# Aerodynamic design optimization of wind turbine rotors under geometric uncertainty

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

...MOPED implements the general layout and the selection techniques of the Non-dominated Sorting Genetic Algorithm II (NSGA-II) [22], but traditional crossover and mutation search approaches of NSGA-II are replaced by sampling of the Parzen model....

[...]

...NSGA-II was chosen as the base for MOPED mainly due to its simplicity, and also for the excellent results obtained for many diverse optimization problems [23, 24]....

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

...MOPED is a multi-objective optimizer for continuous problems that belongs to this class of algorithms and uses the Parzen method [21] to build a probabilistic representation of Pareto solutions, and can handle multivariate dependencies of the variables [16, 17]....

[...]

5,607 citations

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

...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....

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4,273 citations

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##### Frequently Asked Questions (11)

###### 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.