Michael K. McWilliam

The Behavior of Fixed Point Iteration and Newton-Raphson Methods in Solving the Blade Element Momentum Equations

There is a substantial body of ongoing research improving the Blade Element Momentum (BEM) theory and applying it to the optimization of wind turbine rotors. Both of these developments challenge the suitability of fixed point iteration schemes being applied to advanced BEM models. This article explores the mathematical behavior of the BEM equations, with special attention to the application of numerical methods. Under special conditions, multiple solutions will exist when the airfoil is stalled. This situation gives increased uncertainty, where uncertainty in airfoil behavior is already high. This also demonstrates that there could be circumstances where the wake state has weak dependence on blade state. Fixed point iteration and Newton-Raphson numerical methods are investigated in this paper. Both methods will become unstable under certain conditions. The investigation shows that the Newton-Raphson method has well defined conditions for instability in terms of design variables and airfoil properties. By comparison, the fixed point function used here exhibits instability over a larger range.

Towards a Framework for Aero-elastic Multidisciplinary Design Optimization of Horizontal Axis Wind Turbines

Multi-disciplinary Design Optimization (MDO) has been successfully applied in the aerospace industry, so given the similarities to wind turbine design, the application of MDO techniques is a potential opportunity to improve wind turbine design. MDO attempts to solve for optimal design parameters by considering the performance of multiple disciplines simultaneously. This approach differs from sequential optimization in which each discipline is optimized separately. Evaluating the design with a comprehensive approach leads to better balanced designs. This article presents a Multi-Disciplinary Feasible (MDF) framework that incorporates an aerodynamics code based on vortex methods with a nonlinear beam formulation for the blade aerodynamics and structural dynamics, in order to eventually study non-straight blades with arbitrary composite layups. In the current work, the framework is exercised to optimize a conventional design for a 100 m blade. It was found that obtaining accurate coupled gradients for a fully-relaxed wake simulation using explicit aerodynamic solution methods is very challenging. A rigid wake approach enabled more reliable convergence, and suggestions are given for future work in applying MDO to this class of wind turbine analysis methods.

Adjoint Based Sensitivity Analysis for Geometrically Exact Beam Theory and Variational Asymptotic Beam Section Analysis with Applications to Wind Turbine Design Optimization

Many researchers are exploring advanced wind turbine designs that incorporate various passive structural load alleviation strategies. All of these methods rely on physical behavior not resolved in conventional analysis methods. Strategies based on novel blade shapes (sweep and coning) rely on geometric nonlinearity. Other strategies rely on material anisotropy to incorporate various coupling effects. There are two problems encountered when incorporating these concepts into a design optimization process. First sufficiently

Validation of Potential Flow Aerodynamics for Horizontal-Axis Wind Turbines in Steady Conditions using the MEXICO Project Experimental Data

Potential flow methods are a promising alternative to mainstream wind turbine aerodynamics tools such as blade element momentum methods and grid-based computational fluid dynamics approaches. Potential flow methods are relatively easy to setup and robust with respect to geometry. The advent of the fast multipole method and viscous core modelling brings computational speed and robustness. A C++ library employing a Weissinger lifting line model and tailorable potential flow wake models has been developed under the name LibAero. The wake models employ vortex particles, vortex filaments, and vortex quadrilateral elements. Aerodynamic wake models were validated against experimental data from the MEXICO wind tunnel experiments in steady axial wind conditions. Blade forces and flow field data were compared. The experimental blade forces were post-processed from airfoil pressure tap data, whereas flow field data was post-processed from particle image velocimetry data. The results indicate that LibAero is effective at predicting blade forces, power, and thrust. LibAero is similarly effective to blade element momentum methods for modelling the aerodynamics of standard Danish wind rotors, while having the capability to model non-standard wind rotors.

A Corrected Blade Element Momentum Method for Simulating Wind Turbines in Yawed Flow

A new Blade Element Momentum (BEM) model is proposed for yawed wind turbine flows. This method differs from conventional methods in the use of correction factors for the induction. A set of results from potential flow methods is used to define a table of corrections over a wide range of operating conditions and locations within the flow field. The potential flow methods account for the distribution of vorticity in the wake. Applying the resulting corrections give better accuracy than conventional BEM methods. By generating the correction results a priori, the efficiency of the BEM method is preserved. The accuracy of this method and the conventional axial momentum based BEM method are evaluated by comparing results to that of the MEXICO experiment.

Finite Element Based Lagrangian Vortex Dynamics Model for Wind Turbine Aerodynamics

This paper presents a novel aerodynamic model based on Lagrangian Vortex Dynamics (LVD) formulated using a Finite Element (FE) approach. The advantage of LVD is improved fidelity over Blade Element Momentum Theory (BEMT) while being faster than Numerical Navier-Stokes Models (NNSM) in either primitive or velocity-vorticity formulations. The model improves on conventional LVD in three ways. First, the model is based on an error minimization formulation that can be solved with fast root finding algorithms. In addition to improving accuracy, this eliminates the intrinsic numerical instability of conventional relaxed wake simulations. The method has further advantages in optimization and aero-elastic simulations for two reasons. The root finding algorithm can solve the aerodynamic and structural equations simultaneously, avoiding Gauss-Seidel iteration for compatibility constraints. The second is that the formulation allows for an analytical definition for sensitivity calculations. The second improvement comes from a new discretization scheme based on an FE formulation and numerical quadrature that decouples the spatial, influencing and temporal meshes. The shape for each trailing filament uses basis functions (interpolating splines) that allow for both local polynomial order and element size refinement. A completely independent scheme distributes the influencing (vorticity) elements along the basis functions. This allows for concentrated elements in the near wake for accuracy and progressively less in the far-wake for efficiency. Finally the third improvement is the use of a far-wake model based on semi-infinite vortex cylinders where the radius and strength are related to the wake state. The error-based FE formulation allows the transition to the far wake to occur across a fixed plane.

Composite Lay-up Optimization for Horizontal Axis Wind Turbine Blades

Research has shown that design concepts based on advanced lay-ups can improve wind turbine blades. Adding carbon fiber reinforcement at outboard sections can make blades lighter reducing edge-wise bending loads. Adding biased fibers can introduce bend-twist coupling reducing the fatigue damage. Optimizations is a powerful tool that can be used to solve the best configuration. To successfully explore these concepts through optimization this paper introduces a parameterization scheme that reflects the layered nature of these designs. The scheme incorporates span-wise variation of material to accurately model the affect of span-wise transition. This parameterization scheme will be compared with other schemes typically seen in literature. Application of this scheme will be demonstrated by developing an optimal glass/carbon blade and an optimal blade with bend-twist coupling.

An implicit model for Lagrangian vortex dynamics for horizontal axis wind turbine design optimization

This paper presents a new aerodynamic model for simulating horizontal axis wind turbines. The model is based on Lagrangian Vortex Dynamics (LVD) resembles potential ow with the addition of the core model. This model presents a new way of solving the governing equations for a steady state solution. The parametrization scheme is based on shape functions in time. The coupling between space and time in steady state problems obviates the need to apply a time marching algorithm. Instead the system is represented by an error function that can be solved using Newtons algorithm. The model was used to solve the aerodynamics of the MEXICO rotor. Comparisons between experimental measurements showed excellent agreement. A convergence study shows that increasing the resolution improves the agreement, however there still remains some discrepancies that inherent in the LVD method. The aerodynamic model was developted for coupling with a non-linear wind turbine blade structural model, to enable fully-coupled steady state solutions and adjoint based gradients for optimization.

Manufacturing Defect Effects on Bend-Twist Coupled Wind Turbine Blades

New design concepts for large flexible blades are needed to continue the trend towards ever larger turbines and continued economies of scale. Load alleviation based on building bend-twist coupling into the blades is an attractive concept because it can reduce the root bending moments and the fatigue damage in the blades with a negligible affect on the annual energy production. The challenges in quality control associated with fiber laminate lay-up processes present a risk associated with this concept. This study attempts to quantify this risk by evaluating the sensitivity of the blade’s elastic properties to various manufacturing defects. This study will look at the bend-twist coupling coefficient, weight, internal stress and tip deflection. The defects that will be considered are errors in position, size and thickness of the layers along with errors in the fiber angle.