WEC EXPERIMENTS

AND THE EQUATIONS OF MOTION

MORTEN KRAMER

2 0 1 7 M AY N O O T H U N I V E R S I T Y W AV E E N E R G Y W O R K S H O P, 2 0 J A N U A R Y 2 0 1 7

Testing at Aalborg University and DanWEC

Wave basins and flumes (indoor)

Scale: 1:50 to 1:20

Water depth: 0.4 to 1.0 m

Float diameter: ~0.25 m

Nissum Bredning (outdoor)

• Scale: 1:10 to 1:4

• Water depth: 1 to 5 m

• Float diameter: ~1.0 m

Hanstholm North Sea (outdoor)

• Scale: 1:3 to 1:1

• Water depth: 5 to 35 m

• Float diameter: ~5.0 m

Floating Power Plant – key components

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Semisubmersible Platform5-8 MW Wind Turbine

Unique Patented WEC and PTOsDisconnectable Turret Mooring

Experiments with a Floating Power Plant absorber at

Aalborg University

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Motion sensor

Bearing around which the motion takes place

Force sensor

Electrical actuator to

apply any specified

motion or force

Procedure:

1) Wave is generated and measured without the

device in the basin

2) A wave analysis is performed to separate incident

and reflected waves from the beach

3) The same waves are afterwards repeated with

the device in position

Experiments with a Floating Power Plant absorber at

Aalborg University

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Repeatability of waves

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Wave flume experiments with float in position

and power extraction.

Wave generation with active absorption of

reflections:

• Hm0, target = 35 mm

• TP = 1.7 s

• JONSWAP with Gamma = 1.5

• White noise filtering generation method

Actual measurements of incident wave in 11

repeated tests:

= 0.0032*Hm0

Wave height Deviation

H m0 [mm] (1-H m0 /H m0,average )*100 [%]

Target 35.00 2.47

Run 1 35.98 -0.25

Run 2 36.00 -0.30

Run 3 35.70 0.52

Run 4 35.74 0.41

Run 5 35.72 0.47

Run 6 35.86 0.08

Run 7 35.99 -0.28

Run 8 35.96 -0.21

Run 9 35.90 -0.03

Run 10 35.95 -0.17

Run 11 35.98 -0.25

H m0,average,measured 35.89 0

Test

FH

Kinematics and control of the single pitching absorber

The absorbers rotates around a bearing: The cylinder force FH is applying a control moment: Mc = RFH

R

θ

Mc

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Equation of motion for a pitching absorber

Hydrostatic moment: 𝑀ℎ𝑠 = −(𝑀𝑏 −𝑀𝑔) = −𝑘ℎ ∙ (𝜃 − 𝜃0)

Radiation moment: 𝑀𝑟 = −𝑚ℎ∞ −

−∞

𝑡ℎ𝑟 𝑡 − 𝜏 (𝜏)𝑑𝜏

Wave excitation moment: 𝑀𝑒𝑥 = −∞

∞ℎ𝑒𝜂 𝑡 − 𝜏 𝜂 (𝜏)𝑑𝜏

Control moment: 𝑀𝑐

where:

kh: Hydrostatic stiffness coefficient

mh∞ : Hydrodynamic added mass coefficient at infinite frequency

hr: Impulse response function for wave radiation moment

he: Impulse response function for wave excitation moment

𝜂 : Wave elevation

θFixed

support

Moving rigid body

Mc

Mg

Mw

x

z Newton’s second law:

𝐽 𝜃 = 𝑀𝑔 −𝑀𝑤 −𝑀𝑐

By inserting:

𝑀𝑤 = 𝑀𝑏 −𝑀𝑟 −𝑀𝑒𝑥

the equation is expanded to:

𝐽 𝜃 = 𝑀ℎ𝑠 +𝑀𝑟 +𝑀𝑒𝑥 −𝑀𝑐

J: Mass moment of inertia 𝜃: Angular accelerationMg: Gravitational momentMw: Moment from water pressure on hullMc: Control moment from Power Take OffMb: Buoyancy moment (Archimedes)

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Gravity by ”manual measures”

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• Mass is found by weighingAccuracy +/- 1% by judgement and repeated measurements with different

position of cables, and on different days with more or less dry absorber

• Centre of Gravity is found by the “plumb line method” (2D). Accuracy +/- 1 % on CoGx (horizontal) and +/- 5 % on CoGz (vertical). The

absorber is suspended from three positions and vertical lines are drawn.

The intersection of the lines is the center of mass. Due to inaccuracies the

intersection of the three lines form a small triangle. The distance from the

centre of the triangle to the farthest corner is taken as the accuracy.

Gravity by measurements

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𝐽 𝜃 = 𝑀𝑔 −𝑀𝑤 −𝑀𝑐

↔ 0 = 𝑀𝑔 − 0 −𝑀𝑐 , 𝜃 ≅ 𝑀𝑤 ≅ 0

↔ 𝑀𝑔 = −𝑀𝑐

Slow motion in air with un-ballasted float.

Inertia moment by free oscillations

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• Inertia moment by ”pendulum equation”

𝐽 =𝑇

2𝜋

2

𝑚 ∙ 𝑔 ∙ 𝑙

Accuracy +/- 2% by analysis of measured period of free oscillations

J = 4.30 kgm2 = 0.02*J

Inertia moment by measurements

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𝐽 𝜃 = 𝑀𝑔 −𝑀𝑤 −𝑀𝑐

↔ 𝐽 𝜃 = 𝑀𝑔 − 0 −𝑀𝑐 , 𝑀𝑤 ≅ 0

↔ 𝑀𝑔 −𝑀𝑐 = 𝐽 𝜃

Fast motion in air with un-ballasted float.

Hydrostatics

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Slow motion in calm water

𝐽 𝜃 = 𝑀ℎ𝑠 +𝑀𝑟 +𝑀𝑒𝑥 −𝑀𝑐

↔ 0 = 𝑀ℎ𝑠 + 0 + 0 − 𝑀𝑐 , 𝜃 ≅ 𝑀𝑟 ≅ 𝑀𝑒𝑥 ≅ 0↔ 𝑀𝑐 = 𝑀ℎ𝑠

Decay

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𝐽 𝜃 = 𝑀ℎ𝑠 +𝑀𝑟 +𝑀𝑒𝑥 −𝑀𝑐

↔ 𝐽 𝜃 = 𝑀ℎ𝑠 +𝑀𝑟 + 0 − 0, 𝑀𝑐 = 𝑀𝑒𝑥 = 0

Results from flume tests with un-ballasted float

Wave excitation

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𝐽 𝜃 = 𝑀ℎ𝑠 +𝑀𝑟 +𝑀𝑒𝑥 −𝑀𝑐

↔ 0 = 0 + 0 +𝑀𝑒𝑥 − 𝑀𝑐 , 𝜃 ≅ 𝑀ℎ𝑠 ≅ 𝑀𝑟 ≅ 0↔ 𝑀𝑐 = 𝑀𝑒𝑥

Freefloat RAO

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𝐽 𝜃 = 𝑀ℎ𝑠 +𝑀𝑟 +𝑀𝑒𝑥 −𝑀𝑐

↔ 𝐽 𝜃 = 𝑀ℎ𝑠 +𝑀𝑟 +𝑀𝑒𝑥 − 0, 𝑀𝑐 = 0

Results from flume tests with un-ballasted float

Power production

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𝐽 𝜃 = 𝑀ℎ𝑠 +𝑀𝑟 +𝑀𝑒𝑥 −𝑀𝑐

Linear damping control: 𝑀𝑐 = 𝑐𝑐 ∙ 𝜃

Absorbed power: 𝑃𝑎 = 𝑀𝑐 ∙ 𝜃

Results from irregular wave with moderate hight (Hm0 = 2.2 m at full scale). Average values are from test with length of 1000*TP.

Message & recommendations

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• Experiments are necessary and extremely useful when working on development of

numerical models for WECs, especially when the tests are designed to assist the

numerical modelling

• Be aware of limitations in linear potential wave theory and small amplitude

approximations. Linear numerical models tend to generally overestimates the power

production in moderate sea-states.

• When performing experiments it is recommended to compare measurements with

numerical estimates “on the run” and sort out any strange inconsistencies right away.

Document every molecule in the setup in detail, and always measure or estimate the

accuracy of every experimental parameter

Thank you

Our PhD student Pilar Heras is planning on presenting more in-depth comparisons with (non-)linear

models at the WWWFB workshop (April 2017, China) and at OMAE 2017 (June 2017, Norway)