=Paper=
{{Paper
|id=Vol-2485/paper73
|storemode=property
|title=Calculation of the Probabilistic Safety Analysis and Reliability by the Fault Trees and Event Trees Methods
|pdfUrl=https://ceur-ws.org/Vol-2485/paper73.pdf
|volume=Vol-2485
|authors=Maria Berberova,Aleksandr Dmitriev,Aleksandr Golubkov,Aleksandr Elizarov
}}
==Calculation of the Probabilistic Safety Analysis and Reliability by the Fault Trees and Event Trees Methods==
https://ceur-ws.org/Vol-2485/paper73.pdf
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.