Failure Rate Modeling Using

Equipment Inspection Data

Richard E. Brown (SM)*
AbstractSystem reliability models typically use average equipment failure rates. Even if these models are
calibrated based on historical reliability indices, all like
components within a calibrated region remain homogeneous. This paper presents a new method of customizing failure rates using equipment inspection data.
This allows available inspection information to be reflected in system models, and allows for calibration
based on interruption distributions rather than mean
values. The paper begins by presenting a method to
map equipment inspection data to a normalized condition score, and suggests a formula to convert this score
into failure probability. The paper concludes by applying this methodology to a test system based on an actual distribution system, and shows that the incorporation of condition data leads to richer reliability models.
1

Keywordspredictive reliability assessment, equipment

failure rate modeling, inspection-based ranking

I. INTRODUCTION

OWER DELIVERY COMPANIES are under increasing pressure to provide higher levels of reliability for
lower cost. The best way to pursue these goals is to plan,
engineer, and operate power delivery systems based on
quantitative models that are able to predict expected levels
of reliability for potential capital and operational strategies.
Doing so requires both system reliability models and component reliability models.
Predictive reliability models are able to compute system
reliability based on system topology, operational strategy,
and component reliability data. The first distribution reliability model, developed by EPRI in 1978 [1], was not
widely used due to conservative design and maintenance
standards and, to a lesser extent, a lack of component reliability data. Eventually, certain utilities became interested
in predictive reliability modeling and started developing inhouse tools [2-5]. Presently, most major commercial circuit analysis packages offer an integrated reliability module capable of predicting the interruption frequency and
duration characteristics of equipment and customers. Advanced tools have extended this basic functionality to include momentary interruptions [6-7] and risk assessment
[8-9].

*This paper is based on a paper of the same title to be published in IEEE

Transactions on Power Systems. Richard Brown is with KEMA and can be
reached at rebrown@ kema.com.

The application of predictive reliability models has traditionally assigned average failure rate values to all components. Although simplistic, this approach produces useful
results and can substantially reduce capital requirements
while providing the same levels of predicted reliability
[10]. Advanced tools have attempted to move beyond average failure rates by either calibrating failure rates based
on historical system performance [11], or by using multistate weather models [12-13]. A few attempts have been
made to compute failure rates as a function of parameters
such as age [14], maintenance [15], or combinations of
features [16], but these models tend to be system-specific
and are not practicable for a majority of utilities at this
time.
The use of average component failure rates in system reliability models is always limiting and is potentially misleading [17]. Although generally acceptable for capital
planning, the use of average values has two major drawbacks. First, average values cannot reflect the impact of
relatively unreliable equipment and may overestimate the
reliability of customers experiencing the worst levels of
service. Second, average values cannot reflect the impact
of maintenance activities and, therefore, preclude the use
of predictive models for maintenance planning and overall
cost optimization.
Most utilities perform regular equipment inspections and
have tacit knowledge that relates inspection data to the risk
of equipment failure. Integration of this information into
component reliability models can improve the accuracy of
system reliability models and extend their ability to reflect
equipment maintenance in results.
Ideally, each class of equipment could be characterized
by an equation that computes failure rate as a function of
critical parameters. For example, power transformers might
be characterized as a function of age, manufacturer, voltage, size, through-fault history, maintenance history, and
inspection results. Unfortunately, in most cases the sample
size of failed units is far too small to generate an accurate
model, and other approaches must be pursued.
This paper presents a practical method that uses equipment inspection data to assign relative condition rankings.
These rankings are then mapped to a failure rate function
based on worst-case units, average units, and best-case
units. The paper then presents recommended failure rate
models for a broad range of equipment, presents a method
of calibration based on historical customer interruptions,
and concludes by examining the impact of these techniques
on a test system based on an actual distribution system.

II. INSPECTION-BASED CONDITION RANKING

Typical power delivery companies perform periodic inspection on a majority of their electricity infrastructure.
Utilities have various processes for collecting and recording inspection results. Paper forms stored in a multitude of departments make obtaining comprehensive system
inspection results problematic. Many utilities, however,
have migrated their inspection and maintenance programs
to computerized maintenance management systems
(CMMS) and data management systems that can be used
as central warehouses for equipment inspection results.
After a population of similar equipment has been inspected, it is desirable to rank their relative condition. Consider a piece of equipment with n inspection item results,
(r1, r2 , rn). Further suppose that each inspection item
result is normalized so that values correspond to the following:
ri = 0
ri =
ri = 1

; best inspection outcome

; average inspection outcome
; worst inspection outcome

Each inspection item result, ri, is assigned a weight, wi,

based on its relative importance to overall equipment condition. These weights are typically determined by the combined opinion of equipment designers and field service
personnel, and are sometimes modified based on the particular experience of each utility. The final condition of a
component is then calculated by taking the weighted average of inspection item results .By definition, a weighted
average of 0 corresponds to the best possible condition, a
weighted average of corresponds to average condition,
and a weighted average of 1 corresponds to the worst possible condition.

Condition Score =

i =1

wi ri

(1)

i =1

After each piece of equipment is assigned a condition

score between 0 and 1, equipment using the same inspection item weights can be ranked and prioritized for maintenance (typically considering cost and criticality as well as
condition). This approach has been successfully applied to
several utilities by the authors, and inspection forms and
weights for most major pieces of power delivery equipment
have been developed. In addition, inspection items have
guidelines that suggest scores for various inspection outcomes. To illustrate, an inspection form for power transformers is shown in Table 1 and the scoring guideline for
Age is shown in Table 2.
It should also be noted that inspection items can also be
related to external factors. For example, overhead lines can
include inspection items related to vegetation, animals, and
lightning. Scores for these items will reflect both the external condition (e.g., lightning flash density) and system
mitigation efforts (e.g., arrestors, shield wire, and grounding).

Although useful for prioritizing maintenance activities,

relative equipment condition ranking is less useful for rigorous reliability analysis. Since reliability assessment models require equipment failure rates, inspection results
would ideally be mapped into a failure rate through a
closed-form equation derived from regression models. As
mentioned earlier, this is not presently feasible for most
classes of equipment due to limited historical data.
III. FAILURE RATE MODEL
Although there is not enough historical data to map inspection results to failure rates through regression-based
equations, interpolation is capable of providing approximate results. At a minimum, interpolation requires failure
rates corresponding to the worst and best condition scores.
Practically, it requires one or more interior points so that
non-linear relationships can be determined.
After exploring a variety of mapping functions, the authors have empirically found that an exponential model
best describes the relationship between the normalized
equipment condition of Eq. 1 and equipment failure rates.
The specific formula chosen is:

Table 1. Inspection Form for Power Transformers

Criterion

Weight

Age (years of operation)

Condition of internal solid insulation

Oil type

Condition of core

Condition of inaccessible parts

Condition of tank

Condition of cooling system

Condition of tap changer

Condition of accessible parts

Condition of bushings

Experience with this transformer type

Transformer loading history

Number of extraordinary mechanical stresses

Number of extraordinary dielectric stresses

Noise level

Core and winding losses

Gas in oil analysis (current results)

Gas in oil analysis (trend in results)

Oil analysis

Sum

51

Score

Weighted Average

Table 2. Guideline for Power Transformer Age

Age (years of operation)

Score

Less than 1

0.00

1 - 10

0.05

11 - 20

0.10

21-25

0.25

29-31

0.50

32-35

0.60

36-40

0.80

Greater than 40

1.00

( x ) = Ae Bx + C
= failure rate

(2)

x = condition score
Three data pairs are required to solve for the parameters
A, B, and C. The previous section has developed a condition ranking methodology that, by definition, results in
best, average, and worst condition scores of 0, , and 1,
respectively. Therefore, three natural data pairs correspond
to (0), (), and (1). () can be approximated by taking the average failure rate across many components or by
using average failure rates documented in relevant literature. (0) and (1) are more difficult to determine, but can
be derived through benchmarking, statistical analysis, or
heuristics. Given these three values, function parameters
are determined as follows:

[ ( ) (0)]2
(1) 2 ( ) + (0)
( ) + A (0 )
B = 2 ln
A=

C = (0) A

(3)

A detailed benchmarking of equipment failure rates is

found in [18]. These results document low, typical, and
high failure rates corresponding to system averages across
a variety of systems. Assuming that (1) best-condition
equipment have failure rates half that of best system averages, (2) average-condition equipment have failure rates of
typical system averages, and (3) worst-condition equipment have failure rates twice that of best system averages,
parameters for a variety of equipment are shown in Table
3. These parameters, based on historical failure studies
such as [14], are useful in the absence of system specific
data, but should be viewed as initial conditions for calibration, which is discussed in the next section.

1.000

0.100

Ov

ea
erh

nk
T ru

e
Po w

n sf
r Tra

r
or me
ran sf
T
r
e
Po w
A)
5MV
r (>2
orme

)
MVA
(<2 5

0.010
Pr
im
ar
Ca y T
ble r un
k

0.40

Failure Rate (/yr)-

26-29

0.001
0

0.2

0.4

0.6

0.8

Condition (p.u.)
Figure 1. Selected Equipment Failure Rate Functions

Failure rate graphs for some of the equipment in Table 3

are shown in Figure 1. These are simply plots of Eq. 2 using the stated A, B, and C parameters the displayed equipment. It is interesting to see that the range of failure rates
of certain types of equipment is large, while other types
have a more moderate range. This reflects the ranges found
in a broad literature search which forms the basis of Table
3.
IV. MODEL CALIBRATION
After creating a system reliability model, it is desirable
to adjust component reliability data so that predicted system reliability is equal to historical system reliability [11].
This process is called model calibration, and can be generalized as the identification of a set of parameters that minimize an error function.
Traditionally, reliability parameters (such as equipment
failure rates) either remained uncalibrated or were adjusted
based on average system reliability. For example, it may be
known that an analysis area has an average of 1.2 interruptions per customer per year. Based on this number, failure
rates can be adjusted until the predicted average number of
customer interruptions is equal to this historical value. After failure rates are calibrated, switching and repair times
can be adjusted until predicted average interruption duration is also equal to historical values.
Calibrating based on system averages is useful, but does
not ensure that the predicted distribution of customer interruptions is equal to the historical distribution. That is, it
does not ensure that either the most or least reliable customers are accurately represented only that the average
across all customers reflects history. This is a subtle but
important point; since customer satisfaction is largely determined by customers receiving below-average reliability,
calibration of reliability distribution is arguably more important than calibration of average reliability.
A system model with homogeneous failure rates will
produce a distribution of expected customer reliability levels (e.g., customers close to the substation will generally

have better reliability than those at the end of the feeder).

If components on this same system are assigned random
failure rates such that average system reliability remains
the same, the variance of expected customer reliability will
tend to increase. That is, the best customers will tend to get
better, the worst customers will tend to get worse, and
fewer customers can expect average reliability.
The distribution of expected customer reliability is critical to customer satisfaction and should, if possible, be calibrated to historical data. A practical way to accomplish this
objective is to calibrate condition-mapping parameters so
that a distribution-based error function is minimized. Such
an error function can be based on one of three levels of
granularity: (1) individual customer reliability, (2) histograms of customer reliability, or (3) statistical measures of
customer reliability.
An error function can be defined based on the difference
between each customers historical versus predicted reliability. This approach calibrates reliability to the customer
level and utilizes historical data at the finest possible
granu-

Table 3. Representative Failure Rate Model Parameters ( values in failures per year)
Description
Overhead Equipment
Overhead Lines
Primary Trunk*
Lateral Tap*
Secondary & Service Drop*
Pole Mounted Transformer
Disconnect Switch
Fuse Cutout
Line Recloser
Shunt Capacitor
Voltage Regulator
Underground Equipment
Underground Cable
Primary Cable*
Secondary Cable*
Elbow Connectors
Cable Splices and Joints
Padmount Transformers
Padmount Switches
AIS Substation Equipment
Power Transformers
Less than 25 MVA
Bigger than 25 MVA
Circuit Breakers
Disconnect Switches
Instrument Transformers
Air Insulated Busbar
GIS Substation Equipment
GIS Bay (before 1985)
GIS Bay (after 1985)
* Line and cable failure rates are per circuit mile

(0)

()

(1)

0.0100
0.0100
0.0100
0.0020
0.0020
0.0020
0.0025
0.0055
0.0050

0.100
0.160
0.088
0.010
0.014
0.009
0.015
0.020
0.029

0.600
0.600
0.600
0.030
0.280
0.060
0.060
0.170
0.200

0.01976
0.07759
0.01402
0.00533
0.00057
0.00111
0.00481
0.00155
0.00392

3.4295969
2.1522789
3.7632316
1.8325815
6.1971793
3.9718310
2.5618677
4.6729733
3.9272195

-0.009756098
-0.067586207
-0.004018433
-0.003333333
0.001433071
0.000886364
-0.002307692
0.003948339
0.001081633

0.0015
0.0025
3.E-05
3.E-05
0.0005
0.0005

0.070
0.100
6.E-04
0.030
0.010
0.003

1.174
0.300
0.002
0.318
0.100
0.010

0.00453
0.09274
0.00039
0.00348
0.00112
0.00139

5.5597230
1.4369300
1.7971823
4.5255272
4.4970357
2.0592388

-0.003031386
-0.090243902
-0.000361446
-0.003450994
-0.000621118
-0.000888889

0.0075
0.0050
0.0005
0.0020
0.0000
0.0005

0.040
0.030
0.010
0.010
0.010
0.010

0.140
0.120
0.060
0.320
0.060
0.076

0.01565
0.00962
0.00223
0.00021
0.00250
0.00160

2.2478602
2.5618677
3.3214624
7.3142615
3.2188758
3.8767259

-0.008148148
-0.004615385
-0.001728395
0.001788079
-0.002500000
-0.001097345

0.0003
0.0002

0.002
0.001

0.030
0.018

0.00011
0.00004

5.6031525
6.1127138

0.000190114
0.000160494

larity. However, historical customer reliability is stochastic

in nature and will vary naturally from year to year. An error function can be defined based on the difference between each customers historical versus predicted reliability. This approach calibrates reliability to the customer
level and utilizes historical data at the finest possible
granularity. However, historical customer reliability is stochastic in nature and will vary naturally from year to year.
This is especially problematic with frequency measures.
Although customers on average may experience 1 interruption per year, a large number of customers will not experience any interruptions in a given year. Calibrating these
customers to historical data is misleading, making about 10
years of historical data for each customer desirable. Unfortunately, most feeders change enough over 10 years to
make this method impractical.
An error function can also use a histogram of customer
interruptions as its basis. The historical histogram could be
compared to the predicted histogram and parameters adjusted to minimize the chi-squared error (2):

2 =

(h h p )
n

i =1

(4)

Where n is the number of bins, h is the historical bin

value, and p is the predicted bin value. Using the chisquared error is attractive since it emphasized the distribution of expected customer reliability which is strongly correlated to customer satisfaction. Histograms will vary stochastically from year to year, but the large number of customers in typical calibration areas prevent this from becoming a major concern.
Last, an error function can be based on statistical measures such as mean value () and standard deviation ().
The error function will typically consist of a weighted sum
similar to the following:

Error = ( ') + ( ')

2

(5)

Unlike the 2 error, this function allows relative weights

to be assigned to mean and variance discrepancies ( and
). For example, a relatively large a value will ensure that
predicted average customer reliability reflects historical
average customer reliability while allowing relatively large
mismatches in standard deviation.
Once an error measure is defined, failure rate model parameters can be adjusted so that error is minimized. Since
this process is over determined, the authors suggest using
Table 3 for initial parameter values and using gradient de-

scent or hill climbing techniques for parameter adjustment.

Calibration is computationally intensive since error sensitivity to parameters must be computed by actual parameter
perturbation, but calibration need only be performed once.

50
Historical
Uncalibrated

40

Table 4. Calibration Results for Overhead Line Parameters

A
B
C
2 Error (%2)

Uncalibrated
0.01976
3.429597
-0.00976
1148.8

Calibrated
0.0170
2.5981
-0.00528
155.4

30

20

10

0
0.0

0.5

1.0
1.5
2.0
Inte rruptions (per year)

2.5

3.0

Figure 2. Historical Versus Predicted Customer Interruptions

Figure 3. Visualization of Calibrated Results

0.7
Uncalibrated

0.6
Failure Rate (/yr)

The methodologies described in the previous two sections have been applied to a test system derived from an
actual overhead distribution system in the Southern U.S.
This model consists of three substations, 13 feeders, 130
miles of exposure, and a peak load of 100 MVA serving
13,000 customers. The analytical model consists of 4100
components.
Customer historical failures are computed from fouryear historical averages. Equipment condition for this system was not available, and was therefore assigned for randomly for individual components based on a normal distribution with a 0.5 mean and a 0.2 standard deviation.
Calibration for this test system is performed based on
the chi-squared error of customer interruptions. Initial failure rates for all components are assigned based on ()
values in Table 3, and initial failure rates are computed
based on condition and the parameters in Table 3. Calibration is performed by a variable-step local search that guarantees local optimality.
A summary of calibration results for overhead lines is
shown in Figure 2, and a visualization of calibrated results
in shown in Figure 3. The shape of the uncalibrated histogram is similar to the historical histogram, but with a mean
and mode worse than historical values. After calibration,
the modes align, but the predicted histogram retains a
slightly smaller variance. In fact, the historical histogram is
subject to stochastic variance, and the inability of the expected value calibration to match this variance is immaterial and perhaps beneficial.
Uncalibrated and calibrated failure rate parameters are
shown in Table 4, and corresponding failure rate functions
are shown in Figure 4. In effect, the calibration for this
system did not change the failure rates for lines with good
condition (less than 0.2), but drastically reduced the failure
rates for lines with worse-than-average condition (greater
than 0.5). These results are not unexpected, since this particular service territory is relatively homogeneous in both
terrain and maintenance practice, and extremely wide
variations in overhead line failure rates have not been historically observed.

% of Customers

Calibrated

V. APPLICATION TO TEST SYSTEM

Calibrated
0.5
0.4
0.3
0.2
0.1
0
0

0.1

0.2

0.3

0.4 0.5 0.6 0.7

Condition (p.u.)

0.8

0.9

Figure 4. Overhead Line Failure Rate as a Function of Condition

It is important to note that, in this case, equipment conditions were assigned randomly, and some were very high.
Even though actual equipment for this system may never
reach this poor condition state, the calibration process
compensated by ratcheting down the failure rates assigned
to equipment with the highest condition scores.
Once a system has been modeled and calibrated, it can
be used as a base case to explore the impact of issues that
may impact equipment condition such as equipment maintenance. Once the expected impact that a maintenance action will have on inspection items is determined, the system impact of maintenance can be quantified based on the
new failure rate. This allows the cost effectiveness of
maintenance to be determined and directly compared to the
cost effectiveness of system approaches such as new construction, added switching and protection, and system reconfiguration.
VI. CONCLUSIONS
Equipment failure rate models are required for electric
utilities to plan, engineer, and operate their system at the
highest levels of reliability for the lowest possible cost.
Detailed models based on historical data and statistical
regression are not feasible at the present time, but this paper presents an interpolation method based on normalized
condition scores and best/average/worst condition assumptions.
The equipment failure rate model developed in this paper allows condition heterogeneity to be reflected in
equipment failure rates. Doing so more accurately reflects
component criticality in system models, and allows the
distribution of customer reliability to be more accurately
reflected. Further, a calibration method has been presented
that allows condition-mapping parameters to be tuned so
that the predicted distribution of reliability matches the
historical distribution of reliability. Finally, the use of this
condition-based approach allows the impact of maintenance activities on condition to be anticipated and reflected
in system models, enabling the efficacy of maintenance
budgets to be compared with capital and operational budgets.
This model is heuristic by nature, but adds a fundamental level of richness and usefulness to reliability modeling,
especially when parameters are calibrated to historical
data. In the short run, gathering more detailed information
on equipment failure rates and condition will strengthen
this approach. In the long run, this same information can
ultimately be used to develop explicit failure rate models
that eliminate the normalized condition assessment requirement.
VII. REFERENCES
[1]
[2]

EPRI Report EL-2018, Development of Distribution Reliability and

Risk Analysis Models, Aug. 1981.
S. R. Gilligan, A Method for Estimating the Reliability of Distribution Circuits, IEEE Transactions on Power Delivery, Vol. 7, No. 2,
April 1992, pp. 694-698.

[3]

G. Kjolle and Kjell Sand, RELRAD - An Analytical Approach for

Distribution System Reliability Assessment, IEEE Transactions on
Power Delivery, Vol. 7, No. 2, April 1992, pp. 809-814.

[4]

[5]

[6]

[7]

[8]

[9]
[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

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Distribution System Reliability Assessment: Momentary Interruptions
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Inc., 2002.

VIII.BIOGRAPHIES
Richard E. Brown is a principal consultant with KEMA, and specializes in
distribution reliability and asset management. He is the author or co-author
of more than 50 technical papers and the book Electric Power Distribution
Reliability. Dr. Brown received his BSEE, MSEE, and PhD from the University of Washington and his MBA from the University of North Carolina
at Chapel Hill. He is a registered professional engineer, and can be reached
at rebrown@kema.com.