CN106650204A - Product failure behavior coupling modeling and reliability evaluation method - Google Patents

Abstract

A product failure behavior coupling modeling and reliability evaluation method, that is, a product failure behavior coupling modeling and reliability evaluation method based on Vine-Copula and accelerated degradation tests, the steps are as follows: 1: Degradation trajectory modeling; 2 : Failure mechanism consistency test; 3: Extrapolation of the distribution model of performance parameters under normal stress; 4: Coupling modeling of product failure behavior based on Vine-Copula; 5: Reliability evaluation; through the above steps, the failure of the product is completed Behavioral coupling modeling establishes a model of joint distribution of reliability, solves the problem of solving the reliability function by direct integral method, clearly describes the coupling relationship of multiple variables of products, and provides a method for reliability evaluation of products with multi-performance parameter related characteristics.

Description

A Product Failure Behavior Coupling Modeling and Reliability Evaluation Method

technical field

The present invention proposes a product failure behavior coupling modeling and reliability assessment method, in particular to a product failure behavior coupling modeling and reliability assessment method based on Vine-Copula and accelerated degradation tests , which belongs to fault behavior modeling in the field of reliability technology.

Background technique

The safe service of aerospace, aviation, nuclear equipment, power network and other major equipment is of great significance to national economic development and national defense construction, but its operating conditions are complex and the environment is harsh, which can easily lead to malignant accidents. Therefore, a correct understanding of faults, assessment and prediction Its reliability is of great significance to ensure the safe operation of equipment and improve economic benefits. Due to the improvement of system complexity and the in-depth understanding of faults, the research on reliability theory has gradually shifted from "failure time" to "failure process". On the one hand, the research based on the fault process is to study the fault mechanism from a microscopic point of view, that is, fault physics. On the other hand, from a macro point of view, study the occurrence and development of faults and their propagation in the system, that is, the behavior of faults. The product failure behavior coupling modeling and reliability evaluation method based on Vine-Copula and accelerated degradation test is a method for modeling and reliability evaluation of product failure behavior with multiple variables. Through the novelty search of existing technologies, there is no research on the coupling modeling of fault behavior and reliability evaluation of products based on Vine-Copula and accelerated degradation tests at home and abroad.

Contents of the invention

Aiming at the characteristics of high-reliability and long-life complex products having failure mechanism coupling, in terms of constructing a macroscopic failure behavior model for reliability evaluation, a multi-variable product failure behavior modeling and reliability based on Vine-Copula and accelerated degradation tests are proposed. gender assessment method. On the one hand, this method can clearly describe the multivariate variable coupling relationship of the product, and on the other hand, it can evaluate the product reliability within an acceptable time and cost.

In the present invention, before performing product fault behavior coupling modeling and reliability evaluation, the fault behavior description method of the system constituting the product needs to be clarified, and the characterization quantity describing the system fault behavior must be selected. Since the interaction of failure mechanisms among the components that make up the system will be manifested in the state of the components, an available method is to use the state vector Z(t) of the components as a representation to describe the fault behavior of the system and construct a fault behavior coupling model. So as to estimate the reliability of the system. Among them, Z(t)=(D(t), X(t), F(t)) ^T , D(t) is the component damage state variable, X(t) is the component response variable, F(t) is the component The stress variable subjected to. For the system, the mechanism competition or coupling problem among the components or within the components can use the damage state variable in the state vector of each component to represent its degeneration and damage status to build a product failure behavior coupling model; and from a practical point of view, the damage The state variables are generally reflected from the monitoring parameters of the product. Therefore, the present invention starts with the test data of performance parameters at various levels under the accelerated degradation test, and performs modeling and analysis. The specific implementation process is shown in Figure 1.

The present invention is realized by adopting the following technical solutions. The present invention is a product failure behavior coupling modeling and reliability evaluation method, that is, a product failure behavior coupling modeling and reliability evaluation method based on Vine-Copula and accelerated degradation tests, The steps are as follows:

Step 1: Degradation trajectory modeling

The degradation trajectory model of the product can be expressed as a function of the amount of degradation, time and environmental stress; when modeling the degradation trajectory, it is necessary

1. Analyze the characteristics of degradation data, and establish a degradation trajectory model Δy _t = f(t,S), Δy _t represents the amount of degradation at time t, and S represents the level of environmental stress received;

2. Establish the likelihood function f _i represents the probability density function of the i-th degradation test point, and μ, σ represent the parameters of the degradation trajectory model;

3. Substitute the degradation data, use the Nelder-Mead simplex method to solve the maximum value problem of the likelihood function, and obtain the parameter estimation of the degradation trajectory model of each performance parameter under each stress level;

Step 2: Failure Mechanism Consistency Test

In the accelerated degradation test, in order to ensure the validity of the test, the same performance parameter under different accelerated stress levels should comply with the failure mechanism consistency test. In the degradation trajectory modeling, the relationship between the model parameters and the failure mechanism is different depending on the degradation trajectory modeling method. Take the degradation trajectory modeling subject to drifting Brownian motion as an example:

1. Determine the consistency test formula of its acceleration mechanism Among them, μ _i and σ _i are the drift coefficient and diffusion coefficient of the degradation trajectory model under the stress level S _i , respectively;

2. Construct hypothesis test statistics T is from a t distribution with n+m-2 degrees of freedom; where: X obeys a normal distribution with a mean of μ ₁ and a variance of σ ² , Y follows a normal distribution with a mean of μ ₂ and a variance of σ ² , and X ₁ , X ₂ ,...X _n are samples from population X, Y ₁ , Y ₂ ,...Y _m are samples from population Y, and n and m are sample size respectively;

3. Determine the deny domain. Given the significance level α, substitute the data to calculate |T|, and look up the t distribution table to know: the size of t _1-α/2 (n+m-2), if |T|≥t _1-α/2 (n+ m-2), the test results fall within the rejection region, and the failure mechanisms of the parameters under different stresses are not consistent. Otherwise, the consistency test is satisfied;

Step 3: Extrapolating the distribution model of performance parameters under normal stress

The relationship between the product life characteristics and the accelerated stress level can be expressed by the accelerated model, and the degradation model and edge distribution of each variable under normal stress can be deduced by using the accelerated model; in the present invention, the simplified multi-stress Irene accelerated model is used: t is the characteristic life, T and RH represent the temperature and humidity variables respectively, and A, B and C are the parameters to be obtained;

1. Determine the acceleration factor

Let AF _i,0 =t ₀ /t _i be the acceleration factor of the accelerated stress level S _i relative to the normal stress level S ₀ , t _i and t ₀ represent the products under the accelerated stress level S _i and the normal stress level S ₀ respectively, The time required for performance parameters to reach the same amount of degradation;

2. Solving acceleration model parameters

a. Construct the relationship equation between the acceleration factor and the life, drift coefficient and diffusion coefficient between different stresses Among them, AF represents the acceleration factor, L represents the life, μ _i and σ _i represent the drift coefficient and diffusion coefficient of the performance parameter degradation trajectory under the accelerated stress S _i ; the subscript h represents a higher stress level, and the subscript l represents a lower under the stress level;

b. For the performance parameter p, under the k ₁ and k ₂ stress levels, construct the sum of squared deviation function The Nelder-Mead simplex method is used to solve the minimum value problem of the sum of squared deviation function, and the optimal model parameters B and C in the accelerated model are obtained;

3. Estimate the parameters of the degradation trajectory model under normal stress

After obtaining the parameters in the acceleration model, use the least squares method to estimate the parameters of the degradation trajectory model under normal stress, and obtain the deviation sum of squares function and The simplex method is used to solve the parameter values corresponding to the minimum sum of squared deviations, so as to obtain the point estimates of the degradation trajectory model parameters μ _p0 and σ _p0 under normal stress; the final degradation trajectory model under normal stress is: y _p0 (t) = μ _p0 t+σ _p0 B(t);

Step 4: Coupling modeling of product failure behavior based on Vine-Copula

The copula function is a kind of "connection" function, which connects the marginal distribution and the joint distribution. The theoretical basis of Vine-Copula is conditional probability, which is modeled by decomposing the joint distribution into multiple binary copulas and the corresponding multiplication of conditional probabilities. The main process of product failure behavior coupling modeling based on Vine-Copula is as follows:

1. Correlation analysis of performance parameters

Before the coupling modeling of product failure behavior, correlation analysis should be carried out for the degradation of multiple performance parameters of the product; Kendall's τ coefficient is used as the correlation measurement index, and the closer the absolute value of the index is to 1, the stronger the correlation is. And Kendall'sτ coefficient indicates the degree of nonlinear correlation;

2. Use the vine graph model to decompose the joint distribution into a form of multiplication of appropriate conditional probabilities, and use the copula function to construct a multivariate correlation model; specifically,

a. Modeling method based on D-Vine

1) First of all, in the first layer Vine diagram model, each variable is used as a node in the tree diagram, and the nodes are connected by short lines according to their mutual relationship. For example, the first layer relationship of a four-dimensional D-vine is shown in the attached drawing 2. Let c _ij represent the probability density of the copula function connecting node i and node j, then c _ij =c _pq , where p and q are the variables contained in node i and node j respectively.

2) The second-level tree diagram in the Vine diagram is established on the basis of the first-level tree diagram, and the connections in the first-level tree diagram are used as nodes in the second-level tree diagram, and the adjacent connections in the first layer The element formed in the second layer is connected with a connecting line, and the tree diagram of the second layer is constructed. For example, accompanying drawing 3 is based on (1,2), (2,3), (3,4) in accompanying drawing 2 element to build a second-level tree diagram. Still use c _ij to represent the probability density of the copula function connecting node i and node j, c _ij =c _pq|k , {p,k}, {q,k} are the variables contained in node i and node j respectively, where k is a variable common to both nodes.

3) The tree diagram of the third layer is also established on the basis of the tree diagram of the previous layer by a similar method, and so on, when there is only one pair of related relations left in the tree diagram, that is, when there is only one connection line left, it is the last one. layer tree diagram. Similarly, the probability density of the copula function between the third layer treemap and all connected nodes including the last layer treemap has c _ij =c _pq|k , {p,k}, {q,k} is the variable contained in node i and node j, where k is a variable shared by the two nodes.

4) Multiply the copula function probability densities corresponding to the lines in the Vine diagram to obtain the decomposition structure of the copula function based on the Vine diagram. c=∏c _ij . The probability density function of the joint distribution of product reliability can be expressed as x _i represents the i-th performance parameter of the product, r _i represents the reliability probability density function corresponding to the i-th parameter, and the Vine-Copula coupling model has been established so far.

b. Solve the Vine-Copula coupling model

To solve the Vine-Copula coupling model, that is, the probability density of joint reliability distribution, it is necessary to first determine the form and parameters of each copula probability density function in the joint reliability distribution probability density r(x ₁ ,...,x _n ), that is, c _ij form and parameters.

1) Determine c _ij in the tree diagram of the first layer in the Vine model: Substituting the degradation data of the performance parameters (p, q) corresponding to the i-th and j-th nodes. Build likelihood functions from alternative copula functional forms The Nelder-Mead simplex method is used to solve the maximum value of the likelihood function and the corresponding copula function parameter values, and then the most appropriate copula function form and corresponding parameters are selected based on the AIC criterion. Among them, j in the likelihood function L represents the sample number, a total of m samples, i represents the test point serial number, a total of n test points, R _p (t) and R _q (t) represent the p and q performance parameters respectively Reliability distribution function.

2) Determine c _ij in the second layer tree diagram in the Vine model: let {p,k}, {q,k} be the variables contained in node i and node j respectively, then c _ij =c _pq|k , according to Alternative copula function form to build likelihood function The Nelder-Mead simplex method is used to solve the maximum value of the likelihood function and the corresponding copula function parameter values, and then the most appropriate copula function form and corresponding parameters are selected based on the AIC criterion. Among them, j in the likelihood function L represents the sample number, a total of m samples, i represents the test point serial number, a total of n test points, R _p|k (t) and R _q|k (t) respectively represent the joint distribution of reliability The conditional probability expression of , its calculation formula follows:

i. When k is a one-dimensional variable, that is, there is only one same variable between two nodes, the values of R _p|k (t) and R _q|k (t) are according to calculate. where R is the reliability distribution function, Indicates that the copula function is derived with respect to R _j .

ii. When k is a vector with a dimension greater than 1, that is, two nodes contain multiple identical variables, then the values of R _p|k (t) and R _q|k (t) are according to calculate. Where v is a condition vector, v=(v ₁ ,v ₂ ,...,v _d ), v _j is any element in v, and v _-j represents the vector obtained by removing v _j from v.

After determining the c _ij in the second-level dendrogram in the Vine model, and so on, calculate the c _ij in the other dendrograms until all optimal copula function forms in the Vine diagram are determined.

3) After determining the functional forms of all c _ij , re-estimate the parameters of the selected c _ij together. Establish the likelihood function L=r(x ₁ ,...,x _n ), and use the Nelder-Mead simplex method to solve the maximum value of the likelihood function, and the corresponding optimal parameter results can be obtained. Substitute the parameters of all c _ij functions into the decomposition formula of Vine-Copula, and obtain the multivariate correlation model based on Vine-Copula.

Step Five: Reliability Assessment

After using the Vine-Copula function to complete the coupling modeling of product failure behavior, the probability density function of the joint distribution of product reliability can be obtained. However, due to the complex and diverse structure of the density function of the Copula function, and the expression of the reliability probability density function of each performance parameter is also very complicated, the analytical expression of the probability density function of the reliability joint distribution is not only long but also very complicated, and it is difficult to directly integrate method to solve the reliability function. For the exact solution of the joint distribution of product reliability under the condition of non-strict monotone degradation, the present invention adopts the method of conditional probability decomposition to solve the problem.

1. Calculate the conditional probability expression of the reliability joint distribution of binary variables

where R is the reliability distribution function, Indicates that the copula function is derived with respect to R _j .

2. Calculate the conditional probability expression of the joint distribution of multivariate reliability

Where v is a conditional vector, v=(v ₁ ,v ₂ ,...,v _d ), v _j is any element in v, v _-j represents the vector obtained by removing v _j from v, v _-j =( v ₁ ,v ₂ ,...,v _j-1 ,v _j+1 ,...,v _d ).

3. Calculate the multivariate reliability joint distribution expression

Among them, K represents the number of degraded performance parameters, a _k is the code name of the performance parameter, v is the condition vector, v _j is an element in the condition vector v, and v _-j represents the vector obtained by removing v _j from the condition vector v, is the time corresponding to v _j , is the time vector formed by the time corresponding to the elements in vector v _-j .

Wherein, "select the most suitable copula function form based on the AIC criterion" mentioned in step 4 means: the copula corresponding to the minimum value of AIC is considered to be the most suitable choice among the alternative copula forms. The calculation formula of AIC is AIC=-2lnL+2p, where L is the maximum likelihood function value, p is the number of model parameters, M is the sample size, and the corresponding value is the number of observation points of the product.

Among them, in step 5, in order to facilitate the solution of the joint reliability distribution, the principle of decomposing the joint reliability distribution into conditional probabilities is to make the binary Copula functions in the final decomposition result as much as possible that have been used in Vine-Copula and The established binary Copula function, described in the "Conditional Probability Decomposition Method", can follow some principles:

1) The first marginal distribution of reliability in the decomposition formula (ie ), for D-vine, it is advisable to select any end node at both ends of the first layer;

2) The second item in the decomposition formula, namely For D-vine, it is advisable to choose the same-side end node of the second layer;

3) By analogy, it is better to select the corresponding end node for D-vine.

Through the above steps, the coupling modeling of the product's fault behavior is completed, and the model of the joint distribution of reliability is established, which solves the problem of solving the reliability function by the direct integration method, clearly describes the multivariate variable coupling relationship of the product, and provides and methods of product reliability assessment within cost.

The advantages of the present invention are:

For multi-variable high-reliability and long-life products, based on the accelerated degradation test data and drift Brownian motion, and on the basis of the consistency test of the acceleration mechanism, the marginal distribution of variables under normal stress levels is constructed. Based on the hierarchical idea, using the Vine-Copula method, the decoupling problem of the multidimensional joint distribution is described by the binary Copula function, that is, the decoupling modeling of the underlying component correlation mechanism is carried out, and the multivariate Copula function is avoided in high dimensions. numerical simulation to solve the problem. At the same time, in terms of reliability calculation, since the current Copula function modeling method usually uses a strictly monotonic degradation degradation model for life evaluation, the solution method based on conditional probability proposed by the present invention makes it possible for non-strict monotonic degradation cases. solve. Finally, compared with the multidimensional normal distribution modeling method, it overcomes the linear or irrelevant relationship between the degradation parameters, indicating that Vine-Copula can effectively deal with reliable life estimation under nonlinear correlation.

Description of drawings

Fig. 1 is a flowchart for implementing the method of the present invention.

Fig. 2 The tree diagram of the first layer of four-dimensional D-vine.

Fig. 3 The tree diagram of the second layer of four-dimensional D-vine.

Figure 4 D-vine model of the basic error of the smart meter.

Fig. 5 The joint reliability diagram of the four basic errors of the smart meter.

The serial numbers and codes in the figure are explained as follows:

The numbers in Figure 2 and Figure 3 represent the codes of the performance parameters respectively. In Figure 4, BE1, BE5, BE9, and BE10 represent the four basic errors of the smart meter, and CE represents the cumulative error of the smart meter. BE1BE5 represents the connection line connecting node BE1 and node BE5, which represents the correlation between performance parameters BE1 and BE5, and serves as the second layer node in the vine graph. The meanings of BE5BE9 and BE9BE10 are the same as above. BE1BE9|BE5 represents the connection between node BE1BE5 and node BE5BE9, which represents the performance parameter BE5 is related to both BE1 and BE9, and BE1BE9|BE5 is the third layer node in the vine graph. BE5BE10|BE9 has the same meaning as above. BE1BE10|BE5BE9 represents the connection between node BE1BE9|BE5 and node BE5BE10|BE9, representing not only the performance parameter BE5 is related to BE1 and BE10, but also the performance parameter BE9 is related to BE1 and BE10.

detailed description

The present invention will be further described in detail with reference to the accompanying drawings and embodiments. The invention constructs a coupling model for reliability evaluation by analyzing the correlation between product output parameters, selects a smart meter as an application object, performs degradation modeling on the output performance parameters of the smart meter, and evaluates the reliable life of the meter. Smart meters are typical electronic products with multiple performance parameters. The performance parameters that can represent the degradation state of smart meters include 1 item of cumulative error and 4 items of basic error, which are cumulative error CE and basic errors BE1, BE5, BE9, and BE10. In the constant accelerated degradation test of the smart meter, two kinds of accelerated stresses (temperature and humidity) were selected to carry out the test at three stress levels, with 8 samples for each stress level. The specific stress combinations are shown in Table 1. From the accelerated degradation test data, it can be seen that the output performance parameters of the smart meter have degradation characteristics, and its degradation trend gradually increases. According to the design index, there is a clear failure threshold. Under the test conditions, temperature and humidity are the accelerated stresses that promote its degradation.

A product failure behavior coupling modeling and reliability evaluation method of the present invention, that is, a product failure behavior coupling modeling and reliability evaluation method based on Vine-Copula and accelerated degradation tests, as shown in Figure 1, its specific implementation steps as follows:

Step 1: Degradation trajectory modeling

The test data of the five key performance parameters of the smart meter under various stress levels were obtained through the accelerated degradation test. According to the characteristics of the degradation trajectory of each performance parameter of the smart meter, the drift Brownian motion is used to model the degradation trajectory of the performance parameter.

1. The degradation trajectory of performance parameters obeys the model Δy _t = μt+σB(t), where t is time, Δy _t is the degradation amount of performance parameters at time t, μ is the drift coefficient, σ is the diffusion coefficient, and B(t) is Standard Brownian motion.

2. Establish the likelihood function Where j represents the j-th sample, a total of n samples, i represents the i-th test point, a total of m _j test points, t _ji represents the time of the i-th test of the j-th sample, is the degradation amount of the measured performance parameter, μ is the drift coefficient, and σ is the diffusion coefficient.

3. Using the Nelder-Mead simplex method to solve the maximum value problem of the likelihood function, the parameter estimation results of the degradation trajectory model of each performance parameter under each stress level are obtained, as shown in Table 2.

Step 2: Failure Mechanism Consistency Test

For the degradation trajectory obeying the drift Brownian motion model, taking the performance parameter CE of the smart meter as an example, the consistency test of its failure mechanism can be based on

1. For each sample of this parameter under each accelerated stress, use the Nelder-Mead simplex method to estimate the likelihood function Solve the maximum value problem of , and get the parameter value of the degradation model with Where j represents the j-th sample, i represents the i-th test point, a total of m _j test points, t _ji represents the time of the i-th test of the j-th sample, is the degradation amount of the measured performance parameter, μ is the drift coefficient, and σ is the diffusion coefficient.

2. Construct hypothesis test statistics T follows a t-distribution with n+m-2 degrees of freedom. make As the test samples under the three accelerated stresses, the test samples X ₁ , X ₂ , and X ₃ under the three stresses can be obtained, as shown in Table 3. Then T test is _{carried out on the test samples Xi and X j under} any two stresses one by one.

3. Given a significance level of 0.05, look up the t distribution table and know that the rejection range is |T|≥t _0.975 (14) = 2.1448, so that the test samples X _i and X _j are respectively replaced in the T statistics and And bring it into the data calculation, comparing the value of t _0.975 (14), it can be seen that |T|<t _0.975 (14), that is, there is no significant difference between X ₁ and X ₂ and X ₃ , and the parameters under the three stresses The failure mechanism is consistent, and the test results are shown in Table 4.

The failure mechanism consistency test of other performance parameters can repeat steps 1, 2 and 3 until all parameters pass the consistency test.

Step 3: Extrapolating the distribution model of performance parameters under normal stress

After completing the modeling of the degradation trajectory of each performance parameter at each stress level, it is necessary to extrapolate the degradation model parameters at the high stress level to the normal stress.

1. Determine the acceleration factor

According to the accelerated test conditions, the multi-stress Irene acceleration model is selected: make is the acceleration factor of the accelerated stress level _Sh relative to the stress level S _l .

2. Solving acceleration model parameters

For the performance parameter p, under k ₁ and k ₂ stress levels, construct the sum of squared deviation function Using the Nelder-Mead simplex method to solve the problem of the minimum value of the sum of squares function of the deviation, the optimal model parameters B and C in the corresponding acceleration equation can be obtained, as shown in Table 5.

3. Estimate the parameters of the degradation trajectory model under normal stress

After obtaining the parameters in the acceleration model, use the least squares method to estimate the parameters of the degradation trajectory model under normal stress (temperature 20°C; relative humidity 45%), and obtain the dispersion sum of squares function and The simplex method is used to optimize the solution, so as to obtain the point estimates of the degradation trajectory model parameters μ _p0 and σ _p0 under normal stress, as shown in Table 6.

Through the above steps, the degradation trajectory model under normal stress is finally obtained: y _p0 (t) = μ _p0 t + σ _p0 B(t).

Step 4: Coupling modeling of product failure behavior based on Vine-Copula

1. Correlation analysis

Based on the assumption that the correlation relationship remains unchanged under each accelerated stress level, only the test data under one of the accelerated stress levels can be selected for correlation analysis. Here, the test data under the third set of stress levels is selected. Using the correlation measurement index Kendall’sτ coefficient to calculate the correlation between the key performance parameters of the samples is shown in Table 7, and the conclusions are as follows:

1) There is no correlation between the cumulative error CE and the basic error BE

2) The basic errors BE1, BE5, BE9, and BE10 have a strong nonlinear correlation.

2. Use the vine graph model to decompose the joint distribution into a form of multiplication of appropriate conditional probabilities, and use copula to build a multivariate correlation model.

a. Modeling method based on D-Vine

It can be seen from the analysis that the correlation between the four basic errors of the smart meter is relatively close, and the D-vine model among the four basic errors BE can be established, as shown in Figure 4.

1) First, in the Vine graph model of the first layer, the four basic errors BE1, BE5, BE9, and BE10 of the smart meter are coded as 1, 2, 3, and 4 in sequence, and the four basic errors are used as nodes in the tree diagram. Nodes are connected by short lines according to their relationship, as shown in Figure 2. Let c _ij represent the probability density of the copula function connecting node i and node j, there is c _ij ={c ₁₂ ,c ₂₃ ,c ₃₄ }

2) The second-level tree diagram in the Vine diagram is based on the first-level tree diagram, and the connections (1,2), (2,3), and (3,4) in the first-level tree diagram are used as the first-level tree diagram. The nodes in the two-layer tree diagram are connected by connecting lines to construct the tree diagram of the second layer, as shown in Figure 3. Still using c _ij to represent the probability density of the copula function connecting node i and node j, there is c _ij ={c _13|2 ,c _24|3 }

3) The third-level tree diagram in the Vine diagram is established on the basis of the second-level tree diagram, and the connections (13|2) and (24|3) in the second-level tree diagram are used as the connections in the third-level tree diagram Nodes are connected by lines. Still using c _ij to represent the probability density of the copula function connecting node i and node j, there is c _ij ={c _14|23 }

According to the D-vine model, the joint distribution probability density function of the four basic error reliability of the smart meter is:

r(x ₁ ,x ₂ ,x ₃ ,x ₄ )＝c ₁₂ ·c ₂₃ ·c ₃₄ ·c _13|2 ·c _24|3 ·c _14|23 ·r ₁ ·r ₂ ·r ₃ ·r ₄

Among them, the variables x ₁ , x ₂ , x ₃ , and x ₄ represent the basic errors BE1, BE5, BE9, and BE10 respectively; c is the probability density of the corresponding copula function; r _i represents the reliability distribution probability density function of the i-th performance parameter .

b. Solve the Vine-Copula coupling model

To determine the reliability joint distribution function of smart meters, it is necessary to determine the form and parameters of each Copula function in the formula r(x ₁ , x ₂ , x ₃ , x ₄ ). The alternative sets of Copula functions in the present invention are: Gaussian Copula, Frank Copula, Ali-Mikhail-Haq Copula, and Clayton Copula.

1) Determine the c _ij in the first layer tree diagram in the Vine model: c _ij ={c ₁₂ ,c ₂₃ ,c ₃₄ }, and use the Nelder-Mead simplex method to solve the likelihood function according to the alternative copula function form The maximum value of the copula function is obtained, and the parameter estimation and the maximum likelihood function value of the copula function are obtained, and then the most suitable copula function form and corresponding parameters are selected based on the AIC criterion. Among them, (k1, k2)=(1|2,3|2), according to the calculation results (see Table 8), it can be determined that the optimal choice of c ₁₂ , c ₂₃ , and c ₃₄ is Frank Copula.

2) After completing the optimization of the copula function in the first layer of the Vine graph, determine c _{ij in the second layer tree diagram in the Vine model: c ij =} {c _13|2 ,c _24|3 } according to the alternative copula function form Create a likelihood function The Nelder-Mead simplex method is used to solve the maximum value of the likelihood function and the corresponding copula function parameter values, and then the most appropriate copula function form and corresponding parameters are selected based on the AIC criterion. Among them, (p|k, q|k)={(1|2,3|2), (2|3,4|3)}, it is determined that the best choice of c _13|2 is Frank Copula, c ₂₄ The best choice for _|3 is the Clayton Copula. By analogy, to select the function form of c _14|23 , it can be known that the optimal choice of c _14|23 is Frank Copula, so far the copula functions of the formula r(x ₁ ,x ₂ ,x ₃ ,x ₄ ) have been completed Optimizing the form, getting the result

r(x ₁ ,x ₂ ,x ₃ ,x ₄ )=c _Frank (α ₁₂ ,R ₁ ,R ₂ )·c _Frank (α ₂₃ ,R ₂ ,R ₃ )·c _Frank (α ₃₄ ,R ₃ , R ₄ )

·c _Frank (α _13|2 ,R _1|2 R _3|2 )·c _Clayton (α _24|3 ,R _2|3 ,R _4|3 )

·c _Frank (α _14|23 ,R _1|23 ,R _4|23 )·r ₁ ·r ₂ ·r ₃ ·r ₄

Among them, R _1|2 , R _2|3 , and R _1|23 can be obtained according to the conditional probability expression of the joint distribution of multivariate reliability.

3) After determining all the copula function forms, re-estimate the parameters of the selected copula functions together, and establish the likelihood function L=r(x ₁ ,x ₂ ,x ₃ ,x ₄ ), using Nelder-Mead The simplex method solves the maximum likelihood function, and the optimal parameter results can be obtained as shown in Table 9. Substitute the parameters of all copula functions into the decomposition formula of vine-copula, and obtain the multivariate correlation model based on Vine-Copula .

Step 5: Reliability solution

1. Calculate the conditional probability expression of the reliability joint distribution of binary variables

2. Calculate the conditional probability expression of the joint distribution of multivariate reliability

3. According to the multivariate reliability joint distribution expression, the joint reliability distribution of the four basic error couplings can be obtained as:

R _BEs (t)＝R ₁₂₃₄ (t ₁ ,t ₂ ,t ₃ ,t ₄ )

＝R ₄ (t ₄ )·R _3|4 (t ₃ |t ₄ )·R _2|34 (t ₂ |t ₃ ,t ₄ )·R _1|234 (t ₁ |t ₂ ,t ₃ ,t ₄ )

Among them, the values of R _3|4 , R _2|34 , and R _1|234 can be calculated according to R _hj (x _h |x _j ) and R _x|v (x|v).

Finally, the joint distribution function graph of the four basic error reliability is shown in Figure 5. By analyzing the modeling results of the smart meter accelerated degradation test data, it can be seen that when there is correlation among multiple performance parameters, if the correlation is not considered in the degradation modeling and only the series model is used for modeling, the obtained reliability function curve and life evaluation The result will appear conservative. In contrast, modeling approaches that consider correlations can give somewhat well-founded but not overly aggressive assessment results. Therefore, when there is correlation between performance parameters, it is necessary to consider the role of correlation in the process of degradation modeling.

Aiming at the high reliability and long life characteristics of products and the problems related to performance parameters of failure mechanism coupling characterization, the present invention studies the failure behavior modeling and reliability evaluation method of products with multivariate variables based on Vine-Copula and accelerated degradation tests. The modeling method based on Vine-Copula takes advantage of the flexible structure of Vine-Copula, and has its own applicable Vine models for analysis and establishment of correlation models for a variety of complex correlation situations, and regardless of the amount of degradation of performance parameters It can be well applied if the relationship is linear or nonlinear, and it solves the problem of characterization of the relationship between two variables in the multivariate reliability model.

All forms involved in the present invention are as follows:

Table 1 Parameter estimation results of the degradation trajectory model for each performance parameter

Table 2 Parameter estimation of degradation trajectory model under accelerated stress for each performance parameter

Table 3 CE parameter estimates and test samples of each product

Table 4 CE parameter consistency test results

Table 5 Parameter estimation results of each performance parameter acceleration model

Table 6 Model parameters of degradation trajectory of each performance parameter under normal stress

Table 7 Correlation calculation results between performance parameters

Table 8 Function forms and parameter estimation of each copula

Table 9 Copula function optimization results

Claims (3)

1. a kind of product bug behavior Coupling method and reliability estimation method, i.e., a kind of to be moved back based on Vine-Copula and acceleration Change the product bug behavior Coupling method and reliability estimation method of test, it is characterised in that：Its step is as follows：

Step one：Degradation path is modeled

The degradation path model of product is expressed as amount of degradation with time and the function of suffered environmental stress；Build Degradation path is carried out During mould, need

1. degraded data feature is analyzed, degradation path model Δ y is set up_t=f (t, S), Δ y_tRepresent the amount of degradation of t, S tables Show suffered environmental stress horizontal size；

2. likelihood function is set upf_iThe probability density function of i-th amount of degradation test point is represented, μ, σ is represented Degradation path model parameter；

3. degraded data is substituted into, the maximum problem of likelihood function is solved using Nelder-Mead simplex method, obtained The parameter Estimation of each performance parameter degradation path model under each stress level；

Step 2：Failure mechanism consistency check

It is the validity of guarantee test in accelerated degradation test, the same performance parameter under different accelerated stress levels should be accorded with Close failure mechanism consistency check；What is existed between model parameter and failure mechanism are constant in Degradation path modeling contacts according to degeneration Track modeling method different and there is difference, so that the Degradation path for obeying Brownian Motion with Drift is modeled as an example：

1. its acceleration mechanism consistency check formula is determinedWherein μ_iAnd σ_iRespectively stress level S_iUnder The coefficient of deviation and diffusion coefficient of degradation path model；

2. hypothesis testing statistic is builtTs of the T from the free degree for n+m-2 is distributed；Wherein：It is μ that X obeys average₁Side Difference is σ²Normal distribution, Y obey average be μ₂Variance is σ²Normal distribution, and X₁, X₂... X_nIt is the sample from overall X, Y₁,Y₂,…Y_mThe sample of overall Y is taken from, n and m is respectively sample size size；

3. region of rejection is determined；Given level of significance α, substitutes into data and calculates | T |, looks into t distribution tables and knows：

t_1-α/2(n+m-2) size, if | T | >=t_1-α/2(n+m-2), then assay falls in region of rejection, and parameter is answered different Failure mechanism under power does not have uniformity, conversely, then meeting consistency check；

Step 3：The distributed model of performance parameter under extrapolation normal stress

Relation between life of product feature and accelerated stress level is stated by acceleration model, is derived using acceleration model The degradation model and edge distribution of each variable under normal stress；In the present invention, using simplified many stress Aileen acceleration models：T is characterized the life-span, and T and RH represents respectively temperature and humidity variables, and A, B and C are parameter to be asked；

1. accelerated factor is determined

Make AF_i,0=t₀/t_iFor the horizontal S of accelerated stress_iRelative to the horizontal S of normal stress₀Accelerated factor, t_iAnd t₀Represent respectively Product is in the horizontal S of accelerated stress_iS horizontal with normal stress₀Under, performance parameter reaches the time required during identical amount of degradation；

2. acceleration model parameter is solved

A. the relational equation of accelerated factor and life-span, coefficient of deviation, diffusion coefficient between different stress is built Wherein AF represents accelerated factor, and L represents life-span, μ_iAnd σ_iRepresent in accelerated stress S_iThe drift system of lower performance parameter Degradation path Number and diffusion coefficient；Subscript h represents high stress level, and subscript l represents low stress level；

B. to performance parameter p, in k₁And k₂Under stress level, sum of squares of deviations function is constructed The minimum problem of sum of squares of deviations function is solved using Nelder-Mead simplex method, is obtained in acceleration model most Excellent model parameter B, C；

3. degradation path model parameter under normal stress is estimated

After trying to achieve the parameter in acceleration model, degradation path model parameter under normal stress is estimated using least square method Meter, obtains sum of squares of deviations functionWithUsing simplex method Solution make sum of squares of deviations minimum corresponding to parameter value, so as to obtain normal stress under degradation path model parameter μ_p0And σ_p0 Point estimation；Degradation path model is under final normal stress：y_p0(t)=μ_p0t+σ_p0B(t)；

Step 4：Product bug behavior Coupling method based on Vine-Copula

Copula functions are a class " connection " functions, and connection is edge distribution and Joint Distribution；The theoretical base of Vine-Copula Plinth is conditional probability, is carried out by the way that Joint Distribution is resolved into into the form that multiple binary copula and corresponding conditional probability even take advantage of Modeling, the product bug behavior Coupling method main flow based on Vine-Copula is as follows：

1. performance parameter correlation analysis

Before the failure behavior Coupling method for carrying out product, for the polynary performance parameter degradation of product, correlation point is first carried out Analysis；With Kendall ' s τ coefficients as relativity measurement index, the absolute value of index represents that correlation is stronger closer to 1, Kendall ' s τ coefficients represent nonlinear correlation degree；

2. Joint Distribution is decomposed into into the form that suitable conditional probability is multiplied by vine graph models, using copula function structures Build multiple correlation relational model；Specifically,

A. it is based on the modeling method of D-Vine

1) first, in ground floor Vine graph models, each variable as the node in tree graph, by correlation with short between node Line is attached；With c_ijThe probability density of connecting node i and the copula functions of node j is represented, then c_ij=c_pq, p, q are respectively The variable that node i and node j are included；

2) second layer tree graph in Vine figures is set up on the basis of ground floor tree graph, using the connection in ground floor tree graph as the Node in two-layer tree figure, adjacent line is attached in the unit of the formation of the second layer with line in ground floor, builds second The tree graph of layer；Still with c_ijThe probability density of connecting node i and the copula functions of node j is represented, there is c_ij=c_pq|k, { p, k }, { q, k } is respectively the variable that node i and node j are included, and wherein k is the total variable of two nodes；

3) third layer tree graph is also to be set up on the basis of preceding layer tree graph using similar method, by that analogy, works as tree graph In be only left a pair of dependency relations, i.e., be exactly the tree graph of last layer when only remaining a line；In the same manner, third layer tree graph is arrived There is c including the probability density of the copula functions between all connecting nodes including last layer of tree graph_ij=c_pq|k, p, K }, { q, k } be the variable that includes of node i and node j, wherein k is the total variable of two nodes；

4) the corresponding copula function probabilities density of each bar line in Vine figures is multiplied, that is, is obtained based on Vine figures The decomposition texture .c=∏ c of copula functions_ij；Production reliability Joint Distribution probability density function is expressed asx_iRepresent i-th performance parameter of product, r_iRepresent the corresponding reliability probability density of i-th parameter Function, is so far set up based on Vine-Copula coupling models；

B. Vine-Copula coupling models are solved

Solve Vine-Copula coupling models, you can by spending Joint Distribution probability density, first to determine that reliability Joint Distribution is general Rate density r (x₁,...,x_n) in each copula probability density function form and parameter, i.e. c_ijForm and parameter；

1) c in ground floor tree graph in Vine models is determined_ij:Substitute into i-th, j-th node and correspond to the degeneration of performance parameter (p, q) Amount data；Likelihood function is set up according to alternative copula functional formsUsing Nelder- Mead simplex methods solve the maximum and its corresponding copula function parameters value of likelihood function, are then based on AIC criterion choosing Take copula functional forms the most suitable and corresponding parameter；Wherein, j represents sample number, common m sample in likelihood function L This, i represents test point sequence number, common n test point, R_p(t) and R_qT () represents respectively the reliability point of pth and q performance parameters Cloth function；

2) c in second layer tree graph in Vine models is determined_ij:{ p, k }, { q, k } is made to be respectively the change that node i and node j are included Amount, then have c_ij=c_pq|k, likelihood function is set up according to alternative copula functional forms The maximum and its corresponding copula function parameters value of likelihood function, Ran Houji are solved using Nelder-Mead simplex method Copula functional forms the most suitable and corresponding parameter are chosen in AIC criterion, wherein, j represents that sample is compiled in likelihood function L Number, common m sample, i represents test point sequence number, common n test point, R_p|k(t) and R_q|kT () represents respectively reliability Joint Distribution Conditional probability expression formula, its computing formula follows：

I. when k is one-dimensional variable, i.e., an identical variable is only included between two nodes, now R_p|k(t) and R_q|kT the value of () is pressed According toCalculate；Wherein R is reliability distribution function,Represent the copula functions To R_jDerivation；

Ii. when k is vectorial more than 1 of dimension, i.e., multiple identical variables, now R are included between two nodes_p|k(t) and R_q|k(t) Value according toCalculate；Wherein v is conditional vector, v=(v₁,v₂,…, v_d), v_jIt is any one element in v, v_-jRepresent that v removes v_jThe vector for obtaining；

Determine the c in second layer tree graph in Vine models_ijAfterwards, by that analogy, the c of other layer of tree graph is calculated_ij, until determining Vine In figure till all optimum copula functional forms；

3) all c are determined_ijFunctional form after, to selected c_ijParameter estimated again together；Set up likelihood function L =r (x₁,...,x_n), the maximum of likelihood function is solved using Nelder-Mead simplex method method, obtain corresponding optimum Parametric results；By all c_ijIn the breakdown of the parameter back substitution Vine-Copula of function, that is, obtain based on Vine-copula The multiple correlation relational model of foundation；

Step 5：Reliability assessment

The reliability Joint Distribution for completing to be obtained after product bug behavior Coupling method product using Vine-Copula functions is general Rate density function；It is various yet with the density function complex structure of Copula functions, while the reliability of each performance parameter is general Rate density function expression formula is also sufficiently complex, cause the analytic expression of the probability density function of reliability Joint Distribution not only long but also It is extremely complex, it is difficult to the method for passing through direct integral solves Reliability Function；For the product under non-critical dullness degenerate case The accurate solution problem of reliability Joint Distribution, the method that present invention employs conditional probability decomposition is solved；

1. the conditional probability expression formula of the reliability Joint Distribution of binary variable is calculated

$R_{h|j}\left(x_h\right|x_j)=\frac{{\mathrm{dC}}_{hj}\left(R_h\right(x_h),R_j(x_j\left)\right)}{{\mathrm{dR}}_j\left(x_j\right)}$

Wherein R is reliability distribution function,Represent copula function pairs R_jDerivation；

2. the conditional probability expression formula of polytomy variable reliability Joint Distribution is calculated

$R_{x|v}\left(x\right|v)=\frac{{\mathrm{dC}}_{x,v_j|v_{-j}}\left(R\right(x|v_{-j}),R(v_j|v_{-j}\left)\right)}{dR\left(v_j\right|v_{-j})}$

Wherein v is conditional vector, v=(v₁,v₂,…,v_d), v_jIt is any one element in v, v_-jRepresent that v removes v_jObtain Vector, v_-j=(v₁,v₂,…,v_j-1,v_j+1,…,v_d)；

3. polynary reliability Joint Distribution expression formula is calculated

Wherein K represents the performance parameter number for occurring to degenerate, a_kIt is the code name of performance parameter, v is conditional vector, v_jCondition to An element in amount v, v_-jRepresent in conditional vector v and remove v_jThe vector for obtaining,For v_jThe corresponding time,For vector v_-j The time arrow that the middle element corresponding time is constituted；

By above step, the failure behavior Coupling method to product is completed, establish the model of reliability Joint Distribution, solved Immediate integration of having determined solves a difficult problem for Reliability Function, clearly describes product polytomy variable coupled relation, there is provided can hold Method by Reliability Assessment is carried out in time and cost.

2. a kind of product bug behavior Coupling method according to claim 1 and reliability estimation method, i.e. one kind are based on The product bug behavior Coupling method and reliability estimation method of Vine-Copula and accelerated degradation test, it is characterised in that： " choosing copula functional forms the most suitable based on AIC criterion " described in step 4 refers to：With corresponding to AIC minimum of a values Copula be considered selection most widely suited in alternative copula forms；The calculating formula of AIC is AIC=-2lnL+2p, wherein L It is maximum likelihood function value, p is model parameter number, and M is sample size, corresponding value is the observation station number of product.

3. a kind of product bug behavior Coupling method according to claim 1 and reliability estimation method, i.e. one kind are based on The product bug behavior Coupling method and reliability estimation method of Vine-Copula and accelerated degradation test, it is characterised in that： It is to make by the principle that reliability Joint Distribution is decomposed into conditional probability for ease of the solution of reliability Joint Distribution in step 5 The every binary Copula function obtained in final decomposition result is all as far as possible the binary for having used in Vine-Copula and having established Copula functions, described " method that conditional probability is decomposed ", it then follows following some principles：

1) first reliability edge distribution is (i.e. in breakdown), for D-vine preferably selects ground floor two ends arbitrarily to hold Node；

2) Section 2 in breakdown, i.e.,For the homonymy end node that D-vine preferably selects the second layer；

3) by that analogy, for D-vine preferably selects corresponding end node.