Weighted Least Squares Approach for an Adaptive Aerodynamic Engineered Structure With Twist Transformation

2019 ◽  
Vol 141 (5) ◽  
Author(s):  
Fuzhao Mou ◽  
Hamid Khakpour Nejadkhaki ◽  
Aaron Estes ◽  
John F. Hall
Keyword(s):  
Wind Speed ◽  
Least Squares ◽  
Weighted Least Squares ◽  
Twist Angle ◽  
Rayleigh Distribution ◽  
Torsional Stiffness ◽  
Support Design ◽  
Partial Load ◽  
Blade Section ◽  
Least Squares Approach

A design concept for a wind turbine blade with an adaptive twist transformation is presented. The design improves partial-load wind capture by adapting the twist distribution in relation to wind speed. Structural adaptability is enabled by actuating a series of compliant sections that are mounted on a relatively rigid spar. The sections are assumed to have a unique stiffness that is achievable through additive manufacturing technology. The authors' prior work employed an aerodynamic model to establish the theoretical blade twist distribution as a function of wind speed. The work in this paper focuses on a method to optimize the stiffness of each blade section that has been previously defined. A mathematical model is proposed to support design optimization. The model is parameterized in terms of actuator locations and the torsional stiffness ratios of each blade section. These parameters are optimized to allow the blade to adapt its twist distribution to match the prescribed configurations. The optimization is completed using a weighted-least squares approach that minimizes the error between the theoretical and practical design. The selected solution is based upon the configuration that maximizes production. Weights are assigned to bias the performance of the blade toward different operating regimes. Our results indicate that quadratically penalizing twist angle errors toward the blade tip increases power capture. A Rayleigh distribution is used to create three sets of wind data, which vary in average speed. These sets of data are used to evaluate the performance of the proposed blade and design technique.

Weighted-Least Squares Optimization Method for Control and Shape Design of an Adaptive Blade Twist Distribution to Increase Wind Capture

2018 ◽  
Author(s):  
Fuzhao Mou ◽  
Hamid Khakpour Nejadkhaki ◽  
Aaron Estes ◽  
John Hall
Keyword(s):  
Wind Speed ◽  
Least Squares ◽  
Weighted Least Squares ◽  
Twist Angle ◽  
Torque Control ◽  
Shape Design ◽  
Mathematical Framework ◽  
Relative Stiffness ◽  
Partial Load ◽  
And Control

This paper presents a novel wind turbine blade with an actively adaptable twist angle. A weighted-least square technique is proposed to design and control the blade in its application. Controlling the twist distribution provides new capabilities that may not be achievable with blade pitch or rotor torque control. An adaptive twist angle can reduce fatigue loads and improve the efficiency of wind energy conversion. Our previous work established the theoretical blade twist distribution that maximizes wind capture during partial load operation. The twist distribution changes continuously as a function of wind speed. In practice, it is a challenge to design and control the blade to adapt to this range of transformation. Accordingly, a blade concept and engineering design method are proposed to achieve this task. The blade is constructed from additively manufactured sections that are assumed to have tunable stiffness. The sections are mounted on a centralized spar that provides stiffness. The sections are actuated at each end and have two zones of stiffness. A mathematical framework prescribes (1) length of each blade section and (2) the relative stiffness between a pair of compliant shells. Establishing the section length effectively sets the points of actuation, while the relative stiffness establishes a nonlinear twist. These design selections determine the twist distribution. The method employing weighted-least squares is employed to optimize these selections. The approach biases the shape design and control towards the theoretical twist distribution at a range of designated wind speed. This enables a customized solution that maximizes the wind capture based on the wind conditions at a given installation site.

A Weighted Least Squares Approach for the Design of Adaptive Aerodynamic Structures Subjected to an Out-Of-Plane Transformation

2018 ◽  
Author(s):  
Fuzhao Mou ◽  
Hamid Khakpour Nejadkhaki ◽  
Aaron Estes ◽  
John Hall
Keyword(s):  
Least Squares ◽  
Weighted Least Squares ◽  
Turbine Blades ◽  
Operating Conditions ◽  
Torsional Stiffness ◽  
Power Coefficient ◽  
Mathematical Framework ◽  
Complex Needs ◽  
Least Squares Approach ◽  
The Ideal

An optimal design framework for adaptive wind turbine blades is presented. A mathematical framework establishes the topology of actuators and material compliance. These parameters are selected to adapt the blade twist distribution into a range of prescribed blade configurations. Our previous work established the ideal twist distribution configurations. The distributions improve the aerodynamic efficiency for a range of wind speeds in which the system operates at partial production. Within this range the nonlinear blade twist distribution changes in relation to the speed. The possibility of producing adaptively compliant structures is becoming increasingly possible with innovative materials and additive manufacturing (AM) processes. Our overarching goal is to create a comprehensive design infrastructure that integrates manufacturing and materials innovation with the complex needs of adaptive structures. This work proposes a method through which the ideal twist distribution can be actualized in structural implementation. The implementation involves a modular blade composed of flexible sections whose twist is modulated by actuators along the blade. Each flexible blade section is composed of two contiguous segments, each with a different torsional stiffness defined by a stiffness ratio. The stiffness variation within each section allows the blade to assume a nonlinear twist distribution when actuated. Errors relative to an ideal twist distribution are minimized by optimizing the stiffness ratios and twist actuator locations. The optimization is completed using a weighted least squares approach that allows the blade designer to bias blade performance toward different operating conditions. A quadratic weighting scheme that penalizes twist errors toward the blade tip is found to result in a higher power coefficient than other weighting schemes.

Calibration in a closed system: the case of X-ray diffraction analysis

2007 ◽  
Vol 40 (4) ◽  
pp. 694-701
Author(s):  
Thaung Lwin
Keyword(s):  
Diffraction Analysis ◽  
Least Squares ◽  
Functional Relationship ◽  
Weighted Least Squares ◽  
Ordinary Least Squares ◽  
X Ray Diffraction ◽  
Unknown Parameters ◽  
X Ray ◽  
Relationship Model ◽  
Least Squares Approach

Knudsen [X-ray Spectrom.(1981),10, 54–561] proposed and demonstrated a least-squares approach to estimating the unknown parameters of a system of equations required for calibration in X-ray diffraction analysis. The approach is an ordinary least-squares approach which does not incorporate information on the errors of the measured intensities for a set of samples used as standards. The purpose of the present paper is to show that a functional relationship model can be applied to the problem to account for all the variation due to sampling and measurement error in the peak intensities. It is also shown that Knudsen's calibration estimator can be regarded as an approximation to a more general and potentially more efficient weighted least-squares estimator derived from the functional relationship model. The closeness of the approximation depends on the nature of the covariance structure of the intensity measurements.

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