The wind energy industry relies heavily on computational fluid dynamics (CFD)
to analyze new turbine designs. To utilize CFD earlier in the design
process, where lower-fidelity methods such as blade element momentum (BEM)
are more common, requires the development of new tools. Tools that utilize
numerical optimization are particularly valuable because they reduce the
reliance on design by trial and error. We present the first comprehensive 3-D
CFD adjoint-based shape optimization of a modern 10 MW offshore wind
turbine. The optimization problem is aligned with a case study from
International Energy Agency (IEA) Wind Task 37, making it possible to compare
our findings with the BEM results from this case study and therefore allowing
us to determine the value of design optimization based on high-fidelity
models. The comparison shows that the overall design trends suggested by the
two models do agree, and that it is particularly valuable to consult the
high-fidelity model in areas such as root and tip where BEM is inaccurate. In
addition, we compare two different CFD solvers to quantify the effect of
modeling compressibility and to estimate the accuracy of the chosen grid
resolution and order of convergence of the solver. Meshes up to

Wind turbine rotor optimization aims to maximize wind energy extraction and
has been an important area of research for decades. A common metric is to
minimize the levelized cost of energy (LCoE)

A major drawback of naive upscaling is that mass increases with the cube of the rotor radius. The industry avoids the prohibitive mass increase by improving the blade design, which has resulted in blades that are more slender for a given power rating, where the increase in loads (and therefore mass) can be kept low. This further results in blades with increased capacity factors.

Traditionally, the blade design optimization process has been sequential,
where the optimization of airfoils and planform are performed in two distinct
steps. In the present work, we optimize the airfoils and the planform
concurrently using 3-D computational fluid dynamics (CFD). This concurrent
design optimization process is vital for the industry because, as previously
shown, concurrent design processes result in a larger gain compared to
sequential counterparts

The use of 3-D CFD is particularly valuable near the turbine blade root and tip, since the blade element momentum (BEM) method uses empirical models to capture 3-D effects for these regions. The increase in fidelity also allows us to explore out-of-plane features such as blade pre-bend and winglets, which is outside the scope of traditional BEM approaches.

Industry still relies heavily on BEM, given that the 3-D CFD shape design of rotors poses several challenges. One of these challenges is modeling all the load cases that drive the design during an optimization. Much work has been done in steady-state computations with steady uniform inflow, but to truly generate realistic loads, one should transition to turbulent inflow and accurately resolve the time domain. This poses an immense challenge in terms of memory and computation time and is an active area of research.

In this paper, we present results from a high-fidelity aerodynamic shape
optimization of a 10 MW offshore wind turbine rotor. By “high-fidelity”,
we mean a detailed modeling of the rotor in 3-D and the use of
Reynolds-averaged Navier–Stokes (RANS) equations to model the aerodynamics
throughout the optimization. The optimization is based on the case study from
the International Energy Agency (IEA) Wind
Task 37

Ideally, one would include all the relevant disciplines in such an optimization. This has been addressed in previous work using BEM-based aeroelastic tools combined with various cross-sectional analytical or finite-element-based structural tools.

We start the remainder of this paper with a literature review on wind turbine
optimization. We then explain the methodology (Sect.

This literature review on wind turbine optimization is divided into three
overall approaches: those that use low-fidelity and multi-fidelity models
(Sect.

CFD-based aerodynamic shape optimization is still rarely used in wind energy
research, but both the aerospace and the automotive communities have been
using it increasingly often

BEM codes have been used extensively throughout the wind energy community for
aerodynamic optimization. These codes are easy to implement and incur low
computational cost. Robustness has been an issue in BEM codes, as they do not
always converge

It has long been known that the design of wind turbines is inherently a
multidisciplinary endeavor. There have been more than two decades of research
where BEM has been coupled with elastic beam models to account for structural
deflections and material failure

BEM has also been coupled to structural models with different levels of
fidelity. This allowed

As we will detail later, gradient-based optimization algorithms, combined
with an adjoint method for computing the gradients, provide a powerful
approach to address large-scale problems. For multidisciplinary systems, it
is necessary to compute coupled derivatives, which presents
additional challenges

One obstacle in using BEM codes is that the lift and drag data must be at
hand. Typically, one uses data from wind tunnel experiments or low-fidelity
numerical models, such as a panel code

Gradient-based, gradient-free, and hybrid approaches have all been used to
optimize airfoils using panel codes. An example of a gradient-based
optimization approach is the Risø-B1 airfoil family, which currently is in
commercial use by several manufacturers.

Grasso et al. optimized airfoils dedicated to both the blade tip

Medium-fidelity vortex methods are popular aerodynamic models in wind turbine
applications. Vortex theory is based on potential flow, which does not model
the viscous effects modeled in RANS CFD. However, it does provide a more
realistic solution than BEM codes while still keeping the computational cost
low compared to CFD. Well-established vortex codes in the wind energy
community include the GENeral Unsteady Vortex Particle (GENUVP) code

These vortex codes have been widely used in analysis, but applications to
design optimization have been less frequent. Early optimization studies were
performed by

LRP stands for Light Rotor Project.

-30Shape optimization has also been used to optimize turbine blades using 3-D
CFD in conjunction with gradient-free and gradient-based methods.

There has been an increasing interest in blade extensions and winglets for
wind turbines, since they can offer a cost-effective alternative to a
complete blade redesign for site-specific performance enhancements.

To optimize with respect to large numbers of variables, gradient-based
algorithms are the only hope if one wishes to achieve convergence to an
optimum in a reasonable amount of time

The complex-step derivative approximation method is an alternative to finite
differences that is much more accurate but still scales linearly with the
number of variables

For large numbers of variables, the adjoint method provides an efficient way
to compute the required gradients

We now detail previous efforts on RANS CFD-based shape optimization using the
adjoint method, which we also use in the present work. These efforts are
listed in Table

Overview of related work using the adjoint method.

There have been several contributions to 2-D RANS shape optimization that use
the continuous adjoint approach

In a later work,

In a more recent work within unconstrained optimization,

OpenFOAM with a continuous adjoint solver has also been used in 3-D. This was
done by

The above work does not model the rotation, which is important to get the
correct local angle of attack along the blade and thus accurately compute the
forces acting on the blade. Several 3-D adjoint-based optimization efforts
model rotation effects, three of which studied the NREL Phase VI rotor

The optimization was not fully converged, as only three design iterations were performed. One drawback in this early work is the use of the frozen turbulence assumption, which they also identified as an area of future work.

The present work builds on

Overview of differences between the work by

Overview of aerodynamic optimization works of wind turbine rotors using the adjoint method.

Multi: multipoint optimization; turbulence: whether the turbulence model is included in the adjoint solver; deformation: whether the entire blade was allowed to deform; geometry: whether the entire blade was modeled; geometric constraints: whether any geometric constraints were imposed.

Another recent effort is that of

Finally,

In spite of the contributions cited above, many improvements are needed
before we achieve the ultimate goal of providing a “push-button solution”
for wind turbine manufacturers. This paper contributes with some
of these improvements by including
all of the following features in a comprehensive high-fidelity 3-D RANS-based
shape optimization framework:

enforcement of geometric constraints to ensure structural feasibility,

normal operation rotor load constraints limiting thrust and flapwise bending moment,

more precision and stability in the convergence of flow and adjoint solvers,

inclusion of a turbulence model in the adjoint solver,

a comprehensive set of design variables, and

modeling and deformation of the entire blade shape.

As previously mentioned, structural considerations are crucial in wind
turbine design.

We now briefly describe all components of the optimization
framework. The overall workflow is shown in Fig.

Extended design structure matrix (XDSM) showing the optimization framework. Green blocks are iterative solvers, whereas red boxes represent explicit functions. Black lines represent the process flow in the order of the numbers; gray lines represent data dependencies.

To deform the surface geometry and mesh, we use the Python module pyGeo
developed by

The volume deformation tool is called IDWarp and is based on the inverse
distance weighting function

EllipSys3D is an in-house, structured, multi-block, finite volume method (FVM)
flow solver developed at DTU Wind Energy by

In this study, we run EllipSys3D using the third-order quadratic upwind
interpolation for convection kinematics (QUICK) scheme and the

EllipSys3D has been validated against experimental data for the MEXICO RWT

ADflow is a compressible RANS solver based on SUmb

ADflow has been coupled to a structural finite element solver in the MACH
(MDO for aircraft configurations with high fidelity) framework

As previously mentioned, we use an ANK solver, which is implemented in ADflow
to provide robustness. The ANK implementation is important since it is
crucial to properly converge the flow field in order to obtain accurate
gradients. Newton–Krylov (NK) methods are not robust because they might not
converge if the starting point is outside the basin of attraction. ANK
addresses this convergence issue using a globalization method called
pseudo-transient continuation, which starts with the stable but inefficient
backward Euler method with a small time step, and then increases the time
step to approach the higher convergence rate of the NK solver. The ANK method
involves the solution of large linear systems using preconditioners. These
systems are solved in a matrix-free fashion with the generalized minimal residual (GMRES) algorithm

One crucial capability in ADflow is the efficient computation of gradients
through its adjoint solver. Together with the geometry and mesh deformation
tools mentioned above, and the optimization software mentioned in the next
section, this enables aerodynamic shape optimization with respect to hundreds
of design variables

We now derive the adjoint equations and briefly explain how they are
assembled and solved. A detailed description of the implementation is
provided in previous work

The cost of the adjoint method is independent of the number of design
variables because the adjoint equation (Eq.

In the adjoint equation (Eq.

As briefly mentioned in the introduction, there are two modes for automatic
differentiation: the forward mode and the reverse mode.

We use the Sparse Nonlinear OPTimizer (SNOPT)

In this work, we use ADflow as the CFD solver in the design optimization due
to its adjoint gradient computation and integration with geometry
parameterization, mesh deformation, and optimization tools. However,
EllipSys3D has been more thoroughly validated for wind turbine rotor flows,
so in this section, we verify ADflow against EllipSys3D for a three-bladed
pitch-regulated rotor geometry. In this section, we only include a mesh
convergence study for one operational condition. A more detailed flow
comparison is included in Appendix

All simulations are steady-state 3-D RANS of the rotor only, where effects
from both tower and nacelle have been neglected. Since we study a rigid
upwind turbine, neglecting tower and nacelle should have a limited effect. We
also note that we compute the flow field using a co-rotating, non-inertial
reference frame that is attached to the rotor. Therefore, the RANS equations
have additional terms to account for Coriolis and centripetal forces. Just as
for the IEA Wind Task 37 case, the three-bladed pitch-regulated rotor
geometry in the analysis is a design based on the DTU 10 MW RWT

Comparison of chord

The surface mesh consists of three blades, each with 36 blocks. For each
blade, there are 256 cells in the chordwise direction and 128 in the spanwise
direction (tip excluded). The surface mesh is generated using the in-house
Parametric Geometry Library (PGL). The tip was constructed using four blocks
of

The spherical volume mesh has an O–O topology generated with the hyperbolic
in-house mesh generator HypGrid

The mesh we just described above is the finest mesh we use, which we call the
L0 mesh. A coarser (L1) mesh is obtained by coarsening L0 once, i.e., by
removing every second cell in all three directions. Similarly, the L2 mesh is
obtained by coarsening L1. Unless otherwise stated, we use these three meshes
in all the work herein. The turbine geometry and the surrounding spherical
mesh are shown in Fig.

The baseline wind turbine design with the spherical L0 mesh around it. The blade span is 89.166 m, and the spherical mesh stretches to 20 times the blade span.

Baseline geometry used in the flow solver comparison and as starting
point for the optimization. Each blade has a surface mesh with 36 square
blocks. Each block has

Richardson extrapolation (Eq.

To quantify the mesh dependence for each solver, we compute the integrated
metrics – torque and thrust – for the three mesh levels (L0, L1, and L2)
and list them in Table

Mesh convergence study for the compressible solver ADflow and the
incompressible solver EllipSys3D. The operational conditions for the
convergence study correspond to the 8 m s

Table

In Table

It is clear from Fig.

There is a slight increase in error for EllipSys3D in the thrust value on the
finest mesh level, which is unexpected. It is also surprising that the
compressible solver seems to benefit so drastically from an increase in cell
count. Recent studies have suggested this can be the case for some
compressible solvers

This suggests that we could hope to further align the results between ADflow
and EllipSys3D in future work by using classical compressibility corrections.
Based on the grid convergence study above and in Appendix

In this section, we first introduce the design optimization problems for all the CFD and BEM cases we solve. We then explain the FFD parameterization, geometric constraints, and rotor load constraints.

We adapt and extend the design optimization problem from the IEA Wind Task 37
case study, which is to maximize the AEP for a range of wind speeds by
varying chord and twist, while constraining the increase in thrust and
bending moment to be no more than

We solve four different CFD-based optimizations derived from the problem
above:

This is used to maximize torque on the turbine with respect to blade pitch for a single wind speed (which in this case is equivalent to maximizing AEP). The purpose of this case is to validate the newly implemented rotational terms in the adjoint solver.

This is the same as the IEA Wind Task 37 problem (Eq.

This is the same as the single-point planform optimization but with the addition of blade shape variables. This problem takes advantage of the additional design freedom that is not available for BEM-based models.

This is the same as the IEA Wind Task 37 problem (Eq.

The single-point optimizations are all performed for a wind speed of
8 m s

In addition to the CFD-based optimizations, we solve two BEM-based
optimization problems for comparison with the CFD-based planform
optimization:

identical to the single-point planform optimization.

identical to the multipoint full shape optimization, except the shape variables are replaced by spanwise thickness distribution variables.

Overview of baseline geometry and FFD boxes

The baseline design is shown in Fig.

The FFD boxes are used to apply the pitch, twist, chord, and shape variables
to each blade. Since we want all three blades to have the same pitch and
shape, the variables are forced to be the same. Furthermore, the FFD boxes
have two fixed sections close to each other at the root to ensure

In Fig.

There were a few necessary changes we made to the IEA case study, but only
for the full shape optimizations. One such deviation is the dashed segment
connected to the thickness limit curve in
Fig.

The results are split into the four main problems listed in
Table

Overview of optimization problems.

In the pitch optimization, the pitch angle for the seven outer FFD sections
on each blade is controlled by a single design variable. The optimization
result is an increase in torque of

Variation of torque with the pitch design variable.

Figure

As shown in Fig.

Convergence history for the pitch optimizations.

Merit function history as a function of steps taken by the optimizer.

Pitch design variable optimization. All runs used 216 processors. This means that, for example, the L2 optimization had an actual wall-clock time just under 30 min.

For the planform optimization, described in Sect.

As we can see in Fig.

Comparing the results across mesh levels, there is a much larger spread than
for the pitch optimization. The result using the coarsest (L2) mesh is
significantly different from the ones obtained with the finer meshes (L1 and
L0); therefore, the L2 mesh is too coarse to obtain physically representative
results, which is consistent with the mesh convergence study
(Table

Figure

Final chord and twist distributions for the CFD-based planform optimizations.

Convergence history for all three mesh levels.

We now compare our L0 result from the planform optimization to our results
from the BEM1 and BEM2 optimization problems. We obtain the BEM results by
running HAWTOpt2

The BEM optimizations are performed with SNOPT. The baseline and optimized
chord and twist distributions are shown in Fig.

Scaled merit function history for the CFD-based planform optimizations.

Both the BEM1 and the BEM2 results have a steeper, more pronounced increase
in chord values, which we suspect our CFD framework could not reproduce due
to difference in the parameterization. The two innermost fixed FFD sections
ensuring the

Comparison between optimal chord

As for the twist comparison (Fig.

Planform optimization comparison between CFD and BEM results.

Using values for torque from Table

We now solve the full shape optimization problem as a single-point
optimization. As stated in Sect.

Convergence history

Table

Overview of single-point optimization results for the operational
conditions of the 8 m s

We now turn to the shape and pressure (

Comparison of

Comparing the airfoil shapes and corresponding

Another feature of the optimized airfoil shapes is the sharper LE. This is expected due to the fact that we are maximizing the performance at a single wind speed. This shape is not robust to changes in wind speed and would perform poorly at other wind speeds. This issue can be addressed by enforcing the LE radius constraints or by considering the performance for multiple wind speeds in the objective function, as we will see in the next section.

The motivation for this multipoint optimization is to take a whole range of
wind speeds into consideration to achieve a more robust design. We consider
both cases for normal power production and also cases leading to peak loading
conditions. The design optimization problem and model are the same as those
for the single-point optimizations (detailed in
Sect.

When it comes to selecting the wind speeds in a multipoint optimization, it
is important to consider speeds that lie outside the ideal operational range.
Typically, the rotational rate of the wind turbine rotor is controlled to
match a target tip speed ratio,
which is the ratio of the tip speed and the wind speed, given by

Research has shown that, in reality, the angle of attack varies significantly
(more than 4

Rotational rate schedule with wind speed, showing the ideal constant tip speed rate. The green dots are the wind speeds used in the multipoint optimizations.

The history of convergence and merit functions are shown in
Fig.

History of convergence

We first turn to the airfoil shape to assess the effect of adding geometrical
constraints while taking multiple angles of attack into consideration. The
airfoil shapes for the multipoint optimizations are compared to the
single-point ones in Fig.

Comparison of airfoil profiles obtained from single-point and multipoint optimizations. The profiles are taken from 35, 64, and 84 m spanwise positions.

Comparison of airfoil profiles obtained from multipoint optimizations with and without LE geometric constraint.

To obtain more realistic LE shapes, we added an LE thickness constraint to
the optimization problem. The geometric constraint was enforced as a
thickness constraint close to the LE. The resulting shapes are shown in
Fig.

Having verified that the resulting shapes for the multipoint full shape
optimizations are much improved, we now compare the multipoint optimization
results to other optimization results in Table

These results do not show that the industry can necessarily gain a

Overview of optimization results. As further detailed in
Sect.

To analyze the optimized designs from single-point and multipoint shape
optimizations in more detail, we plot the spanwise forces for both optimized
and baseline designs in Fig.

The normal force acts normal to the rotor plane and, integrated over all three blades, yields the rotor thrust. Likewise, the torque can be derived from the driving force by integrating its first moment along all three blades.

For the single-point shape optimization results, we see, as expected, an
overall large increase in tangential loading across the blade, and we observe
that a high loading is achieved in the root region of the blade as well. This
is partially due to the chord increase but also due to the fact that the
blade is optimized based on modeling that accounts for the complex
three-dimensional flow field, which is particularly dominant in the root
region. The thrust constraint and moment constraint were both essential for
the design to be industrially relevant for the single-point result: the
thrust constraint helped lower the overall thrust values to maintain
structural feasibility. The bending moment constraint resulted in a change in
the normal force distribution, where the peak moved farther inboard to reduce
high loads close to the tip region, as one would expect. Based on the
optimization output, we can verify that both constraints are active for the
single-point optimization, meaning that thrust and moment have reached the
upper limits of

Comparison of normal

We find the same overall trends for the multipoint results as we did for the single-point optimization. The relaxed thrust constraint for the multipoint optimization results in a rotor with slightly higher loads, which explains why the more robust design from the multipoint optimization outperforms the single-point result.

The multipoint optimization problem presented in this section is functional but should be further improved in the future to obtain truly practical wind turbines. First, the laminar to turbulent boundary layer transition should be modeled, since this affects the optimal airfoil shapes. In this work, we just assumed the boundary layer to be turbulent throughout. Second, a wider range of operating points should be considered by, for example, varying the rotation rate or pitch setting for a given wind speed.

In this work, we presented results from the high-fidelity RANS-based shape optimization of a 10 MW RWT. Based on our literature review of the high-fidelity shape optimization efforts in wind turbine design, we determined that this was a promising area of research.

We compared two state-of-the-art compressible and incompressible CFD solvers to quantify the mesh dependence and discrepancies across different RANS models applied on the same rotor. The results were compatible, and future work involving classical compressibility corrections was identified.

We investigated the advantage of using higher-fidelity models by comparing our optimization results to low-fidelity BEM results from the same case study. We did this through a planform optimization with chord and twist variables, where shape changes were restricted to keep the design case comparable with the BEM-based optimization. The overall design trends were the same across fidelities, with differences due the parameterizations and models. The same overall amount of improvement was observed.

Finally, full shape optimization was performed with respect to twist, chord, and airfoil shape design variables, which raised the number of design variables from 14 to 154. Here, the planform results were further improved with a factor of 1.44. The improvement was enabled by a decrease in relative thickness as well as the novel airfoil shapes.

While further developments are required to obtain truly practical wind
turbine blade shapes, this work shows that with the right tools, we can model
the entire geometry, including the root, and optimize modern wind turbine
rotors at the cost of

Data are available upon request to the corresponding author.

The following is a continuation of Sect.

Operational conditions for the simulations in the analysis. For the
compressible solver (ADflow), we use velocity, density, and temperature as
input parameters. ADflow then computes the complete thermodynamic conditions.
The density is set to

The case study is defined with a cut-in speed of

Integrated loads, in the form of thrust and torque, have been computed for
each simulation in Table

Figure

Turning to the surface-restricted streamlines in Fig.

In Fig.

The slice at 35 m shows the least consistent comparison, which we suspect is
due to the large amount of separation present both at suction and pressure
side. Given that the solvers use different turbulence models, it would be
surprising to find a perfect match at this position. We also note that the
pressure side separation results in a

Total thrust

Spanwise distribution of the normal force

Surface-restricted streamlines from the EllipSys solution for a wind
speed of 8 m s

Surface-restricted streamlines from the EllipSys solution obtained
using the original DTU 10 MW wind turbine geometry for the 8 m s

MHAM compiled the literature review, implemented the geometry constraint and the algorithmic differentiation contributions, created the FFD boxes and CFD meshes for the test cases, ran the analysis and optimizations, postprocessed the cases, and wrote the bulk of the paper. FZ assisted in work related to CFD meshes, analysis and optimization, and carried out all BEM optimizations. NNS assisted with CFD expertise within mesh generation, CFD solver settings, and issues related to compressibility effects. JRRAM guided the project, secured access to clusters, and gave technical advice on the entire tool chain at the University of Michigan. All authors took part in writing and editing the paper.

The authors declare that they have no conflict of interest.

We would like to thank the members of the MDO Lab for their support. We thank, in particular, Eirikur Jonsson for assistance with programs in the MACH framework, Nicholas Bons for helping with pyGeo, and Anil Yildirim for his assistance with the ANK solver. We also thank Charles A. Mader for consulting on the AD improvements implemented in the adjoint solver. Finally, researcher Michael McWilliam at DTU kindly assisted with IEA material.

This paper was edited by Alessandro Bianchini and reviewed by Joseph Saverin and one anonymous referee.