Forecasting Underground Electric Cable Faults Using the
CrowAMSAA Model
Guest Submission
Yancy Gill, Ph.D., P.E.
Maintenance Engineering, Salt River Project
One of the major economic and reliability challenges facing the Salt River
Project (SRP), a major electric and water utility in the Phoenix, Arizona
metropolitan area, is managing the replacement of 7000 miles of directburied
primary electrical cable that is at or approaching the end of its useful life.
Since cable replacement programs of this magnitude will require 25 years or more to
complete, the ability to model cable faults as a function of cable replacement is
critical to developing a sound cable replacement strategy. The model SRP selected
to forecast faults in aging underground electrical cable is a reliability
growthbased model known as CrowAMSAA [1].
The CrowAMSAA model was originally developed to track and quantify the
reliability growth of preliminary product designs or developmental manufacturing
processes to help establish when a product or process has obtained
adequate reliability
to be put into production. However, over the past several years, the CrowAMSAA
model has found increasing use as a tool to monitor reliability and forecast
failures/faults in fielded mechanical and electrical systems. The advantage of
the CrowAMSAA model is that it models repairable systems, not a failure mode
distribution of replaceable systems such as the Weibull distribution. This is an
important distinction, as CrowAMSAA can model a cable segment that has failed
and been repaired multiple times, while the Weibull distribution can only be used
to model the first failure. The CrowAMSAA model is also capable of handling a
mixture of failure modes whereas the Weibull model works best with one, perhaps
two failure modes only.
Graphically, CrowAMSAA is a loglog plot of cumulative failures versus
cumulative time. If the model applies, the resulting plot will be linear and
an equation of the form
will fit the data, where:
 n(t) is the cumulative number
of failures/faults.
 t is the cumulative time.
 λ is a scale parameter that has no physical
meaning.
 β is a measure of the failure rate. If β is
greater than 1, the failure rate is increasing. Conversely, if β is
less than 1, the failure rate is decreasing. If β equals 1, the
failure rate is considered to be constant or random.
The standard accepted procedure for determining β is the
maximum likelihood estimation (MLE) method. Note that there are several
MLE formulations
for CrowAMSAA and proper formula selection is predicated upon the data type. The
data type commonly encountered with underground electrical cable is grouped data
where the total number of faults over an interval of time are grouped and
subsequently evaluated. The MLE of β for grouped data is the β that
best satisfies the following equation:
where:
 k is the total number of
time intervals.
 T_{k} is the total time
or the cumulative time at the end of
the k^{th} time interval.
 T_{i} and
T_{i1} are the cumulative times at the
end of the i^{th} and
i^{th}  1 time intervals, respectively.
 n_{i} is the number
of failures/faults during the i^{th} time
interval.
By definition, the start time T_{0} is equal
to zero along with the term T_{0} ln
T_{0}. Note the time intervals do not have to be of equal length
to estimate β with the above equation.
Once β has been determined, the scale parameter λ is estimated with
the following equation:
The ChiSquared Goodness of Fit test is used to test the null hypothesis
that the CrowAMSAA model satisfactorily represents the grouped data:
Where e_{i} is the failure/fault estimate
from the CrowAMSAA model:
The null hypothesis is rejected if
the χ^{2} statistic exceeds the
critical value of the chosen significance level at k2 degrees of freedom.
To forecast faults in primary underground cable as a function of cable
replacement, the CrowAMSAA model requires fault and footage data for each
primary cable type by the calendar year of installation. Table 1 presents such
data for underground residential distribution (URD) cable installed in 1977.
The cable year of installation is henceforth referred to as
"vintage."
Table 1: Fault and Footage Data for 1977 Vintage URD Cable, 2000 to
2008
1977 Vintage URD Cable 
Calendar Year 
Faults 
Footage (feet) 
Faults/100 Cable Miles 
2000 
87 
708593 
65 
2001 
91 
708593 
68 
2002 
108 
700530 
81 
2003 
76 
691653 
58 
2004 
98 
690381 
75 
2005 
113 
670307 
89 
2006 
100 
670307 
79 
2007 
116 
632838 
97 
2008 
100 
632838 
83 
Next, the CrowAMSAA parameters λ and β are determined by MLE
from the cumulative faults/100 cable miles versus cumulative time. The fit
of the model to the data is given by
the χ^{2} statistic, as is illustrated
in Figure 1 from ReliaSoft's RGA
7.
Figure 1: LogLog
plot of Cumulative Faults/100 Cable Miles versus Cumulative Time (Years)
including Fit of CrowAMSAA Model for 1977 Vintage URD Cable, 2004 to 2008
Finally, the cumulative faults/100 cable miles are converted back to
discrete faults/100 cable miles by taking the difference between the adjacent
cumulative years. Faults can now be determined by multiplying the discrete
faults/100 cable miles by the actual footage in each of the data years.
Table 2 and Figure 2 show the assumed footage due to cable replacement in
the forecast years.
Table 2: Fault Forecast Using CrowAMSAA by Converting Cumulative
Faults/100 Cable Miles to Discrete Faults/100 Cable Miles and Multiplying by
Cable Length
CrowAMSAA Model Results 1977 Vintage URD Cable 

No Replacement 2009 Forward 
Replace 50K ft/yr 2009 Forward 
Calendar Year 
Cum Faults/100 Cable Miles 
Faults/100 Cable Miles 

Footage 
Faults 
Footage 
Faults 
2004 
77 
77 

690381 
100 
690381 
100 
2005 
160 
83 

670307 
106 
670307 
106 
2006 
246 
86 

670307 
109 
670307 
109 
2007 
334 
88 

632838 
105 
632838 
105 
2008 
423 
89 

632838 
107 
632838 
107 
2009 
513 
90 

632838 
108 
582838 
100 
2010 
604 
91 

632838 
109 
532838 
92 
2011 
696 
92 

632838 
110 
482838 
84 
2012 
789 
93 

632838 
111 
432838 
76 
2013 
882 
93 

632838 
112 
382838 
68 
Figure 2: 1977
Vintage URD Cable Fault Forecast Comparing No Cable Replacement to
50,000 feet/year from 2009 to 2014
In this example, λ and β were determined from the five most recent
years of data, 2004 to 2008. The decision to use only the five most recent years
of data to forecast faults was based upon the use of the CrowAMSAA model in
evaluating new product reliability growth where reliability growth is determined
within a test phase and not across test phases. What this means is that the product
under evaluation is at a fixed design state and no other design changes are
allowed during the evaluation period. For primary underground electrical cable,
the intent is not to evaluate the design but to evaluate only the cable
degradation over time.
As the cable ages, it is conceivable that fault locating processes, operating
procedures or failure modes can change or interact in such a way as to result in
a change in β. By analyzing only the most recent data, we captured the
current state of the cable. We conduct this analysis process on an annual
basis for all vintages of direct buried primary underground cables in the
SRP system.
Conclusions
Although the example presented here is for underground electrical cable,
the method should work for any repairable linear asset provided that the
failure/fault events are adequately modeled by the CrowAMSAA model. For companies
that have Asset Management Systems, the modeling approach and results described
above can be easily integrated into the key strategic elements of the Asset
Management Plan.
References
[1] ReliaSoft Corporation, Reliability Growth &
Repairable System Analysis Reference, Tucson, AZ: ReliaSoft
Publishing, 2009.
