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Conference Paper

NREL/CP-CP-5000-75876

September 2020

Numerical Model Development of a 

Variable-Geometry Attenuator Wave 

Energy Converter 

Preprint 

Nathan Tom,1 Yi Guo,1 Davy Pardonner2

1 National Renewable Energy Laboratory 

2 University of West Florida

Presented at the ASME 2020 39th International Conference on Ocean, 

Offshore and Arctic Engineering 

August 3–7, 2020 
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Office of Energy Efficiency & Renewable Energy

Operated by the Alliance for Sustainable Energy, LLC

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Laboratory (NREL) at www.nrel.gov/publications.

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Conference Paper

NREL/CP-5000-75876

September 2020

Numerical Model Development of a 

Variable-Geometry Attenuator Wave 

Energy Converter

Preprint

Nathan Tom,

1 Yi Guo,1 Davy Pardonner2 

1 National Renewable Energy Laboratory

2 University of West Florida

Suggested Citation

Tom, Nathan, Yi Guo, and Davy Pardonner. 2020. Numerical Model Development of a 

Variable-Geometry Attenuator Wave Energy Converter: Preprint. Golden, CO: National 

Renewable Energy Laboratory. NREL/CP-5000-75876. 

https://www.nrel.gov/docs/fy20osti/75876.pdf
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NUMERICAL MODEL DEVELOPMENT OF A

VARIABLE-GEOMETRY ATTENUATOR WAVE ENERGY CONVERTER

Davy Pardonner

University of West Florida

Pensacola, FL USA

Email: davypardonner@gmail.com

Nathan Tom∗

National Renewable

Energy Laboratory

Golden, CO USA

Email: Nathan.Tom@nrel.gov

Yi Guo

National Renewable

Energy Laboratory

Golden, CO USA

Email: Yi.Guo@nrel.gov

ABSTRACT

Because the wave energy industry is still in its infancy, an

optimal design for wave energy converters (WECs) has yet to

be established; more work is needed to explore various costreduction pathways. The primary cost-reduction pathway considered for this work is the optimization of the geometric profile

on an attenuator WEC to maximize power production while, at

the same time, minimizing capital expenditures through the use

of variable-geometry modules. In this investigation, the variablegeometry modules consist of inflatable bags placed on either side

of a base central steel cylinder that would be inflated in lowmoderate sea states to maximize power capture and then deflated

in moderate-extreme sea states to minimize wave loading. The

numerical model and simulation of the attenuator WEC were developed and completed using WEC-Sim, which is an open-source

code that is appropriate for use in evaluating the dynamic response of the different WEC models in operational seas. The

power production estimates were obtained from the Wave Energy

Prize (WEP) sea states, which are representative of U. S. deployment sites, to calculate the average climate capture width that is

used in the WEP ACE calculation. Preliminary capital expenditure costs were obtained assuming the base central steel cylinder

mass was equal to the fluid displaced mass, minus the mass of

the variable-geometry bags. The additional weight required to

offset the additional buoyancy from the variable-geometry bags

was assumed to come from the addition of seawater ballast. The

variable-geometry attenuator model was found to have a similar

power capture efficiency as a fixed-body model, but is expected

∗Address all correspondence to this author.

to have a lower characteristic capital expenditure given its more

streamlined profile, which demonstrates that variable-geometry

modules may provide a realistic cost-reduction pathway to help

design a more cost-competitive WEC.

INTRODUCTION

The global resource of wave energy potential is estimated

to be around 30,660 TWh/yr [1]; however, despite the large

amount of potential energy available, the wave energy industry

remains in the early stages of development and lacks a wellestablished wave energy converter (WEC) design. Over the past

two decades, the global marine and hydrokinetic energy sector

has experienced a resurgence in the funding and manpower allocated toward research and development. As a whole, the field of

wave energy continues to host a wide diversity of technologies,

ranging from concept- to prototype-scale devices [2]. However,

the cost of energy for WECs remains at the high end of renewable energy technologies, highlighting the need for a structured,

innovative approach to WEC development in hope of achieving an optimal convergence in technologies [3]. As shown in

the U.S. Department of Energy Reference Model Project [4], the

structural cost of WEC designs accounts for approximately 37–

52% of the levelized cost of energy (LCOE), whereas the power

take-off (PTO) is expected to contribute around 8% of LCOE [5].

Therefore, to make a significant impact on LCOE, structural design optimizations must be explored; however, these reductions

will not be obtained until there is a greater load-shedding capability designed into the WEC structure. Recent work has attempted

This report is available at no cost from the National Renewable Energy Laboratory at www.nrel.gov/publications. 1
to incorporate variable-geometry modules into the structural design in hope of providing greater control over the hydrodynamic

properties of the device that define power absorption and structural loading [6].

This study focuses on the hydrodynamic design of an attenuator WEC, which is generally characterized as a floating device

that operates parallel to the wave direction and captures wave

energy from the relative motion between two or more oscillating bodies. There have been several attenuator concepts previously explored, with select prototypes built and tested in the

open ocean, such as the Pelamis Wave Power’s P2 devices [7],

Crestwing Tordenskiold [8], and the Dexa WEC [9]. Pelamis

Wave Power had the first operational commercial WEC farm,

which was deployed off the coast of Portugal starting in 2009.

In 2019, Mocean Energy was awarded Wave Energy Scotland’s

Novel Wave Energy Converter Stage 3 funding to build and deploy a prototype of their Blue Star attenuator WEC [10]. Given

the academic and commercial interest in the attenuator WEC

concept, the WEC design for this study was modeled after the

first-generation Pelamis design.

The goal of this study is to identify properties of a variablegeometry attenuator WEC that could potentially lead to a costcompetitive design for the wave energy industry. Incorporating variable-geometry components is expected to help improve

the power capture efficiency in low-to-moderate sea states while

capping power capture and reducing structural loads in large-toextreme sea states. Improving power capture, while at the same

time reducing costs, is expected to help provide significant reductions in LCOE rather than focusing solely on power maximization. In this work, the variable-geometry components will

be flexible bags that are placed on either side of the main WEC

body and can be inflated or deflated to alter the device profile in

the water. The use of flexible materials in the field of wave energy has also recently gained more attention because plastic or

fabric bags are cheaper to manufacture, transport, and assemble,

although there are additional hydrodynamic modeling challenges

that must be considered [11].

This paper begins with a description of the base design of

the attenuator WEC, its mass properties, and hydrostatic restoring forces that keep the absorber bodies floating at the sea surface. The paper then discusses how the hydrodynamic coefficients are affected by changing the primary dimensions of the

idealized WEC structure in hope of identifying which geometrical dimensions play a greater role in power absorption and

load accumulation. After discussing the WEC hydrodynamics,

the paper introduces the WEC-Sim code, used to simulate the

attenuator dynamics, and the sea states chosen to estimate the

WEC power performance. Next, the variable-geometry flexible

bag concept is introduced, followed by a discussion of the effects of the variable-geometry modules on the hull hydrodynamics. The variable-geometry module description is followed by

an introduction to the ACE calculation [12], which is a simpliFIGURE 1. SCHEMATIC OF THE TWO-BODY ATTENUATOR

WEC-SIM MODEL, INCLUDING THE FREE SURFACE, WAVE DIRECTION, BODY 1, BODY 2, AND PTO 1.

FIGURE 2. SIDE VIEW OF A SINGLE-BODY SECTION.

fied power-to-cost ratio that will be used to evaluate the overall

performance of the variable-geometry WEC concept. The paper concludes with a discussion about the success of the attenuator WEC variable-geometry modules at improving power capture

while reducing the capital costs related to the device structural

design.

TWO-BODY MODEL DESCRIPTION

The two-body attenuator WEC consists of two identical

floating cylindrical hulls with a rotational PTO connecting the

two bodies (Figure 1). The hull mass is assumed to be uniformly

distributed, which sets the center of gravity at the geometric center of the body. The inertial properties of a single body, about the

center of gravity, is defined by the following equations:

Vd =

πD

2L

8

, m = ρ

πD

2L

8

, I55 =

mL2

12 "

1+

3

4



D

L

2

#

, (1)

cb = [0,0,

−2D

3π

], C33 = ρgDL, (2)

C55 = ρgIy +ρgVdzb −mgzg = ρg

DL3

12 "

1−



D

L

2

#

, (3)

where Vd is the displaced volume of the cylinder, D is the diameter of the cylinder, L is the length of the cylinder, m is the mass

of cylinder, ρ is the fluid density, I55 is the pitch mass moment

of inertia of the cylinder, cb is the center of buoyancy, C33 is the

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heave hydrostatic restoring coefficient, C55 is the pitch hydrostatic restoring coefficient, Iy is the second moment of area of the

water plane area, zb is the vertical center of buoyancy, and zg is

the vertical center of gravity. Each body is designed to float with

a draft equal to half the cylinder diameter, and with the mass of

the body equal to the mass of the displaced fluid. The inertial

and geometric properties of a single body are listed in Table 1

and shown in Figure 2. For comparison, the individual bodies

of the Pelamis P1 design had a length, L = 35 m, and diameter,

D = 3.5 m [13]. The dimensions for our attenuator model were

adjusted from the Pelamis P1 to better suit the wave conditions

at U.S. deployment sites. For dynamic modeling purposes, the

system was constrained to oscillate in the surge, heave, and pitch

degrees of freedom (DOF) while the relative pitch rotation between the two bodies drove a rotary PTO to harvest wave power.

TABLE 1. BASE MASS AND GEOMETRIC DIMENSIONS FOR A

SINGLE-BODY SEGMENT.

Variable Symbol Value [unit]

Length L 25 [m]

Diameter D 3 [m]

Draft d 1.5 [m]

Displaced Volume ∀ 88.36 [m3]

Mass m 88357 [kg]

Pitch Mass Moment of Inertia I55 4.65×106[kg· m2]

Center of Gravity cg 0, 0, 0 [m]

Center of Buoyancy cb 0, 0, -0.636 [m]

WEC Hydrodynamics

The first step in this study was the examination of the hydrodynamic coefficients for different geometric hull shapes to

help size the WEC. The surge, heave, and pitch hydrodynamic

coefficients were calculated using WAMIT version 7.2 [14], at

the center of gravity of each body, and were computed before

the power performance and corresponding loading of the WEC

could be estimated. Although the hydrodynamic coefficients for

each DOF contribute to the relative pitch motion between the two

bodies, only the heave hydrodynamic coefficients are discussed

in this section (for brevity).

The length of each body was modeled at four values, in

addition to the base case shown in Table 1, while keeping the diameter, D, constant at 3 m. The heave radiation added mass, μ33,

and wave damping, λ33, for body 1 are plotted in Figure 3, with

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

0.5

0.8

1.1

1.4

1.7

2.0

2.3

2.6

2.9

3.2

3.5

Heave Radiation Added Mass, 33

/ Vd, [-]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Heave Radiation Wave Damping, 33

/ Vd, [-]

33, L 15 m

33, L 25 m

33, L 34 m

33, L 45 m

33, L 55 m

33, L 15 m

33, L 25 m

33, L 34 m

33, L 45 m

33, L 55 m

FIGURE 3. HEAVE RADIATION ADDED MASS, μ33, AND

WAVE DAMPING, λ33, FOR EACH BODY 1 LENGTH. AS THE RADIATION PROBLEM IS SYMMETRIC, THE RADIATION COEFFICIENTS ARE IDENTICAL FOR BODY 2 AND ARE OMITTED.

body 2 values omitted because the radiation problem is symmetric, resulting in the same values for each body. The radiation

hydrodynamic coefficients have been nondimensionalized by the

WEC mass which, in this case, scales linearly with body length.

Therefore, in Figure 3, if the radiation coefficient curves were to

lie on top of one another, the radiation coefficients would scale

linearly with body length; however, at low wave frequencies, the

heave-added mass grows nonlinearly as the body length is increased. In the intermediate frequency range, between 0.8 rad/s

and 1.4 rad/s, the heave-added mass per unit length decreases

with increasing body length before all curves converge at the

highest wave frequencies. A similar trend is observed for radiation wave damping, which has a nonlinear growth with increasing body length at the low and intermediate wave frequencies

before converging at the highest plotted frequencies. The heave

excitation force magnitude, |X3|, and phase φ3 for both bodies

are plotted in Figure 4. In Figure 4a, the wave-excitation force

magnitude has been normalized by the heave hydrostatic restoring coefficient, while in Figure 4b the phase has been normalized

by π. The heave-wave excitation magnitude for body 1 is always

larger than body 2 because of shielding effects; however, this

difference is reduced as the body length is increased. The heave

hydrostatic restoring coefficient scales linearly with body length,

and Figure 4a shows that although the dimensional heave-wave

excitation force is greatest for the longest body length, the increase per unit length is significantly reduced. For each body

length, the heave-wave excitation force drops to 0 at wave frequencies that correspond to wavelengths that are integer multiples of the body length, which also corresponds to the near-zero

crossings of the wave-exciting force phase in Figure 4b.

Next, the diameter of each body was modeled at three additional sizes, in addition to the base case, with the body length

This report is available at no cost from the National Renewable Energy Laboratory at www.nrel.gov/publications. 3
0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Heave Wave-Exciting Force Magnitude, |X3|/ gDL, [-]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Body 1, L 15 m

Body 1, L 25 m

Body 1, L 34 m

Body 1, L 45 m

Body 1, L 55 m

Body 2, L 15 m

Body 2, L 25 m

Body 2, L 34 m

Body 2, L 45 m

Body 2, L 55 m

(a) Heave-Wave Excitation Force Magnitude |X3|

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

-1.0

-0.8

-0.6

-0.4

-0.2

 0.0

 0.2

 0.4

 0.6

 0.8

 1.0

Heave Wave-Exciting Force Phase, 3

/ , [-]

-1.0

-0.8

-0.6

-0.4

-0.2

 0.0

 0.2

 0.4

 0.6

 0.8

 1.0

Body 1, L 15 m

Body 1, L 25 m

Body 1, L 34 m

Body 1, L 45 m

Body 1, L 55 m

Body 2, L 15 m

Body 2, L 25 m

Body 2, L 34 m

Body 2, L 45 m

Body 2, L 55 m

(b) Heave-Wave Excitation Force Phase φ3

FIGURE 4. HEAVE-WAVE EXCITATION FORCE MAGNITUDE

AND PHASE FOR BOTH BODIES AND EACH OF THE FIVE

LENGTH CASES.

kept constant at 25 m. The heave radiation added mass and wave

damping are plotted in Figure 5, and the heave-wave excitation

force magnitude and phase are plotted in Figure 6. Unlike with

increasing body length, increasing the diameter results in a nonlinear reduction in the heave added mass and wave damping relative to the displaced volume of the body. The variation in the

heave-wave excitation force magnitude with body diameter has

a similar trend with increasing body length, where the excitation

per unit width decreases as the diameter increases. Here, all body

geometries have the minimum normalized heave excitation magnitudes at the same wave frequencies because the body length is

constant.

WEC-SIM OPEN-SOURCE CODE

The power performance modeling for the attenuator WEC

was completed in the open-source code WEC-Sim, which is

jointly developed by Sandia National Laboratories and the Na0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

0.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3.0

Heave Radiation Added Mass, 33

/ Vd, [-]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Heave Radiation Wave Damping, 33

/ Vd, [-]

33, D 2 m

33, D 3 m

33, D 4 m

33, D 5 m

33, D 2 m

33, D 3 m

33, D 4 m

33, D 5 m

FIGURE 5. HEAVE RADIATION ADDED MASS, μ33, AND

WAVE DAMPING, λ33, FOR EACH BODY 1 DIAMETER. BECAUSE

THE RADIATION PROBLEM IS SYMMETRIC, THE RADIATION

COEFFICIENTS WILL BE THE SAME FOR BODY 2 AND ARE

THEREFORE OMITTED.

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Heave Wave-Exciting Force Magnitude, |X3|/ gDl, [-]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Body 1, D 2 m

Body 1, D 3 m

Body 1, D 4 m

Body 1, D 5 m

Body 2, D 2 m

Body 2, D 3 m

Body 2, D 4 m

Body 2, D 5 m

(a) Heave-Wave Excitation Force Magnitude |X3|

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

-1.0

-0.8

-0.6

-0.4

-0.2

 0.0

 0.2

 0.4

 0.6

 0.8

 1.0

Heave Wave-Exciting Force Phase, 3

/ , [-]

-1.0

-0.8

-0.6

-0.4

-0.2

 0.0

 0.2

 0.4

 0.6

 0.8

 1.0

Body 1, D 2 m

Body 1, D 3 m

Body 1, D 4 m

Body 1, D 5 m

Body 2, D 2 m

Body 2, D 3 m

Body 2, D 4 m

Body 2, D 5 m

(b) Heave-Wave Excitation Force Phase φ3

FIGURE 6. HEAVE-WAVE EXCITATION FORCE MAGNITUDE

AND PHASE FOR BOTH BODIES AND EACH OF THE FOUR DIAMETER CASES.

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tional Renewable Energy Laboratory, through funding from the

U.S. Department of Energy’s Water Power Technologies Office [15]. The WEC-Sim code is developed in the MATLAB/Simulink environment and uses Simscape Multibody to

solve for the WEC rigid body dynamics. WEC-Sim is a wave

energy converter simulator that models the interaction between

incident waves, device motion, PTO mechanism, and mooring

structures. To solve for the WEC dynamic response, WEC-Sim

uses the time-domain convolution integral formulation based on

the Cummins equation [15].

For this analysis, the Wave Energy Prize (WEP) sea states

were chosen to evaluate the power capture performance of the

two-body fixed-geometry attenuator. The WEP sea states consist

of six representative sea conditions that a WEC deployed along

U.S. coasts would likely encounter [16]. Each sea state is defined by a significant wave height, Hs

, peak wave period, Tp, and

time-averaged wave power flux, Pw, as shown in Table 2, and is

described by a Bretschneider spectrum [15].

TABLE 2. WEP SEA STATES SIGNIFICANT WAVE HEIGHT, HS,

PEAK PERIOD, TP, WAVE POWER FLUX, PW , AND STEEPNESS.

Sea State HS [m] TP [s] Pw [kW/m] λTp

/Hs [-]

1 2.34 7.31 16.7 35.6

2 2.64 9.86 29.0 57.5

3 5.36 11.52 141.1 38.6

4 2.05 12.71 23.1 122.4

5 5.84 15.23 233.5 60.4

6 3.25 16.50 79.8 124.9

PTO POWER AND TORQUE PERFORMANCE MODEL

The WEC PTO system transforms the absorbed mechanical

energy of the device to electricity, making it an important contributor to the power performance of the device. For this analysis, the PTO damping coefficient, Bg, was optimized for each

hull shape and sea state to maximize power capture. Because

of the complexity of the coupled equations of motion for both

bodies, an analytical optimal value of Bg was not derived but

was obtained by completing a parameter sweep. And, because

the dynamic motion of the two bodies changes behavior with the

incident wave characteristics, as well as the PTO damping coefficient, one would expect to see a change in the WEC’s power

capture with Bg, as shown in Figure 7. For this case, the timeaveraged power capture follows a bell curve shape and peaks

when the PTO damping coefficient is near 17 N·s·m/rad, while

the PTO torque continues to increase with Bg.

The time-averaged power capture for each body length

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Damping Coefficient [Nms/rad] 107

0.5

1

1.5

2

2.5

3

3.5

Time-Averaged Power [kW] (left axis), PTO Torque [kNm] (right axis)

104

0

2

4

6

8

10

12

14

16

105

Time-Averaged Power

Torque

FIGURE 7. TIME-AVERAGED POWER CAPTURE (LEFT AXIS)

AND PTO TORQUE (RIGHT AXIS) FOR THE TWO-BODY BASE

MODEL IN WEP SEA STATE 3.

across each WEP sea state, has been plotted in Figure 8 and

shows that the WEC with longer body units is capable of capturing greater power in sea states with the greatest wave steepness (Table 2). The corresponding PTO torque, plotted in Figure 9, maintains a more even distribution across WEP sea states

for the same body length case. The percentage change in PTO

power capture and torque with body length, relative to the base

case dimensions, is plotted in Figure 10 along with a linear

scaling guideline as the structural mass increases linearly with

body length. Figure 10 demonstrates that although the longer

hull lengths have greater power capture, the increase in torque is

much greater and thus drops further below the linear trend line.

The results indicate that the device lengths between 25 m and 35

m have the best PTO power-to-torque ratio.

The effect of varying the hull diameter on the PTO timeaveraged power and torque can be found in Figures 11 and 12, respectively, which indicate that larger diameters generate greater

power, but also experience greater PTO torque. Similar to the

analysis for varying hull length, Figure 13 plots the percent

change in PTO time-averaged power against the percent change

in PTO torque. Because increasing the body diameter results in a

quadratic increase in displaced mass, from a cost perspective the

growth in time-averaged power and torque should lie on or above

the quadratic scaling guideline shown in Figure 13. Points that

appear below the quadratic indicate that either the PTO power

capture has not increased as fast as—or the PTO torque has increased faster than—the hull displaced mass. If the PTO power

This report is available at no cost from the National Renewable Energy Laboratory at www.nrel.gov/publications. 5
1 2 3 4 5 6

WEP Sea States

0

50

100

150

200

250

300

350

Mean Power [kW]

15m

25m

34m

45m

55m

FIGURE 8. POWER COMPARISON ACROSS ALL WEP SEA

STATES AND FOR EACH BODY LENGTH CASE.

1 2 3 4 5 6

WEP Sea States

0

0.5

1

1.5

2

2.5

3

3.5

4

Torque [MN m]

15m

25m

34m

45m

55m

FIGURE 9. TORQUE COMPARISON ACROSS ALL WEP SEA

STATES AND FOR EACH LENGTH CASE.

FIGURE 10. PERCENT CHANGE IN POWER CAPTURE

AGAINST TORQUE FOR EACH LENGTH CASE AND ACROSS

SELECT WEP SEA STATES.

capture lags the growth in hull mass, the developer will have to

invest more in capital costs without receiving an equal return in

power capture. Furthermore, even if the power capture follows

the increase in displaced mass, if the PTO torque requirement increases much faster, the developer will be required to pay more

for a larger-capacity PTO system and may also need to increase

the structural steel thickness to ensure the hull-to-PTO connection does not fail. Therefore, the results from Figure 13 show

that the likely optimum hull diameter will be between 2 m and 4

m and will have the best PTO power-to-torque ratio.

1 2 3 4 5 6

WEP Sea States

0

10

20

30

40

50

60

70

80

90

Mean Power [kW]

2m

3m

4m

5m

FIGURE 11. POWER COMPARISON ACROSS ALL WEP SEA

STATES AND FOR EACH BODY DIAMETER CASE.

1 2 3 4 5 6

WEP Sea States

0

100

200

300

400

500

600

700

800

Torque [kN m]

2m

3m

4m

5m

FIGURE 12. TORQUE COMPARISON ACROSS ALL WEP SEA

STATES AND FOR EACH BODY DIAMETER CASE.

VARIABLE-GEOMETRY ATTENUATOR CONCEPT

The concept of incorporating variable-geometry components into the WEC structure remains a relatively unexplored research topic, with previous investigations considering a few sample WEC architectures. For example, variable-geometry modules were incorporated into the body of an oscillating surge

WEC [17], and were successful at reducing both the PTO and

structural loads on the device. For the analysis of a Pelamis-like

attenuator, the variable-geometry modules consist of inflatable

side bags or floats, as shown in Figure 14, that can inflate or deflate depending on sea conditions. In low-to-moderate seas, the

bags would inflate in hope of increasing power capture, while in

larger-to-extreme sea states the bags would be deflated to reduce

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FIGURE 13. PERCENT CHANGE IN POWER CAPTURE

AGAINST TORQUE FOR EACH DIAMETER CASE AND ACROSS

SELECT WEP SEA STATES.

hydrodynamic loading and allow the device to dive beneath the

waves. The focus of this investigation was to characterize the effect of the variable-geometry modules in normal sea conditions,

thus simulations were not completed in any extreme or survival

sea states. The floats have been assumed to extend the full length

on each side of the hull, along the x-axis, and are located on both

body 1 and body 2. Each float is connected at the top of the hull,

above water, and wraps around half of the circumference to the

bottom where it meets the other float. When the floats are fully

inflated, the cross section of the WEC hull becomes an ellipse

because of the elongation of the width of the device along the

y-axis. The elliptical cross section of the body section will be

defined by a major axis, a, and minor axis, b. The hypothesis

was that the major axis, a, is the dominant parameter in defining the power capture and thus using the floats to increase the

surface area at the calm water surface would help maintain maximum power performance. To test how the variable-geometry

modules affected the WEC power absorption and PTO torque,

the major axis was kept constant, whereas the minor axis, starting at the same length as the major axis, was reduced by a factor

of 3/4, 1/2, and 1/4 to create a set of four cases. A diagram of the

different aspect ratios are shown in Figure 15. From the fixedgeometry analysis, the optimal fixed-geometry diameter lies in

the 2 m to 4 m range, thus the lengths of a considered were 2 m,

3 m, 4 m, and 5 m.

Effect of Variable Geometry on Hull Hydrodynamics

For hydrodynamic modeling purposes, the inflatable bags

have been assumed to withstand the required air pressure to

maintain the idealized elliptical shape. The shape would more

likely take on an inverted curved shape that would be dependent

on the internal pressure, surrounding hydrostatic pressure, and

material stiffness of the bags. However, as a first step at understanding the changing hydrodynamics with WEC shape, the

idealized shape is still valuable in understanding the potential

of this novel feature. Figure 16 plots the heave radiation added

FIGURE 14. SCHEMATIC OF THE WEC CROSS SECTION,

INCLUDING VARIABLE-GEOMETRY MODULES. THE INNER

CYLINDER DIAMETER CORRESPONDS TO MINOR AXIS, B,

WHILE THE FLOATS EXTEND TO MAJOR AXIS, A. EACH FLOAT

COVERS HALF OF THE INNER CYLINDER’S CIRCUMFERENCE.

FIGURE 15. CASES 1-4 FOR A GIVEN MAJOR AXIS LENGTH,

A. MINOR AXIS IS EQUAL TO THE MAJOR AXIS IN CASE 1, 3

4

OF THE MAJOR AXIS FOR CASE 2, 1

2 OF THE MAJOR AXIS FOR

CASE 3, AND 1

4 OF THE MAJOR AXIS FOR CASE 4.

mass and wave damping, nondimensionalized by the displaced

fluid mass, for the variable-geometry aspect ratio cases 1-4, with

a major axis length of 5 m. Increasing the aspect ratio increases

both the nondimensionalized heave radiation coefficients; however, because the WEC displaced volume is also reduced with

increasing aspect ratio, the dimensional added mass is smaller.

Therefore, decreasing the aspect ratio to become more circular

does not lead to increases in the hydrodynamics as compared to

gains in the mass properties. As the minor axis decreases, the

heave-wave excitation force magnitude at each wave frequency

increases slightly, with almost no change in phase (Figure 17).

While the wave-excitation force on body 2 starts at nearly the

same magnitude as body 1, the excitation force quickly reduces,

leading to a maximum reduction of roughly 20%-30% between

wave frequencies 0.7 rad/s and 1.1 rad/s. The resulting heavewave excitation force coefficients with increasing aspect ratio

demonstrate that structural costs can be saved by reducing the

WEC draft while maintaining nearly the same wave-excitation

magnitude.

The second set of hydrodynamic results provided in Figure 18 plot the heave radiation coefficients with varying major

This report is available at no cost from the National Renewable Energy Laboratory at www.nrel.gov/publications. 7
0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

0

1

2

3

4

5

6

7

8

9

10

Heave Radiation Added Mass, 33

/ Vd, [-]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Heave Radiation Wave Damping, 33

/ Vd, [-]

33, a/b = 1.0

33, a/b = 1.3

33, a/b = 2.0

33, a/b = 4.0

33, a/b = 1.0

33, a/b = 1.3

33, a/b = 2.0

33, a/b = 4.0

FIGURE 16. HEAVE RADIATION ADDED MASS, μ33, AND

WAVE DAMPING, λ33, FOR ALL FOUR VARIABLE-GEOMETRY

CASES WITH A 5-M MAJOR AXIS FOR BODY 1. BECAUSE

THE RADIATION PROBLEM IS SYMMETRIC, THE RADIATION

COEFFICIENTS WILL BE THE SAME FOR BODY 2 AND ARE

THEREFORE OMITTED.

axis lengths for case 4. Because the major-to-minor axis ratio

remains the same, the change in hydrodynamics is coming from

the size of the device rather than the elliptical shape. For each

1-m decrease in the major axis, the added mass drops by roughly

30%; however, the displaced volume decreases at a faster rate

and thus the nondimensional radiation coefficients increase with

a smaller major axis. The heave-wave excitation forces, shown

in Fig. 19, decrease anywhere from 20% to 50% for each 1-m reduction in the major axis in the lower wave frequency range. But

when the heave-wave excitation force is normalized by the heave

restoring coefficient, which scales with the water plane area, the

normalized values increase as the major axis is reduced, implying the reduction in the heave-wave excitation force scales nonlinearly with the major axis. As the wave frequency increases,

the excitation force magnitudes for each case begin to converge

toward one another until reaching the wave frequency that results

in a wavelength nearly equal to the body length.

ACE CALCULATIONS

The performance values used to compare the fixed- and

variable-geometry attenuator WECs are the average climate

capture width (ACCW), the characteristic capital expenditure

(CCE), and the ACE metrics that were established for the WEP

[12]. The ACCW is a measure of the power capture efficiency of

a WEC. It is defined as the weighted average of the mechanical

absorbed power across multiple sea states, divided by the incident wave energy flux per meter crest width, CP, as shown in the

following equation:

ACCWj =

∑

n

i=1Ξi jhAP(i)i

hCP(j)i

(4)

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Heave Wave-Excitation Force Magnitude, |X3|/ gDl, [-]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Body 1, a/b = 1.0

Body 1, a/b = 1.3

Body 1, a/b = 2.0

Body 1, a/b = 4.0

Body 2, a/b = 1.0

Body 2, a/b = 1.3

Body 2, a/b = 2.0

Body 2, a/b = 4.0

(a) Heave-Wave Excitation Force Magnitude |X3|

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

-1.0

-0.8

-0.6

-0.4

-0.2

 0.0

 0.2

 0.4

 0.6

 0.8

 1.0

Heave Wave-Excitation Force Phase, 3

/ , [-]

-1.0

-0.8

-0.6

-0.4

-0.2

 0.0

 0.2

 0.4

 0.6

 0.8

 1.0

Body 1, a/b = 1.0

Body 1, a/b = 1.3

Body 1, a/b = 2.0

Body 1, a/b = 4.0

Body 2, a/b = 1.0

Body 2, a/b = 1.3

Body 2, a/b = 2.0

Body 2, a/b = 4.0

(b) Heave-Wave Excitation Force Phase φ3

FIGURE 17. HEAVE-WAVE EXCITATION FORCE MAGNITUDE

AND PHASE FOR BOTH BODIES AND EACH OF THE VARIABLEGEOMETRY ASPECT RATIOS WITH A 5-M MAJOR AXIS.

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

3

4

5

6

7

8

9

10

11

12

13

14

Heave Radiation Added Mass, 33

/ Vd, [-]

0

1

2

3

4

5

6

7

8

9

10

11

Heave Radiation Wave Damping, 33

/ Vd, [-]

33, D 5 m

33, D 4 m

33, D 3 m

33, D 2 m

33, D 5 m

33, D 4 m

33, D 3 m

33, D 2 m

FIGURE 18. HEAVE RADIATION ADDED MASS, μ33, AND

WAVE DAMPING, λ33, FOR VARIABLE-GEOMETRY CASE 4

WITH VARYING MAJOR AXIS LENGTH.

where n is the number of sea states; Ξi j is the scaling factor for

sea states, i, at each location j; hAP(i)i is the average mechanical power absorbed by the WEC for each sea state, i; and hCP(j)i

is the incident average annual wave energy flux for site j. The

seven sites and associated scaling factors used in the WEP are

shown in Table 3 [12].

The CCE is an estimate of the capital cost required to build

This report is available at no cost from the National Renewable Energy Laboratory at www.nrel.gov/publications. 8
0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Heave Wave-Excitation Force Magnitude, |X3|/ gDL, [-]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Body 1, D 5 m

Body 1, D 4 m

Body 1, D 3 m

Body 1, D 2 m

Body 2, D 5 m

Body 2, D 4 m

Body 2, D 3 m

Body 2, D 2 m

(a) Heave-Wave Excitation Force Magnitude |X3|

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

Wave Angular Frequency, , [rad/s]

-1.0

-0.8

-0.6

-0.4

-0.2

 0.0

 0.2

 0.4

 0.6

 0.8

 1.0

Heave Wave-Excitation Force Phase, 3

/ , [-]

-1.0

-0.8

-0.6

-0.4

-0.2

 0.0

 0.2

 0.4

 0.6

 0.8

 1.0

Body 1, D 5 m

Body 1, D 4 m

Body 1, D 3 m

Body 1, D 2 m

Body 2, D 5 m

Body 2, D 4 m

Body 2, D 3 m

Body 2, D 2 m

(b) Heave-Wave Excitation Force Phase φ3

FIGURE 19. HEAVE-WAVE EXCITATION FORCE MAGNITUDE AND PHASE FOR BOTH BODIES FOR THE VARIABLEGEOMETRY ASPECT RATIO EQUAL TO 4 WITH VARYING MAJOR AXIS.

the WEC hull, but does not include other costs with balance of

system, such as mooring lines or PTO. The attenuator model

is considered to be at a low technology readiness level (TRL),

meaning the PTO systems, mooring lines, power cables, energy

storage devices, and other nonload-bearing components are too

variable in cost to estimate the LCOE at this stage of development. Operations and maintenance costs are also not included

because they are difficult to estimate due to the limited availability of deployment data. The CCE is defined as the cost to buy the

raw material and manufacture the load-bearing structure of the

device, which is calculated from the following formula:

CCE =

N

∑

k=1

mk· MMCk (5)

where k is the material index, N denotes the number of key structural materials, mk

is the total mass of material k, and MMCkis

the manufactured material cost per unit mass of material k. The

MMCk values used in this study are shown in Table 4 [12]. Each

hull geometry has a draft equal to half the diameter, meaning it

sits halfway in the water. For a floating body to maintain this poTABLE 3. SCALING FACTORS, ΞIJ , FOR EACH OF THE SEVEN

U.S. SITES USED FOR EVALUATION OF THE ATTENUATOR

WEC. THE SCALING FACTORS ARE WRITTEN AS PERCENTAGES.

Site Location SS1 SS2 SS3 SS4 SS5 SS6

Alaska 24.3 33.2 7.5 20.0 2.4 1.2

Washington 13.7 27.7 4.1 33.8 2.2 4.5

North Oregon 15.5 30.7 5.6 34.4 3.7 4.2

Oregon 17.5 26.8 5.8 29.5 3.4 5.4

North California 20.7 23.0 1.2 46.6 1.6 6.4

South California 15.2 27.0 1.4 39.1 1.0 9.5

Hawaii 32.8 24.5 0.1 13.3 0.0 1.3

sition relative to the still water line, the mass of the device must

be equal to that of the displaced fluid water. The mass of SteelA36 for each device was set equal to the mass of displaced water

from the inner cylinder only.

The ACE metric is obtained by dividing the ACCW by CCE,

ACE =

ACCW

CCE , and is a cost-to-benefit ratio that primarily expresses the power absorption efficiency to the capital cost of

building the device. The ACE metric was developed for the WEP

competition to evaluate low TRL devices at the initial development stage. ACE does not take into account the power conversion losses, PTO loads, fatigue damage, or the costs of any items

beyond the building of the WEC hull.

Variable-Geometry ACCW, CCE, ACE, and PTO Torque

Case 1 for each major axis has no variable-geometry modules, thus the mass of the displaced water equals the total steel

mass of the device. For cases 2–4, for each major axis, the mass

of the variable-geometry modules was calculated by multiplying the circumference of the ellipse by a 20-mm thickness and

the body length, L = 25 m, to estimate the bag volume and was

multiplied by a density of 1,400 kg/m3to estimate the bag mass.

Given the inflatable bags have been assumed to be pressurized

by air, the weight required to offset the additional buoyancy has

been assumed to come from seawater ballast. As the extra weight

is assumed to come from ballast water, which can be pumped in

TABLE 4. MANUFACTURED MATERIAL COST (MMC) OF THE

KEY STRUCTURAL MATERIALS COMPOSING THE HULL.

Material, k MMCk [$/kg] Density, ρk, [kg/m3]

Steel-A36 3.00 7850

Coated Fabric 9.50 1400

This report is available at no cost from the National Renewable Energy Laboratory at www.nrel.gov/publications. 9
and out of ballast chambers, no additional cost has been added to

the CCE. However, for case 4 with major axis, a, equal to 3 m, 4

m, and 5 m, the volume of ballast water required to meet the desired draft was more than the inner cylinder could accommodate.

Therefore, the volumetric ballast water limits for each major axis

have been marked by a red ‘x’ in Figure 20b.

The resulting ACCW and CCE for each geometric configuration has been plotted in Figure 20a, with the resulting ACE

metric plotted in Figure 20b. The ACCW values in Figure 20a

show how efficient each geometric configuration is at capturing

the incident wave power. The resulting ACCW values are nearly

the same for each set of a, which demonstrates the variablegeometry WECs are capturing nearly the same amount of wave

power, while the structural mass and associated CCE are decreasing nonlinearly with aspect ratio. The results for ACCW provide evidence that the major axis is the dominant parameter that

defines the power capture potential, and the variable-geometry

components would be well-suited to maintain the planform area.

The ACE metric compares the power capture efficiency to the

structural capital costs and, as shown in Figure 20b, the ACE

metric increases with the aspect ratio, a/b. In addition, the ACE

metric also has a step increase when decreasing the major axis of

the elliptical body; however, because the power output decreases

with the major axis length (Figure 20a), the only way to have

an increase in ACE is to have a greater reduction in CCE. But,

because ACE only considers up-front capital costs, the perceived

gains from having multiple smaller units may be counteracted

by larger operational and maintenance costs when compared to

having only one or two larger units. The weighted average PTO

torque, across the WEP sea states, is shown in Figure 21. Similar to the ACCW, the average PTO torque stays fairly constant

across the aspect ratios for each major axis; however, the PTO

torque is significantly greater for a = 5 m, and no explanation

has yet to be identified by the authors. Overall, the results do

imply that the variable-geometry cases could potentially provide

the same power capture efficiency as fixed bodies with reduced

capital costs.

CONCLUSIONS

The authors investigated a novel WEC device concept

that combined an attenuator WEC with variable-geometry

modules. The variable-geometry modules consisted of two

identical inflatable bags, or floats, that spanned the length of

each body and could inflate or deflate to change the device

water plane area. The variable-geometry floats were shown to

provide greater control over the device hydrodynamics that can

be used to emphasize either power absorption or load shedding.

However, the shape of the variable-geometry floats was modeled

using rigid body hydrodynamics with an idealized shape, and

further research is required to determine the flexibility of the

bag material, final submerged shape, and the influence of the

1 1.5 2 2.5 3 3.5 4

Aspect Ratio, a/b, [-]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Average Climate Capture Width, ACCW, [m]

0.00

0.15

0.30

0.45

0.60

0.75

0.90

1.05

1.20

1.35

1.50

Characteristic Capital Expenditure, CCE, [$M]

ACCW, a 5 m

ACCW, a 4 m

ACCW, a 3 m

ACCW, a 2 m

CCE, a 5 m

CCE, a 4 m

CCE, a 3 m

CCE, a 2 m

(a) Average Climate Capture Width, ACCW, and Characteristic Capital Expenditure, CCE

(b) ACE Metric Value

FIGURE 20. THE ACCW, CCE, AND ACE METRICS FOR EACH

LENGTH OF A AND ASPECT RATIO A/B.

FIGURE 21. WEIGHTED AVERAGE PTO TORQUE FOR EACH

LENGTH OF A AND ASPECT RATIO A/B.

flexible bag deformation on the dynamic response of the internal

rigid body. The authors expect that improved control over the

operational design loads, provided by the variable-geometry

modules, has the potential to improve the capacity factor of the

device in larger sea states. The current analysis has investigated

device performance only in the irregular waves defined by the

WEP, but additional optimization will be required under a sitespecific joint probability distribution function and survival sea

This report is available at no cost from the National Renewable Energy Laboratory at www.nrel.gov/publications. 10
states. Furthermore, the PTO was modeled as a simple passive

linear damper, but power capture could be further improved if

a more advanced control strategy and PTO combination was

utilized—one that allows for bidirectional energy flow. Future

work remains to evaluate the structural and actuator requirements to determine if the variable-geometry module design can

be effectively controlled without adding significant capital and

operational costs that would result in a minimal reduction in the

LCOE.

ACKNOWLEDGMENT

This work was authored by the National Renewable Energy

Laboratory, operated by Alliance for Sustainable Energy, LLC,

for the U.S. Department of Energy (DOE) under Contract No.

DE-AC36-08GO28308. Funding provided by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Water Power Technologies Office. The views expressed in

the article do not necessarily represent the views of the DOE

or the U.S. Government. The U.S. Government retains and the

publisher, by accepting the article for publication, acknowledges

that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published

form of this work, or allow others to do so, for U.S. Government

purposes.

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