Optimization

Executing FloatingSE by hand is sufficient to explore some simple one-off or comparison analyses between a few runs. OpenMDAO provides extensive optimization capability, which can give yield richer and more insightful analyses.

Design Variables

In WISDEM, via OpenMDAO, any input parameter can be designated a design variable. The design variables used in this study focused on the geometric specification of the floating substructure and mooring subsystem. Slightly different design variables and bounds were used for spar, semisubmersible, and TLP optimizations. The complete listing of the design variables for each optimization configuration is shown in Table 8. Note that the integer design variables were only used in the global optimization with the genetic algorithm, not the local search with the simplex algorithm.

Table 8 Standard design variables, their size, and units used for optimization in FloatingSE. Note that \(n_s\) denotes the number of sections in the column discretization.

Variable	Name	Units	Type	Bounds
Main col section height	main_section_height		Float array (\(n_s\))	0.1–50
Main col outer diameter	main_outer_diameter		Float array (\(n_s+1\))	2.1–40
Main col wall thickness	main_wall_thickness		Float array (\(n_s+1\))	0.001–0.5
Main col freeboard	main_freeboard		Float scalar	0–50
Main col stiffener web height	main_stiffener_web_height		Float array (\(n_s\))	0.01–1
Main col stiffener web thickness	main_stiffener_web_thickness		Float array (\(n_s\))	0.001–0.5
Main col stiffener flange width	main_stiffener_flange_width		Float array (\(n_s\))	0.01–5
Main col stiffener flange thickness	main_stiffener_flange_thickness		Float array (\(n_s\))	0.001–0.5
Main col stiffener spacing	main_stiffener_spacing		Float array (\(n_s\))	0.1–100
Main col permanent ballast height	main_permanent_ballast_height		Float scalar	0.1–50
Main col buoyancy tank diameter	main_buoyancy_tank_diameter		Float scalar	0–50
Main col buoyancy tank height	main_buoyancy_tank_height		Float scalar	0–20
Main col buoyancy tank location (fraction)	main_buoyancy_tank_location		Float scalar	0–1
Number of offset cols	number_of_offset_columns		Integer scalar	3-5
Offset col section height	offset_section_height		Float array (\(n_s\))	0.1–50
Offset col outer diameter	offset_outer_diameter		Float array (\(n_s+1\))	1.1–40
Offset col wall thickness	offset_wall_thickness		Float array (\(n_s+1\))	0.001–0.5
Offset col freeboard	offset_freeboard		Float scalar	2–15
Offset col stiffener web height	offset_stiffener_web_height		Float array (\(n_s\))	0.01–1
Offset col stiffener web thickness	offset_stiffener_web_thickness		Float array (\(n_s\))	0.001–0.5
Offset col stiffener flange width	offset_stiffener_flange_width		Float array (\(n_s\))	0.01–5
Offset col stiffener flange thickness	offset_stiffener_flange_thickness		Float array (\(n_s\))	0.001–0.5
Offset col stiffener spacing	offset_stiffener_spacing		Float array (\(n_s\))	0.01–100
Offset col permanent ballast height	offset_permanent_ballast_height		Float scalar	0.1–50
Offset col buoyancy tank diameter	offset_buoyancy_tank_diameter		Float scalar	0–50
Offset col buoyancy tank height	offset_buoyancy_tank_height		Float scalar	0–20
Offset col buoyancy tank location (fraction)	main_buoyancy_tank_location		Float scalar	0–1
Radius to offset col	radius_to_offset_column		Float scalar	5–100
Pontoon outer diameter	pontoon_outer_diameter		Float scalar	0.1–10
Pontoon wall thickness	pontoon_wall_thickness		Float scalar	0.01–1
Lower main-offset pontoons	lower_attachment_pontoons_int		Integer scalar	0–1
Upper main-offset pontoons	upper_attachment_pontoons_int		Integer scalar	0–1
Cross main-offset pontoons	cross_attachment_pontoons_int		Integer scalar	0–1
Lower offset ring pontoons	lower_ring_pontoons_int		Integer scalar	0–1
Upper offset ring pontoons	upper_ring_pontoons_int		Integer scalar	0–1
Outer V-pontoons	outer_cross_pontoons_int		Integer scalar	0–1
Main col pontoon attach lower (fraction)	main_pontoon_attach_lower		Float scalar	0–0.5
Main col pontoon attach upper (fraction)	main_pontoon_attach_upper		Float scalar	0.5–1
Fairlead (fraction)	fairlead_location		Float scalar	0–1
Fairlead offset from col	fairlead_offset_from_shell		Float scalar	5–30
Fairlead pontoon diameter	fairlead_support_outer_diameter		Float scalar	0.1–10
Fairlead pontoon wall thickness	fairlead_support_outer_thickness		Float scalar	0.001–1
Number of mooring connections	number_of_mooring_connections		Integer scalar	3–5
Mooring lines per connection	mooring_lines_per_connection		Integer scalar	1–3
Mooring diameter	mooring_diameter		Float scalar	0.05–2
Mooring line length	mooring_line_length		Float scalar	0–3000
Anchor distance	anchor_radius		Float scalar	0–5000

Constraints

Due to the many design variables, permutations of settings, and applied physics, there are many constraints that must be applied for an optimization to close. The constraints capture both physical limitations, such as column buckling, but also inject industry standards, guidelines, and lessons learned from engineering experience into the optimization. As described in the Introduction, this is a critically important element in building a MDAO framework for conceptual design that yields feasible results worth interrogating further with higher-fidelity tools. The constraints used in the substructure design optimization and sensitivity studies are listed in Table 9. Where appropriate, some of the constraint values differ from one type of substructure to another. Some additional explanation is provided for a handful of constraints in the subsections below.

Table 9 Optimization constraints used in FloatingSE.

Lower	Variable	Upper	Comments
	Tower / Main / Offset Columns
	Eurocode global buckling	1.0
	Eurocode shell buckling	1.0
	Eurocode stress limit	1.0
	Manufacturability	0.5	Taper ratio limit
120.0	Weld-ability		Diameter:thickness ratio limit
	Main / Offset Columns
	Draft ratio	1.0	Ratio of draft to max value
	API 2U general buckling- axial loads	1.0
	API 2U local buckling- axial loads	1.0
	API 2U general buckling- external loads	1.0
	API 2U local buckling- external loads	1.0
	Wave height:freeboard ratio	1.0	Maximum wave height relative to freeboard
1.0	Stiffener flange compactness
1.0	Stiffener web compactness
	Stiffener flange spacing ratio	1.0	Stiffener spacing relative to flange width
	Stiffener radius ratio	0.50	Stiffener height relative to diameter
	Offset Columns		Semi only
0.0	Heel freeboard margin		Height required to stay above waterline at max heel
0.0	Heel draft margin		Draft required to stay submerged at max heel
	Pontoons		Semi only
	Eurocode stress limit	1.0
	Tower
-0.01	Hub height error	0.01
	Mooring
0.0	Axial stress limit	1.0
	Line length limit	1.0	Loss of tension or catenary hang
	Heel moment ratio	1.0	Ratio of overturning moment to restoring moment
	Surge force ratio	1.0	Ratio of surge force to restoring force
	Geometry
1.0	Main-offset spacing		Minimum spacing between main and offset columns
0.0	Nacelle transition buffer		Tower diameter limit at nacelle junction
-1.0	Tower transition buffer	1.0	Diameter consistency at freeboard point
	Stability
0.10	Metacentric height		Not applied to TLPs
1.0	Wave-Eigenmode boundary (upper)		Natural frequencies below wave frequency range
	Wave-Eigenmode boundary (lower)	1.0	Natural frequencies above wave frequency range
0.0	Water ballast height limit	1.0
0.0	Water ballast mass		Neutral buoyancy

Geometry Constraints

Words Table 10

Table 10 Constraint variables for the geometry in FloatingSE.

Variable	Type	Description
max_draft	Float scalar	Maximum allowable draft for the substructure

Manufacturing Constraints

Manufacturing steel frustum shells requires rolling steel plates into shape and welding along a seam to close the section. To accommodate traditional rolling and welding practices, both the diameter taper over the course of a section and the wall thickness ratio relative to the diameter are capped. Similarly, to facilitate welding the semisubmersible pontoons to the columns, constraints regarding the radio of diameters between the two are enforced. These limits are determined by user parameters in Table 11 and constraints,

Table 11 Constraint variables for the manufacturability in FloatingSE.

Variable	Type	Description
min_taper_ratio	Float scalar	For manufacturability of rolling steel
min_diameter_thickness_ratio	Float scalar	For weld-ability
connection_ratio_max	Float scalar	For welding pontoons to columns

Stress Limits and Code Compliance

The stress and buckling code compliance constraints are formulated as utilization ratios (ratio of actual to maximum values), with a safety factor, which must be less than one. The safety factor parameters are listed in Table 12.

Table 12 Variables specifying the factors of safety within FloatingSE.

Variable	Type	Description
gamma_f	Float scalar	Safety factor on
gamma_b	Float scalar	Safety factor on buckling
gamma_m	Float scalar	Safety factor on materials
gamma_n	Float scalar	Safety factor on consequence of failure
gamma_fatigue	Float scalar	Not currently used

Stability

As described above, surge and pitch stability are enforced through similar approaches. The total force and moment acting on the turbine are compared to the restoring forces and moments applied by the mooring system, buoyancy, or other sources at the maximum allowable point of displacement. These constraints are formulated as ratios with the user specifying the maximum allowable limits via the variables in Table 13.

Table 13 Constraint variables for the mooring system in FloatingSE.

Variable	Type	Units	Description
max_offset	Float scalar	\(m\)	Max surge/sway offset
operational_heel	Float scalar	\(deg\)	Max heel (pitching) angle in operating conditions
max_survival_heel	Float scalar	\(deg\)	Max heel (pitching) angle in parked conditions

Objectives

Different anaylses will emphasize different metrics, requiring different objective functions. Under the default assumption that the user wishes to minimize cost and adhere to stability constraints, the objective function would be total substructure cost (variable name, total_cost) or mass (variable name, total_mass).

Example

Fig. 41 Example of optimized spar.

Fig. 42 Example of optimized semi.

Fig. 43 Example of optimized TLP.