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Contract No. DE-AC36-08GO28308

Balancing Power Absorption 

and Structural Loading for an 

Assymmetric Heave WaveEnergy Converter in Regular 

Waves

Preprint

Nathan M. Tom

National Renewable Energy Laboratory

Farshad Madhi and Ronald W. Yeung

University of California at Berkeley

Presented at the ASME 2016 35th International Conference on 

Ocean, Offshore and Arctic Engineering (OMAE2016)

Busan, South Korea

June 19–24, 2016

Conference Paper

NREL/CP-5000-65656

July 2016
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BALANCING POWER ABSORPTION AND STRUCTURAL LOADING FOR AN

ASYMMETRIC HEAVE WAVE-ENERGY CONVERTER IN REGULAR WAVES

Nathan M. Tom∗

National Renewable Energy Laboratory

National Wind Technology Center

Golden, Colorado, 80401

Email: nathan.tom@nrel.gov

Farshad Madhi†

University of California at Berkeley

Department of Mechanical Engineering

Berkeley, California, 94720

Email: madhi@berkeley.edu

Ronald W. Yeung‡

University of California at Berkeley

Department of Mechanical Engineering

Berkeley, California, 94720

Email: rwyeung@berkeley.edu

ABSTRACT

The aim of this paper is to maximize the power-to-load ratio

of the Berkeley Wedge: a one-degree-of-freedom, asymmetrical,

energy-capturing, floating breakwater of high performance that

is relatively free of viscosity effects. Linear hydrodynamic theory was used to calculate bounds on the expected time-averaged

power (TAP) and corresponding surge restraining force, pitch

restraining torque, and power take-off (PTO) control force when

assuming that the heave motion of the wave energy converter

remains sinusoidal. This particular device was documented to

be an almost-perfect absorber if one-degree-of-freedom motion

is maintained. The success of such or similar future wave energy converter technologies would require the development of

control strategies that can adapt device performance to maximize energy generation in operational conditions while mitigating hydrodynamic loads in extreme waves to reduce the structural mass and overall cost. This paper formulates the optimal

control problem to incorporate metrics that provide a measure

of the surge restraining force, pitch restraining torque, and PTO

control force. The optimizer must now handle an objective function with competing terms in an attempt to maximize power capture while minimizing structural and actuator loads. A penalty

weight is placed on the surge restraining force, pitch restraining

torque, and PTO actuation force, thereby allowing the control fo-

∗Address all correspondence to this author.

†Ph.D candidate UC Berkeley, major field Ocean Engineering.

‡Director, Marine Mechanics Laboratory (MML), University of California at

Berkeley.

cus to be placed either on power absorption or load mitigation.

Thus, in achieving these goals, a per-unit gain in TAP would not

lead to a greater per-unit demand in structural strength, hence

yielding a favorable benefit-to-cost ratio. Demonstrative results

in the form of TAP, reactive TAP, and the amplitudes of the surge

restraining force, pitch restraining torque, and PTO control force

are shown for the Berkeley Wedge example.

INTRODUCTION

The Berkeley Wedge [1] is an asymmetric wave energy converter (WEC) and breakwater. It consists of an asymmetric

floater, a power take-off (PTO) system, and a support structure.

The particular shape of the floater, depicted in Fig. 1, was designed to experience minimal effects from viscosity in heave motion. The mounting structure limits the motion of the floater to

heave only. The PTO system implemented in the design is a linear permanent magnet generator (LPMG) [2] . When the damping of the LPMG is matched with the heave radiation damping of

the floater at resonance, there will be almost no radiated or transmitted waves and almost all of the incident wave energy will be

absorbed by the damping of the LPMG. The Berkeley Wedge

can be used near shore to provide electricity for local communities and act as a breakwater (concurrently) to protect the harbor

with very minimal environmental impact. It can also be attached

to offshore structures and floating platforms to provide electricity and protect the structure. In a recent study [3], the particular asymmetric shape of the Berkeley Wedge was implemented

in a coaxial wave energy converter (consisting of a fixed inner

cylinder and moving outer cylinder) to reduce the viscous effects

1

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on the heave motion of the outer cylinder. The straight-side of

the inner cylinder mimics the backside of the two-dimensional

Berkeley Wedge and its special shape has minimal viscous effects. The experimental testing revealed that the shape of the

Berkeley Wedge reduced the viscous damping on the heave displacement of the outer cylinder by 70%, resulting in an increase

in the heave displacement of the outer cylinder by more than

300%.

The success of such or similar future WEC technologies

will require the development of control strategies that can adapt

device performance to maximize energy generation in operational conditions while mitigating hydrodynamic loads in extreme waves to reduce the structural mass and overall cost [4].

Balancing these objectives offers an interesting design and control challenge. For example, they are in contrast to previous

works that solved the optimal control problem when focused

solely on maximizing the time-averaged power (TAP). The application of state-constrained optimization [5, 6] to WEC control

has gained significant traction recently as it provides the ability to include linear and nonlinear constraints. This optimization

has been pursued using calculus of variations [5], model predictive control [7–9], and pseudo-spectral methods [10–12]. If the

PTO and structural loads are not considered, the optimum WEC

trajectory follows that of complex conjugate control [13], which

is known to require a substantial amount of reactive power when

moving away from the resonance frequency. Suboptimal strategies that eliminate reactive power, notably latching [14] and declutching [15], have been proposed, yet still do not include a load

metric in the optimization. It can be expected that as the controller works to maximize the absorbed mechanical energy, the

growth rate in structural loads may exceed the growth in TAP. To

address this concern, this work incorporates the restraining loads

in the objective function of the optimization routine. As a result,

the optimizer must now balance the opposing contributions in an

attempt to obtain the largest power-to-load ratio.

This paper begins by describing the Berkeley Wedge device

concept. This is followed by construction of the heave timedomain equation of motion to provide the preliminaries for extension into its spectral representation. The upper and lower

bounds on the TAP, surge-restraining force, pitch-restraining

torque, and PTO actuator force are calculated while assuming

that the WEC motion was constrained but remains sinusoidal.

The upper bound was calculated assuming an optimum phase

between the heave wave-exciting force and heave velocity while

the lower bound assumes that the PTO system consists only of

a linear resistive damper, and in both cases the PTO force coefficients are constant and continuous throughout the wave cycle.

Next, pseudo-spectral control theory is reviewed followed by incorporating the surge-restraining force, pitch-restraining torque,

and PTO actuator force into the optimization problem. A penalty

weight is placed on the contributions to the objective function

from the restraining and PTO loads to allow the desired perforTable 1. GEOMETRIC VALUES OF THE BERKELEY WEDGE.

Water Depth, h, 1.5 m Draft, d, 0.7 m

Beam, b, 0.212 m Area, S, 0.0677 m2

Center of Gravity, xg, -0.0848 m Resonance, Tres, 1.25 s

mance to be achieved. The effect of including the restraining

loads on balancing power absorption and load shedding is first

explored by varying the penalty weight magnitudes and comparing against the known performance bounds. The time history of

WEC motion and PTO control force are presented to illustrate

how per-unit increases in TAP can exceed the per-unit increase

in restraining and PTO loads while having a minimal reactive

power requirement.

THE BERKELEY WEDGE

The Berkeley Wedge shape was designed to experience the

smallest effect from viscosity when encountering incident waves

and in motion. The motion of the floater (shown in Fig. 1) is restricted to heave only. The physical dimensions of the asymmetric floater were chosen to fit the model testing facility at the University of California at Berkeley. A detailed theoretical and experimental study [1] confirmed the effectiveness of the design in

reducing the viscous effect on the motion of the device, thereby

capturing almost all of the incident wave energy and providing a

calm water surface leeward of the asymmetric floater.

In this study, the authors examined the Berkeley Wedge with

the physical dimensions given in Table 1. The particular shape

of the floater (Fig. 1) can be obtained from:

F(y ̄)=0.05926(y ̄+1)

2+3.88147(y ̄+1)3−2.94074(y ̄+1)4

(1)

In Eqn. (1),  ̄x = F(y ̄) is a shaping function, and  ̄x = x/b and

y ̄ = y/d are nondimensional scales. In this equation,  ̄y can be

shifted to obtain different drafts. The hydrodynamics coefficients for the asymmetric floater were obtained from the twodimensional (2D) potential-flow code RWYADMXA [16] and

shown in nondimensional form in Fig. 2. The resonance frequency, as reported in Table 1, is the frequency of maximum

motion in regular-wave excitation. The natural frequency is the

frequency of oscillation with an initial displacement or impulsive

velocity. The two are close but not identical, primarily because

the hydrodynamic coefficients are frequency dependent [17].

TIME-DOMAIN-HEAVE EQUATION OF MOTION

The one-degree-of-freedom time-domain-heave equation of

motion is given by:

m ̈ζ2 (t) = fe2 (t) + fr22 (t) + fh (t) + fd (t) + fm (t) (2)

where t is time, m is the mass of the WEC,  ̈ζ2 is the heave acceleration, fe2 is the wave-exciting heave force caused by the incident

2

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−0.25 −0.20 −0.15 −0.10 −0.05 0.00 0.05



−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0.0



Figure 1. 2D SHAPE OF THE BERKELEY WEDGE WITH THE BEAM

(b) = 0.212 m AND DRAFT (d) = 0.7 m.

waves, fr22 is the wave radiation force caused by heave motion,

fh is the hydrostatic restoring force, fd is the drag force caused

by viscous effects, and fm is the mechanical force applied by the

PTO system.

The heave hydrostatic restoring force is given by:

fh(t) = −C22ζ2 (t), with C22 = ρgb (3)

where ρ is the fluid density, g is the gravitational acceleration,

b is the device beam length at the calm water line, and ζ2 is the

time-varying heave displacement.

The linear hydrodynamic wave-radiation heave force will be

represented in the time domain using the Cummins equation [18]

and is written as follows:

fr22(t) = −μ22 (∞)

 ̈ζ2 (t)−

Zt

−∞

Kr22 (t −τ)

 ̇ζ2 (τ)dτ (4)

where μ22 (∞) is the heave-added mass at infinite frequency,

and Kr22 is the heave radiation impulse response function, also

known as the memory function because it represents the wave

1σ ̄ = σ

p

b/g, μ ̄22 = μ22/ρb

2

,

 ̄λ22 = λ22/ρb

2σ, X ̄

2 = X2/ρgb, φ ̄2 =

φ2/π, μ ̄12 = μ12/ρb

2

,

 ̄λ12 = λ12/ρb

2

, X ̄

1 = X1/ρgb, φ ̄1 = φ1/π, μ ̄32 =

μ32/ρb

3

,

 ̄λ32 = λ32/ρb

3σ, X ̄

3 = X3/ρgb2

, φ ̄

3 = φ3/π

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

-0.5

0.1

0.7

1.3

1.9

2.5

σ ̄

Nondimensional Radiation Wave Damping

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

1

2

3

4

5

Nondimensional Radiation Added Mass

λ ̄

2 2

λ ̄

1 2

λ ̄

3 2

μ ̄2 2

μ ̄1 2

μ ̄3 2

3

1

2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

1

2

3

4

5

σ ̄

Nondimensional Wave-Exciting Magnitude

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

-1

-0.6

-0.2

0.2

0.6

1

Nondimensional Wave-Exciting Phase

|X ̄

1|

|X ̄

2|

|X ̄

3|

φ ̄

1

φ ̄

2

φ ̄

3

Figure 2. NONDIMENSIONAL 2D HYDRODYNAMIC RADIATION AND

WAVE-EXCITING COEFFICIENTS1.

radiation memory effect caused by past WEC motions. The relations between the time- and frequency-domain radiation coefficients were derived in [19]:

Kr22 (t) = 2

π

Z∞

0

λ22 (σ) cos(σt)dσ (5)

Kr22 (t) = −

2

π

Z∞

0

σ[μ22 (σ)−μ22 (∞)]sin(σt)dσ (6)

where μ22 (σ) and λ22 (σ) are the frequency-dependent hydrodynamic radiation coefficients commonly known as the added mass

and wave radiation damping.

The wave-exciting heave force can be written in the time

domain as follows:

fe2(t) = Z∞

−∞

Ke2 (t −τ)η(τ)dτ (7)

3

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where Ke2 is the heave wave-excitation kernel, which is noncausal, and η is the wave elevation. The relationship between

the time- and frequency-domain excitation coefficients is given

by:

Ke2 (t) = 1

π

Z∞

0

[R{X2 (σ)}cos(σt)−I{X2 (σ)}sin(σt)]dσ (8)

where X2 is the frequency-dependent complex wave-exciting

heave-force coefficient, R is the real component, and I is the

imaginary component.

The drag force is represented by either of the following:

fd (t) = (

−λvl ̇ζ2 (t)

−λvn

 ̇ζ2 (t)







 ̇ζ2 (t)







(9)

where λvl is the linear-drag coefficient caused by the presence

of viscosity, and λvn is the quadratic-drag coefficient, assuming

they are not negligible. The final one-degree-of-freedom heave

equation of motion can now be written as:

(m+μ22 (∞)) ̈ζ2 (t) = −C22ζ2 (t)−λvl ̇ζ2 (t)

−

Zt

−∞

Kr22 (t −τ)

 ̇ζ2 (τ)dτ

+

Z∞

−∞

Ke2 (t −τ)η(τ)dτ+ fm (t) (10)

where the linear form of the drag force has been used.

Regular Wave Analysis

It is common practice to begin analysis under regular wave

excitation in which the incident wave elevation is described by:

η(x,t) = R



−

1

g

∂φI

∂t









z=h



= R

n

Aei(σt−kx)

o

= Acos(σt −kx)

(11)

where η is the wave elevation, φIis the incident wave potential,

A is the wave amplitude, σ is the wave angular frequency, k is

the wave number, and i =

√

−1 is the imaginary unit. The timeharmonic heave response is then given by:

ζ2 (t) = R



ξ2e

iσt

	

(12)

where ξ2 is the complex amplitude of heave displacement.

Under regular wave excitation, the radiation-convolution integral can be simplified to:

fr22 (t) = −R

−σ

2

μ22 (σ) +iσλ22

ξ2e

iσt

	

(13)

The wave-excitation-convolution integral can be written as:

fe2 (t) = R



AX2 (σ) e

iσt

	

(14)

For the time being, the mechanical force from the PTO system will be described by the following:

fm (t) = −R

Cg −σ

2

μg +iσBg



ξ2e

iσt

	

(15)

where Cg is the linear PTO-restoring coefficient, Bg is the PTO

linear-damping coefficient, and μg is the additional PTO inertia. The frequency-domain expressions can be inserted into

Eqn. (10), leading to the heave-displacement response amplitude

operator:

ξ2

A

=

X2

[C22 −σ2 (m+μ22) +Cg −σ2μg]+iσ[λ22 +Bg]

(16)

where λvl has been set to zero since results as [1] saw minimal

effects from viscosity.

PTO Absorbed Power The TAP absorbed by the PTO

is calculated from:

PT

A2

=

1

2

Bgσ

2









ξ2

A









2

(17)

Equation (16) can be inserted into Eqn. (17) to calculate the optimal PTO damping at each wave frequency. The optimal, unconstrained, time-averaged absorbed power and PTO damping for

each wave frequency is given by:

PT

A2









p

=

1

4

|X2|

2

λ22

1

1+

s

1+



C22+Cg−σ2(m+μ22+μg)

σλ22 2

(18)

Bg|p = λ22s1+



C22 +Cg −σ2 (m+μ22 +μg)

σλ22 2

(19)

where at resonance Bg = λ22, leading to the maximum timeaveraged absorbed power [20]. Because these expressions do not

consider motion constraints, it may be necessary to increase the

PTO damping to remain under a given limit. The required PTO

damping is given by:

Bg|pc =

(

A|X2|

σ|ξ2|max 2

−



C22 +Cg

σ

−σ(m+μ22 +μg)

2

)1/2

−λ22 (20)

where |ξ2|max is the maximum amplitude of heave displacement

[21].

4

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To provide a measure of efficiency for a given device, the

TAP contained within a propagating wave must be known. The

time-averaged wave power per-unit width, Pw, can be obtained

from:

Pw =

1

2

ρgA2Vg =

1

4

ρgA2

r

g

k

tanhkh



1+

2kh

sinh 2kh

(21)

where Vg is the wave group velocity, and h is the water depth.

Maximum Power Under Constrained Motion The

maximum power absorption under motion constraints, while assuming sinusoidal motion, was explored in [22], which led to the

following expression:

PT

A2









mc

=

1

8

|X2|

2

λ22

h

1−H (1−δ)(1−δ)

2

i

(22)

where H(x) is the Heaviside step function, and δ is the ratio between the constrained-to-optimal heave velocity given by:

δ =

σ|ξ2|max

A

2λ22

|X2|

(23)

Equation (22) can be expanded to show the trends in timeaveraged absorbed power for the ranges of δ:

PT |mc =

(

1

8

A

2

|X2|

2/λ22 δ > 1

1

2

A|X2|σ|ξ2|max −λ22σ

2

|ξ2|

2

max δ < 1

(24)

The capture width, defined as the ratio between the TAP absorbed

by the PTO and the incident wave power per-unit width, is a metric used to evaluate the absorption efficiency of the device. The

incident wave power is proportional to the incident wave amplitude squared, see Eqn. (21). For unconstrained motion, which

may also correspond to a very small incident wave amplitude,

the capture width will be invariant to the incident wave height;

whereas for a strongly constrained motion, which may also correspond to a very large incident wave amplitude, the capture width

will be inversely proportional to the incident wave height and become less efficient.

The associated PTO linear-damping coefficient to observe

the motion constraint is given by:

Bg|mc = λ22 1+

2(1−δ)

δ

H (1−δ)



=

(

λ22 δ > 1

A|X2|

σ|ξ2|max

−λ22 δ < 1

(25)

where the PTO spring and inertia coefficients cancel the dynamic

force contribution from the natural body-restoring coefficient,

mass, and hydrodynamic added mass:

Cg −σ

2

μg = −



C22 −σ

2

(m+μ22)



(26)

which is the basis of complex conjugate control [13]. Often in

power quality management it is desirable to have the peak-toaverage power ratio as close as possible to eliminate the need for

advanced signal conditioning. Under complex conjugate control

the peak-to-average power ratio, P±, is given by [21, 23]:

P± = 1±

s

1+



C22 −σ2 (m+μ22)

σBg

2

(27)

When there is no reactive power, Xg = σμg −Cg/σ = 0, the peakto-average power ratio is 2 and the instantaneous power oscillates between 0 and 2PT . The reactive component is eliminated

at the resonance frequency of the isolated floating body and the

peak-to-average power ratio is minimized at 2; however, when

away from the resonant frequency, the peak-to-average power

ratio quickly increases, resulting in large swings in the bidirectional energy flow.

Foundation Reaction Force and Moment

The structural foundation must handle the reaction force and

torque needed to restrain the WEC to heave motion. The reaction

force and torque in surge, Xr1, and pitch, Xr3, are given by:

A(Xr1 +X1) = −σ

2μ12 +iσλ12

ξ2 (28)

A(Xr3 +X3) = −σ

2

(xgm+μ32) +iσλ32ξ2 (29)

where X1 and X3 are the complex surge wave-exciting force and

pitch wave-exciting torque coefficients per unit wave amplitude,

μ12 is the surge-heave added mass, and λ12 is the surge-heave

wave radiation damping, μ32 is the pitch-heave added mass, λ32

is the pitch-heave wave radiation damping, and xg is the horizontal center of gravity. The surge and pitch foundation reaction

force and torque are affected by the heave motion of the WEC,

which can be controlled by the PTO. The time-domain corollary

of Eqns. (28) and (29) is given by:

fr1 (t) = −

Z∞

−∞

Ke1 (t −τ)η(τ)dτ+μ12 (∞)

 ̈ζ2 (t)

+

Zt

−∞

Kr12 (t −τ)

 ̇ζ2 (τ)dτ (30)

fr3 (t) = −

Z∞

−∞

Ke3 (t −τ)η(τ)dτ+ (xgm+μ32 (∞)) ̈ζ2 (t)

+

Zt

−∞

Kr32 (t −τ)

 ̇ζ2 (τ)dτ (31)

5

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Results from Fixed-PTO Coefficients

Maximizing the TAP, as described in previous sections, involves the PTO coefficients to be fixed in time although adapted

for a given wave amplitude and angular frequency. Performance

bounds can be set for the TAP, surge-restraining force amplitude,

pitch-restraining torque amplitude, and PTO control force amplitude, which have been plotted in Fig. 3. A benefit of the current

design can be observed in the bottom plot of Fig. 3, where the

heave amplitude and phase required for elimination of the surgerestraining force and pitch-restraining torque are presented. The

surge and pitch components require a very similar amplitude and

phase for elimination, which will lead to a reduction in both if

only one contribution is heavily penalized in the controller. It is

expected that time-varying PTO coefficients can help optimize

the time-averaged absorbed power while reducing loads, leading to device performance that sits between the maximum constrained and passive curves.

PSEUDO-SPECTRAL CONTROL

The discretization of the control problem is completed by

approximating the heave velocity and PTO force with a linear

combination of basis functions [11, 24]. The heave velocity,  ̇ζ2,

and PTO force, fm, are approximated by a zero-mean truncated

Fourier series with N terms:

 ̇ζ2 (t) ≈

N/2

∑

j=1

ψ

c

j

cos(jσ0t) +ψ

s

j

sin(jσ0t) = Φ(t)ψˆ (32)

fm (t) ≈

N/2

∑

j=1

τ

c

j

cos(jσ0t) +τ

s

j

sin(jσ0t) = Φ(t)τˆ (33)

where

ψˆ =

h

ψ

c

1

,ψ

s

1

,...,ψ

c

N

2

,ψ

s

N

2

i⊤

, τˆ =

h

τ

c

1

,τ

s

1

,...,τ

c

N

2

,τ

s

N

2

i⊤

(34)

Φ(t) = [φ1 (t),φ2 (t),...,φN−1 (t),φN (t)] (35)

=



cos(σ0t),sin(σ0t),...,cos

N

2

σ0t



,sin

N

2

σ0t



with the fundamental frequency given by σ0 = 2π/T and T is

the chosen time duration. The heave equation of motion can be

described as follows:

M22ψˆ = τˆ +eˆ2 (36)

where ˆe2 is the Fourier coefficient vector of the heave waveexciting force. The matrix M22 ∈ R

N×N is block diagonal with

the following structure:

M

j

22 =



λ22 (jσ0) α(jσ0)

−α(jσ0) λ22 (jσ0)



for j = 1,2,...,N/2

α(jσ0) = jσ0 (m+μ22 (jσ0))−C22/(jσ0) (37)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

0.0

0.2

0.4

0.6

0.8

1.0

σ ̄

Power Capture Efficiency, Cw = PT/Pw, [W/m/W/m]

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

0

4

8

12

16

20

Peak-to-Average Power Ratio, P , [W/W]

Cw|

mc

Cw|

p

P+|

mc

|P-|

mc

P+|

p

|P-|

p

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

0

1

2

3

4

5

6

σ ̄

Maximum Constrained Force and Torque Metrics

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

0

1

2

3

4

5

6

Passive Constrained Force and Torque Metrics

f

 ̄

m|mc

f

 ̄

r1|mc

f

 ̄

r3|mc

f

 ̄

m|p

f

 ̄

r1|p

f

 ̄

r3|p

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

0

2

4

6

8

σ ̄

Heave Response Amplitude Operator Magnitude, |ξ

2|/A, [m/m]

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

-1.0

-0.5

0.0 

0.5 

1.0 

Heave Response Amplitude Operator Phase, φ/π, [rad/rad]

|ξ

2

|

n

|ξ

2

|

f

r1

=0

|ξ

2

|

f

r3

=0

|ξ

2

|

p

|ξ

2

|

mc

φ|

n

φ|

f

r1

=0

φ|

f

r3

=0

φ|

p

φ|

mc

Figure 3. PERFORMANCE BOUNDS UNDER A HEAVEDISPLACEMENT AMPLITUDE CONSTRAINT OF 0.1 M AND A

WAVE AMPLITUDE OF 0.02 M. THE SUBSCRIPT p DENOTES

PASSIVE PERFORMANCE AS GIVEN BY EQNS. (17)–(20). THE SUBSCRIPT mc DENOTES MAXIMUM CONSTRAINED PERFORMANCE

AS GIVEN BY EQNS. (22)–(26). THE SUBSCRIPT n DENOTES

THE NATURAL HEAVE MOTION (NO PTO), WHEREAS fr1 = 0 AND

fr3 = 0 DENOTE HEAVE MOTION REQUIRED TO ELIMINATE THE

SURGE-RESTRAINING FORCE AND PITCH-RESTRAINING TORQUE.

THE NONDIMENSIONAL FORCE AND TORQUE VALUES ARE GIVEN

BY:  ̄fm = fm/ρgbA,

 ̄fr1 = fr1/ρgbA, AND  ̄fr3 = fr3/ρgb2A.

6

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.
The heave velocity coefficients can then be determined explicitly

from the control and heave wave-exciting force Fourier coefficients. This representation allows the total absorbed energy, E,

to be written as:

E = −

Z T

0

 ̇ζ2 (t) fm (t)dt = −

T

2

ψˆ

⊤τˆ

= −

T

2

h

τˆ

⊤



M−1

22 ⊤

τˆ +eˆ

⊤

2



M−1

22 ⊤

τˆ

i

(38)

which is in the form of a traditional quadratic problem.

Penalty Terms

Surge Foundation Force Load reduction will consist

of limiting the forces on the WEC structure that are required to

maintain the heave-only constraint. This force has two contributions that arise from the surge wave-exciting force and WEC

motion. The equation for the surge foundation force can be written in a matrix form, similar to Eqn. (36), as follows:

Φ(t)

ˆfr1 = −Φ(t)eˆ1 +μ12 (∞)Γψˆ +Φ(t) (G12 −μ12 (∞)Γ)ψˆ

ˆfr1 = −eˆ1 +G12ψˆ = −eˆ1 +G12M

−1

22 τˆ +G12M

−1

22 eˆ2 (39)

where ˆe1 is the Fourier coefficients of the surge wave-exciting

force, G12 and Γ are block matrices given in Appendix A and

Eqn. (36) has been substituted in the last expression. To maintain the convexity of the quadratic problem, the squared l

2

-norm

of the surge-foundation force vector was added to the objective

function. The objective function is given by:

γ1| fr1|

2 = γ1

Z T

0

ˆf

⊤

r1Φ(t)

⊤ Φ(t)

ˆfr1dt =

T

2

ˆf

⊤

r1

ˆfr1

≈ γ1

T

2



2

h

eˆ

⊤

1 G12M−1

22 −eˆ

⊤

2



M−1

22 ⊤

G

⊤

12G12M−1

22 i

τˆ

−τˆ

⊤



M−1

22 ⊤

G

⊤

12G12M−1

22 τˆ



(40)

where γ1 is a penalty weight applied to the surge foundation

force. In the final expression for the surge-foundation force contribution, there are three constant terms independent of the PTO

control force, which are left out of the optimization. See [12] for

the full expression.

Pitch Foundation Torque Similar to the surgerestraining force, the pitch-restraining torque has two contributions that arise from the pitch wave-exciting torque and WEC

motion. The matrix form of the pitch-restraining torque can be

written as:

ˆfr3 = −eˆ3 +G32ψˆ = −eˆ3 +G32M−1

22 τˆ +G32M−122 eˆ2 (41)

where ˆe3 represents the Fourier coefficients of the pitch waveexciting torque. As with the surge-restraining force, the squared

l

2

-norm of the pitch-restraining torque vector was added to the

objective function. The pitch foundation torque measure is given

by:

γ3| fr3|

2 = γ3

Z T

0

ˆf

⊤

r3Φ(t)

⊤ Φ(t)

ˆfr3dt =

T

2

ˆf

⊤

r3

ˆfr3

≈ γ3

T

2



2

h

eˆ

⊤

3 G32M

−1

22 −eˆ

⊤

2



M

−1

22 ⊤

G

⊤

32G32M

−1

22 i

τˆ

−τˆ

⊤



M−1

22 ⊤

G

⊤

32G32M−1

22 τˆ



(42)

where γ3 is a penalty weight applied to the pitch foundation

torque.

PTO Control Force Magnitude The PTO force is the

only control actuation, and in an effort to reduce computational

time and force spikes, a penalty weight was placed on the squared

l

2

-norm of the PTO force magnitude [7]:

βm|τm|

2 =

Z T

0

βmτm(t)τm(t)dt =

T

2

τˆ

⊤βmINτˆ (43)

where βm is a penalty weight associated with the control force

magnitude, and IN is the identity matrix of size N.

Final Objective Function The objective function will

be the sum of the time-averaged absorbed power, the squared l

2

-

norm of the surge-restraining force, pitch-restraining torque, and

control force magnitude. The four contributions to the objective

function are not of the same units, and the interrelationship between them is complex. Therefore, the final objective function

will consist of the following nondimensional quantities:

J =

E

PwT

+γ1









fr1

ρgbA









2

+βm









fm

ρgbA









2

+γ3









fr3

ρgb2A









2

(44)

PSEUDO-SPECTRAL RESULTS

As a result of the physical restrictions of the model-testing

facility at the University of California at Berkeley [1], only

model-scale waves were used for analysis; however, Froude scaling can be used to estimate device performance at the prototype

level. The work presented in the paper was focused on confirmation of the controller concept and validation of the analysis.

Effect of Penalty Terms

Figure 4 verifies that the pseudo-spectral controller is

achieving the desired results when considering the extremes of

the tested penalty weights. As the control force penalty weight,

βm, is increased the magnitude of the PTO control force and

reactive power is reduced. As shown for the lowest values

of γ1 and βm, the highest TAP is achieved; however, a larger

reactive power component is required. Whereas for the largest

values of γ1 and βm, reduction in the surge-restraining force and

pitch-restraining torque is counterbalanced by an increase in

the PTO control force. The increase in PTO force is a result of

7

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.
0.25

0.5

0.5

0.75

0.75

0.75

1 1

1

1

1

1

1.25

1.25

1.25

1.25

1.25

1.25

1.34

1.34

1.34

1.34

1.34

1.34

β

m

γ

1

0.5

0.5

0.5

1

1

1

1

1.5

1.5

1.5

1.5

4

4

4

4

13

13

13

40 40

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Cw/Cw|

p

Cw/Cw|

reac

1.1

1.1

1.2

1.2

1.2

1.2

1.2

1.4

1.4

1.4

1.6

1.6

1.6

1.8

1.8

2

2

2.2

β

m

γ

1

1

1

1

1.5

1.5

1.5

1.5

2

2

2

2

3

3

3

3

5

5

5

10

10

40

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Cw/Cw|

p

Cw/Cw|

reac

0.3

0.3

0.3

0.3

0.5

0.5

0.5

0.5

0.7

0.7

0.7

0.9

0.9

0.9

β

m

γ

1

0.2

0.2

0.4

0.4

0.4

0.4

0.6

0.6

0.6

0.8

0.8

0.8

1 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

f

r1/fr1|p

f

r3/fr3|p

0.4

0.4

0.5

0.5

0.5

0.5

0.6

0.6

0.6

0.7 0.7

0.7

0.8

0.8

0.8

0.9

β

m

γ

1

0.4

0.4

0.4

0.5

0.5

0.5 0.5

0.6

0.6

0.6 0.6

0.7

0.7

0.7

0.8

0.7

0.8

0.9

1

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

f

r1/fr1|p

f

r3/fr3|p

0.75 1

1

1

1.25

1.25

1.25

1.5

1.5

1.5

1.75

1.75

1.75

2

2

2

2

2.25

2.25

2.25

2.25

2.5

2.5

2.5

2.5

2.75

2.75

β

m

γ

1

1.5

1.5

1.5

1.5

1.6

1.6

1.6

1.8

1.8

1.8

1.8

2

2

2

2

2.2

2.2

2.2

2.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

f

m

/f

m

|

p

|ξ

2

|/|ξ

2

|

p

0.9

1.05

1.05

1.2

1.2

1.4

1.4

1.4

1.6

1.6

1.6

1.8

1.8

1.8 2

2

2

2.2

2.2

2.2

2.4 2.4

2.4

2.6

2.8

3

β

m

γ

1

1.7

2

2

2

2.25

2.25

2.25

2.5

2.5

2.5

2.5

2.75

2.75

2.75

2.75

3

3

3

3.25

3.25

3.5

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

f

m

/f

m

|

p

|ξ

2

|/|ξ

2

|

p

Figure 4. SENSITIVITY OF PERFORMANCE METRICS TO PENALTY WEIGHTS UNDER A HEAVE-DISPLACEMENT AMPLITUDE CONSTRAINT OF

0.1 M AND WAVE AMPLITUDE OF 0.02 M. THE LEFT COLUMN PLOTS RESULTS FOR A WAVE PERIOD OF 1.5 S (σ ̄ = 0.62) AND THE RIGHT

COLUMN FOR A WAVE PERIOD OF 1 S (σ ̄ = 0.92). THE SUBSCRIPT p DENOTES PASSIVE PERFORMANCE AS GIVEN BY EQNS. (17)–(20).

8

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.
the amplitude and phase difference between the unforced (no

PTO) and zero surge-restraining force heave motion (refer to the

bottom plot of Fig. 3). It can be observed that both above and

below the resonance frequency the unforced heave amplitude

of motion is lower than what is required for elimination of the

surge-restraining force. As more emphasis is placed on reducing

the surge-restraining force, greater control forces and reactive

power are required, which implies that complete elimination of

the restraining loads may not be desirable.

In Fig. 4, the region bounded by βm ≥ 0.8 and γ1 ≤ 0.4 is

the most favorable as the capture efficiency can be increased

between 20%–80% with a reactive power contribution comprising only 1/10th of the TAP. Furthermore, the surge-restraining

force (fr1), pitch-restraining torque (fr3), and PTO control

force can be reduced between 10%–30% with respect to the

passive values; however, it is evident that the contours will

vary depending on the incident wave frequency and most likely

on the heave amplitude constraint. The left column of Fig. 4

plots a set of results for a wave frequency below resonance. In

this frequency range the contours follow nearly straight lines

when viewing the γ1 and βm space. Here the greatest difference

in the heave motion amplitude and phase occurs, requiring a

proportionate increase in the PTO control force to decrease

restraining loads. Just below resonance the TAP curve decreases

slower than above resonance; however, a greater control effort

will be needed to reduce restraining loads. These contour

plots provide a clear design space that can be used to optimize

power production, decrease structural loads, or achieve many

combinations in between.

Time History of WEC and PTO

Figure 5 plots the time history of the four points marked in

the plots along the right column of Fig. 4. In region 1 the maximum power absorption is nearly recovered, in region 2 there is

roughly a 50% reduction for all performance metrics compared

to maximum absorption, in region 3 the surge-restraining force

is prioritized at the expense of larger PTO forces and reactive

power, and in region 4 the controller attempts to maximize

TAP with reduced PTO forces at the expense of larger restraint

loads. As the penalty weights are reduced, the PTO control

force moves the heave velocity closer in phase with the heave

wave-exciting force. This phase shift is accompanied by the

greatest amplitudes in PTO control force, surge-restraining force

and pitch-restraining torque, but not reactive power. Marker 3

has the greatest reactive power requirement as the amplitude of

motion to eliminate the surge-restraining force is greater than

the maximum constrained heave profile. As the surge-restraint

and PTO force penalty weights are increased, the controller

will first maintain a near optimum phase while reducing the

amplitude of motion; however, eventually a greater phase shift

is introduced by the controller to eliminate a greater proportion

of the surge-foundation force; refer to Eqn. (28). Further

reduction in the restraint loads will then see an increase in the

heave amplitude of motion and a corresponding increase in

PTO control force and reactive power. The larger reduction in

restraint loads and PTO force can be achieved because of the

ability of the controller to induce a phase shift in the heave

velocity at the expense of bidirectional energy flow, which can

be greatly affected by PTO efficiency [25].

CONCLUSIONS

In this paper, we describe how pseudo-spectral optimal

control was used to optimize the performance of a novel

WEC/breakwater, the Berkeley Wedge. The analysis revealed

that the power capture efficiency increases by 50% for lower frequencies (σ ̄ < 0.74) compared to results obtained from a passive

PTO with a constant linear damper. For frequencies greater than

σ ̄ > 0.74, a capture efficiency of unity can be achieved; however,

as the wave frequency moves away from resonance a greater reactive power component is required. Though the maximum capture efficiency is lower below resonance, the surge-restraining

force and pitch-restraining torque are also lower, in the range of

0.5 ≤ σ ̄ ≤ 0.7, for the maximum constrained heave motion than

the passive. Thus, when operating slightly below the resonance

frequency with a PTO that allows for bidirectional energy flow,

lower restraint forces and torques will be observed, thereby leading to favorable power-to-load ratios. The magnitude and phase

of heave motion required to cancel the surge-restraining force

and pitch-restraining torque were calculated to show that greater

PTO control forces and reactive power is required for achieving

such performance.

The pseudo-spectral optimal control problem was improved

by including the squared l

2

-norm of the surge-restraining force,

pitch-restraining torque, and PTO actuator force in the objective

function. The optimizer performance was found to be adjustable

based on the values chosen for the separate penalty weights

placed on the three load contributions; however, it was found that

because of WEC dynamics reducing either the surge-restraint

force or pitch-restraint torque would lead to a reduction in the

other. Thus, penalizing one of the contributions in the objective

function was sufficient to explore the power-to-load ratios. Two

incident wave periods above and below the resonance frequency,

with a wave amplitude of 0.02 m and maximum allowable heave

displacement of 0.1 m, were used to analyze the pseudo-spectral

controller. When the penalty weights γ1 → 0 and βm → 0, maximum power capture was recovered with minimal reduction in

system loads. The case of γ1 → ∞ and βm → ∞ would significantly reduce restraint loads; however, at the expense of greater

PTO forces and reactive power requirements. If the penalty

weights are kept with the range of βm ≥ 0.8 and γm ≤ 0.4, an

increase in capture efficiency of 20% to 80% is obtainable with

the reactive power comprising no more than 1/10th of the TAP.

9

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.5

-0.4

-0.3

-0.2

-0.1

 0.0

 0.1

 0.2

 0.3

 0.4

 0.5

Time, t/T, [s/s]

Heave Displacement, ζ

2 / |ξ

2

|max, [m/m]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.5

-0.4

-0.3

-0.2

-0.1

 0.0

 0.1

 0.2

 0.3

 0.4

 0.5 ζ

2

|

1

ζ

2

|

2

ζ

2

|

3

ζ

2

|

4

ζ

2

|

mc

ζ

2

|

p

ζ

2

|

f

r1

=0

ζ

2

|

f

r3

=0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1.5

-1.2

-0.9

-0.6

-0.3

 0.0

 0.3

 0.6

 0.9

 1.2

 1.5

Time, t/T, [s/s]

Nondimensional PTO Control and Heave-Exciting Force

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1.5

-1.2

-0.9

-0.6

-0.3

 0.0

 0.3

 0.6

 0.9

 1.2

 1.5

 ̄fm|1

 ̄fm|2

 ̄fm|3

 ̄fm|4

 ̄fm|mc

 ̄fm|p

 ̄fe2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2.25

-1.75

-1.25

-0.75

-0.25

 0.00

 0.25

 0.75

 1.25

 1.75

 2.25

Time, t/T, [s/s]

Nondimensional Surge-Restraining and Surge-Exciting Force

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2.25

-1.75

-1.25

-0.75

-0.25

 0.00

 0.25

 0.75

 1.25

 1.75

 2.25

 ̄fr1|1

 ̄fr1|2

 ̄fr1|3

 ̄fr1|4

 ̄fr1|mc

 ̄fr1|p

−  ̄fe1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2.25

-1.75

-1.25

-0.75

-0.25

 0.00

 0.25

 0.75

 1.25

 1.75

 2.25

Time, t/T, [s/s]

Nondimensional Pitch-Restraining and Pitch-Exciting Torque

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2.25

-1.75

-1.25

-0.75

-0.25

 0.00

 0.25

 0.75

 1.25

 1.75

 2.25

 ̄fr3|1

 ̄fr3|2

 ̄fr3|3

 ̄fr3|4

 ̄fr3|mc

 ̄fr3|p

−  ̄fe3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-4

-3

-2

-1

0

1

2

3

4

5

6

Time, t/T, [s/s]

Instantaneous Power, P/Pw, [W/m/W/m]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-4

-3

-2

-1

0

1

2

3

4

5

6

P|1

P|2

P|3

P|4

P|

mc

P|

p

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Time, t/T, [s/s]

Cumulative Absorbed Energy, E/Ew, [J/m/J/m]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

E|1

E|2

E|3

E|4

E|

mc

E|

p

Figure 5. TIME HISTORY OF WEC MOTION, PTO CONTROL FORCE, PTO POWER, RESTRAINING FORCE, AND RESTRAINING TORQUE. RESULTS FROM APPLYING PSEUDO-SPECTRAL CONTROL WITH T = 1 s, A = 2 cm, AND VARYING PENALTY WEIGHTS. THE NUMBERS 1, 2, 3,

AND 4 IN THE LEGEND REFER TO THE FOUR MARKERS IN THE TOP PLOT OF FIGURE 4. THE SUBSCRIPT p DENOTES PASSIVE PERFORMANCE AS GIVEN BY EQNS. (17)–(20). THE SUBSCRIPT mc DENOTES MAXIMUM CONSTRAINED PERFORMANCE AS GIVEN BY EQNS. (22)–(26).

A HEAVE-DISPLACEMENT AMPLITUDE LIMIT OF 0.1 M WAS USED WHILE THE NUMBER OF FOURIER COEFFICIENTS WAS SET AT N = 100.

Ew IS THE CUMULATIVE ABSORBED ENERGY WHEN ASSUMING PERFECT ABSORPTION.

10 This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.
In the same region, where the surge-restraining force and pitchrestraining torque were reduced between 20%–30% with little to

no increase in PTO force. This work has highlighted some of

the issues that arise when WEC control focuses solely on maximizing power absorption as it is accompanied by proportionately

greater structural and PTO loads that lead to a higher levelized

cost of energy. In the future, pursuit of moderate gains in TAP

from control strategies may be more favorable as the increase in

power absorption may outpace the growth in structural loads.

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 NREL’s Laboratory Directed Research and Development (LDRD) Program.

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A Matrix Expressions

The time-derivative matrix, Γ ∈ R

N×N

, is block diagonal

with the following block structure:

Γ

j =



0 jσ0

−jσ0 0



for j = 1,2,...,N/2 (45)

Using a change of variables, the surge-pitch radiation convolution integral can be represented in matrix form as follows:

fr12 (t) = Zt

−∞

Kr12 (t −τ)

 ̇ζ2 (τ)dτ

= Φ(t) (G12 −μ12 (∞)Γ)ψˆ (46)

11 This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.
where G12 ∈ R

N×N is block diagonal with the following structure:

G

j

12 =



λ12 (jσ0) σμ12 (jσ0)

−jσ0μ12 (jσ0) λ12 (jσ0)



for j = 1,2,...,N/2 (47)

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