Fuzzy fault tree analysis for gas explosion of coal mining and heading faces in underground coal mines

Abstract

During the past decade, gas explosions have been one of the most serious types of disasters in China, threatening the lives of miners and causing significant losses in terms of national property. This article, by constructing the fuzzy fault tree model of gas explosion on the coalface and heading face, deduces the minimum cut sets and minimum path of the fault tree, analyzes the importance of the fault tree structure, and obtains the ratio of gas explosion. The results show that the isolation of gas and heat sources is the most effective way to prevent gas explosion. In addition, a close detection of gas concentration and appropriate treatment can also avoid explosive accidents by reducing the ratio of explosion to below 0.059%, which is the critical value of explosion. The probability of gas explosion occurred in coal working face is about 0%–2.055%, and the most likely probability is 0.059%. However, the probability of gas explosion occurred in heading face is about 0%–8.543%, and the most probability likely to occur is 0.772% which is larger than that in coal working face. The fuzzy fault tree can not only be applied in the analysis of the coal mining gas explosion, but it also provides the theoretical basis for the precaution and prevention of coal mining accidents.

Introduction

In 1962, fault tree analysis (FTA) has been extensively employed as a powerful technique in risk analysis studies at Bell Telephone Laboratories in the United States. ¹ Pan and Yun ² introduced the logic gate into fuzzy theory. In this article, the “and” and “or” gates have been used to establish the fault tree. FTA is widely employed to determine system dependability. ³ Lavasani et al. ⁴ employed fuzzy theory as a solution for analyzing leakages through permanently abandoned (PA) oil and natural gas wells in the drilling industry. Zadeh ⁵ proposed fuzzy set theory to solve imprecise and fuzzy problems, and the method has undergone rapid development.

Since 2006, there have been 29 severe accidents in coal mines in China (Table 1), killing 1317 people, among which there were 14 gas explosions with 741 deaths. The numbers of accidents and deaths accounted for 48.3% and 56.3% of the total. Obviously, gas explosion is one of the most serious accidents in underground coal mines of China. On 3 December 2016, a serious gas explosion occurred in Baoma Coal Supplies Co. Ltd causing 32 deaths. The frequent occurrence of heavy gas explosions in coal mines not only causes great loss of life and property, but also affects the international reputation of China. In order to reduce and prevent the coal mine gas explosion disaster, it is important to scientifically and systematically find out the key influential factors and processes of coal mine gas explosion and take effective measures to cut off the critical segments of accidents.

Table 1.

Statistics for special major accidents occurred in coal mines in China since 2006.

Time	Location	Type of accident	No. of deaths
29 April 2006	Wayaobu coal mine in Yanan, Shanxi	Gas explosion	32
18 May 2006	Xinjing coal mine in Datong, Shanxi	Permeation	56
28 June 2006	Wulong coal mine in Fuxin, Liaoning	Gas explosion	32
5 November 2006	Jiaojiazai coal mine in Tongmei, Shanxi	Gas explosion	47
12 November 2006	Nanshan coal mine in Jinzhong, Shanxi	Gas explosion	34
25 November 2006	Changyuan coal mine in Qujing, Yunnan	Gas explosion	32
16 April 2007	Wangzhuang coal mine in Pingding, Hennan	Gas explosion	31
8 November 2007	Qunli coal mine in Bijie, Guizhou	Coal and gas outburst	35
5 December 2007	Ruiziyuan coal mine in Linfen, Shanxi	Gas explosion	105
13 June 2008	Anxin coal mine in Lvliang, Shanxi	Gunpowder explosion	35
14 July 2008	Lijiawa coal mine in Zhangjiakou, Hebei	Combustion of explosives	35
21 July 2008	Nadu coal mine in Baise, Guangxi	Permeation	36
20 September 2008	Fuhua coal mine in Hegang, Heilongjiang	Conflagration	31
21 September 2008	Xinfeng coal mine in Dengfeng, Henan	Coal and gas outburst	37
22 February 2009	Tunlan coal mine in Xishan, Shanxi	Gas explosion	78
30 May 2009	Tonghua coal mine in Songzao	Coal and gas outburst	30
8 September 2009	Xinhua coal mine in Pingdingshan, Hennan	Gas explosion	76
21 November 2009	Xinxing coal mine in Hegang, Heilongjiang	Gas explosion	108
5 January 2010	Lisheng coal mine in Xiangtan, Hunan	Cable ignition	34
1 March 2010	Luotuoshan coal mine in Nengmenggu	Permeation	32
28 March 2010	Wangjialing coal mine in Huajin, Shanxi	Permeation	38
31 March 2010	Yichuan coal mine in Luoyang, Henan	Coal and gas outburst	48
21 June 2010	Xingdong coal mine in Pingdingshan, Henan	Gunpowder explosion	49
16 October 2010	Pingyu coal mine in Zhongping, Henan	Coal and gas outburst	37
10 November 2011	Sizhuang coal mine in Qujing, Yunnan	Coal and gas outburst	43
29 August 2012	Xiaojiawan coal mine in Panzhihua, Sichuan	Gas explosion	48
29 March 2013	Babao coal mine in Tonghua, Jilin	Gas explosion	53
31 October 2016	Jinshangou coal mine in Chongqing	Gas explosion	33
3 December 2016	Baoma coal mine in Cifeng, Neimenggu	Gas explosion	32

FTA is a qualitative and quantitative method to evaluate the reliability of a complex system. FTA can be applied to both an existing system and a system in design. FTA technique has been widely used in many fields, such as nuclear power, mine explosion, electric power, man–machine robot system, railway system, oil and gas transmission, and so on.^6–11

However, the fuzzy nature and working environment of a system, and the lack of sufficient statistical analysis, all raise difficulties in the assessment of probabilities of basic events.^6,12,13 And this makes quantitative analysis questionable by conventional methods.

The uncertain situations have been taken into consideration to handle imprecise failure information in diverse real applications. Dong and Yu ⁶ used fuzzy theory to evaluate the failure probabilities of basic events. Tanaka et al. ¹⁴ applied fuzzy theory in the FTA technique for safety assessment of a certain system.^15–17

Despite many publications on FTA, few studies have been devoted to the fuzzy fault tree analysis (FFTA) technique for underground coal mine methane explosion. First, in this study the fault tree of coal working face and heading face was used to analyze the influential factors and processes of coal mine gas explosion. Then the probability of coal working face and heading face gas explosion disaster was obtained via the theory and method of fuzzy mathematics to help decision makers determine where and whether to take preventive action on methane explosions in the safety management process of underground coal mines.

Fuzzy fault tree theory and method

Triangular fuzzy number

Definition. Assuming U as a universe of objects, define fuzzy à in U as a fuzzy membership function

$$Ã\left(x\right):U\to [0,1],x\in U$$

(1)

The elements in U mapping to real numbers in [0,1] is Ã = Ã(x)/xdx. Ã(x) is the degree of element x of the domain U belonging to the fuzzy set A, calling x to the degree of membership of A for short. The larger the Ã(x) is, the stronger the degree of belonging to a is.

If the membership function of a fuzzy number is composed of a linear function, it is expressed as

$$P=\{\begin{array}{cc}\frac{x-m+a}{a},\hfill & a\le x\le m\hfill \\ \multicolumn{1}{c}{\frac{m+b-x}{b},}& m\le x\le b\hfill \\ \multicolumn{1}{c}{0,}& \mathrm{others}\hfill \end{array}$$

(2)

where m is the core of Ã(x) and a + b is the range of Ã(x), called triangular fuzzy number, marked as A(a,m,b). Figure 1 shows what this means.

Figure 1.

Triangular fuzzy numbers and the corresponding characteristic functions.

Operation rules of triangular fuzzy numbers

The arithmetic of triangular fuzzy numbers is very similar to that of real numbers. Assuming that the triangular fuzzy numbers q₁ and q₂ are (a₁,m₁,b₁) and (a₂,m₂,b₂), their algebraic algorithms are

$$q_1·q_2=(a_1,m_1,b_1)·(a_2,m_2,b_2)=(a_1{a}_2,m_1{m}_2,b_1{b}_2)$$

(3)

$$q_1+q_2=\left(a_1,m_1,b_1\right)+\left(a_2,m_2,b_2\right)=\left(a_1+a_2,m_1+m_2,b_1+b_2\right)$$

(4)

$$C{q}_1=C(a_1,m_1,b_1)=(C{a}_1,C{m}_1,C{b}_1)$$

(5)

FFTA fuzzy logic gate operation

The calculation formula of fault tree and gate is

$$q=\underset{i=1}{\overset{n}{\Pi }}{q}_i$$

(6)

Likewise, the formula “fuzzy and door” is

$$q=q_1\otimes q_2\otimes \cdots \otimes q_n=(a_1\times \cdots \times a_n,m_1\times \cdots \times m_n,b_1\times \cdots \times b_n)=(\underset{i=1}{\overset{n}{\Pi }}{a}_i,\underset{i=1}{\overset{n}{\Pi }}{m}_i,\underset{i=1}{\overset{n}{\Pi }}{b}_i)$$

(7)

Similarly, the accident tree “or” formula is

$$q=1-\underset{i=1}{\overset{n}{\Pi }}(1-q_i)$$

(8)

Likewise, the formula “fuzzy or door” is

$$q=[1-\underset{i=1}{\overset{n}{\Pi }}(1-a_i),1-\underset{i=1}{\overset{n}{\Pi }}(1-m_i),1-\underset{i=1}{\overset{n}{\Pi }}(1-b_i)]$$

(9)

where $q_i$ represents the occurring probability of the event x_i in the accident chain model (i = 1, 2, 3, …, n).

Representation method of fuzzy number 3σ

In this method, σ represents the standard deviation of all probability values of the expert decision group. When a large number of small-probability events affect one thing, we often treat the event as normal distribution. From the characteristics of gas explosion in this discussion, the estimated probability value is approximate to the normal distribution. According to the small probability of normal distribution, the probability of mean value number m ranging from m – 3σ to m + 3σ is greater than 95%. We can treat the distribution function of the normal distribution as the membership function of probability

$$f(x)=\frac{1}{\sqrt{2\pi \sigma }}\exp [-\frac{{(x-\mu )}^2}{2{\sigma }^2}]$$

(10)

At the same time, assume a = b = 3σ. Then the triangular fuzzy function transforms into (a,m,b) → (3σ,m,3σ). To facilitate the calculation, some formulas of the values about the normal distribution are represented

$$E(x)=\mu =m=\frac{1}{n}(a_1+a_2+\cdots +a_n)$$

(11)

$$\sigma =\sqrt{\frac{\sum _{i=1}^n{(x_i-E(x))}^2}{n}}$$

(12)

FTA of gas explosion

FTA of gas explosion in coal working face

The coal face gas explosion fault tree is shown in Figure 2.

Figure 2.

Gas explosion fault tree in coal working face.

Solution of the minimum path sets

The number of minimum cut sets is 200, but the number of minimum path sets is only 8. So the minimum path set is used to analyze as follows

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{T}\prime ={\mathrm{M}\prime }_1+{\mathrm{X}\prime }_1+{\mathrm{M}\prime }_2+{\mathrm{X}\prime }_{25}}\\ \multicolumn{1}{c}{={\mathrm{M}\prime }_3+{\mathrm{M}\prime }_4+{\mathrm{M}\prime }_5+{\mathrm{X}\prime }_1+{\mathrm{M}\prime }_6{\mathrm{X}\prime }_2{\mathrm{X}\prime }_3{\mathrm{X}\prime }_4{\mathrm{M}\prime }_7+{\mathrm{X}\prime }_{25}}\\ \multicolumn{1}{c}{={\mathrm{X}\prime }_{19}{\mathrm{X}\prime }_{20}{\mathrm{X}\prime }_{21}{\mathrm{X}\prime }_5({\mathrm{X}\prime }_{22}+{\mathrm{X}\prime }_{23}+{\mathrm{X}\prime }_{24})+{\mathrm{X}\prime }_6{\mathrm{X}\prime }_7{\mathrm{X}\prime }_8{\mathrm{X}\prime }_9+{\mathrm{X}\prime }_{10}{\mathrm{X}\prime }_{11}+{\mathrm{X}\prime }_1}\\ \multicolumn{1}{c}{+{\mathrm{X}\prime }_{12}{\mathrm{X}\prime }_{13}{\mathrm{X}\prime }_{14}{\mathrm{X}\prime }_2{\mathrm{X}\prime }_3{\mathrm{X}\prime }_{15}{\mathrm{X}\prime }_{16}{\mathrm{X}\prime }_{17}{\mathrm{X}\prime }_{18}{\mathrm{X}\prime }_4+{\mathrm{X}\prime }_{25}}\\ \multicolumn{1}{c}{={\mathrm{X}\prime }_{19}{\mathrm{X}\prime }_{20}{\mathrm{X}\prime }_{21}{\mathrm{X}\prime }_5{\mathrm{X}\prime }_{22}+{\mathrm{X}\prime }_{19}{\mathrm{X}\prime }_{20}{\mathrm{X}\prime }_{21}{\mathrm{X}\prime }_5{\mathrm{X}\prime }_{23}+{\mathrm{X}\prime }_{19}{\mathrm{X}\prime }_{20}{\mathrm{X}\prime }_{21}{\mathrm{X}\prime }_5{\mathrm{X}\prime }_{24}}\\ \multicolumn{1}{c}{+{\mathrm{X}\prime }_6{\mathrm{X}\prime }_7{\mathrm{X}\prime }_8{\mathrm{X}\prime }_9+{\mathrm{X}\prime }_{10}{\mathrm{X}\prime }_{11}+{\mathrm{X}\prime }_1+{\mathrm{X}\prime }_{12}{\mathrm{X}\prime }_{13}{\mathrm{X}\prime }_{14}{\mathrm{X}\prime }_2{\mathrm{X}\prime }_3{\mathrm{X}\prime }_{15}{\mathrm{X}\prime }_{16}{\mathrm{X}\prime }_{17}{\mathrm{X}\prime }_{18}{\mathrm{X}\prime }_4+{\mathrm{X}\prime }_{25}}\end{array}$$

The combination of the minimum path sets is as follows

$$\begin{array}{c}{\mathrm{P}}_1=\{{\mathrm{X}}_2,{\mathrm{X}}_3,{\mathrm{X}}_4,{\mathrm{X}}_{12},{\mathrm{X}}_{13},{\mathrm{X}}_{14},{\mathrm{X}}_{15},{\mathrm{X}}_{16},{\mathrm{X}}_{17},{\mathrm{X}}_{18}\}\hfill \\ {\mathrm{P}}_2=\{{\mathrm{X}}_6,{\mathrm{X}}_7,{\mathrm{X}}_8,{\mathrm{X}}_9\}\hfill \\ {\mathrm{P}}_3=\{{\mathrm{X}}_{10},{\mathrm{X}}_{11}\}\hfill \\ {\mathrm{P}}_4=\{{\mathrm{X}}_5,{\mathrm{X}}_{19},{\mathrm{X}}_{20},{\mathrm{X}}_{21},{\mathrm{X}}_{22}\}\hfill \\ {\mathrm{P}}_5=\{{\mathrm{X}}_5,{\mathrm{X}}_{19},{\mathrm{X}}_{20},{\mathrm{X}}_{21},{\mathrm{X}}_{23}\}\hfill \\ {\mathrm{P}}_6=\{{\mathrm{X}}_5,{\mathrm{X}}_{19},{\mathrm{X}}_{20},{\mathrm{X}}_{21},{\mathrm{X}}_{24}\}\hfill \\ {\mathrm{P}}_7=\left\{{\mathrm{X}}_1\right\}\hfill \\ {\mathrm{P}}_8=\left\{{\mathrm{X}}_{25}\right\}\hfill \end{array}$$

Structural importance degree

$$\begin{array}{c}{\mathrm{I}}_{\Phi }\left(1\right)={\mathrm{I}}_{\Phi }\left(25\right)=1\hfill \\ {\mathrm{I}}_{\Phi }\left(2\right)={\mathrm{I}}_{\Phi }\left(3\right)={\mathrm{I}}_{\Phi }\left(4\right)={\mathrm{I}}_{\Phi }\left(12\right)={\mathrm{I}}_{\Phi }\left(13\right)={\mathrm{I}}_{\Phi }\left(14\right)\hfill \\ \multicolumn{1}{c}{={\mathrm{I}}_{\Phi }\left(15\right)={\mathrm{I}}_{\Phi }\left(16\right)={\mathrm{I}}_{\Phi }\left(17\right)}\\ ={\mathrm{I}}_{\Phi }\left(18\right)=1-(1-\frac{1}{2^{10-1}})=0.001953125\hfill \\ {\mathrm{I}}_{\Phi }\left(5\right)={\mathrm{I}}_{\Phi }\left(19\right)={\mathrm{I}}_{\Phi }\left(20\right)={\mathrm{I}}_{\Phi }\left(21\right)=1-{(1-\frac{1}{2^{5-1}})}^3\hfill \\ \multicolumn{1}{c}{=0.1760254}\\ {\mathrm{I}}_{\Phi }\left(6\right)={\mathrm{I}}_{\Phi }\left(7\right)={\mathrm{I}}_{\Phi }\left(8\right)={\mathrm{I}}_{\Phi }\left(9\right)=1-(1-\frac{1}{2^{4-1}})=0.125\hfill \\ {\mathrm{I}}_{\Phi }\left(10\right)={\mathrm{I}}_{\Phi }\left(11\right)=1-(1-\frac{1}{2^{2-1}})=0.5\hfill \\ {\mathrm{I}}_{\Phi }\left(22\right)={\mathrm{I}}_{\Phi }\left(23\right)={\mathrm{I}}_{\Phi }\left(24\right)=1-(1-\frac{1}{2^{5-1}})=0.0643\hfill \end{array}$$

The magnitude of the structure is arranged as follows

$$\begin{array}{c}{\mathrm{I}}_{\Phi }\left(1\right)={\mathrm{I}}_{\Phi }\left(25\right)>{\mathrm{I}}_{\Phi }\left(10\right)={\mathrm{I}}_{\Phi }\left(11\right)>{\mathrm{I}}_{\Phi }\left(5\right)={\mathrm{I}}_{\Phi }\left(19\right)\hfill \\ \multicolumn{1}{c}{={\mathrm{I}}_{\Phi }\left(20\right)={\mathrm{I}}_{\Phi }\left(21\right)>{\mathrm{I}}_{\Phi }\left(6\right)={\mathrm{I}}_{\Phi }\left(7\right)={\mathrm{I}}_{\Phi }\left(8\right)}\\ \multicolumn{1}{c}{={\mathrm{I}}_{\Phi }\left(9\right)>{\mathrm{I}}_{\Phi }\left(22\right)}\\ ={\mathrm{I}}_{\Phi }\left(23\right)={\mathrm{I}}_{\Phi }\left(24\right)>{\mathrm{I}}_{\Phi }\left(2\right)={\mathrm{I}}_{\Phi }\left(3\right)={\mathrm{I}}_{\Phi }\left(4\right)={\mathrm{I}}_{\Phi }\left(12\right)\hfill \\ \multicolumn{1}{c}{={\mathrm{I}}_{\Phi }\left(13\right)={\mathrm{I}}_{\Phi }\left(14\right)={\mathrm{I}}_{\Phi }\left(15\right)={\mathrm{I}}_{\Phi }\left(16\right)}\\ \multicolumn{1}{c}{={\mathrm{I}}_{\Phi }\left(17\right)={\mathrm{I}}_{\Phi }\left(18\right)}\end{array}$$

FTA

1. There are 200 sets of minimum cut sets of fault trees, which shows that there are 200 ways of gas explosion accidents in coal face. There are eight sets of fault tree with minimum path sets, which show that there are eight ways to prevent from gas explosion in coal face.

2. It can be seen from the analysis of eight sets of minimum diameter path sets that the basic events in P₆ or P₈ are the minimum, but there is no way to control without oxygen, so we can control the gas accumulation, not to meet the fire source, and deal with the gas in time. The number of minimum cut sets is 200, which is a high quantity, and that of minimum path sets is only 8; therefore, minimum path sets are utilized for the analysis.

FTA of gas explosion in heading face

This study uses the FTA method to analyze the causes of gas explosion in heading faces and prevention measures for gas explosion accidents, in order to prevent accidents from occurring again.

The diagram of gas explosion fault tree in heading face is shown in Figure 3.

Figure 3.

Gas explosion fault tree in heading faces.

The number of minimum cut sets is 320, which is more, and that of minimum path sets is only 6; therefore, minimum path sets are used for the analysis.

The solutions of the minimum path sets are as follows

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{T}\prime ={\mathrm{M}\prime }_1+{\mathrm{X}\prime }_1+{\mathrm{M}\prime }_2+{\mathrm{X}\prime }_{22}}\\ \multicolumn{1}{c}{={\mathrm{M}\prime }_3+{\mathrm{M}\prime }_4+{\mathrm{M}\prime }_5+{\mathrm{X}\prime }_1+{\mathrm{M}\prime }_6{\mathrm{X}\prime }_2{\mathrm{X}\prime }_3{\mathrm{X}\prime }_4{\mathrm{M}\prime }_7+{\mathrm{X}\prime }_{22}}\\ \multicolumn{1}{c}{={\mathrm{X}\prime }_{19}{\mathrm{X}\prime }_{20}{\mathrm{X}\prime }_{21}{\mathrm{X}\prime }_5+{\mathrm{X}\prime }_6{\mathrm{X}\prime }_7{\mathrm{X}\prime }_8{\mathrm{X}\prime }_9+{\mathrm{X}\prime }_{10}{\mathrm{X}\prime }_{11}}\\ \multicolumn{1}{c}{+{\mathrm{X}\prime }_1+{\mathrm{X}\prime }_{12}{\mathrm{X}\prime }_{13}{\mathrm{X}\prime }_{14}{\mathrm{X}\prime }_2{\mathrm{X}\prime }_3{\mathrm{X}\prime }_{15}{\mathrm{X}\prime }_{16}{\mathrm{X}\prime }_{17}}\\ \multicolumn{1}{c}{{\mathrm{X}\prime }_{18}{\mathrm{X}\prime }_4+{\mathrm{X}\prime }_{22}}\end{array}$$

The combination of the minimum path sets is

$$\begin{array}{c}{\mathrm{P}}_1=\{{\mathrm{X}}_{19},{\mathrm{X}}_{20},{\mathrm{X}}_{21},{\mathrm{X}}_5\}\hfill \\ \begin{array}{c}{\mathrm{P}}_2=\{{\mathrm{X}}_6,{\mathrm{X}}_7,{\mathrm{X}}_8,{\mathrm{X}}_9\}\hfill \\ \multicolumn{1}{c}{{\mathrm{P}}_3=\{{\mathrm{X}}_{10},{\mathrm{X}}_{11}\}}\\ \multicolumn{1}{c}{{\mathrm{P}}_4=\left\{{\mathrm{X}}_1\right\}}\\ \multicolumn{1}{c}{{\mathrm{P}}_5=\{{\mathrm{X}}_{12},{\mathrm{X}}_{13},{\mathrm{X}}_{14},{\mathrm{X}}_2,{\mathrm{X}}_3,{\mathrm{X}}_{15},{\mathrm{X}}_{16},{\mathrm{X}}_{17},{\mathrm{X}}_{18},{\mathrm{X}}_4\}}\\ \multicolumn{1}{c}{{\mathrm{P}}_6=\left\{{\mathrm{X}}_{22}\right\}}\end{array}\hfill \end{array}$$

The magnitude of the structure is

$$\begin{array}{c}{\mathrm{I}}_{\Phi }\left(1\right)={\mathrm{I}}_{\Phi }\left(22\right)=1\hfill \\ {\mathrm{I}}_{\Phi }\left(2\right)={\mathrm{I}}_{\Phi }\left(3\right)={\mathrm{I}}_{\Phi }\left(4\right)={\mathrm{I}}_{\Phi }\left(12\right)={\mathrm{I}}_{\Phi }\left(13\right)\hfill \\ \multicolumn{1}{c}{={\mathrm{I}}_{\Phi }\left(14\right)={\mathrm{I}}_{\Phi }\left(15\right)={\mathrm{I}}_{\Phi }\left(16\right)={\mathrm{I}}_{\Phi }\left(17\right)}\\ ={\mathrm{I}}_{\Phi }\left(18\right)=1-(1-\frac{1}{2^{10-1}})=0.001953125\hfill \\ {\mathrm{I}}_{\Phi }\left(5\right)={\mathrm{I}}_{\Phi }\left(19\right)={\mathrm{I}}_{\Phi }\left(20\right)={\mathrm{I}}_{\Phi }\left(21\right)=1-(1-\frac{1}{2^{5-1}})\hfill \\ \multicolumn{1}{c}{=0.0643}\\ {\mathrm{I}}_{\Phi }\left(6\right)={\mathrm{I}}_{\Phi }\left(7\right)={\mathrm{I}}_{\Phi }\left(8\right)={\mathrm{I}}_{\Phi }\left(9\right)=1-(1-\frac{1}{2^{4-1}})=0.125\hfill \\ {\mathrm{I}}_{\Phi }\left(10\right)={\mathrm{I}}_{\Phi }\left(11\right)=1-(1-\frac{1}{2^{2-1}})=0.5\hfill \end{array}$$

The magnitude of the structure is arranged as follows

$$\begin{array}{c}{\mathrm{I}}_{\Phi }\left(1\right)={\mathrm{I}}_{\Phi }\left(22\right)>{\mathrm{I}}_{\Phi }\left(10\right)={\mathrm{I}}_{\Phi }\left(11\right)>{\mathrm{I}}_{\Phi }\left(6\right)={\mathrm{I}}_{\Phi }\left(7\right)\hfill \\ \multicolumn{1}{c}{={\mathrm{I}}_{\Phi }\left(8\right)={\mathrm{I}}_{\Phi }\left(9\right)>{\mathrm{I}}_{\Phi }\left(5\right)={\mathrm{I}}_{\Phi }\left(19\right)={\mathrm{I}}_{\Phi }\left(20\right)}\\ ={\mathrm{I}}_{\Phi }\left(21\right)>{\mathrm{I}}_{\Phi }\left(2\right)={\mathrm{I}}_{\Phi }\left(3\right)={\mathrm{I}}_{\Phi }\left(4\right)={\mathrm{I}}_{\Phi }\left(12\right)={\mathrm{I}}_{\Phi }\left(13\right)\hfill \\ \multicolumn{1}{c}{={\mathrm{I}}_{\Phi }\left(14\right)={\mathrm{I}}_{\Phi }\left(15\right)={\mathrm{I}}_{\Phi }\left(16\right)={\mathrm{I}}_{\Phi }\left(17\right)={\mathrm{I}}_{\Phi }\left(18\right)}\end{array}$$

There are 320 sets of minimum cut sets, which shows that there are 320 ways of top events. There are six sets of minimum path sets of the fault tree, the basic events of any set of minimum path sets do not occur, and top events cannot occur. So this system has six ways to control top events.

It can be seen from the analysis of minimum path sets of six sets that the basic events in P₄ and P₆ are minimum, followed by the basic events of P₃. But oxygen cannot be controlled, so the first choice is to select P₆ as the primary way to control top events, that is, to isolate gas and ignition sources. Second, P₃ can be chosen as the way to control the top events, and the timely discovery of gas overrunning and proper treatment can also avoid the occurrence of gas explosion.

FFTA of gas explosion

FFTA of gas explosion in coal working face

The expert scoring method is used to analyze the risk of gas explosion. Because the expert scoring is influenced by subjective consciousness, the score given is fuzzy; therefore, triangular fuzzy number should be used to transform it. Using the 3σ characterization method, in triangular fuzzy numbers (a,m,b), m is said to be the most possible value, which is the value of the average number of four expert scores, a and b represent the probability range, and the fuzzy probability of basic events is (3σ,m,3σ). Expert scoring is shown in Table 2.

Table 2.

Probability estimation values of basic events of the fault tree for gas explosions in coal working face.

Accident	Expert 1	Expert 2	Expert 3	Expert 4	m	σ
X2	0.046	0.001	0.046	0.091	0.046	0.037
X3	0.054	0.041	0.054	0.067	0.054	0.011
X4	0.041	0.011	0.041	0.071	0.041	0.024
X5	0.049	0.016	0.049	0.081	0.049	0.027
X6	0.375	0.001	0.251	0.749	0.344	0.312
X7	0.061	0.024	0.061	0.098	0.061	0.030
X8	0.044	0.005	0.044	0.083	0.044	0.032
X9	0.044	0.005	0.044	0.083	0.044	0.032
X10	0.101	0.101	0.101	0.101	0.101	0.000
X11	0.307	0.001	0.111	0.613	0.258	0.268
X12	0.099	0.054	0.099	0.143	0.099	0.036
X13	0.051	0.020	0.046	0.083	0.050	0.026
X14	0.077	0.071	0.046	0.083	0.069	0.016
X15	0.044	0.009	0.044	0.078	0.044	0.028
X16	0.859	0.020	0.059	0.098	0.059	0.032
X17	0.037	0.005	0.037	0.070	0.037	0.027
X18	0.044	0.019	0.044	0.068	0.044	0.020
X19	0.034	0.026	0.011	0.041	0.028	0.013
X20	0.055	0.026	0.055	0.084	0.055	0.024
X21	0.039	0.001	0.034	0.078	0.048	0.032
X22	0.084	0.022	0.084	0.145	0.084	0.050
X23	0.081	0.075	0.082	0.104	0.086	0.011
X24	0.126	0.028	0.126	0.224	0.126	0.080

After statistical processing, the fuzzy probability of each basic event is shown in Table 3.

Table 3.

Fuzzy probability values of basic events of the fault tree for gas explosions in coal working face.

Basicevent	Fuzzy probability	Basicevent	Fuzzyprobability
X2	(0.11,0.046,0.11)	X14	(0.049,0.069,0.049)
X3	(0.032,0.054,0.032)	X15	(0.085,0.044,0.085)
X4	(0.073,0.041,0.073)	X16	(0.096,0.059,0.096)
X5	(0.08,0.049,0.08)	X17	(0.08,0.037,0.08)
X6	(0.935,0.344,0.935)	X18	(0.085,0.044,0.085)
X7	(0.091,0.061,0.091)	X19	(0.039,0.028,0.039)
X8	(0.096,0.044,0.096)	X20	(0.071,0.055,0.071)
X9	(0.096,0.044,0.096)	X21	(0.095,0.038,0.095)
X10	(0,0.101,0)	X22	(0.151,0.084,0.151)
X11	(0.805,0.258,0.805)	X23	(0.033,0.086,0.033)
X12	(0.109,0.099,0.109)	X24	(0.24,0.126,0.24)
X13	(0.078,0.050,0.078)

Because there are a small number of minimum path sets with a large number of minimum cut sets, the minimum path set method is adopted

$$\begin{array}{c}\multicolumn{1}{c}{Q_T=(Q_l,Q_m,Q_u)}\\ \multicolumn{1}{c}{Q_l=[1-(1-l_2)(1-l_3)(1-l_4)(1-l_{12})(1-l_{13})}\\ \multicolumn{1}{c}{\cdots (1-l_{18})][1-(1-l_6)(1-l_7)(1-l_8)(1-l_9)]}\\ \multicolumn{1}{c}{[1-(1-l_{10})(1-l_{11})][1-(1-l_5)(1-l_{19})(1-l_{20})}\\ \multicolumn{1}{c}{(1-l_{21})(1-l_{22})]}\\ \multicolumn{1}{c}{[1-(1-l_5)(1-l_{19})(1-l_{20})(1-l_{21})(1-l_{23})]}\\ \multicolumn{1}{c}{[1-(1-l_5)(1-l_{19})(1-l_{20})(1-l_{21})(1-l_{24})]}\end{array}$$

The calculation result is Q_l = 0.02055

$$\begin{array}{c}{Q}_m=[1-(1-m_1)][1-(1-m_2)(1-m_3)(1-m_4)\hfill \\ \multicolumn{1}{c}{(1-m_{12})(1-m_{13})\cdots (1-m_{18})]}\\ [1-(1-m_6)(1-m_7)(1-m_8)(1-m_9)]\hfill \\ \multicolumn{1}{c}{[1-(1-m_{10})(1-m_{11})]}\\ [1-(1-m_5)(1-m_{19})(1-m_{20})(1-m_{21})(1-m_{22})]\hfill \\ [1-(1-m_5)(1-m_{19})(1-m_{20})(1-m_{21})(1-m_{23})]\hfill \\ [1-(1-m_5)(1-m_{19})(1-m_{20})(1-m_{21})(1-m_{24})]\hfill \\ \multicolumn{1}{c}{[1-(1-m_{25})]}\end{array}$$

The calculation result is Q_m = 0.00059

$$\begin{array}{c}{Q}_u=[1-(1-u_1)][1-(1-u_2)(1-u_3)(1-u_4)\hfill \\ \multicolumn{1}{c}{(1-u_{12})(1-u_{13})\cdots (1-u_{18})]}\\ [1-(1-u_6)(1-u_7)(1-u_8)(1-u_9)]\hfill \\ \multicolumn{1}{c}{[1-(1-u_{10})(1-u_{11})]}\\ [1-(1-u_5)(1-u_{19})(1-u_{20})(1-u_{21})(1-u_{22})]\hfill \\ \multicolumn{1}{c}{[1-(1-u_5)(1-u_{19})(1-u_{20})(1-u_{21})(1-u_{23})]}\\ [1-(1-u_5)(1-u_{19})(1-u_{20})(1-u_{21})(1-u_{24})]\hfill \\ \multicolumn{1}{c}{[1-(1-u_{25})]}\end{array}$$

The calculation result is Q_u = 0.02055.

Through the above calculation, we can obtain the values of Q_l, Q_m, and Q_u. Then using Q_l, Q_m, and Q_u, respectively, the probability of gas explosion in coal face (0.02055,0.00059,0.02055) can be calculated. Applying the theory of fuzzy mathematics, we can get the probability of the accident as 0%–2.055%, and the most likely probability to occur is 0.059%. Generally speaking, the possibility of gas explosion in the mine is relatively low, but a lot of risk factors that may cause gas explosion accidents make it difficult to manage and control, which should arouse the attention of the relevant departments.

FFTA of gas explosion in heading face

With the above fault tree, the expert scoring is shown in Table 4.

Table 4.

Probability estimation values of basic events of the fault tree for gas explosions in heading face.

Event	Expert 1	Expert 2	Expert 3	Expert 4	m	σ
X2	0.046	0.001	0.046	0.091	0.046	0.037
X3	0.054	0.041	0.054	0.067	0.054	0.011
X4	0.041	0.011	0.041	0.071	0.041	0.024
X5	0.049	0.016	0.049	0.081	0.049	0.027
X6	0.375	0.001	0.251	0.749	0.344	0.312
X7	0.061	0.024	0.061	0.098	0.061	0.030
X8	0.044	0.005	0.044	0.083	0.044	0.032
X9	0.044	0.005	0.044	0.083	0.044	0.032
X10	0.101	0.101	0.101	0.101	0.101	0.000
X11	0.307	0.001	0.111	0.613	0.258	0.268
X12	0.099	0.054	0.099	0.143	0.099	0.036
X13	0.051	0.020	0.046	0.083	0.050	0.026
X14	0.077	0.071	0.046	0.083	0.069	0.016
X15	0.044	0.009	0.044	0.078	0.044	0.028
X16	0.859	0.020	0.059	0.098	0.059	0.032
X17	0.037	0.005	0.037	0.070	0.037	0.027
X18	0.044	0.019	0.044	0.068	0.044	0.020
X19	0.034	0.026	0.011	0.041	0.028	0.013
X20	0.055	0.026	0.055	0.084	0.055	0.024
X21	0.039	0.001	0.034	0.078	0.048	0.032

In the same way, we can obtain the probability of the accident in heading face as 0%–8.543%, and the most likely probability to occur is 0.772%. It can be seen that the probability of gas explosion occurred in heading face is larger than that in coal working face.

Analyses

1. There are 200 minimum cut sets of gas explosion accidents in coal face and 8 sets of minimum diameter sets. The most effective way is to prevent the gas accumulation from encountering the ignition source and to extract gas in time. Based on the theory of fuzzy mathematics, it can be concluded that the probability of occurrence ranges from 0% to 2.055% and the most likely probability is 0.059%.

2. There are 320 minimum cut sets of gas explosion accidents in heading face and 6 minimum sets. Isolating gas and ignition sources is a way to control heading face gas explosion occurrence; if we handle gas gauge timely and properly, we can also avoid gas explosion. The probability of gas explosion accident in heading face is 0%–8.543%, and the most likely probability to occur is 0.772%. The probability of gas explosion occurred in heading face is larger than that in coal working face.

Conclusion

By comparing the probability of gas explosion in coal face and heading face, it can be seen that the safety management department needs to pay more attention to the safety inspection work and the key factors should be examined daily or weekly in heading face to prevent gas explosion. If necessary, experts should be invited to conduct regular assessment to the working environment and the risk factors. At the same time, risk factors are required to be assessed periodically in coal face. Fuzzy fault tree method can be used well in underground coal mine gas explosion accident analysis, aiming at providing the foundation for coal enterprises to prevent gas explosion accidents.