Research ArticleResearch and Application of FTA and Petri Nets inFault Diagnosis in the Pantograph-Type Current Collector onCRH EMU Trains

Long-long Song1 Tai-yong Wang12 Xiao-wen Song3 Lei Xu3 and De-gang Song3

1School of Mechanical Electronic and Control Engineering Beijing Jiaotong University Beijing 100044 China2School of Mechanical Engineering Tianjin University Tianjin 300192 China3RampD Center CSR Qingdao Sifang Co Ltd Qingdao 266111 China

Correspondence should be addressed to Long-long Song song bjtu163com

Received 28 May 2015 Revised 8 September 2015 Accepted 14 September 2015

Academic Editor Yuanchang Xie

Copyright copy 2015 Long-long Song et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A fault tree is established based on structural analysis working principle analysis and failure mode and effects analysis (FMEA)of the pantograph-type current collector on the Chinese Rail High-Speed Electric Multiple Unit (CRH EMU) train To avoid thedeficiencies of fault tree analysis (FTA) Petri netsmodelling is used to address the problem of data explosion and carry out dynamicdiagnosis Relational matrix analysis is used to solve the minimal cut set equation of the fault tree Based on the established stateequation of the Petri nets initial tokens and enable-transfer algorithms are used to express the fault transfer process mathematicallyand improve the efficiency of fault diagnosis inferences Finally using a practical fault diagnosis example for the pantographs onCRH EMU trains the proposed method is proved to be reasonable and effective

1 Introduction

In recent years high-speed electric multiple unit (EMU)trains have become an important mode of transportation inChina The Chinese Rail High-Speed (CRH) EMU train isa large-scale intelligent system with complex structures Thereliability of an EMU train is directly related to the safety andefficiency of the high-speed heavy-load rail transportationsystem [1] The high-voltage traction drive system is oneof the most critical aspects of an EMU train system As acritical subsystem of the high voltage traction system thepantograph-type current collector plays an important role inproviding power for an EMU train by coupling directly withthe electric catenary wire [2 3] Efficient and accurate faultdiagnosis for the pantograph is vital for the safe and stableoperation of CRH EMU trains

Complexity is defined as a state in which many differentparts (hardware software organizational and human ele-ments) are related to each other in an interconnectedmanner[4] It is defined here in two forms structural complexityand fault complexityThe CRH EMU train system is complex

both in its structure and in its failure modes Though thestructure of the pantograph seems relatively simple theworking principles and failure modes of pantographs aresurprisingly complex For example there are many com-plex potential failure modes for a pantograph which arerelated to mechanical structures electrical facilities pneu-matic transmission network control andmany other aspectsThe operation of the pantograph on a CRH EMU train isautomatically controlled bymicroprocessorsThepantographsystem involves mechanical structures such as guide barselectrical equipment such as the traction converter andrubber products such as the base insulatorThe lifting processis controlled by air pressure The complex failure modes ofthe pantograph can involve cracks breaks fatigue pittingwear and discharge breakdown The relationship betweendifferent failures is complex and uncertain Consequentlythe pantograph system of a CRH EMU train is treatedas a complex system A useful approach combining faulttree analysis and Petri nets theory is applied to deal withthe complexity of fault diagnosis in the pantograph sys-tem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 169731 12 pageshttpdxdoiorg1011552015169731

2 Mathematical Problems in Engineering

Fault tree analysis (FTA) is a useful analytical tool foridentifying and classifying hazards and calculating systemreliability for both simple and complex engineering systems[5] It is a systematic way to assess the reliability of com-plex systems both qualitatively and quantitatively [6] Muchresearch has been conducted and outcomes include extendedfuzzy FTA (FFTA)methodology in the petrochemical processindustry in fuzzy environments [7] the use of a fuzzy-logic-based reliability approach to evaluate basic events infault tree analysis for nuclear power plant probabilistic safetyassessment [8] FFTA applied to reducing uncertainty inexpert judgment in the safety barriers analysis of an offshoredrilling system [9] and the use of fault tree analysis andanalytic hierarchy processes to analyze the risks associatedwith the use of shield tunnel boring machines (TBMs) [10]

However the modelling power of FTA is limited to thestatic evaluation of a single criterion at a time and FTAis not capable of describing dynamic system behavior withredundant components degraded system states and repairor test activities [11] Leveson and Stolzy proposed the safetyanalysis of dynamic systems using time Petri nets [12] Bobbioet al provided an algorithm to convert a parametric faulttree (PFT) into a class of high-level Petri nets to exploit themodelling power and flexibility of the stochastic well-formednets (SWN) formalism [13] Robidoux et al used a frame-work named dynamic reliability block diagrams (DRBD)for modelling the dynamic reliability behavior of computer-based systems and presented an algorithm that automaticallyconverted a DRBD model into a colored Petri net [14] Withincreasing complexity in the relationship between systemstructures and faults fault trees become more difficult toestablish Problems relating to the Nondeterministic Poly-nomial (NP) which can lead to combinatorial explosion asthe numbers of calculations grow exponentially appear whensolving the minimal cut sets Makajic-Nikolic et al proposedan approach to cutting the number of minimal cut sets in afault tree using reverse Petri nets [15] There have also beensome studies on the application of FTA and Petri nets intransportation systems For example Wang et al proposedreliability modelling and evaluation issues of electric vehiclemotors by using FT FTA-based extended stochastic Petrinets [16] Nguyen et al combined a Petri nets (PN) methodwith extensions of FTA to adapt to dynamic systems andapplied the method to a satellite-based railway system [17]and Song et al applied T-S Fuzzy theory and FTA to diagnosepantograph faults in a multisource heterogeneous knowledgeenvironment [18]

Some other mathematicalstatistical methods such asfuzzy reasoning and Bayesian Network (BN) also have beensuccessfully used in the railways risk and safety domainFor example An et al proposed a railway risk managementsystem for railway risk analysis using fuzzy reasoningapproach and fuzzy analytical hierarchy decision makingprocess [19 20] Noori and Jenab developed a fuzzy Bayesiantraction control system for rail vehicles with speed sensorsin intelligent transportation systems [21] Muttram applieda Safety Risk Model combining fault tree analysis andcauseconsequence techniques to predict residual levels ofrailway safety risk [22] Bouillant et al developed a decision

support tool based on Bayesian Networks to evaluatecompare and optimize various operating and maintenancestrategies of Paris metro underground rails [23] Xie etal introduced the Bayesian inference and investigated theapplication of a Bayesian ordered probit (BOP) model indriverrsquos injury severity analysis and verified that BOP modelcould produce more reasonable parameter estimations andbetter prediction performance than the ordered probit (OP)model [24] Bernardi et al generated Repairable Fault Treeand Bayesian Network models for railway modelling by amodel-driven approach called DAM-Rail approach [25]Mahboob and Straub compared fault tree and BayesianNetworks for modeling safety critical components in railwaysystems [26] andWashington andOh incorporated Bayesianmethodology with expert judgment for countermeasureeffectiveness under uncertainty and applied the approach inimproving the safety of railroad crossings [27]

From the above it appears that the conventional analysisapproaches such as FTA Petri nets (PN) fuzzy reasoningand BN have been widely used in the railways risk and safetyanalysis field However each approach has its own advantagesand limitations Fault tree Petri nets and BN have a strongsimilarity in many aspects Fault tree can be establishedmore easily only when the cause-and-effect relationshipsbetween events were clear But it suffers severe limitations ofstatics structure and uncertainty handling Fuzzy reasoningand BN are effective tools for quantitative analysis becausethey are based on probabilistic and uncertain knowledgeBN approach allows dealing with issues such as predictionor diagnosis optimization and data analysis of feedbackexperience [28] It is more often used in prediction becauseof good quantitative analysis ability but the training processof BN is complex And the dependent probabilities amongevents are required in BN approach which may be difficultto obtain in some cases Petri net (PN) is also a powerful andwidely used graphical and mathematical modelling tool forthe description of sequence dependent behaviors of dynamicsystems When the failure status of a system evolves fromone subsystemcomponent to other subsystemscomponentsPetri net modelling is more intuitive and more effectiveto describe the process by carrying mathematical matrixcomputations which is easier to handle by computer Atthe same time it is more suitable for large complex systemsbecause the modelling process is simplified by eliminatingrepeat basic events and reducing the building elementsThrough the above analysis FTArsquos translation into Petri nets ismore suitable in this paper to analyze the dynamic behaviorsof failure statuses of Pantograph-Type Current Collector onCRH EMU trains mathematically as far as the extension fastmodelling and dynamic behavior analysis of the system aremainly concerned

Studies on reliability and failure mode analysis of CRHEMU trains and their subsystems have recently begun Thefocus of the research has turned from design and manu-facture to maintenance management FTA and Petri netsapproaches are applicable and useful analysis tools in the riskmanagement of complex engineering systems but there aresome deficiencies in their application Considerable researchhas been conducted on these problems The main aim of

Mathematical Problems in Engineering 3

this paper is to extend FTA and Petri nets methodology tomaintenance and fault diagnosis in CHR EMU train systemsThis section introduces some existing applications of FTAandPetri nets in a range of industries Structural analysis workingprinciples failure mode and effects analysis (FMEA) andFTAmodelling of pantographs are outlined in Section 2 Petrinetsmodelling of the fault tree and the use of relationalmatrixanalysis in solvingminimal cut sets are provided in Section 3In Section 4 the place mark and enable-transfer algorithm ofthe Petri nets and an actual case study on dynamic transitionand diagnosis of pantograph diagnostics are provided Thelast section emphasises the highlights of this research

2 Structural Analysis and Fault TreeModelling of Pantograph

21 Structural Analysis and Working Principle of PantographThe pantograph of a CRH EMU train is fitted on the roofof the train and is essential to allow the train to get powerfrom the main overhead wires A CRH EMU train is aneight-car multiple unit configured with four motorised carsand four trailer cars There are two power units in all andeach power unit consists of two power cars and two trailercars arranged in T-M-M-T mode DSA250 pantographs arefitted on number 4 and number 6 cars Their maximumlifting height is 3000mm and the width of the head is1990mm In normal operation there will be only one pan-tograph collecting current with another in a folded stateHowever when two CRH EMUs are attached together twopantographs will work simultaneouslyTheworking principleand structural components of pantograph systems for CRHEMUs are shown in Figure 1The structural parameters of thepantograph are shown in Table 1

The pantograph is raised to access high voltage power byallowing the carbon skateboard to contact the catenary wireand descends when its compressed air supply is exhaustedThe 25 kv single-phase power alternating current (AC) fromthe catenary is transferred to the traction transformer fromthe high-voltage electrical equipment by the pantographwhich outputs 1500V single-phase AC power to the tractionconverter A pulse rectifier converts the single-phase AC intodirect current (DC) and outputs 2500ndash3000V DC to thetraction inverter through the DC circuit Then the tractioninverter outputs three-phaseACpowerwhere the voltage andfrequency are all adjustable to drive the traction motor

When the pantograph rises air is compressed into thedrive cylinder through the cushion valve and the cylinderpistonmoves left overcoming the pressure of the reset springThen the lower arm rises and rotates clockwise under theaction of the guide bar and the spring At the same time theupper arm also rises with the drive of the top guide bar

When the pantograph descends the compressed air isremoved from the drive cylinder through the cushion valveThen the piston is pushed to the right with the reset springreleasing pressureThe guide bar alsomoves right to force thelower arm to rotate anticlockwise thus forcing the upper armto descend

The collector head contacts the catenary wire when thepantograph rises Current is led to the bottom frame through

the collector head the upper arm and the pushrod Thepower cable installed on the bottom frame then leads thecurrent to the vehicle Since current will flow through theentire pantograph frame in the power supply state all thepantograph hinges are equipped with bridge connections toprevent the current damaging the bearings

The performance of the pantograph largely dependson contact pressure If there is too little pressure contactresistance will vary easily resulting in poor contact andarcing However toomuch pressure will increase the frictionaggravating wear on the carbon skateboard and wires andreducing the life of the carbon skateboard

22 Failure Mode and Effects Analysis (FMEA) of PantographSystem Pantographs on CRH EMU trains are installed onthe roof exposed to the environment The working environ-ment is complex volatile and sometimes very harsh Therearemany and complex ways in which the pantograph can failwhich are related to mechanical structures electrical facil-ities pneumatic transmission network control and manyother factors The normal operation of the pantograph on aCRH EMU train is automatically controlled by microproces-sors The pantograph system involves mechanical structuressuch as guide bars electrical equipment such as the tractionconverter and rubber products such as the base insulatorThe lifting process is controlled by air pressure Because pan-tographs connect directly with the extra high voltage (EHV)catenary wire they can be easily damaged by partial highvoltage discharges By classifying and analyzing fault trackingrecords the most common pantograph failure modes aresummarized as follows

(i) Failure Mode 1 Pantograph rises to an abnormal position

Probable cause

When two EMUs attach together the spacebetween the two pantographs is less than 190m

(ii) Failure Mode 2 Pantograph rises normally while themonitor (MON) does not show it correctly

Probable causes

(a) The pressure sensors are not operating correctly(b) The pressure switches have failed

(iii) Failure Mode 3 Pantograph cannot rise

Probable causes

(a) Electricity generation system (EGS) is closed(b) Vacuum circuit breaker (VCB) is closed(c) Pressure in the auxiliary air cylinder is too low(d) [PantographsdotVCB] NFB is in the OFF position(e) [PantographRising]NFB is in theOFF position(f) Pantograph pressurised air ducts leak(g) White air ducts connected to the pantograph

leak(h) Air ducts in the pantograph control valve board

leak

4 Mathematical Problems in Engineering

Pantograph bow

Carbon contact strip

Upper arm

Lower arm

Lift device for bow

Top guide bar

Coupling rod

Base frame

System damper

Filter capacitor

Inve

rter

Pulse

re

ctifi

er

Pulse

re

ctifi

erFilter capacitor

Inve

rter

Traction motor

Traction motor

Traction transformer

Main converterMain converter

Vacuum breaker

Pantograph

X4X4

Connection set knee

Pan stop rod

Air bellows

M1M2

Figure 1 Working principle of traction drive systems and pantograph structure in a CRH EMU train

Table 1 Structural parameters of the pantograph on a CRH EMU train

Structural parameters Values Structural parameters ValuesModel DSA250 Vent lifting height 3000mm (including insulator)Structure Single arm pantograph Maximal action height 2800mm (including insulator)Rated voltage 25 kV Minimal action height 888mm (including insulator)Rated current 1000A Folded height 593mm (including insulator)Weight 115 kg Contact pressure 70N plusmn 5NAmbient temperature minus40∘Csim+60∘C Design life 30 years

(i) Pantograph bellows are damaged(j) The pantographrsquos carbon skateboard leaks

(iv) Failure Mode 4 Pantograph cannot be lowered

Probable cause

Contacts in the pantographrsquos rising relay haveadhered

(v) Failure Mode 5 Pantograph cannot pass neutral sectionautomatically

Probable cause

Terminal RXCB board has failed

(vi) Failure Mode 6 Supporting insulator is damaged

Probable causes

(a) Surface defects created during manufacture(b) Insulator being burned by electrical arcing(c) Fog flashovers discharged because of high

humidity or dust(d) Fatigue cracks(e) Insulator being struck by foreign objects

(vii) Failure Mode 7 Pantograph descends automatically

Probable causes

(a) Pressure air ducts in the pantograph leak(b) White air ducts connected with the pantograph

leak(c) Air ducts in the pantograph control valve board

leak(d) Pantographrsquos bellows are damaged

Mathematical Problems in Engineering 5

(e) The pantographrsquos carbon skateboard leaks(f) Incorrect manipulation(g) Pantograph descending loop is instantly ener-

gized by electromagnetic interference(h) Pressure from the catenary is too low(i) There has been discharge between the panto-

graph bracket and the roof

(viii) Failure Mode 8 The catenary voltage transformer isdamaged

Probable causes

(a) Struck by foreign objects(b) Burned by electrical arcing(c) Quality control issues in components

23 Fault TreeModelling of the Pantograph Fault tree analysis(FTA) is one type of FMEA which is integrated withmechanics graph theory optimization theory and artificialintelligence techniques [29] It has been widely applied inmany fields including aerospace systems atomic reactorslarge-scale equipment and electronic computer systems FTAcan indicate causal relationships between complex faults andcan be useful for logical analysis and diagnosis of complexsystem faults It is a systematic way to assess the reliability ofcomplex systems both qualitatively and quantitatively

FTA is a systematic risk analysis method that deals withthe occurrence of an undesired event The fault event thatanalysts do not expect to happen is usually the focus of theFTAmethod Analysts apply top-down logic to find all directand indirect fault events relating to the incident They canthen establish logical relationships between the events form afault tree and undertake quantitative or qualitative analysisIn the process above the focus fault event is the top eventSelection of the top event is crucial to fault tree modelling Ifthe top event is too general it is difficult to analyse the faulttree On the other hand if the event is too specific the faulttree will fail to show the causal relationships of the systemfully In general the failure of the system analyzed will beselected as the top event

Combined with analysis of the structures and workingprinciples of pantographs on CRH-trains a pantograph faulttree has been established as shown in Figure 2 ldquo119879rdquo representsthe top event ldquo119872rdquo represents the intermediate event and ldquo119883rdquorepresents the basic event

Meanings of events in Figure 2 are shown as follows

119879 pantograph fails to work1198721 pantograph rises to an abnormal position1198722 pantograph rises normally but the MON doesnot show it correctly1198723 pantograph cannot rise1198724 pantograph cannot be lowered1198725 pantograph cannot pass neutral section automat-ically1198726 supporting insulators are damaged

1198727 pantograph descends automatically1198728 the catenary voltage transformer is damaged1198729 pressure of the auxiliary air cylinder is too low1198831 two EMUs are attached together1198832 spacing of two pantographs is less than 190m1198833 the pressure sensors do not work properly1198834 the pressure switches have failed1198835 EGS is closed1198836 VCB is closed1198837 in the cab switchboards at both ends[PantographsdotVCB] NFB is in the OFF position1198838 in the running switchboards of the number 4 andnumber 6 cars (CRH-200 CRH-300)number 4 andnumber 13 cars (CRH long-distance seat car CRHlong-distance sleeper car) [Pantograph Rising] NFBis in the OFF position1198839 pressure air ducts of the pantograph leak11988310 air ducts in the pantograph control valve boardleak11988311 pantograph bellows are damaged11988312 pantographrsquos carbon skateboard leaks11988313 white air ducts connected with the pantographleak11988314 contacts in the pantographrsquos rising relay haveadhered11988315 terminal RXCB board has failed11988316 surface defects due to manufacturing errors11988317 burned by electrical arcing11988318 fog flashovers discharge11988319 fatigue cracks11988320 struck by foreign objects11988321 driver error11988322 pantograph descending loop is instantly ener-gized by electromagnetic interference11988323 discharge between the pantograph bracket andthe roof11988324 pressure of the catenary is too low11988325 burned by electrical arcing11988326 air ducts of the air compressor are damaged11988327 the air compressor fails to work11988328 MR ducts are damaged11988329 main air cylinder leaks

The fault tree in Figure 2 shows the causal relationshipsbetween the pantograph fault events improving the accuracyand efficiency of fault diagnosis However the modellingpower of FTA is limited to the static evaluation of a singlecriterion at a time and cannot describe dynamic system

6 Mathematical Problems in Engineering

T

M4 M5 M6M3 M7M2 M8M1

X2X1

X13X5

X14 X15

X20X16

X25X16X4X3

X6 X7 X8 X9 X10 X11 X12M9

X26 X27 X28 X29

X17 X18 X19

X9 X10 X11 X12 X13 X21 X22 X23 X24

X17

Figure 2 FTA modelling of the pantograph-type current collector

behavior It cannot show the transmission and evolution offaults in the system It also needs too much computation tosolve the minimal cut sets of the large-scale fault tree whichmay lead to combinatorial explosion because of Nondeter-ministic Polynomial (NP) problems For example if (119894 =1 2 119899) is set to the 119894th minimal cut set of the fault treethen the top event can be expressed as

119879 =

119899

⋃

119894=1

119862

119894 (1)

The occurrence probability of the top event can beexpressed as

119875 (119879) = 119875(

119899

⋃

119894=1

119862

119894) (2)

The logical relationship between the basic events and theminimal cut set is and If the probabilities of the basic eventsare known the occurrence probability of the minimal cut setcan be expressed as

119875 (119862

119894) = 119875(

119896

⋃

119895=1

119909

119894) (3)

Based on the occurrence probabilities of the minimalcut sets the occurrence probability of the top event can beexpressed as

119875 (119879) =

119899

sum

119894=1

119875 (119862

119894) minus

119899

sum

119894lt119895=2

119875 (119862

119894119862

119895) + sdot sdot sdot

+ (minus1)

119898minus1

119875(

119899

⋃

119894=1

119862

119894)

(4)

According to the equation above the process of deter-mining the probability of the top event consists of 2119899minus1 partsThere are 28 minimal cut sets for the pantograph fault treein Figure 2 so it needs 227 parts to calculate the probabilityof the top event which means 134217728 items in all Thecombinatorial explosion of the calculation will make quanti-tative analysis difficult In addition the FTA method is gen-erally for static analysis and cannot reflect dynamic changesin multiple states in the system

The dynamic and structural properties of Petri nets areused to simplify and analytically calculate the pantographfault tree Relational matrix analysis is used to solve theminimal cut set equation for the fault tree Based on the estab-lished state equation of Petri nets initial token and enable-transfer algorithms are used to express the transfer pro-cess of faults mathematically

Mathematical Problems in Engineering 7

Table 2 Petri net expression of the FTA logic gates

AND logic gate OR logic gateFault tree expression Petri net expression Fault tree expression Petri net expression

3 Petri Nets Modelling and RelationalMatrix Analysis

31 Petri Nets Modelling of the Fault Tree The Petri net wasproposed in 1962 by Petri to express an information flowmodel for reticular structures [30] It is a mathematical andgraphical analysis tool to express the static structures anddynamic changes in a system [31ndash33]

In a Petri net the system state is indicated by ldquoIrdquo Achange in state is indicated by ldquo|rdquo An ordered pair is indicatedby a directed arc ldquorarr rdquo and a token is indicated by ldquoerdquo ThePetri net expressions of the FTA logic gates are shown inTable 2 The transform between Petri net modelling and thefault tree is shown in Figure 3

One Petri net can be defined in a sextuple 119873 =

(119875 119879 119868 119874119872119872

0) The six elements should meet the follow-

ing conditions

(1) 119875 = 119875

1 119875

119899 is a finite set of places 119899 gt 0 is the

number of places

(2) 119879 = 1199051 119905

119898 is a finite set of transactions119898 gt 0 is

the number of transactions 119875 cap 119879 =

(3) 119868 119875 times 119879 rarr 119873 is the input function defining the setof repetitions or the weight of the directed arcs fromset 119875 to set 119879 119873 = 0 1 is a set of nonnegativeintegers

(4) 119874 119879times119875 rarr 119873 is the output function defining the setof repetitions or the weight of the directed arcs fromset 119879 to set 119875 119873 = 0 1 is a set of nonnegativeintegers

(5) 119872 119875 rarr 119873 is the set of the identificationdistribution of every place

(6) 1198720 119875 rarr 119873 is the set of the initial identification dis-

tribution of every place

As shown in Figures 2 and 3 the top event is replaced bythe top place in the Petri net All the probable events thatmay lead to the top place are represented as middle placesor basic places The logical gates of the fault tree are denotedby Transaction andDirected arc Repeating events in the faulttree no longer exist in Petri net modelling

Petri net modelling avoids repeating basic events andcan achieve a 20 decrease in the number of places Thesimplifying effect is more obvious in large-scale systems

32 Relational Matrix Analysis in Solving the Minimal CutSets Besides simplifying the fault tree a Petri net alsoeffectively overcomes the shortcomings of the traditional FTAin solving the minimal cut sets The process of Petri netmodelling can be undertaken more easily on a computer

The structure of the Petri net can be translated into amatrix representation The value of the input function fromplace119875 to transaction 119905 is a nonnegative integer120596 recorded as119868(119875 119905) = 120596 represented by a directed arc from 119875 to 119905 with theside noteThe value of the output function from transaction 119905to place 119875 is a nonnegative integer 120596 recorded as119874(119875 119905) = 120596represented by a directed arc from 119905 to 119875 with the side note120596 The side note is omitted when 120596 = 1 The directed arc isalso omitted when 119868(119875 119905) = 0 or 119874(119875 119905) = 0 119868 and 119874 canboth be represented as 119899 times 119898 nonnegative integer matrixesThe difference between119874 and 119868 is called the relationalmatrixrecorded as 119860 = 119874 minus 119868

The steps for solving the minimal cut sets using therelational matrix method are as follows

Step 1 Find out the row consisting only of ldquo1rdquo and ldquo0rdquo in therelational matrix A This will be assigned the top place withonly inputs and no outputs

Step 2 Search for ldquominus1rdquo by column from 1 in the top placeAny row corresponding to ldquominus1rdquowill be regarded as an input tothe top place If there are multiple ldquominus1rdquo values in the columnthen there will be multiple inputs for the same transactionThe logical relationship between the inputs is AND

Step 3 Search for ldquo1rdquo by row from ldquominus1rdquo determined in Step 2Rows including ldquo1rdquo will be regarded as occupying a middleplaceThen search repeatedly following the second step untila row without ldquo1rdquo is located which will become a basic placeIf there are multiple occurrences of ldquo1rdquo in the row the logicalrelationship between the places corresponding to ldquo1rdquo will beOR

Step 4 Continue to search following the second step and thethird step until all the bottom basic places are located

Step 5 Expand the bottom places according to the logicalrelationships AND and OR Then obtain all the cut sets forthe system

Step 6 Find the minimal cut sets according to the Booleanabsorption rate or the prime number method

8 Mathematical Problems in Engineering

TokensM1

M2

X1

X1

X2

X3

t1 t2

t3 t4

t5

t6

P1 P2

P3P4

P5

P6T

Figure 3 Transform between Petri net modelling and the fault tree

The fault tree in Figure 3 is used as an example toillustrate solving theminimal cut sets with a relationalmatrixRespectively solve the input matrix 119868 the output matrix 119874and the relational matrix 119860 according to the steps above

119868 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 1

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

119874 =

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0 0

0 0 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

119860 = 119874 minus 119868 =

[

[

[

[

[

[

[

[

[

[

minus1 0 0 0 0 minus1

0 minus1 0 0 0 0

1 1 minus1 0 0 0

0 0 0 minus1 0 0

0 0 1 1 minus1 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

(5)

33 Comparison between Petri Net Modelling and Fault TreeModelling The complexity of quantitative analysis dependson the number of nodes and the logical gates in a fault treeComparing Figures 3 and 4 to Figure 2 we can see that thereare two repeating basic events and seven repeating events intwo traditional fault trees The repetition rates can respec-tively reach 17 and 24These repeating events do not existin the corresponding Petri net modelling The building ofthe fault tree needs 6 types of elements which involve basicevents intermediate events top events AND logical gate

OR logical gate and a relation line while the building of thePetri net model needs only three types of elements involvingplaces transactions and directed arcs Though the fault treein Figure 2 seems to provide better visualization than thePetri nets in Figure 4 there are more elements and repeatingevents in the fault tree The relationships among fault eventsin the fault tree are more complex The reduction of buildingelements and repeating events make the Petri net modellingmore convenient and efficient than fault tree modelling

In addition Petri nets are graphical and mathematicaltools for modelling systems and their dynamics while faulttree modelling cannot be updated in a timely manner whenthe system changes The relationship between fault events isdisplayed statically in a fault tree When the failure status of asystem evolves from one subsystemcomponent to other sub-systemscomponents Petri net modelling is more intuitiveand more effective than fault tree modelling because of theuse of token transferring and enable-transferring processesThis process in Petri nets can be described by mathematicalmatrix computations which is easier to handle by computer(as shown in Section 4)

4 Dynamic Transition and Diagnosis ofPantograph Faults

41 PlaceMark and Enable-Transfer Algorithm of the Petri NetA Petri net consists of places ldquoIrdquo transactions ldquo|rdquo directedarcs ldquorarr rdquo and tokens of places ldquoerdquo The places representlogical descriptions of the system states and the transactionsrepresent the arising of system events Relational matrixesand state equations are major tools for Petri net analysis

119872

119896+1= 119872

119896+ 119860

119879

119883

119896 119896 = 1 2 (6)

In (6) 119872119896is the initial identification set of the system

faults before ignition 119872119896+1

is the result identification setof the system faults after ignition 119860119879 is the relationalmatrix and 119883

119870is the transfer sequence for ignition The

Mathematical Problems in Engineering 9

P1 P2 P3 P4

P5 P6 P7 P8 P9 P10 P11P12

P13

P14 P15 P16

P17 P18

P19

P20 P21 P22

P23

P24

P25 P26 P27 P28

P29

P30

P31

P32

P33P34

P35

P36 P37

P38

t1 t2 t3 t4

t5t6

t7

t8 t9 t10 t11 t12 t13 t14 t15

t16

t17

t18t19

t20

t21t22

t23 t24t25

t26t27

t28

t29t30 t31 t32

t33

t34t35

t36

t37

t38

t39

t40

t41

t42 t43

Initial identification 3

Initial identification 2

Initial identification 1

Figure 4 Transfer of tokens in the Petri net of the CRH EMU pantograph

initial identification1198720will convert to119872

119889after the ignition

sequence1198830 119883

119889 The process can be expressed as

119872

119889= 119872

0+ 119860

119879

119889minus1

sum

119896=0

119883

119896 (7)

If the element representing the top place is not less than1 in119872

119889after a series of transfers the fault event represented

by the top place will occur

42 Dynamic Transition and Diagnosis of Pantograph FaultsIn the process of fault diagnosis based on the use of a Petrinet the initial identification of the input place is regardedas the initial symptom of the event occurrence A token isassigned to the input place if the symptom appears otherwisethe place is assigned a null value The future states and finalidentification of the system can be circularly solved throughthe state equations and transfer sequences until the tokenvalue of the target place is found The fault occurs when thenumber of tokens in the target place is not zero

For the AND gate in the Petri net the tokens cannottransfer downward to the next level if there are empty inputplaces That is to say for an AND gate the fault will transferdownward only when all the events of the input places occurFor the OR gate in a Petri net the tokens will transferdownward to the next level place if at least one input placecontains tokens According to the rules of the network theorythe next level place will get two tokens if the two input placesboth contain tokens Regardless of the number of tokensthe occurrence of the fault that is represented by the targetplace is only related to whether there are tokens in the targetplace Transfer of the token in the Petri net of the CRH EMUpantograph is shown in Figure 4

As shown in Figure 4 1198751 represents ldquotwo EMUs attachedtogetherrdquo 1198752 represents ldquospacing of two pantographs of lessthan 190mrdquo 1198758 represents ldquopantograph bellows that aredamagedrdquo 11987526 represents ldquoMR ducts that are damagedrdquo11987529 represents ldquopressure in the auxiliary air cylinder that istoo lowrdquo 11987530 represents ldquopantograph rise to an abnormalpositionrdquo 11987532 represents ldquopantograph that cannot riserdquo11987535 represents ldquopantograph that descends automaticallyrdquoand 11987538 represents ldquopantograph that fails to workrdquo Threerepresentative fault transfer paths were selected to study thetransfer expression of pantograph faults in the Petri net Theinitial identification sets of the three paths were respectivelyrecorded as1198721

0

11987220

and11987230

119872

1

0

= [1 0 0

2le119894le38

]

119879

119872

2

0

= [0 0

1le119894le25

1 0 0

27le119894le38

]

119879

119872

3

0

= [0 0

1le119894le7

1 0 0

9le119894le38

]

119879

(8)

The ignition sequences of the three paths were respec-tively recorded as1198831

0

11988320

and11988330

119883

1

0

= [0 0

1le119894le4

1 0 0

6le119894le43

]

119879

119883

2

0

= [0 1 0 0

3le119894le43

]

119879

119883

3

0

= [0 0

1le119894le11

1 0 0

12le119894le21

1 0 0

23le119894le43

]

119879

(9)

10 Mathematical Problems in Engineering

The input matrix 119868 output matrix 119874 and the relationalmatrix119860119879 of the Petri net are all 38 times 43matrixes which canbe solved using the relational-matrix method

119860

119879

= 119874 minus 119868 = (

119886

11 119886

1119899

d

119886

1198981sdot sdot sdot 119886

119898119899

) (10)

119886

119894119895= 119874

119894119895minus 119868

1198941198951 le 119894 le 38 1 le 119895 le 43 (11)

In (10)119898 = 38 119899 = 43

119872

1

1

= 0

119872

1

2

= 0

(12)

As (12) shows tokens cannot transfer downward to thenext level in the first path Thus there is no token in the topplace at the end of the procedure meaning that the fault eventrepresented by 11987538 does not occur

In the second path11987221

11987222

and11987223

are as follows

119872

2

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le28

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

30le119894le38

]

]

]

119872

2

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

33le119894le38

]

]

]

119872

2

3

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

1

]

]

]

(13)

As (13) shows the tokens in place11987526 are transferred intoplace11987529 after the first ignition and then transferred into11987532after the second ignition Finally the tokens are transferredinto the top place 11987538 after the third ignition meaning thatthe top event occurs in this situation

In the third situation 11987231

can be calculated througha single transfer calculation when the default value of thetokens in place 1198758 is 1

119872

3

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le7

minus1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

9le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(14)

Thus when there are two transfer paths for one token inthe bottom basic place there will be negative values in thenext level state matrix after ignition if the default value of thetokens in the bottom basic place is still 1There is no practicalsignificance to this phenomenon To avoid negative valueschange the default value of the tokens in place 1198758 from 1 to 2to produce1198723

1

1015840

119872

3

1

1015840

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(15)

Then11987232

can be solved according to11987231

1015840

119872

3

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

2

]

]

]

(16)

According to11987232

there will be two tokens in the top place11987528 and the top event will inevitably happen when tokensindicate the fault transferThus the fault event in the top placewill happen if there is at least one token in the top place In thePetri net if an input place is a bottom basic place containingtokens with a value of 119899 (119899 is a positive integer greater than 1)directed arcs out then the default value of the tokens in theinput place is 119899The fault event presented indicated by the topplace would will inevitably happen when there are tokens inthe top place

Failure Mode 1 When two EMUs are attached together andat the same time the spacing of the two pantographs is lessthan 190m the pantograph will rise to an abnormal positionHowever if only one condition of the two is satisfied thepantograph will work normally

Failure Mode 2 Damage to MR ducts causes low pressurein the auxiliary air cylinder which means the pantographcannot rise successfully

FailureMode 3 Damage to the pantograph bellowsmay causethe pantograph to descend automatically or possibly fail torise Both of the two conditions will lead to a fault in thepantograph

The token transfer in the Petri net is consistent withthe logic analysis of the faults It can clearly and effectivelydescribe the dynamic processes of the system faults transferachieving fast and efficient fault diagnosis

5 Conclusions

In this study the working principles and failure modes ofpantographs on CRH EMU trains were analyzed systemati-cally Petri net modelling was used as a graphical modellingtool to simplify the logical relationships and events of thefault tree into a network with places and transactions asnodes The events logical gates and logical relation lines ofFT were transformed to places transitions and directed arcsrespectively The complexity of modelling was reduced by20by avoiding repeating eventsThus the process of solvingthe minimal cut sets was simplified which effectively savedcalculation time for theminimal cut set solutions for complexlarge-scale fault trees

The changes in system states and the evolution of failuresare described well by the dynamic properties of Petri netmodelling A mathematical model for the Petri net of thepantograph fault tree was established The equivalence andcorrectness of the token-transfer description for fault diag-nosis inference in a Petri net were verified Three differentfault paths were used to explain and verify the algorithmThe initial identification sets of the three paths were marked

Mathematical Problems in Engineering 11

The three corresponding ignition sequences were calculatedusing matrix transformations Finally the system status wasassessed correctly using a mathematical method which canbe handled easily in a computerised system

This work analyzes the evolution of failure events ofpantograph system The process of other critical systemsneeds to be further examined Future work will collect andanalyse test data from other systems in CRH EMU trainsto extend the methodology to cover the whole maintenanceprocess for CRH EMU train systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 51475324) The authors wouldlike to thank the reviewers for their constructive commentswhich enhanced the quality of this paper

References

[1] F C Jiao and B J Wang The Management and Maintenanceof Locomotive Rolling Stock Beijing Jiaotong University PressBeijing China 2013

[2] L L Song T Y Wang X W Song L Xu and D Song ldquoFaultdiagnosis of pantograph type current collector of CRH electricmultiple units based on Petri net modelling and fault treeanalysisrdquo Chinese Journal of Scientific Instrument vol 35 no 9pp 1990ndash1997 2014

[3] I Aydin M Karakose and E Akin ldquoAnomaly detection usinga modified kernel-based tracking in the pantographmdashcatenarysystemrdquoExpert SystemswithApplications vol 42 no 2 pp 938ndash948 2015

[4] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2008

[5] D M Shalev and J Tiran ldquoCondition-based fault tree analysis(CBFTA) a newmethod for improved fault tree analysis (FTA)reliability and safety calculationsrdquo Reliability Engineering andSystem Safety vol 92 no 9 pp 1231ndash1241 2007

[6] Y E Senol Y V Aydogdu B Sahin and I Kilic ldquoFaulttree analysis of chemical cargo contamination by using fuzzyapproachrdquo Expert Systems with Applications vol 42 no 12 pp5232ndash5244 2015

[7] S M Lavasani A Zendegani and M Celik ldquoAn extension tofuzzy fault tree analysis (FFTA) application in petrochemicalprocess industryrdquo Process Safety and Environmental Protectionvol 93 pp 75ndash88 2014

[8] J H Purba ldquoA fuzzy-based reliability approach to evaluate basicevents of fault tree analysis for nuclear power plant probabilisticsafety assessmentrdquo Annals of Nuclear Energy vol 70 pp 21ndash292014

[9] N RamzaliM RM Lavasani and J Ghodousi ldquoSafety barriersanalysis of offshore drilling system by employing Fuzzy EventTree Analysisrdquo Safety Science vol 78 pp 49ndash59 2015

[10] K-C Hyun S Min H Choi J Park and I Lee ldquoRisk analysisusing fault-tree analysis (FTA) and analytic hierarchy process(AHP) applicable to shield TBM tunnelsrdquo Tunnelling andUnderground Space Technology vol 49 pp 121ndash129 2015

[11] S Verlinden G Deconinck and B Coupe ldquoHybrid reliabilitymodel for nuclear reactor safety systemrdquo Reliability Engineeringand System Safety vol 101 pp 35ndash47 2012

[12] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 13 no 3 pp386ndash397 1987

[13] A Bobbio G Franceschinis R Gaeta and L Portinale ldquoPara-metric fault tree for the dependability analysis of redundant sys-tems and its high-level Petri net semanticsrdquo IEEE Transactionson Software Engineering vol 29 no 3 pp 270ndash287 2003

[14] R Robidoux H Xu L Xing andM Zhou ldquoAutomated model-ing of dynamic reliability block diagrams using colored Petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 2 pp 337ndash351 2010

[15] DMakajic-Nikolic M Vujosevic andN Nikolic ldquoMinimal cutsets of a coherent fault tree generation using reverse Petri netsrdquoOptimization vol 62 no 8 pp 1069ndash1087 2013

[16] B Wang G Tian Y Liang and T Qiang ldquoReliability modelingand evaluation of electric vehicle motor by using fault tree andextended stochastic petri netsrdquo Journal of Applied Mathematicsvol 2014 Article ID 638013 9 pages 2014

[17] T P K Nguyen J Beugin and JMarais ldquoMethod for evaluatingan extended Fault Tree to analyse the dependability of complexsystems application to a satellite-based railway systemrdquo Relia-bility Engineering and System Safety vol 133 pp 300ndash313 2015

[18] L L Song T Y Wang X W Song et al ldquoFuzzy intelligentfault diagnosis of pantograph type current collector under themulti-source heterogeneous knowledge environmentrdquo ChineseJournal of Scientific Instrument vol 36 pp 1283ndash1290 2015

[19] M An Y Chen and C J Baker ldquoA fuzzy reasoning and fuzzy-analytical hierarchy process based approach to the process ofrailway risk information a railway risk management systemrdquoInformation Sciences vol 181 no 18 pp 3946ndash3966 2011

[20] M An S Huang and C J Baker ldquoRailway risk assessmentmdashthe fuzzy reasoning approach and fuzzy analytic hierarchy pro-cess approaches a case study of shunting at Waterloo depotrdquoProceedings of the Institution of Mechanical Engineers F Journalof Rail and Rapid Transit vol 221 no 3 pp 365ndash383 2007

[21] K Noori and K Jenab ldquoFuzzy reliability-based traction controlmodel for intelligent transportation systemsrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 43 no 1 pp229ndash234 2013

[22] R I Muttram ldquoRailway safetyrsquos safety risk modelrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 216 no 2 pp 71ndash79 2002

[23] L Bouillaut O Francois and S Dubois ldquoA Bayesian network toevaluate underground rails maintenance strategies in an auto-mation contextrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 227 no4 pp 411ndash424 2013

[24] Y Xie Y Zhang and F Liang ldquoCrash injury severity analysisusing Bayesian ordered probit modelsrdquo Journal of Transporta-tion Engineering vol 135 no 1 pp 18ndash25 2009

[25] S Bernardi F Flammini S Marrone et al ldquoEnabling the usageof UML in the verification of railway systems the DAM-railapproachrdquoReliability Engineering and SystemSafety vol 120 pp112ndash126 2013

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Fault tree analysis (FTA) is a useful analytical tool foridentifying and classifying hazards and calculating systemreliability for both simple and complex engineering systems[5] It is a systematic way to assess the reliability of com-plex systems both qualitatively and quantitatively [6] Muchresearch has been conducted and outcomes include extendedfuzzy FTA (FFTA)methodology in the petrochemical processindustry in fuzzy environments [7] the use of a fuzzy-logic-based reliability approach to evaluate basic events infault tree analysis for nuclear power plant probabilistic safetyassessment [8] FFTA applied to reducing uncertainty inexpert judgment in the safety barriers analysis of an offshoredrilling system [9] and the use of fault tree analysis andanalytic hierarchy processes to analyze the risks associatedwith the use of shield tunnel boring machines (TBMs) [10]

However the modelling power of FTA is limited to thestatic evaluation of a single criterion at a time and FTAis not capable of describing dynamic system behavior withredundant components degraded system states and repairor test activities [11] Leveson and Stolzy proposed the safetyanalysis of dynamic systems using time Petri nets [12] Bobbioet al provided an algorithm to convert a parametric faulttree (PFT) into a class of high-level Petri nets to exploit themodelling power and flexibility of the stochastic well-formednets (SWN) formalism [13] Robidoux et al used a frame-work named dynamic reliability block diagrams (DRBD)for modelling the dynamic reliability behavior of computer-based systems and presented an algorithm that automaticallyconverted a DRBD model into a colored Petri net [14] Withincreasing complexity in the relationship between systemstructures and faults fault trees become more difficult toestablish Problems relating to the Nondeterministic Poly-nomial (NP) which can lead to combinatorial explosion asthe numbers of calculations grow exponentially appear whensolving the minimal cut sets Makajic-Nikolic et al proposedan approach to cutting the number of minimal cut sets in afault tree using reverse Petri nets [15] There have also beensome studies on the application of FTA and Petri nets intransportation systems For example Wang et al proposedreliability modelling and evaluation issues of electric vehiclemotors by using FT FTA-based extended stochastic Petrinets [16] Nguyen et al combined a Petri nets (PN) methodwith extensions of FTA to adapt to dynamic systems andapplied the method to a satellite-based railway system [17]and Song et al applied T-S Fuzzy theory and FTA to diagnosepantograph faults in a multisource heterogeneous knowledgeenvironment [18]

Some other mathematicalstatistical methods such asfuzzy reasoning and Bayesian Network (BN) also have beensuccessfully used in the railways risk and safety domainFor example An et al proposed a railway risk managementsystem for railway risk analysis using fuzzy reasoningapproach and fuzzy analytical hierarchy decision makingprocess [19 20] Noori and Jenab developed a fuzzy Bayesiantraction control system for rail vehicles with speed sensorsin intelligent transportation systems [21] Muttram applieda Safety Risk Model combining fault tree analysis andcauseconsequence techniques to predict residual levels ofrailway safety risk [22] Bouillant et al developed a decision

support tool based on Bayesian Networks to evaluatecompare and optimize various operating and maintenancestrategies of Paris metro underground rails [23] Xie etal introduced the Bayesian inference and investigated theapplication of a Bayesian ordered probit (BOP) model indriverrsquos injury severity analysis and verified that BOP modelcould produce more reasonable parameter estimations andbetter prediction performance than the ordered probit (OP)model [24] Bernardi et al generated Repairable Fault Treeand Bayesian Network models for railway modelling by amodel-driven approach called DAM-Rail approach [25]Mahboob and Straub compared fault tree and BayesianNetworks for modeling safety critical components in railwaysystems [26] andWashington andOh incorporated Bayesianmethodology with expert judgment for countermeasureeffectiveness under uncertainty and applied the approach inimproving the safety of railroad crossings [27]

From the above it appears that the conventional analysisapproaches such as FTA Petri nets (PN) fuzzy reasoningand BN have been widely used in the railways risk and safetyanalysis field However each approach has its own advantagesand limitations Fault tree Petri nets and BN have a strongsimilarity in many aspects Fault tree can be establishedmore easily only when the cause-and-effect relationshipsbetween events were clear But it suffers severe limitations ofstatics structure and uncertainty handling Fuzzy reasoningand BN are effective tools for quantitative analysis becausethey are based on probabilistic and uncertain knowledgeBN approach allows dealing with issues such as predictionor diagnosis optimization and data analysis of feedbackexperience [28] It is more often used in prediction becauseof good quantitative analysis ability but the training processof BN is complex And the dependent probabilities amongevents are required in BN approach which may be difficultto obtain in some cases Petri net (PN) is also a powerful andwidely used graphical and mathematical modelling tool forthe description of sequence dependent behaviors of dynamicsystems When the failure status of a system evolves fromone subsystemcomponent to other subsystemscomponentsPetri net modelling is more intuitive and more effectiveto describe the process by carrying mathematical matrixcomputations which is easier to handle by computer Atthe same time it is more suitable for large complex systemsbecause the modelling process is simplified by eliminatingrepeat basic events and reducing the building elementsThrough the above analysis FTArsquos translation into Petri nets ismore suitable in this paper to analyze the dynamic behaviorsof failure statuses of Pantograph-Type Current Collector onCRH EMU trains mathematically as far as the extension fastmodelling and dynamic behavior analysis of the system aremainly concerned

Studies on reliability and failure mode analysis of CRHEMU trains and their subsystems have recently begun Thefocus of the research has turned from design and manu-facture to maintenance management FTA and Petri netsapproaches are applicable and useful analysis tools in the riskmanagement of complex engineering systems but there aresome deficiencies in their application Considerable researchhas been conducted on these problems The main aim of

Mathematical Problems in Engineering 3

this paper is to extend FTA and Petri nets methodology tomaintenance and fault diagnosis in CHR EMU train systemsThis section introduces some existing applications of FTAandPetri nets in a range of industries Structural analysis workingprinciples failure mode and effects analysis (FMEA) andFTAmodelling of pantographs are outlined in Section 2 Petrinetsmodelling of the fault tree and the use of relationalmatrixanalysis in solvingminimal cut sets are provided in Section 3In Section 4 the place mark and enable-transfer algorithm ofthe Petri nets and an actual case study on dynamic transitionand diagnosis of pantograph diagnostics are provided Thelast section emphasises the highlights of this research

2 Structural Analysis and Fault TreeModelling of Pantograph

21 Structural Analysis and Working Principle of PantographThe pantograph of a CRH EMU train is fitted on the roofof the train and is essential to allow the train to get powerfrom the main overhead wires A CRH EMU train is aneight-car multiple unit configured with four motorised carsand four trailer cars There are two power units in all andeach power unit consists of two power cars and two trailercars arranged in T-M-M-T mode DSA250 pantographs arefitted on number 4 and number 6 cars Their maximumlifting height is 3000mm and the width of the head is1990mm In normal operation there will be only one pan-tograph collecting current with another in a folded stateHowever when two CRH EMUs are attached together twopantographs will work simultaneouslyTheworking principleand structural components of pantograph systems for CRHEMUs are shown in Figure 1The structural parameters of thepantograph are shown in Table 1

The pantograph is raised to access high voltage power byallowing the carbon skateboard to contact the catenary wireand descends when its compressed air supply is exhaustedThe 25 kv single-phase power alternating current (AC) fromthe catenary is transferred to the traction transformer fromthe high-voltage electrical equipment by the pantographwhich outputs 1500V single-phase AC power to the tractionconverter A pulse rectifier converts the single-phase AC intodirect current (DC) and outputs 2500ndash3000V DC to thetraction inverter through the DC circuit Then the tractioninverter outputs three-phaseACpowerwhere the voltage andfrequency are all adjustable to drive the traction motor

When the pantograph rises air is compressed into thedrive cylinder through the cushion valve and the cylinderpistonmoves left overcoming the pressure of the reset springThen the lower arm rises and rotates clockwise under theaction of the guide bar and the spring At the same time theupper arm also rises with the drive of the top guide bar

When the pantograph descends the compressed air isremoved from the drive cylinder through the cushion valveThen the piston is pushed to the right with the reset springreleasing pressureThe guide bar alsomoves right to force thelower arm to rotate anticlockwise thus forcing the upper armto descend

The collector head contacts the catenary wire when thepantograph rises Current is led to the bottom frame through

the collector head the upper arm and the pushrod Thepower cable installed on the bottom frame then leads thecurrent to the vehicle Since current will flow through theentire pantograph frame in the power supply state all thepantograph hinges are equipped with bridge connections toprevent the current damaging the bearings

The performance of the pantograph largely dependson contact pressure If there is too little pressure contactresistance will vary easily resulting in poor contact andarcing However toomuch pressure will increase the frictionaggravating wear on the carbon skateboard and wires andreducing the life of the carbon skateboard

22 Failure Mode and Effects Analysis (FMEA) of PantographSystem Pantographs on CRH EMU trains are installed onthe roof exposed to the environment The working environ-ment is complex volatile and sometimes very harsh Therearemany and complex ways in which the pantograph can failwhich are related to mechanical structures electrical facil-ities pneumatic transmission network control and manyother factors The normal operation of the pantograph on aCRH EMU train is automatically controlled by microproces-sors The pantograph system involves mechanical structuressuch as guide bars electrical equipment such as the tractionconverter and rubber products such as the base insulatorThe lifting process is controlled by air pressure Because pan-tographs connect directly with the extra high voltage (EHV)catenary wire they can be easily damaged by partial highvoltage discharges By classifying and analyzing fault trackingrecords the most common pantograph failure modes aresummarized as follows

(i) Failure Mode 1 Pantograph rises to an abnormal position

Probable cause

When two EMUs attach together the spacebetween the two pantographs is less than 190m

(ii) Failure Mode 2 Pantograph rises normally while themonitor (MON) does not show it correctly

Probable causes

(a) The pressure sensors are not operating correctly(b) The pressure switches have failed

(iii) Failure Mode 3 Pantograph cannot rise

Probable causes

(a) Electricity generation system (EGS) is closed(b) Vacuum circuit breaker (VCB) is closed(c) Pressure in the auxiliary air cylinder is too low(d) [PantographsdotVCB] NFB is in the OFF position(e) [PantographRising]NFB is in theOFF position(f) Pantograph pressurised air ducts leak(g) White air ducts connected to the pantograph

leak(h) Air ducts in the pantograph control valve board

leak

4 Mathematical Problems in Engineering

Pantograph bow

Carbon contact strip

Upper arm

Lower arm

Lift device for bow

Top guide bar

Coupling rod

Base frame

System damper

Filter capacitor

Inve

rter

Pulse

re

ctifi

er

Pulse

re

ctifi

erFilter capacitor

Inve

rter

Traction motor

Traction motor

Traction transformer

Main converterMain converter

Vacuum breaker

Pantograph

X4X4

Connection set knee

Pan stop rod

Air bellows

M1M2

Figure 1 Working principle of traction drive systems and pantograph structure in a CRH EMU train

Table 1 Structural parameters of the pantograph on a CRH EMU train

Structural parameters Values Structural parameters ValuesModel DSA250 Vent lifting height 3000mm (including insulator)Structure Single arm pantograph Maximal action height 2800mm (including insulator)Rated voltage 25 kV Minimal action height 888mm (including insulator)Rated current 1000A Folded height 593mm (including insulator)Weight 115 kg Contact pressure 70N plusmn 5NAmbient temperature minus40∘Csim+60∘C Design life 30 years

(i) Pantograph bellows are damaged(j) The pantographrsquos carbon skateboard leaks

(iv) Failure Mode 4 Pantograph cannot be lowered

Probable cause

Contacts in the pantographrsquos rising relay haveadhered

(v) Failure Mode 5 Pantograph cannot pass neutral sectionautomatically

Probable cause

Terminal RXCB board has failed

(vi) Failure Mode 6 Supporting insulator is damaged

Probable causes

(a) Surface defects created during manufacture(b) Insulator being burned by electrical arcing(c) Fog flashovers discharged because of high

humidity or dust(d) Fatigue cracks(e) Insulator being struck by foreign objects

(vii) Failure Mode 7 Pantograph descends automatically

Probable causes

(a) Pressure air ducts in the pantograph leak(b) White air ducts connected with the pantograph

leak(c) Air ducts in the pantograph control valve board

leak(d) Pantographrsquos bellows are damaged

Mathematical Problems in Engineering 5

(e) The pantographrsquos carbon skateboard leaks(f) Incorrect manipulation(g) Pantograph descending loop is instantly ener-

gized by electromagnetic interference(h) Pressure from the catenary is too low(i) There has been discharge between the panto-

graph bracket and the roof

(viii) Failure Mode 8 The catenary voltage transformer isdamaged

Probable causes

(a) Struck by foreign objects(b) Burned by electrical arcing(c) Quality control issues in components

23 Fault TreeModelling of the Pantograph Fault tree analysis(FTA) is one type of FMEA which is integrated withmechanics graph theory optimization theory and artificialintelligence techniques [29] It has been widely applied inmany fields including aerospace systems atomic reactorslarge-scale equipment and electronic computer systems FTAcan indicate causal relationships between complex faults andcan be useful for logical analysis and diagnosis of complexsystem faults It is a systematic way to assess the reliability ofcomplex systems both qualitatively and quantitatively

FTA is a systematic risk analysis method that deals withthe occurrence of an undesired event The fault event thatanalysts do not expect to happen is usually the focus of theFTAmethod Analysts apply top-down logic to find all directand indirect fault events relating to the incident They canthen establish logical relationships between the events form afault tree and undertake quantitative or qualitative analysisIn the process above the focus fault event is the top eventSelection of the top event is crucial to fault tree modelling Ifthe top event is too general it is difficult to analyse the faulttree On the other hand if the event is too specific the faulttree will fail to show the causal relationships of the systemfully In general the failure of the system analyzed will beselected as the top event

Combined with analysis of the structures and workingprinciples of pantographs on CRH-trains a pantograph faulttree has been established as shown in Figure 2 ldquo119879rdquo representsthe top event ldquo119872rdquo represents the intermediate event and ldquo119883rdquorepresents the basic event

Meanings of events in Figure 2 are shown as follows

119879 pantograph fails to work1198721 pantograph rises to an abnormal position1198722 pantograph rises normally but the MON doesnot show it correctly1198723 pantograph cannot rise1198724 pantograph cannot be lowered1198725 pantograph cannot pass neutral section automat-ically1198726 supporting insulators are damaged

1198727 pantograph descends automatically1198728 the catenary voltage transformer is damaged1198729 pressure of the auxiliary air cylinder is too low1198831 two EMUs are attached together1198832 spacing of two pantographs is less than 190m1198833 the pressure sensors do not work properly1198834 the pressure switches have failed1198835 EGS is closed1198836 VCB is closed1198837 in the cab switchboards at both ends[PantographsdotVCB] NFB is in the OFF position1198838 in the running switchboards of the number 4 andnumber 6 cars (CRH-200 CRH-300)number 4 andnumber 13 cars (CRH long-distance seat car CRHlong-distance sleeper car) [Pantograph Rising] NFBis in the OFF position1198839 pressure air ducts of the pantograph leak11988310 air ducts in the pantograph control valve boardleak11988311 pantograph bellows are damaged11988312 pantographrsquos carbon skateboard leaks11988313 white air ducts connected with the pantographleak11988314 contacts in the pantographrsquos rising relay haveadhered11988315 terminal RXCB board has failed11988316 surface defects due to manufacturing errors11988317 burned by electrical arcing11988318 fog flashovers discharge11988319 fatigue cracks11988320 struck by foreign objects11988321 driver error11988322 pantograph descending loop is instantly ener-gized by electromagnetic interference11988323 discharge between the pantograph bracket andthe roof11988324 pressure of the catenary is too low11988325 burned by electrical arcing11988326 air ducts of the air compressor are damaged11988327 the air compressor fails to work11988328 MR ducts are damaged11988329 main air cylinder leaks

The fault tree in Figure 2 shows the causal relationshipsbetween the pantograph fault events improving the accuracyand efficiency of fault diagnosis However the modellingpower of FTA is limited to the static evaluation of a singlecriterion at a time and cannot describe dynamic system

6 Mathematical Problems in Engineering

T

M4 M5 M6M3 M7M2 M8M1

X2X1

X13X5

X14 X15

X20X16

X25X16X4X3

X6 X7 X8 X9 X10 X11 X12M9

X26 X27 X28 X29

X17 X18 X19

X9 X10 X11 X12 X13 X21 X22 X23 X24

X17

Figure 2 FTA modelling of the pantograph-type current collector

behavior It cannot show the transmission and evolution offaults in the system It also needs too much computation tosolve the minimal cut sets of the large-scale fault tree whichmay lead to combinatorial explosion because of Nondeter-ministic Polynomial (NP) problems For example if (119894 =1 2 119899) is set to the 119894th minimal cut set of the fault treethen the top event can be expressed as

119879 =

119899

⋃

119894=1

119862

119894 (1)

The occurrence probability of the top event can beexpressed as

119875 (119879) = 119875(

119899

⋃

119894=1

119862

119894) (2)

The logical relationship between the basic events and theminimal cut set is and If the probabilities of the basic eventsare known the occurrence probability of the minimal cut setcan be expressed as

119875 (119862

119894) = 119875(

119896

⋃

119895=1

119909

119894) (3)

Based on the occurrence probabilities of the minimalcut sets the occurrence probability of the top event can beexpressed as

119875 (119879) =

119899

sum

119894=1

119875 (119862

119894) minus

119899

sum

119894lt119895=2

119875 (119862

119894119862

119895) + sdot sdot sdot

+ (minus1)

119898minus1

119875(

119899

⋃

119894=1

119862

119894)

(4)

According to the equation above the process of deter-mining the probability of the top event consists of 2119899minus1 partsThere are 28 minimal cut sets for the pantograph fault treein Figure 2 so it needs 227 parts to calculate the probabilityof the top event which means 134217728 items in all Thecombinatorial explosion of the calculation will make quanti-tative analysis difficult In addition the FTA method is gen-erally for static analysis and cannot reflect dynamic changesin multiple states in the system

The dynamic and structural properties of Petri nets areused to simplify and analytically calculate the pantographfault tree Relational matrix analysis is used to solve theminimal cut set equation for the fault tree Based on the estab-lished state equation of Petri nets initial token and enable-transfer algorithms are used to express the transfer pro-cess of faults mathematically

Mathematical Problems in Engineering 7

Table 2 Petri net expression of the FTA logic gates

AND logic gate OR logic gateFault tree expression Petri net expression Fault tree expression Petri net expression

3 Petri Nets Modelling and RelationalMatrix Analysis

31 Petri Nets Modelling of the Fault Tree The Petri net wasproposed in 1962 by Petri to express an information flowmodel for reticular structures [30] It is a mathematical andgraphical analysis tool to express the static structures anddynamic changes in a system [31ndash33]

In a Petri net the system state is indicated by ldquoIrdquo Achange in state is indicated by ldquo|rdquo An ordered pair is indicatedby a directed arc ldquorarr rdquo and a token is indicated by ldquoerdquo ThePetri net expressions of the FTA logic gates are shown inTable 2 The transform between Petri net modelling and thefault tree is shown in Figure 3

One Petri net can be defined in a sextuple 119873 =

(119875 119879 119868 119874119872119872

0) The six elements should meet the follow-

ing conditions

(1) 119875 = 119875

1 119875

119899 is a finite set of places 119899 gt 0 is the

number of places

(2) 119879 = 1199051 119905

119898 is a finite set of transactions119898 gt 0 is

the number of transactions 119875 cap 119879 =

(3) 119868 119875 times 119879 rarr 119873 is the input function defining the setof repetitions or the weight of the directed arcs fromset 119875 to set 119879 119873 = 0 1 is a set of nonnegativeintegers

(4) 119874 119879times119875 rarr 119873 is the output function defining the setof repetitions or the weight of the directed arcs fromset 119879 to set 119875 119873 = 0 1 is a set of nonnegativeintegers

(5) 119872 119875 rarr 119873 is the set of the identificationdistribution of every place

(6) 1198720 119875 rarr 119873 is the set of the initial identification dis-

tribution of every place

As shown in Figures 2 and 3 the top event is replaced bythe top place in the Petri net All the probable events thatmay lead to the top place are represented as middle placesor basic places The logical gates of the fault tree are denotedby Transaction andDirected arc Repeating events in the faulttree no longer exist in Petri net modelling

Petri net modelling avoids repeating basic events andcan achieve a 20 decrease in the number of places Thesimplifying effect is more obvious in large-scale systems

32 Relational Matrix Analysis in Solving the Minimal CutSets Besides simplifying the fault tree a Petri net alsoeffectively overcomes the shortcomings of the traditional FTAin solving the minimal cut sets The process of Petri netmodelling can be undertaken more easily on a computer

The structure of the Petri net can be translated into amatrix representation The value of the input function fromplace119875 to transaction 119905 is a nonnegative integer120596 recorded as119868(119875 119905) = 120596 represented by a directed arc from 119875 to 119905 with theside noteThe value of the output function from transaction 119905to place 119875 is a nonnegative integer 120596 recorded as119874(119875 119905) = 120596represented by a directed arc from 119905 to 119875 with the side note120596 The side note is omitted when 120596 = 1 The directed arc isalso omitted when 119868(119875 119905) = 0 or 119874(119875 119905) = 0 119868 and 119874 canboth be represented as 119899 times 119898 nonnegative integer matrixesThe difference between119874 and 119868 is called the relationalmatrixrecorded as 119860 = 119874 minus 119868

The steps for solving the minimal cut sets using therelational matrix method are as follows

Step 1 Find out the row consisting only of ldquo1rdquo and ldquo0rdquo in therelational matrix A This will be assigned the top place withonly inputs and no outputs

Step 2 Search for ldquominus1rdquo by column from 1 in the top placeAny row corresponding to ldquominus1rdquowill be regarded as an input tothe top place If there are multiple ldquominus1rdquo values in the columnthen there will be multiple inputs for the same transactionThe logical relationship between the inputs is AND

Step 3 Search for ldquo1rdquo by row from ldquominus1rdquo determined in Step 2Rows including ldquo1rdquo will be regarded as occupying a middleplaceThen search repeatedly following the second step untila row without ldquo1rdquo is located which will become a basic placeIf there are multiple occurrences of ldquo1rdquo in the row the logicalrelationship between the places corresponding to ldquo1rdquo will beOR

Step 4 Continue to search following the second step and thethird step until all the bottom basic places are located

Step 5 Expand the bottom places according to the logicalrelationships AND and OR Then obtain all the cut sets forthe system

Step 6 Find the minimal cut sets according to the Booleanabsorption rate or the prime number method

8 Mathematical Problems in Engineering

TokensM1

M2

X1

X1

X2

X3

t1 t2

t3 t4

t5

t6

P1 P2

P3P4

P5

P6T

Figure 3 Transform between Petri net modelling and the fault tree

The fault tree in Figure 3 is used as an example toillustrate solving theminimal cut sets with a relationalmatrixRespectively solve the input matrix 119868 the output matrix 119874and the relational matrix 119860 according to the steps above

119868 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 1

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

119874 =

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0 0

0 0 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

119860 = 119874 minus 119868 =

[

[

[

[

[

[

[

[

[

[

minus1 0 0 0 0 minus1

0 minus1 0 0 0 0

1 1 minus1 0 0 0

0 0 0 minus1 0 0

0 0 1 1 minus1 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

(5)

33 Comparison between Petri Net Modelling and Fault TreeModelling The complexity of quantitative analysis dependson the number of nodes and the logical gates in a fault treeComparing Figures 3 and 4 to Figure 2 we can see that thereare two repeating basic events and seven repeating events intwo traditional fault trees The repetition rates can respec-tively reach 17 and 24These repeating events do not existin the corresponding Petri net modelling The building ofthe fault tree needs 6 types of elements which involve basicevents intermediate events top events AND logical gate

OR logical gate and a relation line while the building of thePetri net model needs only three types of elements involvingplaces transactions and directed arcs Though the fault treein Figure 2 seems to provide better visualization than thePetri nets in Figure 4 there are more elements and repeatingevents in the fault tree The relationships among fault eventsin the fault tree are more complex The reduction of buildingelements and repeating events make the Petri net modellingmore convenient and efficient than fault tree modelling

In addition Petri nets are graphical and mathematicaltools for modelling systems and their dynamics while faulttree modelling cannot be updated in a timely manner whenthe system changes The relationship between fault events isdisplayed statically in a fault tree When the failure status of asystem evolves from one subsystemcomponent to other sub-systemscomponents Petri net modelling is more intuitiveand more effective than fault tree modelling because of theuse of token transferring and enable-transferring processesThis process in Petri nets can be described by mathematicalmatrix computations which is easier to handle by computer(as shown in Section 4)

4 Dynamic Transition and Diagnosis ofPantograph Faults

41 PlaceMark and Enable-Transfer Algorithm of the Petri NetA Petri net consists of places ldquoIrdquo transactions ldquo|rdquo directedarcs ldquorarr rdquo and tokens of places ldquoerdquo The places representlogical descriptions of the system states and the transactionsrepresent the arising of system events Relational matrixesand state equations are major tools for Petri net analysis

119872

119896+1= 119872

119896+ 119860

119879

119883

119896 119896 = 1 2 (6)

In (6) 119872119896is the initial identification set of the system

faults before ignition 119872119896+1

is the result identification setof the system faults after ignition 119860119879 is the relationalmatrix and 119883

119870is the transfer sequence for ignition The

Mathematical Problems in Engineering 9

P1 P2 P3 P4

P5 P6 P7 P8 P9 P10 P11P12

P13

P14 P15 P16

P17 P18

P19

P20 P21 P22

P23

P24

P25 P26 P27 P28

P29

P30

P31

P32

P33P34

P35

P36 P37

P38

t1 t2 t3 t4

t5t6

t7

t8 t9 t10 t11 t12 t13 t14 t15

t16

t17

t18t19

t20

t21t22

t23 t24t25

t26t27

t28

t29t30 t31 t32

t33

t34t35

t36

t37

t38

t39

t40

t41

t42 t43

Initial identification 3

Initial identification 2

Initial identification 1

Figure 4 Transfer of tokens in the Petri net of the CRH EMU pantograph

initial identification1198720will convert to119872

119889after the ignition

sequence1198830 119883

119889 The process can be expressed as

119872

119889= 119872

0+ 119860

119879

119889minus1

sum

119896=0

119883

119896 (7)

If the element representing the top place is not less than1 in119872

119889after a series of transfers the fault event represented

by the top place will occur

42 Dynamic Transition and Diagnosis of Pantograph FaultsIn the process of fault diagnosis based on the use of a Petrinet the initial identification of the input place is regardedas the initial symptom of the event occurrence A token isassigned to the input place if the symptom appears otherwisethe place is assigned a null value The future states and finalidentification of the system can be circularly solved throughthe state equations and transfer sequences until the tokenvalue of the target place is found The fault occurs when thenumber of tokens in the target place is not zero

For the AND gate in the Petri net the tokens cannottransfer downward to the next level if there are empty inputplaces That is to say for an AND gate the fault will transferdownward only when all the events of the input places occurFor the OR gate in a Petri net the tokens will transferdownward to the next level place if at least one input placecontains tokens According to the rules of the network theorythe next level place will get two tokens if the two input placesboth contain tokens Regardless of the number of tokensthe occurrence of the fault that is represented by the targetplace is only related to whether there are tokens in the targetplace Transfer of the token in the Petri net of the CRH EMUpantograph is shown in Figure 4

As shown in Figure 4 1198751 represents ldquotwo EMUs attachedtogetherrdquo 1198752 represents ldquospacing of two pantographs of lessthan 190mrdquo 1198758 represents ldquopantograph bellows that aredamagedrdquo 11987526 represents ldquoMR ducts that are damagedrdquo11987529 represents ldquopressure in the auxiliary air cylinder that istoo lowrdquo 11987530 represents ldquopantograph rise to an abnormalpositionrdquo 11987532 represents ldquopantograph that cannot riserdquo11987535 represents ldquopantograph that descends automaticallyrdquoand 11987538 represents ldquopantograph that fails to workrdquo Threerepresentative fault transfer paths were selected to study thetransfer expression of pantograph faults in the Petri net Theinitial identification sets of the three paths were respectivelyrecorded as1198721

0

11987220

and11987230

119872

1

0

= [1 0 0

2le119894le38

]

119879

119872

2

0

= [0 0

1le119894le25

1 0 0

27le119894le38

]

119879

119872

3

0

= [0 0

1le119894le7

1 0 0

9le119894le38

]

119879

(8)

The ignition sequences of the three paths were respec-tively recorded as1198831

0

11988320

and11988330

119883

1

0

= [0 0

1le119894le4

1 0 0

6le119894le43

]

119879

119883

2

0

= [0 1 0 0

3le119894le43

]

119879

119883

3

0

= [0 0

1le119894le11

1 0 0

12le119894le21

1 0 0

23le119894le43

]

119879

(9)

10 Mathematical Problems in Engineering

The input matrix 119868 output matrix 119874 and the relationalmatrix119860119879 of the Petri net are all 38 times 43matrixes which canbe solved using the relational-matrix method

119860

119879

= 119874 minus 119868 = (

119886

11 119886

1119899

d

119886

1198981sdot sdot sdot 119886

119898119899

) (10)

119886

119894119895= 119874

119894119895minus 119868

1198941198951 le 119894 le 38 1 le 119895 le 43 (11)

In (10)119898 = 38 119899 = 43

119872

1

1

= 0

119872

1

2

= 0

(12)

As (12) shows tokens cannot transfer downward to thenext level in the first path Thus there is no token in the topplace at the end of the procedure meaning that the fault eventrepresented by 11987538 does not occur

In the second path11987221

11987222

and11987223

are as follows

119872

2

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le28

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

30le119894le38

]

]

]

119872

2

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

33le119894le38

]

]

]

119872

2

3

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

1

]

]

]

(13)

As (13) shows the tokens in place11987526 are transferred intoplace11987529 after the first ignition and then transferred into11987532after the second ignition Finally the tokens are transferredinto the top place 11987538 after the third ignition meaning thatthe top event occurs in this situation

In the third situation 11987231

can be calculated througha single transfer calculation when the default value of thetokens in place 1198758 is 1

119872

3

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le7

minus1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

9le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(14)

Thus when there are two transfer paths for one token inthe bottom basic place there will be negative values in thenext level state matrix after ignition if the default value of thetokens in the bottom basic place is still 1There is no practicalsignificance to this phenomenon To avoid negative valueschange the default value of the tokens in place 1198758 from 1 to 2to produce1198723

1

1015840

119872

3

1

1015840

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(15)

Then11987232

can be solved according to11987231

1015840

119872

3

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

2

]

]

]

(16)

According to11987232

there will be two tokens in the top place11987528 and the top event will inevitably happen when tokensindicate the fault transferThus the fault event in the top placewill happen if there is at least one token in the top place In thePetri net if an input place is a bottom basic place containingtokens with a value of 119899 (119899 is a positive integer greater than 1)directed arcs out then the default value of the tokens in theinput place is 119899The fault event presented indicated by the topplace would will inevitably happen when there are tokens inthe top place

Failure Mode 1 When two EMUs are attached together andat the same time the spacing of the two pantographs is lessthan 190m the pantograph will rise to an abnormal positionHowever if only one condition of the two is satisfied thepantograph will work normally

Failure Mode 2 Damage to MR ducts causes low pressurein the auxiliary air cylinder which means the pantographcannot rise successfully

FailureMode 3 Damage to the pantograph bellowsmay causethe pantograph to descend automatically or possibly fail torise Both of the two conditions will lead to a fault in thepantograph

The token transfer in the Petri net is consistent withthe logic analysis of the faults It can clearly and effectivelydescribe the dynamic processes of the system faults transferachieving fast and efficient fault diagnosis

5 Conclusions

In this study the working principles and failure modes ofpantographs on CRH EMU trains were analyzed systemati-cally Petri net modelling was used as a graphical modellingtool to simplify the logical relationships and events of thefault tree into a network with places and transactions asnodes The events logical gates and logical relation lines ofFT were transformed to places transitions and directed arcsrespectively The complexity of modelling was reduced by20by avoiding repeating eventsThus the process of solvingthe minimal cut sets was simplified which effectively savedcalculation time for theminimal cut set solutions for complexlarge-scale fault trees

The changes in system states and the evolution of failuresare described well by the dynamic properties of Petri netmodelling A mathematical model for the Petri net of thepantograph fault tree was established The equivalence andcorrectness of the token-transfer description for fault diag-nosis inference in a Petri net were verified Three differentfault paths were used to explain and verify the algorithmThe initial identification sets of the three paths were marked

Mathematical Problems in Engineering 11

The three corresponding ignition sequences were calculatedusing matrix transformations Finally the system status wasassessed correctly using a mathematical method which canbe handled easily in a computerised system

This work analyzes the evolution of failure events ofpantograph system The process of other critical systemsneeds to be further examined Future work will collect andanalyse test data from other systems in CRH EMU trainsto extend the methodology to cover the whole maintenanceprocess for CRH EMU train systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 51475324) The authors wouldlike to thank the reviewers for their constructive commentswhich enhanced the quality of this paper

References

[1] F C Jiao and B J Wang The Management and Maintenanceof Locomotive Rolling Stock Beijing Jiaotong University PressBeijing China 2013

[2] L L Song T Y Wang X W Song L Xu and D Song ldquoFaultdiagnosis of pantograph type current collector of CRH electricmultiple units based on Petri net modelling and fault treeanalysisrdquo Chinese Journal of Scientific Instrument vol 35 no 9pp 1990ndash1997 2014

[3] I Aydin M Karakose and E Akin ldquoAnomaly detection usinga modified kernel-based tracking in the pantographmdashcatenarysystemrdquoExpert SystemswithApplications vol 42 no 2 pp 938ndash948 2015

[4] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2008

[5] D M Shalev and J Tiran ldquoCondition-based fault tree analysis(CBFTA) a newmethod for improved fault tree analysis (FTA)reliability and safety calculationsrdquo Reliability Engineering andSystem Safety vol 92 no 9 pp 1231ndash1241 2007

[6] Y E Senol Y V Aydogdu B Sahin and I Kilic ldquoFaulttree analysis of chemical cargo contamination by using fuzzyapproachrdquo Expert Systems with Applications vol 42 no 12 pp5232ndash5244 2015

[7] S M Lavasani A Zendegani and M Celik ldquoAn extension tofuzzy fault tree analysis (FFTA) application in petrochemicalprocess industryrdquo Process Safety and Environmental Protectionvol 93 pp 75ndash88 2014

[8] J H Purba ldquoA fuzzy-based reliability approach to evaluate basicevents of fault tree analysis for nuclear power plant probabilisticsafety assessmentrdquo Annals of Nuclear Energy vol 70 pp 21ndash292014

[9] N RamzaliM RM Lavasani and J Ghodousi ldquoSafety barriersanalysis of offshore drilling system by employing Fuzzy EventTree Analysisrdquo Safety Science vol 78 pp 49ndash59 2015

[10] K-C Hyun S Min H Choi J Park and I Lee ldquoRisk analysisusing fault-tree analysis (FTA) and analytic hierarchy process(AHP) applicable to shield TBM tunnelsrdquo Tunnelling andUnderground Space Technology vol 49 pp 121ndash129 2015

[11] S Verlinden G Deconinck and B Coupe ldquoHybrid reliabilitymodel for nuclear reactor safety systemrdquo Reliability Engineeringand System Safety vol 101 pp 35ndash47 2012

[12] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 13 no 3 pp386ndash397 1987

[13] A Bobbio G Franceschinis R Gaeta and L Portinale ldquoPara-metric fault tree for the dependability analysis of redundant sys-tems and its high-level Petri net semanticsrdquo IEEE Transactionson Software Engineering vol 29 no 3 pp 270ndash287 2003

[14] R Robidoux H Xu L Xing andM Zhou ldquoAutomated model-ing of dynamic reliability block diagrams using colored Petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 2 pp 337ndash351 2010

[15] DMakajic-Nikolic M Vujosevic andN Nikolic ldquoMinimal cutsets of a coherent fault tree generation using reverse Petri netsrdquoOptimization vol 62 no 8 pp 1069ndash1087 2013

[16] B Wang G Tian Y Liang and T Qiang ldquoReliability modelingand evaluation of electric vehicle motor by using fault tree andextended stochastic petri netsrdquo Journal of Applied Mathematicsvol 2014 Article ID 638013 9 pages 2014

[17] T P K Nguyen J Beugin and JMarais ldquoMethod for evaluatingan extended Fault Tree to analyse the dependability of complexsystems application to a satellite-based railway systemrdquo Relia-bility Engineering and System Safety vol 133 pp 300ndash313 2015

[18] L L Song T Y Wang X W Song et al ldquoFuzzy intelligentfault diagnosis of pantograph type current collector under themulti-source heterogeneous knowledge environmentrdquo ChineseJournal of Scientific Instrument vol 36 pp 1283ndash1290 2015

[19] M An Y Chen and C J Baker ldquoA fuzzy reasoning and fuzzy-analytical hierarchy process based approach to the process ofrailway risk information a railway risk management systemrdquoInformation Sciences vol 181 no 18 pp 3946ndash3966 2011

[20] M An S Huang and C J Baker ldquoRailway risk assessmentmdashthe fuzzy reasoning approach and fuzzy analytic hierarchy pro-cess approaches a case study of shunting at Waterloo depotrdquoProceedings of the Institution of Mechanical Engineers F Journalof Rail and Rapid Transit vol 221 no 3 pp 365ndash383 2007

[21] K Noori and K Jenab ldquoFuzzy reliability-based traction controlmodel for intelligent transportation systemsrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 43 no 1 pp229ndash234 2013

[22] R I Muttram ldquoRailway safetyrsquos safety risk modelrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 216 no 2 pp 71ndash79 2002

[23] L Bouillaut O Francois and S Dubois ldquoA Bayesian network toevaluate underground rails maintenance strategies in an auto-mation contextrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 227 no4 pp 411ndash424 2013

[24] Y Xie Y Zhang and F Liang ldquoCrash injury severity analysisusing Bayesian ordered probit modelsrdquo Journal of Transporta-tion Engineering vol 135 no 1 pp 18ndash25 2009

[25] S Bernardi F Flammini S Marrone et al ldquoEnabling the usageof UML in the verification of railway systems the DAM-railapproachrdquoReliability Engineering and SystemSafety vol 120 pp112ndash126 2013

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

this paper is to extend FTA and Petri nets methodology tomaintenance and fault diagnosis in CHR EMU train systemsThis section introduces some existing applications of FTAandPetri nets in a range of industries Structural analysis workingprinciples failure mode and effects analysis (FMEA) andFTAmodelling of pantographs are outlined in Section 2 Petrinetsmodelling of the fault tree and the use of relationalmatrixanalysis in solvingminimal cut sets are provided in Section 3In Section 4 the place mark and enable-transfer algorithm ofthe Petri nets and an actual case study on dynamic transitionand diagnosis of pantograph diagnostics are provided Thelast section emphasises the highlights of this research

2 Structural Analysis and Fault TreeModelling of Pantograph

21 Structural Analysis and Working Principle of PantographThe pantograph of a CRH EMU train is fitted on the roofof the train and is essential to allow the train to get powerfrom the main overhead wires A CRH EMU train is aneight-car multiple unit configured with four motorised carsand four trailer cars There are two power units in all andeach power unit consists of two power cars and two trailercars arranged in T-M-M-T mode DSA250 pantographs arefitted on number 4 and number 6 cars Their maximumlifting height is 3000mm and the width of the head is1990mm In normal operation there will be only one pan-tograph collecting current with another in a folded stateHowever when two CRH EMUs are attached together twopantographs will work simultaneouslyTheworking principleand structural components of pantograph systems for CRHEMUs are shown in Figure 1The structural parameters of thepantograph are shown in Table 1

The pantograph is raised to access high voltage power byallowing the carbon skateboard to contact the catenary wireand descends when its compressed air supply is exhaustedThe 25 kv single-phase power alternating current (AC) fromthe catenary is transferred to the traction transformer fromthe high-voltage electrical equipment by the pantographwhich outputs 1500V single-phase AC power to the tractionconverter A pulse rectifier converts the single-phase AC intodirect current (DC) and outputs 2500ndash3000V DC to thetraction inverter through the DC circuit Then the tractioninverter outputs three-phaseACpowerwhere the voltage andfrequency are all adjustable to drive the traction motor

When the pantograph rises air is compressed into thedrive cylinder through the cushion valve and the cylinderpistonmoves left overcoming the pressure of the reset springThen the lower arm rises and rotates clockwise under theaction of the guide bar and the spring At the same time theupper arm also rises with the drive of the top guide bar

When the pantograph descends the compressed air isremoved from the drive cylinder through the cushion valveThen the piston is pushed to the right with the reset springreleasing pressureThe guide bar alsomoves right to force thelower arm to rotate anticlockwise thus forcing the upper armto descend

The collector head contacts the catenary wire when thepantograph rises Current is led to the bottom frame through

the collector head the upper arm and the pushrod Thepower cable installed on the bottom frame then leads thecurrent to the vehicle Since current will flow through theentire pantograph frame in the power supply state all thepantograph hinges are equipped with bridge connections toprevent the current damaging the bearings

The performance of the pantograph largely dependson contact pressure If there is too little pressure contactresistance will vary easily resulting in poor contact andarcing However toomuch pressure will increase the frictionaggravating wear on the carbon skateboard and wires andreducing the life of the carbon skateboard

22 Failure Mode and Effects Analysis (FMEA) of PantographSystem Pantographs on CRH EMU trains are installed onthe roof exposed to the environment The working environ-ment is complex volatile and sometimes very harsh Therearemany and complex ways in which the pantograph can failwhich are related to mechanical structures electrical facil-ities pneumatic transmission network control and manyother factors The normal operation of the pantograph on aCRH EMU train is automatically controlled by microproces-sors The pantograph system involves mechanical structuressuch as guide bars electrical equipment such as the tractionconverter and rubber products such as the base insulatorThe lifting process is controlled by air pressure Because pan-tographs connect directly with the extra high voltage (EHV)catenary wire they can be easily damaged by partial highvoltage discharges By classifying and analyzing fault trackingrecords the most common pantograph failure modes aresummarized as follows

(i) Failure Mode 1 Pantograph rises to an abnormal position

Probable cause

When two EMUs attach together the spacebetween the two pantographs is less than 190m

(ii) Failure Mode 2 Pantograph rises normally while themonitor (MON) does not show it correctly

Probable causes

(a) The pressure sensors are not operating correctly(b) The pressure switches have failed

(iii) Failure Mode 3 Pantograph cannot rise

Probable causes

(a) Electricity generation system (EGS) is closed(b) Vacuum circuit breaker (VCB) is closed(c) Pressure in the auxiliary air cylinder is too low(d) [PantographsdotVCB] NFB is in the OFF position(e) [PantographRising]NFB is in theOFF position(f) Pantograph pressurised air ducts leak(g) White air ducts connected to the pantograph

leak(h) Air ducts in the pantograph control valve board

leak

4 Mathematical Problems in Engineering

Pantograph bow

Carbon contact strip

Upper arm

Lower arm

Lift device for bow

Top guide bar

Coupling rod

Base frame

System damper

Filter capacitor

Inve

rter

Pulse

re

ctifi

er

Pulse

re

ctifi

erFilter capacitor

Inve

rter

Traction motor

Traction motor

Traction transformer

Main converterMain converter

Vacuum breaker

Pantograph

X4X4

Connection set knee

Pan stop rod

Air bellows

M1M2

Figure 1 Working principle of traction drive systems and pantograph structure in a CRH EMU train

Table 1 Structural parameters of the pantograph on a CRH EMU train

Structural parameters Values Structural parameters ValuesModel DSA250 Vent lifting height 3000mm (including insulator)Structure Single arm pantograph Maximal action height 2800mm (including insulator)Rated voltage 25 kV Minimal action height 888mm (including insulator)Rated current 1000A Folded height 593mm (including insulator)Weight 115 kg Contact pressure 70N plusmn 5NAmbient temperature minus40∘Csim+60∘C Design life 30 years

(i) Pantograph bellows are damaged(j) The pantographrsquos carbon skateboard leaks

(iv) Failure Mode 4 Pantograph cannot be lowered

Probable cause

Contacts in the pantographrsquos rising relay haveadhered

(v) Failure Mode 5 Pantograph cannot pass neutral sectionautomatically

Probable cause

Terminal RXCB board has failed

(vi) Failure Mode 6 Supporting insulator is damaged

Probable causes

(a) Surface defects created during manufacture(b) Insulator being burned by electrical arcing(c) Fog flashovers discharged because of high

humidity or dust(d) Fatigue cracks(e) Insulator being struck by foreign objects

(vii) Failure Mode 7 Pantograph descends automatically

Probable causes

(a) Pressure air ducts in the pantograph leak(b) White air ducts connected with the pantograph

leak(c) Air ducts in the pantograph control valve board

leak(d) Pantographrsquos bellows are damaged

Mathematical Problems in Engineering 5

(e) The pantographrsquos carbon skateboard leaks(f) Incorrect manipulation(g) Pantograph descending loop is instantly ener-

gized by electromagnetic interference(h) Pressure from the catenary is too low(i) There has been discharge between the panto-

graph bracket and the roof

(viii) Failure Mode 8 The catenary voltage transformer isdamaged

Probable causes

(a) Struck by foreign objects(b) Burned by electrical arcing(c) Quality control issues in components

23 Fault TreeModelling of the Pantograph Fault tree analysis(FTA) is one type of FMEA which is integrated withmechanics graph theory optimization theory and artificialintelligence techniques [29] It has been widely applied inmany fields including aerospace systems atomic reactorslarge-scale equipment and electronic computer systems FTAcan indicate causal relationships between complex faults andcan be useful for logical analysis and diagnosis of complexsystem faults It is a systematic way to assess the reliability ofcomplex systems both qualitatively and quantitatively

FTA is a systematic risk analysis method that deals withthe occurrence of an undesired event The fault event thatanalysts do not expect to happen is usually the focus of theFTAmethod Analysts apply top-down logic to find all directand indirect fault events relating to the incident They canthen establish logical relationships between the events form afault tree and undertake quantitative or qualitative analysisIn the process above the focus fault event is the top eventSelection of the top event is crucial to fault tree modelling Ifthe top event is too general it is difficult to analyse the faulttree On the other hand if the event is too specific the faulttree will fail to show the causal relationships of the systemfully In general the failure of the system analyzed will beselected as the top event

Combined with analysis of the structures and workingprinciples of pantographs on CRH-trains a pantograph faulttree has been established as shown in Figure 2 ldquo119879rdquo representsthe top event ldquo119872rdquo represents the intermediate event and ldquo119883rdquorepresents the basic event

Meanings of events in Figure 2 are shown as follows

119879 pantograph fails to work1198721 pantograph rises to an abnormal position1198722 pantograph rises normally but the MON doesnot show it correctly1198723 pantograph cannot rise1198724 pantograph cannot be lowered1198725 pantograph cannot pass neutral section automat-ically1198726 supporting insulators are damaged

1198727 pantograph descends automatically1198728 the catenary voltage transformer is damaged1198729 pressure of the auxiliary air cylinder is too low1198831 two EMUs are attached together1198832 spacing of two pantographs is less than 190m1198833 the pressure sensors do not work properly1198834 the pressure switches have failed1198835 EGS is closed1198836 VCB is closed1198837 in the cab switchboards at both ends[PantographsdotVCB] NFB is in the OFF position1198838 in the running switchboards of the number 4 andnumber 6 cars (CRH-200 CRH-300)number 4 andnumber 13 cars (CRH long-distance seat car CRHlong-distance sleeper car) [Pantograph Rising] NFBis in the OFF position1198839 pressure air ducts of the pantograph leak11988310 air ducts in the pantograph control valve boardleak11988311 pantograph bellows are damaged11988312 pantographrsquos carbon skateboard leaks11988313 white air ducts connected with the pantographleak11988314 contacts in the pantographrsquos rising relay haveadhered11988315 terminal RXCB board has failed11988316 surface defects due to manufacturing errors11988317 burned by electrical arcing11988318 fog flashovers discharge11988319 fatigue cracks11988320 struck by foreign objects11988321 driver error11988322 pantograph descending loop is instantly ener-gized by electromagnetic interference11988323 discharge between the pantograph bracket andthe roof11988324 pressure of the catenary is too low11988325 burned by electrical arcing11988326 air ducts of the air compressor are damaged11988327 the air compressor fails to work11988328 MR ducts are damaged11988329 main air cylinder leaks

The fault tree in Figure 2 shows the causal relationshipsbetween the pantograph fault events improving the accuracyand efficiency of fault diagnosis However the modellingpower of FTA is limited to the static evaluation of a singlecriterion at a time and cannot describe dynamic system

6 Mathematical Problems in Engineering

T

M4 M5 M6M3 M7M2 M8M1

X2X1

X13X5

X14 X15

X20X16

X25X16X4X3

X6 X7 X8 X9 X10 X11 X12M9

X26 X27 X28 X29

X17 X18 X19

X9 X10 X11 X12 X13 X21 X22 X23 X24

X17

Figure 2 FTA modelling of the pantograph-type current collector

behavior It cannot show the transmission and evolution offaults in the system It also needs too much computation tosolve the minimal cut sets of the large-scale fault tree whichmay lead to combinatorial explosion because of Nondeter-ministic Polynomial (NP) problems For example if (119894 =1 2 119899) is set to the 119894th minimal cut set of the fault treethen the top event can be expressed as

119879 =

119899

⋃

119894=1

119862

119894 (1)

The occurrence probability of the top event can beexpressed as

119875 (119879) = 119875(

119899

⋃

119894=1

119862

119894) (2)

The logical relationship between the basic events and theminimal cut set is and If the probabilities of the basic eventsare known the occurrence probability of the minimal cut setcan be expressed as

119875 (119862

119894) = 119875(

119896

⋃

119895=1

119909

119894) (3)

Based on the occurrence probabilities of the minimalcut sets the occurrence probability of the top event can beexpressed as

119875 (119879) =

119899

sum

119894=1

119875 (119862

119894) minus

119899

sum

119894lt119895=2

119875 (119862

119894119862

119895) + sdot sdot sdot

+ (minus1)

119898minus1

119875(

119899

⋃

119894=1

119862

119894)

(4)

According to the equation above the process of deter-mining the probability of the top event consists of 2119899minus1 partsThere are 28 minimal cut sets for the pantograph fault treein Figure 2 so it needs 227 parts to calculate the probabilityof the top event which means 134217728 items in all Thecombinatorial explosion of the calculation will make quanti-tative analysis difficult In addition the FTA method is gen-erally for static analysis and cannot reflect dynamic changesin multiple states in the system

The dynamic and structural properties of Petri nets areused to simplify and analytically calculate the pantographfault tree Relational matrix analysis is used to solve theminimal cut set equation for the fault tree Based on the estab-lished state equation of Petri nets initial token and enable-transfer algorithms are used to express the transfer pro-cess of faults mathematically

Mathematical Problems in Engineering 7

Table 2 Petri net expression of the FTA logic gates

AND logic gate OR logic gateFault tree expression Petri net expression Fault tree expression Petri net expression

3 Petri Nets Modelling and RelationalMatrix Analysis

31 Petri Nets Modelling of the Fault Tree The Petri net wasproposed in 1962 by Petri to express an information flowmodel for reticular structures [30] It is a mathematical andgraphical analysis tool to express the static structures anddynamic changes in a system [31ndash33]

In a Petri net the system state is indicated by ldquoIrdquo Achange in state is indicated by ldquo|rdquo An ordered pair is indicatedby a directed arc ldquorarr rdquo and a token is indicated by ldquoerdquo ThePetri net expressions of the FTA logic gates are shown inTable 2 The transform between Petri net modelling and thefault tree is shown in Figure 3

One Petri net can be defined in a sextuple 119873 =

(119875 119879 119868 119874119872119872

0) The six elements should meet the follow-

ing conditions

(1) 119875 = 119875

1 119875

119899 is a finite set of places 119899 gt 0 is the

number of places

(2) 119879 = 1199051 119905

119898 is a finite set of transactions119898 gt 0 is

the number of transactions 119875 cap 119879 =

(3) 119868 119875 times 119879 rarr 119873 is the input function defining the setof repetitions or the weight of the directed arcs fromset 119875 to set 119879 119873 = 0 1 is a set of nonnegativeintegers

(4) 119874 119879times119875 rarr 119873 is the output function defining the setof repetitions or the weight of the directed arcs fromset 119879 to set 119875 119873 = 0 1 is a set of nonnegativeintegers

(5) 119872 119875 rarr 119873 is the set of the identificationdistribution of every place

(6) 1198720 119875 rarr 119873 is the set of the initial identification dis-

tribution of every place

As shown in Figures 2 and 3 the top event is replaced bythe top place in the Petri net All the probable events thatmay lead to the top place are represented as middle placesor basic places The logical gates of the fault tree are denotedby Transaction andDirected arc Repeating events in the faulttree no longer exist in Petri net modelling

Petri net modelling avoids repeating basic events andcan achieve a 20 decrease in the number of places Thesimplifying effect is more obvious in large-scale systems

32 Relational Matrix Analysis in Solving the Minimal CutSets Besides simplifying the fault tree a Petri net alsoeffectively overcomes the shortcomings of the traditional FTAin solving the minimal cut sets The process of Petri netmodelling can be undertaken more easily on a computer

The structure of the Petri net can be translated into amatrix representation The value of the input function fromplace119875 to transaction 119905 is a nonnegative integer120596 recorded as119868(119875 119905) = 120596 represented by a directed arc from 119875 to 119905 with theside noteThe value of the output function from transaction 119905to place 119875 is a nonnegative integer 120596 recorded as119874(119875 119905) = 120596represented by a directed arc from 119905 to 119875 with the side note120596 The side note is omitted when 120596 = 1 The directed arc isalso omitted when 119868(119875 119905) = 0 or 119874(119875 119905) = 0 119868 and 119874 canboth be represented as 119899 times 119898 nonnegative integer matrixesThe difference between119874 and 119868 is called the relationalmatrixrecorded as 119860 = 119874 minus 119868

The steps for solving the minimal cut sets using therelational matrix method are as follows

Step 1 Find out the row consisting only of ldquo1rdquo and ldquo0rdquo in therelational matrix A This will be assigned the top place withonly inputs and no outputs

Step 2 Search for ldquominus1rdquo by column from 1 in the top placeAny row corresponding to ldquominus1rdquowill be regarded as an input tothe top place If there are multiple ldquominus1rdquo values in the columnthen there will be multiple inputs for the same transactionThe logical relationship between the inputs is AND

Step 3 Search for ldquo1rdquo by row from ldquominus1rdquo determined in Step 2Rows including ldquo1rdquo will be regarded as occupying a middleplaceThen search repeatedly following the second step untila row without ldquo1rdquo is located which will become a basic placeIf there are multiple occurrences of ldquo1rdquo in the row the logicalrelationship between the places corresponding to ldquo1rdquo will beOR

Step 4 Continue to search following the second step and thethird step until all the bottom basic places are located

Step 5 Expand the bottom places according to the logicalrelationships AND and OR Then obtain all the cut sets forthe system

Step 6 Find the minimal cut sets according to the Booleanabsorption rate or the prime number method

8 Mathematical Problems in Engineering

TokensM1

M2

X1

X1

X2

X3

t1 t2

t3 t4

t5

t6

P1 P2

P3P4

P5

P6T

Figure 3 Transform between Petri net modelling and the fault tree

The fault tree in Figure 3 is used as an example toillustrate solving theminimal cut sets with a relationalmatrixRespectively solve the input matrix 119868 the output matrix 119874and the relational matrix 119860 according to the steps above

119868 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 1

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

119874 =

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0 0

0 0 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

119860 = 119874 minus 119868 =

[

[

[

[

[

[

[

[

[

[

minus1 0 0 0 0 minus1

0 minus1 0 0 0 0

1 1 minus1 0 0 0

0 0 0 minus1 0 0

0 0 1 1 minus1 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

(5)

33 Comparison between Petri Net Modelling and Fault TreeModelling The complexity of quantitative analysis dependson the number of nodes and the logical gates in a fault treeComparing Figures 3 and 4 to Figure 2 we can see that thereare two repeating basic events and seven repeating events intwo traditional fault trees The repetition rates can respec-tively reach 17 and 24These repeating events do not existin the corresponding Petri net modelling The building ofthe fault tree needs 6 types of elements which involve basicevents intermediate events top events AND logical gate

OR logical gate and a relation line while the building of thePetri net model needs only three types of elements involvingplaces transactions and directed arcs Though the fault treein Figure 2 seems to provide better visualization than thePetri nets in Figure 4 there are more elements and repeatingevents in the fault tree The relationships among fault eventsin the fault tree are more complex The reduction of buildingelements and repeating events make the Petri net modellingmore convenient and efficient than fault tree modelling

In addition Petri nets are graphical and mathematicaltools for modelling systems and their dynamics while faulttree modelling cannot be updated in a timely manner whenthe system changes The relationship between fault events isdisplayed statically in a fault tree When the failure status of asystem evolves from one subsystemcomponent to other sub-systemscomponents Petri net modelling is more intuitiveand more effective than fault tree modelling because of theuse of token transferring and enable-transferring processesThis process in Petri nets can be described by mathematicalmatrix computations which is easier to handle by computer(as shown in Section 4)

4 Dynamic Transition and Diagnosis ofPantograph Faults

41 PlaceMark and Enable-Transfer Algorithm of the Petri NetA Petri net consists of places ldquoIrdquo transactions ldquo|rdquo directedarcs ldquorarr rdquo and tokens of places ldquoerdquo The places representlogical descriptions of the system states and the transactionsrepresent the arising of system events Relational matrixesand state equations are major tools for Petri net analysis

119872

119896+1= 119872

119896+ 119860

119879

119883

119896 119896 = 1 2 (6)

In (6) 119872119896is the initial identification set of the system

faults before ignition 119872119896+1

is the result identification setof the system faults after ignition 119860119879 is the relationalmatrix and 119883

119870is the transfer sequence for ignition The

Mathematical Problems in Engineering 9

P1 P2 P3 P4

P5 P6 P7 P8 P9 P10 P11P12

P13

P14 P15 P16

P17 P18

P19

P20 P21 P22

P23

P24

P25 P26 P27 P28

P29

P30

P31

P32

P33P34

P35

P36 P37

P38

t1 t2 t3 t4

t5t6

t7

t8 t9 t10 t11 t12 t13 t14 t15

t16

t17

t18t19

t20

t21t22

t23 t24t25

t26t27

t28

t29t30 t31 t32

t33

t34t35

t36

t37

t38

t39

t40

t41

t42 t43

Initial identification 3

Initial identification 2

Initial identification 1

Figure 4 Transfer of tokens in the Petri net of the CRH EMU pantograph

initial identification1198720will convert to119872

119889after the ignition

sequence1198830 119883

119889 The process can be expressed as

119872

119889= 119872

0+ 119860

119879

119889minus1

sum

119896=0

119883

119896 (7)

If the element representing the top place is not less than1 in119872

119889after a series of transfers the fault event represented

by the top place will occur

42 Dynamic Transition and Diagnosis of Pantograph FaultsIn the process of fault diagnosis based on the use of a Petrinet the initial identification of the input place is regardedas the initial symptom of the event occurrence A token isassigned to the input place if the symptom appears otherwisethe place is assigned a null value The future states and finalidentification of the system can be circularly solved throughthe state equations and transfer sequences until the tokenvalue of the target place is found The fault occurs when thenumber of tokens in the target place is not zero

For the AND gate in the Petri net the tokens cannottransfer downward to the next level if there are empty inputplaces That is to say for an AND gate the fault will transferdownward only when all the events of the input places occurFor the OR gate in a Petri net the tokens will transferdownward to the next level place if at least one input placecontains tokens According to the rules of the network theorythe next level place will get two tokens if the two input placesboth contain tokens Regardless of the number of tokensthe occurrence of the fault that is represented by the targetplace is only related to whether there are tokens in the targetplace Transfer of the token in the Petri net of the CRH EMUpantograph is shown in Figure 4

As shown in Figure 4 1198751 represents ldquotwo EMUs attachedtogetherrdquo 1198752 represents ldquospacing of two pantographs of lessthan 190mrdquo 1198758 represents ldquopantograph bellows that aredamagedrdquo 11987526 represents ldquoMR ducts that are damagedrdquo11987529 represents ldquopressure in the auxiliary air cylinder that istoo lowrdquo 11987530 represents ldquopantograph rise to an abnormalpositionrdquo 11987532 represents ldquopantograph that cannot riserdquo11987535 represents ldquopantograph that descends automaticallyrdquoand 11987538 represents ldquopantograph that fails to workrdquo Threerepresentative fault transfer paths were selected to study thetransfer expression of pantograph faults in the Petri net Theinitial identification sets of the three paths were respectivelyrecorded as1198721

0

11987220

and11987230

119872

1

0

= [1 0 0

2le119894le38

]

119879

119872

2

0

= [0 0

1le119894le25

1 0 0

27le119894le38

]

119879

119872

3

0

= [0 0

1le119894le7

1 0 0

9le119894le38

]

119879

(8)

The ignition sequences of the three paths were respec-tively recorded as1198831

0

11988320

and11988330

119883

1

0

= [0 0

1le119894le4

1 0 0

6le119894le43

]

119879

119883

2

0

= [0 1 0 0

3le119894le43

]

119879

119883

3

0

= [0 0

1le119894le11

1 0 0

12le119894le21

1 0 0

23le119894le43

]

119879

(9)

10 Mathematical Problems in Engineering

The input matrix 119868 output matrix 119874 and the relationalmatrix119860119879 of the Petri net are all 38 times 43matrixes which canbe solved using the relational-matrix method

119860

119879

= 119874 minus 119868 = (

119886

11 119886

1119899

d

119886

1198981sdot sdot sdot 119886

119898119899

) (10)

119886

119894119895= 119874

119894119895minus 119868

1198941198951 le 119894 le 38 1 le 119895 le 43 (11)

In (10)119898 = 38 119899 = 43

119872

1

1

= 0

119872

1

2

= 0

(12)

As (12) shows tokens cannot transfer downward to thenext level in the first path Thus there is no token in the topplace at the end of the procedure meaning that the fault eventrepresented by 11987538 does not occur

In the second path11987221

11987222

and11987223

are as follows

119872

2

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le28

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

30le119894le38

]

]

]

119872

2

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

33le119894le38

]

]

]

119872

2

3

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

1

]

]

]

(13)

As (13) shows the tokens in place11987526 are transferred intoplace11987529 after the first ignition and then transferred into11987532after the second ignition Finally the tokens are transferredinto the top place 11987538 after the third ignition meaning thatthe top event occurs in this situation

In the third situation 11987231

can be calculated througha single transfer calculation when the default value of thetokens in place 1198758 is 1

119872

3

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le7

minus1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

9le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(14)

Thus when there are two transfer paths for one token inthe bottom basic place there will be negative values in thenext level state matrix after ignition if the default value of thetokens in the bottom basic place is still 1There is no practicalsignificance to this phenomenon To avoid negative valueschange the default value of the tokens in place 1198758 from 1 to 2to produce1198723

1

1015840

119872

3

1

1015840

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(15)

Then11987232

can be solved according to11987231

1015840

119872

3

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

2

]

]

]

(16)

According to11987232

there will be two tokens in the top place11987528 and the top event will inevitably happen when tokensindicate the fault transferThus the fault event in the top placewill happen if there is at least one token in the top place In thePetri net if an input place is a bottom basic place containingtokens with a value of 119899 (119899 is a positive integer greater than 1)directed arcs out then the default value of the tokens in theinput place is 119899The fault event presented indicated by the topplace would will inevitably happen when there are tokens inthe top place

Failure Mode 1 When two EMUs are attached together andat the same time the spacing of the two pantographs is lessthan 190m the pantograph will rise to an abnormal positionHowever if only one condition of the two is satisfied thepantograph will work normally

Failure Mode 2 Damage to MR ducts causes low pressurein the auxiliary air cylinder which means the pantographcannot rise successfully

FailureMode 3 Damage to the pantograph bellowsmay causethe pantograph to descend automatically or possibly fail torise Both of the two conditions will lead to a fault in thepantograph

The token transfer in the Petri net is consistent withthe logic analysis of the faults It can clearly and effectivelydescribe the dynamic processes of the system faults transferachieving fast and efficient fault diagnosis

5 Conclusions

In this study the working principles and failure modes ofpantographs on CRH EMU trains were analyzed systemati-cally Petri net modelling was used as a graphical modellingtool to simplify the logical relationships and events of thefault tree into a network with places and transactions asnodes The events logical gates and logical relation lines ofFT were transformed to places transitions and directed arcsrespectively The complexity of modelling was reduced by20by avoiding repeating eventsThus the process of solvingthe minimal cut sets was simplified which effectively savedcalculation time for theminimal cut set solutions for complexlarge-scale fault trees

The changes in system states and the evolution of failuresare described well by the dynamic properties of Petri netmodelling A mathematical model for the Petri net of thepantograph fault tree was established The equivalence andcorrectness of the token-transfer description for fault diag-nosis inference in a Petri net were verified Three differentfault paths were used to explain and verify the algorithmThe initial identification sets of the three paths were marked

Mathematical Problems in Engineering 11

The three corresponding ignition sequences were calculatedusing matrix transformations Finally the system status wasassessed correctly using a mathematical method which canbe handled easily in a computerised system

This work analyzes the evolution of failure events ofpantograph system The process of other critical systemsneeds to be further examined Future work will collect andanalyse test data from other systems in CRH EMU trainsto extend the methodology to cover the whole maintenanceprocess for CRH EMU train systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 51475324) The authors wouldlike to thank the reviewers for their constructive commentswhich enhanced the quality of this paper

References

[1] F C Jiao and B J Wang The Management and Maintenanceof Locomotive Rolling Stock Beijing Jiaotong University PressBeijing China 2013

[2] L L Song T Y Wang X W Song L Xu and D Song ldquoFaultdiagnosis of pantograph type current collector of CRH electricmultiple units based on Petri net modelling and fault treeanalysisrdquo Chinese Journal of Scientific Instrument vol 35 no 9pp 1990ndash1997 2014

[3] I Aydin M Karakose and E Akin ldquoAnomaly detection usinga modified kernel-based tracking in the pantographmdashcatenarysystemrdquoExpert SystemswithApplications vol 42 no 2 pp 938ndash948 2015

[4] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2008

[5] D M Shalev and J Tiran ldquoCondition-based fault tree analysis(CBFTA) a newmethod for improved fault tree analysis (FTA)reliability and safety calculationsrdquo Reliability Engineering andSystem Safety vol 92 no 9 pp 1231ndash1241 2007

[6] Y E Senol Y V Aydogdu B Sahin and I Kilic ldquoFaulttree analysis of chemical cargo contamination by using fuzzyapproachrdquo Expert Systems with Applications vol 42 no 12 pp5232ndash5244 2015

[7] S M Lavasani A Zendegani and M Celik ldquoAn extension tofuzzy fault tree analysis (FFTA) application in petrochemicalprocess industryrdquo Process Safety and Environmental Protectionvol 93 pp 75ndash88 2014

[8] J H Purba ldquoA fuzzy-based reliability approach to evaluate basicevents of fault tree analysis for nuclear power plant probabilisticsafety assessmentrdquo Annals of Nuclear Energy vol 70 pp 21ndash292014

[9] N RamzaliM RM Lavasani and J Ghodousi ldquoSafety barriersanalysis of offshore drilling system by employing Fuzzy EventTree Analysisrdquo Safety Science vol 78 pp 49ndash59 2015

[10] K-C Hyun S Min H Choi J Park and I Lee ldquoRisk analysisusing fault-tree analysis (FTA) and analytic hierarchy process(AHP) applicable to shield TBM tunnelsrdquo Tunnelling andUnderground Space Technology vol 49 pp 121ndash129 2015

[11] S Verlinden G Deconinck and B Coupe ldquoHybrid reliabilitymodel for nuclear reactor safety systemrdquo Reliability Engineeringand System Safety vol 101 pp 35ndash47 2012

[12] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 13 no 3 pp386ndash397 1987

[13] A Bobbio G Franceschinis R Gaeta and L Portinale ldquoPara-metric fault tree for the dependability analysis of redundant sys-tems and its high-level Petri net semanticsrdquo IEEE Transactionson Software Engineering vol 29 no 3 pp 270ndash287 2003

[14] R Robidoux H Xu L Xing andM Zhou ldquoAutomated model-ing of dynamic reliability block diagrams using colored Petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 2 pp 337ndash351 2010

[15] DMakajic-Nikolic M Vujosevic andN Nikolic ldquoMinimal cutsets of a coherent fault tree generation using reverse Petri netsrdquoOptimization vol 62 no 8 pp 1069ndash1087 2013

[16] B Wang G Tian Y Liang and T Qiang ldquoReliability modelingand evaluation of electric vehicle motor by using fault tree andextended stochastic petri netsrdquo Journal of Applied Mathematicsvol 2014 Article ID 638013 9 pages 2014

[17] T P K Nguyen J Beugin and JMarais ldquoMethod for evaluatingan extended Fault Tree to analyse the dependability of complexsystems application to a satellite-based railway systemrdquo Relia-bility Engineering and System Safety vol 133 pp 300ndash313 2015

[18] L L Song T Y Wang X W Song et al ldquoFuzzy intelligentfault diagnosis of pantograph type current collector under themulti-source heterogeneous knowledge environmentrdquo ChineseJournal of Scientific Instrument vol 36 pp 1283ndash1290 2015

[19] M An Y Chen and C J Baker ldquoA fuzzy reasoning and fuzzy-analytical hierarchy process based approach to the process ofrailway risk information a railway risk management systemrdquoInformation Sciences vol 181 no 18 pp 3946ndash3966 2011

[20] M An S Huang and C J Baker ldquoRailway risk assessmentmdashthe fuzzy reasoning approach and fuzzy analytic hierarchy pro-cess approaches a case study of shunting at Waterloo depotrdquoProceedings of the Institution of Mechanical Engineers F Journalof Rail and Rapid Transit vol 221 no 3 pp 365ndash383 2007

[21] K Noori and K Jenab ldquoFuzzy reliability-based traction controlmodel for intelligent transportation systemsrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 43 no 1 pp229ndash234 2013

[22] R I Muttram ldquoRailway safetyrsquos safety risk modelrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 216 no 2 pp 71ndash79 2002

[23] L Bouillaut O Francois and S Dubois ldquoA Bayesian network toevaluate underground rails maintenance strategies in an auto-mation contextrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 227 no4 pp 411ndash424 2013

[24] Y Xie Y Zhang and F Liang ldquoCrash injury severity analysisusing Bayesian ordered probit modelsrdquo Journal of Transporta-tion Engineering vol 135 no 1 pp 18ndash25 2009

[25] S Bernardi F Flammini S Marrone et al ldquoEnabling the usageof UML in the verification of railway systems the DAM-railapproachrdquoReliability Engineering and SystemSafety vol 120 pp112ndash126 2013

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Pantograph bow

Carbon contact strip

Upper arm

Lower arm

Lift device for bow

Top guide bar

Coupling rod

Base frame

System damper

Filter capacitor

Inve

rter

Pulse

re

ctifi

er

Pulse

re

ctifi

erFilter capacitor

Inve

rter

Traction motor

Traction motor

Traction transformer

Main converterMain converter

Vacuum breaker

Pantograph

X4X4

Connection set knee

Pan stop rod

Air bellows

M1M2

Figure 1 Working principle of traction drive systems and pantograph structure in a CRH EMU train

Table 1 Structural parameters of the pantograph on a CRH EMU train

Structural parameters Values Structural parameters ValuesModel DSA250 Vent lifting height 3000mm (including insulator)Structure Single arm pantograph Maximal action height 2800mm (including insulator)Rated voltage 25 kV Minimal action height 888mm (including insulator)Rated current 1000A Folded height 593mm (including insulator)Weight 115 kg Contact pressure 70N plusmn 5NAmbient temperature minus40∘Csim+60∘C Design life 30 years

(i) Pantograph bellows are damaged(j) The pantographrsquos carbon skateboard leaks

(iv) Failure Mode 4 Pantograph cannot be lowered

Probable cause

Contacts in the pantographrsquos rising relay haveadhered

(v) Failure Mode 5 Pantograph cannot pass neutral sectionautomatically

Probable cause

Terminal RXCB board has failed

(vi) Failure Mode 6 Supporting insulator is damaged

Probable causes

(a) Surface defects created during manufacture(b) Insulator being burned by electrical arcing(c) Fog flashovers discharged because of high

humidity or dust(d) Fatigue cracks(e) Insulator being struck by foreign objects

(vii) Failure Mode 7 Pantograph descends automatically

Probable causes

(a) Pressure air ducts in the pantograph leak(b) White air ducts connected with the pantograph

leak(c) Air ducts in the pantograph control valve board

leak(d) Pantographrsquos bellows are damaged

Mathematical Problems in Engineering 5

(e) The pantographrsquos carbon skateboard leaks(f) Incorrect manipulation(g) Pantograph descending loop is instantly ener-

gized by electromagnetic interference(h) Pressure from the catenary is too low(i) There has been discharge between the panto-

graph bracket and the roof

(viii) Failure Mode 8 The catenary voltage transformer isdamaged

Probable causes

(a) Struck by foreign objects(b) Burned by electrical arcing(c) Quality control issues in components

23 Fault TreeModelling of the Pantograph Fault tree analysis(FTA) is one type of FMEA which is integrated withmechanics graph theory optimization theory and artificialintelligence techniques [29] It has been widely applied inmany fields including aerospace systems atomic reactorslarge-scale equipment and electronic computer systems FTAcan indicate causal relationships between complex faults andcan be useful for logical analysis and diagnosis of complexsystem faults It is a systematic way to assess the reliability ofcomplex systems both qualitatively and quantitatively

FTA is a systematic risk analysis method that deals withthe occurrence of an undesired event The fault event thatanalysts do not expect to happen is usually the focus of theFTAmethod Analysts apply top-down logic to find all directand indirect fault events relating to the incident They canthen establish logical relationships between the events form afault tree and undertake quantitative or qualitative analysisIn the process above the focus fault event is the top eventSelection of the top event is crucial to fault tree modelling Ifthe top event is too general it is difficult to analyse the faulttree On the other hand if the event is too specific the faulttree will fail to show the causal relationships of the systemfully In general the failure of the system analyzed will beselected as the top event

Combined with analysis of the structures and workingprinciples of pantographs on CRH-trains a pantograph faulttree has been established as shown in Figure 2 ldquo119879rdquo representsthe top event ldquo119872rdquo represents the intermediate event and ldquo119883rdquorepresents the basic event

Meanings of events in Figure 2 are shown as follows

119879 pantograph fails to work1198721 pantograph rises to an abnormal position1198722 pantograph rises normally but the MON doesnot show it correctly1198723 pantograph cannot rise1198724 pantograph cannot be lowered1198725 pantograph cannot pass neutral section automat-ically1198726 supporting insulators are damaged

1198727 pantograph descends automatically1198728 the catenary voltage transformer is damaged1198729 pressure of the auxiliary air cylinder is too low1198831 two EMUs are attached together1198832 spacing of two pantographs is less than 190m1198833 the pressure sensors do not work properly1198834 the pressure switches have failed1198835 EGS is closed1198836 VCB is closed1198837 in the cab switchboards at both ends[PantographsdotVCB] NFB is in the OFF position1198838 in the running switchboards of the number 4 andnumber 6 cars (CRH-200 CRH-300)number 4 andnumber 13 cars (CRH long-distance seat car CRHlong-distance sleeper car) [Pantograph Rising] NFBis in the OFF position1198839 pressure air ducts of the pantograph leak11988310 air ducts in the pantograph control valve boardleak11988311 pantograph bellows are damaged11988312 pantographrsquos carbon skateboard leaks11988313 white air ducts connected with the pantographleak11988314 contacts in the pantographrsquos rising relay haveadhered11988315 terminal RXCB board has failed11988316 surface defects due to manufacturing errors11988317 burned by electrical arcing11988318 fog flashovers discharge11988319 fatigue cracks11988320 struck by foreign objects11988321 driver error11988322 pantograph descending loop is instantly ener-gized by electromagnetic interference11988323 discharge between the pantograph bracket andthe roof11988324 pressure of the catenary is too low11988325 burned by electrical arcing11988326 air ducts of the air compressor are damaged11988327 the air compressor fails to work11988328 MR ducts are damaged11988329 main air cylinder leaks

The fault tree in Figure 2 shows the causal relationshipsbetween the pantograph fault events improving the accuracyand efficiency of fault diagnosis However the modellingpower of FTA is limited to the static evaluation of a singlecriterion at a time and cannot describe dynamic system

6 Mathematical Problems in Engineering

T

M4 M5 M6M3 M7M2 M8M1

X2X1

X13X5

X14 X15

X20X16

X25X16X4X3

X6 X7 X8 X9 X10 X11 X12M9

X26 X27 X28 X29

X17 X18 X19

X9 X10 X11 X12 X13 X21 X22 X23 X24

X17

Figure 2 FTA modelling of the pantograph-type current collector

behavior It cannot show the transmission and evolution offaults in the system It also needs too much computation tosolve the minimal cut sets of the large-scale fault tree whichmay lead to combinatorial explosion because of Nondeter-ministic Polynomial (NP) problems For example if (119894 =1 2 119899) is set to the 119894th minimal cut set of the fault treethen the top event can be expressed as

119879 =

119899

⋃

119894=1

119862

119894 (1)

The occurrence probability of the top event can beexpressed as

119875 (119879) = 119875(

119899

⋃

119894=1

119862

119894) (2)

The logical relationship between the basic events and theminimal cut set is and If the probabilities of the basic eventsare known the occurrence probability of the minimal cut setcan be expressed as

119875 (119862

119894) = 119875(

119896

⋃

119895=1

119909

119894) (3)

Based on the occurrence probabilities of the minimalcut sets the occurrence probability of the top event can beexpressed as

119875 (119879) =

119899

sum

119894=1

119875 (119862

119894) minus

119899

sum

119894lt119895=2

119875 (119862

119894119862

119895) + sdot sdot sdot

+ (minus1)

119898minus1

119875(

119899

⋃

119894=1

119862

119894)

(4)

According to the equation above the process of deter-mining the probability of the top event consists of 2119899minus1 partsThere are 28 minimal cut sets for the pantograph fault treein Figure 2 so it needs 227 parts to calculate the probabilityof the top event which means 134217728 items in all Thecombinatorial explosion of the calculation will make quanti-tative analysis difficult In addition the FTA method is gen-erally for static analysis and cannot reflect dynamic changesin multiple states in the system

The dynamic and structural properties of Petri nets areused to simplify and analytically calculate the pantographfault tree Relational matrix analysis is used to solve theminimal cut set equation for the fault tree Based on the estab-lished state equation of Petri nets initial token and enable-transfer algorithms are used to express the transfer pro-cess of faults mathematically

Mathematical Problems in Engineering 7

Table 2 Petri net expression of the FTA logic gates

AND logic gate OR logic gateFault tree expression Petri net expression Fault tree expression Petri net expression

3 Petri Nets Modelling and RelationalMatrix Analysis

31 Petri Nets Modelling of the Fault Tree The Petri net wasproposed in 1962 by Petri to express an information flowmodel for reticular structures [30] It is a mathematical andgraphical analysis tool to express the static structures anddynamic changes in a system [31ndash33]

In a Petri net the system state is indicated by ldquoIrdquo Achange in state is indicated by ldquo|rdquo An ordered pair is indicatedby a directed arc ldquorarr rdquo and a token is indicated by ldquoerdquo ThePetri net expressions of the FTA logic gates are shown inTable 2 The transform between Petri net modelling and thefault tree is shown in Figure 3

One Petri net can be defined in a sextuple 119873 =

(119875 119879 119868 119874119872119872

0) The six elements should meet the follow-

ing conditions

(1) 119875 = 119875

1 119875

119899 is a finite set of places 119899 gt 0 is the

number of places

(2) 119879 = 1199051 119905

119898 is a finite set of transactions119898 gt 0 is

the number of transactions 119875 cap 119879 =

(3) 119868 119875 times 119879 rarr 119873 is the input function defining the setof repetitions or the weight of the directed arcs fromset 119875 to set 119879 119873 = 0 1 is a set of nonnegativeintegers

(4) 119874 119879times119875 rarr 119873 is the output function defining the setof repetitions or the weight of the directed arcs fromset 119879 to set 119875 119873 = 0 1 is a set of nonnegativeintegers

(5) 119872 119875 rarr 119873 is the set of the identificationdistribution of every place

(6) 1198720 119875 rarr 119873 is the set of the initial identification dis-

tribution of every place

As shown in Figures 2 and 3 the top event is replaced bythe top place in the Petri net All the probable events thatmay lead to the top place are represented as middle placesor basic places The logical gates of the fault tree are denotedby Transaction andDirected arc Repeating events in the faulttree no longer exist in Petri net modelling

Petri net modelling avoids repeating basic events andcan achieve a 20 decrease in the number of places Thesimplifying effect is more obvious in large-scale systems

32 Relational Matrix Analysis in Solving the Minimal CutSets Besides simplifying the fault tree a Petri net alsoeffectively overcomes the shortcomings of the traditional FTAin solving the minimal cut sets The process of Petri netmodelling can be undertaken more easily on a computer

The structure of the Petri net can be translated into amatrix representation The value of the input function fromplace119875 to transaction 119905 is a nonnegative integer120596 recorded as119868(119875 119905) = 120596 represented by a directed arc from 119875 to 119905 with theside noteThe value of the output function from transaction 119905to place 119875 is a nonnegative integer 120596 recorded as119874(119875 119905) = 120596represented by a directed arc from 119905 to 119875 with the side note120596 The side note is omitted when 120596 = 1 The directed arc isalso omitted when 119868(119875 119905) = 0 or 119874(119875 119905) = 0 119868 and 119874 canboth be represented as 119899 times 119898 nonnegative integer matrixesThe difference between119874 and 119868 is called the relationalmatrixrecorded as 119860 = 119874 minus 119868

The steps for solving the minimal cut sets using therelational matrix method are as follows

Step 1 Find out the row consisting only of ldquo1rdquo and ldquo0rdquo in therelational matrix A This will be assigned the top place withonly inputs and no outputs

Step 2 Search for ldquominus1rdquo by column from 1 in the top placeAny row corresponding to ldquominus1rdquowill be regarded as an input tothe top place If there are multiple ldquominus1rdquo values in the columnthen there will be multiple inputs for the same transactionThe logical relationship between the inputs is AND

Step 3 Search for ldquo1rdquo by row from ldquominus1rdquo determined in Step 2Rows including ldquo1rdquo will be regarded as occupying a middleplaceThen search repeatedly following the second step untila row without ldquo1rdquo is located which will become a basic placeIf there are multiple occurrences of ldquo1rdquo in the row the logicalrelationship between the places corresponding to ldquo1rdquo will beOR

Step 4 Continue to search following the second step and thethird step until all the bottom basic places are located

Step 5 Expand the bottom places according to the logicalrelationships AND and OR Then obtain all the cut sets forthe system

Step 6 Find the minimal cut sets according to the Booleanabsorption rate or the prime number method

8 Mathematical Problems in Engineering

TokensM1

M2

X1

X1

X2

X3

t1 t2

t3 t4

t5

t6

P1 P2

P3P4

P5

P6T

Figure 3 Transform between Petri net modelling and the fault tree

The fault tree in Figure 3 is used as an example toillustrate solving theminimal cut sets with a relationalmatrixRespectively solve the input matrix 119868 the output matrix 119874and the relational matrix 119860 according to the steps above

119868 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 1

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

119874 =

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0 0

0 0 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

119860 = 119874 minus 119868 =

[

[

[

[

[

[

[

[

[

[

minus1 0 0 0 0 minus1

0 minus1 0 0 0 0

1 1 minus1 0 0 0

0 0 0 minus1 0 0

0 0 1 1 minus1 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

(5)

33 Comparison between Petri Net Modelling and Fault TreeModelling The complexity of quantitative analysis dependson the number of nodes and the logical gates in a fault treeComparing Figures 3 and 4 to Figure 2 we can see that thereare two repeating basic events and seven repeating events intwo traditional fault trees The repetition rates can respec-tively reach 17 and 24These repeating events do not existin the corresponding Petri net modelling The building ofthe fault tree needs 6 types of elements which involve basicevents intermediate events top events AND logical gate

OR logical gate and a relation line while the building of thePetri net model needs only three types of elements involvingplaces transactions and directed arcs Though the fault treein Figure 2 seems to provide better visualization than thePetri nets in Figure 4 there are more elements and repeatingevents in the fault tree The relationships among fault eventsin the fault tree are more complex The reduction of buildingelements and repeating events make the Petri net modellingmore convenient and efficient than fault tree modelling

In addition Petri nets are graphical and mathematicaltools for modelling systems and their dynamics while faulttree modelling cannot be updated in a timely manner whenthe system changes The relationship between fault events isdisplayed statically in a fault tree When the failure status of asystem evolves from one subsystemcomponent to other sub-systemscomponents Petri net modelling is more intuitiveand more effective than fault tree modelling because of theuse of token transferring and enable-transferring processesThis process in Petri nets can be described by mathematicalmatrix computations which is easier to handle by computer(as shown in Section 4)

4 Dynamic Transition and Diagnosis ofPantograph Faults

41 PlaceMark and Enable-Transfer Algorithm of the Petri NetA Petri net consists of places ldquoIrdquo transactions ldquo|rdquo directedarcs ldquorarr rdquo and tokens of places ldquoerdquo The places representlogical descriptions of the system states and the transactionsrepresent the arising of system events Relational matrixesand state equations are major tools for Petri net analysis

119872

119896+1= 119872

119896+ 119860

119879

119883

119896 119896 = 1 2 (6)

In (6) 119872119896is the initial identification set of the system

faults before ignition 119872119896+1

is the result identification setof the system faults after ignition 119860119879 is the relationalmatrix and 119883

119870is the transfer sequence for ignition The

Mathematical Problems in Engineering 9

P1 P2 P3 P4

P5 P6 P7 P8 P9 P10 P11P12

P13

P14 P15 P16

P17 P18

P19

P20 P21 P22

P23

P24

P25 P26 P27 P28

P29

P30

P31

P32

P33P34

P35

P36 P37

P38

t1 t2 t3 t4

t5t6

t7

t8 t9 t10 t11 t12 t13 t14 t15

t16

t17

t18t19

t20

t21t22

t23 t24t25

t26t27

t28

t29t30 t31 t32

t33

t34t35

t36

t37

t38

t39

t40

t41

t42 t43

Initial identification 3

Initial identification 2

Initial identification 1

Figure 4 Transfer of tokens in the Petri net of the CRH EMU pantograph

initial identification1198720will convert to119872

119889after the ignition

sequence1198830 119883

119889 The process can be expressed as

119872

119889= 119872

0+ 119860

119879

119889minus1

sum

119896=0

119883

119896 (7)

If the element representing the top place is not less than1 in119872

119889after a series of transfers the fault event represented

by the top place will occur

42 Dynamic Transition and Diagnosis of Pantograph FaultsIn the process of fault diagnosis based on the use of a Petrinet the initial identification of the input place is regardedas the initial symptom of the event occurrence A token isassigned to the input place if the symptom appears otherwisethe place is assigned a null value The future states and finalidentification of the system can be circularly solved throughthe state equations and transfer sequences until the tokenvalue of the target place is found The fault occurs when thenumber of tokens in the target place is not zero

For the AND gate in the Petri net the tokens cannottransfer downward to the next level if there are empty inputplaces That is to say for an AND gate the fault will transferdownward only when all the events of the input places occurFor the OR gate in a Petri net the tokens will transferdownward to the next level place if at least one input placecontains tokens According to the rules of the network theorythe next level place will get two tokens if the two input placesboth contain tokens Regardless of the number of tokensthe occurrence of the fault that is represented by the targetplace is only related to whether there are tokens in the targetplace Transfer of the token in the Petri net of the CRH EMUpantograph is shown in Figure 4

As shown in Figure 4 1198751 represents ldquotwo EMUs attachedtogetherrdquo 1198752 represents ldquospacing of two pantographs of lessthan 190mrdquo 1198758 represents ldquopantograph bellows that aredamagedrdquo 11987526 represents ldquoMR ducts that are damagedrdquo11987529 represents ldquopressure in the auxiliary air cylinder that istoo lowrdquo 11987530 represents ldquopantograph rise to an abnormalpositionrdquo 11987532 represents ldquopantograph that cannot riserdquo11987535 represents ldquopantograph that descends automaticallyrdquoand 11987538 represents ldquopantograph that fails to workrdquo Threerepresentative fault transfer paths were selected to study thetransfer expression of pantograph faults in the Petri net Theinitial identification sets of the three paths were respectivelyrecorded as1198721

0

11987220

and11987230

119872

1

0

= [1 0 0

2le119894le38

]

119879

119872

2

0

= [0 0

1le119894le25

1 0 0

27le119894le38

]

119879

119872

3

0

= [0 0

1le119894le7

1 0 0

9le119894le38

]

119879

(8)

The ignition sequences of the three paths were respec-tively recorded as1198831

0

11988320

and11988330

119883

1

0

= [0 0

1le119894le4

1 0 0

6le119894le43

]

119879

119883

2

0

= [0 1 0 0

3le119894le43

]

119879

119883

3

0

= [0 0

1le119894le11

1 0 0

12le119894le21

1 0 0

23le119894le43

]

119879

(9)

10 Mathematical Problems in Engineering

The input matrix 119868 output matrix 119874 and the relationalmatrix119860119879 of the Petri net are all 38 times 43matrixes which canbe solved using the relational-matrix method

119860

119879

= 119874 minus 119868 = (

119886

11 119886

1119899

d

119886

1198981sdot sdot sdot 119886

119898119899

) (10)

119886

119894119895= 119874

119894119895minus 119868

1198941198951 le 119894 le 38 1 le 119895 le 43 (11)

In (10)119898 = 38 119899 = 43

119872

1

1

= 0

119872

1

2

= 0

(12)

As (12) shows tokens cannot transfer downward to thenext level in the first path Thus there is no token in the topplace at the end of the procedure meaning that the fault eventrepresented by 11987538 does not occur

In the second path11987221

11987222

and11987223

are as follows

119872

2

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le28

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

30le119894le38

]

]

]

119872

2

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

33le119894le38

]

]

]

119872

2

3

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

1

]

]

]

(13)

As (13) shows the tokens in place11987526 are transferred intoplace11987529 after the first ignition and then transferred into11987532after the second ignition Finally the tokens are transferredinto the top place 11987538 after the third ignition meaning thatthe top event occurs in this situation

In the third situation 11987231

can be calculated througha single transfer calculation when the default value of thetokens in place 1198758 is 1

119872

3

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le7

minus1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

9le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(14)

Thus when there are two transfer paths for one token inthe bottom basic place there will be negative values in thenext level state matrix after ignition if the default value of thetokens in the bottom basic place is still 1There is no practicalsignificance to this phenomenon To avoid negative valueschange the default value of the tokens in place 1198758 from 1 to 2to produce1198723

1

1015840

119872

3

1

1015840

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(15)

Then11987232

can be solved according to11987231

1015840

119872

3

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

2

]

]

]

(16)

According to11987232

there will be two tokens in the top place11987528 and the top event will inevitably happen when tokensindicate the fault transferThus the fault event in the top placewill happen if there is at least one token in the top place In thePetri net if an input place is a bottom basic place containingtokens with a value of 119899 (119899 is a positive integer greater than 1)directed arcs out then the default value of the tokens in theinput place is 119899The fault event presented indicated by the topplace would will inevitably happen when there are tokens inthe top place

Failure Mode 1 When two EMUs are attached together andat the same time the spacing of the two pantographs is lessthan 190m the pantograph will rise to an abnormal positionHowever if only one condition of the two is satisfied thepantograph will work normally

Failure Mode 2 Damage to MR ducts causes low pressurein the auxiliary air cylinder which means the pantographcannot rise successfully

FailureMode 3 Damage to the pantograph bellowsmay causethe pantograph to descend automatically or possibly fail torise Both of the two conditions will lead to a fault in thepantograph

The token transfer in the Petri net is consistent withthe logic analysis of the faults It can clearly and effectivelydescribe the dynamic processes of the system faults transferachieving fast and efficient fault diagnosis

5 Conclusions

In this study the working principles and failure modes ofpantographs on CRH EMU trains were analyzed systemati-cally Petri net modelling was used as a graphical modellingtool to simplify the logical relationships and events of thefault tree into a network with places and transactions asnodes The events logical gates and logical relation lines ofFT were transformed to places transitions and directed arcsrespectively The complexity of modelling was reduced by20by avoiding repeating eventsThus the process of solvingthe minimal cut sets was simplified which effectively savedcalculation time for theminimal cut set solutions for complexlarge-scale fault trees

The changes in system states and the evolution of failuresare described well by the dynamic properties of Petri netmodelling A mathematical model for the Petri net of thepantograph fault tree was established The equivalence andcorrectness of the token-transfer description for fault diag-nosis inference in a Petri net were verified Three differentfault paths were used to explain and verify the algorithmThe initial identification sets of the three paths were marked

Mathematical Problems in Engineering 11

The three corresponding ignition sequences were calculatedusing matrix transformations Finally the system status wasassessed correctly using a mathematical method which canbe handled easily in a computerised system

This work analyzes the evolution of failure events ofpantograph system The process of other critical systemsneeds to be further examined Future work will collect andanalyse test data from other systems in CRH EMU trainsto extend the methodology to cover the whole maintenanceprocess for CRH EMU train systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 51475324) The authors wouldlike to thank the reviewers for their constructive commentswhich enhanced the quality of this paper

References

[1] F C Jiao and B J Wang The Management and Maintenanceof Locomotive Rolling Stock Beijing Jiaotong University PressBeijing China 2013

[2] L L Song T Y Wang X W Song L Xu and D Song ldquoFaultdiagnosis of pantograph type current collector of CRH electricmultiple units based on Petri net modelling and fault treeanalysisrdquo Chinese Journal of Scientific Instrument vol 35 no 9pp 1990ndash1997 2014

[3] I Aydin M Karakose and E Akin ldquoAnomaly detection usinga modified kernel-based tracking in the pantographmdashcatenarysystemrdquoExpert SystemswithApplications vol 42 no 2 pp 938ndash948 2015

[4] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2008

[5] D M Shalev and J Tiran ldquoCondition-based fault tree analysis(CBFTA) a newmethod for improved fault tree analysis (FTA)reliability and safety calculationsrdquo Reliability Engineering andSystem Safety vol 92 no 9 pp 1231ndash1241 2007

[6] Y E Senol Y V Aydogdu B Sahin and I Kilic ldquoFaulttree analysis of chemical cargo contamination by using fuzzyapproachrdquo Expert Systems with Applications vol 42 no 12 pp5232ndash5244 2015

[7] S M Lavasani A Zendegani and M Celik ldquoAn extension tofuzzy fault tree analysis (FFTA) application in petrochemicalprocess industryrdquo Process Safety and Environmental Protectionvol 93 pp 75ndash88 2014

[8] J H Purba ldquoA fuzzy-based reliability approach to evaluate basicevents of fault tree analysis for nuclear power plant probabilisticsafety assessmentrdquo Annals of Nuclear Energy vol 70 pp 21ndash292014

[9] N RamzaliM RM Lavasani and J Ghodousi ldquoSafety barriersanalysis of offshore drilling system by employing Fuzzy EventTree Analysisrdquo Safety Science vol 78 pp 49ndash59 2015

[10] K-C Hyun S Min H Choi J Park and I Lee ldquoRisk analysisusing fault-tree analysis (FTA) and analytic hierarchy process(AHP) applicable to shield TBM tunnelsrdquo Tunnelling andUnderground Space Technology vol 49 pp 121ndash129 2015

[11] S Verlinden G Deconinck and B Coupe ldquoHybrid reliabilitymodel for nuclear reactor safety systemrdquo Reliability Engineeringand System Safety vol 101 pp 35ndash47 2012

[12] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 13 no 3 pp386ndash397 1987

[13] A Bobbio G Franceschinis R Gaeta and L Portinale ldquoPara-metric fault tree for the dependability analysis of redundant sys-tems and its high-level Petri net semanticsrdquo IEEE Transactionson Software Engineering vol 29 no 3 pp 270ndash287 2003

[14] R Robidoux H Xu L Xing andM Zhou ldquoAutomated model-ing of dynamic reliability block diagrams using colored Petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 2 pp 337ndash351 2010

[15] DMakajic-Nikolic M Vujosevic andN Nikolic ldquoMinimal cutsets of a coherent fault tree generation using reverse Petri netsrdquoOptimization vol 62 no 8 pp 1069ndash1087 2013

[16] B Wang G Tian Y Liang and T Qiang ldquoReliability modelingand evaluation of electric vehicle motor by using fault tree andextended stochastic petri netsrdquo Journal of Applied Mathematicsvol 2014 Article ID 638013 9 pages 2014

[17] T P K Nguyen J Beugin and JMarais ldquoMethod for evaluatingan extended Fault Tree to analyse the dependability of complexsystems application to a satellite-based railway systemrdquo Relia-bility Engineering and System Safety vol 133 pp 300ndash313 2015

[18] L L Song T Y Wang X W Song et al ldquoFuzzy intelligentfault diagnosis of pantograph type current collector under themulti-source heterogeneous knowledge environmentrdquo ChineseJournal of Scientific Instrument vol 36 pp 1283ndash1290 2015

[19] M An Y Chen and C J Baker ldquoA fuzzy reasoning and fuzzy-analytical hierarchy process based approach to the process ofrailway risk information a railway risk management systemrdquoInformation Sciences vol 181 no 18 pp 3946ndash3966 2011

[20] M An S Huang and C J Baker ldquoRailway risk assessmentmdashthe fuzzy reasoning approach and fuzzy analytic hierarchy pro-cess approaches a case study of shunting at Waterloo depotrdquoProceedings of the Institution of Mechanical Engineers F Journalof Rail and Rapid Transit vol 221 no 3 pp 365ndash383 2007

[21] K Noori and K Jenab ldquoFuzzy reliability-based traction controlmodel for intelligent transportation systemsrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 43 no 1 pp229ndash234 2013

[22] R I Muttram ldquoRailway safetyrsquos safety risk modelrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 216 no 2 pp 71ndash79 2002

[23] L Bouillaut O Francois and S Dubois ldquoA Bayesian network toevaluate underground rails maintenance strategies in an auto-mation contextrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 227 no4 pp 411ndash424 2013

[24] Y Xie Y Zhang and F Liang ldquoCrash injury severity analysisusing Bayesian ordered probit modelsrdquo Journal of Transporta-tion Engineering vol 135 no 1 pp 18ndash25 2009

[25] S Bernardi F Flammini S Marrone et al ldquoEnabling the usageof UML in the verification of railway systems the DAM-railapproachrdquoReliability Engineering and SystemSafety vol 120 pp112ndash126 2013

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

(e) The pantographrsquos carbon skateboard leaks(f) Incorrect manipulation(g) Pantograph descending loop is instantly ener-

gized by electromagnetic interference(h) Pressure from the catenary is too low(i) There has been discharge between the panto-

graph bracket and the roof

(viii) Failure Mode 8 The catenary voltage transformer isdamaged

Probable causes

(a) Struck by foreign objects(b) Burned by electrical arcing(c) Quality control issues in components

23 Fault TreeModelling of the Pantograph Fault tree analysis(FTA) is one type of FMEA which is integrated withmechanics graph theory optimization theory and artificialintelligence techniques [29] It has been widely applied inmany fields including aerospace systems atomic reactorslarge-scale equipment and electronic computer systems FTAcan indicate causal relationships between complex faults andcan be useful for logical analysis and diagnosis of complexsystem faults It is a systematic way to assess the reliability ofcomplex systems both qualitatively and quantitatively

FTA is a systematic risk analysis method that deals withthe occurrence of an undesired event The fault event thatanalysts do not expect to happen is usually the focus of theFTAmethod Analysts apply top-down logic to find all directand indirect fault events relating to the incident They canthen establish logical relationships between the events form afault tree and undertake quantitative or qualitative analysisIn the process above the focus fault event is the top eventSelection of the top event is crucial to fault tree modelling Ifthe top event is too general it is difficult to analyse the faulttree On the other hand if the event is too specific the faulttree will fail to show the causal relationships of the systemfully In general the failure of the system analyzed will beselected as the top event

Combined with analysis of the structures and workingprinciples of pantographs on CRH-trains a pantograph faulttree has been established as shown in Figure 2 ldquo119879rdquo representsthe top event ldquo119872rdquo represents the intermediate event and ldquo119883rdquorepresents the basic event

Meanings of events in Figure 2 are shown as follows

119879 pantograph fails to work1198721 pantograph rises to an abnormal position1198722 pantograph rises normally but the MON doesnot show it correctly1198723 pantograph cannot rise1198724 pantograph cannot be lowered1198725 pantograph cannot pass neutral section automat-ically1198726 supporting insulators are damaged

1198727 pantograph descends automatically1198728 the catenary voltage transformer is damaged1198729 pressure of the auxiliary air cylinder is too low1198831 two EMUs are attached together1198832 spacing of two pantographs is less than 190m1198833 the pressure sensors do not work properly1198834 the pressure switches have failed1198835 EGS is closed1198836 VCB is closed1198837 in the cab switchboards at both ends[PantographsdotVCB] NFB is in the OFF position1198838 in the running switchboards of the number 4 andnumber 6 cars (CRH-200 CRH-300)number 4 andnumber 13 cars (CRH long-distance seat car CRHlong-distance sleeper car) [Pantograph Rising] NFBis in the OFF position1198839 pressure air ducts of the pantograph leak11988310 air ducts in the pantograph control valve boardleak11988311 pantograph bellows are damaged11988312 pantographrsquos carbon skateboard leaks11988313 white air ducts connected with the pantographleak11988314 contacts in the pantographrsquos rising relay haveadhered11988315 terminal RXCB board has failed11988316 surface defects due to manufacturing errors11988317 burned by electrical arcing11988318 fog flashovers discharge11988319 fatigue cracks11988320 struck by foreign objects11988321 driver error11988322 pantograph descending loop is instantly ener-gized by electromagnetic interference11988323 discharge between the pantograph bracket andthe roof11988324 pressure of the catenary is too low11988325 burned by electrical arcing11988326 air ducts of the air compressor are damaged11988327 the air compressor fails to work11988328 MR ducts are damaged11988329 main air cylinder leaks

The fault tree in Figure 2 shows the causal relationshipsbetween the pantograph fault events improving the accuracyand efficiency of fault diagnosis However the modellingpower of FTA is limited to the static evaluation of a singlecriterion at a time and cannot describe dynamic system

6 Mathematical Problems in Engineering

T

M4 M5 M6M3 M7M2 M8M1

X2X1

X13X5

X14 X15

X20X16

X25X16X4X3

X6 X7 X8 X9 X10 X11 X12M9

X26 X27 X28 X29

X17 X18 X19

X9 X10 X11 X12 X13 X21 X22 X23 X24

X17

Figure 2 FTA modelling of the pantograph-type current collector

behavior It cannot show the transmission and evolution offaults in the system It also needs too much computation tosolve the minimal cut sets of the large-scale fault tree whichmay lead to combinatorial explosion because of Nondeter-ministic Polynomial (NP) problems For example if (119894 =1 2 119899) is set to the 119894th minimal cut set of the fault treethen the top event can be expressed as

119879 =

119899

⋃

119894=1

119862

119894 (1)

The occurrence probability of the top event can beexpressed as

119875 (119879) = 119875(

119899

⋃

119894=1

119862

119894) (2)

The logical relationship between the basic events and theminimal cut set is and If the probabilities of the basic eventsare known the occurrence probability of the minimal cut setcan be expressed as

119875 (119862

119894) = 119875(

119896

⋃

119895=1

119909

119894) (3)

Based on the occurrence probabilities of the minimalcut sets the occurrence probability of the top event can beexpressed as

119875 (119879) =

119899

sum

119894=1

119875 (119862

119894) minus

119899

sum

119894lt119895=2

119875 (119862

119894119862

119895) + sdot sdot sdot

+ (minus1)

119898minus1

119875(

119899

⋃

119894=1

119862

119894)

(4)

According to the equation above the process of deter-mining the probability of the top event consists of 2119899minus1 partsThere are 28 minimal cut sets for the pantograph fault treein Figure 2 so it needs 227 parts to calculate the probabilityof the top event which means 134217728 items in all Thecombinatorial explosion of the calculation will make quanti-tative analysis difficult In addition the FTA method is gen-erally for static analysis and cannot reflect dynamic changesin multiple states in the system

The dynamic and structural properties of Petri nets areused to simplify and analytically calculate the pantographfault tree Relational matrix analysis is used to solve theminimal cut set equation for the fault tree Based on the estab-lished state equation of Petri nets initial token and enable-transfer algorithms are used to express the transfer pro-cess of faults mathematically

Mathematical Problems in Engineering 7

Table 2 Petri net expression of the FTA logic gates

AND logic gate OR logic gateFault tree expression Petri net expression Fault tree expression Petri net expression

3 Petri Nets Modelling and RelationalMatrix Analysis

31 Petri Nets Modelling of the Fault Tree The Petri net wasproposed in 1962 by Petri to express an information flowmodel for reticular structures [30] It is a mathematical andgraphical analysis tool to express the static structures anddynamic changes in a system [31ndash33]

In a Petri net the system state is indicated by ldquoIrdquo Achange in state is indicated by ldquo|rdquo An ordered pair is indicatedby a directed arc ldquorarr rdquo and a token is indicated by ldquoerdquo ThePetri net expressions of the FTA logic gates are shown inTable 2 The transform between Petri net modelling and thefault tree is shown in Figure 3

One Petri net can be defined in a sextuple 119873 =

(119875 119879 119868 119874119872119872

0) The six elements should meet the follow-

ing conditions

(1) 119875 = 119875

1 119875

119899 is a finite set of places 119899 gt 0 is the

number of places

(2) 119879 = 1199051 119905

119898 is a finite set of transactions119898 gt 0 is

the number of transactions 119875 cap 119879 =

(3) 119868 119875 times 119879 rarr 119873 is the input function defining the setof repetitions or the weight of the directed arcs fromset 119875 to set 119879 119873 = 0 1 is a set of nonnegativeintegers

(4) 119874 119879times119875 rarr 119873 is the output function defining the setof repetitions or the weight of the directed arcs fromset 119879 to set 119875 119873 = 0 1 is a set of nonnegativeintegers

(5) 119872 119875 rarr 119873 is the set of the identificationdistribution of every place

(6) 1198720 119875 rarr 119873 is the set of the initial identification dis-

tribution of every place

As shown in Figures 2 and 3 the top event is replaced bythe top place in the Petri net All the probable events thatmay lead to the top place are represented as middle placesor basic places The logical gates of the fault tree are denotedby Transaction andDirected arc Repeating events in the faulttree no longer exist in Petri net modelling

Petri net modelling avoids repeating basic events andcan achieve a 20 decrease in the number of places Thesimplifying effect is more obvious in large-scale systems

32 Relational Matrix Analysis in Solving the Minimal CutSets Besides simplifying the fault tree a Petri net alsoeffectively overcomes the shortcomings of the traditional FTAin solving the minimal cut sets The process of Petri netmodelling can be undertaken more easily on a computer

The structure of the Petri net can be translated into amatrix representation The value of the input function fromplace119875 to transaction 119905 is a nonnegative integer120596 recorded as119868(119875 119905) = 120596 represented by a directed arc from 119875 to 119905 with theside noteThe value of the output function from transaction 119905to place 119875 is a nonnegative integer 120596 recorded as119874(119875 119905) = 120596represented by a directed arc from 119905 to 119875 with the side note120596 The side note is omitted when 120596 = 1 The directed arc isalso omitted when 119868(119875 119905) = 0 or 119874(119875 119905) = 0 119868 and 119874 canboth be represented as 119899 times 119898 nonnegative integer matrixesThe difference between119874 and 119868 is called the relationalmatrixrecorded as 119860 = 119874 minus 119868

The steps for solving the minimal cut sets using therelational matrix method are as follows

Step 1 Find out the row consisting only of ldquo1rdquo and ldquo0rdquo in therelational matrix A This will be assigned the top place withonly inputs and no outputs

Step 2 Search for ldquominus1rdquo by column from 1 in the top placeAny row corresponding to ldquominus1rdquowill be regarded as an input tothe top place If there are multiple ldquominus1rdquo values in the columnthen there will be multiple inputs for the same transactionThe logical relationship between the inputs is AND

Step 3 Search for ldquo1rdquo by row from ldquominus1rdquo determined in Step 2Rows including ldquo1rdquo will be regarded as occupying a middleplaceThen search repeatedly following the second step untila row without ldquo1rdquo is located which will become a basic placeIf there are multiple occurrences of ldquo1rdquo in the row the logicalrelationship between the places corresponding to ldquo1rdquo will beOR

Step 4 Continue to search following the second step and thethird step until all the bottom basic places are located

Step 5 Expand the bottom places according to the logicalrelationships AND and OR Then obtain all the cut sets forthe system

Step 6 Find the minimal cut sets according to the Booleanabsorption rate or the prime number method

8 Mathematical Problems in Engineering

TokensM1

M2

X1

X1

X2

X3

t1 t2

t3 t4

t5

t6

P1 P2

P3P4

P5

P6T

Figure 3 Transform between Petri net modelling and the fault tree

The fault tree in Figure 3 is used as an example toillustrate solving theminimal cut sets with a relationalmatrixRespectively solve the input matrix 119868 the output matrix 119874and the relational matrix 119860 according to the steps above

119868 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 1

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

119874 =

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0 0

0 0 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

119860 = 119874 minus 119868 =

[

[

[

[

[

[

[

[

[

[

minus1 0 0 0 0 minus1

0 minus1 0 0 0 0

1 1 minus1 0 0 0

0 0 0 minus1 0 0

0 0 1 1 minus1 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

(5)

33 Comparison between Petri Net Modelling and Fault TreeModelling The complexity of quantitative analysis dependson the number of nodes and the logical gates in a fault treeComparing Figures 3 and 4 to Figure 2 we can see that thereare two repeating basic events and seven repeating events intwo traditional fault trees The repetition rates can respec-tively reach 17 and 24These repeating events do not existin the corresponding Petri net modelling The building ofthe fault tree needs 6 types of elements which involve basicevents intermediate events top events AND logical gate

OR logical gate and a relation line while the building of thePetri net model needs only three types of elements involvingplaces transactions and directed arcs Though the fault treein Figure 2 seems to provide better visualization than thePetri nets in Figure 4 there are more elements and repeatingevents in the fault tree The relationships among fault eventsin the fault tree are more complex The reduction of buildingelements and repeating events make the Petri net modellingmore convenient and efficient than fault tree modelling

In addition Petri nets are graphical and mathematicaltools for modelling systems and their dynamics while faulttree modelling cannot be updated in a timely manner whenthe system changes The relationship between fault events isdisplayed statically in a fault tree When the failure status of asystem evolves from one subsystemcomponent to other sub-systemscomponents Petri net modelling is more intuitiveand more effective than fault tree modelling because of theuse of token transferring and enable-transferring processesThis process in Petri nets can be described by mathematicalmatrix computations which is easier to handle by computer(as shown in Section 4)

4 Dynamic Transition and Diagnosis ofPantograph Faults

41 PlaceMark and Enable-Transfer Algorithm of the Petri NetA Petri net consists of places ldquoIrdquo transactions ldquo|rdquo directedarcs ldquorarr rdquo and tokens of places ldquoerdquo The places representlogical descriptions of the system states and the transactionsrepresent the arising of system events Relational matrixesand state equations are major tools for Petri net analysis

119872

119896+1= 119872

119896+ 119860

119879

119883

119896 119896 = 1 2 (6)

In (6) 119872119896is the initial identification set of the system

faults before ignition 119872119896+1

is the result identification setof the system faults after ignition 119860119879 is the relationalmatrix and 119883

119870is the transfer sequence for ignition The

Mathematical Problems in Engineering 9

P1 P2 P3 P4

P5 P6 P7 P8 P9 P10 P11P12

P13

P14 P15 P16

P17 P18

P19

P20 P21 P22

P23

P24

P25 P26 P27 P28

P29

P30

P31

P32

P33P34

P35

P36 P37

P38

t1 t2 t3 t4

t5t6

t7

t8 t9 t10 t11 t12 t13 t14 t15

t16

t17

t18t19

t20

t21t22

t23 t24t25

t26t27

t28

t29t30 t31 t32

t33

t34t35

t36

t37

t38

t39

t40

t41

t42 t43

Initial identification 3

Initial identification 2

Initial identification 1

Figure 4 Transfer of tokens in the Petri net of the CRH EMU pantograph

initial identification1198720will convert to119872

119889after the ignition

sequence1198830 119883

119889 The process can be expressed as

119872

119889= 119872

0+ 119860

119879

119889minus1

sum

119896=0

119883

119896 (7)

If the element representing the top place is not less than1 in119872

119889after a series of transfers the fault event represented

by the top place will occur

42 Dynamic Transition and Diagnosis of Pantograph FaultsIn the process of fault diagnosis based on the use of a Petrinet the initial identification of the input place is regardedas the initial symptom of the event occurrence A token isassigned to the input place if the symptom appears otherwisethe place is assigned a null value The future states and finalidentification of the system can be circularly solved throughthe state equations and transfer sequences until the tokenvalue of the target place is found The fault occurs when thenumber of tokens in the target place is not zero

For the AND gate in the Petri net the tokens cannottransfer downward to the next level if there are empty inputplaces That is to say for an AND gate the fault will transferdownward only when all the events of the input places occurFor the OR gate in a Petri net the tokens will transferdownward to the next level place if at least one input placecontains tokens According to the rules of the network theorythe next level place will get two tokens if the two input placesboth contain tokens Regardless of the number of tokensthe occurrence of the fault that is represented by the targetplace is only related to whether there are tokens in the targetplace Transfer of the token in the Petri net of the CRH EMUpantograph is shown in Figure 4

As shown in Figure 4 1198751 represents ldquotwo EMUs attachedtogetherrdquo 1198752 represents ldquospacing of two pantographs of lessthan 190mrdquo 1198758 represents ldquopantograph bellows that aredamagedrdquo 11987526 represents ldquoMR ducts that are damagedrdquo11987529 represents ldquopressure in the auxiliary air cylinder that istoo lowrdquo 11987530 represents ldquopantograph rise to an abnormalpositionrdquo 11987532 represents ldquopantograph that cannot riserdquo11987535 represents ldquopantograph that descends automaticallyrdquoand 11987538 represents ldquopantograph that fails to workrdquo Threerepresentative fault transfer paths were selected to study thetransfer expression of pantograph faults in the Petri net Theinitial identification sets of the three paths were respectivelyrecorded as1198721

0

11987220

and11987230

119872

1

0

= [1 0 0

2le119894le38

]

119879

119872

2

0

= [0 0

1le119894le25

1 0 0

27le119894le38

]

119879

119872

3

0

= [0 0

1le119894le7

1 0 0

9le119894le38

]

119879

(8)

The ignition sequences of the three paths were respec-tively recorded as1198831

0

11988320

and11988330

119883

1

0

= [0 0

1le119894le4

1 0 0

6le119894le43

]

119879

119883

2

0

= [0 1 0 0

3le119894le43

]

119879

119883

3

0

= [0 0

1le119894le11

1 0 0

12le119894le21

1 0 0

23le119894le43

]

119879

(9)

10 Mathematical Problems in Engineering

The input matrix 119868 output matrix 119874 and the relationalmatrix119860119879 of the Petri net are all 38 times 43matrixes which canbe solved using the relational-matrix method

119860

119879

= 119874 minus 119868 = (

119886

11 119886

1119899

d

119886

1198981sdot sdot sdot 119886

119898119899

) (10)

119886

119894119895= 119874

119894119895minus 119868

1198941198951 le 119894 le 38 1 le 119895 le 43 (11)

In (10)119898 = 38 119899 = 43

119872

1

1

= 0

119872

1

2

= 0

(12)

As (12) shows tokens cannot transfer downward to thenext level in the first path Thus there is no token in the topplace at the end of the procedure meaning that the fault eventrepresented by 11987538 does not occur

In the second path11987221

11987222

and11987223

are as follows

119872

2

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le28

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

30le119894le38

]

]

]

119872

2

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

33le119894le38

]

]

]

119872

2

3

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

1

]

]

]

(13)

As (13) shows the tokens in place11987526 are transferred intoplace11987529 after the first ignition and then transferred into11987532after the second ignition Finally the tokens are transferredinto the top place 11987538 after the third ignition meaning thatthe top event occurs in this situation

In the third situation 11987231

can be calculated througha single transfer calculation when the default value of thetokens in place 1198758 is 1

119872

3

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le7

minus1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

9le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(14)

Thus when there are two transfer paths for one token inthe bottom basic place there will be negative values in thenext level state matrix after ignition if the default value of thetokens in the bottom basic place is still 1There is no practicalsignificance to this phenomenon To avoid negative valueschange the default value of the tokens in place 1198758 from 1 to 2to produce1198723

1

1015840

119872

3

1

1015840

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(15)

Then11987232

can be solved according to11987231

1015840

119872

3

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

2

]

]

]

(16)

According to11987232

there will be two tokens in the top place11987528 and the top event will inevitably happen when tokensindicate the fault transferThus the fault event in the top placewill happen if there is at least one token in the top place In thePetri net if an input place is a bottom basic place containingtokens with a value of 119899 (119899 is a positive integer greater than 1)directed arcs out then the default value of the tokens in theinput place is 119899The fault event presented indicated by the topplace would will inevitably happen when there are tokens inthe top place

Failure Mode 1 When two EMUs are attached together andat the same time the spacing of the two pantographs is lessthan 190m the pantograph will rise to an abnormal positionHowever if only one condition of the two is satisfied thepantograph will work normally

Failure Mode 2 Damage to MR ducts causes low pressurein the auxiliary air cylinder which means the pantographcannot rise successfully

FailureMode 3 Damage to the pantograph bellowsmay causethe pantograph to descend automatically or possibly fail torise Both of the two conditions will lead to a fault in thepantograph

The token transfer in the Petri net is consistent withthe logic analysis of the faults It can clearly and effectivelydescribe the dynamic processes of the system faults transferachieving fast and efficient fault diagnosis

5 Conclusions

In this study the working principles and failure modes ofpantographs on CRH EMU trains were analyzed systemati-cally Petri net modelling was used as a graphical modellingtool to simplify the logical relationships and events of thefault tree into a network with places and transactions asnodes The events logical gates and logical relation lines ofFT were transformed to places transitions and directed arcsrespectively The complexity of modelling was reduced by20by avoiding repeating eventsThus the process of solvingthe minimal cut sets was simplified which effectively savedcalculation time for theminimal cut set solutions for complexlarge-scale fault trees

The changes in system states and the evolution of failuresare described well by the dynamic properties of Petri netmodelling A mathematical model for the Petri net of thepantograph fault tree was established The equivalence andcorrectness of the token-transfer description for fault diag-nosis inference in a Petri net were verified Three differentfault paths were used to explain and verify the algorithmThe initial identification sets of the three paths were marked

Mathematical Problems in Engineering 11

The three corresponding ignition sequences were calculatedusing matrix transformations Finally the system status wasassessed correctly using a mathematical method which canbe handled easily in a computerised system

This work analyzes the evolution of failure events ofpantograph system The process of other critical systemsneeds to be further examined Future work will collect andanalyse test data from other systems in CRH EMU trainsto extend the methodology to cover the whole maintenanceprocess for CRH EMU train systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 51475324) The authors wouldlike to thank the reviewers for their constructive commentswhich enhanced the quality of this paper

References

[1] F C Jiao and B J Wang The Management and Maintenanceof Locomotive Rolling Stock Beijing Jiaotong University PressBeijing China 2013

[2] L L Song T Y Wang X W Song L Xu and D Song ldquoFaultdiagnosis of pantograph type current collector of CRH electricmultiple units based on Petri net modelling and fault treeanalysisrdquo Chinese Journal of Scientific Instrument vol 35 no 9pp 1990ndash1997 2014

[3] I Aydin M Karakose and E Akin ldquoAnomaly detection usinga modified kernel-based tracking in the pantographmdashcatenarysystemrdquoExpert SystemswithApplications vol 42 no 2 pp 938ndash948 2015

[4] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2008

[5] D M Shalev and J Tiran ldquoCondition-based fault tree analysis(CBFTA) a newmethod for improved fault tree analysis (FTA)reliability and safety calculationsrdquo Reliability Engineering andSystem Safety vol 92 no 9 pp 1231ndash1241 2007

[6] Y E Senol Y V Aydogdu B Sahin and I Kilic ldquoFaulttree analysis of chemical cargo contamination by using fuzzyapproachrdquo Expert Systems with Applications vol 42 no 12 pp5232ndash5244 2015

[7] S M Lavasani A Zendegani and M Celik ldquoAn extension tofuzzy fault tree analysis (FFTA) application in petrochemicalprocess industryrdquo Process Safety and Environmental Protectionvol 93 pp 75ndash88 2014

[8] J H Purba ldquoA fuzzy-based reliability approach to evaluate basicevents of fault tree analysis for nuclear power plant probabilisticsafety assessmentrdquo Annals of Nuclear Energy vol 70 pp 21ndash292014

[9] N RamzaliM RM Lavasani and J Ghodousi ldquoSafety barriersanalysis of offshore drilling system by employing Fuzzy EventTree Analysisrdquo Safety Science vol 78 pp 49ndash59 2015

[10] K-C Hyun S Min H Choi J Park and I Lee ldquoRisk analysisusing fault-tree analysis (FTA) and analytic hierarchy process(AHP) applicable to shield TBM tunnelsrdquo Tunnelling andUnderground Space Technology vol 49 pp 121ndash129 2015

[11] S Verlinden G Deconinck and B Coupe ldquoHybrid reliabilitymodel for nuclear reactor safety systemrdquo Reliability Engineeringand System Safety vol 101 pp 35ndash47 2012

[12] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 13 no 3 pp386ndash397 1987

[13] A Bobbio G Franceschinis R Gaeta and L Portinale ldquoPara-metric fault tree for the dependability analysis of redundant sys-tems and its high-level Petri net semanticsrdquo IEEE Transactionson Software Engineering vol 29 no 3 pp 270ndash287 2003

[14] R Robidoux H Xu L Xing andM Zhou ldquoAutomated model-ing of dynamic reliability block diagrams using colored Petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 2 pp 337ndash351 2010

[15] DMakajic-Nikolic M Vujosevic andN Nikolic ldquoMinimal cutsets of a coherent fault tree generation using reverse Petri netsrdquoOptimization vol 62 no 8 pp 1069ndash1087 2013

[16] B Wang G Tian Y Liang and T Qiang ldquoReliability modelingand evaluation of electric vehicle motor by using fault tree andextended stochastic petri netsrdquo Journal of Applied Mathematicsvol 2014 Article ID 638013 9 pages 2014

[17] T P K Nguyen J Beugin and JMarais ldquoMethod for evaluatingan extended Fault Tree to analyse the dependability of complexsystems application to a satellite-based railway systemrdquo Relia-bility Engineering and System Safety vol 133 pp 300ndash313 2015

[18] L L Song T Y Wang X W Song et al ldquoFuzzy intelligentfault diagnosis of pantograph type current collector under themulti-source heterogeneous knowledge environmentrdquo ChineseJournal of Scientific Instrument vol 36 pp 1283ndash1290 2015

[19] M An Y Chen and C J Baker ldquoA fuzzy reasoning and fuzzy-analytical hierarchy process based approach to the process ofrailway risk information a railway risk management systemrdquoInformation Sciences vol 181 no 18 pp 3946ndash3966 2011

[20] M An S Huang and C J Baker ldquoRailway risk assessmentmdashthe fuzzy reasoning approach and fuzzy analytic hierarchy pro-cess approaches a case study of shunting at Waterloo depotrdquoProceedings of the Institution of Mechanical Engineers F Journalof Rail and Rapid Transit vol 221 no 3 pp 365ndash383 2007

[21] K Noori and K Jenab ldquoFuzzy reliability-based traction controlmodel for intelligent transportation systemsrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 43 no 1 pp229ndash234 2013

[22] R I Muttram ldquoRailway safetyrsquos safety risk modelrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 216 no 2 pp 71ndash79 2002

[23] L Bouillaut O Francois and S Dubois ldquoA Bayesian network toevaluate underground rails maintenance strategies in an auto-mation contextrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 227 no4 pp 411ndash424 2013

[24] Y Xie Y Zhang and F Liang ldquoCrash injury severity analysisusing Bayesian ordered probit modelsrdquo Journal of Transporta-tion Engineering vol 135 no 1 pp 18ndash25 2009

[25] S Bernardi F Flammini S Marrone et al ldquoEnabling the usageof UML in the verification of railway systems the DAM-railapproachrdquoReliability Engineering and SystemSafety vol 120 pp112ndash126 2013

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

T

M4 M5 M6M3 M7M2 M8M1

X2X1

X13X5

X14 X15

X20X16

X25X16X4X3

X6 X7 X8 X9 X10 X11 X12M9

X26 X27 X28 X29

X17 X18 X19

X9 X10 X11 X12 X13 X21 X22 X23 X24

X17

Figure 2 FTA modelling of the pantograph-type current collector

behavior It cannot show the transmission and evolution offaults in the system It also needs too much computation tosolve the minimal cut sets of the large-scale fault tree whichmay lead to combinatorial explosion because of Nondeter-ministic Polynomial (NP) problems For example if (119894 =1 2 119899) is set to the 119894th minimal cut set of the fault treethen the top event can be expressed as

119879 =

119899

⋃

119894=1

119862

119894 (1)

The occurrence probability of the top event can beexpressed as

119875 (119879) = 119875(

119899

⋃

119894=1

119862

119894) (2)

The logical relationship between the basic events and theminimal cut set is and If the probabilities of the basic eventsare known the occurrence probability of the minimal cut setcan be expressed as

119875 (119862

119894) = 119875(

119896

⋃

119895=1

119909

119894) (3)

Based on the occurrence probabilities of the minimalcut sets the occurrence probability of the top event can beexpressed as

119875 (119879) =

119899

sum

119894=1

119875 (119862

119894) minus

119899

sum

119894lt119895=2

119875 (119862

119894119862

119895) + sdot sdot sdot

+ (minus1)

119898minus1

119875(

119899

⋃

119894=1

119862

119894)

(4)

According to the equation above the process of deter-mining the probability of the top event consists of 2119899minus1 partsThere are 28 minimal cut sets for the pantograph fault treein Figure 2 so it needs 227 parts to calculate the probabilityof the top event which means 134217728 items in all Thecombinatorial explosion of the calculation will make quanti-tative analysis difficult In addition the FTA method is gen-erally for static analysis and cannot reflect dynamic changesin multiple states in the system

The dynamic and structural properties of Petri nets areused to simplify and analytically calculate the pantographfault tree Relational matrix analysis is used to solve theminimal cut set equation for the fault tree Based on the estab-lished state equation of Petri nets initial token and enable-transfer algorithms are used to express the transfer pro-cess of faults mathematically

Mathematical Problems in Engineering 7

Table 2 Petri net expression of the FTA logic gates

AND logic gate OR logic gateFault tree expression Petri net expression Fault tree expression Petri net expression

3 Petri Nets Modelling and RelationalMatrix Analysis

31 Petri Nets Modelling of the Fault Tree The Petri net wasproposed in 1962 by Petri to express an information flowmodel for reticular structures [30] It is a mathematical andgraphical analysis tool to express the static structures anddynamic changes in a system [31ndash33]

In a Petri net the system state is indicated by ldquoIrdquo Achange in state is indicated by ldquo|rdquo An ordered pair is indicatedby a directed arc ldquorarr rdquo and a token is indicated by ldquoerdquo ThePetri net expressions of the FTA logic gates are shown inTable 2 The transform between Petri net modelling and thefault tree is shown in Figure 3

One Petri net can be defined in a sextuple 119873 =

(119875 119879 119868 119874119872119872

0) The six elements should meet the follow-

ing conditions

(1) 119875 = 119875

1 119875

119899 is a finite set of places 119899 gt 0 is the

number of places

(2) 119879 = 1199051 119905

119898 is a finite set of transactions119898 gt 0 is

the number of transactions 119875 cap 119879 =

(3) 119868 119875 times 119879 rarr 119873 is the input function defining the setof repetitions or the weight of the directed arcs fromset 119875 to set 119879 119873 = 0 1 is a set of nonnegativeintegers

(4) 119874 119879times119875 rarr 119873 is the output function defining the setof repetitions or the weight of the directed arcs fromset 119879 to set 119875 119873 = 0 1 is a set of nonnegativeintegers

(5) 119872 119875 rarr 119873 is the set of the identificationdistribution of every place

(6) 1198720 119875 rarr 119873 is the set of the initial identification dis-

tribution of every place

As shown in Figures 2 and 3 the top event is replaced bythe top place in the Petri net All the probable events thatmay lead to the top place are represented as middle placesor basic places The logical gates of the fault tree are denotedby Transaction andDirected arc Repeating events in the faulttree no longer exist in Petri net modelling

Petri net modelling avoids repeating basic events andcan achieve a 20 decrease in the number of places Thesimplifying effect is more obvious in large-scale systems

32 Relational Matrix Analysis in Solving the Minimal CutSets Besides simplifying the fault tree a Petri net alsoeffectively overcomes the shortcomings of the traditional FTAin solving the minimal cut sets The process of Petri netmodelling can be undertaken more easily on a computer

The structure of the Petri net can be translated into amatrix representation The value of the input function fromplace119875 to transaction 119905 is a nonnegative integer120596 recorded as119868(119875 119905) = 120596 represented by a directed arc from 119875 to 119905 with theside noteThe value of the output function from transaction 119905to place 119875 is a nonnegative integer 120596 recorded as119874(119875 119905) = 120596represented by a directed arc from 119905 to 119875 with the side note120596 The side note is omitted when 120596 = 1 The directed arc isalso omitted when 119868(119875 119905) = 0 or 119874(119875 119905) = 0 119868 and 119874 canboth be represented as 119899 times 119898 nonnegative integer matrixesThe difference between119874 and 119868 is called the relationalmatrixrecorded as 119860 = 119874 minus 119868

The steps for solving the minimal cut sets using therelational matrix method are as follows

Step 1 Find out the row consisting only of ldquo1rdquo and ldquo0rdquo in therelational matrix A This will be assigned the top place withonly inputs and no outputs

Step 2 Search for ldquominus1rdquo by column from 1 in the top placeAny row corresponding to ldquominus1rdquowill be regarded as an input tothe top place If there are multiple ldquominus1rdquo values in the columnthen there will be multiple inputs for the same transactionThe logical relationship between the inputs is AND

Step 3 Search for ldquo1rdquo by row from ldquominus1rdquo determined in Step 2Rows including ldquo1rdquo will be regarded as occupying a middleplaceThen search repeatedly following the second step untila row without ldquo1rdquo is located which will become a basic placeIf there are multiple occurrences of ldquo1rdquo in the row the logicalrelationship between the places corresponding to ldquo1rdquo will beOR

Step 4 Continue to search following the second step and thethird step until all the bottom basic places are located

Step 5 Expand the bottom places according to the logicalrelationships AND and OR Then obtain all the cut sets forthe system

Step 6 Find the minimal cut sets according to the Booleanabsorption rate or the prime number method

8 Mathematical Problems in Engineering

TokensM1

M2

X1

X1

X2

X3

t1 t2

t3 t4

t5

t6

P1 P2

P3P4

P5

P6T

Figure 3 Transform between Petri net modelling and the fault tree

The fault tree in Figure 3 is used as an example toillustrate solving theminimal cut sets with a relationalmatrixRespectively solve the input matrix 119868 the output matrix 119874and the relational matrix 119860 according to the steps above

119868 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 1

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

119874 =

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0 0

0 0 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

119860 = 119874 minus 119868 =

[

[

[

[

[

[

[

[

[

[

minus1 0 0 0 0 minus1

0 minus1 0 0 0 0

1 1 minus1 0 0 0

0 0 0 minus1 0 0

0 0 1 1 minus1 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

(5)

33 Comparison between Petri Net Modelling and Fault TreeModelling The complexity of quantitative analysis dependson the number of nodes and the logical gates in a fault treeComparing Figures 3 and 4 to Figure 2 we can see that thereare two repeating basic events and seven repeating events intwo traditional fault trees The repetition rates can respec-tively reach 17 and 24These repeating events do not existin the corresponding Petri net modelling The building ofthe fault tree needs 6 types of elements which involve basicevents intermediate events top events AND logical gate

OR logical gate and a relation line while the building of thePetri net model needs only three types of elements involvingplaces transactions and directed arcs Though the fault treein Figure 2 seems to provide better visualization than thePetri nets in Figure 4 there are more elements and repeatingevents in the fault tree The relationships among fault eventsin the fault tree are more complex The reduction of buildingelements and repeating events make the Petri net modellingmore convenient and efficient than fault tree modelling

In addition Petri nets are graphical and mathematicaltools for modelling systems and their dynamics while faulttree modelling cannot be updated in a timely manner whenthe system changes The relationship between fault events isdisplayed statically in a fault tree When the failure status of asystem evolves from one subsystemcomponent to other sub-systemscomponents Petri net modelling is more intuitiveand more effective than fault tree modelling because of theuse of token transferring and enable-transferring processesThis process in Petri nets can be described by mathematicalmatrix computations which is easier to handle by computer(as shown in Section 4)

4 Dynamic Transition and Diagnosis ofPantograph Faults

41 PlaceMark and Enable-Transfer Algorithm of the Petri NetA Petri net consists of places ldquoIrdquo transactions ldquo|rdquo directedarcs ldquorarr rdquo and tokens of places ldquoerdquo The places representlogical descriptions of the system states and the transactionsrepresent the arising of system events Relational matrixesand state equations are major tools for Petri net analysis

119872

119896+1= 119872

119896+ 119860

119879

119883

119896 119896 = 1 2 (6)

In (6) 119872119896is the initial identification set of the system

faults before ignition 119872119896+1

is the result identification setof the system faults after ignition 119860119879 is the relationalmatrix and 119883

119870is the transfer sequence for ignition The

Mathematical Problems in Engineering 9

P1 P2 P3 P4

P5 P6 P7 P8 P9 P10 P11P12

P13

P14 P15 P16

P17 P18

P19

P20 P21 P22

P23

P24

P25 P26 P27 P28

P29

P30

P31

P32

P33P34

P35

P36 P37

P38

t1 t2 t3 t4

t5t6

t7

t8 t9 t10 t11 t12 t13 t14 t15

t16

t17

t18t19

t20

t21t22

t23 t24t25

t26t27

t28

t29t30 t31 t32

t33

t34t35

t36

t37

t38

t39

t40

t41

t42 t43

Initial identification 3

Initial identification 2

Initial identification 1

Figure 4 Transfer of tokens in the Petri net of the CRH EMU pantograph

initial identification1198720will convert to119872

119889after the ignition

sequence1198830 119883

119889 The process can be expressed as

119872

119889= 119872

0+ 119860

119879

119889minus1

sum

119896=0

119883

119896 (7)

If the element representing the top place is not less than1 in119872

119889after a series of transfers the fault event represented

by the top place will occur

42 Dynamic Transition and Diagnosis of Pantograph FaultsIn the process of fault diagnosis based on the use of a Petrinet the initial identification of the input place is regardedas the initial symptom of the event occurrence A token isassigned to the input place if the symptom appears otherwisethe place is assigned a null value The future states and finalidentification of the system can be circularly solved throughthe state equations and transfer sequences until the tokenvalue of the target place is found The fault occurs when thenumber of tokens in the target place is not zero

For the AND gate in the Petri net the tokens cannottransfer downward to the next level if there are empty inputplaces That is to say for an AND gate the fault will transferdownward only when all the events of the input places occurFor the OR gate in a Petri net the tokens will transferdownward to the next level place if at least one input placecontains tokens According to the rules of the network theorythe next level place will get two tokens if the two input placesboth contain tokens Regardless of the number of tokensthe occurrence of the fault that is represented by the targetplace is only related to whether there are tokens in the targetplace Transfer of the token in the Petri net of the CRH EMUpantograph is shown in Figure 4

As shown in Figure 4 1198751 represents ldquotwo EMUs attachedtogetherrdquo 1198752 represents ldquospacing of two pantographs of lessthan 190mrdquo 1198758 represents ldquopantograph bellows that aredamagedrdquo 11987526 represents ldquoMR ducts that are damagedrdquo11987529 represents ldquopressure in the auxiliary air cylinder that istoo lowrdquo 11987530 represents ldquopantograph rise to an abnormalpositionrdquo 11987532 represents ldquopantograph that cannot riserdquo11987535 represents ldquopantograph that descends automaticallyrdquoand 11987538 represents ldquopantograph that fails to workrdquo Threerepresentative fault transfer paths were selected to study thetransfer expression of pantograph faults in the Petri net Theinitial identification sets of the three paths were respectivelyrecorded as1198721

0

11987220

and11987230

119872

1

0

= [1 0 0

2le119894le38

]

119879

119872

2

0

= [0 0

1le119894le25

1 0 0

27le119894le38

]

119879

119872

3

0

= [0 0

1le119894le7

1 0 0

9le119894le38

]

119879

(8)

The ignition sequences of the three paths were respec-tively recorded as1198831

0

11988320

and11988330

119883

1

0

= [0 0

1le119894le4

1 0 0

6le119894le43

]

119879

119883

2

0

= [0 1 0 0

3le119894le43

]

119879

119883

3

0

= [0 0

1le119894le11

1 0 0

12le119894le21

1 0 0

23le119894le43

]

119879

(9)

10 Mathematical Problems in Engineering

The input matrix 119868 output matrix 119874 and the relationalmatrix119860119879 of the Petri net are all 38 times 43matrixes which canbe solved using the relational-matrix method

119860

119879

= 119874 minus 119868 = (

119886

11 119886

1119899

d

119886

1198981sdot sdot sdot 119886

119898119899

) (10)

119886

119894119895= 119874

119894119895minus 119868

1198941198951 le 119894 le 38 1 le 119895 le 43 (11)

In (10)119898 = 38 119899 = 43

119872

1

1

= 0

119872

1

2

= 0

(12)

As (12) shows tokens cannot transfer downward to thenext level in the first path Thus there is no token in the topplace at the end of the procedure meaning that the fault eventrepresented by 11987538 does not occur

In the second path11987221

11987222

and11987223

are as follows

119872

2

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le28

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

30le119894le38

]

]

]

119872

2

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

33le119894le38

]

]

]

119872

2

3

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

1

]

]

]

(13)

As (13) shows the tokens in place11987526 are transferred intoplace11987529 after the first ignition and then transferred into11987532after the second ignition Finally the tokens are transferredinto the top place 11987538 after the third ignition meaning thatthe top event occurs in this situation

In the third situation 11987231

can be calculated througha single transfer calculation when the default value of thetokens in place 1198758 is 1

119872

3

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le7

minus1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

9le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(14)

Thus when there are two transfer paths for one token inthe bottom basic place there will be negative values in thenext level state matrix after ignition if the default value of thetokens in the bottom basic place is still 1There is no practicalsignificance to this phenomenon To avoid negative valueschange the default value of the tokens in place 1198758 from 1 to 2to produce1198723

1

1015840

119872

3

1

1015840

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(15)

Then11987232

can be solved according to11987231

1015840

119872

3

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

2

]

]

]

(16)

According to11987232

there will be two tokens in the top place11987528 and the top event will inevitably happen when tokensindicate the fault transferThus the fault event in the top placewill happen if there is at least one token in the top place In thePetri net if an input place is a bottom basic place containingtokens with a value of 119899 (119899 is a positive integer greater than 1)directed arcs out then the default value of the tokens in theinput place is 119899The fault event presented indicated by the topplace would will inevitably happen when there are tokens inthe top place

Failure Mode 1 When two EMUs are attached together andat the same time the spacing of the two pantographs is lessthan 190m the pantograph will rise to an abnormal positionHowever if only one condition of the two is satisfied thepantograph will work normally

Failure Mode 2 Damage to MR ducts causes low pressurein the auxiliary air cylinder which means the pantographcannot rise successfully

FailureMode 3 Damage to the pantograph bellowsmay causethe pantograph to descend automatically or possibly fail torise Both of the two conditions will lead to a fault in thepantograph

The token transfer in the Petri net is consistent withthe logic analysis of the faults It can clearly and effectivelydescribe the dynamic processes of the system faults transferachieving fast and efficient fault diagnosis

5 Conclusions

In this study the working principles and failure modes ofpantographs on CRH EMU trains were analyzed systemati-cally Petri net modelling was used as a graphical modellingtool to simplify the logical relationships and events of thefault tree into a network with places and transactions asnodes The events logical gates and logical relation lines ofFT were transformed to places transitions and directed arcsrespectively The complexity of modelling was reduced by20by avoiding repeating eventsThus the process of solvingthe minimal cut sets was simplified which effectively savedcalculation time for theminimal cut set solutions for complexlarge-scale fault trees

The changes in system states and the evolution of failuresare described well by the dynamic properties of Petri netmodelling A mathematical model for the Petri net of thepantograph fault tree was established The equivalence andcorrectness of the token-transfer description for fault diag-nosis inference in a Petri net were verified Three differentfault paths were used to explain and verify the algorithmThe initial identification sets of the three paths were marked

Mathematical Problems in Engineering 11

The three corresponding ignition sequences were calculatedusing matrix transformations Finally the system status wasassessed correctly using a mathematical method which canbe handled easily in a computerised system

This work analyzes the evolution of failure events ofpantograph system The process of other critical systemsneeds to be further examined Future work will collect andanalyse test data from other systems in CRH EMU trainsto extend the methodology to cover the whole maintenanceprocess for CRH EMU train systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 51475324) The authors wouldlike to thank the reviewers for their constructive commentswhich enhanced the quality of this paper

References

[1] F C Jiao and B J Wang The Management and Maintenanceof Locomotive Rolling Stock Beijing Jiaotong University PressBeijing China 2013

[2] L L Song T Y Wang X W Song L Xu and D Song ldquoFaultdiagnosis of pantograph type current collector of CRH electricmultiple units based on Petri net modelling and fault treeanalysisrdquo Chinese Journal of Scientific Instrument vol 35 no 9pp 1990ndash1997 2014

[3] I Aydin M Karakose and E Akin ldquoAnomaly detection usinga modified kernel-based tracking in the pantographmdashcatenarysystemrdquoExpert SystemswithApplications vol 42 no 2 pp 938ndash948 2015

[4] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2008

[5] D M Shalev and J Tiran ldquoCondition-based fault tree analysis(CBFTA) a newmethod for improved fault tree analysis (FTA)reliability and safety calculationsrdquo Reliability Engineering andSystem Safety vol 92 no 9 pp 1231ndash1241 2007

[6] Y E Senol Y V Aydogdu B Sahin and I Kilic ldquoFaulttree analysis of chemical cargo contamination by using fuzzyapproachrdquo Expert Systems with Applications vol 42 no 12 pp5232ndash5244 2015

[7] S M Lavasani A Zendegani and M Celik ldquoAn extension tofuzzy fault tree analysis (FFTA) application in petrochemicalprocess industryrdquo Process Safety and Environmental Protectionvol 93 pp 75ndash88 2014

[8] J H Purba ldquoA fuzzy-based reliability approach to evaluate basicevents of fault tree analysis for nuclear power plant probabilisticsafety assessmentrdquo Annals of Nuclear Energy vol 70 pp 21ndash292014

[9] N RamzaliM RM Lavasani and J Ghodousi ldquoSafety barriersanalysis of offshore drilling system by employing Fuzzy EventTree Analysisrdquo Safety Science vol 78 pp 49ndash59 2015

[10] K-C Hyun S Min H Choi J Park and I Lee ldquoRisk analysisusing fault-tree analysis (FTA) and analytic hierarchy process(AHP) applicable to shield TBM tunnelsrdquo Tunnelling andUnderground Space Technology vol 49 pp 121ndash129 2015

[11] S Verlinden G Deconinck and B Coupe ldquoHybrid reliabilitymodel for nuclear reactor safety systemrdquo Reliability Engineeringand System Safety vol 101 pp 35ndash47 2012

[12] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 13 no 3 pp386ndash397 1987

[13] A Bobbio G Franceschinis R Gaeta and L Portinale ldquoPara-metric fault tree for the dependability analysis of redundant sys-tems and its high-level Petri net semanticsrdquo IEEE Transactionson Software Engineering vol 29 no 3 pp 270ndash287 2003

[14] R Robidoux H Xu L Xing andM Zhou ldquoAutomated model-ing of dynamic reliability block diagrams using colored Petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 2 pp 337ndash351 2010

[15] DMakajic-Nikolic M Vujosevic andN Nikolic ldquoMinimal cutsets of a coherent fault tree generation using reverse Petri netsrdquoOptimization vol 62 no 8 pp 1069ndash1087 2013

[16] B Wang G Tian Y Liang and T Qiang ldquoReliability modelingand evaluation of electric vehicle motor by using fault tree andextended stochastic petri netsrdquo Journal of Applied Mathematicsvol 2014 Article ID 638013 9 pages 2014

[17] T P K Nguyen J Beugin and JMarais ldquoMethod for evaluatingan extended Fault Tree to analyse the dependability of complexsystems application to a satellite-based railway systemrdquo Relia-bility Engineering and System Safety vol 133 pp 300ndash313 2015

[18] L L Song T Y Wang X W Song et al ldquoFuzzy intelligentfault diagnosis of pantograph type current collector under themulti-source heterogeneous knowledge environmentrdquo ChineseJournal of Scientific Instrument vol 36 pp 1283ndash1290 2015

[19] M An Y Chen and C J Baker ldquoA fuzzy reasoning and fuzzy-analytical hierarchy process based approach to the process ofrailway risk information a railway risk management systemrdquoInformation Sciences vol 181 no 18 pp 3946ndash3966 2011

[20] M An S Huang and C J Baker ldquoRailway risk assessmentmdashthe fuzzy reasoning approach and fuzzy analytic hierarchy pro-cess approaches a case study of shunting at Waterloo depotrdquoProceedings of the Institution of Mechanical Engineers F Journalof Rail and Rapid Transit vol 221 no 3 pp 365ndash383 2007

[21] K Noori and K Jenab ldquoFuzzy reliability-based traction controlmodel for intelligent transportation systemsrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 43 no 1 pp229ndash234 2013

[22] R I Muttram ldquoRailway safetyrsquos safety risk modelrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 216 no 2 pp 71ndash79 2002

[23] L Bouillaut O Francois and S Dubois ldquoA Bayesian network toevaluate underground rails maintenance strategies in an auto-mation contextrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 227 no4 pp 411ndash424 2013

[24] Y Xie Y Zhang and F Liang ldquoCrash injury severity analysisusing Bayesian ordered probit modelsrdquo Journal of Transporta-tion Engineering vol 135 no 1 pp 18ndash25 2009

[25] S Bernardi F Flammini S Marrone et al ldquoEnabling the usageof UML in the verification of railway systems the DAM-railapproachrdquoReliability Engineering and SystemSafety vol 120 pp112ndash126 2013

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 2 Petri net expression of the FTA logic gates

AND logic gate OR logic gateFault tree expression Petri net expression Fault tree expression Petri net expression

3 Petri Nets Modelling and RelationalMatrix Analysis

31 Petri Nets Modelling of the Fault Tree The Petri net wasproposed in 1962 by Petri to express an information flowmodel for reticular structures [30] It is a mathematical andgraphical analysis tool to express the static structures anddynamic changes in a system [31ndash33]

In a Petri net the system state is indicated by ldquoIrdquo Achange in state is indicated by ldquo|rdquo An ordered pair is indicatedby a directed arc ldquorarr rdquo and a token is indicated by ldquoerdquo ThePetri net expressions of the FTA logic gates are shown inTable 2 The transform between Petri net modelling and thefault tree is shown in Figure 3

One Petri net can be defined in a sextuple 119873 =

(119875 119879 119868 119874119872119872

0) The six elements should meet the follow-

ing conditions

(1) 119875 = 119875

1 119875

119899 is a finite set of places 119899 gt 0 is the

number of places

(2) 119879 = 1199051 119905

119898 is a finite set of transactions119898 gt 0 is

the number of transactions 119875 cap 119879 =

(3) 119868 119875 times 119879 rarr 119873 is the input function defining the setof repetitions or the weight of the directed arcs fromset 119875 to set 119879 119873 = 0 1 is a set of nonnegativeintegers

(4) 119874 119879times119875 rarr 119873 is the output function defining the setof repetitions or the weight of the directed arcs fromset 119879 to set 119875 119873 = 0 1 is a set of nonnegativeintegers

(5) 119872 119875 rarr 119873 is the set of the identificationdistribution of every place

(6) 1198720 119875 rarr 119873 is the set of the initial identification dis-

tribution of every place

As shown in Figures 2 and 3 the top event is replaced bythe top place in the Petri net All the probable events thatmay lead to the top place are represented as middle placesor basic places The logical gates of the fault tree are denotedby Transaction andDirected arc Repeating events in the faulttree no longer exist in Petri net modelling

Petri net modelling avoids repeating basic events andcan achieve a 20 decrease in the number of places Thesimplifying effect is more obvious in large-scale systems

32 Relational Matrix Analysis in Solving the Minimal CutSets Besides simplifying the fault tree a Petri net alsoeffectively overcomes the shortcomings of the traditional FTAin solving the minimal cut sets The process of Petri netmodelling can be undertaken more easily on a computer

The structure of the Petri net can be translated into amatrix representation The value of the input function fromplace119875 to transaction 119905 is a nonnegative integer120596 recorded as119868(119875 119905) = 120596 represented by a directed arc from 119875 to 119905 with theside noteThe value of the output function from transaction 119905to place 119875 is a nonnegative integer 120596 recorded as119874(119875 119905) = 120596represented by a directed arc from 119905 to 119875 with the side note120596 The side note is omitted when 120596 = 1 The directed arc isalso omitted when 119868(119875 119905) = 0 or 119874(119875 119905) = 0 119868 and 119874 canboth be represented as 119899 times 119898 nonnegative integer matrixesThe difference between119874 and 119868 is called the relationalmatrixrecorded as 119860 = 119874 minus 119868

The steps for solving the minimal cut sets using therelational matrix method are as follows

Step 1 Find out the row consisting only of ldquo1rdquo and ldquo0rdquo in therelational matrix A This will be assigned the top place withonly inputs and no outputs

Step 2 Search for ldquominus1rdquo by column from 1 in the top placeAny row corresponding to ldquominus1rdquowill be regarded as an input tothe top place If there are multiple ldquominus1rdquo values in the columnthen there will be multiple inputs for the same transactionThe logical relationship between the inputs is AND

Step 3 Search for ldquo1rdquo by row from ldquominus1rdquo determined in Step 2Rows including ldquo1rdquo will be regarded as occupying a middleplaceThen search repeatedly following the second step untila row without ldquo1rdquo is located which will become a basic placeIf there are multiple occurrences of ldquo1rdquo in the row the logicalrelationship between the places corresponding to ldquo1rdquo will beOR

Step 4 Continue to search following the second step and thethird step until all the bottom basic places are located

Step 5 Expand the bottom places according to the logicalrelationships AND and OR Then obtain all the cut sets forthe system

Step 6 Find the minimal cut sets according to the Booleanabsorption rate or the prime number method

8 Mathematical Problems in Engineering

TokensM1

M2

X1

X1

X2

X3

t1 t2

t3 t4

t5

t6

P1 P2

P3P4

P5

P6T

Figure 3 Transform between Petri net modelling and the fault tree

The fault tree in Figure 3 is used as an example toillustrate solving theminimal cut sets with a relationalmatrixRespectively solve the input matrix 119868 the output matrix 119874and the relational matrix 119860 according to the steps above

119868 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 1

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

119874 =

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0 0

0 0 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

119860 = 119874 minus 119868 =

[

[

[

[

[

[

[

[

[

[

minus1 0 0 0 0 minus1

0 minus1 0 0 0 0

1 1 minus1 0 0 0

0 0 0 minus1 0 0

0 0 1 1 minus1 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

(5)

33 Comparison between Petri Net Modelling and Fault TreeModelling The complexity of quantitative analysis dependson the number of nodes and the logical gates in a fault treeComparing Figures 3 and 4 to Figure 2 we can see that thereare two repeating basic events and seven repeating events intwo traditional fault trees The repetition rates can respec-tively reach 17 and 24These repeating events do not existin the corresponding Petri net modelling The building ofthe fault tree needs 6 types of elements which involve basicevents intermediate events top events AND logical gate

OR logical gate and a relation line while the building of thePetri net model needs only three types of elements involvingplaces transactions and directed arcs Though the fault treein Figure 2 seems to provide better visualization than thePetri nets in Figure 4 there are more elements and repeatingevents in the fault tree The relationships among fault eventsin the fault tree are more complex The reduction of buildingelements and repeating events make the Petri net modellingmore convenient and efficient than fault tree modelling

In addition Petri nets are graphical and mathematicaltools for modelling systems and their dynamics while faulttree modelling cannot be updated in a timely manner whenthe system changes The relationship between fault events isdisplayed statically in a fault tree When the failure status of asystem evolves from one subsystemcomponent to other sub-systemscomponents Petri net modelling is more intuitiveand more effective than fault tree modelling because of theuse of token transferring and enable-transferring processesThis process in Petri nets can be described by mathematicalmatrix computations which is easier to handle by computer(as shown in Section 4)

4 Dynamic Transition and Diagnosis ofPantograph Faults

41 PlaceMark and Enable-Transfer Algorithm of the Petri NetA Petri net consists of places ldquoIrdquo transactions ldquo|rdquo directedarcs ldquorarr rdquo and tokens of places ldquoerdquo The places representlogical descriptions of the system states and the transactionsrepresent the arising of system events Relational matrixesand state equations are major tools for Petri net analysis

119872

119896+1= 119872

119896+ 119860

119879

119883

119896 119896 = 1 2 (6)

In (6) 119872119896is the initial identification set of the system

faults before ignition 119872119896+1

is the result identification setof the system faults after ignition 119860119879 is the relationalmatrix and 119883

119870is the transfer sequence for ignition The

Mathematical Problems in Engineering 9

P1 P2 P3 P4

P5 P6 P7 P8 P9 P10 P11P12

P13

P14 P15 P16

P17 P18

P19

P20 P21 P22

P23

P24

P25 P26 P27 P28

P29

P30

P31

P32

P33P34

P35

P36 P37

P38

t1 t2 t3 t4

t5t6

t7

t8 t9 t10 t11 t12 t13 t14 t15

t16

t17

t18t19

t20

t21t22

t23 t24t25

t26t27

t28

t29t30 t31 t32

t33

t34t35

t36

t37

t38

t39

t40

t41

t42 t43

Initial identification 3

Initial identification 2

Initial identification 1

Figure 4 Transfer of tokens in the Petri net of the CRH EMU pantograph

initial identification1198720will convert to119872

119889after the ignition

sequence1198830 119883

119889 The process can be expressed as

119872

119889= 119872

0+ 119860

119879

119889minus1

sum

119896=0

119883

119896 (7)

If the element representing the top place is not less than1 in119872

119889after a series of transfers the fault event represented

by the top place will occur

42 Dynamic Transition and Diagnosis of Pantograph FaultsIn the process of fault diagnosis based on the use of a Petrinet the initial identification of the input place is regardedas the initial symptom of the event occurrence A token isassigned to the input place if the symptom appears otherwisethe place is assigned a null value The future states and finalidentification of the system can be circularly solved throughthe state equations and transfer sequences until the tokenvalue of the target place is found The fault occurs when thenumber of tokens in the target place is not zero

For the AND gate in the Petri net the tokens cannottransfer downward to the next level if there are empty inputplaces That is to say for an AND gate the fault will transferdownward only when all the events of the input places occurFor the OR gate in a Petri net the tokens will transferdownward to the next level place if at least one input placecontains tokens According to the rules of the network theorythe next level place will get two tokens if the two input placesboth contain tokens Regardless of the number of tokensthe occurrence of the fault that is represented by the targetplace is only related to whether there are tokens in the targetplace Transfer of the token in the Petri net of the CRH EMUpantograph is shown in Figure 4

As shown in Figure 4 1198751 represents ldquotwo EMUs attachedtogetherrdquo 1198752 represents ldquospacing of two pantographs of lessthan 190mrdquo 1198758 represents ldquopantograph bellows that aredamagedrdquo 11987526 represents ldquoMR ducts that are damagedrdquo11987529 represents ldquopressure in the auxiliary air cylinder that istoo lowrdquo 11987530 represents ldquopantograph rise to an abnormalpositionrdquo 11987532 represents ldquopantograph that cannot riserdquo11987535 represents ldquopantograph that descends automaticallyrdquoand 11987538 represents ldquopantograph that fails to workrdquo Threerepresentative fault transfer paths were selected to study thetransfer expression of pantograph faults in the Petri net Theinitial identification sets of the three paths were respectivelyrecorded as1198721

0

11987220

and11987230

119872

1

0

= [1 0 0

2le119894le38

]

119879

119872

2

0

= [0 0

1le119894le25

1 0 0

27le119894le38

]

119879

119872

3

0

= [0 0

1le119894le7

1 0 0

9le119894le38

]

119879

(8)

The ignition sequences of the three paths were respec-tively recorded as1198831

0

11988320

and11988330

119883

1

0

= [0 0

1le119894le4

1 0 0

6le119894le43

]

119879

119883

2

0

= [0 1 0 0

3le119894le43

]

119879

119883

3

0

= [0 0

1le119894le11

1 0 0

12le119894le21

1 0 0

23le119894le43

]

119879

(9)

10 Mathematical Problems in Engineering

The input matrix 119868 output matrix 119874 and the relationalmatrix119860119879 of the Petri net are all 38 times 43matrixes which canbe solved using the relational-matrix method

119860

119879

= 119874 minus 119868 = (

119886

11 119886

1119899

d

119886

1198981sdot sdot sdot 119886

119898119899

) (10)

119886

119894119895= 119874

119894119895minus 119868

1198941198951 le 119894 le 38 1 le 119895 le 43 (11)

In (10)119898 = 38 119899 = 43

119872

1

1

= 0

119872

1

2

= 0

(12)

As (12) shows tokens cannot transfer downward to thenext level in the first path Thus there is no token in the topplace at the end of the procedure meaning that the fault eventrepresented by 11987538 does not occur

In the second path11987221

11987222

and11987223

are as follows

119872

2

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le28

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

30le119894le38

]

]

]

119872

2

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

33le119894le38

]

]

]

119872

2

3

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

1

]

]

]

(13)

As (13) shows the tokens in place11987526 are transferred intoplace11987529 after the first ignition and then transferred into11987532after the second ignition Finally the tokens are transferredinto the top place 11987538 after the third ignition meaning thatthe top event occurs in this situation

In the third situation 11987231

can be calculated througha single transfer calculation when the default value of thetokens in place 1198758 is 1

119872

3

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le7

minus1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

9le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(14)

Thus when there are two transfer paths for one token inthe bottom basic place there will be negative values in thenext level state matrix after ignition if the default value of thetokens in the bottom basic place is still 1There is no practicalsignificance to this phenomenon To avoid negative valueschange the default value of the tokens in place 1198758 from 1 to 2to produce1198723

1

1015840

119872

3

1

1015840

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(15)

Then11987232

can be solved according to11987231

1015840

119872

3

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

2

]

]

]

(16)

According to11987232

there will be two tokens in the top place11987528 and the top event will inevitably happen when tokensindicate the fault transferThus the fault event in the top placewill happen if there is at least one token in the top place In thePetri net if an input place is a bottom basic place containingtokens with a value of 119899 (119899 is a positive integer greater than 1)directed arcs out then the default value of the tokens in theinput place is 119899The fault event presented indicated by the topplace would will inevitably happen when there are tokens inthe top place

Failure Mode 1 When two EMUs are attached together andat the same time the spacing of the two pantographs is lessthan 190m the pantograph will rise to an abnormal positionHowever if only one condition of the two is satisfied thepantograph will work normally

Failure Mode 2 Damage to MR ducts causes low pressurein the auxiliary air cylinder which means the pantographcannot rise successfully

FailureMode 3 Damage to the pantograph bellowsmay causethe pantograph to descend automatically or possibly fail torise Both of the two conditions will lead to a fault in thepantograph

The token transfer in the Petri net is consistent withthe logic analysis of the faults It can clearly and effectivelydescribe the dynamic processes of the system faults transferachieving fast and efficient fault diagnosis

5 Conclusions

In this study the working principles and failure modes ofpantographs on CRH EMU trains were analyzed systemati-cally Petri net modelling was used as a graphical modellingtool to simplify the logical relationships and events of thefault tree into a network with places and transactions asnodes The events logical gates and logical relation lines ofFT were transformed to places transitions and directed arcsrespectively The complexity of modelling was reduced by20by avoiding repeating eventsThus the process of solvingthe minimal cut sets was simplified which effectively savedcalculation time for theminimal cut set solutions for complexlarge-scale fault trees

The changes in system states and the evolution of failuresare described well by the dynamic properties of Petri netmodelling A mathematical model for the Petri net of thepantograph fault tree was established The equivalence andcorrectness of the token-transfer description for fault diag-nosis inference in a Petri net were verified Three differentfault paths were used to explain and verify the algorithmThe initial identification sets of the three paths were marked

Mathematical Problems in Engineering 11

The three corresponding ignition sequences were calculatedusing matrix transformations Finally the system status wasassessed correctly using a mathematical method which canbe handled easily in a computerised system

This work analyzes the evolution of failure events ofpantograph system The process of other critical systemsneeds to be further examined Future work will collect andanalyse test data from other systems in CRH EMU trainsto extend the methodology to cover the whole maintenanceprocess for CRH EMU train systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 51475324) The authors wouldlike to thank the reviewers for their constructive commentswhich enhanced the quality of this paper

References

[1] F C Jiao and B J Wang The Management and Maintenanceof Locomotive Rolling Stock Beijing Jiaotong University PressBeijing China 2013

[2] L L Song T Y Wang X W Song L Xu and D Song ldquoFaultdiagnosis of pantograph type current collector of CRH electricmultiple units based on Petri net modelling and fault treeanalysisrdquo Chinese Journal of Scientific Instrument vol 35 no 9pp 1990ndash1997 2014

[3] I Aydin M Karakose and E Akin ldquoAnomaly detection usinga modified kernel-based tracking in the pantographmdashcatenarysystemrdquoExpert SystemswithApplications vol 42 no 2 pp 938ndash948 2015

[4] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2008

[5] D M Shalev and J Tiran ldquoCondition-based fault tree analysis(CBFTA) a newmethod for improved fault tree analysis (FTA)reliability and safety calculationsrdquo Reliability Engineering andSystem Safety vol 92 no 9 pp 1231ndash1241 2007

[6] Y E Senol Y V Aydogdu B Sahin and I Kilic ldquoFaulttree analysis of chemical cargo contamination by using fuzzyapproachrdquo Expert Systems with Applications vol 42 no 12 pp5232ndash5244 2015

[7] S M Lavasani A Zendegani and M Celik ldquoAn extension tofuzzy fault tree analysis (FFTA) application in petrochemicalprocess industryrdquo Process Safety and Environmental Protectionvol 93 pp 75ndash88 2014

[8] J H Purba ldquoA fuzzy-based reliability approach to evaluate basicevents of fault tree analysis for nuclear power plant probabilisticsafety assessmentrdquo Annals of Nuclear Energy vol 70 pp 21ndash292014

[9] N RamzaliM RM Lavasani and J Ghodousi ldquoSafety barriersanalysis of offshore drilling system by employing Fuzzy EventTree Analysisrdquo Safety Science vol 78 pp 49ndash59 2015

[10] K-C Hyun S Min H Choi J Park and I Lee ldquoRisk analysisusing fault-tree analysis (FTA) and analytic hierarchy process(AHP) applicable to shield TBM tunnelsrdquo Tunnelling andUnderground Space Technology vol 49 pp 121ndash129 2015

[11] S Verlinden G Deconinck and B Coupe ldquoHybrid reliabilitymodel for nuclear reactor safety systemrdquo Reliability Engineeringand System Safety vol 101 pp 35ndash47 2012

[12] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 13 no 3 pp386ndash397 1987

[13] A Bobbio G Franceschinis R Gaeta and L Portinale ldquoPara-metric fault tree for the dependability analysis of redundant sys-tems and its high-level Petri net semanticsrdquo IEEE Transactionson Software Engineering vol 29 no 3 pp 270ndash287 2003

[14] R Robidoux H Xu L Xing andM Zhou ldquoAutomated model-ing of dynamic reliability block diagrams using colored Petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 2 pp 337ndash351 2010

[15] DMakajic-Nikolic M Vujosevic andN Nikolic ldquoMinimal cutsets of a coherent fault tree generation using reverse Petri netsrdquoOptimization vol 62 no 8 pp 1069ndash1087 2013

[16] B Wang G Tian Y Liang and T Qiang ldquoReliability modelingand evaluation of electric vehicle motor by using fault tree andextended stochastic petri netsrdquo Journal of Applied Mathematicsvol 2014 Article ID 638013 9 pages 2014

[17] T P K Nguyen J Beugin and JMarais ldquoMethod for evaluatingan extended Fault Tree to analyse the dependability of complexsystems application to a satellite-based railway systemrdquo Relia-bility Engineering and System Safety vol 133 pp 300ndash313 2015

[18] L L Song T Y Wang X W Song et al ldquoFuzzy intelligentfault diagnosis of pantograph type current collector under themulti-source heterogeneous knowledge environmentrdquo ChineseJournal of Scientific Instrument vol 36 pp 1283ndash1290 2015

[19] M An Y Chen and C J Baker ldquoA fuzzy reasoning and fuzzy-analytical hierarchy process based approach to the process ofrailway risk information a railway risk management systemrdquoInformation Sciences vol 181 no 18 pp 3946ndash3966 2011

[20] M An S Huang and C J Baker ldquoRailway risk assessmentmdashthe fuzzy reasoning approach and fuzzy analytic hierarchy pro-cess approaches a case study of shunting at Waterloo depotrdquoProceedings of the Institution of Mechanical Engineers F Journalof Rail and Rapid Transit vol 221 no 3 pp 365ndash383 2007

[21] K Noori and K Jenab ldquoFuzzy reliability-based traction controlmodel for intelligent transportation systemsrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 43 no 1 pp229ndash234 2013

[22] R I Muttram ldquoRailway safetyrsquos safety risk modelrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 216 no 2 pp 71ndash79 2002

[23] L Bouillaut O Francois and S Dubois ldquoA Bayesian network toevaluate underground rails maintenance strategies in an auto-mation contextrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 227 no4 pp 411ndash424 2013

[24] Y Xie Y Zhang and F Liang ldquoCrash injury severity analysisusing Bayesian ordered probit modelsrdquo Journal of Transporta-tion Engineering vol 135 no 1 pp 18ndash25 2009

[25] S Bernardi F Flammini S Marrone et al ldquoEnabling the usageof UML in the verification of railway systems the DAM-railapproachrdquoReliability Engineering and SystemSafety vol 120 pp112ndash126 2013

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

TokensM1

M2

X1

X1

X2

X3

t1 t2

t3 t4

t5

t6

P1 P2

P3P4

P5

P6T

Figure 3 Transform between Petri net modelling and the fault tree

The fault tree in Figure 3 is used as an example toillustrate solving theminimal cut sets with a relationalmatrixRespectively solve the input matrix 119868 the output matrix 119874and the relational matrix 119860 according to the steps above

119868 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 1

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

119874 =

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0 0

0 0 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

119860 = 119874 minus 119868 =

[

[

[

[

[

[

[

[

[

[

minus1 0 0 0 0 minus1

0 minus1 0 0 0 0

1 1 minus1 0 0 0

0 0 0 minus1 0 0

0 0 1 1 minus1 0

0 0 0 0 1 1

]

]

]

]

]

]

]

]

]

]

(5)

33 Comparison between Petri Net Modelling and Fault TreeModelling The complexity of quantitative analysis dependson the number of nodes and the logical gates in a fault treeComparing Figures 3 and 4 to Figure 2 we can see that thereare two repeating basic events and seven repeating events intwo traditional fault trees The repetition rates can respec-tively reach 17 and 24These repeating events do not existin the corresponding Petri net modelling The building ofthe fault tree needs 6 types of elements which involve basicevents intermediate events top events AND logical gate

OR logical gate and a relation line while the building of thePetri net model needs only three types of elements involvingplaces transactions and directed arcs Though the fault treein Figure 2 seems to provide better visualization than thePetri nets in Figure 4 there are more elements and repeatingevents in the fault tree The relationships among fault eventsin the fault tree are more complex The reduction of buildingelements and repeating events make the Petri net modellingmore convenient and efficient than fault tree modelling

In addition Petri nets are graphical and mathematicaltools for modelling systems and their dynamics while faulttree modelling cannot be updated in a timely manner whenthe system changes The relationship between fault events isdisplayed statically in a fault tree When the failure status of asystem evolves from one subsystemcomponent to other sub-systemscomponents Petri net modelling is more intuitiveand more effective than fault tree modelling because of theuse of token transferring and enable-transferring processesThis process in Petri nets can be described by mathematicalmatrix computations which is easier to handle by computer(as shown in Section 4)

4 Dynamic Transition and Diagnosis ofPantograph Faults

41 PlaceMark and Enable-Transfer Algorithm of the Petri NetA Petri net consists of places ldquoIrdquo transactions ldquo|rdquo directedarcs ldquorarr rdquo and tokens of places ldquoerdquo The places representlogical descriptions of the system states and the transactionsrepresent the arising of system events Relational matrixesand state equations are major tools for Petri net analysis

119872

119896+1= 119872

119896+ 119860

119879

119883

119896 119896 = 1 2 (6)

In (6) 119872119896is the initial identification set of the system

faults before ignition 119872119896+1

is the result identification setof the system faults after ignition 119860119879 is the relationalmatrix and 119883

119870is the transfer sequence for ignition The

Mathematical Problems in Engineering 9

P1 P2 P3 P4

P5 P6 P7 P8 P9 P10 P11P12

P13

P14 P15 P16

P17 P18

P19

P20 P21 P22

P23

P24

P25 P26 P27 P28

P29

P30

P31

P32

P33P34

P35

P36 P37

P38

t1 t2 t3 t4

t5t6

t7

t8 t9 t10 t11 t12 t13 t14 t15

t16

t17

t18t19

t20

t21t22

t23 t24t25

t26t27

t28

t29t30 t31 t32

t33

t34t35

t36

t37

t38

t39

t40

t41

t42 t43

Initial identification 3

Initial identification 2

Initial identification 1

Figure 4 Transfer of tokens in the Petri net of the CRH EMU pantograph

initial identification1198720will convert to119872

119889after the ignition

sequence1198830 119883

119889 The process can be expressed as

119872

119889= 119872

0+ 119860

119879

119889minus1

sum

119896=0

119883

119896 (7)

If the element representing the top place is not less than1 in119872

119889after a series of transfers the fault event represented

by the top place will occur

42 Dynamic Transition and Diagnosis of Pantograph FaultsIn the process of fault diagnosis based on the use of a Petrinet the initial identification of the input place is regardedas the initial symptom of the event occurrence A token isassigned to the input place if the symptom appears otherwisethe place is assigned a null value The future states and finalidentification of the system can be circularly solved throughthe state equations and transfer sequences until the tokenvalue of the target place is found The fault occurs when thenumber of tokens in the target place is not zero

For the AND gate in the Petri net the tokens cannottransfer downward to the next level if there are empty inputplaces That is to say for an AND gate the fault will transferdownward only when all the events of the input places occurFor the OR gate in a Petri net the tokens will transferdownward to the next level place if at least one input placecontains tokens According to the rules of the network theorythe next level place will get two tokens if the two input placesboth contain tokens Regardless of the number of tokensthe occurrence of the fault that is represented by the targetplace is only related to whether there are tokens in the targetplace Transfer of the token in the Petri net of the CRH EMUpantograph is shown in Figure 4

As shown in Figure 4 1198751 represents ldquotwo EMUs attachedtogetherrdquo 1198752 represents ldquospacing of two pantographs of lessthan 190mrdquo 1198758 represents ldquopantograph bellows that aredamagedrdquo 11987526 represents ldquoMR ducts that are damagedrdquo11987529 represents ldquopressure in the auxiliary air cylinder that istoo lowrdquo 11987530 represents ldquopantograph rise to an abnormalpositionrdquo 11987532 represents ldquopantograph that cannot riserdquo11987535 represents ldquopantograph that descends automaticallyrdquoand 11987538 represents ldquopantograph that fails to workrdquo Threerepresentative fault transfer paths were selected to study thetransfer expression of pantograph faults in the Petri net Theinitial identification sets of the three paths were respectivelyrecorded as1198721

0

11987220

and11987230

119872

1

0

= [1 0 0

2le119894le38

]

119879

119872

2

0

= [0 0

1le119894le25

1 0 0

27le119894le38

]

119879

119872

3

0

= [0 0

1le119894le7

1 0 0

9le119894le38

]

119879

(8)

The ignition sequences of the three paths were respec-tively recorded as1198831

0

11988320

and11988330

119883

1

0

= [0 0

1le119894le4

1 0 0

6le119894le43

]

119879

119883

2

0

= [0 1 0 0

3le119894le43

]

119879

119883

3

0

= [0 0

1le119894le11

1 0 0

12le119894le21

1 0 0

23le119894le43

]

119879

(9)

10 Mathematical Problems in Engineering

The input matrix 119868 output matrix 119874 and the relationalmatrix119860119879 of the Petri net are all 38 times 43matrixes which canbe solved using the relational-matrix method

119860

119879

= 119874 minus 119868 = (

119886

11 119886

1119899

d

119886

1198981sdot sdot sdot 119886

119898119899

) (10)

119886

119894119895= 119874

119894119895minus 119868

1198941198951 le 119894 le 38 1 le 119895 le 43 (11)

In (10)119898 = 38 119899 = 43

119872

1

1

= 0

119872

1

2

= 0

(12)

As (12) shows tokens cannot transfer downward to thenext level in the first path Thus there is no token in the topplace at the end of the procedure meaning that the fault eventrepresented by 11987538 does not occur

In the second path11987221

11987222

and11987223

are as follows

119872

2

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le28

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

30le119894le38

]

]

]

119872

2

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

33le119894le38

]

]

]

119872

2

3

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

1

]

]

]

(13)

As (13) shows the tokens in place11987526 are transferred intoplace11987529 after the first ignition and then transferred into11987532after the second ignition Finally the tokens are transferredinto the top place 11987538 after the third ignition meaning thatthe top event occurs in this situation

In the third situation 11987231

can be calculated througha single transfer calculation when the default value of thetokens in place 1198758 is 1

119872

3

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le7

minus1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

9le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(14)

Thus when there are two transfer paths for one token inthe bottom basic place there will be negative values in thenext level state matrix after ignition if the default value of thetokens in the bottom basic place is still 1There is no practicalsignificance to this phenomenon To avoid negative valueschange the default value of the tokens in place 1198758 from 1 to 2to produce1198723

1

1015840

119872

3

1

1015840

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(15)

Then11987232

can be solved according to11987231

1015840

119872

3

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

2

]

]

]

(16)

According to11987232

there will be two tokens in the top place11987528 and the top event will inevitably happen when tokensindicate the fault transferThus the fault event in the top placewill happen if there is at least one token in the top place In thePetri net if an input place is a bottom basic place containingtokens with a value of 119899 (119899 is a positive integer greater than 1)directed arcs out then the default value of the tokens in theinput place is 119899The fault event presented indicated by the topplace would will inevitably happen when there are tokens inthe top place

Failure Mode 1 When two EMUs are attached together andat the same time the spacing of the two pantographs is lessthan 190m the pantograph will rise to an abnormal positionHowever if only one condition of the two is satisfied thepantograph will work normally

Failure Mode 2 Damage to MR ducts causes low pressurein the auxiliary air cylinder which means the pantographcannot rise successfully

FailureMode 3 Damage to the pantograph bellowsmay causethe pantograph to descend automatically or possibly fail torise Both of the two conditions will lead to a fault in thepantograph

The token transfer in the Petri net is consistent withthe logic analysis of the faults It can clearly and effectivelydescribe the dynamic processes of the system faults transferachieving fast and efficient fault diagnosis

5 Conclusions

In this study the working principles and failure modes ofpantographs on CRH EMU trains were analyzed systemati-cally Petri net modelling was used as a graphical modellingtool to simplify the logical relationships and events of thefault tree into a network with places and transactions asnodes The events logical gates and logical relation lines ofFT were transformed to places transitions and directed arcsrespectively The complexity of modelling was reduced by20by avoiding repeating eventsThus the process of solvingthe minimal cut sets was simplified which effectively savedcalculation time for theminimal cut set solutions for complexlarge-scale fault trees

The changes in system states and the evolution of failuresare described well by the dynamic properties of Petri netmodelling A mathematical model for the Petri net of thepantograph fault tree was established The equivalence andcorrectness of the token-transfer description for fault diag-nosis inference in a Petri net were verified Three differentfault paths were used to explain and verify the algorithmThe initial identification sets of the three paths were marked

Mathematical Problems in Engineering 11

The three corresponding ignition sequences were calculatedusing matrix transformations Finally the system status wasassessed correctly using a mathematical method which canbe handled easily in a computerised system

This work analyzes the evolution of failure events ofpantograph system The process of other critical systemsneeds to be further examined Future work will collect andanalyse test data from other systems in CRH EMU trainsto extend the methodology to cover the whole maintenanceprocess for CRH EMU train systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 51475324) The authors wouldlike to thank the reviewers for their constructive commentswhich enhanced the quality of this paper

References

[1] F C Jiao and B J Wang The Management and Maintenanceof Locomotive Rolling Stock Beijing Jiaotong University PressBeijing China 2013

[2] L L Song T Y Wang X W Song L Xu and D Song ldquoFaultdiagnosis of pantograph type current collector of CRH electricmultiple units based on Petri net modelling and fault treeanalysisrdquo Chinese Journal of Scientific Instrument vol 35 no 9pp 1990ndash1997 2014

[3] I Aydin M Karakose and E Akin ldquoAnomaly detection usinga modified kernel-based tracking in the pantographmdashcatenarysystemrdquoExpert SystemswithApplications vol 42 no 2 pp 938ndash948 2015

[4] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2008

[5] D M Shalev and J Tiran ldquoCondition-based fault tree analysis(CBFTA) a newmethod for improved fault tree analysis (FTA)reliability and safety calculationsrdquo Reliability Engineering andSystem Safety vol 92 no 9 pp 1231ndash1241 2007

[6] Y E Senol Y V Aydogdu B Sahin and I Kilic ldquoFaulttree analysis of chemical cargo contamination by using fuzzyapproachrdquo Expert Systems with Applications vol 42 no 12 pp5232ndash5244 2015

[7] S M Lavasani A Zendegani and M Celik ldquoAn extension tofuzzy fault tree analysis (FFTA) application in petrochemicalprocess industryrdquo Process Safety and Environmental Protectionvol 93 pp 75ndash88 2014

[8] J H Purba ldquoA fuzzy-based reliability approach to evaluate basicevents of fault tree analysis for nuclear power plant probabilisticsafety assessmentrdquo Annals of Nuclear Energy vol 70 pp 21ndash292014

[9] N RamzaliM RM Lavasani and J Ghodousi ldquoSafety barriersanalysis of offshore drilling system by employing Fuzzy EventTree Analysisrdquo Safety Science vol 78 pp 49ndash59 2015

[10] K-C Hyun S Min H Choi J Park and I Lee ldquoRisk analysisusing fault-tree analysis (FTA) and analytic hierarchy process(AHP) applicable to shield TBM tunnelsrdquo Tunnelling andUnderground Space Technology vol 49 pp 121ndash129 2015

[11] S Verlinden G Deconinck and B Coupe ldquoHybrid reliabilitymodel for nuclear reactor safety systemrdquo Reliability Engineeringand System Safety vol 101 pp 35ndash47 2012

[12] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 13 no 3 pp386ndash397 1987

[13] A Bobbio G Franceschinis R Gaeta and L Portinale ldquoPara-metric fault tree for the dependability analysis of redundant sys-tems and its high-level Petri net semanticsrdquo IEEE Transactionson Software Engineering vol 29 no 3 pp 270ndash287 2003

[14] R Robidoux H Xu L Xing andM Zhou ldquoAutomated model-ing of dynamic reliability block diagrams using colored Petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 2 pp 337ndash351 2010

[15] DMakajic-Nikolic M Vujosevic andN Nikolic ldquoMinimal cutsets of a coherent fault tree generation using reverse Petri netsrdquoOptimization vol 62 no 8 pp 1069ndash1087 2013

[16] B Wang G Tian Y Liang and T Qiang ldquoReliability modelingand evaluation of electric vehicle motor by using fault tree andextended stochastic petri netsrdquo Journal of Applied Mathematicsvol 2014 Article ID 638013 9 pages 2014

[17] T P K Nguyen J Beugin and JMarais ldquoMethod for evaluatingan extended Fault Tree to analyse the dependability of complexsystems application to a satellite-based railway systemrdquo Relia-bility Engineering and System Safety vol 133 pp 300ndash313 2015

[18] L L Song T Y Wang X W Song et al ldquoFuzzy intelligentfault diagnosis of pantograph type current collector under themulti-source heterogeneous knowledge environmentrdquo ChineseJournal of Scientific Instrument vol 36 pp 1283ndash1290 2015

[19] M An Y Chen and C J Baker ldquoA fuzzy reasoning and fuzzy-analytical hierarchy process based approach to the process ofrailway risk information a railway risk management systemrdquoInformation Sciences vol 181 no 18 pp 3946ndash3966 2011

[20] M An S Huang and C J Baker ldquoRailway risk assessmentmdashthe fuzzy reasoning approach and fuzzy analytic hierarchy pro-cess approaches a case study of shunting at Waterloo depotrdquoProceedings of the Institution of Mechanical Engineers F Journalof Rail and Rapid Transit vol 221 no 3 pp 365ndash383 2007

[21] K Noori and K Jenab ldquoFuzzy reliability-based traction controlmodel for intelligent transportation systemsrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 43 no 1 pp229ndash234 2013

[22] R I Muttram ldquoRailway safetyrsquos safety risk modelrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 216 no 2 pp 71ndash79 2002

[23] L Bouillaut O Francois and S Dubois ldquoA Bayesian network toevaluate underground rails maintenance strategies in an auto-mation contextrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 227 no4 pp 411ndash424 2013

[24] Y Xie Y Zhang and F Liang ldquoCrash injury severity analysisusing Bayesian ordered probit modelsrdquo Journal of Transporta-tion Engineering vol 135 no 1 pp 18ndash25 2009

[25] S Bernardi F Flammini S Marrone et al ldquoEnabling the usageof UML in the verification of railway systems the DAM-railapproachrdquoReliability Engineering and SystemSafety vol 120 pp112ndash126 2013

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

P1 P2 P3 P4

P5 P6 P7 P8 P9 P10 P11P12

P13

P14 P15 P16

P17 P18

P19

P20 P21 P22

P23

P24

P25 P26 P27 P28

P29

P30

P31

P32

P33P34

P35

P36 P37

P38

t1 t2 t3 t4

t5t6

t7

t8 t9 t10 t11 t12 t13 t14 t15

t16

t17

t18t19

t20

t21t22

t23 t24t25

t26t27

t28

t29t30 t31 t32

t33

t34t35

t36

t37

t38

t39

t40

t41

t42 t43

Initial identification 3

Initial identification 2

Initial identification 1

Figure 4 Transfer of tokens in the Petri net of the CRH EMU pantograph

initial identification1198720will convert to119872

119889after the ignition

sequence1198830 119883

119889 The process can be expressed as

119872

119889= 119872

0+ 119860

119879

119889minus1

sum

119896=0

119883

119896 (7)

If the element representing the top place is not less than1 in119872

119889after a series of transfers the fault event represented

by the top place will occur

42 Dynamic Transition and Diagnosis of Pantograph FaultsIn the process of fault diagnosis based on the use of a Petrinet the initial identification of the input place is regardedas the initial symptom of the event occurrence A token isassigned to the input place if the symptom appears otherwisethe place is assigned a null value The future states and finalidentification of the system can be circularly solved throughthe state equations and transfer sequences until the tokenvalue of the target place is found The fault occurs when thenumber of tokens in the target place is not zero

For the AND gate in the Petri net the tokens cannottransfer downward to the next level if there are empty inputplaces That is to say for an AND gate the fault will transferdownward only when all the events of the input places occurFor the OR gate in a Petri net the tokens will transferdownward to the next level place if at least one input placecontains tokens According to the rules of the network theorythe next level place will get two tokens if the two input placesboth contain tokens Regardless of the number of tokensthe occurrence of the fault that is represented by the targetplace is only related to whether there are tokens in the targetplace Transfer of the token in the Petri net of the CRH EMUpantograph is shown in Figure 4

As shown in Figure 4 1198751 represents ldquotwo EMUs attachedtogetherrdquo 1198752 represents ldquospacing of two pantographs of lessthan 190mrdquo 1198758 represents ldquopantograph bellows that aredamagedrdquo 11987526 represents ldquoMR ducts that are damagedrdquo11987529 represents ldquopressure in the auxiliary air cylinder that istoo lowrdquo 11987530 represents ldquopantograph rise to an abnormalpositionrdquo 11987532 represents ldquopantograph that cannot riserdquo11987535 represents ldquopantograph that descends automaticallyrdquoand 11987538 represents ldquopantograph that fails to workrdquo Threerepresentative fault transfer paths were selected to study thetransfer expression of pantograph faults in the Petri net Theinitial identification sets of the three paths were respectivelyrecorded as1198721

0

11987220

and11987230

119872

1

0

= [1 0 0

2le119894le38

]

119879

119872

2

0

= [0 0

1le119894le25

1 0 0

27le119894le38

]

119879

119872

3

0

= [0 0

1le119894le7

1 0 0

9le119894le38

]

119879

(8)

The ignition sequences of the three paths were respec-tively recorded as1198831

0

11988320

and11988330

119883

1

0

= [0 0

1le119894le4

1 0 0

6le119894le43

]

119879

119883

2

0

= [0 1 0 0

3le119894le43

]

119879

119883

3

0

= [0 0

1le119894le11

1 0 0

12le119894le21

1 0 0

23le119894le43

]

119879

(9)

10 Mathematical Problems in Engineering

The input matrix 119868 output matrix 119874 and the relationalmatrix119860119879 of the Petri net are all 38 times 43matrixes which canbe solved using the relational-matrix method

119860

119879

= 119874 minus 119868 = (

119886

11 119886

1119899

d

119886

1198981sdot sdot sdot 119886

119898119899

) (10)

119886

119894119895= 119874

119894119895minus 119868

1198941198951 le 119894 le 38 1 le 119895 le 43 (11)

In (10)119898 = 38 119899 = 43

119872

1

1

= 0

119872

1

2

= 0

(12)

As (12) shows tokens cannot transfer downward to thenext level in the first path Thus there is no token in the topplace at the end of the procedure meaning that the fault eventrepresented by 11987538 does not occur

In the second path11987221

11987222

and11987223

are as follows

119872

2

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le28

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

30le119894le38

]

]

]

119872

2

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

33le119894le38

]

]

]

119872

2

3

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

1

]

]

]

(13)

As (13) shows the tokens in place11987526 are transferred intoplace11987529 after the first ignition and then transferred into11987532after the second ignition Finally the tokens are transferredinto the top place 11987538 after the third ignition meaning thatthe top event occurs in this situation

In the third situation 11987231

can be calculated througha single transfer calculation when the default value of thetokens in place 1198758 is 1

119872

3

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le7

minus1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

9le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(14)

Thus when there are two transfer paths for one token inthe bottom basic place there will be negative values in thenext level state matrix after ignition if the default value of thetokens in the bottom basic place is still 1There is no practicalsignificance to this phenomenon To avoid negative valueschange the default value of the tokens in place 1198758 from 1 to 2to produce1198723

1

1015840

119872

3

1

1015840

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(15)

Then11987232

can be solved according to11987231

1015840

119872

3

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

2

]

]

]

(16)

According to11987232

there will be two tokens in the top place11987528 and the top event will inevitably happen when tokensindicate the fault transferThus the fault event in the top placewill happen if there is at least one token in the top place In thePetri net if an input place is a bottom basic place containingtokens with a value of 119899 (119899 is a positive integer greater than 1)directed arcs out then the default value of the tokens in theinput place is 119899The fault event presented indicated by the topplace would will inevitably happen when there are tokens inthe top place

Failure Mode 1 When two EMUs are attached together andat the same time the spacing of the two pantographs is lessthan 190m the pantograph will rise to an abnormal positionHowever if only one condition of the two is satisfied thepantograph will work normally

Failure Mode 2 Damage to MR ducts causes low pressurein the auxiliary air cylinder which means the pantographcannot rise successfully

FailureMode 3 Damage to the pantograph bellowsmay causethe pantograph to descend automatically or possibly fail torise Both of the two conditions will lead to a fault in thepantograph

The token transfer in the Petri net is consistent withthe logic analysis of the faults It can clearly and effectivelydescribe the dynamic processes of the system faults transferachieving fast and efficient fault diagnosis

5 Conclusions

In this study the working principles and failure modes ofpantographs on CRH EMU trains were analyzed systemati-cally Petri net modelling was used as a graphical modellingtool to simplify the logical relationships and events of thefault tree into a network with places and transactions asnodes The events logical gates and logical relation lines ofFT were transformed to places transitions and directed arcsrespectively The complexity of modelling was reduced by20by avoiding repeating eventsThus the process of solvingthe minimal cut sets was simplified which effectively savedcalculation time for theminimal cut set solutions for complexlarge-scale fault trees

The changes in system states and the evolution of failuresare described well by the dynamic properties of Petri netmodelling A mathematical model for the Petri net of thepantograph fault tree was established The equivalence andcorrectness of the token-transfer description for fault diag-nosis inference in a Petri net were verified Three differentfault paths were used to explain and verify the algorithmThe initial identification sets of the three paths were marked

Mathematical Problems in Engineering 11

The three corresponding ignition sequences were calculatedusing matrix transformations Finally the system status wasassessed correctly using a mathematical method which canbe handled easily in a computerised system

This work analyzes the evolution of failure events ofpantograph system The process of other critical systemsneeds to be further examined Future work will collect andanalyse test data from other systems in CRH EMU trainsto extend the methodology to cover the whole maintenanceprocess for CRH EMU train systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 51475324) The authors wouldlike to thank the reviewers for their constructive commentswhich enhanced the quality of this paper

References

[1] F C Jiao and B J Wang The Management and Maintenanceof Locomotive Rolling Stock Beijing Jiaotong University PressBeijing China 2013

[2] L L Song T Y Wang X W Song L Xu and D Song ldquoFaultdiagnosis of pantograph type current collector of CRH electricmultiple units based on Petri net modelling and fault treeanalysisrdquo Chinese Journal of Scientific Instrument vol 35 no 9pp 1990ndash1997 2014

[3] I Aydin M Karakose and E Akin ldquoAnomaly detection usinga modified kernel-based tracking in the pantographmdashcatenarysystemrdquoExpert SystemswithApplications vol 42 no 2 pp 938ndash948 2015

[4] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2008

[5] D M Shalev and J Tiran ldquoCondition-based fault tree analysis(CBFTA) a newmethod for improved fault tree analysis (FTA)reliability and safety calculationsrdquo Reliability Engineering andSystem Safety vol 92 no 9 pp 1231ndash1241 2007

[6] Y E Senol Y V Aydogdu B Sahin and I Kilic ldquoFaulttree analysis of chemical cargo contamination by using fuzzyapproachrdquo Expert Systems with Applications vol 42 no 12 pp5232ndash5244 2015

[7] S M Lavasani A Zendegani and M Celik ldquoAn extension tofuzzy fault tree analysis (FFTA) application in petrochemicalprocess industryrdquo Process Safety and Environmental Protectionvol 93 pp 75ndash88 2014

[8] J H Purba ldquoA fuzzy-based reliability approach to evaluate basicevents of fault tree analysis for nuclear power plant probabilisticsafety assessmentrdquo Annals of Nuclear Energy vol 70 pp 21ndash292014

[9] N RamzaliM RM Lavasani and J Ghodousi ldquoSafety barriersanalysis of offshore drilling system by employing Fuzzy EventTree Analysisrdquo Safety Science vol 78 pp 49ndash59 2015

[10] K-C Hyun S Min H Choi J Park and I Lee ldquoRisk analysisusing fault-tree analysis (FTA) and analytic hierarchy process(AHP) applicable to shield TBM tunnelsrdquo Tunnelling andUnderground Space Technology vol 49 pp 121ndash129 2015

[11] S Verlinden G Deconinck and B Coupe ldquoHybrid reliabilitymodel for nuclear reactor safety systemrdquo Reliability Engineeringand System Safety vol 101 pp 35ndash47 2012

[12] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 13 no 3 pp386ndash397 1987

[13] A Bobbio G Franceschinis R Gaeta and L Portinale ldquoPara-metric fault tree for the dependability analysis of redundant sys-tems and its high-level Petri net semanticsrdquo IEEE Transactionson Software Engineering vol 29 no 3 pp 270ndash287 2003

[14] R Robidoux H Xu L Xing andM Zhou ldquoAutomated model-ing of dynamic reliability block diagrams using colored Petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 2 pp 337ndash351 2010

[15] DMakajic-Nikolic M Vujosevic andN Nikolic ldquoMinimal cutsets of a coherent fault tree generation using reverse Petri netsrdquoOptimization vol 62 no 8 pp 1069ndash1087 2013

[16] B Wang G Tian Y Liang and T Qiang ldquoReliability modelingand evaluation of electric vehicle motor by using fault tree andextended stochastic petri netsrdquo Journal of Applied Mathematicsvol 2014 Article ID 638013 9 pages 2014

[17] T P K Nguyen J Beugin and JMarais ldquoMethod for evaluatingan extended Fault Tree to analyse the dependability of complexsystems application to a satellite-based railway systemrdquo Relia-bility Engineering and System Safety vol 133 pp 300ndash313 2015

[18] L L Song T Y Wang X W Song et al ldquoFuzzy intelligentfault diagnosis of pantograph type current collector under themulti-source heterogeneous knowledge environmentrdquo ChineseJournal of Scientific Instrument vol 36 pp 1283ndash1290 2015

[19] M An Y Chen and C J Baker ldquoA fuzzy reasoning and fuzzy-analytical hierarchy process based approach to the process ofrailway risk information a railway risk management systemrdquoInformation Sciences vol 181 no 18 pp 3946ndash3966 2011

[20] M An S Huang and C J Baker ldquoRailway risk assessmentmdashthe fuzzy reasoning approach and fuzzy analytic hierarchy pro-cess approaches a case study of shunting at Waterloo depotrdquoProceedings of the Institution of Mechanical Engineers F Journalof Rail and Rapid Transit vol 221 no 3 pp 365ndash383 2007

[21] K Noori and K Jenab ldquoFuzzy reliability-based traction controlmodel for intelligent transportation systemsrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 43 no 1 pp229ndash234 2013

[22] R I Muttram ldquoRailway safetyrsquos safety risk modelrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 216 no 2 pp 71ndash79 2002

[23] L Bouillaut O Francois and S Dubois ldquoA Bayesian network toevaluate underground rails maintenance strategies in an auto-mation contextrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 227 no4 pp 411ndash424 2013

[24] Y Xie Y Zhang and F Liang ldquoCrash injury severity analysisusing Bayesian ordered probit modelsrdquo Journal of Transporta-tion Engineering vol 135 no 1 pp 18ndash25 2009

[25] S Bernardi F Flammini S Marrone et al ldquoEnabling the usageof UML in the verification of railway systems the DAM-railapproachrdquoReliability Engineering and SystemSafety vol 120 pp112ndash126 2013

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

The input matrix 119868 output matrix 119874 and the relationalmatrix119860119879 of the Petri net are all 38 times 43matrixes which canbe solved using the relational-matrix method

119860

119879

= 119874 minus 119868 = (

119886

11 119886

1119899

d

119886

1198981sdot sdot sdot 119886

119898119899

) (10)

119886

119894119895= 119874

119894119895minus 119868

1198941198951 le 119894 le 38 1 le 119895 le 43 (11)

In (10)119898 = 38 119899 = 43

119872

1

1

= 0

119872

1

2

= 0

(12)

As (12) shows tokens cannot transfer downward to thenext level in the first path Thus there is no token in the topplace at the end of the procedure meaning that the fault eventrepresented by 11987538 does not occur

In the second path11987221

11987222

and11987223

are as follows

119872

2

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le28

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

30le119894le38

]

]

]

119872

2

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

33le119894le38

]

]

]

119872

2

3

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

1

]

]

]

(13)

As (13) shows the tokens in place11987526 are transferred intoplace11987529 after the first ignition and then transferred into11987532after the second ignition Finally the tokens are transferredinto the top place 11987538 after the third ignition meaning thatthe top event occurs in this situation

In the third situation 11987231

can be calculated througha single transfer calculation when the default value of thetokens in place 1198758 is 1

119872

3

1

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le7

minus1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

9le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(14)

Thus when there are two transfer paths for one token inthe bottom basic place there will be negative values in thenext level state matrix after ignition if the default value of thetokens in the bottom basic place is still 1There is no practicalsignificance to this phenomenon To avoid negative valueschange the default value of the tokens in place 1198758 from 1 to 2to produce1198723

1

1015840

119872

3

1

1015840

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le31

1 0 0 1 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

36le119894le38

]

]

]

(15)

Then11987232

can be solved according to11987231

1015840

119872

3

2

=

[

[

[

0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119894le37

2

]

]

]

(16)

According to11987232

there will be two tokens in the top place11987528 and the top event will inevitably happen when tokensindicate the fault transferThus the fault event in the top placewill happen if there is at least one token in the top place In thePetri net if an input place is a bottom basic place containingtokens with a value of 119899 (119899 is a positive integer greater than 1)directed arcs out then the default value of the tokens in theinput place is 119899The fault event presented indicated by the topplace would will inevitably happen when there are tokens inthe top place

Failure Mode 1 When two EMUs are attached together andat the same time the spacing of the two pantographs is lessthan 190m the pantograph will rise to an abnormal positionHowever if only one condition of the two is satisfied thepantograph will work normally

Failure Mode 2 Damage to MR ducts causes low pressurein the auxiliary air cylinder which means the pantographcannot rise successfully

FailureMode 3 Damage to the pantograph bellowsmay causethe pantograph to descend automatically or possibly fail torise Both of the two conditions will lead to a fault in thepantograph

The token transfer in the Petri net is consistent withthe logic analysis of the faults It can clearly and effectivelydescribe the dynamic processes of the system faults transferachieving fast and efficient fault diagnosis

5 Conclusions

In this study the working principles and failure modes ofpantographs on CRH EMU trains were analyzed systemati-cally Petri net modelling was used as a graphical modellingtool to simplify the logical relationships and events of thefault tree into a network with places and transactions asnodes The events logical gates and logical relation lines ofFT were transformed to places transitions and directed arcsrespectively The complexity of modelling was reduced by20by avoiding repeating eventsThus the process of solvingthe minimal cut sets was simplified which effectively savedcalculation time for theminimal cut set solutions for complexlarge-scale fault trees

The changes in system states and the evolution of failuresare described well by the dynamic properties of Petri netmodelling A mathematical model for the Petri net of thepantograph fault tree was established The equivalence andcorrectness of the token-transfer description for fault diag-nosis inference in a Petri net were verified Three differentfault paths were used to explain and verify the algorithmThe initial identification sets of the three paths were marked

Mathematical Problems in Engineering 11

The three corresponding ignition sequences were calculatedusing matrix transformations Finally the system status wasassessed correctly using a mathematical method which canbe handled easily in a computerised system

This work analyzes the evolution of failure events ofpantograph system The process of other critical systemsneeds to be further examined Future work will collect andanalyse test data from other systems in CRH EMU trainsto extend the methodology to cover the whole maintenanceprocess for CRH EMU train systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 51475324) The authors wouldlike to thank the reviewers for their constructive commentswhich enhanced the quality of this paper

References

[1] F C Jiao and B J Wang The Management and Maintenanceof Locomotive Rolling Stock Beijing Jiaotong University PressBeijing China 2013

[2] L L Song T Y Wang X W Song L Xu and D Song ldquoFaultdiagnosis of pantograph type current collector of CRH electricmultiple units based on Petri net modelling and fault treeanalysisrdquo Chinese Journal of Scientific Instrument vol 35 no 9pp 1990ndash1997 2014

[3] I Aydin M Karakose and E Akin ldquoAnomaly detection usinga modified kernel-based tracking in the pantographmdashcatenarysystemrdquoExpert SystemswithApplications vol 42 no 2 pp 938ndash948 2015

[4] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2008

[5] D M Shalev and J Tiran ldquoCondition-based fault tree analysis(CBFTA) a newmethod for improved fault tree analysis (FTA)reliability and safety calculationsrdquo Reliability Engineering andSystem Safety vol 92 no 9 pp 1231ndash1241 2007

[6] Y E Senol Y V Aydogdu B Sahin and I Kilic ldquoFaulttree analysis of chemical cargo contamination by using fuzzyapproachrdquo Expert Systems with Applications vol 42 no 12 pp5232ndash5244 2015

[7] S M Lavasani A Zendegani and M Celik ldquoAn extension tofuzzy fault tree analysis (FFTA) application in petrochemicalprocess industryrdquo Process Safety and Environmental Protectionvol 93 pp 75ndash88 2014

[8] J H Purba ldquoA fuzzy-based reliability approach to evaluate basicevents of fault tree analysis for nuclear power plant probabilisticsafety assessmentrdquo Annals of Nuclear Energy vol 70 pp 21ndash292014

[9] N RamzaliM RM Lavasani and J Ghodousi ldquoSafety barriersanalysis of offshore drilling system by employing Fuzzy EventTree Analysisrdquo Safety Science vol 78 pp 49ndash59 2015

[10] K-C Hyun S Min H Choi J Park and I Lee ldquoRisk analysisusing fault-tree analysis (FTA) and analytic hierarchy process(AHP) applicable to shield TBM tunnelsrdquo Tunnelling andUnderground Space Technology vol 49 pp 121ndash129 2015

[11] S Verlinden G Deconinck and B Coupe ldquoHybrid reliabilitymodel for nuclear reactor safety systemrdquo Reliability Engineeringand System Safety vol 101 pp 35ndash47 2012

[12] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 13 no 3 pp386ndash397 1987

[13] A Bobbio G Franceschinis R Gaeta and L Portinale ldquoPara-metric fault tree for the dependability analysis of redundant sys-tems and its high-level Petri net semanticsrdquo IEEE Transactionson Software Engineering vol 29 no 3 pp 270ndash287 2003

[14] R Robidoux H Xu L Xing andM Zhou ldquoAutomated model-ing of dynamic reliability block diagrams using colored Petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 2 pp 337ndash351 2010

[15] DMakajic-Nikolic M Vujosevic andN Nikolic ldquoMinimal cutsets of a coherent fault tree generation using reverse Petri netsrdquoOptimization vol 62 no 8 pp 1069ndash1087 2013

[16] B Wang G Tian Y Liang and T Qiang ldquoReliability modelingand evaluation of electric vehicle motor by using fault tree andextended stochastic petri netsrdquo Journal of Applied Mathematicsvol 2014 Article ID 638013 9 pages 2014

[17] T P K Nguyen J Beugin and JMarais ldquoMethod for evaluatingan extended Fault Tree to analyse the dependability of complexsystems application to a satellite-based railway systemrdquo Relia-bility Engineering and System Safety vol 133 pp 300ndash313 2015

[18] L L Song T Y Wang X W Song et al ldquoFuzzy intelligentfault diagnosis of pantograph type current collector under themulti-source heterogeneous knowledge environmentrdquo ChineseJournal of Scientific Instrument vol 36 pp 1283ndash1290 2015

[19] M An Y Chen and C J Baker ldquoA fuzzy reasoning and fuzzy-analytical hierarchy process based approach to the process ofrailway risk information a railway risk management systemrdquoInformation Sciences vol 181 no 18 pp 3946ndash3966 2011

[20] M An S Huang and C J Baker ldquoRailway risk assessmentmdashthe fuzzy reasoning approach and fuzzy analytic hierarchy pro-cess approaches a case study of shunting at Waterloo depotrdquoProceedings of the Institution of Mechanical Engineers F Journalof Rail and Rapid Transit vol 221 no 3 pp 365ndash383 2007

[21] K Noori and K Jenab ldquoFuzzy reliability-based traction controlmodel for intelligent transportation systemsrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 43 no 1 pp229ndash234 2013

[22] R I Muttram ldquoRailway safetyrsquos safety risk modelrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 216 no 2 pp 71ndash79 2002

[23] L Bouillaut O Francois and S Dubois ldquoA Bayesian network toevaluate underground rails maintenance strategies in an auto-mation contextrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 227 no4 pp 411ndash424 2013

[24] Y Xie Y Zhang and F Liang ldquoCrash injury severity analysisusing Bayesian ordered probit modelsrdquo Journal of Transporta-tion Engineering vol 135 no 1 pp 18ndash25 2009

[25] S Bernardi F Flammini S Marrone et al ldquoEnabling the usageof UML in the verification of railway systems the DAM-railapproachrdquoReliability Engineering and SystemSafety vol 120 pp112ndash126 2013

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

The three corresponding ignition sequences were calculatedusing matrix transformations Finally the system status wasassessed correctly using a mathematical method which canbe handled easily in a computerised system

This work analyzes the evolution of failure events ofpantograph system The process of other critical systemsneeds to be further examined Future work will collect andanalyse test data from other systems in CRH EMU trainsto extend the methodology to cover the whole maintenanceprocess for CRH EMU train systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (no 51475324) The authors wouldlike to thank the reviewers for their constructive commentswhich enhanced the quality of this paper

References

[1] F C Jiao and B J Wang The Management and Maintenanceof Locomotive Rolling Stock Beijing Jiaotong University PressBeijing China 2013

[2] L L Song T Y Wang X W Song L Xu and D Song ldquoFaultdiagnosis of pantograph type current collector of CRH electricmultiple units based on Petri net modelling and fault treeanalysisrdquo Chinese Journal of Scientific Instrument vol 35 no 9pp 1990ndash1997 2014

[3] I Aydin M Karakose and E Akin ldquoAnomaly detection usinga modified kernel-based tracking in the pantographmdashcatenarysystemrdquoExpert SystemswithApplications vol 42 no 2 pp 938ndash948 2015

[4] E Zio ldquoReliability engineering old problems and new chal-lengesrdquo Reliability Engineering and System Safety vol 94 no 2pp 125ndash141 2008

[5] D M Shalev and J Tiran ldquoCondition-based fault tree analysis(CBFTA) a newmethod for improved fault tree analysis (FTA)reliability and safety calculationsrdquo Reliability Engineering andSystem Safety vol 92 no 9 pp 1231ndash1241 2007

[6] Y E Senol Y V Aydogdu B Sahin and I Kilic ldquoFaulttree analysis of chemical cargo contamination by using fuzzyapproachrdquo Expert Systems with Applications vol 42 no 12 pp5232ndash5244 2015

[7] S M Lavasani A Zendegani and M Celik ldquoAn extension tofuzzy fault tree analysis (FFTA) application in petrochemicalprocess industryrdquo Process Safety and Environmental Protectionvol 93 pp 75ndash88 2014

[8] J H Purba ldquoA fuzzy-based reliability approach to evaluate basicevents of fault tree analysis for nuclear power plant probabilisticsafety assessmentrdquo Annals of Nuclear Energy vol 70 pp 21ndash292014

[9] N RamzaliM RM Lavasani and J Ghodousi ldquoSafety barriersanalysis of offshore drilling system by employing Fuzzy EventTree Analysisrdquo Safety Science vol 78 pp 49ndash59 2015

[10] K-C Hyun S Min H Choi J Park and I Lee ldquoRisk analysisusing fault-tree analysis (FTA) and analytic hierarchy process(AHP) applicable to shield TBM tunnelsrdquo Tunnelling andUnderground Space Technology vol 49 pp 121ndash129 2015

[11] S Verlinden G Deconinck and B Coupe ldquoHybrid reliabilitymodel for nuclear reactor safety systemrdquo Reliability Engineeringand System Safety vol 101 pp 35ndash47 2012

[12] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 13 no 3 pp386ndash397 1987

[13] A Bobbio G Franceschinis R Gaeta and L Portinale ldquoPara-metric fault tree for the dependability analysis of redundant sys-tems and its high-level Petri net semanticsrdquo IEEE Transactionson Software Engineering vol 29 no 3 pp 270ndash287 2003

[14] R Robidoux H Xu L Xing andM Zhou ldquoAutomated model-ing of dynamic reliability block diagrams using colored Petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics PartASystems and Humans vol 40 no 2 pp 337ndash351 2010

[15] DMakajic-Nikolic M Vujosevic andN Nikolic ldquoMinimal cutsets of a coherent fault tree generation using reverse Petri netsrdquoOptimization vol 62 no 8 pp 1069ndash1087 2013

[16] B Wang G Tian Y Liang and T Qiang ldquoReliability modelingand evaluation of electric vehicle motor by using fault tree andextended stochastic petri netsrdquo Journal of Applied Mathematicsvol 2014 Article ID 638013 9 pages 2014

[17] T P K Nguyen J Beugin and JMarais ldquoMethod for evaluatingan extended Fault Tree to analyse the dependability of complexsystems application to a satellite-based railway systemrdquo Relia-bility Engineering and System Safety vol 133 pp 300ndash313 2015

[18] L L Song T Y Wang X W Song et al ldquoFuzzy intelligentfault diagnosis of pantograph type current collector under themulti-source heterogeneous knowledge environmentrdquo ChineseJournal of Scientific Instrument vol 36 pp 1283ndash1290 2015

[19] M An Y Chen and C J Baker ldquoA fuzzy reasoning and fuzzy-analytical hierarchy process based approach to the process ofrailway risk information a railway risk management systemrdquoInformation Sciences vol 181 no 18 pp 3946ndash3966 2011

[20] M An S Huang and C J Baker ldquoRailway risk assessmentmdashthe fuzzy reasoning approach and fuzzy analytic hierarchy pro-cess approaches a case study of shunting at Waterloo depotrdquoProceedings of the Institution of Mechanical Engineers F Journalof Rail and Rapid Transit vol 221 no 3 pp 365ndash383 2007

[21] K Noori and K Jenab ldquoFuzzy reliability-based traction controlmodel for intelligent transportation systemsrdquo IEEETransactionson Systems Man and Cybernetics Systems vol 43 no 1 pp229ndash234 2013

[22] R I Muttram ldquoRailway safetyrsquos safety risk modelrdquo Proceedingsof the Institution of Mechanical Engineers Part F Journal of Railand Rapid Transit vol 216 no 2 pp 71ndash79 2002

[23] L Bouillaut O Francois and S Dubois ldquoA Bayesian network toevaluate underground rails maintenance strategies in an auto-mation contextrdquo Proceedings of the Institution of MechanicalEngineers Part O Journal of Risk and Reliability vol 227 no4 pp 411ndash424 2013

[24] Y Xie Y Zhang and F Liang ldquoCrash injury severity analysisusing Bayesian ordered probit modelsrdquo Journal of Transporta-tion Engineering vol 135 no 1 pp 18ndash25 2009

[25] S Bernardi F Flammini S Marrone et al ldquoEnabling the usageof UML in the verification of railway systems the DAM-railapproachrdquoReliability Engineering and SystemSafety vol 120 pp112ndash126 2013

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[26] Q Mahboob and D Straub ldquoComparison of fault tree andBayesian Networks for modeling safety critical components inrailway systemsrdquo inAdvances in Safety Reliability andRiskMan-agement Proceedings of the European Safety and Reliability Con-ference (ESREL 2011) chapter 12 pp 89ndash95 CRC Press 2011

[27] S Washington and J Oh ldquoBayesian methodology incorporat-ing expert judgment for ranking countermeasure effectivenessunder uncertainty example applied to at grade railroad cross-ings in Koreardquo Accident Analysis and Prevention vol 38 no 2pp 234ndash247 2006

[28] P Weber G Medina-Oliva C Simon and B Iung ldquoOverviewon Bayesian networks applications for dependability risk analy-sis andmaintenance areasrdquo Engineering Applications of ArtificialIntelligence vol 25 no 4 pp 671ndash682 2012

[29] N Khakzad F Khan and P Amyotte ldquoSafety analysis in pro-cess facilities comparison of fault tree and Bayesian networkapproachesrdquo Reliability Engineering and System Safety vol 96no 8 pp 925ndash932 2011

[30] M Gao M Zhou X Huang and ZWu ldquoFuzzy reasoning petrinetsrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 33 no 3 pp 314ndash324 2003

[31] G J Tsinarakis N C Tsourveloudis and K P ValavanisldquoModeling analysis synthesis and performance evaluationof multioperational production systems with hybrid timedPetri Netsrdquo IEEE Transactions on Automation Science andEngineering vol 3 no 1 pp 29ndash46 2006

[32] M P Cabasino A Giua S Lafortune and C Seatzu ldquoA newapproach for diagnosability analysis of Petri nets using verifiernetsrdquo IEEE Transactions on Automatic Control vol 57 no 12pp 3104ndash3117 2012

[33] Y-S Huang Y-S Weng and M-C Zhou ldquoDesign of trafficsafety control systems for emergency vehicle preemption usingtimed petri netsrdquo IEEE Transactions on Intelligent Transporta-tion Systems vol 16 no 4 pp 2113ndash2120 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of