Models for lifetime estimation: an overview with focus on applications to wind turbines
Thomas M. Welte
K. Wang Department of Production and Quality Engineering, Norwegian University of Science and Technology
, 7491 Trondheim,
T. M. Welte (&) SINTEF Energy Research
, 7465 Trondheim,
This paper provides an overview of models and methods for estimation of lifetime of technical components. Although the focus in this paper is on wind turbine applications, the major content of the paper is of general nature. Thus, most of the paper content is also valid for lifetime models applied to other technical systems. The models presented and discussed in this paper are classified in different types of model classes. The main classification used in this paper divides the models in the following classes: physical models, stochastic models, data-driven models and artificial intelligence, and combined models. The paper provides an overview of different models for the different classes. Furthermore, advantages and disadvantages of the models are discussed, and the estimation of model parameters is briefly described. Finally, a number of literature examples are given in this paper, providing an overview of applications of different models on wind turbines. An essential question for design and operation of technical components is their useful lives. The estimation of useful life is important for designers and manufacturers for design
improvements and marketing purposes, as well as product
users (customers, operators) for making decisions on which
product to buy. Furthermore, the estimation of remaining
useful life (RUL) after the product has been put into
operation is important for making decisions on
maintenance, repair, reinvestment and new investments. The RUL
estimation is not only important for the users/operators but
also for the manufacturers, especially in guarantee periods
and when there are service agreements between users/
operators and manufacturers.
A large number of different models for estimation of
lifetime and RUL exist. The purpose of this paper is to
provide an overview of different models, classify the
models in different groups and discuss advantages,
disadvantages and methods for parameter estimation.
This paper provides an introduction to the topic of
lifetime modeling by providing a brief overview of
different types of models that are often applied. The paper
may also serve as a good starting point for further reading.
Readers that are familiar with only some models presented
here may also contribute from this paper to become more
familiar with models that they have not been applied
Lifetime models are often an integrated part of more
complex models, for example, maintenance models and
logistic models. The lifetime model is then sub-model in a
more complex and larger main model. Consequently, the
lifetime model is in this case an integrated part of the main
model where the lifetime model processes data and
generates results related to lifetime, and where the other parts
of the main model process the results of the lifetime model
further to results that are useful for making decisions on,
for example, maintenance and logistics.
An important aspect of lifetime modeling is uncertainty.
The lifetime of a technical component is an uncertain
quantity that can never be predicted exactly. Thus, a
lifetime model will result in an estimation rather than a certain
and exact lifetime prediction. The overview in the paper
includes different models and methods that can be used for
handling and expressing uncertainty.
The paper is organized as follows. A classification of
models for lifetime estimation is presented in Sect. 2.
Typical model properties including their advantages and
disadvantages and approaches for parameter estimation are
discussed in Sect. 3. In Sect. 4, examples of different
models are presented. In addition, references to
publications presenting applications to wind turbines are given in
Sect. 5. The paper is summarized in Sect. 6.
2 Classification of models
Models for lifetime estimation may be classified according
to different aspects. The following main classification
representing different main classes or main types of models
is used in this paper. These model classes/types are
physical models, stochastic models, data-driven models and
artificial intelligence (AI), combined models.
Brief general descriptions of these types of models are
given below. Further possibilities of model classification
are discussed in Sect. 2.1.
(i) Physical models
Physical models (also called process-based models or
mechanistic models ) are based on a physical
mechanism (failure mechanism) or process (failure process) that
finally leads to a failure of an item.
(ii) Stochastic models
Stochastic models are based on probability theory and
statistical methods. If a model is purely stochastic it
does not represent a physical process or mechanism, but is
used to make a relation between model inputs and outputs
using any mathematical equations or expressions that
provide a good fit given the data .
Data-driven models and AI
Data-driven models (or data-based models) are based
on methods to identify and abstract information and/or
relationships from large sets of data . Data-driven
methods are in particular suitable for data from
continuous monitoring. Many data-driven models belong to the
field of AI.
AI is a field of research that studies and develops
intelligent machines and software. A more formal
definition of AI is the field that studies the synthesis and
analysis of computational agents act intelligently where an
agent is something that acts in an environment . The
agent is the modeling object, e.g., an animal, a machine,
humans, companies, etc. The interest is in what an agent
does; that is, how it acts. Many different subfields,
techniques and models are associated with AI (see Sect. 4.4).
AI is based on general basic concepts such as search,
learning, reasoning and others.
Combination of models
The combination of different models offers the
possibility to establish new models. Both the combination of
different types of models and the same type of model may
improve a single model.
2.1 Other possibilities of classification
2.1.1 Model quantity
Another possibility of classification is according to the
quantity that is modeled. These quantities that are often the
prediction variables of lifetime models are the number of
failures and failure time, and degradation. Thus, the models
either belong to the class of failure (time) models or to the
class of degradation models.
Failure models are used to model and predict failure
events. The quantity usually modeled is the failure time
and time to failure, the time between failures or the number
of failures (in a given time interval or up to a given point in
time), respectively. It might be more appropriate in some
cases to use other measures of usage than time , e.g.,
revolutions (e.g., for rotating machines), number of
operations (e.g., for switchgear), running times (e.g., for
machines where it must be distinguished between operating
and standstill) or load cycles (e.g., for structural
Degradation models are used to model and predict
degradation of items. Degradation can be expressed and
measured by a quantity which describes in a suitable way
the changing of the technical condition or strength over
time (or another measure of usage). This quantity is
denoted degradation variable. Failure is assumed to occur
when the value of a degradation variable crosses a failure
level (threshold). In some applications, there may be more
than one degradation variables .
2.1.2 Model uncertainty
One could also classify the models according to how they
take into account uncertainty as follows:
(i) Deterministic: uncertainty is not taken into account.
(ii) Stochastic: uncertainty is modeled by probability
(iii) Fuzzy: uncertainty is modeled by fuzzy logic theory.
2.1.3 Model generality
A further possibility of model classification is the
generality/universality of the model. Some models can be
applied to all types of items and failure mechanisms,
whereas others are specific models that only can be applied
to the specific items and failure mechanisms they are
designed for. Usually, specific models are more accurate
than general models. However, the more specific and
accurate a model is, the more complex it usually is.
3 Model properties and parameter estimation
3.1 Physical models
Since physical models are usually based on a physical
mechanism (failure mechanism) or process (failure
process), they are valid for all problems where the process/
mechanism leads to a failure. Sometimes, they are
restricted to specific types of components. Typical for
physical models is that the input parameters have a clear
meaning and represent real (and often measurable)
quantities or natural, physical or material constants. Thus, the
models provide a clear understanding about the model
input and output, resulting in a so-called white-box model.
Therefore, physical models are appealing for those who
wish to get better understanding of the mechanisms and
processes leading to failure . Physical models are in
particular useful for design improvement.
In the first step, a physical model must be established if
a good model is not available. This can be a challenging
work and require good knowledge about the problem that is
modeled. However, once a good model is available, it can
be applied to all comparable problems where good
estimates or measurements of the model input parameters
are available. Since the processes in real world may be
quite complicate and may be affected by many mechanisms
and effects, one usually has not the possibility to take into
account all of them. Thus, a physical model may be
restricted to include the main mechanisms and main effects
Physical models are often empirical, which means they
are based on observation or experiments. Physical models
can basically be used for all kinds of predictions, both
long-term and short-term, depending on what they are
3.2 Stochastic models Most stochastic models are of general nature and can be applied for many different problems. An advantage of stochastic models applied to lifetime prediction is that both
an estimate of the mean lifetime and various estimates of
uncertainty can be established, such as variance of the
lifetime, confidence intervals for parameters and
Parameter estimation in stochastic modeling is based on
the observation of the model output. Thus, observations of
the model output, such as observations of lifetime or
degradation, are usually collected as basis for parameter
estimation. When possible, one should fit different
stochastic models to the data and choose the model that gives
the best prediction. Many techniques exist to choose the
best model and check the goodness of fit (e.g., p-value,
confidence intervals, comparison of the maximum
likelihood values and various graphical methods such as
As alternative to data collection, expert judgement can
be used for parameter estimation. There exist different
techniques for expert judgement (see Refs. [4, 5]).
Stochastic models can basically be used for both
shortterm and long-term predictions. However, for lifetime
prediction, they are mostly used to make medium and
longterm predictions. Furthermore, they are often used in
system modeling or as input in other models (such as
maintenance and optimization models) where the main interest
is in long-term averages (such as failure rates). They can
also successfully be applied for comparing and explaining
the lifetime influence of different designs or other factors
either by looking on the results from different samples or
by incorporating explanatory variables (see Sect. 4.3.4) in
3.3 Data-driven models and AI
Data-driven techniques utilize monitored operational data
related to system health. They can be beneficial when
understanding of the first principles of system operation is
not straightforward or when the system is so complex that
developing an accurate alternative model is prohibitively
expensive. An added value of data-driven techniques is
their ability to transform high-dimensional noisy data into
lower dimensional information useful for decision-making
. Furthermore, recent advances in sensor technology and
refined simulation capabilities enable us to continuously
monitor the health of operating components and manage
the related large amount of reference data.
Many data-driven models can be classified as
blackbox models because the relation of input and output
variables and the model parameters is unclear in such
types of models. Parameter estimation in black-box
models is often based on learning and training. Thus, the
models require data, and often data that covers a time
period where a failure is observed, in order to make a
prediction of the lifetime. Learning can be based on data
from a situation identified as normal. Then, all situations
that are different from the normal situation may be
defined as abnormal and (potentially) erroneous. Such an
approach is appealing for diagnostic applications, because
the observation of failures is not required. However, this
approach is not sufficient for making predictions of the
Data-driven models are mostly suitable for making
short-term predictions when the component reaches the end
of life and when a potential failure becomes apparent in
Since there are many models and methods in the field of
AI that in addition often are quite different (also see
examples of AI models in Sect. 4.4), it is difficult to make
general statements about model properties and the ways of
parameter estimation. Many models can be considered as
black-box models. Some others however, as for example
expert systems, are white-box models where the internal
model logic is based on expert knowledge.
4 Examples of models
4.1 Physical models
When looking on wind turbine applications (also see
Sect. 5), most models that have been applied in the class of
physical models are models for fatigue lifetime prediction.
Two of the basic material fatigue models are
PalmgrenMiner rule (also simply called Miners rule) and Paris law,
where the former can be classified as failure model, and the
latter as degradation model.
4.2 Fatigue and fatigue crack growth models
4.2.1 S-N-curves and Miners rule
S-N-curves and the Miners rule are failure models for
fatigue life assessment of many types of materials.
If a material is subject to a sufficient number of stress
cycles that are above the fatigue limit, a fatigue crack or
damage will develop, finally leading to failure . The
higher the stress level, the less stress cycles are required
until the item fails. This relation is illustrated by
S-Ncurves (stress versus life curves) where N is the number of
cycles to failure. In a situation with constant amplitude
loading, the number of stress cycles leading to failure, and
the remaining lifetime, can be estimated by applying
S-Ncurves. In a situation with variable amplitude loadings, the
Miners rule can be used together with S-N-curves. More
details about the approaches and their applications can be
found in Ref. .
4.2.2 Paris law for fatigue crack growth
Paris law is a physical degradation model that describes
the growing of a fatigue crack. The model can be used to
predict the growing of a fatigue crack and estimate the
RUL until fracture of the material. More details about the
approach can be found in Ref. .
4.3 Stochastic models
4.3.1 Stochastic failure rate models
One of the main concepts in stochastic failure and
reliability modeling is the failure rate (also called hazard rate).
The constant failure rate, i.e., k(t) = k, is probably the
most often applied failure and reliability model. Reasons
for the popularity of using constant failure rates in lifetime
and reliability analyses are as follows:
(i) The constant failure rate model is a simple model that
has only one parameter.
(ii) The model parameter (the failure rate) has a clear
meaning: expected number of failures within a time
interval; the inverse of the failure rate is the mean time
to failure (MTTF) or mean time between failures
(iii) Both data and expert judgment can easily be applied
for estimating the failure rate. Methods exist for
combination of data and expert judgement and
updating of failure rate estimates; e.g., Bayesian methods
(iv) Failure rates for different types of components can be
found in databases and in the literature.
(v) Constant failure rates can often be applied in more
advanced reliability and maintenance models, for
example, in models for systems consisting of several
items, Markov models or models for maintained items
or systems (see Ref. ).
Obviously, the use of the very simple model of constant
failure rates has restrictions. Constant failure rates are often
not very realistic, because the failure rate for real
components will often change during the lifetime, for example,
due to aging or due to variable loadings. One of the
distributions, which offers a time dependent failure rate
function and which is often used in practical applications,
is the Weibull distribution.
Different types of failure rate models exist for both
repairable and non-repairable systems, for example:
models for non-repairable
Failure time distributions (e.g., Weibull distribution)
Stochastic failure rate models for repairable systems
Failure time distributions are not further discussed here.
It is referred to Refs. [3, 8] for further details. The
stochastic processes for repairable systems listed above
belong to the group of counting processes. A counting
process is a general stochastic process for modeling of
phenomena which may be described by the number of
events (e.g., failures) that occur. The counting process is a
stochastic process (i.e., collection of random variables) that
represents the total number of events n which occurred up
to time t. More information about applications of different
failure rate models and stochastic processes can be found in
Refs. [3, 811].
The term failure rate is often ambiguously used for both
non-repairable and repairable systems. Failure rate may
refer in the former case to force of mortality (FOM) and in
the latter case to rate of occurrence of failures (ROCOF).
Whereas the FOM is a relative rate of hazard for a single
item, the ROCOF is the absolute rate of change of an
expected number of failures (see Ref.  for a more
4.3.2 Stochastic degradation models
If degradation data are available, different stochastic
degradation models can be used to model further degradation
and predict the RUL. Some useful models are stochastic
processes (e.g., Markov process, gamma process, Wiener
process) or the general degradation path model presented in
Ref. . Further information about stochastic processes can
be found in Ref. . A good overview of application of the
gamma process in lifetime and maintenance modeling is
given in Ref. .
4.3.3 Other approaches and models
Other stochastic models that might be of interest are
models for imperfect repair (see e.g., Ref. ), additive
regression models of failure rate, relative risk regression,
and frailty models (see Ref. ), and Bayesian belief
networks, also called Bayesian networks (see Ref. ).
4.3.4 Stochastic models with explanatory variables
Stochastic models may include explanatory variables.
Explanatory variables are also called covariates. An
explanatory variable is a variable that may predict changes of the
quantity of interest (i.e., the dependent variable, the lifetime or
deterioration). In contrast to the parameters of the underlying
stochastic model, explanatory variables have a clear/physical
meaning, such as the parameters in physical models. When
using stochastic models with explanatory variables, data must
be collected on the value of the explanatory variables in
addition to the value of the observation.
Examples of explanatory variables are environmental
factors that may influence degradation and the lifetime, or
accelerating factors such as higher stresses (often applied
in accelerated life testing). Different types of technical
designs can also be modeled by explanatory variables.
Important factors for the wind industry one might to
incorporate in the modeling are, wind speed, location of
turbine or turbine concept/design.
4.4 Data-driven models and AI
It is difficult to classify models belonging to this model
class. There exist a number of sub-fields/groups of models
that are partly overlapping. AI includes sub-fields and
groups such as machine learning, pattern recognition,
computational intelligence (CI), expert systems, etc.
Datadriven approaches can be divided into two categories:
statistical techniques (regression methods,
autoregressivemoving-average models, etc.) and CI techniques. The main
focus in this section is on CI. However, other methods are
briefly discussed in Sect. 4.4.2.
4.4.1 Computational intelligence
CI algorithms include artificial neural networks (ANN),
evolutionary computation (EC), swarm intelligence (SI),
artificial immune systems (AIS), and fuzzy systems (FS)
. CI-techniques have their origins in biological
systems: NNs model biological neural systems, EC model
natural evolution (including genetic and behavioural
evolution), SI models the social behaviour of organisms living
in swarms or colonies, AIS models the human immune
system, and FS originates from studies of how organisms
interact with their environment . Detailed information
about the different CI models and their applications can be
found in Refs. [15, 16]. Some CI models and their potential
contributions and applications to RUL estimation are
Artificial neural networks
With respect to AI/CI techniques, the most commonly
used prediction methods are based on ANN (see Refs. [17
19]). For prognostic tasks, the most promising methods are
back-propagation (BP) ANNs and recurrent neural
networks (RNNs) (see Refs. [16, 20]).
For example, an ANN may be trained to predict a
vibration value at one increment ahead of the previously
sampled value. During training process, the previous
sample is taken as training input, and the sample one time
increment ahead is used as the output. The network is then
trained by BP learning algorithm. During querying process
of a trained ANN, we take the samples before the time
interest as input. The output is then the predicted value for
the next time interval.
ANNs are usually not directly used for lifetime
estimation. However, ANN is a powerful tool for fault diagnosis,
and in combination with other models (e.g., stochastic
models or fuzzy logic), which may result in a good
combined model (see Sect. 4.5) for lifetime estimation.
(ii) Fuzzy logic systems
An opportunity for increased transparency and openness
of data-driven model is offered by fuzzy logic methods,
which are increasingly proposed in modern diagnostic
technologies. Based on the principles of Zadehs fuzzy set
theory, fuzzy logic provides a formal mathematical
framework for dealing with the vagueness of everyday
reasoning . As opposed to binary reasoning based on
ordinary set theory, within the fuzzy logic framework
measurement uncertainty and estimation imprecision can
be properly accommodated. Thus, fuzzy logic can be
considered as an alternative to probability theory to
Yan et al.  and Zio and Maio  presented a
similarity-based approach for prognostics of the RUL of a
system. Data from failure dynamic scenarios of the system
are used to create a library of reference trajectory patterns
to failure. Given a failure scenario developing in the
system, the remaining time before failure is predicted by
comparing by the fuzzy similarity analysis of its evolution
data to the reference trajectory patterns and aggregating
their time to failure in a weighted sum which accounts for
their similarity to the developing pattern. The prediction on
the failure time is dynamically updated as time goes by,
and measurements of signals representative of the system
state are collected. The approach allows for the on-line
estimation of the RUL.
4.4.2 Other methods and models
(i) Statistical methods
The most natural data-driven technique for RUL estimation
is to fit a curve of the available data of the component
degradation evolution using regression models and
extrapolate the curve to the criteria (threshold) indicating
failure. However, the available data history sometimes may
be short and incomplete, so that extrapolation may lead to
large errors. The same problem arises when employing
autoregressive-moving-average (ARMA) models, although
the method also can handle situations in which more
runto-failure data are unavailable or insufficient .
(ii) Pattern recognition
Pattern recognition covers models and techniques to
recognize patterns in data. Pattern recognition might be of
interest for lifetime prediction in combination with
casebased reasoning in order to extract an input case in form of
a pattern that has been identified in connection with a
failure. Then, the identification of the same or a similar
pattern may provide a warning for a similar situation.
(iii) Expert methods
Case-based reasoning (CBR) is based on the principle
to identify and solve problems based on experience from
previous situations and similar problems. CBR can mean
adapting old solutions to meet new demands, using old
cases to explain new situations, using old cases to
critique new solutions, or reasoning from precedents to
interpret a new situation (much as lawyers do) or create
an equitable solution to a new problem (much as labor
mediators do) [24, 25]. Obviously, data related to a
potential (future) failure situation may be recognized
from a similar case that has been observed before. The
methods require the building of a data base with relevant
In addition, rule-based expert systems  may be
utilized for lifetime prediction. Expert knowledge and
experience from observed failures may be utilized to establish
suitable rules (e.g., if-then-rules).
4.5 Combined models
Some examples of models that may be classified as
combined models are briefly described in the following.
4.5.1 Physical models with uncertainty
Deterministic physical models, for example, can be
transferred into models that take into account uncertainty by
modeling one or several model parameters as stochastic or
The stress-strength approach, also called load-strength
approach , is based on the general principle that failure
occurs when the load or stress l exceeds the strength s of
the item. Thus, the model is based on the simple
deterministic relation that a failure occurs when l[s. For most
items neither load nor strength is deterministic, but
distributed statistically. This means that the values of both the
load and the strength are uncertain, and this uncertainty is
modeled by assuming that load and strength are modeled as
4.5.2 Finite element methods (FEM)
Modern computer technology allows reducing complicate
problems to simple computational steps that can easily be
solved. These steps are usually based on one or several
simple laws or computational rules, such as physical
models or stochastic models. A repetition of the simple
steps, following by an aggregation and combination of the
calculation results, makes the solution of larger problems
manageable. FEM, for example, utilizes this principle.
4.5.3 Neuro-fuzzy systems Hybrid CI models are able to integrate ANN models and fuzzy logic system models to make a better performance .
5 Applications to wind turbines
This section provides a short overview of applications to
wind turbines (see Table 1). The aim of this section is not
to provide a complete literature review, but a brief
overview of selected applications presented in the literature.
5.1 Physical models
Paris law, the S-N-curve approach and related approaches
have frequently been applied to fatigue life analysis of
wind turbine rotor blades. A comprehensive overview can
be found in Refs. [29, 30]. Other examples with
applications to wind turbines are presented in Refs. .
5.2 Stochastic models The Weibull distribution has been used in Refs. [33, 37, 38] to model failures of wind turbines or different turbine components.
Table 1 Applications to wind turbines
In Refs. [37, 39, 40], the Poisson process is applied to
carry out analyses of failure data available from existing
wind power plants. Rademakers et al.  used the
homogenous Poisson process to model gearbox bearing
failures. The power low process has been used in Refs.
 to model failures of wind turbines and different
wind turbine components. The data for parameter
estimation in these models are taken from different European data
bases (Danish and German data sources such as Windstats
The Markov and Gamma process is applied in Ref. 
to model gearbox failure. In Ref. , the Markov process
is used to model cracks and sudden failure of wind turbine
blades. In Ref. , a general Markov process model is
The use of Bayesian networks in wind turbine reliability
modeling is described in Ref. .
5.3 Data-driven models and AI
Applications of data-driven models and AI models to wind
turbine lifetime prediction are difficult to find. As already
stated in Sect. 3.3, these models are mostly suitable for
making short-term predictions. Thus, they are often applied
to fault detection where the model provides a warning a
short time period before the failure actually occurs, without
providing an estimation of the remaining time to failure.
Case studies using real SCADA and condition monitoring
data from wind turbines show that the models presented are
capable to provide warnings of potential failures either
hours  or days  before they actually occur.
Some of the models and methods that have been applied in
these studies are ANNs, support vector machines, principle
component analysis, auto-associative neural networks and
self-organizing feature maps.
A model for prediction of the RUL can be found in Ref.
. The paper presents a model where an ANN is used for
measurement value prediction and the residuals between
predicted and observed values are the basis for comparing
data related to observed failures with a current fault
evolution. The comparison of the residuals in combination
with fuzzy sets results in a fuzzy remaining time to failure
This paper shows that there are many different models that
may be used for lifetime estimation. All models can be
classified in a large multi-dimensional space of model
classes representing different aspects, e.g., as suggested in this
paper: main class/type of model, the quantity that is modeled,
the way uncertainty that is handled and the model generality.
In addition, this paper discusses different model
properties and parameter estimations. Based on the information
presented in the paper, it should be possible to select one or
several models, or a class of models, that may be
appropriate for lifetime estimation given a specific application,
given the data and knowledge available and given the
preferences regarding handling of uncertainty, prediction
interval (long- or short term) and generality.
Of course, it is not possible to list all available models
and groups of models in such a paper. Thus, the
descriptions are restricted to a general and overview level.
Nevertheless, the paper may be a useful starting point for
getting a general overview of the specific field of lifetime
modeling. Finally, the paper may be helpful in selecting
lifetime models, especially for readers who are not familiar
with the specific field.
Acknowledgement This paper is based on the results from the
research projects Windsense - Add-on instrumentation system for
wind turbines (Grant No. 217607) and NOWITECH - The Norwegian
Research Centre for Offshore Wind Technology (Grant No. 193823).
The financial support by the project participants, the research centre
members and the Research Council of Norway is gratefully