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Integrated Multidisciplinary Constrained Optimization of Offshore Support Structures
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2014 J. Phys.: Conf. Ser. 555 012046
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The Science of Making Torque from Wind 2012
Journal of Physics: Conference Series 555 (2014) 012046
IOP Publishing
doi:10.1088/1742-6596/555/1/012046
Integrated Multidisciplinary Constrained
Optimization of Offshore Support Structures
Rad Haghi1 , Turaj Ashuri2 , Paul L.C. van der Valk3 and David P.
Molenaar1
1
2
3
Offshore Center of Competence, Siemens, the Netherlands
Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA
Faculty of Mechanical Engineering, Delft University of Technology, the Netherlands
E-mail: [email protected]
Abstract. In the current offshore wind turbine support structure design method, the tower
and foundation, which form the support structure are designed separately by the turbine
and foundation designer. This method yields a suboptimal design and it results in a
heavy, overdesigned and expensive support structure. This paper presents an integrated
multidisciplinary approach to design the tower and foundation simultaneously. Aerodynamics,
hydrodynamics, structure and soil mechanics are the modeled disciplines to capture the full
dynamic behavior of the foundation and tower under different environmental conditions. The
objective function to be minimized is the mass of the support structure. The model includes
various design constraints: local and global buckling, modal frequencies, and fatigue damage
along different stations of the structure. To show the usefulness of the method, an existing
SWT-3.6-107 offshore wind turbine where its tower and foundation are designed separately is
used as a case study. The result of the integrated multidisciplinary design optimization shows
12.1% reduction in the mass of the support structure, while satisfying all the design constraints.
1. Introduction
Offshore wind energy is a growing industry, with thousands of megawatts yearly installation
worldwide to enable the transition from a society dependent on fossil fuels to renewable
energy. Offshore wind turbines benefit from higher and steadier winds at sea, but the required
marinization makes them more expensive than onshore wind turbines.
Among several different marinization cost elements, the support structure (foundation and
tower) has the highest cost share [1]. Currently, the most widely used support structure is a
tubular tower that is connected through a transition piece to a monopile, and it is a suitable
concept for water depths of up to 40 m [2]. The common industrial practice of designing such a
concept is to optimize the foundation (monopile and transition piece) and tower independently
by the foundation and wind turbine designer. Few sequential iterations between the wind
turbine and the foundation designer are used to achieve a sound design and meet the integrated
frequency-band requirement of the support structure [3].
Obviously, such an isolated approach results in a suboptimal design for both the tower and
foundation, and it does not offer the required cost reduction needed to make offshore wind energy
competitive with traditional energy resources. An integrated approach where both the tower
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
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Published under licence by IOP Publishing Ltd
1
The Science of Making Torque from Wind 2012
Journal of Physics: Conference Series 555 (2014) 012046
IOP Publishing
doi:10.1088/1742-6596/555/1/012046
and foundation are designed at once can yield a better design and thereby impact the cost of
the support structure more positively.
Previous studies show the advantages of the integrated design with either limited design
constraints or disciplines [4, 5, 6, 7, 8, 9, 10, 11]. The work presented herein addresses some
of the shortcomings of the previous works by developing an integrated design method with
all the relevant disciplines (structure, aerodynamics, hydrodynamics and soil mechanics) and
design constraints (buckling, fatigue damage and natural frequency) involved. Two different
optimization algorithms are used to enhance the speed and convergence rate.
This integrated methodology is used to redesign an existing SWT-3.6-107 wind turbine
support structure of a real offshore wind park where both its tower and foundation are designed
separately. Optimization results are compared with this initial SWT-3.6-107 support structure
to investigate the benefits of the integrated design and its influence on the design constraints
and objective function. Because of data confidentially, results are presented in an abstract and
normalized form, but still clear enough to judge the effectiveness of the method.
The reminder of the paper is organized as follow. First, the integrated design methodology
of the support structure is described. Then, the optimization problem formulation is presented.
Next, results of the integrated design methodology are compared with the initial design. Finally,
recommendation and conclusion are presented.
2. Integrated design methodology
This section presents different disciplines that play an important role in modeling the support
structure. These different disciplines are coupled to enable integrated multidisciplinary
simulation of the support structure and provide the objective function and design constraints
needed for the optimizer. A MATLAB script is used to automate data and process flow, and
prevent the manual intervention of the designer during the optimization iterations [12]. Figure 1
shows the integration and connectivity of different computational models.
3D wind
kinematics
Aerodynamic
loads
Design
variables
Interior
point
Soil-structure
interaction
Structure
Objective
function
SQP
Wave
kinematics
Hydrodynamic
loads
Design
constraints
FD
sensitivity
Figure 1: Optimization problem flowchart showing the building blocks of the integrated
framework and their interaction
2.1. Structural model
The support structure is modeled by Timoshenko beam elements [13]. Using this formulation,
the structure is discretized along the height into different sections. Each section has a constant
thickness, and it has either a tapered or cylindrical shape. In this way, each section is represented
as an element with two nodes as depicted in Figure 2. Structural properties of the support
structure are obtained using an analytical model presented by [14].
2
The Science of Making Torque from Wind 2012
Journal of Physics: Conference Series 555 (2014) 012046
IOP Publishing
doi:10.1088/1742-6596/555/1/012046
Figure 2: A schematic drawing of the support structure (left) and representative finite element
model with lateral springs under the mud line where they represents the soil stiffness (right)
2.2. Soil model
The soil is modeled as a series of linear lateral springs [15]. These springs represent the stiffness
of the soil at different sections as depicted in Figure 2. These sectional soil stiffness matrices
are added to the structural stiffness matrix of the monopile below the mudline to form a unified
stiffness matrix of both the soil and structure [16]. A linear interpolation scheme is performed
to obtain the soil stiffness on the exact location of the finite element nodes of the structure.
2.3. Aerodynamic model
A blade element momentum model is developed to account for aerodynamic loads that the rotor
experiences during operation. The model includes hub and tip loss add-on, as well as dynamic
stall and dynamic inflow correction [17]. These aerodynamic loads (thrust and bending moment)
are applied as nodal forces and moments at the tower top nodes. Using this model, all of the
IEC61400-3 [18] prescribed load cases such as power production (DLC1.2) to evaluate fatigue
loads, and normal shut down combined by an extreme operating gust (DLC4.2) to evaluate
ultimate loads are considered in the design. For this purpose, the relevant 3D turbulent wind
files are generated using TurbSim that is a free and open source code developed by NREL [19].
2.4. Hydromechanic model
Several different methods exist to compute the velocity and acceleration profile of waves [20].
To obtain these wave kinematics needed to calculate the loads on the monopile, irregular waves
are approximated in this work with classical linear (Airy) wave theory [21]. This method
only provides wave kinematics up to the mean water level, and Wheeler stretching is used
to redistribute the velocity and acceleration profiles up to the actual sea surface [22].
To simulate the dynamic response due to the hydrodynamic forces acting on the monopile,
time-series of the hydrodynamic loads for each section are realized using the empirical Morison
3
The Science of Making Torque from Wind 2012
Journal of Physics: Conference Series 555 (2014) 012046
IOP Publishing
doi:10.1088/1742-6596/555/1/012046
equation [23]. In the Morison equation, the drag and inertia forces due to horizontal fluid
particle velocities and accelerations, U and U̇ respectively, are added together to estimate the
total in-line wave force:
πD2
1
U̇
(1)
fM orison = fD + fI = ρCd D |U | U + ρCm
2
4
here ρ and D represent the water density and structural diameter, respectively. The
magnitude of the force components depends on a proper selection of appropriate values for
the added inertia and drag force coefficients, Cm and Cd , and because of data confidentiality are
not presented here.
3. Optimization problem formulation
The aim of this study is to minimize the support structure mass using the integrated design
methodology and compare that with the reference SWT-3.6-107 design where the foundation
and tower are optimized separately by the foundation and wind turbine designer. This mass
minimization procedure is subjected to fatigue damage, local and global buckling, and natural
frequency as the design constraints. The design variables are the wall thicknesses of the sections,
t. The optimization problem can be expressed mathematically as:
min m(t)
and
subject to g(t) ≤ 0
(2)

g1 (t) ≤ 0



 g2 (t) ≤ 0
g(t) ≤ 0 ⇒
..

.


 g (t) ≤ 0
n
where in (2), m(t) is the mass of the support structure and g(t) is the vector of constraints.
3.1. Design variable
The design variables are the wall thickness of the section. As mentioned earlier, the support
structure consists of a number of sections where each section has a constant wall thickness along
its length. From Figure 2, it is observable that there is an overlap between transition piece
and monopile. For that overlap region the thicknesses from monpile and transition piece are
summed up and one element with the larger thickness is considered. All these design variables
are continuous and differentiable. The vector of the design variables is formed as:
t=
t1 t2 t3 . . . tn
T
(3)
The design variable range is limited with upper and lower bounds, i.e. Ll and Ul . These values
are ±300% of the actual wall thickness of the SWT-3.6-107 wind turbine. Other geometrical
parameters of the support structure such as diameter and penetration depth are kept the same
as the initial design to enable a fair comparison.
Ll ≤ t ≤ U l
4
(4)
The Science of Making Torque from Wind 2012
Journal of Physics: Conference Series 555 (2014) 012046
IOP Publishing
doi:10.1088/1742-6596/555/1/012046
3.2. Objective function
Each section’s mass is calculated first, and they are summed up next to calculate the total
support structure mass, which is equal to the objective function to be minimized. The total
mass can be formulated as:
m(t) = πρ
n
2
X
Di2 − (Di − 2ti )2 + Di+1
− (Di+1 − 2ti )2
8
i=1
li
(5)
where
m(t)
n
ρ
Di
Di+1
ti
li
: Mass of support structure [kg]
: Number of sections [-]
: Used material density [kg/m3 ]
: Top diameter of section [m]
: Bottom diameter of section [m]
: Wall thickness of section[m]
: Length of section[m]
Each section’s mass is a function of the wall thickness, therefore by changing the wall thicknesses
(the design variables) the total mass varies too.
3.3. Design constraints
To end up with a feasible and realistic design, several design constraints are introduced to bound
the design space as explained next.
3.3.1. First integrated natural frequency
The first natural frequency is derived from solving an eigenvalue problem based on the finite
element model of the support structure as:
(Kglobal − λM)x = 0
(6)
ωn2
λ=
Kglobal = K + Kg
where Kglobal is the global stiffness matrix, K is the structural stiffness matrix, Kg is the soil
stiffness matrix , M is the mass matrix, λ is an eigenvalue, x is an eigenvector and ωn is a natural
frequency of the structure. Solving this eigenvalue problem yields ω and x which represent the
eigenfrequencies and mode shapes of the structure respectively [24].
Similar to the SWT-3.6-107 wind turbine, the first natural frequency is bounded with a lower
(fl ) and upper bound (fu ) as:
fl ≤ fnat ≤ fu
(7)
therefore, (7) can be formulated as inequality constraints as following.
g1 (t) = fl − fnat ≤ 0
g2 (t) = fnat − fu ≤ 0
5
(8)
(9)
The Science of Making Torque from Wind 2012
Journal of Physics: Conference Series 555 (2014) 012046
IOP Publishing
doi:10.1088/1742-6596/555/1/012046
3.3.2. Buckling constraint
Two kind of bucklings are considered in this study, the local and global buckling. The local
buckling is calculated based on the design code EN1993-1-6-2007 [25], and global buckling is
calculated using the developed finite element model of the support structure.
Global buckling
The eigenvalue problem to calculate global buckling can be formulated as following:
(K − λKG )x = 0
(10)
where in (10) λ is the multiplication of the reference load to make the buckling happen and
x is the buckling mode. The solution of (10), λ, is a multiplier for the reference load P . It
means that global buckling happens when the applied load is equal to λP so:
1−λ<0
therefore,
g3 (t) = 1 − λ ≤ 0
(11)
Local buckling
The local buckling happens when the applied stress on a section is more than the stress
capacity of the section. This means a buckling unity check for every section that should be
smaller than 1 to have structural stability. This unity check along the support structure
can be formulated as:
bloci (Applied stress, t) =
Applied stress
Stress capacity
therefore
bloc =
bloc1
bloc2
bloc3
. . . blocn
T
so
g4 (t) = bloc − 1 ≤ 0
(12)
where g4 (t) is the local buckling constraint vector for the optimization problem.
3.3.3. Fatigue constraint
One of the methods to calculate the fatigue damage of structural members is the S-N
curve [26]. Fatigue properties are determined by the slope of the S-N curves, and the location of
the intercept with the abscissa. A slope of 4 and an intercept of 110 MPa is used to characterize
the support structure steel in this work.
To use these curves, the applied stress on the structure needs to be calculated. Knowing this
value and using the S-N cure, one can obtain the number of cycles that the structure can sustain
under this stress. Since, only one value for stress is accepted, an equivalent stress level based on
von Mises theory is calculated [26].
Fatigue damage calculation is done using rain flow cycle counting of the stress time-signals
and applying Palmgren-Miner’s rule [27].
d=
X ni
Ni
(13)
where d is the total fatigue damage, ni is the actual number of cycle, and Ni the total number
of cycle for a given stress level.
6
The Science of Making Torque from Wind 2012
Journal of Physics: Conference Series 555 (2014) 012046
IOP Publishing
doi:10.1088/1742-6596/555/1/012046
For this work, the stress concentration factors are extracted from DNV-OS-J-101 design
standard [28] in the locations that the support structure sections are welded to each other. The
fatigue damage is a vector with the same size as the number of sections. This vector represents
the damage along the support structure as:
d=
d1 d2 . . . dn
T
In order to keep the structure stable, each member of the vector should be smaller than one.
Reformulating this vector as an inequality constraint yields:
g5 (t) = d − 1 ≤ 0
(14)
3.4. Optimization algorithms
To have a faster and more robust convergence in the optimization procedure, two different
optimization algorithms are successively used. These are the interior point and sequential
quadratic programming (SQP) [29]. First, the interior-point algorithm is used to make an
unfeasible design a feasible design. Then, the SQP algorithm is used to find the optimum
design. The results of the first algorithm serve as the initial value for the second algorithm.
Central finite-difference (FD) method is used to calculate the objective function and design
constraints gradients vector.
4. Results
This section presents the results of the integrated methodology and compares it with the SWT3.6-107 design. First the optimized design variables are presented, followed by the design
constraints. Finally, the optimized objective function is given and compared with the initial
design.
4.1. Design variables
Figure 3a shows the original and optimized wall thickness of the support structure. As the figure
shows, the highest mass reduction is achieved in the transition piece section. The thickness of
the tower increases from the tower bottom to the top except the very near end of the tower top
where some thickness reduction is achieved. Also, the embedded length of the monopile shows
on average an increase in the thickness, but at the mudline a considerable thickness reduction
is obtained.
Figure 3b shows the variation of the support structure mass for every single optimization
iteration . At the beginning of the optimization process, the scattered behavior is the result of
using interior point algorithm to find a feasible design. After finding a feasible design, the SQP
algorithm is used, which uses the design variables obtained by interior point algorithm. SQP
has a good convergence behavior if provided with a good initial feasible design.
4.2. Design constraints
Figure 4 shows the changes in the design constraints. It is noticeable that the buckling
constraints are not active design constraints. In this case, the design is driven by the
fatigue damage and the first natural frequency as the two active constraints. Because of data
confidentiality issues, the initial design constraints are not presented, but it should me noted
that also for the initial design the fatigue damage and first natural frequency are the highest
active constraints that drive the design.
7
The Science of Making Torque from Wind 2012
Journal of Physics: Conference Series 555 (2014) 012046
IOP Publishing
doi:10.1088/1742-6596/555/1/012046
interior point
0.82
0.73
active−set
Tower
0.91
Normalized Mass
Distance from tower top
Mass changes during the optimization process
1
Initial input and the Original thickness
The optimized results with all constraints enabled
sqp
Initial and optimized support structure wall thickness with
all constraints
0
0.64
0.54
TP
Mud line
0.45
MP
0.36
0
0
10
20
30
Thickness
(a) Thickness comparison of both designs
40
50
Iterations [−]
60
70
80
90
(b) Optimized mass per optimization iteration
Figure 3: Figure 3a shows the achieved thickness reduction for the optimized design compared
to the initial design, and Figure 3b shows the convergence behavior of the objective function
using different optimization algorithms per optimization iteration.
Max local buckling based on design standards changes during the
optimization process
Global buckling based on FEM changes during the optimization process
1.4
Local buckling[−]
sqp
active−set
1.8
1
1.6
1.4
active−set sqp
1.2
interior point
Global buckling[−]
2
interior point
2.2
Local buckling limit
0.8
0.6
0.4
1.2
0
0.2
Global buckling limit
1
10
20
30
40
50
60
70
80
0
0
90
10
20
30
(a) Global buckling changes
60
70
80
90
Max fatigue damage changes during the optimization process
First natural frequency changes during the optimization process
10
interior point
FNF upper bound
7
FNF lower bound
sqp
0.28
8
Damage [−]
0.3
0.26
active−set sqp
interior point
9
0.32
6
5
4
3
0.24
active−set
First natural frequency [Hz]
50
(b) Local buckling changes
0.34
0.22
0.2
0
40
Iterations [−]
Iterations [−]
10
20
30
40
50
60
70
80
2
Fatigue limit
1
0
0
90
Iterations [−]
10
20
30
40
50
60
70
Iterations [−]
(c) First natural frequency
(d) Fatigue damage
Figure 4: Constraints change during the optimization procedure
8
80
90
The Science of Making Torque from Wind 2012
Journal of Physics: Conference Series 555 (2014) 012046
IOP Publishing
doi:10.1088/1742-6596/555/1/012046
4.3. Objective function
Table 1 shows the mass reduction percentage of the optimized design compared to the initial
design. A mass reduction of 36% is achieved for the optimized transition piece. Also the
optimized monopile shows a mass reduction of 11.1%. In contrast, a mass increase of 14.6% is
resulted for the tower. In total the integrated mass of the support structure shows 12.1% of
mass reduction.
It should be noted that further mass reduction is achievable if the first natural frequency
constraints would be relaxed. In this case, fatigue damage would probably remain an active
design constraint, but with a different thickness and mass distribution for the support structure
to keep it satisfy.
Table 1: Reduced mass in percentage of the optimized design compared to the initial design
Tower
Transition piece
Monopile
Support structure
Mass changes [%]
14.55
-36.0
-11.1
-12.12
5. Conclusion and future work
The goal of this research was to use the integrated design methodology to simultaneously design
the monopile, transition piece and tower of an existing offshore support structure. This approach
resulted in 12% mass reduction compared to a real design where the monopile, transition piece
and tower were design separately by independent parties. This mass reduction shows that having
a lighter support structure is possible if all disciplines and components are designed at once. It
is therefore recommended to also include the controller design as another discipline [30], and
the rotor design as another component to this multidisciplinary design optimization framework.
Also, including diameter and length of the sections as design variables is recommended to enable
further mass reduction.
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