Optimal selection of autoregressive model coefficients for early damage
detectability with an application to wind turbine blades
Simon Hoell
The LRF Centre for Safety and Reliability Engineering
The University of Aberdeen
Aberdeen, AB24 3UE
UK
Piotr Omenzetter (corresponding author)
The LRF Centre for Safety and Reliability Engineering
The University of Aberdeen
Aberdeen, AB24 3UE
UK
Tel. 44-1224-272529
E-mail: [email protected]
1
ABSTRACT
The constant striving for reduction in operational and maintenance costs and, at the same
time, increase in structural safety and reliability is the main driver behind the progress in
structural health monitoring (SHM) techniques. SHM is especially important and required for
new energy production infrastructure, such as large scale wind turbines, where the increasing
interest in, and demand for, renewable energy clashes with high operational costs. Datadriven vibration-based structural damage detection methods have an edge over other
approaches due to their competitive instrumentation and data analysis costs. The use of
autoregressive model coefficients (ARMCs) as damage sensitive features in combination with
statistical hypothesis testing for making decisions about the structural state and condition is
pursued and enhanced in this study for maximizing early damage detectability. This latter
aspect is the main original contribution of this study. Currently, either the full set of
coefficients or subsets selected by trial-and-error are used as multivariate damage sensitive
features (DSFs). However, these simple approaches to building DSFs are not capable of
systematically searching for the optimal choice of the coefficients and may miss some of
them that are more affected by damage than others. Furthermore, the thresholds of statistical
hypothesis tests used for damage detection increase with the number of coefficients included
in the DSF and incorporating insensitive coefficients will deteriorate the performance of
damage detection algorithms. A methodology for systematic selection of an optimal set of
ARMCs for a multivariate DSF based on their sensitivity to damage versus a statistical
hypothesis testing threshold is therefore proposed and explored in this paper. Two approaches
for the optimal selection, based on either adding or eliminating the coefficients one by one or
alternatively using a genetic algorithm (GA) are proposed. The methods are applied to data
from advanced and realistic numerical simulations of an aerodynamically excited large wind
turbine blade with complex material layup and geometry. To demonstrate the applicability of
the approach, trailing-edge disbonding with varying extent is studied as damage scenarios. It
is demonstrated that the GA outperforms the other selection methods and enables composing
multivariate DSFs that enhance early damage detectability and are insensitive to
measurement noise.
Keywords: autoregressive models; damage detection; hypothesis testing; optimal feature
selection; time series; vibration based damage detection; wind turbine
2
LIST OF SYMBOLS
Roman letters
D
Mahalanobis distance
F
Cumulative probability distribution function; aerodynamic force acting on
blade element
H
Hypothesis
K
Number of time lags
M
Aerodynamic moment acting on blade element
N
Number of surface nodes in blade finite element model
N
Gaussian distribution

Parental population for genetic algorithm
Q
Modified Ljung-Box-Pierce statistic
S
Vector cosine distance in Eq. (21)
T2
Hotelling’s T 2 statistic
a
Autoregressive coefficient; linear aerodynamic load distribution coefficient
across blade
b
Uniform aerodynamic load distribution coefficient
c
Moving average coefficient; linear load distribution coefficient along blade
e
Noise term
f
Fitness function; aerodynamic nodal forces
m
Dimensionality of damage sensitive feature vector
n
Number of samples
p
Autoregressive order
q
Moving average order
r
Autocorrelation function
s
Binary selection variable in genetic algorithm
s
Binary selection vector in genetic algorithm
t
Discrete time
x
Edge-wise coordinate in finite element blade model; edge-wise coordinate in
AeroDyn model
y
Thickness-wise coordinate in finite element blade model; thickness-wise
coordinate in AeroDyn model
z
Zero-mean time series; flap-wise coordinate in finite element blade model;
3
flap-wise coordinate in AeroDyn model
Greek letters
Σ
Variance-covariance matrix
∆
Difference operator
α
Level of significance
β
Number of flipped entries in selection vector in genetic algorithm

Number of parental individuals in genetic algorithm
υ
Damage sensitive feature
υ
Damage sensitive feature vector
μ
Mean value
μ
Mean value vector
ρ
Cross-correlation coefficient
σ
Standard deviation
σ2
Variance
2
Chi-square probability distribution function
Subscripts
0
Null hypothesis
1
Alternative hypothesis
N
Normal to rotor plane
P
Pitching
T
Tangential to rotor plane
c
Current state
d
Damaged state
h
Healthy state
offsp
Offspring
pl
Pooled
r
Blade element in AeroDyn
ref
Reference
rel
Relative
temp
Temporary
x
x direction
4
y
y direction
Superscripts
T
Transpose
ˆ
Estimate
LIST OF ACRONYMS
ACF
Autocorrelation function
AIC
Akaike information criterion
ARMC
Autoregressive model coefficient
AR
Autoregressive
ARC
Autoregressive coefficient
ARMA
Autoregressive moving average
DOF
Degree of freedom
DSF
Damage sensitive feature
EMD
Empirical mode decomposition
FE
Finite element
GA
Genetic algorithm
HHT
Hilbert-Huang transform
IMF
Intrinsic mode function
LE
Leading edge
MA
Moving average
MAC
Moving average coefficient
NBI
Next-Best-In
NREL
National Renewable Energy Laboratory
NWO
Next-Worst-Out
PAC
Partial autocorrelation
SDD
Structural damage detection
SHM
Structural health monitoring
SNL
Sandia National Laboratory
TE
Trailing edge
WT
Wind turbine; wavelet transform
WTB
Wind turbine blade
5
1. INTRODUCTION
The world’s energy infrastructure is undergoing significant changes due to the increasing
interest in, and demand for, renewable energy. For the sector of wind energy, the relentless
strive for more efficient energy harvesting leads to growing numbers and sizes of wind
turbines (WTs) and erections in remote areas, such as offshore, where winds are stronger and
more reliable and predictable. However, the increasing operation and maintenance
expenditure, which can make up to 20% of the total energy production cost [1], affects
adversely the production targets and expected revenues. Knowledge of the current structural
state and condition obtained from interpreting remotely monitored data can counteract this
issue.
The process of continuous monitoring of structures using sensors, extracting
information and knowledge from these observations and determining the structural
performance, condition and reliability is referred to as structural health monitoring (SHM)
[2]. There has been large amount of effort during the past decade to develop effective SHM
methods for application in mechanical, aerospace, civil and other structural systems [3-9].
Several non-destructive testing techniques based on different physical principles, such as
thermal imaging, X-radioscopy, electrical resistance and ultrasonic waves, have been
proposed for structural damage detection (SDD) in wind turbine components [10, 11].
However, the majority of these techniques are not applicable for in-service inspections in
wind turbine blades (WTBs) because of difficult access, complex geometries, and
challenging environmental and operational conditions. The currently available and arguably
more practical methods, such as the acoustic emission and dynamic strain measurement [12],
require dense sensor arrays, which leads to high instrumentation costs.
In contrast, vibration-based methods are less demanding due the use of global
vibrational responses, and thus they received increased attention in the past [13-16]. Under
the premise that damage leads to changes in structural stiffness, mass or energy dissipation
mechanisms of a structure, the use of dynamic response measurements enables to determine
damage sensitive features (DSFs), which depend on the current structural state (i.e. healthy or
damaged). Traditionally, changes in modal properties, e.g. natural frequencies and mode
shapes and their spatial derivatives, have been used for SDD [17-20]. The estimation of these
parameters from large structures like WTBs is commonly done with the help of output-only
data and operational modal analysis techniques [21]. Although different methods have been
proposed for automization of this process [22-25], it can still be hindered by practical
difficulties as well as high computational effort.
6
DSFs defined by time series representations of the underlying stochastic process
represent a non-physics-based, or data-driven, approach. These representations avoid the
estimation of physical parameters, thus their application is less demanding. They can be
classified as non-parametric and parametric [26]. The first class includes SDD methods
working in the frequency domain, e.g. using power spectral densities [27], frequency
response functions [28, 29] and transfer functions [30], or operating in the time domain, e.g.
using cross-correlations [31] and Green’s functions [32]. Even though these stationary nonparametric time series-based DSFs have the advantage of being simple and computationally
efficient, they are usually less parsimonious compared to the parametric ones. This can
complicate the damage decision making process due to increased computational burden and
compromised accuracy. Furthermore, they cannot account for time-varying system dynamics.
Non-parametric SDD methods working in the dual time-frequency domains also
received attention due to their ability to capture non-linear or non-stationary dynamics. The
empirical mode decomposition (EMD) resolves a signal into a finite number of nearly
orthogonal components characterized by different time scales. Chen et al. [33] took
advantage of this decomposition for SDD in a cantilever beam exposed to vibro-impacting
and separated non-smooth interactions and elastodynamic responses. This enabled to indicate
the impact location from the instantaneous mode shapes. Applying the Hilbert transform to
intrinsic mode functions obtained from EMD gives the Hilbert-Huang transform (HHT).
Damage in a numerical bridge model under travelling loads was detected by means of HHT
of accelerations by Roveri and Carcaterra [34]. This approach was found to be sensitive to the
passing speed of the loads. DSFs defined by time averages of instantaneous frequencies and
signal energies from acceleration HHT were applied for SDD in an experimental single
degree of freedom (DOF) system and a wind turbine blade under band limited base excitation
by Carbajo et al. [35]. Damage decision making was done with the help of thresholds defined
online from DSF ranges observed in the healthy state. This enabled to detect severe damage
with frequency changes of more than 15% between the healthy and damage states. The
empirical estimation of thresholds required pre-assigning of several parameters. Lamb waves
from sine burst excitation in plates were analyzed using HHT by Pai et al. [36]. This method
does not utilize ambient vibrations and additional excitation sources are required, which will
be a limitation for real-life applications in large structures. The wavelet transform (WT) can
be used to define DSFs in terms of wavelet energies or entropies [37, 38]. Here, WT
coefficients and scales used to build the DSFs need to be selected in advance. Although, the
7
use of these advanced signal processing techniques for SDD is promising, the review above
indicates there are still a number of challenges to overcome.
Autoregressive (AR) models, as parametric time series representations, have been
studied for SDD in the past [39-48]. In this paper, the term AR is used broadly referring to a
class of models including both pure AR models and autoregressive moving average (ARMA)
models. The term autoregressive model coefficients (ARMCs) is used to refer to all the
coefficients of both pure AR and ARMA models, whereas the term autoregressive
coefficients (ARCs) refers to only the coefficients of the AR part of a model (for pure AR
models this is of course the only part). (The reader is directed to the List of Acronyms at the
beginning of the paper if they wish to remind themselves about the meaning of the various
acronyms used throughout the paper.) ARMA models were utilized for SDD, e.g. by Carden
and Brownjohn [49]. They showed theoretical connections between the parametric model
orders and the number of observable eigenmodes of a physical structure. Nonetheless, due to
the invertability property of AR and moving average (MA) processes [50], pure AR models
can be used to describe the underlying stochastic process instead of ARMA models. Although
they are less parsimonious, the identification and estimation of pure AR models is less
demanding, hence their popularity [43, 51-54]. Nair et al. [55] discussed a theoretical
relationship between the structural stiffness and ARCs and demonstrated that the observation
of ARCs from ARMA models can be used for detecting changes in a structural system.
In order to obtain useful DSFs, the selection of appropriate model orders is an
important step. The influence of model orders on SDD results in a 3-storey laboratory
structure was investigated by Figueiredo et al. [56] with respect to damage detectability and
operational and environmental effects. They found that the conventional approaches to model
order selection, such as those based on the partial autocorrelations (PACs) and the Akaike
information criterion (AIC), lead to robust estimates, while model orders that are too low
adversely affect damage detectability. Decisions about the structural state were made by
examining the model residuals and applying a multivariate test to the DSF vectors composed
of ARCs.
A unified statistical framework for time series-based SDD has been presented by
Fassois and Sakellariou [26], where multivariate hypothesis tests on ARMCs are one case.
These approaches are also consistent with the statistical pattern recognition paradigm by
Farrar et al. [57], where a statistical model for the ARMCs can be estimated in the healthy
state and assumptions of their stochastic distribution enable to define a threshold for
distinguishing between the healthy and the damaged state. Although soft computing
8
techniques for making decisions about the structural state, such as Gaussian mixture models
[58], self-organizing maps [59], fuzzy pattern recognition [60], learning vector quantization
[61], and support vector machines [62], received considerable attention, statistical tests are
invaluable because of their strong theoretical background, conceptual clarity and the
possibility to define detection thresholds systematically.
Identifying an appropriate AR model and testing statistically the full set of the
corresponding ARCs can give reasonably good damage detection results by means of the
statistical tests or control charts [63]. Similarly, approaches based on model residuals
generally require the use of the full ARMCs sets [64]. However, individual ARMCs may be
differently affected by damage, thus the inclusion of less damage-sensitive coefficients in a
DSF vector can adversely affect the SDD performance. Nair et al. [55] proposed a damage
index based on the first three ARCs of an ARMA model and statistical testing of the index
mean. A more systematic selection of ARCs was performed by de Lautour and Omenzetter
[65], where the trends between increasing the numbers of sensors and ARCs and damage
classification errors were explored. In the study, larger ARC sets improved the results of
damage classification by means of an artificial neural network. Dimensionality reduction
techniques, such as principal component analysis, piecewise aggregate approximation and
random projection, were applied by Khoa et al. [66] in order to improve the detectability of
damage and reduce the effects of environmental and operational variations. Principal
component analysis applied to ARCs derived from acceleration measurements of a threestory laboratory structure was investigated by Figueiredo et al. [67]. It was found that
separable DSF clusters could not be identified using only two principal components. Similar
preliminary attempts to select optimal DSFs for SDD were also reported for algorithms
different than those based on AR models. In Žvokelj et al. [68], the selection of intrinsic
mode functions (IMFs) obtained from HHT of accelerations was discussed for optimal
bearing fault detection results. Kurtoses and crest factors of the functions for different time
scales were proposed for automatically selecting the mostly affected scales with respect to a
nominated damage case. Similarly, two IMFs, the first and the third, were chosen due to
significant values of their kurtoses and the corresponding instantaneous frequencies selected
from amongst competing DSFs by Georgoulas et al. [69]. These approaches were developed
based on observing the behavior of IMFs under damage, but they do not directly account for
the mechanics of damage decision making procedures or interactions between multivariate
DSFs. Harvey and Todd [70] proposed an advanced methodology using a genetic algorithm
(GA) for the selection of signal processing routines, from basic, parameter-free time series
9
normalization to more sophisticated wavelet analysis, for DSF extraction from signals to
facilitate damage identification with improved accuracy. However, the results for acceleration
signals from a laboratory structure were still only comparable to those obtained with a pure
AR model with five coefficients, while requiring a demanding feature selection phase due to
the large number of candidate signal processing routines and their related specifications.
However, the approaches reviewed herein do not explicitly evaluate the interplay between the
DSF selection and its dimensionality on the one hand, and the statistical damage detection
thresholds on the other, and the effects of such interactions on damage detectability.
Motivated by this gap in knowledge and practice, the present paper addresses the
unresolved issue of systematic optimal DSF selection for multivariable hypothesis testing to
improve the detectability of early damage. Here, the coefficients of either pure AR or ARMA
models are chosen as representative DSFs for vibration-based SDD but the concept of
optimal selection can be adopted for any other feature set in a straightforward fashion. The
selection is based on the following rationale: Adding another ARMC to the DSF, i.e.
increasing the dimensionality of DSF, will only enhance damage detectability if the
contribution from the candidate ARMC to the stochastic distance between the healthy and
damage DSF is larger than the corresponding increase in the statistical threshold due to the
increased dimensionality, or the number of statistical DOFs. There will thus be an optimal
number of ARMCs to be included in a multidimensional DSF for the best damage detection
results.
Two different approaches are explored to identify the optimal set of ARMCs to be
included in the DSF vector, where decisions about the structural state are made by means of
statistical hypothesis testing of stochastic distances between the DSF vectors from the healthy
and damage state. The first approach utilizes a one-by-one elimination or addition of ARMCs
and observing their contribution to the stochastic distance between the healthy and damage
DSF with respect to a statistical threshold. A binary GA is applied in the second approach as
an evolutionary-inspired global optimization method [71] for the selection of ARMCs. Here,
ARMCs are chosen by random initialization of binary selection vectors, which are
subsequently optimized by recombination, mutation and selection with respect to a damage
detectability threshold.
The proposed methodology is applied to data obtained from advanced numerical
simulations of a large, real-life WTB characterized by a complex material layup and
geometry. Transient aerodynamic calculations are performed under turbulent wind
10
conditions. Several trailing-edge (TE) disbond cases of different extent are introduced as
damage scenarios.
The main advantages of the approach to SDD presented here include using output-only
data, i.e. avoiding the need for artificial excitation which would be very difficult to apply in
practice, describing the dynamic response data using parsimonious parametric models, and
employing statistical hypothesis testing which allows making decisions about the current
structural state automatically and with explicit statistical confidence levels attached to the
decisions. Most importantly, damage detectability is significantly enhanced by retaining only
these DSFs that are most sensitive to detrimental structural changes. The use of statistical
hypothesis testing entails moderate effort for estimating the DSF probability density function
from available monitoring data.
The paper is organized as follows. First, AR time series modelling and statistical
hypothesis testing for SDD are briefly overviewed. Second, the proposed ARMC selection
procedures for improved early damage detectability are presented. The advantage of an
appropriate composition of the DSF is analytically demonstrated for a bivariate DSF vector.
Then, the numerical WTB model with disbonding damage scenarios and transient
aerodynamic simulations are described. Next, the optimal ARMC selection results of the
different procedures considered are presented for identified and validated pure AR and
ARMA models. Furthermore, the effect of artificial noise on the damage detectability is
investigated. Finally, a summary and conclusions round up the paper.
2. THEORY
2.1. Autoregressive time series modelling
Responses from structural systems can be analyzed as stochastic processes using sequentially
measured observations referred to as time series. To use time-invariant AR models for
representing the underlying stochastic process [72], it is assumed that the analyzed vibration
response signals are stationary, i.e. have a constant mean and variance-covariance matrix.
Parametric time series modelling requires usually the following four steps [50]: i) selecting a
parametric model class (herein the pure AR or ARMA model structures have been chosen),
ii) identification of an appropriate model order to capture adequately the underlying system
dynamics, while ensuring computational efficiency and avoiding overfitting, iii) model
parameter estimation, and iv) validation of the model for its appropriateness and accuracy.
11
For an ARMA(p,q) process of AR order p and MA order q, a current value of a zeromean time series z[t] at a time instant t can be expressed as the weighted sum of p previous
time series values, q weighted past noise terms and the current noise term e[t]:
p
q
i 1
i 1
z[t ]   ai z[t  i ]   ci e[t  i ]  e[t ]
(1)
The ARCs, ai (𝑖 = 1, … , 𝑝), the MA coefficients (MACs), ci (𝑖 = 1, … , 𝑞), and the standard
deviation, e, of the zero-mean, normally distributed, independent, random noise term e[t] are
the model unknowns. It should be noted that pure AR models are a subclass of ARMA
models with q=0, where the corresponding process of order p can be abbreviated as AR(p).
Investigating the time series PACs can give a first insight for selecting an adequate AR
order because PACs of an AR(p) process are theoretically zero for lags higher than p.
Similarly, an appropriate MA order can be estimated from a drop in autocorrelation functions
(ACFs) of the time series. A more systematic approach uses measures such as the AIC and
statistics of the identified prediction error. An estimation of model residuals can be obtained
by modifying Eq. (1) as follows:
p
q
i 1
i 1
eˆ[t ]  z[t ]   aˆi z[t  i ]   cˆi eˆ[t  i ]
(2)
where the hat denotes an estimated quantity. Different methods exist for estimating ARMA
models. Here, an iterative search algorithm based on Gauss-Newton minimization is adopted
for solving the non-linear optimization problem defined by a squared prediction error
criterion [73]. This search is terminated by a predefined tolerance threshold for the error
criterion, a defined maximum number of iterations or a threshold for no further improvements
in the minimization function. Initial parameter estimates are obtained from a combined
ordinary least squares optimization and instrumented variable algorithm. Parameters of pure
AR models can be estimated by different methods, such as solving the Yule-Walker
equations or maximum likelihood methods [74]. In the present paper, the Burg algorithm [75]
was used, in which ARCs are recursively estimated up to the preselected model order with
the help of reflection coefficients by minimizing the forward and backward prediction errors
as p linear least squares problems. Although, forward-backward non-linear least squares
approaches are competitors for short time series, the Burg method is computationally simple
and resulting ARCs are guaranteed to be stable [76].
The AIC evaluates candidate models by their likelihood. The sample-size normalized
AIC can be calculated with the help of the estimated noise variance, σ̂ 2e , as [50]:
12
AIC ( p)  ln(ˆ e2 )  2( p  q  1) n
(3)
where n is the sample size. The first term is a measure of the model likelihood, while the
second is a penalty for the model order p+q, and, with it, the model complexity.
If the estimated model is valid, then the residuals have the properties of a Gaussian
white noise sequence. Thus, the residual normality can be inspected by means of normal
probability plots. Furthermore, the residual ACF can be investigated. The unbiased k-th
coefficient of the residual ACF, re, can be calculated as [77]:
re [k ] 
1 nk
 e[i]e[i  k ]
n  k i 1
(4)
For a white noise process, the coefficients should fall within the selected level of confidence
bounds of the appropriate statistic [74]. To test the residual ACF as a whole, a portmanteau
lack-of-fitness test can be utilized. The modified Ljung-Box-Pierce statistic, Q, [78] is
commonly used for the task and it can be defined as:
K
Q  n(n  2) rˆe [k ] (n  k )
(5)
k 1
where K is the maximum ACF lag to be included. The Q statistic follows for a standard
Gaussian white noise process and, with it, for a valid model a  2 distribution with K-p
DOFs, denoted by  K2  p . This fact can be utilized to formulate statistical hypotheses in order
to test the estimated AR models.
2.2. Statistical hypothesis testing
For making decisions about the structural state in SDD applications, statistical hypothesis
testing is widely employed. It enables to distinguish between the healthy and damage state
with the help of DSFs in a systematic manner. Parametric time series models can facilitate
SDD by assessment of their residuals or parameters themselves. The latter approach is
applied in this paper. A DSF vector, υ̂ , of size m can be constructed from the estimated
ARMCs, 𝑣̂𝑖 , as:
υˆ  vˆ1 vˆ2
vˆ p  q 
T
(6)
where superscript T denotes transpose. (Note in the context of this study, neither all the
ARMCs have to be included in the DSF vector, i.e. generally m≤p+q, nor all consecutive
ARMCs for indices smaller than m have to be entered into υ̂ .) It is assumed that DSF vectors
follow a multivariate Gaussian distribution:
13
υˆ
N (μ, Σ)
(7)
where μ and Σ are the true mean vector and variance-covariance matrix, respectively. For
SDD, a statistical model obtained in the healthy state can be used to make decisions about the
current state of a structure. If the structure is undamaged, then newly acquired DSF vectors
should conform to the initial, healthy distribution. This can be tested either by separate
univariate statistical tests applied to each ARMC or by a single multivariate test. Multivariate
statistical tests have the advantage of determining the contribution of each variable in the
presence of other variables and their mutual cross-correlations, preserving the selected level
of significance and having a greater power [79].
The statistical hypotheses for testing the multivariate mean of the current state DSF
against the healthy DSF mean, indicated by the subscripts c and h, respectively, can be
defined as follows:
H 0 : μc  μh
H1 : μ c  μ h
(healthy)
(8)
(damaged)
where the null hypothesis, H0, describes the healthy state and the alternative hypothesis, H1,
the damage state. In practical applications, statistical models can normally only be
constructed with the help of the estimated DSF mean vector, μ̂ , and estimated variancê . The T 2 (m) statistic, as a standardized distance between two mcovariance matrix, Σ
dimensional sample means, can be used for testing the above hypotheses [79]:
T 2  m 
nc nh
T
 μˆ c  μˆ h  Σˆ pl1  μˆ c  μˆ h  Tm2,nc nh 2
nc  nh
(9)
̂ pl, is defined as
The pooled sample variance-covariance matrix, Σ
ˆ  (n  1) Σ
ˆ  (n  1) Σ
ˆ   n  n  2
Σ
pl
c
h
h
c
h
 c
(10)
̂ c and Σ
̂ h are unbiased estimators of the variance-covariance matrices in the current
where Σ
and healthy state, respectively. The numbers of samples used for estimating the mean and
variance-covariance in the current and the healthy state are denoted by nc and nh,
respectively. Alternatively, the T 2 (m) statistic can be estimated without explicitly inverting
̂ pl, as[79]:
the sample pooled variance-covariance matrix, Σ
T ( m) 
2

ˆ  n n (μˆ  μˆ )(μˆ  μˆ )T (n  n )
det Σ
pl
c h
c
h
c
h
c
h
 
ˆ
det Σ
pl
 1
(11)
14
where det() is the determinant. This can increase the stability of the estimate in illconditioned cases, e.g. when high-dimensional DSF vectors are used but only small numbers
of samples are available for variance-covariance matrix estimation.
As indicated in Eq. (9), the T 2 (m) statistic follows Hotelling’s distribution, Tm2,nc nh 2 ,
with m and nc+nh−2 DOFs, where m corresponds to the DSF vector dimensionality. This
2
enables to define a statistical test of the T 2 (m) statistic, by means of the 𝑇𝑚,𝑛
𝑐 +𝑛ℎ −2
cumulative distribution function, FT 2
m ,nc nh 2
T 2  m   FT 2
m ,nc  nh 2
Else
1   
1    , as
 H 0 is accepted
 H 0 is rejected
(12)
where α is the selected level of significance.
For an online SHM, a decision about the current structural state is often required as
early as possible. Thus, testing a single sample of the DSF vector of the current state, υˆ c ,
may be required. The T 2 (m) statistic becomes in such a case the conventional squared
Mahalanobis distance, D2(m) [80]:
D2  m  (υˆ c  μh )T Σh1 (υˆ c  μh )
m2
(13)
where the true, rather than estimated, mean vector and variance-covariance of the healthy
state are assumed to be known or available with high accuracy. The corresponding statistical
test can then be defined as in Eq. (12) with D2(m) instead of T 2 (m) and hypothesis testing
thresholds determined by the cumulative distribution function FX of the  m2 distribution.
2
m
3. AUTOREGRESSIVE MODEL COEFFICIENT SELECTION FOR OPTIMAL
DAMAGE DETECTABILITY
Conventionally, ARMC-based DSF vectors for statistical hypothesis testing in SDD are
constructed by either using the full set of available ARMCs or subsets selected a priori or by
trial and error. Such practices disregard the following two facts. First, the ARMCs may be
differently affected by a damage due to the damage characteristics and its influence on the
structure’s dynamic properties. Additionally, the statistical threshold defined by the relevant
cumulative distribution function, such as FT 2
m ,nc nh 2
1   
used in this study, increases with
the number of DSF vector entries, or stochastic DOFs. Thus, taking all the available ARMCs
or a priori selected subsets into account may not necessarily provide the best damage
15
detectability. In other words, there will be a trade-off between the number of ARMCs to
include in the DSF and overall sensitivity to damage of the so-formed multivariate DSF.
The overall process of optimal selection of ARMCs and their subsequent use for SDD
is schematically shown in Figure 1. The baseline phase is offline and uses normalized time
series segments of accelerations from the healthy structure and from a reference damage
state. The appropriate AR model order is identified with the help of the healthy signals only.
Estimation of ARMCs for the identified model structure is then conducted for time series
segments of both structural states. Next, the parameters of the statistical distributions, i.e.
mean vectors and variance-covariance matrices, are estimated to describe the ARMCs
distributions in both states. With the help of these distributions, the optimal subset of ARMCs
can be identified using the Next-Best-In (NBI), Next-Worst-Out (NWO) or GA-based
selection. In the damage detection phase, the ARMC statistical distribution from the current
structural state is estimated in a similar way. Decisions about the presence of damage in the
current system can then be made online or offline using the selected optimal DSFs via
statistical hypothesis testing.
The following subsection illustrates the central concepts of this work by way of an
example using a bivariate DSF. Then, two different approaches for selecting ARMCs in order
to improve damage detectability are introduced. The first approach adds or discards ARMCs
one by one, and ranks them in the process, according to their contributions to a statistical
distance in relation to a damage detection threshold. The second approach solves the optimal
ARMC selection problem with the help of a binary GA.
3.1. Illustrative example: bivariate DSF
This section illustrates the central concepts of the proposed optimal DSF selection as well as
the importance and benefits of considering jointly the sensitivity of ARMCs to damage and
changes in the statistical threshold in the multivariate DSF selection process by means of an
analytical example. Here, statistical hypothesis testing is done on a single DSF vector sample
and the true mean DSF vector and variance-covariance matrix for the healthy state are
assumed to be known. For the sake of simplicity but without loss of generality, a bivariate
DSF is considered. The squared Mahalanobis distance D2(2) and  22 distribution (Eq. (13))
are therefore used.
The difference between the current and healthy DSF vectors is
16
 vc,1  h,1   v1 
υ  
 
vc,2  h,2  v2 
(14)
and the variance-covariance matrix is
  12
Σ
 12 1 2
12 1 2 

 22 
(15)
where ρ12 is the correlation coefficient between the two ARMCs. Making another simplifying
assumption that the standard deviations σ1=σ2=1 reduces the squared Mahalanobis distance to
D 2  2    v12  v22  2 12 v1v2  1  122 
(16)
For statistical hypothesis testing, damage is indicated with a significance level  if the
squared Mahalanobis distance is equal or exceeds the statistical threshold FX 2 (1   ) :
2
D 2  2   FX 2 (1   )
(17)
2
This enables to identify regions in the parameter space of Δv1 and Δv2 where damage is
detectable, or otherwise, and also explore how the cross-correlation between the two ARMCs
influences detectability. To that end, Figure 2 shows a case without cross-correlation (ρ12=0)
and another one with high cross-correlation (ρ12=0.9), where the statistical threshold is
defined for a 5% level of significance. The enclosed elliptical areas (or circular for ρ12=0 as a
special case) represent the region of undetectability in the parameter space. The shape of the
region depends on the cross-correlations magnitude, and higher values lead to ellipses that are
more ‘stretched’ along their major axis and ‘flattened’ along the minor axis. Additionally, the
case of using only v1 (a univariate DSF) is illustrated, for which the detectability is defined by
D 2 1  FX 2 (1   )
(18)
1
and the corresponding region of undetectability is a horizontal band in the parameter space.
Two paths (Path 1 and Path 2) in the ARMC space are also indicated. They correspond to two
hypothetical damage types and show how the ARMCs are affected by varying damage extent.
The first important observation is that while the regions of undetectability when two
ARMCs are used are bounded by ellipses (as opposed to the unbounded region when only
one ARMC is used), there are types of damage with their associated changes in ARMCs
which will be detected earlier using only one ARMC rather than two. These are the regions
where the ellipses protrude outside the horizontal band. In other words, for such damages
increasing the dimensionality of DSF will actually delay damage detection. (Note ‘early’
means here damage detection when shifts in the ARMCs are small, not necessarily when
damage related physical changes to the system are small, as the sensitivity of the ARMCs to
17
the latter as such is excluded from the discussion). For those damage cases, the statistical
threshold defined by  2 distributions increases faster with additional statistical DOFs than
the contributions to the Mahalanobis distance from additional ARMCs. The analysis can be
extended beyond the bivariate case presented in the example by observing how the threshold
FX 2 1    changes for a wide range of statistical DOFs and, more importantly, what the
m
average contribution to it of each DOFs, FX 2 1    m , is. These are shown in Figure 3 for
m
the case of =5%, where it can be seen that FX 2 1    increase with m is practically linear,
m
and so additional ARMCs to be added to the Mahalanobis distance must at least keep up with
this rate of growth in the threshold to maintain damage detectability. One can then rank the
ARMCs in terms of their contributions to damage detectability, and include in the DSF only
these for which the rate of increase in the Mahalanobis distance outpaces the corresponding
rate of increase in the statistical threshold. This will result in a multivariate DSF which
optimizes damage detectability. Figure 2 demonstrates that such an optimal DSF will be
specific to a given type of damage, because different damage types will shift the DSF in
different ways. There will be damage types for which this optimal DSF will comprise less
than all the available ARMCs, e.g. only one in the illustrative bivariate case.
The second important observation from Figure 2 is concerned with the effect of crosscorrelation between the ARMCs determined from the healthy state. As the cross-correlation
increases, as illustrated by the case with ρ12=0.9, the detectability of the type of damage
associated with Path 1 requires a higher difference in the ARMCs to detect such damage with
confidence. On the other hand, Path 2 illustrates a situation for which a high cross-correlation
facilitates early damage detection as the ARMC shifts can be smaller to detect that type of
damage. Thus, it is shown that while using a multivariate DSF early damage detectability
does not only depend on the sensitivities of individual ARMCs to damage, but their crosscorrelation may have an influence. Hence, the selection of ARMC to form a multivariate DSF
must not be based entirely on the individual sensitivities of ARMCs but needs to take into
account the cross-correlations between the ARMCs. Similarly to the case of choosing the
optimal number of ARMCs discussed earlier, the influence of cross-correlation will depend
on the type of damage, or more precisely on how it affects the ARMCs as illustrated by Paths
1 and 2. However, the dependence of damage detectability on cross-correlation makes the
selection of ARMCs for a DSF more challenging as their individual contributions to a
stochastic distance such as T 2 (m) (Eq. (9)) cannot be easily untangled. While theoretically all
18
possible subsets of ARMCs should be tried for an absolute certainty of finding the optimal
one, there are a total of 2p+q-1 of them and such an all-exhaustive search would be
computationally prohibitive in many cases. Therefore, the following subsections introduce
two iterative methods and one global optimization method for the selection of ARMCs in
order to improve the detectability of early damage.
3.2. Iterative ARMC selection methods: Next-Best-In and Next-Worst-Out
In the presence of cross-correlations between variables, the contributions of individual
ARMCs to the distance between healthy and damage state DSFs are not immediately
accessible. However, by systematically adding or deleting all possible individual ARMCs and
monitoring how the stochastic distance changes with respect to the detection threshold one
can hope to reach the optimal ARMC selection. The NBI procedure starts with trying all
individual ARMCs and retaining the one that gives the largest univariate distance. Then all
the remaining coefficients are added in turn to the DSF and the one that results in the largest
bivariate distance is again retained. The procedure continues until all the ARMCs have been
included in the DSF.
The NWO procedure starts with a DSF composed of all the available p+q ARMCs. It
then eliminates the ARMCs one by one and calculates the distance for the reduced (p+q-1)dimensional DSF. The ARMC that contributes the least to the distance is discarded and the
procedure is repeated until only one ARMC survives.
2
For both procedures, the relative standardized distances, 𝑇𝑟𝑒𝑙
(m), defined as
Tre2l  m   T 2  m FT 2
m ,nc nh 2
1   
(19)
can be plotted as functions of the DSF dimensionality m and the maximum found. This
maximum indicates the optimal number of ARMCs for damage detectability. Furthermore, by
adding or eliminating ARMCs one by one, the two procedures rank them for their usefulness
for damage detection. However, even though these two iterative approaches take the
multivariate statistic into account, they have a fixed starting point and the results are not
guaranteed to be the true optimum in every case. A binary GA, which performs a wider
search, is therefore introduced as an alternative in the next subsection.
3.3. Genetic algorithm-based ARMC selection method
19
GA is a stochastic derivative-free optimization method that is based on evolutionary
strategies as found in the biological principle of evolution [81]. Initially, the parental
population  of  individuals, sj (j=1, … ), is randomly created. The individuals, sj, are
binary selection vectors of dimension p+q, where the i-th ARMC is selected for inclusion in
the DSF vector if sj,i=1, or unselected if sj,i=0. The random initialization is done by setting
each entry of the selection vectors to zero or one using a pseudo random number generator.
Then, the search for the global optimum is performed iteratively until the preset maximum
number of generations is reached. Convergence criteria for terminating the search are not
used, because the binary optimization operates in a discrete solution space defined by the
2p+q-1 possible selection vectors. In each generation, the offspring population offsp of λ
individuals is created. This is done by the dominant recombination of two randomly selected
parents. This means that two binary selection vectors from the parental population resulting
from the previous generation are taken and their entries are randomly combined for creating
the new offspring. Then, these offspring individuals are mutated by means of a flip bit
operation, where  randomly selected vector entries are flipped from zero to one or vice
versa. The number of entries to be flipped is the mutation rate. Next, the fitness function is
evaluated for all the individuals in the union of both populations   offsp. The fitness
2
function is defined using the relative standardized distance, 𝑇𝑟𝑒𝑙
(𝑚) (Eq. (19)), as:
f (s j )  Trel2 (m, s j )  min 1  S (si , s j ) 
(20)
si Ptemp
where temp is the temporary set of already selected individuals. The vector cosine distance
[82], 1  S (si , s j ) , where
S (si , s j )  sTi s j

sTi si sTj s j

(21)
is utilized in Eq. (20) to deter from selecting similar individuals and keep the search space
broad in order to avoid being trapped in local optima. The result is an iterative fitness
2
evaluation. In the first place, only relative standardized distances, 𝑇𝑟𝑒𝑙
(𝑚), are calculated for
each selection vector in the population   offsp. The best individual is added to the
temporary set temp and removed from the population   offsp. Next, the remaining
individuals are again evaluated for their fitness but now considering also the similarity to the
already selected vector in temp using the vector cosine distance. The best individual is finally
selected, added to temp and removed from   offsp. Having two already selected
individuals in temp requires then to evaluate the fitness of the yet unselected vectors with
20
respect to both, which is done separately by taking the minimum cosine distance for each
individual to be assessed (Eq. (20)). The individual with the best fitness is again added to
temp and removed from   offsp. This process continues until the number of individuals in
temp reaches the designated number of individuals in the next generation, κ.
4. SIMULATION OF WIND TURBINE BLADE AERODYNAMIC RESPONSE IN
HEALTHY AND DAMAGE STATES
The structure under study is a large WTB representing the current state-of-the-art. The Sandia
National Laboratory (SNL) defined a reference model of a 61.5 m long WTB [83]. It was
created according to specifications of the National Renewable Energy Laboratory (NREL)
offshore 5-MW baseline wind turbine [84], which is a conventional three-bladed upwind WT
with 90 m hub height and 126 m rotor diameter. The proposed SNL design was made to meet
the basic design criteria as specified in the international standard IEC 61400-1 3rd Ed. [85]
and the NREL specifications. The blade is shown together with its overall dimensions in
Figure 4, where the colors indicate the many different composite layups used. Furthermore, it
can be seen from the cross-section that the high and low pressure caps are supported by two
shear webs. They are made of foam and double bias fiberglass materials.
4.1. Finite element model of wind turbine blade
A finite element (FE) model of the WTB is created for the ANSYS® Mechanical APDL [86]
solver according to the SNL blade specifications and with the help of the SNL software
package NuMAD [87]. According to these specifications, the WTB has a total mass of
approximately 17,700 kg and the sectional flap-wise and edge-wise bending stiffness at the
WTB’s root are 23,380 MNm2 and 23,230 MNm2, respectively. For the analysis in the
present paper, a parked WT case is assumed, which represents a typical condition for
inspections. The cantilevered boundary conditions at the WTB root are introduced. This
assumption ignores the hub and tower flexibility. In order to avoid excessive computational
effort, while assuring the accuracy of the FE model, a FE type and mesh size study was
performed. Assuring a high model quality with respect to the dynamic characteristics is
important because the complex WTB geometry consists of mainly flat thin faces, while the
leading-edge (LE) and TE are strongly curvilinear or even non-smooth. The model accuracy
was assessed by a convergence study for the first ten eigenmodes, where the differences in
natural frequencies and mode shapes were evaluated for FE models of different complexity.
21
The selected FE model has approximately 27,700 kinematic DOFs and uses 1,650 ANSYS®
shell elements of the serendipity type with eight nodes and six DOFs at each node
(SHELL281).
Additionally to the baseline FE model describing the healthy state, numerical models of
the damaged WTB are created. Different damage types were observed in the past by visual
inspections and damage studies [88, 89], such as disbonding damages and surface cracks.
Due to the production process of WTBs, during which the upper and lower shells are bonded
together, bondlines are critical locations for damage initiation and propagation. Especially,
the TE is prone to damage in large WTBs [90] because of the higher risk of TE buckling.
This was also observed by inspections of 99 smaller WTs with rated power between 100 kW
and 300 kW [91]. Therefore, a TE disbonding is selected as the representative damage
scenario.
Sharp geometrical changes are present at the maximum chord location, which lead to
peaks in the stress distribution. Thus, this location is chosen as damage initiation point and it
is assumed that damage grows towards the WTB tip, as shown in Figure 5. The maximum
damage extent is selected to be 7.4% of the blade length or 4.55 m. Separation of nodes is
utilized to introduce the disbonding damage into the numerical model to simulate the loss of
connection between the upper and the lower shell. Different damage extents (disbonding
lengths) are realized by varying the number of separated nodes.
4.2. Aerodynamic loading model
The proposed SDD method utilizes the analysis of acceleration responses. However, for
realistic assessment of vibration-based SDD techniques, the use of excitations and responses
resembling as much as possible these encountered in the real world is paramount, especially
when numerical results are utilized. In the present case, the excitation was assumed to come
from turbulent wind flow. An aerodynamic loading approach is developed to replicate the
important excitation characteristics while balancing computational efforts. The approach
consists of three steps and involves the use of several simulation tools, as presented in Figure
6.
First, the wind field characteristics are defined after the international standard IEC
61400-1 3rd Ed. [85]. The mean inflow wind speed at the hub height is selected to be the
average wind speed for an IEC type I WT as 10 m/s. For the wind category B and normal
turbulence model, the resulting turbulence intensity of the inflow wind velocity component is
18.34%. Additionally, the Kaimal spectrum is selected as the wind power spectrum, and the
22
spatial coherence of the inflow wind velocity component is described by an exponential
coherence model [92]. Then the NREL software TurbSim [93] is used to generate full-field
wind data.
In the second step, using the NREL 5-MW reference WT model aerodynamic loads are
calculated with the help of the NREL software packages FAST [94] and AeroDyn [95]. For
these calculations, the WTB is divided into 17 strip elements, each of average constant
aerodynamic and structural properties defined by the NREL 5-MW reference WT model. The
blade element momentum theory is chosen to model the wake effect. The results are time
series of lift and drag forces, FN and FT, respectively, and pitching moments, MP, at the
element centers (xr, yr, zr). (Note the coordinate system adopted is such that axes x and y run
across the blade width and thickness, respectively, and axis z runs along the blade length.)
The mapping of these loads to nodal forces, fx,i and fy,i, of the N WTB FE model surface
nodes at positions (xi, yi, zi) is the third step. This procedure is based on Berg et al. [96] and is
illustrated in Figure 7. Equations for equivalent forces and moments acting on a single WTB
element and at the related surface nodes can be given as
N
N
FN   f y ,i
FT   f x ,i
i 1
N
M P   ( xi  xr ) f y ,i
i 1
N
zr FN   zi f y ,i
i 1
i 1
N
0   ( yi  yr ) f x ,i
(22)
i 1
N
zr FT   zi f x ,i
i 1
Such a mapping would generally be non-unique, thus it is assumed that the nodal forces in
the x-direction produce zero pitching moments. Additionally, linear spatial distributions of
the nodal forces are imposed:
f x,i  ax ( yi  yr )  bx  cx zi
(23)
f y ,i  a y ( xi  xr )  by  c y zi
where a and b are the unknown coefficients describing the linear distribution in the blade
cross section, and c describes the linear distribution along the blade. Substituting Eq. (23)
into Eq. (22) and solving the resulting system of linear equations enables to calculate these
coefficients in advance for each surface node, which reduces the required data storage and
analysis time.
5. STRUCTURAL DAMAGE DETECTION
23
In the present study, vibration-based SDD is performed by means of ARMCs and statistical
hypothesis testing. Firstly, AR model estimation and validation for acceleration time series is
presented. Then, the methods proposed for the DSF selection for enhanced damage
detectability are applied to the identified ARMCs. Finally, the SDD results are presented for
the optimal DSF considering also the effect of measurement noise.
5.1. Autoregressive time series modelling
For the discussion of AR time series modelling, transient dynamic simulations are performed
for the healthy and the damaged WTB FE models. Flap-wise and edge-wise accelerations at
selected nodes are obtained for a 630 s long, steady state time history of wind excitation and
are sampled at 200 Hz. The maximum duration of these time histories had to be limited due
the high computational demands, where one such simulation corresponding to a single
damage state took approximately 24 hours on a 64-bit desktop PC with the Intel® Core™ i53470 processor and 8 GB read-access memory. However, only flap-wise signals for the node
indicated as “Sensor” in Figure 5 are used in the subsequent analyses. Each time series for the
healthy and different damaged models is divided into 100 overlapping segments of 6,000
samples with a shift of 1,200 samples. At least 6,000 samples per segment were required for
stable estimation of ACFs, and 100 segments for confident estimation of the variancecovariance matrix of AR coefficients. The shift of 1,200 samples was then the result of a
trade-off between the long FE computational time to acquire acceleration time series and the
need to use adequate numbers of samples in AR model identification. The time series
segments are initially low-pass filtered with a Chebyshev Type I filter of order eight and a
cutoff frequency of 20 Hz. Then, the segments are resampled at 25 Hz. To account for
variations of the aerodynamic excitation, each pre-processed segment is normalized by its
estimated mean and standard deviation.
The selection of appropriate model orders is the first step of AR time series modelling.
For illustrative purposes, model order identification and validation is in the following shown
in detail for the pure AR model case, and only abbreviated ARMA time series modelling
results are presented at the end of the section.
Given in Eq. (3), the AIC is widely used for model order selection. Therefore, the AIC
is calculated for all orders from one to 50 for pure AR models and shown in Figure 8. The
AIC’s mean and standard deviation are estimated from the full set of time series segments
generated by simulations for the healthy WTB. Although no clear minimum can be seen, an
AR order of 25 is selected because higher orders do not significantly improve the results.
24
The complementary task to model selection and estimation is its validation, which is
presented in Figure 9 for the residuals of a single time series segment. The residuals should
have the properties of a zero-mean Gaussian white noise process. The normal probability plot
of residuals is constructed in Figure 9a together with 95% confidence bounds defined by the
Kolmogorov-Smirnov test [97]. While the values stray slightly from the reference straight
line at the tails, they nevertheless all firmly remain within the 95% confidence bounds.
Furthermore, the residual ACF is shown in Figure 9b, where only six values, or 2.4%, are out
of the 95% bounds for a Gaussian white noise process. Finally, the modified Ljung-BoxPierce statistics were calculated for selected numbers of ACF coefficients of up to 225 and
the results were all below the corresponding statistical thresholds for 95% confidence levels.
From these tests and the investigations, it is concluded that the selected AR(25) model is
valid.
The selection and validation of the ARMA model were conducted in an analogous way.
An ARMA(12,9) model was selected based on the minimum mean AIC calculated for all
time segments, where candidate models were created for all 2,500 combinations of AR and
MA orders from one to 50. The selected model was validated with the help of the model
residuals of a single time series segment. The residuals showed properties of a zero-mean
Gaussian white noise process when tested using the Kolmogorov-Smirnov test and the
modified Ljung-Box-Pierce statistic concurred. Therefore, the validity of the selected
ARMA(12,9) model is evidenced. It is worth noting that the total ARMA model order of 21
is only slightly smaller than the AR model of 25.
5.2. Autoregressive model coefficient selection for optimal early damage detectability
The proposed ARMC selection methods use data from the aforementioned damage states and
the sensor indicated in Figure 5 in order to build a multivariate DSF with optimal damage
detection power. While it was demonstrated earlier that cross-correlations between ARMCs
may have an effect, useful consideration in the selection process remains the examination of
how the individual ARMCs are affected by the TE disbonding damage (Figure 10).
Parameters of the AR(25) model are estimated from the time series segments generated by
simulations of each damage extent. To account for the effects of ARMC mean values, , and
standard deviations, , Fisher’s criterion, FC, is proposed as a measure of the ARMC
sensitivities to damage:
FC (i)   ˆ d ,i  ˆ h,i  ˆ pl2 ,i
2
(24)
25
where i corresponds to a given ARMC, subscripts d and h to the damaged and healthy states,
respectively, and the pooled standard deviation ˆ pl is calculated using the univariate version
of Eq. (10). A simplified dimensionless measure of the separability between structural states
is the result, however, the effects of cross-correlations are lost. Note, the values presented in
Figure 10 were normalized with respect to the maximum overall FC value (obtained for ARC
a1) to facilitate comparison. Furthermore, coefficients 16 to 25 are omitted in order to save
space and due to their small changes. However, it can be seen in Figure 10 that the
sensitivities do not follow a clear pattern and vary from one ARC to another. For example,
coefficients one, three and five show the most significant changes, while the others are less
affected. Interestingly enough, the FC for larger extents of damage is actually decreasing for
many ARCs. Although a corresponding figure for ARMCs of the ARMA(12,9) model is
omitted, it was observed that the MACs five and seven and ARC one are most strongly
affected by damage.
The ARMC selection procedures, as introduced in Section 3, are first applied to the
ARCs of the pure AR(25) model estimated from the healthy state and systems with four
selected disbond damage extents, namely 2.9%, 3.6%, 5.0% and 7.4%. Figure 11a shows the
identification of the optimum number of ranked ARCs obtained by the NBI procedure. The
2
relative standardized distances 𝑇𝑟𝑒𝑙
(m) for rankings corresponding to the different damage
extents show in all cases peaks for two ARCs as the global optimum. The results of the ARC
selection with the help of the NWO procedure are given in Figure 11b. Similarly to the NBI
results, peaks for two coefficients are present for damage extents 2.9%, 3.6% and 5.0% of the
blade length. Furthermore, the shapes of the plots are comparable to those for NBI. However,
the use of 10 ARCs is indicated as the global optimum for the 7.4% disbond length, and the
corresponding plot is significantly different compared to the previous ones.
For the GA selection process, the number of parental and offspring selection vectors
was set to 10 and 15, respectively. Further, only one entry was allowed to be flipped and the
mutation rate was kept constant throughout the entire search process. The results of the GAbased selection for the four selected damage cases are also given in Figure 11a and b, where
it can be seen that the GA leads in all the cases to the maximum relative distance (which is in
some cases the same as that of either NBI or NWO).
The same selection procedures were also applied to all 21 ARMCs of the ARMA(12,9)
model and the results are shown in Figure 11c and d. Neither the NBI not the NWO
selections indicate one number of ranked ARMCs as optimal across the range of damage
26
extents considered. For example, for NBI it varies between seven ranked ARMCs for 3.6%
disbond and 15 for the disbond of 2.9%, respectively. The peaks in the plots are also much
less pronounced and the relative distances are noticeably smaller compared to the AR(25)
model. Furthermore, generally larger sets of ARMCs are indicated as optimal than for the
AR(25) model.
The GA-based selections are also given. They were obtained with the same parameters
settings for the GA as for the AR(25) model. It can be seen that in all cases the maximum
relative distance (compared to NBI and NWO) is obtained by the GA and only five or six
ARMCs need to be selected. These subsets are significantly smaller than the ones indicated
the NBI and NWO procedures, however, they are still three times larger than the optimal sets
for the AR(25) model for the three smaller damage extents. Furthermore, the values of the
2
𝑇𝑟𝑒𝑙
(𝑚) distances are still substantially smaller. This indicates a lower damage detection
power of the ARMA model compared to the AR model.
Additionally, Table 1 gives more detailed information about the optimization results
2
including the actual selections of the ARMCs and the corresponding 𝑇𝑟𝑒𝑙
(𝑚) distances for
both models. For the AR(25) model, comparing the selection results shows that there is total
lack of agreement between the NBI and NWO method. For damage extents between 2.9%
and 5.0%, NBI consistently picks up ARCs one and two, whereas NWO selects ARCs five
2
and six. However, the resulting best 𝑇𝑟𝑒𝑙
(𝑚) distances are in all cases close, while being
2
obtained with these different selections of ARCs. Based on the maximum distances 𝑇𝑟𝑒𝑙
(𝑚),
one can see that the NWO procedure outperforms the NBI selection for the 2.9%, 3.6% and
7.4% damage extents, while the NBI only leads to a better selection for the 5.0% disbonding.
2
However, the overall winner is the GA, which leads in all the cases to the maximum 𝑇𝑟𝑒𝑙
(𝑚)
distance. For all the damage extents considered, the GA ARC selections are the same as
either those of the NBI procedure or those of the NWO procedure. The 7.4% disbonding case
is different in that the selected coefficients no longer show consistency with the smaller
damage extent cases. As many as 10 the same ARCs are chosen by both NWO and GA for
2
the maximum 𝑇𝑟𝑒𝑙
(𝑚) distance, whereas NBI selects two ARCs, but different than for the
2
smaller damage extents, and gives a slightly smaller 𝑇𝑟𝑒𝑙
(𝑚). This explosion of the number
of ARCs guaranteeing the optimal damage detectability can be explained by looking at
Figure 10: the FCs for many ARCs attain high values for this damage extent and so it is
desirable to include them in the DSF vector for enhanced separability with respect to the
healthy state. Another clear observation is that all the optima in Table 1 are significantly
27
higher than the solutions for the full set of 25 coefficients, notably for damage extents
between 2.9% and 5.0% where they are larger by a factor of approximately 2.5.
The NBI and NWO selections for the ARMA(12,9) model show some overlap for the
different damage extents. Here, the ARCs selected by both procedures are similar, while the
MAC selections are substantially different. For disbonds of 3.6% and 5.0%, the NWO
selections are identical with eight selected ARCs and three selected MACs. The NBI
procedure selects ARCs one, two and five for 3.6% disbond, which are a subset of the NWO
2
selection, but the selected MACs agree only in MAC five. Nevertheless, the 𝑇𝑟𝑒𝑙
(𝑚) is
almost identical with 26 and 25 for the NWO and NBI, respectively. For the 5.0% disbond,
the selected ARMCs of the NWO procedure are mainly a subset of the NBI results, where
only ARC eight is additionally selected and all others appear in the NBI ARMC set. The
2 (𝑚)
𝑇𝑟𝑒𝑙
is approximately 12% higher for the NBI than for the NWO selection. The NBI and
2
NWO procedures for the smallest damage of 2.9% disbond lead to the same 𝑇𝑟𝑒𝑙
(𝑚) and
number of 15 selected ARMCs, but differ in two ARCs and five MACs. The largest damage
extent of 7.4% leads to the selection of most of the ARMCs available, 17, for the NWO
procedure. The NBI algorithm selects nine ARMCs as a subset of the NWO selection, and
2 (𝑚)
only ARC eleven is added. The 𝑇𝑟𝑒𝑙
is significantly higher for the NBI set than for the
NWO. The ARMCs selected by the GA outperform both the NBI and NWO selections for all
the reference damage extents. The GA-based selections for 2.9% and 3.6% disbond are the
same and include ARCs one, six and ten and MACs one to three. Only five ARMCs are
selected by the GA for the 5.0% and 7.4% disbond reference damage. Nonetheless, the
obtained selections agree only in MAC one. The remaining four ARMCs are different.
2 (𝑚)
Finally, comparing 𝑇𝑟𝑒𝑙
for all the selected subsets of ARMA coefficients with the
performance achieved for the full set of available ARMCs highlights that the use of the
selected subsets is advantageous. This confirms the benefits of optimal selection of DSFs
especially for early damage detection. Furthermore, in all cases the GA allows selecting
2 (𝑚)
ARMCs giving the optimal 𝑇𝑟𝑒𝑙
statistic.
5.3. Structural damage detection results
2
The 𝑇𝑚,𝑛
statistic enables to use different numbers of samples to estimate the means
𝑐 +𝑛ℎ −2
and variance-covariance matrices of the current and healthy state (Eq. (9)). Using smaller
numbers of samples may be necessary in the detection phase because this allows making
decisions earlier. However, there is an inevitable trade-off between the number of samples
28
and the SDD performance with respect to the damage detectability. Therefore, the following
analysis is carried out for three different current state sample numbers, namely nc = 1, 5 and
25, randomly selected from all the available 100 acceleration data samples. The healthy state
̂h , are estimated from all the available samples, i.e. nh=100. The effect of
statistics, μ̂ h and 𝚺
measurement noise is also considered.
The SDD results of the pure AR model are shown in Figure 12 as average relative
damage detection rates obtained by performing statistical hypothesis testing on the randomly
generated DSF vector estimates. This is done for the ARC pairs [a1 a2] and [a5 a6],
respectively, as these ARCs were indicated as the optimal selections for disbonding between
2.9% and 5.0%. The full set of 25 coefficients is additionally analyzed.
First, comparing the results for the different selections without noise (Figure 12a, b and
c) demonstrates that the selected subsets of ARCs lead in all the cases to higher detection
rates for small damages than using the full set of available coefficients. However, there is
only slight difference between the performances of the two considered ARC pairs, although
coefficients five and six lead to slightly higher detection rates. Interestingly, the false alarm
rates in the undamaged state are higher for the ARC subsets than for the full coefficient set;
they are, however, always within the tolerable region for the selected level of significance of
5%.
Second, it can be seen that the use of larger numbers of samples in the detection phase
allows detecting smaller damage with higher confidence. For example, using a single DSF
sample (nc=1) enables to detect with at last a 90% success rate a disbond of 3.6% of the blade
length using the two ARC subsets, and only a disbond of 4.3% for the full set of ARCs for
the same success rate. Increasing the sample size to five and 25 allows detection of damages
with extents of 2.9% and 2.2% of blade length, respectively, for all the ARC selections. It can
be furthermore observed that larger numbers of samples in the detection phase lead to smaller
false alarm rates in the healthy state.
Third, the effect of noise is considered by introducing Gaussian random sequences with
a noise-to-signal ratio of 5% into the simulated time series of accelerations. The detection
results for the three different ARC selections are almost unaffected by noise as can be seen by
comparing Figure 12d, e and f to Figure 12a, b and c, respectively. The most significant, but
still small, effects can be observed in the case when a single DSF sample is used. Here, the
detection results using ARCs one and two are higher than with ARCs five and six.
29
Figure 13 shows a similar analysis for the ARMA(12,9) model, where ARMC subsets
from a reference damage of 3.6% disbond are used. Only one case of damage severity was
considered because, as explained earlier, there was no consistency in optimal coefficients
across different damage extents. One selection is obtained by the GA leading to a set of
ARCs one, six and ten and MACs one to three; the NBI set of ARMCs is the second selection
with ARCs one, two and five and MACs one, two, four and five (see Table 1). (The NWO
2 (𝑚).)
selection was not studied as it had a smaller 𝑇𝑟𝑒𝑙
Additionally, the full set of 12 ARCs
and nine MACs is used as reference. Comparing Figures 13a, b and c for SDD without noise
effects shows that similar to the results of the pure AR model increasing the sample size
generally improves the detectability of early damage. Using only single sample estimates for
SDD enables to detect a disbond of 4.3% length, while five and 25 samples allow detecting
3.6% disbond with high confidence of more than 0.9. Furthermore, the false alarm rates of
the selections decrease with increasing the number of samples, but it should be noted that
using single samples leads to exceeding the desired false alarm rate given by the selected
level of significance of 5% by approximately a factor two. The different DSF selections,
however, do not lead to significant different detection results. Another observation is that the
curves in some case are, unlike for the pure AR model, non-monotonic, indicating that the
algorithm is more often confused in the small damage range.
For time series with added artificial noise (noise-to-signal ratio of 5%), shown in
Figures 13d, e and f, the detectability of small damages does improve with increasing sample
sizes as before, but the damages detectable with more than 0.9 relative detection rates
remains unchanged. Nonetheless, differences between the ARMC selections occur, where for
single samples the full set of ARMCs leads to slightly better results for damage of 1.4% and
2.2% disbond. However, for sample sizes of five and 25 samples, ARMC subsets selected by
the GA and the NBI procedure improve the detectability of early damages. These
improvements are similar for both subsets. Noise appears to make the non-monotonic
character of the curves more noticeable.
Finally, comparing the overall SDD performance of the pure AR and the ARMA model
for the different ARMC selections highlights the advantages of the pure AR model. The best
performance is achieved with the help of the ARC subsets of the AR(25), which enable to
detect confidently damage as small as 2.2% disbond using 25 samples from the damage state.
On the other hand, the coefficients of the ARMA(12,9) model only allow detecting
confidently the 3.6% disbond as the smallest damage. However, for both model types damage
detectability is almost unaffected by noise.
30
6. CONCLUSIONS
The problem of optimal DSF selection using ARMCs for multivariate statistical tests for
SDD was discussed and applied to WTB TE disbond. The optimality problem was formulated
as finding such a selection of ARMCs that provides the maximum separability of the damage
and healthy states in terms of a multivariate DSF distance compared to a statistical
classification threshold. ARMCs were estimated using a pure AR and an ARMA model for
comparison. First, analytical considerations were conducted to demonstrate the potential
benefits of such optimal selection, and drawbacks of suboptimal ones, in a bivariate case.
Then, two different approaches to ARMC selection were presented. The first one utilizes
procedures that one by one add ARMCs to, or eliminate them from, the DSF vector. A binary
GA was studied as the second approach. These techniques were applied to data from
advanced, realistic numerical simulations of a large WTB with TE disbonding damage
scenarios of varying extent under aerodynamic excitation. It was shown that the GA
consistently outperformed the alternative DSF selection approaches. The ARC selections of
the pure AR model clearly performed better for SDD than the full set of the available ARCs.
The ARMCs selected from the ARMA model only slightly improved the damage
detectability compared to the full set of ARMA coefficients. The selected ARMCs of the pure
AR model led clearly to the overall best detection results when compared to the ARMA
model.
The influence of the amount of available DSF samples on damage detectability was
explored and trade-offs that exists between the ability to observe damage early and level of
confidence in making correct SDD inferences demonstrated. It was also shown that an
appropriate selection of ARMCs improves the early detectability of damage even in presence
of measurement noise, which had only a limited detrimental effect. Future studies are
underway to explore the optimal selection of various DSFs derived from time series
representations of vibration responses for varying damage locations and types. These will
also be underpinned by experimental data from a small WTB tested in laboratory.
ACKNOWLEDGMENTS
Piotr Omenzetter and Simon Hoell’s work within the Lloyd’s Register Foundation Centre for
Safety and Reliability Engineering at the University of Aberdeen is supported by Lloyd’s
Register Foundation. The Foundation helps to protect life and property by supporting
engineering-related education, public engagement and the application of research.
31
REFERENCES
[1] M.I. Blanco, The economics of wind energy, Renew. Sust. Energ. Rev. 13 (2009) 13721382.
[2] C.R. Farrar, K. Worden, An introduction to structural health monitoring, Philos. Trans. R.
Soc. A Math. Phys. Eng. Sci. 365 (2007) 303-315.
[3] K. Worden, C.R. Farrar, G. Manson, G. Park, The fundamental axioms of structural
health monitoring, Proc. R. Soc. A Math. Phys. Eng. Sci. 463 (2007) 1639-1664.
[4] P.M. Pawar, R. Ganguli, Helicopter rotor health monitoring - A review, Proc. Inst. Mech.
Eng. Part G J. Aerosp. Eng. 221 (2007) 631-647.
[5] Y.Y. Li, Hypersensitivity of strain-based indicators for structural damage identification: A
review, Mech. Syst. Signal Process. 24 (2010) 653-664.
[6] K. Diamanti, C. Soutis, Structural health monitoring techniques for aircraft composite
structures, Prog. Aerospace Sci. 46 (2010) 342-352.
[7] H. Guo, G. Xiao, N. Mrad, J. Yao, Fiber optic sensors for structural health monitoring of
air platforms, Sensors. 11 (2011) 3687-3705.
[8] X. Hu, B. Wang, H. Ji, A wireless sensor network-based structural health monitoring
system for highway bridges, Comput.-Aided Civ. Infrastruct. Eng. 28 (2013) 193-209.
[9] B. Glisic, M.T. Yarnold, F.L. Moon, A.E. Aktan, Advanced visualization and
accessibility to heterogeneous monitoring data, Comput.-Aided Civ. Infrastruct. Eng.
29 (2014) 382-398.
[10] F.P. García Márquez, A.M. Tobias, J.M. Pinar Pérez, M. Papaelias, Condition
monitoring of wind turbines: Techniques and methods, Renew. Energy. 46 (2012) 169178.
[11] J.F. Skov, M.D. Ulriksen, K.A. Dickow, P.H. Kirkegaard, L. Damkilde, On structural
health monitoring of wind turbine blades, Key Eng. Mat. 569-570 (2013) 628-635.
[12] C.C. Ciang, J.R. Lee, H.J. Bang, Structural health monitoring for a wind turbine system:
A review of damage detection methods, Meas. Sci. Technol. 19 (2008) 1-20.
[13] S.W. Doebling, C. R. Farrar, M. B. Prime and D. Shevitz, Damage Identification and
Health Monitoring of Structural and Mechanical Systems from Changes in Their
Vibration Characteristics: A Literature Review, Los Alamos National Laboratory,
Technical Report (LA-13070-MS), 1996.
[14] S.W. Doebling, C.R. Farrar, M.B. Prime, A summary review of vibration-based damage
identification methods, Shock Vib. Dig. 30 (1998) 91-105.
32
[15] H. Sohn, C. R. Farrar, F. M. Hemez, D.D. Shunk, D.W. Stinemates, B.R. Nadler, J.,
Czarnecki, A Review of Structural Health Monitoring Literature: 1996-2001, Los
Alamos National Laboratory, Technical Report (LA-13976-MS), 2004.
[16] J.M.W. Brownjohn, A. de Stefano, Y.L. Xu, H. Wenzel, A.E. Aktan, Vibration-based
monitoring of civil infrastructure: Challenges and successes, J. Civ. Struct. Health
Monit. 1 (2011) 79-95.
[17] E.P. Carden, P. Fanning, Vibration based condition monitoring: A review, Struct. Health
Monit. 3 (2004) 355-377.
[18] Y.Q. Ni, H.F. Zhou, K.C. Chan, J.M. Ko, Modal flexibility analysis of cable-stayed Ting
Kau bridge for damage identification, Comput.-Aided Civ. Infrastruct. Eng. 23 (2008)
223-236.
[19] W. Fan, P. Qiao, Vibration-based damage identification methods: A review and
comparative study, Struct. Health Monit. 10 (2011) 83-111.
[20] M. Bocca, L.M. Eriksson, A. Mahmood, R. Jäntti, J. Kullaa, A synchronized wireless
sensor network for experimental modal analysis in structural health monitoring,
Comput.-Aided Civ. Infrastruct. Eng. 26 (2011) 483-499.
[21] E. Reynders, System identification methods for (operational) modal analysis: Review
and comparison, Arch. Comput. Method. E. 19 (2012) 51-124.
[22] E.P. Carden, J.M.W. Brownjohn, Fuzzy clustering of stability diagrams for vibrationbased structural health monitoring, Comput.-Aided Civ. Infrastruct. Eng. 23 (2008)
360-372.
[23] C. Rainieri, G. Fabbrocino, Automated output-only dynamic identification of civil
engineering structures, Mech. Syst. Signal Process. 24 (2010) 678-695.
[24] E. Reynders, J. Houbrechts, G. de Roeck, Fully automated (operational) modal analysis,
Mech Syst Signal Process. 29 (2012) 228-250.
[25] C. Devriendt, F. Magalhães, W. Weijtjens, G. de Sitter, Á. Cunha, P. Guillaume,
Automatic identification of the modal parameters of an offshore wind turbine using
state-of-the-art operational modal analysis techniques, in: 5th International Operational
Modal Analysis Conference (IOMAC), Guimaraes, Portugal, 2013, pp. 1-12.
[26] S.D. Fassois, J.S. Sakellariou, Time-series methods for fault detection and identification
in vibrating structures, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 365 (2007) 411448.
33
[27] R.P. Kumar, T. Oshima, S. Mikami, Y. Miyamori, T. Yamazaki, Damage identification
in a lightly reinforced concrete beam based on changes in the power spectral density,
Struct. Infrastruct. E. 8 (2012) 715-727.
[28] P. Moreno-García, E. Castro, L. Romo-Melo, A. Gallego, A. Roldán, Vibration tests in
CFRP plates for damage detection via non-parametric signal analysis, Shock Vib. 19
(2012) 857-865.
[29] L. Yu, J.H. Zhu, L.L. Yu, Structural damage detection in a truss bridge model using
fuzzy clustering and measured FRF data reduced by principal component projection,
Adv. Struct. Eng. 16 (2013) 207-217.
[30] S. Beskhyroun, T. Oshima, S. Mikami, Y. Miyamori, Assessment of vibration-based
damage identification techniques using localized excitation source, J. Civ. Struct.
Health Monit. 3 (2013) 207-223.
[31] F.N. Catbas, H.B. Gokce, M. Gul, Nonparametric analysis of structural health
monitoring data for identification and localization of changes: Concept, lab, and reallife studies, Struct. Health Monit. 11 (2012) 613-626.
[32] J.D. Tippmann, F. Lanza di Scalea, Experiments on a wind turbine blade testing an
indication for damage using the causal and anti-causal Green's function reconstructed
from a diffuse field, in: SPIE 9061, Sensors and Smart Structures Technologies for
Civil, Mechanical, and Aerospace Systems 2014, San Diego, USA, 2014, pp. 90641I90641I-7.
[33] H. Chen, M. Kurt, Y.S. Lee, D.M. McFarland, L.A. Bergman, A.F. Vakakis,
Experimental system identification of the dynamics of a vibro-impact beam with a view
towards structural health monitoring and damage detection, Mech. Syst. Signal Process.
46 (2014) 91-113.
[34] N. Roveri, A. Carcaterra, Damage detection in structures under traveling loads by
Hilbert–Huang transform, Mech. Syst. Signal Process. 28 (2012) 128-144.
[35] E. Simon Carbajo, R. Simon Carbajo, C. Mc Goldrick, B. Basu, ASDAH: An automated
structural change detection algorithm based on the Hilbert-Huang transform, Mech.
Syst. Signal Process. 47 (2014) 78-93.
[36] P.F. Pai, H. Deng, M.J. Sundaresan, Time-frequency characterization of Lamb waves for
material evaluation and damage inspection of plates, Mech. Syst. Signal Process. 62
(2015) 183-206.
34
[37] M.M.R. Taha, A. Noureldin, J.L. Lucero, T.J. Baca, Wavelet transform for structural
health monitoring: A compendium of uses and features, Struct. Health Monit. 5 (2006)
267-295.
[38] X. Jiang, H. Adeli, Dynamic wavelet neural network for nonlinear identification of
highrise buildings, Comput.-Aided Civ. Infrastruct. Eng. 20 (2005) 316-330.
[39] C. Bao, H. Hao, Z.X. Li, Integrated ARMA model method for damage detection of
subsea pipeline system, Eng. Struct. 48 (2013) 176-192.
[40] H.Y. Noh, K.K. Nair, A.S. Kiremidjian, C.H. Loh, Application of time series based
damage detection algorithms to the benchmark experiment at the National Center for
Research on Earthquake Engineering (NCREE) in Taipei, Taiwan, Smart Struct. Sys. 5
(2009) 95-117.
[41] M.D. Spiridonakos, S.D. Fassois, An FS-TAR based method for vibration-responsebased fault diagnosis in stochastic time-varying structures: Experimental application to
a pick-and-place mechanism, Mech. Syst. Signal Process. 38 (2013) 206-222.
[42] H. Sohn, J.A. Czarnecki, C.R. Farrar, Structural health monitoring using statistical
process control, J. Struct. Eng. 126 (2000) 1356-1363.
[43] R. Yao, S.N. Pakzad, Autoregressive statistical pattern recognition algorithms for
damage detection in civil structures, Mech. Syst. Signal Process. 31 (2012) 355-368.
[44] A.S. Kiremidjian, G. Kiremidjian, P. Sarabandi, A wireless structural monitoring system
with embedded damage algorithms and decision support system, Struct. Infrastruct. E. 7
(2011) 881-894.
[45] L. Haitao, A. Mita, Damage indicator defined as the distance between ARMA models
for structural health monitoring, J. Struct. Control Health Monit. 15 (2008) 992-1005.
[46] P. Omenzetter, J.M.W. Brownjohn, Application of time series analysis for bridge
monitoring, Smart Mater. Struct. 15 (2006) 129-138.
[47] H. Noh, R. Rajagopal, A.S. Kiremidjian, Sequential structural damage diagnosis
algorithm using a change point detection method, J. Sound Vib. 332 (2013) 6419-6433.
[48] M. Mollineaux, K. Balafas, K. Branner, P. Nielsen, A. Tesauro, A. Kiremidjian, R.
Rajagopal, Damage detection methods on wind turbine blade testing with wired and
wireless accelerometer sensors, in: 7th European Workshop on Structural Health
Monitoring (EWSHM), Nantes, France, 2014, pp. 1863-1870.
[49] E.P. Carden, J.M.W. Brownjohn, ARMA modelled time-series classification for
structural health monitoring of civil infrastructure, Mech. Syst. Signal Process. 22
(2008) 295-314.
35
[50] G.E.P. Box, G.M. Jenkins, G.C. Reinsel, Time Series Analysis Forecasting and Control,
4th ed., John Wiley & Sons, Inc., Hoboken, USA 2008.
[51] H. Sohn, C.R. Farrar, Damage diagnosis using time series analysis of vibration signals,
Smart Mater. Struct. 10 (2001) 446-451.
[52] W. Liu, B. Chen, R.A. Swartz, Investigation of time series representations and similarity
measures for structural damage pattern recognition, The Scientific World Journal. 2013
(2013) 1-13.
[53] H. Zheng, A. Mita, Localized damage detection of structures subject to multiple ambient
excitations using two distance measures for autoregressive models, Struct. Health
Monit. 8 (2009) 207-222.
[54] L.D. Avendaño Valencia, S.D. Fassois, Stationary and non-stationary random vibration
modelling and analysis for an operating wind turbine, Mech. Syst. Signal Process. 47
(2014) 263-285.
[55] K.K. Nair, A.S. Kiremidjian, K.H. Law, Time series-based damage detection and
localization algorithm with application to the ASCE benchmark structure, J. Sound Vib.
291 (2006) 349-368.
[56] E. Figueiredo, J. Figueiras, G. Park, C.R. Farrar, K. Worden, Influence of the
autoregressive model order on damage detection, Comput.-Aided Civ. Infrastruct. Eng.
26 (2011) 225-238.
[57] C.R. Farrar, S.W. Doebling, D.A. Nix, Vibration–based structural damage identification,
Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 359 (2001) 131-149.
[58] K.K. Nair, A.S. Kiremidjian, Time series based structural damage detection algorithm
using Gaussian mixtures modeling, J Dyn. Syst. Meas. Control Trans ASME. 129
(2007) 285-293.
[59] D.A. Tibaduiza, M.A. Torres-Arredondo, L.E. Mujica, J. Rodellar, C.P. Fritzen, A study
of two unsupervised data driven statistical methodologies for detecting and classifying
damages in structural health monitoring, Mech. Syst. Signal Process. 41 (2013) 467484.
[60] E. Altunok, M.M.R. Taha, D.S. Epp, R.L. Mayes, T.J. Baca, Damage pattern recognition
for structural health monitoring using fuzzy similarity prescription, Comput.-Aided
Civ. Infrastruct. Eng. 21 (2006) 549-560.
[61] O.R. de Lautour, P. Omenzetter, Nearest neighbor and learning vector quantization
classification for damage detection using time series analysis, J. Struct. Control Health
Monit. 17 (2010) 614-631.
36
[62] Y. Kim, J.W. Chong, K.H. Chon, J. Kim, Wavelet-based AR-SVM for health monitoring
of smart structures, Smart Mater. Struct. 22 (2013) 1-12.
[63] Z. Wang, K.C.G. Ong, Autoregressive coefficients based Hotelling’s T2 control chart for
structural health monitoring, Comput. Struct. 86 (2008) 1918-1935.
[64] K.C. Lu, C.H. Loh, Y.S. Yang, J.P. Lynch, K.H. Law, Real-time structural damage
detection using wireless sensing and monitoring system, Smart Struct. Syst. 4 (2008)
759-777.
[65] O.R. de Lautour, P. Omenzetter, Damage classification and estimation in experimental
structures using time series analysis and pattern recognition, Mech. Syst. Signal
Process. 24 (2010) 1556-1569.
[66] N.L.D. Khoa, B. Zhang, Y. Wang, F. Chen, S. Mustapha, Robust dimensionality
reduction and damage detection approaches in structural health monitoring, Struct.
Health Monit. 13 (2014) 406-417.
[67] E. Figueiredo, G. Park, J. Figueiras, C. R. Farrar and K. Worden. , Structural Health
Monitoring Algorithm Comparisons Using Standard Data Sets, Los Alamos National
Laboratory, Technical Report (LA-14393), 2009.
[68] M. Žvokelj, S. Zupan, I. Prebil, Non-linear multivariate and multiscale monitoring and
signal denoising strategy using kernel principal component analysis combined with
ensemble empirical mode decomposition method, Mech. Syst. Signal Process. 25
(2011) 2631-2653.
[69] G. Georgoulas, T. Loutas, C.D. Stylios, V. Kostopoulos, Bearing fault detection based
on hybrid ensemble detector and empirical mode decomposition, Mech. Syst. Signal
Process. 41 (2013) 510-525.
[70] D.Y. Harvey, M.D. Todd, Structural health monitoring feature design by genetic
programming, Smart Mater. Struct. 23 (2014).
[71] H.G. Beyer, H.P. Schwefel, Evolution strategies - A comprehensive introduction, Nat.
Comp. 1 (2002) 3-52.
[72] W.W.S. Wei, Time Series Analysis: Univariate and Multivariate Methods, 2nd ed.,
Pearson Education, Boston, USA 2006.
[73] L. Ljung, System Identiation: Theory for the User, 2nd ed., Prentice Hall, Upper Saddle
River, USA 1999.
[74] P.J. Brockwell, R.A. Davis, Introduction to Time Series and Forecasting, SpringerVerlag Inc., New York, USA 1996.
37
[75] S.M. Kay, Modern Spectral Estimation: Theory and Application, Prentice Hall, Upper
Saddle River, USA 1988.
[76] P. Stoica, R. Moses, Spectral Analysis of Signals, Prentice Hall, Inc., Upper Saddle
River, USA 2005.
[77] J.S. Bendat, A.G. Piersol, Random Data Analysis and Measurement Procedures, 4th ed.,
John Wiley & Sons, Inc., Hoboken, USA 2010.
[78] G.M. Ljung, G.E.P. Box, On a measure of lack of fit in time series models, Biometrika.
65 (1978) 297-303.
[79] A.C. Rencher, Multivariate Statistical Inference and Applications, John Wiley & Sons,
Inc., New York, USA 1998.
[80] T. Söderström, P. Stoica, System Identification, Prentice Hall International (UK) Ltd.,
Hemel Hempstead, UK 1989.
[81] O. Kramer, D.E. Ciaurri, S. Koziel, In: Koziel S, Yang XS (Eds.), Computational
Optimization, Methods and Algorithms, Springer Verlag, Berlin, Germany, 2011, pp.
61-83.
[82] A. Singhal, Modern information retrieval: A brief overview, Bulletin of the IEEE
Computer Society Technical Committee on Data Engineering. 24 (2001) 35-42.
[83] B.R. Resor, Definition of a 5MW/61.5m Wind Turbine Blade Reference Model, Sandia
National Laboratories, Technical Report (SAND2013-2569), 2013.
[84] J.M. Jonkman, S. Butterfield, W. Musial and G. Scott, Definition of a 5-MW Reference
Wind Turbine for Offshore System Development, National Renewable Energy
Laboratory, Technical Report (NREL/TP-500-38060), 2009.
[85] IEC Technical Committee 88: Wind turbines, International standard IEC 61400-1, 3rd
ed. (2005).
[86] ANSYS, Inc., ANSYS® Mechanical APDL, Release 14.5, ANSYS, Inc., 2012.
[87] J.C. Berg and B. R. Resor, Numerical Manufacturing and Design Tool (NuMAD v2.0)
for Wind Turbine Blades: User's Guide, Sandia National Laboratories, Technical
Report (SAND2012-7028), 2012.
[88] F.M. Jensen, B.G. Falzon, J. Ankersen, H. Stang, Structural testing and numerical
simulation of a 34 m composite wind turbine blade, Compos.Struct. 76 (2006) 52-61.
[89] S. Ataya, M.M.Z. Ahmed, Forms of discontinuities in 100 kW and 300 kW wind turbine
blades, in: 10th World Wind Energy Conference & Renewable Energy Exhibition 2011,
Cairo, Egypt, 2011, pp. 1-6.
38
[90] F.M. Jensen, A. Kling, J.D. Sørensen, Scale-up of wind turbine blades - changes in
failure type, in: EWEA Annual Event 2012 (EWEA 2012), Copenhagen, Denmark,
2012, pp. 1-6.
[91] S. Ataya, M.M.Z. Ahmed, Damages of wind turbine blade trailing edge: Forms, location,
and root causes, Eng. Failure Anal. 35 (2013) 480-488.
[92] M.O.L. Hansen, Aerodynamics of Wind Turbines, 2nd ed., Earthscan, London, UK
2008.
[93] N. Kelley, B. Jonkman, TurbSim, National Wind Technology Center, v1.06.00 (Last
modified 30-May-2013; accessed 14-October-2013) 2013.
[94] J. Jonkman, FAST, National Wind Technology Center, v7.02.00d-bjj (Last modified 28October-2013; accessed 28-October-2013) 2013.
[95] D.J. Laino, AeroDyn, National Renewable Energy Laboratory, v13.00.02a-bjj (Last
modified 23-February-2013; accessed 15-October-2013.) 2013.
[96] J.C. Berg, J.A. Paquette, B.R. Resor, Mapping of 1D beam loads to the 3D wind blade
for buckling analysis, in: 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural
Dynamics and Materials Conference 2011, Denver, USA, 2011, pp. 1-8.
[97] J.R. Micheal, The stabilized probability plot, Biometrika. 70 (1983) 11-17.
39
Figure 1. Process of optimal ARMC selection and statistical damage decision making.
Hoell and Omenzetter
40
Figure 2. Regions of damage detectability for bivariate DSF.
Hoell and Omenzetter
41
Figure 3. Damage detection threshold at =5% and average contributions of statistical DOFs.
Hoell and Omenzetter
42
a)
b)
Figure 4. Wind turbine blade: a) dimensions and composite material layup, and b) crosssection at 15.17 m from root.
Hoell and Omenzetter
43
Figure 5. Location of TE-disbond with maximum extent and sensor position (width/length
proportions not to scale).
Hoell and Omenzetter
44
Figure 6. Scheme of aerodynamic loading simulation.
Hoell and Omenzetter
45
Figure 7. Aerodynamic loads.
Hoell and Omenzetter
46
Figure 8. Mean and standard deviation of AIC for pure AR models of accelerations.
Hoell and Omenzetter
47
a)
b)
Figure 9. AR(25) model validation: a) normality plot, and b) ACF of residuals.
Hoell and Omenzetter
48
Figure 10. Fisher’s criterion of ARMCs of AR(25) model with increasing damage.
Hoell and Omenzetter
49
a)
b)
c)
d)
2
Figure 11. Relative distances 𝑇𝑟𝑒𝑙
(𝑚) for increasing numbers of ranked ARMCs: a) AR(25):
NBI ranking and GA selection, b) AR(25): NWO ranking and GA selection, c) ARMA(12,9):
NBI ranking and GA selection, and d) ARMA(12,9): NWO ranking and GA selection.
Hoell and Omenzetter
50
a)
b)
c)
d)
e)
f)
Figure 12. Relative rates of structural damage detection of AR(25) model.
Hoell and Omenzetter
51
a)
b)
c)
d)
e)
f)
Figure 13. Relative rates of structural damage detection of ARMA(12,9) model with GA and
NBI selections for 3.6% disbond.
Hoell and Omenzetter
52
Table 1. ARMC selection results.
AR(25)
Damage
extent
Method
2.9%
NBI
No. of
selected
ARCs
2
NWO
3.6%
5.0%
7.4%
ARMA(12,9)
Selected
ARCs
Maximum
𝑇2𝑟𝑒𝑙 (m)
1, 2
95
No. of
selected
ARMCs
15
2
5, 6
102
GA
All
ARMCs
NBI
NWO
2
25
5, 6
1-25
2
2
GA
All
ARMCs
NBI
Selected
ARCs
Selected
MACs
Maximum
𝑇2𝑟𝑒𝑙 (m)
102
37
6
21
1, 6, 10
1-12
1-3, 5-7,
9
1, 4, 5, 79
1-3
1-9
15
15
3-7, 10,
11
2-7, 9-11
1, 2
5, 6
138
144
7
11
1, 2, 4, 5
5, 8, 9
26
25
2
25
5, 6
1-25
144
55
6
21
1, 2, 5
1-3, 5, 6,
8, 10, 11
1, 6, 10
1-12
1-3
1-9
32
20
2
1, 2
376
13
2
5, 6
333
11
1, 4, 5, 79
5, 8, 9
65
NWO
GA
All
ARMCs
NBI
NWO
2
25
1, 2
1-25
376
163
5
21
2, 3, 5, 6,
10-12
1-3, 5, 6,
8, 10, 11
1, 2
1-12
1-3
1-9
83
47
2
10
950
996
9
17
5, 11, 12
1, 4-10,
12
1-3, 6-8
1-4, 6-9
151
117
GA
10
996
5
3, 4, 6, 8
1
186
All
ARMCs
25
1, 19
2, 3, 5, 6,
8-10, 12,
13, 15
2, 3, 5, 6,
8-10, 12,
13, 15
1-25
792
21
1-12
1-9
111
15
20
12
58
Hoell and Omenzetter
53