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Methods for Determining and Processing Probabilities

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Methods for Determining and Processing Probabilities 4 Methods for determining and processing probabilities PUBLICATIEREEKS GEVAARLIJKE STOFFEN Publication Series on Dangerous Substances 4 (PGS 4) Methods for determining and processing probabilities Preface Starting from June 1st 2004, the Advisory Council on Dangerous Substances (Adviesraad Gevaarlijke Stoffen - AGS) was installed by the Cabinet. At the same time the Committee for the Prevention of Disasters (Commissie voor de Preventie van Rampen- CPR) was abolished. CPR issued several publications, the so-called CPR-guidelines (CPR-richtlijnen), that are often used in environmental permits, based on the Environmental Protection Law, and in the fields of of labour safety, transport safety and fire safety. The CPR-guidelines have been transformed into the Publication Series on Dangerous Substances (Publicatiereeks Gevaarlijke Stoffen – PGS). The aim of these publications is generally the same as that of the CPR-guidelines. All CPR-guidelines have been reviewed, taking into account the follo- wing questions: 1. Is there still a reason for existence for the guideline or can the guideline be abolished; 2. Can the guideline be reintroduced without changes or does it need to be updated. This fourth publication in the Series on Dangerous Substances (PGS 4) is not different from the former publication CPR 12E, second edition 1997. Also on behalf of my colleagues at the Ministries of Transport, Social Affairs and of the Interior, The State Secretary of Housing Spatial Planning and the Environment (VROM). Drs. P.L.B.A van Geel [december] 2005 Methods for determining and processing probabilities ‘Red Book’ CPR 12E Principal author J.C.H. Schüller, m.sc. Co-authors J.L. Brinkman, m.sc. P.J. Van Gestel, m.sc. R.W. van Otterloo, m.sc. NRG Utrechtseweg 310, 6812 AR ARNHEM The Netherlands Tel. +31 26 356 8526 Fax. +31 36 445 9035 This report was prepared under the supervision of the Committee for the Prevention of Disasters and is published with the approval of The Director-General for Social Affairs and Employment The Director-General for Environmental Protection The Director-General for Public Order and Security The Director-General for Transport The Hague, 1997 The Director-General for Social Affairs and Employment Committee for Prevention of Disasters Second edition 1997 FOREWORD The report CPR-12E ‘Methods for determining and processing probabilities’ was first issued in 1988. In the following years the “Red Book” has proven to be a useful tool in risk analysis. However it became apparent certain subjects should be added or should be covered more elaborately and that in some cases less cumbersome methods could be introduced. In order to improve the usefulness of the “Red Book” NRG was assigned to prepare a complete revision and update with special emphasis on user friendliness. In this second edition NRG has included many improvements which resulted from the experience it gained in using the “Red Book” in risk analysis. The revision of the “Red Book” was supervised by a committee in which participated: Dr. E.F. Blokker, chairman DCMR Environmental Protection Agency Rijnmond Mr. ir. K. Posthuma, secretary Ministry of Social Affairs and Employment Dr. B.J.M. Ale RIVM National Institute of Public Health and the Environment Drs. R. Dauwe DOW Benelux B.V. Ir. B.W. Moll Province of Limburg Ing. A.J. Muyselaar Ministry of Housing, Spatial Planning and the Environment Ing. A.W. Peters Ministry of Transport, Public Works and Water Management Drs. Ing. A.F.M. van der Staak Ministry of Social Affairs and Employment Dr. P.A.M. Uijt de Haag RVM National Institute of Public Health and the Environment Ir M. Vis van Heemst AKZO Engineering B.V. Ir. J.H. van Zwol Ministry of Transport, Public Works and Watermanagement, Traffic Research Centre With the issue of the second edition of the “Red Book” the Committee for the Prevention of Disasters by Hazardous Materials expects to promote the general use of standardised calculation methods in risk analysis. The Hague, July 4, 1997 THE COMMITTEE FOR THE PREVENTION OF DISASTERS BY HAZARDOUS MATERIALS, Drs. H.C.M. Middelplaats, chairman Contents Contents FOREWORD 1 INTRODUCTION 1.1 Introduction 1.3 1.2 How to use this document 1.5 1.3 References 1.8 2 DEFINITIONS 2.1 Introduction 2.3 2.2 Probability Theory 2.3 2.3 Statistics 2.4 2.4 Reliability Theory 2.6 2.5 Data analysis 2.10 2.6 Methods of Identification of Failure Scenario’s 2.13 2.7 Fault Tree Analysis 2.15 2.8 Event Tree Analysis 2.16 2.9 Markov Processes 2.17 2.10 Accident Sequence Development and Quantification 2.19 2.11 Dependent Failure Analysis 2.20 2.12 Human Failures 2.23 2.13 Reliability Availability and Maintainability Specification 2.27 3 PROBABILITY THEORY 3.1 Introduction 3.3 3.2 Nomenclature 3.4 3.3 Probabilities and events 3.5 3.3.1 Definition of probability 3.5 3.3.2 Set theory and Venn diagrams 3.6 3.3.3 Interpretations of probability 3.7 3.4 Rules for combining events 3.8 3.4.1 Mutually Exclusive Events 3.9 3.4.2 Complementary events 3.9 3.4.3 Independent versus dependent events 3.10 3.4.4 Conditional probabilities 3.11 3.5 Algebraic operations with probabilities 3.12 3.5.1 Simultaneous occurrence of events (Multiplication rule for probabilities) 3.12 3.5.2 Occurrence of at least one of the two events (Addition rule for probabilities) 3.13 3.6 Series-parallel systems 3.14 3.6.1 Series structure of components 3.14 3.6.2 Parallel configuration of components 3.15 i Contents 3.6.3 Example 3.16 3.7 References 3.19 4 STATISTICS 4.1 Introduction 4.3 4.2 Nomenclature 4.4 4.3 Some Important Concepts in Statistics 4.5 4.3.1 Establishment of failure data 4.5 4.3.2 Population, sample, non-randomness, representative 4.5 4.3.3 Correlation 4.7 4.4 Important Probability Distributions 4.8 4.4.1 Definitions 4.8 4.4.2 Mean, median and mode of a distribution 4.9 4.4.3 Spread of a distribution 4.10 4.4.4 Binomial distribution 4.10 4.4.5 Poisson distribution 4.12 4.4.6 Exponential distribution 4.13 4.4.7 Normal distribution 4.15 4.4.8 Standardized normal distribution 4.15 4.4.9 Lognormal distribution 4.16 4.4.10 Uniform and loguniform distributions 4.18 4.4.11 Weibull distribution 4.19 4.4.12 Gamma distribution 4.21 4.4.13 Chi-Square distribution 4.21 4.4.14 F distribution 4.21 4.5 Basic Concepts in Estimation 4.22 4.5.1 Estimation theory 4.22 4.5.2 Maximum likelihood 4.23 4.5.3 Confidence interval 4.25 4.5.4 Estimation of failure parameters 4.26 4.6 Estimation of time dependent reliability characteristics 4.28 4.6.1 Direct methods 4.28 4.6.2 Indirect methods 4.33 4.6.3 Graphic method 4.32 4.6.4 Testing 4.36 4.7 Baysian 4.37 4.7.1 Theoretical background 4.39 4.7.2 Continuous distributions 4.40 4.7.3 Example one (safety valve spurious opening) 4.42 4.7.4 Example two (coolant pump fails to run) 4.44 4.8 References 4.46 ii Contents 5 RELIABILITY THEORY 5.1 Introduction 5.3 5.2 Nomenclature 5.4 5.3 Modelling of component failures 5.5 5.3.1 Time related failures 5.5 5.3.2 Demand related failures 5.7 5.3.3 Bath-tub curve 5.8 5.4 Reliability characteristics 5.9 5.4.1 Reliability and unreliability 5.10 5.4.2 General relation between failure rate, failure density and unreliability 5.11 5.4.3 Availability, unavailability and probability of failure on demand 5.12 5.4.4 Contributors to unavailability 5.15 5.4.5 Expected number of failures 5.15 5.5.6 General relation between failure rate, failure occurrence rate and unavailability 5.16 5.5 Component failure models 5.16 5.5.1 Characteristics of constant failure rate model 5.17 5.5.2 Periodically tested stand-by components 5.19 5.5.3 On-line repairable components 5.21 5.5.4 Mission-type components 5.23 5.5.5 Constant demand model 5.24 5.6 Probability of failure of high-demanded systems 5.24 5.7 References 5.28 6 DATA ANALYSIS 6.1 Introduction 6.3 6.2 Nomenclature 6.5 6.3 Failure classification 6.6 6.3.1 Fault, error and failure 6.6 6.3.2 Primary and secondary faults 6.6 6.3.3 Failure mechanism 6.6 6.3.4 Failure mode 6.7 6.3.5 Critical, degraded and incipient failure 6.7 6.3.6 Random versus systematic failure 6.8 6.4 Generic data analysis 6.8 6.4.1 Sources of generic data 6.9 6.4.2 Uncertainty bounds of generic data 6.12 6.4.3 Generic data analysis procedure 6.13 6.4.4 Failure on demand versus failure rate model 6.16 6.5 Plant-specific data analysis 6.17 6.5.1 Plant-specific data analysis procedure 6.18 6.5.2 Example 6.23 6.6 Use of expert opinion 6.26 6.6.1 Introduction 6.26 6.6.2 Conditions for use of data 6.27 6.6.3 Expert opinion and biases 6.27 iii Contents 6.6.4 The interview session 6.29 6.6.5 Models 6.30 6.6.6 Example 6.33 6.7 Bayesian updating technique 6.35 6.7.1 Bayesian update process procedure 6.38 6.7.2 Example of Bayesian update 6.41 6.8 Assessment of component downtime 6.44 6.9 Assessment of initiating event frequencies 6.46 6.10 References 6.47 Appendix 6-A: Tables Chi-squared distribution and F-distribution 6.48 Appendix 6-B: Generic Component Failure Data Base 6.52 7 METHODS OF IDENTIFICATION OF FAILURE SCENARIO’S 7.1 Introduction 7.
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