Reliability HotWire

Reliability HotWire

Issue 115, September 2010

Hot Topics

Forecasting Underground Electric Cable Faults Using the Crow-AMSAA 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 direct-buried 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 growth-based model known as Crow-AMSAA [1].

The Crow-AMSAA 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 Crow-AMSAA 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 Crow-AMSAA 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 Crow-AMSAA 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 Crow-AMSAA 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, Crow-AMSAA is a log-log plot of cumulative failures versus cumulative time. If the model applies, the resulting plot will be linear and an equation of the form

Equation

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 Crow-AMSAA 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:

Equation

where:

  • k is the total number of time intervals.
  • Tk is the total time or the cumulative time at the end of the kth time interval.
  • Ti and Ti-1 are the cumulative times at the end of the ith and ith - 1 time intervals, respectively.
  • ni is the number of failures/faults during the ith time interval.

By definition, the start time T0 is equal to zero along with the term T0 ln T0. 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:

Equation

The Chi-Squared Goodness of Fit test is used to test the null hypothesis that the Crow-AMSAA model satisfactorily represents the grouped data:

Equation

Where ei is the failure/fault estimate from the Crow-AMSAA model:

Equation

The null hypothesis is rejected if the χ2 statistic exceeds the critical value of the chosen significance level at k-2 degrees of freedom.

To forecast faults in primary underground cable as a function of cable replacement, the Crow-AMSAA 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 Crow-AMSAA 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 χ2 statistic, as is illustrated in Figure 1 from ReliaSoft's RGA 7.

Cumulative Number of Failures Plot
Figure 1: Log-Log plot of Cumulative Faults/100 Cable Miles versus Cumulative Time (Years) including Fit of Crow-AMSAA 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 Crow-AMSAA by Converting Cumulative Faults/100 Cable Miles to Discrete Faults/100 Cable Miles and Multiplying by Cable Length

Crow-AMSAA 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

intage URD Cable Fault Forecast
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 Crow-AMSAA 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 Crow-AMSAA 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.