1. Modeling and Simulation of a Wind
Turbine Generator System
Marko Tanaskovic
Department of Information Technology and Electrical Engineering
Supervisor: Dr. S´ebastien Mari´ethoz
Professor: Prof. Dr. Manfred Morari
March 20, 2011
2.
3. Abstract
For investigating the influence that a wind turbine might have on the power generator or on the grid, a
turbine simulator is needed. In order to build such a simulator wind turbine model needs to be designed
first. Such a model should be simple enough so that it can be used in real time, but also it should be accurate
enough so that it captures all the important aspects of the wind turbine behavior. In this work we propose
a wind turbine model that can be run in real time, but that is more accurate than simple turbine models
already present in the electrical engineering literature. We build this model based on results obtained
turbine simulation software. Our experiments show that the model we propose is very accurate when the
turbine is operating near its optimal line of operation.
I
11. Chapter 1
Introduction
Using wind energy has been recognized as an environmentally friendly, socially beneficial and economi-
cally competitive way of electricity generation. As a result, worldwide installed wind generation capacity
is growing rapidly each year. With such a trend many researchers are focusing their attention to this field.
Consequeltly, the interaction between wind turbines and the grid as well as control of electric generators
for wind turbines are the main focus of electrical engineering community. In these research efforts a wind
turbine simulator is a very important tool. The main reason is that performing experiments on an actual
wind turbine is very expensive and could damage the turbine. Furthermore, since turbine operation greatly
depends on weather conditions, it is often very hard to design a relevant experiment for a real wind turbine
system. Therefore, simulators that emulate the behavior of a real wind turbine need to be used in order to
create a controlled test environment for research. For building such a simulator, a relatively simple and ac-
curate wind turbine model is needed. In figure 1.1 logic behind wind turbine emulator is illustrated. In this
work we propose such a wind turbine model that can be used to build a wind turbine simulator for research
purposes.
The main reason for the interest in the influence that a wind turbine may have on electric generator lays
in the fact that the torque extracted from the wind can be quite unsteady. One of the reasons for torque os-
cillations and pulsations lays in the turbulent nature of the wind and the fact that wind field varies along the
turbine rotor disc. Since wind behavior is very hard to predict during short time intervals, these pulsations
are stochastic in their nature. On the other hand, some of turbine torque oscillations are more predictable.
These oscillations are the result of two effects - termed wind shear and tower shadow. The term wind shear
describes the fact that wind velocity tends to increase with the increase in height, and therefore the upper
part of the turbine rotor receives more wind than the lower one. The term tower shadow stands for the
redirection of wind field around the turbine tower. These effects cause the turbine torque to oscillate even
if the wind field over the turbine rotor is homogenous and wind speed is constant in time. For the three
bladed wind turbines, the torque extracted from the wind is minimal when one of the blades is pointing to
the ground and is aligned with the tower. On the other hand, the torque is maximal when one of the blades
is pointing vertically upwards. Hence the turbine torque will oscillate with the dominant frequency that is
three times higher than the rotational frequency of the turbine. This frequency is often termed 3p frequency
in the literature. All these effects may have a strong impact on the grid or generator part of the turbine sys-
tem. Therefore, their understanding and proper modeling is crucial for further improving the performance
of wind turbine systems.
1
12. CHAPTER 1. INTRODUCTION
Good understanding of wind turbine operation and its precise modeling is essential for wind turbine
construction considerations. That is why a lot of efforts have been made by the mechanical engineering
community to build very detailed and accurate wind turbine models. These models are based on mathe-
matically quite complicated laws of aerodynamics, which makes them difficult to implement in real time.
Moreover, these models require a detailed specification of wind turbine design, which is a disadvantage as
manufacturers are usually reluctant to provide such information. On the other hand, electrical engineers
need a wind turbine model for designing control laws or for emulating an actual wind turbine. For these
applications a model that is less complicated and that can be used in real time is needed. Therefore, models
found in the electrical engineering literature are quite simple. Unfortunately, these models do not often
depict all the important aspects of wind turbine operation. Moreover, torque oscillations, which are very
important for studying the influence that a wind turbine might have on power generator or the grid, are often
neglected. Therefore, this work looks at a way to make a wind turbine model that would be simple enough
to be implemented in real time, but at the same time would be more precise than other existing simplified
models.
Figure 1.1: In emulating wind turbine we are interested in replacing the wind turbine with a DSP and
electric motor where the input to the DSP is apropriate wind description
In our modeling approach we use a wind turbine simulator, which is based on a very detailed aerody-
namic and wind turbine structural dynamic model. In order to build a model, we take a sample wind turbine
already available in the simulator database. For this wind turbine we perform a series of simulations in
which turbine rotational velocity and wind speed values are kept close to the line at which the turbine ex-
tracts maximal power from the wind. The data obtained from the simulations is analyzed and mathematical
model is fitted to it. This model is based on the representation of the turbine mean torque and its oscillations
as the functions of rotational velocity and wind speed where these functions are obtained by fitting the data
obtained from the simulator. The model is validated by performing simulations in closed loop where the tur-
bine rotational velocity is controlled so that the power extracted from the wind is maximized. Experiments
show that the results obtained by the model we propose match well with the results that the turbine simulator
gives when the turbine is operated in a region close to the optimum . The proposed model is suitable for real
time implementation and it models turbine oscillations due to tower shadow and wind shear effects more
precisely than other existing simplified turbine models.
This report first gives a short overview of the existing wind turbine models. Chapter 3 highlights the
main motivation behind the use of a turbine simulator in our modeling. The detailed description of the
model is given in Chapter 4 and in Chapter 5 the model validation procedure is explained. In Chapter 6 the
conclusions, as well as suggestions for further improvements are presented. The description of the turbine
simulator that has been used for modeling is given in Appendix A. In addition, this Appendix explains Mat-
lab functions that have been used to automate the turbine simulator operation. Appendix B lists numerical
2
15. Chapter 2
Magnetic materials overview and
comparison
2.1 Classification of magnetic materials
The history of magnetism begins in the 6th century B.C when the Greek philosopher Thales discovered the
magnetic properties of a mineral called magnetite. However the first scientific study on magnetism was
published in 1600 by William Gilbert [42]. Later the science of electromagnetism was shaped by scientists
like Faraday, Maxwell, Oersted and Hertz.
According to their magnetic properties, all materials can be classified in three groups[40], [41]:
• Diamagnetic materials
• Paramagnetic materials
• Ferromagnetic materials
Diamagnetic materials (Diamagnets) are materials that have magnetic permeability less than µ0 (relative
permeability is less than 1). These materials cause the lines of magnetic flux to curve away from the ma-
terial and hence it appears as they create a magnetic field opposed to an external magnetic field. Such a
behavior is common to most of materials present in nature (and often these materials are referred to as non-
magnetic). However, the effect of repulsion when exposed to external magnetic field is so weak that it is
usually not noticed at all. The only exceptions are superconductors which completely exclude the lines of
magnetic flux and can be considered as perfect diamagnets.
Paramagnetic materials (Paramagnets) have relative permeability close to unity, but slightly higher. These
materials are slightly magnetized in the presence of external magnetic field. However, in the absence of the
external magnetic field these materials retain no magnetization.
Ferromagnetic materials (Ferromagnets) have relative permeability much greater than one (typically from
10 to 100000)[40]. These materials get magnetized in the presence of external magnetic field and unlike
paramagnets do not emediatelly get demagnetized when that external field is removed. Ferromagnetic mate-
rials are the only ones that can be used to produce considerable magnetic forces. These forces can be noticed
5
16. CHAPTER 2. MAGNETIC MATERIALS OVERVIEW AND COMPARISON
Diamagnetic Paramagnetic Ferromagnetic
Superconductor Cesium Cobalt
Graphite Aluminum Iron
Copper Lithium Nickel
Lead Magnesium
Silver Sodium
Water
Table 2.1: Some typical representatives of diamagnetic, paramagnetic and ferromagnetic materials
and felt and they are the ones that are generally associated with the phenomenon of magnetism encountered
in everyday life. Moreover, only these materials are relevant for the design of magnetic components for
power electronics. Therefore in the next section detailed characteristics and further classification of these
materials will be given.
Some typical diamagnetic, paramagnetic and ferromagnetic materials found in the nature are listed in Table
2.1
2.2 Ferromagnetic materials properties and classification
Ferromagnetic materials do not exhibit same properties at all temperatures. If the temperature of the ferro-
magnet increases too much, its relative permeability will drop to 1 and it will start to behave as a paramag-
netic material. This specific temperature, at which the material loses its ferromagnetic properties is called
Curie temperature (Tc) and is an important parameter that describes a material.
In order to understand ferromagnets better, one has to look at the process of magnetization. This process
occurs due to alignment of electron magnetic moments under the influence of external magnetic field. Each
electron of the atoms in ferromagnetic materials has a magnetic (orbital) moment which is a consequence
of its rotation around the atom nucleus. The orientation of these magnetic moments is random in case when
no external magnetic field is applied. However, when the external magnetic field is applied, these magnetic
moments start to orient themselves in the direction parallel to the lines of external magnetic field. As a result
the induction of the material is increased. This increase is nonlinearly dependent on the increase in magnetic
field strength and the exact shape of that nonlinearity is material dependent. When all the magnetic moments
are aligned with the lines of the magnetic field, further increase in the magnetic field strength caused only
minor increase of magnetic induction. At this point it is said that the material is saturated. Magnetic in-
duction at which the material saturates is a very important characteristic of a certain material. It is called
saturation induction and usually denoted as Bsat. In order to demagnetize the material, strength of magnetic
field has to be reduced. However magnetic induction does not decrease in the same way as it increased and
when the magnetic field reaches zero value, there will still be some magnetic induction in the material. This
induction is called remanence and is another important material characteristic (it is often denoted as Br).
In order to drive magnetic induction to zero level, negative magnetic field has to be applied. Negative field
necessary to drive B back to zero is called coercive force and is denoted by Hc. The curve which shows this
nonlinear dependence of magnetic field and magnetic induction is called B-H curve and it is one of the most
important material properties. Although there is a great number of different B-H loop shapes, materials are
often classified into three groups according to their B-H loop shape [40]:
6
17. CHAPTER 2. MAGNETIC MATERIALS OVERVIEW AND COMPARISON
• Materials with square loop
• Materials with round loop
• Materials with elongated loop
Figure 2.2 illustrates these three different types of B-H loops.
Figure 2.1: Three different types of B-H loops
Another way to classify ferromagnetic materials is to divide them according to their coercive force into
[40],[41]:
• Hard magnetic materials
• Soft magnetic materials
According to [40] hard magnetic materials are those that have Hc > 10000 A
m . These materials are often
called permanent magnets. Usually they also have very high value for remanent inductance Br. Therefore,
these materials are very hard to demagnetize (hence the name permanent magnets). Typical applications of
such materials are for electrical motors and generators, sensing devices and mechanical holding.
Soft magnetic materials typically have Hc < 1000 A
m . Therefore they are characterized by much nar-
rower B-H loop compared to hard magnetic materials. Moreover, it is much easier to change magnetic
alignment in the structure of these materials. They are widely used in modern electrical engineering and
electronics. In fact most of magnetic components in power electronics use cores made of these materials.
Materials that have Hc in the range 1000 to 10000 A
m are considered to be somewhat between soft and
hard, however there is no general term that would describe such materials [41]. These materials are mainly
used as recording media.
Since soft magnetic materials are the most relevant for power electronics, we will mainly focus on them.
They can be further divided into iron based soft magnetic materials (ferromagnetic materials in narrow
sense) and ferrimagnetic materials. Here it is important to stress the difference between the terms since of-
ten in literature soft magnetic materials based on iron are referred to as ferromagnetic although they together
with ferrimagnetic materials belong to the larger group of ferromagnetic materials. However it is often said
that soft magnetic materials based on iron are ferromagnetic materials in narrow sense.
7
18. CHAPTER 2. MAGNETIC MATERIALS OVERVIEW AND COMPARISON
Ferrimagnetic materials (ferrites) are ceramic materials made from oxides and carbons of iron and met-
als like manganese, Zink, nickel and cobalt. Their main advantage is low high electrical resistivity and
relatively low losses at high frequencies. However these materials have quite low saturation induction level.
Ferromagnetic materials are made of metal alloys of iron and metals like silicon, nickel, chrome and
cobalt. They have higher saturation induction than ferrites, but also much higher electrical conductivity
(therefore higher losses due to eddy currents). This group of materials can be further divided into several
subgroups based on the manufacturing technology:
• Iron based alloys
• Iron powder
• Amorphous materials
• Nanocrystaline materials
The order in which these different material groups are listed corresponds to chronological order in which
they appeared and started to be manufactured and used in power electronics. Manufacturing process as
well as material characteristics will be discussed in greater detail in following section. Figure ?? illustrates
described classification of magnetic materials
Figure 2.2: Three different types of BH loops
8
19. Chapter 3
Modeling strategy
The main aim of this work is to make a wind turbine model that would be simple enough for a real time
implementation, but at the same time accurate enough so that it captures all the important aspects of wind
turbine operation. Hence, we would like to create a model that would be more accurate than simplified mod-
els already present in the electrical engineering literature. One way to make a simple and accurate model
would be to start with a complex wind turbine model and use some mathematical tools in order to reduce the
complexity of the model without impeding its accuracy too much. However, this would be quite hard since
the equations describing wind turbine aerodynamics are not in closed form. Furthermore, a deep knowledge
of the aerodynamics theory would be necessary. Another way to build a turbine model would be to start
from one of already existing simpler models and try to improve its accuracy. However in this approach the
main difficulty lies in choosing the right model. Also, model built in such a manner would be quite difficult
to validate. This is probably the main reason that for most of the simplified models discussed in previous
chapter no validation was done. However, model validation is a very important part of any modeling effort
and therefore in order to measure the level of accuracy of our model, we would like to validate it. Since we
do not have a real wind turbine at our disposal, we would need to use a wind turbine simulator for model
validation. In doing so, we assume that the model used by the simulator is very accurate and that its results
come close to what we would get with an actual wind turbine.
Using a wind turbine simulator opens another possible way to model a wind turbine. In this approach
a model could be built by fitting the data obtained from the simulator. Performing simulations for various
operational points of the wind turbine allows for thorough analysis of the turbine torque characteristics.
Therefore, in this approach one can identify what are the aspects of wind turbine operation that might have
the greatest impact on electric generator or the grid and focus on those aspects in building a turbine model.
However, with this modeling strategy it is only possible to model a specific wind turbine. Moreover all the
data necessary for running the simulations for that specific wind turbine has to be known.
In our modeling, we adopt the approach in which the turbine simulator is used. As a simulation software
we use FAST (Fatigue, Aerodynamics, Structures and Turbulence) which is an open source code developed
by the USA National Renewable Energy Laboratory. FAST uses blade element momentum theory for mod-
eling turbine aerodynamics and an aero elastic model for representing turbine structural dynamics. The code
allows for turbine simulation with 24 different degrees of freedom. Also, it allows the simulation in various
control or open loop modes and gives a possibility to integrate the simulator with Matlab Simulink package.
One of the great advantages of using FAST as a simulation software is that it contains a couple of sample
9
20. CHAPTER 3. MODELING STRATEGY
Wind turbine characteristics
No. Blades 3
Rated power 1.5MW
Rotor Diameter 70m
Tower Height 82.39m
Gearbox Ratio 87.965
Moment of inertia 34.6 · 103kN/m2
Table 3.1: Some of the most important dimensions of the used sample wind turbine
wind turbine models for which all the data necessary for performing simulations is already available. These
sample turbine models are based on the actual wind turbines that were used to validate the accuracy of the
FAST code. Therefore in this work we model one of those available sample turbines by using simulations
in FAST. Some of the technical data describing this wind turbine is given in table 3.1.
Characteristic of the torque extracted from the wind depend on many parameters. The most important
are wind velocity, the shape of the wind field over the turbine rotor disk and turbine rotational velocity.
Furthermore, pitch angle of the blades as well as turbine nacelle yaw angle can have influence on the turbine
operation. In our modeling we only focus on the wind and turbine rotational velocity and keep the pitch
angle and the nacelle yaw angle constant during our analysis and modeling. Furthermore, FAST uses two
different ways of describing the wind field over the turbine rotor disc. The first one is a full field wind de-
scription where the rotor disc area is discretized and wind velocity and direction in each of the discretization
cells are given. For such a field description wind measurements need to be done on an actual wind turbine
site or some of the software for wind field generation needs to be used. Second one is the description of the
wind field through the wind speed and direction at the hub height and corresponding field share values. In
our modeling we use the later method and take the wind field over the turbine rotor to be homogenous with
only vertical wind shear component. In taking this approach we are not considering wind turbulence in our
modeling.
In building our model we use the turbine simulator to analyze the turbine torque we obtain for differ-
ent turbine rotational and wind velocity pairs. For each of those pairs we perform a simulation in FAST
where we keep both the wind description and the turbine rotational velocity constant during the simulation.
However, performing simulations for all possible combinations becomes a very time consuming and almost
intractable problem. Therefore, the first step in our modeling is to define the relevant simulation area. Af-
ter performing the simulations in this area we analyze the obtained data and come up with a mathematical
model that would fit it.
10
21. Chapter 4
Wind turbine model
4.1 Determining the area of interest
The first step in building our model is to define the relevant area of turbine rotational and wind velocity
values for which we will perform simulations in FAST. In order to define this area we assume that the wind
turbine would be operated so that it extracts maximal power from the wind. For variable speed wind turbines
the power generator torque is regulated so that the turbine rotates at a velocity that maximizes the extracted
power for the given wind velocity. Therefore, it is reasonable to restrict our attention to the neighborhood
of the wind vs. turbine velocity line for which the turbine extracts maximal power from the wind. This line
is often called the optimal line of turbine operation or MPPT line.
In order to find the optimal line, we use the fact that the power extracted from the wind is often given by
the following formula:
Prot =
1
2
ρπR2
Cp(λ, β)v3
(4.1)
Where ρ is the air density, R is the turbine rotor radius, v is the wind velocity and Cp(λ, β) is the power co-
efficient which represents the ratio between the extracted and power available in the wind. Power coefficient
is a nonlinear function of the pitch angle β and the tip speed ratio, which is the ratio of the turbine blade tip
and wind velocity:
λ =
ωrR
v
(4.2)
Where ωr is the rotational velocity of the turbine.
From this equation it is clear that the power extracted from the wind is maximized if the turbine is
operating so that the Cp coefficient is maximized. In our analysis we fix the pitch angle to β = 3◦ and
focus our attention on finding the rotational and wind velocity values that maximize the power coefficient.
Since FAST can give the Cp coefficient as one of its outputs during the simulation, we perform a set of
simulations in which we keep the rotational velocity of the wind turbine constant and gradually change
wind velocity from very low values to higher values. The experiment is repeated for various wind turbine
rotational velocities. For each of these rotational velocities, we determine the wind velocity for which the
Cp coefficient attains its maximum. Putting together these results, for the turbine of table 3.1 we obtain the
MPPT line equation:
v = 0.4ωr + 1.93 (4.3)
11
22. CHAPTER 4. WIND TURBINE MODEL
Where ωr is the rotational velocity of the wind turbine expressed in rotations per minute (rpm) and v is
wind velocity expressed in m
s .
The MPPT line equation we get contradicts the assumption that the Cp coefficient is just a function of the
tip-speed ratio and the pitch angle, since if this was the case the line we obtain would have to pass through
the origin. The fact that we get an offset in this line can be interpreted by loss modeling. Namely, we can
assume that there is a certain threshold wind velocity value below which no energy is extracted from the
wind. Such an assumption would explain the shape of the optimal operational line that we get. In addition
to this, we need to have in mind that the equation 4.1 is a result of momentum theory, whereas FAST uses
blade element momentum theory for modeling turbine aerodynamics. This theory models the interaction of
wind and turbine blades more precisely and takes into consideration various losses such as tip loses and hub
loses. Therefore, a more precise model of the aerodynamic losses is probably the main reason for the slight
discrepancy between the result we obtained and what is to be expected by the equation 4.1.
Having the MPPT line, we define a relevant neighborhood of this line for which we will model the
turbine behavior. In the simulations we look at turbine torque for a number of rotational and wind velocity
pairs of interest. These pairs are defined by the MPPT line, where we do the simulations for rotational
velocities ranging from 10rpm up to 26rpm with the resolution of 0.2rpm. In addition to simulating along
the MPPT line, simulations are done along the lines parallel to the MPPT. These lines are obtained by
increasing and decreasing wind velocities along the MPPT up to value of 1m
s with the resolution of 0.2m
s .
Figure 4.1 illustrates the procedure used to find the MPPT line. It shows the dependence of the power
coefficient on turbine rotational and wind velocity. The figure also shows turbine and wind velocity pairs
for which the simulations are performed.
Figure 4.1: Left figure shows the dependence of the power coefficient on turbine rotational and wind ve-
locities. The line along which Cp attains its maximum is shown in black. In the right figure, the relevant
simulation area is shown. The points represent the turbine-wind velocity pairs for which we perform the
simulations. The points along the MPPT line are marked with red
12
23. CHAPTER 4. WIND TURBINE MODEL
4.2 Turbine simulation
For each of the defined simulation points a 1000s simulation using FAST is performed, where during the
simulation rotational velocity of the turbine and wind velocity are kept constant. In these simulations the
wind field over the turbine rotor is kept homogenous. This means that the turbulence is not taken into
account and that a constant vertical wind shear value is used. Moreover, wind direction is always set to
be perpendicular to the turbine rotor plane and no parallel component is present. In order to force the
wind turbine to rotate at a constant velocity we externally apply the corresponding generator torque. These
simulations are performed by interfacing FAST with Simulink, where FAST takes the generator torque and
power as constant signals from Simulink. In order to calculate the torque and power that we need to apply
to the turbine so as to ensure it is operated at the turbine-wind velocity pair of interest, we first run the
simulation by only using FAST and turning the generator degree of freedom off for each of the simulation
points. The fact that the generator degree of freedom is disabled means that the turbine shaft will always
have zero acceleration no matter which torques are applied to it, and hence the turbine will always rotate
at a constant velocity. Therefore, this method of simulation is convenient for the calculation of the mean
torque values that are generated by the turbine for given values of rotational and wind velocity. However,
the torque oscillations that we observe during such simulations do not correspond to the oscillations that
appear when the rotational velocity of the turbine is regulated. The main reason for this difference is that
the generator degree of freedom is taken into consideration when the turbine is regulated. In this case, when
calculating loads and motions of the turbine shaft, FAST considers the drive train vibration mode to be free
on both sides of the shaft, whereas it is considered to be fixed on one side when the generator degree of
freedom is disabled. Therefore, since we are interested in modeling turbine torque oscillations when the
turbine is regulated, we perform simulations with the generator degree of freedom enabled, where we force
the shaft to rotate at a certain velocity by counteracting the shaft torque with the corresponding generator
torque. Such a way of simulation leads to slight oscillations of the turbine rotational velocity, but since they
are very small we can neglect them and consider that the turbine is rotating at a constant velocity. For each
simulation point we look at the turbine torque at the low speed shaft (the value at the high speed shaft can
be calculated by simply multiplying this value with one over the gearbox ratio).
Since the simulations are performed for a great number of operating points , simulating procedure has to
be automated in order to make it tractable. Since FAST code does not provide a user friendly environment
that enables automation of its operation, we use Matlab to build an environment in which we can easily
create FAST input files. In addition, this environment is used to read FAST output files and prepare the
data for further analysis. Together with FAST, such an environment is a very powerful tool for wind turbine
simulation. It provides a possibility to automate simulations while having a great flexibility when designing
different simulation scenarios. A detailed description of this environment as well as FAST source code is
given in the appendix.
4.3 Data analysis
In order to build a wind turbine model, we first analyze the data obtained from the simulator. This analysis
is done by looking separately at the mean torque values and torque oscillations for each of the defined
simulation points. Simulation results show that the mean torque values change quite regularly with the
change in turbine rotational and wind velocity. Moreover, this change is quadratic with turbine rotational
13
24. CHAPTER 4. WIND TURBINE MODEL
velocity. Such quadratic dependence is already predicted by most of the existing simplified wind turbine
models and is given by equation 2.1.Experimentally obtained curves that relate mean torque values to turbine
rotational velocity are shown in figure 4.4.
In addition to looking at the mean turbine torque values, we are also interested in modeling torque
oscillations. Therefore, we analyze torque oscillations both in time and frequency domain. Figure 4.2
illustrates some of the obtained results.
Figure 4.2: Illustrates the characteristics of torque oscillations. In the upper row the oscillations are shown
for the turbine operating point ωr = 18rpm, v = 9.13m
s and in the lower row for the pair ωr = 24rpm,
v = 11.53m
s . Going from left to the right oscillations in time domain, amplitude spectrum of the oscillations
and amplitude spectrum in logarithmic scale are shown
From these results we may conclude that, in time domain, torque oscillations look like slightly deformed
sinusoidal signals. Looking at the amplitude spectrum of torque oscillations we may conclude that the
dominant frequency in the spectrum is the 3p frequency, which equals three times the frequency of turbine
rotation. Moreover, higher harmonics that are multiples of the 3p frequency also appear, but their amplitudes
are much lower than the amplitude of the basic 3p harmonic. In addition, one may notice the appearance of
1p harmonics, which appear at multiples of the turbine rotational frequency. However, their amplitudes are
extremely small compared to the amplitudes of the 3p harmonics and therefore we do not consider them in
our analysis. The way torque oscillations change with the change in turbine rotational and wind velocity is
quite nonlinear. This change is nonlinear in terms of the shapes of the amplitude and phase spectrum of the
oscillations as well as in the amplitude values of the 3p harmonic.
Although the amplitudes of higher harmonics are quite small compared to the amplitudes of the basic
harmonic, for certain rotational velocities some of the multiples of the basic harmonic can have amplitudes
that are much bigger than the average amplitude values for the higher harmonics. Our assumption is that the
main reason for the amplification of the higher harmonic could be its resonance with one of the structural
14
25. CHAPTER 4. WIND TURBINE MODEL
modes of the turbine. In order to better illustrate this idea, we look at the amplitudes of higher harmonics for
all the simulation points along the MPPT line. Plotting only those that are higher than 1Nm at frequencies
at which they appear, we get figure 4.3. This analysis shows that there are certain frequencies around which
2 3 4 5 6 7 8
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
f [Hz]
amplitude[kNm]
Figure 4.3: Shows the amplitudes of all the higher harmonics that are higher than 1Nm at the frequencies
at which they appear
the selected higher harmonics group. If our assumption about the resonance effects is correct, these could
be the structural modes of the modeled wind turbine.
This nonlinear behavior of the data describing torque oscillations makes our modeling quite difficult.
However, one of the things that we may use to our advantage is the fact that the amplitudes of higher
harmonics reduce significantly with the increase in frequency. Therefore, most of the torque oscillation
power is already captured by couple of low order harmonics. Hence in our modeling it is acceptable to focus
only on these harmonics and not consider the full amplitude spectrum of the oscillations.
4.4 Mathematical model
The fact that the mean torque value change is quadratic with the change of turbine rotational velocity can be
used to express mean turbine torque as a function of turbine rotational velocity and wind speed. Building
such a function, we first model the dependence of the mean torque from turbine rotational velocity. For this
it is enough to look at mean torque values as a function of rotational velocity along the MPPT line and fit a
second order polynomial function to the data. The same procedure can be applied along all the simulation
lines that are parallel to the MPPT line. These are the lines that are formed when the same constant is added
to the wind velocity values along the MPPT line, while the values of turbine rotational velocity stay the
same.
In order to model the dependence of the mean torque values from the wind velocity we look at how
the coefficients used for fitting second order polynomials depend on the distance between the simulation
15
26. CHAPTER 4. WIND TURBINE MODEL
line along which they are fitted and the MPPT line. It turns out that it is possible to express one of these
coefficients as a third order polynomial function of the distance values and the other two as simple linear
functions. Figure 4.4 shows the data obtained along some of the simulation lines and the corresponding fitted
polynomials, in addition to which the dependence of the fitting coefficient values from the line distances and
curves fitted to this data are shown.
10 12 14 16 18 20 22 24 26 28
100
200
300
400
500
600
700
800
900
1000
ω
r
[rpm]
T[kNm]
Obtained mean torque data and curves fitted to the data
v
MPPT
−0.8m/s
v
MPPT
−0.4m/s
v
MPPT
vMPPT
+0.4m/s
vMPPT
+0.8m/s
actual torque values
−1 0 1
13.2
13.4
13.6
13.8
14
14.2
14.4
14.6
14.8
15
15.2
dv [m/s]
coeficient1
−1 0 1
140
150
160
170
180
190
200
dv [m/s]
coeficient2
−1 0 1
350
400
450
500
550
600
dv [m/s]
coeficient3
Figure 4.4: Ilustrates the procedure used to fit mean torque data. In the left plot the actual mean torque
data obtained along different simulation lines is shown by black dots. The colored lines represent the second
order polynomial curves that are fitted to the data. The right figure shows the dependence of polynomial
coefficients used to fit mean torque data along different simulation lines as a function of the distance from
the MPPT line. The coefficient vales are shown as black dots, and the colored lines represent curves fitted
to the data
Therefore, we can express mean torque values as a function of turbine rotational and wind velocities by
using eight coefficients. The fitting function is given by the following equation:
Tmean = (c11d3
v + c12d2
v + c13dv + c14)2
ω2
r + (c22dv + c23)ωr + c32dv + c33 (4.4)
Where dv is the difference between the actual and the optimal wind velocity for a given turbine rotational
velocity:
dv = v − (0.4ωr + 1.93) (4.5)
Alternative, and the most common way to model wind turbine mean torque found in the literature would be
to express the turbine torque as a function of turbine rotational and wind velocity by the following equation:
T =
1
2ωr
ρπR2
Cp(v, ωr)v3
(4.6)
and then numerically fit the Cp coefficient as a function of wind and turbine velocity. However, experiments
show that such a fit would require 12 coefficients. The number of needed coefficients depends on the turbine
that is being modeled and in particular on the nonlinearity of its Cp coefficient. According to other studies
in the literature such a way of fitting mean turbine torque would require polynomials of orders from twelve
to fifteen depending on the turbine being modeled and modeling accuracy. Therefore, the way of fitting the
16
27. CHAPTER 4. WIND TURBINE MODEL
data that we chose here leads to a model that requires smaller number of coefficients and is better suited for
real time implementation.
To model torque oscillations we separately look at different harmonics appearing harmonic as a function
of turbine rotational velocity and wind speed. Here we use an approach that is similar to the one used for
fitting the mean torque values. We look at the amplitude and the phase of the 3p harmonic along the MPPT
line and simulation lines parallel to it, and try to fit curves that would model experimentally obtained ampli-
tude and phase data as a function of turbine rotational velocity. The data analysis shows that polynomials of
sixth degree are enough to fit both the amplitude and the phase data quite accurately. Figure 4.5 illustrates
the data obtained from the simulations and corresponding fitted curves, it also gives a comparison between
the shapes of the fitted polynomial functions for different simulation lines.
10 20 30
0
0.02
0.04
0.06
0.08
0.1
v
MPPT
−0.8m/s
ω
r
[rpm]
|T|[kNm]
10 20 30
0
0.02
0.04
0.06
0.08
0.1
along the MPPT line
ω
r
[rpm]
|T|[kNm]
10 20 30
0
0.02
0.04
0.06
0.08
0.1
v
MPPT
+0.8m/s
ω
r
[rpm]
|T|[kNm]
10 20 30
−40
−20
0
20
ω
r
[rpm]
phase[rad]
10 20 30
−30
−20
−10
0
10
ω
r
[rpm]
phase[rad]
10 20 30
−30
−20
−10
0
10
ω
r
[rpm]
phase[rad]
10 15 20 25 30
0.01
0.02
0.03
0.04
0.05
0.06
0.07
ω
r
[rpm]
|T|[kNm]
fitted curves for the amplitude of the first harmonic
v
MPPT
−0.8m/s
v
MPPT
−0.4m/s
v
MPPT
v
MPPT
+0.4m/s
10 15 20 25 30
−35
−30
−25
−20
−15
−10
−5
0
5
10
15
ω
r
[rpm]
phase[rad]
fitted curves for the phase of the first harmonic
Figure 4.5: The left part of the figure shows the amplitude and phase data for the first 3p harmonic along
different simulation lines and corresponding polynomial curves fitted to the data. The right part compares
curves fitted for the amplitude and phase data of the 3p harmonic along different simulation lines
These results show that the data we obtain changes nonlinearly with switching between different simula-
tion lines. Therefore, trying to fit a surface, as is done for the mean torque values, would require polynomials
of a very high degree. This would open implementation issues and also make the model less suitable for real
time purposes. Therefore, instead of fitting a surface, we opt for a solution which is based on interpolation.
For a given pair of turbine rotational velocity and wind speed we estimate the amplitude and the phase of the
first harmonic by interpolating the values obtained by evaluating polynomial functions describing the data
along two simulation lines between which the given point lies. Therefore, the amplitude and the phase of
the first harmonic can be estimated by using the following equations:
ϕ1 = we1
7
k=1
pϕ
ikω7−k
r + we2
7
k=1
pϕ
jkω7−k
r (4.7)
a1 = we1
7
k=1
pa
ikω7−k
r + we2
7
k=1
pa
jkω7−k
r (4.8)
Where pϕ
ik, pϕ
jk, pa
ik and pa
jk are polynomial coefficients used for fitting the first 3p harmonic phase and
amplitude along the simulation lines i and j between which the operation point of interest lies. Coefficients
17
28. CHAPTER 4. WIND TURBINE MODEL
we1 and we2 are interpolation coefficients that are reversely proportional to the distance of the operation
point of interest from this two lines.
When modeling torque oscillations, higher harmonics also need to be taken into consideration. How-
ever, as already mentioned in previous section, it is not necessary to look at all the higher harmonics. Our
analysis shows that it is enough to consider only the first eight harmonics in order to accurately capture
amplitude spectrum of torque oscillations. In modeling the influence of the higher harmonics we look at
their amplitude and phase values trying to model them as functions of turbine rotational velocity and wind
speed. Looking at the phase values of the first eight higher harmonics along the MPPT line, we conclude
that it is enough to use 10 degree order polynomials to fit this data as a function of turbine rotational ve-
locity. However, modeling the amplitude is not as easy as modeling the phase values. This is mainly the
result of the influence of the resonance which causes the amplitude values of higher harmonics to change
quite nonlinearly. Using the polynomial fitting strategy to fit the amplitude data would lead to polynomial
functions of very high order. One way to overcome this difficulty is to use piecewise polynomial fitting,
where the second order polynomials are sufficient for a good fit. In fitting procedure we divide the rotational
velocity space into intervals for which we do not expect any significant increase in the amplitude values and
intervals where there is a significant increase due to resonance effects. For each interval we fit a polynomial
function. Merging the results we get a function that describes the dependence of the harmonic amplitude on
turbine rotational velocity. For instance, for the second 3p harmonic, we can make a fit with a line and a
second order polynomial, where we use the line for the area in which there are no huge differences between
the amplitude values. The second order polynomial is used here to model a jump in amplitude values for
a narrow interval of rotational turbine velocities. The same technique is used to fit other higher harmonics.
Figure 4.4 compares the real data obtained from the simulation and curves fitted by using the described
method.
In order to model how these fitted characteristics change when the turbine operates away from the MPPT
line, we use the sensitivity of the obtained data. This approach requires looking at how obtained experimen-
tal characteristics change with getting away from the MPPT. Calculating the sensitivity, we subtract the data
obtained along the MPPT line and the simulation line for which the wind velocity is reduced by 0.2m
s and
the data along the simulation line for which the wind velocity is increased by 0.2m
s . Finding the mean value
of these two results and normalizing it by 0.2 we get the sensitivity of the characteristic. This procedure is
done for amplitude and phase characteristics of all the first eight harmonics (apart from the 3p for which
we have a separate method of fitting). To each of these sensitivity characteristics an 8 degree polynomial
function is fitted. This provide us with the full dependence of the higher harmonic phase and amplitude
on the turbine rotational and wind velocity. The main reason we chose to use sensitivity for fitting higher
harmonics is that there is not much need for a very precise model of their behavior since their amplitude
is much smaller than the amplitude of the first 3p harmonic. Therefore, using this method we reduce the
computational burden needed to implement the model. The amplitude and phase of the higher harmonics
can be expressed as a function of turbine rotational velocity and wind speed by the following equations:
aq = pq
pol(ωr) + dv
9
r=1
csa
qrω9−k
r (4.9)
ϕq =
9
u=1
cϕ
qu + dv
9
r=1
csϕ
qr ω9−k
r (4.10)
18
29. CHAPTER 4. WIND TURBINE MODEL
10 20 30
−20
−10
0
10
n=2
ωr
[rpm]
phase[rad]
10 20 30
−20
−10
0
10
n=3
ωr
[rpm]
phase[rad]
10 20 30
−20
0
20
40
n=4
ωr
[rpm]
phase[rad]
10 20 30
0
10
20
30
n=6
ω
r
[rpm]
phase[rad]
10 20 30
−20
−10
0
10
20
n=7
ω
r
[rpm]
phase[rad]
10 20 30
−10
−5
0
5
10
n=8
ω
r
[rpm]
phase[rad]
Figure 4.6: Compares real data for the amplitudes and phases of higher harmonics obtained from the
experiments and the curves fitted to the data. Here n denotes the multiple of the basic 3p harmonic
19
30. CHAPTER 4. WIND TURBINE MODEL
Where pq
pol denotes corresponding piecewise polynomial function used for fitting amplitude values of
the qth harmonic, csa
qr and csϕ
qr are the coefficients used to fit the sensitivity data and cqu are the coefficients
describing the phase data along the MPPT line.Piecewise polynomial fitting denoted by the term pq
pol(ωr) is
given by the following equation:
pq
pol(ωr) =
nq
1
k=1 cha
q1 ω
nq
1−k
r if ωr ∈ [10 ωtr
r1)
...
...
nq
m
k=1 cha
qmωnq
m−k
r if ωr ∈ [ωtr
rm 26]
(4.11)
Where m denotes the number of partitioning regions for the qth harmonic, cha
qi stand for fitting coefficients
in these regions and nq
i denotes the number of coefficients used for fitting in regioni. Borders of the regions
are denoted by ωtr
ri. Values of all these variables are listed in the table B.7 in the appendix.
Putting together the model for mean turbine torque and its oscillations, the turbine torque can be ex-
pressed as a function of turbine rotational velocity and wind speed using the following equation:
T(ωr, v) = Tmean(ωr, v) +
8
i=1
ai(ωr, v)cos(i3ωrt + ϕi(ωr, v)) (4.12)
Where Tmean(ωr, v) is the mean torque value for a given turbine rotational and wind velocity and ai(ωr, v)
and ϕi(ωr, v) are the amplitude and the phase of the ith harmonic for a given turbine operating point. In
order to introduce the dynamics into the turbine model, we use the fact that the difference between the
turbine torque and power generator torque causes the turbine to accelerate. Therefore, we can express
turbine rotational velocity by the following equation:
ωr =
J
s
(T − Tg) (4.13)
Where J is the turbine inertia and Tg is the power generator torque.
The model presented here has wind description at hub height and generator torque values as its inputs.
The output of the model is the wind turbine torque extracted from the wind. Multiple polynomial functions
of turbine rotational velocity and wind velocity are used to calculate this torque The acceleration of the
turbine inertia is used to introduce the dynamics into the model. Figure 4.7 summarizes proposed modeling
procedure and the schematic of its operation.
Since no dynamics is coming from the wind description, the proposed model is quasi-static. This means
that we assume that wind dynamics is not influencing turbine torque, and that the torque changes instan-
taneously with the change in wind velocity. This assumption is also present in all simplified wind turbine
models discussed in chapter 2. Reason for adopting such a model is that experiments show that it is accurate
when the wind velocity is not changing too much. This assumption is also supported by the experiments
performed in order to validate the proposed model that are discussed in the following chapter.
20
31. CHAPTER 4. WIND TURBINE MODEL
Figure 4.7: Schematic illustration of the proposed wind turbine model
21
33. Chapter 5
Model validation
In order to validate the proposed wind turbine model we performed various simulations in closed loop. In
these simulations the wind turbine is regulated so that the power it extracts from the wind is maximized.
In order to keep the extracted power maximal, the rotational velocity of the turbine needs to be controlled
so that the turbine operating point given through the pair of turbine rotational and wind velocity lies on the
MPPT line. Since the turbine rotational velocity can be controlled through the difference between turbine
torque and generator torque, we take the generator torque to be the control input for the regulation. In order
to keep the turbine operating at the MPPT line, a controller in the form of a lookup table can be used. This
controller takes turbine rotational velocity as its input and returns torque values equal to the turbine torque
obtained for the given rotational velocity at the MPPT line. When such a controller is used for a given wind
velocity, the equilibrium point of the closed loop turbine system is adjusted so that it always lies on the
MPPT line. Consequently, as the wind speed changes, the turbine rotational velocity will be altered so that
that it takes the corresponding value at the MPPT line. Although this controller is quite simple and usually
more sophisticated techniques are used to regulate turbine rotational velocity, it captures the main idea lying
behind most control strategies. Therefore, it provides a good framework for testing the accuracy of our wind
turbine model.
In order to check the accuracy of our model, we compare the turbine torque values generated by our
model to the ones obtained by using FAST for various simulation scenarios. For conducting the simulations
we use Matlab Simulink package, where we implement our model as a function that for a given rotational
and wind velocity pair returns corresponding turbine torque values. This function also takes real time as
one of its inputs as it is needed for calculating the contribution of turbine torque oscillations as given by
the equation 4.11. The simulations are also performed by using FAST where the lookup table controller
is implemented in Simulink and the integration between Simulink and FAST is used to simulate the wind
turbine. In our simulations we feed both our model and the FAST software with the same variable wind
description. For our model the input wind is given through a wind signal describing wind velocity at hub
height and for the FAST a special wind file has to be created. In addition to changing the wind description,
we introduce generator torque disturbance into our simulations. This is done by simply adding the same
disturbance signal to the generator torque values coming out of the controller for both the models. The
logic behind introducing this disturbance signal is that actual generator system and the controller are always
imperfect. As we would like to use the proposed model as a real time wind turbine simulator, it is important
to see whether the model is valid in actual operating conditions. Therefore, introducing the error signal
23
34. CHAPTER 5. MODEL VALIDATION
brings the validation experiments closer to real world situation. Our expectation is that this error signal
should not have a great impact on model accuracy which would prove that the proposed model is suitable
for building an actual emulator. Figure 5.1 shows Simulink schematics used for the simulations.
The results obtained from such a simulation setup show that the turbine torque values produced by the
proposed model match very well with what we get from FAST when the turbine is operated close to the
MPPT line. In order to keep the turbine operating close to the MPPT line, wind description must not vary
too much and also the torque disturbances should be kept relatively small. The experiments show that, when
the maximal change in wind velocity is less than 3m
s and the absolute value of the torque disturbance is less
than 15kNm ,the relative error of the proposed model is less than 1%. In addition, the shape of the turbine
torque oscillations is captured very well. Figure 5.2 illustrates the results for such a simulation experiment.
The figure shows the shapes of the wind velocity and generator torque disturbance during the time and
makes a comparison of the results obtained by the proposed model and FAST simulator.
However, in cases when wind description is varying too much or torque disturbance signal is very high,
the results we obtain do not match well any longer. In fact, in cases when the wind speed is changing very
fast and with a big amplitude, the relative errors we get with our model can go up to 50%. Our assumption is
that the main reason for this lies in the fact that great changes in the wind velocity and big generator torque
disturbances cause the turbine to operate far away from the MPPT line. Although the controller we are using
is drawing the turbine operation point to the MPPT line, if the wind velocity changes too dramatically, there
is not enough time to bring the turbine operating point to the MPPT line. As a result the turbine will operate
far away from the MPPT for long periods of time. It is precisely in these periods that the relative error of our
model is high. This is completely expectable, as the simulations using FAST were performed in the close
neighborhood of the MPPT line and based only on the data the turbine model was derived from. Therefore,
it is normal that the model has a significant error when trying to model wind turbine behavior far away from
the MPPT line. However, the distance from the MPPT line may not be the only cause of the model error that
occurs when the wind velocity is changing FA. Another reason could be that such a fast change in the wind
description activates some of the turbine structural modes that were not taken into consideration during the
modeling. Figure 5.3 illustrates one of the simulations scenarios where the results obtained by the proposed
model are significantly different than what we get with FAST. The figure also shows the distance from the
MPPT line in terms of the wind velocity. It is clear from these results that when turbine is operating too far
away from the area which was used to fit the model, the error is significant. In fact, our experiments have
shown that the model gives very good results when the turbine is operating inside the area for which the
model was fitted, also going outside of this area up to a value of 2m
s the obtained results are still acceptable
(relative error is less than 3%. However, when the turbine is operating far away from this area, the model is
no longer valid.
24
35. CHAPTER 5. MODEL VALIDATION
Figure 5.1: Illustration of the Simulink schematic used for simulations
25
36. CHAPTER 5. MODEL VALIDATION
Figure 5.2: The results obtained for one of the simulation experiments are shown. The two upper left figures
show wind velocity at hub height and generator torque disturbance signals during the simulation. The lower
left figure compares turbine torque values that the proposed model and FAST give. The results of FAST are
shown in red and the results of our model in blue. The lower right plot shows torque oscillations more
closely and the upper right one shows the relative error of the model results during the simulation
26
37. CHAPTER 5. MODEL VALIDATION
Figure 5.3: The results obtained for one of the simulation experiments are shown. The upper left figure
shows wind velocity at hub height during the simulation. Lower left figure compares turbine torque values
that the proposed model and FAST give. The results of FAST are shown in red and the results of our model in
blue. The lower right plot shows the distance of turbine operation from the MPPT line during the simulation
in terms of wind speed. The area between the two red lines represents the area which was used to create the
model. The upper right figure shows the relative error of the model results during the simulation
27
39. Chapter 6
Conclusion
In this work we have proposed a wind turbine model that is suitable for real time implementation. The
model is build by fitting the data obtained from the turbine simulator. The data was fitted using polynomials
of different degree. As a result, turbine torque is expressed as a function of turbine rotational velocity and
wind speed through multiple polynomial functions. Therefore, such a model is very easy to program in
DSP as it only incorporates addition and multiplication. In building the model, care was taken to make its
computational expense as low as possible. Hence the proposed model can be used to build a wind turbine
emulator in order to create a controlled test environment for research.
Moreover, the model proposed here is more accurate in terms of modeling tower shadow and wind
shear effects than other simplified turbine models present in literature. As our analysis of the data obtained
through turbine simulation with FAST showed, torque oscillations tend to change quite nonlinearly with the
change in turbine rotational velocity and wind speed. The main reason for this nonlinearity lays in very
complex and extremely nonlinear laws of aerodynamics theory which describes the interaction of wind field
and turbine rotor disc. Moreover, turbine structural modes can apparently have significant influence on the
shape of turbine torque oscillations. Although the nonlinear change in torque oscillations is predicted by
some of the existing turbine models, the functions describing it are quite simple and do not capture fully
the behavior of torque oscillations. The model proposed here, on the other hand, models torque oscillations
very precisely and captures all nonlinearities in their change.
In addition, most simplified turbine models found in literature were not validated. One of the reasons
is that most of these models are too simple to be able to emulate wind turbine properly. Another reason is
the lack of real wind turbine system that could be used for model validation. In our approach this is over-
come by using a turbine simulator. This allows us to better study the turbine behavior and analyze torque
oscillations thoroughly. Furthermore, it enables us to check the level of accuracy for the model we propose
and perform various experiments both on the model and the simulator. These experiments can provide a
valuable insight into the aspects of turbine operation that are modeled well, but also indicate the aspects that
need to be studied further in order to improve the model.
The simulations done with the proposed model show that its results match very well with what is ob-
tained from the wind turbine simulator when the turbine is operating close to the MPPT line. The model
validation shows that the model can be used to simulate a wind turbine in closed loop and under generator
torque disturbance which is what is often encountered in real applications. Moreover, the model captures
the shape of turbine torque oscillations very precisely, which is very important if the model is to be used for
29
40. CHAPTER 6. CONCLUSION
the examination of the influence that turbine torque oscillations may have on power generator operation.
However, the experiments have shown that when the turbine is operating far away from the MPPT line,
the model is not valid any more. The main reason for this is that our model was built by observing only
a small neighborhood around the MPPT line. In addition, since in these experiments wind velocity varies
significantly, the reason of the deterioration of the results could be the excitation of some turbine modes
that were not taken into consideration during model building. Moreover, by taking the wind field over the
turbine rotor to be homogenous, we do not consider the influence that wind turbulence might have on torque
oscillations.
Therefore, studying wind characteristics is a natural next step towards improvement of the proposed
turbine model. This study could look into the highest wind changes encountered by actual wind turbines,
which could help to define relevant area around the MPPT line in which the turbine model should be valid
to ensure its validity in all real operating conditions. In addition, studying the stochastic behavior of the
wind could help understanding the influence that wind turbulence has on torque oscillations. The proposed
model could then be extended so that it also takes into account the influence that wind turbulence may have
on torque oscillations.
30
41. Appendix A
Using FAST for wind turbine simulation
and modeling
FAST (Fatigue, Aerodynamics, Structures and Turbulence) is an open source code for wind turbine simu-
lation developed by USA National Renewable Energy Laboratory. In FAST turbine structural dynamics is
represented by aero-elastic model and blade element momentum theory is used for modeling turbine aero-
dynamics. It is capable of simulating the behavior of two and three bladed horizontal-axis wind turbines.
FAST can operate in two different modes. One is the time marching mode in which the wind turbine is
simulated and the other is linearization mode in which the linearized representation of a complete nonlinear
aero-elastic wind turbine model is extracted. In addition, simulation analysis can be run by using Windows
command line or FAST can be integrated with Simulink. For simulation of three-bladed wind turbines FAST
uses 24 degrees of freedom. Any combination of these degrees of freedom can be enabled or disabled during
simulation.
Installing FAST is quite straightforward. FAST Windows executable program file is distributed in
the FAST archive available at NREL web page: http://wind.nrel.gov/designcodes/simulators/fast/. This
archive is self extracting and contains the FAST executable file, source code, sample input files, installation-
verification test procedure, S-function (for integration with Simulink), and change log. It runs on all 32-bit
Windows platforms. In order to install FAST so that it can be run from command window and any folder, the
folder which contains extracted FAST files has to be added to the system variable list in the control panel.
With this setting, FAST can be run from Windows command line and input and output files can be stored in
folders that are different from the one in which FAST .exe files are stored.
In Order to run FAST several input files need to be created. These files can be edited by using any
standard text editor. Since the input files are quite long and contain a great number of variables that need
to be set, it is useful to start from already existing input files in simulation design. FAST archive contains
17 different simulation scenarios for which all the necessary input files are provided. Table A.1 lists all the
necessary input files needed to run FAST. In case when a sample wind turbine is used all of these files are
already available. In order to perform different experiments with a sample turbine one only needs to modify
the primary input and wind file. In the following section we describe how these files can be altered in order
to create desired simulation scenarios in FAST.
31
42. APPENDIX A. USING FAST FOR WIND TURBINE SIMULATION AND MODELING
input file extension short description
Primary input file .fst This file is used to describe simulation parameters, wind turbine operat-
ing parameters and basic geometry. The blade, tower, furling, aerody-
namic and wind field parameters are read from separate files and all the
other data necessary for simulation is stored in this file. The names of
these separate files to be used are also specified here
Tower input file .dat This file should contain a table of data describing turbine tower charac-
teristics. The file describes the shape and dimensions of turbine tower.
This file is needed even in the case when tower degree of freedom is dis-
abled. In this case the data from the file is not used, but the error occurs
if the file does not exist
Blade input files .dat This file contains three tables describing each turbine blade. It describes
only the dimensions of the blades and not their airfoil data.
Furling input file .dat This file should contain rotor tail dimensions and moment of inertia as
well as other rotor furling data necessary for the simulation. It is needed
even if a turbine without a tail is simulated. In this case the file can be
left out blank, but it has to exist, otherwise the FAST would report an
error
AeroDyn input file .ipt This file contains all the data necessary for the operation of FAST sub-
section which deals with turbine aerodynamics. It contains the airfoil
data of the turbine blades. It describes the tower shadow model to be
used during the simulation and it specifies input wind file to be used in
simulations.
Wind input file .wnd This file contains all the data necessary for the description of the wind
field over the turbine rotor disc. There are two different versions of
wind files. In the first one wind field is described through hub height
wind speed and corresponding wind shear values. The other one uses
the full 3D wind description.
Table A.1: Different input files needed for simulations in FAST
A.1 Primary input file
The primary input file, as all the other files used by FAST, is a simple text file containing multiple rows of
text. The first column in the text file contains the names of variables used by FAST and the second column
contains their corresponding values. In order to illustrate how this file can be modified to arrive at different
simulation scenarios, we give a list of the most important variables and possible ways to set them.
1. Simulation controls
These variables are used to adjust simulation properties in FAST. Among these properties are simula-
tion length and time step as well as other important simulation parameters. Table A.2 gives some of
these variables and explains how they are used.
2. Turbine control variables
FAST allows simulations with a great number of control strategies. Turbine yaw angle, pitch angles
32
43. APPENDIX A. USING FAST FOR WIND TURBINE SIMULATION AND MODELING
Variable Description
AnalMode Variable that describes in which mode FAST is operating. If set to 1 FAST is
in simulation mode, and if set to 2 linearization analysis is performed
NumBI Number of turbine blades (can be 2 or 3)
TMax Simulation time (in seconds)
DT simulation time step (in seconds)
Table A.2: Primary input file simulation control variables
for each of the blades as well as turbine speed can be controlled. The controller can be made by hard-
coding it in FAST or the control strategy can be implemented in Simulink and then used by interfacing
FAST with Simulink. Another possibility that FAST gives is to keep the designed controllers inactive
up to a certain moment in time when they are activated. A combination of these options gives a lot of
designing opportunities for different kinds of experiments in a closed loop. Table A.3 gives some of
the variables used to define which control strategies are used during the simulation.
Variable Description
YCMode This is a switch for the nacelle yaw control. If set to 0 no control strategy is
used, 1 means that a user routine is hardcoded in FAST and 2 that a controller
will be implemented in Simulink
TYOn In case the nacelle yaw control is used, the time moment at which the controller
becomes active can be set by this variable. Time is in seconds
PCMode This is a switch for the pitch control. If set to 0 no control strategy is used and
the pitch angles have initial values during the whole simulation, 1 means that a
user routine is hardcoded in FAST and 2 that a controller will be implemented
in Simulink
TPCon In case the pitch angle control is used, the time moment at which the controller
becomes active can be set by this variable. Time is in seconds
VSControl This is a switch for the variable speed control. If set to 0 no control strategy is
used, 1 means that a user routine is hardcoded in FAST and 2 that a controller
will be implemented in Simulink
Table A.3: Primary input file turbine control variables
3. Simulation degrees of freedom
As previously mentioned, various combinations of different degrees of freedom can be used in simu-
lations. FAST gives the opportunity to turn on or off each of these degrees. Among them are two flap
wise blade bending degrees of freedom as well as edge bending degree of freedom. Also there are
four tower bending modes that can be separately enabled or disabled. Table A.4 lists other degrees of
freedom that are important to consider when designing experiments in FAST
4. Simulation initial conditions
These variables are used to set initial simulation values for the turbine rotor position and velocity. In
cases when the degree of freedom which causes the variable to change is disabled, these variables
present not only initial values, but the values that are constant during the whole simulation. All initial
conditions in FAST are set by initializing corresponding variables in the input file. In case some of the
33
44. APPENDIX A. USING FAST FOR WIND TURBINE SIMULATION AND MODELING
Name Description
DRTrDOF This flag is used to enable or disable torsion flexibility of the drive train. If
set to true, two mass model is used for modeling the turbine shaft. Otherwise
torque values are same for the high and the low speed side of the shaft
GenDOF Setting this flag to true enables generator degree of freedom. If it is set to false
the turbine will rotate at a constant velocity during the whole simulation time.
If turbine velocity is to be controlled, this degree of freedom has to be enabled
YawDOF This flag controls nacelle yaw degree of freedom
CompAero If this flag is set to false turbine aerodynamics is not taken into consideration
and turbine is simulated as operating in vacuum
Table A.4: Primary input file simulation degrees of freedom flags
variables are not initialized, FAST will use the default values. Therefore, user does not need to set the
initial conditions, but can use this option to easily define desired initial configuration of the simulated
turbine. Table A.5 lists some of the most important initial condition variables.
Name Description Unit
OoPDefl Initial out-of-plane blade deflection m
IODefl Initial in-plane blade deflection m
Azimuth Initial turbine rotor angle (initial angle of the first blade) deg
RotSpeed Initial turbine rotational velocity rpm
NacYaw Initial nacelle yaw angle deg
Table A.5: Primary input file simulation initial condition variables
5. Turbine configuration
In case a sample turbine model is used, it is usually not necessary to change the variables describ-
ing turbine configuration. However, these variables contain information on basic turbine dimensions
which can be useful to know when the turbine is modeled. These variables contain the information
on tower height, dimensions of rotor diameter or turbine moment of inertia. In addition in this section
of the primary input file the names of other input files that are used in simulation are specified. Table
A.6 lists some of the configuration variables:
6. Output specification
In this section, the properties of the FAST output file are specified. Usually the default values for
output parameters do not have to be changed. The only variable that is useful is TStart which denotes
the time moment in seconds from which FAST starts writing the results to the output file. This can
be used in cases when one is not interested in turbine startup and would like to skip the results during
this period.
FAST can give more than sixty different values as its output during time simulation. These values
describe loads and motions for each part of the turbine. Having all these values in the output file
would make it hard for storing and processing. Therefore, FAST allows the user to specify which
of these values should be written to the output file. Table A.7 gives some of the most often used
output signals. It is important to stress that this list is far from complete and that detailed list of all
possible output values can be found in FAST users guide. The output file that FAST provides has an
34
45. APPENDIX A. USING FAST FOR WIND TURBINE SIMULATION AND MODELING
Name Description
hline TowerHt Tower height in m
HubRad Rotor radius in m
HubIner Hub moment of inertia in kgm2
GBRation Gearbox ration
DTTorSpr Drive train torsion spring coefficient used for modeling the drive train with a
two mass model (Nm/rad)
DTTorDmp Drive train torsion damping coefficient used for modeling the drive train with
a two mass model (Nm/s)
TwFile Name of the tower input file
FurlFile Name of the furling input file
BldFile Name of the blade characteristic input file
ADFile Name of the AeroDyn input file
Table A.6: Primary input file turbine configuration variables
Name Description Unit
LSSTipPxa Rotor angle deg
LSSTipVxa Rotor angular velocity rpm
LSSTipAxs Rotor angular acceleration deg
s2
LSShftMxa Turbine torque kNm
RotCp Rotor power coefficient none
Table A.7: Variables used for FAST output file specification
extension .out and the same name as the primary input file. This file can be opened and edited by any
text editor. The data is organized in columns where the first column contains time values and other
columns contain variables that were specified in the primary input file. They appear in the same order
as they were listed in the input file. Figure A.1 shows a part of a typical output file
A.2 Wind input file
Wind input file can have two forms. In the first one wind field over the turbine rotor is described through
wind specification at hub height and corresponding share values. This file consists of eight columns. The
first column is used for storing time data. The second one contains information on wind velocity and the
third one on wind direction. This type of input file also allows the description of vertical wind component
which is the wind flowing parallel to the turbine rotor. The velocity values for this wind component are
stored in the fourth column. In addition to wind velocities at hub height, wind shear parameters can be
introduced. Both vertical and horizontal shear can be described. The horizontal wind shear is described
through the value equal to the ratio of the difference between the wind velocities at the blade tip for the
opposite sides of the rotor disc and hub height wind speed. This data is stored in the fifth column. Vertical
wind shear can be described using the following equation:
Vz = Vhub(
z
zhub
)V SHR
(A.1)
35
46. APPENDIX A. USING FAST FOR WIND TURBINE SIMULATION AND MODELING
Figure A.1: Typical form of the FAST output file
Where Vhub is wind speed at hub height, zhub is the hub height and Vz is wind velocity at the height z.
V SHR is the share coefficient that is stored in the sixth column. The vertical wind shear can also be
described using the same method for describing the horizontal wind shear. For this purpose, the seventh
column is used. However, when describing the vertical wind shear only one of these two description types
should be used and the column for the other method should be set to zero. The last column is used to describe
the gust wind which is a constant horizontal wind component across the entire rotor disc. Each row in the
wind file represents the data values for a particular time moment. The values used by FAST are obtained by
interpolating the values present in the input file. Figure A.2 shows a part of FAST input wind file.
Figure A.2: Typical FAST input wind file
Another option to describe the wind field is to use full-field wind data that represents all the three
components of the wind vector varying in space and time. These input files are in a binary format and must
be created by appropriate NREL program. Such a program is freely available from NREL web page. One
way to create such a file is to use measurements from an actual turbine sight and the other is to use one of
the programs for wind simulation.
36
47. APPENDIX A. USING FAST FOR WIND TURBINE SIMULATION AND MODELING
A.3 Automating FAST operation
One way to run FAST is just to type ¡fast primary input file name.fst¿ in Windows command line. When the
simulation is over a message is displayed in the command line and corresponding output text file is created.
This file has the same name as primary input file, but different extension. For further analysis of the results it
is often of interest to process the data obtained by Matlab or a similar program. However, due to the format
of the output file, it cannot be directly loaded by Matlab. The following code can be used to load FAST
output files by Matlab:
As a result, a matrix containing all the simulation output values is created in Matlab. In case when there is
a great number of output files that need to be processed, this code can simply be implemented in a for loop.
In this case Matlab becomes a powerful tool for reading and processing FAST output files.
Matlab can also be used to create FAST wind input files. The main advantage of using Matlab is that
wind profiles can be created very easily. The created profiles can then be transformed into wind input files
by using the following Matlab co
Using this code one can create wind descriptions of arbitrary shape and length in Matlab and then transform
this into wind input files. This becomes very useful when simulating turbine behavior under different wind
37
48. APPENDIX A. USING FAST FOR WIND TURBINE SIMULATION AND MODELING
conditions. Using Matlab enables very fast and efficient creation of the wind input files, whereas to create
the files manually would be almost an intractable task.
Interfacing Matlab and FAST provides a powerful environment for automated wind turbine simulation.
In such an environment FAST input files can be created with ease and the output files can be easily processed
with Matlab. In case FAST is run from Windows command line, batch files can be used to automate the
simulations. These are simple text files with the extension .bat which contains the list of all the primary input
files that need to be run with FAST. In cases where we would like to make a great number of simulations
in which we only change one simulation parameter, such as turbine rotational velocity or wind speed, we
can create an input file for each value of interest. Then a batch file can be used to run all these simulations
consecutively and eventually the results can then be processed by reading FAST output files with Matlab.
Such a way of using FAST enables great flexibility in turbine simulation, while at the same time it automates
the simulation process significantly.
A.4 Interfacing FAST with Simulink
Another way to connect Matlab and FAST is to interface the FAST with Matlab Simulink package. FAST
source files contain all that is necessary for this interface. Moreover, a blank Simulink schematic that can
be used for simulations is also provided. Figure A.bb shows this schematic This schematic should be used
Figure A.3: Simulink schematic used for interface with FAST
as a starting point for the creation of the desired control strategies. Since the control of variable wind speed,
pitch and yaw angle is possible, this interface is a good tool for simulating wind turbines in closed loop.
When running FAST from Simulink, the same input files are needed as in cases when FAST is run
from the command line. The user does not need to worry about Simulink simulation parameters as they
are automatically set to correspond to simulation length and time step specified in the primary input file.
In addition, after simulation a matrix containing all the specified simulation outputs is created in Matlab
workspace. However, the interface cannot be used for any combination of primary input file variables. For
example, Simulink can only be used when a time marching analysis is preformed, but not in case when the
38
49. APPENDIX A. USING FAST FOR WIND TURBINE SIMULATION AND MODELING
model is linearized. The following table lists all the settings that need to be made in the primary input file for
the interface with Simulink to work properly: Using described Matlab functions and interfacing Matlab with
Variable Value
AFAMSPrep 1
AnalMode 1
THSSBrDp >TMax
IPDefl 0
OoPDefl 0
TTDspFA 0
TTDspSS 0
Table A.8: Values that need to be set in primary input file for the integration with Simulink to work properly
Simulink a powerful simulation environment is created. In this environment wind turbine can be simulated
both in open loop and closed loop. In addition a great number of different simulation scenarios can be
created. What is more important, the results can easily be processed and analyzed in Matlab. Therefore such
an environment is ideal for wind turbine modeling.
39
51. Appendix B
Numerical values of model coefficients
This chapter gives numerical values of all the coefficients that are used in the wind turbine model proposed
here. The following tables list these numerical values. All the values are given for the centered and scaled
turbine rotational velocity values in the interval 10 − 26rpm.
1. mean torque
Tmean = (c11d3
v + c12d2
v + c13dv + c14)2
ω2
r + (c22dv + c23)ωr + c32dv + c33 (B.1)
c11 c12 c13 c14 c22 c23 c32 c33
-0.0758 -0.3008 -0.6285 14.5707 19.3080 173.2510 96.2296 477.3400
Table B.1: Coefficients used to fit mean torque values
2. 3p harmonic
a1 = we1
7
k=1
pa
ikω7−k
r + we2
7
k=1
pa
jkω7−k
r (B.2)
ϕ1 = we1
7
k=1
pϕ
ikω7−k
r + we2
7
k=1
pϕ
jkω7−k
r (B.3)
41
52. APPENDIX B. NUMERICAL VALUES OF MODEL COEFFICIENTS
sim. line pa
i1 pa
i2 pa
i3 pa
i4 pa
i5 pa
i6 pa
i7
MPPT-1 0 0.0001 0.0003 0.0010 0.0029 0.0184 0.0434
MPPT-0.8 0.0025 -0.0013 -0.0107 0.0033 0.0110 0.0167 0.0464
MPPT-0.6 -0.0049 -0.0018 0.0227 0.0064 -0.0263 0.0145 0.0561
MPPT-0.4 0.0047 0.0060 -0.0173 -0.0247 0.0107 0.0384 0.0558
MPPT-0.2 -0.0043 -0.0069 0.0087 0.0186 0.0057 0.0164 0.0552
MPPT -0.0015 -0.0001 0.0086 0.0050 -0.0083 0.0155 0.0613
MPPT+0.2 0.0019 0.0002 -0.0060 0.0002 0.0009 0.0201 0.0653
MPPT+0.4 0.0006 -0.0033 -0.0010 0.0094 -0.0050 0.0185 0.0704
MPPT+0.6 0.0018 -0.0014 -0.0035 0.0017 -0.0047 0.0267 0.0741
MPPT+0.8 -0.0027 -0.0033 0.0138 0.0055 -0.0194 0.0275 0.0781
MPPT+1 -0.0089 -0.0017 0.0399 0.0009 -0.0449 0.0307 0.0837
Table B.2: Coefficients used to fit the amplitude of the 3p harmonic
sim. line pϕ
i1 pϕ
i2 pϕ
i3 pϕ
i4 pϕ
i5 pϕ
i6 pϕ
i7
MPPT-1 0.3311 2.3508 -3.2778 -6.3268 15.8946 -6.0411 -18.5689
MPPT-0.8 -0.3732 2.1879 -3.3027 -5.8951 16.3637 -4.3887 -16.9424
MPPT-0.6 -1.0010 0.5035 -6.9930 -2.5940 19.3005 -3.8189 -32.9245
MPPT-0.4 -0.2722 2.1485 -3.9710 -6.0057 18.2634 -1.0159 -14.9457
MPPT-0.2 -0.5617 1.4486 -6.8888 -4.2005 21.0828 -0.6098 -20.2387
MPPT 0.0479 2.0817 -5.5458 -6.0663 20.9100 2.1787 -12.6463
MPPT+0.2 0.1769 1.8833 -6.0801 -5.6014 21.9272 3.6228 -12.0600
MPPT+0.4 0.2917 1.8580 -6.5138 -5.9216 23.0056 5.3368 -12.1544
MPPT+0.6 0.5119 1.2084 -6.3582 -5.4820 23.0421 7.3415 -25.0027
MPPT+0.8 -1.0153 1.2072 -8.9209 -5.1903 26.5390 8.1967 -20.6041
MPPT+1 -0.5908 2.0253 -7.8155 -7.6461 26.7712 10.4958 -18.8779
Table B.3: Coefficients used to fit the phase of the 3p harmonic
3. Higher harmonics
aq = pq
pol(ωr) + dv
9
r=1
csa
qrω9−k
r (B.4)
ϕq =
9
u=1
cϕ
qu + dv
9
r=1
csϕ
qr ω9−k
r (B.5)
42
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46
