PhDThesis_Turaj_Ashuri.
Turaj Ashuri
Integrated Aeroservoelastic Design and Optimization of Large Offshore Wind Turbines
Beyond Classical Upscaling:
Integrated Aeroservoelastic Design
and Optimization of
Large Offshore Wind Turbines
Turaj Ashuri
Beyond Classical Upscaling:
Integrated Aeroservoelastic Design and
Optimization of Large Offshore Wind Turbines
Beyond Classical Upscaling:
Integrated Aeroservoelastic Design and Optimization of
Large Offshore Wind Turbines
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof. ir. K.C.A.M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen
op woensdag 14 november 2012 om 15.00 uur
door
Turaj ASHURI
Master of science in Aerospace Engineering
Sharif University of Technology, Iran
geboren te Iran.
Dit proefschrift is goedgekeurd door de promotoren:
Prof. dr. G.J.W van Bussel
Prof. dr. ir. G.A.M. van Kuik
Samenstelling promotiecommissie:
Rector Magnificus
Prof. dr. ir. G.A.M. van Kuik
Prof. dr. G.J.W van Bussel
Prof. dr. Z. Gürdal
Prof. J.D. Sørensen
Prof. dr. ir. R. Benedictus
P. Jamieson MSc.
Ir. B.H. Bulder
Prof. dr. R. Curran
voorzitter
Technische Universiteit Delft, promotor
Technische Universiteit Delft, promotor
Technische Universiteit Delft
Aalborg Universiteit, Denemarken
Technische Universiteit Delft
University of Strathclyde, UK
Energieonderzoek Centrum Nederland
Technische Universiteit Delft, reservelid
Printed by Wöhrmann Print Service, Zutphen, The Netherlands
Copyright c 2012 by Turaj Ashuri
ISBN: 978-94-6203-210-1
All rights reserved. No part of the material protected by this copyright notice may be
reproduced or utilized in any form or by any means, electronic or mechanical, including
photocopying, recording or by any information storage and retrieval system, without the
prior permission of the author.
Typeset by the author with the LATEX Documentation System.
Author email: [email protected]
Dedicated to the memory of my loving mother, Monir Parsazad (1947-2011) and my father
Iraj Ashuri; taghdim be shoma va mamnon az in hame lotf
and my wife Mali Afshar for her love and support
Acknowledgement
This thesis is submitted as partial fulfillment of the requirements for obtaining the PhD
degree at Delft University of Technology. The work has been carried out at the Aerodynamics and Wind Energy department at the faculty of Aerospace Engineering. The PhD
research was supported by INNWIND (Innovation in Wind Energy) project, funded by
Agentschap NL. I am thankful to the contract monitor, Jaap ’t Hooft, and the INNWIND
Upscaling leader, Bernard Bulder, with this project.
From the beginning of my studies, I already knew that I wanted to do a PhD, therefore I wish to particularly acknowledge three people who trusted me and gave me the
opportunity to do so. Without whom I certainly would not be writing this today:
1. My first promoter Professor Dr. Gerard van Bussel has contributed with a magnificent academic support and made my research at Delft very enjoyable. I would like
to acknowledge with appreciation the help, support, numerous and valuable comments, suggestions and criticism of my work. I never forget how gentle he and his
wife, drs. Anny Beckers, welcomed and treated me and my wife upon first weeks
of arrival in Delft, and this has been repeated at many different occasions where I
needed help and support during these years.
2. I am very grateful for the qualified guidance of my second promoter Professor Dr.
Ir. Gijs van Kuik. I had many inspiring discussions during the progress meetings
with him. He has played a key role in my academic development through his
advice on many topics. Through working with him I have become more mature as
a researcher and learned how to be a well balanced scientist.
3. Michiel Zaaijer, my daily supervisor, for all of his assistance and help. He spent
much time in his office with me confronting interesting aspects of the work that
often appeared in the content of this research. He is truly a remarkable person and
his extraordinary efforts in my success are greatly appreciated.
I owe a special debt of gratitude to my past educational supervisors. Dr. Eng. Sasan
Mohamadi, my BSc supervisor, for his collaboration and friendship over years. His deep
knowledge in many fields has been inspiring and eye opening. Also, Professor Dr. Eng.
Saied Adibnazari, my MSc supervisor, who has done much to further my academic and
professional growth. Dr. Alireza Novinzade, my advanced mathematics professor, for his
unique perspective on life and his great effort to strengthen my scientific background in
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ACKNOWLEDGEMENT
applied mathematics. I am honored to have developed a friendship with him.
I wish to thank my parents who instilled within me a constant desire for knowledge
and mankind. It is difficult to express the profound sense of gratitude that I feel towards
my family who have been unfailingly supportive throughout my educational career.
Also contributing in no small way to my success are the many friends I have made at
Delft over the past several years. I wish to express my deepest thanks to Herdis Heinemann and Carlos Ferreira with whom I have shared so many enjoyable moments. My
many other friends have also been a great help, especially Eeke Mast who helped me
many times with problems I was facing as a foreigner in Delft. She has been always patient and supportive and I could always rely on her on whatever problem I had. I must
also thank my other friends, Jessica Holierhoek, Mostafa Abdalla, Ameya Sathe, Claudia
Hofemann, Thanasis Barlas, Busra Akay, Jaume Bartan, Ali Khani, Ben Geurts, Fanzhong
Meng, Wybren de Vries and Ghanshyam Shrestha for the great moments we had together.
To the staff of the department, Sylvia Willems, Eric van der Pol, Nando Timmer, Wim
Bierbooms and Ruud van Rooij for their support and help. You will be sorely missed but
not forgotten. Also many thanks to my colleagues at Siemens Wind Power, especially Dr.
Ir. David Molenaar for pushing and encouraging me to finish the last two chapters of this
work, while I joined Siemens and giving me the support to finalize it. Also thank you to
Paul van der Valk and Sven Voormeeren for reviewing some chapters of the thesis and
translating the summary to Dutch.
I would like to thank my sons, Ryan and Dorian for putting up with Papa being a
student and saying sorry for many moments that I should have shared and played with
you but I could not. Finally, I wish to express my deepest gratitude to my lovely wife
who has been an extremely valuable support during my MSc and PhD study. Without
her, I would never have finished this thesis. I know the God has blessed me by the gift of
having you as wife and a truly friend, mamnonam azizam.
This thesis is the result of many years of continuous research and during those years, I
experienced many changes. I became a father and have lost my mother, have had the joy
of being a lecturer at university and have learned how to be flexible and strong to deal
with many problems that could potentially prevent finishing a PhD research while being
abroad, but I did manage to finish it and I do praise God for finishing it. As a proverb
says: ’A good PhD is a finished PhD’.
Turaj Ashuri
June 2012
Summary
Issues related to environmental concern and fossil fuel exhaustion has made wind energy the most widely accepted renewable energy resource. However, there are still several challenges to be solved such as the integrated design of wind turbines, aeroelastic
response and stability prediction, grid integration, offshore resource assessment and scaling related problems.
While analyzing the market of wind turbines to find the direction of the future developments, one can see a continuous upscaling of wind turbines. Upscaling is performed
to harness a larger resource and benefit from economy of scale. This will pose several
fundamental implications that have to be identified and tackled in advance.
This research focuses on investigating the technical and economical feasibility and
limits of large scale offshore wind turbines using the current dominant concept, i.e. a
three-bladed, upwind, variable speed, pitch regulated wind turbine installed on a monopile in an offshore wind farm.
Thus, the objective of this research is to investigate how upscaling influences the
offshore wind turbines. Specifically, following questions are of interest:
1. How do the technical characteristics of the larger scales change with size and can
these technical characteristics appear as a barrier?
2. How does the economy of the future offshore wind turbines change with size?
3. What are the considerations and required changes for future offshore wind turbines?
To address these questions, a more sophisticated method than the classical upscaling
method should be employed. This method should provide the detailed technical and
economical data at larger scales and address all the design drivers of such big machines
to identify the associated problems.
However, interdisciplinary interactions among structure, aerodynamics and control
subject to constraints on fatigue, stresses, deflections and frequencies as well as considerations on aeroelastic instability make the development of such a method a cumbersome
and complex task.
Among many different methods, integrated aeroservoelastic design optimization is
found to be the best approach. Therefore, the scaling study of this research is formulated
as an multidisciplinary design optimization problem. This method enables the design
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SUMMARY
of the future offshore wind turbines at the required level of details that is needed to
investigate the effect of size on technical and economical characteristics at larger scales.
Using this method, 5, 10 and 20 MW wind turbines are designed and optimized, including the most relevant design constraints and levelized cost of energy as the objective
function. In addition to the design of these wind turbines, the method itself shows a clear
way forward for the future offshore wind turbine design methodology development.
Based on these optimized wind turbines, scaling trends are constructed to investigate
the behavior of a wind turbine as it scales with size. These trends are formulated as a
function of rotor diameter to properly reflect the scale. Loading, mass, cost and some
other useful trends are extracted to investigate the scaling phenomenon. Blades and
tower as the most flexible load carrying components are examined with more attention.
Using these results, the challenges of very large scale offshore wind turbines up to 20
MW range are explored and identified. These results demonstrate that a 20 MW design
is technically feasible though economically not attractive. Therefore, upscaling of the
current wind turbine configurations seems to be an inappropriate approach for larger
offshore wind turbines.
Samenvatting
Zorgen over milieu en klimaat en het uitputten van fossiele brandstoffen hebben wind
energie de breedst geaccepteerde bron van hernieuwbare energie gemaakt. Echter, er
zijn nog altijd meerdere uitdagingen die om oplossingen vragen, zoals het geïntegreerd
ontwerpen van windturbines, het voorspellen van aero-elastische respons en stabiliteit,
inpassing in het elektriciteitsnetwerk, inschatting van offshore windbronnen en problemen gerelateerd aan opschaling.
Bij het analyseren van de markt voor windturbines, op zoek naar richtingen voor
toekomstige ontwikkelingen, ziet men een continue opschaling van windturbines. Door
dit opschalen kan een grotere hoeveelheid wind energie worden gevangen en kunnen
schaalvoordelen worden behaald. Dit zorgt voor een aantal fundamentele implicaties die
vooraf dienen te worden geïdentificeerd en opgelost.
Dit onderzoek focust op het bepalen van de technische en economische haalbaarheid
en limieten van grootschalige offshore windturbines die gebaseerd zijn op het huidige
dominante concept, i.e. een windturbine met variabele snelheid, drie bladen, tegen de
wind in georiënteerde rotor met pitch-regulering en geïnstalleerd op een monopaal in
een offshore wind park.
Het doel van dit onderzoek is dus om na te gaan welke invloed het opschalen heeft
op offshore windturbines. Daarbij zijn de volgende specifieke vragen van belang:
1. Hoe veranderen de technische karakteristieken met grootte en manifesteren deze
karakteristieken zich als barrière?
2. Hoe veranderen de economische aspecten van toekomstige offshore windturbines
met grootte?
3. Wat zijn de aandachtspunten en vereiste veranderingen voor toekomstige offshore
windturbines?
Om deze vragen te beantwoorden dient een meer vooruitstrevende methode te worden
gebruikt dan de klassieke opschalingmethoden. Deze methode dient gedetailleerde technische en economische data te leveren en de belangrijkste ontwerpinvloeden, evenals de
samenhangende problemen, voor zulke grote machines te kunnen identificeren.
Echter, de interdisciplinaire interacties tussen de structuur, aerodynamica en het regelsysteem die bovendien zijn onderworpen aan beperkingen op het gebied van vermoeiing, spanningen, vervormingen en frequenties alsmede de beschouwing van aeroelastische instabiliteit, maken de ontwikkeling van een dergelijke methode een zeer complexe taak.
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SAMENVATTING
Uit de vele verschillende methoden is een geïntegreerde aero-servo-elastische ontwerpoptimalisatie gekozen als beste aanpak. De studie naar opschaling in dit onderzoek is
om die reden geformuleerd als een probleem van multidisciplinaire ontwerpoptimalisatie. Deze methode maakt het mogelijk toekomstige offshore windturbines te ontwerpen
op het gewenste niveau van detail, om zo het effect van opschaling op de technische en
economische eigenschappen te kunnen onderzoeken.
Met deze methode zijn windturbines van 5, 10 en 20 Megawatt ontworpen en geoptimaliseerd, met inachtneming van de meest relevante ontwerpbeperkingen en de genormaliseerde energiekosten als doelfunctie. Naast het ontwerp van deze windturbines laat de
methode zelf een duidelijk ontwikkelingspad zien voor toekomstige offshore windturbine
ontwerpmethodologieën.
Op basis van deze geoptimaliseerde windturbines zijn schaal-trends gemaakt om het
gedrag van een windturbine te onderzoeken als deze opschalen. Deze trends zijn geformuleerd als een functie van rotordiameter om de schaal adequaat weer te geven. Nuttige
trends zoals de belasting versus diameter, massa versus diameter en kosten versus diameter zijn bepaald om de fenomenen die optreden by opschaling te onderzoeken. De
bladen en toren hebben hierbij extra aandacht gekregen omdat dit de meest flexibele
lastdragende componenten zijn.
Uit deze resultaten zijn de uitdagingen verkend en geïdentificeerd voor zeer grote
offshore windturbines in de orde van 20 Megawatt. Deze resultaten laten zien dat een 20
Megawatt ontwerp technisch haalbaar is maar economisch niet aantrekkelijk. Hierdoor
lijkt het opschalen van huidige windturbine configuraties een ongeschikte aanpak voor
grotere offshore windturbines.
Contents
Acknowledgement
i
Summary
iii
Samenvatting
v
Nomenclature
xiii
1 Introduction
1
1.1 Motivation and goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.2 The main and key research questions . . . . . . . . . . . . . . . . . . . . . . .
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1.3 Research methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Classical upscaling methods
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2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2 Historical development of large scale wind turbines . . . . . . . . . . . . . .
2.2.1 Earlier developments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Recent developments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.3 Linear scaling laws . . .
2.3.1 Blade . . . . . . .
2.3.2 Low speed shaft
2.3.3 Gearbox . . . . .
2.3.4 Generator . . . .
2.3.5 Tower . . . . . . .
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2.4 Upscaling using existing data . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Loading-diameter trends . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Mass-diameter trends . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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viii
CONTENTS
2.4.3 Cost-diameter trends . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.5 Comparison of linear scaling law and existing data . . . . . . . . . . . . . .
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2.6 Why a new approach for upscaling? . . . . . . . . . . . . . . . . . . . . . . . .
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3 Integrated aeroservoelastic design and optimization
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3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.2 Classification of design problems . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Routine design problems . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Nonroutine design problems . . . . . . . . . . . . . . . . . . . . . . . .
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3.3 Difficulties of design problems . . . . . . .
3.3.1 Size of the design space . . . . . . .
3.3.2 Psychological inertia . . . . . . . . .
3.3.3 Uncertainties in the design process
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3.4 Methods of solving design problems . . . . . . . .
3.4.1 Trial-and-error methods . . . . . . . . . .
3.4.2 Inventive based methods . . . . . . . . . .
3.4.3 Knowledge-based engineering methods .
3.4.4 Design search and optimization methods
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3.5 Selection of the design methodology and its overview applicable to upscaling 33
3.6 Architecture of the MDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.7 Optimization problem formulation .
3.7.1 Design variables . . . . . . . .
3.7.2 Design Constraints . . . . . . .
3.7.3 Partial safety factors . . . . . .
3.7.4 Objective function . . . . . . .
3.7.5 Wind turbine simulation tools
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3.8 Design integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.9 Controller design automation . . . . . . . . . . . . . . . . .
3.9.1 Wind turbine models for designing the controller
3.9.2 Control model of the generator torque controller
3.9.3 Control model of the blade pitch controller . . . .
3.9.4 Tower top motion feedback loop . . . . . . . . . .
3.9.5 Controller design and implementation . . . . . . .
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3.10 Automation of extracting structural properties . . . . . . . . . . . . . . . . .
3.10.1 The methodology to extract structural properties . . . . . . . . . . .
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3.11 Wind turbine’s model specification . . . .
3.11.1 Aerodynamic design definition . .
3.11.2 Structural design definition . . . .
3.11.3 Definition of the design load cases
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3.12 Some practical issues about the optimization procedure
3.12.1 Reducing the number of design variables . . . . .
3.12.2 The starting point of design variables . . . . . . .
3.12.3 Decomposition of design variables . . . . . . . . .
3.12.4 Multilevel optimization approach . . . . . . . . . .
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CONTENTS
3.13 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Verification of the integrated aeroservoelastic optimization method
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Purpose of the design optimization of the NREL wind turbine . . . . . .
4.3 MDO of the 5 MW wind turbine . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 The 5MW NREL wind turbine . . . . . . . . . . . . . . . . . . . . .
4.3.2 Design variables of the optimized 5 MW wind turbine . . . . . .
4.3.3 Design constraints of the optimized 5 MW wind turbine . . . . .
4.3.4 Objective function of the optimized 5 MW wind turbine . . . . .
4.3.5 Computational Expense . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Comparison of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Validating the integrated aeroservoelastic optimization method
4.5.2 Improved understanding of the 5 MW wind turbine design . . .
4.5.3 Providing consistent data points to make scaling trends . . . . .
ix
69
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71
71
71
72
72
73
73
74
75
75
79
79
80
80
5 Integrated aeroservoelastic design and optimization of large wind turbines
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 MDO of the 10 and 20 MW wind turbines . . . . . . . . . . . . . . . . . . . .
5.2.1 Design variables of the 10 and 20 MW wind turbines . . . . . . . .
5.2.2 Design constraints of the 10 and 20 MW wind turbines . . . . . . .
5.2.3 Objective function of the 10 and 20 MW wind turbines . . . . . . .
5.3 Properties of the 10 and 20 MW wind turbines . . . . . . . . . . . . . . . . .
5.3.1 Gross properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Blade properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Aerodynamic properties . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.4 Drive train properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.5 Hub and nacelle properties . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.6 Support structure properties . . . . . . . . . . . . . . . . . . . . . . . .
5.3.7 Controller properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
81
81
81
83
85
85
85
85
85
87
89
90
91
6 Aeroelastic stability analysis using eigenvalue method
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Aeroelastic instability in wind turbines . . . . . . . . . . . . . . . . . . . . . .
6.3 Modeling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Structural model of the wind turbine . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Displacement, velocity, acceleration and forces in the inertial frame
6.4.2 Mass, stiffness and damping in the inertial frame . . . . . . . . . . .
6.5 Aerodynamic model of the rotor . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Modification of the wind velocities . . . . . . . . . . . . . . . . . . . .
6.5.2 Modification of the airfoil data . . . . . . . . . . . . . . . . . . . . . .
6.5.3 The inclusion of the drag force . . . . . . . . . . . . . . . . . . . . . .
6.5.4 Discretization of the 3D blade to 2D sections . . . . . . . . . . . . . .
6.5.5 Derivation of the aerodynamic matrices in a FE form . . . . . . . . .
93
93
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95
97
97
99
100
101
102
102
103
104
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x
CONTENTS
6.6 Coupling structural and aerodynamic FE matrices . . . . . . . . . . . . . . . 104
6.7 Eigenvalue analysis of the coupled aeroelastic model . . . . . . . . . . . . . 105
6.7.1 Static instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.7.2 Dynamic instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.8 Implementation of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Verification and aeroelastic stability analysis of the 20 MW wind turbine
111
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2 The finite element model of the wind turbine . . . . . . . . . . . . . . . . . . 111
7.3 Verification of the stability analysis method . . . . . . . . . . . . . . . .
7.3.1 2D analytical stability analysis . . . . . . . . . . . . . . . . . . . .
7.3.2 Undeformed state eigenvalue analysis of the structural model
7.3.3 Full aeroelastic stability analysis . . . . . . . . . . . . . . . . . .
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112
113
116
116
7.4 Stability analysis of the 20 MW wind turbine . . . . . . . . . . . . . . . . . . 118
7.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8 Scaling trends for future offshore wind turbines
121
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.2 Development of scaling trends .
8.2.1 Loading-diameter trends
8.2.2 Mass-diameter trends . .
8.2.3 Cost-diameter trends . .
8.2.4 Other useful trends . . .
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9 Conclusion and recommendation for future work
121
122
128
138
144
147
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.2 Discussion on the methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.2.1 Upscaling context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.2.2 Design context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.3.1 Overall conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.3.2 Conclusions based on research questions . . . . . . . . . . . . . . . . 150
9.4 Contribution to the state of the art . . . . . . . . . . . . . . . . . . . . . . . . 153
9.5 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.5.1 Upscaling context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.5.2 Design context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
A Mass and cost models
157
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
A.2 Mass and cost modeling description . . . .
A.2.1 Blades . . . . . . . . . . . . . . . . .
A.2.2 Hub . . . . . . . . . . . . . . . . . . .
A.2.3 Blade pitch system and its bearing
A.2.4 Nose cone of the hub . . . . . . . .
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158
158
159
159
160
CONTENTS
A.2.5
A.2.6
A.2.7
A.2.8
A.2.9
A.2.10
A.2.11
A.2.12
A.2.13
A.2.14
A.2.15
A.2.16
A.2.17
A.2.18
A.2.19
A.2.20
A.2.21
A.2.22
A.2.23
A.2.24
A.2.25
A.2.26
A.2.27
A.2.28
A.2.29
A.2.30
A.2.31
Low speed shaft . . . . . . . . . . . . . . . . . . . . .
Main bearing and its housing . . . . . . . . . . . . .
Gearbox . . . . . . . . . . . . . . . . . . . . . . . . . .
Mechanical brake and coupling . . . . . . . . . . . .
Generator . . . . . . . . . . . . . . . . . . . . . . . . .
Power electronic . . . . . . . . . . . . . . . . . . . . .
Bed plate . . . . . . . . . . . . . . . . . . . . . . . . . .
Platform and railing . . . . . . . . . . . . . . . . . . .
Hydraulic and cooling system . . . . . . . . . . . . .
Nacelle cover . . . . . . . . . . . . . . . . . . . . . . .
Electrical connections . . . . . . . . . . . . . . . . . .
Yaw system . . . . . . . . . . . . . . . . . . . . . . . .
Control system . . . . . . . . . . . . . . . . . . . . . .
Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Marinization . . . . . . . . . . . . . . . . . . . . . . .
Foundation system . . . . . . . . . . . . . . . . . . . .
Offshore transportation . . . . . . . . . . . . . . . . .
Port and staging equipment . . . . . . . . . . . . . .
Offshore turbine installation . . . . . . . . . . . . . .
Offshore electrical interface and connection . . . .
Offshore permits, engineering, and site assessment
Personnel access equipment . . . . . . . . . . . . . .
Scour protection . . . . . . . . . . . . . . . . . . . . .
Offshore warranty premium . . . . . . . . . . . . . .
Decommissioning . . . . . . . . . . . . . . . . . . . .
Offshore levelized replacement . . . . . . . . . . . .
Operation and maintenance . . . . . . . . . . . . . .
B A theoretical background of the used simulation tools
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xi
160
161
161
162
162
163
163
163
164
164
165
165
166
166
166
167
167
168
168
168
169
169
169
170
170
170
171
173
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.2 TurbSim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.3 AeroDyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
B.4 AirfoilPrep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
B.5 FAST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
B.6 BModes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.7 Crunch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.8 Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
C Verification of the controller design automation
177
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
C.2 Applying the controller design task . . . . . . . . . . . . . . . . . . . . . . . . 177
C.3 Analysis and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
xii
CONTENTS
D Verification of extracting structural properties
181
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
D.2 Verification of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
E Theoretical aspects of optimization algorithms
E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E.2 CONLIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E.3 Lagrange Multiplier (LM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
185
185
185
187
F Scaling laws for initial design variable generation
191
F.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
F.2 Derivation of scaling laws for design studies . . . . . . . . . . . . . . . . . . . 191
F.3 Application of scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
G Properties of the optimized 5 MW wind turbine
G.1 Properties of the optimum 5 MW wind turbine
G.1.1 Blade properties . . . . . . . . . . . . . .
G.1.2 Aerodynamic properties . . . . . . . . .
G.1.3 Drive train properties . . . . . . . . . . .
G.1.4 Hub and nacelle properties . . . . . . . .
G.1.5 Support structure properties . . . . . . .
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195
195
195
196
198
198
199
H Derivation of the aerodynamic forces in a finite element form
H.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H.2 Defining the aerodynamic coordinate system . . . . . . . . . . . . . . . . . .
H.3 Coupling aerodynamic loads with nodal deformations . . . . . . . . . . . .
H.4 Defining aerodynamic loads in terms of functionals . . . . . . . . . . . . . .
H.5 Transforming the functionals domain of integration to coordinate system 1
H.6 Symbolic integral replacement of the deformations . . . . . . . . . . . . . .
H.7 Integrating functionals using Gauss-Legendre technique . . . . . . . . . . .
H.8 Minimizing the functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H.9 Element matrices for the aerodynamic model . . . . . . . . . . . . . . . . . .
H.10 Assembling the global aerodynamic matrices . . . . . . . . . . . . . . . . . .
201
201
201
207
208
209
210
212
213
214
215
Bibliography
224
Curriculum vitae
225
Nomenclature
Latin symbols
Ż
Tower top speed
a
Axial induction factor
A bl d (x)
Blade sectional area distribution
Alss
Low speed shaft area
b
Semi-chord
c
Chord
c
Scale factor
Cd
Drag coefficient
Cl
Lift coefficient
Cm
Moment coefficient
Cp
Aerodynamic power coefficient
C2a
Influence of aging safety factor
C3a
Temperature effect safety factor
C4a
Hand layup laminate safety factor
C5a
Post-cured laminate safety factor
c bl d (x)
Blade chord distribution
Da
Aerodynamic damping matrix
Ds
Ex
Ey
Fa
Ft
Gβ
Gx y
Structural damping matrix
Module of elasticity in fiber direction
Module of elasticity perpendicular to fiber direction
Aerodynamic force
External force
Gain scheduled correction factor
Shear modulus of elasticity in x-y plane
xiii
( ms )
(−)
(m2 )
(m2 )
(m)
(m)
(−)
(−)
(−)
(−)
(−)
(−)
(−)
(−)
(−)
(m)
kg
)
s
kg
( s )
(
(Pa)
(Pa)
(N )
(N )
(−)
(Pa)
xiv
NOMENCLATURE
(Pa)
G yz
Shear modulus of elasticity in y-z plane
Iα
I ar e−lss
I ar e (x)
Id r
I mas (x)
k
k
Mass moment of inertia around elastic axis
Low speed shaft area moment of inertia
Area moment of inertia distribution
Drive train mass moment of inertia
Mass moment of inertia distribution
Reduced frequency
Shape factor
Ka
Aerodynamic stiffness matrix
( s2 )
Ks
Structural stiffness matrix
( s2 )
Kα
Torsional stiffness
KD
Derivative gain
( RPdegM )
Kg
Generator torque proportionality factor
.m
( rNpm
2)
KI
Integral gain
( RPs M )
KP
Proportional gain
( RP M )
Kt
L
L
L lss
L t ow
Tower stiffness
Blade length
Lift force
Low speed shaft length
Tower length
m
Ma
Ms
M bld
M bnd−w g t
M bnd− y aw
Med g
Mf lp
M f or−a f t
Mg b
M gen
Msid−sid
M t rs−lss
M t rs
Pgen
Pr ot
R
Mass per unit length
Aerodynamic mass matrix
Structural mass matrix
Blade mass
Low speed shaft bending moment due to weight
Low speed shaft bending moment due to tilt and yaw
Edgewise moment
Flapwise moment
Tower fore-aft bending moment
Gear box mass
Generator mass
Tower side-to-side bending moment
Low speed shaft torsional moment
Tower torsional moment
Generator power
Rotor power
Rotor radius
(kg.m2 )
(m4 )
(m4 )
(kg.m2 )
(kg.m2 )
(Hz)
(−)
kg
kg
.m
)
( Ndeg
s
deg
deg
(N .m2 )
(m)
(N )
(m)
(m)
kg
(m)
(kg)
(kg)
(kg)
(N .m)
(N .m)
(N .m)
(N .m)
(N .m)
(kg)
(kg)
(N .m)
(N .m)
(N .m)
(W )
(W )
(m)
NOMENCLATURE
Sα
Static moment related to elastic axis
xv
(kg.m2 )
t bl d (x)
Blade thickness distribution
Tg b
Gear box torque
(N .m)
(m)
Tg en
Generator torque
(N .m)
(m)
t lss
Low speed shaft thickness
Tr ot
Rotor torque
t t ow (x)
Tower thickness distribution
(m)
U
Wind velocity
( ms )
u
Nodal displacement
(m)
V
Wind velocity
( ms )
Ve f f
Effective wind velocity
( ms )
Vf
Volumetric fraction
W
Work
w
Shape function
(−)
x
Distributed property
(−)
Z
Tower top displacement
(m)
Zg b
Gear box ratio
(−)
(N .m)
(−)
(W at t)
Greek symbols
α
Angle of attack
α
Wind shear
β
Blade pitch angle
(deg)
βr
Resultant wind velocity angle
(deg)
βt
Aerodynamic twist
(deg)
ω̈
Rotational acceleration
γM
r ev
( min
2)
General material safety factor
(−)
νx y
Poisson ratio in x-y plane
(−)
Ω
Angular velocity
ωϕn
Resonance frequency
ω g en
Generator rotational speed
ωi
Weighting factor
ω r at
Rated rotational speed
(RP M )
ω r ot
Rotor rotational speed
(RP M )
ρ
Material density
θ bl d (x)
Blade twist distribution
ξϕ
Damping ratio
Subscripts
are
area
(deg)
(−)
(RP M )
( 1s )
(RP M )
(−)
kg
( m3 )
(deg)
(−)
xvi
NOMENCLATURE
bld
blade
dr
drive train
edg
edge
flp
flap
for
fore
gb
gear box
gen
generator
lss
low speed shaft
mas
mass
rat
rated
ref
reference
rot
rotor
sid
side
tow
tower
trs
torsion
wgt
weight
Abbreviations
AEP
Annual Energy Production
BEM
Blade Element Method
BOS
Balance Of Station
CFRP
Carbon Fiber Reinforced Plastic
CONLIN
CONvex LINearization
CPU
Central Processing Unit
DLL
Dynamic Link Library
DOP
Design Optimization Process
EA
Elastic Axis
EWM
Extreme Wind Model
FAST
Fatigue Aerodynamic Structure Turbulence
FCR
Fixed Charge Rate
FEA
Finite Element Analysis
FEM
Finite Element Method
FIO
Fully Integrated Optimization
GFRP
Glass Fiber Reinforced Plastic
GII
General Inflation Index
ICC
Initial Capital Cost
LCOE
Levelized Cost Of Energy
LM
Lagrange Multiplier
LRC
Levelized Replacement Cost
NOMENCLATURE
MDO
Multidisciplinary Design Optimization
NAICS
North American Industry Classification System
NREL
National Renewable Energy Laboratory
NTM
Normal Turbulence Model
O&M
Operations and Maintenance
OWT
Offshore Wind Turbine
PDE
Partial Differential Equation
PPI
Producer Price Index
RPM
Revolution Per Minute
SR
Scaling Factor
SR
Scaling Ratio
TCC
Turbine Capital Cost
USD
United States Dollar
xvii
CHAPTER
1
Introduction
1.1
Motivation and goal
The available onshore area for wind energy development in and around Europe is restricted. This is because of the high population density and natural areas that do not
allow any further wind turbine installation for bulk generation of electricity. Going offshore and large scale wind turbines are the two promising solutions that probably enable
further reduction in costs as well.
Offshore winds are typically stronger and more stable than onshore, resulting in significantly higher production per installed turbine. In addition, in many European waters
the water depth increases slowly with the distance from shore which is an important
advantage for the utilization of bottom-mounted offshore wind turbines (OWTs). There
are several large research projects which have been set up to study harvesting the offshore wind energy resources such as: OPTI-OWECS, Kühn et al. (1998), POWER, OnlineReader (2007) and COD, Roth et al. (2005).
The development of large offshore wind turbines is supported by four facts:
1. The energy capture per area land use is higher for larger machines.
2. Many cost elements such as operation and maintenance and infrastructure may
decrease per rated MW capacity with fewer larger machines for the same installed
capacity.
3. The wish to minimize the number of wind turbines required for the bulk generation
of electricity. This means that the wind energy industry thinks on manufacturing
larger machines instead of several smaller machines for the same installed capacity.
4. Size is often considered as a merit index for technology progress and development.
DOWEC, Hendriks and Zaaijer (2004), UPWIND, Online-Reader (2012), ICORASS,
Bulder et al. (2007) and WindPact, Fingersh et al. (2006) are also among the most rel1
2
INTRODUCTION
1.2
evant projects related to doing research on large scale wind turbines, and each of these
projects focuses on one aspect of upscaling.
The DOWEC project focused on the research and development needs for the design of
large-scale offshore wind farms. Although, the project ends up with preliminary design
of a 6 MW OWT and addresses some issues related to larger scale turbines, upscaling has
not been not the direct interest of the project. Both the NREL 5 MW model, Jonkman
et al. (2009), and the UPWIND project utilized data from the DOWEC consortium.
In the ICORASS project, by reducing the number of components and integrate functions as far as possible into single components a 10 MW wind turbine was designed. The
aim of the project was to see how much cost reduction could be achieved by a large scale
wind turbine and less attention is paid to the technical aspects of such a machine.
The WindPact project studied the impact of increased turbine size and configuration
on levelized cost of energy, by developing cost and mass models of wind turbine components. However, the project does not address any of the design constraints and limitations
that a large scale OWT may encounter.
The main objective of the UPWIND project was to identify major technological and
economical barriers of large scale OWTs, with the wind turbine as the essential of a
wind farm. However, to identify the technical and economical characteristics of the large
scale OWTs, linear scaling laws and engineering judgment of the involved experts and
participants were used rather than making realistic wind turbine configurations through
simulation and optimization.
Though significant improvements in upscaling have been made in the past (see chapter
2 for upscaling literature review), there is still much research needed before many of the
technical and economical characteristics associated with upscaling are fully understood
and solved.
In line with the UPWIND project, this research deals with the upscaling of large scale
OWTs. Hence, the goal of this research is to study the challenges which will be faced
during increasing the size of OWTs. The focus is towards a better understanding of
the aeroservoelastic behavior of large OWTs. This understanding is needed because of
following reasons:
1. It leads to accurate load prediction (fatigue and ultimate loads), which is an important factor to determine technical feasibility.
2. It is governing the dynamic behavior of a wind turbine that is primarily responsible
for stability. Especially when the focus is on larger scales, considering the fact
that they are more flexible and therefore their design may suffer from aeroelastic
instabilities.
3. It is the basis for control design which can lead to significant increase in efficiency
and load reduction, and thus influencing the economical characteristics.
Considering all the facts, conceptual and preliminary design activities may reveal
fundamental issues that need to be handled before further progress in upscaling can be
achieved. Therefore, the outcome of the research is the stepping stone toward larger
multi megawatt OWTs up to 20 MW range.
1.3
1.2
THE MAIN AND KEY RESEARCH QUESTIONS
3
The main and key research questions
For a critical evaluation that larger wind turbines are feasible to manufacture and may
eventually be more cost effective, it is needed to analyze how size influences the wind
turbine itself. Therefore, based on the motivation and objective of this research, the main
research question to deal with can be formulated as:
How does upscaling influence the design of current offshore wind turbines?
To plan the construction and utilization of the future OWTs one should know the
probable changes and modifications to the existing wind turbines. In other words, the
specifications and characteristics of the future OWT need to be known well in advance
before taking any action to rely on these giant machines of the future. This is also an
important subject for the society to know how much they can rely on wind as a renewable
source for the future energy planning.
Answering this question is the encompassing subject of this thesis. However, to answer it, three key questions need to be answered first. These are:
How do the technical characteristics of the larger scales change with size and
can these technical characteristics appear as a barrier?
One of the most frequently asked questions that is still open to the wind energy
community is the size limitation. Several attempts have been made to address this issue,
but none of them provides a concrete answer. A reason for that is the huge amount of
time needed to do this investigation for a wind turbine, while facing lots of details and
complexities to fully cover all the aspects.
This research will deal with this question by upscaling a 5 MW wind turbine to larger
sizes up to 20 MW, and investigating the effect of size on the technical characteristics of
these enlarged machines. Based on these machines, scaling trends will be constructed to
identify the size associated problems and find the probable limiting factors.
How does the economy of the future OWTs change with size?
One of the most important factors that can act as a good motivation to utilize large
scale wind turbines is the wish to have a lower cost of energy. Therefore, this research
explores this aspect carefully. It could be the case that there is not a technical barrier but
the economy of larger scales is not in favor of any size increment.
What are the considerations and required changes for future OWTs?
The results of this research should help to identify probable changes and modifications to the existing wind turbines. Upscaling can present many challenges and the early
identification of these challenges helps to make better research planning and come up
with better design solutions and modifications.
1.3
Research methodology
To study the effect of size on characteristics of a wind turbine two different methods are
frequently used. First, the analytical relation between a number of important parameters
that govern the turbine characteristics can be formulated as a function of rotor diameter,
4
INTRODUCTION
1.4
under the assumption that all geometrical parameters scale linearly. This approach is
called linear scaling law (also similarity or scaling rule).
Linear scaling law does not quantify the technical and economical characteristics at
any scale, and the usage of this method is limited to give an overall impression in conceptual design phase. This makes the method less attractive for studying detailed technical
and economical characteristics, and tradeoff studies in which the quantified end results
play an important role.
Second, the relation between rotor diameter and other parameters that govern the
turbine characteristics can be formulated by studying the trends in the existing wind
turbines. In this approach, real data are viewed collectively and by interpolating these
data scaling trends can be extracted. However, to study wind turbines that are larger
than existing ones, an extrapolation must be used. The further the extrapolation goes
outside the data range, the more the uncertainty will be.
To overcome the drawbacks of the first two methods, this research uses a novel
method. In this method, for several given scales of interest a wind turbine will be designed and optimized. Based on these optimized wind turbines the relation between
different parameters and rotor diameter can be extracted and used to develop trends.
Since for every given scale an optimum wind turbine is designed, studying the technical
and economical characteristics of larger OWTs can be made more precisely.
For the upscaling study of this research, the 5 MW NREL wind turbine is used as the
baseline design, Jonkman et al. (2009). This guarantees the use of earlier works and
experiences in the field of wind energy which is gained and evaluated in real situations
for decades and considered in the design of this machine. Also all the data of this machine
are publicly available and has been based on some of the most recent commercial wind
turbine designs in the same class (Multibrid M5000 and the REpower 5M).
To do the upscaling, the 5MW NREL wind turbine is scaled up to 10 and 20 MW
wind turbines without any conceptual change. That is, a 5 MW machine is redesigned
for larger sizes and physical phenomena of these enlarged machines will be analyzed and
evaluated. The analysis and evaluation is based on the results such as: stresses, natural
frequencies, displacements, fatigue, aeroelastic stability and cost of energy. Also based
on these results judgments about feasibility and problems associated to upscaling will be
made, as well as generating trends.
The results of this method are used to identify major technical and economical barriers of larger scale wind turbines. They also help to improve designs with respect to
performance, weight or any other objectives which makes it possible to study the advantage and disadvantage of the given concept for larger scales.
This research ends up with an optimized outline design of 5, 10 and 20 MW wind turbines, which can be used as a reference baseline design for many other research studies.
1.4
Structure of the thesis
This thesis is built up of 9 chapters as follow:
Chapter 1: Introduction
This chapter discusses the research motivation, research questions and scope of the
thesis. At the end of the chapter the thesis outline is presented.
1.4
STRUCTURE OF THE THESIS
5
Chapter 2: Classical upscaling methods
This chapter explains the classical upscaling methods in detail. It also compares them
in terms of advantage and disadvantage. Shortcoming of these methods motivated this
research to develop a more powerful alternative.
Chapter 3: Integrated aeroservoelastic design and optimization
A review of existing design methodologies applicable to wind turbines is made first in
this chapter to find the best approach that is suited for upscaling. This review showed the
integrated multidisciplinary design optimization (MDO) as the best alternative for scaling
study. Then, the formulation of scaling study as a MDO problem is explained together
with the architecture of this method. Based on this setup an integrated design code is
developed that enables the design and optimization of any conventional wind turbine at
any other size of interest.
Chapter 4: Verification of the integrated aeroservoelastic optimization method
Using the developed design code of the previous chapter, the MDO of the 5MW NREL
concept is carried out in this chapter. This is to show the usefulness of this approach and
verify its correct implementation as a computational code by comparing the 5 MW NREL
wind turbine with the optimized 5 MW wind turbine utilized by this research.
Chapter 5: Integrated aeroservoelastic design and optimization of large wind
turbines
The same approach that was used to optimize the 5 MW NREL wind turbine in the
previous chapter is followed in this chapter to design large scale OWTs. This provides
the necessary data that needs to be analyzed in a systematic way to extract technical and
economical trends for large scale OWTs as presented in chapter 8.
Chapter 6: Aeroelastic stability analysis using eigenvalue method
Development of a simulation code for performing aeroelastic stability analysis of wind
turbines is presented in this chapter. The rational behind this code is described in a
general form, as independent of the particular aeroelastic modeling as possible. This
simulation code enables aeroelastic stability analysis of any wind turbine configuration
and at any size since it is made parametric.
Chapter 7: Verification and aeroelastic stability analysis of the 20 MW wind
turbine
The developed code in the previous chapter is verified with the available instability
prediction results in the literature followed by some other test cases. The verified simulation code is used to check the aeroelastic stability boundary of the optimized 20 MW
wind turbine and make sure that its design does not suffer from any unwanted instability
during operation.
Chapter 8: Scaling trends for future offshore wind turbines
Based on the optimized wind turbine designs of the previous chapters, this chapter
constructs scaling trends. All the trends are a function of rotor diameter to reflect size
6
INTRODUCTION
1.4
dependency and they cover both the technical and economical characteristics. These
scaling trends help to identify the design challenges of the future OWTs as they scale in
size.
Chapter 9: Conclusion and future work
This chapter concludes the approach and results of this research in a comprehensive
way. This discussion is followed by highlighting the contribution of the research to the
state of the art knowledge. Based on the results of the research the future work to design
large OWTs is formulated.
The outline of this thesis is visualized in figure 1.1.
1.4
STRUCTURE OF THE THESIS
Aeroservoelastic
Aeroservoelasticdesign
designoptimization
optimizationof
ofwind
windturbines
turbines
Introduction
Introduction&&
literature
literaturereview
review
Aeroservoelastic
Aeroservoelastic
design
design&&optimization
optimization
Chapter 1
Introduction
and outline
Chapter 3
Design
methodology
development
Chapter 2
Classical
upscaling
Chapter 4
Verification &
Optimization of
the 5 MW turbine
Chapter 5
10 & 20 MW
design
Results
Resultsand
andconclusion
conclusion
Chapter 8
Scaling trends
for large turbines
Chapter 9
Conclusion &
future work
Figure 1.1: Thesis outline
Aeroelastic
Aeroelasticstability
stability
analysis
analysis
Chapter 6
Stability
tool
development
Chapter 7
Verification &
20 MW stability
analysis
7
CHAPTER
2
Classical upscaling methods
2.1
Introduction
To study the effect of size on characteristics of a wind turbine two classical scaling methods are used, i.e. the linear scaling law and existing data trends. These two methods are
presented and compared in this chapter together with a review of the historical development of wind turbines in the context of upscaling.
The deficiencies of these two methods motivated this work to search for an alternative to comply fully with the objective of this research. The new methodology should
enable designing larger scale OWTs and providing detailed insight to their technical and
economical characteristics in order to identify size related challenges.
A brief review of existing design methods is made to select the best technique that
is well suited for scaling study. It is found that multidisciplinary design optimization is
the best method to design large scale OWTs. This chapter presents the first two classical
methods, and chapter 3 discusses the multidisciplinary design optimization formulation
for the upscaling problem.
2.2
Historical development of large scale wind turbines
The question regarding the optimal size of a wind turbine is as old as the idea of inventing
such a machine. From a historical point of view, there are contradictory claims about the
origin of wind turbines, but the earliest documented design is a vertical-axis windmill
built in Persia, Spera (1994). However, wind turbines only evolved rapidly in technology
and size, after realizing that they can also be used to generate electricity by James Blyth
in 1887, Charles Brush 1888 and Poul La Cour in 1891, Warne and Calnan (1977).
The development efforts to explore larger scale wind turbines will be presented in this
section. These efforts can be classified in the earlier studies mainly done by individuals,
9
10
CLASSICAL UPSCALING METHODS
2.2
and the more recent national or international plans driven by the oil crisis of 1970s and
more awareness about the negative environmental impact of other conventional energy
resources.
2.2.1
Earlier developments
In 1922, Joseph and Marcellus Jacobs started to develop a 4 meter, three-bladed rotor
directly connected to a low-speed DC generator, Online-Reader (2010). They also established a company in 1925, which is the oldest manufacturer of wind electric systems in
USA. In 1931 a large wind turbine was built in Russia named Wime D-30 with a threebladed rotor of 30 m diameter and a power output of 100 kW, Sektorov (1933). A second
similar wind turbine with the name of Zwei D-30 was installed some years later on the
coast of the Arctic Ocean.
In 1930s the German engineer Hermann Honneff proposed a wind turbine concept
with a tower height of 250 m, a rotor diameter of 160 m and the number of blades
between 3 to 5, Honneff (1932). However, his idea about "the larger the more economical" was far ahead of its time and the concept was never build.
In October 1941, the world’s first really large wind turbine that was based on the
idea of the American engineer Palmer Cosslett Putnam was installed on Grandpa’s Knob,
a hill in the state of Vermont, USA . It had a rotor diameter of 53.3 m, a power output of
1250 kW and a tower height of 35.6 m. This two-bladed downwind turbine was made
of a lattice tower and the speed and power output of the turbine were controlled by
a hydraulic blade-pitching mechanism. This was most probably the first wind turbine
interconnected to conventional power plants within the public utility grid.
In addition, he also tackled the question of the wind turbine size optimality and
concluded that a rotor diameter of 53.3 to 68.5 m, a tower height of 45.7 to 53.3 m and
a power output of 1500 to 2500 kW is the range for an optimum configuration, Putnam
(1947). Also Ulrich Hütter published the similar results carried out by Putnam about size
optimality, Hütter (1942).
In 1942, the German engineer Franz Kleinhenz presented a design of a very large
wind turbine that was improved and refined in many details over 5 years of effort. This
turbine had a rotor diameter of 130 m, a power output of 10 MW and a hub height of
250 m. Unlike Honneff, Kleinhenz knew how to cooperate with famous scientists and
industrial firms of the time. Therefore, his design was confirmed by Albert Betz and the
Maschinenfabrik Augsburg Nürnberg (MAN), but unfortunately the second world war
prevented the actual construction of such a giant machine, Kleinhenz (1942).
2.2.2
Recent developments
After World War II, the prices of fossil fuels dropped again and electricity generation was
relying on conventional power plants. In addition, the subject of environmental protection and air pollution had not yet been considered and as the result there were not
considerable activities to generate electrical power using wind turbines or other renewables.
1970s oil crisis prompted investigation of alternative energies, among them wind
energy as the most promising. However, the high energy cost of wind turbines (compared
2.2
HISTORICAL DEVELOPMENT OF LARGE SCALE WIND TURBINES
11
to fossil fuels) delayed the utility scale realization of electricity generation from wind.
This high demand to wind energy provided more financial support to do research and
development on wind turbines aiming to better solutions, hence technology development
and size increment.
In 1975, the Danish "Tvind Turbine" was erected in Ulfborg. The turbine was designed
with a high level of idealism with a rotor diameter of 52 m and a power output of 2 MW.
After the successful operation of the Tvind turbine in MW scale, the Danish utilities built
the smaller Nibe A and Nibe B wind turbines, each having a rotor diameter of 40 m and
a power output of 630 kW.
In Germany, the Growian project became the largest research program in the late
1970s. The project aimed at a 3 MW wind turbine with a rotor diameter of 113 m
and a hub height of 72 m. However, after the construction, it never completed the test
programme due to numerous severe design faults. The total operation time was 420
hours and the turbine was demolished in 1987.
In 1982, the Swedish research program intended for the first experimental turbine
named WTS-75 (later named Aeolus I). This turbine had a rotor diameter of 75 m and a
power output of 2 MW and it was erected in the island of Gotland. A similar project was
followed few months later by another large turbine with a rotor diameter of 78 m and
a power output of 3 MW and the wind turbine was installed in the South of Sweden in
Marglarp.
In these years, the governmental research institutions in Canada had a special focus
on vertical axis wind turbines. In 1985 one of these large research projects named Eole
was designed with a Darrieus rotor of an equatorial diameter of 64 m, a height of 100 m
and a power output of 4 MW.
The development of large scale wind turbines in the United States started from 1975
to 1987 with a series of large experimental turbines named MOD-0 to MOD-5. However,
the final and largest project, the MOD-5A, never reached completion. The turbine was
intended to have a rotor diameter of 122 m and a rated power of 7.3 MW. In 1993, the
MOD-5A project was canceled in favor of the MOD-5B since it had a better design with a
smaller size. The MOD-5B wind turbine had a rated capacity of 3.2 MW, a rotor diameter
of 100 m and a two-bladed rotor on a 60 m steel tower.
Considering the lessons learned during all these years, at the end of the 1980s, there
were less developed ambitious projects with regard to size. Further, in contrast to the
first attempts initiated by pioneers which had a bottom-up approach and mainly in the
national level, most of the projects were launched from a top-down approach and by
contributing institutions from several countries.
In Europe, the first phase of this action, named WEGA has led to the development of
three experimental large wind turbines with a power output from 1 to 2 MW and rotor
diameters from 55 to 60 m. The first wind turbine, the Tjaereborg 2 MW machine, was
installed at Tjaereborg in Denmark. A few months later, the Spanish-German industrial
consortium AWEC-60 completed the erection of the second turbine at Cabo Villano in
Spain. Finally, third wind turbine, the 1 MW machine at Richborough developed by the
Scottish firm James Howden was installed.
At the same time some more projects that were supported in the frame work of the
THERMIE program have been realized: LS-1 (3 MW) in Britain, GAMMA-60 (1.5 MW)
in Italy, NEWECS-45 (1 MW) in the Netherlands and WKA-60 (1.2 MW) in Germany.
12
CLASSICAL UPSCALING METHODS
2.3
Apart from theoretical approaches relating to the relationship between size and economic feasibility, the WEGA project also contained proposals for the further development
of wind turbines in the MW power range. Therefore, it was followed by the WEGA II
project some years later. In the second program the market’s needs were becoming the
research interest, therefore also many industrial partners were asked to participate. Five
completely new prototypes of large horizontal-axis turbines were developed and built
from 1993 to 1995.
In the mid to late 1990s, the three-bladed upwind rotor with variable-speed-pitchregulated design became the dominant utility-scale configuration. Around the same
period the University of Sunderland made a set of scaling rules for the machines of the
period, Harrison and Jenkins (1993). This report had valuable models to predict the impact of machine size on turbine components. However, after just some few years of this
report the machine size had increased drastically which made the report less accurate for
larger scales.
Beginning in 1999 in the US, the Department of Energy began the WindPACT project.
The focus of this project was on the determination of potential technology pathways that
would lead to more cost-effective wind turbine design. The main goal of this project was
to study the impact of increased turbine size and configuration on the cost of energy.
This was done by setting up several major studies. In each study, the preliminary
design of many concepts at the range of sizes from 750 kW to 5 MW was carried out.
These studies resulted in scaling relations of wind turbine subsystems and components
across the range of sizes. The project ended officially at 2006 and by far this is the most
complete source of scaling relations, Fingersh et al. (2006).
Due to the success of these joint projects, many other research projects were also
launched. These projects can be divided into different categories such as: offshore wind
turbine design, blades and rotors, wind resources forecasting and mapping, wind farm
development and management and integration of wind power to the grid.
UPWIND was the largest wind energy research project funded under the EU’s Sixth
Framework Program (FP6), consisting of 15 scientific and industrial work packages. The
project looked towards the wind power of tomorrow. More precisely, it looked at the
design of very large wind turbines (over 10 MW), both for onshore and offshore application.
UPWIND had 11 different work packages, with upscaling work package dedicated
to identify major technological and economic barriers associated with the development
of future wind turbine technology. Development of scaling trends was supported by
experienced engineering judgment of the partners.
2.3
Linear scaling laws
As explained before, upscaling in the literature is done using linear scaling laws and
existing data. Linear scaling laws provide the fundamental physical and geometrical
relation between a number of important parameters that govern the wind turbine design
and the rotor diameter.
Linear upscaling is realized based on three distinct assumptions:
1. The number of blades, airfoils type, turbine materials, drive train and support struc-
2.3
LINEAR SCALING LAWS
13
ture concepts are the same
2. The tip speed remains constant
3. All other geometrical parameters vary linearly with rotor diameter (except gearbox,
generator and power electronics)
The first documented representation of such a relation goes back to the late 1980s,
and it was presented by Molly (1989) in the form that we see and recognize today.
In 2001 linear scaling laws were further extended and as a case study they were
implemented in the upscaling process of a wind turbine by Nijssen et al. (2001). In the
framework of the UPWIND project, Chaviaropoulos (2007) applied the linear scaling law
on most of the components of the wind turbine to see the effect of wind turbine upscaling
on its loading and operational behavior.
This work can be further improved by including the effect of wind shear, Reynolds
number modification on aerodynamic properties and some other modifications to make
it a more powerful tool for scaling studies. While the inclusion of the wind shear is
straightforward, it is not so easy to make a closed form relation when other modifications
are considered.
To identify the design drivers and critical issues for very large scale wind turbine
blades, Ashuri and Zaayer (2008) used the linear scaling laws and a finite element model
for the analysis. Stresses and displacements due to aerodynamic, gravity and inertial
loadings are analyzed for 5, 10, 15 and 20 MW blades, and based on these data scaling
trends are constructed to examine how upscaling influences the design.
In 2010, Capponi and Ashuri (2010) presented a nonlinear upscaling approach for
the blade based on stresses. In this work, the resultant stress level due to aerodynamic,
weight and centrifugal loads are seen as a design constraint to be fixed as the size increases from a 5 MW reference wind turbine to an upscaled 20 MW wind turbine. To
reach the constant stress level, the geometry of the blade is upscaled nonlinearly (in the
linear scaling law the geometry varies linearly with size).
In the book of Jamieson (2011) a chapter is dedicated to upscaling of wind turbines.
This chapter gives a good overview of upscaling in general with a detailed analysis of
linear scaling laws.
Based on scaling laws assumptions (either linear or nonlinear) and using physical
relations between rotor diameter and other parameters of interest, scaling a wind turbine
to a different size becomes possible. To have a better overview, these physical relations
are studied for the main components of the wind turbine and presented in the following
subsections using linear scaling laws.
2.3.1
Blade
For the blade, a linear scaling relation is used for all the geometrical parameters and
based on that the results are presented in table 2.1. Upscaling the blade increases the
load levels that can be considered as a negative effect. At the same time an increase in
the captured energy can be seen as the result of upscaling.
Making a tradeoff between the increase in the load levels and power production using
this crude model is not possible. Therefore, the overall impact of scaling is not clear using
14
2.3
CLASSICAL UPSCALING METHODS
the linear scaling law and a sophisticated model is needed to precisely address this issue.
This can be considered as a main drawback of using linear scaling laws for upscaling.
Table 2.1: Geometric linear scaling laws for blade, Chaviaropoulos (2007)
Symbol
Description
L
ω r ot
ω r ot · L
c bld (x)
t bld (x)
θ bld (x)
Mf lp
Med g
Pr ot
Tr ot
A bld (x)
I ar e (x)
I mas (x)
M bld
Blade length
Rotor rotational speed
Tip speed
Blade chord distribution
Blade thickness distribution
Blade twist distribution
Flapwise moment
Edgewise moment
Rotor power
Rotor torque
Blade sectional area
Area moment of inertia
Mass moment of inertia
Blade mass
Size dependency
R
R−1
I
R
R
I
R3
R3
R2
R3
R2
R4
R5
R3
R: Linear dependency, I: Size independency
2.3.2
Low speed shaft
The linear scaling law for the low speed shaft is presented in table 2.2. Due to the simple
geometry of the low speed shaft, it is assumed that all the dimensions linearly vary with
size. This geometrical assumption dictates the mass of the low speed shaft to increase
with R3 .
Table 2.2: Linear scaling laws for low speed shaft, Chaviaropoulos (2007)
Symbol
Description
L lss
t lss
Alss
I ar e−lss
M bnd−w g t
M bnd− y aw
M t rs−lss
Shaft length
Shaft thickness
Shaft sectional area
Shaft area moment of inertia
Shaft bending moment due to weight
Shaft bending moment due to tilt and yaw
Shaft torsional moment
Size dependency
R
R
R2
R4
R3
R2
R3
2.4
2.3.3
LINEAR SCALING LAWS
15
Gearbox
For the gearbox a multistage configuration is assumed. Table 2.3 shows the linear scaling
laws for the gearbox. Because of the complex geometry of the gearbox, its mass can not
be formulated as a function of size directly. To overcome that, it is assumed that the mass
of the gearbox changes proportional to the torque that it experiences. Although, this is a
hard assumption, it provides enough insight to see how the gearbox mass changes with
size as it is shown in table 2.3.
Table 2.3: Linear scaling laws for gearbox, Chaviaropoulos (2007)
2.3.4
Symbol
Description
Size dependency
ω r ot
ω gen
Zg b
Tg b
Mg b
Rotor rotational speed
Generator rotational speed
Gearbox ratio
Gearbox torque
Gearbox mass
R−1
I
R
R3
R3
Generator
For this scaling study, a conventional generator is used and the rotational speed of it
is assumed to be independent of scale and therefore remains constant. Similar to the
gearbox, the generator mass is formulated as a function of the generator torque. The
linear scaling laws are presented in table 2.4.
Table 2.4: Linear scaling laws for generator, Chaviaropoulos (2007)
2.3.5
Symbol
Description
Tgen
M gen
Generator torque
Generator mass
Size dependency
R2
R2
Tower
A tubular tower configuration is used for scaling study and based on that the linear
scaling laws are presented in table 2.5. For the tower, upscaling has less negative consequences in an offshore environment than onshore. This is due to a steeper wind shear
profile that offshore wind turbines experience.
This is not the case, however, in onshore environment with a higher value of wind
shear, therefore, it is expected that offshore wind turbines will suffer less from wind shear
induced loads when compared to an onshore wind turbine. This is a benefit when one
considers upscaling in offshore environment.
16
2.4
CLASSICAL UPSCALING METHODS
Table 2.5: Linear scaling laws for tower, Chaviaropoulos (2007)
2.4
Symbol
Description
L t ow
t t ow (x)
A t ow (x)
I ar e (x)
I mas (x)
M f or−a f t
Msid−sid
M t rs
Tower length
Tower thickness distribution
Tower sectional area
Area moment of inertia
Mass moment of inertia
Fore-aft bending moment
Side-to-side bending moment
Torsional moment
Size dependency
R
R
R2
R4
R5
R3
R3
R3
Upscaling using existing data
The most comprehensive work on extracting wind turbines mass and load trends using
certification calculations is carried out by Jamieson (2007) in the framework of the UPWIND project. These data are presented in two parts. In the first part load data are
presented as a function of rotor diameter and in the second part mass of different components. These data are systematically presented again by Jamieson in his resent book,
innovation in wind turbine design, Jamieson (2011).
2.4.1
Loading-diameter trends
To construct loading-diameter trends, Jamieson used a set of 42 pitch regulated, variable
or two-speed controlled wind turbines. However, differences in design configurations and
the wind class result in scattering the data, therefore they should be viewed collectively
to see what may be suggested in terms of scaling trends. Because of the commercial
confidentiality of the data points only a trend line is shown in most of the graphs.
Among the main components of the wind turbine, those that are carrying most of the
loads are of the interest of this research, since they are more relevant from upscaling
point of view. These are the blade, tower and low speed shaft. Unfortunately, in the work
of Jamieson, the low speed shaft is not included. Therefore, the loading-diameter trend
is only presented for the blade and tower :
1. Blade
For the blade the extreme loads at the blade root are presented in figure 2.1 and
2.2. Based on linear scaling law formulation, the flapwise and edgewise loads
should scale with R3 .
However, as it can be seen from these two figures, the edgewise bending moment
scales more rapidly with size than the flapwise moment. This issue will be discussed
in the next subsection.
2. Tower
All the studied towers are of a tubular type and are made of steel. In the original
report, the results are only given for the extreme bending moments at the tower
base and shown for fore-aft and side-to-side bending moments in figure 2.3 and
2.4 respectively.
2.4
UPSCALING USING EXISTING DATA
17
8
Blade flapwise moment (MN.m)
7.2
y = 0.0175x2.86
Q2 = 0.94
6.4
5.6
4.8
4
3.2
2.4
1.6
0.8
0
20
28
36
44
52
60
68
76
84
92
100
Rotor diameter (m)
Figure 2.1: Extreme flapwise bending moment at blade root, Jamieson (2007)
8
Blade edgewise moment (MN.m)
7.2
y = 0.0031x3.25
Q2 = 0.88
6.4
5.6
4.8
4
3.2
2.4
1.6
0.8
0
20
28
36
44
52
60
68
76
84
92
100
Rotor diameter (m)
Figure 2.2: Extreme edgewise bending moment at blade root, Jamieson (2007)
As shown by the graphs, the fore-aft bending moment has the lowest trends exponent, followed by the side-to-side and torsional moments. This is a deviation from
what linear scaling law predicts and the reason will be analyzed and explained in
chapter 8 of this thesis.
18
2.4
CLASSICAL UPSCALING METHODS
Tower fore-aft bending moment (MN.m)
80
72
y = 1.7891x2.33
Q2 = 0.71
64
56
48
40
32
24
16
8
0
20
28
36
44
52
60
68
76
84
92
100
Rotor diameter (m)
Figure 2.3: Extreme fore-aft bending moment at tower base, Jamieson (2007)
80
Tower side-side moment (MN.m)
72
y = 0.0348x3.23
Q2 = 0.74
64
56
48
40
32
24
16
8
0
20
28
36
44
52
60
68
76
84
92
100
Rotor diameter (m)
Figure 2.4: Extreme side-to-side bending moment at tower base, Jamieson (2007)
2.4.2
Mass-diameter trends
The mass-diameter trends are presented in this subsection. Here, the focus will be on
the scaling trends of the blade, the tower top mass and the tower. The tower top mass
refers to all the components atop the tower including the rotor. Studying the mass trend
not only gives useful information on how the mass itself scales, but also provides insight
2.4
UPSCALING USING EXISTING DATA
19
8
Tower torsional moment (MN.m)
7.2
y = 8E-05x4.05
Q2 = 0.76
6.4
5.6
4.8
4
3.2
2.4
1.6
0.8
0
20
28
36
44
52
60
68
76
84
92
100
Rotor diameter (m)
Figure 2.5: Extreme torsional moment at tower base, Jamieson (2007)
on the scaling of gravity driven loads such as the tower side-to-side and blade edgewise
moments.
1. Blade
The blade mass and its scaling behavior is one of the most important parameters
for the dynamic load analysis of the blade and wind turbine. To construct the
blade-diameter trend, 52 data points from 7 manufacturers are used. A power law
fit to the data of figure 2.6 yields an exponent of 2.09 that is less than the predicted
value by the linear scaling law.
This is mainly because of the ongoing development efforts to reduce mass and
associated costs in the recent decade and better manufacturing techniques, as well
as better airfoils and aerodynamic designs.
2. Tower top mass
Generally, there is a big difference in the tower top mass associated with each
manufacturer. This is due to the accumulated effect of adding all the masses atop
the tower, which introduces a big scatter in the data. Furthermore, the addition
of some non-standard components that are only introduced by some manufacturer
(like an internal crane) adds to the scatter.
Figure 2.7 shows the tower top mass of 37 wind turbines in the range of 20 to 120
m rotor diameter. The data trend shows a lower value than the linear scaling law
with an exponent of 2.39. Here, the effect of rotor diameter range is not taken into
account.
As a matter of fact, smaller rotor diameters have a lower trend exponent than
the bigger ones. That is, the more recent wind turbines have a higher tower top
mass trend exponent than the smaller wind turbines when classified in separated
20
2.4
CLASSICAL UPSCALING METHODS
32
29
y = 0.6853x2.09
Q2 = 0.96
Blade mass (ton)
26
22
19
16
13
10
6
3
0
0
18
36
54
72
90
108
126
144
162
180
Rotor diameter (m)
Figure 2.6: Blade mass trend based on blade manufacturers data, data is taken from magazines,
product brochures and company web sites
categories.
This issue represents the stage of the technical development of a wind turbine,
which for larger scales means that they have not yet reached the maturity that the
smaller wind turbines have reached.
290
261
y = 2.348x2.39
2
Q = 0.71
Tower top mass (ton)
232
203
174
145
116
87
58
29
0
0
14
28
42
56
70
84
98
112
126
Rotor diameter (m)
Figure 2.7: Tower top mass trends of different manufacturers, data is taken from magazines, product
brochures and company web sites
2.5
UPSCALING USING EXISTING DATA
21
3. Tower
Figure 2.8 shows the tower mass versus the rotor diameter for 37 wind turbines.
The scatter in the data is mainly because of the variation of the tower heights for a
given wind turbine, since many manufacturers offer a range of tower heights that
best matches the site condition.
As it can be seen from the figure, the tower mass trend exponent has a value that
is close to what linear scaling law predicts. However, the scatter in the data that is
represented by the Q-squared value shows a relatively poor curve fit, and perhaps
it would be better to not make such a judgment based on this figure. Therefore,
it would be always difficult to make comparisons like that when a high scatter is
present in the data.
440
396
y = 0.653x2.78
Q2 = 0.74
Tower mass (ton)
352
308
264
220
176
132
88
44
0
0
13
26
39
52
65
78
91
104
117
130
Rotor diameter (m)
Figure 2.8: Tower mass scaling trend, data is taken from magazines, product brochures and company
web sites
2.4.3
Cost-diameter trends
The cost-diameter trend of the wind turbine per kW installed capacity is presented in
figure 2.9. The cost data includes the turbine units alone i.e., excluding transportation
and installation costs. As it can be seen, there is a region where the specific cost is
minimal. This region shows that the applied technological improvements have reached
maturity after an initial startup and helped the reduction of costs.
This is a typical behavior of a learning curve in many engineering applications, where
the positive sign of improvement becomes apparent only after being implemented and
revised over a period of time. This also explains the slight rise in the right hand side of
the curve where more recent wind turbines are located.
22
2.5
CLASSICAL UPSCALING METHODS
1080
1025
Specific cost (USD/kW)
970
915
860
805
750
695
640
585
530
0
10
20
30
40
50
60
70
80
90
100
Rotor diameter (m)
Figure 2.9: Specific cost trend, Jamieson (2007)
2.5
Comparison of linear scaling law and existing data
Linear scaling laws describe in a simple way how to scale a wind turbine to another size,
and they are often considered as not accurate when compared with existing data. This is
partly true due to the complexity of a wind turbine, which can not be captured accurately
using some simple assumptions on the geometry as the linear scaling laws formulate.
Therefore, these assumptions and simplifications to derive closed form relations of the
problem seem to push the method away from reality.
However, when comparing the linear scaling laws with the current engineering practices, a good match between part of the predicted "square-cube" law with reality exists.
In fact, the "square" part of the prediction has a valid predictive value. This means that
the energy production of the real wind turbines follows well the prediction of the linear
scaling laws.
Figure 2.10 shows the rated power output of 27 wind turbines versus diameter in the
range of 15 to 130 m. As it is shown, the curve fit to the data points shows almost a
"square" relation with size. This is due to the fact that most of the aerodynamic designs
of the wind turbines have reached a level that can be considered as the achievable ideal
efficiency.
As it is known from Betz limit, the energy capture is at its maximum at an axial
induction factor of 13 and this is often the strategy that most of the variable speed wind
turbines follow up to the rated power point. Therefore, the rotor area in the same wind
regime is a good measure to predict the captured energy by the wind turbine as it scales.
Similarly, by looking to figure 2.1 and 2.2 for the flapwise and edgewise bending
moments, a fair match between the trends in existing data and linear scaling law can be
seen. Linear scaling law predicts the increase as R3 for both the flapwise and edgewise
bending moments, and these are R2.86 and R3.25 using the existing data. However, one
2.5
COMPARISON OF LINEAR SCALING LAW AND EXISTING DATA
23
7
Rated power output (MW)
6.3
y = 0.0007x1.85
Q2 = 0.87
5.6
4.9
4.2
3.5
2.8
2.1
1.4
0.7
0
0
14
28
42
56
70
84
98
112
126
140
Rotor diameter (m)
Figure 2.10: Rated power output as a function of rotor diameter, data is taken from magazines,
product brochures and company web sites
should be aware that linear scaling law only predicts the stationary bending moments
and the values in the existing data correspond to the non-stationary loads.
For mass predictions, linear scaling laws sometimes fail at first glance to describe the
behavior of the existing data. This is the second part of the "square-cube" term, which
states that mass scales with R3 as the size increases. Figure 2.6 shows that the mass
scaling of the blade using existing data has an R2.29 relation with size that is far below R3
as predicted by linear scaling law.
This is the influence of the ongoing efforts in airfoil developments as well as having
better manufacturing techniques specially for composite blades in which all the data
points of the figure are based on. If the blade would be made totally from metal then
probably R3 relation could be still a valid prediction. Therefore, the composite blade
should be seen as an exception representing development in technology.
The mass trend of existing towers also does not match with linear scaling laws as
shown in figure 2.8. Here, the trend exponent shows a lower value than what linear
scaling law predicts. This discrepancy has two main reasons. First, by looking to the
tower geometrical dimensions, it can clearly be seen that they do not scale linearly in
practice. This is mainly due to the simple geometry of the tower that allows the use of
design optimization techniques to search systematically the design space, which results
in design solutions that do not follow anymore a linear scaling relation.
Second, the specific tower top mass decreases on average as the size increases. Table
2.6 compares some recent multi-MW wind turbines with the prototypes of the 1970s
and 1980s. This effect clearly is the result of technology development that somehow
counteracts the principles of linear scaling laws.
In spite of often being violated and therefore of limited use in engineering design,
linear scaling laws provide a firm physical basis to understand the scaling phenomenon
24
2.6
CLASSICAL UPSCALING METHODS
Table 2.6: Specific tower head mass of some early and recent multi-MW wind turbines, Martin and
Roux (2011)
Earlier turbines
kg
Value ( m2 )
WTS-75 (Aelous)
WTS-3
Aeolus-II
MOD-2
MOD-5B
Growian
46
41
31
24
35
51
Average
38
Recent turbines
Siemens SWT-3.6
Vestas V90
Nordex N90
WinWind WWD-3
GE 2.5xl
Multibrid M5000
kg
Value ( m2 )
26
17
23
25
16
30
23
in the early stages of design studies. This is of quite importance, since no other design
study aiming at scales should be justified without an underlying physical understanding
that accounts for the general scaling behavior as appears in the trends.
2.6
Why a new approach for upscaling?
To address the research questions of this work a more sophisticated method than the
classical upscaling methods should be employed. Classical upscaling methods are not
capable of making accurate realizations of large scale offshore wind turbines. This is
needed to investigate the technical feasibility, economical characteristic and to construct
reliable scaling trends based on size specific optimized wind turbines.
Linear scaling law is only of a limited direct use. This is due to the formulation (the
inherent assumptions and simplifications) of the linear scaling law that is mainly suitable
in the conceptual design phase and realizing representative wind turbines at larger scales
using this method is not possible. Therefore, the technical feasibility and economical
characteristics of a linearly upscaled wind turbine may not be accurate enough to make
realistic judgment about its design.
Also technical data from manufacturers are obviously not available at larger scales
and extrapolating existing data trends introduces large uncertainties in the design. Additionally, distributed model properties can not be extracted from existing data trends
taking into account the wide range of design solutions from every different manufacturer
present in the trends.
Considering these facts, the goals of this research are beyond the reach of classical
upscaling methods, and this necessitates the search for alternatives. Such an alternative
method should have the following specifications:
• The method should provide the detailed technical data at larger scales and enable
the evaluation of all the design constraints of such big machines to identify the
associated design problems. Only, based on these technical data and their associated design constraints a judgment on the technical feasibility of large scale wind
turbines can be made.
• The method should provide the detailed economical data of larger scales and enable the evaluation of cost of energy to investigate the economical benefit of scale
2.6
WHY A NEW APPROACH FOR UPSCALING?
25
as size increases.
• The method should enable the realization of optimized wind turbines at any specific
size to construct reliable scaling trends.
To meet these specifications, all the involved disciplines and their dynamic interactions, important components relevant to upscaling, design constraints, cost elements and
several other details need to be considered. Based on these considerations, a proper
design methodology needs to be formulated and by having that in place, size specific optimized wind turbines can be realized to construct reliable scaling trends and investigate
the influence of size on the design of large wind turbines.
This will be the subject of next chapter to study different design methods. Integrated
aeroservoelastic design methodology will be selected as the best approach to deal with
the research questions of this work.
CHAPTER
3
Integrated aeroservoelastic design and optimization
3.1
Introduction
In the previous chapter, the two classical upscaling methods (linear scaling law and upscaling using existing data) were discussed. Because of the restrictions associated with
the classical upscaling methods, it was concluded that they cannot be used for addressing
the research questions of this work. This resulted in looking for an alternative method
and the main features of such an alternative method were formulated.
This chapter reviews first the existing design methodologies applicable to engineering
design. Having the formulated specifications in mind the best alternative design methodology will be selected and developed. The developed methodology of this chapter will
be applied on the turbine sizes from 5 to 20 MW in the following chapters to extract
upscaling trends and identify technical barriers and address the economy.
3.2
Classification of design problems
This section explains the features of a good design problem solving technique and lies the
foundation of a good design methodology that is needed for the successful implementation of this research.
A good design methodology is needed to make sound design wind turbines and allow
the identification of their design barriers that may require further attention for. Therefore, one should know different classes of design problems and methods, as well as the
advantages and disadvantages of each to be able to make the best selection.
27
28
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.2.1
3.3
Routine design problems
A design problem is called routine (or engineering) if all the steps to find the solution
are known, and the designer knows in advance that there is at least one feasible solution
(see VanGundy (1992), Pahl et al. (1996), Savransky (2000) and Bayazit (2004)).
There are many methods to solve a routine design problem in different fields such
as aerodynamics or structure, and they can be solved by standardized or automated
procedures. Normally, computer codes are employed to simulate the behavior of the
system under study and based on this behavior a solution can be proposed that satisfies
all the requirements.
The loop of proposing a solution, simulating the behavior of the system and checking
the satisfaction of all the requirements is repeated many times until all the designers are
satisfied. Typically, all the solutions of a routine design problem (also the nonroutine
ones) must satisfy the three following requirements:
1. Physically possible (corresponding to the laws of nature)
2. Technically possible (corresponding to available materials and manufacturing techniques)
3. Economically profitable (the lowest possible cost)
3.2.2
Nonroutine design problems
A design problem is called nonroutine (or inventive) if at least one critical step to a
solution or the solution itself is unknown, Semyon (2000). Inventive problems are often
mistakenly considered as engineering problems. Actually, any inventive problem becomes
at some moment a routine problem, since the recognition of the important steps increases
with time. In short, an inventive problem is usually novel, elusive, ambiguous and poorly
understood.
Based on this classification, this research deals with a nonroutine design problem,
since the solution itself is not known in advance. It could be the case that the design
process of a large scale wind turbine using the existing concept ends up with no feasible
design solution, which then requires a different concept to be studied.
3.3
Difficulties of design problems
Since the design problem of this research is a nonroutine design problem, only the difficulties of this type of design problem are considered and presented below:
3.3.1
Size of the design space
For a design problem, the difficulty can be correlated to the number of variables involved.
While simple design problems involve only a few variables, complicated design problems
involve many variables and are usually solved by a team of designers. The design space
in which a solution exists is defined in terms of the allowable ranges of the involved
variables.
3.4
METHODS OF SOLVING DESIGN PROBLEMS
29
The design space is closed if there is a finite number of correct solutions that satisfy
all the requirements. Rarely, only one acceptable solution exists for a design and usually
there are several possible solutions that make the design space open.
3.3.2
Psychological inertia
The problem-solving process itself depends strongly on the ability of the designer to
enable a search in the entire design space, specially when it deals with an open design
problem. Psychological inertia is the tendency of the designer to preserve the current
stable state or to resist dramatic changes in that state.
It often keeps the designer away from the best solution and complicates decision making. To overcome this problem, in reality many design processes rely on an optimization
algorithm instead of the designer experience to search the entire design space.
3.3.3
Uncertainties in the design process
Uncertainties can be classified based on their origin into two groups: aleatory and epistemic . Aleatory uncertainty is the inherent variation of the physical system under consideration or its operating environment. Epistemic uncertainty arises because of the lack
of knowledge or information in any phase or activity of the design process, and could be
removed if sufficient data is provided, Oberkampf et al. (2004).
Design in the presence of uncertainty consists of three steps, Roy et al. (2009):
1. Identification, modeling and representation of uncertainties into mathematical models based on probability theory or nonprobabilistic approaches
2. Propagating uncertainties (in computer codes) to quantify their impact on the
design problem
3. Selection of an appropriate optimization algorithm to ensure that the optimum
solution obtained is robust against uncertainties
3.4
Methods of solving design problems
In the context of design, a problem is any situation where the designer has the opportunity to make things better using a problem solving method. The effectiveness of such a
method depends on many factors such as: the quality of the solution, time spent on the
problem resolution, and acceptance of a solution. However, any problem solving method
should have at least the following characteristics:
1. It should have a mechanism for directing the solver (or designer) to the most appropriate and strong solutions. This is the heart of the method: it must reduce
quickly the design space and eliminate the nonfeasible solutions.
2. A good problem solver must obtain very high quality solutions with a high level of
recognition in a short time. This is specially very important when one deals with
large scale problems with many variables and different disciplines involved.
30
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.4
3. A good problem solver has to know practically all relevant human knowledge.
However in practice it is almost impossible for a human to meet this requirement,
but a good design methodology needs to have the necessary knowledge for even
difficult and complex nonroutine problems.
4. A good design methodology should "turn off" his psychological inertia. This requirement is a big challenge, therefore many methods to remove psychological
inertia have been proposed for.
Next subsection presents some of the most important solving methods.
3.4.1
Trial-and-error methods
Trial-and-error methods are the oldest solving methods that regardless of the class of the
problem can be applied both on the routine and nonroutine problems. However, they are
more efficient for simple, well-defined, routine and closed problems and only be used for
the open problems, if the possible direction of search is known.
Due to the disadvantages of the trial-and-error method it is not widely used in modern
engineering designs. Some of the disadvantages of this method are:
1. The number of ideas (trials) that succeed per unit of time is small. As a result the
design process lasts a long period of time, which means the method is not time
effective.
2. There is no mechanism for directing the solver’s thinking towards the solution.
Thus the solver is unable to define the direction in which the solution might be
found. This is the crucial disadvantage of the trial-and-error method.
3. There is no mechanism to systematically find the best solutions. As a result, solving
the same design problem might lead to different solutions.
These disadvantages were recognized in the middle of the twentieth century and since
then many attempts have been made to perfect that. These include methods such as:
1. Brainstorming, synectics, lateral thinking, neurolinguistic programming and mind
mapping to overcome the psychological inertia.
2. Morphological analysis, focal objects and forced analogy to expand the search
method.
3. Decision aids such as T-charts and probabilistic decision analysis to make the method
more systematic.
3.4.2
Inventive based methods
Inventive based methods need creativity to solve a design problem that is usually believed
to be beyond comprehension and, thus, methodology. However, they have proven their
benefits for finding original and useful ideas by systematically examining alterations in
existing components within a system, their attributes, functions or internal relationships.
Inventive design enables the designer to change old paradigms, and institute new
feasible design solutions without getting rid of the old ones and create out of the box
solutions. There are three premises on which these methods are based:
3.4
METHODS OF SOLVING DESIGN PROBLEMS
31
1. To make an ideal design with no harmful function.
2. Identifying contradictory behaviors of the system and wholly or partially eliminating them.
3. Giving structure to the design process such that it can be universally applied to any
design problem.
Among many inventive based methods like Quality Function Deployment (QFD), Theory of Constraints (TOC), Taguchi approach and Six Sigma, TRIZ (the theory of inventive
problem solving according to the acronym for the same phrase in Russian) is the most
systematic method for engineering design problems. It is an algorithmic approach, based
on a set of patterns of evolution, which can be used to generate ideas for developing the
next generation of a technological system.
3.4.3
Knowledge-based engineering methods
Knowledge is a key component of engineering design that can be collected from specialists and stored in a knowledge base system. They capture process and product information to make models of engineering design problems and then use the model to automate
all or part of the design process, or make analogies with other engineering design problems to solve a problem. Some of the advantages of knowledge-based methods is as
follow :
1. Simplification of the design process
The design process is simplified for the designer as his/her intervention is required
less in decisions making situations.
2. Expertise base elimination
Eliminates the need for the designer to develop expertise and skills in areas not
necessarily in his/her own field.
3. Innovation encouragement
The simplification of the design process enables the designer to put more time and
efforts for looking to more innovative design solutions.
4. Error-proofing
Complex calculations and tedious database/catalog lookups can be performed in a
quick and error-free manner.
5. Speed
Enables automation of repetitive tasks, which otherwise occupies the designerŠs
productive time.
There are many development methodologies for the knowledge-based engineering
domain such as: MOKA (Stokes (2001)), CommonKADS (Schreiber et al. (2000)) and
47-Step Procedure (Milton (2007)). However, each method has its own advantages and
disadvantages for a given field and application.
32
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.4.4
3.5
Design search and optimization methods
The use of optimization methods is steadily increasing in design. At its simplest form, optimization can be defined as the search for a set of inputs x, design variables, that optimizes the objective function f (x), subject to inequality g j (x) ¾ 0, and equality hk (x) = 0
constraints, Haftka and Gürdal (1992).
Design optimization problems can be characterized in different ways as follows:
1. The mathematical form of equation involved
Both the objective function and constraints can be classified as linear or nonlinear
equations resulting in a linear or a nonlinear optimization problem.
2. The number of objective functions
If the problem under study has only one objective function to optimize, then we are
dealing with a single optimization problem, otherwise we have a multi-objective
function problem.
3. The deterministic nature of design variables
If the design variables have a stochastic distribution rather than a deterministic
value, then we are dealing with a stochastic design optimization problem. Deterministic optimization problems always yield the same optimum if started from the
same point in the design space, stochastic methods make no such guarantees.
4. The presence of constraints
In an unconstrained optimization problem the design space is infinity and the objective function can take any value, whereas in a constrained optimization problem,
either the design variables are bounded by a lower and upper limit or the design
space is bounded by some functional constraints.
5. Type of optimization algorithm
An optimization algorithm is a numerical method that explores the design space to
find a set of design variables, which optimizes (minimize or maximize) the objective function. In this process, if the algorithm uses the derivative of the objective
function or constraints, it is called a gradient based method (sometimes also called
first order), otherwise it is called direct (sometimes also zero order or evolutionary)
method.
6. The permissible value of design variables
The design variables can be continuous or discrete. If they are continuous then
both the zero and first order optimization algorithm can be used to solve them.
However, for discrete values the most common method is integer programming,
although constraint logic programmes are also applicable.
7. The number of involved disciplines
When proper modeling of the system requires a number of disciplines that interact
with each other, then the design optimization problem is called a multidisciplinary
design optimization (MDO).
3.6SELECTION OF THE DESIGN METHODOLOGY AND ITS OVERVIEW APPLICABLE TO UPSCALING
3.5
33
Selection of the design methodology and its overview applicable to upscaling
The common practice in designing a wind turbine is the isolated approach. In this approach, the structural, aerodynamic and control designs do not occur at once. To cope
with more complex future wind turbines, the isolated approach does not seem to be appropriate. An integrated design approach is essential to guarantee the minimum design
time and to cover all the interdisciplinary aspects of design.
Considering all different choices explained before, it is the MDO that offers such an
integrated capability. Additionally, to address both the technical and economical characteristics at the same time and realize size specific optimized wind turbines (as it was
formulated in section 2.6) non of the other methods can be of direct use. Therefore,
the scaling study of this research will be formulated as an MDO. The MDO formulation
enables in a clear way to address the technical feasibility and economical characteristic
of a wind turbine as follows:
• Through the definition of the design variables an optimizing them an accurate
realization of the technical characteristics of the future wind turbines can be made.
These optimized wind turbines are needed to construct reliable scaling trends.
• Through the inclusion of the design constraints all the problems that may result
the design to become nonfeasible or challenging can be covered and by referring
to these design constraints an assessment of the technical feasibility can be made.
• Through the definition of the objective function all the important elements that can
influence the cost of energy such as the cost of all the components, operational and
maintenance cost, installation and the energy yield can be included. This enables
the assessment of the economical characteristic of large scale wind turbines.
For this purpose, all the disciplines influencing the design should be involved. For a
wind turbine, aerodynamic, control and structure are the dominating disciplines and to
preserve their dynamic interactions the wind turbine should be seen as an aeroservoelastic
system. This requires the methodology to be an integration of different computational
models each representing one discipline. Also, the data exchange among all computational models should be consistent and automated. It also should enable the exchanged
data to be processed at any stage before or after passing that among models.
Unfortunately, most of the wind turbine computational models currently in use rely
on an architecture that is not suitable for multidisciplinary design studies (for a review of
existing simulation codes of the wind turbines see Ashuri and Zaaijer (2007)). Therefore,
in this research a large effort has been made to integrate the best candidate out of many
existing simulation tools and automate the data flow among them and finally parameterize the entire design process. Next section discusses the architecture of the MDO used in
this research.
3.6
Architecture of the MDO
A common practice in wind turbine design optimization is to employ a sequential approach to solve multidisciplinary design problems, each focusing on one discipline, for
34
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.6
example, structure, aerodynamic and control. Here, the simplest approach is to first
do an aerodynamic shape optimization of the blade and structural considerations are
not directly entered into the aerodynamic design formulation. However, some bound
constraints that comes from structural consideration can be applied in the aerodynamic
optimization such as the airfoil thickness at several stations along the blade.
Subsequently, knowing the aerodynamic loading and external shape of the blade,
the structural team can do an optimization to find the optimal internal blade geometry.
Unfortunately, this sequential approach neglects the coupling between aerodynamic and
structure known as aeroelasticity, which leads to convergence to a suboptimal solution.
There are however several other different optimization architectures that can be used
at different situations, Martins and Lambe (2011). This study uses a Fully Integrated
Optimization (FIO) architecture. The idea is to directly couple an optimizer to a multidisciplinary computational model, see figure 3.1. This preserves the interaction of
different disciplines. Here, the design variables are passed to the simulation tools and
objective function and constraints are send back by them to be assessed by the optimizer.
This iterative approach continues until the convergence is achieved.
During this process, a chain of computational activities is needed that are classified
as aerodynamic, structure and control evaluation as well as a proper assessment of the
associated costs and design constraints. This classification, while clearly very general,
provides a standard for simulating the behavior of machines that is needed for design
studies (also usable in different fields of machine design, Ashuri et al. (2012)).
Furthermore, in this process many configurations, many alternative shapes and struc-
Optimization engine
Optimization
Optimizationalgorithm
algorithm
{ x1, … , xn }
f(x), gj (x) & hk(x)
Structure
Structure
Cost
Cost
Control
Control
Aerodynamic
Aerodynamic
Integrated multidisciplinary simulation
Figure 3.1: Fully integrated optimization architecture
3.7
OPTIMIZATION PROBLEM FORMULATION
35
tural arrangements including many design variables, design constraints and the objective
function need to be simulated and evaluated iteratively. In order to be able to easily and
effectively develop such a methodology the design optimization formulation should be
defined in advance. This will be explained in the next section.
3.7
Optimization problem formulation
In order to construct an optimization model the design variables, the design constraints
and their relevant safety factors, and the objective function should be defined. In addition, to evaluate the design constraints and the objective function, a set of simulation
tools is required. Then the optimization model can be expressed in the following form:
Find x = x 1 , . . . , x n that minimizes f (x)
(3.1)
subject to side constraints as:
x lower ≤ x ≤ x upper
(3.2)
g j {x} > 0
(3.3)
hk (x) = 0
(3.4)
and functional constraints as:
where x is an n-dimensional vector of design variables to be found with a lower
and upper bound as side constraints, g j (x) as the inequality and hk (x) = 0 as equality
constraints to be satisfied, and f (x) as the objective function to be optimized.
3.7.1
Design variables
A design variable is a changeable parameter by the optimizer, whose influence on the constraints and objective function needs to be evaluated during the design process. Among
many components of a wind turbine, blade and tower are the most important components to optimize. Major reasons for such a conclusion are:
• They are the most flexible components of a wind turbine, thus they require many
design considerations. This is of great importance in larger wind turbines and
upscaling studies, since the larger the wind turbines become the more flexible it
gets.
• Their failure is more critical to the safe state of the entire wind turbine.
• Blade and tower (also the controller) have the highest influence on the energy yield
of the wind turbine. Therefore, to make a cost effective design, which is the target
of any design they need special attention.
• Blades and tower (also the gearbox) have the highest cost share of a wind turbine.
Typically, blade and tower make up 25% to 30% of the capital cost of an offshore
wind turbine.
36
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.7
• To represent the dynamics and economy of the components of a wind turbine other
than the blade and tower, a mass and cost model is used (this will be explained
later in this chapter). In many components, these cost and mass models are a
function of the blade length. Thus, optimizing the blade length provides the mass
and cost of these items in such a way that at the end they result in the lowest LCOE
(Levelized Cost of Energy).
Figure 3.2 shows different stations along the blade and tower in which the wind
turbine parametrization is taking place. Also table 3.1 presents the 23 design variables
and their relevant stations.
Table 3.1: Wind turbine design variables to be optimized
No.
Description of the variable
1
2
3
Blade length
Tower height
Rated rotational speed
4-5
6-9
Blade structural twist at stations 9 and 15
Blade structural chord at stations 3, 7, 15 and 18
10-13
14-16
17-19
Blade skin thickness at stations 1, 3, 6 and 16
Blade shear web thickness at stations 3, 6 and 16
Blade spar cap thickness at stations 3, 6 and 16
20-21
Tower diameter at stations 1 and 22
22-23
Tower thickness at stations 1 and 22
In this research, optimization of the blade is carried out with 17 design variables. For
the external geometry of the blade that is needed for aerodynamic design optimization,
chord at 4 stations along the blade, twist 2 stations along the blade, and the blade length
itself are introduced as design variables.
For the internal thickness distribution of composite lay-ups needed for structural
design optimization of the blade, spar thickness at 3 stations along the blade, shell thickness at 3 stations along the blade, and web thickness at 3 stations along the blade are
introduced as design variables.
Also the rotational speed of the rotor is a design variable, since together with the
blade length they control the tip speed of the blade, which is an important design parameter. In addition, the rotational speed has a big influence in the drive train loads and
thus the costs.
All these design variables are continuous and defined at some stations along the blade
and tower. An interpolation scheme is used to define the values at different stations.
To make a smooth interpolation of the design variables, some other design parameters
are fixed for all different scales from 5 to 20 MW as it is shown in table 3.2. The location
of the shear caps from station 6 to the blade tip are the same as station 6. From station
1 to 6 an interpolation is used to find the location of the shear cap.
3.7
OPTIMIZATION PROBLEM FORMULATION
Figure 3.2: Parametrization of the wind turbine configuration
37
38
3.7
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
Table 3.2: Fixed design parameters of the blade
Description of the parameter
Value (unit)
Shear web thickness at station 1
Spar cap thickness at station 1
0.00 (m)
0.00 (m)
Structural twist at station 20
Structural chord at station 20
0.00 (deg)
0.12 (m)
Nondimensionalized location of the 1st shear cap at station 1
Nondimensionalized location of the 2nd shear cap at station 1
0.00 (-)
1.00 (-)
Nondimensionalized location of the 1st shear cap at station 6
Nondimensionalized location of the 1nd shear cap at station 6
0.25 (-)
0.50 (-)
3.7.2
Design Constraints
A constraint is a condition that needs to be satisfied to have a feasible design. There are
two types of constraints considered in this research; side and functional constraints. The
side constraints are the upper and lower limits of a design variable that bound its value
to be in a range that the designer defines in advance.
Functional constraints are mathematical equations which must be satisfied in an optimization problem. They represent some physical principles which govern the relationship among the design variables. For the functional constraints, in total, 51 inequality
constraints are used to obtain feasible design solution of the blade and tower as shown
in table 3.3 and 3.4.
Table 3.3: Wind turbine blade design constraints
No.
Description of the design constraint
Value (unit)
1-9
Max flapwise tip deflection from 9 to 25
10-15
16-21
Max flapwise fatigue at stations 1, 3, 7, 10, 12 and 14
Max edgewise fatigue at stations 1, 3, 7, 10, 12 and 14
22-27
28-33
Max flapwise stress at stations 1, 3, 7, 10, 12 and 14
Max edgewise stress at stations 1, 3, 7, 10, 12 and 14
34-36
First, second and third natural frequency
37
Tip speed
m
s
Scale dependent
≤ 0.7 (-)
≤ 0.7 (-)
≤ 200 (MPa)
≤ 200 (MPa)
Scale dependent
≤ 120 ( ms )
In these functional constraints, the partial safety factors are also included. However,
the selection of the appropriate partial safety factors that is applied on the design constraints of table 3.3 and 3.4 is treated separately in the next subsections.
For the blade, stresses and cumulative fatigue damage (fatigue damage calculation
is done using rain flow cycle counting of the stress signals and applying Miner’s rule
on these signals) at 5 stations along the blade, tip deflections for different wind speeds
during operation, and the first 3 natural frequencies are the design constraints. While all
the side constraints are scale dependent, only some of the functional constraints are scale
dependent. Therefore, those scale dependent constraints are defined in the subsequent
3.7
OPTIMIZATION PROBLEM FORMULATION
39
chapters for each scale individually.
For the tower, stresses and fatigue damage at 6 stations along the tower and the first
two natural frequencies are the design constraints. In addition to the inequality constraints mentioned above, some side constraints are used to bound each design variable
that are scale dependent and their values will be presented in the following chapters.
Table 3.4: Wind turbine tower design constraints
No.
Description of the design constraint
Value (unit)
1-6
Fore-aft stress at stations 1, 5, 9, 13, 17 and 21
≤ 150 (MPa)
7-12
Fore-aft fatigue at stations 1, 5, 9, 13, 17 and 21
13
14
First natural frequency of the tower
Second natural frequency of the tower
3.7.3
≤ 0.7 (-)
0.3 ≤ ω1n ≤ Scale dependent
0.3 ≤ ω2n ≤ Scale dependent
Partial safety factors
It is a common practise to use partial safety factors in wind turbine design for achieving
sound designs, as prescribed by many standards. The technique comes from an analogy
with IS02394 (General principles on reliability for structures) standard on limit-state
design theory, which describes loads (actions) and strengths (resistances) in a structure.
The ISO2394 standard defines limit-state design procedures that use partial safety
factors to cover uncertainties in the design process. These partial safety factors are dependent on the uncertainties in the analysis methods, the uncertainties in materials,
loads, and the consequences of failure for component classes. To account for all the uncertainties in the design process, partial safety factors are applied to loads, materials and
the failure consequence of different components.
There are many standards that specify the various aspects of the design process, however little agreement exists among them, therefore it is always useful to check at least
two different standards (the approach used in this research).
The well known standard and certification bodies in the wind turbine industry, based
on their popularity are: International Electrotechnical Commission (IEC), Germanischer
Lloyd (GL) and Det Norske Veritas (DNV). Each certification body has its own rules and
requirements by which a specific turbine type is certified, tested, manufactured and documented. One of the major differences among these standards is the treatment of partial
safety factors.
Most design codes address partial safety factors for loads, strength and failure separately. According to IEC 61400-1 standard, a series of partial safety factors for each type
must be used to cover these uncertainties. However, IEC did not always provide sufficient
guideline in the selection of these factors. The GL standard provides an explicit list of
partial safety factors and prescribes all of the individual factors to be applied and non of
the adjustments are left to the designer.
The partial safety factors for materials from the GL standard are shown in table 3.5.
Applying the GL factors as specified implies a combined material factor of 2.94 for composites, which seems to be conservative, and 1.35 for non-composites. In this case the
IEC standard only provides a general material factor of 1.1.
40
3.7
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
Table 3.5: GL partial safety factors for material
Type of safety factor
Symbol
General material factor
Influence of aging
Temperature effect
Hand layup laminate
Post-cured laminate
γM 0
C2a
C3a
C4a
C5a
Total (Π)
Value (-)
1.35
1.5∗
1.1∗
1.2∗
1.1∗
2.94
∗ Only applicable to composites
In both the IEC and GL standard, the load safety factors are classified in an ultimate
and fatigue category. Table 3.6 summarizes the IEC and GL partial safety factors for
ultimate loads, and table 3.7 shows that for fatigue loads. The consequence of failure
is also shown in the same table, since it is classified based on the load type in both
standards.
Table 3.6: Partial safety factors for ultimate load analysis
Type of safety factor
IEC
GL
Loads
Blade consequence of failure
Tower consequence of failure
Materials
1.35
1.0
1.0
1.1
1.35
1.0
1.0
2.45-2.940
1.485
3.307-3.969
Total (Π)
The total partial safety factors for ultimate loads is maximum 1.485 and 3.969 for the
IEC and GL, respectively. A detailed comparison between these two standards is given by
Musial and Butterfield (1997). No adjustment is required under both standards to cover
the consequences of failure for ultimate load analysis for the rotor blade and tower.
Table 3.7: Partial safety factors for fatigue load analysis
Type of safety factor
IEC
GL
Loads
Blade consequence of failure
Tower consequence of failure
Materials
1.0
1.15
1.15
1.1
1.0
1.0
1.0
1.485-1.960
1.265
1.485-1.960
Total (Π)
The IEC fatigue load analysis is 1.265 that excludes the designer-specified material
property adjustments mentioned before. In the GL code, the total partial safety factors
is between 1.485 and 1.96, In both tables the safety factor to cover the consequences of
failure is component dependent and these values are given for rotor blades and tower as
non-fail-safe components.
3.7
OPTIMIZATION PROBLEM FORMULATION
41
These standard partial safety factors are the minimum requirements that a design
should meet, and can be used as a guideline to select representative values. However,
the values used in this work are selected as such as to give consistent values with the 5
MW NREL design, while satisfying the minimum standard requirements. Table 3.8 shows
the selected values of partial safety factor throughout this work.
Table 3.8: Definition of partial safety factors used in this work
Type of safety factor
Material safety factor
Consequence of failure
Ultimate load factors
Fatigue load factor
Value (−)
1.10
Blade 1.0
Tower 1.0
1.48
1.43
Referring to table 3.3 that shows the design constraints of the blade, the followings
partial safety factors for each design constraint is considered:
Maximum flapwise tip deflection of the blade
For item 1 to 9 of table 3.3, the load and the material partial safety should be considered. This combination of safety factor results in a total safety factor of 1.628 (1.1 ×
1.48). That is, if the distance between the unloaded blade at the tip and the tower is for
example 5 m, then the maximum flapwise deflection of the blade when it is loaded should
not be larger than 3.07 m. However, this distance varies at different scales, therefore this
design constraint is scale dependent.
Maximum flapwise and edgewise fatigue of the blade
As it was mentioned before cumulative fatigue damage calculation is done using
Palmgren-Miner rule. Using this rule and usually for design purposes, fatigue failure does
not happen if the total cumulative damage is less than 1. As table 3.8 shows, for item 10
to 21 of table 3.3, a partial safety factor of 1.43 safety for fatigue load is considered. This
means that the total cumulative damage should not be larger than 0.7.
Maximum flapwise and edgewise stress of the blade
In this work, it is assumed that the yield stress of the composite blade is 325 MPa.
However, application of a material safety factor for composite materials is not as straight
and applicable as it is for isotropic and homogeneous materials.
Composite materials have many failure modes, and their yield stress depends strongly
on the layup and the volumetric mixture of the elements that the composite sample is
made of. Exceeding this yield stress in some elements of a composite material does not
necessarily mean the entire structure will fail, since the experienced load of that element
will be distributed among other elements. Considering these facts, the defined value in
this work is just a representative value suitable for the design applications.
For stress calculation, the combination of the material safety factor and ultimate load
safety factor has to be considered. Thus, the total safety factor becomes 1.628 (1.1 ×
42
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.7
1.48). Using this definition, the maximum allowable flapwise and edgewise stresses at
the selected stations along the blade (item 22 to 33 of table 3.3) are 200 MPa.
Allowable natural frequencies of the blade
As it is shown in table 3.3 for item 34 to 36, the natural frequencies are scale dependent. Therefore, after finding the optimum rated rotational speed of the rotor in each scale
(5, 10 and 20 MW) a safety margin of ± 10% for the 1P and 3P excitations is considered.
Similarly, the design constraints of the tower presented in table 3.4 are obtained as
follow:
Maximum fore-aft stress of the tower
In this work, it is assumed that the tower is made of structural steel. Typically, the
yield stress of the structural steel is around 270 MPa. By applying the combination of the
ultimate and material safety factor of 1.628 (1.1 × 1.48) to the defined yield stress, a
maximum allowable fore-aft stress of 150 MPa for the tower is obtained. This maximum
allowable stress is shown at different stations along the tower height (item 1 to 6) in
table 3.4.
Maximum fore-aft fatigue of the tower
Similar to the fatigue damage calculation of the blade, the fatigue damage calculation
of the blade is carried out. As table 3.4 shows, for item 6 to 12 the total cumulative
damage should not be larger than 0.7.
Allowable natural frequencies of the tower
Unlike the blade, the tower has an extra natural frequency constraint. In fact, the
tower is in touch with water and its natural frequency should not coincide with the wave
frequency. The typical wave frequency for the shallow to medium level waters is from
0.1 to 0.3 Hz.
This forces the natural frequencies of the tower to be larger than 0.3 Hz. As it is
shown in table 3.4 for item 13 and 14, the lower bound of the natural frequency is 0.3
Hz and the upper bound is scale dependent. Similar to the blade, a safety margin of ±
10% for the 1P and 3P excitations is considered.
3.7.4
Objective function
When applying any changes to a given design, it is important to know the influence of
this change in the system level. As an example, an increase in blade length to increase
annual energy production (AEP) increases loads and installation costs. If one change
does not balance out the other change, then the proposed improvements end up with a
negative overall impact.
The LCOE is a good indication to see the influence of all these cross couplings in an
MDO, and results in a true assessment of all technical changes. Therefore, it is selected
in this work to be the objective function of the MDO to minimize. For an offshore wind
turbine operating in a wind farm, LCOE is made up of all the system cost elements, which
is explained by Fingersh et al. (2006). This covers the following items:
3.7
OPTIMIZATION PROBLEM FORMULATION
43
• Turbine Capital Cost (TCC): TCC is the cost of all the components of a wind turbine.
These are: blades, hub, pitch mechanism and bearings, nose cone, low speed shaft
and bearings, gearbox, mechanical brake, high speed shaft and coupling, generator, power electronics, yaw drive and bearing, main frame, electrical connections,
hydraulic system, cooling system, nacelle cover, control equipments, safety system,
condition monitoring, tower, marinization (extra cost to protect against marine
environment like salty water).
• Balance of Station (BOS): BOS is the cost of following items: monopile, port and
staging equipment, turbine installation, electrical interface and connections, permits, engineering, site assessment, personnel access equipment, scour protection,
offshore warranty premium and decommissioning. Transportation over land is not
considered as an element in the BOS, since it is assumed that all the manufacturing
facilities will be located near the shore for larger scale offshore wind turbines.
• Initial Capital Cost (ICC): ICC is the summation of the TCC and BOS costs.
• Levelized Replacement Cost (LRC): LRC is the cost of major replacements and overhauls, distributed over the life time of the wind turbine.
• Operations and Maintenance (O&M): O&M is the cost associated to fix failures of
mechanical or electrical components and all the regular and irregular inspection
of the wind turbine. Table 3.9 gives an overview of all the cost elements that are
calculated using these definitions for the 5 MW NREL wind turbine.
LCOE can be calculated using the following equation:
LCOE =
(I C C × F CR) + LRC + O&M
AE P
(3.5)
here: FCR is the fix charge rate and includes construction financing, financing fees,
return on debt and equity, depreciation, income tax, and property tax and insurance.
AEP is the amount of generated electricity in a year for a given wind speed distribution, f (V ). It also includes the mechanical and electrical conversion losses, and machine
availability. Wind speed distribution is modelled as:
V
V
k
f (V ) = ( ) · ( )k−1 · exp −( )k
c
c
c
(3.6)
where V is the wind velocity, k the shape factor, and c the scale factor. In this research,
k is 2 corresponding to the Rayleigh distribution and c is defined to be 9.47. These values
typically represent the condition of the Dutch part of the North Sea, Brand (2008). Then
the AEP production can be expressed as:
AE P ≈ 8760 ×
cut−out
X
P(Vi ) · f (Vi )
(3.7)
i=cut−in
in which 8760 is the number of hours per year, and P(V ) is the power curve of the
wind turbine.
44
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.8
However, these cost models were calculated based on the cost of materials and products
in 2002 and should be updated based on the cost of materials and products in the present
time to account for the inflation of materials, products and labors.
To compensate for this difference a component cost escalation model that is based on
the Producer Price Index (PPI) is used. The PPI is an index that is updated on a monthly
basis by the US Department of Labor, Bureau of Labor Statistics, to track the changes of
costs of products and materials over a wide range of industries and industrial products.
The PPI provides a more accurate update than a general inflation index (GII). As an
example, the increase in the cost of structural steel and copper in 2004 and 2005, were
considerably higher than the GII in the same year. This means that cost of products made
of structural steel and copper increased also rapidly and the GII is unable to track such
changes and using that instead of PPI results in a considerable error.
Table 3.9 is an example of a cost estimate and LCOE calculation from a run of the cost
model (the cost is escalated to 2009 USD) for the offshore (shallow water), 5 MW NREL
wind turbine. The detailed cost models used in this table are presented in appendix A.
3.7.5
Wind turbine simulation tools
Wind turbines are complex systems and to simulate the physical behavior of such a system
computer-based simulations are required. In this work, the NREL series of simulation
tools are used, since they are all open source and free, suitable for research purposes.
The theoretical background of the analysis methods employed in each code is presented in appendix B. Table 3.10 briefly presents each code, and its usage for this research.
Unfortunately, all these simulation tools are in principle independent and do not communicate effectively with other simulation tools, and the data transfer between these
tools has to be done manually by the designer. However, in an MDO problem the manual
intervention of the designer is impossible, since the process needs to run iteratively many
times to find the optimum design solution. To overcome this problem, these tools need
to be integrated. This issue is explained in the next section.
3.8
Design integration
With the available fast computers (especially when used as a cluster), it is possible to
analyze a large scale nonlinear unsteady multidisciplinary design problem. However
the design framework within which the system simulation and analysis should evolve
is itself not a single entity, but a combination of different simulation tools that need
to be integrated in advance and in such a way that it promotes an efficient working
environment.
To develop a responsive and effective design a successful integration of all the simulation tools is needed. This integration should also enable the evaluation of requirements
as the design constraints, and targets as the objective function to be achieved.
Figure 3.3 shows the integration of several simulation tools that are needed to capture the aeroservoelastic behavior of the wind turbine and evaluate the design constraints
and objective function.
3.8
DESIGN INTEGRATION
45
Table 3.9: Cost estimate of the 5 MW NREL wind turbine (2009 USD)
Components
Cost (1000 USD)
Mass (kg)
Blades
Hub
Pitch mechanism and bearing
Nose cone
Low speed shaft
Main bearings
Gearbox
Mechanical brake and coupling
Generator
Power electronics
Yaw drive and bearing
Main frame
Electrical connections
Hydraulic and cooling system
Control,safety and condition monitoring
Tower
Marinization
Turbine capital costs (TCC)
1062.3
130.2
242.0
13.6
166.8
64.4
877.2
11.0
398.0
393.2
146.3
162.7
308.8
77.2
65.3
1491.3
939.1
4722.2
17480
60540
28878
1810
16526
5400
39688
1053
17623
13152
31773
423.7
596520
-
Foundation system
Transportation
Port and staging equipment
Turbine installation
Electrical interface and connection
Permits, engineering and site assessment
Personnel access equipment
Scour protection
Decommissioning
Balance of station costs (BOS)
2174.7
1568.3
144.9
732.8
2063.5
215.5
70.2
403.0
362.8
7373.2
-
Offshore warranty premium
Initial capital cost (ICC)
624.1
13083.0
-
Levelized replacement costs
Operation and maintenance
Fixed charge rate
99.0
561.4
0.1185
-
23911817.1
-
0.0658
-
AEP (kW hr)
LCOE
USD
( kW
)
hr
This integration is carried out by writing a shell program that manages the data flow
between all the codes. The program is used to manage the design optimization process
with the following tasks and features:
• The combination of different simulation tools works effectively and correctly
46
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.9
Table 3.10: Selection of the simulation tools
Tool
Usage
TurbSim
AeroDyn
AirfoilPrep
FAST
BModes
Crunch
Fatigue
Simulating the 3D turbulent wind flow
Simulating the steady and unsteady aerodynamic loads
Modifying 2D airfoil data for 3D effects
Simulating the aeroelastic behavior of the wind turbine
Calculating mode shapes and frequencies
Postprocessing the output data of FAST
Rainflow cycle count of the load histories
• The data exchange among this integrated framework can be done automatically
• All the design changes are reflected in the relevant locations in the framework
• The iterative simulation of the system that is needed for the MDO process is possible
• Parameterize the integrated framework in such a way that simulating different
wind turbine concepts and scales can be done easily
• Giving access to intermediate data at any stage of the design process
• Interfacing with any other optimizer than the one currently in use
• Keeping the architecture of the program modular
• Enabling the integrated framework to run on a cluster of computers
However, the integrated frame work of this research still lacks two important features
that are needed for an MDO of a wind turbine. First, automating the controller design and
updating its parameters in every iteration, second automating the extraction of blade and
tower structural properties. These issues are treated separately in the following sections.
3.9
Controller design automation
One of the main complexities of designing a wind turbine is due to the dynamic interactions of aerodynamic forces, structural flexibility and control actions, known as aeroservoelasticity. A good design optimization process (DOP) should consider these interactions from the first iterations of the system design.
Although the integration of the tools explained in the previous section can simulate
the combined effects of control, structure and aerodynamics, the DOP cannot be done
with all disciplines at a time involved.
Until now, the experience to optimize aerodynamics, structure and control, subject to
many design constraints does not exist in the field of wind energy and there has been no
reported development of a truly aeroservoelastic design optimization capability within
the wind energy industry.
In fact, up to now, the role of the control engineer in wind turbine design activities
has been mainly focused on designing the controller for a given configuration. However,
the shift towards more flexible structures equipped with active control devices operating
3.9
CONTROLLER DESIGN AUTOMATION
Design
Designvariables
variables
Design
Design
variables
variables
Design
Designload
loadcases
cases
Side
Side
constraints
constraints
NTM
NTM
EWM
EWM
Wind
Windturbine
turbinemodel
model
Design
Design
parameters
parameters
Turbine
Turbine
concept
concept
Aeroservoelastic
Aeroservoelasticsimulation
simulation
TurbSim
TurbSim
AeroDyn
AeroDyn
Fatigue
Fatigue
AirfoilPrep
AirfoilPrep
FAST
FAST
Crunch
Crunch
BModes
BModes
Controller
Controller
design
design
AEP
AEP
Stiffness
Stiffness
calculation
calculation
Mass
Mass
calculation
calculation
System
System
costs
costs
Design
Designconstraints
constraints
Fatigue
Fatigue
damage
damage
Modal
Modal
frequencies
frequencies
Stresses
Stresses
Optimization
Optimizationalgorithms
algorithms
Objective
Objective
Blade
Bladetip
tip
deflection
deflection
COE
COE
Sensitivity
Sensitivityanalysis
analysis
Figure 3.3: Integrated Design Optimization Engine
47
48
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.9
in the entire working domain of the wind turbine is changing this role.
As the design becomes more integrated with multidisciplinary design optimization,
the design problem is changing from how do regulate the power or decrease the loads
to how can aerodynamics, structure and active control be optimally combined to build a
wind turbine that has the lowest LCOE, while satisfying all the constraints.
In the context of design and design optimization, most of the DOPs do not take the
controller design as part of the study from the beginning, and those rare cases that include the controller in the DOP are based on a fixed controller setting that is developed
for the initial design, (see for instance Fuglsang et al. (2002), Hansen et al. (2005) and
Xudong et al. (2009)). In both cases, the influence of the controller on the objective
function and design constraints cannot be evaluated properly. This leads to a system that
is not a real optimum.
To properly evaluate the objective function and design constrains, controller design
should be part of the DOP and redesigned within every iterations, with its parameters
introduced either as design variables in the loop, or as fixed variables being updated
in every iteration, Ashuri et al. (2010b). However, this is a challenging problem in an
automated DOPs, in which the manual intervention of a designer is not possible in every
iteration.
This new methodology enables the shifting of the wind turbine design problems from
a single discipline design optimization to a highest level that is aeroservoelastic design
optimization. In this way, control design is also becoming a part of the optimization
process and updated in every iteration.
The implemented controller design contains three control algorithms, a torque controller for below rated region and the transition part, a pitch controller for the above
rated region, and a tower top motion feedback loop to correct the relative wind velocity
that the rotor experiences due to the flexibility of the tower.
For below rated region, the generator torque is assumed to be proportional to the
square of the filtered generator speed. For above rated region, the periodic steady state
operating points are found and the system is linearized around these steady state operating points. Based on this linearized model, the above rated pitch controller can be
designed. Then the controller parameters are updated and provided as a DLL (Dynamic
Link Library) file for every iteration of the DOP.
To find out the usefulness of this methodology and its positive influence in a MDO, it
is applied on the 5 MW NREL wind turbine DOP. To do that, as a case study two different
optimization scenarios are defined. However, to ease the readability and avoid presenting
unnecessary information, these two case studies are explained in detail in appendix C and
not presented here.
In the following subsections, different aspects and features of controller design automation are presented.
3.9.1
Wind turbine models for designing the controller
The model of the entire wind turbine is structured as several interconnected subsystem
models, and represented by a block diagram shown in figure 3.4. The block diagram consists of a wind model, an aerodynamic model, a mechanical model of the drive train and
a control block model for the generator torque controller, tower top motion controller,
3.9
CONTROLLER DESIGN AUTOMATION
49
and the blade pitch controller. These models are explained as follow:
Pitch
Pitch
controller
controller
βref
β
Wind
Wind
Veff
V
Σ
Drive
Drive
train
train
Aerodynamic
Aerodynamic
ωrot
Ż
ωgen
Trot
Generator
Generator
Tgen
P
Ftrst
Tower
Tower
P ref
Generator
Generator
controller
controller
Figure 3.4: Subsystem block diagram of the wind turbine
Wind model
The simulation tool to models the physics of the wind is TurbSim and it is described
in detail in appendix B. The main advantage of this model is its flexility to model a wide
range of problems; from a simple stationary hub wind model to more complex situations.
Because of the changes of wind turbine design variables that influence the rated wind
speed during the optimization process, the simple hub wind model is used to find the
rated wind speed. For the rest of the control design task a wind model on the rotor plane
with a turbulence wind is implemented.
Aerodynamic model
The aerodynamic model transfers the three-dimensional wind speed field into forces
on the blades. The aerodynamic model that is used for the control design task is AeroDyn
and it is explained in appendix B. For the controller design task, it was observed that a
frozen-wake assumption, in which the induced wake velocities are held constant during
the perturbation of the blade-pitch angle, gives a more accurate linearization for heavily
loaded rotors at wind speeds close to rated.
Mechanical model of the drive train
The mechanical model transfers the aerodynamic torque on the blades to the generator shaft. It consists of the rotor, the gearbox and the mechanical parts of the generator.
Only for the purpose of calculating the gains of the blade pitch controller, a one-mass
model with an equivalent mass moment of inertia, stiffness and damping properties of
the drive train is used (as it will be explained later in this section, the linearization of the
wind turbine at different operating points is carried out in FAST that is a wind turbine
aeroelastic code).
50
3.9
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
The mass moment of inertia of the high and low speed shafts and the gearbox wheels
are neglected, since they are very small compared with the huge mass moment of inertia
of the rotor and generator. Using this model, the shaft rotation is the only degree of
freedom of drive train. Thus, for such a model the equation of motion can be described
by:
Tr ot − Z g b · Tgen = I d r · ω̈
(3.8)
Where:
Tr ot
Zg b
Tgen
Id r
ω̈
Rotor torque, (N · m)
Gear box ratio, (−)
Generator torque, (N · m)
Drive train mass moment of inertia (I d r = I r ot + Z g2b · I gen ) in (kg · m2 )
with: I r ot as the rotor mass moment of inertia and I gen as the generator
mass moment of inertia
r ev
Rotational acceleration of the drive train, ( min
2)
3.9.2
Control model of the generator torque controller
As figure 3.5 shows, the generator torque controller consists of five different regions. In
region 1, which is below the cut-in wind speed, the generator torque is zero and the
kinetic energy in the wind is used to accelerate the rotor for the startup.
45
Generator torque (kN.m)
40.5
1
36
1 1/2
2
2 1/2
3
31.5
27
22.5
18
13.5
9
4.5
0
0
180
360
540
720
900
1080
1260
1440
1620
1800
Generator speed (RPM)
Figure 3.5: Generator-torque controller of the 5 MW wind turbine in different regions
Region 2 is the power production region, in which the generator torque is assumed
3.9
CONTROLLER DESIGN AUTOMATION
51
to be proportional to the square of the filtered generator speed to keep a constant (theoretically optimal) tip-speed ratio. That is:
Tgen = K g · ω2gen
(3.9)
with K g as the generator torque proportionality factor.
In region 3, the generator torque changes inversely with generator speed to keep a
constant power output. Region 1 21 is a linear transition between region 1 and 2. Also in
region 2 21 a linear transition between region 2 and 3 occurs, in which its slope is equal
to the slip of an induction generator.
3.9.3
Control model of the blade pitch controller
In full load region, the control objective is to maintain the power production constant. As
the result, the generator torque becomes inversely proportional to the generator speed:
Tgen (Z g b , ω) =
P
(3.10)
Zg b · ω
Also for the rotor in full load region we have:
P
(3.11)
ω
Using Taylor series and expanding equation 3.10 around the rotational DOF of the
drive train ω, where ω = ω r at + ∆ω (note that ∆ω is the rotational speed difference
between the actual, ω, and rated value, ω r at ), and neglecting the second and higher
order terms result in:
Tr ot (ω) =
Tgen '
Pr at
Z g b ω r at
−
Pr at
Z g b ω2r at
· ∆ω
(3.12)
Similarly, expanding equation 3.11 around blade pitch angle β and neglecting the
second and higher order terms, we have:
1
∂P
Pr at
·
· ∆β
(3.13)
+
Tr ot '
ω r at
ω r at
∂β
The small perturbation in rotor speed (∆ω), and blade pitch angle (∆β) can be
correlated with a P I D controller as follow:
∆β = K P · Z g b · ∆ω + K I · Z g b ·
Zt
0
Where:
KP
deg
Proportional gain, ( r pm )
deg
s
K I Integral gain, ( r pm )
r pm
K D Derivative gain, ( deg )
s
∆ω·d t + K D · Z g b · ∆ω̇
(3.14)
52
3.9
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
Replacing ∆β from equation 3.14 into equation 3.13, and substituting the resultant equation together with equation 3.12 into equation 3.8, and setting ∆ω = φ and
rearranging all the terms results in:
Id r +
|
–
|
∂P
· −
· K D φ̈
ω r at
∂β
{z
}
Zg b
+
I
™
Zg b
∂P
Pr at
· −
· KP − 2
φ̇
ω r at
∂β
ω r at
{z
}
+
D
|
∂P
· −
· KI φ = 0
ω r at
∂β
{z
}
Zg b
(3.15)
K
However, experience shows that a satisfactory response can be obtained by choosing
a P I controller instead of a P I D. Therefore, the D controller is ignored in this research.
The P I controller also has been used with success in many research wind turbines such
as the Nibe B and Tjareborg wind turbines.
In equation 3.15, the I, D and K denote the system’s inertia, damping and stiffness,
respectively. Making the assumption that the dynamic characteristic of equation 3.15
behaves as a second order system as it is shown in equation 3.16:
Kd r
I d r · S 2 + Dd r · S + Kd r
=
ω2φn
S 2 + 2 · ξφ · S + ω2φn
(3.16)
Then, the natural frequency (ωϕn ), and the damping ratio (ξϕ ) of such a system can
be found by:
r
ωφn =
Kd r
Id r
(3.17)
and:
ξφ =
Dd r
2 · I d r · ωφn
(3.18)
It is known from the literature that for the transient response analysis of equation
3.15, the proportional gain ,K P , is proportional to the damping ,Dd r , of the system. Also
the integral gain ,K I , is proportional to the stiffness Kd r of the system, and similarly the
derivative term ,K D , is proportional to the inertia I d r of the system. Using this definition
the P I terms used for this work can be expressed as:
2 · I d r · ω r at · ξφ · ωφn
KP β =
Z g b · − ∂∂ βP
and:
(3.19)
3.9
CONTROLLER DESIGN AUTOMATION
2
I d r · ω r at · ωφn
KI β =
Z g b · − ∂∂ βP
53
(3.20)
Where ∂∂ βP is the aerodynamic power sensitivity to the blade pitch angle.
Notice that when this parametrization
is used, the P I control parameters show a de
pendency to the variation of − ∂∂ βP at different operating points in the full load region.
This derivative, − ∂∂ βP , can be obtained by doing a sensitivity analysis of the aerodynamic power of the rotor to the blade pitch angle at different wind speeds above the
rated. It is calculated by doing a central finite difference method with a small perturbation of the pitch angle to each side, and using a frozen wake assumption for the induced
velocities as mentioned before.
The calculated power sensitivity to the blade pitch angle shows a large variation
with wind speed (proportionally increasing with higher winds with a negative sign), so
constant values of gains do not yield the desired results. However, this large variation of
the power sensitivity to the wind speed shows nearly a linear relation with blade-pitch
angle (see figure C.1), and the equation of this line is as follow:
∂P
∂β


=
∂P
∂ ββ=r at
βk


β +
∂P
∂ ββ=r at
(3.21)
Where β r at is the blade pitch angle at rated, and βk is the blade pitch angle at which
the aerodynamic power sensitivity is doubled from its value at the rated. That is:
∂P
∂ βk
=2∗
∂P
∂ β r at
(3.22)
Replacing equation 3.22 into equation 3.21, and substituting that into equation 3.19
and 3.20 results in a closed form solution of the gains as follow:
2 · I d r · ω r at · ξφ · ωφn € Š
KP β =
· Gβ
Z g b · − ∂ ∂βP
(3.23)
2
I d r · ω r at · ωφn € Š
· Gβ
KI β =
Z g b · − ∂ ∂βP
(3.24)
r at
and:
r at
where Gβ is the dimensionless gain-correction factor that is dependent on the blade
pitch angle:


Gβ = 

1
1+
β
βk


(3.25)
54
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.9.4
3.9
Tower top motion feedback loop
To find the optimum value of the design variables, the optimizer needs to search the
design space. As a result, in many iterations the tower natural frequency gets close to the
1P rotation of the rotor, and the wind turbine experiences substantial tower motion parallel to the wind direction resulting in fatigue damage. A tower top feedback controller
is used to minimize this phenomenon during the optimization iterations, with the tower
top acceleration as an input to the pitch demand controller.
While such a controller can minimize the tower top motion, beyond a certain limit
it acts as a cyclic wind speed input to the rotor and results in rotor speed oscillation.
This rotor speed oscillation in turn causes a pitch oscillation. Finding the best trade off
between tower fatigue damage reduction and pitch oscillation reduction is a a matter of
trial and error with many simulations and tunning involved prior to optimization iterations.
To design the tower feedback loop, the tower dynamic is represented as a second
order system with a transfer function from rotor thrust to tower top velocity as presented
in equation 3.26. This is to simplify the tower dynamic by representing its behavior
as a transfer function that only contains the first mode properties (damping and natural
frequency). Such a representation works for the first mode properly and for higher modes
it becomes less effective.
T (S) =
1
Kt
S2
ω2t n
+
·S
2·ξ t ·S
ωt n
+ ω2t n
(3.26)
Where K t is the tower stiffness, ξ t is the tower structural damping with a fixed value
of 4% and ω t n is the tower natural frequency.
Figure 3.6 shows the control flow diagram for the speed controller, including the
tower acceleration as an input to the pitch demand controller. In this figure Q is the rotor
thrust, T the rotor torque, Ż tower top velocity and Z̈ tower top acceleration.
3.9.5
Controller design and implementation
In this research a conventional variable-speed, variable blade-pitch-to-feather controller
is selected. In such a configuration the strategy to control the power-production operation and load reduction is based on the design of three separable control systems:
• A generator torque controller for partial and transition load region
• A full-span rotor-collective blade-pitch controller for the full load region
• A tower top motion feedback controller to account the flexibility of the tower in
both above mentioned controllers.
Generator-torque controller
Maximization of power production in partial load region is the most logical strategy
for the generator torque controller. As discussed in many research papers (see Johnson
et al. (2004)), variable speed operation results in a higher energy capture than constant
3.9
55
CONTROLLER DESIGN AUTOMATION
SS
Pitch
Pitchcontroller
controller
.
Z
Tower
Towerfore-aft
fore-aft
motion
motion
.
Z
..
Z
Q
Lag
Lag
compensator
compensator
dT/dV
dT/dV
dQ/dβ
dQ/dβ
T
ωref
+
ΣΣ
_
ωref - ω
KKP++KKI/S
P
I/S
+ Σ +
Σ
Pitch
Pitchlimits
limits
dT/dβ
dT/dβ
T +
ΣΣ
+
β
Gain
Gain
scheduling
scheduling
ω
Drive
Drive
train
train
Figure 3.6: Controller flow diagram with tower acceleration feedback
speed. Therefore, a variable speed controller is also used here. The aim of the controller
is to find a value of K g such that it maximizes the energy capture.
As discussed before, the generator torque is proportional to the square of generator
speed below rated, then constant (or very slowly changing) above the rated generator
speed. This enables operation at a constant tip speed ratio below rated as long as the
blade pitch remains constant. Thus, the generator torque characteristics must be matched
to the rotor torque in order to actually achieve this constant tip speed ratio operation.
At above rated region, the system power output changes only in proportion to rotor
speed, since the torque should remain almost constant, and power regulation is mainly
dependent on speed regulation.
Blade-pitch controller
The goal of the blade-pitch controller is to regulate the generator speed above the
rated operating point. The blade-pitch controller is designed using a gain scheduled P I
controller. The error between the filtered generator speed and the rated generator speed
is used as the input for the P I controller. The output of this P I controller is the reference
pitch angle for the pitch actuator system.
The pitch demand is also limited to a minimum of 0 ◦ and a maximum of 90 ◦ . The
minimum is the pitch angle at which the optimum rotor aerodynamic performance is
achieved when operating at constant tip speed ratio in below rated region. When the
rotor speed is less than the desired setpoint, the pitch remains fixed at its minimum
value.
56
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.9
The single DOF model of subsection 3.9.2 is used to obtain the proportional, K P ,
and integral, K I gains. However, due to the nonlinear characteristics of the aerodynamic
forces that change during different operating conditions, the K P and K I gains are scheduled as functions of the aerodynamic power sensitivity to the pitch angle.
In addition, because of the changes of blade design variables in the DOP, the value of
the rated wind speed needs to be recalculated in each iteration. This is done using a time
domain simulation with a steady wind at the hub height (the wind hub model).
After finding the rated wind speed, the system is linearized at different wind speeds
above the rated wind speed to find the steady state operating points. This process is
mainly named as periodic steady state linearization, since it is performed at different
blade azimuth angles and then the azimuth averaged values are used to design the pitchcontroller (see Jonkman and Buhl (2004), van Engelen (2007)). Using this process, the
ω r ot -β-V space (with ω r ot as rotor rotational speed, β as blade pitch angle and V as the
wind velocity), in which the system is in dynamic equilibrium can be mapped, Bianchi
et al. (2006).
Based on the mapped values, another linearization is carried out that is mainly named
as initial steady state linearization, with previous obtained points as the inputs. In this
stage the aerodynamic power sensitivity is calculated using the linearized model with the
drive train DOF as the only state variable, blade pitch angle as the system input and the
rotor power as the system output.
Since the system has only one state variable, the derivative of the system output to
the system input is the aerodynamic power sensitivity to the blade pitch angle, which
is equal to the transmission matrix of the linearized system. Based on this strategy and
using equation 3.19 and 3.20 the proportional and integral gains are calculated.
Tower top motion feedback controller
To implementthe tower
controller, the aerodynamic sensitivity of rotor thrust to
∂Q
blade pitch angle ∂ βr ot , and the aerodynamic sensitivity of rotor torque to wind speed
∂ T r ot
should be calculated. These two sensitivities are obtained using the same tech∂V
nique used to calculate the aerodynamic power sensitivity to blade pitch angle. However,
it also should be mentioned that both sensitives have a linear behavior at different operating points.
In addition, a lag compensator is applied for the tower top acceleration to determine
a change in pitch in response to the motion. A lag compensator adds gain at the low
frequencies. This means larger low frequency gain to reduce the steady-state error. In the
context of this research, the lag compensator increases the pitch sensitivity (by adding
gain) to reduce the speed oscillation (steady-state error). This compensator is of the
form:
TLC (S) =
S+
S+
1
T
1
κT
(3.27)
where κ > 1 is a parameter, and T is the time scale of tower motion.
During many simulations, it was found that this feedback reduces tower loads slightly,
improves speed regulation and prevents the simulations from crashing when the tower
natural frequency gets close to 1P. It is obvious that this simplification does not work
3.10
AUTOMATION OF EXTRACTING STRUCTURAL PROPERTIES
57
perfectly in combination with unsteady aerodynamic effects. Therefore, the tower top
motion controller is verified and fine-tunned before level 2 optimization (level 1 and 2
optimization will be discussed later).
However, the selection of the controller concept and all the controller parameters is
a matter of experience that tightly depends on how these parameters lower LCOE and
influence design constraints. This goal can be achieved by introducing all the controller
parameters as design variable in the DOP, however, the computational time becomes
more expensive and unpredicted numerical instabilities may occur. Therefore this is not
considered as an option (despite having the potential of the developed methodology) in
this research due to time restriction.
In addition to the controller design automation, modeling the structural properties of
the blade and tower need to be automated. However, the emphasis will be on the blade,
since the derivation of its properties is more challenging. This issue is explained in next
section.
3.10
Automation of extracting structural properties
The blades of a wind turbine are complex structural components to design. Orthotropic
material properties, nonuniform distribution of different materials and the complex shape
of the blade are the most important reasons for this complexity.
To model this complexity, usually FE models are employed (see Malcolm and Laird
(2007)). These models are of value in investigating the distribution of strains and stresses
within the blade, thus suitable for the final design stage. They are very detailed for an
aeroelastic or aeroservoelastic design optimization, which in general only requires mass
and stiffness distribution along the blade.
To avoid FE models, many design optimization methods use either a linear elastic
I−beam model for the blade like Fuglsang and Madsen (1999), or a prescribed mathematical distribution of design variables along the blade like Benini and Toffolo (2002).
These models can easily represent the stiffness and mass distribution of the blade, but
to ensure their dependability they should be tuned with measurements. Despite their
simplicity, model tuning is a challenge, particularly for upscaling studies where there is
no data available to do any proper tuning.
This work offers a different methodology that enables the extraction of structural
properties of a composite wind turbine blade, while avoiding model tuning. This new
methodology is based on an analytical model of the composite blade, Ashuri et al. (2010a).
The method starts with calculating the second area moment of inertia of different
structural elements (shear webs, spar caps and shell) of a blade section to cope with the
geometrical complexity. Generally, each structural element has a directional property.
That is, the elastic constants depend on the orientation. Since each element consists of
different materials with different thicknesses, the weighting method is used to find the
equivalent directional properties of each section.
Clearly, the complicated nonuniform distribution of materials can be represented by
the bulk representation that has equivalent properties as the original. The actual thickness of each element is maintained, since it is used to extract weights. Combining the
contribution of different elements, the bending centroid of each section is calculated.
58
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.10
Parallel-axis theorem is used to move each second moment of inertia to the section
centroid. Stiffnesses are found by multiplying each element equivalent modulus to its
second moment of inertia and integrated over all elements. Mass per unit length of each
section is found by the same strategy. That is, the weighting method is applied on the
density of each material, and using equivalent density of each element multiplied by its
area, the mass per unit length is calculated. Other properties are also calculated in the
same manner.
This method is verified with the sectional structural properties of a commercial 660
kW wind turbine blade over 20 stations along the blade. However, to ease the readability and avoid presenting unnecessary information, the verification of this method is
presented in appendix D.
3.10.1
The methodology to extract structural properties
In any wind turbine blade design study, structural properties such as area and mass
moment of inertia, stiffness, strength, radius of gyration and mass per unit length are
needed. These properties are used to calculate the mass of the blade, natural frequencies,
deflections, buckling, stresses, fatigue and so on.
This subsection explains how to extract these properties for any arbitrary cross section
of the blade. It starts with explaining different sectional definitions and assumptions
made for the rest of the work. Then, based on these assumptions different sectional
properties relative to the principal axis are calculated. Finally, using a coordinate rotation
matrix, these properties can be transformed to any arbitrary plane.
Sectional description and assumptions
Usually a wind turbine blade cross section consists of two parallel shear webs located
on any point along the chord, two spar cups atop and under the shear webs and a shell
that surrounds the section form leading-edge to the trailing-edge, passing over the spars.
To extract the structural properties for such a section, certain assumptions based on
Euler-Bernoulli theory are made, (see Bauchau and Craig (2009) for more background
information). These assumptions are as follow:
• The strain is proportional to the distance from the neutral surface
• Deflections remain small
• Plane sections of the beam remain plane an perpendicular to the neutral axis
Sectional properties calculation
Since the cross section of a wind turbine blade consists of several elements like shell
and shear webs that have different material properties, the weighting method is used to
extract the equivalent properties of each section.
This helps to represent the complicated nonuniform distribution of materials with a
gross representation that has equivalent properties as the original (see Beer et al. (2006)
for more background information). The actual thickness of each element is maintained,
since it is used to extract weights, see figure 3.7.
shell
For instance the equivalent module of elasticity of the shell, Eequivalent
, consisting of
shell
some ±45◦ layers of composite in top, t shell
t op , and bottom, t bot t om , which is filled with a
3.10
AUTOMATION OF EXTRACTING STRUCTURAL PROPERTIES
59
foam having a modulus of elasticity of E shell
and thickness of t shell
in between (see
f oam
bot t om
figure 3.7) can be calculated as:
n
P
shell
Eequivalent
=
1
Eishell · t ishell
n
P
1
(3.28)
shell
t equivalent
and the numerator is defined as:
n
X
shell
shell
shell
shell
shell
Eishell · t ishell = E tshell
op · t t op + E bot t om · t bot t om + E f oam · t f oam
(3.29)
1
As it can be seen, the denominator of equation 3.28, which is the total thickness of
the shell remains unchanged:
n
X
shell
shell
shell
t equivalent
= t shell
t op + t bot t om + t f oam
(3.30)
1
However, the same approach can be used to extract equivalent densities for each element of the cross section, by simply replacing the modulus of elasticity with the density.
Section centroids in edgewise, X̄ csec t ion , direction is calculated as follow:
n
P
X̄ csec t ion
=
1
i
Eequivalent
· Aiequivalent · X̄ cii
n
P
1
(3.31)
i
Eequivalent
· Aiequivalent
In which the numerator and denominator are defined respectively as:
n
X
i
Eequivalent
· Aiequivalent · X̄ cii
=
shell
shell
Eequivalent
· Ashell
equivalent · X̄ c
+
Eequivalent · Aequivalent · X̄ cspar
+
web
web
Eequivalent
· Aweb
equivalent · X̄ c
1
spar
spar
and:
ttop
EEtop
top
tfoam
EEfoam
foam
tbottom
EEbottom
bottom
tequivalent
Figure 3.7: Equivalent representation of properties
EEequivalent
equivalent
(3.32)
60
3.10
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
n
X
i
Eequivalent
· Aiequivalent
=
shell
Eequivalent
· Ashell
equivalent
+
Eequivalent · Aequivalent
+
web
Eequivalent
· Aweb
equivalent
1
spar
spar
(3.33)
and similarly for Ȳcsec t ion we have:
n
P
Ȳcsec t ion
=
1
i
Eequival
· Aiequivalent · Ȳcii
ent
n
P
1
(3.34)
i
Eequivalent
· Aiequivalent
In which the denominator is the same as equation 3.33, and numerator is defined as:
n
X
i
· Aiequivalent · Ȳcii
Eequivalent
=
shell
shell
· Ashell
Eequivalent
equivalent · Ȳc
+
Eequivalent · Aequivalent · Ȳcspar
+
web
web
Eequivalent
· Aweb
equivalent · Ȳc
1
spar
spar
(3.35)
These section centroids are used to transfer the area moments of inertia that are
calculated relevant to each element centroids to the section centroids using parallel-axis
theorem, which will be explained later in this subsection.
Area moments of inertia can simply be calculated using an integration scheme. In this
scheme, the cross section (X , Y ) coordinates of the shell, spars and webs are separately
used to find the area moment of inertia relevant to their local axis. However, with the
same integration scheme, these integrals also need to be calculated for other elements
like spar and web. These integrals for the shell are as follow:
Z
2
yshell
dx d y
(3.36)
2
x shell
dx d y
(3.37)
x shell yshell d x d y
(3.38)
I xshell
x =
I shell
yy
I xshell
y
=
=
Z
Z
Now the calculated area moments of inertia that are calculated relative to their local
centroids need to be transferred to the section centroids using the parallel-axis theory,
(see Beer et al. (2006)). For the shell, it is as follow (note the difference between shell
and SHELL in the equation where shell refers to local centroids and SHELL refers to
section centroid):
3.10
61
AUTOMATION OF EXTRACTING STRUCTURAL PROPERTIES
€
Š2
shell
I xSHx E L L = I xshell
− X̄ csec t ion
x + Ashell · X̄ c
(3.39)
€
Š2
ELL
shell
I SH
= I shell
− Ȳcsec t ion
yy
y y + Ashell · Ȳc
(3.40)
€
Š €
Š
shell
I xSHy E L L = I xshell
− X̄ csec t ion · Ȳcshell − Ȳcsec t ion
y + Ashell · X̄ c
(3.41)
These area moments of inertia can be rotated around any arbitrary axis. This is
a necessary step to take when one wants to use the in-plane and out-of-plane values
instead of flapwise and edgewise and can be done using the following transformation:
ELL
I xSH
=
0 x0
ELL
I xSH
=
0 x0
ELL
I xSHx E L L + I SH
yy
2
ELL
I xSHx E L L + I SH
yy
2
ELL
I xSH
0 y0
=
+
+
ELL
I xSHx E L L − I SH
yy
2
ELL
I xSHx E L L − I SH
yy
2
ELL
I xSHx E L L − I SH
yy
2
· cos 2θ − I xSHy E L L · sin 2θ
(3.42)
· cos 2θ − I xSHy E L L · sin 2θ
(3.43)
· sin 2θ + I xSHy E L L · cos 2θ
(3.44)
Based on the calculated flapwise area moment of inertia around the section centroid
and the calculated section equivalent modulus of elasticity in flapwise direction for all different elements, section flapwise stiffness can be calculated using the following equation
(using the same strategy, the stiffness in the edgewise direction can also be calculated):
sec t ion
t ion
E tsec
ot al · I x x
=
shell
· I xSHx E L L
Eequivalent
+
Eequivalent · I xSPAR
x
+
W EB
Eequivalent
· I xWxEB
spar
(3.45)
To calculate torsional stiffness following replacement in the above equations needs to
be done:
• Modulus of elasticity with modulus of rigidity
• Flapwise or edgewise second moment of inertia with polar moment of inertia
Mass distribution calculation along the blade can be simply done as follow:
M
L
=
shell
ρequivalent
· Ashell
equivallent
+
ρequivalent · Aequivallent
+
web
ρequivalent
· Aweb
equivallent
st at ion
spar
spar
(3.46)
62
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.11
Where ρ is the density of each cross section, and L is the length of each station along
the blade. The mass of the blade is calculated by using this equation and knowing the
length of each station of the blade in advance as:
M blade =
n
X
Mi · L i
(3.47)
i=1
The same method that is used for the blade can also be used for the tower. However,
with the isotropic material property of the tower and its simple geometry, all the above
equations can be considerably reduced to some simple equations.
3.11
Wind turbine’s model specification
A set of design parameters (airfoil shapes, structural arrangement and materials properties) is selected to be consistent with current commercial wind turbine designs. Using
this baseline configuration, the aerodynamics, structural and control designs are carried
out. The NREL wind turbine is used as the baseline turbine configuration in this research
with the following specifications and assumptions:
• A three bladed, upwind rotor with rigid hub and full-span pitch control
• A maximum tip speed bound of 120
• Air density of 1.225
m
s
kg
m3
• A Rayleigh distribution of wind speed
• A vertical wind shear power exponent of 0.14
• Annual mean wind speed at 90 m height of 8.4
• A cut-in wind speed of 4
m
s
m
s
• A cut-out wind speed of 25
m
s
• A variable speed operation with maximum power coefficient of 0.5
The following subsections describe the design definitions used in this work.
3.11.1
Aerodynamic design definition
Figure 3.2 shows the planform of the blade, with a nonlinear taper from the maximum
chord location at station 6 to the blade tip. The circular blade root (station 1) is located
at 0.0 Rr with R as the blade length. The blade shape is transitioning from a circular cross
section in station 1 to a pure airfoil shape at station 6.
Eight different airfoil types are incorporated for the blade. The three innermost airfoil
types have a circular cross section with a drag coefficients of 0.50, the next two airfoils
have an elliptic cross section with a drag coefficient of 0.35, and no lift. The remaining
six airfoils are a combination of Delft University and NACA types.
As it was explained before, AirfoilPrep is used to modify the airfoil properties. First,
using the Selig and Eggars methods, the lift and drag coefficients from 0 to 90 degree
3.11
WIND TURBINE’S MODEL SPECIFICATION
63
angle of attack are corrected for rotational stall delay. Then the drag coefficient is corrected using the Viterna method for 0 to 90 degree angles of attack with an aspect ratio
which is varying automatically for every optimization iteration. Finally, the BeddoesLeishman dynamic-stall hysteresis parameters are estimated.
Table 3.13 shows a summary of the airfoil type and location along the blade.
Table 3.11: Airfoil shapes and their location along the blade
Airfoil Type
3.11.2
Spanwise location
Pitch axis
Circular
Circular
Circular
0.0000
0.0195
0.0520
0.500
0.500
0.500
Elliptic
Elliptic
0.0845
0.1170
0.460
0.420
DU00W401
DU00W401
DU00W401
DU00W350
DU00W350
DU97W300
DU91W2250
DU93W210
0.1495
0.1821
0.2146
0.2471
0.2959
0.3935
0.4910
0.5886
0.390
0.375
0.375
0.375
0.375
0.375
0.375
0.375
NACA64618
NACA64618
NACA64618
NACA64618
NACA64618
NACA64618
NACA64618
0.6861
0.7837
0.8813
0.9300
0.9544
0.9788
1.0000
0.375
0.375
0.375
0.375
0.375
0.375
0.375
Structural design definition
The structural elements of the blade consist of two shear webs, two spar caps between the
shear webs, and a skin that externally has surrounded the shear webs and the spar caps.
This arrangement is depicted in figure 3.8, where the thickest airfoil section (station 6)
is shown.
Table 3.12 lists the material type of each element. The CDB340 is a triaxial fabric and
has a 25%, 25%, and 50% distribution of +45 ◦ , −45 ◦ , and 0 ◦ fibers, respectively. The
spar cap is made of triaxial and uniaxial (A260) fabric. This layup results in spar cap
laminate with 70% uniaxial and 30% off-axis fibers by weight.
Since the tower is made of steel with isotropic material properties, only a modulus of
kg
elasticity of 210 (GPa), a density of 7800 ( m3 ) and a Poisson ratio of 0.3 are needed to
define its properties.
64
3.11
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
Table 3.12: Material type definition of each blade element
Element
Layer No.
Material
Thickness
Gel coat
Random mat
CBD340
0.5 (mm)
0.4 (mm)
20% Shell thickness
Skin (Shell)
Layer 1
Layer 2
Layer 3
Layer 4:
From 0% − 15%c
From 15% − 50%c
From 50% − 100%c
Layer 5
Balsa
Spar cap mixture
Balsa
CBD340
60% Shell thickness
2 (mm)
60% Shell thickness
20% Shell thickness
Shear webs
Layer 1
Layer 2
Layer 3
Layer 4
Layer 5
Layer 6
Layer 7
Gel coat
Random mat
CBD340
Balsa
CBD340
Random mat
Gel coat
0.2 (mm)
0.4 (mm)
20% Shell thickness
60% Shell thickness
20% Shell thickness
0.4 (mm)
0.2 (mm)
Spar cap
Layer 1
Layer 2
Layer 3
Layer 4
Fill epoxy
A260
Random mat
Gel coat
4 (mm)
100% Spar cap thickness
0.4 (mm)
0.2 (mm)
For the flexible components of the wind turbine (blade and tower in FAST) these
definitions of the material properties are used to calculate the stiffnesses. In addition, a
mass model based on Fingersh et al. (2006) is used for all the other components. These
mass models are needed for a proper modeling of the aeroservoelastic behavior of the
wind turbine and are presented in appendix A.
3.11.3
Definition of the design load cases
As part of the design process and to evaluate the design constraints, a variety of loading
conditions that a wind turbine is experiencing in its lifetime must be analyzed. This
Figure 3.8: Layout of the Blade structural model
3.11
WIND TURBINE’S MODEL SPECIFICATION
65
Table 3.13: Summary of material properties of elements
Material
property
CDB340
A260
Spar
cap
mix.
Balsa
Gel coat
Random
mat
Fill
epoxy
E x (GPa)
E y (GPa)
G x y (GPa)
ν x y (−)
kg
ρ( m3 )
24.2
8.97
4.97
0.39
1670
31.0
7.59
3.52
0.31
1700
27.1
8.35
4.7
0.37
1700
2.07
2.07
0.14
0.22
130
3.44
3.44
1.38
0.3
130
9.65
9.65
3.86
0.3
1670
2.76
2.76
1.10
0.3
120
forces the designer to verify that the turbine will be able to withstand these loads with a
sufficient safety margin. This activity in the design process is standardized by analyzing
the wind turbine for a number of relevant load cases prescribed by standard bodies.
In the IEC standard, the external conditions are site dependent and wind turbine
classes are defined in terms of wind speed and turbulence parameters at the hub-height
with a reference period of 10 min. Three classes of turbulence intensity are defined
according to the Normal Turbulence Model (NTM), A (high), B (medium) and C (low).
The GL and DNV rules also use the same classification as IEC uses, however they
specify the largest number of loading conditions and are the most complete in defining
the design documentation and process, which must be followed to obtain certification of
the turbine.
However, all these standards are common in defining a set of three limit state load
cases covering operational conditions: normal wind and machine states, normal wind
conditions with machine fault states, and extreme wind conditions with normal machine
states, Bossanyi et al. (2001).
According to IEC, wind turbines must be designed to resist for a minimum of 20
years against two types of loads, the fatigue and ultimate. The fatigue loading has two
components, a periodic deterministic and a stochastic. The periodic deterministic loading
is caused either by the rotational associated loads like tower shadow and gravity or the
wind associated loads like yaw errors and wind shear. Wind turbulence is the only source
of the stochastic loading applied on the wind turbine that causes fatigue. For simple
terrains Gaussian statistics of turbulence is used, whereas for more complex terrains
non-Gaussian statistics is preferred.
In civilian structures the ultimate load due to the wind occurs at the extreme wind
speed. However, for wind turbines the ultimate load is more difficult to predict, but most
of the designs assume that the extreme load occurs when the turbine is in parked mode
with an Extreme Wind Model (EWM) of a 50-year recurrence.
It is a common practice in the field of wind energy to apply these loads as a design
load case (DLC) prescribed by the IEC standard. Therefore, this work also follows the IEC
prescription. After checking different DLCs to find those which would likely govern the
design, two primary load cases were selected, each representing either fatigue or extreme
loading.
For fatigue loads, a normal turbulence model (NTM) during the power production
mode is used. This model is applied from the cut-in to cut-out wind speed. For extreme
loads, an extreme wind model (EWM) with a 50-year recurrence period with the turbine
66
3.12
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
in parking condition is used.
This research uses the edition 3 of IEC 61400-1 standard for defining the design load
cases (DLC), and based on section 7.4 of this standard, the design load cases used for
simulations are defined in table 3.14.
Table 3.14: Definition of DLCs based on IEC standard
Design situation
Power production
Parked
DLC
Wind condition
Type of analysis
1.2-00
NTM
Fatigue
6.1
EWM 50-year
Ultimate
EWM: Extreme Wind Model, NTM: Normal Turbulence Model
3.12
Some practical issues about the optimization procedure
There are some issues related to the multidisciplinary design optimization approach in
this research that are unique, therefore an explanation for each of them is given in the
following subsections:
3.12.1
Reducing the number of design variables
The blade and the tower of a wind turbine are long slender bodies, which require the
definition of their properties along their longitudinal axis at several stations. Since this
research is unique in the sense of doing a simultaneous design optimization of the blade,
tower and rated rotational speed, the number of design variables and constraints are
high. This forces another reduction technique to be implemented to further reduce the
size of the problem.
Unlike some other design optimization practices that use a prescribed curve fitting
technique like Bezier (see Benini and Toffolo (2002) and Fuglsang and Madsen (1999)),
to define the design variables at some stations, this research defines the design variables
of the blade and tower at some stations along their longitudinal axes and uses an interpolation scheme to find the design variables at all other stations.
The same technique is also used to define the design variables of the tower. Such
a setup clearly reduces the definition of design variables, and significantly reduces the
computational time. Table 3.1 shows the selected stations for design variables of the
blade. For the tower, the design variables are defined at the tower base and top.
3.12.2
The starting point of design variables
"The closer the starting point to the optimum, the faster the optimization convergence".
This is a general rule in most of the gradient based optimization algorithms. A starting
point that lies outside the feasible design space, not only requires the usage of a different
algorithm suitable to work with unfeasible designs, but also will slow the convergence
3.12
SOME PRACTICAL ISSUES ABOUT THE OPTIMIZATION PROCEDURE
67
speed. Therefore, a good estimation of the initial design variables reduces the number of
optimization iterations and results in less optimization time.
The developed shell program explained in subsection 3.7.5 uses the 5 MW NREL
design as the starting point, and by implementing the linear scaling law described in
chapter 2, provides an initial set of design variables for any different size asked by the
user. The fundamental relations of this technique are explained in detail in appendix F.
Using this technique, all the initial design variables and parameters of the 10 and 20 MW
wind turbines are calculated.
3.12.3
Decomposition of design variables
Since this research deals with the simultaneous design optimization of the blade and
tower, the number of design variables hence the size of the problem is large. To overcome this problem a size reduction technique should be used. System decomposition
is a powerful technique that primarily is involved with dividing the system under study
into subsystems. In many multidisciplinary design optimization problems, it is preferable
to decompose the system based on the disciplines involved. However by doing so the
interaction between the disciplines is lost, which is not desired in this research.
However, system decomposition is a very general technique and can be implemented
in different ways. Here, it is used to decompose design variables instead of disciplines.
This decomposition results in a multilevel optimization problem. Tower height, blade
length and rated rotational speed are the first set of design variables that can change in
the first level of system optimization, while keeping all other design variables fixed.
It is also a common practice in the field of wind energy to define and fix them in
the conceptual design and then do the preliminary design with other design variables involved. Here, the optimization process runs very fast since there are no design constraints
present, and the number of design variables is not too much.
This results in finding a global optimum rather than a local one. This is because of
the reduced degree of feedback among all design variables that results in a monotonic
change of the selected design variables, which in turn helps the optimizer to not trap in
a local optimum.
In the second level, the optimum values of tower height, blade length and rated
rotational speed are fixed and all other design variables can change. Again, LCOE is
minimized but with all the design constraints involved. This process continues until the
point where the specified convergence is satisfied.
Contrary to level one, there is no need to generate a 3D turbulence wind field in level
two in every iteration, which saves a considerable amount of time. This is because of a
fixed blade length and tower height in level two that does not need a different wind grid
size and height in each iteration and once it is generated it can be used until the end of
process. This is one of the main advantages of the multilevel approach implemented in
this work, which saves a huge amount of computational time in the optimization process.
The multilevel approach is explained in the next subsection.
3.12.4
Multilevel optimization approach
As figure 3.9 shows, the design optimization process is done in a multilevel approach. In
the first level, the blade length, tower height and rated rotational speed are optimized,
68
3.12
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
while all other design variables are fixed. From the list of functional constraints of table
3.3, only item 37 that is the blade tip speed is involved. Form the side constraints, a lower
and upper limit is applied on the design variables mentioned before, however, since they
are size dependent they will be explained in the subsequent chapters. This setup results in
an optimization problem with one objective function (LCOE), one functional constraint,
three side constraints, and three design variables.
Level 1
(CONLIN)
No
Level 2
(LM)
Yes
Convergence:
COEi+1 – COEi ≤ ξ
1- Objective function:
COE
1- Objective function:
COE
2- Design variables:
Blade length, tower height and rated rational
speed
2- Design variables:
All except blade length and rated rational speed
3- Constraints:
• Functional: Blade tip speed
• Side: Blade length, tower height and rated
rotational speed
4- Algorithm type: First order
5- Wind file generation in every iteration
3- Constraints:
• Functional: All except blade tip speed
• Side: All except blade length and rated
rotational speed
4- Algorithm type: Second order
5- Running with a fixed wind file generated for the
initial design
Figure 3.9: Multilevel optimization of the wind turbine
After finding the optimum values of the blade length, tower height and rated rotational speed, the second level of optimization takes place. In the second level, the blade
length, tower height and rated rotational speed are fixed and all other design variables
are varying. Here, all the functional and side constraints (except item 37 of table 3.3) are
involved and the optimization problem has one objective function, 20 design variables
and 50 design constraints.
Because of an expensive objective function and design constraints evaluation time,
gradient based algorithms are the best options. However, the nature of the problem in
level one is totally different from level two.
In the first level, since each radial position along the blade where the chord and twist
are defined changes with blade length, the information between each iteration is not
consistent and the gradient based algorithm must be first order (the same thing holds
true for the tower). A first order algorithm does not use the information from a previous
iteration, which is a useful feature for shape optimization problems.
For the first level Convex linearization (CONLIN) is used, which is a first order al-
3.13
CONCLUDING REMARKS
69
gorithm, Fleury (1989). For the second level of the optimization process, the blade
length and tower height are fixed and the topology of the blade does not change from
one iteration to another. Therefore, Lagrange Multiplier (LM) is used, which is a second
order algorithm, Birgin and Martinez (2008). A brief explanation of each algorithm and
the settings used for this research are given in appendix E.
The level one and two optimization process is carried out sequentially in a loop,
until an acceptable convergence on the objective function is achieved. However it was
found that for all different scales only after 3 iterations of a multilevel optimization the
convergence on objective function is achieved and no further optimization is needed.
3.13
Concluding remarks
The approach of this research justifies the purpose of upscaling in the following ways:
• Assessing the technical feasibility of large wind turbines
To examine the technical feasibility of a wind turbine, the design constraints that
can appear as a barrier to the design need to be considered. This will be achieved
by evaluating fatigue, stresses, blade-tower clearance, natural frequencies and aeroelastic instability (as will be explained in the following chapters) as the main
driving constraints on the blade and tower.
Blades and tower are the most dynamic and flexible components of a wind turbine.
Their design is more challenging as the size increases and thereby considered for
upscaling study of this research.
• Characterizing the economy of large wind turbines
To show how the design can benefit from upscaling and characterize it, a trade
off between what can be gained (AEP), and what should be paid (cost elements)
is essential. This is achievable by introducing the LCOE as the objective function
of the design to be optimized. This enables the evaluation of the cost elements
and the AEP at the same time, and the trade off between them defines the overall
economy of large wind turbines.
It should be mentioned that an accurate estimation of the AEP requires all the
influencing variables to be introduced in the design. This includes tower height,
blade chord and twist, rotor diameter and rated rotational speed. Therefore, the
focus on optimizing the blade and tower (as well as the rated rotational speed) has
another strong supportive argument; making an accurate calculation of the AEP
needed to address the economical feature.
• Developing size specific optimized wind turbines to construct scaling trends
The main drawback of the classical upscaling methods is the limitation in realizing
size specific optimized wind turbines. Using the methodology of this research, size
specific optimized wind turbines can be developed. This is achievable by formulating the upscaling research of this work as an MDO. Through the use of the mass
models fed with the optimized design variables and the optimized design variables
themselves, large future offshore wind turbines can be realized.
These optimized wind turbines will be used to construct reliable scaling trends us-
70
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION
3.13
ing a consistent design concept and design assumptions. Therefore, the constructed
trends based on these designs have the least data spread and can identify truly the
design changes as size increases.
CHAPTER
4
Verification of the integrated aeroservoelastic optimization
method
4.1
Introduction
Using the developed design methodology of the previous chapter, the design optimization
of the 5 MW NREL wind turbine is carried out in this chapter. This is to validate the
implementation of the developed design methodology, and illustrate the power of the
MDO for wind turbine design applications.
The objective of the MDO is to minimize the LCOE, which is calculated from the AEP
and the system costs. The design variables of the blade are: chord, twist, thicknesses at
different stations along the blade, and rated rotational speed. For the tower, diameter
and thickness at different stations along the tower, and the tower height are the design
variables. Design constraints are fatigue, stresses and natural frequencies of the blade
and tower, as well as the tower clearance.
Using this design optimization model, the 5 MW NREL wind turbine is optimized and
the results of the original and the optimized wind turbine are compared. The optimized
5 MW design of this chapter also serves as the starting point to make trends for larger
scales. The same methodology of this chapter is used in the next chapter to design 10
and 20 MW wind turbines.
4.2
Purpose of the design optimization of the NREL wind
turbine
There are three main purposes to optimize the 5 MW NREL wind turbine as follow:
1. Validating the integrated aeroservoelastic optimization method Making an optimized 5 MW wind turbine can validate the integrated aeroservoelastic design
71
72
VERIFICATION OF THE INTEGRATED AEROSERVOELASTIC OPTIMIZATION METHOD
4.3
approach by offering a better design solution than the NREL wind turbine that is
based on a conventional design approach relying mainly on a team of experts and
engineering judgments.
2. Improved understanding of the 5 MW wind turbine design By making use of
the integrated design methodology, an improved understanding of the underlying
principles and assumptions of designing the NREL wind turbine can be achieved.
This will be done by evaluating the design constraints and the objective function of
this machine and exploring the boundaries of its design space.
3. Providing consistent data points to make scaling trends The optimized 5 MW
wind turbine provides the first point of the trends toward larger scales. This design
will be in consistency with the 10 and 20 MW wind turbines that will be developed
later, since the same design criteria and assumptions are used to develop all these
wind turbines.
With these particular intentions in mind, the novel integrated aeroservoelastic design
optimization approach is followed, and the results are presented in the following sections.
4.3
MDO of the 5 MW wind turbine
In this section, the concept of integrated aeroservoelastic design optimization is used
to optimize the 5 MW NREL wind turbine. For conciseness, only important results are
presented. Some of these results are used at the end of this chapter to validate the
methodology.
4.3.1
The 5MW NREL wind turbine
As mentioned in the previous chapter, the NREL wind turbine is a wind turbine most
widely used nowadays both in onshore and offshore wind energy research studies. For
the purposes of the present work, this concept is then regarded as the base case, and the
optimized version of this turbine is employed for the upscaling studies.
The optimized 5 MW, and the 10 and 20 MW wind turbines have the same conceptual
design as the original 5 MW NREL wind turbine, proposed by Jonkman et al. (2009). The
NREL 5 MW wind turbine is a 3 bladed, upwind, variable-speed, pitch-controlled machine
with a Doubly Fed Induction Generator (DFIG) and a gearbox. Its main characteristics
are shown in table 4.1.
It should be noted that blades of the 5 MW NREL wind turbine are made from Carbon Fibre Reinforced Plastics (CFRP). However, in this research Glass Fibre Reinforced
Plastics (GFRP) is used as the material of the blade. The main reason for not using CFRP
in this research is the uncertainties in defining representative carbon fibre structural properties (especially the S-N curve), while these properties are well known for GFRP that are
used for many years in the wind turbine industry.
Thus, this machine is used in the MDO of this work and the design variable, design
constrains and objective function are presented in the next subsection. The properties of
4.3
MDO OF THE 5 MW WIND TURBINE
73
Table 4.1: Baseline 5 MW NREL offshore gross properties
Design specification
Value (Unit)
Rated power
Rotor, hub diameter
Rotor orientation, configuration
Hub height, overhang
Cut-in, rated and cut-out wind speed
Cut-in and rated rotor speed
Rated tip-speed
Shaft tilt, precone
Rotor, nacelle, tower mass
Control strategy
Peak power coefficient
Blade-pitch angle at peak power
Rated mechanical power
Rated generator torque
Generator slip in transition region
Maximum blade pitch rate
5 (MW)
126, 3 (m)
Upwind, 3 bladed (-)
90, 5 (m)
3, 11.4, 25 ( ms )
6.9 and 12.1 (RP M )
80 ( ms )
5, 2.5 (deg)
110, 240, 347.46 (t on)
Variable-speed, collective pitch (-)
0.482 (-)
0.0 (deg)
5.297 (M W )
43.093 (kN .m)
10 (%)
deg
8( s )
the optimized 5 MW wind turbine are presented in appendix G.
4.3.2
Design variables of the optimized 5 MW wind turbine
As mentioned in subsection 3.7.2, two types of design constraints are considered in this
work; side and functional constraints. While all the side constraints that are applied on
the design variables are scale dependent, only some of the functional constraints are so.
Table 4.2 shows all the design variables used for the MDO of the 5 MW wind turbine,
with their related lower and upper limits appearing as side constraints in level 1 or 2 of
the optimization process, as well as the final achieved optimum.
4.3.3
Design constraints of the optimized 5 MW wind turbine
The design space in which the optimizer searches for the optimum design variables is
highly constrained with 51 functional constraints as presented in subsection 3.7.2 of
chapter 3. Therefore, only those that have the highest active value are presented in this
work. For the blade, these active design constraints are the tip deflection and natural
frequency. For the tower, fatigue is the highest design constraint. Table 4.3 shows these
constraints for both the blade and tower at the optimized values of the design variables.
As the table shows, the flapwise deflection of the blade reaches a maximum at values
close to the rated wind speed. This peak is due to the high loads that a variable-speed,
pitch-regulated wind turbine experiences at the rated wind speed and as the blade pitches
above the rated operational point the loads on the blade decrease.
Also what is not shown in this table is the blade mass. As explained before the 5 MW
NREL blade has CFRP in its construction. CFRP has a higher stiffness to density ratio,
which means for the same stiffness as a Glass Fiber Reinforced Plastics (GFRP), it weighs
74
4.3
VERIFICATION OF THE INTEGRATED AEROSERVOELASTIC OPTIMIZATION METHOD
Table 4.2: optimized 5 MW wind turbine design variables and side constraints
Description
Opt. level
Min
Optimum
Max
Blade length (m)
Tower height (m)
Rated rotational speed (rpm)
1
1
1
56.0
78.5
10.5
63.5
80.0
12.9
65.0
95.0
13.4
Twist at station 9 (deg)
Twist at station 15 (deg)
2
2
9.0
1.0
10.4
3.0
14.0
4.0
Chord at station 3 (m)
Chord at station 7 (m)
Chord at station 15 (m)
Chord at station 18 (m)
2
2
2
2
3.0
4.0
2.1
1.3
3.1
5.3
3.0
1.5
4.0
5.5
3.0
2.3
Skin thickness at station 1 (cm)
Skin thickness at station 3 (cm)
Skin thickness at station 6 (cm)
Skin thickness at station 16 (cm)
2
2
2
2
8.0
5.0
2.0
1.0
10.0
6.0
2.3
1.5
14.0
10.0
5.0
4.0
Web thickness at station 3 (cm)
Web thickness at station 6 (cm)
Web thickness at station 16 (cm)
2
2
2
1.0
2.0
1.0
1.0
2.0
1.3
5.0
4.0
3.0
Spar thickness at station 3 (cm)
Spar thickness at station 6 (cm)
Spar thickness at station 16 (cm)
2
2
2
1.0
1.0
1.0
1.0
2.5
2.4
4.0
4.0
4.0
Tower diameter at station 1 (m)
Tower diameter at station 22 (m)
2
2
4.0
3.0
6.0
4.1
7.0
5.0
Tower thickness at station 1 (cm)
Tower thickness at station 22 (cm)
2
2
4.0
2.0
4.0
2.4
6.0
5.0
less. This explains why a blade of the 5 MW NREL wind turbine weighs only 17740 kg,
compared to 22851 kg for the optimized design made from GFRP.
4.3.4
Objective function of the optimized 5 MW wind turbine
LCOE is the objective function of the design process and the derivation of that was explained in detail in chapter 3. The calculated LCOE of the 5 MW NREL wind turbine using
USD
, while this value for the optimized wind
the cost functions of this research is 0.0658 kW
·hr
USD
turbine is 0.0630 kW
, which shows 4.2% reduction.
·hr
Table 4.4 shows the cost comparison of the NREL and the optimized 5 MW wind
turbines. All the cost elements in this table are in 1000 USD.
As the table shows, the blades of the 5 MW NREL wind turbine are more expensive,
since they are made from Carbon Fiber Reinforced Plastics (CFRP). Also, there are several
cost elements with same cost for both wind turbines. These cost elements are a function
of rated power output, and since both wind turbines have the same rated power, the
associated costs are the same.
4.4
COMPARISON OF THE RESULTS
75
Table 4.3: Optimized 5 MW wind turbine functional constraints
Description
Opt. level
bound
Optimum
2
2
2
2
2
2
2
2
2
≤ 5.6(m)
≤ 5.6(m)
≤ 5.6(m)
≤ 5.6(m)
≤ 5.6(m)
≤ 5.6(m)
≤ 5.6(m)
≤ 5.6(m)
≤ 5.6(m)
4.5 (m)
4.6 (m)
4.4 (m)
4.3 (m)
3.7 (m)
3.2 (m)
2.9 (m)
2.9 (m)
2.8 (m)
1st blade’s nat. freq. (Hz)
2nd blade’s nat. freq. (Hz)
3 r d blade’s nat. freq. (Hz)
2
2
2
≤ 0.7
≤ 0.7
≤ 0.7
0.70
0.90
2.2
Fatigue at station 1 of the tower
Fatigue at station 5 of the tower
Fatigue at station 9 of the tower
Fatigue at station 13 of the tower
Fatigue at station 17 of the tower
Fatigue at station 21 of the tower
2
2
2
2
2
2
≤ 0.7
≤ 0.7
≤ 0.7
≤ 0.7
≤ 0.7
≤ 0.7
0.66
0.52
0.33
0.18
0.10
0.40
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
4.3.5
9 ms
11 ms
13 ms
15 ms
17 ms
19 ms
21 ms
23 ms
25 ms
Computational Expense
The computational expense of performing optimization in level 1 and 2 is different. This
is due to different number of design variables and constraints as well as differences in
function evaluation. Level 1 optimization takes approximately 50 CPU hours using a
standard HP400 workstation and generally requires 10 to 14 iteration to meet the design
requirements (please refer to appendix E for the list of design requirements). In this level,
about 65% of the iteration time is spent on function evaluation and the rest to perform
optimization.
Level 2 optimization takes approximately 1500 CPU hours. Around 25 iterations are
performed to meet the design requirements. Every function evaluation uses about 40%
of the iteration time and the rest 60% is used by the algorithm to do sensitivity analysis
and providing a new set of design variables to be evaluated in the next iteration.
For optimizing the 5 MW wind turbine, level 1 optimization was performed at first
followed by level 2. However, after doing level 1 again, the differences between the two
level 1 results were found to be negligible and therefore no further optimization was
performed. This means that results of level 2 optimization were considered as the final
optimum results.
4.4
Comparison of the results
As discussed before, the cost models used in this work contain all the relevant components of a wind turbine, as well as the costs of procedures such as decommissioning and
76
VERIFICATION OF THE INTEGRATED AEROSERVOELASTIC OPTIMIZATION METHOD
4.4
Table 4.4: Cost comparison of the NREL and optimized 5 MW wind turbines
Components
NREL
optimized
Blades
Hub
Pitch mechanism and bearing
Nose cone
Low speed shaft
Main bearings
Gearbox
Mechanical brake and coupling
Generator
Power electronics
Yaw drive and bearing
Main frame
Platform and railing
Nacelle cover
Electrical connections
Hydraulic and cooling system
Control,safety and condition monitoring
Tower
Marinization
Turbine capital costs (TCC)
1062.3
130.2
242.0
13.6
166.8
64.4
877.2
11.0
398.0
393.2
146.3
162.7
89.5
73.3
308.8
77.2
65.3
939.1
561.6
4722.4
1057.5
158.3
263.0
14.2
182.6
71.9
877.2
11.0
398.0
393.2
160.5
172.9
95.1
73.3
308.8
77.2
65.3
968.9
722.2
6072.3
Foundation system
Transportation
Port and staging equipment
Turbine installation
Electrical interface and connection
Permits, engineering and site assessment
Personnel access equipment
Scour protection
Decommissioning
Balance of station costs (BOS)
2174.7
1568.3
144.9
732.8
2063.5
215.5
70.2
403.0
362.8
7373.2
2174.7
1568.3
144.9
732.8
2063.5
215.5
70.2
403.0
403.3
7373.2
Offshore warranty premium
Initial capital cost (ICC)
624.1
13083.0
802.5
14651
Levelized replacement costs
Operation and maintenance
Fixed charge rate
99.0
561.4
0.1185
99.0
664.5
0.1185
23911817.1
28396817.4
0.0658
0.0630
AEP (kW hr)
LCOE
USD
)
( kW
hr
O&M. Most of the cost models are a function of the design variables, as used in the optimization process. As an example, an increase in the blade length to capture more energy,
not only increases the blade cost but also increases the loads on the hub and raises its
4.4
COMPARISON OF THE RESULTS
77
cost.
However, the optimizer decides what would be the lowest LCOE among many different design solutions. To show the power and the reliability of the optimization method
proposed in this work a comparison between the 5 MW NREL and the optimized wind
turbine is performed in this section.
Chord, twist angle, blade length, rated rotational speed and the internal thicknesses
are the design variables of the blade to optimize, while, the airfoil shapes remain unchanged in the optimization process.
Figures 4.1 and 4.2 show the chord and twist angle distributions of the NREL and the
optimized blades, respectively.
Optimized
NREL
5.5
4.95
Blade chord (m)
4.4
3.85
3.3
2.75
2.2
1.65
1.1
0.55
0
0
6.35
12.7
19.05
25.4
31.75
38.1
44.45
50.8
57.15
63.5
Blade length (m)
Figure 4.1: Chord distributions of the NREL and the optimized 5 MW wind turbine
As figure 4.1 shows, in the region between 0 and 8.5 m, the optimized rotor has a
smaller chord than the NREL rotor. The chord reduction reaches a maximum value of
17.5% at the rotor radius of 3.3 m. This chord reduction is mainly due to the structural
reason. In fact, because of having a better controller design, the loads on the blade root
of the optimized wind turbine are decreased. Lower loads in the blade root demand less
strength and this results in a smaller chord. From 8.5 m to the tip, the optimized blade
has a larger chord than the NREL, with a maximum value of 16.2% at the rotor radius of
25.0 m. For the optimum twist angle, figure 4.2 shows a reduction in the region between
12 and 28 m, and an increase for the rest compared to the NREL design.
Considering the fact that both rotors have a different chord, twist angle, blade length
and rated rotational speed, a one by one comparison is difficult to make. It is the combination of these properties that makes one design better than another, with respect to
the objective function and the design constraints.
The power curve of both the NREL and the optimized wind turbines is presented in
figure 4.3. As the figure shows, the optimized rotor has a rated wind speed of 11.0 ms ,
78
4.4
VERIFICATION OF THE INTEGRATED AEROSERVOELASTIC OPTIMIZATION METHOD
Optimized
NREL
25.4
38.1
14
12.6
Blade twist (deg)
11.2
9.8
8.4
7
5.6
4.2
2.8
1.4
0
0
6.35
12.7
19.05
31.75
44.45
50.8
57.15
63.5
Blade length (m)
Figure 4.2: Twist distributions of the NREL and the optimized 5 MW wind turbine
compared to 11.4 ms of the NREL. This is mainly because of having a larger rotor radius
for the optimized rotor that results in capturing more energy in the below rated region.
Optimized
NREL
12.4
16.6
Generator power output (kW)
5300
4770
4240
3710
3180
2650
2120
1590
1060
530
0
4
6.1
8.2
10.3
14.5
18.7
20.8
22.9
25
Wind Speed (m/s)
Figure 4.3: Power curve of the NREL and the optimized 5 MW wind turbine
In addition, the optimized wind turbine has a higher hub height of about 2.4 m,
compared with the 80 m hub height of the NREL wind turbine. This is an increase of
2.5% in the tower height that consequently results in a higher AEP.
Considering all the facts, the results show an increase of 15.7% for the AEP, and 22.2%
4.5
CONCLUDING REMARKS
79
for the TCC. However, the higher AEP pays off the higher costs related to having a larger
rotor and tower, and at the end results in a decrease of 4.2% in the LCOE.
The same comparison can also be made for the design constraints. However, the
most significant change with respect to the design of the 5 MW NREL wind turbine is
the system’s natural frequencies. The optimized 5 MW wind turbine is a soft-stiff design,
compared to a soft-soft design of the NREL wind turbine (see Petersen et al. (2010) for a
description of soft-soft and soft-stiff designs). This is mainly related to the difference in
the rated rotational speed of both wind turbines.
The optimized wind turbine has higher rated rotational speed of about 4.2% (12.9
RPM for the optimized and 12.1 for the NREL design). This increase has an influence
on the allowable natural frequencies, and makes the wind turbine from being a soft-soft
(NREL wind turbine) to a soft-stiff (optimized wind turbine) design.
Another influence of this increased rated rotational speed together with a larger rotor
diameter is the tip speed. While the NREL design has a maximum designed tip speed of
80 ms , the optimized design has a tip speed of 87.8 ms . This shows 8.9% increase in the
tip speed of the optimized design.
It is well known in the literature that for the OWTs, there is a clear potential benefit
in higher tip speeds, since there is less constraint on noise emission levels compared to
the 80 ms limit for onshore wind turbines, Malcolm and Hansen (2002). Therefore, it
is not surprising to see a higher tip speed for the optimized wind turbine design of this
work that proofs the idea of having a better design with an increase in the tip speed.
However, with increasing tip speed, the blade solidity usually decreases and this may
result in a more flexible blade. Although, this can be beneficial for system loads, it is
problematic for maintaining the preferred tower clearance in extreme loading conditions. Therefore, there is an optimum for the tip speed that is governed by several design
considerations. In the design of the optimized 5 MW, this is the first natural frequency of
the blade that prevents a further decrease in the blade’s solidity (in this case also blade’s
flexibility) by having a higher rotational speed.
4.5
Concluding remarks
The purpose of optimizing the 5 MW wind turbine was stated in section 4.2. Referring
to these goals, the knowledge gained in the MDO of the 5 MW wind turbine can be
characterized as follows:
4.5.1
Validating the integrated aeroservoelastic optimization method
Aeroservoelastic design optimization of a wind turbine, if used effectively and correctly,
can result in an improved and economical design, and save a huge amount of engineering
Infeasible
Infeasible
Feasible
Feasible
Improved
Improved
Local
Local
Figure 4.4: Different stages of optimization
Global
Global
80
VERIFICATION OF THE INTEGRATED AEROSERVOELASTIC OPTIMIZATION METHOD
4.5
design time. However, it is important to understand the limitation of this technique, and
use it as only one of the many tools at our disposal.
Having this in mind, it can be seen that the aeroservoelastic design optimization can
be applied at different stages of a design process. As figure 4.4 shows, one can use this
tool to make an infeasible design a feasible design. This is exactly the technique that
was used in level 2 of the optimization process to make the results of level 1 feasible.
Improving the quality of a feasible design is another application of this method. This step
was used in the design optimization process by iterating between level 1 and 2 until the
designer or the design criteria are satisfied.
It is already known that the NREL wind turbine is a properly designed wind turbine
and using the integrated aeroservoelastic design optimization comparable results can be
achieved. However, it should be noted that in this research a limited number of design
load cases are used and in practice several other load cases are required to get a wind
turbine certified.
All in all, it can be concluded that the integrated aeroservoelastic design optimization
has the capability to provide feasible design solutions for larger scale OWTs, in which
the scaling trends are going to be based on. Therefore, this tool will be used in the next
chapter to design and optimize 10 and 20 MW wind turbines.
4.5.2
Improved understanding of the 5 MW wind turbine design
In the context of design optimization, defining the design space of a wind turbine is not
an easy task. However, by simulating the physical behavior of the NREL wind turbine, a
good understanding of its design constraints was achieved. This helped to narrow down
the design space and properly fix the lower and upper bounds of the design variables. It
also helped to get an idea about the applied safety factors used to developed this wind
turbine.
This improved understanding can also be helpful for larger scales, since some of these
design constraints can also be scaled using the scaling laws. As an example, the system
natural frequencies are fairly well predictable, and if one knows these values for a 5 MW,
then the initial guess for a 10 to 20 MW wind turbines can be made more accurately.
4.5.3
Providing consistent data points to make scaling trends
To make representative scaling trends of the future wind turbines a consistent set of data
points are beneficial. Therefore, it is needed to use the same design assumptions at all
sizes. In that respect, the same design criteria that are used to optimize the 5 MW wind
turbine will be used for the 10 and 20 MW wind turbines and developing scaling trends
using these three data points has the least inconsistency.
CHAPTER
5
Integrated aeroservoelastic design and optimization of large
wind turbines
5.1
Introduction
The previous chapter has presented the design optimization of the 5 MW wind turbine.
This chapter uses the same design methodology to design the 10 and 20 MW wind turbines. The characteristics of these wind turbines presented in this chapter include the
important properties, as well as all the design variables, design constraints and the objective function.
These wind turbines together with the optimized 5 MW wind turbine act as data
points to make scaling trends and investigate the influence of size on technical and economical characteristics of larger scales wind turbines. These include loading-diameter,
mass-diameter and cost-diameter trends that help to study the influence of scale on
design and identify future design challenges.
5.2
MDO of the 10 and 20 MW wind turbines
This section presents the results of the optimized 10 and 20 MW wind turbines. As it was
proofed before, the concept of aeroservoelastic design optimization serves as a useful
method to find the optimum design solution for such giant wind turbines. However, for
conciseness, only important results are presented.
5.2.1
Design variables of the 10 and 20 MW wind turbines
To start the MDO of the 10 and 20 MW wind turbines, an initial set of design variables
and parameters are needed. Ideally, the initial design variables should be the closest
possible to their optimums to reduce computational time and provide a sound design.
81
82
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION OF LARGE WIND TURBINES
5.2
In this work, linear scaling laws are used to find such an initial set of design variables
and parameters. That is, the linear scaling laws are applied to the optimized 5 MW
wind turbine to find the initial value of the design variables and parameters of the 10
and 20 MW wind turbines with no conceptual change in their design with respect to the
optimized 5 MW wind turbine.
The initial values of the linearly upscaled design variables also help to make an engineering judgment on the upper and lower bounds of the side constraints. Based on
these initial design variables, several test runs were performed to adjust the upper and
lower bounds of the side constraints, while evaluating the functional design constraints
and their sensitivity with respect to the design variables. This results in a trial and error process to find reasonable side constraints to form a design space which is neither
computationally expensive nor narrow bounded.
After setting these initial design variables and their bounds, the MDO of the 10 and 20
MW is carried out. The same process as the MDO of the 5 MW wind turbine is followed
here.
Table 5.1 and 5.2 show the list of design variables for the 10 and 20 MW wind turbines, respectively. These tables also contain the lower and upper limits appearing as side
Table 5.1: 10 MW wind turbine design variables and side constraints
Description
Opt. level
Min
Initial
Optimum
Max
Blade length (m)
Tower height (m)
Rated rotational speed (rpm)
1
1
1
85.0
107.0
8.0
89.8
113.1
9.1
90.0
107.0
8.5
92.0
115.0
10.0
Twist at station 9 (deg)
Twist at station 15 (deg)
2
2
9.0
2.0
10.4
3
10.0
2.6
12.0
4.0
Chord at station 3 (m)
Chord at station 7 (m)
Chord at station 15 (m)
Chord at station 18 (m)
2
2
2
2
3.0
4.0
3.0
0.25
4.4
7.5
4.2
2.1
4.4
6.0
5.2
0.3
5.0
7.7
5.5
2.5
Skin thickness at station 1 (cm)
Skin thickness at station 3 (cm)
Skin thickness at station 6 (cm)
Skin thickness at station 16 (cm)
2
2
2
2
8.0
6.0
3.0
1.6
14.1
8.5
3.3
2.1
12.0
7.0
5.5
2.0
15.0
10.0
7.0
3.0
Web thickness at station 3 (cm)
Web thickness at station 6 (cm)
Web thickness at station 16 (cm)
2
2
2
1.0
2.0
1.0
1.4
2.8
1.8
3.0
3.5
3.0
6.0
5.0
4.0
Spar thickness at station 3 (cm)
Spar thickness at station 6 (cm)
Spar thickness at station 16 (cm)
2
2
2
1.0
1.0
1.0
1.4
3.5
3.4
2.5
3.5
2.0
4.0
4.0
4.0
Tower diameter at station 1 (m)
Tower diameter at station 22 (m)
2
2
7.0
4.0
8.5
5.8
9.4
5.4
10.0
7.0
Tower thickness at station 1 (cm)
Tower thickness at station 22 (cm)
2
2
5.0
3.0
5.7
3.4
8.5
4.0
10.0
5.0
5.2
MDO OF THE 10 AND 20 MW WIND TURBINES
83
constraints for level 1 and 2 optimization, as well as their initial and optimum values.
By looking to the initial and optimized design variables, it can be argued that generally
the linear scaling law provides reasonable initial design variables, though such a set of
design variables do not respect the design constraints.
Table 5.2: 20 MW wind turbine design variables and side constraints
Description
Opt. level
Min
Initial
Optimum
Max
Blade length (m)
Tower height (m)
Rated rotational speed (rpm)
1
1
1
125.0
146.0
6.0
127.0
150.0
6.45
140.1
157.1
6.5
143.0
170
8.0
Twist at station 9 (deg)
Twist at station 15 (deg)
2
2
6.5
2.7
10.4
3.0
7.5
5.5
12.0
6.0
Chord at station 3 (m)
Chord at station 7 (m)
Chord at station 15 (m)
Chord at station 18 (m)
2
2
2
2
6.0
7.0
5.0
1.0
6.2
10.6
6.0
3.0
7.9
10.0
7.0
1.2
8.0
11.0
8.0
3.2
Skin thickness at station 1 (cm)
Skin thickness at station 3 (cm)
Skin thickness at station 6 (cm)
Skin thickness at station 16 (cm)
2
2
2
2
15.0
7.0
4.0
2.0
20.0
12.0
4.6
3.0
19.3
10.0
7.7
2.0
21.0
13.0
10.0
7.0
Web thickness at station 3 (cm)
Web thickness at station 6 (cm)
Web thickness at station 16 (cm)
2
2
2
1.5
2.0
2.0
2.0
4.0
2.6
2.7
2.9
3.3
5.0
4.0
4.0
Spar thickness at station 3 (cm)
Spar thickness at station 6 (cm)
Spar thickness at station 16 (cm)
2
2
2
1.5
1.5
1.0
2.0
5.0
4.8
1.8
2.3
3.2
4.0
6.0
5.5
Tower diameter at station 1 (m)
Tower diameter at station 22 (m)
2
2
14.0
7.0
12.0
8.2
16.1
9.1
18.0
10.0
Tower thickness at station 1 (cm)
Tower thickness at station 22 (cm)
2
2
7.0
4.0
8.0
4.8
10.7
6.0
11.0
7.0
5.2.2
Design constraints of the 10 and 20 MW wind turbines
In total there are 52 design constraints to fully bound the design space. However, only
the highest active constraints are presented here. For the optimum 10 MW blade, the
highest active constraint is the tip deflection. For the optimum 10 MW tower, fatigue is
the highest active constraint.
For the optimum 20 MW blade, the highest active constraints are the tip deflection
and fatigue at the root. For the optimum 20 MW tower, fatigue is the highest active
constraint.
Table 5.3 and 5.4 show these constraints for both the blade and tower at the optimum
values of design variables of the 10 and 20 MW wind turbines, respectively.
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INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION OF LARGE WIND TURBINES
5.2
Table 5.3: 10 MW wind turbine functional constraints
Description
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
Max blade deflection at
9 ms
11 ms
13 ms
15 ms
17 ms
19 ms
21 ms
23 ms
25 ms
Fore-aft fatigue at station 1 of the tower
Fore-aft fatigue at station 5 of the tower
Fore-aft fatigue at station 9 of the tower
Fore-aft fatigue at station 13 of the tower
Fore-aft fatigue at station 17 of the tower
Fore-aft fatigue at station 21 of the tower
Opt. level
bound
Optimum
2
2
2
2
2
2
2
2
2
≤ 7.9(m)
≤ 7.9(m)
≤ 7.9(m)
≤ 7.9(m)
≤ 7.9(m)
≤ 7.9(m)
≤ 7.9(m)
≤ 7.9(m)
≤ 7.9(m)
6.6 (m)
7.9 (m)
7.7 (m)
6.3 (m)
5.1 (m)
4.9 (m)
4.9 (m)
4.5 (m)
3.6 (m)
2
2
2
2
2
2
≤ 0.7
≤ 0.7
≤ 0.7
≤ 0.7
≤ 0.7
≤ 0.7
0.49
0.61
0.70
0.67
0.52
0.23
Table 5.4: 20 MW wind turbine functional constraints
Description
Opt. level
bound
Optimum
Max blade deflection at 9 ms
Max blade deflection at 11 ms
Max blade deflection at 13 ms
Max blade deflection at 15 ms
Max blade deflection at 17 ms
Max blade deflection at 19 ms
Max blade deflection at 21 ms
Max blade deflection at 23 ms
Max blade deflection at 25 ms
2
2
2
2
2
2
2
2
2
≤ 11.8(m)
≤ 11.8(m)
≤ 11.8(m)
≤ 11.8(m)
≤ 11.8(m)
≤ 11.8(m)
≤ 11.8(m)
≤ 11.8(m)
≤ 11.8(m)
11.4 (m)
11.8 (m)
10.4 (m)
9.2 (m)
8.5 (m)
6.8 (m)
6.1 (m)
6.2 (m)
5.5 (m)
Flapwise fatigue at the root of the blade
2
≤ 0.7
0.68
Fore-aft fatigue at station 1 of the tower
Fore-aft fatigue at station 5 of the tower
Fore-aft fatigue at station 9 of the tower
Fore-aft fatigue at station 13 of the tower
Fore-aft fatigue at station 17 of the tower
Fore-aft fatigue at station 21 of the tower
2
2
2
2
2
2
≤ 0.7
≤ 0.7
≤ 0.7
≤ 0.7
≤ 0.7
≤ 0.7
0.58
0.66
0.65
0.54
0.30
0.14
Similar to the optimum 5 MW wind turbine design, the flapwise deflection of the
blade reaches a maximum at values close to the rated operating point for a NTM. This
peak is due to the transition of the controller at rated between the generator-torque and
the blade-pitch control.
5.3
5.2.3
PROPERTIES OF THE 10 AND 20 MW WIND TURBINES
85
Objective function of the 10 and 20 MW wind turbines
The derivation of LCOE and introduction of that as the objective function is explained
in detail in chapter 3. Table 5.5 shows the cost of the optimum 10 and 20 MW wind
turbines. The cost elements in this table are in 1000 USD.
5.3
Properties of the 10 and 20 MW wind turbines
Most of the data presented in this section are summarized for conciseness and clarity.
However, the design methodology used in this research contains a huge amount of data
for the optimized 10 and 20 MW wind turbines, which can not be presented all. Key
data about these optimized wind turbines are selected and presented in the following
subsections.
5.3.1
Gross properties
Table 5.6 shows the gross properties of the optimized 10 and 20 MW wind turbines. To
enable comparison, the optimized 5 MW wind turbine data is presented as well.
5.3.2
Blade properties
The rotor of both the 10 and 20 MW optimum wind turbines is equipped with three
blades of 90 and 140 m length, respectively. In the structural computations of the blade,
20 nonuniformly distributed elements are used. The properties between these elements
are found by a cubic interpolation. Also for both the 10 and 20 MW blades, a structural
damping ratio of 0.48% (critical in all modes of the isolated blade) that is equal to a 3.0%
logarithmic decrement (similar to the 5 MW NREL and the 6 MW DOWEC turbines) is
assumed as a fixed input.
Table 5.7 and 5.8 list the resulting structural properties of the 10 and 20 MW wind
turbines. The first column of both tables labeled Rad is the spanwise locations along the
blade-pitch axis measured from blade root. The second column labeled BldFrac is the
nondimensional distance along the blade-pitch axis from the root to the tip varying form
0 to 1, respectively.
The distributed blade section mass per unit length values labeled as BMsDen are given
in the third column. With this mass distribution, a mass of 53.0 t on and 182.7 t on is
calculated for the optimum 10 and 20 MW wind turbine blade.
The flapwise, BFlpStf, and edgewise, BEdgStf, section stiffnesses are presented in the
4 th and 5 th column respectively. These values are calculated about the principal structural
axes of each cross section.
5.3.3
Aerodynamic properties
The aerodynamic properties of the optimum 10 and 20 MW wind turbines are obtained
by running a series of simulations from the cut-in to cut-out wind speeds. The first 60
86
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION OF LARGE WIND TURBINES
5.3
Table 5.5: Cost estimation of the optimized 10 and 20 MW wind turbines
Components
10 MW
20 MW
Blades
Hub
Pitch mechanism and bearing
Nose cone
Low speed shaft
Main bearings
Gearbox
Mechanical brake and coupling
Generator
Power electronics
Yaw drive and bearing
Main frame
Platform and railing
Nacelle cover
Electrical connections
Hydraulic and cooling system
Control,safety and condition monitoring
Tower
Marinization
Turbine capital costs (TCC)
2474.9
324.2
664.5
21.8
499.9
245.2
2085.0
22.2
796.1
786.4
451.2
341.8
188.1
142.1
617.7
154.5
65.3
3765.6
1842.3
15489.0
8570.7
1037.5
2143.0
36.1
1783.4
1151.5
4955.0
44.4
1592.2
1572.8
1665.5
808.1
444.7
279.6
1235.5
309.0
65.3
12497.0
5042.5
45618.0
Foundation system
Port and staging equipment
Turbine installation
Electrical interface and connection
Permits, engineering and site assessment
Personnel access equipment
Scour protection
Decommissioning
Balance of station costs (BOS)
4349.5
289.9
1465.8
4127.0
431.0
70.2
806.2
810.8
11539.6
8699.0
579.9
2931.5
8253.9
862.1
70.2
1612.3
2058.9
23009.0
Offshore warranty premium
Initial capital cost (ICC)
2047.0
29886.4
6028.8
76714.4
Levelized replacement costs
Operation and maintenance
Fixed charge rate
198.0
1316.8
0.1185
396.1
2873.7
0.1185
56273009.4
122806072.9
0.0641
0.0704
AEP (kW hr)
LCOE
USD
( kW
)
hr
seconds of the simulation lengths were ignored to ensure that all transient behavior were
damped out.
The results are obtained by running a steady wind, and the steady BEM model. Using
this steady model, a rated wind speed of 11.5 ms for the 10 and 10.7 ms for the 20 MW
5.3
PROPERTIES OF THE 10 AND 20 MW WIND TURBINES
87
Table 5.6: Gross properties of the optimized 5, 10 and 20 MW wind turbines
Design specification
Rotor diameter (m)
Rated tip speed (m/s)
Rated rotational speed (RPM)
Hub height (m)
5 MW
10 MW
20 MW
130
88
12.9
82.4
182
82
8.5
110.4
286
98
6.5
162
Table 5.7: Optimized 10 MW wind turbine blade structural properties
kg
Rad. (m)
BldFrac (-)
BMsDen ( m3 )
BFlpStf (N · m2 )
BEdgStf (N · m2 )
0.0
13.5
26.6
44.2
61.7
79.3
90.0
0.000
0.149
0.295
0.491
0.686
0.881
1.000
1067.1
935.0
805.9
530.0
374.6
113.5
4.2
4.16 × 1010
1.63 × 1010
1.13 × 1010
2.33 × 109
8.34 × 108
4.25 × 107
4.26 × 103
4.16 × 1010
3.35 × 1010
2.95 × 1010
1.60 × 1010
9.85 × 109
5.41 × 108
6.00 × 104
wind turbines is obtained. Figures 5.1 and 5.2 show the generator power output curve
and the blade pitch angle of the 10 and 20 MW wind turbines.
5.3.4
Drive train properties
The 10 MW wind turbine has an optimized rated rotor speed of 8.4 rpm that is found in
level 1 optimization. By assuming a rotational speed of 1173.7 rpm similar to the 5 MW
Rotor speed (RPM)
Blade pitch angle (deg)
Generator power output (MW)
24
21.6
19.2
16.8
14.4
12
9.6
7.2
4.8
2.4
0
5
7
9
11
13
15
17
19
21
23
Wind speed (m/s)
Figure 5.1: Aerodynamic properties of the optimized 10 MW wind turbine
25
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INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION OF LARGE WIND TURBINES
5.3
Table 5.8: Optimized 20 MW wind turbine blade structural properties
kg
Rad. (m)
BldFrac (-)
BMsDen ( m3 )
BFlpStf (N · m2 )
BEdgStf (N · m2 )
0.0
16.4
34.7
55.1
82.5
123.5
140.1
0.000
0.117
0.247
0.393
0.588
0.881
1.000
3104.7
2133.8
1695.8
1284.9
1076.5
401.8
4.2
3.96 × 1011
3.20 × 1011
7.20 × 1010
1.99 × 1010
7.72 × 109
4.44 × 108
4.26 × 103
3.96 × 1011
2.77 × 1011
2.09 × 1011
1.40 × 1011
9.86 × 1010
7.17 × 109
6.00 × 104
wind turbine in the generator side, a gearbox ration of 139:1 can be obtained for the
10 MW wind turbine that is used during simulation. The total mechanical-to-electrical
conversion loss is assumed to be 5.6%, which is the same as both the NREL and DOWEC
turbines at the rated power.
Using the properties of the 5 MW wind turbine and applying linear scaling law, an
·m
equivalent drive shaft linear-spring constant of 2.45 × 109 Nr ad
, and an equivalent linear7 N ·m
damping constant of 1.75 × 10 r ad can be calculated and used for the design of the 10
s
MW wind turbine.
For the 20 MW wind turbine an optimum rated rotor speed of 6.5 rpm is found. With
a fixed rated generator speed of 1173.7 rpm, a gearbox ratio of 180:1 is obtained. Similar
to the 10 MW wind turbine, the total mechanical-to-electrical conversion loss is assumed
to be 5.6%.
Using a the similar approach as the 10 MW wind turbine, for the 20 MW wind turbine
·m
an equivalent drive shaft linear-spring constant of 6.94 × 109 Nr ad
, and an equivalent
Rotor speed (RPM)
Blade pitch angle (deg)
Generator power output (MW)
22
19.8
17.6
15.4
13.2
11
8.8
6.6
4.4
2.2
0
5
7
9
11
13
15
17
19
21
23
Wind speed (m/s)
Figure 5.2: Aerodynamic properties of the optimized 20 MW wind turbine
25
5.3
PROPERTIES OF THE 10 AND 20 MW WIND TURBINES
linear-damping constant of 4.97 × 107
N ·m
r ad
s
89
can be calculated and used in simulation.
The drive train gross properties of the optimum 10 and 20 MW wind turbine are
presented in table 5.9.
Table 5.9: Drive train gross properties of the 10 and 20 MW wind turbines
Property (Unit)
Rated rotor speed (RPM)
Gearbox ratio
Low speed shaft mass (ton)
Gearbox mass (ton)
High speed shaft, coupling and brake mass (ton)
Generator mass (ton)
Hydraulic and cooling system mass (ton)
5.3.5
10 MW
20 MW
8.5
139
49.5
84.1
1.98
31.5
0.79
6.5
180
176.7
173.7
3.96
59.7
1.59
Hub and nacelle properties
From the mass models developed for the hub, a mass of 56.2 ton for the 10 MW, and a
mass of 180.0 ton for the 20 MW wind turbines are calculated. It is assumed that the
hub is made from ductile iron castings and has a spherical shape. The diameter of this
sphere is obtained using the scaling laws to be 3 m for the 10 and 6 for the 20 MW wind
turbines.
Since the mass and geometry of the hub are known, with a simple algebraic equation,
the thickness of the hub is found. Based on this thickness, a hub mass moment of inertia
of 2.29 × 105 kg · m2 for the 10 MW, and 1.47 × 106 kg · m2 for the 20 MW wind turbine
are calculated.
The nacelle mass (mass of all tower top components except the rotor and hub) of the
10 MW wind turbine is 379.9 ton. For the 20 MW wind turbine this value is 1026.2 ton.
For both wind turbines, the yaw actuator has a natural frequency of 3 Hz, which is the
same as the NREL wind turbine.
The gross properties of the hub and nacelle for both wind turbines are presented in
table 5.10.
Table 5.10: Hub and nacelle gross properties of the 10 and 20 MW wind turbines
Property (Unit)
Hub mass (ton)
Hub mass moment of inertia (k g · m2 )
Nacelle mass (ton)
Nacelle mass moment of inertia (k g · m2 )
Elevation of yaw bearing from tower base (m)
Yaw bearing to shaft vertical distance (m)
Hub center to yaw axis distance (m)
10 MW
20 MW
56.2
2.29 × 105
379.9
1.35 × 107
107.0
2.77
7.1
180.0
1.47 × 106
1026.2
8.82 × 107
157.1
3.92
10.0
90
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION OF LARGE WIND TURBINES
5.3.6
5.3
Support structure properties
The support structure of the wind turbines consists of the foundation system (monopile
and transition piece) and tower. In this work, the soil mechanics of the monopile is
not modeled and thus it is assumed that all the DOFs of the monopile at the seabed
are constrained, and tower is the only element of the support structure assembly that is
optimized.
However the dynamic and cost of the foundation system is present in the design
process by using engineering models developed in WindPACT project for mass and cost
of these two components (see appendix A. These engineering models provide a basis with
which the integrity of the design is preserved without loosing too much of accuracy in
representing the dynamic and cost.
Table 5.11 and 5.12 give the resulting distributed tower properties of the 10 and 20
MW wind turbines. In both tables, the first column, Height is the vertical locations along
the tower centerline relative to the tower base. It is also assumed that the tower base is
located at the mean sea level.
Table 5.11: Optimized 10 MW wind turbine tower properties
Height (m)
0.0
29.4
77.6
107.0
TowFrac (-)
TowDia (m)
TowTick (cm)
TowStif (N · m2 )
0.000
0.275
0.725
1.000
9.4
8.5
6.7
5.4
8.5
7.3
5.2
4.0
5.6 × 1012
3.3 × 1012
1.2 × 1012
5.1 × 1011
The second column, TowFrac, of the tables is the fractional height along the tower
centerline from the tower base to the tower top that ranges from 0 to 1, respectively.
The third, fourth and fifth columns also represent the diameter, thickness and stiffness of
different tower heights.
As discussed before in chapter 3, only the diameter and thickness at the base and top
of the tower are introduced as design variables, and a linear interpolation is used at other
heights.
The resulting overall tower mass is 1393.1 t on for the 10 and 4623.4 t on for the 20
MW wind turbines.
Table 5.12: Optimized 20 MW wind turbine tower properties
Height (m)
0.0
43.2
113.9
157.1
TowFrac (-)
TowDia (m)
TowTick (cm)
TowStif (N · m2 )
0.000
0.275
0.725
1.000
16.1
14.2
11.0
9.1
10.7
9.4
7.3
6.0
3.6 × 1013
2.2 × 1013
7.9 × 1012
3.6 × 1012
5.3
PROPERTIES OF THE 10 AND 20 MW WIND TURBINES
5.3.7
91
Controller properties
The controller design development is explained in detail in chapter 3. The 10 and 20
MW wind turbines gross controller properties are given in table 5.13.
Table 5.13: Controller gross properties of the 10 and 20 MW wind turbines
Property (Unit)
( ms )
Cut-in, rated and cut-out wind speed
Rated tip-speed ( ms )
Peak power coefficient
Blade-pitch angle at peak power (deg)
Rated rotational speed (rpm)
Rated mechanical power (MW)
Rated generator torque (kN.m)
Generator slip in transition region(%)
deg
Maximum blade pitch rate ( s )
10 MW
20 MW
3, 11.7, 25
82
0.47
0.0
8.5
10.6
119.1
10
5.6
7, 10.7, 25
98
0.47
0.0
6.5
21.2
202.3
10
4.8
Figure 5.3 shows the final aerodynamic power sensitivity to the blade pitch angle at
different steady state operating points for the above rated pitch controller of both wind
turbines. Since the proportional and integral gains are a function of the aerodynamic
power sensitivity, the data presented in this figure are used to design the KP and KI gains.
10 MW
0
2.2
4.4
6.6
8.8
11
20 MW
13.2
15.4
17.6
19.8
22
Aerodynamic power sensitivity to
blade pitch angle (MW/rad)
0
-72
-144
-216
-288
-360
-432
-504
-576
-648
-720
Blade pitch angle (deg)
Figure 5.3: Aerodynamic power sensitivity to the blade pitch of the 10 and 20 MW wind turbines
The variation of the proportional and integral gains to balance out the changes of the
aerodynamic power is shown in figure 5.4 and 5.5 for the 10 and 20 MW wind turbines.
92
INTEGRATED AEROSERVOELASTIC DESIGN AND OPTIMIZATION OF LARGE WIND TURBINES
5.3
The calculated proportional and integral gains are obtained using equation 3.19 and
3.20 of chapter 3.
I Gain
Gain correction factor
0.07
0.9
0.063
0.81
0.056
0.72
0.049
0.63
0.042
0.54
0.035
0.45
0.028
0.36
0.021
0.27
0.014
0.18
0.007
0.09
0
0.5
1.3
7.6
11.2
14.1
16.7
19.2
Gain correction factor
P and I gains
P Gain
0
21.5
Blade pitch angle (deg)
Figure 5.4: PI gains and gain scheduled correction factor of the 10 MW wind turbine
P Gain
I Gain
Gain correction factor
0.12
0.96
0.108
0.72
P and I gains
0.084
0.6
0.072
0.06
0.48
0.048
0.36
0.036
0.24
0.024
Gain correction factor
0.84
0.096
0.12
0.012
0
0.5
5.3
8.1
10.5
12.8
14.8
16.7
0
18.6
Wind speed (m/s)
Figure 5.5: PI gains and gain scheduled correction factor of the 20 MW wind turbine
CHAPTER
6
Aeroelastic stability analysis using eigenvalue method
6.1
Introduction
Historically, aeroelastic instabilities have not been a driving issue in wind turbine design.
Therefore, the aeroelastic instability is rarely addressed in the past while designing a
wind turbine. However, with more lighter and flexible blades the aeroelastic instability
may become a design driver and therefore necessary to consider as a constraint in the
design process.
The previous chapter presented the MDO of the 10 and 20 MW wind turbines. However, these wind turbines should be checked against aeroelastic instabilities to have a
safe design. This chapter presents a method to find the unstable operational points for
these wind turbines using an eigenvalue method and serves as a rough check during the
preliminary wind turbine design process.
For this purpose, the structure and aerodynamics of the wind turbine are modeled
independently and coupled together to account for aeroelasticity. Eigenvalue analysis
of this coupled aeroelastic model provides the unstable operational mode of the wind
turbine.
Since, larger wind turbines are fitted with relatively soft blades and tower, classical
flutter may become a more important instability mode to check. Therefore, the focus of
this chapter is to develop an analysis tool that is well suited for this type of instabilities.
6.2
Aeroelastic instability in wind turbines
Theodorsen (1935) was the first who investigated the aeroelastic instability phenomenon
of a 2D airplane wing section. In Theodorsen’s investigation, an aerodynamic model
with an unsteady lift and moment term was coupled with a simple structural model with
plunging and torsion as the only degrees of freedom. The model used the potential flow
93
94
AEROELASTIC STABILITY ANALYSIS USING EIGENVALUE METHOD
6.2
assumption with an oscillatory motion and no damping was included. He found out that
under certain conditions aerodynamic forces can be coupled with structural deformation
that results in an increase in the amplitude of motion in every cycle called aeroelastic
instability.
Since then, aeroelastic instability has been studied extensively for aircrafts, helicopters and bridges at many different conditions and configurations , Goland (1957),
Dowell (1970), Hong and Chopra (1985), Agar (1989), Patil et al. (2001), Dessi and
Mastroddi (2008) and Su and Cesnik (2010).
However, the literature on the aeroelastic design of wind turbines is relatively new,
with a considerable amount of it being reproduced and adapted from rotary-wing aeroelasticity of helicopters and other rotor blades. Among the early studies in this area,
Ormiston (1973) investigated the effect of size, hub configuration, type of control system
and number of blades on the wind turbine dynamic characteristic and stability. Some
years later he studied the aeroelastic stability of a wind turbine blade using a simple
analytical technique. He modeled the blade as a centrally hinged and spring restrained
element to study the dynamic response of the flapping motion in time domain, Ormiston
(1975).
Friedmann (1976) studied the aeroelastic stability and response of large wind turbine
blades using a set of coupled flap-lag-torsional equations of motion. He derived a general
nonlinear, partial differential equation for determining the aeroelastic envelop. He also
made an excellent survey of helicopter rotor aeroelastic studies of value in wind turbine
design. Also Kaza and Hammond (1976) used a similar approach to study the flap-lag
stability of the blade in the presence of velocity gradients. They used a single centrallyhinged model of the blade without any rotor-tower interaction.
In 1983, Janetzke and Kaza (1983) investigated the possibility of whirl flutter and
formulated the effect of flap-torsion flutter on teetering motion of a 2 bladed wind turbine. The wind turbine had a 3.5 m diameter and they found an unstable point at 77.1
m
wind speed, and 320 RP M angular velocity.
s
Kirchgässner (1983) presented a linear aeroelastic stability analysis method for analyzing the stability of rotorcrafts. However, he also showed how the method could be
used to analyze the stability of a two-bladed wind turbine. In his model, a quasi-steady
aerodynamic model was coupled with a simple structural model to analyze the stability.
At the late 1980’s and early 1990’s when the stall regulated wind turbines were dominating the market, it was observed that the blades of these machines that operate in
a separated flow experience stall-induced vibration, Stiesdal (1994). This resulted in
high loads and instabilities during the operational condition, challenging the designers.
Therefore, it became the concern of many researcher who were attempting to address
and predict it, Björck et al. (1997) and Petersen et al. (1998).
Chaviaropoulos (1999) and Chaviaropoulos et al. (2003) studied the stall-induced
flap and edgewise oscillation in a stall-regulated rotor. Nonlinear aerodynamic loads in
the dynamic stall regime were calculated using a quasi-steady Onera model and NavierStokes solvers respectively. However, the effect of structural nonlinearity was not included in these studies.
In 2003, Lindenburg and Snel showed the importance of finding the aeroelastic
boundaries in the next generation of wind turbines, Lindenburg and Snel (2003). This
work also presents the ECN research projects on the aeroelastic stability of the rotor
6.3
MODELING APPROACH
95
blade vibration. They also showed three different research projects aiming to develop
new tools; STABTOOL-3, DAMPBLADE and STABCON.
Hansen (2004) presented a design tool for performing aeroelastic stability analysis of
wind turbines. In his work, the aerodynamic loads are modelled using the BEM theory
with an extension of Beddoes-Leishman dynamic stall model in a state-space formulation. A finite beam element method is used to model the structure. The multi-blade
transformation was employed to eliminate the periodic coefficients.
Lobitz (2005) studied the sensitivity of two parameters on the flutter speed of a 35
m wind turbine blade. These parameters were the chordwise location of the center of
mass and the ratio of the flapwise to torsional frequency. This study showed how these
parameters can highly influence the flutter speed.
To have a view of the stability characteristics of the wind turbine, the aeroelastic
damping of the operational turbine modes can be used as a good indicator. Therefore,
Hansen et al. (2006) presented two experimental methods for estimating the modal
damping of a wind turbine during operation. In the first method, the turbine modal
mode is excited with a harmonic force at its natural frequency, and damping is obtained
by a decaying response. The second method uses a stochastic subspace identification.
Here, the linear model of the turbine is estimated from the response of measured signals.
For stall-induced flutter, structural damping, blade airfoil characteristics and the effective direction of blade vibrations govern the instability phenomenon, whereas for the
classical flutter the frequency ratio of the two unstable modes, the location of the elastic
axis and the center of gravity and the torsional blade stiffness are the main parameters. These instability characteristics are exemplified by Hansen (2007) in an aeroelastic
stability analysis for different wind turbines.
The main research objective on the next years were to develop new structural and
aerodynamic models to find instabilities for large wind turbines, since it was observed
that they are more susceptible to aeroelastic instabilities. These models should have the
capability of simulating large deformations (non-linear models) with high accuracy.
Currently, many different approaches are used to address the aeroelastic instability of
wind turbines at different conditions; nonlinear structural modeling using superelements
coupled with BEM model to setup equations of motion and find instabilities, Holierhoek
(2008),search for aeroelastic instabilities using reduced order system identification based
on flexible multibody method, Meng et al. (2008), nonlinear aeroelastic models for stallinduced vibration, Sarkar and Bijl (2008), stability analysis for high angle of attack on
parked wind turbine blades, Politis et al. (2009), and study the effect of steady deflections
on the aeroelastic stability of a turbine blade, Kallesoe (2011).
6.3
Modeling approach
The structure of the wind turbine is modeled by articulated nonlinear elements using
Finite Element Method (FEM). For this purpose the commercial FE package, NASTRAN,
is used. This model expresses the structural mass, damping and stiffness matrices in a
noninertial (rotary) reference frame. To account for the rotational effects of the rotor,
these matrices are modified and transformed to an inertial (stationary) reference frame.
The primary application for an inertial (rather than noninertial) frame of reference is
in the field of rotor dynamics where a rotating structure (rotor) is modeled along with
96
6.4
AEROELASTIC STABILITY ANALYSIS USING EIGENVALUE METHOD
a stationary support structure. Examples of such an application include a gas turbine
engine rotor-stator assembly or wind turbine rotor-tower, where the rotor spins with
respect to a fixed supporting structure.
For classical flutter prediction an unsteady aerodynamic model is needed. The Blade
Element Momentum (BEM) model is used as the basis to calculate the axial induction
factors over the rotor plane and find the real wind velocity that each element of the
blade experiences. Then, the unsteady aerodynamic forces are calculated by Theodorsen
function in frequency domain, using the wind velocities obtained from the BEM model.
In the next step, the aeroelastic equations of motion are constructed by coupling the
structural model with the unsteady aerodynamic model. This enables the treatment of
the aeroelastic instability problem of the wind turbine as an eigenvalue analysis problem. Here, the nonlinear aeroelastic model of the wind turbine is linearized about a
steady state equilibrium to set up an eigenvalue problem. This is similar to an eigenvalue
analysis that is performed in structural analysis, except that here all degrees of freedom
of the turbine and all terms of the unsteady aerodynamics are included.
Using this approach, it is possible to evaluate a limited number of eigenvalues of the
coupled aeroelastic matrix using Arnoldi’s iterative algorithm, Jia (1998). Based on this
eigenvalue analysis, the unstable operational points of the wind turbine will be obtained.
Figure 6.1 shows the steps of developing this methodology.
Modeling
Modelingthe
thestructure
structure
ininNASTRAN
NASTRAN
Calculating
Calculatinginduction
induction
factors
factorsusing
usingBEM
BEM
Extracting
ExtractingM,
M,D,
D,KK
matrices
matricesfrom
fromNASTRAN
NASTRAN
Unsteady
Unsteadyaerodynamic
aerodynamic
modeling
modeling(Theodorsen)
(Theodorsen)
Adding
Addingrotational
rotationaleffect
effect
Aeroelastic
Aeroelastic
coupling
coupling
Modifying
ModifyingTheodorsen
Theodorsenfor
for
real
airfoil
real airfoildata
data
Aeroelastic
Aeroelasticmodel
modelofofthe
the
wind
turbine
wind turbine
Eigenvalue
Eigenvalueanalysis
analysis
Figure 6.1: The flowchart of the aeroelastic instability tool
6.4
6.4
STRUCTURAL MODEL OF THE WIND TURBINE
97
Structural model of the wind turbine
The dynamic behavior of a structural body can be represented by the elastodynamic
equations. The elastodynamic equations are a set of Partial Differential Equations (PDE)
and their boundary conditions. However, these equations are only solvable for simple
geometries and for complex engineering problems a closed form solution does not exist.
The Finite Element Analysis (FEA) is a numerical technique for finding approximate
solutions of these PDEs by means of discretization. Two different approaches can be
used to find the solution of a PDE. In the first method the problem can be treated as
a steady state problem by eliminating the differential equation completely. The second
method renders the PDE into an approximating system of ordinary differential equations,
which are then numerically integrated using standard techniques such as Euler’s method,
Runge-Kutta, Newmark, etc.
By using the second method, the continuous equations of an elastodynamic system
can be represented by a set of Ordinary Differential Equations (ODE) as:
[Ms ]
d 2 u(t)
d t2
+ [Ds ]
du(t)
dt
+ [Ks ]u(t) = F(t)
(6.1)
Where:
Ms : structural mass matrix
Ds : structural damping matrix
Ks : structural stiffness matrix
u: vector which contains all the nodes displacement in time
F(t): external forces applied on the structural system
To find the structural matrices, (Ms , Ds and Ks ), NASTRAN finite element software
is used. These matrices contain the structural information of the wind turbine that are
needed to find the instabilities. However, these system of equations (matrices and DOFs)
are in a noninertial reference frame and they need to be modified for the rotational effect.
Next subsections explain the required modifications.
6.4.1
Displacement, velocity, acceleration and forces in the inertial
frame
As explained before, all DOF’s obtained from NASTRAN are in the noninertial reference
frame (labeled as coordinate system 2 in figure 6.2). To express these DOF’s in an inertial reference frame, the relation between the inertial and noninertial reference frames
should be formulated. As figure 6.2 shows, the position of an arbitrary differential mass
can be described by:
u1 = R1 + R2
(6.2)
Here:
u1 : the position vector of the differential mass in the inertial reference frame
R1 : the position vector that relates the origin of the inertial and the noninertial reference
98
6.4
AEROELASTIC STABILITY ANALYSIS USING EIGENVALUE METHOD
2
R2
R1
u1
1
Figure 6.2: Inertia and noninertia reference frames
frames
R2 : the position vector of the differential mass in the noninertial reference frame
This general expression of the differential mass is used as the basis to derive the
velocity and acceleration terms in the inertial reference frame. Using this formulation,
the velocity of the differential mass can be obtained by taking the time derivatives of
equation 6.2 as follow:
du1
dR1
dR2
=
+ Ω × R2 +
(6.3)
dt 1
dt 1
dt 2
Here, Ω = (Ω x , Ω y , Ωz ) is the angular velocity of the noninertial reference frame with
respect to the inertial reference frame. Similarly, the acceleration of the differential mass
is obtained by taking the time derivative of the velocity terms given in equation 6.3 as:
‚
d 2 u1
d t2
Œ
‚
=
1
d 2 R1
Œ
‚
+
d t2
d 2 R2
1
2Ω ×
dR2
dt
d t2
Œ
+
2
dΩ
dt
× R2
+ Ω × Ω × R2
+
(6.4)
2
The equivalent force in the inertial reference frame acting on the differential mass
m, due to the acceleration in the noninertial reference frame is obtained by applying the
Newton’s second law (knowing the acceleration term from equation 6.4) as:
F= m·
d 2 u1
(6.5)
dt
Substituting equation 6.4 in equation 6.5 and integrating over the domain of the
differential mass, the forces on the internal reference frame can be obtained. This is
presented in equation 6.6 as:
6.4
STRUCTURAL MODEL OF THE WIND TURBINE
F=
Z ‚
V
d 2 R1
Œ
‚
d 2 R2
99
Œ
dΩ
× R2
+
+
d t2 1
d t2 2 d t
dR2
2Ω ×
+ Ω × Ω × R2
dm
dt 2
+
(6.6)
Where V is the volume domain of the differential mass. The fourth and fifth terms
of equation 6.6 are called the Coriolis and centrifugal forces. They change the apparent
damping and stiffness of the structure as experienced in an inertial reference frame. This
is explained in the next subsection.
6.4.2
Mass, stiffness and damping in the inertial frame
In the previous subsection, the modification of the displacement, velocity, acceleration
and forces of a finite element mass to account for the changes from a noninertial to
an inertial reference frame was formulated. This subsection explains how these forces
influence the structural matrices.
To have the structural matrices, the wind turbine is modeled in NASTRAN. Hence,
these matrices should be extracted from NASTRAN and modified to account for the rotational effect. Equation 6.1 shows the relation between the mass, stiffness and damping
matrices with the displacement, velocity, acceleration and the external forces applied on
a finite mass element in the noninertial reference frame.
To modify the structural matrices of equation 6.1, additional terms should be added
to this initial FE model that is extracted from NASTRAN as described by Guo et al. (2001).
Thus, the new FE model that includes the equivalent rotational effect should read as:
[Ms ]
d 2u
d t2
+ [Ds + Dr ]
du
dt
+ [Ks − Kr ]u = F(t)
(6.7)
The added terms that represent the equivalent rotational effect in this new formulation are:
Dr as the damping matrix due to the rotation of the element. It is usually called
Coriolis matrix in a noninertial reference frame, and gyroscopic matrix in an inertial
reference frame given by the following equation:

0
 R

2 · Ωz d m
[Dr ]e = 
R V
−2
· Ωy dm
V
R
−2 · Ωz dm
0
R
2
·
Ω x dm
V
V
R
2 · Ω y dm
RV
−2 · Ω x dm
V
0




(6.8)
e
Kr as the stiffness matrix due to the rotation of the element given by the following
equation (spin softening matrix due to the rotation of the structure). It changes the
apparent stiffness of the structure in a rotating reference frame.
100
6.5
AEROELASTIC STABILITY ANALYSIS USING EIGENVALUE METHOD

−

[Kr ]e = 

6.5
R
(Ω2 + Ω2z )d m
RV y
Ω Ω dm
RV x y
Ω x Ωz d m
V
R
Ω Ω dm
R V y x
− V (Ω2z + Ω2x )dm
R
Ω y Ωz dm
V
R
Ω Ω dm
RV z x
Ω Ω dm
R V z y
− V (Ω2x + Ω2y )dm




(6.9)
e
Aerodynamic model of the rotor
As discussed before, for classical aeroelastic flutter prediction an unsteady aerodynamic
model is required. Figure 6.3 shows a sketch of the developed method by Theodorsen,
which addresses the unsteady aerodynamic forces of a 2D flat plate that experiences
oscillatory pitching and plunging motion.
Figure 6.3: Schematic of the Theodorsen unsteady aerodynamic forces on a flat plate
Here:
L: the unsteady lift force
M : the unsteady aerodynamic force
b: semi-chord
U: wind velocity
α: the pitching motion
h: the plunging motion
a: a fraction of semi-chord that defines the location of elastic axis from the semi-chord
d1 : the distance between the aerodynamic center and the elastic axis
The pitching and plunging motions can be represented using a harmonic solution as:
h = h0 e iωt
α = α0 e iωt
(6.10)
Where h0 and α0 are complex constants. Thus, the solution for the lift and moment
(as derived by Theodorsen (1935)) is:
L = 2πρU 2 b
h C(k)
U
ḣ + C(k)α + [1 + C(k)(1 − 2a)]
b
2U
α̇ +
b
2U
ḧ −
2
b2 a i
α̈
2U 2
(6.11)
6.5
AERODYNAMIC MODEL OF THE ROTOR
h
b
C(k)
ḣ + C(k)α + [1 + C(k)(1 − 2a)]
α̇]
M = 2πρU 2 b d1 [
U
2U
i
b3
b2
1
a b2
1
α̇(a − ) +
ḧ +
α̈( − a2 )
2
2
2U
2
8
2U
2U
101
+
(6.12)
Here, C(k) is the Theodorsen function and k = ωb
is the reduced frequency. Thus, the
U
Theodorsen function can be represented as a complex function of the reduced frequency,
which is given by:
(2)
C(k) =
H1 (k)
(2)
(2)
H1 (k) + iH0 (k)
(6.13)
In the above formulation, H denotes the Hankel function. Also, the real and imaginary parts of C are depicted in figure 6.4 as a function of reduced frequency.
Figure 6.4: The Theodorsen function
However, the Theodorsen unsteady forces are developed for the fixed wings of aircrafts, and it need to be modified for the rotating blade of a wind turbine. Therefore,
to make use of the Theodorsen function for a wind turbine several modifications are
needed. These are explained in the following subsections.
6.5.1
Modification of the wind velocities
The wind velocities presented in the Theodorsen lift and moment terms are the freestream wind velocity. However, for a wind turbine the rotor disk causes a flow blockage
resulting in a change of the free-stream velocity. In addition, the rotation of the blade in
its plane causes an extra velocity that each cross section of the blade experiences.
102
6.5
AEROELASTIC STABILITY ANALYSIS USING EIGENVALUE METHOD
In this work, the steady BEM model is used to obtain the velocities that each cross
section of the blade experiences. Since the radial induction factors are much smaller
than the axial induction factors, they are neglected. The influence of the pitch controller
is also taken into account for the above rated region.
These modified wind velocities are used for the calculation of instabilities at different
operational points of the wind turbine from the cut-in to cut-out wind speed.
6.5.2
Modification of the airfoil data
The lift expression of the Theodorsen solution presented in equation 6.11 is developed
∂C
for a flat plate. Therefore, it uses a value of 2π for the lift slope, ∂ αl .
In this work, the 2π value is replaced by the real value of
∂ Cl
∂α
∂ Cl
∂α
from the airfoil data
of each section of the blade. This value,
, is obtained by fitting a linear curve to the
airfoil data between the zero-lift angle and the separation angle. Thus, the resultant lift
expression can be explained by:
L=
∂ Cl
∂α
ρU 2 b
h C(k)
U
ḣ + C(k)α + [1 + C(k)(1 − 2a)]
b
2U
α̇ +
b
2U
ḧ −
2
b2 a i
α̈
2U 2
(6.14)
The expression of the moment given by equation 6.12 is also modified by replacing
∂C
2π, with the real slope of the airfoil, ∂ αm , thus:
M=
6.5.3
∂ Cm
h
C(k)
b
ρU 2 b d1 [
ḣ + C(k)α + [1 + C(k)(1 − 2a)]
α̇]
∂α
U
2U
2
2
3
i
ab
1
b
b
1
α̇(a − ) +
ḧ +
α̈( − a2 )
2
2
2U
2
8
2U
2U
+
(6.15)
The inclusion of the drag force
In the lift and moment expressions of the previous subsection the flow is assumed to
be incompressible, irrotational and inviscid. Using these assumptions, the separation on
the airfoil does not happen and the flow regime remains laminar. Thus a linear relation
between the drag coefficient, Cd , and the angle of attack, α, can be assumed.
∂C
The slope of this line, ∂ αd , is obtained by fitting a linear curve to the airfoil data
before the separation angle of attack. Hence, the drag coefficient is computed using the
following equation:
Cd =
∂ Cd
·α
(6.16)
∂α
The unsteady angle of attack, α, can be obtained from equation 6.14, and by substituting that into equation 6.16 the drag coefficient can be found:
Cd =
∂ Cd ”
L
ḣ
b
b
− −
α̇
· ∂C
ḧ − [1 + C(k)(1 − 2a)]
2
l
∂α
2U C(k)
C(k)ρU 2 b U 2U C(k)
∂α
+
6.5
AERODYNAMIC MODEL OF THE ROTOR
b2 a
2U 2 C(k)
—
α̈
103
(6.17)
Simplifying equation 6.17, the drag force on the airfoil is obtained using the following
equation:
D = ρU 2 b
6.5.4
∂ Cd h
∂α
−
C(k)
U
ḣ − C(k)α + [1 + C(k)(1 − 2a)]
b
2U
α̇ −
b
2U 2
ḧ +
b2 a i
α̈ (6.18)
2U 2
Discretization of the 3D blade to 2D sections
The blade of the wind turbine is discretized in several stations along its longitudinal axis,
with each section having a fixed chord. As figure 6.5 shows, this enables the representation of a 3D blade as a summation of several 2D stations.
1
n
1
n
Figure 6.5: Discretization of a 3D blade to several 2D stations
Using this definition, lift, drag and moment can be calculated using the 2D unsteady
aerodynamic model of Theodorsen. Thus, the lift, drag and moment on each 2D section
of the blade are obtained using the following expressions:
Li = (
∂ Cl
∂α
)i ρUi2 bi
h C(k)
Ui
ḣi + C(k)αi + [1 + C(k)(1 − 2ai )]
Where Ai is the area of each cross section.
bi
α̇i
2Ui
b 2 ai i
bi
ḧ − i 2 α̈i · Ai
2 i
2Ui
2Ui
+
(6.19)
104
6.6
AEROELASTIC STABILITY ANALYSIS USING EIGENVALUE METHOD
Mi = (
∂ Cm
Di = (
6.5.5
h C(k)
bi
ḣi + C(k)αi + [1 + C(k)(1 − 2ai )]
α̇i ]
)i ρUi2 bi di [
∂α
Ui
2Ui
i
bi3
bi2
ai bi2
1
1
α̇i (ai − ) +
ḧ
+
α̈i ( − ai2 )
i
2
2
2Ui
2
8
2Ui
2Ui
∂ Cd
h C(k)
bi
)iρUi2 bi −
ḣi − C(k)αi + [1 + C(k)(1 − 2ai )]
α̇i
∂α
Ui
2Ui
bi2 ai i
bi
ḧ
+
α̈i
i
2Ui2
2Ui2
+
(6.20)
−
(6.21)
Derivation of the aerodynamic matrices in a FE form
As explained in section 6.4, the structural model of the wind turbine is represented by
mass, damping and stiffness matrices. Similarly, lift, drag and moment loads explained
in section 6.5 need to be represented in mass, damping and stiffness matrices form to
construct the aeroelastic model of the wind turbine. This enables the usage of a FE
notation for both the structure and aerodynamics for the subsequent complex eigenvalue
analysis.
The extraction of the aerodynamic matrices is employed using the principle of virtual
work, and it is explained in detail in appendix H. With this principle, the aerodynamic
mass, damping and stiffness matrices can be obtained that represent the complex-valued
Theodorsen function.
6.6
Coupling structural and aerodynamic FE matrices
The coupling of the extracted FE aerodynamic and structural matrices (including the
rotational effect) is presented in this section. The coupled system of equations represents
the dynamic behavior of the turbine. As it was shown before, the coupling between the
structural model and the body forces can be presented as:
[Ms ]
d 2u
d t2
+ [Ds + Dr ]
du
dt
+ [Ks − Kr ]u = F(t)
(6.22)
The external forces, F(t), in this equation are the aerodynamic forces [Fa ] that are
represented by:
[Fa ] = [MA ][ü] + [DA ][u̇] + [KA ][u]
Replacing equation 6.23 on the right hand side of equation 6.22 results in:
(6.23)
6.7
105
EIGENVALUE ANALYSIS OF THE COUPLED AEROELASTIC MODEL
[Ms ]
d 2u
d t2
+ [Ds + Dr ]
du
dt
+ [Ks − Kr ]u = [MA ][ü] + [DA ][u̇] + [KA ][u]
(6.24)
Rearranging this equation by factorizing its displacement, velocity and acceleration
terms results in:
[Ms − MA ]
d 2u
d t2
+ [Ds + Dr − DA ]
du
dt
+ [Ks − Kr − KA ]u = 0
(6.25)
Equation 6.25 represents the aeroelastic coupled matrices and can be written in its
final form as:
[Mt ]
d 2u
d t2
+ [Dt ]
du
dt
+ [Kt ]u = 0
(6.26)
Where:
[Mt ]: the complex mass matrix of the complete model equal to [Ms − MA ].
[Dt ]: the complex damping matrix of the complete model equal to [Ds + Dr − DA ].
[Kt ]: the complex stiffness matrix of the complete model equal to [Ks − Kr − KA ].
6.7
Eigenvalue analysis of the coupled aeroelastic model
The eigenvalues of the coupled aeroelastic matrices of equation 6.26 are solved using
the Arnoldi algorithm with spectral transformation, Saad (1980). This algorithm treats
equation 6.26 as an eigenvalue problem to find the eigenvalues and eigenvectors.
Assuming a simple harmonic motion as u = C · Xeλt , and replacing that in equation
6.26 results in:
([A] − λ[B])X = 0
(6.27)
Here, λ represents the eigenvalues with the corresponding eigenvectors as X. A and
B matrices are also given by the following equations:
I 0
[A] =
(6.28)
0 Kt
0
I
[B] =
(6.29)
−Mt −Dt
106
AEROELASTIC STABILITY ANALYSIS USING EIGENVALUE METHOD
6.7
Based on the sign of the real (Γ), and the imaginary (Θ) parts of the eigenvalues
that are found by the algorithm, different scenarios can happen. Figure 6.6 shows these
different scenarios. As can be seen from the figure, two different instabilities can be identified, static and dynamic. These two cases are explained in more detail in the following
subsections.
6.7.1
Static instability
The first unstable static operational points are computed iteratively by the algorithm in
the following steps:
1. A given set of wind velocities from the cut-in to cut-out are selected.
2. At every given wind velocity the corresponding steady rotational speed and the
relative wind velocities at every section are calculated.
3. The coupled aeroelastic model of the system is constructed for these initial values.
4. A set of values for the reduced frequency, k, to perturb the aeroelastic model are
selected. These values are:
k = [0.001, 0.1429, 0.2857, 0.4286, 0.5714, 0.7143, 0.8571, 1.0000].
5. For the given wind and rotor speed, and a given reduced frequency the algorithm
finds the eigenvalues of the problem.
6. From the obtained eigenvalues, those that have a zero imaginary part are filtered
and selected.
7. From the selected eigenvalues of step 6, those that have a positive real part are
filtered and selected.
8. If in step 7, no unstable point is found the process goes to step 4 and picks the next
reduced frequency.
9. If in step 8, an unstable point is found it will be saved and the process goes to step
1 and picks the next wind velocity.
10. The process stops after checking the last wind velocity.
6.7.2
Dynamic instability
The first unstable dynamic operational points are computed iteratively by the algorithm
in the following steps:
1. A given set of wind velocities from the cut-in to cut-out are selected.
2. At every given wind velocity the corresponding steady rotational speed and the
relative wind velocities at every section are calculated.
3. The coupled aeroelastic model of the system is constructed for these initial values.
4. A set of values for the reduced frequency, k, to perturb the aeroelastic model are
6.7
EIGENVALUE ANALYSIS OF THE COUPLED AEROELASTIC MODEL
Eigenvalue = ± Γ ± Θ
Γ
Θ
Type of motion
>0
=0
Continuous
divergence
Unstable
<0
=0
Continuous
convergence
Stable
≠0
Divergent
oscilation
Unstable
<0
≠0
Convergent
oscilation
Stable
=0
≠0
Simple
harmonic
Stability
boundary
=0
=0
Time
independent
Stability
boundary
>0
Stability characteristic
Shape of motion
Figure 6.6: Types of motion and stability characteristics for various types of eigenvalues
107
108
AEROELASTIC STABILITY ANALYSIS USING EIGENVALUE METHOD
6.8
selected. These values are:
k = [0.001, 0.1429, 0.2857, 0.4286, 0.5714, 0.7143, 0.8571, 1.0000].
5. For the given wind and rotor speed, and a given reduced frequency the algorithm
finds the eigenvalues of the problem.
6. From the obtained eigenvalues, those that have a positive imaginary part are filtered
and selected.
7. From the selected eigenvalues of step 6, those that have a positive real part are
filtered and selected.
8. If in step 7, no unstable point is found the process goes to step 4 and picks the next
reduced frequency.
9. If in step 8, an unstable point is found it will be saved and the process goes to step
1 and picks the next wind velocity.
10. The process stops after checking the last wind velocity.
6.8
Implementation of the method
The implementation of the method is carried out in three main steps: preprocessing,
processing and postprocessing. These steps are illustrated in figure 6.7. For the preprocessing part, following input data needs to be defined by the user.
1. Blade length, chord and twist distribution at each radial position along the blade
2. Material properties of the blade at every section
3. Blade pitch angle, rotational speed and cut-in to cut-out wind speeds
4. Airfoil data at every station along the blade
5. Tower height, tower diameter at the base and top, and tower thickness at the base
and top
6. Mass of the nacelle and its x-y-z location
7. Geometry and density of the hub
The preprocessing part is entirely parametric and given all the input data, it creates
the geometry of the wind turbine. This task is carried out by a shell program that uses
Patran Command Language (PCL) to generate this geometry.
6.8
IMPLEMENTATION OF THE METHOD
Preprocessing
Preprocessing
Modeling
Modelingthe
thewind
windturbine
turbine
(MSC
PATRAN)
(MSC PATRAN)
Processing
Processing
Calling
Callingsolver
solver107
107
(MATLAB)
(MATLAB)
Adding rotational effect to
Adding rotational effect to
the structural FE matrices
the structural FE matrices
(MATLAB)
(MATLAB)
Solving and printing the
Solving and printing the
structural
structuralFE
FEmatrices
matrices
(NASTRAN)
(NASTRAN)
Computing
Computingthe
theaerodynamic
aerodynamic
loads
loadsand
andreforming
reforming
them
themtotoFE
FEshape
shape
(MATLAB)
(MATLAB)
Reading
Readingthe
theFE
FE
structural
matrices
structural matrices
(MATLAB)
(MATLAB)
Aeroelastic
Aeroelasticcoupling
couplingbetween
between
the
structural
the structuraland
and
aerodynamic
aerodynamicFE
FEmatrices
matrices
(MATLAB)
(MATLAB)
Postprocessing
Postprocessing
Finding
Findingthe
thestatic
staticand
and
dynamic instabilities
dynamic instabilities
(MATLAB)
(MATLAB)
Figure 6.7: Steps of implementation the method in PATRAN/NASTRAN/MATLAB
109
110
AEROELASTIC STABILITY ANALYSIS USING EIGENVALUE METHOD
6.8
In the processing part by executing NASTRAN solver 107, the geometry and the material property of the turbine is used to construct the FE model of the wind turbine. In
this phase all the boundary conditions are applied at all the relevant junctions (the joint
between the tower, nacelle, hub and blades is defined as rigid). Here, the nacelle is
modeled as a nodal mass at its center of gravity that is given by the user. Then, the
structural mass, damping and stiffness matrices are calculated and printed out as an output file. A MATLAB code reads these matrices and it automatically modifies the stiffness
matrices to account the rotational effect.
In the next step, BEM theory is used to calculate the resultant wind velocity that
each section of the blade experiences. Using this wind velocity the aerodynamic forces
are calculated. As explained before, these forces are reformed in shape to have them as
mass, damping and stiffness matrix form. Finally, as the last step in the processing part,
the structural and aerodynamic matrices are coupled to construct the aeroelastic model
of the wind turbine.
The postprocessing part starts with reading the aeroelastic model of the turbine and
converting that to an eigenvalue problem. This eigenvalue problem is solved using the
Arnoldi algorithm to find the eigenvectors and eigenvalues of the system. In the processing and postprocessing steps, a combination of MATLAB and DMAP (Direct Matrix
Abstraction Programming) languages is used to do the programming.
CHAPTER
7
Verification and aeroelastic stability analysis of the 20 MW
wind turbine
7.1
Introduction
In the previous chapter the theory behind the development of the aeroelastic stability
method was explained, and it was shown how the method is implemented. The result
was a computer code to find the unstable operational points of a wind turbine.
This chapter shows the verification of the method by finding the stability margin of
the 5 MW NREL wind turbine and compares that with existing studies on the aeroelastic
stability analysis in the literature. It should be noted that the optimized 5 MW wind
turbine can not be used for verification purpose, since there is no available stability data
on this machine and therefore the 5 MW NREL wind turbine is used instead.
After doing the verification of the developed code, the aeroelastic stability analysis of
the 20 MW wind turbine is carried out, since it is a more flexible design comparing to the
5 and 10 MW wind turbines. If the 20 MW wind turbine shows instability problems then
also the smaller designs (5 and 10 MW) are checked against stability criterion. Otherwise
it can be assumed that they are safe designs from an aeroelastic stability point of view.
7.2
The finite element model of the wind turbine
The finite element model of the wind turbine consists of four different submodels. These
are: the tower, the nacelle, the hub and the blades. The developed parametric model is
used to generate the geometry (and mesh) of the turbine.
Both the blade and tower are modeled using the Quad4 elements. For the tower, the
boundary condition is imposed by fixing all the nodes at the tower base. This means that
all the DOFs of the bottom nodes are zero. For the connection between the blades, hub,
111
112
VERIFICATION AND AEROELASTIC STABILITY ANALYSIS OF THE 20 MW WIND TURBINE
7.3
nacelle and tower top, the DOF of all the common nodes is set to equal. This results in a
rigid connection between these components.
For the purpose of simplicity, the nacelle is modeled with an equivalent center of
gravity contributed from all components. Similarly, the hub is modeled as a disk with its
center of gravity located in the geometrical center of the hub. Figure 7.1 illustrates the
modeling approach of the nacelle and hub of the 20 MW wind turbine.
Nacelle
3.8 m
Yaw axis
3.5 m
Shaft axis
10.0 m
g
161.9
m
Hub
Tower top
Tower bottom
Figure 7.1: Nacelle and hub center of mass for the 20 MW wind turbine
7.3
Verification of the stability analysis method
The 5 MW NREL wind turbine is the model to verify the developed aeroelastic stability
method due to the availability of a similar study in the literature on this machine, which
makes the comparison and verification possible, Meng et al. (2008). Before using the
method to find the instabilities, a few verification cases are useful to gain confidence in
this computational tool. This is done by three different case studies as follow:
• 2D analytical stability analysis: To have a feeling about the magnitude of instabilities and get some physical insight of the problem, a 2D analytical flutter analysis
of a section of the 5 MW blade is first carried out. This can give an estimation of
the instability boundary as well as the expected overall behavior of the blade at
different operational points.
• Undeformed state eigenvalue analysis of the structural model: The eigenvalues
of the structural FE model (including the rotational effect) are computed without
Figure 1: Global matrices assembling diagram
7.3
VERIFICATION OF THE STABILITY ANALYSIS METHOD
113
any aerodynamic loads (undeformed state). Then, the results are compared with
an eigenvalue analysis of a rotating blade that is carried out using the FAST aeroelastic code to make sure that the structural formulation of the problem is correct
and has an acceptable accuracy.
• Full aeroelastic stability analysis: The developed structural model is coupled
with two different aerodynamic models. The first model is the original Theodorsen
function with lift and moment terms and the second model is a more advanced
formulation of the first model that includes the drag term as well. The results of
this step can demonstrate up to a great extent the accuracy of the proposed method.
Therefore, a comparison is made with a similar study to say the final words about
its accuracy.
These three different case studies are explained in detail in the following subsections.
7.3.1
2D analytical stability analysis
Today, the majority of the aeroelastic stability analysis methods are complex and involve
the detailed treatment of the stability problem. Although these analyses will probably
yield accurate results, they are cumbersome to implement and the physical understanding
of the stability problem is easily lost.
To overcome this short coming (specially in the first stages of the design), it is a common practice to use a 2D analytical stability analysis, (see for instance Chaviaropoulos
et al. (2003)). In this way more physical insight to the problem can be achieved and
the designer can have a feeling about the order of magnitude of the unstable operational
points to expect.
For the same purpose, the goal of developing this analytical model here is to find the
instabilities and compare the order of their magnitudes with the full aeroelastic model.
This model is 2D and it uses the blade structural properties at the 75% radial position.
The model has two DOFs, h for the plunge motion (corresponding to bending in flapwise
direction), and α for the twist. For the aerodynamic loads, the Theodorsen unsteady
aerodynamic model is used. Figure 7.2 shows this typical section.
For this configuration, the equations of motion can be obtained by applying Newton’s
second law as:
mḧ + Sα α̈ + L(t) + Kh h = 0
(7.1)
Where:
Sα : the static moment related to EA and defined as mx α b
m: mass per unit length
Kh : flapwise stiffness
Similarly, for the moments around elastic axis we have:
Sα ḧ + Iα α̈ + Kα α − M (t) = 0
(7.2)
Here, Iα is the mass moment of inertia around EA and Kα is the torsional stiffness.
Also, L(t) and M (t) are the Theodorsen unsteady aerodynamic lift and moment, as
presented before in chapter 6 by equation 6.14 and 6.15, respectively.
114
VERIFICATION AND AEROELASTIC STABILITY ANALYSIS OF THE 20 MW WIND TURBINE
d1
L
7.3
h
Kh
M
AC
Kα
EA
α
CG
xα b
ab
b
b
Figure 7.2: A 2D aeroelastic model of the blade
Figure 1: Global matrices assembling diagram
These equations are rewritten in the matrix form as:
MẌ + DẊ + KX = 0
(7.3)
Where the vector X and the matrices are defined by:
X = [h α] t


∂C
∂C
2
l
 m + ρ b 2∂ α
M =

∂C
Sα − ρa b3 2∂ αl
(7.4)
Sα − ρab3 2∂ αl
∂C
Iα − ρab3 ( 18 − a2 ) 2∂ αl



(7.5)
1


D=

∂C
ρU bC(k) ∂ αl
∂C
−ρU bd1 C(k) ∂ αl
∂ Cl
ρU b2 [1 +
2∂ α
C(k)(1 − 2a)]



”
—  (7.6)
∂ Cl
b
b
1
2
− ∂ α ρU b d1 [1 + C(k)(1 − 2a) 2U ] + 2U (a − 2 )

 Kh
K =

0

∂C
ρU 2 bC(k) ∂ αl
∂C
Kα − ρU 2 bd1 C(k) ∂ αl



(7.7)
7.3
VERIFICATION OF THE STABILITY ANALYSIS METHOD
115
Table 7.1: 5 MW sectional properties at 75% blade span
Sectional property
Value (Unit)
Chord
Mass
Mass moment of inertia
Flapwise stiffness
Torsional stiffness
Radial position
2.7 (m)
138.9 (kg/m)
27.7 (kg · m2 )
2.7E3 (N · m2 )
3.5E5 (N · m/r ad)
47.7 (m)
To find a solution for equation 7.3, it is assumed that the 2 DOF response of the
section, X = [h α] t , is of the form X = x̂ e pt (simple harmonic motion) with p defined
as p = (σ + iω). The real component of p corresponds to the growth rate of the motion
and its imaginary part corresponds to the frequency of oscillation.
Replacing this harmonic solution in equation 7.3 leads to the characteristic equation
that in this case can only be solved numerically. For non-trivial X the determinant of the
coefficient matrix must be zero at instabilities. Using the values presented in table 7.1,
the instabilities are found numerically and presented in figure 7.3 for the 5 MW blade
section.
The unstable rotational speeds are highly dependent on the reduced frequency value.
Although, this means that the model is sensitive to the input parameters, it can give
reliable estimates of the magnitude of instability as well as the overall behavior of the
blade at different operational points.
As expected, instabilities occur in all operational conditions from the cut-in to cut-out
Analytical model instability boundary
Rotor rotational speed
30
Rotational speed (RPM)
27.5
25
22.5
20
17.5
15
12.5
10
7.5
5
4
6.1
8.2
10.3
12.4
14.5
16.6
18.7
20.8
Wind speed (m/s)
Figure 7.3: Instabilities predicted by the 2D analytical model
22.9
25
116
VERIFICATION AND AEROELASTIC STABILITY ANALYSIS OF THE 20 MW WIND TURBINE
7.3
at low reduced frequency values. The above analysis, apart from qualitatively allowing
the designer to investigate the effect of changing the value of parameters, also enables
him to quantify instabilities and hence the definition of a suitable comparison case for
verification purposes.
7.3.2
Undeformed state eigenvalue analysis of the structural model
In this subsection a verification study is carried out to check the modeling of the structure
and investigate the influence of centrifugal forces in a rotating blade. It is also meant to
verify the correct implementation of the method as a computer code. Therefore, only the
structural model of the rotor is considered and the eigenfrequencies are compared with
a modal analysis study in FAST aeroelastic code.
In the first verification case the rotor speed is set at zero RP M , without any aerodynamic loads. Thus, the modal analysis computes the eigenfrequency of this rotor without
any centrifugal force and the results are compared with FAST on the same basis. In the
second verification case, this approach is repeated with a spinning rotor that rotates at
its rated value (again no aerodynamic loads). The results of these two cases are shown
in table 7.2 and 7.3, respectively.
Table 7.2: Comparison of the 5 MW FE natural frequencies with FAST at zero RP M
Frequency
First blade flapwise
Second blade flapwise
First blade edgewise pitch
FAST (Hz)
FE (Hz)
Deviation (%)
0.6946
2.0175
1.0886
0.6880
2.0521
1.1181
- 0.95
+ 1.71
+ 2.70
Table 7.3: Comparison of the 5 MW FE natural frequencies with FAST at 12.1 RP M
Frequency
First blade flapwise
Second blade flapwise
First blade edgewise pitch
FAST (Hz)
FE (Hz)
Deviation (%)
0.7470
2.0767
1.1158
0.7389
2.1292
1.1448
- 1.08
+ 2.52
+ 2.59
As table 7.2 and 7.3 show, the absolute difference in predicting the first three eigenfrequencies when compared to FAST aeroelastic code is less than 2.7%. Small differences
are due to different ways of modeling the mass and stifness distribution along the blade.
In FAST, both properties are defined as inputs at some discreet stations along the blade,
whereas in the FE model of this work, the ticknesses and external gemotery of the blade
are defined as inputs and mass and stifness distributions are calculated from the element
properties of the continuous blade by the FE code.
7.3.3
Full aeroelastic stability analysis
As a further investigation, the aeroelastic stability computational tool is compared with
another stability study on the 5 MW NREL wind turbine carried out by Meng et al. (2008).
7.3
VERIFICATION OF THE STABILITY ANALYSIS METHOD
117
Comparison with the result of this work is used to demonstrated the accuracy of the
method.
Meng presented a method for aeroelastic modeling of a wind turbine based on time
domain multibody dynamic approach that includes the rotor, nacelle and tower. However, time domain simulation of instabilities gives no information about the blade modal
interactions or system frequencies. Therefore, he applied system identification on the
time domain model to find the frequency that the flap and torsion mode merge together
and cause instability.
At his study only the instabilities at 10 m/s wind speed are considered and the rotational speed of the rotor gradually speeds up from 12 to 19 RP M . The distinguished
feature of flutter is the increasing amplitude of both torsion and flap motions to a point
where in practical terms failure of the blade occurs. Thus, he found that under this
assumption, the flapwise and torsional mode of the 5 MW blade merge together at 19
RP M .
Figure 7.4 shows the computed instabilities for the 5 MW NREL wind turbine. For
the purpose of comparison, this figure also presents the 2D analytical solution and the
instability analysis prediction of Meng et al. (2008).
Without drag
With drag
Rotor rotational speed
Meng
25
Rotational speed (RPM)
23
21
19
17
Meng's prediction
15
13
11
9
7
5
4
6.1
8.2
10.3
12.4
14.5
16.6
18.7
20.8
22.9
25
Wind speed (m/s)
Figure 7.4: Dynamic instabilities for the 5 MW NREL wind turbine
The figure shows two type of solutions. They both use the Theodorsen unsteady
aerodynamic model but one with and the other without the drag term. As it can be seen
from the figure, the aerodynamic model with the drag term predicts higher rotational
speed at which the instabilities occur. This is due to the dissipation of energy from the
system by the drag term that means an increase of the system’s capacity to carry more
energy from the flow at higher rotational speeds, which at the end has an stabilizing
effect.
Another fact in this figure is the overestimation of the rotor speed at which the instabilities occur by the analytical model. Although, the accuracy of this method is ques-
118
VERIFICATION AND AEROELASTIC STABILITY ANALYSIS OF THE 20 MW WIND TURBINE
7.4
tionable, it predicts the physical variation of the system well.
Furthermore, at 10 m/s the instability predicted by the aerodynamic model without
the drag term shows a slightly higher value than Meng’s study. This difference is about
2%. Similarly, the aerodynamic model with both the lift and drag term predicts a higher
value of about 7.8%. However, the difference among these three different methods is
acceptable for a design study in the preliminary stage.
The safety margin of the 5 MW turbine against instabilities is another important
design consideration to address. As the graph shows at the rated wind speed this margin
is the minimum for the aerodynamic model without drag. Thus, using this model a safety
margin of 60.1% can be concluded, which means that this is a safe design.
Considering all the facts, a good agreement between the predicted instabilities based
on the work of Meng et al. (2008) and the developed method in this thesis exists.
7.4
Stability analysis of the 20 MW wind turbine
The previous section presented the verification of the method. Although instabilities
are not expected for the 20 MW wind turbine, an analysis is required to guarantee the
stability of this design. Therefore, this section presents the aeroelastic stability analysis
for the optimized 20 MW wind turbine presented in chapter 5.
The unstable operational points can be divided into static and dynamic, and therefore
also studied here in the same manner (for a definition of the static and dynamic operational points refer to section 6.7 of chapter 6). For both types of instabilities the two
aerodynamic models mentioned before are used.
• Static stability analysis
The static unstable operational points (divergence) of the 20 MW wind turbine are
plotted in figure 7.5. The plot shows two types of solutions. These two use the
Theodorsen unsteady aerodynamic model but one with and the other without the
drag term. As it can be seen the aerodynamic model with the drag term predicts
higher unstable speeds.
Instabilities predicted without the drag term show a lower value at each operational
point. The predicted rotational speeds are higher than the rated rotor speed of the
20 MW wind turbine, with a minimum of 46.0% as safety margin at the rated wind
speed and therefore static instabilities are not expected here.
• Dynamic stability analysis
Figure 7.6 shows the dynamic unstable operational points of the 20 MW wind
turbine. Similar to the static case, the plot presents two type of solutions. Theses
are the Theodorsen aerodynamic model with and without the drag term.
Since the aerodynamic model without the drag term predicts a more conservative
stability margin, it is used to obtain the safety margin. As figure 7.6 shows, at the
rated wind speed this margin is minimum and has a value of 20.4%, while this
value for the static case was higher with a value of 46.0%.
Based on the linear scaling laws presented in chapter 2, both the eigenfrequencies
and the rated rotor speed when upscaling linearly from 5 to 20 MW decrease by
50% (or R−1 with R = 2). This means that the safety margins against instabilities
7.4
STABILITY ANALYSIS OF THE 20 MW WIND TURBINE
Without drag
With drag
119
Rotor rotational speed
12
Rotational speed (RPM)
11.1
10.2
9.3
8.4
7.5
6.6
5.7
4.8
3.9
3
4
6.1
8.2
10.3
12.4
14.5
16.6
18.7
20.8
22.9
25
Wind velocity (m/s)
Figure 7.5: Static instabilities (divergence) for the 20 MW wind turbine
Without drag
With drag
Rotor rotational speed
11.1
Rotational speed (RPM)
10.2
9.3
8.4
7.5
6.6
5.7
4.8
3.9
3
4
6.1
8.2
10.3
12.4
14.5
16.6
18.7
20.8
22.9
25
Wind speed (m/s)
Figure 7.6: Dynamic instabilities (flutter) for the 20 MW wind turbine
remains unchanged.
When looking at the dynamic safety margins of the 5 and 20 MW wind turbines,
one can see that this value decreases from 60.0% to 20.4%. This means that during
the upscaling process the design experiences more reduction in its flexibility (or
stiffness) than what linear scaling laws state.
Considering all the facts, the predicted rotational speeds for both the static and
120
VERIFICATION AND AEROELASTIC STABILITY ANALYSIS OF THE 20 MW WIND TURBINE
7.5
dynamic instabilities are higher than the designed rated rotor speed of the 20 MW
wind turbine and therefore no aeroelastic instabilities are expected for this machine.
7.5
Concluding remarks
The analysis developed here has shown to be useful for predicting instabilities of wind
turbines and it can be concluded that the developed aeroelastic stability method can be
used as a preliminary tool to find the unstable operational points of a given wind turbine
configuration with a reasonable accuracy.
Two observations has been made based on the results of the analysis as follow:
1. By comparing the static and dynamic instability boundary of the 5 and 20 MW
wind turbines, it can be concluded that the static instabilities do not occur before
the dynamics one.
2. Although, the 20 MW wind turbine does not show any instability, the instability
criterion should be seen as a potential design drivers for larger scale wind turbines
and thus the safety margin should be included as a design constraint in the design
optimization problem formulation.
It also should be noted that the analysis is potentially weak in the following aspects:
1. It does not provide any data about the mode shapes at the instability point.
2. The huge amount of DOFs of the system makes it difficult to say which modes at
the instability point are vibrating.
3. The dependency of the aerodynamic model to the reduced frequency makes the
method cumbersome to find, by trial and error, the instabilities.
4. The method does not provide any information about the damping value at the
instability point.
CHAPTER
8
Scaling trends for future offshore wind turbines
8.1
Introduction
The development of the 5, 10 and 20 MW optimized wind turbines was carried out in the
previous chapters. Based on these wind turbines, this chapter constructs scaling trends
to study the influence of size on future offshore wind turbine designs and discusses the
technical and economical barriers associated with them. The identified barriers can help
to formulate the required research and technological changes affecting the development
of very large wind turbines.
Loading-diameter, mass-diameter and cost-diameter as the most important trends to
monitor and examine the scaling behavior are developed and discussed. However, two
other useful trends are addressed as well. Whenever suitable, a comparison is made with
the linear scaling laws and existing data trends presented before in chapter 2 to explain
the scaling phenomenon and the underlying reason for any similarity or discrepancy.
8.2
Development of scaling trends
To evaluate whether large wind turbines are feasible to manufacture and may eventually
be more cost effective, it is needed to analyze how size influences the design. This
can be done by studying the loading, mass, cost and some other useful trends. For the
loading trends, the trend exponent shows how the dynamic behavior of the wind turbine
components (such as blade, tower and low speed shaft) changes with size. For the mass
and cost trends, the trend exponent shows how upscaling impacts every individual mass
and cost element as the size increases. Additionally, by comparing the share of every
mass and cost elements to the total share, it can be seen how upscaling influences the
cost or mass share at different scales.
All the trends are formulated as a function of rotor diameter. Rotor diameter is the
121
122
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
8.2
driver for loads on the wind turbine and it can properly reflect the dependency of the
trends to size. Additionally, the rotor diameter is driving many of the technological innovations and can be used as a key indicator to identify the required technological developments. These trends are constructed using a power curve fit to the data points with
rotor diameter as the independent parameter, i.e: aR b with a as the curve coefficient, R
as the rotor diameter and b as the curve exponent. The reason to use a power curve is
the ease of comparison with the square-cube law as formulated by linear scaling law.
As explained before, no other scaling study should be justified without an underlying physical understanding. This understanding can be achieved by constructing scaling
trends using a power curve and making a comparison on the same basis with linear scaling law (as well as existing data trends that are often presented in this form for the same
reason).
These scaling trends are presented in detail in the following subsections. For this
purpose the optimized 5, 10 and 20 MW wind turbines technical characteristics are used
as the data points.
8.2.1
Loading-diameter trends
This subsection presents the loading-diameter trends of the blade, low speed shaft and
tower as the main load carrying components of a wind turbine, based on the developed
5, 10 and 20 MW machines of the previous chapters. These trends are compared with
the linear scaling laws and existing data trends of chapter 2 to examine and analyze
the scaling behavior of larger wind turbines. In all the figures, the existing data trends
are extrapolated to enable a visual comparison with the optimized data trends at larger
scales.
1. Blade
For the blades of the wind turbine the highest loads are expected at the blade root
where the connection is made with the hub. The linear scaling law predicts the
moments of the blade to scale with R3 for the flapwise, and R4 for the edgewise
component.
The flapwise moment is mainly aerodynamic driven and according to the existing
data trend the corresponding bending moment at the root increases with R2.86 , (figure 2.1 of chapter 2). For the edgewise moment both the gravity and aerodynamic
loads play a role and existing data trend shows that it scales with R3.25 (figure 2.2
of chapter 2).
Figure 8.1 and 8.2 show the extreme blade flapwise and edgewise loading trends
for the optimized wind turbines. For the purpose of comparing the existing data
trends with the optimized wind turbine scaling trends, figure 2.1 and 2.2 are replotted and extrapolated beyond 90 m.
As can be seen the bending moment scales with R2.62 and R3.41 for the flapwise and
edgewise components respectively. This means that the edgewise moment which
is mainly gravity driven increases with size more than the flapwise moment that is
aerodynamic driven. This is in line with linear scaling law that predicts the gravity
driven loads to scale more rapidly than the aerodynamic driven loads as presented
before in chapter 2. Table 8.1 summarizes the blade loads scaling behavior.
8.2
DEVELOPMENT OF SCALING TRENDS
Existing data trend
123
Optimized data trend
Blade flapwise moment (MN.m)
140
126
112
98
84
70
y = 0.0175x2.86
Q2 = 0.94
56
y = 0.0441x2.62
Q2 = 0.99
42
28
14
0
0
30
60
90
120
150
180
210
240
270
300
Rotor diameter (m)
Figure 8.1: Curve fit to the extreme flapwise bending moment at blade root for the wind turbines from
5 to 20 MW and comparison with the extrapolated existing data trends
Existing data trend
Optimized data trend
Blade edgewise moment (MN.m)
140
126
112
98
84
70
y = 0.0031x3.25
Q2 = 0.88
56
42
y = 0.0005x3.41
Q2 = 0.99
28
14
0
0
30
60
90
120
150
180
210
240
270
300
Rotor diameter (m)
Figure 8.2: Curve fit to the extreme edgewise bending moment at blade root for the wind turbines
from 5 to 20 MW and comparison with the extrapolated existing data trends
2. Low speed shaft
Shafts are mechanical components to transfer mainly torsional moments with a
circular cross section that is optimal for this application. However, the low speed
shaft of a wind turbine experiences bending and torsional moments at the same
124
8.2
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
Table 8.1: Comparison of the scaling trends for the blade’s loads
Loads component
Linear scaling law
Existing data
3
Flapwise
Edgewise
Optimized wind turbines
2.86
R
R4
R2.62
R3.41
R
R3.25
time. Bending on the low speed shaft is mainly gravity driven, caused by the rotor
mass, and it is obtained by multiplying the rotor mass with the unsupported length
of the shaft.
Low speed shaft bending moment (MN.m)
In this research, the length of the shaft is linearly scaled with size, but the mass
of the rotor is an optimal mass obtained through optimization. According to the
linear scaling law the bending moment due to the rotor weight scales with R4 .
120
108
96
y = 0.004x3.01
Q2 = 0.99
84
72
60
48
36
24
12
0
120
138
156
174
192
210
228
246
264
282
300
Rotor diameter (m)
Figure 8.3: Curve fit to the extreme low speed shaft bending moment for the wind turbines from 5 to
20 MW
Figure 8.3 shows the bending moment trend for the optimized wind turbines as a
function of rotor diameter. As can be seen the bending moment scales with R3.01
for the optimized wind turbines. Unfortunately, there is no study available to show
how the low speed shaft loads scale with size in practice. Therefore, this result can
only be compared with the linear scaling law, which predicts a size escalation of
R4 .
For the shaft torsional moment, linear scaling law predicts the moment to scale
with R3 . Figure 8.4 shows how the shaft torsional moment scales with size for the
optimized wind turbines. Here, the torsional moment scales with R3.03 , and a good
match can be seen here with linear scaling law.
8.2
DEVELOPMENT OF SCALING TRENDS
125
Low speed shaft torsional moment
(MN.m)
120
108
96
y = 0.0037x3.03
Q2 = 0.99
84
72
60
48
36
24
12
0
120
138
156
174
192
210
228
246
264
282
300
Rotor diameter (m)
Figure 8.4: Curve fit to the extreme low speed shaft torsional moment for the wind turbines from 5 to
20 MW
3. Tower
In this research, all the loading-diameter trends are developed at the tower base
where the moments are at their maximum. Figure 8.5 shows the tower fore-aft
bending moment based on the optimized wind turbines data, which scales with
R2.92 .
When compared with linear scaling law that predicts an increase in the moments
by R3 , a good match can be seen. However, by looking to the existing data trend
with an escalation relation of R2.33 , the bending moment of the optimized wind
turbines shows considerably larger increase as the turbines upscale.
However the scatter in the existing data is something to consider in this comparison. The quality of the curve fit to a data set can be measured with the Q-squared
value, which in this case is 0.71. The Q-squared value is a statistical coefficient that
shows the observer how effective the rotor diameter is at forecasting the fore-aft
bending moment for a wide range of rotor diameters, with a value of one to show
an excellent correlation and a value of zero for no correlation.
Thus, the accuracy of this correlation is questionable with such a low value Qsquared, and comparison of the optimized fore-aft moment with such a high data
scatter may not be that appropriate.
It is also known from chapter 2 that the extreme tower fore-aft bending moment is
mainly aerodynamic driven, and the "square" part of the linear scaling law works
very well for the prediction of aerodynamic loads and performance as discussed
before. Therefore, the good match between the linear scaling law and the data fit
to the bending moment of the optimized wind turbines is more expected than the
mismatch with the existing data trend considering such a poor Q-squared value.
126
8.2
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
Existing data trend
Optimized data trend
Tower fore-aft moment (MN.m)
900
810
720
630
540
450
y = 1.7891x2.33
Q2 = 0.71
360
270
y = 0.0523x2.92
Q2 = 0.99
180
90
0
0
30
60
90
120
150
180
210
240
270
300
Rotor diameter (m)
Figure 8.5: Curve fit to the extreme tower fore-aft bending moment at the base for the wind turbines
from 5 to 20 MW and comparison with the extrapolated existing data trends
For the side-to-side moments, figure 8.6 shows that the bending moment of the
optimized wind turbines scales with R4.19 . This is far above what linear scaling law
and existing data trend predict.
Existing data trend
Optimized data trend
Tower side-side moment (MN.m)
1000
900
y = 4E-05x4.19
Q2 = 0.98
800
700
600
500
400
300
y = 0.0348x3.23
Q2 = 0.74
200
100
0
0
30
60
90
120
150
180
210
240
270
300
Rotor diameter (m)
Figure 8.6: Curve fit to the extreme tower side-to-side bending moment at the base for the wind
turbines from 5 to 20 MW and comparison with the extrapolated existing data trends
8.2
DEVELOPMENT OF SCALING TRENDS
127
To explain that, first the loads that cause the side-to-side bending moment need to
be identified. There are two main loading mechanisms that cause the side-to-side
bending moment. First the asymmetric aerodynamic loads in the plane of rotation
and second the mass imbalance at the tower top. While the former one scales with
R3 , the later scales with R4 based on linear scaling law.
However, these scaling relations are only valid from a stationary point of view.
Therefore, when doing a time domain simulation, the flexibility of the tower causes
a dynamic amplification factor of the loads and this is well recognizable in the sideto-side motion.
Figure 8.7 shows the tower top fore-aft and side-to-side displacement trends. Based
on this figure the tower top displacement in the fore-aft direction decreases with
size, and for the side-to-side motion it shows an increase.
Tower top side-side trend
Tower top fore-aft trend
0.4
Tower top displacment (m)
0.37
0.34
0.31
0.28
y = 0.7583x-0.16
Q2 = 0.91
0.25
0.22
y = 0.001x1.02
Q2 = 0.99
0.19
0.16
0.13
0.1
100
120
140
160
180
200
220
240
260
280
300
Rotor diameter (m)
Figure 8.7: Tower top fore-aft and side-to-side deformation trend for the wind turbines from 5 to 20
MW
The trend exponent for the side-to-side moment increases more rapidly than the
fore-aft, and can become a design challenge as the side-to-side moment will have
almost the same magnitude as the fore-aft but being badly damped. This is an
area to further investigate and to come up with new solutions to prevent such a
phenomenon.
Another important load component at the tower base is the torsional moment. This
is depicted in figure 8.8. Curve fit to the torsional moment of the optimized wind
turbines shows a R3.04 relation. While this is clearly a good match with the linear
scaling law with an R3 relation, it does not match with the existing data trend that
has an exponent of 4.05.
As mentioned by Jamieson (2007), the reason for such a relatively high trend ex-
128
8.2
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
Existing data trend
Optimized data trend
Tower torsional moment (MN.m)
120
108
y = 0.0034x3.04
Q2 = 0.99
96
84
72
60
48
y = 8E-05x4.05
Q2 = 0.76
36
24
12
0
0
30
60
90
120
150
180
210
240
270
300
Rotor diameter (m)
Figure 8.8: Curve fit to the extreme tower torsional moment at the base for the wind turbines from 5
to 20 MW and comparison with the extrapolated existing data trends
ponent is the imbalance of the yaw torque, which might be the effect of turbulence
in creating differential loading across the rotor plane and as the size increases may
become more severe. However, such a phenomenon is not visible for the larger
wind turbines based on the results of this research.
Introduction of a yaw error in the simulation is another mechanism that can result
in a higher torsional moment, and as the size increases it can become more severe.
As requested by the IEC 61400-3 standard, wind turbines need to be simulated
with the addition of ±8 ◦ yaw error for the certification purpose. Therefore, the
optimied wind turbines are simulated again with the addition of ±8 ◦ yaw error.
For this non-aligned case, the result is presented in figure 8.9. When comparing
the curve exponent of 3.10 for the non-aligned case with the aligned case with
an exponent of 3.04, it is clear that the introduction of a yaw error can also not
contribute too much in the torsional moment increase at the tower base and be the
reason for a high trend exponent in the existing data as the turbines upscale.
All in all, the reason to have such a high trend exponent in the existing data is
not clear. However, similar to the tower base fore-aft and side-to-side moments,
the Q-squared value of the torsional moment shows a poor correlation with rotor
diameter with a value of 0.76, therefore, the quality of the presented curve is also
questionable.
8.2.2
Mass-diameter trends
In the previous subsection, the loading-diameter trends of the optimized wind turbines
were presented and analyzed. This subsection develops the mass-diameter trends and
8.2
DEVELOPMENT OF SCALING TRENDS
129
130
Tower torsional moment (MN.m)
117
104
y = 0.0028x3.10
Q2 = 0.98
91
78
65
52
39
26
13
0
120
138
156
174
192
210
228
246
264
282
300
Rotor diameter (m)
Figure 8.9: Curve fit to the extreme tower torsional moment at the base for the wind turbines from 5
to 20 MW with a yaw error
similar to the previous subsection compares that with the linear scaling law and existing
data trend.
1. Blade
The mass of the optimized wind turbine blades is depicted in figure 8.10. As presented by the graph, the mass increase shows a trend exponent of 2.64. Although, this
is lower than the linear scaling law exponent of 3, it is higher than the existing data
with a trend exponent of 2.09.
This shows that upscaling of the existing wind turbines using the same materials,
concept, topology and knowledge to larger scales influences the mass trend exponent in a negative way. Therefore, several modifications and improvements are
needed to keep the trend exponent of the upscaled wind turbines in the same order
(or even lower) as the existing wind turbines to mitigate this negative influence.
2. Tower top mass
As mentioned before, the tower top mass refers to all the components atop the
tower including the rotor. Table 8.2 lists these components with their associated
mass for the 5, 10 and 20 MW optimized wind turbines. In this table, except the
mass of the blade and tower that are optimized for the given size, all the other
masses are obtained using the mass models presented in appendix A. Depending
on the component, these mass models are a function of the blade mass, length or
turbine power output to reflect the size dependency.
According to linear scaling law, the tower top mass increases with R3 . Figure 8.11
shows the tower top mass trend of the optimized wind turbines and existing data.
For the optimized wind turbines the tower top mass increases with an exponent of
130
8.2
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
Existing data trend
Optimized data trend
200
180
y = 0.0571x2.64
Q2 = 0.99
Blade mass (ton)
160
140
120
100
80
60
y = 0.6853x2.09
Q2 = 0.96
40
20
0
0
30
60
90
120
150
180
210
240
270
300
Rotor diameter (m)
Figure 8.10: Curve fit to the single blade mass trend for the optimized wind turbines from 5 to 20
MW and comparison with the extrapolated existing data trends
2.41 and this value using the existing data is 2.39.
The reason for such a good match can be further analyzed by looking to the mass
breakdown of the tower top components of the 5, 10 and 20 MW wind turbines. As
figure 8.12 shows at 5 MW 23.7% of the tower top mass is directly optimized and
Table 8.2: Mass of all the components atop the optimized wind turbines
Component
3 Blades
Hub
Pitch system
Pitch bearing
Nose cone
Low speed shaft
Main bearing
Gearbox
High speed shaft and brake
Generator
Bed plate
Platform and railing
Hydraulic and cooling
Nacelle cover
Yaw system
Tower top mass
5 MW (kg)
10 MW (kg)
20 MW (kg)
68553
27480
12997
9369
1884
18087
3014
36061
990
16646
64552
8069
398
6129
14587
159045
56257
28559
21088
2888
49515
10274
84161
1981
31546
127550
15944
796
11873
46330
548250
180028
95476
71492
4775
176730
48251
17377
3962
59783
301560
37695
1593
23361
199510
Σ =288819
Σ =647808
Σ =1926269
8.2
DEVELOPMENT OF SCALING TRENDS
Existing data trend
131
Optimized data trend
2000
1800
y = 2.304x2.41
Q2 = 0.99
Tower top mass (ton)
1600
1400
1200
1000
800
600
y = 2.348x2.39
Q2 = 0.71
400
200
0
0
30
60
90
120
150
180
210
240
270
300
Rotor diameter (m)
Figure 8.11: Curve fit to the tower top mass of the optimized wind turbines from 5 to 20 MW and
and comparison with the extrapolated existing data trends
the remainig 76.3% is obtained through the mass models. Similar share between
the optimized and modeled mass can be seen for the 10 and 20 MW wind turbines.
Due to the calibration of these mass models by industry data, they have the same
predictive value as the existing data trend. Since the modeled mass has the main
contribution to the tower top mass, it governs the trend behavior and this results
in close match with the existing data trend.
3. Tower
The mass-diameter trend for the optimized towers is shown in figure 8.13. The data
trend represents an exponent of 3.22 that is higher than both the linear scaling law
and existing data trend. However, one should consider the low Q-squared value in
the existing data trend, which should be used for the comparison carefully.
Additionally, in the optimization formulation of this research, the tower diameter
and thickness are optimized only at the tower base and top, resulting in a linear
variation of the diameter and thickness along the tower. This could be improved
by adding more sections along the tower to be optimized and probably helping to
get a lower tower mass as the tower scales and improving the trend exponent.
Having all the mass scaling trends in place, one can also look at the mass trend
exponents and the mass breakdown of different components and the way they scale with
size.
• Mass trend exponents
The mass breakdown is presented in figure 8.14 for the optimized 5, 10 and 20
MW wind turbines. Based on this figure, the mass trend exponent is presented in
table 8.3. As the table shows the mass of the main bearing has the highest trend
132
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
8.2
Optimized part of the
tower top mass
23.7%
Modeled part of the
tower top mass
76.3%
(a) 5 MW
Optimized part of the
tower top mass
24.6%
Modeled part of the
tower top mass
75.4%
(b) 10 MW
Optimized part of the
tower top mass
28.5%
Modeled part of the
tower top mass
71.5%
(c) 20 MW
Figure 8.12: Mass breakdown of the tower top components of the optimized 5, 10 and 20 MW wind
turbines to show the share between the modeled and optimized part
8.2
DEVELOPMENT OF SCALING TRENDS
Existing data trend
133
Optimized data trend
5500
4950
y = 0.0609x3.22
Q2 = 0.99
Tower mass (ton)
4400
3850
3300
2750
2200
1650
y = 0.653x2.78
Q2 = 0.74
1100
550
0
0
30
60
90
120
150
180
210
240
270
300
Rotor diameter (m)
Figure 8.13: Curve fit to the tower mass of the optimized wind turbines from 5 to 20 MW and
comparison with the extrapolated existing data trends
exponent followed by the yaw system, tower, low speed shaft and blade. From
a design point of view, the mass of these components has a direct link with the
mass of the blade. The higher the mass of the blade, the higher the mass of other
components to support it.
In this case a higher trend exponent of the main bearing, yaw system, tower and
low speed shaft may not imply that these components need to undergo a technological improvement. It is mainly the mass increase of the blade that is causing
the mass increase of the other components. This necessitates considering alternative light weight materials for the blade when aiming at larger scales. This issue
is better addressed by looking to the mass breakdown of different components at
different scales as follows next.
r
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3704
4167
4630
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W
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134
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
8.2
Figure 8.14: Mass breakdown of the wind turbine components of the optimized 5, 10 and 20 MW
wind turbines
Mass (ton)
8.2
DEVELOPMENT OF SCALING TRENDS
135
Table 8.3: Mass trend exponent of all the components of a wind turbine as it scales from 5 to 20 MW
in the power output
Component
Main bearing
Yaw system
Tower
Low speed shaft
Blade
Pitch bearing
Pitch system
Hub
Trend exponent
3.52
3.32
3.22
2.89
2.65
2.59
2.54
2.39
Component
Gearbox
Bedplate
Platform and railing
HSS and brake
Hydraulic and cooling
Nacelle cover
Generator
Nose cone
Trend exponent
1.98
1.95
1.95
1.75
1.75
1.69
1.61
1.18
• Mass breakdown at different scales
The mass breakdown shows how upscaling influences the design at different scales.
The results are presented in two separate form for the 5, 10 and 20 MW wind turbines. First, only the mass breakdown of the tower top components is presented
(excluding tower) and second, the mass breakdown of all the wind turbine components (including tower) is presented.
For the 5, 10 and 20 MW wind turbines, the mass breakdown of the tower top
components is presented in figure 8.15. This graph has two important aspects to
discuss. First, upscaling influences equally the tower top components and keeps the
mass share the same with blade as the heaviest component followed by bedplate,
gearbox and hub.
Second, the mass percentage of the blade increases from 23.7% at 5 MW to 28.5%
at 20 MW showing that the blades of the large scale wind turbines will become
progressively heavier. Refering to the results of this research, one can see that the
blade-tower clearence and edgwise loads are the dominant design drivers at larger
scales and keeping them satisfied most probably causes this increase.
Similarly, the mass breakdown of the 5, 10 and 20 MW wind turbine components
is presented in figure 8.16. As the figure shows, the tower mass increases from
55.4% at 5 MW to 70.6% at 20 MW. Taking into account that the tower function is
to supply the tower top mass at the requiered height and withstand the loads, then
it can be argued that the mass increase of the tower top components and higher
loads are the two direct reasons for such an increase in the tower mass share.
136
8.2
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
6.3%
5.8%
9.5%
5.1%
4.5%
3.2%
12.5%
2.8%
2.1%
1.0%
0.7%
0.3%
0.1%
22.4%
23.7%
(a) 5 MW
7.6%
4.9%
7.2%
8.7%
4.4%
3.3%
2.5%
13.0%
1.8%
1.6%
0.4%
0.3%
0.1%
19.7%
24.6%
(b) 10 MW
3.1%
9.2%
10.4%
5.0%
9.3%
3.7%
2.0%
1.2%
9.0%
2.5%
0.2%
0.2%
0.1%
15.7%
28.5%
(c) 20 MW
Blades
Low speed shaft
Pitch bearing
Nose cone
Bedplate
Generator
Platform and railing
HSS and brake
Gearbox
Yaw system
Nacelle cover
Hydraulic and cooling
Hub
Pitch system
Main bearing
Figure 8.15: Mass breakdown of the tower top components of the optimized 5, 10 and 20 MW wind
turbines
6.3%
9.5%
5.8%
5.1%
4.5%
3.2%
12.5%
2.8%
8.2
DEVELOPMENT OF SCALING TRENDS
10.0%
5.6%
137
4.2%
2.8%
10.6%
2.6%
2.3%
2.0%
1.4%
1.2%
0.9%
0.5%
0.3%
0.2%
0.1%
55.4%
(a) 5 MW
7.8%
6.2%
4.1%
2.8%
2.4%
1.5%
2.3%
1.4%
1.0%
0.8%
0.6%
0.5%
0.1%
0.1%
0.0%
68.3%
(b) 10 MW
4.6%
8.4%
2.7%
2.7%
2.7%
0.9%
3.0%
1.5%
1.1%
0.6%
0.4%
0.7%
0.1%
0.1%
0.0%
70.6%
(c) 20 MW
Tower
Hub
Pitch system
Main bearing
Blades
Low speed shaft
Pitch bearing
Nose cone
Bedplate
Generator
Platform and railing
HSS and brake
Gearbox
Yaw system
Nacelle cover
Hydraulic and cooling
Figure 8.16: Mass breakdown of all the components of the optimized 5, 10 and 20 MW wind turbines
5.6%
4.2%
10.0%
2.8%
2.6%
138
8.2.3
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
8.2
Cost-diameter trends
When making any technical improvement to the design of a wind turbine, it is necessary
to evaluate the impact of the design change on the system cost as well. The impact
assessment should not only be limited to the wind turbine itself, but also should consider
the overall impact on the cost elements when considered as an offshore wind farm such
as installation, and operation and maintenance.
As a wind turbine scales, the impact of the scale on all these cost elements and the
design constraints is not clear. While an increase in the blade length results in higher
annual energy production, it also increases the required space between the wind turbines to reduce the farm turbulence effect and this in consequence increases the farm
interconnection cabling cost.
If one positive effect does not balance out all the other negative ones, then the proposed improvement may have a negative overall impact. This necessitates having a merit
index that can measure the overall value of the design change. The LCOE is such a merit
index, and its formulation was explained in chapter 3.
Using LCOE as the objective function and taking all the wind farm cost elements into
account such as installation, offshore cabling, scour protection and decommissioning, this
subsection presents the cost-diameter trends to investigate the economy of larger scales
wind turbines and the impact of size on them.
Figure 8.17 shows the cost of all the components of the optimized wind turbines
together with its associated costs as a 50 MW wind farm with an array spacing of 7 by 7
rotor diameters. Except the cost of the blade and tower that are optimized for the given
size, all the other costs are obtained using the cost models presented in appendix A.
Cost (1000 USD)
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ve
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El
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0
1500
3000
4500
6000
7500
9000
10500
12000
5 MW
10 MW
20 MW
8.2
DEVELOPMENT OF SCALING TRENDS
Figure 8.17: Cost elements of the optimized wind turbines from 5 to 20 MW
139
140
8.2
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
Table 8.4: Cost trend exponent of all the components of a wind turbine as it upscales from 5 to 20
MW in the power output, the ranking is according to the trend exponent
Cost element
Tower
Yaw system
Low speed shaft
Blade
Pitch mechanism
Warranty
Marinization
Main bearing
Hub
Decommissioning
Gearbox
Bedplate
Platform and railing
O&M
Foundation system
Trend exponent
3.22
2.97
2.89
2.66
2.66
2.55
2.55
2.44
2.39
2.22
2.19
1.95
1.95
1.85
1.75
Cost element
Electrical interface
Installation
Scour protection
Generator
Power electronics
Electrical connection
Permits
Port and staging
Levelized replacement
Hydraulic and cooling
HSS and brake
Nacelle cover
Nose cone
Access equipment
Control and safety
Trend exponent
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.69
1.18
Fixed
Fixed
To get more out of this figure, a power trend is fitted to every individual cost element,
and the trend exponents are presented in table 8.4. As the table shows, from cost perspective the tower of larger scale wind turbines has the highest trend exponent followed
by yaw system, low speed shaft and blade.
However, this does not mean that the tower will be the most expensive component,
and it only shows that the tower cost will be highly influenced by the scaling when
compared to other cost elements.
In this table, there are some cost elements that have the same exponent. This is not
a coincidence, since these cost elements are all a function of the rated power output and
therefore have the same exponent. Personnel access equipment, and control and safety
devices are not scale dependent and therefore have a fixed cost at all different scales.
Figure 8.18 shows the cost breakdown of all different elements for the optimized 5,
10 amd 20 MW wind turbines. As the figure shows, the cost share of the blades and
tower increases with size. For the blades, this value is 7.7% at 5 MW and 10.7% at 20
MW meaning that the blades of large scale wind turbine increase in the cost share. For
the tower, the cost share increases from 7.0% at 5 MW to 15.6% at 20 MW.
As these results show, the blade and tower cost increase is in favor of a conceptual
change in the design to compensate this negative impact. As discussed before, load and
mass reduction is vital for large scale wind turbines. Therefore, a 2 bladed downwind
concept has the potential to overcome the problem.
Also figure 8.19 shows how the TCC and BOS change with size. As the figure presents,
at the 5 MW level the TCC and BOS are almost the same, however as the size increases
the TCC is increasing more rapidly than the BOS (almost twice at the 20 MW). Therefore
more emphasis should be put on the TCC when aiming at cost reduction for large scale
offshore wind.
8.2
Foundation system
Warranty
Generator
Permits
P &
i
Electerical interface
Installation
Power electronics
Low speed shaft
L
li d
l
Blades
Marinization
Decommissioning
Bedplate
Pl f
& ili
4.8%
5.2%
Tower
Gearbox TRENDS
DEVELOPMENT
OF SCALING
O&M
Scour protection
Electerical connection
Yaw system
H d li &
li
2.9% 2.9%
5.3%
5.8%
6.4%
Pitch mechanism
Hub
N
ll
2.8%
2.6%
2.2%
1.9%
1.6%
1.3%
1.3%
1.2%
1.1%
1.1%
0.7%
0.7%
0.6%
0.5%
0.5%
0.5%
7.0%
0.5%
0.1%
0.1%
15.8%
7.7%
15.0%
(a) 5 MW
4.2% 2.6% 2.5%
2.5%
2.6%
2.0%
5.9%
4.7%
2.1%
1.4%
1.6%
1.1%
6.5%
1.4%
1.0%
0.9%
6.6%
0.6%
0.6%
0.5%
0.5%
0.8%
0.2%
12.0%
13.9%
7.9%
0.2%
0.1%
0.1%
13.1%
(b) 10 MW
3.6%
6.8%
2.0% 2.0%
2.0%
3.7%
2.6%
1.5%
2.7%
7.5%
1.1%
2.2%
1.0%
2.1%
1.3%
6.2%
0.7%
0.5%
0.6%
0.4%
0.3%
1.4%
0.1%
0.1%
15.6%
10.9%
10.3%
10.7%
0.0%
0.1%
(c) 20 MW
Foundation system
Gearbox
O&M
Decommissioning
Low speed shaft
Port & staging
Nacelle cover
Nose cone
Electerical interface
Warranty
Scour protection
Electerical connection
Bedplate
Levelized replacement
Main bearing
HSS and brake
Blades
Installation
Generator
Pitch mechanism
Yaw system
Platform & railing
Access equipment
Tower
Marinization
Power electronics
Permits
Hub
Hydraulic & cooling
Control & safety
Figure 8.18: Cost breakdown of the optimized 5, 10 and 20 MW wind turbines
141
142
8.2
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
When upscaling a wind turbine the AEP increases, however at the same time the ICC
goes up. As mentioned before, if the positive effect of one change (increase in AEP) does
not balance out the negative impact of the other (increase in ICC) then the proposed
change has a negative overall impact by increasing the LCOE.
The ICC and AEP trends are depicted in figure 8.20 to show this tradeoff phenomenon.
As the figure shows, the AEP has a trend exponent of 1.29, which does not beat the ICC
trend exponent of 1.57 as the size increases. This negatively influences the LCOE as
depicted in figure 8.21 and increases that with a trend exponent of 0.22 at larger scales.
The increase of LCOE over size has also another hidden message. Table 8.5 shows the
increase of LCOE in percentages to decode this message. As the size is getting doubled
from 5 to 10 MW the LCOE increases by 8.6%, and when it is from 10 to 20 MW it
increases by 9.76%. Both scaling steps have the same scaling ratio, but the later shows a
higher increase in the LCOE. This means that the negative impact of upscaling is larger
when the scaling step is more beyond the existing knowledge and technology.
Table 8.5: LCOE percentage increase of a wind turbine as it upscales from 5 to 20 MW in the power
output
Size increase
Percentage (%)
5 to 10 MW
10 to 20 MW
5 to 20 MW
8.60
9.76
19.20
This necessitates to do upscaling with small sequential steps to mitigate the negative
economical impact of scale and benefit from technological developments in every step.
Therefore to make upscaling economically attractive, size should increase with a slower
rate, as also presented by Stiesdal at the conference ’The science of making torque from
Cost (Million USD)
50
45.618
40
23.009
30
15.489
20
10
6.072
11.540
TCC
10 MW
5.804
0
BOS
20 MW
5 MW
Figure 8.19: TCC and BOS of the optimized wind turbines from 5 to 20 MW
8.2
143
DEVELOPMENT OF SCALING TRENDS
AEP trend
ICC trend
80
120
72
56
y = 3E+07x1.29
Q2 = 0.96
100
80
48
40
60
32
1.57
y = 1E+07x
Q2 = 0.96
24
AEP (GWh)
ICC (Milion USD)
64
40
16
20
8
0
0
130
184
286
Rotor diameter (m)
Figure 8.20: ICC and AEP trends of the optimized wind turbines from 5 to 20 MW
wind’ in 2007. He showed that the growth rate of the wind turbines has reached a near
to flat level during the last decade when compared with 1980’s and 1990’s that is mainly
because of economical considerations rather than the technological limitations, and this
may not mean that there is a size limit or even an optimum is achieved.
Upscaling can continue if the negative impacts of every step can be mitigated by the
0.0714
0.07
y = 0.02x0.22
Q2 = 0.99
LCOE (USD/kWh)
0.0686
0.0672
0.0658
0.0644
0.063
0.0616
0.0602
0.0588
0.0574
100
120
140
160
180
200
220
240
260
280
300
Rotor diameter (m)
Figure 8.21: Curve fit to the LCOE of the optimized wind turbines from 5 to 20 MW
144
SCALING TRENDS FOR FUTURE OFFSHORE WIND TURBINES
8.2
use of new technologies in the design. If not so, then the already achieved size might be
considered as an optimum and economically attractive size. However, the new materials
and technologies of the future may also have the potential to improve the economy of
smaller scales (even better than larger scales) and down scaling the wind turbines may
become more economically attractive than upscaling.
8.2.4
Other useful trends
This subsection presents two other useful trends to explain the global behavior of a wind
turbine as it scales and gives the necessary background information when dealing with
size increase.
1. Blade tip-speed
The tip-speed is a key parameter in wind turbine design. It influences at the same
time the driving torque of the drive train, the slenderness of the blade and the
aerodynamic and mechanical noise emission from the rotor and nacelle.
In that respect a higher tip-speed reduces the driving torque of the drive train for
the same power output, which can be considered as a positive impact from loads
perspective. A higher tip-speed also causes the blades to become more slender, a
condition that has its limit due to the large deflections that a slender blade experiences and the extra material needed to avoid this increase. Similarly, it causes a
higher noise emission and as a rule of thumb, the noise emission of a wind turbine
increases with the fifth power of the relative wind speed.
For onshore wind turbines, the most important restriction to design for higher tipspeeds is the aerodynamic noise emission. Therefore, most of the onshore wind
turbines have a maximum of 80 ( ms ) as the tip-speed, a restriction that is set to
not cause annoyance to the close neighborhoods and inhabitants. However, for
the offshore wind turbines such a restriction does not exist and as a consequence
higher tip-speeds are expected.
In this research, the tip-speed is considered as a design constraint with an allowable
upper bound of 120 ( ms ). Table 8.6 shows the designed tip-speed of the optimized
wind turbines. These values are the result of a tradeoff study to meet all the design
constraints and minimize the LCOE. As the table shows, no evidence in the data
can be found to show that an increased or decreased tip-speed is beneficial as the
size increases.
Therefore, no consistent pattern related to size can be formulated for the tip-speed
and this is a global tradeoff study among many design variables, design constraints
and LCOE that determines its value without further interpretation or judgment
related to size. As the result, the 10 MW wind turbine has an optimized tip-speed
that is lower than the 5 and 20 MW wind turbines.
2. Rated power to tower head mass ratio
The rated power to tower head mass is a measure of scaling benefits and it allows to
distinguish light-weight and heavy designs when the same environmental condition
is used. Figure 8.22 shows the ratio for the optimized wind turbines. To see how
negatively upscaling influences the design, table 8.7 shows the relative reduction
8.2
DEVELOPMENT OF SCALING TRENDS
145
Table 8.6: Blade tip-speed of the optimized 5 to 20 MW wind turbines
Power output (MW)
Tip-speed ( ms )
5
10
20
88
82
98
of the ratio in percentage at every scaling step. When the size increase from 5 to
10 the ratio decreases by 10.8%, and 32.7% from 10 to 20 MW.
Power to head mass ratio (Watt/kg)
19
18
17
5 MW
10 MW
16
15
14
13
12
11
20 MW
10
9
100
120
140
160
180
200
220
240
260
280
300
Rotor diameter (m)
Figure 8.22: Curve fit to the power output to tower head mass ratio the of the optimized wind turbines
from 5 to 20 MW
This supports the previous argument that upscaling needs to be done in small steps
with the inclusion of new technological developments to mitigate the negative overall impact in every step. Upscaling using existing materials and design methods
will result in massive wind turbines, which necessities the use of several design
improvements to maintain the same power to tower mass ratio or even improve it.
Therefore, saving weight seems to be the only option for more cost effective large
wind turbines.
Table 8.7: Comparison of power output ratio for every scaling step
Scaling step
5 to 10
10 to 20
5 to 20
Ratio reduction (%)
10.8
32.7
40.0
CHAPTER
9
Conclusion and recommendation for future work
9.1
Introduction
The goal of this PhD research was to investigate the influence of upscaling on future offshore wind turbines by assessing their technical feasibility and economy. To achieve this
goal all the involved disciplines and their dynamic interactions, important components
and design constraints, cost elements, scaling trends and several other considerations
and details had to be considered and addressed.
To show the outcomes of this research, first a discussion on the methodology is made
relevant to upscaling and design. Also the application of the methodology in each context
is described. Then a general conclusion is presented to show the main findings of the
research. A detailed conclusion based on the answers to the research questions presented
in chapter 1 follows afterwards. Next, the contribution to the state of the art is presented
and finally the recommendation for future work is discussed.
9.2
Discussion on the methodology
A discussion on the methodology is presented to give an overview of its advantages,
disadvantages and application. Due to the wide scope of the methodology a distinction
is made between upscaling and design.
9.2.1
Upscaling context
The classical upscaling methods do not have the competence to deal with the detailed
research questions of this work as discussed in chapter 2. This is mainly because of
the limitation to provide size specific integral optimized wind turbines that are needed
to investigate the technical feasibility, economical characteristic and to construct reliable
147
148
CONCLUSION AND RECOMMENDATION FOR FUTURE WORK
9.2
scaling trends. The methodology of this research enables the upscaling study of optimized
wind turbines based on the same set of design assumptions and design concept for all
scales.
The same set of the design assumptions minimizes the scattering of the data points.
The same design concept guarantees the least inconsistency among different data points
that are used to develop scaling trends. In this way the scaling trends become more
confident and accurate for the concept under study. This in turn makes the end results
more reliable for investigating the influence of size on the design.
Additionally, the integrated methodology of this research enables the upscaling study
of any other concept as long as representative mass and cost models can be made available for. In this way one can make a trade off among different concepts to find the
suitability of each at different scales. As an example, it is possible to study a two-bladed
downwind machine at 5, 10 and 20 MW scales and compare its technical and economical
characteristics with the results of the 3-bladed upwind concept as presented in the previous chapters. This can show how another concept could possibly mitigate the negative
influence of upscaling on the present offshore wind turbine design. In a similar way a
direct drive machine can be studied and compared.
With the introduction of new technological developments and design improvements
there is a chance that a smaller scale machine does benefit more from it than a larger
machine. This means that from an economical point of view a smaller machine can
become more attractive, and it motivates downscaling the existing wind turbines. The
methodology of this research allows not only the upscaling but also the downscaling of
machines to be studied.
As an example, the LCOE of the upscaled wind turbines of this research shows an
increase immediately after the 5 MW. This might mean that downscaling the 5 MW wind
turbine may offer a better economy using the integrated methodology as a design improvement. Although not the focus of this research and therefore not considered, the
methodology taken here has the potential to study this phenomenon as well.
Despite several advantages and different applications of the present methodology, it
has some drawbacks as well:
• The increased complexity
The complexity of the developed methodology to deal with the upscaling problem makes it less attractive for quick size assessment studies. Formulation of the
upscaling study as a MDO problem, requires many challenging steps to be taken
before any results can become available. Therefore, an upscaling study based on
the methodology of this research can be a cumbersome and demanding task.
• Dealing with novel concepts
The dependency of the methodology on the mass and cost models makes it less
effective for dealing with novel concepts and technologies. This is due to the fact
that often such cost and mass models are not available or representative enough
for new technologies and concepts at the time of development. This is among
the reasons why the upscaling of the well established existing concept (3-bladed,
upwind, pitch regulated wind turbine) is studied in this research.
9.2
DISCUSSION ON THE METHODOLOGY
9.2.2
149
Design context
Typically, the design of a wind turbine takes place with isolated disciplines. In the isolated
methodology, the structural and aerodynamic design are carried out first. Here, the
role of the control engineer in the wind turbine design is mainly focused on designing
the controller for a given configuration defined by structural and aerodynamic design in
advance. However, the shift towards larger and flexible structures equipped with active
control devices operating in the entire working regime of the wind turbine is demanding
for innovations in design methodology.
The integrated aeroservoelastic design of this research is such an innovation. This
methodology changes the existing design problems from how to increase the power output or decrease the loads to how can aerodynamic, structure and active control be optimally combined to make a wind turbine that has the lowest LCOE, while satisfying all
the design constraints (thereby loosely called ’system-level design’).
This has been achieved by defining the LCOE as a common objective function among
all disciplines to be optimized, rather than optimizing the structure for minimum weight,
aerodynamics for maximum energy output or control for load alleviation and power regulation. By optimizing the LCOE, the integrated design guarantees the optimality of the
wind turbine as an aeroservoelastic system.
Additionally, instead of using the traditional methodology to design every component
of the wind turbine separately, the integrated methodology enables the concurrent design
of the components. In this research, blade and tower are designed at the same time and
this results in a reduced design time and lower LCOE. This also helps the designer to
better understand the overall impact (technical and economic) of every individual component on the design as a whole by doing a sensitivity analysis of the design constraints
and objective function with respect to the design variables.
In this way, a global definition of the objective function, design variables and design
constraints among all the disciplines and the concurrent design of the blades and tower
helps the optimizer to move towards better design solutions. As shown in chapter 4, the
use of the integrated methodology to design the 5 MW NREL wind turbine contributed
to 4.2% reduction in the LCOE. Therefore, the integrated design methodology is vital for
the development of future wind turbines that have to be more optimized than they are
today in order to be cost competitive.
Despite several advantages of the methodology, there are some drawbacks and issues
that require special attention:
• The enlarged design space
To make an aeroservoelastic design optimization happen, in addition to defining
the design variables, design constraints and objective function a huge amount of
additional interdisciplinary design parameters are also needed.
Although it is possible to introduce almost any design parameter as a design variable, it may become computationally very extensive. This can become worse when
the concurrent design of several components has to be considered. Therefore, several design parameters need to be engineered and defined by the designer.
Taking this fact into account wind turbine design still remains a cumbersome task
and the integrated methodology complicates further this task by enlarging the
design space. However, the integrated methodology has the capability of doing
150
CONCLUSION AND RECOMMENDATION FOR FUTURE WORK
9.3
a sensitivity analysis of any design parameter (or design variable) to the objective function and design constraints to show the overall impact of it on the design.
Based on this impact, the designer can judge which design variables or design parameters to eliminate to have a reasonable design space.
• Computational expense
Because of assessing the aeroservoelastic behavior of the wind turbine and the concurrent design of components the simulation and optimization time is expensive.
Therefore, to do an aeroservoelastic design optimization in a reasonable time many
advances such as parallel programmed computational codes are needed.
9.3
Conclusions
In this section an overall conclusion is presented first. Then detailed conclusions are
drawn up based on the formulated research questions that were presented in chapter 1.
9.3.1
Overall conclusions
The developed methodology of this research incorporating an integrated approach for
the simultaneous design of all disciplines and concurrent optimization of components
turned out to be very effective. Each building block of this integrated methodology helps
to address one important aspect; objective function to characterize the economy and optimizing the design, design constraints to identify the technical feasibility and finally the
design variables to realize size specific optimized wind turbines to draw scaling trends.
Therefore, it can be concluded that the approach of this research meets the essential requirements for a thorough analysis of upscaling of current wind turbines; a capability
that does not exist in the classical upscaling methods and is beyond their reach.
The results show the technical feasibility of current wind turbines when upscaled
from 5 to 20 MW. Larger wind turbines suffer mainly from the penalty of weight growth
and mass reduction becomes the main design concern to handle. From cost perspective,
upscaling of a three-bladed, upwind, geared and pitch regulated concept without the
inclusion of several advancements in its design is detrimental beyond the 5 MW range.
The future large offshore wind turbines (similar in concept to the concept of this research) can only be realized without cost increase provided some key innovations are
developed and integrated in the design. These innovations will come with extra initial
costs and they will pose new challenges. This necessitates to evaluate the overall impact
(technical and economical) of these innovations. For this purpose the use of the integrated design methodology (such as this research) is considered essential for the future
offshore wind turbines.
9.3.2
Conclusions based on research questions
How do the technical characteristics of the larger scales change with size and
can these technical characteristics appear as a barrier?
9.3
CONCLUSIONS
151
To assess the technical feasibility of wind turbines up to 20 MW, MDO was used to
do the preliminary design. For this assessment, all the important design constraints of
a wind turbines were considered. These included fatigue, stresses, natural frequencies,
blade-tower clearance and aeroelastic instability.
For both 10 and 20 MW wind turbines, blade-tower clearance and fatigue of the
tower act as active design constraints that are marginally satisfied. From a stress point
of view, both machines can sustain the extreme stresses. Also the natural frequencies
do not appear to pose any design challenge in the context of 1P and 3P frequencies, as
well as the wave spectrum peak frequency. From an aeroelastic stability perspective, the
safety margin against static and dynamic instability shows a considerable reduction for
the 20 MW wind turbine. However, the remaining safety margin is still enough to allow
the turbine to operate safely.
Additionally, based on the designed 5, 10 and 20 MW wind turbines, scaling trends
were constructed to show the behavior of wind turbines as they scale in size. These trends
were used to study the influence of size on important aspects of the wind turbines, and
they were all formulated as a function of rotor diameter to properly reflect the size dependency. Loading-diameter and mass-diameter trends were extracted to investigate the
scaling phenomenon. Blade and tower, as the most dynamic load carrying components,
were investigated with more attention.
For the blade, the loading-diameter trend shows that the edgewise moment increases
more rapidly than the flapwise moment. This is an indication that the gravity loads play
a more important role at larger scales than the aerodynamic loads. Also for the tower,
the side-to-side bending moment shows a higher growth rate at larger scales than the
fore-aft moment. This might pose some complications in the design of future offshore
wind turbines with a misaligned wind-wave loading which primarily can magnify the
side-to-side moment.
Referring to the mass-diameter trends, the design shows to be negatively influenced
by the penalty of weight growth that is caused by the mass of the tower top components.
From a design point of view, the mass of the rotor appears to be the main reason for
such an increase, since it caused the mass of other supporting components to increase
in a progressive way as well. The mass trend exponents reveal that main bearing, yaw
system and tower have the highest trend exponent. For the tower top components, the
mass share of the blades also showed an increase at larger scales.
All in all, the results of this research show the technical feasibility of larger offshore
wind turbines up to 20 MW range with no major technical barrier when referring to the
design constraints and scaling trends, although some minor issues yet need to be solved
as presented in the answer to the last research question.
How does the economy of the future offshore wind turbines change with size?
To evaluate the influence of size on the economy of larger future offshore wind turbines, LCOE was used as the merit index. Using this key index and taking all the wind
farm cost elements into account, the cost-diameter trends were used to investigate the
economy of larger scale wind turbines and study the impact of size on them.
At the component level, the tower of larger scale wind turbines has the highest trend
exponent (and cost share at 20 MW) followed by yaw system, low speed shaft and the
blades, meaning that the tower cost is influenced more by upscaling when compared to
152
CONCLUSION AND RECOMMENDATION FOR FUTURE WORK
9.3
other cost elements. This is in agreement with the conclusion on the mass increase of
the tower top components, taking into account that the tower supports this mass and
therefore gets negatively influenced by size.
The TCC and BOS have almost the same value at 5 MW, but as the size increased
to 20 MW, the TCC increases more rapidly than the BOS (almost twice at the 20 MW),
meaning an increased cost share of the TCC for the future offshore wind turbines.
Finally, the tradeoff between the ICC and AEP trends results in an increase of the
LCOE immediately after the 5 MW. In this case, the extra energy that was captured by
larger wind turbines could not balance out the associated ICC positively and this has
resulted in an increase of LCOE. Overall, the results show that the economy of future
offshore wind turbines is negatively influenced by upscaling with scaling trend exponents
that were against any size increment of the current wind turbines beyond 5 MW.
What are the considerations and required changes for future offshore wind turbines?
Based on the results of this research, following considerations for the future large
scale offshore wind turbine developments can be highlighted:
1. Mass increase of the blade
Referring to the design optimization results and making a root cause analysis of the
blade mass increase for the upscaled wind turbines, prevention to hit the tower and
resistance against the blade edgewise load are observed as the two main reasons
(the first two design drivers for the larger blades).
Extra material to provide sufficient stiffness in longer blades causes the mass of the
rotor to increase (also a progressive course toward mass increase of other components in the nacelle and tower). Future design solutions should solve the conflict
between high stiffness (to have the desired blade-tower clearance) at low mass for
the blade.
2. Mass and cost increase of the tower
The tower design is negatively influenced by upscaling. The two main reasons
causing such an increase of mass and thereby cost of the tower are higher loads and
mass increase of the tower top. Load alleviation and mass reduction are needed to
overcome this problem.
3. Thickness of the blade root
The optimized design variables of the 20 MW blade (see table 5.2 in chapter 5)
show that the blade root has a thickness of 20 cm. Although, it is technically
feasible to construct the blade with such a thickness using the existing materials
and manufacturing techniques, it becomes a complex engineering task to carry
out.
4. Size of the blade and tower
A limiting size can be found for the blade and tower when looking at the existing
land transportation possibilities. Assuming maximum limits of 6 m for diameter
and 80 m for length, then the land transportation limit can be set at around 9 MW
9.4
CONTRIBUTION TO THE STATE OF THE ART
153
for the blade. Using the same limit for the tower diameter, land transportation
becomes impractical for towers corresponding to wind turbines of 5 MW power
output.
5. Nacelle installation difficulties
As a result of upscaling, the weight, height and the volume of the future offshore
wind turbine increase. The increase in the scale of the turbines requires special
installation vessels in terms of carrying and lifting capacities specially relevant for
the tower top components. The 20 MW wind turbine had a hub height of 162 m
and a nacelle weight of more than 1000 tons.
Taking the mass of the nacelle and its installation height together into account, it
becomes apparent how big the challenge is to overcome. At this moment, there is
no lifting device in the world to lift and install such a huge mass atop the tower at
this height. This can become more severe when one considers the installation of
these machines in an offshore environment considering factors such as water depth
variation, seabed condition, wind speed, waves and currents as the disturbing elements causing crane instability.
9.4
Contribution to the state of the art
This PhD research contributes to the state of the art knowledge and presents originality
in three different aspects as discussed below:
1. The integrated aeroservoelastic design optimization method
Experience with integrated design optimization of the structure, aerodynamics and
control subject to constraints on fatigue, stresses, deflections and frequencies with
the LCOE as the objective function and simultaneous design of blade and tower
as well as considerations on aeroelastic instability has not been published in wind
energy to the knowledge of the author (system-level design). This research is the
first work that explores and employs such a capability.
To design future cost effective wind turbines the integrated methodology is useful. Presented results proof the effectiveness of it and they show that through
systematic integrated system optimization studies, the LCOE can be decreased significantly.
In addition, a new aeroelastic stability analysis tool is developed to address aeroelastic stability in the context of a linear eigenvalue analysis using an unsteady
aerodynamic model coupled with a nonlinear FE structural formulation. Although
some aspects of the modeling are not precise, the major physical aspects of the
problem and reasonably accurate results can be obtained that are well suited for
preliminary design purposes.
The application of the tool in predicting the aeroelastic instability margin of the 5
and 20 MW wind turbines provides new insight in the phenomenon of instability
with clear relation with size. The results show how sensitive the aeroelastic stability
safety margin will be and highlight the importance of considering that in the design
154
CONCLUSION AND RECOMMENDATION FOR FUTURE WORK
9.5
process of future wind turbines.
2. Development of the optimized 10 and 20 MW wind turbines
In addition to showing a clear way forward for the future offshore wind energy
design methodology development, its application to design very large wind turbines and identify their technical and economical characteristics is another major
contribution of this research. Using the integrated MDO methodology, the design
characteristics of very large scale offshore wind turbines up to 20 MW range are
explored and identified.
These results can be used as a reference to address and compare any technological
development for larger scales and quantify its added value by using the developed
wind turbines of this research as a baseline design. Hence, the developed wind
turbines of this research can be used in a similar way as the 5 MW NREL wind
turbine is used today by many researchers worldwide, since they have the same
level of design details, content, and format.
3. Development of scaling trends for the ever larger offshore wind turbines
Developing the major technical and economical trends for larger offshore wind
turbines that are based on existing conventional wind turbine designs is a unique
contribution to the state of the art knowledge. The scaling trends enable a critical
assessment of size influence on the design of a wind turbine to determine the engineering feasibility, cost implications, and overall fundamental barriers, as concerns
technology, design tools and concept. Detailed analysis covering all the aspects
of design clearly enables to make a fair judgment about the future research and
technological development needs.
How the design of a wind turbine is influenced by upscaling is an interesting research question in the wind energy community that is covered in this research.
Indeed, the scaling trends of this research demonstrates the technically feasibility
of large scale offshore wind turbines though an economical barrier applies before
a technical barrier.
9.5
Recommendations for future work
Due to the multidisciplinary aspects covered in this research, several directions for improvement and continuation of the research have been identified. The summary of the
most important issues are classified and described below:
9.5.1
Upscaling context
• The blade edgewise load grows rapidly with size and gets more pronounced at large
scales. This is mainly a gravity induced loading due to the mass increase. Therefore
mass increase of the blade was identified as a problem and the design of the blade
clearly dictates the mass reduction as a design solution. Carbon fiber composite is
already employed by some wind turbine blade manufacturers and seems to be the
best lightweight option.
9.5
RECOMMENDATIONS FOR FUTURE WORK
155
• The tower clearance at large scales acts as a design driver. Tilting the rotor and
preconing the blades can be of direct help (preconing also enhances the blade
properties against instability). However, they both have a negative influence on
power production. Therefore, this negative impact has to be evaluated at larger
scales using a trade-off study to see its overall impact on the design.
• Tower side-to-side moment at the base grows more rapidly with size and becomes
the design driver at larger scales. This issue with the combination of a wind-wave
misalignment can result in load levels that can pose many challenges to the designers. A means of load alleviation technique has to be considered for the large
scale towers. Although, it was argued that this is mainly a mass driven problem,
this has to be investigated carefully with the application of lightweight materials to
determine whether tower side-to-side load does not remain the design driver.
• A two-bladed, downwind offshore wind turbine with lightweight materials in its
blade construction has the potential to meet all the requirements at very large
scale. It generally has a higher tip speed and therefore higher noise levels, but this
does not seem to be an issue offshore. The aerodynamic efficiency of a two-bladed
machine may be 1-3 % lower but the reduced weight and loads, ease of installation
and transportation may possitively influence the energy cost calculation.
Compared to a three bladed rotor, a two-blade rotor has lower weight. This directly
reduces the overall system weight that is needed to support the rotor. Therefore,
such a configuration might be the ideal choice at larger scales (taking the mass
problem of large scales into account) and clearly a nice future work in the upscaling
context to mitigate the overall negative impact of size on the design.
9.5.2
Design context
• To overcome the negative impact of an expensive computational time introduced
by multidisciplinary design, reduced order models of the wind turbines are needed.
Model reduction is an efficient means to enable cheap simulation when the computational time tends to become a challenge. The experience of making reduced order
models of wind turbines is very limited and it requires a long run future research
before taking the advantages of it in the multidisciplinary design.
• The results of this research show the benefit of the integrated aeroservoelastic
design (system-level design) over the isolated methodology. Michael Faraday said
’we should begin with the whole, then construct the parts’. Following this idea the
future offshore wind turbines have to be designed at the system-level.
In this way the wind turbine design becomes the task of meeting the design requirements of the multiple components concurrently that build up the system, given only
the requirements of the system to be achieved. This will be the future enabler of
cheap wind turbine design but too complex to be reached soon.
• As the size increases the aeroelastic stability margin decreases. Recently, a 100 m
blade corresponding to a 13.2 MW wind turbine was designed by Sandia, Griffith
and Ashwill (2011), and by doing a stability check it was found that the blade had
almost no safety margin against instability.
156
CONCLUSION AND RECOMMENDATION FOR FUTURE WORK
The results of this research show that the optimized 20 MW blade has a safety
margin of about 20% against instability versus the 60% at 5 MW. This means that
future wind turbine design standards have to address this issue more explicitly and
make clear rules and guidelines to address the aeroelastic instability phenomenon
in the design.
APPENDIX
A
Mass and cost models
A.1
Introduction
The cost and mass models (except the mass model of the blade and tower that are calculated by the DOP) used in this research are based on the WindPACT project, Fingersh
et al. (2006), developed at NREL. The foundation of the developed models come from the
work originally done by University of Sutherland, Harrison and Jenkins (1993), under a
work performed for the US Department of Energy (DOE) under its Wind Partnerships for
Advanced Component Technology (WindPACT) projects.
WindPACT project provides an analytical model to project the cost for wind-generated
electricity based on different scales of both onshore and offshore wind turbines. However,
it should not be confused with turbine pricing, which is a function of volatile market
factors beyond the goal of the original work.
In the WindPACT models, cost and mass scaling functions have been developed for
major components and subsystems. Most of these cost and mass estimates are a function
of wind turbine global parameters. These are the wind turbine rating, rotor diameter and
hub height. However, there are also some few elements that have either another local
key turbine descriptor, or they have a fixed value for all different scales and concepts.
The accuracy of the models are cross checked and improved based on the available
information form the industry. The cost model includes the impact on cost from changes
in economic indicators such as the Gross Domestic Product (GDP) and Producer Price
Index (PPI) that was briefly explained in chapter 3. The product of the WindPACT project
is a set of cost and mass scaling functions that enable work to integrate the economy of
wind turbines with a MDO in consistent way, which has never been done before in the
field.
157
158
APPENDIX A
A.2
Mass and cost modeling description
Increasing the size of a wind turbine is a big challenge for designers over many years. It
is often very difficult to find what the total impact of increasing rotor diameter and the
rating of a turbine will be on all other system parameters. Any changes to the design
of a wind turbine, even if very small, influences the system cost and performance and
therefore essential to the designer to evaluate the impact of these changes.
Therefore, these changes require sophisticated tools and methods to evaluate them
properly. The levelized COE is used for many years in the field of wind energy to evaluate
the total system impact of any changes in the design. This levelized COE is explained in
equation 3.5, and a demonstration of that is presented in table 3.9. Using this definition,
a true assessment of the impact of technical changes on levelized COE for larger scales is
evaluated in this work.
The mass and cost estimates of all the wind turbine elements is based on a mature
design of a 50 MW wind farm installation, matching with the highest design, manufacturing, installation and transportation standards of industry. For cost models, the outputs of
all formulas is in 2009 US dollars, and they contain an escalation using the PPIs or GDP,
as earlier described in chapter 3. The PPI related formulas are sorted by North American
Industry Classification System (NAICS) codes that provide a grouping system of products
and can be found at http://www.bls.gov/ppi/.
A.2.1
Blades
Blade mass
The blade mass is calculated as the result of optimizing its weight to satisfy the design
constraints and minimize the levelized cost of energy in every iteration. The explanation
to find the mass is given in appendix B and not repeated here.
Blade cost
The price of the blade is based on the work of Griffin (2001) as follow:
Blade price = (13.084 × blade mass − 4452.2) × (1 + PPI)Blade
(A.1)
As explained before, the mass of the blade is calculated using equation 3.47 as the
result of running the DOP. Also the PPI category to develop commodity inflation factor is
expressed as the percentages of each commodity assumed to make up the final product,
and can be explained with the following NACIS codes of the blade as:
• Fiberglass fabric (NAICS Code 3272123) with a product make up factor of 60%,
and an inflation rate of 30.8% from 2002 to 2009
• Vinyl type adhesives (NAICS Code 32552044) with a product make up factor of
23%, and an inflation rate of 15.3% from 2002 to 2009
• Other externally threaded metal fasteners (NAICS Code 332722489), including
studs with a product make up factor of 8%, and an inflation rate of 13.8% from
2002 to 2009
• Urethane and other foam products (NAICS Code 326150P) with a product make
up factor of 9%, and an inflation rate of 24.2% from 2002 to 2009
MASS AND COST MODELING DESCRIPTION
159
Using this definition, a PPI of 25.2% is obtained for the blade.
A.2.2
Hub
Hub mass
The mass of the hub is based on the work of Malcolm and Hansen (2002), however,
in this work the mass of the hub is calculated based on static analysis using peak root
moments. Therefore, it underestimates the mass since it does not take into account the
dynamics of the system. To correct that a correction factor of 1.92 is used that brings the
mass closer to the given mass of many existing wind turbines, including the NREL 5 MW
machine. The mass formula is as follow:
Hub mass = 1.92 × (0.954 × blade mass + 5680.3)
(A.2)
Hub cost
The hub cost model is a function of the mass of the hub. It is assumed that the hub
is made of ductile iron castings which every kilogram of its mass will cost 4.25 USD,
including material, manufacturing and labor costs. Thus, its cost can be calculated by:
Hub cost = (4.25 × hub mass) × (1 + PPI)Hub
(A.3)
The PPI has a NAICS code of 3315113, PCU3315113315113 for ductile iron castings
and from year 2002 to 2009 it has an inflation rate of 35.6%.
A.2.3
Blade pitch system and its bearing
The blade pitch mechanism consists of the pitch system and bearings to support its mass.
Pitch mechanism mass
The mass of the pitch system’s bearings is a function of the mass of the blade as:
Pitch bearings mass = 0.1295 × mass of all blades + 491.31
(A.4)
The mass of the pitch system itself is also a function of the mass of bearings. The
pitch system consists if actuators and drives and their mass is estimated as 32.8% of the
bearing mass + 555 kg as it is shown below:
Pitch system mass = 1.328 × pitch bearings mass + 555
(A.5)
Pitch mechanism cost
The pitch mechanism cost is a function of rotor diameter is obtained using the following
equation:
Pitch mechanism cost = (0.4801 × rotor diameter2.6578 ) × (1 + PPI)Pitch
(A.6)
The PPI of the pitch mechanism consists of the following NAICS codes:
• Bearings (NAICS Code 332991P, PCU332991332991P) with a product make up
factor of 50%, and an inflation rate of 35.6% from 2002 to 2009
160
APPENDIX A
• Drive motors (NAICS Code 3353123, PCU3353123353123) with a product make
up factor of 20%, and an inflation rate of 32.9% from 2002 to 2009
• Speed reducer, i.e., gearing (NAICS Code 333612P, PCU333612333612) with a
product make up factor of 20%, and an inflation rate of 28.0% from 2002 to 2009
• Controller and drive (NAICS Code 334513, PCU334513334513) for industrial process control with a product make up factor of 10%, and an inflation rate of 18.9%
from 2002 to 2009
Adding all the inflation rates including the weight of each NAICS code in the final
product make up results in a PPI of 31.87% for the pitch mechanism.
A.2.4
Nose cone of the hub
For the nose cone (or spinner) of the hub, the WindPact has derived a new mass and cost
function based on the work of Poore and Lettenmaier (2003) and its alternative design
study Bywaters et al. (2005), also augmented with data from the Advanced Research
Turbine at the National Wind Technology Center.
Nose cone mass
The mass of the nose cone is a function of rotor diameter as:
Nose cone mass = 18.5 × rotor diamater − 520.5
(A.7)
Nose cone cost
The Nose cone cost model is a function of the mass of the nose cone. It is assumed
that the nose cone is made of the same material as the hub is made. However, some
extra layers of coating is applied on the nose cone, which makes its price per kilogram
more expensive. Thus every kilogram of its mass will cost 5.57 USD, including material,
manufacturing and labor costs. The cost can be calculated by:
Nose cone cost = (5.57 × nose cone mass) × (1 + PPI)Hub
(A.8)
As it can be seen from equation A.8, the same NAICS code as the hub has is used for
the nose cone with an inflation rate of 35.6% for the years of 2002 to 2009.
A.2.5
Low speed shaft
Since the mass and cost models of the low-speed shaft are concept dependent, the data
presented here can not be used for direct drive, single-stage drive, or multi-generator
drives, therefore, these data are only valid for a conventional three stage gearbox with a
doubly fed induction generator (DFIG).
Low speed shaft mass
The mass of the low speed shaft is a function of rotor diameter as follow:
Low speed shaft mass = 0.0142 × rotor diameter2.888
Low speed shaft cost
The low speed shaft cost is also a function of rotor diameter as shown below:
(A.9)
MASS AND COST MODELING DESCRIPTION
Low speed shaft mass = (0.01 × rotor diameter2.887 ) × (1 + PPI)LSS
161
(A.10)
The low speed shaft is made of cast carbon steel castings with an NAICS Code of
3315131 that has a subgroup code of PCU3315133315131. For this NAICS code an
inflation rate of 44.1% is reported during the years of 2002 to 2009.
A.2.6
Main bearing and its housing
The main bearing is used to carry all components of the rotor load except the rotational
degree of freedom, and transfer them to the tower. It consist of a main bearing and its
housing.
Main bearing and housing mass
The bearing mass is a function of rotor diameter, and it is calculated by:
Main bearing mass = 0.0092 × (0.0133 × rotor diameter − 0.033) × rotor diameter2.5
(A.11)
The mass of the housing is also assumed to be the same as the mass of the main
bearing, therefore it shares the same formula.
Main bearing and housing cost
The main bearing cost is a function of its mass as follow:
Main bearing cost = 2 × bearing mass × 17.6 × (1 + PPI)Bearing
(A.12)
The main bearings has an NAICS Code of 332991P, PCU332991332991P. For this
NAICS code an inflation rate of 35.6% is reported during the years of 2002 to 2009.
A.2.7
Gearbox
A proper projection of the mass and cost of the gearbox is difficult to make, since there
are several different gearbox configurations and for each of which several ways to model.
In the WindPACT project four standard designs are considered that can be found on Poore
and Lettenmaier (2003) and Bywaters et al. (2005). These four standard models are a
three-stage planetary/helical gearbox with high-speed generator, single-stage drive with
medium-speed generator, a multi-path drive with multiple generators, and a direct drive
with no gearbox.
In this research a planetary/helical gearbox with a high-speed doubly fed induction
generator is used. This configuration is the dominant concept in use, and it matches
very well with the design assumptions of the 5 MW NREL wind turbine that is used for
upscaling studies of this work.
Gearbox mass
The mass of the gearbox is a function of the low-speed shaft torque and thus adjusts for
differences in rotor diameter and tip speed.
Gear box mass = 70.94 × low speed shaft torque0.759
(A.13)
162
APPENDIX A
Gearbox cost
The cost of the gearbox is a function of machine rating and only changes when the size
of the machine changes from 5 MW to 10 MW and higher.
Gearbox cost = 16.45 × machine rating1.249 × (1 + PPI)Gearbox
(A.14)
The gearbox has a NAICS Code 333612P, PCU333612333612P for industrial highspeed drive and gear. For this NAICS code an inflation rate of 28.6% is reported during
the years of 2002 to 2009.
A.2.8
Mechanical brake and coupling
The mechanical brake system is made of the brake itself that is mounted on the high
speed shaft and a high speed shaft coupling to remove the brake in the case of failure.
Mechanical brake and coupling mass
The mass of the mechanical brake including the coupling is a function of machine rating
as follow:
Mechanical brake & coupling mass = 0.19894 × machine rating
(A.15)
Mechanical brake and coupling cost
The cost model is a function of the mass of the mechanical brake, and it is assumed that
every kilogram of its mass will cost 10 USD, including material, manufacturing and labor
costs. Thus, its cost can be calculated by:
Mechanical brake & coupling cost = 10.0 × mechanical brake mass × (1 + PPI)Brake
(A.16)
The PPI has a NAICS code of 3363401, PCU3363403363401 and from year 2002 to
2009 it has an inflation rate of 12.0%.
A.2.9
Generator
WindPACT project presented several models of the generator concepts that are frequently
used by industry. Among them, the doubly fed induction generator model is selected and
used here.
Generator mass
The generator mass is a function of the machine rating as:
Generator mass = 6.478 × machine rating0.9223
(A.17)
Generator cost
The generator cost is a function of the machine rating as:
Generator cost = 65 × machine rating × (1 + PPI)Generator
(A.18)
A NAICS Code 335312P with a subcategory PCU335312335312P for motor and generator manufacturing is used to predict the inflation rate. For this NAICS code and from
year 2002 to 2009 an inflation rate of 23.0% is reported.
MASS AND COST MODELING DESCRIPTION
A.2.10
163
Power electronic
The generator is equipped with a converter capable of carrying full power output. Mass
of the power electronic is not modeled by the WindPACT project, since it is not heavy
when compared to other components and can simply be ignored. Therefore only a cost
model is presented that is a function of machine rating as below:
Power electronic cost = 79 × machine rating × (1 + PPI)PowElec
(A.19)
The generator power electronic is classified in relay and industrial control manufacturing category with a NAICS code of 335314P, and a subcategory code of PCU335314335314P.
For this item, an inflation rate of 21.5% is reported during the years of 2002 to 2009.
A.2.11
Bed plate
Each drive train design has a different load pattern and hence its own unique load distribution on the bed plate. As the result, bed plate mass and cost models are dependent on
of the type of drive train. Mass and cost for the mainframe are calculated as a function
of the rotor diameter.
Bed plate mass
For a three-Stage drive with high-Speed generator the mass model is presented below.
Bed plate mass = 2.233 × rotor diameter1.953
(A.20)
Bed plate cost
The cost of the bed plate is a function of rotor diameter.
Bed plate mass = (9.489 × rotor diameter1.953 ) × (1 + PPI)BedPlate
(A.21)
The PPI has a NAICS code of 3315113, PCU3315113315113 for ductile iron castings
and from year 2002 to 2009 it has an inflation rate of 35.6%.
A.2.12
Platform and railing
A platform for each component installed on the bed plate is needed. This platform transfers the loads from the component bases to the bed plate. Also a railing guide is installed
on the bed plate to move the gearbox, mechanical brake and generator easily in the
nacelle.
Platform and railing mass
The mass of the platform and railing is assumed to be 12.5% of the mass of the bed plate.
Bed plate mass = 0.125 × bed plate mass
(A.22)
Additional mass of 12.5% is added for platforms and railing for the gearbox, mechanical brake and generator and are calculated on a USD/kg basis.
Platform and railing cost
Costs for the additional platforms and railing equipment installed on the bed plate is
calculated based on the mass of the platform and railing, and it is assumed that every
kilogram of its mass will cost 8.7 USD, including material, manufacturing and labor costs.
164
APPENDIX A
Platform & railing = (8.7 × platform & railing mass) × (1 + PPI)Platform
(A.23)
The NAICS code of 3315113, PCU3315113315113 is used for this item, and from
year 2002 to 2009 it has an inflation rate of 35.6%.
A.2.13
Hydraulic and cooling system
The cooling system is used to keep the temperature of the gear box and mechanical
brake in a certain allowable range, the hydraulic system is used to actuate the blade
pitch mechanism and mechanical brake. The WindPACT project used that mass and cost
models from the LWST project. The mass and cost models presented here show the
summation of hydraulic and cooling system masses and costs.
Hydraulic and cooling system mass
The mass is a function of the machine rating as follow:
Hydraulic & cooling mass = 0.08 × machine rating
(A.24)
Hydraulic and cooling system cost
Similar to the mass formula, the cost of the hydraulic and cooling system is a function of
the machine rating as follow:
Hydraulic & cooling cost = 1.2 × machine rating × (1 + PPI)HydCool
(A.25)
These two system are classified as fluid power cylinder and actuators category with
and NAICS Code of 339954, with a subgroup code of PCU333995333995. For this item,
an inflation rate of 29.3% is reported during the years of 2002 to 2009.
A.2.14
Nacelle cover
The nacelle cover is used to enclose the nacelle from external environment. the WindPact
project has derived a single function for all drivetrain configurations, since data were too
scarce to develop individual formulas for different configurations.
Nacelle cover mass
The nacelle cover is a function of machine rating as follow:
Nacelle cover mass = 0.1 × (11.537 × machine rating + 3849.7)
(A.26)
Nacelle cover cost
The cost of the nacelle cover is a function of nacelle mass, and it is assumed that every
kilogram of its mass will cost 10.0 USD, including material, manufacturing and labor
costs.
Nacelle cover cost = 10 × nacelle cover mass × (1 + PPI)NaclCovr
The PPI of the nacelle cover consists of the following NAICS codes:
(A.27)
MASS AND COST MODELING DESCRIPTION
165
• Fiberglass fabric, NAICS Code 3272123, PCU327212327212P, with a product make
up factor of 55%, and an inflation rate of 30.8% from 2002 to 2009
• Vinyl type adhesives, NAICS Code 32552044, with a product make up factor of
30%, and an inflation rate of 15.3% from 2002 to 2009
• Assembly labor, with a product make up factor of 15%, and general inflation index
of 19.7% from the years 2002 to 2009
Using this definition, and including the product make up factor weight, a PPI of 24.4%
is obtained for the nacelle cover.
A.2.15
Electrical connections
There is no mass model assumed for the electrical connection elements, since their weight
can simply be ignored comparing to the weight of other elements of the wind turbine.
Elements that are categorized in this item are: switchgears, transformer, cables and cabinets. The cost of these elements is a function of rated power.
Electerical connections cost = (40 × machine rating) × (1 + PPI)ElecConc
(A.28)
The PPI of the electrical connections consists of the following NAICS codes:
• Switchgear and apparatus NAICS Code 335313P, PCU335313335313P, with a product
make up factor of 25%, and an inflation rate of 24.7% from 2002 to 2009
• Power wire and cable NAICS Code 3359291, PCU3359293359291 with a product
make up factor of 60%, and an inflation rate of 76.6% from 2002 to 2009
• Assembly labor, with a product make up factor of 15%, and general inflation index
of 19.7% from the years 2002 to 2009
Using this definition, and including the product make up factor weight, a PPI of 55.9%
is calculated for electrical connections.
A.2.16
Yaw system
The yaw system consist of the yaw drive and a bearing installed on top of the tower.
Yaw system mass The mass of the total yaw system (yaw drive and bearing) is a
function of rotor diameter. Mass data are based on calculated moments. These moments
were calculated using ADAMS software. However, they are presented as a function of
rotor diameter:
Yaw system mass = 1.6 × (0.0009 × rotor diameter3.314
(A.29)
Yaw system cost Yaw bearing cost is based on quotes from Avon Bearing company,
and modified based on the size of rotor. Total yaw system cost is twice the bearing cost.
Yaw system cost = (2 × (0.0339 × rotor diameter2.964 ) × (1 + PPI)YawSys
The PPI of the yaw system consists of the following NAICS codes:
(A.30)
166
APPENDIX A
• Drive motor NAICS Code 3353123, PCU3353123353123, with a product make up
factor of 50%, and an inflation rate of 32.9% from 2002 to 2009
• Ball and roller bearings NAICS Code 332991P, PCU332991332991P, with a product
make up factor of 50%, and an inflation rate of 23.9% from 2002 to 2009
The weighted inflation rate based on the product make up factor given above, and
each individual inflation rate is 28.4%.
A.2.17
Control system
The control system consist of safety equipments, condition monitoring equipments, wiring and sensors. The mass of these items is negligible, therefore also ignored in the
WindPACT project. The total cost of these items does not have too much sensitivity to
machine size and rating. Therefore, for offshore systems, a fixed price of 55000 USD,
regardless of machine size or rating is used.
Control system cost = (55000) × (1 + PPI)ConSys
(A.31)
Control system has a NAICS Code of 334513, PCU334513334513 and is categorized
as controller and drive with a subcategory of industrial process control. The PPI for this
class of equipments has an inflation rate of 18.9% during the years 2002 to 2009.
A.2.18
Tower
The tower mass is calculated in every iteration of optimization and using the methodology explained in appendix B. The cost model of the tower uses a cost per kilogram formulation of 1.5 f r acUS Dk g, including material, manufacturing and labor costs. Thus
the cost model can be formulated as:
Tower cost = 1.5 × tower mass × (1 + PPI)Tower
(A.32)
Control system has a NAICS Code of 331221, PCU331221331221 and is categorized
as rolled steel shape manufacturing with a subcategory of primary products. The PPI
has an inflation rate of 80.2% during the years 2002 to 2009. As it can be seen, if one
would use a general inflation index of 19.7% instead of the above PPI, a big error will be
introduced in the cost formula of the tower.
A.2.19
Marinization
The Marinization expenses is used for additional preparation for all components of the
wind turbine, in order to increase their survivability in the extreme sea condition. These
preparations include special paints and coatings, improved seals for gearboxes, generators, electrical components, and electrical connections.
The cost model is calculated as a percentage of all turbine costs, and is mainly based
on the report of DUWIND (2001). However, a range of journals publications also suggest
a marinization factors of between 10% and 15% of the turbine and tower cost. A number
of 13.5% was selected in WindPACT and also used here. However, is also should be
mentioned that this is a rough estimate and it varies with the design.
MASS AND COST MODELING DESCRIPTION
Tower cost = 0.135 × turbine and tower cost
167
(A.33)
Turbine and tower cost, is the participation of all cost elements form equation A.1 to
A.32.
A.2.20
Foundation system
The foundation system consist of the transition piece and monopile. The tower is attached to the monopile that extends from sea bed to mean sea level. There are several
different concepts for the foundation, but the most often used is a driven pile (a steel pile
driven into the sea bed). This allows the tower to be bolted to the top of this structure.
The associated costs to install such a driven pile is greater than the basic concrete
foundation used for onshore wind turbines. The cost model presented in the WindPACT
project is derived from a University of Massachusetts study done by Manwell (2002).
However, this model is augmented with industry communications expressed as a function
of machine rating.
Foundation cost = 300 × machine rating × (1 + PPI)SupStrc
(A.34)
Foundation system has a NAICS Code of BHVY, PCUBHVY-BHVY and is categorized as
heavy construction. A inflation rate of 45.6% is reported for this item during the years
2003 to 2009.
The mass model of the foundation is based on the linearly upscaled offshore version
of the 5 MW NREL wind turbine.
A.2.21
Offshore transportation
There are two cost elements associated with transporting an offshore wind turbine. The
first cost model covers the expenses to get the turbine components to the port staging
and assembly area near the shore. The second cost model is to get the assembled turbine
to the installation site. The second cost model is explained in next subsection.
The first cost model, offshore transportation, covers the expenses to bring the components to the assembly site near the shore. The costs for MW turbines show a significant
increase comparing with smaller turbines. These costs can be significantly reduced by
locating manufacturing facilities close to shore.
The offshore transportation cost model is based on the technical report of Smith
(2001), and is a function of machine rating.
Offshore transportaion cost = machine rating × cost f ac t or × (1 + PPI)OfshrTrnsp (A.35)
In the above formula, the cost factor is a function of machine rating and is given by:
Cost factor = 0.000015machine rating2 + 0.0375 × machine rating + 54.7
(A.36)
Offshore transportation has a NAICS Code of 335312P, PCU484121484121P and is
categorized as general freight trucking, long-distance, truckload. A PPI of 21.2% is reported during the years 2002 to 2009.
168
APPENDIX A
A.2.22
Port and staging equipment
This item covers facilities needed to bring the wind turbine components to the installation
site and maintain a safe operation of wind turbine during its life time. It only covers the
expenses of using special ships and barges to install piles, setting towers and turbines,
laying underwater electrical lines, and providing ongoing servicing, and should not be
confused with the installation expenses, explained in next subsection.
Unfortunately, long-term data on this cost element does not exist and little hard detailed industry data are available. The costs presented in WindPACT project are based on
private industry communications converted to a scaling factor based on machine rating,
and cross-checked with some data published in scientific journals.
Port & staging cost = 20 × machine rating × (1 + PPI)PrtStag
(A.37)
For port and staging cost model a NAICS Code of BHVY, PCUBHVY-BHVY is allocated,
and it is categorized as heavy construction. A PPI of 45.6% is reported during the years
2002 to 2009.
A.2.23
Offshore turbine installation
This cost model covers extra facilities required to do the installation of wind turbine such
as extra tools and fixtures, cranes and hammering machines. Costs in this model are also
based on private industry communications presented as the function of machine rating.
Offshore turbine installation cost = 100 × machine rating × (1 + PPI)Instltn
(A.38)
Offshore transportation has a NAICS Code of BHVY, PCUBHVY-BHVY and it is categorized as heavy construction. A PPI of 47.2% is reported during the years 2003 to
2009.
A.2.24
Offshore electrical interface and connection
This cost model covers expenses to bring the turbine power to shore. The cost model
includes the cost of cabling between turbines and the cable to the grid interconnect at the
shore and switchgears. Costs in this model are based on calculations and data formulated
by Butterfield and Laxson (2004). This cost model should be used with care, since it is
calculated specifically for a distance to shore of about 8 kilometer, a water depth of 10
meters, and an array spacing of 7 by 7.
Offshore elcterical interface cost = 260 × machine rating × (1 + PPI)ElcIntrfc
(A.39)
The PPI of the electrical interface consists of the following NAICS codes:
• An NAICS Code of 3353119, PCU3353113353111 for power and distribution transformers with a product make up factor of 40%, and an inflation rate of 67.3% from
2002 to 2009
MASS AND COST MODELING DESCRIPTION
169
• An NAICS Code of 3335313P, PCU335313335313P for switchgear and apparatus
with a product make up factor of 15%, and an inflation rate of 24.7% from 2002
to 2009
• An NAICS Code of 3359291, PCU3359293359291 for power wire and cable with a
product make up factor of 35%, and an inflation rate of 76.6% from 2002 to 2009
• A general inflation index for assembly and labor with a product make up factor of
10%, and an inflation rate of 19.7% from 2002 to 2009
The weighted inflation rate based on the product make up factor given above, and
the participation of each individual inflation rate is 59.4%.
A.2.25
Offshore permits, engineering, and site assessment
The cost model covers developing detailed engineering plans, measuring wind conditions
for an offshore site, site assessment to locate the wind farm and project management expenses. The costs in this model are based on private industry communications converted
scaling factor.
to a USD
kW
Offshore managment cost = 37 × machine rating × (1 + GDP)
(A.40)
For this item, a general inflation index is used with a growth rate of 17.0% during the
years of 2003 to 2009.
A.2.26
Personnel access equipment
Marine vessels, helicopters or small boats are needed for servicing offshore wind turbines.
During the access operation, the environment will present many hazards to the personnel,
and to improve the safety for these operations, special personnel access equipment is
needed.
In addition, servicing requires special boat access ramps or docking equipment, lifesaving equipment located at each turbine, special tool lifts, and emergency survival equipment in case service personnel are stranded at a turbine. All these items introduce extra
expenses, which are needed to make the working environment safe. The cost model
presented in WindPACT project is developed from private industry communications.
Offshore managment cost = 60000/tur bine × (1 + GDP)
(A.41)
As it can be seen from equation A.41, the cost model is a fixed value regardless of
turbine rating or size. For this item, a general inflation index is used with a growth rate
of 17.0% during the years of 2003 to 2009.
A.2.27
Scour protection
Scour is the removal of granular bed material by hydrodynamic forces in the vicinity of
offshore structures. In this case, currents swirling around the structure have a tendency
to scour bottom material from the base, and increasing the risk of foundation failure.
170
APPENDIX A
To decrease such a risk, graded boulder and rock are placed around the base of the
structure. Cost estimates in the WindPACT model are based on private industry communications converted to a scaling factor by machine rating.
Scour protection cost = 55 × machine rating × (1 + PPI)ScrPrtc
(A.42)
For scour protection a NAICS Code of BHVY, PCUBHVY-BHVY is defined and it is
categorized as heavy construction. A PPI of 47.2% is reported during the years 2003 to
2009.
A.2.28
Offshore warranty premium
Offshore wind turbines are located in remote locations and they operate in an extreme
environment. Therefore, they experience a greater risk of failure than for land-based
installations. As the wind turbine is getting more expensive this risk requires also a higher
warranty premium. The windPACT warranty estimates are based on private industry
communications and the cost model is a function of the wind turbine and tower cost.
Offshore warranty premium cost = 0.15 × turbine and tower cost
(A.43)
Turbine and tower cost, is the participation of all cost elements form equation A.1 to
A.32.
A.2.29
Decommissioning
Offshore installations present long-term navigational hazards. However, there is unlikely
that a good wind farm site will be abounded, but to guarantee funds for removing older
structures this cost element is taken into account. The decommissioning cost model
is developed as a percentage of the initial capital cost, without the offshore warranty
premium.
Decommissioning cost = 0.03 × (ICC − Offshore warranty premium cost)
A.2.30
(A.44)
Offshore levelized replacement
Offshore installations require more frequent overhaul and replacment, since they carry
a higher risk of wear and tear. Levelized replacement cost (LRC) covers the long-term
replacements and overhaul of major turbine components, such as blades, gearboxes, and
generators. The cost model is developed based on private industry communications and
converted to a scaling factor based on a fixed price of 17 USD of machine rating expressed
in kW.
LRC cost = 17 × (machine rating in kW) × (1 + GDP)
(A.45)
The general inflation index has a growth rate of 17.0% during the years of 2003 to
2009.
MASS AND COST MODELING DESCRIPTION
A.2.31
171
Operation and maintenance
Operation and maintenance expenses of offshore wind turbines are higher than those for
onshore, at least during the early period of operation. Initial offshore O&M costs are
expected to be almost three times that of current onshore estimates. In the WindPACT
project costs are estimated on a fixed price of 0.02 USD per kWh basis.
O&M cost = 0.02 × annual energy production × (1 + GDP)
(A.46)
For the general inflation index a growth rate of 17.0% is reported during the years of
2003 to 2009.
APPENDIX
B
A theoretical background of the used simulation tools
B.1
Introduction
As it was explained in chapter 3, a set of simulation tools is needed to model the physical
behavior of a wind turbine. This appendix gives a theoretical background of the simulation tools that are used in this work. All these tools are open source and free to download
and use for research studies from the website of the NREL.
B.2
TurbSim
TurbSim calculates a numerical simulation of a full-field flow, which contains coherent
turbulence structures that reflect the proper spatiotemporal turbulent velocity field relationships. It enables the designer to simulated inflow turbulence environments that
incorporate many of the important fluid dynamic features known to adversely affect turbine aeroelastic response and loading.
TurbSim supports the IEC Kaimal and von Karman Normal Turbulence Models (NTM)
to generate a turbulent inflow and provides the ability to efficiently generate randomized
coherent turbulent structures that are superimposed on the more random background
turbulent field. The code also allows the user to use available observed turbulence information to improve the agreement in specific locations.
The presence of superimposed coherent structures can induce transient loading events
on simulated turbine rotors and supporting structures. The combination of techniques;
i.e., Fourier inversion plus time-domain Poisson and Lognormal modeled events, produce
the highly non-Gaussian flow behavior associated with the presence of coherent structures in actual flows.
The use of the coherent events does require a bit more execution time in the AeroDyn program (where they are superimposed on the background turbulence). The exact
173
174
APPENDIX B
amount of time depends on the number of events and their lengths. The Cholesky factorization algorithm has been rewritten in TurbSim to be more efficient in terms of both
CPU and memory usage.
TurbSim is intended to generate a series of inflow simulations based on a given set of
initial boundary conditions that are employed with the design codes, FAST and YawDyn
to produce response (load) solutions that can be analyzed using ensemble statistics.
B.3
AeroDyn
The AeroDyn code is a set of routines for aerodynamic calculations by Laino and Hansen
(2002), to provide instantaneous aerodynamic loading in an aeroelastic simulation. AeroDyn contains two models for calculating the effect of wind turbine wakes:
• The blade element momentum model (BEM): It is the classical standard used by
many wind turbine designers that has various corrections such as incorporating the
aerodynamic effects of tip losses, hub losses, and skewed wakes.
• The generalized dynamic-wake model: It is a more recent model useful for modeling skewed and unsteady wake dynamics, which represents the induced velocities at the rotor plane with a series solution. It is an expanded form of the Pitt and
Peters dynamic inflow Pitt and Peters (1983), and runs considerably faster than
BEM, since it does not require iteration on the induction factors.
In addition, AeroDyn contains an important model for dynamic stall based on the
semi-empirical Beddoes-Leishman model Leishman and Beddoes (1989). This model is
particularly important for yawed wind turbines. Another aerodynamic model in AeroDyn
is a tower shadow model based on potential flow around a cylinder and an expanding
wake.
When comparing BEM method with other advanced methods, it offers a good prediction of loads and predicts the actual performance in post-stall conditions fairly Duque
et al. (2000), and it runs very fast. For these reasons, modified BEM codes dominate the
arena of wind turbine aerodynamic design tools, and it is used in this work.
Navier-Stokes simulations have numerical stability issues and exceptionally high expense for the unsteady cases, they suffer from inadequate turbulence models, and problems preserving the concentrated vorticity in the wake as it is convected downstream
Robinson et al. (1999). Also CFD modeling using Euler equations yield little to no insight
over vortex methods, at a higher cost, as the relevant inviscid features of the flow are
already captured by LaplaceŠs equation.
In AeroDyn, the blade is modeled as strip elements, with variable length and extend
across an arbitrary disc from the blade root to tip. The blade twist, chord and airfoil
characteristics are specified at the mean radius of each element. The airfoil lift, drag and
moment curves are supplied separately to AeroDyn. The input wind speed is provided as
a time-history rather than a constant velocity, and with a variable yaw angle. Horizontal
and vertical shearing profiles are included as well. The wind field can either be described
by the time-history at the hub height, or by a full grid of histories over the rotor plane
area.
AIRFOILPREP
B.4
175
AirfoilPrep
Wind turbine airfoils frequently operate beyond stall condition at both negative and positive angle of attack. AirfoilPrep is a code to prepare airfoil input files over the full
360-degree data needed by AeroDyn. It uses the Viterna stall equations (see Viterna and
Janetzke (1982)) to expand the full 360-degree data from a limited number of angle of
attacks. It can also apply rotational augmentation corrections for 3-D delayed stall. It
uses Du’s method (see Du and Selig (1998)) to augment the lift, and Eggers’ method to
modify the drag (see Eggers et al. (2003)).
B.5
FAST
FAST is a medium-complexity code for nonlinear aeroservoelastic simulation of wind
turbines. In addition, FAST enables:
• Generation of ADAMS models and all relevant inputs
• Extraction of linear state-space models for control design
• Interfacing with Simulink to implement advanced turbine controls in block diagram
form
FAST models the wind turbine as a combination of rigid and flexible bodies and it
employs a combined modal and multibody dynamics formulation. The rigid bodies are
the earth, hub, nacelle, and optional tip brakes as a point mass. The flexible bodies
include blades, tower, and main shaft.
These bodies are connected using 26 degrees of freedom. These include tower bending, blade bending, nacelle yaw, rotor teeter, rotor speed, and drive shaft torsional flexibility. Blades have two flapwise modes and one edgewise mode per blade, tower has
two modes each in the fore-aft and side-to-side directions, and the main shaft has one
torsional degree of freedom.
FAST uses Kane’s method to set up equations of motion that are solved numerically.
The mode shapes are described by 6th order polynomials, the first two terms of which
must be zero to provide zero deflection and slope at the base of the blade. The coupled
equations of motion are solved using a 4th order Runge-Kutta scheme. The implemented method makes direct use of the generalized coordinates, eliminating the need for
separate constraint equations.
FAST uses the AeroDyn code to generate aerodynamic forces along the blade. The
electrical characteristics of the generator response are also modeled. Two models are
available for the generator: a simple induction motor with a simple torque versus speed
profile, and a Thevenin equivalent circuit model.
FAST also has provisions for active control of the blade pitch angle, generator torque,
tip brakes, and a high-speed shaft brake. The user can make custom control laws, or use
simple switches to turn the various controls on at a certain time, which enables any test
condition from start up through shutdown operations, emergency stop and grid failure.
The model complexity of this code is at the level of current detailed design and verification codes. The outputs of the code are time-histories of every degree of freedom in
various reference frames (inertial, flapwise/edgewise, in-plane/out-of-plane), as well as
176
APPENDIX B
the displacement, velocity and accelerations acting on any component.
For flexible bodies, the bending moments, shear, and normal forces are available,
together with the tip motions used in the modal analysis. Since the source code of FAST
is available, it was modified many times to increase its compatibility to run in an iterative
optimization process for this work.
B.6
BModes
The mode shapes of the blade and tower, and their relevant natural frequencies are
needed as inputs for FAST that can be generated using the BModes code, Bir (2007).
Modes uses the distributed inertial and stiffness properties of the blade and tower along
their longitudinal axis, the rotational speed of the blade to calculate the rotational stiffening and its pitch angle as input. Polynomial expressions of the mode shapes are calculated
from the solution of the associated eigenvalue vectors. In the same way, the natural frequencies are found from the solution of the eigenvalues that include stiffening effects
of rotation. Since for an aeroservoelastic design optimization problem, the blade pitch
and rotational speed change in every iteration, the source code of BModes is modified to
account these changes automatically in every iteration.
B.7
Crunch
Crunch is a post-processing tool to perform: load-roses, rainflow cycle counting of any
load history, extreme event finding for the limit load calculation, probability mass functions (histograms), azimuth averaging, channel combination, moving averages, data filtering and statistical analysis of the data. It is a user-friendly code and enables a wide
variety of data processing from both field tests and time simulations.
B.8
Fatigue
Fatigue is a code to read a series of Crunch rainflow cycled files and using a Rayleigh distribution of wind speed, it calculates cumulative fatigue spectra and damage-equivalent
loads. The source code used in this work is modified to work also with non-Rayleigh
wind distributions.
APPENDIX
C
Verification of the controller design automation
C.1
Introduction
The controller design automation and the integration of that in the design optimization
loop was explained in chapter 3. However, two case studies are presented in this appendix to make sure that the controller design automation is implemented correctly, and
it results in a decrease of the design constraints and objective function in practice.
Two different MDO scenarios are defined and applied on the 5 MW NREL wind turbine. In the first case the torque and pitch controller are designed for the initial design
and their parameters are fixed in the entire DOP. However, at the end of the process both
controllers are redesigned based on the final design variables found by the optimizer.
In the second case, both controllers are redesigned within every optimization iteration. As expected, the objective function and design constraint of case one and two are
different. This shows the influence of a realistic assessment of the objective function and
constraint by the optimizer, which leads to more reduction in the cost of energy and the
constraint.
C.2
Applying the controller design task
To show the fundamental differences of case one and two, costs of energy as the objective
function, and fatigue damage as the design constraint at the blade root are selected for
this investigation. System cost consists of all the elements that a wind turbine is made
of and explained in detail in appendix B and the formula to find the AEP is given by
equation 3.7 of chapter 3, therefore the cost of energy can be calculated as:
LCOE =
System cost
Anual energy production
177
(C.1)
178
APPENDIX C
For calculating fatigue, rain flow cycle counting is applied on the time-series output
of the aeroelastic solver. Based on that, the Miner rule is used to calculate the cumulative
fatigue damage at the blade root, in flapwise direction.
For both cases, the level two optimization technique explained in subsection 3.12.4
of chapter 3 is used, with all the associated design variables in. This new methodology is
implemented as an external master controller dynamic link library (DLL), in the format
of BLADED.
For case one, the the controller is designed two times, the first design is based on the
initial value of all the design variables and parameters before starting the optimization
and then using it in all optimization iterations without applying any changes to it, and
the second design is after the end of the optimization process, based on the final set of
optimism design variables.
Therefore, case one optimization is done with two different DLL files. Contrary to
case one, the controller design in case two is done in every iteration and the DLL file is
automatically generated and directly used in the process.
The DOP is performed using the integrated design optimization tool developed in 3.
The controller design automation is carried out in MATLAB. Therefore, an interface is
developed to fully automate the DOP and guarantee the integrity of all the simulation
tools.
C.3
Analysis and results
To show the influence of the selected strategy on the objective function and constraint,
the final results of case one and two are compared in this section.
Using the generator torque controller, for case one a generator proportional gain of
kN ·m
kN ·m
) is found. This value for case two is 0.02770 ( RP
) that is higher than
0.02710 ( RP
M2
M2
case one. This is due to the fact the optimizer has properly evaluated the objective
function and constraint in every iteration that has resulted in a truly optimum values for
design variables.
For the above rated pitch controller, the final aerodynamic power sensitivity to the
blade pitch angle for different steady state operating points of case one and two is presented in figure C.1. As the figure shows, this sensitivity is different for case one and two,
which demands a different controller design for the pitch controller.
However, the power regulation of case two is better achieved, since in every iteration
the controller used the true chord and twist distribution of the blade. Consequently, this
true assessment of design variables by the controller results in lower loads on the blade
for case two.
For case two, the variation of the proportional and integral gains of the pitch controller to balance out the changes of the aerodynamic power is shown in figure C.2. This is
done using the gain scheduling technique presented before. The calculated proportional
and integral gains are obtained using equation 3.19 and 3.20 respectively.
For both scenarios, the cost of energy and flapwise fatigue at the blade root are calculated and presented in figure C.3. As the figure shows, different design strategies of
implementing the controller in an MDO result in different values of the objective function
and constraint.
179
ANALYSIS AND RESULTS
Case 1
0
2.4
4.8
7.2
9.6
Case 2
12
14.4
16.8
19.2
21.6
24
Aerodynamic power sensitivity to
pitch angle (MW/rad)
-20
-30
-40
-50
-60
-70
-80
-90
-100
-110
-120
Blade pitch angle (deg)
Figure C.1: Aerodynamic power sensitivity to blade pitch of the 5 MW wind turbine
Integral gain
Proportional gain
Gain correction factor
1
0.018
0.9
0.016
0.8
0.014
0.7
0.012
0.6
0.01
0.5
0.008
0.4
0.006
0.3
0.004
0.2
0.002
0.1
0
Gain correction factor
P and I gains
0.02
0
0.0
2.6
5.2
7.8
10.4
13.0
15.6
18.2
20.8
23.4
26.0
Blade pitch angle (deg)
Figure C.2: P I gains and gain scheduled correction factor of case 2
As the results show, a better assessment of the objective function and the constraint
helps the optimizer to move towards better design solutions for the system. This proves
the importance of doing an aeroservoelastic design optimization instead of even an aer-
180
APPENDIX C
oelastic design optimization.
LCOE
Fatigue damage
Non-dimension LCOE and fatigue
damage
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
1
2
Case number
Figure C.3: Nondimension cost of energy and root fatigue damage of the 5 MW NREL wind turbine
APPENDIX
D
Verification of extracting structural properties
D.1
Introduction
In chapter 3 the derivation of an analytical model to extract the blade’s structural properties was explained. However, this method needs to be verified to ensure the correct
implementation of formulas and check the validity of the assumptions.
This task is done in this appendix by doing a comparison with the structural properties
of a 660 kW commercial blade. However, due to the confidentiality of the data used for
the comparison, plots are presented in a nondimensionalized form.
D.2
Verification of the method
A 660 kW wind turbine blade, which all its structural properties are known in advance is
selected for the verification of the method. Three representative structural properties of
the blade, flapwise and edgewise stiffness, and mass per unit length of 20 stations along
the blade are used. However, the same level of accuracy has been found in comparing
other properties (strength, radius of gyration and Ě), but only three of them are presented
here.
Figure D.1 shows the nondimensional flapwise stiffness distribution of the blade. As
the figure shows, there is a good agreement between the calculated flapwise stiffness
using this analytical model and the real data.
Results of the edgewise stiffness are presented in figure D.2. Comparing the accuracy
of the edgewise and flapwise stiffness, the edgewise stiffness shows more discrepancy
between the analytical model and the 660 kW wind turbine blade data.
This is mainly due to manufacturing modifications applied on the leading and trailing
edges of the commercial blade, while in the analytical model the X − Y coordinates of the
airfoil are used for calculating the stiffness. This manufacturing modification has more
181
182
APPENDIX D
660 kW data
Analytical model
Nondimensional flapwise stifness
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Nondimensional blade length
Figure D.1: Comparison of flapwise stiffness of the analytical model with the 660 kW wind turbine
blade
influence on the edgewise stiffness than the flapwise stiffness, since they are more away
from the bending centroid in the edgewise direction.
The comparison of the mass per unit length is presented in figure D.3. As the figure
shows, the predicted values from the analytical model are very close to the values of the
660 kW wind turbine blade.
For different stations along the blade, the analytical model needs the blade geometry,
the thickness of composite laminates including related material properties as inputs. The
geometry should be given in terms of chord, twist, aerodynamic thickness and airfoil
X − Y coordinates along the blade. Since, it is a parametric code it allows any parameter
such as thicknesses and the location of the shear webs to be introduced as design variable
in a design optimization study. The code is very fast and runs in a fraction of a second,
and requires only a little knowledge of composite materials used in wind turbine blades.
Despite the simplicity of the approach, it provides good results. However, there is
no bend-twist coupling that makes the method less accurate for detailed design studies. Therefore the usage of this method should be limited to upscaling and optimization
studies, as it is in this work.
Due to the analytical approach of the method, it is an excellent tool for blade’s internal
layout optimization as well. This can be done for example by playing with the location
and the number of shear webs along the chord or adding ribs to see the influence of that
on the stiffness and mass, thus the associated costs.
VERIFICATION OF THE METHOD
Analytical model
183
660 kW data
Nondimensional edgewise stifness
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Nondimensional blade length
Figure D.2: Comparison of edgewise stiffness of the analytical model with the 660 kW wind turbine
blade
660 kW data
Analytical model
Nondimensional mass per unit
length
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Nondimensional blade length
Figure D.3: Comparison of mass per unit length of the analytical model with the 660 kW wind turbine
blade
APPENDIX
E
Theoretical aspects of optimization algorithms
E.1
Introduction
This appendix explains the theoretical aspects of optimization algorithms used in the
research. A general formulation of the optimization problem was given before in subsection 3.7 (p. 35). Therefore, the focus will be only on the two algorithms that have
demonstrated strong potential for efficient solution of the DOP problems used in this
research.
E.2
CONLIN
In many of the optimization problems the objective function and design constraints are
non-linear and can not be explicitly given as a function of the design variables, which
means that there is not an analytical expression of the problem available. In this type of
problems, it is only possible to compute the values of the objective function and design
constraints, and if possible also their derivatives, for a given set of design variables.
However, all optimization algorithms require several times to do a function evaluation
to solve the problem and find the optimum set of design variables. To do so, it is common
to build explicit approximations of the objective function and design constraints as a
function of design variables and solve the approximated problem. The computational
time and effort of solving the explicit approximation is negligible in practice compared
to the analysis of the real problem.
Using this method, the real objective function and design constraints are replaced
with a series of explicit functions that are obtained by expanding the objective function
and the design constraints in the neighborhood of a given design point. By running the
real analysis the values of the functions and their derivatives are calculated and generally
a linearization process is performed to make the approximation functions. However, this
185
186
APPENDIX E
process is iterative and every iteration requires one analysis and one sensitivity computation.
Usually Taylor series expansion is used in combination with different choices of intermediate design variables to approximate the problem. If design variables are used
directly then the problem is considered as a classical linearization problem. With a reciprocal variables the linearization leads to an inverse variables problem and with the mixed
variables it leads to a linearization in terms of direct and/or inverse variables.
There are several various algorithm for approximation techniques, among them convex approximation as the widely used one. In convex approximation method the linearization process is carried out with mixed variables (either direct or reciprocal) for every
function independently. The signs of the first partial derivatives determines the choice
of the intermediate linearization variables . This linearization scheme leads to a convex
approximation that is in the form of:
€ Š
€ Š
F x ai ' F x 0i +
n
X
i
∂
F0i+
€
x ai −
∂ x 0i
x 0i
Š
+
n
X
i
€ Š2  1
−
∂ F0i− x 0i
xi
0
∂ X 0i
1
x ai
‹
(E.1)
As it can be seen from this equation, a first order Taylor series expansion is used
where the direct linearization is used when the first order term is positive, and the inverse
linearization is used when the first order term is negative. Using this definition, the leastsquare problem, and linear programming problem are both special cases of the general
convex optimization problem.
To illustrate the difference between a convex approximation linearization and a classical linearization approximation a graphical representation of the design space with two
variables can be used. In figure E.1, the continuous curves represent the surface of a
design space, with a feasible domain between these curves. Point x 0 is a given design
point for which the analysis takes place. The dashed curves represent the surfaces of
approximated design constraints.
Figure E.1a illustrates a classical linear approximation, and figure E.1b shows a convex linearization. As it can be seen from these two figures, the convex linearization
results in an approximation of the design constraints (or objective function) that can be
considered as the real problem. In the latter case, depending on a parameter called convexity factor, the feasible approximated domain can be inside, equal or outside the real
feasible domain.
The convex linearization method (CONLIN) is one of these methods that is classified
as a first order convex approximation method that attempts to estimate the curvature of
the problem functions (see Fleury (1989) for detailed information about the algorithm).
Approximation is based on first order derivatives only, and it does not use information of
previous iterations.
This is an extreme useful feature for the level 1 optimization problem of this research,
since the location of the chord and twist changes as a function of blade length, as well
as the tower height. In this case, the external shape of the blade or tower might be
different from one iteration to another and using an algorithm that uses information
from a previous iteration leads to inconsistency of the approximated functions. Using
this feature, iterations are independent of each other.
LAGRANGE MULTIPLIER (LM)
(a) Classical
187
(b) Convex
Figure E.1: 2-D representation of a classical linearization approximation versus convex linearization
approximation
The settings of the algorithm used in the level 1 optimization of this work are given
in the below table:
Table E.1: CONLIN algorithm settings
Property
Type of design variables
Convergence precision
Global perturbation
Move limit
Scaling of design variables
Scaling of functions
Convexity factor
External relaxation of constraints
Internal relaxation of constraints
Discrete design variables round up
E.3
Value
continuous
0.001
0.01 based on range size
100%
enabled (by individual factor)
enabled (by individual factor)
0.001
disabled
enabled
disabled
Lagrange Multiplier (LM)
Lagrange multiplier (sometimes also called augmented Lagrange multiplier or primaldual algorithm) is one of the oldest optimization algorithms, which is still in use by many
people due to its simplicity and powerfulness (for more background information see Rao
(2009), Weisstein (2003), Birgin and Martinez (2008) and Pierre and Lowe (1975)). In
LM, by introducing an augmented function called the Lagrangian, the original problem is
transformed to a new problem that involves both the design constraints, objective function and a parameter called Lagrange multiplier as a single function. This new function
is called the Lagrangian and it is a linear summation of the original objective function
188
APPENDIX E
and the design constraints that are multiplied by the Lagrange multipliers.
The LM was originally developed for equality constrained problems. However, during
the time its reliability and efficiency was increased and it also became possible to study
inequality constraints. The method presented in work is applicable for both the equality
and inequality design constrained problems. As explained before, the formulation of the
LM can be explained as:
m l €
X
X
€
Š
Š
L x, λ, r p = F (x) +
λ j · ψ j + r p · ψ2j +
λk+m · hk (x) + r p · h2k (x)
j=1
(E.2)
k=1
where:
–
ψ j = max g j (x) ,
−λ j
™
2r p
(E.3)
Now, equation E.2 which is the pseudo-objective of the problem needs to be minimized. This equation appears to be a simple modification to the exterior penalty function
method. In this case, equation E.2 is solved sequentially by increasing r p until the optimality criterion is reached. The algorithm starts with assuming an initial value for x, λ,
r p , ψ and r pmax . Based on this assumed values the updated formulas for the Lagrangian
multipliers can ba found as:


(
p )
−λ
j
p+1
p

(E.4)
λ j = λ j + 2r p max  g j (x) ,
2r p
and:
p+1
p
λk+m = λk+m + 2r p ∗ hk (x)
(E.5)
The main advantages of LM are as follow:
• The starting point may be either feasible or infeasible.
• The method is relatively insensitive to the value of r p .
• Working with equality constraints is not difficult to deal with by the algorithm.
The settings that are used for this algorithm are given in table E.2.
LAGRANGE MULTIPLIER (LM)
Table E.2: LM algorithm settings
Property
Type of design variables
Convergence precision
Convergence criteria
Global perturbation
Move limit
Scaling of design variables
Scaling of functions
r pmax
External relaxation of constraints
Internal relaxation of constraints
Discrete design variables round up
Value
continuous
0.001
objective function
0.01 based on range size
100%
enabled (by individual factor)
enabled (by individual factor)
10000
enabled
enabled
disabled
189
APPENDIX
F
Scaling laws for initial design variable generation
F.1
Introduction
The concept of scaling laws was discussed in detail in section 2.3 (p. 12) of chapter 2.
These scaling laws are implemented in this appendix as a case study. All of the scaling
laws explained before are used to get the initial design variables of the 10 and 20 MW
wind turbines. These laws are implemented in a MATLAB code, and this computer code
can be used to extract the properties of any other scale of interest.
F.2
Derivation of scaling laws for design studies
The scaling laws not only describe the dependency of a turbine parameter to rotor diameter, but also they can be used to extract the design parameters of an unknown wind
turbine with respect to the given parameters of a known wind turbine. This can be explained by the following equation:
Px
Pa
=
0.5C px · ρ x · π · r x2 · Vx3
0.5C pa · ρa · π · ra2 · Va3
(F.1)
Where the subscription a and x stand for the known and unknown wind turbines
respectively, with the following parameters:
P
Cp
ρ
r
V
Power output of the wind turbine
Aerodynamic power coefficient
Air density
Rotor radius
Wind velocity at hub height
191
192
APPENDIX F
Assuming the same aerodynamic power sensitivity and air density for both the known
and unknown wind turbines and recalling the dependency of hub height wind velocity to
rotor radius from table 2.1 (p. 14) and replacing that in equation F.1, we have:
Px
=
Pa
r x2 · r x3α
ra2
·
ra3a
r x2+3α
=
ra2+3α
(F.2)
or:
r
r x = ra ·
2+3α
Px
Pa
(F.3)
In equation F.3, the only unknown parameter is the rotor radius of the unknown
wind turbine, since the designer already knows the power output of the unknown wind
turbine of interest, which he is aiming to design. The ratio of the rotor radius of the
unknown turbine to the known turbine is called the scaling ratio, SR. This ratio is used
to calculate the other parameters of interest of the unknown wind turbine using the
following formulation:
Q x = Q a · (SR)SF
(F.4)
Here, Q is a design parameter of the wind turbine, and SF is the scaling factor of the
parameter based on the given scaling relation of table 2.1 (p. 14).
F.3
Application of scaling laws
Imagine that a designer knows the design parameters of a wind turbine and he wants to
upscale this turbine. The known wind turbine has a power output of 5 MW with a rotor
radius of 63 meter. He is interested to design a 10 MW wind turbine, based on the 5 MW
using the scaling laws. Using equation F.3, the rotor radius of the 10 MW wind turbine
for a wind shear of 0.15 can be calculated as:
r
r
P10
2+3×0.15 10
2+3α
r10 = r5 ×
= 63 ×
= 83.6m
(F.5)
P5
5
The scaling ratio can be calculated based on the definition mentioned before as:
SR =
r10
r5
=
83.6
63
= 1.327
(F.6)
Assume a diameter of 3.54 meter at the root of the 5 MW wind turbine blade. The
scaling factor can be found from table 2.1 (p. 14) as the power of the size dependency
of the chord distribution to rotor radius. Thus in this case the scaling factor is equal to
1. Now using equation F.4 the root diameter of the 10 MW blade can be calculated as
follow:
r oot
D10
= D5r oot × (SR)SF = 3.54 × (1.327)1 = 4.698m
(F.7)
Using the same strategy, the designer can generate all design parameters of any size of
interest, providing that he knows these parameters for any known size of a wind turbine.
APPLICATION OF SCALING LAWS
193
This strategy is also used in this work to extract the initial design variables of the 10 and
20 MW wind turbines and hence shrink the design space. This trick helps the optimizer
to search a smaller design space, which consequently reduces the computational time.
APPENDIX
G
Properties of the optimized 5 MW wind turbine
G.1
Properties of the optimum 5 MW wind turbine
Most of the data presented in this appendix are summarized for conciseness and clarity.
However, the design methodology used in this PhD research contains a high level of
detailed about the development of the optimum 5 MW wind turbines. Key data about
this optimum model is presented in the following subsections.
G.1.1
Blade properties
Similar in concept to the 5 MW NREL rotor, the optimum 5 MW wind turbine has three
blades of 63.5 m length. In the structural computations, 20 nonuniformly distributed
blade elements are used. The properties between these elements are found by a cubic
interpolation.
Table G.1 lists the resulting structural properties. The first column of the table labeled
Rad. is the spanwise location along the blade-pitch axis relative to the rotor center (apex).
The second column labeled BldFrac is the nondimensionalized distance along the bladepitch axis from the root to the tip varying form 0 to 1 respectively.
The distributed blade sections mass per unit length values labeled as BMsDen are
given in the third column. With this mass distribution and a 63.5 m blade’s length, a
mass of 22851 kg is calculated for the blade. However, this mass is higher than the
17740 kg mass of the NREL blade. This is due to the fact that in this work, only fiber
glass composites are used, comparing to the mixture of glass and carbon fiber in the
NREL blade.
The main reason for not using carbon fibers here is the uncertainties in defining
representative carbon fiber structural properties (especially the S-N curve). While we
know these properties very well and with a good accuracy for glass fibers that are used
for many years in the wind turbine industry.
195
196
APPENDIX G
Table G.1: optimum 5 MW wind turbine structural properties
kg
Rad. (m)
BldFrac (-)
BMsDen ( m3 )
BFlpStf (N · m2 )
BEdgStf (N · m2 )
0.0
1.2
3.3
5.4
7.4
9.5
11.5
13.6
15.7
18.8
25.0
31.1
37.3
43.6
49.7
56.0
59.0
60.6
62.1
63.5
0.000
0.019
0.052
0.084
0.117
0.149
0.182
0.214
0.247
0.295
0.393
0.491
0.588
0.686
0.783
0.881
0.930
0.954
0.978
1.000
626.4
519.0
416.3
415.1
447.0
392.3
422.1
428.3
426.5
435.7
393.3
384.0
367.5
327.0
280.6
182.9
114.0
79.9
38.3
4.2
1.21 × 1010
1.05 × 1010
7.57 × 109
9.20 × 109
1.60 × 1010
4.81 × 109
5.53 × 109
5.52 × 109
4.48 × 109
4.46 × 109
1.82 × 109
1.11 × 109
6.26 × 108
3.24 × 108
1.92 × 108
7.67 × 107
2.94 × 107
1.38 × 107
2.20 × 106
4.26 × 103
1.21 × 1010
1.05 × 1010
7.83 × 109
8.38 × 109
1.07 × 1010
8.89 × 109
1.05 × 1010
1.07 × 1010
1.04 × 1010
1.06 × 1010
8.60 × 109
7.33 × 109
6.32 × 109
3.97 × 109
2.46 × 109
1.00 × 109
3.94 × 108
1.87 × 108
3.03 × 107
6.00 × 104
The flapwise, BFlpStf, and edgewise, BEdgStf, section stiffnesses are presented in the
4 th and 5 th column respectively. These values are calculated about the principal structural
axes of each cross section.
The structural damping ratio of 0.477465% critical in all modes of the isolated blade
that is equal to a 3% logarithmic decrement used in the NREL and DOWEC study (see
Kooijman et al. (2003)) is also used here without any change.
G.1.2
Aerodynamic properties
The aerodynamic properties of the optimum wind turbine are obtained by running a
series of simulations from the cut-in to cut-out wind speeds. The first 60 seconds of the
simulation lengths were ignored to ensure that all transient behavior were damped out.
The results are obtained by running a steady wind, and the steady BEM code.
Figure G.1 shows the power output curve and the blade pitch angle, which are defined
as follows:
• RotSpd: the rotational speed of rotor
• RotPwr: the power of rotor
• BldPtch: the blade pitch angle
As the graph shows, in the power maximization region, the variable speed controller
is active to guarantee the maximum energy capture. In the above rated region as the
PROPERTIES OF THE OPTIMUM 5 MW WIND TURBINE
197
22
RotPwr (MW)
BldPich (deg)
RotSpd (rpm)
RatSpd (m/s)
20
18
16
14
12
10
8
6
4
2
0
4
6
8
10
12
14
16
Wind speed (m/s)
18
20
22
24
Figure G.1: Aerodynamic properties of the optimum rotor
wind speed increases, by activating the pitch mechanism the power is held constant.
Similar to the blade structural properties, the optimum blade aerodynamic properties
are presented in table G.2. The first column presents the 8 different airfoil types used at
different stations along the blade. The same airfoil types as the NREL are also used in
this study. The two innermost airfoil types represent cylinders with drag coefficients of
0.50 (Cylinder1) and 0.35 (Cylinder2) and no lift.
For the non-cylindrical airfoils a 3D corrections is used to modify the airfoil data.
In this modification, the lift and drag coefficients are corrected for rotational stall delay
and the Beddoes-Leishman model is employed to estimate the dynamic-stall hysteresis
parameters. For this modification, the aspect ratio of the blade is updated for every
iterations during the optimization process.
The second column of this table labeled as RNodes shows the node locations that
are directed along the blade-pitch axis from the rotor apex to the blade tip that are in
total 19. All the aerodynamic loads are calculated and applied on these nodes. The
third column labeled as DRNodes is the distance from one node to the other. The fourth
coulmn, AeroTwst, is the aerodynamic twist of every node, with the highest at the inboard
station and a zero twist at the blade tip. Finally, the last column shows the chord of the
blade at every node.
198
APPENDIX G
Table G.2: optimum 5 MW wind turbine aerodynamic properties
Airfoil
RNodes (m)
DRNodes (m)
AeroTwst (deg)
Chord (m)
Cylinder1
Cylinder1
2.13
3.82
13.30
13.30
1.26
2.11
3.10
3.10
Cylinder2
Cylinder2
Cylinder2
5.93
8.04
10.16
13.30
13.30
13.30
2.11
2.11
2.11
3.27
3.82
4.57
DU00W401
DU00W401
DU00W350
DU00W350
DU97W300
DU91W2250
DU93W210
DU93W210
12.27
14.39
16.50
19.14
23.90
30.24
36.58
42.92
13.01
12.12
11.01
10.02
8.75
7.28
6.02
4.84
2.11
2.11
2.11
3.17
6.34
6.33
6.34
6.33
5.12
5.29
5.27
5.22
5.05
4.73
4.30
3.81
NACA64618
NACA64618
NACA64618
NACA64618
NACA64618
NACA64618
49.26
55.61
60.36
62.74
64.32
65.81
3.63
2.32
1.30
0.80
0.46
0.14
6.34
6.34
3.16
1.5
1.5
1.3
3.27
2.68
2.12
1.10
1.00
0.80
G.1.3
Drive train properties
The NREL 5 MW baseline wind turbine has a rated rotor speed of 12.1 rpm, a rated
generator speed of 1173.7 rpm and a gearbox ratio of 97:1 as the REpower 5M machine.
However, during the design optimization process of this machine the rated rotor speed
was found to be 12.9 rpm instead of 12.1 rpm, which is 6.4% higher.
As a result of this 6.4% increase of rated rotational speed and to keep the rated
generator speed the same, the gearbox ratio was decreased to 91. The total mechanicalto-electrical conversion loss is 5.6%, which is also the same as both the NREL and DOWEC
turbines at the rated power.
The generator inertia about the high-speed shaft is 534.116 kg · m2 , which is the
same inertia used by the NREL and DOWEC wind turbines. The equivalent drive shaft
·m
and the linear-damping constant of 6.2 × 106 Nr ad·m
linear-spring constant of 8.6 × 108 Nr ad
are also selected to be the same as both the NREL and DOWEC wind turbines.
G.1.4
s
Hub and nacelle properties
From the mass models developed for hub, a mass of 27.4 (ton) is obtained. It is assumed
that the hub is made from ductile iron castings and has a spherical shell. This diameter
of this sphere is assumed to be 3 m, which also matches with the hub diameter of the 5
MW NREL wind turbine.
Due to the known mass and geometry of the hub, with the use of a simple algebraic
PROPERTIES OF THE OPTIMUM 5 MW WIND TURBINE
199
equation the thickness of the hub is found. Based on this thickness a mass moment of
inertia of 121351 kg · m2 for the hub can be found.
The nacelle mass of the redesigned wind turbine is 168.8 ton that is 29.8% higher
than the mass of the NREL machine (130 ton). The reason for this increase is the longer
blade which influences also an increase on the other masses as the mass models show.
The nacelle inertia about the yaw axis was taken to be 3386244 (kg · m2 ). This value
is based on the assumption that all the geometrical dimensions inside the nacelle remain
the same for the optimum wind turbine, with respect to the NREL turbine. Because the
mass moment of inertia is proportional to mass, and since there is a mass increase of
29.8% for the nacelle, its mass moment of inertia is also increased by 29.8%.
The nacelle-yaw actuator has a natural frequency of 3 Hz, which is the same as the
NREL wind turbine. As explained in the NREL report, this is equivalent to the highest
full-system natural frequency in the FAST model, and a damping ratio of 2% critical.
The gross properties of the hub and nacelle are presented in table G.3.
Table G.3: Hub and nacelle gross properties of the redesigned wind turbine
Property
Hub mass
Hub mass moment of inertia
Nacelle mass
Nacelle mass moment of inertia
Elevation of yaw bearing above tower base
Vertical distance along yaw axis from yaw bearing to shaft
Distance along shaft from hub center to yaw axis
Distance along shaft from hub center to main bearing
G.1.5
Value (Unit)
27.4 (ton)
121351 kg · m2
168.8 (ton)
3386244 (kg · m2 )
80.0 (m)
1.96 (m)
5.0 (m)
1.9 (m)
Support structure properties
As explained before the support structure of the wind turbines in this work consists of
the foundation system (monopile and transition piece) and tower. Additionally, the soil
mechanics of the monopile is not modeled and thus it is assumed that all the DOFs of
the monopile at the seabed are constrained, and tower is the only element of the support
structure assembly that is optimized.
Table G.4 gives the resulting distributed tower properties of the optimum wind turbine. In this table, the first column, Height is the vertical locations along the tower
centerline relative to the tower base. It is also assumed that the tower base is located at
the mean sea level.
The second column, TowFrac, of the table is the fractional height along the tower
centerline from the tower base to the tower top and ranges from 0 to 1 respectively.
The third, fourth and fifth column also represent the diameter, thickness and stiffness of
stations along the tower. Since the optimization is carried out at the tower base and top,
only the results of these tow stations are given. Using an interpolation, the results at
other stations can also be found.
200
APPENDIX G
The resulting overall tower mass is 358400 kg that is 3% higher than the mass of the
NREL wind turbine (347460 k g). This increase in mass is related to a longer blade of the
optimum turbine that results in higher loads on the tower.
Table G.4: optimum 5 MW wind turbine tower properties
Height (m)
0.0
80.0
TowFrac (-)
TowDia (m)
TowTick (cm)
TowStif (N · m2 )
0.0
1.0
6.0
4.1
4.2
2.4
6.9 × 1011
1.3 × 1011
APPENDIX
H
Derivation of the aerodynamic forces in a finite element form
H.1
Introduction
The deduction task is carried out in nine steps as shown in figure H.1 These steps are
programmed in MATLAB using its symbolic toolbox. These steps are explained in detail
in the following sections.
H.2
Defining the aerodynamic coordinate system
As presented in figure 6.5, each section of the blade has a different chord, twist and radial
position along the blade. However, to construct the aerodynamic matrices, the Degrees
Of Freedom (DOF) of every section should be known with respect to a fixed reference
frame.
A DOF is characterized by a direction and a position. The directions and positions of
the DOF in a cross section are used to describe the energy that an external force (here
aerodynamic forces) applies on a node. Therefore, a finite number of DOF are used to
describe the motion of every element of a section in a local coordinate system. These
local coordinates are then transformed to the reference coordinate system.
201
202
APPENDIX H
2- Coupling aerodynamic loads
with nodal deformations
1- Coordinate system
definition for
aerodynamic loads
4- Changing functional domains
3- Functionals definition:
R 1  

 L  h , h ,h , , ,  du
R 2  

 M  h ,h ,h , , ,  du
R 3  

 D  h , h ,h , , ,  du
A 3 u 3 e1
A3 u 3 e 2
A 3 u 3 e3

 e1 dA 3
3
3
3

 e 2 dA 3

 e3 dA 3
5- Symbolic integration over displacements
7- Minimization of the
functionals
R1 

R2 



 dA 3 
 dA1  du 3  e 2
1 

   M  h , h ,h , , ,  J  dA
u 3 e2 A1
R3 

 dA 3 
 dA1  du 3  e1
1 

   L  h , h ,h , , ,  J  dA
u 3 e1 A1



 dA 3 
 dA1 du 3  e3
1 

   D  h , h ,h , , ,  J  dA
u 3 e3 A1

6- Functional integration using Gauss-Legendre
2D integration quadrature
8- Element matrices of the aerodynamic
model
 F e   M a e  u e   Da e  u e  K a e  u e
9- Assembling aerodynamic global matrices
from element matrices
Figure H.1: Steps to derive the element matrices of the aerodynamic model
DEFINING THE AERODYNAMIC COORDINATE SYSTEM
203
1. Direction of the DOFs:
Figure H.2 illustrates the DOFs to describe the direction of a typical section. The
deformed DOFs of a section are given by the unit vectors e1 , e2 and e3 . These
1
vectors coincide with the direction of the drag, moment and lift, respectively.
ys
e3
βt
βr
βp
e30
βv
v(1 − a)
xs
e2
Ω×r
e1
e10
Figure H.2: Unit vectors of a given section along the blade
Figure 1: Unit vectors of a given section along the blade
Here:
x s : axis parallel to the freestream wind velocity
ys : axis parallel to the plane of rotation
β v : the resultant wind velocity
β p : blade pitch
β t : aerodynamic twist
β r : angle of attack
e10 : unit vector of the undeformed section of the blade in edgewise direction
e1 : unit vector of the deformed section of the blade
e2 and e20 : unit vector of both the deformed and undeformed section of the blade
in spanwise direction
e30 : unit vector of the undeformed section of the blade in flapwise direction
e3 : unit vector of the deformed section of the blade
Ω: angular velocity of the rotor
v: freestream wind velocity
a: axial induction factor
204
APPENDIX H
r: section radius
e10 , e20 and e30 unit vectors are defined by the user (knowing the structural twist
of each section and the blade pitch). However, to calculate the work done by drag,
moment and lift, e1 , e2 and e3 unit vectors are used. As figure H.2 shows, the angle
of attack, β r , can be obtained using the following equation:
βr =
π
2
− β t − β p − arctan
Ω×r v(1 − a)
(H.1)
Unlike the e1 and e3 vectors, the direction of the vector e2 is based on the azimuthal
position of the blade and it does not change with the wind or rotational speed of
the blade. These vectors can be obtained using the following rotation matrix:
 
e
 1   cos(β r + β p )
0
e
=
 2  
e3
− sin(β r + β p )

0
1
0


e
sin(β r + β p )
  10 
0
  e20 
cos(β r + β p )
e30
(H.2)
2. Position of the DOFs:
The undeformed 3D geometry of the blade is discretized using 2D Quad (Quadrilateral Plate Element Connection) elements. By applying an external force to the
nodes of these elements, their nodal positions change as well. Therefore, by tracking the changes of the position of these nodes the deformation of the blade can be
modeled. These deformations together with the forces applied on every node are
used to calculate the energy transformed in every cycle to the structure.
However, the position of these nodes is only known in the element coordinate
system referred as coordinate system 3. To obtain the energy terms, this coordinate
system is first transformed to a local 2D coordinate system referred as coordinate
system 2, and then the reference coordinate system referred as 1.
These transformations are explained below.
• From element coordinate system 3 to local coordinate system 2
The position of the Quad elements on the discretized blade is given in the
element coordinate system, and the position of all the elements is defined
by four nodes in this coordinate system. The transformation that relates the
Quad elements in coordinate system 3 to coordinate system 2 is as follow:
u2 = u32 − u3 · e32
(H.3)
Where:
u2 : nodal position of element u in coordinate system 2
u32 : relative displacement of coordinate system 3 with respect to coordinate
system 2
u3 : nodal position of element u in coordinate system 3
DEFINING THE AERODYNAMIC COORDINATE SYSTEM
205
e32 : rotation matrix from coordinate system 3 to 2
Using equation H.3, the position of the nodes of an element can be transformed from one coordinate system to another. Figure H.3 shows these transformations.
• From local coordinate system 2 to reference coordinate system 1
The Quad elements are transformed from the coordinate system 2 to the reference coordinate system (coordinate system 1), using the bilinear interpolation functions between the corner values of the Quad elements. Equation H.4
shows this transformation.
u1 (x 1 , y1 ) = u12 · w1 (x 1 , y1 ) + u22 · w2 (x 1 , y1 )
+
u32 · w3 (x 1 , y1 ) + u42 · w4 (x 1 , y1 )
(H.4)
The expression on equation H.4 contains 4 bilinear shape functions presented
in equation H.5 to H.8. These shape functions are linear independent as plotted in figure H.4.
1
1 · x1 −
2
2
(H.5)
1 1
w2 (x 1 , y1 ) = − y1 −
· x1 +
2
2
(H.6)
w1 (x 1 , y1 ) =
w3 (x 1 , y1 ) =
y1 −
y1 +
1 1
· x1 +
2
2
(H.7)
1
1 · x1 −
w4 (x 1 , y1 ) = − y1 +
2
2
(H.8)
• The final form of the deformation in the reference coordinate system
The displacements for the 6 DOFs of each node are obtained using the bilinear
shape functions. Thus, by substituting equation H.3 into H.4 and using the
shape functions as presented in equations H.5 to H.8, the final transformation
from coordinate system 3 to 1 can be formulated as:
u1 (x 1 , y1 ) = u13 · y1 −
u33 · y1 +
1 · x1 −
2
1 · x1 +
2
1
− u23 · y1 −
2
1
− u43 · y1 +
2
1 · x1 +
2
1 · x1 −
2
1
2
1
2
+
(H.9)
206
APPENDIX H
A 3D blade model
u43
u33
A Quad element
u13
u23
z3
u43 (x, y, z)
u33 (x, y, z)
Coordinate system 3
x3
u13 (x, y, z)
u23 (x, y, z)
y3
y2
u42 (x, y)
u32 (x, y)
x2
Coordinate system 2
u12 (x, y)
u22 (x, y)
Figure H.3: From a 3D blade geometry to a 2D Quad element coordinate
Figure 1: diagram
COUPLING AERODYNAMIC LOADS WITH NODAL DEFORMATIONS
Shape function at node 1
Shape function at node 2
1
1
0.5
0.5
0
0.5
0
0.5
0.5
0
y −0.5 −0.5 x
0
0
y −0.5 −0.5 x
Shape function at node 3
1
0.5
0.5
0
0.5
0
0.5
0.5
y −0.5 −0.5 x
0.5
0
Shape function at node 4
1
0
207
0
0
y −0.5 −0.5 x
0.5
0
Figure H.4: Shape functions for the bilinear Quad element
Figure 1: diagram
H.3
Coupling aerodynamic loads with nodal deformations
The generalized aerodynamic forces are a function of h, ḣ, ḧ, α, α̇ and α̈. However, the
aerodynamic forces should be formulated as function of the node deformations to see the
influence of structural deformations on loads. Thus, a new arrangement for pitching and
plunging motion is needed to couple the aerodynamic forces with structural deformation.
This coupling can be formulated as:
h(x 1 , y1 ) = h0 + [u31 (x 1 , y1 ), u32 (x 1 , y1 ), u33 (x 1 , y1 ), u34 (x 1 , y1 )] · e31
(H.10)
ḣ(x 1 , y1 ) = [u̇31 (x 1 , y1 ), u̇32 (x 1 , y1 ), u̇33 (x 1 , y1 ), u̇34 (x 1 , y1 )] · e31
”
—
u31 (x 1 , y1 ), u32 (x 1 , y1 ), u33 (x 1 , y1 ), u34 (x 1 , y1 ) · ė31
+
ḧ(x 1 , y1 ) = [ü31 (x 1 , y1 ), ü32 (x 1 , y1 ), ü33 (x 1 , y1 ), ü34 (x 1 , y1 )] · e31
”
—
2 u̇31 (x 1 , y1 ), u̇32 (x 1 , y1 ), u̇33 (x 1 , y1 ), u̇34 (x 1 , y1 ) · ė31
”
—
u31 (x 1 , y1 ), u32 (x 1 , y1 ), u33 (x 1 , y1 ), u34 (x 1 , y1 ) · ë31
+
1
(H.11)
+
(H.12)
208
APPENDIX H
α(x 1 , y1 ) = α0 + [u3θ (x 1 , y1 ), u3θ (x 1 , y1 ), u3θ (x 1 , y1 ), u3θ (x 1 , y1 )] · e31
(H.13)
4
3
2
1
α̇(x 1 , y1 ) = [u̇3θ (x 1 , y1 ), u̇3θ (x 1 , y1 ), u̇3θ (x 1 , y1 ), u̇3θ (x 1 , y1 )] · e31
4
2
3
h 1
i
u3θ (x 1 , y1 ), u3θ (x 1 , y1 ), u3θ (x 1 , y1 ), u3θ (x 1 , y1 ) ė31
+
α̈(x 1 , y1 ) = [ü3θ (x 1 , y1 ), ü3θ (x 1 , y1 ), ü3θ (x 1 , y1 ), ü3θ (x 1 , y1 )] · e31
4
3
2
i
h 1
2 u̇3θ (x 1 , y1 ), u̇3θ (x 1 , y1 ), u̇3θ (x 1 , y1 ), u̇3θ (x 1 , y1 ) ė31
4
3
2
1
i
h
u3θ (x 1 , y1 ), u3θ (x 1 , y1 ), u3θ (x 1 , y1 ), u3θ (x 1 , y1 ) · ë31
+
1
2
3
2
1
(H.14)
4
3
4
+
(H.15)
Where:
h0 : the initial distance from the pitch axis of the section to the center of the given element
in ys direction (please refer to figure H.2)
α0 : the steady angle of attack (equal to β r in figure H.2)
H.4
Defining aerodynamic loads in terms of functionals
To find an approximation for the generalized aerodynamic loads similar to equation 6.1,
the concept of functional minimization is used. Here, the idea is to minimize the integral
of the work done by an external force on a node. The work done by the force can be
defined as:
h
i
W ≈ min < F(u), u >
(H.16)
Where:
W: work done on the node
F: external force vector
u: deformation vector of the node
Integrating this work over its domain results in:
Z
W ≈ min
F(u) · du
(H.17)
V
This way of representing the generalized aerodynamic forces has two main
Pn advantages. First, discretizing the whole domain V into many subdomains, V = j=1 Vj , by
using the property of the integral. Second, using different deformation approximations
for the deformation vector u in each of the subdomains.
The integral part of the right hand side of the equation is called the functional and
in this case represents the work done by the generalized force F on the domain of integration, V . The lift, moment and drag expressions given in equations 6.19, 6.20 and
209
TRANSFORMING THE FUNCTIONALS DOMAIN OF INTEGRATION TO COORDINATE SYSTEM 1
6.21 are now replaced in equation H.17 to obtain an approximated expression for the
aerodynamic loads.
Wli f t = min
ns h
Z X
∂ Cl
V j=1
∂α
ρU 2 b
h C(k)
U
ḣ + C(k)α + [1 + C(k)(1 − 2a)]
b
2U
Wmoment = min
ḧ −
2
b
2U
α̇
i
b2 a i
α̈
·
A
·
du
sec t ion
j
2U 2
ns
Z X
∂ Cm
V j=1
+
(H.18)
h
C(k)
b
ρU 2 b d1 [
ḣ + C(k)α + [1 + C(k)(1 − 2a)]
α̇]
∂α
U
2U
+
b2
i
i
1
a b2
b3
1
2
α̇(a − ) +
−
a
)
·
A
·
du
(H.19)
ḧ
+
α̈(
sec t ion
j
2U
2
2U 2
2U 2 8
Wd r ag
ns h
Z X
∂ Cd h C(k)
b
ρU 2 b
−
ḣ − C(k)α + [1 + C(k)(1 − 2a)]
α̇
= min
∂
α
U
2U
V j=1
b
2U
ḧ +
2
−
i
b2 a i
α̈
·
A
·
du
(H.20)
sec t ion
j
2U 2
Presenting these functionals in a general form and breaking the integral domain results in:
Z Z
R1 =
R2 =
Z Z
A3
R3 =
(H.21)
M (h, ḣ, ḧ, α, α̇, α̈)du3 · e2 dA3
(H.22)
D(h, ḣ, ḧ, α, α̇, α̈)du3 · e3 dA3
(H.23)
u3 ·e2
Z Z
A3
L(h, ḣ, ḧ, α, α̇, α̈)du3 · e1 dA3
u3 ·e1
A3
u3 ·e3
Where:
R1: lift functional
R2: moment functional
R3: drag functional
A3 : domain of integration that is the area of the element in coordinate system 3
H.5
Transforming the functionals domain of integration
to coordinate system 1
The integration domain of the functionals given by equations H.21, H.22 and H.23 are
in coordinate system 3. Therefore, a coordinate system transformation is employed to
210
APPENDIX H
change the domain of integration from coordinate system 3 to 1. This has been done
before for the aerodynamic loads and deformations, and only needs to be repeated again
for the domain of integration of the functionals to have a consistent set of equations.
This results in having the functionals in the reference integration domain 1. The new
functionals now read as :
R1 =
Z Z
A1
R2 =
u3 ·e1
Z Z
A1
R3 =
(H.24)
dA 3
dA1
M (h, ḣ, ḧ, α, α̇, α̈)du3 · e2 J
dA1
(H.25)
dA 3
D(h, ḣ, ḧ, α, α̇, α̈)du3 · e3 J
dA1
dA1
(H.26)
u3 ·e2
Z Z
A1
dA 3
dA1
L(h, ḣ, ḧ, α, α̇, α̈)du3 · e1 · J
dA1
u3 ·e3
Where:
A
area of integration in the coordinate system 1
1 : reference
dA
J dA3 : Jacobian of the coordinate transformation 3 to 1
1
The jacobian of the transformation is given by the expression:
J
H.6
dA 3
dA1
=
∂ x3
∂ x1
∂ y3
∂ x1
∂ x3
∂ y1
∂ y3
∂ y1
(H.27)
Symbolic integral replacement of the deformations
The functional expressions H.24, H.25 and H.26 are integrated over the deformations.
The integration is done using the interpolation functions presented in H.4. The following
variables are defined to integrate these expressions:
a1 = w1 · e1 x
a2 = w2 · e1 x
a3 = w3 · e1 x
a4 = w4 · e1 x
b1 = w1 · e1 y
b2 = w2 · e1 y
b3 = w3 · e1 y
b4 = w4 · e1 y
c1 = w1 · e1z
c2 = w2 · e1z
c3 = w3 · e1z
c4 = w4 · e1z
Thus, for the lift force it reads as:
R1 =
Z
g1 J
A1
Where:
dA 3
dA1
dA1
(H.28)
211
SYMBOLIC INTEGRAL REPLACEMENT OF THE DEFORMATIONS
g1 =
Z
Z
L · a1 · du131 +
L · b1 · du132 +
Z
Z
L · a2 · du231 +
Z
L · c1 · du133 +
L · b2 · du232 +
Z
Z
L · a3 · du331 +
Z
L · c2 · du233 +
L · b3 · du332 +
Z
L · c3 · du333 +
Z
L · a4 · du431
+
L · b4 · du432
+
Z
Z
L · c4 · du433
(H.29)
Similarly, for the moment the following expressions are used:
a1 = w1 · e2 x
a2 = w2 · e2 x
a3 = w3 · e2 x
a4 = w4 · e2 x
b1 = w1 · e2 y
b2 = w2 · e2 y
b3 = w3 · e2 y
b4 = w4 · e2 y
c1 = w1 · e2z
c2 = w2 · e2z
c3 = w3 · e2z
c4 = w4 · e2z
Thus, for the moment it reads as:
R2 =
Z
g2 J
dA 3
dA1
A1
(H.30)
dA1
Where:
g2 =
Z
M · a1 · du13θ +
Z
1
Z
M · b1 · du13θ +
M · c1 · du13θ +
3
M · a3 · du33θ +
1
Z
M · b2 · du23θ +
Z
Z
1
Z
M · b3 · du33θ +
M · c2 · du23θ +
Z
3
M · c3 · du33θ +
M · a4 · du43θ
+
M · b4 · du43θ
+
1
Z
2
2
2
2
Z
M · a2 · du23θ +
Z
Z
3
M · c4 · du43θ (H.31)
3
The following variables are used to define the drag functional:
a1 = w1 · e3 x
a2 = w2 · e3 x
a3 = w3 · e3 x
a4 = w4 · e3 x
b1 = w1 · e3 y
b2 = w2 · e3 y
b3 = w3 · e3 y
b4 = w4 · e3 y
c1 = w1 · e3z
c2 = w2 · e3z
c3 = w3 · e3z
c4 = w4 · e3z
Thus, for the drag it reads as:
R3 =
Z
g3 J
A1
Where:
dA 3
dA1
dA1
(H.32)
212
APPENDIX H
g3 =
Z
Z
D · a1 · du131 +
D · b1 · du132 +
Z
H.7
Z
Z
D · c1 · du133 +
Z
D · a2 · du231 +
D · b2 · du232 +
Z
Z
D · a3 · du331 +
D · b3 · du332 +
Z
D · c2 · du233 +
D · c3 · du333 +
Z
D · a4 · du431
+
D · b4 · du432
+
Z
Z
D · c4 · du433
(H.33)
Integrating functionals using Gauss-Legendre technique
To solve equations H.28, H.30 and H.32, they are integrated numerically using the twopoint Gauss-Legendre integration technique, Golub and Welsch (1967). In this method
the integral of the function f (x) in the domain [−1, 1] is carried out using some weighting functions as follow:
Z
1
f (x)d x ≈
n
X
−1
ωi · f (x i )
(H.34)
i=1
Where:
n: number of evaluation points
ωi : weighting values
x i : evaluation points
The error of this method can be decreased by increasing the number of evaluation
points and choosing the right weighting values. There are several variants of this method
that can be classified based on the number of evaluation points and weighting values.
In the Gauss-Legendre two-point variant, two points and two weighting values are
used. These are presented in table H.1.
Table H.1: Gauss-Legendre two-point method
Points number
ωi
xi
1
2
1.0
1.0
p1
3
−1
p
3
In this work, the reference integration domain is defined between [−1/2, 1/2] for x 1
and y1 axis. For this domain the evaluation points and weighting functions are presented
in table H.2.
Thus, the application of the Gauss-Legendre two-point method on the function f (x, y)
in the defined reference domain reads as:
213
MINIMIZING THE FUNCTIONALS
Table H.2: Gauss-Legendre two-point method in domain [−1/2, 1/2]
Z
1
2
−1
2
Z
1
2
−1
2
Points number
ωi
xi
1
2
2.0
2.0
1
p
2 3
−1
p
2 3
−1 −1
−1
1
f (x, y)d x d y ≈ 4 f ( p , p ) + f ( p , p )
2 3 2 3
2 3 2 3
1
−1
1
1 f( p , p )+ f( p , p )
2 3 2 3
2 3 2 3
+
(H.35)
Using this integration technique, the integral expression for the aerodynamic functionals, defined at the four nodes of the Quad elements can be given by:
R1 = 4 · g1 |( −1
p , −1
p ) +g 1 |( −1
p ,
+g1 |(
1
p
p )
, −1
2 3 2 3
+g1 |(
1
p
, p1 )
2 3 2 3
R2 = 4 · g2 |( −1
p , −1
p ) +g 2 |( −1
p ,
+g2 |(
1
p
p )
, −1
2 3 2 3
+g2 |(
1
p
, p1 )
2 3 2 3
R3 = 4 · g3 |( −1
p , −1
p ) +g 3 |( −1
p ,
+g3 |(
1
p
p )
, −1
2 3 2 3
+g3 |(
1
p
, p1 )
2 3 2 3
2 3 2 3
2 3 2 3
2 3 2 3
1
p
)
2 3 2 3
1
p
)
2 3 2 3
1
p
)
2 3 2 3
(H.36)
(H.37)
(H.38)
Where g1 , g2 and g3 are defined in equations H.29, H.31 and H.33, respectively.
H.8
Minimizing the functionals
To get the aerodynamic loads in a FE form, the functionals, R1 , R2 and R3 are minimized.
These functionals are a function of the four nodes displacements of the Quad elements.
The partial derivative of these functionals with respect to the DOFs of a node gives a FE
form of the loads as follow:
F km =
∂ (R1 + R3 )
M km =
∂ uk3m
∂ R2
∂ uk3θm
(H.39)
(H.40)
Where:
k=1,2,3,4: node numbering
m=1,2,3: the 3 orthogonal axis of a Quad element
Using equations H.39 and H.40, the generalized aerodynamic loads, [F]e , of the element e can be approximated by the following expression:
214
APPENDIX H
h
[F]e = F11 F12 F13 M11 M12 M13 F21 F22 F23 M21 M22 M23
it
F31 F32 F33 M31 M32 M32 F41 F42 F43 M41 M42 M43
e
(H.41)
Using this expression, the generalized aerodynamic loads are represented as a function of the generalized displacements, velocities and acceleration of the nodes.
H.9
Element matrices for the aerodynamic model
Using equation H.41, the generalized aerodynamic forces acting on a Quad element are
defined. These forces are a function of the DOFs of every node and can be resorted in the
form of element matrices, i.e. [MA ]e , [DA ]e and [KA ]e .
This results in having the aerodynamic loads as an equivalent FE representation given
by the following equation:
[F]e = [MA ]e [ü]e + [DA ]e [u̇]e + [KA ]e [u]e
(H.42)
Where:
[F]e : generalized force vector acting on the nodes of the Quad elements
[u, u̇, ü]e : displacement, velocity and acceleration of the nodes of the Quad elements
[MA ]e : mass matrix of the generalized aerodynamic forces due to the acceleration of the
nodes of the Quad elements
[DA ]e : damping matrix of the generalized aerodynamic forces due to the velocity of the
nodes of the Quad elements
[KA ]e : stifness matrix of the generalized aerodynamic forces due to the movement of the
nodes of the Quad elements
The aerodynamic loads acting on the nodes of the Quad elements can be interpreted
as a mechanical system with an equivalent mass [MA ], damping [DA ] and stiffness matrix
[KA ]. However, the equivalent aerodynamic mass, damping and stiffness matrices are
complex and they are not always positive definite matrices.
The complex nature of these matrices introduce some phase shift between the forces
and displacements on the nodes (non-orthogonality), and this phase shift introduces the
possibility of having the instability.
In addition, in the FE representation of a structural system, the mass, damping and
stiffness matrices are only dependent on the local properties of the element. However,
the aerodynamic matrices have more dependency. For the aerodynamic element matrices
of a wind turbine three type of dependencies can be foreseen:
• Positional: The element matrices are dependent on the radial position of the blade.
This also introduces a different velocity that each section of the blade experiences.
• Angle of attack: The aerodynamic element matrices are dependent on the angle of
attack of the blade section in which they lie. All the elements that are in the same
blade section have the same angle of attack.
ASSEMBLING THE GLOBAL AERODYNAMIC MATRICES
215
• Reduced frequency: Although the rotation speed is the same for every section of
the blade, the reduced frequency changes for different blade sections.
H.10
Assembling the global aerodynamic matrices
To couple the aerodynamic element matrices with the structural elements, the aerodynamic matrices of the Quad elements should now be assembled into a global aerodynamic matrix. This global aerodynamic matrix represents the aerodynamic forces over
the nodes of the blade in the global coordinate system. The assembling procedure is
illustrated in figure H.5.
216
APPENDIX H
Element force vector
Element property matrix
Element DOF vector
DOF at node 1
DOF at node 2
DOF at node 3
DOF at node 4
Node 1 force
Node 2 force
Node 3 force
Node 4 force
Forces on the
second node
Force vector
Global matrix
Figure H.5: Global matrix assembling diagram
Figure 1: diagram
1
DOF vector
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Curriculum vitae
Turaj Ashuri was born on 1975 in Kermanshah, Iran (Persian). He obtained his highschool diploma in 1992 with distinction. He proceeded to obtain his bachelor of science
degree in mechanical engineering with a specialization in heat and fluid mechanics at
Tehran Azad University. The title of his bachlor graduation thesis is; ’Modeling, Design
Analysis and Optimization of a Hydropenumatic Suspension System’. He defended his
thesis with the grade 19/20 and obtained his bachelor of science degree in 1999.
Turaj worked in Iranian railway industry for five years, in the same time also functioning as a certified design and inspection engineer in Heating, Ventilating and AirConditioning of buildings (HVAC CEng). In 2002 he successfully passed the national
university exam for higher education in Iran and ranked among the top 1% of the candidates. Based on this ranking he was admitted with a full scholarship at the prestigious
technical university of the middle-east, Sharif University of Technology.
He continued his graduate education in the aerospace engineering faculty of Sharif
with aerostructures as the major and aeroelasticity as the minor track. The title of his
master graduation thesis is; ’A Computational Model for Wind Turbine Composite Blade
Design and Optimization: A Finite Element Approach’. He defended his thesis with the
grade 20/20 and obtained his master of science degree in 2005.
After completing his MSc degree, he worked for two years in TUGA Company as a gas
turbine auxiliary and accessory professional engineer. He joined the department of aerodynamics and wind energy of Delft University of Technology in 2007 as a PhD researcher
to do research in a very exciting and interesting field, wind energy. In his research, he
explores the horizon of larger scale offshore wind turbines using multidisciplinary design
optimization.
He is currently employed by Siemens wind power. He will join the multidisciplinary
design optimization laboratory (MDOLAB) of university of Michigan as a postdoctoral research fellow next year. In his new position in MDOLAB, he further develops the research
that he carried out in his PhD.
Turaj is happily married and has two sons; Ryan (4 years old) and Dorian (1 year
old).
225
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