

Coefficient of air resistance

k t u r d :

Direction coefficient of turnout (1 for reverse turnout, 0.5 for forward turnout)

s f :

Front projected facade area of the rolling car (m2)

H h :

Height of the hump distributing zone (m)

α w i n :

Included angle of the compound direction (wind and rolling direction) and the rolling direction (°)

x j :

x -coordinate of the jth gradient change point

x s t a r t :

Starting x-coordinate of the gradient change point of friction or turnout

L v t a :

Length along the tangent (m)

Δ i :

Value of the adjacent slope change ()

N s l :

Number of slopes

v f r m a x :

Max allowed entry speed of the friction retarder (m/s)

I r e t :

Slope at the friction retarder ()

v e n d h r d :

Speed of the hard rolling car at the end of the distributing zone (m/s)

I a c c m a x :

v e n d s p e :

Specified end speed (m/s)

L m r c :

Lengths of middle rolling car (m)

t t u r :

Turnout working status change time (s)

t m r h t t :

Rolling time of middle rolling car from the top of hump to the first turnout (s)

t m r h t r :

Rolling time of middle rolling car from the top of hump to the first friction retarder (s)

x f p :

x -coordinate of front slope change point

l m i n g :

Minimum slope length (m)

i :