A novel layered fuzzy Petri nets modelling and reasoning method for process equipment failure risk assessment

Journal Pre-proof A novel layered fuzzy Petri nets modelling and reasoning method for process equipment failure risk assessment Weijun Li, Min He, Yibo Sun, Qinggui Cao PII:

S0950-4230(17)30464-3

DOI:

https://doi.org/10.1016/j.jlp.2019.103953

Reference:

JLPP 103953

To appear in:

Journal of Loss Prevention in the Process Industries

Received Date: 16 May 2017 Revised Date:

2 August 2019

Accepted Date: 30 August 2019

Please cite this article as: Li, W., He, M., Sun, Y., Cao, Q., A novel layered fuzzy Petri nets modelling and reasoning method for process equipment failure risk assessment, Journal of Loss Prevention in the Process Industries (2019), doi: https://doi.org/10.1016/j.jlp.2019.103953. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

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Dear A Achuthan, Thank you very much for help us with the publication process. The affiliations are as follows: Weijun Li* a b, Min He a b, Yibo Sun c, Qinggui Cao a b a

b

College of Mining and Safety Engineering, Shandong University of Science and Technology, Qingdao, China

State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province

and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao, China c

Shandong HELP Safety and Environmental Protection Technology Pty Ltd.

* Corresponding author

College of Mining and Safety Engineering

Shandong University of Science and Technology, Qingdao, China

Tel.: +86 17505455080

E-mail address: [email protected]

Best wishes.

Weijun

A novel layered fuzzy Petri nets modeling and reasoning method for process equipment failure risk assessment Abstract: The continuous and stable operations of equipment are the crucial factor for the safety of process industries. Risk assessment has demonstrated its capabilities as a practical method to analyze and prevent process equipment failure. Since the status of equipment is mainly determined by its components and the coupling relationship is complex, a systematical and practicable method is needed to help risk assessment. Petri nets provide a graphical and mathematical representation for risk modelling and reasoning. With the complexity of equipment, there could be lots of diagnosis rules, which makes the network complicated and huge. In order to describe the coupling relationship clearly and make the computational process flexible, a layered Petri nets method is presented in this context to conduct complex rule-based risk analysis and assessment. Moreover, The fuzzy logic is introduced to represent expert knowledge in a semi-quantitative way. The method presented was applied for assessing the risk of reciprocating compressor failure, which is the supercharging equipment widely used in the natural gas transportation process.

Keywords: risk assessment; layered Petri net; fuzzy rules; process equipment failure

1 Introduction Risk assessment for the process industries has been used to prevent failures or accidents. Generally, the safety of the process is mainly determined by the state of equipment, which is further determined by its components. Therefore, it is necessary to analyze the risk of process equipment failure. For instance, hydraulic turbine generator unit risk assessment was conducted for the safe operation of hydropower stations (Xu, et al., 2019), the risk of long-distance pipelines was evaluated for the safe operation of oil and gas transportation system (Guo, et al., 2016). Although some quantitative risk assessment (QRA) methods have been applied extensively in the process industries, uncertainties about equipment failure rates and human factors lead to the derived results (Milazzo, et al., 2015; Pitblado, et al., 2011). Accordingly, human expert knowledge has been introduced to risk assessment for process equipment safety decision making. Knowledge-based risk assessment has been popular due to its advantages of transparent reasoning, uncertainty handling and explicit explanations (Gao, et al., 2015; Venkatasubramanian, et al., 2003; Zhao, et al., 2005). With expert knowledge represented by rules and then described in structured graphs, risk assessment process can be performed explicitly (Kramer and Palowitch, 1987). Two important parts of knowledge-based risk assessment are a) knowledge representation, and b) reasoning mechanisms. In terms of knowledge representation, it almost reaches a consensus that Petri net describes propositions as places and rules as transitions (Cheng, et al. 2015; Silva, 2013; Yang, et al., 2004). In addition, tokens are used in Petri net to represent the dynamic behavior of rule-based reasoning. Another important feature of knowledge-based risk analysis is the ambiguity of propositions and production rules. Therefore, fuzzy logic has been integrated into standard risk assessment methods (Nan, et al., 2008; Shen, et al., 2012). There has been a booming since the 1990s when many research workers focused on fuzzy reasoning Petri net formulas or models (Bugarin and Barro, 1994). Apart from fuzzy Petri net, modifications of the original Petri net include stochastic Petri net (Grunt and Bris, 2015; Talebberrouane et al., 2016), possibilistic Petri net (Lee, et al., 2004), timed Petri net (Mejía, et al., 2016), colored Petri net (Vernez, et al., 2003; Wang, et al., 2016), hybrid Petri net (Zhou and Reniers, 2016), and so on. With the increasing of reasoning data and rules, large-scale and hierarchical Petri nets have been established (Lu, et al., 2002; Saren, et al., 2017; Sheng and Prescott, 2017; Vatani and Doustmohammadi, 2015; Wang, et

al., 2016; Zhou, et al., 2015). Especially in the risk assessment domain, complex relationships exist between symptoms and faults, where a systematical and practicable method is required. The algorithm proposed by Gao (2000) can mapping Petri net into matrices, which make the reasoning and calculation processes structural and easier. Fuzzy Petri net model can formally define reasoning execution rules in algebraic forms. In addition, the formal reasoning algorithm and matrices make it easier to represent complex logical relations and implement calculations with a computer. Because of these advantages, fuzzy Petri net has gained great popularity and acceptance in many domains (Guo, et al., 2016; Wu, et al., 2011; Wu, et al., 2012; Zhong, et al., 2010; Zhou, et al., 2012; Zhou and Reniers, 2017). In this work, we use fuzzy reasoning Petri net as a knowledge-based risk assessment tool. Although a lot of improvements and optimizations have been made to the original Petri net, there remains no generally accepted versions of Petri net in the risk assessment field (Hu, et al., 2011; Liu, et al., 2013; Melani, et al., 2016; Son and Seong, 1997). Petri net has the advantages of concurrent failure analysis and dynamic behavior modeling, but it has gained few attentions in safety-related domains compared with fault tree analysis (FTA) (Yang and Liu, 1997). One of the important reasons may be that FTA provides a more structured way to represent cause-and-effect relationships, which can help managers identify direct causes, indirect causes, and root causes explicitly (Alkhaledi, et al., 2015; Celik, et al., 2010). Although hierarchical Petri net has been widely applied in the literature (Lu, et al., 2002; Saren, et al., 2017; Sheng and Prescott, 2017; Vatani and Doustmohammadi, 2015; Wang, et al., 2016), current models mainly focus more on the parent-and-child relationships of nets rather than cause-and-effect relationships. The hierarchical Petri net is designed to hide details in the form of compound transitions (Manoj, et al., 1998), which is a “double-edged sword”. The feature of hiding details could be advantage as well as disadvantage. For users in the industrial field, for example, the more detailed, the better. One may argue that HFPN is more suitalbe for complex systems since large networks are hard to observe and explain. But for a simple system (e.g. the industrial equipment), a detailed cause analysis of component failure is required so that maintenance measures can be taken. Hence, the objective of this paper is to present a novel layered fuzzy Petri net modeling and reasoning method for process equipment failure risk assessment. By applying the layered fuzzy Petri nets method to a

reciprocating compressor, proper maintenance scheduling and safety design strategies can be developed. The remainder of this paper is organized as follows. Section 2 describes the matrix representation of fuzzy Petri net, as well as the formulas of fuzzy reasoning. Section 3 discusses the stratification principles and the steps of drawing layered fuzzy Petri net. Section 4 introduces a reciprocating compressor risk assessment case to demonstrate the usefulness of the model. The final section is conclusions.

2 Background: matrix representations of Fuzzy Petri net In many cases of risk assessment, accurate statistical data is unavailable. Therefore, knowledge-based risk assessment has received increasing attention. To represent expert knowledge structurally and scientifically, fuzzy rules and graph theories have been proposed and fuzzy Petri net has been one of them. Fuzzy Petri net uses network structure to describe relationships of different statements and firing transitions to describe reasoning process. For example, the basic rules in the knowledge representation domain includes →c,

)a →b AND c,

)a OR b →c, and

)a→b,

)a AND b

)a →b OR c. Next, the production rule Ⅰ)a→b will be

taken as an example to illustrate the fuzzy mechanism and matrix representation process. The production rule Ⅰ)a→b represents that if proposition a is true then proposition b is true. However, uncertainties may exist with the propositions and the logical relation due to lack of accurate data. Therefore, fuzzy production rules and fuzzy Petri net have been introduced. The degree of truth of propositions and the certainty factors of logical relationships are used to describe the fuzzy production rules. The fuzzy Petri net of the fuzzy production rule Ⅰ)a→b is shown in Fig. 1. With Petri net, the propositions a and b are described as places and marked with P1 and P2. The rule Ⅰ is described as transition and marked with T1. Same to the standard Petri net, the circles represent places and bars represent transitions (Diaz, 2009; Peterson, 1981; Reisig, 1998). The strength of the belief in rule Ⅰ and the degree of confidence of the propositions are represented with the value between 0 and 1, marked with S1, D1, and D2. D1

S1

D2

D1

S1

D2

P1

T1

P2

P1

T1

P2

Fig. 1 The Petri net representation of fuzzy production rule Ⅰ)a→b

Whether or not a place contains a token(dot) depends on whether the corresponding propositions occur. During the risk assessment, the places containing a token refer to the occurrence of events that could cause system failures. The transition T1 is enabled if proposition a occurs, i.e. the place P1 contains a token. Then the token is removed from the input place (P1) into the output place (P2) after the transition. Since the aim of fuzzy reasoning is to obtain the possibility of the output place, the degree of confidence for P2 can be calculated by considering the confidence value of input place (P1) and transition (T1), i.e. D2=D1 * S1 (Negoita, 1985). This is the simplest form of a fuzzy Petri net and its reasoning process. If a production rule contains “and” or “or” connectors, it is the composite production rule. The other rules such as

)a

AND b→c, Ⅲ)a→b AND c, Ⅳ)a OR b→c, and Ⅴ)a→b OR c are composite production rules and correspond to four types of Petri net structure, shown in Fig.2. The rule a AND b→c is represented with join structure, which means that one transition has more than one input places. The rule a→b AND c is represented with fork structure, which means that one transition has more than one output places. The rule a OR b→c is represented with attribution structure, which means that one place has more than one input transitions. The rule a→b OR c is represented with choice structure, which means that one place has more than one output transitions (Liao, et al., 2011). For a more comprehensive description of the fuzzy reasoning process of these Petri net structures, the readers are referred to reference (Chen, et al., 1990). P2

P1

P2

T1

P3

(1) join: a AND b→c P1

P1

(2)fork:a→b AND c

T1

P2 T2

P3

T1

T1

P3

(3) attribution:a OR b→c

P1

P2

P3 T2

(4) choice:a→b OR c

Fig. 2 Four types of basic Petri net structure Furthermore, a matrix is needed in order to realize computer automatic reasoning. A unique group of input and output matrices, which corresponds to a particular type of Petri net, is defined, as shown in matrices (1) to (4).

T1 I(a) =

T1

D1 1  D1 0 ， O( a ) =   D 2 1  D 2 0 D 3  0  D3 1 T1

I (b ) =

T1

D1 1  D1 0  ， O(b ) =   D 2 0  D 2 1  D 3 0  D3 1 

T1 T 2 I (c) =

D 1 1 D 2  0 D 3  0 D1  1 D 2  0 D 3  0

(2)

T1 T 2

0 D1 0 0  ， O( c ) =  1 D 2 0 0  0  D3 1 1 

T1 T 2 I (d ) =

(1)

(3)

T1 T 2

1 D1 0 0 ， O( d ) =  0 D 2 1 0 0  D3 0 1

(4)

In many situations, the causal relationship is complex, especially for the process industries. Accordingly, the structure of the Petri net may be large, which increases the calculation account. To address this issue, Gao et al. (2003; 2004) defined a six-tuple

( P, R, I , O,θ , C )

to represent

fuzzy Petri net and developed the fuzzy reasoning algorithm to calculate the possibility of the output place. In the tuple, P refers to the set of places (i.e. propositions).

R refers to the set of transitions

(i.e. rules). I is an M × N matrix representing the input of transitions, where M is the number of places and N is the number of transitions. O is an M × N matrix representing the output of transitions. θ is a M ×1 set representing the certainty factor of propositions.

C is a diagonal matrix representing the confidence of transitions. The possibility value of propositions can be calculated according to equation (5). The symbol ⊕ means outputting the bigger one between two elements and ⊗ means outputting the maximum product of two elements and • is the standard matrix product.

(

)

θ k +1 = θ k ⊕  ( Ok • C k ) ⊗ I kT ⊗ θ k  

(5)

It should be noted that the calculation does not stop until θ k + 1 = θ k , which means that the

possibility value of intermediate places can not be obtained with this approach. Besides, it only provides the quantitative calculation while does not identify the direct causes and root causes explicitly.

3 Fuzzy Petri net based risk assessment methodology 3.1 Model representation and stratification principles Experts knowledge and experience can be represented with production rules. For the complex systems, the production rules may be interrelated and abstract. Therefore, a graphical model is needed to depict complex relationships among these rules. Here an example is presented to illustrate how to convert rules into Petri net. Assume that there are four fuzzy production rules, as shown below: Rule 1: IF P1 THEN P2 AND P6 (R1); Rule 2: IF P2 AND P3 THEN P5 (R2); Rule 3: IF P4 THEN P6 (R3); Rule 4: IF P5 THEN P6 (R4). In the Petri net, P1~P6 can be represented by places and R1~R4 can be represented by transitions. A token is put into the place if the corresponding cause is observed or detected. A transition is fired if the corresponding rule is true. Therefore, the production rules can be represented by the Petri net in Fig. 3. It should be noted that these rules are not for all the fault cases and the rules and Petri net vary with different cause-and-effect structures. R3 P4 P6 P1

R1 P2

R2

P5

R4

P3

Fig. 3 An example of rule-based Petri net We can see from the process that Petri net uses a bottom-up approach of depicting connections among places. By comparison, as another widely used graphical technique, fault tree use a top-down approach of representing the logical relationships between faults and their causes (Liu, et al., 2015). Compared with Petri net, fault tree can explicitly distinguish root causes and direct

causes, as shown in Fig. 4. The events without input are categorized as root causes and those directly connected to the top event are direct causes. The regular structure can help safety analysts understand the causal relationships clearly and each group of input events, logic gate, and output events can be extracted as sub-fault tree intuitively. 4

P1

P4 P1

R1

P6

P2 P5

R2 P3

Fig. 4 Rule-based fault tree To make Petri net more structural and hierarchical, a layered Petri net is developed and stratification principles are established. First, count the number of transitions from each input places to output places. The output places with the same input transitions number, their corresponding input transition bars, and the input places connected to these bars are in the same layer. Take the case of Fig.3 as an example. The numbers of transitions are counted and the Petri net is decomposed, shown in Table 1. Table 1 The numbers of transitions and layers of Petri net Input places to output places

Numbers of transitions

Layers of Petri net

P1→P2

1

The 1st layer

P1→P6

3

*

P2→P5

1

P3→P5

1

P4→P6

1

P5→P6

1

The 2nd layer

The 3rd layer

In particular conditions (marked with * in Table 1)where there are more than one propagation paths from one place to another, auxiliary place(s) and transition(s) will be added to help maintain the layered structure. For example, there are two propagation paths from P1 to P6, which are P1→P2→P5→P6 and P1→P6. Auxiliary place P’ and auxiliary transition R’ should be added. Note

that the possibility of P’ is the same with P1 and the confidence of R’ is 1. According to stratification principles, the Petri net in Fig.3 can be decomposed into three layers, as shown in Fig. 5. The three blocks with different colours represent three layers.

R3 P4 R’ P’ P1

P6

R1

R4 R2

P2

P5

P3

Fig. 5 A case of layered Petri net The layered Petri net provides a higher regular structure compared with the original ones. In addition, each subnet can be conducted further analysis independently and the causal relationships become more intuitive. In Fig.5, P1, P3, and P4 have no input, so they are the root causes. In the domain of equipment failure risk analysis, the root causes are defined as the failures that could be eliminated. Places such as P4, P’(P1), and P5 are directly connected to the final place, so they are included as direct causes. P1 has a dual role as the root cause and the direct cause. It can be seen that the layered Petri net integrates the advantages of both the ordinary Petri net and fault tree model. Apart from the graphical form, the calculation formula is also slightly adjusted to accommodate the layered model. For each layer, compute truth degree vector θ k according to the following equation (adapted from Gao, et al., 2003; 2004):

(

)

θ k = θ k −1 ⊕ ( Ok −1 • C k −1 ) ⊗ I k −1T ⊗ θ k −1  (1 ≤ k ≤ m ) 

(6)

It should be noted that the variable k is a positive integer and m is determined by the number of layers. The steps of inference depend on the number of layers. 3.2 General layered fuzzy Petri net model On the basis of knowledge representation and stratification principles, a general layered fuzzy Petri net model is established, as shown in Fig. 6.

Expert knowledge and experience

Final state calculation for each layer

(

)

θ k = θ k −1 ⊕ ( Ok −1 • Ck −1 ) ⊗ I k −1T ⊗ θ k −1  (1 ≤ k ≤ m )

Petri net representation of



fault diagnosis rules IF... THEN...

Initial state determination for each layer

θ k −1 , I k −1 , Ok −1 , and Ck −1

Petri net decomposition

m layers

For k=1:m k=k+1

Direct causes The loop stops running at any layer

Indirect causes Root causes

Fig. 6 Steps to conduct risk assessment with layered fuzzy Petri net The reasoning algorithm is as follows: 1) Represent fault diagnosis rules into fuzzy Petri net. In the domain of risk assessment, the propositions are described as places and the rules are described as transitions. The strength of the belief in the rules and the degree of confidence of the propositions are represented with the value between 0 and 1. Graphical representations are completed in this step. 2) Decompose a Petri net into m (m≥2) layers according to the stratification principles presented in Section 3.1. Add auxiliary place and transition when necessary. 3) For each layer, confirm the initial state of variables including θ

k −1

,I

k − 1

, Ok−1 , and Ck−1 . The

fuzzy values are represented with the degrees of confidence, which are numbers between 0 and 1. Experts can assign their confidence a value based on their experiences and knowledge. The value range can be obtained according to Table 2 (Chen, 1988). Table 2 Fuzzy linguistic description and fuzzy value Fuzzy linguistic description

Fuzzy values

Always

[1.00, 1.00]

Very strong

[0.95, 0.99]

Strong

[0.80, 0.94]

More or less strong

[0.65, 0.79]

Medium

[0.45, 0.64]

More or less weak

[0.30, 0.44]

Weak

[0.10, 0.29]

Very weak

[0.01, 0.09]

No

[0.00, 0.00]

4) Start the fuzzy reasoning from the first layer, where k =1. Compute θ k according to equation (6). 5) Go to the next Petri net layer and return to step 3. The reasoning stops after k times calculations.

4 Rule-based reciprocating compressor risk assessment To illustrate the application of the reasoning algorithm, risk analysis of a reciprocating compressor is presented in this section. Reciprocating compressors are generally used in petrochemical industries. However, complex structure and poor operation conditions usually lead to reciprocating compressor failure (Corvaro, et al., 2017; Cui, et al., 2009). The sketch of a reciprocating compressor is shown in Fig. 7.

Crosshead Piston ring

Crankshaft

Piston rod

Cylinder

Fig. 7 The sketch of a reciprocating compressor With expert knowledge and experience, some faults can be predicted and diminished before evolving into major accidents. In this work, only parts of equipment failures and causes are considered. In an expert system, expert knowledge is represented by if-then production rules (Amiri, et al., 2017). Three common failures of reciprocating compressor are chosen to be analyzed to construct possible fault propagation paths based on expert knowledge.

4.1 Reciprocating compressor diagnosis rules Faults like deficient discharge volume, high discharge temperature, and low discharge pressure are common in reciprocating compressor failures. The causes of these failures are summarized as rules and their corresponding confidences are listed in Table 3. For the convenience of subsequent Petri net analysis, each proposition is assigned a code P x and each rule is assigned a code R x . The confidence indicates to what degree experts are certain about the rule. Table 3 Reciprocating compressor diagnosis rules Rules

Confidence

Lack of lubrication (P1)  → leakage from stuffing box (P9)  → deficient discharge volume (P15)

R1(0.80) R9(0.95)

2 Wrong installation of stuffing box (P2)  R → leakage from stuffing box (P9) R9  → deficient discharge volume (P15)

R2(0.85) R9(0.95)

R10 3 Piston ring wear (P3)  R deficient → leakage from piston ring (P10) → R11 R12 discharge volume (P15) or → high discharge temperature (P16) or → low discharge pressure (P17)

R3(0.75) R10(0.90) R11(0.80) R12(0.90)

4 Wrong installation of piston ring (P4)  R → leakage from piston ring (P10) R10 R11 → deficient discharge volume (P15) or  → high discharge temperature R12 (P16) or → low discharge pressure (P17)

R4(0.80) R10(0.90) R11(0.80) R12(0.90)

R13 5 Inlet filter clogging (P5)  R deficient → deficient intake volume (P11) → discharge volume (P15)

R5(0.90) R13(0.95)

6 Inlet filter clogging (P5) and deficient cooling water (P6)  R → high discharge temperature (P16)

R6(0.95)

R1

R9

R14 Thermometer malfunction (P12) → high discharge temperature (P16)

R14(0.80)

7 Contaminant between valve seat and gaskets (P7)  R → leakage from suction R15 valves or exhaust valves (P13) → deficient discharge volume (P15) or R16 R17 high discharge temperature (P16) or → low discharge pressure (P17) →

R7(0.80) R15(0.90) R16(0.85) R17(0.90)

8 Wears of suction valves or exhaust valves (P8)  R → leakage from suction R15 valves or exhaust valves (P13) → deficient discharge volume (P15) or R16 R17 high discharge temperature (P16) or → low discharge pressure (P17) →

R8(0.75) R15(0.90) R16(0.85) R17(0.90)

R18 Pressure gauge malfunction (P14) → low discharge pressure (P17)

R18(0.80)

4.2 Knowledge representation and reasoning According to the diagnosis rules, a Petri net is drawn to describe the complex cause-effect relationships, shown in Fig. 8. Based on stratification principles, the net can be divided into two

layers. The arc from R 6 to R16 can be replaced by auxiliary place

P1

R1

P2

R2

P3

R3

P ' and transition R ' .

R9

P9 R10

R11 P4

R4

P10 P11

R5

P15

R12 R13

P5 R6 P’

P6

P12

R’ R14

R15

P16

R7

P7

R16 P13

P8

R8

P14

R17 P17 R18

Fig.8 Layered Petri net of reciprocating compressor risk assessment: a) Network circled by green dotted represents the first layer and corresponds to matrices (8)-(12) and b) Network circled by blue dotted represents the second layer and corresponds to matrices (13)-(17) The initial values are necessary for the quantitative reasoning process. We assume a scenario where the initial state is given by

[ P1, P2 , P3 , P4 , P5 , P6 , P7 , P8 , P12 , P14 ] = [0.6,0.4,0.7,0.5,0.8,0.6,0.9,0.7,0.6,0.6]

(7)

The truth degrees of the other places can be calculated according to the defined algorithms. First, the network circled by green dotted line in Fig.8 is represented by the following matrices. The input

places

include

P1 , P 2 , P3 , P 4 , P5 , P 6 , P 7 , P8

and

the

output

places

include

P 9 , P1 0 , P1 1 , P ', P1 3 . The calculation processes are as follows.

θ0 = [ P1 , P2 , P3 , P4 , P5 , P6 , P7 , P8 , P9 , P10 , P11 , P', P13 ]

T

= [ 0.6,0.4,0.7,0.5,0.8,0.6,0.9,0.7,0,0,0,0,0]

T

(8)

1 0  0  0 0  I 0 = 0 0  0 0  M  0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0  0 0  0 0  1 0  M  013×8

0 0 0 0 0 M 0

0 0 0 0 0 M 0

0 0 0 0 0 M 0

1 0 0 0 0 M 0

1 1 0 0 0 M 0

0 0 1 0 0 M 0

0 M  0  1 O0 =  0  0 0   0

0 0 0 0 0 0 0 M M M M M M M  0 0 0 0 0 0 0  1 0 0 0 0 0 0 0 1 1 0 0 0 0  0 0 0 1 0 0 0 0 0 0 0 1 0 0  0 0 0 0 0 1 1 13×8

(9)

(10)

C 0 = d iag ( 0 .8, 0 .8 5, 0 .7 5, 0 .8 0 , 0 .9 0, 0 .9 5, 0 .8 0 , 0 .7 5 ) 8 × 8

(11)

According to equation (6), θ 1 can be calculated as

(

)

θ1 = θ0 ⊕ ( O0 • C0 ) ⊗ I0T ⊗θ0  

(12)

= [ 0.6, 0.4, 0.7, 0.5, 0.8, 0.6, 0.9, 0.7, 0.48, 0.525, 0.72, 0.57, 0.72]

T

All the output places in this layer are marked with truth degrees, which will be the input of the next layer. Then the network circled by blue dotted line in Fig. 8 is regarded as the second layer and the initial state is given below. The input and output places of this layer are indicated by the updated truth degree set θ 1 .

θ1 = [ P9 ,P10 ,P11 ,P',P12 ,P13 ,P14 ,P15 ,P16 ,P17 ]

T

= [ 0.48, 0.525, 0.72, 0.57, 0.6, 0.72, 0.6, 0, 0, 0 ]

T

(13)

1 0  0  0 0 I1 =  0 0  0 0   0

0 1 0 0 0 0 0 0

0 0  0  0 0  0 1  0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 10×11

(14)

0 M  0 O1 =  1 0  0

0 0 0 0 0 0 0 0 0 0 M M M M M M M M M M  0 0 0 0 0 0 0 0 0 0  1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 0 1 0 0  0 0 1 0 0 0 0 0 1 1 10×11

(15)

0 1 0 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 1 0 0

C 0 = d iag ( 0 .9 5 , 0 .9 0 , 0 .8 0 , 0 .9 0 , 0 .9 5 , 1 .0 0 , 0 .8 0 , 0 .9 0 , 0 .8 5 , 0 .9 0 , 0 .8 0 ) 1 1×1 1

(16)

According to equation (6), θ 2 can be calculated as

(

)

θ2 = θ1 ⊕ ( O1 • C1 ) ⊗ I1T ⊗ θ1  

(17)

= [ 0.48, 0.525, 0.72, 0.57, 0.6, 0.72, 0.6,0.684, 0.612,0.648]

T

It indicates that under existing operation conditions, the truth degrees of deficient discharge volume (P15), high discharge temperature (P16), and low discharge pressure (P17) are 0.684, 0.612, and 0.648, respectively. 4.3 Results Based on the case study of reciprocating compressor risk assessment, it can be concluded that the layered fuzzy Petri net method is able to transform experts knowledge into the graphical model and calculate the probabilities of different equipment failure modes. Regarding the reciprocating compressor risk analysis, the diagnosis rules are obtained from the linguistic description. With Petri net, the cause-and-effect relationships of equipment failure modes can be identified intuitively. In addition, the hierarchical structure makes the calculation process more flexible by decomposing a whole net into several subnets. The places without inputs correspond to the root causes of reciprocating compressor failure and the input places in the second layer correspond to

the direct causes. The layered PN indicates not only the direct causes and root causes from the perspective of logic but also some physical meanings. For the reciprocating compressor case, the input places in the first layer indicate those failures that could be eliminated by taking maintenance measures. The input places in the second layer indicate symptoms of equipment failures, which can not be eliminated directly but can be observed. Take the diagnosis rules (P1→ P9→P15) as an example. The failure type of deficient discharge volume (P15) is the consequence and both “lack of lubrication” (P1) and “leakage from stuffing box (P9)” are causes. But P1 is the cause of P9 and the cause of P1 can be solved by adding lubrication. In this way, P1 is defined as the root cause. Another example, there are two input transitions for P13, which means that a token either in P7 or P8 could move to P15. In other words, contaminant between valve seat and gaskets (P7) and wears of suction valves or exhaust valves (P8) are two root causes for leakage from suction valves or exhaust valves (P13). Therefore, equipment maintenance suggestions are available from the cause-effect relationships. From the quantitative perspective, the fuzzy probability values of leakage from stuffing box (P9), leakage from piston ring (P10), deficient intake volume (P11), and leakage from suction valves or exhaust valves (P13) are 0.48, 0.525, 0.72, and 0.72. This result indicates that the risk of deficient intake volume (P11) and leakage from suction valves or exhaust valves (P13) is high, which should be paid more attention and schedule preventive maintenance. As for the calculation result of the second layer, the fuzzy probability values of deficient discharge volume (P15), high discharge temperature (P16), and low discharge pressure (P17) are 0.684, 0.612, and 0.648. This result indicates that the risk of deficient discharge volume (P15) is higher than the other two failure modes.

5 Conclusions and discussions Petri nets provide a graphical and mathematical representation for risk modelling and reasoning. In this work, the standard Petri net is extended with the layered and fuzzy algorithms in order to process risk assessment rules. From the perspective of qualitative analysis, the places in Petri nets correspond to possible accident causes. With the hierarchical structure, the cause-and-effect relationships of equipment failure modes can be identified. For example, the direct causes can be quickly located on the last layer of the net. From the perspective of quantitative reasoning, it represents risk analysis rules as matrics, which makes reasoning process easier especially for

complex systems. In addition, the hierarchical structure makes the calculation process more flexible by decomposing a whole net into several subnets. The reciprocating compressor case illustrates that the method can help implement complicated diagnosis rules and predict possibilities of failures. One may argue that all the values can be directly calculated with Equation (5) and it is not necessary to conduct the similar calculation twice. But in some situations where only part of the net is required to be calculated or the input values of other subnets are unavailable, it would be time-consuming or impracticable to use the original algorithm to calculate the values of the whole net. Compared with current hierarchical Petri net, the layered fuzzy Petri net in this work provides novel stratification principles to describe cause-and-effect relationships in a more structured way, which enhances the visibility. Besides, experts knowledge and experience can be integrated into the modelling and reasoning process based on fuzzy logic. With the layered structure, users could make decisions more quickly. The field staff can locate the places corresponding to the equipment failures and then take measures. It should be noted that the size of the net is not big and the structure is not complex in this work. So we can conduct the reasoning and calculation process manually. However, a software package would be necessary when the net becomes big and complex. The current work only developed a theoretical approach for analyzing the risk of equipment failure. Future research should include more hazardous scenarios.

Acknowledgements This paper is supported and funded by the program of Shandong Provincial Natural Science Foundation (Grant No. ZR2019BEE018), Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (2017RCJJ002), First-class Discipline of Mineral Engineering construction project of Shandong University of Science and Technology (Grant No. 01AQ01805). The authors would also like to thank the anonymous reviewers and authors of the references.

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Highlights:


A general layered fuzzy Petri net model was established for equipment failure risk assessment.




Root causes and direct causes can be identified intuitively from the layered Petri net.




Layered fuzzy Petri net inference of reciprocating compressor failure was conducted using diagnosis rules