Reliability analysis of power systems EI2452 Lifetime analysis 7 May 2015
Agenda Summary of content: Introduction nonreparable/reparable General information about statistical surveys Lifetime data Nonparametric method TTT-plot Parametric method Weibull analys Hypotesis testing - Goodness-of-fit and Trend
Introduction More litterature about lifetime analysis: Introduction: System Reliability Theory [Rausand, Höyland 2004] Kapitel 11. Advanced Books: Statistical models and methods for lifetime data [Lawless2003], Reliability Modeling, Prediction and Optimization [Blischke, Prabhakar Murthy 2000]
Introduction Prior to a reliability study, the components are classified as either: 1. Nonreparable, or 2. reparable
Introduction 1. Nonreparable components These components are observed until they break Can only fail once This might be because: The component cannot be repaired, It is not economically motivated to do so, or That only the first failure is of interest for the study Is characterized by mean time to failure (MTTF)
Introduction 2. Reparable components These components are repaired upon failure Their lifetime history is dependent on functional states and nonfunctional states Is characterized by mean time between failures (MTBF)
Introduction When you fit a statistical distribution to lifetimes you implicitly make the assumption that the components are nonrepabable. One exception is when a component could be assumed to be brought back to the same condition as before the failure (after being repaired). This could be tested with something called a trend test. Then, this method is also valid for reparable components. There are other models and methods for reparable components that doesn t go back to the same condition after being repaired (stochastic processes, not covered in this course )
Introduction The traditional statistical models are either: Nonparametric (Ex. TTT, median ranger ) Parametric (Ex. Exponential or Weibull distribution) Parametric models are more hands on in advanced analysis Nonparameteric models could be used to test wether or not a parametric model is suitable
Statistical Study A statistical study is performed in four steps: 1. Planning (What problem, hypothesis, population? Good to make reasonable estimations. Define terminology, failure.) 2. Collect data (Ex. experiments or interviews) 3. Processing (Ex. Summarize in tables/diagrams or statistical analysis) 4. Presentation (graphical presentation, results, conclusions, recommendations)
Statistical Study Basic Terminology Population and element/attributes Element/attributes is a certain characteristic of a studied object The population is a group of elements (observations/data) It is important to be clear on what an element is and which elements that are included in the population
Statistical study Basic Terminology Discrete and Continuous variation Ex. Element could be lifetime, color, size, the occurrance of a certain cooling mode One Element could be a combination such as (size, lifetime, cooling mode) Discrete variation is elements that could either be a finite or infitite number (Ex. colors) Continuous variation is elements that could attain any number in an interval (Ex. measurement values)
Statistical study Basic Terminology A complete investigation The entire population is studied Ex. the number of births, total production Expensive and time consuming Sometimes principally or practically unthinkable Does not require any statistical theory (Only analysis, description of data, no inference)
Statistical study Basic Terminology Sample study Considers a part of the population Ex. opinion poll, statistical quality control, reliability prognosis Could give a larger uncertainty than complete investigation Ex. For the control of massproduced units the sample study is often preferable
Statistical study Model for sample study The distribution F is partially unknown, it is dependent of a unknown parameter θ The different values that θ could attain creates the parameter room A. A Sample study is performed to determine θ. The sample test gives the values t 1, t n which i assumed to be observations of the independent stochastic variables T 1,..T n which has the same distribution F
Statistical study Model for sample study 1. Point estimation 2. Interval estimation 3. Hypothesis testing (Qualityy control, goodness-of-fit...) 4. Descision problem
Lifetime data When the reliability of a system should be determined, it is important to know how the system function and included subsystems and components change with time. What is needed is a knowledge of probability distributions.
Lifetime data Models often contain a number of unknown parameters. These parameters needs to be estimated from the collected data. The estimations are based on input data. These input data could be for instance predicted values, experienced-based values or values from simulations
Lifetime data In order to attain information about a certain probaility distribution F(t) for a system/component Perform a trial where n identical, numbered units of a certain component is activated and the lifetime for every unit is observed.
Lifetime data If the test could continue until all units has failed and all lifetimes are observed The resulting material is called complete
Lifetime data Often you will have to work with uncomplete data sets, i.e. the entore sample is not known. The reason for this could be that the trial cannot be completed due to practical or economical reasons In these situations the data set is called censured
Lifetime data Ex. for censured lifetimes Enhet 0 1 2 X Fel 3...... X Fel n 0 t 0 Tid
Lifetime data Right censoring Component i has not failed at time t i which gives that T>t i Right censoring often occur during inspections when no failures has been found.
Lifetime data Left censoring No information is available for when the component was put into operation (or possible earlier failures for reparable components). This situation occurs when for instance the documentation for a component has been lost.
Lifetime data Interval censoring Component i has failed between the times t i-1 and t i which implies that t i-1 <T t i This situation occurs for instance when a component fails between two inspections.
Lifetime data In this course, the lifetimes for the n components is assumed to be stochastically independent with the same distribution and continuous probability distribution F(t). When the studied units are nonreparable, this assumption means that recurring processes, renewable processes, is assumed to be independent and identically distributed lifetimes Läs hit
Total Time at Test - TTT TTT-plott (Total Time at Test plot) Is a nonparametric graphical method Determined an empirical probability distribution F n (t) Could be used to determine if the unknown probaility distribution F(t) have: increasing failure intensity (IFR), descreasing failure intensity (DFR) or constant failure intensity. Could be used for visual goodness-of-fit test Could be used for maintenance optimization (More info: Maintenance, Replacement, and Reliability: Theory and Applications [Jardine, Tsang 2005])
TTT Summary TTT-plot method (cont): Assume that all n units is put into operation at the time t = 0 and that the observation stops at the time t. Total testing time, I(t), is the totally observed lifetime for the n units at time x. The total testing time at failure i is hence: I i ( T( i )) = T( j ) + ( n i) T ( i ) för i = 1,2,..., n j= 1
TTT TTT-plot method (cont): In particular: n ( ) ( n) ( j) I T = T = T n j= 1 j= 1 j Scaling of total testing time at failure i,, could be done by dividing with ( ) ( ) I T n I( T ) ( i ) Hence the relative total, testing time at time t is defined as: I I ( t ) ( T ) ( n )
TTT TTT-plot method (cont): If the following numbers are plotted: ( ) i T( i), I n I( T ) ( n) A so called TTT-plot is created for the data set In order to arrive at conclusions of the underlying lifetime distribution F(t) (from the TTT-plot) one more theorem is needed.
TTT TTT-plot method (cont): If the underlying lifetime distribution is exponentially distributed and n is a large number, then T( X ) ( i) T X ( ) i n för i = 1, 2,..., 1 ( n ) ( n) Hence, it is possible to read directly from the TTT-plot if the probability distribution is exponentially distributed. Since the plot becomes a straight line between origo and one.
Conclusions TTT Summary of the solving procedure using a TTT-plot: 1. Calculate the values and put them in a Table. 2. Plot (i/n,t i /T n ) 3. Study the characteristics of the curve and arrive at conclusions about the failure intensity IFR if concave DFR if convex Constant if straight line
Concave convex Increasing Decreasing, convex
Exercise TTT Example During the development process of an entirely new typ of isolation material, one has tested for stresses five samples and observed the following lifetimes [p.u.]: 5.2 11.8 3.2 7.0 8.8 Make a TTT-plot How does the failure intensity appear? Optimal time for replacement (System Reliability Theory s. 497)
TTT exercise - solution
Weibull analysis f Weibull analysis is a parametric method where the parameters for the Weibull distribution is estimated The Weibull distribution is commonly found in the field of reliability both because of its flexibility but also the interpretation???? The probability density function is defined as: β 1 β t β t η ( t β, η) = e f ( t) 0, β 0, η 0 η η where β is the shape parameter and η the schale parameter (sometimes also the characteristic life) d α α 1 ( λt) f ( t) = F ( t) = αλ ( λt) e för t 0 dt Tidigare 1/n= lambda beta = alfa
Weibull analysis The failure intensity function for a component is a measure of the probability that the component will fail within the next time interval, the time step t+δt, conditioned that it was working at the time t The failure intensity function for the Weibull distribution is defined according to: zt ( βη, ) β t = η η β 1 zt ( ) 0, β 0, η 0
Weibull analysis The density of the Weibull destribution for a constant scale parameter and different shape parameters.
Weibull analysis The failure intensity function for the Weibull distribution for different shape parameters and constant scale parameter
Weibull analysis When β=1, the Weibull distribution is reduced to the exponential distribution expressed as f ( t η) = 1 e η t η Where 1/η=λ is the constant failure intensity
Weibullanalys Weibull paper (alternative to softwares such as Minitab) The parameters β and η could easily be graphically estimated with a so called Weibull paper The axes in the diagram is scales so that the distribution function becomes a straight line For appropriate estimations of the distribution function (read weibull) the points automatically generate a straight line If the point estimates doesnt fit a straight line, this is an indication that data are not from a Weibull distribution
Weibullanalys Weibull paper cont. The assumption that the distribution for the components time to failure is independent is valid in the using of a Weibull paper This is often the case for nonreparable components This could in certain cases also hold for reparable components
Weibullanalys Linearization Rt () = 1 Ft () = e t η Logaritmera båda sidorna t ln[ 1 Ft ( )] = η Mult. båda sidor med -1 och invertera logaritmen ln 1 t 1 Ft () = η Logaritmera båda sidor β β 1 ln ln = βln( t) βln( η) 1 Ft () Vilken är linjär enligt Y = mx + b, där 1 Y = ln ln, m= β, X = ln( t) och b= βln( η) 1 Ft ()
Weibullanalys Median ranger Is used to plot data for the studied variable on the Weibull paper Nonparametric estimation of the distribution function F(t) (non-onesided estimation is better than the empirical estimation i/n) The most common approximation when using the median ranger is the Bernards formula: i 0.3 r i = n + 0.4
Weibullanalys Solving method for a Weibull paper 1. Rank the time to failure with respect to order of magnitude in a table 2. Calculate the medianrangerna with Bernards formula 3. Plot the point estimates on a Weibull paper 4. Estimate the Weibull parameters graphically 5. Solve the assignment
Exercise Weibull analysis Example During the development process of an entirely new typ of isolation material, one has tested for stresses five samples and observed the following lifetimes [p.u.]: 5.2 11.8 3.2 7.0 8.8 Estimate the Weibull parameters β and η with a Weibull paper.
Exercise Weibull analysis Failure (i) Ranked lifetimes (t i ) Median rank % faulty F(t) 1 3.2 0,7/5,4 = 13% 2 5.2 31 3 7.0 50 4 8.8 69 5 11.8 87 r i i 0.3 = n + 0.4
10 2.1 8.2 1. Plot the lifetimes 2. Estimate the shape parameter β 3. Estimate the scale parameter η
Exercise Analysis of Data Example: Today, an electricity distribution company turn to timebased replacements of isolators on all of their 10kV air lines. The isolators is replaced every 10th year when the crack growth (and also failure intensity) is assumed to increase with time. Your assignment is to investigate wether or not this is an optimal replacement strategy. For you help, you have the lifetimes (with respect to crack growth) for this particular isolator that you got from a lifetime test [year]: 12 23 6 11 27 1 3 28 5 4 Use either a TTT-plot or Weibullanalysis.
Hypothesis testing Before one know that they can use a certain probability distribution to model a data set, it is necessary to investigate if the chosen distribution is appropriate for the particular application. The suitability to model data with a certain probability distribution should be analytically tested. Two important tests are: The trend test (valid for reparable components) Test-of-fitness / Goodness-of-fit test
Laplace trend test If you believe that the component will be in the same condition as before the failure has been repaired, you have to test this hypothesis The Laplace trend test is a common test that is used to check for independent failures and for identically distributed lifetimes, i.e. if there is a trend Descriptions and mer trend tests is found in Statistical Methods for Reliability Data [Meeker, Escobar 1998] (s.409)
Goodness-of-fit To test for the compliance between data and a chosen probability distribution a goodness-of-fit test is used. As for instance: χ2 -test Kolmogorov-Smirnov test An overview of these are presented below. More information is found in Modeling, Prediction and Optimization [Blischke, Prabhakar Murthy 2000] (s.396)
Goodness-of-fit χ2 testet is a commonly-used and versatile test which is useful for testing for compliance between data and a number of different probability distributions.
Goodness-of-fit χ2 -test (cont.) The test is performed for data that can be classified and is appropriate only for those cases where sufficiently enough data is available Guidelines to determine of you have sufficiently enough : have at least three classes with at least five data in each class
Goodness-of-fit χ2 -test (cont.) The basic principle for the test is that values within a class is normally distributed around the expected value for the particular class* The number of observed events in each class is compared with the number of events that is expected for the same class for the hypothetical distribution * This means that the test is useful only for testing for normally distributed data.
Goodness-of-fit χ2 -test (cont.) The χ2 test is performed by comparing a calculated value for χ2 with a tabulated value (parameters: the size of the sample data and statistical significance for the test) The test is based on hypothesis testing to determine wether or not a difference is present between observed and expected number of events for each class is statistically significant
Goodness-of-fit Kolmogorov-Smirnovs test (K-S) One benefit with the K-S test is that is could be used for small sample sizes without loss of accuracy The test method is appropriate to use when there is not enough data, i.e. three classes with at least five data samples in each class (according to requirements for the χ2-method) For this method, separate samples is considered individually instead of grouped together (as in the χ2- method)
Goodness-of-fit Kolmogorov-Smirnovs test (cont.) The K-S test has been developed for continuous distributions. When the test is used for discrete distributions one obtain a conservative result which implies that (if the test include the null hypothesis) the hypothesis should be rejected
Goodness-of-fit Comparison between χ2 and K-S test One benefit with K-S is the exact distribution for the test statistic D is known even if the sample size is small. For the χ2 distribution test, the test statistic Q is approximately χ2 distributed and hence gives a bad approximation for small sample sizes.
Goodness-of-fit Comparison between χ2 and K-S test The χ2 distribution test could be used without modification for both continuous and discrete distributions. When the parameters of the distribution are estimated from sample data, the χ2 test is modified with a simple subtraction with the number of degrees of freedoms. For the K-S test there are only tabulated values for those cases when the assumed distribution is exponentially distributed or normally distributed.
Brief summary Check so that the lifetimes are independent and identically distributed (for reparable components, use the Laplace trend test) Adjust one or more distributions (Estimate the parameters with for instance Minitab, or Weibull paper for Weibull analysis) Test-of-Fit: Compare with nonparametric methods Goodness-of-fit