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Abstract

The heavy-haul train has a series of characteristics, such as the locomotive traction properties, the longer length of train, and the nonlinear train pipe pressure during train braking. When the train is running on a continuous long and steep downgrade railway line, the safety of the train is ensured by cycle braking, which puts high demands on the driving skills of the driver. In this article, a driving curve generation method for the heavy-haul train based on a neural network is proposed. First, in order to describe the nonlinear characteristics of train braking, the neural network model is constructed and trained by practical driving data. In the neural network model, various nonlinear neurons are interconnected to work for information processing and transmission. The target value of train braking pressure reduction and release time is achieved by modeling the braking process. The equation of train motion is computed to obtain the driving curve. Finally, in four typical operation scenarios, comparing the curve data generated by the method with corresponding practical data of the Shuohuang heavy-haul railway line, the results show that the method is effective.

Keywords

Heavy-haul train neural network driving curve modeling operation scenarios

Introduction

Automatic train operation (ATO) has been widely used in urban rail transit in recent years. However, there are still some difficulties in applying ATO to the heavy-haul train (HHT) because of some characteristics of HHT, such as the locomotive traction properties, the train is long, and nonlinear train pipe pressure during train brake operation, which are mainly dependent on air braking on the continuous long and steep downgrade and nonlinear train pipe pressure when reducing the speed of vehicles along the train. Due to the nonlinear complexity of dynamic operation, the driver must follow a given driving curve when driving an HHT to ensure the operation safety. The given driving curve is generally obtained from field driving data of many experienced drivers. However, since there are various types of locomotives and marshaling, it is difficult to get a range of driving curves for different scenarios. The driver therefore needs to control the train using his own experience, although he may not always make good choices, for reasons which include the labor intensity of the job and driver fatigue, increasing the risk of accidents. To resolve this issue, a driving curve generation method has been put forward which can generate the driving curve of an HHT based on an artificial neural network (NN).

In recent years, much research on the train operation of rail transit systems has been carried out using different methods. For example, a model selection technique has been introduced to optimize the braking model and identify the parameters of urban rail transit systems in the field of ATO for urban rail transit, but the nonlinearity makes it hard for HHT. Based on system parameter identification theory, the ATO model has been obtained from field data.^1,2 Self-adaptive fuzzy control technique and fuzzy logic self-tuning of scale gene have been applied to the speed of ATO in the urban rail transit system.^3,4 However, the complexity and nonlinearity of dynamic operation make it difficult to directly control the HHT by ATO. In the HHT field, considering the interaction force between vehicles and the control difficulties of HHT, a decentralized control model has been employed to the control of the vehicle’s speed.⁵ But, how to achieve the control speed is not considered. A predictive control model has been introduced to study the whole HHT operation process and control the train speed.^6–8 The control of train speed is realized by the constructed models in the published work, but no specific operational control parameter is given for the driver.^9–12 According to requirements of the speed limits in the different points of slope changing, the backing-off algorithm is used to calculate the HHT operation curve, but it needs to be repeated to calculate the difference value between the train practical speed and the limit speed in the point of slope changing and judging the rationality of driving curve simultaneously.¹³ Therefore, a driving curve generation method for HHT based on an NN is proposed, which mainly focuses on the continuous long and steep downgrade to solve this problem and obtain the train braking pressure reduction (BPR). Providing that the NN model is suitable, the model is trained by field driving data with improved training algorithm, and then the expected target value of train BPR and release time can be obtained by inputting the real-time parameters into the model in the generation process. Combining obtained train braking force with the traction and resistance of the HHT, the driving curve in different railway line scenarios can be achieved by computing the equation of train motion. The curves are compared with different practical driving curves.

Modeling of HHT driving curve based on NN

The train driving curve generation model must ensure both the safety requirements and the efficient operation of the HHT. When an HHT is running on the continuous long and steep downgrade, it is necessary to generate the train driving curve rapidly, efficiently, and accurately based on the synthesized analysis of the real-time information, such as train position, speed, the length, gradient, and the speed limit of the railway line.

Model of train

Both the influence of the air brake function on the continuous long and steep downgrade and the section with difficult operation under a three-aspect block system should be given particular consideration in the operation process of the HHT. The train pipe pressure is 500/600 kPa without using the air brake function. The pressure of the train pipe is reduced to slow the train speed when using air braking control. In addition, in order to ensure the braking ability next time, the pressure of the train pipe should recover to constant by air charging when finishing air brake. Because the whole control process needs sufficient time, improper operation of the driver may cause the train to be out of control with weak braking ability. To cope with this problem, the following scenario will be studied.

Assume that there are two or more long and steep downgrades, and the slope length is not less than 2 km when the train is running on the railway line. The corresponding HHT model and running scenario are described in equations (1) and (2)

{\begin{matrix} {\overset{\cdot}{S}}_{i} = v_{i} \\ {\overset{\cdot}{v}}_{i} = \frac{F_{i}}{m_{i}} + \frac{f_{i - 1}}{m_{i}} - \frac{f_{i}}{m_{i}} - \frac{R_{i}}{m_{i}} \end{matrix}

(1)

where i indicates the ith vehicle and S, v, and m denote the running distance, speed, and mass of the vehicles, respectively. F is traction or braking force, f is the interaction force between the ith and (i + 1)th vehicle and R is the resistance of the ith vehicle. The distance and speed of the train can be calculated by the forces

{\begin{matrix} j_{i} > 0 \cap j_{i + 1} < 0, j_{i + 2} < 0, \dots \\ j_{i} \leq 0 \cap j_{i + 1} < 0, j_{i + 2} < 0, \dots \\ (s_{i + 1} + s_{i + 2} + \dots) \geq 2000 (m) \end{matrix}

(2)

where j(‰) is the gradient, s is the slope length, and i represents the ith slope of the railway line. The equation describes that the downslope number is more than 2, and the slope length is more than 2000 m when the train is on the upslope or downslope.

Equation of train motion

In the assumption scenario, there are three kinds of force, such as traction, braking force, and resistance which always exist in the operation process. There is also rotary motion when the train is in translational motion, in view of running time and distance, equation (1) can also be written as

{\begin{matrix} \int dt = \int \frac{1}{ζ \cdot c} dv \\ \int dS = \int \frac{v}{ζ \cdot c} dv \end{matrix}

(3)

where c is the unit resultant force of the train; S, v, and t denote the distance, speed, and time, respectively; and $ζ$ is the acceleration coefficient which is described as

ζ = \frac{127 M}{M + \sum \frac{I}{R_{h}^{2}}}

(4)

where M is the mass of the whole train, I is the rotational inertia of the rotary motion part, $R_{h}$ is the radius of gyration of the rotary motion part, and $\sum I / R_{h}^{2} / M$ is the rotating mass coefficient, the value of which is 0.06.

Modeling of NN

A NN consists of several neurons that are connected with different weights to process and transfer information. An error backpropagation (BP) NN is easy to implement and widely used in the function approximation, pattern recognition, and classification. The structure of BP NN is so simple, but with proper hidden layers and nodes, the BP NN can approximate any nonlinear mapping relations with good simplicity and generalization ability and keep good simplicity of the algorithm at the same time. The authors use the method of improved BP NN.

The NN model is a multilayer structure and each layer has output vectors, an error vector and weight matrix. So far, there are various training function algorithms, such as gradient descent,¹⁴ conjugate gradient,¹⁵ Levenberg–Marquardt, Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton algorithm¹⁷ and some improved training methods.^18–20 The improved BP NN has a three-layer structure, the f₁ is a tan-sigmoid function, and the f₂ is a linear function.

The output of the NN can be described as

y_{k} = f_{2} [\sum_{j = 1}^{n} w_{jk} \times f_{1} (\sum_{i = 1}^{m} w_{ij} \times x_{i} + b_{1 j}) + b_{2 k}]

(5)

where $x_{i}$ is the neurons input, $w_{ij}$ and $b_{1 j}$ denote weight and deviation of the input layer, and $w_{jk}$ and $b_{2 k}$ denote weight and error of the hidden layer. The outputs of hidden and output layers can be calculated by the inputs, weight, and error. Due to the differences between each training process, the weight and error are not constant, which can cause different results.²¹

In order to make the NN effective, the related data must be normalized for processing, which make the value of all parameters in the range of [0, 1]. The weight values of the NN model are adjusted by the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) quasi-Newton training algorithm, which is an improved learning algorithm of the BFGS algorithm. It does not need to compute the Hessian matrix and it updates the weight values (x_k+₁) by equation (6)

x_{k + 1} = x_{k} - t_{k} \cdot H_{k} \cdot g_{k}

(6)

where $g_{k}$ is the gradient of $x_{k}$ , $t_{k}$ is the step size, and $H_{k}$ is the positive definite matrix, which is easy to calculate and can replace the Hessian inverse matrix. Weight values (x_k+₁) change from (x_k), and the H_k₊₁ updating process is written as

{\begin{matrix} H_{k + 1} = V_{k}^{T} H_{k} V_{k} + ρ_{k} s_{k} s_{k}^{T} \\ ρ_{k} = \frac{1}{y_{k} s_{k}^{T}} \\ V_{k} = I - ρ_{k} y_{k} s_{k}^{T} \\ s_{k} = x_{k + 1} - x_{k} \\ y_{k} = g_{k + 1} - g_{k} \\ H_{k + 1}^{0} = \frac{s_{k}^{T} y_{k}}{y_{k}^{T} y_{k}} I \end{matrix}

(7)

where (s_k, y_k) is the curvature information, ρ_k and V_k are intermediate variables, and H⁰ is the original matrix which is known. The L-BFGS algorithm updates $H_{k}$ by saving the recent m groups’ curvature information one iteration, which is different to the BFGS algorithm and usually m is chosen from 3 to 20. m is set as 3 for keeping the speed of training. Thus, the algorithm is helpful to keep the speed of the Newton algorithm and to reduce the amount of calculation, and at the same time, it can keep strong robustness.

The node number of the hidden layer is determined by the practical scenario and it may affect the precision of the NN. It may have a long training time or it could be over-fitted with too many nodes. The training time and precision will be affected if the node number is too small. To avoid over-fitting, the training sample size N should meet the condition as

N = O (\frac{σ}{ε})

(8)

where $σ$ is the free parameter amount of the network, $ε$ is the allowed classification error part of the test data, and $O ()$ is the order of content in the parenthesis.

The training and validation sample is randomly selected from the whole field data. The node number n of the hidden layer can be determined by the following equation

{\begin{matrix} n < \sqrt{u + z} + a \\ n ≅ C_{f} {(\frac{N}{u \log N})}^{\frac{1}{2}} \\ n ≅ \log_{2} u \end{matrix}

(9)

where a is a constant between 0 and 10, u is the node number of the input layer, z is the node number of the output layer, N is the training sample size, and $C_{f}$ is the first-order absolute momentum. In the process of NN modeling, equation (9) determines the approximate node number of the hidden layer, and the optimal value of n is chosen by trial and error.

In the NN training process, choosing an error as the given standard, the error function is defined by equation (10) to determine whether the training meets the precision requirement

E (W, B) = \frac{1}{2} \times \sum {(t_{k} - y_{k})}^{2}

(10)

where $t_{k}$ is expected output and $y_{k}$ is output of the NN; the error can reflect the training effect of the NN.

Based on the operation and driving characteristics of the HHT, the NN model is established to generate the driving curve. Using the field data to get the mapping relationship, a corresponding NN model can be achieved to control the train using driver’s experience. The modeling data are collected from practical driving data. The modeling data group based on Shuohuang heavy-haul railway line is conducted. The total Shuohuang heavy-haul railway line covers a length of 600 km. The maximum gradient of downslope is 12‰. The maximum speed limit is 80 km/h.

Driving curve generating algorithm of the train

In the process of train driving curve generation, first, it is judged whether the HHT meets the model conditions; if the train is running on the upslope railway line, the train does not need cycle braking. As long as the model satisfies the requirements, the corresponding practical time parameters are put into the trained NN model.

The parameters are railway line speed limit v_lim, gradient j, the weight of the HHT G, length of the train L, train speed v, preceding train distance S, and reaction time of driver t₁. Then, the two outputs’ target value of train BPR r and release time t₂ are obtained by the NN model. The modeling process based on the NN is shown in Figure 1.

Step 1. Collect and process the normalized train driving data, which should be sufficient to make function approximation to reduce the error.

Step 2. Construct the applicable conditions of the generation model, which is based on the analysis of the field operation and characteristics of the HHT. The conditions are shown in equation (2).

Step 3. Confirm and select the parameters that can represent the characteristics of the HHT as the input and output data of the NN model. While decreasing the node number of the hidden layer, the training time will be shorter, but the precision of the fitting will be lower. According to the NN parameters of each layer obtained above, and for the balance of training time and precision of the fitting, the corresponding NN model is built. There are 7 nodes consisting of one input layer and one output layer with 2 nodes, and 11 nodes consisting of one hidden layer by trial and error. So the structure of the NN is (7 × 11 × 2).The training sample and validation sample are selected randomly.

Step 4. Take the railway line speed limit, gradient, the weight of the HHT, length of the train, train speed, preceding train distance, and reaction time of the driver as the input data of the NN; the input parameters are listed as follows:

x ₁: v_lim (railway line speed limit);

x ₂: j (gradient);

x ₃: G (weight of HHT);

x ₄: L (length of the train);

x ₅: v (train speed);

x ₆: S (preceding train distance);

x ₇: t₁ (reaction time of driver).

And take the target value of train BPR and release time as the output data:

y ₁: r (target value of train BPR);

y ₂: t₂ (release time of the train).

Step 5. Using the self-learning and adaptive ability of the NN, the L-BFGS quasi-Newton algorithm is selected to train the NN using the field data.

Figure 1.

Flow of the NN modeling.

The driving curve generation process for the HHT based on the NN is shown in Figure 2.

Figure 2.

Flow of driving curve generation of HHT based on NN.

The driving curve is obtained by the outputs of the NN together with the train motion equation. The final driving curve cannot be obtained until the results meet the requirements, by which the stability and robustness of the process are enhanced, and the error and time delay are reduced.

Step 1. According to the status of HHT operation, if the status meets the constraint conditions of the model, then go to the next step. Otherwise, continue step 1. The constraint conditions of the model are $j_{i + 1} < 0, j_{i + 2} < 0, \dots (s_{i + 1} + s_{i + 2} + \dots) \geq 2000$ .

Step 2. Collect and process the practical time parameters of the train; v_lim, j, G, L, v, S, t₁; like the training sample; judge the kinds of conditions and input the data obtained into the corresponding trained NN model.

In view of the characteristics of HHT and the difficult problem of generating the driving curve of HHT, the authors establish the four typical operation scenarios. The first operation scenario is that a train will stop at the station along a continuous downslope with initial downslope. The second operation scenario is that a train will stop at the station along a continuous downslope with initial upslope. The third operation scenario is that a train will run along a very steep slope with initial downslope. The fourth operation scenario is that a train will run along a very steep slope with initial flat slope.

The data describe the process of cycle braking and train operation. The corresponding railway line speed limit, gradient, the weight of train, the length of the train, driver reaction time, and release time with different train BPRs are shown in Table 1.

Table 1.

Training data of four operation scenarios.

BPR (kPa)	v _lim (km/h)	j (‰)	G (t)	L (m)	t ₁ (s)	t ₂ (s)
50–110	75–80	4–12	10,457–11,352	1694–2043	3.4–3.6	63–115

BPR: braking pressure reduction.

Figure 3 denotes 200 groups of training data of scenario 1 and includes the data of train BPR, speed, and preceding train distance, and Figure 4 denotes that the randomly selected test data have been processed to describe the error of the NN modeling process. With the principle that validation data group should equal or be smaller than the training data group, 180 groups of data are chosen as validation data. The maximum error is 0.72, and the mean error is 2.2%.

Figure 3.

Training data of scenario 1.

Figure 4.

Test data error of scenario 1.

Figure 5 denotes 186 groups of training data of scenario 1 and includes the data of train BPR, speed, and preceding train distance, and Figure 6 denotes that 100 groups of data are chosen as validation data. The maximum error is 0.6, and the mean error is 1.8%.

Figure 5.

Training data of scenario 2.

Figure 6.

Test data error of scenario 2.

Figure 7 denotes 200 groups of training data of scenario 3 and includes the data of train BPR, speed, and preceding train distance, and Figure 8 denotes that 180 groups of data are chosen as validation data. The maximum error is 0.65, and the mean error is 2.8%.

Figure 7.

Training data of scenario 3.

Figure 8.

Test data error of scenario 3.

Figure 9 denotes 203 groups of training data of scenario 4 and includes the data of train BPR, speed, and preceding train distance, and Figure 10 denotes that 100 groups of data are chosen as validation data. The maximum error is 0.5, and the mean error is 1.7%.

Figure 9.

Training data of scenario 4.

Figure 10.

Test data error of scenario 4.

Step 3. Get the outputs r and t₂ from the NN model y₁, y₂ and normalize the parameters. The output is written as

y_{k} = f_{2} [\sum w_{2} \times f_{1} (\sum w_{1} \times x_{i} + b_{1}) + b_{2}]

(11)

where k = 1, 2; i = 1,…, 7. The parameters have the same meaning as equation (5); the subscript indicates the NN layers. Then, calculate the train forces which are important in calculating the equation of train motion.

Step 4. Based on the target value, train BPR r, release time t₂, and other known parameters obtain the train resultant force. Next, calculate the running speed, time, and distance for generating the train driving curve.

When the speed of the locomotive is confirmed, the locomotive traction is set as 90% of the maximum traction with the corresponding speed. Train resistance consists of basic resistance and additional resistance due to the curve and tunnel. Train air braking force is formed by the friction of the brake shoe. Calculation equations of the locomotive traction, train resistance, and braking force can be written as

{\begin{matrix} F = F_{max} \times 0.9 \\ ω = ω_{0} + i_{j} \\ b_{c} = 1000 θ_{h} β_{c} ϕ_{h} \end{matrix}

(12)

where $F_{max}$ is the maximum traction obtained by the locomotive traction characteristic curve, $ω_{0}$ denotes the train’s basic resistance per unit, $i_{j}$ is a thousandth of gradient, $θ_{h}$ represents the equivalent emergency braking ratio of the train, $β_{c}$ is the coefficient of service braking, and $ϕ_{h}$ is the equivalent friction coefficient. Three kinds of forces when calculating the train operation are very important.

The equivalent braking ratio is full and the $β_{c}$ is 1 when the train implements emergency braking. As shown in Table 2, the coefficient is selected according to the constant of train pipe pressure and the value of train pipe pressure reduction when the train implements service braking. Then, the coefficient is multiplied by the full braking ratio as the equivalent braking ratio of service braking.

Table 2.

Value of coefficient of normal braking $β_{c}$ .

Train pipe pressure reduction r (kPa)	$β_{c}$ (train pipe pressure)
Train pipe pressure reduction r (kPa)	600 p (kPa)	500 p (kPa)
50	0.17	0.19
60	0.28	0.32
70	0.37	0.42
80	0.46	0.52
90	0.53	0.60
100	0.60	0.68
110	0.67	0.75
120	0.73	0.82
130	0.78	0.89
140	0.83	0.95
150	0.88	−
160	0.93	−
170	0.96	−

When the train is braking, the train braking distance (BD) can be written as

{\begin{matrix} S = S_{k} + S_{e} \\ S_{k} = \frac{v_{0} \cdot t_{k}}{3.6} \\ S_{e} = \sum \frac{4.17 (v_{1}^{2} - v_{2}^{2})}{b_{c} + ω_{0} + i_{j}} \end{matrix}

(13)

where S is the distance of train braking, which consists of the running distance $S_{k}$ and active BD $S_{e}$ . $S_{k}$ is the running distance during the time $t_{k}$ , which is calculated from the beginning of the braking command with initial speed $v_{0}$ to the time of active braking. $S_{e}$ is the active BD, which represents the running distance of the train with at least 95% of the braking force. The $v_{1}$ and $v_{2}$ are the speed intervals after braking. The r can be used to calculate the b_c which has the same meaning as equation (12).

If the results meet the requirements, record the driving curve; otherwise, repeat the above steps. The constraints of the HHT are integrated as follows

{\begin{matrix} 40 < r < 170 \cap β_{c} \neq 1 \to 40 < r < 100 (kPa) \\ β_{c} = 1 \to v_{0} > 40 (km / h) \\ r \neq 0 \cap v < 30 (km / h) \to r_{i + 1} \geq r_{i} \end{matrix}

(14)

where r is the target value of train BPR by air braking, $β_{c}$ is the coefficient of service braking, $v_{0}$ is the initial speed before braking, and v is the train speed. The subscript represents the two consecutive bracings i and i+1. The r ranges from 40 to 170 kPa, and it is not more than 100 kPa when service braking is triggered. The $v_{0}$ is not less than 40 km/h when emergency braking is triggered. The v is not less than 30 km/h when braking is released by the train.

Step 5. The evaluation indices of the train under different conditions can be calculated, such as expectation and variance of speed difference, BD, and mean speed.

Step 6. Judge the driving curve results and end the process if it meets the requirements; otherwise, rerun the process.

Simulation analysis

Based on four typical operation scenarios, four cases based on Shuohuang heavy-haul railway line are conducted as examples. The generation of the driving curve is achieved by MATLAB simulation software programming. The simulation parameters are shown in Table 3.

Table 3.

Simulation parameters of the model.

Speed limit	Marshaling mode	Type of locomotive	Number of vehicle	Total weight/load	Length of train	Reaction time	Constant of train pipe pressure
75–80 km/h	1 + 1	SS_4B	110	10,457/77,44t	1694 m	3.5 s	600 kPa

The railway line parameters, length of slope, gradient, and cumulative distance of the four operation scenarios are shown in Tables 4 –7, respectively.

Table 4.

Simulation railway line parameters of scenario 1.

Slope length (m)	Gradient (‰)	Distance (m)
900	−10.0	900
1392	−6.0	2292
500	−7.0	2792
1420	−11.0	4212
1550	−9.4	5762
1320	−1.5	7082

Table 5.

Simulation railway line parameters of scenario 2.

Slope length (m)	Gradient (‰)	Distance (m)
1300	3.5	1300
500	−3.5	1800
400	−6.3	2200
600	−6.0	2800
500	−3.0	3300
400	−5.7	3700
1350	−6.0	5050
1000	−10.0	6050
500	−5.7	6550
660	−2.5	7210
1890	0.0	9100
300	−2.8	9400

Table 6.

Simulation railway line parameters of scenario 3.

Slope length (m)	Gradient (‰)	Distance (m)
468	−11.2	468
750	−11.0	1218
850	−12.0	2068
500	−11.0	2568
1650	−10.5	4218
550	−11.5	4768
800	−12.0	5568
900	−11.0	6468
1250	−10.0	7718
1050	−10.5	8768
500	−11.5	9268
380	−12.0	9648
1350	−10.6	10,998
350	−12.0	11,348

Table 7.

Simulation railway line parameters of scenario 4.

Slope length (m)	Gradient (‰)	Distance (m)
700	0.0	700
500	−4.8	1200
500	−11.1	1700
1950	−12.0	3650
450	−11.3	4100
2050	−12.0	6150
1050	−8.0	7200
700	−7.4	7900
375	−4.0	8275

The simulation driving curve (SDC), first practical driving curve (FPDC), and second practical driving curve (SPDC) of four operation scenarios are shown in Figures 11 –14, respectively.

Figure 11.

Simulation and practical HHT driving curves of scenario 1.

Figure 12.

Simulation and practical HHT driving curves of scenario 2.

Figure 13.

Simulation and practical HHT driving curves of scenario 3.

Figure 14.

Simulation and practical HHT driving curves of scenario 4.

The corresponding data of Figures 11 –14 are shown in Tables 8 –11, respectively. The first column in the table denotes the running distance, the second column is the first practical speed of the train, and the third column is the second practical speed of the train. The fourth column denotes the simulation speed. The fifth column denotes twice the practical train BPR. The sixth column represents the train BPR in simulation. The FPDC and SPDC come from the driving data records of Shuohuang heavy-haul railway line, the value of the two curves are used to compare with the simulation results and to evaluate the effectiveness of method.

Table 8.

Simulation and practical driving curves data of scenario 1

Distance (m)	First practical speed (km/h)	Second practical speed (km/h)	Simulation speed (km/h)	Twice practical BPR (kPa)	Simulation BPR (kPa)
0	67	69	68	0/0	0
1200	72	71	72	0/0	0
1300	72	71	72	0/60	55
1500	72	67	70	50/60	55
2000	56	45	51	50/60	55
2100	52	44	44	50/60	0
2200	47	42	45	50/0	0
2300	44	43	46	0/0	0
3500	57	60	59	0/0	0
4500	66	67	66	0/0	0
4600	66	68	67	0/50	0
4800	67	65	67	0/50	51
4900	67	64	67	50/50	51
6000	49	42	44	50/50	51
6770	14	0	5	50/50	51
6840	8	0	0	50/50	51
6980	0	0	0	50/50	51

BPR: braking pressure reduction.

Table 9.

Simulation and practical driving curves data of scenario 2.

Distance (m)	First practical speed (km/h)	Second practical speed (km/h)	Simulation speed (km/h)	Twice practical BPR (kPa)	Simulation BPR (kPa)
0	72	72	71	0/0	0
1500	70	71	70	0/0	0
3000	69	72	70	0/0	0
3500	70	72	70	0/50	0
4400	70	69	70	60/50	0
4500	70	68	68	60/50	55
5850	50	49	48	6050	55
5950	48	49	46	0/50	55
6050	45	45	45	0/0	55
6750	52	51	52	0/0	0
7410	59	58	60	0/0	56
7610	61	60	60	50/0	56
7710	61	60	59	50/60	56
8510	44	40	38	50/60	56
9200	14	8	9	50/60	56
9300	8	0	5	50/60	56
9400	3	0	0	50/60	56
9450	0	0	0	50/60	56

BPR: braking pressure reduction.

Table 10.

Simulation and practical driving curves data of scenario 3.

Distance (m)	First practical speed (km/h)	Second practical speed (km/h)	Simulation speed (km/h)	Twice practical BPR (kPa)	Simulation BPR (kPa)
0	57	53	55	0/0	0
1068	67	63	63	0/0	0
2068	70	68	69	0/0	0
2468	71	70	71	50/0	53
2768	70	71	70	50/50	53
3668	65	67	64	50/50	53
4418	52	62	54	5050	53
4618	54	58	54	50/50	0
5668	65	61	62	0/0	0
6668	70	68	68	0/0	0
7168	71	70	70	50/50	0
7368	70	71	71	50/50	51
8318	64	64	66	50/50	51
9168	54	51	55	50/50	51
9268	55	51	53	0/0	51
10,248	61	57	59	0/0	0
10,948	67	64	63	0/0	0
11,348	69	70	66	0/0	0

BPR: braking pressure reduction.

Table 11.

Simulation and practical driving curves data of scenario 4.

Distance (m)	First practical speed (km/h)	Second practical speed (km/h)	Simulation speed (km/h)	Twice practical BPR (kPa)	Simulation BPR (kPa)
0	66	68	69	0/0	0
1500	67	66	66	0/0	0
2300	69	71	68	0/50	0
2500	70	70	68	50/50	0
3000	69	69	69	50/50	57
4500	62	57	60	50/50	57
4900	55	47	51	50/50	57
5000	52	50	47	50/50	0
5200	45	50	44	50/0	0
5300	44	46	44	0/0	0
6500	58	62	61	0/0	0
7500	67	66	67	0/0	0
8000	69	66	68	0/0	0
8300	69	69	70	0/0	0

BPR: braking pressure reduction.

In order to evaluate the simulation results, the authors define some evaluation indices, including average speed (AS), BD, expectation of speed difference between simulation curve and first practical curve (ESSF), and the corresponding variance between simulation and first practical curve (VSSF), the expectation and variance between simulation and second practical curve (ESSS/VSSS), the expectation and variance between first and second practical curve (ESFS/VSFS). According to the simulation results of four operation scenarios, evaluation indices are shown in Tables 12 –15, respectively.

Table 12.

Simulation results of scenario 1.

Scenario 1	AS (km/h)	BD (m)	FPDC	SPDC
Scenario 1	AS (km/h)	BD (m)	ESSF/VSSF	ESSS/VSSS	ESFS/VSFS
SDC	53.74	6840	1.729/18.543	−1.236/4.38	−
FPDC	55.47	6980	−	−	2.965/32.424
SPDC	52.5	6770	−	−	−

AS: average speed; BD: braking distance; FPDC: first practical driving curve; SPDC: second practical driving curve; ESSF/VSSF: expectation and variance of speed difference between simulation curve and first practical curve; ESSS/VSSS: the expectation and variance between simulation and second practical curve; ESFS/VSFS: the expectation and variance between first and second practical curve; SDC: simulation driving curve.

Table 13.

Simulation results of scenario 2.

Scenario 2	AS (km/h)	BD (m)	FPDC	SPDC
Scenario 2	AS (km/h)	BD (m)	ESSF/VSSF	ESSS/VSSS	ESFS/VSFS
SDC	58.02	6840	1.374/5.027	0.512/5.207	−
FPDC	59.39	6970	−	−	0.862/10.808
SPDC	58.53	6730	−	−	−

Table 14.

Simulation results of scenario 3.

Scenario 3	AS (km/h)	FPDC	SPDC
Scenario 3	AS (km/h)	ESSF/VSSF	ESSS/VSSS	ESFS/VSFS
SDC	62.99	1.025/5.143	−0.160/6.864	−
FPDC	64.02	−	−	1.185/11.372
SPDC	62.83	−	−	−

Table 15.

Simulation results of scenario 4.

Scenario 4	AS (km/h)	FPDC	SPDC
Scenario 4	AS (km/h)	ESSF/VSSF	ESSS/VSSS	ESFS/VSFS
SDC	63.08	0.264/3.355	0.145/3.861	−
FPDC	63.34	−	−	0.119/7.852
SPDC	63.22	−	−	−

Discussion

Based on the four typical operation scenarios of HHT, an NN driving curve generation method for HHT is proposed, which can be acted as a decision-support tool in this research. In the process, a different NN training algorithm is used and combined with the HHT characteristics and the braking operation model is built, the braking parameters are used in the motion equations and the simulation driving curves are achieved. Comparing simulation results with practical driving curves, the generation method is shown to be effective according to the speed, braking, and releasing actions of the corresponding curves, which have a similar target value for air BPR and operation trend.

The analysis of simulation results of four typical operation scenarios are as follows:

Scenario 1. A train stops at the station along a continuous downslope with initial downslope. The simulation results show that the AS differences between SDC and FPDC or SPDC are not more than ±1.73 km/h and the BD differences are not more than ±140 m. The ESSF and ESSS are 1.729 and −1.236, which are less than ESFS 2.965. The VSSF and VSSS are 18.543 and 4.38, which are all less than VSFS 32.424.

Scenario 2. A train stops at the station along a continuous downslope with initial upslope. The simulation results show that the AS differences between SDC and FPDC or SPDC are not more than ±1.37 km/h; the BD differences are not more than ±130 m. The ESSF and ESSS are 1.374 and 0.512, the ESSF is greater than ESFS 0.862, which means that speed difference between the simulation curve and the first curve is bigger than those between the first curve and the second curve, and the ESSS is less than ESFS. The VSSF and VSSS are 5.027 and 5.207, which are all less than VSFS of 10.808. The simulation results of scenario 1 and scenario 2 show that a train can run safely along a continuous downslope with initial slope and confirm that the method is effective.

Scenario 3. A train runs along a very steep slope with initial downslope. The simulation results show that the AS differences between SDC and FPDC or SPDC are not more than ±1.03 km/h. The ESSF and ESSS are 1.025 and −0.16, which are less than ESFS 1.185. The VSSF and VSSS are 5.143 and 6.864, which are all less than VSFS 11.372.

Scenario 4. A train runs along a very steep slope with initial flat slope. The simulation results show that the AS differences between SDC and FPDC or SPDC are not more than ±0.26 km/h. The ESSF and ESSS are 0.264 and 0.145, which are all greater than ESFS 0.119; the results show that speed differences between the simulation curve and the first curve, the simulation curve and the second curve are bigger than those between the first curve and the second curve. The VSSF and VSSS are 3.355 and 3.861, which are all less than VSFS 7.852. The simulation results of scenario 3 and scenario 4 show that a train can run safely under the maximum gradient of downslope with initial slope and confirm that the method is effective.

In order to get the train running curve, the rationality of running curve is judged by the limit speed value of different points of slope changing with the backing-off algorithm, when the calculated difference value between the train running speed and the limit speed exceeds the acceptable range, different backing-off strategies will be used to calculate the train running curve again, and the above process would be repeated until the calculated difference value meets the rationality. For the four kinds of HHT running scenarios in this article, there is no point of slope changing in the scenarios 1, 3, and 4; the rationality of driving curve cannot be judged by the limit speed changing; and the applicable conditions of backing-off algorithm cannot be met. And in the fourth step of NN algorithm proposed in chapter of the driving curve generating algorithm of the train, the conditions of limit speed changing and no limit speed changing conditions are both considered, the driving curve can be generated in the two conditions. So, the NN algorithm is not affected by the limit speed changing. Although the backing-off algorithm can be used in the scenario 2, it needs to iteratively calculate the difference value between the train practical speed and the limit speed in the point of slope changing and judging the rationality of driving curve simultaneously. The efficiency of entire process is low, which is generally suitable for offline calculation but not suitable for dynamic calculation. In comparison, the NN algorithm is intelligent and real-time dynamic calculation algorithm, which can be directly applied in engineering application.

The results of simulation and comparisons with back-off algorithm confirm the method of effectiveness and safety for generating a driving curve of an HHT. According to the modeling method of four typical scenarios, we can build more scenarios of the Shuohuang heavy-haul railway line and generate the driving curve of the whole line, which can be used to guide the driver’s driving operation as a decision-support tool.

For the different characteristics of heavy-haul lines and special operation scenarios, in the fourth step of NN algorithm proposed in chapter of the driving curve generating algorithm of the train, the curvature of the heavy-haul line should be considered to collect as a factor of training data. And some special operation scenarios, such as the auto-passing phase separation scenario, have effect on the driving curve generation, so how to improve the generalization ability of NN model should be considered. And, with the increase in the traction weight and the speed of the HHT, the driving curve is more closely related to the operation safety of the train. On one hand, more suitable modeling methods should be adopted to research the corresponding content to find more accurate driving curve, on the other hand, an automatic train driving method should be researched for a more efficient.

Conclusion

In this article, a driving curve generation method for HHTs based on NN model is proposed. Using the BP NN, the nonlinear braking operation model of an HHT is constructed. In the process of modeling, various nonlinear neurons are interconnected, and the field data are used to train the model. By taking the target value of train BPR as the controlled variable and then computing the train motion equation, the driving curve of an HHT using the NN model is generated finally. Based on four typical operation scenarios of HHT, four cases are conducted as examples, comparing simulation results with corresponding practical driving curves, the results show that the generation method is effective.

Footnotes

Academic Editor: Long Cheng

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the Beijing Jiaotong University basic scientific research funding projects under grant 2015JBM013, the Beijing municipal science and technology plan projects under grant D151100005815001, Shenhua Group scientific research funding projects under grant 20140269 and Beijing Laboratory for Urban Mass Transit.

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A neural network driving curve generation method for the heavy-haul train

Abstract

Keywords

Introduction

Modeling of HHT driving curve based on NN

Model of train

Equation of train motion

Modeling of NN

Driving curve generating algorithm of the train

Simulation analysis

Discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References