surviving to age *a *in the first place. It is the

probability that a piece of equipment will fail to

probability that the failure occurs at age *a*.

perform satisfactorily. The following discussions

explain the terms and their applications used in this

process.

database of equipment histories to establish the

reliability characteristics of various categories of

development of reliability curves.

equipment. The initial work in this area focused on

generator stator windings because there have been a

(1) The reliability of equipment can be

significant number of stator retirements in the form

considered a continuous variable with a

of rewinds (Ayyub, Kaminskiy, and Moser 1996),

probability density function (pdf) of *f*. A pdf is a

but a significant turbine database is also being

theoretical model for the frequency distribution of a

developed. The historical data include many

population of measurements. In this case regarding

attributes such as year installed, age at failure, and

reliability, the pdf is the rate of change of the

rated capacity. Appendix F presents a review of

equipment dependability. Therefore, if the

recent research in hydropower reliability analysis.

dependability of the equipment at age *a *is defined

The raw data are compiled and reduced into annual

as:

summaries of exposures and failures.

any number of means. One method is the

where

application of Iowa Curves developed in the 1930's

by the Engineering Experiment Station at what was

then Iowa State College (Winfrey 1935). Other

distribution functions that may be used include

and

normal, exponential, log-normal, and Weibull. The

Weibull distribution is one of the most widely used

reliability functions. It has been shown that the

Kaminskiy, and Moser 1996)

differences between the Iowa Curves and a Weibull

distribution are statistically insignificant. The

Then the pdf of *D(a) *is

Weibull distribution is much easier to adapt to

computer analysis techniques. Research to develop

new and more refined reliability functions

' *D *(*a*)

continues.

This simply states that the dependability of a piece

been to use the hazard function directly if the

of equipment is equal to the probability that the

condition of the specific equipment in question is

equipment is still functioning at age *a*.

considered average. If, however, the equipment has

exhibited signs of premature or accelerated

(2) The hazard function H(*a*), or incremental

deterioration, the hazard function has been adjusted

failure rate associated with the random variable *A*,

to account for the evident higher probability of

is given by:

failure. Similarly, the hazard function can be

modified to account for lower failure probabilities

for equipment that is in better condition than

'&

average. Contact the Hydroelectric Design Center

(HDC) for the current details on modifying hazard

That is, the incremental failure rate is equal to the

functions.

probability of the equipment life being age *a*

divided by the probability of the equipment

5

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