The formulation is model-independent, in the sense that it does not require knowledge of the equations of motion of the periodic system being analyzed, and it is applicable to an arbitrary number of blades and to any configuration of the machine. In addition, as wind turbulence can be viewed as a stochastic disturbance, the method is also applicable to real wind turbines operating in the field.

The characteristics of the new method are verified first with a simplified analytical model and then using a high-fidelity multi-body model of a multi-MW wind turbine. Results are compared with those obtained by the well-known operational modal analysis approach.

Stability analysis can help address very practical issues, such as assessing
the proximity of flutter boundaries, identifying low-damped modes,
understanding the vibratory content of a machine, evaluating the
effectiveness of control strategies for enhancing modal damping, detecting
incipient failures, and many others. For linear time-invariant (LTI) systems,
the stability analysis is a well-understood problem, and several methods are
available

One popular approach to the stability analysis of rotors in general and of
wind turbines in
particular

In principle, there are at least three issues connected with any Coleman-based
stability analysis approach.

First, the level of approximation implied by the averaging of the remaining periodicity is difficult to assess and quantify a priori. In fact, to the authors' knowledge, there is no theoretical proof yet that the periodicity that remains after the application of the Coleman transformation is small in general nor that this approach amounts to some consistent and bounded approximation of a rigorous Floquet analysis. Given the widespread use of the Coleman transformation, and its generally excellent behavior, such a proof remains a goal very much worth pursuing but, to date, unattained.

Second, the Coleman transformation unfortunately exists only for a number of blades greater than or equal to three. Although this is the most common wind turbine configuration nowadays, a revival of the two-bladed concept is possible.

Third, codes implementing the Coleman transformation require access to the linearized equations of motion of the system. As a consequence, any addition to a simulation code has an impact on the associated stability analysis tool, resulting in extra software maintenance work.

Other possible approaches to the stability analysis of rotors have been
formulated in the frequency domain. For example, the estimation of power
spectra along with modal frequencies and damping ratios of an operating wind
turbine has been addressed by

The operational modal analysis (OMA) has been extended to the periodic case

In the authors' opinion, there are two desirable goals in the stability
analysis of wind turbines that still need further investigation in order to
be fully attained.

First, one would like to account completely rigorously for the periodicity of such systems, without introducing approximations of unknown effects.

Second, one would like to formulate the analysis so that it is system-independent. System independence is here intended to mean that a method can be applied to wind turbine models of arbitrary complexity and topology (e.g., any number of blades and horizontal or vertical axis) and also to real wind turbines operating in the field.

To answer these needs,

Although this approach attains the two goals outlined above, one of its
limits is that it can not be used with measurements obtained on a real wind
turbine operating in the field, since the effects of wind turbulence are not
considered within the PARX model structure. To address this issue, the same
approach was extended to account for the presence of turbulence

The article is organized according to the following plan. The problem of the
identification of PARMAX models is addressed in
Sect.

In accordance with

The stochastic nature of the turbulent wind field violates the assumption of
a deterministic and fully measurable input

Inserting Eq. (

It should be remarked that the present approach does not consider the effects of nonlinearities nor of rotor speed variations induced by turbulence. The former potential problem can be checked a posteriori by looking at the matching between predicted and measured quantities. The latter can be partially solved by averaging the rotor speed over the analyzed time window. Typically, because of the large inertia of wind turbine rotors, angular speed variations are not expected to be highly significant, especially within the short time windows required by the proposed approach.

In order to perform a stability analysis according to Floquet theory (cf.

In the present context, a single-input single-output (SISO) PARMAX model must
be identified from a sequence of

The estimation problem, formalized according to the PEM, is the one of
finding the periodic coefficients

As previously argued, the presence of the MA part in the PARMAX model allows
for a more adequate characterization of the process noise term, at the cost
of a more complex estimation procedure. In fact, the optimal predictor of the
PARMAX process expressed by Eq. (

Moreover, it is easy to verify that
predictor Eq. (

In the literature there are basically two methods to enforce the stability of
the MA part. The first is a heuristic approach in which the coefficients

In this work, an alternative and original method is proposed. The stability
of the predictor is enforced by a nonlinear constraint within the estimation
process, and the resulting constrained optimization is performed by an
interior-point algorithm

The characteristic multipliers that constrain the estimation problem can be
computed from the autoregressive part of
Eq. (

Due to the nonlinear behavior of the predictor, the possible presence of
multiple local minima has to be taken into account. A suitable starting point
for the nonlinear problem can be selected by fitting the recorded data with
simpler models such as ARMAX or
PARX

The OMA is an output-only system identification technique, which has been
widely used to conduct modal analyses of different mechanical systems.
Recently, special attention has been devoted in the literature to the
application of OMA in the field of wind
energy

Consider a strictly proper periodic system and the exponentially modulated
periodic (EMP) expansions of its input and output, noted, respectively,
as

The power spectrum of the output, noted as

The first requirement was analyzed extensively for wind turbine problems
in

If such conditions are verified, the extended input spectrum

Moreover, neglecting again the contribution of overlapping modes, one can also
estimate the participation by evaluating the power spectra at the peak frequency,
since

The POMA technique can then be summarized as follows:

Compute the Fourier transforms of the frequency-shifted copies of the
recorded output

Compute the autospectrum

Extract the related
natural frequency and damping factors from each peak present in

Reconstruct the Fourier coefficients

It is possible to restrict the analysis to the right-half plane just by noting that

As the actual use of POMA and the correct interpretation of all peaks is not a
straightforward exercise in general, a simple Mathieu oscillator is analyzed here in
preparation for the application of this method to the wind turbine problems studied
later on. The dynamics of a Mathieu oscillator is governed by the following
equations:

Harmonic power spectrum of the output of the Mathieu oscillator.

Figure

Starting from this peak and moving to the right, the subsequent higher peaks
are found on the negative-shift curves, first in the

Frequencies and damping factors computed from such spectra using the peak-picking
method are reported in Table

Frequencies and damping factors for the Mathieu oscillator and analytical results.

The output-specific participation factors are displayed in
Table

Most relevant output-specific participation factors for the Mathieu oscillator and related analytical results.

Next, a simplified wind turbine model is used for comparing the results obtained with the PARX and POMA approaches. This is useful because it gives a way of comparing the basic performance of the two methods with respect to a known exact ground truth in the ideal case of zero disturbances. Later on in this work, the two methods will be compared for the case of a higher-fidelity wind turbine model operating in turbulent wind conditions. As no exact solution is known in that case, the preliminary investigation of this section serves the purpose of clarifying whether significant differences exists between the two approaches even at this more fundamental level. Indeed, it will be shown here that some of the underlying hypotheses of POMA are not always fulfilled, and this leads occasionally to some imprecisions in the estimates of the modal quantities of interest.

The analytical model is derived in detail in
Appendix

After having collected all degrees of freedom in vector

Since any mechanical system is linear in

The parameters of the wind turbine analytical model loosely represent a conceptual 6 MW wind turbine, and they are listed in Table

Parameters of the analytical wind turbine model.

The linearized periodic system was first studied using Floquet theory (see
Appendix

Both PARX and POMA estimates were compared with the full Floquet results in terms of
relative errors for frequencies and damping factors and absolute errors for
participation factors. Relative errors are defined as

The blade edgewise mode was excited by imposing the initial edgewise angles of all blades equal to a unique non-zero value, whilst all other states were set to zero at the initial time. This way the blade in-plane mode was excited while avoiding the onset of the whirling modes.

Considering first the POMA approach, the harmonic power spectrum for the second
blade edgewise angle,

Harmonic power spectrum
of the

Analytical results and estimation errors of blade in-plane modal parameters.

Analytical results and estimation errors of tower side–side modal parameters.

Analytical results and estimation errors of in-plane backward whirling modal parameters.

Analytical results and estimation errors of in-plane forward whirling modal parameters.

Clearly, the 0-shift PSD shows a prominent peak at

Next, the PARX analysis was considered. As long as only the blade in-plane
mode is significantly excited, as indicated from the 0-shift curve in
Fig.

Table

Looking at the results, it appears that both the PARX and POMA methods are able to capture the relevant dynamics related to the principal harmonics, as frequencies, damping, and participation factors are of good quality. In particular, damping and participation factors are slightly better estimated by PARX.

The estimation of the super-harmonic modal parameters deserves a special mention. The PARX method is able to provide a good matching for all modal parameters of all harmonics: frequencies and participation factors have negligible errors, whereas damping factors show an error lower than 1 %. On the other hand, the error of the POMA super-harmonic estimates is typically quite large especially for the damping factors, even though the principal harmonic is well captured.

This fact has mainly two possible explanations. First, the hypothesis of well-separated modes is here not fully satisfied, as the side band of the tower
principal harmonic affects all super-harmonic peaks. The lower the rotor
speed, the more pronounced this effect is, as the frequency separations among
super-harmonics coincide with multiples of the rotor frequency. Second, but
more importantly, according to the dynamics of a periodic system all
harmonics belonging to a specific mode descend from a sole characteristic
multiplier. Therefore, their frequencies and damping factors are strictly
connected to each other. This relation is totally ignored by
POMA

The tower side–side and blade in-plane whirling modes were excited by imposing different initial conditions for each blade edgewise angle and a suitable lateral displacement of the tower.

Figure

Harmonic power spectrum
of the

Figure

Comparison between measured (solid line) and predicted (dashed line) normalized blade root edgewise bending moment, in the time (left) and frequency (right) domains.

Considerations similar to ones previously made for the blade in-plane mode can also be stated here for these other three modes. Specifically, the frequency and damping factors of the principal harmonic of all modes are almost perfectly captured by both methods. The PARX method is the one that gives the most accurate results globally for both principal and super-harmonics: damping and participation factor estimates are characterized by small errors, while only the damping factors of the backward whirling mode have errors greater than 10 %. On the other hand, the POMA technique does not provide consistent results for the super-harmonic damping factors, which are characterized by large errors even when the damping factor of the principal harmonic is well captured. Moreover, the participation factors of the whirling modes exhibit non negligible errors for both principal and super-harmonics. This last issue is mainly due to the fact that, especially for the whirling case, the underlying hypothesis of well-separated modes is not completely fulfilled, as previously mentioned.

Periodic Campbell diagram of the first blade edgewise mode obtained from PARMAX identifications. The results of the single identifications along with the confidence level of the fitting curves are shown. Participation factors are computed in the rotating reference frame.

HPSD for the blade in-plane
mode, obtained for a

Periodic Campbell diagram of the first blade edgewise mode obtained from POMA identifications.

A detailed 6 MW wind turbine high-fidelity multi-body model operating in a closed loop, implemented with the aero-servo-elastic simulator

Periodic Campbell diagram for the tower side–side mode obtained from PARMAX (left) and POMA (right) identifications.

HPSD of the

Periodic Campbell diagram of the backward whirling in-plane mode obtained from PARMAX (left) and POMA (right) identifications.

Periodic Campbell diagram of the forward whirling in-plane mode obtained from PARMAX (left) and POMA (right) identifications.

According to the PARMAX-based stability analysis, the system should be
perturbed so as to induce a significant response of one or more modes of
interest. Among the many possible ways of exciting a specific wind turbine
mode, as, for example, the use of pitch and torque
actuators

The selection of the model complexity deserves special care. As the order of
the AR part,

After having performed the estimation for different wind conditions and
therefore at different rotor speeds, the results of the analyses in terms of
frequency, damping, and participation factors were fitted using low-order
polynomials, computed by means of the robust bi-square
algorithm

Two mainly edgewise doublets, applied at mid span and near the tip of the
blade, were used to excite this mode. The PARMAX reduced-order model
considered the following choice of parameters:

The result of an identification executed at the rated rotor speed is shown in
Fig.

To draw the Campbell diagram, eight different identifications were made in
order to cover the entire range of angular speeds of the machine. The results
are shown in Fig.

Similar analyses were conducted by

Much longer portions of the time histories analyzed with PARMAX were then
processed with the POMA method. In
Fig.

The Campbell diagram obtained from POMA is displayed in
Fig.

The tower side–side mode was excited with a chirp-shaped force applied at the
tower top. The frequency band of such signal was set in order to excite only
that single mode. The tower base side–side moment was then recorded and used
as output. As only the tower side–side peak is visible in the PSD of the
response, then

The agreement between the output predicted with the PARMAX reduced model and
the measure, not shown here for the sake of brevity, is very good. The left
plot of Fig.

The Campbell diagram obtained from POMA identifications is shown in the
right-hand plot of Fig.

The backward and forward whirling in-plane modes were excited with a tower
top side–side doublet, whose amplitude and duration were selected such that
the input force spectrum is almost flat in the frequency range of interest.
The three-blade root edgewise bending moments

The PARMAX reduced model was set with the following choice of parameters:

Figures

Once again, the damping of the super-harmonics obtained with the POMA
technique are not well estimated, as already noted in
Sect.

In this paper we have considered a model-independent periodic stability analysis capable of handling turbulent disturbances. The approach is based on the identification of a PARMAX reduced model from a transient response of the machine. The full Floquet theory is then applied to the reduced model, yielding all modal quantities of interest. As only time series of measurements are necessary, the method appears to be suitable for the application to real wind turbines operating in the field.

In order to assess the validity of the proposed method, the well-known POMA was implemented and used for comparison. Tests were performed first with the help of a wind turbine analytical model, whose exact solution can be obtained by the theory of Floquet, and then with a high-fidelity wind turbine multi-body model operating in turbulent wind conditions.

Based on the results obtained in this study, one may draw the following
considerations.

Both methods are able to characterize the relevant behavior of the wind turbine in turbulent wind conditions. However, the results provided by the proposed PARMAX analysis are in general more accurate than those given by the POMA technique, especially if one looks not only at the principal harmonics but also at the super-harmonics.

Often the underlying hypotheses of POMA are not exactly fulfilled, and this
leads to inaccuracies especially in terms of damping and participation
factors. These effects are more visible for the whirling modes, as they are
separated by about 2

A major advantage of PARMAX over POMA is that it requires shorter time histories. This is important in turbulent conditions, where the rotor speed is hardly constant (which, on the other hand, is a fundamental hypothesis of both methods).

The development of the present SISO PARMAX approach suggests a number of
extensions, which are currently under investigation.

The use of multiple outputs in a multiple-input multiple-output (MIMO) PARMAX framework could improve the quality of the results.

Due to the stochastic nature of turbulence, a multi-history PARMAX applied to different realizations of the same experiment could provide more robust modal results, along with the associated variances.

The peak-picking method is rather simple, and it is unable to exploit all the
informational content available in the HPSD, especially in the presence of
noisy peaks. Fitting algorithms have been preliminarily
explored

A generic SISO LTP system in continuous time can be written in state space form as

To study the stability of Eq. (

An important role in the stability analysis of periodic systems is played by
the state transition matrix over one period

In order to determine the frequency content of a periodic system, it is
necessary to introduce the so-called Floquet–Lyapunov transformation. The
Floquet–Lyapunov problem is the one of finding a bounded, periodic, and
invertible state space transformation

Note that there is an infinite number of Floquet factors, and therefore an
infinite number of Floquet–Lyapunov transformations. In fact, one can choose
any invertible initial condition

Given

Consider now, for each mode, one of the infinite solutions of
Eq. (

From Eq. (

For the LTP system, the exponents

Given two-column vectors

The apparent indeterminacy in the computation of the imaginary part of the
logarithm of the characteristic multipliers in
Eq. (

Often, although not always, the harmonic with the highest participation is
very similar in terms of frequency and damping to the one that would result from the invariant analysis of periodic systems based on the Coleman
transformation

In order to understand how each harmonic appears in a specific output of the
system, the output-specific participation factor can be defined. To
this end, consider an output of the autonomous
system Eq. (

The forced response of system Eq. (

In the frequency domain, the input–output relation can be expressed by means
of the HTF

The HTF can also be represented by means of the impulse response of the
system

Inserting Eq. (

From a practical standpoint, the use of the harmonic input–output relation
expressed by the HTF implies that one has to consider a truncated finite
dimensional approximation of

In this section the stability analysis of periodic discrete-time systems is
briefly reviewed. For a more comprehensive treatment, the reader is referred
to

The autonomous dynamic equation of a generic LTP system in discrete time and
its initial conditions are

For reversible discrete-time systems, the state transition matrix

In the discrete-time case, the apparent multiplicity of the characteristic
exponents manifests itself as a phase indetermination since

Following the same approach of the continuous-time case, the transition
matrix can be rewritten as

The simplified upwind horizontal-axis wind turbine model used in this work, depicted
in Fig.

Sketch of the wind turbine analytical model. Only one blade is shown in order to avoid cluttering the figure.

The reference frame used for the derivation of the equations of motions has its
origin located at the hub, the

The contribution of the two blade parts to the total energy can be developed
separately. Thus, let

All springs and gravity contribute to the potential energy of the system as

The damping function

The aerodynamic model is based on a linearized BEM approach with constant
aerodynamic properties along the blade, mostly taken
from

Definitions of the symbols in the aerodynamic loads.

The hub shear force in the fore–aft direction is

The virtual work of the aerodynamic forces and moments results in

Finally, the nonlinear Lagrangian equations of motion of the system are

All equations shown in this section and the system linearization were computed
analytically with Wolfram Mathematica^{®}