Predicting failure behaviour - using mathematical statistics to estimate the bathtub curve.

Helene Seyr, Michael Muskulus
Norwegian University of Science and Technology, Department for Civil and Environmental Engineering, Trondheim, Norway


Knowledge about the failure behaviour in offshore wind turbines is valuable for operations and maintenance. It is beneficial to planning preventive maintenance actions, deciding on a maintenance strategy and is implemented in numerous wind farm models. Using mathematical statistics, we developed a method to estimate the bathtub curve from the observed dates of component failures.
Ziegler et al. showed in their presentation at the Wind Europe summit in Hamburg in 2016 the sensitivity of lifetime extension calculations to the parameters of the bathtub curve. In their analysis, the bathtub curve was fitted using average failure rates over an 8-year period and adding multiple assumptions and constraints to the shape of the curve. The new method presented here does not rely on assumptions about the shape of the curve and uses failure dates instead of averaged failure rates.
This new method has the advantage that it can be used independent of any knowledge about the failure behaviour. On the other hand the model also allows the input of expert opinion in the optimization, were the parameters can be refined by restricting the possible values or awarding likely values a higher probability in the model.


The novel method presented here is able to estimate the bathtub curve from observed failure dates. A function base is chosen and optimisation uses Maximum Likelihood estimation. In addition to failure data, expert opinion can be used as input, restricting the possible parameters for the shape of the bathtub curve using Bayes law.
To test the method, an assumed bathtub curve is used to simulate failure behaviour in different components. The data are then used to estimate the parameters of a bathtub curve with a given function base. This function base consists here of three exponential functions, where the shape parameter is negative, equal to zero and positive respectively. The optimal parameters are chosen, based on their likelihood to result in the observed failure dates.


The quality of the estimation method can be easily assessed, since the failure dates have previously been generated based on a known bathtub curve. The assumed change in failure rates that was previously assumed could be reproduced relatively well. The fitting of parameters and estimation of the failure function is achieved in the very same way for failure data from an offshore wind farm.
The method presented here is not restricted to the offshore wind industry and can be applied to a variety of other industries and research fields.


The novel method presented here enables the user to estimate the bathtub curve from observed failure data and hence makes it possible to predict failure behaviour. The estimation of the bathtub curve can be used for remaining useful life estimation and assessment of the economic viability of lifetime extension.
The use of elaborate mathematical and statistical methods combined with industry experience can lead to an improved understanding of failure behaviour in offshore wind turbines. The method can be extended to other industries by adding industry specific experience and data.


- To understand the challenges of estimating a function describing the failure behaviour from observations.
- To realize the potential of the novel method.
- To create awareness that the used method can be used for problems in operations and maintenance of multiple industries.
- To recognize the importance of combining industry experience and mathematical concepts in order to optimize failure prediction.