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 14×106 cells are used in the optimization whereby flow details are resolved.
The present work shows that it is now possible to successfully optimize
modern wind turbines aerodynamically under normal operating conditions using
Reynolds-averaged Navier–Stokes (RANS) models. The key benefit of a 3-D RANS
approach is that it is possible to optimize the blade planform and
cross-sectional shape simultaneously, thus tailoring the shape to the actual
3-D flow over the rotor. This work does not address evaluation of extreme
loads used for structural sizing, where BEM-based methods have proven very
accurate, and therefore will likely remain the method of choice.
Introduction
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) , which can be
decreased by lowering installation costs and operating expenses or by
increasing the annual energy production (AEP). Simply upscaling the turbine
leads to an increase in swept area, which in turn extracts more energy.
However, a naive upscaling does not capture the complexity of the problem
.
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 , which is the main principle in
multidisciplinary design optimization (MDO) .
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
https://community.ieawind.org/tasks/taskdirectory
(last access: 18 March 2019)
, which allows for a comparison with the
low-fidelity BEM results from this case study. Low-fidelity tools offer a
fast and reliable modeling approach. However, BEM does not capture the
physics as completely as high-fidelity CFD-based tools that solve the RANS
equations. In the present work, we aim to quantify the pros and cons of each
approach.
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.
showed that simultaneous design of the aerodynamic shape
and structural layout of a blade leads to passive load alleviation. This was
achieved through bend–twist coupling, which increased the AEP without
increasing loads and blade mass. The LCoE has been minimized by other
researchers while taking aerodynamics, structures, and controls into account,
thereby truly treating it as an MDO problem both for 5 MW turbines
and for 20 MW turbines . While we could
tackle high-fidelity aerostructural optimization using tools that have
already been demonstrated in aircraft wing design
, we focus solely on aerodynamic
shape optimization in the present work.
We start the remainder of this paper with a literature review on wind turbine
optimization. We then explain the methodology (Sect. ),
followed by a comparison between the compressible flow solver and an
incompressible flow solver (Sect. ). The design
optimization problem is presented in Sect. , followed
by the optimization results in Sect. . We end with our
conclusions in Sect. .
Literature review
This literature review on wind turbine optimization is divided into three
overall approaches: those that use low-fidelity and multi-fidelity models
(Sect. ), approaches that use CFD
models without adjoint sensitivities
(Sect. ), and approaches that
use CFD models with adjoint solvers
(Sect. ).
Low-fidelity and multi-fidelity shape optimization
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 . However, when it comes
to low-fidelity shape optimization, the wind energy community has a large
body of work.
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 . Robustness is critical,
especially when the analysis is part of an optimization cycle. A lack of
robustness will slow down the convergence in the best case, and interrupt the
optimization altogether in the worst case. To address this issue,
re-parameterized the BEM equations using a single
local inflow angle, resulting in guaranteed convergence.
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 , including work
in wind turbine optimization considering site-specific winds
.
BEM has also been coupled to structural models with different levels of
fidelity. This allowed to study possible
configurations to achieve bend–twist coupling resulting in load alleviation.
They found that the highest load reduction is obtained by combining (passive)
bend–twist coupling and (active) individual pitch control instead of using
only a single approach. Another example where BEM is part of a larger
multidisciplinary toolkit applied to the study of load alleviation is that of
, who maximized AEP without exceeding the original
overall loads of a 10 MW reference wind turbine (RWT). They achieved a
8.7 % AEP increase through passive load alleviation without an increase
in the blade mass and only minor increases in the loads, despite blades that
were 9 % longer. The parameterization was comprised of 60 design
variables and just as in the work of , they
computed the gradients with finite differences. After an initial step size
study, they ran a reduced set of design load cases to obtain the final
turbine design, which was then evaluated on the full design load basis. Their
work is a demonstration of the power of integrating design approaches.
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 .
introduced the application of the coupled adjoint method to the MDO of wind
turbines.
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 . Another option is
to use high-fidelity methods such as RANS CFD to generate the lift and drag
coefficients for the BEM code .
combine BEM with both panel and 2-D RANS CFD in a
comparison between two integrated blade design approaches
(“precomputational” and “free-form”) and a sequential approach. They used
a panel code iteratively to converge the BEM residual and then either a panel
code or CFD to generate the final lift and drag coefficients. Like
, they argued for the integrated design approach, but they
found that the precomputational approach achieved most of the benefits
yielded by the free-form approach. This is impressive, since the
precomputational approach took marginally more computation time than the
sequential approach.
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. described the
design and experimental verification process, where they used an in-house MDO
tool. They carried out the numerical design studies using XFOIL
and used the VELUX wind tunnel for 2-D experimental
verification. Due to concerns with XFOIL's accuracy in predicting separation,
they opted to verify the optimization results using the CFD code EllipSys2D,
thus combining fidelities in an attempt to balance speed and accuracy.
Grasso et al. optimized airfoils dedicated to both the blade tip
and the blade root , using
gradient-based and hybrid approaches, respectively, both based on the panel
code RFOIL, which is based on XFOIL. More recent airfoil studies have turned
to large, offshore, pitch-controlled wind turbines, including tests with
vortex generators that resulted in the development of a new airfoil family
.
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
, the Aerodynamic Wind Turbine Simulation Module (AWSM)
, and the method for interactive rotor aeroelastic
simulations (MIRAS) .
These vortex codes have been widely used in analysis, but applications to
design optimization have been less frequent. Early optimization studies were
performed by , , and
. More recent efforts based on vortex codes include
those of , who report on a surrogate-based optimization
methodology, and of , who use the complex-step method to
carry out the gradient-based optimization of a winglet. Researchers have also
developed analytic gradient computation for vortex methods by reformulating
the vortex dynamics using the finite element method (FEM)
. However, BEM is still well entrenched and is
currently the default choice for optimization.
High-fidelity CFD-based shape optimization without adjoint gradients
compared two numerical models of different fidelities (a
panel code and RANS CFD) to wind tunnel data. They found that the choice of
aerodynamic model had a large impact on the optimal design, which thereby
stresses the need for high-fidelity models such as RANS. This agrees with
, who report serious issues with Euler-based aircraft wing
design due to missing viscous effects (compared to RANS-based design). They
found that while Euler-based design yields some insights, the RANS-based
optimization is needed to achieve a realistic design. Therefore, we limit the
discussion in the present section to RANS CFD optimization.
used 2-D RANS with a transition model to carry out
gradient-based optimization using finite differences with nine design
variables and achieved an 11 % increase in torque. Similarly,
used nine design variables and a gradient-free method (a
genetic algorithm) to perform both multi-objective and single-objective
optimizations. By training a surrogate model, they sped up the optimization
process by almost 50 % while achieving similar results.
used two design variables (thickness and camber) to carry
out 2-D shape optimization with a gradient-free method of airfoils (NACA0015)
for vertical axis wind turbines (VAWTs). A subsequent 3-D modeling and CFD
evaluation of the VAWT using the optimized airfoils achieved power
coefficient increases up to 7 %. Finally, carried out
an airfoil optimization and wind tunnel validation. The developed
optimization framework, based on the open-source framework
OpenMDAO , included a combination of panel (XFOIL) and CFD
(EllipSys2D) codes for the analysis, where the turbulence is described using
the k-ω shear-stress transport (SST) turbulence model
and two transition models: γ-Reθ̃
and the eN Drela–Giles transition model described by
chap. 6. They used a total of 21 design variables and
computed the gradients using finite differences. They ran 20 optimizations
under various conditions, and since each optimization involved 2640 CFD
simulations, they split the procedure into two steps of increasing fidelity
to save time. First, they optimized using XFOIL, and then, they used this
intermediate result as a starting point for a subsequent CFD-based
optimization. Such “warm starts” are now common practice, and we also use
them in the present work. Using this framework, completed
the optimization of a 30 % and a 36 % airfoil called
LRP2
LRP stands for Light Rotor
Project.
-30
https://energiteknologi.dk/node/1197 (last
access: 18 March 2019)
and LRP2-36, respectively. Finally, through
experimental results from the Stuttgart Laminar Wind Tunnel for both LRP2-30
and LRP2-36, as well as the FFA-W3 counterparts (FFA-W3-301 and FFA-W3-360),
they demonstrated that the new airfoils exhibit a superior performance
compared to the FFA-W3 airfoils.
Shape optimization has also been used to optimize turbine blades using 3-D
CFD in conjunction with gradient-free and gradient-based methods.
used 3-D RANS and a genetic algorithm to optimize the
shape of wind turbine blades with up to 25 design variables. They concluded
that their gradient-free framework was functional and robust, but also that
many CFD evaluations were needed for the optimizer to converge due to the
high number of variables. As a final example of the use of gradient-free
methods with 3-D CFD models, optimized a winglet, also
using a genetic algorithm. They used two design variables (cant and twist
angle) to optimize the torque, resulting in a 9 % increase in power
production. The results were obtained by training a surrogate model (an
artificial neural network) using 24 CFD samples to reduce computational time.
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.
explore such a design problem. They used 12 design
variables to maximize the energy production while satisfying certain load
constraints from the original blade design. Like , they
also used a surrogate model that they trained using a random sampling
strategy. Here, they seek a more balanced design by using multiple wind
speeds throughout the sampling. Using gradient-based optimization on the
resulting surrogate model, they obtain a power increase of 2.6 % by
adding a winglet, while not increasing the flapwise bending moment at
90 % radius.
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 efficiency of
gradient-based optimization is dependent in large part on the cost and
accuracy of computing the gradients. Finite differences provide a way to
compute gradients that is easy to implement, but they are subject to
numerical errors, and they scale poorly with the number of design variables
.
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 . This method has been widely used,
including in some wind energy applications .
Some efforts tried to reduce the computational cost by using semi-empirical
gradients , surrogate models
, and mixed-fidelity models
.
For large numbers of variables, the adjoint method provides an efficient way
to compute the required gradients , a fact
that has also been verified in the wind energy community
. The adjoint method is the subject of the next section.
High-fidelity CFD-based optimization using the adjoint method
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.
a Number of cells in largest mesh
used for optimization. b Not all papers state the number of
optimization iterations explicitly. In some cases, we report the number of
iterations estimated from the cited figures. As mentioned in
, this number depends on the optimization problem and
optimizer settings, meaning that cross-setup comparison is difficult.
c only gives the number of mesh nodes.
d Reduced geometry where the root section was removed.
e Applied symmetric boundary conditions (BCs) double the mesh size
compared to others. f In cases where a range of Reynolds numbers
were used, we report the maximum values. g We only found
high-fidelity shape optimization for three turbine configurations in the
literature: two smaller turbines – NREL Phase VI and MEXICO
– and the large, commercial-scale IEA 10 MW wind
turbine. We find it reasonable to assume that the simulations for NREL
Phase VI and MEXICO have a Reynolds number on the order of Re=106
(, p. 152; , p. 10), while we
estimate the Reynolds number for the IEA turbine to be on the order of
Re=107p. 15–16.
were the first to use a high-fidelity shape optimization
method with an adjoint solver for wind turbine profiles. They optimize the
lift-to-drag ratio starting from the S809 airfoil using a compressible
solver, a low-Mach preconditioner (both for flow and adjoint solver), and the
Spalart–Allmaras (SA) turbulence model and find a tendency to increase
camber to gain more lift. Finally, they point to the k-ω SST
turbulence model and a transition model as needed improvements.
optimized the same airfoil using an enhanced
framework that included the cited improvements, where they attempted to
postpone the onset of transition. They concluded that both the capability and
accuracy of the discrete adjoint optimization framework improved by including
the new adjoint variables for the transition model.
There have been several contributions to 2-D RANS shape optimization that use
the continuous adjoint approach
. In these
efforts, the continuous adjoint implemented for ducted flows in the flow
solver OpenFOAM was extended to handle external
aerodynamics. First, optimized the lift-to-drag ratio of
the DU 91-W2-250 profile using 720 design variables while constraining
cross-sectional area. They use the “frozen turbulence” assumption, which
means that no adjoint equation is used for the turbulence model. Since each
surface point in this work is a design variable, they smooth the gradient for
stability. The result is a 5.7 % to 59 % increase in lift-to-drag
ratio for angles of attack ranging from 6.15 to 9.66∘.
In a later work, presented a
finite-difference verification of the adjoint gradients. The same group
performed the shape optimization of an upstream leading edge (LE) slat for
the DU 91-W2-250 airfoil and a validation of the framework using wind tunnel
data, showing good agreement below stall . The
optimization, which used 480 design variables, resulted in a 2 % decrease
in drag. As mentioned previously, there have been other efforts in turbine
blade design using 2-D RANS CFD with the adjoint method
that used the open-source compressible
solver SU2 . These works couple the 2-D RANS and adjoint
model to BEM, panel, and beam element analysis codes to arrive at a 3-D
multi-fidelity and multidisciplinary design framework.
present a benchmark of different optimization algorithms
(Nelder–Mead, steepest descent, and quasi-Newton) for unconstrained shape
optimization in 2-D, where the continuous adjoint solver within OpenFOAM is
used. The benchmark optimization problem is to find a target lift coefficient
from any baseline shape. However, they both consider computation time and
ease of use to grade the algorithms. As already mentioned, they point to the
use of the adjoint method to compute the gradient for a large number of
design variables. They recommend that further analysis be done within
constrained optimization and within multidisciplinary optimization.
In a more recent work within unconstrained optimization,
investigated the effect of the “frozen turbulence” assumption in 2-D. They
carried out their investigations on the NACA 0012 and DU 93-W-210 airfoils.
In this single-point study, they concluded that the implementation of adjoint
turbulence models results in better gradients than those obtained through the
frozen turbulence assumption. Finally, they specifically mention thickness
constraints as a future work topic.
OpenFOAM with a continuous adjoint solver has also been used in 3-D. This was
done by , who first performed a 2-D test case with two
design variables. The 3-D test case consisted of an extruded airfoil with a
spanwise length of five chords and a mesh of 2.4×106 cells with an
y+ of 2.5. They investigated both a twist and a bend–twist coupling case
but found that the bending had no discernible effect. This is something they
expect to change for future rotating blades applications.
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
, and another which studied the
MEXICO rotor . used a continuous
adjoint formulation to perform single-point aerodynamic shape optimization
using a compressible RANS model. In 2-D, they reduced drag starting from a
NACA 4412 profile baseline by 4.86 % under imposed thickness constraints.
They used a total of 50 design variables and completed 10 design iterations.
In 3-D, they improved the torque coefficient on a mesh with 7.9 million cells
by 4 % using 84 shape variables with no constraints imposed on geometry
or loads. The free-form deformation (FFD) box covered part of
the blade such that both the trailing edge and the innermost part of the
blade could not deform.
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.
used a discrete adjoint solver to carry out a multipoint
optimization of a two-bladed rotor using a 2.6 million cell mesh, where they
maximized the torque coefficient using up to 252 design variables. They used
pitch, twist, and local shape design variables while constraining the
thickness between 15 % and 50 % of the local blade chord to ensure
adequate space for a structural box. The final multipoint optimization
resulted in a 22.1 % increase in torque coefficient but also increased
the thrust by 69 %. The original design was meant to be a three-bladed
rotor, which explains the low thrust coefficients in the reported results
Table 1. They found the optimized shapes for both single
and multipoint optimization exhibited highly cambered trailing edges at the
root region where the wind speed is reduced. While this does agree with what
has been reported in 2-D cases , it is also exactly what
one would expect when chord is not included as a design variable.
The present work builds on , who used the same CFD solver,
ADflow. Our improvements are summarized in Table . One
major improvement has to do with the adjoint implementation that was used. As
we will explain in more detail later (in
Sect. ), our adjoint implementation uses the
automatic differentiation (AD) technique to compute certain derivatives
. One major improvement is that we implemented the more
memory-efficient reverse automatic differentiation. was
forced to use the less memory-efficient forward automatic differentiation
because the reverse option did not include the rotating terms required to
model wind turbine rotors. We also added constraints on maximum thrust and
flapwise bending moment to align with the IEA case study and enlarged the
design space to include chord design variables. Furthermore,
carried out their studies on the turbine blade excluding
the root because of flow solution convergence issues, whereas we include the
root. This was made possible by the robustness of the new approximate
Newton–Krylov (ANK) solver in ADflow, which also increases its speed
. Finally, we achieve an optimality convergence
tolerance that is up to 5 orders of magnitude lower.
Overview of differences between the work by and
the present work.
Overview of aerodynamic optimization works of wind turbine rotors
using the adjoint method.
ReferenceMultiTurbu-DeformationGeometryLoad constraints GeometricDesign variables lence(✓= full blade)(✓= full blade)ThrustMomentconstraintsTwistChordShape✓✓✓✓✓✓✓✓✓✓✓✓✓✓This study✓✓✓✓✓✓✓✓✓✓
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 , who performed
unconstrained optimization of the NREL Phase VI rotor where they minimized
the thrust by varying up to nine twist design variables using a steepest
descent optimization algorithm. Not surprisingly, they mention convergence
issues, in part due to the turbine being stall regulated and exhibiting
separated flow at some inflow speeds. Vortices at tip and root further
impaired the convergence, which in turn resulted in poor gradient quality.
They addressed this issue by limiting the deformable area to only the outer
50 % of the blade length, which limited the shape design optimization.
Like , they used the frozen turbulence assumption.
However, they differed in choice of turbulence model:
used the k-ω SST model, while used the SA
turbulence model. For future work, they suggested the use of more efficient
optimization algorithms, and mentioned the inclusion of the adjoint
turbulence equations and the study of turbines that are not stall
regulated.
Finally, used a continuous adjoint approach that
included the SA turbulence equations to optimize the MEXICO RWT. The flow was
modeled by the incompressible RANS equations and solved in a co-moving frame
of reference. They maximized the power for a single wind speed of
10 m s-1. Compared to the present work, they used a different
parameterization technique based on volumetric non-uniform rational B splines
(NURBS), which confine the blade in a small volume. NURBS are used both for
the deformation of the surface and the volume meshes, and the outermost NURBS
control points are fixed to keep the outer volume mesh fixed. This only
ensures C0 continuity. They use 385 NURBS, resulting in 135 design
variables, which are only allowed to move in the direction perpendicular to
the rotor plane. This choice of parameterization limits the design space; for
example, no chord increase can be obtained without simultaneously changing
profile shape. The flow and adjoint solvers take advantage of graphics
processing unit (GPU) hardware, resulting in fast solutions. Indeed, they
state that the overall optimization process can run up to 50 times faster on
GPUs than on CPUs. They obtained a 3 % increase in the objective function
and attribute this modest improvement to the limited freedom in the
parameterization.
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.
In Table , the present work is compared to the
above-cited 3-D shape optimization efforts on wind turbine rotors.
As previously mentioned, structural considerations are crucial in wind
turbine design. partially addressed this issue by
coupling the NSU3D RANS solver with the AStrO structural finite element
solver through a fluid–structure interface to converge on realistic,
steady-state loads on the SWiFT RWT. They used Abaqus to make a
finite element model with shell elements. They performed a purely structural
optimization of the composite blade, with the loads computed by the CFD. The
optimization's objective was to, using gradient-based optimization, minimize
the off-axis stress with respect to 16 310 ply orientation variables. They
completed 10 optimization iterations considering five different load cases
and achieved a reduction in the maximum fatigue stress between 40 % and
60 %. They did so without adding any constraints, but they did assume
the material to be a single-ply, unidirectional fiber composite for each
blade section. The logical next step would be to perform the simultaneous
optimization of the structural sizing and aerodynamic shape optimization, as
is already done in aircraft wing design .
Methodology
We now briefly describe all components of the optimization
framework. The overall workflow is shown in Fig. using an
extended design structure matrix (XDSM) diagram . An
initial set of design variable values, x(0), is given to the
optimizer. The optimizer passes the current design variables to the surface
deformation module, prompting it to update the surface mesh (except for the
very first iteration). The surface deformation module also provides analytic
derivatives of the surface mesh with respect to the design variables,
dxs/dx. After the surface mesh has been
updated, it is passed to the volume deformation module, which updates the
volume mesh and computes its analytic derivatives with respect to the surface
mesh, dxv/dxs. Then, the flow solver
computes the flow states, w. These states are passed to the adjoint
solver, which computes the total derivative. Finally, the objective function,
f (e.g., torque), as well as its derivatives, df/dx,
are provided to the optimizer, which computes a new step for another
optimization iteration. Both the surface and volume deformation steps are
fast explicit operations. On the other hand, the flow and adjoint solvers are
costly iterative operations that take up the vast majority of the computation
time. The optimization process involves O(102) major iterations,
which is an absolute minimum bound on the number of CFD solutions and mesh
updates; there are additional CFD solutions within each major iteration.
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.
Geometry and mesh deformation
To deform the surface geometry and mesh, we use the Python module pyGeo
developed by , which implements the FFD
technique. Some of the key features of FFD are
analytic derivatives and a machine-precision representation of the baseline
geometry.
The volume deformation tool is called IDWarp and is based on the inverse
distance weighting function . IDWarp is a fast and
unstructured deformation algorithm that has been demonstrated in aerodynamic
and aerostructural applications .
Flow and adjoint solversEllipSys3D
EllipSys3D is an in-house, structured, multi-block, finite volume method (FVM)
flow solver developed at DTU Wind Energy by
and , and we use it in
the present work to perform the comparison between CFD solvers. It
discretizes the incompressible RANS equations using general curvilinear
coordinates and couples velocity and pressure through the SIMPLE algorithm.
In this study, we run EllipSys3D using the third-order quadratic upwind
interpolation for convection kinematics (QUICK) scheme and the k-ω SST
model to calculate the turbulent eddy viscosity, which
compares favorably to other turbulence models for wind turbine applications
.
EllipSys3D has been validated against experimental data for the MEXICO RWT
and the NREL Phase VI RWT , and
also in a blind comparison . In addition, the unsteady
interaction between tower and blade has been simulated for the NREL Phase VI
RWT with EllipSys3D using overset grid capabilities, and an overall good
agreement was found with experimental data . EllipSys3D has
been used in various rotor applications to perform computations, such as
aerodynamic power and fluid–structure interaction
. The latter work also encompasses a comparison across
fidelities between the CFD-based tool, HAWC2CFD, and the BEM-based HAWC2
solvers where a good agreement was found. EllipSys3D has also been compared
with OpenFOAM for a case with atmospheric flow over complex terrain.
EllipSys3D was found to be 2–6 times faster while producing almost identical
numerical results . More recent sources also show
that these two solvers yield comparable results
.
ADflow
ADflow is a compressible RANS solver based on SUmb , a
structured FVM CFD solver written in Fortran 90 that uses cell-centered
variables on a multi-block grid. Unlike EllipSys3D, ADflow uses the
Spalart–Allmaras (SA) turbulence model and works with
state variables computed using the Jameson–Schmidt–Turkel (JST) scheme.
More recently, implemented overset mesh capability.
ADflow is wrapped with Python to provide a more convenient user interface and
to facilitate integration with optimization algorithms and other components
of an MDO framework.
ADflow has been coupled to a structural finite element solver in the MACH
(MDO for aircraft configurations with high fidelity) framework
, which has been used to perform not only aerostructural
optimization of aircraft configurations
but also
hydrostructural optimization of hydrofoils .
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
using the Portable, Extensible Toolkit for Scientific Computation (PETSc) library
. The adjoint solver linear
systems are solved using the same algorithm. ADflow is considered converged
when the ratio of the L2 norm of the residuals at iteration n and the same
norm of the free stream residual is below a given tolerance, i.e., when
η≤Rn2Rfs2.
For the optimizations presented below, we typically set η=10-9,
whereas the L2 convergence for the adjoint equation is set to 10-7.
These convergence thresholds are not to be confused with the optimality
tolerance, which we set to 10-4.
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 . All the
optimization results in Sect. are obtained with the ADflow
framework.
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 CFD
solver computes the flow field, w, for a given set of design
variables, x, by converging the residuals
R(x,w) of the governing equations to zero. Then, any
function of interest, f(x,w), can be computed. Gradient-based
optimizers require the gradient of the objective and constraint functions
with respect to the design variables. To compute this gradient, we use the
equation for the total derivative:
dfdx=∂f∂x+∂f∂wdwdx.
Here, the partial derivatives correspond to derivatives of explicit
functions, while the total derivative involves the iterative solution of the
governing equations. Thus, the partial derivatives can be found analytically
at a low computational cost, but the direct computation of the total
derivative dw/dx should be avoided. A
similar total derivative equation can be written for the residuals, which
must remain zero for the CFD solution to hold, and thus
dRdx=∂R∂x+∂R∂wdwdx=0.
We can now substitute the solution of the Jacobian given by the above
equation into the total derivative equation (Eq. ) to
obtain
dfdx=∂f∂x-∂f∂w∂R∂w-1︸ΨT∂R∂x,
where we have only partial derivative terms that can be found analytically at
a low computational cost. The linear system in this equation can either be
solved by computing the solution Jacobian,
dw/dx, from the linear system from
Eq. () or by solving the adjoint system:
∂R∂wTΨ=∂f∂wT,
where Ψ is the adjoint vector, which can be substituted into the
total derivative equation (Eq. ), i.e.,
dfdx=∂f∂x-ΨT∂R∂x.
The cost of the adjoint method is independent of the number of design
variables because the adjoint equation (Eq. ) does not
contain x. However, if there are multiple functions of interest f,
we need to solve Eq. () for each f with a different
right-hand side. Given that our problem has O(102) design
variables and only a few functions of interest, the adjoint method is
particularly advantageous.
In the adjoint equation (Eq. ) and total derivative equation
(Eq. ), we need to provide two matrices and two
vectors of partial derivatives. As mentioned above, these derivatives involve
only explicit operations and are in principle cheap to compute. However, they
still require the differentiation of parts of a complex CFD code, and a good
implementation is essential to preserve the accuracy and efficiency of the
adjoint approach. Traditionally, adjoint method developers have derived these
partial derivatives by differentiating the equations or code manually and
programming new functions that compute those derivatives. This process is
labor intensive and prone to programming errors. To address this drawback,
pioneered the use of automatic differentiation to compute
the partial derivatives. Automatic differentiation is a technique that takes
a given code and produces new code that computes the derivatives of the
outputs with respect to the inputs . Using a pure
automatic differentiation approach to compute our derivatives of interest,
df/dx, would mean applying the automatic
differentiation tool to the whole CFD code, including the iterative solver.
While this produces accurate derivatives, it is not an efficient approach. By
selectively using automatic differentiation to produce code that computes
only the partial derivatives, which do not involve the iterative solver, we
lower the adjoint implementation effort while keeping the efficiency of the
traditional adjoint implementation approach. There are still many details
involved in making our adjoint implementation approach efficient; these
details have been presented in previous work .
As briefly mentioned in the introduction, there are two modes for automatic
differentiation: the forward mode and the reverse mode. had
used automatic differentiation in forward mode to compute and store the flow
Jacobian, ∂R/∂w, as well as the other
partial derivatives. Then, these stored matrices are used by the adjoint
solver to compute transpose-matrix-vector products to converge the adjoint
solution, Ψ. Using the reverse mode, no storage of the Jacobian is
needed. Instead, a matrix-free approach is used, where the
transpose-matrix-vector products required to converge the adjoint solution
are computed directly through the reverse mode derivative routines. While the
reverse mode is more efficient in terms of memory usage, the reverse mode
implementation was missing the rotation terms required for wind turbine
modeling. We have fixed this for the implementation in the present work and
use the reverse mode instead. The implemented reverse AD routines may also
lead to speed up depending on the number of Krylov iterations needed to
converge the adjoint system.
Optimizer
We use the Sparse Nonlinear OPTimizer (SNOPT) for all
optimizations herein. SNOPT implements a sequential quadratic programming
(SQP) algorithm. We use it through the open-source Python wrapper
pyOptSparse
https://github.com/mdolab/pyoptsparse (last
access: 18 March 2019)
, which provides a common interface to this and other
optimization software. The convergence in SNOPT is set through the “major
optimality tolerance” setting . We aim at converging all
optimization problems to 10-4.
Flow solver comparison
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 .
Fluid model and computational mesh
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
, where both chord and twist distributions have been altered
to allow for more room for improvement using design optimization. We compare
the twist and thickness distributions for the DTU 10 MW RWT and the IEA Wind
Task 37 baseline in Fig.
Comparison of chord (a) and twist (b) for the DTU
10 MW RWT and the perturbed design used as the starting point for the IEA
optimization case study. Both the chord and twist are reduced. The baseline
blade design is based on the FFA-W3 airfoil family with relative thicknesses
in the range of [24%,36%].
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 32×32 cells each, resulting in a total surface mesh with 110 592
mesh cells.
The spherical volume mesh has an O–O topology generated with the hyperbolic
in-house mesh generator HypGrid . Setting the first boundary
layer cell height to 10-6 m yields a y+ of around 1 for the given
operational conditions, and a total of 128 cell layers are grown from the
surface mesh where the farthest vertices reach a distance of 1740 m. This
results in a total of 432 blocks, each with 32×32×32 cells,
which is equivalent to 14.155776 million cells. Given a span of R=89.166 m, the surrounding spherical mesh expands to about 20 times the
blade span.
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. , and a more detailed view of the
rotor is 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 32×32 cells, resulting in 110 592 surface
mesh cells.
Richardson extrapolation (Eq. ) for the grid convergence
study for thrust (a) and torque (b). Between the two
solvers, the extrapolated continuum values for thrust differ by 3 %,
whereas the errors for the torque values vary by less than 0.7 %.
Mesh convergence study
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 . The operational condition
corresponds to a wind speed of 8 m s-1 and rotor speed of 6.69 rpm at
zero blade pitch, which is one of the conditions listed in
Table in Appendix . As is
evident from the results for meshes L2, L1, and L0 in
Table , ADflow does not produce a sufficiently
mesh-independent solution on mesh L0. This agrees with an earlier mesh
convergence study Table 1, where up to 22 million cells
were used without reaching convergence. Therefore, we generated a finer mesh
with more than 47 million cells called L-1. The L-1 mesh is made exclusively
for the present grid convergence study and will not be used in the ensuing
optimizations.
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-1 case listed in
Table . The error percentages are estimated
using the Richardson extrapolations from Fig. .
Table shows that error reduction from L0 to L-1
for ADflow is much lower (with reductions of about 4 % in thrust and
7 % in torque) than the error reduction from L2 to L1 (15 % and
21 %) or from L1 to L0 (22 % and 41 %). The errors are
computed using the Richardson extrapolation values from Fig. ,
which are based on an estimate of the continuum value (in the limit of an
infinitely fine mesh), given by :
fc≈f1+f1-f2r2-1,
where fc is the continuum value, f1 and f2 are the values
obtained using the L0 and L1 meshes, respectively, and r is the grid
refinement ratio.
In Table , we can also see that the two solvers
tend to converge towards the same thrust and torque continuum values –
0.3 % difference for thrust and 0.7 % difference for torque. Based on
the results in this table, we determine that the L0 mesh represents a
reasonable compromise between accuracy (less than 10 % error) and speed.
It is clear from Fig. that mesh level L2 is very coarse and
yields very different results. As we will demonstrate later, the suggested
design trends from such a coarse mesh can sometimes lead to savings in
computation time and, other times, lead to completely wrong design trends.
Thus, one should use such coarse meshes with care. We report the results
obtained with L2 throughout the presented work to substantiate this claim.
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 . From the expressions for the
Prandtl–Glauert compressibility corrections , one would
expect that compressible effects could be at play, which agrees with our
results. Compressibility effects in wind turbine applications have become
increasingly significant as turbine rotor sizes have increased. One of the
conclusions from the AVATAR project was that compressibility effects play a
role in large wind turbines p. 9. In the AVATAR project,
results from EllipSys3D were compared to results from a compressible CFD
code. Here, they studied a case with an inflow speed of 14 m s-1 and a
Mach number of 0.2457 Fig. 8, where the obtained
Cp curves differed in particular on the suction side close to the
trailing edge (TE). The resulting sectional forces on the blade differed up
to 12.9 % Table 3. The cited Mach number of 0.2457 is
within the Mach number range of the present work, where we have Mach numbers
close to 0.3 at the tip depending on the inflow speed. The effects of
compressibility near the tip region have recently been studied by
. This work also includes results obtained with EllipSys3D.
They find that classical compressibility corrections to incompressible
results can be applied in a post-processing step in order to reduce the lift
and drag error to within 2.5 % for Mach numbers up to 0.3. The cases
studied by include Mach numbers ranging from 0 to 0.5.
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 ,
where we provide more details on the flow phenomena and solver performance,
we conclude that while there are discrepancies due to different turbulence
models, compressibility effects, and numerical scheme order, the trends for
the two solvers largely agree.
Implementation
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.
Design optimization problem
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 14 % and 11 %, respectively.
Thickness constraints are enforced over the blade to ensure structural
integrity. Mathematically, the IEA Wind Task 37 design optimization problem
can be expressed as follows:
maximizeAEPwith respect totwist8chordsubject toT≤1.14⋅TinitMbend≤1.11⋅Mbendinit.
The AEP is computed using a specified Weibull distribution (with scale and
shape parameters A=8 and k=2, respectively) and the power produced for
each wind speed, which is computed from the torque, Q, produced by the
turbine (P=ω⋅Q).
We solve four different CFD-based optimizations derived from the problem
above:Single-point pitch optimization:
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.
Single-point planform optimization:
This is the same as the IEA Wind Task 37 problem (Eq. ),
except with the objective of maximizing torque for a single wind speed. We solve this problem
because it is well suited for comparison with BEM.
Single-point full shape optimization:
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.
Multipoint full shape optimization:
This is the same as the IEA Wind Task 37 problem (Eq. )
but with the addition of blade shape variables.
The single-point optimizations are all performed for a wind speed of
8 m s-1 and rotational rate of 6.69 rpm at zero blade pitch, which is
one of the conditions listed in Table in
Appendix . For the multipoint optimizations, we use the
wind speeds 5, 8, and 11 m s-1, and the relevant operational
conditions can again be found in Table in
Appendix . Furthermore, we use the initial values at
12 m s-1 in the thrust and flapwise bending moment constraint for the
multipoint optimizations because we know from the solver comparison
(Appendix , Fig. ) that the maximum
thrust occurs at that speed.
In addition to the CFD-based optimizations, we solve two BEM-based
optimization problems for comparison with the CFD-based planform
optimization:BEM1:
identical to the single-point planform optimization.
BEM2:
identical to the multipoint full shape optimization,
except the shape variables are replaced by spanwise thickness distribution variables.
The thickness is handled by interpolating between the predefined airfoil
data. While both BEM1 and BEM2 use specified airfoil polar data, BEM2 can
change the relative thickness of the airfoils. The airfoils vary from
72 % to 24 % in relative thickness.
Overview of baseline geometry and FFD boxes (a). Each FFD box has
nine spanwise sections. Each blade (b) has 15 thickness constraints
(blue) and seven LE/TE constraints (red). Thickness
distributions (c) are for the baseline thickness (green) and minimum
allowed thickness (purple). Profile section (d) at 36 m span shows
the shape control points (20), the thickness constraints (10 blue segments)
and LE/TE constraints (two red segments). The LE/TE constraints are only
relevant for the full shape optimizations.
Parameterization
The baseline design is shown in Fig. a, along with the three FFD boxes used
to parameterize the geometry. The FFD boxes have 10×2×9
control points (shown in black), where 10 is the number of control points
from LE to TE, 9 is the number of spanwise sections, and 2 corresponds to the
top and bottom of the FFD box. Our approach to deciding on the number of
control points is to use the largest number possible to provide maximum
freedom in the optimization. However, as the density of control points
approaches that of the CFD mesh, numerical issues occur because the physical
model no longer resolves the effect of the geometry change. We have found
that, as a rule of thumb, we should have no more than one control point for
every four CFD mesh points.
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 C1
continuity there, while the seven outer sections are free to move and deform
the blades. Pitch, xpitch, is achieved by rotating all free
FFD sections by the same amount along the reference axis, which is at
35 % of the chord from the LE. Twist, xtwist, is
achieved by rotating each spanwise section of FFD control points
independently. The chord variables, xchord, are achieved by
scaling each spanwise section in the chord and thickness direction. Thus, the
relative thickness at each section is preserved during the CFD planform
optimization. Only for the full shape optimizations, where the shape
variables are added, can the relative thickness change. The shape variables,
xshape, move each control point independently in the
direction perpendicular to the chord to control the airfoil shape.
In Fig. , the thickness constraints are
highlighted in blue. The thickness constraints in the BEM comparison are only
enforced on the inner 80 % of the blade, as detailed in the definition
of the IEA case study. This is also visualized in
Fig. c.
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. , which prevents negative cell
volumes. Furthermore, there are constraints applied to the LE and TE of the
FFD box. The LE/TE constraints (shown in red in
Fig. ) are only implemented for the
single-point and multipoint full shape optimizations. These constraints force each
pair of points to move exactly the same amount in opposite directions, so
that the midpoint in the segment remains stationary. This ensures that the
individual FFD control points do not apply skewing twist, since they are
meant to control only airfoil profiles. Finally, we mention that the
thickness limit is fully imposed only for the fourth thickness constraint
(counting from the LE), while the remaining nine constraints in a section are
relaxed to not unnecessarily restrict the possible design space.
Results
The results are split into the four main problems listed in
Table . First, we perform a single design variable
optimization where pitch is varied to maximize the torque
(Sect. ). This simple optimization is
included to validate the adjoint formulation for rotating frame of reference
flows. Second, we perform a planform optimization where chord and twist are
varied (Sect. ). This optimization
is well suited for comparison with BEM results because the airfoil shapes do
not change. The two final optimizations are full shape optimization problems
where all variables, including airfoil shape variables, are allowed to
change. First, we solve the problem as a single-point optimization
(Sect. ). Then, we solve it as a multipoint
optimization (Sect. ).
Overview of optimization problems.
Design variables Optimization problemObjectivePitchTwistChordShapeTotalBEM comparisonSingle-point pitchTorque11Single-point planformTorque7714✓Single-point full shapeTorque77140154Multipoint full shapeAEP77140154Pitch optimization
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 25.7 %, 26.1 %, and
23.0 % for mesh levels L2, L1, and L0, respectively.
Figures – summarize
the optimization history for the three mesh levels.
Variation of torque with the pitch design variable.
Figure shows the torque as a function of pitch.
Before optimizing, we performed a sweep of CFD evaluations of the torque for
the whole range of pitch values, for all three mesh levels. These are
represented by black dots in Fig. . The thin
black lines are linearly interpolated from these points. Although the torque
value varies between mesh levels, the trends are consistent, and the maximum
torque is achieved around 7∘ of pitch. The optimization histories for
each mesh are shown in color; they start from the initial pitch, x0,
and end at the optimal one, x*. The purple line shows the
optimization history on the finest (L0) mesh that was obtained by using the
result from a coarser mesh (L2) as a starting point. We use this “warm
start” technique since coarser meshes are much faster to converge. This
technique leads to a reduction of computation time since fewer steps are
taken by the optimizer on the finest mesh level. This is seen in
Table , where only four steps were needed instead of the 16
steps taken in the original optimization. In this case, it reduced the
computation time at approximately 50 %. As expected, the result of this
warm start optimization is identical to the result of optimizing solely on
the finest mesh level. Now that we have introduced (and visualized in
Fig. ) the use of warm starts, we will start
using them regularly. This means that an L1 optimization from now on uses the
result of an L2 optimization, and an L0 optimization uses the result of an L1
optimization.
As shown in Fig. , all optimizations converged to
an optimality of at least 10-4 (black dashed line).
Figure shows the merit function, which combines
the scaled objective function value and constraint feasibility. The merit
function value is equivalent to the scaled objective function value when all
constraints are satisfied towards the end of the optimization process. As we
can see in Fig. , the curves flatten towards the
end, and further iterations are not worthwhile because the optimizer reaches
the limit of what it can achieve with the provided precision of the function
evaluations. The pitch optimizations are summarized in Table .
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.
* Warm start with the L2 optimum, resulting in a total CPU
time of 106.9 h + 6436.3 h = 6543.2 h.
Planform optimization
For the planform optimization, described in Sect. ,
both twist and chord are controlled at the seven outer FFD sections along the
blade, which results in 14 design variables. The high-fidelity planform
optimization results are visualized in
Figs. –, which show the final
chord and twist distributions as well as the history of the convergence and
merit functions.
As we can see in Fig. , the optimized shape for
the finest mesh level has a large increase in chord towards the root and a
decrease in chord towards the tip, just as we would expect for an
aerodynamically optimized blade. The optimized chord distribution is
reminiscent of the DTU 10 MW turbine's chord distribution from
Fig. , which was also designed for maximum
power. However, the DTU 10 MW root chord is not as high due to a constraint
on maximum chord of 6.2 m. Turning to the optimized twist (green curve) in
the lower plot in Fig. , we see it exhibits a
large variation towards the tip compared to its baseline. The result is a
more aggressive twist distribution.
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 ). We cannot rule out that, in some cases,
the L2 result can be useful to perform a warm start sequence, as shown for
the pitch optimization (Fig. and
Table ). However, the planform results certainly show that one
should use the L2 mesh with care and not for final results.
Figure shows the convergence history for the three mesh
levels. Again, all optimizations were converged to at least 10-4. In
Fig. , we see a similar trend to that of the pitch
optimization (Fig. ), where much of the
improvement is gained in the first half of the optimization. Thus, an easy
way to speed up the design process would be to take an intermediate design.
However, one should make sure to check the constraint feasibility, since SQP
methods often explore infeasible regions before fully converging. The sharp
initial decrease for L1 is due to the (infeasible) warm start from L2. Note
that the function is scaled differently for each mesh level to accommodate
all the results in one 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 , which uses HAWCStab2
as the underlying analysis code. Since this
is a comparison between results obtained with completely different models, we
do not expect an exact match, but we expect similar trends. As previously
mentioned, the CFD planform optimization problem and the BEM1 optimization
are completely identical in problem definition, and the relative thickness is
fixed in both optimizations. For the BEM2 optimization, the main difference
is that it is solved as a multipoint optimization and that the relative
thicknesses can be changed through interpolation. We refer to
Sect. for further information.
The BEM optimizations are performed with SNOPT. The baseline and optimized
chord and twist distributions are shown in Fig. . Although
both chord and twist distributions show clear discrepancies for the final
designs, there are several similar traits. When it comes to chord, there is a
large difference in maximum chord. BEM1 converges to a 26 % increase,
BEM2 converges to a 74 % increase, and the CFD optimization converges
to somewhere between these two (43 %). BEM1 is the surprising result of
the three, because it seems that the relation between power and thrust is so
poor that it makes little sense to increase the chord at the root. This is
owed to the fact that BEM1 has fixed relative thickness for all sections. It
makes sense that BEM2 can increase the chord further since it can change the
relative thickness. Given that our CFD-based planform optimization also has
fixed relative thickness, it also makes sense that the BEM2 chord values are
larger than those from the CFD-based planform optimization.
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 C1 mesh continuity make such a steep increase in chord
impossible so close to the root. As a final comment on the discrepancies at
the root, we suspect that BEM profile data for such thick airfoils are far
from precise. Besides, the empirical 3-D correction used on said 2-D profile
data is also likely to be imprecise. Needless to say, the combination of the
two could yield shaky results. To make matters worse, we know from the
comparative analysis (Fig. ) that separation reaches up
to about 37 m span, which further complicates the situation. A more
uniform picture is seen for the tip region where the chord distributions have
converged to a reduced chord, where only minor differences can be seen. In
conclusion, the overall trends in optimal chord distribution are mirrored
across the BEM and CFD models, and the discrepancies are less pronounced
towards the tip.
Comparison between optimal chord (a) and twist (b)
distributions for the IEA Wind Task 37 case study. The three design
optimization problems – (i) single-point planform optimization, (ii) BEM1, and
(iii) BEM2 – are further described in Sect. .
As for the twist comparison (Fig. b), both CFD and
BEM results exhibit the overall trend of decreasing the twist relative to the
baseline, but the BEM twist is consistently 1–2∘ lower than the CFD
result. This difference is likely due to the different modeling. The CFD
parameterization is limited near the root due to the two fixed sections that
enforce C1 continuity, so it cannot match the more abrupt change in twist
for the BEM result in that region. The BEM2 result exhibits an increase in
twist near the root, which is very different from the BEM1 trend. This is
because BEM2 is free to control the chord while lowering the relative
thickness. Thus, BEM2 uses a large chord increase near the root to optimize
the loading, instead of using twist. The planform optimization and BEM
comparison are summarized in Table .
Planform optimization comparison between CFD and BEM results.
1 Relative improvement in torque. 2 Relative improvement in
AEP.
Using values for torque from Table , we can obtain the
power coefficient, CP, defined as
CP=P(1/2)ρV3A,
where P is power, ρ is the air density, V is wind speed, and A is
the area swept by the rotor. The resulting coefficients are CP=[1.04,0.62,0.48] for mesh levels L2, L1, and L0, respectively. Clearly, the
coarser the mesh, the more unphysical the coefficient. The Betz limit for
power coefficients (CPBetz=0.59) is violated for L2 and
L1, which draws the results from coarse mesh levels into doubt. Judging from
the huge spread in these coefficients, it is not surprising that the
optimized designs differ greatly across mesh levels.
Single-point shape optimization
We now solve the full shape optimization problem as a single-point
optimization. As stated in Sect. , this problem is
equivalent to optimizing for torque, when only a single wind speed is used.
Figure shows convergence (left) and scaled merit
function (right) histories for the free-form shape optimizations. Since we
typically request an optimization convergence tolerance that is smaller than
what is possible for the level of the CFD solver convergence, the optimizer
stops before the optimization convergence tolerance is met. Comparing the
convergence history to similar plots for the pitch and planform optimizations
(Figs. and ), we see that as
the mesh is refined, the optimization is better converged, and the finest
mesh level almost meets the requested tolerance (black dashed line). However,
the scaled merit function plots (Fig. b) do seem
flat for L2 and L1 (albeit the latter curve is less smooth), hinting that the
merit function could have plateaued.
Convergence history (a) and scaled merit function
history (b) for the single-point shape optimizations.
Table shows the improvement achieved by the optimization.
The achieved improvement on the finest mesh (15.89 %) is higher than
that of the planform optimization (11.07 %, Table ),
which is expected because this case includes all the planform design
optimization variables plus the additional freedom to optimize the airfoil
shapes. One should not compare these results to the pitch optimization
results since they do not include any thrust constraint. A comparison to the
BEM code results is given farther down in Table once the
multipoint optimization results have been presented.
Overview of single-point optimization results for the operational
conditions of the 8 m s-1 case listed in
Table .
We now turn to the shape and pressure (Cp) distributions for the
baseline and optimized geometries in Fig. . The
optimized blade increases the chord near the root. This design trend agrees
with the planform optimization result.
Comparison of Cp distributions for the baseline and
optimized result from the single-point shape optimization. There is an
increase in TE camber, especially at the root, as well as a less pronounced
suction peak.
Comparing the airfoil shapes and corresponding Cp distributions at
the bottom of Fig. , we can see that the optimization
reduced the thickness and slightly increased the camber. The thickness
reduction is expected when considering only the aerodynamics with no
structural strength constraints. Since we use thickness constraints as a
surrogate for structural feasibility, the optimizer exploits this by
producing the thinnest airfoils that satisfy these constraints. The increased
camber, owed to the physical incentive to generate more lift, is consistent
with the results of , but the increase in camber here is
more modest because the optimizer can increase the torque by tailoring
camber, chord, and twist instead of just camber. The incentive to operate at
high lift coefficient is due to the fact that high Cl/Cd is
most easily achieved by operating at high Cl, especially for
airfoils designed assuming a fully turbulent boundary layer.
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.
Multipoint shape optimization
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. ), except for the objective function. The
objective function here is the AEP estimate, which we describe 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 λ=ωR/V. As long as the tip speed ratio is the same, the blade angle of
attack is the same, and a given design has similar aerodynamic performance.
However, for low wind speeds, the rotational speed has a lower bound to avoid
tower excitation, and at higher wind speeds, the rotational speed is kept
constant, and the turbine starts regulating pitch to maintain rated
mechanical power. In our case, the target tip speed ratio is λ=7.8,
and the rotor speeds corresponding to the minimum and maximum limits are 6.0
and 9.6 rpm, respectively. The variation of rotor speed with wind speed is
shown in Fig. . There are two reasons we consider wind
speeds outside the constant tip speed ratio range. First, the angles of
attack are different at these operating points, which should lead to a more
balanced design. Additionally, we need to consider the load constraints
defined in the optimization case study. For this reason, we choose
5 m s-1 as the lower wind speed, and 11 m s-1, because this
is just below the wind speed at which the rotor reaches rated rotational
speed and rated power and thus peak thrust and flapwise moment.
Research has shown that, in reality, the angle of attack varies significantly
(more than 4∘) over just one rotor revolution
Fig. 5. The explanation for this can be found in the
complex operating conditions for turbines containing, for example, turbulent
inflow and inflow wind shear. To simulate these effects, it would be ideal to
add turbulent inflow and transition from steady-state RANS to unsteady RANS.
A cheaper way could be a multipoint optimization with a fixed rpm for all
turbines operating at slightly different wind speeds. We leave this for
future work.
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. . Just as for the single-point optimization, the
selected threshold is not quite met. However, as before, the scaled merit
function flattens enough that we determined that the design is close enough
to the optimum.
History of convergence (a) and scaled merit
functions (b) for the multipoint shape optimizations.
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. . As we can see, the LE shapes are
somewhat improved but still unrealistically sharp. This points towards the
necessity of including off-design operational cases resulting in wider ranges
of angles of attack, where such a sharp LE would result in deterioration in
performance.
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. , where we compare them to the shape obtained by
the multipoint optimization without the LE constraints. While we choose to
focus solely on the 2-D profile improvement from single-point to multipoint optimizations,
the optimizations are indeed all 3-D rotor optimizations. As we can see,
enforcing the geometric constraint results in a more round LE shape that is
much more similar to previously published wind turbine airfoil shapes.
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 . Whereas the
single-point BEM1 result (8.06 %) is close to the single-point planform
optimization result (11.07 %), the multipoint BEM2 result
(22.46 %) is comparable to the multipoint full shape optimizations
result (23.76 %) since relative thicknesses can change in both cases.
The multipoint result (23.76 %) is somewhat higher than the
single-point full shape optimization result (15.89 %), which can be explained
by the relaxed thrust constraint for multipoint optimizations. Here, we use
the thrust from the 12 m s-1 case instead of the 8 m s-1 case
to define the initial constraint values for thrust and bending moment.
Indeed, the thrust constraint relaxation results in the constraint not being
active at convergence for the CFD-based multipoint full shape optimization,
as seen in Table .
These results do not show that the industry can necessarily gain a
20 % increase simply by using high-fidelity optimization. Indeed, the
amount of improvement depends on the performance of the baseline turbine.
Since we study an intentionally poor baseline design, we therefore get a
large improvement.
Overview of optimization results. As further detailed in
Sect. , the single-point and multipoint
optimizations use the operational conditions for the 8 m s-1 case and
5, 8, and 11 m s-1 cases, respectively. Operational conditions are
listed in Table .
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 14 % and 11 % increase in thrust and moment,
respectively. In the multipoint full shape optimization, the moment
constraint is again active at an 11 % increase in bending moment.
However, the thrust constraint is only at 11 % and is, as mentioned,
not active at convergence due to the relaxed constraint. With these
constraints, we could add span as a design variable in future work.
Comparison of normal (a) and driving (b) forces
for baseline and optimized designs. The shape optimization increases the
normal force, and the peak has also moved further inboard. The driving force
is increased considerably both at the root and close to the tip region.
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.
Conclusions
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 O(102) CFD evaluations.
Data availability
Data are available upon request to the corresponding author.
Extended flow solver comparison
The following is a continuation of Sect. to extend the
comparison between the flow solvers: EllipSys3D and ADflow.
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 1.225 kg m-3, temperature to 15 ∘C,
and dynamic viscosity to 1.784×10-5 kg m-1 s-1.
* Based on a target tip speed ratio of γ=7.8, where
6.0≤rpm≤9.6.
Operational conditions
The case study is defined with a cut-in speed of 4 m s-1 and a
cut-out speed of 25 m s-1. Within this range, we use the eight
operational conditions defined in Table to
compare the solvers.
Integrated loads
Integrated loads, in the form of thrust and torque, have been computed for
each simulation in Table and are visualized
in Fig. . As seen, the ADflow results are
consistently higher than the EllipSys3D results. This trend could partially
be accounted for by applying the mentioned Prandtl–Glauert correction to the
incompressible computations but is also a result of ADflow results on mesh L0
not being fully mesh independent, as shown in
Table . As a low-fidelity reference, we have added
the integrated loads (in gray) from steady-state BEM results using HAWCStab2.
A general agreement between the CFD results and the HAWCStab2 results can be
seen, save for the torque value at 25 m s-1, which could be
corrected with a slight change in pitch setting given in
Table . Agreement is expected between
EllipSys3D and BEM since the airfoil data used in BEM are computed using
EllipSys2D.
Spanwise forces, pressure distribution and flow visualization
Figure shows the spanwise forces and shows
that the difference between solvers is more or less spread out over the
entire span. Not surprisingly, the ADflow values are consistently higher. We
will revisit the distribution of spanwise forces after the optimization to
inspect where performance increase occurs on the blade.
Turning to the surface-restricted streamlines in Fig. ,
we first note the rather large amount of separation. Even the pressure side
shows a distinct area of separation from 19 to 41 m span. Comparing said
area with the pressure side separation for the unperturbed DTU 10 MW rotor
in Fig. , where only a small separation area at the
root is seen, it is clear that the perturbed design we use as a starting
point for the optimization seen in Fig. suffers a more
poor aerodynamic design owed to the reduced chord distribution and increase
in relative thickness. The suction side in Fig. looks
more like one would expect, save for the expanded separation area reaching
just above 37 m in the spanwise direction. Here, the DTU 10 MW only has
separation below the 32 m span, as seen in Fig. .
In Fig. , we compare the obtained Cp curves at
three spanwise positions: 35, 64, and 84 m (positions marked in red in
Fig. ), where the Cp distribution is found
using the dynamic pressure, and the far-field pressure, p∞:
Cp=p-p∞(1/2)ρ(V∞2+(rω)2).
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 Cp curve with a typical flat,
squeezed shape in the 30 % closest to the trailing edge (TE). The
Cp curves for the sections at 64 m span and 84 m span show, in
general, a better likeness to one another. Early investigations showed that
the chordwise distribution of cells has a distinct impact on the solvers'
ability to capture the stagnation point and suction peak. Therefore, we chose
a distribution that seemed to have enough cells close to the stagnation point
while still having an adequate amount of cells to resolve the TE area. In
general, the ADflow suction peaks seem to be more pronounced than those from
EllipSys3D. The same can be said for the blunt TE, where the ADflow
Cp curve again has a more pronounced spike.
Total thrust (a) and torque (b) as a function of
wind speed for the rotor geometry used as the starting point for the
optimization computed using mesh L0. As expected, the torque increases
rapidly from cut-in speed to the rated speed at 12 m s-1, which is
also where the thrust peak occurs. From rated to cut out, the torque curve
flattens. Here, the pitch setting found with steady-state BEM results using
HAWCStab2 (seen in gray) clearly does not result in the CFD solvers tracking
rated power accurately due to the model changes. ADflow consistently
overshoots the EllipSys results, which is consistent with the trend seen in
Table . Operational conditions for the eight
simulations are given in Table .
Spanwise distribution of the normal force (a) and driving
force (b) for the 8 m s-1 case listed in
Table .
Surface-restricted streamlines from the EllipSys solution for a wind
speed of 8 m s-1, both for the pressure side (a) and the
suction side (b) for the perturbed design we use as a starting point
for the optimization. The operational conditions are listed in
Table .
Surface-restricted streamlines from the EllipSys solution obtained
using the original DTU 10 MW wind turbine geometry for the 8 m s-1
case in Table both for the pressure
side (a) and the suction side (b).
Cp curves for 35 m (a), 64 m (b),
and 84 m (c).
Author contributions
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.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
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.
Review statement
This paper was edited by Alessandro Bianchini and reviewed
by Joseph Saverin and one anonymous referee.
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