This article qualitatively shows the yaw stability of a free-yawing downwind turbine and the ability of the turbine to align passively with the wind direction using a model with 2 degrees of freedom. An existing model of a Suzlon S111 upwind 2.1 MW turbine is converted into a downwind configuration with a 5

With the increase in wind turbine rotor size and the increase in rotor blade flexibility, downwind concepts where the rotor is placed behind the tower are re-experiencing an increase in research efforts. The downwind concept potentially comes with the option of a passive yaw alignment. A passive yaw concept could save costs on the yaw system, decrease maintenance, and reduce the complexity of the yaw system. In situations where one side of a rotor under yawed inflow is loaded higher than the other, the resulting forces on the blades create a restorative yaw moment and could potentially align the rotor with the wind direction.

These passive yaw systems have been investigated already in the 1980s and the
early 1990s.

In further tests on the MOD-0 100 kW turbine,

Simple equations of motion for the aerodynamic yaw moment were used by

In 1986, the University of Utah and the Solar Energy Research Institute in
the US started to develop and validate a model for the prediction and
understanding of yaw behaviour. In a time domain modelling approach, they
coupled the flapwise blade motion to the yaw motion. In several studies

Other modelling approaches were chosen for example by

In this article the equilibrium yaw position of a free-yawing, pitch-regulated 2.1 MW downwind turbine is investigated. The influence of geometrical parameters such as cone angle, tilt angle, and shaft length on the equilibrium yaw position are considered. Further, a simple model with 2 degrees of freedom with free-yaw and tower side–side motion is developed to calculate the damping of the free-yaw mode. The influence of cone, shaft length, and the centre of gravity position of the nacelle on the damping of the free-yaw mode are regarded. It is shown that a full alignment with the wind direction is only achievable without tilt angle of the turbine and inclination angle of the wind field. It is shown that large cone angles increase the alignment with the wind direction and the damping of the free-yaw mode. Finally, it is shown that flapwise blade flexibility needs to be added to the 2 degree of freedom model, as the flapwise flexibility will significantly reduce the damping and the yaw equilibrium could become unstable.

The total moment on the yaw bearing is determined by different mechanisms
creating the yaw loading around the tower longitudinal axis. The following
estimation identifies the main contributors to the total yaw moment

Aerodynamic yaw moment for the tilt angle of a downwind rotor
sketched in

There are two yaw moment contributions due to the tilt angle. The first one
is a projection of the main shaft torque,

Aerodynamic mechanisms for yaw stiffness of a downwind
rotor

The moment due to induction variation over the rotor plane

The positive yaw displacement, as shown in Fig.

In the case of coning, there is a difference in the projected wind speed
between the left and the right side of the rotor when the rotor is yaw
misaligned, resulting in a difference in angle of attack. From the difference
in loading, a restoring yaw moment is created (Fig.

Compared to the mechanical stiffness of a spring, the aerodynamic stiffness term does not necessarily create a restorative yaw moment. Negative force coefficient slopes over the angle of attack can create a negative stiffness term. In this case, any disturbance from the equilibrium point would increase the force moving the system away from the equilibrium point. An example would be the operation of the turbine during stall.

Aerodynamic mechanism for yaw damping for a downwind rotor.

The damping mechanism for the free-yaw motion is shown in
Fig.

The stability of the equilibrium position of the yaw mode can be determined
from the eigenvalue analysis of the system matrices. If the resulting real
part of the eigenvalue

Flutter instability is given as

This study focuses on two aspects. Firstly, the equilibrium yaw angle of a
free-yawing turbine model which can align passively with the wind direction,
and secondly the dynamic stability of the free-yaw mode. This study uses a
simplified model of the Suzlon 2.1 MW turbine S111 (wind class IIIA). The
original turbine has a three-bladed upwind rotor with a diameter of 112 m and a
tower of 90 m height. The rotor is tilted 5

Schematics of turbine model and the according coordinate systems,
front view

Flow chart for the implemented BEM code to compute the equilibrium yaw angle.

The equilibrium yaw angle, where the aerodynamic forces are in balance, is
calculated with MATLAB (version 2018a). From a blade element momentum
(BEM) code with yaw and tilt model, the forces on the rotor are calculated
and the yaw angle associated with the zero-mean yaw moment on the yaw bearing is
interpolated between the loading for different yaw angles, assuming that the
effect of inertial terms is negligible. The BEM code is based on the
aerodynamic module of the aeroelastic code HAWC2

For the original turbine configuration, this method is validated with a HAWC2 simulation with a free-yawing turbine model without bearing friction. Thus, the rotor can align freely with the wind field. The wind field is steady, without shear, veer, inclination angle, or tower shadow model. The dynamic stall effects are neglected. The validated BEM code is then used for a parameter study, investigating the influence of tilt and cone angle, as well as the shaft length onto the equilibrium yaw angle of the turbine over wind speed. The operational conditions of the turbine are purely based on the free wind speed, neglecting any loss in power production due to skewed inflow.

To evaluate the dynamic stability of the free-yaw mode, a simple 2-DOF model
is set up in Maple software (MapleSoft, version 2016.2). The 2 degrees of freedom (2-DOF) model is based on an
existing 15-DOF model without cone angle, described by

The governing equations of motion are set up from the Lagrange equation
without structural damping

and

The potential energy

The resulting forces in the global coordinate frame can be read as

Inserting the time derivative of Eq. (

Here, the velocity triangle in the steady state is inserted with the
components of

From the upper equations (Eqs.

A steady simple BEM code (referred to as the “simple BEM code”) is used in MATLAB (R2018a), to determine the force coefficients along the blade span and to include the induction in the inflow velocity on the airfoil. The simple BEM code does not include skewed inflow models due to yaw or tilt. The induction is calculated along the rotor radius only since there is no dependency of the induction on the azimuth position. The structural stiffness of the tower is tuned to account for the neglected mass distribution of the tower. Eigenanalysis of the system matrix is performed in MATLAB over a range of wind speeds, and the real parts of the eigenvalue of the yaw mode are evaluated.

For the turbine configuration with the original cone, length, and mass
distribution, the 2-DOF model is imitated in the aeroelastic modal analysis
tool HAWCStab2, described by

The validated model is used for a parameter study to investigate the influence of geometrical turbine parameter on the real part of the yaw mode eigenvalue. The varied parameter are the cone angle, the shaft length, and the position of the centre of gravity of the nacelle along the shaft.

Finally, HAWCStab2 is used to investigate if the stability limit of the yaw mode would occur within the normal operational wind speed range of the turbine and which further degrees of freedom, additional to the tower side–side and yaw, would be needed to predict instability.

The following section shows the results for the equilibrium yaw angle and the stability of the yaw mode.

Figure

Comparison of the equilibrium yaw angle over wind speed from HAWC2
and the BEM code for the original turbine configuration with 5

The figure shows that the equilibrium angle is not zero. The equilibrium yaw
angle is constant at

The analysis shows that there will be a yaw moment even with a perfect
alignment of the rotor with the wind direction, which drives the rotor to the
non-zero equilibrium angle. This yaw moment is due to the tilt angle.
Including a tilt angle has two effects. Aerodynamically, the projection of
the global wind speed leads to a yaw moment as illustrated in
Fig.

Figure

Equilibrium yaw angle (

A zero tilt angle will give a zero equilibrium yaw angle, which means a full alignment of the rotor with the wind direction. Negative tilt angles show a positive equilibrium yaw angle and positive tilt angles show a negative equilibrium yaw angle. The dependency of the equilibrium yaw angle on the tilt angle is stronger for higher wind speeds. The relative power difference shows the highest losses for extreme tilt angles and high wind speeds. There is zero relative power difference at zero tilt angle.

There is no yaw moment for yaw alignment of the rotor plane with the wind
direction if there is a zero tilt angle. As a yaw moment due to a tilt angle
is dependent on the sine of the tilt angle, the equilibrium yaw angle is
anti-symmetric around the line of full alignment (0

Figure

Equilibrium yaw angle (

Figure

Varying the cone angle for the tilted turbine configuration has an effect on
the torque. A larger cone angle reduces the torque, which leads to a reduced
projected yaw moment due to the tilt

Figure

Equilibrium yaw angle (

Figure

As discussed previously the shaft length acts as the moment arm for the
summed in-plane shear forces on the hub. For the in-plane shear forces to be
significantly large, a large yaw angle and high wind speeds are required to
create an imbalance on the angle of attack between the upper and the lower
rotor halves (see Fig.

The following section discusses the stability of the free-yaw mode of the
turbine for a tilt angle of 0 and 3.5

Comparison of the frequency

It can be seen at the top of the figure that there is a yaw frequency of zero
up to a wind speed of 42 m s

Including the aerodynamic forces to the mechanical system has two effects.
Firstly, there is an aerodynamic stiffness, due to the mechanisms of wind
speed projection as shown in Fig.

The main influence can be observed from the slope of the lift coefficients if the outputs from the simple BEM code are manually manipulated for the eigenanalysis. As the yaw moment for moderate pitch angles is dominated by the flapwise forces, the drag and the slope of the drag coefficient are of minor influence. As the projection of the forces changes with the pitch angle, a clear dependency on the wind speed can be observed and also the change in the slope of the real part of the eigenvalue can be observed at the rated wind speed. Further, the operational point changes so the force coefficients and their slopes are expected to change the eigenvalues over wind speed.

Figure

Real part of the first

The figure shows that the real part of the first eigenvalue changes sign
and becomes positive for cone angles of

The cone angle mainly affects the aerodynamic stiffness, as shown in
Fig.

Figure

Real part of the first

It can be seen that the real parts of the eigenvalues hardly change with
variation in the shaft length. A lower shaft length slightly increases the
real part of the eigenvalue. High shaft length slightly decreases the minimum
of the second eigenvalue at around 14 m s

Large shaft lengths can increase the projected wind speed for the damping
term, as the centre of rotation is far away from the rotor plane. However, as
realistic values of the shaft length are always much lower than the blade
length, the influence on the damping is very low. Also, the influence on the
stiffness can hardly be observed, as the effect of in-plane forces is very
small for small yaw angles (linearization point of 0

A figure for the real parts of the eigenvalue changing with the position of the centre of gravity is not shown. The distance of the centre of gravity only effects the rotational inertia for the yaw motion. As the stiffness and damping are not effected, the real part of the eigenvalues are hardly changing from eigenanalysis of the system matrix.

Figure

Comparison of the frequency

The figure shows that the frequency is zero for all models within the
investigated wind speed range. The figure shows at the bottom the characteristic
behaviour of the real part of the eigenvalue of the 2-DOF model imitation with
HAWCStab2, already compared in Fig.

Flapwise flexibility introduces the flapwise motion of the blades. The
asymmetric flapwise motion of the forward and backward whirling mode could
be stabilizing or destabilizing the yaw equilibrium, depending on the phase
difference between the yaw motion and the asymmetric flapwise motion. The
phase difference between the flapwise modes and the free-yaw mode is observed
to be around 180

The free-yawing behaviour of the Suzlon S111 2.1 MW turbine in a downwind configuration has been investigated. A BEM code was used to show the equilibrium yaw angle and the parameters creating a yaw loading on the rotor. A small analytical model with only 2 degrees of freedom was developed. It was used for a brief overview and understanding of the parameters influencing the stability of the passive yaw equilibrium position, exemplified on the Suzlon turbine.

It was observed that the original tilt angle of 5

The yaw misalignment introduces a restoring yaw moment from the flapwise
blade moments due to induction variation over the rotor plane. This restoring
yaw moment can be increased with an increasing cone angle, as the combination
of cone and yaw angles creates a favourable wind speed projection and therefore
increases the yaw stiffness as predicted by

With a significantly large yaw angle, the wind speed projection leads to an in-plane force imbalance that increases the restoring yaw moment. In
conclusion, an in-plane force due to load imbalance will also be created from
the tower shadow and wind shear. In contrast to the previous effect, this
force imbalance will also exist when the rotor is fully aligned with the wind
direction and it will vary with varying wind conditions. Such a negative
effect from vertical wind shear and tower shadow has already been observed by

An eigenanalysis of a 2-DOF model of a turbine without tilt angle was conducted. It was observed that the cone angle can significantly increase the real part of the eigenvalue of the yaw mode and therefore stabilize the yaw equilibrium as it increases to a positive stiffness term. It was further observed that cone angles that are too small can give a negative stiffness term and therefore leads to a positive real part of the eigenvalue and an instability in the yaw mode.

Modelling the free-yawing motion with 2-DOF has been seen to be insufficient as flapwise blade motion changes the stiffness and the damping of the free-yaw mode. The comparison with HAWCStab2 showed that flapwise blade flexibility significantly increases the real part of the eigenvalue and destabilized the yaw equilibrium. The phase difference between yaw and asymmetric flapwise blade mode significantly decreases the damping of the free-yaw mode. The stiffness is mainly influenced by flapwise blade deformation as the steady-state blade deflection decreases the effective cone angle.

Overall, this analysis showed clearly that the S111 turbine in downwind configuration will not align with the wind direction and the power loss is significant. Further, changing wind conditions such as inclination angle or wind shear will lead to a yaw misalignment that will change with the environmental conditions. As the free-yaw mode further becomes unstable for high wind speeds, it will not be possible to run the S111 turbine in a free-yawing downwind configuration. Stabilizing the free-yaw mode and increasing the alignment with high cone angles might be possible. Yaw bearings could potentially be designed for a lower yaw load. Yaw drives will always be needed as a cable unwinding mechanism. Since there will be a power loss associated with either a yaw misalignment or a larger cone angle, it is highly doubtful that the passively free-yawing downwind turbine can be a more cost efficient solution than a yaw-controlled downwind turbine in terms of levelized cost of energy.

The data is not publicly accessible, since the research is based on a commercial turbine and the data is not available for disclosure by Suzlon.

The mass matrix results in

The mass matrix, on the other hand, is fully populated. In the first element is the total mass of the turbine that will be moved with the tower side–side motion. In the second diagonal element there is the mass moment of inertia for the rotation around the yaw centre. This includes the mass moment of inertia of blades, hub and nacelle-shaft assembly, as well as their respective Steiner radii to the centre of rotation. The coupling terms in the off-diagonal are the mass elements times the respective radii to the rotational axis.

The resulting stiffness matrix

The coupling coefficient

where the subscript

GW and MHH set up the 2-DOF model and validated the model. GW and TJL have set up and validated the BEM code to calculate equilibrium yaw angles. GW carried out the calculations. All authors have interpreted the obtained data. GW prepared the paper with revisions of all co-authors.

This project is an industrial PhD project funded by the Innovation Fund Denmark and Suzlons Blade Science Center. Gesine Wanke is employed at Suzlons Blade Science Center.

This research has been supported by the Innovation Fund Denmark (grant no. 5189-00180B).

This paper was edited by Carlo L. Bottasso and reviewed by Vasilis A. Riziotis and one anonymous referee.