Uncertainty in wind turbine life equivalent load due to variation of site conditions

M.Sc. Thesis Project

Technical University of Denmark Fluid Mechanics Section

Raymond Patrick Downey

19 May 2006

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Dedication Before I sat down to write this dedication I expected it to be written quite quickly, only to discover that there are so many people who deserve so much recognition for the way they have influenced me, both on a personal and professional level. Being unable to include all these people by name or even to give a reason why I am so indebted to them, I will simply dedicate this work to my family, friends, my loving wife and beautiful son.

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Abstract Wind turbine load calculations can be checked against measurements and (to some extent) be tuned to reflect reality; one may assume that loads can be predicted accurately if the conditions in a particular the 10 minute period are known. However the site conditions are always estimated from site measurements or from nearby meteo stations and even if long term averages are exactly known the 20 year average values will have some distribution around these. The inherent uncertainty contained in the estimation procedure to some extent has lead to the conservative over designing of wind turbine. A parametric study of the site parameters is used to investigate the influence of the uncertainty of the site conditions on the wind turbine equivalent loads. By using a wind turbine modelled in Flex5 each of the parameters can be isolated and its influence on the equivalents can be determined. By applying Miners linear damage rule to the outputs the 20 year equivalent load can be determined, and from which the probability of failure can be found using the structural design LRFD (Load Resistance Factor Design) method.

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Acknowledgements There are many people who contributed in a large or a small way to this work they are probably too numerous to mention. But of course there are the obvious ones such as my supervisors Kurt Hansen (Danmarks Tekniske Universitet) and Dick Veldkamp (TU Delft and VESTAS wind systems), who between the two of them know everything there is to know about wind energy (of that I’m sure), and who have been a pleasure to know and work with. Thanks to Kurt for always being available whenever called upon for advice and guidance, especially for his help with the winddata.com database and for working so hard to get the Aalborg NM92 data uploaded in time for the analysis. Thanks to Dick who was also available whenever needed and has to be the fastest replier to emails in the world. His ability to express the seemingly complicated in an uncomplicated way has been essential in the progression and development of the project from the original concept to the final draft. A special thanks to the people who have contributed in some way or another to the project, to the people at VESTAS especially Erik Miranda who helped me immensely on the finer points Flex5 and various other programs, it was a lot of work in one day; Stig Øye for allowing me to use a copy of Flex5 and for his input on some of the more abstract aspects of wind energy, I now know what a wind seed is; Ari Bronstein for reading this report and for his constructive comments. Of course this Thesis would have never been possible without the hard work of Martin Hansen and Jens Sorensen from the fluid mechanics department for firstly being the ones responsible for the wind energy masters programme, and secondly for being so helpful through out my time here in Denmark. I guess the only way to show gratitude to an institute, is to say it has been a pleasure studying at DTU and without any doubt it can count itself amongst the best universities in Europe. Last Note to Dick, Good luck with your Thesis.

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TABLE OF CONTENTS

Dedication.......................................................................................................................... ii

Abstract ............................................................................................................................ iii

Acknowledgements ...........................................................................................................iv

1 Introduction................................................................................................................1 1.1 Pollution and the role of renewable energy .........................................................1 1.2 Wind turbines a brief history ...............................................................................2 1.3 Contribution of wind energy industry to the acceptance of wind power.............3 1.4 Objectives ............................................................................................................4

2 Wind turbine design procedure................................................................................5 2.1 Introduction..........................................................................................................5 2.2 Procedure .............................................................................................................5 2.3 Structural reliability .............................................................................................8

2.3.1 Partial safety factors ....................................................................................9 2.3.2 Limit state ..................................................................................................10

2.4 Site admission....................................................................................................11 2.4.1 Site parameters (vector of dependents)......................................................11

3 Determine the site parameter distributions ..........................................................13 3.1 Introduction........................................................................................................13 3.2 Wind speed ........................................................................................................13

3.2.1 Wind prediction .........................................................................................15 3.2.2 WAsP Method ...........................................................................................15 3.2.3 MCP method..............................................................................................16 3.2.4 Initial MCP analysis ..................................................................................17 3.2.5 Distribution of Danish results....................................................................20 3.2.6 Pastoral and coastal data............................................................................20 3.2.7 Complex Terrain........................................................................................22 3.2.8 Yearly wind speed variation ......................................................................24 3.2.9 Wind speed Summary................................................................................25

3.3 Wind speed profile/shear ...................................................................................26 3.3.1 Wind shear exponent .................................................................................27 3.3.2 Wind speed profile estimation ...................................................................31

3.4 Air density .........................................................................................................38 3.5 Turbulence intensity ..........................................................................................39

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3.5.1 Normal turbulence model ......................................................................... 39 3.5.2 Turbulence Intensity profile...................................................................... 40

4 Determine the Load Resistance of the wind turbine............................................ 48 4.1 Introduction....................................................................................................... 48 4.2 Wind turbine aeroelastic model ........................................................................ 49

4.2.1 Verification of the wind turbine model..................................................... 49 4.3 Fatigue............................................................................................................... 50 4.4 Wohler (SN) curve............................................................................................ 50 4.5 Cycle Counting ................................................................................................. 52 4.6 Linear damage rule ........................................................................................... 55

4.6.1 Stress correction........................................................................................ 56 4.6.2 Stress reduction factor............................................................................... 57

4.7 Equivalent loads................................................................................................ 58

5 Design calculations and parameter distribution .................................................. 60 5.1 Introduction....................................................................................................... 60 5.2 Limit state function ........................................................................................... 60 5.3 Site equivalent fatigue loads ............................................................................. 62 5.4 Resistance equivalent fatigue loads .................................................................. 65 5.5 Example ............................................................................................................ 67

5.5.1 Site conditions........................................................................................... 67 5.5.2 Site loads................................................................................................... 67 5.5.3 Resistance ................................................................................................. 69

6 Results ...................................................................................................................... 73

7 Conclusions and further work ............................................................................... 78 7.1 Conclusions....................................................................................................... 78 7.2 Further work...................................................................................................... 81

References........................................................................................................................ 83

List of Figures.................................................................................................................. 86

List of Tables ................................................................................................................... 88

Appendix.......................................................................................................................... 90

Appendix A Parameter estimation results ............................................................. 91

A.1 Wind speed profile............................................................................................ 91 A.2 Mean wind speed yearly variation ................................................................... 95

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A.3 Wind shear exponent .........................................................................................96 A.4 Turbulence intensity profile...............................................................................99 A.5 New method ....................................................................................................101

Appendix B Database..............................................................................................106 B.1 A Database insight...........................................................................................106 B.2 Relational database ..........................................................................................106 B.3 Database architecture (tables)..........................................................................107

Appendix C Wind turbine and site description....................................................109

Appendix D Load verification results....................................................................111

Appendix E Monte Carlo simulation ....................................................................115 E.1 Hit and miss integration...................................................................................115 E.2 Two parameter Monte Carlo............................................................................116 E.3 Monte Carlo MatLab example code ................................................................119

Appendix F Beaufort scale .....................................................................................122

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1

Chapter 1 1 Introduction

1.1 Pollution and the role of renewable energy

With more and more evidence coming to light about the adverse effects of concentrated amounts of CO2 in the atmosphere we would know that we need to change the way in which we produce the majority of our energy, i.e. burning of fossil fuels. This need for change led to introduction of the Kyoto accord; a legally binding agreement dedicated to reduction of CO2 emissions. This agreement has had the effect of spurring on governments to take proactive measures to try and tackle the problem. Although the transport and the energy sector have been the biggest culprits, most focus has been on the energy sector. This is perhaps due to the presence of already established and emerging clean energy alternatives such as wind, solar, wave, and hydro power. The suitability of some of these methods is dependent on the geographic location, for instance there doesn’t seem to be many places where hydro dams and reservoirs can be installed in Denmark or the Netherlands, the same goes for solar where the northern European climate will limit its impact. As for wave power; well it hasn’t proved itself yet despite the fact that it has been around for some time now. That leaves wind; wind has been seen as the fastest way to make an impact on increasing a countries share of renewable energy and reducing emissions, making wind the renewable energy of choice.

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1.2 Wind turbines a brief history

There has been a lot of development of the wind turbine during the last century, this development however was somewhat sporadic with the most interest being shown in the development of the wind turbine at times of high energy prices but when prices decreased the interest in wind turbines decreased along with them. This was the pattern up until the oil crisis during the yon kippur war in 1973, where Arabic states refused to ship oil to countries that supported Israel in the conflict. This lead to a sudden increase in the price of oil, which in turn stimulated a number of government funded programs of research into alternative energy sources. This was the beginning of a concerted effort to produce utility scale machines in many countries such as Britain, Denmark, Germany and the USA. The remarkable growth in the output of the wind turbine is directly attributed to this period of sustained research born out of the oil crisis.

0

1000

2000

3000

4000

5000

6000

1980 1985 1990 1995 2000 2003

Year

Pow

er [k

w]

Figure 1.1 Increasing size of output of wind turbines

One of the anticipated outcomes of these research programs was to enhance the so called security of supply, i.e. a country should as much as possible have its own source of energy and not be dependent upon another party, especially not if that other party is in the extremely unstable Middle East. This was the main driving force and still plays a significant part in policy on energy, as mentioned earlier, in recent years the prominence of the environmental lobby has given the industry the injection it needed to take the size of wind turbines into multi megawatts. With countries now clambering to reduce their CO2 emissions to meet their obligations signed up to in the Kyoto accord.

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1.3 Contribution of the wind energy industry to the acceptance of wind power

Issues such as security of supply and environmental concerns may be the external driving force in the wind energy industry, but internally it has been the evolution of the wind turbine. Over the last 30 years and most notably in the last 10 this evolution has driven down the cost of producing electricity. In terms of technology the modern wind turbine is hardly recognisable when compared to its forerunner from 30 years ago, yes the concept is the same, steel tower and 3 blades, but the technical advances have been extraordinary with the introduction of sophisticated manufacturing processes, new materials and control systems all combine to enable the wind turbine to fulfil its full potential.

0.000

0.020

0.040

0.060

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1992 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Pric

e pe

r kW

h [e

ur]

0

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15000

20000

25000

30000

35000

40000

45000

50000

Inst

alle

d w

ind

pow

er [M

W]

Price per kWh Installed capacity

Figure 1.2 Falling price of wind energy alongside the installed capacity of wind energy

(source BWEA). These advances in technology have combined to create more cost effective wind turbines and although they are not entirely responsible for the falling costs, they are a considerable contributory factor. This fall in price has a knock on effect and it is no coincidence that the trend of decreasing cost corresponds directly to an increase in the installed capacity. From Figure 1.2 it would seem that the reduction in the cost1 of wind energy is quite a strong driving force in its acceptance and is quickly becoming an energy source to rival the so called conventional energy sources. Thus, it is essential that the wind energy industry works towards the reduction of manufacturing costs to facilitate further expansion. This will be possible through the introduction of new design methods, or the optimisation of existing ones. 1 These costs are base on the bid price for the Non Fossil Fuel Obligation for a 15 year generation contract [source BWEA].

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1.4 Objectives

A complete implementation to quantify all the uncertainties is not possible due to time constraints. Therefore, an investigation of the fatigue loads due to the uncertainty in site conditions will be carried out on the following:

1. Wind speed estimation 2. Wind shear estimation 3. Turbulence intensity estimation 4. Errors in load calculations (accuracy of the wind turbine model)

Finally, having quantified the uncertainties two example case studies will be used to demonstrate the procedure of estimating the probability of failure of some selected components. The results from such a procedure will demonstrate the level of conservatism, allowing in some cases the opportunity to redesign the component to a more appropriate (lower) probability of failure. This in turn will lead to a reduction in manufacturing costs and ultimately reduce the price of electricity generated. In a nut shell, this project aims to reduce the costs of manufacture by achieving a greater understanding of the operating conditions of the wind turbine, allowing the possibility for a more sensible design.

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Chapter 2 2 Wind turbine design procedure

2.1 Introduction

In order to understand the effects of the loads on a wind turbine it will first be necessary to discuss conventional design procedure. The wind turbine industry as with any other manufacturing industries will have a specific set of design procedures. The core of this procedure is standard across the whole manufacturing industry and in most cases each industry will have a product standard it has to achieve before it can come into service in many countries in the world. There are several certification bodies and standards and the standards which will be used in this thesis are the IEC 61400-1, standard on safety.

2.2 Procedure

The procedure can be described best with the aid of a flow chart as shown in Figure 2.1.

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Load Calculations

Prototype only 1st time

Load measurements

only 1st time

Design loads according to some standard

Difference between calculated and measured

loads

Adjust aeroelastic model

Small enough

Too big

Turbine can be certified

Figure 2.1 Simplified Flow chart of the design procedure of a wind turbine (Veldkamp) The life of a wind turbine begins in the design office, where the criteria are always the same, i.e. the initial condition of the design is that the turbine will extract as much energy as possible while withstanding the applied loads for a specified lifetime. Due to the complexity of a wind turbine it is not possible to design a site specific machine but rather a range of site conditions which a particular turbine can operate in. These ranges, known as classes are laid down in 61400-1 and are shown in Table 2-1. The downside to this procedure is that a turbine can be over designed for a site and thus is not as economic to produce; on the other hand a standardised manufacturing approach drastically reduces the cost of production.

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WTG Classes I II III S Vrel (m/s) 50 42.5 37.5 A Iref(-) 0.16 B Iref(-) 0.14 C Iref(-) 0.12

Values to be specified by the

designer

Table 2-1 Basis parameters for WTG classes2

As can be seen in table above the WTG’s are defined by Vrel (50 yr max wind speed), and Iref is the expected turbulence intensity at 15 m/s. The standard wind average wind speed is given as

refavg VV 2.0= (2.1) These are the main parameters that will cause the ultimate destruction of the WTG. There are 3 classes, but if the A, B and C parameters for the turbulence intensity are taken into account we can say that there are in fact 9 classes. The following is a long version of the process sketched in Figure 2.1

• A wind turbine’s design loads are defined by the Class in which it is to operate for example if the proposed site is classified as Class II with turbulence parameter A, then the turbine will have be designed with these specifications as a minimum. There are other site related loads, but for now we will just concentrate on the ones in Table 2-1. The turbine designed to these specifications is now a Class II A wind turbine;

• Before any prototype can be built the aeroelastic model will have to be made to enable the necessary load calculations;

• When the model demonstrates that the design loads can be resisted with a satisfactory margin of safety for the complete duration of its lifetime (20 years); the prototype can be made;

• This prototype will be fully instrumented to enable a direct comparison between the actual measurements and the model. If this model shows a satisfactory similarity to the measured loads. The turbine can be put into production otherwise the model will be tuned until the results of the comparison are close enough; around 10%.

2 All values given the table are to be applied to hub height

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2.3 Structural reliability

As mentioned previously the wind turbine is to be designed to withstand the specified conditions based on its class. There are several methods to achieve this, but an increasingly common method in structural reliability has been the Load Resistance Factor Design (LRFD). This method in its simplest form can be written as

DD RL < (2.2) Where LD is the design load and RD is the design resistance to the load, for structural integrity LD should always be less than RD. Both of these are related to the characteristic loads and resistance,

CfD LL γ= (2.3)

m

CD

RR

γ= (2.4)

Where the partial safety factors fγ and mγ are the load factor and material factor respectively3 and the characteristic values for the load and the resistance are Lc and Rc. The characteristic load and resistance are defined as the value of some conservative fractile, for example in the case of the turbulence intensity the value is the mean + 1.25 standard deviations (90% fractile). The same applies for the fatigue strength, except that the assumption is of a reduced resistance, i.e. for the fatigue strength the value is the mean – 2 standard deviations (2.3 % fractile), meaning that 97.7% of all samples will be stronger than this.

3 The value for the load and material factors are based on the uncertainty of the resistance and the loads, the reader is referred to IEC 61400-1 for further information.

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Figure 2.2 Probability density functions for loads and resistance

The design difference or margin of safety will be given by the partial safety factors as in Figure 2.2. It is these safety factors which bring failure probability down to an acceptable level, meaning that the wind turbine can just withstand the site characteristic loads. Figure 2.2 illustrates a situation where the uncertainty in the loads and resistance are distributed around the respective design mean values, and the region where failure can occur is where the two sets overlap, this does not mean that the overlapping region is equivalent to the probability of failure.

2.3.1 Partial safety factors Although the LRFD method can calculate the partial safety factors the wind turbine still has to be designed to the level of safety prescribed in the standards. The IEC has defined the partial safety factors for the loads and materials in 61400-1, and these are designed to take account of the following: Loads

• The possibility of unfavourable deviations from the load from the characteristic value

• Uncertainties in the loading model (e.g. Wind turbine modelled in Flex5) Materials

• The possibility of unfavourable deviations from the strength of the material from the characteristic value.

• Possibility if inaccurate assessment of the resistance of the strength of the sections or load carrying capacity of the parts of the structure

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• Uncertainty of the geometrical parameters. • Uncertainty in the relation between the material properties in the structure

and those measured by tests on control specimens. These different uncertainties are combined to give the two partial safety factors

fγ and mγ .

2.3.2 Limit state Even when these partial safety factors are applied there is still the possibility of failure as can be seen in Figure 2.2. This failure probability can be written in terms of a so called limit state function.

DD LRZ −= (2.5) where failure will occur when Z<0. This probability of failure can be evaluated using several methods including first and second order reliability methods (FORM, SORM) and other simulation methods. The method of choice here is the Monte Carlo simulation methods details of which can be found in Appendix E. Since in most cases the probability of failure can never be eliminated there are guidelines suggested for the probability of failure depending on the failure type and the consequence of failure and these are shown in Table 2-2. Failure consequence

Failure type Less serious LOW SAFETY

CLASS (small possibility for personal injuries and

pollution, small economic

consequences, negligible risk to life)

Serious NORMAL SAFETY

CLASS possibilities for

personal injuries, fatalities,

pollution, and significant economic

consequences)

Very serious HIGH SAFETY

CLASS (large possibilities for

personal injuries, fatalities, significant pollution, and very

large economic consequences)

Ductile failure with reserve capacity

(redundant Structure)

PF=1e-3 bT=3.09

PF=1e-4 bT=3.72

PF=1e-5 bT=4.26

Ductile failure with no reserve

capacity (significant warning before

occurrence of failure in

nonredundant

PF=1e-4 bT=3.72

PF=1e-5 bT=4.26

PF=1e-6 bT=4.75

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structure) Brittle failure

(no warning before occurrence of failure in

nonredundant structure)

PF=1e-5 bT=4.26

PF=1e-6 bT=4.75

PF=10e-7 bT=5.20

Table 2-2 Target annual failure probabilities PF and corresponding reliability indices βT (source DNV/Risø [6] )

2.4 Site admission

Even though a wind turbine has received certification, this certification is subject to the wind turbine operating at or below the loads which it has been designed for. When a wind turbine is to be installed at a particular site, the so called site admission rule has to be adhered to. A common version of this rule states that a wind turbine can be installed ‘if the expected loads due to the combination of site parameters are smaller than assumed’. Where the assumed loads are those defined by the basic IEC Class requirements, i.e. mean wind speed, turbulence intensity, wind shear, etc. These parameters make up the vector of dependents and for any site load calculations these parameters will need to be known to establish the probability of failure.

2.4.1 Site parameters (vector of dependents) The site parameters themselves are dependent upon the location, for instance if the site were offshore there would be parameters for the waves and currents, or if the site were a wind farm, the wind farm turbulence (wake turbulence) parameter would have to be included. A simple example of a set of site parameters is the vector of site dependents xs, defined for a solitary wind turbine at an onshore site and given in (2.6)

),,,( airavgs TiUx ρα= (2.6) Where

Uavg Mean wind speed Ti Mean Turbulence intensity a Wind shear exponent ρair Density of air

In the majority of cases the values of site parameters are not known, and have to be estimated using historical wind/weather data from a nearby met station, or extrapolated from lower heights, e.g. if the measured wind data is taken at 10m

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instead of hub height, then it will have to be extrapolated. This will introduce some uncertainties and these will have to be accounted for in any design calculations.

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Chapter 3

3 Determine the site parameter distributions

3.1 Introduction

As mentioned earlier the site parameters will make up the fatigue loads part of the LRFD analysis. In this section we are going to look at how accurately we can predict the mean values for: wind speed, turbulence intensity and wind shear for the following terrain types:

• Pastoral • Coastal • Complex • Offshore

For each parameter the variation in the mean will be determined and the uncertainty quantified.

3.2 Wind speed

Probably the most important site parameter, which affects the equivalent loads that a turbine encounters over its lifetime, is the wind speed. For almost any site the 10 min mean wind speed can be fitted to a Weibull distribution as described by (3.1) and shown in Figure 3.1

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⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−−=

k

AuuP exp1)( (3.1)

Where the parameters k and A are site and height dependent. A special case of the Weibull distribution is the Rayleigh distribution, with a shape parameter k ≈ 2, which is a quite typical value for many locations (North West Europe). The scale parameter A is related to Uavg, by the relationship

)/11( kAU avg +Γ= (3.2) And Γ is the gamma function.

Figure 3.1 Wind speed distribution Measured and Weibull fit for Horns rev (Denmark

offshore) Although the Weibull fit doesn’t fit all the measurements exactly it is quite close. Thus we can assume that Weibull is valid fit for calculating the loads on the basis that the fit will over estimate the occurrences of some wind speeds and under estimate others. The error arising out of this assumption is not likely to cause much deviation from the actual loads. As seen here it is relatively easy to determine the distribution of the wind, the more difficult thing to do is determine the mean wind speed as will be discussed in the next section.

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3.2.1 Wind prediction As seen above the wind speed distribution can be accurately predicted, but only if the site dependent parameters are known. IEC 61400-1 recommends a shape factor k=2 as mentioned above is quite close to the value found at most on land sites (North West Europe). This is not the case for the mean wind speed, which cannot be known since it is the mean value over the next 20 years. However a good approximation of the future wind speeds can be made based on historical wind data, which can be directly applied if the historical data is available for that particular site. This is very seldom the case and in the absence of long term measured wind data for a site it will be necessary to estimate the wind climate using data from somewhere else in the region. For example a meteorological station some kilometres away may have wind data for 20-30 years, by applying a relevant forecasting method to the data the wind climate for the site can be determined with reasonable accuracy. Over the years there have been two methods of forecasting have proved to be most popular in the wind energy industry, these being Wind Atlas and applications program (WAsP) and the Measure Correlate predict (MCP) Method. Before going any further it will be necessary to give a short explanation of these two methods just to give an understanding of the forecasting process.

3.2.2 WAsP Method WAsP was developed at Risø national laboratory, Roskilde, Denmark. It works on the principle of predicting the site wind climate from a met station somewhere in the same region by utilising the winds in the geostrophic layer. This is possible due to the fact that changes local wind speeds can be assumed to be caused entirely by surface site conditions, e.g. roughness, coastal effects etc. This means that over a sizeable region the geostrophic winds can be assumed constant. So the only difference between the winds in two locations in the same region will be due to the local terrain. The downside to this method is that an accurate description of the terrain is needed to yield results with a reasonable precision.

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Figure 3.2 Mean Geostrophic winds over northern Europe

3.2.3 MCP method Measure Correlate Predict is a simple and effective tool used in the prediction of wind speeds at a potential wind farm site (predictor site) and is based on the relationship of the predictor site and a site where long term wind data is available (reference site). Unlike WAsP this method requires that wind data be available from both the predictor and the reference sites. The period which data must be available from the predictor site has been suggested to be in the region of 10-12 months to rule out any seasonal influences. Periods of time longer than this have been studied but the results don’t show any great increase in accuracy. Another important feature of this method is that no knowledge of the site terrain is needed other than that both predictor and reference sites must have similar terrain and also that the wind direction from the predictor site is the same as that for the reference site. The method is applicable when the following criteria are adhered to:

• Predictor site and reference site must have similar terrain • Wind direction at the both reference and predictor site is the same • There is a distinct correlation between the two wind speeds

Concurrent wind data from the reference site and the predictor site will then be compared and the relationship determined. In the case of this thesis the concurrent period is 12 months of hourly data. Now that this relationship (correlation) has been established it should be applied to historic wind data from the reference site; usually in the region of 10 to 20 years.

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As with WAsP the validity of this method is based on statistics that show that the mean wind speed will not change much from one 20 year period to the next. In fact we are predicting the next 20 years using the previous 20 years. There are several different methods for determining the correlation. Most use a direction sectored linear regression using 2 parameters, or just a straight forward ratio. In this thesis both are used in a direction sectored and a complete (360o) site analysis.

3.2.4 Initial MCP analysis An initial analysis to determine the accuracy of the MCP method for flat coastal terrain was carried out using data from the winddata.com database. This database is compiled and maintained at the Technical University of Denmark. The data is in the form of 10 min statistics and is not suitable for this type of analysis when dealing with the correlation of wind speeds over long distances, thus it will be necessary to convert the data to hourly averaged statistics based on the 10 min mean. For the initial analysis Denmark was chosen as the country of reference, and from Denmark three sites suitable for an MCP analysis were selected. The main criteria for suitability were that the sites had an adequate concurrent duration (several years) and that they were situated reasonably far apart (to demonstrate the extent of the correlation over distance). These sites along with their duration can be seen in Table 3-1.

From Until Duration [yrs] Kegnæs 1-1-1991 31-12-2001 10 Sprogø 1-1-1977 31-12-1999 22 Risø 1-1-1996 31-12-2001 5

Table 3-1 Sites used in the MCP analysis including their measurement time period Samples of the results are shown in Figure 3.3 below. These results show that the criteria for an acceptable MCP analysis are satisfied. That is, that it is acceptable to say that the wind direction from both sites can be assumed to be the same and that there is a definite correlation between the two sets of wind data.

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Figure 3.3 Correlations between hourly mean values for Kegnæs and Risø

Using the prediction methods mentioned earlier a prediction was made for Risø and Sprogø with Kegnæs as the reference site. The results for the site pairs are shown in Table 3-2. Where the relation between the reference and the predictor site for each year is used to predict the mean for the duration of the analysed data; in this case a 6 year mean for both site pairs.

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Kegnæs Sprogø 0.964 0.998 0.988 1.039 0.994 1.012 --- --- --- ---

Kegnæs Risø --- --- --- --- 0.977 0.985 1.021 0.985 1.053 0.990

Table 3-2 Ratio of predicted mean wind speeds with actual mean wind speeds. Example Say a mast was erected in 1997 at Sprogø with the purpose of predicting the 6 year mean for the site. The ratio between the predictor and the reference for this year would be directly used to predict the 6 year mean by applying this ratio to historic data from Kegnæs for 1997 and the previous 5 years.

97,

97,97

k

s

VV

R = yrkpred VRV 6,97= (3.3)

Where yrkV 6, is the 6 year mean wind speed for Kegnæs.

In this case, we happen to have the previous 6 years for the predictor site and can determine its 6 year mean. By comparing the actual 6 year mean to the predicted 6 year mean we get a ratio ≈ 1.012 or 1.2% of an error. If in fact we used another year other than 1997 we would get a different result as can be seen in Table 3-2.

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012.16,

6 ==yrs

predyr V

VR (3.4)

These results show the method to be quite accurate. To get some perspective on these results they were compared to results obtained from a WAsP analysis. These results are available for several site pairs available from the European Wind Atlas. The site pairs chosen were ones which corresponded as near as possible with sites already analysed using MCP. All the sites can be seen in the map in Figure 3.4.

Skr Bel Vær Skrydstrup 1.00 1.114 1.098

Beldringe 0.868 1.00 0.978

Værlose 0.902 1.021 1.00 Table 3-3 Ratio of estimated mean wind speed to actual mean wind speed for a period of

approximately 7 years Table 3-3 shows the ratio of results for the WAsP predictions, with the predicted stations written in full and the predicting stations by 3 letter abbreviations. The shaded sections are where the site has predicted itself and will be equal to 1. From the comparison it can be seen that the results for the MCP method are more accurate than that found using WAsP.

Figure 3.4 Location of sites used in the MCP analysis along with the sites used in the WAsP

analysis.

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3.2.5 Distribution of Danish results The results for the Kegnæs – Sprogø analysis are plotted as a cumulative probability as shown in Figure 3.5 below, It demonstrates that the results will possibly follow a normal distribution4.

Figure 3.5 Cumulative distribution of the ratio for the predicted and the actual site mean

wind speed It would be extremely optimistic to determine anything from the resulting distribution other than it shows the possibility of obeying a normal distribution. However for a more conclusive set of results it will be necessary to have many more years of concurrent data than were used in the Danish analysis. It can be quite difficult to obtain data (for free) due to the fact that selling wind data can be a good business; however there are some agencies that provide wind data for free, one such institution is the Royal Netherlands Meteorological Institute (KNMI). They provide quite a comprehensive set of data covering in some cases up to 50 years, with the data in the form of hourly-normalised5 values.

3.2.6 Pastoral and coastal data The terrain in the Netherlands is quite similar to that found in Denmark and can be described as Coastal and Pastoral. Consequently the results obtained can be said to be representative of these types of terrain, which cover WTG Classes I to III. Several site pairs from this database were investigated and their results 4 The central limit theorem explains why many distributions tend to be close to the normal distribution. The key ingredient is that the random variable being observed should be the sum or mean of many independent distributed random variables as long as no one distribution is dominant. 5 Potential wind speed is derived measured wind speeds and the local roughness. The wind speeds here are transformed to potential wind speeds at standard height 10m and standard roughness length 0.03m. For further information see KNMI website link.

21

tabulated in Table 3-5. Figure 3.6 shows the cumulative distribution of the ratio of the prediction compared to the actual mean value. From this it is clear that ratios are distributed normally.

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Leeuwarden predicting Ijmuiden

U predict / U Ijmuiden

Cum

ulat

ive

Pro

babi

lity

Normal distributionPrediction

Figure 3.6 Ratio between the prediction and the actual site mean wind speed

To try an increase the accuracy of the predictions the range of the data used was censored i.e. not including the lower wind speeds in the analysis. This was expected to increase the accuracy, since there it assumed there would be no correlation between the lower wind speeds. In fact it was thought that no correlation would exist. An example of this could be two concurrent wind speeds V1 = 2 m/s at one site and V2 = 0.5 m/s at another site this would give a ratio of 1:4. This would not be representative of the site ratio and might as well be left out. The table below is for IJmuiden calculated from the reference site Leeuwarden WS > 0.5 m/s WS > 2 m/s WS > 3.0 m/s WS > 3.5 m/s COV 0.070 0.066 0.062 0.058 Mean V 1.014 0.991 0.976 0.968

Table 3-4 IJmuiden calculated from the reference site Leeuwarden Where COV is the coefficient of variation and is defined as

µσ

=COV (3.5)

22

Where s and m are the standard deviation and the mean of the predictions respectively. On the evidence from the table above it can be seen that censoring the data has very little benefit, i.e. although the COV will be reduced the accuracy of the predictions also reduces. However there will be an optimal cut off / censoring point, a value of 3m/s as a cut off point has been suggested, and this agrees with the results in Table 3-4.

Site COV [%]

Mean [-]

Ijmuiden 7.0 1.00 Hoek van Holland 10.7 1.00 Schipol 5.3 1.00 Average 7.66 1.00

Table 3-5 Coefficient of variation and means for data from 1962-2002 using Leeuwarden as the reference site

The results above are in agreement with a similar but far more comprehensive study carried out by Veldkamp [24], where he shows that the occurrence a COV of 10.6 is not a common occurrence. Since the average for such a small sample is hugely affected by the inclusion of the COV from Hoek van Holland it will be left out, giving a new average of 6.15%. From the results the important thing to note is that the mean predictions will be approximately equal to 1, but the COV is quite large, meaning that for some pairs there is the possibility of under/over estimating by as much as 15 % as is the case for IJmuiden. This demonstrates the need for standards that are designed to these conditions.

3.2.7 Complex Terrain It is becoming more common to site wind turbines in complex terrain because there is in most cases far better wind resources. The data for a complex terrain analysis was taken from the Deutscher Wetterdienst (German weather service). This database has a comprehensive collection of weather data that is supplied in the standard WMO format. The format is quite different from that which is normally encountered, i.e. m/s and degrees. The measurements/observations are only taken 3 time daily and have the wind direction in the form of 32 sectors and the wind in the form of a force; the Beaufort scale. A conversion table for this scale can be seen in Appendix F (not very important but it is a little interesting) Since there was some apprehension about using this data an analysis was carried out using sites in the flat pastoral region in the north of Germany to see if the results compared with those found for the Netherlands. The results from the prediction pair Hamburg/Bremen was mean =1.00 and COV = 0.058, for a period of 20 years from 1960-1980. These compare well with the Dutch and

23

Danish results and show that the method of conversion was accurate. A point to note is that the recording of the data is somewhat subjective, and thus will contain inherent errors. Southern Germany The terrain from southern Germany can definitely be termed as complex as can be seen in Figure 3.7 below, also included in this figure are the site locations used in the implementation of the MCP method6.

Figure 3.7 Measurement locations in Southern Germany

The mean wind speeds found for these sites are shown in Table 3-6, the interesting thing about these is that the wind speeds from the lower lying stations have such a low mean wind speed, making any comparison with these completely useless. This leaves only 1 useful site pair; Hohen – Zugspitz.

Site V [m/s] Hohen’ 4.69 Zugspitze 7.79 Kempten 1.89 Stuttgart 2.99 Augsburg 3.00

Table 3-6 Mean wind speeds for southern German sites

6 It is unsure whether or not the MCP criterion are adhered to in complex terrain

24

Site COV cross prediction

COV (1 yr site measure) Mean

Hohen’ 0.12 0.108 1.00 Zugspitze 0.12 0.118 1.00

Table 3-7 Coefficient of variation and means for data from 1960-1999 As in the previous analysis the mean is quite close to unity, but the COV is greater, much larger coefficients of variation were calculated but these were from predictions using the now obsolete lower lying sites. From Table 3-7 it can be seen that using a single measured yearly mean instead of using MCP will give a lower COV, but will still be rather high. A more detailed investigation is needed before any concrete recommendations can be made regarding whether or not the using the 1 year measured mean is better then making a prediction.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Hohen predicting Zugspitze

U predict / U Zugspitz

Cum

ulat

ive

prob

abili

ty

Normal distributionPrediction

Figure 3.8 Difference between the prediction and the actual site mean wind speed

3.2.8 Yearly wind speed variation Even if the 20 year mean was exactly known the next 20 years will not be exactly the same due to the yearly variation. The COV of the 20 year mean is related to the COV of the 1 year mean by

201

20year

yearCOV

COV = (3.6)

25

No. of years Average U(10) COV 1 yr COV 20 yr Netherlands Ijmuiden 40 6.5 7.4 1.66 Leeuwarden 40 5.4 4.3 0.95 Hoek van Holland 40 6.4 10.6 2.37 Schipol 40 5.3 4.1 0.91 Average 5.9 6.6 1.47

Table 3-8 Yearly variation of mean wind speed for the Netherlands Example A mast is located at IJmuiden on the west coast of the Netherlands it is proposed to erect a wind turbine at this site. There are 40 years of measurements available at this site. Find the 20 year COV. In this case the 20 year mean wind speed can be exactly know for the site, this however cannot be assumed to be the identical to the next 20 year mean. In fact like the 1 year mean the 20 year mean will also have a COV, this 20 year COV will be a factor of 20 smaller than the 1 year COV and is calculated as follows

66.1204.7

20 ==yearCOV (3.7)

A limited number of sites have been investigated in northern Europe and southern Germany and the results have been tabulated in Appendix A.2 . These results show the average for Northern European region º 1.37.

3.2.9 Wind speed Summary As discussed earlier, using the MCP method will introduce errors, with these errors being distributed about a mean of 1.00. The COV will be different for different site pairs, with an average for the Dutch sites of 6.15%. If we consider a northern European region we can include the site pair prediction from northern Germany, doing so gives an average of 6%. Applying MCP to the complex terrain in southern Germany was hindered by the extremely low mean wind speeds in the lower lying regions, leaving only a single site pair to analyse. This site pair produced a COV of 12 % in such a case using the yearly measured mean might as well be used and in this case it would prove to be more accurate. Even if the long term mean was estimated exactly the 20 year average value will still be distributed about this. The COV of the 20 year averages can be

26

determined using (3.7). Applying this to the values calculated for northern Europe we get C0V20 year = 1.37%.

3.3 Wind speed profile/shear

The methods of the previous section can be used to accurately predict the wind speed at the measurement height. A common height for measurements is approximately 10 m, but we need to know what the wind speed is at Hub height. The wind speed profile describes the wind speed as a function of height above the surface for any particular site. This method can be used to extrapolate the wind speed from 10 m to hub height. Before we do this we have to identify the wind speed range that is of concern to us. Since this project is dealing with fatigue loading, it will only be necessary to a look at the wind speed that contributes to these loads. These are in the range of 10 – 20 m/s, although it can be said that wind speeds above 20 m/s will also contribute to the fatigue loads, but the probability of wind speeds greater than 20 m/s means that any contribution to fatigue loads will be limited. The method that is recommended by IEC 61400-1 for the calculation of the normal wind profile (NWP) for the standard WTGS classes is the power law7:

α

⎟⎟⎠

⎞⎜⎜⎝

⎛=

refref z

zVzV )( (3.8)

Where; α is given as 0.2. This profile is assumed to define the average vertical wind shear in the rotor plane. An example of the use of the power law is shown in Figure 3.9 below for a Class II site; with Zref = 20m. It shows that profile is highly dependent on the exponent. The error in mean wind speed when using α = 0.2 as suggested is in IEC 61400-1 is in order of ~11%, and ~6% at heights of 140m and 40m respectively.

7 The power law will be used to define the wind speed profile in Flex5.

27

5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200Wind profile for Cabauw

Wind speed [m/s]

Ver

tical

hei

ght [

m]

measured dataalpha = 0.1alpha = 0.2alpha = 0.3

Figure 3.9 Wind profile for Cabauw (The Netherlands) for 10 m/s < V < 25 m/s

These values when put in context of this thesis are not acceptable, and it clearly does not describe the vertical wind speed profile over the entire rotor plane. In general it seems that this value for α is far too conservative.

3.3.1 Wind shear exponent The importance of the using the correct shear exponent is evident making it necessary to be able to calculate/derive it accurately. There are 2 methods for calculating the shear exponent,

1. Calculating using the wind speed measured at several heights; 2. Deriving/estimating the exponent from knowledge of the site terrain

roughness length. In the case of a met mast with several instrumented heights the shear exponent can be easily calculated using the relation between the wind speeds measured at separate heights, thus by manipulating (3.8) we get the following expression for α

)/ln()/ln(

zzVV

ref

refz=α (3.9)

However in reality this is very seldom the case and more often there is just one set of measurements available at around 10m. In such a situation the previous method to determine the wind shear exponent is not possible and therefore it will have to be estimated from the surface roughness as follows,

28

⎟⎠⎞

⎜⎝⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=

HzzHzz

o

o

ln

ln

lnln

α (3.10)

Where zo surface roughness

H reference height z height

Where zo will be determined in the following 3 ways

1. Derived from the turbulence (See section 3.5 on turbulence) 2. Offshore / Coastal-sea is calculated using Charnock’s formula (see

(3.12)). 3. From a table of typical values for several site types based on the work of

Petersen (Risø, 1980).

Terrain type Roughness length zo [m] Open sea without waves 0.0001 Open sea without waves 0.0001 – 0.003 Coastal areas with onshore wind 0.001 Open country without significant vegetation or buildings 0.01

Cultivated land with scattered buildings 0.05 Forests and suburbs 0.3 City centres 1-10

Table 3-9 Roughness length for various site terrain types (Petersen, Risø, 1980) Using (3.9) the wind shear exponent a was calculated using 10 min average wind speed measurements taken at several heights for selected met masts in a number of different terrain classes. At all times Zref was based on the lowest height on any particular met mast. Result for a selected number of sites are shown tabulated in Table 3-10 below. The full table can be seen in Appendix A.

29

Site Terrain Height [m] Mean α [m/s] Std. dev α [m/s]

Cabauw Flat / Pastoral 140 80 40

0.15 0.14 0.14

0.08 0.08 0.04

Horns Rev Offshore 62 45 30

0.10 0.09 0.09

0.05 0.04 0.05

Tjæreborg Coastal 90 60

0.17 0.15

0.08 0.08

Oak creek Complex 79 65 50

0.08 0.09 0.09

0.04 0.04 0.04

Skipheia Coastal (Sea direction)

101 72 41

0.10 0.10 0.11

0.03 0.03 0.03

Table 3-10 Shear exponent mean and standard deviation for several sites and heights.

From this table it can be seen that the mean value of a varies quite a bit from site to site. Cabauw and Tjæreborg are the only sites that can be defined by a standard Class as set down in 61400-1 and shown in Table 2-1. The exponents in both cases are reasonably close to the value of 0.2 suggested in the standards. The others belong to a special class that the turbine designer has to specify and are of Class S.

30

5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200Wind profile for Cabauw

Wind speed [m/s]

Ver

tical

hei

ght [

m]

measured dataalpha = 0.1alpha = 0.2alpha = 0.3alpha best fit

Figure 3.10 Wind profile for Cabauw (The Netherlands) for 10 m/s < V < 25 m/s

The mean value of the shear exponent calculated in this way can be assumed to be exact and since in most cases we have 3-4 mean values the obvious thing to do is to use the best fit (average) mean value. Now the error in the estimation when using (3.10) can be determined by the difference form the bestfit a. In Table 3-11 the results are presented for the 3 methods used to estimate zo for the offshore sites.

Site Læsø Skipheia Horns Rev Egmond Terrain Hub height Mean a Std. s

Offshore 62

0.12 0.06

Coastal (sea dir) 72

0.10 0.03

Offshore 62

0.10 0.05

Offshore 65

0.09 0.04

Derived from Ti Reference height [m] Measured Ti [-] Derived roughness [m] Shear exponent a [-]

15

0.091 2.3E-4 0.075

20.5 0.13 0.003 0.11

15

0.10 5.8E-4 0.09

21

0.08 9.1E-5 0.07

Petersen roughness [m] Shear exponent a [-]

0.001 0.08

0.001 0.10

0.001 0.10

0.001 0.09

Charnock roughness [m] Shear exponent a [-]

2.2E-4 0.08

2.8E-4 0.08

2.4E-4 0.08

2E-4 0.08

Table 3-11 Measured and estimated wind shear exponent for offshore sites at approximately Hub height, the estimate names refer to the method used to estimate zo

31

The results in this format are not very useful, what we need to know is the variation of the errors and the bias for the different sites, where the bias is defined as the difference of the estimated value from the expected. In our case the expected value will always be zero.

0

0.2

0.4

0.6

0.8

1

1.2

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

Relative shear exponent error [-]

Cum

ulat

ive

prob

abili

ty [-

]

PredictedNormal fit

Mean =-0.156Std = 0.147Sample size =20

Figure 3.11 Distribution of the relative error for the shear exponent estimated at

approximately Hub height using the methods 1 and 2 mentioned earlier (Petersen, derived zo).

Estimate type Bias

[-] Relative Bias

[-] Standard deviation

[-] Petersen method m = -0.018 m = -0.140 σ = 0.175 Derived zo method m = -0.019 m = -0.146 σ = 0.148 Total (including Charnock) m = -0. 02 m = -0.156 σ = 0.147

Table 3-12 Bias and relative bias and standard deviation for the shear exponent error In Table 3-12 and Figure 3.11 it is easy to see the bias in the estimation procedure where in both cases the error is biased by ≈14%. The estimations in all but 3 cases will have the effect of underestimating the mean wind speed when extrapolating from a lower height to Hub height. The error is represented as a relative error (error/aref). Both methods have proven to be as good as each other.

3.3.2 Wind speed profile estimation Aside from the Power law there is also the widely used Log law, which in actual fact the power law is based on.

32

ozzuzU ln)( *

κ= (3.11)

Where

u* is the friction velocity k is Von Kármán constant = 0.4

zo is the roughness length

For coastal and offshore applications it may be more appropriate to define the profile using the Charnock formula8 for roughness.

gu

Azo

2*= (3.12)

Where g is the acceleration due to gravity = 9.81 and A is the Charnock constant ≈ 0.03 for coastal sites and ≈ 0.01. By combining (3.11) and (3.12) we get an implicit equation for the friction velocity.

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

2*

*

ln

)(

uAzg

zUu κ

(3.13)

A comparison for the log law was made using data from Cabauw and the resulting profiles are shown in Figure 3.12. For the best fit power law profile only the values for a up to height 80 were used to see how accurate the profile could predict the wind speed at a height of 140m. As can be seen both methods perform extremely well. The log law profile is based on roughness calculated at Zref = 20m. This demonstrates the advantages of the log law in that it is not necessary to have several heights in order to achieve an acceptable level of accuracy.

8 For the sake of simplicity the profile derived from this equation will now be called the Charnock profile.

33

6 8 10 12 14 16 180

20

40

60

80

100

120

140

160

180

200Wind profile for cabauw

Wind speed [m/s]

Ver

tical

hei

ght [

m]

Measured dataalpha best fitLog law,Power law zref = 20

Figure 3.12 Wind profile for Cabauw calculated using Log law and Power law (best fit 20m-

40m-80m) Figure 3.13 shows all 4 methods applied to Horns Rev; an offshore site on the west coast of Denmark. All three show a good fit of the data with the best fit power law performing the best. This is not surprising since all the data at all the measurement heights are used in the calculation of the best fit, however only Zref = 15m was used for the other two methods.

9 10 11 12 13 14 15 16 170

20

40

60

80

100

120

140

160

180

200Wind profile for Horns rev

Wind speed [m/s]

Ver

tical

hei

ght [

m]

Measured dataalpha best fitlog law CharnockPower law using estimated alpha

Figure 3.13 Wind speed profiles for Horns rev

34

Table 3-13 below shows how each method performs in determining the wind speed profile for some selected sites. The results from further log law and power law analyses using the various estimation techniques can be seen in Appendix A.

Log Law (derived zo)

Charnock Site Terrain Height

[m] Measured

[m/s] U DU U DU

Cabauw Pastoral z0 = 0.05 I20 = 0.130

140 80 40 20

17.2 15.73 14.16 13.1

17.4 16.1 14.6 13.1

0.2 0.4 0.5 -

- -

Horns Rev Offshore z0 = 0.001 I15 = 0.10

62 45 30 15

14.7 14.1 13.6 12.7

14.6 14.1 13.6 12.7

-0.2 0.0 0.0 -

14.5 14.1 13.5 12.7

-0.3 -0.1 0.0 -

Skipheia Offshore z0 = 0.001 I20 = 0.130

62 45 30

20.5

16.6 15.9 15.3 14.4

16.6 16.1 15.3 14.4

0.1 0.2 0.0 -

17.0 16.5 15.6 14.4

0.5 0.6 0.3 -

Oak Creek Complex z0 = 0.05 I10 = 12

79 65 50 10

16.6 16.7 16.3 14.3

16.3 16.1 15.8 14.3

0.3 0.6 0.4 -

- -

Table 3-13 Comparison of the measured and estimated wind speeds using the log law. And the value of zo corresponds to terrain roughness derived from Iz

The table shows that all methods presented are quite accurate. Though there were some inaccuracies with the coastal and complex terrains that needed some remedial steps to correct. Some precautions to take when dealing with these terrains are as follows

• Coastal - Sea direction It is important to disregard any measurements taken in the mixed boundary layer that occurs due to the land sea interface. There is a rule of thumb regarding the height of this layer, and it is based on the distance of the mast to the shore. To avoid using data measure in this layer the Zref for Skipheia was taken at 20m instead of 11m.

• Complex terrain

Take account of what measurements may be affected adversely by the terrain and minimise there affect on the profile calculations by eliminating them, i.e. wind speeds measured at low heights (10m in this

35

instance) are highly dependent on the terrain, and will contribute to an inaccurate profile.

36

Petersen Derived from Ti

Site Terrain Height [m]

Measured [m/s] U

[m/s] DU

[m/s] U

[m/s] DU

[m/s] Cabauw Pastoral

z0 = 0.05 I20 = 0.130

140 80 40 20

17.2 15.73 14.16 13.1

17.4 16.1 14.6 13.1

0.2 0.4 0.5 -

16.4 15.5 14.3 13.1

-0.7 -0.3 0.1 -

Skipheia Offshore z0 = 0.001 I15 = 0.10

62 45 30

20.5

16.6 15.9 15.3 14.4

16.6 16.1 15.3 14.4

0.1 0.2 0.0

17.0 16.5 15.6 14.4

0.5 0.6 0.3 -

Horns Rev Offshore z0 = 0.001 I20 = 0.130

101 72 41 15

14.7 14.1 13.6 12.7

14.6 14.1 13.6 12.7

-0.2 0.0 0.0 -

14.5 14.1 13.5 12.7

-0.3 -0.1 0.0 -

Oak Creek Complex z0 = 0.05 I10 = 12

79 65 50 10

16.6 16.7 16.3 14.3

16.3 16.1 15.8 14.3

0.3 0.6 0.4 -

17.9 17.6 17.1 14.3

1.3 0.9 0.8 -

Table 3-14 Difference between the extrapolations (Power Law) and the measured wind speeds at several heights for all the common terrain types

0

0.2

0.4

0.6

0.8

1

1.2

-0.1 -0.05 0 0.05 0.1

Relative wind speed error [-]

Cum

ulat

ive

prob

abili

ty [-

]

PredictedNormal Fit

Mean = 0.007Std = 0.039Sample size = 28

Figure 3.14 Relative wind speed error calculated using all 3 methods in Table 3-15 below.

37

Estimate type Bias [m/s]

Relative Bias [-]

Standard deviation [-]

Petersen method m = -0.03 m = -0.008 s = 0.040 Derived zo method m = -0.07 m = -0.007 s = 0.046 Log Law (derived zo) m = -0.3 m = -0.019 s = 0.031

Table 3-15 Bias and standard deviation for the relative wind speed error.

Unlike the error distribution for the wind shear exponent there is little or no Bias in the estimation using the 3 methods in Table 3-15, and also the standard deviation is in the error is quite acceptable. No one method out performs the others, but it must the noted that the Petersen method of determining the roughness zo is quite subjective. in the sense that the roughness must be estimated by visual inspection and an approximation made based on the Petersen diagrams. For this reason the other methods are to be preferred. All in all the various methods used to determine the wind speed profile all performed quite well, however some methods are more suitable to particular sites, with the power law being the most robust. The log law is also quite robust if the Charnock roughness is used for offshore, but it doesn’t fair so well in complex terrain.

38

3.4 Air density

The IEC defines a standard air density, 1.225 kg / m2; and is calculated as follows

TRP

=ρ (3.14)

With Pressure: P = 1013.25 hpa

Temperature: T = 15±C (calculations are in Kelvin = 15 + 273.15) Gas constant: R = 287.05

The pressure over Northern Europe is variable but compared to the variation in temperature it is negligible. Thus the changing temperature is the dominating influence over the air density.

Figure 3.15 Air density at Hamburg as a function of temperature and time, (1960-1969)

Figure 3.15 shows the air density over a 10 year period from 1960-1969, the seasonal fluctuations are quite noticeable, which is not surprising since these fluctuations can be directly contributed to the seasonal variations in temperature. To conclude the yearly average temperature in northern Europe will not change by any great amount resulting in an almost constant annual Air density. This has been proven by the calculation of the COV of the yearly density found for Hamburg ≈ 0.003 (negligible) for years 1960-1999.

39

3.5 Turbulence intensity

Turbulence intensity is defined as the ratio between the standard deviation and the mean wind speed over a 10 min period. This deviation in the wind speed is almost entirely due to the terrain roughness length zo at any particular site.

10UI u

Tσ

=

(3.15)

This is an extremely important site parameter to be aware of since it is the one of the main contributors to fatigue loading of the wind turbine (blade, nacelle, and tower). The figure below illustrates the turbulence for an onshore site at Delapole, U.K.

Figure 3.16 Ti, mean Ti and Tichar for Delapole Ti char is for a WTG Class II A.

The wind speed that is of most concern for fatigue reasons is 10 m/s < Vmean < 20 m/s. Thus, the mean turbulence intensities for wind speeds not within this range can be neglected.

3.5.1 Normal turbulence model For design / accreditation purposes the IEC 61400-1 has defined an expression for the characteristic standard deviation for the wind speed based on the WTG classes.

)75.0(, bVI hubrefcu +=σ (3.16)

40

Where

Iref = Turbulence intensity at a wind speed of 15 m/s b = 5.6 m/s Vhub = Mean wind speed at hub height.

By substituting the su,c into (3.15) we get the Characteristic turbulence intensity profile, shown in Figure 3.16. This value is based on the definition of the mean value of su plus 1.25 standard deviations of su to give the 90% quantile.

3.5.2 Turbulence Intensity profile The turbulence intensity profile as with the wind speed profile is the relation between the turbulence and height z above the surface. Figure 3.17 shows the mean turbulence for several heights over the wind speed range from 2-25 m/s. This clearly shows that mean turbulence is a function of height.

Figure 3.17 Mean turbulence profile for Skipheia

When measurements are made at hub height we can assume that the measured mean turbulence measured is exact and will not change. But if the measurements are made at heights less than hub height they will have to be extrapolated. An example of a turbulence intensity profile is shown in Figure 3.18 below along with two common methods for predicting the profile.

41

Figure 3.18 Turbulence intensity profile for the offshore sector of Skipheia

There are several methods of predicting the turbulence profile, but not all take the wind speed into account and assume that the turbulence intensity is independent of the wind speed. This is not the case in reality and certainly not the case when working offshore. A simple method of describing the turbulence profile is (3.17). The setback with this method is that as mentioned above it is assumes the turbulence is independent of wind speed and thus has no real application for offshore or coastal sites.

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ozz

zIln

1 (3.17)

Another method is that based on (3.13) where the standard deviation can be approximated as

**5.2 uchar =σ (3.18) Again substituting this into (3.15) will yield the turbulence intensity. The important point to note about this method is that when zo is calculated using the Charnock roughness formula the turbulence intensity increases with wind speed. Which is the case in reality, as can be seen in the plot of the turbulence intensity for Horns rev in Figure 3.19.

42

Figure 3.19 Turbulence intensity for Horns rev including the mean and the estimated mean. Another two new methods will also be considered, New method 1

po

zz

zI ⎟⎠

⎞⎜⎝

⎛=)4.0exp(*

4.2*100 (3.19)

• Proposed by Hansen and Larsen in [9].

The values for zo and exponent p are shown in Table 3-16 below. An explanation of this method is can be found in Appendix A.

Site zo P Skipheia 0.005 0.39 Vindeby 0.003 0.38 Læso 0.003 0.38 Horns Rev 0.003 0.38 Cabauw 0.03 0.43 Tobøl 0.04 0.43 Tjæreborg 0.03 0.45 North sea 0.003 0.38

Table 3-16 Exponents and roughness values used with Hansen/Larsen method

43

New method 2

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

gzzURCu new ln)(*1*,

κ(3.20)

• Proposed in this thesis and based on the Charnock formula, and from here

on will be referred to as the New Method.

Where C and R are constants defined for different terrains and based on a roughness obtained at approximately 10m. 1. Complex/Onshore zC 0005.0−=

( ) 1244.1ln0891.0 += ozR

2. Offshore zC 002.0−=

( ) 7788.0ln0357.0 += ozR

In both instances the roughness is derived from the turbulence at reference height (~10m) by rearranging (3.17), then by applying (3.18) we can get the standard deviation of the wind speed. Finally the turbulence intensity can be determined by

ref

zu

UIz

,10

,σ= (3.21)

A full explanation of how this method was derived is given in Appendix A. A plot of the turbulence profile and the predictions can be seen in Figure 3.20 below. All methods shown agree quite well with the measured values, with the Hansen/Larsen and the new method performing the best.

44

Figure 3.20 Turbulence intensity profile for Skipheia including 4 predicted profiles

The two tables below contain a comparison of the predicted and the actual turbulence measured at 15 m/s and at specified heights and sites.

Name

Terrain

Height I15

(measured)

I15

Simple Method

I15 Hansen/ Larsen

I15

Charnock New

Method

Horns rev Offshore 62 45 30 15

7.03 7.78 8.51 9.85

8.64 8.89 9.22 9.85

6.40 7.23 8.44 10.98

7.82 8.06 8.39 9.02

7.01 7.66 8.48 9.97

Læsoe Offshore 62 45 30 15

6.59 6.98 7.77 9.03

8.00 8.21 8.50 9.03

6.40 7.23 8.44 10.98

7.87 8.11 8.45 9.09

6.56 7.17 7.94 9.33

Skipheia Offshore9 101 72 41 20 11

6.48 7.18 8.56

10.35 12.19

8.88 9.15 9.65 10.35 11.06

5.87 6.70 8.35 10.84 13.95

7.63 7.87 8.31 8.95 9.58

6.52 7.35 8.63 10.25 11.77

Egmond (NSW)

Offshore 116 70 21

5.35 6.05 8.10

7.12 7.38 8.10

5.05 6.11 9.66

7.36 7.69 8.65

4.98 5.90 8.20

Table 3-17 Measured and predicted Turbulence intensity for offshore sites. Egmond (Near Shore Windfarm) is an offshore mast in the north sea of the coast of the Netherlands

9 Only the wind coming form the sea direction is used, so this coastal can be termed offshore

45

Name

Terrain

Height I15

(measured)

I15

Simple Method

I15 Hansen/ Larsen

New Method

Cabauw Onshore 200 140 80 40 20

7.38 8.61

10.67 12.83 13.03

10.03 10.40 11.04 11.95 13.03

6.47 7.54 9.59

12.92 17.40

8.40 9.11 10.19 11.61 13.27

Tobøl Onshore 62 49 30 15

12.56 13.89 15.79 17.41

14.17 14.65 15.79 17.73

12.11 13.40 16.54 22.29

12.90 13.49 14.80 16.98

Tjæreborg Onshore 90 60 30

8.35 9.62

11.93

10.55 11.02 11.93

7.83 9.40

12.83

9.76 10.55 12.02

Oak creek Complex 79 65 50 10

8.76 9.43 9.92

12.20

9.49 9.67 9.92 11.81

8.30 9.06 10.1

21.02

8.40 8.72 9.16 12.63

Table 3-18 Measured and predicted Turbulence intensity for onshore sites As can be seen all 4 methods fit with reasonable success, with the two site specific methods performing the best, and the Charnock and Simple method fitting the offshore/coastal and onshore sites respectively.

Height I15

Simple Method

I15 Hansen/ Larsen

I15

Charnock New Method

Horns Rev 1.61 -0.63 0.79 0.32 Læsø 1.41 -0.19 1.28 0.28 Skipheia 1.97 -0.48 0.69 0.17 North sea 1.33 0.06 1.62 0.95 Cabauw -0.37 0.09 ----- -0.98 Tobøl 1.61 -0.45 ----- 0.84 Tjæreborg 1.40 -0.23 ----- 0.93 Oak Creek 0.24 -0.37 ----- -0.71

Table 3-19 Turbulence intensity error (%Ti) for all sites at approximately Hub height

46

Estimate type Bias [%]

Relative Bias [-]

Standard deviation [-]

Simple method m = 1.29 m = 0.18 s = 0.010 New method m = -0.13 m = -0.016 s = 0.04 Hansen/Larsen m = -0.45 m = -0.05 s = 0.04

Table 3-20 Bias and standard deviation for the turbulence and relative turbulence intensity error.

Hansen/Larsen performs quite well, but there is an inherent problem in that the exponent p used in the profile needs to be tuned for every site. This was not a problem for the offshore sites where the same value for p and zo were used throughout. On the other hand a comparison for different onshore sites has proven to be problematic due to the uncertainty in the exponent. This method also has a bias 0.05, and a standard deviation of 0.04.

0

0.2

0.4

0.6

0.8

1

1.2

-0.15 -0.1 -0.05 0 0.05 0.1

Relative turbulence intensity error

Cum

ulat

ive

dist

ribut

ion

Hansen / LarsenNormal fit

Mean = -0.05Std = 0.042Sample size = 16

Figure 3.21 Distribution of the relative turbulence intensity error using Hansen/Larsen,

sample size = 16, m = -0.05 and s = 0.042

47

0

0.2

0.4

0.6

0.8

1

1.2

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

Relative turbulence intensity error [-]

Cum

ulat

ive

dist

ribut

ion

[-]

New MethodNormal fit

Mean = -0.016Std = 0.040Sample size = 14

Figure 3.22 Distribution of the relative turbulence intensity error using the new method,

sample size = 14, m = -0.016 and s = 0.04. Unlike the Hansen/Larsen method the new method has no variables other than the terrain roughness. The new method fits the turbulence at lower heights much better than any of the other methods and the bias for this method is only -0.016, with a standard deviation of 0.04.

48

Chapter 4

4 Determine the Load Resistance of the wind turbine

4.1 Introduction

Now that the site loads have been discussed we can now concentrate on the resistance of the wind turbine to these loads. The main aim of this project is to determine the probability of failure of the wind turbine from fatigue loading. Thus we will not concern ourselves with problems arising from extreme loads or non-fatigue related catastrophic failure. It is difficult to quantify the resistance of a wind turbine as such, since it is a combination of many components. So it is necessary to determine the resistance of each of the important components, which will be subject to fatigue loads of sufficient size to be of concern. Table 1 shows the components along with material, which will be investigated10. For the sake of simplicity and time to compute, only the root moments on the tower and blade will be analysed. The validity for this is that the greatest moment will be at the root and any failure due to fatigue will occur at the root first. Also only 1 blade need be analysed since all three blades are assumed the same.

10 Although the Hub is not taken into account it is not because it isn’t important, it is just that the results from flex could not be verified because there was no corresponding measurement from the NM 92.

49

Component Name Material Wohler exp. M Main shaft torque MxNr Cast Iron 6.5 Tilt moment MyNf Cast Iron 6.5 Yaw moment MzNf Cast Iron 6.5 Blade edgewise moment Mx11 Composite 12 Blade Flapwise moment My11 Composite 12 Tower root moment Mxt10 Steel 4

Table 4-1 Components and their material Again to further simplify the process we will only include the loads on the turbine during a constant production duty cycle with no stop start sequences. This will yield acceptable results since fatigue is in the main, a problem associated with wind speeds between 10 and 20 m/s and the wind turbine spends the vast majority of its time in production in this wind speed range.

4.2 Wind turbine aeroelastic model

Before the turbine can be built a certain degree of confidence/assurance in the behaviour of the turbine needs to be attained. This assurance comes from the utilisation of a wind turbine model, known as an aeroelastic model. Aeroelastic models are essential in the design process and development of new wind turbines. Working with a model enables the designer to determine the loads and responses of the many constituent components of a wind turbine using the load conditions specified by 61400-1. However unlike in reality these loads are user defined, meaning the model can be subjected to almost any conditions of the designer’s choice. This allows a more comprehensive study of the behaviour of the turbine than is possible from testing. Therefore, a valid model is a necessity when designing a turbine to withstand the prescribed aerodynamic loads while also having a controllable response.

4.2.1 Verification of the wind turbine model A valid model is one, which has been validated from measurements and before we can draw any conclusions from the simulation results our model must first be validated. This validation takes the form of running simulations at a number pre determined wind speeds and conditions, with similar input conditions to the conditions experienced in the measurements, i.e. wind speed, shear, turbulence etc. The results of the simulation will then be post processed and analysed both in the time and the frequency domain. If the loads are within say 10 % - 20 % of the measurements then the model is assumed to be accurate enough.

50

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Equivalent load comparison for My11

Normailised measured equivalent load [-]

Nor

mai

lised

sim

ulat

ed e

quiv

alen

t loa

d [-]

data 1 linear

Figure 4.1 Comparison of calculated and measured 1 Hz equivalent loads with for blade

root flap moment (m=12); R2 = 0.69 and the slope is x = 0.82y

If this accuracy has not been achieved then the model will have to be tuned until the desired precision is achieved.

4.3 Fatigue

Fatigue is the damage process of a component produced by cyclic loading. The fatigue process begins by first initiating a crack in the surface of the component. Crack propagation is a cumulative process, with each cyclic load causing some permanent plastic deformation. Under these conditions the crack will propagate until some point where the stress is too great and catastrophic failure will occur.

4.4 Wohler (SN) curve

Modern fatigue research began with the work of August Wohler, a German railway engineer, who investigated the failure of train axles which occured well below the ultimate static loading stress. This failure he attributed to the cyclic loading, and after many experiments to try and classify the fatigue strength of material, Wohler developed a set of curves. These are now known as S-N or Wohler curves, these curves describe the number of cycles to failure of a component as a function of stress (load). This is usually on a log-log plot as can be seen in Figure 4.2.

51

Figure 4.2 Log-Log plot of a Wohler (S-N) curve including stress and cycles to failure for 3

different stress ranges The Wohler curve will only be represented as a straight line on a log-log plot this is because the power law describing it is assumed to be valid. In general the experimental fatigue data will not sit perfectly on the Wohler curve, but will be distributed about it. This effect is termed the scatter. A scatter number has been defined as the number of cycles achieved by 10 % of the test specimens divided by the number of cycles achieved by 90 % of the test specimens.

90

10

NN

Tn = (4.1)

The distribution of this data is assumed to be log-normal with

⎟⎟⎠

⎞⎜⎜⎝

⎛Φ=

scatterSxxF )ln()( (4.2)

Where the scale parameter Sscatter is the inverse of the normal distribution

563.2ln

)9.0(2ln N

inv

Nscatter

TTS ≈

Φ= (4.3)

The coefficient of variation is

)1exp( 2 −= scatterscatter SCOV (4.4)

52

To transform from life scatter to stress, we can use the slope of the Wohler curve to give

mNTT /1=σ (4.5)

mCOV

COV scatter≈σ (4.6)

4.5 Cycle Counting

To establish the load ranges from a varying load history it is necessary to use a cycle counting method. One such method is Rain-flow counting method was developed by Tatsuo Endo and M. Matsuiski (1968). This method is analogous to the path followed by rain drops flowing down from one pagoda roof to another, hence the name Rainflow counting. Before the Rainflow path can be defined the time series first has to be converted to a series of tensile peaks and compressive troughs. Where the peaks and troughs represent the maxima and minima of the cycle and the distance between them is the range. An example of how to implement this using the range pair criterion is shown Figure 4.3.

Figure 4.3 Range pair counting criterion (IEC)

These criteria are applied to the first 100 seconds of a 10 min load time history shown in Figure 4.4, where the actual time history is the continuous line and the maxima and minima are the defined by the circles.

53

0 10 20 30 40 50 60 70 80 90 100

-100

0

100

200

Edgewise bending moment

Time [s]M

omen

t [kN

m]

0 50 100 150 200 250 3000

100

200

300

400Load ranges and occurrences for edgewise bending moment

Moment [kNm]

No

of o

ccur

renc

es

Figure 4.4 Time series and the varying load ranges output from the Rainflow counting

Now that the range pairs have been defined the load ranges and their occurrences seen in the histogram in Figure 4.4 can be determined. As mentioned earlier the load ranges are analogous to the Rainflow paths and these paths are defined according to the rules set out by Wirsching and Shehtata (1977) as follows with reference to Figure 4.5: A rain flow is started at each peak and trough. With the peaks being even numbered. When a rain-flow path started at a trough comes to the tip of the roof, the flow stops if the opposite trough is more negative than that at the start of the path under consideration (7-8, 9-10). A path started at a peak is stopped by a peak which is more positive than that at the start of the rain path (not shown but just think of it as the same as for the path started at a trough). If the rain flowing down a roof intercepts flow from a previous path, the present path is stopped if its trough is less negative or positive or its peak is less positive (5–5a, 8–8a). A new path is not started until the path under consideration is stopped.

54

Figure 4.5 Rainflow path according to Wirsching and Shehtata [6] The magnitude of the half cycle load is the projected distance along the load axis, e.g. 2-3, 3-12, 6-11. For a sufficiently long time series any tensile half cycle will be followed by a corresponding compressive half cycle of the same range. By pairing these together the full cycles (load ranges) can be defined. These half and full cycles can be seen for the same load history in Figure 4.6.

0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

1. Cycle, up

2. Half-cycle, down

3. Cycle, up

4. Half-cycle, up

5. Cycle, down

6. Half-cycle, down

7. Half-cycle, up

peaks, counted from 0

Load

Rainflow cycles extracted from signal

peaks from signal

Figure 4.6 Load ranges determined from the Rainflow counting method

55

4.6 Linear damage rule

To get the results from our load history we have to be able to relate our load ranges and occurrences to the Wohler curve. This is achieved using the linear damage rule. The linear damage sum was first proposed by Palmgren for application to the ball bearing industry in 1924. In 1945 Miner from Douglas Aircraft applied the linear damage rule to tension-tension axial fatigue data. Miner demonstrated excellent11 agreement between the linear damage rule and experimental results. It states that for a number of life a cycles Ni that a cycle ni consumes ni/Ni of the fatigue life of a component for constant amplitude loading.

( )∑=i i

ii SN

nD (4.7)

Where ni is the number of stress cycles12 with stress range Si, and N(Si) is the number of cycles to failure at the stress level Si. The sum is over all stress ranges Si. This concept can be seen in Figure 4.7.

Figure 4.7 S1,S2, and S3 are the stress range cycle of equal amplitude.

However since Miners work was conducted, the linear damage rule has been shown to be unreliable. The main reasons for this are attributed The Palmgren-Miner rule does not consider sequence effects, i.e. it makes no distinction between high stress cycles followed by low stress cycles and vice

11 It must be noted that these results were obtained for constant amplitude loading for a limited number of tests. 12 In the case of this thesis the cycles are load cycles and not stress.

56

versa. Sequence effects have been shown to give differing damage levels which the rule completely ignores, leading to some un-conservative results. The rule says that damage accumulation is independent of stress level. This means that the rule is dependent on the stress amplitude and doesn’t take account of the mean stress level. For example if the mean stress is not zero it could very well be already heavily stressed and failure could occur with very small stress amplitudes. Both of these failings have to some extent been tackled and by the use of correction methods/ factors the error can be reduced.

4.6.1 Stress correction The rates of crack propagation in tests have shown to be related to the mean stress from the load as well as the stress load range. Since cracks can only propagate under tensile loads there will be no damage caused if the cycle induces compressive stresses. On the other had if the load cycle is tensile then damage will occur. Three mean stress correction methods13 have been developed to go some way in dealing with the mean stress. Goodman method - generally suitable for brittle materials. Gerber method - generally suitable for ductile materials. Soderberg method - generally the most conservative.

Figure 4.8 Mean stress correction curves

All three of the methods apply only when the associated Wohler curves are based on fully reversed loading. The size of these corrections become significant where the fatigue cycle has a large mean stress compared with the stress range. 13 It must be noted that the stress correction methods were not implemented and are only included to give a general impression of the failings of the Miner rule.

57

4.6.2 Stress reduction factor A factor used in the reduction of the error caused by the sequence effects is the stress reduction factor qo, shown in Figure 4.9. This factor is best explained by following the procedure used in its determination as shown below.

Figure 4.9 Stress factor qo derived from tests on steel with welded seams [Heuler]

4.6.2.1 Procedure to determine qo From constant amplitude S-N curve derived from testing we have the relation;

mAD

m NN σσ ∆=∆ (4.8) Where DsA is the fatigue and ND is knee number of cycles. Now for example take the simplified variable amplitude situation as shown in Figure 4.7. The life curve for a component subjected to these conditions can be predicted by applying the Palmgren-Miner to each block based on constant amplitude loading. Usually this will not be successful as this method tends to over predict the cycles to failure. However it is possible to alter the prediction to reflect reality by correcting with qo. Firstly the life prediction must be made

mAD

mii NN σσ ∆=∆ (4.9)

58

Where Ni is the cycles allowed for the stress iσ∆ and the subscript i denotes the load block. The partial damage can now defined as:

m

A

i

D

i

i

ii N

nNn

d ⎟⎟⎠

⎞⎜⎜⎝

⎛∆∆

==σσ

(4.10)

With the total damage

∑= idD (4.11)

Now we will most likely find that failure occurred at a number of cycles less than predicted, corresponding to a total damage of D<1. However we want the total damage to be D=1. This is achieved by reducing the S-N curve by the factor qo, giving the new allowed cycles iN ′ .

m

i

ADi

qNN ⎟⎟

⎠

⎞⎜⎜⎝

⎛∆

∆=′

σσ0 (4.12)

m

A

i

D

i

i

ii qN

nNn

d ⎟⎟⎠

⎞⎜⎜⎝

⎛∆∆

=′

=′σσ

0 (4.13)

With qo <1, the new damage id ′ will be greater than di and we can set the values of qo such that we get D=1 at failure.

4.7 Equivalent loads

In this project we are concerned with the load spectrum of the operational mode of the wind turbine. To simplify the load spectrum it is often necessary to represent these loads as a single damage-equivalent load So. This is a constant load range which in Neq cycles will produce the same amount of damage as the actual load spectrum with different load ranges Si and cycles ni. The equivalent load So is defined as;

59

m

eq

mi

ii

o N

SnS

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=∑

(4.14)

The equivalent number of cycles Neq can be set to any value. For instance if the load spectrum is from a 10 min time history then a choice of Neq = 600 would give the 1 Hz equivalent load. This is quite useful when comparing measured data to simulated data. Another choice of Neq can be 5 x 106, which is the standard number of cycles for fatigue tests, this choice will enable direct comparison to standards.

60

Chapter 5

5 Design calculations and parameter distribution

5.1 Introduction

Now that all our distributions are known we can finally calculate the fatigue damage on the wind turbine. The LRFD (Load Resistance Factor Design) methodology will be applied to the data we have collated so far. To simplify the calculations the results and data are normalised to their respective Characteristic values based on an IEC Class II site and turbine (61400-1). That is, the loads obtained form an aeroelastic simulation when using the IEC parameters.

5.2 Limit state function

Whether the wind turbine will fail in service is defined by the so called limit state function Z(x), defined by

In our case the limit state function is dependent on the vector of parameters (dependents) x, this vector contains the variables we have been discussing up to now and can be seen in (5.3).

⎪⎩

⎪⎨

⎧

<=>

=failed

stateLimittheonisfailednot

xZ000

)(

61

What we are interested in is the probability of failure, and this is defined as

[ ]0)( ≤= xZPPf (5.1) The limit state function Z can be divided up in to site loads and resistance as follows

( ) ( )SR xSxRxZ −=)( (5.2)

Where R(xR) is the resistance and S(xS) is the site loads. We should now define our vector of dependents x, this vector has many variables much more than were investigated in this thesis. So the following vector of dependencies will contain just the ones that were used.

),,,,,,,( dim seedfemoAavg xxxqxTiUx σα ∆= (5.3) Where,

avgU Mean wind speed

Ti Mean turbulence intensity α Shear exponent from the power law

Ax σ∆ Material fatigue strength divided by a characteristic value

qo load sequence reduction factor on fatigue strength

femx Factor for FEM calculation errors

dimx Dimension factor for deviations in bending resistance

seedx Factor for random wind field seed

The site loads and resistance can be functions of all of the components of x, however it is easier to define them as it would be expected with the site loads being dependent on the wind regime and the resistance being dependent on the structural variables. Thus we can define them as follows ( ) ),,( αTiUSxS avgS = (5.4)

62

( ) ),,,,( dim seedfemoAR xxxqxRxR σ∆= (5.5) If we are to determine the limit state function, both S(xS) and R(xR) will have to be evaluated.

5.3 Site equivalent fatigue loads

The site loads S(xS) are determined from an aeroelastic analysis and are in the form of a varying load time series. As mentioned in the section 4.5 this time series has to be converted to load ranges and occurrences as in the histogram in Figure 4.4. The Rainflow counting method is employed to determine these load ranges and occurrences and from these the site equivalent loads can be established. This is done in approximately the same way as for the equivalent loads, but with some minor adjustments. As mentioned earlier the choice of Neq is arbitrary, but with regard to site loads it is common to define Neq = Nlife, where Nlife is the number of 1Hz cycles in the expected life of the wind turbine (20 yr). To calculate the 20 year equivalent load the wind regime at the site must be known, i.e. the distribution of the wind speed. As mentioned earlier the wind speed at a site is always modelled using a two parameter Weibull distribution,

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−−=

k

AUUP exp1)( (5.6)

and the wind climate is related to the 20 year equivalent load by

[ ]m

i

miiyr USopSo

/1

20 )( ⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑ (5.7)

Where: m Wohler exponent [-]

pj Probability (time fraction) [-]

For example, by applying (5.7) to the load spectrum shown in Figure 4.4, a varying load spectrum can be reduced to a single equivalent load. For a complete analysis the loads with respect to all the components of S(xS) need to be known, such that all possible combinations of S(xS) can be evaluated. This would necessitate a full simulation for every possible outcome and since x

63

contains 9 variables this is totally impractical. For this reason an estimation of the loads has to be made. A number of aeroelastic simulations are carried out to determine the load at a selected number of intervals over the parameter range (see Table 5-1). By only having one variable and holding the rest constant we can get the derivative of the loads with respect to that variable. Thus we can interpolate between these values to obtain any combination of S(xs).

Component Characteristic Value Calculation range14 Interval

Uavg15 [m/s] 8.5 3.5-11.5 1 Ti [%] 16.0 5-25 5 a [-] 0.2 0-1 0.25 Y [deg] 6.0 0-12 2

Table 5-1 Parameters used in the aeroelastic analysis A number of simulations were carried out in Flex5 over a range of wind speeds and using the characteristic values shown in Table 5-1. The results form these simulations were post processed by Rainflow counting and the 20 yr equivalent load calculated using (5.7). This equivalent load is the characteristic (nominal) load Schar(xS). Figure 5.1 shows a series of these 20 yr equivalent loads, in which the change in wind speed is the change in Uavg. Notice that Myt10 as with most of the other components16 can be easily fitted to a linear or 2nd order line polynomial.

14 The wide range for the wind speed calculations is just for demonstration purposes, ordinarily a wind turbine would not be sited where Uavg < 7 m/s. 15 Uavg is the average wind speed in a Weibull distribution 16 MxNr is fitted to a 3rd order polynomial.

64

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40.2

0.4

0.6

0.8

1

1.2

1.4

1.6Myt10

Normalised wind speed [-]

Nor

mai

lised

load

[-]

y = - 0.18*x2 + 1.5*x - 0.3

Measurements quadratic

Figure 5.1 Results for tower root bending Vs wind speed, both normalized to their

characteristic Values (Table 5-1) There are several methods to estimate the Site loads S(xS) from functions produced by these limited simulations, one such method is Taylor expansion, with a base point xo and in this case an obvious choice for xo would be xo = xchar

xcharxjjjcharjchar x

SxxSxS=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−+= ∑ )()( , (5.8)

A linear approximation of this formula is obtained by putting the derivative in terms of slope

xcharxjcharj

jcharj

jjcharjchar xx

SSxxSxS

=⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−−+= ∑ )(

)()()(

,

,, (5.9)

And this reduces to a simple difference

∑ −+=j

jcharjchar SSSxS )()( , (5.10)

Although this method gives reasonable results it is dependent upon the base value such that its accuracy depends on the distance of x from xo. A more accurate approach would be to fit the simulation results to polynomials directly as follows.

65

Remove the offset17 values from the polynomials and set all x values to their respective characteristic value xo (5.11). Sum all the polynomials and find the difference from Schar, this is now the offset (5.12). Now if all the values of x were equal to xo then S(x) = Schar, and any change in S(x) is directly due to a change in any of the values of x. This in effect is creating a new polynomial to describe all of x.

)(...)()()()( ,3,2,1, nooooo xSxSxSxSxS +++= (5.11)

∑−=j

jocharx xSSoffset )( ,, (5.12)

)(...)()()()( 321 nS xSxSxSxSOffsetxS ++++= (5.13)

An additional load factor is added to Sx,char for additional safety.

)()( SfdesignS xSxS γ= (5.14) The load safety factor is defined in the 61400-1 and has a value of fγ ¥1.

5.4 Resistance equivalent fatigue loads

To obtain the resistance of the fatigue strength, additional information will be required to determine a stress from the bending moments output from the Flex5. We will use some concepts adopted from design procedures. If we define the resistance of a material to fatigue as

)()( dim AR WxxR σ∆= (5.15) Where xdim is a factor owing to the uncertainty in the dimensions of the component, W is the nominal bending resistance and Aσ∆ is the fatigue strength of the material. xdim is an unknown quantity, but as with most processes containing limited data it will be assumed to be normally distributed (central limit theorem).

17 Polynomial = x3+x2+x +offset

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To find the nominal bending stress W we will use the aforementioned design procedure of setting the bending strength equal to the characteristic design strength, such that

)( charxSW = (5.16) An added complication to the definition of S(xchar) is that the characteristic load is dependent upon the wind field seed. This so called wind field seed is used in the process of modelling the wind field and is the base input for the random wind field generator used in the model. In layman’s terms, the wind random generator has to start somewhere, and that somewhere is the wind seed. It can be said that the output values of the random generator are not entirely random and that are related to the wind seed. A number of these different random seeds should be used to obtain more realistic load calculations, however this leads to somewhat arbitrary load calculations. Therefore to get the average characteristic load we need to multiply by the factor xseed. This factor is found from carrying out a number of simulations using an identical load spectrum but with a varying random wind seed. The variation in the resulting simulated loads can then be directly attributed to the wind seed. In Veldkamp [24] 100 such simulations were carried out and the results show that all components have a different response to the change in wind seed.

seedcharchar xxSxS )()( = (5.17) As with xdim there will be uncertainty on the fatigue strength as discussed in (Palmgren-Miner). This uncertainty is owed to the sequence effects and general discontinuities in the material.

charAoA Axq ,σσ σ ∆=∆ ∆ (5.18)

Where,

Ax σ∆ ratio of the characteristic fatigue and the nominal fatigue strength.

qo factor depending on the load sequence effects. As with the design loads an additional safety factor is introduced, a material factor mγ . A stress reserve factor SRF is also used, in an ideal situation this factor would be equal to 1, but in most design cases an air of caution is used and the more likely value is between 1 and 1.1. The design resistance can now be defined by substituting (5.16) and back into (5.15) and including the additional factors.

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)(, charavgseedmcharAofemD xSxSRFxqWxR

Aγσσ ∆= ∆ (5.19)

To solve for the Z(x) it will be necessary to apply a simulation technique to evaluate the probabilities of failure. The method to be used here is the Monte Carlo method (see Appendix E for an explanation of this method)

5.5 Example

A CLASS II wind turbine is to be erected on a pastoral site. There is some worry about the loads on the blade in the flap wise direction. Therefore a limited parameter investigation of the failure of probability will to be carried out.

5.5.1 Site conditions Uavg 8.5 [m/s] The analysis will be carried out using the limit state function to determine the probability of failure

( ) ( )xSxRxZ −=)( (5.20)

With the vector of parameters limited to x = (Uavg, qo, xDsA) (5.21)

5.5.2 Site loads Using (5.4) we define the site loads as ( ) )(1 avgUSxS = (5.22)

From before, it was mentioned that the site loads18 are in fact a function of a number of parameters, and the change in site loads caused by the variation of 1 parameter can be easily found by allowing the parameter under consideration to be variable, while holding the rest constant. This results in the following polynomial function describing the site loads S(x1) = -0.6x2 + 1.9x - 0.309 (5.23)

18 Where site loads are mentioned they refer to 20 yr equivalent site loads

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where S(x1) will give the site equivalent load as a function of the wind speed normalised to the characteristic site value. The true values can be seen in Figure 5.2 below, it is noticeable that the error arising out of using an estimation of this kind will be quite small. Wind speed In Chapter 3 we found that the average wind speed cannot be assumed to be constant but will be distributed around some mean m and with standard deviation of s. For example a typical pastoral site is IJmuiden, Netherlands. The coefficient of variation calculated with MCP for this site was 0.07. Yearly variation in wind speed was COV1,yr = 0.07, this can be transformed into a life time variation using the relation

20,1

,20yr

yrCOV

COV = (5.24)

The max error assume when extrapolating from 10m to hub height for this region is 0.4 m/s, taking this as 3s, then s = 0.4/3 = 0.133, and COV = 0.133/8.5 = 0.015. The combined coefficient of variation of the wind speed is

061.0015.02006.006.0 2

22

, =+=avgUCOV (5.25)

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Figure 5.2 Equivalent fatigue loads for the blade in the flapwise direction as a function of

wind speed.

5.5.3 Resistance From the example the Resistance is ( ) )(3,2 oA qxxR σ∆= (5.26)

But before we deal with R as a function of x we will firs take care of the additional factors. For added safety due to the uncertainty the IEC have recommended values for the safety factors gm and gf. The factors are then multiplied into R(x) to increase the resistance. gm = 1.1 gf = 1.15 SRF = 1.05 Adding in the factors gives ( ) )( SRFqxxR fmoA γγσ∆= (5.27)

Fatigue strength ratio Fatigue strength ratio defined as

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charA

AAx

,σσ

σ ∆∆

=∆ (5.28)

From Veldkamp [24] we find that the coefficient of variation of Aσ∆ is 0.08, and with the fact that the site characteristic load is set to 1, Ax σ∆ is equal to Aσ∆ . From the standards an extra safety precaution suggested in the standards is to use the 0.023 fractile. Since the mean = 1, then the standard deviation = COV, and the 0.023 fractile can be defined as

)08.0(211

−=∆ Ax σ (5.29)

Stress factor qo Up to now there is still not enough data or testing on composites to make an accurate estimation, so conservative values are nearly always used. In our case we will have a scatter number Tn = D90/ D10 = 10 (Veldkamp [24]). Assuming a miner curve slope = 12, the scatter number can be put in terms of stress using (4.5) Ts = 10 1/12 and for a mean of 1. The COV is

08.05632.2

ln=≈ σT

COVqo (5.30)

Parameter Dist Mean Fractile Std. Dev COV Design Uavg N 8.5 0.5 0.061 8.5 qo LN 1 0.5 0.07 1 XDsA LN 1 0.023 0.04 1

Table 5-2 Summary of the parameters used in the example Example results

Pfail Beta (20 yr) My11 3.6E-4 3.78

Table 5-3 Probability of failure and reliability index The Monte Carlo simulation does not output the influence factors, but they can be easily calculated by summing the contributions to the uncertainty. This is achieved by taking the difference of each calculated parameter from their

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respective mean, summing these values and then dividing the components by the total.

)]()()([var Ameanmeanmean xRabsqoRabsUloadSabsInf σ∆−−−= (5.31)

Where Rmean is Smean multiplied by the 3 safety parameters gm, gf , and SRF

∑= varInfInftot (5.32)

tot

j

InfInf

xInf var,)( = (5.33)

Alternatively the variances can be used in much the same way by replacing the load expressions in Infvar with the variance of the variables as in (5.34).

)]var()var()[var(var AxqoUloadInf σ∆= (5.34) In other words if we say that the sum of all the variances is equal to 1, then the contribution (influence) of the mean wind speed on the load is

100)var()var(

)( xAll

UU load

load =α (5.35)

Parameter Influence [%] Uavg 6 qo 47 xDsA 47

Table 5-4 Contribution to uncertainty

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0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

0.01

0.02

0.03

0.04

0.05

0.06My11

Normalised loads [-]

Pdf

[-]

Site loadsResistance

Figure 5.3 Results from the Monte Carlo simulation

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Chapter 6

6 Results

The following tables and graphs are the results from the Monte Carlo simulation for two specific cases.

1. Aalborg site a. With Weibull A = 8.2, k=2.25 b. Turbulence intensity = 12% c. Shear exponent a = 0.17

2. The base case, i.e. a simulation based on an IEC Class II turbine and site. Take notice of the variables included in the analysis. Not all possible variables are included and thus the results will reflect this. The parameters used are all summarised in Table 6-1. Also when ever loads are referred to they mean the 20 year equivalent loads. The safety factors were as specified in 61400-1 with

gm = 1.1 gf = 1.15 Following common design practice a reasonably conservative stress reserve factor was chosen SRF = 1.05 These safety factors were applied to all components.

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Parameter Dist Mean Fractile Std COV Design Site IEC Aal Uavg

Ti a

N N N

8.5 0.16 0.2

8.2 0.12 0.17

0.5 0.5 0.5

0.06 0.03

0.061

8.5 16 0.2

Material XDsA Cast Iron

Blade Welded steel

LN LN LN

1.0 1.0 1.0

0.023 0.023 0.023

0.07 0.08 0.10

1.0 1.0 1.0

qo Cast Iron Blade Welded steel19

LN LN LN

1.0 1.0 0.8

0.5 0.5 0.5

0.13 0.08 0.10

1.0 1.0 1.0

Seed MxNr

MyNf MzNf Mx11 My11 Mxt10

N N N N N N

1.0 1.0 1.0 1.0 1.0 1.0

0.5 0.5 0.5 0.5 0.5 0.5

0.016 0.01 0.009 0.002 0.014 0.031

Geometry xdim N 1 0.5 0.03 Stress xfem N 1 0.5 0.03

Table 6-1 Summary of the parameters and their statistical properties As shown earlier the loads can be normalised to the base case, i.e. the loads as a calculated using the parameters identical to the IEC parameters will be equal to 1. Table below show the difference in the loads for the IEC site and the Aalborg due to the difference in their parameters mean values.

19 The value of 0.8 is that suggested for added safety in welded steel components.

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IEC loads

(normalised)Aalborg loads (normalised)

MxNr 1 0.84 MyNf 1 0.68 MzNf 1 0.75 Mx11 1 0.95 My11 1 0.72 Myt10 1 0.76

Table 6-2 IEC class II and Aalborg 20 year equivalent loads normalised to the IEC class II loads

The reduction in the loads shown can be explained by the difference in the site parameters, it is obvious that the decrease in the mean turbulence intensity is the main factor for the reduction in loads, since the difference in the mean wind speed and shear exponent is quite small.

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30.4

0.5

0.6

0.7

0.8

0.9

1

1.1MzNf

Turbulence intensity [%]

Nor

mai

lised

Loa

d [-]

MeasurementsFit

Figure 6.1 Normalised Aalborg 20 year equivalent load as a function of normalised Aalborg

turbulence intensity, both are normalised to IEC class II. An example of this is can be seen in Figure 6.1 where a reduction of 25% in the turbulence causes a corresponding reduction in the loads ≈ 17%, i.e. the difference between the loads at Ti =1 and Ti =0.75. From Table 6-3 it can be seen that the full reduction in loads is 25%, meaning that the remaining 8% is made up of a combination of the other parameters; shear exponent, wind speed and the Weibull shape factor.

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The calculated probability of failure based on 1 x 107 simulations can be seen for the two sites in table Table 6-3 below.

Pfail (20 yr) Beta (20 yr) Beta(1 year)

IEC Aal IEC Aal IEC Aal

MxNr 8.6E-04 2.0E-05 3.13 4.11 3.64 4.46 MyNf 1.4E-03 1.0E-07 2.98 5.20 3.52 5.44 MzNf 1.1E-03 3.0E-06 3.06 4.53 3.59 4.83 Mx11 2.3E-04 5.3E-05 3.50 3.88 3.94 4.26 My11 4.1E-04 0 3.35 - 3.81 - Mxt10 1.6E-02 1.5E-04 2.13 3.61 2.92 4.04

Table 6-3 Probability of failure and reliability index for selected components

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06Myt10

Normalised loads [-]

Pdf

[-]

Site loads AalSite loads IECResistance

Figure 6.2 Results from the Monte Carlo simulation for the tower bottom bending moment

showing the distributions of the loads and the resistance for IEC and Aalborg site. The distributions of the loads for the tower bottom at the two sites are shown in Figure 6.2. Note that the normalised site loads can be seen to deviate from 1. This is due to the introduction of the offset for the wind direction20, which will in effect reduce the mean loads by 0.1. This offset is also included in the calculation for the tower loads in Aalborg, which reduces the mean shown in Table 6-2 to ≈ 0.66. 20 Since we assume that the wind only comes from one direction, this will lead to an over estimation of the tower loads. This over estimation has shown be approximately 10%

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Since the same COV’s were used for both sites the influence factors for the IEC and Aalborg sites will be the same and are shown in Table 6-4.

qo DsA Ti Uavg Shear xdim xfem xseed

Total [%]

MxNr 67 20 5 0 0 4 4 1 100 MyNf 59 18 10 6 1 3 3 0 100 MzNf 64 20 6 4 0 3 3 0 100 Mx11 43 43 1 0 0 6 6 0 100 My11 36 36 10 5 1 6 6 1 100 Myt10 37 37 6 9 0 4 4 4 100 Table 6-4 Influence factors determined from the Monte Carlo simulation for Aalborg in

percent

0%

20%

40%

60%

80%

100%

MxNr MyNf MzNf Mx11 My11 Myt10

seedfemdimsigmaAqoalphatiwind

Figure 6.3 Results from Table 6-4 in graphical form

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Chapter 7

7 Conclusions and further work

7.1 Conclusions

It was the intention of this report to investigate the effect of the uncertainty the site parameter have on the fatigue loads. The parameters that were investigated were: • Mean Wind speed at 10 m height • Wind shear exponent and speed profile • Turbulence intensity profile • Accuracy of the wind turbine model for load estimation

By applying the Measure Correlate Predict technique to the vast amount of data available from the windadta.com, KNMI and the German weather service a clear picture of the uncertainty in wind speed prediction was obtained for NW Europe. The method used proved to be an unbiased estimator with m≈0, with Coefficient of Variation, V≈ 0.06, and from a comparison with WAsP results the MCP method proved to be more accurate. It must be mentioned though that unlike WAsP this method requires a Met mast to be placed on the site for a duration of at least 12 months. Several methods of determining the wind shear a were used, one of which was exact (fit based on measurements) and the other two were based on the estimation of terrain roughness zo. The exact method used concurrent wind data from several heights, while the others used wind speed measurements from the lowest height on the measurement mast ≈10m, as is the typical case in reality. The second method of prediction was based on the work of Petersen (Risoe) where the terrain roughness can be estimated visually. The third method was to

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derive the roughness from turbulence intensity measurements. The accuracy of these methods was determined by finding the error between these two and the so called exact (measured) value for a. The results for several sites in all the common terrains showed that there was significant bias in the calculation of error with the relative mean error, m =-0.14 and a standard deviation of s = 0.148. Although the both methods provided more or less the same accuracy the subjectivity of the Petersen method could prove to be unreliable. The opposite can be said for any method derived from measurements and thus the method derived from the turbulence intensity is the method of choice. The Power law and the Logarithmic Law were used to investigate the uncertainty in the mean wind speed prediction when extrapolated to hub height from lower heights. As with the wind shear calculations the site roughness was determined using the same methods. Both Petersen and the derived zo were used with the power law, while only the derived zo was used with the Log Law. The accuracy for all 3 (2 Power Law and 1 Log Law) was quite similar with a bias close to zero and a relative standard deviation ≈ 0.4. Over all the different terrains the Power Law using both the derived zo and Petersen proved to be the most robust, followed by the Log Law (also derived zo). However, since the results are quite close to each other the Log Law is to be preferred over the Power Law calculated using Petersen. In calculating the uncertainty in the turbulence intensity, a number of methods were used, two of which were new methods.

1. Established Methods o Charnock – based on the Logarithmic Law using the Charnock

roughness formula o Simple method - Iz = 1/ ln(z/zo) where zo is derived from

turbulence at the lowest available measurement height.

2. New Methods o Hansen/Larsen (DTU, Risoe) – base on the power law o New Method (this report) – empirical fit base on the Charnock

method for Iz. Both of the established methods are of a very general nature and thus were not as accurate as the two new methods which are site specific in nature. The Simple method had significant bias with a relative mean error; m ≈ 0.18 while the Hansen/Larsen (H/L) and the New method had a relative mean error; m ≈ -0.05, m ≈ -0.016 respectively. All 3 methods had a standard deviation s ≈ 0.04. Although the H/L method was quite accurate this accuracy was the result of tuning the exponent for each site, there were no such problems in the new method where the only variable was the roughness zo. For the following reasons we can conclude that the New method proposed in this thesis is best;

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1. Average turbulence is unbiased 2. No tuning is necessary 3. Standard deviation is the same

The results obtained from the analysis of the site parameters have allowed for a more accurate description of the various uncertainties than was previously possible using rules of thumb and educated guesses. To reduce the uncertainty arising out of the extrapolation methods described above the obvious thing to do would be to require all sites to have measurements at hub height. The parametric study has shown that the dominating influence on the failure comes from the uncertainty in the fatigue resistance parameters. The influence of derivatives of the site parameters on the failure has shown that for some components the influence is much higher than others; with the yaw, tilt, blade root flapwise and tower bending moments being the most affected by the site parameters. For these components the uncertainty in the wind is the most important followed by the turbulence intensity and the shear exponent. The difference in the loads for the IEC site and the Aalborg site show that if Aalborg is assumed to be a Class II A site then there is a high degree of conservatism in terms of loads, with reductions as much as 32% in the 20 year equivalent loads for some components. Regarding the choice of the normal distribution for the uncertainty, it seems to be acceptable based on the central limit theorem and in absence of any other contradictory data. It must be noted however that the choice of distribution will have an effect on results. The Monte Carlo method of simulating loads cases is a little heavy on computational time, but has proved to be an invaluable tool in the implementation of this project. In the introduction, it was mentioned that the goal of the project was to quantify the uncertainties and determine if the costs of designing a wind turbine could be reduced by implementing a LRFD design. This investigation has shown that the uncertainties in the influence parameters can be quantified making it possible to design to a target failure probability (reliability).

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The probabilistic approach taken here will prove to be a valid method in the design of wind turbines, with the advantages of this method over conventional design being that a target failure probability can be achieved. The structural reliability analysis example of the Class II A turbine placed at Aalborg is a clear example of the usefulness of this approach, in the sense that any ‘over/under designed’ components can easily be identified by their failure probability in accordance with some prescribed failure probabilities such as those shown in Table 2-2. Any deviation from the prescribed values can be corrected by adjusting the design partial safety factors accordingly.

7.2 Further work

Wave loads For offshore sites the effects of wave loads will need to be considered. In a previous Thesis [12] Petersen investigated the effects of linear and non-linear wave loads on the wind turbine. A combination of this thesis and the one by Petersen would be a good starting point for the investigation. Wind farm wake At present the most economical application of wind energy is in a wind farm configuration, this is due to the reduction in costs of the construction, logistics, grid connection etc. the down side to wind farm configurations is that they lead to an increase in the wind turbine loads due to the effect of one wind turbine operating it the wake of another. An investigation into the effect of this wake on the uncertainty in the turbulence intensity (according to some simple wake formula) should be made and its effect on the probability of failure quantified. Tuning an aeroelastic model Before a wind turbine can be certified a valid model has to be produced. A valid model is one which has been verified using measurements from the prototype turbine. In such a case the loads calculated from each of the components of the model will have to be within 10 -15 % of the actual measured values. If the model is not within this range the wind turbine components will have to be “tuned” to resemble reality. This can be carried out using some generic model and wind turbine but the procedure will have to be in cooperation with a manufacturer. Parametric study allowing the comparison of Monte Carlo simulation and Reliability methods During this thesis it was necessary to carry out a number of Monte Carlo simulations for each component. Although the Monte Carlo method is computationally expensive, it becomes even more so when using MatLab, which

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on an average takes 1 hour per component for 1 x 10 7 simulations. The reason for so many simulations is that to find a probability of 1 x 10-7 we need to do 1 x 10 7 simulations. This makes the Monte Carlo method a bit too expensive on computing time. Reliability methods such as the First and Second Order Reliability Methods (FORM and SORM) are also widely used in structural analyses. It is suggested to investigate these methods to determine their suitability for use with MatLab. They are said to be much quicker than Monte Carlo but at the expense of the accuracy. It is proposed that a limited parametric study is carried out, having some of the focus on quantifying the difference in accuracy and speed between these methods, and coming to some conclusions regarding which method is the most appropriate to be used with MatLab. Alternatively a software program could be designed in another computing language, which does not have the problem of slow computation time associated with MatLab.

Aside Determine if any improvements/modifications can be made to the Hansen/Larsen turbulence profile method.

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References

[1] C. Berger, K.-G. Eulitz, P. Heuler, K.-L. Kotte, H. Naundorf, W. Schuetz, C.M. Sonsino, A. Wimmer, H. Zenner, Betriebsfestigkeit in Germany — an overview.

[2] M. Anderson. A review of MCP techniques. Technical Report 01327R00022 issue 02, Renewable Energy systems Ltd, July 2004.

[3] M. Anderson. MCP errors. Technical Report 01327 R 00025, Renewable Energy Systems Ltd, January 2005.

[4] R. Barthelmie, O. Hansen, K. Enevoldsen, J. Højstrup, S. Larsen, S. Frandsen, S. Pryor, M. Motta, and P. Sanderhoff. Ten years of meteorological measurements for offshore wind farms. Journal of Solar Energy Engineering, 12(2):170–176, 2005.

[5] H. Braam, J.J.D. van Dam, C.J. Christensen, M.L. Thøgersen, G.C. Larsen, and K.O. Ronold. Methods for probabilistic design of wind turbines. Technical Report R-1082(EN), Risø National Laboratory, Roskilde, December 1998.

[6] Det Norske Veritas and Risø National Laboratory. Guidelines for Design of Wind Turbines. DNV/Risø, second edition, 2002.

[7] Martin O.L. Hansen, Aerodynamics of wind turbines, James and James, 2000

[8] Kurt S. Hansen, Gunner C. Larsen, Database on Wind Characteristics, Analyses of Wind Turbine Design Loads, Risø National Laboratory, Roskilde June 2004

[9] K.S. Hansen and G.C. Larsen. Parameterisation of turbulence intensity. In European Wind Energy Conference Madrid. EWEA, EWEA, June 2003.

[10] K.S. Hansen and G.C. Larsen. Parameterisation of turbulence intensity. In European Wind Energy Conference Madrid. EWEA, EWEA, June 2003.

[11] International Electrotechnical Committee. IEC 61400-1: Wind turbine generator systems part 1: Safety Requirements (draft). IEC, 2nd edition, 1998.

[12] K. S. Correa Bomholt Pedersen. Relative damage on offshore wind turbines induced by linear and nonlinear wave loadings. Master’s thesis, Denmark’s Technical University, Lyngby, August 2005.

[13] Yung-Li Lee, Jwo Pan, Richard, B. Hathaway, Mark E. Barkey, Fatigue and testing analysis (theory and practice), Elsevier Butterworth–Heinemann, 2005

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[14] Bernhard Lange, Rebecca Barthelmie, Jørgen Højstrup, Description of the Rødsand field measurement

[15] Clifford H. Lange, Probabilistic Fatigue Methodology and Wind Turbine Reliability, Civil Engineering Department Stanford University, May 1996

[16] Gunner C. Larsen, Kurt S. Hansen, Database on Wind Characteristics Users Manual, winddata.com, November 2001

[17] James F. Manwell, Anthony F. Ellis, Mohit Dua Wind Resourse Data Interpretation Report for Ipswich town, December 17, 2004

[18] Theory and Problems of Probability, Random Variables & Random Processes, Schaum's Outlines, McGraw-Hill

[19] Fundamentals Of Probability And Statistics For Engineers, John Wiley And Sons

[20] Anthony L. Rogers, Comparison of the Performance of Four Measure-Correlate-Predict Algorithms.

[21] Murray R. Spiegel, John Liu, Schaums Mathematical handbook of formulas and tables, McGraw-Hill, 1000.

[22] Uwe Schmidt Paulsen, Description of the data acquisition system installed on the NM92/2750 kW Wind Turbine at Aalborg East, Risø 2004

[23] Keneth thomsen, Wind turbine loads, power point Risø

[24] D. Veldkamp, Chances in wind energy: A probabilistic approach to wind turbine fatigue design, PhD thesis, Delft University Wind Energy Research Institute, (unpublished)30 January 2006

Data Websites [25] www.winddata.com

[26] Deutscher Wetterdienst (German weather service) www.dwd.de/en/FundE/Klima/KLIS/daten/online/nat/index_standardformat.htm

[27] Royal Netherlands Meteorological Institute (KNMI) www.knmi.nl/samenw/hydra/register/index.html

Websites [28] Monte Carlo simulation

www.visionengineer.com/mech/monte_carlo_simulation.shtml

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[29] Monte Carlo simulation http://www.riskglossary.com/link/monte_carlo_method.htm

[30] Potential wind speed (KNMI) http://www.knmi.nl/samenw/hydra/faq/upot.htm

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List of Figures

Figure 1.1 Increasing size of output of wind turbines ............................................ 2 Figure 1.2 Falling price of wind energy alongside the installed capacity of wind energy (source BWEA)........................................................................................... 3 Figure 2.1 Simplified Flow chart of the design procedure of a wind turbine (Veldkamp) ............................................................................................................. 6 Figure 2.2 Probability density functions for loads and resistance .......................... 9 Figure 3.1 Wind speed distribution Measured and Weibull fit for Horns rev (Denmark offshore)............................................................................................... 14 Figure 3.2 Mean Geostrophic winds over northern Europe.................................. 16 Figure 3.3 Correlations between hourly mean values for Kegnæs and Risø ........ 18 Figure 3.4 Location of sites used in the MCP analysis along with the sites used in the WAsP analysis. ............................................................................................... 19 Figure 3.5 Cumulative distribution of the ratio for the predicted and the actual site mean wind speed................................................................................................... 20 Figure 3.6 Ratio between the prediction and the actual site mean wind speed ... 21 Figure 3.7 Measurement locations in Southern Germany .................................... 23 Figure 3.8 Difference between the prediction and the actual site mean wind speed............................................................................................................................... 24 Figure 3.9 Wind profile for Cabauw (The Netherlands) for 10 m/s < V < 25 m/s27 Figure 3.10 Wind profile for Cabauw (The Netherlands) for 10 m/s < V < 25 m/s............................................................................................................................... 30 Figure 3.11 Distribution of the relative error for the shear exponent estimated at approximately Hub height using the methods 1 and 2 mentioned earlier (Petersen, derived zo). ............................................................................................................ 31 Figure 3.12 Wind profile for Cabauw calculated using Log law and Power law (best fit 20m-40m-80m)........................................................................................ 33 Figure 3.13 Wind speed profiles for Horns rev .................................................... 33 Figure 3.14 Relative wind speed error calculated using all 3 methods in Table 3-15 below. ........................................................................................................... 36 Figure 3.15 Air density at Hamburg as a function of temperature and time, (1960-1969) ..................................................................................................................... 38 Figure 3.16 Ti, mean Ti and Tichar for Delapole Ti char is for a WTG Class II A................................................................................................................................ 39 Figure 3.17 Mean turbulence profile for Skipheia................................................ 40 Figure 3.18 Turbulence intensity profile for the offshore sector of Skipheia....... 41 Figure 3.19 Turbulence intensity for Horns rev including the mean and the estimated mean...................................................................................................... 42

87

Figure 3.20 Turbulence intensity profile for Skipheia including 4 predicted profiles .................................................................................................................. 44 Figure 3.21 Distribution of the relative turbulence intensity error using Hansen/Larsen, sample size = 16, m = -0.05 and s = 0.042 ................................. 46 Figure 3.22 Distribution of the relative turbulence intensity error using the new method, sample size = 14, m = -0.016 and s = 0.04. ............................................ 47 Figure 4.1 Comparison of calculated and measured 1 Hz equivalent loads with for blade root flap moment (m=12); R2 = 0.69 and the slope is x = 0.82y ................ 50 Figure 4.2 Log-Log plot of a Wohler (S-N) curve including stress and cycles to failure for 3 different stress ranges ....................................................................... 51 Figure 4.3 Range pair counting criterion (IEC).................................................... 52 Figure 4.4 Time series and the varying load ranges output from the Rainflow counting ................................................................................................................ 53 Figure 4.5 Rainflow path according to Wirsching and Shehtata [6] .................... 54 Figure 4.6 Load ranges determined from the Rainflow counting method............ 54 Figure 4.7 S1,S2, and S3 are the stress range cycle of equal amplitude. ............... 55 Figure 4.8 Mean stress correction curves ............................................................. 56 Figure 4.9 Stress factor qo derived from tests on steel with welded seams [Heuler].............................................................................................................................. 57 Figure 5.1 Results for tower root bending Vs wind speed, both normalized to their characteristic Values (Table 5-1).......................................................................... 64 Figure 5.2 Equivalent fatigue loads for the blade in the flapwise direction as a function of wind speed.......................................................................................... 69 Figure 5.3 Results from the Monte Carlo simulation ........................................... 72 Figure 6.1 Normalised Aalborg 20 year equivalent load as a function of normalised Aalborg turbulence intensity, both are normalised to IEC class II. ... 75 Figure 6.2 Results from the Monte Carlo simulation for the tower bottom bending moment showing the distributions of the loads and the resistance for IEC and Aalborg site........................................................................................................... 76 Figure 6.3 Results from Table 6-4 in graphical form ........................................... 77

88

List of Tables

Table 2-1 Basis parameters for WTG classes ......................................................... 7 Table 2-2 Target annual failure probabilities PF and corresponding reliability indices βT (source DNV/Risø [6] ) ....................................................................... 11 Table 3-1 Sites used in the MCP analysis including their measurement time period .................................................................................................................... 17 Table 3-2 Ratio of predicted mean wind speeds with actual mean wind speeds. . 18 Table 3-3 Ratio of estimated mean wind speed to actual mean wind speed for a period of approximately 7 years ........................................................................... 19 Table 3-4 IJmuiden calculated from the reference site Leeuwarden .................... 21 Table 3-5 Coefficient of variation and means for data from 1962-2002 using Leeuwarden as the reference site .......................................................................... 22 Table 3-6 Mean wind speeds for southern German sites...................................... 23 Table 3-7 Coefficient of variation and means for data from 1960-1999 .............. 24 Table 3-8 Yearly variation of mean wind speed for the Netherlands ................... 25 Table 3-9 Roughness length for various site terrain types (Petersen, Risø, 1980)28 Table 3-10 Shear exponent mean and standard deviation for several sites and heights. .................................................................................................................. 29 Table 3-11 Measured and estimated wind shear exponent for offshore sites at approximately Hub height, the estimate names refer to the method used to estimate zo ............................................................................................................. 30 Table 3-12 Bias and relative bias and standard deviation for the shear exponent error....................................................................................................................... 31 Table 3-13 Comparison of the measured and estimated wind speeds using the log law. And the value of zo corresponds to terrain roughness derived from Iz.......... 34 Table 3-14 Difference between the extrapolations (Power Law) and the measured wind speeds at several heights for all the common terrain types.......................... 36 Table 3-15 Bias and standard deviation for the relative wind speed error. ......... 37 Table 3-16 Exponents and roughness values used with Hansen/Larsen method.. 42 Table 3-17 Measured and predicted Turbulence intensity for offshore sites. Egmond (Near Shore Windfarm) is an offshore mast in the north sea of the coast of the Netherlands ................................................................................................. 44 Table 3-18 Measured and predicted Turbulence intensity for onshore sites ........ 45 Table 3-19 Turbulence intensity error (%Ti) for all sites at approximately Hub height..................................................................................................................... 45 Table 3-20 Bias and standard deviation for the turbulence and relative turbulence intensity error. ....................................................................................................... 46 Table 4-1 Components and their material............................................................. 49 Table 5-1 Parameters used in the aeroelastic analysis .......................................... 63

89

Table 5-2 Summary of the parameters used in the example................................. 70 Table 5-3 Probability of failure and reliability index ........................................... 70 Table 5-4 Contribution to uncertainty .................................................................. 71 Table 6-1 Summary of the parameters and their statistical properties ................. 74 Table 6-2 IEC class II and Aalborg 20 year equivalent loads normalised to the IEC class II loads .................................................................................................. 75 Table 6-3 Probability of failure and reliability index for selected components ... 76 Table 6-4 Influence factors determined from the Monte Carlo simulation for Aalborg in percent ................................................................................................ 77

90

Appendix

91

Appendix A Parameter estimation results

A.1 Wind speed profile

Log Law

(derived zo) Charnock

Site Terrain Height [m]

Measured [m/s] U DU U DU

Laesø Offshore z0 = 0.001 I15 = 0.09

62 45 30 15

14.5 13.9 13.2 12.2

13.8 13.4 12.9 12.2

0.7 0.5 0.2 -

13.7 13.4 12.9 12.2

0.74 0.48 0.24

- Skipheia Offshore

z0 = 0.001 I20 = 0.130

101 72 41

20.5

16.6 15.9 15.3 14.4

16.6 16.1 15.3 14.4

0.1 0.2 0.0 -

17.0 16.5 15.6 14.4

0.5 0.6 0.3 -

Horns Rev Offshore z0 = 0.001 I15 = 0.10

62 45 30 15

14.7 14.1 13.6 12.7

14.6 14.1 13.6 12.7

-0.2 0.0 0.0 -

14.5 14.1 13.5 12.7

-0.3 -0.1 0.0 -

Egmond Offshore z0 = 0.001 I15 = 0.09

116 70 21

13.3 13.0 12.2

13.9 13.4 12.2

-0.5 -0.4

-

14.0 13.5 12.2

-0.6 -0.4

- Table A.1 Measure and calculated wind speed measurements for offshore sites

Log Law

(derived zo) Site Terrain Height [m]

Measured [m/s] U DU

Cabauw Pastoral z0 = 0.05 I20 = 0.130

140 80 40 20

17.2 15.73 14.16 13.1

17.4 16.1 14.6 13.1

0.2 0.4 0.5 -

Tobøl Pastoral z0 = 0.05 I15 = 0.17

64.5 49 30 15

16.2 15.2 13.9 12.4

15.6 15.0 13.9 12.4

0.6 0.2 0 -

92

Tjareborg Coastal z0 = 0.05 I30 = 0.12

90 60 30

16.0 14.7 13.2

14.8 14.2 13.2

1.2 0.5 -

Oak Creek Complex z0 = 0.05 I10 = 12

79 65 50 10

16.6 16.7 16.3 14.3

16.3 16.1 15.8 14.3

0.3 0.6 0.4 -

Table A.2 Measure and calculated wind speed measurements for onshore sites

Petersen Derived from Ti Site Terrain Height

[m] Measured

[m/s] U DU U DU

Cabauw Pastoral z0 = 0.05 I20 = 0.130

140 80 40 20

17.2 15.73 14.16 13.1

17.4 16.1 14.6 13.1

0.2 0.4 0.5 -

16.4 15.5 14.3 13.1

-0.7 -0.3 0.1 -

Tobøl Pastoral z0 = 0.05 I15 = 0.17

64.5 49 30 15

16.2 15.2 13.9 12.4

15.3 14.6 13.8 12.4

-0.9 -0.6 -0.1

-

15.5 14.8 13.9 12.4

-0.7 -0.4 0.0 -

Tjareborg Coastal z0 = 0.05 I30 = 0.12

90 60 30

16.0 14.7 13.2

15.0 14.3 13.2

-1.1 -0.4

-

14.9 14.3 13.2

-1.1 -0.5

- Oak Creek Complex

z0 = 0.05 I10 = 12

79 65 50 10

16.6 16.7 16.3 14.3

16.3 16.1 15.8 14.3

0.3 0.6 0.4 -

17.9 17.6 17.1 14.3

1.3 0.9 0.8 -

Table A.3 Measure and calculated wind speed measurements for onshore sites using Power law with zo derived using two different methods

Petersen Derived from Ti

Site Terrain Height [m]

Measured [m/s] U DU U DU

Laesø Offshore z0 = 0.001 I15 = 0.09

62 45 30 15

14.5 13.9 13.2 12.2

14.0 13.6 13.1 12.2

-0.5 -0.3 -0.1

-

13.8 13.4 13.0 12.2

-0.7 -0.5 -0.2

-

93

Skipheia Offshore z0 = 0.001 I20 = 0.130

101 72 41

20.5

16.6 15.9 15.3 14.4

16.6 16.1 15.3 14.4

0.1 0.2 0.0 -

17.0 16.5 15.6 14.4

0.5 0.6 0.3 -

Horns Rev Offshore z0 = 0.001 I15 = 0.10

62 45 30 15

14.7 14.1 13.6 12.7

14.6 14.1 13.6 12.7

-0.2 0.0 0.0 -

14.5 14.1 13.5 12.7

-0.3 -0.1 0.0 -

Egmond Offshore z0 = 0.001 I15 = 0.09

116 70 21

13.3 13.0 12.2

14.5 13.8 12.3

0.9 0.8 -

14.1 13.5 12.3

0.4 0.5 -

Table A.4 Measure and calculated wind speed measurements for offshore sites using Power law with zo derived using two different methods

0

0.2

0.4

0.6

0.8

1

1.2

-0.150 -0.100 -0.050 0.000 0.050 0.100 0.150

Relative wind speed error [-]

Cum

ulat

ive

prob

abilt

iy [-

]

MeasuredNormal fit

Figure A.1 Distribution of the relative wind speed error using a combination of Log Law

(derived from Iz and Charnock), Power Law (Petersen, derived from Iz) with m = -0.009, s = 0.035, for a sample size of 28

94

Site No. of 10 min

Samples Offshore Horns Rev Laesoe Egmond (NSW)21 Skipheia Vindeby

40361 13705 3456

24410 2932

Onshore Cabauw Tjæreborg Tobøl Oak creek

1356 20862 2336 9217

Table A.5 This tables shows the number of 10 min samples in the wind speed range 10 < V < 20 at hub height.

21 The number of data points only corresponds to the a single boom direction, NE

95

A.2 Mean wind speed yearly variation

No. of years

Average U(10)

COV 1 yr

COV 20 yr

Netherlands IJmuiden Leeuwarden Hoek van Holland Schipol

40 40 40 40

6.5 5.4 6.4 5.3

7.4 4.3 10.6 4.1

1.66 0.95 2.37 0.91

Average 5.9 6.6 1.47 Northern Germany Bremen Hamburg

40 40

4.3 4.0

5.1 6.3

1.14 1.40

Average 4.2 5.7 1.27 Southern Germany Zugspitze Hohen'

40 40

7.1 4.75

9.64 7.97

2.16 1.78

Average 5.9 8.8 1.97 Table A.6 Yearly variation of mean wind speed for northern Europe and southern Germany

No. of years Average U(z) COV 1 yr COV 20 yr Denmark Sprogø Kegnæs Risø

22 10 6

8.28 7.05 7.82

3.22 5.06 2.98

0.72 1.13 0.67

Average 7.7 3.8 0.8 Table A.7 Yearly variation of mean wind speed for Denmark, the measurement heights are

different and in order as follows 60m, 23m, 125m Since the Danish data is measured at different heights and for a much shorter duration it is difficult to make any direct comparisons with Table A.4, although it can be said that they are all of the same magnitude.

96

A.3 Wind shear exponent

Site Terrain Height [m] Mean α [-] Std. dev α [-] Laesoe Offshor

62 45 30

0.12 0.12 0.11

0.06 0.06 0.06

Skipheia Offshore 101 72 41

0.10 0.10 0.11

0.03 0.03 0.03

Horns Rev Offshore 62 45 30

0.10 0.09 0.09

0.05 0.04 0.05

Egmond Offshore

116 70

0.08 0.09

0.05 0.05

Table A.8 Measured wind shear exponent for offshore sites

Site Terrain Height [m] Mean α [-] Std. dev α [-] Cabauw Pastoral

140 80 40

0.15 0.14 0.14

0.08 0.08 0.04

Tobol Pastoral 64 49 30

0.19 0.18 0.17

0.02 0.02 0.03

Tjareborg Coastal 90 60

0.17 0.15

0.08 0.08

Oak Creek Complex 79 65 50

0.08 0.09 0.09

0.04 0.04 0.04

97

Table A.9 Measure wind shear exponent for offshore sites

98

Site Laesoe Skipheia Horns Rev Egmond Terrain Hub height Mean a Std. s

Offshore 62

0.12 0.06

Coastal (sea dir) 72

0.10 0.03

Offshore 62

0.10 0.05

Offshore 65

0.09 0.04

Derived from Ti Reference height [m] Measured Ti [-] Derived roughness [m] Shear exponent a [-]

15

0.091 2.3E-4 0.075

20.5 0.13 0.003 0.11

15

0.10 5.8E-4 0.09

21

0.08 9.1E-5 0.07

Petersen roughness [m] Shear exponent a [-]

0.001 0.08

0.001 0.10

0.001 0.10

0.001 0.09

Charnock roughness [m] Shear exponent a [-]

2.2E-4 0.08

2.8E-4 0.08

2.4E-4 0.08

2E-4 0.08

Table A.10 Estimated wind shear exponent for offshore sites using 3 methods to determine the terrain roughness zo

Site Cabauw Tobøl Tjæreborg Oak Creek Terrain Hub height Mean a Std. s

Pastoral 80

0.14 0.08

Coastal 64

0.18 0.03

Coastal 60

0.16 0.08

Complex 65

0.09 0.04

Derived from Ti Reference height [m] Measured Ti [-] Derived roughness [m] Shear exponent a [-]

20

0.13 0.009 0.12

15

0.17 0.04 0.16

30

0.12 0.005 0.11

10

0.12 0.003 0.10

Petersen roughness [m] Shear exponent a [-]

0.005 0.15

0.03 0.14

0.003 0.13

1E-5 0.06

Table A.11 Estimated wind shear exponent for onshore sites using 3 methods to determine the terrain roughness zo

99

A.4 Turbulence intensity profile

Error when extrapolating turbulence using various methods

Site I15

Simple Method

I15 Hansen/ Larsen

I15

Charnock New Method

Horns Rev 0.31 0.59 0.58 0.29 Læsoe 0.26 0.22 1.20 0.40 Skipheia 0.82 0.43 0.70 0.09 North sea 1.55 0.18 1.80 0.71 Cabauw 0.62 0.59 ----- 0.62 Toeboel 0.38 0.63 ----- 0.27 Tjæreborg 1.80 0.38 ----- 1.17 Oak Creek 0.12 0.27 ----- 0.73

Table A.12 Average absolute error for all points above the ref height 40m, note the accuracy Hansen\Larsen method was achieved using trial and error.

The choice of the reference height at 40m was due to the fact that it was first used to develop an offshore turbulence profile because it was a common height over the range of offshore sites. It was later adapted for use in an onshore situation and the reference height was never changed. It can be used with sensors at higher or lower but it will have a loss in accuracy.

Height I15

Simple Method

I15 Hansen/ Larsen

I15

Charnock New Method

Horns Rev 0.37 -0.63 0.84 0.32 Læsoe 0.33 -0.19 1.28 0.28 Skipheia 0.99 -0.48 0.69 0.17 North sea 1.33 0.06 1.62 0.95 Cabauw -0.87 0.09 ----- -0.98 Toeboel 1.61 -0.45 ----- 0.84 Tjæreborg 1.40 -0.23 ----- 0.93 Oak Creek 0.24 -0.37 ----- -0.71 Table A.13 Turbulence intensity error for a hub height at approximately 60m

Sample = 16 Mean = -0.003 Std = 0.054

100

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Relative turbulence intensity error [-]

Cun

ulat

ive

dist

ribut

ion

[-]

Simple methodNormal fit

Mean = -0.18Std = 0.102Sample size = 16

Figure A.2 Distribution of the relative turbulence intensity error using the Simple method

Explanation of the Hansen Larsen method (Gunner Larsen)

po

zz

zI ⎟⎠

⎞⎜⎝

⎛=)4.0exp(*

4.2*100

The formula is derived by

1. Expressing the U standard deviation as function of the friction velocity. 2. Dividing this result with the mean wind speed formulated as a power law. 3. Deriving/eliminating the reference height corresponding to the friction

velocity by assuming the wind speed profile very close to the ground (i.e. the region from the sought z=z_* to z=z_0; z_0 being the roughness length) to be described as a logarithmic profile.

And the multiplication factor 2.4 is the proportion between the standard deviation of the along wind turbulence component and the friction velocity (cf. e.g. Panofsky&Dutton: Atmospheric Turbulence)

101

Site Zo P Skipheia 0.005 0.39 Vindeby 0.003 0.38 Læso 0.003 0.38 Horns Rev 0.003 0.38 Cabauw 0.03 0.43 Tobøl 0.04 0.43 Tjæreborg 0.03 0.45 North sea 0.003 0.38

Table A.14 Exponents and roughness values used with Hansen/Larsen method

A.5 New method 22

For the reasons stated above it there was too much uncertainty in using the new method proposed by Hansen/Larsen. Therefore a new method was developed which would be based on turbulence measurements form a reference height. The reference height of 40m to 45m was chosen because many of the offshore and coastal sites included data measured within this height range. It was expected that a better impression of the accuracy of the method would be achieved, if more sites were included. As can be seen in Table 3-17 the Charnock Method is not as accurate as the Hansen/Larsen method but it is quite consistent, with an error of around 1 - 2% Turbulence intensity. This was interesting since no fitting of the data was required for this accuracy. Thus it was believed that this method if adjusted to account for site specific conditions would yield a more accurate fit. To do this it important to identify the need for a change in the equations making up the Charnock profile, this was simply derived from the plot of the Charnock method in Figure A.3.

22 This method was originally developed for offshore application, and this procedure is for offshore only.

102

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200

Turbulence Intensity [%]

Hei

ght [

m]

Turbulence intensity profile for ski with wind speed [15 m/s]

Mean Ti (15 m/s)CharnockHansen/LarsenModified Hansen/Larsen1./log(z./zo)

Figure A.3 Turbulence intensity profiles for mean wind speed of 15 m/s for Skipheia

In the figure above it is clearly visible that the curvature in the Charnock is far too small to fit the data, this was common for all the sites with the wind speed at 15 m/s. It behaves similarly to that simple method used onshore. Indicating that the friction velocity ∗u calculated using the log profile with the Charnock formula may not be the most appropriate, since it seems to be over-estimating the friction velocity. The following formula was derived, which would create a larger curvature.

RgzzUu

)(ln)(κ

=∗ (A.1)

Where R is a multiplication factor obtained by fitting the curve to the data. The resulting profile is shown below alongside Charnock,

103

6 8 10 12 14 16 18 20 22 240

20

40

60

80

100

120

140

160

180

200

Turbulence Intensity [%]

Hei

ght [

m]

Turbulence intensity profile for ski with wind speed [15 m/s]

Mean Ti (15 m/s)CharnockNew method

Figure A.4 Turbulence intensity profiles showing the new method and Charnock

The important thing to note about Charnock is that it underestimates as z increases from zero until a point where it then overestimates the turbulence intensity. This is the reason why it gives consistent results. Thus the equation above was to converge with Charnock after fitting the lower data points, this is illustrated in Figure A.4. From this it was obvious to see that a slight modification would pick up the rest of the data points. This was achieved by applying a negative slope to the profile. The resulting equation is as below.

RgzzUCu

)(ln)()1( κ

+=∗ (A.2)

Where the constant C = -0.00225 z

104

0 5 10 15 20 250

20

40

60

80

100

120

140

160

180

200

Turbulence Intensity [%]

Hei

ght [

m]

Turbulence intensity profile for Skiphia with wind speed [15 m/s]

Mean Ti (15 m/s)CharnockNew method

Figure A. 5 Comparison of the New method and the Charnock method

Since the profile fitted each site in Table 1 with very little error, it was obvious the multiplication factor R could be directly related to some site specific parameter. Both wind speed and roughness were investigated, and a clear trend was visible from a plot of roughness length zo versus R. As illustrated below a log curve was fitted to the points and the resulting equation was combined with (A.2) to yield

0.574)ln(z0.023)(ln)()1( 0 ++=∗ gz

zUCu κ(A.3)

105

Constant for the new turbulence profile Vs roughness

y = 0.0237Ln(x) + 0.7524

0.45

0.47

0.49

0.51

0.53

0.55

0.57

0.59

0.61

0.63

0.65

0 0.0005 0.001 0.0015 0.002 0.0025

Roughness (at 40m > z < 45m) [m]

Series1Log. (Series1)

Figure A. 6 Multiplication factor R Vs roughness height calculated at 40m < z < 45m

Note the roughness height was calculated using the measured turbulence intensity using (3.17) with zref between 40 and 45 m.

Constant for the new turbulence profile Vs roughness

y = 0.0237Ln(x) + 0.7524

0.45

0.47

0.49

0.51

0.53

0.55

0.57

0.59

0.61

0.63

0.65

0.00001 0.0001 0.001 0.01

Roughness (at 40m > z < 45m) [m]

Series1Log. (Series1)

Figure A. 7 Constant for new turbulence method plotted on a semi log plot

106

Appendix B Database

B.1 A Database insight

The data base used in this project is a MySQL database. MySQL is a relational database management system which uses the SQL (structured query language) to manipulate, create and display data. MySQL is a program that manages data bases much like Microsoft’s Excel manages spreadsheets, and SQL is the programming language that is used by MySQL to accomplish tasks within the database just as Excel uses Visual Basic.

B.2 Relational database

For this project it is necessary to avoid clutter and have a database where the data can be manipulated, displayed, stored quickly, easily and in a methodical manner. The MySQL database offers this because of its relational abilities. A relational database is simply defined as tables and columns that relate to each other. These relationships are based on a key value that is contained in a column (field). For example in the ‘scourse’ database the run_id is used as a primary key which will allow the data for a wind speed and a wind direction for a particular 10 minute series to be retrieved simultaneously even though they are in two different tables. In addition to the run_id there is also a channel_id key. This is used to distinguish between two sensors (outputs) which have the same run_id. For example the following SQL query will use the 2 field keys to retrieve 2 sets of concurrent wind speed and direction data from specially created tables for the Dutch data used in the analysis. SELECT di. mean, si. mean, ds. mean, ss. mean FROM `dutch_dirs` di, ´dutch_speeds´ si, `dutch_dirs` ds, ´dutch_speeds´ ss Where di. channel_id = 2000 and si. channel_id = 2001 ds. channel_id = 3000 and ss. channel_id = 3001 and di. run_id = si. run_id and di. run_id = ds. run_id and di. run_id = ss. run_id

107

This query retrieves concurrent wind speed and direction hourly means for IJmuiden and Schipol. By simply selecting the mean and making a condition that it must have the same run_id, the run_id is a unique time stamp given to all data stored in the database. The handles (joins) di, si, ds, ss are arbitrary and in this case they represent d for direction and s for speed, the second letter is just the first letter of the site name. The channel_id just specifies the sensor, again the number attributed to the sensor is arbitrary but it is good practice to use a distinct number range. That is, for the IJmuiden the sensors will be in 2000 range and Schipol will be in the 3000 range. This example is made clearer by viewing the tables below.

B.3 Database architecture (tables)

The following tables are an example of some of the tables that were created to allow the efficient retrieval and storage of the data. Tables

1. Channels: the channels table store information about the sensors 2. Runs: stores information about the time and date of the data 3. Wind direction: stores information about the wind direction 4. Speeds: Stores information about the wind speeds

1. Channels

Channel_id Channel_name units description rel_table

1 --- --- --- --- --- 2 --- --- --- --- --- 3 --- --- --- --- --- 4 --- --- --- --- --- 5 --- --- --- --- ---

Table B. 1 Channels table

108

2. Dutch runs

Run_id channel_id Year_no Month Day hour Minute

1 --- --- --- --- --- --- --- 2 --- --- --- --- --- --- --- 3 --- --- --- --- --- --- --- 4 --- --- --- --- --- --- --- 5 --- --- --- --- --- --- ---

Table B. 2 Runs 4. Dutch dirs

Run_id channel_id mean stdev min Max

1 --- --- --- --- --- --- 2 --- --- --- --- --- --- 3 --- --- --- --- --- --- 4 --- --- --- --- --- --- 5 --- --- --- --- --- ---

Table B. 3 Wind direction 6. Dutch speeds

Run_id channel_id Mean stdev Ti min Max

1 --- --- --- --- --- --- --- 2 --- --- --- --- --- --- --- 3 --- --- --- --- --- --- --- 4 --- --- --- --- --- --- --- 5 --- --- --- --- --- --- ---

Table B. 4 Wind speeds

109

Appendix C Wind turbine and site description

C.1 Wind turbine description Turbine name/ Manufacturer NM92 / NEG Micon Power regulation Pitch regulated variable speed pitch

No. of blades 3

Rotor diameter 92 m

Rotor area 6648 m2

Hub height 70 m

RPM Max 15.6 RPM

Tilt-angle 5°

Cone-angle 0°

Revolution direction Clockwise when seen from upwind

Start wind 3 m/s

Stop wind 25 m/s

Nominal power 2750 kW C.2 Site description The test site is flat in directions except for the Limfjord running NW-SE. The terrain is flat agricultural soil with some natural fetches, and isolated residences to other directions than NNW. Aalborg city and industrial area is located NNW with some tall buildings. All disturbances are more than 550 m away from the turbine. The met mast is placed in direction 277° distance L= 207 metres from the wind turbine, equivalent to 2.3 times the rotor diameter [22]. The wind climate is described by the Weibull distribution as shown in Figure C. 1, with the measurements taken at a height of 70m. The duration of the data is 23 days in December to January has the following max and min wind speeds. Min wind speed 0.6 m/s Max wind speed 18.57 m/s

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0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

Winspeed [m/s]

PD

F [-]

Weibull distributiuon for Aalborg

Figure C. 1 Weibull distribution of the wind climate for Aalborg, with A = 8.21, k = 2.25

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Appendix D Load verification results

The following plots show the results from a flex simulation of the NM92 and actual measurements. All conclusions are based on the assumption that the structural load measurements are correct. Thus any discrepancies can be explained by inaccuracies in the Flex model.

Name

20 yr equivalent load

Measured [kNm]

20 yr equivalent load Flex

[kNm]

20 yr Equiv load Ratio Measure/

Flex

Wohler exponent

Tower Mxt59 Myt59 Mzt59 Mxt15 Myt15 Mxt10 Myt10

1.62 0.88 0.82 0.42 0.66 0.71 1.08

4 4 4 4 4 4 4

Blade Mx11 My11 Mx21 My21 Mx31 My31

0.91 0.82 0.91 0.87 0.91 0.86

12 12 12 12 12 12

Main shaft MxNr MyNr MzNr

0.99 0.65 0.67

6 6 6

Tilt/Yaw moment MyNf MzNf

0.79 0.74

6 6

Table D. 1 Ratio of the 20 yr equivalent loads for Flex and the measurements The verification of the Nm92 wind turbine was made using FLEX5. The results show that if tuning takes place the error in prediction will be reduced. An example of this is that the blade data has been tuned and the error lies in the

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region of approximately 10-20%. This is opposed to no structural tuning of the shaft or tower where the some of the errors are significantly greater. In conclusion tuning of the turbine parts should be carried out and it is expected that the FLEX model will simulate the turbine accurately. The tuning of the blade seems to need a little more fine tuning. A couple of examples are illustrated in the next section, it can be seen that in most cases Flex will model the behaviour of the turbine quite well. In the case of the tower loads the loads in x and y-direction will be transformed into the nacelle coordinates.

Figure D. 1 Mean, Max and Min loads for the tower bottom transformed to nacelle

coordinates

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Figure D. 2 1 Hz Equivalent load for the tower bottom bending moment

Figure D. 3 Power curve for the NM 92

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Figure D. 4 Power spectral density for the tower bottom

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Appendix E Monte Carlo simulation

The Monte Carlo simulation technique is actually very general and Monte Carlo (MC) methods are just stochastic techniques, meaning that they are based on the use of random numbers and probability statistics to investigate problems. These problem solving methods find applications in most of the scientific disciplines, from nuclear physics to traffic flow problems. The basis of the MC method is the use of random numbers to examine some problem.

E.1 Hit and miss integration

"Hit and miss" integration is the simplest type of MC method to understand. The following example will demonstrate how the MC method can calculate the value of pi based on a "hit and miss" integration. Example If you are a very bad dart player is to throw darts at the board as in Figure E.1, it is easy to imagine that each dart thrown is thrown randomly and can hit anywhere. With this in mind it would be safe to say that the number of darts in the square and the number of darts in the circle would be proportional to their areas.

Figure E. 1 Circle circumscribed by a square

In other words

ππ414/

2

2===

squareofAreracircleofArera

ddP (E.1)

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If we say our circle's radius is 1.0, for each throw we can generate two random numbers, an x and a y coordinate, which we can then use to calculate the distance from the origin (0,0) using Pythagoras theorem (x2 + y2 = hyp2). If the distance from the origin is less than or equal to 1.0, it is within the shaded area and counts as a hit. Do this thousands (or millions) of times, and you will wind up with an estimate of the value of pi. How good it is will depend on how many iterations (throws) are done. But the convergence is quite quick. This is shown implemented in MatLab in the code below.

E.2 Two parameter Monte Carlo

A famous "needle-dropping" experiment first proposed by Buffon in 1777 provides a good example of probabilistic modelling. In this experiment a needle was dropped between two parallel lines, from which we can determine the probability that the needle would cross the parallel lines. Experiment setup

φLd

y

Figure E. 2 Buffon’s needle experiment

Tow parallel lines are drawn on a surface, at distance d apart; the needle of length L is dropped randomly, with the centre of the needle a distance y from the closest of the parallel lines. Procedure Define random variables Identify joint sample space Determine the joint probability of the sample space

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Work within the sample space to determine the answers to any question about the experiment. Define the random variables From the diagram it can be seen the distance of the tip of the needle will be a function of two variables: y centre of the needle to the nearest parallel line φ the angle the needle measured with respect to the parallel lines With the needle position defined in this way we can see that 0 ≤ y ≤ d/2, and 0 ≤ φ ≤ π. These variables are chosen due to the symmetry in the experiment. Joint probability distribution To determine the joint probability density function fy,φ (y, φ), we must define what is meant by random variable. In the absence of any other information it is acceptable to assume that the angular position of the needle is uniformly distributed. Where the PDF of a uniformly distributed variable is

⎪⎩

⎪⎨⎧ ≤≤

−=elsewhere

bxaforabxf x

0

1)( (E.2)

Thus

πφπ

φϕ ≤≤= 01)( forf (E.3)

Similarly, it is reasonable to assume that the location of the centre of the needle is uniformly distributed, implying that

202)( dyfor

dyf y ≤≤= (E.4)

To obtain the joint PDF it would be ideal if f and y were independent, then all we would have to do is simply get the product of the two PDF’s. In fact this is the case since knowing the needles angle does not give us any information on where

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its centre position is or vice versa, and thus the assumption of independence is valid, and

πφφ d

yf y2),(, = (E.5)

What we need do is identify the set of points in the joint (Y, ) sample space that corresponds to "intersection of a parallel line" and integrate the joint PDF over that set of points to obtain the desired probability. To achieve this we must define the intersection of the line. From Figure E.2 it is clear that the intersection will occur if y is small enough so that at some angle f the tip of the needle crosses the line. Where coordinate of the lower end of the needle is y – L/2 sinf. If this is negative intersection will occur. Thus the needle will cross the lines for all points in the joint Pdf satisfying the y ≤ (L/2) sin f, and by integrating over this range we get

dLddy

dP

l

exact πφ

π

φπ 22sin2/

00

== ∫∫ (E.6)

For a Monte Carlo simulation the probability can be described as;

trialscrossPest =

Where,

cross denotes the times the needle point crosses the parallel line trials is the number of trial or simulations carried out

As mentioned earlier a cross occurs when y < (L/2) sinf, This is implemented by creating a uniform distribution for the variables y and f using their respective ranges and comparing them for every loop of the simulation. The value cross is then incremented for every y < (L/2) sinf. The comparison of the estimated and the exact value for the Probability is shown below, and as with the previous example the value of π can also be estimated in this way. Pexact = 0.4775

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Pest = 0.4775 Working examples of these simple codes (MatLab) are also given in this appendix. Final Note:

The Monte Carlo simulation used in the project was implemented in much the same way as these examples.

E.3 Monte Carlo MatLab example code

close all clear all clc %xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx % State which example to compute 1 or 2 %xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx example =1; %xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx % 1 paramter example - hit and miss integration %xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx if example == 2 inside = 0;% Initialise the hits inside for i = 1:100000 x = rand(1); % randomly generates a normally distributed number between 0-1 y = rand(1); hyp = sqrt(y^2 + x^2);% Using pytagoras theorem to define the position if hyp<=1 inside=inside +1; % Increment by one when the if statement is satisfied end end

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sample_size =i;% Sample size or total area pi_est = 4*inside/i% Estimate pi else %xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx % 2 paramter example - buffons needle experiment %xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx cross = 0; sim_length = 10000000; %--------- Generate a series of simulations with lenght sim_length ----- for i = 1:sim_length % ------- Set constants ----- L = 3; d = 4; % -------- Boundary conditons ------- phimin = 0; phimax = pi/2; dmin = 0; dmax = d/2; % ------ Generate uniformly distributed numbers ----- y = dmin + (dmax-dmin)*rand(1); phi = phimin + (phimax-phimin) * rand(1); %----- Determine vertical distance of the needle point ---- opp = L*0.5*sin(phi); % ------ If the distance opp is greater than y the needle has crossed ----- if y<opp cross = cross+1; end end % ---- Probabiltiy ------ prob = (2*L)/(d*pi); prob_est = cross/i; %-------- Estimate Pi -------

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pi_est = i*2*L/(d*cross); pi_exact =2*L/(prob*d); end

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Appendix F Beaufort scale

A wind force scale devised in 1805 by Commander Francis Beaufort of the British Navy for observing and classifying wind force at sea. The Beaufort scale as originally drawn up made no reference to the speed of the wind, however correlations have since been made between the force scale and the wind speed and are tabulated below.

Beaufort wind force

Limits ofwind speed [m/s]

Descriptive term

0 Less than 0.5 Calm

1 0.5-1.5 Light air

2 2.1-3.1 Light breeze

3 3.6-5.1 Gentle breeze

4 5.7-8.2 Moderate breeze

5 8.7-11 Fresh breeze

6 11-14 Strong breeze

7 14-17 Near gale

8 17-21 Gale

9 21-24 Strong gale

10 25-28 Storm

11 29-32 Violent storm

12 33 Hurricane

Table F. 1 Beaufort scale The wind speed was calculated using the following formula

2/3837.0 FV = (F. 1) Where F is the Beaufort wind force.

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Beaufort wind force

Land and sea criterion

0 Smoke rises vertically Sea like a mirror

1 Direction of wind shown by smoke drift, but not by wind vanes Ripples with appearance of scales; no foam crests

2 Wind felt on face; leaves rustle; ordinary vanes moved by wind Small wavelets; crests of glassy appearance; not breaking

3 Leaves and small twigs in constant motion; wind extends light flags Large wavelets; crests begin to break; scattered whitecaps

4 Raises dust and loose paper; small branches move Small waves, becoming longer; numerous whitecaps

5 Small trees in leaf begin to sway; crested wavelets form on inland water Moderate waves, taking longer to form; many whitecaps; some spray

6 Large branches in motion; whistling heard in telegraph wires Larger waves forming; whitecaps everywhere; more spray

7 Whole trees in motion; inconvenience felt when walking against wind Sea heaps up; white foam from breaking waves begins to be blown in streaks

8 Breaks twigs off trees; generally impedes progress Moderately high waves of greater length; edges of waves begin to break into spindrift; foam is blown into well marked streaks

9 Slight structural damage occurs; slates blown from roofs High waves; sea begins to roll; dense streaks of foam; spray may reduce visibility

10 Seldom experienced on land; trees broken or uprooted; structural damage occurs Very high waves with overhanging crests; sea takes white appearance as foam is blown in very dense streaks; rolling is heavy and visibility is reduced

11 Seldom experienced on land; trees broken or uprooted; considerable structural damage Exceptionally high waves; sea covered with white foam patches; visibility still further reduced

12 Very rarely experienced on land; usually accompanied by widespread damageAir filled with foam; sea completely white with driving spray; visibility greatly reduced