Sea Level Compensation System for Wave Energy Converters

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UNIVERSITATISACTA UPSALIENSIS

UPPSALA

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1386

Sea Level Compensation System for Wave Energy Converters

VALERIA CASTELLUCCI

ISSN 1651-6214 ISBN 978-91-554-9613-5

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Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 9 September 2016 at 09:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Ronan Costello (National University of Ireland Maynooth).

Abstract

Castellucci, V. 2016. Sea Level Compensation System for Wave Energy Converters. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1386. 71 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9613-5.

The wave energy converter developed at Uppsala University consists of a linear generator at the seabed driven by the motion of a buoy on the water surface. The energy absorbed by the generator is negatively affected by variations of the mean sea level caused by tides, changes in barometric pressure, strong winds, and storm surges.

The work presented in this doctoral thesis aims to investigate the losses in energy absorption for the present generation wave energy converter due to the effect of sea level variations, mainly caused by tides. This goal is achieved through the modeling of the interaction between the waves and the point absorber. An estimation of the economic cost that these losses imply is also made.

Moreover, solutions on how to reduce the negative effect of sea level variations are discussed.

To this end, two compensation systems which adjust the length of the connection line between the floater and the generator are designed, and the first prototype is built and tested near the Lysekil research site.

The theoretical study assesses the energy loss at about 400 coastal points all over the world and for one generator design. The results highlight critical locations where the need for a compensation system appears compelling. The same hydro-mechanic model is applied to a specific site, the Wave Hub on the west coast of Cornwall, United Kingdom, where the energy loss is calculated to be about 53 %. The experimental work led to the construction of a buoy equipped with a screw jack together with its control, measurement and communication systems.

The prototype, suitable for sea level variations of small range, is tested and its performance evaluated. A second prototype, suitable for high range variations, is also designed and is currently under construction.

One main conclusion is that including the compensation systems in the design of the wave energy converter will increase the competitiveness of the technology from an economic point of view by decreasing its cost per kWh. The need for a cost-effective wave energy converter with increased survivability emphasizes the importance of the presented research and its future development.

Keywords: Ocean energy, Tides, Linear generator, Point absorber, Offshore experiment, Hydro-mechanic modeling, Power absorption, Control system, Communication system, Measurement system, Lysekil research site, Wave Hub.

Valeria Castellucci, Department of Engineering Sciences, Electricity, Box 534, Uppsala University, SE-75121 Uppsala, Sweden.

ISBN 978-91-554-9613-5

urn:nbn:se:uu:diva-295603 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-295603)

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Dedicated to my husband

Exceptional tides call for exceptional measures - Revised proverb -

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List of papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Castellucci, V., García-Terán, J., Eriksson, M., Padman, L., and Waters, R., “Influence of Sea State and Tidal Height on Wave Power Absorption”. Accepted for publication by IEEE Journal of Oceanic Engineering, July 2016.

II Castellucci, V., Eriksson, M., and Waters, R., “Impact of Tidal Level Variations on Wave Energy Absorption at Wave Hub”. Submitted to Energies, July 2016.

III Castellucci, V., Waters, R., Eriksson, M., and Leijon, M., “Tidal effect compensation system for point absorbing wave energy converters”, Renewable Energy, 51, 247-254, 2013.

DOI:10.1016/j.renene.2012.09.043

IV Castellucci, V., Abrahamsson, J., Svensson, O., and Waters, R.,

“Algorithm for the Calculation of the Translator Position in Permanent Magnet Linear Generators”, Journal of Renewable and Sustainable Energy, 6, 063102, 2014. DOI:/10.1063/1.4900553.

V Castellucci, V., Eriksson, M., Waters, R., Ferhatovic, S., and Leijon, M., “Wireless System for Tidal Effect Compensation in the Lysekil Research Site”, Proceedings of the ASME 2012, 31st International Conference on Ocean, Offshore and Arctic Engineering, Rio de Janeiro, Brazil, 1-6 July, OMAE 2012.

VI Castellucci, V., Kamf, T., Hai, L., and Waters, R., “Control System for Compensator of Mean Sea Level Variations at the Lysekil Research Site”, Proceedings of the 2nd Asian Wave and Tidal Energy

Conference, Tokyo, Japan, 28-31 July, AWTEC 2014.

VII Castellucci, V., Abrahamsson, J., Kamf, T., and Waters, R., “Nearshore Tests of the Tidal Compensation System for Point-Absorbing Wave Energy Converters", Energies, 8(4), 3272-3291, 2015.

DOI:10.3390/en8043272.

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VIII Ayob, M.N., Castellucci, V., Terzi, M., and Waters, R., “Tidal Effect Compensation System Design for High Range Sea Level Variations”, Proceedings of the 11th European Wave and Tidal Energy Conference, Nantes, France, 6-11 Sept., EWTEC 2015.

Reprints were made with permission from the publishers.

The author has contributed to the following papers, not included in the thesis.

1. Ekergård, B., Castellucci, V., Savin, A., Leijon, M., “Axial Force Damper in a Linear Wave Energy Converter”, Development and Applications of Oceanic Engineering, 2(2), 33-38, 2013.

2. Hong, Y., Hultman, E., Castellucci, V., Ekergård, B., Sjökvist, L., So- man, D. E., Krishna, R., Haikonen, K., Baudoin, A., Lindblad, L., Lejer- skog, E., Käller, D., Rahm, M., Strömstedt, E., Boström, C., Waters, R., and Leijon, M., “Status Update of the Wave Energy Research at Uppsala University”, Proceedings of the 9th European Wave and Tidal Energy Conference, Aalborg, Denmark, 2-5 Sept., EWTEC 2013.

3. Sjökvist, L., Krishna, R., Rahm, M., Castellucci, V., Hagnestål, A., and Leijon, M., “On the Optimization of Point Absorber Buoys”, Journal of Marine Science and Engineering, 2(2), 477-492, 2014.

4. Hai, L., Svensson, O., Castellucci, V., Lejerskog, E., Waters, R., and Leijon, M., “Force in the connection line for a wave energy converter:

simulation and experimental setup”, Proceedings of the 33rd Interna- tional Conference on Ocean, Offshore and Arctic Engineering, San Fran- cisco, USA, June 8-13, 2014.

5. Widén, J., Carpman, N., Castellucci, V., Lingfors, D., Olauson, J., Re- mouit, F., Bergkvist, M., Grabbe, M., and Waters, R., “Variability assessment and forecasting of renewables: A review for solar, wind, wave and tidal resources”, Renewable and Sustainable Energy Reviews, 44, 356-375, 2015, ISSN 1364-0321. DOI: 10.1016/j.rser.2014.12.019.

6. Parwal, A., Remouit, F., Hong, Y., Francisco, F., Castellucci, V., Hai, L., Ulvgård, L., Li, W., Lejerskog, E., BauDOIn, A., Ayob, N., Chatzi- giannakou, M., Haikonen, K., Ekström, R., Boström, C., Göteman, M., Waters, R., Svensson, O., Sundberg, J., Rahm, M., Strömstedt, E., En- gström, J., Savin, A., and Leijon, M., “Wave Energy Research at Uppsala University and The Lysekil Research Site, Sweden: A Status Update”, Proceedings of the 11th European Wave and Tidal Energy Conference, Nantes, France, 6-11 Sept., EWTEC 2015.

7. Hong, Y., Eriksson, M., Castellucci, V., Boström, C., and Waters, R.,

“Linear generator-based wave energy converter model with experimen- tal veriﬁcation and three loading strategies”, IET Renewable Power Gen- eration, 10(3), 349-359, 2016. DOI: 10.1049/iet-rpg.2015.0117.

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Nomenclature and Abbreviations

Symbol SI Unit Quantity

C m/s Wave celerity

C_i Euro Cash ﬂow

d_c Ns/m Damping coefﬁcient of the WEC capsule

d_w Ns/m Damping coefﬁcient of the connection line

E_loss % Energy loss

E^∗ MWh/y Annual energy absorption

E_nt^∗ MWh/y Annual energy absorption without tide E_t^∗ MWh/y Annual energy absorption with tide

F_d N Damping force

F_e N Excitation force

F_es N End-stop force

F_gb N Gravity force of the buoy

F_gt N Gravity force of the translator

F_h N Hydrostatic force

F_r N Radiation force

F_w N Connection line force

f_c N Force on the WEC capsule

f_e N/m Transfer function of the excitation force

f_l N Lower end-stop force

f_u N Upper end-stop force

g m/s² Gravity force

H m Wave height

H_s, Hm0 m Signiﬁcant wave height

H_t m Tidal range

h m Water depth

k 1/m Wave number

k_c N/m Spring constant of WEC capsule

k_e N/m Spring constant of the end-stops

k_w N/m Spring constant of the connection line

L(t) - Radiation impulse response function

l_i m Max reversible compression of the end-stop

l_s m Stroke length of the translator

m_a kg Added mass

m^∞_a kg Added mass at the inﬁnite limit

m_b kg Buoy mass

m_k m²/s^k Spectral moment of order k

m_t kg Translator mass

P kW/m Power per meter of wave crest

P^∗ kW Optimum power absorption

r m Radius of the cylindrical buoy

r_d % Discount rate

S_bs m²s/rad Bretschneider spectrum

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Symbol SI Unit Quantity

T s Period

T_e, T_m0−1 s Energy period

U V Instantaneous voltage

w_p m Magnetic pole width

x m Displacement of the translator

˙x m/s Speed of the translator

¨x m/s² Acceleration of the translator

x_m m Mean translator position

y m Displacement of the buoy

˙y m/s Speed of the buoy

¨y m/s² Acceleration of the buoy

γ Ns/m Damping coefﬁcient of the generator

γ^∗ Ns/m Optimum damping factor

η m Water surface

λ m Wavelength

ρ kg/m³ Density

φ m²/s Velocity potential

φd m²/s Velocity potential of the diffracted wave φi m²/s Velocity potential of the incident wave φr m²/s Velocity potential of the radiated wave

ω rad/s Angular frequency

ωm rad/s Modal frequency

Abbreviation Description

CAD Computer-Aided Design

CFE Swedish Centre for Renewable Electric Energy Conversion

cRIO CompactRIO

ECMWF European Centre For Medium-Range Weather Forecast

EMEC European Marine Energy Centre

FPGA Field-programmable gate array

FTP File Transfer Protocol

GSM Global System for Mobile Communications IGBT Insulated-gate bipolar transistor

IPCC Intergovernmental Panel on Climate Change

IRR Internal Rate of Return

NPV Net Present Value

OWC Oscillating water columns

PAC Programmable automation controller

PMW Pulse width modulation

RE Renewable energy

RPi Raspberry Pi

SMS Short Message Service

TMD Tidal Model Driver

WEC Wave energy converter

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1. Introduction

The IPCC [1] defines renewable energy (RE) as the energy harvested from solar, geophysical or biological sources, and which is replenished by natural processes at a rate that is equal or grater than its rate of use. The potential of RE sources exceeds the present and future global energy demand, but what makes it challenging is the ability to capture a reasonable share of this potential in a cost-effective and environmentally friendly way. REs like hydropower, wind and solar energy, geothermal energy and biomass are technically mature and they are at a commercial stage, while other sources of RE, e.g. wave energy, are at an early phase of technical readiness. Both public institutions and private companies are involved in the research and development of these technologies, and their commitment will eventually bring innovation in areas which lack maturity.

The thesis presented here will deal with the wave energy resource and will provide insight into one of the challenges that keeps the researchers at Uppsala University busy.

1.1 Wave energy

Ocean waves are generated by winds blowing across the open sea, by the gravitational pull from the Moon and Sun, by changes in atmospheric pressure, by earthquakes, etc. Wind generated waves are the most common ones and the most relevant for wave energy conversion. Waves, compared to winds, are more uniformly distributed in time, i.e. less intermittent: even if the wind ceases, waves will keep rolling in for some time. In general, the long term temporal variability of the waves make them a more attractive source of energy than wind, solar and tidal energy [2]. Fig. 1.1 shows that the variability of wave power tends to be quasi-independent from the time of the day, as also suggested in [3].

Waves are a powerful source of energy, but they are difficult to harness and convert into electricity in large quantities. The power available at sea is usu- ally expressed as kilowatts per meter of wave crest [kW/m] and it is a function of the energy period, Te, and the square of the significant wave height, Hs, at the site [4]. The average annual wave power around the globe is presented in Fig. 1.2. The total theoretical energy of the waves is approximately 32,000 TWh/y [1], but the technical potential is substantially lower, because of technological and societal limitations, and associated costs. For example,

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'HF

:LQG :DYH 7LGDO

-DQ

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Figure 1.1. Variability of the energy sources in Hawaii (above) and Sweden (below) during one year at different times of the day. The results are normalized measurement data. Figure taken from [2].

Figure 1.2. Mean wave power at coastal areas calculated over 10 years of ECMWF modeled data.

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the wave energy converters (WECs) cannot be placed too far from shore without cabling, commissioning and maintenance becoming too expensive. Ad- ditionally, the WECs can only be deployed until certain depths because of limitations and costs of the mooring systems. The major difﬁculty is to build systems economically viable and capable to survive harsh weather and storms.

A wide range of operating principles have been investigated, and many WECs have been conceived and tested. Wave energy technologies can be classified as oscillating bodies, oscillating water columns (OWC), and overtopping devices, as shown in Figs. 1.3, 1.4 and 1.5 (idea taken from [5]). Each category is characterized by devices having a fixed, floating or submerged structure; they can be mounted offshore, nearshore or on the shoreline. Some examples of WECs that have reached a prototype stage are shown in the tree diagrams. One of the WECs classified as submerged oscillating bodies is the Uppsala University wave energy converter. The work presented in this thesis focuses on this WEC typology, and more details regarding the working principle are given in the following section.

Oscillating bodies (Point absorber,

Attenuator, Terminator)

Floating

Translation

AquaBuOY

Finavera Renewables Ltdc

Rotation

cPelamisWave Power Ltd.

Submerged

Translation

Uppsala University

Rotation

OysterAquamarine Power^c

Figure 1.3. Tree diagram showing some examples of oscillating devices.

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OWC (Air chamber)

Floating

Mighty Whale

JAMSTECc

Fixed

In breakwaters

Mutriku

Ente Vasco de la Energíac

Isolated

LimpetWavegen Ltd.^c

Figure 1.4. Tree diagram showing some examples of OWC technologies.

Overtopping (Reservoir)

Floating

cWave Dragon^Ltd.

Fixed

In breakwaters

SSGWave Energy AS^c

Shoreline

TapChanNorwave AS^c

Figure 1.5. Tree diagram showing some examples of overtopping devices.

1.2 Lysekil Project

1.2.1 The concept

The WEC developed by Uppsala University, together with the Swedish Centre for Renewable Electric Energy Conversion (CFE), consists of a linear generator placed at the seabed and driven by the motion of a buoy ﬂoating at the water surface (see Fig. 1.6). Inside the generator, the moving part called the translator is made of magnets, and it is connected to the buoy via a steel

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Figure 1.6. Illustration of a wave energy converter (courtesy of Erik Lejerskog).

wire. The present generation WEC uses ferrite magnets, while the previous generations were made of neodymium-iron-boron magnets. The need for a cost-effective WEC solution marked the transition between the two translator types [6]. In both cases, the translator moves vertically along a ﬁxed part, the stator, which is wound with copper wires. The motion of the translator induces current in the windings of the stator and the mechanic energy of the waves is converted into electric energy. The translator is driven upwards until it hits a spring, called the upper end-stop, and the gravity force due to its weight brings it downwards to the lower end-stop.

This wave energy converter is particularly suitable for areas characterized by moderate sea states and depths in the range from 20 to 200 meters. The mechanical design of the WECs can be modiﬁed depending on the needs, e.g.

according to the design wave at the chosen site. This implies a variation of costs due to the size of the converters, and a variation in the power production of the single WEC.

The advantages of this technology are many, for example, the modularity (a wave energy farm is dimensioned according to the energy demand), the low visual and environmental impact, and the protection of the generator from biofouling and mechanical stresses due to the impact from the waves at the surface. On the other hand, there are many challenges to face, such as cor- rosion, effects due to tides, maintenance− all important topics that keep the researchers at Uppsala University busy and engaged with the project.

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1.2.2 The research site

The researchers at Uppsala University are able to test their prototypes at the Lysekil research site, one of the few pre-commercial test sites in Europe, alongside the Wave Hub, located west of Cornwall, and the European Ma- rine Energy Centre (EMEC), in the north of Scotland. The Lysekil research site takes its name from the nearby town of Lysekil, located on the west coast of Sweden. The experimental site, in use since 2004, is situated 2 km off the island of Härmanö. The sea ﬂoor is characterized by sandy silts and the depth is 25 m [7]. The wave climate has been described in [8] and the last status update of the project has been presented in [9].

During many years of research and development of this technology, twelve generators, called L1 to L12, have been designed, built and tested. On average, the stroke length of the translators is 2 m and the height of the generators from the foundation to the funnel is 8 m. The WECs are three-phase voltage generators, except for the one-phase L7 and L8 [9]. As the design of the generator evolved, the number of sides of the translator, thereby the number of sides of the stator, increased from 4 to 8 or 9 in order to make the generators more compact and reduce the amount of material used. The weight of the translators have increased from a couple of tonnes to about 10 tonnes, partly due to an increase in energy absorption and partly to decrease costs by avoiding the need for large retractive springs used in the oldest full-scale prototypes.

Moreover, buoys of different shapes and sizes have been designed and tested offshore. The toroidal shape has been chosen because it minimizes the added mass, i.e. the inertia of the system [10]. Other theoretical studies demonstrated that the energy absorption increases considerably by using a submerged buoy in resonance with the traditional one [11].

As for the buoys, different types of connection lines have been evaluated, in order to reduce the risk of breakage of this component of vital importance for the WEC. A solution to minimize the friction between the line and the funnel down to the generator is to use Dyneemaﬁbers, which cover the steel wire clad in a cross-linked polyethylene pipe [12].

One of the highest costs of the generator is the translator, due to the high price of the permanent magnets. For this reason, the present generation WECs are designed using ferrite magnets, which are much cheaper, but more fragile and less magnetic. Consequently, the size of the magnets has to be bigger in order to reach the same level of magnetic energy as for the neodymium magnets, and consequently the weight of the translator increases [6]. The L12 generator is the only one with these characteristics.

A single generator is not able to provide a suitable input for the electric grid:

the linear generator produces a voltage that varies in amplitude and frequency, hence, it is necessary to regulate the voltage before the grid connection [13].

The marine substation accomplishes this task by rectifying the output from the single generators, connecting several WECs in parallel, and converting

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DC voltage to AC by means of an inverter. With the purpose of stabilizing the power production, the use of multilevel inverters has been investigated [14], but the solution adopted at the present time is a transformer with a tap changer.

Two substations have been realized so far: the ﬁrst was used to connect three generators in parallel, while the second can connect up to seven generators.

Connection to the Swedish grid was achieved in November 2015. Before then the energy produced by the WECs at the Lysekil research site was dissipated in dump loads mounted at a measurement station on land, where data is collected and sent to Uppsala University via internet [15].

Recently, modeling of wave power farms and survivability tank tests in ex- treme wave conditions have been performed [16, 17]. The use of remote op- erated vehicle tools for the automatized connection of underwater cables is under investigation.

Finally, environmental studies have been carried out since the beginning of the project for the purpose of monitoring the marine environment [18].

1.3 Research question

After many years of offshore tests [9] it has essentially been conﬁrmed that the studied technology works in Swedish waters. However, there are sig- niﬁcant differences between Swedish and international waters: the wave climate and the tidal range in the Baltic Sea are moderate compared to many other locations around the world. We do not have to look too far to en- counter more powerful seas with higher wave energy resource and tides, e.g.

the neighbouring country Norway which has an estimated commercial poten-

Figure 1.7. Illustration of the effect of tides on the stroke length of the translator.

During a high tide the connection line is too short and the translator hits the upper end of its stroke. During a low tide the line is too long and the translator drops towards the lower end of its stroke.

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tial of 400 TWh/y against the 5-10 TWh/y of Sweden [19], and a tidal range up to about 2 m against approximately 0.2 m of Sweden (source TMD, Tidal Model Driver [20]).

Even though offshore sites with greater resources, like Norway, can make higher energy absorption possible, the inﬂuence of sea level variations, mainly due to tides, will negatively affect the optimum power absorption of the WEC.

In fact, the average position of the translator with respect to the stator shifts with the tide and may cause limitations of the motion of the translator, as illustrated in Fig. 1.7.

The research presented in this thesis will mostly focus on the potential negative effect of tides on the energy absorption and survivability of the WECs, that can compromise the competitiveness of the technology. To this end, a theoretical study that quantiﬁes the negative impact of tides has been com- plimented with large scale experimental work: the energy losses have been estimated at different locations around the world and a compensation system for small sea level variations (approximately 1.5 m high) has been built and tested, while a system for higher tidal ranges has been proposed.

In conclusion, the research question that this doctoral thesis aims to answer is: How much do tides effect the energy absorption of an Uppsala WEC and how can the impact of sea level variations be mitigated?

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2. Method

This chapter describes the theory of waves and tides in sections 2.1 and 2.2 respectively, and the method employed to model the response of the WEC to the displacement of the free surface in section 2.3. This method is used to conduct the studies presented in Paper I and Paper II. Section 2.4 describes the wave climate and tidal range at a global scale and at the Wave Hub site, while section 2.5 deals with the economic modeling of the tidal compensation system.

2.1 Linear theory of ocean waves

The linear theory presented in this section is also known as the Airy wave theory from the the scientist that first published it, as commented in [21]. An extensive literature describes the derivation of the theory (e.g. see [22–24]), that is here briefly discussed. The theory applies to gravity waves, i.e. surface waves that have gravity as restoring force, characterized by small amplitudes and freely propagating in a two-dimensional space. Furthermore, the assump- tions of ideal (or inviscid), incompressible fluid and irrotational flow are made.

Once the velocity vector, v, is deﬁned and assuming incompressible ﬂuid (ρ = const.), the continuity equation reduces to:

∇ · v = 0. (2.1)

The assumption of irrotational flow is verified ifv can be expressed by a potential function, φ(x,y,z,t), whose gradient coincides with the velocity field, i.e.:

v= ∇φ. (2.2)

Combining Eq. 2.1 and Eq. 2.2, the continuity equation is further simpliﬁed to the Laplace equation:

∇²φ = 0. (2.3)

Consider the Cartesian coordinate system (x,z) with the origin laying on the line that represents the free surface at rest, z pointing upwards and x pointing at the direction of motion of the wave, as shown in Fig. 2.1. The free surface is given by z= η(x,t) and the depth is given by z = −h(x) [22]. In order to solve Eq. 2.3, the following boundary conditions are required:

1. sea ﬂoor boundary condition (z= −h and hx= 0)

−φz= 0 (2.4)

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2. kinematic surface boundary condition (z= η(x,t))

φz= ηxφx− ηt (2.5)

3. dynamic surface boundary condition (z= η(x,t)) η + 1

2g(φ_x²+ φ_z²) −1

gφt = 0 (2.6)

where the subscripts indicate the derivative of the function in space or time.

Since the value ofη is unknown and the conditions are not linear in φ and η, the problem can be simpliﬁed by neglecting non linear terms, which are small compared to the order of magnitude of the linear terms. After linearization, the boundary conditions at the sea ﬂoor and at the surface become respectively:

φz= 0 (2.7)

φz= −ηt

η =¹_gφt

φz+1

gφtt= 0. (2.8)

Combining Eq. 2.7 and Eq. 2.8 with the Laplace equation, a solution forφ and η can be found:

φ =gH 2ω

cosh k(h + z)

cosh(kz) sin(kx − ωt) (2.9) η =H

2 tanh(kh) cos(kx − ωt) (2.10) where g is the gravity force, H is the wave height,ω is the angular frequency (²_T^π), and k is the wave number (²_λ^π).

Figure 2.1. Illustration of the Cartesian reference system.

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Finally, the dispersion relation at the free surface can be calculated by dif- ferentiating Eq. 2.9 and combining it with the surface boundary condition (Eq. 2.8):

ω²= kg tanh(kh). (2.11)

Eq. 2.11 implies that ω and k cannot be chosen at will. For a given k = 0, only the two frequencies, ±ω, which satisfy the equation are allowed, i.e.

the dispersion relation tells us how the frequency and the wave number are connected [25].

The equations for η, φ and the dispersion relation expressed in Eqs. 2.9, 2.10 and 2.11 represent the core of linear wave theory.

When the water depth is greater then one-half of the wavelength, _λ^h > ¹₂, the wave is classiﬁed as a deep water wave. Water particles inside this type of wave move in a circular orbit whose diameter decreases with depth until it disappears at the wave base. When the depth is less then one twentieth of the wavelength, _λ^h < ₂₀¹, waves are said to be in shallow water. In this case, the water particle orbits inside the wave become elliptical rather then circular:

the up-down component of the motion is ‘squeezed’ by the presence of the bottom [26].

The speed at which an individual wave propagates is called wave celerity, deﬁned as C= ^λ_T. For a deep water wave the celerity is directly proportional to the wave period, T . The formula for deep water celerity is:

C=

g

k ≈ 1.56 T (2.12)

while the celerity of an individual shallow water wave is given by:

C=

gh. (2.13)

Note that the deep water wave celerity does not depend on water depth, while the shallow water celerity depends on the depth rather then the wave period.

Tides behave as shallow water waves [27], and their wavelength is:

λ = TC = 3.13 T√

h. (2.14)

The dominant period of the semidiurnal tide is roughly 12 hours and the great- est ocean depths are about 12 km. The corresponding tidal wavelength would be roughly 15,000 km and one twentieth of this value is 750 km. In other words, tidal waves have an extremely high length-to-depth ratio and the verti- cal motion of the water particles is insigniﬁcant compared to their horizontal motion, which represents the tidal current.

2.2 Theory of tides

Tides are the periodic rise and fall of the water level in seas and oceans, and they are the result of the gravitational pull of the Moon and Sun on the Earth,

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Figure 2.2. Illustration of the high tides, represented by the blue bulges, created by the attraction of the Moon and the Sun on the Earth.

as well as the perpetual spinning rotation of the Earth itself. In this section an overview of the phenomenon will be given; for a pedagogic and thorough explanation see e.g. [27] and [28].

The gravitational effect of the Moon is what inﬂuences the mean sea level variation the most: the water is pulled towards the Moon itself and a bulge is created on the surface of the ocean at the side of the Moon. Furthermore, the spinning of the Earth-Moon system causes a centrifugal force creating a second bulge at the opposite side of the Moon. These bulges, or high tides, are represented in Fig. 2.2. As the Moon rotates around the Earth the bulges shift with it. The Moon rotates in the same direction the Earth rotates around its axis, thus the Moon takes a little more then a day, 24 hours and 50 minutes, to fully rotate around the Earth. Moreover, the effect of the Moon is the same both in the zenith and nadir (see Fig. 2.2), so one tide cycle takes about 12 hours and 25 minutes and the time between a high tide and a low tide is, on average, 6 hours and 12.5 minutes.

The tidal effect due to the Sun that exercises a gravitational attraction on the Earth is less powerful than the lunar effect. Although the distance between Earth and Sun is three orders of magnitude greater then the distance Earth- Moon, the mass of the Sun is ﬁve orders of magnitude higher then the mass of the Moon, so that the contribution of the Sun is not negligible.

Approximately twice a month, the Sun, Moon and Earth will more or less align to form either a Full Moon or a New Moon. During each phase of a

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Figure 2.3. Tidal regimes (idea taken from [30]).

New Moon or a Full Moon, the sum of the two tidal effects results in higher high tides and lower low tides, both called Spring Tides. Twice each month the Sun and Moon are at right angles to the Earth and opposing each other (First and Third Quarter Moons). In that case the tidal ranges are less then normal and they are called Neap Tides. Because of these periodic ﬂuctuations in gravitational pulls from the Sun and Moon, the height of the tides varies from day to day.

The theory just discussed is true for all the water basins on Earth, but only if our planet were a perfect sphere without big continents. In that case all the seas and oceans would experience two equally proportioned high and low tides every lunar day [29]. The continents, however, block the westward passage of the tidal bulges as the Earth rotates, hindering the tides from moving freely around the globe. Three basic tidal patterns occur along the coasts, as shown in Fig. 2.3: a diurnal pattern, when only one high and one low tide each day occur; a semidiurnal pattern, when two high tides and two low tides each day occur; a mixed pattern, when both semidiurnal and diurnal components are important, resulting in a signiﬁcant difference between the height of high and low tides during the day.

2.3 Hydro-mechanic modeling

In order to be able to comprehend the interaction between the waves and the WEC, some notion of ocean wave-structure interaction are given in subsection 2.3.1. Subsection 2.3.2 describes how the wave energy converter is modeled, while subsection 2.3.3 explains how the time series of the sea level has been generated. Extensive literature can be found in engineering books, e.g. [23, 24, 31], and scientiﬁc papers, e.g. [32, 33].

2.3.1 Wave-buoy interaction

Imagine the ﬂoater of the WEC being a cylinder, moving in heave when hit by the waves.

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The velocity potential is the sum of three contributions:

φ = φi+ φd+ φr (2.15)

whereφi is the incident wave velocity potential,φd is the diffraction velocity potential andφris the radiation velocity potential. This means that the buoy is affected by the waves and the waves are affected by the buoy: in fact, the incident waves excite the buoy and diffract behind it (scattering problem), while the buoy, oscillating, creates radiated waves (radiation problem). The hydrodynamic forces which result from these two problems are called excitation force, F_e, and radiation force, F_r, respectively. In time domain these forces can be expressed as:

F_e(t) = fe(t) ∗ η(t) (2.16)

Fr(t) = m^∞_a ÿ+ L(t) ∗ ˙y(t) (2.17) where f_e is the transfer function proportional to the sumφi+ φd,∗ is the con- volution operator, m^∞_a is the added mass at the infinite frequency, ÿ and ˙y are the acceleration and speed of the buoy in the heave direction, and L is the impulse response function defined as:

L(t) = 2 π

_∞

0 ω [m^∞_a − ma(ω)]sin(ωt)dω. (2.18)

2.3.2 WEC model

The hydro-mechanic model used to describe the behavior of the WEC couples the buoy hydrodynamic equation of motion (2.19) with the generator electro- mechanical equation (2.20):

m_b ¨y= Fe− Fr+ Fh− Fgb+ Fb− Fw (2.19)

m_t ¨x= Fw− Fd− Fgt+ Fes (2.20) where m_b and m_t are the masses of the buoy and the translator, ¨y and ¨x are the accelerations of the buoy and the translator, F_h is the restoring force, F_gb and F_gt are the gravity forces due to the buoy and translator masses, F_b is the buoyancy force, F_wis the connection line force between the buoy and the translator, F_d is the damping force, and F_esis the end-stop force.

The restoring force is calculated as:

F_h= −ρgπr²y (2.21)

where r is the radius of the cylinder and y the displacement of the buoy.

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The connection line is modeled as damped single harmonic oscillator:

Fw= (

kw(y − x) + dw( ˙y− ˙x), if y > x

0, otherwise (2.22)

where kwis the spring constant and dwis the damping coefficient of the wire.

The end-stop force is the sum of the contributions of the upper end-stop, fu, the lower end-stop, fl, and the upper wall of the WEC capsule, fc. The force exerted by the hull of the WEC, fc, is modeled as a stiff damped single harmonic oscillator, which the translator hits when the stresses on the upper end-stop spring start to provoke irreversible deformations, i.e. once the upper end-stop is compressed by the length li. The following equations describe each contribution to the end-stop force:

fu=

(−ke(x − ls/2), if x > ls/2

0, otherwise (2.23)

fl =

(−ke(x + ls/2), if x < −ls/2

0, otherwise (2.24)

fc=

(−kc(x − (ls/2 + li)) + dcx,˙ if x ≥ ls/2 + li

0, otherwise. (2.25)

The upper and lower end-stops are springs having the same spring constant ke, while kcis a fictitious constant for the wall, as well as the damping factor dc. The stroke length of the translator is ls, and the middle point of the translator when centered with respect to the stator is found at x = 0.

The damping of the generator is defined as:

F_d= γ ˙x (2.26)

where the damping factor γ changes with the position of the translator. How- ever, it will be considered constant in the study discussed within this thesis.

The assumption of constant γ as been made by other researchers, e.g. in [34], and has been proven to be true if an highly resistive load is connected to the generator [35].

The equations presented above, even though slightly modified, are thor- oughly described and used in literature, e.g. in [35] and [36].

2.3.3 Sea level time series

In order to be able to calculate the excitation force of Eq. 2.16, a time series of the waves, η, has to be generated. The sea state of a specific patch of sea is characterized by a representative wave height and period, called the

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significant wave height, H_s, and the energy period, T_e. The former is defined as the wave height of the highest third of the waves (H₁_/3), or as four times the square root of the area under the wave spectrum curve, H_m0= 4√m0. For the energy period, many definitions apply. In this study, the period calculated as the function of the spectral moments of order 0 and -1 is used: T_m0−1= ^m_m⁻¹₀ .

Given H_m0 and T_m0₋₁, Eq. 2.27 is used to calculate the available power per meter of wave crest [kW/m] at one location, assuming the deep water approximation to be valid [37].

P= ρg²

64πH_m0² T_m0−1. (2.27)

A time series characterized by a speciﬁc wave climate, i.e. a combination of H_s and T_e, can be generated using the Bretschneider spectrum [31]. This spectrum is a function of the signiﬁcant wave height and energy period, as Eq. 2.28 shows:

S_bs(ω) = 5ω_m⁴

16ω⁵ H_s²e^{−5 ω}^m⁴^{/4 ω}⁴ (2.28) whereω is the frequency in radians per second, and ωmis the modal frequency.

T_m is calculated iteratively until it matches T_e. In conclusion, for each combination of significant wave height and period a time series of polychromatic waves reflecting the specified wave climate can be obtained by using the S_bs spectrum.

In order to add the contribution of the tidal component, a low frequency wave having the same height as the tidal range, H_t, and the same period as the tidal cycle is superimposed to the time series obtained with the Bretschneider spectrum.

2.4 Data sources of sea level variations

2.4.1 Global study

The model described in subsection 2.3.2 will here be used to estimate the energy absorption of a single WEC deployed at different locations around the world. In order to do so, a combination of (H_s, T_e, H_t) for each site is selected.

The data source for H_s and T_e is the European Centre For Medium-Range Weather Forecast (ECMWF), which produces the global atmospheric reanalysis dataset ERA-Interim [38]. Fig. 2.4 illustrates the signiﬁcant wave height at the chosen points.

Ht is generated from the TPXO7.2 global inverse tide model [20], using the Tidal Model Driver (TMD) Matlab toolbox developed by Earth & Space Research and Oregon State University. The mean range at the selected coastal areas is shown in Fig. 2.5.

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Figure 2.4. Mean signiﬁcant wave height for coastal locations selected in a 3x3 de- grees grid.

Figure 2.5. Mean tidal range for the same coastal locations selected in Fig. 2.4.

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Table 2.1. Condensed wave climate scatter diagram at the Wave Hub site.

T_e[s]

4 5 6 7 8

3 - - - 6.0 7.4

H_s[m] 2 - - 14.0 14.6 9.7

1 7.4 15.7 15.5 9.7 -

2.4.2 Case study: the Wave Hub site

The Wave Hub, located off the coast of Cornwall, is the largest and most technically advanced site in the world for offshore renewable energy development [39]: it offers companies and institutions engaged with marine energy projects the opportunity to deploy and to plug in their prototypes to the grid. Conse- quently, the Wave Hub is a site of interest for many offshore technology devel- opers, including Uppsala University and its spin-off company, Seabased AB.

The characteristic wave climate of the south-west coast of England is thor- oughly described in [40], that presents the wave resource assessment for the Wave Hub based on a 23-year period dataset, obtained by using a high resolution regional SWAN model fed with reanalysis wave and wind data from larger ocean models. The wave climate scatter plot, which shows the occur- rence of different combinations of signiﬁcant wave height and energy period, is presented in Fig. 11 of Ref. [40]. A condensed diagram is extracted, as described in Paper II, and shown in Tab. 2.1. The signiﬁcant wave height calculated over 23 years is found to be 2 m and the corresponding energy period, calculated from the spectral moments m0and m₋₁, is equal to 7 s.

The water level variations due to tides at the west coast of Great Britain are among the highest in the world. Fig. 2.6 shows the tidal levels during 2014 with a temporal resolution of 1 h at Newlyn Tidal Observatory, about 22 km south of the Wave Hub. The peak to peak variation was registered to be about 6.6 m and the tidal cycle to follow a semidiurnal trend, i.e. two tidal cycles per day occur, corresponding to 730 cycles per year.

2.5 Economic analysis

Nowadays, wave energy technologies are still at a pre-commercial stage and this is manly due to the low cost-efﬁciency of the converters, implying an excessive cost of the device per kWh. Among the factors that keep the price of the WECs high and their lifetime low, we ﬁnd the manufacturing costs due to the lack of a systematic production line, the offshore deployment and maintenance costs, as well as the harsh nature of the waves.

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Figure 2.6. Sea level variation at Newlyn during 2014. Credits go to the Sea Level Data Portal hosted by the British Oceanographic Data Centre [41].

A solution to increase the survivability and energy absorption of the point absorber discussed in this thesis is to include in the WEC design a compensation system for sea level variation. Chapter 3 will describe in detail this system and its beneﬁts will be presented in section 5.2. However, including a new mechanical and electrical system in the existing WEC design involves an investment cost that this economic analysis aims to estimate. Note that the goal of the analysis is not to optimize the cost of the full WEC, but primarily to investigate what the viable cost of a compensation system should be in order to improve the performance of the WEC without having repercussions on the cost of the technology.

In order for the compensation system to be proﬁtable, the Net Present Value (NPV) deﬁned by Eq. 2.29 has to be positive:

NPV=∑^N

i=0

C_i

(1 + rd)ⁱ (2.29)

where C_iis the cash ﬂow at year i, and r_dis the discount rate. With the purpose of assessing the desirability of a project, another commonly used parameter is the Internal Rate of Return (IRR), calculated implicitly as follows:

∑N i=0

Ci

(1 + IRR)ⁱ = 0. (2.30)

Sea Level Compensation System for Wave Energy Converters

Sea Level Compensation System for Wave Energy Converters

List of papers

Contents

Nomenclature and Abbreviations

1. Introduction

2. Method