Aerodynamic design optimization of wind turbine rotors under geometric uncertainty

TLDR: 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.

Figures

Figure 2: Analysis of AEP [kWh] PDFs. Left subplot: non-optimal nominal rotor. Right subplot: optimal nominal rotor.

Figure 6: Rotor aerodynamics of deterministic and robust designs at U = 12 m/s. Left subplot: angle of attack α [deg] against radius r [m]. Right subplot: lift coefficient CL against radius.

Table 1: Range of design variables.

Table 2: Expectations and standard deviations of AEP [kWh] and BM [kNm] of a non-optimal nominal rotor based on MC and URQ sampling.

Table 3: Expectations and standard deviations of AEP [kWh] and BM [kNm] of an optimal nominal rotor based on MC and URQ sampling.

Figure 4: Rotor performance of deterministic and robust designs. Left subplot: mechanical power [kW ] against wind speed U [m/s]. Middle subplot: proportion of AEP [kWh] accounted for by each wind speed. Right subplot: blade root bending moment [kNm] against wind speed.

Full text

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.
Ω 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

Frequently asked questions

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. 

Q2. How is the shape of the blade reconstructed?

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

Q3. What is the way to reduce the sensitivity of the turbine to rotor geometry errors?

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. 

Q4. What is the way to optimize a stall-regulated rotor?

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. 

Q5. What is the simplest way to propagate uncertainty through an analysis system?

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

Q6. Why did the authors use evolution-based optimization in this study?

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

Q7. What is the sensitivity of the rotor to stochastic geometry 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. 

Q8. What is the effect of the aerodynamic loading on the blades of the probabilistic rot?

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. 

Q9. What is the way to propagate uncertainty in the multi-disciplinary analysis system?

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

Q10. What is the pitch angle of the probabilistically optimal rotor?

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. 

Q11. How does the latter approach reduce the cost of each robust analysis?

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.