A Finite Element Approach for Aeroelastic Instability Prediction of Wind Turbines P.A Castillo Capponi A Finite Element Approach for Aeroelastic Instability Prediction of Wind Turbines Thesis dissertation, submitted in partial fulfillment of the requirements for the degree of the Master Program of Aerodynamic Pablo A. Castillo Capponi October, 2010 Graduation committee Prof.dr.ir. G.J.W van Bussel Ir. T. Ashuri Dr.ir. J. Holierhoek Delft University of Technology Faculty of Aerospace Engineering Wind Energy Research Group Contents Nomenclature 1 1 Introduction 5 1.1 WindEnergy .............................. 5 1.2 Motivation ............................... 6 1.3 Goalofthethesis............................ 9 1.4 Outline ................................. 9 2 Literature review 13 2.1 LiteratureReview ........................... 13 2.1.1 Aeroelastic Instability in Airplanes . 14 2.1.1.1 Historic Approaches to Predict Aeroelastic Insta- bilities in Airplanes . 14 2.1.1.2 Instability Prediction for Airplanes . 16 2.1.2 Aeroelasticity Instabilities in Wind Turbines . 18 2.1.3 State of Art - Aeroelastic Codes for Wind Turbines Insta- bilities . 22 2.2 Multibody and Finite Element Method . 24 3 The Finite Element Approach for Aeroelastic Instability Prediction 29 3.1 FormulationoftheStructuralModel . 30 3.1.1 The Finite Element Method for Structural Components (FEM).............................. 30 3.1.2 Generalized Body Forces in a Non-Inertial Reference Frame 31 3.1.3 Mass, Damping and Stiffness Matrices for Forces due Ac- celerationsinaNon-InertialFrame . 34 3.1.4 Coupling the Generalized Body Forces to the FEM Method 36 3.2 FormulationoftheAerodynamicModel . 39 i ii CONTENTS 3.2.1 AerodynamicModel ...................... 39 3.2.1.1 Basis of the Aerodynamic Model: The Theodor- sen Solution for a Flat Plate . 39 3.2.1.2 Drag Model for the Theodorsen Solution . 41 3.2.1.3 The Aerodynamic Model used in this Thesis . 41 3.2.2 The Finite Element Method applied to the Aerodynamic Model .............................. 45 3.2.2.1 The Basis idea: Minimization of the Aerodynamic EnergyFunctional . 45 3.2.2.2 Variables Definition of the Aerodynamic Element 49 3.2.2.3 Deduction of the Aerodynamic Element Matrices . 50 3.2.2.4 Assembling of Global Aerodynamic Matrices . 66 3.2.2.5 A Non True Finite Element Matrices for the Ae- rodynamicModel .................. 68 3.3 Integration of the Structural & Aerodynamic Models . 69 3.3.1 Complete Formulation of the Aeroelastic Method . 69 3.3.2 Methodology to find the First Unstable Operational Point . 70 3.4 ImplementationoftheMethod . 78 4 Verification of the Method 81 4.1 Finite element model for the 5MW Reference Wind Turbine . 82 4.1.1 Tower .............................. 82 4.1.2 HubandNacelleModel .................... 83 4.1.3 Blades.............................. 84 4.1.4 Boundary Conditions and Connection between the Blades, Nacelle,HubandTower . 84 4.1.5 Pitch, Rotational and Wind Speed Control Curves . 86 4.1.6 ParametricModelinPATRAN . 86 4.1.7 Modelsummary ........................ 88 4.1.8 Simple Model for Stability Analysis . 89 4.2 Unstable Operational Points for the 5MW Wind Turbine . 92 4.2.1 Static Unstable Points . 92 4.2.2 Dynamic Unstable Points . 93 5 Application: Analysis of a 20MW Wind Turbine 95 5.1 Upscaling process of the 5 MW NREL wind turbine to an optimum 20MW.................................. 95 5.2 Finite Element Model for the 20MW Reference Wind Turbine . 96 5.2.1 Tower .............................. 96 5.2.2 HubandNacelleModel .................... 97 5.2.3 Blades.............................. 98 5.2.4 Boundary conditions and connection between the blades, nacelle,hubandtower. 98 CONTENTS iii 5.2.5 Pitch, Rotational and Wind Speed Control Curves . 101 5.2.6 ParametricModelinPATRAN . 101 5.2.7 Modelsummary ........................ 102 5.3 Unstable Operational Scenarios for the 20MW Wind Turbine . 104 5.3.1 Static Unstable Points . 104 5.3.2 Dynamic Unstable Points . 105 6 Conclusions and Recommendations 107 6.1 Conclusions ............................... 107 6.2 Recommendations ........................... 109 A Appendix I: Blade layout for the 5MW Wind Turbine 111 Bibliography 115 Nomenclature List of Symbols A Sectional blade area [m2] a Acceleration [m/s2] b Diameter ratio between the upscaled and the reference blade [-] C Theodorsen function or tensor notation [ ] − Cl Aerodynamic lift coefficient [ ] − Cm Aerodynamic moment coefficient [ ] D Sectional drag [N/m− ] d1 Distance from the elastic axis to 1/4 of the airfoil chord [m] d2 Distance from the elastic axis to 3/4 of the airfoil chord [m] Cd Aerodynamic drag coefficient [ ] − Da Aerodynamic damping matrix * Dr Damping matrix due the rotational frame * Ds Structural damping matrix * e1 Unitary vector which defines the direction of the h DOF * e Lift direction without rotational speed of the blade [ ] 10 − e2 Unitary vector which defines the direction of the s DOF * e3 Unitary vector which defines the direction of the α DOF * e30 Drag direction without rotational speed of the blade [ ] F Generalized force *− Fpkm Force acting on the node k in the direction m [N] Fr Generalized force due the rotational frame * h DOF defined on the lift direction [m] H Hankel function [ ] − h0 Value around h is linearized or plunging amplitude [m] i Imaginary number unit [ ] k Reduced frequency or index [−] − Ka Aerodynamic stiffness matrix * 1 2 CONTENTS 0.0 Kr Stiffness matrix due the rotational frame * Ks Structural stiffness matrix * L Sectional lift [N/m] m Mass [kg] M Sectional moment [N/m] Mpkm Moment acting on the node k in the direction m [Nm] Ma Aerodynamic mass matrix * Mri Equivalent mass matrix due the rotational frame * Ms Structural mass matrix * n Direction cosines [Pa] p Position vector [m] R Position vector [m] R1 Lift functional [Nm] R2 Moment functional [Nm] R3 Drag functional [Nm] s DOF defined on the drag direction [m] S Surface [m2] s0 Value around s is linearized [m] t Time or surface traction [s], [pa] u Generalized displacements * uk3m Displacement on the node number k in the direction m [m] uk3θm Angular rotation on the node k in the direction m [rad] V Volume domain [m3] x Coordinate x in space [m] X External forces [N/m3] x1 First basis component of the reference coordinate system [m] x2 First basis component of the second coordinate system [m] x3 first basis component of the global coordinate system [m] xs x axis parallel to the wind speed [ ] y Coordinate y in space [m−] y1 Second basis component of the reference coordinate system [m] y2 Second basis component of the second coordinate system [m] y3 Second basis component of the global coordinate system [m] ys y axis parallel to the rotational wind speed [ ] z Coordinate z in space [m−] z3 third basis component of the global coordinate system [m] Greek symbols α Angle of attack [rad] α0 Value around α is linearized or pitching amplitude [m] Ω Angular velocity [rad/s] 0.0 CONTENTS 3 ω Frequency or weight function [rad/s], [ ] ρ Density [kg/m3] − ε Mechanical deformation [ ] σ Mechanical stress [Pa− ] Subscripts 0 Configuration initial a Refers to aerodynamic blade Complete blade domain e Element matrix i Element index, direction index j Counter index, direction index n Identification number or coordinate component of a vector ns Number of sections on the blade p Pitch of the blade section r Angle of attack of the blade section section Refers to a section of the blade t Twist of the blade section up Upper side of the blade us Lower side of the blade V Wind speed Superscripts (1) Hankel function of first order (2) Hankel function of second order Notations bold Vector or matrix ()n Respect to the n coordinate system DOF Degree of freddom DOFs Degree of freddoms Determinant <| · |, > Inner product · · The symbol * means the vector or matrix has the unit of its contained equations. Chapter 1 Introduction 1.1 WindEnergy The development of the human society is influenced by the use of energy. The energy helps the society to manage the natural resources doing easier the adaptation to a new environments. This is the reason why managing the energy is inevitable in any society. The development of energy resources is essential for transportation, agriculture, waste collection and communications, which play an important role in a developed society. The energy consumption has been increasing since the industrial revolution and this brought with it a number of serious problems. These problems are re- lated to a critical damage of natural environments. One example is the global warming which present potentially grave risks to the world. Today the consumption per capita is 115 times higher than the energy consump- tion for an primitive human (See figure 1.1). The energy becomes every day an important subject. Therefore, different types of renewable energy are in development and under investigation, with wind energy as one of those. Wind energy promises to be one of the affordable green alterna- tive energies. The mechanism of energy generation of wind turbines is the conversion of the kinetic energy of the free streaming air to a mechanical power, which in turn can be used to rotate a generator to produce electricity. The increment use of wind energy to obtain electricity is presented in figure 1.2. At the end of 2009, the energy generated by wind was 2% of worldwide electri- city usage. However, still the electricity produced by other technologies is cheaper than the electricity produced by wind. Therefore, many scientist and engineers are attempting every day to developed technologies which decreases the costs of 5 6 INTRODUCTION 1.2 Estimated Daily Consumption of Energy per Capita at Different Historical Points 250 Transportation Industry and Agriculture Home and Commerce 200 Food 150 Kcal 1000 100 50 0 1 2 3 4 5 6 Figure 1.1: Estimated Daily Consumption of Energy per Capita at Different Historical Points Adapted from: E. Cook, “The Flow of Energy in an Industrial Society” Scientific American, 1971 p.