final report niall madden 11132175
TRANSCRIPT
NAME: N. MADDEN
I.D: 11132175
SUPERVISOR: DR. PAT WALSH
COURSE: B.Sc. ENERGY
PROJECT TITLE: OPTIMISATION OF WIND FARM
LAYOUTS WITH CONSIDERATION TO
WAKE INTERACTION
DATE: March 2015
i
Declaration
I declare that this dissertation hereby submitted to the department of Physics and
Energy in the University of Limerick for the partial fulfilment of the degree of
Bachelor of Science in Energy has not been submitted to this or any other University
by me or any other person. I further declare that it is my entirely own work
throughout except where reference is made in the text.
_____________________________
Niall Madden 27th
March 2015
ii
Acknowledgements
There are a number of people who I would like to thank for making this dissertation
possible, my supervisor, Dr. Pat Walsh for his help and patience over the course of the
academic year, the deli staff in spar for keeping me nourished and fuelled, the staff in
the universities main building for cleaning around me over the last seven months and
finally my parents for both their financial and emotional support throughout the year.
iii
Abstract
Wind Energy is the largest growing source of renewable energy in the world. Large
amounts of wind farms are being constructed across the world each year and in order
for these farms to achieve a high annual energy production and maximize profit an
appropriate wind turbine layout is essential. In this dissertation the optimisation of
wind farm layouts with consideration to turbine wake interaction is investigated by
varying wind farm layouts in order to find the minimum cost per kilowatt hour of
energy produced. A model was created using matlab to imitate the wake effects
downstream of a turbine. The model was verified using measured data from Horns
Rev wind farm and implemented for over 1500 different layouts with the optimum
array found to be a staggered layout with a spacing of 3 diameters crosswind and 10
diameters downstream. The LPC for this configuration was found to be .0901€/kWh.
iv
Nomenclature
Symbol Definition Unit
Axial induction factor -
A Area
Area of Wake
AEP Annual Energy Produced kWh
Power Coefficient -
Thrust Coefficient -
Total cost €
CuP Cost of Copper €
CuRo Density of Copper Kg/
D Turbine Diameter m
I Turbulence Intensity -
k Wake decay constant -
Wake growth rate -
LS Length of string m
Length of connecting cable m
LPC Levelised production cost €/kWh
Number of Turbines -
Free stream velocity m/s
U Velocity at hub height m/s
x Distance downstream m
Roughness length m
Hub height m
v
Table of Contents
Declaration ...................................................................................................................... i
Acknowledgements ........................................................................................................ ii
Abstract ........................................................................................................................ iii
Nomenclature ................................................................................................................ iv
Table of Contents ........................................................................................................... v
Table of Figures ......................................................................................................... viii
Chapter 1: Introduction .............................................................................................. 1
Chapter 2: Literature Review..................................................................................... 7
2.1 Characteristics of Turbines................................................................................... 7
2.1.1 Thrust coefficient ........................................................................................... 7
2.1.2 Power Curve .................................................................................................. 9
2.2 Wake models ................................................................................................. 10
2.3 Near and Far wake.............................................................................................. 13
2.4 Objective Functions for Optimization: .......................................................... 14
2.5 Multiple wakes ................................................................................................... 17
2.6 Partial effect of wakes ........................................................................................ 19
2.7 Wind Shear. ........................................................................................................ 20
2.8 Turbulence Intensity ........................................................................................... 21
2.9 Horns Rev data ................................................................................................... 22
Chapter 3: Experimental .......................................................................................... 24
3.1 Jensen wake model ............................................................................................. 24
vi
3.2 Bastankhah wake model ..................................................................................... 26
Chapter 4: Results, Analysis and Discussion .......................................................... 29
4.1 Introduction ........................................................................................................ 29
4.2 Development of the model: ................................................................................ 29
4.2.1 Define Site and Turbine Parametres ............................................................ 30
4.2.2 Turbine Layout ............................................................................................ 31
4.2.3 Thrust Coefficient ........................................................................................ 31
4.2.4 Wake Speed Deficit ..................................................................................... 32
4.2.5 Wind Speed at Turbine ................................................................................ 34
4.2.6 Turbine Power Curve ................................................................................... 35
4.3 Validation of Model ........................................................................................... 36
4.4 Rayleigh Distribution ......................................................................................... 42
4.5 Cost Model. ........................................................................................................ 43
4.6 Rotation of Model .............................................................................................. 44
4.7 Randomly Generated Layouts ............................................................................ 47
4.8 Optimisation Using Rayleigh ............................................................................. 49
4.10 Optimisation Using Rotation ............................................................................ 52
4.10.1 Non Staggered ........................................................................................... 52
4.10.2 Staggered ................................................................................................... 55
4.10.3 Random ...................................................................................................... 57
Chapter 5: Conclusions ................................................................................................ 60
vii
Bibliography ................................................................................................................ 61
Appendix A .................................................................................................................. 64
Appendix B .................................................................................................................. 66
Appendix C .................................................................................................................. 68
Appendix D .................................................................................................................. 72
viii
Table of Figures
Figure 1: Wake effects at horns rev wind farm (tudelft, 2007) ................................................................ 1
Figure 2: Cost of energy as a function of turbine separation distance (Guillen 2010) ............................. 3
Figure 3: Layout of the Middelgrunden offshore wind farm: (a) actual, (b) optimized with symmetrical
constraints, (c) optimized. (González, et al., 2014) ........................................................................ 4
Figure 4: vertical profiles of mean velocity (above) and velocity deficit (bottom) assuming (a) top hat
model and (b) Gaussian model (Bastankhah & Porte-Agel, 2014) ................................................. 6
Figure 5: Thrust coefficient curve for Vestas V80 Turbine and fitted equation, equation provide by
(Frohboese & Schmuck, 2010) ....................................................................................................... 9
Figure 6: Park model used by Jensen, (Manwell, 2009) ......................................................................... 11
Figure 7: Prediction of wake behaviour compared to wind tunnel measurements (Bastankhah & Porte-
Agel, 2014) ................................................................................................................................... 12
Figure 8: Vertical profiles of velocity deficit at different distance downstream with varying surface
roughness (Bastankhah & Porte-Agel, 2014). .............................................................................. 13
Figure 9: Transition between near and far wake, (Bloomhoff, 2012) .................................................... 14
Figure 10: Mosetti's total cost function taken for a 4 million euro Turbine ........................................... 15
Figure 11: Multiple interaction of wakes F. González et al (2011) ....................................................... 18
Figure 12: Validation of Jensens Model for Multiple Wakes ................................................................. 19
Figure 13: Plan and elevation view of the partial effect of wakes F. González et al (2011) .................. 20
Figure 14: Velocity ratio at different distances downstream of a turbine , (Manwell, 2009) ................. 21
Figure 15: Wind velocity due to wind shear at different heights with varying Roughness Length Zo .. 21
Figure 16: Power Curve/ Power Coefficient/ Thrust coefficient curve for Vestas V80 wind turbine ... 23
Figure 17: Jensen's wake model implemented using excel. ................................................................... 24
Figure 18: Velocity deficit at hub height for Jensen model .................................................................... 25
Figure 19: Velocity deficit at hub height for various distances downstream for Jensen Model. ............ 25
Figure 20: Jensen’s model used to investigate the interaction of multiple wakes .................................. 26
Figure 21: Bastankhah et al wake model modelled using Excel. .......................................................... 27
Figure 22: Velocity deficit at hub height for various distances downstream of a turbine using
Bastankhah model ......................................................................................................................... 28
ix
Figure 23 Velocity deficit at hub height for different distances offset from centre at various distances
downstream ................................................................................................................................... 28
Figure 24:Matlab Flow chart .................................................................................................................. 30
Figure 25: Thrust coefficient values passing through a row of turbines ................................................. 32
Figure 26 Single Turbine matrix ............................................................................................................ 33
Figure 27: Horns Rev layout Jensen Model ........................................................................................... 34
Figure 28: Horns Rev layout Bastankhah Model ................................................................................... 34
Figure 29: Power Curve and Equation for Vestas V80 .......................................................................... 36
Figure 30: Relative power from Horns Rev Wind Farm Garrad Hassan (2010) .................................... 37
Figure 31 Layout of Horns Rev Wind Farm (Hassan, 2010).................................................................. 38
Figure 32: Validation of models, Column 4, wind approaching from 240 degrees ................................ 38
Figure 33: Validation of model Column 10............................................................................................ 39
Figure 34: Validation of model column 4 for varying k Values ............................................................. 40
Figure 35: Validation of Model Column 10 for varying k values .......................................................... 41
Figure 36: Validation of model for wind approaching from 270 degrees, varying k values .................. 42
Figure 37: Method for finding TProt ...................................................................................................... 44
Figure 38: Rotation of model in increments of 30 degrees..................................................................... 46
Figure 39: Sample wind rose .................................................................................................................. 46
Figure 40: Position matrices and grid layout for random placement ...................................................... 48
Figure 41: Optimum separation distance downstream Rayleigh distribution ......................................... 49
Figure 42: AEP & Cost Vs Seperation Distance downstream Rayleigh distribution ............................. 50
Figure 43: Optimum separation distance Rayleigh distribution ............................................................. 51
Figure 44: Optimum array layout for non-staggered rotational model ................................................... 53
Figure 45: Array efficiency for optimum non –staggered rotation ......................................................... 54
Figure 46: Array efficiency for worst performing non–staggered rotation ............................................ 54
Figure 47: Optimum array layout rotation staggered ............................................................................. 56
Figure 48: Array efficiency for optimum array layout rotation staggered .............................................. 57
Figure 49: Best performing turbine layout for random generation ......................................................... 59
Figure 50: Sample array efficiency for random array ............................................................................ 59
x
List of Tables:
Table 2-1: Cable details (Guillen, 2010) ................................................................................................ 16
Table 4-1: Optimum separation distance Rayleigh distribution ............................................................. 50
Table 4-2: Optimum separation distances rotation non staggered .......................................................... 52
Table 4-3: Optimum separation distances rotation staggered ................................................................. 55
Table 4-4: LPC values for varying number of turbines, Random........................................................... 58
1
Chapter 1: Introduction
This dissertation looks at the problems involving the interaction of wakes when
deciding on the optimal layout for wind farms. As the number of wind farms being
constructed world-wide gets larger and larger each year, the wakes of wind turbines
and how they affect the output performance of the farm is a very important topic of
study. An adequate layout is paramount to ensure sufficient performance from the
farm throughout its life span. In a wind farm, turbines which are downwind from
other turbines are exposed to lower wind speeds due to the disturbances in the wake
and higher turbulence levels. This can be noticed in fig1 where each turbine in the
foreground is affecting several turbines downwind. If the wake effects of turbines are
not taken into consideration during construction of wind farms large calculation errors
may occur while predicting the wind farm yield. In turn this could lead to a
significant loss in anticipated profits.
Figure 1: Wake effects at horns rev wind farm (tudelft, 2007)
2
Currently there are several different commercially available software used for
modelling wind farms and there wake effects. Examples of these include WindFarmer
and WindPRO which process site and turbine information and produce a wind farm
that maximizes energy consumption and account for construction issues and
restrictions.
Wind turbine wakes don’t become negligible until approximately 15 to 20 diameters
(of turbine blades swept area) downstream of a turbine. However the space between
turbines is usually far smaller than this as other factors come into play. These factors
include the cost of land, cost of installation (cables/logistics), land or sea area
available, favourable wind conditions in areas of site, connection to electrical grid and
environmental issues. In fig 2 the cost of energy was plotted against turbine separation
distance by (Guillen, 2010). While the turbines are spaced close together the wake
effects are high and as a result of this so too is the cost of energy and when the
turbines are spaced far apart the cost of land and installation become too high which
also drives up the cost of energy. The optimum point can be found marked in the
middle of the graph where the cost of energy is at a minimum. In large farms the
wakes of turbines can cause a considerable effect on the annual energy produced.
Previous studies suggest that wake losses could range from 5 to 15% (Barthelmie, et
al., 2004), bearing this in mind it is necessary to minimize the wake effects in order to
maximize the energy produced.
Discovering new high quality methods of optimising wind farm layouts could
effectively give high profits for wind farm developers.
3
Figure 2: Cost of energy as a function of turbine separation distance (Guillen 2010)
In the past little thought was put into optimising the layout of wind farms with regard
to the wake interaction as few studies had been done on the effect of wakes. This led
to many turbines being placed in square or conveniently shaped grids to minimise
installation costs. Recently however many papers have shown that irregular layouts of
turbines are often a better solution in order to maximise the energy produced by the
farm. This can be seen clearly in fig 3 below where the actual layout of the wind farm
is in a conveniently shaped grid modelled around the prevailing direction on Wind
Energy Rose for the site. The problem that accompanies this however is that when the
wind blows from the north or the south the wake effects on the array will be
substantial. According to (Nuebert, et al., 2009) this wind farms Annual Energy
Production (AEP) could be increased by 5% and 6% for the layouts seen in (b) and (c)
respectfully. The layout in (b) may still prove to be more profitable however because
the installation costs of (c) would be significantly larger due to the irregular layout.
The lifetime costs of the wind farm would have to be taken into account to figure this
out.
4
Figure 3: Layout of the Middelgrunden offshore wind farm: (a) actual, (b) optimized with symmetrical
constraints, (c) optimized. (González, et al., 2014)
Various parameters of wind turbines effect the way the wake downstream of the
turbine behaves. Throughout this project various relationships will be reviewed
between these parameters and how they affect the wake.
The thrust coefficient ( is the proportion of available thrust extracted from the
wind when wind passes through the blades of a turbine. The lower the thrust
coefficient the lower the amount of energy extracted from the wind, which means the
lower the wind speed deficit between the free stream velocity and the wake of the
turbine.
The Power curve of a wind turbine varies for different types of turbines and shows the
power produced for any given wind speed, it also takes the cut in/out speeds into
account which determine the speeds at which the turbine blades begin spinning and
stop spinning respectively. The hub height of a turbine also affects the wake as
different wind speeds occur at different heights due to the wind shear profile caused
by the roughness of the terrain upwind of the turbine.
5
Wind speeds begin to recover immediately after passing through the turbine. Wake
recovery depends on different factors such as turbulence intensity along with
parameters of the turbine which include the thrust coefficient and the hub height. For
wind farms, factors such as wind farm size and layout come into play as the
interaction of wakes slows down wake recovery while producing turbulence in the
wind which also would affect the output and reliability of a downwind turbine.
Different wake models have been proposed which describe the behaviour of the wake
for given distances downstream of the wind turbines. These models fall into different
categories including Top hat models, Gaussian models, Empirical models from
measured data and Navier stokes models. This project will not focus on Navier stokes
models as there are very high computational costs involved which are out of scope of
this project. Top hat models are often used due to low computational cost while still
giving relatively accurate results once the wake is fully developed (approximately 2
diameters downstream). In these models velocity is a constant at a given distance
downstream which can be seen in fig 4(a). This causes the velocity deficit to be
underestimated in the core region of the wake and overestimated towards the edges of
the wake. The most popular model is a top hat model which was proposed (Jensen,
1983). This model will be referred to as the Jensen model for the rest of the study.
This was one of the first wake models proposed, it assumes a linearly expanding wake
and was derived using the laws of conservation of momentum. The Jensen model
owes its popularity to its low computational cost while still giving quite an accurate
result for a given distance downstream of the wake. Gaussian Models give a more in
depth physical representation than top hat models as wind speed deficit tends to
follow a Gaussian curve at a given distance downstream with the wind speed deficit at
the core of the wake greater than that towards the edges as seen in fig 4(b).
6
Figure 4: vertical profiles of mean velocity (above) and velocity deficit (bottom) assuming (a) top hat model
and (b) Gaussian model (Bastankhah & Porte-Agel, 2014)
For any optimisation problem an objective function must be chosen. The term
objective function refers to the parameter you have chosen to maximize or minimise.
In terms of Wind farm layouts several different objective functions can be considered.
Popular choices include maximizing energy produced, maximizing profit, and
maximising AEP/cost. Different objective functions are taken into consideration
throughout this project as any could be implemented depending on the desires of the
wind farm developer.
The objectives for this project are to investigate the theory behind the behaviour and
formation of wakes to, develop a model using matlab to imitate wake behaviour in a
wind farm and to, analyse a range of wind turbine layouts and find the optimum array
which minimises cost while producing a large amount of energy.
7
Chapter 2: Literature Review
The study of wakes from wind turbines has become a very intensively researched
area. In order to get a good understanding of how wakes effect the performance of a
wind farm it is vital to first review and compare different papers which have been
published on the topic.
2.1 Characteristics of Turbines
2.1.1 Thrust coefficient
The larger the thrust coefficient the larger the wind speed deficit between the wake
and the free stream velocity, this implies that turbines downstream will be more
heavily affected. The thrust coefficient of turbines begins to decrease as the turbine
begins to reach its nominal speed as can be seen in fig 5 meaning that when the wind
surpasses the nominal speed of the turbine the wind farms efficiency will go up due to
the thrust co-efficient reducing. This is due to the reduction in the turbines power
coefficient (the percentage of wind energy extracted from the wind). The power
coefficient is reduced as the energy in the wind is proportional to so the energy in
the wind continues to rise rapidly while the energy extracted stays at a constant
driving the thrust coefficient towards 0. Both the thrust coefficient and the power
coefficient can be related using the axial induction factor (the fractional decrease in
wind speed from the free stream velocity to right after the turbine) using the following
equations.
Equation 2-1
8
Equation 2-2
Giving the following:
Equation 2-3
This is useful as most manufacturer manuals only provide Power Coefficient graphs.
The axial induction factor is often taken as a constant of 0.33 which gives a thrust
coefficient of 0.89. This approach is taken by most authors including (Grady, et al.,
2005) , (Mosetti, et al., 1994) and (Kusiak & Song, 2010). However taking the thrust
coefficient as a constant can lead to noticeable errors in calculations. Few papers have
used varying thrust coefficients which are determined by different wind speeds and
data provided by the manufacturer.
(Frohboese & Schmuck, 2010) used the following:
Equation 2-4
This equation gives the following curve which is plotted alongside the thrust
coefficient curve for a Vestas V80 wind turbine. A more gradual decrease in the thrust
coefficient can be seen using equation 2.4 compared to the measured curve for the
Vestas V80 above where a steep drop can be seen as the blades begin to change pitch
at approximately 13m/s. An equation was fitted to the Vestas V80 thrust coefficient
curve
9
Figure 5: Thrust coefficient curve for Vestas V80 Turbine and fitted equation, equation provide by
(Frohboese & Schmuck, 2010)
The equation fitted to the vestas V80 Curve was given as:
Equation 2-5
For velocities below 12m/s the coefficient was taken as a constant of .805
2.1.2 Power Curve
Power curves show the amount of power generated by the selected turbine at any
given wind speed, (Mosetti, et al., 1994), (Kusiak & Song, 2010) and (Grady, et al.,
2005)) all use theoretical curves which are found by using betz’ laws and
conservation of momentum. (Serrano-González, et al., 2010) used an experimental
power curve. For this project a power curve from a turbine manual provided by
manufacturers will be used.
0
0.2
0.4
0.6
0.8
1
1.2
4 5 6 7 8 9 10111213141516171819202122232425
Ct
Wind speed (m/s)
Thrust-Coefficient Curve
Frohboese & Schmuck
Vestas V80 Equation
Vestas V80
10
2.2 Wake models
Many different wake models have been proposed to describe the behaviour of a wake
at a given distance downstream of a turbine. The most popular wake model is a
model proposed by (Jensen, 1983) which is used by (Mosetti, et al., 1994), (Kusiak &
Song, 2010) and (Grady, et al., 2005) this model assumes that the wake expands
linearly. It is derived by conservation of momentum downstream and the velocity
deficit depends on the distance behind the rotor the thrust coefficient, the wake decay
constant and the diameter of the turbine. The velocity in the wake for a given distance
x downstream is given as:
[ √
( )
]
Equation 2-6
Where is the wake decay constant which is determined by how turbulent the
surrounding air is. Small k values tend to go hand in hand with small roughness
lengths as the air is not as turbulent as areas with large roughness lengths. Larger k
values produce wider wakes with the velocity recovering quicker downstream as the
turbulent air in the wake mixes with the surrounding air. For simplicity a value of
0.75 is widely used for onshore wind while a value between 0.04 and 0.05 is used for
offshore farms (Hassan, 2010). The value can also be calculated using the following
equation.
(
)
Equation 2-7
11
Figure 6: Park model used by Jensen, (Manwell, 2009)
(Frandsen, 1992) proposed a top hat model similar to the Jensen model which predicts
the velocity at a given distance downstream as a constant for all radial positions
(Hassan, 2010). This model tends to underestimate the effects of the wake at any
given distance downstream. The velocity downstream is given as:
[ √
]
Equation 2-8
(Bastankhah & Porte-Agel, 2014) proposed a new wake model in 2014. This model
falls into the Gaussian category and gives a more in depth physical representation
than the previously mentioned models proposed by Jensen and Frandsen. The model
was derived in Appendix A. The wind speed deficit tends to follow a Gaussian curve
at a given distance downstream and is given by:
12
∆
(
√
8 ( √𝛽)
)
× exp
(
( 𝑑
√𝛽) {(
𝑧 𝑧
)
(𝑦
)
}
)
Equation 2-9
Where: 𝛽 √
√
Equation 2-10
𝑦 is taken as the span wise co-ordinate, 𝑧 the vertical co-ordinate, 𝑧 the hub height
of the turbine. This model was the compared to wind tunnel measurements by
(Bastankhah & Porte-Agel, 2014) in fig 7 and proved to be very accurate for the given
conditions. This figure shows the model with the wind shear taken into account
whereas the equation above produces symmetrical Gaussian curves.
Figure 7: Prediction of wake behaviour compared to wind tunnel measurements (Bastankhah & Porte-Agel,
2014)
13
In fig 8 it can be seen that the model was then compared to Jensen and Frandsen
models proving to give more accurate velocity profiles downstream of the turbines
when all three were compared to data from a Large Eddy Simulation.
Figure 8: Vertical profiles of velocity deficit at different distance downstream with varying surface
roughness (Bastankhah & Porte-Agel, 2014).
For all situations shown above the model proposed by (Bastankhah & Porte-Agel,
2014) delivers a good quality prediction on the behaviour of the wake. The model
appears to show it superiority over the top hat models it is compared against.
2.3 Near and Far wake
Wakes are divided into different parts, the near wake, an area between the turbine and
approximately 1-2 diameters downstream, where the pressure is lower than that of the
ambient and the influence of each blade of the turbine on the wake can be
14
distinguished and the far wake where pressure has materially returned to ambient and
the effect of each individual turbine blade cannot be determined. The model proposed
by (Bastankhah & Porte-Agel, 2014) does not model near wake conditions while the
top hat models proposed by (Jensen, 1983) and (Frandsen, 1992) do include this
region but do not include any differences between the near and far wake regions. For
this project it will be acceptable to use models which don’t include the near wake
region as in all wind farms turbines will be spaced out further than the near wake area
due to general requirements given by the manufacturers of the turbines. An example
of how the wake changes from the near to far region can be seen in fig 9 below.
Figure 9: Transition between near and far wake, (Bloomhoff, 2012)
2.4 Objective Functions for Optimization:
Many different objective functions have been proposed to optimise the layout of wind
farms. Most objective functions take the total cost of the wind farm into account a
popular means of defining this was proposed by (Mosetti, et al., 1994) which takes
into account the fact that the cost of installation will go down per turbine when the
number of wind turbines installed increases as can be seen in fig 10 below. This cost
model is given as:
15
× (
× ×
)
Equation 2-11
Where is the number of turbines installed in the given wind farm.
Figure 10: Mosetti's total cost function taken for a 4 million euro Turbine
The slope of this model can be seen to reduce as more turbines are purchased this
means the more turbines bought the cheaper each one will cost. However the slope
can be seen to increase slightly after approximately 30 turbines but then continues on
as a constant.
(U Aytun Ozturk, 2004) chose maximum profit as the objective function to be used in
there study. Where is the selling price of energy and is as defined above.
×
Equation 2-12
(Grady, et al., 2005) again used mosetti’s cost model, minimizing Cost/AEP:
0
20
40
60
80
100
120
140
160
0 20 40 60 80 100 120
Co
st(m
€)
No. of Turbines
Total Cost Vs No. Of Turbines
Total Cost
16
Equation 2-13
In order to account for the increase in price alongside the increase of land/cable usage
for the optimisation process a suitable cost model is needed. (Guillen, 2010) have put
forward a model which takes the amount of cables used into consideration. The model
is a Levelised Production Cost (LPC) model, which means the constant price per unit
of energy that causes the investment to just break even.
((
8 ) )
×
Equation 2-14
Where:
Equation 2-15
The annuity factor is used to calculate the present value of the lifetime costs. r is the
interest rate (taken as 7%) and T is the lifetime of the farm taken to be 25 years.
Table 2-1 Cable details (Guillen, 2010)
Copper Density (CuRo) 8940
Copper Price (CuP) €/tonne 2782
Cross Sectional Area 1000
17
The Cost of copper is given by:
× × × × ×
Equation 2-16
Where profit is the manufacturer profit per cable as % of copper cost, this was kept at
a constant of 200% by (Guillen, 2010).
×
Equation 2-17
Mosetti’s cost function was added to the equation along with the cost of land taken as
an estimated 500000€/km.
((
8 ) 𝑑 )
×
Equation 2-18
2.5 Multiple wakes
(González-Longatt, et al., 2012) states that in a wind farm every wake that interacts
with a downstream turbine will have an effect on the performance of said turbine. As
can be seen in fig 11 the output of turbine j is affected by turbines 1, 2, 3 and i.
However some papers including (Hassan, 2011) say that a downstream turbine is
mainly affected by the turbine that is closest upstream of it and that the effect of the
other wakes interacting with the turbine are negligible. In relation to fig 11 this would
mean that the output of turbine j is only affected by the wake of turbine i and the
wakes from turbines 1, 2 and 3 can be neglected. Garrad Hassan’s model was
compared to actual data from offshore wind farms and proved to be very accurate. A
paper compiled by (Samorani, 2013) backs up Garrad Hassan’s model stating that the
18
presence of other upwind turbines (apart from the closest) does not have a substantial
effect on the downstream turbine. However (Jensen, et al., 1986) proposed a method
for the summation of multiple wakes using the sum of squares of velocity deficits as
shown below.
(
)
∑(
)
Equation 2-19
Figure 11: Multiple interaction of wakes F. González et al (2011)
The two methods were tested against one another and compared to measured data in
the fig 12. It can be seen that both models are quite accurate however the single wake
models speed deficit remains constant after the second column. In the measured data
and the multiple wake data a slight decrease in speed can be seen after each column
showing the Jensen model is more accurate and will be used for this project.
19
Figure 12: Validation of Jensens Model for Multiple Wakes
2.6 Partial effect of wakes
The partial effect of a wake refers to when a turbine downstream is not fully
immersed in the wake of an upstream turbine but is still affected. Whether or not the
effect of multiple wakes is to be taken into account the partial effect of wakes can be
an important part of the analysis as depending on wind direction a large number of
turbines may be affected by a small portion of their swept area being overlapped by
the wake of an upwind turbine. The effect of a partial wake is taken into account in
order to not overestimate the wake effect. However few papers take this effect into
consideration. (González-Longatt, et al., 2012) used the following equation to
describe the partial effect on a downwind turbine.
[ ( )] (
( ))
(𝑑
( )) 𝑑 𝑧
Equation 2-20
With the velocity in turbine j given as follows:
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12
Vel
oci
ty d
efic
it
Column No
Multiple Wakes
Jensen Model
Single Wake Model
Measured Data
20
[ ∑ √
(
( ))
]
Equation 2-21
Figure 13: Plan and elevation view of the partial effect of wakes F. González et al (2011)
2.7 Wind Shear.
Wake models sometimes decipher the velocity deficit at each point in the flow field
by incorporating the wind shear into the calculations for each point on a field. In fig
14 a model proposed by Smith and Taylor (1991) is compared to experimental results
which showing the model to be very accurate.
21
Figure 14: Velocity ratio at different distances downstream of a turbine , (Manwell, 2009)
Fig 15 shows a plotted graph of the wind shear profile for varying roughness lengths.
As can be seen the larger the roughness length ( the higher the altitude it takes the
wind velocity to reach a free stream velocity.
Figure 15: Wind velocity due to wind shear at different heights with varying Roughness Length Zo
2.8 Turbulence Intensity
Turbulence intensity is taken into account by few papers. (Bastankhah & Porte-Agel,
2014) account for the effects of this to verify the accuracy of their model. It can be
0
50
100
150
200
0 5 10 15 20
He
igh
t (m
)
wind velocity (m/s)
Wind Shear Profile
Zo=0.5
Zo=.1
Zo=.02
22
used with the wind shear data to give more accurate readings of how the wake will
behave for the wind farms local area. The turbulence intensity is given by the
following (Thomsen & Sorsensen, 1999):
(𝑧𝑧
)
Equation 2-22
The wake decay constant can then be related to the turbulence intensity by (Thomsen
& Sorsensen, 1999)
Equation 2-23
2.9 Horns Rev data
Horns Rev is an offshore wind farm off the coast of Denmark. The wind farm began
operating in 2002 and covers an area of roughly 20km², it has a square grid layout
with a spacing of 560m between each turbine in each direction. There are 80 turbines
in the wind farm all of which are Vestas V80 2.0MW turbines. Horns Rev wind farm
has been used in many studies over the past decade to verify the accuracy of wind
turbine models including (Hassan, 2010) and (Wu & Porté-Agel, 2009). The Expected
annual output for the wind farm is 600GWh whereas the actual output each year is
noticeably smaller. These values can then be compared with values obtained when
considering the losses when using the models. Fig 16 shows a power curve of the
aforementioned turbines along with thrust and power co-efficient curves.
23
Figure 16: Power Curve/ Power Coefficient/ Thrust coefficient curve for Vestas V80 wind turbine
In order to find the k value for the site a roughness length for the site was needed.
According to (Ramli, et al., 2009) for open sea with a fetch greater than 5km a
roughness length of 0.0002 can be taken for the site.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
500
1000
1500
2000
2500
0 5 10 15 20 25
Cp
an
d C
t
Po
wer
(k
W)
Wind speed (m/s)
Vestas v80 Power/Power Coefficient/Thrust
Coefficient Curve
Power curve
Thrust coefficient
Power Coefficient
24
Chapter 3: Experimental
3.1 Jensen wake model
To begin Jensen’s wake model was implemented using Excel as it is the most
commonly used model. In this model which can be seen in fig 17 the velocity deficit
for a given distance downstream of a wind turbine is shown. The free stream velocity
(the outer blue region) has a velocity deficit of 0 whereas the deficit in the orange
region is between 0.2 and 0.4. This means the velocity in this region would be
between 60 and 80 per cent of the free stream velocity.
Figure 17: Jensen's wake model implemented using excel.
Fig 18 Shows the velocity deficit at hub height for any given distance downstream of
the turbine for Jensen’s model. The deficit initially falls at a relatively quick pace and
then begins to level off while still gradually heading towards zero. This velocity
deficit is the same all parts of the turbines swept area.
Dis
tan
ce A
cro
ss G
rid
Distance Down Grid
Jensen Wake Model
0.8-1
0.6-0.8
0.4-0.6
0.2-0.4
0-0.2
25
Figure 18: Velocity deficit at hub height for Jensen model
The velocity deficit at hub height is a constant value across the entire wake for any
given distance downstream. In fig 19 it can be seen that the velocity deficit drops to a
value of zero at the edges of the wake. In reality there would be a more gradual
reduction in the deficit towards the outer edges of the wake with a higher peak in the
centre.
Figure 19: Velocity deficit at hub height for various distances downstream for Jensen Model.
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
Ve
loci
ty d
efi
cit
Diametres downstream
Velocity Deficit at Hub Height
0
0.05
0.1
0.15
0.2
0.25
-10 -5 0 5 10
Ve
loci
ty D
efi
cit
Diametres Offset From Centre
Velocity Deficit At Hub Height Various Distances Downstream
15 Diametres
5 Diametres
9 Diametres
26
In fig 20 three turbines are spaced 2.5 diameters apart with a 1.5 diameter distance
downwind to make the graph clearer. As can be seen in areas where the wakes
intersect the velocity is lowered. The values of the areas of intersection are got by
finding the average velocity of each individual wake at that point. Jensen’s method for
the summation of multiple wakes was used for the model.
Figure 20: Jensen’s model used to investigate the interaction of multiple wakes
3.2 Bastankhah wake model
It can be seen in fig 21 that the velocity follows a Gaussian path downstream of the
turbine. In the below image the model is plotted using a thrust coefficient of 0.89 as
this is a far wake model the near wake is not shown in the fig 21 below. This can be
seen clearer in fig 23 where the wake can be seen to start at approximately 3 diameters
downstream.
Dis
tan
ce A
cro
ss G
rid
Distance Down Grid
Jensens Model For Multiple Wakes
80-100
60-80
40-60
20-40
0-20
27
Figure 21: Bastankhah et al wake model modelled using Excel.
Fig 22 shows the velocity deficit at hub height for various distances downstream. As
can be seen the deficit decreases the further downstream travelled while the area
affected is spread over a larger area. This is due to the turbulent mixing at the edges of
the wake which causes the wake to spread out and recover to free stream velocity. At
15 diameters downstream the maximum velocity deficit is given as .01 i.e. for a free
stream velocity of 10 m/s the velocity in line with the centre of the turbine at 15
diameters downstream would be 9m/s. However for measurements taken at an offset
of 2 diameters from centre the velocity the velocity deficit is very close to zero.
Compared to the Jensen model above there is a clear difference in the deficit as you
move towards the edges of the wake.
Dis
tan
ce A
cro
ss G
rid
Distance Down Grid
Bastankah Model
0.8-1
0.6-0.8
0.4-0.6
0.2-0.4
0-0.2
28
Figure 22: Velocity deficit at hub height for various distances downstream of a turbine using Bastankhah
model
Unlike the Jensen model the velocity is not taken as a constant for a given distance
downstream. The fig 23 below shows the varying velocity across the wake at hub
height for any given diameter downstream for the far wake region.
Figure 23 Velocity deficit at hub height for different distances offset from centre at various distances
downstream
0
0.1
0.2
0.3
0.4
0.5
0.6
-6 -4 -2 0 2 4 6 8
Ve
loci
ty D
efi
cit
Diametres from Centre
Velocity Deficit At Hub Height, Various Distances Downstream
5Diameters9Diameters15Diameters
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20
Ve
loci
ty D
efi
cit
Diametres Downstream
Velocity Deficit At Hub Height
0 Diameter
0.25Diameter
0.5 Diameter
29
Chapter 4: Results, Analysis and Discussion
4.1 Introduction
In order to model the theory discussed in the previous section a computer programme
was needed as calculations by hand would be far too time consuming. At the
beginning of the project Microsoft Excel was used due to its familiarity. This was
mainly to help visualise how the different models behave. However it became clear at
an early stage that Excel would not be sufficient to complete the calculations due to
iterative calculations and varying conditions which would need to be completed
manually. The programme chosen to be learnt which would be effective and efficient
for the projects calculations was matlab.
4.2 Development of the model:
In order to model a wind farm a flowchart was developed to use as a guide for
creating the matlab code. In this section a step by step explanation of the horns rev
model will be given.
30
Figure 24:Matlab Flow chart
4.2.1 Define Site and Turbine Parametres
To begin with a site or grid is needed. This is in the form of a matrix with the required
dimensions for grid width and grid depth. In the case of the horns rev model a matrix
of size 750 by 750 cells was used. This relates to an actual site width and depth of
7.5km so every cell in the matrix represented 10 metres by 10 metres or 100 . The
wind speed of the site at hub height was then defined. This would mean that if no
turbines were to be present in the grid the result would be a matrix of 750 by 750 with
each cell containing the specified wind speed. Next the turbine radius and diameter
are defined. When writing code in matlab is important to remember to link variables
31
together if they are dependent on one another. In this case the diameter will always be
twice the value of the radius and so should be written as two times the radius or
Dt=Rt*2 where Dt is the diameter and Rt is the radius. In this way if the radius value
is changed so too will the diameters value all the way through the code. Next the hub
height of the turbine and the roughness length on site are defined. These values are
then used to decipher the wake decay constant on site.
4.2.2 Turbine Layout
The Turbine positions were calculated by hand as the data used to validate the model
was recorded when the wind was blowing from an angle of 240 degrees off North at
the horns rev wind farm. For simplicity the wind was taken to be moving from left to
right on the grid in matlab at all times and the turbine positions vary in accordance.
The positions were then recorded in an excel workbook. To save time this workbook
was imported from excel using the “xlsread” function in matlab. This allows matrices
stored in Excel to be read and used in programmes in matlab.
4.2.3 Thrust Coefficient
The thrust coefficient is dependent on the wind speed at each point of the grid. For an
accurate representation of the thrust coefficient a matrix the same size as the original
grid is needed with a thrust coefficient for each cell on the grid. However the wind
speed deficit behind each turbine depends on the thrust coefficient so an iterative loop
is needed as the thrust coefficient will change the value of the wind speed behind the
first turbines which will then change the thrust coefficient for the next turbine in the
first turbines wake which will in turn change the speed deficit behind the next turbine
and the cycle continues until the last turbines. In order to achieve this a while loop
was created. A while loop is a loop that continues to re-evaluate its body while a
specific condition holds true. In this case the condition was while the change in thrust
32
coefficient was greater than × run the loop again using the new value. For the
loop to run through the first time the thrust coefficient was predefined using an initial
guess and once the loop has run through the value was changed using the equation
outline in the literature review. The fig 26 shows the changing value of the thrust
coefficient as it passes through a row of turbines.
Figure 25: Thrust coefficient values passing through a row of turbines
4.2.4 Wake Speed Deficit
Once the thrust coefficient and the turbine positions are defined the wake speed
deficit matrix can be established as these are the only two variables that are needed
for the wake models all other parameters are kept constant. The wake model matrix is
put inside the thrust coefficient while loop so that it updates with each iteration also.
A three dimensional matrix was created with a sheet for each turbine on the grid in a
for statement. A for statement loops a specified number of times and keeps track of
33
each iteration with a coinciding index variable. The wake model was then applied to
each sheet for the position of each separate turbine. As can be seen in the below
Image.
Figure 26 Single Turbine matrix
After the for loop all the separate sheets are combined using equation 2.19 to create a
single two dimensional matrix called ‘wakes’ containing the wake speed deficit of all
80 turbines in the farm which can be seen below. As the model proposed by
(Bastankhah, 2014) does not model the near wake and therefore no change to the grid
would be seen until approximately 2 diameters downwind the Jensen model was
implemented in this area to clearly show the turbine positions. This has no effect on
the output of the matlab model as no turbines are positioned in the near wake region.
34
Figure 27: Horns Rev layout Jensen Model
Figure 28: Horns Rev layout Bastankhah Model
4.2.5 Wind Speed at Turbine
The wind speed at each of the turbines to be used to calculate power output was got
by taking the average wind speed in front of the turbine. This was done using a for
loop which created an 80 by 8 matrix (Number of turbines by diameter). Where each
35
row in the matrix represents the wind speed across the face of a turbine. The average
of each of these rows was then got by using the mean function on matlab.
4.2.6 Turbine Power Curve
To analyse the power output of the farm the power curve of the Vestas V80 was used.
Before the wind speed reaches 4 and when it exceeds 25 the power output
of the turbines is 0 and from 14-25 the turbine is at nominal power or 2000kW.
In between 4 and 14 the power output curves upwards this can be seen in the
following graph. A sixth order polynomial trend line with the following equation was
fitted to the curve in excel
8
8
Equation 4-1
This equation was then applied when the wind speed was between these values on
matlab using if and else if statements in a for loop. For each turbine if the speed is in
between 4 and 14 the aforementioned equation is applied or else if the wind
speed at the turbine position is in between 14-25 and if the
windspeed is below 4 or above 25 .
36
Figure 29: Power Curve and Equation for Vestas V80
A matrix containing the for each of the turbines was created and then this value
for each turbine was divided by the from the first turbine in the grid which was
unaffected by other turbines to get the relative power output. These output powers
were plotted in a graph for the validation of the model.
4.3 Validation of Model
The model created in matlab was validated using measured data from Horns Rev wind
farm provided by (Hassan, 2010). The data was taken from the below graph using plot
digitizer. The fourth and tenth rows of the farm were analysed to check the accuracy
of the model through the full length of the farm. The code used for the verification of
the model can be found in Appendix B.
y = 0.0311x6 - 1.5609x5 + 30.92x4 - 310.81x3 + 1702.7x2 - 4767.6x + 5340.8
0
500
1000
1500
2000
2500
0 5 10 15
Po
we
r (k
W)
Wind Speed (m/s)
Vestas V80 Power Curve
Vestas V80
37
Figure 30: Relative power from Horns Rev Wind Farm Garrad Hassan (2010)
The two different wake models being analysed were tested against the measured data.
The conditions under which the data was measured was recreated. The wind speed
was kept as a constant at 9 and the turbine positions were rotated so that the wind
hit the farm at an angle of 240 from North. A roughness length of 0.0002 was taken
for the site as the farm is in open sea with a fetch greater than 5km (Ramli, et al.,
2009). Using Equation 2.7 this relates to k value of 0.04 for the Jensen Model and
from (Bastankhah & Porte-Agel, 2014) a k* value of 0.03. Horns rev wind farm is
compiled of 80 Vestas V80 2MW turbines which are arranged in 8 rows and 10
columns as can be seen in the figure below. Each turbine is spaced 7 diameters
downwind and across from the last however there is a slight angle so that it not a
perfectly square grid. Exact dimensions were taken from (tudelft, 2007)
38
Figure 31 Layout of Horns Rev Wind Farm (Hassan, 2010)
Both models used in this study were plotted against the measured data and two
models used in (Hassan, 2010) for columns 4 and 10 which can be seen marked C4
and C10 respectively in fig 31 above.
Figure 32: Validation of models, Column 4, wind approaching from 240 degrees
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
24 26 28 30 32 34
Rel
ati
ve
Po
wer
Turbine Number
Validation of Model (Column 4)
Measured Data
Jensen Model
Bastankah Model
Model A, Hassan
2010
Model B, Hassan
(2010)
39
In fig 32 both models can be seen to underestimate the wake effects. The Jensen
model gives a more accurate depiction than the Bastankhah model. A slight rise in
power output can be seen in turbines number 27 to 29 for both models. This can be
put down to wake not spreading out enough from upstream turbines and not affecting
these turbines due to the small roughness length assumed. This is more noticeable on
the Bastankhah model due to the lower velocities on the outskirt of the wake. The
Jensen model gives similar results to model A used by (Hassan, 2010) while model B
gives a very similar output to the measured data. Model B is similar to model A but
with an added internal boundary layer condition which increases the roughness length
at an area downstream of a turbine for each wake it is immersed in. This reduces the
velocity at hub height which reduces the power output of the downstream turbines.
Figure 33: Validation of model Column 10
The wake effects in column 10 are far worse than predicted by either models put
forward. The Bastankhah model varies noticeably along the column from turbines 73-
78 while both the Jensen model and model A remain almost constant along these
turbines. Model B once again gives very similar results to the measured data.
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
72 74 76 78 80 82
Rel
ati
ve
Po
wer
Turbine Number
Validation of Model (Column 10)
Measured data
Jensen Model
Bastankah
Model A, Hassan (2010)
Model B, Hassan (2010)
40
Unfortunately no method attempted to apply the internal boundary layer condition to
the models was successful and so other factors were looked at to improve results. The
Jensen model was chosen above the Bastankhah model as the model to continue
testing due to more accurate results.
As the thrust coefficient and the output power were taken from the turbines manual
they are assumed to be accurate so the wake decay constant was varied in order to
increase the models accuracy. The value of the decay constant was varied from 0.4 to
0.005 and the results plotted in the below graph. The relative power continues to drop
towards the measured data until a value of 0.005 is used the increase in power output
at this value is again due to the wakes not spreading out enough to effect turbines that
are not directly downstream of the turbine creating the wake. Similar peaks can be
noticed for all values under .04 tested.
Figure 34: Validation of model column 4 for varying k Values
For row 10 the model was at its most accurate with a decay constant of .01. Similar
results are obtained when k values of .02 and .03 are used. However under real
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
24 26 28 30 32 34
Rela
tiv
e P
ow
er
Turbine Number
Validation of Model (Column 4) Varying k values
Measured Data
k=.04
k=0.03
k=.02
k=.01
k=0.005
41
circumstances values lower than .03 are unrealistic as turbulence levels on any site
would be too high as a roughness length of × would be needed.
Figure 35: Validation of Model Column 10 for varying k values
The model was then compared to data provided by (Stevens, et al., 2014) which
shows the relative power output per column when the wind is blowing from 270⁰ at
8 . This surprisingly showed the model was overestimating the effects of wakes
for turbines aligned directly downwind of one another.
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
72 74 76 78 80 82
Rel
ati
ve
Po
wer
Turbine Number
Validation of Model (Column 10) Varying k
values
Measured Data
k=.04
k=.03
k=.02
k=0.01
42
Figure 36: Validation of model for wind approaching from 270 degrees, varying k values
As a result of this it was decided that a k value of .04 would be used for the rest of the
study as it was felt that the over estimation in the direct wake may compensate for the
underestimation in the indirect cases.
4.4 Rayleigh Distribution
The Rayleigh distribution was used to create a more realistic interpretation of the
wind speed ad provide an accurate AEP as a constant speed would be used for the
entire year which would lead to unrealistic results. A distribution with a shape factor
of 2 and an average speed of 8.5 was created using Excel with a maximum wind
speed of 30 . The wake speed deficit matrix was then multiplied by each of the
possible wind speeds (1-30) creating a 3D matrix 30 layers deep. The power output
was then found for each wind speed and then multiplied by the probability of
occurrence of that wind speed. The sum of these values was then found and multiplied
by 8760 hours to give the AEP of the wind farm.
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15
Rel
ati
ve
Po
wer
Column No
Validation of Model 270⁰ Varying k value
Measured Data
Model Data k=.04
Model Data k=.03
43
4.5 Cost Model.
Firstly the cable cost model was found using equation 2.17. For the non-staggered
layout the total length of the strings was found by finding the max value in the Y co-
ordinates of the turbine position matrix and multiplying it by the number of columns
in the array. For the staggered layout the model was implemented by dividing the grid
into a specified number of sub sections with all the turbines in each connected to a
single cable or string. This was done using a for loop which divided the turbine
position matrix into the specified number of matrices for each sub section. The
maximum distance in each matrix was then taken as the length of that string. This
method works well for the staggered and non-staggered grid layouts as the string
lengths are repetitive but when the random array was used the code needed to be
altered with the number of strings. As this model was to be run in a loop for hundreds
of iterations this was far too time consuming. Instead each string in the matrix is
assumed to be half the distance between the minimum and maximum Y co-ordinates.
Next the length of the cable connecting all the strings together was calculated. This
was done by finding the turbines with the maximum and minimum X coordinates and
taking the distance between the two as the length of the joining cable. The cost of
cable per km was calculated next using equation 2.16 this value was then multiplied
by the total length of cable used on the model which resulted in a cost of 1.49m€/km
which matches an estimated value by (Martinez, 2009) of 1.5m€/km. Next the turbine
cost was calculated using Mosetti’s cost function, equation 2.11. Finally the LPC of
energy was calculated for the model using equation 2.18.
44
4.6 Rotation of Model
In order to model the wind effects from varying directions the model was rotated
around 360 degrees and tested. The code used for this model can be found in
Appendix C. Fig 37 shows how turbine positions were edited to account for the
change in wind direction. The X and Y Co-Ordinates of each turbine were calculated
along with the distance from the origin. When the model was rotated the distance
from the origin would remain the same and the new X and Y Co-Ordinates were
calculated using the equations below.
Figure 37: Method for finding TProt
Equation 4-2
√ √
Equation 4-3
45
(
)
Equation 4-4
Equation 4-5
×
Equation 4-6
×
Equation 4-7
Equation 4-8
As the turbine positions rotate some of the dimensions become too large for the
defined grid size. To prevent an error from occurring the grid dimensions were placed
in the for loop and change in accordance with the turbine position matrix by taking
the max values from the X and Y column of TProt on each rotational angle iteration
and letting them equal the width and depth of the grid respectively. Fig 38 below
shows the rotation model rotating from 0 to 90 degrees in 30 degree increments. The
rotational angle can be seen in the top left corner of each box. It can be seen that the
grid dimensions change in accordance with the rotation angle by taking note of the
axis values.
46
Figure 38: Rotation of model in increments of 30 degrees
A sample wind rose seen in fig 39 below was developed then divided into 30 degree
bins to be used for the AEP calculation.
Figure 39: Sample wind rose
0
0.05
0.1
0.15
0.20
30
60
90
120
150
180
210
240
270
300
330
Wind Rose
Probability of
Occurance
47
The number of hours spent the wind blowing in each direction was found by
multiplying 8760 by the probability of occurrence for each directional bin. These
values were then multiplied by the power output from each direction and summed up
to find the AEP for the farm.
4.7 Randomly Generated Layouts
To appropriately generate random layouts the randi function was used in matlab
which generates random integers between specified limits. Firstly a minimum
distance between turbines was established in each direction. The width and depth of
the grid was then divided by these distances to see how many turbines could be placed
in the grid. These are also the positions the turbines can be placed into ensure turbines
don’t overlap once multiplied back by the grid width and depth. A number of turbines
is then defined before the randi function is used. Two different matrices called Px and
Py were created for each direction in the grid with the number of turbines to be input
into the grid equal to the number of rows in each matrix. In the image below an
example can be seen where 10 turbines are placed in a grid which can fit 16 turbines
in both the X and Y directions. The 256 cells in the table below represent positions on
the grid once multiplied back by the minimum separation distance between the
turbines giving matrices Px2 and Py2 seen in fig 40.
48
Figure 40: Position matrices and grid layout for random placement
When smaller grids are being used it is quite common that more than one turbine can
be assigned to a single position in the grid. In order to prevent this from occurring a
while loop was used. Both matrices Px2 and Py2 were combined to make a co-
ordinate matrix called P. The matrix P was then checked for duplicates using the
unique function on matlab to remove them. If this was the case not the specified
number of turbines would not be allocated positions on the grid. The while loop was
then created stating while the length of matrix P is less than the number of turbines
specified another matrix that’s length is taken as the difference between the length of
P and the number of turbines specified. This new matrix was then added on to the
bottom of P. This was once again checked for duplicates and the code cannot exit the
loop until each row is unique and the number of rows is equal to the number of
turbines specified.
49
4.8 Optimisation Using Rayleigh
4.8.1 Varying Grid size
Firstly a set number of turbines was analysed varying the size of the grid using 42
turbines in a non-staggered array consisting of 7 rows and 6 columns for each test.
The distance between the columns was varied 11 times from 2 diameters to 30
diameters with a constant cross wind spacing of 4 diameters. As can be seen in the
graph below the lowest value for the LPC occurred when the separation distance
between the columns was 7 diameters.
Figure 41: Optimum separation distance downstream Rayleigh distribution
The next graph shows clearer why the lowest LPC occurs at this distance. As the AEP
of the wind farm rises rapidly initially when the columns are spread out but eventually
the AEP begins to level off while the land and cable costs continue to rise causing the
LPC to rise continuously.
0.15
0.16
0.17
0.18
0.19
0.2
0.21
0 5 10 15 20 25 30 35
LPC
Seperation (D)
Optimum Seperation Distance
50
Figure 42: AEP & Cost Vs Seperation Distance downstream Rayleigh distribution
4.8.2 Constant Grid size
Next a square grid of 300 by 300 cells was created with each cell representing a 10m
by 10m area giving an area of 9 . Various non-staggered layouts were then tested
to see which array had the lowest LPC. The minimum spacing downwind and cross
wind was changed each time the model was run to vary the number of turbines on the
grid. In table xx the lower LPC values are coloured green and the higher values are
coloured red this colour method will be used throughout the rest of the analysis.
Table 4-1: Optimum separation distance Rayleigh distribution
Number of Columns
Diameters Downwind
Diameters Across
2 4 6 8
2 25 0.1386 0.1909 0.2178 0.2636
3 15 0.1365 0.1812 0.2096 0.2558
4 10 0.1474 0.1825 0.2114 0.2616
5 7 0.1686 0.1993 0.2274 0.2833
6 6 0.189 0.2181 0.2436 0.3019
7 5 0.2083 0.2378 0.2612 0.319
-1.80E+08
-1.70E+08
-1.60E+08
-1.50E+08
-1.40E+08
-1.30E+08
-1.20E+08
-1.10E+08
-1.00E+08
1.10E+08
1.15E+08
1.20E+08
1.25E+08
1.30E+08
1.35E+08
1.40E+08
1.45E+08
0 10 20 30 40
AEP
Seperation Distance (D)
AEP&Cost Vs Seperation Distance
AEP
Cost
51
The Optimum layout was found when the turbines are placed in 3 columns spaced 15
diameters downstream of one another and 2 diameters or 16 rows in the crosswind
direction. This equates to a total of 48 turbines and the layout can be seen in fig 43
plotted using matlab.
Figure 43: Optimum separation distance Rayleigh distribution
An interesting fact about these results is that the minimum LPC for each cross-wind
spacing occurs when there are 3 columns or a spacing of 15 diameters downwind in
the grid. This spacing allows the wake to recover before reaching the next turbine.
These results agree with results found by (Guillen, 2010) and may be put down to the
small k value due to the small roughness length. The spacing in the cross wind
direction does not correspond to the same literature however as having this many
turbines spaced so close together in a row would prolong the wakes recovery due to
overlapping wake and also have a major impact on the turbulence in the farm which
according to (Hassan, 2010) would cause the boundary layer to change effecting the
52
output of turbines downstream. This layout would also cause environmental concerns
such as interfering with wildlife.
4.10 Optimisation Using Rotation
The grid was kept at a constant size for the rotation analysis. Once again a grid width
and depth of 300 cells were used with each cell representing a 10m by 10m area. A
constant free stream velocity was taken for this part of the experiment due to
difficulty combining the Rayleigh with the rotational and iterative thrust coefficient
model. The wind rose shown in fig 39 was used along with a free stream velocity of
8.5 for all three types of array layout.
4.10.1 Non-Staggered
The same variations were tested for the rotational models as the Rayleigh model with
the addition of one extra column. The model was run 28 times for a non-staggered
layout giving the results seen in the table below.
Table 4-2: Optimum separation distances rotation non-staggered
Number of Columns
Diameters Downwind
Diameters Across
2 4 6 8
2 25 0.0994 0.1197 0.1262 0.1408
3 15 0.1014 0.1156 0.1189 0.1359
4 10 0.1129 0.1127 0.1255 0.1323
5 7 0.1302 0.1272 0.118 0.1281
6 6 0.1457 0.1274 0.1205 0.1383
7 5 0.1635 0.1324 0.1192 0.1341
8 4 0.2 0.1449 0.1339 0.1292
The minimum LPC was found for when there are 2 columns spaced 25 diameters
downwind with the turbine spacing only 2 diameters across. This layout can be seen
in fig 44.
53
Figure 44: Optimum array layout for non-staggered rotational model
This result does not match results from (Guillen, 2010). However upon looking at the
array efficiency in fig 45 for the layout through the 360 degrees and taking into
consideration that only two strings are needed which reduces cost it becomes clearer
as to why this layouts performance is so good. There are 16 rows of turbines and 2
columns summing to a total of 32 turbines and although the efficiency drops to
approximately 10 % when the wind is blowing from the North or South, due to the
small cross wind spacing, for the rest of the angles the efficiency is above 80%.
54
Figure 45: Array efficiency for optimum non –staggered rotation
The worst preforming array was found with a separation distance of 4 diameters
downstream and 2 diameters across. In this case the wake is not given enough time to
recover which reduces the array efficiency which can be seen in fig 46 not to exceed
50% from any direction.
Figure 46: Array efficiency for worst performing non–staggered rotation
55
4.10.2 Staggered
The staggered layout showed similar results to the non-staggered however the
optimum in this case was found at a spacing of 3 diameters across and 10 diameters
downstream. Consisting of 4 columns with 11 and 10 turbines every second column.
This comes to a total of 42 turbines.
Table 4-3: Optimum separation distances rotation staggered
Number of Columns
Diameters Downwind
Diameters Across
3 5 7 9
2 25 0.0996 0.11 0.1217 0.1308
3 15 0.0909 0.1015 0.1105 0.1192
4 10 0.0901 0.0963 0.1074 0.1239
5 7 0.0998 0.0955 0.1157 0.1217
6 6 0.1025 0.0987 0.1211 0.109
7 5 0.1097 0.1152 0.1075 0.1066
8 4 0.1209 0.1268 0.1029 0.1079
It is important to note that the spacing across is taken as the distance between turbines
in the same column the staggered turbines in the following row are in the middle of
this and so in the optimum layout each downstream turbine is offset 1.5 Diameters
from the turbine upstream this can be seen in the image below.
56
Figure 47: Optimum array layout rotation staggered
Again the efficiency drops almost as low as 10% when the wind is blowing from the
North/South Directions but in general stays around 80% the low LPC in this case can
also be put down to the high AEP due to the large number of turbines on the grid.
57
Figure 48: Array efficiency for optimum array layout rotation staggered
In general the staggered array layout preformed better than the non staggered layouts
with only one staggered layout having an LPC of greater than 0.13€/kWh whereas 11
of the non staggered layouts tested exceeded this value. This is due to less interaction
of wakes because of the offset.
4.10.3 Random
The random array layouts can be seen in table 4.4 below the model was run 75 times
changing number of turbines on the grid was varied in increments of 10 from 20 to 60
with 15 runs for each amount. This was done to get a rough estimate of LPC values
for each number of turbines. A minimum spacing of 2 diameters across and 2
diameters downwind was put in place between each turbine. This allowed for a total
of 256 different possible locations for the turbines. The best performance was found
when 40 turbines were placed on the grid during the 10th
run.
58
This relates to an average of roughly 4.5 turbines per although it can be seen in
the below image of the optimum layout that this is not how they are dispersed. A plot
of the grid was created on excel as each turbine position matrix was saved but the
matlab plots close before the final result is given.
Table 4-4: LPC values for varying number of turbines, Random.
Iteration Number
Number of Turbines
20 30 40 50 60
1 0.1196 0.1214 0.1239 0.1296 0.1403
2 0.1163 0.1219 0.1149 0.1294 0.1387
3 0.117 0.1188 0.1163 0.1312 0.1489
4 0.1184 0.1207 0.1182 0.1335 0.1508
5 0.1183 0.1236 0.1199 0.1318 0.1451
6 0.1217 0.116 0.1155 0.1343 0.1399
7 0.1151 0.1157 0.1186 0.1349 0.1456
8 0.1161 0.1194 0.1196 0.1352 0.1423
9 0.1186 0.1267 0.1204 0.1287 0.1559
10 0.1149 0.1242 0.1144 0.1328 0.1522
11 0.1194 0.1176 0.1196 0.1399 0.145
12 0.1168 0.1189 0.1291 0.1295 0.1493
13 0.1243 0.1158 0.1331 0.1344 0.1482
14 0.1184 0.117 0.1219 0.1328 0.1444
15 0.1156 0.1193 0.1246 0.1372 0.1467
The code was then placed in a loop which runs the model 500 times saving each of
the LPC values and the corresponding turbine position matrices. This was repeated 3
times summing to a total of 1500 runs. The optimum LPC was found to be
.1118€/kWh the layout was plotted using excel and can be seen in fig 49 and the
turbine position matrix can be seen in Appendix D.
59
Figure 49: Best performing turbine layout for random generation
This optimum layout performed worse than both the non-staggered and staggered
layouts and from the grid layout this is mainly due to the lack of an optimisation
algorithm which would take arrays with good performances and alter them slightly
until performance is improved and continue doing so until an optimum is found. This
method is far more efficient than running a random generating model on repeat. The
high LPC value may also be due to the inaccuracy of the cable model or a low
average array efficiency similar to the sample taken which can be seen in fig 50.
Figure 50: Sample array efficiency for random array
0
50
100
150
200
250
300
0 50 100 150 200 250 300
60
Chapter 5: Conclusions
In total 1655 Turbine layouts were tested for a 9 site mainly using rotational
analysis with free stream wind velocity taken as a constant at 8.5 . Three types of
layouts were tested including non-staggered, staggered and randomly generated
arrays. The optimum array found to be a staggered layout with a spacing of 3
diameters crosswind and 10 diameters downstream. The LPC was found to be
.0901€/kWh. The worst performance was found using a non-staggered layout with a
separation distance of 4 diameters downstream and 2 diameters crosswind this in-
efficient performance was due to the wake not having enough time to recover
highlighting the need for further study in this area.
There are a number of areas of this dissertation that could have been further
developed:
A model including the internal boundary layer condition discussed in Section
4.3 which could enhance the accuracy of results obtained.
A combination of the two models developed in this dissertation allowing for
both rotational and varying wind speed analysis.
A more effective cable cost model which will automatically adjust for random
arrays.
Varying turbine parameters such as hub height and turbine diameter and
investigating effects on optimum layouts.
61
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64
Appendix A
The equation below can be found for the wake by neglecting viscous and pressure
terms in the momentum equation and applying mass and momentum conservation.
∫ 𝑑 8
Total Force acting on the turbine is given as:
9
Normalised velocity deficit:
∆
10
Due to the Gaussian shape the above equation can be written as:
∆
11
The wake velocity is then given by:
(
) 12
Putting Equations 9 and 12 into 8 and integrating from 0 to∞ gives
8 (
𝑑 )
(
𝑑 )
Solving 13 gives two values while only one is physically acceptable:
√
8 ( 𝑑
)
65
Assuming a linear expansion in the wake region
𝑑
𝑑
Where
(
)
Inserting equations 14 and 15 into equation 11 gives:
∆
(
√
8 ( 𝑑
)
)
× exp
(
( 𝑑
) {(
𝑧 𝑧 𝑑
)
(𝑦
𝑑 )
}
)
66
Appendix B
clear all clc close all GW=550;%Grid Width GD=450;%Grid Depth U0=9;%Free Stream Wind Velocity D=zeros(GD,GW);%Grid Rt=4;%Radius of Turbine Dt=(Rt*2);%Diametre of Turbine zh=7;%hub height Zo=.003;%Roughness length k=(0.5/log(zh*10/Zo));%Wake Decay Constant
%Importing Turbine Position Matrix TP=xlsread('pos270.xlsx'); Nt=length(TP);%Number of Turbines
%Wake speed deficit and thrust coefficient loop % Predefined variables to ensure loop can execute on its first run diff_max=1; counter=1; Ct=ones(GD,GW)*.805; Ct_old=Ct; D=zeros(GD,GW,Nt); while diff_max>=1e-3 && counter<=100 for l=1:Nt for m=1:GW if m>=TP(l,2) for n=1:GD if n>TP(l,1)-Rt-k*(m-TP(l,2)) &&
n<TP(l,1)+Rt+k*(m-TP(l,2)); Ct_local=Ct(TP(l,1),TP(l,2)-1); D(n,m,l)=((1-sqrt(1-Ct_local))/((1+(2*k*((m-
TP(l,2))/Dt)))^2)); end end end end end
wakes=U0*(1-sqrt((sum(D.^2,3))));
Ct_old=Ct; % Save Ct values to old matrix before calculating updated
values
Ct=1094.3*wakes.^-3.094; Ct(Ct>.5)=.805; diff=abs(Ct_old-Ct); % Calculating difference between old and new Ct
for while loop termination diff_max=max(max(diff));
X=linspace(0,5500,GW); Y=linspace(0,4500,GD); pcolor(X,Y,wakes); hold on; shading interp;
67
end
%Calculating Speed Before Turbine Ub=zeros(length(TP),Dt+1);
for n=1:length(TP) Ub(n,:)=wakes(TP(n,1)-Rt:TP(n,1)+Rt, TP(n,2)-1); end Um=mean(Ub,2);
%Calculating Power Output
N=length(TP); Pmax=2000;
Pout=zeros(length(TP),1);
for n=1:length(TP) if Um(n)<=4 && Um(n)>25; Pout(n,1)=0; elseif Um(n)>4 && Um(n)<14; Pout(n,1)=0.0311*(Um(n))^6 - 1.5609*(Um(n))^5 +
30.92*(Um(n))^4 - 310.81*(Um(n))^3 + 1702.7*(Um(n))^2 -
4767.6*(Um(n)) + 5340.8; else (Um(n))>=14 && (Um(n))<=25; Pout(n,1)=Pmax; end end
RelPout=Pout(:,1)/max(Pout);
%Plotting Relative Power Output
figure (2) scatter(1:Nt,RelPout(:,1)) Energy_Production=sum(Pout(1:length(TP),1))
68
Appendix C
clear all clc close all GW1=300;%Grid Width GD1=300;%Grid Depth U0=8.5;%Free Stream Wind Velocity D=zeros(GD1,GW1);%Grid Dt=10;%Diametre of Turbine Rad=round(Dt/2);%Radius of Turbine k=0.04;%Wake Decay Constant
%Importing Wind Rose rose=xlsread('wind_rose');
count=1;
%Minimum Distance between Turbines GW_min=4*Dt; GD_min=8*Dt; Nx= floor(GW1/GW_min)+1;%No. of Turbines Across Ny= floor(GD1/GD_min)+1;%No. of Turbines Down
% Rotational Loop
for r=0:pi/6:(11/6)*pi %resetting the grid size GW=GW1; GD=GD1;
%Possible Positions of Turbines
X=round(linspace(Dt/2+1,GW-Dt,Nx)); Y=round(linspace(Dt/2+1,GD-Dt,Ny));
%Turbine Position Matrix
TPx=ones(Ny*Nx,2); for i=1:Nx TPx((i-1)*Ny+1:i*Ny,1)=X(i); TPx((i-1)*Ny+1:i*Ny,2)=Y'; end TP=TPx;
Nt=length(TP); %No. of Turbines
%Finding Angle And Distance to Origin TPrt=zeros(Nt,2); TPrt(:,2)=atan((TP(:,1))./TP(:,2)); TPrt(:,1)=sqrt(TP(:,1).^2+TP(:,2).^2);
ro=r;
%New X and Y Positions X1=round(TPrt(:,1).*cos(TPrt(:,2)+ro)); Y1=round(TPrt(:,1).*sin(TPrt(:,2)+ro));
69
TP1=[X1,Y1]; Offset=(min(TP1));
%Removing Negatives TProt(:,1)=TP1(:,1)-Offset(1,1)+Dt; TProt(:,2)=TP1(:,2)-Offset(1,2)+Dt;
TP=TProt; %Changing Grid Size GD=max(TProt(:,1))+Dt; GW=max(TProt(:,2))+Dt;
% Predefined variables to ensure loop can execute on its first
run diff_max=1; counter=1; Ct=ones(GD,GW)*.805; Ct_old=Ct; D=zeros(GD,GW,Nt); while diff_max>=1e-3 && counter<=100 for l=1:Nt for m=1:GW if m>=TP(l,2) for n=1:GD if n>TP(l,1)-Rad-k*(m-TP(l,2)) &&
n<TP(l,1)+Rad+k*(m-TP(l,2)); Ct_local=Ct(TP(l,1),TP(l,2)-1); D(n,m,l)=((1-sqrt(1-
Ct_local))/((1+(2*k*((m-TP(l,2))/Dt)))^2)); end end end end end
wakes=U0*(1-sqrt(sum(D.^2,3)));
Ct_old=Ct; % Save Ct values to old matrix before calculating
updated values Ct=1094.3*wakes.^-3.094; Ct(Ct>.5)=.805; diff=abs(Ct_old-Ct); % Calculating difference between old and new
Ct for while loop termination diff_max=max(max(diff)); counter=counter+1;
end
%Plotting Wake Matrix figure(2) contourf(wakes,l,'LineStyle','none') colorbar
%Calculating Speed Before Turbine
Ub=zeros(length(TP),Dt+1); for n=1:length(TP)
70
Ub(n,:)=wakes(TP(n,1)-Rad:TP(n,1)+Rad, TP(n,2)-1); end Um=mean(Ub,2);
%Calculating Power Output Pmax=2000; Pout=zeros(length(Um),1); for n=1:length(Um) if Um(n)<=5 || Um(n)>25; Pout(n)=0; elseif Um(n)>5 && Um(n)<14; Pout(n)=0.0311*(Um(n))^6 - 1.5609*(Um(n))^5 +
30.92*(Um(n))^4 - 310.81*(Um(n))^3 + 1702.7*(Um(n))^2 -
4767.6*(Um(n)) + 5340.8; else (Um(n))>=16 && (Um(n))<=25; Pout(n)=Pmax; end end
%Array Efficiency
Array_Efficiency(count,1)=r*360/(2*pi); Array_Efficiency(count,2)=sum(Pout)/Nt(1,1)/max(Pout);
%Annual Energy Produced Per Directional Bin AEPs(count,1)=rose(count,4)*sum(Pout(1:length(Pout),1)); count=count+1; end
AEP=sum(AEPs);
%Plotting Array Efficiency
figure (2) plot(Array_Efficiency(:,1),Array_Efficiency(:,2),'--')
% Length of Strings Pos2=max(TP(:,2))+1; Pos1=min(TP(:,2))-1;
for n=1:2 counter=1; for i=1:length(TP) if TPx(i,2)>=Pos1+(n-1)*(Pos2-Pos1)/2 &&
TPx(i,2)<Pos1+n*(Pos2-Pos1)/2 Pos(counter,:,n)=TPx(i,:); counter=counter+1; end end end
Pos_1=Pos(:,:,1); Pos_1( ~any(Pos_1,2), : )=[]; LOS1=max(Pos_1(:,1)); LSc= max(TP(:,2))-min(TP(:,2));
% Cable cost. CuP=2782;%Copper Cost CuD=8940;%Copper Density
71
AAcross=1000;%Cross Sectional Area profit=2; CCu=profit*CuP*3*CuD*AAcross*(1/100000);%Cost of Cable Per km LoC=(LOS1*Nx+LSc)/100;%Length of Cable CaC=CCu*LoC;%Total Cost
% annuity factor LT=20;%Lifetime of Farm r=.07;%Interest Rate a=(1-(1+r)^-(LT))/r;
% Cost of turbines CPT=2500000; TTC=Nt*CPT*(2/3+1/3*exp(-0.00174*(Nt^2)));%%Mosetti Cost Function
% Cost of land Ckm=500000;%Cost Per km CoL=Ckm*((GD/10)*(GW/10));%Cost of Grid
% Levelised production Cost LPC=(((TTC*(1/.8)*.97)+CaC+CoL)/(a*AEP))*(1/.53) Cost=(TTC*(1/.8)*.97)+CaC+CoL%Total Cost of Farm
72
Appendix D
Optimum layout found for random Array:
X Co-Ordinate Y Co-Ordinate X Co-Ordinate Y Co-Ordinate
20 80 200 180
20 160 200 240
20 260 220 20
40 100 220 60
60 180 220 160
60 300 220 260
80 120 220 280
80 180 240 120
120 20 240 180
120 120 240 220
120 220 240 280
120 260 260 80
140 20 260 200
140 300 260 220
140 80 280 20
160 20 280 80
160 180 280 220
180 40 300 80
180 60 300 100
180 180 300 300