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Contract No. DE-AC36-08GO28308

Application of the Most Likely 

Extreme Response Method for 

Wave Energy Converters

Preprint

Eliot Quon, Andrew Platt, Yi-Hsiang Yu,

and Michael Lawson

National Renewable Energy Laboratory

Presented at the 35th International Conference on Ocean, Offshore 

and Arctic Engineering (OMAE 2016)

Busan, South Korea

June 19‒24, 2016

Conference Paper

NREL/CP-5000-65926

July 2016 
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APPLICATION OF THE MOST LIKELY EXTREME RESPONSE METHOD FOR WAVE

ENERGY CONVERTERS

Eliot Quon

National Renewable Energy Laboratory

Golden, CO 80401

Email: Eliot.Quon@nrel.gov

Andrew Platt

National Renewable Energy Laboratory

Golden, CO 80401

Email: Andrew.Platt@nrel.gov

Yi-Hsiang Yu∗

National Renewable Energy Laboratory

Golden, CO 80401

Email: Yi-Hsiang.Yu@nrel.gov

Michael Lawson

National Renewable Energy Laboratory

Golden, CO 80401

Email: Michael.Lawson@nrel.gov

ABSTRACT

Extreme loads are often a key cost driver for wave energy converters (WECs). As an alternative to exhaustive Monte

Carlo or long-term simulations, the most likely extreme response

(MLER) method allows mid- and high-fidelity simulations to be

used more efficiently in evaluating WEC response to events at

the edges of the design envelope, and is therefore applicable

to system design analysis. The study discussed in this paper

applies the MLER method to investigate the maximum heave,

pitch, and surge force of a point absorber WEC. Most likely

extreme waves were obtained from a set of wave statistics data

based on spectral analysis and the response amplitude operators

(RAOs) of the floating body; the RAOs were computed from a

simple radiation-and-diffraction-theory-based numerical model.

A weakly nonlinear numerical method and a computational fluid

dynamics (CFD) method were then applied to compute the shortterm response to the MLER wave. Effects of nonlinear wave and

floating body interaction on the WEC under the anticipated 100-

year waves were examined by comparing the results from the

linearly superimposed RAOs, the weakly nonlinear model, and

CFD simulations. Overall, the MLER method was successfully

applied. In particular, when coupled to a high-fidelity CFD analysis, the nonlinear fluid dynamics can be readily captured.

∗Address all correspondence to this author.

KEYWORDS

Wave energy; extreme conditions; most likely extreme response;

computational fluid dynamics

INTRODUCTION

The extreme wave-induced loads that wave energy converters (WECs) experience during both operational and survival sea

states are often the key cost driver for WEC designs [1, 2]. As

a result, it is essential to find an efficient pathway to reduce the

cost of energy for the WEC industry to be successful. In particular, the effect of nonlinear complex wave and floating body

interactions significantly influences the prediction of survival sea

state design loads. The prediction of those loads is a critical

step in the design process, and the application of mid- and highfidelity numerical methods (e.g., weakly-nonlinear-based models

and computational fluid dynamics [CFD] simulations [3]) and

experimental wave tank tests are often needed. Because the extreme load is a stochastic event and its occurrence is a matter

of chance caused by the body’s instantaneous position and a series of random waves, statistical analyses are commonly used to

determine the extreme design load [4].

The rapid improvement of computing technology has made

mid- and high-fidelity numerical methods available for design

applications; however, the required computational time is still

long for statistical analysis to search through all essential wave

environments in which extreme loads are most likely to occur.

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 1

Proceedings of the 35th International Conference on Ocean, Offshore and Arctic Engineering

OMAE 2016

June 19-24, 2016, Busan, South Korea

OMAE2016-54751
To improve the model efficiency and reduce required computational time, the shipping and offshore industry often perform

their mid- and high-fidelity simulations and experimental wave

tank tests using an equivalent design wave profile generated from

(linear) low-fidelity model solutions [5]. The equivalent design waves represent the sea state at which the extreme loads

are most likely to occur. Several methods have been used for

numerical and experimental naval architecture studies, including the most likely wave (MLW), most likely extreme response

(MLER), most likely response wave (MLRW), conditional random response wave (CRRW) [6,7], and design load generator [8]

approaches. These methods assume the nonlinear response of a

floating vessel can be approximated by a linearized response, and

the nonlinear response is a small perturbation from the linear solution.

Yu et al. [9] applied a similar approach to develop a framework for analyzing WEC extreme design loads. The study included a series of Monte-Carlo-type linear time-domain simulations to determine the sea state at which extreme loads occur and

a set of CFD simulations in regular waves with a statistically determined 100-year maximum wave height. Still, the method did

not account for the influence from the body instantaneous position and the series of random waves.

The objective of this paper is to improve the modeling

framework and methodology used for predicting the WEC design load under survival wave conditions. The MLER method

was applied in the study to model a floating ellipsoid. MLER

waves were generated from the extreme wave statistics data and

the linear response amplitude operators (RAOs) were obtained

from a simple radiation-and-diffraction-method-based numerical

model known as the Wave Energy Converter Simulator, or WECSim. The resulting waves were then used in the nonlinear WECSim and CFD simulations. For this study, the high-fidelity CFD

analysis was performed using an unsteady Reynolds-averaged

Navier-Stokes (URANS) methodology, Star-CCM+. The maximum heave and pitch and the peak surge force were analyzed.

We compared the results from linear RAO-based solutions to

those from nonlinear WEC-Sim and Star-CCM+ simulations to

examine the influence of nonlinear wave-body interaction on the

floating body dynamics. Finally, a discussion on further applications of the MLER method for extreme condition modeling is

presented.

MODEL GEOMETRY AND TEST CONFIGURATIONS

The body tested in this study is an ellipsoid with a cross

section characterized by semimajor and semiminor axes of 5.0 m

and 2.5 m in the wave propagation and normal directions, respectively (Fig. 1). The ellipsoid is at its equilibrium position with its

origin located at the mean water surface. The mass of the body is

set to 1.342×105 kg, and the center of gravity is located 2 m below the origin. As shown in Fig. 1, two test configurations were

considered in the study. In Configuration A, we only allowed the

body to move freely in heave. In Configuration B, the body is

allowed to rotate around its center of gravity in addition to the

FIGURE 1. SIDE VIEW OF THE NOTIONAL POINTABSORBER WEC EVALUATED IN THIS STUDY; CONFIGURATIONS A AND B AND THEIR ASSOCIATED DEGREES

OF FREEDOM ARE ILLUSTRATED IN THE TOP LEFT AND

RIGHT BOXES.

FIGURE 2. ONE-HUNDRED-YEAR CONTOUR FROM THE

COASTAL DATA INFORMATION PROGRAM STATION

AND NATIONAL DATA BUOY CENTER BUOY (ADOPTED

FROM [10]).

heave motion specified in Configuration A. Heave and pitch degrees of freedom were considered because these displacements

are typically limited with an end stop in power take-off (PTO)

systems. For simplicity, the behavior of the PTO system was not

modeled.

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 2
TABLE 1. WAVE ENVIRONMENTS ALONG THE 100-YEAR

CONTOUR FROM THE COASTAL DATA INFORMATION

PROGRAM STATION AND NATIONAL DATA BUOY CENTER BUOY.

Case Significant Energy Peak

Number Wave Height Hs Period Te Period Tp

1 5 m 7.6 s 8.2 s

2 7 m 10.1 s 11.7 s

3 9 m 13 s 15.1 s

4 10.5 m 17.7 s 20.5 s

5 8 m 22.3 s 25.9 s

EXTREME SEA STATES

For the extreme wave environment, we used the wave statistics data based on the measurements from the Coastal Data Information Program #128 station and the National Data Buoy Center

#46212 buoy near Humboldt Bay, California. The latter is one of

the potential wave energy test sites in the United States. Figure 2

shows the sea state measurements and the estimated 100-year

contour [10]. We then selected five representative wave environments along the 100-year contour to perform our short-term

response analysis using the MLER method, listed in Tab. 1.

METHODOLOGY

To generate the wave profile that produces the largest likely

response of the WEC device, the RAO is calculated using linear

WEC-Sim for each of the degrees of freedom of interest. The

RAO is then combined with the Brettschneider spectrum, which

corresponds to each of the five extreme wave environments listed

in Tab. 1, using the MLER method to produce the underlying

wave profile that leads to the largest response in each degree of

freedom. The resulting wave profile is then used in nonlinear

WEC-Sim and high-fidelity URANS simulations to fully model

the system’s nonlinear hydrodynamics. This section describes all

the numerical methods used in the study, including the MLER,

WEC-Sim, and URANS methods.

Most Likely Extreme Response

The MLER method uses the linear RAO of the device and

the spectrum for the sea state of interest to produce a wave profile

that gives the largest response. Because it uses linear theory, the

underlying assumption is that the higher-order effects are small

in comparison to the linear effects. Starting with a wave time

series described by

Z(t) =

N

∑

n=1

az,n [Vn cos(ωnt −knx) +Wn sin(ωnt −knx)], (1)

where az,n is the spectral amplitude, p

S(ωn) ∆ω, and the coefficients Vn and Wn are independent standard normal random variables containing white Gaussian noise and phase randomization.

The corresponding response can be written as

ζ (t) =

N

∑

n=1

az,n |ΦR,n|[Vn cos(ωnt −knx+θR,n)

+Wn sin(ωnt −knx+θR,n)] (2)

where |ΦR,n| and θR,n are the RAO magnitude and phase angle,

respectively.

The incident wave profile that yields the largest response of

the floating body can then be derived using information from the

wave spectrum and the response spectrum to replace the values

of Vn and Wn. The response spectrum is calculated by combining

the wave spectrum with the RAO of the body as

SR(ωn) = |ΦR(ωn)|

2

S(ωn) (3)

where ΦR(ωn) is the RAO for a given degree of freedom. Using this information and the derivations presented in Dietz [6]

and Drummen [11], a set of equations are derived by way of a

Slepian process to describe the reaction of a vessel to an incoming wave train. This yields a set of equations that describe the

maximum response of the vessel conditioned by the RAO. New

values for Vn and Wn conditioned with the vessel response spectrum are given as

V R,n =|ΦR,n|Mdaz,n

cos(θR,n)



m2m0 −m2

1



×[(m2 −ωnm1) +ωR (ωnm0 −m1)] (4)

WR,n =|ΦR,n|Mdaz,n

sin(θR,n)



m2m0 −m2

1



×[(m2 −ωnm1) +ωR (ωnm0 −m1)] (5)

where Md is the desired response amplitude, ωR = m1/m0 of the

response spectrum, and the response spectral amplitude defined

as aR,n ≡ |ΦR,n|az,n. The spectral moments of the response, m0,

m1, and m2, are given by

mi =

N

∑

n=1

ω

i

nSR(ωn)∆ω where i = 0,1,2,3,... (6)

Replacing Vn and Wn in Eqn. (1) with Eqns. (4) and (5), and

simplifying significantly, yields the underlying wave profile that

will produce the largest response

ZR,c(t) =

N

∑

n=1

az,nMdAR,n cos(ωnt −knx−θR,n) (7)

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where

AR,n ≡

|ΦR,n|az,n

m0m2 −m2

1

[(m2 −m1ωn) +ωR (m0ωn −m1)] (8)

The corresponding response from Eqn. (2) becomes

ζR,c(t) =

N

∑

n=1

Mdaz,n |ΦR,n|AR,n cos(ωnt −knx) (9)

Note that the phase term present in the wave profile of Eqn. (7) is

perfectly canceled out by the RAO phase giving a maximum response amplitude at t = 0. It is important to note that this method

produces a shaped wave profile that is independent of the desired

magnitude of the response, Md. Therefore, it can be rescaled

to waves of different amplitudes for the purposes of generating

short-term statistical loads. In this case, the peak values of the

underlying wave amplitudes from Eqn. (7) were rescaled to simulate the maximum 100-year individual wave height, H100, for

the starting sea state, which is most likely 1.9 times the significant wave height for the 100-year wave given in Tab. 1, assuming

a storm lasts for 3 hours with 1000 waves [12].

WEC-Sim

WEC-Sim is an open-source system dynamics modeling

tool developed for simulating multibody WECs [13]. The dynamic response is calculated by solving the equation of motion

about its center of gravity, which can be given as

(m+A∞)X ̈ =−

Zt

0

K(t −τ)X ̇(τ)dτ

+Fext +Fvis +Fres +FPTO +Fmoor

(10)

where A∞ is the added mass matrix at infinite frequency, X is the

(translational and rotational) displacement vector of the body, m

is the mass matrix, Fext is the wave excitation force, FPTO is the

force from the PTO system, Fmoor is the mooring force, Fvis is the

viscous damping force vector, and Fres is the net buoyancy restoring force. The convolution integral term, −

R t

0 K(t −τ)X ̇(τ)dτ is

calculated following the Cummins’ equation [14], where K is the

impulse response function.

In this study, the added-mass matrix, impulse response function, and diffraction component in Fext were computed using

WAMIT [15], a frequency-domain potential flow solver. Linear WEC-Sim was used to compute the RAOs that were used in

the MLER method. In linear WEC-Sim simulations, Fres and the

Froude-Krylov force component in Fext, generated by the undisturbed incident waves, were also obtained from WAMIT. Nonlinear WEC-Sim was used to simulate the focused wave scenario

and model the body system dynamics using the MLER method.

TABLE 2. URANS NUMERICAL SETTINGS.

Pressure-velocity coupling Transient SIMPLE algorithm

Time-marching scheme Second-order implicit

Convection scheme Second-order upwind

Free surface capturing Volume of fluid model

Wave absorber Sponge-layer damping zone

Device motion Overset scheme

The nonlinear WEC-Sim followed a weakly nonlinear approach,

in which the Fres and Froude-Krylov force were computed by

integrating the theoretical static and dynamic components of the

hydrodynamic pressure for each panel along the wetted body surface at each time step [16]. Wheeler stretching was also applied

through coordinate mapping for wave kinematics to satisfy the

boundary condition on the water surface based on the instantaneous wave elevation [17].

URANS Simulations

A finite-volume URANS solver, StarCCM+ [18], was used

to simulate focused waves defined by the MLER approach and

model the body system’s dynamic response. The method discretizes the time-accurate Navier-Stokes equations of motion

over the computational domain and solves the system of linear

equations in the time domain. This solver and numerical model

were successfully applied previously for a heaving device in extreme regular waves [9]. The numerical settings used in this

study are listed in Tab. 2. For simplicity, turbulence modeling

was not employed and the fluid flow was assumed to be laminar. An overset methodology was applied to simulate device

motion wherein two layers of meshes were implemented, one

for the floating body subject to kinematic constraints and a second for simulating the background wave field. The two meshes

are solved simultaneously to capture unsteady wave-body interactions. Figure 3 illustrates the present overset configuration.

The stationary background mesh cells have been colored as inactive (cell type=-1), no solution performed; active (0), solution

performed at each time step; donors (1), active, and provides a

solution to the moving mesh; intermediate (2); and receptors (3),

active, and receiving a solution from the moving mesh.

In the case of a focused wave, a single reference length (e.g.,

the wavelength) is not well defined. For this work, a characteristic length (L) was chosen as the wavelength of the largest

amplitude wave in the wave train, which has a frequency corresponding to the peak period. Spatial resolution near the water

surface is 80 cells per L in the downwave direction (Nx = 80) and

10 cells per Hsin the normal direction (Nz = 10). This resulted

in an aspect ratio of 4 (Fig. 3), and additional refinement was not

performed in the normal direction to keep the aspect ratio ∼O(1).

Grid resolution near the device was set to 0.25 m, providing apThis report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 4
FIGURE 3. OVERSET MESH CONFIGURATION.

proximately 100 points along a surface cross-sectional curve. A

water depth (d) of 70 m was modeled. To absorb outgoing and

reflected waves without generating additional numerical disturbances, a 1.5 L-long damping zone was also included. The entire

computational domain extended from -600 m to 600+1.5L m in

the downwave (x) direction, from -L/2 to L/2 in the lateral (y)

direction, and from −d to d in the vertical direction (z).

The MLER wave spectra were specified as components of

a linear superposition wave on the inflow boundary and used

to initialize the solution on the interior of the computational

domain. Time marching is performed with a second-order algorithm and a relatively small time-step size chosen to minimize temporal error. Using the reference length L and the peak

period Tp, a reference wave speed (U) is obtained. An estimate of the maximum Courant number is therefore provided by

C = U∆t/∆x = (L∆t)/(Tp∆x) and a time-step size is selected to

achieve C ≤ 0.25. For this case, ∆t = 0.02 was chosen for an

estimated C ≈ 0.12, or 15,000 steps per Tp.

SHORT-TERM RESPONSE

The results from the MLER application study are presented

in this section and include two sets of URANS simulations. First,

a series of simulations for a single sea state (Case 3 - Configuration A) were performed without the float to assess mesh resolution requirements. Then, simulations for all five sea states selected along the 100-year contour (Tab. 1) were conducted using

both Configurations A and B. The URANS results were compared to those from weakly nonlinear WEC-Sim simulations and

theoretical solutions from linear RAOs.

URANS Mesh Resolution Study

A convergence study was performed to verify that the solution was not sensitive to discretization error and that the specified

wave solution on the inlet boundary is able to propagate through

the entire computational domain without accumulating significant numerical errors. Sea state 3 (Configuration A), characterized by a relatively steep, large amplitude, focused wave was

FIGURE 4. SENSITIVITY TO GRID REFINEMENT OF SEA

STATE 3 (CONFIGURATION A) SIMULATED WITH URANS

WITHOUT A DEVICE IN THE WATER.

TABLE 3. URANS MESH CONVERGENCE.

Nx Zmax rel. error [%] M0 rel. error [%]

40 7.90 m - 158.3 -

80 8.73 m 10.5% 161.3 1.90%

160 9.06 m 3.78% 161.2 0.06%

simulated without a device in the water. URANS solutions for

the propagated wave at three levels of grid refinement are plotted in Fig. 4 in comparison with the theoretical MLER result.

The three URANS results show that numerical simulations are

convergent (Nx = 80 and 160 curves are visually indistinguishable) and demonstrate excellent resolution of the peak location

and amplitude. Although the linear MLER predicts a symmetric wave elevation profile, the URANS method permits asymmetry from the nonlinearity of steep waves. Table 3 shows the

convergence of the maximum wave elevation (Zmax) and spectral

content in terms of the zeroth spectral moment (M0), with the relative error defined as the change in a given quantity compared to

the previous level of refinement. When using a baseline resolution of 80 cells per reference length (Nx = 80), the spectral content is nearly invariant. Further refinement (Nx = 160) reduces

the relative error in wave height to 4%, which does not justify

the accompanying two-fold increase in computational cost. The

baseline grid has been used for all subsequent simulations.

MLER Applications

The five sea states identified in Tab. 1 were simulated with

the ellipsoid float geometry. For each case, Configurations A and

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 5
TABLE 4. SUMMARY OF URANS SIMULATED RESULTS.

Zmax ζmax θmax Fx,min Fx,max

[m] [m] [deg] [kN] [kN]

Config.

A

Sea 1A 3.78 2.70 0 -504 366

Sea 2A 6.44 5.52 0 -386 702

Sea 3A 8.73 7.89 0 -460 607

Sea 4A 10.9 9.85 0 -390 465

Sea 5A 8.96 8.49 0 -301 219

Config.

B

Sea 1B 3.69 3.37 29.0 -602 690

Sea 2B 4.82 5.11 23.1 -392 564

Sea 3B 7.88 9.71 52.1 -648 1238

Sea 4B 10.5 10.4 37.2 -527 744

Sea 5B 7.41 6.92 17.1 -492 155

B (with one and two degrees of freedom, respectively) were considered. The simpler case included only vertical motion (heave),

whereas the latter permitted both heaving and pitching motion

around the body’s center of gravity. Two different sets of MLERfocused waves were generated for the two float configurations. In

Configuration A, the RAO was conditioned to achieve the maximum heave response, whereas in Configuration B the RAO was

conditioned for maximum pitch response. The focused waves

in the latter cases were steeper, with a maximum wave elevation

occurring prior to the instant of maximum response.

Results for all calculations are summarized in Tab. 4. With

only heave motion, the largest displacement (ζmax) was observed

in Sea State 4, whereas the largest surge force (Fx) was observed

in Sea State 2. In all cases, the float acted as a wave follower

and the vertical displacement of the device never exceeded the

surface elevation; however, when pitching motion was permitted the float dynamics changed significantly, and both the largest

pitch (θmax) and maximum load were observed in Sea State 3.

Configuration B saw significantly higher surge forces in nearly

all cases because the device was constrained in the surge direction, and any nonzero pitch increased the frontal area of the float.

Overall trends for the different methodologies are illustrated

in Figs. 5–7. The heave and pitch responses are normalized by

the significant wave height (Hs) and maximum wave slope angle, respectively. The maximum wave angle was directly evaluated from the MLER coefficients using the spatial derivative of

Eqn. (7) and corresponds to the wave profile in the absence of

the device.

In the heave-only configuration (Configuration A), the

weakly nonlinear WEC-Sim and fully nonlinear URANS simulations showed that the amplitude of the device response was reduced for steep and short-period focused MLER waves (Fig. 5).

FIGURE 5. COMPARISON OF (NORMALIZED) HEAVE RESPONSE FOR CONFIGURATION A.

For peak periods ≤ 15 s, the URANS response is between 65–

92% of the weakly-nonlinear response. For peak periods larger

than 20 s, the heave responses of the device are similar to

those obtained from the linear RAO and nonlinear WEC-Sim approaches.The larger URANS heave response in Sea State 5 corresponds to an increased wave crest height associated with relatively shallow water. The nondimensional depth, d/gT2

p

, was an

order of magnitude smaller in Sea State 5 compared to Sea State

1, which lies in the deep water regime.

Figure 6 illustrates the heave response for Configuration

B, which permits both heaving and pitching motion. The normalized heave responses predicted by the linear RAO and nonlinear WEC-Sim are very similar. As before, the linear RAO

slightly overpredicts the heave response for short peak periods.

In contrast, the URANS calculations predict significantly different trends. The three regions that have been highlighted in

Fig. 6 correspond to three different dominant physical behaviors. In Region 1, the relatively short peak period corresponds

to a very steep wave that exceeds the breaking wave limit of

Hs/Lre f ≈ 0.14. The URANS simulations for Seas 1B and 2B

capture the wave breaking, which disrupts the wave train and alters both the wave peak amplitude and location; these responses

cannot be directly compared to the linear RAO and nonlinear

WEC-Sim. Simulations in Region 2 do not break but exhibit

strong fluid-body interactions that are captured by URANS; in

contrast, the linear RAO and nonlinear WEC-Sim do not account

for the deformation of the wave surface caused by the device motion. Finally, Region 3 simulations approximate wave-follower

behavior and similar results between all methods are expected.

The most notable differences between the various simulation methods were observed in the pitch response for Configuration B (Fig. 7). Because the float orientation is expected to approximately track the wave elevation profile, pronounced maxima and minima in pitch were observed prior to and after the

focused wave passage; both follow distinct trends and therefore

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 6
FIGURE 6. COMPARISON OF (NORMALIZED) HEAVE RESPONSE FOR CONFIGURATION B.

FIGURE 7. COMPARISON OF (NORMALIZED) PITCH RESPONSE FOR CONFIGURATION B; (+) AND (-) DENOTE

THE MAXIMUM POSITIVE AND NEGATIVE RESPONSES.

have been presented. In the linear case, the maximum pitch response is approximately 1.5 times the maximum wave surface

elevation slope, and nearly invariant across all sea states. The

secondary response is a maximum for Sea State 1B, which has

the steepest focused wave; this response decreases linearly with

the peak period. As with the heave response in Configuration A,

the weakly nonlinear WEC-Sim model tends to reduce the amplitude of the maximum response for shorter-period sea states. Both

the primary and secondary weakly nonlinear WEC-SIM pitch responses are larger than predicted by the linear RAO, but maintain

the same trend for longer peak periods.

To further elucidate the float response under the various sea

states, traces of the wave surface elevation at the device location

and the device responses are plotted for Sea States 3 and 5 in

(a) Heave response - configuration A

(b) Heave response - configuration B

(c) Pitch response - configuration B

FIGURE 8. RESULTS FOR SEA STATE 3 (Hs=9m; Tp=15.1s).

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 7
(a) Heave response - Configuration A

(b) Heave response - Configuration B

(c) Pitch response - Configuration B

FIGURE 9. RESULTS FOR SEA STATE 5 (Hs=8m; Tp=25.9s).

Figs. 8 and 9. For Configuration A in both sea states, the float

response never deviated significantly from the simulated wave elevation, acting as a wave follower. In Configuration B, the input

wave spectra were designed to produce the peak pitch response

at t = 150 s. As expected, the linear RAO and nonlinear WECSim simulations reflected this for both cases, predicting a maximum response where the wave slope was a maximum; however,

from both the heave and pitch responses in Sea State 3 (Fig. 8),

the URANS responses for Configuration B deviated significantly

from wave-follower behavior because of nonlinear fluid-body interactions. The peak pitch response occurred about 3 s earlier, coinciding with the passage of the focused wave crest. As shown in

Fig. 10, the float was partially out of the water and the body pitch

response exceeded the range of linear approximation. These differences between methodologies are expected because the wave

field was altered by the motion of the body, a situation that is not

considered in the nonlinear WEC-Sim model.

In Sea State 5, the URANS response shows good agreement with the other methodologies (Fig. 9). The float exhibits

wave-follower behavior at all stages of the wave-train passage

(Fig. 11): the maximum and minimum heave response correspond to the passage of the focused wave crest and trough, respectively, and the maximum pitch response occurs near the instant at which the focused wave slope is a maximum.

DISCUSSION

The study demonstrated the use of the MLER method to

evaluate the dynamic response of an ellipsoid float under 100-

year extreme waves. Figure 12 shows the snapshots of water surface and the pressure distribution on the float for Configuration

B in Sea State 3. The application of the MLER method provided

a simple and efficient approach to evaluate the nonlinear extreme

response for the floating body, particularly when coupled with a

high-fidelity CFD simulation, in which highly nonlinear effects

on the system dynamics can be captured. High-resolution pressure distribution on the body surface data can then be utilized for

high-fidelity structural analyses.

In this study, Configuration A was used for verification, because the float heave response generally followed the wave elevation. The nonlinear wave-body interaction effect was essential

only in Sea State 1, where waves were relatively steep. On the

other hand, the pitch motion in Configuration B resulted in significant nonlinear interactions between the waves and the float.

The MLER waves maximized the wave slope so that the maximum pitch motion for the float was most likely to occur. Because

the waves along the 100-year wave contour were steep, particularly for shorter period sea states, test cases need to be carefully

selected to avoid the occurrence of wave breaking if that is not

essential to the response of interest.

Note that the maximum pitch angle was greater than 30

degrees for Configuration B in Sea States 1-3, which was beyond the limitations for linear theory. The inclusion of nonlinear restoring and Froude-Krylov forces in nonlinear WEC-Sim,

based on the instantaneous free surface, were effective for capturThis report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 8
(a) Maximum heave and pitch response (t = 146.8 s)

(b) Minimum heave response (t = 149.6 s)

(c) Subsequent pitching response (t = 151.6 s)

FIGURE 10. CONTOURS OF THE VOLUME FRACTION OF

WATER FOR SEA STATE 3, CONFIGURATION B.

ing the weakly nonlinear wave-body interaction. Still, the nonlinear WEC-Sim model did not consider the influence of body

motion on the water surface and failed to capture its effect on the

body dynamics.

The MLER method was applied successfully in this study;

however, this type of equivalent design wave method has been

shown to be less accurate when the load response depends heavily on the instantaneous wave train, vessel position, and its deformation, unless a random background wave was used [7]. Ultimately, these methods may work well for some responses of

interest but not for others, depending on the WEC design and

working principle. Further analysis is needed to evaluate the accuracy of the equivalent design wave approach, particularly for

cases in which complex nonlinear wave-body interactions are essential.

(a) Maximum heave response (t = 146.0 s)

(b) Minimum heave response (t = 149.2 s)

(c) Minimum heave response (t = 152.8 s)

FIGURE 11. CONTOURS OF THE VOLUME FRACTION OF

WATER FOR SEA STATE 5, CONFIGURATION B.

CONCLUSIONS

The MLER approach demonstrates a viable methodology

for identifying and analyzing extreme loads within the design

process. When coupled to a high-fidelity CFD analysis, nonlinear fluid dynamics may be readily captured even with simplified kinematics. Moreover, the device response to events at the

edges of the design envelope may be evaluated using a short-term

analysis. This analysis can potentially reduce the need for longterm simulations or tank tests that require the generation of thousands of wave events, thereby providing significant cost savings.

Future work will include longer simulations to further validate

and motivate application of the MLER/URANS approach. Additional investigations into the accuracy of the URANS methodology will be performed, particularly in Region 2 where strong

fluid-body interactions were observed. Finally, the influence

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 9
(a) Maximum positive surge force (t = 145.6 s)

(b) Float out of water (t = 147.2 s)

(c) Maximum negative surge force (t = 148.8 s)

(d) Maximum heave force (t = 149.2 s)

FIGURE 12. VISUALIZATION OF THE SEA SURFACE IN

SEA STATE 3, CONFIGURATION B; FLOAT SURFACE IS

COLORED BY PRESSURE.

of including additional degrees of freedom will be considered

and more detailed structural response analysis will be performed

through coupling with finite-element analysis methods.

ACKNOWLEDGMENT

This work was supported by the U.S. Department of Energy

under Contract No. DE-AC36-08GO28308 with the National

Renewable Energy Laboratory. Funding for the work was provided by the DOE Office of Energy Efficiency and Renewable

Energy, Wind and Water Power Technologies Office.

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