Calculation of the Probabilistic Safety Analysis and Reliability by the Fault Trees and Event Trees Methods M.A. Berberova1, A.V.Dmitriev1, A.V.Golubkov1, A.I.Elizarov2 maria.berberova@gmail.com | avdv@list.ru | sgo@mail.ru | eiao8lzrv@mail.ru 1 International Nuclear Safety Center, Moscow, Russia 2 Open Joint Stock Company «All-Russian Research Institute for Nuclear Power Plants Operation», Moscow, Russia One of the main requirements for ensuring a high level of safety and economic efficiency of nuclear power units at all stages of the life cycle - designing new ones, operating existing power units and decommissioning them - is a probabilistic safety analysis of nuclear power units. The most widely used method for probabilistic safety analysis is the fault tree method. NPP power units are a complex system consisting of a large number of units of equipment, systems and units that are interconnected functionally and affect each other. In addition, to increase the adequacy of the developed probabilistic model of a power unit, it is necessary to take into account equipment failures for general reasons and the human factor. The resulting in-depth probabilistic models of power units can contain tens of thousands of fault trees and, as a result, hundreds or more of thousands of minimum sections and require lengthy calculations to obtain acceptable accuracy of the results. This complicates the application of this method, especially when monitoring risk in real time, when it is necessary to promptly make changes to the model and assess the impact of these changes on the current risk. The novelty of the project is the use of a modified modularization method, which significantly accelerates the generation of many minimal sections. Keywords: probabilistic safety analysis, NPP, fault trees, event trees. Required Parameters: λ, µ(r,TR). 1. Introduction Optional parameters: q. 2. Periodically checked item (type 2) Probabilistic safety analysis of a nuclear power plant (PSA) Required Parameters: λ, TI(r,TI). is a system safety analysis of a nuclear power plant unit, during Optional parameters: q, TR, TF. which probabilistic models are developed and probabilistic The required parameters characterize the traditional model of safety indicators are determined, and the results of which are a periodically controlled element. For such model, the used for qualitative and quantitative assessments of the level of unavailability of this type element Q(t) is calculated by the safety of a nuclear power plant unit and development of decisions formula during design and operation unit of a nuclear power plant [1]. 𝑄(𝑡) = 1 − 𝑒 −λ(𝑡−𝑇𝐼) , TI = 0, 𝑇𝐼, 2𝑇𝐼, … (2) The main requirements for the implementation of PSA are 3. An element with constant unavailability over time, given in [2-6]. characterized by a refusal of a requirement (type 3). This is the A detailed description of the «Risk» and «RISK- simplest and most frequently used model, using the only q SPECTRUM» Software tools is given, respectively, in [7] and parameter - the probability of the request failure. In this case, the [8]. formulas are used To determine the unavailability of primary events in the PSA 𝑄(𝑡) = 𝑞, 𝑄𝑚𝑒𝑎𝑛 = 𝑞, W(t)=0. (4) model development, probabilistic reliability models of elements 4. Element with a fixed working time (type 4) of the following types are used: Required Parameters: λ, TM.  constantly monitored, restored element (type 1), Optional parameter: q.  periodically checked item (type 2), The following formulas are used  an element with constant unavailability over time, 𝑄(𝑡) = 𝑞 + 1 − 𝑒 −λ𝑇𝑀 , 𝑄𝑚𝑒𝑎𝑛 = 𝑞 + 1 − 𝑒 −λ𝑇𝑀 , 𝑊(𝑡) = 0. (5) characterized by a refusal of a requirement (type 3), 5. An event characterized by a constant frequency (type 5).  element with a fixed working time (type 4), This model is used when the event is well described by the  an event characterized by a constant frequency (type 5), Poisson process, i.e. when events occur at a constant frequency.  non-recoverable item (type 6). In this case, the only parameter f. Table 1 shows the parameters used as input data, and the 𝑄(𝑡) = 0, 𝑄𝑚𝑒𝑎𝑛 = 0, W(t)=f. (6) corresponding parameters of the formulas used to calculate the 6. Non-recoverable item (type 6). unavailability of elements. Required Parameter: λ(r). Table. 1. Parameters used as input Optional parameter: q. Formula Options Description Codes 𝑄(𝑡) = 𝑞 + 1 − 𝑒 −λ𝑡 , W(t) = λ(1-Q(𝑡)). (7) Q The probability of failure on demand q For each calculation option, an analysis of the minimum λ Failure rate r cross sections is carried out. F Frequency f The uncertainty analysis is carried out in addition to the point W Failure Flow Parameter W µ Recovery flow parameter (frequency) 1/TR estimate obtained in the analysis of the minimum cross sections. TR Average recovery time TR The uncertainty analysis is based on a simple version of the TI Test Interval TI Monte Carlo method. TF First check time TF The parameters of the reliability models of primary events TM Work time TM have their own (regardless of primary events) record in which the developer sets a point (average) value of the reliability parameter 2. Calculation models and, if the uncertainty of the parameter is taken into account, the distribution of uncertainty. Distributions are used such as: 1. Constantly monitored, restored element (type 1).  Lognormal – fig. 1; Unavailability Q (t) of this type element is calculated by the  Gamma – fig. 2; formula λ  Beta – fig. 3; 𝑄(𝑡) = 𝑞𝑒 −𝜇𝑡 + ( ) · (1 − 𝑒 −(𝜆+𝜇)𝑡 ). (1)  Normal – fig. 4; 𝜆+𝜇 Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).  Uniform – fig. 5;  Log-uniform – fig. 6;  Discrete - fig. 7. Fig. 7. Discrete Distribution Example 3. Unavailability indicators calculation for Fig. 1. Lognormal Distribution Example simple structures 1. Logical operator «OR». In terms of fault trees, such a structure corresponds to logic of the «OR» type, i.e. at least one input event occurs (Fig. 8). In mathematical expressions, the operator «OR» is indicated by the symbol «« or the sign «+». Fig. 2. Gamma Distribution Example Fig. 8. Example of a fault tree with the logical operator «OR» According to the formula of total probability, the probability of the event AB (P(AB)) will be equal to: 𝑃(𝐴𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴𝐵). (8) 2. Logical operator «AND». In terms of fault trees, such a structure corresponds to logic of the «AND» type, i.e. all input events occur (Fig. 9). In mathematical expressions, the operator Fig. 3. Beta Distribution Example «AND» is indicated by the symbol «» or the sign «». Fig. 4. Normal Distribution Example Fig. 9. Example of a fault tree with the logical operator «AND» By the multiplication theorem, the probability of the event AB (P(AB)) will be equal to: 𝑃(𝐴𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵). (9) 3. Logical operator «K from N» (K / N). For such a system, the failure criterion is the failure of any K elements from N (for example, two elements from three). In this case, a logical operator of type K / N is used in the fault tree (Fig. 10). Fig. 5. Uniform Distribution Example Fig. 10. Example of a fault tree with the logical operator «K/N» According to the formula of total probability and the multiplication theorem, the probability of the event ABC (P(ABC)) will be equal to: 𝑃(𝐴𝐵𝐶) = 𝑃(𝐴𝐵) + 𝑃(𝐵𝐶) + 𝑃(𝐴𝐶) − Fig. 6. Log-uniform Distribution Example 𝑃(𝐴𝐵𝐵𝐶) − 𝑃(𝐵𝐶𝐴𝐶) − 𝑃(𝐴𝐵𝐴𝐶) − 𝑃(𝐴𝐵𝐵𝐶𝐴𝐶). (10) 4. Logical operator «NOR». In terms of fault trees, such a According to the formula of total probability and the structure corresponds to the logic of the «Not OR» type, i.e. multiplication theorem, the probability of the event AB (P(AB)) denial of OR or none of the events occur (Fig. 11). will be equal to: 𝑃(𝐴𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵) + (𝑃(𝐴) ∙ (1 − 𝑃(𝐵))) + +((1 − 𝑃(𝐴)) ∙ 𝑃(𝐵) − 𝑃(𝐴𝐵). (14) 8. Logical operator NXOR. In terms of fault trees, such a structure corresponds to the logic of the «NON-EXCLUSIVE OR» (NXOR) type, i.e. no event takes place (denial of XOR) (Fig. 15). Fig. 11. Example of a fault tree with the logical operator «NOR» According to the multiplication theorem, the probability of the event 𝐴𝐵 (P(𝐴𝐵)) will be equal to 𝑃(𝐴𝐵) = (1 − 𝑃(𝐴)) ∙ (1 − 𝑃(𝐵)). (11) 5. Logical operator NAND. In terms of fault trees, this Fig. 15. Example of a fault tree with the logical operator structure corresponds to the logic of the «Not and» (NAND) type, «NXOR» i.e. denial of AND or not all events occur (Fig. 12). According to the formula of total probability and the multiplication theorem, the probability of the event AB (P(AB)) will be equal to: 𝑃(𝐴𝐵) = (1 − 𝑃(𝐴)) ∙ 𝑃(𝐵) + (𝑃(𝐴) ∙ (1 − 𝑃(𝐵))) − 𝑃(𝐴𝐵). (15) 9. Logical operator NXAND. In terms of fault trees, such a structure corresponds to the logic of the type «NON- EXCLUSIVE AND» (NXAND), i.e. only one event is realized Fig. 12. Example of a fault tree with the logical operator (negation of XAND) (Fig. 16). «NAND» According to the formula of total probability, the probability of the event 𝐴𝐵 (P(𝐴𝐵)) will be equal to: 𝑃(𝐴𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴𝐵). (12) 6. Logical operator XOR. In terms of fault trees, this structure corresponds to the logic of the type «OR only» (XOR), i.e. strictly one of the input events occurs (with the exception of the OR operator) (Fig. 13). Fig. 16. Example of a fault tree with the logical operator «NXAND» According to the formula of total probability and the multiplication theorem, the probability of the event AB (P(AB)) will be equal to: 𝑃(𝐴𝐵) = (1 − 𝑃(𝐴)) + (1 − 𝑃(𝐵)) ∙ (1 − 𝑃(𝐴)) + Fig. 13. Example of a fault tree with the logical operator +𝑃(𝐵)) ∙ (1 − 𝑃(𝐵)) + 𝑃(𝐴) − 𝑃(𝐴𝐵). (16) «XOR» 10. Logical operator NOT. NOT - the operator «NOT» (ie, the negation operator) (Fig. 17). According to the formula of total probability and the multiplication theorem, the probability of the event AB (P(AB)) will be equal to: 𝑃(𝐴𝐵) = 𝑃(𝐴) ∙ (1 − 𝑃(𝐵)) + (1 − 𝑃(𝐴)) ∙ 𝑃(𝐵) − −((1 − 𝑃(𝐴)) ∙ (1 − 𝑃(𝐵)). (13) 7. Logical operator XAND. In terms of fault trees, such a structure corresponds to the logic of the «And Only» (XAND) type, i.e. exactly one event does not occur (Fig. 14). Fig. 17. Example of a fault tree with the logical operator «NOT» 4. Calculation of unavailability indicators for systems of medium complexity An example of a fault tree is shown in Fig. 18. Comparative Fig. 14. Example of a fault tree with the logical operator results and calculation results are given in tables 2-4 and in fig. «XAND» 19-21. Fig. 18. Fault Tree Example Table. 4. Sequences. Event frequency Table. 2. Fault trees. Average unavailability rate Top event The number of RISK РИСК The number of RISK minimal cutset SPECTRUM Top event РИСК ALOCA-02 92 5.802E-007 5.811e-007 minimal cutset SPECTRUM SYS-DPS 52 1.364E-002 1.364e-002 ALOCA-03 82 6.098E-007 6.112e-007 SYS-ECCS 124 6.137E-003 6.161e-003 ALOCA-04 201 1.098E-008 1.092e-008 SYS-EFWS 125 6.898E-003 6.932e-003 F-EC001-01 1 1.000E-006 1.000E-006 SYS-MFWS 7 3.265E-002 3.265e-002 F-EC001-06 11 5.248E-010 5.248E-010 SYS-RHRS 147 5.843E-003 5.861e-003 F-EC001-08 0 0.000E+000 0.000E+000 F-EC001-09 0 0.000E+000 0.000E+000 F-EC001-10 3 4.434E-012 4.495E-012 F-RB001-06 26 3.378E-010 3.323E-010 F-RB001-08 0 0.000E+000 0.000E+000 F-RB001-09 0 0.000E+000 0.000E+000 F-RB001-10 0 0.000E+000 0.000E+000 F-RB002-06 13 2.440E-010 2.432E-010 F-RB002-09 0 0.000E+000 0.000E+000 F-RB002-10 0 0.000E+000 0.000E+000 TRANS-04 1776 9.761E-006 9.817E-006 TRANS-05 1654 1.014E-005 1.021E-005 TRANS-06 3017 1.825E-004 1.825E-004 Fig. 19. Fault trees. Average unavailability rate TRANS-08 4967 1.325E-007 1.304E-007 TRANS-09 5270 1.376E-007 1.355E-007 Table. 3. Consequences. Event frequency TRANS-10 1819 2.496E-006 2.496E-006 The number RISK Top event of minimal РИСК SPECTRUM cutset CD-ALOCA 199 1.196E-006 1.198E-006 CD-TRANS 141 2.021E-004 2.022E-004 CORE DAMAGE TOTAL 3480 2.033E-004 2.034E-004 CD-FIRES 3820 2.404E-009 2.305E-009 Fig. 21. Sequences. Event frequency 5. Results In this article, a comparative calculation of the probabilistic Fig. 20. Consequences. Event frequency analysis of safety and reliability using the RISK [7] and RISK- SPECTRUM [8]. The sets of minimum sections completely coincided for all fault trees. The probabilities of all the corresponding minimal cross sections also coincided. In accordance with [9], the error in the PSA calculation results does not exceed 0.15. 6. Conclusions To significantly reduce the calculation time of existing and developed codes and to increase the accuracy of probabilistic safety assessments, including when monitoring the safety of the current state of the power unit in real time (risk monitoring), it is necessary to develop methods and algorithms that accelerate the process of constructing a set of minimum sections for assessing the reliability and safety parameters of complex probabilistic models of nuclear power plants with a large number of fault trees. 7. Acknowledgments The study was carried out within the framework of grants 19-07- 00455, 20-07-00577 and 17-07-01475. 8. References: [1] The main recommendations for the development of a probabilistic safety analysis of level 1 for a nuclear power plant unit at initiating events caused by external influences of natural and technogenic origin. Safety Guide RB-021-14, Rostekhnadzor, 2014. [2] Recommendations on the procedure for performing the reliability analysis of systems and elements of nuclear plants important for safety and their functions. Safety Guide RB- 100-15, Rostekhnadzor, 2015. [3] General provisions for the safety of nuclear power plants. Federal norms and rules in the field of atomic energy use NP 001-15, Rostekhnadzor, 2015. [4] NUREG/CR-2300, «PRA Procedures Guide», US NRC, January 1983. [5] NUREG/CR-2815, «Probabilistic Safety Analysis Procedures Guide», US NRC, August 1985. [6] NUREG/CR-4550. Analysis of Core Damage Frequency from Internal Events: Methodology Guidelines. Volume 1. US NRC, September 1987. [7] Development of guidelines for the implementation of tasks within the PSA of levels 1 and 2 for all operational states and categories of initiating events of power units of RBMK- 1000 NPPs. Guidelines for the development of PSA-1. Guidelines for the development of a database for VAB-1 NPPs with RBMK-1000, taking into account data for equipment aging models. Guidelines for the analysis of personnel reliability. Guidelines for the analysis of the uncertainty, significance and sensitivity of the results of the PSA-1 nuclear power plant with RBMK-1000: research report reg. No. 1562MY09 / Dmitriev A.V., Golubkov A.V., Elizarov A.I., Berberova M.A., Derevyankin A.A. - M .: International Center for Nuclear Safety, 2009. - 287 p. [8] RiskSpectrum. [Electronic resource] – URL: http://www.riskspectrum.com/ [9] On ensuring the uniformity of measurements: [Federal Law No. 102-FZ dated 06/26/08: adopted by the State Duma on June 11, 2008]. - M., 2008 . - 16 p.