# Research on the Calculation Model and Control Method of Initial Supporting Force for Temporary Support in the Underground Excavation Roadway of Coal Mine

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## Abstract

**:**

## 1. Introduction

## 2. Calculation Model for Initial Support Strength and Prediction Model for Supporting Strength

#### 2.1. Calculation Model of Initial Support Force

_{0}of the brace controlling the equivalent direct top must fulfill the following condition:

_{1}is the lower direct top self-weight, kN; R

_{2}is the upper equivalent direct top self-weight, kN; h

_{1}is the thickness of the lower direct roof, m; h

_{2}is the upper equivalent direct roof thickness, m; h

_{3}is the basic roof key layer thickness, m; ∑h is the equivalent direct roof total thickness, m; l

_{s}is the width of the support, m; l is the upper direct roof breaking length, m; l

_{k}is the control roof distance of the support, m; l

_{x}is the direct roof leading coal wall breaking distance; α is the direct roof breaking angle, °; and γ is the average bulk density of bedrock, kN/m

^{3}.

_{0}required to control the equivalent direct top in the unstable state as follows:

#### 2.2. Classification of Initial Support Force Levels

_{0}= 800 kN), the integration of time can yield a specific value for the pressure in the lower chamber of the hydraulic cylinder, denoted as P

_{t}= 53 MPa·s. Similarly, the corresponding interval nodes for each level of initial bracing force for the bracket can be determined (1100, 1300, and 1600 kN) and this parameter can be used to establish Table 1 as the reference standard for the bracket’s initial bracing force.

_{t}, is utilized as the input with a range of [53, 110] MPa·s. The initial bracing force rating, denoted as Q, is given as the output and has a value range of [1, 3]. Under these circumstances, a single-input single-output fuzzy identification controller (F

_{1}) is established. A linear relationship exists between the initial bracing force and the pressure in the hydraulic cylinder’s lower cavity during the support process. Additionally, there is a proportional coefficient adjustment between the initial bracing force and the rated working resistance, which yields a numerical range indicating the level of the initial bracing force. Consequently, the Trapmf type affiliation function curve is selected, and its corresponding function is depicted in Figure 6.

#### 2.3. Prediction of Brace Initial Support Strength Based on Grey System

^{(0)}(k) = [X

^{(0)}(1),X

^{(0)}(2),…,X

^{(0)}(n)], (k = 1,2,…,n)}. Due to the influence and constraints of multiple factors, it is challenging to consider the behavioral sequence of the system in the general mine roof envelope as having a straightforward exponential growth characteristic. Therefore, the gray system prediction model, specifically the DNGM (1,1) model, is applied using the direct estimation method of whitening differential equation parameters to predict and estimate the support strength within the controlled roof distance of the excavated tunnel [24].

^{(0)}be a sequence defined as X

^{(0)}= (x

^{(0)}(1), x

^{(0)}(2),···, x

^{(0)}(n), where x(0)(k) ≧ 0, k = 1,2,···,n. The sequence X

^{(0)}is then cumulatively summed once to produce the sequence X

^{(1)}, and the adjacent mean value generation sequence Z

^{(1)}of X

^{(1)}is given by:

^{(1)}(t + 1) with the analog value ${\widehat{x}}^{\left(1\right)}\left(t+1\right)=\widehat{\alpha}{x}^{\left(1\right)}\left(t\right)+\widehat{\beta}t+\widehat{\gamma}$; the sum of squared errors can be obtained:

## 3. Design of SAPSO-PID Initial Support Force Controller

#### 3.1. PSO-PID Controller

_{P}is the proportional gain; K

_{I}is the integral gain; and K

_{D}is the differential gain.

_{i}= [x

_{i}

_{1},x

_{i}

_{2},···,x

_{iD}]

^{T}(i = 1,2,···,N) is formed by N particles in a D-dimensional target search space, and each particle’s own state is described by a set of position vectors p

_{i}= [p

_{i}

_{1},p

_{i}

_{2},···,p

_{iD}]

^{T}(i = 1,2,···,N) and velocity vectors V

_{i}= [v

_{i}

_{1},v

_{i}

_{2},···,v

_{iD}]

^{T}(i = 1,2,···,N) are described, in which the velocity of the particle directly affects the search distance of the particle at each step in the search space and can be adjusted according to the advantages and disadvantages of other particles and its own fitness. The optimal individual extremum searched by each particle is recorded as p

_{best}= [p

_{bi}

_{1},p

_{bi}

_{2},···,p

_{biD}]

^{T}(i = 1,2,···,N), and the global optimal extremum searched by the whole particle population is recorded as g

_{best}= [g

_{1},g

_{2},···,g

_{D}]

^{T}. Then, the particle properties are updated by Equation (20) as follows:

_{ij}(t) denotes the j-th dimensional flight velocity component of particle i when evolving to generation t; x

_{ij}(t) denotes the j-th dimensional position component of particle i when evolving to generation t; p

_{bij}(t) denotes the j-th dimensional individual optimal position p

_{best}component of particle i when evolving to generation t; p

_{bij}(t) denotes the j-th dimensional component of the optimal position g

_{best}of the whole particle population at evolution to generation t; c

_{1},c

_{2}are acceleration factors; r

_{1},r

_{2}are random numbers of [0, 1]; T

_{max}denotes the maximum evolutionary generation; $\varpi $

_{max}denotes the maximum inertia weight; and $\varpi $

_{min}denotes the minimum inertia weight.

- (1)
- Generate a search particle swarm (either initialized or updated).
- (2)
- Assign particles in the swarm to the middle control parameters K
_{P}, K_{I}, and K_{D}of the PID controller sequentially. - (3)
- Link the Particle Swarm algorithm to the PID control model of the initial bracing force control system and calculate the performance indices associated with the control parameters.
- (4)
- Subsequently, the performance metric is used as the fitness value for each particle in the control algorithm, followed by a determination of whether it is possible to exit the algorithm by reaching the optimal value and breaking out of the loop.

#### 3.2. Establishment of SAPSO Algorithm

_{old}) represents the fitness value of the particle in its initial position state, f(p

_{new}) represents the fitness value of the particle at its new position, and P denotes the acceptance probability for the system transitioning from the initial state to the new position state.

- (i)
- If f(p
_{old}) < f(p_{new}), indicating a decrease in energy after the state change, the particle accepts p_{new}as its current state. - (ii)
- If f(p
_{old}) < f(p_{new}), it denotes an increase in energy after the state change, indicating that the particle diverges from the optimal position. Subsequently, the control algorithm employs the rand (0, 1) function to generate a random number for comparison with the P value. If the P value is greater, the control algorithm accepts p_{new}as the current state. In contrast, if the P value is smaller, the current state remains as p_{old}.

#### 3.3. The Composition of the Initial Bracing Force Control System

## 4. Simulation Model Building and Result Analysis of Initial Support Force Control System

#### 4.1. Modeling and Stability Verification of Temporary Support Hydraulic Control System

#### 4.1.1. Establishment of the System Model

- (1)
- Pilot Operated Proportional Relief Valve Modeling

_{pa}is the gain coefficient of the proportional amplifier, A/V; I(s) is the current of the proportional amplifier, A; and U(s) is the input voltage of the proportional amplifier, V.

_{ps}is the current-force gain coefficient of the proportional electromagnet, N/A; F(s) is the corresponding output thrust of the electromagnet, N.

_{4}is the outlet pressure of the pilot valve, N/m

^{3}; A

_{4}is the effective area of the pilot valve spool, m

^{2}; P

_{4}A

_{4}is the algebraic sum of the hydraulic pressure on the effective area, N; m

_{sp}is the mass of the pilot valve spool, kg; x is the displacement of the spool, m; B

_{s}is the equivalent damping factor of the valve, N/(m·s

^{−1}); K

_{es}is the equivalent spring stiffness, N/m, which is the sum of the mechanical spring the sum of stiffness and steady-state hydrodynamic stiffness; F

_{ss1}is the steady-state hydrodynamic force, N; and F

_{cf}is the Coulomb friction force on the spool, N. Where P

_{4}A

_{4}, F

_{ss}

_{1}, and F

_{cf}are usually negligible, the above equation is collated and the Laplace transformation is completed, the displacement of the pilot valve spool under the action of electromagnetic force can be obtained as:

_{1}is the pressure at the lower end of the main valve, N/m

^{2}; x is the displacement of the spool, m; Q

_{4}is the flow rate of the lower volume chamber of the flow master valve, m

^{3}/s; ω

_{rv}is the turning frequency of the main valve spool movement, Hz; and K

_{rs}, K

_{r0}, ξ

_{r0}and ω

_{r0}are coefficients related to the intrinsic frequency of the main valve of the flow valve, the flow coefficient of each structure of the valve, and the frequency associated with the valve motion.

- (2)
- Hydraulic Cylinder Modeling

_{H}is the driving force generated by the hydraulic cylinder, N; A

_{H}is the effective pressure cross-sectional area of the hydraulic cylinder, m

^{2}; PH is the cavity pressure of the hydraulic cylinder, N/m

^{2}, because the relief valve is closer to the hydraulic cylinder, P

_{H}can be approximated as equal to P

_{1}; mp is the mass of the hydraulic cylinder plunger, kg; Bp is the equivalent damping factor of the hydraulic cylinder, N/(m·s

^{−1}); K

_{mp}is the equivalent spring stiffness, N/m; z is the displacement of the hydraulic cylinder plunger, m; and F

_{ef}is the equivalent external load pressure, N, mainly including the gravity of the top beam of the support, etc.

_{Hc}is the hydraulic cylinder flow rate, m3/s; C

_{hc}is the internal leakage coefficient of the hydraulic cylinder, L/min/bar; V

_{t}is the working volume cavity of the hydraulic cylinder, m

^{3}; and β

_{ec}is the effective volume modulus of elasticity of the hydraulic cylinder, MPa.

- (3)
- Pressure Feedback Sensor Modeling

_{f}is the amplification factor of the pressure sensor, V/N.

#### 4.1.2. Control System Model Stability Analysis

_{r0}= 18.54 Hz, ω

_{rv}= 187.5 Hz, ω

_{rA}= 0.244 Hz, and ξ

_{r0}= 1.832. These parameters are then substituted into the transfer function shown in Figure 11. Utilizing the Matlab programming environment, the system’s Nyquist diagram, Bode diagram, pole-zero diagram, and unit step response and unit impulse response curves are plotted, as depicted in Figure 12.

#### 4.2. Simulation Model Building

#### 4.3. Analysis of Simulation Results

## 5. Experiment

#### 5.1. Experimental Protocol

#### 5.2. Experimental Results

## 6. Conclusions

- (1)
- Developed a temporary support design for underground coal mine excavation tunnels and investigated the roof control mechanism of the support system. Created calculation and prediction models for the initial bracing force levels of the temporary support in the excavation tunnels, enabling accurate forecasting of bracing force requirements at different locations within the tunnels.
- (2)
- Designed an initial bracing force controller based on the SAPSO-PID approach and established a control system model specifically for the initial bracing force. Constructed a mathematical model to represent the pressure system of the pilot-operated relief valve-controlled hydraulic cylinder and subjected it to analysis and stability verification.
- (3)
- An integrated simulation platform combining AMESim and Matlab was employed to develop a simulation model for the initial bracing force control system utilizing the SAPSO-PID methodology. The simulation results demonstrate the outstanding performance of the SAPSO-PID initial bracing force control system in terms of response time and error control, highlighting its effectiveness in achieving efficient and adaptive control of the initial bracing force.
- (4)
- Carried out experiments within underground coal mine excavation tunnels where structurally similar temporary support systems based on the SAPSO-PID initial bracing force control were employed. The experimental findings demonstrate that the implemented bracing force control system effectively and adaptively manages the initial bracing force of the temporary support structures in the excavation tunnels. The temporary bracing control system utilized in the experiment demonstrates its universal applicability in coal mine excavation tunnels. It enables the adjustment of temporary bracing dimensions based on the tunnel cross-section dimensions in different projects while preserving the effectiveness of the control system remains intact.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Schematic diagram of temporary support structure for underground roadway excavation. 1—The top beam, 2—moving cylinder, 3—balance jack, 4—base, 5—Horizontal telescopic beam, 6—Column hydraulic cylinder, 7—Guard board.

**Figure 4.**Model for calculating the initial bracing force of the support at the excavation working face.

**Figure 5.**Schematic diagram of the initial support force range based on the pressure in the hydraulic cylinder chamber.

**Figure 11.**Block diagram of relief valve-controlled hydraulic cylinder pressure system transfer function.

**Figure 12.**Control system performance curves and step response and impulse curves. (

**a**) Nyquist diagram; (

**b**) Pole-zero diagram; (

**c**) Step response; (

**d**) Impulse response;(

**e**) Bode diagram.

**Figure 17.**Different signal input response. (

**a**) Sine wave signal input response; (

**b**) Sine wave signal input response; (

**c**) Square wave signal input response.

**Figure 19.**Pressure dynamic response curves for different initial bracing force levels. (

**a**) Pressure dynamic response process at higher initial support level; (

**b**) Pressure dynamic response curve at medium level of initial bracing force; (

**c**) Pressure dynamic response curve at medium level of initial bracing force.

P_{0}/(kN) | 800 | 1100 | 1300 | 1600 |

P_{t}/(MPa·s) | 53 | 73 | 86 | 110 |

Number | Actual Data | Forecast Data | Residual | Relative Error |
---|---|---|---|---|

1 | 60.5 | 60.371 | 0.129 | 0.213% |

2 | 60.1 | 60.2837 | 0.184 | 0.306% |

3 | 60 | 60.042 | 0.042 | 0.070% |

4 | 59.8 | 59.7454 | −0.055 | 0.092% |

5 | 59.9 | 59.3813 | −0.519 | 0.866% |

6 | 58.5 | 58.9344 | 0.434 | 0.741% |

7 | 58.4 | 58.3859 | −0.014 | 0.024% |

**Table 3.**The relief valve control hydraulic cylinder pressure system parameters to take the value of the table.

Parameters | Unit | Value |
---|---|---|

K_{pa} | A/V | 25 |

K_{ps} | N/A | 6.2 |

K_{es} | N/m | 2910 |

m_{sp} | kg | 0.002 |

A_{4} | mm^{2} | 19.625 |

K_{ss} | N/m | 12,000 |

ω_{rv} | Hz | 3.4 |

A_{H} | mm^{2} | 4.3 |

m_{p} | kg | 250 |

B_{p} | N/(m·s^{−}^{1}) | 300 |

K_{mp} | N/m | 3.2 × 10^{6} |

Algorithm | Input Signal | Overshoot (kN) | Response Time (s) | |
---|---|---|---|---|

PID | signal level | 1 | 0.934 | 1.04 |

2 | 0.167 | 0.59 | ||

3 | 0.199 | 0.49 | ||

Sine wave | 0.597 | 1.98 | ||

Sawtooth | 0.314 | 4.02 | ||

Square | 0.848 | 0.41 | ||

SAPSO-PID | signal level | 1 | 0.847 | 0.15 |

2 | 0.088 | 0.11 | ||

3 | 0.106 | 0.09 | ||

Sine wave | 0.443 | 0.18 | ||

Sawtooth | 0.040 | 0.55 | ||

Square | 0.152 | 0.16 |

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## Share and Cite

**MDPI and ACS Style**

Wang, D.; Li, R.; Cheng, J.; Zheng, W.; Shen, Y.; Zhao, S.; Wu, M.
Research on the Calculation Model and Control Method of Initial Supporting Force for Temporary Support in the Underground Excavation Roadway of Coal Mine. *Axioms* **2023**, *12*, 948.
https://doi.org/10.3390/axioms12100948

**AMA Style**

Wang D, Li R, Cheng J, Zheng W, Shen Y, Zhao S, Wu M.
Research on the Calculation Model and Control Method of Initial Supporting Force for Temporary Support in the Underground Excavation Roadway of Coal Mine. *Axioms*. 2023; 12(10):948.
https://doi.org/10.3390/axioms12100948

**Chicago/Turabian Style**

Wang, Dongjie, Rui Li, Jiameng Cheng, Weixiong Zheng, Yang Shen, Sihai Zhao, and Miao Wu.
2023. "Research on the Calculation Model and Control Method of Initial Supporting Force for Temporary Support in the Underground Excavation Roadway of Coal Mine" *Axioms* 12, no. 10: 948.
https://doi.org/10.3390/axioms12100948