This paper studies the effect of the tower dynamics upon the wind turbine model by using mixed sets of rigid and/or nodal and/or modal coordinates within multibody system dynamics approach. The nodal model exhibits excellent numerical properties, especially in the case where the rotation of the rotor-blade is extremely high, and therefore, the geometric stiffness effect can not be ignored. However, the use of nodal models to describe the flexibility of large multibody systems produces huge size of coordinates and may consume massive computational time in simulation. On the other side, the dynamics of the tower as well as other components of wind turbine remain exhibit small deformations and can be modeled using Cartesian and/or reduced set of modal coordinates. The paper examines a method of using mixed sets of different coordinates in the same model, although there are differences in the scale and the physical interpretation. The equations of motion of the wind-turbine model is presented based on the floating frame of reference formulation. The mixed coordinates vector consists of three sets: Cartesian coordinates set to present the rigid body motion (nacelle and rotor bodies), elastic nodal coordinates for rotating blades, and reduced-order modal coordinates for low speed components and those that deflect by simple motion shapes (circular Tower). Experimental validation has been carried out successfully, and consequently, the proposed model can be utilized for design process, identification and health monitoring aspects.

Computational modeling of wind turbines is an important tool in design and
control of these dynamic systems. The presence of wind turbines in highly
dynamic environment reduces the assumptions that may facilitate the dynamic
model, and therefore, the use of multibody system dynamic approach is
inevitable. Multibody systems are characterized by two distinguishing
features: the system components undergo finite relative rotations, and these
components are connected by mechanical joints that impose restrictions on
their relative motion

It is certainly that the accuracy of the FFR depends on the number of
elements used to construct the model. Therefore, a large number of elements
must be used and thus huge size of elastic coordinates. Consequently, the use
of finite element models to describe flexibility may consume large
computational time. Despite the increasing computational capabilities of
digital processors, the need remains for using coordinate reduction methods,
especially in the case of large multibody systems. A truly straightforward
and computationally efficient way of describing deformations is the use of
linear deformation modes of the body

Small-size wind turbine as a multibody system.

Unfortunately, in the case of high-speed rotary machines, such as small-size
wind turbines, the use of the reduced-model and the associated modal
coordinates face with numerical difficulty

On the other hand, the tower of the turbine, remains exhibit small deformations and there is no rotations of its local frame; and therefore the modal coordinates can be easily used. The use of modal transformation reduce the number of degrees of freedom and consequently decrease the computational time. But the issue remains how to combine different coordinates in the same model, although there are differences in the scale and the physical interpretation, and their effect on the numerical integration of the mathematical model.

An additional aspect of this study is dealing with the nature of the tower
body – most of which are circular shapes – whether solid or hollow. Several
publications have presented the finite-element model of rod, which is mostly
subjected to high speed rotations and torque applications

This investigation proposes FFR model of wind turbines using three sets of coordinates: Cartesian coordinates plus the Euler parameters to present the rigid body motion (Nacelle and rotor bodies). Elastic nodal coordinates for rotating blades, and modal coordinates for non-rotating bodies (Tower). The paper examines the effectiveness of the proposed FFR formulation in modeling small-size wind turbines as well as the effect of the tower dynamics on the rotor speed.

Beam element of circular cross section within FFR formulation.

The general dynamic equations of the flexible multibody systems that consist
of rigid and flexible can be written as

In the FFR formulation, element deformation can be described with respect to
a frame of reference, this frame is used to describe the large displacements
and rotations of body motion. The global position of an arbitrary point

Using the displacement field of Eq. (

For the two-nodes 3D-beam element used in this investigation, the nodal
coordinates

The Cartesian coordinates are related to the cylindrical coordinates, see
Fig.

Interconnected bodies of wind turbine.

Using the modal analysis techniques; a reduced set of eigenvectors of the
free vibration discrete equations of motion as flexible modal coordinates.
The reduction is achieved by eliminating the high frequency modes, which
carry little energy. Modal reduction offers an efficient way to reduce the
number of elastic degrees of freedom, i.e,

The modal transformation matrix

The next section presents the multibody model of wind turbine with tower
dynamics. Once the modal transformation matrix of the tower is obtained, the
body model can be integrated to the rotor-blade system constructed in

The complete model of wind turbine consist of flexible tower

Frequency response function of the tower.

As shown in Fig.

Equation (21) can be solved for the mixed generalized
coordinates vector

Eigen-Frequencies of tower body.

First three modes of tower in

Test-rig: (1) wind turbine, (2) wind generator, (3) Pitot tube, (4) Data acquisition, (5) wind speed controller, (6) datalogging display.

Transverse displacement of the Nacelle at

The wind turbine prototype is constructed using copper rod of

The wind turbine prototype is fixed in the front of the wind generator as
shown in Fig.

Comparison of rotor velocity at

Comparison of rotor velocity at

Figure

Moreover, the comparisons between the output rotor velocity of the FFR model
and the experimental benchmark are shown in
Figs.

It is concluded that the comparison show a very good agreement which encourage to utilize the wind-turbine model based on the FFR formulation and by using the suggested mixed coordinates model.

In this paper, an efficient procedure is developed based on Floating Frame of Reference formulation (FFR) for modeling a complete system components of wind turbine. The dynamic of the tower, nacelle, rotor and blades are included with different sets of coordinates. Modal coordinates are assigned for the tower body, Cartesian coordinates set are assigned for the nacelle and the rotor, and elastic nodal coordinates sets are used for the rotating blades.

The paper describes an experimental test-rig of small size wind turbine in
front of wind velocity of 8 m s

No data sets were used in this article.

A beam element of 12 degrees of freedom, assuming that the element
displacements in the

The theory, model, and mathematical manipulations are contributed by AN. Both authors contributed together to the experimental validation.

The author declares that he has no conflict of interest.

The authors acknowledge the Deanship of scientific research – Jazan University for providing the financial support to carry out the research work reported in this paper against the limited grant project number 146-7-37. Also, we are grateful to Automatic Control Lab, Jazan University, for generous support of license agreement of MATLAB software package and technical support. Edited by: Lotfi Romdhane Reviewed by: two anonymous referees