ETL 1110-2-550
30 May 97
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
b. The Corps is continuing to assemble a large
process.
database of equipment histories to establish the
a. The following two functions are used in 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.
D(a) = P(A > a)
c. The raw retirement data can be fitted using
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
A = age of the equipment at retirement
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
P(A > a) = probability that A > a (Ayyub,
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
dD(a)
f(a) '
' D (a)
continues.
da
d. The practice in Corps evaluation reports has
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
d In D(a)
D (a)
for equipment that is in better condition than
H(a) '
'&
average. Contact the Hydroelectric Design Center
da
D(a)
(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
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