Definition

1. The Basic Reliability Calculations Definition Reliability is a general quality of an object – an ability to perform a desired function, sustaining the values of rated operational indicators in given limits and time according to given technical conditions. Reliability is probability that an activity of an appliance in given time and given operation conditions will be adequate to its purpose. EIA (Electronic Industry Association, USA)

2. Reliability Calculations • Reliability of single parts of networks in the time of production of project documentation • Reliability of already operated networks • Reliability in the area of control of electric power system operation

3. Numerical Representation of Reliability (Classical - reliability of elements) • failure rate  [ year-1] • mean time of failure  [ h ] • probability of failure-free run R [ - ] • probability of failure Q [ - ] • mean time between failures tS [ h ] Restoredx Notrestored objects Mean time between failuresxMean time to failure

4. Global Indices of Reliability (Reliability of electric energy supply) • Outage rate- SAIFIaverage system outage rate • (number of outages/year/consumer) • Total time of all outages - SAIDIaverage system outage time • (min/year/consumer) • Time of one outage - CAIDIaverage outage time at a customer • (min/outage)

5. l Early failure period Wear-out failure period Constant failure rate period I II III t Bathtub curve

6. The relation between the function of reliability and failure rate is: For failure rate it is valid:

7. Division of Probability of Failure Exponential division Exponential rule of failure

8. Poisson’s division If k = 0, there is probability of no failure, therefore probability of failure-free running. Weibull’s deal

9. Calculation of Reliability in Electricity Industry Obtaining of input values for reliability calculations A priori reliability– determination of reliability quantities from data of a producer. Empirical reliability–monitoring of failures in electricity industry. The empirical method is mostly used for obtaining the input values for reliability calculations, because an application of a priori reliability method requires different attitude to every element of electricity system.

10. Analysis of Distribution Network Failure Databases Exclusive outage databases had been on the rise since 1975 in the former Czechoslovakia. Unfortunately, database building stopped in1990 because of political and social changes. Thanks to the expert group CIRED Czech distributors opted for unified monitoring of global reliability indices and the reliability of selected pieces of equipment in 1999 again. Data for the reliability computation is centrally processed and analyzed at the VSB - Technical University of Ostrava since the year 2000. Collected data are often heterogeneous. It is necessary to solve the storage, indexing, and also transformation of such data. We need to create a common relational scheme for the storage of the data, a new relation makes the querying and analysis possible.

11. Database range

14. Heterogeneous Data We developed a common relational schema.

15. Results • Failure rate (year-1) • N = number of failures (-) • Z = number of elements of the given type in the network (-) • P = the considered period (year)

16. Results • Mean duration of the failure (h) • N = number of failures (-) • ti = the considered period (h)

17. Results The value tendency of reliability indices of the 22 kV cable

18. Results Comparison with methodology ČEZ 22/80

19. Results Division of failures according to their causes

20. The Main Calculation Methods of Reliability Department of electrical power engineering Markov‘s processes Method of reliabilty schemes Simulative methods

21. Method of Reliability Schemes -       make-up of reliability diagram, -         assignment of relevant reliability quantities to single elements, -         simplification of reliability diagram towards one element, Advantages: -         considered systems do not have to really exist as yet, -         procedure of solving is well-arranged and not exacting concerning mathematics, -         mathematical procedure does not require iterative calculation, -       accuracy of results depends only on the accuracy of input parameters of calculation. Disadvantages: -      it is impossible to pursue power balance of network, -         „T“ type bay can be modelled only approximately.

22. Rule of Multiplication of Probabilities : P(A) probability of occurrence of A P(B)probability of occurrence ofB Series systems A failure of one element leads to a failure of a system. Probability of failure-free run:

23. Parallel Systems A failure of a system occurs when all elements have a failure Probability of a failure: Probability of failure-free run:

24. Simplified Probability of failure-free run:

25. Methodology of Calculation of Reliability according to ČEZ 22/80Regulation • Advantages: • - takes into account maintenance outages, • -enables to include manipulation into calculation as well, takes so-called cold reserves into account. • The disadvantage is that this methodology does not include so-called coordination of maintenance. • These operating states are considered with calculation: • - operation, • - failure outage, • - maintenance outage. • Supposition and simplification: • - the effect of weather on failure rate and repair rate is not taken into account, • exponential division of distributive function of time of failures • and repairs for all elements of electric network is taken into account, • - average data are started from.

26. Series connection of elements For this circuit with two elements it holds: P ... Failure rate [year-1] U ... Maintenance rate [year-1] ... Outage rate (maintenance + repair) [year-1]  Mean times of outages of a two-elements system:

27. Parallel connection of elements – hot reserve Failure rate: Mean time of failure: Maintenance outage cannot occur at this connection, because at the failure of one element maintenance of another element will not begin.

28. Parallel circuit of elements – cold reserve …. Manipulation time [h]

29. Simulation Methods of Calculation of Reliability • It is necessary to know the intensity of outages and mean time of outages of all the elements of a system. • Simulation - numerical method which resides in experimenting with mathematical models of real systems on numerical computers. • Advantages: • - considered systems do not have to really exist as yet, • considered systems can be too complicated for using analytical methods, • simulation makes possible study of behaviour of systems in real, accelerated, or retarded time. The second possibility is the most important in thiscase, because the processes of outage of elementsand their re-introduction into operation are very slow. It would be very inefficient to study them in any other time but accelerated. • with simulation it is possible to verify results obtained by other independant processes, • possibility of modelling „T“ type bays • simple power balance of a diagram is carried out, outage is always simulated at overloaded elements.

30. Disadvantages of simulation methods: • construction of a useful simulation model is very time-consuming. Mostly several variants of a model are needed. • - simulation is a numerical method, so a solution of certain problem cannot be generally transferred on analogous problems. • the results obtained from stochastic simulation models are values of • accidental quantities, and it would be very computer time-consuming if their accuracy should be increased • - precision of results depends on the number of iterations, • - the needed number of iterations depends on the extent of the solved network and on the required precision.