Simulation-based multiobjective optimization of tail beam control strategy for top-coal caving

Abstract

In longwall top-coal caving, the motion control strategy of the hydraulic support tail beam directly affects the caving performance. To reduce gangue admixture and improve the efficiency of top-coal discharge, this study proposes an offline parameter-selection method for tail-beam caving control based on DEM simulation and weighted multi-objective optimization. A discrete element model of top-coal caving is established to analyze the effects of tail-beam opening angle, swing frequency, and scraper-conveyor chain speed on the caving process. Based on multi-objective optimization theory, the tail-beam motion parameters are optimized with the objectives of enhancing caving efficiency, reducing residual coal, and minimizing gangue content in the discharged material. The results show that scraper-chain speed, tail-beam swing amplitude, and swing frequency all have significant influences on caving performance. Under the selected weighting preference, the optimized parameter combination was determined as a scraper-chain speed of 0.9042 m/s, a tail-beam swing amplitude of 40.00°, and a swing frequency of 0.0594 Hz. DEM simulation of the optimized scheme showed a conveying efficiency of 255.21 kg/s and a residual coal ratio of 88.62%, while no gangue discharge was observed within 20 s. These results indicate that the proposed method provides a practical compromise among conveying efficiency, residual coal recovery, and gangue control for tail-beam caving operations.

Introduction

Top-coal caving is a core technique for underground extraction of thick coal seams, and the “one-cut, three-cave” cyclic operation mode is currently widely adopted. In this method, the lower section of the working face is first extracted in a single pass. Subsequently, the overlying top coal is fragmented under mine pressure and transformed into a flowable granular medium. It is then discharged in an orderly manner through the coal outlets located at the rear of the hydraulic supports, ultimately enabling full-height extraction of the thick coal seam in a single operation. ¹ This technique significantly enhances the recovery rate of coal resources and improves mining efficiency. However, the top-coal caving process involves complex multi-body interactions among the support, top coal, and false roof, forming a highly dynamic and nonlinear system. ² Specifically, the top coal undergoes fracturing and breakage under the load of the overlying strata, descends through the coal outlet controlled by the tail beam of the hydraulic support, and is transported away by the rear scraper conveyor. After caving is completed, the hydraulic support advances, and the immediate roof collapses to fill the goaf. Throughout this process, the top-coal caving control strategy—including the caving initiation timing, the duration of caving, and the motion pattern of the caving mechanism—serves as the key determinant of caving performance. If improperly controlled, the process can result in insufficient recovery of top coal and an increased dilution ratio due to excessive gangue inflow, which may further lead to support instability, large-scale roof weighting, and other safety hazards. ³ Therefore, formulating an appropriate control strategy to effectively prevent gangue from entering the coal transportation system is essential for achieving efficient top-coal caving with low gangue content.^4,5

In current research, numerous scholars have conducted in-depth investigations into top-coal caving control technologies from various perspectives. For example, Yang et al. employed a deep reinforcement learning method to optimize the caving decision-making process and proposed a batch Q-value updating mechanism; simulation results indicated that this approach can effectively improve the coal–gangue separation state. ⁶ Liu and Li conducted simulation studies on intermittent caving using CDEM software and found that a spacing of 1.2 m yields the optimal caving performance. ⁷ Huo et al. introduced radar detection technology to monitor the top-coal thickness in real time, providing essential data support for precise control of the caving process. ⁴ Wang et al. employed PFC numerical simulations to reveal the evolution characteristics of the coal–rock interface during the caving process and analyzed the influence of support-induced disturbances on the interface morphology. ⁸ Bui et al. used FLAC3D to examine the distribution characteristics of abutment pressure ahead of a top-coal caving face, and found that caving operations lead to an expansion of the stress concentration zone and a forward shift of the peak stress. ⁹ In terms of coordinated control, Shi et al. developed a collaborative caving algorithm model for multiple coal outlets, and simulation results indicated that this approach can increase the top-coal recovery rate by approximately 17%. ¹⁰ Yang et al. proposed an intelligent caving strategy based on Q-learning, which demonstrated significant advantages in reducing gangue contamination and enhancing economic efficiency. ⁶ In addition, Huo et al. integrated coal flow monitoring, top-coal thickness identification, and coal–gangue interface information to establish an overall control framework for an intelligent top-coal caving system. ⁴

Although previous studies have provided valuable insights into top-coal caving decision-making, monitoring, and numerical simulation, several limitations remain. First, most existing studies focus on either coal–gangue identification, caving timing, or the influence of a single process parameter, while the coordinated effect of scraper-conveyor speed and tail-beam motion parameters has not been sufficiently addressed. Second, the conflicting objectives of conveying efficiency, residual coal ratio, and gangue-admixture ratio are rarely incorporated into a unified optimization framework. Third, the linkage between DEM-based flow analysis and practical parameter selection for tail-beam caving control remains insufficiently developed. To address these gaps, this study combines DEM simulation with weighted multi-objective optimization to determine a practical offline parameter combination for coordinated tail-beam caving control.

Based on the above considerations, a discrete element model of top-coal caving was established to investigate the effects of scraper-conveyor speed, tail-beam swing amplitude, and swing frequency on coal-flow behavior. A comprehensive evaluation system was then constructed using conveying efficiency, residual coal ratio, and gangue-admixture ratio as performance indicators. On this basis, a weighted-sum objective function was formulated and solved using a genetic algorithm to determine a practical compromise parameter combination for tail-beam caving. The study is intended to provide a quantitative basis for offline parameter setting and coordinated control of top-coal caving operations.

Construction of a discrete element model for top coal caving

Caving performance is fundamentally governed by the mechanical behavior of the top-coal–roof system under mining-induced conditions. As the working face advances, the overlying strata undergo displacement, fracturing, and collapse, generating support pressure within the top coal. This pressure induces crack initiation, propagation, and eventual fragmentation of the top coal. The degree of fragmentation, block size distribution, and the interface characteristics with the immediate roof collectively determine the initial conditions for caving. If the immediate roof can advance synchronously with mining and form a well-defined coal–rock interface, it is conducive to top-coal recovery; conversely, excessive top-coal fragmentation or a mixed coal–rock interface can easily result in premature gangue contamination.

The top-coal discharge process can be described using the “discharged body” theory from coal flow mechanics. According to this theory, when granular coal is released from the caving outlet, the particles within the flow field form an approximately ellipsoidal flow body. The morphology, development height, and volume of the discharged body are influenced by factors such as the size of the caving outlet, the fragmentation and expansion characteristics of the top coal, and the caving process parameters. In practical caving operations, phenomena such as arching and jamming often occur in the coal flow, where large coal–gangue blocks form stable structures above the outlet, obstructing subsequent coal discharge and significantly reducing recovery. Understanding the interaction mechanisms between coal and rock, as well as the flow behavior of coal, provides the theoretical basis for establishing accurate discrete element models and optimizing the hydraulic support caving mechanisms.

As the direct actuator controlling top-coal discharge, the tail beam of the hydraulic support plays a decisive role in caving performance. The motion pattern, trajectory, velocity, and acceleration of the tail beam directly affect the opening size and duration of the caving outlet, as well as the degree of disturbance imparted to the overlying coal mass. The interaction between the tail beam and the granular coal–rock mass represents a typical fluid–solid coupling problem. Investigating coal flow responses under different motion parameters is of significant importance for establishing accurate discrete element models and optimizing tail-beam motion control strategies.

In recent years, with the advancement of computer technology and numerical simulation methods, researchers have begun to employ more refined approaches to analyze and optimize the top-coal caving process. Among these approaches, a deep understanding of coal–rock interactions and the dynamic characteristics of hydraulic supports forms the foundation for building accurate models and effective control strategies. The discrete element method (DEM) provides a powerful tool for simulating top-coal fragmentation and flow, ¹¹ while multi-objective optimization algorithms, such as evolutionary algorithms, offer a theoretical framework for finding an optimal balance between conflicting objectives, such as high recovery and low gangue content. ¹²

Theoretical description of coal–gangue impact on the tail beam

Based on contact theory and impact dynamics, the interaction between the tail beam and the coal–rock mass during top-coal caving is extremely complex. The top coal is released freely through column compression, tail-beam swing, and gate plate extension, exhibiting dynamic behaviors such as impact, rolling, and sliding. ¹³ The overall working model of the top-coal caving support is shown in Figure 1.

Figure 1.

Hydraulic support caving coal diagram.

To facilitate the description of the loads acting on the tail beam during caving, the dynamic behavior of coal–gangue impacting the tail beam is equivalently modeled as coal–rock particles with mass M colliding with the tail-beam metal plate at a certain impact velocity V, based on the collapse characteristics of coal and gangue. The simplified model is shown in Figure 2.^14,15 A portion of the energy of the falling coal–rock particles is converted into elastic energy associated with particle collision and contact deformation, while another portion is transformed into the deformation energy of the metal plate. When the impact velocity of the coal–rock particles exceeds their initial yield collision velocity, plastic strain occurs.

$$S=\frac{F_N^2{X}^3}{3\mathrm{EI}}$$

(1)

Figure 2.

Simplified model of coal rock impact tail beam.

In the equation: S represents the deformation energy of the tail beam metal plate, J; X denotes the impact position of the collapsing material, m; EI represents the bending stiffness, Pa.

When the collision velocity is lower than the initial yield velocity of the coal–gangue, according to the law of energy conservation, the impact energy of the falling coal–gangue is composed of the deformation energy of the tail-beam metal plate and the elastic energy of contact deformation in the coal–gangue:

$$\frac{1}{2}M{V}^2=\underset{0}{\overset{\delta }{\int }}{F}_N(\delta )>\mathrm{d}\delta +\frac{F_N^2{X}^3}{3}$$

(2)

In the equation: M represents the mass of the fallen coal and gangue, kg; V denotes the contact collision velocity, m/s. By the principle of conservation of kinetic energy, we obtain:

$$\mathrm{MgH}=\underset{0}{\overset{\delta }{\int }}{F}_N(>\delta )>\mathrm{d}\delta +\frac{{F_N}^2{X}^3}{3\mathrm{EI}}$$

(3)

In the equation: H represents the height between the initial position of the coal gangue and the contact point, m.

When the maximum contact stress exceeds the material yield strength, the relationship between the elasto-plastic contact force and the compression deformation is expressed as:

$$F_f=F_N+2\pi R{F}_N(\delta -{\delta }_q)$$

(4)

In the equation: $F_f$ is the normal collision contact force, N; ${\delta }_q$ is the yield compressive deformation, m.

When the collision velocity exceeds the initial yield velocity of the coal–gangue, according to the law of energy conservation, it can be expressed as:

$$\frac{1}{2}M{V}^2=\underset{0}{\overset{{\delta }_q}{\int }}{F}_N(\delta )\mathrm{d}\delta +\underset{{\delta }_q}{\overset{{\delta }_{max}}{\int }}{F}_f(\delta )\mathrm{d}\delta +\frac{{F_f}^2{X}^3}{3}$$

(5)

In the formula: ${\delta }_{max}$ represents the maximum normal deformation, m.

Equations (1)–(5) are introduced to describe the mechanical background of coal–gangue impact on the tail beam and to provide a theoretical interpretation of the impact process during caving. These equations were not directly embedded into the DEM contact law in the EDEM simulations. In the actual numerical implementation, particle interactions were governed by the linear spring–dashpot contact model together with Coulomb friction, as described in Section 1.2.

Development of a discrete element model

The Discrete Element Method (DEM) is a numerical approach in discontinuum mechanics, particularly suitable for simulating the motion and interactions of granular materials, such as fragmented coal–rock. It discretizes the simulated object into a series of independent rigid or deformable particle units, and by tracking the motion of each unit along with the contact forces between them, it reproduces the overall macroscopic mechanical behavior. ¹⁶

The core principle of DEM is based on Newton’s second law. At each time step, the acceleration of every particle in the system is calculated according to all the forces acting on it, including contact forces and body forces. The velocity and position of each particle are then updated through time integration. ¹⁷ This iterative calculation process is repeated continuously, thereby simulating the dynamic evolution of the entire system.

For any particle i, its translational and rotational motion is described by the following equations:

$$\begin{array}{c}{m}_i\frac{d^2{u}_i}{d{t}^2}={\sum }_j{F}_{\mathrm{ij}}^c+F_i^g\hfill \\ I_i\frac{d{w}_i}{\mathrm{dt}}={\sum }_j{M}_{\mathrm{ij}}\hfill \end{array}$$

(6)

In the equation: $m_i$ and $I_i$ are the mass and rotational inertia of particle i, respectively. $u_i$ and $w_i$ are the displacement and angular velocity vectors of particle i, respectively. $F_{\mathrm{ij}}^c$ is the contact force exerted on particle i by particle j. $F_i^g$ is the body force acting on particle i (such as gravity). $M_{\mathrm{ij}}$ is the moment generated by the contact force $F_{\mathrm{ij}}^c$ .

The calculation of contact forces is fundamental to DEM. When two particles come into contact, a repulsive force proportional to the amount of overlap is generated. The most commonly used contact model is the linear spring–dashpot model. ¹⁸ The contact force $F^c$ is typically decomposed into a normal component $F_n$ and a tangential component $F_s$ .

The normal force is typically modeled using a normal spring and a normal dashpot, representing the elasticity between particles and the associated energy dissipation.

$$F_n=K_n·u_n+C_n·v_n$$

(7)

In the equation: $K_n$ is the normal stiffness, $u_n$ is the normal overlap, $C_n$ is the normal damping coefficient, and $v_n$ is the normal relative velocity. In software implementations, a more commonly used form is $F_i(t+\mathrm{\Delta }t)=F_i(t)-K_n\mathrm{\Delta }{u}_n$ ; this incremental formulation facilitates numerical computation. ¹⁹

The tangential force models the friction between particles. It is also composed of a tangential spring and dashpot, but is constrained by Coulomb’s law of friction. ¹⁷

The tangential force is first calculated using the elastic increment:

$$\mathrm{\Delta }{F}_s=-K_s·\mathrm{\Delta }{u}_s$$

(8)

In the equation: $K_s$ is the tangential stiffness, and $\mathrm{\Delta }{u}_s$ is the tangential displacement increment. The Coulomb friction criterion is then applied to determine the final tangential force:

If $\left|F_S\right|\le \mu ·\left|F_n\right|$ , the particles are in a sticking state, and the tangential force is equal to $F_S$ .

If $\left|F_S\right|>\mu ·\left|F_n\right|$ , sliding occurs between the particles, and the tangential force is adjusted to $F_s=\mu ·\left|F_n\right|·\mathrm{sign}(\mathrm{\Delta }{u}_s)$ , where $\mu$ is the coefficient of friction.

In numerical implementation, DEM is solved using explicit time integration schemes, such as the central difference method. Each computational time-step loop includes the following steps: first, contact detection is performed based on the current particle positions to determine the contact states between particles and between particles and boundaries; next, all contact forces are calculated according to the force–displacement law. Commonly used contact models include the linear spring–dashpot model and the parallel bond model, which accounts for bonding strength and can effectively simulate the transition of coal–rock from continuous to discontinuous fracture; finally, the equations of motion are used to update the velocities and positions of the particles. The computational stability of this method strongly depends on the choice of time step, which must be smaller than the system’s critical time step.^20,21 For fully mechanized top-coal caving simulations, a coupled continuum–discrete approach is commonly employed, in which the continuous strata are simulated using the finite element method, while the fragmented top coal is modeled with the discrete element method, balancing computational efficiency and simulation accuracy.

During the top-coal caving process of a hydraulic support, the main beam and canopy beam primarily serve a supporting function, while the rotation angle of the tail beam is the key structural factor controlling coal discharge. Throughout the caving operation, the tail beam and scraper conveyor are the primary moving components, and optimizing their coordinated interaction is crucial for improving caving performance.

As shown in Figure 3, a simplified discrete element model of hydraulic support top-coal caving was established using EDEM (Engineering Discrete Element Method), a commercial software platform for simulating the motion and interaction of bulk granular materials. The model consists of three hydraulic supports combined with one scraper conveyor. The hydraulic supports have a width of 1.5 m, with an initial tail-beam angle of 30°, and the scraper conveyor has a width of 0.73 m. The coal seam above the top beam of the hydraulic supports has a thickness of 2 m, with a 1 m thick gangue layer. Coal is discharged by controlling the tail-beam rotation and transported using the scraper conveyor.

Figure 3.

Discrete element model of top coal caving.

The hydraulic supports maintain their load-bearing function, while coal discharge is simulated through tail-beam rotation. Coal–gangue particles were generated with random diameters ranging from 40 to 60 mm, with the upper gangue layer representing the immediate roof. It should be noted that this particle-size range does not represent the full in-situ fragmentation distribution of top coal and gangue. Instead, it is adopted as an equivalent DEM particle-size range to balance computational efficiency, numerical stability, and the ability to capture the main macroscopic flow characteristics of the caving process. Accordingly, the present study focuses on the comparative influence of control parameters on caving performance, rather than on an exact reconstruction of the full field particle-size distribution. A mesh size of 2 R is used, the simulation time step is set to 25% of the critical step, and data are recorded at intervals of 0.1 s. The materials of the hydraulic supports, scraper conveyor, and coal baffles are set as steel. The material properties used in the constructed discrete element model are listed in Table 1 ²² :

Table 1.

Material property.

Material	Density/kgm−3	Poisson’s ratio	Elastic modulus/Pa
Coal	1373	0.25	1.0 × 108
Gangue	2240	0.22	1.4 × 109
Steel	7800	0.20	7.0 × 1010

The contact parameters of the materials used in the top-coal caving DEM model are presented in Table 2.

Table 2.

Material contact parameter.

Material	Coefficient of restitution	Static friction coefficient	Dynamic friction coefficient
Coal-Coal	0.10	0.65	0.10
Gangue-Gangue	0.50	0.55	0.10
Coal-Gangue	0.30	0.55	0.25
Coal-Steel	0.60	0.80	0.35
Gangue-Steel	0.85	0.45	0.15

After the DEM model was established, simulations were conducted by adjusting the motion parameters of the tail beam and the scraper conveyor. Upon completion of the simulations, the variations in coal and gangue mass under different operating conditions were analyzed to investigate the influence of various factors on the top-coal caving process.

It should be noted that the present DEM model is primarily intended for comparative analysis of different tail-beam control strategies under a unified simulation framework, rather than for exact prediction of all field-scale quantities. Because direct field measurements and dedicated laboratory calibration data are not available in the present study, the applicability of the model is assessed mainly by whether it can reproduce the main flow characteristics reported in previous top-coal caving studies, including the evolution trend of discharge rate, the coal–gangue interface behavior, and the qualitative influence of control parameters on caving performance. Therefore, the conclusions of this study are focused on the relative differences among control schemes under the same modeling assumptions.

Analysis of coal flow characteristics

Establish a comprehensive evaluation system

The tail beam opening angle and the chain speed of the scraper conveyor are the primary factors influencing the caving performance of the hydraulic support. The vertical oscillation of the tail beam directly affects the spatial distribution of the coal and gangue layers during the caving process. A comprehensive evaluation system for top-coal caving is established using scraper-conveying efficiency, residual-coal ratio, and gangue-admixture ratio as the assessment indicators.

The scraper conveying efficiency $k_a$ reflects the mass of discharged material transported per unit time, and its evaluation model is expressed as:

$$k_a=(M_{c,0}-M_{c,r})/M_{c,0}-M_{c,r})tt$$

(9)

In the equation: M_c,0 denotes the initial mass of the coal seam, kg; M_c,r represents the residual coal mass at the end of the caving process, kg; and t is the caving duration, s.

The residual coal ratio $k_b$ reflects the caving performance of the hydraulic support. A high residual coal ratio indicates that a substantial portion of top coal remains unextracted, implying poor caving efficiency and economic performance. The evaluation model is expressed as:

$$k_b=M_{c,r}/M_{c,0}$$

(10)

The gangue-admixture ratio $k_c$ reflects the mass ratio of gangue to coal blocks transported by the scraper conveyor during caving. A lower gangue-admixture ratio indicates less gangue content in the discharged coal, corresponding to better economic efficiency. The evaluation model is expressed as:

$$k_c=(M_{g,0}-M_{g,r})/(M_{c,0}-M_{c,r})$$

(11)

In the equation: M_g,0 denotes the initial mass of the gangue layer, kg; and M_g,r represents the residual gangue mass at the end of the caving process, kg.

Analysis of the impact of different control schemes on the coal feeding process

The number of tail-beam openings and the opening mode both affect the caving process. Simulations were conducted for individual tail-beam opening, simultaneous multi-beam opening, and tail-beam oscillation modes. In this context, a direct tail-beam opening refers to the tail beam rotating to a certain angle and remaining stationary, allowing the top coal to fall freely under gravity. Tail-beam oscillation refers to the tail beam rotating to a certain angle and then oscillating vertically, adjusting the state of the top coal through its interaction. The simulation schemes are listed in Table 3.

Table 3.

Tail beam opening scheme design.

No.	Number of beams opened	Opening mode	Swing angle/°
1	Single beam	Direct opening	45
2	Single beam	Swinging	45
3	Dual beams	Direct opening	45
4	Dual beams	Swinging	45

Simulations were conducted for different caving control schemes, as shown in Figure 4(a) and (c). Under the same time conditions, the top gangue in the single tail-beam caving scenario reaches the scraper conveyor earlier than in the multi-beam case. The simultaneous caving of multiple tail beams exhibits a higher discharge capacity than a single tail beam, resulting in a greater amount of coal discharged within the same time period. The tail-beam oscillation mode directly affects the distribution of the top gangue layer and the amount of coal discharged. The caving volume under the tail-beam oscillation mode is lower than that under the direct tail-beam opening mode. As shown in Figure 4(c) and (d), tail-beam oscillation results in a smoother coal–gangue interface, which is beneficial for controlling the gangue content in the discharged coal and reducing the gangue-admixture ratio.

Figure 4.

Changes of coal flow under different control schemes at 20 s: (a) single tail beam direct opening, (b) single tail beam swinging, (c) multiple tail beams direct opening, and (d) multiple tail beams swinging.

As shown in Figure 5, the variations in top-coal mass during the four tail-beam caving schemes are compared. Once stabilized, the discharge rate in the direct-opening mode remains essentially constant, and the unit-time caving capacities of the single- and double-beam configurations are comparable. Within the interval from 12 to 17 s, the single-beam direct-opening mode achieves a discharge rate of 301.4 kg/s, whereas the double-beam direct-opening mode reaches 353.8 kg/s, with the latter being only 17.38% higher than the former.

Figure 5.

Change of coal flow quality under different control schemes.

The velocity vector state of coal–gangue motion during the caving process is shown in Figure 6. At the onset of caving, the top coal falls vertically through the tail-beam opening and accumulates near the tail beam, after which the scraper conveyor transports the discharged coal. At 20 s, the velocity of the top coal decreases, and the number of moving particles is reduced; the upper coal layer continues to cave in a flared, funnel-like pattern. The velocity-vector contour maps indicate that the caving velocity near the tail beam consistently remains around 0.3 m/s. Although the number of moving particles decreases, a small portion of coal particles exhibits an increase in velocity.

Figure 6.

Material velocity vector distribution: (a) 10 s and (b) 20 s.

The tail-beam opening mode of the hydraulic support has a notable influence on the conveying performance of the scraper conveyor. As the number of opened tail beams increases, the mass of material conveyed during the initial conveying period rises; however, the incremental increase per unit time remains relatively small. Consequently, the scraper conveyor’s conveying capacity per unit time becomes the primary factor limiting the overall caving rate.

Patterns of influence of different factors on top coal

The analysis in this section adopts a one-factor-at-a-time strategy to identify the primary influence trend of each control parameter under the present simulation framework. This treatment is intended as a preliminary parametric analysis for revealing the individual effects of scraper-chain speed, tail-beam swing amplitude, and swing frequency. It should be noted that such an approach does not fully capture possible interaction effects among multiple factors.

The scraper chain speed, tail-beam swing amplitude, and swing frequency all exert significant influence on the top-coal caving performance. To enhance caving efficiency and optimize the caving control strategy, numerical simulations were conducted for each individual factor, thereby identifying the caving patterns corresponding to different operating conditions. Formulating an appropriate caving scheme based on the observed caving patterns is beneficial for avoiding excessively high gangue content in the discharged material and improving the overall economic efficiency of mining operations. To clarify the role of each parameter, a one-factor-at-a-time simulation design was adopted, and the parameter settings of Schemes 1–8 are summarized in Table 4. Specifically, Schemes 1–3 were designed to examine the effect of scraper-chain speed, with the chain speed varied from 0.8 to 1.2 m/s while the tail-beam swing amplitude was kept at 55° and no swing frequency was imposed. Schemes 2, 4, and 5 were used to examine the effect of tail-beam swing amplitude, in which the swing amplitude was varied from 35° to 55° while the scraper-chain speed was kept at 1.0 m/s. Schemes 6–8 were designed to examine the effect of tail-beam swing frequency, in which the swing frequency was varied from 0.03 to 0.10 Hz while the scraper-chain speed and swing amplitude were fixed at 1.2 m/s and 55°, respectively. Thus, Table 4 reports the input parameter settings of the simulation schemes, whereas Table 5 presents the corresponding performance results of these schemes.

Table 4.

Simulation scheme design.

No.	Scraper chain speed/ms−1	Tail beam swing amplitude/°	Tail beam swing frequency/Hz
1	0.8	55	-
2	1.0	55	-
3	1.2	55	-
4	1.0	35	-
5	1.0	45	-
6	1.2	55	0.03
7	1.2	55	0.05
8	1.2	55	0.10

Table 5.

Effect analysis of different coal drawing schemes in the same time.

No.	Conveying efficiency/kgs−1	Residual coal rate/%	Gangue mixing rate/%
1	211.65	90.39	0.00
2	274.59	87.89	3.00
3	338.06	85.97	9.36
4	202.12	90.83	0.00
5	255.53	88.58	1.61
6	280.12	87.29	0.94
7	191.59	91.31	0.00
8	203.24	90.78	0.00

Note. “0.00” indicates that the discharged gangue mass was below the adopted reporting precision during the 20 s simulation period.

As shown in Figure 7, an increase in scraper conveyor chain speed leads to a greater number of moving coal particles. At lower chain speeds, a single high-velocity coal-particle region forms between the caving support and the adjacent non-caving support. At higher chain speeds, two distinct high-velocity peak regions of coal-particle motion emerge. A larger swing amplitude results in a greater caving thickness transported by the scraper conveyor and thus improves the caving performance, while producing no marked change in the overall coal-flow characteristics. The swing frequency directly alters the velocity distribution of the coal flow, causing the particle-motion region to become more concentrated and producing a distinct stratification interface, which is advantageous for controlling coal–gangue separation.

Figure 7.

Coal flow simulation results under different factors: (a) chain speed 0.8 m/s, (b) chain speed 1.2 m/s, (c) swing amplitude 35°, (d) swing amplitude 55°, (e) swing frequency 0.03 Hz, and (f) swing frequency 0.10 Hz.

As shown in Figure 8(a), the mass of top coal discharged per unit time increases with the chain speed. The caving of gangue also occurs earlier as it is transported together with the top coal. The mixed conveying of gangue and top coal results in a higher gangue content in the material stream. As shown in Figure 8(c), the mass of top coal discharged per unit time increases with the tail beam swing angle, but the rate of increase gradually diminishes. As shown in Figure 8(e), as the tail beam swing frequency increases, the period of discharge obstruction caused by tail beam oscillation decreases while its occurrence frequency increases. Within the swing-frequency range of 0.03–0.10 Hz, the duration of reduced gangue mass first increases and then decreases.

Figure 8.

Quality change of top coal: (a)effect of chain speed on coal seam mass, (b) effect of chain speed on gangue seam mass, (c) effect of swing angle on coal seam mass, (d) effect of swing angle on gangue seam mass, (e) effect of swing frequency on coal seam mass, and (f) effect of swing frequency on gangue seam mass.

A quantitative analysis of different caving control schemes within the first 20 s was performed, and the variation in top-coal mass is shown in Figure 8. For Scheme 3, which adopts a chain speed of 1.2 m/s, a tail-beam swing amplitude of 55°, and a direct-opening mode, the caving rate is the highest. The instantaneous caving rate reaches 338.06 kg/s, while the gangue content rises to 9.36%. Schemes 1, 4, 7, and 8 show a gangue-admixture ratio of 0.00 in Table 5. This indicates that the discharged gangue mass was not observed within the adopted reporting precision during the first 20 s of simulation. Under Scheme 7, characterized by a scraper chain speed of 1.2 m/s, a tail beam swing angle of 55°, and a swing-opening mode, the discharge rate of top coal is the lowest, with a mass flow rate of 191.59 kg/s.

It should be noted that the results in Table 5 are obtained from deterministic simulation cases under the same modeling settings, and repeated-run statistical variability was not evaluated in the present study. Therefore, the discussion is focused on comparative trends among schemes rather than formal statistical significance.

Multi-objective optimization model for coal discharge from the tail beam

The control of top-coal caving in hydraulic supports involves multiple conflicting objectives, including improving conveying efficiency while reducing residual coal ratio and gangue-admixture ratio. Therefore, the problem is multi-objective in nature from an engineering decision-making perspective. ²³

In the present study, instead of constructing the full Pareto front, a weighted-sum scalarization method was adopted to transform the multi-objective problem into a single scalar objective function. The resulting optimization problem was then solved using a genetic algorithm to obtain a compromise solution under predefined engineering preferences. This treatment is suitable for identifying a practical parameter combination for tail-beam control under the selected objective priorities.

For the present optimization problem, scraper-chain speed, tail-beam swing amplitude, and swing frequency were taken as decision variables, while conveying efficiency, residual coal ratio, and gangue-admixture ratio were used to construct the scalarized objective function. First, regression-based surrogate functions were established for the three performance indicators using the simulation data. Then, the weighted-sum objective function defined in equation (15), together with the constraints in equations (17)–(19), was optimized using a genetic algorithm. In this way, the parameter combination corresponding to the minimum weighted objective value was obtained. Therefore, the result reported in Table 6 should be interpreted as a weighted compromise solution under the selected engineering preference, rather than as a complete Pareto-optimal solution set.

Table 6.

Parameter optimization results.

Scraper chain speed/ms−1	Tail beam swing amplitude/°	Tail beam swing frequency/Hz
0.9042	40.00	0.0594

Regression analyses were conducted to construct surrogate functions for conveying efficiency k _a , residual coal ratio k _b , and gangue-admixture ratio k _c using the simulation results from Table 4. Quadratic polynomial models were employed to capture the nonlinear relationship between the performance indicators and the decision variables: scraper chain speed v tail-beam swing amplitude A, and tail-beam swing frequency f.

The regression models are expressed as follows:

$$\begin{array}{c}\multicolumn{1}{c}{k_a=-488.56+131.41v+19.08A-3792.52f}\\ \multicolumn{1}{c}{+101.82v^2-0.17A^2+23355.21f^2}\end{array}$$

(12)

$$\begin{array}{c}\multicolumn{1}{c}{k_b=125.26-16.48v-0.85A+127.51f}\\ \multicolumn{1}{c}{+2.21v^2+0.008A^2-736.46f^2}\end{array}$$

(13)

$$\begin{array}{c}\multicolumn{1}{c}{k_c=8.62-54.77v+0.25A-305.47f}\\ \multicolumn{1}{c}{+38.73v^2-0.001A^2+2156.12f^2}\end{array}$$

(14)

The regression models achieved good fits to the simulation data, with the coefficient of determination R² being 0.95 for conveying efficiency, 0.92 for residual coal ratio, and 0.89 for gangue-admixture ratio. These R² values indicate that the surrogate models can reasonably represent the trends observed in the DEM simulation data and provide a reliable basis for subsequent multi-objective optimization.

With the objectives of maximizing conveying efficiency and minimizing residual coal ratio and gangue-admixture ratio, the weighted-sum objective function is defined as:

$$minF(X)=K_1[k_a]+K_2[k_b]+K_3[k_c]$$

(15)

In the equation, $K_1$ denotes the weight coefficient for conveying efficiency, $K_2$ denotes the weight coefficient for residual coal ratio, and $K_3$ denotes the weight coefficient for gangue-admixture ratio.

Obtain the weights K1, K2, and K3:

$$K_1=0.4/k_{amax},K_2=0.3/k_{bmax},K_3=0.3/k_{cmax}$$

(16)

In this study, the weighting coefficients were assigned to reflect engineering preference in practical production. Conveying efficiency was given a slightly higher weight because it directly affects production capacity and operational efficiency, whereas residual coal ratio and gangue-admixture ratio were both treated as important indicators related to resource recovery and product quality. Accordingly, the weights were set to 0.4, 0.3, and 0.3, respectively. It should be noted that these values represent an engineering preference rather than a unique universally optimal choice.

The scraper chain speed, tail-beam swing amplitude, and tail-beam swing frequency are used as constraint conditions:

• (1) Scraper Chain Speed Constraint

The scraper conveyor chain speed is typically stable because the motor drives the sprocket through a reducer. Therefore, the chain speed is set to vary within the following range:

$$0.8\le v\le 1.2$$

(17)

• (2) Tail-Beam Swing Amplitude Constraint

The tail beam has an initial angle of 30°, and the maximum opening angle does not exceed 90°. Therefore, the tail-beam swing amplitude is constrained as follows:

$${40}^{°}\le A\le {60}^{°}$$

(18)

• (3) Tail-Beam Swing Frequency Constraint

The tail-beam swing frequency facilitates the application of forces on the top coal, promoting the breakage of large coal blocks and adjusting the coal-flow distribution. However, excessively high swing frequencies increase the occurrence of caving obstructions, which is unfavorable for continuous coal discharge:

$$0.02\le f\le 0.2$$

(19)

The tail-beam caving model was optimized using a genetic algorithm (GA). For reproducibility, the main GA settings are reported as follows: the population size was set to 20, the crossover probability was set to 0.4, the mutation probability was set to 0.01, and the maximum number of iterations was set to 50. These parameter settings were selected to balance search efficiency and computational cost in the present engineering optimization problem. The convergence behavior of the GA is shown in Figure 9, indicating that the optimization process reached a stable solution within the prescribed number of iterations.

Figure 9.

Genetic algorithm convergence curve.

The optimal motion parameters reported in Table 6 were not selected directly from the discrete simulation cases in Table 4. Instead, they were obtained through the following procedure. First, the simulation results under the parameter schemes listed in Table 4 were used to establish the regression-based surrogate functions for conveying efficiency, residual coal ratio, and gangue-admixture ratio. Second, these surrogate models were incorporated into the weighted-sum objective function defined in equation (15), together with the weighting coefficients in equation (16) and the parameter constraints in equations (17)–(19). Third, a genetic algorithm was employed to search the feasible parameter space and identify the parameter combination corresponding to the minimum weighted objective value. The resulting optimized values of scraper-chain speed, tail-beam swing amplitude, and tail-beam swing frequency are listed in Table 6.

The optimized parameter combination in Table 6 should be interpreted as a weighted compromise solution rather than as the best value for every individual indicator. Since the optimization objective in this study was formulated as a weighted-sum scalarization of conveying efficiency, residual coal ratio, and gangue-admixture ratio, the final solution was selected according to the minimum overall weighted objective value under the predefined engineering preference, rather than according to the best performance in any single metric alone. For example, although Scheme 5 in Table 5 shows a slightly higher conveying efficiency (255.53 kg/s), it still involves a nonzero gangue-mixing ratio (1.61%). By contrast, the optimized solution sacrifices only a very small amount of conveying efficiency to achieve stronger gangue control within the same 20 s evaluation window. Therefore, the advantage of the optimized solution lies in the overall weighted trade-off among the three objectives, rather than in outperforming every discrete scheme in each individual index.

Figure 10 shows the EDEM simulation result of the optimized scheme. The left panel presents the material distribution in the caving zone under the optimized parameter combination, while the right panel shows the corresponding velocity-field distribution. It can be observed that, within the first 20 s, the top coal is discharged continuously through the tail-beam opening and transported by the scraper conveyor, whereas the upper gangue layer remains above the outlet and does not enter the conveying stream. Meanwhile, the velocity distribution indicates that particle motion is mainly concentrated in the lower coal-flow region near the caving outlet and scraper conveyor, while the gangue layer remains relatively stable. These results are consistent with the quantitative indicators in Table 6, demonstrating that the optimized parameter combination can maintain coal discharge while effectively suppressing gangue mixing within the selected evaluation time window.

Figure 10.

Optimization result.

Conclusion

• (1) Under the same time conditions, in a single tail-beam caving scenario, the top gangue reaches the scraper conveyor earlier than in a multi-beam scenario. However, the simultaneous operation of multiple tail beams exhibits a higher discharge capacity, resulting in a greater amount of coal discharged within the same time frame.

• (2) As the scraper chain speed increases, the mass of top coal discharged per unit time rises, and the caving of gangue occurs earlier along with the top coal. With an increase in the tail-beam swing angle, the unit-time caving efficiency improves, but the incremental increase gradually diminishes. As the tail-beam swing frequency increases, the period of caving obstruction caused by the oscillation decreases, while the duration of gangue discharge first increases and then decreases.

• (3) The multi-objective optimization model for tail-beam caving effectively controls gangue behavior within the specified time, preventing gangue from falling onto the scraper conveyor and mixing with coal, thereby demonstrating a significant improvement in overall caving performance.