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June 2020

Revisiting Theoretical Limits for OneDegree-of-Freedom Wave Energy 

Converters

Preprint

Nathan Tom

National Renewable Energy Laboratory

Presented at the 14th International Conference on Energy Sustainability

June 17–18, 2020 
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Conference Paper

NREL/CP- 5000-75701

June 2020

Revisiting Theoretical Limits for OneDegree-of-Freedom Wave Energy 

Converters

Preprint

Nathan Tom

National Renewable Energy Laboratory

Suggested Citation

Tom, Nathan. 2020. Revisiting Theoretical Limits for One-Degree-of-Freedom Wave 

Energy Converters: Preprint. Golden, CO: National Renewable Energy Laboratory. 

NREL/CP-5000-75701. https://www.nrel.gov/docs/fy20osti/75701.pdf
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REVISITING THEORETICAL LIMITS FOR ONE-DEGREE-OF-FREEDOM

WAVE ENERGY CONVERTERS

Nathan M. Tom ∗

National Wind Technology Center

National Renewable Energy Laboratory

Golden, Colorado, USA

E-mail: Nathan.Tom@nrel.gov

ABSTRACT

This work revisits the theoretical limits of one-degree-offreedom wave energy converters. This paper considers the floating sphere used in the Ocean Energy Systems Task 10 Wave

Energy Converter modeling and verification effort for analysis.

Analytical equations are derived to determine bounds on the motion amplitude, time-averaged power, and power-take-off (PTO)

force. A unique result was found that shows the time-averaged

power absorbed by a wave energy converter can be defined solely

by the inertial properties and radiation hydrodynamic coefficients. In addition, a unique expression for the PTO force amplitude was derived that has provided upper and lower bounds when

resistive control is used to maximize power generation. For complex conjugate control, this same expression can only provide a

lower bound, as there is theoretically no upper bound. These

bounds are used to compare the performance of a floating sphere

if it were to extract energy using surge or heave motion. The

analysis shows that because of the differences in hydrodynamic

coefficients of each oscillating mode, there will be different frequency ranges that provide better power capture efficiency. The

influence of a motion constraint on power absorption while also

utilizing a nonideal power take-off is examined and found to reduce the losses associated with bidirectional energy flow. The

expression to calculate the time-averaged power with a nonideal

PTO is modified by the mechanical-to-electrical efficiency and

the ratio of the PTO spring and damping coefficients. The PTO

∗Address all correspondence to this author.

spring and damping coefficients were separated in the expression, which allows for limits to be set on the possible values of

PTO coefficients to ensure a net flow of power to the grid.

INTRODUCTION

Over the last 45 years since the publication of Salter’s foundational paper on wave power [1], research and development

in wave energy technologies has pushed forward in hopes of

bringing the industry closer to commercialization. Over the past

two decades, the worldwide marine and hydrokinetic energy sector has seen a resurgence in investment and research funding;

however, it can be argued that the industry as a whole remains

in the early stages of technology development. To date, there

are still no commercially grid-connected wave energy converters

(WECs) installed in the United States, with only a few megawatts

installed worldwide [2]. A significant challenge with WEC technology is the conversion of slow, irregular, and high-force oscillatory motion to drive a generator with output quality that is

acceptable for a utility grid network [3].

As a whole, the field of wave energy continues to host a

wide diversity of technologies ranging in scale from paper concept to oceangoing prototype. Therefore, development funds

can be spread thin across a diverse research portfolio that highlights the need for a structured innovation approach to facilitate

an optimal convergence in WEC design and operation [4]. Although there are many different forms, WECs can generally be

divided into three predominant classifications: 1) attenuator, 2)

point absorber, and 3) terminator. Within these classifications,

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WEC devices can be further defined based on the mode of motion used to capture energy, such as point absorbers, oscillating

wave surge converters, oscillating water columns, overtopping

devices, and bulge wave devices [3]. Of all of the WEC concepts,

the most common and well-studied design is the point absorber,

which typically has a narrow-banded response; therefore, control

is generally required to improve energy capture [5].

Often the point absorber is assumed to oscillate in heave

with the wave elevation profile given the common buoy-like and

bottom-referenced device concepts. However, as discussed in

[6], the hydrodynamic radiation and diffraction forces between

surging, similar for pitch, and heaving devices have different

frequency responses that influence the effective energy capture

bandwidth. For example, it is well known that the theoretical

maximum power that can be captured by a surging or pitching

device is twice that of a heaving device because of the asymmetric wave radiation pattern [7].

This paper builds off the work of [6] by extending the performance metric comparison to the required displacement amplitude and power-take-off (PTO) force, and considering nonideal

conversion efficiency. After introducing the floating WEC used

in this analysis, the author provides a discussion of the differences in the hydrodynamic and radiation coefficients. This is

followed by deriving the upper bound on the amplitude of displacement for each mode of motion, highlighting how constraints

such as the PTO stroke length can reduce the energy capture potential. Next, we discuss the influence of the hydrodynamic radiation coefficients on maximum power capture using complex

conjugate [5] or resistive PTO control. This is followed by deriving the upper and lower bounds of the PTO force requirement,

which depends on the control strategy implemented. Finally, we

provide a discussion on how improvements in maximum power

capture can be degraded by nonunity PTO efficiency, which can

be aggravated based on the mode of oscillation.

FLOATING BODY DESCRIPTION

For this study, we selected the floating sphere used for the

Ocean Energy Systems Task 10 Wave Energy Converter modeling and verification effort [8]. The floating sphere has a radius of

5.0 m, and its origin is located on the mean water surface at the

center of the spherical body, with a summary of the model parameters provided in Table 1. The mass of a floating half-submerged

sphere is given by:

m =

2ρπr

3

3

(1)

where ρ is the fluid density, and r is the sphere radius. The linearized heave hydrostatic restoring force coefficient is given by:

C33 = ρgAwp = ρgπr

2

(2)

Table 1. General Properties of the Floating Sphere [8]

Parameters Variable Assigned Values

Radius of Sphere r 5 m

Center of Gravity cg 0.0, 0.0, -2.0 m

Center of Buoyancy cb 0.0, 0.0, -1.875 m

Mass of Sphere m 261.8x103 kg

Water Depth m Infinite m

Water Density ρ 1000 kg/m3

where Awp is the water plane area. A free floating body will not

have a hydrostatic restoring coefficient in surge; however, for station keeping an external spring, such as mooring or PTO control

strategies, it will need to be coupled to the surge degree of motion. For this analysis, we assumed that the floating sphere could

be connected to a fixed reference frame by an external spring

with a magnitude equal to the heave hydrostatic restoring coefficient (i.e., C11 = C33).

The hydrodynamic coefficients used in the analysis to estimate the displacement and power absorption of the sphere were

obtained from WAMIT [9] and are plotted in Fig. 1. As shown

in Fig. 1(a), for heave oscillation the radiation wave damping

peaks at a lower wave frequency than surge and the same for the

added mass; however, for surge, the wave damping peak corresponds with a greater drop from the added mass peak than

heave, which will be highlighted in a later section. For the waveexcitation force shown in Fig. 1(b), in heave the force is maximum at the lowest frequencies and continues to decrease as the

wave frequency increases. While in surge, the wave-excitation

force tends to zero at the upper and lower frequencies while peaking in the intermediate frequency regime. As will be discussed

in a future section, as the WEC motion becomes more heavily

constrained, the peak in power capture centers on the peak in the

wave-excitation force rather than at the resonant frequency.

REGULAR WAVE ANALYSIS

In floating body dynamics, it is common practice to understand the frequency response of the device by analyzing the response under regular wave excitation, in which the incident wave

elevation is described by:

η(x,t) = Acos(ωt −kx) = R



Aeiωt

	

(3)

where η is the wave elevation, A is the wave amplitude, ω is

the wave angular frequency, and k is the wave number. For the

time being, the mechanical force from the PTO system will be

described by:

fPTO j = −R



(λg −iCg/ω)iωξje

iωt

	

(4)

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0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Nondimensional Wave Angular Frequency, ω(r/g)−1/2, [−]

Nondimensional Radiation Added Mass, μjj / Mjj, [−]

0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

Nondimensional Radiation Wave Damping, λjj / (ω Mjj

μ

), [−]

11

μ

33

λ

11

λ

33

(a) Radiation Added Mass μj j and Wave Damping λj j

0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Nondimensional Wave Angular Frequency, ω(r/g)−1/2, [−]

Nondimensional Exciting Force Magnitude, |Xj|/C33, [−]

0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1

−1.0

−0.8

−0.6

−0.4

−0.2

 0.0

 0.2

 0.4

 0.6

 0.8

 1.0

Nondimensional Exciting Force Phase, 

∠ Xj

/π, [−]

|X1|

|X3|

φ

1

φ

3

(b) Wave Exciting Force Magnitude



Xj



 and Phase φj

Figure 1. Nondimensional hydrodynamic surge and heave radiation and

wave-excitation force coefficients for the floating sphere.

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

linear-damping coefficient, and ξjis the complex amplitude of

motion for the j

th degree of freedom.

The frequency-domain expressions for the hydrostatic, radiation, diffraction, and PTO forces can be inserted into the onedegree-of-freedom equation of motion to derive the displacement

and velocity response amplitude operators of the floating sphere:

ξj

A

=

Xj

[Cj j +Cg −ω2 (Mj j +μj j)]+iω[λj j +λg]

(5)

iωξj

A

=

Xj

[λj j +λg] +i[−(Cj j +Cg)/ω+ω(Mj j +μj j)] (6)

where Xjis the complex wave-exciting force per unit wave amplitude, Cj j is the restoring coefficient in the j

th degree of freedom, Mj j is the mass or moment of inertia in the j

th degree of

freedom, μj j is the radiation added mass or added moment of inertia in the j

th degree of freedom, and λj j is the radiation wave

damping in the j

th degree of freedom.

In order to obtain bounds on the motion amplitude for a

given device, the magnitude of Eqn. (5) will be taken as follows:









ξj

A







 =



Xj





√

2ωλj j

1

Ξj

q

1+

1

Ξj

(7)

=

4ερgVg

√

2ωk



Xj





1

Ξj

q

1+

1

Ξj

, ε =

(

1 for j = 3

2 for j = 1, 5

where Vg is the wave group velocity, and the Haskind expression

[10] has been substituted in the second line of Eqn. (7). Ξ is a

measure of the ratio between the inertial and resistive forces that

arise from WEC oscillation and is defined by:

Ξj =

s

1+



Cj j +Cg −ω2 (Mj j +μj j)

ωλj j 2

(8)

Because the bracketed term in Eqn. (8) is squared, Ξ will be

bounded between 1 and ∞; see Fig. 2. The lower bound will be

obtained when the spring forces cancel the acceleration forces,

and if no phase control is implemented, Cg = 0 will occur only at

the natural frequency of the isolated floating body. Furthermore,

it has also been assumed that the PTO linear damping coefficient

has been selected, such that λg = λj jΞ. The reason behind this

selection will be described in the following section.

If deep water is assumed, kh ≥ π, then Eqn. (7) can be

simplified to the following form:









ξj

A







 =

2ερg

√

2k

2



Xj





1

Ξj

q

1+

1

Ξj

| {z }

Π

=

2ερg

3

√

2ω4



Xj





Π (9)

=

ερgλ

2

w

2

√

2π

2



Xj





Π , ε =

(

1 for j = 3

2 for j = 1, 5

where λw is the wavelength, and h is the water depth. Because

Eqn. (8) is bounded between [1,∞], the term, Π, in Eqn. (9) is

bounded between h0,1/

√

2

i

; see Fig. 3. The upper bound on the

motion response is set by the following expression:









ξj

A









max

=

ερgλ

2

w

4π

2



Xj





, ε =

(

1 for j = 3

2 for j = 1, 5

(10)

Therefore, the maximum displacement amplitude, in deep water,

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0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

0

1

2

3

4

5

6

7

8

9

10

Nondimensional Wave Angular Frequency, ω(r/g)−1/2, [−]

Nondimensional Ratio, Ξ, [−]

Ξ

m1

Ξ

r1

Ξ

m3

Ξ

r3

Figure 2. The values of Ξ, given by Eqn. (8), which is a measure of

the ratio between inertial/spring and resistive forces that are generated

because of WEC oscillations. The subscript m refers to maximum power

absorption, and the subscript r refers to the maximum power absorption

under resistive control when setting Cg = 0.

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Nondimensional Wave Angular Frequency, ω(r/g)−1/2, [−]

Nondimensional Ratio, 

Π, [−]

Πm1

Πr1

Πm3

Πr3

Figure 3. The values of Π, defined in Eqn. (9). The subscript m refers

to maximum power absorption, and the subscriptr refers to the maximum

power absorption under resistive control when setting Cg = 0.

is proportional to the wavelength squared and inversely proportional to the wave-exciting force/torque.

If shallow water is assumed, kh → 0, then Eqn. (7) can be

simplified to the following form:









ξj

A









max

=

2ερg

2h

ω2



Xj





, ε =

(

1 for j = 3

2 for j = 1, 5

(11)

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Nondimensional Wave Angular Frequency, ω(r/g)−1/2, [−]

Magnitude Complex Velocity Amplitude, |iωζ

j| / (gr)−1/2, [−]

|iωζ1|

m

|iωζ1|

r

|iωζ3|

m

|iωζ3|

r

Figure 4. Nondimensional surge and heave velocity response amplitude

operators for the floating sphere. The subscript m refers to maximum

power absorption, and the subscript r refers to the maximum power absorption under resistive control when setting Cg = 0.

The maximum motion displacement amplitude in shallow water is proportional to the water depth, and inversely proportional

to the wave-exciting force and to the wave angular frequency

squared.

When comparing the velocity amplitudes between surge and

heave in deep water for maximum power capture, see Fig. 4; for

wave frequencies below the resonance frequency, surge outpaces

heave. Above the resonance frequency, the surge oscillation requires a smaller amplitude compared to heave that is consistent

with Eqn. (10), as the wave-excitation force is one variable that

differs between the two oscillation modes. Referring back to

Fig. 1(b), one can see that the surge wave-excitation force is

larger than heave for higher frequencies and will reduce the required displacement amplitude.

Time-Averaged Power Absorption

The maximum time-averaged power absorbed by a WEC oscillating in a single degree of freedom [7] can be modeled by the

following:

PT j

A2

=

λg



iωξj





2

2

=

1

4

|Xj|

2

λj j

1

1+

s

1+



Cj j+Cg−ω2(Mj j+μj j)

ωλj j 2

=

1

4

|Xj|

2

λj j

1

1+Ξj

(12)

Therefore, it is beneficial to minimize the variable Ξ ratio, which

can be described by the following inequality:

(Cj j +Cg)/ω−ω(Mj j +μj j) << λj j (13)

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As described earlier in the floating body description section,

given that the hydrostatic restoring coefficient is static, then the

power capture potential of the WEC will be greater when the radiation added mass is minimized and the wave damping is maximized. Thus, it may be possible to adjust the shape of a WEC

at each wave frequency to meet the inequality by controlling the

radiation added mass and wave damping coefficients.

The previously mentioned expressions require unconstrained motion, and the PTO linear damping coefficient takes

the following value:

λg = λj js1+



Cj j +Cg −ω2 (Mj j +μj j)

ωλj j 2

= λj jΞ (14)

Eqn. (12) can be simplified further by substituting the Haskind

expression, as follows:

PT j

A2

=

ερgVg

k

1

1+Ξj

| {z }

χ

, ε =

(

1 for j = 3

2 for j = 1, 5

(15)

As shown by Eqn. (15), the WEC time-averaged power can be

defined solely by the inertial properties and the radiation hydrodynamic coefficients. The expression for the time-averaged

power now requires fewer hydrodynamic coefficients to evaluate

the maximum power potential of a given WEC. The power absorption of a floating body can be estimated from the value of χ,

which is plotted in Fig. 5. In addition, the theoretical maximum

power absorption limit for an axisymmetric device, which is oscillating in pitch or surge, is twice that of the heaving case [7,11].

The maximum power absorption from Eqn. (15) is obtained

when Ξj = 1, which occurs when the inertial and spring forces

cancel. This condition is naturally met when the wave frequency

matches the resonance frequency of the floating body [7]. As

the wave frequency moves either above or below the resonance

frequency, an external positive or negative PTO spring, equal to

Cg = −



Cj j −ω

2

(Mj j +μj j)



, must be added to the WEC system to obtain maximum power absorption and is often referred

to as complex conjugate control [11]. Incorporating this controllable spring remains a current research interest in the wave

energy field, with several efforts implementing the spring either

mechanically or through control strategies. Because of the increased complexity related to incorporating a controllable spring,

many developers have relied on resistive control by adjusting the

PTO damping coefficient, λg, which is equal to Eqn. (14), when

setting Cg = 0.

The power absorption obtained from complex conjugate

and resistive control is equal at the resonance frequency of the

floating body; however, it is of interest to develop a ratio between

these two control strategies at other wave frequencies. The ratio

can be calculated by taking the ratio of Eqn. (15) when setting

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

Nondimensional Wave Angular Frequency, ω(r/g)−1/2, [−]

Nondimensional Ratio, χ, [−]

χ

m1

χ

r1

χ

m3

χ

r3

Figure 5. The values of χ, defined in Eqn. (15). The subscript m refers to

maximum power absorption, the subscript r refers to the maximum power

absorption under resistive control when setting Cg = 0.

Ξj = 1 and with Ξj when Cg = 0, which provides the following

expression:

PT m j

PTr j

=

1+Ξr j

2

=

1+

s

1+



Cj j−ω2(Mj j+μj j)

ωλj j 2

2

(16)

where the subscript m refers to maximum power absorption,

whereas the subscript r refers to the maximum power absorption under resistive control, and Ξr j refers to setting Cg = 0 in

Eqn. (8). Equation (16) has been plotted in Fig. 6 and is unique

in that it remains dependent only on the spring, inertial, and

radiation coefficients. Furthermore, as shown in Fig. 6, when

above the resonance frequencies, the ratio between complex conjugate and resistive control strategies is much lower for surge

than in heave, whereas below resonance surge will quickly outpace heave. This performance can be attributed to the natural

hydrodynamic bandwidth, which helps tune the power capture,

as discussed in [6].

Incident Wave Power and Capture Width

To provide a measure of the capture efficiency for a given

WEC, the time-averaged power contained within a propagating

wave must be calculated. The time-averaged wave-power-perunit width, Pw, can be calculated from the following equation:

Pw

A2

=

1

2

ρgVg =

1

4

ρg

r

g

h

tanhkh1+

2kh

sinh 2kh

kh→∞

z}|{

=

1

4

ρg

2

ω

=

ρg

2T

8π

(17)

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0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Wave Angular Freqnuency, ω, [rad/s]

Power Absoprtion Ratio, PTmj / PTrj, [−]

PTm1/PTr1

PTm3/PTr3

Figure 6. Ratio of power absorption between complex conjugate and

resistive control for the floating sphere.

where T is the wave period. The capture width is normally defined as the ratio of the time-averaged WEC absorbed to the wave

power and given as:

lj =

PT j

Pw

(18)

Equations (15) and (17) can be substituted into Eqn. (18) to calculate the capture width of a given WEC:

lj =

ερgVg

k

1

1+Ξj

1

2

ρgVg

=

2ε

k

1

1+Ξj

(19)

max lj =

ε

k

=

ελw

2π

, ε =

(

1 for j = 3

2 for j = 1, 5

(20)

The maximum capture width is related to the wavelength, which

theoretically implies that a WEC can absorb more incident wave

power than contained within the WEC length that is perpendicular to the wave crest (see Fig. 7); however, it is well known

that the required amplitudes of motion may not be physically

achieved [7]. The results from Eqn. (19) are limited to axisymmetric devices, but if the device characteristic length is

small compared to the wavelength, the device could be treated

as a point absorber with antisymmetrical wave radiation patterns [14].

Maximum Absorbed Power Under Motion Constraints

The theoretical maximum WEC power absorption, given by

Eqn. (12), requires unconstrained motion and is known to require

unrealistic amplitudes of motion. As observed from Eqn. (10),

the required displacement amplitude will quickly increase in

longer wave periods [7], where the wavelength can be several

0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Nondimensional Wave Angular Frequency, ω(r/g)−1/2, [−]

Nondimensional Capture Width, l/2r, [−]

l

m1

l

r1

l

m3

l

r3

Figure 7. The capture width of complex conjugate and resistive control

nondimensionalized by the diameter of the floating sphere.

hundred meters. Therefore, the maximum power absorption under motion amplitude constraints, while assuming sinusoidal motion, was explored in [12], which provides the following expression:

PT j =

(

1

8

A

2



Xj





2

/λj j δj ≥ 1

1

2

A



Xj



ω



ξj





max −λj jω

2



ξj





2

max

δj < 1

(21)

where



ξj





max

is the maximum motion displacement amplitude,

and δ is the ratio between the constrained-to-optimal velocity

given by:

δj =

ω



ξj





max

A

2λj j



Xj





=

ω



ξj





max

A

k



Xj





2ερgVg

(22)

To match the power output described earlier, for δj < 1, the required linear PTO coefficients are given by:

λg

λj j

=







1 , δj ≥ 1

A|Xj|

ω|ξj|maxλj j

−1 , δ < 1

(23)

Cg = −



Cj j −ω

2

(Mj j +μj j)



(24)

If Eqn. (21), with δj < 1, was substituted into the numerator

of Eqn. (18), the capture width would be inversely proportional

to the wave amplitude. For unconstrained motion, which may

correspond to a very small incident wave amplitude, the capture width will be invariant to the incident wave height; whereas

for 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 from the hydrodynamic perspective. In taking the

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limit, δ → 0, the absorbed power is maximized at the frequency

with the greatest wave-exciting force/torque rather than at the

resonance frequency [13].

Peak-to-Average Power Ratio

The instantaneous power absorbed by an ideal power-takeoff unit [15] can be modeled as follows:

Pj(t) = λg





iωξj





2

2

+





iωξj





2

2

|Z| cos(2(ωt +φ)+ν) (25)

where Z = λg − iCg/ω, ν is the phase angle of Z, and φ is the

phase angle of iωξj. As observed in the second term in Eqn. (25),

if Cg 6= 0, the instantaneous power will fluctuate between negative and positive values, indicating a reversal of the energy flow.

The peaks in the PTO instantaneous absorbed power (P+j) and

the reactive input power (P−j) are given by:

P±j =

λg|iωξj|

2

2



1±

s

1+



Cg

ωλg

2



 (26)

The peak-to-average power ratio for the WEC can be obtained

by dividing Eqn. (26) by the time-averaged power, as shown in

Eqn. (12), leading to the following expression:

PA±j = 1±

s

1+



Cg

ωλg

2

(27)

Therefore, the PTO peak-to-average power ratio is defined only

by the PTO force coefficients and can be tuned to stay within the

power capacity of the PTO. As observed from Eqn. (27), there

will be no bidirectional energy flow whenCg = 0, preventing any

negative instantaneous power. This condition is often referred to

as resistive control, and the peak-to-average power ratio will be

2. The input power requirement, Cg 6= 0, is also naturally eliminated at the resonance frequency of the isolated floating body;

however, as the wave frequency moves away from resonance, the

peak-to-average power ratio quickly increases, resulting in large

swings in bidirectional energy flow.

If bidirectional energy flow is allowed and the PTO force

coefficients, Cg and λg, are selected such that the phase and amplitude conditions required for optimum power extraction are

achieved [11]—refer to Eqn. (23) and Eqn. (24)—then the peakto-average power ratio can be calculated from the following expression:

PA±j = 1±

s

1+



Cj j −ω2 (Mj j +μj j)

ωλj j 2

= 1±Ξr j (28)

Equation (28) requires unconstrained motion and has been plot0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10

Nondimensional Wave Angular Frequency, ω(r/g)−1/2, [−]

Peak−to−Average Power Ratio, PA±, [−]

PA+1

|PA−1|

PA+3

|PA−3|

PA+r

|PA−r|

Figure 8. The peak-to-average power ratio for complex conjugate and

resistive control of the floating sphere.

ted in Fig. 8, but can be adapted to account for motion constraints, similar to Eqn. (21). If the expression for the linear PTO

damping coefficient, when δj < 1, is inserted into Eqn. (27), the

following expression is obtained:

PA±j = 1±

vuuuuut

1+









Cj j −ω2 (Mj j +μj j)

ω



A|Xj|

ω|ξj|max

−λj j









2

= 1±

vuuuut

1+







Cj j −ω2 (Mj j +μj j)

A



1−

δj

2



|Xj|

|ξj|max







2

(29)

where, as the wave amplitude increases, all terms are constant

except for A and δj. However, as A increases, δj decreases,

which provides a nonlinear increase in the denominator under

the square root. As the linear PTO damping coefficient is increased to meet a given displacement constraint, the peak-toaverage power ratio will be reduced, and the reduction is nonlinearly proportional to the increase in wave amplitude.

Power-Take-Off Force

For the WEC system to capture energy, a PTO unit must

be included to provide a resistive force that acts against WEC

motion. The complex PTO force amplitude, αj, in the frequency

domain can be expressed as:

αj

A

=

[Cg +iωλg]Xj

[Cj j +Cg −ω2 (Mj j +μj j)] +iω[λg +λj j]

= −[λg −iCg/ω]iωξj = Zu jiωξj (30)

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 7
where αj

is the complex PTO force/torque amplitude depending

on the degree of freedom used to extract the incident wave power,

and Zu j is the PTO force-to-velocity transfer function. In order

to estimate the magnitude of the force amplitude required by the

PTO, we take the absolute value of Eqn. (30) and divide it by

ωλj j:







αj

A





 =

rh

Cg

ωλj j i2

+

h

λg

λj j i2

s

Cj j+Cg−ω2(Mj j+μj j)

ωλj j 2

+

h

λg

λj j

+1

i2



Xj



 (31)

=

r

1+

h

Cg

ωλg

i2

r

1+2

h

λj j

λg

i

+Ξ2

h

λj j

λg

i2



Xj





(32)

If bidirectional energy flow is allowed, then the optimum phase

and amplitude conditions required for maximum power extraction can be met [11], and the complex PTO force/torque amplitude can be simplified as the following expression:







αj

A





 =



Xj





2

s

1+



Cj j −ω2 (Mj j +μj j)

ωλj j 2

=



Xj



Ξr j

2

(33)

The second term under the square root in Eqn. (33) will always

be greater than or equal to 0, resulting in the square root having a lower bound of 1 and an unconstrained upper bound. The

lower bound will only be met when the wave frequency is equal

to the resonance frequency of the isolated floating body. Therefore, the expected PTO force under the optimum conditions for

power extraction will always be greater than half of the waveexciting force.

If bidirectional energy flow is prohibited, Cg = 0, and the

PTO damping coefficient is selected to optimize power absorption (refer to Eqn. (8), the required PTO force will be expressed

as follows:







αj

A





 =

√

2

2



Xj





s

Ξj

1+Ξj

=

(|Xj|

2

as Ξj → 0

√

2

2



Xj



 as Ξj → ∞

(34)

For resistive control, the PTO force amplitude will vary between

1/2 and √2/2 of the wave-exciting force; see Fig. 9. Equations

(33) and (34) converge to the same value when the wave frequency matches the resonant frequency of the isolated floating

body, as the conditions for optimum power extraction are met by

both control schemes.

The magnitude and phase relationship of the PTO force-to0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

Nondimensional Wave Angular Frequency, ω(r/g)−1/2, [−]

Nondimensional PTO Force Magnitude, |α

j|/|Xj|, [−]

α

m1

α

r1

α

m3

α

r3

2

−1/2 Limit

1/2 Limit

Figure 9. The nondimensional power take-off force amplitude for complex conjugate and resistive control of the floating sphere.

velocity transfer function can be expressed as follows:

∠Zu j = arctan 

I



Zu j	

R



Zu j	

!

= arctan−

Cg

ωλg



(35)

whereas under complex conjugate control, the phase angle will

take the following form:

∠Zu j = arctan−

ω[Mj j +μj j]−Cj j/ω

λj j 

(36)

The phase angle between the PTO force and WEC velocity is

bounded between π/2 (ω → 0) and 3π/2 (ω → ∞) and crosses

π at the resonance frequency of the WEC, as shown in Fig. 10.

As shown in Fig. 11, the phase angle of the optimal PTO forceto-velocity transfer function deviates from π, which corresponds

to resistive control. The linear trace between the WEC velocityto-PTO force for resistive control is no longer optimum and becomes more oblong as the ratio of the imaginary-to-real component of Zu j. At the limits of ∠Zu j = ±π/2, the control command has the largest force demand when the WEC is moving

at the lowest speed. Requiring high forces at low speeds can

lead to issues with conversion efficiency, as the PTO systems are

not designed for optimum performance in this operating regime.

Therefore, the gains over resistive control when applying complex conjugate control can be degraded with potential negative

energy capture [16]. As observed in Fig. 10, at frequencies above

resonance the phase angle for the heave oscillation mode decays

more quickly to 3π/2 than surge. Therefore, even when off resonance the surge oscillation mode requires less of a control action

to tune the response of the WEC for optimum power capture, as

long as the PTO is able to generate the required force.

Nonideal Power-Take-Off Efficiency

Complex conjugate control requires a two-way energy flow

between the oscillating body and an energy storage system,

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 8
0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

Nondimensional Wave Angular Frequency, (r/g)-1/2, [-]

0

1

2

3

4

5

6

7

8

9

10

Magnitude Force-to-Velocity Transfer Function, |Zuj|/ jj, [-]

-1.0

-0.8

-0.6

-0.4

-0.2

 0.0

 0.2

 0.4

 0.6

 0.8

 1.0

Phase Force-to-Velocity Transfer Function, Zuj / , [-]

|Zu1|

|Zu3|

 Zu1

 Zu3

Figure 10. The magnitude and phase of the PTO force-to-velocity transfer function for complex conjugate control.

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Normalized WEC Velocity, i j

, [-]

-1.0

-0.8

-0.6

-0.4

-0.2

 0.0

 0.2

 0.4

 0.6

 0.8

 1.0

Normalized Power-Take-Off Force, 

j, [-]

 Zuj = 

 Zuj = 15 /18

 Zuj = 12 /18

 Zuj = /2

Figure 11. Plot of the changing relationship between velocity and force

with changing phase angle.

which will have losses associated with the energy flux-reversal

process if the PTO has a mechanical-to-electrical efficiency less

than unity. If a PTO unit is selected, with a time-invariant

mechanical-to-electrical efficiency defined by ηe, the resulting

PTO output time-averaged power [16, 17] is given by:

PO = ηe

λg|iωξj|

2

2

[1+e

∗

g

∗

], e

∗ =

1−η

2

e

η2

e

, (37)

g

∗ =

2G

∗ −sin 2G∗ −2G



1−cos2 G

∗



2π

, (38)

G =









Cg

ωλg









, G

∗ = arctanG (39)

where PO is the time-averaged power, after accounting for the efficiency steps in the power conversion chain, which can be sent to

the grid. The nonideal PTO peak-to-average power ratios, PAηe±,

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

6

7

8

9

10

η

e

 and G*/(π/2)

e

*

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

g

*

 & −G

e

*

g

*

−G

Figure 12. The effect on e

∗

and g

∗

for a range of ηe and G values.

can now be calculated from:

PAηe+ =

1+

√

1+G2

[1+e

∗g∗

]

, PAηe− =

1−

√

1+G2

η2

e

[1+e

∗g∗

]

(40)

The variation of the terms e

∗

and g

∗

in Eqn. (37), with respect

to the PTO coefficient ratio, are plotted in Fig. 12. It can be observed from Eqn. (39) that the variable, G, is bounded between

[0, ∞] and that G

∗

is bounded between [0, π/2]. As observed

in Fig. 12, the variable, e

∗ ≥ 0, while g∗ ≤ 0, resulting in the

bracketed term in Eqn. (37) being bounded by [1,−∞]. When

the value in brackets falls below 0, there is a net flow of energy

to the device rather than the grid. Inclusion of the mechanicalto-electrical efficiency can greatly reduce the time-averaged output power if care is not taken in limiting G. A new ratio of the

time-averaged output power, when considering nonideal PTO efficiency between the two control strategies, can be calculated by

taking the ratio of Eqn. (37) when setting Ξj = 1 and with Ξj

when Cg = 0, leading to the following expression:

POm j

POr j

=

1+Ξr j

2

[1+e

∗

g

∗

] (41)

As e

∗

and g

∗

are independent and can be separated in

Eqn. (37), it is possible to set maximum limits on G to ensure

net flow of power to the grid (tabulated in Table 2). Table 2

shows that, as the PTO mechanical-to-electrical efficiency is reduced, the maximum G value for a net power output is reduced

at a much greater rate. Equation (41) only requires the radiation

coefficients, and the natural hydrodynamic bandwidth of the device will play a role in tempering the negative effects of PTO

efficiency. Therefore, we can expect the surge oscillation mode

to be less sensitive than heave when considering PTO efficiency.

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 9
Table 2. Limits on G as a function of ηe

ηe e

∗ Ge

∗g∗=−1/4 Ge∗g∗=−1/2 Ge

∗g∗=−1

0.70 1.04 1.82 2.73 4.36

0.80 0.56 2.60 4.12 7.01

0.90 0.23 4.71 8.14 14.89

0.95 0.11 8.73 16.04 30.60

Conclusion

This paper explores the theoretical one-degree-of-freedom

limits for a demonstrative floating spherical WEC. The dynamic

response of the WEC considered both surge and heave modes of

oscillation, as the differences in the hydrodynamic coefficients

for different frequency ranges may lead to improved power capture efficiency. This follows the concept of a natural hydrodynamic bandwidth that has been shown to improve tuning of the

WEC system for maximum power capture for the surge mode

of motion compared to heave; however, this work does not suggest that heaving WECs are poorly designed, only that additional

consideration should be taken when designing the floating body.

As the work in this paper shows, the benefits of proper hydrodynamic tuning can significantly reduce the control strategy burden to maximize power capture, which can be degraded further

when considering the PTO efficiency. Furthermore, to the best of

our knowledge, a unique analytical expression for the PTO force

amplitude was derived that provides upper and lower bounds

that depend only on the wave-exciting force. Such bounds allow for rapid design iteration, as they eliminate the need to complete higher-fidelity simulations and can be calculated using frequency domain techniques. Furthermore, analytical bounds have

been placed on the ratio of the PTO spring and damping coefficients, when considering PTO efficiency, to ensure a net power

output that is also strongly influenced by the natural hydrodynamic bandwidth. The final decision in selecting the WEC mode

of oscillation will be completed based on the designer vision and

must also consider other external factors, such as station keeping, extreme loading, power-take-off selection, and maintenance

windows.

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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This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 10