Analysis and Evaluation of the Wavebox Wave Energy Converter

UPTEC F11 018

Examensarbete 30 hp Mars 2011

Analysis and Evaluation of the

Wavebox Wave Energy Converter

Björn Gotthardsson

Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Analysis and Evaluation of the Wavebox Wave Energy Converter

Björn Gotthardsson

Increasing attention to climate change in combination with ever-growing energy consumption worldwide has boosted the demand for new green energy sources.

Wave power is developing in many different branches to become part of the new era of electricity production. This thesis deals with a wave power system in its primary stages of development. The system was investigated in order to estimate its potential to produce electric power from sea waves. It is a system consisting of a moored buoy to which the energy is transferred when the wave tilts the buoy in the pitch direction.

Due to the increased pitch angle, an amount of liquid contained inside the buoy is allowed to flow via ramps to an upper container, from where it flows down through a hydroelectric turbine. A computer program was used to calculate the properties of the buoy in sea waves. Another program was written in MATLAB to simulate the movements in sea waves and from a set of given parameters calculate the power output. A brief economic study was made to determine if the power output was large enough for the concept to be of financial interest to any future investors. The results show that the wave power system produced 0.9 kW in a wave climate equal to that off the coast of Hanstholm, Denmark, and 1.6 kW in a wave climate off the coast of San Diego, USA. The economic study shows that the power output needed to be improved by a factor of at least five to have a chance of being economically viable. A number of enhancements were suggested to increase the power output of the system, and further investigation could be of use to improve the concept. The created computer simulation model, as well as the results in this thesis could be valuable in any future research on the concept.

Sponsor: Vattenfall AB

ISSN: 1401-5757, UPTEC F11 018 Examinator: Tomas Nyberg Ämnesgranskare: Magnus Rahm Handledare: Richard Bensow

Sammanfattning

Energibehovet i världen växer, samtidigt som medvetenheten om klimatförändringarnas orsaker och verkan blir större, vilket har lett till ett ökat globalt intresse för förnyelsebar energi. Vågkraft är ett av de nya gröna energislagen som hoppas kunna bidra till morgondagens energisamhälle. I det här examensarbetet presenteras en analys av ett nytt vågkraftkoncept. En utredning gjordes för att försöka ta reda på konceptets potential att omvandla vågornas energi till elenergi, och en kortare undersökning av dess ekonomiska potential utfördes. Vågkraftverket är pitch-baserat och består av en bottenförankrad boj, till vilken energiöverföringen sker när vågen får bojen att vrida sig kring en axel vinkelrät mot vågens utbredningsriktning. En innesluten vätskemängd, i botten av bojen, fås av denna vridning att flöda till en övre reservoar inne i vågkraftverket, varifrån det rinner ner genom en lågtrycksturbin kopplad till en generator, för att sedan nå botten på kraftverket igen.

Datorprogrammet SCORES användes för att ta reda på bojens sjöegenskaper, och ett annat program skrevs i MATLAB för att simulera bojens rörelser till havs och från detta räkna ut vilken effekt som kan levereras. Resultaten visar att vågkraftverket kan leverera 0,9 kW i en vågsituation motsvarande den i havet utanför Hanstholm i Danmark och 1,6 kW i en vågsituation motsvarande den i havet utanför San Diego i USA. Den ekonomiska undersökningen visar att effekten behöver förbättras och för att ha en chans att vara ekonomiskt gångbar krävs en ökning av effekten med en faktor fem. En rad olika förändringar föreslås för att förbättra effekten och ytterligare utredning är en förutsättning för att konceptet ska kunna tillgodogöra sig förbättringarna. För framtida undersökning och utveckling av konceptet kan, förutom resultaten i denna rapport, även simuleringsprogrammet användas.

ii

Nomenclature

Symbol Quantity Unit

a Wave amplitude m

 Wave length m

 Angular frequency rad/s

f Wave frequency s^-1

k Angular wave number rad/m

c Phase velocity of waves m/s

cg Group velocity of waves m/s

d Depth m

u Velocity m/s

 Velocity potential m²/s

 Wave elevation m

F Force N

M Mass kg

 Density kg/m³

E Energy J

P Power W

J Energy transport W/m

T Wave period s

Tm Mean wave period s

Tp Peak wave period s

H Wave height m

Hs Significant wave height m

HRMS Root mean square wave height m

S Spectral density m²s

m Spectral moment

L Length of Wavebox m

B Breadth of Wavebox m

H_WB Height of Wavebox m

D Draft of Wavebox m

 Lower ramp angle of Wavebox rad

 Upper ramp angle of Wavebox rad

RG Radius of gyration m

h Head of Wavebox m

q Mean flow of water inside Wavebox m³/s

 Efficiency

i Transfer function component

Yi RAO component

 Phase angle rad

Gn Discount factor

NPV Net present value €

Tl Life time years

r Interest

Cinv Investment cost €

p Electricity price €

t Running hours per year hours/year

Kn Maintenance costs €

Table of contents

1 Introduction ... 1

1.1 Wavebox ... 1

1.2 Objective ... 3

1.3 Methods ... 3

1.4 Limitations ... 3

2 Background... 5

2.1 Existing wave power techniques ... 5

2.1.1 Wave Dragon ... 5

2.1.2 Pelamis ... 6

2.1.3 The Lysekil project ... 6

3 Theory ... 8

3.1 Airy wave theory ... 8

3.2 Wave spectrum ... 11

3.3 Motion of floating bodies in sea waves ... 14

3.4 Software ... 15

3.4.1 SCORES... 16

3.4.2 MATLAB ... 17

3.5 Net present value ... 17

4 Wave data ... 18

5 Analysing Wavebox ... 20

5.1 The hull ... 20

5.1.1 Different hull shapes... 23

5.1.2 Radius of gyration ... 25

5.1.3 Eigenfrequency in pitch ... 26

iv

5.2 MATLAB simulation ... 27

5.2.1 Head ... 30

5.2.2 Flow rate ... 32

5.3 The turbine ... 34

5.4 Mooring ... 35

6 Results ... 36

6.1 Hull shape ... 36

6.2 Power output ... 36

7 Economic perspective ... 42

8 Discussion ... 43

9 Conclusions... 45

Acknowledgements ... 46

References ... 47

1 Introduction

With a world facing climate change due to overuse of fossil fuels, it is not hard to motivate clean, zero-emission electricity production. When energy demands are rising worldwide and the need for environmentally friendly technologies is becoming greater, wave power is aspiring to become a part of the new green electricity producing technology spectrum. The wave resource is enormous with an estimated magnitude as large as the world demand for electricity [1]. Even if just a few percent of this huge resource could be harvested, wave power would contribute significantly to global electricity demands. However, there are many challenges to overcome for wave power to be a part of tomorrow’s even more energy craving world. Most importantly, time and money needs to be invested in new technology, in order to determine which of the different approaches are sufficiently effective without being too costly.

1.1 Wavebox

The Wavebox Wave Energy Converter is a concept proposed by inventor Richard Nilsson. It is in its very first stages of development. No prototype has been built as of December 2010, but a Swedish patent is pending.

The Wavebox WEC is a pitch-based wave power system, consisting of a moored buoy with an amount of water contained inside it, as depicted in Figure 1.1. The energy is transferred from the wave when it causes the buoy to lean in the pitch direction, i e, a rotation around an axis perpendicular to the waves’ direction of propagation. If the buoy leans at a sufficient pitch angle the water contained inside the buoy flows to a separate container located at the side end of the buoy, which can be seen in Figure 1.2. The water is prevented from flowing back to the bottom by a hatch. When the wave crest has past the buoy, it will start to lean in the opposite way causing the water in the side container to flow to the upper middle of the buoy where it will fall into a third container, as shown in Figure 1.3. From this upper container the water is allowed to flow down through a hydroelectric turbine, after which it reaches the bottom container of the buoy from where it started.

Figure 1.1: Cross section of Wavebox.

2 Figure 1.2: Cross section of Wavebox at a pitch angle large enough to make water flow from the bottom container to the right side container.

Figure 1.3: Cross section of Wavebox at a pitch angle large enough to make water flow from the bottom container to the left side container and from the right side container to the upper container. Water in the upper container flows through a hydroelectric turbine back to the bottom container.

The energy transfer from wave to electricity can be divided into different steps of energy conversion. Firstly, the wave transfers energy to the buoy by rotating it in the pitch direction causing the enclosed water to gain kinetic energy when flowing to the side container, as in Figure 1.2. In the next step, the wave rotates the buoy in the opposite direction, as in Figure 1.3. This movement lifts the water, causing it to gain potential energy. The water then flows from the upper container down through the turbine converting its potential energy to kinetic energy which is transferred to the turbine. The turbine is connected to a generator located at the top of the construction. Conversion from kinetic energy to electric energy makes up the final step. The generated electricity is transferred to the mooring at the bottom, and from there on to the power grid on land.

The Wavebox WEC consists of relatively few components, of which most are well-known, e g, generator, turbine and mooring devices. This could be an advantage when trying to reach high reliability and low production costs. The hatches, seen in Figures 1.1, 1.2 and 1.3 are not known to be used in any other application.

In existing wave power techniques such as overtopping systems, e g, Wave Dragon, the turbine has to be designed to withstand the impurities and high corrosion of sea water. The liquid inside Wavebox can be clean fresh water or any other suitable liquid. The contained liquid will be referred to as water in this thesis, unless otherwise stated.

1.2 Objective

The objective of this thesis is to investigate and evaluate the Wavebox concept. It aims to answer whether or not the concept is worth doing more research on, by examining different Wavebox designs and calculating their potential of producing electric power from sea waves.

1.3 Methods

Calculations of hull movements in sea waves are done at SSPA AB (former Statens Skeppsprovningsanstalt) to be able to predict the movements of Wavebox and its properties at sea. The computer program SCORES, described in Chapter 3.4, is used for this task. Different hull configurations are tested in the software at SSPA and the configurations with the most wanted properties are selected for further investigation. This investigation is done in MATLAB, which is used to create a computer simulation model for the selected hull configurations. The main tasks for the MATLAB simulation model will be to show the pitch angles of Wavebox as a function of time and calculate the power output of a given Wavebox configuration. An important parameter is the water flow which is calculated by a geometrical approach. In order to synthesise a realistic wave climate in both SCORES and the MATLAB simulation model, wave data is needed. This data is gathered from meteorological institutes around the world. The results of the MATLAB simulation model are analysed, and if needed, changes are made to the configuration to optimise its performance. The changes are applied in the model, forming the steps of this iterative process. If changes are made to the hull, the configuration is analysed in SCORES to acquire the needed input in the MATLAB simulation model.

In addition, a brief economic analysis is made by calculating the net present value of the investment in Wavebox. The analysis aims to calculate the total income generated during the estimated lifetime of Wavebox and in this way find out the amount of money that can be spent on building it.

1.4 Limitations

The Wavebox WEC has many variable parameters, which have to be determined when going from concept to finished product. These include, but are not limited to, angles of the planes inside the box, water mass inside the box, turbine placement as well as all spatial parameters defining length, breadth and height of the different parts of the box. The parameter space is obviously very large, which leads to a need to simplify the concept, especially since many of the parameters are linked together in advanced ways. The movement of the WEC is caused not only by the sea waves but by the internal water flow. Additionally, the waves are affected by the WEC itself, making the interaction rather complex. Limitations are needed and choosing them is a balance between the quality of the result and the total amount of time spent.

The limitations of this thesis are the following:

4

▪ The movements of Wavebox caused by the movements of the water inside Wavebox are not considered. This limitation is done due to the complexity of this movement and for the reason that SCORES offers no such input when calculating the properties of the hull in sea waves.

▪ Only three shapes of the hull are investigated: the box shape, formed like a straight box; the banana shape, with curved bottom; and the H-shape, formed like an H when seen from above.

▪ The water movement of the contained water is simplified based on the assumption that Wavebox moves smoothly and slowly in the waves. The surface of the contained water can then be assumed to be horizontal at every time.

▪ The turbine and its performance are dependent on the shape and size of its water outlet. The design of this outlet and its impact on the hull design are not considered in the calculation of the hull movement in sea waves. However, the problem of fitting the outlet inside the hull is discussed in Chapter 5.3. The water level in the bottom container is assumed to always be above the bottom of the outlet during operation, eliminating air inside the outlet.

▪ Only unidirectional waves are used to simulate the movements of Wavebox. The placement of the wire used to moor Wavebox will automatically align the construction to the dominating direction of the incoming waves, which motivates this assumption.

▪ The effect of the mooring on the pitch movement is not investigated, mainly due to the lack of this input in SCORES.

▪ Control systems for the turbine and the contained water might be needed but are not investigated.

▪ The hatches inside the Wavebox preventing water from flowing in the wrong direction are not investigated. They are assumed to act as check valves, letting water flow in one direction only.

2 Background

Wave power has been used for electricity production for a hundred years but never in a large scale. The oil crisis in the seventies gave economic motivation for the development of many new wave power concepts, some of which did not survive the dropping oil prices in the eighties. Increasing attention to climate change has again spurred research in the field of wave power, and many new ideas have seen the light in recent years. Today, there are a number of competing approaches to producing electricity from sea waves and it is still uncertain, if any concept has the potential to make it into the commercial stage.

2.1 Existing wave power techniques

Different techniques are often categorised by the method they use to convert the energy of the waves, e g, point absorber, surface following, oscillating water column and overtopping. In this chapter some of the most prominent systems will be presented in order to have something to relate the Wavebox WEC to.

2.1.1 Wave Dragon

Wave Dragon is of the overtopping type and consists of a floating reservoir with an inlet shaped as a curved ramp, into which the waves are directed by two large reflector arms. The water surface in the reservoir can in this way be held above sea level, which implies that the waves’ energy has gone from kinetic energy to potential energy. The water leaves the reservoir back to the sea through outlet holes in the bottom, passing a hydroelectric turbine.

To optimise the output, the reservoir’s height above sea level needs to be adapted to accommodate the height of the incoming waves. This is done by a control system that can adjust the amount of air held in pockets underneath the hull. The speed of the turbines can also be adjusted, controlling the outflow of water from the reservoir.

The Wave Dragon project was developed in Denmark and is a joint EU project with funding from several countries within the EU. A prototype has been built in scale 1:4.5, shown in Figure 2.1, weighing 237 ton with a power output rated at 20 kW.

Figure 2.1: The Wave Dragon prototype tested at sea [2].

6 When going to full scale, Wave Dragon aims to produce between 12 and 35 GWh/year, depending on wave climate, with corresponding power output rating of 4 to 12 MW [3].

However, recent problems with the reflector arms have slowed development and it is unclear when a full scale device will be built [4].

2.1.2 Pelamis

Pelamis is of the surface following type and is made up of cylindrical segments linked together by hinged joints, as depicted in Figure 2.2. The first segment is moored causing the snake-like shape to align to the wave direction. The waves cause the joints to bend; a movement that is resisted by hydraulic rams, which pump fluid through hydraulic motors linked to generators, producing electricity. Pelamis was the first to build a multi-unit wave farm. This was made in Aguçadoura, outside Portugal, even though the concept was developed and tested in Scotland [5]. Three units rated at 750 kW each were installed and connected to the grid in September 2008, but technical problems arose only after two months, and the units were towed to shore. Pelamis claims to have solved the problem but their main investor, Babcock & Brown, has gone into voluntary administration, causing the project to be halted until a new investor has been found [6]. There are a couple of other Pelamis projects that are in their first stages, some of which are aiming to deploy Pelamis wave farms in Scotland as big as 26 units [7].

Figure 2.2: Pelamis prototype being tested at EMEC, Orkney [8].

2.1.3 The Lysekil project

The Lysekil project started 2002 and is operated by the Division for Electricity at Uppsala University. The concept is of the point absorbing type and consists of a buoy at the water surface, attached via a rope to a linear generator at the seabed, as shown in Figure 2.3. The motions of the waves cause the buoy to move up and down, i e, in the heave direction, causing the piston inside the generator to move up and down, generating electricity. Due to the variation in both amplitude and frequency of the induced voltage, it is transferred to a marine

substation where it is rectified, filtered and alternated [9]. This substation will be able to connect to several buoys, combining their input. Each buoy has a diameter of 3 m and a generator with power rated at 10 kW, assumed to match a significant wave height of 2 m. The project has so far gained permission from Swedish authorities to place 10 generators outside Orust, Sweden until 2014. Such a system has a total power output of 100 kW and is expected to generate 0.3 GWh/year [10]. However, the aim of the project in the near future is not to set up as many buoys and generators as possible, rather than to do research on those placed to improve the concept.

Figure 2.3: The Lysekil project: A buoy attached to its linear generator [11].

8

3 Theory

This chapter describes the theory of the dynamics of sea waves, wave spectra and an introduction to the analysis of the motions of floating bodies in sea waves as well as the concept of net present value. The discussions and calculations made in this thesis are based upon this. The computer programs used are also described in this chapter. In some of the other chapters, e g, Chapter 5, additional theory is incorporated due to its specific nature. For further reading on the theory of the dynamics of sea waves and their interaction with oscillating systems, see [12], and for the theory of motions of floating bodies in sea waves, [13] is recommended.

3.1 Airy wave theory

Airy wave theory is often referred to as linear wave theory and is a branch of fluid dynamics.

It describes the linearised propagation of any gravity wave in any homogeneous fluid under the assumptions that the fluid has a uniform mean depth and that the flow is incompressible, irrotational and has no viscosity. These assumptions work very well for water and describing sea waves.

The surface of the fluid is the described by

(3.1) with z the vertical coordinate. The mean elevation of the surface is z = 0, whereas the impermeable bottom is at z = -d. The flow velocity components in the horizontal and vertical directions, ux and uz, are related to the velocity potential (x,z,t) by

(3.2) and

(3.3)

The velocity potential has to satisfy the Laplace equation, which is

(3.4)

Three boundary conditions are needed. Firstly, the bottom is impermeable, leading to

(3.5)

at z = -d. The second boundary condition requires the trick of introducing infinitesimal waves on the surface, for which the vertical motion of the flow has to be equal to the velocity of the free surface, leading to

(3.6)

 

u_x x





 

z . u_z







.

2 0

2 2

2 













z x

0





 z

z t 



 





at z = (x,t). Bernoulli’s equation for an unsteady potential flow gives the third boundary condition

(3.7) at z = (x,t). g is the acceleration due to gravity. A propagating wave of a single frequency is described by the elevation of the surface with a sinusoidal function. The horizontal position is x and time is t according to:

(3.8) where a is the amplitude, k is the angular wave number and  is the angular frequency. The wave number k is defined from the wavelength , according to

(3.9)

The angular frequency  is related to the period T and the frequency f by

(3.10)

The phase velocity c describes the speed at which the waves propagate and is defined

(3.11)

While the wave at the surface propagates in one direction, the particles in the fluid moves in closed elliptical orbits, as illustrated in Figure 3.1. In deep water the orbits are circular and when approaching the bottom the orbits become flatter with their diameter reduced. The mean velocity of the particles is zero but they still transfer the energy of the wave in the propagation direction through the oscillation around their average position.

Figure 3.1: Orbital motion of fluid particles under linear waves.

0

 



 g t

 





   

2 .



  k

. 2 2

T f

 

k . c T 





10 The velocity potential, which is a solution to the partial differential equation (3.4) and is satisfying the first two boundary conditions, is

(3.12)

Both  and  have to satisfy the third boundary condition, which they do only if the dispersion relation is satisfied. The dispersion relation can be written

(3.13)

This means that the angular frequency and the wave number are related and cannot be chosen independently. The same holds true for the period and the wavelength, making the description of wave propagation an eigenproblem. The dispersion relation also states that waves with longer wavelength propagate faster. For deep water, i e, d bigger than half the wavelength which very often is the case at sea, the dispersion relation can be simplified to

(3.14) and the phase velocity can be rewritten to

(3.15)

The velocity at which a group of waves travel across the ocean is called the group velocity and is defined according to

(3.16)

Using the deep water approximation for the dispersion relation, the group velocity can be rewritten:

(3.17)

The Airy theory is a linear theory describing waves to the first order but many second-order wave properties can be derived from it, e g, the wave energy. The energy of a regular progressive wave is the sum of its kinetic energy and potential energy

(3.18) and can be calculated by integrating over one wavelength. Beginning with the potential energy, which is calculated as the deformation work needed to give form to the wave:

(3.19)

where  is the fluid density. The kinetic energy is the total kinetic energy contained in the water mass from the free water surface to the bottom of the sea. The part above the mean

 

 

 

 

sinh

cosh kx t

kd d z k

ak  _





 

2

gk  kd



2 ,

gk



k. c  g

k,

p E

E E 



 ^  

 ₀ 4 ^2,

1 2

1 g dx ga

E_p

k . c_g



 

2. 2

c c_g  g 



water level cannot be included in first order calculations and is thus not included in the integral:

(3.20)

which turns out to be equal to the potential energy. The wave energy is then from Equation (3.18) calculated as

(3.21)

and has the unit J/m². This shows that the energy of a wave is proportional to the square of the amplitude of the wave. A more common way to describe the energy in a wave in engineering is the power per meter wave front. It can be calculated by vertically integrating the work done per unit time at level z. The energy transport becomes

(3.22)

after the integration. This states that the wave energy travels across the ocean with speed equal to that of the group velocity. Note also that a is the wave amplitude as opposed to the wave height, denoted H, which is the distance measured between the crest and the trough of the wave, making it twice the size of the wave amplitude. Rewriting Equation (3.22) with the wave height leads to the following formula for the power per meter wave front:

(3.23)

The unit for J is W/m.

3.2 Wave spectrum

The origin of sea waves are the forces acting on the water, most essential being the wind blowing across the surface. Other forces acting can be planetary forces, such as the moon producing tidal waves, or earthquakes producing tsunamis. Waves at sea are not adequately described as monochromatic, harmonic oscillations. The sea often shows a seemingly random pattern of waves with different wave heights and frequencies which are superpositioned. An example of this randomness can be seen in Figure 3.2.

4 , 1 2

1 2

1 2

2

1 ⁰ 2 2 2

2







 



 



  ga

a k dz

T ae

E_k   ^kz  

2 ,

1 2

ga E

E

E _p  _k  

2 , 4

1 2

E c cE ga k

J      _g

32 .

1 2 2

TH g

J 

 

12 Figure 3.2: Measured wave height of sea waves [14].

An important concept is the significant wave height. It is denoted Hs and is defined as the mean of the highest third of the waves in a time series like the one in Figure 3.2. Other central quantities are the mean period, Tm, and the peak period Tp. The mean period is defined as the mean of all wave periods in a time series, and the peak period is the most energetic period. To describe how the wave components are distributed over different frequencies, a wave spectrum is needed. Several spectrum types have been proposed as models to fit wave data.

One of the most common wave spectra is the Pierson-Moscowitz spectrum [15], also called PM spectrum:

(3.24)

where PM and PM are constants and 0 is a factor depending on the wind speed. This is often referred to as the standard spectrum and assumes that the wind has blown steadily for a long time over a large area, causing the waves to come into equilibrium with the wind. This state is called fully arisen sea or fully developed sea. Another spectrum type was proposed, designed to fit measured wave data better, called the JONSWAP spectrum [16]:

(3.25)

where JONSWAP is a constant, p is a factor depending on the wind speed,  is the peak enhancement factor and  is a frequency dependent parameter. The JONSWAP spectrum is similar to the PM spectrum except that waves continue to grow with time (or distance) and the sea is thus never fully developed. Another important difference is that the peak in the spectrum is more pronounced, which is specified by the -term. Examples of the PM spectrum and the JONSWAP spectrum are compared in Figure 3.3.

 

4 0 5

2











 



 



 

 

 



 ^PM _PM

PM

S g

 

4 exp 5

4 5

2 







  















 



 



 ^{JONSWAP p}

JONSWAP

S g

Figure 3.3: Pierson-Moscowitz spectrum and JONSWAP spectrum [17].

The spectra used in this thesis are of ISSC type, which is a JONSWAP spectrum with  = 1.

A number of valuable quantities can be estimated from a wave spectrum through the use of spectral analysis. The spectral moments m_n with n = 0,1,2,… are calculated according to:

(3.26) The significant wave height and the root mean square wave height Hrms are estimated as

(3.27) The mean period Tm is for a broad spectrum estimated as

(3.28) and for a narrower spectrum

(3.29)

Several other quantities can be calculated through the spectral moments. This is usually done directly at the buoy or weather station performing the wave measurements.

 

0





  S  d

m_n ⁿ

. 2

4 ₀

0 rms s

m m H H

H   

, 2

1 0

m T_m   m

. 2

2 0

m T_m   m

14

3.3 Motion of floating bodies in sea waves

A rigid floating body has six motional degrees of freedom, denoted 1,2,…,6: three translational, i e, in the x- y- and z-directions, and three rotational around each of the axes.

There are names for each degree of freedom as shown in Figure 3.4.

Figure 3.4: Motional degrees of freedom of a floating body [18].

The equation of motion of a floating body can be written

(3.30) where M is a 6 × 6 mass matrix,  is the vector of positions in the six degrees of freedom, differentiated twice in time in the equation forming the body acceleration, and F is the vector of forces and moments acting on the body. The forces can be split according to

(3.31) where Fe denotes the wave-excited forces, Fr is the hydrodynamic reaction forces from the water and Frs is the reaction forces from the mooring system. The hydrodynamic reaction force can be written

(3.32) where Ar is a 6 × 6 matrix containing hydrodynamic mass or added mass, Br is a 6 × 6 matrix containing hydrodynamic damping coefficients and Cr is a 6 × 6 matrix containing the hydrostatic stiffness. Combining Equations (3.30), (3.31) and (3.32) produces the following expression:

(3.33) ,

F M

rs,

r

e F F

F

F  

,



 _{r r}

r

r A B C

F   





This equation has to be solved when investigating the movement of ships in waves. Its matrix form produces six different differential equations to be solved and different techniques are used for each of the six.

The response amplitude operator, with acronym RAO, is frequently used when trying to predict the behaviour of floating objects in waves, for example ships. It is obtained when solving Equation (3.33) and gives information about the ratio between the response amplitude of the studied variables and the wave amplitude. The ratio is calculated for different values of the frequency and the RAO is thus a vector when calculating it for one degree of freedom or a matrix when calculating it for two or more degrees of freedom. An example of a pitch RAO plotted as a function of frequency can be seen in Figure 3.5.

Figure 3.5: Response amplitude operator for the pitch of a box.

In Figure 3.5, the RAO is close to one for small wave frequencies (long periods) meaning that the pitch angle of the box follows the angle of the water surface, which can be understood intuitively. For higher wave frequencies the box follows the water surface less accurately and above some maximum frequency value it shows no ambition to do so, and the RAO has a value close to zero.

3.4 Software

This chapter describes the computer programs used when analysing the Wavebox WEC:

SCORES and MATLAB. SCORES was used unmodified, whereas MATLAB was used to create a simulation model for different Wavebox configurations. Both SCORES and the created simulation model are in part based upon the theory in Chapters 3.1, 3.2 and 3.3.

MATLAB is briefly described in this chapter, whereas the created simulation model is described more thoroughly in Chapter 5.2.

16

3.4.1 SCORES

SCORES [19] is a Fortran-based program available to SSPA AB (a maritime consultancy and towing tank facility) and is used to calculate ship movements in sea waves. The input data needed by the program is information about the dimensions of the hull, displacement of the hull, radius of gyration of the hull (describes the way the mass is distributed around the rotation axis in the pitch direction) and one or more wave spectra. The wave spectra used in SCORES are two parameter spectra, i e, the spectra is fully characterised by significant wave height and mean period. SCORES gives several outputs, of which the most important are the RAO of the hull and the significant pitch angle for each input wave spectrum. The significant pitch angle is an analogue to the significant wave height, thus defined as the mean of the highest third in a time series. It is, in contrast to the significant wave height, defined with the single amplitude and not with the double amplitude. SCORES also transforms the input spectra through discretisation, going from the two input parameters into a spectrum consisting of 48 components. A discrete spectrum can be seen in Figure 3.6.

Figure 3.6: A wave spectrum discretised in SCORES into 48 components.

SCORES does not compute the movements of the hull in the time domain but the RAO can be used, and is in fact essential, when calculating the movements explicitly. The RAO is also used when calculating the significant pitch angle in SCORES by computing the scalar product of the square of the RAO and the wave spectrum.

The inputs to SCORES are given in the form of a text-file that needs to be created, one for each hull configuration. However, all direct operation of SCORES is done by the contact persons at SSPA, who were kind enough to commit themselves to doing it for free.

3.4.2 MATLAB

MATLAB was used to create a computer model that simulates the pitch movement of any given Wavebox configuration. The created simulation model is described in Chapter 5.2.

Version 7.5.0 (R2007b) of MATLAB was used in this thesis.

3.5 Net present value

When calculating the economic potential of any electricity producing unit, the method of net present value can be used. In this method all future income and expenses are reduced by multiplication with the discount factor, G. The discount factor is time dependent and its value goes from 1 for transactions made in the present, and nears 0 for transactions made in the distant future. It is determined according to

(3.34)

where r is the interest and n is number of years in the future. The net present value is then calculated

(3.35)

where Cinv is the investment cost, Tl is the lifetime, p is the electricity price, tr is the number of running hours per year, P is the mean power output and K_n is the year’s maintenance costs. If C_inv and K_n are set to zero in Equation (3.35), NPV can be interpreted as the lifetime income of the electricity producing unit.

) , 1 (

1

n n

G r

 

 



















 ^Tl

n

n n n r

inv

NPV C p t P G K G

1

,

18

4 Wave data

The computer program SCORES requires wave data in the form of wave spectra characterised by significant wave height and mean period. Due to the early stage of the Wavebox WEC project, no suitable location has yet been decided, and because of this it was desirable to use wave data that was as universal and general as possible. The maximum number of spectra that can be run at each time in the program is nine, which led to the decision to use nine spectra with significant wave height ranging between 0.5 m and 4.5 m. Relevant mean periods for these significant wave heights were chosen. The values for the nine spectra used are presented in Table 4.1. The names S1 to S9 will be used later in this thesis. Spectra S1 to S9 were also used in the MATLAB simulation, in the discretised form output by SCORES. Mean wave length in Table 4.1 was calculated by combining Equations (3.9), (3.10) and (3.14), solving for .

Table 4.1: Specifications of spectra used in SCORES.

Spectrum name

Hs

(m)

Tm

(s)

Tp

(s)

 (mean) (m)

S1 0.50 3.0 4.2 13.8

S2 1.00 3.8 5.4 23.0

S3 1.50 4.6 6.5 33.3

S4 2.00 5.3 7.5 43.8

S5 2.50 5.9 8.4 54.0

S6 3.00 6.4 8.9 63.4

S7 3.50 6.8 9.7 71.5

S8 4.00 7.1 10.0 78.1

S9 4.50 7.3 10.3 82.7

When having gained knowledge about the performance of Wavebox in different wave scenarios, through SCORES and MATLAB simulation, the plan was to calculate the mean power output when placing Wavebox at one specific location. Historical wave data for the location has to be acquired to do this calculation. Most meteorological institutes charge a fee for making wave data available whereas some provide it for free. Free data was searched for and found for the following locations: San Diego in USA and Hanstholm in Denmark. In the San Diego case only the raw data was available which required some data processing. A good way to present wave data is in the form of a scatter diagram, which displays the distribution of different wave conditions at one specific location. An example of this can be seen in Table 4.2.

Table 4.2: Scatter diagram showing the distribution of waves at Hanstholm, Danmark [20].

Hs

[m]

Tm [s]

2.25-3.25

Tm [s]

3.25-4.25

Tm [s]

4.25-5.25

Tm [s]

5.25-6.25

Tm [s]

6.25-7.25

Tm [s]

7.25-8.25 Sum

0-0.25 0.50% 1.71% 0.65% 0.07% 0.02% 0.02% 2.96%

0.25-0.75 2.40% 12.90% 7.99% 2.53% 0.45% 0.22% 26.48%

0.75-1.25 0.26% 8.44% 14.36% 5.94% 0.73% 0.05% 29.78%

1.25-1.75 0.01% 1.08% 11.32% 5.07% 0.80% 0.08% 18.36%

1.75-2.25 0.00% 0.08% 3.95% 6.01% 0.64% 0.11% 10.78%

2.25-2.75 0.00% 0.00% 0.16% 4.33% 0.71% 0.07% 5.28%

2.75-3.25 0.00% 0.00% 0.05% 1.10% 1.49% 0.11% 2.75%

3.25-3.75 0.00% 0.00% 0.02% 0.05% 0.88% 0.13% 1.08%

3.75-4.25 0.00% 0.00% 0.00% 0.03% 0.21% 0.24% 0.48%

4.25-4.75 0.00% 0.00% 0.00% 0.02% 0.01% 0.17% 0.19%

Sum 3.17% 24.21% 38.49% 25.14% 5.94% 1.19% 98.14%

20

5 Analysing Wavebox

This chapter describes different design possibilities of Wavebox with the aim to make it an as effective energy producer as possible. Investigating the parameter space is an iterative process, in the way that each design is evaluated and redesigned until a set of parameters make out a satisfying configuration of Wavebox. One way to make an analysis of the concept would be to build a prototype, which could be tested, evaluated and if needed, redesigned and evaluated again. This method was early discarded due to the many uncertainties regarding the design parameters, making it impossible to decide one single prototype design without first investigating the concept in a more theoretical way. The investigation was therefore done by computer simulation, which is a good and cheap way to analyse new technology. The hull design and its parameters were investigated in cooperation with SSPA, whereas the simulation and calculation of power output of the selected configurations were done in a program written in MATLAB.

Figure 5.1 shows the components of Wavebox. The hull parameters, i e, component F, is investigated in Chapter 5.1, whereas the others are discussed in Chapters 5.2, 5.3 and 5.4.

Figure 5.1: Components of Wavebox. A) side containers, B) bottom container, C) mooring wire, D) upper container, E) hydroelectric turbine, F) extension of the hull used in SCORES, G) generator, H) hatches.

5.1 The hull

The properties of different hull configurations in sea waves were investigated at SSPA in the simulation software SCORES; see section 3.4.1 for information about the computer program.

The most important output from SCORES when determining shape and dimensions of the hull is the significant pitch angle, since the primary idea of Wavebox is to use its pitch angle to increase the potential energy of the contained water. Simply put, the higher the significant pitch angle, the more energy is likely to be extracted from the wave. This argument led to the straightforward approach to try different hulls, varying one parameter at a time and selecting the one with the highest significant pitch angle.

The approach was modified to be less mathematical and more engineering-oriented when it was realised that it might generate hull configurations at fault when looking at them from the

perspective of energy production, e g, faults like very low water flow or very small head (head being the distance between the upper water surface and the lower water surface, i e, the distance in height for the water to go when flowing through the turbine). The configurations could also not be overly heavy. This raised the question: How heavy is a configuration allowed to be? The output power is needed to answer this, since the energy produced is to pay for construction and material costs, which will be increased for a heavier configuration. Since the power output is not available until much further ahead in the process and in fact one of the primary goals of the investigation, it was suggested to search for a configuration with high significant pitch angle and low weight, if possible less than 50 tonnes, and at the same time taking into account parameters that was probable to seriously impact the power output.

Every configuration needed to be run separately in SCORES, which meant that one input file had to be created for each configuration and then sent via e-mail to SSPA for a run through the program. This made the investigation of the parameter space a bit time-consuming and the configurations needed to be chosen carefully in order to get the most out of SCORES in the limited time that it was available through SSPA.

The first proposed shape of the hull was naturally the simplest: a box shape. Thus, four variables had to be examined, of which three were the space variables, length, breadth, and draft, as depicted in Figure 5.2, and the fourth was the radius of gyration in the pitch direction.

All input variables in SCORES are the ones measured under the still waterline, which is the reason why draft has to be given instead of height. The side containers, seen in Figure 5.1, can be left out provided that they are located at a sufficient height above the water surface.

Figure 5.2: The simplest hull shape: box shape.

To begin with, the length was conveniently set at 10 m in order to vary the breadth for two different drafts. The radius of gyration was set to a quarter of the length, assuming the mass of the configuration to be evenly distributed across its box-shaped body. The draft was set to 1.17 m and 0.98 m, respectively and the breadth was varied between 2.2 m and 2.9 m. The result of the breadth variation in wave spectrum S4, with significant wave height equal to 2 m, can be seen in Figure 5.3.

22 ),

tan(



L H_WB

Figure 5.3: Significant pitch angle as function of breadth for two different drafts. A wave spectrum with significant wave height equal to 2 m is used.

The significant pitch angle increases a little when the breadth decreases, and has a similar dependence for both drafts. This implies choosing a narrow box. However, if chosen too narrow, the balance of the construction is at stake and the risk of keeling over is big, especially for a construction with large height. An estimation of the height was needed to be able to figure out how narrow the hull should be. The planes inside Wavebox transporting the contained water could probably be designed to have an inclination somewhere around the significant pitch angle. For these planes to fit inside the hull, the box needed to have a height at least the height of the planes, which can be expressed

(5.1) where  would be chosen equal to the biggest significant pitch angle, which as expected occurred for the spectrum with the biggest significant wave height, i e, spectrum S9. With

 = 10.59°, the minimal height is HWB = 1.87 m, indicating that less than one meter of the height would be over the water surface with the chosen drafts. A box as narrow as 2.2 m could probably be chosen but the narrowness has one more drawback, which is simple: The narrower the box, the less water can be contained inside. More contained water would probably imply bigger water flow inside and thus higher power output. Having considered the advantages and drawbacks of choosing a narrow box, a breadth of 2.5 m was chosen compromisingly to maintain stability and amount of contained water, while trying to optimise pitch. Since the estimated height of the box is dependent on the length, the breadth had to be increased accordingly if choosing a bigger length.

The investigation of the hull’s parameter space continued with the variation of length and draft. Four different lengths were tried and the associated breadths were chosen according to the reasoning described above. The draft was varied between 0.9 m and 1.35 m while the

radius of gyration was kept at a quarter of the length. The result of the draft variation, again in wave spectrum S4, can be seen in Figure 5.4.

Figure 5.4: Significant pitch angle as function of draft for four different lengths and breadths.

A wave spectrum with significant wave height equal to 2 m is used.

The significant pitch angle increases noticeably as the length decreases. It is also increased for increased draft, which suggests that a short hull with large draft should be chosen. Limitations on draft at this point was again weight, since a large draft inevitably leads to a big displacement which coincides with the weight of the configuration. Choosing a shorter hull would inescapably lead to a smaller height and the head of the contained water would thus be smaller. However, the exact consequences of the length on the power output was not known at this point and therefore the configuration with the highest significant pitch angle was chosen, i e, the one with length equal to 10 m, breadth equal to 2.5 m and draft equal to 1.35 m. From here on this hull will be referred to as Hull 1 in this thesis.

5.1.1 Different hull shapes

A brief investigation of the impact of different hull shapes was done. Two hull shapes apart from the box shape were tried: the banana shape and the H-shape. The banana shape has a curved bottom, which is depicted in Figure 5.5, while the rest of the configuration is similar to the box configuration. The H-shape has an enlarged section on each end connected by the narrower middle, forming an H, as can be seen in Figure 5.6.

24 Figure 5.5: Hull with banana shape.

Figure 5.6: Hull with H-shape seen from above.

Wanting to take advantage of the good properties of Hull 1, the banana shape length and breadth was set to the same as Hull 1 and the draft was set to 2 m giving it a displacement similar to Hull 1: 32.8 tonnes for the banana shape and 34.6 tonnes for the box shape. This gave a configuration close enough to Hull 1 to call it Hull 1 with banana shape. Similarly, the parameters of the H-shape were set to imitate Hull 1, giving it the same length and draft.

Breadth 1 and breadth 2, as shown in Figure 5.6, was set to 2 m and 5 m, respectively. The length of side was set to 1.9 m, all parameters resulting in a displacement equal to 44.3 ton.

Hull 1 with banana shape and H-shape was compared to Hull 1 with box shape in all nine wave spectra, which can be seen in Figure 5.7.

Figure 5.7: Significant pitch angle as function of significant wave height. Hull shapes banana, box and H-shape are compared in nine wave spectra.

The H-shape proved to have lower values of the significant pitch angle than the box shape in all nine wave spectra, while the banana shape had higher values than the box shape for all nine wave spectra. Because of this no further investigation was done on the H-shape.

5.1.2 Radius of gyration

The mass of the configuration had been assumed to be evenly distributed in the body; a temporary simplification done to keep the radius of gyration at a quarter of the length.

However, this is likely to be untrue almost all of the time, partly because the water moving inside constantly changes the weight distribution and thus the radius of gyration and partly because the configuration is likely to have inhomogeneous structures inside, distributing the mass unevenly. This called for an investigation of the radius of gyration and its effect on the significant pitch angle. Hull 1 was used with box shape and banana shape varying the radius of gyration. The result for wave spectrum S4 is plotted in Figure 5.8.

26 Figure 5.8: Significant pitch angle as function of radius of gyration. Hull shapes banana and box are shown. A wave spectrum with significant wave height equal to 2 m is used.

The significant pitch angle increases with increasing radius of gyration. The behaviour is similar for box shape and banana shape. This implies to try to allocate weight to the ends of the Wavebox configuration, when it is possible, to increase the radius of gyration.

5.1.3 Eigenfrequency in pitch

Every hull configuration has an eigenfrequency in pitch and it was suggested that it would be beneficial for the significant pitch angle if this frequency was near the dominant frequency in the wave spectrum. A good assumption is that the dominant frequency is the same as the peak frequency, which is the most energetic in the spectrum. An investigation of the eigenfrequencies and corresponding eigenperiods of 5 hull configurations was done to see if any of them matched or was close to any of the nine peak periods of the nine wave spectra.

The eigenperiods are presented in Table 5.1. The radius of gyration was set to a quarter of the length for all configurations.

Table 5.1: Eigenperiod in pitch for 5 hull configurations.

Configuration 1 2 3 4 5

Hull type Box Box Box Box Banana

Length (m) 10 12 14 16 10

Breadth (m) 2,5 2,8 3,0 3,2 2,5

Draft (m) 1,35 1,35 1,35 1,35 2,00

Displacement (t) 34,59 46,49 58,12 70,85 32,78

Eigenperiod in pitch (s) 2,94 3,06 3,31 3,72 3,09

The peak periods of the nine used spectrums range from 4 seconds for a spectrum with significant wave height equal to 0.5 m (spectrum S1) to 10 seconds for a spectrum with significant wave height equal to 4.5 m (spectrum S9). A full table of these periods can be seen in Chapter 4, Table 4.1. The hull with length equal to 16 m (configuration 4) and spectrum S1

are the pair that are closest to matching the eigenperiod to the peak period. This implies that it takes large constructions to match the periods even of small waves. To verify this, a very large configuration was tried: length equal to 30 m, breadth equal to 10 m and draft equal to 3 m giving it a displacement of 923 tonnes. It had an eigenperiod in pitch equal to 5.4 seconds, which approximately matches the peak period of a spectrum with significant wave height equal to 1 meter (spectrum S2). A match had been found but such a spectrum is relatively low in energy, and the construction would be too costly due to its large size and weight. This result was considered to rule out the possibility of matching the periods in a realistic way, taking economy into account. To further substantiate this, an argument was made that the eigenperiod is time dependent due to the movement of the water inside Wavebox, and also the dominating period is varying in time due to the ever-changing wave conditions at sea. If choosing a configuration with the only motivation to have an eigenperiod matching one dominating wave period, it would still only be likely to be close to the actual dominating period part of the time.

5.2 MATLAB simulation

The main tasks for the MATLAB simulation program was to show the pitch angle of a given Wavebox configuration as a function of time and from that information calculate the power output. To simulate the pitch angle of a Wavebox configuration as a function of time the RAO of the pitch of the configuration is needed as well as a discretised wave spectrum. Both are outputs from SCORES. A time series can be created for each discretised wave spectrum by summation of its components, which can be written

(5.2) where ai, i and i are the amplitude, angular frequency and phase of the ith component, respectively. The angular frequencies are given in SCORES and the phases should be chosen randomly. The amplitudes are chosen as

(5.3) where S is the spectral density and i is the mean difference between the ith component of  and its two neighbouring components. The wave time series in Equation (5.2) is plotted in Figure 5.9 using wave spectrum S4, with significant wave height 2 m and mean period 5.3 seconds.

 







i

i i

i t

a

t  



 

2 _{i i}

i S

a    

28 Figure 5.9: Time series synthesised from the parameters describing wave spectrum S4. Green lines mark significant wave height.

Multiplying each wave component with the RAO of the pitch produces the time series of the pitch which can be written

(5.4)

i is the ith component of the transfer function of the pitch and can be written

(5.5) where Yi is the ith component of the RAO of the pitch. The pitch time series in Equation (5.4) is plotted in Figure 5.10 using the same waves as plotted in Figure 5.9 and hull parameters equal to that of Hull 1, a configuration defined in the end of Chapter 5.1. The movement in pitch is as expected linked with the incoming waves, which can be seen when comparing Figure 5.9 with Figure 5.10.

 









i

i i i

i a

t   

i,

i

i Y

g 





Figure 5.10: Pitch time series for a Wavebox configuration with Hull 1 in waves synthesized from spectrum S4.

There are several parameters in the Wavebox concept that are independent of the hull parameters. These inner parameters are the inclinations of the lower and upper ramps, the length of the side containers and the water mass in the bottom container, as shown in Figure 5.11. The side containers do not need to be counted as part of the hull in the case that they are located high enough above the water surface to never come in contact with it. This is assumed to always be the case in this thesis, with motivation that SCORES cannot take into account parts of the hull that are above the water surface in still water but might come into water contact when waves are present.

Figure 5.11: Inner parameters of Wavebox.

The mean power output for a hydroelectric turbine is

(5.6) where h is the head, q is the mean flow rate through the turbine,  is the density of the liquid, g is the acceleration due to gravity and  is the combined coefficient of efficiency for the

,

 





h q g P