CN107967545B - Method for calculating Birnbaum importance and structure importance of subsystem component in probability safety evaluation - Google Patents
Method for calculating Birnbaum importance and structure importance of subsystem component in probability safety evaluation Download PDFInfo
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Abstract
The invention belongs to the field of probabilistic security evaluation (PSA), and particularly relates to a method for calculating Birnbaum importance and structural importance of a subsystem component in PSA. The Birnbaum importance and the structural importance of each component in the object subsystem to the system are obtained through the Birnbaum importance and the structural importance of the object subsystem to the system and the Birnbaum importance and the structural importance of each component in the object subsystem. The method reveals an in-depth rule of the Birnbaum importance and the structure importance, and provides the Birnbaum importance and the structure importance information of the required specific parts for the importance analysis of PSA.
Description
Technical Field
The invention belongs to the technical field of probability safety evaluation, and particularly relates to a method for calculating Birnbaum importance and structure importance of a subsystem component in probability safety evaluation.
Background
The Importance (Importance) is defined as: the conditional probability of a top event occurring under a certain condition is a function of time, component reliability parameters, and system architecture.
The importance analysis is an indispensable analysis item in probabilistic security evaluation (PSA), and possible uses of importance are as follows: improving the system design; determining a part needing to be monitored during system operation; and establishing a check list for system fault diagnosis.
Birnbaum importance and structural importance are two of the generally required importance.
The Birnbaum importance is defined as: the change in system probability is caused by the sensitivity of system inefficiency (unreliability) to changes in component failure probability or by 1 unit change in the probability of component i. Corresponding to the two definitions, the mathematical definitions are shown in formula (1) and formula (2), respectively.
In the formula (I), the compound is shown in the specification,
phi is a structure function;
p (phi) is the probability of occurrence (time t);
q is the probability of a fundamental event;
x is a basic event;
BM is the importance of Birnbaum;
the upper label of BM is a component i of the importance of the considered Birnbaum;
the subscript of BM is a system with relative importance of the considered Birnbaum;
sys is the system.
The mathematical definition of the structural importance is shown in formula (3):
in the formula (I), the compound is shown in the specification,
str is the structural importance.
Further, since the mathematical expression of the structural importance may be formula (4):
in the formula (I), the compound is shown in the specification,
j is a part number different from part i;
other symbols have the same meaning as before.
From the above, the structure importance can be regarded as a special case of the Birnbaum importance.
A typical engineering system may be divided into subsystems, which may be further divided into subsystems consisting of components or equipment, and in this case there are a total of four levels from system to component or equipment (component or equipment belongs to the first level, and the system is the highest level). The number of layers from system to component or device can be defined according to actual needs. When PSA is carried out on the system, the splitting idea is also adopted. However, the following problems arise: while the Birnbaum importance and structural importance of the component in the subsystem and the Birnbaum importance and structural importance of the subsystem in the system are obtained, the Birnbaum importance and structural importance of the component in the system is unknown, which is exactly what is needed in PSA applications.
Disclosure of Invention
The invention aims to provide a novel calculation method aiming at the defects of a method for calculating the Birnbaum importance and the structure importance of a subsystem component in probability safety evaluation. The method can fully utilize the known information of the relevant importance, can save a large amount of computing resources, and can be used for computing the Birnbaum importance and the structural importance when subsystems exist in probability safety evaluation.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a method for calculating Birnbaum importance and structure importance of a subsystem component in probabilistic security evaluation is disclosed, wherein the Birnbaum importance of a component i in M subsystems containing the component i and the Birnbaum importance of the subsystems to a system are known, and then the Birnbaum importance of the component i to the system is calculated according to the following formula (5):
in the formula (I), the compound is shown in the specification,
BM is the importance of Birnbaum;
the upper label of BM is a component i or a subsystem sub of the importance of the considered Birnbaum;
the subscript of BM is a subsystem sub or a system sys with relative importance of the considered Birnbaum;
sys is a system;
sub is a subsystem, M is the number of the subsystems containing the component i, and the number of the subsystems is M;
knowing the structural importance of component i in the M subsystems containing component i, and the structural importance of the subsystems to the system, the structural importance of component i to the system is calculated by the following equation (6):
in the formula (I), the compound is shown in the specification,
str is the structural importance;
str is labeled as component i or subsystem sub of structural importance under consideration;
the Str subscripts the subsystem sub or system sys with the relative structural importance of interest.
Further, a system for probability safety evaluation analysis aimed at by the method is a two-state system, and a system analysis method for probability safety evaluation can be a fault tree analysis method or a GO method.
The invention has the following beneficial effects: the invention provides a method for calculating the two importance degrees by using the known information of the related importance degrees aiming at the problems that the known Birnbaum importance degree and the structural importance degree of a component in a subsystem, the Birnbaum importance degree and the structural importance degree of the subsystem in the system and the Birnbaum importance degree and the structural importance degree of the component in the system are unknown. Compared with the method for directly utilizing the definition to solve the two importance degrees of the components in the subsystem, the method can fully utilize the known information and greatly reduce the waste of computing resources.
Detailed Description
The invention provides a method for calculating the Birnbaum importance and the structure importance of a subsystem component in probability safety evaluation, which is based on the Birnbaum importance of a known component i in M subsystems containing the component i and the Birnbaum importance of the subsystems to the system; the structural importance of component i in the M subsystems containing component i, and the structural importance of the subsystems to the system, are known.
Then the Birnbaum importance of component i to the system is calculated by the following equation (5):
in the formula (I), the compound is shown in the specification,
BM is the importance of Birnbaum;
the upper label of BM is a component i or a subsystem sub of the importance of the considered Birnbaum;
the subscript of BM is a subsystem sub or a system sys with relative importance of the considered Birnbaum;
sys is a system;
sub is a subsystem, and M is the number of the subsystems containing the component i, and the number of the subsystems is M.
This is well understood in the physical sense of the importance of Birnbaum: assuming that component i is present in only one subsystem, the conclusion is clear; assuming that component i appears in M subsystems containing component i, it is obvious that the Birnbaum importance of component i to each subsystem M and the Birnbaum importance of subsystem M to the system are related, and the definition of Birnbaum importance can only be shown by summing up as in equation (5).
It is obvious that the same holds true for structural importance for formula (5), namely for formula (6):
examples
Taking 2/3 system (the system has three components, if two of them are normal, the system is normal) as an example, specifically explaining the implementation process of equation (5), equation (7) is a calculation expression of the occurrence probability of 2/3 system.
Here, X is1X2As a structural function of sub-subsystem sub 1; handle X1X3As a structural function of the sub-subsystem 2, the component to be considered is component 1, and first, the Birnbaum importance of the component 1 in the system is directly calculated, according to equation (1), there are:
secondly, according to the formula (5), the Birnbaum importance in the subsystem is calculated respectively. Consider that in sub1, there are:
then, consider that component 1 is in sub 2:
adding formula (11) and formula (14):
(11)+(14)＝Q2+Q3-2Q2Q3 (15)
comparing the results of equations (8) and (15), it was found that they were equal. This also proves that the Birnbaum importance of the component i to the system calculated by equation (5) is consistent with the result calculated by the definitional equation (1). This example is both a concrete embodiment and a proof of the advantage of the proposed inventive method.
Because the structural importance can be regarded as a specific example of the importance of Birnbaum, the implementation process and the proof of the formula (5) are also applicable to the formula (6), and are not described again.
It will be apparent to those skilled in the art that various changes and modifications may be made in the present invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is intended to include such modifications and variations.
Claims (3)
1. A Birnbaum importance and structure importance calculation method of a subsystem component in probability safety evaluation divides an engineering system into subsystems, the subsystems are divided into the subsystems, the subsystems are composed of components, and the method is characterized in that: knowing the Birnbaum importance of the component i in M subsystems containing the component i and the Birnbaum importance of the subsystems to the engineering system, the Birnbaum importance of the component i to the engineering system is calculated by the following formula:
in the formula (I), the compound is shown in the specification,
BM is the importance of Birnbaum;
the upper label of BM is a component i or a subsystem sub of the importance of the considered Birnbaum;
the subscript of BM is a subsystem sub or an engineering system sys with relative importance of the considered Birnbaum;
sys is an engineering system;
sub is a subsystem, M is the number of the subsystems containing the component i, and the number of the subsystems is M;
knowing the structural importance of the component i in the M subsystems containing the component i and the structural importance of the subsystems to the engineering system, the structural importance of the component i to the engineering system is calculated by the following formula:
in the formula (I), the compound is shown in the specification,
str is the structural importance;
str is labeled as component i or subsystem sub of structural importance under consideration;
the Str subscripts the subsystem sub or the engineering system sys with the relative structural importance under consideration.
2. The method for calculating Birnbaum importance and structural importance of a subsystem component in probabilistic security evaluation as recited in claim 1, wherein: the engineering system is a two-state system.
3. The method for calculating Birnbaum importance and structural importance of a subsystem component in probabilistic security evaluation according to claim 1 or 2, wherein: the system analysis method for probability safety evaluation is a fault tree analysis method or a GO method.
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