Aero-Structural Optimization of a 5 MW Wind Turbine Rotor THESIS

Transcription

1 Aero-Structural Optimization of a 5 MW Wind Turbine Rotor THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Richard W. Vesel Jr. Graduate Program in Mechanical and Aerospace Engineering The Ohio State University 2012 Master s Examination Committee: Professor Jack J. McNamara, Advisor Professor Mei Zhuang

2 Copyright by Richard W. Vesel Jr. 2012

3 Abstract A 5 MW wind turbine rotor blade based on the NREL 5 MW reference turbine is optimized for maximum efficiency and minimum flapwise hub bending moment. Eighty three total design variables are considered, encompassing: airfoil shapes modeled by Bézier curves, defined near the root, mid-span, and tip; chord and twist distributions; and the amount of bend-twist coupling in the blade. Optimization is achieved with a genetic algorithm. A relatively new method requiring significantly less computation than finite element analysis is utilized for planning and predicting the bend-twist coupling behavior of the rotor. Airfoil performance is predicted with XFOIL, and wind turbine simulations are performed in FAST. The objective function is cost of energy rotor cost ($) (COE), defined as, where AEP is annual energy production. Reductions in flapwise bending loads and blade surface area are assumed to correspond AEP (MWh/yr) to decreases in rotor cost due to material savings. As a result of the optimization, hub flapwise bending loads and blade surface area are each reduced by about 15%, without any decrease in AEP, yielding a 6.8% reduction in COE. ii

4 Dedication This thesis is dedicated to MU. iii

5 Acknowledgements I would like to extend my thanks to all who read this thesis. Also, my college life would have been substantially more expensive and less interesting without the mentorship of my advisor, Professor Jack McNamara, to whom I am grateful. I would also like to thank Professor Mei Zhuang for her participation on the examination committee, and for allowing me to serve as a teaching assistant in the Department of Mechanical and Aerospace Engineering. I also thank Sandia National Laboratory s D. Todd Griffith for contributing helpful information regarding large wind turbines, which aided in the completion of this work. iv

6 Vita B.S. Aeronautical and Astronautical Engineering, The Ohio State University Graduate Teaching Assistant, The Ohio State University, Aerospace Engineering Fields of Study Major Field: Aeronautical and Astronautical Engineering v

7 Contents Abstract ii Dedication iii Acknowledgements iv Vita Table of Contents List of Figures List of Tables v vi viii xii Nomenclature xiii 1 Introduction and Objectives Introduction Literature Review Chord and Twist Optimization Airfoil Optimization Structural Geometry and Material Selection Bend-Twist Coupling Research Objectives Key Novel Contributions in this Dissertation Geometric Model Airfoil Shapes Bézier Curve Airfoil Representation Airfoil Constraints Chord and Twist Distributions Geometric Model Construction Aerostructural Model Airfoil Aerodynamics vi

8 3.2 Wind Turbine Aerodynamics and Performance Aerodynamics Power Capture Bend-Twist Coupling Model Cost Model Annual Energy Production Rotor Cost Genetic Algorithm Introduction to Genetic Algorithms Variable Encoding Selection Crossover and Mutation Replacement Diversity Measure Results Baseline Blade Baseline Blade Geometry Bend-Twist Coupling in the Baseline Blade Baseline Blade Performance Case 1: Chord and Twist Only Case 2: Chord, Twist, and Bend-Twist Coupling Case 3: Full Geometry and Bend-Twist Coupling Concluding Remarks Conclusions Recommendations for Future Research Bibliography vii

9 List of Figures 1.1 Global installed wind energy capacity Wind turbine diagram Wind turbine size increase over time Comparison of an American football field with a 5 MW wind turbine rotor Optimized airfoils from Li et al Sample of optimized airfoils from Sun and Lee Sample of optimized airfoils from Endo Possible composite layup pattern for bend-twist coupling Stability boundaries with variation of coupling coefficient Original and modified twist distributions from Lobitz and Veers Flapwise root bending moment from Lobitz and Veers Schematic of wind turbine blade model including geometry, aerodynamics, structures, and cost estimation Airfoil defined by an erratic distribution of Bézier curve control points, from Xuan Discretization concealing the smaller features of an undulating curve Sample Bézier curve airfoil Normalized and rotated airfoil Airfoil demonstrating undulation on the upper surface and selfintersection near the trailing edge Geometric rotor model before application of chord or twist distributions Geometric rotor model after application of chord distribution Geometric rotor model after application of chord and twist distributions Geometric model of the baseline rotor, to scale Schematic of wind turbine blade model including geometry, aerodynamics, structures, and cost estimation Lift coefficient vs. angle-of-attack for the S818 airfoil Drag coefficient vs. angle-of-attack for the S818 airfoil Full extrapolation of lift coefficient vs. angle-of-attack for the S818 airfoil viii

10 3.5 Full extrapolation of drag coefficient vs. angle-of-attack for the S818 airfoil Lift to drag ratio vs. angle-of-attack for the NREL S818 airfoil Locations of the eight airfoils tested in XFOIL Diagram of 1D airflow through a wind turbine rotor Swept area of a wind turbine blade element Diagram of air flow in blade element theory Forces on a blade element Power coefficients up to rated wind speed for the NREL 5 MW reference turbine Power curve for the NREL 5 MW reference turbine Hub bending moment distributions at different operating conditions, from Maheri Quasi-static reference calculation flowchart Quasi-static performance calculation flowchart Weibull wind speed distribution Flowchart of genetic algorithm Simple example of a population of four individuals, sorted by fitness. Each box in an individual represents a binary digit Parents undergoing crossover and mutation as part of a genetic algorithm Population of solutions after replacement has occurred. Green has been replaced by the child of orange Roulette wheel for a simple population Example of crossover for a binary gene Example of a Hamming distance calculation for two binary strings Three dimensional view of the baseline blade Baseline chord distribution Baseline twist distribution Baseline airfoils Flapwise bending stiffness and torsional stiffness for the NREL 5 MW reference turbine Baseline bending moment distribution Induced twist distribution for baseline blade from integration of Equation Normalized β from integration calculation and model β distribution for γ = Error between integration calculation for induced twist and the model with γ = Best COE versus generation, Case ix

11 6.12 Initial and final populations, Case 1, after 9000 generations Initial and final populations, Case 1, after 9000 generations Rotor power curve, Case Hub bending moment versus wind speed, Case Hub bending moment reduction versus wind speed, Case Optimized chord distribution, Case Optimized twist distribution, Case Axial force distribution, Case Axial induction factor, Case Tangential force distribution, Case Angle of attack distribution, Case Best COE versus generation, Case Initial and final populations, Case 2, after 5000 generations Initial and final populations, Case 2, after 5000 generations Hub bending moment versus wind speed, Case Hub bending moment reduction versus wind speed, Case Rotor power curve, Case Optimized chord distribution, Case Optimized twist distribution, Case Axial force distribution, Case Axial induction factor, Case Tangential force distribution, Case Angle of attack distribution, Case Diversity and best COE versus generation, Case Initial and final populations, Case 3, after 2700 generations Initial and final populations, Case 3, after 2700 generations Rotor power curve, Case Optimized rotor blade, alternate view, Case Hub bending moment versus wind speed, Case Hub bending moment reduction versus wind speed, Case Optimized rotor blade, Case Optimized root airfoil, Case Optimized mid-span airfoil, Case Optimized tip airfoil, Case Optimized root airfoil lift coefficient versus angle-of-attack Optimized root lift-to-drag ratio versus angle-of-attack Optimized mid-span airfoil lift coefficient versus angle-of-attack Optimized mid-span airfoil lift-to-drag ratio versus angle-of-attack Optimized tip airfoil lift coefficient versus angle-of-attack Optimized tip airfoil lift-to-drag ratio versus angle-of-attack Optimized thickness distribution, Case Optimized chord distribution, Case x

12 6.54 Optimized twist distribution, Case Axial force distribution, Case Axial induction factor, Case Tangential force distribution, Case Angle-of-attack distribution, Case xi

13 List of Tables 2.1 Radial location of aerodynamic shapes. The shape at any intermediate location is the linear interpolation of the two adjacent shapes Chord and twist distribution control point locations Minimum and maximum chord values at each control point Blade production cost multipliers Blade costs over production run Rotor cost model parameters Diversity calculation for a population of five members (rows) and eight digits (columns). This population has relatively low diversity, since several columns contain only one value Diversity calculation for a sample population with five members, eight bits, and high diversity Baseline blade properties. Bold values are invariant Results of optimization for Case 1. Only chord and twist distributions were optimized. Airfoils were fixed and no bend-twist coupling was permitted Results of optimization Case 2: fixed airfoils only Results of optimization Case 3: full geometry and bend-twist coupling Comparison of the results from all three optimizations xii

14 Nomenclature Symbols A AR a s a a b b c C D C fixed C L C M C mm C P C tool D D i EI f obj f w Rotor swept area Aspect ratio Shear exponent in power law Tangential induction factor Axial induction factor Binary digit in a genome Column centroid Blade chord Airfoil drag coefficient Fixed costs Airfoil lift coefficient Airfoil moment coefficient Cost for master blades and molds Rotor power coefficient Cost for tooling Drag Diversity measure Flapwise bending stiffness Objective function Weibull wind speed probability distribution xiii

15 G g GJ I K L L B L/D M M hub M hub,ref N C N M P Q R r T t T h v e ip v e op V avg V d V V TOTAL Decoded gene integer value Bend-twist coupling term Torsional stiffness Moment of inertia Response surface model coefficients Lift Blade load parameter Lift-to-drag ratio Bending moment Hub bending moment Hub bending moment at reference operating condition Number of digits (columns) in a genome Number of members (rows) in a population Rotor power Rotor torque Rotor radius Radial location Twisting moment Airfoil thickness Rotor thrust Velocity due to in-plane deflection Velocity due to out-of-plane deflection Average wind velocity Downstream wind velocity Upstream wind velocity Total wind speed xiv

16 V z z α α g β β β T β T,d ε η Γ γ κ Ω ϕ ϕ 0 Φ τ θ ϑ ζ Velocity at height z Height Angle-of-attack Bending-torsion coupling coefficient Induced twist Normalized induced twist Induced tip twist Induced tip twist at design condition Error Wind turbine efficiency Gamma function Bend-twist coupling curve fit exponent Curvature Rotor angular velocity Blade twist Unloaded blade twist Bézier curve Bézier curve parametric variable Inflow angle Flapwise blade slope Bézier curve control point Acronyms AEP BEM CAS Annual energy production Blade element momentum Coupled aero-structure simulation xv

17 CFD COE EA FEA FEM GA MOOP PSO RPM RSM Computational fluid dynamics Cost of energy Evolutionary algorithm Finite element analysis Finite element method Genetic algorithm Multi-objective optimization Particle swarm optimization Rotations per minute Response surface model Subscripts avg D hub Average Designed or design condition Property at the rotor hub i, j Indices n ref TOT Normalized Property at the reference condition Total xvi

18 Chapter 1 Introduction and Objectives 1.1 Introduction Wind power is an increasingly important segment of energy strategy worldwide [1], demonstrated by the rapid global growth of installed wind energy capacity shown in Figure 1.1. Furthermore, in 2008, the U.S. Department of Energy set forth the goal of 20% of the United States electricity produced by wind by the year 2030 [2]. Wind turbines are used to convert the energy of moving air into electrical power. See Figure 1.2 for a diagram showing the main components of a wind turbine. The wind turbine rotor directly interacts with the wind, converting air movement into mechanical energy. The gearbox and generator, stored in the nacelle, then serve to convert the rotor s mechanical energy into electrical energy. The gearbox scales up Figure 1.1 Global installed wind energy capacity [1]. 1

19 the rotational velocity, which is only on the order of 10 rotations per minute (RPM) for large wind turbines, to a level usable by the generator, where the mechanical-toelectrical energy conversion process is completed. Not pictured are the foundation and the remaining electrical components and grid connections necessary for the transportation of electricity. As wind power generation has proliferated over the last three decades, the trend within the industry has been toward larger rotors, as shown in Figure 1.3. To appreciate the scale of a modern 5 MW wind turbine, see Figure 1.4, where an entire American football field fits within the rotor swept area. Historically, larger blade radius has led to a more rapid increase in lifetime energy production than in lifetime turbine costs, resulting in a reduction in lifetime cost of energy (COE), which can be defined as COE = Capital Cost + Operations Cost + Maintenance Cost. Lifetime Energy Production Note that a different formulation of COE is used later in this work. However, as blade lengths surpass the 60 m range, rotor costs are beginning to increase nearly as quickly or quicker than energy capture, primarily due to weight growth within the blade and other components. One way to approach this problem is to minimize blade weight and loading, which has a multiplier effect, reducing loads throughout the entire turbine [3]. Advanced design optimization, incorporating improved technologies, has the potential to accomplish this goal, and allow for further reductions in COE for large wind turbines. 2

20 Figure 1.2 Wind turbine diagram. 3

21 Figure 1.3 Wind turbine size increase over time. Figure 1.4 Comparison of an American football field with a 5 MW wind turbine rotor. 4

22 1.2 Literature Review Many wind turbine optimization studies are available in the literature, encompassing different approaches to improving performance and reducing COE. Since optimization of the rotor blade is the focus of the present work, prior work relating to optimization of rotor geometry and structure will be reviewed, as will prior work relating to the inclusion of structural coupling effects in wind turbine rotors. Gaps in the existing literature and opportunities to further advance state-of-the-art in wind turbine optimization and design will be identified. Prior work relating to wind turbine optimization can be divided into roughly 5 categories, in order of increasing modeling complexity: 1) Wind turbine properties such as diameter, hub height, number of blades, rotor speed, regulation type, and tip speed [4, 5, 6]. The objective function in this case may be a function to estimate rotor cost and AEP. It may incorporate a blade element momentum (BEM) code or an inverse design code such as NREL s WT_Perf. 2) Rotor blade chord and/or twist distributions [5, 7, 8]. BEM analysis is often used to determine the effect of design changes, although more sophisticated methods, including computational fluid dynamics (CFD), may be used. 3) Airfoil shape(s) [9, 10, 11, 12]. An airfoil analysis program can be employed, with subsequent BEM analysis if the overall rotor performance is sought. Alternatively, higher fidelity CFD may be applied to the airfoils or the blade as a whole. 4) Structural geometry/material selection [13, 14, 15]. This typically necessitates finite element analysis (FEA) and coupled aero-structural analysis using an 5

23 aerodynamic solver. 5) Bend-twist or sweep-twist coupling behavior [16, 3, 17, 18]. This is an extension of category four, and creates an even greater burden on the accuracy of FEM models to correctly capture coupling effects [19]. To the author s knowledge, no optimization has been performed for bend-twist or sweep-twist coupling incorporating full FEM evaluations as part of the objective function. Using structural coupling to improve wind turbine performance and reduce loads is an active area of interest within the industry [19, 20]. Since the present work engages in geometric optimization of the rotor geometry with respect to aerodynamic and structural considerations, the literature strictly within the first category is omitted from further discussion. A survey of wind turbine optimization efforts in categories 2-4 is conducted below, followed by a discussion of the literature related to bend-twist coupling in wind turbines Chord and Twist Optimization Liu et al. [7] utilized an extended compact genetic algorithm (ECGA), which offers faster convergence, to optimize a 30 m, 1.3 MW stall regulated wind turbine blade for maximum power capture. Design variables were chord and twist distributions, and aerodynamics were modeled via a custom BEM code. The optimization produced small changes to the blade geometry, resulting in a net improvement of about 10% throughout the operating range. The most substantial change was to the pitch distribution, where it appeared that a simple change in pitch setting angle would have had a similar effect to the optimization. Benini and Toffolo [5] used a genetic algorithm to optimize a 600 kw, 800 kw, and 1 MW stall regulated wind turbine for two objectives: minimum cost of energy 6

24 and maximum AEP per square meter of wind park. There were 14 design variables, incorporating tip speed, r hub /R ratio, chord distribution, and twist distribution. Cost of energy was a function of blade mass and turbine rated power, while AEP was calculated via BEM analysis. They concluded that large blades with low loading offer the best opportunity to minimize COE. In 2009, Xudong, Shen and Jin [8] optimized three different wind turbines, including the NREL 5 MW reference turbine, for minimum COE. They combined a BEM method with an aeroelastic solver utilizing the general dynamic equations. Design variables were chord and twist distributions, relative thickness distribution, and tip pitch angle. The MATLAB function fmincon was used to perform the optimization. Constraints were imposed on the problem, including that the total rotor thrust was less than or equal to the maximum thrust of the original rotor, and the shaft torque is less than or equal to the maximum of the original rotor. The optimized power curve for the NREL virtual rotor was nearly unchanged from the original, while the cost of energy was reduced by around 2.6% due to reductions in the chord. Blade weight was assumed to be directly proportional to the chord distribution, an approach which neglects the dependency of blade mass on structural loads. The above three studies were limited in their approach, in that they only addressed some aspects of blade geometry, primarily chord and twist distributions, while neglecting airfoil shapes or rotor size. Furthermore, loading was neglected in all but the last study. Finally, each work represents a relatively small refinement of a baseline design, as opposed to a more substantial deviation. The motivation in the present work to produce a more unique design solution is discussed in Section

25 1.2.2 Airfoil Optimization Several studies have been conducted regarding wind turbine airfoil optimization. Li et al. [10] used a response surface technique combined with fmincon to optimize an airfoil for wind turbine use, by means of maximizing lift to drag ratio, L/D. A penalty for reduced surface area was applied in order to allow room for the internal structure. In response surface techniques, the objective function, f obj, is modeled as a quadratic function of n design variables, as shown below: f obj = K 0 + n K j x j + j=1 n K jj x 2 j + j=1 n 1 n i=1 j=i+1 K ij x i x j. The above equation forms the response surface model (RSM). After the objective function has been evaluated over a sample set of design variable values via experimentation or calculation, the coefficients, K, can be determined from a least-squares regression analysis. To reduce computational cost, Li used eight design variables and 28 sample evaluations to model the response surface, while ignoring the interaction terms, K ij. Airfoil performance was calculated via the Reynolds-averaged Navier-Stokes (RANS) equations in Fine/TURBO, using a Spalart-Allmaras turbulence model. After RSM construction, the MATLAB function fmincon was used to optimize variable selection for maximum L/D, resulting in an improvement of 7-15% for different baseline airfoils. A second optimization loop was performed using refined variable ranges, resulting in a net improvement of 10-24%. See Figure 1.5 for a sample of baseline and optimized airfoils. While not directly for wind turbine optimization, Sun and Lee [9] combined a response surface model with a genetic algorithm in order to optimize helicopter rotor airfoil shape. Improvements between 0-5% were obtained for rotor coefficient of power 8

26 or coefficient of torque, depending on the objective function. A sample of the baseline and optimized airfoils from this work can be seen in Figure 1.6. In this work and the previous airfoil optimization studies discussed above, similarly to the studies presented in Section 1.2.1, small refinements are obtained over the baseline designs. Similarity between baseline and design solution can be explained partly by a high quality baseline. However, there is also the risk that the objective function model is not sufficiently robust to predict the performance for designs that deviate substantially from the baseline. Additionally, there should be some concern over whether one has optimized a design for real improvements in practice, or simply for slightly better performance in a particular numerical model, without achieving a significant difference in the physical world. In 2010, Grasso [11] used a gradient-based algorithm to optimize a tip-region wind turbine airfoil for maximum L/D at 7 angle-of-attack. In contrast to the other airfoil studies discussed in this section, several geometric and aerodynamic constraints were Figure 1.5 Optimized airfoils from Li et al. [10]. 9

27 applied to ensure the optimized airfoil was suitable for wind turbine use. RFOIL, a modified version of XFOIL, was used to calculate airfoil performance. The optimized airfoil was compared to several existing wind turbine airfoils, demonstrating increased C Lmax from 1.5 to over 2.0, with higher maximum L/D in most cases. Finally, in a 2011 MS thesis, Endo [12] used a particle swarm optimization (PSO) technique to maximize lift and minimize drag on a wind turbine airfoil. Aerodynamics were computed via the RNG κ-ɛ model in the CFD code FLUENT. In PSO, individual solutions are represented as having a position and velocity in the design space, influenced by the locations and fitnesses of surrounding solutions, and moving collectively toward higher performance. The only constraint on airfoil shape was to prevent intersection of the upper and lower surfaces. This lack of constraints, combined with a highly flexible airfoil representation scheme and optimization mechanism, produced many optimized airfoils with unusual geometry, as shown in Figure 1.7. In addition to the lack of deviation from baseline designs exhibited in some of the above studies, two other areas for concern can be pointed out. First, the airfoils are Figure 1.6 Sample of optimized airfoils from Sun and Lee [9]. 10

28 evaluated in isolation, without respect to any change their design might have on actual wind turbine performance. To find an optimal wind turbine airfoil, given particular objectives, e.g. maximum power, minimum bending, etc., the beneficial effect of the airfoil optimization should be calculated directly rather than assumed. Second, in a large wind turbine, different airfoil shapes are blended across the rotor span, meaning that at most positions, the airfoil performance is a result of the interaction between two or more airfoil shapes. If those airfoils were optimized separately, there is no guarantee that the performance of the blended airfoils is optimal as well. Figure 1.7 Sample of optimized airfoils from Endo [12] where the baseline airfoil is a dashed line, and optimized airfoils are solid lines. 11

29 1.2.3 Structural Geometry and Material Selection Regarding structural optimization, Jureczko, Pawlak and Mężyk [13] used a genetic algorithm to optimize the material selection and structural properties for a 45 m wind turbine blade using ANSYS. Five design variables were considered, including shell thickness, web thickness, number of stiffening ribs, and the location of stiffening ribs. Objective criteria was minimum weight, while satisfying constraints on blade stress, maximum deflection, and natural frequency. Optimized results showed substantial improvement in blade weight throughout the course of the algorithm, although no baseline was offered for comparison. Although aerodynamic loads were calculated, no aerodynamic optimization of the blade took place. Grujicic et al. [15] created an optimization scheme for minimizing COE via structural and aerodynamic optimization. The optimization consists of two loops - a loop that optimizes the structure for minimal mass with constraints on stress, deflection, and fatigue life, and a separate loop that optimizes chord and twist distributions for maximum AEP. JavaFoil was utilized for airfoil analysis, and wind turbine performance was predicted by the NREL design code PropID. The NREL S818 airfoil was used over the entire rotor. Around a 3.5% improvement in COE was obtained for the optimized blade, due to a large increase in AEP, although the blade mass actually increased. The final design was not compared to an existing wind turbine blade. Some areas for improvement in the above studies can be identified. First, the optimizations were limited to design spaces of five and eight design variables, respectively, in order to avoid excessive computation. Thoroughness of the aerodynamic optimization was also lacking. In Grujicic, a single airfoil was used for the entire rotor, which is not efficient for a large blade, and in Jureczko, no aerodynamic optimization took place. Finally, no consideration was made of structural coupling effects, although 12

30 that was not a stated goal in either study Bend-Twist Coupling The structural coupling effect of interest in this work is bend-twist coupling, which is the tendency for a wind turbine blade to twist when loaded in bending. This is accomplished by taking advantage of orthotropic material properties, typically by means of altering the layup pattern of the composite materials in the blade skin and/or structure, as shown in Figure 1.8. This effect, when combined with a pitch control system, has the potential to increase power capture by 1-3%, as well as reduce extreme and cyclic loads [21, 16]. These benefits, however, come at the risk of reduced stability margins, as illustrated in Figure 1.9, as well as potentially increased structural complexity and cost of manufacture. Despite these potential drawbacks, bend-twist coupling represents an advance in wind turbine technology that can contribute toward COE reduction. The literature relating to bend-twist coupling in wind turbine blades will now be reviewed. In 1999, de Goeij, van Tooren, and Beukers [22] modeled bend-twist coupling via FEM for a 26.3 m blade by implanting angled glass and carbon fibers in the skins. For a [25/-65/25] S configuration, 3.91 of tip twist was observed in the blade at a wind speed of 13 m/s. A 1.2 m spar test section was constructed, without the aerodynamic skin, that achieved 1.9 of twist, showing good agreement with FEM calculations. They concluded that to using bend-twist coupling to mitigate loads rather than increase power capture was more practical, due to the lower necessary coupling. Lobitz and Veers [16] examined load mitigation via bend-twist coupling for a 14.9 m blade. The study encompassed several different control strategies, including a variable speed, pitch-regulated turbine. The blade was designed with about 5 of 13

31 Figure 1.8 Possible composite layup pattern for bend-twist coupling. Fibers are angled in a mirror type configuration, where the direction on the upper surface matches that of the lower surface. tip twist at a design condition of 13 m/s. The initial twist distribution of the bendtwist coupled rotor was modified, as shown in Figure 1.10, such that, at the design condition of 13 m/s, the original twist distribution was recovered due to the bendtwist coupling effect. The blade s aeroelastic response to turbulent wind loading was then modeled in ADAMS. As shown in Figure 1.10, the effect of bend-twist coupling was a reduction in the amplitude of cyclic flapwise root bending moment loads by 10-20%. Fatigue damage reductions were even greater, due to the nonlinear relationship between fatigue damage and load amplitude. Maheri, Noroozi and Vinney [23] developed a simplified method for predicting bend-twist coupling behavior in wind turbine blades. They showed that the bendinduced twisting response of the blade could be modeled as a function of thickness distribution and flapwise bending moment distribution only, and that the normalized 14

32 Figure 1.9 Stability boundaries with variation of coupling coefficient, α [21]. A typical blade has a coupling coefficient close to zero. flapwise bending moment distribution was nearly independent of operating condition. They applied this method to the optimization of a 13.7 m radius, stall regulated wind turbine for maximum power capture. They obtained an average power increase of 15%, some of which was accounted for by a 10% increase in swept area. Other than the level of bend-twist coupling, only rotor radius and blade pre-twist were modified by the optimization. Deilmann [17] found the desired twist values for an optimally adaptive blade, in which adjacent blade elements are allowed to freely rotate against one another in order to seek the most efficient operating condition for each given wind speed. Then, the closest approximation to those values was sought for a bend-twist coupled, pitch controlled blade. Unfortunately, in addition to the lack of any interaction between 15

33 Figure 1.10 Original and modified twist distributions from Lobitz and Veers [16]. the real aerodynamic loads and the structural response during the optimization, the final blade design twisted opposite the desired direction. Although bend-twist coupling offers potential reductions in cyclic loading, to the author s knowledge, the effect of reductions in mean load levels, combined with cyclic load reductions, on wind turbine fatigue life have not yet been examined in the literature. Similarly, no rotor optimization studies, including all main aspects of geometry such as airfoils, chord, and twist, as well as bend-twist coupling effects, were found in the literature. 16

34 Figure 1.11 Flapwise root bending moment from Lobitz and Veers [16]. 17

35 1.3 Research Objectives In light of the literature review conducted above, several opportunities can be identified for advancing the state-of-the-art in large wind turbine optimization: Airfoil optimizations where the objective function is a measure of performance of the wind turbine rather than the airfoil Simultaneous optimization of the different airfoil shapes defining a rotor blade Optimization of a rotor for minimum overall blade loading in combination with high energy output Comprehensive aerodynamic optimization of the rotor, including chord, twist, and airfoil shape distribution Comprehensive aerostructural optimization of the rotor, addressing structural effects such as bend-twist coupling in addition to aerodynamics as described above Examination of the combined effect of reduced mean and cyclic load values on wind turbine fatigue life Rotor designs that differ substantially from the baseline, to avoid trivial refinements produced by a particular numerical model Optimized designs demonstrating a substantial improvement over high quality baseline wind turbines The wind turbine operates as an interrelated system, and optimizing particular components in isolation is less likely to yield the global optimal solution. To the author s 18

36 knowledge, a comprehensive optimization of rotor blade geometry and structure, accounting for varying airfoil shape, chord distribution, twist distribution, and bendtwist coupling effects, has not been performed in the literature. Although the problem is complex, it is not intractable. An optimization of this kind is the primary focus of the current work. As such, three primary research objectives can be defined: 1. Develop a wind turbine blade geometric and structural model that incorporates: a highly flexible airfoil shape that is variable across the rotor; chord distribution; twist distribution; and a simplified model for bend-twist coupling effects 2. Optimize the blade model for minimum COE by means of simultaneous minimization of blade loads and maximization of energy output, where COE = Rotor Cost Annual Energy Output 3. Compare the optimized solution(s) to a high quality baseline 1.4 Key Novel Contributions in this Dissertation Achieving the stated objectives will represent an important advance in the field of multidisciplinary wind turbine optimization, particularly with regard to load reduction and structural coupling in large blades. The following contributions are unique to this study: 1. Several primary components of rotor geometry are optimized simultaneously: Three airfoil shapes, encompassing the inboard, mid-span, and tip Chord distribution Twist distribution 19

37 Bend-twist coupling behavior 2. A comprehensive rotor blade design with significantly reduced flapwise bending loads, surface area, and no reduction in energy output is produced. 3. A unique rotor blade design that differs substantially from conventional designs and demonstrates improved performance over a high quality baseline is produced. 20

38 Chapter 2 Geometric Model The wind turbine blade model is composed of three main components, shown schematically in Figure 2.1: 1. Geometric model, including airfoil shapes, chord distribution and twist distribution 2. Aerostructural model, including airfoil and wind turbine aerodynamics, as well as the interaction between the bend-twist coupling model and wind turbine aerodynamic loading 3. Cost model, where the overall rotor cost is estimated This chapter contains a detailed description of the geometric model. The aerostructural model and cost model are addressed in Chapters 4 and 5, respectively. 2.1 Airfoil Shapes Including the cylindrical sections near the hub, a total of four aerodynamic shapes are used to define the airfoil distribution across the blade. Only the last three airfoil shapes, after the cylindrical sections have tapered into airfoil shapes, are treated as 21

39 Figure 2.1 Schematic of wind turbine blade model including geometry, aerodynamics, structures, and cost estimation. Arrows represent interaction between the bend-twist coupling model and the wind turbine aerodynamic simulation. part of the optimization. The radial locations of the four aerodynamic shapes are shown in Table Bézier Curve Airfoil Representation The primary requirements of the airfoil representation scheme are the flexibility to model virtually any airfoil shape, and validity of the airfoils, avoiding unphysical and impractical solutions. In this study, Bézier curves [24] are used for airfoil parameterization. Bézier curves are defined by a series of control points, where the first 22

40 Table 2.1 Radial location of aerodynamic shapes. The shape at any intermediate location is the linear interpolation of the two adjacent shapes. Radius (m) Aerodynamic Shape 2.87 Cylinder 5.6 Cylinder 11.7 Root Airfoil 37.8 Mid-Span Airfoil 63 Tip Airfoil and last control point serve as the beginning and end of the curve, respectively. In this case, these two points are placed at the origin, which locates the trailing edge of the airfoils. The intermediate control points exist as guides, and are not typically intersected by the curve. For sufficiently many control points, Bézier curves provide a flexible representation scheme for airfoil optimization. Steps are taken to ensure validity, as described in Section For a series of control points ζ i for i = 1, 2,..., P, where each ζ i = {x i, y i }, the Bézier curve, Φ(τ) = {x(τ), y(τ)}, can be defined parametrically over the range τ = [0, 1] as: Φ(τ) = where D i = k ζ i D i i=1 and k = P 1 k! i!(k i)! τ i ( 1 τ k i). In the above, D i is a scalar. In the script that computes the Bézier curves, the factorials 1 through k are calculated and stored in advance in order to improve efficiency. In the present work, 26 total control points are used, where the first and last points are fixed at the origin as described above, and the remaining 24 points are constrained 23

41 in the x-direction. Thus, every airfoil is defined by 24 design variables, designating the y-coordinate of the movable control points. Fixing the x-coordinate helps avoid erratic control point distributions, which leads to difficulty refining the airfoil shape due to small changes in control point locations resulting in severe and unexpected changes to the Bézier curve. An example of an undesirable control point distribution is shown in Figure 2.2. The control points are also slightly grouped toward the leading edge, leading to higher resolution of the leading edge in the airfoil definition. This benefits the aerodynamic analysis of the airfoil, where good resolution of the leading edge is required. Figure 2.2 Airfoil defined by an erratic distribution of Bézier curve control points, from Xuan [25]. The actual airfoil is defined by 201 points, corresponding to the number of divisions of the interval over which τ is defined. This number of points was chosen for two reasons. First, it gives high enough resolution of the airfoil for aerodynamic analysis in the panel code (XFOIL - see Section 3.1). Second, it reduces the risk that smaller features of the Bézier curve are obscured by the discretization, as shown in Figure 2.3. This is important for determining airfoil validity, discussed below. A sample airfoil and the corresponding Bézier curve control points are shown in Figure 2.4. Two normalizations are applied to the Bézier curves, as shown in 24

42 Figure 2.3 Discretization concealing the smaller features of an undulating curve. Figure 2.5. First, the airfoil is normalized to unit chord. Second, it is rotated such that the leading edge is placed on the x-axis. This is important to isolate the effect of design variables on the chord and twist distributions. Figure 2.4 Sample Bézier curve airfoil Airfoil Constraints Given the above airfoil definition, two criteria are enforced as constraints. The first is intersection between the upper and lower surfaces, referred to as self-intersection, and 25

43 Figure 2.5 Normalized and rotated airfoil. the second is more than one change in the sign of curvature of the airfoil, resulting in an undulating surface. An airfoil displaying both self-intersection and undulation is shown in Figure 2.6. Self-intersection must be avoided since the resulting shape is unphysical. The constraint on undulation, however, may not be necessary with a higher fidelity aerodynamic model. Ideally, the optimization process should be allowed to self-select the best airfoil shape, but this is impossible in a design process with unmodeled physics. Therefore, it is up to the analyst to enforce intuitive constraints, in this case a restriction against overly unconventional airfoil designs containing undulation. To check for self-intersection, the airfoil is divided into upper and lower surfaces. Then, the y-coordinate for each point on the lower surface is subtracted from that of the corresponding point on the upper surface. If any of these values are negative, the surfaces cross, and the airfoil is considered invalid. The check for excessive changes in curvature is slightly more involved. For both the upper and lower surfaces, the signed curvature, κ, is calculated at each point from the following equation [26]: κ i = x iy i y ix i (x 2 i + y. i 2)3/2 26

44 Figure 2.6 Airfoil demonstrating undulation on the upper surface and selfintersection near the trailing edge. The first derivative, x i, is calculated by applying the diff function in MATLAB to the vector of x-coordinates, and the second derivative, x i, is calculated by applying diff again. The proper derivatives would be scaled by 1 200, since x = x, but since t only the sign of curvature is of importance here, this is neglected. The y derivatives are calculated in the same manner. After the curvature is calculated, the number of sign switches in the curvature is counted for each surface separately. If either count totals more than one, there are three or more total directions of curvature on that surface, which indicates undulation, and is defined here as an invalid airfoil shape. 2.2 Chord and Twist Distributions The chord and twist distributions are each modeled by 5 control points, at fixed radial locations, shown in Table 2.2. For practical purposes, the chord is set to vary linearly between the control points. A nonlinear chord distribution may be more difficult to fabricate, in addition to producing high inertial or gravity-induced 27

45 stresses, for example, in a blade that tapers toward the mid-span and then gets larger (and heavier) toward the tip. Since no structural or aeroelastic analysis, other than consideration of bend-twist coupling effects, is performed in this work, these risks are minimized by enforcing a minimum and maximum permissible chord value at each control point, as shown in Table 2.3. This guarantees a relatively conventional blade planform that tapers linearly toward the tip. The twist distribution, however, is allowed to vary nonlinearly. The control points are connected via Bézier curve segments, similar to a spline curve, using a script obtained from the MATLAB file exchange [27]. Samples of control points and the resulting chord and twist distributions are provided in Chapter 3. Table 2.2 Chord and twist distribution control point locations. Chord Twist Control Point Location (m) Location (m) Table 2.3 Minimum and maximum chord values at each control point. Control Point Location (m) c min (m) c max (m)

46 2.3 Geometric Model Construction The blade is divided into 25 sections, which serve as the elements for blade element analysis (see Section 3.2.1): two cylindrical sections near the root, three sections transitioning from a cylinder to the first airfoil shape, and 20 sections over the remainder. The overall blade shape is generated by the following steps: 1. Connect the aerodynamic shapes to form the blade basis with unit chord, as shown in Figure Apply the chord distribution, which scales the shape at each spanwise station by the local chord value, as shown in Figure Apply the twist distribution, which rotates each section about the 1/4 chord point toward feather (leading edge into the oncoming wind direction) by the local pre-twist value, as shown in Figure 2.9 The geometric model of the baseline rotor resulting from the process described above is shown to scale in Figure Figure 2.7 Geometric rotor model before application of chord or twist distributions. 29

47 Figure 2.8 Geometric rotor model after application of chord distribution. Figure 2.9 Geometric rotor model after application of chord and twist distributions. 30

48 Figure 2.10 Geometric model of the baseline rotor, to scale. 31

49 Chapter 3 Aerostructural Model The wind turbine aerostructural model, shown with respect to overall blade model in Figure 3.1, consists of three main components: airfoil aerodynamics, wind turbine aerodynamics, and the bend-twist coupling model. This chapter provides a detailed description of these components. 3.1 Airfoil Aerodynamics Airfoil performance was calculated primarily though the 2D panel code XFOIL. XFOIL is a viscid-inviscid airfoil analysis code [28]. Inviscid calculations are via a linear-vorticity panel method with a Karman-Tsien compressibility correction. The boundary layer and transition equations are solved simultaneously with the inviscid flowfield by a global Newton method. XFOIL is convenient and efficient for airfoil performance prediction. However, it may overestimate lift coefficient [11], and it is not intended to make predictions beyond stall. Other methods must be employed in order to estimate airfoil behavior at high positive and negative angles of attack. In the present work, since the wind turbine is optimized for maximum efficiency as well as minimum loads, it is expected that most of airfoil operation will be in the linear aerodynamic range, where high 32

50 Figure 3.1 Schematic of wind turbine blade model including geometry, aerodynamics, structures, and cost estimation. Arrows represent interaction between the bend-twist coupling model and the wind turbine aerodynamic simulation. angles of attack are not of concern. However, a complete table of airfoil coefficients, spanning from -180 to 180 is needed, since, while searching for the point of optimal efficiency, the airfoils may encounter extreme angles of attack. A simple method is used for predicting the actual stall point from the XFOIL output, which compensates for over-prediction of C Lmax. A line is drawn from 5 to the maximum angle-of-attack from XFOIL, and the elbow in the C Lα found the point furthest from the line. This will be at or before the C Lmax curve is found by XFOIL. 33

51 A method based on the Viterna equations, as presented by Tangler and Kocurek [29], is used for extrapolating airfoil data into the post-stall regime. The extrapolation of drag coefficient is calculated for α stall α 90 from C D = B 1 sin 2 α + B 2 cos α (3.1) Figure 3.2 Lift coefficient vs. angle-of-attack for the S818 airfoil. Figure 3.3 Drag coefficient vs. angle-of-attack for the S818 airfoil. 34

52 where B 1 = C dmax = AR B 2 = C D stall C Dmax sin 2 α stall cos α stall R AR =. chord 0.8R Similarly, the lift coefficient is extrapolated over α stall α 90 from C L = A 1 sin 2α + A 2 cos 2 α sin α (3.2) where A 1 = B 1 /2 A 2 = (C Lstall C Dmax sin α stall cos α stall ) sin α stall cos 2 α stall For angles of attack between 90 and 180 positive or negative, lift and drag coefficients are approximated as the flat plate values: C Lplate = 2 sin α cos α (3.3) C Dplate = 2 sin 2 α (3.4) Examples of lift and drag coefficients resulting from XFOIL analysis combined with the extrapolation methods described above are shown in Figures , respectively. The airfoil tested is the NREL S818, which is a member of an airfoil family for large wind turbines that includes the NREL S827 and S828 airfoils [30]. These three airfoils are the used as the baseline airfoil shapes in this study, see Section Since the optimization was conducted on an eight core desktop PC running Windows, eight XFOIL and/or FAST computations could be run simultaneously. The blade was divided into 20 spanwise elements, and airfoils properties were calculated at the eight locations shown in Figure 3.7, corresponding to elements 1, 4, 7, 9, 11, 35

53 14, 17, and 20. Airfoil properties at intermediate locations were taken as the linear interpolation of the properties for the two adjacent airfoils. The cylindrical and semi-cylindrical airfoils preceding the first pure airfoil near the rotor hub are also modeled in FAST. The cylindrical sections have a lift coefficient of 0, and a drag Figure 3.4 Full extrapolation of lift coefficient vs. angle-of-attack for the S818 airfoil. Figure 3.5 Full extrapolation of drag coefficient vs. angle-of-attack for the S818 airfoil. 36

54 coefficient of 0.5 [31]. The aerodynamic coefficients are interpolated between the last pure cylindrical section and the first pure airfoil section, as with the rest of the rotor. Figure 3.6 Lift to drag ratio vs. angle-of-attack for the NREL S818 airfoil. Figure 3.7 Locations of the eight airfoils tested in XFOIL. 37

55 3.2 Wind Turbine Aerodynamics and Performance Aerodynamics Wind turbine blades are similar in shape to airplane propellers, but instead of adding energy to the air in order to propel the aircraft forward, they are used to extract energy from the wind. Figure 3.8 shows an idealized picture of the airflow created by a wind turbine rotor. The rotor is treated as an infinitely thin surface at which the pressure in the flow instantaneously changes, called an actuator disc. The energy ] Figure 3.8 Diagram of 1D airflow through a wind turbine rotor. captured by the wind turbine is thus the difference in energy between the upstream and downstream flows, which, when divided by the total available energy in the upstream flow, is expressed as the turbine s efficiency [32]: η = 1 2 ( 1 V ) ( d 1 + V ) 2 d V V 38

56 It can easily be shown that this efficiency is maximum when the ratio of downstream to upstream velocity is 1/3, corresponding to a net axial induction factor of a = 1/3. Axial and tangential induction factors are quantities that quantify the impact of the rotor on the relative flow velocity, as shown in Figure The efficiency at this condition is 59.3%, commonly known as the Betz limit. The Betz limit serves as a limit for potential wind turbine efficiency. Although the actuator disc model is useful for understanding the general ideas behind flow through a rotor, a more detailed treatment is necessary when it comes to analysis of actual designs. Wind turbine rotors are commonly studied with a combination of blade element theory and momentum theory. The resulting blade element momentum (BEM) theory, when combined with various corrections that serve to improve its flexibility and accuracy, offers a reasonable compromise between computational speed and predictive capability. Since BEM theory is not the focus of the present work, only an overview is provided here. Figure 3.9 Swept area of a wind turbine blade element [33] In blade element theory, the blade is divided into small, radial elements, as in 39

57 Figure 3.9, over which inflow properties and airfoil characteristics are assumed to be constant. The forces that act on the airfoil section are then calculated from the coefficients of lift, drag, and moment for the local airfoil shape, C L, C D, and C M, respectively: L = 1 2 ρv 2 cc L dr D = 1 2 ρv 2 cc D dr and M = 1 2 ρv 2 c 2 C M dr where ρ is air density, V is local relative air velocity, and c is local blade chord. The aerodynamic coefficients are dependent on the angle-of-attack at the blade element, α, shown in Figure The inflow velocity and direction is dependent on the wind speed and rotor speed, as well as the rotor s effect on the axial and rotational air velocity. These effects are quantified by the axial induction factor, a, and the tangential induction factor, a [33]. Axial and tangential velocity components produced by the elastic deformations of the blade are represented by v e-op and v e-ip, respectively. The elemental lift and drag can be resolved into differential contributions to rotor torque, dq, and rotor thrust, dt h, as shown in Figure The induction factors are solved for using an iterative approach at each individual blade element. The NREL design code FAST is commonly used to perform aeroelastic simulations of wind turbines, and is used in this study. FAST employs AeroDyn for solving the aerodynamics, which combines utilizes both BEM theory and a generalized dynamic wake model, and applies several other corrections accounting for tip and hub losses, high axial induction factors, dynamic stall, and skewed wake. For more details, please refer to the AeroDyn Theory Manual [33]. 40

58 Figure 3.10 Diagram of air flow in blade element theory, adapted from [33]. Figure 3.11 Forces on a blade element, adapted from [33]. 41

59 3.2.2 Power Capture In this study, the power capture of the wind turbine is measured at four wind velocities: 4.5, 7.5, 10, and 12 m/s. The resulting efficiency curve, shown in Figure 3.12, is extrapolated to the cut-in (minimum operational) wind speed, and interpolated everywhere else up to rated wind speed. If rated power is made at a wind speed above 12 m/s, the efficiency curve is extrapolated accordingly. Power production and blade forces are assumed to remain constant from rated wind speed to cut-out. Cut-in and cut-out wind speeds are 3 m/s and 25 m/s, respectively. Figure 3.12 Power coefficients up to rated wind speed for the NREL 5 MW reference turbine. The reported efficiencies for the reference turbine are slightly better than the calculated values, which can likely be attributed to the selection of airfoils. The baseline rotor in this work is defined by the NREL airfoils S818, S827, and S828 at root, mid-span, and tip, respectively, while retaining the original twist distribution from the reference turbine, which was intended for different airfoils. Definitions for the Delft University airfoils used in the reference turbine were not readily available. 42

60 The actual rotor power, ignoring gearbox and generator efficiencies, is found by multiplying C P by the available energy in the wind: P = 1/2ρAV 3 C P Once the rotor reaches rated power, in this case MW [31], the power stays at that level as a result of the pitch regulation control system. The resulting power curve calculated for the baseline turbine is shown in Figure The power output is 1% lower than NREL s data on average. Figure 3.13 Power curve for the NREL 5 MW reference turbine. 3.3 Bend-Twist Coupling Model Incorporating FEM prediction of bend-twist coupling behavior into a wind turbine blade optimization can be prohibitively time consuming, due to the number of evaluations required. In Section 1.2.4, a simplified method for predicting bend-twist coupling developed by Maheri, Noroozi and Vinney [34, 23] was introduced, which 43

61 significantly reduces the computational burden required for predicting bend-twist coupling effects. This section contains a description of the bend-twist coupling model and its application, following [34, 23]. The force-displacement relations for bend-twist coupled blades are given by [16]: EI g g ϑ/ r = GJ β/ r M T where ϑ is blade flapwise slope, β is blade twist, M is flapwise bending moment, T is twisting moment, and g = α g EI GJ. The coupling parameter, αg, is constrained to 1 < α g < 1 to ensure positive definiteness. Edgewise bending and axial forces are neglected in the formulation. Solving the above for β/ r and substituting in for g yields β r = α g M EI GJ(1 α 2 g ) + T GJ(1 α 2 g). (3.5) This equation for β/ r can be rewritten as: β r = h 1h 2 M + h 3 h 4 T (3.6) where h 1 h 2 = α EI GJ(1 α2 ) h 3 h 4 = 1 GJ(1 α 2 ). Here, h 1 and h 3 are used to represent blade material properties, including shell thicknesses, and h 2 and h 4 represent cross-sectional properties [34]. In general, h 1, h 2, h 3, and h 4 all vary with radius r, as do M and T. Since the aerodynamic twisting force is typically small compared to the bending moment, T can be neglected. Furthermore, if the material properties (layup, shell thicknesses) are constant along the blade span, 44

62 h 1 is constant. From Equation 3.6, the induced twist along the rotor can then be expressed as the integral β(r) = h 1 r (h 2 M) dr 0 The normalized induced twist, defined as the induced twist distribution divided by the induced twist at the tip, is given by β (r) = r 0 R 0 h 2 M dr h 2 M dr Finally, the assumption is made that h 2 is mainly dependent on airfoil thickness [34]: ( 1 h 2 t max ) γ (3.7) where a good fit was obtained for the induced twisting behavior of several bend-twist coupled beams [34], including a wind turbine rotor, for γ = 3. This yields β (r) = r 0 R 0 M(r) t 3 max(r) dr M(r) t 3 max(r) dr. (3.8) The result is that the normalized induced twist depends only on the airfoil thickness distribution and bending moment distribution. Dividing the top and bottom of 45

63 Equation 3.8 by the hub bending moment yields β (r) = β(r) β T = r 0 R 0 M (r) t 3 max(r) dr M (r) t 3 max(r) dr (3.9) where M (r) is the normalized hub bending moment distribution. Observing that M (r) can be considered independent of operating condition, as shown in Figure 3.14, indicates, from Equation 3.9, that the normalized induced twist distribution is also independent of operating condition. Now, let the normalized twist distribution, hub bending moment, and tip twist value at a reference operating condition be β (r), M hub,ref, and β T,ref, respectively, and let β T,ref be treated as a design variable, β T,ref β T,d. The inducted twist distribution is then calculated from M hub β(r) = β (r)β T,d (3.10) M hub,ref Equations 3.9 and 3.10 represent a model to compute induced twist at any operating condition for any blade obeying the assumptions above (constant spanwise material properties, negligible aerodynamic twist), without the use of FEA. For the purposes of this study, it represents a convenient mechanism for introducing bend-twist coupling as a design variable. The application of this model depends on two quasi-static, iterative computations. The first is conducted at a reference operating condition, and is shown in Figure The algorithm is: 1) Make an initial guess for M 2) Calculate β(r) from Equations 3.9 and 3.10 and update the blade twist, ϕ(r) 46

64 Figure 3.14 Hub bending moment distributions at different operating conditions, from Maheri [23]. 47

65 Figure 3.15 Quasi-static reference calculation flowchart. 3) Simulate the wind turbine, extracting M and M hub,ref 4) If M hub,new M hub,old ε, return to step 2, otherwise output β. In the present work, since the tested wind turbine blades can have significant variation from one another, defining a reference operating condition in terms of blade pitch is difficult, since a certain pitch setting for one blade may result in a very different load condition for another. Therefore, to provide consistency between different blade designs, M hub,ref is set at in all cases, at a wind speed of 12 m/s and 10 RPM. This differs from [23], where M hub,ref is determined by the reference calculation. The wind turbine simulation routine seeks the pitch setting with maximum flapwise hub bending moment under at the reference wind speed and RPM, in order to avoid calculating β when M hub is close to zero or negative, which represent unusual loading. Once β (r) is known from the reference calculation, the induced twist at any other operating condition can be calculated from the second quasi-static calculation, shown in Figure The algorithm is: 48

66 Figure 3.16 Quasi-static performance calculation flowchart. 1) Guess M hub 2) Calculate β(r) and the corrected blade twist 3) Run the wind turbine simulation with the ϕ(r) found above, extracting the hub bending moment 4) If the error M hub,new M hub,old ε, return to Step 2, otherwise, output power coefficient, hub bending moment, and other pertinent information 49

67 Chapter 4 Cost Model The objective of the present work is to minimize cost of energy. Since only the rotor itself is optimized with respect to loads and efficiency, without consideration of the rest of the wind turbine, the COE calculation considers only the cost of the rotor. Note that the definition of cost of energy given here is different from the one presented in Chapter 1. Cost of energy is defined as: COE = Rotor Cost ($) AEP(MWh/yr). (4.1) This chapter contains a detailed description of the rotor cost and AEP calculations needed to find COE. 4.1 Annual Energy Production As discussed in Section 3.2.1, part the output of the wind turbine simulation is the power curve over the range of wind speeds. To calculate net energy output, the power curve is applied to a wind speed probability distribution, for example, a Weibull wind speed distribution, or an actual wind speed histogram for a potential wind turbine site. Following [35], a Weibull distribution is used in this study. 50

68 The Weibull wind speed distribution is calculated by first finding the average hub height wind velocity, V z, from a power law: ( ) as z V z = V ref z ref where V ref is wind speed at height z ref, taken to be 5.8 m/s for a height of 10 m, and a s is the shear exponent, taken to be 1/7. If the hub height is 1.3*D = 81.9 m, then the reference hub height wind speed is 8.65 m/s. The likelihood of a given normalized wind speed, S(V n ), is described by a Weibull probability distribution [36]: f w (V n ) = kγ(k) [V n Γ(K)] k 1 e (VnΓ(K))k where V n = V V z, k = 2, Γ is the gamma function, and K = For the actual k wind speeds, f w (V ) = f w (V n )/V hub. The final plot of the wind speed probability distribution is shown in Figure 4.1. The average power output can be calculated by integrating the product of rotor power and probability density with respect to wind speed, as in P avg = V out V in f w P dv. (4.2) For the baseline rotor, P avg is 2.49 MW, which, assuming 100% availability, yields an AEP of 21.8 GWh. 4.2 Rotor Cost The rotor cost is estimated following the procedure in [35], where the cost of the wind turbine components other than the rotor is neglected. In the present work, load reduction is assumed to correlate to reductions in structural mass within the blade. 51

69 Figure 4.1 Weibull wind speed distribution The baseline mass is taken to be kg [35], of which 30% (roughly 7500 kg) is assumed to be structural mass that can be lightened via the reduction of flapwise hub bending moment. For example, if flapwise hub bending moments were reduced by 50% overall, the corresponding blade mass reduction would be (0.5)(7500 kg) = 3750 kg. The assumption of 30% reducible structural mass is arbitrary, and was made so that blade load reduction would be reflected in the COE, which is used as the objective function of the optimization. Load reductions can create other benefits, such as improving blade fatigue life, as well as reducing loads throughout the other wind turbine components [3]. A different objective function might incorporate blade load reductions differently, without assuming a direct affect on blade mass. Since the blade operates over a range of conditions and hub bending moments, a load function is needed to characterize the net change in hub bending moment. Several different methods for calculating the load function, L B, were applied in this 52

70 study, where the fractional load reduction is L Bopt L Bbaseline. In the above, L Bopt and L Bbaseline are the load measure for the optimized and baseline blades, respectively. The first method sets L B equal to the maximum hub bending moment obtained over the tested wind conditions. However, this measure resulted in too much bias toward load reduction at the highest tested wind speed of 12 m/s, leading to significantly reduced power near the original rated wind speed of 11.4 m/s. The second method calculates L B from the integral of hub bending moment versus wind speed, as shown below. This promotes a load reduction throughout the operating range, without emphasizing a particular wind speed. L B = 12 m/s M hub (V ) dv 4.5 m/s For the third method, L B is the sum of the hub bending moments weighted by the corresponding wind speed: n L B = M hubi V i i=1 where n is the number of wind speeds tested. Load reduction across the operating range is accounted for, but emphasis is given to the loads at higher wind speeds, which, due to their larger magnitude, have more influence on the wind turbine s structural requirements. The third method was adopted for the optimizations presented in this study. Blade mass can also be reduced via surface area reduction. Ten percent of the baseline mass (2487 kg) is assumed to be skin mass, based on information provided 53

71 by Sandia National Laboratory s D.T. Griffith [personal communication, October 31, 2011]. This nonstructural mass is assumed to vary in direct proportion to surface area. For example, if blade surface area was reduced by 50%, the corresponding mass reduction would be (0.5)(2487 kg) = kg. The remaining 60% of blade mass is considered nonreducible. Once the mass of the blade has been adjusted due to load and surface area changes, the blade cost is calculated by multiplying the blade mass by a cost of $10.45/kg. The cost of hub attachment material is also added, taken to be $20/kg for an unvarying mass of 851 kg. For the baseline blade, the total mass-based cost is ($10.45/kg)( kg) + ($20/kg)(851 kg) = $ The cost model includes the additional costs incurred during a blade production run for master blades, molds, and tooling [35]. Costs for master blades and molds are calculated from C mm = $1880S where S is blade surface area. Tooling costs are calculated from C tool = $4300S ( ) 1/2 R 35 m where R is blade radius. These two costs added together make up the fixed cost, C fixed = C mm + C tool, which is independent of the number of blades produced. The baseline blade has a surface area of 449 m 2 and radius of 63 m, yielding an initial fixed cost of $ The fixed cost and the blade cost are combined to give an average blade cost when the schedule and duration of the overall production run is considered. In this study, 54

72 a production run of 600 blades over the course of five years, requiring production of 10 blades per month, is assumed. The affect of a learning curve is incorporated into blade cost, where blades produced early in the production run have an increased cost, as shown in Table 4.1. The fixed cost is affected by interest rate: C fixedtot = C fixed M [ ] i i + (1 + i) M 1 where i is the monthly interest rate, and M is the number of months in the production run. This yields a total fixed cost for the baseline of $ For the baseline blade cost of $ , the blade costs throughout the production run are shown in Table 4.2. The overall average cost per blade is then calculated from Average Blade Cost ($) = C fixed tot + C TBP. # Blades where C TBP is the total blade production cost. For the baseline blade, the total cost of the production run is C fixedtot + C TBP = $ million, resulting in a final average blade cost of $ For the calculated AEP of MWh/yr, the baseline COE, $44.46 calculated from Equation 4.1, is. This value compares well to the COE MWh/yr calculated in [35] for a 5 MW rotor. The above method is used for calculating the COE for the optimized blades, which is used as the objective function of the optimization. A summary of the rotor cost model parameters is given in Table 4.3. One shortcoming of the cost model described above is that it does not take into account the cost increase due to building the blade with bend-twist coupling. The effect on cost is difficult to predict, given the different approaches for implementing bend-twist coupling, such as whether it is implemented in the internal structure, skin, or both. However, it is expected that, as technology improves over time, the influence 55

73 of bend-twist coupling on blade cost will diminish. Table 4.1 Blade production cost multipliers. Blade Position in Production Run Multiplier on Production Cost Table 4.2 Blade costs over production run. # Blades Cost Multiplier Cost per Blade($) Cost for Group ($) Total 600 $

74 Table 4.3 Rotor cost model parameters. Parameter Value Reducible Blade Mass 40% Blade Skin Mass 10% (counted as reducible) Blade Weight $/kg $10.45 Hub Material $/kg $20 Hub Connection Weight 851 kg Baseline Blade Weight kg Length of Production Run 5 years / 60 months Blades in Production Run 600 Blades Produced per Year 120 Annual Interest Rate (%) 10 Monthly Interest Rate (%) Masters & mmolds Cost $1880 S Tooling Cost $4300 ( R 35 m) 1/2 S 57

75 Chapter 5 Genetic Algorithm A genetic algorithm is used to optimize the wind turbine model for minimum COE. Genetic algorithms operate by mimicking the processes by which natural evolution takes place. Primarily, these are selection, crossover, mutation, and survival of the fittest. They have the advantages over gradient-based optimization methods of being less susceptible to local convergence, as well as being better suited to large design spaces. This chapter contains a description of the genetic algorithm employed in this work. 5.1 Introduction to Genetic Algorithms In genetic algorithms, many individual solutions, together making up a population, interact with each other and evolve over time. Each individual is assigned a fitness based on its performance in the objective function. The population evolves by selecting parents, creating offspring, and then making replacements in the parent population based on fitness level and possibly other criteria. A variety of approaches may be used for selection, crossover, and replacement, each with various advantages and disadvantages that guide the progress of the optimization. The general flowchart for a genetic algorithm is shown in Figure 5.1. The process is: 58

76 Figure 5.1 Flowchart of genetic algorithm. 1) Initialize a population of N members by randomizing a baseline individual and evaluating the resulting individuals 2) Select two parents 3) Combine the parents in order to produce two offspring via crossover and mutation 4) Evaluate the resulting offspring 5) Choose which individuals, if any, are replaced in the population by the offspring. Track fitness, diversity, and any other values as needed. 6) Increase the generation count by one. Repeat steps 2-5 until stopping criteria are met, such as maximum generation number or minimum acceptable fitness level A simple example of a population undergoing selection, crossover, mutation, and replacement is shown in Figures Two parents, blue and orange, are selected from the initial population of four members. Offspring are created by replacing digits 59

77 in orange by the corresponding ones from blue and vice-versa, as shown in Figure 5.3. This process is referred to as crossover. A small probability of the digits in the offspring being flipped is applied, referred to as mutation. Finally, the green member of the parent population is replaced by one of the offspring. This process is repeated with each generation, until convergence criteria, such as sufficient fitness or generation number, are met. Figure 5.2 Simple example of a population of four individuals, sorted by fitness. Each box in an individual represents a binary digit. In order to facilitate the use of the GA, a population class of variables is implemented in MATLAB, containing convenient mechanisms for storing populations and fitness metrics for individuals, replacing and deleting members, combining populations, and more. The bulk of the data kept for a population is composed of the matrix of members, where each row contains all the binary digits composing a single genome. An additional matrix containing miscellaneous information about every member, such as the performance in different objective criteria, as well as other information such as age in generations and overall fitness, is also stored. Lastly, the population class provides for matrices containing the best fitness, average fitness, and population diversity level (explained in Section 5.5) versus generation number. 60

78 Figure 5.3 Blue and orange have been selected. Offspring are created by the exchange of digits between blue and orange, combined with a small probability of the offsprings digits mutating into the opposite value (0 or 1), shown in pink. Figure 5.4 Population of solutions after replacement has occurred. Green has been replaced by the child of orange. 61

79 5.2 Variable Encoding In genetic algorithms, single design variable values are called genes. Taken together, all the genes compose a set of all the design values for an individual. This set of all genes is called a genome. A common practice in genetic algorithms, followed in this work, is to encode each gene into a string of binary digits [37]. To construct the actual geometry of a candidate for evaluation, the genome must be decoded, where the genes are converted into design variable values, and the values are assembled into the blade geometry. The geometric assembly was discussed previously in Section 2. Eight bits are used to represent every design variable, which consequently has 256 equally spaced potential values. The 8-bit genome is directly converted into an integer, G, from 0 to 255. If D = {d 1, d 2,, d n } is the vector of n design variables, each having a range d imin d i d imax, then the decoded value for d i is: d i = d imin + G (d i max d imin ) (5.1) 5.3 Selection There are many methods for selecting parents from among the individuals in a population. The GA should seek high performers, but not neglect other members who might add valuable diversity to future solutions. In fact, if the selection and replacement schemes are overly biased toward performance, the algorithm may exhibit early convergence. The selection scheme used here is Roulette-wheel selection. Each member is given a slot on a virtual roulette wheel, where an individual s chance of being selected is proportional to their fitness level. The Roulette wheel for a simple sample population of ten members with fitness levels one through ten is shown in 62

80 Figure 5.5. The members with higher fitness have a greater chance of selection. Figure 5.5 Roulette wheel for a simple population. 5.4 Crossover and Mutation Once the parents are selected, one referred to as the mother and the other as the father, the offspring are created by combining the two genomes in a process known as crossover. As with selection, various methods for applying crossover are available. However, 70% probability of swapping for every individual bit is chosen for this study. A sample crossover scenario for a single gene is demonstrated in Figure 5.6. The highlighted bits have been randomly selected for crossover. The son, then, contains the original unselected bits from the father, mixed with the mother s crossover bits, and vice-versa for the daughter. When this process has been applied to every gene in the genome, crossover is complete. After crossover has taken place, mutation is decided. Each bit in the offspring is given a small chance, usually below 2%, to flip its value to the opposite bit. This 63

81 helps ensure a more thorough search of the design space. Mutation occurs regardless of whether the bit was affected by crossover or not. Figure 5.6 Example of crossover for a binary gene. 5.5 Replacement Replacement strategies often containing more complexity than the selection and crossover components. The features of the replacement algorithm can have a significant effect on the convergence time and features of the solution. Replacement schemes can be tailored to accelerate convergence, by replacing the worst members, or to promote stable, diverse subpopulations, through means of tournament selection. They can also guide a population toward more diverse solutions spanning the Pareto optimal front, or to converge on a single optimal solution. The actual replacement scheme used in the present work can be described as a Replace Oldest scheme with elitism, combined with Replace-Worst [38, 37]. In one generation, up to two members of the parent population are replaced. The replacement algorithm is: 1) From the current population, not including offspring, select the oldest member, or a random member from among the equally oldest, if there are more than one. 64

82 2) Select any other member of the population at random to serve as an opponent. 3) If the oldest member has equal or lesser fitness than the opponent, replace the oldest. Otherwise, replace the opponent. 4) Replace the least fit member of the population with the second offspring. In this way, diversity is ensured by forcing replacement every generation, as well as attempting to replace old members, while ensuring quality via the fitness-based tournament. Following replacement, average fitness, maximum fitness, and population diversity are calculated and stored in the population for that generation. The technique for measuring diversity is described in the following section. 5.6 Diversity Measure The concept of diversity in a genetic algorithm varies by application [37]. Pareto ranking might be used, or another common measure, the Hamming distance, that counts the number of replacements needed to convert one string into another. This is particularly suited to the binary coded genetic algorithm, consisting of only ones and zeros. An example of a Hamming calculation is shown in Figure 5.7, where the number of bits different between the two genes, and hence the Hamming distance, is three. For a group of more than two strings, the overall Hamming distance is Figure 5.7 Example of a Hamming distance calculation for two binary strings. obtained by comparing each string to every other string. The result is a somewhat 65

83 computationally expensive process for large populations. An alternative to Hamming distance proposed by Morrison and De Jong [39] considers the distance of the bits in a genome from an average value, and calculates the equivalent of the moment of inertia about that average value. The computation required for the moment of inertia method, described below, is linear with population size, whereas Hamming distance is quadratic with population size. First the diversity calculation is described, and then examples are given for populations with low and high diversity in Tables If a population matrix consists of rows of genomes, where each column represents one bit contributing to the gene of a particular design value, the measure of diversity is found by calculating the moment of inertia of each column. For a population with N C columns and N M members, where b i,j is the value of the j th bit for the i th member, the average value, or centroid, for a given bit/column is b j = N C b i,j i=1 N C. The moment of inertia for the j th column is then calculated from I j = N M i=1 ( bi,j b j ) 2 where for every element in the population matrix, the quantity ( ) 2 b i,j b j represents that bit s contribution to overall diversity. Thus, the summation of one row yields one member s total contribution to diversity a quantity that will be used if sorting members by diversity level. Subsequently, the total moment of inertia is the sum of 66

84 the column moments of inertia: N C Itot = which can be normalized to account for different genome sizes by dividing by N C. Thus, the final value for population diversity is j=1 I j N C N M ( bi,j b j ) 2 D i = j=1 i=1 N C or D i = I tot N C. The economy of this diversity measure is complemented by its convenience, in that it is easily converted to the Hamming distance via multiplication by the number of members, N M, where H = ItotN M = D i N C N M. Population diversity can be used to determine replacement, as well as track the convergence of the population. Table 5.1 Diversity calculation for a population of five members (rows) and eight digits (columns). This population has relatively low diversity, since several columns contain only one value. Diversity Parameter Population Matrix Column Centroid, b j Column MOI, I j I TOT 4 MOI Diversity Measure, D 0.5 Hamming Distance 20 67

85 Table 5.2 Diversity calculation for a sample population with five members, eight bits, and high diversity. Diversity Parameter Population Matrix Column Centroid, b j Column MOI, I j I TOT 8.4 MOI Diversity Measure, D 1.05 Hamming Distance 42 68

86 Chapter 6 Results 6.1 Baseline Blade This section contains a discussion of the geometry, bend-twist coupling behavior, and performance of the baseline blade. Note that the NREL 5 MW reference turbine has no built-in bend-twist coupling. Rather, bend-twist coupling was assumed to be implemented in the blade for this analysis Baseline Blade Geometry The baseline blade is modeled after the NREL 5 MW reference turbine [31]. It is intended to be geometrically identical to the reference blade except for the airfoil definitions, which were not readily available. Instead, the NREL S818, S827, and S828 airfoils are used to define the aerodynamic shape at the root, mid-span, and tip, respectively. These airfoils are part of a family of airfoils developed by the National Renewable Energy Laboratory, and are recommended for use on wind turbines rated up to 1 MW [30]. Although they were not developed for use in a 5 MW wind turbine, they are found to provide good performance in the baseline blade, see Section The chord and twist distributions, hub radius, radial locations of the cylindrical shell sections near the root, radial location of the first pure airfoil, and tip radius are all 69

87 identical to the reference turbine. The baseline blade is shown in Figure 6.1, and the baseline chord distribution, twist distribution, and airfoil shapes are provided in Figures Figure 6.1 Three dimensional view of the baseline blade. Figure 6.2 Baseline chord distribution. 70

88 Figure 6.3 Baseline twist distribution. 71

89 Figure 6.4 Baseline airfoils. 72

90 6.1.2 Bend-Twist Coupling in the Baseline Blade One potential area of concern in the bend-twist coupling model is that the material properties are assumed constant over blade span. In a large blade such as that of the 63 m radius, 5 MW reference turbine, the material properties are likely to vary due to ply drops, changing shell thicknesses, etc. Since NREL [31] provides spanwise bending (EI) and torsional (GJ) stiffnesses for the reference blade, plotted in Figure 6.5, the force-displacement equation for induced twist can be integrated directly (neglecting aerodynamic twist) for an assumed coupling coefficient of α g = 0.3, yielding the true induced twist: β(r) = α g 1 α 2 g r 0 M EI GJ dr. Note that this assumes that the level of coupling, α g, remains constant across the blade span. Furthermore, the actual baseline blade has no bend-twist coupling; this analysis assumes the bend-twist coupling to be built into the blade. Figure 6.5 Flapwise bending stiffness and torsional stiffness for the NREL 5 MW reference turbine [31]. 73

91 The true induced twist can then be compared to the normalized induced twist distribution predicted by the bend-twist coupling model from the Equation 3.9, restated below, as well as to the maximum allowed tip twist in the design optimization. β (r) = β(r) β T = r 0 R 0 M (r) t 3 max(r) dr M (r) t 3 max(r) dr The comparison is made for a test condition of 7.5 m/s. For the bending moment distribution in Figure 6.6, the true induced twist distribution is given in Figure 6.7. The maximum allowed value of 10 of tip twist at M hub = Nm assumed in this work is reasonable, since α g = 0.3 is only 30% of the maximum theoretical coupling coefficient value of ±1, and the tip induced twist is nearly 6 for a hub bending moment of Nm. Recalling that, in the model, the induced twist is directly proportional to hub bending moment, β(r) = β (r)β T,d M hub M hub,ref the expected tip twist at the reference hub bending moment of Nm is 10.23, demonstrating good agreement with the maximum allowed value. Figure 6.8 compares the true normalized induced twist to that resulting from the bend-twist coupling model. The error between the predicted and calculated induced twist is fairly large, exceeding 20% of tip twist toward the middle of the blade. This may be the result of material factors varying throughout the blade, or from the general inaccuracy of the model. Reconsideration of Equation 3.7 leads to a superior fit of the induced twist when the exponent on the maximum thickness in Equation 3.9 is 74

92 Figure 6.6 Baseline bending moment distribution. Figure 6.7 Induced twist distribution for baseline blade from integration of Equation

93 increased from 3 to 4, as shown in Figures 6.9 and This adjustment brings the maximum error between the model and the true value to around 7%. Figure 6.8 Normalized β from integration calculation and model. Figure 6.9 β distribution for γ = 4. Unfortunately, this improvement was not incorporated into the optimizations discussed in this study. However, it was determined that this error in the induced twist distribution actually has a very small effect on the predicted performance of a ro- 76

94 tor. For one of the optimized rotors, changing the exponent in Equation 3.7 from γ = 3 to γ = 4 results in up to a 2.7% change in hub bending moment, leading to a 1.9% reduction in the overall blade loading parameter, L B. The difference in annual energy production is 0.1%, and the change to COE is only 0.4%. Therefore, it is assumed that this correction would not have substantially affected the outcomes of the optimizations. Regardless, this correction should be applied in any future studies applying this model to the NREL 5 MW reference turbine. Figure 6.10 Error between integration calculation for induced twist and the model with γ = Baseline Blade Performance The blade properties are summarized in Table 6.1, where values that are fixed throughout the optimization are displayed in bold. 77

95 Table 6.1 Baseline blade properties. Bold values are invariant. Parameter Value Hub radius 2.87 m Radius at root airfoil 11.7 m Tip radius 63 m Root airfoil NREL S818 Mid-span airfoil NREL S827 Tip airfoil NREL S828 Chord distribution Same as ref. Twist distribution Same as ref. Bend-twist coupling, β T,d 0 Surface Area m 2 Mass kg AEP MWh/yr 6.2 Case 1: Chord and Twist Only The first results presented are those for an optimization where airfoil shapes were fixed and bend-twist coupling was set to zero. In this way, the effect of incorporating different blade properties into the optimization can be examined. Figure 6.11 illustrates the best COE and diversity level vs. generation for the optimization. Resulting changes in blade properties are shown in Table 6.2. The objective function values of the initial and final populations, i.e. the AEP and load parameter, are compared in Figure 6.12, and the COE vs. AEP in Figure The final population is highly converged, with all 30 members collapsing into a small area of the objective space, corresponding to the point of lowest COE. The population has almost reached its best value by around 1000 generations, although improvements continue beyond 3000 generations. This is expected for such a large design space; even when the population is highly refined, incremental improvements are possible late in the optimization. Since airfoil coefficients were determined a priori for this optimization, average timer per generation (creating and evaluating two offspring, and conducting replace- 78

96 Figure 6.11 Best COE versus generation, Case 1. Figure 6.12 Initial and final populations, Case 1, after 9000 generations. 79

97 Figure 6.13 Initial and final populations, Case 1, after 9000 generations. ment) was about 130 seconds on an 8 core, 2.33 GHz desktop computer with 4 GB of RAM running Windows. Thus, the optimization took about two weeks of computer time for 9000 generations. Table 6.2 Results of optimization for Case 1. Only chord and twist distributions were optimized. Airfoils were fixed and no bendtwist coupling was permitted. Parameter Baseline Optimized % Difference COE $44.50 $ AEP (MWh/yr) Mass (kg) Net Flap Load Surface Area (m 2 ) The optimization achieved a decrease in COE of 4.5%, accompanied by a reduction in energy output by about 0.8%, as indicated by the rotor power curve in Figure The COE reduction is mainly due to a net decrease in flapwise loads of around 12%, 80

98 illustrated in Figures 6.15 and 6.16, as well as a surface area reduction of nearly 15%, suggesting that the algorithm minimized surface area as a means of minimizing weight, at apparently little cost to power generation. If the larger initial chord values of the baseline, presented in Figure 6.17, are not necessary for structural or other reasons, then they could probably be reduced for additional material savings. Figure 6.14 Rotor power curve, Case 1. The optimized twist distribution is presented in Figure 6.18, and is nearly constant at around 15 throughout much of the span. This is distinct from the baseline twist distribution, which decreases almost linearly from 13 to 0 from root to tip. Relative to the baseline, this translates to a twist toward feather (reduced lift) from the root to the tip. The apparent strategy of reducing loading over the outboard blade span is confirmed by the aerodynamic loading results discussed next. Consideration of the blade s aerodynamic behavior offers insight into how power output is virtually unchanged at reduced loads. Blade loading is examined at 7.5 m/s for this and the other two cases discussed later in this chapter. For Case 1, the most obvious difference from the baseline is that the optimized blade experiences relatively higher forces over the inboard section of the rotor, as demonstrated by the 81

99 Figure 6.15 Hub bending moment versus wind speed, Case 1. Figure 6.16 Hub bending moment reduction versus wind speed, Case 1. 82

100 Figure 6.17 Optimized chord distribution, Case 1. Figure 6.18 Optimized twist distribution, Case 1. 83

101 Figure 6.19 Axial force distribution, Case 1. Figure 6.20 Axial induction factor, Case 1. 84

102 Figure 6.21 Tangential force distribution, Case 1. axial (flapwise) force distribution in Figure The axial induction factor, provided in Figure 6.20, supports this idea, indicating that the rotor is acting more to slow the airflow in the axial direction over the inboard part of the blade and less over the outboard span. Figure 6.21 indicates a reduction in tangential forces across the blade. Since power is the product of rotor torque and angular velocity, and the rotor is producing nearly the same power, it operates at a higher RPM to compensate for the reduced torque. The angle-of-attack distribution (Figure 6.22) offers deeper insight into the aerodynamic behavior of the blade, since it incorporates the effects of twist, airfoil performance, and the induction factors. Additionally, it gives a direct comparison between the behavior of the baseline and optimized blades, since the airfoils are the same in both cases. The angles of attack confirm the changes in axial force and induction, in that α is higher inboard than baseline, and lower outboard. When the angle for (L/D) max is plotted for the 20 airfoils distributed across the blade, it becomes clear that the baseline rotor operates just below (L/D) max for most of the inboard span, and is above (L/D) max over the outboard part. The optimized rotor, on the other 85

103 hand, operates above maximum lift-to-drag on the inboard span, and operates near or below (L/D) max over the outboard part. Figure 6.22 Angle of attack distribution, Case Case 2: Chord, Twist, and Bend-Twist Coupling For the second optimization, the airfoils remained frozen, but bend-twist coupling was included. The progression of the optimization, COE vs. AEP, and load parameter vs. AEP are presented in Figures Table 6.3 give the resulting change in rotor parameters. As expected, the additional complexity of bend-twist coupling in the optimization allowed for improvement in COE reduction over Case 1, but only about 0.3%. Changes in individual parameters were more substantial - surface area was reduced by an additional 2.5%, and hub bending moment by an additional 1.1%. The hub bending moments and hub bending moment reductions are shown in 86

104 Figures Figure 6.23 Best COE versus generation, Case 2. Table 6.3 Results of optimization Case 2: fixed airfoils only. Parameter Baseline Optimized % Difference COE $44.5 $ AEP (MWh/yr) Mass (kg) Net Flap Load Surface Area (m 2 )

105 Figure 6.24 Initial and final populations, Case 2, after 5000 generations. Figure 6.25 Initial and final populations, Case 2, after 5000 generations. 88

106 Figure 6.26 Hub bending moment versus wind speed, Case 2. Figure 6.27 Hub bending moment reduction versus wind speed, Case 2. 89

107 Interestingly, AEP actually decreased further, dropping another 0.4% to -1.2%, as indicated by the power curve in Figure 6.28, counter to the prediction made in [21]. This is most likely due to the objective function favoring low loads as a more effective means of decreasing COE than maximizing AEP. Thus, it is important to note the importance of an accurate COE model to ensure reliable optimization. Further note that tuning of the objective function could increase the weighting towards higher energy capture, if such an outcome was desired. Figure 6.28 Rotor power curve, Case 2. The optimized chord in Figure 6.29 is nearly identical to that of Case 1. Thus, the chord distribution is close to the optimal chord schedule for minimum surface area without substantial energy loss for this wind turbine. The optimized twist distribution, shown in Figure 6.30, illustrates how the blade balances power capture with load reduction. The best blade contained the maximum amount of bend-twist coupling available, approximately 10 of tip twist for a hub bending moment of Nm. The unloaded rotor pre-twist is similar to that of the baseline, which has good efficiency. As the induced twist increases under loading, however, the overall blade twist becomes more similar to the optimized pre-twist from Case 1, which, as discussed in 90

108 Section 6.1.3, results in low loading at the higher wind speeds. Figure 6.29 Optimized chord distribution, Case 2. Figure 6.30 Optimized twist distribution, Case 2. As indicated by Figures , the aerodynamic loads show similar trends to Case 1. At a wind speed of 7.5 m/s, the axial force demonstrates the same increase throughout the middle portion and reduction toward the blade tip. Figure 6.32 provides the axial induction factor, which also follows the trend of axial force, but with 91

109 a less extreme increase over the inboard span than in Case 1. Tangential blade force was nearly identical to Case 1, but Case 2 shows that some torque was recovered near the root. Figure 6.31 Axial force distribution, Case 2. Figure 6.32 Axial induction factor, Case 2. The angle-of-attack distribution, as Figure 6.34 demonstrates, is similar to that of Case 1, while being relatively lower near the root and higher over the middle portion of the blade. Near the tip, angle-of-attack is nearly identical between the two 92

110 Figure 6.33 Tangential force distribution, Case 2. optimizations. The trend of operating above (L/D) max prior to the tip and below (L/D) max toward the tip is continued in this case. Figure 6.34 Angle of attack distribution, Case 2. 93

111 6.4 Case 3: Full Geometry and Bend-Twist Coupling The final case considered is for optimization of the blade simultaneously considering all design variables: airfoil shapes, chord and twist distribution, and bend-twist coupling. Computational time for this optimization problem was longer than the others, due to airfoil evaluations being required for every rotor. Average time per generation was about 320 seconds, resulting in 10 days of computer time for 2700 generations. Figure 6.23 suggests that the bulk of improvement is achieved in the first 1500 generations, but it is likely that incremental improvements would have continued well beyond 3000 generations. The plot of initial and final load parameters vs. AEP and Figure 6.35 Diversity and best COE versus generation, Case 3. COE vs. AEP are shown in Figures 6.36 and Some of the initial randomized members of this population have load and AEP values close to the optimized blades from Case 1. Note, however, that this does not translate to equality in COE, as 94

112 demonstrated in Figure The initial blades have not benefited from a comprehensive optimization involving chord and surface area reduction, which significantly affects mass and therefore COE. Figure 6.36 Initial and final populations, Case 3, after 2700 generations. The change in blade parameters is shown in Table 6.4, and a comparison of all three optimizations is given in Table 6.5. The fully optimized blade achieves a further reduction in COE, loads, and blade mass, while bringing AEP above the baseline level. Optimized rotor power is shown in Figure 6.38, and the blade is shown in Figure 6.39, and the three optimized airfoils are highlighted where they appear in the cross-section. The hub bending moment reduction across the operating range is an improvement over Case 2, and is shown in Figures 6.40 and Airfoil optimization clearly played a substantial role in producing the best performing blade, since the COE reduction grew by over 30% from the previous two optimizations. Optimized airfoil shapes and performance are presented in Figures The optimization sought airfoils with high maximum lift particularly for the 95

113 Figure 6.37 Initial and final populations, Case 3, after 2700 generations. Table 6.4 Results of optimization Case 3: full geometry and bend-twist coupling Parameter Baseline Optimized % Difference COE $44.5 $ AEP (MWh/yr) Mass (kg) Net Flap Load Surface Area (m 2 ) Table 6.5 Comparison of the results from all three optimizations. % Change Parameter Case 1 Case 2 Case 3 COE AEP (MWh/yr) Mass (kg) Net Flap Load Surface Area

114 Figure 6.38 Rotor power curve, Case 3. Figure 6.39 Optimized rotor blade, alternate view, Case 3. 97