# Failure Probability

The Failure Probability curve represents the annual probability of failure of an asset as a function of its age. This is a conditional probability, meaning that it reflects the chance of, say, a transformer failing at age 30 given that it has already reached age 30. This is also known as the hazard rate.

Conceptual Curve

The failure probability curve is commonly referred to as a “bathtub curve” since its shape, at least in concept, resembles the cross-section of a bathtub, as shown in the figure below. An asset whose failure rate followed this curve would be relatively likely to fail at the beginning of its service life when errors in manufacture or installation come to light. This would be followed by a relatively flat period of unlikely and nearly random failures. The final state is when aging and deterioration reach the point where the probability of failure increases rapidly.

Curves Used in Asset Tools

Asset tools are concerned mainly with aging equipment, so we are primarily interested in the right hand side of the curve, after the burn-in period of infant mortality. The figure below shows the typical shape of failure probability used by the asset tools: a continuously rising curve.

Interpretation of the Failure Probability Curve

The failure curve in this form is often called a “hazard rate.” It is a conditional probability curve, defining the annual probability of failure assuming the asset has already reached a given age. For example, the curve above shows that the probability of failure at age 20 is 2 percent. What this means is that if we have a length of URD cable at age 20, we expect a 2 percent probability of failure before age 21. This says nothing about the likelihood that any asset will actually reach a given age in the first place.

Variation in Failure Curves within an Asset Class

Failure curves are general to an asset type. For example, we assume that all medium-voltage vacuum circuit breakers follow the same curve. This is because we expect all of these breakers to follow more or less the same pattern of degradation over time. A key exception is where condition data (such as health index) is available, indicating that some particular asset degrading faster than expected and may have a higher-than-expected failure probability. In these cases, the asset tools may apply a failure probability multiplier.

Failure Probability can vary within an asset class. For example, different types of breakers (e.g., air, SF6, etc.) may have different failure probability curves based on their different expected patterns of aging and deterioration. It is common for the failure curve of a proposed new asset to have a different shape than the asset it replaces. For example, direct-bury cable is replaced by cable in conduit, which has a different, presumably lower, failure probability curve. Because of this, the failure probability curve, and hence Risk Cost, for an asset may be different before replacement than after.

Developing Failure Curves

Reliable sources of failure probability information are rare, so utilities must develop their own failure curves for use in the asset tools. This process varies depending on the robustness of the available data:

• Extensive data. The most desirable approach is to perform a regression or other statistical technique to fit failure curves over failure data compiled by the utility or its peers. This is a time-consuming process, and it requires a large amount of reliable failure data. A very powerful technique for fitting failure curves is the maximum likelihood estimator (MLE) parameter estimation method. An example application of this to develop failure curves for sewer pipes is described here.
• Limited data. Where limited failure data is available (e.g., total number of failures last year) failure curves can be developed by assuming a curve shape, then calibrating it up or down to match the available data. An example of this is described in calibrating failure curves.
• Little or no data. When little or no hard data is available, it is possible to create failure curves based on the experience and educated guesses of your engineers and O&M personnel. If they are able to provide an estimate of the mean time to failure for the asset class, it is possible to generate a reasonable failure curve. Even when more data is available, it is strongly advisable that the analytical results be compared with the failure curves your personnel expect based on their experience.

For an example of curve fitting with good data, see a failure curve fitting case study for underground cable.

Continue to generate a reasonable failure curve.

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