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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 3, AUGUST 2002
Incorporating Aging Failures in Power System
Reliability Evaluation
Wenyuan Li, Fellow, IEEE
Abstract—The paper presents a method to incorporate aging
failures in power system reliability evaluation. It includes development of a calculation approach with two possible probability
distribution models for unavailability of aging failures and implementation in reliability evaluation. The defined unavailability of
aging failures has a consistent form which is the same as that for repairable failures. This allows aging failures to be easily included in
existing reliability evaluation techniques and tools. Differences between the two models using normal and Weibull distributions have
been discussed. The BC Hydro North Metro system was used as
an example to demonstrate an application of the proposed method
and models. The results indicate that aging failures have significant
impacts on system reliability, particularly for an “aged” system.
Ignoring aging failures in reliability evaluation of an aged power
system will result in an overly underestimation of system risk and
most likely a misleading conclusion in system planning.
Index Terms—Aged system, aging failure, power system reliability, repairable failure, unavailability.
I. INTRODUCTION
C
ONSIDERABLE efforts have been devoted to power
system reliability evaluation in the past two decades
[1]–[7]. Many considerations have been incorporated in the
evaluation. These include independent failures, common cause
failures, dependent failures, weather effects, load curve models,
bus load uncertainty and correlation, multiarea application,
sensitivity of failure and repair rates, noncoherence and security
limits, etc. [1], [2], [8]–[14].
There are two failure modes for a power system component:
1) repairable and 2) nonrepairable. The repairable failures are
characterized by average failure frequency and repair time. Almost all of the methods and tools presented so far have only
considered the repairable failure mode. In other words, nonrepairable failures, although an important failure mode, have not
been modeled in power system reliability assessment. The aging
failure is a nonrepairable failure event. Once a component fails
due to aging, it will die forever. There is no concept of repair
time associated with it. The aging failures of system components
such as transformers, cables, breakers, capacitors and reactors,
etc. have been a major concern and a driving factor in system
planning of many utilities since more and more system components are approaching their end-of-life stage. In fact, aging is
a general phenomenon in real life. Excluding aging failures in
power system reliability evaluation will definitely lead to underestimation of the system risk. If some key components in a
Manuscript received July 30, 2001.
The author is with Grid Operations, BC Hydro, Burnaby, BC V3N 4X8,
Canada.
Publisher Item Identifier 10.1109/TPWRS.2002.800989.
system are aged, aging failures could become a dominant factor
of system unreliability. In this case, only considering repairable
failures will most likely result in a misleading conclusion in
system planning.
The traditional reliability theory [15] has provided an approach to calculating probability to an aging failure in a given
time period. Unfortunately, it is a transition probability and does
not have a consistent concept as unavailability which is used
for repairable failures. This probability cannot be directly incorporated into the existing methods and tools for reliability assessment of generation or transmission systems. As a matter of
fact, the calculation approach for unavailability of aging failures
has not been developed. This may be a main reason why aging
failures have not been considered in reliability assessment of
generation, transmission, or distribution systems.
This paper proposes a method to incorporate aging failures in
system reliability evaluation. The paper is organized as follows.
Section II presents a definition and a calculation approach with
two possible probability distribution models for unavailability of
aging failures. Section III describes implementation techniques
in system reliability evaluation. Section IV offers an actual
application to BC Hydro North Metro system. Although this
is an application to a transmission system, the proposed
calculation approach, models, and implementation techniques
to include unavailability of aging failures are general and can
be applied to reliability assessment of generation, distribution,
or any other engineering systems. Conclusions are given in
Section V.
II. DEFINITION AND CALCULATION APPROACH
UNAVAILABILITY OF AGING FAILURES
FOR
A. Definition
According to the traditional reliability theory [15], probability to an aging failure is defined as a conditional probability
that an aging failure of a component will take place within a
specified period given that it has survived for years. This is
a concept of transition probability (i.e., the likelihood of a component transiting from a survival state to an aging failure state).
According to the author, unavailability of aging failure can
be defined as an average probability that a component is found
unavailable due to aging failures during a specified time period
given that it has survived for years. The component can fail
at any point within , but with different probabilities. Therefore,
the unavailability should be a conditional mathematical expectation of time length not available due to aging failures during
divided by the period considered ( ). The condition is the age .
0885-8950/02$17.00 © 2002 IEEE
LI: INCORPORATING AGING FAILURES IN POWER SYSTEM RELIABILITY EVALUATION
It can be seen from the above definitions that the probability
of transition to aging failure and the unavailability of aging
failure are completely different concepts although both are a
conditional probability. There are two time parameters in the
definition: 1) the age ( ) and 2) the subsequent time period to
consider ( ). There is always a reference year and a specified
period in power system reliability evaluation. For example, we
often assess reliability of a system on a yearly basis and calculate annual reliability indexes in power system planning. Once
the reference year is selected, the age of each component becomes known and the subsequent one year is the time period
to consider. In the case of only considering repairable failures,
although the concept of the “subsequent” year is not explicitly
emphasized, we do calculate annual indexes based on the system
state (system configuration, load level, and generation patter)
for that year. Therefore, two time parameters in the definition
will not create any problem for system reliability assessment.
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The continuous integral from
following discrete sum:
to
in (3) becomes the
(5)
where
(6)
If the aging failure is modeled by a posteriori normal distribution, the integration in (6) does not have an explicit analytical
expression. A polynomial approximation can be used for (6) as
follows:
B. Calculation Approach
According to the definition of reliability function and the
conditional probability concept, the probability of transition to
aging failure of a component in a subsequent period after
having survived for years can be calculated by
(1)
(7)
where and are the mean and standard deviation of normal
distribution and the function is calculated by
if
if
is the failure density probability function (Weibull
where
or normal distribution).
From (1), the aging failure probability in a small interval
at any point within can be calculated by
(2)
If the component fails at the point , the unavailable duration
. Since could be any point between [0, ], the
within is
average unavailability can be mathematically expressed using
the following integral:
If the aging failure is modeled by a posteriori Weibull distribution, (6) can become
(8)
(3)
Unfortunately, it is impossible to implement an analytical resolution to (3) although the expression is mathematically accurate
against the proposed definition of the unavailability of aging
failures. An alternate method is discretization. The period is
divided into
equal intervals, each having a length
. It
is small enough so that the failure probais assumed that
bility at any point within
is approximately constant. If the
component fails within the th interval, the average unavailable
duration within is given by
(4)
where and are the scale and shape parameters, respectively,
for Weibull distribution.
It should be noted that it is better to use a “functional age,”
which reflects the degree of “wear-out,” for in the model. The
functional age depends on usage history, maintenance situation,
and actual status of equipment, and sometimes is difficult to be
determined. In most cases, the natural age of equipment is used
and can still produce an acceptable and reasonable result for the
system planning purpose.
III. USING UNAVAILABILITY OF AGING FAILURES IN POWER
SYSTEM RELIABILITY EVALUATION
We use transmission reliability evaluation as an example
to explain how to incorporate unavailability of aging failures.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 3, AUGUST 2002
Transmission system reliability evaluation generally includes
the following basic steps.
Step 1) Select a system state. This includes a load level
and states of all system components (up, down, or
derated states).
Step 2) Conduct power flow and contingency analysis for
the selected system state to check if there is a system
problem (overloading, voltage limit violations, isolated buses, islands, etc.)
Step 3) If there is no system problem, go back to Step 1 to
select a new system state. Otherwise, go to Step 4.
Step 4) Perform remedial actions. This is often an optimal
power flow model. The purpose of the model is to
reschedule generations, alleviate line overloading or
voltage limit violations, and avoid load curtailments
if possible, or minimize the total load curtailments if
unavoidable.
Step 5) Update reliability indexes based on results in Step 4.
Step 6) Repeat steps 1-5 until a stopping rule is met.
There are three basic methods for state selection in system reliability evaluation. They are as follows:
1) the state enumeration;
2) the state sampling (nonsequential Monte Carlo);
3) the state duration sampling (sequential Monte Carlo).
The first two methods use unavailability of system components
while the third one uses failure and repair rates of components.
Since the unavailability of aging failures derived above has a
consistent form as the unavailability of repairable failures, aging
failures can be easily incorporated in the first two methods. The
additional effort is only associated with Step 1 to include unavailability of both repairable and aging failures. All other steps
are exactly the same as the traditional method. It looks like it
is extremely difficult for the state duration sampling method to
capture aging failures because it focuses on simulation of many
failure and repair events but there is no concept of repair at all
for aging failures.
In a traditional method, each component only considers the
unavailability due to repairable failures. Once aging failures are
incorporated, the two unavailability parameters for repairable
and aging failures have to be considered.
A. Using Unavailability of Aging Failures in the State
Enumeration Technique
In principle, two unavailability parameters for each component can still be enumerated. However, this will increase the
number of system states dramatically leading to a heavy burden
in calculation efforts. An appropriate method is to calculate an
equivalent total unavailability first using the following union
concept:
(9)
and
are the unavailability for repairable and aging
where
failures, respectively.
Fig. 1.
Sampling two unavailability parameters.
B. Using Unavailability of Aging Failures in the Monte Carlo
Simulation
The component state sampling method is more flexible to
simulate the two unavailability parameters for repairable and
aging failures.
Two independent random numbers are drawn for each component, the repairable and aging failures of which are considand another for . The concept of sampling
ered; one for
and
will
is shown in Fig. 1. The union relation between
be captured automatically.
should be calculated separately
It is worthy to note that
for each year over a system planning time frame (such as 10
years) since unavailability of aging failures depends on the age
which increases as the time advances. On the other hand, the
unavailability of repairable failures ( ) will be the same for
different ages.
IV. CASE STUDY
The BC Hydro North Metro system shown in Fig. 2 was used
as an example of application. The main links in the system rely
on eight underground cables which have been highlighted in the
figure. The ages of four cables are near to the mean life and
therefore their aging failures become a concern of system risk.
In the study, both repairable and aging failures were considered
for the eight cables, while only repairable failures were considered for overhead lines.
A. Unavailability of Aging Failures for the Cables
The unavailability of repairable failures of the eight cables
and their natural ages are given in Table I. Repairable failure
data is based on statistics over the past 15 years. The mean life
for the 230-kV cables was estimated to be 45 years with a standard deviation of 15 years. A computer program has been developed to calculate unavailability of aging failures using the
proposed method. Both normal and Weibull distribution models
were considered. As indicated earlier, the normal distribution is
characterized by a mean life and a standard deviation, but the
Weibull distribution by scale and shape parameters. For comparability between the two models, the scale and shape parameters for the Weibull distribution must be calculated from the
same mean life and standard deviation. The calculation method
is given in the Appendix.
Unavailability values of aging failures for the eight cables
from 2001 to 2006 using the normal and Weibull models are
shown in Tables II and III, respectively.
In order to show a more clear comparison between the unavailability values of repairable failures and aging failures as
LI: INCORPORATING AGING FAILURES IN POWER SYSTEM RELIABILITY EVALUATION
Fig. 2.
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BC Hydro North Metro system.
TABLE I
AGE AND UNAVAILABILITY OF REPAIRABLE FAILURE
TABLE II
UNAVAILABILITY OF AGING FAILURE (NORMAL MODEL)
TABLE III
UNAVAILABILITY OF AGING FAILURE (WEIBULL MODEL)
well as those between normal and Weibull models, annual unavailability values of the three cables with typical ages from
2001 to 2006 have been plotted in Figs. 3–5.
The following observations can be obtained from Tables I–III
and Figs. 3–5.
1) The unavailability of aging failures increases with the
age while the unavailability of repairable failures keeps
constant.
Fig. 3.
Unavailability of cable 2L31 (age
= 16 y).
Fig. 4.
Unavailability of cable 2L50 (age
= 28 y).
Fig. 5.
Unavailability of cable 2L40 (age
= 44 y).
2) The unavailability of aging failures at a “young” age is
smaller than the unavailability of repairable failures. The
unavailability of aging failures at a “midage” is in an order
of magnitude comparable with the unavailability of repairable failures. The unavailability of aging failures at
an “old age” is much larger than the unavailability of repairable failures.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 3, AUGUST 2002
3) The normal distribution model provides a lower estimation of unavailability at a “young or midage” but a higher
estimation at an “old age” compared to the Weibull model
if they have the same mean life and standard deviation.
However, more investigations indicate that when the age
is “very old,” the situation will change; the Weibull model
will lead to a higher estimation of unavailability than the
normal model, particularly in the case of a small standard
deviation. In other words, there are two intersection points
on the two curves of unavailability vis-à-vis age for the
normal and Weibull models. Further discussions on this
are beyond the scope of this paper.
B. Reliability Evaluation of North Metro System
The Monte Carlo state sampling method using separate
random numbers for repairable and aging failures has been
incorporated into a composite system reliability evaluation
program and was used to assess reliability of the BC Hydro
North Metro system from 2001 to 2006. It was assumed that
each year has 2% of load growth with the same shape of an
annual load curve. The expected energy not supplied (EENS)
indexes for two cases are shown in Tables IV and V and Fig. 6.
In Case 1, only repairable failures for all system components
were considered and in Case 2, aging failures for the eight
cables were also considered on top of Case 1.
It can be seen that the system EENS indexes of incorporating cable aging failures are much larger than those of only
considering repairable failures. The contributions due to aging
failures of the eight cables are dominant on the results. This
is because the cables are the main transmission links in the
system and four of them are close to end-of-life with high aging
failure unavailability. This suggests that for an aged system,
if aging failures are ignored in modeling, as we usually do,
reliability indexes will be overly underestimated and this will
most likely lead to a misleading conclusion in system planning.
The EENS index for the case of only considering repairable
failures has a slight increase with the years because of the
assumption of 2% load growth each year. On the other hand, the
EENS index for the case of incorporating cable aging failures
increases dramatically as the time advances. This obviously is
due to the fact that the unavailability of cable aging failures
has a relatively large increase with the years.
V. CONCLUSIONS
The paper presents a method to incorporate aging failures in
power system reliability evaluation. It includes development of
the calculation approach with the two possible probability distribution models for unavailability of aging failures and implementation in reliability evaluation. The defined unavailability of
aging failures has a consistent form as that for repairable failures and therefore it is simple and straightforward to include it
in system reliability assessment. This modeling is easily added
in existing reliability evaluation techniques and tools.
The differences between the two models for unavailability of
aging failures using normal and Weibull distributions have been
discussed. Generally, under the condition of the same mean life
Fig. 6.
EEENS of North Metro system with and without cable aging failures.
TABLE IV
SYSTEM EENS (MWh/y)—NORMAL MODEL FOR CASE 2
TABLE V
SYSTEM EENS (MWh/y)—WEIBULL MODEL FOR CASE 2
and standard deviation, the normal model provides a lower estimation of unavailability at a young or midage and a higher estimation at an age around the mean life compared to the Weibull
model. Which model should be used in an actual application
depends on the match between the parameters of the model and
failure data statistics.
The BC Hydro North Metro system was used as an example to
demonstrate an application of the proposed method and models.
The results indicate that aging failures have significant impacts
on the system reliability, particularly for an “aged” system. Ignoring aging failures in reliability evaluation of an aged power
system will result in an overly underestimation of system risk
and most likely a misleading conclusion in system planning.
APPENDIX
THE METHOD OF CALCULATING AND FROM THE MEAN
AND STANDARD DEVIATION FOR WEIBULL DISTRIBUTION
The mean and the standard deviation can be easily obtained
from statistic data, but a Weibull distribution is parameterized
by scale ( ) and shape ( ) parameters. The following method
was developed by the author to calculate the and from the
mean ( ) and standard deviation ( ).
LI: INCORPORATING AGING FAILURES IN POWER SYSTEM RELIABILITY EVALUATION
According to the definition of the mean and the standard deviation, we can obtain the following equations from the mathematical expression of the Weibull density distribution function:
(A1)
(A2)
where
is called the gamma function which is defined as
(A3)
By eliminating
from (A1) and (A2), we have
(A4)
Using an approximate expression of the gamma function, (A4)
is approximated by
(A5)
Equation (A5) can be solved to obtain using a bifurcation
algorithm. Then is found from (A2) using .
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Wenyuan Li (F’02, SM’89) received the Ph.D. degree in electrical engineering
at Chongqing University, Chongqing, China, in 1987.
Currently, he is a Specialist Engineer of the Control Center Technologies Department, Grid Operations, BC Hydro, Burnaby, BC, Canada. He is coauthor
of the book Reliability Assessment of Electrical Power Systems Using Monte
Carlo Methods, (New York: Plenum Press, 1994).
Dr. Li was the winner of the 1996 “Outstanding Engineer Award” of
IEEE Canada for “contributions in power system reliability and probabilistic
planning.”