## Livrables

WOBBE is a tool that can be used to investigate the aero-elastic stability of wind turbines. It can also be used for helicopters, but the development is currently focussing on wind turbines. The tool can run fully non-linear time simulations of rigid body models where these rigid bodies are interconnected by hinges and springs. The equations of motion do not need to be derived, enabling the user a.o. to easily change the models complexity.
BEM is used to calculate the aerodynamic forces. The induced velocity is calculated using dynamic inflow and it is assumed constant over the rotor swept area. The option of dynamic stall is included in the programme. Prandtl's tip correction can also be used.
To analyse the results an identification tool such as AerID can be used.

Operational Modal Analysis is used to extract the real modal frequencies and damping of the same tower modes and the two first edgewise-whirling modes.
The OMA method was chosen as a supplement to the exciter method to reach the objective of measuring the modal damping of all lower order turbine modes. This method only requires that long time-series of vibrational turbine responses are measured under almost steady operational conditions. The time-series selected for the analysis were signals from strain gauges mounted on the tower, main shaft, and rotor blades and used for load assessment measurements. The natural frequencies and modal damping extracted from strain-gauge signals in the fixed frame of reference showed to be most robust, and this frame of reference has furthermore the advantage that the extracted natural frequencies are directly comparable to the theoretical predictions with the new stability tools.
The OMA results gave new estimates of the first two tower modes, and estimates of the first two whirling modes that involve edgewise blade vibrations. These four modes are sufficiently low damped by the aerodynamic forces, whereas modes involving flapwise blade vibrations are highly damped by the aerodynamic forces. The latter do therefore not appear in the strain-gauge signals, which disables their extraction.
The advantage of OMA is that it does not require an extensive investigation of the turbine prior to the measurements as described in the Task-2 Report. Most importantly, the results of OMA are real estimates of both natural frequencies and modal damping of the aeroelastic turbine modes. The disadvantage is that it requires long-term measurements, which means that the controller has to be active in closed-loop operation, and this can affect the estimated modal properties.

The computer program TURBU Offshore has been developed for frequency domain analysis of offshore wind turbines. Next to load calculation via power spectral density functions, TURBU Offshore allows for:
(i) the computation and analysis of modal properties, and
(ii) time-domain simulation with feedback loops for control included under stochastic excitation from a wind field and sea waves.
The structural dynamics are catered for by multi-body models of the blade and tower structure, which consist of mass/spring/damper-elements with up to six degrees of freedom (DOF). Lumped DOFs are included for:
(i) yaw, roll and tilt of the nacelle,
(ii) bending and torsion of the rotor shaft,
(iii) control of the generator counter torque, and
(iv) flap and lead oriented hinges in the rotor blades and pitch actuation.
The aerodynamic forces are uniformly distributed over a blade element while tip and root losses are taken into account via Prandtl's correction factor; unsteady aerodynamic forces can be included in accordance with the ECN dynamic stall model The dynamic inflow is included as the linearised version of the ECN cylindrical wake model.
The equations of motion are derived in accordance with Newton's first and second law. The thus formulated equations of motion (EOM), together with the strongly automated generation of the implementation of the EOMs, allows for convenient incorporation of uncommon distributed actuation mechanisms on the rotor blades. The linear equations of motion without periodic coefficients are derived by transforming the rotating blade and shaft DOFs into coordinates in a fixed frame of reference. These fixed-frame coordinates also apply for input and output variables related to the rotor blades, such as the wind speed as experienced by the blades, the pitch angles and the blade bending loads. This coordinate transformation allows for traditional Eigen value analysis as well as for time-domain simulation with a linear time-invariant (LTI) system. In this LTI time-domain simulation the rotor azimut dependent operations are moved to the boundaries of the system: the blade inputs are demodulated before they enter the LTI-system and the rotor blade outputs from the LTI-system are remodulated in order to get the meaning of variables on the rotating blades.
After further development, the TURBU code will be part of a control development software environment in which controller synthesis, stability analysis and time-domain evaluation are integrated. It is pursued to offer this software environment when cooperating with medium to large wind turbine manufacturers for identifying and addressing mid- and long-term research items. This cooperation intends to make wind turbine manufactures conscious of the potential of control and of interweaved control/aero-elasticity problems. The software will be offered in a 'laboratory version' without so called help desk function. Some kind of license agreement will apply. A user should be moderately to high experienced in the field of wind turbine control and aero-elastics.

An aeroelastic software (GAST), for the analysis of rotor and rotor-pylon configurations such as wind turbines, helicopters, propeller aircrafts etc. is developed at NTUA.
This tool can be used for:
- the eigen value analysis in vacuum and steady state conditions,
- the stability analysis for time-constant and time periodic conditions,
- time domain aeroelastic simulations.
It consists of a dynamic/structural model based on a multi-body approach, where all flexible bodies are considered as linear beams and a rotor aerodynamics module based on the blade element momentum method. For the discretization of the dynamic equations of motion the finite element method is applied. In the latest version of GAST, the concept of the "aeroelastic element" is introduced.
The dynamic equations of the beam are supplemented by the equations of the ONERA model for the aerodynamic d.o.f (lift, drag and moment). In this way the equations of every beam element, apart from the elastic d.o.f. (elastic deflections and rotations), also include the aerodynamic d.o.f. The complete set of the coupled aeroelastic equations of the beam is solved simultaneously using FEM.

A comprehensive non-linear aeroelastic tool is developed at NTUA. With this model, the aeroelastic problem of the complete wind turbine configuration is considered, within the context of advanced aerodynamic and structural modeling. Aerodynamics is accounted for by a 3D free-wake vortex method instead of the commonly used blade element momentum model, while a 2nd order non-linear beam model is used in formulating the structural dynamic equations instead of the linear classical beam theory. The two are non-linearly coupled and time-domain simulations are performed which through signal processing can provide information on the stability characteristics of the complete construction. Assessment of the significance non-linearities have on stability is carried out through comparisons with results obtained from different linear models while comparisons with full scale measurements on a multi MW machine validate the theoretical developments.

AerId is a linear system identification tool used to build a model and perform the stability analysis of a dynamic system, in the present case a wind turbine, treated as a black box.
It is based on a linear least square algorithm to determine a state space representation for the system to be identified. The system is considered as autonomous, i.e. no inputs are applied, and every state must be known and measurable. Coleman transformations are used to transform the rotor degrees of freedom into the non-rotating frame.
AerId can perform both time-varying, periodic, model identification and time-constant model identification. Due to the small differences between the constant and time-varying model identification results for a wind turbine, the identification and the stability analysis are performed using the time-constant model for the wind turbine.

Vibrations of large flexible wind turbines are highly dependent on its modal properties: Natural frequencies, damping factors and mode shapes. HAWCStab is a tool for computing and analysing these modal properties of a wind turbine at any operation condition with, or without the unsteady aerodynamic forces. The inclusion of the aerodynamic forces enables stability analysis for predicting negative damped aeroelastic modes of the turbine.
The structural model of HAWCStab uses Timoshenko beam elements with six degrees of freedom (DOFs) per node. The aerodynamic forces are assumed to follow a parabolic distribution per element and they are obtained using the BEM method and a modified Beddoes--Leishman dynamic stall model combined with Prandlt's tip correction. The linear equations of motion without periodic coefficients are derived analytically by introducing multi-blade coordinates for the blade DOFs into linearized Lagrange's equations. Because the aeroelastic equations of motion are linear and autonomous, traditional eigenvalue analysis can be performed to obtain natural frequencies, damping, and mode shapes for structural and aerodynamic DOFs.

ARLIS is the acronym for Aero-elastic analysis of Rotating Linear Systems. The program-system many years ago was specifically developed for the stability analysis of horizontal axis wind turbines. In the framework of the Project it was adopted to state-of-the-art computers and further refined. Applying Floquet's Theory it is capable to handle wind turbines in operation with one, two and more blades. It is, however, restricted to constant r.p.m. of the wind turbine.
For the dynamic analysis of the coupled rotor-tower-system the separated systems are described by finite element models, using any appropriate Finite Element Program System. The eigen modes are calculated for both systems separately, which are then reduced via condensation to small systems with only a few generalised degrees of freedom (eigen modes) which are assumed to be sufficient for the description of the dynamics and the stability of the system. The displacements are assumed to be small enough to work with linearised equations of motions. The validity of this assumption can be checked by the dynamic response of the system. In the case of large stationary displacements the stability analysis can be conducted around the equilibrium state of the deformed structure.
Tower and rotor are connected via one nodal point (coupling point) with 6 degrees of freedom. The drive train is modelled by a simple FE-model allowing a ``stiffness and/or damping coupling to ground' of the generator, thus allowing a modelling of synchronous and asynchronous generators. ARLIS takes over the matrices of the condensed systems and builds up the matrices of the linearised-coupled rotor-tower-system, including quasisteady aerodynamics. Since, in general, the matrices of the coupled system have periodic coefficients the stability analysis is carried out using Floquet's theory.
Furthermore, steady state and transient response of the coupled system can be calculated taking into account loads due to deadweight, wind shear, gusts, unbalance, etc. The generalised displacements are given back to the Finite Element System, where nodal point displacements, stresses, forces etc. are computed in the usual manner.

An aeroelastic stability tool (Stab-Turb) for predicting the natural frequencies, the damping and the mode shapes of a complete wind turbine in operation has been developed. It employs two node beam elements with six DOFs per node. A Lagrange formulation is used for deriving the equation of motion, where rigid body and elastic motion are separated; the former being explicitly handled.
The BEM method combined with a constant distribution of forces along the elements and Prandlt's tip correction is used for the aerodynamics. In an unsteady implementation, the Extended Onera lift and Drag model is used. To contemporary treat the aerodynamics and the dynamics, aerodynamic DOFs are introduced and they are combined with the structural ones in the so-called 'Aeroelastic Beam Element' (ABE), that comprises up to nine DOFs per node. In order to perform the stability analysis of the complete wind turbine in operation, a multi-blade transformation is introduced.
The method is capable for linear and non-linear integration of the equations of motion in the time domain, from which the stability limits are calculated through a modal analysis of the time simulations. With the stability tool is possible to perform a large number of parameter variations and even numerical optimisation in this process because these tools are based on eigenvalue analysis.
The tool has been extended to support closed loop operation of wind turbines under various control concepts.