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Journal ArticleDOI

Aerodynamic design optimization of wind turbine rotors under geometric uncertainty

TL;DR: In this paper, a robust optimization strategy for the aerodynamic design of horizontal axis wind turbine rotors including the variability of the annual energy production due to the uncertainty of the blade geometry caused by manufacturing and assembly errors is presented.
Abstract: Presented is a robust optimization strategy for the aerodynamic design of horizontal axis wind turbine rotors including the variability of the annual energy production due to the uncertainty of the blade geometry caused by manufacturing and assembly errors. The energy production of a rotor designed with the proposed robust optimization approach features lower sensitivity to stochastic geometry errors with respect to that of a rotor designed with the conventional deterministic optimization approach that ignores these errors. The geometry uncertainty is represented by normal distributions of the blade pitch angle, and the twist angle and chord of the airfoils. The aerodynamic module is a blade-element momentum theory code. Both Monte Carlo sampling and the univariate reduced quadrature technique, a novel deterministic uncertainty analysis method, are used for uncertainty propagation. The performance of the two approaches is assessed in terms of accuracy and computational speed. A two-stage multi-objective evolution-based optimization strategy is used. Results highlight that, for the considered turbine type, the sensitivity of the annual energy production to rotor geometry errors can be reduced by reducing the rotational speed and increasing the blade loading. The primary objective of the paper is to highlight how to incorporate an efficient and accurate uncertainty propagation strategy in wind turbine design. The formulation of the considered design problem does not include all the engineering constraints adopted in real turbine design, but the proposed probabilistic design strategy is fairly independent of the problem definition and can be easily extended to turbine design systems of any complexity.

Summary (3 min read)

1 Introduction

  • The growing availability of large computational resources and progress of design optimization technologies offer the means to automate significant portions of product design.
  • This paper focuses on the development and the demonstration of a general methodology to incorporate uncertainty in HAWT design.
  • It should be noted that, even when only one objective function is considered, robust design optimization problems can be viewed as multi-objective optimization problems.
  • The main objective of the robust turbine design optimization exercise presented herein, however, is to explore the potential of robust design optimization for improving general multi-disciplinary HAWT design technologies [19], rather than proposing new design solutions for a particular turbine type.
  • The following section presents a comparative numerical study of the MC and URQ approaches to uncertainty propagation aimed at assessing their accuracy and computational cost in view of their use within the global evolution-based optimizers used in this study.

2 Optimizers

  • Evolutionary Algorithms (EAs) solve optimization problems by making a generation of individuals evolve subject to selection and search operators.
  • The Parzen method allocates Nind identical kernels (where Nind is the number of individuals of the current population), each centered on a different element of the sample.
  • In the unconstrained DE algorithm [26], and also in the unconstrained IDEA algorithm [18], each parent solution is compared with its offspring, and the solution with a better value of the objective function is passed to the next generation.
  • When the cp values of parent and offspring are the same, the selection is performed on the basis of the objective function.
  • This algorithm ’pushes’ the individuals of a sub-population of the MOPED front towards a better local approximation of the sought Pareto front.

3.1 design problem definition

  • In all HAWT rotor design optimizations reported below, the objective function to be maximized is the annual energy production (AEP) of a three-blade turbine, and this output is taken to be the yearly amount of mechanical energy at the turbine shaft.
  • The rotational speed Ω associated with each wind speed is also a design variable.
  • The adopted modeling choice will also enable accounting for such variability in HAWT design optimization.
  • The stochastic blade shape geometry errors may induce some uncertainty in the dependence of the power coefficient on TSR, and, hence, also on the optimal values of the latter parameter, which determine the optimal rotational speeds for each wind speed.
  • WINSTRIP has been validated against the experimental data of the NREL 2-blade UAE phase-VI test turbine [32], and a good agreement has been observed.

3.2 modeling aspects

  • To analyze both the algorithmic and engineering aspects of the proposed probabilistic design optimization framework, the robust optimization problem presented above has been formulated and solved in several different manners.
  • On the engineering side, two different definitions of the problem have been considered.
  • The latter scenario is more realistic, but the former has also been considered because it enables a clearer identification of the effects of stochastic geometry variability on the AEP variability.
  • In the MC3 case, the performance of each ’real’ rotor is obtained by taking the arithmetic average of the performance of three different ’fictitious’ rotors, each with identical blades affected by a different pattern of geometry errors.
  • As the deterministic sampling of URQ requires 2n + 1 evaluations of the functional of interest [11], with n being the number of uncertain design variables, each robust analysis requires 27 computations of AEP, namely 189 WINSTRIP runs.

4 Uncertainty propagation

  • Here the URQ and MC sampling techniques are assessed and compared both in terms of accuracy and computational costs and accuracy.
  • The MC and URQ expectations and standard deviations of AEP and BM of the optimal nominal rotor are reported in Table 3.
  • In other robust optimizations making use of URQ for propagating uncertainty [9, 34], the discrepancies between the URQ and MC estimates of the standard deviations of the output functionals has been found to be substantially smaller than in the present case, presumably due to smaller nonlinearities of the output functionals.
  • The deviation of the parzen fit from the gaussian curve can be taken as a measure of the nonlinearity of the BEM-based analysis.
  • These findings account for the observed discrepancies between the URQ- and MC-based estimates of σAEP .

5.1 robust optimizations

  • The adopted formulation of the robust design optimization problem leads to a Pareto front arising from the trade-off between the expectation and the standard deviation of AEP.
  • The solution labeled ‘URQ1 ref.’ is the optimal rotor considered in URQ/MC cross-comparison in the preceding section.
  • One notes that also in the MC3 and URQ3 cases a wide range of Pareto-optimal values of µAEP exists, and that a substantially smaller range of σAEP corresponds to this µAEP range.
  • These are the main reasons why evolution-based optimization has been selected for this study.
  • This require 10, 000 probabilistic function evaluations.

5.2 robust and deterministic optima

  • The optimal rotor obtained by solving the deterministic optimization problem described in section 3 has a nominal AEP of 96, 170 kWh, and µAEP = 89, 974 kWh.
  • The performance of these two rotors is compared in greater detail in the three subplots of Fig.
  • One sees that the sectional pitch angle of the probabilistically optimal rotor blade is smaller than that of the deterministically optimal blade over most part of the blade length, featuring a maximum reduction of about 16 % at 4 m blade length.
  • More realistic design specifications would consider this type of structural constraint applied to extreme load conditions.
  • Therefore, the variation of the lift coefficient caused by a given variation of α is smaller for the ‘URQ1 ref.’ blade.

6 Conclusions

  • A novel robust optimization framework for the design of variable-speed HAWT rotors has been presented.
  • This multi-objective optimization problem has been solved with an effective two-stage multi-objective evolution-based optimization strategy.
  • Two alternative methods for propagating uncertainty through the design chain have been considered: a standard Monte Carlo approach and the Univariate Reduced Quadrature.
  • The comparative analysis of a rotor design obtained by considering the stochastic geometry errors, and a rotor design obtained by neglecting all uncertainties shows that the AEP standard deviation of the former rotor is less than 40 % that of the latter.
  • These verifications were beyond the scope of the paper, and are ground for extensions of this work.

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Content maybe subject to copyright    Report

This is the peer reviewed version of the following article “Aerodynamic design optimization of
wind turbine rotors under geometric uncertainty”, by M.S. Campobasso, E. Minisci, M. Caboni,
Wind Energy, 2016, which has been published in final form at https://doi.org/10.1002/we.1820.
This article may be used for non-commercial purposes in accordance with Wiley Terms and
Conditions for Use of Self-Archived Versions.
Aerodynamic design optimization of wind turbine rotors under geometric
uncertainty
M. Sergio Campobasso, E. Minisci, M. Caboni
Wind Energy, Vol. 19, no. 1, 2016, pp. 51-66.
First published: 14 November 2014

AERODYNAMIC DESIGN OPTIMIZATION OF WIND
TURBINE ROTORS UNDER GEOMETRIC UNCERTAINTY
M. Sergio Campobasso
Lancaster University
Department of Engineering
Engineering Building
Lancaster LA1 4YR, United Kingdom
m.s.campobasso@lancaster.ac.uk
Edmondo Minisci
University of Strathclyde
Department of Mechanical and
Aerospace Engineering
75 Montrose Street
Glasgow, G1 1XJ, United Kingdom
edmondo.minisci@strath.ac.uk
Marco Caboni
University of Glasgow
School of Engineering
James Watt Building South
University Avenue
Glasgow, G12 8QQ, United Kingdom
Phone: +44 (0) 7918876704
m.caboni.1@research.gla.ac.uk
November 16, 2014
Address all correspondence to this author.
1

Abstract
Presented is a robust optimization strategy for the aerodynamic design of horizontal axis wind
turbine rotors including the variability of the annual energy production due to the uncertainty
of the blade geometry caused by manufacturing and assembly errors. The energy production
of a rotor designed with the proposed robust optimization approach features lower sensitivity
to stochastic geometry errors with respect to that of a rotor designed with the conventional
deterministic optimization approach that ignores these errors. The geometry uncertainty is
represented by normal distributions of the blade pitch angle, and the twist angle and chord of the
airfoils. The aerodynamic module is a blade-element momentum theory code. Both Monte Carlo
sampling and the univariate reduced quadrature technique, a novel deterministic uncertainty
analysis method, are used for uncertainty propagation. The performance of the two approaches
is assessed in terms of accuracy and computational speed. A two-stage multi-objective evolution-
based optimization strategy is used. Results highlight that, for the considered turbine type, the
sensitivity of the annual energy production to rotor geometry errors can be reduced by reducing
the rotational s peed and increasing the blade loading. The primary objective of the paper is
to highlight how to incorporate an efficient and accurate uncertainty propagation strategy in
wind turbine design. The formulation of the considered design problem does not include all
the engineering constraints adopted in real turbine design, but the proposed probabilistic design
strategy is fairly independent of the problem definition and can be easily extended to turbine
design systems of any complexity.
KEYWORDS: wind turbine rotor design; sto chastic geometry errors; manufacturing toler-
ances; probabilistic design optimization.
2

Nomenclature
AEP Annual energy production.
BM Root bend ing moment.
DE Differential evolution.
EA Evolutionary algorithm.
HAW T Horizontal axis wind turbine.
IDEA Inflationary differential evolution algorithm.
MC Monte Carlo.
MOP ED Multi-obj ective Parzen-based estimation of distribution.
P DF Probability density function.
N
d
Number of design variables.
R Tip radius.
SSC Stochastic simplex collocation.
T SR Tips speed ratio.
U Freestream wind velocity.
U
rel
Relative wind velocity.
URQ Univariate reduced quadrature.
n Number of uncertain design variables.
r Radius along the blade.
x Array of design variables.
α Angle of attack.
θ
p
Section pitch angle.
θ
p,0
Blade pitch angle.
θ
T
Blade twist angle.
µ Expectation.
σ Standard deviation.
σ
2
Variance.
φ Angle of relative wind.
Rotational speed.
3

1 Introduction
The growing availability of large computational resources and progress of design optimization tech-
nologies offer the means to automate significant portions of product design. In the past few years,
several studies on the us e of diverse optimization techniques for the preliminary design of hor-
izontal axis wind turbines (HAWTs) have appeared. Some of these applications have focused
on the optimization of existing blades by means of local search approaches [1, 2, 3, 4], utilizing
low- to medium-fidelity models. Global multi-objective evolution-based search methods have also
been used, often to optimize HAWT conceptual designs, and investigate the choice of fundamental
HAWT design parameters, such as its rotor diameter, on the economy of whole wind farm s [5].
One way in which modern HAWT d esign could be further improved is by accounting for the
effects of environmental, operational and engineering uncertainty throughout the design process.
The use of uncertainty management an d quantification tools increases computational costs, and
this motivates the efforts to develop new approaches allowing these technologies to be efficiently
integrated in HAWT design. Accounting for the impact of uncertainty in HAWT design requires the
use of numerical methods which can reliably propagate uncertainty throughout the design system
without keeping HAWT design computationally unaffordable. The conceptually simplest way to
propagate uncertainty through an analysis system is to sample the design space using Monte Carlo
(MC) methods [6]. Unfortunately, MC methods are computationally expensive, requiring a large
number of function evaluations to converge. Therefore, researchers have been developing alterna-
tive, computationally cheaper approaches to uncertainty propagation. The main difficulty is to
reduce computational costs with respect to MC methods while maintaining an acceptable accuracy
of the probabilistic parameters of the output values. The techniques that have been pr oposed to ac-
complish these two conflicting requir ements range from the Taylor-based m ethod of moments [7, 8]
to quadrature methods [9] and polynomial chaos expansion [10]. Among the proposed alternatives,
an appealing one is the Univariate Reduced Quadrature (URQ) approach [11], which has b een suc-
cessfully used for the robus t shape optimization of a transonic airfoil by means of a local gradient
based search. The use of this deterministic sampling technique in robust design optimization based
on global search methods is appealing and promising, but so far the URQ uncertainty propagation
technique has n ot been used in global design optimization. As shown below, the use of URQ in the
context of robust design optimization of HAWT rotors b ased on global search method is one of the
novel elements of this paper.
This p aper focuses on the development and the demonstration of a general methodology to
incorporate uncertainty in HAWT design. To the best of the authors’ knowledge, this issue has so
far received little attention despite the significant implications it may have on HAWT design, turbine
energy production and, ultimately, cost of energy. One of the sources of engineering u ncertainty
is the effect of b lade geometry errors caused by finite manufacturing and assembly tolerances on
the power and, for a selected site, the energy production of the tur bine. The problems associated
with deviations of the actual blade geometry from its nominal shape is mentioned in [12], and a f ew
preliminary investigations of this matter are reported in [13]. Petrone et al. [14] have studied th e
impact of blade twist errors due to finite manufacturing tolerances, and also wind speed, turbulence
intensity and wind direction variability, and blade surface roughness variations caused by insect
contamination on the mean power coefficient and acoustic emissions of a stall-regulated HAWT.
These authors have used Latin Hypercu be Sampling and the Stochastic Simplex Collocation (SSC)
method to propagate uncertainties throughou t the multi-disciplinary analysis system. Petrone et
al. [15] have also optimized the blade geometry of a stall-regulated rotor for maximum mean power
co efficient and minimum acoustic emissions considering th e uncertainty on laminar-to-turbulent
transition caused by uncertain blade surface roughness levels. Uncertainty has been propagated
4

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References
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"Aerodynamic design optimization of ..." refers methods in this paper

  • ...The corrections implemented in WINSTRIP include: a) Prandtl’s tip and hub loss corrections [28], b) Glauert’s correction [28] improved by Buhl [29] to account for axial induction factors exceeding the maximum theoretical limit of 1/2, and c) Snel’s [30] and the AERODAS [31] corrections to account for the rotational effect known as Himmelskamp or centrifugal pumping effect, occurring in the presence of large areas of stalled flow....

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TL;DR: This paper suggests a non-dominated sorting-based MOEA, called NSGA-II (Non-dominated Sorting Genetic Algorithm II), which alleviates all of the above three difficulties, and modify the definition of dominance in order to solve constrained multi-objective problems efficiently.
Abstract: Multi-objective evolutionary algorithms (MOEAs) that use non-dominated sorting and sharing have been criticized mainly for: (1) their O(MN/sup 3/) computational complexity (where M is the number of objectives and N is the population size); (2) their non-elitism approach; and (3) the need to specify a sharing parameter. In this paper, we suggest a non-dominated sorting-based MOEA, called NSGA-II (Non-dominated Sorting Genetic Algorithm II), which alleviates all of the above three difficulties. Specifically, a fast non-dominated sorting approach with O(MN/sup 2/) computational complexity is presented. Also, a selection operator is presented that creates a mating pool by combining the parent and offspring populations and selecting the best N solutions (with respect to fitness and spread). Simulation results on difficult test problems show that NSGA-II is able, for most problems, to find a much better spread of solutions and better convergence near the true Pareto-optimal front compared to the Pareto-archived evolution strategy and the strength-Pareto evolutionary algorithm - two other elitist MOEAs that pay special attention to creating a diverse Pareto-optimal front. Moreover, we modify the definition of dominance in order to solve constrained multi-objective problems efficiently. Simulation results of the constrained NSGA-II on a number of test problems, including a five-objective, seven-constraint nonlinear problem, are compared with another constrained multi-objective optimizer, and the much better performance of NSGA-II is observed.

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"Aerodynamic design optimization of ..." refers methods in this paper

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TL;DR: This volume explores the differential evolution (DE) algorithm in both principle and practice and is a valuable resource for professionals needing a proven optimizer and for students wanting an evolutionary perspective on global numerical optimization.
Abstract: Problems demanding globally optimal solutions are ubiquitous, yet many are intractable when they involve constrained functions having many local optima and interacting, mixed-type variables.The differential evolution (DE) algorithm is a practical approach to global numerical optimization which is easy to understand, simple to implement, reliable, and fast. Packed with illustrations, computer code, new insights, and practical advice, this volume explores DE in both principle and practice. It is a valuable resource for professionals needing a proven optimizer and for students wanting an evolutionary perspective on global numerical optimization.

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TL;DR: The differential evolution (DE) algorithm is a practical approach to global numerical optimization which is easy to understand, simple to implement, reliable, and fast as discussed by the authors, which is a valuable resource for professionals needing a proven optimizer and for students wanting an evolutionary perspective on global numerical optimisation.
Abstract: Problems demanding globally optimal solutions are ubiquitous, yet many are intractable when they involve constrained functions having many local optima and interacting, mixed-type variables.The differential evolution (DE) algorithm is a practical approach to global numerical optimization which is easy to understand, simple to implement, reliable, and fast. Packed with illustrations, computer code, new insights, and practical advice, this volume explores DE in both principle and practice. It is a valuable resource for professionals needing a proven optimizer and for students wanting an evolutionary perspective on global numerical optimization.

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Frequently Asked Questions (11)
Q1. What are the contributions mentioned in the paper "Aerodynamic design optimization of wind turbine rotors under geometric uncertainty" ?

The primary objective of the paper is to highlight how to incorporate an efficient and accurate uncertainty propagation strategy in wind turbine design. 

The shape of the blade is reconstructed by using the MATLAB R© shape-preserving piecewise cubic interpolation function pchip over the six radial stations. 

Results highlight that, for the considered turbine type, the sensitivity of the annual energy production to rotor geometry errors can be reduced by reducing the rotational speed and increasing the blade loading. 

Petrone et al. [15] have also optimized the blade geometry of a stall-regulated rotor for maximum mean power coefficient and minimum acoustic emissions considering the uncertainty on laminar-to-turbulent transition caused by uncertain blade surface roughness levels. 

The conceptually simplest way to propagate uncertainty through an analysis system is to sample the design space using Monte Carlo (MC) methods [6]. 

one of the main reasons for using evolution-based optimization in this study was that this technology can easily handle multi-objective problems. 

The energy production of a rotor designed with the proposed robust optimization approach features lower sensitivity to stochastic geometry errors with respect to that of a rotor designed with the conventional deterministic optimization approach that ignores these errors. 

This increment of the aerodynamic loading is achieved by increasing the angle of attack (to a large extent through lower values of the sectional pitch) and the chord of the blade over most part of the blade length. 

These authors have used Latin Hypercube Sampling and the Stochastic Simplex Collocation (SSC) method to propagate uncertainties throughout the multi-disciplinary analysis system. 

The pitch angle of the probabilistically optimal rotor and, to a minor extent, that of the deterministically optimal rotor, are negative over part of the blade height. 

The latter approach reduces the cost of each robust analysis by up to three orders of magnitude with respect to the case in which standard Monte Carlo sampling is adopted.