WESWind Energy ScienceWESWind Energ. Sci.2366-7451Copernicus PublicationsGöttingen, Germany10.5194/wes-1-177-2016Periodic stability analysis of wind turbines operating in turbulent wind conditionsRivaRiccardohttps://orcid.org/0000-0002-9364-3050CacciolaStefanohttps://orcid.org/0000-0002-5370-1105BottassoCarlo Luigicarlo.bottasso@tum.dehttps://orcid.org/0000-0002-9931-4389Dipartimento di Scienze e Tecnologie
Aerospaziali, Politecnico di Milano, Milano, ItalyWind Energy Institute, Technische Universität München, Garching
bei München, GermanyCarlo Luigi Bottasso (carlo.bottasso@tum.de)20October20161217720319December201521January201628July20169September2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://wes.copernicus.org/articles/1/177/2016/wes-1-177-2016.htmlThe full text article is available as a PDF file from https://wes.copernicus.org/articles/1/177/2016/wes-1-177-2016.pdf
The formulation is model-independent, in the sense that it does not require
knowledge of the equations of motion of the periodic system being analyzed,
and it is applicable to an arbitrary number of blades and to any
configuration of the machine. In addition, as wind turbulence can be viewed
as a stochastic disturbance, the method is also applicable to real wind
turbines operating in the field.
The characteristics of the new method are verified first with a simplified
analytical model and then using a high-fidelity multi-body model of a
multi-MW wind turbine. Results are compared with those obtained by the well-known operational modal analysis approach.
Introduction
Stability analysis can help address very practical issues, such as assessing
the proximity of flutter boundaries, identifying low-damped modes,
understanding the vibratory content of a machine, evaluating the
effectiveness of control strategies for enhancing modal damping, detecting
incipient failures, and many others. For linear time-invariant (LTI) systems,
the stability analysis is a well-understood problem, and several methods are
available e.g..
However, it is unfortunately not possible to ignore the periodic nature of
wind
turbines .
In fact, blades experience different wind conditions in their travel around
the rotor disk, for example due to shears and wind misalignment, so that
the aerodynamically induced damping and stiffness vary cyclically. Furthermore,
the blade structural stiffness also varies periodically under the effects of
its own weight, while couplings among tower and blades depend on the
azimuthal position of the rotor. Additionally, the use of individual pitch
control (IPC) may introduce yet a further source of periodicity in the system
dynamics. The design of future, very large wind turbines, principally for the
exploitation of off-shore wind resources, will highlight even further the
importance of a rigorous treatment of the periodic nature of the system when
studying its stability. In fact, the system dynamics will be complicated by
the hydro-elastic characteristics of the submerged – possibly floating –
structure, including the excitation caused by periodic waves.
One popular approach to the stability analysis of rotors in general and of
wind turbines in
particular see is to use
the multi-blade coordinate (MBC) transformation
of . Given the dynamical system equations of
motion, this periodic transformation expresses the model rotating degrees of
freedom in a new set of coordinates, in this way achieving a significant
reduction, but in general not an exact cancellation, of the periodic content
of the state matrix. The remaining periodicity is typically removed by
averaging, and the resulting LTI model is finally analyzed using standard
time-invariant techniques.
In principle, there are at least three issues connected with any Coleman-based
stability analysis approach.
First, the level of approximation implied by the averaging of the remaining
periodicity is difficult to assess and quantify a priori. In fact, to the
authors' knowledge, there is no theoretical proof yet that the periodicity
that remains after the application of the Coleman transformation is small in
general nor that this approach amounts to some consistent and bounded
approximation of a rigorous Floquet analysis. Given the widespread use of the
Coleman transformation, and its generally excellent behavior, such a proof
remains a goal very much worth pursuing but, to date, unattained.
Second, the Coleman transformation unfortunately exists only for a number of
blades greater than or equal to three. Although this is the most common wind
turbine configuration nowadays, a revival of the two-bladed concept is
possible.
Third, codes implementing the Coleman transformation require access to the
linearized equations of motion of the system. As a consequence, any addition
to a simulation code has an impact on the associated stability analysis tool,
resulting in extra software maintenance work.
Other possible approaches to the stability analysis of rotors have been
formulated in the frequency domain. For example, the estimation of power
spectra along with modal frequencies and damping ratios of an operating wind
turbine has been addressed by . That
paper considered several parametric and non-parametric methods and their
application to experimental data, including the periodic autoregressive (PAR)
model. In addition, periodic autoregressive moving average (PARMA) models
have been considered by . Two
subspace algorithms for periodic systems have been presented
by and ,
one being used for numerically generated time series and the other for
experimentally measured ones.
The operational modal analysis (OMA) has been extended to the periodic case
by using the concept of the harmonic transfer function (HTF). In , the simple peak-picking
method was used for extracting relevant properties from the spectra, while
more specialized fitting algorithms were proposed
by . Subsequent applications and
developments can be found
in . Although the method is
general, the estimation of the quantities of interest for a stability
analysis from noisy spectra remains a somewhat delicate operation, as will
be shown later on in the following pages.
In the authors' opinion, there are two desirable goals in the stability
analysis of wind turbines that still need further investigation in order to
be fully attained.
First, one would like to account completely rigorously for the periodicity of
such systems, without introducing approximations of unknown effects.
Second, one would like to formulate the analysis so that it is
system-independent. System independence is here intended to mean that a method
can be applied to wind turbine models of arbitrary complexity and topology
(e.g., any number of blades and horizontal or vertical axis) and also to
real wind turbines operating in the field.
To answer these needs, proposed a periodic
stability analysis formulated in terms of input–output discrete-time
responses. Such time histories could come from “virtual” experiments
performed on a given model, from simplified ones to the more advanced
contemporary comprehensive multi-body-based aero-hydro-servo-elastic models.
Using this approach, a reduced periodic autoregressive with exogenous input
(PARX) model is first identified from a recorded response of the system and
then used for conducting a stability analysis according to Floquet theory. On
the practical side, this implies that the analysis respects the periodic
nature of the problem. Furthermore, one can easily replace the model with a
different one, without having to modify or adjust in any way the stability
analysis procedure.
Although this approach attains the two goals outlined above, one of its
limits is that it can not be used with measurements obtained on a real wind
turbine operating in the field, since the effects of wind turbulence are not
considered within the PARX model structure. To address this issue, the same
approach was extended to account for the presence of turbulence
. Using this new technique, one first
identifies a periodic autoregressive moving average with exogenous input
(PARMAX) model, whose stability is then analyzed according to Floquet.
showed only one example related to the first blade edgewise mode of a
wind turbine rotor. The goal of the present paper is to expand and formulate in
detail the PARMAX-based method originally proposed
by . A second goal of this paper is to
compare the PARMAX method with the periodic operational modal analysis
(POMA) see, which is taken here to
represent the accepted state of the art for the stability analysis of wind
turbines operating in turbulent wind conditions.
The article is organized according to the following plan. The problem of the
identification of PARMAX models is addressed in
Sect. . Here, a newly developed algorithm that has
its basis in the prediction error method (PEM) is formulated, with particular
emphasis on the guaranteed stability of the PARMAX predictor.
Section is devoted to POMA theory. After reviewing the
concept of HTFs, the treatment proceeds by discussing the method and its use
for conducting periodic stability analyses. As the authors are not aware of a
reference collecting together all useful background information on Floquet
theory and the signal analysis tools needed for POMA, this material is
synthetically reviewed in Appendix , to ease reading.
The accuracy of the PARX and POMA identification techniques is then compared
to an exact reference in
Sect. . To this purpose, first
a nonlinear wind turbine analytical model is developed. Then, the stability
of its linearized version is studied according to Floquet theory, providing a
reference ground truth used for comparing PARX and POMA. The equations of
such an analytical model are derived in
Appendix . In
Sect. , a procedure to obtain the
Campbell diagram of a rotor with the PARMAX method is described. PARMAX and
POMA techniques are then used to identify the first low-damped modes of a
high-fidelity wind turbine model, operating in the partial load region in
turbulent winds. Conclusions and recommendations are then given in the final
section of the paper.
The PARMAX modelModeling of wind turbine behavior in turbulent wind conditions using
the PARMAX sequence
showed that the relevant dynamics of a wind
turbine output can be accurately captured by a PARX sequence. Stability is
then verified by applying Floquet theory to the PARX reduced model. The
resulting process is model-independent and fully compliant with the periodic
nature of the problem. However, the use of PARX models must be restricted to
systems subjected to deterministic inputs, as their structure does not
consider the presence of process noise, such as atmospheric
turbulence. As a step towards the application of this periodic stability
analysis concept to real wind turbines, a PARMAX sequence is considered here.
In accordance with , the deterministic
behavior of a wind-turbine-measured output z can be modeled with a PARX
sequence as
A(q;k)z(k)=B(q;k)ut(k),
where k is the time index and q the back-shift operator, such that
z(k)q-i=z(k-i). The autoregressive and exogenous parts are defined, respectively, by polynomials A(q;k) and B(q;k) as
A(q;k)=1-∑i=1Naai(k)q-i,B(q;k)=∑j=0Nbbi(k)q-i,
both being characterized by periodic coefficients ai(k)=ai(k+K) and
bj(k)=bj(k+K), where Na and Nb indicate the order of the AR and
X part, respectively, while K is the period of the system. Finally,
ut is the input, assumed here to be the turbulent wind.
The stochastic nature of the turbulent wind field violates the assumption of
a deterministic and fully measurable input ut. To account for
this, the actual wind is viewed as a sum of two distinct contributions: a
mean wind u(k) and a turbulence-induced perturbation δut(k). As the spectrum of the atmospheric turbulence is far from
being constant, δut(k) is modeled by means of a shape
filter F(q;k) such that
ut(k)=u(k)+F(q;k)e(k),
where e(k) is a zero-mean, white, and Gaussian noise, with periodic variance
σ(k)2.
Inserting Eq. () into
Eq. (), the following is derived:
A(q;k)z(k)=B(q;k)u(k)+G(q;k)e(k),
where C(q;k)=B(q;k)F(q;k).
Equation () is a PARMAX model whose MA part
is represented by polynomial G(q;k), defined as
G(q;k)=1+∑i=1Nggw(k)q-i,
where gw(k)=gw(k+K) are the MA periodic coefficients and Ng the MA
order. The overall order of the system is defined as
n=max(Na,Nb,Ng). The resulting PARMAX sequence is then
z(k)=∑i=1Naai(k)z(k-i)+∑j=0Nbbj(k)u(k-j)+∑w=1Nggw(k)e(k-w)+e(k).
It should be remarked that the present approach does not consider the effects
of nonlinearities nor of rotor speed variations induced by turbulence. The
former potential problem can be checked a posteriori by looking at the
matching between predicted and measured quantities. The latter can be
partially solved by averaging the rotor speed over the analyzed time window.
Typically, because of the large inertia of wind turbine rotors, angular speed
variations are not expected to be highly significant, especially within the
short time windows required by the proposed approach.
State space representation of PARMAX sequences
In order to perform a stability analysis according to Floquet theory (cf.
and the review reported in
Appendix ), it is necessary to realize the PARMAX
Eq. () in an equivalent state space
representation. To this end, consider a linear discrete-time system with
time-varying coefficients in observable canonical form:
x(k+1)=A(k)x(k)+B(k)u(k)+E(k)e(k),y(k)=C(k)x(k)+D(k)u(k)+F(k)e(k),
where x(k)=(x1(k),…,xn(k))T, while the system matrices are given by
A(k)B(k)E(k)C(k)D(k)F(k)=00⋯0αn(k)βn(k)γn(k)10⋯0αn-1(k)βn-1(k)γn-1(k)01⋯0αn-2(k)βn-2(k)γn-2(k)⋮⋱⋱⋮⋮⋮⋮00⋯1α1(k)β1(k)γ1(k)00⋯01β0(k)1.
Including the presence of the MA part, the input–output sequence of system
Eq. () can be derived as
y(k)=∑i=1nαi(k-i)y(k-i)+∑i=1n(βi(k-i)-β0(k-i)αi(k-i))u(k-i)+β0u(k)+∑i=1n(γi(k-i)-αi(k-i))e(k-i)+e(k).
Comparing Eq. () with
Eq. (), the following equivalence relations
are obtained
αi(k)=ai(k+i)∀i=(1,…,Na),β0(k)=b0(k),βi(k)=bi(k+i)+ai(k+i)b0(k)∀i=(1,…,Nb),γi(k)=gi(k+i)+ai(k+i)∀i=(1,…,Ng),
which readily give the state space system matrices. Once these are known,
stability is assessed according to Floquet theory as described in
Appendix .
Identification through the prediction error method
In the present context, a single-input single-output (SISO) PARMAX model must
be identified from a sequence of N measurements. Among the plethora of
existing estimation methods, which may range from time to frequency domain
and from optimization-based to subspace algorithms, the
PEM is chosen here. This method has been
frequently used for rotating systems, such as rotorcraft vehicles and wind
turbines. For example, the periodic equation-error method was used for
identifying a reduced-order model of a helicopter rotor
by , whereas
proposed a periodic output-error method for
the identification of reduced wind turbine models.
The estimation problem, formalized according to the PEM, is the one of
finding the periodic coefficients ai(k), bj(k), and gw(k) that
minimize the cost function J defined as the mean value of the square of the
prediction error, i.e.,
J=1N∑k=1Nε2(k).
Here ε(k)=z(k)-z^(k|k-1) is the prediction error at time
instant k, being z^(k|k-1) (hereafter more concisely written as
z^(k)), the optimal one-step-ahead prediction of z(k) based on
knowledge of all data until time step k-1. According
to and , the optimal
one-step-ahead predictor of process () is
z^(k)=-∑i=1ngi(k)z^(k-i)+∑j=1n(aj(k)+gj(k))z(k-i)+∑w=1nbi(k)u(k-i).
As previously argued, the presence of the MA part in the PARMAX model allows
for a more adequate characterization of the process noise term, at the cost
of a more complex estimation procedure. In fact, the optimal predictor of the
PARMAX process expressed by Eq. () is
nonlinear in the parameters, as any z^(k) is a function of its
previous values z^(k-w), which in turn depend on the parameters.
Consequently, the minimization of cost
function Eq. () involves an iterative optimization.
If the MA part in Eqs. () and () is neglected, a PARX sequence is obtained
and the estimation problem reduces to the so-called equation-error
approach .
Moreover, it is easy to verify that
predictor Eq. () is by itself a PARX dynamic
system, in which the autoregressive part is described by coefficients
-gw(k), whereas coefficients aj(k)+gj(k) and bj(k) define two X
parts with inputs z(k) and u(k), respectively. This fact is not
surprising, since it often happens that the poles of the predictor coincide
with the zeros of the system to be predicted. As a consequence, it may happen
that, during the optimization, coefficients gw define an unstable
predictor, jeopardizing the entire identification
process see.
In the literature there are basically two methods to enforce the stability of
the MA part. The first is a heuristic approach in which the coefficients
gw(k) are perturbed (for example, halved) repeatedly until the achievement
of a stable predictor. This method actually corresponds to a
re-initialization of the parameters with unpredictable effects on the
convergence of the estimation. The second approach is based on the
computation of a new predictor, with different coefficients gw but the
same autocorrelation of the unstable one. For the time-invariant case, this
new canonical model can be obtained using Bauer's
algorithm , whereas for the periodic case it can be obtained by
solving a suitable periodic Riccati
equation or through the multivariate
Rissanen
factorization .
In this work, an alternative and original method is proposed. The stability
of the predictor is enforced by a nonlinear constraint within the estimation
process, and the resulting constrained optimization is performed by an
interior-point algorithm cf.. The
estimation problem is then reformulated as
p=argminpJ(ε(k);p),s.t.:P(p)<1,
where p is the vector of the unknown coefficients and
P(p) are the characteristic multipliers of the PARMAX
predictor.
The characteristic multipliers that constrain the estimation problem can be
computed from the autoregressive part of
Eq. (), i.e., y^(k)=∑w-gw(k)y^(k-w), which can be realized as a state space form according to
Eqs. ()–(),
leading to the following dynamic matrix
N(k)=00⋯0-gNg(k+Ng)10⋯0-gNg-1(k+Ng-1)01⋯0-gNg-2(k+Ng-2)⋮⋱⋱⋮⋮00⋯1-g(k+1).
The periodic coefficients ai(k), bj(k), and gw(k) are approximated by using
truncated Fourier expansions, i.e.,
ai(k)=ai0+∑l=1NFaailccoslψ(k)+ailssinlψ(k),bj(k)=bj0+∑m=1NFbbjmccosmψ(k)+bjmssinmψ(k),gw(k)=gw0+∑r=1NFggwrccosrψ(k)+gwrssinrψ(k),
where ψ(k) is the rotor azimuth. The unknown amplitudes of such
expansions are collected in the vector of parameters pp=…,ai0,ailc,ails,…,bj0,bjmc,bjms,…,gw0,gwrc,gwrs,…T,
where i=(1,…,Na), j=(1,…,Nb), w=(1,…,Ng), l=(1,…,NFa), m=(1,…,NFb), and r=(1,…,NFg), NFa, NFb, and NFg being the
number of Fourier harmonics of the periodic coefficients for the AR, X, and
MA parts, respectively.
Due to the nonlinear behavior of the predictor, the possible presence of
multiple local minima has to be taken into account. A suitable starting point
for the nonlinear problem can be selected by fitting the recorded data with
simpler models such as ARMAX or
PARX or by
using a recursive extended least-squares
algorithm . In the present work, convergence to the
global minimum is ensured by performing several optimization trials from a
randomly chosen set of initial conditions.
Theory of periodic operational modal analysis
The OMA is an output-only system identification technique, which has been
widely used to conduct modal analyses of different mechanical systems.
Recently, special attention has been devoted in the literature to the
application of OMA in the field of wind
energy and to the related underlying
hypotheses .
An output-only technique specifically tailored to time periodic systems was
developed by . This technique,
called periodic OMA (POMA), exploits the particular behavior of a linear time periodic (LTP)
system
in the frequency domain, as described by the HTF (see
Sect. for details). In the present paper, POMA will be
briefly reviewed and then compared to the PARMAX-based stability analysis
proposed here.
Consider a strictly proper periodic system and the exponentially modulated
periodic (EMP) expansions of its input and output, noted, respectively,
as U and Y, as described in
Sect. . The input–output behavior of the system can be
analyzed through the HTF G as
U(s)=G(s)Y(s),
with s∈C and G(s) defined according to
Eq. ().
Projecting Eq. () onto the
imaginary axis, each element of the EMP expansion of
Y and U can be computed as
the Fourier transform of frequency-shifted copies of y(t) and u(t) as
yk(ω)=∫-∞∞y(t)eıω+ıkΩtdt,uk(ω)=∫-∞∞u(t)eıω+ıkΩtdt.
As reported in and briefly reviewed
in , the input–output behavior
in the frequency domain can be expressed as
Y(ω)=G(ω)U(ω),
where
Y(ω)=⋯y-1(ω)y0(ω)y1(ω)⋯T,U(ω)=⋯u-1(ω)u0(ω)u1(ω)⋯T.
Accordingly, the harmonic frequency response function (HFRF)
G(ω) is given by
G(ω)=∑j=1Ns∑w=-∞∞C‾j,wB‾j,wTıω-(ηj+ıwΩ),
where C‾j,w and C‾j,w are defined in
Eqs. () and
() of Sect. .
The power spectrum of the output, noted as SYY(ω),
can be written in terms of the HFRF G(ω) and the
power spectrum of the input SUU(ω) as
SYY(ω)=G(ω)SUU(ω)G(ω)H,
where (⋅)H denotes the complex-conjugate transpose. Inserting Eqs. () into
(), the following
expression is derived:
SYY(ω)=∑j=1Ns∑w=-∞∞∑p=1Ns∑q=-∞∞C‾j,wW(ω)j,w,p,qC‾p,qHıω-(ηj+ıwΩ)ıω-(ηp+ıqΩ)H,
where Wj,w,r,t=B‾j,rSUUB‾w,tH.
Equation () can be simplified
first by considering a flat expanded input power spectrum
Wj,r(ω)=B‾j,rSUUB‾j,rH, at least in the band of
interest of a specific mode, and secondly by assuming that all modes of the
system are “suitably separated”.
The first requirement was analyzed extensively for wind turbine problems
in . There the authors pointed out
that the extended input spectrum could be significantly colored, a problem
that requires particular care with simplified output-only methods. The second
requirement deserves special attention as well. In fact, not only is the
separation of the principal harmonics of two modes required, but it is also
necessary that all super-harmonics with significant participation are
well separated. For rotary wing systems, this requirement has to be
considered especially carefully when looking at the whirling modes, as the
principal harmonics of backward and forward modes are typically separated by
about 2Ω. This typically creates a crisscrossing of modes in the
frequency–rotor-speed plane, leading to frequent frequency encounters.
If such conditions are verified, the extended input spectrum W
loses its dependency on ω, and the contribution of mode ηp+ıqΩ to mode ηj+ıwΩ can be neglected when p≠j and
q≠w. Hence, Eq. () is
simplified to
SYY(ω)≈∑j=1Ns∑w=-∞+∞C‾j,wWj,wC‾j,wH(ıω-(ηj+ıwΩ))(ıω-(ηj+ıwΩ))H.
From Eq. () one can see that the
peak related to any super-harmonic of a given mode can be viewed as the peak of a
linear time-invariant mode. Accordingly, one is allowed to use a standard LTI
frequency domain identification technique (e.g., peak picking, curve fitting) to
compute frequencies, damping factors, and modal shapes from the measured spectra.
Moreover, neglecting again the contribution of overlapping modes, one can also
estimate the participation by evaluating the power spectra at the peak frequency,
since
C‾j,wC‾j,wH∝SYY(ωj+wΩ).
Expressing the product
C‾j,wC‾j,wH, one gets
C‾j,wC‾j,wH=⋱⋮⋮⋮⋰⋯cj-2cj*-1cj-1cj*0cj0cj*1⋯⋯cj-1cj*-1cj0cj*0cj1cj*1⋯⋯cj0cj*-1cj1cj*0cj2cj*1⋯⋰⋮⋮⋮⋱,(⋅)* being the complex conjugate. From
Eq. (),
one could envision several criteria for extracting the participation factors
for each harmonic belonging to the jth mode. The simplest one is to
compute the central column of the HTF and to pick the amplitudes of the
spectra at the frequency of interest. The participation factors are then
extracted according to
Eq. (), reported in
Sect. , as
ϕjyn=cjn∑ncjn=cjncj*0∑ncjncj*0=cjncj*0∑ncjncj*0.
One can also perform multiple estimations of the participation factors by looking
again at the central column of SYY. In fact, from
Eq. (), it
appears that the amplitudes picked from the ℓth column at frequency
ωj+wΩ are equivalent to those picked from the central column at
ωj+(w+ℓ)Ω. This also means that computing the central column could
be sufficient for having an estimation of frequencies, damping, and participation
factors, as already noted in .
The POMA technique can then be summarized as follows:
Compute the Fourier transforms of the frequency-shifted copies of the
recorded output y(t), yk(ω)=FFTy(t)e-ıkΩt and collect them in vector
Y(ω)=(…,yk(ω),…)T.
Compute the autospectrum SYY(ω) using a standard
frequency domain analysis method; in the present paper the method of Welch was
employed for this purpose.
Extract the related
natural frequency and damping factors from each peak present in SYY(ω) using any standard LTI frequency domain
estimation tool . In this paper the
straightforward peak-picking method was used, as also done
by .
Reconstruct the Fourier coefficients cjn, and in turn the
participation factors, by evaluating the spectrum in correspondence to each
peak.
It is possible to restrict the analysis to the right-half plane just by noting that
yn(-ω)=y-n*(ω).
Equation () is particularly useful for
identifying the Fourier coefficients from the peaks of the “reflected
super-harmonics”, since according to
Eq. () one can demonstrate that
cjncorrectpeak=cj*-nreflectedpeak.
Application of periodic operational modal analysis to the Mathieu oscillator
As the actual use of POMA and the correct interpretation of all peaks is not a
straightforward exercise in general, a simple Mathieu oscillator is analyzed here in
preparation for the application of this method to the wind turbine problems studied
later on. The dynamics of a Mathieu oscillator is governed by the following
equations:
x˙x¨=01-ω02-ω12cos(Ωt)-2ξω0xx˙,y=10xx˙.
The parameters in Eq. () were set,
following , as ω02=1,
ω12=0.4, ξ=0.04, and Ω=0.8. The system was numerically
integrated from x(0)=(1000,0)T, and studied by means of POMA. The
results were then compared with those obtained by the full Floquet theory
described in Sect. .
Harmonic power spectrum of the
output of the Mathieu oscillator.
Figure shows the power spectra of the
central column of SYY, yk(ω)y0H(ω) for
k=-4,…,4. The fundamental peak (i.e., the highest one) is found on the
0-shift curve at 0.16 Hz and corresponds to the amplitude cj0cj0H. At such a frequency, all curves show a prominent peak, from which
one may also easily compute the damping factors using, for example, the
standard half-power bandwidth method. The participation factors are then
extracted by looking at the amplitudes of the power spectra using
Eq. ().
Starting from this peak and moving to the right, the subsequent higher peaks
are found on the negative-shift curves, first in the -1-shift one at
0.28 Hz and then in the -2-shift one at 0.41 Hz, etc. The opposite happens
when moving to the left. Peaks located at negative frequencies appear as
reflected in the positive frequency range but with opposite shifts. This is
clear if one looks at the peak located at -0.10 Hz, which has the -2-shift
curve as the one with the highest amplitude, whereas the reflected peak at
0.10 Hz is associated with the 2-shift curve. This complex behavior is
easily explained by means of
Eq. (), which also states that the
information in the negative frequency range can be reconstructed by looking
at the curve with the opposite shift in the positive frequency plane.
Frequencies and damping factors computed from such spectra using the peak-picking
method are reported in Table .
The same table also displays the results obtained from the full Floquet analysis of
the system. The comparison shows good accuracy, especially for frequencies and
damping factors of the first highest super-harmonics.
Frequencies and damping factors for the Mathieu oscillator and
analytical results.
The output-specific participation factors are displayed in
Table . Multiple estimates have
been computed from each spectrum peak in the positive frequency plane. The last
column also shows the analytical results. As expected, in general super-harmonics
with lower participation factors are associated with higher estimation errors.
Most relevant
output-specific participation factors for the Mathieu oscillator and related
analytical results.
0.35 Hz0.23 Hz0.10 Hz0.03 HzPeak at0.28 Hz0.41 Hz0.54 Hz0.67 HzExact(-4Ω)(-3Ω)(-2Ω)(-1Ω)0.16 Hz(+1Ω)(+2Ω)(+3Ω)(+4Ω)ϕ1-4x0.01740.01670.01640.01620.0163––––4.961 × 10-4ϕ1-3x0.03520.03460.03160.03230.03280.0323–––0.0097ϕ1-2x0.06590.05870.06600.06260.06180.06520.0667––0.0477ϕ1-1x0.15090.14050.14190.14090.14100.14330.14730.1560–0.1583ϕ10x0.70000.66140.64450.64990.65370.65620.67530.72340.85270.7160ϕ11x–0.06420.06870.06660.06610.06850.07030.07310.08620.0655ϕ12x––0.01340.01310.01300.01340.01370.01440.01700.0023ϕ13x–––0.00850.00850.00860.00890.00950.01114.401 × 10-5ϕ14x––––0.00670.00670.00690.00740.00875.325 × 10-7Stability analysis of a model wind turbine problem
Next, a simplified wind turbine model is used for comparing the results
obtained with the PARX and POMA approaches. This is useful because it gives a
way of comparing the basic performance of the two methods with respect to a
known exact ground truth in the ideal case of zero disturbances. Later on in
this work, the two methods will be compared for the case of a higher-fidelity
wind turbine model operating in turbulent wind conditions. As no exact
solution is known in that case, the preliminary investigation of this section
serves the purpose of clarifying whether significant differences exists
between the two approaches even at this more fundamental level. Indeed, it
will be shown here that some of the underlying hypotheses of POMA are not
always fulfilled, and this leads occasionally to some imprecisions in the
estimates of the modal quantities of interest.
The analytical model is derived in detail in
Appendix , which
also gives a schematic sketch of the system in
Fig. . The model considers the coupled
motion of tower and blades subjected to aerodynamic and gravitational forces.
The fore–aft and side–side flexibility of the tower is rendered by two
equivalent linear springs, whereas each blade is represented as a rigid body
connected to the hub through two coincident linear torsional springs,
allowing, respectively, the blade flap-wise and edgewise rotations. The
characteristics of each element in the model are chosen so as to match the
first tower fore–aft and side–side modes and the first blade flap-wise and
edgewise modes in vacuo of a reference 6 MW wind turbine, as computed using
a high-fidelity multi-body model. The aerodynamic formulation is inspired by
the treatment of , in which the
aerodynamic forces and moments at the blade hinges are computed assuming
linear aerodynamics, small flap and lag angles, uniform inflow over the rotor
disk, and constant rotor speed. The aerodynamic forces induced by tower
motion, not present in the treatment of ,
are additionally considered in this paper. The model represents the complete
lower spectrum of a wind turbine, including the first side–side and fore–aft
tower modes, the first in-plane and out-of-plane blade modes as well as their
related whirling modes.
After having collected all degrees of freedom in vector ξ=(β1,…,βB,ζ1,…,ζB,yH,zH)T, B being the number of blades, and the inputs in vector ν=(θp1,…,θpB)T, θk being the pitch angle of
the kth blade, the resulting nonlinear second-order implicit system writes
f(ξ,ξ˙,ξ¨,ν,t)=0.
System Eq. () can be integrated in time using any
suitable numerical scheme, starting from a consistent set of initial
conditions. This was done for generating the time histories used for PARX and
POMA, paying attention not to excite the system nonlinearities, as the
reference solution is based on the Floquet analysis of the linearized
problem.
Since any mechanical system is linear in ξ¨, one may
compute the mass matrix
M^(ξ,ξ˙,t) and
rewrite the system as M^(ξ,ξ˙,t)ξ¨=g(ξ,ξ˙,ν,t).
System Eq. (), if asymptotically stable, converges
to a periodic trajectory ξ̃(t) when subjected to a
periodic input ν̃(t). In such a regime, the
linearized periodic equations of motion write
M(t)ξ^¨+T(t)ξ^˙+K(t)ξ^+W(t)ν^=0,
where the new state ξ^(t) and input
ν^(t) are defined as
ξ^(t)=ξ(t)-ξ̃(t),ν^(t)=ν(t)-ν̃(t),
and the periodic mass, damping, stiffness, and input matrices are defined as
M(t)=∂f∂ξ¨ξ̃,ξ̃˙,ξ̃¨,ν̃,T(t)=∂f∂ξ˙ξ̃,ξ̃˙,ξ̃¨,ν̃,K(t)=∂f∂ξξ̃,ξ̃˙,ξ̃¨,ν̃,W(t)=∂f∂νξ̃,ξ̃˙,ξ̃¨,ν̃.
Note that M(t) is equal to
M^(ξ,ξ˙,t), evaluated on the periodic trajectory ξ̃. These
linearized equations of motion about a periodic orbit were then used for
developing the analysis according to Floquet, yielding the ground truth
solution.
Stability analysis of a wind turbine analytical model
The parameters of the wind turbine analytical model loosely represent a conceptual 6 MW wind turbine, and they are listed in Table . The stability of the model is studied in a uniform axial
wind of 9 m s-1 for a collective pitch angle of -0.54∘,
corresponding to operation towards the end of the partial load region.
Parameters of the analytical wind
turbine model.
ParameterSymbolValueNumber of bladesB3Rotor radiusR75 mRotor speedΩ11.5 rpmHinge offsete25.651 % RMass of hubmH7.500 × 104 kgBlade mass (movable part)mD1.448 × 104 kgBlade mass (fixed part)mU1.087 × 104 kgBlade CG after hingerGD18.72 mBlade moment of inertiaJD7.488 × 106 kg m2Edgewise spring stiffnessKζ2.119 × 108 N mEdgewise spring damperCζ1.756 × 106 N m sFlap-wise spring stiffnessKβ5.215 × 107 N mFlap-wise spring damperCβ1.756 × 106 N m sTower SS spring stiffnessKy7.312 × 105 N m-1Tower SS spring damperCy1.329 × 104 N s m-1Tower FA spring stiffnessKz6.581 × 105 N m-1Tower FA spring damperCz1.329 × 104 N s m-1Lock numberγ20Wind shear gradientK10.018 s-1
The linearized periodic system was first studied using Floquet theory (see
Appendix ) in order to get the exact natural frequencies,
damping, and output-specific participation factors. Next, the model was used for
generating all outputs needed for performing the PAR(MA)X and POMA analyses by
integrating the system forward in time starting from suitable initial non-zero
conditions, chosen in order to excite the modes of interest. In this exercise, the
wind was considered as stationary, so that the PARMAX identification reduces to the
simpler PARX one as the MA part is not necessary.
Both PARX and POMA estimates were compared with the full Floquet results in terms of
relative errors for frequencies and damping factors and absolute errors for
participation factors. Relative errors are defined as vE/vR-1,
while absolute errors are defined as vE-vR, where v is a specific modal
parameter and the subscripts E and R refer, respectively, to
an estimated and a real (exact) quantity.
Identification of the blade edgewise mode
The blade edgewise mode was excited by imposing the initial edgewise angles of all
blades equal to a unique non-zero value, whilst all other states were set to zero at
the initial time. This way the blade in-plane mode was excited while avoiding the
onset of the whirling modes.
Considering first the POMA approach, the harmonic power spectrum for the second
blade edgewise angle, ζ2, was computed with frequency shifts from -2Ω
to +2Ω. The results obtained this way are reported in
Fig. .
Harmonic power spectrum
of the ζ2 output of the wind turbine analytical model. The peak of the
n=0 curve is caused by the blade in-plane mode, while spikes are due to
the rotational frequency and its multiples.
Analytical results and estimation
errors of blade in-plane modal parameters.
Clearly, the 0-shift PSD shows a prominent peak at
ωE=0.86 Hz, related to the blade in-plane mode, from which
one may easily extract the frequency and damping factor of the principal
harmonic. The peak-picking method could in principle be applied to any of the
peaks displayed in the figure; however, one may observe that most of the
peaks are of a low amplitude and often barely noticeable from the side band
of the principal harmonic. For example, the super-harmonic at 0.67 Hz, even
if visible within the 0-shift curve, does not have enough energy to allow one to
estimate its modal quantities to any reasonable accuracy. Therefore, it was
preferred to compute frequency and damping factors only by looking at the
highest peaks: the frequency and damping factor of the super-harmonic at
ωE+Ω were extracted from the peak at 1.05 Hz of the
-1-shift curve, while those of the super-harmonic at
ωE+2Ω were extracted from the peak at 1.24 Hz of -2-shift curve,
and similarly for the other super-harmonics. For the same reason,
participation factors were obtained only by looking at the PSD amplitude at
ωE. In fact, at this frequency all curves show peaks that
are prominent and distinct enough to compute the participation factors
according to Eq. ().
Next, the PARX analysis was considered. As long as only the blade in-plane
mode is significantly excited, as indicated from the 0-shift curve in
Fig. , the order of the AR part may
be set as Na=2. A first-order X part (Nb=1) was considered as the
inputs (wind speed and pitch angle) are constant in this case. Finally, the
number of harmonics for the Fourier series expansion of both the AR and X parts,
NFa and NFb, were both set equal to 1. The matching
between predicted and simulated output, not reported here for the sake of
brevity, showed excellent correlation, proof of the fact that the identified
model captures the dynamics of interest very well.
Table reports the Floquet modal parameters,
assumed as ground truth, as well as the errors obtained by the two methods
considered here.
Looking at the results, it appears that both the PARX and POMA methods are
able to capture the relevant dynamics related to the principal harmonics, as
frequencies, damping, and participation factors are of good quality. In
particular, damping and participation factors are slightly better estimated
by PARX.
The estimation of the super-harmonic modal parameters deserves a special
mention. The PARX method is able to provide a good matching for all modal
parameters of all harmonics: frequencies and participation factors have
negligible errors, whereas damping factors show an error lower than 1 %. On
the other hand, the error of the POMA super-harmonic estimates is typically
quite large especially for the damping factors, even though the principal
harmonic is well captured.
This fact has mainly two possible explanations. First, the hypothesis of well-separated modes is here not fully satisfied, as the side band of the tower
principal harmonic affects all super-harmonic peaks. The lower the rotor
speed, the more pronounced this effect is, as the frequency separations among
super-harmonics coincide with multiples of the rotor frequency. Second, but
more importantly, according to the dynamics of a periodic system all
harmonics belonging to a specific mode descend from a sole characteristic
multiplier. Therefore, their frequencies and damping factors are strictly
connected to each other. This relation is totally ignored by
POMA cf., as it considers
each peak in the frequency response as a stand-alone mode.
Identification of other low-damped modes
The tower side–side and blade in-plane whirling modes were excited by
imposing different initial conditions for each blade edgewise angle and a
suitable lateral displacement of the tower.
Figure shows the harmonic power
spectral density (HPSD) for the tower side–side displacement yH, with
frequency shifts from -2Ω to +2Ω. Here again, the 0-shift
curve shows three distinct peaks: the tower side–side mode and the backward and
forward in-plane whirling modes, respectively, at 0.34, 0.68 and 1.1 Hz.
Accordingly, the PARX complexity was set as Na=6, Nb=1, NFa=1, and NFb=1. As for the previous case, the matching between
predicted and simulated output, not reported here, is excellent. Comparisons among the exact and identified modal parameters are displayed in
Table through Table .
Harmonic power spectrum
of the yH output of the wind turbine analytical model. Three modes are
visible on the n=0 curve, along with the rotational frequency and its
harmonics.
Figure clearly shows that a good
mode separation is not fully achieved here, as whirling super-harmonics
interact with each other. This is not due to the specific wind turbine or
condition considered here, as in fact any rotating blade system will always
have the principal harmonics of its whirling modes separated by about
2Ω. In addition, it also appears that the second super-harmonic of the
tower mode at 0.73 Hz is very close to the second super-harmonic of the
forward (FW) whirling mode at 0.71 Hz; additionally, both harmonics are
close to the backward (BW) whirling mode at 0.68 Hz. For this reason, there
are missing values in Table
through Table , wherever it was not
possible to pick all peaks for all modes of interest using POMA.
Comparison between
measured (solid line) and predicted (dashed line) normalized blade root
edgewise bending moment, in the time (left) and frequency (right) domains.
Considerations similar to ones previously made for the blade in-plane mode can also be stated here for these other three modes. Specifically, the
frequency and damping factors of the principal harmonic of all modes are
almost perfectly captured by both methods. The PARX method is the one that
gives the most accurate results globally for both principal and
super-harmonics: damping and participation factor estimates are characterized
by small errors, while only the damping factors of the backward whirling mode
have errors greater than 10 %. On the other hand, the POMA technique does
not provide consistent results for the super-harmonic damping factors, which
are characterized by large errors even when the damping factor of the
principal harmonic is well captured. Moreover, the participation factors of
the whirling modes exhibit non negligible errors for both principal and
super-harmonics. This last issue is mainly due to the fact that, especially
for the whirling case, the underlying hypothesis of well-separated modes is
not completely fulfilled, as previously mentioned.
Periodic Campbell
diagram of the first blade edgewise mode obtained from PARMAX
identifications. The results of the single identifications along with the
confidence level of the fitting curves are shown. Participation factors are
computed in the rotating reference frame.
HPSD for the blade in-plane
mode, obtained for a 3 m s-1 average wind speed.
Periodic Campbell
diagram of the first blade edgewise mode obtained from POMA
identifications.
PARMAX-based damping estimation using a high-fidelity multi-body model
A detailed 6 MW wind turbine high-fidelity multi-body model operating in a closed loop, implemented with the aero-servo-elastic simulator
Cp-Lambda, was then used for a
comparison of the POMA and the proposed PARMAX stability analysis techniques
in a more sophisticated setting. Blades and tower are modeled with
geometrically exact beam elements, discretized in space using the finite-element method, whereas the classical blade element momentum (BEM) theory is
used to model the aerodynamics, with the usual inclusion of wake swirl, tip
and hub losses, unsteady corrections, and dynamic stall. The total number of
degrees of freedom in the resulting finite-element multi-body model is about
2500. A pitch–torque controller complements the aero-servo-elastic model.
Wind histories compliant with IEC-61400 design guidelines were generated
through TurbSim. The considered wind
fields are characterized by a 5 % turbulence intensity and
10 min averaged wind speeds ranging from 3 to 10 m s-1, an upflow of
8∘, and an atmospheric boundary layer power law exponent equal to
0.2.
Periodic Campbell diagram
for the tower side–side mode obtained from PARMAX (left) and POMA (right)
identifications.
HPSD of the Md load,
obtained for a 7 m s-1 average wind speed.
Periodic Campbell diagram
of the backward whirling in-plane mode obtained from PARMAX (left) and POMA
(right) identifications.
Periodic Campbell diagram
of the forward whirling in-plane mode obtained from PARMAX (left) and POMA
(right) identifications.
According to the PARMAX-based stability analysis, the system should be
perturbed so as to induce a significant response of one or more modes of
interest. Among the many possible ways of exciting a specific wind turbine
mode, as, for example, the use of pitch and torque
actuators or of eccentrical
masses , impulsive forces
were used in this work. Such forces could be realized in practice by
pyrotechnic exciters. The rotor angular speed is averaged over the length of
the recorded history and used to compute the system period. Afterwards, the
signal is resampled in order to have an integer number of steps within a
period.
The selection of the model complexity deserves special care. As the order of
the AR part, Na, is strictly related to the number of system modes, it can
be estimated by looking at the number of principal-harmonic peaks present in
the output PSD. This heuristic approach for the problem at hand turned out to
be simple and effective and was preferred to more sophisticated
criteria .
As described in Sect. , the input wind
speed was considered as the sum of two contributions, a constant
deterministic part and a turbulence-induced one. As long as the deterministic
input is considered to be constant, one is allowed only to estimate an X part
with order Nb=1. The MA-part order (noted as Ng) as well as the number of
harmonics used to model the periodicity of the coefficients (noted as NFa,
NFb and NFg) were set with a trial an error approach, until the
achievement of satisfactory results.
After having performed the estimation for different wind conditions and
therefore at different rotor speeds, the results of the analyses in terms of
frequency, damping, and participation factors were fitted using low-order
polynomials, computed by means of the robust bi-square
algorithm . The fitting process
was applied only to the frequency and damping of the principal harmonic,
indicated with the subscript (⋅)0. The corresponding characteristic
exponent was then computed as
ηj0=-ωj0ξj0+ıωj01-ξj02.
The super-harmonics were finally obtained by means of
Eq. (). On the other hand, the
participation factors of all super-harmonics were fitted with the same
bi-square algorithm.
Blade edgewise mode
Two mainly edgewise doublets, applied at mid span and near the tip of the
blade, were used to excite this mode. The PARMAX reduced-order model
considered the following choice of parameters: Na=6, NFa=1, Nb=1,
NFb=1, Ng=2, and NFg=0. This setting allows for the modeling of
three periodic modes.
The result of an identification executed at the rated rotor speed is shown in
Fig. . The excellent
superposition of the curves indicates a reduced-order PARMAX model of very
good quality, capable of modeling the signal behavior despite the small
nonlinearities and rotor speed variations characterizing the system that
generated the data.
To draw the Campbell diagram, eight different identifications were made in
order to cover the entire range of angular speeds of the machine. The results
are shown in Fig. , where red
dots indicate each specific identification, whereas lines refer to their
quadratic fits. The gray bands are the 2σ nonsimultaneous functional
prediction bounds, and measure the confidence level of the fitting curves.
From the gray bands one can infer that each frequency and damping factor
identified at a specific rotor speed is coherent with the others, as all the
estimates define a clear trend. On the other hand, a significant but
acceptable uncertainty still characterizes the participation factors.
Similar analyses were conducted by ,
where a different turbulence intensity (IEC level “B”, instead of a uniform
5 % over the whole wind speed range) was used, caeteris paribus.
As the Campbell diagram is similar in both works, one may conclude that the
PARMAX-based analysis does not appear to be significantly influenced by
turbulence level.
Much longer portions of the time histories analyzed with PARMAX were then
processed with the POMA method. In
Fig. , the HPSD obtained for a wind
field with a 3 m s-1 average speed is shown (note the similarities
with Fig. ). For this case the
turbulence intensity was quite low, and the HPSD lines present well-defined
peaks. However it was found that, for increasing wind speed, while the n=0 lines remain well defined, the quality of the peaks associated with the
super-harmonics progressively degrades, making the estimation of damping
(and, in some cases, also of frequency) increasingly more difficult.
The Campbell diagram obtained from POMA is displayed in
Fig. . Comparing this figure
with the PARMAX plot shows that frequencies are well identified, but the high
dispersion of damping factors masks the expected trend. Several differences
may also be seen between the plots with respect to the participation
factors. While both approaches indicate that the principal harmonic is the
most important in the response, they do, however, detect a markedly different
behavior as a function of rotor speed. In addition, POMA overestimates the
participation factors of the ±2 super-harmonics.
Tower side–side mode
The tower side–side mode was excited with a chirp-shaped force applied at the
tower top. The frequency band of such signal was set in order to excite only
that single mode. The tower base side–side moment was then recorded and used
as output. As only the tower side–side peak is visible in the PSD of the
response, then Na was set equal to 2. The other coefficients were set as
NFa=1, Nb=1, NFb=1, Ng=2, and NFg=1.
The agreement between the output predicted with the PARMAX reduced model and
the measure, not shown here for the sake of brevity, is very good. The left
plot of Fig. shows the Campbell
diagram obtained with the PARMAX approach. In this diagram the results of the
identifications are approximated with straight lines. Looking at this plot,
it appears that at 0.8Ωr the principal harmonic intersects the
2 × Rev. For the PARMAX identification this is not particularly
problematic, and in fact only the participation factor has been slightly
underestimated. On the other hand, this poses a major problem for POMA. In
fact, when the signal is frequency-shifted by +2Ω, its average value
is transported over the principal peak, making it difficult to estimate the
mode shape and the damping of the tower side–side mode.
The Campbell diagram obtained from POMA identifications is shown in the
right-hand plot of Fig. . The plot clearly
shows that the damping of the principal harmonic estimated with the
half-power bandwidth is double the one estimated by PARMAX.
Backward and forward whirling in-plane modes
The backward and forward whirling in-plane modes were excited with a tower
top side–side doublet, whose amplitude and duration were selected such that
the input force spectrum is almost flat in the frequency range of interest.
The three-blade root edgewise bending moments M1, M2, and M3 were
recorded, and the multi-blade coordinate transformation
M0MdMq=131112cos(ψ1)2cos(ψ2)2cos(ψ3)2sin(ψ1)2sin(ψ2)2sin(ψ3)M1M2M3
was used to yield the direct and quadrature moments, noted, respectively, as Md
and Mq. The spectra of Md, displayed in
Fig. , show well-defined peaks.
The PARMAX reduced model was set with the following choice of parameters:
Na=8, NFa=1, Nb=1, NFb=1, Ng=2, and NFg=1. Both
the backward and forward whirling in-plane modes, as well as the side–side
tower mode, were nicely visible in the frequency plot of the perturbed time
histories. Thus, for each wind speed, only one reduced model capable of
representing the behavior of all these three modes was identified.
Figures
and show on the left the periodic
Campbell diagram obtained using the PARMAX approach and on the right the one
computed with POMA, respectively, for the backward and the forward whirling
in-plane modes. It should be noted that both approaches provide the same
results in terms of frequencies. The overall trend of the principal-harmonic
damping factors as functions of the rotor speed is similarly captured. In
particular, the PARMAX results are characterized by a lower uncertainty for
the backward mode and a higher uncertainty for the forward one. The rising of
the damping factors with the angular speed, for these two modes, has been
observed also in , although for an isotropic
condition.
Once again, the damping of the super-harmonics obtained with the POMA
technique are not well estimated, as already noted in
Sect. . Moreover, the
participation factors of the ±2 super-harmonics are typically too high:
for example, in the right-hand plot of Fig. , one can see that the
participation of super-harmonic +2 of the forward whirling mode is higher
than that of the principal one. This strongly overestimated participation is
due to the nearly 2Ω spacing of the whirling modes, which causes their
super-harmonics to nearly overlap.
Conclusions
In this paper we have considered a model-independent periodic stability
analysis capable of handling turbulent disturbances. The approach is based on
the identification of a PARMAX reduced model from a transient response of the
machine. The full Floquet theory is then applied to the reduced model,
yielding all modal quantities of interest. As only time series of
measurements are necessary, the method appears to be suitable for the
application to real wind turbines operating in the field.
In order to assess the validity of the proposed method, the well-known POMA
was implemented and used for comparison. Tests were performed first with the
help of a wind turbine analytical model, whose exact solution can be obtained
by the theory of Floquet, and then with a high-fidelity wind turbine
multi-body model operating in turbulent wind conditions.
Based on the results obtained in this study, one may draw the following
considerations.
Both methods are able to characterize the relevant behavior of the wind
turbine in turbulent wind conditions. However, the results provided by the
proposed PARMAX analysis are in general more accurate than those given by the
POMA technique, especially if one looks not only at the principal harmonics
but also at the super-harmonics.
Often the underlying hypotheses of POMA are not exactly fulfilled, and this
leads to inaccuracies especially in terms of damping and participation
factors. These effects are more visible for the whirling modes, as they are
separated by about 2Ω, which means that there will always be a perfect
overlap between the super-harmonics of these two modes at some angular
velocity. The PARMAX analysis is less prone to such problems.
A major advantage of PARMAX over POMA is that it requires shorter
time histories. This is important in turbulent conditions, where the rotor
speed is hardly constant (which, on the other hand, is a fundamental
hypothesis of both methods).
The development of the present SISO PARMAX approach suggests a number of
extensions, which are currently under investigation.
The use of multiple outputs in a multiple-input multiple-output (MIMO)
PARMAX framework could improve the quality of the results.
Due to the stochastic nature of turbulence, a multi-history PARMAX applied
to different realizations of the same experiment could provide more robust
modal results, along with the associated variances.
The peak-picking method is rather simple, and it is unable to exploit all the
informational content available in the HPSD, especially in the presence of
noisy peaks. Fitting algorithms have been preliminarily
explored see, but their application
to the multiple output case has not yet been attempted.
Review of linear time periodic systemsFloquet theory in continuous time
A generic SISO LTP system in continuous time can be written in state space form as
x˙=A(t)x+B(t)u,y=C(t)x+D(t)u,
where t is time and x, u, and y the state, input, and
output vectors, respectively, while A(t), B(t),
C(t), and D(t) are periodic system matrices such that
A(t+T)=A(t),B(t+T)=B(t),C(t+T)=C(t),D(t+T)=D(t),
for any t. The smallest T satisfying
Eq. () is defined as the system period.
Scalar u can be any of the wind turbine control inputs (i.e., blade pitch angles,
electrical torque, possibly the yaw angle) as well as exogenous inputs
related to the wind states (e.g., wind speed, vertical or lateral shears,
cross-flow).
To study the stability of Eq. (), its autonomous version
is considered together with the associated initial conditions:
x˙=A(t)x,x(0)=x0.
The state transition matrix Φ(t,τ) maps the state at time
τ, x(τ), onto the state at time t, x(t),
x(t)=Φ(t,τ)x(τ),
and it obeys a similar equation with its associated initial conditions
Φ˙(t,τ)=A(t)Φ(t,τ),Φ(τ,τ)=I,
where I is the identity matrix. It can be shown that in the continuous-time
case the transition matrix is always invertible .
An important role in the stability analysis of periodic systems is played by
the state transition matrix over one period
Ψ(τ)=Φ(τ+T,τ), termed
monodromy matrix. By definition, the monodromy matrix relates two
states separated by a period; consequently, a generic state that is sampled
at every period, noted as x̃τ(k)=x(τ+kT), obeys
the following linear-invariant discrete-time equation
x̃τ(k+1)=Ψ(τ)x̃τ(k).
The system is asymptotically stable if all the eigenvalues of the monodromy
matrix, characteristic multipliers and noted as θj,
belong to the open unit disk in the complex plane. It can be shown that the
eigenvalues of the monodromy matrix and their multiplicity are time-invariant
even if the monodromy matrix is periodic .
For this reason, one can ignore the time lag τ when referring to the
characteristic multipliers. The eigenvalues θj and associated
eigenvectors sj are obtained by the spectral factorization of the
monodromy matrix, i.e.,
Ψ(τ)=Sdiag(θj)S-1,
with S=[…,sj,…].
In order to determine the frequency content of a periodic system, it is
necessary to introduce the so-called Floquet–Lyapunov transformation. The
Floquet–Lyapunov problem is the one of finding a bounded, periodic, and
invertible state space transformation z(t)=Q(t)x(t)
such that the resulting governing equation
z˙=Rz
is time-invariant, i.e., the Floquet factor matrix R is
constant. Since R=Q(t)A(t)Q-1(t)+Q˙(t)Q-1(t), the periodic transformation
Q(t) must obey the following matrix differential equation
Q˙(t)=RQ(t)-Q(t)A(t),
whose solution is
Q(t)=eR(t-τ)Q(τ)Φ-1(t,τ).
Exploiting the periodicity condition Q(τ+T)=Q(τ), one gets the relationship between monodromy matrix and
Floquet factor, which writes
Ψ(τ)=Q(τ)-1eRTQ(τ).
The eigenvalues of the Floquet factor, called characteristic
exponents and noted as ηj, are computed by the spectral factorization of
R:
R=Vdiag(ηj)V-1,
with V=[…,vj,…]. Inserting
Eqs. () and ()
into Eq. (), the following result is
derived
diag(θj)=S-1Q(τ)-1Vdiag(eηjT)V-1Q(τ)S,
which shows that V=Q(τ)S and, more
importantly, that characteristic multipliers and characteristic exponents are
related as
θj=eηjT.
Note that there is an infinite number of Floquet factors, and therefore an
infinite number of Floquet–Lyapunov transformations. In fact, one can choose
any invertible initial condition Q(τ). In addition, computing
characteristic exponents from multipliers by inverting
Eq. () leads to a multiplicity of
solutions, as in fact
ηj=1Tln(θj)=1T(lnθj+ı(∠(θj)+2ℓπ)),
where ℓ∈Z is an arbitrary integer. This indeterminacy, however, does
not affect the real frequency content of the response, since the transition matrix
is uniquely defined. This aspect of the problem will be further analyzed later on in
these notes.
Given Q(τ) and R, the transition matrix is readily
obtained from Eq. () as
Φ(t,τ)=P(t)eR(t-τ)P(τ)-1,
where the periodic matrix P(t)=Q(t)-1 is termed
periodic eigenvector.
Consider now, for each mode, one of the infinite solutions of
Eq. (), for example, the one with ℓ=0,
noted as η^j. Introducing Ω=2π/T, any other characteristic
exponent ηj could be computed from η^j as
ηj=η^j+ınΩ,n∈Z.
Inserting Eq. ()
into Eq. (), one can express the state
transition matrix as the following modal sum:
Φ(t,τ)=∑j=1NsZj(t,τ)eη^j(t-τ),
where Zj(t,τ)=P(t)VIjjV-1Q(τ),
while Ijj is a matrix with the sole element (j,j) equal to 1
and all others equal to 0. Because of the particular definition of
Ijj, matrix Zj(t,τ) is of unitary rank
∀(t,τ), and it is also equal to
ψj(t)Lj(τ)T, where
ψj(t)=colj(Ξ(t)),Lj(τ)T=rowj(Ξ-1(τ)),
with Ξ(t)=P(t)V.
Equation () can be now reformulated as
Φ(t,τ)=∑j=1Nsψj(t)Lj(τ)Teη^j(t-τ).
Exploiting the periodicity of ψj(t),
Eq. () becomes
Φ(t,τ)=∑j=1Ns∑n=-∞+∞ψjnLj(τ)Te(η^j+ınΩ)(t-τ)eınΩτ,
where ψjn is the amplitudes of the harmonics of the
Fourier expansion of ψj(t). This expression of the state
transition matrix can also be found in
, , and .
From Eq. () it appears that, for each mode, an
infinite number of exponents (playing the role of eigenvalues of the LTI
system) participates in the response of the system. Furthermore, a single
frequency is not sufficient for completely characterizing that mode. All
exponents have imaginary parts that differ by integer multiples of Ω
and have the same real part; thus, all exponents of a given mode are either
stable or unstable. This fact is not surprising, as the stability of the
system is just determined by the characteristic multipliers, which are
uniquely defined.
For the LTP system, the exponents η^j+ınΩ play the
role of the eigenvalues of the LTI case, as they yield the frequencies
ωjn=η^j+ınΩ and damping
factors ξjn=-Re(η^j)/ωjn of each mode.
To describe this situation, this infinite multiplicity of frequencies is
termed a fan of modes cf.. Each
harmonic in a fan contributes to the overall response according to its
associated “modal shape” ψjn. The relative
contribution of the nth harmonic to the jth mode is measured through its participation factor, defined as
ϕjn=ψjn∑nψjn.
The triads {ωjn,ξjn,ϕjn} describe completely the
behavior of a periodic mode. The participation factors can be defined also as
functions of the Frobenius norm of the harmonics of
Zj(t,τ), Zjn=ψjnLj(τ)T, as shown in :
ϕjn=ZjnF∑nZjnF.
The two definitions are exactly equivalent as, in this specific case,
ZjnF=ψjnLj(τ) and
Lj(τ) stays the same for all harmonics.
Given two-column vectors v=(…,vi,…)T and
w=(…,wj,…)T, the square of the Frobenius norm
of the product vwT can be expressed as
vwTF2=∑i∑j(viwj)2=∑ivi2∑jwj2=v2w2.
The apparent indeterminacy in the computation of the imaginary part of the
logarithm of the characteristic multipliers in
Eq. () is then understood. In fact, all
the exponents that satisfy Eq. () are
present in the response of the system, as it can be seen from
Eq. (). Since the transition matrix is uniquely
defined, any choice of the integer ℓ in
Eq. () would act as a shift in the
frequency content of Zj, such that all triads
{ωjn,ξjn,ϕjn} remain exactly the same, as first
observed by and later discussed
by .
Often, although not always, the harmonic with the highest participation is
very similar in terms of frequency and damping to the one that would result from the invariant analysis of periodic systems based on the Coleman
transformation . As suggested
by , such a harmonic may be called the
principal one, while the others may be termed super-harmonics.
Furthermore, any one of these harmonics could resonate with external
excitations.
In order to understand how each harmonic appears in a specific output of the
system, the output-specific participation factor can be defined. To
this end, consider an output of the autonomous
system Eq. (),
y(t)=C(t)Φ(t,τ)x(τ)=Φy(t,τ)x(τ).
Inserting Eq. ()
into Eq. () the following is derived:
Φy(t,τ)=∑j=1NsC(t)ψj(t)Lj(τ)Teη^j(t-τ).
Exploiting now the periodicity of the product
C(t)ψj(t),
Eq. () can be rearranged as
Φy(t,τ)=∑j=1Ns∑n=-∞+∞cjne(η^j+ınΩ)(t-τ)Lj(τ)TeınΩτ,
where cjn is the harmonics of the Fourier expansion of
C(t)ψj(t). The output-specific
participation factor can finally be defined as
ϕjyn=cjn∑ncjn.
The harmonic transfer function and the harmonic frequency response function
The forced response of system Eq. (), called yF(t), can be computed as
yF(t)=∫0th(t,σ)u(σ)dσ=∫0t(C(t)Φ(t,σ)B(σ)+D(σ)δ(t-σ))u(σ)dσ,
where
h(t,τ)=C(t)Φ(t,τ)B(τ)+D(τ)δ(t-τ)
is the impulse response. From Eq. (), it
appears that the periodicity of C(t), B(t), and
Φ(t,τ) results in an input–output behavior of an LTP system that is far from being describable as an LTI-like one. In particular, it can be
shown that an LTP system subjected to an input at a given frequency may
respond at an infinite number of frequencies, which in addition to the input
frequency itself include also the integer multiples of the system
frequency . This is also the
reason why any output of a wind turbine subjected to a constant-in-time wind
(i.e., at the zero frequency) is characterized by frequencies at the multiples
of the rotor speed (i.e., 0×Rev, 1×Rev, 2×Rev, …).
In the frequency domain, the input–output relation can be expressed by means
of the HTF cf., which can
be interpreted as the extension to periodic systems of the standard
time-invariant transfer function. To this end, the so-called exponentially modulated periodic (EMP) signal is defined as
v(t)=∑k∈Zvke(s+ıkΩ)t,
where s∈C. According to definition
Eq. (), any vk can be also
viewed as the Laplace transformation of v(t) evaluated at s+ıkΩ as
vk(s)=∫-∞∞v(t)e-(s+ıkΩ)tdt.
It can be shown that a periodic system subjected to an EMP admits an EMP
regime and that in such a regime its states are EMP
signals. In order to exploit this property, one has first to define two doubly
infinite-dimensional vectors containing, respectively, the EMP harmonics u(t) and
y(t), as
Y(s)=⋯y-1(s)y0(s)y1(s)⋯T,U(s)=⋯u-1(s)u0(s)u1(s)⋯T.
Next, the doubly-infinite Toeplitz matrices A,
B, C, and
D, containing the Fourier expansions Ak,
Bk, Ck, and Dk of the
corresponding system matrices, are defined as
A=⋱⋮⋮⋮⋰⋯A0A-1A-2⋯⋯A1A0A-1⋯⋯A2A1A0⋯⋰⋮⋮⋮⋱,
and similarly for the B, C, and
D matrices. Finally, by inserting the EMP expansions of y
and u and the Fourier expansions of the system matrices into
Eq. (), summing up all terms at the same frequency, the
input–output relationship is derived as
Y(s)=G(s)U(s),
where the HTF is defined as
G(s)=C(sI-(A-N))-1B+D,
with N=blkdiag{ıkΩI,k∈Z} and I and
I being identity matrices of suitable dimensions.
The HTF can also be represented by means of the impulse response of the
system . From
Eq. (), it is easily verified that function
h(t,t-r) for a fixed time lag r is periodic and, consequently, that it
can be expanded in a Fourier series as
h(t,τ)=∑k=-∞∞hk(t-τ)eıkΩt.
The output equation can then be written according to the following
convolution:
y(t)=∑k=-∞∞∫0thk(t-τ)eıkΩ(t-τ)u(τ)eıkΩτdτ,
which leads to the input–output relation in the Laplace domain:
Y(s)=∑k=-∞∞Hk(s-ıkΩ)U(s-ıkω),
where Y(s), U(s), and Hk(s) are, respectively, the Laplace transforms of
y, u, and hk.
Equation () can be
evaluated for each element of the EMP output signal
Y by substituting the complex number s with the
exponentially modulated periodic one s+ıkΩ with
k∈Z, leading to the following relationship:
Y(s+ıkΩ)=∑n=-∞Hk-n(s+ınΩ)U(s+ınΩ).
Consequently, since Y(s+ıkΩ)=yk(s) and U(s+ıkΩ)=uk(s)
because of Eq. (), the HTF can be written as
G(s)=⋱⋮⋮⋮⋰⋯H0(s-ıΩ)H-1(s)H-2(s+ıΩ)⋯⋯H1(s-ıΩ)H0(s)H-1(s+ıΩ)⋯⋯H2(s-ıΩ)H1(s)H0(s+ıΩ)⋯⋰⋮⋮⋮⋱.
Inserting Eq. () into Eq. (), one
can derive the following expression
h(t,τ)=∑n=-∞∞∑j=1Ns∑m=-∞∞cjneηj+ınΩt-τljmeı(n+m)Ωτ+∑k=-∞∞dkeıkΩtδt-τ,
where the product LjT(τ)B(τ) and
D(τ) have been expanded in Fourier series, ljm
and dk being the related amplitudes. After some manipulations see
also, the Laplace transformation of
hk(t-τ)e-ınΩ(t-τ) can be finally written as
Hk(s+ınΩ)=∑j=1Ns∑m=-∞∞cjk+mlj-ms-(ηj+ı(m-n)Ω)+dk.
Consider now the row index ℓ∈Z and the column index r∈Z
of the HTF, defined such that the element with ℓ=r=0 (noted as G0,0)
corresponds to the median element H0(s) and the element with ℓ=r=-1 (noted
as G-1,-1) to H0(s-iΩ). Hence, according to such definitions and thanks to
Eq. (),
the following holds:
Gℓ,r(s)=Hℓ-r(s+ırΩ)=∑j=1Ns∑w=-∞∞cjℓ+wlj-r-ws-(ηj+ıwΩ)+dℓ-r.
Consequently, the HTF can be computed as
G(s)=∑j=1Ns∑w=-∞∞C‾j,wB‾j,wTs-(ηj+ıwΩ)+D‾,
where
C‾j,r=⋯cj-1+wcjwcj1+w⋯T,B‾j,m=⋯lj1-wlj-wlj-1-w⋯T,
and D‾=D.
From a practical standpoint, the use of the harmonic input–output relation
expressed by the HTF implies that one has to consider a truncated finite
dimensional approximation of G(s), which corresponds
to the use of truncated versions of the EMP input and output signals. The
convergence of truncated HTFs has been discussed
in .
The discrete-time case
In this section the stability analysis of periodic discrete-time systems is
briefly reviewed. For a more comprehensive treatment, the reader is referred
to and .
The autonomous dynamic equation of a generic LTP system in discrete time and
its initial conditions are
x(k+1)=A(k)x(k),x(0)=x0,
where k is a generic time instant and A(k) is a periodic matrix
of period K such that A(k+K)=A(k),∀k.
Similarly, the transition matrix obeys the following equation with its
initial conditions:
Φ(k+1,κ)=A(k)Φ(k,κ),Φ(κ,κ)=I.
In this work we consider only reversible systems, i.e., those for which
(Φ(k,κ))≠0,∀(k,κ).
For reversible discrete-time systems, the state transition matrix
Φ(k,κ) can be decomposed into periodic and contractive
parts as
Φ(k,κ)=P(k)R(k-κ)P(κ)-1,
where P(k) is periodic and R is constant. Here again, the
system is stable if the characteristic multipliers θj, i.e., the
eigenvalues of the monodromy matrix
Ψ(κ)=P(κ)R(K)P(κ)-1,
belong to the open unit disk in the complex plane. The relationship between
characteristic multipliers and characteristic exponents is
θj=ηjK.
In the discrete-time case, the apparent multiplicity of the characteristic
exponents manifests itself as a phase indetermination since
ηj=θjKexpı∠(θj)+2ℓπK,
where ℓ=0,…,K-1 is an arbitrary integer. As in the
continuous-time case, this does not in reality generate any inconsistency as
frequencies, damping, and participation factors of the various harmonics are
unaffected by this apparent arbitrariness.
Following the same approach of the continuous-time case, the transition
matrix can be rewritten as
Φ(k,κ)=∑j=1Nsψj(k)Lj(κ)Teη^j(k-κ),
where Ξ(k)=P(k)V and
ψj(k)=colj(Ξ(k)),Lj(κ)T=rowj(Ξ-1(κ)).
After having expanded ψj(k) in Fourier series, one gets
Φ(k,κ)=∑j=1Ns∑n=0K-1ψjnLj(κ)Tη^jexp(ı(∠(η^j)+n2πK))k-κ,
where ψjn is now the amplitudes of the harmonics of the
Fourier expansion of ψj(k). Coherently, the multiplication of
Eq. () with C(k) leads to
Φy(k,κ)=∑j=1Ns∑n=0K-1cjnLj(κ)Tη^jexp(ı(∠(η^j)+n2πK))k-κ,cjn being the harmonics of the Fourier expansion of
C(k)ψj(k). This shows that the jth mode is
characterized by K exponents with the same modulus and different phases. Each
exponent can be transformed into the continuous one using the following
expression cf.:
ηjc=1Δtlnηjd,
where Δt is the sampling time and subscripts (⋅)c and (⋅)d
refer, respectively, to the continuous- and discrete-time cases. Once the
continuous-time exponents are computed, frequencies, damping, and participation
factors can be readily obtained as in the continuous-time case.
Derivation of the equations of motion for a wind turbine analytical model
The simplified upwind horizontal-axis wind turbine model used in this work, depicted
in Fig. , considers the coupled motion of tower
and blades. The tower fore–aft and side–side flexibility are rendered by two
equivalent linear springs and dampers. Each blade is modeled as two rigid bodies
connected to each other by means of two equivalent revolute joints, which allow, respectively, the blade flap and edgewise rotations. The inner part of the blade is
rigidly connected to the hub. Each joint is associated with a rotational spring and a
rotational damper. The inertial and structural characteristics of each element are
chosen so as to match the first tower fore–aft and side–side mode and the first
blade flap-wise and edgewise modes in vacuo, computed using a high-fidelity
multi-body model of the wind turbine.
Sketch of the wind turbine
analytical model. Only one blade is shown in order to avoid cluttering the figure.
The reference frame used for the derivation of the equations of motions has its
origin located at the hub, the x axis directed downward, the z axis directed
from the tower to the rotor, and the y axis selected so as to form a right-handed
triad. To simplify the notation, in the following subscript k, denoting the blade
number, will be dropped together with the time dependence whenever possible.
The contribution of the two blade parts to the total energy can be developed
separately. Thus, let rU and rD indicate, respectively, the dimensional abscissa
along the inner and the movable parts of the blade. The position of a
generic blade point is given by
rU=rUcosψyH+rUsinψzH
when the point belongs to the inner part of the blade and by
rD=ecosψ+rDcosβcos(ψ+ζ)yH+esinψ+rDcosβsin(ψ+ζ)zH+rsinβ
when it belongs to the movable part. The kinetic energy of the whole rotor is
obtained by summing up the kinetic energy of the hub, TH, and of both the inner
and the movable parts of the kth blade, respectively, noted as TDk and
TUk, resulting in
T=TH+∑k=1BTUk+TDk,
where
TH=12mH(y˙H2+z˙H2)
and
TUk=12∫0eρ(r)r˙U(r)⋅r˙U(r)dr,TDk=12∫eRρ(r)r˙D(r)⋅r˙D(r)dr,ρ(r) being the blade mass per unit span.
All springs and gravity contribute to the potential energy of the system as
V=VyH+VzH+∑k=1BVβk+Vζk+VUk+VDk,
where the potential energy of the side–side and fore–aft springs is defined, respectively, as VyH=1/2KyyH2 and VzH=1/2KzzH2, while that
of the flap-wise and edgewise
springs is defined as Vβk=1/2Kββk2 and
Vζk=1/2Kζζk2. Finally the contribution of gravity can be
expressed as
VU=-mUgxCGU=-mUgrGUcosψ,VD=-mDgxCGD=-mDgrGDcosβcos(ψ+ζ).
The damping function D follows a rather similar procedure, where
D=DyH+DzH+∑k=1BDβk+Dζk.
The aerodynamic model is based on a linearized BEM approach with constant
aerodynamic properties along the blade, mostly taken
from , with the addition of the hub
velocity (y˙H,z˙H) to the inflow and cross-flow terms but
neglecting the yaw rate. Table
gives the meaning of some symbols used in the following equations.
Definitions of the
symbols in the aerodynamic loads.
The hub shear force in the fore–aft direction is
Sβaero=12γJDΩ2R{λ2+θp3-β˙/Ω3-sinψ[U¯0β2]-cosψ[U¯0(λ-β˙/Ω2-θp)+K1V¯03]}.
The hinge out-of-plane moment is
Mβaero=12γJDΩ2{λ3+θp4-β˙/Ω4-sinψ[U¯0β3]-cosψ[U¯0(λ2-β˙/Ω3+2θp3)+K1V¯04]}.
The hub shear force in the direction parallel to the chord of the blade, and
pointing towards the leading edge, is
Sζaero=12γJDΩ2R{λ(λ+θp2)-β˙Ω(λ+θp3)-cosψ[K1V¯0(λ+θp3)+U¯0θpλ-β˙Ω(23K1V¯0+U¯0θp2)]-sinψ[βU¯0(2λ+θp2-β˙Ω)]}.
The hinge moment in the edgewise direction is
Mζaero=12γJDΩ2{λ(λ2+θp3)-β˙Ω(23λ+θp4)-cosψ[K1V¯0(23λ+θp4)+U¯0θpλ2-β˙Ω(K1V¯02+U¯0θp3)]-sinψ[βU¯0(λ+θp3-23β˙Ω)]}.
This aerodynamic model assumes that the wind velocity varies linearly over the rotor
disc, and therefore it is not suited to simulate turbulent wind fields.
The virtual work of the aerodynamic forces and moments results in
δWaero=∑k=1BSζkaerocos(ψk+ζk)δyH+SβkaeroδzH+Mβkaeroδβk+Mζkaeroδζk.
The generalized forces follow directly from the previous expression.
Finally, the nonlinear Lagrangian equations of motion of the system are
JDβ¨+Cββ˙+Kββ=Mβaero-JD(Ω+ζ˙)2cosβsinβ-mDrGD(gcos(ψ+ζ)sinβ+eΩ2cosζsinβ-y¨Hsin(ψ+ζ)sinβ+z¨Hcosβ),JDcos2βζ¨+Cζζ˙+Kζζ=Mζaero+2JD(Ω+ζ˙)β˙cosβsinβ-mDrGDcosβ(gsin(ψ+ζ)+eΩ2sinζ+y¨Hcos(ψ+ζ)),(mH+B(mU+mD))z¨H+Czz˙H+KzzH=∑k=1B(Sβkaero-mDrGD(β¨kcosβk-β˙k2sinβ)),(mH+B(mU+mD))y¨H+Cyy˙H+KyyH=∑k=1B(Sζkaerocos(ψk+ζk)+mDrGD(Ω2cosβksin(ψk+ζk)+β˙k2cosβksin(ψk+ζk)+2β˙kζ˙ksinβkcos(ψk+ζk)+ζ˙k2cosβksin(ψk+ζk+2Ω(β˙ksinβkcos(ψk+ζk)+ζ˙kcosβksin(ψk+ζk))+β¨ksinβksin(ψk+ζk)-ζ¨kcosβkcos(ψk+ζk))).
All equations shown in this section and the system linearization were computed
analytically with Wolfram Mathematica®.
Nomenclature
A(q;k)Periodic autoregressive polynomialB(q;k)Periodic exogenous polynomialNaOrder of the autoregressive partNbOrder of the exogenous partNgOrder of the moving average partF(q;k)Shape filter polynomialG(q;k)Periodic moving average polynomialKDiscrete-time system periodTContinuous-time system periodJCost functionNTotal number of samples used for identificationNsNumber of statesP(p)Characteristic multipliers of the PARMAX predictor pNFaNumber of harmonics of the autoregressive coefficientsNFbNumber of harmonics of the exogenous coefficientsNFgNumber of harmonics of the moving average coefficientsCComplex number setZInteger number setBNumber of bladesY(s)Laplace transformation of the outputU(s)Laplace transformation of the inputHk(s)Laplace transformation of the kth harmonic of the impulse responsezMeasured outputz^Predicted outputqOne-step-ahead shift operatorkTime indexutTurbulent wind inputuMean wind speednOrder of the system, n=max(Na,Nb,Ng)ySystem outputeProcess noiseyk(ω)Fourier transformation of the kth shifted copy of the outputuk(ω)Fourier transformation of the kth shifted copy of the inputyk(s)Laplace transformation of the kth shifted copy of the outputuk(s)Laplace transformation of the kth shifted copy of the inputtTimeyFForced responseh(t,τ)Impulse responsehk(t)kth harmonic of the impulse responseıImaginary unitsLaplace variableA(t)State matrixB(t)Input matrix
E(t)Process noise input matrixC(t)Output matrixD(t)Direct transition matrixF(t)Measurement noise matrixN(t)State matrix of the PARMAX predictorGHarmonic transfer functionUExponentially modulated periodic expansion of the inputYExponentially modulated periodic expansion of the outputY(ω)Vector of Fourier transformations of all shifted copies of the outputU(ω)Vector of Fourier transformations of all shifted copies of the inputG(ω)Harmonic frequency response functionSYY(ω)Harmonic power spectrum of the outputSUU(ω)Harmonic power spectrum of the inputΦState transition matrixIIdentity matrixΨMonodromy matrixSEigenvector matrix of the monodromy matrixQ(t)Floquet–Lyapunov transformationRFloquet factorVEigenvector matrix of the Floquet factorP(t)Periodic eigenvectorxState vectorpUnknown vector of model coefficientsx0Initial state vectorx̃τState vector sampled at every periodsjjth eigenvector of the monodromy matrixzFloquet–Lyapunov transformed state vectorΩRotor speedδutTurbulent perturbation of the windαiith coefficient of canonical system matrix Aβiith coefficient of canonical input matrix Bγiith coefficient of canonical process noise input matrix EψAzimuth angleεPrediction errorωGeneric frequencyηjjth characteristic exponentτTime lagθjjth characteristic multiplierϕjnParticipation factor of the nth harmonic of the jth modeϕjynOutput-specific participation factor of the nth harmonic of the jth mode(⋅)*Complex conjugate(⋅)sSine amplitude(⋅)cCosine amplitude(⋅)TTranspose(⋅)HComplex conjugate transpose(⋅)˙Time derivative
IPCIndividual pitch controlHTFHarmonic transfer functionHFRFHarmonic frequency response functionMBCMulti-blade coordinateLTILinear time-invariantLTPLinear time periodicARMAXAutoregressive moving average with exogenous inputPARMAXPeriodic ARMAXPOMAPeriodic operational modal analysisPEMPrediction error methodEMPExponentially modulated periodicSISOSingle-input single-outputPSDPower spectral densityHPSDHarmonic PSDBEMBlade element momentumSSState-spaceCGCenter of gravity
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