Fuzzy Reliability Based Bow Tie Analysis of Coal Mine Water Inrush Risks

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1 Proceedigs of the 8 th Iteratioal Coferece o Sustaiable Developmet i the Mierals Idustry press@camdemia.ca Editors: Z.X. Li, Z. Agioutatis ad D.H. Zou ISB: SDIMI 207: Fuzzy Reliability Based Bow Tie Aalysis of Coal Mie Water Irush Risks Cog Zhag, Huayig Wag * ad ixi Wei Departmet of Mieral Resources Egieerig, Uiversity of Sciece ad Techology Beijig, Beijig, Chia Abstract: Water irush is a serious geological accidet i udergroud coal mies. This paper focuses o the method of the bow tie aalysis based o fuzzy reliability to assess the risk of water irush i coal mies. First, post failure probability ad repair probability of iitiatig evets are divided ito 5 levels. The triagular fuzzy umbers are adopted to quatify experts laguage ad to acquire the reliability rate which close to the actual values by defuzzificatio ad calculatio. Accordig to dyamic bow-tie aalysis model, usig the method of momets to fit the reliability time of evets by Weibull distributio ad offerig the radom samplig rule of failure ad repair time for illustratig the feasibility of fuzzy reliability evaluatio. To demostrate the effectiveess of the proposed assessmet approach, bow tie aalysis model of water irush i coal mies is take as a example. Keywords: water irush i coal mies, bow tie aalysis, fuzzy reliability, Weibull distributio. Itroductio Dyamic bow tie (DBT aalysis is composed of dyamic fault tree ad dyamic evet tree aalysis, ad cosiders the impact of the time factor upo risk assessmet of a system. There are may algorithms applied to calculatio of the probabilities of ode accidet ad accidet cosequece of dyamic bow tie aalysis: Hidde Markov Model, Bayesia etwork (Khakzad et al 203, Itegral Algorithm (Amari et al 2003 ad Mote Carlo Simulatio (Kim et al 206. I terms of simulated time ad practicability, the Mote Carlo Simulatio has a advatage over other methods. DBT requires mass data i quatitative aalysis, icludig the failure rates ad the repair rates of the basic evets of the Fault Tree (FT ad the subsequet evets of the Evet Tree (ET at differet time poits, i.e., the distributio of failure time ad repair time. I practical aalysis, it is hard to obtai a large amout of exact historical statistical data, which is a major problem of risk assessmet. Due to the difficulty of obtaiig the exact data ad historical failure data of the basic evets ad the subsequet evets, the FT ad ET aalysis is stuck with the study of miimal cut sets, miimal diameter sets, structural importace ad qualitative aalysis (Liu ad Zhag 200, a et al 202, Wag et al 2009, Li ad Wag 20. This paper will use fuzzy umbers ad the defuzzificatio method to trasform the experts laguage ito the failure rates ad repair rates of basic evets ad subsequet evets, solvig the problem that it is difficult to obtai a large amout of historical statistical data, ad will use the method of momet to coduct Weibull fittig of the reliability time to provide the radom samplig criteria of evet failure ad repair time i Mote Carlo simulatio. 2. Fuzzy Reliability Calculatio System reliability is defied as the ability of the product to perform its iteded fuctio uder the specified coditios ad time. This ability is assessed by the failure or repair situatio. For example, failure time ad repair time are itroduced to reliability evaluatio of a power grid (Zhao et al 203. This paper uses failure rate, failure time, repair rate ad repair time to describe the reliability, ad uses triagular fuzzy umbers to represet probability values of occurrece of the evets which do ot allow the acquisitio of the exact historical data. 2. Establishmet of reliability evaluatio level To assess the basic ad subsequet evets of risks, experieced experts are eeded. However, it is hard for eve the experts to give accurate mathematical represetatio for failure evets ad repair rate. The laguage of the experts' subjective descriptio is fuzzy, like very low ad higher. The actual reliability probabilities reflected by the accidet fuzzy reliability evaluatio laguages of differet idustries are ot the same. For istace, the failure rate of a coal mie water irush accidet ad related evets is lower tha accidets (like electrical appliace accidets of other idustries. The probability rage is differet i differet types of accidets. I the uclear idustry, very low ca represet the failure rate which is smaller tha 0-8. The triagular fuzzy umbers are represeted by three parameters, ad the membership fuctio thereof is: The triagular fuzzy umbers is represeted by three parameters A (a,c,b, ad membership fuctio is: ( x a / ( c a ( a x c A(x ( b x / ( b c (c x b 0 otherwise ( * Correspodig Author: H.. Wag, @63.com, phoe: Copyright 207 Caamaple Academia Services, DOI: /gree

2 SDIMI 207: Authors: C. Zhag et al where A is the fuzzy set i the domai, A(x is the membership fuctio of x to the fuzzy set A. I order to facilitate expert evaluatio, the fuzzy laguage is divided ito 5 levels: Very Low, Low, Medium, High ad Very High. Each level has o clear boudaries. The failure rate ad the repair rate of evets are less likely to be Very High ad Very Low, followed by High ad Low, ad are most likely to be Medium. First of all, the triagular fuzzy umbers of Medium are determied as (0.3, 0.5, 0.7. The, based o symmetry ad the priciple that the iterval width decreases i sequece, other levels of fuzzy umbers are determied. The fialized triagular fuzzy umbers are as follows: very high x x 0.00,0.05,0.0 ( , 0.23, ,0.50, , 0.77, 0.92 x x low (3 x x medium (4 high x x (5 very high x 5 x 0.90,0.95,.00 (6 Suppose m evets eed to be assessed by experts i a bow tie aalysis model, the m* matrix is as follows: A (7 2 m m m Where is the triagular fuzzy umbers correspodig to the evaluative laguage of a first evet by a first expert. For example, if the evaluative laguage is low, 2 ( x (0.08, 0.23, Calculatio of fuzzy reliability probability Accordig to the experts' fuzzy evaluative laguage ad the correspodig triagular fuzzy umbers, defuzzificatio is used to calculate the score that best represets the fuzzy laguage. The fuzzy score is trasformed ito the reliability probability that is close to the true value, usig the empirical formula Defuzzificatio fuzzy evaluatio laguage There are a umber of defuzzificatio methods for triagular fuzzy umbers, icludig: max-average, ceter of area ad so o. This paper adopts the cetroid poit defuzzificatio method, wherei the fuzzy evaluatio value is r ( a b c. Cetroid poit defuzzificatio is coducted 3 o (7 to lead to the score R of m evets by experts. 2 2 r r r r( r( r( 2 2 r2 r2 r2 r( 2 r( 2 r( 2 R 2 2 rm rm rm r( m r( m r( m r Where deotes the fuzzy evaluatio value of a first evet r( by a first expert, ad is the defuzzificatio fuctio of the fuzzy evaluatio value of the first evet by the first expert. ( Calculatio of actual probability of reliability The fuzzy value (0~ oly deotes the compariso value i the rage of evet failure rate or repair rate, ad is ot the real reliability probability value. r 0 y p 0 (9 r 0 0 where /3 r y 2.30 (0 r ad p is the actual probability value of the evet reliability, r is the fuzzy evaluatio value. From Eq. (-9 we fid the actual reliability probability close to the real value. The fuzzy evaluatio value is used to trasform Eq. (- 9 ito the actual evet failure rate ad repair rate, ad the matrix obtaied is as follows: 2 2 p p p p(r p(r p(r 2 2 p2 p2 p2 p(r 2 p(r 2 p(r 2 P 2 2 pm pm pm p(r m p(rm p( rm p ( where is the actual reliability probability of a first evet evaluated by a first expert; p(r is a fuctio to obtai the actual reliability probability from the fuzzy evaluatio value of the first evet value by the first expert. 3. Reliability Time Distributio The traditioal bow tie aalysis just eeds the reliability probability value of a evet. Through the FT ad ET aalysis, the probabilities of ode accidet ad accidet cosequece are obtaied. However, the dyamic bow tie aalysis ot oly icludes Boolea logic relatio but also icludes dyamic logic gates, ad some evets' reliability chages over time. I order to simulate the dyamic bow tie aalysis model, we eed to obtai the reliability time distributio of the reliability variable evets, i.e., failure time ad repair time distributio. 3. Calculatig reliability time base o reliability probability A evet's failure rate fp at time t meas: the probability fp of occurrece of the accidet described by the evet i a uit time after a time t whe the system has operated till the time t (Xu et al 202. I other words, the accidet described by the evet occurs fp times i oe uit time. Usually, fp <=, which meas that the accidet described by the evet occurs oce i /f uit time. I the same way, the accidet repair described by the evet occurs oce i /rp (rp is the repair rate uit time. Accordig to above reasoig, the system reliability time is calculated as follows: 90

3 Fuzzy Reliability Based Bow Tie Aalysis of Coal Mie Water Irush Risks SDIMI 207: T 2 2 t t t 2 t2 t2 t2 2 p2 p2 p 2 t t t 2 m m m t p p p 2 p m pm p m (2 where deotes the reliability time of a first evet evaluated by a first expert. 3.2 Weibull distributio fittig of reliability time The hypothetic distributio of reliability aalysis icludes: expoetial distributio, Gamma distributio ad Weibull distributio. The expoetial distributio is most commoly used. However, the expoetial distributio has memory (whe it is kow that the basic evet keeps the origial state for s hours, f(t=f(s+t, wherei the failure rate is a costat ad the applicatio thereof is limited. I recet years, research shows that the rule of product failure complies with the Weibull distributio (Behia et al 206. A bathtub curve comprises three parts: early failures, radom failures ad wear-out failures, which correspod to shape parameters, ad of Weibull, respectively. The usafe factors ca be divided ito Huma, Machie, Eviromet ad Maagemet. The Huma ad Machie hazards resemble the rule of Machie wherei the failure rate decreases i the early stage, remais substatially costat durig the middle stage, ad fially icreases. Therefore, the failure ad repair time of evets is i accordace with Weibull distributio fittig i this paper. The probability desity formula of 2-parameters Weibull distributio is: x ( x f ( x ( e (3 the distributio fuctio is writte as: x ( F( x e (4 is the Weibull distributio scale parameter, is Weibull distributio shape parameter. There are may parameter estimatio methods for 2- parameters Weibull distributio, such as graphic method, maximum likelihood estimatio, momet method, least square method ad empirical approach (Cheg ad Sheu 206, Saleh et al 202. Therefore, this paper uses the momet method (MoM to get the evet reliability time based o parameter estimatio of Weibull distributio. The MoM estimates Weibull distributio parameters i the followig way: ( t t ( / Where: ti i.086 ( 0 (5 (6 t (7 2 (t i t (8 i Weibull parameter estimate Matrix T i Eq. (2- to get the distributio parameter of failure ad repair time. Shape parameter determiat is B (,, T m, scale parameter determiat is (,, T m. 4. Fuzzy Reliability Evaluatio The umber of deaths i water irush accidets is secod i all coal mie accidets, ad the umber of occurreces is third. Therefore, risk assessmet of water irush is very importat. However, quatitative aalysis eeds mass data, but it is impossible to obtai all the accurate data i reality or theoretically. 4. Buildig the dyamic bow tie model The bow tie aalysis model is built with the water irush poit i mie as the ode of bow tie aalysis, usig FT to idetify hazards of water irush risks ad ET to show the evolutio of water irush accidet. Base o the occurrece mechaism of water irush accidet i mie, water irush accidet i coal mie is a state that draiig ability of a mie caot cotrol the mie water i a reasoable rage whe abormal water icreases, ad the emergecy measures fail. Accordig to the characteristics of water irush i coal mie ad survey of the literature, the 3-level FT model is built (Gao 200. I the 3-level FT model, sufficiet aggregatio of the source of water irush is the prerequisite of water irush occurrece. I Chia, the sources of water irush are maily surface water, pore water, fracture water, karst ad goaf water (Wu 203, State Admiistratio of Work Safety 20a, State Admiistratio of Work Safety 20b. Expert advice, accidet ivestigatio ad literature review are adopted to aalyze a water irush poit accidet by ET. There are usually five key evets after accidet occurrece, icludig: breakig through the irush zoe, floodig the horizotal tuel, floodig the local lowest poit, floodig the shaft statio ad floodig the shaft or iclied shaft. The local lowest poit meas that this poit has o other adjacet poits except the iflow poit or the elevatios of the other adjacet poits are higher tha that of this poit (Li 202. Dyamic aalysis is coducted o the FT ad ET model, ad a dyamic bow tie aalysis model is built. The model icludes oe SEQ ad oe PAD. I Figure, there are 22 hazards of water irush i coal mie. ie factors (X6 X7 X8 X2 X3 X5 Xc7 X9 X2 are huma factors, two (X8 X20 are machie factors, seve (X X2 X3 X4 X5 X9 X0 are eviromet factors ad four (X X4 X9 X22 are maagemet factors. Oce the water irush happes, there will be six differet cosequeces (C C2 C3 C4 C5 C Reliability time Weibull distributio The hazards of eviromet factors X X2 X3 X4 X5 X9 X0 ca be obtaied from statistical data, but the 9

4 SDIMI 207: Authors: C. Zhag et al X2 Pore water accumulatio X3 Fissure water accumulatio X4 Karst collapse water accumulatio X Surface water accumulatio B B Draiage ditch, flood-dam, artificial riverbed B Surveyig water, dewaterig B3 Groutig ad blockig water, waterproof coal pillar B4 Steel gate/wall X5 X8 X0 X2 Gob water accumulatio Floor failure form udergroud Fault Karst collapse forms udergroud X7 X9 X Roof cavig forms udergroud Karst collapse colum Drillig close badly X4 Illegal operatio X6 X3 Groudwater accumulatio Roadway excavatio forms udergroud Stopig forms udergroud water course Udergroud cocetratio Ureasoable miig desig B2 Source of irush water accumulatio Udergroud activatio Hazards of water irush B3 Iflow of water is sharply icreased B4 Mie water irush Break irush water zoe B Flood horizotal tuel B2 Flood local lowest poits B3 Flood shaft statio B4 Flood shaft/iclied shaft B5 C C2 C3 C4 C5 C6 X5 Shortage of sump capacity E X6 X8 Cleaig up sump ot i time Equipmet fault X7 Shortage of pre displacemet Exceedig sump capacity Exceedig equipmet capacity Lack of draiage capacity C Mie floodig for log time C2 MIe/shaft statio floodig for short time C3 Mie floodig for short time C4 Shaft floodig C5 Irush water i irush water zoe C6 Irush water i irush water poit X9 Mie iformatio uspecified X20 Lack of emergecy equipmet Techical problems X2 Emergecy pla ureasoable Failure of emergecy measures SEQ PAD Safety barrier Ad gate Ro gate X22 Improper operatio Maagemet problems Figure. The bow-tie graph of mie water irush. 92

5 Fuzzy Reliability Based Bow Tie Aalysis of Coal Mie Water Irush Risks SDIMI 207: data of other hazards ad subsequet evets caot be obtaied or accurate data thereof is hard to obtai. Accordig the logical relatioship betwee SEQ ad PAD, with the treatmet process i Part ad Part 2, fuzzy reliability evaluatio is coducted to obtai parameter values of failure time ad repair time Weibull distributio. Accordig to the fuzzy reliability evaluatio approach, te experts of water prevetio ad cotrol i coal mie are first ivited to assess the failure rate ad repair rate of basic evets ad subsequet evets of a water irush part at a coal mie water irush poit. Accordig to formulas (2 - (6, the experts assessmet is trasformed ito matrix A, to obtai the correspodig triagular fuzzy umbers. ext, experts' triagular fuzzy evaluatio umbers are defuzzified. A umber of defuzzificatio methods are available, icludig cetroid method, ceter of gravity, mea max membership ad so o. Through the pilot calculatio, it is foud that ceter of gravity (COG is the best for water irush risk assessmet i coal mie. Through defuzzificatio of matrix A, experts' scores ca be obtaied, i.e., matrix R (8. I order to make the fuzzy rate close to the real failure rate ad repair rate, (0 is used to obtai the real failure rate ad repair rate matrix P of basic evets ad subsequet evets, like (. I dyamic bow tie aalysis, the failure time ad repair time of basic evets ad subsequet evets eed to be obtaied, ad the bow tie model is simulated by Mote Carlo. Hece, we eed failure time ad repair time distributio of basic evets ad subsequet evets to provide the failure time ad repair time radom samplig priciple for Mote Carlo simulatio. I order to coduct Weibull distributio fittig of failure time ad repair time, the matrix method is used to estimate the shape parameter ad the scale parameter. Formulas (5 ad (8 are used to estimate the failure time ad the repair time of basic evets, respectively. We ca obtai shape parameter determiat B ad scale parameter determiat. I the dyamic bow tie graph of water irush i coal mie, SEQ appears i the first level of FT o the left of the ode. Thus, failure time distributio eeds to be cosidered i all the basic evets of water irush i mie. The logical relatioship of subsequet evets is similar to PAD i the part of FT o the right of the ode, so the failure time distributio also eeds to be cosidered. The failure rates of X, X2, X3, X4, X5, X9 ad X0 are costats ad ca be obtaied through statistical data. The above evets, oce happeig, caot be repaired, ad the repair rate is zero. Accordig the logical relatioship betwee SEQ ad PAD, oe oly eeds to get the repair time distributio of X5, X6, X7 ad X8 to coduct Mote Carlo simulatio o a bow tie aalysis model of water irush i coal mie. Based o above aalysis, Weibull distributio parameters of failure time ad repair time of some basic evets ad subsequet evets i bow tie aalysis of water irush i coal mie are obtaied, as show i Table. I Table, scale parameters of failure time ad repair time Weibull distributio are greatly differet. For example, the scale parameter of X poor drillig sealed off is , which meas most of the failure time is i the uit time. The scale parameter of X2 karst collapse by mie draiage is , which meas most of the failure time is i the uit time. The above results demostrate that the failure rates ad repair rates of differet evets at differet time poits are very differet. Past static bow tie aalysis is o the premise that the evet reliability is costat, which is ot i coformity with the actual situatio. Dyamic bow tie aalysis cosiders the dyamic effect of the time factor o the system, which is more accordat with the actual situatio. Table. Weibull distributio parameter. Basic/subsequet evets Failure time Repair time X X X X X X X X X X X X X X X B B B B B Coclusios It is hard to obtai the accurate historical data i actual risk maagemet. The bow tie aalysis of water irush i coal mie eeds the qualitative descriptio from experts, for example. Fuzzy reliability evaluatio calculates the reliability data of dyamic bow tie aalysis based o basic evet ad subsequet evet reliability probabilities i a dyamic bow tie aalysis model characterized by triagular fuzzy umbers, ad failure time ad repair time of basic evets ad subsequet evets fitted by Weibull distributio. To verify the feasibility of fuzzy reliability evaluatio, the bow tie aalysis of water irush i coal mie is take as a example. The result shows that the scale parameters of failure time ad repair time Weibull distributio of basic evets ad subsequet evets differ greatly. The case study shows that dyamic bow tie aalysis is more practical tha traditioal bow tie aalysis, ad that fuzzy reliability evaluatio is a effective method for calculatig the evet reliability value whe the historical data is hard to obtai. 93

6 SDIMI 207: Authors: C. Zhag et al Refereces Amari, S., G. Dill ad E. Howald, A ew approach to solve dyamic fault trees. Reliability ad Maitaiability Symposium. Aual. IEEE, DOI: 0.09/RAMS Behia, A.,. Rajbar, H. K. Chai, & M. Masaeli, 206. Failure predictio ad reliability aalysis of ferrocemet composite structures by icorporatig machie learig ito acoustic emissio moitorig techique. Costructio ad Buildig Materials, 22: DOI: 0.06/j.cobuildmat Cheg,. ad S. Sheu, 206. Robust estimatio for weibull distributio i partially accelerated life tests with early failures. Quality ad Reliability Egieerig Iteratioal. 32: DOI: 0.002/qre.928 Gao,., 200. Applicatio of fault tree aalysis to establish safety checklist for groudwater irush i udergroud coalmies. Coal Geology & Exploratio. 06: Khakzad,., F. Kha ad P. Amyotte, 203. Dyamic safety aalysis of process systems by mappig bow-tie ito bayesia etwork. Process Safety ad Evirometal Protectio, 9(: DOI: 0.06/j.psep Kim, J.,. oh, K. Chag ad D. Chag, 206. Determiatio of hydrate ihibitor ijectio rate for flowlies based o Mote Carlo method. Joural of Loss Prevetio i the Process Idustries, 44: DOI: 0.06/j.jlp Li, K. ad Z. Wag, 20. The applicatio of fault tree of gas i mie. Eergy Techology ad Maagemet. 03: 0 -. DOI: /j.iss Li, X., 202. Dyamic visualizatio of water irushes ad emergecy resposes i udergroud mies. Ph.D. Thesis, Uiversity of Sciece ad Techology, Beijig. Liu, G. ad L. Zhag, 200. The applicatio of fault tree of tuel egieerig i geological survey durig flood accidet about lead-zic deposit i huajiaoshupig, shelogjia forestry regio. Resources Eviromet & Egieerig, 02: DOI: /j.cki.iss Saleh, H., E.A. Aly ad S. Abdel-Hady, 202. Assessmet of differet methods used to estimate weibull distributio parameters for wid speed i zafaraa wid farm, Suez Gulf, Egypt. Eergy, 44(: DOI: 0.06/j.eergy State Admiistratio of Work Safety, 20a. Provisios o prevetio ad cotrol of water i coal mies. Chia Coal Idustry Publishig, Beijig. State Admiistratio of Work Safety, 20b. Safety regulatios i coal mie. Chia Coal Idustry Publishig, Beijig. Wag, C.,. Su ad. Hag, Applicatio of fault tree aalysis to risk assessmet of potetial waterirush hazards i coal miig. Chiese Joural of Rock Mechaics ad Egieerig. 02: Wu, Q., 203. Prevetio ad cotrol of coal mie water. Chia Coal Idustry Publishig, Beijig, Xu, J., D. Su,. Fa ad Q. Re, 202. Accidet risk probability for three highs gas fields based o fault tree aalysis. Systems Egieerig - Theory & Practice, 4: a, C., J. Ji, W. Liu ad S. Che, 202. Fault tree aalysis of water irush i coal mie. Moder Miig, : Zhao,.,. Guo ad K. Xie, 203. Research o probability distributio characteristics of bulk power system reliability cosiderig parameter ucertaity. Power System Techology, 08: DOI: /j pst

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