Research on the Design of Multi-Rope Friction Hoisting System of Vertical Shaft Gravity Energy Storage System

Abstract

Renewable energy generation methods such as wind power and photovoltaic power have problems of randomness, intermittency, and volatility. Gravity energy storage technology can realize the stable and controllable conversion of gravity potential energy and electric energy by lifting and lowering heavy loads. The hoisting system is an important component of a gravity energy storage system, and its lifting capacity and speed seriously restrict its energy storage capacity, energy conversion efficiency, and operational safety and reliability. In this paper, a design method for a multi-rope friction hoisting system of a vertical shaft gravity energy storage system is proposed. The parameter design and calculation of the hoisting rope, balance rope, and friction wheel of the friction hoisting system under typical conditions were carried out. The static and dynamic anti-slip capabilities of the friction hoisting system under the typical condition were explored. The results show that the maximum acceleration and deceleration speed of the compacted strand wire rope scheme is the largest, and the lifting and lowering time is the shortest. The maximum acceleration and deceleration speed of the triangular strand wire rope scheme is the lowest, and the lifting and lowering time is the longest. The dynamic tension of the hoisting rope at the heavy-load end is positively correlated with the acceleration, and the maximum value occurs in the accelerated lifting stage and decelerated lowering stage of the heavy load. The static anti-slip safety factor between the hoisting rope and the friction lining and the specific pressure between the hoisting rope and the friction lining comply with the requirements of China’s Safety Regulations for Coal Mines. The dynamic anti-slip safety factor of the hoisting system under different rope selection schemes is greater than the minimum value of 1.25 stipulated in the Safety Regulations for Metal and Nonmetal Mines. The research results are of great significance for the safety, reliability, and stable and efficient energy storage of a gravity energy storage system.

1. Introduction

In order to realize the goal of “carbon peaking and carbon neutrality”, the construction of a new type of power system has become the main direction of China’s energy structure transformation [1,2,3,4]. However, the randomness, intermittency, and volatility of renewable energy generation, such as that from wind power and photovoltaic power, pose challenges when integrating large-scale renewable energy into the power system. This results in issues like insufficient power consumption and resource waste [5]. Compared with wind power and photovoltaic power generation methods, gravity energy storage technology has the advantages of large application scale, high conversion efficiency, low cost, long service life, not being affected by the environment, long energy storage time, and no self-discharge. It can realize stable and controllable conversion between gravitational potential energy and electric energy by lifting and lowering heavy loads cyclically through a hoisting system [6], which is suitable for grid peaking and realizing day and night transfer of electric energy, and is of great significance for maintaining the stability of a high proportion of renewable energy power systems [7].

A vertical shaft gravity energy storage system (Figure 1) mainly includes a weight block, a hoisting system, an energy conversion system, and a power grid connection system. The hoisting system realizes the reciprocal lifting and lowering of weights in a wide range of transport distances for energy storage and release. At the same time, the energy conversion system achieves the mutual conversion of mechanical energy and electric energy, and then the electrical energy is integrated into the power grid. The hoisting system is the core component of the gravity energy storage system, and its lifting capacity and lifting speed seriously restrict its capacity, energy conversion efficiency, and operational safety and reliability. Therefore, the design of a friction hoisting system for gravity energy storage system is crucial.

In the process of energy storage, the permanent magnet motor drives the friction wheel to rotate, and the friction between the steel wire rope and the friction lining on the friction wheel drives the loaded cage and the unloaded cage to rise and fall alternately. When the loaded cage is hoisted to the wellhead for unloading, the unloaded cage is lowered to the bottom of the mine for loading. Then, the loaded cage is hoisted again by friction to the wellhead for unloading, while the original loaded cage becomes empty and is lowered to the bottom of the mine for the next loading. In the process of energy release, the loaded cage is lowered under the motor assistance and the action of gravity, and the friction between the steel wire rope and the friction lining on the friction wheel drives the friction wheel to rotate. When the loaded cage is lowered to the bottom of the mine for unloading, the unloaded cage is hoisted to the wellhead for loading. Subsequently, the loaded cage is again lowered to the bottom of the mine for unloading under the motor assistance and the action of gravity, while the original loaded cage becomes empty and raised to the wellhead for the next loading.

The gravity energy storage system is still in the preliminary research stage at home and abroad. Research teams in Europe and the United States have carried out in-depth exploration in the mechanical structure innovation of the hoisting system, such as the use of advanced materials and manufacturing processes to design high-strength and lightweight hoisting parts to reduce energy loss. Part of the research focuses on the development of novel driving mechanisms, such as the use of superconducting magnetic levitation technology to reduce friction resistance and improve lifting efficiency. In addition, many universities and research institutes have carried out theoretical analysis and calculation for the mechanical model and energy conversion efficiency of the hoisting system. By establishing mathematical models, researchers have discussed the system performance and optimization methods under different lifting methods to improve the traditional hoisting technology, such as optimizing the performance of the motor and improving the structure of the transmission device so as to improve the reliability and stability of the system, as well as conducting basic research on new hoisting technologies, such as exploring the use of linear motors or electric cylinders to achieve hoisting functions. In 2013, Fraenkel Weight in the UK proposed a gravity energy storage scheme based on a single winch hoist [8]. In 2018, Gezhouba Zhongke Energy Storage Technology Co., Ltd. proposed a gravity energy storage scheme using a mine elevator to hoist a single heavy load [9]. In 2019, the China Coal Energy Research Institute proposed a gravity energy storage scheme using a mine elevator to hoist multiple heavy loads [10]. In 2020, Gravitricity in the UK proposed a gravity energy storage scheme based on multiple winch hoists to hoist single or multiple heavy loads [11], and completed a 250 kW principle demonstration system test with a 15 m-deep drilling platform [12]. In 2021, American scholars proposed a gravity energy storage scheme for hoisting multiple heavy loads in a vertical shaft using two independent winch hoists [13]. In 2021, the Institute of Electrical Engineering of the Chinese Academy of Sciences proposed a gravity energy storage scheme based on hoisting multiple weights with a gantry hoist [14,15,16]. In the scheme, automatic hoisting devices or cages are used to achieve the hoisting of multiple heavy loads, and multiple heavy objects are placed horizontally in the roadway or vertically stacked in the shaft. Xiao et al. [17] put forward the scheme of building an energy storage tower in a vertical or inclined shaft, and listed their respective application scenarios.

The existing research is still in the conceptual stage of designing a vertical shaft gravity energy storage system with light load and low lifting height, and there has been no research on the design of a vertical shaft gravity energy storage friction hoisting system under kilometer-level, large transportation distance, large load, and high lifting speed conditions. Therefore, a design method for a vertical shaft gravity energy storage friction hoisting system is proposed herein. The parameters of the hoisting rope, balance rope, friction wheel, and derrick of the friction hoisting system under the typical hoisting condition were calculated. The static and dynamic anti-slip capabilities of the friction hoisting system under typical hoisting conditions were investigated. The research results have definite guiding significance for the design of a multi-rope friction gravity energy storage system and ensuring the safety and stability of the system.

2. Design Theory of Multi-Rope Friction Hoisting System with Double Cages in Vertical Shaft

The multi-rope friction hoisting system is an important component of the shaft gravity energy storage system, which is mainly responsible for lifting and lowering heavy loads in the process of energy storage and release. The system is mainly composed of a sheave wheel, hoisting rope, derrick, friction wheel, double cages, and balance rope (Figure 2).

2.1. Calculation Theory and Selection of the Structural Parameters of the Hoisting System

2.1.1. Calculation of Chord Length and Elevation Angle of Upper and Lower Ropes

The calculation formulas for the chord lengths of the upper rope and lower rope L_s and L_x are shown in Equations (1) and (2).

$$L_s=\sqrt{{\left(H_j-E_0\right)}^2+{\left(L_0+S-{\displaystyle \frac{D_t}{2}}\right)}^2-{\left({\displaystyle \frac{D_g-D_t}{2}}\right)}^2}$$

(1)

$$L_x={\left(H_j-H_s-E_0\right)}^2+{\left(L_0-{\displaystyle \frac{D_t}{2}}\right)}^2-{\left({\displaystyle \frac{D_t+D_g}{2}}\right)}^2$$

(2)

The calculation formulas for the elevation angle of the upper rope and lower rope β_s and β_x are shown in Equations (3) and (4):

$${\beta }_s=\arctan \left({\displaystyle \frac{H_j-E_0}{L_0+S-\frac{D_t}{2}}}\right)-\arctan ({\displaystyle \frac{D_g-D_t}{2L_s}})$$

(3)

$${\beta }_x=\arctan \left({\displaystyle \frac{H_j-H_s-E_0}{L_0-\frac{D_t}{2}}}\right)-\arctan ({\displaystyle \frac{D_g+D_t}{2L_x}})$$

(4)

where E₀ is the vertical height from the center of the friction wheel to the level of the wellhead lock (rail surface), H_j is the vertical height from the center of the friction wheel to the lock of the wellhead, L₀ is the horizontal distance from the center of the friction wheel to the right vertical rope, S is the center distance of the left and right vertical rope, D_t is the diameter of the upper sheave wheel and lower sheave wheel, and D_g is the diameter of the drum.

2.1.2. Calculation of Parameters of Friction Wheel

The calculation formula for the wrapping angle of the hoisting wire rope on the friction lining α is shown in Equation (5):

$$\alpha =180°+{\beta }_x-{\beta }_s$$

(5)

where β_s is the elevation angle of the upper rope and β_x is the elevation angle of the lower rope.

According to Article 417 of China’s Safety Regulations for Coal Mines [18], the ratio of the minimum diameter of the sheave wheel, drum, friction wheel, and guide wheel to the diameter of the steel wire rope should meet the requirements in

Table 1. Requirements for the ratio of the minimum diameter of the sheave wheel, drum, friction wheel, and guide wheel to the diameter of the steel wire rope.

Use Classification		Minimum Ratio
Friction wheel and the sheave wheel of the floor-standing friction hoisting system, and the friction wheel of the tower friction hoisting system with a wrapping angle greater than 180°	Aboveground	90
	Underground	80
Friction wheel of the tower friction hoisting system with a wrapping angle of 180°	Aboveground	80
	Underground	70
Guide wheel of friction hoisting system		80
Drum and the sheave wheel with a wrapping angle greater than 90° of the ground winding hoisting system		80
Sheave wheel with a wrapping angle smaller than 90° of the ground winding hoisting system		60
Drum of the underground winding hoist and drilling hoist, the leading wheel and tail guide wheel and the sheave wheel with a wrapping angle greater than 90° of the underground overhead passenger device		60
Sheave wheel with a wrapping angle smaller than 90° of the underground winding hoist, the drilling hoist and the underground overhead passenger device		40

.

In addition, according to Safety Inspections—Testing Specifications of In-Service Friction Hoist System for Coal Mines [19], the ratio of the diameter of the sheave wheel, friction wheel, and guide wheel to the diameter of the thickest steel wire in the steel wire rope should not be less than 1200.

A floor-standing multi-rope friction hoisting system was designed and the method of installation on the well was adopted. The ratio of the diameter of the friction wheel, sheave wheel, and hoisting rope in this system should all meet Equations (6) and (7):

$${\displaystyle \frac{D_1}{d}}\ge 90$$

(6)

$${\displaystyle \frac{D_1}{\delta }}\ge 1200$$

(7)

where D₁ is the diameter of the friction wheel and sheave wheel, d is the nominal diameter of the hoisting rope, and $\delta$ is the diameter of the thickest steel wire in the hoisting rope.

According to the requirements of Formulas (6) and (7) and combined with the basic parameters of the floor-standing multi-rope friction elevator in GB/T10599-2023 (Multi-rope friction hoist) [20], the appropriate diameter of friction wheel and the rope groove spacing of the friction wheel is selected.

2.2. Calculation Theory and Selection of Hoisting Rope and Tail Rope

2.2.1. Initial Overhang Height of Hoisting Rope and Tail Rope

The initial position of the cage in this system is at the bottom of the vertical shaft, and there is a certain height difference between the left and right sheave wheels. Therefore, it is necessary to calculate the initial overhang height of the hoisting ropes and tail ropes on both sides separately.

When the heavy-load end is the left cage, the initial overhang height of the hoisting rope at the heavy-load end h₁, the initial overhang height of the hoisting rope at the light-load end h₂, the initial overhang height of the tail rope at the heavy-load end h₃, and the initial overhang height of the tail rope at the light-load end h₄ are shown in Equations (8)–(11).

$$h_1=H+H_{t1}-h$$

(8)

$$h_2=H_{t2}-h$$

(9)

$$h_3=H_0+H_g+H_h$$

(10)

$$h_4=H+H_g+H_h-h-H_0$$

(11)

When the heavy-load end is the right cage, the initial overhang height of the hoisting rope at the heavy-load end h₁, the initial overhang height of the hoisting rope at the light-load end h₂, the initial overhang height of the tail rope at the heavy-load end h₃, and the initial overhang height of the tail rope at the light-load end h₄ are shown in Equations (12)–(15):

$$h_1=H+H_{t2}-h$$

(12)

$$h_2=H_{t1}-h$$

(13)

$$h_3=H_g+H_h$$

(14)

$$h_4=H+H_g+H_h-h$$

(15)

where H is the target hoisting height, H_t1 is the height of the sheave wheel at the heavy-load end, H_t2 is the height of the sheave wheel at the light-load end, h is the height of the cage, H₀ is the hoisting height at a certain time, H_g is the height of overwind, and H_h is the height of the tail rope ring.

2.2.2. Approximate Unit Weight and Minimum Breaking Force of Hoisting Rope

According to GB/T 8918-2006 (Steel wire ropes for important purposes) [21], the approximate unit weight of wire rope p_k and the minimum breaking force of wire rope F₀ are calculated through Formulas (16) and (17):

$$P_k={\displaystyle \frac{K_1{d}^2}{100}}$$

(16)

where d is the nominal diameter of the wire rope and K₁ is the mass coefficient of the wire rope and:

$$F_0=K_2{d}^2{R}_0$$

(17)

where K₂ is the minimum breaking tension coefficient of the wire rope, d is the nominal diameter of the wire rope, and R₀ is the nominal tensile strength of the wire rope.

2.2.3. Minimum Nominal Diameter of Hoisting Wire Rope

According to an empirical formula [22] of wire rope diameter selected in the design manual, the calculation formula for the minimum diameter of the hoisting wire rope d_min is shown in Equation (18):

$$d_{\min }=\sqrt{{\displaystyle \frac{100Q_d}{n_1{K}_1\left(K_z\frac{R_0}{m_a}-h_1\right)}}}\pi$$

(18)

where Q_d is the end load of the rope, N₁ is the number of hoist ropes, K₁ is the mass coefficient of the wire rope, K_z is the comprehensive coefficient of the wire rope, and its formula is $K_z={\displaystyle \frac{100K_2{K}_3}{K_{1}g}}$, R₀ is the nominal tensile strength of the wire rope, m_a is the minimum safety coefficient of wire rope stipulated in the Safety Regulations for Coal Mines, and h₁ is the initial overhang height of the hoisting rope at the heavy-load end.

2.2.4. Safety Coefficient of Hoisting Wire Rope

According to Article 408 of the Safety Regulations for Coal Mines, the safety coefficient of hoisting steel wire ropes must meet the requirements in

Table 2. Minimum safety coefficient of wire rope under different working conditions.

Service Classification			Minimum Safety Coefficient
Single rope winding hoisting system	Specially designed for hoisting personnel		9
	Hoist personnel and materials	Hoist personnel	9
		Mixed hoist	9
		Hoist materials	7.5
	Specially designed for hoisting materials		6.5
Friction wheel hoisting system	Specially designed for hoisting personnel		9.2–0.0005H
	Hoist personnel and materials	Hoist personnel	9.2–0.0005H
		Mixed hoist	9.2–0.0005H
		Hoist materials	8.2–0.0005H
	Specially designed for hoisting materials		7.2–0.0005H
Cage guide rope, anti-collision rope and hoisting rope			6

.

The multi-rope friction hoisting system (floor-standing) for a vertical shaft gravity energy storage system was adopted, and was specially designed for lifting materials. The calculation formula of the minimum safety coefficient of the wire rope m_a is shown in Equation (19):

$$m_a=7.2–0.0005H$$

(19)

where H is the lifting height.

According to the Safety Regulations for Coal Mines, the safety coefficient of the steel wire rope is the ratio of the sum of breaking force of the qualified steel wire to the maximum static tension F_max (Equation (20)), so the calculation formula of the safety coefficient of the steel wire rope is shown in Equation (21):

$$F_{\max }=\left(Q_d+n_1{p}_k{h}_1+n_2{q}_k{h}_3\right)g$$

(20)

$$m={\displaystyle \frac{n_1{K}_3{F}_0}{F_{\max }}}\ge m_a$$

(21)

where Q_d is the end load of the rope, n₁ is the number of hoist ropes, n₂ is the number of tail ropes, p_k is the unit weight of the hoisting rope, q_k is the unit weight of the tail rope, h₁ is the initial overhang height of the hoisting rope at the heavy-load end, h₃ is the initial overhang height of the tail rope at the heavy-load end, m is the safety coefficient of the steel wire rope, m_a is the minimum safety coefficient of wire rope stipulated in the Safety Regulations for Coal Mines, and K₃ is the conversion factor of the minimum breaking force of the wire rope.

2.2.5. Type Selection of Hoisting Wire Rope

According to the calculation results of the minimum diameter of the hoisting wire rope (Equation (18)) and the minimum safety coefficient stipulated in the Safety Regulations for Coal Mines (Equation (19)), combined with the selection standards of the wire rope in GB/T 8918-2006 (Steel wire ropes for important purposes) and YB/T 5359-2020 (Compact-ed strand ropes) [23], the appropriate type and diameter of the wire rope is selected and the safety coefficient according to the Equation (21) is checked to ensure that the safety coefficient of the hoisting wire rope is greater than the minimum value specified in the Safety Regulations for Coal Mines.

2.2.6. Type Selection of Balance Tail Rope

There are three main types of balance tail rope. The advantage of multi-layer profiled-strand (nonrotating) steel wire rope is that it has a small rotational torque during operation, but the disadvantage is that it is difficult to check for broken wires in the inner strand, and the outer strand is prone to loosening during operation. The advantages of flat steel wire rope are small transverse vibration amplitude, no rotation, and stable and reliable operation. The disadvantage is that the flat steel wire rope is generally manually woven, so it is expensive. Ordinary point contact around strand steel wire ropes can also be used as balance tail ropes, such as 6 × 19, 6 × 37, which have the advantages of low price and sufficient supply. The disadvantage is that the round strand steel wire rope is prone to knotting at the tail rope ring [24].

The selection of balance tail ropes is based on the equal weight tail rope balance scheme [5], which means that the total weight of n₁ hoisting steel wire ropes with unit length is equal to the total weight of n₂ balance tail ropes with unit length, as shown in Equation (22):

$$q_k={\displaystyle \frac{n_1}{n_2}}{p}_k$$

(22)

where q_k is the unit weight of a tail rope, p_k is the unit weight of a hoisting rope, n₁ is the number of hoisting ropes, and n₂ is the number of tail ropes.

According to the calculation result based on Equation (22) and the selection standard of a balance tail rope in GB/T 20119-2023 (Steel wire ropes for balance) [25], the appropriate balance tail rope is selected.

2.3. Static Calculation Theory of Hoisting System

2.3.1. Static Tension Calculation of Hoisting Wire Rope

The hoisting system of the shaft gravity energy storage system is a multi-rope friction hoisting system with double cages (floor-standing). The structure diagram of the hoisting system in the static state is shown in Figure 3. Since the system adopts a double-cage hoisting scheme, and the initial overhang heights of the hoisting wire ropes and tail ropes on both sides of the wheel are different, it is necessary to consider the force conditions of the hoisting system when the heavy load is loaded in different cages. Additionally, due to the use of an equal-weight balanced tail rope scheme (n₁p_k = n₂q_k), the static tension at the tangent point of the friction wheel remains approximately constant throughout the hoisting process (assuming the overhang height of hoisting wire rope at the heavy-load end is h_c at a certain moment, the overhang height of tail rope is h_w, and the hoisting height is Δh, then n₁p_k(h_c − Δh) + n₂q_k(h_w + Δh) = n₁p_kh_c + n₂q_kh_w, with the same principle applying to the light-load end). Therefore, when calculating the static tension of hoisting wire rope, it is only necessary to calculate the static tension at the tangent point of the friction wheel of the hoisting system in its initial position.

When the left-side cage is the heavy-load end, the static tension at the tangent point of the friction wheel at the heavy-load end F_jw and the static tension at the tangent point of the friction wheel at the light-load end F_jl can be calculated through Equations (23) and (24).

$$F_{jw}=\left[n_1{q}_k\left(h_1-L_s\sin {\beta }_s\right)+n_2{p}_k{h}_3+1000\cdot (Q+Q_z)\right]g$$

(23)

$$F_{jl}=\left[n_1{q}_k\left(h_2-L_x\sin {\beta }_x\right)+n_2{p}_k{h}_4+1000\cdot Q_z\right]g$$

(24)

When the right-side cage is the heavy-load end, the static tension at the tangent point of the friction wheel at the heavy-load end F_jw and the static tension at the tangent point of the friction wheel at the light-load end F_jl can be calculated through Equations (25) and (26):

$$F_{jw}=\left[n_1{q}_k\left(h_1-L_x\sin {\beta }_x\right)+n_2{p}_k{h}_3+1000\cdot (Q+Q_z)\right]g$$

(25)

$$F_{jl}=\left[n_1{q}_k\left(h_2-L_s\sin {\beta }_s\right)+n_2{p}_k{h}_4+1000\cdot Q_z\right]g$$

(26)

where n₁ and n₂ are the number of hoisting ropes and tail ropes, p_k is the unit weight of the hoisting rope, q_k is the unit weight of the tail rope, h₁ is the initial overhang height of the hoisting rope at the heavy-load end, h₂ is the initial overhang height of the hoisting rope at the light-load end, h₃ is the initial overhang height of the tail rope at the heavy-load end, h₄ is the initial overhang height of the tail rope at the light-load end, L_s and L_x are the chord lengths of the upper rope and lower rope, β_s and β_x are the elevation angles of the upper rope and lower rope, Q is the lifting load, Q_z is the weight of the hoisting container, and g is the acceleration due to gravity.

The values h₁, h₂, h₃, h₄ are calculated by Equations (8)–(11) or (12)–(15) according to the position of the heavy-load.

2.3.2. Calculation of the Static Anti-Slip Safety Coefficient between Hoisting Rope and Friction Lining

The multi-rope friction hoisting system utilizes the friction between the wire rope and the friction linings on the friction wheel to transmit motion. According to Euler’s formula for flexible body friction transmission, the ratio of the wire rope tensions on both sides of the friction wheel should satisfy Equation (27) to prevent slipping:

$${\displaystyle \frac{F_{jl}}{F_{jw}}}\le e^{\mu \alpha }$$

(27)

where F_jw is the static tension at the tangent point of the friction wheel at the heavy-load end, F_jl is the static tension at the tangent point of the friction wheel at the light-load end, μ is the friction coefficient between the friction lining and the wire rope, and α is the wrapping angle of the hoisting rope on the friction lining.

According to GB 16423-2020 (Safety regulation for metal and nonmetal mines) [26], the static anti-slip safety coefficient δ is defined as the ratio of the force preventing the wire rope from slipping (static friction force) to the difference in tension of the wire rope on both sides of the friction wheel. Thus, the static anti-slip safety coefficient δ can be calculated using Equation (28).

$$\delta ={\displaystyle \frac{F_{jl}\left(e^{\mu \alpha }-1\right)}{F_{jw}-F_{jl}}}$$

(28)

Additionally, GB 16423-2020 (Safety regulation for metal and nonmetal mines) stipulates that the static anti-slip safety coefficient for a multi-rope friction hoisting system should not be less than 1.75, thus δ ≥ 1.75.

2.3.3. Calculation of Contact Pressure between Hoisting Rope and Friction Lining

The contact pressure between the hoisting rope and the friction lining can be calculated using Equation (29):

$$P={\displaystyle \frac{F_{jw}+F_{jl}}{n_{1}d{D}_g}}$$

(29)

where F_jw is the static tension at the tangent point of the friction wheel at the heavy-load end, F_jl is the static tension at the tangent point of the friction wheel at the light-load end, n₁ is the number of hoisting ropes, d is the nominal diameter of the hoisting rope, and D_g is the diameter of the friction wheel.

Additionally, the contact pressure between the wire rope and the friction lining P should not exceed 2 Mpa according to the Safety Regulations for Coal Mines.

2.4. Dynamic Calculation Theory of Hoisting System

2.4.1. Equivalent Mass of the Hoisting System

The hoisting system is a complex motion system. The d’Alembert principle is used to simplify the calculation of the system’s inertial forces. The mass of moving components (wire ropes, sheave wheels, friction wheels, motors, cages, heavy loads) are replaced with a concentrated mass at the friction wheel. This equivalent mass is referred to as the hoisting system’s equivalent mass.

The total equivalent mass ΣM of the system can be calculated using Equation (30) [27].

$${\displaystyle \sum M}=Q+2Q_z+n_1{p}_k{L}_c+n_2{q}_k{L}_w+N{m}_t+m_j+m_d$$

(30)

When the left-side cage is the heavy-load end and located at the initial bottom position of the shaft, the maximum equivalent mass at the heavy-load end m_w and the minimum equivalent mass at the light-load end m_l can be calculated using Equations (31) and (32).

$$m_w=Q+Q_Z+n_1{p}_k(L_x+h_1)+n_2{q}_k{h}_3+m_t$$

(31)

$$m_l=Q_Z+n_1{p}_k(L_s+h_2)+n_2{q}_k{h}_4+m_t$$

(32)

When the right-side cage is the heavy-load end and located at the initial bottom position of the shaft, the maximum equivalent mass at the heavy-load end m_w and the minimum equivalent mass at the light-load end m_l can be calculated using Equations (33) and (34):

$$m_w=Q+Q_Z+n_1{p}_k(L_x+h_1)+n_2{q}_k{h}_3+m_t$$

(33)

$$m_l=Q_Z+n_1{p}_k(L_s+h_2)+n_2{q}_k{h}_4+m_t$$

(34)

where Q is the lifting load, Q_z is the weight of the hoisting container, L_c is the total length of the hoisting rope, L_w is the total length of the tail rope, p_k is the unit weight of the hoisting rope, q_k is the unit weight of the tail rope, n₁ and n₂ are the number of hoisting ropes and tail ropes, N is the number of sheave wheel, m_t is the equivalent mass of the sheave wheel, m_j is the equivalent mass of the friction wheel, m_d is the equivalent mass of the motor, L_s is the chord length of the upper rope, L_x is the chord length of the lower rope, h₁ is the initial overhang height of the hoisting rope at the heavy-load end, h₂ is the initial overhang height of the hoisting rope at the light-load end, h₃ is the initial overhang height of the tail rope at the heavy-load end, and h₄ is the initial overhang height of the tail rope at the light-load end.

2.4.2. Limiting Acceleration/Deceleration

The maximum acceleration/deceleration of the hoisting system can be calculated using Equation (35). Since only the inertial force caused by equivalent mass is considered in the calculation process and the mine resistance is ignored, a certain safety margin should be left for the calculation result, that is, a_max = 0.8·a.

$$a\le {\displaystyle \frac{(e^{\mu \alpha }-1)F_{jl}-{\delta }_{d\mathit{min}}(F_{jw}-F_{jl})}{\left(e^{\mu \alpha }-1\right)m_l+{\delta }_{d\mathit{min}}\left(m_l+m_w\right)}}$$

(35)

where δ_dmin is the minimum dynamic anti-slip safety coefficient for the multi-rope friction hoisting system, 1.25, F_jw is the static tension at the tangent point of the friction wheel at the heavy-load end, F_jl is the static tension at the tangent point of the friction wheel at the light-load end, m_w is the maximum equivalent mass at the heavy-load end, m_l is the minimum equivalent mass at the light-load end, μ is the friction coefficient between the friction lining and the wire rope, and α is the wrapping angle of the hoisting rope on the friction lining.

2.4.3. Dynamic Tension of the Hoisting Rope

During the hoisting and lowering processes of heavy loads, several sources of resistance affect the operation of the system, including air resistance, resistance between the cage and the guide rail, and bending resistance between the rope and sheave wheels or friction wheels. The specific resistance coefficients for each condition cannot be precisely determined due to these resistances. Therefore, the resistances can be neglected in the calculations, but a safety margin of 10% to 20% should be added to the computed parameters.

In this study, the dynamic tension of the hoisting rope during the energy storage and release of the shaft gravity energy storage system is calculated considering only the inertial forces caused by the equivalent mass. The influence of mine resistance and elastic vibration of wire ropes is neglected. Since there are multiple stages during the energy storage and release processes, as shown in Figure 4 and Figure 5, and the hoisting system employs a double-cage hoisting scheme, dynamic tension analysis must be performed for the hoisting rope at different loading positions and stages.

According to the dynamic model in Figure 4 and Figure 5, the stress of the hoisting rope at the sheave wheel and friction wheel is analyzed, as shown in Figure 6. Under different working conditions, the dynamic tension at the left tangent points between the rope and the upper and lower sheave wheels is calculated by Equations (36) and (37). The dynamic tension at the right tangent points between the rope and the upper and lower sheave wheels is calculated by Equations (38) and (39). The dynamic tension at the tangent points on both sides of the friction wheel is shown in Equations (40) and (41).

$$F_{dw}^{\prime }=\left[(Q+Q_z)+(n_1{p}_k{h}_1+n_2{q}_k{h}_3)/1000\right]·\left[g+a\right]$$

(36)

$$F_{dl}^{\prime }=\left[Q_z+(n_1{p}_k{h}_2+n_2{q}_k{h}_4)/1000\right]·\left[g-a\right]$$

(37)

$$F_w^{\prime }=(1+{\displaystyle \frac{a}{g}})F_{dw}+k_1{F}_c+{\displaystyle \frac{a}{g}}\times G_t$$

(38)

$$F_l^{\prime }=(1-{\displaystyle \frac{a}{g}})F_{dl}+k_1{F}_c+{\displaystyle \frac{a}{g}}\times G_t$$

(39)

$$F_{dw}=F_w^{\prime }-{\displaystyle \frac{m_s}{\sin {\beta }_s}}a+{\displaystyle \frac{m_j+m_d}{2}}a$$

(40)

$$F_{dl}=F_l^{\prime }+{\displaystyle \frac{m_x}{\sin {\beta }_x}}a-{\displaystyle \frac{m_j+m_d}{2}}a$$

(41)

where n₁ and n₂ are the number of hoisting ropes and tail ropes, p_k is the unit weight of the hoisting rope, q_k is the unit weight of the tail rope, h₁ is the initial overhang height of the hoisting rope at the heavy-load end, h₂ is the initial overhang height of the hoisting rope at the light-load end, h₃ is the initial overhang height of the tail rope at the heavy-load end, h₄ is the initial overhang height of the tail rope at the light-load end, L_s is the chord length of the upper rope, L_x is the chord length of the lower rope, β_s is the elevation angle of the upper rope, β_x is the elevation angle of the lower rope, g is the acceleration due to gravity, a is the acceleration of hoisting system, F_c is the static tension difference on both sides of the friction wheel, calculated by F_c = F_jw − F_jl, k₁ is the resistance coefficient, taken as 0.1, Q is the lifting load, Q_z is the weight of the hoisting container, G_t is the equivalent mass of the sheave wheel, m_s is the mass of the upper rope, m_x is the mass of the lower rope, m_j is the equivalent mass of the friction wheel, m_d is the equivalent mass of the motor, m_w is the maximum equivalent mass of the heavy-load end, and m_l is the maximum equivalent mass of the light-load end.

2.4.4. Dynamic Anti-Slip Safety Coefficient

According to GB 16423-2020 (Safety regulation for metal and nonmetal mines), the dynamic anti-slip safety coefficient δ_d for a multi-rope friction hoisting system can be calculated using Equation (42):

$${\delta }_d={\displaystyle \frac{F_{dl}\left(e^{\mu \alpha }-1\right)}{F_{dw}-F_{dl}}}$$

(42)

where F_dw is the dynamic tension at the tangent point of the friction wheel at the heavy-load end, F_dl is the dynamic tension at the tangent point of the friction wheel at the light-load end, μ is the friction coefficient between the friction lining and the wire rope, and α is the wrapping angle of the hoisting rope on the friction lining.

Additionally, GB 16423-2020 (Safety regulation for metal and nonmetal mines) stipulates that the dynamic anti-slip safety coefficient for a multi-rope friction hoisting system should not be less than 1.25, thus δ ≥ 1.25.

2.4.5. Elongation of Hoisting Wire Rope

The elastic modulus E_S of the hoisting wire rope is calculated by Equation (43):

$$E_s={\displaystyle \frac{{\sigma }_z}{\epsilon }}$$

(43)

where σ_z is the nominal stress of rope or strand and ε is the axial strain.

The axial strain ε of the hoisting wire rope is calculated by Equation (44):

$$\epsilon ={\displaystyle \frac{\Delta L}{L}}$$

(44)

where ΔL is the axial elongation of the hoisting wire rope and L is the initial length of the hoisting wire rope.

The nominal stress of rope or strand is calculated by Equation (45):

$${\sigma }_z={\displaystyle \frac{F}{A}}$$

(45)

where F is the dynamic tension of the hoisting wire rope and A is the cross-sectional area of the hoisting wire rope.

Combining the results from Equations (43)–(45), the elongation of the hoisting rope can be determined as Equation (46):

$$\Delta L={\displaystyle \frac{FL}{A{E}_s}}$$

(46)

Substituting the calculated results of dynamic tension under different operating conditions into Equation (46) can provide the elongation of the hoisting rope for each specific condition.

3. Design and Calculation Results of Vertical Shaft Multi-Rope Friction Hoisting System

3.1. Operational and Structural Parameters of Typical Hoisting System

Three different types of hoisting wire ropes (round strand, triangular strand, and compacted strand) have been selected for design calculations. Their specific performance parameters are detailed in

Table 3. Operational and structural parameters of typical hoisting system.

Parameters			Hoisting Rope
			Round Strand	Triangular Strand	Compacted Strand
Operational Parameters	Maximum operating speed, vmax/(m/s)		18
	Hoisting acceleration/deceleration, a/(m/s2)		0.74	0.61	0.75
	Operating time, T/(s)		85.9	90.7	85.7
Structural parameters	Upper rope length, Ls/(m)		62.2
	Lower rope length, Lx/(m)		49.49
	Upper rope angle, βs/(°)		53.04
	Lower rope angle, βx/(°)		54.39
	Wrapping angle of hoisting rope on friction lining, α/(rad)		181.35
	Sheave wheel diameter ratio	The calculation result of Equation (6)/(mm)	≥5580	≥5220	≥5400
		The calculation result of Equation (7)/(mm)	≥3936	≥3804	≥4400
	Friction wheel-groove spacing/(mm)		425

. The hoisting system employs a trapezoidal five-stage acceleration curve, as shown in Figure 7, Figure 8 and Figure 9. Taking Figure 7a,b as examples, the hoisting system first undergoes a trapezoidal acceleration process, and the calculation formulas for its acceleration slope can be found in Equations (47)–(51). Subsequently, the system maintains the maximum speed for uniform operation until it reaches the predetermined deceleration position. Next, the system undergoes a trapezoidal deceleration process, and the speed gradually slows down to a crawling speed (v = 0.5 m/s). After maintaining for 3 s in the crawling stage, the system enters the parking stage and finally smoothly reduces the speed to zero. The design of this five-segment acceleration curve aims to enhance the stability and efficiency of the overall operation. The maximum hoisting acceleration/deceleration of three different types of hoisting wire ropes are 0.74 m/s², 0.61 m/s², and 0.75 m/s² respectively. The maximum hoisting speed is 18 m/s and the hoisting distance is 1000 m. The running time of a single hoisting/lowering of three different types of hoisting wire ropes is 85.9 s, 90.7 s, and 85.7 s, respectively. The diameter of the sheave wheel and the spacing of the friction wheel groove are 7000 mm and 425 mm, respectively.

The propagation velocity j of the elastic wave can be calculated by Equation (47):

$$j=\sqrt{{\displaystyle \frac{EF}{P_k}}}$$

(47)

where E is the elastic modulus of the hoisting rope, F is the cross-sectional area of the hoisting rope, and P_k is the unit weight of the hoisting rope.

Calculate the ratio of the weight of the first rope to the load at the end of the rope, and construct the transcendental equation, as shown in Equation (48):

$$\begin{array}{l}\alpha ={\displaystyle \frac{N_1{P}_k{H}_c}{Q_d}}\\ \alpha ={\lambda }_1\tan {\lambda }_1\end{array}$$

(48)

where N₁ is the number of the hoisting rope, P_k is the unit weight of the hoisting rope, H_c is the overhang height of the first rope, Q_d is the load at the end of the rope, and λ₁ is the root of the transcendental equation.

Fundamental vibration frequency ω₁ can be calculated by Equation (49):

$${\omega }_1={\displaystyle \frac{j{\lambda }_1}{H_C-H_h-H_R}}$$

(49)

where j is the propagation speed of elastic wave, H_c is the overhang height of the first rope, λ₁ is the root of the transcendental equation, H_h is the height of the tail rope ring, and H_R is the height of the container.

The vibration period of elastic wave T_j1 can be calculated by Equation (50):

$$T_{j1}={\displaystyle \frac{2\pi }{{\omega }_1}}$$

(50)

where, ω₁ is the fundamental vibration frequency.

The impact limit value can be calculated by Equation (51):

$$J_1={\displaystyle \frac{a_1}{T_{j1}}}$$

(51)

where a₁ is the maximum acceleration (deceleration) speed of the hoisting system and T_j1 is the vibration period of the elastic wave.

3.2. Selection Parameters of Hoisting Wire Rope and Balancing Tail Rope

The selection parameters of hoisting wire rope and balancing tail rope are shown in

Table 4. Selection design of hoisting rope and balancing tail rope.

Design Parameter		Selection Scheme
Selection design of hoisting rope	Structure	Round strand	Triangular strand	Compacted strand
		6 × 37S + IWR	6V × 37 + FC	35W × K19S
	Number of hoisting rope, n1	6	6	6
	Minimum safety coefficient, ma specified by Regulations	6.7	6.7	6.7
	Minimum nominal diameter, dmin/(mm)	60	67.2	54
	Actual selected nominal diameter, d1/(mm)	62	58	60
	Weightlift factor, K1	0.418	0.405	0.48
	Minimum breaking force factor, K2	0.356	0.36	0.395
	Conversion factor, K3	1.321	1.177	1.287
	Nominal tensile strength, R0/(MPa)	1770	1770	1960
	Unit weight, pk/(kg/m)	16.6	13.6	17.3
	Cross-sectional area, A/(mm2)	3019	2642	2827
	Elastic modulus, ES/(GPa)	210	206	120
	Minimum breaking force, F0/(kN)	2420	2140	2787
	Calculated minimum breaking force, F0/(kN)	2422	2144	2787
	Safety coefficient, m (Calculated value)	6.9	5.8	7.6
	Load ratio = Effective lifting load/Rope weight	1.54	1.88	1.48
Selection design of balance tail rope	Number of tail rope, n2	4	4	3
	Unit weight, qk/(kg/m)	24.9	20.4	34.2
	Elastic modulus, Ew/(GPa)	120	120	120

Note: For the selection of the 6V × 37 + FC type hoisting rope, since this model does not have a large diameter of 67.2 mm, the largest diameter available in the selection table, 58 mm, is chosen as the calculation case.

.

3.3. Static Analysis and Calculation of the Hoisting System

The static calculation results of the hoisting system with different rope selection schemes are shown in

Table 5. Static calculations of different rope selection schemes in hoisting system.

Heavy-Load End Position	Calculation Parameter	Round Strand	Triangular Strand	Compacted Strand
Left cage as heavy-load end	Static tension at friction wheel cut point at heavy-load end, Fj1/(kN)	2752	2573	2793
	Static tension at friction wheel cut point at light-load end, Fj2/(kN)	1776	1596	1805
	Ratio of static tension, Fj1/Fj2	1.55	1.61	1.54
	Static anti-slip safety coefficient of the rope-lining, δ	2.19	1.97	2.20
	Pressure ratio of the rope-lining, P/(MPa)	1.74	1.71	1.82
Right cage as heavy-load end	Static tension at friction wheel cut point at heavy-load end, Fj3/(kN)	2756	2576	2797
	Static tension at friction wheel cut point at light-load end, Fj4/(kN)	1771	1593	1802
	Ratio of static tension, Fj1/Fj2	1.56	1.62	1.55
	Static anti-slip safety coefficient of the rope-lining, δ	2.17	1.96	2.18
	Pressure ratio of the rope-lining, P/(MPa)	1.74	1.71	1.82

Note: According to the Safety Regulations for Coal Mines, the static anti-slip safety coefficient of wire ropes in multi-rope friction hoisting systems must not be less than 1.75. Additionally, the ratio pressure between the wire rope and the friction lining should not exceed 2 MPa.

. The maximum static tension at the tangent point of the friction wheel for the three rope selection schemes is consistently observed at the heavy-load end when the right side is the heavy-load end, with values of 2756 kN, 2576 kN, and 2797 kN and the ratio of static tensions is 1.56, 1.62, and 1.55 respectively. The static anti-slip safety coefficient and the pressure ratio between the hoist rope and the friction lining all meet the requirements of the Safety Regulations for Coal Mines. Also, there is little difference in the calculation results when lifting heavy loads on the left and right sides, primarily due to the small vertical distance between the sheave wheels on both sides, resulting in similar calculation results of initial overhang height on both sides of the hoist rope and tail rope.

3.4. Dynamic Analysis and Calculation of the Hoisting System

3.4.1. Equivalent Mass and Limit Acceleration/Deceleration

Table 6. Dynamic calculations of hoisting system with different rope selection schemes (displacement mass and limiting acceleration/deceleration calculation).

Parameters	Position of the Heavy-Load End	Calculation Parameter	Hoisting Rope
			Round Strand	Triangular Strand	Compacted Strand
Displacement mass of hoisting system	Left cage	Maximum displacement mass at heavy-load end, m1/(t)	332	312	335
		Minimum displacement mass at light-load end, m2/(t)	229	210	234
	Right cage	Maximum displacement mass at heavy-load end, m3/(t)	329	310	334
		Minimum displacement mass at light-load end, m4/(t)	232	212	235
	Total displacement mass of hoisting system, ∑M/(t)		736	696	744
Limiting acceleration/deceleration of hoisting system	Left cage	Limiting acceleration/deceleration, amax/(m/s2)	0.94	0.78	0.95
	Right cage		0.93	0.76	0.94
	Actual acceleration/deceleration value = 0.8·Min [Left, Right amax]		0.74	0.61	0.75

shows the total equivalent mass and limit acceleration/deceleration of the hoisting system for three rope selection schemes. The total equivalent mass of the hoisting system for different rope selection schemes is 736 t, 696 t, and 744 t. The maximum equivalent masses and the minimum equivalent masses are similar when lifting heavy loads on the left and right sides, respectively. The maximum accelerations/decelerations of the hoisting system are 0.74 m/s², 0.61 m/s², and 0.75 m/s², respectively.

3.4.2. Dynamic Tension and Dynamic Anti-Slip Safety Coefficient of Hoisting Rope

The dynamic tension of the hoisting rope at different stages in different rope selection schemes is shown in Figure 10, Figure 11 and Figure 12. The dynamic anti-slip safety coefficients of the hoisting rope at different stages in different rope selection schemes is shown in Figure 13, Figure 14 and Figure 15. In Figure 10, 1 represents the stage of heavy load accelerating upward, 2 represents the stage of heavy load moving upward at a constant speed, 3 represents the stage of heavy load decelerating upward, 4 represents the stage of crawling, 5 represents the stage of parking, 6 represents the stage of heavy load accelerating downward, 7 represents the stage of heavy load moving downward at a constant speed, 8 represents the stage of heavy load decelerating downward, 9 represents the stage of crawling, 10 represents the stage of parking.

As can be seen from Figure 10, Figure 11 and Figure 12, the changing trends of the dynamic tension exerted on the hoisting wire rope at the heavy-load end in different stages for different rope selection schemes are basically consistent with the changing trend of the acceleration control curve, both exhibiting a five-stage variation. This indicates that the dynamic tension on the hoisting wire rope at the heavy-load end is positively correlated with the acceleration. The greater the acceleration, the greater the dynamic tension it undergoes. The dynamic tension of the hoisting wire rope at the light-load end has an opposite changing trend to the acceleration control curve. This is because the movement direction of the light-load end is opposite to that of the heavy-load end, resulting in an opposite changing trend of the dynamic tension. Further observations reveal that the differences in dynamic tension on both sides of pulley for three rope selection schemes are the greatest in the heavy-load accelerated hoisting stage and the heavy-load decelerated lowering stage, which are 1556.3 kN, 1448.9 kN, and 1576 kN respectively. Among them, the second scheme that adopts the fiber core wire rope has the smallest tension difference. This is attributed to the relatively light unit weight of the fiber core wire rope, which has a relatively small influence on the tension difference.

The anti-slip safety was evaluated based on the dynamic tension (Figure 13, Figure 14 and Figure 15). The dynamic anti-slip safety coefficients in each stage for three rope selection schemes all meet the minimum requirement of 1.25 stipulated in the Safety Regulations for Metal and Nonmetal Mines. It is worth noting that in the heavy-load accelerated hoisting and heavy-load decelerated lowering stages, although the dynamic anti-slip safety coefficient meets the specification standard, the anti-slip safety coefficient is relatively low due to the large tension difference on both sides of the pulley, increasing the risk of slippage. Conversely, in the heavy-load decelerated hoisting and heavy-load accelerated lowering stages, due to the opposite changing trends of the dynamic tension on both sides of the pulley, the tension difference decreases, and the dynamic anti-slip safety coefficient significantly increases to the maximum value, reducing the possibility of slippage.

Through comparison, it is found that although the second rope selection scheme performs exceptionally well in reducing the tension difference, its dynamic anti-slip safety coefficient is not significantly increased. Instead, it is similar to other schemes. This is because the dynamic anti-slip safety coefficient of 1.25 is selected when calculating the limit acceleration/deceleration. Therefore, regardless of the calculation method, the dynamic anti-slip safety coefficient under this acceleration will be greater than 1.25. Hence, future research should consider the comprehensive influence of multiple parameters such as speed, acceleration, and hoisting load on the anti-slip safety performance more comprehensively to provide more accurate evaluation and optimization strategies.

3.4.3. Elongation of Hoisting Wire Rope

The elongation of the steel wire rope at the connection between container and rope at different stages in different rope selection schemes is shown in Figure 16, Figure 17 and Figure 18.

As can be seen from Figure 16a, Figure 17a and Figure 18a, in the heavy-load lifting phase, the elongation of the hoisting ropes at the heavy-load end presents a downward trend, and the variation ranges are 0.1 mm–0.8 mm, 0.07 mm–0.86 mm, and 0.05 mm–1.68 mm, respectively. The reason is that with the lifting of heavy-load, the overhang height and dynamic tension of the hoisting rope at the heavy-load end decrease continuously, and the elongation of the rope decreases when the elastic modulus and cross-sectional area of rope are unchanged. On the contrary, the elongation of the hoisting rope at the light-load end continues to increase, and their variation ranges are 0.05 mm–0.5 mm, 0.06 mm–0.55 mm, 0.02 mm–1.04 mm, respectively.

As can be seen from Figure 16b, Figure 17b and Figure 18b, in the heavy-load lowering phase, the elongation of the hoisting ropes at the heavy-load end presents an upward trend, and the variation ranges are 0.09 mm–0.8 mm, 0.1 mm–0.76 mm, and 0.18 mm–1.53 mm, respectively. The reason is that with the lowering of heavy-load, the overhang height and dynamic tension of the hoisting rope at the heavy-load end increase continuously, and the elongation of the rope increases when the elastic modulus and cross-sectional area of rope are unchanged. On the contrary, the elongation of the hoisting rope at the light-load end continues to decrease, and the variation ranges are 0.03 mm–0.52 mm, 0.04 mm–0.54 mm, 0.07 mm–0.98 mm, respectively.

In addition, it is found that the elongation of the compacted strand wire rope is much larger than the other two kinds of steel wire rope, because its elastic modulus is smaller and it deforms easily.

4. Conclusions

In this paper, a design method for a multi-rope friction hoisting system of a vertical shaft gravity energy storage system was proposed. A complete set of calculation and selection theories about the structural parameters, the hoisting rope and the tail rope, and static and dynamic analysis of the hoisting system were proposed. In addition, the parameter calculations of the hoisting rope, balance rope, and friction wheel of the friction hoisting system under typical lifting conditions were carried out. The static and dynamic anti-slip capabilities of the friction hoisting system under the typical condition were explored. The design scheme has definite guiding significance for the design of a hoisting system for a gravity energy storage system.

• The results show that the maximum acceleration/deceleration speed of the compacted strand steel wire rope is 0.75 m/s², and the time of lifting/lowering is the shortest, which is 85.7 s. The maximum acceleration/deceleration speed of the triangular strand steel wire rope is the smallest, which is 0.61 m/s², and the time of lifting/lowering is the longest, which is 90.7 s.
• The maximum static tension at the tangent point of the friction wheel in different rope selection schemes was all at the heavy-load end when the right side was the heavy-load end, which was 2756 kN, 2576 kN, and 2797 kN, respectively, and the ratio of static tension was 1.56, 1.62, and 1.55, respectively. The static anti-slip safety coefficient between the hoisting rope and friction lining and the specific pressure between the hoisting rope and friction lining meet the requirements of the Safety Regulations for Coal Mines.
• The dynamic tension of the hoisting rope at the heavy-load end is positively correlated with the acceleration, and the maximum values are at the acceleration lifting stage and the deceleration lowering stage of the heavy load: 3252.8 kN, 3000 kN, and 3302.6 kN, respectively. At this time, the dynamic tension of the hoisting rope at the light-load end is the lowest: 1696.5 kN, 1551.1 kN, and 1726.6 kN, respectively. The minimum values appear at the deceleration lifting stage and the acceleration lowering stage of the heavy load: 2573.7 kN, 2400.6 kN, and 2614.2 kN, respectively. At this time, the dynamic tension of the hoisting rope at light-load end is the largest: 2132.8 kN, 1900.6 kN, and 2168.1 kN, respectively.
• The dynamic anti-slip safety coefficient of the hoisting system in the different rope selection schemes is greater than the minimum value of 1.25 stipulated in the Safety Regulations for Metal and Non-Metal Mines. The elastic modulus of the hoisting rope has a great influence on the elongation of the wire rope. Considering that the elongation of the wire rope will cause the cages on both sides to fail to reach the designated parking position at the same time, it is recommended that hoisting ropes with large elastic moduli be used.

In future study, we will continue to explore the transmission reliability of the hoisting system under different environments and operating parameters and investigate the impact of different lifting parameters (lifting height, lifting speed, and lifting capacity) on the energy storage efficiency of the gravity energy storage system. Finally, we will comprehensively consider the energy storage efficiency, lifting efficiency, and operating reliability of the gravity energy storage system, and come up with an optimal hoisting scheme.