Rotational sampling in wind turbines: Difference between revisions

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{{Further|Wind turbine design}}

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{{More citations needed|date=October 2013}}

The loads on both [[horizontal -axis wind turbinesturbine]]s (HAWTs) and [[vertical -axis wind turbinesturbine]]s (VAWTs) are cyclic; that is, the [[thrust]] and [[torque]] acting on the blades is dependentdepend on where the blade is. In a horizontal axis wind turbine, both the apparent [[wind speed]] seen by the blade and the [[angle of attack]] dependdepends on the blade's position of the blade. This phenomenon is described as '''rotational sampling'''. This article will provide an insight into the cyclic nature of the loads that arise because of rotational sampling for a horizontal axis wind turbine.

Rotational sampling can be divided into two parts: [[Deterministic system|deterministic]] and [[Stochastic process|stochastic]]. Deterministic processes present themselves as spikes on a [[Power spectral density|power spectrum]], whereaswhere as stochastic processes are broader i.e. spread over a wider [[frequency range]].

The loads on both horizontal axis wind turbines (HAWTs) and vertical axis wind turbines (VAWTs) are cyclic; that is, the thrust and torque acting on the blades is dependent on where the blade is. In a horizontal axis wind turbine, both the apparent wind speed seen by the blade and the angle of attack depend on the position of the blade. This phenomenon is described as rotational sampling. This article will provide an insight into the cyclic nature of the loads that arise because of rotational sampling for a horizontal axis wind turbine.

== Background ==

Rotational sampling can be divided into two parts: deterministic and stochastic. Deterministic processes present themselves as spikes on a power spectrum, whereas stochastic processes are broader i.e. spread over a wider frequency range.

[[Analysis]] of the loads on a [[wind turbine]] arecan usuallybe carried out through the use of [[power spectra]]. A power spectrum is defined as the power [[spectral density]] function of a signal plotted against frequency. The power spectral density function of a plot is defined as the [[Fourier transform]] of the [[covariance function]].<ref>Remote sensing: models and methods for image processing, R. a. Schowengerdt</ref><ref>Remote Sensing: Models and Methods for Image Processing, Robert A. Schowengerd</ref> Regarding the analysis of loads, the analysisit involves [[time series]], in which case the covariance function becomes the [[autocovariance]] function. In the signal processing sense, the [[autocovariance]] can be related to the [[autocorrelation]] function.

== Deterministic processes ==

== Background ==

=== Sources of deterministic processes ===

Upon completing a single revolution, a blade has produced an ever -changing [[torque]], and so power. Some of these changes are due to deterministic processes, i.e., processes that can be determined and do not require statistical methods. Examples of deterministic processes are listed below:

* [[Mass|Gravitational loading]]

Analysis of the loads on a wind turbine are usually carried out through use of power spectra. A power spectrum is defined as the power spectral density function of a signal plotted against frequency. The power spectral density function of a plot is defined as the Fourier transform of the covariance function.<ref>Remote sensing: models and methods for image processing, R. a. Schowengerdt</ref><ref>Remote Sensing: Models and Methods for Image Processing, Robert A. Schowengerd</ref> Regarding analysis of loads, the analysis involves time series, in which case the covariance function becomes the [[autocovariance]] function. In the signal processing sense, the autocovariance can be related to the [[autocorrelation]] function.

== Deterministic processes ==

=== Sources of deterministic processes ===

Upon completing a single revolution, a blade has produced an ever changing torque, and so power. Some of these changes are due to deterministic processes i.e. processes that can be determined and do not require statistical methods. Examples of deterministic processes are listed below:

* Gravitational loading

* Tower shadow <ref>Time-Domain Modeling of Tower Shadow and Wind Shear in Wind Turbines,

Swagata Das, Neeraj Karnik, and Surya Santoso</ref>

* [[Wind shear ]]<ref>Time-Domain Modeling of Tower Shadow and Wind Shear in Wind Turbines,

Swagata Das, Neeraj Karnik, and Surya Santoso</ref>

==== Gravitational loading ====

As a blade sweeps through each cycle, [[gravity]] is acting on the blade. Depending on the part of the cycle, gravity might be acting to accelerate the blade, or decelerate it. The additional torque that arises on a blade due to gravity is given by

As a blade sweeps through each cycle, gravity is acting on the blade. Depending on the part of the cycle, gravity might be acting to accelerate the blade, or decelerate it. The additional torque that arises on a blade due to gravity is given by

<math>

Line 29 ⟶ 25:

</math>

where ''r'' is the length of the blade, ''m'' is the mass of the blade, ''g'' is the [[Gravitational constant|gravitational field strength]], ''t'' is the time, and <math>\Omega_0</math> is the angular [[velocity]] of the blade.

==== Tower shadow ====

In [[fluid dynamics]], the flow of a [[fluid]] is dependent upon [[Boundary conditions in fluid dynamics|boundary conditions]]. Boundary conditions are influenced by the presence of solid bodies. In a wind turbine, the presence of the tower results in a reduction of the wind [[speed]] directly in front of it; that is, the blades experience a reduced wind speed when they pass in front of the tower.

====Wind shear====

In fluid dynamics, the flow of a fluid is dependent upon boundary conditions. Boundary conditions are influenced by the presence of solid bodies. In a wind turbine, the presence of the tower results in a reduction of the wind speed directly in front of it; that is, the blades experience a reduced wind speed when they pass in front of the tower.

In fluid dynamics, there exists the [[No-slip condition|no slip]] boundary condition. This states that the velocity of a fluid at the surface of a [[solid]] body, such as the [[Earth]], is zero. A consequence of that is that the wind speed varies with height above ground. This effect is known as [[wind shear]]. As a result, a blade at the highest part of its cycle will experience a greater [[wind speed]] than that of one at the lowest part of its cycle.

====Power Windspectral sheardensity =functions===

==== Drivetrain components ====

The drive train[[drivetrain]] of a wind turbine comprises the hub, the low speed shaft, the [[Transmission (mechanical device)|gearbox]], the high speed shaft, and the generator. The torque at the hub is strongly influenced by the [[Rotordynamics|rotor dynamics]]. The instantaneous hub torque is found by summing all the torques from all the blades of the wind turbine at any instant in time.

Consider an <math>n</math> bladed wind turbine. Each [[blade]] is separated angularly from a neighbouringneighboring blade by <math>360/n</math> degrees. That is, for a 3-bladed wind turbine, the blades are 120 degrees apart.

In fluid dynamics, there exists the [[No-slip condition|no slip]] boundary condition. This states that the velocity of a fluid at the surface of a solid body, such as the Earth, is zero. A consequence of that is that the wind speed varies with height above ground. This effect is known as wind shear. As a result, a blade at the highest part of its cycle will experience a greater wind speed than that of one at the lowest part of its cycle.

The [[torque]] acting on the blade is defined as the z-component of <math>\textbf{r}\times\mathbf{F}</math>, where '''r''' is the radius from the axis of rotation (in this case the hub), and '''F''' is the force acting on the blade. If the torque is defined as the z-component of this cross product, then the torque is simply ''rF''_perp where ''F''_perp is the force perpendicular to the radius vector, or tangential to the instantaneous velocity of the blade (See figure below)

=== Power spectral density functions ===

==== Drivetrain components ====

The drive train of a wind turbine comprises the hub, the low speed shaft, the gearbox, the high speed shaft, and the generator. The torque at the hub is strongly influenced by the rotor dynamics. The instantaneous hub torque is found by summing all the torques from all the blades of the wind turbine at any instant in time.

Consider an <math>n</math> bladed wind turbine. Each blade is separated angularly from a neighbouring blade by <math>360/n</math> degrees. That is, for a 3-bladed wind turbine, the blades are 120 degrees apart.

The torque acting on the blade is defined as the z-component of <math>\textbf{r}\times\mathbf{F}</math>, where '''r''' is the radius from the axis of rotation (in this case the hub), and '''F''' is the force acting on the blade. If the torque is defined as the z-component of this cross product, then the torque is simply ''rF''_perp where ''F''_perp is the force perpendicular to the radius vector, or tangential to the instantaneous velocity of the blade (See figure below)

From the figure above, it can be seen that the torque, ''T'', due to gravitational forces acting on a single blade is given by the following expression:

Line 69 ⟶ 61:

</math>

The [[covariance]] function of a sum of [[Sinusoidal model|sinusoids]] is itself a sum of [[sinusoidal functionsfunction]]s. Thus, the power spectral density function is a set of Dirac delta functions. The locations of these are at multiples of ''n''. Thus, on a power spectrum, deterministic processes such as gravitational loading manifest themselves as spikes. This can be seen from analysing generator torque.

==== Blades ====

For analysis of torque on a single blade, the spikes occur at <math>k'\Omega_0</math> where ''k''' is 1,2,3,..<ref>Aerodynamics of Wind Turbines, Martin O. L. Hansen</ref> This can be seen from taking the autocovariance of equation 1, and then taking the [[Fourier transform]] of this result.

== Structural loads ==

For analysis of torque on a single blade, the spikes occur at <math>k'\Omega_0</math> where ''k''' is 1,2,3,..<ref>Aerodynamics of Wind Turbines, Martin O. L. Hansen</ref> This can be seen from taking the autocovariance of equation 1, and then taking the Fourier transform of this result.

[[Spectral analysis (disambiguation)|Spectral analysis]] of component loading is useful in [[fatigue analysis]].

== References ==

==Stochastic processes ==

{{Reflist}}

{{Improve categories|date=June 2023}}

Stochastic processes are those belonging to turbulence. These require a purely probabilistic approach. A stochastic process such as turbulence is stationary. Analysis of stochastic processes reveals that they present themselves as broad peaks on a power spectrum.

== Structural loads ==

Spectral analysis of component loading is useful in fatigue analysis.

== References ==

{{Reflist}}

[[Category:Wind turbines]]