Matlab'sWind Turbine Blade Design Optimization Massachusetts Institute of Technology, Cambridge, MA, 02139, USA We develop a methodology for analyzing wind turbine blade geometries and pitch con-trol schemes over a range of incoming wind speeds. We use this model for an orthogonal array design of experiments, a gradient-based sequential quadratic programming optimiza-tion, and a multi-objective genetic algorithm to maximize the expected power output while minimizing the blade volume and structural stress violations. Design of experiments gen-erates good results with little expense. Sequential quadratic programming with Hessian re-scaling and multiple starting points generates good design vectors with a large compu-tational expense, and heuristic algorithms such as the genetic algorithm are not suited to this problem. The best design point achieves between 60 and 70% of the Betz limit for efficiency for a large range of incoming wind speeds. Nomenclature R Blade radius, m Qmax Maximum generator torque, N-m t Blade shell thickness, m k Cut-off velocity in standard deviations above the mean wind speed T Twist distribution vector, radians F Foil shape parameter distribution, non-dimensional C Chord length distribution, m β Pitch control curve, radians x Design vector J(x, param) Objective function PE Expected power output, W P Turbine power at a particular wind speed, W Q Turbine torque at a particular wind speed, N-m Vblades Blade material volume, m3 ξ(x, param) Penalty function σr max Maximum blade stress, MPa σallowable Maximum allowable blade stress, MPa cweibull Scale parameter of the Weibull distribution of incoming wind speeds kweibull Shape parameter of the Weibull distribution of incoming wind speeds ω Angular velocity, rad/s Ft/ΔR,Ft/ΔR Tangential and axial aerodynamic load distributions on the blade, N/m v0 Incoming wind speed ahead of the blade, m/s H Hessian matrix, matrix of second derivatives I. Motivation As renewable energies become a growing part of the energy portfolio, focus is being put on increased performance and efficiency of proven sources such as horizontal axis wind turbines. Modern commercial power wind turbines are predominantly horizontal-axis, three-bladed behemoths, with a fixed blade design ∗Graduate Student, Computation for Design and Optimization 1 of 11 American Institute of Aeronautics and Astronautics Anonymous MIT Studentsthat is adapted to varying wind conditions by a blade pitch control mechanism. They are complex systems whose design requires the integration of many engineering disciplines including aerodynamics, structures, controls, and electrical engineering. Previous wind turbine design optimization techniques have focused on specific regions of interest, including optimal control,1 optimal blade shape, and site-specific performance increases.2 Our goal was to combine all of these with a simplified analysis model to make computation tractable, but to design a blade from scratch with optimal performance over a site-specific wind profile probability distribution. II. Problem Formulation A. Goal The goal of our project was to design an electricity-generating wind turbine in a manner that maximizes power output over all expected wind conditions while minimizing construction costs. The predominant expense of wind turbine construction is the manufacture of the turbine blades.3 As such, we focus on minimizing the amount of material required to make the blade, while maximizing power output. B. System Boundary The considered disciplines include aerodynamics, structures, and control. We consider a range of incoming velocities that represent the possible operating conditions of the turbine, calculating the expected power output and extreme structural load over this range. We model the generator as a resistive torque and a constraint on the maximum torque allowed, and design a control system to keep the blades at the optimal pitch angle for a given incoming wind velocity. We do not consider tower design, nor do we consider nacelle shape. However, we assume there is no wind within 20% of the blade radius, roughly the expected nacelle size. To further limit our design space, we assumed a three-bladed design with aluminum blades. We choose three blades because an initial design of experiments (DOE) test showed that three blades outperformed both four and five-bladed designs in almost all cases. Although modern blades are often being constructed with composites, our knowledge of structural behavior in composites is limited, and we therefore elected to assume aluminum blades. Finally, we assume that our wind turbine has a control system that allows it to feather the blades in a manner that the structural stresses are alleviated when the wind speed is too high. C. Design Vector In order to perform numerical optimization, the design must be broken down into a list of decision variables and parameters that completely define the design. In order to reduce the order of our model, we used a minimal number of design variables to define the blade shape but then mapped the design vector into a higher dimensional discretization for analysis by using Piecewise Cubic Hermite Interpolating Polynomials (PCHIP). The bounds were chosen to mirror the physical constraints of the wind turbine design. For example, the blade radius is limited to 16.15 m, because this is the standard flatbed truck length of 52 feet, and each blade would need to be transported to the site. Also, the sum of the bounds for the twist angle and pitch control angles is equal to a range from zero to ninety degrees. This ensures a realistic range of feasible geometric blade angles. The design vectors and bounds are shown in Table 1. In addition, a set of parameters that include physical constants such as air density, or fixed design parameters such as the number of turbine blades, are defined in Table 2. For the twist distribution, the twist at the hub is assumed to be zero, so only two decision variables for the value of twist at the midpoint and tip of the blade specify the entire distribution. Similarly, the foil shape parameter specifies a linear combination of two airfoil shapes and