Modeling, Simulation and Dynamic
control of a Wave Energy Converter
MARIA
BÅNKESTAD
Master of Science Thesis
Stockholm, Sweden 2013
Modeling, Simulation and Dynamic
control of a Wave Energy Converter
MARIA
BÅNKESTAD
Master’s Thesis in Numerical Analysis (30 ECTS credits)
Degree Progr. in Engineering Physics 300 credits
Royal Institute of Technology year 2013
Supervisor at CorPower Ocean was Patrik Möller
Supervisor at KTH was Christina Carlsund Levin
Examiner was Michael Hanke
TRITA-MAT-E 2013:55
ISRN-KTH/MAT/E--13/55--SE
Royal Institute of Technology
School of Engineering Sciences
KTH SCI
SE-100 44 Stockholm, Sweden
URL: www.kth.se/sci
Abstract
The energy in ocean waves is a renewable energy resource not yet fully
exploited. Research in converting ocean energy to useful electricity has
been ongoing for about 40 years, but no one has so far succeed to do
it at sufficiently low cost. CorPower Ocean has developed a method,
which in theory can achieve this. It uses a light buoy and a control
strategy called Phase Control.
The purpose of this thesis is to develop a mathematical model of this
method—using Linear Wave Theory to derive the hydrodynamic forces—
and from the simulated results analyze the energy output of the method.
In the process we create a program that will help realizing and improving the method further.
The model is implemented and simulated in the simulation program
Simulink. On the basis of the simulated results, we can concludes that
the CorPower Ocean method is promising. The outcome shows that
the energy output increases—up to five times—compared to conventional methods.
Modellering, simulering och dynamisk
kontroll av ett vågkraftverk
Sammanfattning
Vågenergi är en förnyelsebar energikälla som ännu inte utnyttjas fullt
ut. Forskning inom konvertering av vågenergi till användbar elektricitet
har pågått i cirka 40 år, men ingen har hittills lyckas att göra det tillräckligt kostnadseffektivt. CorPower Ocean har utvecklat en metod,
som i teorin kan uppnå detta. De använder en lätt boj och en kontrollstrategi kallad Phase Control.
Syftet med detta examensarbete är att utveckla en matematisk modell
av metoden—genom att använda Linear Wave Theory för att härleda de
hydrodynamiska krafterna—och från de simulerade resultaten analysera
energiutbytet. Under arbetets gång skapades också ett simuleringsprogram som hjälpmedel till att realisera och förbättra metoden.
Modellen implementeras och simuleras i programmet Simulink. Utifrån
de simulerade resultaten kan vi dra slutsatsen att CorPower Oceans
metod är lovande. Resultatet visar att energiutbytet ökar—upp till fem
gånger—jämfört med konventionella metoder.
Acknowledgments
I would like to thank my colleges Patrik Möller and Gunnar Ásgeirsson
on CorPower Ocean for their support during the project. I also want
the thank Marco Alves from WacEC for helping me with his expertise.
My warmest gratitude goes to my supervisor Ninni Carlsund for her
enthusiasm, interest end support during writing this thesis.
Contents
1 Introduction
2 Theoretical background
2.1 Basic concepts . . . . . . . . . .
2.1.1 Buoy motion . . . . . . .
2.2 Wave theory . . . . . . . . . . . .
2.2.1 Linear Wave Theory . . .
2.2.2 Ocean Waves . . . . . . .
2.3 Hydrodynamic Forces . . . . . .
2.3.1 Radiation Force . . . . . .
2.3.2 Excitation Force . . . . .
2.3.3 Hydrostatic Force . . . .
2.3.4 WAMIT . . . . . . . . . .
2.3.5 Drag Force . . . . . . . .
2.3.6 The equations of motion .
2.4 The Power Take Off (PTO) . . .
2.5 The system of Partial Differential
2.6 Phase control . . . . . . . . . . .
2.6.1 Resonant buoy oscillation
2.6.2 Latching strategy . . . . .
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3 Modeling
3.1 Modeling Approach . . . . . . . . .
3.2 Modeling of the buoy . . . . . . . .
3.2.1 Simulink implementation .
3.2.2 Radiation Force . . . . . . .
3.2.3 Excitation Force . . . . . .
3.2.4 Hydrostatic and Drag Force
3.3 Implementing the PTO . . . . . .
3.4 Linear damping . . . . . . . . . . .
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the WEC
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4 Simulation results
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5 Discussion
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5.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Bibliography
45
Appendices
46
A Radiation Force Solution
47
B State-Space approximations
49
Nomenclature
h
H
Hs
T
Te
λ
ω
η
r
q
g
ρ
p
φ
φ0
φd
φrad
water depth
wave height
significant wave height
wave period
energy wave period
wave length
angular wave frequency
surface elevation
position vector
generalized position vector
acceleration of gravity
water density
pressure
total velocity potential
initial potential
diffraction potential
radiation potential
φ̂(r)
Sb
n
ñ
Frad
Fexc
Fhyst
complex potential amplitude
buoy wetted surface
normal vector to the surface Sb
generalized normal vector to the surface Sb
radiation force
excitation force
hydrostatic force
Mrad
Mexc
Mhyst
radiation torque
excitation torque
hydrostatic torque
τ rad
µrad
m∞
A(ω)
generalized radiation force
convolution part of the radiation force
infinite added mass
radiation added mass matrix
B(ω)
radiation damping matrix
z(t)
state vector in state-space approximation
Ã
constant matrix in state-space approximation
B̃
constant matrix in state-space approximation
1
CONTENTS
C̃
constant matrix in state-space approximation
D̃
τ exc
τ hyst
constant matrix in state-space approximation
generalized excitation force
generalized hydrostatic force
τ drag
generalized drag force
CD
Sc
mb
mosc
xosc
ξ˙j
drag coefficient
cross section area of the buoy
buoy mass
oscillator mass
oscillator position vector
Jf w
flywheel inertia
Pf rac,j
energy absorption variable for energy device j
Tgen,j
u
rpin
generator torque for energy device j
gear ratio
pinion radius
Ftrans
p0u
p0l
V0u
V0l
dcyl
transmission force for energy devices in PTO
initial pressure in upper gas reservior
initial pressure in lower gas reservior
initial volume in upper gas reservior
initial volume in lower gas reservior
cylinder diameter in gas spring
lstoke
Fgas
length from oscillator bottom to oscillator equilibrium position
force from gas spring in PTO
Fmosc g
gravitational force from oscillatorin PTO
Fµ
force due to oscillator friction in PTO
FP T O
ωN
WT
total PTO force
natural period of the buoy in heave
wire tension variable
velocity of flywheel j
2
Chapter 1
Introduction
In today’s energy demanding world and with increasing greenhouse gases from fossil
fuel in the atmosphere, the need for renewable energy sources is greater than ever
before. One that has not yet been fully utilized is the energy in ocean surface waves.
Wave energy originates from the wind, which in turn originates from the sun. When
the wind blows over the ocean, friction gives rise to water movement and waves are
generated. Even though the total energy of waves on earth is much lower than the
total solar energy, it is much more dense, specifically wave energy is about five times
more dense than solar energy; with power density up to 2–3 kW/m2 on the surface
[8]. This makes wave energy a good renewable energy source, but only if we can
find an efficient way to extract it. The study of converting the mechanical energy
in the ocean waves to useful electricity began in the 1970’s [8], but still no one has
found a way to make it profitable.
The device that converts ocean wave energy to electricity is called a Wave Energy Converter (WEC). A WEC have various shapes and sizes. The one we model
in this thesis is a so called point absorber: it has a buoy that moves with the wave
motion. The buoy is connected to an energy converter called Power Take Off (PTO),
so that the movement of the buoy is transmitted to the PTO. Inside the PTO, a
generator converts the mechanical energy to electrical energy and transmits it to
the grid (see Figure 1.1).
A mathematical model of a product, in our case a WEC, assists during the developing process. With the mathematical model we can test the behavior of the
product before we have a physical product to test on. The objective of this thesis
is to develop a simulation tool of a WEC developed by the company CorPower
Ocean.
CorPower Ocean has developed a WEC of the point absorbing type that aims to
have higher energy output per capital cost, than those on the market today. The
1
CHAPTER 1. INTRODUCTION
Figure 1.1: A sketch of a Wave Energy Converter of point absorber type. A buoy is
connected with a wire to an energy converter unit which converts the mechanical buoy
movement to electricity.
special characteristics of the WEC is that the buoy is light compared with similar
WEC and uses a control strategy called Phase Control. Figure 1.2 shows a photo
of the CorPower buoy in scale one to thirty. The final buoy will have a height of
around 9 m, a width of around 8 m and weigh around 25 tons.
Figure 1.2: A photo of the CorPower buoy. The buoy in the photo is thirty times smaller
than the final buoy.
The highest power output is achieved when the buoy moves in resonance (phase)
with the incident wave. The CorPower WEC uses Phase Control, which makes
it possible to use it in an optimized way in almost all sea states (waves of various
lengths and amplitudes).
The Phase Control is achieved by locking and unlocking the buoy motion at certain
times. This is called latching since we are latching and unlatching the buoy.
2
The mathematical model of the WEC is developed in the numerical computing
program MATLAB and the graphical simulation program Simulink. It consists
of the mathematical model of the buoy and the mathematical model of the PTO
merged into one system of partial differential equations. A controlling block is implemented in the Simulink model, where the latching algorithm for Phase Control
is located. A graphical user interface is developed where the user can set the input
parameters, start the simulation and view the simulation result.
The thesis consists of four different chapters following the introduction chapter.
In turn, they cover the theoretical background, the modeling and implementation,
the results and finally a discussion.
Chapter 2 - Theoretical background
The second chapter provides the theoretical background of a WEC: the buoy motion
the PTO and the control algorithm- phase control.
Chapter 3 - Modeling
The third chapter describes, using the theory from Chapter 2, the mathematical
modeling of the WEC and the implementation in Simulink.
Chapter 4 - Simulation results
The forth chapter presents results and model verification.
Chapter 5 - Discussion
The final chapter discusses the result from the WEC model and its limitations. It
also covers future areas of improvement.
3
Chapter 2
Theoretical background
This chapter goes through the theoretical background needed to understand the
modeling of surface waves on the oceans.
The chapter covers four parts: the basic theory of ocean surface waves; the derivation of main forces acting on the buoy, calculated from the theory of surface waves;
the Power Take Off (PTO), which converts mechanical energy to electricity; the
theory of power absorption and the control strategy Phase Control.
Before we start with the theory of waves, we first define some basic concepts for the
WEC modeling.
2.1
Basic concepts
In this thesis we only consider waves acting in two dimension, the x−z-plane. These
waves are called plane waves, since their wave front moves in parallel planes. The
coordinate system for the wave is defined such that the x-axis is in the direction of
the wave propagation and the z-axis is vertical to the wave motion. The z-axis is
zero where the water surface is located when the wave is at restl (see figure 2.1). The
buoy is symmetric, so the (x, z)−axes can rotate to coincide with the plane movement of the waves. Hence, it is sufficient to model the buoy as two–dimensional.
We need some definitions to describe the waves: the wave length λ, the wave angular frequency ω and the wave period T . The wave height H defines as twice the
wave amplitude (figure 2.1).
Waves can be classified as shallow water waves, intermediate waves or deep water waves, depending on the relationship between the wave length λ and the water
depth h (see Figure 2.1). In this thesis we only consider deep water waves that
occur when h/λ >> 1 [9].
5
CHAPTER 2. THEORETICAL BACKGROUND
λ
z
η(x, t)
z = 0
H
h
z = −h
Figure 2.1: Drawing of the main parameters of the wave: the wave length λ, the wave
height H, the sea bottom depth h, and the surface elevation η(x, t).
To describe an ocean wave we need an expression for the free surface movement in
the z-direction. The free surface is the surface between the air and the water and
is by definition zero at z = 0. The free surface elevation, located at x at time t, is
denoted η(x, t) (figure 2.1).
2.1.1
Buoy motion
The general buoy motion is defined by six degrees of freedom; the movement in
x, y, z directions and the rotations around the x, y, z axes. By common notation
we call these six degrees of freedom: surge, sway, heave, roll, pitch and yaw (see
Figure 2.2).
z (heave)
yaw
pitch
roll
y (swey)
x (surge)
Figure 2.2: The coordinates used when simulation the movement of a buoy in a surface
wave. The six degrees of freedom are (surge , sway , heave , roll , pitch , yaw).
In this thesis, only three degrees of freedom are considered, since the waves are
6
2.2. WAVE THEORY
moving in two dimensions and the buoy is symmetric. These are surge, heave
and pitch, where surge is the x-movement, heave is the z-movement and pitch
is the rotation around the y-axis. The derivations following this chapter could be
done in the same way for six degrees of freedom, but we limit the derivation to three.
The position vector, (x, z) is denoted r. If we also include the motion in pitch
we get the generalized position vector q, representing the position in three degrees
of freedom.
2.2
Wave theory
We have to understand the ocean waves and their interaction with the buoy to
model the buoy movement. This section covers the theory of surface waves and
their interaction with an oscillating buoy.
Ocean waves arise when the wind is blowing over the ocean, forcing the water
surface to move. The gravity is then working as a restoring force. Ocean waves
can also originate from other sources such as earthquakes and Coriolis force. We
only consider the wind generated waves since these are the waves with a frequency
spectrum interesting for a Wave Energy Converter.
In this thesis, we consider both regular and irregular surface waves. A regular
wave is sinusoidal and occurs if a steady wind blows during a long time period.
Ocean waves mostly consist of super-positioned regular waves, since the wind speed
varies. A wave that consist of a range of regular waves with varying shapes is called
an irregular wave.
The first part of this section goes through Linear Wave Theory. The second part
describes how regular and irregular surface waves are modeled.
2.2.1
Linear Wave Theory
In this section we derive the equations that describes the motion and the pressure
of the fluid, using Linear Wave Theory. The theory, also called Airy Wave Theory
[5], seeks to describe the wave motion as a velocity potential with a few assumptions
and approximations.
We assume the fluid to be incompressible (constant density) and inviscid (zero
viscosity). This is an approximation of reality and to get a more correct model, a
viscous force called drag force is later implemented. Furthermore, we assume the
flow to be irrotational (no local rotation of fluid elements). We linearize the equation by assuming higher order terms to be negligible.
We start by looking at two basic fluid dynamic equations: the Continuity equa7
CHAPTER 2. THEORETICAL BACKGROUND
tion and the Navier-Stokes equation [9]. For an incompressible flow the Continuity
equation is
∇ · ṙ = 0,
(2.1)
where r is the position vector (x, z). We use a simplified version of the NavierStokes equation—assuming an incompressible, inviscid and irrotational flow—called
the Euler equation [9]
Dṙ
∂ ṙ
ρ
=ρ
+ (ṙ · ∇)ṙ = −∇p + f .
Dt
∂t
(2.2)
ṙ
The term D
Dt is the material derivative of the velocity, which is the rate of change of
the velocity for a particle moving with the fluid. The term p is the total pressure of
the fluid and f is the external force acting on the fluid, in our case the gravitational
restoring force, f = −ρgez = −∇ρgz.
Velocity Potential
If a flow is both irrotational and incompressible; its velocity can be described with
a velocity potential φ.
( ∂φ
ṙ = ∇φ ⇔
∂x
∂φ
∂z
= ẋ
= ż.
(2.3)
When the velocity potential is used in the Continuity equation (2.1), we get the
Laplace equation
∇ · ṙ = ∇2 φ = 0.
(2.4)
The Linearized Bernoulli equation
W replace the velocities with the velocity potential (2.3) in the Euler equation, (2.2)
∂φ (∇φ)2 p
∇
+
+ + gz
∂t
2
ρ
!
= 0,
(2.5)
and then integrates the equation to get the Bernoulli equation [9],
∂φ (∇φ)2 p
+
+ + gz = constant.
∂t
2
ρ
(2.6)
We linearize the Bernoulli equation by eliminating the nonlinear term (∇φ)2
∂φ p
+ + gz = constant,
∂t
ρ
(2.7)
and then rearrange it to
p = −ρ
∂φ
− ρgz + constant.
∂t
8
(2.8)
2.2. WAVE THEORY
We now look at the static case, which occurs when the water is still and ∂φ
∂t = 0.
The pressure at the free surface η(0, t) = 0 is patm (atmospheric pressure). This
gives us that the constant in (2.8) is patm .
p = −ρ
∂φ
− ρgz + patm
∂t
(2.9)
Here, −ρ ∂φ
∂t = dynamic pressure, −ρgz = hydrostatic pressure, and p0 = atmospheric pressure. We later use these pressures to calculate the forces acting on the
buoy.
In conclusion, we have derived the Laplace equation of the velocity potential (2.4),
and the linearized Bernoulli equation (2.9). Both these equations must be satisfied
throughout the fluid, so they need to satisfy conditions on the boundaries.
Boundary conditions
We have four different boundaries to consider: the boundary at the sea bottom
z = −h, the wetted surface of the oscillating buoy Sb (the buoy surface in contact
with water), the sea bottom z = −h, and the free surface of the water η(x, t) (see
figure 2.3).
η(x, t)
Sb
n
vn
∂η(x,t)
∂t
•
∂φ
∂z
=
•
∂φ
∂n
= vn on Sb (moving buoy).
•
∂φ
∂n
= 0 on Sb (fixed buoy).
•
∂φ
∂z
= 0 at z = −h.
on η(x, t)
z = −h
Figure 2.3: Sketch of the boundaries for the velocity potential with its corresponding
boundary conditions.
Let us first look at the free water surface η(x, t). There we have a so called kinematic
boundary condition, which means that the fluid velocity component normal to the
free surface, must be the same as the free surface velocity in the same direction [9].
It can be approximated to [5]
∂η
∂φ
= ż =
∂t
∂z
9
on η(x, t).
(2.10)
CHAPTER 2. THEORETICAL BACKGROUND
The pressure of the fluid is constant at the free surface (atmospheric pressure, patm ).
If we use the relation in equation 2.9 we get the boundary condition,
∂p
∂η
∂2φ
=0⇒ 2 +g
= 0 at η(x, t).
∂t
∂t
∂t
(2.11)
Now consider the wetted surface of the buoy Sb (see figure 2.3). No flow is permitted
through the buoy surface. The fluid velocity perpendicular to the buoy surface is
therefore equal to the normal component of the buoy velocity vn (see figure 2.3) [5].
This gives us the boundary conditions
∂φ
= vn ,
∂n
on Sb ,
(2.12)
when the buoy is moving, and a special case when the buoy is fixed,
∂φ
= 0,
∂n
on Sb .
(2.13)
The fluid velocity is zero at the sea bottom z = −h, so we have the boundary
condition
∂φ
= 0, when z = −h.
(2.14)
∂z
These boundary conditions will be extended with one at infinite distance from the
buoy.
ż =
Summary of the linearized wave equations
To summarize, we have the linearized Bernoulli equation and the Laplace equation
of the velocity potential,
p
= −ρ
∇2 φ
= 0,
∂φ
− ρgz + patm ,
∂t
(2.15)
(2.16)
with boundary conditions,
∂2φ
∂t2
∂φ
∂n
∂φ
∂n
∂φ
∂z
= −g
∂φ
,
∂t
at the free surface η(x, t),
(2.17)
on Sb , when the buoy is moving,
(2.18)
= 0,
on Sb , when the buoy is fixed,
(2.19)
= 0,
at the water depth z = −h.
(2.20)
= vn ,
These equations are used when we derive the forces acting on the buoy. Before we
derive the forces, the mathematical background for ocean waves is covered.
10
2.2. WAVE THEORY
2.2.2
Ocean Waves
Waves on an ocean are typically irregular and consists of multiple regular waves
with varying frequencies and amplitudes, traveling on top on each other. Figure
2.4 illustrates the free surface elevation as a function of time for a regular and an
irregular wave.
Figure 2.4: Example of a regular and an irregular wave.
First we describe the behavior of regular waves. The theory of regular waves is then
used to describe irregular waves.
Regular Waves
Regular surface waves arise when the wind blows with constant speed for a long
time over a large area. We let x be the direction of wave propagation and since
we assume the waves to be two dimensional, x together with the time t are the
only coordinates we need to describe the wave elevation. The surface elevation of a
regular wave can be expressed with a sinusoidal function
η(x, t) =
H
cos (ωt − kx)
2
(2.21)
where H/2 is the wave amplitude, ω is the angular frequency, k = 2π/λ is the wave
number and λ in turn is the wave length.
Irregular Waves
The period T and the wave height H are important properties of a wave. For
irregular waves we use corresponding properties called significant wave height Hs —
calculated by taking the average value of the highest one-third heights of the incoming waves—and energy period, Te , which is the wave period of a regular wave,
carrying the same amount of energy as the irregular wave (more information is
found in [6]).
11
CHAPTER 2. THEORETICAL BACKGROUND
Figure 2.5: The surface elevation of two regular waves with different wave height an ,
angular frequency θn and phase shift ωn and the two regular waves super-positioned to an
irregular wave.
The wave elevation for an irregular wave is described as a sum of waves of different
shapes.
η(x, t) =
M
X
an cos(ωn t − kn x + θn )
(2.22)
n=1
Here, M is the number of waves added together, an is the wave amplitude, and θn is
the phase shift of the nth wave. The phase shift takes a random value in the range
[0, 2π]. Figure 2.5 shows an example of two super-positioned waves. The angular
frequencies ωn , the energy period Te and the significant wave height Hs depends on
the sea state at the geographic location we wish to simulate.
To describe a sea state mathematically, we use a so called energy density spectrum,
which describes the amount of energy transported in the wave per wave angular
frequency ωn . The spectrum depends on Hs and Te . An example of an energy
spectrum is seen in figure 2.6, which shows the power density S(ω) as a function of
the wave angular frequency ω. The surface elevation, calculated from the spectra is
also shown, where the wave height an is obtained from the value S(ωn ), taken from
the spectrum.
In this thesis the JONSWAP (Joint North Sea Wave Project) spectrum [2], is used
to model the irregular waves. It is derived from theory together with data from real
measurements. The JONSWAP spectrum is commonly used for deep water waves
(the interested reader can read more in [2]).
2.3
Hydrodynamic Forces
In this section we derive the different forces acting on the buoy, by using the equations for the velocity potential and the pressure summarized in section 2.2.1.
12
2.3. HYDRODYNAMIC FORCES
Figure 2.6: To the left we see the energy density spectrum for a wave with significant wave
height, Hs = 2.26 m and energy period, Te = 8.0 s. To the right we see the surface elevation
of a wave, derived from the spectrum by equation (2.22), where an depends on the value
S(ωn ).
In section 2.2.1, we saw that the fluid motion is described by a velocity potential φ.
The velocity potential can be separated into three different potentials.
φ = φ0 + φd + φrad
(2.23)
The initial potential φ0 is the velocity potential in an incident wave if no buoy is
present and the diffraction potential, φd , compensates for the presence of a fixed
buoy in the incident wave. These two together represent the potential of a fixed
buoy in an incident wave. The radiation potential, φrad , is due to the oscillation of
a buoy in still water. The potential partition gives us a way to separate the forces
acting on the buoy.
We assume when we calculate the different hydrodynamic forces that the oscillating buoy is initially at equilibrium position. We also assume that the sea bottom
is sufficiently deep not to affect the free surface elevation. Irregular waves are superpositioned regular waves (2.22); thus, the derivations are limited to regular waves.
The velocity potential can be expressed as [5],
h
i
φ(r, t) = Re φ̂(r)eiωt ,
(2.24)
for a sinusoidal time varying wave. Here φ̂(r) is the complex potential amplitude, r
is the location vector, ω is the angular frequency of the incoming wave and t is the
time. From now on we will use an accentˆ for all complex entities.
The forces on the buoy are derived from the fluid pressure on the buoy wetted
surface Sb . The pressure is obtained from the Bernoulli equation (2.15) and by
using the complex potential (2.24).
p = −ρ
h i
∂φ
− ρgz = −ρRe iω φ̂0 + φ̂d + φ̂rad eiωt − ρgz.
∂t
13
(2.25)
CHAPTER 2. THEORETICAL BACKGROUND
Here, the term patm is canceled out, since we assume that the buoy initially is in its
equilibrium position. The forces are obtained by integrating the pressure over the
surface Sb ,
ZZ
F=
pndS,
Sb
(2.26)
where n is the normal vector to the surface Sb . The torque acting on the buoy is
calculated with
ZZ
M=
p(r × n)dS.
(2.27)
Sb
We introduce a generalized force, which is a force vector in surge, heave and pitch.
By introducing the vector
τ = (F, M)
(2.28)
ñ = (n, (r × n)),
(2.29)
we can write the generalized force τ , as
τ =
ZZ
p ñdS.
Sb
(2.30)
The three different potentials and the hydrostatic term −ρgz cause three different
forces acting on the buoy: τ rad , τ exc and τ hyst . We assume the fluid to be inviscid,
which is a simplification of reality. Therefore, one additional force is added, to get
a more accurate model. The force is called the drag force τ drag and is due to the
friction in the water [9]. The forces are listed in table 2.1.
Table 2.1: The four hydrodynamic and hydrostatic forces.
Name
Hydrostatic force
Abbreviation
τ hyst
Origin
−ρgz
Description
hydrostatic restoring force.
Radiation force
τ rad
φ̂rad
due to the radiating wave the oscillating buoy creates.
Excitation force
τ exc
φ̂0 , φ̂d
Drag force
τ drag
due to a fixed buoy in an incident
wave.
due to the water friction.
The five following sections covers the calculation of these forces.
2.3.1
Radiation Force
Consider a still ocean. If we let a buoy oscillate in it, a wave will emerge; this wave
gives rise to the radiation wave potential φrad .
14
2.3. HYDRODYNAMIC FORCES
The complex radiation potential is a superposition of the wave potentials in the
three degrees of freedom [5]. This lets us write the radiation potential as
3
X
φ̂rad =
ûk ϕ̂k = iω
k=1
3
X
(2.31)
q̂k ϕ̂k ,
k=1
where ϕ̂k is the so called unit-amplitude radiation potential, ûk is the generalized
complex velocity vector and q̂k is the generalized complex position in the k’th degree of freedom. The expression comes from the fact that the velocity in complex
notation is written as
ûk = iω q̂k eiωt .
(2.32)
The boundary condition ∂φ/∂n = vn on the boundary Sb for a moving buoy (2.18),
can—using complex units— be written as
∂ φ̂rad
= ∇φ̂rad · ñ = iω q̂ · ñ = v · ñ,
∂n
(2.33)
where ñ is defined in (2.29). This gives us a boundary condition on Sb for ϕ̂k :
∂ ϕ̂k
= ñk
∂n
(2.34)
The potential ϕ̂ will also satisfy the Laplace equation (2.4) along with the free water
surface boundary condition (2.17) and the sea bottom boundary condition (2.20).
The radiation potential needs one more boundary to be completed that says that a
wave from the oscillating buoy goes to infinity.
The generalized radiation force can then be written as (2.30)
τ rad = (Frad , Mrad ) = −ρ
ZZ
ñ
Sb
∂φrad
dS = ρ
∂t
ZZ
h
Sb
i
ñ Re ω 2 ϕ̂T q̂ eiωt dS.
This equation is simplified into (see Appendix A for a detailed calculation),
τ rad = −B(ω)q̇ − A(ω)q̈,
(2.35)
(2.36)
where A(ω) and B(ω) ∈ R3×3 , are two frequency depending symmetric matrices,
and q̇ and q̈ are the generalized velocity and acceleration vectors. The matrix A(ω)
is called the added mass matrix and B(ω) is called the damping matrix. If we let
the frequency go to infinity in A(ω), we get the so call infinitely added mass m∞ ,
which is a matrix depending on the infinity added mas in the different degrees of
freedom. This equation represent the radiation force in frequency domain. If we
instead look at the same force in time domain we get [12]
τ rad = −m∞ q̈ −
Z t
0
K(t − t0 )q̇(t0 )dt0 = −m∞ q̈ + µrad .
15
(2.37)
CHAPTER 2. THEORETICAL BACKGROUND
The matrix K(t) in the convolution term represents a memory effect due to the
waves from the oscillating buoy.
The Fourier transform of the convolution integral µrad in equation (2.37) is
µ̂rad (ω) = −û(ω)K̂(ω).
(2.38)
where û is the Fourier transform of the generalized velocity q̇. This is an inputouput system, where K̂(ω) is the transfer function. The transfer function is [12]
K̂(ω) = B(ω) + iω [A(ω) − m∞ ] .
(2.39)
To get an approximation of (2.38) in time domain a state-space approximation is
used [3].
ż(t) = Ãz(t) + B̃q̇
(2.40)
µrad = C̃z(t) + D̃q̇.
(2.41)
Here, z(t) is the state vector, and Ã, B̃, C̃ and D̃ are constant matrices (See
Appendix B for a more detailed explanation). The accuracy of the state-space
approximation depends on the number of states included in the state vector z(t).
The approximation tends to the true solution as the number of states goes to infinity.
2.3.2
Excitation Force
The excitation force is the force acting on a buoy fixed in an incident wave. To
calculate the excitation force we consider the incident wave potential φ0 and the
diffraction potential φd . The generalized excitation force is calculated by
τ exc = −ρ
ZZ
h
Sb
i
ñRe iω φ̂0 + φ̂d eiωt dS.
(2.42)
From the boundary condition (2.19) we know that the velocity potential has a
∂φ
Neumann boundary condition ∂n
= 0 on the wetted surface Sb , when the buoy is
fixed.
∂ φ̂0
∂ φ̂d
=−
,
∂n
∂n
on the boundary, Sb .
(2.43)
We also have the boundary condition for the potential at the sea bottom (2.20) and
the boundary condition at the water surface (2.17).
16
2.3. HYDRODYNAMIC FORCES
2.3.3
Hydrostatic Force
The hydrostatic force is also called the gravitational restoring force, since it always
strives to reach the equilibrium of the buoy, at z = 0. The generalized hydrostatic
force is calculated as
τ hyst = −ρg
ZZ
z ñdS.
Sb
(2.44)
Further calculations on the hydrostatic force will not be done in this thesis, the
interested reader can find more information in [5].
The generalized hydrostatic force can be written as
τ hyst = Sq,
(2.45)
where S is the 3 × 3 hydrostatic stiffness matrix and q is the displacement from the
equilibrium position.
We now have covered the three hydrodynamic forces acting on the buoy. The
excitation force depends on the surface elevation and the shape of the buoy. The
remaining hydrodynamic forces, the radiation force τ rad and the hydrostatic force
τ hyst , depends on the shape and movement of the buoy.
The forces are calculated by solving the velocity potential. The velocity potential is frequency dependent, as evident from the complex velocity potential (2.24).
There are standard tools used to calculate the hydrodynamic forces. One of them
is WAMIT [1], which is the one we use in this thesis.
2.3.4
WAMIT
WAMIT (Wave Analysis At Massachusetts Institute of Technology) is a computer
program used to analyze submerged bodies in presence of waves and is based on
Linear Wave Theory [1].
The input to WAMIT is the shape and equilibrium point of the buoy. WAMIT
calculates the velocity potentials in equation (2.23), with the associated boundary
conditions (2.17-2.20). The frequency dependent forces are calculated using the
velocity potentials as explained earlier in this chapter.
The excitation force is an array with the forces for the frequencies in a range of
choice. The outputs for the radiation force are the matrices A(ω), B(ω) and m∞
(explained in section 2.3.1) and the output for the hydrostatic force is the stiffness
matrix S (explained in section 2.3.3).
17
CHAPTER 2. THEORETICAL BACKGROUND
2.3.5
Drag Force
The drag force is due to the fluid resistance and its magnitude depends on the velocity of the buoy. The fluid resistance depends on the viscous friction in the fluid,
which was neglected when the hydrodynamic forces were calculated.
The force is calculated by [9]
1
τ drag,k = ρq̇k2 Sc,k CD,k
2
(2.46)
where ρ is the water density, Sc,k is the cross section area of the buoy perpendicular
to the motion in the k:th degree of freedom, and CD,k is the corresponding drag
coefficient, which depends on the buoy shape and Reynolds number [9]. The drag
coefficients are obtained from empirical measurements.
2.3.6
The equations of motion
The motion of the buoy is calculated by adding up the hydrodynamic and hydrostatic forces.
mb q̈ = τ exc + τ hyst + τ rad + τ drag ,
(2.47)
Here, mb is the buoy mass. By using the relation in (2.37) it can be rewritten as
(Imb + m∞ ) q̈ = τ exc + τ hyst + µrad + τ drag ,
(2.48)
where I is the identity matrix. This relation is used when we model the complete
WEC.
The second part of the WEC is the Power Take Off (PTO) device, which is described in the following section.
2.4
The Power Take Off (PTO)
This section describes the different parts of the Power Take Off (PTO) device: the
oscillator, two energy converting devices and the gas spring.
The motion of the buoy is transferred to the PTO through a wire, where an oscillator moves with the buoy. The oscillator is connected to two energy converter
devices with a gear (a cogwheel). One of the devices is used for the up motion and
one is used for the down motion. The up motion energy converter is engaged during
the oscillator up motion and free during the down motion, and the down motion
energy converter is engaged during the down motion and free during the up motion
(see figure 2.7).
18
2.4. THE POWER TAKE OFF (PTO)
Wire connected with buoy.
Oscillator.
Energy converter 2, engaged
when oscillator is moving down.
Energy converter 1, engaged
when oscillator is moving up.
Figure 2.7: Drawing of the oscillator with the energy conversion units. The wire is connected to the oscillator, transmitting the motion of the buoy to the PTO. The energy
converter device 1 is engaged during oscillator up motion and energy converter device 2 is
engaged during oscillator down motion.
The energy converter consists of a flywheel and a generator. The generator extracts energy from the flywheel—which works as an energy storage—when best
suited. The largest share of the energy is extracted from the flywheel to the generator when the energy conversion unit is free. This decreases the damping on the
buoy motion, since the flywheels can move almost freely during engagement. Figure
2.7 shows a sketch of the energy conversion devices and the oscillator.
The last parts of the PTO is an upper and a lower gas reservoir, which work as a
gas spring for the oscillator (see figure 2.8). This gas spring protects the PTO from
large forces, and works as a pre-tension force that puts the initial position for the
buoy at its midpoint.
The free PTO (not connected with the buoy) is modeled by adding up the forces
acting on the oscillator.
mosc ẍosc = k · Ftrans + Fgas + Fmosc g + Fµ = FP T O
(2.49)
Here, mosc is the oscillator mass, ẍosc is the oscillator acceleration, Fgas is the force
from the gas spring, Ftrans is a force from the energy conversion devices, Fmosc g is
the gravitational force and Fµ is a friction force. The variable k = 1 when an energy
converting device is engaged, and k = 0 otherwise.
The energy conversion devices, when engaged to the oscillator, gives rise to the
transmission force
ξ¨j u Pf rac,j Tgen,j · u
= −sign(ẋosc ) Jf w
+
,
rpin
rpin
!
Ftrans
(2.50)
where the first term is the inertial force from the flywheel and the second term is the
resistant force from the generator. Here, j = 1 when the energy conversion device
one is engaged, and j = 2 when energy conversion device two is engaged. ẋosc is
the velocity of the oscillator, Jf w is the flywheel inertia, u is the gear ratio, rpin
19
CHAPTER 2. THEORETICAL BACKGROUND
Wire connected to buoy.
pu
Vu
xosc = 0
dcyl
Vl pl
lstoke
Wire connected to ground.
Figure 2.8: Drawing of the gas spring of the oscillator. The oscillator movement (up and
down) changes the pressure and volume in the upper and lower reservoir and works as a
gas spring.
is the pinion radius (the pinion connects the oscillator with the flywheel), Tgen,j is
the torque from the generator j and Pf rac,j is a controller factor for generator j,
deciding the amount of energy to be transferred from the energy conversion device
to the grid. The energy conversion device, disengaged to the oscillator, is modeled
with the equation
Jf w ξ¨j = −Pf rac,j Tgen,j ,
(2.51)
where ξ¨j is the flywheel acceleration.
The gas spring gives rise to a gas force Fgas . It is modeled with the equation

Fgas = p0l 

γ

V0l

V0l +
πd2cyl
4
xosc +
lstoke
2
γ 



 − p0u 
V0u
V0u −
πd2cyl
4
xosc +
lstoke
2
2
  πdcyl
.
 
4
(2.52)
The first term is the force from the lower reservoir acting upward on the oscillator
and the second term is the force form the upper reservoir, acting downward on the
oscillator (see figure 2.8). The terms p0u , p0l , V0u and V0l are the gas pressures and
volumes when the oscillator is at the bottom position −lstoke ; lstoke is the length
from the bottom position to the equilibrium position for the oscillator. dcyl is the
diameter of cylinder in which the piston moves. The term xosc is the position of the
oscillator and γ is the specific heat ratio. The pressure of the lower gas reservoir is
adjusted when the WEC is installed, so that the oscillator moves from −lstoke to its
20
2.5. THE SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS FOR THE WEC
equilibrium position at xosc = 0.
The weight of the oscillator gives rise to a gravitational force
Fmosc g = −mosc · g,
(2.53)
where g is the acceleration due to gravity. The last PTO force is the friction force
from the oscillator
Fµ = −µ · ẋosc ,
(2.54)
where ẋosc is the oscillator velocity and µ is a friction constant determined from
empirical measurements.
We have now the equations for the buoy and the PTO. These are in the following section merged to one system of Partial Differential Equations.
2.5
The system of Partial Differential Equations for the
WEC
We have derived the hydrodynamic forces acting on the buoy and the mechanical
forces from the PTO. These forces together constitute a system of partial differential equations that describes the motion of the whole WEC. The buoy is modeled
in three degree of freedom, while the PTO is modeled in one degree of freedom.
The PTO and the buoy are connected by a wire. This gives us two different situations: one when there is wire tension and the PTO is connected to the buoy, and
one when there is no wire tension, so that the buoy and the PTO are two different
systems modeled separately. The two equations (2.48) and (2.49) for the buoy and
the PTO are when there is no tension:
(Imb + m∞ ) q̈ = τ exc + τ hyst + µrad + τ drag ,
(2.55)
mosc ẍosc = FP T O .
(2.56)
The PTO is connected to the buoy when the wire is tense and the movement of the
oscillator and the buoy is the same. Then the system is modeled as
(I(mb + mosc ) + m∞ ) q̈ = τ exc + τ hyst + µrad + τ drag + FP T O
ẍosc = ẍbuoy ,
(2.57)
(2.58)
where FP T O is the PTO force, split into the components of the coordinate system
of the buoy, and ẍbuoy is the acceleration of the buoy projected on the direction of
the wire.
The last component we add to the model is a strategy to control the buoy motion, with the aim to increase the power output. This strategy is called Phase
Control.
21
CHAPTER 2. THEORETICAL BACKGROUND
2.6
Phase control
The goal for a WEC is to absorb as much of the energy in the waves as possible
and convert it to electricity. This section describes the strategy we use to achieve
optimal power absorption.
One way of achieving this is to use the control strategy Phase Control by latching. This control strategy acts by locking and unlocking the motion of the buoy at
certain moments, to force the buoy to move in phase with the incident wave. This
maximizes the buoy velocity and thereby maximizes the energy output. Figure 2.9
illustrates the motion of the buoy when Phase Control with latching is used.
Figure 2.9: The surface elevation and the buoy motion when latching is used, when only
looking at heave.
2.6.1
Resonant buoy oscillation
A buoy oscillating in an ocean has a natural frequency. The natural frequency is
the frequency with which the buoy oscillates in still water. The natural angular
frequency of a buoy in heave (2.2) is
s
ωN =
ρgA
,
mb + m∞,heave
(2.59)
where g is the acceleration of gravity, A is the buoy plane area, mb is the buoy mass
and m∞ is the infinitely added mass for the heave coordinate (due to the movement
of the surrounding fluid as discussed in section 2.3.1).
The optimal movement of a buoy (maximal velocity and position) occurs when
the natural frequency of the buoy is the same as the frequency of the incoming
wave [5], so that the buoy is at resonance with the wave. This typically requires a
22
2.6. PHASE CONTROL
large and heavy buoy to match the relatively low frequency of the wave. Since the
frequency of the waves vary, a buoy must regulate its mass mb or the buoy plane
area A to be in resonance.
There is an alternative way to achieve the resonance effect, without regulating
the buoy mass and shape. The idea is to have a small buoy with natural frequency
higher than most wave frequencies, and force the buoy to move in phase with the
wave by locking the motion of the buoy at certain times. The buoy can then move
with its natural frequency, even though it is much higher than the frequency of the
incoming wave. The control strategy of locking the buoy motion is called Phase
Control by latching. Figure 2.10 shows the movement of a buoy having the same
natural frequency as the one of the incoming wave, together with a buoy with higher
natural frequency than the incoming wave, but forced to move with its natural frequency using latching.
b)
c)
a)
a) Surface elevation of incident wave
b) Movement of light buoy,
with phase control.
c) Movement of heavy buoy, with
optimal mass and size.
t
Figure 2.10: Drawing adapted from [5] showing the movement of a heavy buoy, with the
same natural period as for the incident wave, along with the movement of a small buoy,
using the control strategy latching to achieve the same phase as the incident wave.
If we choose a buoy light and small enough we can force the buoy to move with
its natural frequency for almost all incoming waves, since the frequency always is
larger than the wave frequency.
2.6.2
Latching strategy
For a regular steady sinusoidal wave it is an easy task to predict when to lock and
unlock the motion to get the optimal power absorption. For an unsteady or irregular wave on the other hand, we need a way to predict when this should occur.
The locking of the motion occur when the buoy velocity is zero. The unlocking
is more trickier to decide, so we need a way to predict when the optimal timing is.
One way to do this is to have a threshold for the surface elevation. When the wave
passes the threshold, we unlock the buoy. The optimized threshold for the surface
elevation was calculated from tests results in [10] to
H
π
TN
=
1−
sin
2
2
T
ηthres
23
,
(2.60)
CHAPTER 2. THEORETICAL BACKGROUND
where TN is the natural period of the buoy, H is the wave height and T the wave
period. For irregular waves we can replace the wave period T with the average
energy period Te and the wave height H with the significant wave height Hs . In
the model the wave height is read from time series vectors of the surface elevation
and in real life the wave height is estimated by sensors placed close to or on the buoy.
The method of wave elevation threshold works well for steady waves where the
average energy period Te and the significant wave height Hs stays constant. For
waves changing much with time, a way to predict the wave appearance—before it
reaches the buoy—is needed. There are several ideas on how to estimate the wave
shape before the actual wave reaches the buoy. One of them is to use sensors to
sample the wave shape at a few location before the wave reaches the buoy. The wave
shape at the buoy can then be estimated e.g, by a Kalman filter. Wave prediction is
not included in this thesis, so we assume that Te and Hs are constant when irregular
waves are modeled.
24
Chapter 3
Modeling
This chapter covers the numeric modeling of the WEC where the three sections
covers: the modeling approach and some assumption that was made to simplify the
model; the modeling of the buoy, with the hydrodynamic forces acting on the buoy;
the model extension to also include the PTO system, giving us the final model of
the whole WEC.
3.1
Modeling Approach
The model is restricted to ocean waves moving with two degrees of freedom, x and
z, where the surface elevation is denoted η(x, t). Because of the symmetry of the
buoy it can be considered to move in two dimensions as well. When modeling the
buoy we use the degrees of freedom: surge, heave and pitch (see figure 2.2), where
surge and heave are the motions in x and z-direction and pitch is the rotation
around the y-axis (see figure 3.1).
Figure 3.1: A sketch of the degrees of freedom used when simulating the buoy movement
in a wave (see figure 2.2).
The hydrodynamic forces acting on the buoy are derived from Linear Wave Theory (see section 2.2). We use generalized forces to represent the forces acting on
the buoy, which include the forces in surge and heave, and the torque acting in
pitch. The generalized hydrodynamic forces are obtained using the computer pro25
CHAPTER 3. MODELING
gram WAMIT (see section 2.3.4).
We use Simulink [14] as the simulation program to model the WEC. Simulink is
preferred to regular MATLAB code since it better overviews the model, which in
turn makes it easy for other users to understand the model and develop it further.
To build a GUI (graphical user interface), we use the MATLAB user interface tool
GUIDE [11], which communicates with the Simulink model. The GUI is developed
to make the model easy to use.
The numerical solver we use is ode45, since it is both robust and relatively fast.
ode45 is provided by MATLAB and is based on the Dormand-Prince Runga-Kutta
(4-5) formula [13].
Overview of the Simulink model
The mathematical model of the WEC is implemented in Simulink. The model
includes a block calculating the hydrodynamic forces (see section 2.3), a block calculating the PTO forces (see section 2.4) and a controller block that handles the
PTO settings and the Phase Control. Figure 3.2 overviews the complete Simulink
implementation.
Switching states
block
ẍ
FP T O
FP T O
block
τ hydro
τ hydro
block
1
s
ẋ
1
s
x
Controller
Figure 3.2: A sketch of the complete WEC model in SIMULINK. The position vector x
includes the buoy q, the oscillator xosc and the flywheels ξi . The 1/s block represent an
integration.
In the following section the implementation of the buoy with the hydrodynamic and
hydrostatic forces is covered.
26
3.2. MODELING OF THE BUOY
3.2
Modeling of the buoy
The modeling of the buoy comes down to the generalized hydrodynamic and hydrostatic forces acting on the buoy: τ rad , τ exc , τ hyst and τ drag (see section 2.3 ).
In this section we describe the model implementation of these forces. Before we
go into detail, we first overview the complete Simulink implementation of the buoy.
3.2.1
Simulink implementation
Simulink is a graphical modeling and simulation program, where blocks are used
to create a block diagram of a model. Blocks included in Simulink are e.g. the
integrator block, used for integration and the gain block, used to multiply an input
with a constant or matrix. The equation of motion for a buoy oscillating in a wave
(2.48),
(m∞ + Imb )q̈ = τ exc + τ hyst + µrad + τ drag ,
(3.1)
is implemented in Simulink. An outline of the Simulink model is seen in figure 3.3
where the excitation force is the input and the driving force of the model.
Figure 3.3: An overview of the Simulink model of the buoy. The buoy acceleration q̈ is
calculated by dividing the force vector with the buoy mass. The acceleration vector is then
integrated one time to get the velocity q̇ and then one more time to get the position vector
q. The position and velocity vector is then used to calculate the forces (see equation 3.1).
In the following sections the modeling of the forces are explained in more detail.
27
CHAPTER 3. MODELING
3.2.2
Radiation Force
The derivation of the generalized radiation force was done in section 2.3.1, where
we saw that the radiation force can be written as (2.37)
τ rad = −m∞ q̈ + µrad
(3.2)
The infinitely added mass m∞ is constant, and it can therefore be seen as a part
of the buoy mass mb . The infinitely added mass is an output from the computer
program WAMIT.
The computer program WAMIT gives the added mass matrix A(ω) and the damping
matrix B(ω) explained in (2.36), as an output. The frequency dependent transfer
function K̂(ω) can then be calculated by (2.39).
The output-input system in (2.38), with the transfer function K̂(ω), can in time
domain be approximated with a state-space approximation [3] (see Appendix B for
a detailed explanation).
ż(t) = Ãz(t) + B̃q̇(t)
(3.3)
µrad = C̃z(t) + D̃q̇(t)
(3.4)
The state-space approximation, for all three degrees of freedom, is calculated with
two toolboxes: the SS Fitting toolbox [4] that provides a state-space model from
the WAMIT data and the MSS FSI (Marine Systems Simulator) [7] that identifies
the fluid memory effects of the buoy. The inputs to the toolbox are the WAMIT
data: m∞ , A(ω) and B(ω) .
When we have the state-space matrices in equation (3.3-3.4), we calculate the radiation force with the Simulink state-space block (see figure 3.4).
Figure 3.4: The Simulink State-Space block which takes the buoy velocity q̇ as an input and
calculates the convolution part of the radiation force µrad by a state-space approximation
(see equation (3.3-3.4).
28
3.2. MODELING OF THE BUOY
3.2.3
Excitation Force
The generalized excitation force τ exc (see section 2.3.2) does not depend on the
motion of the buoy itself but instead on the buoy shape and the wave elevation
η(x, t) (see equation (2.22)). The excitation force in the k:th degree of freedom is
calculated by [8]:
τexc,k = Re
"N
X
#
F̂k,i (ωi )η̂i (x)e
jωi t
.
(3.5)
i
where N are the number of regular weaves included in η(x, t), η̂i ejωi t is the surface
elevation in frequency domain and F̂k,i (ω) is the frequency dependent excitation
force in the k:th degree of freedom (output from WAMIT).
The excitation force is a pre-calculated time dependent vector1 , since the free surface
elevation is pre-calculated. WAMIT provides the frequency dependent excitation
force for a range of frequencies. The excitation force is then calculated using the
frequencies included in the wave. For a regular wave, only one of the frequencies is
used in the excitation force vector. An irregular wave consists of a range of frequencies, calculated from the JONSWAP energy density spectrum (see section 2.2.2).
Since the surface elevation η(x, t) depends on the significant wave height Hs and
the energy period Te then so does the excitation force.
The excitation force is modeled as a pre-calculated time dependent vector by using
the SIMULINK pre-lookup table and interpolation block. The lookup table locates
the correct location in the excitation force vector and the interpolation block finds
the excitation force at that position (see figure 3.5).
Figure 3.5: The excitation force in heave—pre-calculated by (3.5)—is calculated at the
current time t using a pre-lookup table together with an interpolation block.
1
This part of the implementation is partially based on code developed by Marco Alves at
WavEC.
29
CHAPTER 3. MODELING
3.2.4
Hydrostatic and Drag Force
The hydrostatic stiffness matrix S from equation (2.45), is an output from WAMIT.
The stiffness matrix is modeled as a gain block in SIMULINK, with the position
vector q as an input.
The drag force is calculated with equation (2.46). The drag coefficient CD is an
input to the model. It is now set to 0.2 but will be modified later with test data
from real experiments. The drag force is modeled in Simulink with a gain block,
with q̇2 as input.
After the implementation of the hydrodynamic and hydrostatic forces, it is now
time to implement the PTO.
3.3
Implementing the PTO
In this section the PTO (see section 2.4) is merged into the model of the WEC.
This gives us a complete model of the WEC.
The motion of the complete WEC is derived in section 2.5, where we explained
that buoy is connected to the PTO with a wire. A tense wire entails that the buoy
and the PTO are connected and can be modeled as one system, while a loose wire
entails that the buoy and the PTO must be modeled as two different systems. We
introduce a new variable W T :
(
WT =
1,
0,
when wire is tense,
when wire is loose.
(3.6)
The equations describing the system differs depending on the value of W T . The
equations (2.55)–(2.58) can using W T , be written as
(I(mb + W T · mosc ) + m∞ ) q̈ = τ exc + τ hyst + µrad + τ drag + W T · FP T O . (3.7)
and
ẍosc = FP T O (1 − W T )/mosc + ẍbuoy · W T.
(3.8)
where ẍbuoy is the acceleration of the buoy projected to the direction of the wire.
Energy conversion devices
The energy conversion devices, explained in (2.51) and (2.50), are either engaged
or disengaged. When they are engaged, their velocities and acceleration are equal
to the ones of the oscillator.
u
u
ξ¨j = ẍosc
, and ξ˙j = ẋosc
,
rpin
rpin
when flywheel j is engaged.
30
(3.9)
3.3. IMPLEMENTING THE PTO
When they are disengaged they moves freely with the equation
Pf rac,j Tgen,j
ξ¨j = −
= ξ¨j,f ree ,
Jf w
when flywheel j is disengaged.
(3.10)
The energy converting devices are disengaged when the free acceleration ξ¨j is higher
than the acceleration of the oscillator. For an engaged energy converter we have
the relation
when |ξ¨j,f ree | =< |ẍosc
Stay engaged,
u
|
rpin
u
|.
when |ξ¨j,f ree | > |ẍosc
rpin
Disengage,
Phase Control
The Phase Control algorithm with latching is the last part we include in our system. The control algorithm locks the movement of the oscillator at certain times.
No motion in the oscillator occurs which means that the length of the wire together
with the PTO stays constant. The buoy can still move, but not in the direction of
the wire. The real PTO has latch valves, which are closed when the WEC latched.
This does not require any extra force. To model this scenario we add an imaginary
reaction force to stop the oscillator motion.
All forces except the ones perpendicular to the wire cancel out due to the reaction force in the wire. We model this by including a reaction force inside the power
take, and we call this force the latching force Flatch . This force includes a force to
cancel the hydrodynamic forces in the direction of the wire Freaction (see figure 3.6)
and a PD (proportional-derivative) regulator which forces the oscillator to stay in
the locked position.
The force Flatch is calculated with
Flatch = Freaction − P (xosc − xosc,ref ) − Dẋosc ,
(3.11)
where P and D are constants in the PD regulator, and xosc,ref is the reference
position, where the oscillator was locked.
The buoy is latched when the heave-velocity is zero (see section 2.6). The unlatching timing is decided by the surface elevation η(x, t), where the threshold ηthres is
calculated by equation (2.60). The buoy is unlatched when the magnitude of the
surface elevation is larger than the threshold ηthres .
31
CHAPTER 3. MODELING
ez
Fheave
ex
α
Reaction force on buoy during latching
Freaction = − (Fsurge,proj + Fheave,proj )
α
= − (Fsurge sin α + Fheave cos α) ,
Fsurge
α
Wire
Figure 3.6: A sketch showing the resulting forces acting on the buoy during latching. The
hydrodynamic forces in the direction of the wire are canceled out by the reacting force from
the wire.
Final model of the WEC
We now have the information we need to finalize the model of the WEC. Figure 3.7
shows a sketch of the complete model of the system.
3.4
Linear damping
Linear damping is an alternative way to extract energy from a WEC and we use it
to evaluate the simulation output. Linear damping is achieved by adding a damping
force Fdamp to the oscillator, proportional to the oscillator velocity ẋosc ,
Fdamp = µdamp ẋosc
(3.12)
while the energy converting devices with the flywheels are turned of.
Since the generators are turned off we need another way to calculate the power
output. We use the mechanical power output, which is calculated as force times
velocity. Thus, the power output from linear damping is
P = Fdamp ẋosc = µdamp ẋ2osc .
(3.13)
We call it optimal linear damping since we adjust the damping coefficient µdamp to
find the maximal power output. When Phase Control with the energy converting
devices are used the mechanical power output is calculated by
P = Fwire ẋosc
where Fwire is the force the wire performs on the oscillator.
32
(3.14)
3.4. LINEAR DAMPING
WEC is latched
Yes
Is the buoy
latched ?
(mb + m∞ + W T · mosc ) q̈ = τ hydro + W T · (FP T O + Flatched )
ẍosc = 0
ξ¨1 = ξ¨1,f ree
ξ¨2 = ξ¨2,f ree
No
Up motion, EC 1 engaged
(I(mb + W T · mosc ) + m∞ ) q̈ = τ hydro + W T · FP T O .
EC
1
ẍosc = FP T O (1 − W T )/mosc + ẍbuoy · W T
ξ¨1 = ξ¨1,f ree
u
ξ¨2 = ẍosc
rpin
Down motion, EC 2 engaged
Which energy
EC 2
converting unit
is engaged?
(I(mb + W T · mosc ) + m∞ ) q̈ = τ hydro + W T · FP T O .
ne
No
ẍosc = FP T O (1 − W T )/mosc + ẍbuoy · W T
u
ξ¨1 = ẍosc
rpin
¨
¨
ξ2 = ξ2,f ree
Up and down motion, EC1 & EC2, disengaged
(I(mb + W T · mosc ) + m∞ ) q̈ = τ hydro + W T · FP T O .
ẍosc = FP T O (1 − W T )/mosc + ẍbuoy · W T
ξ¨1 = ξ¨1,f ree
ξ¨2 = ξ¨2,f ree
Figure 3.7: A sketch showing the system of the WEC, including the different states of the
system. EC1 and EC2 stand for energy converter one and two.
33
Chapter 4
Simulation results
This chapter contains results from the model simulations. We start by showing
the buoy motion modeled without the PTO. Then we add the PTO to the model
to see the motion of the complete WEC. This is followed by checks on modeling
approximations and numerical accuracy. Last the effects of Phase Control on the
power output is shown.
Velocity and position
The buoy motion when the buoy is modeled alone and with the complete WEC
is shown in figure 4.1 and figure 4.2. Both regular waves and irregular waves are
shown, with a significant wave height of 2.26 m and an energy period of 8 s.
Figure 4.1: Position of the buoy in heave in a regular and an irregular wave plotted with
the surface elevation, when only the buoy is modeled.
35
CHAPTER 4. SIMULATION RESULTS
Figure 4.2: Position of the buoy in heave when Phase Control is used in a regular and an
irregular wave, plotted with the surface elevation.
The velocities of the oscillator and the flywheels are important because they are
directly proportional to the power output. Figure 4.3 shows the oscillator and flywheel velocities in a regular wave with a wave height of 2.26 m and an energy period
of 8 s. The oscillator velocity is zero when the buoy is latched.
Figure 4.3: The velocities of the oscillator ẋosc and flywheels ξ˙j when Phase Control is
used. The flywheel velocities are scaled to match the oscillator velocity.
36
Modeling approximations and numerical accuracy
The numerical accuracy of the model is tested by calculating the root mean square
error of the approximated oscillator velocity vector ẋosc,appr . The error measures
the deviation from the true solution ẋosc by
v
u
n
u1 X
error = t
(ẋosc,appr,i − ẋosc,i )2
n
(4.1)
i=1
where n is the number of time steps in the vectors. The vector ẋosc,appr is first
interpolated, so that both vectors have the same length.
The true solution is approximated by the numerical when the relative tolerance
is set to 1e − 9 and the absolute tolerance is set to 1e − 10, which are inputs in
Simulink. The simulation time for the varying accuracies is also noted. The results
are presented in figure 4.4.
The accuracy of the radiation force depends on the number of states included in
the state-space approximation. The force is modeled in a regular wave when Phase
Control is used (see figure 4.5). In the simulation it is modeled with a two–state
approximation. For comparison a 21–state approximation —which gives an almost
exact fit to the convolution part of the radiation force µrad —is used as ground truth.
Power absorption and Phase Control
The model is tested with and without the Phase Control algorithm. The buoy position and velocity in heave for the two cases are plotted in figure 4.6. The average
electrical power output from the generator is calculated for the two cases. The
power is obtained by using a generator power curve, which is a curve—unique for
the specific generator—mapping the generator rotation speed to a certain power
output.
We now compare the Phase Control algorithm with the PTO settings to optimal
linear damping explained in section 3.4. Figure 4.7 shows the result for different
wave heights and wave periods. For each wave state, the mechanical power output
is optimized (by changing the damping coefficient) to get the highest possible power
output.
37
CHAPTER 4. SIMULATION RESULTS
(a) Regular wave
(b) Irregular wave
Figure 4.4: Measured root mean square error and simulation time for a solution of the
oscillator velocity ẋosc in, (a) a regular wave and (b) an irregular wave. The relative
tolerance is set to absolute tolerance divided by 10. The solutions are compared with a
solution with absolute tolerance = 1e − 10.
38
Figure 4.5: The radiation force in heave modeled with 2 and 21 states in the state space
approximation. The 21–state approximation is interpreted as the true solution.
Figure 4.6: Motion of the buoy when Phase control is used and when it is turned off. The
average electrical power output is 170 kW and 9 kW respectively.
39
CHAPTER 4. SIMULATION RESULTS
Figure 4.7: The mechanical power output depending on wave period T (H is set to 2m)
and wave height H (T is set to 8s) for a WEC using optimal linear damping and Phase
Control. The mechanical power is calculated as the force acting on the oscillator from the
wire times the oscillator velocity ẋosc .
40
Chapter 5
Discussion
The WEC model was successfully implemented in Simulink. The model calculates
the buoy and the PTO positions and velocities along with the power. The Phase
Control algorithm successfully optimizes the buoy motion, thereby increasing the
power output. Our simulations indicated that CorPower Ocean’s method works.
Modeling approximations and numerical accuracy
In figure 4.4 we investigated the numerical accuracy of the model. The curves show
that the simulation error smoothly increases—while the simulation time smoothly
decreases—with increasing accuracy. The smoothness implies that there is no critical limit in the tolerance that must be met. The accuracy is a compromise we both
want an accurate model and a fast program. We settle for a relative tolerance of
1e − 4 and an absolute tolerance of 1e − 5, which gives a reasonably high numerical
accuracy and a relatively fast simulation time.
The radiation force is approximated with a state-space approximation per degree of
freedom. Hence, we use a two-state approximation when we model only heave and
four-state approximation when we model surge and heave.
Figure 4.5 shows the the simulation result for radiation force in heave. The twostate approximation is here plotted together with a 21-state approximation—which
is close to the true solution.
We see an under-shoot in the two-state curve compared to the 21-state curve, which
occurs when the buoy is being latched. This is because the two-state approximation
cannot completely resolve the irregularity of the buoy movement at that instant.
Apart from that, the two curves are close to equal and since the simulation time is
longer with a large state-vector; our choice falls on the two-state approximation.
41
CHAPTER 5. DISCUSSION
Power absorption and Phase Control
We can see that Phase Control has a positive effect on the buoy motion—as suggested in [5]—by looking at figure 4.6. By adding the Phase Control algorithm, the
buoy amplitude increases from 0.8 to 1.6 m and the peak velocity increases from
0.8 m/s to 3.2 m/s. This shows how Phase Control increases the power output.
A way to evaluate the Phase Control algorithm together with the PTO is to compare
it to optimal linear damping, where energy is extracted from the WEC by adding a
damping coefficient to the oscillator velocity; as done in figure 4.7. We can see that
Phase Control with the PTO is much more effective at absorbing wave energy. In
the same figure we see that the power output increases with wave height and wave
period. This is expected since the energy transported in the wave increases with
these parameters.
Model limitations
The WEC model uses linearized hydrodynamic forces. WAMIT linearizes these
around the buoy equilibrium point; thus the simulation results are the most reliable
for small buoy amplitudes. It is a difficult task to draw a line when the model is no
longer valid; but one limit can be drawn.
This occurs when the buoy is completely submerged (the buoy is under the water
surface). Since WAMIT assumes that this never happens, data from this point are
no longer valid. It is therefore important to keep track of this during the simulation.
Future studies
The Phase Control algorithm benefit from being extended to include wave prediction (discussed briefly in section 2.6). This can be done e.g by collecting wave data
before the wave reaches the buoy, and then estimate the wave elevation at the buoy
position using a Kalman filter.
The limitation on the buoy amplitude, due to the linearized hydrodynamic forces
(discussed in Model limitations), is the main restriction of the model. Further
research on ways to overcome this is needed. An idea could be to get WAMIT data
using different equilibrium positions as input, including the case when the buoy is
completely submerged. The model can then switch between different WAMIT data,
depending on its position.
The pitch motion is not yet implemented due to time limitation. This is presumed
to have a modest effect on the result, since the pitch motion is assumed to be small.
Finally, the model must be validated with test data from real measurements. This
is an activity ongoing, but outside the scope of this thesis project.
42
5.1. SUMMARY
5.1
Summary
Wave energy is a promising renewable energy resource because of its high energy
density. No one has yet found a way to extract this energy efficiently enough, even
though many have tried. The company CorPower Ocean has developed a Wave
Energy Converter (WEC) of point absorbing type, which might be able to achieve
this. It uses a relatively small buoy, a special type of Power Take Off (PTO) and a
control strategy: Phase Control by latching.
Phase Control aims at making the buoy move in phase with the incident wave.
This optimizes the velocity of the buoy, which leads to an optimized power output.
In this thesis we develop a mathematical model of this WEC. The buoy is modeled
using hydrodynamic forces, derived from Linear Wave Theory. The forces are calculated using the program WAMIT; which returns the frequency dependent forces
for a certain buoy shape. The models of the buoy and the PTO are merged to get
a complete model of the WEC.
The reason for developing the model is to provide a tool, which can be used by
CorPower Ocean during the WEC development. We also want to evaluate the
effect of Phase Control on the buoy motion and the WEC power output.
The implementation is done in the graphical simulation program Simulink, where
different blocks represent different parts of the model: one block calculates the hydrodynamic forces, one block calculates the PTO forces and one block merges the
two blocks into a single system of PDEs. A fourth controlling block is also implemented that controls the PTO settings and the Phase Control algorithm. The
model is controlled by a graphical user interface that sets parameters and shows the
simulation results.
The results from the simulations show that the buoy motion increases when the
PTO and the Phase Control are turned on. In particular, the velocity increases
significantly, which in turn increases the power output. To evaluate the WEC we
compare it with linear damping (which extracts wave energy by adding a damping
force onto the oscillator). By looking at the results, we see that the CorPower
WEC outperforms linear damping at extracting wave energy.
In conclusion, the CorPower WEC holds great promise and we hope that its
realization will turn out just as good.
43
Bibliography
[1]
WAMIT, 2010. WAMIT User Manual, http://www.wamit.com.
[2]
Ocean wave spectra, 2012.
Spectra.
[3]
M. Alves, M. Vicente, A. Sarmento, and M. Guerine. Implementation and
verification of a time domain model to simulate the dynamics of OWCs. In
Proceedings of the 9th European Wave and Tidal Energy Conference (EWTEC
2011), Southampton, UK, 2011.
[4]
Tiago Durate. NWTC Computer-Aided Engineering Tools (SS Fitting by
Tiago Duarte), 2013. http://wind.nrel.gov/designcodes/preprocessors/
SS_Fitting.
[5]
J. Falnes. Ocean Waves and Oscillating Systems. Cambridge University Press,
2002.
[6]
J. Falnes. A revire of wave-energy extraction. Marine Structures, Volume 20,
2007.
[7]
Thor I. Fossen and Tristan Perez. A Matlab Tool for Parametric Identification
of Radiation-Force Models of Ships and Offshore Structures. Modelling, Identification and Control, MIC-30(1):1-15, 2009. http://www.marinecontrol.org.
[8]
Jorgen Hals. Modelling a phase control of wave-energy converters. PhD thesis,
NTNU-Norwegian University of Sience and Technology, 2010.
[9]
Pijush K. Kundu, Ira M. Cohen, and David R. Dowling. Fluid Mechanics.
Academic Press, 5th edition.
http://www.wikiwaves.org/Ocean-Wave_
[10] M.F.P. Lopes, J. Hals, R.P F. Gomes, T. Moan, L.M.C. Gato, and A.F.de O.
Falcão. Experimental and numerical investigation of non-predictive phasecontrol strategies for a point-absorbing wave energy converter. Ocean Engineering, Volume 36, 2009.
[11] MATLAB. version 8.1 (R2013a). The MathWorks Inc, Natick, Massachusetts,
2013.
45
BIBLIOGRAPHY
[12] Tristan Perez and Thor I. Fossen. Joint Identification of Infinite-Frequency
Added Mass and Fluid-Memory Models of Marine Structures. Modeling, Identification and Control, Volume 29, 2008.
[13] Lawrence F. Shampine and Mark W. Reichelt. The MATLAB ODE suite.
SIAM Journal on Scientific Computing, Volume 18, 1997.
[14] Simulink. version 8.1 (R2013a). The MathWorks Inc., Natick, Massachusetts,
2013.
[15] A. Vretblad. Fourier Analysis and its applications. Springer, 2000.
46
Appendix A
Radiation Force Solution
The complex radiation potential φ̂rad can be written as
φ̂rad = iω ϕ̂Trad q̂
(A.1)
where q̂ is the generalized complex position vector and ϕ̂rad is the complex unit
amplitude radiation potential. The generalized radiation force derived from the
radiation potential is
τ rad = ρ
ZZ
i
h
Sb
(A.2)
ñ Re ω 2 ϕ̂Trad q̂ eiωt dS
The potential ϕ̂rad can be divided into an imaginary and a complex part.
ϕ̂rad = ϕrad,re + i ϕrad,im .
(A.3)
where ϕrad,re and ϕrad,im are real. This expression can be used in A.2.
ZZ
τ rad =
=
ρ
Sb
ZZ
ρ
Sb
ñ Re
h
ñ Re
ϕTrad,re ω 2 q̂eiωt
i
h
dS +
i
ϕTrad,re + iϕTrad,im ω 2 q̂eiωt dS
ZZ
h
Sb
i
ñ Re ϕTrad,im iω 2 q̂eiωt dS
where q̂ does not depend on the buoy wetted surface Sb . The equation can therefore
be written as
τ rad = −ρ
ZZ
h
Sb
i
ñϕTrad,re dS Re −ω 2 q̂eiωt + ρ ω
ZZ
The generalized velocity can be expressed as
h
i
q̇ = Re iω q̂eiωt ,
and the generalized acceleration can be expressed as
h
i
q̈ = Re −ω 2 q̂eiωt .
47
h
Sb
i
ñϕTrad,im dS Re iω q̂eiωt .
APPENDIX A. RADIATION FORCE SOLUTION
so the expression for the generalized radiation force τ rad is
τ rad =
ZZ
− ρ
Sb
ñϕTr,re dS q̈ − −ρ ω
=
ZZ
Sb
ñϕTr,im dS q̇
−A(ω)q̈ − B(ω)q̇.
where A(ω) and B(ω) ∈ R3×3 , since the we model in three degrees of freedom.
The matrices are two frequency dependent symmetric matrices that depend on the
shape of the buoy and the radiation potential.
ZZ
A(ω) = ρ
B(ω) = −ρω
Sb
ñϕTr,re dS
ZZ
Sb
ñϕTr,im dS
The matrix A(ω) is called the added mass matrix while B(ω) is called the damping
matrix. If we let the frequency go to infinity we get the so call infinitely added mass
m∞ .
48
Appendix B
State-Space approximations
The fluid memory in the radiation force term is expressed as
µrad = −
Z t
0
K(t − t0 )q̇dt0 .
(B.1)
The goal is to express this equation with a state-space representation.
We start by looking at some properties of the Fourier transformation. Let u(t)
be the input signal, y(t) the output signal and H(t) the convolution kernel of a
system.
Z
y(t) =
t
0
H(t − τ )u(τ )dτ
(B.2)
The Fourier transform of the kernel H(t), the output signal y(t) and the input signal
is u(t) are,
Z
Ĥ(ω) = F {H(t)} =
ŷ(ω) = F {y(t)} =
û(ω) = F {u(t)} =
∞
−∞
Z ∞
−∞
Z ∞
H(t)e−iωt dt,
(B.3)
y(t)e−iωt dt
(B.4)
u(t)e−iωt dt
(B.5)
−∞
The Fourier transform of the convolution integral in equation B.2 is the same as
taking the product of the Fourier transform of the two convolution factors [15].
Z t
F
0
H(t − τ )u(τ )dτ
= Ĥ(ω)û(ω),
(B.6)
In the frequency domain the integral in equation B.2 is
ŷ(ω) = Ĥ(ω)û(ω).
(B.7)
A state-space representation is of the form
ż(t) = Az(t) + Bu(t)
(B.8)
y(t) = Cz(t) + Du(t)
(B.9)
49
APPENDIX B. STATE-SPACE APPROXIMATIONS
where z(t) ∈ Rn×1 , A ∈ Rn×n , B ∈ Rn×1 , C ∈ R1×n , D ∈ R and n is the number of
state the convolution integral is approximate with. By taking the Fourier transform
of the state space representation, we get
iωẑ(ω) = Aẑ(ω) + Bû(ω)
(B.10)
ŷ(ω) = Cẑ(ω) + Dû(ω)
(B.11)
We can re-write the state-space approximation on an input-output form.
ŷ(ω) = C (iωI − A) B−1 + D û(ω).
(B.12)
For the state-space approximation to approximate the input-output system in equation B.7 we need
Ĥ(ω) ≈ C (iωI − A) B−1 + D.
(B.13)
Let us now go back the function B.1. We can now take the Fourier transform of the
convolution integral,
µ̂(ω) = −K̂(ω)û(ω).
(B.14)
where K̂(s) is a matrix of transfer functions and û(s) is the Laplace transform of
the velocity vector q̇. We can now find a state space representation for all transfer
functions K1..n (s), where n is the number of degrees of freedom in the system dof
plus the coupling of the degrees of freedom. We then take the direct sum of these
state-space approximation to get a global state space representation for the whole
system [3].

A1



à = 


0
..
.
0
A2
..
.
0
···
..
.
..
.
0
.. 
. 


· · · An

,


(B.15)

B1
 
 B2 

B̃ = 
 ..  ,
 . 
(B.16)
Bn
h
i
C̃ = C1 C2 · · · Cn ,

D1
(B.17)



 D2 

D̃ = 
 ..  .
 . 
(B.18)
Ddof
The global state-space representation for the radiation force is
ż(t) = Ãz(t) + B̃q̇
(B.19)
µrad = C̃z(t) + D̃q̇.
(B.20)
This is the expression used in the simulation to approximate the radiation force.
50
TRITA-MAT-E 2013:55
ISRN-KTH/MAT/E—13/55-SE
www.kth.se