Calculating Probability of Failure

Set out below is a basis for determining the Probability of Failure (PoF) characteristics, which is used to calculate risk cost.

We have assumed that the asset has a dominant failure mode (e.g. winding failure in a transformer).

Two types of PoF behaviors have been recognized:

• Random failures have a constant probability with time. Examples are electronic components and complex electrical assets.
• For decaying assets the PoF increases with age or use as it deteriorates.

Variance to these behaviors occur, for example, 'infant mortality', changes in loading, and 'bedding in'.

Failure statistics can be obtained from:

• Known internal and external historical data
• Estimation from experience and expert knowledge
• Actuarial data
• Accelerated tests.

There is a mathematical relationship between the instantaneous probability of failure r(t), the failure frequency f(t), and the mean time between failures. If we can determine one of these characteristics we can therefore calculate the remainder to determine the current and future risk costs.

(a) Instantaneous probability failure versus age r(t)

We can determine the instantaneous PoF of an asset type by asking experienced and expert staff the question, "for an average asset of age x, what is the PoF in year x".

A curve of best fit can then be drawn to represent the 'standard curve' for the instantaneous PoF of that type of asset.

This standard is a general guide and each asset needs to be assessed as to whether it has a higher or lower probability of failure than the standard. That is, does the actual curve for a particular asset fall above or below the standard.

(b) Failure frequencies f(t)

We can determine the failure frequency f(t) by asking, "if we had a hundred of these assets in this condition how many do you think would fail this year". The question should be then repeated for subsequent years.

A curve of best fit can then be drawn to represent the 'standard curve' for the failure frequency of that type of asset.

Again we need to consider how the failure frequency of a particular asset may differ from the standard.

Mean time between failures (MTBF)

We can determine the mean time between failure by asking the question, if you had a hundred of these assets, what would you expect to be the average time between failures.

As many assets are not allowed to fail, this characteristic is more accurate if calculated from other characteristics or collected data.

Instantaneous probability failure versus age — generic curve

Generic curves used in the SCAR and SMEAR studies provide a general approximation, which is adequate for initial risk assessments.

The current PoF r(t) is determined by asking the question, "in how many years are you 100% sure this asset will have failed". i.e. r(t) = 1, S f(t) = 1.

From the generic curve we can then determine the current PoF.

Failure characteristics

The most commonly understood characteristic is the current instantaneous PoF and how it changes with time as an asset decays.

Experienced engineers who have an understanding of probability theory are required to estimate the frequency. It is a more difficult concept to understand but does provide suitable results.

From these two methods, the frequency curve can be determined to calculate risk cost for use in the ORDM Process.

Data

The following can be used to assist in predicting the probability of failure:

• Historical data
• Records from other authorities, manufacturers and other sources
• Conditioning monitoring programs
• Predictive modeling
• Weibull failure distribution analysis.

A detailed electronic inventory system greatly assists asset managers to model the effective life of assets, based on:

• Their own experience
• Their experience with similar assets in other positions
• The experience of other authorities
• The experience and knowledge of the manufacturer of the asset.

Asset owners cannot predict asset failures with 100% accuracy. However an inventory system with the best available estimate of effective life can help identify the assets that are approaching the end of their effective life.

The process therefore becomes:

• Record the asset construction date
• Give each asset the best effective life estimate
• Check the actual condition of the assets, starting with the most critical.

This approach justifies condition monitoring activities.

Evaluation

The probability of failure is generally related to the condition or performance of the asset, with the key modes of failure being:

• Capacity failure
• Structural integrity (catastrophic failure) or condition failure.

To accurately estimate the current probability of failure, each of these modes of failure needs to be considered.

The greatest risk cost for most service authorities is structural integrity or physical asset failure, and the probability will be directly related to the physical condition of the asset.

An example of asset effective life and determination of probability of failure is shown in below:

Alternatively, a preliminary assessment can be made on asset condition.

Assets at risk are identified through a general assessment of their structural integrity. The probability of failure can be assigned a ranking according to the structural integrity:

If field staff collect data, only four condition ratings need to be used :

Setting probability factors is arbitrary, however, it is the relative order that is important. The accuracy and reliability of results from the analysis is only as good as the data input. However, if a consistent approach is taken in the condition assessment, useful information can still be derived to assist in determining assets at risk.

A typical decay curve probability of failure relationship is shown below.

Practical issues

In predicting the probability of failure we very often have to rely on failure histories and derive the failure pattern in relation to the actual age of the assets when failure occurs.

If sufficient and reliable failure records are available, we can predict the likely number of failures as assets age and hence the probability of failure. As more failure records are collected and we understand the causes and modes of failure, we are then able to develop the failure profile and predict with confidence the probability of failure. In the meantime, we may have to rely on some anecdotal failure evidence, past experience and local knowledge of field staff.

Knowing the time when failure will ultimately occur, we can then plot the likely probability of failure during the interim.

Depending on the operating environment, it is sometimes possible two similar assets may have different failure profiles, although they would be considered to have failed at the same time. In this case we can either adopt the appropriate failure profile, or we can adopt a combination of the failure profiles with the confidence level attached to each.