Reliability parameter processing method for PSA equipment of nuclear power plant
The reliability parameters of the PSA equipment of the nuclear power plant through the Bayesian method are selected and the Bayesian renewal is selected to solve the problem of errors and fitting when the reliability parameters of nuclear power plants are processing nuclear power plants. A more accurate and stable post -test data processing is implemented.
Patent Information
- Application Number
- CN202310919670.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2023-07-25
- Publication Date
- 2025-01-28
AI Technical Summary
The prior art is difficult to effectively handle the reliability parameters of PSA equipment in nuclear power plants, especially in sample data sets, where traditional methods have problems of error and poor fit.
Use the Bayesian method to process the reliability parameters of the nuclear power plant PSA device, and select the appropriate priority distribution (such as Gamma, Beta, Popularity and information without information), determine the like -minded function, and via Bayesh through Bayesh Update and calculate the distribution after the calculation, and then estimate the post -test parameters.
The specific data of general data and nuclear power plants are realized, and post -test data that are more adaptable to specific conditions with nuclear power plants are obtained, which improves the accuracy and stability of parameter processing.
Smart Images
Figure CN119379470A_ABST
Abstract
Description
Technical Field
[0001] The invention belongs to the field of nuclear power parameter processing, and in particular relates to a method for processing reliability parameters of PSA equipment in a nuclear power plant. Background Art
[0002] There are two commonly used estimation methods for uncertain reliability parameters: classical estimation and Bayesian estimation. Classical estimation regards the estimated parameter as an unknown constant and mainly relies on sampled power plant operation data. Bayesian estimation assumes that the uncertainty of the estimated parameter follows a certain distribution, and then uses a priori information and observed data to obtain posterior data.
[0003] Although the classical estimation is obtained from the operation data of the power plant for many years, it often cannot fully reflect the characteristics of the power plant; and the sample size of the Bayesian estimation data is small, and the estimated parameters are often different from the actual situation. Therefore, for nuclear power plants, the most reliable data processing method is to use general data as prior data and the historical data of specific power plants as sample data, and then perform Bayesian processing to obtain posterior data.
[0004] For the operation failure with Gamma distribution as prior distribution, the demand failure with Beta distribution as prior distribution, and the operation and demand failure with no information prior distribution, the current common practice is to directly perform Bayesian update, which is relatively simple to handle. The present invention describes its derivation process.
[0005] The traditional approach to operating and demand failures with lognormal as prior distribution is to convert them into Gamma or Beta distribution and then perform Bayesian update. Common transformation methods include EF value transformation method, 95% quantile transformation method and variance transformation method, which have been maturely applied in the engineering field. However, practice shows that this method will cause certain errors in the transformation process, so the method of Bayesian update using numerical integration is proposed. However, when the sample data of nuclear power plants are relatively concentrated, the existing methods are not well applicable to engineering practice.
[0006] Based on this, it is urgent to develop a reliability parameter processing method suitable for practical engineering fields. Summary of the invention
[0007] The object of the present invention is to provide a method for processing reliability parameters of PSA equipment in a nuclear power plant. The method solves the reliability parameter processing problems of operation failure with Gamma distribution as prior distribution, demand failure with Beta distribution as prior distribution, operation and demand failure with lognormal distribution as prior distribution and uninformative prior distribution through Bayesian method.
[0008] The technical solution to achieve the purpose of the present invention is:
[0009] A method for processing reliability parameters of PSA equipment in a nuclear power plant, the method comprising:
[0010] Step 1: Select the prior distribution;
[0011] Step 2: Determine the likelihood function;
[0012] Step 3: Determine the posterior distribution of the parameters;
[0013] Step 4: Estimate the posterior parameters.
[0014] In step 1, the prior distribution includes: for operational failure, the prior distribution types include Gamma, lognormal, and uninformative prior distribution; for demand failure, the prior distribution types include Beta, lognormal, and uninformative prior distribution.
[0015] In step 2, for operational failure, the likelihood function of the prior distribution of Gamma distribution is Poisson distribution, and the likelihood function of the prior distribution of lognormal distribution is Poisson distribution; for demand failure, the likelihood function of the prior distribution of Beta distribution is Binomial distribution, and the likelihood function of the prior distribution of lognormal distribution is Binomial distribution.
[0016] The posterior distribution function of the parameters in step 3 is:
[0017]
[0018] Where x is the sample, θ is the parameter, p(θ|x) is the posterior distribution function, f(θ) is the prior distribution function, and L(x|θ) is the likelihood function.
[0019] In step 3, for the failure of operation whose prior distribution is Gamma distribution, the posterior distribution is Gamma(α+x, β+t), where
[0020] α: called shape parameter, dimensionless;
[0021] β: called the scale parameter, the unit is time;
[0022] x: the cumulative number of operation failures during the operation of the equipment;
[0023] t: The cumulative number of operating hours during which the equipment is in operation.
[0024] In step 3, for demand failures whose prior distribution is Beta distribution, the posterior distribution is Beta(α+x, β+nx), where
[0025] α: called shape parameter, dimensionless;
[0026] β: called the scale parameter, the unit is time;
[0027] x: cumulative number of demand failures;
[0028] n: cumulative number of requests.
[0029] In step 3, the prior distribution is a lognormal distribution, and the posterior distribution is in,
[0030] θ: random variable;
[0031] μ post , σ post : are the lognormal distribution parameters of the posterior distribution.
[0032] In step 3, the requirement that the prior distribution is a lognormal distribution fails, and the posterior distribution is in,
[0033] θ: random variable;
[0034] μ post , σ post : are the lognormal distribution parameters of the posterior distribution.
[0035] The step 4 is specifically as follows: selecting N samples of the posterior distribution, calculating the sample mean and the sample variance, and solving the parameters of the lognormal distribution according to the lognormal distribution parameter estimation method.
[0036] The beneficial technical effects of the present invention are:
[0037] 1. The method for processing reliability parameters of PSA equipment in a nuclear power plant provided by the present invention obtains a posteriori data that is more suitable for the specific conditions of the nuclear power plant by applying the Bayesian method and combining general data with specific data of the nuclear power plant.
[0038] 2. A method for processing reliability parameters of a nuclear power plant PSA equipment provided by the present invention adopts a multi-sample sampling method to fit a posteriori parameters, with a high and stable fitting degree, thereby ensuring the accuracy of the posteriori data. BRIEF DESCRIPTION OF THE DRAWINGS
[0039] Figure 1 It is a Bayesian processing process diagram of the operation failure reliability parameter whose prior distribution is Gamma distribution in a specific implementation manner of the present invention;
[0040] Figure 2 It is a Bayesian processing process diagram of demand failure reliability parameters whose prior distribution is Beta distribution in a specific implementation manner of the present invention;
[0041] Figure 3 Graph showing the Bayesian processing of operational and demand failure reliability parameters whose prior distribution is a log-normal distribution in a specific implementation manner of the present invention. DETAILED DESCRIPTION
[0042] The present invention is further described in detail below with reference to the accompanying drawings and embodiments.
[0043] The present invention provides a method for processing reliability parameters of a nuclear power plant PSA equipment, which specifically comprises the following steps:
[0044] Step 1: Choose a prior distribution
[0045] For operational failures, the prior distribution types include Gamma, lognormal, and uninformative prior distribution; for demand failures, the prior distribution types include Beta, lognormal, and uninformative prior distribution.
[0046] Step 2: Determine the likelihood function
[0047] For operational failures, the likelihood function when the prior distribution is the Gamma distribution is the Poisson distribution, and the likelihood function when the prior distribution is the lognormal distribution is the Poisson distribution; for demand failures, the likelihood function when the prior distribution is the Beta distribution is the Binomial distribution, and the likelihood function when the prior distribution is the lognormal distribution is the Binomial distribution.
[0048] Step 3: Determine the posterior distribution of the parameters
[0049] The joint distribution of sample x and parameter θ is:
[0050] C(x,θ)=f(θ)L(x|θ)
[0051] The marginal density function of x is:
[0052] Z(x)=∫f(θ)L(x|θ)dθ
[0053] According to the Bayesian theory formula Assume A1,…,A k are mutually incompatible events. A contains event B, then:
[0054]
[0055] a) The prior distribution of θ is f(θ) and the posterior distribution is p(θ|x), where x is the sample after sampling.
[0056] b) Joint density function of the prior distribution of sample x and parameters:
[0057] C(x1,…,x n ,θ)=L(x1,…,x n |θ)f(θ)
[0058] c) In the joint density function, given samples x1,…,x n, θ is an unknown parameter, and the posterior distribution of θ can be estimated through samples and prior distribution. Event B is equivalent to x1,…,x n The distribution of , through integration, the distribution function of the posterior distribution is:
[0059]
[0060] Step 4: Estimate posterior parameters
[0061] According to the posterior distribution of the parameter, the Bayesian estimate of the parameter is obtained by fitting method.
[0062] Select N sample values of the posterior distribution, calculate their sample means and sample variances, and solve the parameters of the lognormal distribution according to the lognormal distribution parameter estimation method.
[0063] The lognormal distribution density function is:
[0064] Here, μ and σ are the mean and standard deviation respectively.
[0065] expect
[0066] variance
[0067] The parameters μ and σ of the lognormal distribution can be found.
[0068] According to different prior distribution types, the present invention provides a method for processing reliability parameters of PSA equipment in a nuclear power plant, which is specifically as follows:
[0069] (1) For operational failures whose prior distribution is the Gamma distribution and demand failures whose prior distribution is the Beta distribution, the posterior distribution of the parameters is consistent with the prior distribution. The Bayesian processing process is as follows:
[0070] a) Gamma distribution
[0071]
[0072] Gamma prior β prior ) distribution probability density function is:
[0073]
[0074] The Poisson distribution describing the sample information is:
[0075] L(x|λ)=e -λt (λ t ) x / x!
[0076] The posterior distribution of the estimated parameter λ after Bayes update is:
[0077]
[0078]
[0079] The formula simplifies to:
[0080]
[0081] Right now,
[0082] Referring to the calculation process of the lognormal distribution above, the marginal density function of the Gamma distribution x is:
[0083]
[0084] Comparing the posterior distribution p(λ|x) with the probability density function of the Gamma distribution, we can obtain:
[0085] The parameters of the posterior distribution are:
[0086] α post =α prior +x,β post =β prior +t
[0087] Therefore, the posterior distribution of λ is Gamma(α+x, β+t).
[0088] in,
[0089] α: called shape parameter, dimensionless;
[0090] β: called the scale parameter, the unit is time;
[0091] x: the cumulative number of operating failures during the operation of the equipment; t: the cumulative number of operating hours during the operation of the equipment;
[0092] α post , β post : posterior parameters α, β;
[0093] α prior , β prior : Prior parameters α, β.
[0094] b) Beta distribution
[0095]
[0096] The probability density function of the Beta distribution is:
[0097]
[0098] The binomial distribution describing the sample information is:
[0099]
[0100] The posterior distribution of the estimated parameter λ after Bayes update is:
[0101] The formula simplifies to:
[0102]
[0103] Right now
[0104] Referring to the calculation process of the lognormal distribution above, the marginal density function of the Beta distribution x is:
[0105]
[0106] Comparing the posterior distribution p(λ|x) with the probability density function of the Beta distribution, we can obtain:
[0107] The parameters of the posterior distribution are:
[0108] α post =α prior +x,β post =β prior +(nx)
[0109] Therefore, the posterior distribution of λ is Beta(α+x, β+nx).
[0110] in,
[0111] α: called shape parameter, dimensionless;
[0112] β: called the scale parameter, the unit is time;
[0113] x: cumulative number of demand failures;
[0114] n: cumulative number of requests;
[0115] nx: cumulative number of successes;
[0116] λ: failure rate;
[0117] α post , β post : posterior parameters α, β;
[0118] α prior , β prior : Prior parameters α, β.
[0119] (2) For the operation and demand failure of the log-normal distribution as the prior distribution, the Bayesian update is mainly performed through the numerical integration method. The process is as follows:
[0120] The probability density function of the lognormal is:
[0121]
[0122] in,
[0123] λ: failure rate, parameter to be estimated
[0124] μ: hyperparameter in the prior distribution;
[0125] σ: A hyperparameter in the prior distribution.
[0126] a) Operational failure
[0127] The prior distribution is the running failure sample information of log-normal distribution, which obeys Poisson distribution:
[0128] L(x|λ)=e -λt (λt) x / x!
[0129] in,
[0130] x: number of failures in the sample;
[0131] t: Cumulative running time in the sample.
[0132] The posterior distribution of the estimated parameter λ after Bayes update is:
[0133]
[0134] The formula simplifies to:
[0135]
[0136] Right now
[0137] The marginal density function of x is:
[0138]
[0139] The parameters of the posterior distribution are:
[0140]
[0141] δ post =ln(1+D(x) / E(x)) 2 )
[0142] in,
[0143] E(x) is the expected value;
[0144] D(x) is the variance value.
[0145] The posterior distribution of λ is:
[0146]
[0147] b) Demand failure
[0148] The prior distribution is the demand failure sample information of log-normal distribution, which obeys binomial distribution.
[0149]
[0150] in,
[0151] x: cumulative number of demand failures;
[0152] n: cumulative number of requests;
[0153] nx: cumulative number of successes;
[0154] λ: failure rate.
[0155] The posterior distribution of the estimated parameter λ after Bayes update is:
[0156]
[0157] The formula simplifies to:
[0158]
[0159] Right now
[0160] The marginal density function of x is:
[0161]
[0162] The parameters of the posterior distribution are:
[0163]
[0164] δ post =ln(1+D(x) / (E(x)) 2 )
[0165] in,
[0166] E(x) is the expected value;
[0167] D(x) is the variance value.
[0168] The posterior distribution of λ is:
[0169]
[0170] For operational failures and demand failures whose prior distribution is lognormal distribution, the Bayesian solution process of the posterior distribution is to calculate the marginal density function Z(x) after Bayesian update by using the trapezoidal integration method with adjustable step size and adjustable upper and lower limits of integration. By using the numerical calculation method, the integration interval is divided into several parts, and a set of maximum value data (λ) of the true posterior distribution is found by recording the maximum value in the integration interval. i ,y i ), j groups of data smaller than the maximum value and k groups of true posterior data larger than the maximum value.
[0171] Through the multi-sample sampling estimation method of j+1+k groups of true posterior distribution data, the fitted posterior distribution parameter μ is obtained post and σ post .
[0172] The lognormal distribution density function is:
[0173] Here, μ and σ are the mean and standard deviation respectively.
[0174] expect
[0175] variance
[0176] The parameter μ of the lognormal distribution can be obtained post and σ post .
[0177] The posterior distribution is:
[0178] Goodness of fit
[0179] The closer the R value is to 1, the better the fit between the fitted posterior distribution and the true posterior distribution.
[0180] (3) For the operational failure of the uninformative prior distribution, according to the characteristics of Jeffreys uninformative prior distribution, the uninformative prior distribution Gamma (1 / 2, 0) based on Jeffreys is selected, that is, α prior =1 / 2,β prior =0, according to the above derivation of Gamma distribution, the corresponding posterior parameter is: α post =1 / 2+x,β post =t, so the posterior distribution is Gamma(x+1 / 2, t).
[0181] For the failure of the requirement of uninformative prior distribution, according to the characteristics of Jeffreys uninformative prior distribution, we select the uninformative prior distribution Beta(1 / 2, 1 / 2) based on Jeffreys, that is, α prior =1 / 2,β prior =1 / 2, according to the above derivation of Beta distribution, the corresponding posterior parameter is: α post =1 / 2+x,β post =n-x+1 / 2, so the posterior distribution is Beta(x+1 / 2,n-x+1 / 2).
[0182] Example 1: Taking the operation failure of a certain electric pump (MDP) equipment as an example, its prior distribution type is log-normal distribution (LN), parameter mean = 3.06E-6, error factor EF = 10, cumulative operation time t = 2759400h, and number of failures x = 0.
[0183] After the Bayesian method is updated in this paper, the calculation of the method is realized by programming. By inputting the corresponding electric pump operating parameters and prior distribution parameters, the posterior mean = 3.07E-7, the error factor EF = 3.46, R = 0.64 can be obtained, and the posterior distribution type is lognormal distribution.
[0184] The calculation of the degree of fit is based on the following formula:
[0185]
[0186] The Bayesian method used in this paper is compared with the moment matching method and the error factor transformation method. The comparison results are shown in Table 1.
[0187] Table 1 Fitting results of different methods
[0188] method Method of the present invention Moment Matching Method Error Factor Conversion Method Goodness of fit 0.64 -11.67 0.12
[0189] The results in Table 1 show that the method used in this paper has a higher degree of fit than the moment matching method and the error factor transformation method, and the calculation results are more accurate and the error is smaller.
[0190] The present invention is described in detail above with reference to the accompanying drawings and embodiments, but the present invention is not limited to the above embodiments, and various changes can be made within the knowledge of ordinary technicians in the field without departing from the purpose of the present invention. The contents not described in detail in the present invention can adopt the existing technology.
Claims
1. A method for processing reliability parameters of PSA equipment in a nuclear power plant, characterized in that: The method comprises: Step 1: Select the prior distribution; Step 2: Determine the likelihood function; Step 3: Determine the posterior distribution of the parameters; Step 4: Estimate the posterior parameters.
2. A method for processing reliability parameters of PSA equipment in a nuclear power plant according to claim 1, characterized in that: In step 1, the prior distribution includes: for operational failure, the prior distribution types include Gamma, lognormal, and uninformative prior distribution; for demand failure, the prior distribution types include Beta, lognormal, and uninformative prior distribution.
3. A method for processing reliability parameters of PSA equipment in a nuclear power plant according to claim 1, characterized in that: In step 2, for operational failure, the likelihood function of the prior distribution of Gamma distribution is Poisson distribution, and the likelihood function of the prior distribution of lognormal distribution is Poisson distribution; for demand failure, the likelihood function of the prior distribution of Beta distribution is Binomial distribution, and the likelihood function of the prior distribution of lognormal distribution is Binomial distribution.
4. A method for processing reliability parameters of PSA equipment in a nuclear power plant according to claim 1, characterized in that: The posterior distribution function of the parameters in step 3 is: Where x is the sample, θ is the parameter, p(θ|x) is the posterior distribution function, f(θ) is the prior distribution function, and L(x|θ) is the likelihood function.
5. A method for processing reliability parameters of PSA equipment in a nuclear power plant according to claim 4, characterized in that: In step 3, for the failure of operation whose prior distribution is Gamma distribution, the posterior distribution is Gamma(α+x, β+t), where α: called shape parameter, dimensionless; β: called the scale parameter, the unit is time; x: the cumulative number of operation failures during the operation of the equipment; t: The cumulative number of operating hours during which the equipment is in operation.
6. A method for processing reliability parameters of PSA equipment in a nuclear power plant according to claim 4, characterized in that: In step 3, for demand failures whose prior distribution is Beta distribution, the posterior distribution is Beta(α+x, β+nx), where α: called shape parameter, dimensionless; β: called the scale parameter, the unit is time; x: cumulative number of demand failures; n: cumulative number of requests.
7. A method for processing reliability parameters of PSA equipment in a nuclear power plant according to claim 4, characterized in that: In step 3, the prior distribution is a lognormal distribution, and the posterior distribution is in, θ: random variable; μ post , σ post : are the lognormal distribution parameters of the posterior distribution.
8. A method for processing reliability parameters of PSA equipment in a nuclear power plant according to claim 4, characterized in that: In step 3, the requirement that the prior distribution is a lognormal distribution fails, and the posterior distribution is in, θ: random variable; μ post , σ post : are the lognormal distribution parameters of the posterior distribution.
9. A method for processing reliability parameters of PSA equipment in a nuclear power plant according to claim 1, characterized in that: The step 4 is specifically as follows: selecting N samples of the posterior distribution, calculating the sample mean and the sample variance, and solving the parameters of the lognormal distribution according to the lognormal distribution parameter estimation method.