CN103971024A - Method for evaluating reliability of relaying protection systems under small sample failure data - Google Patents

Abstract

The invention discloses a method for evaluating the reliability of relaying protection systems under small sample failure data, and belongs to the field of power system relay protection. The problems existing in study of the power system relay protection field at present are solved. The method includes the steps that first, failure samples of the relaying protection systems are integrated; then, a model using Weibull distribution as failure distribution is established by testing the fit goodness, a Monte Carlo method is used for sampling to expand data samples, the data samples and small sample original data are used as priori information together, and a Bayes-Monte Carlo evaluation model is established; finally, reliability evaluation is conducted on each relaying protection system by combining the Bayes-Monte Carlo evaluation model. By the adoption of the method, the limitations that the number of needed samples is large, and the accuracy of reliability parameters is low in practical application in a traditional reliability evaluation method based on large failure samples are eliminated, and reliability evaluation of the relaying protection systems under small sample failure data is effectively achieved.

Description

Relay protection system reliability estimation method under small sample fail data

Technical field

The present invention relates to field of relay protection in power, particularly relay protection system reliability estimation method under small sample fail data.

Background technology

As the first line of defence of electric system, the reliability level of relay protection system is for the safe and stable operation important in inhibiting of electric system.Statistics from current serial document to protective relaying device practical operation situation, within its length of service, seldom there is the even defect situation of failure conditions such as " malfunction ", " tripping " in protective relaying device.In addition the protective device replacement cycle shorten, current still codes and standards very not of relay protection system defect and fail message record, in certain region and time range, specific to the complete failure logging (record) sample of device model, failure mode still less.Up to now, reliability of relay protection appraisal procedure is also seldom considered the feature that this small sample lost efficacy, and main taking routine the reliability estimation method based on analytical method and simulation as main.Analytical method is mainly set up reliable probability model according to the structure of system, function or the logical relation of the two as Markov state-space method, GO method, Fault Tree, venture analysis etc., by the solving model such as recursion or iteration, parameter.Simulation as Monte-Carlo Simulation be selection and the estimation by probability distribution being sampled to carry out state, utilize statistical method to obtain reliability index.Consider that relay protection system is difficult to obtain enough fail datas (even no-failure data) in finite time; even if defective data sample is not abundant yet; therefore the validity that depends on traditional reliability estimation method of large inefficacy sample can be given a discount; and reduce to a certain extent the confidence level of its assessment result, may further affect the reference value of Strategies of Maintenance taking reliability assessment as foundation etc.Also have method to consider protective relaying device operation characteristic; adopt Weibull distribution as its Lifetime Distribution Model; directly determine that by regretional analysis form parameter, scale parameter in Weibull distribution calculate the reliability indexs such as fiduciary level, crash rate; but because the fail data of protective relaying device is few, utilize the accuracy of limited fail data sample analysis result lower.

For the problems referred to above, the present invention proposes relay protection system reliability estimation method under small sample fail data.The present invention relates to following two background technologies.

1 relay protection system small sample fail data

The fail data of relay protection is at present mainly derived from other data sources such as failure information system, maintenance service bulletin, dispatching center's operational report, failure wave-recording and account system.Along with improving constantly of electric system and protective relaying device operation level, seldom there is tripping and maloperation situation in protection equipment, only has only a few fail data can supply dependent evaluation used.In addition the cycle that renewal of the equipment is at present regenerated is shorter, and a lot of protective devices did not occur that failure conditions made originally more to lack with regard to the data of same model device in not abundant inefficacy sample in-service at all.On the whole, the on-the-spot fail data obtaining presents more typical small sample feature.

The protective relaying device fail data obtaining for scene, generally can be divided into following three kinds of data types: partial data, left censored data and Type I censoring data.In fact,, due to protective device put into operation differing greatly of time and the difference of on-the-spot ruuning situation, the data of reflection protective device ruuning situation more meet the feature of Type I censoring data.For the repair time in censored data, because it is extremely short with respect to operation hours, in addition during as analytic target, after having repaired, can reenter the measurement period of working time, therefore can ignore repair time in this alanysis taking the inefficacy that occurred or defective data.

Choosing of 2 Bayesian formulas and prior distribution thereof

The information that classical statistics only provides with sample is done statistical inference under certain statistical model, and the larger situation of sample size is had to good statistical inference effect.And bayes method is before obtaining sample observations, the unknown parameter θ in parametric statistics model is had to some priori, these knowledge embody with the form of prior distribution.And obtaining sample observations x (this method refers to time between failures) afterwards, and the information being provided by x and prior distribution obtains posteriority and distributes, and it is the basis that Bayesian statistics is inferred that posteriority distributes.Owing to having utilized all kinds of prioris, such as historical data, expert info and l-G simulation test information etc., reduced the degree of dependence of classical way to on-the-spot service data sample, thereby often also very effective to Small Sample Size.

Bayesian formula is the direct embodiment of bayesian theory, is written as:

$g\left(\mathrm{\θ}\right|x)=\frac{f\left(x\right|\mathrm{\θ})g\left(\mathrm{\θ}\right)}{f\left(x\right)}---\left(1\right)$

Wherein, θ is the unknown parameter in population distribution, this method is specially form parameter m and the scale parameter η in Weibull distribution, wherein form parameter m is used for the different failure types of the equipment of distinguishing, in the time of m>1, crash rate temporal evolution is increment type, and in the time of m=1, crash rate is very fixed, do not change over time, in the time of m<1, crash rate temporal evolution is decrescendo; Scale parameter η plays the effect that zooms in or out coordinate scale; X is the observed reading of sample X, f (x)=∑ f (x| θ) g (θ), f (x _i| θ) conditional probability distribution while being given parameters θ, be called likelihood and distribute.G (θ | the joint probability density while x) being given x, posteriority distributes.Obtaining before sample X, about the understanding of parameter being summarized in g (θ), in sample, the new packets of θ is contained in likelihood function f (x| θ).Through Bayesian formula obtain than more comprehensive posterior information g (θ | x).

Summary of the invention

The object of the invention is to, propose relay protection system reliability estimation method under small sample fail data, in order to solve the problem existing in current relay protection system Research on Reliability Evaluation.

For achieving the above object, the present invention propose technical scheme be, relay protection system reliability estimation method under small sample fail data,

Suppose: relay protection system fault causes because of hardware unaccelerated aging or inefficacy;

It is characterized in that described method comprises the following steps:

Step 1: relay protection system inefficacy sample is integrated, calculates uptime interval t ₁, t ₂..., t _n, choosing the maximum time interval is T closing time _s, wherein, n is inefficacy total sample number;

Step 2: build Bayes-Monte Carlo assessment models, wherein build model and specifically comprise:

Step 21: according to the life distribution type of device, adopt Weibull distribution to distribute as estimating, and carry out the inspection of fitting of distribution goodness, whether tally with the actual situation to confirm that the life-span of Selection Model distributes,

The probability density of failure of Two-parameter Weibull Distribution is

Reliability Function is R (t)=1-F (t)=exp[-(t/ η) ^m]

Crash rate function is λ (t)=mt ^m-1/ η ^m

Wherein, t is time between failures, the form parameter that m is Two-parameter Weibull Distribution, and η is scale parameter, F (t) is unreliable degree function;

Step 22: the estimated value of utilizing small sample fail data to carry out fitting of distribution to obtain one group of form parameter m and the scale parameter η of Weibull distribution;

Step 23: by above-mentioned parameter estimated value substitution Weibull distribution model, and utilize inverse function method to carry out Monte Carlo sampling to obtain Bootstrap increment, that is, obtain M and organize the bootstrap that every group of sample size is n, carry out respectively fitting of distribution and obtain the form parameter m of M group Weibull distribution _iwith scale parameter η _i, and the representation g (m=m of definite prior distribution _i, η=η _j)=1/M ², wherein, i=1 ..., M, j=1 ..., M;

Step 24: utilize small sample fail data and the prior distribution of relay protection system, obtain Bayes-Monte Carlo assessment models according to bayesian theory,

$g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j|x)=\frac{f\left(x\right|m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_i)}{\&Integral;f\left(x\right|m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)}$

Wherein, f (x|m=m _i, η=η _j) be likelihood estimation function, g (m=m _i, η=η _j) be prior density function, x is sample observations, is protective relaying device time between failures;

Step 3: in conjunction with Bayes-Monte Carlo assessment models, each relay protection system is carried out to reliability assessment.

The method of choosing increment given figure M in described step 23 is with " M-fiduciary level value " the consequent stability criterion of slope of a curve under a certain operation hours, be under a certain operation hours, when M is [20, 300] in scope, if for different M values and corresponding fiduciary level value R, | Δ R/ Δ M| keeps minimum and close to 0 o'clock, assessment result is subject to M value to affect minimum, now using M the minimum value in this segment limit as stabilized reference value, choose this value as final M value, wherein Δ M is the changing value of increment given figure, desirable fixed value, represent that M is [20, 300] change interval in scope, Δ R is fiduciary level during with increment given figure changes delta M, the changing value of fiduciary level.

In described step 3, in conjunction with Bayes-Monte Carlo assessment models, each relay protection system is carried out to reliability assessment, specifically comprises:

Step 31: set reliability foundation according to industrial actual conditions, provide fiduciary level reference threshold;

Step 32: for each relay protection system, calculate the indexs such as its fiduciary level, mean time between failures, crash rate and probability density of failure by Bayes-Monte Carlo assessment models,

The point estimation of Reliability Function is:

$R\left(t\right)=\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{R}_{\mathrm{ij}}\left(t\right)g\left[i\right]\left[j\right]=\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}g\left[i\right]\left[j\right]\exp [-{(t/{\mathrm{\&eta;}}_j)}^{m_i}]$

The point estimation of fault probability function is:

$f\left(t\right)\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{f}_{\mathrm{ij}}\left(t\right)g\left[i\right]\left[j\right]=\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}g\left[i\right]\left[j\right]\frac{m_i{t}^{m_i-1}}{{\mathrm{\&eta;}}^{m_i}}\exp [-{(t/{\mathrm{\&eta;}}_j)}^{m_i}]$

Point estimation that can gain and loss efficiency function by Reliability Function and fault probability function is:

λ(t)＝f(t)/R(t)

Mean time between failures:

$\mathrm{MTBF}=\underset{0}{\overset{\&infin;}{\&Integral;}}R\left(t\right)\mathrm{dt}$

Wherein, R _ij(t) represent m=m _i, η=η _j(i=1 ..., M; J=1 ..., M) time corresponding Reliability Function value, g[i] [j] be g (m=m _i, η=η _j), be stochastic variable m=m _i, η=η _j(i=1 ..., M; J=1 ..., M) time joint probability density value;

Step 33: fiduciary level is less than a certain threshold value, keeps in repair or safeguards, and the reference frame that the mean time between failures can be used as protective device life prediction and formulates optimal repair cycle.And crash rate and probability density of failure index can be used as the basic data of other reliability estimation method.

The present invention has realized relay protection system reliability estimation method under small sample fail data, and its beneficial effect is as follows:

The 1 actual motion feature in conjunction with protective relaying device " reliability is high, fail data is few ", Bayes-Monte Carlo appraisal procedure of research protective relaying device reliability.Compare the reliability estimation method of tradition based on large inefficacy sample; the method has fully utilized Monte Carlo method and has been subject to problem dimension to affect little feature and bayes method effectively in conjunction with the prior imformation of multiple source, various ways; obtain the advantage of more complete posterior information; do not need very large inefficacy sample can obtain better probability estimate value, more meet the operation reality of relay protection system.

2 distribute because Two-parameter Weibull Distribution does not exist conjugation, can not directly carry out Bayesian formula calculating by using the conjugate prior of Weibull distribution to distribute, the present invention utilizes the joint probability density of the estimates of parameters that Bootstrap increment draws as prior distribution, both fully utilize prior imformation, solved again the problem that under Weibull distribution, Bayesian formula is difficult to solve.In addition, adopt the Monte Carlo method Bootstrap increment obtaining of sample can reduce the dependence to small sample data as Bayesian prior distribution, the robustness of raising parameter estimation.

The index that 3 methods obtain can be used for the reference frame of time between overhauls(TBO) as MTBF, and crash rate etc. can be used as the basic data in reliability estimation method.

Brief description of the drawings

Fig. 1 is fiduciary level temporal evolution curve.

Fig. 2 is crash rate temporal evolution curve.

Fig. 3 is failure probability density temporal evolution curve.

Embodiment

Below in conjunction with accompanying drawing, preferred embodiment is elaborated.Should be emphasized that, following explanation is only exemplary, instead of in order to limit the scope of the invention and to apply.

The thinking that the present invention deals with problems is relay protection system reliability estimation method under small sample fail data: first, integrate relay protection system inefficacy sample; Then, establish using Weibull distribution as inefficacy distributed model by the test of fitness of fot, and utilize Monte Carlo method sampling to expand data sample, jointly as prior imformation, build Bayes-Monte Carlo assessment models with small sample raw data; Finally, in conjunction with Bayes-Monte Carlo assessment models, each relay protection system is carried out to reliability assessment.

Set forth relay protection system reliability estimation method under small sample fail data below in conjunction with accompanying drawing and example.

Suppose: relay protection system fault causes because of hardware unaccelerated aging or inefficacy.Do not comprise the human factor such as connection error, design problem.

The method comprises the steps:

Step 1: relay protection system inefficacy sample is integrated, calculates uptime interval, and choosing the maximum time interval is closing time.

Be provided with original failure data t ₁, t ₂..., t _n, i.e. t ₁, t ₂..., t _nfor time between failures, can calculate and obtain according to the difference of equipment failure moment and equipment investment time of running.T _sfor closing time is (as time between failures t>T _stime time between failures t will be not counted in sample), and t ₁≤ t ₂≤ ...≤t _nfor censored data.Capacity n (n is the fail data number of record) is generally less than 30.

Step 2: build Bayes-Monte Carlo assessment models, wherein build model and specifically comprise:

Step 21: according to the life distribution type of device, adopt Weibull distribution to distribute as estimating, and carry out the inspection of fitting of distribution goodness.

The probability density of failure of Two-parameter Weibull Distribution:

$f\left(t\right)=\frac{m{t}^{m-1}}{{\mathrm{\&eta;}}^m}\exp [-{(t/\mathrm{\&eta;})}^m]---\left(2\right)$

Reliability Function:

R(t)＝1-F(t)＝exp[-(t/η) ^m] (3)

Crash rate function:

λ(t)＝mt ^m-1/η ^m(4)

Wherein, t is time between failures, the form parameter that m is Two-parameter Weibull Distribution, and η is scale parameter, F (t) is unreliable degree function.Fiduciary level refers to equipment in official hour t and under defined terms, completes the probability of predetermined function.The Reliability Function R (t) of general relay protection represents that working as operation hours is the probability that t can complete predetermined function.Crash rate refers to the equipment that equipment work was not yet lost efficacy to the t moment, and after this moment, in the unit interval, the frequency losing efficacy occurs product, is the instantaneous failure rate of equipment.

First utilize existing fail data to carry out fitting of distribution by present conventional least square method etc. and determine one group of form parameter and scale parameter.

For ensureing the credibility of fail-safe analysis result, relay protection fail data is carried out to the test of fitness of fot, to verify whether sample meets Weibull distribution.The conventional fitting of distribution method of inspection mainly contains χ ²method of inspection, W method of inspection, normal distribution-test method and Andrei Kolmogorov-Si Mil Lip river husband (K-S) method of inspection etc.χ ²method of inspection is applicable to large sample (sample size is greater than 50) situation.W inspection, for checking overall normality, is applicable to the data of sample size between [8,50].K-S test rules is applicable to the inspection of the less situation lower probability of sample number distribution pattern, and checks the deviation between empirical distribution function and the theoretic distribution function of institute's matching.Therefore adopt K-S method of inspection to carry out the inspection of Weibull Distribution goodness.

First K-S method of inspection presses n fail data sequence from small to large, according to the distribution of hypothesis, calculates null hypothesis distribution function F0 (x corresponding to each data _i), then by itself and empirical distribution function F _n(x _i) compare, wherein the absolute value of difference maximum is test statistics D _nobserved value, by D _nwith critical value D _n,acompare, satisfy condition and accept null hypothesis, otherwise refusal null hypothesis:

D _n＝sup|F _n(x)-F ₀(x)|＝maxd _i≤D _n,a(5)

In formula, a is given confidence level, generally gets a=0.05.

$F_n\left(x\right)=\left\{\begin{array}{c}0,x<x_i\\ i/n,x_i\&le;x<x_{i+1}\\ 1,x\&GreaterEqual;x_n\end{array}\right.---\left(6\right)$

$d_i=\max \{F_0\left(x_i\right)-\frac{i-1}{n},\frac{i}{n}-F_0\left(x_i\right)\}---\left(7\right)$

In the time of a=0.05, look into tables of critical values known:

Step 22: utilize small sample fail data to carry out fitting of distribution and obtain one group of form parameter of Weibull distribution and the estimated value of scale parameter.

This method is carried out calculation of parameter based on fail data, therefore by pressing the uptime length sequence of equipment after the data filtering of correct operation, obtain time series t _i, as raw data.The Weibull distribution Reliability Model of utilizing formula (3) to set up, by least square fitting, obtains estimating form parameter and the scale parameter estimated value of Weibull distribution.Specifically can the narration of table 1 and table 2 part in " embodiment 1 " analysis referring to below.

Particularly, first formula (3) being got to twice logarithm continuously obtains

Order: $y=\ln \ln \frac{1}{1-F\left(t\right)},x=\ln t,a=-m\ln \mathrm{\&eta;},b=m,$ There is y=a+bx.

Then according to observation data utilize least square method can obtain regression coefficient m, η.And then can obtain the expression formula of failure rate function and Reliability Function etc.The F (ti) is here empirical distribution function, can choose the function shown in formula (6), also can calculate by approximate meta order formula.

Step 23: by above-mentioned parameter estimated value substitution Weibull distribution model, and utilize inverse function method to carry out Monte Carlo sampling to obtain Bootstrap increment, that is, obtain M and organize the bootstrap that every group of sample size is n, carry out respectively fitting of distribution and obtain the form parameter m of M group Weibull distribution _iwith scale parameter η _i, and the representation g (m=m of definite prior distribution _i, η=η _j)=1/M ², wherein, i=1 ..., M, j=1 ..., M.

The method of choosing increment given figure M in step 23 is with " M-fiduciary level value " the consequent stability criterion of slope of a curve under a certain operation hours, be under a certain operation hours, when M is [20, 300] in scope, if for different M values and corresponding fiduciary level value R, | Δ R/ Δ M| keeps minimum and close to 0 o'clock, assessment result is subject to M value to affect minimum, now using M the minimum value in this segment limit as stabilized reference value, choose this value as final M value, wherein Δ M is the changing value of increment given figure, desirable fixed value, represent that M is [20, 300] change interval in scope, Δ R is fiduciary level during with increment given figure changes delta M, the changing value of fiduciary level.

Definite Weibull distribution parameters m, η, sets it as given value above, and substitution formula (8) is sampled, and produces M and organizes the bootstrap that every group of sample size is n (equaling original sample number).M recommends to get 100, and it is for little inefficacy sample, and sample number enough meets accuracy requirement.Sampling first obtains the upper equally distributed pseudorandom array e[n in interval (0,1)].Must be obeyed the stochastic variable of Weibull distribution by inverse function method, sampling formula is as follows again:

t[n]＝η(-ln(e[n])) ^1/m(8)

Above formula is the inverse function of Reliability Function (3), by the pseudorandom array e[n that sampling is obtained] alternative Reliability Function value, and utilize in the hope of parameter m, η try to achieve sampling fail data t[n], required t[n] be the analog data sample that has similar Changing Pattern to original failure data sample.

By t[n] by arranging from small to large, and every group of data from the sample survey and closing time are compared, for example, for l data from the sample survey, if t[l] >T _s, t[l] and=T _s, be met the data from the sample survey of the truncation feature of relay protection basic data.Then utilize regression analysis to obtain the parameter m of Weibull distribution to every group of data from the sample survey respectively _i, η _i, its numerical value is stored in respectively to array m[i], in η [i], (can go out one group of parameter m according to Weibull distribution model the Fitting Calculation for every group of bootstrap fail data _i, η _i).

Suppose m _i, η _iseparate, the discrete priori joint probability density that can obtain m and η distributes:

$\left|\begin{array}{c}(m_1,{\mathrm{\&eta;}}_1),(m_1,{\mathrm{\&eta;}}_2)......(m_1,{\mathrm{\&eta;}}_M)\\ (m_2,{\mathrm{\&eta;}}_1),(m_2,{\mathrm{\&eta;}}_2)......(m_2,{\mathrm{\&eta;}}_M)\\ ......\\ (m_M,{\mathrm{\&eta;}}_1),(m_M,{\mathrm{\&eta;}}_2)......(m_M,{\mathrm{\&eta;}}_M)\end{array}\right|$

By the known stochastic variable m of probability theory _i, η _ijoint probability density be constant g (m=m _i, η=η _j)=1/M ², i=1 ..., M, j=1 ..., M.Set it as the prior distribution of Bayesian formula.

Step 24: utilize small sample fail data and the prior distribution of relay protection system, obtain Bayes-Monte Carlo assessment models according to bayesian theory,

$g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j|x)=\frac{f\left(x\right|m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_i)}{\&Integral;f\left(x\right|m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)}$

Wherein, f (x|m=m _i, η=η _j) be likelihood estimation function, g (m=m _i, η=η _j) be prior density function, x is sample observations, is protective relaying device time between failures.

Fault probability function using Two-parameter Weibull Distribution distributes as the condition of the sample value under given parameters, that is:

$f\left(t_i\right|m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_i)=\frac{m_{i}t{\left[i\right]}^{m_i-1}}{{{\mathrm{\&eta;}}_j}^{m_i}}\exp [-{\left(t\right[i]/{\mathrm{\&eta;}}_j)}^{m_i}]---\left(9\right)$

Likelihood function can be expressed as:

$\begin{array}{c}f\left(x\right|m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)=\underset{i=1}{\overset{n}{\mathrm{\&Pi;}}}f\left(x_i\right|m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)\\ ={m_i}^r{{\mathrm{\&eta;}}_j}^{-r{m}_i}{U}^{m_i-1}\exp (-X/{{\mathrm{\&eta;}}_j}^{m_i})\end{array}---\left(10\right)$

Wherein, $U=\underset{i=1}{\overset{r}{\mathrm{\&Pi;}}}t\left[i\right],X=\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}t{\left[k\right]}^{m_i}.$

Prior distribution is constant:

g(m＝m _i,η＝η _j)＝1/M ²(11)

, according to Bayesian formula, can obtain m and η posterior probability and distribute suc as formula shown in (12).In formula, i, j=1,2 ..., M.Required result is in conjunction with prior imformation and Monte Carlo sampling sample information, through m and the η probability distribution of Bayesian formula correction.

$\begin{array}{c}g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j|x)=\frac{f\left(x\right|m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)}{f\left(x\right|m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)}\\ =\frac{{m_i}^r{{\mathrm{\&eta;}}_j}^{-r{m}_i}{U}^{m_i-1}\exp (-X/{{\mathrm{\&eta;}}_j}^{m_i})}{\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{m_i}^r{{\mathrm{\&eta;}}_j}^{-r{m}_i}{U}^{m_i-1}\exp (-X/{{\mathrm{\&eta;}}_j}^{m_i})}\end{array}---\left(12\right)$

Step 3: in conjunction with Bayes-Monte Carlo assessment models, each relay protection system is carried out to reliability assessment, specifically comprise:

Step 31: set reliability foundation according to industrial actual conditions, provide fiduciary level reference threshold.

Step 32: for each relay protection system, calculate the indexs such as its fiduciary level, mean time between failures, crash rate and probability density of failure by Bayes-Monte Carlo assessment models;

The parameter m of trying to achieve according to bayesian theory and η probability distribution, the point estimation that can try to achieve Reliability Function is:

$R\left(t\right)=\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{R}_{\mathrm{ij}}\left(t\right)g\left[i\right]\left[j\right]=\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}g\left[i\right]\left[j\right]\exp [-{(t/{\mathrm{\&eta;}}_j)}^{m_i}]---\left(13\right)$

The point estimation of fault probability function is:

$f\left(t\right)\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{f}_{\mathrm{ij}}\left(t\right)g\left[i\right]\left[j\right]=\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}g\left[i\right]\left[j\right]\frac{m_i{t}^{m_i-1}}{{\mathrm{\&eta;}}^{m_i}}\exp [-{(t/{\mathrm{\&eta;}}_j)}^{m_i}]---\left(14\right)$

Point estimation that can gain and loss efficiency function by Reliability Function and fault probability function is:

λ(t)＝f(t)/R(t) (15)

Mean time between failures:

$\mathrm{MTBF}=\underset{0}{\overset{\&infin;}{\&Integral;}}R\left(t\right)\mathrm{dt}---\left(16\right)$

Wherein, R _ij(t) represent m=m _i, η=η _j(i=1 ..., M; J=1 ..., M) time corresponding Reliability Function value, g[i] [j] be g (m=m _i, η=η _j), be stochastic variable m=m _i, η=η _j(i=1 ..., M; J=1 ..., M) time joint probability density value.

Step 33: fiduciary level is less than a certain threshold value, keeps in repair or safeguards, and the reference frame that the mean time between failures can be used as protective device life prediction and formulates optimal repair cycle.And crash rate and probability density of failure index can be used as basis (input) data of other reliability estimation method.

Embodiment 1:

Economize the failure logging (record) of protective relaying device of grid company and each attribute information as example taking certain below, specifically introduce relay protection system reliability estimation method under small sample fail data.

Table 1 logout table

Table 1 is to economize 67 certain model protective relaying devices of grid company under approximately uniform maintenance levels and operating mode from certain, in the failure logging (record) of 2005～2007 years, filters out the failure logging (record) of relatively complete relay system, and each attribute information.

The inefficacy here specifically refers to tripping and the malfunction of relay protection system." malfunction " and " tripping " is the proper noun in relay protection field.Corresponding is the correct operation of relay protection system with it; be inside, protected object inside/protection zone while breaking down, corresponding protective relaying device/system should automatically be sent out trip signal and drive corresponding circuit breaker trip with isolated fault (i.e. so-called action).And " malfunction " when referring to protected object non-fault or protected object external fault, the relay protection system action of protected object, belongs to " misoperation "; Tripping refers to that there is fault protected object inside, and the relay protection system of protected object should be moved but it does not have auto-action.Cause the reason of both of these case more, have hardware aging, fault, wiring error, design problem etc., the data that the present invention chooses are all because hardware unaccelerated aging or inefficacy cause.

The method comprises the steps:

Step 1: relay protection system inefficacy sample is integrated, calculates uptime interval, and choosing the maximum time interval is closing time.

According to table 1, choosing last out-of-service time 2007-12-25 is closing time, i.e. T _s=20407h.

Step 2: build Bayes-Monte Carlo assessment models, wherein build model and specifically comprise:

Step 21: according to the life distribution type of device, adopt Weibull distribution to distribute as estimating, and carry out the inspection of fitting of distribution goodness.

The K-S method of inspection describing in detail before utilizing has carried out goodness inspection, meets the requirements.

This method is carried out calculation of parameter based on fail data, therefore by pressing the time length sequence of the normal operation of equipment after the data filtering of correct operation, as shown in table 2.Set up Weibull distribution model, through least square fitting, must estimate Weibull distribution parameters m=1.486, η=79359.988.

Table 2 fault data

Step 22: utilize small sample fail data to carry out fitting of distribution and obtain one group of form parameter of Weibull distribution and the estimated value of scale parameter.

Set up Weibull distribution model, through least square fitting, must estimate Weibull distribution parameters m=1.486, η=79359.988.

Step 23: by above-mentioned parameter estimated value substitution Weibull distribution model, and utilize inverse function method to carry out Monte Carlo sampling to obtain Bootstrap increment, carry out fitting of distribution respectively and obtain the parameter estimation of every group of increment, as prior imformation, determine the representation of prior distribution.

Using this estimated parameter value as known parameters, by formula (8) sampling, producing M group sample size is n (equal'sing original sample number) bootstrap.

Step 24: utilize small sample fail data and the prior distribution of relay protection system, obtain Bayes-Monte Carlo assessment models according to bayesian theory.

The parameter estimation of every group of Bootstrap sample of table 3

Every group of data from the sample survey and closing time are compared, be met the data from the sample survey of the truncation feature of relay protection basic data.To every group of data from the sample survey utilization, the Weibull distribution model based on least square fitting is obtained corresponding parameter array m[i respectively], η [i] result is as shown in table 3.Utilize m _i, η _jtry to achieve m, the probability distribution of η by Bayesian formula shown in formula (9).

Step 3: in conjunction with Bayes-Monte Carlo assessment models, each relay protection system is carried out to reliability assessment, specifically comprise:

Step 31: set reliability foundation according to industrial actual conditions, provide fiduciary level reference threshold MTBF;

Step 32: for each relay protection system, calculate the indexs such as its fiduciary level, mean time between failures, crash rate and probability density of failure by Bayes-Monte Carlo assessment models;

Near protection operation hours reaches mean time between failure time, should strengthen monitoring, or be overhauled or change, trouble-saving generation as early as possible, improves the operational reliability of protection system and even electric system.In addition in the time that the fiduciary level of protecting is lower, also should strengthen monitoring.

And then can draw the index such as fiduciary level, probability density of failure and crash rate of equipment operation, the index value of typical case's point working time is as shown in table 4.

Obtain mean time between failure=13016 by formula (16).Each reliability index function curve respectively as shown in FIG. 1 to 3.Wherein, Fig. 1 is fiduciary level temporal evolution curve, and Fig. 2 is crash rate temporal evolution curve, and Fig. 3 is failure probability density temporal evolution curve.

Table 4 reliability index

From table 4 and Fig. 1-Fig. 3: first, because not existing conjugation, Two-parameter Weibull Distribution distributes, distribute for Bayesian formula by the condition using the fault probability function of Weibull distribution as the sample value under given parameters, make the Weibull distribution model of original hypothesis after Bayesian formula correction, it is no longer strict Weibull distribution, from function curve, can obtain preliminary identification, but main trend is roughly the same.The actual contrast of operation of its variation tendency and reliability classical theory and protective device is also effective to a certain degree having proved that this method utilizes small sample data to carry out reliability assessment.Secondly, probability density of failure and crash rate are also non-vanishing in the short period after putting equipment in service, but reduce rapidly with the increase of working time, have reflected to a certain extent the initial failure of protective device.

Step 33: fiduciary level is less than a certain threshold value, keeps in repair or safeguards, and the reference frame that the mean time between failures can be used as protective device life prediction and formulates optimal repair cycle.And crash rate and probability density of failure index can be used as basis (input) data of other reliability estimation method.

For contrasting with classical Weibull distribution model, adopt Weibull distribution model to carry out parameter estimation and obtain m=1.4860, η=79359.9882, its MTBF is about 40000h.According to this method raw data, utilize MTBF definition and engineering calculating method, have: MTBF=(4399+5862+9582+9606+13327+16158+17622+20407)/8=12120.375h.Visible the former more meets the failure conditions that raw data reflects, this method can adapt to the reliability assessment problem that small sample lost efficacy to a certain extent better.

The parameters such as the crash rate of calculating by the method can be used as the underlying parameter of the analytic method conventional such as the relay protection system such as Markov model, fault tree reliability assessment; higher than the ethical basis parameter accuracy of directly calculating according to original small sample data statistics, be more conducive to field operator and grasp the true operation conditions of protective relaying device.

The above; only for preferably embodiment of the present invention, but protection scope of the present invention is not limited to this, is anyly familiar with in technical scope that those skilled in the art disclose in the present invention; the variation that can expect easily or replacement, within all should being encompassed in protection scope of the present invention.Therefore, protection scope of the present invention should be as the criterion with the protection domain of claim.

Claims (3)

1. a relay protection system reliability estimation method under small sample fail data,

Suppose: relay protection system fault causes because of hardware unaccelerated aging or inefficacy;

It is characterized in that described method comprises the following steps:

Step 1: relay protection system inefficacy sample is integrated, calculates uptime interval t ₁, t ₂..., t _n, choosing the maximum time interval is T closing time _s, wherein, n is inefficacy total sample number;

Step 2: build Bayes-Monte Carlo assessment models, wherein build model and specifically comprise:

Step 21: according to the life distribution type of device, adopt Weibull distribution to distribute as estimating, and carry out the inspection of fitting of distribution goodness, whether tally with the actual situation to confirm that the life-span of Selection Model distributes,

The probability density of failure of Two-parameter Weibull Distribution is

Reliability Function is R (t)=1-F (t)=exp[-(t/ η) ^m]

Crash rate function is λ (t)=mt ^m-1/ η ^m

Wherein, t is time between failures, the form parameter that m is Two-parameter Weibull Distribution, and η is scale parameter, F (t) is unreliable degree function;

Step 22: the estimated value of utilizing small sample fail data to carry out fitting of distribution to obtain one group of form parameter m and the scale parameter η of Weibull distribution;

Step 23: by above-mentioned parameter estimated value substitution Weibull distribution model, and utilize inverse function method to carry out Monte Carlo sampling to obtain Bootstrap increment, that is, obtain M and organize the bootstrap that every group of sample size is n, carry out respectively fitting of distribution and obtain the form parameter m of M group Weibull distribution _iwith scale parameter η _i, and the representation g (m=m of definite prior distribution _i, η=η _j)=1/M ², wherein, i=1 ..., M, j=1 ..., M;

Step 24: utilize small sample fail data and the prior distribution of relay protection system, obtain Bayes-Monte Carlo assessment models according to bayesian theory,

$g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j|x)=\frac{f\left(x\right|m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_i)}{\&Integral;f\left(x\right|m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)g(m=m_i,\mathrm{\&eta;}={\mathrm{\&eta;}}_j)}$

Wherein, f (x|m=m _i, η=η _j) be likelihood estimation function, g (m=m _i, η=η _j) be prior density function, x is sample observations, is protective relaying device time between failures;

Step 3: in conjunction with Bayes-Monte Carlo assessment models, each relay protection system is carried out to reliability assessment.

2. relay protection system reliability estimation method under a kind of small sample fail data according to claim 1, the method that it is characterized in that choosing in described step 23 increment given figure M is with " M-fiduciary level value " the consequent stability criterion of slope of a curve under a certain operation hours, be under a certain operation hours, when M is [20, 300] in scope, if for different M values and corresponding fiduciary level value R, | Δ R/ Δ M| keeps minimum and close to 0 o'clock, assessment result is subject to M value to affect minimum, now using M the minimum value in this segment limit as stabilized reference value, choose this value as final M value, wherein Δ M is the changing value of increment given figure, desirable fixed value, represent that M is [20, 300] change interval in scope, Δ R is fiduciary level during with increment given figure changes delta M, the changing value of fiduciary level.

3. relay protection system reliability estimation method under a kind of small sample fail data according to claim 1, is characterized in that in described step 3, in conjunction with Bayes-Monte Carlo assessment models, each relay protection system being carried out to reliability assessment, specifically comprises:

Step 31: set reliability foundation according to industrial actual conditions, provide fiduciary level reference threshold;

Step 32: for each relay protection system, calculate the indexs such as its fiduciary level, mean time between failures, crash rate and probability density of failure by Bayes-Monte Carlo assessment models,

The point estimation of Reliability Function is:

$R\left(t\right)=\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{R}_{\mathrm{ij}}\left(t\right)g\left[i\right]\left[j\right]=\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}g\left[i\right]\left[j\right]\exp [-{(t/{\mathrm{\&eta;}}_j)}^{m_i}]$

The point estimation of fault probability function is:

$f\left(t\right)\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{f}_{\mathrm{ij}}\left(t\right)g\left[i\right]\left[j\right]=\underset{i=1}{\overset{M}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}g\left[i\right]\left[j\right]\frac{m_i{t}^{m_i-1}}{{\mathrm{\&eta;}}^{m_i}}\exp [-{(t/{\mathrm{\&eta;}}_j)}^{m_i}]$

Point estimation that can gain and loss efficiency function by Reliability Function and fault probability function is:

λ(t)＝f(t)/R(t)

Mean time between failures:

$\mathrm{MTBF}=\underset{0}{\overset{\&infin;}{\&Integral;}}R\left(t\right)\mathrm{dt}$

Wherein, R _ij(t) represent m=m _i, η=η _j(i=1 ..., M; J=1 ..., M) time corresponding Reliability Function value, g[i] [j] be g (m=m _i, η=η _j), be stochastic variable m=m _i, η=η _j(i=1 ..., M; J=1 ..., M) time joint probability density value;

Step 33: fiduciary level is less than a certain threshold value; keep in repair or safeguard; and the reference frame that the mean time between failures can be used as protective device life prediction and formulates optimal repair cycle, and crash rate and probability density of failure index can be used as the basic data of other reliability estimation method.