Fuzzy fault tree analysis for fire and explosion of crude oil tanks

Journal of Loss Prevention in the Process Industries xxx (2013) 1e9

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Fuzzy fault tree analysis for fire and explosion of crude oil tanks Daqing Wang a, *, Peng Zhang b, Liqiong Chen a a b

School of Petroleum Engineering, Southwest Petroleum University, 610500 Chengdu, PR China School of Civil Engineering and Architecture, Southwest Petroleum University, 610500 Chengdu, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 January 2013 Received in revised form 16 March 2013 Accepted 28 August 2013

Crude oil tank fire and explosion (COTFE) is the most frequent type of accident in petroleum refineries, oil terminals or storage which often results in human fatality, environment pollution and economic loss. In this paper, with fault tree qualitative analysis technique, various potential causes of the COTFE are identified and a COTFE fault tree is constructed. Conventional fault tree quantitative analysis calculates the occurrence probability of the COTFE using exact probability data of the basic events. However, it is often very difficult to obtain corresponding precise data and information in advance due to insufficient data, changing environment or new components. Fuzzy set theory has been proven to be effective on such uncertain problems. Hence, this article investigates a hybrid approach of fuzzy set theory and fault tree analysis to quantify the COTFE fault tree in fuzzy environment and evaluate the COTFE occurrence probability. Further, importance analysis for the COTFE fault tree, including the FusselleVesely importance measure of basic events and the cut sets importance measure, is performed to help identifying the weak links of the crude oil tank system that will provide the most cost-effective mitigation. Also, a case study and analysis is provided to testify the proposed method.
2013 Elsevier Ltd. All rights reserved.

Keywords: Crude oil tank Fire and explosion Fuzzy fault tree analysis Occurrence probability Importance analysis

1. Introduction Recent years see sustainable economic growth of China, while the turbulence in international crude oil market has stimulated China’s need for much larger strategic oil reserves. More and more largescale crude oil storage tanks have been designed and constructed presently. Although most companies follow strict engineering guidelines and standards for the construction, material selection, design and safe management of storage tanks and their accessories, there is always the possibility of fire or explosion for various causes. According to statistics, the crude oil tank fire and explosion (COTFE) is the most frequent type of accident in petroleum refineries, oil terminals or storage (Fan, 2005). Besides China, yearly losses due to the COTFE are substantial all over the world (Chang & Lin, 2006). Fault tree analysis (FTA) is a systematic approach to estimate safety and reliability of a complex system, qualitatively as well as quantitatively. FTA can be applied both to an existing system and a system in designation. For system in design, FTA can provide an estimate of the failure probability and contributors using generic data and also can be used as a supporting tool of a performance-based design. In an existing system, FTA can identify weaknesses, evaluate possible upgrades, monitor and predict behavior. For those merits, FTA technique has been extensively used in many fields, such as nuclear power, electric * Corresponding author. Tel.: þ86 13658001455. E-mail address: [email protected] (D. Wang).

power, chemical process, oil and gas transmission, etc (Dong & Yu, 2005; Prugh, 1992; Sadiq, Saint-Martin, & Kleiner, 2008). In traditional FTA, the failure probabilities of the basic events (BEs) are expressed by exact values (Dong & Yu, 2005; Ferdous, Khan, Sadiq, Amyotte, & Veitch, 2009; Sadiq et al., 2008). However, in reality, the vagueness nature of a system, the working environment of a system, and the lack of sufficient statistical inference, all raise difficulties in the estimation of occurrence probabilities of components or BEs (Dong & Yu, 2005; Liang & Wang, 1993; Pan & Yun, 1997). And this makes quantitative analysis of a fault tree of a system questionable by conventional methods. In order to handle inevitable imprecise failure information in diversified real applications, many researches have taken the uncertain situations into consideration. Fuzzy set theory has been proven to be effective on solving problems where there are no sharp boundaries and precise values, while it is also efficient (Onisawa, 1990; Suresh, Babar, & Raj, 1996; Zadeh, 1965). Chen (1994) and Mon and Cheng (1994) carried out system reliability analysis by using fuzzy set theory. Dong and Yu (2005) applied fuzzy theory to estimate the failure probabilities of BEs. Tanaka, Fan, Lai, and Toguchi (1983), Pan et al. (2007), Suresh et al. (1996), and Miri Lavasani, Wang, Yang, and Finlay (2011) implemented fuzzy theory into the FTA technique for certain system safety assessment. In this paper, the imprecise failure data of BEs of the COTFE fault tree are replaced with fuzzy numbers and an approach of fuzzy based fault tree analysis (FFTA) is introduced to estimate the probability of occurrence of the COTFE. Further, the

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D. Wang et al. / Journal of Loss Prevention in the Process Industries xxx (2013) 1e9 2

Tank fire and explosion

AND

OR

AND

OR

OR

Electrified railway

Electric leakage nearby

Cathodic protection

Non-explosion proof monitor or detector

Other non-explosion proof electrical equipments

Audio-visual or photographic equipment Mobile telephone

Match

Lighter

Vehicles without flame arresters Fire work Smoking

Wearing iron nail-shoes

Without installing lightning protection facilities Lightning induction

Lightning invasion along pipelines Direct lightning flash

Deflector damaged

Ground rod damaged

A

Air terminal damaged

Breathing valve open by breakdown

Gauge hatch often open

Flexible connection pipe rupture

High degree corrosion of tank wall

Poor seal around manhole

Floating metal debris on oil surface

Oil lashing against metal materials

Rough inner wall of pipeline

High oil flow velocity

Excessive loading

Wrong valve opened

Tank wall broken by external force

Tank top unattended

Friction between fiber and human body Operator close to a conductor

Friction between splashing oil and air

Not enough standing time

Non-standard apparatus

Non-standard ground resistance

Broken ground wire

Without installing anti-static grounding device

X27

X26

X29

X24

Stray current

X43

X28

X30

X36

X35

X32

X31

X42

X41

OR

X25

X34

X37

Measuring operational error

Operational error X33

Oil leakage Oil spill

X39

OR

X15 X14 X13

Human body electrostatic discharge

X22 X6

Collision of metal tools and tank wall during maintenance operation Using non-explosion proof tools

OR

X38

OR

X10

OR Electrostatic accumulation

X12 X9

Static sparks

OR X11

B

OR

X19 X2

X21

OR

Arrester faults

X20 X16

X18

Imperfect earth

X8

Lightning stroke

X7

AND Bad grounding

A

X40

OR

X23

AND AND

OR

Vapor-air mixtures within explosive range Oil tank electrostatic discharge

Vapor-air mixtures within explosive range

X17 X1

Electrical apparatus sparks

OR

X5 X4 X3

Lightning sparks

B

Open fires Static sparks Impact sparks

Ignition sources

OR

Fig. 1. Schematics of the COTFE fault tree.

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D. Wang et al. / Journal of Loss Prevention in the Process Industries xxx (2013) 1e9

proposed approach is used to perform importance analysis of the COTFE fault tree in order to help decision maker determine whether and where to take preventive or corrective action on the crude oil tank system in the risk management process. 2. Traditional FTA of COTFE 2.1. Construction of COTFE fault tree FTA is a deductive method for identifying ways in which hazards can lead to accidents. The approach starts with a top undesired event and work backwards towards the various scenarios that can cause the accident. In a fault tree, the top, intermediate and basic events are connected together by logic gates. The gates show relationships of input events needed for the occurrence of a fault at the output of the gate. AND gates combine input events, all of which must exist simultaneously for the output to occur. OR gates also combine input events, but any one is sufficient to cause the output. In present paper, the COTFE accident is considered as the top event (TE). Two intermediate events must occur together for the COTFE: ‘ignition sources’ and ‘vapor-air mixtures within explosive range’, so they must be connected to the TE by an AND gate. Various ignition sources may exist in the tank park: impact sparks, static sparks, open fires, lighting sparks, electrical apparatus sparks and stray current. Any one of them could ignite an explosive mixture if contacted, so these must be connected by an OR gate. Also there are many causes that can lead to the explosive mixtures, such as oil spill, gauge hatch often open, breathing valve open by breakdown or oil leakage. Next, consider these sub-events as the new intermediate events, and then each of them will be substituted by the lower events. Continue developing the fault tree until its all branches have been terminated by basic or undeveloped events. Finally, a complete fault tree of the COTFE is constructed as shown in Fig. 1. The proposed fault tree includes 43 BEs that contribute to the occurrence of the COTEF accident.

After the COTFE fault tree is fully drawn, both qualitative and quantitative evaluation can be performed. The aim of qualitative analysis of a fault tree is to find out the minimal cut sets (MCSs). The MCSs relate the TE directly with the basic event causes and a MCS is the smallest combination of BEs which if they all fail will cause the occurrence of the undesired event. The MCSs are very useful for determining the various ways in which a top undesired event could occur. In this study, the MCSs of the COTFE fault tree are obtained by using the combination of Fussell-Vesely algorithm and the rules of Boolean algebra (Fussell & Vesely, 1972; Wang, 1999). The proposed fault tree yields 392 MCSs for just 43 BEs, including 90 MCSs of order 2, 234 MCSs of order 3 and 68 MCSs of order 4. The MCSs equation is as follows:

XXX m

þ

þ

n

k

XXX s

þ

Xm Xk Xn X37 þ

r

XX i

n

i

j

XX

X

! Xs Xr Xn X37

n

Xi Xn X37 þ

n

X33 X34 Xn X37

0

þ@

XXX m

X j

3. Fuzzy based FTA of COTFE 3.1. Fuzzy numbers to define probabilities of the BEs The concept of fuzzy set theory was introduced by L.A Zadeh (1965) to deal with uncertain or vague information. A fuzzy set defined on a universe of discourse (U) is characterized by a membership function, m(x), which takes values from the interval [0, 1]. A membership function provides a measure of the degree of similarity of an element in U to the fuzzy subset. Fuzzy sets are defined for specific linguistic variables. Each linguistic term can be represented by a triangular, trapezoidal or Gaussian shape membership function. The selection of a membership function essentially depends on the variable characteristics, available information and expert’s opinion. Here, triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (ZFNs) are employed for their simplicity and efficiency to quantify the probabilities of the BEs. The triangular representation shows the fuzzy possibility of a BE can be denoted by a triplet (a1, a2, a3) and the corresponding membership function is written as (Wang, 1997):

0 ðx  a1 Þ=ða2  a1 Þ mA~ ðxÞ ¼ > > ða3  xÞ=ða3  a2 Þ : 0

; x  a1 ; a1  x  a2 ; a2  x  a3 ; x  a3

(2)

A ZFN denoted by a quadruple (a1, a2, a3, a4) is defined as follows:

8 0 > > > > < ðx  a1 Þ=ða2  a1 Þ mA~ ðxÞ ¼ 1 > > > ða4  xÞ=ða4  a3 Þ > : 0

; x  a1 ; a1  x  a2 ; a2  x  a3 ; a3  x  a4 ; x  a4

(3)

3.2. Aggregation of fuzzy numbers of the BEs

T ¼ MCS1 þ MCS2 þ . þ MCSN ¼

where 1  i  8 and 16  i  22, 38  j  43, 26  k  32, 23  m  25, 35  n  36, 12  r  15, 9  s  11; N is the serial number of MCS, 1  N  392; X represents BE. The aim of quantitative analysis of a developed fault tree is to provide a measure of the probability of occurrence of the TE and the major faults contributing to the TE. The quantitative evaluation requires the gathering of exact failure data of BEs for input to the fault tree. However, for the COTFE fault tree, it is difficult to have a precise estimation of the BE probability due to insufficient data; in fact many BEs of the COTFE fault tree may not have quantitative data at all due to its inherent uncertainty and imprecision. Therefore, it is not possible to assign a single value of probability to each BE. In order to overcome such limitations in traditional FTA, a fuzzy based approach is developed and discussed in the following sections.

8 > > <

2.2. Evaluation of COTFE fault tree

3

X33 X34 Xj þ

k

Xm Xk Xj

j

XXX s

r

1 Xs Xr Xj A

j

Xi Xj (1)

Since each expert may have a different opinion about the same BE according to his/her experience and expertise in the relevant field, in order to achieve agreement among experts’ conflicted views, the fuzzy numbers assigned by different experts should be aggregated to a single one. A consistency aggregation method (Wei, Qiu, & Wang, 2001) is proposed in this paper. This methodology is a revised version of the Hsu and Chen’s algorithm (Hsu & Chen, 1996), which overcomes the assumed restriction that the opinions of all experts represented by fuzzy numbers should have a common intersection. The proposed method is described as follows: ~ ;A ~ Þ of the opinions A ~ and A ~ (1) Calculate the similarity degree sðA i j i j of a pair of experts Ei and Ej.

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D. Wang et al. / Journal of Loss Prevention in the Process Industries xxx (2013) 1e9

   ~ ;A ~ ¼ EVi =EVj ; s A i j EVj =EVi ;

EVi  EVj EVj  EVi

(4)

~ ;A ~ Þ ˛[0,1] is the similarity function; A ~ and A ~ are two where sðA i j i j standard fuzzy numbers, respectively; EVi and EVj separately ~ or A ~ . The EV of a traprepresent the expectancy evaluation for A i j ~ ¼ ða ; a ; a ; a Þ is defined as: ezoidal fuzzy number A 1 2 3 4

h    i   ~ þ Eþ A ~ ~ ¼ 1 E A EV A 2 

n Y

¼

1

n Y

1 n

n X

1

(7)

(3) Calculate the relative agreement degree (RAD) of each expert.

RADi ¼ AðEi Þ=

(11)

~i Þ ð1Qp

ð1  ai2 Þ;

ð1  ai3 Þ; 1 

(12) !

ð1  ai4 Þ

i¼1

Q where denotes fuzzy multiplication; subtraction. For triangular fuzzy number (ai1, ai2, ai3): n Y

n Y

~i ¼ p

i¼1

isj j ¼ 1

n X

n Y

i¼1

~ c ¼ ANDFðp ~1 ; p ~ 2 ;:::; p ~n Þ ¼ P

  ~ ;A ~ sij A i j

n Y

i¼1 n Y

~ ;A ~ Þ, if i ¼ j, then sij ¼ 1. A(Ei) is defined as: where sij ¼ sðA i j

AðEi Þ ¼

ai4

i¼1

ð1  ai1 Þ; 1 

i¼1 n Y

(6)

ai3 ;

i¼1

!

n Y

i¼1

¼

1

ai2 ;

i¼1

n Y

~ c ¼ ORFðp ~1 ; p ~2 ; .; p ~ n Þ ¼ 1Q P

where E (A) ¼ (a1 þ a2)/2, E (A) ¼ (a3 þ a4)/2.

/ B 1 s12 / s1n C B M ¼ @ s21 1 s C 1 12 A sn1 sn2 1 /

n Y

ai1 ;

i¼1

þ

0

~i p

i¼1

(5)

(2) Construct the consensus matrix M and calculate the average agreement degree A(Ei) of the experts.

n Y

~ c ¼ ANDFðp ~1 ; p ~ 2 ; .; p ~n Þ ¼ P

~ c ¼ ORFðp ~1 ; p ~ 2 ; :::; p ~ n Þ ¼ 1Q P

ai1 ;

i¼1 n Y

Q denotes fuzzy

n Y

ai2 ;

i¼1

n Y

! ai3

~i Þ ð1Qp

i¼1

¼

1

n Y

ð1  ai1 Þ; 1 

i¼1

n Y

(13)

i¼1

ð1  ai2 Þ; 1 

i¼1

n Y

! ð1  ai3 Þ

i¼1

(14) AðEi Þ

(8)

i¼1

(4) The aggregation weight (wi) of each expert Ei is the combination of the RADi and the importance degree (EIDi) of experts Ei.

~ TE ) can be calculated Hence, the fuzzy possibility of the COTFE (P using the following equation:

~ TE ¼ 1  P

n  Y

~ 1P ci



i¼1

wi ¼ a$EIDi þ ð1  aÞ$RADi

i ¼ 1; 2; .; n

(9)

where a (0a  1) is a relaxation factor which shows the imporP tance EIDi over RADi; EIDi (0  EIDi  1 and EIDi ¼ 1) can be determined by using Delphi method (Dong & Yu, 2005) or analytic hierarchy process (Bryson & Mobolurin, 1994).

¼ 1

    i h ~ ~ ~ 1P c1 5 1  P c2 5.5 1  P cn

(15)

~ TE ~c1 ; p ~c2 ; .; p ~cn denote the fuzzy possibilities of all MCSs; P where p is the fuzzy COTFE possibility.

3.4. Defuzzification of the fuzzy COTFE possibility (5) The aggregation result of the experts’ opinions can be obtained as follows:

~j ¼ p

n X

~ ij wi 5p

j ¼ 1; 2; .; m

(10)

i¼1

~j is the aggregated fuzzy number of BEj; p ~ij is the fuzzy where p number of BEj assigned by expert Ei; m is the number of experts; n is the number of BEs; wi is a weighting factor of the expert Ei. 3.3. The fuzzy COTFE possibility estimation To minimize the error due to uncertainty in BE probability data, the present algorithm uses fuzzified possibility data of BE for quantification of a fault tree. Fuzzy arithmetic operations rules (Liang & Wang, 1993; Tanaka et al., 1983) are employed to estimate the fuzzy possibility of the MCSs and the same for the COTFE. ~1 ; p ~2 ; .; p ~n , the fuzzy Giving the fuzzy possibilities for all BEs, p ~ c ) are estimated using following possibilities of the MCSs (P expressions. For trapezoidal fuzzy number (ai1, ai2, ai3, ai4):

To provide a useful outcome for decision making, the fuzzy possibility of the COTFE must be first mapped to crisp possibility score (CPS) through defuzzification. A number of defuzzification methods (Ross, 2004; Wang, 1997) are available, including mean max membership, centroid method, weighted average method, center of largest area, center of sums and so on. In this paper, the center of area defuzzification technique (Miri Lavasani et al., 2011; Wang, 1997) is ~ ¼ adopted for its simplicity and usefulness. Defuzzification of TFN A ða1 ; a2 ; a3 Þ can be obtained by the following expression:

Z * PTE ¼ Z

Za2 xmA~ ðxÞdx

mA~ ðxÞdx

¼

a1

x  a1 xdx þ a2  a1

Za2 a1

x  a1 dx þ a2  a1

Za3 a2 Za3 a2

a3  x xdx a3  a2 a3  x dx a3  a2

1 ¼ ða1 þ a2 þ a3 Þ 3

(16)

* is the defuzzified output; x is the output variable. where PTE ~ ¼ ða ; a ; a ; a Þ is: Defuzzification of ZFN A 1 2 3 4

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D. Wang et al. / Journal of Loss Prevention in the Process Industries xxx (2013) 1e9

Za2 * PTE ¼

a1

x  a1 xdx þ a2  a1

Za2 a1

¼

Za3

Za4 xdx þ

a2

x  a1 dx þ a2  a1

a3

Za3

Za4 dx þ

a2

a3

5

a4  x xdx a4  a3

MCS importance is estimated by calculating the ratio of the MCS probability to the COTFE probability. The calculation is performed as follows:

a4  x dx a4  a3

IjCS ¼

1 ða4 þ a3 Þ2  a4 a3  ða1 þ a2 Þ2 þ a1 a2 $ 3 a4 þ a3  a2  a1

(17)

j PMCS PTE

(21)

j where IjCS is the CS-I index of the jth MCS; PMCS is the occurrence probability of the jth MCS.

3.5. Convert crisp possibility score (CPS) into probability value (PV)

4. A case study

In traditional FTA the final result is an exact probability value. In fuzzy based FTA, however, the output is crisp possibility score (CPS) because the occurrence probability of each BE is represented by fuzzy numbers. There is inconsistency between the real probability data and the possibility score. This issue can be solved by transforming the CPS into the form of probability of occurrence. The following conversion function (Onisawa, 1988, 1990) is proposed:

The COTFE accident erupted in Hunan oil depot (Fan, 2005), which result in four people died and two injured, is selected here as a case study. The proposed fuzzy based FTA is performed to evaluate the occurrence probability of the COTFE accident. And the implementation of the proposed approach also provides an opportunity to reinvestigate the causes of accident, which is helpful to prevent or reduce the occurrence of such accidents in the future. One of the other aims of the study is to compare the results obtained by the fuzzy FTA with the results reached by traditional FTA. Considering the COTFE as an undesired top event, the COTFE tree has been constructed as shown in Fig. 1. Once the fault tree has been developed, it can be evaluated to identify the possible basic causes (or BEs) and pathways (minimum combination of BEs) that would lead to the undesired event. As previously concluded, quantitative analysis shows there are 43 BEs and 392 MCSs in the COTFE tree. Subsequently, using the failure probability data of the BEs, those pathways can be further evaluated to estimate the COTFE occurrence probability and find out the most vulnerable pathways and BEs.

( PV ¼

1 10m ;

CPSs0

0;

CPS ¼ 0

(18)

where

 m ¼

1 1  CPS 3  2:301 CPS

(19)

* ¼ CPS, P and PTE TE ¼ PV; PTE is the probability of occurrence of the CCOTFE.

3.6. Importance analysis of the COTFE fault tree At the time of decision making process, it is useful to have the events sorted according to some criteria. This ranking is enabled by importance analysis. In this study, the importance analysis of the COTFE fault tree is carried out based on the investigation of the importance measures of the BEs and the MCSs in the proposed tree. 3.6.1. FusselleVesely importance of BEs The FusselleVesely importance (FV-I) is employed to evaluate the contribution of each BE to the occurrence probability of the COTFE. This importance measure is sometimes called the top contribution importance. It provides a numerical significance of all the BEs in the COTFE fault tree and allows them to be prioritized. The FV-I of a BE is calculated by the following equation (Vinod, Kushwaha, Verma, & Srividya, 2003):

IxFVi ¼

xi ¼0 PTE  PTE PTE

(20)

xi ¼0 where IxFVi is the FV-I index of ith BE; PTE is the occurrence probability of the COTFE by setting the probability of ith BE to 0. Decision makers use this importance index to improve the safety features of the analyzed crude oil tanks.

3.6.2. Cut sets importance Cut sets importance (CS-I) is used to evaluate the contribution of each MCS to the TE occurrence probability. This importance measure provides a method for ranking the impact of each MCS and identifying the most likely path that leads to the TE. In order to measure the CS importance, the output fuzzy possibility of each MCS of the COTFE fault tree needs to be converted into a probability value using the methods described in Section 3.4e3.5. Then the

4.1. Fuzzy-based approach 4.1.1. Fuzzy numbers defining probabilities of BEs Due to lack of the precise probability data of BEs of the COTFE tree, the approach synthesizing the fuzzy set theory and experts’ linguistic judgments is proposed to quantify the occurrence possibilities of the BEs. In this study, three experts, including a reliability analyst and two senior field engineers, are invited to perform the assessments. In order to capture experts’ linguistic notions of probabilities for the BEs, a seven level linguistic rating scale, i.e. {Very Low (VL), Low (L), Mildly Low (ML), Medium (M), Mildly High (MH), High (H) and Very High (VH)}, has been proposed. Then, the linguistic expressions are transformed into fuzzy numbers using a numerical approximation system as shown in Fig. 2 (Chen, Hwang, & Hwang, 1992). The result of the expert judgments for all the BEs is shown in Table 1. 4.1.2. Aggregating fuzzy numbers assigned by different experts Aggregation provides an agreement among the conflicted knowledge provided by different experts. Here the proposed consistency aggregation method is adopted to achieve it. In addition, for the ease of analysis, the TFNs defining the BE probabilities should first be converted into the corresponding ZFNs; for example, a TFN (a1, a2, a3) can be expressed as a ZFN (a1, a2, a2, a3). Then, according to Eqs. (4)e(10), the aggregated fuzzy possibility values for each BEs involved in COTFE tree are obtained (see Table 1), which will be taken as the input data for fuzzy COTFE probability calculation. As an example, the detailed aggregation calculations for BE36 are given in Table 2, which include the calculations such as ~ ;A ~ Þ, average agreement degree A(Ei), relative similarity degree sðA i j agreement degree (RAD), aggregation weight (wi), etc.

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D. Wang et al. / Journal of Loss Prevention in the Process Industries xxx (2013) 1e9

Membership function, µ

Very Low

Low

Mildly Low

Medium

Mildly High

High

estimated, which is also a continuous ZFN (0.523, 0.897, 0.995, 1.000). Obviously, the fuzziness of the COTFE event is determined by that of the BEs. And the information about the real state of the crude oil tank system is revealed sufficiently when such fuzzy description is reserved.

Very High

1.0 0.8 0.6 0.4 0.2 0

0.1

0.2

0.3

0.4

0.5 0.6 Possibility, p

0.7

0.8

0.9

1.0

Fig. 2. Fuzzy numbers represent linguistic value.

4.1.3. Estimating fuzzy possibility of the COTFE Quantitative analysis of the COTFE tree attempts to calculate occurrence probability of the top event. In this study, the probabilities for all BEs are represented in a form of ZFNs, so calculations of the fuzzy possibility of the COTFE and the MCSs must follow the fuzzy arithmetic operation rules (Liang & Wang, 1993; Tanaka et al., 1983). According to Eqs. (11)e(15) and the fuzzy possibilities of the BEs in Table 1, the fuzzy possibility of occurrence of the COTFE is

4.1.4. Defuzzifying fuzzy possibility of the COTFE The result obtained above is a fuzzy variable, which needs to be further converted into a crisp possibility score (CPS) by defuzzification. The CPS is a single crisp numeric value, which represents the most likely score that an event may occur (Dong & Yu, 2005). Here, the centre of area defuzzification method Eq. (17) is adopted to achieve it. The crisp defuzzfied result (as shown in Fig. 3) allows displaying the percentage contribution of the COTFE fuzzy possibility number in fuzzy set representing fuzzy possibility range. The CPS value of the COTFE is 0.833 and belongs to two sets: High (H) with the membership degree of 67% and Very High (VH) in 33%. The result makes the decision-making in risk assessment more convenient. 4.1.5. Converting CPS of the COTFE into a probability value In order to ensure compatibility between the CPS and the exact probability data obtained from sufficient statistical inference, the

Table 1 Fuzzy possibility values and FV-I measures for BEs in fuzzy COTFE FTA. BE

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 X24 X25 X26 X27 X28 X29 X30 X31 X32 X33 X34 X35 X36 X37 X38 X39 X40 X41 X42 X43

Description

Using non-explosion proof tools Collision of metal tools and tank wall during maintenance operation Wearing iron nail-shoes Smoking Fire work Vehicles without flame arresters Match Lighter Direct lightning flash Lightning invasion along pipelines Lightning induction Without installing lightning protection facilities Air terminal damaged Deflector damaged Ground rod damaged Mobile telephone Audio-visual or photographic equipment Non-explosion proof monitor or detector Other non-explosion proof electrical equipments Cathodic protection Electrified railway Electric leakage nearby Without installing anti-static grounding device Non-standard ground resistance Broken ground wire Non-standard apparatus Not enough standing time Friction between splashing oil and air Floating metal debris on oil surface Oil lashing against metal materials Rough inner wall of pipeline High oil flow velocity Operator close to a conductor Friction between fiber and human body Excessive loading Wrong valve opened Tank top unattended Breathing valve kept open since broke down Gauge hatch often open Tank wall broken by external force Poor seal around manhole High degree corrosion of tank wall Flexible connection pipe rupture

Linguistic judgments of experts

Aggregation of fuzzy numbers

FV-I measure

L L

(0.065, 0.130, 0.165, 0.265) (0.073, 0.147, 0.173, 0.273)

0.0634 0.0711

9 8

VL ML M VL VL L L VL VL VL L L ML VL VL VL L VL VL VL VL L L L VL VL L VL VL L VL VL L ML L ML ML VL L VL L

(0.000, (0.167, (0.319, (0.030, (0.064, (0.136, (0.034, (0.040, (0.037, (0.030, (0.138, (0.213, (0.132, (0.035, (0.000, (0.000, (0.065, (0.000, (0.000, (0.040, (0.038, (0.034, (0.132, (0.026, (0.038, (0.030, (0.026, (0.038, (0.037, (0.065, (0.037, (0.030, (0.062, (0.089, (0.065, (0.250, (0.171, (0.000, (0.136, (0.030, (0.060,

0.0017 0.1536 0.2873 0.0298 0.0623 0.1279 0.0204 0.0238 0.0288 0.0045 0.0182 0.0261 0.0117 0.0346 0.0017 0.0017 0.0634 0.0017 0.0017 0.0396 0.0131 0.0119 0.0414 0.0067 0.0095 0.0077 0.0067 0.0095 0.0094 0.0158 0.0015 0.0015 0.0089 0.0121 0.0212 0.3811 0.2768 0.0046 0.2262 0.0539 0.1044

35 5 2 15 10 6 20 18 16 34 21 17 26 14 35 35 9 35 35 13 23 25 12 32 27 31 32 28 29 22 36 36 30 24 19 1 3 33 4 11 7

Expert 1

Expert 2

Expert 3

L L

VL VL

VL ML M L L ML VL L VL L ML M L VL VL VL L VL VL L L VL ML L L L VL L VL L VL L VL VL L M ML VL ML L ML

VL L L VL L L VL VL L VL L L L L VL VL VL VL VL VL VL VL L VL VL VL VL VL L VL L VL L L VL L L VL L VL L

Index

0.000, 0.267, 0.419, 0.059, 0.128, 0.236, 0.068, 0.080, 0.074, 0.059, 0.238, 0.313, 0.232, 0.070, 0.000, 0.000, 0.130, 0.000, 0.000, 0.080, 0.076, 0.068, 0.232, 0.051, 0.076, 0.059, 0.051, 0.076, 0.074, 0.130, 0.074, 0.059, 0.124, 0.147, 0.130, 0.350, 0.271, 0.000, 0.236, 0.059, 0.120,

0.100, 0.333, 0.419, 0.130, 0.164, 0.273, 0.134, 0.140, 0.137, 0.130, 0.277, 0.313, 0.265, 0.135, 0.100, 0.100, 0.165, 0.100, 0.100, 0.140, 0.138, 0.134, 0.265, 0.126, 0.138, 0.130, 0.126, 0.138, 0.137, 0.165, 0.137, 0.130, 0.162, 0.221, 0.165, 0.386, 0.341, 0.100, 0.273, 0.130, 0.160,

0.200) 0.433) 0.519) 0.230) 0.264) 0.373) 0.234) 0.240) 0.237) 0.230) 0.377) 0.413) 0.365) 0.235) 0.200) 0.200) 0.265) 0.200) 0.200) 0.240) 0.238) 0.234) 0.365) 0.226) 0.238) 0.230) 0.226) 0.238) 0.237) 0.265) 0.237) 0.230) 0.262) 0.321) 0.265) 0.486) 0.441) 0.200) 0.373) 0.230) 0.260)

Ranking

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D. Wang et al. / Journal of Loss Prevention in the Process Industries xxx (2013) 1e9 Table 2 The aggregation calculations for the BE36.

Table 3 The CS-I ranking of top 50 MCSs in fuzzy COTFE FTA.

~ A 1 ~ A 2 ~ A

VL (0, 0, 0.1, 0.2) L (0.1, 0.2, 0.2, 0.3) ML (0.2, 0.3, 0.4, 0.5)

EV(1) EV(2) EV(3)

0.0750 0.2000 0.3500

S12 S13 S21 S23 S31 S32

2.6667 4.6667 2.6667 0.5714 4.6667 0.5714

A(E1) A(E2) A(E3) RAD1 RAD2 RAD3

3.6667 1.6190 2.6190 0.4639 0.2048 0.3313

EID1 EID2 EID3

0.38 0.32 0.30

w1 w2 w3

0.42 0.26 0.32

3

MCSs

CS-I Index

MCSs Ranking

CPS must be converted into the form of probability data. This can be achieved by using Eqs. (18) and (19). The corresponding probability of occurrence for the COTFE is 4.514  102.

Membership function, µ

4.1.6. Importance measure for the COTFE fault tree An important aim of many reliability and risk analyses is to identify the most important BEs and MCSs from a reliability or risk viewpoint so that they can be given priority for improvements. The most crucial BEs in the COTFE fault tree for causing the occurrence of the COTFE can be justified through FV importance (FVeI) measures. Using Eq. (20), The FV-I indexes of all BEs in the COTFE tree are calculated and ranked as shown in Table 1. The result helps to conclude that particular attention must be given to the events X38, X5, X39, X41, X4, X6, X43, X2, {X1; X19} and X7 as these BEs have the greatest potential to cause the COTFE accident. The MCS represents the smallest collection of BEs whose failures are necessary and sufficient to result in the COTFE accident. The most crucial MCSs for the undesired COTFE event can be measured by ranking of their CS-I index. Here, the MCS X4X38 is taken as an example to illustrate the calculation procedure of the CS-I index. First, the fuzzy possibility of the MCS X4X38 is calculated based on fuzzy arithmetic operations Eq. (11), which is also a fuzzy number of (0.042, 0.093, 0.128, 0.210). Next, the CPS of X4X38 is estimated as 0.120 by the deffuzification technique of Eq. (17). Then, this FPS is substituted into Eq. (18) and (19) to calculate PV and the PV of X4X38 is 3.395  105. Finally, using Eq. (21), the CS-I index of X4X38 is 7.521  104. The CS-I indexes of other MCSs are calculated using the same procedures and the results of ranking top 50 MCSs are provided in Table 3. As shown in Table 3, the MCSs ranked the top ten crucial contributions to the COTFE probability are X5X38, X5X39, X4X38, X5X41, X4X39, X8X38, X8X39, X4X41, X8X41 and X5X43 respectively. This reveals that these MCSs are the weakest links of the crude oil tank system. The path MCS X5X39 (‘Fire work’ and ‘Breathing

Very Low

Low

0.1

0.2

Mildly Low

Medium

Mildly High

High

Very High

0.6

0.8

0.9

0.67

0.5 0.33

0

0.3

0.4

0.5

Possibility, p

0.7

0.833

Fig. 3. Fuzzy possibility of the COTFE event on fuzzy scale.

1.0

Ranking

X8X43 X16X39 X6X39 X4X42 X22X41 X5X40 X16X41 X6X41; X8X42 X3X38; X17X38; X18X38; X20X38; X21X38 X3X39; X17X39; X18X39; X20X39; X21X39 X2X43 X4X40

1.083E-05 1.023E-05 8.217E-06 7.421E-06 5.118E-06 4.217E-06 4.089E-06 3.243E-06 2.912E-06

26 27 28 29 30 31 32 33 34

1.664E-06

35

1.507E-06 1.487E-06

36 37

1.152E-06 1.103E-06 5.941E-07

38 39 40

4.401E-07 4.233E-07 4.004E-07 3.497E-07 3.282E-07 3.153E-07 2.762E-07 1.852E-07

41 42 43 44 45 46 47 48

1.346E-07 1.092E-07

49 50

2.300E-03 1.229E-03 7.521E-04 6.165E-04 4.088E-04 4.007E-04 2.126E-04 1.955E-04 9.897E-05

1 2 3 4 5 6 7 8 9

X5X43

7.895E-05

10

X2X38 X1X38; X19X38 X7X38 X2X39 X1X39; X19X39

7.563E-05 5.928E-05

11 12

5.703E-05 3.862E-05 3.033E-05

13 14 15

2.918E-05 2.433E-05 2.410E-05 2.338E-05 1.956E-05 1.663E-05 1.561E-05 1.290E-05

16 17 18 19 20 21 22 23

X1X43; X19X43 X7X43 X3X41; X8X40; X17X41; X18X41; X20X41; X21X41 X22X43 X2X42 X11X15X38 X16X43 X1X42; X19X42 X7X42 X6X43 X11X15X39

1.264E-05 1.238E-05

24 25

X22X42 X16X42

X7X39 X22X38 X5X42 X4X43 X16X38 X2X41 X6X38 X1X41; X19X41 X22X39 X7X41

CS-I Index

X5X38 X5X39 X4X38 X5X41 X4X39 X8X38 X8X39 X4X41 X8X41

~ 36 ¼ ð0:089; 0:147; 0:221; 0:321Þ p

1.0

7

valve kept open since broke down’ occur simultaneously) has maximum probability of occurrence for this COTFE accident, which corresponds to the official investigation results. In addition, it can be seen that these weakest MCSs are mainly composed of the top five BEs by their FV-I values. The ranking results together open up the critical importance of the BEs including X38 (Breathing valve kept open since broke down), X5 (Fire work), X39 (Gauge hatch often open), X41 (Poor seal around manhole), X4 (Smoking) and X43 (Flexible connection pipe rupture). Such results can help decisionmaker take the targeted preventive measures, such as more strictly management regulation, security check and maintenance, to eliminate or mitigate the identified safety deficiencies, and hence prevent or reduce the occurrence of such COTFE accidents. 4.2. Traditional-based approach During the process of the traditional FTA of the COTFE, due to absence of accurate probability data for BEs, the generic data are used to roughly estimate of the COTFE occurrence probability. In general, BE generic probability data can be derived from reliability data handbook (SINTEF Industrial Management, 2002), expert judgments and statistical data in oil depots. These probability data are used in single-point form and are inherently uncertainty and imprecise. In this case study, the probability data for some of the BEs could hardly be obtained from reliability data handbook or statistical data, such as the BEs X4, X7, X16, X32, X37, etc. Hence, in order to ensure the consistency among all the BE probabilities and reasonable comparison with the fuzzy-based approach, the generic data for the BEs in the COTFE fault tree are also obtained from expert judgments, but each BE probability data is represented by a single possibility score as shown in Table 4.

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D. Wang et al. / Journal of Loss Prevention in the Process Industries xxx (2013) 1e9

Since there is no repeated BEs among all MCSs, the possibility of occurrence of the COTFE is achieved as follows (Wang, 1999):

Table 4 The generic data and FV-I measures for BEs in traditional COTFE FTA. BE

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 X24 X25 X26 X27 X28 X29 X30 X31 X32 X33 X34 X35 X36 X37 X38 X39 X40 X41 X42 X43

Expert judgment by a single possibility score Expert 1

Expert 2

Expert 3

0.14 0.15 0.02 0.26 0.35 0.12 0.14 0.25 0.05 0.10 0.05 0.14 0.26 0.40 0.20 0.06 0.04 0.02 0.16 0.05 0.04 0.15 0.16 0.06 0.30 0.06 0.15 0.16 0.04 0.12 0.05 0.12 0.05 0.14 0.05 0.05 0.12 0.38 0.26 0.02 0.28 0.15 0.05

0.08 0.06 0.05 0.15 0.24 0.06 0.16 0.16 0.05 0.08 0.15 0.05 0.16 0.16 0.20 0.12 0.02 0.04 0.06 0.03 0.05 0.05 0.05 0.04 0.15 0.05 0.05 0.05 0.05 0.05 0.15 0.05 0.15 0.05 0.15 0.12 0.08 0.14 0.15 0.02 0.12 0.05 0.15

0.12 0.12 0.02 0.24 0.30 0.06 0.06 0.15 0.12 0.06 0.06 0.05 0.14 0.14 0.35 0.08 0.04 0.03 0.12 0.02 0.02 0.08 0.06 0.15 0.20 0.15 0.06 0.03 0.15 0.05 0.04 0.16 0.04 0.05 0.12 0.28 0.15 0.25 0.25 0.03 0.15 0.04 0.14

Aggregated possibility score 0.115 0.112 0.030 0.219 0.300 0.083 0.122 0.191 0.071 0.082 0.085 0.084 0.192 0.245 0.245 0.085 0.034 0.029 0.116 0.035 0.037 0.097 0.095 0.081 0.222 0.084 0.091 0.086 0.076 0.077 0.079 0.110 0.079 0.084 0.103 0.141 0.116 0.264 0.116 0.023 0.190 0.085 0.109

FV-I measures

Y

NG

Index

Ranking

0.1232 0.1177 0.0323 0.2299 0.3284 0.1138 0.1549 0.2248 0.0836 0.0922 0.0750 0.0460 0.0718 0.0845 0.0662 0.0906 0.0373 0.0337 0.1280 0.0385 0.0419 0.1039 0.0749 0.0746 0.1433 0.0327 0.0385 0.0579 0.0543 0.0544 0.0554 0.0667 0.0327 0.0327 0.0498 0.0588 0.0669 0.4608 0.4712 0.0548 0.3952 0.2113 0.2469

12 13 42 6 4 14 9 7 19 16 20 34 23 18 26 17 38 39 11 37 35 15 21 22 10 41 36 28 32 31 29 25 40 40 33 27 24 2 1 30 3 8 5

PTE ¼ W

r¼1

q xi ˛Gr i

(22)

where PTE denotes the possibility of top event; NG represents the number of all MCSs; r denotes the ordinal numbers of MCSs; xi ˛ Gr represents the ith BE belongs to rth MCS; qi denotes the possibility score of the ith BE. Using Eq. (22), the occurrence possibility of the COTFE event is 0.850. Then, according to the probability conversion formulas of Eqs. (18) and (19), the occurrence probability of the COTFE is obtained and the result is 5.141  102. The importance of each BE is also measured based on their FV-I index using Eq. (20) and results are also shown in Table 4. It shows that the top ten critical BEs are X39, X38, X41, X5, X43, X4, X8, X42, X7 and X25 respectively. The FV-I index of all the MCSs are also calculated by Eq. (21) and the CS-I ranking of top 50 MCSs is listed in Table 5. As shown in Table 5, the MCSs ranked the top ten leading contributions to the COTFE probability are X5X38, X5X39, X4X38, X5X41, X8X38, X4X39, X8X39, X4X41, X8X41 and X5X43 respectively. According to the FV-I and FV-I ranking result by the traditional-based approach, the most critical BEs which have to be given utmost attention are X39, X38, X41, X5, X4 and X8. 4.3. Results and discussion The calculations have been carried out by fuzzy-based approach and traditional approach. Table 6 presents the final important results for comparison between the two approaches. The results show that: 1) the occurrence probability value of the COTFE by the fuzzy approach is about 12% lower than the value by the traditional approach; 2) The fuzzy FTA provides the detailed information about the contribution of linguistic rating scale (H and VH) to the COTFE probability, whereas such information is unknown from traditional FTA; 3) there is slight difference in the most critical BEs and big difference in the ranking of these BEs. The main reason for the differences mentioned above is that the fuzzy FTA approach distributes all BE data uncertainty in the whole triangular or trapezoidal region and thus attempts to represent a more realistic scenario as compared to the traditional approach. In reality, it is unreasonable to evaluate the occurrence of each BE by using a single-point estimate without considering

Table 5 The CS-I ranking of top 50 MCSs in traditional COTFE FTA. MCSs

X5X38 X5X39 X4X38 X5X41 X8X38 X4X39 X8X39 X4X41 X8X41 X5X43 X7X38 X19X38 X1X38 X2X38 X7X39 X19X39 X22X38

CS-I

MCSs

Index

Ranking

1.192E-04 5.472E-05 2.851E-05 2.644E-05 1.483E-05 1.215E-05 6.099E-06 5.466E-06 2.655E-06 1.483E-06 1.397E-06 1.028E-06 9.683E-07 8.478E-07 5.045E-07 3.649E-07 3.562E-07

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

X5X42 X1X39 X2X39 X4X43 X7X41 X16X38 X19X41 X6X38 X1X41 X22X39 X2X41 X8X43 X16X39 X4X42 X22X41 X6X39 X8X42

CS-I

MCSs

Index

Ranking

3.441E-07 3.424E-07 2.976E-07 2.286E-07 1.940E-07 1.589E-07 1.380E-07 1.324E-07 1.291E-07 1.190E-07 1.114E-07 9.685E-08 5.070E-08 4.553E-08 4.252E-08 4.181E-08 1.796E-08

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

X16X41 X6X41 X7X43 X19X43 X1X43 X2X43 X22X43 X7X42 X19X42 X21X38 X1X42 X2X42 X16X43 X20X38 X6X43 X17X38

CS-I Index

Ranking

1.734E-08 1.416E-08 4.272E-09 2.844E-09 2.626E-09 2.201E-09 6.955E-10 6.102E-10 3.924E-10 3.821E-10 3.600E-10 2.973E-10 2.377E-10 2.079E-10 1.865E-10 1.618E-10

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

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9

References

Table 6 The important results for two approaches. Fuzzy approach

Traditional approach

The COTFE probability

4.514  102 (H ¼ 67%, VH ¼ 33%)

5.141  102

The most critical BEs

X38 X5 X39 X41 X4 X43

X39 X38 X41 X5 X4 X8

the inherent uncertainty and imprecision a state has. Overall the noteworthy attributes of the fuzzy FTA approach, including the resilience towards lack of precision in the BE data and more detailed probability information provided, confirm that the fuzzy approach enables better probability assessment of the COTFE accident and more reliable identification of the most critical BEs, and hence provides effective help for risk management and decision making.

5. Conclusions According to the results of this study, the following conclusions are drawn: (1) The fault tree of crude oil tank fire and explosion (COTFE) is constructed, and the qualitative analysis of the tree shows that it totally includes 43 basic events and 392 minimal cut sets possibly leading to the accident. (2) The proposed approach which incorporates the fuzzy set theory and the conventional FTA technique is demonstrated as a viable and effective method for estimation of the COTFE occurrence probability when encountered with basic data uncertainty. (3) The approach can be used to perform the importance analysis of the COTFE fault tree which can provide valuable information for decision maker to improve the safety performance of the crude oil tank system.

Acknowledgments Authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (Grant No. 50974105) and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20105121110003).

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