Downloaded 28 times
![List of Figures
2.1 Out-of-plane and in-plane with respect to the pitch angle. Source [1] . . . 8
2.2 Example of standstill asymmetric modes and Campbell diagram. Source [2] 9
2.3 Terminology applied to wind turbines DOF. Source [2] . . . . . . . . . . . 11
2.4 Asymmetric rotor motion indicating reaction forces. Source [2] . . . . . . 13
2.5 Example of signal divided into m segments with 50% overlap. Source [3] . 17
2.6 Beam element model in HAWCStab2 (only one blade). Source [4] . . . . . 20
3.1 Summary of the methodology . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Aerial view of V27 site location . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Sensors location and orientation. Source: [5] . . . . . . . . . . . . . . . . . 26
3.4 Nacelle sensors in the V27. Source [6] . . . . . . . . . . . . . . . . . . . . 27
3.5 View of V27 and met mast from neighbouring Nordtank . . . . . . . . . . 28
3.6 V27 and met combined data . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.7 Wind profile for low and high rotor speed data sets . . . . . . . . . . . . . 32
3.8 Mode shapes no. 1 to 8 in HAWC2 . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Frequencies vs. wind speed . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.10 Damping ratios vs. wind speed. All modes . . . . . . . . . . . . . . . . . . 39
3.11 Damping ratios vs. wind speed. Low damped modes . . . . . . . . . . . . 39
3.12 Operational data assumed in HS2 . . . . . . . . . . . . . . . . . . . . . . . 40
4.1 Tri-axial accelerometer coordinates. Representation of the nacelle view from the top 44
4.2 PSD nacelle x-direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
ix](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-13-2048.jpg)
![x LIST OF FIGURES
4.3 PSD nacelle y-direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 PSD blades signals V27 outer section, low rotor speed . . . . . . . . . . . 48
4.5 PSD blades signals H2 outer section, low rotor speed . . . . . . . . . . . . 48
4.6 PSD blades signals V27 inner section, low rotor speed . . . . . . . . . . . 49
4.7 PSD blades signals H2 inner section, low rotor speed . . . . . . . . . . . . 50
4.8 PSD blades signals V27 outer section, high rotor speed . . . . . . . . . . . 50
4.9 PSD blades signals H2 outer section, high rotor speed . . . . . . . . . . . 51
4.10 PSD blades signals V27 inner section, high rotor speed . . . . . . . . . . . 51
4.11 PSD blades signals H2 inner section, high rotor speed . . . . . . . . . . . 52
4.12 SVD applied on edgewise signals at low rotor speed . . . . . . . . . . . . . 53
4.13 SVD applied on flapwise signals at low rotor speed . . . . . . . . . . . . . 53
4.14 SVD applied on edgewise signals at high rotor speed . . . . . . . . . . . . 55
4.15 SVD applied on flapwise signals at high rotor speed . . . . . . . . . . . . 55
4.16 MBC applied on edgewise signals at low rotor speed . . . . . . . . . . . . 56
4.17 MBC applied on flapwise signals at low rotor speed . . . . . . . . . . . . . 57
4.18 MBC applied on edgewise signals at high rotor speed . . . . . . . . . . . . 58
4.19 MBC applied on flapwise signals at high rotor speed . . . . . . . . . . . . 59
4.20 TSA procedure. Source [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.1 Simple geometrical representation of the V27 . . . . . . . . . . . . . . . . 62
5.2 Tower modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Tower modes stability diagram at low rotor speed . . . . . . . . . . . . . . 63
5.4 Tower modes stability diagram at high rotor speed . . . . . . . . . . . . . 64
5.5 Geometrical representation of the MBC coordinates . . . . . . . . . . . . 66
5.6 Edgewise modes stability diagram at low rotor speed . . . . . . . . . . . . 67
5.7 Phase angle difference between asymmetric modes . . . . . . . . . . . . . 67
5.8 Edgewise modes stability diagram at high rotor speed . . . . . . . . . . . 68
5.9 Smooth anisotropic effect on the model . . . . . . . . . . . . . . . . . . . 70
5.10 Flapwise modes stability diagram at low rotor speed . . . . . . . . . . . . 72
5.11 Flapwise modes filtered stability diagram at low rotor speed . . . . . . . . 72
5.12 Flapwise modes stability diagram at high rotor speed . . . . . . . . . . . . 73
5.13 Impulse loading spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.14 Spectra of the random function . . . . . . . . . . . . . . . . . . . . . . . . 75
5.15 SVD on the blade signals for impulse and random cases . . . . . . . . . . 76](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-14-2048.jpg)
![2 Introduction
as input excitation and reading output measurements at different locations of the struc-
ture. Based on the input/output transfer function, the dynamic characteristics of the
structure can be identified. This methodology can be used for relatively small structures
subjected to a known forcing, but the challenge arises when huge structures, subjected
to stochastic loading distributed over a substantial part of the structure, are brought to.
When structures are large enough, as is the case with current wind turbines, introducing
an artificial excitation could mean expensive costs associated to the test implementation
if possible at all. In the 1990’s, James et al. [8] laid down the foundations of Opera-
tional Modal Analysis (OMA), also known as Output-Only Modal Analysis or Ambient
Modal Identification. This method refers to techniques where the excitation forces are not
measured, and modal parameters are estimated only from the output responses. It has
attracted much attention in the civil, mechanical and aerospace engineering because of its
advantages for identifying modal properties of a structure solely based on its measured
responses during operating conditions. Main advantages are:
Unlike the classical modal testing approach, it does not need to measure the exci-
tation force;
The measured response is representative of the real operating conditions of the
structure;
The operating structure does not need to be interrupted to perform the test;
The OMA approach can deal with inevitable unknown (stochastic) external loading;
The method can be also used for other purposes such as damage detection and
health monitoring of structures.
Due to these reasons, one may think that OMA is an excellent technique to identify
modal parameters. However, some requirements must be fulfilled:
the structure is linear;
the structure is time invariant;
the operational excitation forces must in turn fulfil the following requisites
a) they have broadband frequency spectra;
b) they are uncorrelated;
c) they are distributed over the entire structure.
Despite that an operating wind turbine may violate all of these, some approximations
must be made in order to alleviate these requirements and find an scenario where the
method could be applied with a degree of confidence.
The linearisation of the system is achieved by assuming equilibrium at operating points,
e.g. pitch angle, torque or rotational speed at a particular wind speed. Wind turbines
are not time invariant systems: they are composed of various substructures that moves
one with respect to the other - for instance, the pitching of the blades or the yawing of](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-20-2048.jpg)
![1.2 Damping 3
the nacelle. Selecting time stamps where there is none or minor activity of these motions
is a manner to partially cope with this issue. The major drawback, however, is the
rotor rotation, which makes the mass, stiffness and damping matrices in the equations
of motion to be time dependent. One way to overcome this matter is by means of the
so-called Coleman transformation method [9, 10], which converts the time-variant system
into time-invariant under particular conditions. When it comes to the excitation forces,
the turbulent wind seems to suit the third OMA premise very well, but the rotation of the
rotor changes the aerodynamic forces and the broadband spectra of the turbulent wind
displays peaks at the rotor harmonics. This effect is pointed out by Tcherniak et al. in
[11].
From the different algorithms OMA embraces, the Stochastic Subspace Identification
(SSI) technique, described by Overschee and De Moor [12], appeals most to this work
based on the success of previous research studies as pointed out in the coming Section 1.3.
In the present work, a commercial software package from Br¨uel & Kjær is used that has
already implemented this method.
1.2 Damping
Damping estimation appears as of great importance among the modal properties one
may want to estimate using OMA. Its definition may be stated as the dissipation of
energy from a structure that is vibrating. The accurate prediction of damping has a large
effect on the lifetime and dynamic response of the system. Moreover, in wind turbines
damping consists in structural and aerodynamic contributions (from the aerodynamic
forces). Damping also plays a relevant role in the stability of wind turbines. In this sense,
Hansen [2] commented on the aeroelastic instabilities that have occurred and may still
occur for commercial wind turbines.
Modal parameters of the vibration modes of operating wind turbines can be computed
with or without accounting for the aerodynamic forces, since aerodynamic damping is a
main source of uncertainty. The unloaded turbine defines the basis of its modal dynamics,
with the structural modes as the foundation of the aeroelastic dynamic behaviour. When
the turbine is loaded, the vibration modes define its aeroelastic stability properties by the
damping of these modes. However, damping is always difficult to estimate with accuracy,
and even more when aerodynamic forces are included. In other words, it poses a challenge
to differentiate both damping contributions (structural and aerodynamic) from the total
damping.
In the present work, the total damping in the rotor plane and out of the rotor plane is
investigated, i.e. lateral and longitudinal to the wind direction respectfully. The first case
includes tower side-to-side mode and rotor edgewise modes, and the second case refers
to tower fore-aft mode and rotor flapwise modes - meaning lowly pitched blades when
referring to the rotor modes. While there is some optimism in identifying reasonable
damping ratios in the in-plane modes from previous research studies [6], it is foreseen
a demanding task to identify the out of plane modes due to the high contribution of
aerodynamic damping in that direction.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-21-2048.jpg)
![4 Introduction
1.3 State of the Art
Being a hot topic nowadays, several research studies have been conducted previously
about OMA applied to wind turbines, of which only a few have used the SSI technique,
and even less of them, have identified modal parameters from real measurements. Below
is listed research studies which match these conditions under particular cases at the time
of writing this thesis:
Hansen et al. [13] presented an estimation of aeroelastic damping on a NM80 2.75
MW wind turbine operating prototype using strain gauges. It was concluded that
the SSI method can handle the deterministic excitation from wind, and the modal
frequencies and damping of the first tower and first edgewise whirling modes were
extracted.
Tcherniak et al. [1] applied the technique to an operating ECO 100 Alstom wind
turbine with accelerometers in the nacelle and tower, reporting that is possible to
identify some rotor modes using only these signals.
Andersen and Rosenow [14] investigated the dynamics of a parked Fuhrl¨ander AG
2.5 MW with sensors in the main frame. It was also reported that some rotor modes
can be identified only using sensors in the nacelle.
Tcherniak and Larsen [5] described the technical challenges regarding blade instru-
mentation and data acquisition, the processing of data to convert the system to
time-invariant and assessed preliminary results on three cases: parked, idle and
normal operation.
Yang et al.[6] compared the Coleman transformation followed by the SSI technique
with the harmonic power spectrum (HPS), method based on an extension of modal
analysis to linear time periodic systems, using blades accelerations of an operating
wind turbine. The paper focuses on a comparison of results for the first edgewise
modes, concluding that the Coleman transformation could lead to erroneous results
due to rotor anisotropy [5].
Van Der Valk and Ogno [15] identified modal parameters in an idling Siemens SWT-
3.6 MW offshore wind turbine with several strain gauges and one accelerometer. The
four first global eigenfrequencies were identified, but also concluded that the best
results came from the accelerometer located in the nacelle.
Marinone et al. [16] compared experimental modal results of two parked Vestas V27
wind turbines, where OMA is only applied to one of them. The modes found using
OMA were highly correlated with the experimental modal results.
Apparently, there are only two research studies gathering information both from the
sensors mounted in the non-rotating frame (tower) and the rotating frame (blades). These
works shared the same data for similar purposes, with a special emphasis on the edgewise
modes.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-22-2048.jpg)
![1.4 Motivation of the Work and Goals 5
1.4 Motivation of the Work and Goals
This thesis may be seen as natural continuation of the works covered in [5, 6]. Based on
the same real measurements and methods (although not HPS), it is intended to go one
step ahead and identify not only the first edgewise modes, but also the first tower and
flapwise modes.
Some questions, nonetheless, has to be addressed based on the previous research studies:
1) Is it possible to apply the Coleman transformation to anisotropic rotors successfully?
2) Can all rotor modes be identified by only using nacelle sensors?
3) To what extent will the aerodynamic loads influenced by the rotation of the rotor
allow to identify modal parameters?
4) And in connection to the previous question, is it possible to identify the flapwise
modes, with a certain confidence?
There are two main reasons motivating the work in this thesis. To the author’s knowl-
edge, there is not yet presented any research of modal analysis on a full-scale operating
wind turbine combining both tower and blade signals for the identification of tower and
rotor modes. This thesis attempts to identify such modes with the limitation that tor-
sional modes are excluded. This fact leads to a very attractive topic to be investigated
and is the major motivation for this thesis.
The second motivation relies on the comparison of the OMA results from the V27 with
numerical models. A model will be implemented in HAWC2 attempting to simulate the
dynamic characteristics of the V27. Besides, a theoretical prediction of modal parameters
is conducted using HAWCStab2, which is an aeroelastic stability tool that includes the
Coleman transformation and solves the corresponding eigenvalue problem under symmet-
ric rotor assumption. If the resulting modal parameters are similar among them, it may
suggest these tools as a helpful support for OMA in wind turbines.
The three-fold comparison may help to understand the challenges that modal parame-
ters estimation poses when dealing with real life operating wind turbines.
Despite the Vestas V27 model is an old-fashioned wind turbine, it is physically similar
to modern wind turbines in the sense that it features pitch- (limited) and yaw-control.
All the attention is paid on the frequency range (0 to 5 Hz) from the accelerations output
spectra that, not only embraces all the modes under investigation, but also correlates
with the first modes encountered in modern wind turbines (typically within the range 0
to 2-3 Hz).
1.5 Thesis Structure
This thesis is the result of the work done over the last five months. The introduction
to the topic and the related theory are covered in the following chapters, reflecting the](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-23-2048.jpg)
![Chapter 2
Background Knowledge
This chapter introduces the reader to the fundamental concepts and methods dealt within
this thesis. A review on modal dynamics of wind turbines is presented first. Here, the
Coleman transformation is introduced. Then, the methods to be applied on the experi-
mental and simulated data are presented. Lastly, the tools employed for the implementa-
tion of the model and the modal parameters prediction under symmetric rotor assumption
are described.
This chapter will be useful to review while going through Chapter 4 and Chapter 5
where the main concepts are applied to real and simulated data.
2.1 Wind Turbine Modal Dynamics
The typical first global modes of a wind turbine are presented in this section. This
information is relevant to following future chapters, when the modal frequencies and
damping ratios for each mode are discussed.
First part describes wind turbine mode shapes inspired from Hansen [2]. Next, the
Coleman transformation is explained based on [17, 18], and its premises are discussed for
the application on this work.
2.1.1 Dynamics of the Main Components
The dynamics of wind turbines are governed by three main substructures: tower, drive-
train and rotor.
Tower
The tower deflects longitudinally and laterally with respect to the wind direction. These
two bending modes also interact due to the gyroscopic coupling of towertop and rotor,
7](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-25-2048.jpg)
![8 Background Knowledge
that may lead to elliptical motion of the tower. Also, the tower torsion is noticed in the
yaw motion of nacelle and rotor.
Drivetrain
The drivetrain rotation makes the blades to rotate around its axis. Taking one blade
as the reference, the azimuth angle is the particular angular position of that blade at
that precise instant. The drivetrain consists of a main shaft, a gearbox and a generator,
and introduces a torsional mode between rotor hub and generator coupled with blade
simultaneous edgewise bending (or flapwise bending dependent on the pitch angle).
Rotor
A single blade cantilevered at the hub has three family modes: flapwise, edgewise and
torsion; where the modes of a blade can be a mixture of these families. The term rotor
is referred as the ensemble of blades attached to the same hub. The dynamics of blades
are normally expressed in the rotating frame, however, the excitations on the rotor occur
in the non-rotating frame, in the same way that the tower-nacelle substructures feel the
dynamics of the rotor and not the individual blades effect. Hence, the rotor dynamics are
considered as a whole and all modes of the individual blades contribute to the dynamics
of the entire wind turbine.
The dynamics of a 3-bladed rotor are described depending on the flapwise, edgewise and
torsional modes of the blades, and can be differentiated by 3 rotor modes: one symmetric
and two asymmetric. The symmetric mode refers to the case where all three blades deflect
symmetrically (flapwise, edgewise or torsion). The asymmetric modes refer to the case
where two blades deflect in the same direction and one opposite to them. Besides, the
flapwise and edgewise modes are dependent of the pitch angle. Figure 2.1 is provided to
support the following explanation.
Figure 2.1: Out-of-plane and in-plane with respect to the pitch angle. Source [1]
For instance, if the pitch angle is set to minimum, e.g. 0◦ (typically in operating
conditions, before pitch controller action), the blade flapwise motion is out of the rotor
plane, while the edgewise mode is in the rotor plane. If the pitch angle is maximum
(typically in parked conditions), the motion of the flapwise mode is rotor in-plane, while
the edgewise is out of the rotor plane.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-26-2048.jpg)
![2.1 Wind Turbine Modal Dynamics 9
2.1.2 Whirling Modes
At standstill, the system is time-invariant and all OMA assumptions are fulfilled, for which
one can expect reliable results. When the wind turbine is under operation the system
is time-variant and therefore modal analysis cannot be applied. The so-called Coleman
transformation is used to transform the rotating blade coordinates to the non-rotating
frame, and under certain assumptions, the system can be treated as time-invariant.
Figure 2.2 displays the nomenclature of the asymmetric modes at standstill conditions
and what happens when the rotor rotates. It can be seen that the torsional modes are
not displayed in the figure since they are typically at higher frequencies. In this work,
the torsional modes are not considered, since it is assumed they are out of the frequency
range of interest. Therefore, it is natural to explain from now on only the flapwise and
edgewise modes.
Figure 2.2: Example of standstill asymmetric modes and Campbell diagram. Source [2]
The global mode shapes basically consists of a combination of tower and blades bending
modes. The order of the very first modes is often the same: first tower modes, then, first
flapwise and edgewise modes. The first longitudinal tower mode may have slightly lower
frequency than the lateral mode due to larger inertia from the tilting of the rotor. There
are also modes coupling between tower and rotor. The longitudinal and lateral tower
modes are translated to tilting and rolling of the nacelle, respectively, due to blades
flapwise bending and the gyroscopic effect of the drivetrain plus the blades edgewise
bending.
As Hansen describes in greater depth [2], the flapwise and edgewise modes form pairs
and their natural frequencies are slightly different at standstill. This occurs because of
the different vertical and horizontal flexibilities acting on the rotor support. The yaw
component has lower frequency than the tilt component since the tower is stiffer in tilt
than in yaw. Similarly, the flapwise modes have slightly lower frequency that the edgewise](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-27-2048.jpg)
![10 Background Knowledge
modes since these latter are stiffer. The sequence of the edgewise modes is given by the
vertical and horizontal stiffness of the rotor support.
When the rotor is rotating, these pairs of frequencies split in the ground-fixed frame by
±Ω where the center is approx. the blade frequency in the considered deflection shape.
The splitting occurs due to the inverse transformation of the frequencies in the ground-
fixed frame to the rotating frame. This means that an observer in the ground-fixed frame,
e.g. towertop, will measure the same frequency ω for each of the split components while
an observer in a blade will measure ω − Ω or ω + Ω depending on the component.
This phenomena is directly linked to the Coleman transformation in Section 2.1.4,
where the equations may clarify the aforementioned explanation.
2.1.3 Terminology
To improve readability, a brief description of each mode is shown using typical nomencla-
ture for the modes under discussion in this thesis. Note that the asymmetric components
of modes denote two different nomenclatures, depending on the state of the turbine, if at
standstill (first term on brackets/abbr.) or operating (second term on brackets/abbr.):
Tower longitudinal bending (tower fore-aft): tower moves fore and aft with respect
to the wind direction. Abbr.: TFA
Tower lateral bending (tower side-side): tower moves side to side with respect to
the wind direction. Abbr.: TSS
Asymmetric flapwise (yaw/backward whirling): nacelle and rotor move around the
yaw axis. Abbr.: YF/BWF
Asymmetric flapwise (tilt/forward whirling): rotor and nacelle tilt up and down.
Abbr.: TF/FWF
Symmetric flapwise (collective): blades deflect same direction simultaneously. Abbr.:
CF
Asymmetric edgewise (vertical/backward whirling): asymmetric blades regressive
deflection. Abbr.: VE/BWE
Asymmetric edgewise (horizontal/forward whirling): asymmetric blades progressive
deflection. Abbr.: HE/FWE
Drivetrain torsion (collective edgewise): blades deflect edgewise symmetrical and
simultaneously. Abbr.: DT/CE
A graphical representation of the degrees of freedom (DOF) defined by Hansen [2] is
found in Figure 2.3 to back up the previous nomenclature.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-28-2048.jpg)
![2.1 Wind Turbine Modal Dynamics 11
Figure 2.3: Terminology applied to wind turbines DOF. Source [2]
2.1.4 Coleman Transformation
A wind turbine is a time-variant system. The tower and nacelle are subjected to the
ground-fixed frame of reference (non-rotating) while the blades are rotating, and thus,
blade dynamic characteristics are azimuthally time dependent. The interaction of the
blades rotating frame with the tower-nacelle non-rotating frame introduces periodicity in
the equations of motion of the wind turbine. Also, the rotor responds as a whole to the
excitation on the non-rotating frame. Thus, it seems appropriate to work with DOF that
reflect this behaviour, in order to simplify both the analysis and the understanding of the
dynamic behaviour of the wind turbine [9].
In addition, conventional modal analysis is only applicable to time-invariant systems,
ending up with the necessity of converting the wind turbine from time-variant to time-
invariant system.
The Coleman transformation, also known as multiblade coordinate transformation
(MBC) for bladed rotors [9, 10], is a method to describe the motion of the individual
blade coordinates in the non-rotating frame. Provided that all coordinates of the system
are defined in the same frame of reference, this transformation yields the system to be
time-invariant as all periodicity is removed, thus in turn allowing the application of modal
analysis.
However, the fundamental assumption of this method is that the rotor is isotropic, i.e.
all blades identical and mounted symmetrically on the hub.
This section explains the method for transforming the blade coordinates to the non-
rotating reference frame, and ends up with a discussion on the benefits and the different
viewpoints from previous works.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-29-2048.jpg)
![12 Background Knowledge
Transformation of Blade Coordinates
For a 3-bladed rotor, with the blades equally spaced, MBC transformation is defined as :
a0 =
1
3
3
k=1
qk
a1 =
2
3
3
k=1
qk cos ψk
b1 =
2
3
3
k=1
qk sin ψk
(2.1)
where ψk = Ωt+ 2π
3 (k−1) is the azimuth angle of blade k = 1, 2, 3. The three multiblade
coordinates a0, a1 and b1 replace the blade coordinates q1, q2 and q3.
The inverse transformation back to the blade coordinates is also possible by the formu-
lation:
qk = a0 + a1 cos ψk + b1 sin ψk (2.2)
Equation (2.2) permits the change from multiblade coordinates to blade coordinates.
The transformed blade coordinates in the non-rotating frame can be categorised in one
symmetric (collective) and two asymmetric components, as explained in Section 2.1.
As an example, let us assume a flapwise deflection (in the wind direction) of the blade
coordinates qk. The multiblade coordinate a0 describes a collective flapwise deflection (all
blades deflect symmetrically), while a1 and b1 describes tilt and yaw motions respectively.
If it were an edgewise deflection, a1 and b1 would describe horizontal and vertical motions
whereas a0 would describe a collective edgewise motion of the blades that may interact
with the drivetrain torsional mode. The different motions are represented in Figure 2.4
for ease of follow-up.
According to Hansen [2], the mode shape in MBC coordinates is defined as
a0 = A0 sin(ωt + φ0)
a1 = Aa sin(ωt + φa)
b1 = Ab sin(ωt + φb)
(2.3)
and plugging 2.3 into Equation (2.2) gives the mode shape in blade coordinates:
qk(t) = A0 sin(ωt + φ0)+
+ABW sin (ω + Ω)t +
2π
3
(k − 1) + φBW +
+AF W sin (ω − Ω)t −
2π
3
(k − 1) + φF W
(2.4)](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-30-2048.jpg)
![2.1 Wind Turbine Modal Dynamics 13
with the first term indicating the symmetric component, and the subscripts in the
second and third terms indicating the backward (BW) and forward (BW) whirling com-
ponents of the rotor blades amplitudes and phases.
Therefore, the modal response of blade k is composed of three components, recalling
Section 2.1. If the frequency is measured in the fixed reference system, the natural
frequency is ω for all three components. Whereas if the frequency is measured in the
rotating frame, the natural frequency is ω for the symmetric component and ω + Ω and
ω − Ω for the BW and FW components, respectively, i.e. the BW and FW components
are shifted ±Ω in the rotating frame. However, it must be pointed out that pure modes do
rarely exist. If the rotor is isotropic, an (averaged) spectra of the accelerations measured
on the blades should present the same magnitudes, and the phase of the signals equals to
±120◦, where all phases sum up to 0±360◦. Once the signals are transformed, the phase
difference between a1 and b1 is ±90◦, where a1 leads b1 in the case of FW, i.e. +90◦, and
a1 lags b1 in the case of BW, i.e. -90◦.
Figure 2.4: Asymmetric rotor motion indicating reaction forces. Source [2]
Discussion
The fundamental assumption of the MBC is that the rotor is isotropic. Nevertheless,
there are different criteria concerning the assumptions on which the method succeeds.
One the one hand, Hansen defines in [2, 4, 17] the following assumptions to correctly
apply the MBC:
1) the number of blades must be odd
2) the inflow to the rotor must be uniform
3) the rotor must be isotropic (blades are identical and symmetrically mounted)
In that case, the periodic terms in the aeroelastic equations of motion are removed and
the MBC converts the system to time-invariant.
On the other hand, Bir [18] does not deal with the number of blades nor the flow, but
states that all attempts at MBC thus far assumed constant rotor speed (which rarely is
the case) and identical blades (typically blades have no identical structural or dynamical
properties). Bir shows that the MBC can be applied to variable-speed turbines because
the varying rotor speed is only associated to the stiffness matrix [18], but also mentions
that:](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-31-2048.jpg)
![2.2 Methods 15
Formulation
To describe the PSD, it is necessary to introduce the definition of autocovariance, denoted
as:
RXX(τ) =< X(t)X(t + τ) > −µ2
X (2.5)
where X is the signal, t is the time, τ is the time lag, <> is the mean value operator
and µ is the mean of the signal. The autocovariance defines how much similar a signal
is with the time-shifted version of itself. This quality is useful to determine repeated
patterns or trends in a signal.
Similarly, the crosscovariance defines the similarity between two signals:
RXY (τ) =< X(t)Y (t + τ) > −µXµY (2.6)
The autocovariance, therefore, could be seen as a special case of crosscovariance.
Commonly, the autocorrelation function is the autocovariance of the signal divided by
the variance:
ρX(τ) =
RX(τ)
σ2
X
(2.7)
Often named as the autocorrelation coefficient. However, in signal analysis context, the
autocovariance without normalization is referred to autocorrelation. Similarly, from the
crosscovariance one can derive the crosscorrelation coefficient.
The relation between the PSD and the autocovariance is that they form Fourier trans-
form pairs. Hence, the Fourier transform of the autocovariance, as defined in Equa-
tion (2.5), is the PSD. Mathematically, the area under the PSD is equal to the variance
of the signal.
One may recall at this point that the variance is the square of the standard deviation
or averaged of the square differences to the mean. Besides, the covariance is just the
variance for two different variables, but generally, the covariance is defined as the variance
of any two time series separated by a time lag τ [19]. From this definition of covariance,
the previous descriptions of autocovariance and crosscovariance apply (Equations (2.5)
and (2.6)).
The forward Fourier transform of the covariance is the double-sided PSD, denoted as:
SXX(f) =
∞
−∞
RXX(τ)e−i2πfτ
dτ
SXY (f) =
∞
−∞
RXY (τ)e−i2πfτ
dτ
(2.8)](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-33-2048.jpg)
![16 Background Knowledge
where SXX is the autospectral power density (PSD), and SXY is the cross-spectral
power density (CSD), which describes how the autocovariance and crosscovariance are
distributed on different frequencies.
The double-sided term mentioned above refers to the property that half of the physical
frequency content appears at positive frequencies and half at negative frequencies. Hence,
a single-sided interpretation is required assuming the following properties:
GXX (f) = 2SXX(f) for f>0
GXX (0) = SXX(0)
GXY (f) = 2SXY (f) for f>0
GXY (0) = SXY (0)
Real measurements are not continuous in time but discrete values separated by a sam-
pling period ∆t that leads to a sampling frequency fs = 1/∆t. For a time series sampled
at fs the highest frequency to estimate the spectrum is the Nyquist frequency fN = fs/2.
The spectrum is hence estimated at the frequencies:
fl = fs
l
N
=
l
N∆t
(2.9)
with l = 1, ...N/2, as the frequency lines or bins and N the length of the parts in which
time series are split.
Spectral Analysis
If the spectrum is estimated at the frequencies fl, and each frequency line is a frequency
(and not a range of frequencies), there are gaps between each frequency lines. These gaps
are referred to the so-called spectral leakage phenomena, meaning that the energy leaks
between the frequency lines.
In order to correct this phenomena the spectral estimation is performed using the
modified averaged periodogram method (Welch’s technique), with an overlap in this work
of 67% and a Hanning [20] weighting function, likewise the OMA Type 7760 software
package from Br¨uel & Kjær [21] used in this work. This ensures that all data are equally
weighted in the averaging process, minimizing leakage and picket fence effects [22]. A
rough description of the Welch method [23] is that the signal is divided into m segments,
where each segment overlaps the next one.
The Hanning window is then applied to each segment weighing the mid point most and
decaying exponentially towards the sides, and the modified periodogram - an estimate
of the spectral density signal - is averaged for each frequency line. Then, the Welch’s
method is computed. The Welch’s method is fully described by Brandt in [3].
Figure 2.5 shows the segment based averaging process.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-34-2048.jpg)
![2.2 Methods 17
Figure 2.5: Example of signal divided into m segments with 50% overlap. Source [3]
Spectral Density Matrices
Once the PSD and CSD of the signals time histories are estimated, the spectral density
matrices can be formed. The size of the matrices is n x n, with n being the number of
selected signals.
The diagonal terms are real valued elements with the magnitudes of the spectral den-
sities between a response and itself; the off-diagonal terms are complex valued elements
which carry the phase information between two measurements [22]. For the corresponding
sensor in each blade e.g. edgewise outer section, the spectral density matrix looks like:
GYY(f) =
GXX (f)11 GXY (f)12 GXY (f)13
GXY (f)21 GXX (f)22 GXY (f)23
GXY (f)31 GXY (f)32 GXX (f)33
(2.10)
Note that these matrices are Hermitian.
2.2.3 Singular Value Decomposition
The singular value decomposition technique (SVD) is explained in this section and its
practical application to the signals of this work can be found in Chapter 4. Although
it is a well-known technique with broad fields of applications, only the more relevant
properties to the concerns of this work are commented here.
Description
The SVD allows to break up a rectangular matrix into the product of three matrices,
where two contain the singular vectors and one the singular values of the original matrix.
In very plain words, the SVD for modal parameter estimation consists in extracting the](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-35-2048.jpg)
![18 Background Knowledge
singular values - the square roots of eigenvalues in the diagonal matrix - from the original
matrix and performing a curve that indicates with a peak where a mode is located.
If only one mode exists at a particular location, it will only be described by one set
of singular values. If two sets of singular values curves are needed to describe one mode,
it means that there are actually two modes, probably indiscernible. This property is
especially useful when modes are really close.
Provided the spectral density matrices in the form of Equation (2.10), the SVD is
performed for each of the matrices at each frequency and measurement. Specifically to
this report, the SVD will return 3 singular value curves.
The result is the determination of modes and a estimation of the modal frequencies prior
to OMA, which is considered a helpful method to validate the modes when identified in
OMA.
Formulation
Through the SVD, a matrix of m x n dimension can be decomposed into the product
of three matrices: an orthogonal unitary matrix, a diagonal matrix and the transpose of
another orthogonal unitary matrix as
A = USVH
(2.11)
with A being one of the spectral matrices, i.e. GYY(f), where the expansion of this
matrix yields
A = [{u1}{u2}{u3} . . . ]
s1
s2
s3
...
[{v1}T
{v2}T
{v3}T
. . . ] (2.12)
A = {u1}s1{v1}T
+ {u2}s2{v2}T
+ {u3}s3{v3}T
+ . . . (2.13)
where the diagonal terms of S are the square roots of the eigenvalues from U and
V sorted in descending order, named singular values. The columns of U and V are
orthonormal eigenvectors of AAT and AT A respectively, and H denotes Hermitian, which
in case of real matrices is simply the transposed.
The singular vectors are the estimated mode shapes and the singular values are the
spectral densities. The descending order to which the singular values are sorted in the
matrix makes the first singular value curve be the largest.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-36-2048.jpg)
![2.3 Tools 19
2.3 Tools
This section describes the tools that are used further to simulate the dynamics of the
V27:
HAWC2
HAWCStab2
OMA Type 7760
2.3.1 HAWC2
HAWC2 (Horizontal Axis Wind turbine Code 2nd generation) is a nonlinear aeroelastic
code intended for calculating wind turbine response in the time domain. It was origi-
nally developed by Petersen (HAWC) [24, 25], while the second generation was mainly
developed by Larsen and Hansen [26]. However, many individuals have contributed to
the enhancement of the possibilities of the code.
The structural part of the code is based on a multibody formulation applied to the
Timoshenko beam element, i.e. each body is an assembly of Timoshenko beam elements.
A typical configuration is tower, towertop, shaft, hub and blades, considered as main
bodies. Each of these bodies are divided into different sub-bodies and every sub-body is
break down into Timoshenko beam elements with its own reference system and described
by 6 DOF. Each element consists of two nodes that specify the geometrical position in
space. Stiffness, mass and inertia properties are constant along the beam element. Since
the deformations are assumed small at a single sub-body, bodies of interest have to be
divided into different sub-bodies to represent real behaviour with sufficient accuracy. An
example are the blades bodies, which uses sub-bodies to better describe the aeroelastic
blade behaviour. The structural properties are called from a file that gathers all the
structural information required. The turbine, then, is modelled by an arbitrary number
of bodies connected with constraint equations, where a constraint could be e.g. a rigid
coupling or a bearing.
The aerodynamic part of the code is based on the blade element momentum (BEM)
theory, extended from the classic approach to handle dynamic inflow, dynamic stall, skew
inflow, shear effects on the induction and effects from large deflections. The aerodynamic
properties of the airfoil are called from two files: one providing the blade planform -
defining the chord length and profile thickness related to the chord length at each cross-
section of the airfoil - and the other providing the profile coefficients of the airfoil - lift,
drag and moments in the angle of attack range from -180 to 180◦, identified by the
thickness over chord ratio (link to the blade planform properties).
Two turbulence formats can be used: one based on Veer’s model (polar grid) and
the other based on Mann model formulation (creates spatial vector field in Cartesian
coordinates).
Control of the turbine is performed through one or more DLLs (Dynamic Link Library).
The format for these DLLs is also very general, which means that any possible output](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-37-2048.jpg)
![20 Background Knowledge
sensor normally used for data file output can also be used as a sensor to the DLL. This
allows the same DLL format to be used whether a control of a bearing angle, an external
force or moment is imposed on the structure.
This and more information about HAWC2 can be found in [26].
2.3.2 HAWCStab2
HAWCStab2 is a linear aeroelastic stability tool, developed by Hansen [4], that predicts
structural and aeroelastic modal frequencies, damping ratios and mode shapes, through
open- and closed-loop aero-servo-elastic eigenvalue and frequency-domain analysis. The
aeroelastic model accounts for nonlinear kinematics based on Timoshenko elements.
The underlying structural model used in HAWCStab2, is the same as for HAWC2. The
beam element model used in HAWCStab2 is shown in Figure 2.6.
The linearisation of equations of motion are done about a steady-state equilibrium -
at a given operating point defined by constant wind speed, rotor speed and pitch angle
- that approximates the mean of periodic steady state, considering uniform inflow. The
periodicity associated with an operating wind turbine is then eliminated by means of
the MBC, previously explained in Section 2.1.4, based on the fundamental assumption of
isotropic rotors.
The aerodynamic loads are based on the BEM theory coupled with a Beddoes-Leishman
type dynamic stall model in a state-space formulation [27]. The distribution of aerody-
namic forces are a parabolic approximation based on 3 aerodynamic calculation points,
placed at the nodes and center of each blade element. The aerodynamic forces and
moments in the calculation points are explained and derived in [4], finally yielding the
coupled equations of motion of the linear aeroelastic model of wind turbines.
Figure 2.6: Beam element model in HAWCStab2 (only one blade). Source [4]
These equations of motion enable an eigenvalue analysis to determine the aeroelastic
natural frequencies, damping ratios and mode shapes at each operating point.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-38-2048.jpg)
![2.3 Tools 21
Further information on HAWCStab2 is provided in [2, 4, 17].
2.3.3 Br¨uel & Kjær Operational Modal Analysis Type 7760
Operational Modal Analysis Type 7760 is a software package from Br¨uel & Kjær for
experimental identification of modal parameters with the SSI algorithm already imple-
mented.
The SSI technique is presented in a discrete time state space formulation. The response
data is collected in the so-called Hankel matrix, which structure is related to covariance
estimation. The subsequent projection of the Hankel matrix is explained in terms of
covariances and leads to a set of free responses for the system. Lastly, a SVD is then
applied to the projection matrix in order to obtain the estimated system matrices, and
by extension, the modal parameters.
As mentioned in the Section 1.1, there are some assumptions that need to be fulfilled:
the structure is linear;
the structure is time invariant;
the operational excitation forces must
a) have broadband frequency spectra;
b) be uncorrelated;
c) be distributed over the entire structure.
The SSI algorithm is fully described in [12]. A detailed explanation of the SSI technique
implemented in the Operational Modal Analysis Type 7760 software package is found in
[28, 29].](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-39-2048.jpg)
![24 Implementation of the Numerical Model
To initiate the mentioned procedure, a natural step is to implement a numerical model
of V27 using HAWC2, capable of mimicking the conditions of the V27 matching measure-
ments. The structural properties of the H2 model are also used in the HS2 model from
which the modal parameters are extracted.
On this basis, this chapter relies on three key points:
a) to describe the experimental setup and environmental characteristics.
b) to create a valid numerical model (H2 model) able to be run in HAWC2, assuming
structural properties, wind site conditions and output channels being in agreement
with the measurement campaign.
c) to use the structural properties of the H2 model to perform an aeroelastic modal
analysis in HAWCStab2, ending up with estimated natural frequencies and damping
ratios to which compare with in Chapter 5. These estimations are referred in the
following to ”HS2”.
3.1 Experimental Data
This section describes the measurement setup conducted to collect the experimental data.
The description is limited to the data related to the signals used in this work for the sake
of simplicity. The complete description can be found in [5], with details regarding devices
and challenges involving the instrumentation of the turbine.
3.1.1 The Wind Turbine
The V27 is owned by DTU Wind Energy (former Risø National Laboratory for Sustainable
Energy) and was erected at the Risø test site in March 1989. This turbine has supported
many research projects so far, and it is well instrumented and well known. Despite it
is an old-fashioned machine compared to modern multi-MW turbines, it may provide a
physical insight of the dynamics of current wind turbines, since they have in common
similar construction design as well as controllers, such as yaw and pitch controls. Details
of the turbine are given below in Table 3.1.
According to [25], this wind turbine has a three-bladed upwind rotor with pitch-
regulated cantilevered blades mounted on cast iron hub. The disc brake is mounted
on the high speed shaft. The gearbox is mounted behind the shaft on the nacelle frame,
and the generator is connected to the gearbox by a stiff coupling. The wind turbine yaws
by an electrical motor controlled by a wind vane, mounted on the top of the nacelle. The
electrical control system is mounted on the nacelle and in the tower bottom. The tower
is a tube tower of a single section.
A particular feature of the V27 is that the rotor has two different speeds depending on
the number of poles connected to the generator, which can be 6 or 8 poles.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-42-2048.jpg)
![26 Implementation of the Numerical Model
3.1.3 Measurement Campaign
The measurement campaign was conducted between October 2012 and May 2013 partly
under the EUDP (Danish Energy Technology Development and Demonstration Pro-
gramme) project frame, ”Predictive Structure Health monitoring of Wind Turbines”,
with grant number 64011-0084.
Measurements from the V27
The instrumentation of the V27 consists in a total 51 channels providing data sampled
at 4096 Hz. Among all channels, the following are particularly interesting for this work:
10 accelerometers per blade
3 tri-axial accelerometers at the nacelle
1 tachometer
1 pitch angle signal sensor
The blade sensors are displayed for each blade section in Figure 3.3.
Figure 3.3: Sensors location and orientation. Source: [5]
The blade sensors comprise 10 accelerometers oriented to the flapwise direction of which
two are biaxial, i.e. also oriented to the edgewise direction. These accelerometers are
used to read the acceleration signals of all three blades. They are distributed along the](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-44-2048.jpg)
![3.1 Experimental Data 27
blade span at specific sections of the blade, with 5 in the leading edge and 5 located in
the trailing edge, where the bi-axial accelerometers are located. To define the optimal
sensor location on the blades, simulations in H2 were carried out [30]. The eigenvectors
of the operating blade were approximated with eigenvectors associated to a blade with
fixed support (e.g. no rotational effects), assuming that the operating blade eigenvectors
difference is negligible with respect to those. Thus, the maximum modal deflections were
found at particular cross-sections and, in consequence, the accelerometers were decided
to be installed in the locations shown in Figure 3.3. In the present work, only the bi-
axial accelerometers are used, located close to the blade tip (96% of blade span) and the
intermediate section (67%). These sections are hereafter called outer and inner sections,
respectively. The reason is that, first, it is assumed that these sensors are sufficient to
represent with confidence the dynamic behaviour of the V27, and second, to reduce the
potential risk of misalignment by using sensors that form between them a 90◦ angle.
The nacelle sensors are tri-axial accelerometers, which monitor the nacelle accelerations
in the x, y and z direction. They are expected to identify not only tower modes, but also
the rotor modes. In 2010, Tcherniak et al. showed that rotor modes can be extracted
from nacelle sensors [1]. In the present study, one nacelle sensor is mounted in the yaw
bearing and two in the rear of the nacelle (left and right corners). As a result, the
rotor modes are expected to be extracted from the nacelle signals and validated against
those extracted from the blades signals, reinforcing the findings in the above mentioned
investigation (2010).
Figure 3.4: Nacelle sensors in the V27. Source [6]
The tachometer is used to determine the rotor azimuth angle - the position of a blade
as a function of time - with the turbine under operation. It was found in [6] that the
tachometer provided the most accurate measurement of the rotating frequency among](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-45-2048.jpg)
![3.2 Defining the Model in HAWC2 31
Criteria for the Selection of Data Sets
A wind turbine may be modelled as a time periodic system if the fluctuation of particular
variables is minimized, i.e. rotor speed, blade pitch and nacelle yaw, so as to assume that
these variations are constant. Hence, the criteria for selecting data sets is based on a low
standard deviation of these parameters.
Besides, OMA requires long enough time series for better performance of the algorithm.
A rough rule of thumb is that the time histories should at least be 500 times longer than
the period of the lowest mode of interest [31]. The lowest mode of interest is expected
around 0.95 Hz, therefore, a rough minimum is about 526 seconds. Nonetheless, the larger
the time series, the better the algorithm performs, and 20 minutes is thought as a good
balance between the computational time expense and the operational modal analysis time
history demand. This time span length, translated to the recorded measurements, means
to create data sets comprising 4 chunks of V27 measurements linked to 2 chunks of met
measurements. Further, it is desired to investigate time series at both low (33 RPM) and
high rotor speed (43 RPM).
Provided these considerations, two main data sets are created with successive measure-
ments, which are picked conveniently for low rotor speed (December 16th, 11:10-11:30)
and high rotor speed (December 15th, 05:10 to 05:30). Detailed information about these
data sets is shown in Table 3.3.
Parameter Low rotor speed High rotor speed
Time stamp 16/12/12 11:10-11:30 15/12/12 05:10-05:30
Std. Dev. tacho (RPM) 0.54 1.28
Mean rotor speed (RPM) 32.20 43.10
Std. Dev. pitch sensor (◦) 0.09 0.59
Std. Dev. yaw (◦) 0.17 0.02
Mean wsp (m/s) 5 11
Std. Dev. wsp (m/s) 0.56 1.37
Max./min. power (kW) 0/0 265/56.3
Table 3.3: Selected data sets
It is to be noticed that the standard deviation of the tachometer refers to 1 pulse or
RPM with respect to the HSS, i.e. 750 or 1000 RPM, for low and high rotor speed time
series, respectively. With the entire collection of tacho ”events”, the mean is found, and
the rotational speed is derived for the low speed shaft (LSS, rotor speed).
3.2 Defining the Model in HAWC2
Once the time spans are selected from the experimental data, this section defines the im-
plementation of the numerical model to simulate the behaviour of the V27 at the selected
time spans. The external excitation in the H2 model is defined based on the recorded
meteorological data during the selected time spans. The structural and aerodynamic
properties of the H2 model are given from previous Risø research studies.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-49-2048.jpg)
![3.2 Defining the Model in HAWC2 33
The wind shear is calculated using mean wind speeds recorded at different heights from
anemometers at the met mast anemometers at the particular time frames. The power
law method is employed to estimate the wind shear exponent, and thus, the mean wind
speed at the hub height. The hub height was assumed at 30 meters above ground level to
be consistent with the provided measurements file (Section 3.1.4), despite that the real
hub height is apparently 31.5 meters. Figure 3.7 shows the wind profile resulting in wind
shear exponents of 0.23 and 0.13 for low and high rotor speed.
Turbulence Field
The turbulence box is generated using the Mann model for the simulation time and wind
speed in each case. The turbulence box resolution is set as 32 x 32 x 8192 and the
turbulence parameters are assumed to be those by default in HAWC2, i.e. according to
the IEC standard [32], except that the length scale is not the usual 29.4 but 15.484. This
correction is done according to the standard for hub heights lower than 60 meters. Thus,
the standard is followed to meet the new length scale and the specific values are shown
below:
αǫ2/3
= 1
Γ = 3.9
L = 15.484
where
L = 0.7Λ1
Λ1 = 0.7z
with L, αǫ2/3 and Γ being the turbulence parameters, Λ1 being the wavelength and z
as the hub height.
Besides, aerodynamic load calculation is done at 30 sections for each blade, including
Beddoes-Leishman dynamic stall model, in accordance with HAWCStab2. Moreover,
aerodynamic drag is set for the tower and nacelle.
Controller
In HAWC2, the V27 model displays two control algorithms: one for the generator and
one for the blade pitch. The most relevant features of the generator controller are the
gearbox exchange ratio, set to 23.33335, the nominal electric power (225 kW) and the
synchronous generator speed (750 or 1000 RPM, depending on the data set). The pitch
controller permits the blades to pitch from 0 to 90◦ in a collective manner.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-51-2048.jpg)
![3.2 Defining the Model in HAWC2 35
HAWC2 contains an embedded eigenvalue solver, which is capable of providing modal
parameters with the wind turbine at standstill. The assumption for such analysis is
that there is no wind forcing, and that the wind turbine is in standstill, i.e. rotor with
fix bearing (braked rotor). HAWC2 offers two methods to calculate the frequencies and
damping at standstill: one involves including the mass and stiffness proportional damping
parameters and the other only the stiffness terms. There are some issues related to the
damping model in HAWC2 that makes the mass proportional damping terms to only be
correctly applied in the special case when the main body is fixed to the ground and it
consists in a single body (such as the platform in this H2 model). If this condition is not
fulfilled, it yields a different damping if the number of sub-bodies is higher than one [26].
A blade could be an example, where it is required to have multiple sub-bodies within
the main body (blade) to represent accurately deflections. Since it is important to use
the same number of bodies in the structural eigenanalysis as well as in the simulations,
and the blade main body is subdivided into five bodies, the selected method in this work
only uses the stiffness proportional damping parameters, which enables to consider the
response of the entire structure.
In order to obtain the acceptable level of damping mentioned in the beginning of this
section, a tuning of the stiffness proportional damping parameters is done. The final
coefficients of the stiffness contribution to the damping of the structure are presented in
Table 3.5.
Body Mx [-] My [-] Mz [-] Kx [-] Ky [-] Kz [-]
Platform 0 0 0 1.7e-3 1.7e-3 4.5e-4
Tower 0 0 0 0.85e-3 0.85e-3 2.5e-7
Shaft 0 0 0 1.5e-3 1.2e-3 8.5e-4
Blade 0 0 0 4e-4 6.06e-4 1e-7
Table 3.5: Stiffness contribution coefficients to damping
The results of this tuning are presented in Table 3.6.
Mode no. Freq. [Hz] Log. decr. [%] Damp. [%] Nomenclature
1 0.949 1.982 0.301 1st TSS
2 0.953 1.990 0.302 1st TFA
3 2.044 3.003 0.431 1st YF
4 2.095 3.054 0.437 1st TF
5 2.159 3.032 0.435 1st DT
6 2.436 2.932 0.423 1st CF
7 3.590 2.986 0.429 1st VE
8 3.644 3.007 0.432 1st HE
Table 3.6: Results of structural eigenanalysis with H2
Graphical representations of these modes can be seen in Figure 3.8.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-53-2048.jpg)
![3.3 Estimation of Modal Parameters in HAWCStab2 37
3.3.1 Tool Setup
Structural Model
The structural model used in HAWCStab2 is exactly the same as in HAWC2. Therefore,
it holds the same damping characteristics (cf. Table 3.5).
Operating Points
HAWCStab2 requires operational data where the operating points are defined (steady-
state equilibrium, see Section 2.3.2). The operational data is computed provided a wind
speed range, number of operational points, rotation speed range, minimum pitch angle,
tip speed ratio and maximum rated aerodynamic power, which are based on the V27
experimental data. The output is the number of operational points related to wind speed,
pitch angle, rotor speed, aerodynamic power and aerodynamic thrust.
For the concerned cases, i.e. wind speeds of 5 m/s (low rotor speed) and 11 m/s (high
rotor speed), Table 3.7 presents the operating points.
Variable Low rotor speed High rotor speed
Mean rotor speed (RPM) 32.14 35.02
Mean wsp (m/s) 5 11
Mean pitch angle (◦) 0.41 1.57
Table 3.7: HS2 operating points
3.3.2 Comparison of the H2 Model and HAWCStab2
A structural modal analysis is performed in HS2. Considering only the results at 0 RPM,
corresponding to a standstill condition, modal parameters extracted are shown in Table 3.8
where they are compared to the modal parameters from the eigensolver in HAWC2.
Mode no.
HS2 H2
Nomenclature
Freq. [Hz] Damp. [%] Freq. [Hz] Damp. [%]
1 0.992 0.332 0.949 0.301 1st TSS
2 0.995 0.333 0.953 0.302 1st TFA
3 2.045 0.476 2.044 0.431 1st YF
4 2.096 0.485 2.095 0.437 1st TF
5 2.155 0.482 2.159 0.435 1st DT
6 2.434 0.469 2.436 0.423 1st CF
7 3.590 0.473 3.590 0.429 1st VE
8 3.643 0.480 3.644 0.432 1st HE
Table 3.8: HS2 vs. H2 modal frequencies and damping in standstill](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-55-2048.jpg)
![38 Implementation of the Numerical Model
It can be observed that frequencies are practically the same, while damping ratios do
not greatly differ (around 10% of difference). The main disagreement is found on the
tower modes, which is attributed to the modelling of the tower plus platform. Based on
this comparison, it is assumed that the HS2 is correctly implemented so as to perform an
aeroelastic modal analysis to benchmark against the V27 and H2 model OMA results in
Chapter 5.
3.3.3 Structural and Aeroelastic Modal Analyses
The results shown in the previous section refer to a structural modal analysis at standstill
condition. Next, a structural modal analysis for the entire wind speed range is presented,
together with an aeroelastic modal analysis. The aim of comparing both is to observe
if the frequencies and dampings change when aerodynamic effects are included (always
under HAWCStab2 perspective, i.e. symmetric rotor and inflow, no turbulence). In both
analysis the wind turbine is rotating, but the reader may associate the first type of modal
analysis as the wind turbine rotating in vacuum.
In Figure 3.9 the structural and aeroelastic frequencies are plotted. The associated
nomenclature is joined to each mode. The structural modes are denoted with solid lines,
while the aerodynamic modes are denoted with dashed-point lines.
According to the figure, the frequencies hardly vary including or not the aerodynamics.
The largest difference (in this context) is found in the components of the flap mode,
because of the impact of the aerodynamics. The in-plane modes, such as edgewise rotor
mode or shaft mode, remain almost equal. It can be also noticed how the 1st collective,
FW flap and BW edge components couple within the frequency range from 2.5 to 3 Hz.
The damping ratios of the respective analyses are shown in Figure 3.10. The structural
modes are less damped, because only the internal loads and the rotor speed act on the
structure. It can be further noted from Figure 3.11, that the damping behaviour is almost
invariant with the wind speed.
0 5 10 15 20 25
1
1.5
2
2.5
3
3.5
4
4.5
wind speed [m/s]
modalfrequency[Hz]
1st TFA
1st TSS
1st BWF
1st CF
1st FWF
1st BWE
1st DT
1st FWE
struct.modes
aero modes
Figure 3.9: Frequencies vs. wind speed](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-56-2048.jpg)
![3.3 Estimation of Modal Parameters in HAWCStab2 39
The aeroelastic damping decrease, in general for all modes, because the blades are not
pitching yet, and there is a change in the angle of attack. This fact is especially visible in
the flapwise direction, since the flap components are the most affected by the wind, and
thus, by the aerodynamic damping.
0 5 10 15 20 25
0
5
10
15
20
25
30
wind speed [m/s]
dampingratio[%]
1st BWF
1st CF
1st FWF
1st TFA
struct.modes
aero modes
Figure 3.10: Damping ratios vs. wind speed. All modes
However, the zoom view in Figure 3.11 reveals that the least damped modes also de-
crease for the same reason, in a much lower degree though, because the aerodynamic
damping influence is not relevant.
0 5 10 15 20 25
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
wind speed [m/s]
dampingratio[%]
1st TSS
1st BWE
1st DT
1st FWE
struct.modes
aero modes
Figure 3.11: Damping ratios vs. wind speed. Low damped modes
The BWE component, as mentioned previously, couples with some of the flapwise
components. This coupling apparently introduces an increase in damping contributed by
these flapwise components, as it can be perceived from Figure 3.11. Contrary, the DT](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-57-2048.jpg)
![40 Implementation of the Numerical Model
and FWE components have lower damping with respect to the BWE component, more
likely for the same coupling with the edgewise component.
When the blades start to pitch - according to the HS2 operational data, slightly at
9 m/s, and considerably at 12 m/s - the angle of attack is corrected, and the aeroelas-
tic damping ratios change the pattern and increase. At 14 m/s the rated wind speed
is achieved, and the controller takes over to secure a proper damping for the rest of
operational points.
To support these statements, the operational data can be checked in Figure 3.12. The
operational data is not extremely accurate, since the rotor speed takes the wind speed
to increase from 10 to 14 m/s to achieve the high rotor speed. One may recall that the
selected data sets from the V27 measurements assumes about 43 RPM at 11 m/s (from
average values). Also, the pitch angle in the mentioned data sets is specified as 0◦ and 0.6◦
for 5 m/s and 11 m/s (Table 3.3), whereas HS2 computes 0.41◦ and 1.57◦ respectively.
5 10 15 20 25
0
50
100
150
200
250
kW
Power (kW)
Rotor speed (RPM)
Pitch (º)
0 5 10 15 20 25
0
10
20
30
40
50
wind speed [m/s]
RPM/ºFigure 3.12: Operational data assumed in HS2
Finally, the aeroelastic frequencies and damping corresponding to the operating points
5 m/s and 11 m/s are shown in Table 3.9.
Mode no.
Low rotor speed High rotor speed
Nomenclature
Freq. [Hz] Damp. [%] Freq. [Hz] Damp. [%]
1 0.999 2.621 0.993 1.994 1st TFA
2 1.004 0.345 1.004 0.291 1st TSS
3 1.694 15.884 1.608 15.034 1st BWF
4 2.586 11.924 2.548 11.305 1st CF
5 2.733 11.357 2.726 10.177 1st FWF
6 3.108 0.490 3.061 0.273 1st BWE
7 4.099 0.869 4.101 0.818 1st DT
8 4.169 0.606 4.225 0.508 1st FWE
Table 3.9: Aeroelastic frequencies and damping in HS2 at 5 and 11 m/s](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-58-2048.jpg)
![Chapter 4
Signal Analysis
In this chapter, the methods described in Chapter 2 are applied on the V27 and H2
signals.
Recalling Chapter 2, modal analysis requires the system to be time-invariant. This
assumption is expected to be fulfilled by means of the MBC transformation. This can
work for the H2 model, fulfilling the isotropic condition, but may not work for the V27
signals, because real life rotors are never isotropic. In addition, OMA is based on several
assumptions that are mainly not fulfilled, thus, it is not assured that the application of
OMA is successfully possible to the V27 measurements, meaning that applying OMA
right away may lead to erroneous results.
Signal analysis is not based on any assumptions, and therefore it can be seen as a
natural step to analyse the signals prior to OMA. Methods such as the PSD or SVD
(as described in Figure 3.1) are hence used to provide insight on peaks arising in the
frequency domain, that could provide a hint about their meaning, e.g. if the peak refers
to a harmonic or to a mode.
The sequence of the analysis is structured as follows:
1) PSD of the signals
2) SVD of the spectral density matrices (formed with PSD and CSD of the signals)
3) Transformation of the signals from the rotating to the non-rotating frame using the
MBC (converting to time-invariant system)
4) After the MBC, an averaging technique can be used as stated by Bir [18]. The
TSA technique is employed to remove the presence of harmonics in the transformed
signals (according to Bir, all harmonics but the multiples of 3 times the rotational
frequency). This is achieved by removing the deterministic periodic excitation (har-
monics) from the stochastic excitation (wind), which satisfies OMA leading to more
reliable results.
43](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-61-2048.jpg)
![4.1 Dynamic System Behaviour I: PSD 47
peak in the y signals. A wide peak around 0.95 Hz may suggest the first tower mode,
since it was also found in Figure 4.2. Similar trend of the signals is found at high rotor
speed in Figure 4.3(b).
4.1.2 Blades Signals
The PSD of the blade signals are plotted in this section. While the PSD of the tower
signals left some traces that one may think of as modes, in particular, the first tower
modes, one cannot dare such assumptions here. First, we are dealing with a time-variant
system, and therefore modal analysis cannot be applied. Secondly, as reviewed in Chap-
ter 2, blades modes rely on components (two asymmetric and one symmetric), hence, only
traces of components could be suggested in the best case scenario.
However, it is interesting to look further into those double peaks appearing in the tower
PSD, and also, at which frequency ranges there is more power content.
The section is divided in two subsections: signals at low and high rotor speed. Each of
the parts compares the V27 with the H2 signals, sorted from outer to inner section of the
blade. Figures include all signals, edge and flap.
Low Rotor Speed
Figure 4.4 and Figure 4.5 represent the blades signals of V27 and H2 at low rotor speed
in the outer section of the blades. As expected, the flap signals content more energy than
the edge signals.
Peaks are denoted at all rotor harmonics except at the 1P frequency, which shows a
different shape than the expected from a harmonic (narrow and sharp peak). This peak
is fairly sharp at the crest but wide at the trough. Actually, the harmonic peaks in the
V27 signals are hardly recognizable in the low frequency range. For instance, the 4P
harmonic shape is similar to a mode but features a small peak in the harmonic frequency.
In general, the energy in the signal increases nearby of the harmonics, showing the wide
trough effect. The H2 data shows similar behaviour on all signals, but the small peaks
are more noticed (perhaps with the exception of the 2P). The peak at 1P shares the same
characteristic as in the V27 signals. Tcherniak et al. commented on this effect in [11],
where the spectra of the aerodynamic forces was analysed. They conclude that when the
blade passes regions with higher or lower wind speed, periodicity is generated, adding up
that increasing the tip speed ratio generates higher peaks and deeper troughs. It is also
mentioned that the wide troughs or ”thick tails” could emerge due to the convolution of
two autocorrelation functions, one in the root (flat spectra) and the other at higher radius
(periodic peaks growing as a function of the radius). Indeed, Figure 4.4 (a) appears close
to the normalized PSD of the autocorrelation functions shown in this study.
Around 3.6 Hz a double peak appears in the V27 data - single peak in the H2 data.
This double peak is magnified in Figure 4.4 (b), revealing that all blades manifest this
duplicity. Rotor anisotropy might be beneath this double peak behaviour. In any event,
this peak contents more energy than the flap signals in this frequency range. Since it is
not related to a harmonic and shows a ”thick tail”, one may suspect that it is a mode.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-65-2048.jpg)
![4.3 Dynamic System Behaviour III: MBC 57
From Figure 4.16 several observations can be done:
1) The asymmetric coordinates are well correlated one to each other, while the sym-
metric shows a different trend, meaning that the MBC effectively transforms from
individual blades to rotor blades, and divides the collective from the asymmetric
rotor behaviour.
2) Two peaks show up at 3 and 4.2 Hz. According to the theory, the whirling com-
ponents of a mode are associated with ±Ω (difference between them 2Ω), where
Ω is the rotational speed. One may recall that the suspected mode in Figure 4.12
was found around 3.5 Hz. At low rotor speed Ω equals close to 0.55 Hz (33 RPM
approx.), and if it is subtracted and added to 3.5 Hz approx., it ends up with these
two asymmetric peaks. Hence, still pending of a modal analysis, these two peaks
may be considered as BW and FW components of a rotor mode.
3) It is also characteristic that both peaks keep the double peak in the crest, meaning
that this fact is not related to two modes, it rather refers to anisotropic consequences.
4) According to Bir [18], the MBC filters out all harmonics but the multiple of 3Ω in
the case of isotropic rotor (3-bladed), as seen in (c) and (d). Bir also states that
if the rotor is anisotropic, the MBC still works, but cannot filter the harmonics, as
can be checked in (a) and (b).
5) A peak arises between the asymmetric edge components, separated by approx. Ω.
The acting aerodynamic loads mask any trace of mode in Figure 4.17.
0 1 2 3 4 5
10
−6
10
−4
10
−2
10
0
10
2
10
4
Hz
(m/s
2
)
2
/Hz
1P 2P 3P 4P 5P 6P 7P 8P 9P
a0
a1
b1
(a) V27 outer section
0 1 2 3 4 5
10
−6
10
−4
10
−2
10
0
10
2
10
4
Hz
(m/s
2
)
2
/Hz
1P 2P 3P 4P 5P 6P 7P 8P 9P
a0
a1
b1
(b) V27 inner section
0 1 2 3 4 5
10
−6
10
−4
10
−2
10
0
10
2
10
4
Hz
(m/s
2
)
2
/Hz
1P 2P 3P 4P 5P 6P 7P 8P 9P
a0
a1
b1
(c) Model outer section
0 1 2 3 4 5
10
−6
10
−4
10
−2
10
0
10
2
10
4
Hz
(m/s
2
)
2
/Hz
1P 2P 3P 4P 5P 6P 7P 8P 9P
a0
a1
b1
(d) Model inner section
Figure 4.17: MBC applied on flapwise signals at low rotor speed](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-75-2048.jpg)
![4.4 Time Synchronous Averaging 59
0 1 2 3 4 5
10
−6
10
−4
10
−2
10
0
10
2
10
4
Hz
(m/s
2
)
2
/Hz
1P 2P 3P 4P 5P 6P
a0
a
1
b1
(a) V27 outer section
0 1 2 3 4 5
10
−6
10
−4
10
−2
10
0
10
2
10
4
Hz
(m/s
2
)
2
/Hz
1P 2P 3P 4P 5P 6P
a0
a
1
b1
(b) V27 inner section
0 1 2 3 4 5
10
−6
10
−4
10
−2
10
0
10
2
10
4
Hz
(m/s
2
)
2
/Hz
1P 2P 3P 4P 5P 6P
a0
a1
b1
(c) Model outer section
0 1 2 3 4 5
10
−6
10
−4
10
−2
10
0
10
2
10
4
Hz
(m/s
2
)
2
/Hz
1P 2P 3P 4P 5P 6P
a0
a1
b1
(d) Model inner section
Figure 4.19: MBC applied on flapwise signals at high rotor speed
4.4 Time Synchronous Averaging
The time synchronous averaging (TSA) technique is applied to the signals after MBC
transformation in order to remove the sharp peaks in the harmonics. The purpose of
this task is to avoid misinterpretations of the algorithm on the data (e.g. identifying
a harmonic like a mode), thus leading to more reliable results. According to Bir, the
averaging must follow the MBC.
In general words, the TSA removes the deterministic (periodic) component of the signals
due to the rotation of the rotor, maintaining only the stochastic component of the signal
due to random events, e.g. wind turbulence. A rough explanation of how the method
works is provided in the following based on Jacob et al. [7]. A tachometer is used to
identify the periodic component. The acceleration time series are then chopped for each
period and averaged. An enhanced signal of the periodic component results is replicated
along the entire time series and subtracted from the original signal. The remaining part
may be called the residual signal. This residual is the stochastic component of the signal,
which satisfies the OMA assumption in this aspect. Further information may be found in
[7].
In Figure 4.20 the procedure is shown for better understanding.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-77-2048.jpg)
![60 Signal Analysis
Figure 4.20: TSA procedure. Source [7]
The TSA technique is consequently applied to the MBC signals, and their corresponding
PSD are plotted in Appendix B. It is advisable to compare MBC figures against TSA that
with the exception of the high rotor speed V27 data, apparently works properly.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-78-2048.jpg)
![64 Identifying Modal Parameters
The frequencies and damping of the FWE component are also similar, particularly
among H2 and HS2. The assumed FWE component displays the double peak feature
(described in previous sections), and the algorithm identifies two components at the same
peak.
V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature
0.96 0.95 0.99 1.91 2.15 2.62 1st TFA
- 0.96 1 - 1.68 0.35 1st TSS
- 3.09 3.11 - 0.84 0.49 1st BWE
4.08 4.14 4.17 0.82 0.75 0.61 1st FWE
Table 5.1: Modal parameters comparison for the tower sensors at low rotor speed
5.1.2 High Rotor Speed
A similar analysis is shown in Figure 5.4 for the high speed operational case. The tower
and edgewise modes are found straightforwardly both in the V27 and H2 data, unlike
in the low rotor speed case, where the TSS and BWE are missing. A set of modes are
found in the frequency range close to the 3P, similar as in the low rotor speed case, that
could be related to the flap components. Since they have similar frequencies and cannot
be decomposed in symmetric or asymmetric components, they are disregarded in the
analysis. Some other modes respond to a harmonic, for instance the first mode located
at the 1P. From all the modes, only those associated to the tower and BWE/FWE are
considered, because are clearly identified.
(a) V27 Tower modes
(b) H2 model tower modes
Figure 5.4: Tower modes stability diagram at high rotor speed
According to Table 5.2, the tower results from V27 and H2 are more similar compared
to the HS2 results, where the TSS mode is still very low damped. Analogously, the](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-82-2048.jpg)
![5.2 Rotor Modes 65
presumed BWE and FWE components also display similar values in the V27 and H2
cases. However, the BWE component in the V27 case seems to be more damped than the
FWE, contrary to what is observed in the H2 case. An important characteristic is that
the difference in frequency between the BWE and FWE components is 2Ω (43 RPM; the
HS2 rotor speed at this operating point is 35 RPM, cf. Table 3.7), indicating a whirling
mode according to the theory.
V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature
0.94 0.95 0.99 2.51 2.11 1.99 1st TFA
0.95 0.95 1 1.23 1.05 0.29 1st TSS
2.88 2.91 3.06 0.41 0.59 0.27 1st BWE
4.31 4.34 4.23 0.69 0.45 0.51 1st FWE
Table 5.2: Modal parameters comparison for the tower sensors at high rotor speed
5.1.3 Tower Modes Identification Discussion
Despite that the sensors used returned noise at low frequencies in the x direction, tower
modes are well identified. Since the low rotor speed case is apparently more sensitive to
the noise issue, the discussion embraces the high rotor speed case for the tower modes.
The agreement between the V27 and H2 data is quite good showing the same range in
frequencies and damping ratios. The frequencies of the first tower modes (TFA and TSS)
are about the same, and the damping ratios difference in % is low (2.51 vs 2.11, 1.23 vs.
1.05, for each tower mode respectively). The greater challenge perhaps is that the rotor
harmonics could not be removed in the V27 data, and therefore, peaks related to them
are also identified. However, the animation tool guides one to select the proper mode.
Moreover, edgewise modes were identified with apparent success because the whirling
components are clearly shown in the tower spectra (especially at high rotor speed). How-
ever, the modal parameters found must be assessed against those ones extracted from the
blade edgewise sensors for higher confidence.
The flapwise modes seemed to be identified around the 3P at both low and high rotor
speed, where 3P is different for each case. This fact added to that they could not be
traced neither in PSD or SVD analysis, lead to predict a difficult identification of these
modes.
5.2 Rotor Modes
In this section the edgewise and flapwise components are independently analysed. The
rotor motion is represented in the MBC transformed coordinates, i.e. a0 (symmetric
component), a1 and b1 (asymmetric components), similarly to [6], as shown in Figure 5.5.
The nodes where the first digit in the notation starts with 1 denotes the collective com-
ponent, while if starting with 2 or 3, it denotes the FW and BW components, respectively.
Besides, the third digit equal to 3 or 8 denotes the outer or inner section in the edgewise.
Similarly, with the third digit equal to 1 and 6, respectively, for the flapwise case.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-83-2048.jpg)
![68 Identifying Modal Parameters
available results. Therefore, the first pair (components 2 and 4 the table) is disregarded.
In general, the frequencies display good agreement. The damping ratios in V27 and HS2
are very close, but the BWE is more damped in the V27 results than in the HS2 results.
V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature
2.98 - - 1.38 - - 1st BWE
3.06 3.09 3.11 1.12 0.61 0.49 1st BWE
4.04 - - 0.41 - - 1st FWE
4.11 4.16 4.17 0.62 0.91 0.61 1st FWE
Table 5.3: Modal parameters comparison for rotor edge sensors at low rotor speed
High Rotor Speed
Figure 5.8 shows the identification results from the blade sensors on the edgewise direction.
Contrary to the low rotor speed case, 2 more modes are found in the V27 high speed
analysis, related to harmonic frequencies 1P and 5P. Besides, the same type of results is
found with 4 peaks identified for the V27 data and 2 for the H2 referring to asymmetric
components.
The animation provides the same phase difference as in the low rotor speed case, and
thus connecting first-third modes and second-fourth peaks as pairs.
For the high speed case the frequencies of the double peaks in V27 are much closer to
the H2 and HS2 frequencies, complicating the task of correlating them with the simulated
results. However, for this analysis either peak no.1 and 4 or peak no. 2 and 4 can become
a pair. The difference is the same with respect to the rotational speed, approx. 2Ω. To
be consistent with the low rotor speed case, the second pair is selected.
(a) V27 edgewise modes
(b) H2 model edgewise modes
Figure 5.8: Edgewise modes stability diagram at high rotor speed](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-86-2048.jpg)
![5.2 Rotor Modes 69
The HS2 frequencies disagree with the V27 and H2 frequencies, which in turn are quite
similar, especially for the BWE. The difference between BWE and FWE is 2Ω for each
of the results. The damping ratios are quite close in the FWE, but not in the BWE. In
addition, the highest damped modes are the BWE in the V27 and H2 data, contrary to
the HS2 data. Results can be observed in Table 5.4.
V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature
2.87 - - 0.35 - - 1st BWE
2.93 2.91 3.06 1.15 0.62 0.27 1st BWE
4.29 - - 0.69 - - 1st FWE
4.33 4.34 4.23 0.70 0.5 0.51 1st FWE
Table 5.4: Modal parameters comparison for rotor edge sensors at high rotor speed
5.2.2 Rotor Anisotropic Effects Induced in the H2 Model
The double peak effect is shortly analysed in this section. The aim is to understand
why this phenomena occurs and how much it affects the identification performance. The
edgewise components are chosen for this analysis, since they clearly show this effect.
For this analysis it has intentionally been specified different stiffnesses for the blades to
resemble the V27 double peak, with the aim of confirming that this behaviour is due to
anisotropic effects. The stiffness of the blades is changed as follows: 5% less stiffness in
blades 1 and 2, and 10% more stiffness in blade 3, trying to maintain the mean stiffness
in the rotor. Simulations are done for three cases: (a) low rotor speed, (b) high rotor
speed, and (c) high rotor speed with the damping ratio limited to 2.3%, which is slightly
larger than in the previous analysis.
The reason of increasing the damping ratio limit imposed in the previous cases is
because the algorithm is not capable of identifying each of the double peaks, as it is
intended, therefore indicating that these modes have a larger damping, probably due to
the change in structural properties of the blades that was assumed for the investigation
of the anisotropic effect. Results are shown in Figure 5.9.
With this change, the double peaks are now more separated. In case (b), only two modes
are identified with the usual damping limit of 1.5%. However, the difference between them
is notably less than 2Ω. If the damping limit is extended, 4 peaks appear identified as
modes, and the difference turns out to be around 2Ω. It is worth to recall that the mean
rotor speed is 43.22 RPM in the high rotor speed case, and the pair of peaks shows a
difference of 42 and 42.6 RPM, respectively.
One may also recall that in the low speed case the rotor mean speed is 32.25 RPM.
The frequency difference of the peaks found is 32.43 and 32.01 RPM, for each respective
pair.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-87-2048.jpg)
![70 Identifying Modal Parameters
(a) Low rotor speed
(b) High rotor speed, 1.5% max. damping ratio
(c) High rotor speed, 2.3% max. damping ratio
Figure 5.9: Smooth anisotropic effect on the model
5.2.3 Edgewise Modes Identification Discussion
If one compares the identification results from the tower sensors and from the blade
edgewise sensors, they appear to be very close. Table 5.5 displays the frequencies and
damping of the edgewise components from the identification in the tower and in the blades
for the best approach found at high rotor speed:
Tower V27 Blades V27 Tower H2 Blades H2 Nomenclature
Frequencies [Hz]
2.88 2.87 2.91 2.91 1st BWE
4.31 4.33 4.34 4.34 1st FWE
Damping ratios [%]
0.41 0.35 0.59 0.62 1st BWE
0.69 0.70 0.45 0.50 1st FWE
Table 5.5: Edge frequencies and damping from tower and blade sensors
It is recalled from Section 5.2.1 that two pairs of BWE-FWE were obtained from the
double peak behaviour. They were selected upon some assumptions in the low rotor speed](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-88-2048.jpg)
![5.2 Rotor Modes 73
90◦ at 2.79 Hz (FWF). The collective component is not clearly identified. In (b), BWF
components are identified at 1.49 and 1.9 Hz, and a collective component is identified at
2.73 Hz. The animation of the remaining modes do not provide clear arguments to define
them as symmetric or asymmetric components.
There are discrepancies in the flap identification as shown in Table 5.6, where the re-
sults from the filtered analysis are given. Two BWF and two CF components are found
in the H2 and V27 data, respectively, where the frequencies of the BWF is different com-
pared to the BWF frequencies from HS2. The damping ratios from HS2 are higher than
those emerging from V27 and H2, and the largest similarity is found between V27 and H2.
V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature
- 1.49 - - 6.39 - 1st BWF
1.84 1.9 1.69 10.15 9.51 15.88 1st BWF
2.18 - - 6.08 - - 1st CF
2.63 2.73 2.59 7 7.64 11.92 1st CF
2.79 2.8 2.73 9.3 5.54 11.36 1st FWF
Table 5.6: Modal parameters comparison for rotor flap sensors at low rotor speed
High Rotor Speed
The identification for the high rotor speed case provides good agreement between the
V27 and H2 data, despite that the harmonics are not removed as seen in Figure 5.12
(a), because the TSA is not applied. The identified modes at frequencies above 4 Hz are
disregarded.
(a) V27 flapwise modes
(b) H2 model flapwise modes
Figure 5.12: Flapwise modes stability diagram at high rotor speed
From the identified components, there is only agreement regarding the BWF component](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-91-2048.jpg)
![74 Identifying Modal Parameters
between the V27 and H2, identified at 1.69 and 1.7 Hz, respectively, with very similar
damping ratios and an acceptable difference with respect to HS2. If the data is filtered
such as in Section 5.2.4, only the BWF is identified, at very similar frequencies (1.72 and
1.73 Hz) but with about half the damping ratios (9.83 and 7.65 %). No clear CF of FWF
components are found in the analysis, and thus, are disregarded in the collected results,
as shown in Table 5.7.
V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature
1.7 1.69 1.61 19.24 18.84 15.03 1st BWF
- - 2.55 - - 11.31 1st CF
- - 2.73 - - 10.18 1st FWF
Table 5.7: Flapwise modal parameters comparison at low rotor speed
5.2.5 Model Simplification
The flap identification yields uncertain results regarding the frequency of the components.
Several modes are identified but, based on their animation, only the BWF components
seemed to be identified properly. Therefore, a different approach is launched to provide a
more accurate identification of the flap modes, consisting in performing two simulations
with the H2 model - one with impulse excitation and the other using a random excitation.
It is recalled that OMA relies basically on three assumptions: 1) the system is linear;
2) the system is invariant with respect to time; and 3) forces applied are white noise
excitation. To investigate what the basic challenges with the flapwise mode identification
are, a simplified model is set up, where:
1) the geometry is simplified, as tilt is removed;
2) wind loading is replaced with a white noise excitation.
The new loading cases agrees well with the third OMA assumption. Subsequently,
the wind loading is gradually implemented to observe where the algorithm experiences
identification problems, thus shedding some light on what is the challenge regarding the
identification of flapwise modes. The low rotor speed model is used during this approach,
since it seemed the most challenging for OMA at first instance. The rotational speed has
been set to a constant equal to 32.24 RPM for the first simulations.
Impulse Loading
The impulse loading is applied in the flapwise direction to excite the blades. A force
magnitude of 100 N is used. The spectra of the impulse force is shown in Figure 5.13. As
seen, the spectra is fairly flat within the frequency range of 0-5 Hz.
An impulse excitation is applied every 150 s at the outer section of a blade. In total,
2 excitations per blade are imposed during the simulation time. The underlying idea is
to mimic a hammer impact at each blade with the turbine in operation. Each of the](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-92-2048.jpg)
![5.2 Rotor Modes 75
impulses lasts 0.05 seconds to achieve a flat spectrum, which agrees well with the white
noise assumption in the frequency range of interest.
0 1 0 2 0 3 0 4 0 5 0
Fre q u e n c y (Hz)
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
Psd
im p u ls e [-]
Figure 5.13: Impulse loading spectra
Random Loading
The random loading is set up to simulate white noise excitation. Compared to the im-
pulse loading, this case is probably better fulfilling the third OMA assumption, and it is
considered as a double-check for the identification of the flap modes.
The random loading is applied to the tower at different sections (towertop and 18
meters above ground) and on the blades at the outer and inner sections, both in the x
and y directions with respect to the wind. However, the tower loading was specified 20
times lower than the blade loading. This setup is required to avoid iteration issues in the
HAWC2 simulation. Figure 5.14 shows the spectra of the blade loading.
0 10 20 30 40 50
Frequency (Hz)
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
Psd
Random [-]
Figure 5.14: Spectra of the random function
Flapwise Signal Analysis
In Figure 5.15, the SVD of both cases are shown. One may observe that the impulse case
provides a clearer visualization of the peaks. However, the mutual agreement between
the cases is good.
In (a), two possible modes at 2.16 and 3.61 Hz are shown, where the last coincides with
the edgewise mode found in previous section. The same peaks are found in (b), with a](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-93-2048.jpg)
![78 Identifying Modal Parameters
wind/turbulence effect. With this premise, the impulse case is continued in the following
assuming that the OMA excitation assumption is fulfilled.
(a) Impulse case
(b) Random case
Figure 5.17: Flapwise modes stability diagram for simple model
I [Hz] R [Hz] HS2 [Hz] I [%] R [%] HS2 [%] Nomenclature
1.63 1.63 1.69 0.72 0.65 15.88 1st BWF
2.52 2.51 2.59 0.51 0.69 11.92 1st CF
2.68 2.68 2.73 0.45 0.46 11.36 1st FWF
Table 5.8: Comparison of frequencies and damping for simple approach
Detecting the Flapwise Problem
In the previous stage of the simplified model excitation procedure the rotor aerodynamics
were not included. The results showed that the flapwise components can be identified,
but also additional modes such as the tower modes.
The next step involves adding aerodynamic effects to the H2 model. Two cases are
also performed: the impulse case including deterministic wind excitation (IW) and only
deterministic wind excitation (W). A shear factor (power law) of 0.23 is set for the wind
conditions. No tilt angle, tower shadow or turbulence is included yet. The rotor speed is
still kept constant.
The modal identification in the IW case is able to identify almost the same frequencies
and damping ratios as shown in Figure 5.18 (a). However, no modes are identified in the
W case, similar to Figure 5.10 (b). Subsequently, tilt and tower shadow are included in
the simulation. Even with these changes, the identification algorithm cannot identify any
mode yet. Therefore, the wind speed is increased to 5.4 m/s, and with this new change
some components are identified, as shown in Figure 5.18 (b).](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-96-2048.jpg)
![5.2 Rotor Modes 79
(a) Impulse + wind 5 m/s
(b) Only wind 5.4 m/s
Figure 5.18: Stability diagrams of impulse+wind and only wind
The results show that the modal frequencies emerging from the W case in general
increase, whereas in damping ratio decreases, as observed in Table 5.9. Note that the
abbreviations refer to the impulse plus wind case (I+W) and to the only wind case (W).
I+W [Hz] W [Hz] HS2 [Hz] I+W [%] W [%] HS2 [%] Nomenclature
1.67 1.80 1.69 15.37 14.52 15.88 1st BWF
2.55 2.58 2.59 11.38 10.45 11.92 1st CF
2.70 2.88 2.73 11.83 10.65 11.36 1st FWF
Table 5.9: Comparison of frequencies and damping including wind
When the wind is included the effect of the aerodynamic damping can be clearly dis-
tinguished, varying from less than 1% to about 15%. The difference between damping
ratios might be a good hint to estimate the aerodynamic damping.
Bearing in mind these results, the next stage is to introduce turbulence loading. A
turbulence intensity of 2% is included in the simulation using the same turbulence box as
used in the low rotor speed model in Section 5.2.4. With the introduction of turbulence
in the simulation, the impulse does not significantly affect the results, and no components
can be identified. Therefore, a new impulse with larger magnitude is implemented (10
kN), this time exciting both flapwise and edgewise directions.
On the one hand, with this new impulse excitation the results show that components
can be identified, as shown in Figure 5.19 (a). On the other hand, the soft slope denoting
CF and FW has disappeared in (b), and therefore, they are not identified, similar to what
happened in Figure 5.10 and therefore motivating further investigation on this phenom-
ena. Including the turbulence seemingly means that the flapwise components start to be
masked, and hence difficult to identify. This fact pin downs that the turbulence loading
is the point of inflexion.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-97-2048.jpg)
![80 Identifying Modal Parameters
(a) Impulse + wind
(b) Only wind
Figure 5.19: Stability diagrams with low turbulence
The frequencies and damping ratios extracted from this last analysis are shown in
Table 5.10. It is observed that with the impulse excitation the results acceptably agree
with those from HS2.
I+W [Hz] W [Hz] HS2 [Hz] I+W [%] W [%] HS2 [%] Nomenclature
1.63 1.77 1.69 11.34 8.28 15.88 1st BWF
2.54 - 2.59 11.05 - 11.92 1st CF
2.71 - 2.73 10.91 - 11.36 1st FWF
Table 5.10: Comparison of frequencies and damping for turbulent cases
5.2.6 Flapwise Modes Identification Discussion
The flapwise analysis showed the difficulties of identifying flap modes with the current
method.
A minimum wind speed seems to be required to excite properly the flapwise modes if
the system, as seen in the low rotor speed case, where the modal frequencies and damping
resulted from filtering and assuming a specific frequency range for flap modes. The results
between V27 and H2 data are acceptable, but the damping ratios with respect to the HS2
are somehow different. It is also shown that the damping ratios are higher in the high
rotor speed case (such as in the HS2 results), but also that the SSI algorithm identified
several modes, from which only the BWF was identified from the V27 and H2 data,
leading to the simplified approach. It has to be taken into account that the assumptions
in HS2 are different from V27 and H2 cases.
With respect to the simplified model, the results in the most simple case are successful
for each of the two approaches, giving the same frequencies and very close damping ratios.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-98-2048.jpg)
![Bibliography
[1] Tcherniak D., Chauhan S., Basurko J., Salgado O., Carcangiu C. E., and Rossetti M.
Application of OMA to Operational Wind Turbine. Proceedings - 4th International
Operational Modal Analysis Conference (IOMAC’11), 2011.
[2] Hansen M. H. Aeroelastic instability problems for wind turbines. Wind Energy, Vol.
10, No. 6, 2007, p. 551-577, 2007.
[3] Brandt A. Noise and Vibration Analysis: Signal Analysis and Experimental Proce-
dures. John Wiley & Sons, Ltd., 2010.
[4] Hansen M. H. Aeroelastic stability analysis of wind turbines using an eigenvalue
approach. Wind Energy, Vol. 7, No. 2, 2004, p. 133-143, 2004.
[5] Tcherniak D. and Larsen G. C. Application of OMA to an Operating Wind Tur-
bine: now including Vibration Data from the Blades. Proceedings - 5th International
Operational Modal Analysis Conference (IOMAC’13), 2013.
[6] Yang S., Tcherniak D., and Allen M. S. Modal Analysis of Rotating Wind Turbine
Using Multiblade Coordinate Transformation and Harmonic Power Spectrum. Topics
in Modal Analysis I, Volume 7, p. 77-92 Proceedings of the 32nd IMAC, A Conference
and Exposition on Structural Dynamics., 2014.
[7] Jacob T., Tcherniak D., and Castiglione R. Harmonic Removal as a Pre-processing
Step for Operational Modal Analysis: Application to Operating Gearbox Data. Br¨uel
& Kjær Conference Paper, 2014.
[8] James G. H., Carne T. G., and Lauffer J. P. The natural excitation technique
(NExT) for modal parameter extraction from operating wind turbines. Technical
Report SAND92-1666, Sandia National Laboratories, 1993.
[9] Johnson W. Helicopter theory. Princeton University Press, New Jersey, 1980.
[10] Peters D. A. Fast Floquet theory and trim for multi-bladed rotorcraft. Journal of the
American Helicopter Society; 39: 82-89, 1994.
85](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-103-2048.jpg)
![86 BIBLIOGRAPHY
[11] Tcherniak D., Chauhan S., and Hansen M. H. Applicability Limits of Operational
Modal Analysis to Operational Wind Turbines. Structural Dynamics and Renewable
Energy. Vol. 1 Society for Experimental Mechanics, 2011. p. 317-327 (Conference
Proceedings of the Society for Experimental Mechanics Series; No. 10), 2010.
[12] Van Overschee P. and De Moor B. Subspace identification for linear systems: theory,
implementation, applications. Kluwer Academic Publishers, 1996.
[13] Hansen M. H., Thomsen K., Fuglsang P., and Knudsen T. Two Methods for Esti-
mating Aeroelastic Damping of Operational Wind Turbine Modes from Experiments.
Wind Energy, Vol. 9, No. 1-2, 2006, p. 179-191, 2006.
[14] Palle P. and Rosenow S. E. Operational Modal Analysis of a Wind Turbine Mainframe
using Crystal Clear SSI. Structural Dynamics and Renewable Energy, Volume 1.
Proceedings of the 28th IMAC, A Conference on Structural Dynamics, 2010.
[15] Van Der Valk P. L. C. and Ogno M. G. L. Identifying Structural Parameters of an
Idling Offshore Wind Turbine Using Operational Modal Analysis. Dynamics of Civil
Structures, Volume 4, p. 271-281 Proceedings of the 32nd IMAC, A Conference and
Exposition on Structural Dynamics., 2014.
[16] Marinone T., Cloutier D., LeBlanc B., Carne T., and Palle P. Artificial and Natural
Excitation Testing of SWiFT Vestas V27 Wind Turbines. Proceedings of the 32th
International Modal Analysis Conference (IMAC) Orlando, Florida USA, 2014.
[17] Hansen M. H. Improved modal dynamics of wind turbines to avoid stall-induced
vibrations. Wind Energy, Vol. 6, 2003, p. 179-195, 2003.
[18] Bir G. Multiblade coordinate transformation and its application to wind turbine
analysis. ASME Wind Energy Simposium, Reno, Nevada, 2008.
[19] Berg J., Mann J., and Nielsen M. Notes for DTU course 46100: Introduction to
micro meteorology for wind energy. DTU Wind Energy E-0009 (EN) ISBN 978-87-
92898-15-5, 2013.
[20] MATLAB. version 8.2 (R2013b). Natick, Massachusetts: The MathWorks Inc.,
2013.
[21] Batel M. Operational Modal Analysis - Another Way of Doing Modal Testing. Journal
of Sound and Vibration, 2002.
[22] Hansen M. H., Gaunaa M., and Aagaard Madsen H. A Beddoes-Leishman type
dynamic stall model in state-space and indicial formulations. Forskningscenter Risoe.
Risoe-R, no. 1354(EN), 2004.
[23] Br¨uel & Kjær. Product Data Operational Modal Analysis Type 7760. www.bksv.com
- Accessed August, 2014.
[24] Petersen J. T. The aeroelastic code HawC - model and comparisons. In Pedersen
BM, editor, State of the Art of Aeroelastic Codes for Wind Turbine Calculations,
volume Annex XI, pages 129-135, Lyngby. International Energy Agency, Technical
University of Denmark, 1996.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-104-2048.jpg)
![BIBLIOGRAPHY 87
[25] S. M. Petersen. Wind Turbine Test Vestas V27-225 kW. Technical Report Risø-M-
2861, Risø National Laboratory, DK-4000 Roskilde, Denmark, 1990.
[26] Larsen T. J. and Hansen A. M. How 2 HAWC2, the user’s manual. Ris-R-1597(ver.
4-4)(EN), 2013.
[27] Petersen J. T. Kinematically Nonlinear Finite Element Model of a Horizontal Axis
Wind Turbine. PhD Thesis, Risø National Laboratory, DK-4000 Roskilde, Denmark,
1990.
[28] Brincker R. and Andersen P. Understanding stochastic subspace identification. Con-
ference Proceedings : IMAC-XXIV : A Conference & Exposition on Structural Dy-
namics. Society for Experimental Mechanics, 2006.
[29] Brincker R. and Andersen P. The stochastic subspace identification techniques.
www.svibs.com, 2001.
[30] Larsen G. C. Aeroelastic simulations as basis for V27 accelerometer instrumentation.
Internal Report. Ris DTU Wind Energy. Aeroelastic Design Section, 2012.
[31] Jacobsen N. J. and Andersen P. Operational Modal Analysis on Structures with
Rotating Parts. ISMA Conference Paper no. 268, 2008.
[32] IEC technical committee 88. IEC 61400-1 Ed.3: Wind turbines - Part 1: Design
requirements. International Electrotechnical Commission, 2005.](https://image.slidesharecdn.com/241a832d-f56c-4326-9575-fec098bad6cf-150707200632-lva1-app6891/75/MSc_Thesis_ORR-105-2048.jpg)
MSc_Thesis_ORR
- 1. Identification of ModalParameters Applying Operational Modal Analysis on a Full Scale Operating Vestas V27 Wind Turbine ´Oscar Ram´ırez Reques´on DTU Wind Energy MSc Thesis September 28th , 2014
- 2. Title: Identification ofModal Parameters Applying Operational Modal Analysis on a Full Scale Operating Vestas V27 Wind Turbine Author: ´Oscar Ram´ırez Reques´on Resume (max. 2000 char.): This thesis deals with modal analysis of a full scale Ves- tas V27 wind turbine under operation using a stochas- tic subspace identification algorithm. The wind tur- bine acceleration data used originates from a measure- ment campaign conducted at Risø between late 2012 and early 2013. Using meteorological data as well as data from sensors located in the wind turbine, an aeroelastic model is implemented in the HAWC2 plat- form to mimic the behaviour of the V27 wind turbine at specific recording periods. Identification of the wind turbine modal parameters is hence performed on both the real and the synthetic data, and finally compared to the analogue results from a linear stability tool. Re- sults and findings are discussed, and conclusions are drawn. September 28th , 2014 Project Period: 2014.04.08 2014.09.28 ECTS: 30 Education: Master of Science Division: Wind Energy Supervisor: Gunner C. Larsen Supervisor (external): Dmitri Tcherniak Ivan Sønderby Comments: This report is submitted as partial fulfillment of the requirements for graduation in the above education at the Technical University of Den- mark. Cover: Vestas V27 wind turbine at Risø test site. Danmarks Tekniske Universitet DTU Vindenergi Frederiksborgvej 399 Bygning 118 4000 Roskilde Danmark www.vindenergi.dtu.dk
- 3. Identification of ModalParameters Applying Operational Modal Analysis on a Full Scale Operating Vestas V27 Wind Turbine ´Oscar Ram´ırez Reques´on
- 5. Abstract This thesis dealswith the modal analysis of a full scale Vestas V27 wind turbine under operation using a stochastic subspace identification algorithm. The wind turbine acceleration data used originally comes from an EUDP project, in which the measurement campaign was conducted at Risø between late 2012 and early 2013. Using meteorological data in conjunction with data from sensors located in the wind turbine, an aeroelastic model is implemented in the HAWC2 platform to simulate the behaviour of the V27 wind turbine at specific recording periods. Operational modal analysis together with the Coleman transformation is hence per- formed to identify the wind turbine modal parameters on both the real and the simulated data, and finally compared to the analogue results from a linear stability tool. This comparison reveals that the identification of modes is possible, despite the diffi- culties that certain modes present, and that simulations using aeroelastic codes and theo- retical predictions obtained from aeroelastic modal analysis tools could be recommended. However, challenges are also found, which are presumably related to rotor anisotropy according to a tested simulation-based case. i
- 7. Acknowledgements I would liketo thank all the individuals from Risø, DTU and friends that, even not being mentioned here, have contributed to this work in a major or minor extent. I really appreciate their support during these last two years that have formed my education in the wind energy master. In particular, my most sincere gratitude to Gunner C. Larsen, main advisor from Risø DTU Wind Energy, who kindly embraced this thesis in a period of uncertainty, and provided guidance, ideas, fruitful discussions and encouragement along the whole project. I am also very thankful to Dmitri Tcherniak, advisor from Br¨uel & Kjær, for his com- mitment and involvement in the project, providing codes, data, experience and advices during this productive period. In extension, to Br¨uel & Kjær Innovation group for the license of the Type 7760 and for hosting me so many times always in an excellent envi- ronment. I would also like to thank Ivan Sønderby, advisor from Vestas Wind Systems, for orig- inally suggesting the topic and for his availability in our multiple supervisory meetings despite the distance. Finally, thank you to the AED group, from Risø DTU Wind Energy, for offering me a place to work, the opportunity to meet world-class top researchers and professionals in the field of wind energy and the help provided on the specific issues encountered along this work. Last but not least, to Esther for her endless love and support. Risø, Denmark ´Oscar Ram´ırez Reques´on September 28th, 2014 iii
- 9. Contents Abstract i Acknowledgements iii Listof Figures ix List of Tables xiii 1 Introduction 1 1.1 Operational Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Motivation of the Work and Goals . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Background Knowledge 7 2.1 Wind Turbine Modal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Dynamics of the Main Components . . . . . . . . . . . . . . . . . . 7 2.1.2 Whirling Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.4 Coleman Transformation . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Time Domain and Frequency Domain . . . . . . . . . . . . . . . . 14 v
- 10. vi Contents 2.2.2 PowerSpectral Density . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . 17 2.3 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 HAWC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 HAWCStab2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.3 Br¨uel & Kjær Operational Modal Analysis Type 7760 . . . . . . . 21 3 Implementation of the Numerical Model 23 3.1 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.1 The Wind Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.2 Site Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.3 Measurement Campaign . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.4 Selection of Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Defining the Model in HAWC2 . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.1 Simulation Setup in HAWC2 . . . . . . . . . . . . . . . . . . . . . 32 3.2.2 Eigenvalue Analysis of the H2 model . . . . . . . . . . . . . . . . . 34 3.3 Estimation of Modal Parameters in HAWCStab2 . . . . . . . . . . . . . . 36 3.3.1 Tool Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3.2 Comparison of the H2 Model and HAWCStab2 . . . . . . . . . . . 37 3.3.3 Structural and Aeroelastic Modal Analyses . . . . . . . . . . . . . 38 4 Signal Analysis 43 4.1 Dynamic System Behaviour I: PSD . . . . . . . . . . . . . . . . . . . . . . 44 4.1.1 Nacelle Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1.2 Blades Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Dynamic System Behaviour II: SVD . . . . . . . . . . . . . . . . . . . . . 52 4.2.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 Dynamic System Behaviour III: MBC . . . . . . . . . . . . . . . . . . . . 56 4.3.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4 Time Synchronous Averaging . . . . . . . . . . . . . . . . . . . . . . . . . 59
- 11. Contents vii 5 IdentifyingModal Parameters 61 5.1 Tower Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1.1 Low Rotor Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.1.2 High Rotor Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.1.3 Tower Modes Identification Discussion . . . . . . . . . . . . . . . . 65 5.2 Rotor Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2.1 Edgewise Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.2.2 Rotor Anisotropic Effects Induced in the H2 Model . . . . . . . . . 69 5.2.3 Edgewise Modes Identification Discussion . . . . . . . . . . . . . . 70 5.2.4 Flapwise Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2.5 Model Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2.6 Flapwise Modes Identification Discussion . . . . . . . . . . . . . . 80 6 Conclusions 83 6.1 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Bibliography 85 Appendices 89
- 12.
- 13. List of Figures 2.1Out-of-plane and in-plane with respect to the pitch angle. Source [1] . . . 8 2.2 Example of standstill asymmetric modes and Campbell diagram. Source [2] 9 2.3 Terminology applied to wind turbines DOF. Source [2] . . . . . . . . . . . 11 2.4 Asymmetric rotor motion indicating reaction forces. Source [2] . . . . . . 13 2.5 Example of signal divided into m segments with 50% overlap. Source [3] . 17 2.6 Beam element model in HAWCStab2 (only one blade). Source [4] . . . . . 20 3.1 Summary of the methodology . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Aerial view of V27 site location . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Sensors location and orientation. Source: [5] . . . . . . . . . . . . . . . . . 26 3.4 Nacelle sensors in the V27. Source [6] . . . . . . . . . . . . . . . . . . . . 27 3.5 View of V27 and met mast from neighbouring Nordtank . . . . . . . . . . 28 3.6 V27 and met combined data . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.7 Wind profile for low and high rotor speed data sets . . . . . . . . . . . . . 32 3.8 Mode shapes no. 1 to 8 in HAWC2 . . . . . . . . . . . . . . . . . . . . . . 36 3.9 Frequencies vs. wind speed . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.10 Damping ratios vs. wind speed. All modes . . . . . . . . . . . . . . . . . . 39 3.11 Damping ratios vs. wind speed. Low damped modes . . . . . . . . . . . . 39 3.12 Operational data assumed in HS2 . . . . . . . . . . . . . . . . . . . . . . . 40 4.1 Tri-axial accelerometer coordinates. Representation of the nacelle view from the top 44 4.2 PSD nacelle x-direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ix
- 14. x LIST OFFIGURES 4.3 PSD nacelle y-direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 PSD blades signals V27 outer section, low rotor speed . . . . . . . . . . . 48 4.5 PSD blades signals H2 outer section, low rotor speed . . . . . . . . . . . . 48 4.6 PSD blades signals V27 inner section, low rotor speed . . . . . . . . . . . 49 4.7 PSD blades signals H2 inner section, low rotor speed . . . . . . . . . . . . 50 4.8 PSD blades signals V27 outer section, high rotor speed . . . . . . . . . . . 50 4.9 PSD blades signals H2 outer section, high rotor speed . . . . . . . . . . . 51 4.10 PSD blades signals V27 inner section, high rotor speed . . . . . . . . . . . 51 4.11 PSD blades signals H2 inner section, high rotor speed . . . . . . . . . . . 52 4.12 SVD applied on edgewise signals at low rotor speed . . . . . . . . . . . . . 53 4.13 SVD applied on flapwise signals at low rotor speed . . . . . . . . . . . . . 53 4.14 SVD applied on edgewise signals at high rotor speed . . . . . . . . . . . . 55 4.15 SVD applied on flapwise signals at high rotor speed . . . . . . . . . . . . 55 4.16 MBC applied on edgewise signals at low rotor speed . . . . . . . . . . . . 56 4.17 MBC applied on flapwise signals at low rotor speed . . . . . . . . . . . . . 57 4.18 MBC applied on edgewise signals at high rotor speed . . . . . . . . . . . . 58 4.19 MBC applied on flapwise signals at high rotor speed . . . . . . . . . . . . 59 4.20 TSA procedure. Source [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.1 Simple geometrical representation of the V27 . . . . . . . . . . . . . . . . 62 5.2 Tower modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.3 Tower modes stability diagram at low rotor speed . . . . . . . . . . . . . . 63 5.4 Tower modes stability diagram at high rotor speed . . . . . . . . . . . . . 64 5.5 Geometrical representation of the MBC coordinates . . . . . . . . . . . . 66 5.6 Edgewise modes stability diagram at low rotor speed . . . . . . . . . . . . 67 5.7 Phase angle difference between asymmetric modes . . . . . . . . . . . . . 67 5.8 Edgewise modes stability diagram at high rotor speed . . . . . . . . . . . 68 5.9 Smooth anisotropic effect on the model . . . . . . . . . . . . . . . . . . . 70 5.10 Flapwise modes stability diagram at low rotor speed . . . . . . . . . . . . 72 5.11 Flapwise modes filtered stability diagram at low rotor speed . . . . . . . . 72 5.12 Flapwise modes stability diagram at high rotor speed . . . . . . . . . . . . 73 5.13 Impulse loading spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.14 Spectra of the random function . . . . . . . . . . . . . . . . . . . . . . . . 75 5.15 SVD on the blade signals for impulse and random cases . . . . . . . . . . 76
- 15. LIST OF FIGURESxi 5.16 MBC on the blade signals for impulse and random cases . . . . . . . . . . 77 5.17 Flapwise modes stability diagram for simple model . . . . . . . . . . . . . 78 5.18 Stability diagrams of impulse+wind and only wind . . . . . . . . . . . . . 79 5.19 Stability diagrams with low turbulence . . . . . . . . . . . . . . . . . . . . 80
- 16. xii LIST OFFIGURES
- 17. List of Tables 3.1Technical specification of the V27 . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Example of combined V27 and met measurements . . . . . . . . . . . . . 29 3.3 Selected data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 H2 operating points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 Stiffness contribution coefficients to damping . . . . . . . . . . . . . . . . 35 3.6 Results of structural eigenanalysis with H2 . . . . . . . . . . . . . . . . . 35 3.7 HS2 operating points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.8 HS2 vs. H2 modal frequencies and damping in standstill . . . . . . . . . . 37 3.9 Aeroelastic frequencies and damping in HS2 at 5 and 11 m/s . . . . . . . 40 5.1 Modal parameters comparison for the tower sensors at low rotor speed . . 64 5.2 Modal parameters comparison for the tower sensors at high rotor speed . 65 5.3 Modal parameters comparison for rotor edge sensors at low rotor speed . 68 5.4 Modal parameters comparison for rotor edge sensors at high rotor speed . 69 5.5 Edge frequencies and damping from tower and blade sensors . . . . . . . . 70 5.6 Modal parameters comparison for rotor flap sensors at low rotor speed . . 73 5.7 Flapwise modal parameters comparison at low rotor speed . . . . . . . . . 74 5.8 Comparison of frequencies and damping for simple approach . . . . . . . . 78 5.9 Comparison of frequencies and damping including wind . . . . . . . . . . 79 5.10 Comparison of frequencies and damping for turbulent cases . . . . . . . . 80 xiii
- 18. xiv LIST OFTABLES
- 19. Chapter 1 Introduction The growingworld energy consumption demand added to the climatic change effects call on a sustainable future based on renewable energies. Among the different renewable energy technologies, wind energy appears as the more promising to compete with fossil fuel based energy systems. Since wind turbines size is constantly increasing, technical developments are required to achieve competitive machines able to lower the energy price in the market and to face traditional energy systems. The structural design is one of the challenges associated with wind turbines. If the size increases, relatively less material has to be used in order to keep the costs down, and subsequently, higher dynamic response might potentially occur in the structure. Hence, designers are required to understand wind turbine dynamics to succeed in a better balance between materials, performance and costs. The dynamic characterization is generally done in terms of modal parameters - modal frequencies, damping and mode shapes - where a proper estimation is essential, for instance, to avoid inconvenient cases such as coupling of modal frequencies with multiples of the rotational speed or to predict the fatigue loads the structure suffers from. This work presents the results found on the identification of modal parameters on a full-scale Vestas V27 wind turbine (hereafter, V27) using particular data sets originated from a measurement campaign initiated in 2012. These modal parameters are compared to those resulting from a nonlinear aeroelastic model of this turbine to which the same identification procedure is applied. This model will be called ”H2” in the following for simplicity. Both results will be checked against modal parameters theoretically predicted using a linear aeroelastic stability tool. 1.1 Operational Modal Analysis Modal analysis is the field of measuring and analysing dynamic properties of structures. Classically, a vibration test could be conducted on a structure by e.g. using a hammer 1
- 20. 2 Introduction as inputexcitation and reading output measurements at different locations of the struc- ture. Based on the input/output transfer function, the dynamic characteristics of the structure can be identified. This methodology can be used for relatively small structures subjected to a known forcing, but the challenge arises when huge structures, subjected to stochastic loading distributed over a substantial part of the structure, are brought to. When structures are large enough, as is the case with current wind turbines, introducing an artificial excitation could mean expensive costs associated to the test implementation if possible at all. In the 1990’s, James et al. [8] laid down the foundations of Opera- tional Modal Analysis (OMA), also known as Output-Only Modal Analysis or Ambient Modal Identification. This method refers to techniques where the excitation forces are not measured, and modal parameters are estimated only from the output responses. It has attracted much attention in the civil, mechanical and aerospace engineering because of its advantages for identifying modal properties of a structure solely based on its measured responses during operating conditions. Main advantages are: Unlike the classical modal testing approach, it does not need to measure the exci- tation force; The measured response is representative of the real operating conditions of the structure; The operating structure does not need to be interrupted to perform the test; The OMA approach can deal with inevitable unknown (stochastic) external loading; The method can be also used for other purposes such as damage detection and health monitoring of structures. Due to these reasons, one may think that OMA is an excellent technique to identify modal parameters. However, some requirements must be fulfilled: the structure is linear; the structure is time invariant; the operational excitation forces must in turn fulfil the following requisites a) they have broadband frequency spectra; b) they are uncorrelated; c) they are distributed over the entire structure. Despite that an operating wind turbine may violate all of these, some approximations must be made in order to alleviate these requirements and find an scenario where the method could be applied with a degree of confidence. The linearisation of the system is achieved by assuming equilibrium at operating points, e.g. pitch angle, torque or rotational speed at a particular wind speed. Wind turbines are not time invariant systems: they are composed of various substructures that moves one with respect to the other - for instance, the pitching of the blades or the yawing of
- 21. 1.2 Damping 3 thenacelle. Selecting time stamps where there is none or minor activity of these motions is a manner to partially cope with this issue. The major drawback, however, is the rotor rotation, which makes the mass, stiffness and damping matrices in the equations of motion to be time dependent. One way to overcome this matter is by means of the so-called Coleman transformation method [9, 10], which converts the time-variant system into time-invariant under particular conditions. When it comes to the excitation forces, the turbulent wind seems to suit the third OMA premise very well, but the rotation of the rotor changes the aerodynamic forces and the broadband spectra of the turbulent wind displays peaks at the rotor harmonics. This effect is pointed out by Tcherniak et al. in [11]. From the different algorithms OMA embraces, the Stochastic Subspace Identification (SSI) technique, described by Overschee and De Moor [12], appeals most to this work based on the success of previous research studies as pointed out in the coming Section 1.3. In the present work, a commercial software package from Br¨uel & Kjær is used that has already implemented this method. 1.2 Damping Damping estimation appears as of great importance among the modal properties one may want to estimate using OMA. Its definition may be stated as the dissipation of energy from a structure that is vibrating. The accurate prediction of damping has a large effect on the lifetime and dynamic response of the system. Moreover, in wind turbines damping consists in structural and aerodynamic contributions (from the aerodynamic forces). Damping also plays a relevant role in the stability of wind turbines. In this sense, Hansen [2] commented on the aeroelastic instabilities that have occurred and may still occur for commercial wind turbines. Modal parameters of the vibration modes of operating wind turbines can be computed with or without accounting for the aerodynamic forces, since aerodynamic damping is a main source of uncertainty. The unloaded turbine defines the basis of its modal dynamics, with the structural modes as the foundation of the aeroelastic dynamic behaviour. When the turbine is loaded, the vibration modes define its aeroelastic stability properties by the damping of these modes. However, damping is always difficult to estimate with accuracy, and even more when aerodynamic forces are included. In other words, it poses a challenge to differentiate both damping contributions (structural and aerodynamic) from the total damping. In the present work, the total damping in the rotor plane and out of the rotor plane is investigated, i.e. lateral and longitudinal to the wind direction respectfully. The first case includes tower side-to-side mode and rotor edgewise modes, and the second case refers to tower fore-aft mode and rotor flapwise modes - meaning lowly pitched blades when referring to the rotor modes. While there is some optimism in identifying reasonable damping ratios in the in-plane modes from previous research studies [6], it is foreseen a demanding task to identify the out of plane modes due to the high contribution of aerodynamic damping in that direction.
- 22. 4 Introduction 1.3 Stateof the Art Being a hot topic nowadays, several research studies have been conducted previously about OMA applied to wind turbines, of which only a few have used the SSI technique, and even less of them, have identified modal parameters from real measurements. Below is listed research studies which match these conditions under particular cases at the time of writing this thesis: Hansen et al. [13] presented an estimation of aeroelastic damping on a NM80 2.75 MW wind turbine operating prototype using strain gauges. It was concluded that the SSI method can handle the deterministic excitation from wind, and the modal frequencies and damping of the first tower and first edgewise whirling modes were extracted. Tcherniak et al. [1] applied the technique to an operating ECO 100 Alstom wind turbine with accelerometers in the nacelle and tower, reporting that is possible to identify some rotor modes using only these signals. Andersen and Rosenow [14] investigated the dynamics of a parked Fuhrl¨ander AG 2.5 MW with sensors in the main frame. It was also reported that some rotor modes can be identified only using sensors in the nacelle. Tcherniak and Larsen [5] described the technical challenges regarding blade instru- mentation and data acquisition, the processing of data to convert the system to time-invariant and assessed preliminary results on three cases: parked, idle and normal operation. Yang et al.[6] compared the Coleman transformation followed by the SSI technique with the harmonic power spectrum (HPS), method based on an extension of modal analysis to linear time periodic systems, using blades accelerations of an operating wind turbine. The paper focuses on a comparison of results for the first edgewise modes, concluding that the Coleman transformation could lead to erroneous results due to rotor anisotropy [5]. Van Der Valk and Ogno [15] identified modal parameters in an idling Siemens SWT- 3.6 MW offshore wind turbine with several strain gauges and one accelerometer. The four first global eigenfrequencies were identified, but also concluded that the best results came from the accelerometer located in the nacelle. Marinone et al. [16] compared experimental modal results of two parked Vestas V27 wind turbines, where OMA is only applied to one of them. The modes found using OMA were highly correlated with the experimental modal results. Apparently, there are only two research studies gathering information both from the sensors mounted in the non-rotating frame (tower) and the rotating frame (blades). These works shared the same data for similar purposes, with a special emphasis on the edgewise modes.
- 23. 1.4 Motivation ofthe Work and Goals 5 1.4 Motivation of the Work and Goals This thesis may be seen as natural continuation of the works covered in [5, 6]. Based on the same real measurements and methods (although not HPS), it is intended to go one step ahead and identify not only the first edgewise modes, but also the first tower and flapwise modes. Some questions, nonetheless, has to be addressed based on the previous research studies: 1) Is it possible to apply the Coleman transformation to anisotropic rotors successfully? 2) Can all rotor modes be identified by only using nacelle sensors? 3) To what extent will the aerodynamic loads influenced by the rotation of the rotor allow to identify modal parameters? 4) And in connection to the previous question, is it possible to identify the flapwise modes, with a certain confidence? There are two main reasons motivating the work in this thesis. To the author’s knowl- edge, there is not yet presented any research of modal analysis on a full-scale operating wind turbine combining both tower and blade signals for the identification of tower and rotor modes. This thesis attempts to identify such modes with the limitation that tor- sional modes are excluded. This fact leads to a very attractive topic to be investigated and is the major motivation for this thesis. The second motivation relies on the comparison of the OMA results from the V27 with numerical models. A model will be implemented in HAWC2 attempting to simulate the dynamic characteristics of the V27. Besides, a theoretical prediction of modal parameters is conducted using HAWCStab2, which is an aeroelastic stability tool that includes the Coleman transformation and solves the corresponding eigenvalue problem under symmet- ric rotor assumption. If the resulting modal parameters are similar among them, it may suggest these tools as a helpful support for OMA in wind turbines. The three-fold comparison may help to understand the challenges that modal parame- ters estimation poses when dealing with real life operating wind turbines. Despite the Vestas V27 model is an old-fashioned wind turbine, it is physically similar to modern wind turbines in the sense that it features pitch- (limited) and yaw-control. All the attention is paid on the frequency range (0 to 5 Hz) from the accelerations output spectra that, not only embraces all the modes under investigation, but also correlates with the first modes encountered in modern wind turbines (typically within the range 0 to 2-3 Hz). 1.5 Thesis Structure This thesis is the result of the work done over the last five months. The introduction to the topic and the related theory are covered in the following chapters, reflecting the
- 24. 6 Introduction research doneduring the first stage of the thesis. The project itself starts at Chapter 3 and runs up to Chapter 5. More specifically, the thesis is structured as follows: In Chapter 2, the theory behind the methodology followed to describe system behaviour is reviewed to introduce the reader to the concepts and methods used along the report. Chapter 3 gives details regarding the measurement campaign, model implementation in HAWC2 and provides theoretical predictions in HAWCStab2. In Chapter 4 the sensor signals are analysed in a step-by-step procedure useful to consistently describe traces of modes prior to OMA. The SSI identification technique is applied in Chapter 5 and results are discussed. Finally, conclusions are drawn and further research is suggested.
- 25. Chapter 2 Background Knowledge Thischapter introduces the reader to the fundamental concepts and methods dealt within this thesis. A review on modal dynamics of wind turbines is presented first. Here, the Coleman transformation is introduced. Then, the methods to be applied on the experi- mental and simulated data are presented. Lastly, the tools employed for the implementa- tion of the model and the modal parameters prediction under symmetric rotor assumption are described. This chapter will be useful to review while going through Chapter 4 and Chapter 5 where the main concepts are applied to real and simulated data. 2.1 Wind Turbine Modal Dynamics The typical first global modes of a wind turbine are presented in this section. This information is relevant to following future chapters, when the modal frequencies and damping ratios for each mode are discussed. First part describes wind turbine mode shapes inspired from Hansen [2]. Next, the Coleman transformation is explained based on [17, 18], and its premises are discussed for the application on this work. 2.1.1 Dynamics of the Main Components The dynamics of wind turbines are governed by three main substructures: tower, drive- train and rotor. Tower The tower deflects longitudinally and laterally with respect to the wind direction. These two bending modes also interact due to the gyroscopic coupling of towertop and rotor, 7
- 26. 8 Background Knowledge thatmay lead to elliptical motion of the tower. Also, the tower torsion is noticed in the yaw motion of nacelle and rotor. Drivetrain The drivetrain rotation makes the blades to rotate around its axis. Taking one blade as the reference, the azimuth angle is the particular angular position of that blade at that precise instant. The drivetrain consists of a main shaft, a gearbox and a generator, and introduces a torsional mode between rotor hub and generator coupled with blade simultaneous edgewise bending (or flapwise bending dependent on the pitch angle). Rotor A single blade cantilevered at the hub has three family modes: flapwise, edgewise and torsion; where the modes of a blade can be a mixture of these families. The term rotor is referred as the ensemble of blades attached to the same hub. The dynamics of blades are normally expressed in the rotating frame, however, the excitations on the rotor occur in the non-rotating frame, in the same way that the tower-nacelle substructures feel the dynamics of the rotor and not the individual blades effect. Hence, the rotor dynamics are considered as a whole and all modes of the individual blades contribute to the dynamics of the entire wind turbine. The dynamics of a 3-bladed rotor are described depending on the flapwise, edgewise and torsional modes of the blades, and can be differentiated by 3 rotor modes: one symmetric and two asymmetric. The symmetric mode refers to the case where all three blades deflect symmetrically (flapwise, edgewise or torsion). The asymmetric modes refer to the case where two blades deflect in the same direction and one opposite to them. Besides, the flapwise and edgewise modes are dependent of the pitch angle. Figure 2.1 is provided to support the following explanation. Figure 2.1: Out-of-plane and in-plane with respect to the pitch angle. Source [1] For instance, if the pitch angle is set to minimum, e.g. 0◦ (typically in operating conditions, before pitch controller action), the blade flapwise motion is out of the rotor plane, while the edgewise mode is in the rotor plane. If the pitch angle is maximum (typically in parked conditions), the motion of the flapwise mode is rotor in-plane, while the edgewise is out of the rotor plane.
- 27. 2.1 Wind TurbineModal Dynamics 9 2.1.2 Whirling Modes At standstill, the system is time-invariant and all OMA assumptions are fulfilled, for which one can expect reliable results. When the wind turbine is under operation the system is time-variant and therefore modal analysis cannot be applied. The so-called Coleman transformation is used to transform the rotating blade coordinates to the non-rotating frame, and under certain assumptions, the system can be treated as time-invariant. Figure 2.2 displays the nomenclature of the asymmetric modes at standstill conditions and what happens when the rotor rotates. It can be seen that the torsional modes are not displayed in the figure since they are typically at higher frequencies. In this work, the torsional modes are not considered, since it is assumed they are out of the frequency range of interest. Therefore, it is natural to explain from now on only the flapwise and edgewise modes. Figure 2.2: Example of standstill asymmetric modes and Campbell diagram. Source [2] The global mode shapes basically consists of a combination of tower and blades bending modes. The order of the very first modes is often the same: first tower modes, then, first flapwise and edgewise modes. The first longitudinal tower mode may have slightly lower frequency than the lateral mode due to larger inertia from the tilting of the rotor. There are also modes coupling between tower and rotor. The longitudinal and lateral tower modes are translated to tilting and rolling of the nacelle, respectively, due to blades flapwise bending and the gyroscopic effect of the drivetrain plus the blades edgewise bending. As Hansen describes in greater depth [2], the flapwise and edgewise modes form pairs and their natural frequencies are slightly different at standstill. This occurs because of the different vertical and horizontal flexibilities acting on the rotor support. The yaw component has lower frequency than the tilt component since the tower is stiffer in tilt than in yaw. Similarly, the flapwise modes have slightly lower frequency that the edgewise
- 28. 10 Background Knowledge modessince these latter are stiffer. The sequence of the edgewise modes is given by the vertical and horizontal stiffness of the rotor support. When the rotor is rotating, these pairs of frequencies split in the ground-fixed frame by ±Ω where the center is approx. the blade frequency in the considered deflection shape. The splitting occurs due to the inverse transformation of the frequencies in the ground- fixed frame to the rotating frame. This means that an observer in the ground-fixed frame, e.g. towertop, will measure the same frequency ω for each of the split components while an observer in a blade will measure ω − Ω or ω + Ω depending on the component. This phenomena is directly linked to the Coleman transformation in Section 2.1.4, where the equations may clarify the aforementioned explanation. 2.1.3 Terminology To improve readability, a brief description of each mode is shown using typical nomencla- ture for the modes under discussion in this thesis. Note that the asymmetric components of modes denote two different nomenclatures, depending on the state of the turbine, if at standstill (first term on brackets/abbr.) or operating (second term on brackets/abbr.): Tower longitudinal bending (tower fore-aft): tower moves fore and aft with respect to the wind direction. Abbr.: TFA Tower lateral bending (tower side-side): tower moves side to side with respect to the wind direction. Abbr.: TSS Asymmetric flapwise (yaw/backward whirling): nacelle and rotor move around the yaw axis. Abbr.: YF/BWF Asymmetric flapwise (tilt/forward whirling): rotor and nacelle tilt up and down. Abbr.: TF/FWF Symmetric flapwise (collective): blades deflect same direction simultaneously. Abbr.: CF Asymmetric edgewise (vertical/backward whirling): asymmetric blades regressive deflection. Abbr.: VE/BWE Asymmetric edgewise (horizontal/forward whirling): asymmetric blades progressive deflection. Abbr.: HE/FWE Drivetrain torsion (collective edgewise): blades deflect edgewise symmetrical and simultaneously. Abbr.: DT/CE A graphical representation of the degrees of freedom (DOF) defined by Hansen [2] is found in Figure 2.3 to back up the previous nomenclature.
- 29. 2.1 Wind TurbineModal Dynamics 11 Figure 2.3: Terminology applied to wind turbines DOF. Source [2] 2.1.4 Coleman Transformation A wind turbine is a time-variant system. The tower and nacelle are subjected to the ground-fixed frame of reference (non-rotating) while the blades are rotating, and thus, blade dynamic characteristics are azimuthally time dependent. The interaction of the blades rotating frame with the tower-nacelle non-rotating frame introduces periodicity in the equations of motion of the wind turbine. Also, the rotor responds as a whole to the excitation on the non-rotating frame. Thus, it seems appropriate to work with DOF that reflect this behaviour, in order to simplify both the analysis and the understanding of the dynamic behaviour of the wind turbine [9]. In addition, conventional modal analysis is only applicable to time-invariant systems, ending up with the necessity of converting the wind turbine from time-variant to time- invariant system. The Coleman transformation, also known as multiblade coordinate transformation (MBC) for bladed rotors [9, 10], is a method to describe the motion of the individual blade coordinates in the non-rotating frame. Provided that all coordinates of the system are defined in the same frame of reference, this transformation yields the system to be time-invariant as all periodicity is removed, thus in turn allowing the application of modal analysis. However, the fundamental assumption of this method is that the rotor is isotropic, i.e. all blades identical and mounted symmetrically on the hub. This section explains the method for transforming the blade coordinates to the non- rotating reference frame, and ends up with a discussion on the benefits and the different viewpoints from previous works.
- 30. 12 Background Knowledge Transformationof Blade Coordinates For a 3-bladed rotor, with the blades equally spaced, MBC transformation is defined as : a0 = 1 3 3 k=1 qk a1 = 2 3 3 k=1 qk cos ψk b1 = 2 3 3 k=1 qk sin ψk (2.1) where ψk = Ωt+ 2π 3 (k−1) is the azimuth angle of blade k = 1, 2, 3. The three multiblade coordinates a0, a1 and b1 replace the blade coordinates q1, q2 and q3. The inverse transformation back to the blade coordinates is also possible by the formu- lation: qk = a0 + a1 cos ψk + b1 sin ψk (2.2) Equation (2.2) permits the change from multiblade coordinates to blade coordinates. The transformed blade coordinates in the non-rotating frame can be categorised in one symmetric (collective) and two asymmetric components, as explained in Section 2.1. As an example, let us assume a flapwise deflection (in the wind direction) of the blade coordinates qk. The multiblade coordinate a0 describes a collective flapwise deflection (all blades deflect symmetrically), while a1 and b1 describes tilt and yaw motions respectively. If it were an edgewise deflection, a1 and b1 would describe horizontal and vertical motions whereas a0 would describe a collective edgewise motion of the blades that may interact with the drivetrain torsional mode. The different motions are represented in Figure 2.4 for ease of follow-up. According to Hansen [2], the mode shape in MBC coordinates is defined as a0 = A0 sin(ωt + φ0) a1 = Aa sin(ωt + φa) b1 = Ab sin(ωt + φb) (2.3) and plugging 2.3 into Equation (2.2) gives the mode shape in blade coordinates: qk(t) = A0 sin(ωt + φ0)+ +ABW sin (ω + Ω)t + 2π 3 (k − 1) + φBW + +AF W sin (ω − Ω)t − 2π 3 (k − 1) + φF W (2.4)
- 31. 2.1 Wind TurbineModal Dynamics 13 with the first term indicating the symmetric component, and the subscripts in the second and third terms indicating the backward (BW) and forward (BW) whirling com- ponents of the rotor blades amplitudes and phases. Therefore, the modal response of blade k is composed of three components, recalling Section 2.1. If the frequency is measured in the fixed reference system, the natural frequency is ω for all three components. Whereas if the frequency is measured in the rotating frame, the natural frequency is ω for the symmetric component and ω + Ω and ω − Ω for the BW and FW components, respectively, i.e. the BW and FW components are shifted ±Ω in the rotating frame. However, it must be pointed out that pure modes do rarely exist. If the rotor is isotropic, an (averaged) spectra of the accelerations measured on the blades should present the same magnitudes, and the phase of the signals equals to ±120◦, where all phases sum up to 0±360◦. Once the signals are transformed, the phase difference between a1 and b1 is ±90◦, where a1 leads b1 in the case of FW, i.e. +90◦, and a1 lags b1 in the case of BW, i.e. -90◦. Figure 2.4: Asymmetric rotor motion indicating reaction forces. Source [2] Discussion The fundamental assumption of the MBC is that the rotor is isotropic. Nevertheless, there are different criteria concerning the assumptions on which the method succeeds. One the one hand, Hansen defines in [2, 4, 17] the following assumptions to correctly apply the MBC: 1) the number of blades must be odd 2) the inflow to the rotor must be uniform 3) the rotor must be isotropic (blades are identical and symmetrically mounted) In that case, the periodic terms in the aeroelastic equations of motion are removed and the MBC converts the system to time-invariant. On the other hand, Bir [18] does not deal with the number of blades nor the flow, but states that all attempts at MBC thus far assumed constant rotor speed (which rarely is the case) and identical blades (typically blades have no identical structural or dynamical properties). Bir shows that the MBC can be applied to variable-speed turbines because the varying rotor speed is only associated to the stiffness matrix [18], but also mentions that:
- 32. 14 Background Knowledge 1)the MBC does not eliminate the periodic terms of the equations of motion, but acts as a filter, letting through the multiples of 3Ω, provided the operating conditions are time-invariant and the blades are identical 2) for blades that are not similar the MBC can still be applied, with these dissimilarities included in the system matrices. The major drawback is that the rotor harmonics not multiples of 3Ω will not be filtered out 3) the only restriction to the MBC is the blades to be mounted spaced equally around the rotor azimuth In this work, the MBC is applied to both anisotropic (V27) and isotropic (H2) cases, aiming to evaluate these statements. 2.2 Methods The methods applied to the signals from measurements and simulations are described in this section. The scope is not to develop the complete formulation of the methods, but to give an overview about what the inputs and the outputs of each method are. This section is helpful to understand observations and figures in Chapter 4. 2.2.1 Time Domain and Frequency Domain The primary data to deal with in this work are acceleration time histories. It means that the magnitude of accelerations registered from a particular sensor are recorded over time. Actually, the signal is registered using a sampling period, meaning that the data is stored equally spaced and discretized in time. It is typically useful to observe the signals in time domain since one may detect the variation of the signal over time. However, there might be components of the signal that cannot be appreciated in the time domain. Converting to the frequency domain allows to identify these masked components. Therefore, it is often convenient to observe the signal in the frequency domain, where the energy of the signal is represented versus frequency. Roughly speaking, time domain and frequency domain are two ways of looking at the same signal, where the signal can be converted from time domain to frequency domain back and forth without losing information. For the dynamic characterization, the frequency domain is very useful, since peaks that gives information about modal parameters of the system are easily identified. 2.2.2 Power Spectral Density The Fourier transform is a mathematical tool that allows to transform signals between time and frequency domain. In the frequency domain, the Power Spectral Density (PSD) describes the distribution of the power content of a signal at each frequency. The PSD is a powerful tool in signal analysis to characterize the dynamic behaviour of a system.
- 33. 2.2 Methods 15 Formulation Todescribe the PSD, it is necessary to introduce the definition of autocovariance, denoted as: RXX(τ) =< X(t)X(t + τ) > −µ2 X (2.5) where X is the signal, t is the time, τ is the time lag, <> is the mean value operator and µ is the mean of the signal. The autocovariance defines how much similar a signal is with the time-shifted version of itself. This quality is useful to determine repeated patterns or trends in a signal. Similarly, the crosscovariance defines the similarity between two signals: RXY (τ) =< X(t)Y (t + τ) > −µXµY (2.6) The autocovariance, therefore, could be seen as a special case of crosscovariance. Commonly, the autocorrelation function is the autocovariance of the signal divided by the variance: ρX(τ) = RX(τ) σ2 X (2.7) Often named as the autocorrelation coefficient. However, in signal analysis context, the autocovariance without normalization is referred to autocorrelation. Similarly, from the crosscovariance one can derive the crosscorrelation coefficient. The relation between the PSD and the autocovariance is that they form Fourier trans- form pairs. Hence, the Fourier transform of the autocovariance, as defined in Equa- tion (2.5), is the PSD. Mathematically, the area under the PSD is equal to the variance of the signal. One may recall at this point that the variance is the square of the standard deviation or averaged of the square differences to the mean. Besides, the covariance is just the variance for two different variables, but generally, the covariance is defined as the variance of any two time series separated by a time lag τ [19]. From this definition of covariance, the previous descriptions of autocovariance and crosscovariance apply (Equations (2.5) and (2.6)). The forward Fourier transform of the covariance is the double-sided PSD, denoted as: SXX(f) = ∞ −∞ RXX(τ)e−i2πfτ dτ SXY (f) = ∞ −∞ RXY (τ)e−i2πfτ dτ (2.8)
- 34. 16 Background Knowledge whereSXX is the autospectral power density (PSD), and SXY is the cross-spectral power density (CSD), which describes how the autocovariance and crosscovariance are distributed on different frequencies. The double-sided term mentioned above refers to the property that half of the physical frequency content appears at positive frequencies and half at negative frequencies. Hence, a single-sided interpretation is required assuming the following properties: GXX (f) = 2SXX(f) for f>0 GXX (0) = SXX(0) GXY (f) = 2SXY (f) for f>0 GXY (0) = SXY (0) Real measurements are not continuous in time but discrete values separated by a sam- pling period ∆t that leads to a sampling frequency fs = 1/∆t. For a time series sampled at fs the highest frequency to estimate the spectrum is the Nyquist frequency fN = fs/2. The spectrum is hence estimated at the frequencies: fl = fs l N = l N∆t (2.9) with l = 1, ...N/2, as the frequency lines or bins and N the length of the parts in which time series are split. Spectral Analysis If the spectrum is estimated at the frequencies fl, and each frequency line is a frequency (and not a range of frequencies), there are gaps between each frequency lines. These gaps are referred to the so-called spectral leakage phenomena, meaning that the energy leaks between the frequency lines. In order to correct this phenomena the spectral estimation is performed using the modified averaged periodogram method (Welch’s technique), with an overlap in this work of 67% and a Hanning [20] weighting function, likewise the OMA Type 7760 software package from Br¨uel & Kjær [21] used in this work. This ensures that all data are equally weighted in the averaging process, minimizing leakage and picket fence effects [22]. A rough description of the Welch method [23] is that the signal is divided into m segments, where each segment overlaps the next one. The Hanning window is then applied to each segment weighing the mid point most and decaying exponentially towards the sides, and the modified periodogram - an estimate of the spectral density signal - is averaged for each frequency line. Then, the Welch’s method is computed. The Welch’s method is fully described by Brandt in [3]. Figure 2.5 shows the segment based averaging process.
- 35. 2.2 Methods 17 Figure2.5: Example of signal divided into m segments with 50% overlap. Source [3] Spectral Density Matrices Once the PSD and CSD of the signals time histories are estimated, the spectral density matrices can be formed. The size of the matrices is n x n, with n being the number of selected signals. The diagonal terms are real valued elements with the magnitudes of the spectral den- sities between a response and itself; the off-diagonal terms are complex valued elements which carry the phase information between two measurements [22]. For the corresponding sensor in each blade e.g. edgewise outer section, the spectral density matrix looks like: GYY(f) = GXX (f)11 GXY (f)12 GXY (f)13 GXY (f)21 GXX (f)22 GXY (f)23 GXY (f)31 GXY (f)32 GXX (f)33 (2.10) Note that these matrices are Hermitian. 2.2.3 Singular Value Decomposition The singular value decomposition technique (SVD) is explained in this section and its practical application to the signals of this work can be found in Chapter 4. Although it is a well-known technique with broad fields of applications, only the more relevant properties to the concerns of this work are commented here. Description The SVD allows to break up a rectangular matrix into the product of three matrices, where two contain the singular vectors and one the singular values of the original matrix. In very plain words, the SVD for modal parameter estimation consists in extracting the
- 36. 18 Background Knowledge singularvalues - the square roots of eigenvalues in the diagonal matrix - from the original matrix and performing a curve that indicates with a peak where a mode is located. If only one mode exists at a particular location, it will only be described by one set of singular values. If two sets of singular values curves are needed to describe one mode, it means that there are actually two modes, probably indiscernible. This property is especially useful when modes are really close. Provided the spectral density matrices in the form of Equation (2.10), the SVD is performed for each of the matrices at each frequency and measurement. Specifically to this report, the SVD will return 3 singular value curves. The result is the determination of modes and a estimation of the modal frequencies prior to OMA, which is considered a helpful method to validate the modes when identified in OMA. Formulation Through the SVD, a matrix of m x n dimension can be decomposed into the product of three matrices: an orthogonal unitary matrix, a diagonal matrix and the transpose of another orthogonal unitary matrix as A = USVH (2.11) with A being one of the spectral matrices, i.e. GYY(f), where the expansion of this matrix yields A = [{u1}{u2}{u3} . . . ] s1 s2 s3 ... [{v1}T {v2}T {v3}T . . . ] (2.12) A = {u1}s1{v1}T + {u2}s2{v2}T + {u3}s3{v3}T + . . . (2.13) where the diagonal terms of S are the square roots of the eigenvalues from U and V sorted in descending order, named singular values. The columns of U and V are orthonormal eigenvectors of AAT and AT A respectively, and H denotes Hermitian, which in case of real matrices is simply the transposed. The singular vectors are the estimated mode shapes and the singular values are the spectral densities. The descending order to which the singular values are sorted in the matrix makes the first singular value curve be the largest.
- 37. 2.3 Tools 19 2.3Tools This section describes the tools that are used further to simulate the dynamics of the V27: HAWC2 HAWCStab2 OMA Type 7760 2.3.1 HAWC2 HAWC2 (Horizontal Axis Wind turbine Code 2nd generation) is a nonlinear aeroelastic code intended for calculating wind turbine response in the time domain. It was origi- nally developed by Petersen (HAWC) [24, 25], while the second generation was mainly developed by Larsen and Hansen [26]. However, many individuals have contributed to the enhancement of the possibilities of the code. The structural part of the code is based on a multibody formulation applied to the Timoshenko beam element, i.e. each body is an assembly of Timoshenko beam elements. A typical configuration is tower, towertop, shaft, hub and blades, considered as main bodies. Each of these bodies are divided into different sub-bodies and every sub-body is break down into Timoshenko beam elements with its own reference system and described by 6 DOF. Each element consists of two nodes that specify the geometrical position in space. Stiffness, mass and inertia properties are constant along the beam element. Since the deformations are assumed small at a single sub-body, bodies of interest have to be divided into different sub-bodies to represent real behaviour with sufficient accuracy. An example are the blades bodies, which uses sub-bodies to better describe the aeroelastic blade behaviour. The structural properties are called from a file that gathers all the structural information required. The turbine, then, is modelled by an arbitrary number of bodies connected with constraint equations, where a constraint could be e.g. a rigid coupling or a bearing. The aerodynamic part of the code is based on the blade element momentum (BEM) theory, extended from the classic approach to handle dynamic inflow, dynamic stall, skew inflow, shear effects on the induction and effects from large deflections. The aerodynamic properties of the airfoil are called from two files: one providing the blade planform - defining the chord length and profile thickness related to the chord length at each cross- section of the airfoil - and the other providing the profile coefficients of the airfoil - lift, drag and moments in the angle of attack range from -180 to 180◦, identified by the thickness over chord ratio (link to the blade planform properties). Two turbulence formats can be used: one based on Veer’s model (polar grid) and the other based on Mann model formulation (creates spatial vector field in Cartesian coordinates). Control of the turbine is performed through one or more DLLs (Dynamic Link Library). The format for these DLLs is also very general, which means that any possible output
- 38. 20 Background Knowledge sensornormally used for data file output can also be used as a sensor to the DLL. This allows the same DLL format to be used whether a control of a bearing angle, an external force or moment is imposed on the structure. This and more information about HAWC2 can be found in [26]. 2.3.2 HAWCStab2 HAWCStab2 is a linear aeroelastic stability tool, developed by Hansen [4], that predicts structural and aeroelastic modal frequencies, damping ratios and mode shapes, through open- and closed-loop aero-servo-elastic eigenvalue and frequency-domain analysis. The aeroelastic model accounts for nonlinear kinematics based on Timoshenko elements. The underlying structural model used in HAWCStab2, is the same as for HAWC2. The beam element model used in HAWCStab2 is shown in Figure 2.6. The linearisation of equations of motion are done about a steady-state equilibrium - at a given operating point defined by constant wind speed, rotor speed and pitch angle - that approximates the mean of periodic steady state, considering uniform inflow. The periodicity associated with an operating wind turbine is then eliminated by means of the MBC, previously explained in Section 2.1.4, based on the fundamental assumption of isotropic rotors. The aerodynamic loads are based on the BEM theory coupled with a Beddoes-Leishman type dynamic stall model in a state-space formulation [27]. The distribution of aerody- namic forces are a parabolic approximation based on 3 aerodynamic calculation points, placed at the nodes and center of each blade element. The aerodynamic forces and moments in the calculation points are explained and derived in [4], finally yielding the coupled equations of motion of the linear aeroelastic model of wind turbines. Figure 2.6: Beam element model in HAWCStab2 (only one blade). Source [4] These equations of motion enable an eigenvalue analysis to determine the aeroelastic natural frequencies, damping ratios and mode shapes at each operating point.
- 39. 2.3 Tools 21 Furtherinformation on HAWCStab2 is provided in [2, 4, 17]. 2.3.3 Br¨uel & Kjær Operational Modal Analysis Type 7760 Operational Modal Analysis Type 7760 is a software package from Br¨uel & Kjær for experimental identification of modal parameters with the SSI algorithm already imple- mented. The SSI technique is presented in a discrete time state space formulation. The response data is collected in the so-called Hankel matrix, which structure is related to covariance estimation. The subsequent projection of the Hankel matrix is explained in terms of covariances and leads to a set of free responses for the system. Lastly, a SVD is then applied to the projection matrix in order to obtain the estimated system matrices, and by extension, the modal parameters. As mentioned in the Section 1.1, there are some assumptions that need to be fulfilled: the structure is linear; the structure is time invariant; the operational excitation forces must a) have broadband frequency spectra; b) be uncorrelated; c) be distributed over the entire structure. The SSI algorithm is fully described in [12]. A detailed explanation of the SSI technique implemented in the Operational Modal Analysis Type 7760 software package is found in [28, 29].
- 40.
- 41. Chapter 3 Implementation ofthe Numerical Model Once the basic background and the tool have been presented, the core of the thesis starts in this chapter. The procedure used in the succeeding is shortly described as follows. The MBC and other methods exposed in Chapter 2 are applied to the measurements, collected in the measurement campaign, to investigate the peaks that may be associated with modes. Observations of the modified signals are provided in Chapter 4. Then, OMA is applied to the signals and the eigenvalues of the system are extracted. Besides, a numerical model is implemented in HAWC2 based on the experimental data with the scope of simulating the full-scale V27. The mentioned procedure above is also employed with the output emerging from the H2 model simulation. Based on the H2 model, an aeroelastic analysis is performed using HS2, tool in which the MBC is already implemented, solving directly the eigenvalue problem as mentioned in Section 2.3.2. The resulting modal parameters from the V27 and H2 data are finally compared against the HS2 predictions, and all results are compiled in Chapter 5, where a comparative is performed. The procedure is schematically illustrated in Figure 3.1. Figure 3.1: Summary of the methodology 23
- 42. 24 Implementation ofthe Numerical Model To initiate the mentioned procedure, a natural step is to implement a numerical model of V27 using HAWC2, capable of mimicking the conditions of the V27 matching measure- ments. The structural properties of the H2 model are also used in the HS2 model from which the modal parameters are extracted. On this basis, this chapter relies on three key points: a) to describe the experimental setup and environmental characteristics. b) to create a valid numerical model (H2 model) able to be run in HAWC2, assuming structural properties, wind site conditions and output channels being in agreement with the measurement campaign. c) to use the structural properties of the H2 model to perform an aeroelastic modal analysis in HAWCStab2, ending up with estimated natural frequencies and damping ratios to which compare with in Chapter 5. These estimations are referred in the following to ”HS2”. 3.1 Experimental Data This section describes the measurement setup conducted to collect the experimental data. The description is limited to the data related to the signals used in this work for the sake of simplicity. The complete description can be found in [5], with details regarding devices and challenges involving the instrumentation of the turbine. 3.1.1 The Wind Turbine The V27 is owned by DTU Wind Energy (former Risø National Laboratory for Sustainable Energy) and was erected at the Risø test site in March 1989. This turbine has supported many research projects so far, and it is well instrumented and well known. Despite it is an old-fashioned machine compared to modern multi-MW turbines, it may provide a physical insight of the dynamics of current wind turbines, since they have in common similar construction design as well as controllers, such as yaw and pitch controls. Details of the turbine are given below in Table 3.1. According to [25], this wind turbine has a three-bladed upwind rotor with pitch- regulated cantilevered blades mounted on cast iron hub. The disc brake is mounted on the high speed shaft. The gearbox is mounted behind the shaft on the nacelle frame, and the generator is connected to the gearbox by a stiff coupling. The wind turbine yaws by an electrical motor controlled by a wind vane, mounted on the top of the nacelle. The electrical control system is mounted on the nacelle and in the tower bottom. The tower is a tube tower of a single section. A particular feature of the V27 is that the rotor has two different speeds depending on the number of poles connected to the generator, which can be 6 or 8 poles.
- 43. 3.1 Experimental Data25 Rotor diameter 27 m Swept area 573 m2 Low rotational speed 33 RPM High rotational speed 43 RPM Blade length 13.0 m Hub height 31.5 m Tilt angle 4 deg Cone angle 0 deg. Gear box ratio 1:23.4 Nominal power 225/50 kW Generator speed 750/1000 RPM Cut-in wind speed 4 m/s Cut-out wind speed 25 m/s Rated wind speed 14 m/s Table 3.1: Technical specification of the V27 3.1.2 Site Characteristics The V27 is located at the Risø test site as Figure 3.2 shows. The meteorological mast (met mast) is located next to it at approx. 73 meters with an azimuth angle of 283◦, the most common direction of the incoming wind. Figure 3.2: Aerial view of V27 site location The V27 is located in the field, facing towards the sea on the west side. The terrain appears mostly flat with a non-abrupt hill at the south-west. There are some farms and small buildings at a relative far distance from the turbine. Some crops are found to the east and some medium size trees toward the sea side. Next to the turbine there is a Nordtank 500/41 wind turbine, which is slightly larger, and in addition, two met masts for this turbine.
- 44. 26 Implementation ofthe Numerical Model 3.1.3 Measurement Campaign The measurement campaign was conducted between October 2012 and May 2013 partly under the EUDP (Danish Energy Technology Development and Demonstration Pro- gramme) project frame, ”Predictive Structure Health monitoring of Wind Turbines”, with grant number 64011-0084. Measurements from the V27 The instrumentation of the V27 consists in a total 51 channels providing data sampled at 4096 Hz. Among all channels, the following are particularly interesting for this work: 10 accelerometers per blade 3 tri-axial accelerometers at the nacelle 1 tachometer 1 pitch angle signal sensor The blade sensors are displayed for each blade section in Figure 3.3. Figure 3.3: Sensors location and orientation. Source: [5] The blade sensors comprise 10 accelerometers oriented to the flapwise direction of which two are biaxial, i.e. also oriented to the edgewise direction. These accelerometers are used to read the acceleration signals of all three blades. They are distributed along the
- 45. 3.1 Experimental Data27 blade span at specific sections of the blade, with 5 in the leading edge and 5 located in the trailing edge, where the bi-axial accelerometers are located. To define the optimal sensor location on the blades, simulations in H2 were carried out [30]. The eigenvectors of the operating blade were approximated with eigenvectors associated to a blade with fixed support (e.g. no rotational effects), assuming that the operating blade eigenvectors difference is negligible with respect to those. Thus, the maximum modal deflections were found at particular cross-sections and, in consequence, the accelerometers were decided to be installed in the locations shown in Figure 3.3. In the present work, only the bi- axial accelerometers are used, located close to the blade tip (96% of blade span) and the intermediate section (67%). These sections are hereafter called outer and inner sections, respectively. The reason is that, first, it is assumed that these sensors are sufficient to represent with confidence the dynamic behaviour of the V27, and second, to reduce the potential risk of misalignment by using sensors that form between them a 90◦ angle. The nacelle sensors are tri-axial accelerometers, which monitor the nacelle accelerations in the x, y and z direction. They are expected to identify not only tower modes, but also the rotor modes. In 2010, Tcherniak et al. showed that rotor modes can be extracted from nacelle sensors [1]. In the present study, one nacelle sensor is mounted in the yaw bearing and two in the rear of the nacelle (left and right corners). As a result, the rotor modes are expected to be extracted from the nacelle signals and validated against those extracted from the blades signals, reinforcing the findings in the above mentioned investigation (2010). Figure 3.4: Nacelle sensors in the V27. Source [6] The tachometer is used to determine the rotor azimuth angle - the position of a blade as a function of time - with the turbine under operation. It was found in [6] that the tachometer provided the most accurate measurement of the rotating frequency among
- 46. 28 Implementation ofthe Numerical Model three redundant sensors, and thus, the average rotating frequency over the measurement duration is used to compute the azimuth angle of each blade. It is installed at the high speed shaft (HSS) to improve the estimation of the azimuth angle providing one pulse per revolution of the HSS. The pitch angle sensor was installed in the hub to detect pitch activity. These readings are useful to select measurements where there is none or low pitch activity. Pitch activity is not desired, since it also introduces time-variant components in the system. All measurements are collected in 5 minutes chunks and transmitted via an Ethernet cable to a computer located at the bottom of the V27 tower, where they are recorded to an external hard disk. Measurements from the Met Mast The meteorological readings are taken from the met mast nearby the V27. It is provided with wind sonic sensors, cup anemometers and wind vanes at different heights. Others devices exist to measure pressure, temperature and precipitation. Figure 3.5: View of V27 and met mast from neighbouring Nordtank Above all, the most relevant are the cup anemometers located in the met mast at 18.5, 30 and 32 meters. The yaw variation is also measured to determine to what direction the wind turbine faces the wind during a particular span of time. Raw met data is not used in this work; the raw measurements are averaged in a 10- minute scheme. Finally, a data acquisition unit (DAU) is required at the met mast to collect the signals and send them to the central acquisition system.
- 47. 3.1 Experimental Data29 3.1.4 Selection of Data Sets From the entire measurement campaign (October 2012 to May 2013), a collection of measurements dating from December 13th to December 22nd was provided and used in this work for simplicity. These measurements were provided in a file with the V27 raw measurements and the met averaged ones already combined. Since the V27 recordings are based on a 5-minute scheme, each two successive chunks are linked to one 10-minute met measurement. There- fore, averaged data from one measurement chunk of the met mast is repeated in two successive chunks from the V27 measurements, as illustrated in Table 3.2. Time stamp V27 measurements Met measurements 16/12/12 11:05 5-minute raw data 10-minute averaged data 16/12/12 11:10 5-minute raw data Table 3.2: Example of combined V27 and met measurements Correction on the Original Data The provided file showed that data was shifted 3 minutes. The first recorded readings of the V27 measurements started December 13th, 11:17, which were linked to the first met measurements in the file, December 13th, 11:20. This fact means that the V27 readings were initialized at different time with respect to the met readings. Moreover and more seriously, a mismatch of about 12 hours between the met and V27 measurements was identified on the provided file. This issue was corrected moving the met data to fit the sensors measurements, as it can be appreciated in Figure 3.6, that in turn led to adjust the 3-minute initial shift to 2-minute, therefore improving the difference between time stamps. The correction was double-checked with the original recordings from the met mast database. For the H2 model implementation, is assumed that the 2-minute difference is acceptable. In Figure 3.6 a chart is presented with the selected period of 9 days chosen for simplicity, where the original (light red) and corrected mean wind speed (solid red) can be observed. From Figure 3.6, and recalling that data consists in mean values, it can be observed that: Full production is achieved during a short time, and production usually starts around 6 m/s The peak mean wind speed is about 13 m/s The turbine seems to produce energy only at high rotor speed According to the low mean wind speed registered, the pitch activity is low, but it shows a, perhaps, idling event, since the blades are pitched about 30◦ some hours around December 19th It should be noted that the above observations are based on mean data. Improvements could be done if dealing with raw data instead.
- 48. 30 Implementation ofthe Numerical Model Figure3.6:V27andmetcombineddata
- 49. 3.2 Defining theModel in HAWC2 31 Criteria for the Selection of Data Sets A wind turbine may be modelled as a time periodic system if the fluctuation of particular variables is minimized, i.e. rotor speed, blade pitch and nacelle yaw, so as to assume that these variations are constant. Hence, the criteria for selecting data sets is based on a low standard deviation of these parameters. Besides, OMA requires long enough time series for better performance of the algorithm. A rough rule of thumb is that the time histories should at least be 500 times longer than the period of the lowest mode of interest [31]. The lowest mode of interest is expected around 0.95 Hz, therefore, a rough minimum is about 526 seconds. Nonetheless, the larger the time series, the better the algorithm performs, and 20 minutes is thought as a good balance between the computational time expense and the operational modal analysis time history demand. This time span length, translated to the recorded measurements, means to create data sets comprising 4 chunks of V27 measurements linked to 2 chunks of met measurements. Further, it is desired to investigate time series at both low (33 RPM) and high rotor speed (43 RPM). Provided these considerations, two main data sets are created with successive measure- ments, which are picked conveniently for low rotor speed (December 16th, 11:10-11:30) and high rotor speed (December 15th, 05:10 to 05:30). Detailed information about these data sets is shown in Table 3.3. Parameter Low rotor speed High rotor speed Time stamp 16/12/12 11:10-11:30 15/12/12 05:10-05:30 Std. Dev. tacho (RPM) 0.54 1.28 Mean rotor speed (RPM) 32.20 43.10 Std. Dev. pitch sensor (◦) 0.09 0.59 Std. Dev. yaw (◦) 0.17 0.02 Mean wsp (m/s) 5 11 Std. Dev. wsp (m/s) 0.56 1.37 Max./min. power (kW) 0/0 265/56.3 Table 3.3: Selected data sets It is to be noticed that the standard deviation of the tachometer refers to 1 pulse or RPM with respect to the HSS, i.e. 750 or 1000 RPM, for low and high rotor speed time series, respectively. With the entire collection of tacho ”events”, the mean is found, and the rotational speed is derived for the low speed shaft (LSS, rotor speed). 3.2 Defining the Model in HAWC2 Once the time spans are selected from the experimental data, this section defines the im- plementation of the numerical model to simulate the behaviour of the V27 at the selected time spans. The external excitation in the H2 model is defined based on the recorded meteorological data during the selected time spans. The structural and aerodynamic properties of the H2 model are given from previous Risø research studies.
- 50. 32 Implementation ofthe Numerical Model The first step is to define the simulation and the external loads in a turbulent scenario using the experimental data for two different scenarios, low and high rotor speed. Further, a tuning of the structural damping is required in the H2 model, in order to agree with some limitations in HAWC2. Finally, the structural part of the H2 model is used for an aeroelastic modal analysis using HAWCStab2, and thus, to extract the theoretical predictions of the aeroelastic frequencies and dampings. 3.2.1 Simulation Setup in HAWC2 Time Parameters According to the selected data sets in Table 3.3, the time span for simulations is set to 1250 seconds, of which 50 seconds are removed due to transients (i.e. a total of 1200 seconds, 20 minutes). The time step is set to 0.01 seconds. Wind Loading The reader may remember the approximation for time stamp between the V27 sensors and the met mast data mentioned in 3.1.3, and that these four data sets comprise two 10-minute meteorological data. An average wind is found by taking the mean of the two mean wind speeds, 5 m/s for low rotor speed and 11 m/s for high wind speed, while the turbulence intensity is found by assuming that the mean of two wind speeds standard deviation is applicable, which results in 0.12 and 0.10 for low and high rotor speed respectively: Iu = σu ¯u (3.1) where Iu is the turbulence intensity, σu is the wind speed standard deviation, and ¯u is the mean wind speed. 0 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 wind speed (m/s) height(m) mean values low rotor speed power law low rotor speed mean values high rotor speed power law high rotor speed Figure 3.7: Wind profile for low and high rotor speed data sets
- 51. 3.2 Defining theModel in HAWC2 33 The wind shear is calculated using mean wind speeds recorded at different heights from anemometers at the met mast anemometers at the particular time frames. The power law method is employed to estimate the wind shear exponent, and thus, the mean wind speed at the hub height. The hub height was assumed at 30 meters above ground level to be consistent with the provided measurements file (Section 3.1.4), despite that the real hub height is apparently 31.5 meters. Figure 3.7 shows the wind profile resulting in wind shear exponents of 0.23 and 0.13 for low and high rotor speed. Turbulence Field The turbulence box is generated using the Mann model for the simulation time and wind speed in each case. The turbulence box resolution is set as 32 x 32 x 8192 and the turbulence parameters are assumed to be those by default in HAWC2, i.e. according to the IEC standard [32], except that the length scale is not the usual 29.4 but 15.484. This correction is done according to the standard for hub heights lower than 60 meters. Thus, the standard is followed to meet the new length scale and the specific values are shown below: αǫ2/3 = 1 Γ = 3.9 L = 15.484 where L = 0.7Λ1 Λ1 = 0.7z with L, αǫ2/3 and Γ being the turbulence parameters, Λ1 being the wavelength and z as the hub height. Besides, aerodynamic load calculation is done at 30 sections for each blade, including Beddoes-Leishman dynamic stall model, in accordance with HAWCStab2. Moreover, aerodynamic drag is set for the tower and nacelle. Controller In HAWC2, the V27 model displays two control algorithms: one for the generator and one for the blade pitch. The most relevant features of the generator controller are the gearbox exchange ratio, set to 23.33335, the nominal electric power (225 kW) and the synchronous generator speed (750 or 1000 RPM, depending on the data set). The pitch controller permits the blades to pitch from 0 to 90◦ in a collective manner.
- 52. 34 Implementation ofthe Numerical Model Output Channels The H2 model is provided with all the required sensors at the particular locations in accordance to the V27 instrumentation. The only disagreement is found at blade radius 10.5 meters, where a biaxial accelerometer is placed in the V27, while the equivalent simulated sensor in HAWC2 is placed at blade radius 11 meters instead (see Figure 3.3) to not modify the original blade sections of the V27 numerical model in HAWC2 (the blade is divided in a number of sections where structural and aerodynamic properties are applied). However, this disagreement is assumed as acceptable. Among the various output channels, it is interesting to highlight the rotor speed, the wind speed and the pitch angle mean values. They can be compared with the V27 ones, and further, with the HS2 operating points in the coming section. Table 3.4 shows these values for H2. Variable Low rotor speed High rotor speed Mean rotor speed (RPM) 32.24 43.22 Mean wsp (m/s) 5 11 Mean pitch angle (◦) 0 0.74 Table 3.4: H2 operating points 3.2.2 Eigenvalue Analysis of the H2 model The aim of performing an eigenvalue analysis of the H2 model in standstill is to obtain a suitable structural damping in the tower and rotor modes at standstill conditions. In this sense, it is found an acceptable damping of about 2% logarithmic decrement for the tower modes and 3% for the rotor modes within the frequency range of interest (0 to 5 Hz). First, it is described the structural part of the H2 model. Next, the eigensolver in HAWC2 and the tuning procedure are described. The structural part of the H2 model is divided in different bodies represented in HAWC2 as beam elements (as described in Section 2.3.1): platform, tower, shaft and blades. The platform is the base body and links the rest of the elements to the fix ground reference frame. The tower connects with the platform in the bottom and to the shaft in the top. The yaw bearing is located at the second-to-last node of the tower. The shaft is located in the top of the tower and holds the rotor. It presents a bearing that allows the rotor to rotate. The rotor is composed of three cantilevered blades that can pitch around the blade longitudinal axis. The structural damping for each of these bodies is expressed in terms of constants proportional to mass and stiffness contribution of the damping matrix. Generally, the mass proportional damping parameter affects the mean damping level, and the stiffness proportional damping parameter affects the high frequency vibrations. In practice, these parameters consists of 6 constants (Mx, My, Mz, Kx, Ky and Kz - contribution in each direction) that multiply the mass and stiffness contribution of the Rayleigh damping model.
- 53. 3.2 Defining theModel in HAWC2 35 HAWC2 contains an embedded eigenvalue solver, which is capable of providing modal parameters with the wind turbine at standstill. The assumption for such analysis is that there is no wind forcing, and that the wind turbine is in standstill, i.e. rotor with fix bearing (braked rotor). HAWC2 offers two methods to calculate the frequencies and damping at standstill: one involves including the mass and stiffness proportional damping parameters and the other only the stiffness terms. There are some issues related to the damping model in HAWC2 that makes the mass proportional damping terms to only be correctly applied in the special case when the main body is fixed to the ground and it consists in a single body (such as the platform in this H2 model). If this condition is not fulfilled, it yields a different damping if the number of sub-bodies is higher than one [26]. A blade could be an example, where it is required to have multiple sub-bodies within the main body (blade) to represent accurately deflections. Since it is important to use the same number of bodies in the structural eigenanalysis as well as in the simulations, and the blade main body is subdivided into five bodies, the selected method in this work only uses the stiffness proportional damping parameters, which enables to consider the response of the entire structure. In order to obtain the acceptable level of damping mentioned in the beginning of this section, a tuning of the stiffness proportional damping parameters is done. The final coefficients of the stiffness contribution to the damping of the structure are presented in Table 3.5. Body Mx [-] My [-] Mz [-] Kx [-] Ky [-] Kz [-] Platform 0 0 0 1.7e-3 1.7e-3 4.5e-4 Tower 0 0 0 0.85e-3 0.85e-3 2.5e-7 Shaft 0 0 0 1.5e-3 1.2e-3 8.5e-4 Blade 0 0 0 4e-4 6.06e-4 1e-7 Table 3.5: Stiffness contribution coefficients to damping The results of this tuning are presented in Table 3.6. Mode no. Freq. [Hz] Log. decr. [%] Damp. [%] Nomenclature 1 0.949 1.982 0.301 1st TSS 2 0.953 1.990 0.302 1st TFA 3 2.044 3.003 0.431 1st YF 4 2.095 3.054 0.437 1st TF 5 2.159 3.032 0.435 1st DT 6 2.436 2.932 0.423 1st CF 7 3.590 2.986 0.429 1st VE 8 3.644 3.007 0.432 1st HE Table 3.6: Results of structural eigenanalysis with H2 Graphical representations of these modes can be seen in Figure 3.8.
- 54. 36 Implementation ofthe Numerical Model (a) 1st TSS (b) 1st TFA (c) 1st YF (d) 1st TF (e) 1st DT (f) 1st CF (g) 1st VE (h) 1st HE Figure 3.8: Mode shapes no. 1 to 8 in HAWC2 The H2 model is now ready for simulating the characteristics of the V27 at the particular time spans. 3.3 Estimation of Modal Parameters in HAWCStab2 The scope of this section is to extract an estimation of the modal parameters using the aeroelastic stability tool HAWCStab2. As mentioned earlier, this tool shares the same structural properties as the H2 model. The estimated values are being compared with those from the V27 and H2 model resulting after the OMA analysis, later in the report. Firstly, the setup of the tool is described. Secondly, the modal parameters at standstill in HS2 are compared with those from Table 3.6 in H2. It is expected that the structural modal parameters are to be (almost) the same since only the internal loads of the structure interact. Lastly, the estimated modal parameters resulting from the aeroelastic modal analysis are presented.
- 55. 3.3 Estimation ofModal Parameters in HAWCStab2 37 3.3.1 Tool Setup Structural Model The structural model used in HAWCStab2 is exactly the same as in HAWC2. Therefore, it holds the same damping characteristics (cf. Table 3.5). Operating Points HAWCStab2 requires operational data where the operating points are defined (steady- state equilibrium, see Section 2.3.2). The operational data is computed provided a wind speed range, number of operational points, rotation speed range, minimum pitch angle, tip speed ratio and maximum rated aerodynamic power, which are based on the V27 experimental data. The output is the number of operational points related to wind speed, pitch angle, rotor speed, aerodynamic power and aerodynamic thrust. For the concerned cases, i.e. wind speeds of 5 m/s (low rotor speed) and 11 m/s (high rotor speed), Table 3.7 presents the operating points. Variable Low rotor speed High rotor speed Mean rotor speed (RPM) 32.14 35.02 Mean wsp (m/s) 5 11 Mean pitch angle (◦) 0.41 1.57 Table 3.7: HS2 operating points 3.3.2 Comparison of the H2 Model and HAWCStab2 A structural modal analysis is performed in HS2. Considering only the results at 0 RPM, corresponding to a standstill condition, modal parameters extracted are shown in Table 3.8 where they are compared to the modal parameters from the eigensolver in HAWC2. Mode no. HS2 H2 Nomenclature Freq. [Hz] Damp. [%] Freq. [Hz] Damp. [%] 1 0.992 0.332 0.949 0.301 1st TSS 2 0.995 0.333 0.953 0.302 1st TFA 3 2.045 0.476 2.044 0.431 1st YF 4 2.096 0.485 2.095 0.437 1st TF 5 2.155 0.482 2.159 0.435 1st DT 6 2.434 0.469 2.436 0.423 1st CF 7 3.590 0.473 3.590 0.429 1st VE 8 3.643 0.480 3.644 0.432 1st HE Table 3.8: HS2 vs. H2 modal frequencies and damping in standstill
- 56. 38 Implementation ofthe Numerical Model It can be observed that frequencies are practically the same, while damping ratios do not greatly differ (around 10% of difference). The main disagreement is found on the tower modes, which is attributed to the modelling of the tower plus platform. Based on this comparison, it is assumed that the HS2 is correctly implemented so as to perform an aeroelastic modal analysis to benchmark against the V27 and H2 model OMA results in Chapter 5. 3.3.3 Structural and Aeroelastic Modal Analyses The results shown in the previous section refer to a structural modal analysis at standstill condition. Next, a structural modal analysis for the entire wind speed range is presented, together with an aeroelastic modal analysis. The aim of comparing both is to observe if the frequencies and dampings change when aerodynamic effects are included (always under HAWCStab2 perspective, i.e. symmetric rotor and inflow, no turbulence). In both analysis the wind turbine is rotating, but the reader may associate the first type of modal analysis as the wind turbine rotating in vacuum. In Figure 3.9 the structural and aeroelastic frequencies are plotted. The associated nomenclature is joined to each mode. The structural modes are denoted with solid lines, while the aerodynamic modes are denoted with dashed-point lines. According to the figure, the frequencies hardly vary including or not the aerodynamics. The largest difference (in this context) is found in the components of the flap mode, because of the impact of the aerodynamics. The in-plane modes, such as edgewise rotor mode or shaft mode, remain almost equal. It can be also noticed how the 1st collective, FW flap and BW edge components couple within the frequency range from 2.5 to 3 Hz. The damping ratios of the respective analyses are shown in Figure 3.10. The structural modes are less damped, because only the internal loads and the rotor speed act on the structure. It can be further noted from Figure 3.11, that the damping behaviour is almost invariant with the wind speed. 0 5 10 15 20 25 1 1.5 2 2.5 3 3.5 4 4.5 wind speed [m/s] modalfrequency[Hz] 1st TFA 1st TSS 1st BWF 1st CF 1st FWF 1st BWE 1st DT 1st FWE struct.modes aero modes Figure 3.9: Frequencies vs. wind speed
- 57. 3.3 Estimation ofModal Parameters in HAWCStab2 39 The aeroelastic damping decrease, in general for all modes, because the blades are not pitching yet, and there is a change in the angle of attack. This fact is especially visible in the flapwise direction, since the flap components are the most affected by the wind, and thus, by the aerodynamic damping. 0 5 10 15 20 25 0 5 10 15 20 25 30 wind speed [m/s] dampingratio[%] 1st BWF 1st CF 1st FWF 1st TFA struct.modes aero modes Figure 3.10: Damping ratios vs. wind speed. All modes However, the zoom view in Figure 3.11 reveals that the least damped modes also de- crease for the same reason, in a much lower degree though, because the aerodynamic damping influence is not relevant. 0 5 10 15 20 25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 wind speed [m/s] dampingratio[%] 1st TSS 1st BWE 1st DT 1st FWE struct.modes aero modes Figure 3.11: Damping ratios vs. wind speed. Low damped modes The BWE component, as mentioned previously, couples with some of the flapwise components. This coupling apparently introduces an increase in damping contributed by these flapwise components, as it can be perceived from Figure 3.11. Contrary, the DT
- 58. 40 Implementation ofthe Numerical Model and FWE components have lower damping with respect to the BWE component, more likely for the same coupling with the edgewise component. When the blades start to pitch - according to the HS2 operational data, slightly at 9 m/s, and considerably at 12 m/s - the angle of attack is corrected, and the aeroelas- tic damping ratios change the pattern and increase. At 14 m/s the rated wind speed is achieved, and the controller takes over to secure a proper damping for the rest of operational points. To support these statements, the operational data can be checked in Figure 3.12. The operational data is not extremely accurate, since the rotor speed takes the wind speed to increase from 10 to 14 m/s to achieve the high rotor speed. One may recall that the selected data sets from the V27 measurements assumes about 43 RPM at 11 m/s (from average values). Also, the pitch angle in the mentioned data sets is specified as 0◦ and 0.6◦ for 5 m/s and 11 m/s (Table 3.3), whereas HS2 computes 0.41◦ and 1.57◦ respectively. 5 10 15 20 25 0 50 100 150 200 250 kW Power (kW) Rotor speed (RPM) Pitch (º) 0 5 10 15 20 25 0 10 20 30 40 50 wind speed [m/s] RPM/ºFigure 3.12: Operational data assumed in HS2 Finally, the aeroelastic frequencies and damping corresponding to the operating points 5 m/s and 11 m/s are shown in Table 3.9. Mode no. Low rotor speed High rotor speed Nomenclature Freq. [Hz] Damp. [%] Freq. [Hz] Damp. [%] 1 0.999 2.621 0.993 1.994 1st TFA 2 1.004 0.345 1.004 0.291 1st TSS 3 1.694 15.884 1.608 15.034 1st BWF 4 2.586 11.924 2.548 11.305 1st CF 5 2.733 11.357 2.726 10.177 1st FWF 6 3.108 0.490 3.061 0.273 1st BWE 7 4.099 0.869 4.101 0.818 1st DT 8 4.169 0.606 4.225 0.508 1st FWE Table 3.9: Aeroelastic frequencies and damping in HS2 at 5 and 11 m/s
- 59. 3.3 Estimation ofModal Parameters in HAWCStab2 41 These values will be compared in Chapter 5 with the results from the identification of the V27 and the H2 model data, respectively. As a last comment, it is important to keep in mind that only one set of first flap and edge modes are found. Collective, FW and BW are just components of the flap or edge modes, i.e. one mode is decomposed in three components. Sometimes it is easy to be confused with the terms when dealing with the modes, especially from now on.
- 60. 42 Implementation ofthe Numerical Model
- 61. Chapter 4 Signal Analysis Inthis chapter, the methods described in Chapter 2 are applied on the V27 and H2 signals. Recalling Chapter 2, modal analysis requires the system to be time-invariant. This assumption is expected to be fulfilled by means of the MBC transformation. This can work for the H2 model, fulfilling the isotropic condition, but may not work for the V27 signals, because real life rotors are never isotropic. In addition, OMA is based on several assumptions that are mainly not fulfilled, thus, it is not assured that the application of OMA is successfully possible to the V27 measurements, meaning that applying OMA right away may lead to erroneous results. Signal analysis is not based on any assumptions, and therefore it can be seen as a natural step to analyse the signals prior to OMA. Methods such as the PSD or SVD (as described in Figure 3.1) are hence used to provide insight on peaks arising in the frequency domain, that could provide a hint about their meaning, e.g. if the peak refers to a harmonic or to a mode. The sequence of the analysis is structured as follows: 1) PSD of the signals 2) SVD of the spectral density matrices (formed with PSD and CSD of the signals) 3) Transformation of the signals from the rotating to the non-rotating frame using the MBC (converting to time-invariant system) 4) After the MBC, an averaging technique can be used as stated by Bir [18]. The TSA technique is employed to remove the presence of harmonics in the transformed signals (according to Bir, all harmonics but the multiples of 3 times the rotational frequency). This is achieved by removing the deterministic periodic excitation (har- monics) from the stochastic excitation (wind), which satisfies OMA leading to more reliable results. 43
- 62. 44 Signal Analysis Therotor harmonics are often referred to 1P, 2P, 3P..., where P is the given rotor frequency (per-revolution harmonics). Of particular importance are the 1P (rotational frequency) and the 3P (blade passing frequency for a 3-bladed turbine) frequencies. For practical considerations, all harmonics are plotted as dotted vertical lines in the figures. It is worth to recall that, as shown in Figure 3.4, the x direction defines the in-plane motion, and the y direction refines the out-of-plane motion, i.e. lateral and longitudinal directions with respect to the wind direction. Also, that the frequency range of interest in this work is 0 to 5 Hz, where the first tower and rotor modes are expected. This may be seen as an analogy to modern wind turbines, where the first tower and rotor modes are typically found in the range of 0 to 2-3 Hz. Tower and blade torsional modes are not considered within the scope of this work. Finally, this chapter will also serve to check how accurate the H2 model is with respect to the V27. If successful, the spectra of the signals should have a very similar appearance. 4.1 Dynamic System Behaviour I: PSD 4.1.1 Nacelle Signals In this section the PSD of the nacelle signals are plotted. The main goal is to provide an understanding of the wind turbine dynamic behaviour by simple observation of the peaks. Both the V27 and H2 signals are presented in the same figures for ease of comparison. The V27 front signal must be directly confronted towards the towertop H2 signal, since both sensors are located in the yaw bearing. In Figure 4.2, the PSD for the in-plane signals are shown. To start with, it is pointed out the noise at low frequencies originally from the time history acceleration. The reader might remember that tri-axial accelerometers where used in the nacelle. Apparently, the type of accelerometer presented noisy characteristics at low frequencies in the y and z direction of the sensor coordinates. However, in this work always the x, y and z directions refer to the wind coordinates. The sensor coordinates are shown below for clarity in Figure 4.1. Figure 4.1: Tri-axial accelerometer coordinates. Representation of the nacelle view from the top Therefore, one may need to bear mind that the signals up to 2 Hz approx. are contam- inated with noise. Despite the noise in Figure 4.2(a), peaks at the harmonic frequencies
- 63. 4.1 Dynamic SystemBehaviour I: PSD 45 are noticed in the V27 signals. The H2 signal only presents peaks at 3P, 6P and 9P, meaning that the tower spectra is not influenced by the rotational rotor but only by the multiple of the blade passing frequencies. In general, the V27 signals are larger in ampli- tude than the H2 signal, which seems to be moved a little to the right from 3 Hz onwards. In addition, there are three major attributes to discuss: 0 1 2 3 4 5 10 −8 10 −6 10 −4 10 −2 10 0 10 2 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P V27 front V27 rear left V27 rear right H2 towertop (a) Low rotor speed 0 1 2 3 4 5 10 −8 10 −6 10 −4 10 −2 10 0 10 2 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P V27 front V27 rear left V27 rear right H2 towertop (b) High rotor speed Figure 4.2: PSD nacelle x-direction - Around 2.5 Hz, the rear signals greatly split from the front signal. This could be due to a torsion in the nacelle or tower, since the front signal follow the same trend along the frequency range - Around 4.1 Hz a peak appears with double peak in the crest - The H2 signal features a peak at about 3.1 Hz, which is hardly detected in the V27 front end signal
- 64. 46 Signal Analysis Athigher rotor speed shown in Figure 4.2(b), 5 peaks are noticed. Two small peaks (probably due to the noise) coincide with 1P and 3P. They seem to refer to the rotor and blade passing frequencies. The rest of peaks have in common that they are wider, with one located out of rotor harmonics (0.95 Hz approx.). The last two peaks, even wider, coincide with the 4P and 6P. The correlation between the V27 front signal and the H2 signal is similar, but the H2 signal is again somehow displaced to the right for the last two peaks. In Figure 4.3, not only all the V27 signals are seemingly correlating well together, but also with the H2 signal. 0 1 2 3 4 5 10 −8 10 −6 10 −4 10 −2 10 0 10 2 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P V27 front V27 rear left V27 rear right H2 towertop (a) Low rotor speed 0 1 2 3 4 5 10 −8 10 −6 10 −4 10 −2 10 0 10 2 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P V27 front V27 rear left V27 rear right H2 towertop (b) High rotor speed Figure 4.3: PSD nacelle y-direction Now there is much less noise (up to 0.5 Hz, with respect to the x direction) and the 1P harmonic is clearly noted. The previous double peak is noticed again at the same frequency, which is absent in the H2 signal. This may suggest a contribution of the x
- 65. 4.1 Dynamic SystemBehaviour I: PSD 47 peak in the y signals. A wide peak around 0.95 Hz may suggest the first tower mode, since it was also found in Figure 4.2. Similar trend of the signals is found at high rotor speed in Figure 4.3(b). 4.1.2 Blades Signals The PSD of the blade signals are plotted in this section. While the PSD of the tower signals left some traces that one may think of as modes, in particular, the first tower modes, one cannot dare such assumptions here. First, we are dealing with a time-variant system, and therefore modal analysis cannot be applied. Secondly, as reviewed in Chap- ter 2, blades modes rely on components (two asymmetric and one symmetric), hence, only traces of components could be suggested in the best case scenario. However, it is interesting to look further into those double peaks appearing in the tower PSD, and also, at which frequency ranges there is more power content. The section is divided in two subsections: signals at low and high rotor speed. Each of the parts compares the V27 with the H2 signals, sorted from outer to inner section of the blade. Figures include all signals, edge and flap. Low Rotor Speed Figure 4.4 and Figure 4.5 represent the blades signals of V27 and H2 at low rotor speed in the outer section of the blades. As expected, the flap signals content more energy than the edge signals. Peaks are denoted at all rotor harmonics except at the 1P frequency, which shows a different shape than the expected from a harmonic (narrow and sharp peak). This peak is fairly sharp at the crest but wide at the trough. Actually, the harmonic peaks in the V27 signals are hardly recognizable in the low frequency range. For instance, the 4P harmonic shape is similar to a mode but features a small peak in the harmonic frequency. In general, the energy in the signal increases nearby of the harmonics, showing the wide trough effect. The H2 data shows similar behaviour on all signals, but the small peaks are more noticed (perhaps with the exception of the 2P). The peak at 1P shares the same characteristic as in the V27 signals. Tcherniak et al. commented on this effect in [11], where the spectra of the aerodynamic forces was analysed. They conclude that when the blade passes regions with higher or lower wind speed, periodicity is generated, adding up that increasing the tip speed ratio generates higher peaks and deeper troughs. It is also mentioned that the wide troughs or ”thick tails” could emerge due to the convolution of two autocorrelation functions, one in the root (flat spectra) and the other at higher radius (periodic peaks growing as a function of the radius). Indeed, Figure 4.4 (a) appears close to the normalized PSD of the autocorrelation functions shown in this study. Around 3.6 Hz a double peak appears in the V27 data - single peak in the H2 data. This double peak is magnified in Figure 4.4 (b), revealing that all blades manifest this duplicity. Rotor anisotropy might be beneath this double peak behaviour. In any event, this peak contents more energy than the flap signals in this frequency range. Since it is not related to a harmonic and shows a ”thick tail”, one may suspect that it is a mode.
- 66. 48 Signal Analysis 01 2 3 4 5 10 −5 10 0 10 5 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P Blade A 3 e Blade B 3 e Blade C3 e Blade A1 f Blade B 1 f Blade C 1 f (a) PSD blades V27, all signals (b) Zoom view double peaks Figure 4.4: PSD blades signals V27 outer section, low rotor speed 0 1 2 3 4 5 10 −5 10 0 10 5 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P Blade A3 e Blade B3 e Blade C 3 e Blade A 1 f Blade B1 f Blade C 1 f Figure 4.5: PSD blades signals H2 outer section, low rotor speed
- 67. 4.1 Dynamic SystemBehaviour I: PSD 49 The double peak seen in Figure 4.4 (a) is shown as a single peak in Figure 4.5, supporting the rotor anisotropy conjecture. Figure 4.6 and Figure 4.7 presents very much the same behaviour as in the outer section, apart from the difference in energy content of the signals. 0 1 2 3 4 5 10 −5 10 0 10 5 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P Blade A3 e Blade B3 e Blade C 3 e Blade A 1 f Blade B1 f Blade C1 f (a) All signals (b) Zoom view 1P Figure 4.6: PSD blades signals V27 inner section, low rotor speed Nonetheless, a different feature one may observe is that one of the flap signals is di- verging from the neighboured signals, being lower in magnitude. An analysis of the PSD flap signals in the leading and trailing edge at that particular blade and section reveals that the trailing edge is more excited and its magnitude is about twice the magnitude of the leading edge signal. This may be caused either by the higher deflection of this blade when subjected to the same loading, or by an asymmetric mounting of the sensor. It must be emphasized that the anisotropy in wind turbines rotors may originate from physical asymmetry among the blades or from the measurement system.
- 68. 50 Signal Analysis InFigure 4.7, likewise Figure 4.5, the signals correlate well without any further char- acteristic to be mentioned, apart from the lower energy content of the signals. 0 1 2 3 4 5 10 −5 10 0 10 5 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P Blade A 3 e Blade B 3 e Blade C3 e Blade A1 f Blade B 1 f Blade C 1 f Figure 4.7: PSD blades signals H2 inner section, low rotor speed High Rotor Speed At high rotor speed, the signals not only presents an increase in overall energy content, but also become more flat between harmonics, as it can be distinguish in Figure 4.8 and in Figure 4.9. At low rotor speed, generally, between harmonic peak to peak a trough was found. Now, this trough is not discernible easily. As an example, between 3P and 4P there is no trough at all, and this behaviour repeats all along the highest frequencies. 0 1 2 3 4 5 10 −5 10 0 10 5 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P Blade A3 e Blade B3 e Blade C 3 e Blade A 1 f Blade B1 f Blade C1 f Figure 4.8: PSD blades signals V27 outer section, high rotor speed When it comes to the peaks, a similar picture is seen when compared to the low rotor speed case. It is worth to mention that the suspected mode, around 3.6 Hz, is now placed
- 69. 4.1 Dynamic SystemBehaviour I: PSD 51 at the 5P frequency. The double peak in the crest is now only appreciated in one of the edge signals, probably due to the excitation of the harmonic frequency. Similarly, the same peaks are found in the H2 signals in a more clear visualization due to the perfect isotropy of the blades and sensors. 0 1 2 3 4 5 10 −5 10 0 10 5 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P Blade A 3 e Blade B3 e Blade C3 e Blade A 1 f Blade B 1 f Blade C1 f Figure 4.9: PSD blades signals H2 outer section, high rotor speed Besides the features discussed in the PSD of the outer section, Figure 4.10 may suggest torsional effects in two blades, both in the edge and flap signals, at the 1P frequency. Again, this effect results, perhaps, for the larger loading due to high winds or misalignment of the sensors. It also occurs between the 4P and 5P frequencies for an edge signal, while the two others show good agreement. 0 1 2 3 4 5 10 −5 10 0 10 5 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P Blade A 3 e Blade B3 e Blade C3 e Blade A 1 f Blade B 1 f Blade C1 f Figure 4.10: PSD blades signals V27 inner section, high rotor speed The simulated data in Figure 4.11 presents similar behaviour as in the V27 data with all signals in perfect agreement.
- 70. 52 Signal Analysis 01 2 3 4 5 10 −5 10 0 10 5 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P Blade A 3 e Blade B 3 e Blade C3 e Blade A1 f Blade B 1 f Blade C 1 f Figure 4.11: PSD blades signals H2 inner section, high rotor speed 4.1.3 Discussion Measurements at particular data sets from the real turbine and output from the H2 model have been presented. The tower signals presented peaks that one may relate to modes, especially to the first tower modes - peaks close to 0.95 Hz. PSD on V27 blade signals show that, at some frequencies, blades are excited differently leading to different magnitudes (and probably phases). Also, it is assumed that the double peaks found at rotor modes are due to the anisotropy effect presented in real turbines, even that this conjecture needs a backup. At higher rotor speed the accelerations magnitude increases as expected, but the information one can get remains practically the same. 4.2 Dynamic System Behaviour II: SVD Recalling the theory in Chapter 2, the rotational speed is associated with the whirling mode phenomena. There was found only little information about rotor modes in the raw data, but before applying the MBC to the blade signals, a SVD can be used to investigate if two modes appear in the peaks found, especially at the blade edge PSD. The SVD is performed on the spectral density matrices formed by the spectral density functions representing each of the curves in the previous section. The importance of the SVD is that it allows to identify closely spaced or repeated modes, which cannot be distinguished from the spectral density functions. If only one mode is dominating at a frequency, a singular value (SV) will be dominating. If more than one SV is dominating, then it will have as many SV as modes at the particular frequency. For instance, if two SV shows the same trend, the SVD may determine that there are two modes - closely spaced. If a harmonic peak appears, all singular values in the spectra might show a peak at that harmonic frequency. Figure 4.12 and Figure 4.13 show the SVD applied to the signals at low rotor speed.
- 71. 4.2 Dynamic SystemBehaviour II: SVD 53 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s2 )2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P 1st SV 2nd SV 3rd SV (a) V27 outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P 1st SV 2nd SV 3rd SV (b) V27 inner section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s2 )2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P 1st SV 2nd SV 3rd SV (c) Model outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P 1st SV 2nd SV 3rd SV (d) Model inner section Figure 4.12: SVD applied on edgewise signals at low rotor speed 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s2 )2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P 1st SV 2nd SV 3rd SV (a) V27 outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P 1st SV 2nd SV 3rd SV (b) V27 inner section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s2 )2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P 1st SV 2nd SV 3rd SV (c) Model outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P 1st SV 2nd SV 3rd SV (d) Model inner section Figure 4.13: SVD applied on flapwise signals at low rotor speed
- 72. 54 Signal Analysis AnalysingFigure 4.12 some observations are done: 1) There is a double peak formed in the first SV, which is followed by the second SV. This datum not only reinforces the previous suspicion that it is about a mode, but indicates two modes instead, since this peak needs to be described with two SV. This particular repeats in all the SVD, irrespective of the section or data, with the only difference that in the H2 model SVD, the double peak is just single - connecting to the anisotropy issue again. 2) All first and almost all the second and third SV denotes a peak at the harmonic frequencies, particularly in the V27 plots. In the H2 plots, the first SV shows all peaks as well, but larger with respect to the V27 plots, where the second and third SV hardly denotes any peak. This could mean that in the H2 representation the system behaviour is well described only with the first SV. 3) Very small peaks can be found involving one or more SV. Some examples are at 0.95 Hz and 1.5 Hz (in all subplots), or 2.5 Hz (V27 SVD). It may give a hint of where more modes can be located. 4) In Figure 4.12 (b) the second harmonic frequency denotes a larger peak (compared with the rest of plots), which could mean a torsional effect, since it also appears in its analogous flap SVD in Figure 4.13 (b). Some of these observations are also applicable to Figure 4.13. For instance, points 1 and 5, that are associated to Figure 4.12 (a,b), are also reflected in Figure 4.13 (b), which may support the blade torsional effect mentioned earlier. It can be observed how the ”thick tails” are more prominent at low frequencies, becoming a flat shape with sharp peaks at higher frequencies. In Figure 4.14, besides the prominent peaks described by the first and second SV at the 5P frequency, some smaller peaks appear in the rest of figures. As an example, at 0.95 Hz all figures present this small peak, sometimes with two SV tracking each other. Other small interesting peaks are found at 1.6 Hz and 2.2 Hz, denoted as double peaks in Figure 4.14 (a,b) and as single peak in Figure 4.14 (c,d). One supposition could be that the small size of these peaks might be related to flap modes. None of these peaks can be observed in Figure 4.15, mostly due to the aerodynamic forces subjected to the rotational frequency and its harmonics. 4.2.1 Discussion From the previous analysis, the first impression is that it has sustained the hypothesis that two modes are located in those double peaks (single peaks in the H2 model), which could be associated to the rotor edgewise components, since they are observed in the edgewise signals. One cannot define these modes yet, and a modal analysis is required to identify them properly. Besides these possible modes, there are small peaks appearing in almost all the SVD figures. Especially noticed is that at 0.95 Hz, in agreement with the peak found in the PSD of the nacelle signals performed in Section 4.1.1. So far, no trace of flapwise modes could be observed neither at low nor high rotor speed.
- 73. 4.2 Dynamic SystemBehaviour II: SVD 55 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s2 )2 /Hz 1P 2P 3P 4P 5P 6P 1st SV 2nd SV 3rd SV (a) V27 outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 1st SV 2nd SV 3rd SV (b) V27 inner section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s2 )2 /Hz 1P 2P 3P 4P 5P 6P 1st SV 2nd SV 3rd SV (c) Model outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 1st SV 2nd SV 3rd SV (d) Model inner section Figure 4.14: SVD applied on edgewise signals at high rotor speed 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s2 )2 /Hz 1P 2P 3P 4P 5P 6P 1st SV 2nd SV 3rd SV (a) V27 outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 1st SV 2nd SV 3rd SV (b) V27 inner section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s2 )2 /Hz 1P 2P 3P 4P 5P 6P 1st SV 2nd SV 3rd SV (c) Model outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 1st SV 2nd SV 3rd SV (d) Model inner section Figure 4.15: SVD applied on flapwise signals at high rotor speed
- 74. 56 Signal Analysis 4.3Dynamic System Behaviour III: MBC In the previous section some hypothesis referring to modes at 3.5 Hz and 0.95 Hz were addressed. The latter is more likely related to the tower due to the low frequency, and subsequently, it is not expected to appear in a blade signal analysis, at least in a relevant extent. Now, the system is to be transformed from time-variant to time-invariant by transforming the acceleration time histories from the sensors mounted on the blades using the MBC transformation. Since the rotor modes are split into three components, one may expect to find two asymmetric and one symmetric component for each of the rotor modes, in the flapwise and edgewise directions. One may recall from the blades spectra that some peaks featured a double peak in the crest. If this behaviour refers to two closely space modes, a side effect of the MBC is that the closely spaced peaks can be separated and clearly visualize. So far, this double effect was shown both in the PSD and SVD sections. In this sense, the MBC will shed some light in the eventual case that the double peaks actually represent two modes. The new coordinates after the transformation refer to three modal components: one symmetric (collective, a0) and two asymmetric (forward and backward whirling, a1 and b1). The reader may refer the theory explained in Chapter 2 for better understanding. Following, a PSD of these new coordinates is applied to each of the different cases: low and high rotor speed, edgewise and flapwise. The results are analysed in the following. Figure 4.16 shows the MBC applied to the edgewise signals at low rotor speed. 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P a0 a1 b1 (a) V27 outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P a0 a1 b1 (b) V27 inner section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P a0 a1 b1 (c) Model outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P a0 a1 b1 (d) Model inner section Figure 4.16: MBC applied on edgewise signals at low rotor speed
- 75. 4.3 Dynamic SystemBehaviour III: MBC 57 From Figure 4.16 several observations can be done: 1) The asymmetric coordinates are well correlated one to each other, while the sym- metric shows a different trend, meaning that the MBC effectively transforms from individual blades to rotor blades, and divides the collective from the asymmetric rotor behaviour. 2) Two peaks show up at 3 and 4.2 Hz. According to the theory, the whirling com- ponents of a mode are associated with ±Ω (difference between them 2Ω), where Ω is the rotational speed. One may recall that the suspected mode in Figure 4.12 was found around 3.5 Hz. At low rotor speed Ω equals close to 0.55 Hz (33 RPM approx.), and if it is subtracted and added to 3.5 Hz approx., it ends up with these two asymmetric peaks. Hence, still pending of a modal analysis, these two peaks may be considered as BW and FW components of a rotor mode. 3) It is also characteristic that both peaks keep the double peak in the crest, meaning that this fact is not related to two modes, it rather refers to anisotropic consequences. 4) According to Bir [18], the MBC filters out all harmonics but the multiple of 3Ω in the case of isotropic rotor (3-bladed), as seen in (c) and (d). Bir also states that if the rotor is anisotropic, the MBC still works, but cannot filter the harmonics, as can be checked in (a) and (b). 5) A peak arises between the asymmetric edge components, separated by approx. Ω. The acting aerodynamic loads mask any trace of mode in Figure 4.17. 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P a0 a1 b1 (a) V27 outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P a0 a1 b1 (b) V27 inner section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P a0 a1 b1 (c) Model outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P 7P 8P 9P a0 a1 b1 (d) Model inner section Figure 4.17: MBC applied on flapwise signals at low rotor speed
- 76. 58 Signal Analysis Oneof the scarce traces is a peak, mainly collective, with a light contribution of an asymmetric component, presumably a tower mode. A contribution of one of the edge asymmetric components is found around 4 Hz. In Figure 4.18, the asymmetric peaks do not show double peak due to lying at a harmonic frequency. In the H2 data, it is noticed that the FW peak is larger than the BW, thus perhaps the model demands finer tuning. In Figure 4.19 (b), two prominent 1P and 2P peaks can be seen, presumably adduc- ing blade torsional effects. Also, it can be observed that the two edgewise asymmetric components (comment no.2 above) and the peak in between them (comment no.5 above) contribute to the flapwise signals, as seen in Figure 4.18(b). 4.3.1 Discussion The presumed two modes reasoned from the SVD analysis are confirmed here. The MBC separated the closely spaced modes, leading to an effective identification of the asymmetric components of edgewise mode both in the V27 and H2 model. Moreover, and according to the theory, it removes all the harmonics but the multiples of 3P. This is accomplished in the model, which is isotropic, but not in the V27, because the rotor is anisotropic, thus so far satisfying Bir’s statements. The larger magnitude of the asymmetric peaks in the H2 model makes one to think either about imperfection in the structural properties, or on the proximity to the 6P frequency. 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P a0 a1 b1 (a) V27 outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P a0 a1 b1 (b) V27 inner section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P a0 a1 b1 (c) Model outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P a0 a1 b1 (d) Model inner section Figure 4.18: MBC applied on edgewise signals at high rotor speed
- 77. 4.4 Time SynchronousAveraging 59 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P a0 a 1 b1 (a) V27 outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P a0 a 1 b1 (b) V27 inner section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P a0 a1 b1 (c) Model outer section 0 1 2 3 4 5 10 −6 10 −4 10 −2 10 0 10 2 10 4 Hz (m/s 2 ) 2 /Hz 1P 2P 3P 4P 5P 6P a0 a1 b1 (d) Model inner section Figure 4.19: MBC applied on flapwise signals at high rotor speed 4.4 Time Synchronous Averaging The time synchronous averaging (TSA) technique is applied to the signals after MBC transformation in order to remove the sharp peaks in the harmonics. The purpose of this task is to avoid misinterpretations of the algorithm on the data (e.g. identifying a harmonic like a mode), thus leading to more reliable results. According to Bir, the averaging must follow the MBC. In general words, the TSA removes the deterministic (periodic) component of the signals due to the rotation of the rotor, maintaining only the stochastic component of the signal due to random events, e.g. wind turbulence. A rough explanation of how the method works is provided in the following based on Jacob et al. [7]. A tachometer is used to identify the periodic component. The acceleration time series are then chopped for each period and averaged. An enhanced signal of the periodic component results is replicated along the entire time series and subtracted from the original signal. The remaining part may be called the residual signal. This residual is the stochastic component of the signal, which satisfies the OMA assumption in this aspect. Further information may be found in [7]. In Figure 4.20 the procedure is shown for better understanding.
- 78. 60 Signal Analysis Figure4.20: TSA procedure. Source [7] The TSA technique is consequently applied to the MBC signals, and their corresponding PSD are plotted in Appendix B. It is advisable to compare MBC figures against TSA that with the exception of the high rotor speed V27 data, apparently works properly.
- 79. Chapter 5 Identifying ModalParameters In this chapter the operational modal analysis software package OMA Type 7760 from Br¨uel & Kjær (hereafter, OMA) is used for identification of the modal parameters ex- pressed in the V27 and H2 transformed coordinates, respectively. The identification is done using the SSI algorithm. Besides the structural modes, the algorithm unfortunately may also detect spurious modes due to noise. The expected outcome are the identified tower and rotor modes with the corresponding modal frequencies and damping ratios. The identification is arranged in the following manner: 1) Tower modes identification; 2) Rotor edge modes identification. A note on the effect of anisotropy is moreover included in this section; 3) Rotor flap modes identification. Due to the difficulties in identifying these modes, a simplified model is used to try to pin the problem down. The procedure first restricts the accelerations to the 0-5 Hz frequency range. Further, stabilization diagrams are used in the identification process to provide understanding of which modes are consistent or stable as the order of the model increases, and to discrim- inate between physical and computational modes. The modal parameter identification is limited to 1.5% damping ratio for in-plane modes and to 20% for out-of-plane modes, according to the HS2 results (cf. Table 3.9), meaning that all modes identified with damping ratios above these thresholds are neglected. The identification is supported with animations of the modes found, which specify the order of the modes and provide modal frequencies and damping. 5.1 Tower Modes Among the three acceleration sensors in the nacelle, only the front sensor data is used in the identification to compare with the H2 towertop channel, because they are geometri- cally the most similar. In Figure 5.1 a simple geometrical representation of the turbine is 61
- 80. 62 Identifying ModalParameters sketched, and the sensor x and y directions are indicated. It is intended to identify the tower modes and to verify if the algorithm is capable of detecting the rotor modes from one sensor only. Figure 5.1: Simple geometrical representation of the V27 The aforementioned displays the motion of the tower with respect to the wind direc- tion, which can be in the x-plane (in-plane) or the y-plane (out-of-plane). The tower modes are identified depending on the observed type of motion. For the rotor modes it is assumed that out-of-plane refers to flapwise and in-plane to edgewise, however, they are investigated in more detail in the next sections. A snapshot of the animation of the tower modes is provided in Figure 5.2 to visualize how the tower modes are identified. (a) TFA (b) TSS Figure 5.2: Tower modes 5.1.1 Low Rotor Speed First, the low speed rotor is investigated. Figure 5.3 shows a stability diagram with the identified modes from the V27 and H2 model signals.
- 81. 5.1 Tower Modes63 (a) V27 tower modes (b) H2 model tower modes Figure 5.3: Tower modes stability diagram at low rotor speed From the results presented in Figure 5.3, the following observations emerge: 1) The expected tower modes identified in Section 4.1.1 are also identified here (around 0.95 Hz). 2) Further, some modes with frequencies 3.09 Hz and 4.14 Hz are identified, which are related to the peaks found in the PSD tower signals and MBC blade signals. As for the latter, which from the latter, these were to be potentially related to the BWE and FWE components, respectively. 3) A group of modes are identified around 3P. Their frequencies match the expected frequencies for the flapwise modes, being between the tower and edge modes in this context. However, there is no hint about the character of these modes. 4) The noise at low frequencies leads to erroneous identification of two modes (left- most), in accordance to what was commented on in Chapter 4, In general, the identification based on H2 results shows a good agreement with the iden- tification based on V27 full-scale data. The identified modal frequencies and damping for the tower modes and the presumed BWE and FWE mode components are ratios presented in Table 5.1. The flapwise com- ponents are not included, since they are still unknown based on the analyses in previous sections. Apparently, neither the first TSS nor the BWE component of the V27 cannot be identified. The frequencies of the identified tower modes show a good agreement in the mutual comparison, and an acceptable deviation among the damping ratios. The HS2 shows the highest damping in the TFA, the lowest in the TSS and the highest frequencies.
- 82. 64 Identifying ModalParameters The frequencies and damping of the FWE component are also similar, particularly among H2 and HS2. The assumed FWE component displays the double peak feature (described in previous sections), and the algorithm identifies two components at the same peak. V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature 0.96 0.95 0.99 1.91 2.15 2.62 1st TFA - 0.96 1 - 1.68 0.35 1st TSS - 3.09 3.11 - 0.84 0.49 1st BWE 4.08 4.14 4.17 0.82 0.75 0.61 1st FWE Table 5.1: Modal parameters comparison for the tower sensors at low rotor speed 5.1.2 High Rotor Speed A similar analysis is shown in Figure 5.4 for the high speed operational case. The tower and edgewise modes are found straightforwardly both in the V27 and H2 data, unlike in the low rotor speed case, where the TSS and BWE are missing. A set of modes are found in the frequency range close to the 3P, similar as in the low rotor speed case, that could be related to the flap components. Since they have similar frequencies and cannot be decomposed in symmetric or asymmetric components, they are disregarded in the analysis. Some other modes respond to a harmonic, for instance the first mode located at the 1P. From all the modes, only those associated to the tower and BWE/FWE are considered, because are clearly identified. (a) V27 Tower modes (b) H2 model tower modes Figure 5.4: Tower modes stability diagram at high rotor speed According to Table 5.2, the tower results from V27 and H2 are more similar compared to the HS2 results, where the TSS mode is still very low damped. Analogously, the
- 83. 5.2 Rotor Modes65 presumed BWE and FWE components also display similar values in the V27 and H2 cases. However, the BWE component in the V27 case seems to be more damped than the FWE, contrary to what is observed in the H2 case. An important characteristic is that the difference in frequency between the BWE and FWE components is 2Ω (43 RPM; the HS2 rotor speed at this operating point is 35 RPM, cf. Table 3.7), indicating a whirling mode according to the theory. V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature 0.94 0.95 0.99 2.51 2.11 1.99 1st TFA 0.95 0.95 1 1.23 1.05 0.29 1st TSS 2.88 2.91 3.06 0.41 0.59 0.27 1st BWE 4.31 4.34 4.23 0.69 0.45 0.51 1st FWE Table 5.2: Modal parameters comparison for the tower sensors at high rotor speed 5.1.3 Tower Modes Identification Discussion Despite that the sensors used returned noise at low frequencies in the x direction, tower modes are well identified. Since the low rotor speed case is apparently more sensitive to the noise issue, the discussion embraces the high rotor speed case for the tower modes. The agreement between the V27 and H2 data is quite good showing the same range in frequencies and damping ratios. The frequencies of the first tower modes (TFA and TSS) are about the same, and the damping ratios difference in % is low (2.51 vs 2.11, 1.23 vs. 1.05, for each tower mode respectively). The greater challenge perhaps is that the rotor harmonics could not be removed in the V27 data, and therefore, peaks related to them are also identified. However, the animation tool guides one to select the proper mode. Moreover, edgewise modes were identified with apparent success because the whirling components are clearly shown in the tower spectra (especially at high rotor speed). How- ever, the modal parameters found must be assessed against those ones extracted from the blade edgewise sensors for higher confidence. The flapwise modes seemed to be identified around the 3P at both low and high rotor speed, where 3P is different for each case. This fact added to that they could not be traced neither in PSD or SVD analysis, lead to predict a difficult identification of these modes. 5.2 Rotor Modes In this section the edgewise and flapwise components are independently analysed. The rotor motion is represented in the MBC transformed coordinates, i.e. a0 (symmetric component), a1 and b1 (asymmetric components), similarly to [6], as shown in Figure 5.5. The nodes where the first digit in the notation starts with 1 denotes the collective com- ponent, while if starting with 2 or 3, it denotes the FW and BW components, respectively. Besides, the third digit equal to 3 or 8 denotes the outer or inner section in the edgewise. Similarly, with the third digit equal to 1 and 6, respectively, for the flapwise case.
- 84. 66 Identifying ModalParameters (a) Edgewise collective and asymmetric (b) Flapwise collective and asymmetric Figure 5.5: Geometrical representation of the MBC coordinates The reason for such a scheme is that can be useful to define if the mode found has an evident symmetric or asymmetric component, even if this geometry has not a physical meaning. Accordingly, the nodes only move in the direction of the arrow. In the case of whirling modes, if the phase between a1 and b1 is -90◦, it refers to a BW component, while if it is +90◦, it refers to a FW component. However, modes are never pure symmetric or asymmetric, and there is expected some contribution from collective to asymmetric and vice versa. In addition, if both nodes in a same component moves in-phase, it refers to a first bending mode, otherwise it is a second bending mode. Since only the first bending modes are treated here, the analysis only considers in-phase motion of the nodes. 5.2.1 Edgewise Analysis Low Rotor Speed Figure 5.6 shows the results from identification of modes in the edgewise data associated with the low rotor speed operational case. It can be observed that the double peak behaviour makes OMA to identify one component for each of the peaks in the V27 data, leading to 4 components instead of 2. The H2 analysis only shows 2 modes. In H2, the difference between the peaks is 2Ω, hence, one may assume that they refer to the asymmetric components. In V27, the difference between the first peak and third peak is also 2Ω, and similarly for the second and fourth peaks, meaning that they form pairs of asymmetric modes. Apparently, one of the pairs is not valid, since the rotor response only has one first edgewise mode (with two asymmetric components). The collective component is not identified.
- 85. 5.2 Rotor Modes67 (a) V27 edgewise modes (b) H2 model edgewise modes Figure 5.6: Edgewise modes stability diagram at low rotor speed The animation of these modes is used to check the phase between the asymmetric components. It is observed that a1 lags b1 in the first and third mode, indicating BWE. Contrary, a1 leads b1 in the second and fourth modes, indicating FWE. In addition, it can be seen in Figure 5.7 that when b1 reaches maximum amplitude, the phase angle is 180◦, and when a1 is at maximum the phase angle is 270◦, hence, a difference in phase of 90◦, where a1 lags b1. Figure 5.7: Phase angle difference between asymmetric modes According to Table 5.3, the second pair (components 2 and 4 in the table) of identified modes in the V27 data are closer to the frequencies and damping ratios from H2 and HS2 data. The frequency difference between BWE and FWE is in all cases about 1.05 Hz, i.e. 2Ω. It is assumed that, from the above, the second pair fits better the rest of
- 86. 68 Identifying ModalParameters available results. Therefore, the first pair (components 2 and 4 the table) is disregarded. In general, the frequencies display good agreement. The damping ratios in V27 and HS2 are very close, but the BWE is more damped in the V27 results than in the HS2 results. V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature 2.98 - - 1.38 - - 1st BWE 3.06 3.09 3.11 1.12 0.61 0.49 1st BWE 4.04 - - 0.41 - - 1st FWE 4.11 4.16 4.17 0.62 0.91 0.61 1st FWE Table 5.3: Modal parameters comparison for rotor edge sensors at low rotor speed High Rotor Speed Figure 5.8 shows the identification results from the blade sensors on the edgewise direction. Contrary to the low rotor speed case, 2 more modes are found in the V27 high speed analysis, related to harmonic frequencies 1P and 5P. Besides, the same type of results is found with 4 peaks identified for the V27 data and 2 for the H2 referring to asymmetric components. The animation provides the same phase difference as in the low rotor speed case, and thus connecting first-third modes and second-fourth peaks as pairs. For the high speed case the frequencies of the double peaks in V27 are much closer to the H2 and HS2 frequencies, complicating the task of correlating them with the simulated results. However, for this analysis either peak no.1 and 4 or peak no. 2 and 4 can become a pair. The difference is the same with respect to the rotational speed, approx. 2Ω. To be consistent with the low rotor speed case, the second pair is selected. (a) V27 edgewise modes (b) H2 model edgewise modes Figure 5.8: Edgewise modes stability diagram at high rotor speed
- 87. 5.2 Rotor Modes69 The HS2 frequencies disagree with the V27 and H2 frequencies, which in turn are quite similar, especially for the BWE. The difference between BWE and FWE is 2Ω for each of the results. The damping ratios are quite close in the FWE, but not in the BWE. In addition, the highest damped modes are the BWE in the V27 and H2 data, contrary to the HS2 data. Results can be observed in Table 5.4. V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature 2.87 - - 0.35 - - 1st BWE 2.93 2.91 3.06 1.15 0.62 0.27 1st BWE 4.29 - - 0.69 - - 1st FWE 4.33 4.34 4.23 0.70 0.5 0.51 1st FWE Table 5.4: Modal parameters comparison for rotor edge sensors at high rotor speed 5.2.2 Rotor Anisotropic Effects Induced in the H2 Model The double peak effect is shortly analysed in this section. The aim is to understand why this phenomena occurs and how much it affects the identification performance. The edgewise components are chosen for this analysis, since they clearly show this effect. For this analysis it has intentionally been specified different stiffnesses for the blades to resemble the V27 double peak, with the aim of confirming that this behaviour is due to anisotropic effects. The stiffness of the blades is changed as follows: 5% less stiffness in blades 1 and 2, and 10% more stiffness in blade 3, trying to maintain the mean stiffness in the rotor. Simulations are done for three cases: (a) low rotor speed, (b) high rotor speed, and (c) high rotor speed with the damping ratio limited to 2.3%, which is slightly larger than in the previous analysis. The reason of increasing the damping ratio limit imposed in the previous cases is because the algorithm is not capable of identifying each of the double peaks, as it is intended, therefore indicating that these modes have a larger damping, probably due to the change in structural properties of the blades that was assumed for the investigation of the anisotropic effect. Results are shown in Figure 5.9. With this change, the double peaks are now more separated. In case (b), only two modes are identified with the usual damping limit of 1.5%. However, the difference between them is notably less than 2Ω. If the damping limit is extended, 4 peaks appear identified as modes, and the difference turns out to be around 2Ω. It is worth to recall that the mean rotor speed is 43.22 RPM in the high rotor speed case, and the pair of peaks shows a difference of 42 and 42.6 RPM, respectively. One may also recall that in the low speed case the rotor mean speed is 32.25 RPM. The frequency difference of the peaks found is 32.43 and 32.01 RPM, for each respective pair.
- 88. 70 Identifying ModalParameters (a) Low rotor speed (b) High rotor speed, 1.5% max. damping ratio (c) High rotor speed, 2.3% max. damping ratio Figure 5.9: Smooth anisotropic effect on the model 5.2.3 Edgewise Modes Identification Discussion If one compares the identification results from the tower sensors and from the blade edgewise sensors, they appear to be very close. Table 5.5 displays the frequencies and damping of the edgewise components from the identification in the tower and in the blades for the best approach found at high rotor speed: Tower V27 Blades V27 Tower H2 Blades H2 Nomenclature Frequencies [Hz] 2.88 2.87 2.91 2.91 1st BWE 4.31 4.33 4.34 4.34 1st FWE Damping ratios [%] 0.41 0.35 0.59 0.62 1st BWE 0.69 0.70 0.45 0.50 1st FWE Table 5.5: Edge frequencies and damping from tower and blade sensors It is recalled from Section 5.2.1 that two pairs of BWE-FWE were obtained from the double peak behaviour. They were selected upon some assumptions in the low rotor speed
- 89. 5.2 Rotor Modes71 case. However, in the edgewise mode identification at high rotor speed, it was difficult to judge which pair was the correct whirling mode. It can be seen from Table 5.5, that the pair formed with components no.1 and no.4 (cf. Table 5.4) best matches the identification of edgewise modes from the tower sensors, displaying very close results. Hence, the major challenge in the edgewise components is related to the rotor anisotropy (either in dissimilarities of the blades or sensors position/orientation). The two peaks appearing at each component are confusing because they make pairs both with approx. 2Ω between each pair peak. According to the anisotropy case simulated in Section 5.2.2, it seems that the more dissimilar are the blades, the largest is the difference in frequency between the two peaks. This may suggest that the pairs are formed by peak no. 1 and 3 (first pair) and peak no.2 and 4 (second pair). However, at high rotor speed, this affirmation is not supported. Besides this issue, the edgewise frequencies and damping ratios are quite similar among the V27, H2 and HS2 results. The main difference may come by the assumed operational data in HS2, which stands for a different rotational speed, especially at the high rotor speed case. Thereof the difference in frequencies. Also, the lower damping in HS2 could be associated to the different pitch angle in the operating points, with respect to the pitch angle in V27 and H2 time stamps. It is observed that the FWE damping in V27 is lower than in BWE, contrary to H2 and HS2, at low rotor speed. It can be also noticed that the BWE peaks are larger at low rotor speed, while the FWE are larger at high rotor speed, especially in the H2 results. This effect may be caused by the proximity to the 3P and 6P frequencies respectively, that makes increasing the energy in the peak noticeably. More simulations may be needed with different turbulence seeds in order to discuss uncertainties about what component has higher damping, which is not covered in this thesis. 5.2.4 Flapwise Analysis Low Rotor Speed The same procedure that was applied with the edgewise data is now carried out with the flapwise data. The damping ratios are limited to be between 5% and 20% to avoid considering the least damped tower or edgewise modes. As for the H2 flap analysis no modes are detected. Contrary, the V27 analysis detects several modes. The first mode coincides with the 2P frequency. Using animations, the next three modes identified at 1.54, 1.62 and 1.67 Hz, respectively, are apparently the same, since the phase between the asymmetric components is always -90◦, indicating a BWF component. In the fifth and sixth identified modes, the animation shows only the collective component excited, leading to interpret this despite the considerable difference in frequency among them (2.2 and 2.67 Hz). The seventh mode at 3.05 Hz has a phase angle of +90◦, close to the BWE frequency in agreement with the results from the edgewise analysis section. The remaining modes have symmetric and asymmetric components excited and they are thus considered noise modes. Figure 5.10 shows the identified modes.
- 90. 72 Identifying ModalParameters (a) V27 flapwise modes (b) H2 model flapwise modes Figure 5.10: Flapwise modes stability diagram at low rotor speed By-pass filtering is now applied to both the V27 and H2 data. The filter is limiting the frequency range to between 1.4 and 3 Hz in an attempt to improve the results by focusing on the flap mode frequency range according to the HS2 results. After filtering and using the animation tool, modes are identified from the H2 data as shown in Figure 5.11. (a) V27 flapwise modes (b) Model flapwise modes Figure 5.11: Flapwise modes filtered stability diagram at low rotor speed In (a), a1 lags b1 by a phase of 90◦ in the identified component at 1.84 Hz (BWF); two collective components are found at 2.18 and 2.63 Hz; and a1 leads b1 by a phase of
- 91. 5.2 Rotor Modes73 90◦ at 2.79 Hz (FWF). The collective component is not clearly identified. In (b), BWF components are identified at 1.49 and 1.9 Hz, and a collective component is identified at 2.73 Hz. The animation of the remaining modes do not provide clear arguments to define them as symmetric or asymmetric components. There are discrepancies in the flap identification as shown in Table 5.6, where the re- sults from the filtered analysis are given. Two BWF and two CF components are found in the H2 and V27 data, respectively, where the frequencies of the BWF is different com- pared to the BWF frequencies from HS2. The damping ratios from HS2 are higher than those emerging from V27 and H2, and the largest similarity is found between V27 and H2. V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature - 1.49 - - 6.39 - 1st BWF 1.84 1.9 1.69 10.15 9.51 15.88 1st BWF 2.18 - - 6.08 - - 1st CF 2.63 2.73 2.59 7 7.64 11.92 1st CF 2.79 2.8 2.73 9.3 5.54 11.36 1st FWF Table 5.6: Modal parameters comparison for rotor flap sensors at low rotor speed High Rotor Speed The identification for the high rotor speed case provides good agreement between the V27 and H2 data, despite that the harmonics are not removed as seen in Figure 5.12 (a), because the TSA is not applied. The identified modes at frequencies above 4 Hz are disregarded. (a) V27 flapwise modes (b) H2 model flapwise modes Figure 5.12: Flapwise modes stability diagram at high rotor speed From the identified components, there is only agreement regarding the BWF component
- 92. 74 Identifying ModalParameters between the V27 and H2, identified at 1.69 and 1.7 Hz, respectively, with very similar damping ratios and an acceptable difference with respect to HS2. If the data is filtered such as in Section 5.2.4, only the BWF is identified, at very similar frequencies (1.72 and 1.73 Hz) but with about half the damping ratios (9.83 and 7.65 %). No clear CF of FWF components are found in the analysis, and thus, are disregarded in the collected results, as shown in Table 5.7. V27 [Hz] H2 [Hz] HS2 [Hz] V27 [%] H2 [%] HS2 [%] Nomenclature 1.7 1.69 1.61 19.24 18.84 15.03 1st BWF - - 2.55 - - 11.31 1st CF - - 2.73 - - 10.18 1st FWF Table 5.7: Flapwise modal parameters comparison at low rotor speed 5.2.5 Model Simplification The flap identification yields uncertain results regarding the frequency of the components. Several modes are identified but, based on their animation, only the BWF components seemed to be identified properly. Therefore, a different approach is launched to provide a more accurate identification of the flap modes, consisting in performing two simulations with the H2 model - one with impulse excitation and the other using a random excitation. It is recalled that OMA relies basically on three assumptions: 1) the system is linear; 2) the system is invariant with respect to time; and 3) forces applied are white noise excitation. To investigate what the basic challenges with the flapwise mode identification are, a simplified model is set up, where: 1) the geometry is simplified, as tilt is removed; 2) wind loading is replaced with a white noise excitation. The new loading cases agrees well with the third OMA assumption. Subsequently, the wind loading is gradually implemented to observe where the algorithm experiences identification problems, thus shedding some light on what is the challenge regarding the identification of flapwise modes. The low rotor speed model is used during this approach, since it seemed the most challenging for OMA at first instance. The rotational speed has been set to a constant equal to 32.24 RPM for the first simulations. Impulse Loading The impulse loading is applied in the flapwise direction to excite the blades. A force magnitude of 100 N is used. The spectra of the impulse force is shown in Figure 5.13. As seen, the spectra is fairly flat within the frequency range of 0-5 Hz. An impulse excitation is applied every 150 s at the outer section of a blade. In total, 2 excitations per blade are imposed during the simulation time. The underlying idea is to mimic a hammer impact at each blade with the turbine in operation. Each of the
- 93. 5.2 Rotor Modes75 impulses lasts 0.05 seconds to achieve a flat spectrum, which agrees well with the white noise assumption in the frequency range of interest. 0 1 0 2 0 3 0 4 0 5 0 Fre q u e n c y (Hz) 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 Psd im p u ls e [-] Figure 5.13: Impulse loading spectra Random Loading The random loading is set up to simulate white noise excitation. Compared to the im- pulse loading, this case is probably better fulfilling the third OMA assumption, and it is considered as a double-check for the identification of the flap modes. The random loading is applied to the tower at different sections (towertop and 18 meters above ground) and on the blades at the outer and inner sections, both in the x and y directions with respect to the wind. However, the tower loading was specified 20 times lower than the blade loading. This setup is required to avoid iteration issues in the HAWC2 simulation. Figure 5.14 shows the spectra of the blade loading. 0 10 20 30 40 50 Frequency (Hz) 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Psd Random [-] Figure 5.14: Spectra of the random function Flapwise Signal Analysis In Figure 5.15, the SVD of both cases are shown. One may observe that the impulse case provides a clearer visualization of the peaks. However, the mutual agreement between the cases is good. In (a), two possible modes at 2.16 and 3.61 Hz are shown, where the last coincides with the edgewise mode found in previous section. The same peaks are found in (b), with a
- 94. 76 Identifying ModalParameters very little variation in frequency, which is considered negligible. In this last case, it can be appreciated that the first SV is noticeable higher than the other two SV, which can be attributed to the different force setup required to troubleshoot the iteration issues in HAWC2, as mentioned above. 0 1 2 3 4 5 10 −12 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 1P 2P 3P 4P 5P 6P 7P 8P 9P X: 3.613 Y: 0.0001179 Hz N2 /Hz X: 2.163 Y: 0.1432 1st SV 2nd SV 3rd SV (a) Impulse case 0 1 2 3 4 5 10 −10 10 −5 10 0 1P 2P 3P 4P 5P 6P 7P 8P 9P X: 2.169 Y: 0.01859 X: 3.613 Y: 6.879e−05 Hz N2 /Hz 1st SV 2nd SV 3rd SV (b) Random case Figure 5.15: SVD on the blade signals for impulse and random cases Similarly, in Figure 5.16 the MBC of the blade signals for both cases are presented. In both cases, three flap components at 1.63, 2.51 and 2.68 Hz are identified, with which an association may be done to the BW, collective and FW flap components. Some other peaks are also found that could be related to the TFA (0.95 Hz), edgewise asymmetric components (3.05 Hz and 4.14 Hz) and an asymmetric mode-shape peak close to 5 Hz. If one takes the presumed flap mode in the SVD (2.16 Hz), and subtract/add the rotational speed ( 32.24 RPM), it turns out to be the asymmetric components found in the MBC diagrams, which is in agreement with the MBC theory.
- 95. 5.2 Rotor Modes77 0 1 2 3 4 5 10 −12 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 1P 2P 3P 4P 5P 6P 7P 8P 9P X: 2.688 Y: 0.05002 X: 2.513 Y: 0.06652 X: 1.638 Y: 0.04044 Hz N2 /Hz a 0 a 1 b1 (a) Impulse case 0 1 2 3 4 5 10 −10 10 −5 10 0 1P 2P 3P 4P 5P 6P 7P 8P 9P X: 1.638 Y: 0.008039 Hz N2 /Hz X: 2.513 Y: 10.29 X: 2.681 Y: 0.01666 a0 a1 b 1 (b) Random case Figure 5.16: MBC on the blade signals for impulse and random cases Flap Mode Identification In Figure 5.17 the identification of both excitation cases is shown. All flap components and the TFA mode can be observed in both cases. An additional mode is found in (a) that could be related to the peak close to 5 Hz in Figure 5.16 (a). The frequencies obtained from both cases are very close to the HS2 results as can be seen in Table 5.8. This shows the good match between the two different cases. The damping ratios are very low, since aerodynamic effects are not included, and therefore, the aerodynamic damping does not contribute to the flapwise damping. Note the abbre- viations, i.e. impulse case (I) and random case (R). It is seen that the frequencies are are basically the same among the two cases. The damping ratios are influenced by the absence of aerodynamics in the simple approach, and therefore it is senseless to compare with the HS2 results. However, between the two approaches, the damping is almost the same. It is conjectured that the impulse may clear things up along this process, while the random excitation is more similar to the
- 96. 78 Identifying ModalParameters wind/turbulence effect. With this premise, the impulse case is continued in the following assuming that the OMA excitation assumption is fulfilled. (a) Impulse case (b) Random case Figure 5.17: Flapwise modes stability diagram for simple model I [Hz] R [Hz] HS2 [Hz] I [%] R [%] HS2 [%] Nomenclature 1.63 1.63 1.69 0.72 0.65 15.88 1st BWF 2.52 2.51 2.59 0.51 0.69 11.92 1st CF 2.68 2.68 2.73 0.45 0.46 11.36 1st FWF Table 5.8: Comparison of frequencies and damping for simple approach Detecting the Flapwise Problem In the previous stage of the simplified model excitation procedure the rotor aerodynamics were not included. The results showed that the flapwise components can be identified, but also additional modes such as the tower modes. The next step involves adding aerodynamic effects to the H2 model. Two cases are also performed: the impulse case including deterministic wind excitation (IW) and only deterministic wind excitation (W). A shear factor (power law) of 0.23 is set for the wind conditions. No tilt angle, tower shadow or turbulence is included yet. The rotor speed is still kept constant. The modal identification in the IW case is able to identify almost the same frequencies and damping ratios as shown in Figure 5.18 (a). However, no modes are identified in the W case, similar to Figure 5.10 (b). Subsequently, tilt and tower shadow are included in the simulation. Even with these changes, the identification algorithm cannot identify any mode yet. Therefore, the wind speed is increased to 5.4 m/s, and with this new change some components are identified, as shown in Figure 5.18 (b).
- 97. 5.2 Rotor Modes79 (a) Impulse + wind 5 m/s (b) Only wind 5.4 m/s Figure 5.18: Stability diagrams of impulse+wind and only wind The results show that the modal frequencies emerging from the W case in general increase, whereas in damping ratio decreases, as observed in Table 5.9. Note that the abbreviations refer to the impulse plus wind case (I+W) and to the only wind case (W). I+W [Hz] W [Hz] HS2 [Hz] I+W [%] W [%] HS2 [%] Nomenclature 1.67 1.80 1.69 15.37 14.52 15.88 1st BWF 2.55 2.58 2.59 11.38 10.45 11.92 1st CF 2.70 2.88 2.73 11.83 10.65 11.36 1st FWF Table 5.9: Comparison of frequencies and damping including wind When the wind is included the effect of the aerodynamic damping can be clearly dis- tinguished, varying from less than 1% to about 15%. The difference between damping ratios might be a good hint to estimate the aerodynamic damping. Bearing in mind these results, the next stage is to introduce turbulence loading. A turbulence intensity of 2% is included in the simulation using the same turbulence box as used in the low rotor speed model in Section 5.2.4. With the introduction of turbulence in the simulation, the impulse does not significantly affect the results, and no components can be identified. Therefore, a new impulse with larger magnitude is implemented (10 kN), this time exciting both flapwise and edgewise directions. On the one hand, with this new impulse excitation the results show that components can be identified, as shown in Figure 5.19 (a). On the other hand, the soft slope denoting CF and FW has disappeared in (b), and therefore, they are not identified, similar to what happened in Figure 5.10 and therefore motivating further investigation on this phenom- ena. Including the turbulence seemingly means that the flapwise components start to be masked, and hence difficult to identify. This fact pin downs that the turbulence loading is the point of inflexion.
- 98. 80 Identifying ModalParameters (a) Impulse + wind (b) Only wind Figure 5.19: Stability diagrams with low turbulence The frequencies and damping ratios extracted from this last analysis are shown in Table 5.10. It is observed that with the impulse excitation the results acceptably agree with those from HS2. I+W [Hz] W [Hz] HS2 [Hz] I+W [%] W [%] HS2 [%] Nomenclature 1.63 1.77 1.69 11.34 8.28 15.88 1st BWF 2.54 - 2.59 11.05 - 11.92 1st CF 2.71 - 2.73 10.91 - 11.36 1st FWF Table 5.10: Comparison of frequencies and damping for turbulent cases 5.2.6 Flapwise Modes Identification Discussion The flapwise analysis showed the difficulties of identifying flap modes with the current method. A minimum wind speed seems to be required to excite properly the flapwise modes if the system, as seen in the low rotor speed case, where the modal frequencies and damping resulted from filtering and assuming a specific frequency range for flap modes. The results between V27 and H2 data are acceptable, but the damping ratios with respect to the HS2 are somehow different. It is also shown that the damping ratios are higher in the high rotor speed case (such as in the HS2 results), but also that the SSI algorithm identified several modes, from which only the BWF was identified from the V27 and H2 data, leading to the simplified approach. It has to be taken into account that the assumptions in HS2 are different from V27 and H2 cases. With respect to the simplified model, the results in the most simple case are successful for each of the two approaches, giving the same frequencies and very close damping ratios.
- 99. 5.2 Rotor Modes81 The introduction of the wind makes the estimations to slightly deviate in the case without impulse, but the results are still within an acceptable range with similar damping ratios. When the turbulence is introduced, the BWF is the only component identified. The cases with impulse loading apparently identify modes with success, even with turbulence. The general trend is that the damping ratios decrease as the model increase complexity, while the damping ratios are higher without turbulence. Besides, even that the wind is seen as an excellent excitation force to OMA, the in- teraction of the aerodynamics with the rotational rotor (correlated forces by multiples of 1P) avoids proper identification. Indeed, it seems that with a deterministic excitation (wind shear) the algorithm fulfills better than when the stochastic excitation (turbulence) is added.
- 100. 82 Identifying ModalParameters
- 101. Chapter 6 Conclusions The workpresented in this thesis included identification of modal parameters on a full- scale Vestas V27 wind turbine based on experimentally obtained acceleration signals, at low and high rotor speed. Concurrently, a HAWC2 model was implemented to simulate the behaviour of the real V27 turbine, and this allowed to compare predicted results in the H2 model with the measured results in the real V27. Also, a theoretical prediction of modal parameters based on uniform inflow, without turbulence and gravity loads was performed with HAWCStab2. The first result obtained showed that it is possible to extract modal parameters from experimental data applying the MBC in conjunction with OMA. The tower and edgewise modes were possible to extract from only the tower sensors. However, the flapwise modes were more difficult to extract. They required data from the blades and special treatment, in this case, a band-pass filter. A simplified numerical model was also used to improve the identification results. This simple approach matched the predictions from HAWCStab2. It has been also found that an impulse excitation is helpful for the identification of these modes. The impulse force may vary depending on the characteristics of the wind, where a turbulent wind overlaps a low impulse excitation. Secondly, the reason why the flapwise modes were difficult to identify could be the turbulence. The algorithm could even perform satisfactorily with the deterministic com- ponent of the wind (wind shear), however, the interaction of the aerodynamic loading with the rotation of the rotor seemed to interfere in the flapwise identification, especially at low wind speed. This obstruction was apparently higher with turbulence, despite that turbulence is seen as a perfect excitation for OMA. Finally, it has been also shown that the numerical model implemented in HAWC2, and the subsequent modal analysis in HAWCStab2, were able to provide comparable results with the full-scale experiment. Besides, some challenges were detected. In particular, the double peak feature found in the experimental data, such as in Figure 5.6 (a), complicated the proper identification of edgewise mode components. A hypothesis was considered that the double peak feature 83
- 102. 84 Conclusions regards tothe rotor anisotropy since the numerical model did not show this behaviour, which may be associated to dissimilarity on the blades or in the sensor mounting. This hypothesis was tested introducing a smooth stiffness difference among the blades, which confirmed that this could be a possible reason of the observed behaviour. 6.1 Future Research This work has been based on only two time series that gave some estimation of modal parameters. Therefore, more time series from the experimental measurements from the V27 as well as more simulations in HAWC2 might be needed to trigger more questions especially in the damping ratios. This could be helpful for e.g. to determine which whirling component is actually more damped, or the uncertainties regarding the damping values. The anisotropy mentioned on the small note could be also applied in a different way. For instance, sensors might be intentionally misaligned in order to confirm that the double peak behaviour is also shown. The identification could also be done using another method different to the MBC, which might not consider the rotor anisotropy, and then results could be compared to pin down the differences.
- 103. Bibliography [1] Tcherniak D.,Chauhan S., Basurko J., Salgado O., Carcangiu C. E., and Rossetti M. Application of OMA to Operational Wind Turbine. Proceedings - 4th International Operational Modal Analysis Conference (IOMAC’11), 2011. [2] Hansen M. H. Aeroelastic instability problems for wind turbines. Wind Energy, Vol. 10, No. 6, 2007, p. 551-577, 2007. [3] Brandt A. Noise and Vibration Analysis: Signal Analysis and Experimental Proce- dures. John Wiley & Sons, Ltd., 2010. [4] Hansen M. H. Aeroelastic stability analysis of wind turbines using an eigenvalue approach. Wind Energy, Vol. 7, No. 2, 2004, p. 133-143, 2004. [5] Tcherniak D. and Larsen G. C. Application of OMA to an Operating Wind Tur- bine: now including Vibration Data from the Blades. Proceedings - 5th International Operational Modal Analysis Conference (IOMAC’13), 2013. [6] Yang S., Tcherniak D., and Allen M. S. Modal Analysis of Rotating Wind Turbine Using Multiblade Coordinate Transformation and Harmonic Power Spectrum. Topics in Modal Analysis I, Volume 7, p. 77-92 Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics., 2014. [7] Jacob T., Tcherniak D., and Castiglione R. Harmonic Removal as a Pre-processing Step for Operational Modal Analysis: Application to Operating Gearbox Data. Br¨uel & Kjær Conference Paper, 2014. [8] James G. H., Carne T. G., and Lauffer J. P. The natural excitation technique (NExT) for modal parameter extraction from operating wind turbines. Technical Report SAND92-1666, Sandia National Laboratories, 1993. [9] Johnson W. Helicopter theory. Princeton University Press, New Jersey, 1980. [10] Peters D. A. Fast Floquet theory and trim for multi-bladed rotorcraft. Journal of the American Helicopter Society; 39: 82-89, 1994. 85
- 104. 86 BIBLIOGRAPHY [11] TcherniakD., Chauhan S., and Hansen M. H. Applicability Limits of Operational Modal Analysis to Operational Wind Turbines. Structural Dynamics and Renewable Energy. Vol. 1 Society for Experimental Mechanics, 2011. p. 317-327 (Conference Proceedings of the Society for Experimental Mechanics Series; No. 10), 2010. [12] Van Overschee P. and De Moor B. Subspace identification for linear systems: theory, implementation, applications. Kluwer Academic Publishers, 1996. [13] Hansen M. H., Thomsen K., Fuglsang P., and Knudsen T. Two Methods for Esti- mating Aeroelastic Damping of Operational Wind Turbine Modes from Experiments. Wind Energy, Vol. 9, No. 1-2, 2006, p. 179-191, 2006. [14] Palle P. and Rosenow S. E. Operational Modal Analysis of a Wind Turbine Mainframe using Crystal Clear SSI. Structural Dynamics and Renewable Energy, Volume 1. Proceedings of the 28th IMAC, A Conference on Structural Dynamics, 2010. [15] Van Der Valk P. L. C. and Ogno M. G. L. Identifying Structural Parameters of an Idling Offshore Wind Turbine Using Operational Modal Analysis. Dynamics of Civil Structures, Volume 4, p. 271-281 Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics., 2014. [16] Marinone T., Cloutier D., LeBlanc B., Carne T., and Palle P. Artificial and Natural Excitation Testing of SWiFT Vestas V27 Wind Turbines. Proceedings of the 32th International Modal Analysis Conference (IMAC) Orlando, Florida USA, 2014. [17] Hansen M. H. Improved modal dynamics of wind turbines to avoid stall-induced vibrations. Wind Energy, Vol. 6, 2003, p. 179-195, 2003. [18] Bir G. Multiblade coordinate transformation and its application to wind turbine analysis. ASME Wind Energy Simposium, Reno, Nevada, 2008. [19] Berg J., Mann J., and Nielsen M. Notes for DTU course 46100: Introduction to micro meteorology for wind energy. DTU Wind Energy E-0009 (EN) ISBN 978-87- 92898-15-5, 2013. [20] MATLAB. version 8.2 (R2013b). Natick, Massachusetts: The MathWorks Inc., 2013. [21] Batel M. Operational Modal Analysis - Another Way of Doing Modal Testing. Journal of Sound and Vibration, 2002. [22] Hansen M. H., Gaunaa M., and Aagaard Madsen H. A Beddoes-Leishman type dynamic stall model in state-space and indicial formulations. Forskningscenter Risoe. Risoe-R, no. 1354(EN), 2004. [23] Br¨uel & Kjær. Product Data Operational Modal Analysis Type 7760. www.bksv.com - Accessed August, 2014. [24] Petersen J. T. The aeroelastic code HawC - model and comparisons. In Pedersen BM, editor, State of the Art of Aeroelastic Codes for Wind Turbine Calculations, volume Annex XI, pages 129-135, Lyngby. International Energy Agency, Technical University of Denmark, 1996.
- 105. BIBLIOGRAPHY 87 [25] S.M. Petersen. Wind Turbine Test Vestas V27-225 kW. Technical Report Risø-M- 2861, Risø National Laboratory, DK-4000 Roskilde, Denmark, 1990. [26] Larsen T. J. and Hansen A. M. How 2 HAWC2, the user’s manual. Ris-R-1597(ver. 4-4)(EN), 2013. [27] Petersen J. T. Kinematically Nonlinear Finite Element Model of a Horizontal Axis Wind Turbine. PhD Thesis, Risø National Laboratory, DK-4000 Roskilde, Denmark, 1990. [28] Brincker R. and Andersen P. Understanding stochastic subspace identification. Con- ference Proceedings : IMAC-XXIV : A Conference & Exposition on Structural Dy- namics. Society for Experimental Mechanics, 2006. [29] Brincker R. and Andersen P. The stochastic subspace identification techniques. www.svibs.com, 2001. [30] Larsen G. C. Aeroelastic simulations as basis for V27 accelerometer instrumentation. Internal Report. Ris DTU Wind Energy. Aeroelastic Design Section, 2012. [31] Jacobsen N. J. and Andersen P. Operational Modal Analysis on Structures with Rotating Parts. ISMA Conference Paper no. 268, 2008. [32] IEC technical committee 88. IEC 61400-1 Ed.3: Wind turbines - Part 1: Design requirements. International Electrotechnical Commission, 2005.
- 106.
- 107. Appendix A The accelerationstime histories from the V27 measurements and H2 model channels are reviewed in this chapter. First, an overview of all sensors in the blade is presented in Figure A.1 and Figure A.2. For ease of follow-up, the legend indicates the orientation of the sensor, blade number and sensor according to Figure 3.3 in Chapter 3. Blade 3 signals from sensor no. 2 appears to be faulty in both data sets. In the bottom right corner, one of the two edgewise sensors, no. 3, is clearly showing a strong in-plane excitation - gravity loads. Its magnitude is therefore around 10 m/s2 as indicated. The other edgewise sensor, no. 8, has not the same behaviour as no. 3, since it is located at 67% of the blade length (so-called inner section in the report). Therefore, it seems to have a flapwise contribution. In the same way, sensors no. 6, 9 and 11 have a close appearance to no. 8, but it is interesting that they are installed to measure flapwise accelerations, not edgewise. Hence, there is also a contribution of the gravity loads on these flapwise sensors. It is relevant to remember that all these sensors are located in the trailing edge of the blade and sorted as following: 1, 3, 4, 6, 8, 9 and 11. Indeed, the shape of signal in sensors 6, 8 and 9 is basically the same, with the exception of the magnitude: only 5 m/s2 for sensor no. 9. Similarly, the gravity loads are also found in the leading edge, where all sensors are measuring flapwise motion. However, the influence is not as evident as in the trailing edge. 89
- 108. APPENDICES90 278 280 282284 −10 0 10 Leading edge flap1 2 flap2 2 flap3 2 278 280 282 284 −5 0 5 flap1 5 flap2 5 flap3 5 278 280 282 284 −5 0 5 acceleration(m/s 2 ) flap1 7 flap2 7 flap3 7 278 280 282 284 −5 0 5 flap1 10 flap2 10 flap3 10 278 280 282 284 −5 0 5 flap1 12 flap2 12 flap3 12 278 280 282 284 −10 0 10 time (s) edge1 8 edge2 8 edge3 8 278 280 282 284 −10 0 10 Trailing edge flap1 1 flap2 1 flap3 1 278 280 282 284 −5 0 5 10 flap1 4 flap2 4 flap3 4 278 280 282 284 −10 0 10 acceleration(m/s 2 ) flap1 6 flap2 6 flap3 6 278 280 282 284 −5 0 5 flap1 9 flap2 9 flap3 9 278 280 282 284 −10 0 10 flap1 11 flap2 11 flap3 11 278 280 282 284 −10 0 10 time (s) edge1 3 edge2 3 edge3 3 Figure A.1: Acceleration time histories low rotor speed
- 109. APPENDICES91 278 280 282284 −20 0 20 Leading edge flap1 2 flap2 2 flap3 2 278 280 282 284 −10 0 10 20 flap1 5 flap2 5 flap3 5 278 280 282 284 −10 0 10 acceleration(m/s 2 ) flap1 7 flap2 7 flap3 7 278 280 282 284 −10 0 10 flap1 10 flap2 10 flap3 10 278 280 282 284 −10 0 10 flap1 12 flap2 12 flap3 12 278 280 282 284 −10 0 10 time (s) edge1 8 edge2 8 edge3 8 278 280 282 284 −20 0 20 Trailing edge flap1 1 flap2 1 flap3 1 278 280 282 284 −10 0 10 flap1 4 flap2 4 flap3 4 278 280 282 284 −10 0 10 acceleration(m/s 2 ) flap1 6 flap2 6 flap3 6 278 280 282 284 −10 0 10 flap1 9 flap2 9 flap3 9 278 280 282 284 −10 0 10 20 flap1 11 flap2 11 flap3 11 278 280 282 284 −20 0 20 time (s) edge1 3 edge2 3 edge3 3 Figure A.2: Acceleration time histories high rotor speed
- 110. APPENDICES 92 V27 BladesAccelerations at Low Rotor Speed 0 200 400 600 800 1000 1200 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (a) Outer section edgewise 277 278 279 280 281 282 283 284 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (b) Outer section edgewise. Zoom view 0 200 400 600 800 1000 1200 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (c) Inner section edgewise 277 278 279 280 281 282 283 284 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (d) Inner section edgewise. Zoom view 0 200 400 600 800 1000 1200 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (e) Outer section flapwise 277 278 279 280 281 282 283 284 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (f) Outer section flapwise. Zoom view 0 200 400 600 800 1000 1200 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (g) Inner section flapwise 277 278 279 280 281 282 283 284 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (h) Inner section flapwise. Zoom view Figure A.3: Time series of blade accelerations
- 111. APPENDICES 93 V27 BladesAccelerations at High Rotor Speed 0 200 400 600 800 1000 1200 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (a) Outer section edgewise 277 278 279 280 281 282 283 284 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (b) Outer section edgewise. Zoom view 0 200 400 600 800 1000 1200 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (c) Inner section edgewise 277 278 279 280 281 282 283 284 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (d) Inner section edgewise. Zoom view 0 200 400 600 800 1000 1200 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (e) Outer section flapwise 277 278 279 280 281 282 283 284 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (f) Outer section flapwise. Zoom view 0 200 400 600 800 1000 1200 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (g) Inner section flapwise 277 278 279 280 281 282 283 284 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (h) Inner section flapwise. Zoom view Figure A.4: Time series of blade accelerations
- 112. APPENDICES 94 H2 BladesAccelerations at Low Rotor Speed 0 200 400 600 800 1000 1200 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (a) Outer section edgewise 692 694 696 698 700 702 704 706 708 710 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (b) Outer section edgewise. Zoom view 0 200 400 600 800 1000 1200 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (c) Inner section edgewise 692 694 696 698 700 702 704 706 708 710 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (d) Inner section edgewise. Zoom view 0 200 400 600 800 1000 1200 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (e) Outer section flapwise 692 694 696 698 700 702 704 706 708 710 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (f) Outer section flapwise. Zoom view 0 200 400 600 800 1000 1200 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (g) Inner section flapwise 692 694 696 698 700 702 704 706 708 710 −25 −20 −15 −10 −5 0 5 10 15 20 25 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (h) Inner section flapwise. Zoom view Figure A.5: Time series of blade accelerations
- 113. APPENDICES 95 H2 BladesAccelerations at High Rotor Speed 0 200 400 600 800 1000 1200 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (a) Outer section edgewise 674 676 678 680 682 684 686 688 690 692 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (b) Outer section edgewise. Zoom view 0 200 400 600 800 1000 1200 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (c) Inner section edgewise 674 676 678 680 682 684 686 688 690 692 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (d) Inner section edgewise. Zoom view 0 200 400 600 800 1000 1200 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (e) Outer section flapwise 674 676 678 680 682 684 686 688 690 692 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (f) Outer section flapwise. Zoom view 0 200 400 600 800 1000 1200 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s2 ) blade 1 blade 2 blade 3 (g) Inner section flapwise 674 676 678 680 682 684 686 688 690 692 −50 −40 −30 −20 −10 0 10 20 30 40 50 time (s) acceleration(m/s 2 ) blade 1 blade 2 blade 3 (h) Inner section flapwise. Zoom view Figure A.6: Time series of blade accelerations
- 114.
- 115. Appendix B Time SynchronousAveraging 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 Hz (m/s2 )2 /Hz a0 a1 b1 (a) V27 outer section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 Hz (m/s2 )2 /Hz a0 a1 b1 (b) V27 inner section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 Hz (m/s2 )2 /Hz a0 a1 b1 (c) Model outer section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 Hz (m/s2 )2 /Hz a0 a1 b1 (d) Model inner section Figure A.7: TSA applied on MBC edgewise signals at low rotor speed 97
- 116. APPENDICES 98 0 0.51 1.5 2 2.5 3 3.5 4 4.5 5 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 Hz (m/s2 )2 /Hz a0 a1 b1 (a) V27 outer section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 Hz (m/s2 )2 /Hz a0 a1 b1 (b) V27 inner section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 Hz (m/s2 )2 /Hz a0 a1 b1 (c) Model outer section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 Hz (m/s2 )2 /Hz a0 a1 b1 (d) Model inner section Figure A.8: TSA applied on MBC flapwise signals at low rotor speed 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −4 10 −2 10 0 10 2 10 4 10 6 Hz (m/s2 )2 /Hz a0 a1 b1 (a) V27 outer section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −4 10 −2 10 0 10 2 10 4 10 6 Hz (m/s 2 ) 2 /Hz a0 a1 b1 (b) V27 inner section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −4 10 −2 10 0 10 2 10 4 10 6 Hz (m/s2 )2 /Hz a0 a1 b1 (c) Model outer section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −4 10 −2 10 0 10 2 10 4 10 6 Hz (m/s 2 ) 2 /Hz a0 a1 b1 (d) Model inner section Figure A.9: TSA applied on MBC edgewise signals at high rotor speed
- 117. APPENDICES 99 0 0.51 1.5 2 2.5 3 3.5 4 4.5 5 10 −4 10 −2 10 0 10 2 10 4 10 6 Hz (m/s2 )2 /Hz a 0 a 1 b1 (a) V27 outer section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −4 10 −2 10 0 10 2 10 4 10 6 Hz (m/s 2 ) 2 /Hz a 0 a 1 b1 (b) V27 inner section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −4 10 −2 10 0 10 2 10 4 10 6 Hz (m/s2 )2 /Hz a0 a1 b1 (c) Model outer section 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −4 10 −2 10 0 10 2 10 4 10 6 Hz (m/s 2 ) 2 /Hz a0 a1 b1 (d) Model inner section Figure A.10: TSA applied on MBC flapwise signals at high rotor speed
- 118.
- 119. DTU Wind Energyis a department of the Technical University of Denmark with a unique integration of research, education, innovation and public/private sector consulting in the field of wind energy. Our activities develop new opportunities and technology for the global and Danish exploitation of wind energy. Research focuses on key technical-scientific fields, which are central for the development, innovation and use of wind energy and provides the basis for advanced education at the education. We have more than 240 staff members of which approximately 60 are PhD students. Research is conducted within nine research programmes organized into three main topics: Wind energy systems, Wind turbine technology and Basics for wind energy. Danmarks Tekniske Universitet DTU Vindenergi Frederiksborgvej 399 Bygning 118 4000 Roskilde Danmark www.vindenergi.dtu.dk