Optimization Study of Pneumatic–Electric Combined Braking Strategy for 30,000-ton Heavy-Haul Trains

Abstract

The normalized operation of 30,000-ton heavy-haul trains is of significant importance for enhancing the transportation capacity of heavy-haul railways. However, with the increase in train formation size, traditional braking strategies result in excessive longitudinal impulse when combined pneumatic and electric braking is applied on long, steep gradients. This presents a serious challenge to the braking safety of the train. To this end, this paper establishes a longitudinal dynamic model of a 30,000-ton heavy-haul train based on vehicle system dynamics theory, and validates the model’s effectiveness through line test data. On this basis, the influence of two braking parameters, namely, the distribution of the magnitude of the electric braking force and the matching time of pneumatic braking and electric braking, on the longitudinal dynamic behavior of heavy-haul trains is studied. Thereby, an optimized combined pneumatic and electric braking strategy is formulated to reduce the longitudinal impulse of the trains. The results show that setting reasonable braking parameters can effectively reduce the longitudinal impulse, with the braking matching time having a significant impact on the longitudinal impulse. Specifically, when using a strategy where the electric braking forces of three locomotives are set to 90 kN, 300 kN, and 300 kN, with a 30 s delay in applying the electric braking force, a better optimization effect is achieved. The two proposed braking strategies reduce the maximum longitudinal forces by 20.27% and 47.83%, respectively, compared to conventional approaches. The research results provide effective methods and theoretical guidance for optimizing the braking strategy and ensuring the operational safety of 30,000-ton heavy-haul trains.

1. Introduction

With the rapid economic development, the demand for heavy-haul railway freight capacity has been increasing. Developing heavy-haul trains with larger axle loads and longer formations is an effective measure to solve this problem [1,2,3]. Currently, China has achieved the regular operation of 20,000-ton heavy-haul trains on key railways such as the Daqin and Shuohuang lines. To further improve railway transport efficiency and reduce train operating density, the operation of 30,000-ton heavy-haul trains has become an urgent task. However, given that a 30,000-ton heavy-haul train can reach a total length of nearly 4 km and requires at least three locomotives, brake safety issues become particularly prominent and must be addressed [4]. During train braking, the different vehicles, depending on the varying track conditions, along with the coordination between the master and slave locomotive, can affect the braking synchronization of the front, middle, and rear cars. This can lead to significant interaction forces between the vehicles, posing a safety risk to the operation of heavy-haul trains [5]. In addition, the rational utilization of control strategies and optimization methods can enhance the stability of mechanical systems [6,7]. Therefore, it is urgent to carry out research on the braking strategies for 30,000-ton heavy-haul trains, in order to understand the longitudinal dynamic behavior of train braking and develop appropriate braking strategies to ensure the braking safety of 30,000-ton heavy-haul trains.

Currently, due to the high cost of track testing for heavy-haul trains, numerous researchers have focused their studies on the longitudinal dynamics of heavy-haul trains primarily through simulation-based research [8,9]. Chang and colleagues developed a numerical model of the composite draft gear and analyzed the impact of factors such as track gradient, locomotive control synchronization delay, and braking methods on the longitudinal forces of super long heavy-haul trains weighing 30,000 tons or more [10,11,12]. Zhang et al. studied the impact of differentiated coordinated control of the locomotive’s electric braking time and magnitude on the longitudinal impulse of 20,000-ton heavy-haul trains under conditions of cyclic braking on long and steep gradients on the Daqin Railway [13]. Their research provides guidance for optimizing the control strategy of 20,000-ton heavy-haul trains. Wang et al. [14] and Ma et al. [15] conducted a study using a combined simulation system of train pneumatic braking and longitudinal dynamics to investigate the impact of the signal propagation method of the electronically controlled pneumatic (ECP) braking system on the longitudinal dynamic performance of a 30,000-ton heavy-haul train. The study provides valuable guidance for the development of the ECP system and the formulation of braking strategies for heavy-haul trains. Serajian et al. [16] developed a longitudinal dynamics model to analyze the impact of the number of vehicle sections on the tensile and compressive coupling forces during the braking process of heavy-haul trains.

The above research shows that although the research on the longitudinal impulse mechanism and braking strategy of 20,000-ton heavy-haul trains is relatively mature, it is still hard to avoid relatively large longitudinal impulses generated during braking on long steep slopes. In contrast, studies on 30,000-ton heavy-haul trains are primarily focused on factors such as influencing parameters, braking methods, and train composition. There is limited literature on their braking strategies. In particular, 30,000-ton heavy-haul trains have a greater number of locomotives and a much higher number of cars in the formation than existing trains in China. Their operation is more complicated, and more locomotives are required to participate in braking during the running process. Moreover, on sections with continuous small curve radii, the slave locomotives will receive more severe impacts, and the risk of train derailment will be further increased. Therefore, the existing braking strategies for 20,000-ton heavy-haul trains are not fully applicable to 30,000-ton heavy-haul trains.

Therefore, this paper establishes a longitudinal dynamic model for 30,000-ton heavy-haul trains and investigates the impact of combined pneumatic and electric braking on the longitudinal impulse on long and steep slopes. In response to the significant longitudinal impulse observed in traditional braking strategies, this study explores the effects of different braking parameters on the longitudinal dynamic behavior of heavy-haul trains while maintaining a constant total braking force. A synchronous operation with asynchronous braking collaborative control strategy is proposed for the combined pneumatic and electric braking of 30,000-ton heavy-haul trains. This strategy aims to reduce the longitudinal coupler force of heavy-haul trains, thereby reducing the occurrence of coupling failures, train separations, derailments, and other related accidents. It also alleviates the wear and tear on couplers and related components, leading to a reduction in the frequency of replacements and maintenance. By implementing this braking strategy, the operational safety of the 30,000-ton heavy freight train is enhanced.

2. Longitudinal Dynamic Model of Heavy-Haul Trains

A 30,000-ton heavy-haul train is hauled by multiple locomotives, utilizing a locomotive wireless synchronous control (Locotrol) system for distributed power control, which enables synchronized operation of the locomotives. Each locomotive can independently serve as both a power source and a braking air source, significantly reducing the longitudinal impacts experienced by the 30,000-ton heavy-haul train. Based on the principles of multibody dynamics, the longitudinal dynamic model of heavy freight trains retains only the longitudinal degrees of freedom for the locomotive and freight cars. The vertical and lateral interactions are neglected, and each vehicle is treated as a point mass. The vehicles are interconnected through buffers, simplifying the system to a spring–damper model. Based on the experiments of 30,000-ton heavy-haul trains on the Daqin Railway presented in reference [4], as well as the simulation studies on the formation of 30,000-ton heavy-haul trains in reference [17], and with reference to the actual formation used in the operation of 20,000-ton heavy-haul trains, the 30,000-ton heavy-haul trains in this study adopt the “1 + 1 + 1 + 1” formation. Three HXD₁ locomotives and one SS4 locomotive are used, with each HXD₁ locomotive hauling 105 C80 freight cars, while the tail locomotive is an SS4 electric locomotive, as shown in Figure 1, where v represents the train’s running speed, m represents the vehicle mass, and the subscript denotes the vehicle number.

Based on the actual operating conditions of the line, the single vehicle force analysis is shown in Figure 2, and the longitudinal dynamic equation is given by

$$m_i\ddot{x}=F_{Ti}+F_{Ci}+F_{pi}-F_{Di}-F_{Bi}-F_{C(i+1)}-F_{Zi}$$

(1)

where $\ddot{x}$ represents the vehicle acceleration, i denotes the vehicle index, m is the vehicle mass, F_T is the locomotive’s tractive force, F_p is the gradient resistance (negative for uphill and positive for downhill), F_Ci is the coupling force from the trailing vehicle, F_C(i+1) is the coupling force from the leading vehicle, FB is the aerodynamic drag force, F_D is the locomotive’s dynamic braking force, F_Z is the running resistance of the vehicle, and $\alpha$ is the gradient in per mille (‰).

2.1. Coupler Force Model

The coupler is the most important part of the longitudinal dynamics model of heavy-haul trains. The C80 freight car uses the MT-2 spring-friction draft gear, and its characteristic curve is obtained by fitting drop hammer test data. Due to the obvious hysteresis characteristics and strong nonlinearity of the draft gear, in order to improve simulation efficiency, researchers both domestically and internationally have simplified the buffer model into a numerical model, the principle of which is shown in Figure 3 [18,19,20,21,22]. The equation is as follows:

$$F_C(x_t)=\left\{\begin{array}{c}{f}_l(x) x_t\ge x_{t-\Delta t}\\ f_u(x) x_t<x_{t-\Delta t}\end{array}\right.$$

(2)

$$F_C(x_t)=F_C(x_{t-\Delta t})+k(x_t-x_{t-\Delta t})$$

(3)

$$\left|f_u(x)\right|\le F_C(x_t)\le \left|f_l(x)\right|$$

(4)

The coupler force $F_{\mathrm{C}}(x_t)$ is mainly divided into three stages. When $x_t\ge x_{t-\Delta t}$, meaning the current draft gear stroke is greater than or equal to the previous draft gear stroke, the draft gear is in the loading stage, and interpolation is performed using the loading curve. When $x_t<x_{t-\Delta t}$, meaning the current step draft gear stroke is less than the previous step’s draft gear stroke, the draft gear is in the unloading stage, and interpolation is performed using the unloading curve. When the buffer transitions from the loading curve to the unloading curve, it is in the transition stage. Direct transition may cause an integration discontinuity, leading to non-convergence in the calculation. In this paper, the stiffness transition method is used to form the hysteresis loop.

2.2. Pneumatic Braking System Model

Existing pneumatic braking system models are primarily categorized into empirical braking models and fluid dynamics-based pneumatic braking models [23]. The fluid dynamics pneumatic braking model formulates differential equations based on the principles of fluid dynamics, with the associated boundary condition calculations being computationally expensive [24]. This paper adopts an empirical braking model, where the brake cylinder pressure, based on the mathematical methods and experimental data presented in [25], $P_{\mathrm{z}}$, can be expressed as

$$P_z=\left\{\begin{array}{l}0 t\le t_{di}\hfill \\ {({\displaystyle \frac{t-t_{di}}{t_1+t_{\Delta i}}})}^{{\displaystyle \frac{\lambda }{t-t_i}}}P t_{di}\le t<t_m\hfill \\ P_{\max } t\ge t_m\hfill \end{array}\right.$$

(5)

where t represents the vehicle running time, $t_{di}$ is the initial inflation time of the brake cylinder for the ith vehicle, $t_1$ is the inflation start time of the brake cylinder for the first vehicle, $t_{\Delta i}$ is the inflation time difference between the ith vehicle and the first vehicle, $\lambda$ is the characteristic parameter of the brake control valve, $P_{\max }$ is the maximum brake cylinder pressure, and $t_m$ is the time to reach the maximum brake cylinder pressure. Figure 4 shows the brake cylinder pressurization curve for a 30,000-ton heavy-haul train, with a typical brake pressure reduction of 170 kPa.

Currently, the pneumatic braking systems for heavy-haul trains in China primarily consists of disc brakes on the locomotives and block brakes on the freight cars. The braking parameters are shown in

Table 1. Braking parameters.

Models	${\mathit{d}}_{\mathbf{z}}$	${\mathit{\gamma }}_{\mathbf{z}}$	${\mathit{\eta }}_{\mathbf{z}}$	${\mathit{n}}_{\mathbf{z}}$	${\mathit{n}}_{\mathbf{k}}$
C80	203	8.5	0.9	2	8
HXD1	176	2.41	0.9	12	24
SS4	178	2.85	0.85	8	16

.

According to the train traction calculation regulations [26], the braking pressure can be expressed as

$$F_w={\displaystyle \frac{\frac{\pi }{4}\cdot {d_z}^2\cdot P_z\cdot {\gamma }_z\cdot {\eta }_{\mathrm{z}}\cdot n_z}{n_k\cdot {10}^6}}$$

(6)

$$F_l={\displaystyle \frac{\frac{\pi }{4}\cdot {d_z}^2\cdot P_z\cdot {\gamma }_z\cdot n_z\cdot r_z}{{10}^6\cdot R_c}}$$

(7)

where $F_w$ is the freight car tread brake shoe pressure, $F_l$ is the locomotive disc brake pad pressure, $d_{\mathrm{z}}$ is the brake cylinder diameter (mm), ${\eta }_{\mathrm{z}}$ is the transmission efficiency, ${\gamma }_z$ is the braking force amplification factor, $n_z$ is the number of brake cylinders, $n_{\mathrm{k}}$ is the number of brake shoes, $P_{\mathrm{z}}$ is the brake cylinder pressure (kPa), $r_z$ is the friction radii of the brake disc (mm), and $R_c$ is the wheel radius (mm).

The braking friction coefficient of heavy-haul trains is determined by the brake shoe material, train operating speed, and the contact pressure between the brake shoes. According to the empirical method presented in reference [26], the friction coefficient can be expressed as

$$\phi ={\phi }_i\cdot {\displaystyle \frac{F+200}{4F+200}}\cdot {\displaystyle \frac{2v+150}{3v+150}}$$

(8)

where F is the braking force (kN) and v is the vehicle speed (km/h). The friction coefficients for locomotive disc brakes and freight car brake shoes differ; therefore, for the locomotive, ${\phi }_i$ is taken as 0.391, and for the freight car it is 0.481. The braking force can be expressed as

$$F_{\mathrm{B}}=F\cdot \phi \cdot n_k$$

(9)

2.3. Locomotive Electric Braking and Resistance Model

Electric braking is a local braking force that can only be applied by the locomotive. It can be used to supplement the pneumatic braking force of the heavy-haul train, reducing wear on the basic braking system and ensuring the braking safety of the heavy-haul train. The electric braking force of the HXD₁ locomotive can be expressed as [26]

$$F_{\mathrm{D}}=\left\{\begin{array}{l}461\times v/5 v\le 5\mathrm{km}/\mathrm{h}\hfill \\ 461 5\mathrm{km}/\mathrm{h}\le v\le 75\mathrm{km}/\mathrm{h}\hfill \\ 9600\times 3.6/v 75\mathrm{km}/\mathrm{h}\le v\le 120\mathrm{km}/\mathrm{h}\hfill \end{array}\right.$$

(10)

The electric resistance braking force of the SS4 locomotive can be expressed as

$$F_D=\left\{\begin{array}{l}f(v)\times \eta \times t/\Delta t t\le \Delta t\hfill \\ f(v)\times \eta t>\Delta t\hfill \end{array}\right.$$

(11)

where $f(v)$ is the maximum electric resistance braking force, $\Delta t$ is the time required to reach the maximum braking force, and $\eta$ is the utilization coefficient of the electric resistance braking.

The unit running resistance of the train can be expressed as

$$F_z=A+Bv+C{v}^2$$

(12)

In the equation, A, B, and C are the train running resistance coefficients. The gradient resistance F acting on the train along the track, caused by the component of the gravitational force along the incline, can be expressed as

$$F_{\mathrm{p}}=\alpha \cdot m\cdot g$$

(13)

where $\alpha$ is the gradient (in ‰), m is the mass of the vehicle, and g is the gravitational acceleration.

3. Model Verification

To validate the accuracy of the established longitudinal dynamics model, a simulation analysis was conducted based on the comprehensive test data from the heavy-load combined train trials on the Daqin Railway [3]. The simulation verification was carried out using 10,000-ton heavy-haul trains. Its marshalling form was an HXD1 locomotive + 105 C80 freight cars. All the freight cars were equipped with MT-2 buffers and carried out emergency braking and service full braking on a straight and level track with an initial speed of 80 km/h. The distribution of the maximum coupler force along the train is shown in Figure 5. A comparison of the maximum coupler force and its position under both braking conditions between the experimental and simulated results is presented in

Table 2. Comparison and validation of maximum coupler force for 10,000-ton heavy-haul train.

Braking Conditions	Tested Max Coupler Force (kN)	Tested Max Coupler Force Position	Simulated Max Coupler Force (kN)	Simulated Max Coupler Force Position
Emergency braking	−1160	67	−1118	72
Service braking	−630	52	−638	46

. The positions of the maximum coupler force under both emergency braking and standard full braking conditions were found to be nearly identical in both the experimental and simulation results. The maximum coupler force values differed by 3.62% and 1.27%, respectively. Therefore, the longitudinal dynamics model developed in this study demonstrates good accuracy and reliability.

4. Optimization of Pneumatic–Electric Combined Braking Strategy

When heavy-haul combination trains traverse a long and steep slope, electric braking is primarily used for speed regulation. However, when the gradient exceeds 6‰, electric braking alone is insufficient to overcome the resistance caused by the slope, and additional pneumatic braking must be applied for speed control [27]. Although air braking can meet the speed regulation requirements, cyclic braking on long and steep slopes increases the burden on the air brake system, which may negatively affect vehicle safety. Therefore, it is important to optimize the combination of locomotive electric braking and pneumatic braking to improve braking efficiency and reduce energy consumption. However, the transition between pneumatic braking and electric braking can cause a sudden change in braking force, leading to a discontinuity in braking force. This phenomenon can exacerbate longitudinal impacts, and the excessive electric braking force may cause wheel skidding or damage to the locomotive’s wheels. This section focuses on the analysis of the longitudinal dynamic characteristics of a 30,000-ton heavy-haul train with an initial speed of 80 km/h, a train line pressure reduction of 50 kPa, and on a 12‰ gradient slope under pneumatic–electric combined braking conditions. Based on statistical data from the Daqin Railway on locomotive wireless synchronization response delays, a 3 s delay for the slave locomotive was used in this analysis. The train’s brake activation time under these conditions is shown in Figure 6.

The traditional braking strategy synchronizes the electric braking force of each locomotive, with all locomotives applying both electric and pneumatic braking simultaneously. According to the basic principles of pneumatic–electric combined braking [28], 50% of the total electric braking force is applied with a braking time of 100 s. To distinguish between the buff and draft forces, the positive value of the coupling force is defined as the draft force, and the negative value as the buff force. The maximum coupling force distribution along the train is shown in Figure 7 under the traditional braking strategy. From Figure 7, it can be observed that the maximum compressive force is 962 kN, occurring at the 118th coupler between the no. 1 and no. 2 slave locomotives. The maximum tensile force is 575 kN, occurring at the 106th coupler of the no. 1 slave locomotive.

Under traditional strategies, a significant discontinuity in the coupler force is observed at the locomotive position. To optimize the maximum coupler force, two approaches are proposed based on the braking strategy for 20,000-ton heavy-haul trains and some basic train handling practices of railway operators: (1) adjusting the distribution of the electric braking force, and (2) regulating the synchronization time between electric and pneumatic braking. This study investigates the impact of these two approaches on the longitudinal forces of 30,000-ton heavy-haul trains, aiming to develop an appropriate pneumatic–electric combined braking strategy.

4.1. Optimization of Electric Braking Force Distribution Strategy

Due to the pneumatic braking delay characteristics of 30,000-ton heavy-haul trains and the response characteristics of the Locotrol system, the pneumatic braking force is initially applied by the master locomotive and then transmitted to the slave locomotives. After a delay of 3 s, the slave locomotives begin to respond by applying a pneumatic braking force to both sides. The electric braking force is also influenced by the Locotrol system, with the slave locomotives applying an electric braking force 3 s after the master locomotive. The significant electric braking force exerted by the master locomotive can result in an excessive deceleration, leading to an intensified compressive interaction between the front vehicles of the master locomotive and the no. 1 slave locomotive. Conversely, the no. 3 slave locomotive, positioned at the rear of the train and lacking any interactive forces from subsequent freight cars, will also experience excessive deceleration due to the substantial dynamic braking. This situation causes the vehicles between the no. 2 slave locomotive and the no. 3 slave locomotive to experience increased tensile stress. When the electric braking force of the locomotive is large, it can cause discontinuities in the braking force at the locomotive, leading to significant fluctuations in the coupler force near the locomotives. Since all slave locomotives have the same response time, when distributing the electric braking force it is necessary to ensure that the electric braking forces of the no. 1 and no. 2 slave locomotives remain consistent in order to minimize the impact of any desynchronization of electric braking between the slave units.

Based on the aforementioned analysis, this section primarily explores the influence of the dynamic braking force distribution across each locomotive on the longitudinal impact forces of the heavy-haul trains. While maintaining a constant total braking force for the train, the distribution of electric braking forces among the individual locomotives is varied to identify the optimal allocation strategy for electric braking forces. The HXD₁ is an AC locomotive, capable of ensuring that the electric braking force at any given speed matches the desired braking force set by the operator. In contrast, the SS4 is a DC locomotive, which cannot provide the electric dynamic braking force at all speeds. Therefore, in the simulation of pneumatic–electric braking, the braking force applied by the SS4 locomotive is kept along its braking characteristic curve, while only the electric braking force of the HXD₁ locomotive is varied. To maintain a constant total electric braking force and a pneumatic braking pressure reduction of 50 kPa for the heavy-haul trains, the electric braking force of the master locomotive is either increased or decreased, with the distribution of electric braking forces among the three locomotives as shown in

Table 3. Decreasing the electric braking force of the master locomotive.

Strategy Number	Master Locomotive (kN)	No. 1 Slave Locomotive (kN)	No. 2 Slave Locomotive (kN)
1	230	230	230
2	150	270	270
3	90	300	300
4	0	345	345

and

Table 4. Increasing the electric braking force of the master locomotive.

Strategy Number	Master Locomotive (kN)	No. 1 Slave Locomotive (kN)	No. 2 Slave Locomotive (kN)
1	230	230	230
2	300	195	195
3	380	155	155
4	460	115	115

.

The distribution of the maximum coupler force along the train when the electric braking force of the master locomotive is reduced and the electric braking forces of slave locomotives 1 and 2 are increased is shown in Figure 8. From the electric braking force distribution strategy described above, it can be observed that as the electric braking force of the master locomotive decreases, the maximum compressive coupler force for all vehicles decreases accordingly. As the electric braking forces of slave locomotives 1 and 2 increase, the maximum tensile coupler force between the lead locomotive and auxiliary locomotive 1 significantly increases, while the change in the maximum tensile coupler force of vehicles following auxiliary locomotive 1 is relatively small. Furthermore, this phenomenon becomes more pronounced as the difference in electric braking forces between the master locomotive and slave locomotives 1 and 2 increases.

The distribution of the maximum coupler force along the train when the electric braking force of the master locomotive is increased while the electric braking forces of slave locomotives 1 and 2 are reduced is shown in Figure 9. From the electric braking force distribution strategy described above, it can be observed that as the electric braking force of the master locomotive increases, the maximum compressive coupler force at all coupler positions shows an increasing trend. As the electric braking forces of slave locomotives 1 and 2 decrease, the maximum tensile coupler force between the master locomotive and slave locomotive 1 significantly decreases. However, the maximum tensile coupler force of the vehicles following slave locomotive 1 shows an increasing trend. Moreover, this phenomenon becomes more pronounced as the difference in electric braking forces between the master locomotive and slave locomotives 1 and 2 increases.

In order to compare the overall coupler force optimization effects under different strategies, the maximum coupler forces and their positions under the aforementioned strategies are shown in

Table 5. Comparison of maximum coupler forces and positions with reduced electric braking of the master locomotive.

Strategy Number	Max Compressive Coupler Force (kN)	Max Compressive Coupler Force Position	Max Tensile Coupler Force (kN)	Max Tensile Coupler Force Position
1	962	118	575	106
2	824	111	672	106
3	767	108	735	106
4	688	112	850	106

and

Table 6. Comparison of maximum coupler forces and positions with increased electric braking of the master locomotive.

Strategy Number	Max Compressive Coupler Force (kN)	Max Compressive Coupler Force Position	Max Tensile Coupler Force (kN)	Max Tensile Coupler Force Position
1	962	118	575	106
2	1106	119	499	106
3	1218	120	451	295
4	1312	118	456	296

. When the electric braking force of the lead locomotive is decreased, compared to the traditional braking strategy, strategies 2, 3, and 4 show a reduction in the maximum compressive coupler force by 14.35%, 20.27%, and 28.48%, respectively. The maximum compressive forces are concentrated in the front vehicles between slave locomotives 1 and 2. In contrast, the maximum tensile coupler forces increase by 16.87%, 27.83%, and 47.83%, respectively, with the maximum tensile forces occurring at coupler position 106. Conversely, when the electric braking force of the master locomotive is increased, strategies 2, 3, and 4 show an increase in the maximum compressive coupler force compared to the traditional braking strategy, with increases of 14.97%, 26.61%, and 36.38%, respectively. Again, the maximum compressive forces are concentrated in the front vehicles between slave locomotives 1 and 2. In contrast, the maximum tensile coupler forces decrease by 16.87%, 27.83%, and 47.83%, with the maximum tensile forces primarily occurring at coupler position 106 and near slave locomotive 3.

This is due to the pneumatic braking delay characteristics and the response dynamics of the Locotrol system, which result in an asymmetric distribution of braking forces between the front and rear vehicles during the brake cylinder charging phase. Specifically, the overall braking force on the front vehicle is greater than that on the rear vehicles. When the electric braking force from locomotives 1 and 2 is increased, it compensates for the braking force differential between the front and rear vehicles. This reduces braking desynchronization between the vehicles, alleviates the compressive interaction between the front and rear vehicles, and consequently reduces the maximum compressive force. On the other hand, this would reduce the maximum compressive force. However, when the electric braking force of the slave locomotives is increased, due to the rapid application of electric braking and the fact that the electric braking force is greater than the pneumatic braking force, the speed of slave locomotives 1 and 2 decreases more quickly. This results in increased impact on the intermediate vehicles between the master locomotive and slave locomotive 1, as the faster deceleration of the front and rear locomotives leads to a more pronounced shock to the intermediate vehicles, thereby increasing the maximum tensile force. On the other hand, this would reduce the maximum tensile force. When the electric braking forces of slave locomotives 1 and 2 approach that of slave locomotive 3, which is at the rear and lacks the interaction from trailing freight cars, the relatively large electric braking force will cause it to decelerate too quickly. This leads to an increased stretching of the vehicles between slave locomotives 2 and 3, resulting in a higher coupler force. Consequently, the distribution of the maximum tensile force is altered.

4.2. Optimization of Braking Time-Matching Strategy

Under the traditional braking strategy, both pneumatic braking and electric braking commands are applied simultaneously. However, as shown in Figure 4 and Figure 6, the pneumatic braking system of 30,000-ton heavy-haul trains exhibits a delayed response. When the train pipe pressure is reduced by 50 kPa, the pressure buildup time of the brake cylinders is within 30 s. The electric braking system has a faster response time and can quickly generate electric braking force, allowing for an immediate reduction in the train’s speed. When both electric braking force and pneumatic braking force are applied simultaneously, there is a significant discrepancy between the locomotive’s electric braking force and the pneumatic braking force before the brake cylinder pressure reaches its maximum value. This can lead to a more pronounced braking force imbalance, where the front of the train experiences a greater braking force than the rear, causing the rear vehicles to compress against the front vehicles more noticeably. At the same time, the response delay of the slave locomotives exacerbates the dynamic impact between the vehicles. Based on this, this section primarily analyzes the effect of the time-matching relationship between the application of electric and pneumatic braking on the longitudinal forces of heavy-haul trains. The braking force conditions remain consistent with those described in Section 2.1, with the electric braking force of each locomotive being uniformly distributed. The matching times between electric braking and pneumatic braking are shown in

Table 7. Electric and pneumatic braking matching times.

Strategy Number	Electric Braking Command Delay Time (s)	Strategy Number	Pneumatic Braking Command Delay Time (s)
1	0	1	0
2	10	2	10
3	20	3	20
4	30	4	30

.

The distribution of maximum coupler forces along the train for electric braking and pneumatic braking at different matching times is shown in Figure 10 and Figure 11. As observed from Figure 10, delaying the application of the electric braking force significantly reduces the longitudinal impact between trains, with both tensile and compressive coupler forces being noticeably decreased. Figure 11 demonstrates that pneumatic braking, like electric braking with delayed application, also leads to an improvement in longitudinal impact. As the delay in pneumatic braking time increases, the maximum tensile and compressive coupler forces across all vehicles tend to decrease. When the delay time of electric braking or pneumatic braking exceeds 20 s, the optimization effect on the tensile and compressive coupler forces diminishes, and the variation in the distribution of these forces along the train positions becomes more gradual.

The maximum coupling force and its location for electric and pneumatic braking at different matching times are shown in

Table 8. Maximum coupler force and location for electric and pneumatic braking at different matching times.

Delay Time	Max Compressive Coupler Force (kN)	Max Compressive Coupler Force Position	Max Tensile Coupler Force (kN)	Max Tensile Coupler Force Position
0	962	118	575	106
10	813	108	454	211
20	616	108	409	211
30	502	127	361	106
−10	846	108	505	106
−20	823	112	354	106
−30	722	108	378	301

, where positive values represent a delay in the application of electric braking and negative values represent a delay in the application of pneumatic braking. Compared with the traditional braking strategy, the maximum coupler compressive forces of the electric braking force delayed application strategies 2, 3, and 4 are reduced by 15.49%, 35.97%, and 47.82%, respectively, and the maximum compressive coupler forces all occur near the no. 1 slave locomotive. The maximum tensile coupler forces are reduced by 21.04%, 28.87%, and 37.22%, respectively, with the maximum tensile coupler forces occurring near either the no. 1 or no. 2 slave locomotive. In contrast, compared with the traditional braking strategy, the maximum compressive coupler forces of the pneumatic braking force delayed application strategies 2, 3, and 4 are reduced by 12.06%, 14.45%, and 24.95%, respectively, and the maximum compressive coupler forces also all occur near the no. 1 slave locomotive. The maximum tensile coupler forces are reduced by approximately 12.17%, 38.43%, and 34.26%, respectively, with the maximum tensile coupler force occurring near the no. 2 or no. 3 slave locomotive.

This is due to the delayed application of electric or pneumatic braking. Initially, under a single braking mode, the sudden change in braking force is relatively small. As the electric or pneumatic braking gradually takes effect with delay, the vehicles compress against each other, causing the coupler to exert a substantial restraining force on the vehicles. However, when the electric and pneumatic braking are applied with delay, the differences in braking forces caused by the asynchronous application of pneumatic and electric braking are much smaller than the forces exerted by the coupler on the vehicles. As a result, the impulse between the vehicles is reduced, leading to a decrease in longitudinal impact.

The aforementioned study indicates that adjusting the matching times of electric braking and pneumatic braking is conducive to optimizing the coupler force. However, a delay in the application of the braking force results in insufficient braking during the initial phase, making it difficult to counteract the gradient force from the track’s incline, thereby increasing the risk of overspeed. The speed characteristics under different matching times of electric braking and pneumatic braking are shown in Figure 12. As shown in Figure 12, during the process of a 10–30 s delay in electric braking, the increase in the maximum speed of the heavy-haul train becomes smaller as the delay time increases. With a 30 s delay, the maximum speed reaches 81.97 km/h, which is a 2.46% increase compared to the initial speed, indicating a low risk of overspeed. However, in the process of a 10–30 s delay in pneumatic braking, the maximum speed of the train increases significantly as the delay time increases. With a 10 s delay, the maximum speed reaches 84.04 km/h, a 5.05% increase compared to the initial speed. And with a 30 s delay, the maximum speed reaches 89.64 km/h, a 12.05% increase compared to the initial speed, significantly increasing the risk of overspeed.

5. Conclusions

This paper establishes a longitudinal dynamics model for 30,000-ton heavy-haul trains with the formation of ‘1 × HXD1 + 1 × HXD1 + 1 × HXD1 + SS4’. After validating the effectiveness of the model, the impact of combined pneumatic and electric braking parameters on the longitudinal impulse between vehicles is studied.

The main conclusions are as follows. On a long and steep slope with a gradient of 12‰, reducing the electric braking force of the master locomotive and increasing the electric braking force of the no. 1 and no. 2 slave locomotives decreases the maximum compressive coupler force and increases the maximum tensile coupler force. Conversely, when the operation is reversed, the effect is also reversed. Both the electric braking delay and the pneumatic braking delay can effectively reduce the maximum tensile and compressive coupler forces, but there is a certain risk of train overspeed. For the electric braking delay strategy, when the delay time is close to the pressure rise time of the air brake cylinder, a better overall optimization effect on the coupler force and speed can be achieved. Particularly, under the circumstances of avoiding the risks of braking skidding and overspeed of locomotives, both the strategy of applying electric braking forces of 90 kN, 300 kN, and 300 kN, respectively, to the three locomotives and the strategy of delaying the application of the electric braking force by 30 s achieve good optimization effects. Compared with the traditional braking strategy, the maximum coupler forces are reduced by 20.27% and 47.82%, respectively. Moreover, in the combined pneumatic and electric braking strategies for super-heavy-haul trains (such as 30,000-ton and 40,000-ton ones), due to the prevalent existence of response delays in slave locomotives, the research conclusions of this paper can provide reasonable electric braking force distribution and time-matching strategies to guide the safe running of the trains.

This paper optimizes the braking strategy solely from the perspective of longitudinal coupler force, without accounting for its impact on wheel–rail forces and vehicle vibrations. Furthermore, the distribution of the electric braking force and the coordination timing between electric and pneumatic brakes for locomotives in 30,000-ton heavy-haul trains have not been addressed. Future work will focus on investigating brake control strategies under synchronous and asynchronous operations to achieve optimal braking performance. Intelligent optimization algorithms will be employed to minimize longitudinal impulses and enhance overall train control.